Wind-driven Circulation

Our discussion has been limited to models based on Sverdrup dynamic. The western boundary currents were just matched to the interior solution. Within this theoretical framework, the maximal streamfunction is totally determined by a zonal integration of the Ekman pumping velocity, started from the eastern boundary of the basin. 

Observations, however, indicate that the maximal volume flux in the Gulf Stream is about 150Sv, which is several times larger than the value calculated from the Sverdrup relation. 

The discrepancy between the linear theory and observations is due to two factors:

1. Inertial terms in the horizontal momentum equations have been neglected for analyitical simplicity in the commonly used linear theories. 
2. There is a strong interaction with stratified flow over topography, which is called bottom pressure torque or JEBAR (joint effect of baroclinicity and bottom relief)

When the ocean bottom is not flat, the JEBAR term arises when the curl is taken of the vertically-averaged horizontal momentum equations. 

In fact, JEBAR comes directly from taking the curl of the veritically-averaged horizontal pressure gradient term. 

\(J(\Psi,\frac{f}{H})=J(\Phi,\frac{1}{H})+\frac{1}{\rho_0}[\frac{\partial }{\partial x}(\frac{\tau_s^y}{H})-\frac{\partial }{\partial y}(\frac{\tau_s^x}{H})]\)

The first term on the right-hand side is called the JEBAR term, and the second term on the right-hand side is the contribution due to wind-strss curl.

The equation states that changes of the barotropic streamfunction along the \(f/H\) contours are due to contributions from these two terms. 

The JEBAR term is associated with bottom pressure torque, and is the result of interaction between the stratification and bottom  topography. It can be readily seen that, for the case of a flat bottom, this terms vanishes. 

Greatbatch et al. (1991) analyzed the circulation on the North Atlantic Ocean. The most crucial contributor to the volume transport in the recirculation regime is the JEBAR term. 

While geostrophic motion refers to the wind that would result from an exact balance between the Coriolis force and horizontal pressure-gradient forces,[1] quasi-geostrophic (QG) motion refers to flows where the Coriolis force and pressure gradient forces are almost in balance, but with inertia also having an effect.

This effect of inertia is reffered to the ageostrophic part of the velocity, which is under the following assumptions much smaller than the geostrophic part:

1. The magnitude of accelartion is small compared to the magnitude of the Coriolis force:
   1. Then: \(|v_g|>>|v_a|\)
   2. This is the case if the Rossby number is small:
      1. \(R_0=\frac{U^2/L}{f_0U}=\frac{U}{f_0L}\)
      2. \(\frac{Du}{Dt}=fv-\frac{1}{\rho}\frac{\partial p}{\partial x}=f(v-v_g)\)
      3. if \(R_0\)is small -> \(\frac{Du}{Dt}\) is small -> \(v-v_g\)is small -> \(|v_g|>>|v_a|\)
2. The \(\beta\)-plane approximation is valid. 
   1.  justifies letting the Coriolis parameter have a constant value  in the geostrophic approximation and approximating its variation in the Coriolis force term by .^[4] However, because the acceleration following the motion, which is given in (1) as the difference between the Coriolis force and the pressure gradient force, depends on the departure of the actual wind from the geostrophic wind, it is not permissible to simply replace the velocity by its geostrophic velocity in the Coriolis term.

The most important implification is that vertical displacements are small compared to the background stratification! 

At highest order of simplification the flow is purely geostrophic:This means also that on a f-plane the flow is non-divergent. 

To have vertical motion we need to consider the ageostrophic flow. 

The ageostrophic flow has 4 parts:

1. \(\beta\)-part: \(u_a\)and \(v_a\) are parallel to \(u_g\)and \(v_g\), reinforcing (\(u_g,v_g\)) for y<0, weakening \((u_g,v_g)\) for y>0.
2. Isallobaric part: is normal to the isallobars and flows towards the isallobaric low (where pressure is falling most rapidly) -> Convergence in regions where the pressure is falling most rapidly. 
3. Non-linear part: 
   1. flow around the through is "subgeostrophic" (i.e. weaker than geostrophic), so that the pressure gradient force towards the centre of curvature is slightly stronger than the opposing Coriolis force, enabling the flow to take a curved path. 
   2. Likewise, around the ridge, the flow is slightly "supergeostrophic". 
   3. This leads to divergence ahead of the trough and convergene behind the trough (i.e. ahead of the ridge). Regions ahead of an upper level trough (i.e. ahead of the ridge).
   4. Regions ahead of an upper level trough are favourable for low level storm development. 
4. Ekman part: ageostrophic flow to the right of the windstress. 

