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Fenster schliessen

Give the equation for the linear vorticity balance and explain it!

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 

 

Fenster schliessen

Give the Sverdrup relation.

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: \(\frac{D}{Dt}[f\rho_z]=0\)

Fenster schliessen

Sketch the Stommel Model! (alt)

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: 

Fenster schliessen

Quasi-Geostrophic Dynamics: Derive and discuss the quasi-geostrophic equations for a (non-linear) \(1\frac{1}{2}\) layer model. 

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: 

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

Fenster schliessen

What is the advantage of reduced gravity models? 

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. 

Fenster schliessen

Why do we need a western boundary? 

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. 

 

Fenster schliessen

Explain the interior solution. 

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. 
Fenster schliessen

Common features of the Western boundary layer regardless to the specific dynamic balance assumptions (Stommel, Munk,...)

To a very good approximation the cross-stream pressure gradient is in balance with the Coriolis force associated with the downstream velocity. 

However, the downstream momentum is in ageostrophic balance, i.e. the downstream momentum balance must include downstream pressure. 

Thus, the boundary layer is said to be in semi-geostrophic balance. 

Two important features for the general solution of the western boundary layer: 

  1. The layer thickness declines toward the wall. This is caused by the strong western boundary current and is required for balancing the model's mass, vorticity, and energy. 
  2. Layer thickness along the western wall decreases northward. This meridional pressure gradient is the result of the geostrophic constraint agross the western boundary layer, so this feature holds for all types of western boundary layer. 

(Note that layer thickness in the reduced-gravity model can be interpreted as either pressure or free surface elevation.)