Notation $z$ 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. 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: \[\boldsymbol{\nabla}_z = \left( \left. \partial_x \right|_z, \left. \partial_y \right|_z, 0 \right) .\]