Lecture 23. Hydrological cycle and haline circulation 5/9/ :07 PM 1. Dynamics role of hydrological cycle in climate Poleward heat flux in the

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1 Lecture 23. Hydrological cycle and haline circulation 5/9/26 12:7 PM 1. Dynamics role of hydrological cycle in climate Poleward heat flux in the climate system has been one of the main focuses of climate study. A study by Trenberth and Caron (21) showed that the atmospheric component of the poleward heat flux dominates the total poleward heat flux at high latitudes. As stated by the authors in the abstract, "At 35 o latitude, at which the peak total poleward transport in each hemisphere occurs, the atmosphere transport accounts for 78% of the total in the Northern Hemisphere and 92% in the Southern Hemisphere. In general, a much greater portion of the required poleward transport is contributed by the atmosphere than the ocean, as compared with previous estimates." The oceans seem unimportant for high-latitude climate, in terms of carrying the heat flux much needed for warming up high latitudes. Is that true? In particular, the Southern Hemisphere is covered primarily by the oceans, especially between 35 o S and 7 o S. Therefore, it seems incomprehensible how can the oceans play such a minor role in poleward heat flux. A. Definition of poleward heat flux Oceanic heat flux has been discussed in many papers and textbooks. As both instruments and numerical models improved, poleward heat fluxes are better diagnosed. For the most update information the readers are referred to the review by Bryden and Imawaki (21). The main point is that although heat flux data may be further improved, there seems a fundamental problem in the definition of poleward heat fluxes that gives rise to misconception and similar mistakes have been made by many people. From thermodynamics, it is well known that heat flux cannot be well defined for a system that has net mass gain (or loss) through the lateral boundary of the system. For example, Trenberth and Caron (21) made a point that the poleward heat fluxes in the South Pacific and Indian are not well-defined because the existence of the Indonesian Throughflow. The same caution should apply to both the oceans and atmosphere because both the ocean and atmosphere are not closed systems in terms of mass -- there is water exchange between them. In fact, the hydrological cycle in forms of evaporation and precipitation is an essential ingredient for the heat flux in atmosphere -- the latent heat flux. The mass flux associated with evaporation and precipitation has been traditionally ignored in heat flux calculation by oceanographers. This mass flux is not accounted for in many previous calculations because the follows. First, such a small mass flux is rather hard to be identified from the classical dynamic calculation based on either a reference level or other means. Second, most oceanic general circulation models have been based on the Boussinesq Approximations, thus, the dynamic effects associated with evaporation and precipitation is simulated in terms of the virtual salt flux condition on the upper surface; while the mass flux associated with evaporation and precipitation is totally ignored. As the oceanic general circulation models improved, the new generation of models is formulated under the natural boundary condition (Huang, 1993). As a result, the mass flux associated with evaporation and precipitation is exactly accounted for. A more accurate definition of heat flux is, thus, necessary. A simple definition of heat flux is defined in the Appendix, which can be used for an oceanic section with a net mass flux exchange. Strictly speaking, the poleward heat flux in the climate system should be defined as three terms, as shown in Fig. 1: 1) Sensible heat flux in the oceans; 2) Sensible heat flux in the atmosphere; 3) Latent heat flux in the atmosphere-ocean coupled system. 1

2 Over the latitudinal range from the equator to pole there is a continuous exchange of heat between these three loops. Fig. 1. Sketch of the atmosphere-ocean coupled climate system. As an example, we calculate the freshwater flux through the air-sea interface. We use the evaporation and precipitation rate over the world oceans by Da Silva et al. (1994), Fig. 2. This dataset has taken into consideration of river run-off, so the globally integrated evaporation and precipitation rates are balanced. Near the equator precipitation dominates, especially between o N to 1 o N. However, evaporation dominates over the subtropics and thus produces a net water vapor flux that is transported by the atmosphere to high latitudes. At high latitudes (roughly 4 o off the equator) precipitation dominates. 2

3 Fig. 2. Water vapor source and sink due to evaporation and precipitation, integrated zonally, from Da Silva et al. (1994) However, this dataset seems to give rise a poleward freshwater flux that is too large in the southern hemisphere. As an alternative we use the water vapor flux calculated from atmospheric circulation models, Gaffen et al. (1997). The freshwater flux reported in their study is based on average of 25 atmospheric general circulation models. This poleward water vapor flux M w is the major mechanism of poleward heat flux in the climate system, Fig. 3. The poleward heat flux associated with the water vapor cycle is closely related with the latent heat content of water vapor H f = ( Lh hw) Mw (1) where Lh = 25 J / g is the latent heat content for water vapor, h w is the enthalpy of the return water in the ocean, which is much smaller than L L h and neglected in the calculation. The heat flux associated with the water vapor flux should be attributed to the ocean-atmosphere-land coupled system. However, the mass flux associated with the water vapor flux is two orders of magnitude smaller than other mass fluxes in the climate system (Table 1). This is one of the reasons why this mass flux has been overlooked in many papers discussing climate. On the other hand, this seemingly tiny mass flux is associated with a large heat flux. One kilogram of water vapor can deliver J of heat. When one kilogram of water is cooled down by 1 o C, the heat released is J. Thus, water vapor is 6 times more efficient than the water in transporting heat. Here we use Sverdrup as a unit for mass flux. Sverdrup has been widely used as a volume flux unit in oceanography, and 1Sv=1 6 m 3 s -1. Since sea water density is very close to 1 3 kg/m 3, as a mass flux unit we have 1Sv=1 9 kg/s. As shown in Table 1, Sverdrup is a very convenient unit for climate system because mass fluxes in both the oceans and atmosphere have the same order of magnitude, although the density of water is one thousand times larger than air (Held, personal communication, 21). The mass flux in the Jet Stream is inferred from the angular momentum. According to Peixoto and Oort (1992), the maximum angular momentum in the North Hemisphere is about kgm 2 s -1. Assuming the mean latitude is at 3 o N, this gives a rough estimate of 5 Sv for the mass flux associated with the westerlies in the Northern Hemisphere. Current Systems Mass flux (Sv) Gulf Stream Kuroshio ACC Meri. Cell in atmosphere Jet Steam Poleward water vapor ~5 1 Table 1. Mass transport maxima in the climate system. The source of the values: Gulf Stream, Hogg and Johns (1995); Kuroshio, Wijffels et al. (1998); ACC (Antarctic Circumpolar Current), Rintoul et al. (1998); meridional overturning rate and the Jet Stream in the atmosphere, Peixoto and Oort (1992); poleward water vapor flux, Gaffen et al. (1997). 3