\(\psi \) is the quasi-geostrophic stream function.

\(\frac{D_g}{Dt}(\nabla^2\psi +\beta y-\frac{f_0^2}{c^2}\psi )=\frac{1}{\rho_0H_1}(\frac{\partial }{\partial x}\tau_s^y-\frac{\partial}{\partial y} \tau_s^x)\)

\(\nabla ^2 \psi \)... is the relative vorticity 

\(f_0+\beta y\)... is the planetary vorticity 

\(\frac{f_0^2}{c^2 }\psi\)... is the stretching term

\(\frac{\partial }{\partial x}\tau_s^y-\frac{\partial}{\partial y}\tau_s^x\)... is the wind stress curl

If the wind stress curl is 0 then: 

\(\frac{D_q}{Dt}=0\;\;where\;\;q=\nabla^2\psi+\beta y-\frac{f_0^2}{c^2}\psi \)

q is the quasi-geostrophic potential vorticity;

\(\frac{Dq}{Dt}=0\) states the Conservation of Quasi-Geostrophic Potential Vorticity 

If q is conserved, then changes of the planetary vorticity are balanced bx changes of the relative vorticity and vortex stretching. 

It is assumed, that vertical displacements are not so large as to disturb the background stratification. In the \(1\frac{1}{2}\) layer model, this corresponds to the assumption that the interface displacements are small compared to the undisturbed depth \(H_1\).

Under the assumption of conservation of the quasi-geostrophic potential vorticity \(Dq/Dt=0\) and a solution of the form:

\(\psi=\psi_0e^{i(kx+ly-\omega t)}\)

We find the dispersion relation: 

\(\omega=-\frac{\beta k }{(k^2+l^2+\frac{f_0^2}{c^2})}\)

For fixed l the solution has the following form as shown in the graphic below. 

long waves are: \(\frac{\partial \omega}{\partial k}<0 \) -> westward group velocity 

short waves are: \(\frac{\partial \omega}{\partial k }>0\) -> eastward group velocity

Western intensification: short waves trapped near western boundary; long waves escape westwards. 

The limit for the long waves is:

\(k^2+l^2<<\frac{f_0^2}{c^2}\)

... horizontal length scales are much larger than the radius of deformation 

• typical for baroclinic modes in the ocean. For example, for the first baroclinic mode (\(c=1-3ms^{-1}\))

for long waves the dispersion relation reduces to: 

\(\omega =-\beta\frac{c^2}{f_0^2}k\)

-> These waves are non-dispersive with westward phase and group velocity given by \(\beta\frac{c^2}{f_0^2}\).

thinking of the shallow water equations:

\(u_t-fv=-g'h_x\\ v_t+fu=-g'h_y\\ h_t+H_1(u_x+v_y)=0\)

The long wave limit is equivalent to neglecting \(u_t,v_t\) and so:

\(h_t-\frac{\beta g'H_1}{f^2}h_x=0\;\;with\;\;\beta=\frac{df}{dy}\)

assuming wave solution \(h=h_0e^{i(\omega t-kx-ly)}\) we get to the dispersion relation (see above). 

Divergence of the ageostrophic flow due to \(\beta\) is balanced by "vortex stretching", i.e. pumps the thermocline up and down via the \(h_t \) term. 

To cross the North Atlantic Ocean, these waves need roughly 25 years! Although the propagation speed increases toward equator, where it takes only 1 year. 

The limit of short waves is:

\(k^2+l^2>>\frac{f_0^2}{c^2}\)

-> length scales are much smaller than the Radius of Deformation!

Usually appropriate for the barotropic mode in the ocean: \(c=200m/s\)

The dispersion relation reduces to: 

\(\omega=-\frac{\beta k}{(k^2+l^2)}\)

These waves are highly dispersive (the phase and group velocity are far from being equal).

