10.2: Geostrophic Equations

The geostrophic balance requires that the Coriolis force balance the horizontal pressure gradient. The equations for geostrophic balance are derived from the equations of motion assuming the flow has no acceleration, $du/dt = dv/dt = dw/dt = 0$; that horizontal velocities are much larger than vertical, $w << u, v$; that the only external force is gravity; and that friction is small. With these assumptions equations $(7.6.5-7)$ become $$\frac{\partial p}{\partial x} = \rho fv; \quad \frac{\partial p}{\partial y} = -\rho fu; \quad \frac{\partial p}{\partial z} = -\rho g \nonumber$$

where $f = 2\Omega \sin \varphi$ is the Coriolis parameter. These are the geostrophic equations.

The equations can be written: $$u = \frac{1}{f \rho} \frac{\partial p}{\partial y}; \quad v = \frac{1}{f \rho} \frac{\partial p}{\partial x} \nonumber$$

$$p = p_{0} + \int_{-h}^{\zeta} g(\varphi, z) \rho(z) \ dz \nonumber$$ where $p_{0}$ is atmospheric pressure at $z = 0$, and $\zeta$ is the height of the sea surface. Note that I have allowed for the sea surface to be above or below the surface $z = 0$, and the pressure gradient at the sea surface is balanced by a surface current $u_{s}$ as follows:

Substituting $(\PageIndex{3})$ into $(\PageIndex{2})$ gives: $$\begin{align} u &= -\frac{1}{f \rho} \frac{\partial}{\partial y} \int_{-h}^{0} g (\varphi, z) \rho(z) \ dz - \frac{g}{f} \frac{\partial \zeta}{\partial y} \nonumber \\[4pt] u &= -\frac{1}{f \rho} \frac{\partial}{\partial y} \int_{-h}^{0} g (\varphi, z) \rho(z) \ dz - u_{s} \end{align} \nonumber$$

where I have used the Boussinesq approximation, retaining full accuracy for $\rho$ only when calculating pressure. In a similar way, we can derive the equation for $v$. $$\begin{align} v &= \frac{1}{f \rho} \frac{\partial}{\partial x} \int_{-h}^{0} g(\varphi, z) \rho(z) \ dz + \frac{g}{f} \frac{\partial \zeta}{\partial x} \nonumber \\[4pt] v &= \frac{1}{f \rho} \frac{\partial}{\partial x} \int_{-h}^{0} g(\varphi, z) \rho(z) \ dz + v_{s} \end{align} \nonumber$$

If the ocean is homogeneous and density and gravity are constant, the first term on the right-hand side of $(\PageIndex{4})$ and $(\PageIndex{5})$ is equal to zero; and the horizontal pressure gradients within the ocean are the same as the gradient at $z = 0$. This is barotropic flow, further described in Section 10.4.

If the ocean is stratified, the horizontal pressure gradient has two terms, one due to the slope at the sea surface, and an additional term due to horizontal density differences. These equations include baroclinic flow, also discussed in Section 10.4. The first term on the right-hand side of $(\PageIndex{4})$ and $(\PageIndex{5})$ is due to variations in density $\rho(z)$, and it is called the relative velocity. Thus calculation of geostrophic currents from the density distribution requires the velocity $(u_{0}, v_{0})$ at the sea surface or at some other depth.