Well, for any $y\in\mathbb{R}^{4}$,

$By\cdot y=y_{i}\langle\hat{x}^{i},\,z\rangle y_{j}\langle
x^{j},\,x\rangle=y\cdot{\left(\langle\hat{x}^{i},\,z\rangle\right)}_{i=1}^{4}\leqslant{\lvert
y\rvert}^{2}\left(\sum_{i=1}^{4}{\langle\hat{x}^{i},\,z\rangle}^{2}\right).$

Since ${\left(\hat{x}^{i}\right)}_{i=1}^{4}$ is an orthonormal set, we have
that

$\sum_{i=1}^{4}{\langle\hat{x}^{i},\,z\rangle}^{2}\leqslant{\left\lvert\left\lvert
z\right\rvert\right\rvert}^{2},$

which verifies that $B\leqslant{\left\lvert\left\lvert
z\right\rvert\right\rvert}^{2}I$. Finally we show that
$B<{\left\lvert\left\lvert z\right\rvert\right\rvert}^{2}I$. This follows from
the fact that ${\left(\hat{x}^{i}\right)}_{i=1}^{4}$ span $\mathcal{I}$, but
not $\mathbb{R}^{5}$, and so

$\sum_{i=1}^{4}{\langle\hat{x}^{i},\,z\rangle}^{2}={\left\lvert\left\lvert\text{proj}_{\mathcal{I}}z\right\rvert\right\rvert}^{2}<{\left\lvert\left\lvert
z\right\rvert\right\rvert}^{2}$

since $z=S-I\notin\mathcal{I}$ since $I\in\mathcal{I}$ but
$S\notin\mathcal{I}$. In other words: since $S\notin\mathcal{I}$ we have that
$B<{\left\lvert\left\lvert z\right\rvert\right\rvert}^{2}I$, and so
$\nabla^{2}f>0$. ∎

Now we verify that the distance function introduced in Definition 5.14 is
indeed a metric. We will also show that this Snell-type distance also agrees
with the energy, when they are comparable.

######  Lemma 5.16 (The Snell-type distance is a metric).

As introduced in Definition 5.14, $d_{\text{Snell}}$ is a metric on
$\mathbb{R}^{5}$. Moreover it agrees with the energy density in the sense
that, for any $S_{1},\,S_{2}\in\mathbb{R}^{5}$, if the signs of $q_{1}$ and
$q_{2}$ agree (i.e. if the states $S_{1}$ and $S_{2}$ are in the same phase)
then

${d_{\text{Snell}}(S_{1},\,S_{2})}^{2}={\lvert
u_{1}-u_{2}\rvert}^{2}+{(\theta_{1}-\theta_{2})}^{2}+{(q_{1}-q_{2})}^{2}h+{((\theta_{1}+q_{1})-(\theta_{2}-q_{2}))}^{2}$

where

$h=\left\\{\begin{aligned} &1&&\text{ if }q_{1},\,q_{2}<0\text{ and }\\\
&0&&\text{ if }q_{1},\,q_{2}\geqslant 0.\end{aligned}\right.$

###### Proof.

Positive-definiteness and symmetry are immediate so we only need to prove that
the triangle inequality holds. Indeed: the agreement with the energy follows
immediately from the definition of
${\left\lvert\left\lvert\,\cdot\,\right\rvert\right\rvert}_{u}$ and
${\left\lvert\left\lvert\,\cdot\,\right\rvert\right\rvert}_{s}$ in Definition
5.12.

For $S_{1},\,S_{2},\,S_{3}\in\mathbb{R}^{5}$ we therefore wish to show that

$d_{\text{Snell}}(S_{1},\,S_{2})\leqslant
d_{\text{Snell}}(S_{1},\,S_{3})+d_{\text{Snell}}(S_{3},\,S_{2}).$

Clearly the triangle inequality holds when all three states belong to the same
phase (either $\mathcal{U}$ or $\mathcal{S}$). So we need only consider the
case where two of the states $S_{1}$, $S_{2}$, and $S_{3}$ are in one phase
and the last state is in the other. Since all that matters is whether or not
the intermediate point $S_{3}$ is alone in its phase, there are without loss
of generality two cases to consider.

  * •

Case 1: $S_{1},\,S_{2}\in\mathcal{S}$ and $S_{3}\in\mathcal{U}$. Let us denote
by $I_{13}$ the point intersecting the geodesic between $S_{1}$ and $S_{3}$
and the state-space interface $\mathcal{I}$, and similarly $I_{23}$ for the
point arising from the geodesic between $S_{2}$ and $S_{3}$. We use the
triangle inequality in $\mathcal{U}$, then the triangle inequality in $S$:

$\displaystyle d_{\text{Snell}}(S_{1},\,S_{3})\,+$
$\displaystyle\,d_{\text{Snell}}(S_{3},\,S_{2})$
$\displaystyle=d_{\text{Snell}}(S_{1},\,I_{13})+d_{\text{Snell}}(I_{13},\,S_{3})+d_{\text{Snell}}(S_{3},\,I_{23})+d_{\text{Snell}}(I_{23},\,S_{2})$
$\displaystyle\geqslant
d_{\text{Snell}}(S_{1},\,I_{13})+d_{\text{Snell}}(I_{13},\,I_{23})+d_{\text{Snell}}(I_{23},\,S_{2})$
$\displaystyle\geqslant d_{\text{Snell}}(S_{1},\,S_{2}).$

$S_{1}$$S_{2}$$S_{3}$$I_{13}$$I_{23}$

$S_{1}$$S_{2}$$S_{3}$$I_{13}$$I_{23}$

Figure 3: A pictorial depiction of the two steps used in case 1 of the proof
of Lemma 5.16.

  * •

Case 2: $S_{1},\,S_{3}\in\mathcal{S}$ and $S_{2}\in\mathcal{U}$. Let $I_{12}$
and $I_{23}$ be defined similarly to in case 1. We use the triangle inequality
in $\mathcal{S}$ and the definition of $d_{\text{Snell}}$ as a minimization
problem when crossing the interface to deduce that

$\displaystyle
d_{\text{Snell}}(S_{1},\,S_{3})+d_{\text{Snell}}(S_{3},\,S_{2})$
$\displaystyle=d_{\text{Snell}}(S_{1},\,\
S_{3})+d_{\text{Snell}}(S_{3},\,I_{23})+d_{\text{Snell}}(I_{23},\,S_{2})$
$\displaystyle\geqslant
d_{\text{Snell}}(S_{1},\,I_{23})+d_{\text{Snell}}(I_{23},\,S_{2})$
$\displaystyle\geqslant d_{\text{Snell}}(S_{1},\,S_{2}).$

$S_{1}$$S_{3}$$S_{2}$$I_{12}$$I_{23}$

$S_{1}$$S_{3}$$S_{2}$$I_{12}$$I_{23}$

Figure 4: A pictorial depiction of the two steps used in case 2 of the proof
of Lemma 5.16.

∎

We now seek to determine whether or not this metric, once ported over to
measure the distance between spatially-varying states, agrees with the
balanced-unbalanced decomposition of Theorem 3.5.

To do so we will need to understand this Snell-type metric in the pointwise
case (which we have discussed so far) in more detail. In particular we will
rely on the explicit computation of geodesic trajectories in the case where
$u_{1}=u_{2}=0$. In that case, the states live in the $(\theta,\,q)$–plane. We
consider two states in that plane living on either side of the state-space
interface $\mathcal{I}$, i.e. where one state is saturated and the other is
not. The geodesic between these two points is then piecewise affine, changing
direction when it reaches the interface. Crucially: in this simplified setting
where $u_{1}=u_{2}=0$ we can compute _explicitly_ where the geodesic
intersects the interface.

######  Lemma 5.17 (Explicit computation of the intersection point).

For any $S_{1}\in\mathcal{S}$ and any $S_{2}\in\mathcal{U}$, if
$u_{1}=u_{2}=0$ then, for
$I(\theta)\vcentcolon=(0,\,0,\,0,\,\theta,\,0)\in\mathcal{I}\subseteq\mathbb{R}^{5}$,

$\operatorname*{arg\,min}_{\theta\in\mathbb{R}}{\left\lvert\left\lvert
S_{1}-I(\theta)\right\rvert\right\rvert}_{s}+{\left\lvert\left\lvert
S_{2}-I(\theta)\right\rvert\right\rvert}_{u}\in\left\\{\frac{\alpha-\sqrt{\delta}}{\beta},\,\frac{\alpha+\sqrt{\delta}}{\beta}\right\\}$

for

$\displaystyle\alpha$
$\displaystyle=4q_{1}^{2}(2\theta_{2}+q_{2})-6q_{2}^{2}(2\theta_{1}+q_{1}),$
$\displaystyle\beta$ $\displaystyle=8q_{1}^{2}-24q_{2}^{2},\text{ and }$
$\displaystyle\delta$
$\displaystyle=12q_{1}^{2}q_{2}^{2}{(2\theta_{1}+q_{1}+4\theta_{2}+2q_{2})}^{2}.$

###### Proof.

The key idea is that the first derivative condition for minimizers may be
squared to produce a fourth-order polynomial equation. Solving this equation
using symbolic computation software then produces the claim. More precisely,
let us define $f:\mathbb{R}\to\mathbb{R}$ via

$f(\theta)\vcentcolon={\left\lvert\left\lvert
S_{1}-I(\theta)\right\rvert\right\rvert}_{s}+{\left\lvert\left\lvert
S_{2}-I(\theta)\right\rvert\right\rvert}_{u}.$

Then, proceeding as in the proof of Lemma 5.15 we see that

$f^{\prime}(\theta)=\frac{{\langle
I-S_{1},\,\partial_{\theta}I\rangle}_{s}}{{\left\lvert\left\lvert
S_{1}-I\right\rvert\right\rvert}_{s}}+\frac{{\langle
I-S_{2},\,\partial_{\theta}I\rangle}_{u}}{{\left\lvert\left\lvert
S_{2}-I\right\rvert\right\rvert}_{u}}$

and so, since $\partial_{\theta}I=e_{\theta}$, we see that if
$f^{\prime}(\theta)=0$ then

${\langle I-S_{2},\,e_{\theta}\rangle}_{u}^{2}{\left\lvert\left\lvert
S_{1}-I\right\rvert\right\rvert}_{s}^{2}={\langle
I-S_{1},\,e_{\theta}\rangle}_{s}^{2}{\left\lvert\left\lvert
S_{2}-I\right\rvert\right\rvert}_{u}^{2}.$ (5.12)

Since

${\langle I-S_{i},\,e_{\theta}\rangle}_{u}={\langle
I-S_{i},\,e_{\theta}\rangle}_{u}=2\theta-2\theta_{i}-q_{i}$

while

$\displaystyle{\left\lvert\left\lvert
S_{1}-I\right\rvert\right\rvert}_{s}^{2}$
$\displaystyle={(\theta-\theta_{1})}^{2}+{(\theta-\theta_{1}-q_{1})}^{2}\text{
and }$ $\displaystyle{\left\lvert\left\lvert
S_{2}-I\right\rvert\right\rvert}_{u}^{2}$
$\displaystyle=\frac{1}{2}{(\theta-\theta_{2}+q_{2})}^{2}+\frac{3}{2}{(\theta-\theta_{2}-q_{2})}^{2},$

where we have used the identity

$\theta^{2}+q^{2}h=(1-h/2){(\theta-qh})^{2}+(h/2){(\theta+q)}^{2},$

the equation (5.12) comes down to a polynomial equation of order four for
$\theta$. We may solve this equation explicitly using symbolic computation
software, thus producing the claim. ∎

$\theta=-3,\,q=1$$\theta=1,\,q=-1$$\theta_{*}^{(1)}=-4+\frac{3}{2}\sqrt{2}\approx-1.4$$\theta_{*}^{(2)}=-4-\frac{3}{2}\sqrt{2}\approx-6.6$
Figure 5: This shows an example where the two possible intersection points are
computed from Lemma 5.17. In practice, it is very straightforward to identify
the correct intersection point: the lengths of both paths may readily be
computed, and the intersection point corresponding to the shortest path is
then the correct one.

We now discuss how to extract balanced components in this pointwise setting
where the state space is $\mathbb{R}^{5}$ (and not $\mathbb{L}^{2}_{\sigma}$).
The system of interest is

$\displaystyle\frac{du_{h}}{dt}+\frac{1}{\varepsilon}u_{h}^{\perp}=0,$ (5.13a)
$\displaystyle\frac{dw}{dt}+\frac{1}{\varepsilon}(\theta-{\min}_{0}\,q)=0,$
(5.13b) $\displaystyle\frac{d\theta}{dt}+\frac{1}{\varepsilon}w=0,\text{ and
}$ (5.13c) $\displaystyle\frac{dq}{dt}-\frac{1}{\varepsilon}w=0,$ (5.13d)

which fits into our generic framework as it can be written in the form, for
$S=(u_{h},\,w,\,\theta,\,q)$, as

$\frac{dS}{dt}+\frac{1}{\varepsilon}N(S)=0\text{ for
}N(S)=\begin{pmatrix}u_{h}^{\perp}\\\ \theta-{\min}_{0}\,q\\\ w\\\
-w\end{pmatrix}.$

As detailed in Section 3, the first step in identifying an appropriate
decomposition for this system is to study the kernel / zero set of $N$ and the
image of $N$. We see that the kernel of $N$ may be characterized (akin to
Proposition 3.1) as a zero set of appropriate measurements or may be explictly
parameterised:

$\displaystyle N(S)=0$ $\displaystyle\iff S\in\ker N=\vcentcolon B$
$\displaystyle\iff u_{h}=0,\,w=0,\text{ and
}b(S)\vcentcolon=\theta-{\min}_{0}\,q=0$ $\displaystyle\iff
S=\begin{pmatrix}0\\\ 0\\\ \frac{1}{2}{\min}_{0}\,M\\\
M-\frac{1}{2}{\min}_{0}\,M\end{pmatrix}$

while, akin to Proposition 3.3, the image of $N$ may be characterised as
follows:

$S\in\operatorname{im}N=\vcentcolon W\iff M(S)\vcentcolon=\theta+q=0\iff
S=\begin{pmatrix}u_{h}\\\ w\\\ \eta\\\ -\eta\end{pmatrix}.$

This leads to the following decomposition of $\mathbb{R}^{5}$:

$S=\begin{pmatrix}u_{h}\\\ w\\\ \theta\\\
q\end{pmatrix}=\underbrace{\begin{pmatrix}0\\\ 0\\\
\frac{1}{2}{\min}_{0}\,M\\\ M-\frac{1}{2}{\min}_{0}\,M\end{pmatrix}}_{\in\ker
N=B}+\underbrace{\begin{pmatrix}u_{h}\\\ w\\\ \eta\\\
-\eta\end{pmatrix}}_{\in\operatorname{im}N=W}.$ (5.14)

Now let us restrict our attention once again to the $(\theta,\,q)$–plane. We
seek to determine whether or not the Snell-type metric agrees with the
decomposition (5.14). In the $(\theta,\,q)$–plane, the balanced component
takes the form

$\begin{pmatrix}\theta_{B}\\\
q_{B}\end{pmatrix}=\begin{pmatrix}\frac{1}{2}{\min}_{0}\,M\\\
M-\frac{1}{2}{\min}_{0}\,M\end{pmatrix}.$

It turns out that this decomposition _fails to agree_ with the Snell-type
energy.

######  Proposition 5.18 (The Snell-type distance does not agree with the
pointwise decomposition).

As introduced in Definition 5.14, $d_{\text{Snell}}$ does _not_ agree with the
extraction of a balanced component, in the sense that there exists
$\mathcal{S}\in\mathbb{R}^{5}$ and $T_{*}\in B$ such that

$d_{\text{Snell}}(S,\,T_{*})<d_{\text{Snell}}(S,\,S_{B}).$

In other words:

$S_{B}\neq\operatorname*{arg\,min}_{T\in B}d_{\text{Snell}}(S,\,T).$

###### Proof.

Choose $S=(-3,\,1)$ such that $M=-3+1=-2$ and so its balanced component is
$S_{B}=(-1,\,-1)$ and choose $T_{*}=(-1.1,\,-1.1)$. Note that $S$ is saturated
whereas $S_{B}$ and $T_{*}$ are unsaturated. And, of course, both $S_{B}$ and
$T_{*}$ belong to the balanced set $B=\ker N$ since they have vanishing $M$.

On the one hand, equipped with Lemma 5.17 we see that

$d_{\text{Snell}}(S,\,S_{B})={\left\lvert\left\lvert
S-I_{1}\right\rvert\right\rvert}_{s}+{\left\lvert\left\lvert
S_{B}-I_{1}\right\rvert\right\rvert}_{u}$

where

$I_{1}\vcentcolon=I(\theta_{1})\text{ for }\theta_{1}=-3+\frac{\sqrt{3}}{2}$

and hence

$d_{\text{Snell}}(S,\,S_{B})=\sqrt{\frac{5}{2}-\sqrt{3}}(1+\sqrt{3})>2.394.$

On the other hand, for $I_{2}\vcentcolon=I(\theta_{2})$ where
$\theta_{2}=\frac{11}{526}(-135+17\sqrt{3})$,

$d_{\text{Snell}}(S,\,T_{*})={\left\lvert\left\lvert
S-I_{2}\right\rvert\right\rvert}_{s}+{\left\lvert\left\lvert
T_{*}-I_{2}\right\rvert\right\rvert}_{u}=(\sqrt{101488-31790\sqrt{3}}(10+11\sqrt{3}))/2630<2.381.$

Since

$d_{\text{Snell}}(S,\,T_{*})<d_{\text{Snell}}(S,\,S_{B})$

this proves the claim. ∎

$q=0$$b=0$$S$$S_{B}$$T_{*}$$I_{1}$$I_{2}$ Figure 6: A graphical depiction of
Proposition 5.18. Note that the location of the points is not exact: this is
an artistic rendering of the setup of the proof of Proposition 5.18 used
because it makes it easier to see what is going on.

We conclude this section by extending the Snell-type distance discussed so far
_pointwise_ to a distance between spatially-varying states.

######  Definition 5.19 (Snell-type metric, spatially-varying states).

We define
$d_{S}:\mathbb{L}^{2}_{\sigma}\times\mathbb{L}^{2}_{\sigma}\to[0,\,\infty)$ as
follows. For any $\mathcal{X},\,\mathcal{Y}\in\mathbb{L}^{2}_{\sigma}$,

$d_{S}(\mathcal{X},\,\mathcal{Y})\vcentcolon={\left\lvert\left\lvert
d_{\text{Snell}}(\mathcal{X},\,\mathcal{Y})\right\rvert\right\rvert}_{L^{2}}={\left(\int
d_{\text{Snell}}^{2}(\mathcal{X}(x),\,\mathcal{Y}(x))dx\right)}^{1/2}.$

This distance is indeed a metric, and it agrees with the conserved energy when
comparable to it.

######  Proposition 5.20.

As introduced in Definition 5.19, $d_{S}$ is a metric on
$\mathbb{L}^{2}_{\sigma}$ which agrees with the energy in the sense that, if
$\mathcal{X}_{1},\,\mathcal{X}_{2}\in\mathbb{L}^{2}_{\sigma}$ have the same
phases, meaning that the signs of $q_{1}$ and $q_{2}$ agree everywhere, then

$\displaystyle{(d_{S}(\mathcal{X}_{1},\,\mathcal{X}_{2}))}^{2}$
$\displaystyle=E_{H}(\mathcal{X}_{1}-\mathcal{X}_{2})$
$\displaystyle=\int{\lvert
u_{1}-u_{2}\rvert}^{2}+{(\theta_{1}-\theta_{2})}^{2}+{(q_{1}-q_{2})}^{2}H+{((\theta_{1}+q_{1})-(\theta_{2}+q_{2}))}^{2}$

for $H\vcentcolon=\mathds{1}(q_{1}<0)=\mathds{1}(q_{2}<0)$.

###### Proof.

First we verify that $d_{S}$ is a metric. The positive-definiteness and
symmetry follow immediately from the fact that $d_{\text{Snell}}$ is a metric,
since the $L^{2}$ norm is also positive-definite.

