$$ \leq c (\|f\|_1 + \|g\|_3) \|f - g\|_3 $$

having applied (5). From the linearity of $vA$ and $\frac{1}{2} \sum_{i=1}^{\infty} \mathcal{P}B_i^2$ then the corresponding result follows immediately from (74) and this subsequently justifies (75). Additionally (76) follows immediately from the already justified Assumption 5.4.

Assumption 5.8: (26) and (28) will be our basis of showing (77). The task is to control

$$ 2 \left\langle \mathcal{P}_n \left( -\mathcal{P}\mathcal{L} - vA + \frac{1}{2} \sum_{i=1}^{\infty} \mathcal{P}B_i^2 \right) f_n, f_n \right\rangle_2 + \sum_{i=1}^{\infty} \| \mathcal{P}_n \mathcal{P}B_i f_n \|_2^2 $$

which we rewrite as

$$ -2\langle \mathcal{P}_n \mathcal{P}\mathcal{L}_{f_n} f_n, f_n \rangle_2 - 2v\langle \mathcal{P}_n A f_n, f_n \rangle_2 \\ + \sum_{i=1}^{\infty} \left( \langle \mathcal{P}_n \mathcal{P}B_i^2 f_n, f_n \rangle_2 + \| \mathcal{P}_n \mathcal{P}B_i f_n \|_2^2 \right). \quad (29) $$

Recalling the assumption (21) and (26), (28), we have that for any $\varepsilon > 0$,

$$ \begin{aligned} (29) \le {}& -2v\langle \mathcal{P}_n A f_n, f_n \rangle_2 + c(\varepsilon) \|f_n\|_2^4 + \varepsilon \|f_n\|_3^2 \\ & + \sum_{i=1}^{\infty} \left( c(\varepsilon) \|\xi_i\|_{W^{3,\infty}}^2 \|f_n\|_2^2 + \varepsilon \|\xi_i\|_{W^{3,\infty}}^2 \|f_n\|_3^2 \right) \\ = {}& -2v\langle A f_n, f_n \rangle_2 + \left[ c(\varepsilon) \|f_n\|_2^4 + c(\varepsilon) \sum_{i=1}^{\infty} \|\xi_i\|_{W^{3,\infty}}^2 \|f_n\|_2^2 \right] \\ & + \varepsilon \left[ 1 + \sum_{i=1}^{\infty} \|\xi_i\|_{W^{3,\infty}}^2 \right] \|f_n\|_3^2 \\ \le {}& -2v\langle A^{1/2} A f_n, A^{3/2} f_n \rangle_2 + c(\varepsilon) \left[ 1 + \|f_n\|_2^2 \right] \|f_n\|_2^2 \\ & + \varepsilon \left[ 1 + \sum_{i=1}^{\infty} \|\xi_i\|_{W^{3,\infty}}^2 \right] \|f_n\|_3^2 \\ = {}& -2v \|f_n\|_3^2 + c(\varepsilon) \left[ 1 + \|f_n\|_2^2 \right] \|f_n\|_2^2 + \varepsilon \left[ 1 + \sum_{i=1}^{\infty} \|\xi_i\|_{W^{3,\infty}}^2 \right] \|f_n\|_3^2 \end{aligned} $$

where we have embedded the $\sum_{i=1}^{\infty} \|\xi_i\|_{W^{3,\infty}}^2$ into the constant $c(\varepsilon)$. Therefore by choosing
$$ \epsilon := \frac{\nu}{1 + \sum_{i=1}^{\infty} \|\xi_i\|_{W^{3,\infty}}^2} $$

then (77) is satisfied for $\kappa := \nu$. Moving on to (78), we are interested in the term

$$ \sum_{i=1}^{\infty} \langle \mathcal{P}_n \mathcal{P} B_i f_n, f_n \rangle_2^2. $$

Observe that

$$ \begin{aligned} \langle \mathcal{P}_n \mathcal{P} B_i f_n, f_n \rangle_2^2 &= \langle \mathcal{P} B_i f_n, f_n \rangle_2^2 \\ &= \langle \mathcal{P}[\Delta, B_i] f_n + \mathcal{P} B_i \Delta f_n, A f_n \rangle^2 \\ &\le 2 \langle \mathcal{P}[\Delta, B_i] f_n, A f_n \rangle^2 + 2 \langle \mathcal{P} B_i \Delta f_n, A f_n \rangle^2 \\ &= 2 \langle [\Delta, B_i] f_n, A f_n \rangle^2 + 2 \langle B_i A f_n, A f_n \rangle^2. \end{aligned} $$

The first of these terms is dealt with through a simple Cauchy-Schwarz, as

$$ \langle [\Delta, B_i] f_n, A f_n \rangle^2 \le \|[\Delta, B_i] f_n\|^2 \|A f_n\|^2 \le c \|\xi_i\|_{W^{3,\infty}}^2 \|f_n\|_2^4 $$

using Proposition 5.2, and the second comes directly from (20) as

$$ \langle B_i A f_n, A f_n \rangle^2 \le c \|\xi_i\|_{W^{1,\infty}}^2 \|A f_n\|^4 \le c \|\xi_i\|_{W^{3,\infty}}^2 \|f_n\|_2^4. $$

Summing up the two terms and over all $i$ gives that

$$ \sum_{i=1}^{\infty} \langle \mathcal{P}_n \mathcal{P} B_i f_n, f_n \rangle_2^2 \le \left( c \sum_{i=1}^{\infty} \|\xi_i\|_{W^{3,\infty}}^2 \right) \|f_n\|_2^4 $$

which justifies (78) and Assumption 5.8.

Assumption 5.9: For (79) we must bound the term

$$ 2 \left\langle \left( -\mathcal{P}\mathcal{L} - \nu A + \frac{1}{2} \sum_{i=1}^{\infty} \mathcal{P}B_i^2 \right) f - \left( -\mathcal{P}\mathcal{L} - \nu A + \frac{1}{2} \sum_{i=1}^{\infty} \mathcal{P}B_i^2 \right) g, f - g \right\rangle_1 \\ + \sum_{i=1}^{\infty} \|\mathcal{P}B_i f - \mathcal{P}B_i g\|_1^2 $$

which we simply rewrite as

$$ -2\langle \mathcal{P}\mathcal{L}_f f - \mathcal{P}\mathcal{L}_g g, f - g \rangle_1 - 2\nu\langle A(f - g), f - g \rangle_1 $$
$$+ \sum_{i=1}^{\infty} \left( \langle \mathcal{P}B_i^2(f-g), f-g \rangle_1 + \| \mathcal{P}B_i(f-g) \|_1^2 \right)$$

and inspect the distinct items individually. Firstly from Lemma 3.8 we have that for any $\epsilon > 0$,

$$-2\langle \mathcal{P}L_f f - \mathcal{P}L_g g, f-g \rangle_1 \le c(\epsilon) \left( \|g\|_1^4 + \|f\|_2^2 \right) \|f-g\|_1^2 + \epsilon \|f-g\|_2^2. \quad (30)$$

Similarly to the justification of Assumption 5.8 we also see that

$$-2\nu\langle A(f-g), f-g \rangle_1 = -2\nu\|f-g\|_2^2. \quad (31)$$

Shifting focus to the final term, note that in (27) we in fact showed that

$$\langle \mathcal{P}B_i^2 f_n, f_n \rangle_1 + \langle \mathcal{P}B_i f_n, \mathcal{P}B_i f_n \rangle_1 \le c(\epsilon) \|\xi_i\|_{W^{3,\infty}}^2 \|f_n\|_1^2 + \epsilon \|\xi_i\|_{W^{3,\infty}}^2 \|f_n\|_2^2$$

and scanning the proof we see that all arguments hold for arbitrary $f_n \in W_{\sigma}^{3,2}(\mathbb{T}^3; \mathbb{R}^3)$ so we can deduce directly the bound

$$\langle \mathcal{P}B_i^2(f-g), f-g \rangle_1 + \|\mathcal{P}B_i(f-g)\|_1^2 \le c(\epsilon) \|\xi_i\|_{W^{3,\infty}}^2 \|f-g\|_1^2 \\ + \epsilon \|\xi_i\|_{W^{3,\infty}}^2 \|f-g\|_2^2. \quad (32)$$

Summing over (30), (31) and all $i$ in (32), we deduce a bound by

$$-2\nu\|f-g\|_2^2 + c(\epsilon) \left[ \|g\|_1^4 + \|f\|_2^2 + \sum_{i=1}^{\infty} \|\xi_i\|_{W^{3,\infty}}^2 \right] \|f-g\|_1^2 \\ + \epsilon \left[ 1 + \sum_{i=1}^{\infty} \|\xi_i\|_{W^{3,\infty}}^2 \right] \|f-g\|_2^2$$

so again a choice of

$$\epsilon := \frac{\nu}{1 + \sum_{i=1}^{\infty} \|\xi_i\|_{W^{3,\infty}}^2} \quad (33)$$

ensures (79) is satisfied for $\kappa := \nu$. Moving on to (80), we are interested in the term

$$\sum_{i=1}^{\infty} \langle \mathcal{P}B_i(f-g), f-g \rangle_1^2, \quad (34)$$

noting that
$$ \begin{aligned} \langle \mathcal{P} B_i(f-g), f-g \rangle_1^2 &= \langle \mathcal{A} \mathcal{P} B_i(f-g), f-g \rangle^2 \\ &\le N \sum_{k=1}^3 \langle \partial_k B_i(f-g), \partial_k(f-g) \rangle^2. \end{aligned} $$

We have

$$ \partial_k B_i(f-g) = B_{\partial_k \xi_i}(f-g) + B_i \partial_k(f-g) $$

so

$$ \begin{aligned} \langle \partial_k B_i(f-g), \partial_k(f-g) \rangle^2 &\le 2 \langle B_{\partial_k \xi_i}(f-g), \partial_k(f-g) \rangle^2 \\ &\quad + 2 \langle B_i \partial_k(f-g), \partial_k(f-g) \rangle^2. \end{aligned} $$

Now from (18),

$$ \begin{aligned} \langle B_{\partial_k \xi_i}(f-g), \partial_k(f-g) \rangle^2 &\le \|B_{\partial_k \xi_i}(f-g)\|^2 \|\partial_k(f-g)\|^2 \\ &\le c \|\xi_i\|_{W^{3,\infty}}^2 \|f-g\|_1^4 \end{aligned} $$

and from (15),

$$ \begin{aligned} \langle B_i \partial_k (f-g), \partial_k (f-g) \rangle^2 &= \langle \mathcal{T}_{\xi_i} \partial_k (f-g), \partial_k (f-g) \rangle^2 \\ &\le c \|\xi_i\|_{W^{3,\infty}}^2 \|f-g\|_1^4. \end{aligned} $$

By summing both terms, over all $k = 1, \dots, N$ and $i \in \mathbb{N}$, we have shown that

$$ \sum_{i=1}^{\infty} \langle \mathcal{P} B_i(f-g), f-g \rangle_1^2 \le \left( c \sum_{i=1}^{\infty} \|\xi_i\|_{W^{3,\infty}}^2 \right) \|f-g\|_1^4 \quad (35) $$

demonstrating (80) and hence Assumption 5.9.

Assumption 5.10: For (81) we must bound the term

$$ 2 \left\langle \left( -\mathcal{P}\mathcal{L} - \nu A + \frac{1}{2} \sum_{i=1}^{\infty} \mathcal{P}B_i^2 \right) f, f \right\rangle_1 + \sum_{i=1}^{\infty} \|\mathcal{P}B_i f\|_1^2 $$

which we simply rewrite as

$$ -2\langle \mathcal{P}\mathcal{L}_f f, f \rangle_1 - 2\nu \langle Af, f \rangle_1 + \sum_{i=1}^{\infty} \left( \langle \mathcal{P}B_i^2 f, f \rangle_1 + \|\mathcal{P}B_i f\|_1^2 \right). \quad (36) $$
The nonlinear term can be controlled precisely as seen in Lemma 3.8 to deduce that for any $\varepsilon > 0$,

$$|\langle \mathcal{P}\mathcal{L}_f f, f \rangle_1| \leq c(\varepsilon) \|f\|_1^6 + \varepsilon \|f\|_2^2.$$

Meanwhile across (31) and (32) we have that

$$-2\nu \langle Af, f \rangle_1 + \sum_{i=1}^{\infty} \left( \langle \mathcal{P}B_i^2 f, f \rangle_1 + \|\mathcal{P}B_i f\|_1^2 \right) \\ \leq -2\nu \|f\|_2^2 + c(\varepsilon) \left[ \sum_{i=1}^{3} \|\xi_i\|_{W^{3,\infty}}^2 \right] \|f\|_1^2 \\ + \varepsilon \left[ \sum_{i=1}^{\infty} \|\xi_i\|_{W^{3,\infty}}^2 \right] \|f\|_2^2$$

so with the familiar choice of $\varepsilon$ (33) we see that

$$(36) \leq c \left( 1 + \|f\|_1^6 \right) - \nu \|f\|_2^2$$

which is more than sufficient to show (81). (82) follows immediately from (35), concluding the justification.

Assumption 5.11: For any $\eta \in W_{\sigma}^{2,2}(\mathbb{T}^3; \mathbb{R}^3)$ we must bound the term

$$\left\langle \left( -\mathcal{P}\mathcal{L} - \nu A + \frac{1}{2} \sum_{i=1}^{\infty} \mathcal{P}B_i^2 \right) f - \left( -\mathcal{P}\mathcal{L} - \nu A + \frac{1}{2} \sum_{i=1}^{\infty} \mathcal{P}B_i^2 \right) g, \eta \right\rangle_1$$

which we simply rewrite as

$$-2\langle \mathcal{P}\mathcal{L}_f f - \mathcal{P}\mathcal{L}_g g, \eta \rangle_1 - 2\nu\langle A(f-g), \eta \rangle_1 + \sum_{i=1}^{\infty} \langle \mathcal{P}B_i^2(f-g), \eta \rangle_1$$

and further by

$$-2\langle \mathcal{P}\mathcal{L}_f f - \mathcal{P}\mathcal{L}_g g, A\eta \rangle - 2\nu\langle A(f-g), A\eta \rangle + \sum_{i=1}^{\infty} \langle \mathcal{P}B_i^2(f-g), A\eta \rangle.$$

Through Cauchy-Schwarz this is controlled by

$$\|\eta\|_2 \left( 2\|\mathcal{P}\mathcal{L}_f f - \mathcal{P}\mathcal{L}_g g\| + 2\nu\|A(f-g)\| + \sum_{i=1}^{\infty} \|\mathcal{P}B_i^2(f-g)\| \right).$$
so our problem is reduced to bounding the bracketed terms. The linear terms are trivial when recalling (18), and for the nonlinear term we revert back to (4) to see that

$$ \|\mathcal{P}L_f f - \mathcal{P}L_g g\| \le \|\mathcal{L}_{f-g} f\| + \|\mathcal{L}_g(f-g)\| \le c (\|f\|_2 + \|g\|_1) \|f-g\|_2 $$

comfortably justifying the assumption.

**Proof of Theorem 3.1** The new space X will again be $L_{\sigma}^2(\mathbb{T}^3; \mathbb{R}^3)$ as laid out in Sect. 3.3. We choose the bilinear form $\langle \cdot, \cdot \rangle_{X \times H}$ to be given by

$$ \langle f, g \rangle_{X \times H} := \langle f, Ag \rangle \quad (37) $$

noting that the property (86) follows from (12). Noting also that the system $(a_k)$ is an orthogonal basis of $W_{\sigma}^{1,2}(\mathbb{T}^3; \mathbb{R}^3)$, and that the operators were shown to be measurable into the relevant spaces in Sect. 3.3, we are in the setting of Sect. 5.4. We now proceed to justify the assumptions required to apply Theorem 5.22.

**Assumption 5.16:** This follows identically to Assumption 5.7, referring again to Sect. 3.3 and (4).

**Assumption 5.17:** Continuing to consider the distinct terms, we have that

$$ -2\nu\langle A(f-g), f-g \rangle = -2\nu\|f-g\|_1^2 $$

and

$$ \langle PB_i^2(f-g), f-g \rangle + \|PB_i(f-g)\|^2 \\ \le \langle B_i^2(f-g), f-g \rangle + \|B_i(f-g)\|^2 \le c\|\xi_i\|^2_{W^{2,\infty}} \|f-g\|^2. $$

from (19). With these components in place, the proof of (89) then follows identically to that of (79). (90) is a direct consequence of (20), concluding the justification.

**Assumption 5.18:** This stronger Assumption was in fact already verified in the address of Assumption 5.10.

# 4 The Vorticity Equation on a Bounded Main

In order to address the well-posedness problem of the SALT Navier-Stokes Equations on bounded domains, we now pose it in vorticity form. The analysis conducted in Sect. 3.4 was done with reference to the properties derived across Sects. 2.2 and 2.3, applicable to the bounded domain as well as the torus. The issue in studying
the velocity form is that our operators do not map into the correct spaces in order to use these properties: in particular, the Leray Projector does not preserve the zero trace property and so the operators do not map into the necessary $W_{\sigma}^{k,2}(\mathcal{O}; \mathbb{R}^3)$ spaces (see Remark 2). The motivation behind the vorticity form is to circumvent the necessity of Leray Projection.

Our attentions shall be decidedly on the bounded domain $\mathcal{O}$, though the results for the vorticity form carry over seamlessly to the torus $\mathbb{T}^3$. For this section we impose new constraints on the $\xi_i$, which are such that for each $i \in \mathbb{N}$, $\xi_i \in W_{\sigma}^{1,2}(\mathcal{O}; \mathbb{R}^3) \cap W_0^{2,2}(\mathcal{O}; \mathbb{R}^3) \cap W^{3,\infty}(\mathcal{O}; \mathbb{R}^3)$ and they collectively satisfy

$$ \sum_{i=1}^{\infty} \|\xi_i\|_{W^{3,\infty}}^2 < \infty. \qquad (38) $$

## 4.1 Deriving the Equation

The vorticity form of the equation is derived through taking the curl of the velocity form, where the curl operator is defined for $f \in W^{1,2}(\mathcal{O}; \mathbb{R}^3)$ by

$$ \mathrm{curl} f := \begin{pmatrix} \partial_2 f^3 - \partial_3 f^2 \\ \partial_3 f^1 - \partial_1 f^3 \\ \partial_1 f^2 - \partial_2 f^1 \end{pmatrix}. $$

We introduce the Lie Bracket operator $\mathcal{L}$ defined on sufficiently regular functions $f, g : \mathcal{O} \to \mathbb{R}^3$ by

$$ \mathcal{L}_f g := \mathcal{L}_f g - \mathcal{L}_g f. \qquad (39) $$

In [44] Subsection 12.1 it is shown that, with notation $\phi := \mathrm{curl} f$,

$$ \mathrm{curl} (\mathcal{L}_f f) = \mathcal{L}_f \phi $$

where it is also observed that the curl of elements of $W^{1,2}(\mathcal{O}; \mathbb{R}^3) \cap L_{\sigma}^{2,\perp}(\mathcal{O}; \mathbb{R}^3)$ is null. It is clear that the Laplacian commutes with the curl operation, and we now consider how the curl operation interacts with the stochastic term.

**Lemma 4.1** We have that

$$ \mathrm{curl}(B_i f) = \mathcal{L}_{\xi_i} \phi \qquad (40) $$

where once more $\phi := \mathrm{curl} f$.

**Proof** We shall show only that the identity (40) holds in its first component, with the others following similarly. To this end we calculate the first component of the
left hand side of (40):

$$
\begin{align*}
[\mathrm{curl}(B_i f)]^1 &= \partial_2[B_i f]^3 - \partial_3[B_i f]^2 \\
&= \partial_2 \left( \sum_{j=1}^3 \xi_i^j \partial_j f^3 + f^j \partial_3 \xi_i^j \right) - \partial_3 \left( \sum_{j=1}^3 \xi_i^j \partial_j f^2 + f^j \partial_2 \xi_i^j \right) \\
&= \sum_{j=1}^3 \left( \partial_2 \xi_i^j \partial_j f^3 + \xi_i^j \partial_2 \partial_j f^3 + \partial_2 f^j \partial_3 \xi_i^j + f^j \partial_2 \partial_3 \xi_i^j \right) \\
&\quad - \sum_{j=1}^3 \left( \partial_3 \xi_i^j \partial_j f^2 + \xi_i^j \partial_3 \partial_j f^2 + \partial_3 f^j \partial_2 \xi_i^j + f^j \partial_3 \partial_2 \xi_i^j \right) \\
&= \sum_{j=1}^3 \left( \partial_2 \xi_i^j \partial_j f^3 + \xi_i^j \partial_2 \partial_j f^3 + \partial_2 f^j \partial_3 \xi_i^j - \partial_3 \xi_i^j \partial_j f^2 \right. \\
&\qquad \left. - \xi_i^j \partial_3 \partial_j f^2 - \partial_3 f^j \partial_2 \xi_i^j \right) \\
&= \sum_{j=1}^3 \xi_i^j \partial_j (\partial_2 f^3 - \partial_3 f^2) + \sum_{j=1}^3 \left( \partial_2 \xi_i^j \partial_j f^3 + \partial_2 f^j \partial_3 \xi_i^j \right. \\
&\qquad \left. - \partial_3 \xi_i^j \partial_j f^2 - \partial_3 f^j \partial_2 \xi_i^j \right) \\
&= [\mathcal{L}_{\xi_i} \phi]^1 + \sum_{j=1}^3 \left( \partial_2 \xi_i^j \partial_j f^3 + \partial_2 f^j \partial_3 \xi_i^j - \partial_3 \xi_i^j \partial_j f^2 - \partial_3 f^j \partial_2 \xi_i^j \right).
\end{align*}
$$

Therefore it remains to show that

$$
\sum_{j=1}^{3} \left( \partial_2 \xi_i^j \partial_j f^3 + \partial_2 f^j \partial_3 \xi_i^j - \partial_3 \xi_i^j \partial_j f^2 - \partial_3 f^j \partial_2 \xi_i^j \right) = -[\mathcal{L}_{\phi} \xi_i]^1. \quad (41)
$$

We expand the sum in (41) to

$$
\begin{align*}
& \left( \partial_2 \xi_i^1 \partial_1 f^3 + \partial_2 f^1 \partial_3 \xi_i^1 - \partial_3 \xi_i^1 \partial_1 f^2 - \partial_3 f^1 \partial_2 \xi_i^1 \right) \\
& + \left( \partial_2 \xi_i^2 \partial_2 f^3 + \partial_2 f^2 \partial_3 \xi_i^2 - \partial_3 \xi_i^2 \partial_2 f^2 - \partial_3 f^2 \partial_2 \xi_i^2 \right) \\
& + \left( \partial_2 \xi_i^3 \partial_3 f^3 + \partial_2 f^3 \partial_3 \xi_i^3 - \partial_3 \xi_i^3 \partial_3 f^2 - \partial_3 f^3 \partial_2 \xi_i^3 \right)
\end{align*}
$$

achieving some immediate cancellation in the second two brackets to
$$ (\partial_2 \xi_i^1 \partial_1 f^3 + \partial_2 f^1 \partial_3 \xi_i^1 - \partial_3 \xi_i^1 \partial_1 f^2 - \partial_3 f^1 \partial_2 \xi_i^1) + (\partial_2 \xi_i^2 \partial_2 f^3 - \partial_3 f^2 \partial_2 \xi_i^2) + (\partial_2 f^3 \partial_3 \xi_i^3 - \partial_3 \xi_i^3 \partial_3 f^2). $$

We now simply rewrite the above by combining like terms, into

$$ \partial_2 \xi_i^1 (\partial_1 f^3 - \partial_3 f^1) + \partial_3 \xi_i^1 (\partial_2 f^1 - \partial_1 f^2) + (\partial_2 \xi_i^2 + \partial_3 \xi_i^3) (\partial_2 f^3 - \partial_3 f^2) $$

or more succinctly as

$$ - \partial_2 \xi_i^1 \phi^2 - \partial_3 \xi_i^1 \phi^3 + (\partial_2 \xi_i^2 + \partial_3 \xi_i^3) \phi^1 $$

to which we add and subtract $\partial_1 \xi_i^1 \phi^1$ to arrive at

$$ - \sum_{j=1}^{3} \phi^j \partial_j \xi_i^1 + \sum_{j=1}^{3} (\partial_j \xi_i^j) \phi^1. $$

The first term is precisely $-[\mathcal{L}_\phi \xi_i]^1$ as we wished to show, appreciating that the second term is zero given the divergence free condition on $\xi_i$. $\square$

From this point forwards we adopt the notation $\mathcal{L}_i := \mathcal{L}_{\xi_i}$. Writing the Stratonovich integral of (1) in its component form over the basis vectors of $\mathfrak{A}$, and introducing the notation $w := \text{curl}u$, at a heuristic level we can take the curl of (1) to obtain

$$ w_t - w_0 + \int_0^t \mathcal{L}_{u_s} w_s \, ds - v \int_0^t \Delta w_s \, ds + \sum_{i=1}^{\infty} \int_0^t \mathcal{L}_i w \circ dW_s^i = 0. \quad (42) $$

Having already rigorously demonstrated the Itô conversion of the velocity form (22) we shall make the conversion without explicit reference to the conditions of Sect. 5.2, though this can be precisely shown in the same way. At least again then at the heuristic level, the Itô form is

$$ w_t = w_0 - \int_0^t \mathcal{L}_{u_s} w_s \, ds + v \int_0^t \Delta w_s \, ds + \frac{1}{2} \int_0^t \sum_{i=1}^{\infty} \mathcal{L}_i^2 w_s \, ds - \sum_{i=1}^{\infty} \int_0^t \mathcal{L}_i w \, dW_s^i $$

which can again be projected<fnref n="3" /> to the equation

<footnote n="3">³ Although the pressure term is eliminated, we still consider the Leray Projection to ensure that all terms belong to $L_\sigma^2(\mathcal{O}; \mathbb{R}^3)$.</footnote>
$$ w_t = w_0 - \int_0^t \mathcal{P}\mathcal{L}_{u_s} w_s ds - \nu \int_0^t A w_s ds \\ + \frac{1}{2} \int_0^t \sum_{i=1}^\infty \mathcal{P}\mathcal{L}_i^2 w_s ds - \sum_{i=1}^\infty \int_0^t \mathcal{P}\mathcal{L}_i w dW_s^i. \quad (43) $$

Having motivated this section by an avoidance of the Leray Projection this may seem counter intuitive, however we shall shortly show that the projection is not felt in the noise (where it becomes problematic in velocity form); that is to say that for sufficiently regular $w$, $\mathcal{P}\mathcal{L}_i w = \mathcal{L}_i w$. The goal is to deduce the existence of a unique maximal solution of (43), a task which requires some clarification. Having reached (43) from the velocity form, we now look to solve the equation for vorticity which demands a representation of the velocity $u$ in terms of the vorticity $w$. For this we quote Theorem 1 of [20] (or in fact, a slightly relaxed version).