The time to cross the North Atlantic for a barotropic wave is typically about one week. So, much fast than a baroclinic wave. 

The shallow water equations are: 

\(u_t-fv=-g\eta_x\\ v_t+fu=-g\eta_y\\ \eta_t+H(u_x+v_y)=0\)

The short wave limit is equivalent to negelcting the \(\eta_t\) term in the continuity equation. 

This enables us to work with a streamfunction for which: \(u=-\psi_y,\;v=\psi_x\):

\(\frac{\partial }{\partial t}(\nabla ^2\psi )+\beta\psi_x=0\)

The horizontal divergence is zero. This means that the dvergence flow caused by \(\beta\) must be absorbed by the convergence of the isallobaric flow arising from the \(u_t,\;v_t\)terms. 

Note that the isallobaric part of the ageostrophic flow flows towards the isallobaric low, where pressure drops most rapidly. The only way the resulting convergence can be absorbed by the \(\beta\) part is when the waves propagate westward! (is unclear for me) 

The gorverning equations and boundary conditions from a Sturm-Liouville problem so that there is a complete set of (orthogonal) vertical structure functions \(\widehat{p}_n(z),\;\widehat{w}_n(z)\) called vertical normal modes, enabling us to write: 

\(u(x,y,z,t)=\sum^\infty_{n=0}\tilde{u_n}(x,y,t)\widehat{p}_n(z)\)

the horizontal structure associated with each mode satisfies the shallow water equations!

\(c_n\)ist the wave speed associated with the n'th mode and \(H_n\) is the equivalent depth. 

\(c_0>c_1>c_2>...\)

n=0 is called the barotropic mode. For this mode, \(c=\sqrt{gH}\) (H is the total depth), \(\widehat{p}_0(z)=1\) and \(\widehat{w}_0(z)=\frac{z+H}{H}\) are a good approximation. for the barotropic mode, the vertical velocity varies essentially linearly with depth from zero at the bottom to a maximum at the top and, to good appr., the barotropic velocities \(\tilde{u_0}\) and \(\tilde{v_0}\) are the vertically averaged horizontal velocities. 

For \(n\geq1\)we have the baroclinic modes; typically \(c_1=1-3ms^{-1}\)(the first baroclinic mode). For these modes the rigid-lid approximation can be applied, i.e. we can put \(\widehat{w}_n(0)=0\) (i.e. w=0 at Z=0) to a good approximation and all displacements are internal. For the baroclinic modes \(\int^0_{-H}udz=0\) (to a good approximation) and all displacements are internal. For the first mode, \(\widehat{p}_1(z)\) has one zero crossing, for the n'th mode \(\widehat{p}_n(z)\) has n zero crossings. 

Note that all modes are "orthogonal", with the implication that each mode varies independently of the others. 

the shallow water equations can be obtained from the 3-D governing equations by projecting the 3-D governing equations on to each vertical mode.

• Equastion linearised about a state of rest for a stratified ocean. 
• Seperation into vertical modes, since the bottom is flat. 

\(u_t-fv=-g\eta_x+\frac{\tau_s^x}{\rho_0H}\\ v_t+fu=-g\eta_y+\frac{\tau_s^y}{\rho_0H}\\ \eta_t+H(u_x+v_y)=0\)

For a quasi-geostrophic flow these equations reduce to:

\(\frac{\partial }{\partial t}(\nabla^2\psi-\frac{f_0^2}{c^2}\psi)+\beta\psi_x=\frac{1}{\rho_0H}(\frac{\partial }{\partial x}\tau^y_s-\frac{\partial }{\partial y}\tau_s^x)\;where\;\psi=\frac{g\eta}{f_0 }\)

take \(\tau_s^y=0\;and\;\frac{\tau_s^x}{\rho_0H}=X_0sin(ly)\)

look for solutions of the form: \(\psi=\psi'cos(ly)\)

we become:

\(\frac{\partial }{\partial t}(\frac{\partial ^2\psi'}{\partial x^2}-(l^2+\frac{f_0^2}{c^2})\psi')+\beta\psi'_x=-lX_0\)

first we discuss the situation for no boundaries. Since the wind stress has no dependence on the zonal coordinate, x, neither will the solution. 

