MSCI-3001 PHYSICAL OCEANOGRAPHY and MSCI-5004 OCEANOGRAPHIC PROCESSES COURSE NOTES

Transcription

1 MSCI-3001 PHYSICAL OCEANOGRAPHY and MSCI-5004 OCEANOGRAPHIC PROCESSES COURSE NOTES Session 2, 2008

2 These notes have been developed over the past 10 years by many members of the School of Mathematics and Statistics at UNSW. They were updated most recently by Dr Moninya Roughan with help from Caroline Ummenhofer and Maxwell Gonzalez. Copyright The University of New South Wales 2008.

3 Contents 1 Description of the Oceans and Definitions Geography Properties of Sea Water Salinity S Temperature T Pressure P Density ρ Other properties of composition Light Transmission Units Summary: Coordinate system and notation Gradients and derivatives Stratification, Stability and the Ocean s Thermohaline Circulation Stability, the mixed layer, and turbulent mixing The Onset of Turbulence Mass, Heat and Salt Fluxes Total Heat and Salt Fluxes Properties and circulation of water masses in the ocean The Surface Mixed Layer The Main Thermocline Surface T-S Properties Water-Mass Profiles and T-S Diagrams The Physics of Ocean Flow Vertical Movement Barotropic Motion Horizontal motion and currents Pressure effects The Earth s rotation effects - The Coriolis Force The Geostrophic Balance The Effects of Friction and Incompressibility The Conservation of Mass The Thermal Wind Balance i

4 4 Tides, Estuaries and Coastal Processes Tides Description of tides The Tide-Generating Forces Strength of the tidal force Shape of the Tidal Wave Co-oscillation tides Estuaries Dynamics of estuaries Classification of Estuaries Salt wedge estuaries Highly stratified estuaries Partially Mixed Estuaries Vertically mixed estuaries Inverse estuaries Intermittent estuaries Regional classification: a survey of Australian estuaries Flushing times Definition of flushing time Unforced Motions Wave Refraction, Diffraction and Shoaling Refraction Shoaling Wave Breaking Shallow and Deep Water Waves Long waves affected by the Earth s Rotation Internal Waves Potential and Relative Vorticity Geostrophic Adjustment Wind Forced Motion and Large-Scale Gyre Circulations Ekman Layer The Ekman Velocity Depth Averaged Ekman Layer Storm Surge, Downwelling and Upwelling Storm Surge Downwelling Upwelling Ekman Pumping A Model for the Large Scale Ocean Gyre Circulation The Ocean Gyre Conserving potential vorticity (PV) in an ocean gyre Western Boundary Currents Gulf Stream Circulation along the Equator ii

5 7 Measuring and modelling the oceans Measuring water masses, T S and tracers Conductivity-Temperature-Depth (CTD) instruments The Expendable Bathythermograph (XBT) The Expendable Conductivity-Temperature-Depth (XCTD) probe Niskin bottles Measuring ocean currents directly Surface drifters Subsurface floats Current meters Acoustic Doppler Current profiling (ADCP) ARGO floats High Frequency Radar Remote Sensing What can we measure from space? Analysis of Satellite Data Remote Sensing Applications Other data sources Surface Meteorological Data In situ sea level data Australia s Integrated Marine Observing System Ocean modelling - an introduction

6 Introduction What is physical oceanography? Briefly it may be regarded as comprising a systematic quantitative description of ocean currents and the resultant distribution of properties such as heat, salt and dissolved gases and biological scalars such as nutrients, phyto/zooplankton and fish eggs. In this course we shall examine a number of aspects of physical oceanography including waves, tides, ocean currents, water-masses, mixing, upwelling, and both coastal and large-scale flow patterns. We begin with the basic geography of the sea and the properties of seawater. We go on to study the surface mixed layer, the oceanic thermocline, and deep water circulation. The course is a broad introduction to physical oceanography. We study how the oceans are affected by winds, atmospheric heating/cooling, the earth s rotation, as well as understanding the role of bottom topography and coastal boundaries. At every opportunity the relationship between the physics and biology of the sea is explored; e.g., coastal upwelling, the transport of larvae, distribution of nutrients. The course does contain some mathematics and physics which allows us to easily describe the concepts and models for ocean circulation. However, if you don t feel overly confident in this area don t worry. The emphasis here is on obtaining a physical understanding of the equations and the systems they represent. This course will give you a solid basis from which to move forward into dynamical descriptions in your area of marine science. 2

7 1 Description of the Oceans and Definitions 1.1 Geography A typical zonal slice through an ocean basin is shown below. Figure 1.1: Diagrammatic cross section of an ocean basin, showing the various geographic features. (Not to scale; large vertical exaggeration.) (Taken from Nybakken, 2001) The diagram is not drawn to scale and indeed the oceans are very much wider than they are deep with an approximate aspect ratio: depth width 4 km 6000 km < However, as we will see, both the horizontal and vertical gradients (or variability) of properties such as density can be just as important in determining the global distributions of heat and salt. 3

8 Figure 1.2: Area, volume, and mean depth of oceans and seas. (Taken from Libes, 1992) 4

9 1.2 Properties of Sea Water The principal constituents by weight apart from water are Na 30% Cl 55% Mg 3.7% SO 4 7.7% Ca 1.1% HCO 3 0.4% K 1.1% Br 0.2% Other (e.g., O 2, 14 C, CFCs, etc.) < 1% The total concentration of salts varies from place to place but the ratios of the more abundant components remain almost constant. Only man-made compounds (e.g. CFCs, radioactive fallout) exhibit large deviations from these ratios Salinity S Salinity is the total amount of dissolved material in seawater. Old Defn: Salinity is total amount of solid materials (in grams) contained in one kg of seawater when all carbonate is converted to oxides, bromine and iodine are replaced by chlorine and all organic matter is completely oxidized. New Defn: Salinity is now determined through a determination of the electrical conductivity of seawater. A complicated empirical expression gives salinity as a function of conductivity, temperature and pressure. Salinity used to be expressed as g/kg or as o / oo (parts per thousand or ppt) However, as salinity is not actually a unit of measure (more a ratio) we now do not use units (although some people still prefer to use practical salinity units (psu)). 1gm/kg = 1 o / oo = 1 ppt = 1 psu The average salinity of the world oceans is about S = 35 psu although the range is psu. For the Baltic Sea S 8 psu, while for the Persian Gulf/Red Sea S 40 psu. Measurement Accuracies: Old: through titration ±0.02 ppt New: through conductivity measurement and standardization ±0.003 psu Resolution: New profiling sensors (CTDs) now give ± psu 5

10 Figure 1.3: Global distribution of the annual-mean salinity at the sea surface (from Levitus, 1982). (Taken from Peixoto and Oort, 1992) Temperature T Temperature is a measure of the heat content of water and at the surface varies from T 0 C at the poles to T > 28 C at the equator. At abyssal depths a nearly constant temperature of 0 4 C pertains. Since temperature is dependent upon pressure, we sometimes use potential temperature θ which is the temperature of a water mass if it were brought to a reference pressure level (usually 1 atmosphere; the surface). Temperature T is sometimes called in situ temperature and is warmer that θ: since water is slightly compressible, a sample brought from depth will expand then cool. Potential temperature θ is calculated from temperature, salinity and pressure using a complicated empirical formula. In situ temperature T is that which is measured locally (e.g., the thermometer reading at depth), potential temperature θ is the temperature this water would have if it were at sealevel pressure. Measurement Accuracies: Old T: Thermometers (still used to calibrate electronic equipment) ±0.001 C New T: Measured electronically ± C 6

11 Figure 1.4: Global distribution of the annual-mean temperature ( C) at the sea surface (from Levitus, 1982). (Taken from Peixoto and Oort, 1992) Pressure P Pressure is the force (or weight) per unit area, so pressure in the ocean is just the weight of water (and atmosphere) above a given water mass (per m 2 ). Usually pressure is measured in decibars (db, dbar) in the ocean, where 1 db = 100mb = 10 5 dyne/cm 2 = 10kPa. The pressure in the ocean increases by approximately 1 atmosphere per 10 m (1 atmos = mb = kPa). Hence 1 db is the pressure due to 1m of seawater and 1 κpa is the pressure due to 10cm of seawater. If the ocean were of uniform density the pressure would just increase linearly with depth. Variations in density lead to subtle variations in ocean pressure, which we will see can generate important deep water currents Density ρ Density is the mass of water per unit volume, and is measured in gm/cc (old) or in kg/m 3 (SI) and depends on water temperature T (colder water is denser), salinity S (saltier water is denser) and pressure p (water is compressible). Some examples of density values for different S,T,p are included in Figure 1.5. The relation between density and pressure is called the equation of state and tells us about the nature of the fluid. For an incompressible fluid such as pure water at ordinary pressures and temperatures, density is constant. In the ocean, however, water density is a complicated function of pressure, temperature and salinity. Details can be found in (Gill, 1982, Appendix 3), but for most applications, (where T and S vary little) it can be assumed that the density of seawater is independent of pressure ( Incompressibility) and linearly dependent upon both 7

12 Figure 1.5: Values of density in situ for fresh and seawater (kg m 3 ). (Taken from Pond and Pickard, 1983) temperature (warmer waters are lighter) and salinity (saltier waters are denser), according to: ρ ρ 0 [1 α(t T 0 ) + β(s S 0 )] (1.1) where T is the temperature (in degrees Celsius or Kelvin) and S the salinity. The constants ρ 0,T 0, and S 0 are reference values of density, temperature, and salinity, respectively, whereas α is the coefficient of thermal expansion and β is called, by analogy, the coefficient of saline contraction. Typical seawater values are ρ 0 = 1028 kg m 3 T 0 = 10 C = 283 K, S 0 = 35 psu, α = K 1, and β = (Cushman-Roisin, 1994). The more general dependence of ρ on T and S is given in Figure 1.6 which shows curves of constant density on a T-S diagram. Density (ρ) in the ocean is affected by pressure for two reasons: (i) seawater is compressible (the ocean s weight can squash a water parcel at depth into a smaller volume), and (ii) because ρ depends on temperature which is itself affected by pressure (as described above). It is therefore useful to distinguish between in situ density which is the density of water in its local environment, and potential density which is the density this water would have at some reference depth, normally taken to be the sea surface. In summary, in situ density is a function of local (T,S,p) whereas potential density is corrected for pressure effects, so depends on (θ,s,p = p ref ), where p ref is normally 0 (i.e. atmospheric pressure). At the sea surface, density varies from around kg m 3 i.e. by less than 1%. Thus it is often convenient to subtract out the 1000 kg m 3 and deal with the residual. This quantity is referred to as σ t ( sigma t ), where σ t ρ(s,θ, 0) 1000 and reference pressure p has been taken as zero (i.e. atmospheric pressure). 8

13 Figure 1.6: Values of density (as σ t ) and thermosteric anomaly ( s,t ) as functions of temperature and salinity over ranges appropriate to most of the oceans. (90% of the world ocean has temperature and salinity values within the shaded rectangle.) (Taken from Pond and Pickard, 1983) Other properties of composition The concentrations of nutrients and other chemicals are sometimes useful in determining the origin and thus movement of water masses. This includes natural biogeochemical tracers such as oxygen and silcate, as well as anthropogenic compounds and pollutants. Examples include: Dissolved Oxygen (mll 1 ) which can indicate if the water has been in recent contact with the atmosphere. Dissolved Phosphate, Silicate and Nitrate/Nitrite (µmol L 1 ) of biological importance. Fluorescence or Chlorophyll-a (mg m 3 ). Tritium ( 3 H), Carbon-14, CFCs, Strontium-90 resulting from industry, bomb testing, and so on. These chemicals occur in relatively low concentrations but turn out to be extremely useful as tracers. See corresponding diagrams below. 9

14 Figure 1.7: Zonal-mean cross section of sigma-t for the top ocean layer m and the layer below 1000 m depth in kg m 3 for annual-mean conditions (after Levitus, 1982). Vertical profiles of the hemispheric and global mean sigma-t are shown on the right. (Taken from Peixoto and Oort, 1992) CFCs were introduced into the atmosphere from the 1930s, and until very recently, have increased in concentration quite rapidly. They are weakly soluble compounds in seawater, and so ever increasing concentrations are detected in the ocean. In the interior, recently overturned waters can be detected as they carry with them a higher signature of CFC. The below diagram demonstrates such an example at about 3000-m depth in the Atlantic Ocean Light Transmission The vertical penetration of light decreases with depth due to scattering by molecules and absorption by molecules and particulate matter. Typically the intensity I of light decreases as a function depth z as I(z) = I 0 exp( kz) where I 0 is the surface intensity and k the attenuation coefficient. The length scale z = k 1 is called the e folding scale of penetration since I(k 1 )/I 0 = e

15 Figure 1.8: Latitude-depth section of oxygen concentration in the global ocean For seawater, the coefficient k varies considerably with wavelength. For clear ocean water (Figure 1.9 (a), curve 1) k has its minimum value at about 0.45 µm wavelength so that blue light is attenuated least (penetrates best) while at shorter wavelengths (toward the ultra violet) and at longer wavelengths (in the red and infra red) the attenuation is much greater and penetration correspondingly less. The increased attenuation in the ultra violet is not important to the heat budget of the oceans because the amount of energy reaching sea level at such short wavelengths is small. The increased absorption is more important in and beyond the red end of the spectrum where more energy is present in the sun s radiation. Virtually all of the energy shorter than the visible is absorbed in the top metre of water, while the energy of wavelength 1.5 µm or greater is absorbed in the top one centimetre or less. In Figure 1.9 (b) (full lines) are shown the relative amounts of energy penetrating clear ocean water to 1, 10 and 50 m as a function of wavelength. 11

16 Figure 1.9: [Left] (a) Attenuation coefficient k λ as a function of wavelength λ for clearest ocean water (full line) and turbid coastal water (dashed line). (b) Relative energy reaching 1, 10 and 50 m depth for clearest ocean water and reaching 1 and 10 m for turbid coastal water. (Taken from Pickard and Emery, 1990) [Right] The sloping lines depict the light intensity versus depth for clear coastal and clearest open ocean waters, as well as for moonlight. Note that the horizontal scale is logarithmic. Also shown on the diagram are vertical lines depicting the light intensity cutoff for various biological processes. (Taken from Sarmiento and Gruber, 2006). 1.3 Units Summary: Always calculate quantities using kg, meters, seconds and Pascals and the answers will be in one of the units below. 1.4 Coordinate system and notation In oceanography the Cartesian (x, y, z) coordinate system is used to denote position in space. That is, x denotes east-west position, y north-south position, and z depth. Standard units are in metres. Position in the horizontal is also often quoted in degrees longitude and latitude. The corresponding velocity of water circulating in the ocean is denoted by (u, v, w), where u is the east-west speed, v the north-south speed, and w the vertical velocity. Note that by convention, postitive directions are, respectively, eastwards, northwards, and downwards. 12

17 Quantity Symbol Units Units Conversion Time seconds s 1 day s= 10 5 s Mass kilograms kg Length meters m Velocity u, v, w meters per sec ms ms 1 = 10 km h 1 Pressure p Pascals (force Pa 1 hectopascal per unit area) =100 Pa Salinity S kg salt per psu 1000 kg water Temperature T Degrees C (ocean) Centigrade Temperature T Kelvin K T ( C) =T(K) 273 (atmos) Density ρ kg per unit kg m 3 volume Density σ σ = ρ 1000 Coriolis f = 2Ω 1/seconds s 1 Parameter sin(latitude) Windstress τ Pascals (force Pa 1Pa = 1N m 2 per unit area) 1N = 1 kgm s 2 Transport Sv Sverdrups m 3 s 1 1 Sv= m 3 s Gradients and derivatives In most applications in oceanography the gradient terms in equations, such as dt dx, dρ dt can be approximated as follows: dt dx T x where means the change in a certain property (temperature, time, and so on). This is generally valid for small x. etc. 13

18 Figure 1.10: Schematic of coordinate system and grid boxes across the earth 14

19 15 Figure 1.11: Topography of the Pacific, Atlantic and Indian Oceans. The 1000, 3000, and 5000 m isobaths are shown, and regions less than 3000 m deep are stippled. (Taken from Tomczak and Godfrey, 1994)

20 2 Stratification, Stability and the Ocean s Thermohaline Circulation In this chapter we will learn about the ocean s Thermohaline Circulation (THC). This implicitly includes the density, stratification, and stability in the ocean including the formation of the oceanic mixed layer and the maintenance of an ocean thermocline. The THC is simply that part of the circulation driven by the ocean s temperature (thermo ) and salinity ( haline): in Chapter 1 we saw how T-S determines density; thus, the ocean s THC is the density-driven circulation in the ocean. In principle, it is difficult to separate the so-called wind-driven and density-driven components of the ocean s circulation, as there are interactions between these two (e.g., wind-driven flow can establish a densitydriven circulation). However, it is convenient to first consider these two aspects of ocean circulation separately. In most regions of the ocean the density of seawater increases with depth. This is due to the fact that temperature T generally decreases with depth owing to the general pole to equator overturning circulation shown in Figure 2.1. Figure 2.1: Schematic diagram showing the global overturning of water on an idealised ocean covered planet with uniform salinity and no winds. The real situation on earth is complicated by salinity variations and geography such as subsurface bathymetry and continental land masses. For example, Figure 2.2 shows the Atlantic Ocean T, S, σ and oxygen on a north-south vertical section. In the Atlantic, temperature T and salinity S vary with depth and position for example, 16

21 17 Figure 2.2: Left: Atlantic Ocean south-north vertical sections of θ, S, σ and O 2 along the western trough. (Taken from Pickard and Emery, 1990) Right: Subsurface water masses and circulation patterns in the three major oceans. (a) Atlantic Ocean, (b) Pacific Ocean, (c) Indian Ocean. (Taken from Nybakken, 2001)

22 Antarctic bottom water (AABW) water formed in the Antarctic (T < 2 C) moves equatorward at depth and is found north of the equator, North Atlantic Deep Water (NADW) intrudes to the south at m depth, and low-salinity Antarctic Intermediate Water (AAIW) flows to the north near 1000 m depth. Note that in spite of the interleaving of these different water masses at different depths, the density increases with depth everywhere. From common experience it seems reasonable that density should increase with depth or else the fluid would overturn leading to mixing. To understand the distributions of ρ, T and S we must first consider the ideas of stability and turbulent mixing. Figure 2.3: Major water masses of the world ocean. (Taken from Libes, 1992) 18