**Theorem 4.2** There exists a mapping $K: \mathcal{O} \times \mathcal{O} \to \mathbb{R}^3$ such that for every $\phi \in W_\sigma^{1,2}(\mathcal{O}; \mathbb{R}^3) \cap W^{k,p}(\mathcal{O}; \mathbb{R}^3)$ where $k \in \mathbb{N} \cup \{0\}$, $1 < p < \infty$, the function $BS_K\phi: \mathcal{O} \to \mathbb{R}^3$ defined $\lambda - a.e. by

$$ (BS_K\phi)(x) = \int_{\mathcal{O}} K(x, y)\phi(y)dy \quad (44) $$

is such that:

1. $BS_K\phi \in L_\sigma^2(\mathcal{O}; \mathbb{R}^3) \cap W^{k+1,p}(\mathcal{O}; \mathbb{R}^3)$;

2. $\mathrm{curl}(BS_K\phi) = \phi$;

3. There exists a constant $C$ independent of $\phi$ (but dependent on $k, p$) such that

$$ \|BS_K\phi\|_{W^{k+1,p}} \leq C\|\phi\|_{W^{k,p}}. $$

It should immediately be noted that such a $K$ is not claimed to be unique, and in [20] is explicitly shown to be non-unique, however it does allow us to identify a velocity from a given vorticity satisfying the divergence-free and boundary conditions (2). From this point forwards we fix a specific $K$ from the class of admissible integral kernels postulated in Theorem 4.2. We thus understand the nonlinear term as a mapping

$$ \phi \mapsto \mathcal{L}_{BS_K}\phi. $$

This mapping shall at times be simply denoted by $\mathcal{L}_{BS_K}$. The Eq. (43) is thus closed in $w$.
## 4.2 Definitions and Results

We now state and prove the existence, uniqueness and maximality results for (43). We recall that to solve the velocity form (23) we used the extended criterion of Theorem 5.22, requiring the space $V := W_{\sigma}^{3,2}(\mathbb{T}^3; \mathbb{R}^3)$ to prove Theorem 3.6. This arose naturally in first showing Theorem 3.1, where we considered solutions explicitly in terms of the original Stratonovich form. For (43), however, solutions can be obtained for the natural choice of $w_0 \in W_{\sigma}^{1,2}(\mathcal{O}; \mathbb{R}^3)$ with an application only of Theorem 5.15 in Sect. 5.3 (see Remark 5). We note that such a choice is natural as the identity is satisfied in $L^2(\mathcal{O}; \mathbb{R}^3)$. Thus we do not take the detour of considering a fourth Hilbert Space to rigorously define solutions of the Stratonovich form (42), although this can be done similarly. Notions of local strong solution, maximal strong solution and pathwise uniqueness can then be defined identically to Definitions 3.3, 3.4 and 3.5 for the new identity (43). The result is then the following.

**Theorem 4.3** For any given $\mathcal{F}_0$- measurable $w_0 : \Omega \to W_{\sigma}^{1,2}(\mathcal{O}; \mathbb{R}^3)$, there exists a unique maximal strong solution $(w, \Theta)$ of the equation (43). Moreover at $\mathbb{P}-\text{a.e. } \omega$ for which $\Theta(\omega) < \infty$, we have that

$$ \sup_{r \in [0, \Theta(\omega))} \|w_r(\omega)\|_1^2 + \int_0^{\Theta(\omega)} \|w_r(\omega)\|_2^2 dr = \infty. $$

## 4.3 Operator Bounds

Just as in Sect. 3.2, we state some intermediary results regarding the operators involved. In the following and throughout this section $c$ will represent a generic constant changing from line to line, $c(\varepsilon)$ will be a generic constant dependent on a fixed $\varepsilon$, $\phi$ and $\psi$ will be arbitrary elements of $W_{\sigma}^{2,2}(\mathcal{O}; \mathbb{R}^3)$ and $\phi_n \in \text{span}\{a_1, \dots, a_n\}$. The mapping $\mathcal{L}_i$ satisfies the same boundedness as (18), and also the following.

**Lemma 4.4** There exists a constant $c$ such that for each $i$ and for all $f \in W^{k+2,2}(\mathcal{O}; \mathbb{R}^3)$ with $k = 0, 1$, we have the bounds

$$ \langle \mathcal{L}_i^2 f, f \rangle_{W^{k,2}} + \|\mathcal{L}_i f\|_{W^{k,2}}^2 \le c \|\xi_i\|_{W^{k+2,\infty}}^2 \|f\|_{W^{k,2}}^2, \quad (45) $$

$$ \langle \mathcal{L}_i f, f \rangle_{W^{k,2}}^2 \le c \|\xi_i\|_{W^{k+1,\infty}}^2 \|f\|_{W^{k,2}}^4, \quad (46) $$

$$ \|[\Delta, \mathcal{L}_i] f\|^2 \le c \|\xi_i\|_{W^{3,\infty}}^2 \|f\|_{W^{2,2}}^2 \quad (47) $$

where $[\Delta, \mathcal{L}_i]$ is the commutator

$$ [\Delta, \mathcal{L}_i] := \Delta \mathcal{L}_i - \mathcal{L}_i \Delta. $$
**Proof** See Sect. 5.1.

**Lemma 4.5** For any $\varepsilon > 0$ we have the bound

$$ \langle P_n P \mathcal{L}_i^2 \phi_n, \phi_n \rangle_1 + \langle P_n P \mathcal{L}_i \phi_n, P_n P \mathcal{L}_i \phi_n \rangle_1 \\ \leq c(\varepsilon) \|\xi_i\|_{W^{3,\infty}}^2 \|\phi_n\|_1^2 + \varepsilon \|\xi_i\|_{W^{3,\infty}}^2 \|\phi_n\|_2^2. $$

**Proof** This now follows precisely as for (27).

**Lemma 4.6** For any $\varepsilon > 0$ we have the bound

$$ |\langle P_n P \mathcal{L}_{BS_K} \phi_n, \phi_n \rangle_1| \leq c(\varepsilon) \|\phi_n\|_1^4 + \varepsilon \|\phi_n\|_2^2. $$

**Proof** See Sect. 5.1.

**Remark 5** Recalling Remark 4, the nonlinear term in velocity form fails this estimate. It is satisfied in the vorticity form due to the additional regularity that $f_n$ has compared to $\phi_n$. This difference is what enables us to apply Theorem 5.15 directly in the case $H := W_{\sigma}^{1,2}$ for the vorticity form, whereas for the velocity form the appropriate estimate is only satisfied for $H := W_{\sigma}^{2,2}$ (see Assumption 5.8) hence the need for Theorem 5.22 in velocity form.

In the following $g$ is defined by

$$ g(x) = \int_{\mathcal{O}} K(x, y) \psi(y) dy $$

as in Theorem 4.2. The subsequent lemma is in analogy with Lemma 3.9.

**Lemma 4.7** For any $\varepsilon > 0$, we have that

$$ |\langle P \mathcal{L}_{BS_K} \phi \phi - P \mathcal{L}_{BS_K} \psi \psi, \phi - \psi \rangle| \leq c(\varepsilon) [\|\phi\|_1^2 + \|\psi\|_1^2] \|\phi - \psi\|^2 + \varepsilon \|\phi - \psi\|_1^2 $$

**Proof** See Sect. 5.1.

## 4.4 Proof of Theorem 4.3

As discussed we apply Theorem 5.15, which we do for the spaces

$$ V := W_{\sigma}^{2,2}(\mathcal{O}; \mathbb{R}^3), \quad H := W_{\sigma}^{1,2}(\mathcal{O}; \mathbb{R}^3), \quad U := L_{\sigma}^2(\mathcal{O}; \mathbb{R}^3). $$

**Proof of Theorem 4.3** The density relations are clear as $C_{0,\sigma}^{\infty}(\mathcal{O}; \mathbb{R}^3) \subset W_{\sigma}^{2,2}(\mathcal{O}; \mathbb{R}^3)$ is dense in both $W_{\sigma}^{1,2}(\mathcal{O}; \mathbb{R}^3)$ and $L_{\sigma}^2(\mathcal{O}; \mathbb{R}^3)$. The bilinear form (70) is simply again (37). Now we shift attentions to checking that the operators
are measurable into the correct spaces. We note that $\mathcal{L}_{BS_K}$ has improved regularity properties over $\mathcal{L}$ given item 3 of Theorem 4.2, so retains the continuity with measurability following. There is no change to the Stokes Operator from Sect. 3. As for $\mathcal{P}\mathcal{L}_i$, we in fact first show that $\mathcal{L}_i \in C(W_\sigma^{2,2}(\mathcal{O}; \mathbb{R}^3); W_\sigma^{1,2}(\mathcal{O}; \mathbb{R}^3))$ (and hence is invariant under $\mathcal{P}^4$). This consists of three parts: showing that it is continuous as a mapping into $W^{1,2}(\mathcal{O}; \mathbb{R}^3)$, showing the divergence free property and then the zero trace property. In fact with the appropriate regularity, it follows identically to (18) that we again have

$$ \|\mathcal{L}_i \phi\|_{W^{k,2}}^2 \le c \|\xi_i\|_{W^{k+1,\infty}}^2 \|\phi\|_{W^{k+1,2}}^2 \quad (48) $$

which addresses the continuity. The fact that $\mathcal{L}_i \phi$ is divergence free comes immediately from the relation $\mathcal{L}_i \phi = \text{curl}(B_i[BS_K \phi])$ and the well established fact the divergence of a curl is zero. For the zero trace property it is sufficient to show the existence of a sequence of compactly supported $\eta_n \in W^{1,2}(\mathcal{O}; \mathbb{R}^3)$ which converge to $\mathcal{L}_i \phi$ in $W^{1,2}(\mathcal{O}; \mathbb{R}^3)$. By definition of the property that $\xi_i \in W_0^{2,2}(\mathcal{O}; \mathbb{R}^3)$ there is a sequence $(\gamma_n)$, $\gamma_n \in C_0^\infty(\mathcal{O}; \mathbb{R}^3)$ such that $\gamma_n \to \xi_i$ in $W^{2,2}(\mathcal{O}; \mathbb{R}^3)$. Evidently $\eta_n := \mathcal{L}_{\gamma_n} \phi$ is again compactly supported, and observe that

$$ \|\mathcal{L}_{\gamma_n} \phi - \mathcal{L}_i \phi\|_{W^{1,2}} = \|\mathcal{L}_{\gamma_n - \xi_i} \phi\|_{W^{1,2}} \le c \|\gamma_n - \xi_i\|_{W^{2,2}} \|\phi\|_{W^{2,2}} $$

from (6), which converges to zero as required to justify the zero trace property. The fact that $\mathcal{P}\mathcal{L}_i^2 \in C(W_\sigma^{2,2}(\mathcal{O}; \mathbb{R}^3); L_\sigma^2(\mathcal{O}; \mathbb{R}^3))$ again follows from the linearity, (48) and continuity of $\mathcal{P}$. We now proceed to justify the numbered assumptions of Sect. 5.3.

Assumption 5.6: We use the system $(a_k)$ of eigenfunctions of the Stokes Operator.

Assumption 5.7: Items (74), (75) follow identically to the justification of (87), (88) for (23) given the increased regularity of $\mathcal{L}$ over $\mathcal{L}$ and the corresponding boundedness of the noise term (48). With the linearity of $\mathcal{L}_i$ then (76) follows trivially from (48).

Assumption 5.8: (77) now follows from Lemmas 4.5 and 4.6 in the same manner as for the velocity equation. (78) is shown exactly as (82) was for the velocity equation, using that the $P_n$ are orthogonal projections in $W_\sigma^{1,2}(\mathcal{O}; \mathbb{R}^3)$.

Assumption 5.9: The justification now comes together exactly as in the proof for Assumption 5.17 in the velocity case, noting again that (19) holds for $\mathcal{L}_i$ as well, and using Lemma 4.7.

Assumption 5.10: There is very little to demonstrate here, as the linear terms follow from Assumption 5.9 so we just briefly address the nonlinear term. Through the same process as in Lemma 4.7, we have that

<footnote n="4">⁴ As $\mathcal{P}\mathcal{L}_i$ is equal to $\mathcal{L}_i$ on $W_\sigma^{2,2}(\mathcal{O}; \mathbb{R}^3)$, then $(\mathcal{P}\mathcal{L}_i)^2$ is equal to $\mathcal{P}\mathcal{L}_i^2$ on this space too, justifying the consistency between taking the Leray Projector and then converting from Stratonovich to Itô and vice versa as seen for (23).</footnote>
$$ \begin{aligned} |\langle \mathcal{P}\mathcal{L}_f\phi, \phi \rangle| & \le |\langle \mathcal{L}_f\phi, \phi \rangle + \langle \mathcal{L}_\phi f, \phi \rangle| \\ & \le c [\|f\|_2 \|\phi\|_1 + \|\phi\|_1 \|f\|_2] \|\phi\| \\ & \le c \|\phi\| \|\phi\|_1^2 \end{aligned} $$

where the rest simply follows as in Assumption 5.9.

Assumption 5.11: We consider the different operators in turn, starting with the nonlinear term and using that

$$ \begin{aligned} |\langle \mathcal{P}\mathcal{L}_f\phi - \mathcal{P}\mathcal{L}_g\psi, \eta \rangle| &= |\langle \mathcal{L}_f\phi - \mathcal{L}_\phi f - \mathcal{L}_g\psi + \mathcal{L}_\psi g, \eta \rangle| \\ &= |\langle \mathcal{L}_{f-g}\phi + \mathcal{L}_g(\phi - \psi) - \mathcal{L}_{\phi-\psi}f - \mathcal{L}_\psi(f-g), \eta \rangle| \end{aligned} $$

where exactly as in Lemma 4.7 we have that

$$ \| \mathcal{L}_{f-g}\phi + \mathcal{L}_g(\phi - \psi) - \mathcal{L}_{\phi-\psi}f - \mathcal{L}_\psi(f-g) \| \le c [\|\phi\|_1 + \|\psi\|_1] \|\phi - \psi\|_1 $$

so in particular

$$ |\langle \mathcal{P}\mathcal{L}_f\phi - \mathcal{P}\mathcal{L}_g\psi, \eta \rangle| \le \|\eta\| (c [\|\phi\|_1 + \|\psi\|_1] \|\phi - \psi\|_1). \quad (49) $$

For the Stokes Operator we simply apply Proposition (12) to see that

$$ |\langle A\phi - A\psi, \eta \rangle| = |\langle A(\phi - \psi), \eta \rangle| = \langle \phi - \psi, \eta \rangle_1 \le \|\phi - \psi\|_1 \|\eta\|_1 \quad (50) $$

and for the $\mathcal{L}_i^2$ term we use the adjoint characterisation to observe that

$$ \begin{aligned} |\langle \mathcal{L}_i^2\phi - \mathcal{L}_i^2\psi, \eta \rangle| &= |\langle \mathcal{L}_i^2(\phi - \psi), \eta \rangle| = |\langle \mathcal{L}_i(\phi - \psi), B_i^*\eta \rangle| \\ & \le c \|\xi_i\|_{W^{1,\infty}}^2 \|\phi - \psi\|_1 \|\eta\|_1. \end{aligned} \quad (51) $$

Combining (49), (50) and (51) gives the result.

□

# 5 Appendices

## 5.1 Proofs from Sects. 2.3, 3.2, and 4.3

We begin with the proofs from Sect. 2.3.

**Proof of Proposition 2.6** We begin with showing (19), tasked to control

$$ \langle B_i^2 f, f \rangle_{W^{k,2}} + \|B_i f\|_{W^{k,2}}^2 \quad (52) $$
and do so with each derivative in the sum for the inner product: that is, we are looking at

$$ \langle D^\alpha B_i^2 f, D^\alpha f \rangle + \langle D^\alpha B_i f, D^\alpha B_i f \rangle \quad (53) $$

where we simplify the derivatives by writing

$$ \begin{aligned} D^\alpha B_{\xi_i} f &= \sum_{\alpha' \le \alpha} B_{D^{\alpha-\alpha'} \xi_i} D^{\alpha'} f \\ &= \sum_{\alpha' < \alpha} B_{D^{\alpha-\alpha'} \xi_i} D^{\alpha'} f + B_{\xi_i} D^\alpha f. \end{aligned} \quad (54) $$

Plugging this result in, we also see that

$$ \begin{aligned} D^\alpha B_{\xi_i}^2 f &= D^\alpha B_{\xi_i} (B_{\xi_i} f) \\ &= \sum_{\alpha' < \alpha} B_{D^{\alpha-\alpha'} \xi_i} D^{\alpha'} B_{\xi_i} f + B_{\xi_i} D^\alpha B_{\xi_i} f \end{aligned} $$

which will use in our analysis of (53), reducing the expression to

$$ \left\langle \sum_{\alpha' < \alpha} B_{D^{\alpha-\alpha'} \xi_i} D^{\alpha'} B_{\xi_i} f + B_{\xi_i} D^\alpha B_{\xi_i} f, D^\alpha f \right\rangle + \langle D^\alpha B_{\xi_i} f, D^\alpha B_{\xi_i} f \rangle $$

which we further break up in terms of the adjoint $B_{\xi_i}^*$:

$$ \left\langle \sum_{\alpha' < \alpha} B_{D^{\alpha-\alpha'} \xi_i} D^{\alpha'} B_{\xi_i} f, D^{\alpha'} f \right\rangle + \langle D^{\alpha} B_{\xi_i} f, B_{\xi_i}^* D^{\alpha'} f \rangle + \langle D^{\alpha} B_{\xi_i} f, D^{\alpha} B_{\xi_i} f \rangle \quad (55) $$

requiring that $D^\alpha f \in W^{1,2}(\mathcal{O}; \mathbb{R}^3)$, which is satisfied by the assumption $f \in W^{k+2,2}(\mathcal{O}; \mathbb{R}^3)$. By summing the second and third inner products and using (54), this becomes

$$ \left\langle \sum_{\alpha' < \alpha} B_{D^{\alpha-\alpha'} \xi_i} D^{\alpha'} B_{\xi_i} f, D^{\alpha'} f \right\rangle + \left\langle D^{\alpha} B_{\xi_i} f, B_{\xi_i}^* D^{\alpha'} f + \sum_{\alpha' < \alpha} B_{D^{\alpha-\alpha'} \xi_i} D^{\alpha'} f + B_{\xi_i} D^{\alpha'} f \right\rangle $$

which we look to simplify by combining $B_{\xi_i}^*$ and $B_{\xi_i}$, noting that

$$ B_i^* + B_i = \mathcal{L}_{\xi_i}^* + \mathcal{T}_i^* + \mathcal{L}_{\xi_i} + \mathcal{T}_i = \mathcal{T}_i^* + \mathcal{T}_i. $$

Indeed we arrive at the expression

$$ \left\langle \sum_{\alpha' < \alpha} B_{D^{\alpha-\alpha'} \xi_i} D^{\alpha'} B_{\xi_i} f, D^{\alpha'} f \right\rangle + \left\langle D^{\alpha} B_{\xi_i} f, (\mathcal{T}_{\xi_i} + \mathcal{T}_{\xi_i}^*) D^{\alpha'} f + \sum_{\alpha' < \alpha} B_{D^{\alpha-\alpha'} \xi_i} D^{\alpha'} f \right\rangle $$
As we are looking to achieve control with respect to the $W^{k,2}(O; \mathbb{R}^3)$ norm of $f$, then it is the terms with differential operators of order greater than $k$ that concern us. Of course this was the motivating factor behind combining $B_{\xi_i}$ and its adjoint, nullifying the additional derivative coming from $\mathcal{L}_{\xi_i}$. There are more higher order terms to go though, and the strategy will be to write these in terms of commutators with a differential operator of controllable order. This will involve considering different aspects of our sum in tandem, which will be helped with (54) reducing our expression again to

$$ \left\langle \sum_{\alpha' < \alpha} B_{D^{\alpha-\alpha'} \xi_i} D^{\alpha'} B_{\xi_i} f, D^{\alpha} f \right\rangle + \left\langle \sum_{\beta < \alpha} B_{D^{\alpha-\beta} \xi_i} D^{\beta} f + B_{\xi_i} D^{\alpha} f, (T_{\xi_i} + T_{\xi_i}^*) D^{\alpha} f \right. \\ \left. \qquad + \sum_{\alpha' < \alpha} B_{D^{\alpha-\alpha'} \xi_i} D^{\alpha'} f \right\rangle. $$

Ultimately the terms in the summand are split up into

$$ \langle B_{\xi_i} D^{\alpha} f, (T_{\xi_i} + T_{\xi_i}^*) D^{\alpha} f \rangle \quad (56) $$

$$ + \left\langle \sum_{\beta < \alpha} B_{D^{\alpha-\beta} \xi_i} D^{\beta} f, (T_{\xi_i} + T_{\xi_i}^*) D^{\alpha} f + \sum_{\alpha' < \alpha} B_{D^{\alpha-\alpha'} \xi_i} D^{\alpha'} f \right\rangle \quad (57) $$

$$ + \sum_{\alpha' < \alpha} \left( \langle B_{D^{\alpha-\alpha'} \xi_i} D^{\alpha'} B_{\xi_i} f, D^{\alpha} f \rangle + \langle B_{\xi_i} D^{\alpha} f, B_{D^{\alpha-\alpha'} \xi_i} D^{\alpha'} f \rangle \right) \quad (58) $$

with the intention of controlling each one individually. Firstly for a treatment of (56),

$$ \begin{aligned} (56) &= \langle (\mathcal{L}_{\xi_i} + \mathcal{T}_{\xi_i}) D^{\alpha} f, (\mathcal{T}_{\xi_i}^* + \mathcal{T}_{\xi_i}) D^{\alpha} f \rangle \\ &= \langle \mathcal{L}_{\xi_i} D^{\alpha} f, \mathcal{T}_{\xi_i}^* D^{\alpha} f \rangle + \langle \mathcal{L}_{\xi_i} D^{\alpha} f, \mathcal{T}_{\xi_i} D^{\alpha} f \rangle \\ &\quad + \langle \mathcal{T}_{\xi_i} D^{\alpha} f, \mathcal{T}_{\xi_i}^* D^{\alpha} f \rangle + \langle \mathcal{T}_{\xi_i} D^{\alpha} f, \mathcal{T}_{\xi_i} D^{\alpha} f \rangle \\ &= \langle (\mathcal{T}_{\xi_i} \mathcal{L}_{\xi_i} D^{\alpha} f, D^{\alpha} f) + \langle \mathcal{L}_{\xi_i} D^{\alpha} f, \mathcal{T}_{\xi_i} D^{\alpha} f \rangle \rangle \\ &\quad + \langle \mathcal{T}_{\xi_i}^2 D^{\alpha} f, D^{\alpha} f \rangle + \langle \mathcal{T}_{\xi_i} D^{\alpha} f, \mathcal{T}_{\xi_i} D^{\alpha} f \rangle \rangle. \end{aligned} $$

