Module 8 Ocean Circulation and Surface Processes 8.1 Introduction Winds are produced in response to radiative heating of the atmosphere.

Transcription

1 Module 8 Ocean Circulation and Surface Processes 8.1 Introduction Winds are produced in response to radiative heating of the atmosphere. These winds constitute an important forcing for ocean currents, which are generated due to momentum transfer into the ocean by winds. The pressure gradients generated by radiative heating could produce wind speeds of about 1 ms 1 in the atmosphere just above the ocean. Yet, there will be no momentum transfer by winds to ocean layers if there were no friction at the surface. Because of the frictional contact, no slip condition will be satisfied by the airflow at solid surface boundary; that is, the air in immediate contact with the boundary attains zero velocity. This will set up a velocity gradient (or shear) near the solid boundary. The shear flow set up in this manner is not stable at higher windspeeds because small disturbances can grow at the expense of mean motion to turn the flow turbulent. The turbulent eddies are responsible for the gusty nature of the flow, modify shear for a well-defined mean velocity structure to develop after sufficiently long time. The flow velocity is a function of z, i.e. the distance from the surface in vertical direction. The shear depends on mean stress τ, density ρ and distance z from the ground, and one obtains a logarithmic mean velocity profile. Near the ground, wind shear varies as ( u / ) 1/ z ; that is, the inverse law holds only sufficiently close to the ground. The turbulent eddies in the shear flow close to the surface affect the transfer of momentum, heat and moisture at the interface of ocean and atmosphere. What are the other processes that affect the evolution of the surface layer besides the dynamical forcing of winds? How are their effects included in the analysis? There exists an extensive literature for an elaborate response to these questions. The other thermodynamically mediated processes are primarily the solar heating deposited in a few tens of meters in the upper ocean; evaporation at the sea surface; cooling by evaporation and sensible heat transfer at the surface of the ocean. Atmospheric circulations of different temporal and spatial scales produce horizontal gradients in the seawater properties in the open ocean affecting the current strengths below the interface. Such horizontal gradients, in conformity with nonlinearity of the equation of state of seawater, could impact the ocean stratification in different latitudes. The ocean waters are inherently stratified with higher density waters arranged at the ocean bottom where temperatures are cold and salinity is low. The uniform distribution of potential temperatures in deep ocean signifies constant mixing forced by friction in the bottom Ekman layer. At the ocean surface, atmospheric processes constantly change the distribution of sea surface temperature and salinity through horizontal transport and stirring by winds, rain and evaporation, while moving air parcels pick up moisture from the sea surface and distribute it around the globe. For the same solar heating rate, land temperatures rise faster in summer than the sea temperatures and also land cools rapidly during winter with the diminishing radiative heating. As a result of this contrast, low-pressure systems during summers and high pressures during winters develop over land areas. Such a flip-flop in atmospheric pressures over land will result in the alteration of pressure systems over the oceans as well. For example, there is a high-pressure system (the Azores high) in the north Atlantic in summer, which is replaced by the Icelandic low during winter. Since the winds will be almost parallel to isobars over the oceanic regions, the wind induced surface ocean 1

2 currents (Fig. 8.1) will have anticyclonic orientation below the atmospheric high pressure cell and cyclonic below the low pressure cell. Due to Ekman transport, light waters are on the right relative to the direction of the wind in the northern hemisphere, a lens of warm water (Fig. 8.1) will form the core of anticyclonic surface current cell, and a cold core for the cyclonic cell. In this manner, interaction forced by atmospheric circulation at the ocean surface contributes to variability of the climate system, which will finally affect the transfer processes at the ocean surface wherever air interacts with the sea. High Pressure Cell Atmosphere Low Pressure Cell Atmosphere sea level Warm lens convergence sea level divergence Thermocline Cold Ocean Thermocline Warm Cold Ocean Fig. 8.1 Development of a warm lens on the sea surface under the action of anticyclonic wind stress and cooling of surface waters under cyclonic wind stress. Note the vertical movement of thermocline under the influence of a circulation associated with a low and a high in the atmosphere. These processes also play a dominant role in changing the density of seawater, and any loss of buoyancy would engender convective overturning in the ocean depth. From the equation of the state of seawater, it may be inferred that temperature and salinity generally play an opposing role on density of ocean waters. In the context of dynamics of oceans and atmosphere, it may be said that atmospheric winds drive ocean circulation; and evaporation from sea surface contributes water vapour to the atmosphere, which drives atmospheric circulation in combination with the radiative heating and convective overturning in the troposphere. Therefore, interaction of the oceans and atmosphere is key to the understanding of climate and its variability at shorter time scales and to climate change at longer time scales. 8.2 Drag Coefficient The atmosphere and ocean exchange momentum, heat and mass including aerosols/tracers, which have a bearing on biological productivity of the oceans. The dynamical interaction between the ocean surface and wind is incorporated through roughness length, z, which depends on the wind at standard height (1 m) above the surface, wave formation and breaking and momentum dissipation at the surface. In the absence of waves all the momentum will be transferred to the sea surface. However, for momentum transfer, viscosity of a fluid defines its diffusive capacity. Therefore at the ocean-atmosphere interface, the ratio of the diffusive capacities of air and water directly gives the surface velocity in terms of the geostrophic wind U g. For given density and 2

