Reynolds}\doteq\fint\left[\frac{\widehat{\bm{p}}\otimes\widehat{\bm{p}}}{\overline{\rho}}-\frac{\widehat{\bm{p}}\otimes\overline{\bm{p}}}{\overline{\rho}}\frac{\widehat{\rho}}{\overline{\rho}}-\frac{\overline{\bm{p}}\otimes\widehat{\bm{p}}}{\overline{\rho}}\frac{\widehat{\rho}}{\overline{\rho}}+\frac{\overline{\bm{p}}\otimes\overline{\bm{p}}}{\overline{\rho}}\left(\frac{\widehat{\rho}}{\overline{\rho}}\right)^{2}\right]d\theta.$
(178)

Finally, substituting the expression for $\widehat{\bm{p}}$ in Eq. (170) into
$T_{\rm Reynolds}$ leads to Eq. (175). ∎

With the above equations, we can now deduce a dynamical equation for the wave
action density $\mathcal{I}$. This is given in the next proposition.

######  Proposition 8.

The governing equation to leading order for the wave action density
$\mathcal{I}$ is

$\displaystyle\partial_{t}\mathcal{I}+\bm{\nabla}\cdot(\bm{v}_{g}\mathcal{I})=0,$
(179)

where $\bm{v}_{\rm
g}\doteq\overline{\bm{p}}/\overline{\rho}+c_{s}\bm{e}_{\bm{k}}$ is the wave
group velocity.

######  Remark 9.

One can in principle prove this result by calculating the time derivative of
$\mathcal{I}$ and then substituting the governing equations for the mean and
fluctuating quantities [Eqs. (168), (169), (174), and (175)], as well as the
dispersion relation (171). The ensuing calculation is tedious since one must
take into account corrections to the leading-order solution for the momentum
density [Eq. (170)]. Alternatively, the proof may be constructed as a
straightforward corollary of results presented in Sec. IV and later in the
present Section. Because this alternative approach is simpler, we will
postpone the proof until we have proved Proposition 9.

We may now summarize the results obtained so far as the following set of
equations governing the leading-order wave-mean-flow interaction between a
sound wave and a bulk isothermal flow.

######  Definition 13 (Wave–mean-flow equations).

The governing equations describing a high-frequency, small-amplitude acoustic
wave interacting with a slowly-varying, isothermal bulk fluid are

$\displaystyle\partial_{t}\overline{\rho}+\bm{\nabla}\cdot\overline{\bm{p}}=0$
(180)
$\displaystyle\partial_{t}\overline{\bm{p}}+\bm{\nabla}\cdot\left(\frac{\overline{\bm{p}}\otimes\overline{\bm{p}}}{\overline{\rho}}+\epsilon^{2}\mathcal{I}\frac{\bm{\nabla}S\otimes\bm{\nabla}S}{|\bm{\nabla}S|}+c_{s}^{2}\,\overline{\rho}\,\mathbb{I}\right)=0$
(181)
$\displaystyle\partial_{t}\mathcal{I}+\bm{\nabla}\cdot(\bm{v}_{g}\mathcal{I})=0$
(182)
$\displaystyle\partial_{t}S+\frac{\overline{\bm{p}}}{\overline{\rho}}\cdot\bm{\nabla}S+c_{s}|\bm{\nabla}S|=0,$
(183)

where $\mathcal{I}$ is the wave action density (176) and $\bm{v}_{\rm
g}\doteq\overline{\bm{p}}/\overline{\rho}+c_{s}\bm{e}_{\bm{k}}$ is the wave
group velocity.

As written, Eqs. (180)–(183) are closed in the sense that they possess a
(formally) well-posed initial value problem. Perhaps surprisingly, these
equations also follow from a variational principle! This is shown in the
theorem below.

######  Theorem 5 (Effective action for wave–mean-flow interactions).

Let $\overline{\mathcal{C}}_{0}=\mathcal{C}_{0}\times C^{\infty}_{+}(Q)\times
C^{\infty}(Q,S^{1})$. (The space $\mathcal{C}_{0}$ is introduced in Definition
7.) That is, $\overline{\mathcal{C}}$ comprises maps
$Q\ni\bm{x}\mapsto(\overline{\bm{h}}(\bm{x}),\overline{\bm{p}}(\bm{x}),\overline{\rho}(\bm{x}),\overline{\chi}(\bm{x}),\mathcal{I}(\bm{x}),S(\bm{x}))\in
Q_{0}\times\mathbb{R}^{3}\times\mathbb{R}\times\mathbb{R}\times\mathbb{R}\times
S^{1}$, where $\overline{\bm{h}}$ is a diffeomorphism and
$\overline{\rho}(\bm{x})>0$, $\mathcal{I}(\bm{x})>0$ for all $\bm{x}\in Q$.
Consider the action $\overline{\mathsf{A}}_{\rm T}$ defined on the space of
paths $[t_{1},t_{2}]\rightarrow\overline{\mathcal{C}}_{0}$ by the formula

$\overline{\mathsf{A}}_{\rm T}=\int_{t_{1}}^{t_{2}}\overline{\mathsf{L}}_{\rm
T}(\overline{\bm{h}},\dot{\overline{\bm{h}}},\overline{\bm{p}},\dot{\overline{\bm{p}}},\overline{\rho},\dot{\overline{\rho}},\overline{\chi},\dot{\overline{\chi}},\mathcal{I},\dot{\mathcal{I}},S,\dot{S})dt,$
(184)

where the Lagrangian $\overline{\mathsf{L}}_{\rm T}$ is given by

$\displaystyle\overline{\mathsf{L}}_{\rm
T}(\overline{\bm{h}},\dot{\overline{\bm{h}}},\overline{\bm{p}},\dot{\overline{\bm{p}}},\overline{\rho},\dot{\overline{\rho}},\overline{\chi},\dot{\overline{\chi}},\mathcal{I},\dot{\mathcal{I}},S,\dot{S})$
$\displaystyle=\int_{Q}\left(\overline{\bm{p}}\cdot\overline{\bm{v}}+\overline{\rho}\big{(}\partial_{t}\overline{\chi}+\overline{\bm{v}}\cdot\bm{\nabla}\overline{\chi}\big{)}\right)\,d^{3}\bm{x}-\int_{Q}\mathcal{H}_{\rm
T}(\overline{\bm{p}},\overline{\rho})\,d^{3}\bm{x}$
$\displaystyle\quad-\epsilon^{2}\int_{Q}\mathcal{I}\left(\partial_{t}S+\overline{\bm{v}}\cdot\bm{\nabla}S+c_{s}|\bm{\nabla}S|\right)\,d^{3}\bm{x}.$
(185)

Here the mean velocity $\overline{\bm{v}}$ is defined as
$\overline{\bm{v}}\doteq-\partial_{t}\overline{\bm{h}}\cdot(\bm{\nabla}\overline{\bm{h}})^{-1}$,
and the Hamiltonian for the isothermal fluid is given by

$\mathcal{H}_{\rm
T}(\overline{\bm{p}},\overline{\rho})\doteq\frac{|\overline{\bm{p}}|^{2}}{2\overline{\rho}}+c_{s}^{2}\overline{\rho}\ln\left(\frac{\overline{\rho}}{\rho_{0}}\right).$
(186)

The parameters $c_{s}$ and $\rho_{0}$ are the sound speed and reference mass
density, respectively. Equations (180)–(183) are embedded in the
Euler–Lagrange equations obtained when varying the action
$\overline{\mathsf{A}}_{\rm T}$ with respect to $\overline{\bm{h}}$,
$\overline{\bm{p}}$, $\overline{\rho}$, $\overline{\chi}$, $\mathcal{I}$, and
$S$.

###### Proof.

Since $\overline{\mathsf{A}}_{\rm T}$ is functional of paths
$[t_{1},t_{2}]\rightarrow\overline{\mathcal{C}}_{0}$, we can vary the fields
$(\overline{\bm{h}},\overline{\bm{p}},\overline{\rho},\overline{\chi},\mathcal{I},S)$
independently. Moreover, varying the action with respect to the fields
$(\overline{\bm{h}},\overline{\bm{p}},\overline{\rho},\overline{\chi})\in\mathcal{C}_{0}$
follows an almost identical procedure as that given in the proof of
Proposition 1. Varying the action with respect to $\overline{\bm{p}}$ leads to
$\overline{\bm{v}}=\overline{\bm{p}}/\overline{\rho}$. Varying the action with
respect to the scalar field $\overline{\chi}$ gives

$\partial_{t}\overline{\rho}+\bm{\nabla}\cdot(\overline{\bm{v}}\overline{\rho})=0.$
(187)

Substituting $\overline{\bm{v}}=\overline{\bm{p}}/\overline{\rho}$ into the
equation above trivially leads to Eq. (180). Varying the action with respect
to $\overline{\rho}$ gives

$\partial_{t}\overline{\chi}+(\overline{\bm{v}}\cdot\bm{\nabla})\overline{\chi}=-\frac{|\overline{\bm{p}}|^{2}}{2\overline{\rho}^{2}}+c_{s}^{2}\ln\left(\frac{\overline{\rho}}{\rho_{0}}\right)+c_{s}^{2}.$
(188)

Varying $\overline{\bm{h}}$ leads to

$\partial_{t}(\bm{p}+\overline{\rho}\bm{\nabla}\overline{\chi}-\mathcal{I}\bm{\nabla}S)+\bm{\nabla}\cdot(\bm{v}\otimes(\bm{p}+\overline{\rho}\bm{\nabla}\overline{\chi}-\mathcal{I}\bm{\nabla}S))+(\bm{\nabla}\bm{v})\cdot(\bm{p}+\overline{\rho}\bm{\nabla}\overline{\chi}-\mathcal{I}\bm{\nabla}S)=0.$
(189)

By substituting Eqs. (187) and (188) into Eq. (189) and following a similar
algebraic manipulation as in the proof of Proposition 1, one can recover Eq.
(181) from Eq. (189). Finally, varying the action $\overline{\mathsf{A}}_{\rm
T}$ with respect to $\mathcal{I}$ and $S$ leads to Eqs. (182) and (183),
respectively. Thus, we have shown that Eqs. (180)–(183) are embedded in the
Euler–Lagrange equations associated to the action $\overline{\mathsf{A}}_{\rm
T}$. ∎

###  VI.2 Why are the wave–mean-flow equations variational?

The proof of Theorem 5 gave no indication as to why the leading-order wave-
mean-flow equations arise from a variational principle. We now want to give a
principled explanation for this result using the machinery developed in this
paper. One indication that the wave–mean-flow equations might be variational
is that the parent isothermal fluid equations (163) and (164), where we
started our asymptotic analysis, also come from a variational principle. This
can be easily proven because the isothermal fluid equations comprise a special
case of the LBEP equations discussed in Sec. III. Hence, we can readily write
a variational principle for Eqs. (163) and (164), which is given below.

######  Corollary 2.

Let $\mathsf{A}_{\rm T}$ be the action functional defined on the space of
paths $[t_{1},t_{2}]\rightarrow\mathcal{C}_{0}$ such that

$\displaystyle\mathsf{A}_{\rm T}=\int_{t_{1}}^{t_{2}}\mathsf{L}_{\rm
T}(\bm{h}(t),\dot{\bm{h}}(t),\bm{p}(t),\dot{\bm{p}}(t),\rho(t),\dot{\rho}(t),\chi(t),\dot{\chi}(t))\,dt,$
(190)

where

$\displaystyle\mathsf{L}_{\rm
T}(\bm{h},\dot{\bm{h}},\bm{p},\dot{\bm{p}},\rho,\dot{\rho},\chi,\dot{\chi})=\int_{Q}\bm{p}\cdot\bm{v}\,d^{3}\bm{x}+\int_{Q}(\dot{\chi}+\bm{v}\cdot\bm{\nabla}\chi)\,\rho\,d^{3}\bm{x}-\int_{Q}\mathcal{H}_{\text{T}}(\bm{p},\rho)\,d^{3}\bm{x}$
(191)

and the Hamiltonian $\mathcal{H}_{\text{T}}$ is defined in Eq. (186).

A path $t\mapsto(\bm{h}(t),\bm{p}(t),\rho(t),\chi(t))\in\mathcal{C}_{0}$ is a
critical point of the action functional (190) for isothermal fluids if and
only if $\bm{h}$, $\bm{p}$, $\rho$, and $\chi$ satisfy the following system of
PDEs:

$\displaystyle\partial_{t}\bm{h}=-\frac{\bm{p}}{\rho}\cdot\bm{\nabla}\bm{h}$
(192)
$\displaystyle\partial_{t}\bm{p}+\bm{\nabla}\cdot\left(\frac{\bm{p}\otimes\bm{p}}{\rho}\right)=-c_{s}^{2}\bm{\nabla}\rho$
(193) $\displaystyle\partial_{t}\rho+\bm{\nabla}\cdot\bm{p}=0$ (194)
$\displaystyle\partial_{t}\chi+\frac{\bm{p}}{\rho}\cdot\bm{\nabla}\chi=-\frac{|\bm{p}|^{2}}{2\rho^{2}}+c_{s}^{2}\ln\left(\frac{\rho}{\rho_{0}}\right)+c_{s}^{2}.$
(195)

###### Proof.

This is directly verified by substituting Eqs. (190)–(191) into Eqs. (75)–(78)
in Proposition 1. ∎

######  Remark 10.

It is clear that the isothermal fluid equations (163) and (164) are simply
embedded into the LBEP equations above.