\(\frac{\partial }{\partial t}(-(l^2+\frac{f_0^2}{c^2})\psi')=-lX_0\)

for baroclinic modes: \(l^2<< f_0^2/c^2\) it follows:

\(\frac{\partial }{\partial t}(-\frac{f_0^2}{c^2}\psi')=-lX_0\) or 

\(\frac{\partial \eta}{\partial t}=\frac{1}{f_0\rho_0}(-\frac{\partial \tau_s^x}{\partial y})\)

-> Vertical displacement of the thermocline by Ekman pumping. 

Ekman transport is part of the ageostrophic flow. The dominant part is the geostrophic flow. When there are no boundaries there is no x-dependency and \(v_g=0\). The geostrophic flow is zonal and in the same direction as the wind. Because the thermocline is being continually pumped, the zonal geostrophic flow increases with time. 

Rossby waves:

1. long waves -> westward group velocity 
2. short waves -> eastward group velocity 

-> need only to consider long Rossby Waves - the waves with westward group velocity. 

-> Work with the baroclinic mode for which: \(k^2+l^2 << \frac{f_0^2}{c^2}\) (horizontal length scales are long compared to the radius of deformation.

Returning to the quasi-geostrophic equation, using the long wave limit and putting \(\psi=\psi'cos(ly)\) we come to the same solution as for no boundaries:

\(-\frac{f_0^2}{c^2}\frac{\partial \psi'}{\partial t}=-lX_0\)

This means that the solution is as for the case with no boundaries, until the (long Rossby) wave arrives from the eastern boundary and introduces x-dependences to the solution. 

1. In the interior of the ocean, first there is the Ekman pumping of the thermocline by the wind stress curl. This is the same as the solution with no boundaries. -> the thermocline moves uniformly downward through Ekman pumping. 
   1. the balance is: \(-\frac{f_0^2}{c^2}\frac{\partial \psi'}{\partial t}=-lX_0\)
2. When the waves from the eastern boundary arrive, the solution switches to the Sverdrup balance:
   1. \(\beta\psi'_x=-lX_0\)

For geostrophic flow on an f-plane we have: 

\(-fv=-\frac{1}{\rho_0 }p_x;\;\; fu=-\frac{1}{\rho_0}p_y\)

since f is per definition uniform on a f-plane, it follows that: \(u_x+v_y=0\)

for continuity we have: \(u_x+v_y+w_z=0\)

It follows that vertical motion is severely restricted, since for example \(w=0\) at the bottom of the ocean. 

A flow that feels the effect of Earth's rotation (e.g. flows in the atmosphere and the ocean that evolve on time scales longer than a day) tend to be very horizontal. Rotation acts to inhibit vertical motion! 

On a rotating planet, flow tends to be along lines of latitude (cf. Sverdrup balance).

The equation for the vertically integrated transport stream function \(\psi \):

\(J(\Psi,f)=\frac{1}{\rho_0}J(p_b,H)+\frac{1}{\rho_0}[\frac{\partial \tau_s^y}{\partial x}-\frac{\partial \tau_s^x}{\partial y}]\) 

or

\(\beta\Psi_x=\frac{1}{\rho_0}J(p_b,H)+\frac{1}{\rho_0}[\frac{\partial \tau_s^y}{\partial x}-\frac{\partial \tau_s^x}{\partial y}]\)

If the total torque on the right hand side is zero, then: 

\(\beta \Psi_x=0\)

from which it follows that \(\Psi \) depends only on latitude and the flow is entirely zonal, i.e. along lines of latitude. Hence the result!

Note that when the ocean has variable bottom topography, then if the flow extends to the bottom of the ocean, and feels the effect of the variable bottom topography, then, in general, the bottom pressure torque isn't zero. Indeed, in an unstratified ocean, the flow tends to follow \(f/H\) contour lines.

Beginning with the linearized density equation:

\(\frac{\partial \rho'}{\partial t}+w\overline{\rho}=0\)

In steady state, this reduces to: 

\(w\overline{\rho}_z=0\)

It follows that in a stratified ocean we must have \(w=0\). This constraint strongly encourages motion in the horizontal. 