3 kinematic viscosity ρ a, ν a (= µ a / ρ a ) of air and ρ w, ν w (= µ w / ρ w ) of seawater, the surface drift velocity U on the ocean side of the interface is given by.5 U = ρ a ν a 1 ρ w ν w U g = 2 U g =.5U g (8.1) U has the same direction as that of geostrophic velocity U g. However, the Ekman theory predicts the interface velocity in the direction of the resultant of geostrophic wind and the current vectors. This amounts to a contradiction between the two views and further studies are necessary. Below the interface into the sea, the mean velocity is much smaller than the interface drift velocity. However, for air-sea interaction, it is the 1-m wind (U 1 ), which is used in the calculations. Moreover, because surface roughness of the ocean waters changes with windspeed, a drag coefficient C D may therefore be defined to deal with the changing winds and wave interaction. For windspeeds exceeding 8 ms 1 we have u 2 C D = (U 1 U ) ; U = 2 u2 x + u2 y (8.2) where u * is the friction velocity, U 1 and U are respectively the windspeeds at 1m height and at the earth surface. The drag coefficient is an appropriate parameter for the momentum transfer under different atmospheric conditions and the state of the sea surface stable, unsteady or dominated by waves. Once C D is known, then from the boundary layer theory, the wind stress (τ ) at the surface is defined as, τ = ρ a u 2 = ρ a C D (U 1 U ) 2 2 = ρ a C D U 1 as U << U 1. (8.3) That is, the wind stress τ is directly related to the 1m windspeed from (8.2). The formula (8.2) is one of the bulk aerodynamic formulae for computing the fluxes of momentum, heat and moisture at the atmosphere-ocean and atmosphere-land interface. On the ground U =, but over the ocean U is the velocity of the wind at the sea surface. Though ocean is not a solid surface, yet the surface velocity U is sufficiently small (typically U.3U 1 ). This is basically due to the density differences between the air and water and the same momentum of air masses at air-sea interface can be carried with much smaller velocities of water masses. Hence the turbulence production over ocean is similar to land because the shear over the oceans is as large as that over the land. The drag coefficient C D increases with wind speed (U) for the ocean surface as well, thus 1 3 C D = 1.1 U 1 6. ms U 1, 6. < U 1 < 22. ms 1 Charnock (1981) gave the following expression for the drag coefficient: C D = κ ln ρgz aτ 1 ; κ =.4, a =.185 (Smith 198) u v 1 2 is the windspeed at 1 m The parameter κ is the von Karman constant. U 1 = height and U is the component of ocean surface velocity along the wind direction. U o is 3

4 generally neglected in comparison to U 1 ; but it should not be neglected in the regions of ocean where surface currents are strong and winds are weak. 8.3 Bulk parameterization of heat and moisture The expression (8.3) relates the wind stress τ with windspeed at 1 m height, which can be further simplified by neglecting U, which gives the following expression for the friction velocity u 2 = C U 2 D 1 (8.4) The bulk aerodynamic formulae for the sensible heat flux Q H and the moisture flux E are given by Q H = ρ a C p C H U 1 (T T ) (8.5) E = ρ a C E U 1 [q(t ) q(t )] (8.6) The latent heat flux is then calculated as Q E = L v E (8.7) These parameterizations are applicable both for land and sea with the coefficients C H for sensible heat and C E for water vapour. These parameters must be known for stable, neutral and unstable atmospheric conditions. Both C H (Stanton number) and C E (Dalton number) are the dimensionless aerodynamic exchange coefficients for temperature and humidity transfer respectively. When one uses these formulae for air-sea exchange to compute the fluxes of sensible heat and evaporation, then T and q in (8.5) and (8.6) are respectively replaced by the sea surface temperature T s and saturated specific humidity q s. Since the primary goal is to estimate the fluxes of momentum, heat and moisture using mean observations at one height, the formulae (8.4) (8.6) fulfil this requirement. The value of both C H and C E in neutral conditions is which remains independent of windspeed within 5 2 ms 1, for windspeeds exceeding 2 ms 1 possibly C E may increase due to enhanced evaporation. Under unstable conditions, the transfer coefficients have large values for light wind conditions, which decrease with increasing windspeeds. For C H and C E, Smith (198) found a good fit of data with the following values.83 1 C H = C E = for stable conditions for unstable conditions, The bulk aerodynamic formulae are based on the premise that the near-surface turbulence arises from mean wind shear over the surface and that turbulent fluxes of heat and moisture are proportional to their gradients just above the sea surface. Like the shear, the temperature and humidity gradients also increase, as the ocean surface is approached, in inverse proportion (~ 1/ z) with the distance (z) from the surface. Both heat and moisture are vertically transferred at the air-sea interface by bodily movement of air parcels. Hot and moist air parcels move upward and relatively cold and dry air would be transferred downward. 4

