# 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:

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 \(\boldsymbol{v}\)

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:

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