Doing the hamiltonian myself

We assume that the derivation of the prognostic equations should be fine, aside from the definition of pseudodensity. We use Dubos and Tort TD14 to explain how to get a correct pseudodensity for our equation set in Lagrangian mass coordinates.

Useful stuff from TD14:

Pseudodensity in Lagrangian coordinates

We work continuously for the moment. Let $\hat{\mu}$ be the pseudodensity in Eulerian coordinates. The paragraph between Eqn (6) and Eqn (7) gives that $\hat{\mu} = \frac{r^2 \cos\phi}{\alpha} = \rho r^2 \cos\phi.$ Let $\eta$ be an arbitrary vertical coordinate. Let $\xi^3$ be the vertical Eulerian vertical coordinate. The aforementioned paragraph gives that for spherical geometry $\xi^3 \equiv r$.

The mass budget, which is quite closely related to the question we want to answer, is governed in Lagrangian coordinates by $$\begin{align*} 0 &= \partial_t \mu + \partial_l (\mu u^l) \\ \mu &= J\hat{\mu} \\ J &= \det{\partial_l \xi^k } \end{align*}$$

Eqn (24) in TD14 states that pseudodensity under an arbitrary vertical coordinate $\eta$ is $$\begin{align*} \mu &= \hat{\mu} \partial_\eta \xi^3 \\ &= \hat{\mu} \partial_\eta r, \end{align*}$$ which follows from the expression for $J$ and the fact that $\lambda, \phi$ are still Eulerian. Therefore the Jacobian is diagonal, with all entries equal to 1 except for the $\partial_\eta r$ term.

Now suppose that $\eta$ is specifically the mass-based eta coordinate used by HOMME. It is Lagrangian, i.e. $u^3 = \dot{\eta} = 0.$

Let $\eta \in [0, 1]$. This is defined in terms of $$\begin{align*} M(\xi^{1},\xi^2, t) &= \int_0^1 \mu(\xi^{1}, \xi^{2}, \eta) \,\mathrm{d}\eta\\ \mu &= \partial_\eta [A]M + \partial_\eta[B] \end{align*}$$

The rather peculiar dependence of $\mu$ on $M$ leads TD14 to claim that $M$ is the quantity which should be used to determine the prognostic equations. TD14 prognoses $M$. This should be equivalent to calculating it within levels, since we are prognosing $\int_{\eta_i}^{\eta_{i+1}} \mu \, \mathrm{d}\eta$?

TD14 makes the definition $\mathcal{H}'[v_i, r, \mu v_3, M, S] = \mathcal{H}[v_i, r, \mu v_3, A'M + B', S].$ Why is this a problem?}

Note: Chris's appendix in Tea20 attempts to derive the equations of motion using $$\begin{align*} \frac{\delta \mathcal{H}}{\delta \frac{\partial \pi}{\partial s}} \end{align*}$$ I suspect that the "energetically neutral transport term" of Tea20 $\mathcal{V} \cdot \dot{s}$ rectifies the difference. In any case,

Miscellany from appendix

$$\begin{align*} M(\xi^{1},\xi^2, t) &= \int_{\textrm{bottom}}^{\textrm{top}} \mu \, \mathrm{d}\eta = \int_{\textrm{bottom}}^{\textrm{top}} \hat{\mu} \, \mathrm{d} r \end{align*}$$

This uniquely defines the position of $r(\xi^1, \xi^2, \eta)$ according to $$\begin{align*} A(\eta) M(\xi^1, \xi^2) + B(\eta) = \int_{r_0}^{r(\eta)}\hat{\mu} \, \mathrm{d}r \end{align*}$$

Oksana's notes

For the purposes of disambiguation, I'll stick with calling pseudodensity $\mu$ and refer to the pressure adjustment factor ($\mu$ in Oksana's notes) as $\psi$.

A pseudodensity is simply a quantity which evolves according to a flux-form continuity equaiton. The simplest such density is a rewrite of the continuity equation $$\begin{align*} \rho_t + \nabla \cdot (\rho \mathbf{v}) = 0 \end{align*}$$ and the notes state that from this we could derive $(r_sr^2\rho)_\tau + \nabla_s (r^{-1} r_s r^2 \mathbf{u}) + (\dot{s}r_sr^2\rho)_s.$ The definition above $\mu = \hat{\mu} \partial_s r$ combined with $\hat{\mu} = r^2\rho$ gives $\mu = r_s r^2\rho.$

The trouble with defining $$\begin{align*} \partial_s p_h &= \partial_s [p_h] \partial_r [s] \\ &= -g\rho r_s \hat{r}^2 \end{align*}$$ is that $g = g_0 \hat{r}^{-2}.$ This implies that the actual pressure value per gridpoint in the deep atmosphere would be essentially the same, but it would represent a fundamentally different quantity.

