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Fenster schliessen

Argue why geostrophic flow on an f-plane is horizontally non-divergent. The Taylor-Proudman theorem! 

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! 

Fenster schliessen

In what horizontal direction tends flow to be on a rotating planet? 

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.

 

Fenster schliessen

Density stratification tends to confine the flow near the surface and insulate the flow from the influence of the variable bottom topography. 

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.