The triangle inequality of $d_{S}$ then follows from the triangle inequalities
of $d_{\text{Snell}}$ and of $L^{2}$ and from the fact that $s\mapsto\sqrt{s}$
and $s\mapsto s^{2}$ are increasing:

$\displaystyle d_{S}(\mathcal{X}_{1},\,\mathcal{X}_{2})$
$\displaystyle={\left(\int
d_{\text{Snell}}^{2}(\mathcal{X}_{1},\,\mathcal{X}_{2})\right)}^{1/2}$
$\displaystyle\leqslant{\left(\int{\left(d_{\text{Snell}}(\mathcal{X}_{1},\,\mathcal{X}_{3})+d_{\text{Snell}}(\mathcal{X}_{3},\,\mathcal{X}_{2})\right)}^{2}\right)}^{1/2}$
$\displaystyle={\left\lvert\left\lvert
d_{\text{Snell}}(\mathcal{X}_{1},\,\mathcal{X}_{3})+d_{\text{Snell}}(\mathcal{X}_{3},\,\mathcal{X}_{2})\right\rvert\right\rvert}_{L^{2}}$
$\displaystyle\leqslant{\left\lvert\left\lvert
d_{\text{Snell}}(\mathcal{X}_{1},\,\mathcal{X}_{3})\right\rvert\right\rvert}_{L^{2}}+{\left\lvert\left\lvert
d_{\text{Snell}}(\mathcal{X}_{3},\,\mathcal{X}_{2})\right\rvert\right\rvert}_{L^{2}}$
$\displaystyle=d_{S}(\mathcal{X}_{1},\,\mathcal{X}_{3})+d_{S}(\mathcal{X}_{3},\,\mathcal{X}_{2}).$

Finally we prove that $d_{S}$ agrees with the energy, when comparable. So
suppose that $q_{1}(x)$ and $q_{2}(x)$ have the same phases, i.e. their signs
agree everywhere in the spatial domain. Then, by virtue of Lemma 5.16, we have
that

$\displaystyle d_{S}^{2}(\mathcal{X}_{1},\,\mathcal{X}_{2})$
$\displaystyle=\int
d_{\text{Snell}}^{2}\left(\mathcal{X}_{1}(x),\,\mathcal{X}_{2}(x)\right)dx$
$\displaystyle=\int{\lvert
u_{1}-u_{2}\rvert}^{2}+{(\theta_{1}-\theta_{2})}^{2}+{(q_{1}-q_{2})}^{2}H+{((\theta_{1}+q_{1})-(\theta_{2}+q_{2}))}^{2}$
$\displaystyle=E_{H}(\mathcal{X}_{1}-\mathcal{X}_{2})$

for $H\vcentcolon=\mathds{1}(q_{1}<0)=\mathds{1}(q_{2}<0)$. ∎

######  Remark 5.21.

Finally we note that, although this distance agrees with the energy, the
evidence from the restricted case of spatially constant cases strongly
suggests that this Snell-type distance does _not_ agree with the extraction of
a balanced component, especially in light of Lemma A.23 which tells us that
spatially constant solutions are honest-to-goodness solutions of the moist
Boussinesq system considered here.

While this Snell-type distance may not agree with the balanced–unbalanced
decomposition, it could be useful for other applications such as data-driven
methods that identify nonlinear features in dynamical systems and utilize a
metric, distance function, and/or kernel function [CL06, GM12, BH16, DG19,
Gia19].

###  5.3 Metric-less approach

To make the approach of this section easy to follow we must be clear about
what the last two sections, Sections 5.1 and 5.2, achieved.

  * •

In Section 5.1 we devised a metric which agreed with the decomposition, but
which failed to agree with the energy (it agreed only _conditionally_).

  * •

In Section 5.2 we devised a metric which agreed with the energy, but which
failed to agree with the decomposition.

In this section we will see that it is possible to agree with both the
decomposition and the energy, at a price. That price is that we lose the
triangle inequality. Instead of a metric, we will therefore be dealing with a
statistical divergence.

######  Definition 5.22 (Statistical divergence).

Let $\mathcal{H}$ be a Hilbert space. A _statistical divergence_ on
$\mathcal{H}$ is a map $D:\mathcal{H}\times\mathcal{H}\to\mathbb{R}$
satisfying the following properties.

  1. 1.

Non-negative: $D(x,\,y)\geqslant 0$ for every $x,\,y\in\mathcal{H}$.

  2. 2.

Definite: $D(x,\,y)=0$ if and only if $x=y$.

  3. 3.

Limiting quadratic: for every $x\in\mathcal{H}$ there is a positive-definite
quadratic form $\mathcal{Q}_{x}$ such that, for every $y\in\mathcal{H}$,

$\frac{1}{\varepsilon^{2}}D(x,\,x+\varepsilon y)\to\mathcal{Q}_{x}(y)$

as $\varepsilon\downarrow 0$. Since this quadratic form gives rise to a
positive-definite inner product, $\mathcal{Q}_{x}$ is often referred to as the
_Riemann metric induced by the divergence_.

The prototypical example of a statistical divergence is the relative entropy,
or Kullback–Leibler divergence, from information theory [CT12, BM12, CMT14].
Another reminiscent notion from the PDE literature is the _relative energy_
[Daf79, FJN12, FKNZ16, BFH17].

The statistical divergence here is built in a simple way: we use the metric
$d_{H}$ introduced in Section 5.1, but now let $H$ vary depending on the
states whose divergence is being measured.

######  Definition 5.23.

For any $\mathcal{X}_{1},\,\mathcal{X}_{2}\in\mathbb{L}^{2}_{\sigma}$ we
define

$D(\mathcal{X}_{1};\,\mathcal{X}_{2})\vcentcolon=\frac{1}{2}{d_{H_{B_{1}}}(\mathcal{X}_{1},\,\mathcal{X}_{2})}^{2}$

where $d_{H}$ is as in Definition 5.10, where
$H_{B_{1}}\vcentcolon=\mathds{1}(q_{B,\,1}<0)$ denotes the _balance_ interface
of the first state $\mathcal{X}_{1}$.

######  Remark 5.24.

We detail how to compute the statistical divergence $D$ introduced in
Definition 5.23 (similarly to how we detailed the computation of the metric
$d_{H}$ in Definition 5.10). So let
$\mathcal{X}_{1}=(u_{1},\,\theta_{1},\,q_{1})$ and
$\mathcal{X}_{2}=(u_{2},\,\theta_{2},\,q_{2})$ be states in
$\mathbb{L}^{2}_{\sigma}$.

First we compute $\mathcal{X}_{B,\,1}$ in order to determine the indicator
$H_{B_{1}}$. This is done by setting

$M_{1}\vcentcolon=M(\mathcal{X}_{1})=\theta_{1}+q_{1}\text{ and
}{PV}_{1}=PV(\mathcal{X}_{1})=\nabla^{\perp}_{h}\cdot
u_{h,\,1}+\partial_{3}\theta_{1}$

and finding the solution $p_{1}\in\mathring{H}^{1}$ of

$\Delta
p_{1}+\frac{1}{2}\partial_{3}{\min}_{0}\,(M_{1}-\partial_{3}p_{1})={PV}_{1}.$

The phases of the balanced component of the first state are then encoded in

$H_{B_{1}}\vcentcolon=\mathds{1}(q_{B,\,1}<0)=\mathds{1}(\partial_{3}p_{1}<M_{1}).$

Second we proceed as in Defintion 5.10 and compute
$d_{H_{B_{1}}}(\mathcal{X}_{1},\,\mathcal{X}_{2})$. This means that we let

$\displaystyle
PV\vcentcolon=PV(\mathcal{X}_{1})-PV(\mathcal{X}_{2}),\,M\vcentcolon=M(\mathcal{X}_{1})-M(\mathcal{X}_{2}),$
$\displaystyle
j\vcentcolon=j(\mathcal{X}_{1})-j(\mathcal{X}_{2}),\,w\vcentcolon=w(\mathcal{X}_{1})-w(\mathcal{X}_{2}),\text{
and }a\vcentcolon=a(\mathcal{X}_{1})-a(\mathcal{X}_{2}),$

and that we let $p\in\mathring{H}^{1}$ and $\sigma\in{(L^{2})}^{3}_{\sigma}$
solve

$\nabla\cdot(A_{H_{B_{1}}}\nabla p)=PV-\frac{1}{2}\partial_{3}(HM)$

and

$\nabla\times(A_{H_{B_{1}}}^{-1}\sigma)=j+(\partial_{3}w)e_{3}\text{ subject
to }\fint A_{H_{B_{1}}}^{-1}\sigma=a\text{ and }\nabla\cdot\sigma=0$

for $A$ and $A^{-1}$ as in Definition 5.5.

So finally we have that

$D(\mathcal{X}_{1},\,\mathcal{X}_{2})=\frac{1}{2}{d_{H_{B_{1}}}(\mathcal{X}_{1},\,\mathcal{X}_{2})}^{2}=\int
Q_{H}^{+}(p)+\left(1+\frac{H}{2}\right)M^{2}+Q_{H}^{-}(\sigma)+w^{2}$

for $Q_{H}^{\pm}$ as introduced in Definition 5.5. We note that, in the more
concise notation of Definition 5.10 we may write this as

$D(\mathcal{X}_{1},\,\mathcal{X}_{2})=\widetilde{E}_{H_{B_{1}}}(\mathcal{C}_{H_{B_{1}}}[\mathcal{M}(\mathcal{X}_{1})-\mathcal{M}(\mathcal{X}_{2})])$

######  Remark 5.25.

We saw in Remark 5.24 that we may write the statistical divergence in terms of
$\mathcal{C}_{H}$. This map is an analog of $\mathcal{C}$, which is the map
from measurement space to coordinate space which associates to a state’s
measurements its corresponding coordinates (see Remark 3.9). The analogous map
$\mathcal{C}_{H}$ proceeds similarly, except that now the three
transformations (between state space, measurement space, and coordinate space)
are _linear_. The linearity comes from treating a particular indicator
function $H$ as fixed.

More precisely, given an indicator function $H$, we have the following.

  * •

The “fixed $H$” coordinate-to-state map
$\mathcal{S}_{H}:\mathfrak{C}\to\mathbb{L}^{2}_{\sigma}$ is given by

$\mathcal{S}_{H}(p,\,M,\,\sigma,\,w)=(u,\,\theta,\,q)\iff\left\\{\begin{aligned}
u&=\nabla_{h}^{\perp}p+\sigma_{h}^{\perp}\\\
\theta&=\partial_{3}p+\frac{1}{2}H(M-\partial_{3}p)+\sigma_{3}\\\
q&=M-\partial_{3}p-\frac{1}{2}H(M-\partial_{3}p)-\sigma_{3},\end{aligned}\right.$

where recall that $\mathfrak{C}$ defined in Remark 3.9 is the coordinate
space.

  * •

The “fixed $H$” state-to-measurement map
$\mathcal{M}_{H}:\mathbb{L}^{2}_{\sigma}\to\mathfrak{M}$, where $\mathfrak{M}$
denotes the measurement space as in Proposition 3.7, is given by

$\mathcal{M}_{H}(u,\,\theta,\,q)=(PV,\,M,\,j_{H},\,w,\,a_{H})$

where

$j_{H}(u,\,\theta,\,q)=\partial_{3}u_{h}-\nabla_{h}^{\perp}(\theta-qH)\text{
and }a_{H}=-\fint u_{h}^{\perp}+\fint(\theta-qH)e_{3}.$

This means that only $j$ and $a$ are modified into $j_{H}$ and $a_{H}$,
respectively, but the remaining measurements are unchanged. This is because
they are the only nonlinear measurements in the first place since they are the
only measurements involving the buoyancy.

  * •

The “fixed $H$” measurement-to-coordinate map
$\mathcal{C}_{H}:\mathfrak{M}\to\mathfrak{C}$ is as defined in Lemma 5.8.

These maps will all be used below to compute the Riemann metric induced by the
statistical divergence introduced in Definition 5.23.

######  Proposition 5.26.

The map
$D:\mathbb{L}^{2}_{\sigma}\times\mathbb{L}^{2}_{\sigma}\to[0,\,\infty)$
introduced in Definition 5.23 is a statistical divergence in the sense of
Definition 5.22. In particular, for any
$\mathcal{X},\,\mathcal{Y}\in\mathbb{L}^{2}_{\sigma}$,

$\frac{1}{\varepsilon^{2}}D(\mathcal{X};\,\mathcal{X}+\varepsilon\mathcal{Y})\to
E_{B}\left({\left[\mathcal{Y}\right]}_{\mathcal{X}}\right)$

where
${\left[\mathcal{Y}\right]}_{\mathcal{X}}\vcentcolon=\left(\mathcal{S}_{B}\circ\mathcal{C}_{B}\circ\mathcal{M}_{H}\right)(\mathcal{Y})$
is known as the $\mathcal{X}$–representative of $\mathcal{Y}$ (such that if
$H_{\mathcal{X}}=H_{\mathcal{Y}}$ and
$H_{B,\,\mathcal{X}}=H_{B,\,\mathcal{Y}}$ then
${\left[\mathcal{Y}\right]}_{\mathcal{X}}=\mathcal{Y}$). Moreover

  1. 1.

this statistical divergence agrees with the conserved energy in the sense
that, if $H_{1}=H_{2}=H_{B,\,1}=H_{B,\,2}$ then

$D(\mathcal{X}_{1};\,\mathcal{X}_{2})=E_{H}(\mathcal{X}_{1}-\mathcal{X}_{2})$

where $H=\mathds{1}(q_{1}<0)=\mathds{1}(q_{2}<0)$ and

  2. 2.

this statistical divergence agrees with the extraction of a balanced component
in the sense that

$\mathcal{X}_{B}=\operatorname*{arg\,min}_{\mathcal{Y}\in\mathcal{B}}D(\mathcal{X};\,\mathcal{Y}).$

###### Proof.

The fact that $D$ is non-negative and definite follows immediately from the
positive-definiteness of $d_{H}$, which is established in Proposition 5.11.
The fact that $D$ agrees with the energy and is compatible with the extraction
of a balanced component also follows from the proof of Proposition 5.11. So
finally we turn our attention to the Riemann metric induced by this
statistical divergence. Let
$\mathcal{X},\,\mathcal{Y}\in\mathbb{L}^{2}_{\sigma}$, which we write as
$\mathcal{X}=(u,\,\theta,\,q)$ and $\mathcal{Y}=(v,\,\phi,\,r)$, and let
$\varepsilon>0$.

We note that

$\displaystyle j(\mathcal{X}+\varepsilon\mathcal{Y})-j(\mathcal{X})$
$\displaystyle=\partial_{3}(u_{h}+\varepsilon
v_{h})-\nabla_{h}^{\perp}(\theta+\varepsilon\phi)+\nabla_{h}^{\perp}({\min}_{0}\,(q+\varepsilon
r))$
$\displaystyle\quad-\partial_{3}u_{h}+\nabla_{h}^{\perp}\theta-\nabla_{h}^{\perp}{\min}_{0}\,q$
$\displaystyle=\varepsilon\partial_{3}v_{h}-\varepsilon\nabla_{h}^{\perp}\phi+\nabla_{h}^{\perp}\left[{\min}_{0}\,(q+\varepsilon
r)-{\min}_{0}\,q\right]$
$\displaystyle=\varepsilon\partial_{3}v_{h}-\varepsilon\nabla_{h}^{\perp}\phi+\nabla_{h}^{\perp}\left[\varepsilon
r\mathds{1}(q+\varepsilon r<0)+R_{\varepsilon}(q;\,r)\right]$
$\displaystyle=\varepsilon
j_{H(\mathcal{X}+\varepsilon\mathcal{Y})}(\mathcal{Y})+\nabla_{h}^{\perp}R_{\varepsilon}(q;\,r)$

for

$\displaystyle R_{\varepsilon}(q;r)$
$\displaystyle\vcentcolon=q\left[\mathds{1}(q+\varepsilon
r<0)-\mathds{1}(q<0)\right]$
$\displaystyle=\varepsilon\cdot\underbrace{\frac{q}{\varepsilon}\left[\mathds{1}\left(\frac{q}{\varepsilon}+r<0\right)-\mathds{1}\left(\frac{q}{\varepsilon}<0\right)\right]}_{\to
q(\delta_{0}\circ q)r}$

where $\delta_{0}$ denotes the Dirac delta distribution at zero such that,
since $q(\delta\circ q)=0$, $\frac{1}{\varepsilon}R_{\varepsilon}(q;\,r)\to 0$
as $\varepsilon\downarrow 0$.

Therefore, since all other measurements in $\mathcal{M}$ besides $j$ are
linear,

$\displaystyle\mathcal{M}(\mathcal{X}+\varepsilon\mathcal{Y})-\mathcal{M}(\mathcal{X})=\begin{pmatrix}\varepsilon
PV(\mathcal{Y})\\\ \varepsilon M(\mathcal{Y})\\\ \varepsilon
j_{H(\mathcal{X}+\varepsilon\mathcal{Y})}(\mathcal{Y})+\nabla_{h}^{\perp}R_{\varepsilon}(q;\,r)\\\
\varepsilon w(\mathcal{Y})\\\ \varepsilon
a(\mathcal{Y})\end{pmatrix}=\varepsilon\begin{pmatrix}PV(\mathcal{Y})\\\
M(\mathcal{Y})\\\ j_{H(\mathcal{X}+\varepsilon\mathcal{Y})}(\mathcal{Y})\\\
w(\mathcal{Y})\\\ a(\mathcal{Y})\end{pmatrix}+\varepsilon\begin{pmatrix}0\\\
0\\\ \nabla_{h}^{\perp}\frac{1}{\varepsilon}R\\\ 0\\\ 0\end{pmatrix}$
$\displaystyle=\vcentcolon\varepsilon\mathcal{M}_{H_{\varepsilon}}(\mathcal{Y})+\varepsilon\mathcal{R},$

where as noted above $\mathcal{R}\to 0$ when $\varepsilon\downarrow 0$.

So finally we have that

$\displaystyle D(\mathcal{X};\,\mathcal{X}+\varepsilon\mathcal{Y})$
$\displaystyle=E_{B}\left(\mathcal{C}_{B}\left[\mathcal{M}(\mathcal{X}+\varepsilon\mathcal{Y})-\mathcal{M}(\mathcal{X})\right]\right)$
$\displaystyle=E_{B}\left(\mathcal{C}_{B}\left[\varepsilon\mathcal{M}_{H_{\varepsilon}}(\mathcal{Y})+\varepsilon\mathcal{R}\right]\right)$
$\displaystyle=\varepsilon^{2}E_{B}\left({\left[Y\right]}_{\mathcal{X}+\varepsilon
Y}+\mathcal{C}_{B}\mathcal{R}\right)$

such that indeed, since $H(\mathcal{X}+\varepsilon\mathcal{Y})\to
H(\mathcal{X})=H$,

$\frac{1}{\varepsilon^{2}}D(\mathcal{X};\,\mathcal{X}+\varepsilon\mathcal{Y})\to
E_{B}\left({\left[\mathcal{Y}\right]}_{\mathcal{X}}\right),$

as claimed. ∎

###  5.4 Summary

The takeaway message from this section is the following: if we wish to
quantify the discrepancy between two states in $\mathbb{L}^{2}_{\sigma}$ in a
manner which is both consistent with the decomposition and with the conserved
energy then we must give up on the triangle inequality. As Sections 5.1 and
5.2 show, we are unable to obtain a metric which has both of these properties.
We therefore identify, in Section 5.3, a _statistical divergence_ which fails
to satisfy the triangle inequality but which is consistent with both the
decomposition and the conserved energy. This is summarized in the table below.

Approach | Agrees with | Agrees with | Satisfies the
---|---|---|---
| the decomposition | the energy | triangle inequality
Parseval-type metric | Yes | No | Yes
Snell-type metric | No | Yes | Yes
Statistical divergence | Yes | Yes | No

##  6 Iterative methods for the decomposition and their geometry

In this section we discuss how to compute the decomposition of a state into
its slow and fast components in _practice_. The only computationally non-
trivial step in that process is to extract the balanced component, since this
requires the inversion of the nonlinear elliptic PDE

$\Delta p+\frac{1}{2}\partial_{3}{\min}_{0}\,(M-\partial_{3}p)=PV$

where $PV$ and $M$ are given and we solve for $p$.

Our ability to solve this nonlinear elliptic PDE comes from identifying a
_variational form_ of this equation. That is the observation underpinning
[RTSS24]. We recall this variational formulation here.

######  Theorem 6.1 (Variational formulation of $PV$-and-$M$ inversion).

For any $M\in L^{2}$, $PV\in H^{-1}$, and $p\in\mathring{H}^{1}$, $p$ is a
global minimizer of $E_{\text{var}}$, defined as

$E_{\text{var}}\vcentcolon=\int_{\mathbb{T}^{3}}\frac{1}{2}{\lvert\nabla
p\rvert}^{2}-\frac{1}{4}{{\min}_{0}\,(M-\partial_{3}p)}^{2}+{\langle
PV,\,p\rangle}_{H^{-1}\times\mathring{H}^{1}},$

if and only if $p$ is an $H^{1}$–weak solution of

$\Delta p+\frac{1}{2}\partial_{3}{\min}_{0}\,(M-\partial_{3}p)=PV$

in the sense that

$\int_{\mathbb{T}^{3}}\nabla
p\cdot\nabla\phi+\frac{1}{2}{\min}_{0}\,(M-\partial_{3}p)\partial_{3}\phi=-{\langle
PV,\,\phi\rangle}_{H^{-1}\times\mathring{H}^{1}}\text{ for every
}\phi\in\mathring{H}^{1}.$ (6.1)

###### Proof.

See Lemma 4.9 in [RTSS24]. ∎

With this variational formulation of $PV$-and-$M$ inversion in hand, we can
now leverage iterative methods from optimization in order to find
computational methods. We discuss such methods in Section 6.1 below, obtaining
convergence guarantees. In Section 6.2 we then relate one of these iterative
methods (arising from Newton’s method) to the geometry of the balanced set
$\mathcal{B}$, obtaining a geometric interpretation of this descent method.