4 Fig. 3. Northward water vapor flux (Sv) and associated latent heat flux (PW) due to evaporation and precipitation, as calculated from atmospheric general circulation models. In the Northern the water vapor flux due to oceanic process peaks at 4 o N, with a volume flux of.84sv and the corresponding poleward heat flux of 2.1PW. In the Southern Hemisphere, the poleward water vapor flux reaches the maximum at 4 o S, with a volume flux of 1.5Sv and the corresponding poleward heat of 2.64PW. It is clear that the latent heat flux associated with the evaporation and precipitation over the world oceans consists of a substantial portion of the poleward heat flux in the current climate system. From the hydrological cycle alone, we can estimate the poleward heat flux associated with latent heat, and this is a major component of the poleward heat flux, e.g., Schmitt (1995). Such a large poleward heat flux is, of course, intimately connected to the general circulation in the oceans. In the events of climate change, surface temperature and salinity change will certainly affect the latent heat flux, and thus the entire climate system. After introducing the poleward heat flux associated with water vapor cycle, the poleward heat fluxes in the current climate system are shown in Fig. 4. The sum of these three fluxes is the total poleward heat flux diagnosed from the satellite measurement. The oceanic sensible heat flux is the oceanic heat flux from the NCEP data (Trenberth and Caron, 21). Although the atmospheric sensible heat flux is still the largest component in the both hemisphere, it is not much larger than the other components. 4

5 Fig. 4. Northward heat flux in PW; RT indicates the heat flux required for radiation balance as diagnosed from satellite data, AT indicates the atmospheric sensible heat flux, OT indicates the oceanic sensible heat flux, and LT indicates the latent heat flux associated with the hydrological cycle in the atmosphere-ocean coupled system. B. Heat flux divergence Caution must be taken in the interpretation of the poleward heat flux. Heat flux itself may not be very important -- what really matters is the divergence of the heat flux. For example, at 65 o S atmospheric sensible heat flux is stronger than the latent heat flux. However, a close examination reveals that the latent heat flux divergence is much stronger than that of the atmospheric sensible heat flux, Fig. 5. Similarly, in the Northern Hemisphere oceanic sensible heat flux divergence is the dominating source of heat south of 4 o N, north of this latitude the divergence of latent heat flux becomes more important, and north of 5 o N the atmospheric sensible heat flux dominates. 5

6 Fig. 5. Heat flux divergence. Atmos indicates the atmospheric sensible heat flux divergence, Ocean indicates the oceanic sensible heat flux divergence, and Water Vapor indicates the latent heat flux divergence associated with the hydrological cycle in the atmosphere-ocean coupled system. In both hemispheres, poleward heat flux is carried by three components and it works like a relay team. In subtropics oceanic sensible heat flux is the dominating contributor to the poleward heat flux divergence, and in mid latitudes the latent heat flux divergence is the dominating contributor. Finally, in high latitudes the atmospheric sensible heat flux divergence dominates, Fig. 5. The situation in the Southern Hemisphere is an excellent example demonstrating the roles of these three components. The most noticeable term is the latent heat flux divergence that dominates over mid latitudes. The atmospheric sensible heat flux divergence plays major role south of 7 o S, where the Southern Ocean meets the continent. The heat flux balance for a given water column is Short waveradiation + divergenceof oceanic sensibleheat flux = latent heat + long waves radiation + sensible heat It is important to note that divergence of the oceanic sensible heat flux is a relatively small term in the oceanic heat flux balance at any given location, Fig. 6. In comparison, latent heat released to the atmosphere is much larger than the divergence of the oceanic sensible heat flux at any given location. The latent heat flux released to the atmosphere is, thus, a continuous source of heat for the atmospheric circulation. Although the atmosphere may play an important role in sending heat to higher latitudes, without the continuous heat support from the oceans, the warm climate we enjoyed now may not be realized at all. 6