The next indication is that, in order to explicitly introduce a scale
separation into the fluid equations, the next step we used in the analysis of
Sec. VI.1 was passing to the NL-WKB extension of the isothermal fluid
equations [see Eqs. (165) and (166)]. Following our results from Sec. IV,
these equations are variational as well! The precise statement of this
observation is as follows.

######  Corollary 3.

Let $\gamma:[t_{1},t_{2}]\rightarrow\ell\mathcal{C}_{0}\times
C^{\infty}(Q,S^{1})$ be a smooth curve with components
$\gamma=(\widetilde{\bm{h}},\widetilde{\bm{p}},\widetilde{\rho},\widetilde{\chi},S)$.
Let the functional
$\widetilde{\mathsf{A}}(\gamma)=\int_{t_{1}}^{t_{2}}\widetilde{\mathsf{L}}_{\rm
T}(\gamma(t),\partial_{t}\gamma(t))\,dt$ be defined such that

$\displaystyle\widetilde{\mathsf{L}}_{\rm
T}(\widetilde{\bm{h}},\widetilde{\bm{p}},\widetilde{\rho},\widetilde{\chi},S,\dot{\widetilde{\bm{h}}},\dot{\widetilde{\rho}},\dot{\widetilde{\chi}},\dot{S})=$
$\displaystyle\fint\int_{Q}\left(\widetilde{\bm{p}}\cdot\widetilde{\bm{v}}+\widetilde{\rho}\,\dot{\widetilde{\chi}}+\epsilon^{-1}\widetilde{\rho}\,\dot{S}\partial_{\theta}\widetilde{\chi}+\widetilde{\rho}\widetilde{\bm{v}}\cdot\bm{\nabla}^{S/\epsilon}\widetilde{\chi}\right)\,d^{3}\bm{x}\,d\theta$
$\displaystyle-\fint\int_{Q}\mathcal{H}_{\rm
T}(\widetilde{\bm{p}},\widetilde{\rho})\,d^{3}\bm{x}\,d\theta,$ (196)

where $\widetilde{\bm{v}}$ is defined in Eq. (102). Then, the curve $\gamma$
is a (fixed-endpoint) critical point of $\widetilde{\mathsf{A}}(\gamma)$ if
and only if the component functions
$(\widetilde{\bm{h}},\widetilde{\bm{p}},\widetilde{\rho},\widetilde{\chi},S)$
satisfy

$\displaystyle\partial_{t}^{S/\epsilon}\widetilde{\bm{h}}=-\frac{\widetilde{\bm{p}}}{\widetilde{\rho}}\cdot\bm{\nabla}^{S/\epsilon}\widetilde{\bm{h}}$
(197)
$\displaystyle\partial_{t}^{S/\epsilon}\widetilde{\bm{p}}+\bm{\nabla}^{S/\epsilon}\cdot\left(\frac{\widetilde{\bm{p}}\otimes\widetilde{\bm{p}}}{\widetilde{\rho}}\right)=-c_{s}^{2}\bm{\nabla}^{S/\epsilon}\widetilde{\rho}$
(198)
$\displaystyle\partial_{t}^{S/\epsilon}\widetilde{\rho}+\bm{\nabla}^{S/\epsilon}\cdot\widetilde{\bm{p}}=0$
(199)
$\displaystyle\partial_{t}^{S/\epsilon}\widetilde{\chi}+\frac{\widetilde{\bm{p}}}{\widetilde{\rho}}\cdot\bm{\nabla}^{S/\epsilon}\widetilde{\chi}=-\frac{|\widetilde{\bm{p}}|^{2}}{2\widetilde{\rho}^{2}}+c_{s}^{2}\ln\left(\frac{\widetilde{\rho}}{\rho_{0}}\right)+c_{s}^{2}.$
(200)

###### Proof.

This is immediately verified by substituting Eq. (196) into the result in
Theorem 3. ∎

The question that now remains to be answered is whether the variational
structure underlying the NL-WKB extension of the isothermal fluid equations is
somehow compatible with the asymptotics leading to the wave–mean-flow
equations (180)–(183). A geometrically satisfying way to address this question
is through the application of a dynamical systems tool known as _slow-manifold
reduction_.

The concept of slow manifolds originated from the theory of fast-slow
dynamical systems, which essentially are singularly perturbed dynamical
systems.Fenichel (1979); Verhulst (2005) Before explaining the role played by
slow manifolds in our example, we will first give a quick overview of slow
manifold theory.

######  Definition 14 (Fast-slow dynamical system).

Let $X,Y$ be Banach spaces and $\epsilon\ll 1$. A fast-slow dynamical system
is an ODE on $X\times Y$ of the form

$\epsilon\dot{y}=f_{\epsilon}(x,y),\qquad\dot{x}=g_{\epsilon}(x,y),$ (201)

with $D_{y}f_{0}(x,y)\colon Y\to Y$ an isomorphism when $(x,y)\in
f_{0}^{-1}(\\{0\\})$. The functions $f_{\epsilon}$ and $g_{\epsilon}$ are
required to depend smoothly on $\epsilon$ in such a manner that
$f_{\epsilon},g_{\epsilon}=O(1)$ as $\epsilon\rightarrow 0$.

By convention, the variable $y$ is called the “fast” variable, while the
variable $x$ is called the “slow” variable. For fast-slow dynamical systems,
it then follows that invariant manifolds given as graphs over the slow
variables satisfy a nonlinear (functional) PDE. This is illustrated below.

######  Lemma 5.

Suppose a fast-slow dynamical systems admits an invariant manifold
$I_{\epsilon}$ of the form $I_{\epsilon}=\\{(x,y)\in X\times
Y|y=y_{\epsilon}^{\star}(x)\\}$ for some smooth map
$y_{\epsilon}^{\star}\colon X\to Y$. Then,

$\epsilon
Dy^{\star}_{\epsilon}(x)[g(x,y^{\star}_{\epsilon}(x))]=f(x,y^{\star}_{\epsilon}(x)),$
(202)

for each $x\in X$.

###### Proof.

Supposing $y=y_{\epsilon}^{\star}(x)$, one then inserts this into
$\epsilon\dot{y}=f(x,y)$. Using the chain rule and substituting the time-
evolution equation for $x$ leads to the claimed result. ∎

######  Definition 15 (Slow manifold).

If $I_{\epsilon}$ is an invariant manifold given as the graph of
$y^{\star}_{\epsilon}\colon X\to Y$, $I_{\epsilon}$ is a slow manifold when
$y^{\star}_{\epsilon}(x)$ is a formal power series solution of Eq. (202).

Of the invariant manifolds given as graphs, slow manifolds play a special role
for several reasons. First of all, slow manifolds are unique; i.e., if
$I_{\epsilon}$ and $I_{\epsilon}^{\prime}$ are two slow manifolds, then
$I_{\epsilon}=I_{\epsilon}^{\prime}$. Moreover, the formal power series
expansion of the graphing function $y^{\star}_{\epsilon}(x)$ may be obtained
using explicit formulas. In addition, dynamics restricted to the slow manifold
is indeed slow; it is simple to check that the time derivatives of both the
fast and slow variables are $O(1)$ on the slow manifold. The slow manifold may
therefore be interpreted intuitively as the region in phase space where the
fast degrees of freedom are not excited.

For the purposes of the present discussion, a crucial result on slow-manifold
dynamics is that they inherit Hamiltonian structure from the parent fast-slow
system whenever the larger system has such a structure. One way to state this
fact precisely is as follows.

######  Theorem 6 (Inheritance of Hamiltonian structure).

Consider a fast-slow system satisfying the variational principle

$\delta
A=\delta\int_{t_{1}}^{t_{2}}\bigg{(}\Theta_{\epsilon}(x(t),y(t))[\dot{x}(t),\dot{y}(t)]-H_{\epsilon}(x(t),y(t))\bigg{)}dt=0,$
(203)

with $\delta(x(t_{1}),y(t_{1}))=\delta(x(t_{2}),y(t_{2}))=0$. Here
$\Theta_{\epsilon}$ is an $\epsilon$-dependent one-form on $X\times Y$, and
$H_{\epsilon}$ is an $\epsilon$-dependent smooth function on $X\times Y$.
Suppose a slow manifold exists where $y=y^{\star}_{\epsilon}(x)$. Then, the
slow dynamics for the variable $x\in X$ satisfy the variational principle

$\delta A_{\rm slow}=\delta\int_{t_{1}}^{t_{2}}\left(\Theta_{\rm
slow}(x(t))[\dot{x}(t)]-H_{\rm slow}(x(t))\right)dt=0,$ (204)

with $\delta x(t_{1})=\delta x(t_{2})=0$. Here $\Theta_{\rm slow}(x)[\delta
x]\doteq\Theta(x,y_{\epsilon}^{\star}(x))[\delta
x,Dy_{\epsilon}^{\star}(x)[\delta x]]$, and $H_{\rm slow}(x)\doteq
H(x,y_{\epsilon}^{\star}(x))$.

###### Proof.

With the given boundary conditions, it is clear that $\delta A=0$ holds even
if the trajectory $t\mapsto(x(t),y(t))$ that is subject to variations lies in
the slow manifold, i.e. $y(t)=y^{\star}_{\epsilon}(x(t))$, since $\delta
y(t_{1,2})=Dy^{\star}_{\epsilon}(x(t_{1,2}))\,\delta x(t_{1,2})=0$. In
particular, $\delta A=0$ when variations are constrained to lie along the slow
manifold. This is equivalent to saying that the first variation of
$A_{\text{slow}}$, which is $A$ restricted to paths contained in the slow
manifold, is zero along a solution contained in the slow manifold. After
restriction to the slow manifold, the two terms in the integrand of $A$ may be
written

$\Theta_{\epsilon}(x,y_{\epsilon}^{\star}(x))\left[\dot{x},\frac{d}{dt}y^{\star}_{\epsilon}(x)\right]=\Theta_{\epsilon}(x,y_{\epsilon}^{\star}(x))\left[\dot{x},Dy^{\star}_{\epsilon}(x)\,\dot{x}\right]=\Theta_{\rm
slow}(x)[\dot{x}],$ (205)

and $H_{\epsilon}(x,y_{\epsilon}^{\star}(x))=H_{\text{slow}}(x)$. (We have
omitted writing the time dependence explicitly.) Thus, $A_{\text{slow}}$ may
be written as in the Theorem statement. Moreover, we have already argued
$\delta A_{\text{slow}}=0$ along any solution of the fast-slow system
contained in the slow manifold. This completes the proof. ∎

We will now argue that Theorem 6 may be used to systematically derive the
variational principle for the leading-order wave–mean-flow equations
(180)–(183). The first thing to be verified is that the NL-WKB extension of
the isothermal fluid equations (197)–(200), together with the dispersion
relation (183) for specifying the dynamics of $S$, indeed form a fast-slow
system. If this is the case, then by Theorem 6, slow-manifold reduction will
allow us to construct a variational principle for the slow, wave–mean-flow
system. Our argument will then be complete if we can show that the variational
principle given by Theorem 6 reproduces the variational principle from Theorem
5. The rest of this subsection will be devoted to establishing that Eqs.
(197)–(200), together with the dispersion relation (183), comprise a fast-slow
system. The following subsection will sketch the details of manipulating
$A_{\text{slow}}$ from Theorem 6 in order to produce
$\overline{\mathsf{A}}_{\rm T}$ from Theorem 5.

As in the previous section, we consider only high-frequency, small-amplitude
waves. Hence, we adopt the parameterization given in Eqs. (167) for the fields
$\widetilde{\rho}$ and $\widetilde{\bm{p}}$. Although it is not technically
necessary for the slow-manifold analysis, we shall also parameterize the back-
to-labels map $\widetilde{\bm{h}}$ by following the generalized-Lagrangian-
mean (GLM) approach proposed by Andrew and McIntyre.Andrews and McIntyre
(1978) For more information on this approach, we recommend reading as well the
works by Holm and Gjaja,Holm (2002a, b); Gjaja and Holm (1996); Holm (2002c)
as well as Buhler’s accessible book.Bühler (2009) In GLM theory, one
introduces a space $\overline{Q}$ that is diffeomorphic to $Q$ and that is
interpreted as the collection of “mean” Eulerian positions. Then
$\widetilde{\bm{h}}$ is written as the composition of a mean component
$\overline{\bm{h}}\in\text{Diff}(\overline{Q},Q_{0})$ and a fluctuating
component $\widehat{\bm{\tau}}\in\text{Diff}(Q,\overline{Q})$, i.e.
$\widetilde{\bm{h}}=\overline{\bm{h}}\circ\widehat{\bm{\tau}}$. Additionally,
and in order to uniquely specify $\widehat{\bm{\tau}}$, we consider
$\widehat{\bm{\tau}}$ to be a near-identity transformation of the form

$\widehat{\bm{\tau}}(\bm{x})=\bm{x}+\epsilon^{2}\widehat{\bm{\alpha}}(\bm{x}),$
(206)

where $\widehat{\bm{\alpha}}:Q\rightarrow\mathbb{R}^{3}$ satisfies
$\fint\widehat{\bm{\alpha}}\,d\theta=0$. Finally, we shall parameterize the
Lagrange multiplier $\widetilde{\chi}$ according to

$\widetilde{\chi}=\overline{\chi}+\epsilon^{2}\widehat{\chi},$ (207)

where $\fint\widehat{\chi}\,d\theta=0$.

