Primitive Equations

Overview

Primitive equations refer to the set of complete mathematical equations that describe the heat, salt, mass, and momentum at any given location in the ocean. They are formulated using conservation laws for energy, momentum, and mass. Further, they are linked with the equation of state. Here, the equations are collected in one place to provide a concise overview. The next three subsections for Momentum, Thermodynamics, and Equation of State describe the terms of these equations with some details about how they are derived.

The Equations

Under Construction.

\[\begin{split}\frac{D\textbf{v}}{Dt} &= - \frac{1}{\rho}\nabla p - 2\boldsymbol\Omega\times\textbf{v} + \textbf{g} + \textbf{F}_r\\ \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} + \frac{\partial w}{\partial z} &= 0\\ \frac{\partial S}{\partial t} + \textbf{u} \cdot \nabla S &= J_S\\ \frac{\partial \Theta}{\partial t} + \textbf{u} \cdot \nabla \Theta &= \frac{J_H}{\rho C_p}\\ \rho &= f(T,S,p)\end{split}\]

Notation

Under Construction.

Here we define the symbology used in the above equations and subsequent sections:

Symbol	Variable	Typical Units
x, y, z	Cooridinates for zonal, meridional, and veritical directions	meters
u, v, w	Fluid velocity in the zonal, meridional, and vertical directions	meters per second
\(\rho\)	Density	kg/m\(^3\)
g	Gravitational Acceleration	m/s\(^3\)
\(\Theta\)	Potential Temperature	\(^{\circ}\)C
S	Practical Salinity	psu

The Total Derivative

Under Construction.