Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 333 / 2 / a

Table of contents
- Solution a

Solution

Assume a steady, small-Rossby number surface boundary layer, neglect horizontal viscosity and nonlinear acceleration, take the pressure gradient to be independent of depth, impose no normal flow at the surface, and let the viscous stress vanish at

z = - h

. Subtract the depth-independent geostrophic balance from horizontal momentum. For the ageostrophic velocity,

- f v_{a} = ν u_{zz}, f u_{a} = ν v_{zz} .

Integrating from

- h

to

0

and using

τ = ρ ν (u_{z}, v_{z}) ∣_{z = 0}

gives the Ekman transport

M_{E} = \int_{- h}^{0} (u_{a}, v_{a}) d z = (\frac{τ _{y}}{ρ f}, - \frac{τ _{x}}{ρ f}) = \frac{τ \times z}{ρ f} .

Depth-integrated mass conservation, with

w (0) = 0

, gives

\nabla_{h} \cdot M_{E} - w (- h) = 0.

Hence the vertical velocity entering the ocean interior is the Ekman pumping velocity

w_{E} \equiv w (- h) = z \cdot \nabla_{h} \times (\frac{τ}{ρ f}) .

For constant

f

this reduces to

w_{E} = \frac{τ _{y, x} - τ _{x, y}}{ρ f} .

 Ancestors (10)

 View article source

 Discussion (0)

There are no discussions about this article yet.

 Articles by others on the same topic (0)

There are currently no matching articles.

  See all articles in the same topic Create my own version