23 Figure 2.4: Meridional cross-section of the Atlantic Ocean, showing movement of the major water masses; NADW = North Atlantic Deep Water; AAIW = Antarctic Intermediate Water; AABW = Antarctic Bottom Water. Note how the low-salinity tongue of AAIW extends northwards from the Antarctic Polar Frontal Zone, to overlie the more saline NADW. The M at about 35 N indicates the inflow of water from the Mediterranean. The Subtropical Convergences correspond to the centres of the subtropical gyres. (Taken from OpenUniversity, 2002) Figure 2.5: Global deep ocean circulation is often simplified as a loop-like conveyor. Evidence of North Atlantic Deep Water (NADW) formation in three sites shown in the North Atlantic region is more complex. Southern Ocean deep water formation of Antarctic Bottom Water (AABW) is well known from the Weddell Sea, but also occurs in the Ross Sea. (Taken from Kershaw, 2000) 19

24 Figure 2.6: The region of the Southern Ocean around the Weddell Sea is an important site of deepocean water flow. Cold water adjacent to the Filchner Ice Shelf sinks down the continental margin into deep water to form Antarctic Bottom Water (AABW); further north, cold surface water meeting the warmer subantarctic surface waters of the South Atlantic gyre (at the Antarctic Convergence) sinks to form Antarctic Intermediate Water (AAIW). These two sinking water masses are replaced by upwelling of Circumpolar Deep Water (CPDW), which brings nutrients to the surface to stimulate the high productivity of these cold waters. Ocean bottom waters are oxygenated, and so little organic storage occurs in sediments in this region. Surface water flows in opposing directions either side of 60 S, because of the polar high pressure zone, and forms the West Wind Drift (WWD) and East Wind Drift (EWD). (Taken from Kershaw, 2000) 2.1 Stability, the mixed layer, and turbulent mixing Gravity acts on vertical density gradients in the ocean to either stabilise or destabilise the water column. Here we shall quantify how changes in density with depth can act to stabilise vertical movement of water parcels. Consider an ocean with density dependent only on depth: ρ = ρ(z), with z zero at the surface and increasing with depth. For convenience, let us assume that ρ increases linearly with depth according to: ρ(z) = ρ 0 + kz = ρ 0 + z dρ dz (2.1) where k = dρ/dz is a constant. 20

25 z =0 o (z) z z Figure 2.7: Adjustment of a fluid parcel. Now if a parcel is moved from z = 0 to a depth z as shown in Figure 2.7 then Archimedes tells us that the force on the parcel will be equal to the mass displaced times gravity g (= 9.8 m s 2 ) i.e. the acceleration a is a = g(ρ 0 ρ(z))/ρ 0 (2.2) If dρ/dz > 0 then ρ(z) > ρ 0 so that density at depth is larger than at the surface. In this case a < 0 and the acceleration (force/unit mass) is upward. Thus, the tendency is for the parcel to be forced back to its initial position. Now with z the position of the parcel, the acceleration is a = d 2 z/dt 2 (t is time) and putting ρ(z) from (2.1) into (2.2) we get ( ) d 2 z g dt = a = z dρ. 2 ρ 0 dz Let us define N 2 = g ρ 0 dρ dz (2.3) (we will see why below), so that d 2 z dt 2 = N2 z (2.4) 21

26 This is a differential equation for the motion of the fluid parcel. A solution is z = A sin(nt) (2.5) where the amplitude A is unknown. However, we do know that the parcel will move like a spring with frequency N, the buoyancy or Brunt-Vaisala frequency. The period T is given by T = 2π/N (2.6) so that if dρ/dz is large (i.e., the stratification strong), then N is large and T small i.e. fast oscillations in a strongly stratified ocean. If dρ/dz is small (e.g., in the deep ocean), however, then N is small and T is large. These oscillations are called internal waves. Note we are assuming that N is positive so that from (2.3) dρ dz > 0 i.e. density increases with depth. This is stable stratification, and a common way of representing the increase with ρ with depth is to plot the buoyancy frequency N as a function of depth as shown below. As is evident from the figure N can vary from around 2π/(10 minutes) to less than 2π/(7 hrs), so that internal wave period varies greatly as a function of stratification. In this profile salinity effects on density happen to be small, so we only need sketch T, ρ, and N against depth Depth (m) Temperature ( C) Density (ρ) B V Freq (N Cycles hour 1 ) Figure 2.8: Profiles of temperature ( C), density (kg m 3 ) and buoyancy frequency N (cph), from Levitus data for the North Atlantic Ocean. What happens if ρ decreases with depth (i.e., dρ/dz < 0)? In this case N 2 < 0 and a solution to Equation 2.4 is z = Ae Nt (2.7) 22

27 so that the fluid parcel continues to move in the direction pushed convective overturning. The solution (2.7) is very simplistic. In reality the overturning and mixing only takes place over a limited depth until the density profile is stable again. This mechanism is very important as it leads to the formation of the winter mixed layer. Figure 2.9: The thermal structure and extent of mixing in temperate, tropical, and polar seas during the four seasons of the year (Taken from Nybakken, 2001). The above also raises the question of turbulent mixing. Water can be directly overturned in convective plumes when gravitational instability occurs (i.e., dρ/dz < 0), as above. But it can also be mixed due to turbulent diffusion, such as wind-driven mixing in the upper ocean. To address this problem let us define w as a vertical eddy velocity, S(z) as the mean salinity (that varies with depth), and s as a departure from the mean due to the eddy motions. The total salinity is s + S(z). Now consider the situation where S increases with z and fluid parcels move randomly up and down carrying salt with a velocity w. The flux down is ws, while the flux up is w(s + S). The net flux is them ws. Hence the mean salinity profile will change. Ultimately we would expect the mean distribution of salt to be the same at all depths through the homogenisation effect of the turbulence. This is how a wind-driven mixed layer is formed, quite unlike the mechanisms described above for the deep winter mixed layer formed by convective overturn. The mechanism for salt transport is characterised by a flux 23

28 Γ = ws (2.8) where the overbar denotes an average. The flux is down gradient; that is, carries salt from regions of high concentration to regions of low concentration. A simple representation for this is the flux gradient law Γ = K ds dz (2.9) where K is called the vertical diffusivity with units m 2 s 1. K is a property of the turbulence and can be estimated from w and L, where L is the depth-scale of the turbulent motion and w the effective vertical exchange rate: K wl. In the deep ocean K 10 4 m 2 s 1 while near the surface K 10 1 m 2 s 1. Deep ocean mixing occurs near rough bottom bathymetry whereas surface layer turbulence can be caused by wind mixing. To get a feel for the above conduct the following experiment. Fill a container with marbles where those on one side are white, on the other blue. You now have a gradient for S. Now each marble represents an eddy. By shaking the container what happens? A velocity w exists along with a diffusivity K which will be larger if you shake the container more rapidly. A flux of blue marbles is set up from one side of the container to the other. Ultimately the distribution of blue marbles will be uniform across the container. Now the equation which describes how (salt) concentration is changed in time t is given by ds dt + dγ dz = 0 (2.10) or ds dt = K d2 S (2.11) dz 2 Equation: 2.10 shows that if the flux Γ is the same at all depths then dγ/dz = 0 and ds/dt = 0 so that S remains unchanged. The same basic relationship holds for lateral (or horizontal) turbulence in the ocean; namely ds dt = K H( d2 S dx 2 + d2 S dy 2 ) (2.12) where K H is the horizontal diffusivity - typically 1000 m 2 s 1 near strong ocean currents. The same equations hold for temperature T. As an example read Widespread Intense Turbulent Mixing in the Southern Ocean Garabato et al. Science 9 January 2004: 210 DOI: /science An example Consider the temperature structure in the Gulf Stream drawn in Figure Diffusive effects will be strongest in the regions of greatest temperature gradient. 24

29 Figure 2.10: Upper panel: the depth, in hundreds of meters, of the 15 isothermal surface, showing the Gulf Stream, nine cyclonic rings and three anticyclonic rings. The contours are based on data obtained between March 16 and July Lower panel: a temperature section through the Gulf Stream and two cyclonic (cold core) rings south of the Stream. The section is a dog-leg from 36 N, 75 W to 35 N, 70 W and then to 37 N, 65 W. (Taken from Mellor, 1996) Summary: We have seen that the stratification, where density increases with depth, leads to a restoring force and oscillatory motions called internal waves with a characteristic frequency given by N. The restoring force will act to damp out any existing turbulence. However, where gravitational instability occurs, convective overturn takes place. We saw how this can lead to mixed layer deepening in winter. Mixing can also be activated by turbulence driven by, say, wind forcing at the surface. The distributions of T-S can be changed by the down gradient flux (Eq 2.9) and material is dispersed over a characteristic length scale (2Kt) 1 2 where K is the diffusivity and t is time. The diffusivity K is dependent on the turbulence 25

30 itself (e.g., whether wind-driven, or driven by flow over rough bathymetry, and so on). 2.2 The Onset of Turbulence It turns out that the Richardson Number Ri can be used to characterise the onset of turbulence in the ocean, where Ri = N2 ( du dz )2 (2.13) In the above, N is the Brunt-Vaisala frequency (given in Equation 2.3), and u is a horizontal current speed of the mean or average flow. If Ri < 0 then density variations enhance the turbulence (as Ri < 0 N 2 < 0). If Ri > 0 (i.e., stable stratification) then the mechanics of the flow are important in determining whether turbulence is found. In particular, the vertical velocity shear du/dz must be sufficiently large to generate turbulence in spite of the stability provided by the stratification (N 2 > 0). It turns out that: if Ri 1 no turbulence is found, whereas 4 if Ri < 1 turbulence may be found. 4 An example Determine whether turbulence will occur in the following two scenarios: Figure 2.11: Uppper Ocean (Left) and Deep Ocean (right) examples of velocity shear and the Buoyancy Frequency. 26

31 2.3 Mass, Heat and Salt Fluxes Important quantities in physical oceanography are the transports of heat, salt, and mass by the oceans. These are known as fluxes and each is computed in a logical manner. The mass flux or mass transport is the volume of water passing through a given area per second, normally expressed in units of 10 6 m 3 s 1 or Sverdrups (Sv). One Sverdrup equals one million cubic metres of water passing by a given region per second! The mass flux associated with an ocean current u through an area A is just ua. In Figure 2.12, if u is the typical ocean current through the channel drawn, then the mass transport is given by Mass transport = ua uhl Heat (Q) and salt (S f ) fluxes are a little more complicated to measure, requiring a knowledge of temperature-salinity distributions and ocean currents. In the simplest case of no turbulence they are calculated as a flux per unit area, namely: Q = ρc p ut (2.14) S f = ρus (2.15) where ρ is density, c p the specific heat of seawater (c p 4000 J C 1 kg 1 ) and S is salinity in absolute units, namely around 35/1000 gm of salt per kg of seawater Total Heat and Salt Fluxes To obtain the total heat and salt fluxes through a channel (e.g. Figure 2.12) we must multiply the flux by the area of the channel, HL (units are then Watts for heat and kgs 1 for salt). H (depth) u L (width) Area = H x L Total Flux = flux x Area Figure 2.12: Diagram showing area for calculating the total heat (salt) flux through a channel. An example Using the values in Figure 2.13 calculate the total mass, heat, and salt fluxes 27

32 a) past Tasmania and b) through the Drake Passage. Assume that the ocean currents persist over the upper 1000 m and that below this depth there is negligible motion. The T S values shown are the mean T S over the regions south of Tasmania and in the Drake Passage (see also Chapter 1). Figure 2.13: Map of the Pacific Ocean showing idealised values of temperature, salinity and velocity. c) Noting that the area of the Pacific Ocean is about km 2 (Chapter 1), and that the salt flux (S f ) can be converted into an equivalent freshwater flux (FW f ) where FW f = S f /(SρA), calculate what the average heat and freshwater fluxes must be over the Pacific Ocean to maintain this T S gradient from Australia to the Drake Passage. d) Why might the mass flux be different at the two locations? 2.4 Properties and circulation of water masses in the ocean In the following we will detail some of the observed larger scale features of density, temperature, and salinity as well as the factors which affect their distributions. 28

33 2.4.1 The Surface Mixed Layer Near the surface of the ocean, T-S and other water-mass properties are well mixed, both through wind-driven turbulence and diurnal and/or seasonal convective overturn (as discussed previously). This yields zero vertical T-S and density gradients in the surface mixed layer. The mixed layer depth varies from only a metre or so (on a sunny day with little wind) to m (in a wintertime convective plume in polar waters). Figure 2.14: Global ocean mixed layer depth (m) during March The Main Thermocline At mid- to low- latitudes a region of warm water, typically 500 m deep is found to overlie deeper cool water (0 4 C) at depths of m. The transition region is known as the thermocline. Continued heating of the warm surface water should, through turbulent diffusion, result in a warming of the lower layers. It is thought however that this downwards heat transfer is balanced by an upwards advection of cool water Surface T-S Properties Factors which affect the distribution of heat (T) and salt (S) at the surface of the ocean include: (i) Radiation (T) Short-wave radiation by sunlight (infra-red) is mostly absorbed in the first meter or so even though the visible light extends much further. The incident radiation 29

34 on the ocean surface has a diurnal and seasonal cycle that is a function of latitude and cloud cover. The ocean emits longwave radiation into the atmosphere. At night the ocean surface emits radiation but only the few top centimeters cool resulting in cold (heavy) water over relatively warm (light) water. This density profile is unstable and overturning or mixing occurs. (ii) Evaporation (T and S) Evaporation takes water molecules from the surface of the water but leaves the salt molecules behind so that the surface becomes slightly saltier (and therefore denser). Evaporation also cools the surface waters through the latent evaporative heat flux. Evaporation is highest when the wind is strong, when the airtemperature/sea temperature difference is high, and when air humidity is low. (iii) Precipitation (S) Precipitation of rain, snow or hail all act to decrease the surface density by adding freshwater. River run-off also injects very low salinity water into the oceans at specific coastal locations. (iv) Formation and Melting of Sea Ice. (T and S) Sea-ice is formed when cold air takes sufficient heat out of the sea surface. The ice lattice does not incorporate the Sodium or Chlorine ions easily so that salt is extruded into the water making it denser. Note that after the formation of sea-ice and the associated leaching of freshwater from the ocean, the cool saline surface waters may overturn to depths of m in the Arctic and Antarctic. Conversely, when sea ice melts it releases fresh water which acts to make the surface layers lighter and thus more stable. Note also that sea-ice has a high albedo (i.e., is very reflective) which isolates the sea beneath from any atmospheric warming. (v) Input of Mechanical Energy. (T and S) The wind through wave generation and breaking and directly through forcing motions in the surface layers of the ocean can put energy into the ocean. Such energy often takes the form of turbulent eddies that can homogenise (or mix) the top 100 m as described previously in this Chapter. (vi) Convective Overturn, Ocean Currents. Ocean Currents move heat and salt around the globe. e.g. the EAC transports warm water southward from the equator to the pole. Later in the course we learn about other processes internal to the ocean that can affect the mixed layer T S, such as surface currents and upwelling. Typical mean profiles of T and S in the surface mixed layer are shown below. Note that both local and seasonal effects may be very important at any given site. The factors (i)-(ii) and (iv)-(v) above can each contribute to the formation of mixed layers in the top m or so of the water column. All (scalar) water properties (T, S, chemical tracers, etc.) are well-mixed and therefore uniform over the mixed layer water column. The mixing by turbulent eddies can also take nutrients up towards the surface as well as moving phytoplankton down below the euphotic zone: thus primary production can be increased or decreased. 30

35 Figure 2.15: Annual mean net shortwave radiation flux (W m 2 ) (top), annual mean surface latent heat flux (W m 2 ) (middle) annual mean surface sensible heat flux (W m 2 ) (bottom), for the period N.B.: Positive is upward. (Based on NCEP/NCAR reanalysis data and provided by NOAA/OAR/ESRL PSD, Boulder, Colorado, USA, through their website 31

36 Figure 2.16: Average boundaries of pack ice for winter (September) and summer (March) in the ocean around Antarctica. (Taken from Ingmanson and Wallace, 1995) 2.5 Water-Mass Profiles and T-S Diagrams In the above discussion we have concentrated on the near surface regions. Within the ocean, the (ρ,t,s) properties will vary from site to site. However, by measuring (ρ,t,s) from conductivity temperature depth (CTD) profiles at many sites, an idea of the large scale ocean circulation can be obtained. Once away from the surface, water parcels can only have their (ρ, T, S) characteristics altered by mixing with neighbouring water masses. However, without turbulence, the mixing process is very slow and tends to occur mainly along surfaces of constant potential density. The reason for this is that a parcel of density will only mix with one of greater density if the latter lies above the former and overturning can occur. Such overturning results in stable stratification so that no further mixing occurs. This is the case for most regions below the thermocline. Evidence for the conservation of the (ρ, T, S) properties of a water parcel is shown in the north south CTD section of the North Atlantic in Figure 2.2. Warm salty water from the Mediteranean is apparent over great depths while North Atlantic Deep water (formed north of 60 N) extends south of the equator. Above rough topography and near confluence regions of currents, diapycnal (cross-isopycnal) mixing is intensified, and neighbouring water masses will diffuse (ρ, T, S) properties rather efficiently. The (ρ, T, S) properties in the deep ocean will also change through similar processes - albeit more slowly than at the surface. In the following we examine how a given (ρ,t,s) sample can be shown to be a mixture of various water types through the construction of 32

37 Figure 2.17: Changes in coastal salinity, and temperature (top row) can be caused by the input of freshwater runoff (a), by dry offshore winds causing a high rate of evaporation (b), or by both (c). Changes in coastal temperature (bottom row) depend on latitude. In high latitude (d), the temperature of coastal water remains uniformly near freezing. In low latitudes (e), coastal water may become uniformly warm. In the mid-latitudes, coastal surface water is significantly warmed during summer (f) and cooled during the winter (g). (Taken from Thurman and Trujillo, 2004) mixing triangles. Consider Figure 2.20 where the small circles A, B and C represent arbitrary water types while the straight dashed lines AB, BC and CA represent water masses made up of mixtures of A and B,B and C, and C and A respectively. The curved dotted line represents one example of a water mass made up of the three types A,B and C. Lines within the mixing triangle ABC represent all possible masses from types A,B and C. This interpretation implies that temperature and salinity are conservative quantities in the ocean, i.e. that no processes exist for generating or removing heat or salt. This is a reasonable assumption within the body of the ocean, but near the surface it is not so. Here, the sun may heat the water or through evaporation cool it, rain may decrease the salinity or evaporation increase it. Therefore in plotting the T S diagram it is usual to regard the points corresponding to 33

38 Figure 2.18: (a) Schematic illustration of terminology used to describe various features of the thermocline. (b) Seasonal behavior of the mixed layer at 50 N, 145 W in the eastern North Pacific. (Taken from Sarmiento and Gruber, 2006) Figure 2.19: (a) Density variation with depth for high- and low-latitude regions. The zone of rapidly changing density with depth in low latitudes is the pycnocline. (b) Temperature variation with depth for high- and low-latitude regions. The zone of rapidly changing temperature with depth in low latitudes is the thermocline. (c) A typical ocean profile showing the variation of temperature and density with depth, showing the inverse relationship between temperature and density. (Taken from Thurman and Trujillo, 2004) 34

