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# Primitive equations

## Definitions

• u is the zonal velocity (velocity in the east/west direction tangent to the sphere).
• v is the meridional velocity (velocity in the north/south direction tangent to the sphere).
• ω is the vertical velocity
• T is the temperature
• φ is the geopotential
• f is the term corresponding to the Coriolis force, and is equal to sin(φ), where Ω is the angular rotation rate of the Earth (2π / 24 radians/hour), and φ is the latitude.
• R is the gas constant
• p is the pressure
• cp is the specific heat
• J is the heat flow per unit time per unit mass
• π is the exner function
• θ is the potential temperature

## Forms of the primitive equations

The precise form of the primitive equations depends on the vertical coordinate system chosen, such as pressure coordinates, log pressure coordinates, or sigma coordinates. Furthermore, the velocity, temperature, and geopotential variables may be decomposed into mean and perturbation components using Reynolds decomposition.

### Vertical pressure, cartesian tangential plane

In this form pressure is selected as the vertical coordinate and the horizontal coordinates are written for the cartesian tangential plane (i.e. a plane tangent to some point on the surface of the Earth). This form does not take the curvature of the Earth into account, but is useful for visualizing some of the physical processes involved in formulating the equations due to its relative simplicity.

Note that the capital derivatives are the material derivatives.

• the geostrophic momentum equations
$\frac{Du}{Dt} - f v = -\frac{\partial \phi}{\partial x}$
$\frac{Dv}{Dt} + f u = -\frac{\partial \phi}{\partial y}$
• the hydrostatic equation, a special case of the vertical momentum equation in which there is no background vertical acceleration and with substitution of the Ideal Gas Law (Equation of State).
$0 = -\frac{\partial \phi}{\partial p} - \frac{R T}{p}$
$\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} + \frac{\partial \omega}{\partial p} = 0$
$\frac{\partial T}{\partial t} + u \frac{\partial T}{\partial x} + v \frac{\partial T}{\partial y} + \omega \left( \frac{\partial T}{\partial p} + \frac{R T}{p c_p} \right) = \frac{J}{c_p}$

When a statement of the conservation of water vapor substance is included, these six equations form the basis for any numerical weather prediction scheme.

### Primitive equations using sigma coordinate system, polar stereographic projection

• According to the National Weather Service Handbook No. 1 - Facsimile Products, the primitive equations can be simplified into the following equations:
• Temperature: ∂T/∂t = u (∂Tx/∂X) + v (∂Ty/∂Y) + w (∂Tz/∂Z)
• Wind in E-W direction: ∂u/∂t = ηv - ∂Φ/∂x – Cp θ (∂π/∂x) – z (∂u/∂σ) – [∂(u2 + y) / 2] / ∂x
• Wind in N-S direction: ∂v/∂t = -η(u/v) - ∂Φ/∂y – Cp θ (∂π/∂y) – z (∂v/∂σ) – [∂(u2 + y) / 2] / ∂y
• Precipitable water: ∂W/∂t = u (∂Wx/∂X) + v (∂Wy/∂Y) + z (∂Wz/∂Z)
• Pressure Thickness: ∂(∂p/∂σ)/∂t = u [(∂p/∂σ)x /∂X] + v [(∂p/∂σ)y /∂Y] + z [(∂p/∂σ)z /∂Z]
• These simplifications make it much easier to understand what is happening in the model. Things like the temperature (potential temperature), precipitable water, and to an extent the pressure thickness simply move from one spot on the grid to another with the wind. The wind is forecasted slightly differently. It uses geopotential, specific heat, the exner function π, and change in sigma coordinate.

## Solution to the primitive equations

The analytic solution to the primitive equations involves a sinusoidal oscillation in time and longitude, modulated by coefficients related to height and latitude.

$\begin{Bmatrix}u, v, \phi \end{Bmatrix} = \begin{Bmatrix}\hat u, \hat v, \hat \phi \end{Bmatrix} e^{i(s \lambda + \sigma t)}$

s and σ are the zonal wavenumber and angular frequency, respectively. The solution represents atmospheric waves and tides.

When the coefficients are separated into their height and latitude components, the height dependence takes the form of propagating or evanescent waves (depending on conditions), while the latitude dependence is given by the Hough functions.

This analytic solution is only possible when the primitive equations are linearized and simplified. Unfortunately many of these simplifications (i.e. no dissipation, isothermal atmosphere) do not correspond to conditions in the actual atmosphere. As a result, a numerical solution which takes these factors into account is often calculated using general circulation models and climate models.

### How the wind vectors, u and v, are calculated

• Normally, wind consists of two components: speed and direction
• In the model, wind still consists the speed in N-S direction and speed in E-W direction
• Speed in E-W direction is the variable u.
• Speed in N-S direction is the variable v.
• Wind the E-W direction is:
• u = W (cos d)
• Wind in the N-S direction is:
• v = W (sin d)
• Where W is the wind speed and d is the direction in degrees
• It should be noted that although the wind in meteorology is measured in degrees from the north (0 is north, 90 is east, 180 is south, 270 is west), in trigonometry degrees are measured starting at east at 0, north is 90, west is 180, south is 270. Therefore the wind direction must be converted to this standard to be calculated on a computer or calulator.
• This can be corrected by using the equation -θ + 90. If the resulting number is below 0, add 360. If the resulting number is above 360, subtract 360. You should get a number between 0 and 360.