Geostrophic wind

Geostrophic currents

Flow of ocean water is also largely geostrophic. Just as multiple weather balloons that measure pressure as a function of height in the atmosphere are used to map the atmospheric pressure field and infer the geostrophic wind, measurements of density as a function of depth in the ocean are used to infer geostrophic currents. Satellite altimeters are also used to measure sea surface height anomaly, which permits a calculation of the geostrophic current at the surface.

Limitations of the geostrophic approximation

The effect of friction, between the air and the land, breaks the geostrophic balance. Friction slows the flow, lessening the effect of the Coriolis force. As a result, the pressure gradient force has a greater effect and the air still moves from high pressure to low pressure, though with great deflection. This explains why high-pressure system winds radiate out from the center of the system, while low-pressure systems have winds that spiral inwards.

The geostrophic wind neglects frictional effects, which is usually a good approximation for the synoptic scale instantaneous flow in the midlatitude mid-troposphere. Although ageostrophic terms are relatively small, they are essential for the time evolution of the flow and in particular are necessary for the growth and decay of storms. Quasigeostrophic and semi geostrophic theory are used to model flows in the atmosphere more widely. These theories allow for a divergence to take place and for weather systems to then develop.

Formulation

Newton's second law can be written as follows if only the pressure gradient, gravity, and friction act on an air parcel, where bold symbols are vectors:

${\displaystyle \frac{D\mathit{U}}{Dt}=-\frac{1}{\rho }\mathrm{\nabla }p-2\mathbf{\Omega }\times \mathit{U}+\mathit{g}+{\mathit{F}}_r}$

Here U is the velocity field of the air, Ω is the angular velocity vector of the planet, ρ is the density of the air, P is the air pressure, F_r is the friction, g is the acceleration vector due to gravity and ⁠D/Dt⁠ is the material derivative.

Locally this can be expanded in Cartesian coordinates, with a positive u representing an eastward direction and a positive v representing a northward direction. Neglecting friction and vertical motion, as justified by the Taylor–Proudman theorem, we have:

${\displaystyle \frac{Du}{Dt}=-\frac{1}{\rho }\frac{\mathrm{\partial }P}{\mathrm{\partial }x}+fv+0+0}$

${\displaystyle \frac{Dv}{Dt}=-\frac{1}{\rho }\frac{\mathrm{\partial }P}{\mathrm{\partial }y}-fu+0+0}$

${\displaystyle \frac{Dw}{Dt}=-\frac{1}{\rho }\frac{\mathrm{\partial }P}{\mathrm{\partial }z}+0-g+0}$

With f = 2Ω sin φ the Coriolis parameter (approximately 10⁻⁴ s⁻¹, varying with latitude).

Assuming geostrophic balance, the system is stationary and the first two equations become:

${\displaystyle fv=\frac{1}{\rho }\frac{\mathrm{\partial }P}{\mathrm{\partial }x}}$

${\displaystyle fu=-\frac{1}{\rho }\frac{\mathrm{\partial }P}{\mathrm{\partial }y}}$

${\displaystyle -g-\frac{1}{\rho }\frac{\mathrm{\partial }P}{\mathrm{\partial }z}=0}$

By substituting using the third equation above, we have:

${\displaystyle \begin{array}{cc}fv& =\frac{-g}{\frac{\mathrm{\partial }P}{\mathrm{\partial }z}}\frac{\mathrm{\partial }P}{\mathrm{\partial }x}:=\frac{-g}{c}a\\ fu& =-\frac{-g}{\frac{\mathrm{\partial }P}{\mathrm{\partial }z}}\frac{\mathrm{\partial }P}{\mathrm{\partial }y}:=+\frac{g}{c}b\end{array}}$

with z the geopotential height of the constant pressure surface, satisfying

${\displaystyle \mathrm{d}P=\frac{\mathrm{\partial }P}{\mathrm{\partial }x}\mathrm{d}x+\frac{\mathrm{\partial }P}{\mathrm{\partial }y}\mathrm{d}y+\frac{\mathrm{\partial }P}{\mathrm{\partial }z}\mathrm{d}z:=a\mathrm{d}x+b\mathrm{d}y+c\mathrm{d}z=0}$

Further simplify those formulae above:

${\displaystyle \begin{array}{cc}fv& =\frac{-g}{c}a=+g(\frac{\mathrm{d}z}{\mathrm{d}x}{)}_{\mathrm{d}y=0}\\ fu& =+\frac{g}{c}b=-g(\frac{\mathrm{d}z}{\mathrm{d}y}{)}_{\mathrm{d}x=0}\end{array}}$

This leads us to the following result for the geostrophic wind components:

${\displaystyle v_g=\frac{g}{f}\frac{\mathrm{d}z}{\mathrm{d}x}}$

${\displaystyle u_g=-\frac{g}{f}\frac{\mathrm{d}z}{\mathrm{d}y}}$

The validity of this approximation depends on the local Rossby number. It is invalid at the equator, because f is equal to zero there, and therefore generally not used in the tropics.

Other variants of the equation are possible; for example, the geostrophic wind vector can be expressed in terms of the gradient of the geopotential Φ on a surface of constant pressure:

${\displaystyle {\mathbf{V}}_{\mathrm{g}}=\frac{\hat{\mathbf{k}}}{f}\times {\mathrm{\nabla }}_p\mathrm{\Phi }}$