5 8.4 Ekman layers One may notice from Fig. 8.1, that low-level winds have a direct influence on the surface currents; besides this, the net effect of wind action on the surface is seen in the vertical movement of thermocline. The momentum transfer to the ocean surface can be calculated if the components of the wind stress are known. For this purpose, the wind stress components from (8.3) can be written as τ = (τ x, τ y ) = ρ a C D U 1 (u 1, v 1 ) (8.8) Here (u 1, v 1 ) are the components of the wind at 1 m height. Another simplification arises if the Rossby number is small: one can neglect the acceleration terms in the equation of motion. In this situation, the Coriolis force, the horizontal pressure gradient and force due to wind stress would balance in the top layer of the ocean, which reads as fv = 1 p x + 1 τ x (8.9) fu = 1 p y + 1 τ y (8.1) In the above equations (= 127 kg m 3 ) is the reference density of seawater, (u, v) are the components of horizontal velocity of ocean currents. The effect of surface wind stress will propagate downwards by the action of turbulence and wind induced stirring at the sea surface. The balance of wind stress and the frictional force arising from vertical shear-induced turbulence will happen in a thin boundary layer which is known as the Ekman layer in the ocean. This suggests that wind stress can be modelled using fluxgradient relation by employing an eddy viscosity coefficient K as τ x = K u and τ y = K v. (8.11) In view of (8.11), the governing equations (8.9) and (8.1) of the Ekman layer take the following form f (v v g ) = K u (8.12) f (u u g ) = K v (8.13) The above equations underscore the role of ageostrophic components of the wind in the Ekman layer dynamics. The boundary conditions for the above set of equations are (u, v) as z (at the bottom) (8.14) K u, K v = τ x (), τ y() at z = (at the sea surface) (8.15) Note that in (8.12) and (8.13), the pressure gradient terms have been replaced by the velocity components of the geostrophic current. Equations (8.12) and (8.13) can be further simplified if the velocity components (u, v) are decomposed in to a geostrophic part and an ageostrophic part as follows: 5

6 (u, v) = (u g + u ag, v g + v ag ) (8.16) Using (8.16) in (8.12) and (8.13), we obtain the following equations for the ageostrophic velocity components of the ocean currents fv ag = K u ag (8.17) fu ag = K v ag (8.18) The boundary conditions for the above system now read as (u ag, v ag ) as z or z δ (Ekman layer depth) (8.19) τ x K u ag, K v ag =, τ y at z = (assuming u g = ; v g = ) (8.2) The Ekman layer thickness in the ocean is about 3m, i.e. δ = 3m. The condition (8.19) requires ageostrophic velocity to vanish at the lower limit of the Ekman layer and (8.2) does not allow shear in the geostrophic currents. An important derivation: Before attempting the solution of the Ekman boundary value problem ( ) we derive an important relation. Vertically integrating the governing equations for the ageostrophic components (8.17) and (8.18), we get f δ v ag dz = K u ag z= z= δ z= = K u ag () = τ x f u ag dz = K v ag = K v ag δ () = τ y ρ z= δ Since the wind stress vanishes at the lower boundary ( z = δ ) of the Ekman layer, the above equations can be written in terms of integrated mass transport through the Ekman layer as f M y = τ x and f M x = τ y ; M x = u ag dz and M y = v ag dz (8.21a) δ Hence, we write the mass transport M as follows ˆk τ M = ; M = (M x, M y ) (8.21b) f In (8.21b), ˆk is the unit vector along the vertical. The important inference from (8.21b) is that the lateral mass transport is directed at 9 on the right to the direction of wind stress in the northern hemisphere. Note that the direction of the wind stress is same as that of the wind. Determination of Ekman solution and Ekman spiral: We now present the solution of the Ekman boundary value problem (BVP) described by (8.17) (8.2). The equations (8.17) and (8.18) can be combined into one equation by multiplying by i (= 1) the eq. (8.18) and then adding the result to (8.17) and assuming constant eddy viscosity coefficient K, we get the following steady state equation, δ 6

7 K d 2 (u ag + i v ag ) dz 2 = i f (u ag + i v ag ) (8.22) with boundary conditions u ag + i v ag as z δ (8.23) K d(u ag + iv ag ) dz = τ x () + iτ y () at z = (8.24) The solution of Ekman BVP (8.22) (8.24) is straightforward but involves some algebraic and trigonometrical manipulations. The steady solution of the above BVP is u ag = v ag = e λz f K τ y()cos λz + π 4 +τ ()cos λz π x 4 (8.25) e λz f K τ y()cos λz + π 4 τ ()cos λz π x 4 (8.26) f 1 In the above solution, λ = ; also λ has dimension of length, therefore, for every 2K increase in depth equal to λ 1, the magnitude of the velocity decreases by a factor e 1. The hodograph of the steady solution (8.25) and (8.26) depicts the Ekman spiral, with maximum current speed at the surface inclined at angle of 45 on the right to the direction of wind stress vector in the northern hemisphere. Indeed, E.W. Ekman (194) produced this solution overnight when Fridtjof Nensen suggested this problem to him while Ekman was a student of Vilhelm Bjerknes, the famous Swedish meteorologist who first recognized weather forecasting as an initial value problem in mathematical physics in 194 and also proclaimed that the system of equations to be solved were already known, at least in general form. (The Bjerknes family has made important contributions to meteorology in its recognition as a science. C. Bjerknes, father of V. Bjerknes, formulated the equations of motion with rotation; and, this knowledge helped V. Bjerknes to make several landmarks in meteorology along with his son J. Bjerknes, who is famous for providing a deeper understanding of El Niño Southern Oscillation). Ekman pumping and suction: Away from the geographical boundaries and equator, the wind stress forcing can produce vertical motions in the open ocean, when horizontal gradients in wind stress are present. A simple assumption that seawater may be considered incompressible in the upper layer, then the continuity equation reads, u x + v y + w =. Integrating the above equation over the depth of the Ekman layer, one obtains u w() w( δ ) = ag x + v ag y dz = x δ δ u ag dz + x δ v ag dz 7

