Math 2.0

Fluid mechanics Definition

Geostrophic balance

Builds on Geophysical fluid dynamicsRossby number

Definition

Consider a fluid of density ρ>0\rho > 0 in a frame rotating with a planet, with local Cartesian coordinates xx (east), yy (north) and zz (up). Let f≠0f \ne 0 be the Coriolis parameter, f=2∣Ω∣sin⁡φf = 2 \abs{\Omega} \sin \varphi at latitude φ\varphi. Let eze_z be the upward unit vector and ∇h=(∂x,∂y)\nabla_h = (\partial_x, \partial_y) the horizontal gradient. A horizontal velocity uh=(u,v)u_h = (u, v) is in geostrophic balance with the pressure pp when f ez×uh=−1ρ∇hp,f \, e_z \times u_h = -\frac{1}{\rho} \nabla_h p, that is, when u=−1ρf∂p∂yu = -\dfrac{1}{\rho f} \dfrac{\partial p}{\partial y} and v=1ρf∂p∂xv = \dfrac{1}{\rho f} \dfrac{\partial p}{\partial x}. Such a velocity is called the geostrophic velocity, or the geostrophic wind in the atmosphere.

Remarks

The relation is the horizontal momentum equation of geophysical fluid dynamics with the acceleration and friction dropped, which is justified when the Rossby number and the Ekman number are small. Since uh⋅∇hp=0u_h \cdot \nabla_h p = 0, the geostrophic velocity is parallel to the isobars.

Example: with f=10−4f = 10^{-4} s−1^{-1}, ρ=1.2\rho = 1.2 kg/m3^3 and a pressure that increases northward by 11 hPa per 100100 km, ∂yp=10−3\partial_y p = 10^{-3} Pa/m, so u=−10−3/(1.2⋅10−4)≈−8.3u = -10^{-3} / (1.2 \cdot 10^{-4}) \approx -8.3 m/s, a wind from east to west.

Non-example: near the equator f→0f \to 0, and the balance cannot hold for a nonzero pressure gradient; geostrophy breaks down in the tropics.

When ρ\rho and ff are constant, the geostrophic velocity is horizontally nondivergent and has the stream function ψ=p/(ρf)\psi = p / (\rho f), with u=−∂yψu = -\partial_y \psi and v=∂xψv = \partial_x \psi.

Reading path

To understand this page from the ground up, read these 3 pages in order. The foundations come first.

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Geophysical fluid dynamics, concept, no evidence levelGeophysical fluid dyn…conceptRossby number, concept, no evidence levelRossby numberconceptGeostrophic balance, definition, no evidence levelGeostrophic balancedefinition
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  1. Geophysical fluid dynamics concept
  2. Rossby number concept
  3. Geostrophic balance definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Geostrophic_wind
About this page
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Kinddefinition
WritingW0 stub
MSC76U60, 86A10, 86A05
DefinesGeostrophic balance
UsesGeophysical fluid dynamics, Rossby number
Changed3 November 2026

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