######  Proposition 9.

With the parameterizations given in Eqs. (167), (206), and (207), Eqs.
(197)–(200), together with the dispersion relation (183), are equivalent to a
fast-slow dynamical system.

###### Proof.

We begin by inspecting the equations of motion for the mean fields. Upon
$\theta$-averaging Eqs. (197)–(200), we immediately obtain

$\displaystyle\partial_{t}\overline{\bm{h}}=-\frac{\overline{\bm{p}}}{\overline{\rho}}\cdot\bm{\nabla}\overline{\bm{h}}+\mathcal{O}(\epsilon^{2})$
(208)
$\displaystyle\partial_{t}\overline{\bm{p}}=-\bm{\nabla}\cdot\overline{\left(\frac{\widetilde{\bm{p}}\otimes\widetilde{\bm{p}}}{\widetilde{\rho}}\right)}-c_{s}^{2}\bm{\nabla}\overline{\rho}$
(209)
$\displaystyle\partial_{t}\overline{\rho}=-\bm{\nabla}\cdot\overline{\bm{p}}$
(210)
$\displaystyle\partial_{t}\overline{\chi}=-\overline{\left(\frac{\widetilde{\bm{p}}}{\widetilde{\rho}}\right)}\cdot\bm{\nabla}\overline{\chi}-\epsilon\overline{\frac{\widetilde{\bm{p}}}{\widetilde{\rho}}\cdot\bm{\nabla}S\partial_{\theta}\widehat{\chi}}-\overline{\frac{|\widetilde{\bm{p}}|^{2}}{2\widetilde{\rho}^{2}}}+c_{s}^{2}\overline{\ln\left(\frac{\widetilde{\rho}}{\rho_{0}}\right)}+c_{s}^{2},$
(211)

where in the first equation we used
$\overline{\partial_{t}^{S/\epsilon}\widetilde{\bm{h}}}=\partial_{t}\overline{\bm{h}}+\mathcal{O}(\epsilon^{4})$
and
$\overline{(\widetilde{\bm{p}}/\widetilde{\rho})\cdot\bm{\nabla}^{S/\epsilon}\widetilde{\bm{h}}}=-(\overline{\bm{p}}/\overline{\rho})\cdot\bm{\nabla}\overline{\bm{h}}+\mathcal{O}(\epsilon^{2})$.
We have also introduced the shorthand notation $\overline{Q}=\fint Q\,d\theta$
for denoting averages over $\theta$. When comparing Eqs. (208)–(211) to
Definition 14, it so far seems that the variables
$(\overline{\bm{h}},\overline{\bm{p}},\overline{\rho},\overline{\chi})\in\mathcal{C}_{0}$
should be included amongst the slow variables. Additionally, according to the
dispersion relation (183), the time derivative $\partial_{t}S=O(1)$,
suggesting that $S$ should be a slow variable.

Let us next examine the dynamical equations for the fluctuating quantities. A
straightforward calculation leads to

$\displaystyle\epsilon\partial_{t}\widehat{\bm{\alpha}}=-\left(\partial_{t}S+\frac{\overline{\bm{p}}}{\overline{\rho}}\cdot\bm{\nabla}S\right)\partial_{\theta}\widehat{\bm{\alpha}}-\left(\frac{\widehat{\bm{p}}}{\overline{\rho}}-\frac{\overline{\bm{p}}}{\overline{\rho}}\frac{\widehat{\rho}}{\overline{\rho}}\right)+\mathcal{O}(\epsilon)$
(212)
$\displaystyle\epsilon\partial_{t}\widehat{\bm{p}}=-\left[\left(\partial_{t}S+\frac{\overline{\bm{p}}}{\overline{\rho}}\cdot\bm{\nabla}S\right)\mathbb{I}+\frac{\overline{\bm{p}}\otimes\bm{\nabla}S}{\overline{\rho}}\right]\cdot\partial_{\theta}\widehat{\bm{p}}+\left(\frac{\bm{\nabla}S\cdot\overline{\bm{p}}}{\overline{\rho}^{2}}\,\overline{\bm{p}}-c_{s}^{2}\bm{\nabla}S\right)\partial_{\theta}\widehat{\rho}+\mathcal{O}(\epsilon)$
(213)
$\displaystyle\epsilon\partial_{t}\widehat{\rho}=-\partial_{t}S\,\partial_{\theta}\widehat{\rho}-\bm{\nabla}S\cdot\partial_{\theta}\widehat{\bm{p}}+\mathcal{O}(\epsilon)$
(214)
$\displaystyle\epsilon\partial_{t}\widehat{\chi}=-\left(\partial_{t}S+\frac{\overline{\bm{p}}}{\overline{\rho}}\cdot\bm{\nabla}S\right)\partial_{\theta}\widehat{\chi}-\left(\frac{\widehat{\bm{p}}}{\overline{\rho}}-\frac{\overline{\bm{p}}}{\overline{\rho}}\frac{\widehat{\rho}}{\overline{\rho}}\right)\cdot\bm{\nabla}\overline{\chi}-\frac{\overline{\bm{p}}\cdot\widehat{\bm{p}}}{\overline{\rho}^{2}}+\frac{|\overline{\bm{p}}|^{2}}{\overline{\rho}^{2}}\frac{\widehat{\rho}}{\overline{\rho}}+c_{s}^{2}\,\frac{\widehat{\rho}}{\overline{\rho}}+\mathcal{O}(\epsilon),$
(215)

where we have omitted $\mathcal{O}(\epsilon)$ terms related to nonlinearities
in the fluctuations and $\mathcal{O}(\epsilon)$ terms involving spatial
derivatives. These omissions are motivated by the fact that, in order to prove
the singularly-perturbed dynamical system $\epsilon\dot{y}=f_{\epsilon}(x,y)$,
$\dot{x}=g_{\epsilon}(x,y)$ is in fact a fast-slow system, it is enough to
check that $f_{\epsilon},g_{\epsilon}=O(1)$ and that $D_{y}f_{0}$ is
invertible along the zero level of $f_{0}$.

At first glance, Eqs. (212)–(215) seem to suggest that
$\widehat{\bm{\alpha}},\widehat{\bm{p}},\widehat{\rho}$ and $\widehat{\chi}$
should be fast variables. Indeed, the time derivative of each of these fields
is generically $O(\epsilon^{-1})$. However, there happens to be a non-trivial
combination of these quantities whose time derivative is
$\mathcal{O}(\epsilon)$. It is straightforward to verify that the field
$\widehat{\lambda}\colon Q\times S_{1}\to\mathbb{R}$ given by

$\displaystyle\widehat{\lambda}=\widehat{\rho}+\frac{\widehat{\bm{p}}\cdot\nabla
S}{c_{s}|\nabla S|-\overline{\bm{p}}\cdot\nabla S/\overline{\rho}}$ (216)

satisfies $\partial_{t}\widehat{\lambda}=O(1)$. This suggests that a viable
set of slow variables might be
$x=(\overline{\bm{h}},\overline{\bm{p}},\overline{\rho},\overline{\chi},\widehat{\lambda})$
with corresponding fast variables
$y=(\widehat{\bm{\alpha}},\widehat{\bm{p}},\widehat{\chi})$. The rest of the
proof will be devoted to showing that, when expressed in terms of $x$ and $y$,
Eqs. (197)–(200), together with the dispersion relation (183), do in fact
comprise a fast-slow dynamical system.

In order to write Eqs. (197)–(200) and the dispersion relation (183) in terms
of $x$ and $y$, it is only necessary to exchange the dependent variable
$\widehat{\rho}$ with the new dependent variable $\widehat{\lambda}$. Because
this change of dependent variables is independent of $\epsilon$, our
calculations so far already demonstrate that $dx/dt=O(\epsilon)$. In order to
prove that we have identified the correct fast and slow variables, we
therefore only have to show that $\epsilon dy/dt=f_{0}(x,y)+O(\epsilon)$ and
that $D_{y}f_{0}(x,y)$ is invertible along the zero level of $f_{0}$.

In order to identify $f_{0}(x,y)$, we substitute the definition of
$\widehat{\lambda}$ given by Eq. (216) into Eqs. (212), (213), and (215),
thereby obtaining

$\displaystyle\epsilon\partial_{t}\widehat{\bm{\alpha}}=$ $\displaystyle
c_{s}|\nabla
S|\partial_{\theta}\widehat{\bm{\alpha}}-\left[\mathbb{I}+\frac{1}{1-\bm{e}_{\bm{k}}\cdot{(\overline{\bm{p}}/\overline{\rho})}/{c_{s}}}\frac{(\overline{\bm{p}}/\overline{\rho})}{c_{s}}\otimes\bm{e}_{\bm{k}}\right]\cdot\frac{\widehat{\bm{p}}}{\overline{\rho}}+\frac{\widehat{\lambda}}{\overline{\rho}}\frac{\overline{\bm{p}}}{\overline{\rho}}+O(\epsilon)$
(217) $\displaystyle\epsilon\partial_{t}\widehat{\bm{p}}=$
$\displaystyle\left[c_{s}|\nabla
S|\left(\mathbb{I}+\bm{e}_{\bm{k}}\otimes\bm{e}_{\bm{k}}\right)-\frac{(\mathbb{I}-\bm{e}_{\bm{k}}\otimes\bm{e}_{\bm{k}})\cdot(\overline{\bm{p}}/\overline{\rho})\otimes\nabla
S}{1-\bm{e}_{\bm{k}}\cdot{(\overline{\bm{p}}/\overline{\rho})}/{c_{s}}}\right]\cdot\partial_{\theta}\widehat{\bm{p}}$
$\displaystyle-\nabla
S\cdot\left[c_{s}^{2}\mathbb{I}-\frac{\overline{\bm{p}}\otimes\overline{\bm{p}}}{\overline{\rho}^{2}}\right]\partial_{\theta}\widehat{\lambda}+O(\epsilon)$
(218) $\displaystyle\epsilon\partial_{t}\widehat{\chi}=$ $\displaystyle
c_{s}|\nabla
S|\partial_{\theta}\widehat{\chi}-\left[\frac{\overline{\bm{p}}}{\overline{\rho}}+\nabla\overline{\chi}+\frac{\left[(\overline{\bm{p}}/\overline{\rho})\cdot\nabla\overline{\chi}+|\overline{\bm{p}}/\overline{\rho}|^{2}+c_{s}^{2}\right]}{c_{s}-(\overline{\bm{p}}/\overline{\rho})\cdot\bm{e}_{\bm{k}}}\bm{e}_{\bm{k}}\right]\cdot\frac{\widehat{\bm{p}}}{\overline{\rho}}$
$\displaystyle+\left[(\overline{\bm{p}}/\overline{\rho})\cdot\nabla\overline{\chi}+|\overline{\bm{p}}/\overline{\rho}|^{2}+c_{s}^{2}\right]\frac{\widehat{\lambda}}{\overline{\rho}}.$
(219)

These expressions show that $f_{0}(x,y)$ is of the form
$f_{0}(x,y)=A(x)[y]+C(x)$, where $A(x):Y\rightarrow Y$ is a linear map and
$C(x)\in Y$ is independent of $y$. In particular, the derivative of $f_{0}$
with respect to $y$ is given by $D_{y}f_{0}(x,y)=A(x)$. Thus, for any
$(x,y)\in X\times Y$, $D_{y}f_{0}(x,y)$ is invertible if and only if $A(x)$ is
invertible.