We now bound the brackets in terms of $\|D^{\alpha} f\|^2$ separately, starting with the latter one as

$$ \begin{aligned} \langle \mathcal{T}_{\xi_i}^2 D^{\alpha} f, D^{\alpha} f \rangle &\le \| \mathcal{T}_{\xi_i}^2 D^{\alpha} f \| \| D^{\alpha} f \| \le c \| \xi_i \|_{W^{1,\infty}} \| \mathcal{T}_{\xi_i} D^{\alpha} f \| \| D^{\alpha} f \| \\ &\le c \| \xi_i \|_{W^{1,\infty}}^2 \| D^{\alpha} f \|^2 \end{aligned} $$

and similarly
$$ \langle \mathcal{T}_{\xi_i} D^\alpha f, \mathcal{T}_{\xi_i} D^\alpha f \rangle \le \| \mathcal{T}_{\xi_i} D^\alpha f \| \| \mathcal{T}_{\xi_i} D^\alpha f \| \le c \| \xi_i \|_{W^{1,\infty}}^2 \| D^\alpha f \|^2. $$

Now for the first bracket, we add and subtract a term to have an expression through the commutator of the operators:

$$ \begin{aligned} & \langle \mathcal{T}_{\xi_i} \mathcal{L}_{\xi_i} D^\alpha f, D^\alpha f \rangle + \langle \mathcal{L}_{\xi_i} D^\alpha f, \mathcal{T}_{\xi_i} D^\alpha f \rangle \\ &= \langle (\mathcal{T}_{\xi_i} \mathcal{L}_{\xi_i} - \mathcal{L}_{\xi_i} \mathcal{T}_{\xi_i}) D^\alpha f, D^\alpha f \rangle + \langle \mathcal{L}_{\xi_i} \mathcal{T}_{\xi_i} D^\alpha f, D^\alpha f \rangle + \langle \mathcal{L}_{\xi_i} D^\alpha f, \mathcal{T}_{\xi_i} D^\alpha f \rangle \\ &= \langle (\mathcal{T}_{\xi_i} \mathcal{L}_{\xi_i} - \mathcal{L}_{\xi_i} \mathcal{T}_{\xi_i}) D^\alpha f, D^\alpha f \rangle + \langle \mathcal{T}_{\xi_i} D^\alpha f, \mathcal{L}_{\xi_i}^* D^\alpha f \rangle + \langle \mathcal{L}_{\xi_i} D^\alpha f, \mathcal{T}_{\xi_i} D^\alpha f \rangle \\ &= \langle (\mathcal{T}_{\xi_i} \mathcal{L}_{\xi_i} - \mathcal{L}_{\xi_i} \mathcal{T}_{\xi_i}) D^\alpha f, D^\alpha f \rangle. \end{aligned} $$

The commutator term is given explicitly through

$$ \begin{aligned} \mathcal{T}_{\xi_i} \mathcal{L}_{\xi_i} D^\alpha f &= \mathcal{T}_{\xi_i} \left( \sum_{j=1}^3 \xi_i^j \partial_j D^\alpha f \right) \\ &= \sum_{k=1}^3 \left( \sum_{j=1}^3 \xi_i^j \partial_j D^\alpha f \right)^k \nabla \xi_i^k \\ &= \sum_{k=1}^3 \sum_{j=1}^3 \xi_i^j \partial_j D^\alpha f^k \nabla \xi_i^k \end{aligned} $$

and

$$ \begin{aligned} \mathcal{L}_{\xi_i} \mathcal{T}_{\xi_i} D^\alpha f &= \mathcal{L}_{\xi_i} \left( \sum_{k=1}^3 D^\alpha f^k \nabla \xi_i^k \right) \\ &= \sum_{j=1}^3 \xi_i^j \partial_j \left( \sum_{k=1}^3 D^\alpha f^k \nabla \xi_i^k \right) \\ &= \sum_{j=1}^3 \sum_{k=1}^3 \xi_i^j \partial_j (D^\alpha f^k \nabla \xi_i^k) \\ &= \sum_{j=1}^3 \sum_{k=1}^3 \left( \xi_i^j \partial_j D^\alpha f^k \nabla \xi_i^k + \xi_i^j D^\alpha f^k \partial_j \nabla \xi_i^k \right) \end{aligned} $$

such that

$$ (\mathcal{T}_{\xi_i} \mathcal{L}_{\xi_i} - \mathcal{L}_{\xi_i} \mathcal{T}_{\xi_i}) D^\alpha f = - \sum_{j=1}^3 \sum_{k=1}^3 \xi_i^j D^\alpha f^k \partial_j \nabla \xi_i^k. $$
Therefore

$$
\begin{aligned}
\|(T_{\xi_i} \mathcal{L}_{\xi_i} - \mathcal{L}_{\xi_i} T_{\xi_i}) D^\alpha f\| &\le c \sum_{j=1}^3 \sum_{k=1}^3 \|\xi_i^j D^\alpha f^k \partial_j \nabla \xi_i^k\| \\
&\le c \sum_{j=1}^3 \sum_{k=1}^3 \sum_{l=1}^3 \|\xi_i^j D^\alpha f^k \partial_j \partial_l \xi_i^k\|_{L^2(\mathcal{O};\mathbb{R})} \\
&\le c \sum_{j=1}^3 \sum_{k=1}^3 \sum_{l=1}^3 \|\xi_i^j \partial_j \partial_l \xi_i^k\|_{L^\infty(\mathcal{O};\mathbb{R})} \|D^\alpha f^k\|_{L^2(\mathcal{O};\mathbb{R})} \\
&\le c \|\xi_i\|_{W^{2,\infty}}^2 \sum_{j=1}^3 \sum_{k=1}^3 \sum_{l=1}^3 \|D^\alpha f^k\|_{L^2(\mathcal{O};\mathbb{R})} \\
&\le c \|\xi_i\|_{W^{2,\infty}}^2 \|D^\alpha f\|
\end{aligned}
$$

and

$$
\begin{aligned}
\langle (T_{\xi_i} \mathcal{L}_{\xi_i} - \mathcal{L}_{\xi_i} T_{\xi_i}) D^\alpha f, D^\alpha f \rangle &\le \|(T_{\xi_i} \mathcal{L}_{\xi_i} - \mathcal{L}_{\xi_i} T_{\xi_i}) D^\alpha f\| \|D^\alpha f\| \\
&\le c \|\xi_i\|_{W^{2,\infty}}^2 \|D^\alpha f\|^2.
\end{aligned}
$$

Combining these inequalities we determine the bound

$$
(56) \leq c \|\xi_i\|_{W^{2,\infty}}^2 \|D^\alpha f\|^2.
$$

As for (57) we look to use Cauchy-Schwarz and bound each item in the inner
product. Indeed straight from (18) in the $L^2(\mathcal{O}; \mathbb{R}^3)$ setting, by simply replacing
$\xi_i$ with $D^{\alpha-\beta}\xi_i$, we have that

$$
\| B_{D^{\alpha-\beta} \xi_i} D^{\beta} f \|^2 \le c \| D^{\alpha-\beta} \xi_i \|_{W^{1,\infty}}^2 \| D^{\beta} f \|_{W^{1,2}}^2 \le c \| \xi_i \|_{W^{k+1,\infty}}^2 \| f \|_{W^{k,2}}^2.
$$

Moreover,

$$
\left\| \sum_{\beta<\alpha} B_{D^{\alpha-\beta}\xi_i} D^\beta f \right\|^2 \le c \sum_{\beta<\alpha} \| B_{D^{\alpha-\beta}\xi_i} D^\beta f \|^2 \le c \|\xi_i\|^2_{W^{k+1,\infty}} \|f\|^2_{W^{k,2}}.
$$

In addition to this,

$$
\|(T_{\xi_i} + T_{\xi_i}^*) D^\alpha f\| \le \|T_{\xi_i} D^\alpha f\| + \|T_{\xi_i}^* D^\alpha f\| \le c \|\xi_i\|_{W^{k+1,\infty}} \|D^\alpha f\|
$$

using the equivalence in operator norm of the adjoint. Together this amounts to
$$
\begin{align*}
(57) & \le \left\| \sum_{\beta < \alpha} B_{D^{\alpha-\beta} \xi_i} D^\beta f \right\| \cdot \left\| (T_{\xi_i} + T_{\xi_i}^*) D^\alpha f + \sum_{\alpha' < \alpha} B_{D^{\alpha-\alpha'} \xi_i} D^{\alpha'} f \right\| \\
& \le \left\| \sum_{\beta < \alpha} B_{D^{\alpha-\beta} \xi_i} D^\beta f \right\| \cdot \left( \left\| (T_{\xi_i} + T_{\xi_i}^*) D^\alpha f \right\| + \left\| \sum_{\alpha' < \alpha} B_{D^{\alpha-\alpha'} \xi_i} D^{\alpha'} f \right\| \right) \\
& \le c \|\xi_i\|_{W^{k+1,\infty}} \|f\|_{W^{k,2}} (c \|\xi_i\|_{W^{k+1,\infty}} \|D^\alpha f\| + c \|\xi_i\|_{W^{k+1,\infty}} \|f\|_{W^{k,2}}) \\
& \le c \|\xi_i\|_{W^{k+1,\infty}}^2 \|f\|_{W^{k,2}}^2.
\end{align*}
$$

Let's now turn our attentions to (58), which for each $\alpha'$ in the sum we rewrite as

$$ \langle D^\alpha f, B_{D^{\alpha-\alpha'} \xi_i} D^{\alpha'} B_{\xi_i} f + B_{\xi_i}^* B_{D^{\alpha-\alpha'} \xi_i} D^{\alpha'} f \rangle \quad (59) $$

and employing (54) again we see this becomes

$$
\begin{align*}
\langle D^\alpha f, B_{D^{\alpha-\alpha'} \xi_i} \left( \sum_{\beta < \alpha'} B_{D^{\alpha'-\beta} \xi_i} D^\beta f + B_{\xi_i} D^{\alpha'} f \right) + B_{\xi_i}^* B_{D^{\alpha-\alpha'} \xi_i} D^{\alpha'} f \rangle \\
&= \langle D^\alpha f, \sum_{\beta < \alpha'} B_{D^{\alpha-\alpha'} \xi_i} B_{D^{\alpha'-\beta} \xi_i} D^\beta f \rangle \\
&\quad + \langle D^\alpha f, B_{D^{\alpha-\alpha'} \xi_i} B_{\xi_i} D^{\alpha'} f + B_{\xi_i}^* B_{D^{\alpha-\alpha'} \xi_i} D^{\alpha'} f \rangle.
\end{align*}
$$

We have split up these terms to make our approach clearer, as the two right hand sides of the inner products will be considered separately. For the first inner product, two applications of (18) give that

$$
\begin{align*}
\| B_{D^{\alpha-\alpha'} \xi_i} B_{D^{\alpha'-\beta} \xi_i} D^\beta f \|^2 &\le c \| D^{\alpha-\alpha'} \xi_i \|_{W^{1,\infty}}^2 \| B_{D^{\alpha'-\beta} \xi_i} D^\beta f \|_{W^{1,2}}^2 \\
&\le c \| D^{\alpha-\alpha'} \xi_i \|_{W^{1,\infty}}^2 (c \| D^{\alpha'-\beta} \xi_i \|_{W^{2,\infty}}^2 \| D^\beta f \|_{W^{2,2}}^2) \\
&\le c \| \xi_i \|_{W^{k+1,\infty}}^4 \| f \|_{W^{k,2}}^2
\end{align*}
$$

Moreover,

$$
\begin{align*}
\left\| \sum_{\beta < \alpha'} B_{D^{\alpha-\alpha'} \xi_i} B_{D^{\alpha'-\beta} \xi_i} D^\beta f \right\|^2 &\le c \| B_{D^{\alpha-\alpha'} \xi_i} B_{D^{\alpha'-\beta} \xi_i} D^\beta f \|^2 \\
&\le c \| \xi_i \|_{W^{k+1,\infty}}^4 \| f \|_{W^{k,2}}^2.
\end{align*}
$$

As for the second inner product, we rewrite the right side as

$$ B_{D^{\alpha-\alpha'} \xi_i} ((\mathcal{L}_{\xi_i} + \mathcal{T}_{\xi_i}) D^{\alpha'} f) + (\mathcal{L}_{\xi_i}^* + \mathcal{T}_{\xi_i}^*) B_{D^{\alpha-\alpha'} \xi_i} D^{\alpha'} f $$
and further

$$ (B_{D^{\alpha-\alpha'} \xi_i} \mathcal{L}_{\xi_i} - \mathcal{L}_{\xi_i} B_{D^{\alpha-\alpha'} \xi_i}) D^{\alpha'} f + B_{D^{\alpha-\alpha'} \xi_i} \mathcal{T}_{\xi_i} D^{\alpha'} f + \mathcal{T}_{\xi_i}^* B_{D^{\alpha-\alpha'} \xi_i} D^{\alpha'} f. \quad (60) $$

Starting with the latter two terms,

$$ \begin{aligned} \| B_{D^{\alpha-\alpha'} \xi_i} \mathcal{T}_{\xi_i} D^{\alpha'} f \|^2 &\le c \| D^{\alpha-\alpha'} \xi_i \|_{W^{1,\infty}}^2 \| \mathcal{T}_{\xi_i} D^{\alpha'} f \|_{W^{1,2}}^2 \\ &\le c \| D^{\alpha-\alpha'} \xi_i \|_{W^{1,\infty}}^2 (c \| \xi_i \|_{W^{2,\infty}}^2 \| D^{\alpha'} f \|_{W^{1,2}}^2) \\ &\le c \| \xi_i \|_{W^{k+1,\infty}}^4 \| f \|_{W^{k,2}}^2 \end{aligned} $$

and likewise

$$ \begin{aligned} \| \mathcal{T}_{\xi_i}^* B_{D^{\alpha-\alpha'} \xi_i} D^{\alpha'} f \|^2 &\le c \| \xi_i \|_{W^{1,\infty}}^2 \| B_{D^{\alpha-\alpha'} \xi_i} D^{\alpha'} f \|^2 \\ &\le c \| \xi_i \|_{W^{1,\infty}}^2 (c \| D^{\alpha-\alpha'} \xi_i \|_{W^{1,\infty}}^2 \| D^{\alpha'} f \|_{W^{1,2}}^2) \\ &\le c \| \xi_i \|_{W^{k+1,\infty}}^4 \| f \|_{W^{k,2}}^2. \end{aligned} $$

Now we show explicitly that the commutator in

$$ (B_{D^{\alpha-\alpha'} \xi_i} \mathcal{L}_{\xi_i} - \mathcal{L}_{\xi_i} B_{D^{\alpha-\alpha'} \xi_i}) D^{\alpha'} f \quad (61) $$

from (60) is of first order (so of $k^{\text{th}}$ order when composed with $D^{\alpha'}$), through the expressions

$$ \begin{aligned} B_{D^{\alpha-\alpha'} \xi_i} \mathcal{L}_{\xi_i} D^{\alpha'} f &= \sum_{j=1}^{3} \left( D^{\alpha-\alpha'} \xi_j^j \partial_j \left( \sum_{k=1}^{3} \xi_k^k \partial_k D^{\alpha'} f \right) \right. \\ &\qquad \left. + \left( \sum_{k=1}^{3} \xi_k^k \partial_k D^{\alpha'} f \right)^j \nabla D^{\alpha-\alpha'} \xi_j^k \right) \\ &= \sum_{j=1}^{3} \sum_{k=1}^{3} \left( D^{\alpha-\alpha'} \xi_j^k \partial_j \xi_k^k \partial_k D^{\alpha'} f \right. \\ &\qquad \left. + D^{\alpha-\alpha'} \xi_j^j \xi_k^j \partial_j \partial_k D^{\alpha'} f + \xi_k^j \partial_k D^{\alpha'} f^j \nabla D^{\alpha-\alpha'} \xi_j^k \right) \end{aligned} $$

and

$$ \mathcal{L}_{\xi_i} B_{D^{\alpha-\alpha'} \xi_i} D^{\alpha'} f = \sum_{k=1}^{3} \xi_i^k \partial_k \left( \sum_{j=1}^{3} D^{\alpha-\alpha'} \xi_j^j \partial_j D^{\alpha'} f + D^{\alpha'} f^j \nabla D^{\alpha-\alpha'} \xi_i^j \right) $$
$$ = \sum_{j=1}^{3} \sum_{k=1}^{3} \left( \xi_i^k \partial_k D^{\alpha-\alpha'} \xi_j^j \partial_j D^{\alpha'} f + \xi_i^k D^{\alpha-\alpha'} \xi_j^j \partial_k \partial_j D^{\alpha'} f \right. \\ \left. + \xi_i^k \partial_k D^{\alpha'} f^j \nabla D^{\alpha-\alpha'} \xi_j^j + \xi_i^k D^{\alpha'} f^j \partial_k \nabla D^{\alpha-\alpha'} \xi_j^j \right) $$

such that

$$ (61) = \sum_{j=1}^{3} \sum_{k=1}^{3} \left( D^{\alpha-\alpha'} \xi_i^j \partial_j \xi_i^k \partial_k D^{\alpha'} f - \xi_i^k \partial_k D^{\alpha-\alpha'} \xi_i^j \partial_j D^{\alpha'} f \right. \\ \left. - \xi_i^k D^{\alpha'} f^j \partial_k \nabla D^{\alpha-\alpha'} \xi_i^j \right) $$

and in particular

$$ \begin{aligned} \|(61)\|^2 &\le c \sum_{j=1}^3 \sum_{k=1}^3 \left( \|D^{\alpha-\alpha'} \xi_i^j \partial_j \xi_i^k \partial_k D^{\alpha'} f\|^2 + \|\xi_i^k \partial_k D^{\alpha-\alpha'} \xi_i^j \partial_j D^{\alpha'} f\|^2 \right. \\ &\qquad \left. + \|\xi_i^k D^{\alpha'} f^j \partial_k \nabla D^{\alpha-\alpha'} \xi_i^j\|^2 \right) \\ &= c \sum_{j=1}^3 \sum_{k=1}^3 \sum_{l=1}^3 \left( \|D^{\alpha-\alpha'} \xi_i^j \partial_j \xi_i^k \partial_k D^{\alpha'} f^l\|^2_{L^2(\mathcal{O};\mathbb{R})} \right. \\ &\qquad \left. + \|\xi_i^k \partial_k D^{\alpha-\alpha'} \xi_i^j \partial_j D^{\alpha'} f^l\|^2_{L^2(\mathcal{O};\mathbb{R})} + \|\xi_i^k D^{\alpha'} f^j \partial_k \partial_l D^{\alpha-\alpha'} \xi_i^j\|^2_{L^2(\mathcal{O};\mathbb{R})} \right) \\ &\le c \|\xi_i\|_{W^{k+2,\infty}}^4 \sum_{j=1}^3 \sum_{k=1}^3 \sum_{l=1}^3 \left( \|\partial_k D^{\alpha'} f^l\|^2_{L^2(\mathcal{O};\mathbb{R})} \right. \\ &\qquad \left. + \|\partial_j D^{\alpha'} f^l\|^2_{L^2(\mathcal{O};\mathbb{R})} + \|D^{\alpha'} f^l\|^2_{L^2(\mathcal{O};\mathbb{R})} \right) \\ &\le c \|\xi_i\|_{W^{k+2,\infty}}^4 \left( \sum_{j=1}^3 \|f\|_{W^{k,2}}^2 + \sum_{k=1}^3 \|f\|_{W^{k,2}}^2 + \sum_{j=1}^3 \sum_{k=1}^3 \|D^{\alpha'} f\|^2 \right) \\ &\le c \|\xi_i\|_{W^{k+2,\infty}}^4 \|f\|_{W^{k,2}}^2. \end{aligned} $$

Finally now we can piece these four inequalities together to produce a bound on (59):
$$
\begin{align*}
|(59)| & \le \|D^\alpha f\| \left\| \sum_{\beta < \alpha'} B_{D^{\alpha-\alpha'} \xi_i} B_{D^{\alpha'-\beta} \xi_i} D^\beta f + (60) \right\| \\
& \le \|D^\alpha f\| \left( \left\| \sum_{\beta < \alpha'} B_{D^{\alpha-\alpha'} \xi_i} B_{D^{\alpha'-\beta} \xi_i} D^\beta f \right\| + \|B_{D^{\alpha-\alpha'} \xi_i} T_{\xi_i} D^{\alpha'} f\| \right. \\
& \qquad \left. + \|T_{\xi_i}^* B_{D^{\alpha-\alpha'} \xi_i} D^{\alpha'} f\| + \|(61)\| \right) \\
& \le c \|D^\alpha f\| \left( \|\xi_i\|_{W^{k+1,\infty}}^2 \|f\|_{W^{k,2}} + \|\xi_i\|_{W^{k+1,\infty}}^2 \|f\|_{W^{k,2}} \right. \\
& \qquad \left. + \|\xi_i\|_{W^{k+1,\infty}}^2 \|f\|_{W^{k,2}} + \|\xi_i\|_{W^{k+2,\infty}}^2 \|f\|_{W^{k,2}} \right) \\
& \le c \|D^\alpha f\| \left( \|\xi_i\|_{W^{k+2,\infty}}^2 \|f\|_{W^{k,2}} \right) \\
& \le c \|\xi_i\|_{W^{k+2,\infty}}^2 \|f\|_{W^{k,2}}^2
\end{align*}
$$

and subsequently of (58):

$$
(58) = \sum_{\alpha' < \alpha} (59) \le c \sum_{\alpha' < \alpha} \|\xi_i\|_{W^{k+2,\infty}}^2 \|f\|_{W^{k,2}}^2 \le c \|\xi_i\|_{W^{k+2,\infty}}^2 \|f\|_{W^{k,2}}^2.
$$

We can now conclude the proof of (19) by observing that

$$
\begin{align*}
(52) &= \sum_{|\alpha'| \le k} (53) \\
&= \sum_{|\alpha'| \le k} (56) + (57) + (58) \\
&\le c \sum_{|\alpha'| \le k} \left( \|\xi_i\|_{W^{2,\infty}}^2 \|D^\alpha f\|^2 + \|\xi_i\|_{W^{k+1,\infty}}^2 \|f\|_{W^{k,2}}^2 + \|\xi_i\|_{W^{k+2,\infty}}^2 \|f\|_{W^{k,2}}^2 \right) \\
&\le c \sum_{|\alpha'| \le k} \|\xi_i\|_{W^{k+2,\infty}}^2 \|f\|_{W^{k,2}}^2 \\
&= c \|\xi_i\|_{W^{k+2,\infty}}^2 \|f\|_{W^{k,2}}^2.
\end{align*}
$$

As for (20), using (54) once more, we see that for each $\alpha$ in the sum for the inner product,

$$
|(D^\alpha B_i f, D^\alpha f)| = \left| \left\langle \sum_{\alpha' < \alpha} B_{D^{\alpha-\alpha'} \xi_i} D^{\alpha'} f + B_{\xi_i} D^\alpha f, D^\alpha f \right\rangle \right|
$$
$$
\begin{align*}
&= \left| \left\langle \sum_{\alpha' < \alpha} B_{D^{\alpha-\alpha'} \xi_i} D^{\alpha'} f, D^{\alpha} f \right\rangle + \langle B_{\xi_i} D^{\alpha} f, D^{\alpha} f \rangle \right| \\
&\le \left| \left\langle \sum_{\alpha' < \alpha} B_{D^{\alpha-\alpha'} \xi_i} D^{\alpha'} f, D^{\alpha} f \right\rangle \right| + \left| \langle T_{\xi_i} D^{\alpha} f, D^{\alpha} f \rangle \right|
\end{align*}
$$

using the cancellation from (15) to dispose of the order $k + 1$ term. In our treatment of (57) in (19), we showed the bound

$$
\left\| \sum_{\beta < \alpha} B_{D^{\alpha-\beta} \xi_i} D^{\beta} f \right\| \le c \|\xi_i\|_{W^{k+1,\infty}} \|f\|_{W^{k,2}}
$$

and therefore

$$
\begin{align*}
\left| \left\langle \sum_{\alpha' < \alpha} B_{D^{\alpha-\alpha'} \xi_i} D^{\alpha'} f, D^{\alpha} f \right\rangle \right| &\le c \|\xi_i\|_{W^{k+1,\infty}} \|f\|_{W^{k,2}} \|D^{\alpha} f\| \\
&\le c \|\xi_i\|_{W^{k+1,\infty}} \|f\|_{W^{k,2}}^2
\end{align*}
$$

whilst a simple bound on the second term yields the result.