###  6.1 Convergence guarantees

In this section we introduce and compare several gradient descent methods that
can be employed to minimise the variational energy introduced in Theorem 6.1,
and thus find solutions of $PV$-and-$M$ inversion. We will discuss three
descent methods: $\mathring{H}^{1}$-gradient descent, $L^{2}$-gradient
descent, and Newton descent (which, as we will discuss, can be interpreted
locally as a gradient descent), recording corresponding convergence rates for
each method. Ultimately, the convergence rates of interest are those obtained
for _backtracking line search_ (see Definition 6.18) which are recorded in
Theorem 6.20 but we also record analogous convergence rates in Theorem 6.16
for _exact_ line search. This is done to make the rates obtained in the main
result of Theorem 6.20 easier to contextualise, by comparing them with what
can be obtained in the idealized scenario of exact line search.

We then conclude this section with a consistency result that guarantees that
minimizers of the variational energy restricted to an (appropriate) finite-
dimensional subspace of $\mathring{H}^{1}$ do converge to the true minimizer
of the energy. Put together, the convergence rates of Theorem 6.20 and the
consistency result of Theorem 6.22 tell us that we have a ready-to-implement
method for finding approximate minimizers which are guaranteed to converge to
the true minimizer.

Before we get started we recall results from [RTSS24] that will be used in
this section.

######  Theorem 6.2 (Properties of the variational energy).

The variational energy $E_{var}$ defined in Theorem 6.1 has the following
properties.

  1. 1.

$E_{var}$ is $\frac{1}{2}$-strongly convex, weakly lower semi-continuous over
$\mathring{H}^{1}$, and coercive over $\mathring{H}^{1}$ in the sense that
$E_{var}(p)\geqslant\frac{1}{16}{\left\lvert\left\lvert
p\right\rvert\right\rvert}_{\mathring{H}^{1}}^{2}-\frac{3}{4}{\left\lvert\left\lvert
M\right\rvert\right\rvert}_{L^{2}}^{2}-4{\left\lvert\left\lvert
PV\right\rvert\right\rvert}_{H^{-1}}^{2}$ for every $p\in\mathring{H}^{1}$.

  2. 2.

We may write

$E_{var}(p)=\int_{\mathbb{T}^{3}}e_{M}\left(x,\,\nabla p(x)\right)dx+\langle
PV,\,p\rangle$ (6.2)

for $e_{M}(x,\,u)\vcentcolon=\frac{1}{2}{\lvert
u\rvert}^{2}-\frac{1}{4}{\min\left(M(x)-u_{3},\,0\right)}^{2}$.

  3. 3.

$E_{var}$ is Gâteaux differentiable on $\mathring{H}^{1}$ with Gâteaux
derivative at any $p\in\mathring{H}^{1}$, denoted by $DE_{var}(p)$, given by

$DE_{var}(p)\phi=\int_{\mathbb{T}^{3}}\nabla
p\cdot\nabla\phi+\frac{1}{2}\min(M-\partial_{3}p,\,0)\partial_{3}\phi+\langle
PV,\,\phi\rangle\text{ for every }\phi\in\mathring{H}^{1}.$ (6.3)

  4. 4.

The Gâteaux derivative $E_{var}$ is Lipschitz, with specifically the estimate
${\left\lvert\left\lvert
DE_{var}(p_{1})-DE_{var}(p_{2})\right\rvert\right\rvert}_{H^{-1}}\leqslant\frac{3}{2}{\left\lvert\left\lvert
p_{1}-p_{2}\right\rvert\right\rvert}_{\mathring{H}^{1}}$ for every
$p_{1},\,p_{2}\in\mathring{H}^{1}$, and so $E_{var}$ is Fréchet
differentiable.

  5. 5.

Suppose that $E_{var}$ has a minimizer $p^{*}\in\mathring{H}^{1}$. For any
$p\in\mathring{H}^{1}$ the following estimate holds: ${\left\lvert\left\lvert
p-p^{*}\right\rvert\right\rvert}_{\mathring{H}^{1}}^{2}\leqslant
4\left(E_{var}(p)-E_{var}(p^{*})\right)=4\left(E_{var}(p)-\min
E_{var}\right).$

###### Proof.

See Proposition 4.4, Lemma 4.6, Corollary 4.7, and Lemma 4.8 of [RTSS24]. ∎

In particular [RTSS24] establishes the well-posedness of nonlinear
$PV$-and-$M$ inversion.

######  Theorem 6.3 (Existence and uniqueness).

For every $M\in L^{2}$, and $PV\in H^{-1}$ the variational energy $E_{var}$
introduced in Theorem 6.1 has a unique global minimizer in $\mathring{H}^{1}$
which is also the unique weak solution of (6.1).

Moreover we have the following $H^{2}$ regularity result ([RTSS24] proves
higher-order regularity, but this $H^{2}$ is all that is required below). Let
$M\in H^{1}$ and $PV\in L^{2}$. Any $\mathring{H}^{1}$-weak solution $p$ of
(6.1) belongs to $H^{2}$ and satisfies ${\left\lvert\left\lvert
p\right\rvert\right\rvert}_{\dot{H}^{2}}\leqslant{\left\lvert\left\lvert
M\right\rvert\right\rvert}_{\dot{H}^{1}}+2{\left\lvert\left\lvert
PV\right\rvert\right\rvert}_{L^{2}}.$

###### Proof.

This is Theorem 5.1 and Lemma 5.2 in [RTSS24]. ∎

Now to kick things off in earnest we define the gradient of a functional,
taking particular care to highlight the role payed by the choice of the
underlying inner product.

######  Definition 6.4 (Gradient).

Consider a functional $\mathcal{F}:B\to\mathbb{R}$ on a Banach space $B$
equipped with an inner product $\left(\,\cdot\,,\,\cdot\,\right)$ which is not
necessarily complete and suppose that there exists a map $D:B\to B$ such that
$D\mathcal{F}(u)\phi=\left(D(u),\,\phi\right)$ for every $u,\,\phi\in B$. We
call $D$ the _gradient_ of $\mathcal{F}$ with respect to the inner product
$\left(\,\cdot\,,\,\cdot\,\right)$. Moreover, if
$\left(\,\cdot\,,\,\cdot\,\right)$ is known as the “$X$–inner product” then we
write $D=\nabla_{X}\mathcal{F}$ and call it the _$X$ -gradient_ of
$\mathcal{F}$.

With Definition 6.4 in hand we may now turn our attention towards two
gradients of the variational energy that we will use throughout: the
$\mathring{H}^{1}$ and $L^{2}$-gradients. We record expressions for both of
these below.

######  Lemma 6.5 ($\mathring{H}^{1}$ and $L^{2}$-gradients for our
variational energy).

Consider the variational energy $E_{var}$ introduced in Theorem 6.1. The
$\mathring{H}^{1}$-gradient of $E_{var}$ is given by

$\left(\nabla_{\mathring{H}^{1}}E_{var}\right)(p)={\left(-\Delta\right)}^{-1}\left[-\Delta
p-\frac{1}{2}\partial_{3}\left(\min\left(M-\partial_{3}p,\,0\right)\right)+PV\right],$

meaning that $u=\left(\nabla_{\mathring{H}^{1}}E_{var}\right)(p)$ is the
unique weak solution of

$-\Delta u=-\Delta
p-\frac{1}{2}\partial_{3}\left(\min\left(M-\partial_{3}p,\,0\right)\right)+PV$
(6.4)

satisfying $\fint_{\mathbb{T}^{3}}u=0$, i.e. $u\in\mathring{H}^{1}$ such that,
for every $\phi\in\mathring{H}^{1}$,

$\int_{\mathbb{T}^{3}}\nabla u\cdot\nabla\phi=\int_{\mathbb{T}^{3}}\nabla
p\cdot\nabla\phi+\frac{1}{2}\min\left(M-\partial_{3}p,\,0\right)\partial_{3}\phi+\langle
PV,\,\phi\rangle$ (6.5)

Moreover, if $p\in H^{2}$, $M\in H^{1}$, and $PV\in L^{2}$ then the
$L^{2}$-gradient of $E_{var}$ is given by

$\left(\nabla_{L^{2}}E_{var}\right)(p)=-\Delta
p-\frac{1}{2}\partial_{3}\left(\min\left(M-\partial_{3}p,\,0\right)\right)+PV.$
(6.6)

###### Proof.

Item 3 of Theorem 6.2 tells us that $u=\nabla_{\mathring{H}^{1}}E_{var}(p)$
satisfies

$\displaystyle\int_{\mathbb{T}^{3}}\nabla
u\cdot\nabla\phi={\left(u,\,\phi\right)}_{\mathring{H}^{1}}={\left(\nabla_{\mathring{H}^{1}}E_{var}(p),\,\phi\right)}_{\mathring{H}^{1}}=DE_{var}(p)\phi$
$\displaystyle=\int_{\mathbb{T}^{3}}\nabla
p\cdot\nabla\phi+\frac{1}{2}\min\left(M-\partial_{3}p,\,0\right)\partial_{3}\phi+\langle
PV,\,\phi\rangle,$

which is precisely (6.5), the weak formulation of (6.4). In order to establish
(6.6) it suffices to integrate by parts the expression for the Gâteaux
derivative of $E_{var}$ obtained in item 3 of Theorem 6.2. This yields

${\left(\nabla_{L^{2}}E_{var}(p),\,\phi\right)}_{L^{2}}=DE_{var}(p)\phi=\int_{\mathbb{T}^{3}}\left[-\Delta
p-\frac{1}{2}\partial_{3}\left(\min\left(M-\partial_{3}p,\,0\right)\right)+PV\right]\phi,$

from which the claim follows. ∎

######  Remark 6.6 (Comparing the $L^{2}$ and $\mathring{H}^{1}$ gradients).

It is not a coincidence that in Lemma 6.5 above the $L^{2}$ and
$\mathring{H}^{1}$ gradients are related via
$\nabla_{\mathring{H}^{1}}E_{var}={(-\Delta)}^{-1}(\nabla_{L^{2}}E_{var})$.
This identity actually follows from the following fact. For any Hilbert space
$H\subseteq L^{2}$ and any Fréchet differentiable $F:L^{2}\to\mathbb{R}$, if
we denote by $R_{H}:R\to R^{*}$ the Riesz representation map on $H$ then
$\nabla_{H}F=R_{H}(\nabla_{L^{2}}F)$. Indeed, for any $u$, $\phi\in H$:
$DF(u)\phi={\left(\nabla_{L^{2}}F(u),\,\phi\right)}_{L^{2}}={\langle\nabla_{L^{2}}F(u),\,\phi\rangle}_{H^{*}\times
H}={\left(R_{H}(\nabla_{L^{2}}F)(u),\,\phi\right)}_{H}.$

In our case we have that $R_{\mathring{H}^{1}}={(-\Delta)}^{-1}$ in the sense
that, for any $f\in H^{-1}$, $u=R_{\mathring{H}^{1}}F$ if and only if $u$ is
the $\mathring{H}^{1}$-weak solution of $-\Delta u=f$ in $\mathbb{T}^{3}$. The
general fact above thus tells us that
$\nabla_{\mathring{H}^{1}}E_{var}=R_{\mathring{H}^{1}}(\nabla_{L^{2}}E_{var})={(-\Delta)}^{-1}(\nabla_{L^{2}}E_{var})$,
as observed in Lemma 6.5.

Besides gradient descent methods using the $\mathring{H}^{1}$ and $L^{2}$
gradients, the third descent method we will discuss is _Newton descent_. This
comes from using, locally, the inner product generated by the _Hessian_ of the
variational energy to define a gradient direction. This idea is now made
precise.

######  Definition 6.7 (Newton descent).

Let $\mathcal{F}:H\to\mathbb{R}$ be a twice Fréchet differentiable map over a
Hilbert space $H$. Suppose there exists $C>0$ such that, for every $u\in H$,
$D^{2}_{u}\mathcal{F}(\,\cdot\,,\,\cdot\,)$ induces an inner product, written
${\left(\,\cdot\,,\,\cdot\,\right)}_{D_{u}^{2}\mathcal{F}}$ and called the
_Hessian inner product_ , such that
$\frac{1}{C}{\left\lvert\left\lvert\phi\right\rvert\right\rvert}_{H}\leqslant{\left\lvert\left\lvert\phi\right\rvert\right\rvert}_{D^{2}_{u}\mathcal{F}}\leqslant
C{\left\lvert\left\lvert\phi\right\rvert\right\rvert}_{H}\text{ for every
}\phi\in H$ (i.e.
${\left\lvert\left\lvert\,\cdot\,\right\rvert\right\rvert}_{D_{u}^{2}\mathcal{F}}$
is _uniformly_ comparable to
${\left\lvert\left\lvert\,\cdot\,\right\rvert\right\rvert}_{H}$). The negative
of the gradient of $\mathcal{F}$ with respect to the Hessian inner product is
called the _Newton descent direction_.

Having defined Newton descent we now record its form for our variational
energy of interest.

######  Proposition 6.8 (Newton descent for our variational energy).

Let $E_{var}$ be the variational energy introduced in Theorem 6.1. The Hessian
inner product induced by $E_{var}$ is, for any $p\in\mathring{H}^{1}$ and
$M\in L^{2}$,

${\left(\phi,\,\psi\right)}_{D_{p}^{2}E_{var}}=\int_{\mathbb{T}^{3}}\nabla\phi\cdot\nabla\psi-\frac{1}{2}\mathds{1}(M<\partial_{3}p)(\partial_{3}\phi)(\partial_{3}\psi)$
(6.7)

and so the Newton descent direction $\pi$ is the $\mathring{H}^{1}$-weak
solution of

$-\nabla\cdot\left[\left(I-\frac{1}{2}\mathds{1}(M<\partial_{3}p)e_{3}\otimes
e_{3}\right)\nabla\pi\right]=\Delta
p+\frac{1}{2}\partial_{3}\left(\min\left(M-\partial_{3}p,\,0\right)\right)-PV,$
(6.8)

for $PV\in H^{-1}$, i.e., for all $\phi\in\mathring{H}^{1}$,

$\displaystyle\int_{\mathbb{T}^{3}}\nabla\pi\cdot\nabla\phi-\frac{1}{2}\mathds{1}(M<\partial_{3}p)(\partial_{3}\pi)\partial_{3}\phi$
$\displaystyle=\int_{\mathbb{T}^{3}}-\nabla
p\cdot\nabla\phi-\frac{1}{2}\min\left(M-\partial_{3}p,\,0\right)\partial_{3}\phi-\langle
PV,\,\phi\rangle.$ (6.9)

###### Proof.

Note that
${\left(\phi,\,\psi\right)}_{D_{p}^{2}E_{var}}=D^{2}E_{var}(\phi,\,\psi)$ and
so (6.7) follows from differentiating the expression for $DE_{var}$ recorded
in item 1 of Theorem 6.2, since $\min(\,\cdot\,,0)$ is Lipschitz with
derivative $\min(\,\cdot\,,0)^{\prime}=\mathds{1}(\,\cdot\,<0)$. Since the
Newton descent direction $\pi$ is characterised by the fact that
${\left(\pi,\,\phi\right)}_{D^{2}_{p}E_{var}}=-DE_{var}(p)\phi$ for every
$\phi\in\mathring{H}^{1}$, (6.9) then follows immediately from (6.7) and the
expression for $DE_{var}$ of item 3 of Theorem 6.2. To see that (6.8) is
indeed the strong form of (6.9) we integrate by parts. ∎

######  Remark 6.9 (Interpretation of Newton descent).

Section 6.2 below details how the Newton descent iteration, recorded in
Proposition 6.8 above, may be interpreted geometrically.

Another interpretation is already available to us now. First: note that once
Newton descent enters the so-called “quadratic regime”, backtracking line
search will systematically select steps of unit size. Given a current iterate
$p_{k}$, the next iterate $p_{k+1}$ will thus be given by $p_{k+1}=p_{k}+\pi$
where $\pi$ solves (6.8), which we may rewrite as

$-\nabla\cdot\left(A_{H_{k}}\nabla\pi\right)=\nabla\cdot\left(A_{H_{k}}\nabla
p_{k}\right)+\frac{1}{2}\partial_{3}\left(H_{k}M\right)-PV$

We may rearrange this equation and deduce that the new iterate
$p_{k+1}=p_{k}+\pi$ solves

$\nabla\cdot\left(A_{H_{k}}\nabla
p_{k+1}\right)=PV-\frac{1}{2}\partial_{3}\left(H_{k}M\right).$

In other words: the next iterate $p_{k+1}$ is obtained by solving the
linearised $PV$-and-$M$ inversion where the Heaviside characterising the
linearisation comes from the previous step. Crucially: this is precisely the
method that was used in previous works [HESS21, TS24] to numerically solve the
nonlinear $PV$-and-$M$ inversion. This means that the convergence rates
presented here for Newton descent are applicable to previous works. Moreover
this means that the convergence _guarantees_ for Newton descent are also the
first convergence guarantees of this numerical method which is already used in
practice.

The remainder of this section is devoted to the obtention of convergence rates
for each of the three descent methods discussed above. There are two key
assumptions that must be verified about our variational energy in order for
such convergence rates to be obtained. The first comes from convexity and
takes the form of a quadratic lower bound on the variational energy. The
second, which takes the form of a quadratic _upper_ bound on the variational
energy, is that to which we turn our attention now.

The basic ideas are that (1) we will introduce a simpler energy density for
which, due to its piecewise quadratic nature, we can easily find such a
quadratic upper bound, and (2) we will show that this simpler energy density
can be turned into our true variational energy density up to some affine
transformations. The result will then follow from the fact that such affine
transformations will only interact with these quadratic upper bounds in
predictable ways. In order to carry out this argument it is particularly
convenient to introduce the notion of a _quadratic remainder_ , which we
define below.

######  Definition 6.10 (Quadratic remainder).

Consider a continuously differentiable function
$f:\mathbb{R}^{d}\to\mathbb{R}$. For any $x\in\mathbb{R}^{d}$ we call function
$T_{x}^{f}:\mathbb{R}^{d}\to\mathbb{R}$ defined as
$T_{x}^{f}(y)\vcentcolon=f(y)-f(x)-Df(x)(y-x)$ for every $y\in\mathbb{R}^{d}$
the _quadratic remainder of $f$ at $x$_.

We now show that the aforementioned “simpler” energy density – introduced in
the result below – does indeed have a quadratic upper bound.

######  Lemma 6.11 (Quadratic upper bound for a piecewise quadratic function).

Consider $f:\mathbb{R}^{3}\to\mathbb{R}$ defined by $f(x)=\frac{1}{2}{\lvert
x\rvert}^{2}-\frac{1}{4}\min{\left(x_{3},\,0\right)}^{2}$ for all
$x\in\mathbb{R}^{3}$. For every $x\in\mathbb{R}^{3}$, $f(y)\leqslant
f(x)+Df(x)(y-x)+\frac{1}{2}{\lvert y-x\rvert}^{2}.$

###### Proof.

To make the computations easier we introduce the following function:
$A(x)\vcentcolon=I-\frac{1}{2}\mathds{1}(x_{3}<0)e_{3}\otimes e_{3}$, such
that $f(x)=\frac{1}{2}A(x)x\cdot x,$ and hence $\nabla f(x)=A(x)x$. In
particular note that, for every $x$, the largest eigenvalue of $A(x)$ is equal
to $1$, and so $A(x)\xi\cdot\xi\leqslant{\lvert\xi\rvert}^{2}$ for all
$\xi\in\mathbb{R}^{3}$.

In terms of the quadratic remainder $T_{x}^{f}$, we may write the target
inequality as $T_{x}^{f}(y)\leqslant\frac{1}{2}{\lvert y-x\rvert}^{2}$, so we
begin by using the fact that $A$ is pointwise symmetric to rewrite $T_{x}^{f}$
in a more convenient way. We obtain

$T_{x}^{f}(y)=\frac{1}{2}\left[A(y)-A(x)\right]y\cdot
y+\frac{1}{2}A(x)(y-x)\cdot(y-x).$

Since $A(y)-A(x)=\frac{1}{2}\left[\mathds{1}(x_{3}<0\leqslant
y_{3})-\mathds{1}(y_{3}<0\leqslant x_{3})\right]e_{3}\otimes e_{3}$ we compute
that

$T_{x}^{f}(y)=\left\\{\begin{aligned}
&-\frac{y_{3}^{2}}{4}+\frac{1}{2}A(x)(y-x)\cdot(y-x)&&\text{if
}y_{3}<0\leqslant x_{3},\\\
&\frac{x_{3}^{2}}{4}+\frac{1}{2}A(x)(y-x)\cdot(y-x)&&\text{if
}x_{3}<0\leqslant y_{3},\text{ and }\\\
&\frac{1}{2}A(x)(y-x)\cdot(y-x)&&\text{otherwise.}\end{aligned}\right.$

Only one of these three cases is troublesome since, if $y_{3}<0\leqslant
x_{3}$ or if $x_{3}$ and $y_{3}$ have the same sign then
$T_{x}^{f}(y)\leqslant\frac{1}{2}A(x)(y-x)\cdot(y-x)\leqslant\frac{1}{2}{\lvert
y-x\rvert}^{2}.$ In the troublesome case, i.e. if $x_{3}<0\leqslant y_{3}$,
then we need to be more careful. In this case writing
$z_{h}=\left(z_{1},\,z_{2}\right)$ for any $x\in\mathbb{R}^{3}$ tells us that

$\displaystyle
T_{x}^{f}(y)=\frac{1}{2}A(x)(y-x)\cdot(y-x)+\frac{1}{4}x_{3}^{2}=\frac{1}{2}{\lvert
x_{h}-y_{h}\rvert}^{2}+\frac{1}{4}{(x_{3}-y_{3})}^{2}+\frac{1}{4}x_{3}^{2}.$

Crucially: if $x_{3}<0\leqslant y_{3}$ then $x_{3}y_{3}\geqslant 0$ and hence
$\frac{1}{4}{(x_{3}-y_{3})}^{2}+\frac{1}{4}x_{3}^{2}\leqslant\frac{1}{2}{(x_{3}-y_{3})}^{2},$
which proves that $T_{x}^{f}(y)\leqslant\frac{1}{2}{\lvert x-y\rvert}^{2}$
even in this case, concluding the proof. ∎

With Lemma 6.11 in hand we may now use the fact that quadratic upper bounds on
quadratic remainders behave nicely under transformations (see Lemmas
A.16–A.18) to deduce that the true variational energy density of interest also
has a quadratic upper bound.