Furthermore, the linear vorticity balance is: 

\(\beta v=fw_z+\frac{G(z)}{\rho_0H_E}[\frac{\partial \tau_s^y}{\partial x}-\frac{\partial \tau_s^x}{\partial y}]\)

Since it  is a prediction of linear dynamics that in steady state \(w=0\), it follows that a further prediction of linear dynmics is that:

\(\beta v=\frac{G(z)}{\rho_0H_E}[\frac{\partial \tau_s^y}{\partial x}-\frac{\partial \tau_s^x}{\partial y}]\)

in which case the horizontal flow field has the same vertical structure as \(G(z)\)and hence is confined entirely to the surface mixed layer. The flow is then insulated from the effect of the variable bottom topography and the bottom pressure torque term is zero. 

• A natural way to simulate the ocean circulation is to assume a two-layer fluid. The main thermocline is used as the interface. The lower layer is very thick and the water in the lower layer moves much slower than in the upper. 
  • reduced gravity model: the lower layer is in stagnation-> One active layer only! (also called the 1 1/2- layer model)
• If the model has two levels and both are in motion than its the 2 layer model... 

S.262

The upper boundary condition of the ocean model is moved from the free surface \(z=\zeta\) to a flat surface \(z=0\).

If we assume the lowest layer to be stagnant then this leads to a drop out of the atmospheric pressure in the horizontal pressure gradient terms. 

\(p_a=p_{a,0}+\rho g \zeta\)

This leads for the 1 1/2-layer model again to the shallow water equations: 

\(u_t-fv= -g'h_x \\ v_t+fu=-g'h_y \\ h_t+H_1(u_x+v_y)=0\)

g' is the reduced gravity:

\(g'=g(\rho_1-\rho_2)/\rho_0\)

1. We find prevailing westerlies at mid-latitudes and easterlies at low and high latitudes. 
2. This wind stress pattern drives poleward Ekman flows at low and high latitudes but an equatorward Ekman flow at mid-latitudes. 
3. The meridional convergence of the Ekman flux in the upper ocean gives rise to the Ekman pumping (compression of the water column) and upwelling (expansion) below the base of the Ekman layer.
4. In the basin interior, relative vorticity is negligible, so the potential vorticity can be written as  \(f/h\) .

• A compressed water column needs to move equatorward and a lengthened water column needs to move polewards to maintain potential vorticity.

=> Ekman pumping in the subtropical basin drives an equatorward flow in the ocean interior.

=> Ekman upwelling in the subpolar basin drives a poleward flow in the ocean interior. 

figure in Huang et al. 2010 S.286

\(u_t-fv=-\frac{p'_x}{\rho_0}+\frac{\partial X}{\partial z} \\ v_t+fu=-\frac{p'_y}{\rho_0}+\frac{\partial Y}{\partial z}\)

with

\(X=\nu u_z,\;\;Y=\nu v_z\)

where \(\nu \) is a vertical eddy viscosity (but could also be something else)

The connection between vertical eddy viscosity and the surface wind stress is:

\(\nu u_z=\tau^x_s/\rho_0\;\;\;\nu v_z=\tau^y_s/\rho_0\)

This means: \(\rho_0(X,Y)\) is the horizontally acting stress at each depth 

In this course we often set: 

\(\nu=\left\{ \begin{array}{c} \nu_0\;\;\;for\;-H_E\leq z\leq0\\0\;\;\; otherwise \end{array} \right.\)

The balance 

\(-fv=(\nu u_z)_z\\fu=(\nu v_z)_z\)

implies, for uniform \(\nu\) the scaling

\(fU=\frac{\nu U}{h^2}=>h=\sqrt{\frac{\nu}{f}}\)  (h is the vertical scale)

i.e. for uniform \(\nu\) variations in the vertical take place in a scale \(\sqrt{\frac{\nu}{f}}\) . If this scale is much larger than the depth over which \(\nu\) is non-zero, then the horizontal velocity will be essentially independent of depth over the region where \(\nu\) is non-zero, in this case the depth  \(H_E\) .

For this case we can treat wind stress as a body force over the depth \(H_E\) . Which means the wind stress forcing is constant for each layer whithin the Ekman Layer.

