ESCI 485 Air/sea Interaction Lesson 5 Oceanic Boundary Layer

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1 ESCI 485 Air/sea Interaction Lesson 5 Oceanic Boundar Laer References: Descriptive Phsical Oceanograph, Pickard and Emer Introductor Dnamical Oceanograph, Pond and Pickard Principles of Ocean Phsics, Apel DENSITY AND SALINITY The densit of the ocean is determined b two factors: temperature and salinit. Densit increases with decreasing temperature Densit increases with increasing salinit. Usuall the temperature is the dominant factor in controlling densit. Therefore, a warmer temperature above cooler temperatures usuall is a sign of stabilit. ο However, in areas with large salinit gradients it is possible to have colder fresh water over warmer, salt water and still be stable. MIXED LAYER The ocean can be divided into three laers ο Mied (or surface) laer ο Thermocline ο Deep laer The mied laer is akin to the atmospheres planetar boundar laer. The mied laer gets its name from the fact that it tends to be well mied, with the temperature being nearl isothermal with depth. The depth of the mied laer varies with location and season. Tpical ranges are from 5 to 5 meters. The depth is determined primaril b how rough the seas are. The rougher the seas, the deeper the miing. ο Since seas are generall rougher in winter, the mied laer depth is usuall deeper in winter than in summer.

2 THERMOCLINE At the bottom of the mied laer is the beginning of the thermocline. The thermocline is characterized b a decrease in temperature with depth. The thermocline is a ver stable laer. Because of this, vertical miing in the ocean at depths below the mied laer is ver slow. Because the ocean tpicall has a strong thermocline that inhibits miing between the mied laer and the deep laer, it is sometimes conceptuall and mathematicall convenient to model the ocean as a two-laer fluid. EQUATIONS GOVERNING OCEAN DYNAMICS The equations that govern the dnamics of the ocean are nearl identical to those for the atmosphere. The are the three momentum equations, the continuit equation, the thermodnamic energ equation, and an equation of state. There is no equation for the continuit of water vapor, but there is an equation for the continuit of salinit. Like the atmosphere, for large-scale flow the ocean can be considered to be in hdrostatic balance. The ocean can also usuall be considered as incompressible (even more so than the atmosphere). Ecept in a thin laer right near a boundar, viscous forces can be neglected. The three momentum equations and the continuit equation for the ocean are therefore Du Dt Dv Dt 1 p = + ρ 1 p = ρ p = ρ g u v w + + = 1 τ fv + ρ 1 τ fu + ρ where the stress terms are due to the Renolds, or vertical turbulent momentum flues. (1)

3 THE EKMAN SPIRAL An oceanographer named Nansen, around 1898, came up with a qualitative argument as to wh icebergs tend to blow at an angle to the right of the wind. His argument was based on a balance of the wind stress (the wind force on the iceberg), the Coriolis force, and friction. A few ears later, Nansen s assistant (Ekman) formulated a quantitative argument. Ekman began with the two momentum equations above. However, he assumed no pressure gradient force, and also assumed stead motion (so the time derivatives become zero). This leaves the following two equations 1 τ fv + ρ 1 τ fu ρ = () = The Renolds stresses were parameterized in terms of an edd viscosit, K, such that u τ = ρk (3) v τ = ρk so that the equations governing the flow are d u f v + = (4) K d v f u = (5) K (we ve changed the partial derivatives to regular derivatives since z is the onl independent variable). Equations (4) and (5) are a set of coupled, nd -order ODE s. We can resort to a little trick to solve them. We define a comple velocit so that w = u + iv. (6) If we multipl Eq. (5) b i and then add it to Eq. (4) we get 3

4 d ( u + iv) + Manipulating the comple number we can show so that our equation becomes f K ( v iu) =. (7) v iu = i( v i u) = i( iv u) = i( u + iv) d ( u + iv) i 4 f K or ( u + iv) = d w f i w =. (8) K ο Our trick turned the coupled set of ODE s into a single ODE of a comple variable. But, a single ODE is easier to solve than a sstem of ODE s. The general solution to Eq. (8) is ( z if K ) + B ( z if K ) w( z) = Aep ep. (9) B another identit of comple numbers we can show that can write the solution as where ( γ ) ( γ ) ( γ ) ( γ ) +1 i = i, so that we w( z) = Aep z ep i z + B ep z ep i z (1) γ f K (11) To find the constants A and B we have to appl the boundar conditions at the surface and at depth. ο We require the velocit to vanish as depth increases (z becomes more negative). This implies that B =. ο At the surface we require that w = W, the surface current. This implies that A = W. The solution to the equation is then ( γ z) ep( iγ z) Using Euler s formula we can write Eq. (1) as w( z) = W ep. (1) ( γ z) [ cos( γ z) isin( γ z) ] u + iv = ( U + iv )ep. (13) + ο Separating the real and imaginar parts, and writing each separatel, we get

5 u( z) = ep v( z) = ep ( γ z) [ U cos( γ z) V sin( γ z) ] ( γ z) [ V cos( γ z) + U sin( γ z) ]. The characteristics of this current profile are more easil seen if we assume the surface current is strictl zonal (V = ). Then Eqs. (14) become u( z) = U v( z) = U ep ep ( γ z) cos( γ z) ( γ z) sin( γ z). If the current is plotted on a hodograph it traces a decaing clockwise spiral with depth. This is known as the Ekman spiral. The depth of the Ekman laer is taken to be that point at which the current has decaed b a fraction of Ekman laer is 1 e (the e-folding scale). Therefore, the depth of the (14) (15) 1 K D E = =. (16) γ f THE SURFACE CURRENT We still haven t eplained the fact that the surface current is to the right of the wind speed. To do this we have to look at another boundar condition at the surface. At the surface the stress must be continuous (i.e., the stress in the air must equal that in the water). The stress in the air at the surface is just the wind stress, and so is in the direction of the surface wind. The stress in the water at the surface is that due to the Renolds stresses, and is given b u τ = ρk v τ = ρk z= z= (17). Taking the case of our purel zonal current, and appling the boundar conditions above we get 5