Ok so the hypsometric equation is hidden in here (assume shallow atm for the moment): $$\begin{align*} &\partial_s \phi = -R_d \partial_s[\pi] \theta_v \frac{\Pi}{p}\\ \implies& \int \partial_s [\phi] \, \mathrm{d}s = -R_d \int \partial_s[\pi] \theta_v \frac{\Pi}{p} \, \mathrm{d}s\\ \implies& \Delta \phi = -R_d \int \partial_s[\pi] \theta_v \frac{\Pi}{p} \, \mathrm{d}s\\ \implies& \Delta \phi = -R_d \int \frac{T}{p} \partial_s[\pi]\, \mathrm{d}s\\ \implies& \Delta \phi = -R_d \int \frac{T}{p}\, \mathrm{d}\pi\\ \end{align*}$$ Assuming $T=T_0$ and $p=\pi,$ then we get the standard hydrostatic hypsometric equation.

Following Quasi-hamiltonian derivation in Tea20

The definition for the mass coordinate is $$\begin{align*} \pi \equiv A(\eta) p_0 + B(\eta) p_s \end{align*}$$

which allows us to back out that $$\begin{align*} \dot{\eta} \partial_\eta \pi = B(\eta) \int_{\eta_{\textrm{top}}}^{1} \nabla_{\eta}\cdot \partial_{\eta} [\pi ] \mathbf{u} \intd{\eta} - \int_{\eta_{\textrm{top}}}^{\eta} \nabla_\eta \cdot \partial_\eta [\pi] \mathbf{u} \intd{s} \end{align*}$$

Assuming that $\pi$ is defined with respect to the deep atmosphere, I see no problems with this so far.

The components to assemble the equation of state include $$\begin{align*} \partial_{\eta}[\pi] \partial_{z} [\eta] &= -\rho g\\ &= -\partial_{\eta} [\pi] \left(\partial_{\eta} [gz] \right)^{-1} \cdot g \end{align*}$$

Mass integrals:

Shallow:

$$\begin{align*} \int_{z_\textrm{surf}}^{z_{\textrm{top}}} \rho X \intd{z} &= \int_{z_\textrm{surf}}^{z_{\textrm{top}}} \partial_{\eta} [\pi] \left(\partial_{\eta} [g_0z] \right)^{-1} X \intd{z}\\ &= g_0^{-1} \int_{z_\textrm{surf}}^{z_{\textrm{top}}} \partial_{\eta} [\pi] \partial_{z} [\eta] X \intd{z}\\ &= g_0^{-1} \int_{\eta_\textrm{top}}^{\eta_{\textrm{surf}}} \partial_{\eta} [\pi] X \intd{\eta} \end{align*}$$ which demonstrates the shallow atmosphere identity from the paper.

Deep:

Let's do this for the deep atmosphere in a naive way: $$\begin{align*} \int_{z_\textrm{surf}}^{z_{\textrm{top}}} \rho X \intd{z} &= \int_{z_\textrm{surf}}^{z_{\textrm{top}}} \partial_{\eta} [\pi] \left(\partial_{\eta} \left[g_0\frac{zR_0^2}{(R_0+z)^2}\right] \right)^{-1} X \intd{z} \end{align*}$$ which is garbage. But can we get further if we don't expand geopotential (it's still monotonic in $\eta$ so we can use the inverse function theorem) $$\begin{align*} \int_{z_\textrm{surf}}^{z_{\textrm{top}}} \rho X \intd{z} &= \int_{z_\textrm{surf}}^{z_{\textrm{top}}} \partial_{\eta} [\pi] \partial_{\phi} \left[\eta \right] X \intd{z} \end{align*}$$ which invites the question whether the physical integral on the LHS can be restated in terms of geopotential. $$\begin{align*} \int_{z_{\textrm{bot}}}^{z_{\textrm{top}}} \rho X \intd{z} &= \int_{\phi_{\textrm{bot}}}^{\phi_{\textrm{top}}} \rho \left(\partial_{z} [\phi]\right)^{-1} X \intd{\phi} \\ &= \int_{\phi_{\textrm{bot}}}^{\phi_{\textrm{top}}} \partial_{\eta} [\pi] \partial_{\phi} [\eta] \left(\partial_{z} [\phi]\right)^{-1} X \intd{\phi} \\ &= \int_{\eta_{\textrm{top}}}^{\eta_{\textrm{bot}}} \partial_{\eta} [\pi] \left(\partial_{z} [\phi]\right)^{-1} X \intd{\eta} \\ \end{align*}$$ where $$\begin{align*} \partial_z [\phi] &= g'(z) z + g(z) \end{align*}$$ where $$\begin{align*} &g'(z)z = -R_0^2 z\frac{2}{(R_0+z)^3}\\ \implies& |g'(z) z| \approx 0.024 \textrm{~m~s}^{-2} \end{align*}$$ so in assuming that $\left(\partial_z \phi\right)^{-1} = \frac{1}{g_0}$ we incur a 0.3% multiplicative error.