\(\left(\frac{\partial X}{\partial z},\frac{\partial Y}{\partial z}\right)=\frac{G(z)}{\rho_0H_E}(\tau_s^x,\tau^y_s)\) 

with

\(G(z)=\left\{ \begin{array}{c} 1\;\;-H_E\leq z\leq0\\0\;\;otherwise \end{array} \right.\)

for the 1 1/2 layer model follows with wind forcing: 

\(u_t-fv=-g'h_x+\frac{\tau^x_s}{\rho_1H_1}\\v_t+fu=-g'h_y+\frac{\tau_s^y}{\rho_1H_1}\\h_t+H_1(u_x+v_y)=0\)

Consider the Ekman part of the flow: 

\(u_t-fv=-\frac{p'_x}{\rho_0}+\frac{\partial X}{\partial z}\\ v_t+fu=-\frac{p'_y}{\rho_0}+\frac{\partial Y}{\partial z}\)

is going to:

\(u_t-fv=+\frac{\partial X}{\partial z}\\ v_t+fu=-\frac{\partial Y}{\partial z}\)

1. Assuming that the flow is al confined above a depth \(H_E\) , the EKMAN TRANSPORT, (\((U_E,V_E)\), is defined as:

\((U_E,V_E)=\int\limits_{-H_E}^0(u_E,v_E)dz\)

\((u_E,v_E) \)i is the Ekman Velocity.

2. Assuming \((X,Y)=0\)below the Ekman depth, it follows that 

\(U_{Et}-fV_E=\frac{\tau^x_s}{\rho_0H_E}\\V_{Et}+fU_E=\frac{\tau^y_s}{\rho_0H_E}\)

-> In steady state, the Ekman transport is at right angles to the surface wind stress, i.e. 

\(-fV_E=\frac{\tau^x_s}{\rho_0H_E}\\fU_E=\frac{\tau^x_s}{\rho_0H_E}\)

The geostrophic balance is expressed by:

\(-fv_g=-\frac{1}{\rho_0}p_x\\ fu_g=-\frac{1}{\rho_0}P_y\)

where \((u_g,v_g)\) is the geostrophic flow. It follows that:

\(\frac{\partial u_g}{\partial x}+\frac{\partial v_g}{\partial y}=-\frac{\beta v}{f}\)

On a f-plane,  \(\beta=0\) and hence 

\(\frac{\partial u_g}{\partial x}+\frac{\partial v_g}{\partial y}=0\)

Using \(u_x+v_y+w_z=0 \) it follows for the f-plane:

\(w_z=0\)

This means that if vertical velocity is somewhere zero (as it is for the bottom the case), than it is everywhere zero! It follows that flow will tend to go around topography and not over topography!

Furthermore, with the thermal wind equations it can be shown that for a fluid with uniform density, that there is no variation in the geostrophic flow in the vertical (no geostrophic shear!) which means that above a bump, the flow will be stagnant at all depths!

The Taylor-Proudman Theorem states that flow tends to go around bumps rather than over them!

Combine the geostrophic balance 

\(-fv=-\frac{1}{\rho_0}p_x\\fu=-\frac{1}{\rho_0}p_y\)

with the hydrostatic balance \(0=-p_z-g\rho\) to give

\(fv_z=-\frac{g}{\rho_0}\rho_x\\fu_z=\frac{g}{\rho_0}\rho_y\)

The steady-state zonal and meridional momentum equations (linearised about a state of rest) are:

\(-fv=-\frac{1}{\rho_0}p_x+\frac{\tau^x_s}{\rho_0H_E}G(z)\\ fu=-\frac{1}{\rho_0}p_y+\frac{\tau^y_s}{\rho_0H_E}G(z)\)

Taking  \(\frac{\partial 2}{\partial x}-\frac{\partial 1}{\partial y}\)  and using  \(u_x+v_y+w_z=0\) gives

\(\beta v=fw_z+\frac{G(z)}{\rho_0H_E}[\frac{\partial \tau^y_s}{\partial x}-\frac{\partial \tau^x_s}{\partial y}]\)

\(\beta v\) ...advection of planetary vorticity

\(fw_z\) ...vortex stretching -> locally stretching \((w_z>0)\)leads to an increase of vorticity and likewise 

\(\frac{G(z)}{\rho_0H_E}[\frac{\partial \tau^y_s}{\partial x}-\frac{\partial \tau^x_s}{\partial y}]\)...vorticity input by the curl of the surface wind stress 