We will now complete the proof by showing that $A(x)$ is invertible for all
$x\in X$ that satisfy $\nabla S(\bm{x})\neq 0$ for all $\bm{x}\in Q$. Fix
$\delta
y=(\delta\widehat{\bm{\alpha}},\delta\widehat{\bm{p}},\delta\widehat{\chi})\in
Y$. If there is a $y=(\widehat{\bm{\alpha}},\widehat{\bm{p}},\widehat{\chi})$
that solves the equation $A(x)[y]=\delta y$, then, by Eqs. (217)-(219), $y$
must satisfy

$\displaystyle\delta\widehat{\bm{\alpha}}=$ $\displaystyle c_{s}|\nabla
S|\partial_{\theta}\widehat{\bm{\alpha}}-\left[\mathbb{I}+\frac{1}{1-\bm{e}_{\bm{k}}\cdot{(\overline{\bm{p}}/\overline{\rho})}/{c_{s}}}\frac{(\overline{\bm{p}}/\overline{\rho})}{c_{s}}\otimes\bm{e}_{\bm{k}}\right]\cdot\frac{\widehat{\bm{p}}}{\overline{\rho}}$
(220) $\displaystyle\delta\widehat{\bm{p}}=$ $\displaystyle\left[c_{s}|\nabla
S|\left(\mathbb{I}+\bm{e}_{\bm{k}}\otimes\bm{e}_{\bm{k}}\right)-\frac{(\mathbb{I}-\bm{e}_{\bm{k}}\otimes\bm{e}_{\bm{k}})\cdot(\overline{\bm{p}}/\overline{\rho})\otimes\nabla
S}{1-\bm{e}_{\bm{k}}\cdot{(\overline{\bm{p}}/\overline{\rho})}/{c_{s}}}\right]\cdot\partial_{\theta}\widehat{\bm{p}}$
(221) $\displaystyle\delta\widehat{\chi}=$ $\displaystyle c_{s}|\nabla
S|\partial_{\theta}\widehat{\chi}-\left[\frac{\overline{\bm{p}}}{\overline{\rho}}+\nabla\overline{\chi}+\frac{\left[(\overline{\bm{p}}/\overline{\rho})\cdot\nabla\overline{\chi}+|\overline{\bm{p}}/\overline{\rho}|^{2}+c_{s}^{2}\right]}{c_{s}-(\overline{\bm{p}}/\overline{\rho})\cdot\bm{e}_{\bm{k}}}\bm{e}_{\bm{k}}\right]\cdot\frac{\widehat{\bm{p}}}{\overline{\rho}}.$
(222)

By decomposing Eq. (221) into components parallel and perpendicular to $\nabla
S$, it is straightforward to show that $\partial_{\theta}\widehat{\bm{p}}$
must be given by

$\displaystyle\partial_{\theta}\widehat{\bm{p}}=$
$\displaystyle\frac{1}{2c_{s}|\nabla
S|}\left[2\mathbb{I}-\bm{e}_{\bm{k}}\otimes\bm{e}_{\bm{k}}+[\mathbb{I}-\bm{e}_{\bm{k}}\otimes\bm{e}_{\bm{k}}]\cdot\frac{(\overline{\bm{p}}/\overline{\rho})\otimes\bm{e}_{\bm{k}}}{c_{s}-\bm{e}_{\bm{k}}\cdot(\overline{\bm{p}}/\overline{\rho})}\right]\cdot\delta\widehat{\bm{p}}$
$\displaystyle\equiv$ $\displaystyle\frac{1}{c_{s}|\nabla
S|}{\mathbb{T}}\cdot\delta\widehat{\bm{p}},$ (223)

which implies $\widehat{\bm{p}}=(c_{s}|\nabla S|)^{-1}\mathbb{T}\cdot
I[\delta\widehat{\bm{p}}]$, where $I[\delta\widehat{\bm{p}}]$ is the unique
$\theta$-antiderivative of $\delta\widehat{\bm{p}}$ with zero mean, i.e.

$\displaystyle
I[\delta\widehat{\bm{p}}](\theta)=\int_{0}^{\theta}\widehat{\bm{p}}(\overline{\theta})\,d\overline{\theta}-\fint\left(\int_{0}^{\theta}\widehat{\bm{p}}(\overline{\theta})\,d\overline{\theta}\right)\,d\theta.$
(224)

By substituting this expression for $\widehat{\bm{p}}$ into Eqs. (220) and
(222), it follows that $\widehat{\bm{\alpha}}$ and $\widehat{\chi}$ must be
given by

$\displaystyle\widehat{\bm{\alpha}}=$ $\displaystyle\frac{1}{c_{s}|\nabla
S|}I[\delta\widehat{\bm{\alpha}}]+\frac{1}{(c_{s}|\nabla
S|)^{2}}\left[\mathbb{I}+\frac{(\overline{\bm{p}}/\overline{\rho})\otimes\bm{e}_{\bm{k}}}{c_{s}-\bm{e}_{\bm{k}}\cdot(\overline{\bm{p}}/\overline{\rho})}\right]\cdot\mathbb{T}\cdot
I^{2}[\delta\widehat{\bm{p}}/\overline{\rho}]$ (225)
$\displaystyle\widehat{\chi}=$ $\displaystyle\frac{1}{c_{s}|\nabla
S|}I[\delta\widehat{\chi}]+\frac{1}{(c_{s}|\nabla
S|^{2})}\left[\frac{\overline{\bm{p}}}{\overline{\rho}}+\nabla\overline{\chi}+\frac{\left[(\overline{\bm{p}}/\overline{\rho})\cdot\nabla\overline{\chi}+|\overline{\bm{p}}/\overline{\rho}|^{2}+c_{s}^{2}\right]}{c_{s}-(\overline{\bm{p}}/\overline{\rho})\cdot\bm{e}_{\bm{k}}}\bm{e}_{\bm{k}}\right]\cdot\mathbb{T}\cdot
I^{2}\left[\frac{\delta\widehat{\bm{p}}}{\overline{\rho}}\right],$ (226)

where
$I^{2}[\delta\widehat{\bm{p}}/\overline{\rho}]=I[I[\delta\widehat{\bm{p}}/\overline{\rho}]]$
denotes the antiderivative operator applied two times. Thus, if there is a $y$
satisfying $A(x)[y]=\delta y$, then that $y$ is unique. Conversely, by
substituting the above expressions for $y$ back into $A(x)[y]=\delta y$, we
conclude that a solution $y$ exists for any $\delta y$. Therefore $A(x)$ is
invertible as claimed. ∎

######  Remark 11.

In the above Proposition, the dispersion relation (183) plays two important
roles. First, it supplies an evolution equation for $S$. This is necessary for
the Proposition to work because if the dispersion relation was not imposed,
then Eqs. (197)–(200) would not specify a dynamical system, let alone a fast-
slow dynamical system. Second, it ensures that the system supports wave motion
whose asymptotic behavior is captured by the NL-WKB ansatz.

######  Remark 12.

As is true of all fast-slow systems, Eqs. (197)–(200), together with the
dispersion relation (183), admit a slow manifold. In terms of the fast and
slow variables identified in the proof of the Proposition, the slow manifold
is a subset of $X\times Y$ of the form $I_{\epsilon}=\\{(x,y)\in X\times Y\mid
y=y^{\star}_{\epsilon}(x)\\}$, where $y^{\star}_{\epsilon}$ is the so-called
slaving function. Using the expressions for the inverse of $A(x)$ from the
proof, it is straightforward to find the leading-order term in slaving
function
$y^{\star}_{0}(x)=(\widehat{\bm{\alpha}}_{0}^{\star},\widehat{\bm{p}}_{0}^{\star},\widehat{\chi}_{0}^{\star})$.
We have

$\displaystyle\widehat{\bm{\alpha}}_{0}^{\star}=$
$\displaystyle\,\frac{1}{2|\nabla
S|}\left[1-\frac{\bm{e}_{\bm{k}}\cdot(\overline{\bm{p}}/\overline{\rho})}{c_{s}}\right]I[\widehat{\lambda}/\overline{\rho}]\bm{e}_{\bm{k}}$
(227) $\displaystyle\widehat{\bm{p}}_{0}^{\star}=$
$\displaystyle\,\frac{1}{2}\widehat{\lambda}c_{s}\left[1-\left(\frac{\bm{e}_{\bm{k}}\cdot(\overline{\bm{p}}/\overline{\rho})}{c_{s}}\right)^{2}\right]\bm{e}_{\bm{k}}+\frac{1}{2}\widehat{\lambda}c_{s}\left[1-\left(\frac{\bm{e}_{\bm{k}}\cdot(\overline{\bm{p}}/\overline{\rho})}{c_{s}}\right)\right][\mathbb{I}-\bm{e}_{\bm{k}}\otimes\bm{e}_{\bm{k}}]\cdot\frac{\overline{\bm{p}}}{c_{s}\overline{\rho}}$
(228) $\displaystyle\widehat{\chi}_{0}^{\star}=$
$\displaystyle\,\frac{1}{2|\nabla
S|}\left[\bm{e}_{\bm{k}}\cdot(\overline{\bm{p}}/\overline{\rho}+\nabla\overline{\chi})-c_{s}\right]\left[1-\frac{\bm{e}_{\bm{k}}\cdot(\overline{\bm{p}}/\overline{\rho})}{c_{s}}\right]I[\widehat{\lambda}/\overline{\rho}].$
(229)

While it was convenient to introduce the dependent variable
$\widehat{\lambda}$ for the sake of showing equivalence with a fast-slow
system, now that the existence of the slow manifold has been established, we
are free to express the slow manifold in terms of $\widehat{\rho}$ instead of
$\widehat{\lambda}$. By a slight abuse of notation, the slow manifold may
written in terms of $\widehat{\rho}$ as

$\displaystyle
I_{\epsilon}=\\{(\overline{\bm{h}},\overline{\bm{p}},\overline{\chi},\widehat{\rho},S,\widehat{\bm{\alpha}},\widehat{\bm{p}},\widehat{\chi})\mid\widehat{\bm{\alpha}}=\widehat{\bm{\alpha}}^{\star}_{\epsilon},\,\widehat{\bm{p}}=\widehat{\bm{p}}^{\star}_{\epsilon},\,\widehat{\chi}=\widehat{\chi}_{\epsilon}^{\star}\\},$
(230)

where now
$y_{\epsilon}^{*}=(\widehat{\bm{\alpha}}^{\star}_{\epsilon},\widehat{\bm{p}}^{\star}_{\epsilon},\widehat{\chi}_{\epsilon}^{\star})$
is a function of
$(\overline{\bm{h}},\overline{\bm{p}},\overline{\chi},\widehat{\rho},S)$. In
this alternate representation, the leading-order terms in the slaving
functions are given by:

$\displaystyle\widehat{\bm{\alpha}}_{0}^{\star}=$
$\displaystyle\frac{\bm{e}_{\bm{k}}}{|\bm{\nabla}S|}I[\widehat{\rho}/\overline{\rho}]$
(231) $\displaystyle\widehat{\bm{p}}_{0}^{\star}=$
$\displaystyle\overline{\bm{p}}\,\frac{\widehat{\rho}}{\overline{\rho}}+c_{s}\widehat{\rho}\,\bm{e}_{\bm{k}}$
(232) $\displaystyle\widehat{\chi}_{0}^{\star}=$
$\displaystyle\frac{1}{|\bm{\nabla}S|}\left(\bm{e}_{\bm{k}}\cdot\bm{\nabla}\overline{\chi}+\frac{\overline{\bm{p}}}{\overline{\rho}}\cdot\bm{e}_{\bm{k}}-c_{s}\right)I[\widehat{\rho}/\overline{\rho}].$
(233)

We remind the reader that the antiderivative operator $I$ was defined in Eq.
(224).

######  Remark 13.

We are now in a good position to prove Proposition 8.

###### proof of Proposition 8.

In Corollary 3, we demonstrated that the NL–WKB extension of the isothermal
fluid equations (165) and (166) is variational. Additionally, Theorem 3 shows
that all extLBEP fluid equations imply a wave-action conservation equation
(121). Upon substituting the action (196) and the Hamiltonian (186) into Eq.
(121), we obtain

$\partial_{t}\fint\widetilde{\mathcal{I}}\,d\theta+\bm{\nabla}\cdot\fint\frac{\widetilde{\bm{p}}}{\widetilde{\rho}}\,\widetilde{\mathcal{I}}\,d\theta=0.$
(234)

The specific wave action density $\widetilde{\mathcal{I}}$ is given by Eq.
(116), which we rewrite below for clarity:

$\widetilde{\mathcal{I}}=\widetilde{\rho}\,\partial_{\theta}\widetilde{\chi}+(\widetilde{\bm{p}}+\widetilde{\rho}\,\bm{\nabla}^{S}\widetilde{\chi})\cdot\widetilde{\bm{\zeta}}.$
(235)

Here
$\widetilde{\bm{\zeta}}=-(\partial_{\theta}\widetilde{\bm{h}})\cdot(\bm{\nabla}\widetilde{\bm{h}}+\epsilon^{-1}\bm{\nabla}S\otimes\partial_{\theta}\widetilde{\bm{h}})^{-1}$.
Let us now calculate the terms in Eqs. (234) and (235) by substituting the
leading-order slaving functions (231)–(233). Specifically, when inserting
$\widetilde{\bm{h}}=\overline{\bm{h}}\circ\widehat{\bm{\tau}}$ and
$\widehat{\bm{\tau}}(\bm{x})\simeq\bm{x}+\epsilon^{2}\widehat{\bm{\alpha}}_{0}^{\star}(\bm{x})$
into $\widetilde{\bm{\zeta}}$, we obtain

$\widetilde{\bm{\zeta}}=-\epsilon^{2}\partial_{\theta}\widehat{\bm{\alpha}}_{0}^{\star}+\epsilon^{3}(\partial_{\theta}\widehat{\bm{\alpha}}_{0}^{\star}\cdot\bm{\nabla}S)\partial_{\theta}\widehat{\bm{\alpha}}_{0}^{\star}+\mathcal{O}(\epsilon^{4}).$
(236)

We then substitute this result as well as the parameterizations (167), (206),
and (207) and the leading-order slaving functions (231)–(233) into the first
term in Eq. (234). We obtain