######  Lemma 6.12 (Quadratic upper bound for our variational energy density).

Consider $e:\mathbb{R}^{3}\to\mathbb{R}$ defined by
$e(x)\vcentcolon=\frac{1}{2}{\lvert
x\rvert}^{2}-\frac{1}{4}{\min(r-x_{3},\,0)}^{2}$, where $r$ is fixed. For
every $x,\,y\in\mathbb{R}^{3}$, $e(y)\leqslant
e(x)+De(x)(y-x)+\frac{1}{2}{\lvert x-y\rvert}^{2}$.

###### Proof.

We note that, for $f$ as in Lemma 6.11, $f(ru_{3}-x)=e(x)+a(x)$ for an affine
function $a(x)=-rx_{3}+\frac{1}{2}r^{2}$. Introducing the affine
transformation $\Phi(x)=me_{3}-x$ this means that $e=f\circ\Phi-a$. Therefore
we may use Lemmas A.16, A.17, and A.18 to see that the quadratic remainder of
$e$ is given by
$T^{e}_{x}=T_{x}^{f\circ\Phi}-T_{x}^{a}=T^{f}_{\Phi(x)}\circ\Phi.$ In
particular we deduce from Lemma 6.11 that
$T_{x}^{e}(y)\leqslant\frac{1}{2}{\lvert\Phi(y)-\Phi(x)\rvert}^{2}=\frac{1}{2}{\lvert
y-x\rvert}^{2},$ which is precisely the desired inequality. ∎

Our search for a quadratic upper bound on the variational energy now concludes
as we leverage the quadratic upper bound on the variational energy density
obtained in Lemma 6.12 to deduce that the variational energy itself does
indeed have a comparable quadratic upper bound.

######  Lemma 6.13 (Quadratic upper model for our variational energy).

Let $E_{var}$ be the variational energy introduced in Theorem 6.1. For every
$p,\,r\in\mathring{H}^{1}$, the following inequality holds:
$E_{var}(r)\leqslant
E_{var}(p)+DE_{var}(p)(r-p)+\frac{1}{2}{\left\lvert\left\lvert
r-p\right\rvert\right\rvert}_{\mathring{H}^{1}}^{2}.$

###### Proof.

We write $E_{var}$ as in (6.2), abusively writing $e_{M}(\nabla p)$ to mean
$e_{M}(x,\,\nabla p(x))$ in the sequel, and note that item 3 of Theorem 6.2
tells us that

$DE_{var}(p)\phi=\int_{\mathbb{T}^{3}}\nabla_{u}e_{M}(\nabla
p)\cdot\nabla\phi+\langle PV,\,\phi\rangle.$ (6.10)

Similarly we may reinterpret Lemma 6.12 as telling us that

$e_{M}(\nabla r)\leqslant e_{M}(\nabla p)+\nabla_{u}e_{M}(\nabla
p)\cdot\nabla(r-p)+\frac{1}{2}{\lvert\nabla(r-p)\rvert}^{2}.$ (6.11)

Putting (6.2), (6.10), and (6.11) together then yields the claim. ∎

We now have all of the ingredients we need to prove the convergence rates for
our three descent methods in the context of _exact_ line search. First we
state a generic version of that result in Proposition 6.14 before specialising
it to our energy afterwards in Theorem 6.16 . Note that the result below is
classical – see for example [BV04]. We record it here, along with its proof,
in order to ensure that good care is taken of tracking the constants arising
from comparable norms involved in the descent method. This is particularly
important for us since ultimately it will lead to significant differences
between the convergence rate of the $L^{2}$-gradient descent compared with the
other descent methods.

######  Proposition 6.14 (Key inequality for the convergence rate of gradient
descent with exact line search in Hilbert spaces).

Let ${\left(\,\cdot\,,\,\cdot\,\right)}_{X}$ be a complete inner product on a
Hilbert space $H$ for which there exists a constant $C>0$ such that

${\left\lvert\left\lvert u\right\rvert\right\rvert}_{H}\leqslant
C{\left\lvert\left\lvert u\right\rvert\right\rvert}_{X}\text{ for every }u\in
H.$ (6.12)

Let $\mathcal{F}:H\to\mathbb{R}$ be a continuously Fréchet differentiable
functional satisfying, for some constants $m,\,M>0$,

$\frac{m}{2}{\left\lvert\left\lvert
v-u\right\rvert\right\rvert}_{H}^{2}\leqslant\mathcal{F}(v)-\mathcal{F}(u)-D\mathcal{F}(u)(v-u)\leqslant\frac{M}{2}{\left\lvert\left\lvert
v-u\right\rvert\right\rvert}_{H}^{2}$ (6.13)

for every $u,\,v\in H$, and let $u^{*}$ denote the unique minimizer of
$\mathcal{F}$. Then

$\mathcal{F}\left(u-\frac{1}{C^{2}M}\nabla_{X}\mathcal{F}(u)\right)\leqslant\mathcal{F}(u)-\frac{1}{2C^{2}M}{\left\lvert\left\lvert\nabla_{X}\mathcal{F}(u)\right\rvert\right\rvert}_{X}^{2}$
(6.14)

and

$\mathcal{F}(u)-\mathcal{F}(u^{*})\leqslant\frac{1}{2m}{\left\lvert\left\lvert\nabla_{H}\mathcal{F}(u)\right\rvert\right\rvert}_{H}^{2}$
(6.15)

such that

$\mathcal{F}\left(u-\frac{1}{C^{2}M}\nabla_{X}\mathcal{F}(u)\right)-\mathcal{F}(u^{*})\leqslant\left(1-\frac{m}{C^{4}M}\right)\left(\mathcal{F}(u)-\mathcal{F}(u^{*})\right).$
(6.16)

######  Remark 6.15 (On the importance of the key inequality).

The so-called “key inequality” (6.16) warrants such a lofty name since the
actual convergence rate follows from that inequality immediately. Indeed, if
we denote let $x_{0}\in\mathbb{T}^{3}$ and denote by $x_{k}$ the iterates
obtained via gradient descent with exact line search, i.e.

$x_{k}\vcentcolon=\operatorname*{arg\,min}_{t\in\mathbb{R}}\mathcal{F}\left(x_{k-1}-t\nabla\mathcal{F}(x_{k-1})\right),$

then inducting on the key inequality tells us that

$\mathcal{F}(x_{k-1})-\mathcal{F}(x^{*})\leqslant{\left(1-\frac{m}{C^{4}M}\right)}^{k}\left(\mathcal{F}(x_{0})-\mathcal{F}(x^{*})\right),$

which guarantees that indeed $\mathcal{F}(x_{k})$ converges to
$\mathcal{F}(x_{0})$.

###### Proof of Proposition 6.14.

Plugging $v=u-t\nabla_{X}\mathcal{F}(u)$ into the second inequality in (6.13)
tells us that

$\displaystyle\mathcal{F}\left(u-t\nabla_{X}\mathcal{F}(u)\right)\leqslant\mathcal{F}(u)-t{\left\lvert\left\lvert\nabla_{X}\mathcal{F}(u)\right\rvert\right\rvert}_{X}^{2}+\frac{Mt^{2}}{2}{\left\lvert\left\lvert\nabla_{X}\mathcal{F}(u)\right\rvert\right\rvert}_{H}^{2}$
$\displaystyle\leqslant\mathcal{F}(u)+\left(\frac{C^{2}Mt^{2}}{2}-t\right){\left\lvert\left\lvert\nabla_{X}\mathcal{F}(u)\right\rvert\right\rvert}_{X}^{2}.$

Since the quadratic $q(t)\vcentcolon=\frac{C^{2}Mt^{2}}{2}-t$ is minimized at
$t^{*}=\frac{1}{C^{2}M}$, where it attains the value
$q(t^{*})=-\frac{1}{2C^{2}M}$, we deduce (6.14) from the inequality above.

To establish (6.15) we use the first inequality in (6.13). We begin by writing
$D\mathcal{F}(u)(v-u)={\left(\nabla_{H}\mathcal{F}(u),\,v-u\right)}_{H}$,
which tells us that with respect to $v$ the right-hand side of that inequality
is minimized at $v^{*}=u-\frac{1}{m}\nabla_{H}\mathcal{F}(u)$. Therefore, for
any $u,\,v\in H$,

$F(v)\geqslant\left[\mathcal{F}(u)+{\left(\nabla_{H}\mathcal{F}(u),\,v-u\right)}_{H}+\frac{m}{2}{\left\lvert\left\lvert
v-u\right\rvert\right\rvert}_{H}^{2}\right]|_{v=v^{*}}=\mathcal{F}(u)-\frac{1}{2m}{\left\lvert\left\lvert\nabla_{H}\mathcal{F}(u)\right\rvert\right\rvert}_{H}^{2}.$

Setting $v=u^{*}$ produces (6.15).

Finally we turn our attention to the key inequality (6.16). (6.12) and (6.15)
tell us that
${\left\lvert\left\lvert\nabla_{X}\mathcal{F}(u)\right\rvert\right\rvert}_{X}^{2}\geqslant\frac{2m}{C^{2}}\left(\mathcal{F}(u)-\mathcal{F}(u^{*})\right)$
and so combining this inequality with (6.14) yields (6.16). ∎

We now specialise the generic result proved above to the case of our
variational energy.

######  Theorem 6.16 (Convergence rates for our variational energy with exact
line search).

Let $E_{var}$ be the variational energy introduced in Theorem 6.1, let $V$ be
a non-empty subspace of $\mathring{H}^{1}$ (possibly equal to
$\mathring{H}^{1}$), and let $p^{*}$ be the unique minimizer of $E_{var}$ in
$V$.

  1. 1.

Let $p\in V$ and let $\pi=-\nabla_{\mathring{H}^{1}}E_{var}|_{V}(p)$ denote
the $\mathring{H}^{1}$-gradient descent direction for $E_{var}|_{V}$ at $p$.
Then
$E_{var}(p+\pi)-E_{var}(p^{*})\leqslant\frac{1}{2}\left(E_{var}(p)-E_{var}(p^{*})\right).$

  2. 2.

Let $p\in V$ and let $\pi$ denote the Newton descent direction for
$E_{var}|_{V}$ at $p$. Then
$E_{var}\left(p+\frac{1}{4}\pi\right)-E_{var}(p^{*})\leqslant\frac{31}{32}\left(E_{var}(p)-E_{var}(p^{*})\right).$

  3. 3.

Suppose that $V\subseteq H^{2}$, suppose that there exist a constant $C_{n}>0$
such that ${\left\lvert\left\lvert
p\right\rvert\right\rvert}_{\mathring{H}^{1}}\leqslant
C_{n}{\left\lvert\left\lvert p\right\rvert\right\rvert}_{L^{2}}$ for every
$p\in V$, and suppose that $M\in H^{1}$ and $PV\in L^{2}$. Let $p\in V$ and
let $\pi=-\nabla_{L^{2}}E_{var}|_{V}(p)$ denote the $L^{2}$-gradient descent
direction for $E_{var}|_{V}$ at $p$. Then
$E_{var}\left(p+\frac{1}{C_{n}^{2}}\pi\right)-E_{var}(p^{*})\leqslant\left(1-\frac{1}{2C_{n}^{4}}\right)\left(E_{var}(p)-E_{var}(p^{*})\right).$

###### Proof.

Since all of these descent directions are gradient descents, we need to verify
that the hypotheses of Proposition 6.14 are verified. Item 4 of Theorem 6.2
guarantees that $E_{var}$ is continuously Fréchet differentiable, item 1 of
Theorem 6.2, Lemma 6.13, and Lemma A.15 show that (6.13) is verified with
$m=\frac{1}{2}$ and $M=1$, and Theorem 6.3 verifies that $E_{var}$ does indeed
have a unique minimizer. The only hypothesis left to check is (6.12), and the
value of the constant $C$ varies from case to case since the inner product
${\left(\,\cdot\,,\,\cdot\,\right)}_{X}$ used to define the gradient differs
in each case.

  1. 1.

In this case
${\left(\,\cdot\,,\,\cdot\,\right)}_{X}={\left(\,\cdot\,,\,\cdot\,\right)}_{\mathring{H}^{1}}$
and so $C=1$.

  2. 2.

In this case we may use the Hessian inner product recorded in Proposition 6.8
to see that
${\left\lvert\left\lvert\phi\right\rvert\right\rvert}_{D^{2}_{p}E_{var}}^{2}=\int_{\mathbb{T}^{3}}{\lvert\nabla\phi\rvert}^{2}-\frac{1}{2}\mathds{1}(M<\partial_{3}p)\geqslant\frac{1}{2}{\left\lvert\left\lvert\phi\right\rvert\right\rvert}_{\mathring{H}^{1}}^{2},$
i.e. $C=2$.

  3. 3.

In this case $C=C_{n}$, by assumption.

∎

######  Remark 6.17 (On the convergence rate of $L^{2}$-gradient descent).

We comment on the constant $C_{n}$ which appears in the third item of Theorem
6.16 above. Note that in practice such a constant exists since we work with
$V$ a finite-dimensional subspace of $\mathring{H}^{1}$, in which case all
norms on $V$ are equivalent. Moreover, say that in such an instance the
dimension of $V$ is given by $n$ (typically commensurate with the number of
elements in a finite-element scheme). Then, as $n$ approaches infinity, the
constant $C_{n}$ would necessarily grow unboundedly, which would in turn mean
that $1-\frac{1}{2C_{n}^{4}}\uparrow 1$. This means that in the case of
$L^{2}$-gradient descent the convergence rate remains linear but the constant
involved worsens (i.e. approaches one) as the dimension of the subspace we
work with increases (e.g. as the mesh is refined).

Note that this remark also apply when backtracking is used instead of exact
line search – c.f. Theorem 6.20 below.

We now turn our attention towards the second, and arguably more important, set
of convergence rates recorded in this section: those arising from
_backtracking_ line search. For we define backtracking line search below.
Recall that, by contrast with exact line search discussed so far, backtracking
line search can be implemented immediately in practical situations (which is
typically not the case for exact line search) while retaining enough
favourable structure such that provable convergence rates are still in play
with backtracking line search.

######  Definition 6.18 (Backtracking line search).

Consider a functional $\mathcal{F}:B\to\mathbb{R}$ on a Banach space $B$, let
$x\in B$ be a current iterate, let $\Delta x\in B$ be a search direction, and
consider two parameters $\alpha\in(0,\,1/2)$ and $\beta\in(0,\,1)$.
Backtracking line search proceeds as follows: initiate the step size $t_{0}=1$
and, for $i\geqslant 0$, if

$f\left(x+t_{i}\Delta x\right)>f(x)+\alpha tDf(x)\Delta x$

then set $t_{i+1}\vcentcolon=\beta t_{i}$, while if

$f\left(x+t_{i}\Delta x\right)\leqslant f(x)+\alpha tDf(x)\Delta x$ (6.17)

then set the step size $t_{*}\vcentcolon=t_{i}$ and terminate.

As we did above in the context of exact line search when specialising from the
generic result of Proposition 6.14 to Theorem 6.16 which applies to the
variational energy of interest, we first record a generic result for the
convergence rate of gradient descent methods with _backtracking_ line search.
As with Proposition 6.14 above, such a result is classical (see [BV04]) and
its proof is provided here in order to carefully track the constants that
appear due to the presence of comparable norms.

######  Proposition 6.19 (Key inequality for the convergence rate of gradient
descent with exact line search in Hilbert spaces).

Consider the same notation and assumptions as Proposition 6.14 and let
$\alpha\in(0,\,1/2)$ and $\beta\in(0,\,1)$. For any $u\in H$ and any
$t\in\left[0,\,\frac{1}{C^{2}M}\right]$,

$\mathcal{F}\left(u-t\nabla_{X}\mathcal{F}(u)\right)\leqslant\mathcal{F}(u)-\alpha
t{\left\lvert\left\lvert\nabla_{X}\mathcal{F}(u)\right\rvert\right\rvert}_{X}^{2},$
(6.18)

i.e. the backtracking exit condition (6.17) holds. Moreover, for any $u\in H$
if $t_{*}$ denotes the step size obtained via backtracking line search with
parameters $\alpha$ and $\beta$ and search direction
$-\nabla_{X}\mathcal{F}(u)$ then

$t_{*}\geqslant\min\left(1,\,\frac{\beta}{C^{2}M}\right)$ (6.19)

and

$\mathcal{F}\left(u-t_{*}\nabla_{X}\mathcal{F}(u)\right)-\mathcal{F}(u^{*})\leqslant\left(1-\frac{2\alpha
m}{C^{2}}\min\left(1,\,\frac{\beta}{C^{2}M}\right)\right)\left(\mathcal{F}(u)-\mathcal{F}(u^{*})\right).$

###### Proof.

Proceeding as in the proof of Proposition 6.14 we deduce from (6.13) that

$\mathcal{F}\left(u-t\nabla_{X}\mathcal{F}(u)\right)\leqslant\mathcal{F}(u)+\left(\frac{C^{2}Mt^{2}}{2}-t\right){\left\lvert\left\lvert\nabla_{X}\mathcal{F}(u)\right\rvert\right\rvert}_{X}^{2}.$

In particular we note that if $t\in\left[0,\,\frac{1}{C^{2}M}\right]$ then
$\frac{C^{2}Mt^{2}}{2}-t\leqslant-\frac{t}{2}<-\alpha t$, and so indeed (6.18)
holds. As a consequence this means that either $t_{*}=t_{0}=1$ or $t_{*}=\beta
t_{i}$ for some $t_{i}$ which did _not_ satisfy the backtracking exit
condition, and so which must satisfy $t_{i}>\frac{1}{C^{2}M}$. Putting these
two options together establishes (6.19). Finally we note that $t_{*}$ must
satisfy the backtracking exit condition (6.18) and so combining that
inequality with (6.13), (6.15), and (6.19) allows us to obtain the main
inequality. ∎

We now specialise Proposition 6.19 above to our variational energy.

######  Theorem 6.20 (Convergence rates for our variational energy with
backtracking line search).

Let $E_{var}$ be the variational energy introduced in Theorem 6.1, let $V$ be
a non-empty subspace of $\mathring{H}^{1}$ (possibly equal to
$\mathring{H}^{1}$), let $p^{*}$ denote the unique minimizer of $E_{var}$ in
$V$, and let $\alpha\in(0,\,1/2)$ and $\beta\in(0,\,1)$. In each case below
let $t_{*}$ denote the step size obtained via backtracking line search with
direction $\pi$ (chosen differently in each case) and parameters $\alpha$ and
$\beta$.

  1. 1.

Let $p\in V$ and let $\pi=-\nabla_{\mathring{H}^{1}}E_{var}|_{V}(p)$ denote
the  $\mathring{H}^{1}$-gradient descent direction for $E_{var}|_{V}$ at $p$.
Then

$t_{*}\geqslant\beta\text{ and
}E_{var}(p+t_{*}\pi)-E_{var}(p^{*})\leqslant(1-\alpha\beta)\left(E_{var}(p)-E_{var}(p^{*})\right).$

  2. 2.

Let $p\in V$ and let $\pi$ denote the Newton descent direction for
$E_{var}|_{V}$ at $p$. Then

$t_{*}\geqslant\frac{\beta}{4}\text{ and
}E_{var}(p+t_{*}\pi)-E_{var}(p^{*})\leqslant\left(1-\frac{\alpha\beta}{16}\right)\left(E_{var}(p)-E_{var}(p^{*})\right).$

  3. 3.

