Coordinate system and notation

Oceananigans.jl is formulated in a Cartesian coordinate system $\mathit{x}=(x,y,z)$ with unit vectors $\hat{\mathit{x}}$, $\hat{\mathit{y}}$, and $\hat{\mathit{z}}$, where $\hat{\mathit{x}}$ points east, $\hat{\mathit{y}}$ points north, and $\hat{\mathit{z}}$ 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 $\mathbf{\nabla }\equiv {\partial }_x\hat{\mathit{x}}+{\partial }_y\hat{\mathit{y}}+{\partial }_z\hat{\mathit{z}}$. Horizontal gradients are denoted with ${\mathbf{\nabla }}_h\equiv {\partial }_x\hat{\mathit{x}}+{\partial }_y\hat{\mathit{y}}$.

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