□

**Proof of Lemma 2.7** For $\nabla g \in L_{\sigma}^{2,\perp}(O; \mathbb{R}^3) \cap W^{1,2}(O; \mathbb{R}^3)$,

$$
\begin{align*}
B_i(\nabla g) &= \mathcal{L}_{\xi_i}(\nabla g) + \mathcal{T}_{\xi_i}(\nabla g) \\
&= \sum_{j=1}^{3} \xi_i^j \partial_j (\nabla g) + \sum_{j=1}^{3} \partial_j g \nabla \xi_i^j \\
&= \sum_{j=1}^{3} \xi_i^j (\nabla \partial_j g) + \sum_{j=1}^{3} (\nabla \xi_i^j) \partial_j g \\
&= \nabla \sum_{j=1}^{3} \xi_i^j \partial_j g \\
&\in L_{\sigma}^{2,\perp}(O; \mathbb{R}^3)
\end{align*}
$$

using in the last line the assumption that $\nabla g \in W^{1,2}(O; \mathbb{R}^3)$ and that $\xi_i \in W^{1,\infty}(O; \mathbb{R}^3)$. We now make a distinction between the settings of $\mathbb{T}^3$ and $O$. For the bounded domain $O$, we take any $f \in W^{1,2}(O; \mathbb{R}^3)$ and use the representation (9) to see that

$$
\mathcal{P} B_i f = \mathcal{P} B_i (\mathcal{P} f + \nabla g) = \mathcal{P} B_i \mathcal{P} f + \mathcal{P} (B_i \nabla g) = \mathcal{P} B_i \mathcal{P} f
$$
as required, using again the $W^{1,2}(\mathcal{O}; \mathbb{R}^3)$ regularity of both components of the decomposition (9). In the case of the Torus we must address the constant term in the decomposition (10), appreciating that

$$B_i c = \mathcal{T}_{\xi_i} c = \sum_{j=1}^{3} c^j \nabla \xi_i^j = \nabla \sum_{j=1}^{3} c^j \xi_i^j \in L_{\sigma}^{2, \perp}(\mathbb{T}^3; \mathbb{R}^3)$$

so the result follows in the same manner.

We now state two results used in proving the Lemmas of Sects. 3.2 and 4.3. The first is derived from the Gagliardo-Nirenberg Inequality, whilst the second is proved below.

**Proposition 5.1** There exists a constant $c$ such that for any $f \in W_{\sigma}^{1,2}(\mathbb{T}^3; \mathbb{R}^3)$ and $g \in W_{\sigma}^{2,2}(\mathbb{T}^3; \mathbb{R}^3)$,

$$\|\mathcal{L}_f g\| \le c \|f\|_1 \|g\|_1^{1/2} \|g\|_2^{1/2}. \qquad (62)$$

Meanwhile for $f \in W_{\sigma}^{1,2}(\mathcal{O}; \mathbb{R}^3)$ and $g \in W_{\sigma}^{2,2}(\mathcal{O}; \mathbb{R}^3)$ we have that

$$\|\mathcal{L}_f g\| \le c \|f\|_1 (\|g\|_1^{1/2} \|g\|_2^{1/2} + \|g\|_1). \qquad (63)$$

**Proposition 5.2** There exists a constant $c$ such that for every $f \in W^{3,2}(\mathcal{O}; \mathbb{R}^3)$,

$$\|[\Delta, B_i]f\|^2 \le c\|\xi_i\|^2_{W^{3,\infty}} \|f\|^2_{W^{2,2}}$$

where $[\Delta, B_i]$ is the commutator

$$[\Delta, B_i] := \Delta B_i - B_i \Delta.$$

**Proof** We fix any such $f$ and first show that

$$[\Delta, B_i]f = \sum_{k=1}^{3} \sum_{j=1}^{3} \left( \partial_k^2 \xi_i^j \partial_j f + 2\partial_k \xi_i^j \partial_k \partial_j f + 2\partial_k f^j \partial_k \nabla \xi_i^j + f^j \partial_k^2 \nabla \xi_i^j \right). \qquad (64)$$

Indeed

$$\begin{aligned} \Delta B_i f &= \sum_{k=1}^{3} \partial_k^2 \left( \sum_{j=1}^{3} (\xi_i^j \partial_j f + f^j \nabla \xi_i^j) \right) \\ &= \sum_{k=1}^{3} \partial_k \left( \sum_{j=1}^{3} (\partial_k \xi_i^j \partial_j f + \xi_i^j \partial_k \partial_j f + \partial_k f^j \nabla \xi_i^j + f^j \partial_k \nabla \xi_i^j) \right) \end{aligned}$$
$$ = \sum_{k=1}^{3} \sum_{j=1}^{3} \left( \partial_k^2 \xi_i^j \partial_j f + 2 \partial_k \xi_i^j \partial_k \partial_j f + \xi_i^j \partial_k^2 \partial_j f + \partial_k^2 f^j \nabla \xi_i^j \right. \\ \left. + 2 \partial_k f^j \partial_k \nabla \xi_i^j + f^j \partial_k^2 \nabla \xi_i^j \right) $$

and

$$ B_i \Delta f = \sum_{j=1}^{3} \left( \xi_i^j \partial_j \left( \sum_{k=1}^{3} \partial_k^2 f \right) + \left( \sum_{k=1}^{3} \partial_k^2 f \right)^j \nabla \xi_i^j \right) \\ = \sum_{k=1}^{3} \sum_{j=1}^{3} \left( \xi_i^j \partial_k^2 \partial_j f + \partial_k^2 f^j \nabla \xi_i^j \right) $$

therefore

$$ [\Delta, B_i] f = \Delta B_i f - B_i \Delta f \\ = \sum_{k=1}^{3} \sum_{j=1}^{3} \left( \partial_k^2 \xi_i^j \partial_j f + 2 \partial_k \xi_i^j \partial_k \partial_j f + 2 \partial_k f^j \partial_k \nabla \xi_i^j + f^j \partial_k^2 \nabla \xi_i^j \right) $$

justifying (64). The result then follows with direct calculation:

$$ \begin{aligned} \| [\Delta, B_i] f \|^2 &= \left\| \sum_{k=1}^{3} \sum_{j=1}^{3} \left( \partial_k^2 \xi_i^j \partial_j f + 2 \partial_k \xi_i^j \partial_k \partial_j f + 2 \partial_k f^j \partial_k \nabla \xi_i^j + f^j \partial_k^2 \nabla \xi_i^j \right) \right\|^2 \\ &\le c \sum_{k=1}^{3} \sum_{j=1}^{3} \left\| \partial_k^2 \xi_i^j \partial_j f + 2 \partial_k \xi_i^j \partial_k \partial_j f + 2 \partial_k f^j \partial_k \nabla \xi_i^j + f^j \partial_k^2 \nabla \xi_i^j \right\|^2 \\ &= c \sum_{k=1}^{3} \sum_{j=1}^{3} \sum_{l=1}^{3} \left\| \partial_k^2 \xi_i^j \partial_j f^l + 2 \partial_k \xi_i^j \partial_k \partial_j f^l + 2 \partial_k f^j \partial_k \partial_l \xi_i^j \right. \\ &\qquad \left. + f^j \partial_k^2 \partial_l \xi_i^j \right\|^2_{L^2(\mathcal{O}; \mathbb{R})} \\ &\le c \sum_{k=1}^{3} \sum_{j=1}^{3} \sum_{l=1}^{3} \left( \| \partial_k^2 \xi_i^j \partial_j f^l \|^2_{L^2(\mathcal{O}; \mathbb{R})} + \| 2 \partial_k \xi_i^j \partial_k \partial_j f^l \|^2_{L^2(\mathcal{O}; \mathbb{R})} \right. \\ &\qquad \left. + \| 2 \partial_k f^j \partial_k \partial_l \xi_i^j \|^2_{L^2(\mathcal{O}; \mathbb{R})} + \| f^j \partial_k^2 \partial_l \xi_i^j \|^2_{L^2(\mathcal{O}; \mathbb{R})} \right) \end{aligned} $$
$$
\begin{aligned}
&\le c \|\xi_i\|_{W^{3,\infty}}^2 \sum_{k=1}^3 \sum_{j=1}^3 \sum_{l=1}^3 \left( \|\partial_j f^l\|_{L^2(\mathcal{O};\mathbb{R})}^2 + \|\partial_k \partial_j f^l\|_{L^2(\mathcal{O};\mathbb{R})}^2 \right. \\
&\qquad \left. + \|\partial_k f^j\|_{L^2(\mathcal{O};\mathbb{R})}^2 + \|f^j\|_{L^2(\mathcal{O};\mathbb{R})}^2 \right) \\
&\le c \|\xi_i\|_{W^{3,\infty}}^2 \|f\|_{W^{2,2}}^2.
\end{aligned}
$$

□

We now prove Lemma 3.7 through the properties (26), (27) and (28) indepen-
dently.

**Proof of (26)** We use the established result that for $k = 2$ then the Sobolev Space $W^{k,2}(\mathcal{O}; \mathbb{R})$ is an algebra (a result first presented in [46]), to deduce that

$$
\| \mathcal{L}_{f_n} f_n \|_{W^{2,2}} = \left\| \sum_{j=1}^{3} f_n^j \partial_j f_n \right\|_{W^{2,2}} \le c \|f_n\|_2 \|f_n\|_3.
$$

From here we simply use that $\mathcal{P}_n$ is self-adjoint and Young's Inequality to see that

$$
|\langle \mathcal{P}_n \mathcal{P} \mathcal{L}_{f_n} f_n, f_n \rangle_2| = |\langle \mathcal{P} \mathcal{L}_{f_n} f_n, f_n \rangle_2| \le c(\epsilon) \|f_n\|_2^4 + \epsilon \|f_n\|_3^2.
$$

□

*Proof of (27)* As the *P*<sub>*n*</sub> are self-adjoint we can readily justify the inequality

$$
\langle \mathcal{P}_n \mathcal{P} B_i^2 f_n, f_n \rangle_1 + \langle \mathcal{P}_n \mathcal{P} B_i f_n, \mathcal{P}_n \mathcal{P} B_i f_n \rangle_1 \le \langle \mathcal{P} B_i^2 f_n, f_n \rangle_1 + \langle \mathcal{P} B_i f_n, \mathcal{P} B_i f_n \rangle_1
$$

and moreover from (12) that this is just

$$
\langle \mathcal{P} B_i^2 f_n, A f_n \rangle + \langle \mathcal{P} B_i f_n, A \mathcal{P} B_i f_n \rangle.
$$

We rewrite this as

$$
\langle (\mathcal{P} B_i)^2 f_n, A f_n \rangle + \langle \mathcal{P} B_i f_n, A \mathcal{P} B_i f_n \rangle
$$

and further as

$$
\langle \mathcal{P} B_i f_n, B_i^* A f_n \rangle - \langle \mathcal{P} B_i f_n, \Delta B_i f_n \rangle \quad (65)
$$

for the adjoint $B_i^* = \mathcal{L}_{\xi_i} + \mathcal{T}_{\xi_i}^*$. We look to commute the Laplacian and $B_i$, using
Proposition 5.2 and subsequently the cancellation of the derivative in $B_i$ when
considered with the adjoint. Indeed,
$$
\begin{align*}
- \langle \mathcal{P} B_i f_n, \Delta B_i f_n \rangle &= - \langle \mathcal{P} B_i f_n, ([\Delta, B_i] + B_i \Delta) f_n \rangle \\
&= - \langle \mathcal{P} B_i f_n, [\Delta, B_i] f_n \rangle - \langle \mathcal{P} B_i f_n, \mathcal{P} B_i \Delta f_n \rangle \\
&= - \langle \mathcal{P} B_i f_n, [\Delta, B_i] f_n \rangle + \langle \mathcal{P} B_i f_n, \mathcal{P} B_i A f_n \rangle \\
&= - \langle \mathcal{P} B_i f_n, [\Delta, B_i] f_n \rangle + \langle \mathcal{P} B_i f_n, B_i A f_n \rangle.
\end{align*}
$$

Thus (65) becomes

$$
\langle \mathcal{P} B_i f_n, B_i^* A f_n \rangle - \langle \mathcal{P} B_i f_n, [\Delta, B_i] f_n \rangle + \langle \mathcal{P} B_i f_n, B_i A f_n \rangle
$$

or simply

$$
\langle \mathcal{P} B_i f_n, (\mathcal{T}_{\xi_i} + \mathcal{T}_{\xi_i}^*) A f_n - [\Delta, B_i] f_n \rangle \quad (66)
$$

which we look to bound through Cauchy-Schwarz and the results of (18) and
Proposition 5.2 to see that

$$
\begin{align*}
(66) \le \| \mathcal{P} B_i f_n \| \left( \| (\mathcal{T}_{\xi_i} + \mathcal{T}_{\xi_i}^*) A f_n \| + \| [\Delta, B_i] f_n \| \right) & \\
&\le c \| \xi_i \|_{W^{1,\infty}} \| f_n \|_{W^{1,2}} ( \| \xi_i \|_{W^{1,\infty}} \| A f_n \| + \| \xi_i \|_{W^{3,\infty}} \| f_n \|_{W^{2,2}} ) \\
&\le c \| \xi_i \|_{W^{1,\infty}} \| f_n \|_1 ( \| \xi_i \|_{W^{1,\infty}} \| f_n \|_2 + \| \xi_i \|_{W^{3,\infty}} \| f_n \|_2 ) \\
&\le c(\epsilon) \| \xi_i \|_{W^{3,\infty}}^2 \| f_n \|_1^2 + \epsilon \| \xi_i \|_{W^{3,\infty}}^2 \| f_n \|_2^2
\end{align*}
$$

as required.

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**Proof of (28)** As with (27) we can immediately say that

$$
\begin{align}
& \langle \mathcal{P}_n \mathcal{P} B_i^2 f_n, f_n \rangle_2 + \langle \mathcal{P}_n \mathcal{P} B_i f_n, \mathcal{P}_n \mathcal{P} B_i f_n \rangle_2 \nonumber \\
& \le \langle \mathcal{P} B_i^2 f_n, f_n \rangle_2 + \langle \mathcal{P} B_i f_n, \mathcal{P} B_i f_n \rangle_2 \tag{67}
\end{align}
$$

which we again manipulate to give

$$
\begin{align*}
(67) &= \langle \mathcal{A} \mathcal{P} B_i^2 f_n, A f_n \rangle + \langle \mathcal{A} \mathcal{P} B_i f_n, \mathcal{A} \mathcal{P} B_i f_n \rangle \\
&= -\langle \mathcal{P} \Delta B_i^2 f_n, A f_n \rangle - \langle A B_i f_n, \mathcal{P} \Delta B_i f_n \rangle \\
&= -\langle \mathcal{P}[\Delta, B_i] B_i f_n + \mathcal{P} B_i \Delta B_i f_n, A f_n \rangle - \langle A B_i f_n, \mathcal{P}[\Delta, B_i] f_n + \mathcal{P} B_i \Delta f_n \rangle \\
&= -\langle \mathcal{P}[\Delta, B_i] B_i f_n, A f_n \rangle + \langle \mathcal{P} B_i A B_i f_n, A f_n \rangle - \langle A B_i f_n, \mathcal{P}[\Delta, B_i] f_n \rangle \\
&\quad + \langle A B_i f_n, \mathcal{P} B_i A f_n \rangle \\
&= \langle B_i A B_i f_n, A f_n \rangle + \langle A B_i f_n, B_i A f_n \rangle - \langle \mathcal{P}[\Delta, B_i] B_i f_n, A f_n \rangle \\
&\quad - \langle A B_i f_n, \mathcal{P}[\Delta, B_i] f_n \rangle
\end{align*}
$$
$$ \begin{aligned} &= \langle AB_i f_n, (B_i + B_i^*) A f_n \rangle - \langle \mathcal{P}[\Delta, B_i] B_i f_n, A f_n \rangle - \langle AB_i f_n, \mathcal{P}[\Delta, B_i] f_n \rangle \\ &= \langle AB_i f_n, (\mathcal{T}_{\xi_i} + \mathcal{T}_{\xi_i}^*) A f_n \rangle - \langle \mathcal{P}[\Delta, B_i] B_i f_n, A f_n \rangle - \langle AB_i f_n, \mathcal{P}[\Delta, B_i] f_n \rangle. \end{aligned} $$

We shall treat each term individually using Cauchy-Schwarz, Young's Inequality and Proposition 5.2 in the same manner as the proof of (27):

$$ \begin{aligned} \langle AB_i f_n, (\mathcal{T}_{\xi_i} + \mathcal{T}_{\xi_i}^*) A f_n \rangle &\le \|AB_i f_n\| \|(\mathcal{T}_{\xi_i} + \mathcal{T}_{\xi_i}^*) A f_n\| \\ &\le c(\varepsilon) \|\xi_i\|_{W^{3,\infty}}^2 \|f_n\|_2^2 + \frac{\varepsilon}{3} \|\xi_i\|_{W^{3,\infty}}^2 \|f_n\|_3^2 \end{aligned} $$

as well as

$$ \begin{aligned} - \langle \mathcal{P}[\Delta, B_i] B_i f_n, A f_n \rangle &\le \|\mathcal{P}[\Delta, B_i] B_i f_n\| \|A f_n\| \\ &\le c \|[\Delta, B_i] B_i f_n\| \|f_n\|_2 \\ &\le c \|\xi_i\|_{W^{3,\infty}} \|B_i f_n\|_{W^{2,2}} \|f_n\|_2 \\ &\le c \|\xi_i\|_{W^{3,\infty}}^2 \|f_n\|_{W^{3,2}} \|f_n\|_2 \\ &\le c(\varepsilon) \|\xi_i\|_{W^{3,\infty}}^2 \|f_n\|_2^2 + \frac{\varepsilon}{3} \|\xi_i\|_{W^{3,\infty}}^2 \|f_n\|_3^2 \end{aligned} $$

and finally

$$ \begin{aligned} - \langle AB_i f_n, \mathcal{P}[\Delta, B_i] f_n \rangle &\le \|AB_i f_n\| \|\mathcal{P}[\Delta, B_i] f_n\| \\ &\le c \|B_i f_n\|_{W^{2,2}} \|[\Delta, B_i] f_n\| \\ &\le c \|\xi_i\|_{W^{3,\infty}}^2 \|f_n\|_{W^{3,2}} \|f_n\|_{W^{2,2}} \\ &\le c(\varepsilon) \|\xi_i\|_{W^{3,\infty}}^2 \|f_n\|_2^2 + \frac{\varepsilon}{3} \|\xi_i\|_{W^{3,\infty}}^2 \|f_n\|_3^2 \end{aligned} $$

Summing these up completes the proof. □

**Proof of Lemma 3.8** Observe that

$$ \begin{aligned} \langle \mathcal{P}\mathcal{L}_f f - \mathcal{P}\mathcal{L}_g g, f - g \rangle_1 &= \langle \mathcal{P}\mathcal{L}_f f - \mathcal{L}_g g, A(f - g) \rangle \\ &= \langle \mathcal{L}_{f-g} f + \mathcal{L}_g(f - g), A(f - g) \rangle \end{aligned} $$

and so it is sufficient to control the terms

$$ \begin{aligned} |\langle \mathcal{L}_{f-g} f, A(f-g) \rangle| &\le \|\mathcal{L}_{f-g} f\| \|A(f-g)\| \\ &\le c(\varepsilon) \|f\|_2^2 \|f-g\|_1^2 + \frac{\varepsilon}{2} \|f-g\|_2^2 \end{aligned} $$

and
$$
\begin{aligned}
|\langle \mathcal{L}_g(f-g), A(f-g) \rangle| &\le \|\mathcal{L}_g(f-g)\| \|A(f-g)\| \\
&\le c \|g\|_1 \|f-g\|_1^{1/2} \|f-g\|_2^{1/2} \|f-g\|_2 \\
&\le c(\varepsilon) \|g\|_1^4 \|f-g\|_1^2 + \frac{\varepsilon}{2} \|f-g\|_2^2
\end{aligned}
$$

using (62) and Young's Inequality with conjugate exponents 4 and 4/3. □

**Proof of Lemma 3.9** As in Lemma 3.8, we use the inequality

$$
|\langle \mathcal{P}\mathcal{L}_f f - \mathcal{P}\mathcal{L}_g g, f-g \rangle| \le |\langle \mathcal{L}_{f-g} f, f-g \rangle| + |\langle \mathcal{L}_g(f-g), f-g \rangle|.
$$

For the first term, appealing to (4), observe that

$$
|\langle \mathcal{L}_{f-g} f, f-g \rangle| \le \|\mathcal{L}_{f-g} f\| \|f-g\| \le c(\varepsilon) \|f\|_2^2 \|f-g\|^2 + \varepsilon \|f-g\|_1^2.
$$

The second term is null due to (15), which concludes the proof. □

**Proof of Lemma 4.4** We consider the proofs individually.

Assumption (45): We look to use the same approach as the one for (19), and given
the similar structure of $\mathcal{L}_i$ to $B_i$ we provide only a sketch of the proof addressing
the key properties. The essential component of the proof is in removing the
derivative dependency coming from the transport type operator. Inspecting the
proof of (19), there are three key properties that should be shown.

1. $\mathcal{L}_i^* + \mathcal{L}_i$ is of zeroth order.

2. Defining $Q_i$ by $Q_i : f \mapsto \mathcal{L}_f \xi_i$, then the commutator $[Q_i, \mathcal{L}_{\xi_i}]$ is of zeroth order.

3. For $|\alpha| \le 1$, the commutator $[\mathcal{L}_{D^\alpha \xi_i}, \mathcal{L}_{\xi_i}]$ is of first order.

We note that property 3 can be shown for arbitrary $\alpha$ if we assume sufficient regularity for $\xi_i$, as we did in Proposition 2.6. Property 1 is clear from the same structure $\mathcal{L}_i^* = -\mathcal{L}_{\xi_i} + \mathcal{Q}_i^*$ where $\mathcal{Q}_i$, $\mathcal{Q}_i^*$ are of zeroth order. We calculate the commutator in 2 explicitly, acting on $f \in W^{1,2}(\mathcal{O}; \mathbb{R}^3)$:

$$
\begin{align*}
Q_i \mathcal{L}_{\xi_i} f &= \sum_{k=1}^{3} (\mathcal{L}_{\xi_i} f)^k \partial_k \xi_i \\
&= \sum_{k=1}^{3} \left( \sum_{j=1}^{3} \xi_i^j \partial_j f \right)^k \partial_k \xi_i \\
&= \sum_{k=1}^{3} \sum_{j=1}^{3} \xi_i^j \partial_j f^k \partial_k \xi_i
\end{align*}
$$

and
$$ \begin{aligned} \mathcal{L}_{\xi_i} Q_i f &= \sum_{j=1}^{3} \xi_i^j \partial_j (Q_i f) = \sum_{j=1}^{3} \xi_i^j \partial_j \left( \sum_{k=1}^{3} f^k \partial_k \xi_i \right) \\ &= \sum_{j=1}^{3} \sum_{k=1}^{3} \xi_i^j \left( \partial_j f^k \partial_k \xi_i + f^k \partial_j \partial_k \xi_i \right) \end{aligned} $$

hence

$$ [Q_i, \mathcal{L}_{\xi_i}] = - \sum_{j=1}^{3} \sum_{k=1}^{3} \xi_i^j f^k \partial_j \partial_k \xi_i $$

which is of zeroth order. As for 3, the term which needs to be addressed is the one $[\mathcal{L}_{D^{\alpha}\xi_i}, \mathcal{L}_{\xi_i}]$ which was already attended to in the original proof, so the result is concluded here.

Assumption (46): Comparing to the proof of (20), this is a consequence of the property (15) once more and the same boundedness of (18).

Assumption (47): Once more the critical term is $[\Delta, \mathcal{L}_{\xi_i}]$ which was addressed in the proof of Proposition 5.2.

