Deriving Ekman Transport in the Southern Hemisphere

[ATY]

Hey guys, I got this problem:
We had the derivation of the ekman transport today in class. And what I wondered about is this:
Usually the equation for the ekman transport looks similar to this (depends on the author)
$$u = V_0 e^{az} cos(\frac{\pi}{4}+az)$$
$$v = V_0 e^{az} sin(\frac{\pi}{4}+az)$$
$$a= \sqrt{\frac{f}{2 A_z}}$$

This is fine for the northern part of the earth, but what happens when I go to the southern hemisphere ? the coriolis parameter f should become negative (since f is $$f = 2 \Omega sin(\phi)$$)
So I can not use the equations above. I am really confused because none of the derivations that I found talked about this. Or am I missing a really obvious point ?
best wishes
ATY

 

[Andy Resnick]

Hey guys, I got this problem:
<snip> I am really confused because none of the derivations that I found talked about this. Or am I missing a really obvious point ?

The derivation I have is in Pozrikidis ("Introduction to Theoretical and Computational Fluid Mechanics"), and there is no ambiguity- I wonder if you have a choice-of-coordinates sign change hidden somewhere in your derivation. The Coriolus force can written as -2Ω×u, where Ω is the rotation rate (Ω=Ωe_z) of the fluid and u=(u_x(z),u_y(z),0) is the "horizontal" velocity. In the end, the velocity components of u are found to be:

u_x+iu_y=(U_x+iU_y)exp(-(1+i) |z|/δ)

where U_x and U_y are the horizontal velocity components on the fluid surface (taken to be z = 0) and δ is the Ekman layer thickness.

Does this help?