Math 2.0

Fluid mechanics Definition

Quasi-geostrophic equations

Builds on Geostrophic balanceRossby numberBrunt–Väisälä frequencyBoussinesq approximation for buoyancyStream function

Definition

Consider a stably stratified Boussinesq fluid on a β\beta-plane, with Coriolis parameter f=f0+βyf = f_0 + \beta y for constants f0≠0f_0 \ne 0 and β\beta. Let N(z)>0N(z) > 0 be the Brunt–Väisälä frequency of the background state. Let ψ(x,y,z,t)\psi(x, y, z, t) be the geostrophic stream function, with horizontal velocity (u,v)=(−∂yψ,∂xψ)(u, v) = (-\partial_y \psi, \partial_x \psi). The quasi-geostrophic equations are ∂q∂t+J(ψ,q)=0,q=Δhψ+βy+∂∂z(f02N2∂ψ∂z),\frac{\partial q}{\partial t} + J(\psi, q) = 0, \qquad q = \Delta_h \psi + \beta y + \frac{\partial}{\partial z} \left( \frac{f_0^2}{N^2} \frac{\partial \psi}{\partial z} \right), where Δh=∂x2+∂y2\Delta_h = \partial_x^2 + \partial_y^2 and J(a,b)=∂xa ∂yb−∂ya ∂xbJ(a, b) = \partial_x a \, \partial_y b - \partial_y a \, \partial_x b. The quantity qq is the quasi-geostrophic potential vorticity. At horizontal boundaries z=z0z = z_0 with no vertical velocity, the buoyancy b=f0∂zψb = f_0 \partial_z \psi satisfies ∂tb+J(ψ,b)=0\partial_t b + J(\psi, b) = 0.

Remarks

They are derived from the Boussinesq equations by an expansion in the Rossby number. The expansion also assumes ∣βy∣≪∣f0∣\abs{\beta y} \ll \abs{f_0} and small deviations of the stratification from the background. The leading order is geostrophic balance with ψ=p/(ρ0f0)\psi = p / (\rho_0 f_0). Given qq and boundary data, ψ\psi is found by inverting an elliptic operator, and the velocity follows.

Example: if ψ\psi does not depend on zz, the equations reduce to the barotropic vorticity equation ∂t(Δhψ)+J(ψ,Δhψ)+β∂xψ=0\partial_t (\Delta_h \psi) + J(\psi, \Delta_h \psi) + \beta \partial_x \psi = 0, whose linearisation about rest has the Rossby waves as solutions.

Non-example: gravity waves and sound waves are not described by the model. In the interior the only time derivative in the quasi-geostrophic equations is that of qq, and these fast waves are filtered out.

The shallow water and two-layer forms are also called quasi-geostrophic equations. They replace the stretching term by −ψ/Ld2-\psi / L_d^2, with a deformation radius LdL_d, or by an interaction between layers.

Reading path

To understand this page from the ground up, read these 6 pages in order. The foundations come first.

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Boussinesq approximation for buoyancy, concept, no evidence levelBoussinesq approximat…conceptBrunt–Väisälä frequency, concept, no evidence levelBrunt–Väisälä frequen…conceptGeostrophic balance, concept, no evidence levelGeostrophic balanceconceptRossby number, concept, no evidence levelRossby numberconceptStream function, concept, no evidence levelStream functionconceptQuasi-geostrophic equations, definition, no evidence levelQuasi-geostrophic equ…definition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
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  1. Boussinesq approximation for buoyancy concept
  2. Brunt–Väisälä frequency concept
  3. Geostrophic balance concept
  4. Rossby number concept
  5. Stream function concept
  6. Quasi-geostrophic equations definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Quasi-geostrophic_equations
About this page
TagXQZ69H4PZ9
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WritingW0 stub
MSC76U60, 35Q86, 86A10
DefinesQuasi-geostrophic equations
UsesGeostrophic balance, Rossby number, Brunt–Väisälä frequency, Boussinesq approximation for buoyancy, Stream function
Changed3 November 2026

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