□

**Proof of Lemma 4.6** Note that

$$ |\langle P_n P \mathcal{L}_{BS_K} \phi_n, \phi_n \rangle_1| = |\langle P_n P \mathcal{L}_{BS_K} \phi_n, A\phi_n \rangle| \le c(\epsilon) \|\mathcal{L}_{BS_K} \phi_n\|^2 + \epsilon \|\phi_n\|^2_2 $$

and

$$ \| \mathcal{L}_{BS_K} \phi_n \phi_n \|^2 \le 2 \left( \| \mathcal{L}_{BS_K} \phi_n \phi_n \|^2 + \| \mathcal{L}_{\phi_n} BS_K \phi_n \|^2 \right) $$

so we look to control these two terms. Indeed,

$$ \begin{aligned} \| \mathcal{L}_{BS_K} \phi_n \phi_n \|^2 &\le c \sum_{j=1}^3 \sum_{k=1}^3 \| BS_K \phi_n^j \|_{L^\infty(\mathcal{O}; \mathbb{R})} \| \partial_j \phi_n^k \|_{L^2(\mathcal{O}; \mathbb{R})}^2 \\ &\le c \sum_{j=1}^3 \sum_{k=1}^3 \| BS_K \phi_n^j \|_{W^{2,2}(\mathcal{O}; \mathbb{R})}^2 \| \partial_j \phi_n^k \|_{L^2(\mathcal{O}; \mathbb{R})}^2 \\ &\le c \| BS_K \phi_n \|_{W^{2,2}}^2 \| \phi_n \|_{W^{1,2}}^2 \\ &\le c \| \phi_n \|_{W^{1,2}}^4 \\ &\le c \| \phi_n \|_1^4 \end{aligned} $$
using the Sobolev Embedding of $W^{2,2}(O; \mathbb{R}) \hookrightarrow L^{\infty}(O; \mathbb{R})$ and item (3) of
Theorem 4.2. Likewise observe that

$$
\begin{align*}
\| \mathcal{L}_{\phi_n} B S_K \phi_n \|^2 &\le c \sum_{j=1}^3 \sum_{k=1}^3 \| \phi_n^j \|_{L^4(O; \mathbb{R})}^2 \| \partial_j B S_K \phi_n^k \|_{L^4(O; \mathbb{R})}^2 \\
&\le c \sum_{j=1}^3 \sum_{k=1}^3 \| \phi_n^j \|_{W^{1,2}(O; \mathbb{R})}^2 \| \partial_j B S_K \phi_n^k \|_{W^{1,2}(O; \mathbb{R})}^2 \\
&\le c \| \phi_n \|_{W^{1,2}}^2 \| B S_K \phi_n \|_{W^{2,2}}^2 \\
&\le c \| \phi_n \|_{W^{1,2}}^4 \\
&\le c \| \phi_n \|_1^4.
\end{align*}
$$

Summing these terms completes the proof.

□

**Proof of Lemma 4.7** We write out the left hand side in full:

$$
\begin{align*}
& \left| \langle \mathcal{P} \mathcal{L}_{BS_K} \phi \phi - \mathcal{P} \mathcal{L}_{BS_K} \psi \psi, \phi - \psi \rangle \right| \\
&= \left| \langle \mathcal{L}_{BS_K} \phi \phi - \mathcal{L}_{\phi} B S_K \phi - \mathcal{L}_{BS_K} \psi \psi + \mathcal{L}_{\psi} B S_K \psi, \phi - \psi \rangle \right| \\
&= \left| \begin{aligned}[t]
& \langle \mathcal{L}_{BS_K} \phi - B S_K \psi \phi + \mathcal{L}_{BS_K} \psi (\phi - \psi) - \mathcal{L}_{\phi-\psi} B S_K \phi \\
& \quad - \mathcal{L}_{\psi} (B S_K \phi - B S_K \psi), \phi - \psi \rangle
\end{aligned} \right| \\
\end{align*}
$$

from which we shall split up the terms and control them individually. Firstly,

$$
\begin{align*}
|\langle \mathcal{L}_{BS_K \phi - BS_K \psi} \phi, \phi - \psi \rangle| &\le \| \mathcal{L}_{BS_K \phi - BS_K \psi} \phi \| \| \phi - \psi \| \\
&\le c \| BS_K (\phi - \psi) \|_2 \| \phi \|_1 \| \phi - \psi \| \\
&\le c \| \phi - \psi \|_1 \| \phi \|_1 \| \phi - \psi \| \\
&\le c(\epsilon) \| \phi \|_1^2 \| \phi - \psi \|^2 + \frac{\epsilon}{3} \| \phi - \psi \|_1^2
\end{align*}
$$

using (4) and that $[BS_K\phi - BS_K\psi](x) = \int_O K(x, y)[\phi - \psi](y)dy$ is the solution
specified by $BS_K(\phi - \psi)$ in Theorem 4.2 for $\phi - \psi$. Even more directly we have
that

$$
\langle \mathcal{L}_{BS_K} \psi (\phi - \psi), \phi - \psi \rangle = 0
$$

owing to (15), and for the final two terms the bounds

$$
|\langle \mathcal{L}_{\phi-\psi} B S_K \phi, \phi - \psi \rangle| \le c \|\phi - \psi\|_1 \|B S_K \phi\|_2 \|\phi - \psi\|
$$
$$ \leq c(\epsilon) \|\phi\|_1^2 \|\phi - \psi\|^2 + \frac{\epsilon}{3} \|\phi - \psi\|_1^2 $$

and

$$ \begin{aligned} |\langle \mathcal{L}_{\psi}(BS_K\phi - BS_K\psi), \phi - \psi \rangle| &\leq c\|\psi\|_1 \|BS_K\phi - BS_K\psi\|_2 \|\phi - \psi\| \\ &\leq c(\epsilon)\|\psi\|_1^2 \|\phi - \psi\|^2 + \frac{\epsilon}{3}\|\phi - \psi\|_1^2. \end{aligned} $$

Summing these terms concludes the proof. □

## 5.2 A Conversion from Stratonovich to Itô

This theory is taken from [30] Subsections 2.2 and 2.3, and is provided here for simplicity to apply in Sect. 3.3. We work with a quartet of embedded Hilbert Spaces

$$ V \hookrightarrow H \hookrightarrow U \hookrightarrow X $$

where the embedding is assumed as a continuous linear injection. We start from an SPDE

$$ \Psi_t = \Psi_0 + \int_0^t Q \Psi_s ds + \int_0^t G \Psi_s \circ dW_s. \quad (68) $$

where the mappings $Q, G$ satisfy the following conditions, with the general operator $\tilde{K}: H \to \mathbb{R}$ defined by

$$ \tilde{K}(\phi) := c (1 + \|\phi\|_U^p + \|\phi\|_H^q) $$

for any constants $c, p, q$ independent of $\phi$.

**Assumption 5.3** $Q: V \to U$ is measurable and for any $\phi \in V$,

$$ \|Q\phi\|_U \leq \tilde{K}(\phi)[1 + \|\phi\|_V^2]. $$

**Assumption 5.4** $G$ is understood as a measurable mapping

$$ G: V \to \mathcal{L}^2(\mathcal{U}; H), \quad G: H \to \mathcal{L}^2(\mathcal{U}; U), \quad G: U \to \mathcal{L}^2(\mathcal{U}; X) $$

defined over $\mathcal{U}$ by its action on the basis vectors

$$ G(\cdot, e_i) := G_i(\cdot). $$
In addition each $\mathcal{G}_i$ is linear and there exists constants $c_i$ such that for all $\phi \in V$, $\psi \in H$, $\eta \in U$:

$$ \|\mathcal{G}_i\phi\|_H \le c_i\|\phi\|_V, \quad \|\mathcal{G}_i\psi\|_U \le c_i\|\psi\|_H, \quad \|\mathcal{G}_i\eta\|_X \le c_i\|\eta\|_U, \quad \sum_{i=1}^{\infty} c_i^2 < \infty. $$

In this setting, we have the following result ([30] Theorem 2.3.1 and Corollary 2.3.1.1).

**Theorem 5.5** Suppose that $(\Psi, \tau)$ are such that: $\tau$ is a $\mathbb{P}$-a.s. positive stopping time and $\Psi$ is a process whereby for $\mathbb{P}$-a.e. $\omega, \Psi.(\omega) \in C([0, T]; H)$ and $\Psi.(\omega)1_{\le \tau(\omega)} \in L^2([0, T]; V)$ for all $T > 0$ with $\Psi.1_{\le \tau}$ progressively measurable in $V$, and moreover satisfy the identity

$$ \Psi_t = \Psi_0 + \int_0^{t \wedge \tau} \left( Q + \frac{1}{2} \sum_{i=1}^{\infty} \mathcal{G}_i^2 \right) \Psi_s ds + \int_0^{t \wedge \tau} \mathcal{G} \Psi_s dW_s $$

$\mathbb{P}$-a.s. in $U$ for all $t \ge 0$. Then the pair $(\Psi, \tau)$ satisfies the identity

$$ \Psi_t = \Psi_0 + \int_0^{t \wedge \tau} Q \Psi_s ds + \int_0^{t \wedge \tau} \mathcal{G} \Psi_s \circ dW_s $$

$\mathbb{P}$-a.s. in $X$ for all $t \ge 0$.

The mapping $\frac{1}{2} \sum_{i=1}^{\infty} \mathcal{G}_i^2$ is understood as a pointwise limit, which is justified in [30] Subsection 2.3.

**Remark 6** Practically, Theorem 5.5 provides an Itô equation from a Stratonovich one in the sense that solving this Itô equation is sufficient to satisfy the identity in Stratonovich form. To discuss an equivalence between the equations we would need to formally define a solution of the Stratonovich equation, which we do not do here. To make sense of the Stratonovich integral one would have to impose that the solution is a local semimartingale in $U$, in which case the two notions are genuinely equivalent.

## 5.3 Abstract Solution Criterion I

The result is given in the context of an Itô SPDE

$$ \Psi_t = \Psi_0 + \int_0^t \mathcal{A}(s, \Psi_s) ds + \int_0^t \mathcal{G}(s, \Psi_s) dW_s. \quad (69) $$

We state the assumptions for a triplet of embedded Hilbert Spaces
$$V \hookrightarrow H \hookrightarrow U$$

and ask that there is a continuous bilinear form $\langle \cdot, \cdot \rangle_{U \times V} : U \times V \to \mathbb{R}$ such that for $\phi \in H$ and $\psi \in V$,

$$\langle \phi, \psi \rangle_{U \times V} = \langle \phi, \psi \rangle_H. \qquad (70)$$

The mappings $\mathcal{A}, \mathcal{G}$ are such that for any $T > 0$, $\mathcal{A}: [0, T] \times V \to U, \mathcal{G}: [0, T] \times V \to \mathcal{L}^2(\Omega; H)$ are measurable. We assume that $V$ is dense in $H$ which is dense in $U$.

**Assumption 5.6** There exists a system $(a_n)$ of elements of $V$ such that, defining the spaces $V_n := \text{span}\{a_1, \dots, a_n\}$ and $\mathcal{P}_n$ as the orthogonal projection to $V_n$ in $U$, then:

1. There exists some constant $c$ independent of $n$ such that for all $\phi \in H$,

$$\|\mathcal{P}_n \phi\|_H^2 \le c \|\phi\|_H^2. \qquad (71)$$

2. There exists a real valued sequence $(\mu_n)$ with $\mu_n \to \infty$ such that for any $\phi \in H$,

$$\|(I - \mathcal{P}_n)\phi\|_U \le \frac{1}{\mu_n} \|\phi\|_H \qquad (72)$$

where $I$ represents the identity operator in $U$.

These conditions are of course supplemented by a series of assumptions on the mappings. We shall use general notation $c_t$ to represent a function $c: [0, \infty) \to \mathbb{R}$ bounded on $[0, T]$ for any $T > 0$, evaluated at the time $t$. Moreover we define functions $K, \tilde{K}$ relative to some non-negative constants $p, \tilde{p}, q, \tilde{q}$. We use a generic notation to define the functions $K: U \to \mathbb{R}, K: U \times U \to \mathbb{R}, \tilde{K}: H \to \mathbb{R}$ and $\tilde{K}: H \times H \to \mathbb{R}$ by

$$K(\phi) := 1 + \|\phi\|_U^p, \quad K(\phi, \psi) := 1 + \|\phi\|_U^p + \|\psi\|_U^q,$$

$$\tilde{K}(\phi) := K(\phi) + \|\phi\|_H^{\tilde{p}}, \quad \tilde{K}(\phi, \psi) := K(\phi, \psi) + \|\phi\|_H^{\tilde{p}} + \|\psi\|_H^{\tilde{q}}$$

Distinct use of the function $K$ will depend on different constants but in no meaningful way in our applications, hence no explicit reference to them shall be made. In the case of $\tilde{K}$, when $\tilde{p}, \tilde{q} = 2$ then we shall denote the general $\tilde{K}$ by $\tilde{K}_2$. In this case no further assumptions are made on the $p, q$. That is, $\tilde{K}_2$ has the general representation

$$\tilde{K}_2(\phi, \psi) = K(\phi, \psi) + \|\phi\|_H^2 + \|\psi\|_H^2 \qquad (73)$$

and similarly as a function of one variable.
We state the subsequent assumptions for arbitrary elements $\phi, \psi \in V$, $\phi^n \in V_n$, $\eta \in H$ and $t \in [0, \infty)$, and a fixed $\kappa > 0$. Understanding $\mathcal{G}$ as a mapping $\mathcal{G}: [0, \infty) \times V \times \mathfrak{U} \to H$, we introduce the notation $\mathcal{G}_i(\cdot, \cdot) := \mathcal{G}(\cdot, \cdot, e_i)$.

**Assumption 5.7**

$$ \|\mathcal{A}(t, \boldsymbol{\phi})\|_{U}^{2} + \sum_{i=1}^{\infty} \|\mathcal{G}_{i}(t, \boldsymbol{\phi})\|_{H}^{2} \leq c_{t} K(\boldsymbol{\phi}) [1 + \|\boldsymbol{\phi}\|_{V}^{2}], \quad (74) $$

$$ \|\mathcal{A}(t, \boldsymbol{\phi}) - \mathcal{A}(t, \boldsymbol{\psi})\|_{U}^{2} \leq c_{t} \left[ K(\phi, \psi) + \|\phi\|_{V}^{2} + \|\psi\|_{V}^{2} \right] \|\phi - \psi\|_{V}^{2}, \quad (75) $$

$$ \sum_{i=1}^{\infty} \|\mathcal{G}_i(t, \boldsymbol{\phi}) - \mathcal{G}_i(t, \boldsymbol{\psi})\|_{U}^{2} \leq c_t K(\phi, \psi) \|\phi - \psi\|_{H}^{2}. \quad (76) $$

**Assumption 5.8**

$$ 2\langle \mathcal{P}_n \mathcal{A}(t, \boldsymbol{\phi}^n), \boldsymbol{\phi}^n \rangle_H + \sum_{i=1}^{\infty} \|\mathcal{P}_n \mathcal{G}_i(t, \boldsymbol{\phi}^n)\|_H^2 \leq c_t \tilde{K}_2(\boldsymbol{\phi}^n) [1 + \|\boldsymbol{\phi}^n\|_H^2] - \kappa \|\boldsymbol{\phi}^n\|_V^2, \quad (77) $$

$$ \sum_{i=1}^{\infty} \langle \mathcal{P}_n \mathcal{G}_i(t, \boldsymbol{\phi}^n), \boldsymbol{\phi}^n \rangle_H^2 \leq c_t \tilde{K}_2(\boldsymbol{\phi}^n) [1 + \|\boldsymbol{\phi}^n\|_H^4]. \quad (78) $$

**Assumption 5.9**

$$ \begin{aligned} & 2\langle \mathcal{A}(t, \boldsymbol{\phi}) - \mathcal{A}(t, \boldsymbol{\psi}), \boldsymbol{\phi} - \boldsymbol{\psi} \rangle_U + \sum_{i=1}^{\infty} \|\mathcal{G}_i(t, \boldsymbol{\phi}) - \mathcal{G}_i(t, \boldsymbol{\psi})\|_U^2 \\ & \leq c_t \tilde{K}_2(\boldsymbol{\phi}, \boldsymbol{\psi}) \|\boldsymbol{\phi} - \boldsymbol{\psi}\|_U^2 - \kappa \|\boldsymbol{\phi} - \boldsymbol{\psi}\|_H^2, \end{aligned} \quad (79) $$

$$ \sum_{i=1}^{\infty} \langle \mathcal{G}_i(t, \boldsymbol{\phi}) - \mathcal{G}_i(t, \boldsymbol{\psi}), \boldsymbol{\phi} - \boldsymbol{\psi} \rangle_U^2 \leq c_t \tilde{K}_2(\boldsymbol{\phi}, \boldsymbol{\psi}) \|\boldsymbol{\phi} - \boldsymbol{\psi}\|_U^4. \quad (80) $$

**Assumption 5.10**

$$ 2\langle \mathcal{A}(t, \boldsymbol{\phi}), \boldsymbol{\phi} \rangle_U + \sum_{i=1}^{\infty} \|\mathcal{G}_i(t, \boldsymbol{\phi})\|_U^2 \leq c_t K(\boldsymbol{\phi}) [1 + \|\boldsymbol{\phi}\|_H^2], \quad (81) $$

$$ \sum_{i=1}^{\infty} \langle \mathcal{G}_i(t, \boldsymbol{\phi}), \boldsymbol{\phi} \rangle_U^2 \leq c_t K(\boldsymbol{\phi}) [1 + \|\boldsymbol{\phi}\|_H^4]. \quad (82) $$
**Assumption 5.11**

$$ \langle \mathcal{A}(t, \phi) - \mathcal{A}(t, \psi), \eta \rangle_U \le c_t (1 + \|\eta\|_H) [K(\phi, \psi) + \|\phi\|_V + \|\psi\|_V] \|\phi - \psi\|_H. \quad (83) $$

With these assumptions in place we state the relevant definitions and results, first announced in [31] and proven in [32]. Definition 5.12 is stated for an $\mathcal{F}_0$-measurable $\Psi_0 : \Omega \to H$.

**Definition 5.12** A pair $(\Psi, \tau)$ where $\tau$ is a $\mathbb{P}-a.s.$ positive stopping time and $\Psi$ is a process such that for $\mathbb{P}-a.e. \omega, \Psi.(\omega) \in C([0, T]; H)$ and $\Psi.(\omega)1_{\le \tau(\omega)} \in L^2([0, T]; V)$ for all $T > 0$ with $\Psi.1_{\le \tau}$ progressively measurable in $V$, is said to be an $H$-valued local strong solution of the Eq. (69) if the identity

$$ \Psi_t = \Psi_0 + \int_0^{t \wedge \tau} \mathcal{A}(s, \Psi_s)ds + \int_0^{t \wedge \tau} \mathcal{G}(s, \Psi_s)dW_s \quad (84) $$

holds $\mathbb{P}-a.s.$ in $U$ for all $t \ge 0$.

**Definition 5.13** A pair $(\Psi, \Theta)$ such that there exists a sequence of stopping times $(\theta_j)$ which are $\mathbb{P}-a.s.$ monotone increasing and convergent to $\Theta$, whereby $(\Psi_{\wedge \theta_j}, \theta_j)$ is a $V$-valued local strong solution of the Eq. (69) for each $j$, is said to be an $H$-valued maximal strong solution of the Eq. (69) if for any other pair $(\Phi, \Gamma)$ with this property then $\Theta \le \Gamma \mathbb{P}-a.s.$ implies $\Theta = \Gamma \mathbb{P}-a.s.$.

**Definition 5.14** An $H$-valued maximal strong solution $(\Psi, \Theta)$ of the equation (69) is said to be unique if for any other such solution $(\Phi, \Gamma)$, then $\Theta = \Gamma \mathbb{P}-a.s.$ and for all $t \in [0, \Theta)$,

$$ \mathbb{P}(\{\omega \in \Omega : \Psi_t(\omega) = \Phi_t(\omega)\}) = 1. $$

We now state the main theorem in this setting.

**Theorem 5.15** Suppose that Assumptions 5.6–5.11 are satisfied in this framework. Then for any given $\mathcal{F}_0$-measurable $\Psi_0 : \Omega \to H$, there exists a unique $H$-valued maximal strong solution $(\Psi, \Theta)$ of the equation (69). Moreover at $\mathbb{P}-a.e. \omega$ for which $\Theta(\omega) < \infty$, we have that

$$ \sup_{r \in [0, \Theta(\omega))} \|\Psi_r(\omega)\|_H^2 + \int_0^{\Theta(\omega)} \|\Psi_r(\omega)\|_V^2 dr = \infty. \quad (85) $$

**Proof** See [32] Theorem 3.15. □
## 5.4 Abstract Solution Criterion II

We extend the framework of Sect. 5.3, introducing now another Hilbert Space $X$ which is such that $U \hookrightarrow X$. We ask that there is a continuous bilinear form $\langle \cdot, \cdot \rangle_{X \times H} : X \times H \to \mathbb{R}$ such that for $\phi \in U$ and $\psi \in H$,

$$ \langle \phi, \psi \rangle_{X \times H} = \langle \phi, \psi \rangle_U. \qquad (86) $$

Moreover it is now necessary that the system $(a_n)$ from Assumption 5.6 forms an orthogonal basis of $U$. We state the remaining assumptions now for arbitrary elements $\phi, \psi \in H$ and $t \in [0, \infty)$, and continue to use the $c, K, \tilde{K}, \kappa$ notation of Assumption Set 1. We now further assume that for any $T > 0$, $\mathcal{A} : [0, T] \times H \to X$ and $\mathcal{G} : [0, T] \times H \to \mathcal{L}^2(\Omega; U)$ are measurable.

**Assumption 5.16**

$$ \|\mathcal{A}(t, \boldsymbol{\phi})\|_{X}^{2} + \sum_{i=1}^{\infty} \|\mathcal{G}_{i}(t, \boldsymbol{\phi})\|_{U}^{2} \leq c_{t} K(\boldsymbol{\phi}) \left[1 + \|\boldsymbol{\phi}\|_{H}^{2}\right], \qquad (87) $$

$$ \|\mathcal{A}(t, \boldsymbol{\phi}) - \mathcal{A}(t, \boldsymbol{\psi})\|_{X} \leq c_t [K(\phi, \psi) + \|\phi\|_{H} + \|\psi\|_{H}] \|\phi - \psi\|_{H} \qquad (88) $$

**Assumption 5.17**

$$ 2\langle \mathcal{A}(t, \boldsymbol{\phi}) - \mathcal{A}(t, \boldsymbol{\psi}), \boldsymbol{\phi} - \boldsymbol{\psi} \rangle_{X} + \sum_{i=1}^{\infty} \|\mathcal{G}_{i}(t, \boldsymbol{\phi}) - \mathcal{G}_{i}(t, \boldsymbol{\psi})\|_{X}^{2} \leq c_{t} \tilde{K}_{2}(\boldsymbol{\phi}, \boldsymbol{\psi}) \|\boldsymbol{\phi} - \boldsymbol{\psi}\|_{X}^{2}, \qquad (89) $$

$$ \sum_{i=1}^{\infty} \langle \mathcal{G}_i(t, \boldsymbol{\phi}) - \mathcal{G}_i(t, \boldsymbol{\psi}), \boldsymbol{\phi} - \boldsymbol{\psi} \rangle_X^2 \leq c_t \tilde{K}_2(\boldsymbol{\phi}, \boldsymbol{\psi}) \|\boldsymbol{\phi} - \boldsymbol{\psi}\|_X^4 \qquad (90) $$

**Assumption 5.18** For every $\phi \in V$, it holds that

$$ 2\langle \mathcal{A}(t, \boldsymbol{\phi}), \boldsymbol{\phi} \rangle_U + \sum_{i=1}^{\infty} \|\mathcal{G}_i(t, \boldsymbol{\phi})\|_U^2 \leq c_t K(\boldsymbol{\phi}) - \kappa \|\boldsymbol{\phi}\|_H^2, \qquad (91) $$

$$ \sum_{i=1}^{\infty} \langle \mathcal{G}_i(t, \boldsymbol{\phi}), \boldsymbol{\phi} \rangle_U^2 \leq c_t K(\boldsymbol{\phi}). \qquad (92) $$

**Remark 7** Note that Assumption 5.18 is stronger than Assumption 5.10, as we are bounding the same terms but we are not afforded a control in the $H$ norm of $\phi$ in addition to its $U$ norm. Thus in applying Theorem 5.22 it is sufficient to only demonstrate Assumption 5.18.
Analogously to Susbsection 5.3, we state the relevant definitions and the resulting theorem in this context (again proved in [32]). Definition 5.19 is stated for an $\mathcal{F}_0$-measurable $\Psi_0 : \Omega \to U$.

**Definition 5.19** A pair $(\Psi, \tau)$ where $\tau$ is a $\mathbb{P}-a.s.$ positive stopping time and $\Psi$ is a process such that for $\mathbb{P}-a.e. \omega, \Psi.(\omega) \in C([0, T]; U)$ and $\Psi.(\omega)1_{\le \tau(\omega)} \in L^2([0, T]; H)$ for all $T > 0$ with $\Psi.1_{\le \tau}$ progressively measurable in $H$, is said to be a $U$-valued local strong solution of the Eq. (69) if the identity

$$ \Psi_t = \Psi_0 + \int_0^{t \wedge \tau} \mathcal{A}(s, \Psi_s)ds + \int_0^{t \wedge \tau} \mathcal{G}(s, \Psi_s)dW_s \quad (93) $$

holds $\mathbb{P}-a.s.$ in $X$ for all $t \ge 0$.