The Hamiltonian derivation

Kinetic, internal, and potential energy are supposedly given by $$\begin{align*} K = \frac{1}{2} \partial_{\eta} [\pi] \mathbf{v}^2, \qquad I = c_p \Theta \Pi - \partial_{\eta} [\pi] \frac{1}{\rho} p + p_{\textrm{top}} \phi_{\textrm{top}} \qquad P = \partial_{\eta} [\pi] \phi \end{align*}$$ and note that $p_{\textrm{top}}$ is really a hydrostatic $p$. This section simply notes that $\intd{A}$ is an "area" metric. This has no radial dependence in the shallow atmosphere but (in the most naive formulation) gains a radial dependence in the deep atmosphere. Note: no functional derivatives with respect to non-hydrostatic pressure $p$ are sought. Due to the EOS $p$ is subjugated by$\phi$.

Therefore define $$\begin{align*} \mathcal{H} = \iint \textcolor{#2a3d45}{\stackrel{(1)}{\frac{1}{2} \partial_{\eta} [\pi](\langle \mathbf{u}, \mathbf{u}\rangle + w^2)}} + \textcolor{#DDC9B4}{\stackrel{(2)}{c_p \Theta \Pi + \partial_{\eta} [\phi] p + p_{\textrm{top}} \phi_{\textrm{top}}}} + \textcolor{#C17C74}{\stackrel{(3)}{\partial_{\eta} [\pi] \phi}} \intd{A} \intd{\eta} \end{align*}$$ and we do the typical algebraic shenanigans and discard second-order terms: $$\begin{align*} \textcolor{#2a3d45}{(1)}:&\frac{1}{2} (\partial_{\eta} [\pi] + \delta \left[\partial_{\eta} [\pi]\right])(\langle \mathbf{u} + \delta \mathbf{u}, \mathbf{u} + \delta \mathbf{u} \rangle + (w + \delta w)^2)\\ =& \frac{1}{2} (\partial_{\eta} [\pi] + \delta \left[\partial_{\eta} [\pi]\right])(\langle \mathbf{u}, \mathbf{u} \rangle + 2 \langle \delta \mathbf{u}, \mathbf{u} \rangle + \langle \delta \mathbf{u}, \delta \mathbf{u} \rangle + w^2 + 2w\delta w + \delta w^2)\\ =& \frac{1}{2} (\partial_{\eta} [\pi] + \delta \left[\partial_{\eta} [\pi]\right])(\langle \mathbf{u}, \mathbf{u} \rangle + 2 \langle \delta \mathbf{u}, \mathbf{u} \rangle + w^2 + 2w\delta w ) + \mathcal{O}(\varepsilon^2)\\ =& \frac{1}{2} \partial_{\eta}[\pi]( \langle \mathbf{u}, \mathbf{u} \rangle + w^2) + \partial_{\eta} [\pi] (\langle \delta \mathbf{u}, \mathbf{u} \rangle + w \delta w) + \frac{1}{2} \delta \left[\partial_{\eta} [\pi]\right] (\langle \mathbf{u}, \mathbf{u}) + w^2) + \delta \left[\partial_{\eta} [\pi]\right] (\langle \delta \mathbf{u}, \mathbf{u} \rangle + w\delta w) + \mathcal{O}(\varepsilon^2)\\ =& K + \partial_{\eta} [\pi] (\langle \delta \mathbf{u}, \mathbf{u} \rangle + w \delta w) + \frac{1}{2} \delta \left[\partial_{\eta} [\pi]\right] (\langle \mathbf{u}, \mathbf{u}) + w^2) +\mathcal{O}(\varepsilon^2) \\ =& K + \langle \partial_{\eta} [\pi] \mathbf{u}, \delta \mathbf{u} \rangle + \partial_{\eta} [\pi] w \delta w + \frac{1}{2} (\langle \mathbf{u}, \mathbf{u}\rangle + w^2) \delta \left[\partial_{\eta} [\pi]\right] +\mathcal{O}(\varepsilon^2) \\ \end{align*}$$ and $$\begin{align*} \textcolor{#DDC9B4}{(2)}:& c_p (\Theta + \delta \Theta) \Pi + (\partial_{\eta} [\phi] + \delta [\partial_{\eta} [\phi]]) p + p_{\textrm{top}} (\phi_{\textrm{top}} + \delta \phi_{\textrm{top}}) \\ =& c_p \Theta \Pi + \partial_{\eta} [\phi] p + p_{\textrm{top}}\phi_{\textrm{top}} + c_p \Pi \delta \Theta + p \delta[\partial_{\eta}[\phi]] + p_{\textrm{top}} \delta[\phi_{\textrm{top}}] \\ =& I + c_p \Pi \delta \Theta + p \delta[\partial_{\eta}[\phi]] + p_{\textrm{top}} \delta[\phi_{\textrm{top}}] \end{align*}$$ and $$\begin{align*} \textcolor{#C17C74}{(3)}:& (\partial_\eta [\pi] + \delta [\partial_{\eta}[\pi]])(\phi + \delta \phi) \\ =& \partial_{\eta}[\pi] \phi + \partial_{\eta} [\pi] \delta \phi + \phi \delta [\partial_{\eta}[\pi]] + \delta[\partial_{\eta}[\pi]] \delta \phi \\ =& P + \partial_{\eta} [\pi] \delta \phi + \phi \delta [\partial_{\eta}[\pi]] +\mathcal{O}(\varepsilon^2) \end{align*}$$ giving $$\begin{align*} \delta \mathcal{H} &= \lim_{\varepsilon \to 0} \frac{\mathcal{H}(\mathbf{u} + \varepsilon \delta \mathbf{u}, w + \varepsilon \delta w, \phi + \varepsilon \delta \phi, \Theta + \delta \Theta, \partial_{\eta}[\pi] + \delta[\partial_{\eta}[\pi]] )- \mathcal{H}(\mathbf{u}, w, \phi , \Theta , \partial_{\eta}[\pi] )}{\varepsilon}\\ &= \iint \langle \partial_{\eta} [\pi] \mathbf{u}, \delta \mathbf{u} \rangle + \partial_{\eta} [\pi] w \delta w + \frac{1}{2} (\langle \mathbf{u}, \mathbf{u}\rangle + w^2) \delta \left[\partial_{\eta} [\pi]\right] + c_p \Pi \delta \Theta + p \delta[\partial_{\eta}[\phi]] + p_{\textrm{top}} \delta[\phi_{\textrm{top}}] + \partial_{\eta} [\pi] \delta \phi + \phi \delta [\partial_{\eta}[\pi]] \intd{A} \intd{\eta} \\ &= \iint \langle \partial_{\eta} [\pi] \mathbf{u}, \delta \mathbf{u} \rangle + \partial_{\eta} [\pi] w \delta w + \left( \frac{\langle \mathbf{u}, \mathbf{u}\rangle + w^2}{2} + \phi \right) \delta \left[\partial_{\eta} [\pi]\right] + c_p \Pi \delta \Theta + p \delta[\partial_{\eta}[\phi]] + p_{\textrm{top}} \delta[\phi_{\textrm{top}}] + \partial_{\eta} [\pi] \delta \phi \intd{A} \intd{\eta}. \end{align*}$$ and we rewrite $$\begin{align*} \int \int p\delta[\partial_{\eta}[\phi]] + p_{\textrm{top}} \delta \phi_{\textrm{top}} \intd{A} \intd{\eta} &= \int \int p\delta[\partial_{\eta}[\phi]] \intd{\eta} + p_{\textrm{top}} \delta \phi_{\textrm{top}} \intd{A} \\ &= \int [p \delta \phi]_{\eta = \eta_{\textrm{top}}}^{\eta = 1} - \int \partial_{\eta} [p] \delta \phi \intd{\eta} + p_{\textrm{top}} \delta \phi_{\textrm{top}} \intd{A} \\ &= \int p_{\textrm{bot}}\delta\phi_{\textrm{bot}} - p_{\textrm{top}}\delta\phi_{\textrm{top}} + p_{\textrm{top}} \delta \phi_{\textrm{top}} - \int \partial_{\eta} [p] \delta \phi \intd{\eta} \intd{A} \\ \end{align*}$$ and we now note that $\delta \phi_{\textrm{bot}} = 0$ due to the stationary topography at the lower boundary condition. Therefore $$\begin{align*} \int \int p\delta[\partial_{\eta}[\phi]] + p_{\textrm{top}} \delta \phi_{\textrm{top}} \intd{A} \intd{\eta} &= \int p_{\textrm{bot}}\delta\phi_{\textrm{bot}} - \int \partial_{\eta} [p] \delta \phi \intd{\eta} \intd{A} \\ &= \int \int -\partial_{\eta} [p] \delta \phi \intd{\eta} \intd{A} \\ \end{align*}.$$ where we have relied on the fact that $\partial_{\eta} \delta \phi = \delta \partial_{\eta} \phi$. We can return to the total functional differential to find $$\begin{align*} \delta \mathcal{H} &= \iint \langle \partial_{\eta} [\pi] \mathbf{u}, \delta \mathbf{u} \rangle + \partial_{\eta} [\pi] w \delta w + \left( \frac{\langle \mathbf{u}, \mathbf{u}\rangle + w^2}{2} + \phi \right) \delta \left[\partial_{\eta} [\pi]\right] + c_p \Pi \delta \Theta + (\partial_{\eta} [\pi] - \partial_{\eta} [p]) \delta \phi \intd{A} \intd{\eta}. \end{align*}$$ which gives $$\begin{align*} \fder{\mathcal{H}}{\mathbf{u}} &= \partial_{\eta}[\pi] \mathbf{u}\\ \fder{\mathcal{H}}{w} &= \partial_{\eta} [\pi] w \\ \fder{\mathcal{H}}{\phi} &= \partial_{\eta} [\pi] - \partial_{\eta} [p]\\ \fder{\mathcal{H}}{\Theta} &= c_p \Pi \\ \fder{\mathcal{H}}{\partial_{\eta} [\pi]} &= \frac{\mathbf{u}^2 + w^2}{2} + \phi \end{align*}$$ which agrees precisely with Tea20. However: does the integration by parts trickery work if we do the pseudodensity trick instead of modifying $\mathrm{d}A$?

