Geostrophic wind

A useful heuristic is to imagine air starting from rest, experiencing a force directed from areas of high pressure toward areas of low pressure, called the pressure gradient force. If the air began to move in response to that force, however, the Coriolis force would deflect it, to the right of the motion in the Northern Hemisphere or to the left in the Southern Hemisphere. As the air accelerated, the deflection would increase until the Coriolis force's strength and direction balanced the pressure gradient force, a state called geostrophic balance. At this point, the flow is no longer moving from high to low pressure, but instead moves along isobars. Geostrophic balance helps to explain why, in the Northern Hemisphere, low-pressure systems (or cyclones) spin counterclockwise and high-pressure systems (or anticyclones) spin clockwise, and the opposite in the Southern Hemisphere.

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.

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.^[4] 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.

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:

${\displaystyle {D{\boldsymbol {U}} \over Dt}=-{1 \over \rho }\nabla p-2{\boldsymbol {\Omega }}\times {\boldsymbol {U}}+{\boldsymbol {g}}+{\boldsymbol {F}}_{r}}$

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:

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

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

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

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:

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

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

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

By substituting using the third equation above, we have:

${\displaystyle {\begin{aligned}fv&={\frac {\;-g\;}{\;{\frac {\partial P}{\partial z}}\;}}{\frac {\partial P}{\partial x}}:={\frac {\;-g\;}{\;c\;}}a\\[5px]fu&=-{\frac {\;-g\;}{\;{\frac {\partial P}{\partial z}}\;}}{\frac {\partial P}{\partial y}}:=+{\frac {\;g\;}{\;c\;}}b\end{aligned}}}$

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

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

Further simplify those formulae above:

${\displaystyle {\begin{aligned}fv&={\frac {\;-g\;}{\;c\;}}a=+g{\biggl (}{\frac {{\rm {d}}z}{{\rm {d}}x}}{\biggr )}_{{\rm {d}}y=0}\\[5px]fu&=+{\frac {\;g\;}{\;c\;}}b=-g{\biggl (}{\frac {{\rm {d}}z}{{\rm {d}}y}}{\biggr )}_{{\rm {d}}x=0}\end{aligned}}}$

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

${\displaystyle v_{g}={g \over f}{{\rm {d}}z \over {\rm {d}}x}}$

${\displaystyle u_{g}=-{g \over f}{{\rm {d}}z \over {\rm {d}}y}}$

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:

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

• Geostrophic current
• Thermal wind
• Gradient wind
• Prevailing winds