**Definition 5.20** A pair $(\Psi, \Theta)$ such that there exists a sequence of stopping times $(\theta_j)$ which are $\mathbb{P}-a.s.$ monotone increasing and convergent to $\Theta$, whereby $(\Psi_{\wedge \theta_j}, \theta_j)$ is a $U$-valued local strong solution of the Eq. (69) for each $j$, is said to be an $H$-valued maximal strong solution of the Eq. (69) if for any other pair $(\Phi, \Gamma)$ with this property then $\Theta \le \Gamma \mathbb{P}-a.s.$ implies $\Theta = \Gamma \mathbb{P}-a.s.$.

**Definition 5.21** A $U$-valued maximal strong solution $(\Psi, \Theta)$ of the Eq. (69) is said to be unique if for any other such solution $(\Phi, \Gamma)$, then $\Theta = \Gamma \mathbb{P}-a.s.$ and for all $t \in [0, \Theta)$,

$$ \mathbb{P}(\{\omega \in \Omega : \Psi_t(\omega) = \Phi_t(\omega)\}) = 1. $$

**Theorem 5.22** Suppose that Assumptions 5.6–5.11 and 5.16–5.18 are satisfied in this framework. Then for any given $\mathcal{F}_0$-measurable $\Psi_0 : \Omega \to U$, there exists a unique $U$-valued maximal strong solution $(\Psi, \Theta)$ of the Eq. (69). Moreover at $\mathbb{P}-a.e. \omega$ for which $\Theta(\omega) < \infty$, we have that

$$ \sup_{r \in [0, \Theta(\omega))] \|\Psi_r(\omega)\|_U^2 + \int_0^{\Theta(\omega)} \|\Psi_r(\omega)\|_H^2 dr = \infty. \quad (94) $$

**Proof** See [32] Theorem 4.9. □

**Acknowledgments** Daniel Goodair was supported by the Engineering and Physical Sciences Research Council (EPSRC) Project 2478902. Dan Crisan was partially supported by the European Research Council (ERC) under the European Union's Horizon 2020 Research and Innovation Programme (ERC, Grant Agreement No 856408).

We would like to express our sincere gratitude to the anonymous reviewers of this paper, whose detailed and insightful comments have greatly contributed to this publication.
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<figure><img src="image_35.png" /></figure>
# On the Interactions Between Mean Flows and Inertial Gravity Waves in the WKB Approximation

<figure><img src="image_36.png" /></figure>

Check for
updates

Darryl D. Holm, Ruiao Hu, and Oliver D. Street

**Abstract** We derive a Wentzel-Kramers-Brillouin (WKB) closure of the generalised Lagrangian mean (GLM) theory by using a phase-averaged Hamilton variational principle for the Euler-Boussinesq (EB) equations. Following Gjaja and Holm 1996, we consider 3D inertial gravity waves (IGWs) in the EB approximation. The GLM closure for WKB IGWs expresses EB wave mean flow interaction (WMFI) as WKB wave motion boosted into the reference frame of the EB equations for the Lagrangian mean transport velocity. We provide both deterministic and stochastic closure models for GLM IGWs at leading order in 3D complex vector WKB wave asymptotics. This paper brings the Gjaja and Holm 1996 paper at leading order in wave amplitude asymptotics into an easily understood short form and proposes a stochastic generalisation of the WMFI equations for IGWs.

## 1 Introduction

Inertial gravity waves (IGWs), also known as internal waves, comprise a classical form of wave disturbances in fluid motions under gravity that propagate in three-dimensional stratified, rotating, incompressible fluid and involve nonlinear dynamics among inertia, buoyancy, pressure gradients and Coriolis forces [23, 17, 24].

**Satellite Images and Field Data** Satellite Synthetic Aperture Radar (SAR) is a powerful sensor for ocean remote sensing, because of its continuous capabilities and high spatial resolution. The spatial resolution of the state-of-the-art satellite SAR images reaches 20–30 m, and the swath width reaches 100–450 km. Figure 1 shows a typical representation of the range of SAR field data and Fig. 2 shows a typical SAR image of IGWs on the ocean surface.

D. D. Holm · R. Hu · O. D. Street (✉)

Department of Mathematics, Imperial College London, London, UK

e-mail: d.holm@imperial.ac.uk; ruiao.hu15@imperial.ac.uk; o.street18@imperial.ac.uk
<figure><img src="image_37.png" /></figure>

**Fig. 1** The distribution of observed IGW packets and bathymetry in the South China Sea courtesy of [25]. Bold lines represent crest lines of leading waves in IGW packets interpreted from SAR images. The rectangular box on the right of this figure outlines the IGW generation source region. Looking closely near the center of this figure, one sees the crescent shape of the Dongsha atoll whose diameter is 25 km. Details of SAR images of waves near Dongsha atoll are shown in Fig. 2

**Fig. 2** A satellite image showing the strong surface signatures of IGWs in the South China Sea near Dongsha atoll. Notice also the pronounced roughness of the surface due to surface gravity waves through which the IGW surface signatures propagate. For discussion of other observations, see, e.g., [16]

<figure><img src="image_38.png" /></figure>

**Theoretical Basis of the Present Work** The paper [5] derived a hierarchy of approximate models of wave mean-flow interaction (WMFI) for IGWs by using asymptotic expansions and phase averages. Two different derivations of the same WMFI IGW equations were given. The first derivation was based on Fourier projections of the Euler-Boussinesq equations for a stratified rotating inviscid incompressible fluid. The second derivation was based on Hamilton's principle for these equations. Two small dimensionless parameters were used in the asymptotic
expansions. One small parameter was the ratio of time scales between internal waves at most wavenumbers and the mesoscale mean flow of the fluid. This “adiabatic ratio” is small and is comparable to the corresponding ratio of space scales for the class of initial conditions that support internal waves. The other small parameter used in the asymptotic expansions was the ratio of the amplitude of the internal wave to its wavelength. An application of Noether’s theorem to the phase-averaged Hamilton’s principle showed that the resulting equations conserve the wave action, convect a potential vorticity and can, depending on the order of approximation, convect wave angular momentum. Legendre transforming from the phase-averaged Hamilton’s principle to the Hamiltonian formulation brought the WMFI theory into the Lie-Poisson framework in which formal and nonlinear stability analysis methods are available [15]. The Hamiltonian framework also revealed an analogy between the two-fluid model of the interaction of waves and mean flow with the interaction of the superfluid and normal fluid components of liquid He⁴ without vortices. The relations to similar results for the Charney-Drazin non-acceleration theorem, Whitham averaging, WKB stability theory, Craik-Leibovich theory of Langmuir circulations as well as the generalised Lagrangian-mean (GLM) fluid equations for prescribed wave displacements were also discussed in [5].

**Goal of the Present Work** Our goal here is to derive 3D IGW equations in the class of wave mean flow interaction (WMFI) derived in [5] as a mutual interaction of the mean fluid flow and the slowly varying envelope of fluctuation dynamics that is consistent with IGWs in the full 3D Euler–Boussinesq fluid flow. Physically, we take nonhydrostatic pressure effects on the wave dispersion relation into account and derive consistent nonlinear feedback effects of the internal of waves on the generation of fluid circulation based on a dynamic version of the well-known Craik-Leibovich theory of Langmuir circulation [3]. Mathematically, we introduce the two WMFI degrees of freedom by factorising the full 3D Euler–Boussinesq flow map into the composition of two smooth invertible maps in Hamilton’s principle for Eulerian fluid dynamics [14].

The present work derives 3D equations for wave mean flow interaction (WMFI) as WKB wave motion boosted into the reference frame of the fluid equations for the Lagrangian mean transport velocity. The final equations derived here are consistent with traditional approaches such as Craik-Leibovich (CL) theory [3] except that the Stokes drift velocity in the CL formulation has its own dynamics in the present formulation. The present formulation can also be considered as a WKB closure for the GLM approach [1], similar also to the oscillation centre ponderomotive closure in magnetohydrodynamics [18, 19]. Namely, the present formulation uses a combination of asymptotic expansion and phase resonance to close the GLM equations derived by the composition-of-maps approach and obtaining explicit formulas for the wave polarisation parameters and dispersion relation for the Doppler-shifted frequency.

Finally, the present work also formulates stochastic equations of motion for 3D WMFI dynamics, permitting a statistical representation of the uncertainty present in observational data of geophysical flows.
# 2 Deterministic 3D Euler-Boussinesq (EB) Internal Gravity Waves

## 2.1 Lagrangian Formulation of the WMFI Equations at Leading Order

**GLM Theory** The Generalised Lagrangian Mean (GLM) theory of wave mean flow interaction (WMFI) is derived in Andrews and McIntyre [1] by taking the time mean (·) at a fixed position x of the Eulerian fluid velocity, U(x, t), shifted to a rapidly fluctuating position, xξ := x + αξ(x, t) with constant scale factor α ≪ 1 and zero Eulerian mean ξ̄(x, t) = 0. The Lagrangian mean velocity uL(x, t) at Eulerian position x is then defined via the following calculation,

$$ U(x^\xi, t) := U(x + \alpha\xi(x, t), t) = u^L(x, t) + \alpha \frac{d}{dt} \xi(x, t) \\ \text{where } \overline{U(x + \alpha\xi(x, t))} =: u^L(x, t), \qquad (2.1) $$

with

$$ \frac{d}{dt} \xi(\mathbf{x}, t) = \partial_t \xi + (\mathbf{u}^L \cdot \nabla)\xi =: \mathbf{u}^\ell, \quad \overline{\mathbf{u}^\ell} = 0 \quad \text{and} \quad \overline{\mathbf{u}^L} = \mathbf{u}^L. \qquad (2.2) $$

Consequently, the Kelvin circulation integral for GLM in a rotating frame with constant Coriolis parameter 2Ω may be derived; see, e.g., [1, 5, 6, 7, 9, 10] and the appendix for details,

$$ \begin{align} I_{GLM}(\mathbf{u}^L) &= \oint_{C(u^L)} \overline{(\mathbf{u}(\mathbf{x}^\xi, t) + \mathbf{\Omega} \times \mathbf{x}^\xi) \cdot d\mathbf{x}^\xi} \nonumber \\ &= \oint_{C(u^L)} (\mathbf{u}^L(\mathbf{x}, t) + \mathbf{\Omega} \times \mathbf{x}) \cdot d\mathbf{x} \tag{2.3} \\ &\quad + \alpha^2 \overline{(\mathbf{\Omega} \times \xi(\mathbf{x}, t) + \mathbf{u}^\ell(\mathbf{x}, t)) \cdot d\xi(\mathbf{x}, t)}. \nonumber \end{align} $$

The Lagrangian transport velocity for GLM in (2.3) is indeed uL. However, the Eulerian momentum per unit mass in the integrand of the GLM circulation integral in (2.3) acquires an order O(α²) shift due to the mean effects of the quadratic nonlinearity in the last fluctuating displacement terms in (2.3).

**Choice of GLM Closure** Gjaja and Holm [5] studied the dynamics of 3D IGWs by closing the GLM theory for the case that the fluctuation displacement αξ(x, t) in (2.1) is given by a single-frequency travelling wave ℜ(a(εx, εt)eimφ(εx, εt)/ε) with slowly varying complex vector amplitude a(εx, t) and slowly varying, but rapid phase φ(εx, εt)/ε, with ε ≪ 1; so that the time averaged Lagrangian mean of the displacement field αξ(x, t) would be negligible.
We choose to represent the fluctuation displacement field $\xi(\mathbf{x}, t)$ in the following form

$$ \xi(\mathbf{x}, t) = \mathbf{a}(\epsilon\mathbf{x}, \epsilon t)e^{i\phi(\epsilon\mathbf{x}, \epsilon t)/\epsilon} + \mathbf{a}^*(\epsilon\mathbf{x}, \epsilon t)e^{-i\phi(\epsilon\mathbf{x}, \epsilon t)/\epsilon}, \quad (2.4) $$

and the total pressure decomposes into

$$ p(\mathbf{X}, t) = p_0(\mathbf{X}, t) + \sum_{j \ge 1} \alpha^j \left( b_j(\epsilon \mathbf{X}, \epsilon t) e^{ij\phi(\epsilon \mathbf{X}, \epsilon t)/\epsilon} + b_j^*(\epsilon \mathbf{X}, \epsilon t) e^{-ij\phi(\epsilon \mathbf{X}, \epsilon t)/\epsilon} \right). \quad (2.5) $$

Here the adiabatic parameter $\epsilon$ is defined as the ratio between space and time scales of the wave oscillations and mean flow respectively. Thus, quantities that are functions of $\mathbf{x}$ and $t$, for example $\xi(\mathbf{x}, t)$, have fast dependence on $\mathbf{x}$ and $t$. Likewise, quantities which are functions of $\epsilon\mathbf{x}$ and $\epsilon t$, for example $\mathbf{a}(\epsilon\mathbf{x}, \epsilon t)$, have slow dependence of the space and time coordinates. Thus, in the fluctuation displacement $\xi$ in (2.4), the fast phase dynamics is represented by $\exp(i\phi(\epsilon\mathbf{x}, \epsilon t)/\epsilon$ which is slowly modulated by the complex vector amplitude $\mathbf{a}(\epsilon\mathbf{x}, \epsilon t)$.

We will apply the GLM closure in Eqs. (2.4) and (2.5) to the 3D Euler-Boussinesq equations, which can be derived from Hamilton's principle with the following reduced Lagrangian

$$ 0 = \delta \int_{t_0}^{t_1} \int_{\mathcal{M}} \mathcal{D} \left( \frac{1}{2} |\mathbf{U}|^2 + \mathbf{U} \cdot \mathbf{\Omega} \times \mathbf{X} - g\rho Z \right) + p(1 - \mathcal{D}) d^3 X dt, \quad (2.6) $$

where $\mathcal{D}d^3X = d^3x_0 \in \text{Den}(\mathbb{R}^3)$ is the fluid density, $\rho \in \mathcal{F}(\mathbb{R}^3)$ is the fluid buoyancy, and $\mathcal{M}$ is the spatial domain. Substitution of (2.1), (2.4) and (2.5) into the Euler-Boussinesq Lagrangian in (2.6) followed by asymptotic expansion in $\alpha \ll 1$ and $\epsilon \ll 1$ at order $O(\alpha^2)$ neglecting corrections at orders $O(\alpha^2\epsilon)$ and $O(\alpha^4)$ and phase averaging (i.e., keeping coefficients of resonant phase factors only) produces the following wave mean flow interaction (WMFI) closure for Hamilton's principle in Eulerian fluid variables, which splits into the sum of the average mean-flow action $\bar{L}_{MF}$ and the average wave action $\bar{L}_W$, given by [5] and derived in the appendix as, cf. Eq. (A.8),

$$ \begin{aligned} 0 &= \delta(S_{MF} + S_W) = \delta \int_{t_0}^{t_1} (\bar{L}_{MF} + \alpha^2 \bar{L}_W) dt \\ &= \delta \int_{t_0}^{t_1} \int_{\mathcal{M}} D \left[ \frac{1}{2} |\mathbf{u}^L|^2 + \mathbf{u}^L \cdot \mathbf{\Omega} \times \mathbf{x} - \rho g z + \alpha^2 \tilde{\omega}^2 |\mathbf{a}|^2 + 2i \alpha^2 \tilde{\omega} \mathbf{\Omega} \cdot (\mathbf{a} \times \mathbf{a}^*) \right. \\ & \qquad \left. - \alpha^2 i (\mathbf{b}\mathbf{k} \cdot \mathbf{a}^* - \mathbf{b}^*\mathbf{k} \cdot \mathbf{a}) - \alpha^2 a_i^* a_j \frac{\partial^2 p_0}{\partial x_i \partial x_j} \right] + (1-D)p_0 + \mathcal{O}(\alpha^2 \epsilon) d^3 x dt. \end{aligned} \quad (2.7) $$
The averaged fluid quantities $\mathbf{u}^L(\epsilon\mathbf{x}, \epsilon t)$, $D(\epsilon\mathbf{x}, \epsilon t)$ and $\rho(\epsilon\mathbf{x}, \epsilon t)$ are defined to have slow dependence on $\mathbf{x}$ and $t$ in the averaging procedure. To see the construction of slow dependence from Lagrangian labels, see section (2.1) of Gjaja and Holm [5]. In the averaged wave Lagrangian $\bar{L}_W$, the wave vector and wave frequency are defined in terms of the wave phase $\phi(\epsilon\mathbf{x}, \epsilon t)$, as

$$ \mathbf{k}(\epsilon\mathbf{x}, \epsilon t) := \nabla_{\epsilon\mathbf{x}}\phi(\epsilon\mathbf{x}, \epsilon t) \quad \text{and} \quad \omega(\epsilon\mathbf{x}, \epsilon t) := -\frac{\partial}{\partial\epsilon t}\phi(\epsilon\mathbf{x}, \epsilon t). \qquad (2.8) $$

The Doppler-shifted oscillation frequency, $\tilde{\omega}$, due to the coupling to the mean flow $\mathbf{u}^L$ is defined through the advective time derivative $\frac{d}{d\epsilon t} := \frac{\partial}{\partial\epsilon t} + \mathbf{u}^L \cdot \nabla_{\epsilon\mathbf{x}}$ and the wave phase as

$$ \tilde{\omega} := -\frac{d}{d\epsilon t}\phi = -\left(\frac{\partial}{\partial\epsilon t}\phi + \mathbf{u}^L \cdot \nabla_{\epsilon\mathbf{x}}\phi\right) = \omega - \mathbf{u}^L \cdot \mathbf{k}. \qquad (2.9) $$

Upon introducing the Doppler-shifted oscillation $\tilde{\omega}$ into $\bar{L}_W$ in (2.7) and pairing its definition in (2.9) with a Lagrange multiplier, $N$, one arrives at the following variational principle

$$ \begin{aligned} 0 &= \delta(S_{MF} + S_W) = \delta \int_{t_0}^{t_1} (\bar{L}_{MF} + \alpha^2 \bar{L}_W) dt \\ &= \delta \int_{t_0}^{t_1} \int_{\mathcal{M}} D \left[ \frac{1}{2} |\mathbf{u}^L|^2 + \mathbf{u}^L \cdot \mathbf{\Omega} \times \mathbf{x} - \rho g z + \alpha^2 \tilde{\omega}^2 |\mathbf{a}|^2 + 2i \alpha^2 \tilde{\omega} \mathbf{\Omega} \cdot (\mathbf{a} \times \mathbf{a}^*) \right. \\ & \qquad \left. - \alpha^2 i (b \mathbf{k} \cdot \mathbf{a}^* - b^* \mathbf{k} \cdot \mathbf{a}) - \alpha^2 a_i^* a_j \frac{\partial^2 p_0}{\partial x_i \partial x_j} \right] + (1-D) p_0 d^3 x \\ & \qquad + \alpha^2 \left\langle N, -\frac{\partial}{\partial \epsilon t} \phi - \mathbf{u}^L \cdot \nabla_{\epsilon \mathbf{x}} \phi - \tilde{\omega} \right\rangle + \mathcal{O}(\alpha^2 \epsilon) dt. \end{aligned} \qquad (2.10) $$

Since, it may not be immediately clear how to take variations of the action (2.7). The inclusion of the Lagrange multiplier, $N$, imposes the relationship among the Doppler-shifted frequency $\tilde{\omega}$, the Lagrangian mean velocity $\mathbf{u}^L$, and the phase $\phi$, thereby facilitating the variations. Namely, the forms of the constrained variations of the velocity field $\mathbf{u}^L$ and its advected quantities, $D$ and $\rho$, are shown in (2.14). All other variables have arbitrary variations. The Euler-Poincaré theorem can then be applied to the variational derivatives with respect to $\mathbf{u}^L$, $D$, and $\rho$, to obtain an equation for the total momentum of the system, and stationarity of the action with respect to the remaining variables implies a collection of equations for the remaining dynamics. This procedure results in a closed system of equations for both waves and mean flow, and describes their mutual interaction. For Hamilton's principle of least action to apply to an asymptotically expanded action, we make use of the following definition to formalise the idea of Hamilton's principle to a given order, in the situation where the action is expanded asymptotically.
**Definition 2.1 (Variational Derivatives in an Asymptotically Expanded Action)**

When making an asymptotic expansion in Hamilton's principle, the Lagrangian in terms of any new variables, $ℓ(\mathbf{u}^L, D, \rho)$ for example, becomes an infinite sum. Variational derivatives are then defined under the integral up to some order, i.e.

$$ 
\begin{aligned}
0 = \delta S &= \delta \int \ell(\mathbf{u}^L, D, \rho) dt \\
&= \int \left\langle \frac{\delta \ell_{\alpha^2}}{\delta \mathbf{u}^L}, \delta \mathbf{u}^L \right\rangle + \left\langle \frac{\delta \ell_{\alpha^2}}{\delta D}, \delta D \right\rangle + \left\langle \frac{\delta \ell_{\alpha^2}}{\delta \rho}, \delta \rho \right\rangle + \mathcal{O}(\alpha^2 \epsilon),
\end{aligned} 
\quad (2.11) $$

where the truncated Lagrangian, $ℓ_{\alpha^2}$, is defined as the part of the Lagrangian which corresponds to these variations

$$ \ell(\mathbf{u}, D, \rho) = \ell_{\alpha^2}(\mathbf{u}, D, \rho) + H.O.T.. $$

Note that we have declined to use the 'big O' notation in the above equation, since $ℓ_{\alpha^2}$ is defined to include all terms of order less than $\alpha^2 \epsilon$ as well as any higher order terms which produce lower order terms after integrating by parts to take variational derivatives.