7 Fig. 6. Heat fluxes through from the ocean to the atmosphere, integrated zonally, and the divergence of the oceanic sensible heat flux, labeled as Ocean Current. C. The hydrological cycle In the climate system, the solar radiation is primarily absorbed by the ocean first. The atmosphere is primarily heated through evaporation and latent heat release in the atmosphere, plus the so-called greenhouse gases absorption of heat in the atmosphere itself. As shown in Fig. 4, the amount of latent heat flux from the ocean to the atmosphere is huge. The total amount of evaporation is 12.45Sv, so the latent heat associated with evaporation is about 31.1 PW. As the oceanic circulation changes, the latent heat flux will be substantially modified; thus, we should study how this is going to happen. In previous studies, the important role of the atmosphere-ocean hydrological cycle associated with evaporation and precipitation has been overlook in oceanographic literature. In oceanic heat flux calculations, the poleward heat flux associated with the hydrological cycle has been neglected, and it was incorrectly attributed to the atmosphere alone. Thus, what most oceanographers calculated in the past should be called oceanic sensible heat flux. On the other hand, meteorologists have claimed the poleward heat flux associated with the hydrological cycle as entirely due to the atmospheric processes, and it has very little connection with the ocean circulation. Such an entire separation of heat fluxes is inconsistent with the physics. Without the oceanic currents to bring heat northward, there would be no high surface temperature in the Gulf Stream (or Kuroshio) region. As a result, there would be no strong latent heat release in these areas. It is well known that the latent heat flux is the dominating form of poleward heat flux in the atmospheric circulation; however, this component has long been thought of primarily an 7

8 atmospheric process. The reason of such a misconception is, at least partially, due to the seemingly small amount of mass associated with this water vapor flux. The global integrated meridional water 9 vapor flux is on the order of 1 kg / s, which is much smaller than the mass flux of other components of the oceanic general circulation and the atmospheric circulation. Thus, the return branch of the global hydrological cycle, which takes place in the oceans, has been overlook. This misconception is widely spread in many documents, papers, and books about climate. First, although the main branch of the global hydrological cycle is through the oceans, many existing scientific programs related to hydrological cycle completely ignore the oceanic component of the cycle. Second, the thermohaline circulation in many existing climate models is not realistically simulated. For example, the oceanic component in many climate models is represented in terms of the so-called swap ocean, i.e., a very shallow layer of water (5 m or slightly deeper) with no current. In such models the atmosphere takes up water vapor at low latitudes and transports it to high latitudes. After releasing the latent heat from the water vapor, the water has been discarded from the model, and not too many people thought about dynamic consequence of this seemingly small amount of water flux. Obviously, an ocean without current would be unable to transport the fresh water from high latitudes, where precipitation prevails, back to lower latitudes, where the evaporation prevails. Thus, without the oceanic currents the subtropical ocean will eventually dry up, with all water piles up at high latitudes -- an equilibrium state impossible to reach. It is very unlikely that when a quasi-steady state is reached such models can reproduce the strong thermohaline circulation in the oceans. Thus, the sea surface conditions, such as surface temperature, surface salinity, and air-sea heat fluxes, from such a model may be quite different from the current climate. Many of the existing climate models have an oceanic component based on the so-called Boussinesq approximations. In such models, the ocean is treated as an incompressible fluid environment, and the salinity balance in the model is controlled by the so-called virtual salt flux condition at the sea surface, or the traditional technique of requiring the surface salinity relaxation to the climatological mean surface salinity. Changes in the water vapor cycle, thus, in such models have no dynamic consequence at all. As the oceanic general circulation models improved, the dynamic role of freshwater will be simulated more realistically. Among many other features, evaporation and precipitation can drive the so-called Goldsbrough--Stommel circulation (Huang and Schmitt, 1993) that is a barotropic circulation with a horizontal circulation on the order of a Sverdrup. This circulation is much smaller than the wind-driven or thermohaline circulation, so that it has been neglected from most discussion about ocean circulation and climate. However, the seemingly small freshwater flux associated with evaporation and precipitation is responsible for driving the haline circulation in the oceans. In fact, evaporation and precipitation alone (without wind stress and heating) can drive a three-dimensional baroclinic circulation, two orders of magnitudes stronger than that of the barotropic Goldsbrough--Stommel circulation, i.e., the same order of strength as the wind-driven circulation or the thermally-driven circulation (Huang, 1993). Thus, evaporation and precipitation is one of the major forces in driving the oceanic general circulation. As the climate system shifts to a new state, the freshwater flux through the oceanatmosphere system will change in response. As a result, the thermohaline circulation will be in a different state, and the sea surface temperature distribution will be different from the present-day distribution. Consequently, the surface air-sea heat fluxes shown in Fig. 6 can be dramatically 8

