10: Geostrophic Currents

Scaling the Equations: The Geostrophic Approximation

We wish to simplify the equations of motion to obtain solutions that describe the deep-sea conditions well away from coasts and below the Ekman boundary layer at the surface. To begin, let’s examine the typical size of each term in the equations in the expectation that some will be so small that they can be dropped without changing the dominant characteristics of the solutions. For interior, deep-sea conditions, typical values for distance \(L\), horizontal velocity \(U\), depth \(H\), Coriolis parameter \(f\), gravity \(g\), and density \(\rho\) are:

\[\begin{array}{l} &L \approx 10^{6} \ \text{m} \quad\quad\quad &H_{1} \approx 10^{3} \ \text{m} \quad\quad\quad &f \approx 10^{-4} \ \text{s}^{-1} \quad\quad\quad &\rho \approx 10^{3} \ \text{kg/m}^{3} \nonumber \\ &U \approx 10^{-1} \ \text{m/s} &H_{2} \approx 1 \ \text{m} &\rho \approx 10^{3} \ \text{kg/m}^{3} &g \approx 10 \ \text{m/s}^{2} \end{array} \nonumber \]

where \(H_{1}\) and \(H_{2}\) are typical depths for pressure in the vertical and horizontal. From these variables we can calculate typical values for vertical velocity \(W\), pressure \(P\), and time \(T\):

\[\begin{align*} \frac{\partial W}{\partial z} &= - \left(\frac{\partial U}{\partial x} + \frac{\partial v}{\partial y}\right) \\ \frac{W}{H_{1}} &= \frac{U}{L}; \quad W = \frac{U H_{1}}{L} = \frac{10^{-1} \cdot 10^{3}}{10^{6}} \text{m/s} = 10^{-4} \ \text{m/s} \\ P &= \rho g H_{1} = 10^{3} \cdot 10^{1} \cdot 10^{3} = 10^{7} \ \text{Pa}; \quad \partial p/\partial x = \rho g H_{2}/L = 10^{-2} \ \text{Pa/m} \\ T &= L/U = 10^{7} \ \text{s} \end{align*} \]

The momentum equation for vertical velocity is therefore:

\[\begin{align*} \frac{\partial w}{\partial t} + u \frac{\partial w}{\partial x} + v \frac{\partial w}{\partial y} + w \frac{\partial w}{\partial z} &= -\frac{1}{\rho} \frac{\partial \rho}{\partial z} + 2 \Omega \ u \cos \varphi - g \\ \frac{W}{T} + \frac{UW}{L} + \frac{UW}{L} + \frac{W^{2}}{H} &= \frac{P}{\rho H_{1}} + fU - g \\ 10^{-11} + 10^{-11} + 10^{-11} + 10^{-11} &= 10 + 10^{-5} + 10 \end{align*} \]

and the only important balance in the vertical is hydrostatic: \[\frac{\partial p}{\partial z} = -\rho g \quad \text{Correct to } 1:10^{6} \nonumber \]

The momentum equation for horizontal velocity in the \(x\) direction is \[\begin{align*} \frac{\partial u}{\partial t} + u \frac{\partial u}{\partial x} + u \frac{\partial u}{\partial y} + u \frac{\partial u}{\partial z} &= -\frac{1}{\rho} \frac{\partial p}{\partial x} + fv \\ 10^{-8} + 10^{-8} + 10^{-8} + 10^{-8} &= 10^{-5} + 10^{-5} \end{align*} \]

Thus the Coriolis force balances the pressure gradient within one part per thousand. This is called the geostrophic balance, and the geostrophic equations are: \[\frac{1}{\rho} \frac{\partial p}{\partial x} = fv; \quad \frac{1}{\rho} \frac{\partial p}{\partial y} = -fu; \quad \frac{1}{\rho} \frac{\partial p}{\partial z} = -g \nonumber \]

This balance applies to oceanic flows with horizontal dimensions larger than roughly 50 km and times greater than a few days.