Rectifying pseudodensity using TD14:

A fundamental property of our pseudodensity is that $$\begin{align*} m_s = \frac{p_s}{g} &= \int_0^1 \mu \intd{\eta} \end{align*}$$ and TD14 suggests that we define $\mu = \partial_{\eta} (A) M + \partial_{\eta} (B) M_0$. Let's validate that this works: $$\begin{align*} \int_0^1 \partial_{\eta} (A) M + \partial_{\eta} (B) \intd{\eta} &= \left[A\right]_{\eta=0}^{\eta =1} M + \left[B\right]_{\eta=0}^{\eta =1} \\ &= M \end{align*}$$ so this constrains that the $\textrm{dp3d} = p_s\Delta A + p_0 \Delta B = g_0(m_s \Delta A + m_0 \Delta B)$. Dividing by $g_0$, even in the deep atmosphere, gives the mass located between particular model levels. This means that the correct generalization to the deep atmosphere is $$\textrm{dp3d} = g_0 \frac{a^2}{(a+z)^2} (m_s \Delta A + m_0 \Delta B)$$ This would make the value of $\sum \textrm{dp3d}$ coincide with the mass-weighted integral. However, this means that the $g$ correction really must be inside the integral.

Curl form for mass coordinates in TD14:

In TD14 notation we have $$\begin{align*} \partial_t M + \partial_i \int \fder{\mathcal{H}'}{v_i} \intd{\eta} &= 0 \\ \partial_t S + \partial_i \left(s \fder{\mathcal{H}'}{v_i} \right) + \partial_{\eta} \left( S \dot{\eta} \right) &= 0 \\ \partial_{t} v_i + (\partial_\eta v_i - \partial_i v_3) \dot{\eta} + \frac{\partial_j v_i - \partial_i v_j}{\mu} \fder{\mathcal{H'}}{v_j} + \partial_i \left( \fder{\mathcal{H}}{\mu} + \dot{\eta} v_3 \right) + s \partial_i \left(\fder{\mathcal{H}'}{S} \right) &= 0\\ \partial_t V_3 + \partial_{\eta} (V_3 \dot{\eta}) + \fder{\mathcal{H}'}{\xi^3} = 0\\ \partial_t \xi^3 + \dot{\eta} \partial_{\eta} \xi^3 - \fder{\mathcal{H}'}{V_3} = 0 \end{align*}$$ where $$\begin{align*} V_3 &\equiv \mu \hat{v}_3 \\ v_i &\equiv \hat{v}_i + \hat{v}_3 \partial_i \xi^3 \\ \hat{v_l} &\equiv \pder{\hat{L}}{\hat{u}^l}. \end{align*}$$ We note that $\xi^3 = r$.