$\fint\widetilde{\mathcal{I}}\,d\theta=-\epsilon^{3}\fint\overline{\rho}\,\frac{c_{s}}{|\bm{\nabla}S|}\left(\frac{\widehat{\rho}}{\overline{\rho}}\right)^{2}+O(\epsilon^{4})=-\epsilon^{3}\mathcal{I}+\mathcal{O}(\epsilon^{4}),$
(237)

where $\mathcal{I}$ is the wave action density defined in Eq. (176). For the
second term in Eq. (234), a similar calculation leads to

$\displaystyle\fint\frac{\widetilde{\bm{p}}}{\widetilde{\rho}}\,\widetilde{\mathcal{I}}\,d\theta$
$\displaystyle=\fint\left(\frac{\overline{\bm{p}}}{\overline{\rho}}+\epsilon\frac{\widehat{\bm{p}}^{\star}_{0}}{\overline{\rho}}-\epsilon\frac{\overline{\bm{p}}}{\overline{\rho}}\frac{\widehat{\rho}}{\overline{\rho}}+\mathcal{O}(\epsilon^{2})\right)\widetilde{\mathcal{I}}\,d\theta$
$\displaystyle=\frac{\overline{\bm{p}}}{\overline{\rho}}\fint\widetilde{\mathcal{I}}\,d\theta+\epsilon\fint\left(\frac{\widehat{\bm{p}}^{\star}_{0}}{\overline{\rho}}-\frac{\overline{\bm{p}}}{\overline{\rho}}\frac{\widehat{\rho}}{\overline{\rho}}+\mathcal{O}(\epsilon)\right)\widetilde{\mathcal{I}}\,d\theta$
$\displaystyle=-\epsilon^{3}\left(\frac{\overline{\bm{p}}}{\overline{\rho}}+c_{s}\bm{e}_{\bm{k}}\right)\mathcal{I}+\mathcal{O}(\epsilon^{4}).$
(238)

Finally, inserting Eqs. (237) and (238) into Eq. (234) leads to our claim in
Proposition 8. ∎

###  VI.3 Calculation of the effective action $\overline{\mathsf{A}}_{\rm T}$

In the previous section Sec. VI.2, we gave the general arguments explaining
why the wave–mean-flow equations (180)–(183) are variational. We also broadly
discussed how to calculate the effective action (184) for wave–mean-flow
interactions by using slow-manifold reduction (see Theorem 6). In this
section, we present some of the technical details in obtaining Eq. (184).

We start from the NL–WKB extended action (196) for the isothermal fluid
equations. As was done in the proof of Theorem 4, we apply a phase shift to
the fields. This gives

$\displaystyle\widetilde{\mathsf{L}}_{\rm T}$
$\displaystyle=\int_{Q}\fint\left(\widetilde{\bm{p}}^{S/\epsilon}\cdot\widetilde{\bm{v}}^{S/\epsilon}+\widetilde{\rho}^{S/\epsilon}\big{(}\partial_{t}\widetilde{\chi}^{S/\epsilon}+\widetilde{\bm{v}}^{S/\epsilon}\cdot\bm{\nabla}\widetilde{\chi}^{S/\epsilon}\big{)}\right)\,d\theta\,d\bm{x}-\int_{Q}\fint\mathcal{H}_{\rm
T}(\widetilde{\bm{p}}^{S/\epsilon},\widetilde{\rho}^{S/\epsilon})\,d\theta\,d\bm{x},$
(239)

where the superscript “$S/\epsilon$” denotes that $\theta$ angle is shifted by
$S/\epsilon$ (see Definition 11).

As it was explained in Theorem 6, after a slow manifold $I_{\epsilon}$ has
been identified, one can restrict the action of the parent fast-slow system
onto the slow manifold $I_{\epsilon}$ in order to obtain an effective action
for the slow variables only. Following this same procedure, we substitute the
parameterizations (167), (206), and (207) into Eq. (239). We then restrict the
fast variables $(\widehat{\bm{\alpha}},\widehat{\bm{p}},\widehat{\chi})$ to
the slow manifold by using Eqs. (231)–(233). This leads to

$\displaystyle\widetilde{\mathsf{L}}_{\rm T}$
$\displaystyle=\int_{Q}\fint\left(\widetilde{\bm{p}}^{\star
S/\epsilon}_{0}\cdot\widetilde{\bm{v}}^{\star
S/\epsilon}_{0}+\widetilde{\rho}^{S/\epsilon}\big{(}\partial_{t}\widetilde{\chi}^{\star
S/\epsilon}_{0}+\widetilde{\bm{v}}^{\star
S/\epsilon}_{0}\cdot\bm{\nabla}\widetilde{\chi}^{\star
S/\epsilon}_{0}\big{)}\right)\,d\theta\,d\bm{x}-\int_{Q}\fint\mathcal{H}_{\rm
T}(\widetilde{\bm{p}}^{\star
S/\epsilon}_{0},\widetilde{\rho}^{S/\epsilon})\,d\theta\,d\bm{x},$ (240)

where $\widetilde{\bm{p}}^{\star
S/\epsilon}_{0}=\overline{\bm{p}}+\epsilon\widehat{\bm{p}}^{\star
S/\epsilon}_{0}$ and similarly for the rest of the variables restricted to the
slow manifold. From hereon, we shall consider the Lagrangian (240) restricted
to the lowest-order slaving functions. Hence, to simplify our notation, we
shall omit the “0” subscript when referring to the lowest-order slaving
functions (231)–(233).

Before explicitly substituting the expressions for
$\widehat{\bm{\alpha}}^{\star}$, $\widehat{\bm{p}}^{\star}$, and
$\widehat{\chi}^{\star}$ into Eq. (240), it convenient to first perform a
variable transformation. (The following transformation is closely related to
the well-known oscillation-center transform used in kinetic theories for
plasma–wave interactions.Dewar (1973)) First, we note that the velocity

$\widetilde{\bm{v}}^{\star S/\epsilon}=-\partial_{t}\widetilde{\bm{h}}^{\star
S/\epsilon}\cdot(\bm{\nabla}\widetilde{\bm{h}}^{\star
S/\epsilon})^{-1}=-\partial_{t}(\overline{\bm{h}}\circ\widetilde{\bm{\tau}}^{\star
S/\epsilon})\cdot[\bm{\nabla}(\overline{\bm{h}}\circ\widetilde{\bm{\tau}}^{\star
S/\epsilon})]^{-1}$ (241)

can be written as

$\widetilde{\bm{v}}^{\star S/\epsilon}=(\widetilde{\tau}^{\star
S/\epsilon})^{*}(\overline{\bm{v}}-\widetilde{\bm{\nu}}^{\star S/\epsilon}),$
(242)

where $(\widetilde{\tau}^{\star S/\epsilon})^{*}$ is the pullback associated
to $\widetilde{\bm{\tau}}^{\star S/\epsilon}$, $\overline{\bm{v}}$ is the mean
Lagrangian velocity

$\overline{\bm{v}}\doteq-\partial_{t}\overline{\bm{h}}\cdot(\bm{\nabla}\overline{\bm{h}})^{-1},$
(243)

and $\widetilde{\bm{\nu}}^{\star S/\epsilon}$ is the velocity associated to
$\widetilde{\bm{\tau}}^{\star S/\epsilon}$:

$\widetilde{\bm{\nu}}^{\star
S/\epsilon}\doteq\partial_{t}(\widetilde{\bm{\tau}}^{\star
S/\epsilon})\circ(\widetilde{\bm{\tau}}^{\star S/\epsilon})^{-1}.$ (244)

In order to simplify the expression for the velocity (242), it is convenient
to apply the pushforward $\widetilde{\tau}^{\star S/\epsilon}_{*}$ to the
symplectic part of the Lagrangian (240). This leads to

$\displaystyle\widetilde{\mathsf{L}}_{\rm T}$
$\displaystyle=\int_{Q}\fint\left(\widetilde{\tau}^{\star
S/\epsilon}_{*}\widetilde{\bm{p}}^{\star
S/\epsilon}\cdot\left(\overline{\bm{v}}-\widetilde{\bm{\nu}}^{\star
S/\epsilon}\right)+\widetilde{\tau}^{\star
S/\epsilon}_{*}\widetilde{\rho}^{S/\epsilon}\big{(}\partial_{t}\widetilde{\varphi}^{\star
S/\epsilon}+\overline{\bm{v}}\cdot\bm{\nabla}\widetilde{\varphi}^{\star
S/\epsilon}\big{)}\right)\,d\theta\,d^{3}\bm{x}$
$\displaystyle\qquad-\int_{Q}\fint\mathcal{H}(\widetilde{\bm{p}}^{\star
S/\epsilon},\widetilde{\rho}^{S/\epsilon})\,d\theta\,\mathrm{d}^{3}\bm{x},$
(245)

Some remarks should be given on the symbols appearing in Eq. (245). First, the
term $\widetilde{\tau}^{\star S/\epsilon}_{*}\widetilde{\bm{p}}^{S/\epsilon}$
is understood as the $\widetilde{\tau}^{\star S/\epsilon}_{*}$ acting on
$\widetilde{\bm{p}}^{\star S/\epsilon}$ which is treated as a one-form density
in the domain $Q$. Similarly, $\widetilde{\tau}^{\star
S/\epsilon}_{*}\widetilde{\rho}^{\star S/\epsilon}$ is interpreted as the
pull-back $\widetilde{\tau}^{\star S/\epsilon}_{*}$ acting on
$\widetilde{\rho}^{S/\epsilon}$ treated as a three-form in the domain $Q$. We
also introduced a new variable

$\widetilde{\varphi}^{\star S/\epsilon}=\widetilde{\tau}^{\star
S/\epsilon}_{*}\widetilde{\chi}^{\star S/\epsilon},$ (246)

which is a scalar in the domain $Q$. Finally, we used the Lie derivative
theorem to establish the identities $\partial_{t}\widetilde{\chi}^{\star
S/\epsilon}=\partial_{t}[(\widetilde{\tau}^{\star
S/\epsilon})^{*}\widetilde{\varphi}^{\star
S/\epsilon}]=(\widetilde{\tau}^{\star
S/\epsilon})^{*}\partial_{t}\widetilde{\varphi}^{\star
S/\epsilon}+(\widetilde{\tau}^{\star
S/\epsilon})^{*}\mathfrak{L}_{\widetilde{\bm{\nu}}^{\star
S/\epsilon}}\widetilde{\varphi}^{\star S/\epsilon}$ and
$\mathfrak{L}_{\widetilde{\bm{v}}^{\star S/\epsilon}}\widetilde{\chi}^{\star
S/\epsilon}=(\widetilde{\tau}^{\star
S/\epsilon})^{*}\mathfrak{L}_{\overline{\bm{v}}-\widetilde{\bm{\nu}}^{\star
S/\epsilon}}\widetilde{\varphi}^{\star S/\epsilon}$. Note that, by applying
the pushforward $\widetilde{\tau}^{\star S/\epsilon}_{*}$, we were able to
replace the oscillating velocity $\widetilde{\bm{v}}^{\star S/\epsilon}$
appearing in the $\widetilde{\bm{v}}^{\star
S/\epsilon}\cdot\bm{\nabla}\widetilde{\chi}^{\star S/\epsilon}$ term of Eq.
(239) with the mean Lagrangian velocity $\overline{\bm{v}}$. This was
originally the main motivation for the transformation.

