eta-to-eta initialization

Let $\eta_{\textrm{d}}, \eta_{\textrm{s}}$ be the deep and shallow mass coordinates, respectively. We define Analytic test cases, as well as, e.g., initializing from data, specify an invertible map $\eta_{\textrm{s}} \mapsto z$. When we read $\eta$ values from a file in DA HOMME, these are $\eta_{\textrm{d}}$ values. We define the new mass coordinate such that $$\frac{\partial \pi_\textrm{s}}{\partial \eta_{\textrm{s}}} = \rho g_0 \textrm{ becomes } \frac{\partial \pi_\textrm{d}}{\partial \eta_{\textrm{d}}} = \left(\frac{a+z}{a}\right)^2\rho g_0 = \left(\frac{r}{a}\right)^2 \rho g_0 = \hat{r}^2 \rho g_0$$ and Wood and Staniforth find that height can be constructed as $$\begin{align*} r^3 = r_s^3 - \frac{3a^2}{g} \int_{\pi'_\textrm{d} = \pi_\textrm{d}}^{\pi'_\textrm{d} = \pi_0} \frac{R_d T}{p} \,\mathrm{d}\pi_\textrm{d}\\ r^3 = r_s^3 - \frac{3a^2}{g} \int_{\pi'_\textrm{s} = \pi_\textrm{s}}^{\pi'_\textrm{s} = \pi_0} \hat{r}^2 \frac{R_d T}{p} \,\mathrm{d}\pi_\textrm{s} \end{align*}$$ At the topmost boundary, $\pi_\textrm{d} = \pi_{\textrm{s}} = \pi_0$, though the integral that defines how $\pi_0$ is interpreted is different in each model! Since we are using an isobaric upper boundary condition $p = \pi_0$ at the top boundary in both models, this allows us to determine $z(\eta_{\textrm{d, top}}) = z(\eta_{\textrm{s, top}})$

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Initializing by translating $\eta_{\textrm{deep}}$ to $\eta_{\textrm{shallow}}$

For simplicity of prototyping I assume that $p_{\textrm{top}} = \pi_{\textrm{top}}$ (eventually this will be $p_{\textrm{top}} = \pi_{\textrm{top}}$) and that the initial data allows me to determine $z_{\textrm{top}}$ based on this pressure value. To initialize the atmosphere, first calculate $$\pi_{\textrm{deep, surf}} = \int_{z_\textrm{surf}}^{z_\textrm{top}} \hat{r}^2 \frac{p}{R_d T} \intd{z}$$. In practice, this integral proceeds from the top of the atmosphere. I do the integral in $z$ coordinates because I find it notationally easier to tell what's going on. That said, the integrals in Wood and Staniforth provide the analogous integral in $\eta_{\textrm{shallow}}$ coordinates (though height must be determined in any case). and then use that to calculate $\pi_{\textrm{deep}}(\eta_{\cdot,k}) = A(\eta_{\cdot, k}) \pi_0 + B(\eta_{\cdot, k}) \pi_{\textrm{deep, surf}}$, for $\eta_{\textrm{int}, k}, \eta_{\textrm{model}, k}$ Then perform a scan from the top of the atmosphere to get $z(\eta_{\cdot, k})$ that satisfy $\pi_{\textrm{deep}}(\eta_{\cdot,k}) = \int_{z(\eta_{\cdot, k})}^{z_{\textrm{top}}} \hat{r}^2 \frac{p}{R_d T} \intd{z}$ Then set $T_v, u, v, w,$ based on either

Notes:

• The preliminary integral to calculate the total mass in the column cannot be done at the same time as the scan to determine $z(\eta_{\cdot, k})$, because determining the new $\pi_{\textrm{deep}}(\eta_{\cdot,k})$ relies on knowledge of $\pi_{\textrm{deep, surf}}$
• Note that $\pi_{\textrm{deep, surf}}$ becomes ps_v in the code, and may be totally different from the physical pressure at the surface!