hydro_prim_eqn.rst

Hydrostatic Primitive Equations for the Atmosphere in Pressure Coordinates

The hydrostatic primitive equations (HPE’s) in p−coordinates are:

$$\frac{D\vec{\mathbf{v}}_{h}}{Dt}+f\hat{\mathbf{k}}\times \vec{\mathbf{v}}_{h}+\mathbf{\nabla }_{p}\phi = \vec{\mathbf{\mathcal{F}}}$$

$$\frac{\partial \phi }{\partial p}+\alpha = 0$$

$$\mathbf{\nabla }_{p}\cdot \vec{\mathbf{v}}_{h}+\frac{\partial \omega }{\partial p} = 0$$

pα = RT

$$c_{v}\frac{DT}{Dt}+p\frac{D\alpha }{Dt} = \mathcal{Q}$$

where $\vec{\mathbf{v}}_{h}=(u,v,0)$ is the ‘horizontal’ (on pressure surfaces) component of velocity, $\frac{D}{Dt}=\frac{\partial}{\partial t}+\vec{\mathbf{v}}_{h}\cdot \mathbf{\nabla }_{p}+\omega \frac{\partial }{\partial p}$ is the total derivative, f = 2Ωsin φ is the Coriolis parameter, ϕ = gz is the geopotential, α = 1/ρ is the specific volume, $\omega =\frac{Dp }{Dt}$ is the vertical velocity in the p−coordinate. Equation atmos-heat is the first law of thermodynamics where internal energy e = c_vT, T is temperature, Q is the rate of heating per unit mass and $p\frac{D\alpha }{Dt}$ is the work done by the fluid in compressing.

It is convenient to cast the heat equation in terms of potential temperature θ so that it looks more like a generic conservation law. Differentiating atmos-eos we get:

$$p\frac{D\alpha }{Dt}+\alpha \frac{Dp}{Dt}=R\frac{DT}{Dt}$$

which, when added to the heat equation atmos-heat and using c_p = c_v + R, gives:

$$c_{p}\frac{DT}{Dt}-\alpha \frac{Dp}{Dt}=\mathcal{Q}$$

Potential temperature is defined:

$$\theta =T(\frac{p_{c}}{p})^{\kappa }$$

where p_c is a reference pressure and κ = R/c_p. For convenience we will make use of the Exner function Π(p) which is defined by:

$$\Pi (p)=c_{p}(\frac{p}{p_{c}})^{\kappa }$$

The following relations will be useful and are easily expressed in terms of the Exner function:

$$c_{p}T=\Pi \theta \;\;;\;\;\frac{\partial \Pi }{\partial p}=\frac{\kappa \Pi }{p}\;\;;\;\;\alpha =\frac{\kappa \Pi \theta }{p}=\frac{\partial \ \Pi }{ \partial p}\theta \;\;;\;\;\frac{D\Pi }{Dt}=\frac{\partial \Pi }{\partial p} \frac{Dp}{Dt}$$

where $b=\frac{\partial \ \Pi }{\partial p}\theta$ is the buoyancy.

The heat equation is obtained by noting that

$$c_{p}\frac{DT}{Dt}=\frac{D(\Pi \theta )}{Dt}=\Pi \frac{D\theta }{Dt}+\theta \frac{D\Pi }{Dt}=\Pi \frac{D\theta }{Dt}+\alpha \frac{Dp}{Dt}$$

and on substituting into eq-p-heat-interim gives:

$$\Pi \frac{D\theta }{Dt}=\mathcal{Q}$$

which is in conservative form.

For convenience in the model we prefer to step forward potential-temperature-equation rather than atmos-heat.

Boundary conditions

The upper and lower boundary conditions are:

$$\begin{aligned}\mbox{at the top:}\;\;p=0 &\text{, }\omega =\frac{Dp}{Dt}=0\end{aligned}$$

$$\begin{aligned}\mbox{at the surface:}\;\;p=p_{s} &\text{, }\phi =\phi _{topo}=g~Z_{topo}\end{aligned}$$

In p−coordinates, the upper boundary acts like a solid boundary (ω = 0 ); in z−coordinates the lower boundary is analogous to a free surface (ϕ is imposed and ω ≠ 0).

Splitting the geopotential

For the purposes of initialization and reducing round-off errors, the model deals with perturbations from reference (or ‘standard’) profiles. For example, the hydrostatic geopotential associated with the resting atmosphere is not dynamically relevant and can therefore be subtracted from the equations. The equations written in terms of perturbations are obtained by substituting the following definitions into the previous model equations:

θ = θ_o + θ^′

α = α_o + α^′

ϕ = ϕ_o + ϕ^′

The reference state (indicated by subscript ‘o’) corresponds to horizontally homogeneous atmosphere at rest (θ_o, α_o, ϕ_o) with surface pressure p_o(x, y) that satisfies ϕ_o(p_o) = g Z_topo, defined:

$$\begin{aligned} \theta _{o}(p) = f^{n}(p) \\\ \end{aligned}$$

$$\begin{aligned} \alpha _{o}(p) = \Pi _{p}\theta _{o} \\\ \end{aligned}$$

ϕ_o(p) = ϕ_topo − ∫_p0^pα_odp

The final form of the HPE’s in p−coordinates is then:

$$\frac{D\vec{\mathbf{v}}_{h}}{Dt}+f\hat{\mathbf{k}}\times \vec{\mathbf{v}} _{h}+\mathbf{\nabla }_{p}\phi ^{\prime } = \vec{\mathbf{\mathcal{F}}}$$

$$\frac{\partial \phi ^{\prime }}{\partial p}+\alpha ^{\prime } = 0$$

$$\mathbf{\nabla }_{p}\cdot \vec{\mathbf{v}}_{h}+\frac{\partial \omega }{ \partial p} = 0$$

$$\frac{\partial \Pi }{\partial p}\theta ^{\prime } = \alpha ^{\prime }$$

$$\frac{D\theta }{Dt} = \frac{\mathcal{Q}}{\Pi }$$