39 Figure 2.20: a) T-S diagram showing three water types and four water masses. (b) Example of realistic T-S water mass diagram (Atlantic) and basic water types. (Taken from Pickard and Emery, 1990) shallow depths within the influence of the surface effects as less conservative, or even to omit them altogether. The T S diagram turns out to be a powerful tool for the study of ocean waters. The shape of the T S curve is often characteristic of water from a particular locality in the ocean, and individual features of the curve may indicate mixtures of different types of water. For example, the Atlantic water masses could be represented to a close approximation as a mixture of four types as shown schematically in Figure 2.20 b. Salinity minima or maxima such as B and C in Figure 2.20 a are common on T S diagrams but, except for the surface and bottom values, temperature minima or maxima are uncommon. The reason is that except in polar or coastal regions, density depends chiefly on temperature and less on salinity. A temperature minimum, such as in Figure 2.21 b, would imply that the water below the minimum, i.e. toward H, was less dense than that at the temperature minimum and therefore that the water between G and H was unstable. Such instabilities are found occasionally, such as in the complicated water masses at the northern side of the Gulf Stream where the Labrador Current joins it, but only over very limited depth ranges and certainly only as a transient phenomenon. 35

40 Figure 2.21: Temperature and salinity profiles and corresponding T-S diagram. (Taken from Pickard and Emery, 1990) Figure 2.22: Left: A temperature-salinity diagram showing the effect of mixing three water types (I, II and III) to give a mixture R with T = 3 C and S = 35. The method used to determine the relative proportions of I, II and III in the mixture is shown here graphically. Right: A completed temperature-salinity diagram. (Taken from OpenUniversity, 2002) 36

41 3 The Physics of Ocean Flow To understand the circulation of the oceans and atmosphere it is necessary to examine the underlying equations which govern their motion. In the following, the equations are all derived from a consideration of Newton s law F = ma where F denotes the applied force on a fluid parcel of mass m and a the resultant acceleration. Fluid mechanics in oceanography is based on newtonian mechanics where we consider conservation of Mass (continuity equation), Conservation of Energy (conservation of heat, heat budgets, mechanical energy, and wave equations) Conservation of Momentum (Navier Stokes Equations) and Conservation of Angular Momentum (conservation of vorticity). We will use the primitive equations, which are a version of the Navier-Stokes equations that describe hydrodynamic flow on the sphere (the earth) under the assumptions that vertical motion is much smaller than horizontal motion (hydrostatic approximation) and that the fluid layer depth is small compared to the radius of the sphere. Thus, they are a good approximation of global ocean and atmospheric flow and are used in most ocean and atmospheric models. In general, nearly all forms of the primitive equations relate the five variables u =(u,v,w), T, p (where we can infer ρ from p and T) and their evolution over space and time. 3.1 Vertical Movement In the vertical direction, the two most important forces are those that result from gravity and changes in pressure. If gravity had its way, water would move downward, whereas if pressure forces win out, water would flow from high to low pressure (i.e., would rise). The net result is a balance of these two forces. Consider a small cylinder of fluid of mass m as shown in Figure 3.1. z 1 p(z 1 ) -m/.dp/dz Pressure h m z 2 p(z 2 ) mg Gravity Figure 3.1: The Hydrostatic Balance 37

42 The force downwards due to gravity g is mg. The mass of the cylinder also leads to an increase in pressure (force/unit area) from z 1 to z 2 = z 1 + h. This increase in pressure leads to a net force upwards given by [ ] p(z1 ) p(z 2 ) V dp h dz V (3.1) where V is the volume of the cylinder. Now since m = ρv, the acceleration of the cylinder a = dw/dt is equal to the sum of all the forces F/m so that If the vertical motion w is not changing rapidly in time, i.e., dw dt = 1 dp ρ dz + g (3.2) dw dt << g the balance of forces equation Eq 3.2 reduces to dp dz = ρg (3.3) This is known as the Hydrostatic Balance, where the gradient of pressure supports the fluid against the force of gravity. Even in locations of strong vertical motion, the hydrostatic balance is a very good model for long period motions. For example, consider upwelling along the equatorial Pacific Ocean, which is about 100 Sv on average (Figure 3.2). Variations in the surface winds (as we will see later in the course) over periods of a few weeks can lead to a near shut-down in this equatorial upwelling during El Niño. Even then, the magnitude of dw dt << g. Exercise Show that the hydrostatic approximation holds true for equatorial upwelling noting the equatorial Pacific is about 10, 000 km wide and upwelling is concentrated in a 50 km wide band centered at the Equator (Figure 3.2). 38

43 SE Trade Winds NE Trade Winds 0m Depth (m) 1000m 2000m Equatorial Upwelling ~ 100 Sv z 3000m south O o north(y) Figure 3.2: Idealised equatorial upwelling in the Pacific Ocean. Another example is coastal upwelling, which might see isotherms raised by about 30 m during a 3-day period. In this case, the vertical velocity goes from w = 0 to w ms ms 1 during 3 days. This means we have a magnitude for dw dt w t = days = m s 2 This term may be compared with g = 9.8 m s 2 and is thus very small. Gravity cannot be balanced by dw dt and must be balanced by 1 dp so that Eq 3.3 is a very good approximation ρ dz for these motions. It turns out that almost all ocean circulation of period a day or more satisfies the hydrostatic balance. 0m Tuesday τy Friday τy Depth (m) 50m 18 o C 17 o C 16 o C 18 o C 17 o C 16 o C 100m 2km 2km Distance Offshore Figure 3.3: Idealised upwelling along the NSW coastline. 39

44 3.2 Barotropic Motion A major simplification can be made if we assume that the density of the ocean is constant everywhere so that ρ = ρ 0. In this case the hydrostatic relation Eq 3.3 reduces to so that dp dz = ρ 0g p = ρ 0 gz + f(x,y,t) = ρ 0 g(z + η) (3.4) Here f(x,y,t) = ρ 0 gη is a function independent of depth (z). As we see below, η turns out to be the sealevel height, which varies about zero, and changes in time due to surface waves, tides, and so on. z=0 p'= o g z p=p o +p' p o = o gz Figure 3.4: Perturbations in Sealevel As the ocean surface is not fixed in position (because of swell, tides, and other factors) z = 0 is only the mean sealevel and η is the variation about this height. Equation 3.4 can then be rewritten p = p 0 + p = ρ 0 gz + ρ 0 gη where p 0 and is called the static pressure and p is the pressure associated with variations in the sealevel height due to transients, such as tides, waves and other deviations in the sea surface height. 40

45 The static pressure is only related to the constant density ρ 0 = 1000 kg m 3 and serves to support the bulk of the water mass between z = 0 and the depth z (see above). The component p is of more interest since it is related to variations in the sealevel height and therefore the ocean circulation as we shall see later. Since p 0 supports the mass from the surface to a depth z, p supports the residual mass from the surface to z = η (Figure 3.4) i.e. p = ρ 0 gη (3.5) Note that while η may vary with x,y and t it and thus p are independent of depth. An ocean in which density is constant is called barotropic and on shelves where the water is well mixed this is a very good approximation. In fact, measurements of bottom pressure can be used to obtain p from p = p ρ 0 gh (h is bottom depth) and thus sea level variations η. Finally, it is worth noting that by measuring pressure on an instrument that the depth may be very accurately obtained. Since in general we have p 0 p so that p p 0 = ρ 0 gz. 3.3 Horizontal motion and currents Pressure effects In the horizontal (x, y) plane pressure gradients will also result in forces on fluid parcels. In the following we will consider a barotropic ocean so that these forces may be thought to act over the entire column depth. The forces due to horizontal pressure gradients may be written as 1 dp ρ 0 dx and 1 dp (3.6) ρ 0 dy and we note that only p actually contributes to horizontal gradients in pressure as p = p 0 +p, with the static pressure p 0 = ρ 0 gz only a function of z The Earth s rotation effects - The Coriolis Force The second force that must be considered is that which arises from the rotation of the Earth. To illustrate this force consider the rotating table as sketched below with two observers B, at rest and A, spinning with the table. Now if A rolls a ball towards B, the observer B will see the ball move in the straight line indicated. Observer A however will say that the ball was deflected to the left. The force which does this only appears to the observer in the rotating reference frame and only acts on moving objects. If Ω denotes the angular speed of the turntable (radians/sec) then the force is given by 2ΩUm where U and m are the velocity and mass of the ball. For the Earth, the local 41

46 B fv A A v Figure 3.5: The Coriolis Force angular speed varies with latitude. At the equator it is zero, while at the poles it is largest (2π/ s 1 ). A convenient way of expressing this is through the Coriolis parameter f = 2Ω sin(latitude) and the acceleration due to the Coriolis force is given by fv and fu in the x and y directions respectively. Note v and u are the velocities in the y and x directions. In the northern hemisphere the Coriolis force acts to deflect to the right and f is positive. In the southern hemisphere the reverse holds. Now collecting the forces we have the following horizontal equations (a = F/m) and in the vertical equation as given in Eq 3.3. du dt = 1 dp + fv ρ 0 dx (3.6) dv dt = 1 dp fu ρ 0 dy (3.7) dp dz = ρg For a barotropic ocean where ρ = ρ 0 and p = p 0 + p with p = ρ 0 gη the above simplify to du dt dv dt = g dη + fv dx (3.8) = g dη fu dy (3.9) 42

47 3.4 The Geostrophic Balance The above equations may be simplified further for motions with periods of about 10 days or more. To see this we again perform a scaling analysis on Equations 3.6 and 3.7 and choose a period T 10 days = s. (Remember we always work in mks - metres, kilograms and seconds). With f 10 4 s 1 (latitude 40 ) and assuming that u and v U where U is a typical scale velocity then while du dt, dv dt U T U 10 6 s 1 fv,fu fu U 10 4 s 1. Thus, the acceleration terms are of order 1 / 100 of the Coriolis terms and negligible for these long period motions. In this case Equations reduce to 1 dp ρ 0 dx 1 dp ρ 0 dy = fv (3.10) = f u (3.11) the geostrophic balance. The deflection by the Coriolis force is now balanced by the force due to the gradients of pressure. As an example consider the atmospheric High Low system sketched in Figure 3.6. The fluid parcel shown does not rush from High to Low (the pressure gradient force) but instead is balanced by a Coriolis force associated with motion in the y direction. L PG CF H v SH Example Plan View x Figure 3.6: Geostrophic Balance in the Southern Hemisphere. Coriolis Force (CF) =fv and the Pressure Gradient Force (PG) = 1 ρ o dp dx A second way to view the geostrophic balance in an approximately barotropic ocean is to replace the pressure gradients with sealevel gradients, remembering that p = p 0 + p with 43

48 p = ρ 0 gη. The geostrophic balance is then: g dη dx g dη dy = fv (3.12) = f u (3.13) As a second example consider the sea level slope as sketched in Figure 3.7. η z=0 gdη dx v fv SH Example x Figure 3.7: Sealevel Slope and Geostrophy Since η increases with x, dη > 0 and the force gdη is negative as shown. The geostrophic dx dx balance is 0 = g dη dx + fv so that fv is negative and the fluid column must move out of the page (the y direction). An example Cold and warm core eddies Sketch the sealevel shape and derive the sense of rotation for the southern hemipshere eddies shown in Figure 3.8. Do the same for a northern hemisphere example. 44

49 0m Depth (m) 50m 18 o C 16 o C 100m Cold Core Eddy 14 o C Cold Core Eddy 0m 18 o C Depth (m) 50m 16 o C 14 o C 100m Warm Core Eddy Warm Core Eddy Southern Hemisphere Northern Hemisphere Figure 3.8: Sketch the sealevel shape and derive the sense of rotation for the Southern Hemisphere cold and warm core eddies (Left) and for the northern hemisphere example (Right). 3.5 The Effects of Friction and Incompressibility In the above we have neglected the effects of frictional drag on the oceans, which becomes important near the bottom. A simple model for the frictional force due to the sea floor is given by ru/h and rv/h in the x and y directions where h is the ocean depth and r a coefficient given by r = C D v C D is a non dimensional drag coefficient and v denotes a typical bottom turbulent or tidal velocity. The equations of motion Equations 3.6 and 3.7 then become du dt dv dt = g dη + fv ru/h dx (3.14) = g dη fu rv/h. dy (3.15) The effects of friction are readily seen if we consider the simpler balance A solution is du dt = ru/h. u = u 0 e rt/h so that at t = 0 u = u 0 while as t becomes large, u becomes small. Clearly (h/r) plays the role of an e folding time scale of frictional spin down. 45

50 An example Bass Strait In Bass Strait the typical ocean depth is h = 70 m and flow through the strait can reach speeds of v 2.5 cms 1 at the bottom after the passage of a storm. Calculate the time-scale over which the storm surge current is damped by friction. r = C D v = ms 1 and h = 70 m gives h/r = 16.2 days. Friction thus acts to damp out this motion over about a fortnight. 3.6 The Conservation of Mass Finally, we must include an additional equation that results from the (near) incompressibility of sea water. This equation simply says that flow into a given region must be balanced elsewhere by flow leaving the region; that is, mass is conserved. For the barotropic ocean it may be written as dη dt + d(hu) dx + d(hv) = 0 dy and expresses the fact that as sea level changes, water must flow in or out in compensation (see Figure 3.9). up down up h x u u Figure 3.9: Continuity: As sea level changes water must flow in or out to compensate. In this case h is constant and v = 0 so that mass conservation requires dη dt + hdu dx = 0 The generalised mass conservation law or continuity equation for the ocean is given by du dx + dv dy + dw dz = 0 46

51 3.7 The Thermal Wind Balance In the above we have assumed that density ρ is constant. However, small horizontal changes in density can result in large vertical changes in currents, particularly near fronts and eddies. For a geostrophic balance we have v = 1 dp ρ 0 f dx u = 1 ρ 0 f dp dy (3.16) (3.17) Note that density is in the denominator in the geostrophic balance, so we can take it to be constant (ρ 0 ), whereas it is in the numerator in the hydrostatic balance (see Equation 3.20), so its variations may be important (ρ). Now differentiating Equations 3.16 and 3.17 with respect to z (depth) we get dv = 1 ( ) d dp = 1 ( ) d dp dz ρ 0 f dz dx ρ 0 f dx dz du = 1 ( ) d dp = 1 ( ) d dp dz ρ 0 f dz dy ρ 0 f dy dz (3.18) (3.19) The hydrostatic balance tells us the vertical pressure gradient is balanced by ρg, that is So we can substitute this into Equations 3.18 and 3.19 to get: dp dz = ρg (3.20) dv dz du dz = g dρ ρ 0 f dx = g ρ 0 f dρ dy (3.21) (3.22) These two equations are known as the thermal wind balance. This name comes from the fact that the balance was first noticed in the atmosphere; wherein horizontal density gradients (dominated by temperature gradients in the atmosphere) could explain the vertical profile of winds. Nowadays the balance is widely used in oceanographic applications. An example a cold-core eddy Suppose we want to work out the ocean circulation associated with the eddy shown below, for example, the vertical velocity shear at 500 m depth near x = 50 km. The ocean section 47

52 Figure 3.10: Distribution of potential density σ θ (kg m 3 ) for a CTD section through Eddy Bob (Olson, 1980). shows potential density σ θ (kg m 3 ) through eddy Bob measured just south of the Gulf Stream in 1980: Right at the sea surface there is no longer a sign of colder water (probably because the cold eddy has been warmed by air-sea fluxes since it was first formed). However, we can see a dramatic shallowing of density surfaces in the centre of the eddy, so we know the eddy has either a cold or saline core (and since it is south of the Gulf Stream, it must be a cold core eddy). So the eddy must be a few months old, with its surface character capped off by air-sea heating. Because the density surfaces are sloped in the interior (dρ/dx 0), the eddy must be spinning to balance this (to satisfy the thermal wind balance). To work out the vertical velocity shear at 500 m depth we have dv dz = g dρ ρ 0 f dx g ρ ρ 0 f x To work out the value of ρ/ x at x = 50 km we note that ρ changes from kg m 3 (towards the edge of the eddy) to about kg m 3 (towards the eddy centre) over a distance of about 25 km, so that Converting km to m we get ρ x km ρ x

53 Putting this into the thermal wind balance above we have dv dz g ρ 0 f and taking g = 10 m s 2, ρ kg m 3 and f = 10 4 s 1 we get dv dz This means, for example, that at z = 500 m, the velocity in the eddy v changes by about v dv dz z = ( ) 100 over 100-m depth ( z) in the water column. That is, v ms 1 = ms 1 This means the velocity shear at x = 50 km is about 15.6 cms 1 per 100 m, which is quite a lot. As we go to greater depth in the eddy, the density surfaces flatten out, implying that dρ dx 0 and therefore dv dx 0 We can then assume a depth of no motion where the density surfaces are flat. Then using our calculation for dρ/dx above, we can infer the velocity field in a cold core eddy in the Northern Hemisphere. 0m x Depth (m) 1000m ρ 1 ρ m ρ 3 ρ 4 z 3000m dρ > 0 dx dρ < 0 dx Figure 3.11: Density surfaces in an eddy. 49

54 Note that at the other side of the eddy, the density surfaces slope in the opposite direction (i.e., dρ/dx < 0), so the circulation must also be in the opposite sense: Other applications of the thermal wind balance and geostrophy Eddies Derive the rotation sense of cold- and warm-core eddies in the Southern Hemisphere using the thermal wind balance dv dz = g dρ ρ 0 f dx 0m 18 o C Depth (m) 50m 18 o C 17 o C 17 o C 16 o C 16 o C 100m Cold Core Eddy Warm Core Eddy Figure 3.12: Density surfaces in an eddy. Referring to Figure 3.12, sketch the shape of the sealevel over the eddies from a knowledge of the interior density distribution shown. Use the geostrophic balance fv = g dη to confirm dx the rotation sense derived above. Upwelling Consider the scenario in Figure 3.13 of upwelling along the NSW coast. What is the sign of each of the terms in the thermal wind equations (Eq 3.21)? Sketch the velocity profile. 0m τy Depth (m) 50m ρ=27.0 ρ=27.1 ρ=27.2 ρ= m 2 Distance Offshore (km) 4 Figure 3.13: Coastal Upwelling density surfaces. 50