The next step is to explicitly calculate the terms
$\widetilde{\bm{\nu}}^{\star S/\epsilon}$, $\widetilde{\tau}^{\star
S/\epsilon}_{*}\widetilde{\bm{p}}^{\star S/\epsilon}$,
$\widetilde{\tau}^{\star S/\epsilon}_{*}\widetilde{\rho}^{S/\epsilon}$, and
$\widetilde{\varphi}^{\star S/\epsilon}$ appearing in Eq. (245). Let us first
start by calculating $\widetilde{\bm{\nu}}^{\star S/\epsilon}$ in Eq. (244).
Since $(\widetilde{\bm{\tau}}^{\star S/\epsilon})^{-1}$ is a near-identity
transformation, one can verify that $(\widetilde{\bm{\tau}}^{\star
S/\epsilon})^{-1}(\bm{x})=\bm{x}-\epsilon^{2}\widehat{\bm{\alpha}}^{\star
S/\epsilon}(\bm{x})+\mathcal{O}(\epsilon^{3})$. Substituting this into Eq.
(244) gives

$\displaystyle\widetilde{\bm{\nu}}^{\star S/\epsilon}$
$\displaystyle=[\epsilon(\partial_{t}S)\partial_{\theta}\widehat{\bm{\alpha}}^{\star
S/\epsilon}+\epsilon^{2}(\partial_{t}\widehat{\bm{\alpha}})^{\star
S/\epsilon}]\circ(\bm{id}-\epsilon^{2}\widehat{\bm{\alpha}}^{\star
S/\epsilon})+\mathcal{O}(\epsilon^{3})$
$\displaystyle=(1-\epsilon^{2}\widehat{\bm{\alpha}}^{\star
S/\epsilon}\cdot\bm{\nabla})\left(\epsilon(\partial_{t}S)\partial_{\theta}\widehat{\bm{\alpha}}^{\star
S/\epsilon}+\epsilon^{2}(\partial_{t}\widehat{\bm{\alpha}})^{\star
S/\epsilon}\right)+\mathcal{O}(\epsilon^{3})$
$\displaystyle=\epsilon(\partial_{t}S)\partial_{\theta}\widehat{\bm{\alpha}}^{\star
S/\epsilon}+\epsilon^{2}(\partial_{t}\widehat{\bm{\alpha}})^{\star
S/\epsilon}-\epsilon^{2}(\bm{\nabla}S\cdot\widehat{\bm{\alpha}}^{\star
S/\epsilon})(\partial_{t}S)\partial_{\theta}^{2}\widehat{\bm{\alpha}}^{\star
S/\epsilon}+\mathcal{O}(\epsilon^{3})$
$\displaystyle=\epsilon(\partial_{t}S)\partial_{\theta}\widehat{\bm{\alpha}}^{\star
S/\epsilon}+\epsilon^{2}(\bm{\nabla}S\cdot\partial_{\theta}\widehat{\bm{\alpha}}^{\star
S/\epsilon})(\partial_{t}S)\partial_{\theta}\widehat{\bm{\alpha}}^{\star
S/\epsilon}$
$\displaystyle\quad+\epsilon^{2}(\partial_{t}\widehat{\bm{\alpha}}^{\star})^{S/\epsilon}-\epsilon^{2}\partial_{\theta}\left((\bm{\nabla}S\cdot\widehat{\bm{\alpha}}^{\star
S/\epsilon})(\partial_{t}S)\partial_{\theta}\widehat{\bm{\alpha}}^{\star
S/\epsilon}\right)+\mathcal{O}(\epsilon^{3})$
$\displaystyle=\epsilon(\partial_{t}S)\partial_{\theta}\widehat{\bm{\alpha}}^{\star
S/\epsilon}+\epsilon^{2}(\bm{\nabla}S\cdot\partial_{\theta}\widehat{\bm{\alpha}}^{\star
S/\epsilon})(\partial_{t}S)\partial_{\theta}\widehat{\bm{\alpha}}^{\star
S/\epsilon}+\epsilon^{2}\mathrm{Osc}.$ (247)

Here the term “$\epsilon^{2}\mathrm{Osc}$” means that we have neglected
$\mathcal{O}(\epsilon^{3})$ terms and that we have omitted writing fluctuating
terms that are $\mathcal{O}(\epsilon^{2})$ whose $\theta$-average is zero.
Since we are only calculating the effective Lagrangian up to
$\mathcal{O}(\epsilon^{2})$, it is safe to omit those terms since they will
not contribute anything once the Lagrangian (245) is explicitly
$\theta$-averaged. More specifically, the term
$\epsilon(\partial_{t}S)\partial_{\theta}\widehat{\bm{\alpha}}^{\star
S/\epsilon}$ is kept because it could later multiply another
$\mathcal{O}(\epsilon)$ term in the Lagrangian (245). The term
$\epsilon^{2}(\bm{\nabla}S\cdot\partial_{\theta}\widehat{\bm{\alpha}}^{\star
S/\epsilon})(\partial_{t}S)\partial_{\theta}\widehat{\bm{\alpha}}^{\star
S/\epsilon}$ is also kept because it is quadratic in $\widehat{\rho}$, so it
has a non-zero $\theta$-average. The term
$\epsilon^{2}(\partial_{t}\widehat{\bm{\alpha}}^{\star})^{S/\epsilon}$ is
omitted because it is oscillatory so its $\theta$-average is zero. Also, the
term
$\epsilon^{2}\partial_{\theta}((\bm{\nabla}S\cdot\widehat{\bm{\alpha}}^{\star
S/\epsilon})(\partial_{t}S)\partial_{\theta}\widehat{\bm{\alpha}}^{\star
S/\epsilon})$ is omitted because it is as a total derivative in $\theta$,
which will vanish when integrating over $\theta$. Finally, substituting the
expression for $\widehat{\bm{\alpha}}^{\star S/\epsilon}$ in Eq. (231) gives

$\widetilde{\bm{\nu}}^{\star
S/\epsilon}=\epsilon\bm{e}_{\bm{k}}\frac{\partial_{t}S}{|\bm{\nabla}S|}\frac{\widehat{\rho}^{S/\epsilon}}{\overline{\rho}}+\epsilon^{2}\bm{e}_{\bm{k}}\frac{\partial_{t}S}{|\bm{\nabla}S|}\left(\frac{\widehat{\rho}^{S/\epsilon}}{\overline{\rho}}\right)^{2},$
(248)

where $\bm{e}_{\bm{k}}\doteq\bm{\nabla}S/|\bm{\nabla}S|$.

Let us now proceed by calculating the term $\widetilde{\tau}^{\star
S/\epsilon}_{*}\widetilde{\rho}^{S/\epsilon}$ appearing in Eq. (245).
Remembering that the density should be considered as a 3-form, we obtain

$\displaystyle\widetilde{\tau}^{\star
S/\epsilon}_{*}(\widetilde{\rho}^{S/\epsilon}d^{3}\bm{x})$
$\displaystyle=\widetilde{\rho}^{S/\epsilon}\circ(\widetilde{\bm{\tau}}^{\star
S/\epsilon})^{-1}\,\det(\bm{\nabla}(\widetilde{\bm{\tau}}^{\star
S/\epsilon})^{-1})\,d^{3}\bm{x}$
$\displaystyle=\widetilde{\rho}^{S/\epsilon}\circ(\bm{id}-\epsilon^{2}\widehat{\bm{\alpha}}^{\star
S/\epsilon})\,\det(\mathbb{I}-\epsilon^{2}\bm{\nabla}\widehat{\bm{\alpha}}^{\star
S/\epsilon})\,d^{3}\bm{x}+\mathcal{O}(\epsilon^{3})$
$\displaystyle=(\widetilde{\rho}^{S/\epsilon}-\epsilon^{2}\widehat{\bm{\alpha}}^{\star
S/\epsilon}\cdot\bm{\nabla}\widetilde{\rho}^{S/\epsilon})(1-\epsilon^{2}\bm{\nabla}\cdot\widehat{\bm{\alpha}}^{\star
S/\epsilon})\,d^{3}\bm{x}+\mathcal{O}(\epsilon^{3})$
$\displaystyle=\widetilde{\rho}^{S/\epsilon}d^{3}\bm{x}-\epsilon^{2}\bm{\nabla}\cdot(\widehat{\bm{\alpha}}^{\star
S/\epsilon}\widetilde{\rho}^{S/\epsilon})\,d^{3}\bm{x}+\mathcal{O}(\epsilon^{3})$
$\displaystyle=\overline{\rho}\,d^{3}\bm{x}+\epsilon\widehat{\rho}^{S/\epsilon}d^{3}\bm{x}-\epsilon^{2}\bm{\nabla}\cdot(\widehat{\bm{\alpha}}^{\star
S/\epsilon}\overline{\rho})\,d^{3}\bm{x}-\epsilon^{3}\bm{\nabla}\cdot(\widehat{\bm{\alpha}}^{\star
S/\epsilon}\widehat{\rho}^{S/\epsilon})\,d^{3}\bm{x}+\mathcal{O}(\epsilon^{3})$
$\displaystyle=\overline{\rho}\,d^{3}\bm{x}+\epsilon\widehat{\rho}^{S/\epsilon}d^{3}\bm{x}-\epsilon\bm{\nabla}S\cdot(\partial_{\theta}\widehat{\bm{\alpha}}^{\star
S/\epsilon}\overline{\rho})\,d^{3}\bm{x}-\epsilon^{2}\bm{\nabla}S\cdot\partial_{\theta}(\widehat{\bm{\alpha}}^{\star
S/\epsilon}\widehat{\rho}^{S/\epsilon})\,d^{3}\bm{x}+\mathcal{O}(\epsilon^{3})$
$\displaystyle=\overline{\rho}\,d^{3}\bm{x}+\epsilon^{2}\mathrm{Osc},$ (249)

where in the last line, we substituted Eq. (231) so that
$\bm{\nabla}S\cdot(\partial_{\theta}\widehat{\bm{\alpha}}^{\star
S/\epsilon}\overline{\rho})=\widehat{\rho}^{S/\epsilon}$. We also used the
well-known formula for the determinant of a near-identity matrix:

$\displaystyle\det(\mathbb{I}$
$\displaystyle-\epsilon^{2}\bm{\nabla}\widehat{\bm{\alpha}}^{\star
S/\epsilon})$
$\displaystyle=1-\epsilon^{2}\mathrm{Tr}(\bm{\nabla}\widehat{\bm{\alpha}}^{\star
S/\epsilon})-\frac{\epsilon^{4}}{2}\left(\mathrm{Tr}^{2}(\bm{\nabla}\widehat{\bm{\alpha}}^{\star
S/\epsilon})-\mathrm{Tr}(\bm{\nabla}\widehat{\bm{\alpha}}^{\star
S/\epsilon})^{2}\right)+\mathcal{O}(\epsilon^{6})$
$\displaystyle=1-\epsilon^{2}\bm{\nabla}\cdot\widehat{\bm{\alpha}}^{\star
S/\epsilon}-\frac{\epsilon^{4}}{2}\left((\bm{\nabla}\cdot\widehat{\bm{\alpha}}^{\star
S/\epsilon})^{2}-(\bm{\nabla}\widehat{\bm{\alpha}}^{\star
S/\epsilon}:\bm{\nabla}\ \widehat{\bm{\alpha}}^{\star
S/\epsilon})\right)+\mathcal{O}(\epsilon^{6})$
$\displaystyle=1-\epsilon^{2}\bm{\nabla}\cdot\widehat{\bm{\alpha}}^{\star
S/\epsilon}-\frac{\epsilon^{2}}{2}\left((\bm{\nabla}S\cdot\partial_{\theta}\widehat{\bm{\alpha}}^{\star
S/\epsilon})^{2}-(\bm{\nabla}S\otimes\partial_{\theta}\widehat{\bm{\alpha}}^{\star
S/\epsilon}:\bm{\nabla}S\otimes\partial_{\theta}\widehat{\bm{\alpha}}^{\star
S/\epsilon})\right)+\mathcal{O}(\epsilon^{4})$
$\displaystyle=1-\epsilon^{2}\bm{\nabla}\cdot\widehat{\bm{\alpha}}^{\star
S/\epsilon}+\mathcal{O}(\epsilon^{4}).$ (250)

where at the end, the terms in parentheses cancel because
$\widehat{\bm{\alpha}}^{\star}$ is parallel to $\bm{\nabla}S$.

In a similar manner, we can calculate the term $\widetilde{\tau}^{\star
S/\epsilon}_{*}\widetilde{\bm{p}}^{\star S/\epsilon}$. Note, however, that we
should consider $\widetilde{\bm{p}}^{\star S/\epsilon}$ as a one-form density
so that the above is written as
$\widetilde{\tau}^{S\star/\epsilon}_{*}(\widetilde{\bm{p}}^{\star
S/\epsilon}\cdot d\bm{x}\otimes d^{3}\bm{x})$. A direct calculation leads to

