Primitive Equations

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Earth sciences · Topic

Primitive Equations

partial differential equations · differential geometry · dynamical systems · variational calculus

The primitive equations are a simplified form of the Navier-Stokes equations applied to a thin rotating spherical shell of atmosphere. They invoke the hydrostatic approximation and treat the Coriolis effect explicitly, yielding a tractable yet accurate system for large-scale flow. Nearly every operational numerical weather prediction model solves some variant of these equations.

Momentum Equations on a Rotating Sphere

In pressure coordinates $(x, y, p, t)$ the horizontal momentum equations are:

\[\frac{Du}{Dt} - fv = -\frac{\partial \Phi}{\partial x}\] \[\frac{Dv}{Dt} + fu = -\frac{\partial \Phi}{\partial y}\]

where $f = 2\Omega\sin\phi$ is the Coriolis parameter, $\Omega = 7.292 \times 10^{-5}\ \text{rad s}^{-1}$ is Earth’s rotation rate, $\phi$ is latitude, and $\Phi = gz$ is geopotential.

Hydrostatic and Thermodynamic Equations

The hydrostatic approximation replaces the vertical momentum equation:

\[\frac{\partial \Phi}{\partial p} = -\frac{RT}{p} = -\alpha\]

The thermodynamic energy equation in pressure coordinates is:

\[\frac{DT}{Dt} - \frac{\omega \alpha}{c_p} = \frac{Q}{c_p}\]

where $\omega = Dp/Dt$ is vertical pressure velocity and $Q$ is diabatic heating.

Continuity and Closure

Mass continuity in pressure coordinates takes the elegant divergence form:

\[\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} + \frac{\partial \omega}{\partial p} = 0\]

Together with the ideal gas law $p = \rho RT$ and a moisture equation for specific humidity $q$, this closes the system. The $\beta$-plane approximation linearizes the Coriolis parameter as $f \approx f_0 + \beta y$ with $\beta = \partial f/\partial y$, enabling analytical wave solutions that illuminate large-scale dynamics before full numerical integration is applied.