9 different. A comprehensive understanding will require much effort toward the ocean-atmosphere coupled system. D. The implication of a swap ocean of no current: The role of the oceans in the ocean-atmosphere coupled system is illustrated in Fig. 7. In the current climate, there are three components in this climate heat conveyor, and they work together like a relay team. Although the atmosphere is responsible for sending heat to high latitudes, it might be a misleading statement in saying the atmosphere heat transport alone is responsible for the high latitudes poleward heat flux. If there were no water vapor cycle that is closely related to the thermohaline circulation in the oceans, there would be no strong latent heat flux in the climate system. Thus, the whole Earth would be cooled down substantially. Since there is no evaporation and precipitation, there will be no strong salinity gradient in the oceans. Due to less heat flux transport from low latitudes, temperature at high latitudes should be lower than at current climate condition. Due to the strongerthan-current meridional temperature difference and almost zero salinity difference, the oceanic sensible heat flux would be intensified, which can partially compensate the loss of poleward heat flux due to lack of the hydrological cycle, as shown in experiment B. In some climate models, the oceanic component has been treated as a swap, i.e., without the current. Although it is clear that such a model would not allow heat flux transport associated with current, a far more strong constraint on the climate system has not been clearly spelled out. A close examination reveals the following constraints for such a model: 1) No current nor meso-scale eddies, so there is no horizontal heat flux -- molecular heat diffusion is negligible. 2) No current to transport freshwater horizontally. As a result: i) Evaporation and precipitation must be locally balanced. ii) No latent heat flux in the atmosphere. Consequences: Although the ocean can still play the role of absolving short wave radiation from the sun and heat up the atmosphere through latent heat release associated with evaporation, long wave radiation and sensible heat flux, there is no oceanic sensible heat flux nor latent heat flux in the atmosphere, and the only mechanism to transport heat is the dry atmosphere circulation. 9

10 Fig. 7. Sketches for (a) climate model with current and (b) climate model with a swap ocean. Summary (1) It is more appropriate to separate the poleward heat flux into three components: the atmospheric sensible heat flux, the oceanic sensible heat flux, and the atmosphere-ocean coupled latent heat flux. (2) It is important to examine the divergence of these fluxes and their interaction. (3) It is very important to explore how changes in the oceanic circulation affect the surface conditions, such as surface temperature and the air-sea heat fluxes, and the global climate conditions on the Earth. (4) Hydrological cycle plays important role in setting up the sea surface temperature and salinity; thus, the so-called swamp ocean model cannot simulate the oceanic role in climate correctly. I hope this note can promote people's new interest in study the ocean-atmosphere coupled system from a slightly different angle. 2. Haline Circulation induced by Evaporation and Precipitation Historically, evaporation/precipitation was the first mechanism explored as the driving force for the oceanic general circulation. However, the haline circulation in the ocean has been overlooked for many decades. In fact, in most previous oceanic circulation models the haline circulation was driven by either salt relaxation condition or the virtual salt flux condition; thus, the real physics of haline circulation remained obscure for long time. Even before 199, there was only a very few papers published, in which the oceanic circulation was driven by evaporation and precipitation. It is important to note that surface freshwater flux does not provide the mechanic 1

11 energy required for sustaining the circulation against friction; thus, evaporation and precipitation along cannot drive haline circulation energetically. In the following discussion we keep use this terminology for historical reason, and sometime we will use the term induce instead. A. Classical circulations driven by evaporation and precipitation (1) Hough (1897) solution Evaporation and precipitation can drive oceanic circulation, and it was first explored by Hough in his tidal paper published in For simplicity, it was assumed precipitation over Northern hemisphere and evaporation over the southern hemisphere. (Hough originally assumed 2 the E-P pattern to be a second Legendre polynomial P2( x) = ( 3x 1 )/2, i.e., precipitation over high latitudes and evaporation over low latitudes.) Hough did not know how to parameterize friction, so there was no friction in his model. In fact, his solution may be valid only for the initial stage of the problem, when the solution can be treated as infinitesimal, so that the nonlinear effect can be neglected. As time goes on, the free surface elevation and the zonal velocity would be unbounded, so the solution is not valid. Hough did not include any figures in his paper; his solution became much clearer after it was illustrated graphically by Stommel (1957), Fig. 8. Fig. 8. Two successive stages of the Hough-type circulation pattern, driven by precipitation distributed over the northern hemisphere (P) and evaporation distributed over the Southern hemisphere. The hovering arrows indicate the distribution of precipitation-evaporation. The arrows drawn on the surface of the spheres are velocity components. The zonal currents grow with time. The central solid portion of the earth is shown as shaded in cut-away mid-section. (Hough, 1897; Stommel, 1957) (2) Goldsbrough (1933) solution Goldsbrough discussed the case of the steady flow driven by evaporation and evaporation. By choosing the E-P pattern in a special form, he was able to construct a steady solution without involving the friction terms. Most importantly, he first derived the vorticity constraint for largescale circulation 11

12 2.6.3 β hv = f ( E P) (2) This is now called the Goldsbrough relation, according to this relation evaporation and precipitation work as "pumping" in driving large-scale oceanic circulation. Since precipitation drives equatorward flow, to close the circulation evaporation is needed at the same latitudes to bring the water back poleward. Thus, the trick in finding the Goldsbrough circulation is to assume a pattern of E-P that is zonally alternating so that the zonal integration of E- P is zero along any latitude circle. To illustrate the solution, we choose the following evaporation and precipitation pattern for a basin 6 o by 3 o (Fig. 9): 1 π x cos, x Δ π ( y y ) 2 n Δ Δ E P = ( e p) sin, e p= (3) ( yn ys) 1 π x cos, Δ x 1 1 Δ 2(1 Δ) where the unit is m/yr, and x is the non-dimensional coordinate for each zonal circle, Δ=.6. Therefore, the zonal sum of evaporation and precipitation is zero, as proposed by Goldsbrough. This freshwater flux induces a circulation that is closed within the basin, without requesting the help of other dynamic processes. It is also important to note that barotropic circulation induced by evaporation and precipitation is on the order of 1Sv, which is too small compared with the circulation observed in the oceans; thus, this circulation cannot be used to explain the much stronger circulation observed in the oceans. Latitude a) Evaporatin and precipitation (m/yr) Longitude Latitude b) Streamfunciton (Sv) Longitude Fig. 9. Goldsbrough's solution for a closed basin, with precipitation and evaporation balanced for each latitudinal circle (3) Stommel (1957, 1984) solution Goldsbrough's main contribution is to derive the vorticity constraint for the ocean interior. He did not know how to treat the friction; thus, in order to find a steady solution he used a distribution of evaporation minus precipitation that is quite unrealistic. The general case of arbitrary evaporation minus precipitation was solved by Stommel (1957). Stommel pointed out that, in principle, circulation driven by evaporation minus precipitation in the ocean interior can be closed by western boundary currents. Therefore, the fresh-water-driven circulation should be called the