55 4 Tides, Estuaries and Coastal Processes 4.1 Tides Tides are long waves, either progressing or standing. They therefore always travel with a phase speed of c = gh (4.1) as well as obeying other properties of long waves (e.g., shortening wavelength and increasing amplitude as h decreases on the shelf and near the coast). The dominant period usually is 12 hours 25 minutes, which is 1/2 of a lunar day. Tides are generated by the gravitational potential of the moon and the sun. Their propagation and amplitude are influenced by friction, the rotation of the earth (Coriolis force), and resonances determined by the shapes and depths of the ocean basins and marginal seas. The most obvious expression of tides is the rise and fall in sea level. Equally important is a regular change in current speed and direction; tidal currents are among the strongest in the world ocean Description of tides High water: a water level maximum ( high tide ) Low water: a water level minimum ( low tide ) Mean Tide Level: the mean water level, relative to a reference point (the datum ) when averaged over a long time Tidal range: the difference between high and low tide Daily inequality: the difference between two successive low or high tides Spring tide: the tide following full and new moon Neap tide: the tide following the first and last quarter of the moon phases. The result of alternate spring and neap tides is a half monthly inequality in tidal heights and currents. Its period is days, which is half the synodic month. (Synodic: related to the same phases of a planet or its satellites. A synodic period or synodic month is thus the time that elapses between two successive identical phases. In tidal theory synodic always refers to the moon, so a synodic month is the time that elapses between successive identical phases of the moon, for example between successive new moons.) There are other inequalities with similar and longer periods. 51

56 4.1.2 The Tide-Generating Forces As the earth revolves around the gravitational centre of the sun/earth system, the orientation of the earths axis in space remains the same. This is called revolution without rotation. The tide generating force is the sum of gravitational and centrifugal forces. In revolution without rotation the centrifugal force is the same for every point on the earths surface, but the gravitational force varies (Figure 4.1). It follows that the tide generating force varies in intensity and direction over the earth s surface. Its vertical component is negligibly small against gravity; its effect on the ocean can be disregarded. Its horizontal component produces the tidal currents, which result in sea level variations. Figure 4.1: The tide generating force as the resultant of centrifugal and gravitational forces. The lower panel in Figure 4.1 shows the earth s movement as revolution without rotation. The centrifugal force experienced by all points on the earth (as well as inside it) is the same, 52

57 in magnitude and direction. The gravitational force exerted by the sun always points to the centre of the sun. The effect of this force experienced by points on the earth s surface therefore varies with position around the earth, in both magnitude and direction. The resulting balance of forces results in a horizontal component which is the tide generating force. The same principle applies to the interaction between the earth and its moon. Both bodies revolve around their common centre of gravity, which in this case is inside the earth (but not at its centre). The earth again revolves around this centre without rotation, so that the centrifugal force is the same everywhere but the gravitational force exerted by the moon varies over the earth s surface. Note again that in addition to revolving around the sun without rotation, the earth spins around its axis. This rotation around its own axis is an entirely different issue and does not invalidate the findings about the balance between gravitational and centrifugal force with respect to the earth s revolution around the sun. Its only effect on the tides is that it moves the entire tide-generating force field around the earth once every day Strength of the tidal force The gravitational force exerted by a celestial body (moon, sun or star) is proportional to its mass but inversely proportional to the square of the distance. The larger distance between the sun and earth, compared to the distance between the moon and earth, means that the sun s gravitational force (and hence its tide-generating force) is only about 46% of that from the moon. Other celestial bodies do not exert a significant tidal force. Main tidal periods Tides produced by the moon M2 (semidiurnal lunar) 1/2 lunar day = 12h 25min O1 (diurnal lunar) 1 lunar day = 24h 50min Tides produced by the sun S2 (semidiurnal solar) 1/2 solar day = 12h K1 (diurnal solar) 1 solar day = 24h The tides can be represented as the sum of harmonic oscillations with these periods, plus harmonic oscillations of all the other combination periods (such as inequalities). Each oscillation, known as a tidal constituent, has its amplitude, period and phase, which can be extracted from observations by harmonic analysis. Hundreds of such oscillations have been identified, but in most situations and for predictions over a year or so it is sufficient to include only M2, S2, K1 and O1. Practical predictions produced on computers for official 53

58 tide tables use significantly more terms than these four; for example, the Australian National Tidal Facility uses 115 terms to produce the official Australian Tide Tables. Tidal Classification The form factor F is used to classify tides. It is defined as F = (K1 + O1) (M2 + S2) (4.2) where the symbols of the constituents indicate their respective amplitudes. Four categories are distinguished and are shown in Figure 4.2. Figure 4.2: Tidal types defined by the tidal form number. (Taken from Boon, 2004) Four examples of tidal records are shown in Figure 4.3, corresponding to each of the main tidal categories as outlined in Figure 4.2. Immingham: semidiurnal; two high and low waters each day. San Francisco: mixed, mainly semidiurnal; two high and low waters each day during most of the time, only one high and low water during neap tides. Manila: mixed, mainly diurnal, one dominant high and low water each day, two high and low waters during spring tide. Do San: diurnal; one high and low water each day. Note that in each case there is a pattern of increasing and decreasing tidal range over the lunar month. The tide producing forces arise as a consequence of the gravitational 54

59 Figure 4.3: Tidal records through March 1936 at four coastal stations illustrating variations in the amplitudes of the semi-diurnal (M 2 + S 2 ) and diurnal constituents (K 1 + O 1 ). (Taken from Mann and Lazier, 1996) attraction between the earth/moon and earth/sun systems. Since the earth, moon and sun are constantly revolving relative to each other, the magnitude of the tide producing forces varies in time. It turns out that this variation is periodic and that these forces are at their strongest when the sun and moon are either on the same side as the earth or on opposite sides. Tidal variations at this time are largest and these tides are known as spring tides. When the sun and moon are at 90 out of phase with respect to each other (during socalled half-moons), the tide producing forces are at their minimum and the resultant tidal variations are small. These tides are referred to as neap tides. Successive spring or neap tides occur approximately every 15 days. 55

60 4.1.4 Shape of the Tidal Wave The scales of variations in the forcing field are of global dimensions. Only the largest water bodies can accommodate directly forced tides. On a non-rotating earth the tides would be standing waves; they would have the form of seiches, that is, a back and forth movement of water across lines of no vertical movement (nodes). On a rotating earth the tidal wave is transformed into movement around points of no vertical movement known as amphidromic points. At amphidromic points the tidal range is zero. Co-range lines (lines of constant tidal range) run around amphidromic points in quasicircular fashion. Co-phase lines (lines of constant phase, or lines which connect all places where high water occurs at the same time) emanate from amphidromic points like spokes of a wheel. Note that on a rotating earth the tides take the shape of propagating waves, and the wave propagates around the amphidromic point in clockwise or anti-clockwise fashion. The shape Figure 4.4: The M 2 tide computed from a numerical model, showing amphidromic points (open circles), the phase is shown by solid lines marked in Greenwich hours, and the corange is shown by dashed lines in centimeters. (Taken from Mellor, 1996) of the tidal wave depends on the configuration of ocean basins and are difficult to evaluate. 56

61 Computer models can give a description of the wave on an oceanic scale (Figure 4.4). Their results have to be verified against observations of tidal range and times of occurrence of high and low water. Distortions of the wave on the continental shelf caused by shallow water make it difficult to assess results for the open ocean. In deep water the tidal range rarely exceeds 0.5 m. Figure 4.4 shows the M2 tidal wave. The complete solution consists of superposition of this result with the results for S2, K1 and O1 (and other constituents). Solid lines are cophase lines indicating the crest of the tidal wave at 0, 1, 2, 3,.10, 11 lunar hours after the moon s transit through the meridian of Greenwich (12 lunar hours correspond to 12 hours 25 minutes). Dashed lines indicate the tidal range (cm) due to the M2 tide. In the open ocean the tidal range is generally small ( m; it is zero at amphidromic points). The model indicates large tidal ranges closer to the coast; but the coarse model resolution does not reproduce the known extreme tides in the Bay of Fundy or on the French coast. (These regions show up with very large tidal ranges, though) Co-oscillation tides Tides in marginal seas and bays cannot be directly forced; they are co-oscillation tides generated by tidal movement at the connection with the ocean basins. Depending on the size of the sea or bay they take the shape of a seiche or rotate around one or more amphidromic points. If the tidal forcing is in resonance with a seiche period for the sea or bay, the tidal range is amplified and can be enormous. This produces the largest tidal ranges in the world ocean (14 m in the Bay of Fundy on the Canadian east coast; 10 m at St. Malo in France, 8 m on the North West Shelf of Australia and at the extreme north of the Gulf of California in Mexico; all are mainly semidiurnal tides). Tidal range is then largest at the inner end of the bay, in accordance with the dynamics of seiches in open basins. Modest amplification is exerienced in Spencer Gulf of South Australia where the tidal range at spring tide is 3 m at the inner end, while it is less than 1 m at the entrance to the Gulf. Figure 4.5 shows an example of a co-oscillation tide in Spencer Gulf. The tide is forced from the open end by the oceanic tide, which has a maximum tidal range (at spring tide) of about 1 m. Because of the width of the basin the Coriolis force is able to shape the wave, producing amphidromic points around which the wave propagates. Horizontal tidal currents Much of the above discussion relates to the vertical motion of the sea surface as tides propagate around the world or in regional seas and harbours. However, the propagation of tidal currents implies significant horizontal currents, particularly near the coast or in narrow straits. The horizontal flow associated with tidal oscillations is often represented by a tidal ellipse. Tidal ellipses for the M2 tide are shown in Figure

62 Figure 4.5: The M2 co-range (dashed contours; units cm) and co-phase (solid contours; units degrees relative to U.T.C.) obtained numerically for Spencer Gulf with boundary forcing (J.F Middleton Per. Comm.) The 4-5 fold increase in tidal range within the Gulf is likely the result of a 1/4 wave resonance mechanism. Figure 4.6: The M2 tidals ellipses, Red is counter clockwise, blue is clockwise. 58

63 A final example of tidal currents is given below. Tidal velocities in Sydney Harbour No wind scenario Tidal velocities in Sydney Harbour No wind scenario Tidal cycle: Tidal cycle: 10.5 School of Mathematics UNSW School of Mathematics UNSW Scale: 0.5m/s S.B 7.5 Scale: 0.5m/s S.B Distance (km) 6 M.H N.H Distance (km) 6 M.H N.H S.H S.H 4.5 P.R 4.5 P.R C.I. C.I. 3 3 S.H.B S.H.B Distance (km) D.B Distance (km) D.B Figure 4.7: Tidal velocities in Sydney Harbour on the flood (left) and ebb (right) tide, from a barotropic model simulation (Roughan et al., 2008). Figure 4.7 shows horizontal tidal currents on the flood tide (incoming high tide) and ebb tide (outgoing high tide) in Sydney Harbour from a computer model. Notice the acceleration of the tidal currents within the narrow parts of the Harbour. The coriolis force can also act to deflect flow to the left in this circulation pattern. 4.2 Estuaries Estuaries can be defined in a number of different ways, depending on your immediate point of view. As far as oceanographers are concerned, we define an estuary as a semi-enclosed coastal body of water having free connection to the open sea at least intermittently, and within which the salinity is measurably different from the salinity in the adjacent open sea (Tomczak and Godfrey, 1994). Most estuaries are found at river mouths, thus they are generally long and narrow, resembling a channel. They are regions where the variations in salinity are so large that the circulation is determined by the resultant density differences (thermohaline circulation). Estuaries tend to be regions of rich biological activity. The combined effects of high nutrient concentrations from freshwater runoff, and sufficient sunlight penetration (due to the shallow nature of most estuaries) promote vigorous plant growth. Plants provide food and shelter for juvenile fish, molluscs, crustaceans and larvae of many species. Furthermore, estuaries are an example of a delicately balanced ecosystem, which are sensitive to human activities. The study of estuarine circulation is necessary if we are to minimise the human impact, by a better understanding of such issues as water quality, sediment transport and biological activity. 59

64 4.2.1 Dynamics of estuaries Estuaries are regions of the coastal ocean where salinity variations in space are so large that they determine the circulation. They are related to mediterranean seas in the sense that the major driving forces for their circulation are thermohaline processes. They differ from mediterranean seas mainly in size and configuration. Most estuaries are found at river mouths; they are thus long and narrow, resembling a channel. Compared to the flow in the direction of the estuary axis, cross-channel motion is very restricted, and the estuarine circulation is well described by a two-dimensional current structure. Tides are a dominant physical process in estuaries, forcing strong currents which in turn feed energy to turbulent dissipation and mixing. The repetitive mixing of saline oceanic water with fresh river water creates a gradient of salinity (hence density) along the length of the estuarine channel. This along-channel salinity gradient has a tendency to flatten out (re-stratify). Hence there is competition between the restratification and the tidally-driven de-stratification by vertical mixing. Hence the balance of forces establishes a steady state in most estuaries (types 1-4 above) involving advection of freshwater from a river and introduction of sea water through turbulent mixing and tidal currents. The fluid mechanics of estuarine circulation is complex, however, it is possible to make some careful asumptions to find useful analytical results. A first step is to develop tidally-averaged ( sub-tidal ) equations for the evolution of the sub-tidal circulation and salinity structure. These help us to understand the basic estuarine properties such as the average stratification, the length of the salt intrusion and the residence time in the estuary. A rule of thumb about tidal averaging is that the tidal excursion should be small compared with the length of the salt intrusion. Starting with the equations for mass, momentum and density we can make various simplifications. Typically in an estuary we can express density in terms of salinity only since this (usually) dominates density changes. We assume that x is in the along channel direction and that in a narrow, straight channel, variations in the y direction (across-channel) are negligible, and hence terms involving v are negligible. We can eliminate pressure, expressing it in terms of η and salinity instead, The vertical turbulent eddy viscosity term is important, and is a complex tidally-averaged mix of shear and turbulence. Thus our simplified along channel momentum equation in an estuary is given by du dt = gdη d s + gβ dx dx z + du (KT M dz ) z (4.3) As previously, we can assume a steady state and hence eliminate the time varying component of u. By taking the vertical derivative of equation 4.3 we can eliminate the η x term, (for K T M constant in depth) to get an expression for the velocity: u zzz = gβ s x K T M (4.4) 60

65 Figure 4.8: Schematic diagram showing the basic processes that maintain the tidally averaged salinity and circulation structures in an estuary. Hence through integration, with appropriate boundary conditions, we can find that the steady sub-tidal velocity is given by the cubic: u = 3 2ū(1 ζ2 ) + u E (1 9ζ 2 8ζ 3 ) (4.5) 61

66 where the bar indicates the depth averaged componentand where u E gβ s x H 3 /(48K T M ) is the estuarine velocity and ζ z/h is the dimensionless vertical co-ordinate, which goes from -1 at the bottom to 0 at the surface. The way we have written the velocity in Equation 4.5 is the sum of two velocity profiles: a quadratic which carries the riverflow in a frictional channel, and a cubic which carries the exchange flow (where the subscript E in u E is for exchange ) as shown in Figure 4.9. Figure 4.9: Schematic diagram showing the two components of velocity (river flow and exchange flow) that contribute to the sub-tidal velocity profile in an estuary. Thinking more physically about the exchange flow, Figure 4.10 shows the physical interaction between the pressure forces due to surface height tilt and the salinity gradient which makes the exchange flow work. The fundamental equation governing salt storage in an estuary can be interpreted as a sum of the time rate of change of net salt in the estuary (landward of a point x) loss of salt due to the section averaged river flow advecting the section averages salinity gain of salt due to the exchange flow times the stratification (surface outflow has lower salinity and the deeper inflow generally has higher salinity) gain of salt due to all other processes such as tidal eddies, stirring the mean salinity gradient. which can be written as: x sdx + ū s = u t s + K H s x (4.6) head where the overbar denotes depth averaged and the prime denotes depth-varying components. 62

67 Figure 4.10: Schematic diagram showing the forces (excluding friction) that drive the alongchannel exchange flow in an estuary Classification of Estuaries Estuaries can be classified by their tidal range (e.g microtidal, meso tidal, macrotidal), by their topography (e.g drowned river valleys and fjords), and their morphological classification. As an oceanographer, we like to classify estuaries based on their salinity structure, i.e. the ratio of freshwater input to sea water mixed in by the tides Dyer (1997). Generally tidal mixing of freshwater from river input and seawater from the coastal ocean drives the circulation within and estuary. The majority of the motion is up and down the 63

68 channel, whilst cross channel motion is very restricted. Moreover, estuaries can be grouped according to the ratio of freshwater input to mixed seawater, which determines the properties and salinity distribution of each estuary. The most important estuary types are: 1. Salt Wedge Estuary 2. Highly stratified Estuary 3. Slightly Stratified Estuary 4. Vertically mixed Estuary 5. Inverse Estuary 6. Intermittent Estuary One way of quantifying this is by comparing the volume R of freshwater that enters from the river during one tidal period, with the volume V of water brought into the estuary and removed over each tidal cycle. R is sometimes called the river volume, while V is known as the tidal volume. It is important to note that it is only the ration R : V that determines the estuary type, not the absolute values of R or V. In other words, estuaries can be of widely different size and still belong to the same type. Salt wedge estuaries for example can be produced by a small creek in a nearly tide-free bay, or they can be of the scale of the Mississippi and Amazon rivers, which carry so much water that even strong tidal mixing is insignificant in comparison Salt wedge estuaries The salt wedge estuary is the simplest case, where the river volume is very much larger than the tidal volume. Low salinity river water flows in at the head of the river and spreads over the top of the more dense seawater that has come in through the mouth of the bay. The two layers are separated by a marked density discontinuity (Figure 4.11). Mixing occurs due to wind and wave action which results in entrainment of the underlying salty water up into the fresher surface waters. Vertical salinity profiles therefore show zero salinity at the surface and oceanic salinity near the bottom all along the estuary. Examples of salt wedge estuaries include the Mississippi and the Congo Rivers. Other examples may be as small as only a few kilometres long in regions with little tidal flow Highly stratified estuaries In a highly stratified estuary, river volume R is moderate to strong compared to tidal volume V. Strong velocity shear at the interface produces internal wave motion at the trnasition between the two layers. The waves break and topple over in the upper layer, causing entrainment of salt water upward. Entrainment is a one-way process, so no fresh water is mixed downward. This results in a salinity increase for the upper layer, while the salinity in the lower layer remains unchanged, provided the lower layer volume is significantly larger than the river volume R and can sustain an unlimited supply of salt water (Figure 4.12). Examples of this type of estuary are fjords, which are usually very deep and have a large salt water reservoir below the upper layer. 64

