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1 Blackbody radiation Everything with a temperature above absolute zero emits electromagnetic radiation. This phenomenon is called blackbody radiation. The intensity and the peak wavelength of the radiation are functions of the body s temperature. Intensity is Q = σt 4, (1) where Q is energy flux (or heat flux), T is temperature in Kelvin, and σ is the Stefan-Boltzman constant (σ = W/m 2 /K 4 ) C Tungsten melting point sun Intensity (nondimensional) wavelength (nm) FIG. 1. Nondimensional spectra for blackbody radiation The sun emits as a blackbody, with maximum intensity in the visible range. Incandescent lightbulbs emit as blackbodies, with maximum intensity at the red/infrared boundary The ocean emits as a blackbody, with peak intensity in the infrared. The ocean also absorbs blackbody radiation from the sun and from the atmosphere Greenhouse If a layer of glass is suspended over the earth it blocks some (or all) of the outgoing longwave radiation (Lp) (figure 2). The glass radiates both upwards and downwards (Lg), warming the earth below. At a steady state, the total radiation outwards must equal the total radiation inwards. The emissivity (e) is the fraction of longwave energy that the glass absorbs. Shortwave reflected Shortwave in Glass (1-e)Lp Lp Lg Lg FIG. 2. A greenhouse earth 1

2 Structure of the upper ocean The upper ocean exchanges heat with the sun and atmosphere. Most of the heat does not make it into the deep ocean immediately. There is strong turbulent mixing in the surface boundary layer because it is stirred by the wind and the density gradient is weak. The density gradient in the pycnocline suppresses turbulent mixing T z Thermocline (strong temperature gradient). Often associated with halocline (salt) and pycnocline (density). O(10 m) - O(100 m) thick Surface boundary layer ( mixed layer) O(10 m) - O(100 m) thick (can be thinner in summer and thicker in deep convection) FIG. 3. thermal structure of the upper ocean The units of fluxes Flux is transport of any quantity per unit area. It is written as an amount of stuff moving through a unit area in a given amount of time or as a concentration times a speed. Dissolved oxygen: mol m 2 s = mol m m 3 s (2) eat: J m 2 s = W m 2 (3) Temperature ( concentration of heat): T f = temperature flux = heat flux/ρc p 1 W kg/m 3 J/kg C m 2 = m3 kg C J Jkg m 2 s = C m s (4) 2

3 Diffusion eat flux is Q = κ T = κ T x κ T y κ T z Warming of a point through the divergence of molecular heat flux is (5) if κ is constant. T = Q = ( κ T ) = κ 2 T (6) Temperature Temperature Position Position Gaining eat Losing eat FIG. 4. temperature profiles Continuity equation (conservation of mass), Boussinesq approximation, and material derivative Given a box of uniform density and flow only in the x-direction. The box s mass is M = ρ x y z. (7) This allows us to use ρ as a proxy for mass. Last time we established that the rate of change of the mass in the box is M = 1 t t ( ρ x y z) = (ρ 2u 2 y z ρ 1 u 1 y z), (8) and since the box s volume doesn t change, x, y, and z are constant. We divide by ρ1u1 3 Δx FIG. 5. A box of fluid Δy Δz ρ2u2