13 Goldsbrough-Stommel circulation. Fig. 1 shows Stommel's generalization of Goldsbrough's for a single hemisphere basin. Fig. 1. Left: The Goldsbrough gyre driven by evaporation-precipitation and presented by him in 1933 as a model of the North Atlantic; right panel: Effects of using a more realistic distribution of evaporation-precipitation, and including western boundary currents. (Stommel, 1985) D. Goldsbrough-Stommel circulation in the world oceans. Using the existing data sets of evaporation minus precipitation, Huang and Schmitt (1992) calculated the Goldsbrough-Stommel circulation in the world oceans. The solution is based on zonally uniform evaporation and precipitation. The western boundary currents that play the role of closing the meridional mass flux at each latitude in each basin is then attached to the interior solution. 13

14 Fig. 11. The Goldsbrough-Stommel circulation of the world oceans. Each arrow indicates the o o horizontal mass flux integrated over a 5 5 box, in Sv. Along the western boundary of each 6 3 basin, there is a curve indicting the northward mass flux (in 1 m / s within the western boundary, which is required to close the circulation. In this calculation, the inter-basin transports are neglected (Huang and Schmitt, 1993). B. Is there something missed? The circulation discussed above is only the barotropic mass flux, which is tiny compared with the wind-driven circulation. There is no salinity involved in the dynamic analysis, were the salinity included the entire picture would be totally different. A. Fresh water flux induced circulation in a salty estuary The circulation in an estuary depends on many factors, such as the amount of river run-off, the mean salinity, and the tidal mixing. If there were no vertical mixing, the fresh water from river run-off would overflow on top of the salty water in the estuary, and this is the only circulation pattern -- the salty water below would be stagnant, if we neglect the tidal flow, Fig. 1.5a. However, with the tidal mixing, a small amount of river run-off can induce a huge return flow in an estuary, Fig. 1.5b. In our discussion we will assume that there is always energy source available for mixing, such as the barotropic and internal tides, internal waves, and other sources; however, the exact nature of the energy source is not our concern here. 14

15 Fig. 12. Sketch of models with freshwater-driven circulation in an estuary and an open ocean. a) and c) Model without vertical mixing; b) and d) Model with vertical mixing (Huang, 1993). C. The North Atlantic as an estuary Similar to the case discussed above, the North Atlantic can be treated as a huge estuary, with precipitation at high latitudes that is exactly balanced by precipitation at low latitudes. First, let us assume there is absolutely no mixing. At time t=, rain starts to come into the subpolar basin, and evaporation starts at low latitudes. Precipitation at high latitudes builds up the free surface, and water starts to flow toward low latitudes (rotation would modifies the path). At low latitudes, in the beginning evaporation would make some water saltier and sinks to depth, and this would give rise to motion in the salty water. However, as fresh water arrives the low latitude and gradually covers up the entire upper surface of the basin, evaporation can only affect the plain water, but not the salty water. As the residual motion in the deep water gradually loses their kinetic energy, the only motion will be the equatorward flow of plain water on top of the stagnant deep and salty water, Fig. 12c. Second, if there is vertical mixing, there will be a very strong return flow induced by vertical mixing, Fig. 12d. The overturning rate is S R = F F (4) Δ S Because Δ S is much smaller than S, a small amount of precipitation can drive a strong meridional circulation. 3. Upper boundary conditions for the salinity balance 15