69 Fresh Water Inflow Brackish Water S = 0 S = 10 S = 35 Figure 4.11: Salinity in a salt wedge estuary. Fresh Water Inflow S = 0 psu S = 10 S = 20 S = 30 Figure 4.12: Salinity in a highly stratified estuary Partially Mixed Estuaries In a partially mixed estuary, the river volume R is small compared to tidal volume V. The tidal flow is turbulent through the entire water column (where the turbulence is induced mainly at the bottom). As a result, salt water is stirred into the upper layer and fresh water into the lower layer. Salinity therefore changes along the estuary axis not only in the upper layer (as was the case in the highly stratified estuary) but in both layers (Figure 4.13). There is some increase in surface velocity and upper layer transport towards the sea but not nearly as dramatic as in the highly stratified case. This type of estuary is widespread in temperate and subtropical climates; many examples are found around the world. 65

70 Fresh Water Inflow S = 0 S = 10 S = 20 S = 30 S = 35 Figure 4.13: Salinity in a partially mixed estuary Vertically mixed estuaries River volume R is insignificant compared with tidal volume V. Tidal mixing dominates the entire estuary. Locally it achieves complete mixing of the water column between surface and bottom, erasing all vertical stratification. As a result, vertical salinity profiles show uniform salinity but a salinity increase from station to station as the outer end of the estuary is approached (Figure 4.14). This type of estuary is found in regions of particularly strong tides; an example is the River Severn in England. Fresh Water Inflow S = 0 S = 10 S =2 0 S = 30 S = 35 Figure 4.14: Salinity in a vertically mixed estuary Inverse estuaries These estuaries have no fresh water input from rivers and are in a region of high evaporation. Surface salinity does not decrease from the ocean to the inner estuary, but water loss from 66

71 evaporation in the shallower waters of the inverse estuary leads to an increase in salinity. (Figure 4.15). This water is more dense and thus sinks to the bottom at the inner end of the estuary. The resulting circulation is an inflow in the surface waters and an outflow of heavier more dense water near the bottom. Inverse estuaries occur in some parts of Australia where evaporation is large and there is little freshwater inflow. Evapora on S = 38 S = 35 S = 37 S = 36 Figure 4.15: Salinity in an inverse estuary Intermittent estuaries Many estuaries change their classification type because of highly variable rainfall over the catchment area of their river input. River input may be small, but as long as some fresh water enters the estuary, the estuarine character is maintained (in the form of a salt wedge estuary). If the river input dries up completely during the dry season, estuaries loose their identity and turn into oceanic embayments. An example is the South West arm of Port Hacking south of Sydney which turns into a highly stratified estuary for a few weeks after heavy rains. The high environmental variability in an intermittent estuary has a huge effect on marine life, as very few plants or animals can cope with the dramatic salinity changes that occur between the estuarine and seawater phases. 4.3 Regional classification: a survey of Australian estuaries The basis for any estuarine classification is the ratio R/V of river volume per tidal cycle R over tidal volume V (the volume of water imported into the estuary and removed by the tide during one tidal cycle). Where this ratio is not know for a given estuary it can with reasonable accuracy be estimated from tidal and rainfall data, which are usually available. Tide tables can give a very good idea of the tidal range that can be expected along a coast, 67

72 and rainfall data can give an idea of the river discharges, especially when they are used in combination with information on catchment areas. When this principle is applied to Australia, it leads to several natural subdivisions of the coastline. The diagram below shows the tidal range for all major Australian coastal locations. It shows that the tides are extremely large on the North West Shelf, across the Timor Sea and in the Gulf of Carpentaria. The tides are of moderate range along the east coast and in the south east (Bass Strait and Tasmania) and decrease further to small tidal range in the south. The tidal range is very small to insignificant in the southwest and west. Figure 4.16: Left) Tides around Australia. The outer colours show the tide type, based on the form factor F (see the section on tides). More important in the present context is the tidal range, measured as the sum of the amplitudes of the four most important constituents M2 + S2 + K1 + O1 and shown by the inner colours. Right) Annual Mean Rainfall over Australia. Note how the majority of the rainfall occurs by the coast. The second piece of information required is rainfall. Annual mean rainfall can be used as a first approximation, but it can be misleading where the rainfall is highly seasonal. The annual average rainfall, shown below, indicates a subdivision of the Australian coast into a high to extreme rainfall region in the north, moderate rainfall in the east, south east and south west, and very low rainfall in the south and north west. If this is further detailed into summer rainfall and winter rainfall it is seen that the high rainfall in the north falls exclusively during summer, when the northwest monsoon brings moist air to the coast and with it daily thunderstorms; rainfall in the south west is predominantly winter rainfall; and rainfall along the east coast and in the south east is rather evenly distributed throughout the year. Additional information that needs to be taken into account include the extreme levels of evaporation in the south due to its proximity to the desert-like climate of Australia s interior; and the Great Dividing Range, a mountain system that stretches from north to south in close proximity of the eastern coast and limits the catchment areas of rivers flowing into estuaries along the eastern coast. 68

73 The information can be combined into a classification of Australian estuaries, shown in the following table: Region Tidal Range Rainfall Estuary Type Example East and Moderate Moderate Slightly Stratified to Hawkesbury River, Derwent South East Intermittent River (slightly stratified), Port Hacking (Intermittent) South Small Nearly none; Inverse Spencer Gulf, St Vincent Gulf High evaporation South West Small A little Slightly Stratified Swan River North and Large to Wet Season: Very Large Wet Season: Slightly South Alligator River Northeast extreme Dry Season: nearly Stratified Wenlock River none; high evaporation Dry Season: Salt Plug Figure 4.17: Classification of Australian Estuaries 4.4 Flushing times The ocean has always been a convenient place to dispose of unwanted material. Estuaries are always under pressure to serve as waste disposal areas, particularly for fluid waste. Sewage disposal into estuaries has been a practice for centuries without major adverse effects on the ecosystem; it becomes a serious problem when today s megacities continue the practice on a much larger scale. The introduction of new industrial production methods has greatly increased the list of potentially harmful waste products for which estuaries serve as dumping grounds. Managing the health of the estuarine ecosystem has therefore become a necessity. One of the tools of estuarine management is the flushing time concept. It is often used to determine how much of a potentially harmful substance an estuary can tolerate before its ecosystem is adversely affected to significant degree. While the flushing time concept is a legitimate scientific tool, it has to be understood as Harbours and bays and exactly that: a scientific aid for decision makers. The basic decision whether a particular substance should be disposed of and introduced into an estuary has to be made before the flushing time concept comes into play. The motivation for the flushing time concept stems from two problems of modern estuary management. The harmful effects of a potentially harmful substance is usually a function of its concentration, and knowledge of the flushing time can assist in determining allowable disposal loads for a particular estuary. In addition, knowledge of the flushing time can provide some guidance how to handle accidental spills of harmful or toxic material and design emergency procedures for industrial disaster situations. 69

74 4.4.1 Definition of flushing time Transport timescales within an estuary are useful quantities for comparing both regions within the estuary and comparing one estuary to another. Monsen et al. (2002) clearly identifies three transport timescales: flushing time, residence time and age of the water. These timescales are all limited by underlying assumptions, eg. steady flow or spatial homogeneity, or even the omission of important processes such as tidal diffusivity. However as a first order approximation estimates of residence and flushing timescales are useful. Flushing time describes the exchange in an estuary by way of bulk parameterizations, but does not identify the underlying physical processes (Monsen et al., 2002) and can be defined as: T f = V Q (4.7) where V is the total volume of the bay/estuary and Q is the volumetric flow rate through the system. In a tidal system with known basin geometry and tidal range the Tidal Prism approach can be utilised to estimate flushing time. The assumption here is that tides alone flush the system (ie there is no river inflow) and is given as: T f = V T (1 b)p (4.8) In this case P is the tidal prism i.e. the volume of water between high and low tide (with no river input), and is T the tidal period. The return flow factor b is a quantification of the fraction of water returning to the system each flood tide. This approximation is appropriate during dry periods where there is no freshwater (river) inflow. When we add river input to an estuary the flushing timescale changes. The Tidal Prism method for determining the flushing time of an estuary can be expanded to include freshwater inflow. The addition of freshwater acts to change the salinity of the estuary, and the flushing timescale also changes. T f = V T (4.9) (V t + V r ) Where the combined tidal prism value P = V t + V r represents the difference between high and low water (tidal volume plus river volume). The tidal prism estimate of flushing assumes that the salt water volume is fully mixed with the freshwater volume over the tidal cycle, before the water leaves the Estuary. The salinity of the freshwater entering the system is 0 psu. If the salinity of the salt water brought in from the rising tide is S o, the salinity S of the mixed water in the volume V t + V r is calculated from 70

75 S = V ts o (V t + V r ) (4.10) Regardless which method is used to derive a flushing time for a real estuary, the flushing time is only a very basic description of the flushing process. A single number may give a reasonable estimate of the time it takes to flush a bay, but it is not necessarily representative for the time required to remove an introduced substance from all parts of the estuary. Many estuaries have a complicated topography, which includes isolated depressions and secluded embayments with little water exchange. While most of a potential pollutant may be flushed from the estuary in the time estimated from any of the three methods, high pollutant levels can remain in pockets of stagnant water for much longer. Proper estuary management requires estimates of flushing times for all parts of the estuary. Another method that requires only verification with data from selected locations has become available as a result of the development of numerical modelling techniques. Numerical models derive flushing times for every location in an enclosed water body by calculating the time a water particle needs to reach the estuary mouth. This time is determined by particle tracking. The procedure results in a flushing time map, which gives flushing time as a function of location. Such maps highlight problem areas in estuaries where the circulation stagnates and can be used to improve the water movement through engineering measures. 71

76 5 Unforced Motions In this section we will consider unforced motions that satisfy the equations of motion described in Chapter 3 We will begin with surface gravity waves where the effects of the Earth s rotation are unimportant and then consider waves for which this is not the case. This will lead to the concept of the potential vorticity (or spin) of a fluid column which is vital in understanding large scale ocean circulation. Examples of such waves include those seen at the beach in the region before the surf zone. Other examples include tidal motions and tsunamis, although very long waves such as these are sometimes also influenced by rotational effects (discussed later). To model long surface gravity waves that are unaffected by rotation, we will consider motion in the (x,z) plane as shown below. Figure 5.1: Schematic diagram showing the basic properties of a long surface gravity waves In Figure 5.1, λ is the wavelength, η is the sealevel, A is the wave s amplitude, and h is the depth of the ocean. For long surface waves, the wavelength λ is much larger than the water depth h (so that the hydrostatic approximation will be valid). The effects of bottom friction and of the Earth s rotation may also be ignored. 72

77 Figure 5.2: (top) Types of progressive waves. (bottom) A diagrammatic view of an idealized ocean wave showing its characteristics. (Taken from Thurman and Trujillo, 2004) Exercise: Show that the hydrostatic approximation is valid for long surface waves using a scaling argument with T 10 s and h = 10 m. For surface gravity waves, where the waves effectively propagate on the interface between water and air, density variations within the ocean are small compared to the air-sea density difference. This means that we can assume a barotropic ocean (i.e., constant ρ). The momentum equation (Eq 3.14) and the equation for continuity of mass in Section 3.6 can then be written as du = g dη (5.1) dt dx dη dt + hdu dx = 0 (5.2) We can solve for η (eliminating u) by differentiating Eq 5.2 with respect to time t and Eq 5.2 with respect to distance x we obtain the wave equation d 2 η dt 2 ghd2 η dx 2 = 0 (5.3) 73

78 Let us consider a plane wave solution of the form η = A cos(kx ωt) (5.4) where ω = 2π/T is the frequency and k = 2π/λ is the wavenumber. Note that T is the period of the wave (i.e., the time between crests, at one location). Substituting (5.4) into (5.3) we get or ω 2 A cos(kx ωt) + hgk 2 A cos(kx ωt) = 0 ( ω 2 + ghk 2 )A cos(kx ωt) = 0 Dividing both sides by A cos(kx ωt) results in the condition or ω 2 /k 2 = gh c = ω/k = gh (5.5) The quantity c ω/k is called the phase speed and is the rate at which wave crests (or troughs) pass a fixed point. This may be seen from Eq5.4 where for η = A (a crest) we must have kx ωt = 0 (say) so that we must change x by the amount to keep up with the crest (Figure 5.3). x = ω k t x=ct c=ω/k Figure 5.3: Displacement of the crest of a wave 74

79 Now returning to Eq 5.5, the wave is only a solution if c = gh, so that if h = 2 m then c = 4.4 ms 1. If we know the wave period, say T = 10 s, then since c = ω/k = λ/t we have λ = ct so that λ = 44 m. Figure 5.4: As wind blows across the sea, wave size increases with increasing wind speed, duration, and fetch. As waves advance beyond their area of origination, they advance across the ocean surface and become sorted into uniform, symmetric swell. (Taken from Thurman and Trujillo, 2004) 5.1 Wave Refraction, Diffraction and Shoaling Refraction If you look out from any headland you will see that wave crests tend to become parallel to the shore as the wave moves inshore. Exercise A wave has a frequency ω = 2π/4s 1 in 100 m of water. Calculate c and λ. Calculate c and λ after the wave has reached a depth of 10 m of water. Explain what has happened. Now consider two parts I and J of a wave crest. Offshore, the crests move at a speed c I = c J = gh ms 1. As the wave moves further in, the crest J is in deeper water so that c J = gh J > c I = gh I since h J > h I. Thus, the wave crest at J moves faster and the wave tends to become more parallel with the shoreline. 75

80 I J 150m C I I C J 100 m J 50 m 20 m Coast Figure 5.5: Wave refraction as it approaches shallow water Shoaling Shoaling is the term for the changes in wave characteristics that occur when a wave reaches shallow water. As well as slowing, waves steepen as the depth decreases. This observation may be explained using the following formulae for the flux of energy Γ: where c is the wave phase speed and Γ = ce (5.6) E = 1 2 ρ 0gA 2 (5.7) is the total kinetic and potential energy of the wave (due to motion and sea level variations) for η = A cos(kx ωt). Now consider a profile of the beach shown in Figure 5.7. In the absence of friction and wave breaking the energy flux at I is equal to that at J so that and From Eq 5.7 we then have Γ I = c I E I = Γ J = c J E J (5.8) E J = (c I /c J )E I (5.9) A 2 J = (h I /h J ) 1 2 A 2 I so that with h J < h I, the amplitude of the wave A J is larger than A I, i.e. the wave steepens. 76

81 Figure 5.6: Wave refraction along a straight shoreline. Waves approaching the shore at an angle first feel bottom close to shore. This causes the segment of the wave in shallow water to slow, causing the crest of the wave to refract or bend so that the waves arrive at the shore nearly parallel to the shoreline. The arrows represent direction and speed of the wave. (b) Wave refraction along an irregular shoreline. As waves first feel bottom in the shallows off the headlands, they are slowed, causing the waves to refract and align nearly parallel to the shoreline. Evenly spaced orthogonal lines (black arrows) show that wave energy is concentrated on headlands (causing erosion) and dispersed in bays (resulting in deposition). (Taken from Thurman and Trujillo, 2004) 77

82 Ci Cj i Flux j Figure 5.7: Steepening / Shoaling of a wave as it approaches the shore. Figure 5.8: Co-cumulative wave spectra as significant wave height (H S ) and wave energy against frequency (f) and period (T) for three wind speeds (full lines), four fetches (dotted lines) and four durations (dashed lines). (Taken from Pond and Pickard, 1983) 78

83 An example Tsunamis A 2 m seismic shift in the earth s crust occurs north-east of Papua New Guinea in 5000 m depth of water forming a 1 m amplitude tsumani. Calculate the amplitude of the tsumani when it encounters depths of 100 m (the continental shelf) and 2 m (the beach zone). Also calculate the phase speed of the tsunami at its formation site (h = 5000 m) and at the continental shelf and beach zones. Figure 5.9: Abrupt vertical movement along a fault on the sea floor raises or drops the ocean water column above a fault, creating a tsunami that travels from deep to shallow water where it is experienced as alternating surges and withdrawals of water at the shore. (Taken from Thurman and Trujillo, 2004) In summary the decreasing depth causes: (a) An increase in wave height - The conservation of energy results in more energy forced into a smaller area. Since wave energy is proportional to wave height squared, this increases wave height as it propagates toward shore even though some of the energy is dissipated by bottom friction. (b) A decrease in wave speed - Remember that waves in shallow water have speeds that are dependent on the square root of water depth. As the depth decreases, so too will the wave speed. Refraction is the change in direction of a wave due to a change in its speed. This is most commonly seen when a wave passes from one medium to another, or in the case of the ocean, moves from deep water into shallow water. Refraction of ocean waves generally occurs as they approach the shore. This process unevenly distributes wave energy along the shoreline, and results in erosion at headlands and deposition on beaches. Diffraction refers to the various phenomena associated with wave propagation, such as the 79

84 bending, spreading and the interference of waves passing by an object. In the ocean wave diffraction results from wave energy being transferred around or away from barriers impeding its forward motion. e.g. waves move past barriers into harbours because their energy moves laterally along the crest of the wave, and the wave behind the barrier goes out in all directions (Thurman and Trujillo, 2004) Wave Breaking As the wave moves into shallower water, shoaling affects the wave form by slowing its base while having less effect on the crest. At some point, the crest of the wave is moving too fast for the bottom of the wave form to keep up. The wave then becomes unstable and breaks. For a wave of the form η = A cos(kx ωt) observations show that steepening will occur and the wave will break if the wave height gets sufficiently large and H λ/12. For tsunamis, where the wavelength is initially very long (hundreds of kilometres), this criteria will not be met, even when the wavelength shortens in shallower water. In these instances, theoretical derivations demonstrate that breaking occurs when. H > 0.8h. 5.2 Shallow and Deep Water Waves Because the phase speed c = ω/k depends on the wave number, wave components of different wavelengths travel at different speeds, and the wave is said to be dispersive. The phenomenon is called dispersion hence the name dispersion relation is given to the following equation. ω 2 = gk tanh(kh) (5.10) This important equation determines the frequency and hence the phase speed of waves of a given wave number. This means that waves of different wavelengths starting at the same place will move away at different speeds and thus will disperse or spread out. The waves described above are restricted to the case where λ h, or kh 1. More generally, it can be shown that c = ω k = [ g k tanh(kh) ]1 2 (5.11) 80

85 Figure 5.10: As waves approach the shore and encounter water depths of less than one-half wavelength, the waves feel bottom. The wave speed decreases and waves stack up against the shore, causing the wavelength to decrease. This results in an increase in wave height to the point where the wave steepness is increased beyond the 1:7 ratio, causing the wave to pitch forward and break in the surf zone. (Taken from Thurman and Trujillo, 2004) noting that tanh(x) tends to x for small x and tanh(x) tends to 1 for large x and if λ h(kh 1) this reduces to shallow water waves above where c = ω k = gh. Where λ < 2h we have deep water waves and Eq 5.11 becomes c = ω k = g k = gλ 2π which is the dispersion relation for deep water waves (Cushman-Roisin, 1994). In reality there is a transition between these two types of surface waves, known as the transition zone (Figure 5.12). 81

