Plan March 2023

Current plan for Deep Atmosphere HOMME:

  • By next week:

    • Verify derivation in MT overleaf document myself (Use python notebook with toy coefficients/profiles to ensure that there aren't typos)
    • Verify lines in prim_advance_mod.F90 which correspond to each of these.
    • Optional: Identify lines in EOS.f90 which need to be modified, and list of variables that might accidentally assume EOS elsewhere in code
    • No actual modifications

  • Probably can be done by next month?

  • Staniforth/white:

    • Need to separate out centrifugal force term in both HPE and 3d Euler.
    • But: I'm already modifying the equations of state. Can just add a correction term to prim_advance_mod.F90

Breadcrumb: How is hydrostatic pseudodensity calculated?

There's something funky in the calculation of π(s)\pi(s). Do the definitions in MT's document account for spatial variation in gg?

I'm gonna have to look really foolish because theres a simplified form for gg that you can derive. How does that work? Newton's law of gravitation (assuming a sufficiently nice earth) gives us Fm1,m2=Gm1m2r2F_{m_1,m_2} = G\frac{m_1m_2}{r^2} Therefore the acceleration for m1 m_1 is g=Gm2(R0+z)−2=Gm2R02⋅(R0R0+z)2=g0(R0R0+z)2 \begin{align*} g = Gm_2 (R_0 + z)^{-2} &= \frac{Gm_2}{R_0^2} \cdot \left(\frac{R_0}{R_0+z}\right)^{2} \\ &= g_0 \left(\frac{R_0}{R_0+z} \right)^2 \end{align*} and so defining r^=R0+zR0 \hat{r} = \frac{R_0 + z}{R_0} we find g=g0r^−2. g = g_0 \hat{r}^{-2}.

Hydrostatic pressure in shallow atmosphere

We define πs(z)=ptop+∫zztopρg dz=(1)ptop∫zztopρg0 dz=ptop−∫zztop dp=ptop−(ptop−π(z)) \begin{align*} \pi_s(z) &= p_{\tm{top}} + \int_{z}^{z_{\tm{top}}} \rho g \intd{z} \\ &\stackrel{(1)}{=}p_{\tm{top}} \int_{z}^{z_\tm{top}} \rho g_0 \intd{z} \\ &= p_{\tm{top}} - \int_{z}^{z_\tm{top}} \intd{p}\\ &= p_{\tm{top}} - (p_{\tm{top}} - \pi(z))\\ \end{align*} where (1) holds only under the shallow atmosphere assumption. Done this a hundred times.

Hydrostatic pressure in deep atmosphere

πd(z)=ptop+∫zztopρg dz=ptop+∫zztopρg0r^−2 dz=ptop+g0R02∫zztopρ(R0+z)2 dz \begin{align*} \pi_d(z) &= p_{\tm{top}} + \int_{z}^{z_{\tm{top}}} \rho g \intd{z} \\ &= p_{\tm{top}} + \int_{z}^{z_{\tm{top}}} \rho g_0\hat{r}^{-2} \intd{z} \\ &= p_{\tm{top}} + \frac{g_0}{R_0^2}\int_{z}^{z_{\tm{top}}} \rho (R_0+z)^2 \intd{z} \\ \end{align*} So it turns out that this hydrostatic integral (and the definition of arbitrary vertical quantities) incorporate gravitational variation. But gg in the MT document is g0g_0 here.

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