If we set for the linear vorticity equation wind stress zero, then we get the Sverdrup relation:

\(\beta v=fw_z\)

By vertically integrating we can gain the conservation of potential vorticity:

layers of uniform density: \(\frac{D}{Dt}\frac{f}{h}=0\)

continuous stratification: \(\frac{D}{Dt}[f\rho_z]=0\)

we start from a flat-bottomed ocean with uniform density. The linearized equations of motions for a steady flow are:

\(-fv=-g\eta_x+(vu_z)_z\\ fu=-g\eta_y+(vv_z)_z\\ u_x+v_y+w_z=0\)

vertical averaging and paramterization of the bottom stressin terms of the vertical averaged flow brings: 

\(-f\bar{v}=-g\eta_x+\frac{\tau_s^x}{\rho_0H}-\frac{r\bar{u}}{H}\\ f\bar{u}=-g\eta_y+\frac{\tau_s^y}{\rho_0 H}-\frac{r\bar{v}}{H}\)

since \((\bar{u}H)_x+(\bar{v}H)_y=0\) ,there exists a streamfunction 

Taking \(\frac{\partial }{\partial x}(2)-\frac{\partial }{\partial y}(2)\) gives 

\(.\\ \frac{r}{H}[\frac{\partial \bar{v}}{\partial x}-\frac{\partial \bar{u}}{\partial y}]+\beta\bar{v}=\frac{1}{\rho_0H}[\frac{\partial }{\partial x}(\tau_s^y)-\frac{\partial }{\partial y} (\tau_s^x)]\)

or as stream function: \(\gamma \nabla^2\Psi+\beta \Psi_x=F\)

where \(\gamma=\frac{r}{H}\) is the friction parameter and \(F=\frac{1}{\rho_0}[\frac{\partial }{\partial x}(\tau_s^y)-\frac{\partial }{\partial y} (\tau_s^x)]\) is the curl of the wind stress. 

Coming to the model geometry:

x=0 ... western boundary; x=\(\alpha\)L ... eastern boundary

\(y=y_0\)... runs across the middle of the domain. 

Parameterization of the surface wind stress:

\(\tau_s^x=\tau_0sin(\frac{y-y_0}{L})\\ \tau_s^y=0\)

it follows: \(F=-\frac{1}{\rho_0}[\frac{\tau_0}{L}cos(\frac{y-y_0}{L})]\)

Seek a solution of the form \(\Psi=\psi(x)cos(\frac{y-y_0}{L})\) where \(\psi=\psi(x)\) . Then

\(-\frac{\gamma}{L^2}\psi+\gamma \frac{d^2\psi}{dx^2}+\beta\frac{d\psi}{dx}=-\frac{\tau_0}{\rho_0L}\)

We now can non-dimensionalise the equation by defining: \(x'=x/L\) and \(\psi'=\frac{\psi}{\tau_0/(\rho_0\beta)}\) . Then: 

The governing equations are: 

1. \(u_t+uu_x+vu_y-fv=-g'h_x+\frac{\tau_s^x}{\rho_0(H_1+h)}\\\)
2. \(v_t+uv_x+vv_y+fu=-g'h_y+\frac{\tau_s^y}{\rho_0(H_1+h)}\)
3. \(h_t+uh_x+vh_y+(H_1+h)(u_x+v_y)=0\)

Since we aspect the dominant balance to be geostrophy, i.e. 

\(-fv=-g'h_x\\fu=-g'h_y\)

this suggests: \(f_0U=\frac{g'D}{L}\)

Now non-dimensionalise as follows: 

\(u,v \leftrightarrow U\\x,y\leftrightarrow L\\t\leftrightarrow T\\\tau_s^x,\tau_s^y\leftrightarrow\tau_0\\h\leftrightarrow \frac{f_0UL}{g'}\)

The non-dimensionalising brings the following: 