86 Figure 5.11: Wave speed versus water depth for various wavelengths. (Taken from Pond and Pickard, 1983) Figure 5.12: (a) Deep-water waves, showing the diminishing size of the circular orbits with increasing depth. (b) Shallow-water waves, where the ocean floor interferes with circular orbital motion, causing the orbits to become more flattened. (c) Transitional waves, which are intermediate between deep-water and shallow-water waves. All diagrams are not to scale. (Taken from Thurman and Trujillo, 2004) 82

87 Figure 5.13: Short and long-wave formulae and sample values. (Taken from Pond and Pickard, 1983) Figure 5.13 shows a table of some key formulae. Examples are also given for typical wavelength, phase speed, and period of wind waves ( chop ), swell, and tsunamis. 83

88 5.3 Long waves affected by the Earth s Rotation In the above the effects of the Earth s rotation have been ignored. However, if the geostrophic terms are included, then it may be shown that the wave equation (Eq 5.3) is modified to be d 2 η dt η 2 ghd2 dx + 2 f2 η = 0 (5.12) To see the effect of f 0 on a wave, again substitute the solution into Eq In this case we obtain η = A cos(kx ωt) ω 2 = f 2 + k 2 gh (5.13) This is called the dispersion relation for long waves affected by rotation. Let us denote c 0 = gh the wave speed if f=0. In this case we may write Eq 5.13 as ω 2 = f 2 + k 2 c 2 0 (5.14) ω 2 /k 2 = c f 2 /k 2 or ω 2 /k 2 = c 2 0 (1 + λ 2 /r 2 e) (5.15) where r e is a length scale called the (external) deformation radius given by r e = 2πc 0 / f (5.16) Clearly from equation Eq 5.15, when the deformation radius is much larger than the wavelength of the wave, i.e., r e >> λ, we get c = ω/k = c 0 = gh, the dispersion relation for long waves unaffected by rotation. Thus, the deformation radius is a critical parameter that tells us whether or not the wave in question is long enough to feel the effects of rotation (Gill, 1982). An example Consider Botany Bay where h 20 m and f 10 4 s 1. In this case r e = 2π gh/ f 890 km, and is much larger than the bay. For a wave λ 100 m it is clear that λ 2 << r 2 e so that (Eq 5.15) reduces to ω 2 /k 2 = c 2 0 Since c = ω/k is the speed at which the wave moves, the above shows that c = c 0 i.e. the wave moves at the same speed obtained for the case where the Earth s rotation was 84

89 unimportant. This would be the case for any waves within the bay as r e is much longer than the width of the bay (about km). Offshore, however, long tidal waves can be thousands of kilometres in wavelength, so rotational effects would become important. In this case, λ cannot be neglected compared to the deformation radius r e. Then (Eq 5.14) may be written as c 2 = c f 2 /k 2 This shows that the effect of the Earth s rotation is to make the waves travel faster; c > c 0. This effect gets stronger as you go to higher latitudes, (towards the poles, f 2 increases). What these results show is that the deformation radius is a fundamental length scale for the importance of the Earth s rotation. Another way of writing Eq 5.15 is as ω 2 = f 2 (1 + r 2 e/λ 2 ) (5.17) or ω f = 1 + r 2 e/λ 2 This is shown in the dispersion plot in Figure For k 2 r 2 e >> 1 (λ << r e ) we have ω = c 0 k (surface gravity waves) While for k 2 r 2 e << 1, we get ω f. Such waves are called inertial waves; they have a frequency that equals the inertial frequency f of the Earth s rotation. In general, long waves affected by rotation are called Poincaré waves. The most common example are the tides that propagate around the earth. 5.4 Internal Waves The above may also be simply extended to consider the motion of internal waves in a stratified fluid. Let us consider a two layer fluid as a model for stratification. The top layer is light of density ρ 1 (say oil) while the bottom layer is heavy of density ρ 2 (say water). Now provided that the lower layer depth h 2 is much larger than h 1, and if (ρ 2 ρ 1 )/ρ 2 << 1 then the equation for displacements η of the interface is d 2 η dt 2 g h 1 d 2 η dx 2 + f2 η = 0 (5.18) and identical to Eq:5.12 except that h and g have been replaced by h 1 (the top layer depth) and ( ) g ρ2 ρ 1 = g << g (5.19) ρ 2 85

90 Figure 5.14: (a) A wave progressing from left to right on the density interface showing the instantaneous particle streamlines, the resultant surface wave, the zones of convergence and divergence, and the region where slicks are likely to form on the surface. (b) The particle orbits over one cycle due to the passing internal wave. The diameter decreases away from the interface and the direction of the trajectory reverses across the interface as indicated by the double arrows on the central orbit. (Taken from Mann and Lazier, 1996) the reduced gravity. This quantity reflects the fact that the restoring force at the interface depends on the density difference of the two layers. If ρ 2 = ρ 1 then there is no restoring force. If ρ 2 >> ρ 1 as for the case of the air sea interface then g g. Solutions to (Eq 5.19) are again waves but now the dispersion relation (Eq 5.17) becomes where c 2 = ω 2 /k 2 = c 2 i (1 + λ 2 /r 2 i )ω 2 = f 2 (1 + r 2 i /λ 2 ) (5.20) c i = g h 1 is the internal wave speed when rotation effects are unimportant (shown below) and is called the internal deformation radius. r i = 2π g h 1 / f An example Consider an internal wave in Botany Bay, propagating on a sharp summer-time thermocline where the surface density is 1027 kg m 3 and the deeper cold water has density 1029 kg m 3. Calculate g and r i for this internal wave if the upper warm layer is h 1 = 5 m thick. Compare 86

91 the scale of r i (the internal radius) with r e (the external radius) and comment on the role of rotation on waves in Botany Bay. Hopefully it is clear from the dispersion relations (Eq 5.20) that the length scale r i is important in determining the effects of the Earth s rotation for internal waves. For waves of small scale compared to r i we have λ 2 << r 2 i and Eq 5.20 reduces to c 2 = ω 2 /k 2 g h 1 or c = ω k = g h 1 The internal waves move at a speed g h 1 (not gh) and for the example above c = 0.31 ms 1. Since g << g and h 1 << h (total depth) the internal waves move much slower than those on the surface. This can be demonstrated in a wave tank using fluids of 2 different densities. For much larger scale waves where the Earth s rotation is important and the term λ 2 /r 2 i cannot be neglected, we have c 2 = c 2 i + f 2 /k 2 (4.19) where c i = g h 1 is called the internal phase speed when rotational effects are unimportant. Note that rotation again tends to speed up the wave s speed. Note also that the speed at which the waves move is always given by c = ω/k (for internal and external waves with or without rotation). 5.5 Potential and Relative Vorticity It turns out that an important conservation law governing flow in the ocean (and atmosphere) dictates that the total spin of a fluid column remain constant in proportion to the column s height. To introduce this relationship, we first need to understand what the spin or vorticity is for fluids flowing on a rotating planet. Relative vorticity The relative vorticity ζ is related to the spin directly associated with the fluid motion. It is defined as ζ = dv dx du (5.21) dy This is clearly related to the (horizontal) velocity shear, as without any contrast in horizontal motion (i.e., u,v), the relative vorticity ζ = 0. As shown in Figure 5.15 a cork near a bathtub wall will spin since at the wall the velocity v = 0 (friction) while further out v 0. Since dv/dx > 0, ζ > 0 and the spin is positive: we use a right hand rule as shown. 87

92 =2Xspin = dv/dx>0 x v y wall Figure 5.15: Spin of a cork due to velocity shear at a wall. Planetary vorticity The planetary vorticity f, that we have seen in the geostrophic equations, and so on, is just the spin on the fluid due to the earth s rotation. That is, if the fluid is at rest or has no relative vorticity, it will still have some spin on it due to the fact that it is sitting on a rotating sphere (the Earth). This vorticity would be detectable from space, for example. Figure 5.16 shows the value of f as a function of latitude. =f/2 North Pole Figure 5.16: The rotation of the earth relative to the north pole. 88

93 Total vorticity The total or absolute vorticity is just ζ + f. Potential vorticity The potential vorticity Q is the total vorticity divided by the height of the fluid column, namely: Q = ζ + f (5.22) h + η In looking at the effect of the Earth s rotation on waves an additional term f 2 η appeared in Eq 5.12 that was not present before. This term may be shown to arise from a requirement that the potential vorticity (or spin) Q = (ζ + f)/(h + η) be conserved following a fluid column. As we will see, the fact that Q is conserved (unchanged) following a fluid column will permit us to understand ocean fronts and large scale ocean circulation. Now to understand Eq 5.22 remember that the Coriolis parameter f may be regarded as the spin of a fluid column as seen from outer space in the absence of any other motion. Indeed, at the poles f = 2Ω where Ω = 2π radians/day. The total spin of a fluid column will consist of both ζ +f as observed from space. The length h + η of the fluid column becomes important because the potential vorticity or total spin is conserved (unchanged) following a fluid column i.e. d dt ( ζ + f h + η ) = 0. (5.23) This result may be understood by considering the skater who with arms outstretched has a spin f and Q = f/h. As she brings her arms in her effective height increases. Now since her initial and final potential vorticity remain the same Q i = f h i = Q f = f + ζ h f and h f > h i we have an increase in her relative vorticity: ( ) hf ζ = f > 0 (4.23) i.e. she spins faster, (right hand rule). h i The same is also true for a fluid column, cyclonic vorticity is acquired (ζ/f > 0) for vortex stretching and anticyclonic vorticity (ζ/f < 0) for vortex squashing: the column of fluid might be regarded as a vortex. Figure 5.18 shows this for flow in the Northern Hemisphere. 89

94 f f+ h i h f Q i = f/h i = Q f = (f+ )/h f Figure 5.17: Column squashing and stretching with rotation. Figure 5.18: Change of absolute vorticity associated with (a) convergence, (b) divergence (both for Northern Hemisphere). (Taken from Pond and Pickard, 1983) 90

95 5.6 Geostrophic Adjustment Suppose that water has been piled up against the coast as shown in Figure 5.19 but that there is no motion. Now to infer the final sea-level and velocity field we expect that the water will move away from the coast (due to the pressure gradient), but that ultimately a sea-level slope may exist that is in geostrophic balance fv = g dη dx In addition we may consider the conservation of potential vorticity Q = (ζ + f) h(total) of three columns (a), (b) and (c) as shown during the adjustment process and also note that ζ = dv dx At column (c) nothing happens since the depth does not change, dη/dx = 0 and so v = 0. h a b c x x Figure 5.19: Left: Piling of water against the coast, Right: Geostrophic adjustment of three columns of water and subsequent change in sea level. The velocity u in the x direction will also everywhere zero since we assume there is no pressure gradient in the y direction. For the other columns Eq 5.22 can be rewritten as ζ f f = h f h i 1, where the subscripts i and f refer to the initial and final states. For column (b) stretching occurs so ζ f /f > 0 and the spin will be as for the right hand. In this case we then have dv/dx > 0 so that v must be increasing. For the column (a), squashing occurs and ζ f /f < 0, left hand spin, and dv/dx < 0 so that v is decreasing. Putting this information together we get the following picture of the velocity field (as seen from above): 91

96 v>0 0 a b c x Figure 5.20: Plan view of velocity vectors v Exercise: A fluid column or eddy at the north pole is observed by a spy satellite to be rotating at a rate of 2(2π π) radians per day. (1 day= s). Given f = 2(2π/86400) s 1 calculate ζ. (a) If the fluid column of depth h I = 4000 m passes over a ridge (h f = 3500 m) describe the increase in ζ using the concept of conservation of potential vorticity. (b) If ζ = dv/dx is constant within the eddy of radius 100 km, calculate v at x = 100 km both before and after it passes over the ridge (assume v = 0 at x = 0, the center of the eddy). (c) If v is in geostrophic balance calculate the sea-level field of the eddy both before it crosses the ridge. Assume ζ = 0 at the edge of the eddy. 92

97 6 Wind Forced Motion and Large-Scale Gyre Circulations The wind blowing on an ocean produces a force per unit area called a wind stress τ. The surface wind stress can be related to the wind velocity through a relationship known as the bulk formula. (τ x,τ y ) = ρ air c D u 10 (u a,v a ) (6.1) where τ x and τ y are respectively the zonal (x direction) and meridional (y direction) components of wind stress, c D is a bulk transfer coefficient for momentum (typically c D = ), ρ air is the density of air at the sea surface ( 1.3 kg m 3 ) and u 10 is the speed of the wind at a height of 10 m above the ocean. 6.1 Ekman Layer The effects of a wind stress on the ocean surface are transmitted down through the water column by the action of turbulent eddies that are themselves generated by the wind, breaking waves and boundary shear stresses. The depth to which the effects of wind are felt is called the Ekman layer thickness or Ekman depth (H e ), where H E = (2K/ f ) 1 2 (6.2) where K is the eddy diffusivity and K WL where W and L denote a characteristic eddy velocity and size. Typically K m 2 s 1 so that with f = 10 4 s 1, H E 20 m. However in regions of high wind-driven turbulence, K can be up to 0.5 m 2 s 1, so that H e can reach 100 m. Now to understand the effects of wind consider the steady equations of motion for a barotropic ocean of depth h. In the absence of wind and if the bottom stress is weak, these equations of motion would be the geostrophic equations plus the effects of wind forcing, which then enter as an extra frictional term, that is fv = g dη dx + τ x ρh fu = g dη dy + τ y ρh (6.3) (6.4) where τ x and τ y are the x and y direction components of the wind stress. Now for simplicity, let s split the velocity field u and v into geostrophic and wind-driven components; that is: u = u g + u e (6.5) v = v g + v e (6.6) 93

98 where (u g,v g ) denote geostrophic velocities such that there is a balance between the pressure gradient and the Coriolis force: fv g = g dη dx fu g = g dη dy (6.7) (6.8) Hence the remainder of velocity (u e, v e ) must balance the force of the wind: fv e = τ x ρh fu e = τ y ρh (6.9) (6.10) The velocities (u e,v e ) are called the (depth-averaged) Ekman velocities and while driven by the wind stress (τ x, τ y ), the velocities (u e, v e ) are at right angles to it. 6.2 The Ekman Velocity To understand the effects of the wind let us assume it blows in the y direction only with a force per unit area (a stress) denoted by τ y. Assuming there are no sea level gradients, the equations of motion, averaged over the Ekman layer depth may be written as du E fv E = 0 (6.11) dt dv E + fu E = τ y (6.12) dt ρh E The horizontal velocity averaged over the Ekman layer is denoted by (U E,V E ). Initially there is no motion and the wind accelerates the column shown in Figure 6.1 in the y-direction according to dv E /dt = τ y /ρh E. As the column moves faster, the Coriolis force fv E gets larger and the column is deflected to the right (Northern Hemisphere): du E /dt = fv E. This continues until V E vanishes and a steady state is reached fu E = τ y ρh E (6.13) and the Coriolis force and wind stress balance.the column moves at right angles to the wind. Hence the force balance on the fluid column is in this case between the Coriolis force and the force of the wind. 94

99 y/ HE U E fu E y V E y fv E x Figure 6.1: Plan view of the motion of a column of water subject to an initial acceleration by the wind. An example: Calculate the Ekman velocity U E given a wind stress τ y = 0.1 Pa (0.1 Nm 2 ) Assume an Ekman layer depth of H E = 20 m, and f = 10 4 s 1 (Equation 6.13). Compare this to the conditions off northern California (Figure 6.8). 6.3 Depth Averaged Ekman Layer Now to see what happens with depth we note that the wind only directly affects motions over the depth H e, not the entire depth h. Thus the velocity u e,v e only occurs in the Ekman layer and its value here must be (h/h e ) times the full depth-averaged velocity u e,v e. This is called the Ekman layer velocity and is given by U e = h H e u e = τ y ρh e f τ x V e = h v e = H e ρh e f (6.14) (6.15) since u e and v e represent velocity averaged over depth h. As shown in Figure 6.2, a northsouth wind stress drives a velocity U e in the east-west direction - that is, at right angles to the wind. Checking the signs in Equations 6.14 and 6.15 we can see that flow is to the right in the northern hemisphere, and to the left in the southern hemisphere. 95

100 Side View y H E UE Ekman Layer Figure 6.2: Side view of the movement of a column of water in the Ekman Layer in the Northern Hemisphere. Note: There is no motion below the Ekman layer as there is no slope in sea level, hence the geostrophic velocities are zero. An example a) Find the Ekman layer depth H e and the Ekman velocities u e and v e when a wind stress of 0.5 Pa (0.5 Nm 2 ) blows in a north-easterly direction in the mid-latitudes of the northern hemisphere, where the depth of the ocean h = 1000 m. Take the vertical diffusivity to be K = 0.1 ms 1 and f = 10 4 s 1. b) What is the Ekman layer velocity U e,v e? c) Sketch the resultant flow. 6.4 Storm Surge, Downwelling and Upwelling Storm Surge Let us now consider the effects of an alongshore wind stress on coastal circulation. Two situations can arise depending on the direction of the alongshore winds. The first situation is where a southward wind blows along the east coast of a land mass in the northern hemisphere, shown in Figure 6.3. The effect of the coast is to block the onshore Ekman flux resulting in the raising of sea level. Since η now changes with x, an alongshore geostrophic velocity v g = g f must exist to balance the force due to dη/dx. dη dx < 0 96

101 Sea level raised τy Vg Ue Figure 6.3: Idealised model of a storm surge (Northern Hemisphere). x Thus wind forcing results in the transport of water towards the coast in a thin layer H e, which can in turn raise sea level and drive an alongshore velocity Downwelling Where the ocean is stratified (density increases with depth), the onshore Ekman flux also pushes water down along the continental shelf leading to mixing and the growth of thick bottom boundary layers. The thermal wind shear near the shelf opposes, the alongshore current v g = (g/f)dy/dx and an undercurrent flowing in the opposite direction can result. Exercise: Use the thermal wind relation (3.21 and 3.22) and equation of state (1.1) to show this is so. τy Vg U E Inc. Density Figure 6.4: Idealised downwelling scenario in the Northern Hemisphere. 97

