QHE-SE shallow

The idea here is to try the equations given in Tort and Dubos adapted for CAM-SE's hybrid η\eta coordinate.

The equations given in that paper read dudt−[2Ω(1+2za)+ucos⁡ϕ]vsin⁡ϕ+2Ωwcos⁡ϕ+1ρacos⁡ϕ∂p∂λ=0dvdt+[2Ω(1+2za)+uacos⁡ϕ]usin⁡ϕ+1ρa∂p∂ϕ=0−2Ωcos⁡ϕ+g+1ρ∂p∂z=0 \begin{align*} \der{u}{t} - \left[2\Omega \left(1 + \frac{2z}{a} \right) + \frac{u}{\cos\phi} \right]v\sin \phi + 2 \Omega w \cos \phi + \frac{1}{\rho a \cos \phi} \pder{p}{\lambda} &= 0 \\ \der{v}{t} + \left[2 \Omega \left( 1 + \frac{2z}{a}\right) + \frac{u}{a\cos\phi} \right] u \sin\phi + \frac{1}{\rho a} \pder{p}{\phi} &= 0 \\ -2\Omega \cos\phi + g + \frac{1}{\rho} \pder{p}{z} &= 0 \end{align*} where the factor of two comes from an asymptotic expansion of the Lagrangian used to derive these equations. Theoretically, the derivation in Tort and Dubos 2014 could be used to prove the desired continuum conservation properties, but the notational differences would make this a substantial project. If this becomes publishable it would be worth doing this derivation.

For now we adapt the White and Bromley form of the quasi-hydrostatic equation The original W&B equation reads rs(p)=a+∫ppsurfRdTs(p)gp′ dp′∂∂p[gz]=Rdp(Tv+2Ωurscos⁡ϕ+u2+v2rsgTs). \begin{align*} r_s(p) &= a + \int_p^{p_\textrm{surf}} \frac{R_d T_s(p)}{gp'} \intd{p'}\\ \pder{}{p} \left[ gz\right] &= \frac{R_d}{p} \left(T_v + \frac{2 \Omega u r_s \cos \phi + u^2 + v^2}{r_s g}T_s \right). \end{align*} The resulting equations are dudt−[2Ω(1+2za)+ucos⁡ϕ]vsin⁡ϕ−2Ωωρgcos⁡ϕ+1ρacos⁡ϕ∂p∂λ=0dvdt+[2Ω(1+2za)+uacos⁡ϕ]usin⁡ϕ+1ρa∂p∂ϕ=0∂∂p[gz]+RdTvp(1+2Ωucos⁡ϕg)=0 \begin{align*} \der{u}{t} - \left[2\Omega \left(1 + \frac{2z}{a} \right) + \frac{u}{\cos\phi} \right]v\sin \phi - 2 \Omega \frac{\omega}{\rho g} \cos \phi + \frac{1}{\rho a \cos \phi} \pder{p}{\lambda} &= 0 \\ \der{v}{t} + \left[2 \Omega \left( 1 + \frac{2z}{a}\right) + \frac{u}{a\cos\phi} \right] u \sin\phi + \frac{1}{\rho a} \pder{p}{\phi} &= 0 \\ \pder{}{p} \left[ gz\right] + \frac{R_dT_v}{p} \left(1 + \frac{2 \Omega u \cos \phi}{g}\right) &= 0 \end{align*}

Hydrostatic balance in CAM-SE reads ∂Φ∂η=−RdTvp∂p∂η, \pder{\Phi}{\eta} = -\frac{R_d T_v }{p} \pder{p}{\eta}, discretized as Φk+12=Φsurf+Rd∑j=knlev[(Tv)jpjΔpj] \Phi_{k+\half} = \Phi_{\textrm{surf}} + R_d \sum_{j=k}^{\textrm{nlev}} \left[ \frac{ (T_v)_j}{p_j} \Delta p_j\right] which we adapt to Φk+12=Φsurf+Rd∑j=knlev[(Tv)jpj(1+2Ωucos⁡ϕ)Δpj] \Phi_{k+\half} = \Phi_{\textrm{surf}} + R_d \sum_{j=k}^{\textrm{nlev}} \left[ \frac{ (T_v)_j}{p_j} \left(1 + 2\Omega u\cos \phi \right) \Delta p_j\right] where the Lorenz staggering removes the need to decide on an averaging for uu

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