Dynamic Equations for Weather and Climate

“I could have done it in a much more complicated way”,
said the Red Queen, immensely proud. — Lewis Carroll.

Books on dynamic meteorology and oceanography usually have a full chapter devoted to the basic dynamical equations. Since the Earth’s fluid envelop is approximately a thin spherical shell, spherical coordinates ${(\lambda,\varphi, r)}$ are convenient. Here ${\lambda}$ is the longitude and ${\varphi}$ the latitude. In Figure 1 we show the momentum equations as presented in the monograph of Lorenz (1967):

Fig 1. The momentum equations, as in Lorenz (1967). The metric terms are boxed.

The left-hand sides are the eastward, northward and upward accelerations. The terms with ${\Omega}$ are the Coriolis terms, arising from the Earth’s rotation, and the final two terms are the pressure gradient term and friction. We are interested here in the first two terms on the right side of each equation (the terms in red boxes). Each of these involves a quadratic expression in the velocity components. But where do these mysterious terms come from?

Origin of the Metric Terms

We can represent the velocity of a “parcel” of air by a vector

$\displaystyle \mathbf{v} = u \mathbf{i} + v \mathbf{j} + w \mathbf{w}$

where ${u}$ , ${v}$ and ${w}$ are the components eastward, northward and upward and ${\{ \mathbf{i}, \mathbf{j}, \mathbf{k} \}}$ is a unit orthogonal triad. The acceleration, which is required to write down Newton’s equations, is the time derivative of ${\mathbf{v}}$ . But we must take account of the variation of the unit vectors ${\{ \mathbf{i}, \mathbf{j}, \mathbf{k} \}}$ from place to place as the parcel moves. So we get

$\displaystyle \mathbf{a } = \frac{\mathrm{d}\mathbf{v} } {\mathrm{d} t} = \left( \frac{\mathrm{d}u } {\mathrm{d} t} \mathbf{i} + \frac{\mathrm{d}v } {\mathrm{d} t} \mathbf{j} + \frac{\mathrm{d}w } {\mathrm{d} t} \mathbf{k} \right) + \left( u \frac{\mathrm{d}\mathbf{i} } {\mathrm{d} t} + v \frac{\mathrm{d}\mathbf{j} } {\mathrm{d} t} + w \frac{\mathrm{d}\mathbf{k} } {\mathrm{d} t} \right)$

The first group of three terms comprises the accelerations as they appear on the left sides of Lorenz’s equations. The second group comprises the metric terms, which are the boxed terms on the right side of the equations in Figure 1.

Textbooks, such as Holton (2004) give a full derivation of the metric terms. The treatment is purely kinematic. This approach is elementary but not entirely transparent. We consider below how the metric terms arise in a completely general context.

Acceleration in Terms of Tensors

The position vector in Cartesian coordinates with origin at the Earth’s centre is

$\displaystyle \mathbf{x} = x \mathbf{I} + y \mathbf{J} + z \mathbf{K}$

We consider a change of reference frame from the Cartesian coordinates to a completely general curvilinear system denoted ${\{q^i\}}$ :

$\displaystyle \mathbf{x} = \mathbf{x}(q^i) \,.$

We can define a set of basis vectors in the new frame

$\displaystyle \mathbf{g}_i := \frac{\partial\mathbf{x}}{\partial q^i}$

and also a reciprocal basis ${\mathbf{g}^j}$ such that ${\mathbf{g}^j\mathbf{\cdot}\mathbf{g}_i = \delta_i^j}$ . The velocity components in the new frame are ${v^i = \dot q^i}$ , so that ${\mathbf{v} = v^i \mathbf{g}_i}$ . As is customary in tensor analysis, we use the summation convention.

To compute the acceleration, we must take the time derivative of the velocity, noting that both the coefficients ${v^i}$ and the base vectors ${\mathbf{g}_i}$ vary:

$\displaystyle \mathbf{a} := \frac{\mathrm{d}\mathbf{v}}{\mathrm{d}t} = \dot v^i \mathbf{g}_i + v^i \mathbf{\dot g}_i \,.$

Since ${\mathbf{g}_i}$ varies with position, we can write ${\mathbf{\dot g}_i = q^j \mathbf{g}_{i, j}}$ where ${(\ )_{,j}}$ indicates a partial derivative. And ${\mathbf{g}_{i, j}}$ can be expanded in the base vectors as

$\displaystyle \mathbf{g}_{i, j} = \Gamma^k_{i j} \mathbf{g}_{k} \,.$

The quantities ${\Gamma^k_{i j}}$ are the Christoffel symbols. They are awkward expressions involving derivatives of the metric tensor, but they can easily be computed automatically. A table of these coefficients for the case of spherical coordinates, from Ehrendorfer (2012), is shown in Figure 2.

Fig 2. The Christoffel symbols for standard spherical coordinates (Ehrendorfer, 2012). The nine non-vanishing symbols are boxed.

With some relatively straightforward manipulation, we can now write the acceleration as

$\displaystyle \mathbf{a} = [ \dot v^k + v^i v^j \Gamma^k_{i j} ] \mathbf{g}_k \,,$

from which we can immediately read off the components of acceleration, ${a^k = \dot v^k + v^i v^j \Gamma^k_{i j}}$ . The terms ${\dot v^k}$ are those on the left side of Lorenz’s system. The terms ${v^i v^j \Gamma^k_{i j}}$ are the metric terms; each is a sum of nine separate terms, but many of these are zero as all but nine of the ${3^3 = 27}$ Christoffel symbols vanish. Using the values of the coefficients in Figure 2, we can confirm that the equations of Lorenz emerge.

With the expression for acceleration, we can now write Newton’s Second Law as

$\displaystyle f^k = m[ \dot v^k + v^i v^j \Gamma^k_{i j} ]$

For a particle moving freely (no applied forces), we get the equations for a geodesic:

$\displaystyle \frac{\mathrm{d}^2 q^k}{\mathrm{d}t^2} + \Gamma^k_{i j}\frac{\mathrm{d}q^i}{\mathrm{d}t}\frac{\mathrm{d}q^j}{\mathrm{d}t} = 0 \,.$

As a bonus, this is also formally identical to the equation for a (four-dimensional) world line of a particle in general relativity.

Sources

${\bullet}$ Lorenz, Edward N., 1967: The Nature and Theory of the General Circulation of the Atmosphere. Monograph of the World Meteorological Organization.

${\bullet}$ Holton, J. R., 2004: An Introduction to Dynamic Meteorology. 4th Edition, Elsevier Academic Press, xii+535~pp.

${\bullet}$ Ehrendorfer, Martin, 2012: Spectral Numerical Weather Prediction Models. Soc. Ind. Appl. Math. (SIAM), xxv+482 pp. ISBN: 978-1-61197-198-9

${\bullet}$ Simmonds, James G., 1994: A Brief on Tensor Analysis. Springer, ISBN: 978-0-3879-4088-5.

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