4 x, y, and z to get ρ t = (ρu) x. (9) In differential form this is ρ = (ρu), x (10) which can be expanded to three dimensions as separated into and rearranged as ρ = x (ρu) y (ρv) (ρw), (11) z ( ρ u = u ρ x v ρ y w ρ z ρ x + v y + w ), (12) z ( ρ u + u ρ x + v ρ y + w ρ z = ρ x + v y + w ). (13) z The term in parentheses on the right hand side of the equation is associated with squeezing the water and making it smaller (more dense). It turns out that this is hard to do. If you slow down the water coming from one side of the box, it speeds up in another direction and squirts out a different side of the box. Put another way, water is incompressible, and to a good approximation, u x + v y + w 0. (14) z This is the common form of the continuity equation. It allows us to write the flux terms in the evolution of density (or salinity or temperature etc) in a simpler way. This approximation is part of what is called the Boussinesq Approximation. Using (14), (13) becomes ρ + u ρ x + v ρ y + w ρ z = ρ + u ρ 0. (15) This says that if we follow a fluid parcel (whatever that is), its mass will not change. We know from the demonstration of the water thermometer that that is not quite true, but these density changes are usually very small compared to the total density so they are safe to ignore. The left hand side of (15) is called the material derivative, and it describes the evolution of a single parcel of water. The material derivative is often written Dρ Dt = ρ + u ρ x + v ρ y + w ρ z = ρ + u ρ. (16) The material derivative is used to transform the Eulerian view (examining fluid at fixed locations) to the Lagrangian view (examining fluid parcels as they evolve). We have left out diffusion, in part because the idea of diffusion of density is a little strange. So let s think about diffusion in terms of salinity. We can follow similar reasoning as above to write a similar equation for salinity + u S = κ 2 S. (17) This equation says that if you have a box of salty water, the only way to change the salinity of that box is by adding or subtracting salt through molecular diffusion. 4

5 Turbulence and Reynolds averaging Bodies of water have turbulence (fluctuating velocities) superimposed on a mean flow. Some examples are: Entrainment at the edges of fluid jets Mixing caused by wind stress (or a hairdryer) Stirring a coffee cup Smoke or steam billows from a chminey Wind gusts All quantities can be decomposed into mean and fluctuating parts. We usually think of the mean as averaging over time. ow long? Times long compared to turbulent fluctuations (seconds to minutes) and short compared to the evolution of the mean. The time or space scales over which you average to separate the turbulent and mean quantities depend on your perspective and interests. In tidal flows we average over 20 minutes to find the mean. In climate studies we average over years Decomposition and definitions u = u + u (18) v = v + v (19) w = w + w (20) u = u + u (21) T = T + T (22) S = S + S (23) u = 0 (24) u = u (25) u + u = u + u = u (26) ut = ut = 0 (27) u T = covariance (28) T T = T = T For salinity we have + u S = κ 2 S. (31) Let s separate this into average and fluctuating parts. + + u S + u S + u S + u S = κ 2 S + κ 2 S. (32) (29) (30) 5

6 Now take the average of the entire equation Several terms average to zero and leave + + u S + u S + u S + u S = κ 2 S + κ 2 S. (33) + u S + u S = κ 2 S. (34) It is conventional to add back in the continuity equation to the turbulent part (adding in a quantity equal to zero). u S = u S + S u = u S (35) and to move the turbulence term to the right hand side (essentially as a transport mechanism for the mean salinity) to get + u S = u S + κ 2 S. (36) (1) + (2) = (3) + (4) In words, these terms are (1) the time rate of change of salinity at a point; (2) the advection of changes in salinity by mean flow along the mean salinity gradient; (3) the advection of changes in salinity by fluctuating flow along the fluctuating salinity gradient; and (4) diffusion by molecular proceses. Even though turbulence happens on small scales, it affects the mean flow. Properly measuring or modeling turbulence requires observations on spatial scales of meters or less and timescales of seconds. This is impractical. So what do we do? The most common method is to assume that turbulence acts as a downgradient diffusion process, much like molecular diffusion. Starting with we substitute u S = u S + v S + w S (37) u S = K x v S = K v (38) (39) w S = K v z. (40) K v,k are much larger than the molecular equivalent, κ. K v,k are properties of the flow and need to be determined. K v,k are variable in space and time. They are hard to work with, but are much simpler than working directly with turbulent quantities. If we put everything into a conventional form we get + u S = [ (K + κ) ] + [ (K + κ) ] + [ (K v + κ) ]. (41) x x y x z x 6