$\displaystyle\widetilde{\tau}^{\star
S/\epsilon}_{*}(\widetilde{\bm{p}}^{\star S/\epsilon}\cdot d\bm{x}\otimes
d^{3}\bm{x})$ $\displaystyle=d\bm{x}\cdot\bm{\nabla}((\widetilde{\tau}^{\star
S/\epsilon})^{-1})\cdot(\widetilde{\bm{p}}^{\star
S/\epsilon}\circ(\widetilde{\bm{\tau}}^{\star
S/\epsilon})^{-1})\otimes\det(\bm{D}(\widetilde{\bm{\tau}}^{\star
S/\epsilon})^{-1})\,d^{3}\bm{x}$
$\displaystyle=d\bm{x}\cdot(\mathbb{I}-\epsilon^{2}\bm{\nabla}\widehat{\bm{\alpha}}^{\star
S/\epsilon})\cdot(\widetilde{\bm{p}}^{\star
S/\epsilon}\circ(\bm{id}-\epsilon^{2}\widehat{\bm{\alpha}}^{\star
S/\epsilon}))\otimes\det(\mathbb{I}-\epsilon^{2}\bm{\nabla}\widehat{\bm{\alpha}}^{\star
S/\epsilon})\,d^{3}\bm{x}+\mathcal{O}(\epsilon^{3})$
$\displaystyle=d\bm{x}\cdot(\mathbb{I}-\epsilon^{2}\bm{\nabla}\widehat{\bm{\alpha}}^{\star
S/\epsilon})\cdot(\widetilde{\bm{p}}^{\star
S/\epsilon}-\epsilon^{2}(\widehat{\bm{\alpha}}^{\star
S/\epsilon}\cdot\bm{\nabla})\widetilde{\bm{p}}^{\star
S/\epsilon})\otimes(1-\epsilon^{2}\bm{\nabla}\cdot\widehat{\bm{\alpha}}^{\star
S/\epsilon})\,d^{3}\bm{x}+\mathcal{O}(\epsilon^{3})$
$\displaystyle=d\bm{x}\cdot(\mathbb{I}-\epsilon\bm{\nabla}S\otimes\partial_{\theta}\widehat{\bm{\alpha}}^{\star
S/\epsilon})\cdot(\widetilde{\bm{p}}^{\star
S/\epsilon}-\epsilon(\widehat{\bm{\alpha}}^{\star
S/\epsilon}\cdot\bm{\nabla}S)\partial_{\theta}\widetilde{\bm{p}}^{\star
S/\epsilon})\otimes(1-\epsilon\bm{\nabla}S\cdot\partial_{\theta}\widehat{\bm{\alpha}}^{\star
S/\epsilon})\,d^{3}\bm{x}+\epsilon^{2}\mathrm{Osc}$
$\displaystyle=\bigg{(}\overline{\bm{p}}+\epsilon\widehat{\bm{p}}^{\star
S/\epsilon}-\epsilon(\overline{\bm{p}}\cdot\partial_{\theta}\widehat{\bm{\alpha}}^{\star
S/\epsilon})\bm{\nabla}S-\epsilon(\bm{\nabla}S\cdot\partial_{\theta}\widehat{\bm{\alpha}}^{\star
S/\epsilon})\overline{\bm{p}}+\epsilon^{2}\partial_{\theta}((\widehat{\bm{\alpha}}^{\star
S/\epsilon}\cdot\bm{\nabla}S)\widehat{\bm{p}}^{\star S/\epsilon})$
$\displaystyle\quad-\epsilon^{2}(\widehat{\bm{p}}^{\star
S/\epsilon}\cdot\partial_{\theta}\widehat{\bm{\alpha}}^{\star
S/\epsilon})\bm{\nabla}S+\epsilon^{2}(\overline{\bm{p}}\cdot\partial_{\theta}\widehat{\bm{\alpha}}^{S/\epsilon})(\bm{\nabla}S\cdot\partial_{\theta}\widehat{\bm{\alpha}}^{\star
S/\epsilon})\bm{\nabla}S\bigg{)}\cdot d\bm{x}\otimes
d^{3}\bm{x}+\epsilon^{2}\mathrm{Osc}$
$\displaystyle=\bigg{(}\overline{\bm{p}}+\epsilon
c_{s}\widehat{\rho}^{S/\epsilon}\bm{e}_{\bm{k}}-\epsilon(\overline{\bm{p}}\cdot\bm{e}_{\bm{k}})(\widehat{\rho}^{S/\epsilon}/\overline{\rho})\,\bm{e}_{\bm{k}}-\epsilon^{2}c_{s}\overline{\rho}(\widehat{\rho}^{S/\epsilon}/\overline{\rho})^{2}\bm{e}_{\bm{k}}\bigg{)}\cdot
d\bm{x}\otimes d^{3}\bm{x}+\epsilon^{2}\mathrm{Osc},$ (251)

where in the last line, we substituted the expressions for
$\widehat{\bm{\alpha}}^{\star}$ and $\widehat{\bm{p}}^{\star}$ in Eqs. (231)
and (232).

Finally, a far simpler calculation of $\widetilde{\varphi}^{\star S/\epsilon}$
introduced in Eq. (246) gives

$\displaystyle\widetilde{\varphi}^{\star S/\epsilon}$
$\displaystyle=(\widetilde{\tau}^{\star
S/\epsilon})_{*}\widetilde{\chi}^{\star S/\epsilon}$
$\displaystyle=\widetilde{\chi}^{\star
S/\epsilon}\circ(\widetilde{\bm{\tau}}^{\star S/\epsilon})^{-1}$
$\displaystyle=\widetilde{\chi}^{\star
S/\epsilon}\circ(\bm{id}-\epsilon^{2}\widehat{\bm{\alpha}}^{\star
S/\epsilon})+\mathcal{O}(\epsilon^{3})$ $\displaystyle=\widetilde{\chi}^{\star
S/\epsilon}-\epsilon^{2}\widehat{\bm{\alpha}}^{\star
S/\epsilon}\cdot\bm{\nabla}\widetilde{\chi}^{\star
S/\epsilon}+\mathcal{O}(\epsilon^{3})$
$\displaystyle=\overline{\chi}+\epsilon^{2}\widehat{\chi}^{\star
S/\epsilon}-\epsilon^{2}\widehat{\bm{\alpha}}^{\star
S/\epsilon}\cdot\bm{\nabla}(\overline{\chi}+\epsilon^{2}\widehat{\chi}^{\star
S/\epsilon})+\mathcal{O}(\epsilon^{3})$
$\displaystyle=\overline{\chi}+\epsilon^{2}\mathrm{Osc}.$ (252)

We now insert Eqs. (247), (249), (251), and (252) into the Lagrangian (245).
Starting from the first integral in Eq. (245), we substitute the obtained
expressions for
$\widetilde{\tau}^{S/\epsilon}_{*}\widetilde{\bm{p}}^{S/\epsilon}$ and
$\widetilde{\bm{\nu}}^{S/\epsilon}$. We then Whitham average, or $\theta$
average, the Lagrangian and only keep terms up to $\mathcal{O}(\epsilon^{2})$.
We obtain

$\int_{Q}\fint(\widetilde{\tau}^{\star
S/\epsilon}_{*}\widetilde{\bm{p}}^{\star
S/\epsilon})\cdot(\overline{\bm{v}}-\widetilde{\bm{\nu}}^{\star
S/\epsilon})\,d\theta\,d^{3}\bm{x}=\int_{Q}\left(\overline{\bm{p}}\cdot\overline{\bm{v}}-\epsilon^{2}\mathcal{I}(\partial_{t}S+\overline{\bm{v}}\cdot\bm{\nabla}S)\right)\,d^{3}\bm{x}+\mathcal{O}(\epsilon^{3}),$
(253)

where $\mathcal{I}$ is the wave action density introduced in Eq. (176). For
the next term of the Lagrangian (245), we substitute Eqs. (249) and (252).
This leads to

$\int_{Q}\fint\widetilde{\tau}^{\star
S/\epsilon}_{*}\widetilde{\rho}^{S/\epsilon}\big{(}\partial_{t}\widetilde{\varphi}^{\star
S/\epsilon}_{\theta}+\overline{\bm{v}}\cdot\bm{\nabla}\widetilde{\varphi}^{\star
S/\epsilon}_{\theta}\big{)}\,d\theta\,d^{3}\bm{x}=\int_{Q}\overline{\rho}\left(\partial_{t}\overline{\chi}+\overline{\bm{v}}\cdot\bm{\nabla}\overline{\chi}\right)\,d^{3}\bm{x}+\mathcal{O}(\epsilon^{3}).$
(254)

In a similar manner, substituting Eqs. (167) and (232) gives the following for
the $\theta$-averaged Hamiltonian:

$\displaystyle\int_{Q}\fint\bigg{(}\frac{|\widetilde{\bm{p}}^{\star
S/\epsilon}|^{2}}{2\widetilde{\rho}^{S/\epsilon}}+c_{s}^{2}\widetilde{\rho}^{S/\epsilon}\ln\bigg{(}\frac{\widetilde{\rho}^{S/\epsilon}}{\rho_{0}}\bigg{)}\bigg{)}d\theta\,d^{3}\bm{x}=\int_{Q}\left(\frac{|\overline{\bm{p}}|^{2}}{2\overline{\rho}}+c_{s}^{2}\overline{\rho}\ln\left(\frac{\overline{\rho}}{\rho_{0}}\right)+\epsilon^{2}c_{s}\mathcal{I}|\bm{\nabla}S|\right)\,\mathrm{d}^{3}\bm{x}+\mathcal{O}(\epsilon^{3}).$
(255)

When combining the results in Eqs. (253)–(255), we obtain the effective action
given in Eq. (184), which we rewrite below for convenience:

$\displaystyle\overline{\mathsf{L}}_{\rm
T}(\overline{\bm{h}},\dot{\overline{\bm{h}}},\overline{\bm{p}},\dot{\overline{\bm{p}}},\overline{\rho},\dot{\overline{\rho}},\overline{\chi},\dot{\overline{\chi}},\mathcal{I},\dot{\mathcal{I}},S,\dot{S})$
$\displaystyle=\int_{Q}\left(\overline{\bm{p}}\cdot\overline{\bm{v}}+\overline{\rho}\big{(}\partial_{t}\overline{\chi}+\overline{\bm{v}}\cdot\bm{\nabla}\overline{\chi}\big{)}\right)\,d^{3}\bm{x}-\int_{Q}\mathcal{H}_{\rm
T}(\overline{\bm{p}},\overline{\rho})\,d^{3}\bm{x}$
$\displaystyle\quad-\epsilon^{2}\int_{Q}\mathcal{I}\left(\partial_{t}S+\overline{\bm{v}}\cdot\bm{\nabla}S+c_{s}|\bm{\nabla}S|\right).$
(256)

In summary, in this section we have presented additional details for
calculating the effective action for wave–mean-flow interactions in an
isothermal fluid. Our method was primarily based on slow-manifold reduction,
whose general ideas were presented in Sec. VI.2. Our derivation followed to
two main steps. First, we restricted the variational principle to the slow
manifold. This is essentially done by substituting the expressions obtained
for the fast variables. Second, we identified a transformation that
facilitated computations of the wave–mean-flow action.

##  VII Discussion

In this article we have identified the variational structure underlying the
nonlinear WKB methodWhitham (1965a); Miura and Kruskal (1974) as it applies to
ideal fluid equations in the Eulerian frame. This work therefore compliments
previous studies on variational nonlinear WKB in the _mean_ Eulerian
frame.Dewar (1970); Bretherton (1971); Gjaja and Holm (1996) Our main results
concern what we have termed the nonlinear WKB extension procedure, which is
the technique used for generating a system of equations governing the profile
functions appearing in the nonlinear WKB ansatz. Our results may be summarized
as follows. (i) Given Eulerian fluid equations arising from an Euler-Poincaré
variational principle,Holm, Marsden, and Ratiu (1998) we have shown that the
enlarged system resulting from the nonlinear WKB extension procedure also
arises from a variational principle. (ii) This new variational principle
inherits a “looped” version of the original system’s symmetry group. After
recognizing that a subgroup of this looped group comprises a looped version of
the particle relabeling group, we have used Noether’s theorem to identify a
family of circulation invariants parameterized by $S^{1}$. (iii) By combining
the newly discovered class of variational principles with ideas from the
theory of slow manifold reduction, we have presented an example of a
systematic procedure for identifying variational principles governing the
self-consistent interaction between (possibly nonlinear) locally-plane waves
and mean flows.

Our analysis made use of several technical assumptions that are
straightforward to relax. In particular, we restricted our attention to
barotropic fluid equations that arise from a local Lagrangian. A more general
equation of state involving an advected entropy could readily be incorporated
into our discussion by an enterprising reader. Similarly, it would not be
prohibitively difficult to allow for spatial non-locality in the Lagrangian.
(On the other hand, temporal nonlocality would not be simple to include.) More
generally, extensions of our work to fluid systems not discussed in this paper
may readily be accommodated as long as the proof of Theorem 4 remains in tact.

Two key technical features that distinguish our work from much of the previous
work on variational fluid mechanics are (i) our use of the inverse of the
Lagrangian configuration map $\bm{h}=\bm{g}^{-1}$, and (ii) our use of the
fluid phase-space Lagrangian (akin to $L=p\dot{q}-H(q,p)$). The use of
$\bm{h}(\bm{x})$ instead of $\bm{g}(\bm{x}_{0})$ allowed us to reformulate the
Euler-Poincaré approach to fluid variational principles in terms of
conventional classical field theory, which in turn enabled us to apply
Whitham’s averaged Lagrangian technique in the Eulerian frame. Our inspiration
for this shift in perspective came from Ref. Beig and Schmidt, 2003, which
explains the use of $\bm{h}$ within the theory of relativistic elastic solids.
Using the phase-space Lagrangian allowed us to apply Theorem 6 on the
inheritance of Hamiltonian structure in order to explain the variational
principle underlying the interaction between small-amplitude acoustic waves
and a compressible barotropic mean flow. This same idea was used in Refs.
Burby, 2017 and Burby and Sengupta, 2018 to explain the Hamiltonian structures
underlying magnetohydrodynamics and kinetic magnetohydrodynamics,
respectively.

