Coordinate system and notation

Oceananigans.jl is formulated in a Cartesian coordinate system $\boldsymbol{x} = (x, y, z)$ with unit vectors $\boldsymbol{\hat x}$, $\boldsymbol{\hat y}$, and $\boldsymbol{\hat z}$, where $\boldsymbol{\hat x}$ points east, $\boldsymbol{\hat y}$ points north, and $\boldsymbol{\hat z}$ points 'upward', opposite the direction of gravitational acceleration.

We denote time with $t$, partial derivatives with respect to time $t$ or a coordinate $x$ with $\partial_t$ or $\partial_x$, and denote the gradient operator $\boldsymbol{\nabla} \equiv \partial_x \boldsymbol{\hat x} + \partial_y \boldsymbol{\hat y} + \partial_z \boldsymbol{\hat z}$. Horizontal gradients are denoted with $\boldsymbol{\nabla}_h \equiv \partial_x \boldsymbol{\hat x} + \partial_y \boldsymbol{\hat y}$.

We use $u$, $v$, and $w$ to denote the east, north, and vertical velocity components, such that $\boldsymbol{v} = u \boldsymbol{\hat x} + v \boldsymbol{\hat y} + w \boldsymbol{\hat z}$. We reserve $\boldsymbol{v}$ for the three-dimensional velocity field and use $\boldsymbol{u}$ to denote the horizontal components of flow, i.e., $\boldsymbol{u} = u \boldsymbol{\hat x} + v \boldsymbol{\hat y}$.