\(\epsilon_Tu_t+R_0(uu_x+vu_y)-(1+\epsilon_\beta y)v=-h_x+\frac{\epsilon_E\tau_s^x}{(1+FR_0h)}\\\epsilon_Tv_t+R_0(uv_x+vv_y)-(1+\epsilon_\beta y)u=-h_y+\frac{\epsilon_E\tau_s^y}{(1+FR_0h)}\)

where 

\(\epsilon_T=\frac{1}{f_0T}\;\;\;\epsilon_\beta=\frac{\beta L}{f_0}\\R_0=\frac{U}{f_0L}\;\;(Rossby\;number)\\\epsilon_E=\frac{\tau_0}{\rho_0H_1f_0U}\;\;(Ekman\;number)\\F=(\frac{L}{R})^2\;\;where\;\;R=\frac{c}{f_0}\;(radius\;of\;deformation)\;\;c=\sqrt{g'H_1}\)

For geostrophic balance we assume: \(F\approx1\;and\;\epsilon_T=R_0=\epsilon_\beta =\epsilon_E=\epsilon\ll1\)

At lowest order in \(\epsilon\), we have \(u=u_g,\;v=v_g\)

The full velocity is then (with the additional 4 parts):

\(u=u_g+\epsilon u_a\\v=v_g+\epsilon v_a\)

where \(\epsilon u_a\;and\;\epsilon v_a\) is the ageostrophic flow.

The ageostrophic part of the \(U=U_g+U_a\)velocity , approximated all terms of order 1 in respect to  \(\epsilon\) , is 

back to the dimensionalized form: 

\(f_0u_a=-\beta yu_g-v_{gt}-(u_gv_{gx}+v_gv_{gy})+\frac{\tau_s^y}{\rho_0H_1}\\f_0v_a=-\beta yv_g+u_{gt}+(u_gu_{gx}+v_gu_{gy})+\frac{\tau_s^x}{\rho_0H_1}\)

The wind-driven circulation has been described in terms of the quasi-geostrophic model derived from the shallow water equation in many textbooks. 

Although, the wind-driven gyres in mid latitudes have horizontal scales much larger than the synoptic scale assumed in the quasi-geostrophic approximation. 

As a result, one of the basic assumptions in the traditional quasi-gesostrophic approximation , that deviations from the mean stratification are small, is no longer valid. 

Although the quasi-geostrophic theory remains a useful tool for describing the circulation, the strong nonlinearity due to the meridional change of stratification can be handled much more accurately by using simple reduced-gravity models. 

The wind-driven circulation in the ocean interior can be described in terms of the balance between the Coriolis force, the pressure gradient and the wind stress. 

From both the energy and potential vorticity equations, it is clear that a purely inertial western boundary current cannot satisfy the energy and potential vorticity balance in a closed basin. No matter how small the friction is, it plays an essential role in balancing the energy and potential vorticity in a closed basin by dissipating the potential vorticity and energy input from the wind stress. 

The following models for the western boundary do exist: 

1. Stommel boundary layer: balance between planetary vorticity advection and the interfacial friction torque. 
2. Munk boundary layer: balance between planetary vorticity advection and lateral friction. 

Assumptions:

1. In the ocean interior, frictional and inertial terms are negligible 
2. fror simplicity we assume the following wind stress: \(\tau^x=\tau^x(y),\;\;\tau^y=0\)

the momentum equations are reduced to: 

\(-fhv=-g'hh_x+\tau^x/\rho_0\\ fhu=-g'hh_y\)

Cross-differentiating and subtraction leads to the vorticity equation:

\(\beta hv=-\tau^x_y/\rho_0 \)

This equation is called the Sverdrup relation!

Substituting the Sverdrup relation back to the first moemntum equation gives the first-order ordinary differential equation:

\(hh_x=-\frac{f^2}{g'\rho_0\beta}(\frac{\tau^x}{f})_y\)

Integrating this leads to the interior solution. To start the integration we need to choose the eastern boundary to balance the vorticity in the basin. 

Using  \(\psi_x=hv,\;\;\psi_y=-hu\) leads to:

\(\psi =\frac{1}{\rho_0 \beta}\tau_y^x(x_e-x)\)

This volume is called the Sverdrup transport!

Two results:

1. easterlies prevail in the low latitudes and westerlies prevail in the mid latitudes: the wind-stress curl is negative: \(curl\; \tau=-\tau^x_y\;<\;0\)
   1. equatorward flow in the interior
2. The depth of the main thermocline increases westward.