It is most interesting to compare the approach we have introduced here for
variational modeling of wave-mean-flow interaction with earlier
approachesDewar (1970); Bretherton (1971); Gjaja and Holm (1996); Holm (2002a,
b); Gjaja and Holm (1996); Holm (2002c) based on generalized Lagrangian mean
(GLM) theory.Andrews and McIntyre (1978); Bühler (2009) As an intuitively
appealing way of representing waves superimposed on a mean flow, previous
authors have decomposed the Lagrangian configuration map $\bm{g}$ as the
composition of a mean configuration map with a fluctuating configuration map.
This decomposition forms the foundation of GLM theory. The mean configuration
map takes values in (and in fact defines) the mean Eulerian frame, while the
fluctuating configuration map takes values in the conventional Eulerian frame.
When the averaging operation is identified with WKB phase averaging, the
prevailing trend has then been to express the fluctuating configuration map in
terms of the WKB ansatz. This effectively amounts to applying the WKB method
within the mean Eulerian frame. While this approach obfuscates the connection
between wave-mean-flow dynamics and the conventional Eulerian-frame WKB
method, especially at higher orders in asymptotic expansions, it is compatible
with variational formulations of fluid dynamics in a simple manner. Indeed, it
is straightforward to decompose the Lagrangian configuration map in an Euler-
Poincaré variational principle using the GLM ansatz, and then apply WKB phase
averaging to the result.Dewar (1970); Bretherton (1971); Gjaja and Holm (1996)
In contrast, the perspective taken in our new approach is that, in principle,
there is no need to introduce the mean Eulerian frame in order to identify
wave-mean-flow variational principles. Instead, one can start from our new
variational principle for the nonlinear WKB extension of the Eulerian-frame
fluid equations, and then apply slow manifold reduction to obtain the desired
reduced variational principle for wave-mean-flow interaction. Aside from
maintaining a clear link with the Eulerian-frame WKB procedure, a benefit of
this approach is that it systematically incorporates the closure (a.k.a.
“slaving” or “balance”) relations needed to express the rapidly-varying
fluctuations in terms of slowly-varying mean quantities, thereby eliminating
the risk of unwanted fast modes creeping into the variational principle. (For
an example of the latter phenomenon, see the Hamiltonian models in Refs. Burby
_et al._ , 2015 and Brizard and Tronci, 2016 for low-frequency dynamics of
strongly-magnetized plasmas. Those models support high-frequency
electromagnetic waves that must be handled with care.) Interestingly however,
our calculations have revealed that it is practically expedient to express our
variational principle in terms of mean-Eulerian frame quantities. In so doing,
the Lagrangian simplifies dramatically. In fact, it is not at all clear that
Lagrangian expressed in terms of conventional Eulerian frame quantities
behaves well with respect to truncation, i.e. when high-order terms in the
asymptotic expansions of the slaving functions are dropped. Thus, GLM theory
plays an important practical role in our new formalism, even though it is not
a necessary ingredient at a conceptual level.

Given the dichotomy between our new method and the established mean-Eulerian
frame approach, it is also interesting to ask how the family of circulation
invariants given in Corollary 1 relates to the mean circulation invariants of
Refs. Bretherton, 1971 and Gjaja and Holm, 1996. Because the family of
circulation invariants identified in Corollary 1 is parameterized by the angle
$\theta\in S^{1}$, it may be averaged over $\theta$ to obtain a mean
circulation invariant for the nonlinear WKB extension of the ideal barotropic
fluid equations. In particular, the averaged circulation is constant along
solutions that lie in the slow manifold. Thus, the $S^{1}$-mean of our family
of circulation invariants restricted to the slow manifold is a circulation
invariant for the wave-mean-flow dynamics. It is in this manner that mean
circulation invariants of the types found in Refs. Bretherton, 1971 and Gjaja
and Holm, 1996 emerge from our formalism. As an illustration of this point, we
prove in Appendix A that the average of our family of circulation invariants
restricted to the slow manifold is equivalent to the circulation invariant
associated with mean particle relabeling symmetry of the wave–mean-flow
Lagrangian (196).

In the future, we plan to use the tools developed in this article to capture
the effects of harmonic generation and corrections to ray trajectories caused
by space-dependent wave polarizationLittlejohn and Flynn (1991); Ruiz and
Dodin (2017) in wave-mean-flow problems arising in fluids and plasmas.

###### Acknowledgements.

The authors would like to thank Richard Montgomery and Cesare Tronci for a
number of helpful discussions of this work at MSRI. This material is based
upon work supported by the National Science Foundation under Grant No.
DMS-1440140 while one of the authors (JWB) was in residence at the
Mathematical Sciences Research Institute in Berkeley, California, during the
Fall 2018 semester. In addition, research presented in this article was
supported by (1) the Laboratory Directed Research and Development program of
Los Alamos National Laboratory under project number 20180756PRD4, and (2)
Sandia National Laboratories. Sandia National Laboratories is a multimission
laboratory managed and operated by National Technology and Engineering
Solutions of Sandia, LLC., a wholly owned subsidiary of Honeywell
International, Inc., for the U.S. DOE National Nuclear Security Administration
under contract DE-NA-0003525. This paper describes objective technical results
and analysis. Any subjective views or opinions that might be expressed in the
paper do not necessarily represent the views of the U.S. DOE or the U.S.
Government.

##  Appendix A Proof of mean circulation theorem

######  Corollary 4 (Kelvin’s theorem for wave–mean-flow system).

Given a closed curve $C_{0}\subset Q_{0}$ and a solution
$(\overline{\bm{h}},\overline{\bm{p}},\overline{\rho},\overline{\chi},\mathcal{I},S)$
of the wave–mean-flow equations (180)–(183), then

$\frac{\mathrm{d}}{\mathrm{d}t}\oint_{\overline{C}}\left(\frac{\overline{\bm{p}}}{\overline{\rho}}-\frac{\mathcal{I}}{\overline{\rho}}\bm{\nabla}S\right)\cdot\mathrm{d}\bm{x}=0.$
(257)

where $\overline{C}=\overline{\bm{h}}^{-1}(C_{0})$.

Thus, in the wave–mean-flow framework developed here, the circulation theorem
(257) is now a closed contour integral of the fluid momentum minus a term
related to the wave momentum. The modification of Kelvin’s circulation theorem
due to wave effects has been noticed before.Bretherton (1971) In essence, this
result shows that waves can affect the vorticity of the bulk fluid. The last
term in Eq. (257) is sometimes referred as “wave pseudomomentum.”Gjaja and
Holm (1996); Bühler (2009)

###### Proof.

Equation (257) can be proven by following a similar procedure as that used in
Lemma 4. The only difference is that now the simplectic form associated to
$\overline{\mathcal{C}}_{0}$ is $-\bm{\mathrm{d}}\overline{\Theta}$, where the
1-form $\overline{\Theta}$ is given by

$\displaystyle\overline{\Theta}[\delta\overline{\bm{h}},\delta\overline{\bm{p}},\delta\overline{\rho},\delta\overline{\chi},\delta\mathcal{I},\delta
S]=\int_{Q}\overline{\bm{p}}\cdot\overline{\bm{\xi}}\,d^{3}\bm{x}+\int_{Q}\overline{\rho}(\delta\overline{\chi}+\overline{\bm{\xi}}\cdot\bm{\nabla}\overline{\chi})\,d^{3}\bm{x}-\int_{Q}\mathcal{I}(\delta
S+\overline{\bm{\xi}}\cdot\bm{\nabla}S)\,d^{3}\bm{x}$ (258)

and
$\overline{\bm{\xi}}\doteq-\delta\overline{\bm{h}}\cdot(\bm{\nabla}\overline{\bm{h}})^{-1}$.
Alternatively, we can also show the result in Eq. (257) by using the general
Kelvin’s theorem for Eulerian WKB obtained in Corollary 1. Indeed, since
solutions along the slow manifold [see, e.g., Eqs. (231)–(233)] are solutions
of the NL–WKB isothermal fluid equations (197)–(200), then Eq. (257) can be
obtained by restricting Eq. (153) onto the slow manifold and averaging over
the phase $\theta$. Specifically, we have

$0=\frac{\mathrm{d}}{\mathrm{d}t}\overline{\left(\oint_{C_{\theta}}\frac{\widetilde{\bm{p}}^{\star
S/\epsilon}}{\widetilde{\rho}^{\star S/\epsilon}}\cdot
d\bm{x}\right)}=\frac{\mathrm{d}}{\mathrm{d}t}\oint_{\overline{C}}\overline{\left(\widetilde{\tau}^{\star
S/\epsilon}_{*}\frac{\widetilde{\bm{p}}^{\star
S/\epsilon}}{\widetilde{\rho}^{\star S/\epsilon}}\cdot d\bm{x}\right)}.$ (259)

Following a similar calculation as in Eq. (251), we can calculate the
pushforward appearing in the integral above:

$\displaystyle\widetilde{\tau}^{\star
S/\epsilon}_{*}\frac{\widetilde{\bm{p}}^{\star
S/\epsilon}}{\widetilde{\rho}^{\star S/\epsilon}}\cdot d\bm{x}$
$\displaystyle=d\bm{x}\cdot\bm{\nabla}((\widetilde{\bm{\tau}}^{\star
S/\epsilon})^{-1})\cdot\left(\frac{\widetilde{\bm{p}}^{\star
S/\epsilon}}{\widetilde{\rho}^{\star
S/\epsilon}}\right)\circ(\widetilde{\bm{\tau}}^{\star S/\epsilon})^{-1}$
$\displaystyle=d\bm{x}\cdot(\mathbb{I}-\epsilon^{2}\bm{\nabla}\widehat{\bm{\alpha}}^{\star
S/\epsilon})\cdot\bigg{(}\frac{\overline{\bm{p}}}{\overline{\rho}}+\epsilon\frac{\widehat{\bm{p}}^{\star
S/\epsilon}}{\overline{\rho}}-\epsilon\frac{\overline{\bm{p}}}{\overline{\rho}}\frac{\widehat{\rho}^{\star
S/\epsilon}}{\overline{\rho}}-\epsilon^{2}\frac{\widehat{\bm{p}}^{\star
S/\epsilon}}{\overline{\rho}}\frac{\widehat{\rho}^{\star
S/\epsilon}}{\overline{\rho}}$
$\displaystyle\qquad+\epsilon^{2}\frac{\overline{\bm{p}}}{\overline{\rho}}\bigg{(}\frac{\widehat{\rho}^{\star
S/\epsilon}}{\overline{\rho}^{\star
S/\epsilon}}\bigg{)}^{2}\bigg{)}\circ(\bm{id}-\epsilon^{2}\widehat{\bm{\alpha}}^{\star
S/\epsilon})+\mathcal{O}(\epsilon^{3})$
$\displaystyle=d\bm{x}\cdot(\mathbb{I}-\epsilon\bm{\nabla}S\otimes\partial_{\theta}\widehat{\bm{\alpha}}^{\star
S/\epsilon})\cdot\bigg{(}\frac{\overline{\bm{p}}}{\overline{\rho}}+\epsilon\frac{\widehat{\bm{p}}^{\star
S/\epsilon}}{\overline{\rho}}-\epsilon\frac{\overline{\bm{p}}}{\overline{\rho}}\frac{\widehat{\rho}^{\star
S/\epsilon}}{\overline{\rho}}-\epsilon^{2}\frac{\widehat{\bm{p}}^{\star
S/\epsilon}}{\overline{\rho}}\frac{\widehat{\rho}^{\star
S/\epsilon}}{\overline{\rho}}$
$\displaystyle\qquad+\epsilon^{2}\frac{\overline{\bm{p}}}{\overline{\rho}}\bigg{(}\frac{\widehat{\rho}^{\star
S/\epsilon}}{\overline{\rho}^{\star
S/\epsilon}}\bigg{)}^{2}-\epsilon^{2}\bm{\nabla}S\cdot\widehat{\bm{\alpha}}^{\star
S/\epsilon}\partial_{\theta}\frac{\widehat{\bm{p}}^{\star
S/\epsilon}}{\overline{\rho}}+\epsilon^{2}\bm{\nabla}S\cdot\widehat{\bm{\alpha}}^{\star
S/\epsilon}\partial_{\theta}\frac{\overline{\bm{p}}}{\overline{\rho}}\frac{\widehat{\rho}^{\star
S/\epsilon}}{\overline{\rho}}\bigg{)}+\epsilon^{2}\mathrm{Osc},$
$\displaystyle=\bigg{(}\frac{\overline{\bm{p}}}{\overline{\rho}}-\epsilon^{2}\frac{\widehat{\bm{p}}^{\star
S/\epsilon}}{\overline{\rho}}\frac{\widehat{\rho}^{\star
S/\epsilon}}{\overline{\rho}}+\epsilon^{2}\frac{\overline{\bm{p}}}{\overline{\rho}}\bigg{(}\frac{\widehat{\rho}^{\star
S/\epsilon}}{\overline{\rho}^{2}}\bigg{)}^{2}-\epsilon\partial_{\theta}\widehat{\bm{\alpha}}^{\star
S/\epsilon}\cdot\frac{\overline{\bm{p}}}{\overline{\rho}}\bm{\nabla}S\bigg{)}\cdot
d\bm{x}+\epsilon^{2}\mathrm{Osc},$ (260)

where in the last line, some $\mathcal{O}(\epsilon^{2})$ terms were written as
a total derivative of $\theta$. Since their $\theta$-average is zero, we
omitted writing them and placed them under the symbol
“$\epsilon^{2}\mathrm{Osc}$”. Finally, averaging over $\theta$ and
substituting the expressions for $\widehat{\bm{p}}^{\star S/\epsilon}$ and
$\mathcal{I}$ leads to

$\overline{\left(\widetilde{\tau}^{\star
S/\epsilon}_{*}\frac{\widetilde{\bm{p}}^{\star
S/\epsilon}}{\widetilde{\rho}^{\star S/\epsilon}}\cdot
d\bm{x}\right)}=\frac{\overline{\bm{p}}}{\overline{\rho}}\cdot
d\bm{x}-\frac{\mathcal{I}}{\overline{\rho}}\bm{\nabla}S\cdot d\bm{x}.$ (261)

Inserting this into Eq. (259) finishes the proof. ∎

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