8 Assuming that w() = and writing w( δ ) = w e as the vertical velocity at the bottom of the Ekman layer, the above equation can be written as w e = x M x + y M y = τ y x f + y τ x f = ˆk. τ x f Hence we have the following expression for Ekman pumping and suction w e = ˆk. τ f (8.27) Thus, in the open ocean the vertical velocity at the bottom of the Ekman layer is proportional to wind stress at the surface divided by the Coriolis parameter. As an example of air-sea interaction where surface wind induced Ekman pumping and suction play a key role, is the triggering of El Niño Southern Oscillation (ENSO) event in the Pacific Ocean. It is one of the most outstanding phenomena in the tropics that produce global impacts such as large-scale droughts and flooding in many parts of the world. The El Niño (warm SST) anomalies have been explained as the response of the ocean to surface winds. The westerly winds induce downwelling over the equator and upwelling on the northern and southern latitudes (Fig. 8.2a) in accordance to the Ekman dynamics. The Ekman layer is thought to be infinitely thin on the ocean surface where waters are collected horizontally by Ekman drift; and these surface waters sink down by Ekman pumping. As a result, thermocline will deepen allowing charging of warm waters on the equator in the west Pacific. Further, the westerly wind anomalies depress the thermocline in the east and reduce equatorial upwelling in the Pacific. The warm waters from the western Pacific subsequently set out eastwards under the forcing of the westerlies over the equatorial region. Warmer than normal SSTs, developed in the eastern Pacific, induce upward atmospheric motions accompanied by moisture convergence, which increases the conditional instability of the column above. The instability is accompanied by the release of large amount of latent heat in the deep cloud columns, which will increase atmospheric heating. This is the Bjerknes positive feedback mechanism of the anomalies in the ocean and the atmosphere (J. Bjerknes, Atmospheric teleconnections from the equatorial Pacific. Monthly Weather Rev., vol.97, 1969). Thus, ocean dynamics changes sea-surface temperatures in the equatorial region, which change atmospheric heating and induce changes in the circulation; and the altered atmospheric circulation, in turn, changes the ocean dynamics., τ y f Westerlies Westerlies (a) Equatorial downwelling Upwelling Ekman drift Sinking Ekman drift Upwelling 1N Equator 1S (b) Equatorial upwelling Downwelling Easterlies Ekman drift Upwelling Ekman drift Easterlies Downwelling 1N Equator 1S Fig. 8.2 Illustration of downwelling and upwelling in the equatorial latitudes This mechanism has been successfully verified in the Cane-Zebiak model, a simple couple ocean-atmosphere model (SE Zebiak and MA Cane, A model of El Niño Southern 8

9 Oscillation, Monthly Weather Rev., vol.115, 1987). The model could also demonstrate that prior to the onset of ENSO, the equatorial heat content increases which, after the onset of ENSO, actually decreases. The Cane-Zebiak model could also predict, in the Pacific, return to normal conditions (non-el Niño case) in much agreement with the observations. Once the normal conditions are established in the Pacific, the easterlies reappear and, as shown in Fig. 8.2b, the associated upwelling of cold waters over the equator produce a tongue of cold water in east Pacific extending westward from the Peru coast. The winds force a coastal current that also brings cold Southern Ocean waters to the eastern Pacific equatorial region. 8.5 Western boundary currents The western boundary currents have been mentioned earlier in Module 2 while discussing the global current systems in the ocean. These currents also arise in response to the forcing due to wind stress and flow northward in the northern hemisphere. The current speeds exceed 1cm s 1 and both types of currents, i.e. warm and cold, are driven by the winds. The Gulf Stream in the North Atlantic and Kuroshio Current in Western Pacific are the warm currents and transport large amounts of heat poleward. However, the Somali Current produces strong upwelling on the east coast of Somalia and therefore it is a cold current that gives rise to significant Arabian Sea cooling during monsoon months. To discuss the large-scale response of the ocean to winds, it is pertinent to begin with the vorticity equation as discussed in Module 2. It was pointed out that divergent motions produce significant amount of relative vorticity which will give rise to meridional movement of rotating column as explained in Fig of Module 2. The equation (2.13) can be reduced to a simpler form where stretching of the vortex tube (planetary vorticity) by divergent motion is balanced by the meridional advection of planetary vorticity; that is, v df dy = f.v Since β = df w and.v =, V is the mean horizontal velocity; the above equation dy can be written in the following form βv = f w (8.28) The eq. (8.28) is the most simplest equation, known as the Sverdrup relation, which has a deeper physical interpretation: production of vorticity by stretching of the planetary vorticity ( f w ) equals to loss due to advection of planetary vorticity (βv) by the meridional velocity v. This means that positive vorticity production will force northward movement of water masses to reach the latitude (φ) with correct value of Coriolis parameter ( f = 2Ωsinφ ). The western boundary currents can be explained using the vorticity model of winddriven circulation. In this model it is recognised that the torque of the atmospheric wind field would generate motion of similar kind in the ocean. Thus, between trades (easterlies) and westerlies, anticyclonic vorticity is generated in the oceanic cells; analogously, cyclonic vorticity in cells between westerlies and polar easterlies. Between 9