Suppose that $V\subseteq H^{2}$, suppose that there exist a constant $C_{n}>0$
such that ${\left\lvert\left\lvert
p\right\rvert\right\rvert}_{\mathring{H}^{1}}\leqslant
C_{n}{\left\lvert\left\lvert p\right\rvert\right\rvert}_{L^{2}}$ for every
$p\in V$, and suppose that $M\in H^{1}$ and $PV\in L^{2}$. Let $p\in V$ and
let $\pi=-\nabla_{L^{2}}E_{var}|_{V}(p)$ denote the $L^{2}$-gradient descent
direction for $E_{var}|_{V}$ at $p$. Then
$t_{*}\geqslant\min\left(1,\,\frac{\beta}{C_{n}^{2}}\right)$ and

$E_{var}(p+t_{*}\pi)-E_{var}(p^{*})\leqslant\left(1-\frac{\alpha}{C_{n}^{2}}\min\left(1,\,\frac{\beta}{C_{n}^{2}}\right)\right)\left(E_{var}(p)-E_{var}(p^{*})\right).$

###### Proof.

This follows from Proposition 6.19 just as Theorem 6.16 follows from
Proposition 6.14, noting that the appropriate values of the constant $C$ in
each case were identified in the proof of Theorem 6.16. ∎

With the convergence rates in hand we change gear and turn our attention
towards a consistency result. We seek to prove that minimizers of the
variational energy restricted to (appropriate) finite-dimensional subspaces of
$\mathring{H}^{1}$ do converge to the true minimizer. In order to do so we
first prove the following elementary estimate.

######  Lemma 6.21 (Explicit local modulus of continuity of the variational
energy).

Let $E_{var}$ be the variational energy introduced in Theorem 6.1. For any
$p_{1},\,p_{2}\in\mathring{H}^{1}$ the following estimate holds.

$E_{var}(p_{1})-E_{var}(p_{2})\leqslant\left(\frac{1}{2}{\left\lvert\left\lvert
M\right\rvert\right\rvert}_{L^{2}}+\frac{3}{4}{\left\lvert\left\lvert\nabla
p_{1}\right\rvert\right\rvert}_{L^{2}}+\frac{3}{4}{\left\lvert\left\lvert\nabla
p_{2}\right\rvert\right\rvert}_{L^{2}}\right){\left\lvert\left\lvert\nabla(p_{1}-p_{2})\right\rvert\right\rvert}_{L^{2}}.$
(6.20)

###### Proof.

We write $g_{0}(s)\vcentcolon=\min(s,\,0)$ such that

$E_{var}(p_{1})-E_{var}(p_{2})=\int_{\mathbb{T}^{3}}\frac{1}{2}\left({\lvert\nabla
p_{1}\rvert}^{2}-{\lvert\nabla
p_{2}\rvert}^{2}\right)-\frac{1}{4}\left[g_{0}(M-\partial_{3}p_{1})-g_{0}(M-\partial_{3}p_{2})\right].$

We then estimate

$\left|{\lvert\nabla p_{1}\rvert}^{2}-{\lvert\nabla
p_{2}\rvert}^{2}\right\rvert=\left|\nabla(p_{1}+p_{2})\right\rvert\;\left|\nabla(p_{1}-p_{2})\right\rvert\leqslant\left(\lvert\nabla
p_{1}\rvert+\lvert\nabla p_{2}\rvert\right)\lvert\nabla(p_{1}-p_{2})\rvert$

and, since $g_{0}$ is $1$-Lipschitz, we may proceed in the same way to
estimate

$\left|{g_{0}(M-\partial_{3}p_{1})}^{2}-{g_{0}(M-\partial_{3}p_{2})}^{2}\right\rvert\leqslant\left(2\lvert
M\rvert+\lvert\partial_{3}p_{1}\rvert+\lvert\partial_{3}p_{2}\rvert\right)\lvert\partial_{3}(p_{1}-p_{2})\rvert.$

Putting it all together produces the desired inequality. ∎

We are now equipped to prove the last main result of this section, which
pertains to numerical consistency.

######  Theorem 6.22 (Convergence of the minimizers of the approximate
variational energy to the minimizer of the true variational energy).

Let $V_{n}\subseteq\mathring{H}^{1}$ be a closed subspace of
$\mathring{H}^{1}$, define $E_{n}\vcentcolon=E_{var}|_{V_{n}}$, and let
$p^{*}\vcentcolon=\operatorname*{arg\,min}E_{var}\text{ and
}p^{*}_{n}\vcentcolon=\operatorname*{arg\,min}E_{n}.$ For any
$p^{\varepsilon}_{n},\,p_{n}\in V_{n}$,

$\displaystyle{\left\lvert\left\lvert
p_{n}^{\varepsilon}-p^{*}\right\rvert\right\rvert}_{\mathring{H}^{1}}$ (6.21)
$\displaystyle\leqslant 4\left[E_{n}(p_{n}^{\varepsilon})-\min
E_{n}\right]+\left(8{\left\lvert\left\lvert
M\right\rvert\right\rvert}_{L^{2}}+12{\left\lvert\left\lvert
PV\right\rvert\right\rvert}_{H^{-1}}+3{\left\lvert\left\lvert
p_{n}-p^{*}\right\rvert\right\rvert}_{\mathring{H}^{1}}\right){\left\lvert\left\lvert
p_{n}-p^{*}\right\rvert\right\rvert}_{\mathring{H}^{1}}.$

In particular, if $p^{\varepsilon}_{n}$ and $p_{n}$ are chosen to satisfy, for
some $\alpha,\,C>0$,

$E_{n}(p_{n}^{\varepsilon})-\min E_{n}\leqslant\varepsilon\text{ and
}{\left\lvert\left\lvert
p_{n}-p^{*}\right\rvert\right\rvert}_{\mathring{H}^{1}}\leqslant
Cn^{-\alpha}\text{ for all }n$ (6.22)

then, for $n$ sufficiently large to ensure $Cn^{-\alpha}\leqslant 1$,

${\left\lvert\left\lvert
p_{n}^{\varepsilon}-p^{*}\right\rvert\right\rvert}_{\mathring{H}^{1}}\lesssim\varepsilon+\left(1+{\left\lvert\left\lvert
M\right\rvert\right\rvert}_{L^{2}}+{\left\lvert\left\lvert
PV\right\rvert\right\rvert}_{H^{-}1}\right)n^{-\alpha},$ (6.23)

i.e. $\varepsilon$-minimizers of the discretised variational energy $E_{n}$
can be made arbitrarily close to the minimizer of the true variational energy
$E_{var}$.

###### Proof.

First note that the $p_{n}^{*}$’s are well-defined since the properties of
$E_{var}$ parlay over to properties of $E_{n}$ and the proof of Theorem 6.3
can be reproduced for $E_{n}$.

Since (6.23) follows from (6.21) under the additional assumptions of (6.22) it
suffices to establish (6.21). First we note that, since $p^{*}$ and
$p^{*}_{n}$ are minimizers of $E_{var}$ and $E_{n}$, respectively, item 5 of
Theorem 6.2 (which holds _mutatis mutandis_ for $E_{n}$) tells us that

$\displaystyle{\left\lvert\left\lvert
p_{n}^{\varepsilon}-p^{*}\right\rvert\right\rvert}_{\mathring{H}^{1}}\leqslant{\left\lvert\left\lvert
p_{n}^{\varepsilon}-p_{n}^{*}\right\rvert\right\rvert}_{\mathring{H}^{1}}+{\left\lvert\left\lvert
p_{n}^{*}-p^{*}\right\rvert\right\rvert}_{\mathring{H}^{1}}$
$\displaystyle\leqslant
4\left[E_{n}(p_{n}^{\varepsilon})-E_{n}(p_{n}^{*})\right]+4\left[E_{var}(p_{n}^{*})-E_{var}(p^{*})\right].$
(6.24)

By virtue of Theorem 6.3 and 6.21 we may estimate the second term above via

$\displaystyle E_{var}(p_{n}^{*})-E_{var}(p^{*})$
$\displaystyle=E_{n}(p_{n}^{*})-E_{var}(p^{*})=\min
E_{n}-E_{var}(p^{*})\leqslant E_{var}(p_{n})-E_{var}(p^{*})$
$\displaystyle\leqslant\left(\frac{1}{2}{\left\lvert\left\lvert
M\right\rvert\right\rvert}_{L^{2}}+\frac{3}{4}\left[{\left\lvert\left\lvert
p_{n}\right\rvert\right\rvert}_{\mathring{H}^{1}}+{\left\lvert\left\lvert
p^{*}\right\rvert\right\rvert}_{\mathring{H}^{1}}\right]\right){\left\lvert\left\lvert
p_{n}-p^{*}\right\rvert\right\rvert}_{\mathring{H}^{1}}$
$\displaystyle\leqslant\left(2{\left\lvert\left\lvert
M\right\rvert\right\rvert}_{L^{2}}+3{\left\lvert\left\lvert
PV\right\rvert\right\rvert}_{H^{-1}}+\frac{3}{4}{\left\lvert\left\lvert
p_{n}-p^{*}\right\rvert\right\rvert}_{\mathring{H}^{1}}\right){\left\lvert\left\lvert
p_{n}-p^{*}\right\rvert\right\rvert}_{\mathring{H}^{1}}.$ (6.25)

Putting (6.24) and (6.25) together then produces (6.21). ∎

###  6.2 Geometry

In this section we do two things.

  * •

Foreshadowed by Proposition 3.1 we see that $p$ and $M$ may be viewed as
_coordinates_ along the balanced set $\mathcal{B}$. Moreover $PV$ and $M$ then
arise naturally as _dual coordinates_ (with respect to some natural inner
product arising from the energy).

  * •

Equipped with this geometric language we interpret the Newton descent
iteration of Section 6.1 above in geometric terms. We explain how the Newton
iteration can be viewed as a flow along the balanced set $\mathcal{B}$. This
flow follows the coordinate direction along $M$ while driving $PV$ towards its
true value.

We begin by recalling a definition from Section 3.

######  Definition 6.23 (Global chart for the balanced set).

For $\mathbb{H}\vcentcolon=\mathring{H}^{1}\times L^{2}$ we define
$\Phi:\mathbb{H}\to\mathcal{B}$ via

$\Phi(p,\,M)=\begin{pmatrix}\nabla_{h}^{\perp}p\\\
\partial_{3}p+\frac{1}{2}{\min}_{0}\,(M-\partial_{3}p)\\\
M-\partial_{3}p-\frac{1}{2}{\min}_{0}\,(M-\partial_{3}p)\end{pmatrix}.$

Figure 7: A depiction of the global chart for the balanced set introduced in
Definition 6.23.

We now collect some more properties of this map.

######  Proposition 6.24.

$\Phi$ is bi–Lipschitz, i.e. Lipschitz with Lipschitz inverse.

In particular, since the parametrisation $\Phi$ of $\mathcal{B}$ is bi-
Lipschitz it follows that the balanced set is an honest-to-goodness manifold.
This is depicted in Figure 7.

######  Corollary 6.25.

The balanced set $\mathcal{B}$ is a Lipschitz-regular Hilbert manifold,
modeled after the Hilbert space $\mathbb{H}$, with global chart $\Phi$.

###### Proof of Corollary 6.25.

This follows from immediately from Proposition 6.24. ∎

###### Proof of Proposition 6.24.

First we note that the inverse of $\Phi$ is given by

$\Phi^{-1}(u,\,\theta,\,q)=\begin{pmatrix}\Delta^{-1}\left[\nabla_{h}^{\perp}\cdot
u_{h}+\partial_{3}(\theta-{\min}_{0}\,q)\right]\\\ \theta+q\end{pmatrix}.$

Indeed, suppose that $(u,\,\theta,\,q)=\Phi(p,\,M)$. We see that

$\operatorname{sign}q=\operatorname{sign}(M-\partial_{3}p)$

and so Lemmas A.6 and A.5 tell us that

$\displaystyle(u,\,\theta,\,q)=\Phi(p,\,M)$
$\displaystyle\iff\left\\{\begin{aligned} &u=\nabla_{h}^{\perp}p\\\
&\theta=\partial_{3}p+\frac{1}{2}{\min}_{0}\,(M-\partial_{3}p)\\\
&q=M-\partial_{3}p-\frac{1}{2}{\min}_{0}\,(M-\partial_{3}p)\end{aligned}\right.$
$\displaystyle\iff\left\\{\begin{aligned} &u=\nabla_{h}^{\perp}p\\\
&\theta+q=M\\\ &\theta-{\min}_{0}\,q=\partial_{3}p\end{aligned}\right.$
$\displaystyle\iff\left\\{\begin{aligned}
&-u_{h}^{\perp}+(\theta-{\min}_{0}\,q)e_{3}=\nabla p\\\ &u_{3}=0\\\
&\theta+q=M\end{aligned}\right.$ $\displaystyle\iff\left\\{\begin{aligned}
&p=\Delta^{-1}\nabla\cdot\left[-u_{h}^{\perp}+(\theta-{\min}_{0}\,q)e_{3}\right]\\\
&u_{3}=0\\\ &M=\theta+q\end{aligned}\right.$
$\displaystyle\iff\left\\{\begin{aligned}
&p=\Delta^{-1}\left[\nabla_{h}^{\perp}\cdot
u_{h}+\partial_{3}(\theta-{\min}_{0}\,q)\right]\\\ &M=\theta+q\\\
&u_{3}=0\end{aligned}\right.$
$\displaystyle\iff(p,\,M)=\Phi^{-1}(u,\,\theta,\,q).$

The fact that both $\Phi$ and $\Phi^{-1}$ are Lipschitz then follows from the
fact that $s\mapsto{\min}_{0}\,s$ is Lipschitz, that
$\Delta:\mathring{H}^{1}\to H^{-1}$ is an isometry, and from Lemma A.14.
Indeed, for $\Phi$ we have that

$\displaystyle{\left\lvert\left\lvert\Phi(p_{1},\,M_{1})-\Phi(p_{2},\,M_{2})\right\rvert\right\rvert}_{L^{2}}^{2}$
$\displaystyle\lesssim{\left\lvert\left\lvert\nabla p_{1}-\nabla
p_{2}\right\rvert\right\rvert}_{L^{2}}^{2}+{\left\lvert\left\lvert
M_{1}-M_{2}\right\rvert\right\rvert}_{L^{2}}^{2}$
$\displaystyle={\left\lvert\left\lvert(p_{1},\,M_{1})-(p_{2},\,M_{2})\right\rvert\right\rvert}_{\mathring{H}^{1}\times
L^{2}}^{2}$

while for $\Phi^{-1}$ we have that

$\displaystyle{\left\lvert\left\lvert\Phi^{-1}(u_{1},\,\theta_{1},\,q_{1})-\Phi^{-1}(u_{2},\,\theta_{2},\,q_{2})\right\rvert\right\rvert}_{\mathring{H}^{1}\times
L^{2}}^{2}$
$\displaystyle={\left\lvert\left\lvert\Delta^{-1}\left[\nabla_{h}^{\perp}\cdot
u_{h,\,1}+\partial_{3}(\theta_{1}-{\min}_{0}\,q_{1})-\nabla_{h}^{\perp}\cdot
u_{h,\,2}-\partial_{3}(\theta_{2}-{\min}_{0}\,q_{2})\right]\right\rvert\right\rvert}_{\mathring{H}^{1}}^{2}$
$\displaystyle\qquad+{\left\lvert\left\lvert(\theta_{1}+q_{1})-(\theta_{2}+q_{2})\right\rvert\right\rvert}_{L^{2}}^{2}$
$\displaystyle={\left\lvert\left\lvert\nabla_{h}^{\perp}\cdot
u_{h,\,1}+\partial_{3}(\theta_{1}-{\min}_{0}\,q_{1})-\nabla_{h}^{\perp}\cdot
u_{h,\,2}-\partial_{3}(\theta_{2}-{\min}_{0}\,q_{2})\right\rvert\right\rvert}_{H^{-1}}^{2}$
$\displaystyle\qquad+{\left\lvert\left\lvert(\theta_{1}+q_{1})-(\theta_{2}+q_{2})\right\rvert\right\rvert}_{L^{2}}^{2}$
$\displaystyle\lesssim{\left\lvert\left\lvert
u_{h,\,1}-u_{h,\,2}\right\rvert\right\rvert}_{L^{2}}^{2}+{\left\lvert\left\lvert(\theta_{1}-{\min}_{0}\,q_{1})-(\theta_{2}-{\min}_{0}\,q_{2})\right\rvert\right\rvert}_{L^{2}}^{2}+{\left\lvert\left\lvert(\theta_{1}+q_{1})-(\theta_{2}+q_{2})\right\rvert\right\rvert}_{L^{2}}^{2}$
$\displaystyle\lesssim{\left\lvert\left\lvert(u_{1},\,\theta_{1},\,q_{1})-(u_{2},\,\theta_{2},\,q_{2})\right\rvert\right\rvert}_{L^{2}}^{2}.\qed$

######  Remark 6.26 (Alternate form of $\Phi^{-1}$).

We typically write $\Phi^{-1}$ in terms of the nonlinear $PV$-and-$M$
inversion, and not in terms of the inversion of a Laplacian as is done in the
proof of Proposition 6.24 above. This is because the Laplacian inversion does
not use $PV$, but instead uses a potential vorticity based on the _buoyancy_
(by contrast with $PV$ which is based on the equivalent potential temperature
$\theta$) given by

${PV}_{b}=\nabla_{h}^{\perp}\cdot u_{h}+\partial_{3}(\theta-{\min}_{0}\,q).$

Typically we do _not_ want to use ${PV}_{b}$ since it is not slow. That is why
$PV$ is usually preferable. However, for the purpose of the proof of
Proposition 6.24, ${PV}_{b}$ is preferable to $PV$ since it lets us prove more
easily that $\Phi^{-1}$ is Lipschitz.

So far we have been able to endow the balanced set $\mathcal{B}$ with the
structure of a Lipschitz-regular manifold. Unfortunately, that is as regular
as the balanced set gets due to the presence of ${\min}_{0}\,q$ in the
buoyancy term, which is Lipschitz but not differentiable.

Nonetheless we can formally differentiate ${\min}_{0}\,q$ and thus endow the
balanced set $\mathcal{B}$ with a formal differentiable structure. In order to
do so, we introduce notation for its formal tangent space, for the formal
derivative of the chart $\Phi$, and for the inner product we will use (which
comes from the conserved energy).

######  Definition 6.27.

Fix an indicator function $H$.

  1. 1.

We define

$\displaystyle
T_{H}\mathcal{B}\vcentcolon=\bigg{\\{}(u,\,\theta,\,q)\in\mathbb{L}^{2}_{\sigma}:$
$\displaystyle\;\partial_{3}u_{h}-\nabla_{h}^{\perp}(\theta-
qH)=0,\,u_{3}=0,\,$ $\displaystyle\fint u_{h}=0,\text{ and }\fint(\theta-
qH)=0\bigg{\\}}.$

  2. 2.

We define $D_{H}\Phi:\mathbb{H}\to T_{H}\mathcal{B}$ via

$D_{H}\Phi(p,\,M)\vcentcolon=\begin{pmatrix}\nabla_{h}^{\perp}p\\\
\partial_{3}p+\frac{H}{2}(M-\partial_{3}p)\\\
M-\partial_{3}p-\frac{H}{2}(M-\partial_{3}p)\end{pmatrix}.$

  3. 3.

For any $(u,\,\theta,q)\in\mathbb{L}^{2}_{\sigma}$ we define

${\left\lvert\left\lvert(u,\,\theta,q)\right\rvert\right\rvert}_{L^{2}_{H}}^{2}\vcentcolon=\int{\lvert
u\rvert}^{2}+\theta^{2}+q^{2}H+{(\theta+q)}^{2}$

and let ${\langle\,\cdot\,,\,\cdot\,\rangle}_{L^{2}_{H}}$ denote the
corresponding innner product.

As expected, the formal derivative of $\Phi$ completely characterizes the
formal tangent space. (This would be trivially true if the balanced set
$\mathcal{B}$ was endowed with an honest-to-goodness differentiable structure
by its chart $\Phi$.)

######  Lemma 6.28.

$T_{H}\mathcal{B}=\operatorname{im}D_{H}\Phi$.

