# Notation for equations¶

## Symbols for variables¶

$$z$$ refers to elevation (or height), increasing upward so that for much of the ocean $$z$$ is negative.

$$x$$ and $$y$$ are the Cartesian horizontal coordinates.

$$\lambda$$ and $$\phi$$ are the geographic coordinates on a sphere (longitude and latitude respectively).

Horizontal components of velocity are indicated by $$u$$ and $$v$$ and vertical component by $$w$$.

$$p$$ is pressure and $$\Phi$$ is geo-potential:

$\Phi = g z .$

The thermodynamic state variables are usually salinity, $$S$$, and potential temperature, $$\theta$$ or the absolute salinity and conservative temperature, depending on the equation of state. $$\rho$$ is in-situ density.

## Vector notation¶

The three-dimensional velocity vector is denoted $$\boldsymbol{v}$$

$\boldsymbol{v} = \boldsymbol{u} + \widehat{\boldsymbol{k}} w ,$

where $$\widehat{\boldsymbol{k}}$$ is the unit vector pointed in the upward vertical direction and $$\boldsymbol{u} = (u, v, 0)$$ is the horizontal component of velocity normal to the vertical.

The gradient operator without a suffix is three dimensional:

$\boldsymbol{\nabla} = ( \boldsymbol{\nabla}_z, \partial_z ) .$

but a suffix indicates a lateral gradient along a surface of constant property indicated by the suffix:

$\boldsymbol{\nabla}_z = \left( \left. \partial_x \right|_z, \left. \partial_y \right|_z, 0 \right) .$