10 the western and eastern boundaries of the ocean, the motion will be symmetric about the centre of the cell. Now, consider an anticyclonic oceanic cell on the west side of the basin, which is displaced by the curl of the wind stress poleward. Due to this action the anticyclonic relative vorticity of fluid column will further intensify to conserve its absolute vorticity. While on the eastern side the anticyclonic vorticity will be balanced by the positive relative vorticity as the fluid columns move equatorward. Essentially, on the western side of the ocean, there is an excess of anticyclonic vorticity, which continues to get augmented as the fluid columns move further poleward. This cannot continue and a breaking mechanism is necessary. The friction at the west boundary could provide brakes to the current. Therefore, when the strong poleward moving current interacts with western boundary, a strong shearing flow will be produced with equal and opposite component of frictionally generated cyclonic vorticity; consequently, the current will become narrow and strong along the west coast. In this manner the lateral boundaries provide a balancing mechanism, but strong asymmetry in the motion of water masses will be noticeable: strong northward moving current on the west side of the basin and much weaker equatorward current on the eastern side, though the torque of the wind driving the currents is symmetrical. The above mechanism explains all those very strong and narrow currents in the major ocean basins, viz., the Gulf Stream in the North Atlantic, Kuroshio Current in the western Pacific near Japan, Somali Current in the Indian Ocean, Brazil Current in the South Atlantic, Agulhas Current in the Indian Ocean and the Australian Current. Henry Stommel (1951) called these currents as the western boundary currents. Based on eq. (8.28), Stommel also explained the abyssal circulation in the global ocean. There exist western boundary currents in the deep ocean also. The sources of deep water formation are the North Atlantic Ocean and the Weddell Sea in the South Atlantic Ocean. The theory explaining abyssal circulation is not included here but one can follow it from the standard textbooks on oceanography. 8.6 Mixed Layers in the ocean In the shallow top layer, directly in contact with the atmosphere, both salinity and temperature are well mixed due to stirring of the ocean surface by the action of wind gusts and turbulence. The seawater properties have practically uniform profiles there. This region is known as the mixed layer with near-neutral stratification (N = ) in the ocean. An oceanic mixed layer is bounded below by seasonally changing thermocline. One of the key features of this layer is that it responds strongly to seasonal changes. Incident solar radiation of all wavelengths is absorbed in the top 1m with nearinfrared and infrared being absorbed in just a centimetre thick layer on the sea surface. The action of wind at the surface enhances evaporation, which cools the surface; and cold parcels will then sink and would be replaced by lighter fluid from below. Hence in the mixed layer, there is net upward flux of energy due to convective overturning which mixes the fluid parcels from different depths. Turbulent mixing is another mechanism, which produces a uniform profile in the mixed layer. During summers, intense solar heating of seawater at the interface produces a warm layer. This will reduce the buoyancy of parcels, which will result in a shallow mixed layer. On the contrary, if surface layer is cooled at a faster rate, the buoyancy induced convective overturning will be strong because the cold parcels could sink as deep as one kilometre. This will result in a deeper 1