###### Proof.

Proceeding similarly to the proof of Proposition 3.1 we introduce the “good
unknown” (see Remark 3.2)

$v_{H}\vcentcolon=-u_{h}^{\perp}+(\theta-qH)e_{3}.$

For $\mathcal{X}=(u,\,\theta,\,q)$ we have on one hand that, as per Lemma A.5,

$\mathcal{X}\in T_{H}\mathcal{B}\iff\nabla\times v_{H},\,\fint v_{H}=0,\text{
and }u_{3}=0.$

On the other hand, proceeding as in the proof of Proposition 6.24 and using
Lemma A.6 tells us that

$\displaystyle\mathcal{X}=D_{H}\Phi(p,\,M)$
$\displaystyle\iff\left\\{\begin{aligned} &-u_{h}^{\perp}+(\theta-
qH)e_{3}=\nabla p\\\ &u_{3}=0\\\ &\theta+q=0\end{aligned}\right.$
$\displaystyle\iff\left\\{\begin{aligned} &v_{H}=\nabla p\\\ &u_{3}=0\\\
&\theta+q=M\end{aligned}\right.$

The Helmholtz decomposition (Corollary A.2) allows us to conclude:

$\displaystyle\mathcal{X}\in T_{H}\mathcal{B}$ $\displaystyle\iff\nabla\times
v_{H}=0,\,\fint v_{H}=0,\text{ and }u_{3}=0$ $\displaystyle\iff\nabla\times
v_{H}=0,\,\fint v_{H}=0,\,u_{3}=0,\text{ and }\theta+q=\vcentcolon M$
$\displaystyle\iff v_{H}=\nabla p\text{ for some
}p\in\mathring{H}^{1},\,u_{3}=0,\text{ and }\theta+q=M$
$\displaystyle\iff\mathcal{X}=D_{H}\Phi(p,\,M)\text{ for some
}(p,\,M)\in\mathbb{H}$
$\displaystyle\iff\mathcal{X}\in\operatorname{im}D_{H}\Phi.\qed$

Figure 8: The formal tangent space to the balanced set introduced in
Definition 6.27 is the image of the formal derivative of the global chart.

We now remark that, as suggested by the notation we chose to employ,
$D_{H}\Phi$ may be viewed formally as the derivative of $\Phi$. Since the
balanced set $\mathcal{B}$ is the image of the chart $\Phi$, it follows that
the image of its formal derivative $D_{H}\Phi$ may be interpreted formally as
the tangent space to that same balanced set. Since Lemma 6.28 above tells us
that the image of $D_{H}\Phi$ is precisely $T_{H}\mathcal{B}$, we see that
once again the notation was chosen to be suggestive: $T_{H}\mathcal{B}$ may be
viewed formally as the tangent space to the balanced set $\mathcal{B}$. This
is made precise in Remark 6.29 below and is depicted in Figure 8.

######  Remark 6.29.

Formally, we may compute that, for $H\vcentcolon=\mathds{1}(q<0)$ and for any
derivative $\partial$,

$\partial({\min}_{0}\,q)=(\partial q)\mathds{1}(q<0)=(\partial q)H.$

This shows that $D_{H}\Phi$ is, formally, the derivative of $\Phi$. Since
$\Phi$ is a global chart for the Hilbert manifold $\mathcal{B}$ this means
that we may formally view $T_{H}\mathcal{B}$, which is the image of
$D_{H}\Phi$, as the tangent space of $\mathcal{B}$.

An alternative justification, more direct albeit still formal, is as follows.
Let $\mathcal{X}=(u,\,\theta,\,q):(-1,\,1)\to\mathcal{B}$ be a curve of
balanced states and let $\mathcal{Z}$ denote its derivative at zero, i.e.

$\mathcal{Z}=(v,\,\phi,\,r)\vcentcolon=\frac{d}{ds}\mathcal{X}(s)\Big{|}_{s=0}.$

Since $\mathcal{X}\in\mathcal{B}$ it follows that

$j(\mathcal{X})=0,\,w(\mathcal{X})=0,\text{ and }a(\mathcal{X})=0.$

Both $w$ and $a$ are linear, so we deduce immediately that

$w(\mathcal{Z})=0\text{ and }a(\mathcal{Z})=0.$

What about $j$? Using the formal identity $\partial({\min}_{0}\,q)=(\partial
q)H$ discussed above we compute that

$0=\frac{d}{ds}j(\mathcal{X})\Big{|}_{s=0}=\frac{d}{ds}\left[\partial_{3}u_{h}-\nabla_{h}^{\perp}(\theta-{\min}_{0}\,q)\right]\Big{|}_{s=0}=\partial_{3}v_{h}-\nabla_{h}^{\perp}(\phi-
rH)=\vcentcolon j_{H}(\mathcal{Z}).$

This means that $\mathcal{Z}$ belongs to $T_{H}\mathcal{B}$. In other words:
if $\mathcal{X}$ is a curve of balanced states then its formal tangent vector
$\mathcal{Z}$ lies in $T_{H}\mathcal{B}$. This warrants treating formally
$T_{H}\mathcal{B}$ as the tangent space of the balanced set $\mathcal{B}$.

Note that since the chart $\Phi$ is invertible and since $D_{H}\Phi$ is its
formal derivative, we may expect $D_{H}\Phi$ to inherit this invertibility.
This is indeed the case. We record this result here since it will be of use
later.

######  Lemma 6.30.

$D_{H}\Phi$ is invertible.

###### Proof.

This follows as in the proof that $\Phi$ is invertible (Proposition 6.24)
since $(1+H)(1-H/2)=1$ and so

$\displaystyle(u,\,\theta,\,q)=D_{H}\Phi(p,\,M)$
$\displaystyle\iff\left\\{\begin{aligned} &u=\nabla_{h}^{\perp}p\\\
&\theta=\partial_{3}p+\frac{H}{2}(M-\partial_{3}p)\\\
&q=M-\partial_{3}p-\frac{H}{2}(M-\partial_{3}p)\end{aligned}\right.$
$\displaystyle\iff\left\\{\begin{aligned} &-u_{h}^{\perp}+(\theta-
qH)e_{3}=\nabla p\\\ &u_{3}=0\\\ &\theta+q=M\end{aligned}\right.$
$\displaystyle\iff\left\\{\begin{aligned}
&p=\Delta^{-1}\nabla\cdot\left[-u_{h}^{\perp}+(\theta-qH)e_{3}\right]\\\
&u_{3}=0\\\ &M=\theta+q\end{aligned}\right.$
$\displaystyle\iff\left\\{\begin{aligned}
&p=\Delta^{-1}\left[\nabla_{h}^{\perp}\cdot u_{h}+\partial_{3}(\theta-
qH)\right]\\\ &u_{3}=0\\\ &M=\theta+q\end{aligned}\right.$

and so the inverse is given by

${(D_{H}\Phi)}^{-1}(u,\,\theta,\,q)=\begin{pmatrix}\Delta^{-1}\left[\nabla_{h}^{\perp}\cdot
u_{h}+\partial_{3}(\theta-qH)\right]\\\ \theta+q\end{pmatrix}.\qed$

Note that in particular this allows us to view $D_{H}\Phi$ as recording the
_coordinate vectors_ arising from the chart $\Phi$. This is made precise in
Remark 6.31 below and depicted in Figure 9.

######  Remark 6.31 (Coordinate vectors).

Since $D_{H}\Phi$ is the formal derivative of $\Phi$ and since $D_{H}\Phi$ is
an invertible linear map whose image is precisely the formal tangent space
$T_{H}\mathcal{B}$, we may view $D_{H}\Phi$ as recording the coordinate
vectors corresponding to $\Phi$. Indeed, for any
$\mathcal{Z}=(u,\,\theta,\,q)\in T_{H}\mathcal{B}$ there is a unique
$(\pi,\,\mu)\in\mathbb{H}$ such that

$\mathcal{Z}=(D_{H}\Phi)(\pi,\,\mu)=\underbrace{\begin{pmatrix}\nabla_{h}^{\perp}\pi\\\
\left(1-\frac{H}{2}\right)\partial_{3}\pi\\\
-\left(1-\frac{H}{2}\right)\partial_{3}\pi\end{pmatrix}}_{=\vcentcolon
e_{p}(\pi)}+\underbrace{\begin{pmatrix}0\\\ \frac{H}{2}\mu\\\
\left(1-\frac{H}{2}\right)\mu\end{pmatrix}}_{=\vcentcolon e_{M}(\mu)}$

where $e_{p}$ and $e_{M}$ denote the coordinate “vectors” along the $p$ and
$M$ coordinates, respectively. Note that these are not actually vectors, but
rather coordinate _operators_ , namely $e_{p}:\mathring{H}^{1}\to
T_{H}\mathcal{B}$ and $e_{M}:L^{2}\to T_{H}\mathcal{B}$ such that
$T_{H}\mathcal{B}=\operatorname{im}e_{p}+\operatorname{im}e_{M}$. This is
depicted in Figure 9.

Figure 9: The “columns” of the formal derivative $D_{H}\Phi$, denoted by
$e_{p}$ and $e_{M}$, may be viewed as coordinate vectors on the formal tangent
space.

A minor miracle now occurs: while $p$ and $M$ are coordinates along the
balanced set $\mathcal{B}$, their dual coordinates turn out to be,
essentially, $PV$ and $M$! We see this by computing the $L^{2}_{H}$–adjoint of
$D_{H}\Phi$.

######  Lemma 6.32.

Note that the dual of $\mathbb{H}$ is $\mathbb{H}^{*}=H^{-1}\times L^{2}$.
Equipping $T_{H}\mathcal{B}$ with the inner product $L^{2}_{H}$, the adjoint
${(D_{H}\Phi)}^{*}:T_{H}\mathcal{B}\to\mathbb{H}^{*}$ is given by

${(D_{H}\Phi)}^{*}(u,\,\theta,\,q)=\begin{pmatrix}-\nabla_{h}^{\perp}\cdot
u_{h}-\partial_{3}\theta+\frac{1}{2}\partial_{3}\left[H(\theta+q)\right]\\\
\left(1+\frac{H}{2}\right)(\theta+q)\end{pmatrix}=\begin{pmatrix}-PV+\frac{1}{2}\partial_{3}(HM)\\\
\left(1+\frac{H}{2}\right)M\end{pmatrix}.$

###### Proof.

For any $(p,\,M)\in\mathbb{H}$ and any $(u,\,\theta,\,q)\in T_{H}\mathcal{B}$
we compute by integrating by parts that

$\displaystyle{\left\langle
D_{H}\Phi(p,\,M),\,(u,\,\theta,\,q)\right\rangle}_{L^{2}_{H}}$
$\displaystyle=\int
u\cdot\nabla_{h}^{\perp}p+\theta\left[\partial_{3}p+\frac{H}{2}(M-\partial_{3}p)\right]$
$\displaystyle\qquad+qH\left[M-\partial_{3}p-\frac{H}{2}(M-\partial_{3}p)\right]+(\theta+q)M$
$\displaystyle=\int
u\cdot\nabla_{h}^{\perp}p+\theta\partial_{3}p-\frac{H}{2}(\theta+q)\partial_{3}p+\left(1+\frac{H}{2}\right)(\theta+q)M$
$\displaystyle=\int-(\nabla_{h}^{\perp}\cdot
u_{h})p-(\partial_{3}\theta)p+\frac{1}{2}\partial_{3}\left[H(\theta+q)\right]+\left(1+\frac{H}{2}\right)(\theta+q)M$
$\displaystyle={\left\langle-\nabla_{h}^{\perp}\cdot
u_{h}-\partial_{3}\theta+\frac{1}{2}\partial_{3}[M(\theta+q)],\,p\right\rangle}_{H^{-1}\times\mathring{H}^{1}}$
$\displaystyle\qquad+{\left\langle\left(1+\frac{H}{2}\right)(\theta+q),\,M\right\rangle}_{L^{2}}$
$\displaystyle={\left\langle\begin{pmatrix}-\nabla_{h}^{\perp}\cdot
u_{h}-\partial_{3}\theta+\frac{1}{2}\partial_{3}[H(\theta+q)]\\\
\left(1+\frac{H}{2}\right)M\end{pmatrix},\,\begin{pmatrix}p\\\
M\end{pmatrix}\right\rangle}_{H^{*}\times H}.\qed$

######  Remark 6.33 (Dual coordinate vectors).

Just as in Remark 6.31 we viewed $D_{H}\Phi$ as recording the coordinate
vectors corresponding to the chart $\Phi$, we can view ${(D_{H}\Phi)}^{*}$ as
recording their dual coordinate one-forms. Indeed, if we define, for any
$(u,\,\theta,\,q)\in T_{H}\mathcal{B}$,

$\displaystyle e_{p}^{\flat}(u,\,\theta,\,q)$
$\displaystyle\vcentcolon=-PV+\frac{1}{2}\partial_{3}(HM)\text{ and }$
$\displaystyle e_{M}^{\flat}(u,\,\theta,\,q)$
$\displaystyle\vcentcolon=\left(1+\frac{H}{2}\right)(\theta+q)$

then we see that, for any $(u,\,\theta,\,q)\in T_{H}\mathcal{B}$ and any
$(\pi,\,\mu)\in\mathbb{H}$,

$\displaystyle{\langle(u,\,\theta,\,q),\,e_{p}(\pi)\rangle}_{L^{2}_{H}}$
$\displaystyle={\langle
e_{p}^{\flat}(u,\,\theta,\,q),\,p\rangle}_{\mathring{H}^{1}}\text{ and }$
$\displaystyle{\langle(u,\,\theta,\,q),\,e_{M}(\mu)\rangle}_{L^{2}_{H}}$
$\displaystyle={\langle e_{M}^{\flat}(u,\,\theta,\,q),\,M\rangle}_{L^{2}}.$

This is depicted in Figure 10.

Figure 10: A depiction of the dual coordinates
$e_{M}^{\flat}=\left(1+\frac{H}{2}\right)M$ and
$e_{p}^{\flat}=-PV+\frac{1}{2}\partial_{3}(HM)$ on the formal tangent space
$T_{H}\mathcal{B}$.

We can see that these coordinates $p$ and $M$ are actually orthogonal. This is
recorded below, in a result that will also come in handy when interpreting the
Newton descent iteration geometrically.

######  Lemma 6.34.

For any $(p,\,M)\in\mathbb{H}$,

${(D_{H}\Phi)}^{*}(D_{H}\Phi)(p,\,M)=\begin{pmatrix}-\nabla\cdot(A_{H}\nabla
p)\\\ \left(1+\frac{H}{2}\right)M\end{pmatrix}$

for $A_{H}\vcentcolon=I-\frac{H}{2}e_{3}\otimes e_{3}$ (as usual).

######  Remark 6.35.

Note that ${(D_{H}\Phi)}^{*}(D_{H}\Phi)=(-\Delta)\oplus 1$ when $H=0$, which
is the canonical isometry from $\mathbb{H}=\mathring{H}^{1}\times L^{2}$ to
its dual $H^{*}=H^{-1}\times L^{2}$. This is because when $H=0$ the dry case
may be viewed as a special case of the moist Boussinesq system which
corresponds to neglecting $q$ (i.e. setting $q=0$).

###### Proof of Lemma 6.34.

Let $(p,\,M)\in\mathbb{H}$ and let us write
$(u,\,\theta,\,q)\vcentcolon=D_{H}\Phi(p,\,M)$. Then
$M(u,\,\theta,\,q)\vcentcolon=\theta+q=M$ while

$PV(u,\,\theta,\,q)\vcentcolon=\nabla_{h}^{\perp}\cdot
u_{h}+\partial_{3}\theta=\nabla\cdot(A_{H}\nabla
p)+\frac{1}{2}\partial_{3}(HM).$

Therefore

$\displaystyle{(D_{H}\Phi)}^{*}(D_{H}\Phi)(p,\,M)$
$\displaystyle=\begin{pmatrix}-PV+\frac{1}{2}\partial_{3}(HM)\\\
\left(1+\frac{H}{2}\right)M\end{pmatrix}$
$\displaystyle=\begin{pmatrix}-\nabla\cdot(A_{H}\nabla
p)-\frac{1}{2}\partial_{3}(HM)+\frac{1}{2}\partial_{3}(HM)\\\
\left(1+\frac{H}{2}\right)M\end{pmatrix}$
$\displaystyle=\begin{pmatrix}-\nabla\cdot(A_{H}\nabla p)\\\
1+H/2\end{pmatrix}$

as claimed. ∎

######  Remark 6.36.

Written in block form,

${(D_{H}\Phi)}^{*}(D_{H}\Phi)=\begin{pmatrix}-\nabla\cdot(A_{H}\nabla)&0\\\
0&1+H/2\end{pmatrix}$

which is block-diagonal. This means that the coordinates $p$ and $M$ of the
Hilbert manifold $\mathcal{B}$, which arise from the global chart $\Phi$, may
formally be viewed as $L_{H}^{2}$–orthogonal (but not orthonormal).

In other words, for any $\mathcal{Z}\in T_{H}\mathcal{B}$, which may be
written uniquely as

$\mathcal{Z}=e_{p}(\pi)+e_{M}(\mu)$

for $(\pi,\,\mu)\in\mathbb{H}$ and $e_{p}$ and $e_{M}$ as in Remark 6.31, we
have that

$\displaystyle{\langle e_{p}(\pi),\,e_{M}(\mu)\rangle}_{L^{2}_{H}}=\int
0+\left(1-\frac{H}{2}\right)(\partial_{3}\pi)\cdot\frac{H}{2}\mu-\left(1-\frac{H}{2}\right)(\partial_{3}\pi)\cdot\left(1-\frac{H}{2}\right)\mu\cdot
H+0$
$\displaystyle=\int\frac{H}{4}(\partial_{3}\pi)\mu-\frac{H}{4}(\partial_{3}\pi)\mu=0.$

This allows us to refine our comment of Remark 6.31: the decomposition
$T_{H}\mathcal{B}=\operatorname{im}e_{p}+\operatorname{im}e_{M}$ is actually
$L^{2}_{H}$–orthogonal.

Moreover, for the dual one-forms $e_{p}^{\flat}$ and $e_{M}^{\flat}$ as
defined in Remark 6.33 we have that

$\mu=\theta+q\text{ and }\pi\text{ solves
}\nabla\cdot(A_{H}\nabla\pi)=\nabla_{h}^{\perp}\cdot
u_{h}+\partial_{3}\theta.$ (6.26)

Indeed:

$\left(1+\frac{H}{2}\right)(\theta+q)=e_{M}^{\flat}(\mathcal{Z})=e_{M}(\flat(e_{p}(\pi))+e_{M}^{\flat}(e_{M}(\mu))=0+\left(1+\frac{H}{2}\right)\mu.$

Similarly: on one hand

$e_{p}^{\flat}(\mathcal{Z})=-\nabla_{h}^{\perp}\cdot
u_{h}-\partial_{3}\theta+\frac{1}{2}\partial_{3}[H(\theta+q)]=-\nabla_{h}^{\perp}\cdot
u_{h}-\partial_{3}\theta+\frac{1}{2}\partial_{3}(HM)$ (6.27)

while on the other hand

$e_{p}^{\flat}(e_{p}(\pi))+e_{p}^{\flat}(e_{M}(\mu))=-\nabla\cdot(A_{H}\nabla\pi)+\frac{1}{2}\partial_{3}(H\mu)=-\nabla\cdot(A_{H}\nabla\pi)+\frac{1}{2}\partial_{3}(HM).$
(6.28)

Equating (6.27) and (6.28) yields the equation for $\pi$ recorded in (6.26),
as desired.

######  Remark 6.37 (Comparison of the $L^{2}_{H}$–orthogonal decomposition
with well-known orthogonal decompositions).

In Remark 6.33 above we have explained how any formal tangent vector may be
written in a unique way in terms of the “coordinate vectors” $e_{p}$ and
$e_{M}$.

More precisely, we know that for any formal tangent vector
$\mathcal{Z}=(u,\,\theta,\,q)\in T_{H}\mathcal{B}$ there exists a unique pair
$(\pi,\,\mu)\in\mathbb{H}$ such that

$\displaystyle\mathcal{Z}=e_{p}(\pi)+e_{M}(\mu)$ $\displaystyle\text{ where
}\left\\{\begin{aligned}
&e_{M}^{\flat}(e_{M}(\mu))=e_{M}^{\flat}(\mathcal{Z})\text{ and }\\\
&e_{p}^{\flat}(e_{p}(\pi))=e_{p}^{\flat}(\mathcal{Z})\end{aligned}\right.$
(6.29) $\displaystyle\iff\left\\{\begin{aligned} &\mu=\theta+q\text{ and }\\\
&\pi\text{ solves }\nabla\cdot(A_{H}\nabla\pi)=PV.\end{aligned}\right.$

We will now help the reader make better sense of this by showing how their
favourite orthogonal decomposition, namely that of Fourier series, may be
written in this way.

Fourier series tell us that every $f\in L^{2}(\mathbb{T}^{3})$ has uniquely
determined coordinates ${(a_{k})}_{k}\in l^{2}(\mathbb{Z}^{3})$ such that

$f(x)=\sum_{k}a_{k}e^{2\pi ik\cdot x}.$ (6.30)

Given a function $f$, the process producing its Fourier series can be split
into two steps.

  1. 1.

We compute the amplitudes $a_{k}$.

  2. 2.

We reconstruct $f$ from its amplitudes.

Let us rewrite each of these two steps in a manner which will make it easier
to directly compare with the $L^{2}_{H}$–orthogonal decomposition discussed
above.

  * •

Step 1: The computation of each amplitude corresponds to a map
$a_{k}:L^{2}(\mathbb{T}^{3})\to\mathbb{C}$ defined by

$a_{k}(f)=\int_{\mathbb{T}^{3}}f(x)e^{-2\pi ik\cdot x}dx.$

  * •

Step 2: We write the reconstruction step in an admittedly unusual manner. This
will make it easier to compare Fourier series to the $L^{2}_{H}$–orthogonal
decomposition of our formal tangent vectors.

