Quasi-geostrophic equations

While geostrophic motion refers to the wind that would result from an exact balance between the Coriolis force and horizontal pressure gradient forces,Quasi-geostrophic (QG) motion refers to flows where the Coriolis force and pressure gradient forces are almost in balance, but with inertia also having an effect.

Atmospheric and oceanographic flows take place over horizontal length scales which are very large compared to their vertical length scale, and so they can be described using the shallow water equations. The Rossby number is a dimensionless number which characterises the strength of inertia compared to the strength of the Coriolis force. The quasi-geostrophic equations are approximations to the shallow water equations in the limit of small Rossby number, so that inertial forces are an order of magnitude smaller than the Coriolis and pressure forces. If the Rossby number is equal to zero then we recover geostrophic flow.

In Cartesian coordinates, the components of the geostrophic wind are

where ${\displaystyle {\Phi }}$ is the geopotential height.

The geostrophic vorticity

can therefore be expressed in terms of the geopotential as

Equation (2) can be used to find ${\displaystyle {\zeta _{g}(x,y)}}$ from a known field ${\displaystyle {\Phi (x,y)}}$. Alternatively, it can also be used to determine ${\displaystyle {\Phi }}$ from a known distribution of ${\displaystyle {\zeta _{g}}}$ by inverting the Laplacian operator.

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