11 Depth (m) mixed layer. However, in the tropics mixed layer depth (MLD) is generally taken as the depth of the 2 C isotherm in the ocean. The mixed layer depth in tropics varies in the range 5 1 m as seasons change. z = 3 1 cooling cooling Evap/Precip SW LW heating Turbulence Wind mixing T BLD / MLD Thermocline Mixed layer Barrier layer Entrainment Fig. 8.3 Temperature profile, different layers of the ocean, surface and vertical processes, and the coordinate system are depicted here. In the upper part of the mixed layer both temperature and density are well mixed. But in the barrier layer part of the mixed layer, density gradients are dominant (curved line illustrates density gradients in BL). Abyssal Waters The situation in the Bay of Bengal is rather unique as it receives large volumes of freshwater discharge from the Brahmaputra River. Besides the river runoff, the region also receives excess precipitation during the monsoons. The freshwater flux makes the near surface waters less saline and creates strong salinity stratification in the isothermal mixed layer. As a consequence, an internal layer forms within the mixed layer above the thermocline. This intermediate layer is referred to as the barrier layer (BL), which inhibits turbulent entrainment of cool thermocline water at the bottom of the mixed layer (Girishkumar, Ravichandran, McPhadan and Rao, JGR, vol.116, C39, 211). The formation of barrier layer would thus impact the vertical heat flux and thereby the sea surface temperature (SST). Ekman pumping and suction will modulate the barrier layer depth (BLD) and it would vary between 4 1 m. Thicker BL will make the wellmixed layer shallow (MLD ~ 1 2 m) at the top. Thus, barrier layer is bounded above by a region of mixed layer with no vertical gradients of temperature and density, and below by thermocline or the layer dominated by strong temperature gradients. Note that salinity stratification could as well give rise to temperature inversions in the mixed layer. In tropics, depth of the isothermal layer could be less than the depth of the 2 C isotherm. This observation could easily be confirmed from the Argo Data of the Bay of Bengal. There are various models of oceanic mixed layer, which are based on physical processes of entrainment of fluid from surrounding stable layers. Along with the temperature profile, we have schematically shown in Fig. 8.3 the surface vertical processes and locations of mixed layer and the barrier layer in the interior of the oceans. The thickness of the barrier layer and therefore of mixed layer depends much on the large-scale variability of the ocean. The curved line represents BLD if the barrier layer present, else the mixed layer depth (MLD). Besides entrainment from bottom of mixed layer, other important processes that determine the deepening and shoaling of mixed layer are the shortwave heating, longwave cooling, turbulent mixing induced by wind stirring, momentum transfer by winds, evaporation, freshwater flux and the large scale 11

12 dynamics that produces Ekman pumping and suction. In our discussion, both the top layer and the barrier layer (if present) are the two regions of the composite mixed layer with different salinity structures that could even give rise to temperature inversions in the mixed layer. To summarize, in mixed layer both temperature and salinity are well mixed (or uniform), while salinity is stratified and temperature is uniform in the barrier layer. This is a distinct feature in the salinity and temperature profiles of the mixed layer that allows identification of the barrier layer. In the coordinate system chosen here, z = is at the surface and decreases downward as shown in Fig Let h represent the mixed layer depth, then the local changes in h are determined from the following equation h t = h.v + w e (8.29) where V is mean horizontal velocity and w e is the entrainment velocity at the base of the isothermal mixed layer. The first term in (8.29) on the right hand side, represents the changes in the mixed layer depth arising from processes such as the Ekman pumping and internal gravity waves. The second term in the above equation represents the changes in h arising from turbulent entrainment of the fluid in the mixed layer from below. That is, the changes in the mixed layer depth (h) are caused by entrainment (turbulent) of fluid from its bottom and convective overturning caused by sinking plume from surface cooling. One of the characteristics of the ML is that both temperature (T ) and salinity (S) are well mixed. Therefore, due to thorough mixing in the ML, the vertical density gradients will also vanish. The density gradients will produce buoyancy of fluid parcels, which is needed to compute local changes in h. Using the Boussinesq approximation (i.e. density variations are important only when occur in combination with acceleration due to gravity), the buoyancy b per unit volume can be calculated as b = g ρ ρ = g[α T (T T ) β(s S )] (8.3a) α T T and β T ( T > 5 C ) (8.3b) In writing the right hand side of (8.3a), we have used the equation of state (6.11) for seawater. The quantities with subscript are the reference values; α T (T, S) is the thermal expansion and β(t, S) is the haline contraction. Linear approximations (8.3b) for α T and β have been found to be useful for mixing studies at atmospheric pressure. The evolution of b is described by a conservation equation of the form F pen db dt + wb' = gα T = S b (8.31) ρc p where F pen (z) is the penetrating solar radiation flux, C p is the specific heat of seawater. Hence the term S b / represents the source of buoyancy arising from solar radiation absorbed in the interior of the mixed layer. Since the vertical gradients of b will vanish ( b / = ) if density is well mixed; and if the horizontal variations of b are assumed to be negligibly small, then the Lagrangian time derivative in (8.31) can be replaced by the local derivation; and (8.31) can be written as 12

13 b t + wb' = S b (8.32) On assuming z increasing downward and integrating (8.32) vertically from to h, we get the following result h b t + wb'(h) wb'() = S (h) S b b () (8.33) Set wb'() = B, the buoyancy flux at the interface in the ocean; neglect the small amount of penetrating flux S b (h) ; and represent mathematically wb'(h) = w e Δb, then (8.33) takes the following form h b t = B w Δb S e b () (8.34) In equation (8.33), the surface buoyancy flux B can be parameterized as B = g α T Q w + βs(e P) (8.35) ρ C p where C p is the specific heat of seawater; and the surface heat flux Q w is given in terms of net radiative flux ( F net ), sensible heat flux (Q s ) and latent heat flux ( L v E ) as Q w = F net + Q s + L v E (8.36) It may be noted that B has different numerical values in the ocean and the atmosphere but it has the same sign. The buoyancy flux at the bottom ( w e Δb ) is indeed the power required for seawater at the bottom to drive it into the mixed layer, that is w e Δb = w e (b b h ) (8.37) Thus Δb is the buoyancy discontinuity at z = h. The evolution equation of b h reads b h b(z) = w e (8.38) t From turbulent kinetic energy (TKE) considerations, Krauss and Businger (Atmosphere- Ocean Interaction, Oxford University Press, 1994) derived the following expression for the entrainment velocity w e, which reads m w e = 2 1 u 3 + m w 3 * 2 * c q 2 m 3 (ΔU) (8.39) 2 where q 2 = u j u j is the mean squared turbulent velocity, c 1 is the velocity of the internal gravity wave along the density discontinuity at z = h and it is possible to estimate it as hδb = hg ρ ρ h c 2 1. (8.4) In (8.39) ΔU is the horizontal velocity change across h which arises due to shear. If w e is positive, then the mixed layer will grow as time derivative of h will be positive. This is deepening of the mixed layer. However, if the denominator and the numerator in (8.39) are positive, then entrainment will occur from the bottom of the mixed layer as w e < (because downward direction is positive). This means the thermocline waters will be forced into the mixed layer and shoaling of the mixed layer will happen. The values of constants m 1, m 2 and m 3 (Kraus and Businger 1994) are as follows: 13

