\( \newcommand\D{\mathrm{d}} \newcommand\E{\mathrm{e}} \newcommand\I{\mathrm{i}} \newcommand\bigOh{\mathcal{O}} \newcommand{\cat}[1]{\mathbf{#1}} \newcommand\curl{\vec{\nabla}\times} \newcommand{\CC}{\mathbb{C}} \newcommand{\NN}{\mathbb{N}} \newcommand{\QQ}{\mathbb{Q}} \newcommand{\RR}{\mathbb{R}} \newcommand{\ZZ}{\mathbb{Z}} \)
UP | HOME

General Circulation Model

Table of Contents

1. Overview

Broadly speaking, "the" general circulation model for the atmosphere is "just" a "shallow" Navier-Stokes fluid around a rotating sphere, relative to an observe on the sphere.

Typically the viscousity is neglected (since its contribution is on the order of \(10^{-7}\) compared to dominant effects), so we could extend the notion of a general circulation model to include Euler fluids around a rotating sphere, relative to an observer on the spehere. At least, John Holton describes the negligibility of viscosity in his book An Introduction to Dynamic Meteorology; it's not the defining quality of the atmosphere, anyways.

Most general circulation models work in spherical coordinates, so extra care must be taken. For example, the vector Laplacian is defined by the identity \(\nabla^{2}\mathbf{v} = \vec{\nabla}(\nabla\cdot\mathbf{v})-\nabla\times(\nabla\times\mathbf{v})\) and not just the sum of squares of partial derivatives.

The notation for spherical coordinates in Earth Sciences appears to be — when expressed in radians! — the \(\lambda\in[-\pi,\pi)\) is used for longitude measuring the "azimuthal" angle East-West, latitude \(\phi\in[-\pi/2,\pi/2]\) measuring the "polar" angle North-South with the equator being located at \(\phi_{\text{eq}}=0\). This needs some correction to account for the Earth being a little "short and fat" rather than a perfect sphere, but it's a good approximation.

We change coordinates to spherical coordinates, taking \(\lambda\) to describe the longitude (Eastwards angle), \(\phi\) the latitude (angular distance polewards), and \(r\) the radial distance from the center of the Earth. In these coordinates, any scalar quantity \(Q\) has its material derivative be

\begin{equation} \frac{\mathrm{D}Q}{\mathrm{D} t} = \frac{\partial Q}{\partial t} + \frac{u}{r\cos(\phi)}\frac{\partial Q}{\partial\lambda} + \frac{v}{r}\frac{\partial Q}{\partial\phi} + w\frac{\partial Q}{\partial r}. \end{equation}

Mass conservation in spherical coordinates is

\begin{equation} \frac{\partial\rho}{\partial t} + \frac{1}{r\cos(\phi)}\frac{\partial(u\rho)}{\partial\lambda} + \frac{1}{r\cos(\phi)}\frac{\partial}{\partial\phi}(v\rho\cos(\phi)) + \frac{1}{r^{2}}\frac{\partial}{\partial r}(r^{2}w\rho) = 0 \end{equation}

where we have \(\lambda\) be the latitude coordinate, \(\phi\) the longitude coordinate, \(r\) the vertical/radial coordinate; and \(\vec{u}=(u,v,w)\) is the velocity pointing North \(u\), East \(v\), and outward \(w\).

Momentum conservation in a rotating frame using spherical coordinates are (neglecting the viscosity term)

\begin{equation} \frac{\mathrm{D} u}{\mathrm{D} t} -\left(2\Omega + \frac{u}{r\cos(\phi)}\right)(v\sin(\phi)-w\cos(\phi)) = \frac{-1}{\rho r\cos(\phi)}\frac{\partial p}{\partial\lambda} \end{equation} \begin{equation} \frac{\mathrm{D} v}{\mathrm{D} t} -\left(2\Omega + \frac{u}{r\cos(\phi)}\right)u\sin(\phi) = \frac{-1}{\rho r}\frac{\partial p}{\partial\phi} \end{equation} \begin{equation} \frac{\mathrm{D} w}{\mathrm{D} t} - \frac{u^{2}+v^{2}}{r} - 2\Omega u\cos(\phi) = \frac{-1}{\rho}\frac{\partial p}{\partial r}-g. \end{equation}

Here \(g\approx 9.8 \mathrm{m}/\mathrm{s}^{2}\) is the Earth's gravitational force (treated as a constant). The quadratic terms on the left-hand side involving factors of \(1/r\) are usually called metric terms, and those involving factors of \(\Omega\) are Coriolis terms. Empirically, the Earth's rotation is approximately

\begin{equation} \Omega=7.2921150\times10^{-5}\,\mathrm{rad}\cdot\mathrm{s}^{-1}. \end{equation}

The "full blown" GCM equations are first-order quasilinear partial differential equations. They are under-determined, since we have now 4 equations for 5 unknowns (the 3 velocity components, the pressure, and the density). We need one more equation, and some initial conditions, to have a unique solution.

(1) What extra equation can we add to this system?

(2) Can we solve these equations exactly?