Hamilton's action principle (2.10) yields the following variations up to order $\mathcal{O}(\alpha^2)$

$$ 
\begin{aligned}
0 = \delta \int_{t_2}^{t_1} (\bar{L}_{MF} + \alpha^2 \bar{L}_W) dt \\
= \int_{t_1}^{t_2} \left\langle \delta \mathbf{u}^L, D\mathbf{u}^L + D\mathbf{\Omega} \times \mathbf{x} - \alpha^2 N \nabla_{\epsilon} \mathbf{x} \phi \right\rangle + \langle \delta\rho, -Dgz \rangle + \left\langle \delta b, -\alpha^2 i \mathbf{k} \cdot \mathbf{a}^* \right\rangle \\
+ \left\langle \delta b^*, \alpha^2 i \mathbf{k} \cdot \mathbf{a} \right\rangle \\
+ \left\langle \delta \mathbf{a}, \alpha^2 (D\tilde{\omega}^2 \mathbf{a}^* + 2i\tilde{\omega}\mathbf{a}^* \times \mathbf{\Omega} + ib^*\mathbf{k} - (\mathbf{a}^* \cdot \nabla)\nabla p_0) \right\rangle \\
+ \left\langle \delta \mathbf{a}^*, \alpha^2 (D\tilde{\omega}^2 \mathbf{a} + 2i\tilde{\omega}\mathbf{a} \times \mathbf{\Omega} - ib\mathbf{k} - (\mathbf{a} \cdot \nabla)\nabla p_0) \right\rangle \\
+ \left\langle \delta\tilde{\omega}, 2\alpha^2 D(\tilde{\omega}|\mathbf{a}|^2 + i\mathbf{\Omega} \cdot (\mathbf{a} \times \mathbf{a}^*)) - \alpha^2 N \right\rangle + \langle \delta D, \tilde{\omega} \rangle \\
+ \left\langle \delta N, -\frac{\partial}{\partial\epsilon} \phi - \mathbf{u}^L \cdot \nabla_{\epsilon} \mathbf{x} \phi - \tilde{\omega} \right\rangle \\
+ \left\langle \delta\phi, \frac{\partial}{\partial\epsilon} N + \operatorname{div}_{\epsilon}\mathbf{x}(\mathbf{u}^L N) + i\operatorname{div}_{\epsilon}\mathbf{x}(Db\mathbf{a}^* - Db^*\mathbf{a}) \right\rangle dt \\
+ \langle \delta p_0, 1 - D \rangle + \mathcal{O}(\alpha^2 \epsilon).
\end{aligned} 
\quad (2.12) $$
where we have

$$ 
\begin{aligned}
\bar{\omega} := & \delta(\bar{L}_{MF} + \alpha^2 \bar{L}_W)/\delta D \\
= & \frac{1}{2} |\mathbf{u}^L|^2 - \rho g z + \mathbf{u}^L \cdot \mathbf{\Omega} \times \mathbf{x} - p_0 \\
& + \alpha^2 \left( \tilde{\omega}^2 |\mathbf{a}|^2 + 2i \tilde{\omega} \mathbf{\Omega} \cdot (\mathbf{a} \times \mathbf{a}^*) - i(b \mathbf{k} \cdot \mathbf{a}^* - b^* \mathbf{k} \cdot \mathbf{a}) - a_i^* a_j \frac{\partial^2 p_0}{\partial x_i \partial x_j} \right).
\end{aligned} 
\quad (2.13) $$

The constrained variations in (2.12) take the Euler-Poincaré form [14]

$$ \delta \mathbf{u}^L = \frac{\partial}{\partial \epsilon t} \mathbf{v} + \mathbf{u}^L \cdot \nabla_{\epsilon \mathbf{x}} \mathbf{v} - \mathbf{v} \cdot \nabla_{\epsilon \mathbf{x}} \mathbf{u}^L, \quad \delta \rho = -\mathbf{v} \cdot \nabla_{\epsilon \mathbf{x}} \rho, \quad \delta D = -\operatorname{div}_{\epsilon \mathbf{x}} (\mathbf{v} D), \quad (2.14) $$

where the $\epsilon$ appears in the derivatives of the constrained variations due to their slow dependence on space and time. Note that when isolating the arbitrary variations, $\mathbf{v}$, through integration by parts, $\nabla_{\epsilon \mathbf{x}}$ does not generate higher order terms when operating on $\bar{\omega}$. From the constrained variations, one has that $\rho$ and $D$ are advected by the flow which then satisfies the following advection equations

$$ \frac{\partial}{\partial \epsilon t} D + \operatorname{div}_{\epsilon \mathbf{x}} (\mathbf{u}^L D) = 0, \quad \frac{\partial}{\partial \epsilon t} \rho + \mathbf{u}^L \cdot \nabla_{\epsilon \mathbf{x}} \rho = 0. \quad (2.15) $$

The total momentum of the mean and fluctuating parts of the flow is defined through the variational derivative w.r.t to $\mathbf{u}^L$, which is given by

$$ \mathbf{M} := D\mathbf{u}^L + D\mathbf{\Omega} \times \mathbf{x} - \alpha^2 N \nabla_{\epsilon\mathbf{x}} \phi, \quad (2.16) $$

which through the Euler-Poincaré theorem [14], satisfies the Euler-Poincaré equation

$$ \begin{aligned} & \frac{\partial}{\partial \epsilon t} \left( \frac{\mathbf{M}}{D} \right) - \mathbf{u}^L \times \operatorname{curl}_{\epsilon \mathbf{x}} \left( \frac{\mathbf{M}}{D} \right) + \nabla_{\epsilon \mathbf{x}} \left( \frac{1}{2} |\mathbf{u}^L|^2 + p_0 \right) + \frac{1}{\epsilon} g \rho \hat{\mathbf{z}} \\ & \qquad + \alpha^2 \nabla_{\epsilon \mathbf{x}} \left( -\omega \frac{N}{D} + \tilde{\omega}^2 |\mathbf{a}|^2 + a_i a_j^* \frac{\partial^2 p_0}{\partial x_i \partial x_j} \right) = 0, \end{aligned} \quad (2.17) $$

where $\hat{\mathbf{z}} := \nabla_{\epsilon \mathbf{x}} \mathbf{z}$. Stationarity of the sum of actions $S_{MF} + \alpha^2 S_W$ in (2.10) under variations with respect to the fluid variables ($\mathbf{u}^L$, $D$, $\rho$) has produced the equations for the mean flow, with order $O(\alpha^2)$ wave forcing which arises from the dependence of $\alpha^2 \bar{L}_W$ on the fluid variables $D$ and $\rho$. We note from the variation in $p_0$ that incompressibility of the Lagrangian mean velocity holds only within the asymptotic regime, and does not hold in an exact form. Indeed,
$$D = 1 - \alpha^2 \epsilon^2 \frac{\partial^2}{\partial \epsilon x_i \partial \epsilon x_j} (Da_i^* a_j) = 1 + \mathcal{O}(\alpha^2 \epsilon^2) \implies \mathrm{div}_{\epsilon \mathbf{X}}(\mathbf{u}^L) = \mathcal{O}(\alpha^2 \epsilon). \quad (2.18)$$

**Conservation of Wave Action Density** Keeping only resonant combinations in the Lagrangian $\bar{L}_W$ in (2.29) has eliminated any explicit dependence on the phase, $\phi$. Hence, a symmetry of the Lagrangian under constant phase shift, $\phi \to \phi + \phi_0$, has arisen. Consequently, one expects that Noether's theorem will yield a conservation law for the conjugate momentum $N$ under variations in $\phi$ of the average wave Lagrangian, $\bar{L}_W$. The arbitrary variation $\delta\tilde{\omega}$ in (2.12) reveals the definition of $N$ as

$$N := \frac{\delta \bar{L}_W}{\delta \tilde{\omega}} = 2D(\tilde{\omega}|\mathbf{a}|^2 + i\mathbf{\Omega} \cdot \mathbf{a} \times \mathbf{a}^*), \quad (2.19)$$

and the arbitrary variation $\delta\phi$ in (2.12) produces the following wave action conservation law,

$$\frac{\partial N}{\partial \epsilon t} + \mathrm{div}_{\epsilon \mathbf{X}}(N(\mathbf{u}^L + \mathbf{v}_G)) = 0, \quad \text{where} \quad \mathbf{v}_G := \frac{iD}{N}(\mathbf{a}^*b - \mathbf{a}b^*) = \frac{2D}{N}\Im(\mathbf{a}b^*). \quad (2.20)$$

Thus, the wave action $N$ is transported in an Eulerian frame by the sum of the Lagrangian mean velocity $\mathbf{u}^L$ and the group velocity of the waves, $\mathbf{v}_G$, defined above in (2.20). The evolution equation of $\phi$ in (2.9) can be written in terms of $N$ as follows

$$\frac{\partial}{\partial \epsilon t} \phi + \mathbf{u}^L \cdot \nabla_{\epsilon \mathbf{X}} \phi = \frac{1}{2D|\mathbf{a}|^2} (N - 2Di\mathbf{\Omega} \cdot \mathbf{a} \times \mathbf{a}^*), \quad (2.21)$$

thus removing the explicit dependence on $\tilde{\omega}$. The Eqs. (2.20) and (2.21) are in fact canonical Hamilton's equations boosted to the reference frame of the mean flow $\mathbf{u}^L$ which is discussed in Sect. 2.2.

**Remark 2.1 (Boundary Conditions for Integrations by Parts)** In taking variations of wave properties, we are not considering a free upper boundary. Instead, we have set

$$(\hat{\mathbf{n}} \cdot \delta\mathbf{a}^*)\mathbf{a} \cdot \frac{\partial p}{\partial \mathbf{x}} = 0 \quad \text{and} \quad \delta\phi \hat{\mathbf{n}} \cdot N(\mathbf{u}^L + \mathbf{v}_G) = 0, \quad (2.22)$$

on the boundary, when integrating by parts. This means that the displacement of the wave amplitude and the flux of wave action density are both taken to be tangential to the boundary.

Combining the evolution equation of wave action density $N$ (2.20) and wave phase $\phi$ (2.9), one has the evolution equation of the internal wave momentum $\mathbf{p}/D := \alpha^2 N \nabla_{\epsilon \mathbf{X}} \phi / D$.
$$ \frac{\partial}{\partial \epsilon t} \frac{\mathbf{p}}{D} - \mathbf{u}^L \times (\nabla_{\epsilon} \mathbf{x} \times \frac{\mathbf{p}}{D}) + \nabla_{\epsilon} \mathbf{x} (\mathbf{u}^L \cdot \frac{\mathbf{p}}{D}) = -\frac{\alpha^2}{D} (N \nabla \tilde{\omega} + \mathbf{k} \operatorname{div}_{\epsilon} \mathbf{x} (N \mathbf{v}_G)). \quad (2.23) $$

The Euler-Poincaré equations for the total momentum (2.17) and wave momentum (2.23) may be assembled into the Euler-Poincaré equation for the mean flow momentum, $\mathbf{m} = D\mathbf{u}^L + D\mathbf{\Omega} \times \mathbf{x}$. Dividing this through by the advected mass density, $D$, gives the following equation for $\mathbf{u}^L$

$$ \begin{aligned} & \frac{\partial}{\partial \epsilon t} \mathbf{u}^L - \mathbf{u}^L \times \operatorname{curl}_{\epsilon} \mathbf{x} (\mathbf{u}^L + \mathbf{\Omega} \times \mathbf{x}) + \nabla_{\epsilon} \mathbf{x} \left( \frac{1}{2} |\mathbf{u}^L|^2 + p_0 \right) + \frac{1}{\epsilon} g \rho \hat{\mathbf{z}} \\ & = -\alpha^2 \nabla_{\epsilon} \mathbf{x} \left( -\tilde{\omega} \frac{N}{D} + \tilde{\omega}^2 |\mathbf{a}|^2 + a_i \alpha_j^* \frac{\partial^2 p_0}{\partial x_i \partial x_j} \right) \\ & \quad - \frac{\alpha^2}{D} (N \nabla_{\epsilon} \mathbf{x} \tilde{\omega} + \mathbf{k} \operatorname{div}_{\epsilon} \mathbf{x} (N \mathbf{v}_G)). \end{aligned} \quad (2.24) $$

**Remark 2.2 (Hydrostatic and Geostrophic Balances)** As explained in section 2 of Gjaja and Holm [5], at leading order $O(1/\epsilon)$ the motion equation (2.24) establishes hydrostatic and geostrophic balances, namely

$$ 2\mathbf{\Omega} \times \mathbf{u}^L(\epsilon\mathbf{x}, \epsilon t) + g\rho(\mathbf{x}, \epsilon t)\hat{\mathbf{z}} + \frac{\partial p_0(\mathbf{x}, \epsilon t)}{\partial \mathbf{x}} = 0. \quad (2.25) $$

In order to provide the restoring force for internal waves, the advected relative density (or, buoyancy) $\rho(l^A(\mathbf{x}, t))$ must have one derivative of order $O(1)$ with respect to the vertical coordinate $z$. In order to contribute to the wave component of the pressure gradient at order $O(\alpha^2)$ in the motion equation (2.24), the mean pressure $p_0$ must have two derivatives of order $O(1)$ with respect to the vertical coordinate $z$.

**Remark 2.3 (Kelvin's Circulation Theorem for WMFI)** The two Euler-Poincaré equations for the total momentum $\mathbf{M}$ and mean flow momentum $\mathbf{m}$ readily implies their respective Kelvin-circulation theorems. Namely, for the mean flow momentum $\mathbf{m}$, (2.24) implies the following

$$ \begin{aligned} & \frac{d}{d\epsilon t} \oint_{c(\mathbf{u}^L)} (\mathbf{u}^L + \mathbf{\Omega} \times \mathbf{x}) \cdot d\mathbf{x} + \oint_{c(\mathbf{u}^L)} \frac{1}{\epsilon} \rho g \hat{\mathbf{z}} \cdot d\mathbf{x} \\ & = -\alpha^2 \oint_{c(\mathbf{u}^L)} D^{-1} (N \nabla_{\epsilon} \mathbf{x} \tilde{\omega} + \mathbf{k} \operatorname{div}_{\epsilon} \mathbf{x} (N \mathbf{v}_G)) \cdot d\mathbf{x}, \end{aligned} \quad (2.26) $$

in which one notes that the internal wave terms contribute to the creation of circulation of the mean flow at order $O(\alpha^2)$. For the total momentum $\mathbf{M}$, Eq. (2.17) implies that
$$ \frac{d}{d\epsilon t} \oint_{c(\mathbf{u}^L)} (\mathbf{u}^L + \mathbf{\Omega} \times \mathbf{x} - \alpha^2 D^{-1} N \mathbf{k}) \cdot d\mathbf{x} + \oint_{c(\mathbf{u}^L)} \frac{1}{\epsilon} \rho g \hat{\mathbf{z}} \cdot d\mathbf{x} = 0. \quad (2.27) $$

Thus, just as for the introduction of Stokes drift velocity into the integrand of Kelvin's circulation theorem in Craik-Leibovich theory [3], one may regard the additional non-inertial force of the internal waves on the mean flow circulation as arising from a shift in the momentum per unit mass in the Kelvin circulation integrand, performed to include the internal wave degree of freedom.

**Legendre Transforming Wave Lagrangian $\tilde{L}_W$ Into Canonical Phase Space Variables** By using the definitions of $N$ and $\tilde{\omega}$, one can compute the Legendre transform of $\tilde{L}_W$ to obtain the following WMFI Hamiltonian $\tilde{H}_W$,

$$ \begin{aligned} \tilde{H}_W := \langle N, \tilde{\omega} \rangle - \tilde{L}_W &= \int_{\mathcal{M}} D (\tilde{\omega}^2 |\mathbf{a}|^2 + i (b\mathbf{k} \cdot \mathbf{a}^* - b^*\mathbf{k} \cdot \mathbf{a}) \\ & \qquad + a_i^* a_j \frac{\partial^2 p_0}{\partial x_i \partial x_j}) \, d^3 x \\ &= \int_{\mathcal{M}} \frac{1}{4D|\mathbf{a}|^2} (N - 2i D \mathbf{\Omega} \cdot \mathbf{a} \times \mathbf{a}^*)^2 + i D (b\mathbf{k} \cdot \mathbf{a}^* - b^*\mathbf{k} \cdot \mathbf{a}) \\ & \qquad + D a_i^* a_j \frac{\partial^2 p_0}{\partial x_i \partial x_j} \, d^3 x, \end{aligned} \quad (2.28) $$

where we have dropped the dependence on higher order terms $O(\alpha^2 \epsilon, \alpha^4)$ in the asymptotic expansion. Inserting the expression (2.28) for $\tilde{H}_W$ into (2.10) yields the phase space expression of $\tilde{L}_W$ as

$$ \begin{aligned} \tilde{L}_W = & \int_{\mathcal{M}} -N \left( \frac{\partial \phi}{\partial \epsilon t} + \mathbf{u}^L \cdot \nabla_{\epsilon} \mathbf{x} \phi \right) - \frac{1}{4D|\mathbf{a}|^2} (N - 2i D \mathbf{\Omega} \cdot \mathbf{a} \times \mathbf{a}^*)^2 \\ & - iD (b\mathbf{a}^* - b^*\mathbf{a}) \cdot \nabla \phi - D a_i^* a_j \frac{\partial^2 p_0}{\partial x_i \partial x_j} + \mathcal{O}(\alpha^2 \epsilon) \, d^3 x. \end{aligned} \quad (2.29) $$

**Remark 2.4 (Physical Interpretation of GLM WMFI)** The variations of the WKB mean wave Lagrangian $\tilde{L}_W$ in the variables $N$ and $\phi$ recover canonical Hamiltonian WKB wave equations (2.20) and (2.21) for $N$ and $\phi$. These canonical equations have been boosted into the reference frame of the Lagrangian mean transport velocity $\mathbf{u}^L$. Moreover, the Lagrangian mean transport velocity $\mathbf{u}^L$ satisfies the Euler-Boussinesq equations on the left-hand side of Eq. (2.26). Thus, the phase space expression of the wave Lagrangian $\tilde{L}_W$ provides the physical interpretation of the WKB mean wave motion in GLM. Namely, GLM expresses WMFI as WKB wave motion boosted into the reference frame of the Euler-Boussinesq equations satisfied by the Lagrangian mean transport velocity, $\mathbf{u}^L$, and its corresponding pressure, $p_0$, and density, $\rho$. The dependence of the wave Lagrangian $\alpha^2 \tilde{L}_W$ on the fluid variables $D$ and $\rho$ implies from its variation in $p_0$ that incompressibility
of the Lagrangian mean transport velocity, $\mathbf{u}^L$, no longer holds exactly. Indeed, Eq. (2.18) shows that the divergence of $\mathbf{u}^L$ is of order $O(\alpha^2\epsilon)$, which would need to be considered when going beyond the order of asymptotics $O(\alpha^2)$ considered here.

**Remark 2.5 (Potential Vorticity (PV) Advection Theorem for WMFI)** Rewriting the indicated operations in the Kelvin circulation theorem for WMFI after applying the Stokes thereom gives us

$$ (\partial_t + \mathcal{L}_{uL}^\epsilon) d(D^{-1}\mathbf{M} \cdot d\mathbf{x}) + \frac{1}{\epsilon} g d\rho \wedge dz = 0, \quad (2.30) $$

where $\mathcal{L}^\epsilon$ denotes the Lie-derivative taken w.r.t to the rescaled basis $\epsilon\mathbf{x}$. Since $D$ and $\rho$ are advected, i.e. they satisfies the advection equations (2.15), one finds

$$ \begin{aligned} & (\partial_t + \mathcal{L}_{uL}^\epsilon) (d(D^{-1}\mathbf{M} \cdot d\mathbf{x}) \wedge d\rho) \\ & = (\partial_t + \mathcal{L}_{uL}^\epsilon) (D^{-1}\nabla_{\epsilon\mathbf{x}}\rho \cdot \mathrm{curl}_{\epsilon\mathbf{x}}(D^{-1}\mathbf{M}) D d^3x) = 0. \end{aligned} \quad (2.31) $$

Consequently, one finds the following total advective conservation law for WMFI potential vorticity PV,

$$ \frac{\partial}{\partial \epsilon t} q + \mathbf{u}^L \cdot \nabla_{\epsilon \mathbf{x}} q = 0 \quad \text{where } q := D^{-1} \nabla_{\epsilon \mathbf{x}} \rho \cdot \mathrm{curl}_{\epsilon \mathbf{x}} (\mathbf{u}^L + \mathbf{\Omega} \times \mathbf{x} + \alpha^2 D^{-1} N \mathbf{k}). \quad (2.32) $$

**Solving for Wave Polarisation Parameters/Lagrange Multipliers p, b, b*, a and a*** The quantities p and b in (2.10) are Lagrange multipliers which impose the incompressibility constraints for volume preservation $D = 1$ and transversality of the wave vectors $\mathbf{k} \cdot \mathbf{a} = 0$, respectively. The complex vector wave amplitudes $\mathbf{a}$ and $\mathbf{a}^*$ are also Lagrange multipliers whose variations impose a linear relationship among most of the wave variables. In particular, stationarity of wave action $S_W$ under variations of wave polarisation parameters b and $\mathbf{a}^*$ gives, respectively,

$$ \mathbf{k} \cdot \mathbf{a} = 0 \quad \text{and} \quad \tilde{\omega}^2 \mathbf{a} - 2i \tilde{\omega} \mathbf{\Omega} \times \mathbf{a} - (\mathbf{a} \cdot \nabla) \frac{\partial p_0}{\partial \mathbf{x}} = ib\mathbf{k}, \quad (2.33) $$

from which b follows easily from the first constraint, upon taking the dot product of $\mathbf{k}$ with the second constraint,

$$ |\mathbf{k}|^2 ib = -2i \tilde{\omega} (\mathbf{\Omega} \times \mathbf{a}) \cdot \mathbf{k} - \mathbf{k} \cdot (\mathbf{a} \cdot \nabla) \nabla p_0 = -k^l (2i \tilde{\omega} \hat{\mathbf{\Omega}}_{lj} + (p_0)_{lj}) a^j, \quad (2.34) $$

where $\hat{\mathbf{\Omega}}_{ij} = -\epsilon_{ijk} \mathbf{\Omega}^k$ and the complex vector amplitude $\mathbf{a}$ is found from the $3 \times 3$ Hermitian matrix inversion,
$$ \begin{bmatrix} \tilde{\omega}^2 - (p_0)_{11} & i\tilde{\omega}2\hat{\Omega}_{12} - (p_0)_{12} & i\tilde{\omega}2\hat{\Omega}_{13} - (p_0)_{13} \\ i\tilde{\omega}2\hat{\Omega}_{12} - (p_0)_{12} & \tilde{\omega}^2 - (p_0)_{22} & i\tilde{\omega}2\hat{\Omega}_{23} - (p_0)_{23} \\ i\tilde{\omega}2\hat{\Omega}_{13} - (p_0)_{13} & i\tilde{\omega}2\hat{\Omega}_{23} - (p_0)_{23} & \tilde{\omega}^2 - (p_0)_{33} \end{bmatrix} \mathbb{P}_{\perp} \begin{bmatrix} a_1 \\ a_2 \\ a_3 \end{bmatrix} = ib \begin{bmatrix} k_1 \\ k_2 \\ k_3 \end{bmatrix}, \qquad (2.35) $$

in which the 3 × 3 matrix $\mathbb{P}_{\perp}$ given by

$$ \mathbb{P}_{\perp ij} := \left( \delta_{ij} - \frac{k_i k_j}{|\mathbf{k}|^2} \right) $$

projects out the component along $\mathbf{k}$ of the complex vector amplitude $\mathbf{a} \in \mathbb{C}^3$.

**An Index Operator Form of the Polarisation Constraints** The wave polarisation constraints in (2.33) and (2.35) may be rewritten in index form as

$$ \begin{gathered} a^i k_i = 0 \quad \text{and} \quad D_{ij} a^j = ib k_i \quad \text{with} \quad D_{ij} = \tilde{\omega}^2 \delta_{ij} + i \tilde{\omega} 2 \hat{\Omega}_{ij} - \frac{\partial^2 p_0}{\partial x^j \partial x^i} \\ \text{so} \quad a^{*i} D_{ij} a^j = 0. \end{gathered} \qquad (2.36) $$

The index operator form in (2.36) of the polarisation relations for ($\mathbf{a}, \mathbf{b}$) in (2.33) suggests a more compact representation of the wave Lagrangian, $\tilde{L}_W$, as we discuss next.

**Representing the Wave Polarisation Parameters a and b as a Complex Four-Vector Field** After an integration by parts using the boundary conditions in (2.22), the Eulerian action principle in (2.10) may be expressed equivalently as

$$ \begin{align} 0 = \delta(S_{MF} + \alpha^2 S_W) &= \delta \int_{t_0}^{t_1} (\bar{L}_{MF} + \alpha^2 \bar{L}_W) dt \nonumber \\ &:= \delta \int_{t_0}^{t_1} \int_{M} \left( \frac{D}{2} |\mathbf{u}^L|^2 + D \mathbf{u}^L \cdot \mathbf{\Omega} \times \mathbf{x} - g D \rho z - p(D-1) \right. \tag{2.37} \\ & \qquad \left. + \alpha^2 D F^{\mu*} D_{\mu\nu} F^{\nu} + O(\alpha^2 \epsilon, \alpha^4) \right) d^3 x \, dt, \nonumber \end{align} $$

where, for notational convenience, the fields $\mathbf{a}$ and $\mathbf{b}$ comprise a complex “four-vector field”,

$$ F^{\mu} = (\mathbf{a}, \mathbf{b})^{T}, $$

with $\mu = 1, 2, 3, 4$, and the Hermitian dispersion tensor $D_{\mu\nu} = D_{\nu\mu}^*$ is given by

$$ D_{ij} = \tilde{\omega}^2 \delta_{ij} + i \tilde{\omega} 2 \hat{\Omega}_{ij} - \frac{\partial^2 p_0}{\partial x^i \partial x^j}, \quad D_{4j} = ik_j = -D_{j4}, \quad D_{44} = 0. $$
It is clear from the decomposition of the WMFI action in (2.37) that stationarity of $S_W$ with respect to variations of the fields $F = (a, b)^T$ yields linear relations among the wave parameters $(a, b)$ that recover the polarisation relations in (2.33)

$$D_{\mu\nu}F^{\nu} = 0. \qquad (2.38)$$

Equation (2.38) recovers the linear constraints in (2.33) on the polarization eigendirections of the field $F^\mu$ up to an overall complex constant that can be set at the initial time.

**Doppler-Shifted Dispersion Relation** The solvability condition $\det(D_{\mu\nu}) = 0$ for (2.38) now produces the dispersion relation for the Doppler-shifted frequency of internal gravitational waves (IGW),

$$ \begin{aligned} \tilde{\omega}^2 &:= (\omega - \mathbf{u}^L \cdot \mathbf{k})^2 = \left(-\frac{\partial}{\partial \epsilon t}\phi - \mathbf{u}^L \cdot \nabla_{\epsilon} \mathbf{x}\phi\right)^2 \\ &= \frac{(2\Omega \cdot \mathbf{k})^2}{|\mathbf{k}|^2} + \left(\delta_{ij} - \frac{k_i k_j}{|\mathbf{k}|^2}\right) \frac{\partial^2 p_0}{\partial x^i \partial x^j}, \end{aligned} \qquad (2.39) $$

which is independent of the magnitude $|\mathbf{k}|$ of the wave vector $\mathbf{k}$, except for the Doppler shift due to the fluid motion. Formula (2.39) updates the phase $\phi$ of the wave at each time step. The complex vector amplitude $\mathbf{a}$ is then found from inversion of the $3 \times 3$ Hermitian matrix in (2.35). The remaining wave quantity $b$ is then determined from (2.34) at a given time step.

**Remark 2.6** Under conditions of hydrostatic balance and equilibrium stratification, when $\mathbf{u}^L = 0$ and the pressure Hessian $p_{ij}$ has only the $p_{33}$ component, Eq. (2.39) reduces to the well-known dispersion relation for linear internal waves [23]. For non-equilibrium flows, though, Eq. (2.39) shows the sensitivity of the propagation of internal waves to the pressure Hessian.