Tea20 makes the definition $\mu = \partial_{\eta} [A] M + \partial_{\eta} [B]$. With this definition $$\begin{align*} \partial_t M + \partial_i \int \fder{\mathcal{H}'}{v_i} \intd{\eta} &= \partial_t \int \mu \intd{\eta} + \partial_i \int \mu u_i \intd{\eta} \end{align*}$$ which is quite straightforwardly the continuity equation in weak form. It remains to show that (43) and (44) in TD14 are satisfied by this definition, but I suspect they are.

Let's examine $$\begin{align*} \int \partial_{\eta} \end{align*}$$

The Hamiltonian derivation for Oksana's notes

The idea for this hack is to essentially redefine $\mathrm{d}A$ as $\hat{r}^2 \intd{A}$ by defining $\mathrm{dp3d}$.

Kinetic, internal, and potential energy are supposedly given by $$\begin{align*} K = \frac{1}{2} \partial_{\eta} [\pi]_{\hat{r}^2} \mathbf{v}^2, \qquad I = c_p \Theta \Pi - \partial_{\eta} [\pi]_{\hat{r}^2} \frac{1}{\rho} p + p_{\textrm{top}} \phi_{\textrm{top}} \qquad P = \partial_{\eta} [\pi]_{\hat{r}^2} \phi \end{align*}$$ and note that $p_{\textrm{top}}$ is really a hydrostatic $p$. In mass-based coordinates, it is also time-invariant. Since $\Theta = \partial_{\eta} [\pi] \theta_v$, terms containing $\Theta$ gain a correction. This time we are rather more careful, noting that if $$\begin{align*} \hat{r}^2 \partial_{\eta} \phi &= -R_d \partial_{\eta} [\pi]_{\hat{r}^2} \theta_v \frac{\Pi}{p} \\ &= -\frac{R_d}{p} \theta_v\frac{T_v}{\theta_v} \partial_{\eta} [\pi]_{\hat{r}^2} \\ &= -\frac{1}{\rho} \partial_{\eta} [\pi]_{\hat{r}^2}. \end{align*}.$$