If we view each $a_{k}$ as corresponding to a coordinate direction in
$L^{2}(\mathbb{T}^{3})$ we may then encode this coordinate direction as a map
$e_{k}:\mathbb{C}\to L^{2}(\mathbb{T}^{3})$ given by

$[e_{k}(a)](x)=ae^{2\pi ik\cdot k}.$

Combining these formulations of the two steps used to produce Fourier series
we may write (6.30) in the form

$f=\sum_{k}e_{k}(a_{k})\text{ where }a_{k}={\langle f,\,e^{2\pi ik\cdot
x}\rangle}_{L^{2}}.$ (6.31)

In particular, in the language we used to describe the formal tangent space
$T_{H}\mathcal{B}$, each “coordinate vector” $e_{k}$ has a “dual coordinate
vector” $e_{k}^{\flat}$, which is really just its adjoint and is a map
$e_{k}^{\flat}:L^{2}(\mathbb{T}^{3})\to\mathbb{C}$. Using the $L^{2}$ inner
product on $L^{2}(\mathbb{T}^{3})$ and the standard complex inner product on
$\mathbb{C}$ defined by ${\langle x,\,y\rangle}_{\mathbb{C}}=x\bar{y}$ for
every $x,\,y\in\mathbb{C}$ we may compute these dual coordinates vectors: for
any $k\in\mathbb{Z}^{3}$, any $f\in L^{2}(\mathbb{T}^{3})$, and any
$a\in\mathbb{C}$,

$\displaystyle{\langle e_{k}^{\flat}(f),\,a\rangle}_{\mathbb{C}}={\langle
f,\,e_{k}(a)\rangle}_{L^{2}}=\int_{\mathbb{T}^{3}}f(x)\overline{ae^{2\pi
ik\cdot x}}dx=\bar{a}\int_{\mathbb{T}^{3}}f(x)e^{-2\pi ik\cdot
x}dx=\bar{a}\,a_{k}(f)={\langle a_{k}(f),\,a\rangle}_{\mathbb{C}}.$

In other words: $e_{k}^{\flat}=a_{k}$.

We may thus write (6.31), already an alternate formulation of (6.30), in a
manner even more reminiscent of our treatment of coordinate vectors and dual
coordinate vectors in $T_{H}\mathcal{B}$, namely

$f=\sum_{k}e_{k}(a_{k})\text{ where }a_{k}=e_{k}^{\flat}(f).$ (6.32)

Finally, to turn this expression into an exact parallel of our treatment of
$T_{H}\mathcal{B}$ we must use the orthonormality of the Fourier basis. That
is, we use the fact that

$e_{k}^{\flat}(e_{k}(a))=a_{k}(e_{k}(a))=\int_{\mathbb{T}^{3}}ae^{2\pi ik\cdot
x}e^{-2\pi ik\cdot x}dx=a.$

We may therefore rewrite (6.32) in its final form as

$f=\sum_{k}e_{k}(a_{k})\text{ where
}e_{k}^{\flat}(e_{k}(a_{k}))=e_{k}^{\flat}(f).$ (6.33)

This is exactly the same form as (6.29)!

This section is concerned with the geometry of the balanced set $\mathcal{B}$
on its own, and not with how it is embedded in the broader state space
$\mathbb{L}^{2}_{\sigma}$. Nonetheless, at this stage it is worth commenting
on how the $L^{2}_{H}$ inner product endowed on the formal tangent space
$T_{H}\mathcal{B}$ of $\mathcal{B}$ interacts with the wave set. The result
below is depicted in Figure 11.

######  Lemma 6.38.

The wave set $\mathcal{W}$ is $L^{2}_{H}$–orthogonal to the formal tangent
space $T_{H}\mathcal{B}$ in the sense that, for any $\mathcal{Z}\in
T_{H}\mathcal{B}$ and any $\mathcal{Y}\in\mathcal{W}$,
${\langle\mathcal{Z},\,\mathcal{Y}\rangle}_{L^{2}_{H}}=0$.

###### Proof.

Lemma 6.28 tells us that $\mathcal{Z}=D_{H}\Phi(\pi,\,\mu)$ for some
$(\pi,\,mu)\in\mathbb{H}$ while Proposition 3.3 tells us that
$\mathcal{Y}=\Psi(\sigma,\,w)$ for some $\sigma\in\mathring{H}^{1}_{\sigma}$
and $w\in L^{2}$. Then

$\displaystyle{\langle\mathcal{Z},\,\mathcal{Y}\rangle}_{L^{2}_{H}}$
$\displaystyle={\langle
D_{H}\Phi(\pi,\,M),\,\Psi(\sigma,\,w)\rangle}_{L^{2}_{H}}$
$\displaystyle=\int\nabla_{h}^{\perp}\pi\cdot\sigma_{h}^{\perp}+\left(1-\frac{H}{2}\right)(\partial_{3}\pi)\cdot\sigma_{3}+\left(1-\frac{H}{2}\right)(\partial_{3}\pi)\cdot\sigma_{3}\cdot
H+0$
$\displaystyle=\int\nabla_{h}\pi\cdot\sigma_{h}+(\partial_{3}\pi)\sigma_{3}=\int\nabla\pi\cdot\sigma=0$

since $\sigma$ is divergence-free. ∎

$\mathcal{B}$$p$$M$$T_{H}\mathcal{B}$$\mathcal{Z}=D_{H}\Phi(\pi,\,\mu)$$\mathcal{W}$$\mathcal{Y}=\Psi(\xi,\,w)$$\mathcal{X}$
Figure 11: Depiction of Lemma 6.38: the wave set is $L^{2}_{H}$–orthogonal to
the formal tangent space to the balanced set.

We are now ready to use the geometric language introduced in this section in
order to provide a geometric interpretation of the Newton descent iteration
presented in Section 6.1.

First, we note that we can rewrite the defining equation of the Newton descent
search direction, namely (6.8), as follows.

######  Remark 6.39.

The Newton descent direction $\pi$ solves, for
$H\vcentcolon=\mathds{1}(M<\partial_{3}p)$,

$-\nabla\cdot(A_{H}\nabla\pi)=PV(\Phi(p,\,M))-PV.$

This follows from (6.8) upon observing that
$\min_{0}(M-\partial_{3}p)=H(M-\partial_{3}p)$.

In other words, and this will play a key role in the geometric interpretation
of Newton descent: $\pi$ is determined by the potential vorticity discrepancy,
i.e. by how far the potential vorticity at the current guess is from the true
potential vorticity.

We are now ready to describe the geometric interpretation of Newton descent.

######  Remark 6.40 (Geometric interpretation of Newton descent).

Consider the following formal flow on the balanced set $\mathcal{B}$ and its
formal tangent space $T_{H}\mathcal{B}$.

$\left\\{\begin{aligned} &\dot{\mathcal{X}}(s)={(D_{H}\Phi)}^{-*}\left(PV-
PV(\mathcal{X}),\,0\right)\text{ and }\\\
&\mathcal{X}(0)=\Phi(p_{0},\,M)\end{aligned}\right.$ (6.34)

where $PV$ and $M$ are given and where the starting point
$p_{0}\in\mathring{H}^{1}$ may be freely chosen.

We claim that, formally, the Newton descent iteration may be viewed as a
discretisation of this flow. First: what does this flow even do?

Well, since

$e_{p}^{\flat}=-PV+\frac{1}{2}\partial_{3}(HM)\text{ and
}e_{M}^{\flat}=\left(1+\frac{H}{2}\right)M$

we may view $PV$ and $M$ as dual coordinates along the balanced set
$\mathcal{B}$. The map ${(D_{H}\Phi)}^{-*}:\mathbb{H}^{*}\to T_{H}\mathcal{B}$
thus associates to $(PV,\,M)\in\mathbb{H}^{*}$ the corresponding tangent
vector in $T_{H}\mathcal{B}$. This means that the flow follows the
$PV$–direction, keeping the value of $M$ fixed, so as to approach the given
target value of $PV$. This is depicted in Figure 12. (An analog of this flow
is presented in Remark 6.41 below in a much simpler setting in order to make
the intuition easier to follow.)

To see that the Newton iteration is a discretisation of this flow we rewrite
the Newton iteration in a different way.

The Newton iteration proceed as follows.

  * •

$PV$ and $M$ are given.

  * •

Initialize the iteration by choosing an arbitrary $p_{0}\in\mathring{H}^{1}$.

  * •

Iterate, for $k\geqslant 0$,

    1. 1.

Let $H_{k}\vcentcolon=\mathds{1}(M<\partial_{3}p_{k})$ and
${PV}_{k}\vcentcolon=\Delta
p_{k}+\frac{1}{2}\partial_{3}{\min}_{0}\,(M-\partial_{3}p_{k})$.

    2. 2.

$\pi$ solves $\nabla\cdot(A_{H_{k}}\nabla\pi)=PV-{PV}_{k}$.

    3. 3.

Let $p_{k+1}\vcentcolon=p_{k}+\pi$.

Note that step 3 is written this way because, once the Newton iteration
reaches the so-called “quadratic regime” (which is guaranteed to occur once
iterates are sufficiently close to the true solution), then exact line search
always produces a unit step size. This is why we can update
$p_{k+1}=p_{k}+t\pi$ with $t=1$.

We are now ready to relate the Newton iteration to the flow above. We may
interpret (really, rewrite) the Newton iteration geometrically as follows.

  * •

$PV$ and $M$ are given.

  * •

Initialize the iteration by choosing an arbitrary $p_{0}\in\mathring{H}^{1}$,
i.e. choosing an arbitrary $\mathcal{X}_{0}=\Phi(p_{0},\,M)\in\mathcal{B}$
with the correct $M$–coordinate.

  * •

Iterate, for $k\geqslant 0$.

    1. 1.

Let $(p_{k},\,M)\vcentcolon=\Phi^{-1}(\mathcal{X}_{k})$ and
${PV}_{k}\vcentcolon=PV(\mathcal{X}_{k})$.

    2. 2.

$\pi$ solves
${(D_{H}\Phi)}^{*}(D_{H}\Phi)(\pi,\,0)=\left(PV-{PV}_{k},\,0\right)$.

    3. 3.

Let $\mathcal{X}_{k+1}\vcentcolon=\Phi\left((p_{k},\,M)+(\pi,\,0)\right)$.

Note that step 2 is the same as above, merely written differently (actually:
every step coincides!) Indeed Lemma 6.34 tells us that

${(D_{H}\Phi)}^{*}(D_{H}\Phi)(\pi,\,0)=\left(-\nabla\cdot(A_{H}\nabla\pi),\,0\right).$

In other words the Newton iteration may be written in one line as

$\mathcal{X}_{k+1}=\Phi\left(\Phi^{-1}(\mathcal{X}_{k})+(\pi,\,0)\right)\text{
where $\pi$ solves
}{(D_{H}\Phi)}^{*}(D_{H}\Phi)(\pi,\,0)=\left(PV-{PV}_{k},\,0\right).$

When $\mathcal{X}_{k+1}$ is close to $\mathcal{X}_{k}$, which equivalently
means that $\pi$ is small, we may thus formally approximate

$\mathcal{X}_{k+1}\sim\mathcal{X}_{k}+(D_{H}\Phi)(\pi,\,0).$

We compute that

$\displaystyle(D_{H}\Phi)(\pi,\,0)=(D_{H}\Phi){\left[{(D_{H}\Phi)}^{*}(D_{H}\Phi)\right]}^{-1}\left(PV-{PV}_{k},\,0\right)$
$\displaystyle={(D_{H}\Phi)}^{-*}\left(PV-PV(\mathcal{X}_{k}),\,0\right)$

In other words:

$\mathcal{X}_{k+1}\sim\mathcal{X}_{k}+{(D_{H}\Phi)}^{-*}\left(PV-
PV(\mathcal{X}_{k}),\,0\right)$

which verifies that indeed the Newton iteration is a formal discretisation of
the flow in (6.34) above.

Figure 12: The Newton descent method follows curves of constant $M$ until the
appropriate value of $PV$ is reached: at a point $\mathcal{X}$ on the balanced
set, the flow corresponding to Newton descent follows the direction of the
formal vector $\mathcal{Z}$ whose $p$–coordinate is given by the Newton
descent search direction $\pi$.

######  Remark 6.41 (A simple flow).

We discuss here a very simple analog of the flow presented in Remark 6.40.
Consider the following flow on the two-dimensional plane:

$\displaystyle\dot{x}(s)=x_{*}-x(s)$ $\displaystyle\dot{y}(s)=0$

with initial conditions

$\displaystyle x(0)=x_{0}$ $\displaystyle y(0)=y_{*}$

where $x_{*}$ and $y_{*}$ are given (they play the role of the given target
values of $PV$ and $M$) and where $x_{0}$ (just like $p_{0}$) may be freely
chosen. This means that the $x$–coordinate plays the role of the
$PV$–coordinate whereas the $y$–coordinate plays the role of the
$M$–coordinate.

Then the new unknown

$\tilde{x}(s)\vcentcolon=x_{*}-x(s)$

satisfies

$\dot{\tilde{x}}(s)=-\dot{x}(s)=-\dot{\tilde{x}}(s)$

and so we deduce that the solution is given by

$\displaystyle\tilde{x}(s)=\tilde{x}(0)e^{-s}=(x_{*}-x_{0})e^{-s}$
$\displaystyle y(s)=y_{*}.$

In terms of the original unknowns $x$ and $y$ this yields

$\displaystyle x(s)=x_{*}-\tilde{x}(s)=(1-e^{-s})x_{*}+e^{-s}x_{0}$
$\displaystyle y(s)=y_{*}.$

Therefore the solution $(x,\,y)(s)$ approaches the point with coordinates
$(x_{*},\,y_{*})$ exponentially fast, along a trajectory of constant $y$.

##  7 Summary and comparison with Helmholtz and the dry case

We are approaching the conclusion of this paper. Before we conclude, we recall
in this section that the decomposition we introduced for the moist Boussinesq
system under consideration has many different facets. We summarize these many
facets here. In order to make it as easy as possible for the reader to follow
this discussion, we first recall the many facets of the Helmholtz
decomposition and of the vortical-wave decomposition of the dry Boussinesq
system.

This is not merely a recapitulation of the discussion of these two classical
systems done in the introduction. Throughout the paper we have introduced a
variety of concepts and terminology for the sake of the moist decomposition,
but we see that many of these have counterparts in these classical systems. It
is these counterparts that we summarize first, in Sections 7.1 and 7.2 below,
before summarizing the various facets of the moist decomposition in Section
7.3.

We could more prosaically call this section: “the twelve aspects of a
decomposition found in both Helmholtz, dry, and moist Boussinesq”.

###  7.1 Helmholtz

We first summarize the various facets of the Helmholtz decomposition.

  1. 1.

For any three-dimensional vector field $u$ the Helmholtz decomposition is

$u=\nabla p+\sigma$

where $\sigma$ is a divergence-free vector field.

  2. 2.

In the context of the Helmholtz decomposition, if we wish to use terminology
from atmospheric science and talk about _balanced states_ , a state $u$ will
be balanced precisely when it is a solenoidal vector field. In other words the
balance in question is

$u_{B}=\sigma,$

where the subscript $B$ denotes the balanced component of $u$.

While introducing this terminology in the context of the Helmholtz
decomposition may appear overzealous, it does serve to make the parallels with
the dry Boussinesq system, and ultimately with the _moist_ Boussinesq system,
more evident.

  3. 3.

Since the second component of the Helmholtz decomposition is the balanced
component, by analogy with the terminology of this paper we refer to the first
component as the _wave component_ 222 This terminology may also be justified
more directly as follows, without appealing to how the Helmholtz decomposition
is similar to the dry and moist Boussinesq decompositions. Indeed, the
Helmholtz decomposition may be interpreted as a slow-fast decomposition for
the low Mach limit of the compressible Euler equations (see Remark A.3) and in
this context the first component of the decomposition will precisely obey a
wave equation. . By analogy with Proposition 3.3 we may identify the _slow
measurement_ in this setting as the measurement which annihilates the wave
set. Since the wave set is the set of potential vector fields this means that
the slow measurements are precisely the _curl_ and the _spatial average_ :
$\\\ nablap$ is the wave component of $u$ and

$\nabla\times\nabla p=0\text{ and }\fint\nabla p=0.$

  4. 4.

In this case there is no subtlety to identifying the “good unknown” since it
is the vector field $u$ itself. (See Remark 3.2 for a discussion of what is
meant by the term “good unknown” and what role they play.)

  5. 5.

The elliptic PDE used to reconstruct the balanced component can be viewed as a
div-curl system. Indeed $\sigma$ solves

$\nabla\times\sigma=\nabla\times u\text{ subject to }\nabla\cdot\sigma=0\text{
and }\fint\sigma=\fint u.$

Note that the forcing is precisely the slow measurements: here the curl and
the spatial average are the analogs of the potential vorticity in the dry case
since they are slow measurements which characterise the balanced component.

Alternatively we can view this div-curl system as a curl-squared system for
the vector potential of $\sigma$. Writing $\sigma=c+\nabla\times\psi$ we see
that $c=\fint u$ while $\psi$ solves
$\nabla\times(\nabla\times\psi)=\nabla\times u$. (This curl-squared system may
be written as a Laplacian equation since we may without loss of generality
impose that the vector potential $\psi$ be divergence-free, and in that case
the curl-squared operator acting on $\psi$ reduces to a Laplacian.)

  6. 6.

The elliptic PDE used to reconstruct the wave component is Laplace’s equation:

$\Delta p=\nabla\cdot u.$

Note that it is not the fast measurement, the divergence, which plays the role
of a forcing term for the wave component.

  7. 7.

Using the decomposition we can obtain a couple of global changes of
coordinates. We can fully describe the vector field $u$ in three ways.

    1. (a)

Describe it using $u$ itself.

    2. (b)

Describe it using the measurements $\nabla\cdot u$, $\nabla\times u$, and
$\fint u$.

    3. (c)

Describe it using the coordinates $p$ and $\sigma$.

  8. 8.

Recall that the Helmholtz decomposition may be viewed as a slow-fast
decomposition for the low Mach number limit of the compressible Euler
equations. In that case the leading-order operator of the dynamics is linear,
and it is recorded in Remark A.3. We bring this up now in order to later
compare the leading order operators in the case of dry and moist Boussinesq.
For now it suffices to note that, as is discussed in more detail in Remark
A.3, the Helmholtz decomposition is a decomposition into the kernel and the
image for that operator: the balanced set is its kernel and the wave set is
its image.

  9. 9.

As discussed in Remark A.3, the wave component of the Helmholtz decomposition
can be seen to obey a wave equation. Once again, this is not necessarily
worthy of note by itself, but we record it here to compare it with the dry and
moist Boussinesq decompositions in Sections 7.2 and 7.3 below.

  10. 10.

When viewed as a slow-fast decomposition, the Helmholtz decomposition inherits
the additional feature of being orthogonal with respect to the conserved
energy, which happens to simply be the $L^{2}$ norm.

  11. 11.

We note that the Helmholtz decomposition and its induced changes of
coordinates discussed above yield Parseval–type identities which allow us to
rewrite the energy in three different ways: in terms of the state $u$, in
terms of the coordinates, and in terms of the measurements. We obtain that

$\displaystyle\int{\lvert u\rvert}^{2}=\int{\lvert\nabla
p\rvert}^{2}+{\lvert\sigma\rvert}^{2}=\int{\lvert\nabla\Delta^{-1}(\nabla\cdot
u)\rvert}^{2}+{\left\lvert{\nabla\times}^{-1}\left(\nabla\times u,\,\fint
u\right)\right\rvert}^{2}.$

Here we write $\sigma={(\nabla\times)}^{-1}(\omega,\,a)$ to mean that $\sigma$
solves

$\nabla\times\sigma=\omega\text{ subject to }\nabla\cdot\sigma=0\text{ and
}\fint\sigma=a.$

  12. 12.

Finally we observe that the extraction of the balanced component of a vector
field $u$ may be viewed as a projection. Indeed, if we define the balanced
set, in the context of the Helmholtz decomposition, to be
$\mathcal{B}_{Hel.}\vcentcolon={(L^{2})}^{3}_{\sigma}$, the space of
divergence-free three-dimensional vector fields, then we see that

$u_{B}=\operatorname*{arg\,min}_{v\in\mathcal{B}_{Hel.}}{\left\lvert\left\lvert
u-v\right\rvert\right\rvert}_{L^{2}}.$

###  7.2 Dry Boussinesq

We now turn our attention to the various facets of the vortical-wave
decomposition used for the dry Boussinesq system. Note that the numbering
below is the same as the numbering in the discussion of the Helmholtz
decomposition in Section 7.1 above. That is not an accident: it serves to
highlight the parallels between the two decompositions. Then again we caution
the reader that the roles played by the potential and solenoidal pieces are
now flipped: $\nabla p$ appeared in the fast waves in Section 7.1, it now
appears in the slow balanced components; $\sigma$ appeared in the slow
balanced components in Section 7.1, it now appears in the fast waves.

  1. 1.