102 6.4.3 Upwelling Now consider the two layer coastal model in Figure 6.5. With the wind stress directed out of the page (SH example) the Ekman flux is directed off shore so that the coastal sea level must drop. However, we might expect the interface to rise as well and water (and nutrients) to be drawn from the deep ocean as shown. 0m Sea level is lowered Ue τy Ue τy Depth (m) 50m 18 o C 17 o C 16 o C Upwelling in BBL Vg 18 o C 17 o C 16 o C 100m Upwelling in BBL Figure 6.5: Schematic diagram of wind driven upwelling in the Southern Hemisphere. Figure 6.6: Ekman transport. While coastal upwelling occurs in only about 1% of the oceans area, the increased biological productivity accounts for about 50% of the worlds fish catch. Figure 6.12 shows the distribution of winds over the ocean, Note that the coast of Peru, west Africa and California are subject to strong upwelling favourable winds. Ekman layer depths vary with wind forcing (strength and duration). An example is found in the Northern Californian upwelling region during the WEST study (Largier et al., 2006; 98

103 Figure 6.7: Ekman transport and associated upwelling and downwelling resulting from wind blowing parallel to shore. (Taken from Sarmiento and Gruber, 2006) Roughan et al., 2006). Figure 6.8 from (Dever et al., 2006) shows data from a mooring in 90 m of water, showing along and across shelf wind (ms 1 ), temperature ( C) and cross-shore and along shore velocities throughout the water column. The coldest (warmest) temperatures correspond with the periods of strong (weak) alongshore wind forcing, and offshore (onshore) transport in the surface waters. 99

104 Figure 6.8: An example of upwelling favourable winds and the ocean response at the 90 m isobath off the Northern Californian coast from (Dever et al., 2006). (A) along-shelf (solid) and cross-shelf (dashed) wind velocity, (B) temperature with surface and bottom mixed layers (black lines), temperature logger depths (black triangles) (c) cross-shelf velocity with surface and bottom mixed layers (black lines), and Pollard Rhines and Thompson scaling (magenta lines) and (D) along-shelf velocity with surface and bottom mixed layers (black lines). 6.5 Ekman Pumping In the above we have implicitly assumed that the wind stress τ y and thus Ekman flux U E are constant. What if τ y changes in space (e.g with x). In the schematic in Figure 6.9 it is apparent that the flux U E will converge at the origin leading to a flux of water w E out of the Ekman layer. 100

105 y U E U E x W E z Figure 6.9: Ekman Pumping, resulting from a curl (change) in the wind stress. Figure 6.10: (a) Equatorial divergence and upwelling resulting from Ekman transport driven by trade winds. (b) Ekman downwelling resulting from convergent Ekman transport driven by the westerlies and trade winds. (Taken from Sarmiento and Gruber, 2006) The velocity w e is called the Ekman pumping velocity and given by or w e (z = H e ) = H e du e dx (6.16) w e = d dx (τ y/ρf) (6.17) The Ekman pumping velocity corresponds to a velocity of fluid leaving or entering the base of the Ekman layer so as to compensate the horizontal variations in the horizontal flow due 101

106 to the Ekman flux U E. These result in either convergence or divergence depending on the direction of the Ekman transport. Figure 6.11: The effect of a cyclonic wind in the Northern Hemisphere (a) on surface waters, (b) on the shape of the sea-surface and thermocline. Diagrams (c) and (d) show the effects of an anticyclonic wind in the Northern Hemisphere. (Remember that in the Southern Hemisphere, cyclonic = clockwise and anticyclonic = anticlockwise.) (Taken from OpenUniversity, 2002) 102

107 Figure 6.12: Map showing the annual mean distribution of winds around the world. Which regions are upwelling favourable? Figure 6.13: Surface convergence and divergence zones. 103

108 An example: a hurricane Consider the single layer ocean in Figure 6.14 that is subject to the wind stress field shown (a hurricane). τy τy Ue Ue V We x z Figure 6.14: Schematic Diagram of the sea surface response to a Hurricane. Note that the Ekman layer thins leading to vortex stretching of the lower layer (ζ/f 0). For the northern hemisphere U e is directed to the right of the wind τ so that there is a loss of fluid in the Ekman layer. However, this fluid is replaced by the upward Ekman pumping velocity w e. Note that not all the fluid is replaced and we might expect sea level to dip in the centre as shown. The circulation in the ocean in response to the cyclone may then be inferred from this gradient of sea level since it will adjust by geostrophy, that is: v g = (g/f) dη dx so that v g > 0 (into the page) on the right of the origin 0 and v g < 0 on the left of the origin. For regions far from the cyclone, v g = 0 (as sea-level flattens eventually), and at x = 0, v g = 0 since dη/dx is zero. Thus the hurricane or cyclone induces a circulation similar to itself in the ocean. A second way to understand the circulation is in terms of vortex stretching. The lower layer is effectively stretched by the wind-driven upwelling as shown. Thus the stretching will lead to the production of cyclonic vorticity, ζ/f > 0 as shown above. 104

109 The Ekman layer displacement η e is directly related to thepumping velocity w e w e = dη e dt Remember that by definition we take z positive downwards throughout the course, so that w e is negative when upwelling occurs. Remember that sea level displacements are the exception to this rule, that is, η and η e are positive when sea-level rises; hence the negative sign in Eq Since w e is negative (upwelling) η e will be positive so that the vortex column in the interior will be stretched. Figure 6.15: Two sections showing isopycnals as observed several days after the passage of a Hurricane across the Gulf of Mexico. The offshore Ekman transport and resultant Ekman upwelling has raised the isopycnals by more than 60 m near the hurricane centre. Exercise Use the thermal wind relations to estimate the change in velocity over a depth of m at the position indicated by the arrow for section C-C in Figure

110 6.6 A Model for the Large Scale Ocean Gyre Circulation A summary of the observed circulation in the Atlantic Ocean is presented below. In the northern hemisphere, the flow is dominated by a clockwise gyre, in the southern hemisphere, by an anti-clockwise gyre. These gyres are analogous to large-scale oceanic eddies or vortices, only they are stationary and in steady-state (i.e., d dt 0). Focusing on the example of the Gulf Stream, weak equatorward flow persists over most of the gyre, which is returned poleward in an intense western boundary current (WBC). In fact, the same occurs in the southern hemisphere, only the WBC is the southward flowing Brazil Current. Similar subtropical gyres dominate the flow patterns of all three ocean basins. A schematic summary of the northern hemisphere gyre is included below, showing the circulation along with the eastward wind field τ x and its gradient dτ x /dy. As we will see, the circulation can be understood in terms of vortex stretching that arises from an Ekman pumping velocity w e which is driven by the wind. 106

111 Figure 6.16: Surface currents of the Pacific Ocean. Abbreviations are used for the Mindanao Eddy (ME), the Halmahera Eddy (HE), the New Guinea Coastal (NGCC), the North Pacific (NPC), and the Kamchatka Current (KC). Other abbreviations refer to fronts: STF: Subtropical Front, SAF: Subantarctic Front, PF: Polar Front, CWB/WGB: Continental Water Boundary/Weddell Gyre Boundary. The shaded region indicates banded structure (Subtropical Countercurrents). In the western South Pacific Ocean the currents are shown for April-November when the dominant winds are the Trades. During December-March the region is under the influence of the northwest monsoon, flow along the Australian coast north of 18 S and along New Guinea reverses, the Halmahera Eddy changes its sense of rotation and the South Equatorial Current joins the North Equatorial Countercurrent east of the eddy. (Taken from Tomczak and Godfrey, 1994) 107

112 Figure 6.17: Surface currents of the Indian Ocean. Late Northeast Monsoon season (March- April). Abbreviations are used for the East Arabian (EAC), South Java (SJC), Zanzibar (ZC), East Madagascar (EMC), and Somali (SC) Currents. Other abbreviations denote fronts: STF: Subtropical Front, SAF: Subantarctic Front, PF: Antarctic Polar Front, WGB: Weddell Gyre Boundary. (Taken from Tomczak and Godfrey, 1994) The Ocean Gyre In the case above where the wind field τ x changes with y (north south distance) the Ekman pumping velocity in the gyre is given by w e = dτ x (6.18) ρf dy

113 Figure 6.18: A schematic of wind distribution and the resulting current system in the Atlantic Ocean, the western boundary current is the Gulf Stream whereas, in the Pacific, it is the Kuroshio Current. (Taken from Mellor, 1996) where w e is the vertical velocity at the Ekman layer base w e = dη e dt (6.19) If w e is positive we get vortex squashing with the production of anticyclonic vorticity, that is, (ζ +f) must decrease following a water parcel in the gyre. Using geostrophy or conservation of potential vorticity it is clear that the gyre flow matches the direction of the wind circulation (as with the tropical cyclone example above). From these examples, it is hopefully clear that w e will be of opposite sign to the rate of change in total vorticity in an ocean gyre or eddy. For example, w e < 0 accompanies vortex stretching which (by conservation of PV = vorticity ) means vorticity must increase. The height opposite holds for the Gulf Stream gyre above. Formally, the equation expressing the way in which total vorticity (ζ +f) changes below the Ekman layer turns out to be d dt (ζ + f) = f h w e (6.20) where it is assumed that sea level variations η are unimportant, and h the depth is taken as constant. Therefore, using (6.18) for w e we obtain d 1 dτ x (ζ + f) = dt ρh dy 109 (6.21)

114 We can see that for the North Atlantic gyre dτ x /dy > 0 so that w e > 0 (Eq. 6.18), i.e. the Ekman transport is towards the center of the gyre leading to a positive pumping velocity. The tendency is towards the production of anticyclonic vorticity, d(ζ + f)/dt < 0, or vortex squashing. However, in the gyre region, away from the intense western boundary current, observations show that u and v are very small (of the order of cm/sec) and their spatial variation (or velocity shear) is weak. That is, ζ = dv dx du dy 0 (6.22) What happens? If ζ 0 how can the total vorticity (ζ + f) decrease to compensate for the squashing in the gyre? The answer turns out to be that the fluid flows in a way to change its planetary vorticity f, as we shall see below. Figure 6.19: A simple ocean basin. For the Atlantic Ocean, the boundaries, x = 0 and x = a, would roughly correspond to 75 W and 10 W whereas y = 0 and y = b would correspond to 15 N and 50 N. Labels on the streamlines are values of the transport streamfunction in units of 10 6 m 3 s 1. (Taken from Mellor, 1996) Conserving potential vorticity (PV) in an ocean gyre The Coriolis parameter is given by f = 2Ω sin φ where φ= latitude (in degrees). As shown below f is zero at the equator (φ = 0) and f = 2Ω at the north pole (φ = 90 N). Over the region of the gyre 10 to 50 N we may approximate the variation in f by the straight line f = f 0 + βy (6.23) where f 0 = 2Ω sin(10 ) and β = (ms) 1. The distance north y is in metres. Now returning to our model for the ocean gyre we have ζ 0 from (6.22) so that (6.21) becomes df dt = f h w e (6.24) 110

115 If we now take f = f 0 + βy in the right hand side and note that d means the time rate of dt change following a fluid column then df dt = βdy dt = βv (6.25) where v = dy/dt is the northwards velocity of the fluid column. Combining (6.24) and (6.25) we get i.e. the Ekman pumping acts to drive a southward drift or using the wind-driven formula for w e from equation (6.18) βv = f h w e < 0 (6.26) v = f βh w e (6.27) v = 1 dτ x βρh dy (6.28) This velocity in the gyre interior is called the Sverdrup velocity. The explanation is that as the fluid column in the gyre gets squashed, this causes a reduction in (ζ + f) due to conservation of PV. As ζ 0 away from the western boundary current, the gyre has to move in a way so that f decreases except in the WBC, i.e., general movement towards the equator. So, the fluid column acquires anticyclonic vorticity (of the planetary form) by moving southward. Looking at the observed circulation it is clear that this is the case: a general southward drift occurs over much of the gyre. Using dτ x dy 0.1 Pascal 2000 km β = (ms) 1, h = 4000 m we get a Sverdrup velocity of v 10 3 ms 1 = 0.1 cms 1 This is rather small, and supports our assumption that changes in ζ are small compared to changes in f over the scale of the gyre. In the above we have assumed that η and ζ are negligible. In reality they are relatively small (compared to h and f respectively), but we will expect sea level to be raised (order 1 metre height) in the gyre center and ζ/f to be negative (vortex squashing). 111

116 6.7 Western Boundary Currents The Sverdrup transport (6.28) moves southward and at lower latitudes towards the west where it returns northward as an intense boundary current. A chart of sealevel in the vicinity of the Gulf Stream shows the narrow scale of the Western Boundary Current (order 100 km): Figure 6.20: Upper panel: the depth, in hundreds of meters, of the 15 isothermal surface, showing the Gulf Stream, nine cyclonic rings and three anticyclonic rings. The contours are based on data obtained between March 16 and July Lower panel: a temperature section through the Gulf Stream and two cyclonic (cold core) rings south of the Stream. The section is a dog-leg from 36 N, 75 W to 35 N, 70 W and then to 37 N, 65 W. (Taken from Mellor, 1996) For the Gulf stream the volume of water transported to the north has been observed to be about m 3 s 1 or 100 Sverdrups. (1 Sverdrup = 10 6 m 3 s 1 ). This estimate may be compared to that based on (6.28) where v 10 3 ms 1. For an ocean of width 112

117 L = 10, 000 km and depth h = 4000 m, the volume transported south per second is vlh 40 Sverdrups. We are out by a factor of 2 or so with our model of the wind-driven gyre. The missing 60 Sv can be explained by the fact that our model of the gyre overlooks smallerscale circulations. A feature of all western boundary currents is that they form mesoscale eddies which have velocities of order 1 ms 1 and diameters km. Both cyclonic (low pressure cold core) and anticyclonic (high pressure warm core) eddies can be formed as the warm water from the equator interacts with the colder waters at higher latitudes. This accounts for the missing 60 Sv above. Some data are presented below Gulf Stream In Figure 6.20, both warm (light) and cold (dense) eddies are shown to be imbedded in the Gulf Stream. Cold core km diameter approximately 9 formed per year Warm core km diameter approximately 5 formed per year Lifetime months years 6.8 Circulation along the Equator There is a complicated current structure near the equator which is associated with the wind driven circulation in each hemisphere. The North and South Equatorial Currents (NEC and SEC) are westward currents that are associated with the wind gyres with speeds of cms 1 and cms 1 respectively, (the SEC is larger due to stronger trade winds in the south). In addition, an eastward North Equatorial Counter Current (NECC) exists in the region of the doldrums, (4 10 N). It is highly variable cms 1 and can extend to 1500 m in depth. To understand these currents, consider the Ekman transport, V E = τ x ρfh E The SE trades create a southward Ekman transport in the southern hemisphere but a northward transport between 0 4 N since f changes sign. In addition from 10 S to the Equator, the sea level is lowered and the thermocline is raised. North of the equator sea level rises because τ x 0 in the doldrums. Further north the winds again blow to the west and again sea level is raised by the Ekman transport. Geostrophy leads to the observed NECC and the intensification of the SEC and NEC that are associated with the wind driven gyre. 113

118 τ x η W E V E V E V E W E W E W E W E Doldrums South Latitude North Figure 6.21: Schematic diagram showing the circulation in the Equatorial Pacific Figure 6.22: Schematic diagram showing the circulation in the Equatorial Pacific (a) plan view of the winds and east-west surface currents. (b) vertical south -north section showing surface slopes (exaggerated), thermocline shape and directions of east and westward currents. An additional feature is the Equatorial Undercurrent, some 300 km wide and 200m deep extending across the Pacific (14, 000 km) with speeds of up to 1.7 ms 1 and transport of 40 Sv. The current arises from the westward wind and blocking due to the islands in the Western Pacific. With x in the directed eastward the barotropic momentum equation is given by du dt fv = gdη dx + τ x ρh ru h. 114

119 Figure 6.23: Cross sections of temperature, salinity, oxygen, phosphate and velocity at 140 o W in the Pacific. In the mixed layer above the thermocline, the water is high in oxygen and low in phosphate. The reverse is true below. Tongues of high salinity extend equatorward. The slope of the thermocline is consistent with a north and south equatorial current separated by a counter current between 5 10 N (confined to the mixed layer). The Equatorial Undercurrent and its effect on the thermocline can be seen at the equator. From Knauss (2000). Right at the equator f = 0 and if the flow does not accelerate du/dt = 0 so that dη dx = τ x ρgh ru gh (6.29) and water is piled up in the west by the wind. A flow u to the east exists which acts to drain this reservoir of water: the flow is the Equatorial Undercurrent. The model (Equation 6.29) makes no allowance for how things vary with depth. Within the 115

120 surface Ekman layer, the current is in the direction of the wind ru τ x ρ < 0 Below this layer, the direct effects of the wind vanish and the Undercurrent is driven by the sea level gradient: ru gh dη dx > 0 At depths of more than 300 m, the thermal wind effect due to the depressed isotherms in the western Pacific, cancels out the pressure gradient due to the sea level. The Undercurrent vanishes. Figure 6.24: Velocity (left) and temperature (right) transect through the equatorial undercurrent. A doming in the isotherms is associated with the eastward flowing undercurrent. Away from the equator f is non-zero so that a north-south current v may exist. However, north of the equator f > 0 so that the geostrophic current v s = (g/f)dη/dx is negative since dη/dx is also. Thus, the geostrophic current would tend to push the EUC back along the equator. A similar argument applies to the velocity v g south of the equator. 116

121 Figure 6.25: a) west-east section along the equator of thermosteric anomaly (b,c) Schematic of south-north sections across currents in the western and eastern Pacific (d) south-north section for salinity in west (e) Schematic of south - north section for temperature in the east Pond and Pickard (1983). 117

122 7 Measuring and modelling the oceans The goal of oceanographic measurement is to describe the flow and variability of the ocean at all depths and in three dimensions, and to quantify property distributions (heat, salinity, oxygen, nutrients, carbon, trace gases). These property distributions help deduce ocean circulation, as well as being important for the transports and distributions of the properties themselves (e.g., heat transport, CO 2 fluxes, and so on). 7.1 Measuring water masses, T S and tracers Temperature and salinity are measured to provide density profiles, which can then be used to compute the vertical shear of geostrophic currents perpendicular to the line connecting a station pair. Inferences of ocean circulation can be made by mapping various properties, for example, along vertical cross-sections, or on level maps. Tracers with independent sources and sinks are the most useful - these include salinity and temperature themselves, nutrients, oxygen, chlorofluorocarbons, tritium, helium-3, carbon-14, and other tracers. T S are relatively easy to measure, whereas most chemical tracers (apart from O 2 ) require direct water samples sometimes quite a large volume taken at pre-determined depths and later analysed in a chemistry laboratory Conductivity-Temperature-Depth (CTD) instruments A Conductivity-Temperature-Depth (CTD) instrument measures salinity, temperature and depth (the latter via pressure). The T S sensors provide very high accuracy readings. A CTD system is quite large and heavy (typically a metre in height and weighing about 40 kg). Typically large CTD instruments also measure the following: Dissolved Oxygen Alkalinity (ph) Turbidity (with a Transmissometer ) Par (Light) Fluoresence (ie., Chlorophyll-a) 118

