Primitive equations

Definition

Consider a thin layer of fluid on a rotating planet, in local coordinates $x$, $y$ (horizontal) and $z$ (up). Let $f$ be the Coriolis parameter and $g$ the gravitational acceleration. The unknowns are the horizontal velocity $u_h = (u, v)$, the vertical velocity $w$, the pressure $p$, the density $\rho$ and a thermodynamic variable $T$. The primitive equations are the equations of geophysical fluid dynamics with the vertical momentum equation replaced by hydrostatic balance. In the Boussinesq form, with reference density $\rho_0$, the horizontal momentum equation is $$\frac{\partial u_h}{\partial t} + (u_h \cdot \nabla_h) u_h + w \frac{\partial u_h}{\partial z} + f e_z \times u_h = -\frac{1}{\rho_0} \nabla_h p + \mathcal{F}.$$ The hydrostatic balance and the conservation of mass are $$\frac{\partial p}{\partial z} = -\rho g, \qquad \nabla_h \cdot u_h + \frac{\partial w}{\partial z} = 0.$$ The thermodynamic equation is $$\frac{\partial T}{\partial t} + u_h \cdot \nabla_h T + w \frac{\partial T}{\partial z} = Q.$$ An equation of state $\rho = \rho(T)$ closes the system. In the ocean $\rho$ depends also on salinity, which obeys an equation like that of $T$. Here $\nabla_h = (\partial_x, \partial_y)$, $e_z$ is the upward unit vector, and $\mathcal{F}$ and $Q$ are the friction and heating terms. A typical friction term is $\mathcal{F} = \nu_h \Delta_h u_h + \nu_v \partial_z^2 u_h$ with viscosities $\nu_h, \nu_v \ge 0$.