Therefore define $$\begin{align*} \mathcal{H} = \iint \textcolor{#2a3d45}{\stackrel{(1)}{\frac{1}{2} \partial_{\eta} [\pi]_{\hat{r}^2}(\langle \mathbf{u}, \mathbf{u}\rangle + w^2)}} + \textcolor{#DDC9B4}{\stackrel{(2)}{c_p \Theta_{\hat{r}^2} \Pi + \partial_{\eta} [\phi]_{\hat{r}^2} p + \hat{r}_{\textrm{top}}^2p_{\textrm{top}} \phi_{\textrm{top}}}} + \textcolor{#C17C74}{\stackrel{(3)}{\partial_{\eta} [\pi]_{\hat{r}^2} \phi}} \intd{A} \intd{\eta} \end{align*}$$ and the same algebraic manipulations gives $$\begin{align*} \delta \mathcal{H} &= \lim_{\varepsilon \to 0} \frac{\mathcal{H}(\mathbf{u} + \varepsilon \delta \mathbf{u}, w + \varepsilon \delta w, \phi + \varepsilon \delta \phi, \Theta_{\hat{r}^2} + \delta \Theta_{\hat{r}^2} , \partial_{\eta}[\pi]_{\hat{r}^2} + \delta[\partial_{\eta}[\pi]_{\hat{r}^2} ] )- \mathcal{H}(\mathbf{u}, w, \phi , \Theta_{\hat{r}^2} , \partial_{\eta}[\pi]_{\hat{r}^2} )}{\varepsilon}\\ &= \iint \langle \partial_{\eta} [\pi]_{\hat{r}^2} \mathbf{u}, \delta \mathbf{u} \rangle + \partial_{\eta} [\pi]_{\hat{r}^2} w \delta w + \frac{1}{2} (\langle \mathbf{u}, \mathbf{u}\rangle + w^2) \delta \left[\partial_{\eta} [\pi]_{\hat{r}^2} \right] + c_p \Pi \delta \Theta_{\hat{r}^2} + p \delta[\partial_{\eta}[\phi]_{\hat{r}^2}] + p_{\textrm{top}} \delta[\phi_{\textrm{top}}] + \partial_{\eta} [\pi]_{\hat{r}^2} \delta \phi + \phi \delta [\partial_{\eta}[\pi]_{\hat{r}^2} ] \intd{A} \intd{\eta} \\ &= \iint \langle \partial_{\eta} [\pi]_{\hat{r}^2} \mathbf{u}, \delta \mathbf{u} \rangle + \partial_{\eta} [\pi]_{\hat{r}^2} w \delta w + \left( \frac{\langle \mathbf{u}, \mathbf{u}\rangle + w^2}{2} + \phi \right) \delta \left[\partial_{\eta} [\pi]_{\hat{r}^2} \right] + c_p \Pi \delta \Theta_{\hat{r}^2} + p \delta[\partial_{\eta}[\phi]_{\hat{r}^2}] + p_{\textrm{top}} \delta[\phi_{\textrm{top}}] + \partial_{\eta} [\pi]_{\hat{r}^2} \delta \phi \intd{A} \intd{\eta}. \end{align*}.$$ We rewrite $$\begin{align*} \int \int p\delta[\partial_{\eta}[\phi]_{\hat{r}^2}] + p_{\textrm{top}} \delta \phi_{\textrm{top}} \intd{\eta} \intd{A} &= \int \int p\delta[\partial_{\eta}[\phi]_{\hat{r}^2}] \intd{\eta} + p_{\textrm{top}} \delta \phi_{\textrm{top}} \intd{A} \\ &= \int [p \delta \phi]_{\eta = \eta_{\textrm{top}}}^{\eta = 1} - \int \partial_{\eta} [p] \delta \phi \intd{\eta} + \int p_{\textrm{top}} \delta \phi_{\textrm{top}} \intd{\eta} \intd{A} \\ &= \int p_{\textrm{bot}}\delta\phi_{\textrm{bot}} - p_{\textrm{top}}\delta\phi_{\textrm{top}} + p_{\textrm{top}} \delta \phi_{\textrm{top}} - \int \partial_{\eta} [p] \delta \phi \intd{\eta} \intd{A} \\ \end{align*},$$ which relies on the fact that reconstructing $\phi$ from $\partial_{\eta} \phi_{\hat{r}^2}$ is a mass-weighted integral, which consumes the $\hat{r}^2$ factor. We now note that $\delta \phi_{\textrm{bot}} = 0$ due to the stationary topography at the lower boundary condition. Therefore $$\begin{align*} \int \int p\delta[\hat{r}^2\partial_{\eta}[\phi]] + p_{\textrm{top}} \delta \phi_{\textrm{top}} \intd{A} \intd{\eta} &= \int p_{\textrm{bot}}\delta\phi_{\textrm{bot}} - \int \partial_{\eta} [p] \delta \phi \intd{\eta} \intd{A} \\ &= \int \int -\partial_{\eta} [p] \delta \phi \intd{\eta} \intd{A} \\ \end{align*}$$ where we have relied on the fact that $\partial_{\eta} \delta \phi = \delta \partial_{\eta} \phi$. We can return to the total functional differential to find $$\begin{align*} \delta \mathcal{H} &= \iint \langle \partial_{\eta} [\pi]_{\hat{r}^2} \mathbf{u}, \delta \mathbf{u} \rangle + \partial_{\eta} [\pi]_{\hat{r}^2} w \delta w + \left( \frac{\langle \mathbf{u}, \mathbf{u}\rangle + w^2}{2} + \phi \right) \delta \left[\partial_{\eta} [\pi]_{\hat{r}^2} \right] + c_p \Pi \delta \Theta_{\hat{r}^2} + (\partial_{\eta} [\pi]_{\hat{r}^2} - \partial_{\eta} [p]) \delta \phi \intd{A} \intd{\eta}. \end{align*}$$ The only problem term in this equation is in the differential for $\delta \phi$, namely $\partial_{\eta} [p] \delta \phi$. The lack of a $\hat{r}^2$ correction in this term dictates the modification of $\mu \equiv \partial_{\eta} [p] \left(\partial_{\eta} [\pi] \right)^{-1}$ so we rewrite $$\begin{align*} \delta \mathcal{H} &= \iint \langle \partial_{\eta} [\pi]_{\hat{r}^2} \mathbf{u}, \delta \mathbf{u} \rangle + \partial_{\eta} [\pi]_{\hat{r}^2} w \delta w + \left( \frac{\langle \mathbf{u}, \mathbf{u}\rangle + w^2}{2} + \phi \right) \delta \left[\partial_{\eta} [\pi]_{\hat{r}^2} \right] + c_p \Pi \delta \Theta_{\hat{r}^2} + \partial_{\eta} [\pi]_{\hat{r}^2}( 1 - \partial_{\eta} [p] \left( \partial_{\eta} [\pi]_{\hat{r}^2}\right)^{-1}) \delta \phi \intd{A} \intd{\eta} \\ &= \iint \langle \partial_{\eta} [\pi]_{\hat{r}^2} \mathbf{u}, \delta \mathbf{u} \rangle + \partial_{\eta} [\pi]_{\hat{r}^2} w \delta w + \left( \frac{\langle \mathbf{u}, \mathbf{u}\rangle + w^2}{2} + \phi \right) \delta \left[\partial_{\eta} [\pi]_{\hat{r}^2} \right] + c_p \Pi \delta \Theta_{\hat{r}^2} + \partial_{\eta} [\pi]_{\hat{r}^2}( 1 - \partial_{\eta} [p] \left( \partial_{\eta} [\pi]_{\hat{r}^2}\right)^{-1}) \delta \phi \intd{A} \intd{\eta}. \end{align*}$$ which dictates that $\mu_{\hat{r}^2} = \hat{r}^2 \partial_{\eta} [p] \left( \partial_{\eta} [\pi]_{\hat{r}^2}\right)^{-1}$. $$\begin{align*} \fder{\mathcal{H}}{\mathbf{u}} &= \partial_{\eta}[\pi] \mathbf{u}\\ \fder{\mathcal{H}}{w} &= \partial_{\eta} [\pi] w \\ \fder{\mathcal{H}}{\phi} &= \partial_{\eta} [\pi]_{\hat{r}^2}(1 - \mu_{\hat{r}^2})\\ \fder{\mathcal{H}}{\Theta} &= c_p \Pi \\ \fder{\mathcal{H}}{\partial_{\eta} [\pi]} &= \frac{\mathbf{u}^2 + w^2}{2} + \phi \end{align*}$$ which illustrates that the integration by parts trick works Let us check if this is consistent by returning to the implicit hypsometric equation that defines our EOS.