7 Because in most cases κ (K h,k v ), molecular diffusion can usually be ignored, leaving + u S = x (K x ) + y (K x ) + z (K v ). (42) x If we are not measuring turbulence, often we don t distinguish that the measurements are of mean quantities, so we don t write the overbars. This leaves us with equations for mass conservation (continuity), temperature conservation, and salinity conservation: u = 0 (43) T + u T = x (K T x ) + y (K T y ) + z (K T v y ) + 1 I ρc p z. (44) Evolution of momentum + u S = x (K x ) + y (K y ) + z (K v ). (45) y Momentum evolves in the same way that other tracers do. The molecular processes are a little different, which is why molecular viscosity is larger than molecular diffusivity. Momentum is mass velocity. A heavy object moving slowly and a light object moving quickly can have the same momentum. Because we look at concentrations in fluids, we replace the mass by density to get momentum per unit volume. And because the density of the ocean is nearly uniform we can divide out that part and look at velocity. The mass in a box of fluid is Call momentum in the x-direction N. The momentum in a box of fluid is Divide by x y z to find the momentum per unit volume, ρu. Newton s Second Law M = ρ x y z (46) N = ρu x y z. (47) For the speed of any moving object to change, a force must be applied to that object. Newton tells us that F = Ma, (48) where F is force and a is acceleration. To speed up an object, apply a force in the direction it is moving. To slow an object down, apply a force in the opposite direction. To turn an object, apply a force perpendicular to its path. In keeping the analogy with the evolution of temperature and salt, think of forces as sources or sinks of momentum in the ocean. We can write N = (divergence of transport by mean currents) + (divergence of transport by turbulence) + (forces). Forces in the ocean Pressure gradient forces Friction Gravity Coriolis 7

8 PRESSURE If you squeeze a fluid, it moves from high pressure to low pressure. In the ocean, pressure can be increased by raising the sea surface or by increasing the density of the overlying water (see figure 7). Pressure forces act in the direction opposite the pressure gradient (fluid accelerates down the pressure gradient). z Sea Surface η dp dx > 0 To compute the pressure at any point in a body of water that is not moving, simply add up the weight of the water overlying that point. Z η P(η) P(z) = dz ( ρg) (49) z FIG. 6. Pressure gradient caused by sea surface slope For a fluid of constant density, this gives P(z) = P atm + ρg(η z), (50) where P atm is the atmospheric pressure at the sea surface. We often ignore the atmospheric pressure and read gauge pressure, which is simply absolute pressure minus atmospheric pressure. for a level surface (η = 0) this reduces to P(z) = P atm ρgz, (51) FRICTION Friction is a source or sink of momentum at the boundaries, not an internal source or sink. Water in the ocean knows about boundary friction only through the transport terms. Turbulence is often the important player in transporting momentum to or from the boundaries from or to the interior. Example: A river. The East Branch of the Penobscot at Grindstone is about 100 m higher in elevation than the river is here in Orono. If you drop a ball straight down from 100 m, how fast would it go when it hit the ground? Fast. Is the river moving that fast? Is the river accelerating here in Orono, or is the speed relatively steady? It s steady. ow does that fit with Newton s Second Law? so F must also be zero. F = Ma (52) a = 0 (53) River surface River bed FIG. 7. Profile of a river dp dx < 0 F = F pressure +?? (54) x 8