For the dry Boussinesq system discussed in this paper, the decomposition of a
state consisting of a vector velocity field $u$ and a scalar potential
temperature field $\theta$ takes the form

$\begin{pmatrix}u\\\ \theta\end{pmatrix}=\begin{pmatrix}\nabla_{h}^{\perp}p\\\
\partial_{3}p\end{pmatrix}+\begin{pmatrix}\sigma_{h}^{\perp}+we_{3}\\\
\sigma_{3}\end{pmatrix},$

where $\sigma$ is divergence-free.

  2. 2.

In the dry case, the balanced component satisfies the geostrophic and
hydrostatic balances:

$\displaystyle u_{B}$ $\displaystyle=\nabla_{h}^{\perp}p\text{ and }$
$\displaystyle\theta_{B}$ $\displaystyle=\partial_{3}p.$

  3. 3.

The measurement characterising the wave set is, in the dry case, the potential
vorticity. Indeed in this case the wave set is precisely the set of states
whose potential vorticity vanishes:

$\displaystyle PV(\sigma_{h}^{\perp}+we_{3},\,\sigma_{3})$
$\displaystyle=\nabla_{h}^{\perp}\cdot{(\sigma_{h}^{\perp}+we_{3})}_{h}+\partial_{3}\sigma_{3}$
$\displaystyle=\nabla_{h}\cdot\sigma_{h}+\partial_{3}\sigma_{3}=0$

since $\sigma$ is divergence-free.

  4. 4.

In the dry case the good unknown is the following three-dimensional vector
field combining both the horizontal velocity and the potential temperature:

$-u_{h}^{\perp}+\theta e_{3}.$

  5. 5.

The elliptic equation used to recover the pressure $p$, and hence the balanced
component, is the Laplace/Poisson equation

$\Delta p=PV.$

  6. 6.

The wave component is recovered by setting $w=u_{3}$ and finding $\sigma$ to
be the solution of the following div-curl system:

$\nabla\times\sigma=\partial_{3}u_{h}-\nabla_{h}^{\perp}\theta+(\partial_{3}w)e_{3}=j+(\partial_{3}w)e_{3}\text{
subject to }\nabla\cdot\sigma=0\text{ and }\fint\sigma=-\fint
u_{h}^{\perp}+\fint\theta e_{3}=a.$

  7. 7.

We may describe states of the dry Boussinesq system in three equivalent ways.

    1. (a)

Describe them using $u$ and $\theta$.

    2. (b)

Describe them using the measurements $PV$, $j$, $w$, and $a$.

    3. (c)

Describe them using the coordinates $p$, $\sigma$, and $w$.

  8. 8.

The leading-order operator of the dynamics of the dry Boussinesq system
considered in this paper is given by

$\mathcal{L}\begin{pmatrix}u\\\
\theta\end{pmatrix}=\begin{pmatrix}\mathbb{P}_{L}\left(u_{h}^{\perp}-\theta
e_{3}\right)\\\ u_{3}\end{pmatrix}.$

The decomposition is adapted to this operator since the balanced set is its
kernel and the wave set is its image.

  9. 9.

We may compute immediately that the wave component of a state undergoes linear
oscillations under the leading-order operator since, when restricted to the
wave set, the square of the operator behaves like multiplication by $-1$:

$\mathcal{L}^{2}\begin{pmatrix}\sigma_{h}^{\perp}+we_{3}\\\
\sigma_{3}\end{pmatrix}=-\begin{pmatrix}\sigma_{h}^{\perp}+we_{3}\\\
\sigma_{3}\end{pmatrix}.$

  10. 10.

The conserved energy in the dry case is simply the $L^{2}$ norm given by

$E=\int{\lvert u\rvert}^{2}+\theta^{2}.$

  11. 11.

The Parseval-type identity in the dry case lets us rewrite the energy in terms
of the coordinates or the measurements as follows:

$\int{\lvert u\rvert}^{2}+\theta^{2}=\int{\lvert\nabla
p\rvert}^{2}+{\lvert\sigma\rvert}^{2}+w^{2}=\int{\lvert\nabla\Delta^{-1}PV\rvert}^{2}+{\left\lvert{\nabla\times}^{-1}\left(j+(\partial_{3}w)e_{3},\,a\right)\right\rvert}^{2}.$

  12. 12.

For the dry Boussinesq case the balanced set takes the form

$\mathcal{B}_{dry}\vcentcolon=\left\\{(u,\,\theta)\in{(L^{2})}^{3}_{\sigma}\times
L^{2}:u=\nabla_{h}^{\perp}p,\,u_{3}=0,\text{ and }\theta=\partial_{3}p\text{
for some }p\in\mathring{H}^{1}\right\\}.$

Therefore the extraction of a balanced component once again has a projection
perspective:

$(u_{B},\,\theta_{B})=\operatorname*{arg\,min}_{\mathcal{Y}\in\mathcal{B}_{dry}}{\left\lvert\left\lvert(u,\,\theta)-\mathcal{Y}\right\rvert\right\rvert}_{L^{2}}.$

###  7.3 Moist Boussinesq

Finally we perform the same exercise as in Sections 7.1 and 7.2: we list the
twelve aspects of the moist decomposition presented in this paper, in the same
order as the above sections to make the comparison easy to follow.

Recall that here, in a manner consistent with the rest of this paper (barring
the introduction) we write $\theta$ to denote the _equivalent_ potential
temperature (denoted $\theta_{e}$ in the introduction).

  1. 1.

The decomposition of a state $(u,\,\theta,q)$ is given by

$\begin{pmatrix}u\\\ \theta\\\
q\end{pmatrix}=\begin{pmatrix}\nabla_{h}^{\perp}\\\
\partial_{3}p+\frac{1}{2}{\min}_{0}\,(M-\partial_{3}p)\\\
M-\partial_{3}p-\frac{1}{2}{\min}_{0}\,(M-\partial_{3}p)\end{pmatrix}+\begin{pmatrix}\sigma_{h}^{\perp}+we_{3}\\\
\sigma_{3}\\\ -\sigma_{3}\end{pmatrix},$

where $p$ has vanishing average and $\sigma$ is divergence-free.

  2. 2.

In the moist case the balanced component satisfies the geostrophic and
hydrostatic balances, meaning that

$\displaystyle u_{B}$ $\displaystyle=\nabla_{h}^{\perp}p\text{ and }$
$\displaystyle\theta_{B}-{\min}_{0}\,q_{B}$ $\displaystyle=\partial_{3}p.$

To fully characterise the balanced component we must also include the moist
variable $M$ and the following “auxiliary” balance:

$\theta_{B}+q_{B}=M.$

  3. 3.

The slow measurements characterising the wave set are now $PV$ and $M$, and
indeed

$PV(\sigma_{h}^{\perp}+we_{3},\,\sigma_{3},\,-\sigma_{3})=\nabla\cdot\sigma=0\text{
and
}M(\sigma_{h}^{\perp}+we_{3},\,\sigma_{3},\,-\sigma_{3})=\sigma_{3}-\sigma_{3}=0.$

  4. 4.

There are _two_ good unknowns in the moist case, owing to the fact that the
leading-order operator is _not_ skew-symmetric (this is discussed in more
detail in Remark 3.8). One good unknown is

$-u_{h}^{\perp}+(\theta-{\min}_{0}\,q)e_{3},$

which is related to $j$ and the null set of the leading-order operator, while
its “adjoint” good unknown is

$-u_{h}^{\perp}+\theta e_{3},$

which is related to $PV$ and the image of the leading-order operator.

  5. 5.

In order to reconstruct the balanced component we set $M=\theta+q$ and then
let $p$ be the solution of nonlinear $PV$-and-$M$ inversion

$\Delta p+\frac{1}{2}\partial_{3}{\min}_{0}\,(M-\partial_{3}p)=PV.$

  6. 6.

To reconstruct the wave component we let $w=u_{3}$ and let $\sigma$ solve

$\displaystyle\nabla\times\left(A_{B}^{-1}\sigma\right)=\partial_{3}u_{h}-\nabla_{h}^{\perp}(\theta-\theta
H_{B})+(\partial_{3}w)e_{3}=\vcentcolon j_{B}+(\partial_{3}w)e_{3}$
$\displaystyle\text{ subject to }\nabla\cdot\sigma=0\text{ and }\fint
A_{B}^{-1}\sigma=-\fint u_{h}^{\perp}+\fint(\theta-qH_{B})e_{3},$

where $A_{B}^{-1}=I+H_{B}e_{3}\otimes e_{3}$ for
$H_{B}\vcentcolon=\mathds{1}(M<\partial_{3}p)$.

  7. 7.

We can describe moist Boussinesq states in three ways.

    1. (a)

Describe them using $u$, $\theta$, and $q$.

    2. (b)

Describe them using the measurements $PV$, $M$, $j$, $w$, and $a$.

    3. (c)

Describe them using the coordinates $p$, $M$, $\sigma$, and $w$.

  8. 8.

The leading-order operator of the dynamics in the moist case is

$\mathcal{N}\begin{pmatrix}u\\\ \theta\\\
q\end{pmatrix}=\begin{pmatrix}\mathbb{P}_{L}\left[u_{h}^{\perp}-(\theta-{\min}_{0}\,q)e_{3}\right]\\\
u_{3}\\\ -u_{3}\end{pmatrix}.$

  9. 9.

As detailed in Section 4 the operator of interest acting on the wave set,
denoted by $\mathcal{L}_{W,\,H_{B}}$, is the linearisation of $\mathcal{N}$
about a balanced state with phases characterised by $H_{B}$. We then observe
that the restriction of this operator to the wave set produces oscillatory
dynamics. By contrast with the case of the Helmholtz decomposition and of the
dry Boussinesq system, the frequencies of these oscillations are _phase-
dependent_ :

$\mathcal{L}_{W,\,H_{B}}^{2}\begin{pmatrix}\sigma_{h}\\\ 0\\\
0\end{pmatrix}=-\begin{pmatrix}\sigma_{h}\\\ 0\\\ 0\end{pmatrix}\text{ and
}\mathcal{L}_{W,\,H_{B}}^{2}\begin{pmatrix}we_{3}\\\ \sigma_{3}\\\
-\sigma_{3}\end{pmatrix}=-(1+H_{B})\begin{pmatrix}we_{3}\\\ \sigma_{3}\\\
-\sigma_{3}\end{pmatrix}.$

  10. 10.

In the moist case the conserved energy is no longer quadratic. It is now
nonlinear and given by

$E=\int{\lvert u\rvert}^{2}+\theta^{2}+{\min}_{0}^{2}\,q+{(\theta+q)}^{2}.$

  11. 11.

In the moist case there is no simple Parseval-type identity which holds: the
entirety of Section 5 is devoted to finding an alternative! Ultimately we
identify in Section 5.3 a _statistical divergence_ given by

$D(\mathcal{X}_{1};\,\mathcal{X}_{2})=\frac{1}{2}\int A_{H_{1}}\nabla
p\cdot\nabla
p+\left(1+\frac{H_{1}}{2}\right){(M_{1}-M_{2})}^{2}+A_{H_{1}}^{-1}\sigma\cdot\sigma+{(w_{1}-w_{2})}^{2}$

where, for $H_{1}\vcentcolon=\mathds{1}(q_{B,\,1}<0)$,

     * •

$p$ solves $\nabla\cdot\left(A_{H_{1}}\nabla
p\right)={PV}_{1}-{PV}_{2}-\frac{1}{2}\partial_{3}\left[H_{1}\left(M_{1}-M_{2}\right)\right]$
and

     * •

$\sigma$ solves
$\nabla\times\left(A_{H_{1}}^{-1}\sigma\right)=j_{1}-j_{2}+\partial_{3}(w_{1}-w_{2})e_{3}$
subject to $\nabla\cdot\sigma=0$ and $\fint A_{H_{1}}^{-1}\sigma=a_{1}-a_{2}$.

  12. 12.

The projection perspective in the moist case takes the following form:

$\mathcal{X}_{B}=\operatorname*{arg\,min}_{\mathcal{Y}\in\mathcal{B}}D(\mathcal{X};\;\mathcal{Y}).$

Above we have summarized in detail how the many aspects of the Helmholtz, dry,
and moist Boussinesq decompositions are analogous to one another, highlighting
their differences. Below we provide a table summarizing Sections 7.1–7.3 and
which may serve either as a reference or as a more visual way to compare and
contrast the various aspects of each decomposition.

| Helmholtz | Dry Boussinesq | Moist Boussinesq
---|---|---|---
Decomposition | $u=\nabla p+\sigma$ | $\begin{pmatrix}u\\\ \theta\end{pmatrix}=\begin{pmatrix}\nabla_{h}^{\perp}p\\\ \partial_{3}p\end{pmatrix}+\begin{pmatrix}\sigma_{h}^{\perp}+we_{3}\\\ \sigma_{3}\end{pmatrix}$ | $\begin{pmatrix}u\\\ \theta\\\ q\end{pmatrix}=\begin{pmatrix}\nabla_{h}^{\perp}p\\\ \partial_{3}p+\frac{1}{2}{\min}_{0}\,(M-\partial_{3}p)\\\ M-\partial_{3}p+\frac{1}{2}{\min}_{0}\,(M-\partial_{3}p)\end{pmatrix}+\begin{pmatrix}\sigma_{h}^{\perp}+we_{3}\\\ \sigma_{3}\\\ -\sigma_{3}\end{pmatrix}$
Balanced set | $\mathcal{B}_{Hel.}={(L^{2})}^{3}_{\sigma}$ | $\mathcal{B}_{dry}=\left\\{\begin{aligned} &u=\nabla_{h}^{\perp}p,\,u_{3}=0,\text{ and }\\\ &\theta=\partial_{3}p\text{ for some }p\in\mathring{H}^{1}\end{aligned}\right\\}$ | $\mathcal{B}=\left\\{\begin{aligned} &u_{h}=\nabla_{h}^{\perp}p,\,u_{3}=0,\,\\\ &\text{and }\theta-{\min}_{0}\,q=\partial_{3}p\\\ &\text{ for some }p\in\mathring{H}^{1}\end{aligned}\right\\}$
Slow measurements, which | $\nabla\times u$ | $PV=\nabla_{h}^{\perp}\cdot u_{h}+\partial_{3}\theta$ | $PV=\nabla_{h}^{\perp}\cdot u_{h}+\partial_{3}\theta$
characterise balanced components |  and $\fint u$ |  |  and $M=\theta+q$
Fast measurements, which | $\nabla\cdot u$ |  $j=\partial_{3}u_{h}-\nabla_{h}^{\perp}\theta$, |  $j=\partial_{3}u_{h}-\nabla_{h}^{\perp}(\theta-{\min}_{0}\,q)e_{3}$,
characterise wave components |  |  $a=-\fint u_{h}^{\perp}+\fint\theta e_{3}$, |  $a=-\fint u_{h}^{\perp}+\fint(\theta-{\min}_{0}\,q)e)_{3}$,
|  |  and $w=u_{3}$ |  and $w=u_{3}$
Elliptic PDE for the | $\nabla\times\sigma=\nabla\times u$ | $\Delta p=PV$ | $\Delta p+\frac{1}{2}\partial_{3}{\min}_{0}\,(M-\partial_{3}p)=PV$
balanced component |  subject to $\nabla\cdot\sigma=0$ |  |
|  and $\fint\sigma=\fint u$ |  |
Elliptic PDE for the | $\Delta p=\nabla\cdot u$ | $\nabla\times\sigma=j+(\partial_{3}w)e_{3}$ | $\nabla\times\left(A_{B}^{-1}\sigma\right)=j_{B}+(\partial_{3}w)e_{3}$
wave component |  |  subject to $\nabla\cdot\sigma=0$ and |  subject to $\nabla\cdot\sigma=0$ and
|  | $\fint\sigma=-\fint u_{h}^{\perp}+\fint\theta e_{3}$ | $\fint A_{B}^{-1}\sigma=-\fint u_{h}^{\perp}+\fint(\theta-qH_{B})e_{3}$
Projection perspective | $u_{B}=\operatorname*{arg\,min}\limits_{v\in\mathcal{B}_{Hel.}}{\left\lvert\left\lvert u-v\right\rvert\right\rvert}_{L^{2}}$ | $(u_{B},\,\theta_{B})=\operatorname*{arg\,min}\limits_{\mathcal{Y}\in\mathcal{B}_{dry}}{\left\lvert\left\lvert(u,\,\theta)-\mathcal{Y}\right\rvert\right\rvert}_{L^{2}}$ | $\mathcal{X}_{B}=\operatorname*{arg\,min}\limits_{\mathcal{Y}\in\mathcal{B}}D(\mathcal{X};\;\mathcal{Y})$
Global changes of coordinates |  State: $u$ |  State: $u$ and $\theta$ |  State: $u$, $\theta$, and $q$
|  Measurements: $\nabla\cdot u$, $\nabla\times u$, and $\fint u$ |  Measurements: $PV$, $j$, $w$, and $a$ |  Measurements: $PV$, $M$, $j$, $w$, and $a$
|  Coordinates: $p$ and $\sigma$ |  Coordinates: $p$, $\sigma$, and $w$ |  Coordinates: $p$, $M$, $\sigma$, and $w$
Leading-order operator |  See Remark A.3 | $\mathcal{L}\begin{pmatrix}u\\\ \theta\end{pmatrix}=\begin{pmatrix}\mathbb{P}_{L}(u_{h}^{\perp}-\theta e_{3})\\\ u_{3}\end{pmatrix}$ | $\mathcal{N}\begin{pmatrix}u\\\ \theta\\\ q\end{pmatrix}=\begin{pmatrix}\mathbb{P}_{L}\left[u_{h}^{\perp}-(\theta-{\min}_{0}\,q)e_{3}\right]\\\ u_{3}\\\ -u_{3}\end{pmatrix}$
Conserved energy |  See Remark A.3 | $E=\int{\lvert u\rvert}^{2}+\theta^{2}$ | $E=\int{\lvert u\rvert}^{2}+\theta^{2}+{\min}_{0}^{2}\,q+{(\theta+q)}^{2}$

##  8 Conclusion

In this section we discuss the main contributions of this paper and the main
takeaways for the reader.

The core contribution of this paper is the following. We provided a blueprint
for decomposing the solutions of $\partial_{t}+\mathcal{N}$, specifically when
$\mathcal{N}$ is nonlinear. Recall that otherwise, if $\mathcal{N}$ is linear,
then we can proceed via spectral methods and identify eigenvalues and
eigenvectors of that operator.

This blueprint requires a good understanding of the null set and the image of
$\mathcal{N}$, which we refer to as the balanced set and the wave set in the
context of this paper.

We actually provide _two_ descriptions of each of these sets: (1) an explicit
parametrisation and (2) a characterisation as the joint zero set of some
measurements. Note that the measurements annihilating the wave set are known
as slow measurements while those annihilating the balanced set are known as
fast measurements.

A crucial condition to check in order to follow this general blueprint is that
the slow measurements be characteristic of balanced states. In other words:
there must be only one balanced set with any given $PV$ and $M$. Geometrically
this is asking that the wave set (i.e. the joint zero set of $PV$ and $M$) be
_transversal_ to the balanced set.

This allows us to define a global decomposition of state space:

$\mathbb{L}^{2}_{\sigma}=\mathcal{B}+\mathcal{W}.$

In particular this decomposition gives broader meaning to nonlinear
$PV$-and-$M$ inversion. Previously, this inversion was understood in the
context of balanced states: given $PV$ and $M$, the inversion was used to
produce the corresponding balanced state. Now this inversion has relevance for
_any_ state: it is used to produce its balanced _component_.

We then discuss three aspects of this decomposition.

  1. 1.

The first aspect we discuss is its interpretation as a slow-fast
decomposition. We obtain the three-part decomposition

$\mathcal{X}=\overunderbraces{&\br{1}{\text{balanced}}\br{3}{\text{wave}}}{&\mathcal{X}_{B}+\overline{\mathcal{X}}_{W}+\left(\mathcal{X}_{W}-\overline{\mathcal{X}}_{W}\right)}{&\br{3}{\text{slow}}\br{1}{\text{fast}}}$
(8.1)

where $\overline{\mathcal{X}}_{W}$ denotes the time average of the wave
component.

Contrast this with the dry case: in that case the wave component undergoes
linear oscillations, their time averages thus vanish, and so they do _not_
contribute to the slow component at lowest order (they do contribute, a
little, when the small parameter $\varepsilon$ is finite). Here, in the moist
case, the wave component undergoes nonlinear oscillations and so its time
average contributes to the slow component.

  2. 2.

The second aspect of the decomposition we discuss is its connection to the
conserved energy. (Recall: when the operator $\mathcal{N}$ is linear, the
eigendecomposition obtained often has favourable orthogonality properties,
e.g. with respect to the inner product generated by the conserved energy as is
the case in the case for the dry Boussinesq system.) We obtain a statistical
divergence which quantifies the discrepancy between states in a manner
consistent with both the conserved energy and the decomposition.

This statistical divergence is inpired by the Parseval-type identities which
hold for the Helmholtz decomposition and the dry Boussinesq decomposition. For
example in the dry case we ca write the energy either in terms of the state
$(u,\,\theta)$ or in terms of the measurements $(PV,\,j,\,w,\,a)$. We proceed