16 A. The rigid-lid approximation The physical boundary condition on the free surface is Dη w= + ( e p), atz = η Dt (5) where w is the vertical velocity, D = + u + v Dt t x y is the total derivative, η is the free surface elevation, e-p is the evaporation minus precipitation. For the motions in the ocean interior, the following scaling can be used ( x, y) = L( x', y'), ( u, v) = U( u', v') (6) t = Tt', η = flu η ' g (7) ( ) w= δuw', e p= λδu e p ' (8) where δ = H / L 1 is the aspect ration, and λ >.1 1. An assumption has been made that the time scale is determined by the horizontal advection. Substituting these relations into the boundary condition and dropping the primes, one obtains η w= εt F + εfu η+ λ( e p), atz = εfη (9) t where ε = T 1 ft is the Kible number, U ε = fl 1 f L is the Rossby number, and F = > O(1). gh If the advection time scale L/U is chosen as the time scale, the first two terms are of the same magnitude. For motions with horizontal scale of 1 km, the non-dimensional number ε F is 4 about 1, which is much smaller than λ. Thus, the upper boundary condition can be linearized as w= e p, atz = (1) The rigid-lid approximation is not accurate for motions with time scale shorter than decadal, motions on the planetary scale, or motions in the shallow seas. For example, for motions of 7 7 planetary scale, L= 1 m, for wave motion on seasonal time scales or shorter, T 1 s, so ε T >.2. Thus, the vertical velocity due to wave motions of the free surface elevation has the same magnitude as that of the evaporation minus precipitation. Under such circumstance, the rigidlid approximation, in which the free surface motion is neglected, is inaccurate. For the large scale circulation on climate time scale the rigid-lid approximation is valid. However, in the past the rigid-lid approximation has been equal to set w= on the upper surface. Since no water can go through this so-called rigid lid, to simulate the salinity change a virtual salt flux across the air-sea interface is introduced. This virtual salt flux formulation has become a "real" flux simply because it has been used almost in every oceanographic books and papers. Some time people forget that this salt flux is not real. B. Three types of boundary conditions In this section the focus is on different types of boundary conditions for the haline circulation in the x-z plane. For convenience, Cartesian coordinates will be used in this section. The salinity balance in a surface box, Fig. 13, can be written as S 1 + ( ) ( ) 1 ( ) + κ ( ) H + κ us us ws ws ( S ) V = x Sx + ( Sz Sz ) (11) t Δx Δz Δx Δz 16

17 where ( us ) + and ( us ) are the salt flux advected across the right and left boundaries, S + x are the horizontal salinity gradient at the right and left boundaries; similarly, ( ws ) + and ( ) the salt flux advected through the upper and low boundaries, and S + z + salinity gradient at the upper and lower boundary; Sf κvsz and S z are the vertical and S x = is the salt flux cross the air-sea interface. In addition, vertical velocity is prescribed on the upper surface + w= w, atz = (12) 1) Relaxation Condition (a) w + =, so ( ws ) + vanishes. (13) This boundary condition is usually called the rigid-lid condition. As will be discussed above, the rigid-lid approximation does not necessarily require the vertical velocity to be zero at the upper surface. ws * (b) Sf =Γ( S S), (14) * where S is specified according to climatological mean surface salinity. 2) Mixed Boundary Condition (a) w + =, so ( ws ) + vanishes. (15a) (b) Sf = ( E P) S, (15b) where E and P are the evaporation and precipitation, S is the salinity in the box. This is the virtual salt flux required for the salinity balance. 3) Natural Boundary Condition (a) w + = E P, (16a) and this is the fresh water flux controlling the salinity balance. + + (b) ( ws ) κvsz =. (16b) Note that within the water column there is always turbulent salt flux κ VSz and advective salt flux ws. However, in the air there is no salt, so both these terms are identically zero, although w is not zero. At the air-sea interface the turbulent flux exactly cancels the advective flux so that there is absolutely no salt flux crossing the air-sea interface, as required by the physics. Thus, we may call this turbulent flux Sf = κ vsz an anti-advective salt flux. As seen from discussion above, ( ) Sf = E P S at the air-sea interface; however, this flux is identically zero above the water because there is no salt advection and no need for the anti-advective salt flux. are 17

18 Fig. 13. A finite difference grid box in the x-z plane (Huang, 1993). Accordingly, the salinity balance for a surface box is reduced to S κ H + κv + ( us ) ( us ) ( ws ) = ( Sx Sx ) Sz (17) t Δx Δz Δx Δz If the mixing terms are negligible, (17) is further reduced to DS Dt = (18) This equation means that the total salt is conserved. The local salinity change is due to fresh water dilution/concentration, and there is no need for the virtual salt flux. In many previous mixed layer models and primitive equation models for the oceanic general circulation the so-called ``rigid-lid'' approximation has been interpolated as equivalent to setting the vertical velocity equal to zero at the sea surface. Without a vertical flux of plain water through the upper surface, the salinity change due to evaporation and precipitation has to be simulated by the virtual salt flux as discussed above. Therefore, the so-called anti-advective salt flux has been substituted as a forcing to simulate the salt balance. However, the essential assumption of the rigid lid approximation is to move the upper boundary from the free surface z = η to the rigid lid z=, and it is not inconsistent with adding a small vertical velocity at the air-sea interface. As long as we are only concern about large-scale geostrophic motions of climate time scale, the rigid-lid approximation is valid. In fact, the rigid-lid approximation has been used in many models with non-zero vertical velocity at z=. For example, in many quasi-geostrophic models the dynamic effect of wind stress curl is simulated by imposing the equivalent Ekman pumping velocity at the upper surface. In general, the Ekman pumping velocity is about 3 times larger than the vertical velocity associated with precipitation and evaporation. Since all these quasi-geostrophic numerical models have been used extensively without any trouble associated with the non-zero vertical velocity on the upper surface, this should be true for the primitive equation models too. For motions of planetary scale or 18