(3) If we pretend the material derivative \(\mathrm{D}/\mathrm{D} t\) are just partial derivatives with respect to time \(\partial/\partial t\), then what are the solutions to that system of PDEs?

Presumably one family of boundary conditions we could impose would use the fact that at "the top" of the atmosphere, we expect:

  • the Solar constant or effective temperature of the Earth would be used if we use the ideal gas Law,
  • the density of the atmosphere \(\rho(a+z_{\text{max}})=0\) where \(a\) is the Earth's radius and \(z_{\text{max}}\) is the maximum altitude (the "top" of the atmosphere),
  • and so on.

2. Primitive Equations

There are three simplifications to the equations, the first two of them come as a package-deal (i.e., must be taken together or not at all):

  1. Shallow fluid: \(r=a+z\) where \(a\) is the radius of the Earth and \(z\) is the altitude. In this approximation, \(z\ll a\), so \(1/(a+z)= a^{-1}/(1+z/a)=a^{-1} - a^{-2}z+\dots\approx a^{-1}\). Partial derivatives with respect to \(r\) are then taken with respect to \(z\);
  2. Hydrostatic approximation: the radial component of the Navier-Stokes equation may be approximated as \(\partial_{z}p=\rho g\); and
  3. Traditional approximation: we can discard the Coriolis terms in the horizontal equations of motion, as well as terms \(uw/r\) and \(vw/r\) since they are negligibly small.

Taken altogether, the Navier-Stokes equations — with \(u\) being the atmospheric fluid's velocity North-wards, \(v\) is the East-wards component, and \(w\) is the altitude "outward" component — simplifies to:

\begin{equation} \frac{\mathrm{D} u}{\mathrm{D} t} - 2\Omega v\sin(\phi) - \frac{uv}{a}\tan(\phi) = \frac{-1}{\rho a\cos(\phi)}\frac{\partial p}{\partial\lambda} \end{equation} \begin{equation} \frac{\mathrm{D} v}{\mathrm{D} t} - 2\Omega u\sin(\phi) - \frac{u^{2}\tan(\phi)}{a} = \frac{-1}{\rho a}\frac{\partial p}{\partial\phi} \end{equation} \begin{equation} 0 = \frac{-1}{\rho}\frac{\partial p}{\partial z}-g. \end{equation}

In this approximation, the material derivative is

\begin{equation} \frac{\mathrm{D}}{\mathrm{D} t}=\frac{\partial}{\partial t} + \frac{u}{a\cos(\phi)}\frac{\partial}{\partial\lambda} + \frac{v}{a}\frac{\partial}{\partial\phi} + w\frac{\partial}{\partial z}. \end{equation}

We have 3 first-order partial differential equations in 5 unknowns (the 3 components of the atmospheric fluid velocity, the pressure, and the denity). We need 2 more equations and initial conditions to have a unique solution. At first, it may be tempting to use the ideal gas Law for an additional constraint — after all, the atmosphere is remarkably close to an ideal gas. But then we introduce another unknown, the temperature.

The ideal gas Law relevant for the atmosphere would be \(p=\rho RT\) where \(R = 287\; \mathrm{J}\cdot\mathrm{kg}^{-1}\cdot\mathrm{K}^{-1}\) is the gas constant for dry air, and \(T\) is the atmosphere's temperature (another unknown scalar function which we need to fix by means of another equation).

Is there an exact solution to the primitive equations? It seems like, if we work in the Lagrange description, there should be a unique solution we could find fairly easily?

The mass conservation equation for a shallow atmosphere becomes

\begin{equation} \frac{\partial\rho}{\partial t} + \frac{u}{a\cos(\phi)}\frac{\partial\rho}{\partial\lambda} + \frac{v}{a}\frac{\partial\rho}{\partial\phi} + w\frac{\partial\rho}{\partial z} + \rho\left(\frac{1}{a\cos(\phi)}\frac{\partial u}{\partial\lambda} + \frac{1}{a\cos(\phi)}\frac{\partial(v\cos(\phi))}{\partial\phi} + \frac{\partial w}{\partial z} \right) = 0 \end{equation}

or equivalently

\begin{equation} \frac{\partial\rho}{\partial t} + \frac{1}{a\cos(\phi)}\frac{\partial(u\rho)}{\partial\lambda} + \frac{1}{a\cos(\phi)}\frac{\partial(v\rho\cos(\phi))}{\partial\phi} + \frac{\partial(w\rho)}{\partial z} = 0. \end{equation}

What is the rigorous "limiting process" relating the solutions of the full GCM equations to the solutions to the primitive equations? I sense there is some difficult discontinuity here, but it is not at all "immediately obvious" at the moment…

3. References

  • James R. Holton and Gregory J. Hakim,
    An Introduction to Dynamic Meteorology.
    Academic Press, 5th ed., 2012.
  • G. K. Vallis,
    Atmospheric and Oceanic Fluid Dynamics: Fundamentals and Large-Scale Circulation.
    Cambridge University Press, 2 ed., 2017.

Last Updated 2022-04-12 Tue 07:36.