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.
Origin 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. The quasi-geostrophic equations were first formulated by Jule Charney.
Derivation of the single-layer QG equations In Cartesian coordinates, the components of the geostrophic wind are
f 0 v g = ∂ Φ ∂ x {\displaystyle {f_{0}}{v_{g}}={\partial \Phi \over \partial x}} (1a)
f 0 u g = − ∂ Φ ∂ y {\displaystyle {f_{0}}{u_{g}}=-{\partial \Phi \over \partial y}} (1b) where Φ {\displaystyle {\Phi }} is the geopotential. The geostrophic vorticity
ζ g = k ^ ⋅ ∇ × V g {\displaystyle {\zeta _{g}}={{\hat {\mathbf {k} }}\cdot \nabla \times \mathbf {V_{g}} }}
can therefore be expressed in terms of the geopotential as
ζ g = ∂ v g ∂ x − ∂ u g ∂ y = 1 f 0 ( ∂ 2 Φ ∂ x 2 + ∂ 2 Φ ∂ y 2 ) = 1 f 0 ∇ 2 Φ {\displaystyle {\zeta _{g}}={{\partial v_{g} \over \partial x}-{\partial u_{g} \over \partial y}={1 \over f_{0}}\left({{\partial ^{2}\Phi \over \partial x^{2}}+{\partial ^{2}\Phi \over \partial y^{2}}}\right)={1 \over f_{0}}{\nabla ^{2}\Phi }}} (2) Equation (2) can be used to find ζ g ( x , y ) {\displaystyle {\zeta _{g}(x,y)}} from a known field Φ ( x , y ) {\displaystyle {\Phi (x,y)}} . Alternatively, it can also be used to determine Φ {\displaystyle {\Phi }} from a known distribution of ζ g {\displaystyle {\zeta _{g}}} by inverting the Laplacian operator. The quasi-geostrophic vorticity equation can be obtained from the x {\displaystyle {x}} and y {\displaystyle {y}} components of the quasi-geostrophic momentum equation which can then be derived from the horizontal momentum equation
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