14 .55 < m 1 <.7 ; m 2 =.2 ; m 3 =.65 In (8.39), w * and u * are estimated as w 3 * = 1 2 hb ; u 3 * = hb (8.41) 2m 1 Another expression for calculating w e : The turbulent entrainment also velocity w e reads B C 1 u 3 * C 2 ρ h w e = h g(α T ΔT β ΔS) ; C = 2. and C (8.42) In (8.42), the entrainment velocity depends on surface buoyancy flux B, friction velocity u * and the jumps in temperature ΔT and salinity ΔS at the base of the mixed layer (i.e. at the depth h ). If mixed layer surface cools, then convective instability will provide mixing energy; on the other hand, heating would cause density to decrease. Hence, some of the mechanical energy (C 1 u 3 * ) will be used to overcome the stabilizing effect of surface heating. Therefore, h will increase with increasing surface windspeeds and decreasing surface buoyancy flux B. There are two regions in the World Ocean where thermodynamic extremes are present: (i) the ocean below the sea ice cover in the Arctic Ocean; and (ii) the Warm Pool in the Western Pacific and. The mixed layer characteristics in these regions are now described in what follows. (i) Mixed layer in the Arctic Sea: The presence of sea ice cover in the Arctic Sea totally prevents the transfer of momentum by winds; therefore, only buoyancy fluxes maintain the mixed layer in this basin. The buoyancy fluxes arise from the processes of ice growth and melting. The salinity of the mixed layer below the Arctic ice cover is between 3 31 psu due to large amount of river runoff in the Arctic basin. Maximum salinity reaches in early spring and a minimum in late summer. Thus the annual cycle of salinity in this region is dominated by the freeze-melt cycle of the Arctic sea ice. The mixed layer temperatures do not depart significantly from freezing temperature, but relative velocities of ice (zero velocity) and ocean may cause mechanical mixing, which together with buoyancy effects could modulate the mixed layer depth. During freezing, brine rejection is also significant, which increases salinity during autumn. Brine drainage continues with ice cover growth in the Arctic Sea. The rise in salinity will increase density; consequently, static instability shall be generated which will enhance mixing (turbulent) in the layer. This causes the mixed layer depth to grow (i.e. mixed layer deepens) to a depth of 6 m in mid-may. On the contrary, melting snow and ice during the melt season will add fresh water and density would reduce in the top layer. This will cause stabilization (i.e. reduces turbulent mixing). Therefore the mixed layer in the Arctic Sea will collapse continuously to reach a minimum depth of 15 meters due to combined effect of stabilization arising from increased freshening of saline waters and layer warming due to increased flux of solar radiation. The strong increase in salinity with depth in the Arctic Ocean produces a very stable density structure up to a depth of 2 meters. Consequently, heat and salt from layers below 2 meters do not upwell to 14

15 the surface. Because of this reason, the surface waters, augmented by river runoff in Arctic Sea, remain relatively fresh and cold. The development of a simplified, convective, oceanic mixed layer in winter in the Arctic could serve as an example of a simple model of mixed layer (Marshal and Plumb 26). The key paradigm of such a model is that the water column responds to heat loss at the sea surface by adjusting its heat content. That is, by matching the changing heat content of water column and heat lost from surface, deepening of the mixed layer can be calculated from prescribed surface temperature T s and lapse rate of temperature T. Both T s and the lapse rate change due to cooling caused by the loss of heat. Though ocean surface cooling (heating) happens due to (i) emission (absorption) of radiation and (ii) exchange (loss/gain) of sensible heat between ocean surface and atmosphere, yet both can be combined into a prescribed rate of cooling so that the mixed layer model could be be easily understood. Convection sets in due to sinking of colder water as the ocean surface cools mixes the waters that results in the deepening of the mixed layer. To develop a mathematical model of the mixed layer based on above paradigm, assume a temperature profile in the ocean at the start of winter as T z ( ) = T s + Λ z, (8.43) where z is depth (which is zero at the surface and increase upwards) and the gradient Λ >. Also, suppose that heat is lost from the surface at a prescribed rate Q W/m 2 during the winter. As surface cools, convection sets in and mixes the developing cold mixed layer of depth h m (t) to reach a uniform temperature T m (t). Further assume that temperature is continuous across the base of the developing mixed layer. Since the depth h m (t) is constantly increasing due to surface cooling, ocean waters arranged in the column according to (8.43) would mix up to a depth h m (t). Hence the mean temperature T m (t) of the layer is h m (t ) h m (t ) T m (t) = T (z)dz / dz = 1 h m (t ) T (z)dz. (8.44) h(t) The density of the well-mixed waters ignoring the salinity changes in the layer can be obtained from density-temperature relationship of the following form ρ m = [1 α T (T m T s )]. (8.45) Substituting T(z) from (8.43) into (8.44) and integrating the resulting expression, we get T m (t) = T m () + Λ 2 h m(t) = T s + Λ 2 h m(t) (8.46) The heat content of the layer of depth h m (t) at any time level t is H (t) = ρ m h m (t)c w T m (t). (8.47) The heat content of the layer at the next time level t + Δt will become H (t + Δt), hence by equating the change in heat content ΔH = H (t + Δt) H (t) to cooling of the sea surface, the requite equation can be formulated as ΔT ΔH = QΔt or ρ m h m (t)c m w = Q (8.48) Δt In the above equation, it is assumed that h m remains constant within the interval Δt. Taking the limit Δt, the discrete form (4.48) is written in the analytic form as 15