19 motions associated with seasonal time scale (or shorter), the rigid-lid assumption may not be valid, and a free surface should be included in the calculation. C. Remark: The natural boundary condition discussed above is based on the rigid lid approximation, which was used extensively in the past. However, in the new generation of numerical models, the free surface is a preferred choice. The suitable boundary condition for salinity balance is a simple mathematical statement that (P-E) is a source of freshwater at the upper surface of the ocean, with no salinity flux through the air-sea interface. It is to note that the upper surface of the ocean is neither an Eulerian surface nor a Lagrangian surface because freshwater moves across this surface, as discussed in Lecture 4. In a pressure-sigma coordinate, the vertical coordinate is defined as σ = ( p p )/ p, pbt = pb pt, where p b and p t are pressure on the bottom and the sea surface, which can change with time. The upper boundary condition is p btσ = gρ f ( e p) (19) where ρ f is density of freshwater through the air-sea interface. This boundary condition means that due to precipitation minus evaporation the pressure in each grid point in the water column is increased. The corresponding salinity condition at the upper surface is that the next salt flux due to salt advection and vertical salt diffusion exactly cancel each other, i.e., S f = S f, adv + S f, diffu =, at the surface (2) In the traditional z-coordinate, the vertical velocity at the sea surface satisfies η w= + u η + e p ρ ρ (21) ( ) / f s t where ρ s is the sea surface density. Vertical integration of this equation leads to a prognostic equation for the free surface η η η = udz + vdz + ( p e) + RT t x H y H (22) where RT means rest of terms associated with thermohaline processes in the water column. Thus, precipitation tends to increase the sea level. However, the local sea level is also closely related to the horizontal convergence/divergence of the vertically integrated velocity field. For the details of sea surface elevation change induced by precipitation/evaporation, the reader is referred to Huang and Jin (22). D. The Pitfalls of the Relaxation and Mixed Conditions These three boundary conditions are quite different in their physical meaning and numerical stability. Since the relaxation condition is a negative feedback, the solutions are mostly stable. Although the ability to match the observed surface salinity may be an advantage for simulating the present climate, it may not be suitable for simulating the oceanic circulation for general situations. For example, the surface salinity for the world oceans in the remote past and in the future is unknown, so the relaxation condition cannot apply. Similarly, using the salinity relaxation condition for oceanic forecasting is questionable. Neither the reference salinity * distribution S nor the suitable relaxation constant is clear. t bt 19

20 There are essential difficulties associated with the first and second types of boundary conditions. First, to balance the salt in the oceans a huge virtual salt flux through the air-sea interface and the atmosphere is required for taking up the salt from the subpolar basin, transporting it equatorward, and dumping it there, Fig The virtual salt flux required in the North Atlantic can be estimated as ( E P) S > 1 kg/ s > kg/ s (23) 1 It is obvious that such a huge virtual salt flux is unphysical and should be avoided. Second, using the ( E P) S to force a model can cause the total salt in a basin to increase unboundedly. The argument is very simple indeed. At the present day, the global integration of precipitation minus evaporation should be zero: ( E P) da= (24) However, the global integration of the virtual salt flux is larger than zero: ( E P) SdA > (25) This is due to a positive correlation between evaporation and high salinity. For example, evaporation is very strong in the subtropical North Atlantic that gives rise to a salinity of more than 37 psu; meanwhile excess precipitation in the subpolar Pacific gives rise to a surface salinity lower than 32 psu. Thus, to avoid the salinity explosion one must use the virtual salt flux defined by f ( ) s S = E P S where S s is the mean sea surface salinity averaged over the whole domain of the model. Notice that the original definition of virtual salt flux includes a positive (negative) feedback wherever evaporation is stronger (weaker) than precipitation. However, in this new form the virtual salt flux has to be entirely fixed and it has nothing to do with the local salinity. Apparently, the feedback due to the local salinity is all gone. Even if the virtual salt flux can be accepted as a parameterization, the physics associated with the virtual salt flux is distorted. In addition, this constraint can introduce large errors wherever the local salinity is much different from the basin or global mean. For example, surface salinity in the North Pacific can be lower than 33psu, and the surface salinity in the North Atlantic can be higher than 37psu. Suppose S s is defined as 35psu for a world ocean circulation mode, a systematic bias of 1% will be introduced through the upper boundary condition for the salinity. Remark: Although the virtual salt flux has been used in general circulation models only within the past few years, it has been used in many books and papers involving air-sea interaction for a long time. For example a virtual salt flux, ( P E) S, appeared in the salinity equation in the classic paper by Niiler and Kraus (1977). As discussed above, this virtual salt flux is unphysical. In addition, the global salinity conservation requires replacement of the local S with the global mean S s ; thus, errors are introduced through use of the virtual salt flux. The natural boundary conditions for the salinity balance apply to all other cases involving fresh water flux cross the air-sea interface, including mixed layer models, ice-ocean coupling models, and atmosphere-ocean coupling models. E. A virtual natural boundary condition 2