6 τ = ρk τ = ρk d d ( U ( U ep ep ( γ z) cos( γ z) z= ( γ z) sin( γ z) = ρkγu. z= This gives us two important pieces of information. = ρkγu ο It tells us that for our purel zonal current that there had to be a non-zonal component of wind stress. So, the current isn t flowing in the direction of the wind, but at an angle to the wind. ο Since the magnitudes of the and -components of the wind stress are equal, this means that the surface current should be flowing at eactl 45 to the right of the surface wind. This is a quantitative description of the effect that Nansen observed! ο It tells us the speed of the surface current in terms of the magnitude of the wind stress, since (18) τ = ρkγu = ρu Kf or τ U =. (19) ρ Kf EKMAN TRANSPORT AND EKMAN PUMPING The net transport of water in the Ekman laer can be found b integrating the Ekman equations through the depth of the laer, ( ) ( ) M = ρ u( z) = ρu ep γ z cos γ z = ρu γ ( ) ( ) M = ρv( z) = ρu ep γ z sin γ z = ρu γ. This result shows that the net transport is directed at 45 to the right of the surface current. Since the surface current itself is directed at 45 to the right of the surface wind we have shown that the net transport in the oceanic Ekman laer is directed at 9 to the right of the surface wind. The total Ekman transport is 6 ()

7 M = M + M = ρ U γ = τ f. (1) ο Since the Ekman transport is directed at 9 to the right of the wind stress, we can write the following vector equation 1 M = kˆ τ. () f EKMAN TRANSPORT AND OCEAN CIRCULATION Ekman transport has important implications for the ocean circulation. ο The anticlonicall rotating wind-driven gres in the ocean basins will have a net Ekman transport toward the center of the gre. ο The surface convergence in the center of the gre results in elevated sea level heights in the gre s center. ο The surface convergence also pushes the colder, deeper water to even greater depths. ο The surface convergence must be compensated b downward vertical motion (called downwelling). ο This entire process is sometimes referred to as Ekman pumping. ο Ekman pumping results in a secondar circulation superimposed on the gre, with convergence and downward motion in the middle of the gre, and divergence deeper in the gre. The vertical velocities associated with this downwelling can be found b integrating the continuit equation through the depth of the Ekman laer From Eq. () we have w = D e D e 1 V = M. ρ D e w 1 = ρ The integral on the left-hand-side is 1 kˆ 1 τ = f ρ f τ. De w = w() w E = w E (the vertical velocit at the surface is zero). Therefore, we get the result 7

8 1 w E = ρ f τ. (3) ο This result sas that the vertical velocit at the bottom of the Ekman laer is proportional to the curl of the wind stress. ο A cclonic wind stress will give upwelling, and an anticclonic wind stress will give downwelling. Ekman transport ma be important in the response of the ocean to individual storms. The wind stress from a cclonicall rotating storm will induce Ekman transport awa from the center of the storm, and can result in local upwelling near the center of the circulation. EKMAN TRANSPORT AND THE SEASONAL CLIMATE OF THE WEST COAST Ekman transport also has dramatic and important effects on the climate of the West Coasts of North and South America. ο Along the west coast of North America in the spring and summer the Pacific High moves further offshore. The prevailing winds during this period are from the northwest to north-north west. ο Ekman transport is therefore directed off-shore, and pulls surface water awa from the coast. ο This horizontal divergence at the surface is compensated b upwelling, which brings colder water from the deep ocean up to the surface. ο This upwelling causes the waters off the west coast of North America to be colder in the spring and summer, than at other times of the ear, and eplains wh the coastal climate of Central and Northern California, Oregon, and Washington is so cool and often fogg in the summer. ο In the fall and winter the Pacific high is farther to the north and west, and the upwelling is not present. ο It is the upwelling associated with the Ekman transport that is the reason the Pacific Coast often has ver nice weather in the fall, but is usuall cold and fogg in spring and summer. 8

9 ο This same phenomena is observed off of the West Coast of South America also. EQUATORIAL CURRENTS The surface currents in the Pacific and Atlantic have a similar structure, and can be eplained at least in part b convergence and divergence associated with Ekman transport. The diagram above shows the air flow (open arrows) and the resultant Ekman transport (dark arrows). The DIV and CONV denote regions of divergence and convergence in the Ekman transport. Regions of divergence will result in a lowering of the sea-surface, while regions of convergence will raise the sea surface. 9

10 Other than within a degree or so of the Equator, the ocean flow will be parallel to the sea-surface contours with low heights to the left in the Northern Hemisphere, and to the right in the Southern Hemisphere. The ocean currents and their directions are indicated b the W and E annotations, with W indicating a westward current, while E indicates an eastward current. The resultant surface currents are the North Equatorial Current Westward flowing Equatorial Counter Current Eastward flowing, more-or-less aligned with the ITCZ. South Equatorial Current Westward flowing, and in both hemispheres. There is also an Eastward flowing Equatorial Undercurrent that more-or-less flows along the Equator at depth. The equatorial currents in the tropical Indian Ocean differ in that since the atmospheric flow switches directions seasonall due to the monsoon, so do the currents. 1