Returning to EOS

$$\begin{align*} &\partial_\eta \phi = -R_d \partial_\eta[\pi]_{\hat{r}^2} \theta_v \frac{\Pi}{p}\\ \implies& \int \partial_\eta [\phi] \, \mathrm{d}\eta = -R_d \int \partial_\eta[\pi]_{\hat{r}^2} \theta_v \frac{\Pi}{p} \, \mathrm{d}\eta\\ \implies& \Delta \phi = -R_d \int \partial_\eta[\pi]_{\hat{r}^2} \theta_v \frac{\Pi}{p} \, \mathrm{d}\eta\\ \implies& \Delta \phi = -R_d \int \frac{T}{p} \partial_\eta [\pi]_{\hat{r}^2} \, \mathrm{d}\eta\\ \implies& \Delta \phi = -R_d \int \frac{T}{p}\, \mathrm{d}\pi\\ \end{align*}$$ and so we see that this holds if $$\int [\cdot]\, \partial_\eta [\pi]_{\hat{r}^2} \intd{\eta} = \int [\cdot]\, \hat{r}^2 \partial_\eta [\pi] \intd{\eta} = \int [\cdot] \intd{\pi}$$ which is precisely what we constructed $\partial_{\eta} [\pi]_{\hat{r}^2}$ to satisfy!

The integration by parts:

$$\begin{align*} &\int \int p\hat{r}^2\delta[\partial_{\eta}[\phi]] + \textrm{BC} +\hat{r}^2 \partial_{\eta} [p] \delta \phi \intd{\eta} \intd{A} = \int \left[p\hat{r}^2 \phi\right]_{\eta = 1}^{\eta = 0} + \textrm{BC} + \int -\partial_{\eta} [p\hat{r}^2\delta \phi] + \hat{r}^2 \partial_{\eta} [p] \delta \phi \intd{\eta} \intd{A}\\ \end{align*}$$and so ^