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Let us consider a wind blowing parallel to the surface of a sea. Due to viscosity, deeper layers of the sea will move, with the magnitude of the velocity decreasing with depth. In a rotating frame, like on a spinning planet, the velocity is also affected by the Coriolis force. The governing equations for the two components of the velocity parallel to the surface $ u $, $ v $ are

$ -f v = D \frac{d^2 u}{d z^2} $

$ f u = D \frac{d^2 v}{d z^2} $

where $ D $ is the viscosity or diffusion coefficient, $ f = 2 \Omega \sin \phi $ is the Coriolis parameter, $ \Omega $ is the spin frequency and $ \phi $ is the latitude. We can express the two components of the velocity as a single complex variable $ w = u + i v $

$ D \frac{d^2 w}{d z^2} = i f w \Rightarrow w = w_0 \exp \left(- z \sqrt{\frac{f}{2 D}} \right) \left(1+i \right) $

Hence the velocity rotates and declines as the depth $ z $ increases. The characteristic distance over which this happens is $ \sqrt{D/f} $.