Ekman laers, friction & eostrohic flow The homework roblem (ucks_on_ice) illustrates several imortant oints: 1. Particles move erendicular to the alied force, to the riht in the northern hemishere, to the left in the southern hemishere 2. Particles do not move steadil but oscillate with an inertial frequenc, equal to 2Ωsin(θ), where θ is the latitude 3. After the eternal force is removed, article motion decas ecet when close to the bum where the continue to rotate around the bum with hih elevation on the riht (left) in the northern (southern) hemishere 4. Due to friction, articles near the bum slowl slide down the sloe Point 1 is an illustration of Ekman laers in the ocean. A force (wind stress) ushes them at riht anles in the rotatin sstem. One miht wonder how it is ossible in such a world to drive from California to Seattle: one would have to head westward in our car (towards the Pacific Ocean) to et there! But in fact, friction for automobiles is rather lare: the tires constantl imeded motion ecet in the direction the car is ointed. So don t tr this eeriment!
The variation of the article motion with ever-increasin friction is shown in the fiure above. As friction is increased (relative to f ) the article moves more in the direction of the alied force (blue arrow). The oscillation amlitude also decreases. In the limit in which friction is much larer than f, the motion is in the direction of the alied force. The dnamics behind the Pucks on Ice roblem is all contained in the equation for the chane in momentum of a article of mass m movin in a rotatin sstem about a frozen, bum surface h(,) under the action of a bod force, F and weak friction r. As we have seen, when friction is weak enouh (in one limit discussed above), the ucks can do some thins that are initiall surrisin when the bod force is switched on and then off after a short time (3 das in the roblem). The dnamical balance is iven as dv m[ + 2 Ω v] = mrv m h+ F dt 1 2 3 4 5 ( ) ( ) ( ) ( ) ( ) Each of the terms has been numbered to facilitate some discussion. One feature of the solution is that shortl after the bod force is either switched on or off, articles undero oscillations in a circle. This oscillation is an inertial oscillation and is characterized as a balance between terms (1) & (2). Inertial oscillations: terms (1) & (2) For those articles near the bum toorah, their motion will be such as to move with velocities that have the hih toorah on their riht (in the Northern Hemishere, left in the S. Hemishere). This balance is called the eostrohic balance. Geostrohic flow: terms (2) & (4) Under the action of the bod force, articles will move at riht anles to the force, to the riht in the N. hemishere (left in S. Hemishere). This is reresents a balance between terms (2) and (5) and is the fundamental balance in the wind-driven, Ekman transort (after V. Walfrid Ekman). Ekman flow: terms (2) & (5) [&(3)] Weak friction will modif all of the above, and in the roblem, the above balances are never eactl followed, onl aroimatel, in different reions
and at different times. Yet these balances lie at the core of wind-driven motion in the ocean. In all of these cases, the Coriolis force (2) is central. Ekman flow in the ocean Now consider a non-frozen ocean forced b an alied wind stress in the direction of the blue vector. For the surface laer, turbulence is lare and the effective friction is lare: so a article will move downwind but slihtl to the riht (in the N. hemishere, which we will assume in what follows). The laer below, which is causin the dra on the surface laer then wants to move too. It will move to the riht of the overlin surface laer for the same reasons. This continues in the vertical until the turbulence, which coules the laers in the water column, eventuall dies awa and there is no subsequent motion. The resultin velocit vector looks like a siral: oin from slihtl downwind but to the riht of the wind, rotatin to the clockwise and decain with increasin deth. This was first ointed out b Ekman with a ver simle model of the frictional coulin in 1905, and convincinl demonstrated in the ocean b Price, Weller and Schudlich (Science, vol. 238, 1534-1538) in 1987. The net transort in this frictional laer is almost eactl to the riht of the stress as redicted! We call this surface laer of wind stress influence the Ekman laer. In the fiure below, we can see this almost eact balance of net transort interated over the surface Ekman laer, which is about 25m deth, and the wind stress, such that the net transort is the riht of the wind stress (in the N. Hemishere).