16 ρ m C w h m (t) dt m dt = Q (8.49) On integrating (8.49), on obtains Qt T m (t) = T s + ρ m C w h m (t) (8.5) However, the main aim of this mathematical model is to obtain the mixed layer depth, which be obtained by substituting the expression (8.46) of T m in (8.5). After some rearrangement, the expression the h m is obtained as h 2 = 2Qt m ρ m C wm Λ (assuming ρ m = = const) (8.51) If density of the mixed water is calculated from the T m, then combining (8.45) and (4.46), we get the expression of seawater density as ρ m = [1 α T Λ 2 h ] m (8.52) Substituting ρ m from (8.52) in (8.51), we obtain the following cubic equation in h m α T Λ 2 h3 h 2 + 2Qt C w Λ = (8.53) However, the magnitude of the first term is much less that the second one in (8.53), hence the computation of h m (t) is sufficiently accurate, and it is thus estimated from 2Qt h m = (8.54) ρ m C wm Λ Note that in the estimate of mixed layer depth, surface temperature does not appear in the above expression but the lapse rate of temperature Λ does. This means the growth rate of the mixed layer depends inversely upon the square root of Λ and directly to the square root of the cooling rate Q. We have already described a very elaborate model earlier, but a simple model like this helps comprehension of an evolving complex phenomenon. Though the model applies to regions like Artic, yet the formula (8.54) offers some deeper insights on the depth of mixed layer in global oceans. If Q and Λ are specified, then one can calculate the time that is required for the mixed layer to grow to a given depth. For example, with ρ m = 125kgm 3 and specific heat capacity C w = 398J kg 1 K 1, if Q = 25Wm 2 and Λ = 1 C km 1, then mixed layer would become 1m deep in 91 days. That is, the mixed layer develops 1 m deep everyday with a given rate of cooling 25 Wm -2 at the surface of the ocean. Further, this simple model explains that when lapse rate of temperature increases in the ocean, the mixed layer depth reduces. A consequence of this fact is that in regions of appreciable lapse rates (i.e. warm surface waters) mixed layers are shallow. (ii) Mixed layer in the Pacific warm pool: In the tropical warm pool, skin temperatures (i.e. temperatures at the interface between the atmosphere and ocean) could reach as high as 34 C, when light winds (5 ms 1 ) prevail. Due to deep convection, precipitation is also high in the warm pool region P E ~ 2 m yr 1 ( ). Due to net surface heating, the 16

17 mixed layer would be relatively shallow ( ~ 3 m deep). However, large amounts of precipitation could produce a steady, constant density mixed layer ~ 4 m deep ( ) and below the salinity profile shows sudden increase with depth (i.e. presence of barrier layer below 4 m ). However, the temperature profile remains almost uniform further down in the deeper layer. It is also observed that the nearly isothermal layer extends down in the warm pool up to a depth of 1 m around 155 E in the equatorial Western Pacific region. This discussion clarifies that the layer above the barrier layer (i.e. top layer) satisfies condition of well-mixedness. The presence of barrier layer below the well-mixed layer affects the heat flux and mixing of water further down. The warm pool mixed layer responds strongly to daily weather conditions. Under low surface wind conditions, both surface temperatures and heat content of the mixed layer rise significantly in response to surface heating because of the presence a relatively fresh isohaline layer ( ~ 4 m deep) produced by heavy precipitation. A diurnal cycle with amplitude of 1 2 C has been observed in the surface temperatures. During TOGA- COARE, it has also been observed that if windspeeds decrease (~ 4 ms 1 ), mixed layer becomes very shallow (~ 2 m deep); on the contrary, if the windspeeds increase, mixed layer depth also increases. This implies that the action of the winds at the interface is mainly responsible for the deepening and shoaling of mixed layer when an isohaline layer is present just below the surface. 17