21 Evaporation and precipitation data is hard to collect, and there is no reliable historical data. As a compromise, one can use the sea surface salinity as forcing field to reconstruct the salinity balance in the past, using the virtual natural boundary condition as following. First, the equivalent evaporation and precipitation field can be inferred from the historical sea surface salinity data from the model as * E P =Γ( S S)/ Ss, * where S is the observed salinity at a given time in the past, and S s is the global mean sea surface salinity. Second this equivalent evaporation and precipitation field is used the freshwater flux through the air-sea interface in the model. 4. Thermohaline Circulation under Freshwater Forcing Alone o o o o The example shown here is for a 4 4 resolution model ocean of 6 6 basin, driven by a "linear" profile of E-P, shown in Fig. 14, with w = 1 m/ year. Note that the model's only forcing is the fresh water flux through the upper surface. The total meridional mass flux (indicated by the dashed line) associated with E-P is about.3 Sv, this is really very small, compared with the wind-driven circulation. Fig. 14. Meridional distribution of mass fluxes: E-P in the evaporation minus precipitation (in 5 1 cm / s ), M int erior is the poleward mass flux in the interior, M wbc is the mass flux of the western boundary current, M total is the total poleward mass flux; all these mass fluxes are in Sverdrups. The model reaches a state of periodic oscillation with a regular period of 18 years, not shown here. Fig. 5.2 shows the barotropic streamfunction (upper panel) and the total barotropic o o meridional mass flux within each 4 4 grid-box (lower panel). These two figures show the southward flow driven by precipitation in the subpolar basin, the poleward flow driven by evaporation in the subtropical basin, and the western boundary currents required for closing the 21

22 circulation, as suggested by Stommel. Fig. 15. (a) Barotropic streamfunction and (b) integrated meridional mass flux through each box, in Sv. The meridional and the zonal overturning cells are shown in Fig. 16. a) and b). In the vertical direction, there are two strong baroclinic gyres circulate in the opposite direction. Fig. 16. Time-mean circulation patterns, depth in km. a) Meridional streamfunction in Sv. Note that contour interval is 2 Sv for the unlabeled curves. b) Longitudinal streamfunction. c) Zonally mean meridional salinity distribution (deviation from the 22

23 mean salinity). d) Meridionally mean zonal salinity distribution (deviation from the mean salinity). Interval is.5 for the solid lines and.5 for the dashed lines. In this idealized numerical experiment, surface freshwater flux is the only driving force. Nevertheless, the model demonstrated that the salinity distribution in the oceans in the results of the freshwater flux condition specified on the sea surface, Fig. 17. Fig. 17. Salinity (deviation from the mean) distribution at 15, 32, 64, and 123 m. Under the freshwater forcing, there is a complicated three-dimensional haline circulation, Fig. 18. Precipitation tends to be very irregular in nature. Thus, in human s mind s eyes, there is not a clear connection between the tiny bits of rain drops in precipitation and large-scale organized haline circulation in the ocean. However, precipitation and evaporation are responsible for creating the salinity difference in the oceans, and thus regulating the haline component of the oceanic general circulation. 23

24 Fig. 18. Schematic structure of the saline circulation driven by precipitation at high latitudes and evaporation at low latitudes. In some previous studies, the salt flux diagnosed from models driven by the virtual salt flux has been misinterpreted as real salt flux in the oceans. As shown in Fig. 19, the net meridional salt flux approaches a quasi-equilibrium, as the circulation itself approaches the final equilibrium, Fig. 19a. The large advective salt flux diagnosed from models based on the virtual salt flux condition should be re-interpreted, and after such a reinterpretation the meridional salt flux diagnosed from models based on the virtual salt flux should also reaches a quasi-equilibrium. Note that for timedependent problem, the so-called salt flux diagnosed from models based on virtual salt flux condition is meaningless. 24

25 Fig. 19. Time evolution of the meridional salt fluxes at 28 o 6 N, in 1 kg / s : a) The new model; b) the old model based on virtual salt flux condition. 5. Salinity contribution to stratification and horizontal pressure gradient in the oceans Density is a nonlinear function of temperature, salinity, and pressure. In particular, the thermal expansion coefficient strongly depends on temperature. As a result, density at warm temperature is primarily controlled by temperature; however, at high altitudes, thermal expansion coefficient is so small (Fig. 2) that density is primarily controlled by salinity. 25

26 Fig. 2. Thermal expansion coefficient at the sea surface in the world ocean, in units of 3 1 / o C At low temperature salinity can made important contribution to the stratification in the ocean, and this can be illustrated through the following diagnosis based on climatological mean T, S properties in the world oceans. First, we will study how much salinity contributes to the vertical stratification in the water column. As an example, we shown the stratification ratio, which is defined as follows. The commonly used stratification can be extended as 2 g d 2 g d ( T, S) N = ρθ ρθ, NT, DN N NT ρ dz = ρ dz = (26) 2 where N T is defined as the equivalent stratification due to the vertical distribution of temperature only, with the salinity set to a constant value of S = 35. The difference between these two terms is denoted as DN, thus DN / N indicates salinity s contribution to the stratification. For example, the zero contour indicates the place where vertical salinity gradient does not contribute to the local stratification. As shown in Fig. 21a), in the upper ocean, stratification is primarily due to density gradient associated with temperature profile, while salinity gradient tends to weaken the stratification. At the level of nearly 1.5km below the sea level, the stratification is primarily controlled by the relatively fresh water of the Antarctic Intermediate Water, Fig. 21b). 26

27 Fig. 21. Contribution of temperature and salinity to stratification along 3.5 o W, inferred from climatology. Left panel: N (black line), N due to temperature only (red line), and N due to salinity only (blue line); right panel stratification ratio. In the Atlantic Ocean, salinity control of the stratification can be clearly identified from depth below the upper kilometer, Fig. 22b. In particular, both salinity gradient associated with masses of Antarctic Intermediate Water and the North Atlantic Deep Water clearly plays major role in shaping up the stratification at middle depth. 27