Imae removed due to coriht concerns. Frictional or Ekman laers can eist near the ocean bottom as well. This was also illustrated in the homework roblem as the ucks slowl slid down the sloin toorah because of friction. The main motion of the ucks was clockwise around the toorahic bum. This reflects the eostrohic balance between motion and the sloe in surface elevation. Since frictional coulin is aain stron when there is flow in near contact with the ocean bottom, this aeostrohic (non-eostrohic) motion, aain deflects the trajectories of articles near to the bottom to the left of those further awa from the bottom. This creates a siral in the oosite direction with net frictional flow to the left of the overlin fluid (in the N. hemishere). A similar result is obtained for eostrohic flow around a dimle. Near the bottom, friction usets this balance and there is aain a flow down the ressure radient and a turnin of the flow to the left of the interior motion.
Notes on the Geostrohic Balance Consider the diaram at the riht. The ocean consists of two densit laers of uniform densit ρ 1 & ρ 2. The heiht of the free surface is iven b h(), that of the interface b H() and of the flat lower surface H 0 : it needn t be the ρ 2 ocean bottom, onl a level surface. We will use the hdrostatic relation to determine the ressure on the lower surface. It is iven b the followin: P( z = H ) = ρ 1( h + H ) + ρ 2 ( H 0 0 H ) ρ 1 Z=h Z=0 Z= -H Z= -H 0 If at this surface there is no horizontal ressure radient, P = 0 and we et the followin (after cancelin out, which aears in both terms): P h = 0 0 = ρ ( h ρ = H ρ 1, 1 + H ) ρ H where ρ ρ ρ 2 2 1 Since ρ/ρ 1 << 1, the interface thickens and deeens much more than the free surface rises. For the Gulf Stream, the above icture is what one would see standin near Cae Hatteras and lookin to the east. The surface rises about 1 meter across the Gulf Stream due to the stron flow at the surface, but this flow decas raidl with deth. The densit contrast across the cnocline makes ρ/ρ 1 ~ 210-3 so the cnocline sloes down to the south (riht) about 500m across the Gulf Stream. Of course, the above situation is onl an idealization of the actual situation (the class will have a homework roblem usin real data), but this illustrates the deree to which the densit field can comensate for a non-level free surface eression., or
We will now look at the equations eressin the eostrohic balance for baroclinic motion. In our rand equations, we can write the balance for eostrohic motion [ (u,v)=(u,v )] as ρ fv ρ fu = = where we have noted that the ressure radients in (,) are calculated on a constant eootential surface for clarit. Now recall from the definition of eootential surfaces that there is no chane in otential ener as a article is moved in a surface of constant, and that ((,),,) elicitl deends on ressure as well as horizontal osition. We can use this to make a chane of variables in the above. For eamle, if =( (, ),, ), then δ= δ + δ+. δ For δ, δ = 0,,,, δ = /. δ,,, So we can write / ρ = =, / ρ. = =, And our eostrohic balance becomes
(0) D( ) fv ( ) = = + fu (0) D( ) ( ) = = + where we have used the definition of eootential resented earlier in terms of dnamic heiht and also made use of the fact that there is no horizontal variation of one of the terms in the definition of eootential which is roortional to the reference secific volume anomal α 0. Τhe last air of equations can be further simlified reconizin that for = 0, D = 0, thus f [ v f [ u (0) v (0) u D( ) ( )] = D( ) ( )] = This is the form of the eostrohic equations most commonl used since the radient of eootential at the free surface is not easil measured. It eresses how velocit will chane with ressure based on horizontal radients of dnamic heiht. Without an other information, on cannot determine the eostrohic velocit at a oint, onl its variation with ressure (or deth): there is an unknown constant, which can be determined b fiat: [sain that the velocit at some reference ressure must be zero] or b other means [such as equatin the velocit at some deth with direct measurements of velocit and then requirin that measured velocit to be eostrohic ]. [reference for some of this is Fofonoff in The Sea, Vol. 1]. In a later homework roblem, we will calculate eostrohic currents usin actual oceanorahic data. Use of the above air of equations will be essential as well as understandin how to use the concet of a dee reference ressure as a level of no motion. On an f-lane, contours of constant dnamic heiht are equivalent to streamlines of the eostrohic flow relative to some assumed level of no motion. While this is a convenient fact and of much use in eaminin satial mas of dnamic heiht from hdrorahic data, one must alwas be aware of the limitations of this aroimation. Because f can var with latitude, this aroimation breaks down for whole ocean basins.