123 7.1.2 The Expendable Bathythermograph (XBT) The Expendable Bathythermograph (XBT) has been used by oceanographers for many years to obtain information on the temperature structure of the ocean to depths of up to 1500 meters. The XBT, shown next page, is a probe which is dropped from a ship and measures the temperature as it falls through the water. Two very small wires transmit the temperature data to the ship where it is recorded for later analysis. The probe is designed to fall at a constant rate, so that the depth of the probe can be inferred from the time since it was launched. XBTs are cheap (about 100$US each), easy to operate (can be launched from merchant ships), and do not require a stationary ship for launching. They are widely used to analyse upper ocean temperature structure The Expendable Conductivity-Temperature-Depth (XCTD) probe XCTDs are a more expensive version of the XBT, in that a conductivity probe is also engineered onto the instrument. Cost is order 500$US each Niskin bottles Niskin bottles are used to take water samples at pre-determined depths, normally for biological or chemical sampling (e.g., tracers, plankton, and so on). 7.2 Measuring ocean currents directly Surface drifters The earliest maps of ocean circulation came from ship drift calculations, based on speed through the water and heading. Drift bottles and drift cards - released in large quantities in early part of the century through WWII, combined with ship drift calculations, were used to estimate surface ocean flow. Nowadays, ship drift calculations are still used quite successfully given the current excellent state of navigation using GPS satellites. Surface drifters with drogues below the surface ( parachutes ) follow the current just below the surface with minimum windage problems (recall the field trip) Subsurface floats Subsurface floats are either tracked acoustically (SOFAR floats) or are tracked periodically by satellite navigation when they pop to the surface (ALACE floats). Global deployments 119

124 Figure 7.1: Lagrangian free-drifting buoy elements: surface float with radio transmitter (top right), and three types of drogues. (Taken from Pickard and Emery, 1990) for WOCE are concentrating on the metre level. Often subsurface floats are built with a conductivity and temperature sensor to sample T S as the drifter moves along its trajectory. See section on Argo floats below Current meters Current meters are normally deployed on fixed moorings. Less often they are used as a profiling instrument (a profiling current meter or PCM), since high variability in ocean currents due to tides or internal waves require longer time scales of sampling than that provided by a single vertical profile. Most current meters use a rotor or vane (or pair) and a compass to derive current speed and direction: 120

125 Figure 7.2: Schemes for mooring current meters, data loggers, etc. with surface and subsurface upper floats. (Taken from Pickard and Emery, 1990) Acoustic Doppler Current profiling (ADCP) Acoustic Doppler Current profilers measure the speed of flow by an acoustic doppler shift of particles in the ocean. There are generally several beams at angles to each other. This permits a measure of flow direction. The range of an ADCP is about 300 meters, depending on the frequency and efficiency of scattering. ADCP s are used in ship mountings, on lowered instrument packages and on moorings as current meters. 121

126 7.3 ARGO floats Argo is an international program (started in 1999) that uses autonomous floats to measure temperature and salinity in the ice-free oceans, which can then be used to infer current velocities. Figure 7.3: Float Description The floats are designed to drift with the ocean currents at a depth of 1000 m. After 10 days, the floats descend to a depth of around 2000 m, then rise to the surface where they transmit via satellite the salinity and temperature of the vertical profile. To save energy, not all floats profile to 2000 m. 66% of the profiles extend deeper than 1500 m and 46% to around 2000 m. After transmitting the data, the floats remain at the surface for around 10 hours after which they sink again and repeat their mission. 122

127 Figure 7.4: Park and Profile Mission Operation The nominal lifetime of any given float is about 5 years. During their life span, they will yield important information about large-scale ocean water property distributions and currents. 23 countries contribute floats to the Argo program. The program reached its target of 3000 floats distributed worldwide in early 2008, creating a global array of profiling floats spaced every 3 through the ice-free oceans. Figure 7.5: Argo Array Status Before Argo, most of our knowledge of the interior of the ocean came from research ship measurements and from temperature probes (Expendable Bathythermographs, aka XBTs) dropped from merchant ships. These observations were sparse and seasonally biased towards summer and merchant ship highways. Combining satellite and float data to define the state of the ocean requires the use of data 123

128 assimilation; a technique used in weather forecasting that was applied to the oceans during the World Ocean Circulation Experiment (WOCE). Assimilation combines observations with detailed computer models in a way that ensures results agree with the fundamental physical laws that govern ocean dynamics. Since 2002, Argo has been the largest single source of ocean profile data. As well as being more numerous, the Argo data extend deeper than the 750 m XBTs, measure temperature more accurately and also collect salinity and ocean current data. Temperatures are accurate to ±0.005 C and depths to ±5 m. Argo has a data management network of national centres that provide data in real time (90% of profiles available within 24 hours) and delayed mode from two global hubs: one in France (Coriolis) and the other by GODAE (USA). Some useful webpages are: Argo portal: Argo General information: Coriolis: GODAE: Uses of Argo Data Centres in Australia, France, Japan, the UK and USA produce global and regional analyses of subsurface properties using Argo data. These are published on the web and give early warning of significant temperature and salinity anomalies and changes in the ocean circulation. In the gulf of Alaska and around Japan, Argo data is being used to aid the monitoring of environmental conditions that affect fish stocks and biological productivity. In the UK, the Met Office issues forecasts for the winter based on the data collected by Argo floats on the previous winter. 7.4 High Frequency Radar Through a series of strategically placed radar poles installed along the back edge of beaches that send radio signals pulsing across the sea surface and receive the altered signal bits that scatter back, the radars can track the movements of coastal currents in a pie-shaped swath up to 150km out over the shore. When you send out a radio wave to a moving target, and you get a different frequency returning back to you as it bounces off that moving target (Doppler Shift). This then enables you to calculate the surface current speed and direction. It is possible to obtain a continuous, detailed map of the surface current (updated every hourly) covering a 1 km x 1 km grid up to 150 km offshore. A single HF Radar site consists of transmit antenna and a receive antenna, separated by about 30 m. A single site measures radial currents by transmitting a radio signal at a specific frequency out over the surface of the ocean. The radio waves scatter off of the waves on the 124

129 surface of the ocean, and are then recorded by the receive antenna. These backscattered radio waves are used to compute currents moving toward or away from the site. Two sites in close proximity to one another complement each other in such a way that total surface currents over their region of overlap are computed. Radial speed maps from each radar site alone are not a complete depiction of the surface current flow, which is two-dimensional. This is why at least two radars are normally used to construct a total vector from each site s radial components. At the central data combining station, the radial vector maps from multiple radar stations are merged to create a total velocity vector current map. Figure 7.6: Example of HF Radar derived surface velocities overlayed on Satellite derived SST off the coast of San Diego Southern California USA.(Roughan et al., 2005). An HF Radar system can measure Surface currents Wind direction 125

130 Wave height Directional wave spectrum Over a large area As often as every 10 minutes 7.5 Remote Sensing The understanding of ocean processes depends upon being able to measure a wide range of variables and parameters which describe the oceans. With the introduction of satellites, became possible to measure the oceans remotely from satellites (and aircraft) which led to a new branch in oceanography. Until recently, oceanographers relied on measurements taken by ships and buoys and were therefore constrained to those areas of the ocean with shipping traffic. Satellites, have enabled spatially detailed measurements to be made almost instantly over wide areas and provide a new perspective of the ocean. Thanks to satellites, we have now repeated monitoring of the ocean on a global scale. This revolution in ocean measurements has enabled us to better monitor the state of the oceans and detect of some of the changes induced by climate change. Satellite Oceanography started in 1978 when sensor technology had developed to a stage where oceanographers could begin to make scientific measurements of the sea. In 1978, 3 satellites were launched with sensors capable of recording a variety of ocean processes: Seasat, Tiros-N and Nimbus

131 Figure 7.7: NASA s Illustration of its experimental satellite Seasat. Seasat, launched by NASA s Jet Propulsion Laboratory, was the first Earth-orbiting satellite designed for remote sensing of the Earth s oceans. It had a payload of five sensors designed to return the maximum information from ocean surfaces: Radar altimeter to measure spacecraft height above the ocean surface. Microwave scatterometer to measure wind speed and direction. Microwave radiometer to measure sea surface temperature. Visible and infrared radiometer to identify cloud, land and water features. Synthetic Aperture Radar (SAR) L-band, HH polarization, fixed look angle to monitor the global surface wave field and polar sea ice conditions. Seasat remained operational for 105 days until October 10, 1978, when a massive short circuit in the satellite electrical system ended the mission. Although only approximately 42 hours of real time data was received, the mission demonstrated the feasiblity of using remote sensing to monitor ocean conditions. Since then, new sensors with higher performances have been built and new methods for processing the data have been developed to improve the accuracy with which ocean parameters can be measured. The space agencies of Europe (which launched its first ocean remote sensing satellite in 1991: ERS-1), North America and 127

132 Japan have a continuing program of Earth Observing Satellites with a number of sensors dedicated to oceanographic applications. Figure 7.8: NASA s satellite scientific missions How does it work? Remote sensing relies on measuring electromagnetic radiation from the sea. Most electromagnetic radiation is absorbed or scattered by the atmosphere. However, there are distinct wavelength bands (windows) at which rays can penetrate the atmosphere with less interference: Visible wavelength nm. Infrared wavelength around 3.7 mm and mm. Microwaves which include radar and are longer than 10mm. Both visible and Infra-red cannot penetrate clouds without being scattered or absorbed, but microwaves are less affected and permit an all-weather monitoring of the sea. Satellites carry onboard a payload of different sensor types designed to measure one or more ocean parameters. Sensors may be either passive or active. 128

133 Passive sensors measure naturally occurring radiation, using either the sun s rays scattered from beneath the ocean surface, or energy emitted by the ocean itself in the infra-red or microwave part of the spectrum. NB: Visible waveband sensors that rely on solar illumination can only operate during daylight, while all others can operate both day and night. Active sensors emit their own source of electromagnetic energy which is directed towards the sea and measured after reflection. These sensors are restricted to micro waves. These provide more options as they can measure not only the magnitude of the returned pulse relative to that emitted, but also its timing, the shape of the pulse and any shift in frequency What can we measure from space? Despite there being a wide range of instruments, remote sensing is able to measure just four basic properties of the sea: 1. Sea surface temperature (top 10µm of the water column). 2. Sea surface colour. 3. Surface roughness at short length scales. 4. The slope of the surface averaged over tens of kilometres. It is from these primary parameters that a range of other properties can be derived depending on the sensor. i.e: Colour can be used to estimate the chlorophyl concentration and hence infer the primary productivity on the surface of the oceans. Surface slope can be used to derive barotropic ocean currents. Surface roughness can be interpreted as a factor of wind speed. SST Infra-red sensors measure the radiation emitted from the sea surface in the wavebands µm, µm and µm. The emitted radiation increases as SST increases, but is reduced as it travels through the atmosphere. The atmospheric effect varies with wavelength and so the difference between brightness temperatures at different wavebands can be used as a basis for atmospheric correction. 129

134 Figure 7.9: Diagram showing the type of sensors, the parameters they measure and the parameters that can be derived. From Oceanography, (Summerhayes and Thorpe, 1996). Ocean Colour The apparent colour of the water is affected by the combination of scattering and absorption of the Sun s light. The red end of spectrum is absorbed by water and so light scattered by deep water consequently appears blue. However, chlorophyll in the water will absorb blue light, so that the water-leaving light will appear greener in the presence of phytoplankton, providing there is enough particulate matter to scatter the green light before it is absorbed by the water itself. Other substances can influence the water colour such as dissolved inorganic material (yellow substance) and red tide phytoplankton blooms. Again, the brightness of scattered light also gives a qualitative indication of the amount of suspended particulate matter in the water. Remote Sensing Sampling The type of sensor and the satellite carrying it will affect the size of the area that can be viewed simultaneously, the spatial resolution of the resulting image and the frequency with which it can be revisited. Sensors record individual point measurements of a property of the ocean corresponding to an average over the instantaneous field of view (IFOV), which can be as small as a few metres 130

135 or as large as hundreds of kilometres, depending on the sensor. Some sensors, such as an altimeter, simply view in a single direction following the groundtrack beneath the moving satellite. Placed on a polar orbiting satellite, these sensors perform 15 orbits per day. Over many days, the gaps between the single day paths can be filled, providing a global picture of the oceans. Figure 7.10: Earth viewing by a non scanning sensor (Summerhayes and Thorpe, 1996). If instead the sensor scans several times per second in the direction of travel, an array of data is collected which corresponds to an image of the ocean surface composed of individual picture elements or pixels, each being an independent measurement. The swath width will determine the spatial resolution. The wider the swath, the lower the resolution but coverage of successive orbits overlaps and coverage of the whole earth can be achieved daily. The narrower the swath, the higher the resolution but a single point on Earth may then be scanned only once every days. Thus, there is a trade-off between spatial resolution and frequency of coverage and this must be matched to the characteristics of the ocean phenomena to be observed. Earth observation satellites in near-polar orbits fly at about km above the Earth and their orbit is arranged to proceed slowly so that the orbit plane remains fixed relative to the Sun. This Sun-synchronous orbit ensures that the local (solar) time is always the same at the time of sampling. This ensures uniformity of solar illumination for ocean colour scanners and helps to eliminate diurnal heating cycle aliasing. The other type of satellite used for remote sensing of the ocean is that in a geostationary orbit. Placed at about km above the Earth, it orbits once per day and remains stationary to a fixed point on the ground. From this satellite, sensors can scan the whole visible disc of the Earth and scan at any frequency. Existing meteorological sensors can scan the full view in 30 minutes but have a spatial resolution larger than 5 km. These sensors cannot view latitudes above 55 because of the curvature of the Earth. 131

136 7.5.2 Analysis of Satellite Data Before the raw data that is transmitted from the sensors to the the ground stations can be used by marine scientists, it needs to be processed in a series of complex steps: Sensor calibration converts the numbers transmitted from the satellites into a measure of electromagnetic radiation reaching the sensor. Atmospheric correction makes the necessary corrections to allow for the effect of the atmosphere signal to be removed from the data. For data derived from infra-red and visible wavelength, screening for clouds detection is applied in which case all pixels containing cloud are rejected, leaving blanks in the final image. Geophysical calibration converts the water leaving electromagnetic radiation into one of the primary ocean variables and then into a derived ocean parameter. Calibration algorithms are generally derived empirically using in situ data and calibration or validation experiments. Geolocation defines the geographical location of each pixel on the image. For many purposes, it is convenient to combine the data from several different satellite overpasses into a single map representing average conditions over a single time span forming a composite image. i.e. global SST maps are derived from all the overpasses from several days enabling gaps due to cloud cover on some days to be filled from clear overpasses on others. Calibrated image of ocean parameter Remote Sensing Applications Satellite data analyses lead to images or maps of a particular ocean parameter. These then need to be analysed or interpreted in order to fully understand ocean processes. One of the main aims of creating these sensors was to obtain regular global maps of SST distribution and its seasonal variations. With a long time-series, it is possible to identify trends and anomalies that may indicate changes in climate, such as the irregular phenomenon of El Nino-Southern Oscillation (ENSO) events. Images derived from synthetic aperture radar (SAR) are not as easily interpreted as colour or thermal images but, show the variation of radar backscatter over the sea. Swell waves can be seen by the way in which back-scatter varies between trough, crest leading and trailing face and even the refraction of wave pattern is sometimes imaged. SAR images also show phenomena that appear beneath the water surface but which alter surface roughness, such as bathymetry or internal waves. Radar images can also be strongly influenced by the presence of surface films which generate slicks of smoother water having low back-scatter; this is used for locating oil spills. 132

137 Satellite altimetry can over time build evidence of sea surface topography which can be interpreted in terms of ocean currents. Until there is an independent method to quantify the shape of the geoid due to variations of gravity, only deviations of slope from the mean can be obtained, but these can give a good estimate of variability in ocean currents. Also, the ability of satellite altimeters to estimate wave heights from the shape of the reflected radar pulse plays an important role in forecasting. Finally, the scatterometer, which measures the average roughness over large areas of sea surface and enables local wind speed and direction to be derived, has a valuable use in meteorology. Remote sensing technologies can be used to measure the following: Sealevel Satellite altimetry provides a measure of the sea surface height relative to the earth s geoid. The sea surface height measurement is directly related to the pressure and hence to the geostrophic currents at the sea surface. Sea Surface Temperature When the ocean is not covered by cloud, infra-red sensors can detect the surface ocean temperature to high levels of accuracy, resolving small-scale features in ocean currents. The infra-red sensors can be used on satellites without loss of accuracy. (See Figure 7.6) Sea Surface Salinity The ocean emits an electromagentic field (EMF) which is dependent on its conductivity, that is, its salinity S. Presently, EMF sensors can detect conductivity in the ocean from a distance of about 1 km. This means these sensors can be mounted to special aircraft flying over the ocean. Measured S has an accuracy of about 0.2 psu. Wave height As for sealevel, altimetry can measure wave height to high accuracy (order 2-3 cm). Estimates of sealevel simply filter out the short time-scale variations associated with surface waves (such as swell). 133

138 Wind stress The amount of backscatter of microwaves emitted from a satellite can be used to determine the roughness of the ocean, which can be directly related to the near-surface wind field (both speed and direction). Sea Ice The reflectivity or albedo measured from ligh reflecting off the earth s surface can be used to estimate the level of sea-ice coverage over the oceans. 7.6 Other data sources Surface Meteorological Data Weather data (land-based) Weather data (ship based) Weather estimates (from models) Weather data (surface ocean weather stations) In situ sea level data Sea level data can be recorded directly using tide guages at islands and coastal stations. These measurements, although scattered in space, can be used to determine long time series records of sea-level or for calibrating against satellite data. 7.7 Australia s Integrated Marine Observing System IMOS is a distributed set of equipment and data-information services which collectively contribute to meeting the needs of marine climate research in Australia. The observing system provides data in the open oceans around Australia out to a few thousand kilometres as well as the coastal oceans. The IMOS Office coordinates the deployment of a wide range of equipment and assembles the data through 11 Facilities distributed around the country. The data are made available to researchers through the electronic Marine Information Infrastructure (emii) located at the University of Tasmania. The IMOS infrastructure also contributes to Australia s role in international programs of ocean observing. 134

139 IMOS was planned through extensive consultation with the Australian marine research community through Nodes, including a bluewater open ocean node and five regional nodes around the country IMOS is coordinated and managed nationally by staff at the University of Tasmania supported by CSIRO Marine and Atmospheric Research. IMOS is an NCRIS funded project. Initially the project will run from Figure 7.11: Schematic Diagram showing the distribution of IMOS nodes and facilities. 135