9 The answer to all these questions is that friction matters. The pressure gradient force wants to accelerate the river downstream. But friction balances that so the net force (and net acceleration) is zero. In fluids we use shear stress to describe friction. Shear stress is the force parallel to a fluid surface divided by the area of that surface. In the ocean, wind imparts stress at the surface, driving currents. Bottom friction slows down the currents. In solid body mechanics, the shear stress between two objects is usually computed as the product of the coefficient of friction and the normal force pushing the objects together 2 m τ = 0.1 Pa = 0.1 N/m τ τ F = τa 2 m 2 2 F = 0.1 N/m x 4 m = 0.4 N 1 m 1 m 2 2 F = 0.1 N/m x 1 m = 0.1 N FIG. 8. Stress and is independent of the relative speed of the objects. In fluids, the shear stress at the boundary is a function of the relative speeds of the fluid and the boundary. In fact, shear stress is proportional to the square of the velocity. We usually think of fluids having a no-slip condition at their boundaries. That is, the fluid velocity at the boundary is the same as the velocity of the boundary. If the wind is blowing 3 m/s at the sea surface, then the uppermost layer of water molecules is also moving at 3 m/s. Viscosity and then turbulence allow there to be large gradients in velocity at the boundaries. Similarly, at the sea floor, the no-slip condition specifies that the molecules right at the boundary have zero velocity. 2 GRAVITY AND CORIOLIS Gravity exerts a force on everything, pulling perpendicular to equipotential surfaces. We usually define vertical as being parallel to the direction that gravity acts. The equipotential surfaces are what we think of as horizontal. Because earth is spinning, and because of topography and density variations within the earth, the horizontal surfaces have the shape of a bumpy ellipsoid called the geoid. Even if the ocean weren t moving, the ocean surface would have topographic variations of tens of meters. Reference ellipsoid Dense mass Ocean surface An object moving in a circle has velocity tangential to the circle. Its direction is continually FIG. 9. An equipotential surface changing, however, so it must be continually accelerated. To maintain a circular arc it must be accelerated towards the circle s center, perpendicular to the direction of travel. This pull can t slow down or speed up the object because it is never pulling parallel to the direction of travel. We are not interested in what initially made this object begin to spin in a circle; we are interested in the consequences of it continuing to do so. On the surface of the earth, we are spinning. Ignoring the yearly cycle, we spin around once per day. Because 9

10 gravity acts only to pull us towards the center of the earth, and does not pull east or west, it is unable to speed up or slow down our travel around the planet. On earth s surface, water is trying to satisfy two criteria. If no forces are acting on it, moving water wants to travel in a straight line. It is also trying to stay on earth s surface. We already know that the second criterion wins and that gravity keeps the water from flying into space. The combination of gravity pulling down and the sea floor pushing up provides the resultant force that accelerates the fluid and makes it travel in a new direction. Because we live on earth we want to look at things in a reference frame fixed to the earth s surface. Because earth rotates, this is a noninertial reference frame. That is, the reference frame itself accelerates. Ultimately, this leads to new terms in equations for momentum evolution to stand in for the fact that the reference frame itself is accelerating. Force pushing straight up (perpendicular to ellipsoid) Resultant force keeps us in circular orbit (on surface) Force due to gravity FIG. 10. Force balance for objects at rest on spinning earth. Looking from a fixed reference frame. Resulting equations Even though the earth is curved and its curvature and rotation have important effects on fluid motion, ocean dynamics are often examined as if the ocean were a flat plane. The coordinate system can be arbitrarily defined, but the custom is to align it with latitude and longitude or with a coastline or river. In the Gulf of Maine, for example, we might define a right handed x-y-z coordinate system such that x points to the east, y points to the north, and z is positive upwards. z = 0 is usually (but not always) at the sea surface. Going back to our equation for momentum, N = ρu x y z. (55) Dividing by x y z gives the momentum per unit volume, ρu, and dividing further by ρ gives the momentum per unit mass. For convenience, equations for conservation of momentum in the ocean are usually written as equations for conservation of velocity. 10

11 Combining all the forces, apparent forces, and transport terms gives three momentum conservation equations, one for each component. u + u u = x (K u x ) + y (K u y ) + z (K u v z ) 1 P + f v. (56) ρ 0 x v + u v = x (K v x ) + y (K v y ) + z (K v v z ) 1 P f u. (57) ρ 0 y w + u w = x (K w x ) + y (K w y ) + z (K w v z ) 1 P ρ 0 z gρ. (58) ρ 0 ere g is the component of gravity perpendicular to the geoid (a positive number). The Coriolis parameter, f, is f = 2Ωsinθ, (59) where θ is latitude and Ω is the radian frequency of earth s rotation (Ω = 2π/86164 s = s 1, with the number of seconds in a stellar day) Because vertical velocities are often small compared to the pressure and gravity terms, the hydrostatic approximation is often made. This eliminates all but two terms in (58) to give which is consistent with (49). P z = gρ, (60) 11