## 2.2 Hamiltonian Structure for the WMFI Equations at Leading Order

Thus far, we have considered a Legendre transform within the wave Lagrangian (see Eq. (2.28)). It remains to perform the same calculation for the mean flow to see the full Hamiltonian structure of the model. We define the momentum of the entire flow by

$$ \begin{aligned} \mathbf{M} &:= D\mathbf{u}^L + D\Omega \times \mathbf{x} - \alpha^2 N \nabla_{\epsilon} \mathbf{x} \phi := \mathbf{m} - \mathbf{p} \\ \text{with } \mathbf{m} &:= D\mathbf{u}^L + D\Omega \times \mathbf{x}, \quad \text{and} \quad \mathbf{p} := \alpha^2 N \nabla_{\epsilon} \mathbf{x} \phi. \end{aligned} \qquad (2.40) $$
In the above definition, the momenta **m** and **p** are the mean and wave parts of the momentum respectively and the total momentum, **M**, is the variational derivative of the constrained Lagrangian (2.10) with respect to the Lagrangian mean velocity. We perform both the wave and mean flow Legendre transforms concurrently as

$$ 
\begin{aligned}
h &= \langle \mathbf{M}, \mathbf{u}^L \rangle + \alpha^2 \langle N, \omega \rangle - \bar{L}_{MF} - \alpha^2 \bar{L}_W \\
&= \langle D\mathbf{u}^L + D\mathbf{\Omega} \times \mathbf{x}, \mathbf{u}^L \rangle + \alpha^2 \langle N, \tilde{\omega} \rangle - \bar{L}_{MF} - \alpha^2 \bar{L}_W.
\end{aligned} 
\quad (2.41) 
$$

The resulting WMFI Hamiltonian then follows as

$$ 
\begin{aligned}
h(\mathbf{M}, D, \rho, \mathbf{p}, N) = & \int \left\{ \frac{1}{2D} |\mathbf{M} + \mathbf{p} - D(\mathbf{\Omega} \times \mathbf{x})|^2 \right. \\
& \left. \qquad + D\rho gz + \frac{\alpha^2 D}{4|\mathbf{a}|^2} \left( \frac{N}{D} - 2i\mathbf{\Omega} \cdot (\mathbf{a} \times \mathbf{a}^*) \right)^2 \right. \\
& \left. \qquad + \frac{iD}{N} (b\mathbf{p} \cdot \mathbf{a}^* - b^*\mathbf{p} \cdot \mathbf{a}) \right. \\
& \left. \qquad + \alpha^2 D a_i^* a_j \frac{\partial^2 p_0}{\partial x_i \partial x_j} + (D-1)p_0 \right\} d^3x.
\end{aligned} 
\quad (2.42) 
$$

The variational derivatives of the constrained Hamiltonian (2.42) may be determined from the coefficients in the following expression,

$$ 
\begin{aligned}
\delta h = & \int \left\{ -\bar{\omega} \delta D + Dgz \delta \rho + \mathbf{u}^L \cdot \delta \mathbf{M} - (1-D) \delta p_0 \right. \\
& \left. \qquad + \alpha^2 \left[ \tilde{\omega} - \frac{iD}{N} (b \mathbf{k} \cdot \mathbf{a}^* - b^* \mathbf{k} \cdot \mathbf{a}) \right] \delta N \right. \\
& \left. \qquad + \left[ \mathbf{u}^L + \mathbf{v}_G \right] \cdot \delta \mathbf{p} + i\alpha^2 D (\delta b \mathbf{k} \cdot \mathbf{a}^* - \delta b^* \mathbf{k} \cdot \mathbf{a}) \right. \\
& \left. \qquad - \alpha^2 \left[ \delta \mathbf{a}^* \cdot \left( D \tilde{\omega}^2 \mathbf{a} + 2i D \tilde{\omega} (\mathbf{\Omega} \times \mathbf{a}) \right) \right. \right. \\
& \left. \left. \qquad -i D b \mathbf{k} - D \left( \mathbf{a} \cdot \frac{\partial}{\partial \mathbf{x}} \right) \frac{\partial p_0}{\partial \mathbf{x}} \right) + \text{c.c.} \right] \right\} + \mathcal{O}(\alpha^2 \epsilon) d^3 x,
\end{aligned} 
\quad (2.43) 
$$

**Remark 2.7 (Discussion)**

* The quantity $\bar{\omega} = -\delta h/\delta D = \delta(\bar{L}_{MF} + \alpha^2 \bar{L}_W)/\delta D$ is the Bernoulli function given by Eq. (2.13) now expressed as a variational partial derivative holding fixed the other variables in $h$ arising in the Legendre transform.

* The Hamiltonian $h$ in (2.42) is stationary for variations in the diagnostic variables $(\mathbf{a}, \mathbf{a}^*, b, b^*, p_0)$. The stationary variations in these variables written in (2.43)
generate the constraints on the prognostic variables and the relations among the diagnostic variables. The solvability condition for these relations among the diagnostic variables determines the dispersion relation for the WKB IGWs.

• The $N$ and $\mathbf{p}$ equations can combine to yield

$$ \partial_{\epsilon t} \mathbf{k} + \nabla_{\epsilon \mathbf{x}} \omega = 0. $$

This is the so-called ‘conservation of waves’ equation, which imposes equality of cross derivatives of the phase function, $\phi(\epsilon \mathbf{x}, \epsilon t)$.

The above variational derivatives can be assembled into the following untangled Lie-Poisson Hamiltonian form which separates the dynamics of the total momentum $\mathbf{M}$ in (2.40) and the advected fluid variables, $D$ and $\rho$, from the wave momentum $\mathbf{p}$ and wave action density $N$,

$$ \frac{\partial}{\partial \epsilon t} \begin{bmatrix} M_j \\ D \\ \rho \\ p_j \\ N \end{bmatrix} = - \begin{bmatrix} M_k \partial_{\epsilon j} + \partial_{\epsilon k} M_j & D \partial_{\epsilon j} - \rho_{,\epsilon j} & 0 & 0 \\ \partial_{\epsilon k} D & 0 & 0 & 0 \\ \rho_{,\epsilon k} & 0 & 0 & 0 \\ 0 & 0 & 0 & p_k \partial_{\epsilon j} + \partial_{\epsilon k} p_j \\ 0 & 0 & 0 & \partial_{\epsilon k} N & 0 \end{bmatrix} \times \begin{bmatrix} \delta h / \delta M_k = u^L k \\ \delta h / \delta D = -\omega \\ \delta h / \delta \rho = D gz \\ \delta h / \delta p_k = (\mathbf{u}^L + \mathbf{v}_G)^k \\ \delta h / \delta N = \alpha^2 \tilde{\omega} \end{bmatrix}. \quad (2.44) $$

Here, we are using a shorthand notation for the derivatives, $\partial_{\epsilon j} = \partial / \partial \epsilon x_j$ for example, and we have used the constraint that $\mathbf{k} \cdot \mathbf{v}_G = 0$ in taking the variations in $b$ and $b^*$.

**Remark 2.8** The untangled Lie-Poisson Hamiltonian form in (2.44) of the ideal wave mean flow system of equations derived in the previous section from the GLM Hamilton’s principle represents a constrained Lie-Poisson Hamiltonian fluid system. Its Lie-Poisson bracket is defined on the dual of the direct sum of two semidirect-product Lie algebras

$$ \mathcal{X}_{TOT} \circled (\mathcal{F}_{MF} \oplus \mathrm{Den}_{MF}) \oplus (\mathcal{X}_{W} \circled \mathcal{F}_{W}). $$

Dual variables in $L^2(\mathbb{R}^3)$ pairing are the following, whose definitions also explain the geometric meanings of the standard calculus notation for the (MF) and (W) variables.

• The total momentum 1-form density $\tilde{M} = \mathbf{M} \cdot d\mathbf{x} \otimes d^3x \in \Lambda^1(\mathbb{R}^3) \otimes \mathrm{Den}(\mathbb{R}^3)$ is dual to the vector fields $\mathcal{X}_{TOT}(\mathbb{R}^3)$.

• The density $\tilde{D} = Dd^3x \in \mathrm{Den}_{MF}(\mathbb{R}^3)$ is dual to scalar functions $\mathcal{F}_{MF}(\mathbb{R}^3)$.
• The scalar function $\rho \in \mathcal{F}_{MF}(\mathbb{R}^3)$ is dual to densities $\mathrm{Den}_{MF}(\mathbb{R}^3)$.

• The wave momentum 1-form density $\tilde{p} = \mathbf{p} \cdot d\mathbf{x} \otimes d^3\mathbf{x} \in \Lambda^1(\mathbb{R}^3) \otimes \mathrm{Den}(\mathbb{R}^3)$ is dual to the vector fields $\mathcal{X}_W(\mathbb{R}^3)$.

• The wave action density $\tilde{N} = Nd^3\mathbf{x}$ is dual to scalar functions $\mathcal{F}_W(\mathbb{R}^3)$.

**Remark 2.9 (Preservation of PV Casimir s)** Notice that the Casimir functions for the Hamiltonian structure of GLM WMFI in the upper left block diagonal of the Lie-Poisson operator in Eq. (2.44) are in the same form as for the Euler-Boussinesq fluid, except they have been modified to accommodate the wave momentum. Consequently, no Casimir functions have been gained or lost in coupling the mean flow to the fluctuations.

**Canonical Structure of the Wave Dynamics**

The wave dynamics above are written in their Lie-Poisson Hamiltonian structure. Should we return to the canonical variables, $N$ and $\phi$, then the standard canonical structure emerges. Indeed, substituting $\mathbf{p} = \alpha^2 N \nabla \phi$ into the Hamiltonian (2.42) and taking variations gives<fnref n="1" />

$$ \alpha^2 \frac{\partial \phi}{\partial \epsilon t} = - \frac{\delta h}{\delta N} = - \alpha^2 \mathbf{u}^L \cdot \nabla_{\epsilon} \mathbf{x} \phi - \alpha^2 \tilde{\omega}, \quad (2.45) $$

$$ \alpha^2 \frac{\partial N}{\partial \epsilon t} = \frac{\delta h}{\delta \phi} = -\alpha^2 \mathrm{div}_{\epsilon} \mathbf{x} (N \mathbf{u}^L) - \alpha^2 i \mathrm{div}_{\epsilon} \mathbf{x} (D(b\mathbf{a}^* - b^*\mathbf{a})) . \quad (2.46) $$

**Tangled Version of the Lie-Poisson Hamiltonian Structure** By writing the Hamiltonian in terms of the mean flow momentum, **m**, rather than the total momentum, **M**, we recover the tangled version of the Lie-Poisson Hamiltonian form of the equations. Above, as in [13], we have presented wave-current interaction in its untangled form. In a previous work [12], the authors presented both the tangled and untangled forms, and an analogous calculation is also possible for this model of WMFI. Indeed, the Hamiltonian $h(\mathbf{M}, D, \rho, \mathbf{p}, N)$ in (2.42) becomes

$$ \begin{aligned} h'(\mathbf{m}, D, \rho, \mathbf{p}, N) = & \int \left\{ \left[ \frac{1}{2D} |\mathbf{m} - D(\mathbf{\Omega} \times \mathbf{x})|^2 \right. \right. \\ & \left. \left. + D\rho gz + \frac{\alpha^2 D}{4|\mathbf{a}|^2} \left( \frac{N}{D} - 2i \mathbf{\Omega} \cdot (\mathbf{a} \times \mathbf{a}^*) \right)^2 \right] \right. \\ & \left. + \frac{iD}{N} (b \mathbf{p} \cdot \mathbf{a}^* - b^* \mathbf{p} \cdot \mathbf{a}) \right. \\ & \left. + \alpha^2 D a_i^* a_j \frac{\partial^2 p_0}{\partial x_i \partial x_j} + (D-1)p_0 \right\} d^3\mathbf{x}. \end{aligned} \quad (2.47) $$

<footnote n="1">¹ The constant factor of $\alpha^2$ appearing within the canonical structure has emerged due to the choice of multiplying the constraints in Hamilton's principle by the same constant.</footnote>
The variational derivatives are largely the same, with differences only in the variation with respect to **p**. The tangled form of the Hamiltonian equations in the Hamiltonian $h(\mathbf{m}, D, \rho, \mathbf{p}, N)$ in (2.47) is

$$ \frac{\partial}{\partial \epsilon t} \begin{bmatrix} m_j \\ D \\ \rho \\ p_j \\ N \end{bmatrix} = - \begin{bmatrix} m_k \partial_{\epsilon j} + \partial_{\epsilon k} m_j & D \partial_{\epsilon j} - \rho_{,\epsilon j} p_k \partial_{\epsilon j} + \partial_{\epsilon k} p_j & N \partial_{\epsilon j} \\ \partial_{\epsilon k} D & 0 & 0 \\ \rho_{,\epsilon k} & 0 & 0 \\ p_k \partial_{\epsilon j} + \partial_{\epsilon k} p_j & 0 & 0 \\ \partial_{\epsilon k} N & 0 & 0 \end{bmatrix} \times \begin{bmatrix} \delta h'/\delta m_k = u^L k \\ \delta h'/\delta D = -\omega \\ \delta h'/\delta \rho = D gz \\ \delta h'/\delta p_k = \mathbf{v}_G^k \\ \delta h'/\delta N = \alpha^2 \tilde{\omega} \end{bmatrix}. \quad (2.48) $$

Instead of the direct sum in the untangled case in Remark 2.8, this tangled Lie-Poisson bracket is defined on the dual of two nested semidirect-product Lie algebras

$$ (\mathcal{X}_{MF}\mathbb{S}(\mathcal{F}_{MF} \oplus \text{Den}_{MF}))\mathbb{S}(\mathcal{X}_W\mathbb{S}\mathcal{F}_W). $$

Corresponding dual variables in $L^2(\mathbb{R}^3)$ pairing are similar to those explained in Remark 2.8.

## 3 Stochastic WMFI

Stochastic equations of motion may be used in fluid dynamics to model uncertainty, and such equations may be derived through Hamilton's principle [8]. Such stochastic terms can be used to parametrise unresolved 'subgridscale' dynamics absent in computational simulations, and as such are particularly relevant to geophysical applications.

Motivated by the fact that, due to computational limitations, the mean flow may only be solved for on a coarse grid when considering large scale geophysical flows, we apply the method of stochastic advection by Lie transport [8] to the mean flow map, $\bar{g}_t$. This may be done as

$$ d\bar{g}_t x_0 = (\mathbf{u}^L \circ \bar{g}_t)x_0 dt + \sum_i (\zeta_i \circ \bar{g}_t)x_0 \circ dW_t^i, \quad (3.1) $$
where $W_t^i$ are independent and identically distributed Brownian motions and $\circ dW_t^i$ denotes Stratonovich integration.<fnref n="2" /> This is equivalent to

$$d\bar{g}_t\bar{g}^{-1}(\mathbf{x}_t) = \mathbf{u}^L(\mathbf{x}_t) dt + \sum_i \zeta_i(\mathbf{x}_t) \circ dW_t^i := d\mathbf{x}_t,$$

and we see that the Lagrangian mean velocity, $\mathbf{u}^L$, has been stocastically perturbed. By an application of the Kunita-Itô-Wentzell formula [4], we see that

$$d\mathbf{X}_t = d\dot{g}_t g^{-1} \mathbf{X}_t = d\mathbf{x}_t + \alpha^2 (d\xi_t(\mathbf{x}_t) + d\mathbf{x}_t \cdot \nabla \xi_t(\mathbf{x}_t)). \quad (3.2)$$

Should we assume that the entire motion, corresponding to $\mathbf{U}_t = \dot{g}_t g^{-1}$, also has a stochastic part, corresponding to $\zeta_i^\xi$, then we have

$$\mathbf{U}_t \, dt + \sum_i \zeta_i^\xi \circ dW_t^i = d\mathbf{x}_t + \alpha^2 (d\xi_t(\mathbf{x}_t) + d\mathbf{x}_t \cdot \nabla \xi_t(\mathbf{x}_t)). \quad (3.3)$$

The uniqueness of the Doob-Meyer decomposition then indicates that each $\zeta_i^\xi$ decomposes into a part corresponding to the mean flow, $\zeta_i$, and a part corresponding to the wave motion, which we call $\sigma_i...$ [21].

**Remark 3.1** Following Street and Crisan, [21], by the compatibility of $\xi_t$ with the driving semimartingale, we have a representation $d\xi = A_0 dt + \sum_i A_i \circ dW_t^i$. The uniqueness of the Doob-Meyer decomposition then gives $\mathbf{U}_t = \mathbf{u}^L + \alpha^2 (A_0 + \mathbf{u}^L \cdot \nabla_{\mathbf{u}} \mathbf{x})$ and $\sigma_i = \zeta_i + \alpha^2 (A_i + \zeta_i \cdot \nabla_{\mathbf{u}} \mathbf{x})$.

WMFI is not limited to temporally averaged terms. The variability of WMFI must also be considered. This consideration results inevitably in differential equations for the slow components of the climate system, which include stochastic transport and forcing terms. There are many ways of introducing stochasticity into the WMFI system. Some guidance in this matter can be found, e.g., in [10].

In this section, we will consider two distinct framework of introducing stochasticity into Hamiltonian fluid systems. The first option laid out here in this section enables wave and fluid dynamics to possess different stochastically fluctuating components in their transport and phase velocities, as follows, in which variations of the deterministic Hamiltonian below are the as those in Eq. (2.43). The introduction of the stochastic vector fields to the WMF evolution equations can be accomplished by making the deterministic Hamiltonian to the WMFI a semimartingale in each degree of freedom. The chosen augmentation of the Hamiltonian is based on coupling noise by $L^2$ pairings of spatially varying noise ‘modes’ with the momentum maps dual to the respective velocities for each degree of freedom, **m**, **p**, and **N**. The coupling is done such that the variational derivatives with respect to the momentum variables

<footnote n="2">² The notation $\circ$ may be used to denote both composition and Stratonovich integration.</footnote>
will add stochastic transport terms to each of the corresponding dual velocities, as follows,

$$
\begin{equation}
\begin{aligned}
dh = \int & \left\{ \left[ \frac{1}{2D} |\mathbf{m} - D(\mathbf{\Omega} \times \mathbf{x})|^2 + D\rho g z + \frac{\alpha^2 D}{4|\mathbf{a}|^2} \left( \frac{N}{D} - 2i \mathbf{\Omega} \cdot (\mathbf{a} \times \mathbf{a}^*) \right)^2 \right] \right. \\
& \left. + \frac{iD}{N} (b \mathbf{p} \cdot \mathbf{a}^* - b^* \mathbf{p} \cdot \mathbf{a}) + \alpha^2 D a_i^* a_j \frac{\partial^2 p_0}{\partial x_i \partial x_j} \right\} d^3 x \, d\epsilon t \\
& + \int (D-1) d p_0 d^3 x \\
& + \sum_i \int \mathbf{m} \cdot \zeta_i(\mathbf{x}) \circ dW_{\epsilon t}^i d^3 x + \sum_i \int \mathbf{p} \cdot \sigma_i(\mathbf{x}) \circ dB_{\epsilon t}^i d^3 x.
\end{aligned}
\tag{3.4}
\end{equation}
$$

where $dW_t^i$ and $dB_t^i$ are chosen to be distinct Brownian motions and $\zeta_i(\mathbf{x})$ and $\sigma_i(\mathbf{x})$ in principle need to be determined from calibration of transport data of each type, and leading eventually to uncertainty quantification. We have introduced a stochastic component of the pressure, thus introducing the notation $d\rho_0$, following the framework of semimartingale driven variational principles [21]. The influence of the stochastic terms on the Lie-Poisson Hamiltonian dynamics can then be easily revealed, as

$$
\begin{equation}
\begin{split}
d \begin{bmatrix} m_j \\ D \\ \rho \\ p_j \\ N \end{bmatrix} = - & \begin{bmatrix} m_k \partial_{\epsilon j} + \partial_{\epsilon k} m_j & D \partial_{\epsilon j} - \rho_{,\epsilon j} & p_k \partial_{\epsilon j} + \partial_{\epsilon k} p_j & N \partial_{\epsilon j} \\ \partial_{\epsilon k} D & 0 & 0 & 0 \\ \rho_{,\epsilon k} & 0 & 0 & 0 \\ p_k \partial_{\epsilon j} + \partial_{\epsilon k} p_j & 0 & 0 & p_k \partial_{\epsilon j} + \partial_{\epsilon k} p_j & N \partial_{\epsilon j} \\ \partial_k N & 0 & 0 & \partial_{\epsilon k} N & 0 \end{bmatrix} \\
& \times \begin{bmatrix} \delta dh / \delta m_k = u^L k \, d\epsilon t + \zeta_i^k(\mathbf{x}) \circ dW_{\epsilon t}^i \\ \delta dh / \delta D = \bar{\pi} \, d\epsilon t + d p_0 \\ \delta dh / \delta \rho = Dgz \, d\epsilon t \\ \delta dh / \delta p_k = \mathbf{v}_G^k \, d\epsilon t + \sigma_i^k(\mathbf{x}) \circ dB_{\epsilon t}^i \\ \delta dh / \delta N = \alpha^2 \tilde{\omega} \, d\epsilon t \end{bmatrix}.
\end{split}
\tag{3.5}
\end{equation}
$$

where $\bar{\pi}$ is given by

$$
\bar{\pi} = -\bar{\omega} - p_0,
$$

for $\bar{\omega}$ as defined in Eq. (2.13). The Hamiltonian variables are as defined in the
deterministic case,

$$
\mathbf{m} := D(\mathbf{u}^L + \mathbf{\Omega} \times \mathbf{x}), \quad \mathbf{p} := \alpha^2 N \mathbf{k}, \quad \mathbf{v}_G := \frac{iD}{N}(\mathbf{a}^* b - \mathbf{a} b^*) = \frac{2D}{N} \Im(\mathbf{a} \mathbf{b}^*).
$$
These variables have already appeared in the integrand of Kelvin's circulation theorem in (2.27). The stochastic version of the GLM Kelvin circulation theorem for Euler-Boussinesq incompressible flow in Eq. (2.27) thus becomes

$$ 
\begin{aligned}
d \oint_{c(\mathbf{d}\mathbf{x}_t)} D^{-1} \mathbf{M} \cdot d\mathbf{x} &= d \oint_{c(\mathbf{d}\mathbf{x}_t)} \left( \mathbf{u}^L + \mathbf{\Omega} \times \mathbf{x} - D^{-1} \mathbf{p} \right) \cdot d\mathbf{x} \\
&= -\frac{1}{\epsilon} g \oint_{c(\mathbf{d}\mathbf{x}_t)} \rho \, dz \, d\epsilon t,
\end{aligned} 
\quad (3.6) $$

in which the material loop $c(\mathbf{d}\mathbf{x}_t)$ moves along stochastic Lagrangian trajectories given by the characteristics of the following stochastic vector field

$$ \mathbf{d}\mathbf{x}_t = \mathbf{u}^L(\mathbf{x}_t, t)dt + \sum_{a=1}^{N} \zeta_a(\mathbf{x}_t) \circ dW_t^a. \quad (3.7) $$

### A Stochastic Canonical Structure in the Wave Dynamics

The canonical structure between the wave variables $N$ and $\phi$, noted in equations (2.45) and (2.46), now becomes stochastic. Indeed, substituting $\mathbf{M}$ and $\mathbf{p} = \alpha^2 N \nabla_{\epsilon} \mathbf{x} \phi$ into the action and taking variations gives

$$ 
\begin{aligned}
\alpha^2 d\phi = -\frac{\delta d\mathbf{h}}{\delta N} = & -\alpha^2 \mathbf{u}^L \cdot \nabla_{\epsilon} \mathbf{x} \phi \, d\epsilon t - \alpha^2 \sum_i \zeta_i \cdot \nabla_{\epsilon} \mathbf{x} \phi \circ dW_{\epsilon t}^i - \alpha^2 \tilde{\omega} \, d\epsilon t \\
& - \alpha^2 \sum_i \nabla_{\epsilon} \mathbf{x} \phi \cdot \boldsymbol{\sigma}_i \circ dB_{\epsilon t}^i,
\end{aligned} 
\quad (3.8) $$

$$ 
\begin{aligned}
\alpha^2 dN = \frac{\delta d\mathbf{h}}{\delta\phi} = & -\alpha^2 \operatorname{div}_{\epsilon}\mathbf{x}(N\mathbf{u}^L) \, d\epsilon t - \alpha^2 \sum_i \operatorname{div}_{\epsilon}\mathbf{x}(N\zeta_i) \circ dW_{\epsilon t}^i \\
& - \alpha^2 i \operatorname{div}_{\epsilon}\mathbf{x} (D(b\mathbf{a}^* - b^*\mathbf{a})) \, d\epsilon t - \alpha^2 \sum_i \operatorname{div}_{\epsilon}\mathbf{x} (N\boldsymbol{\sigma}_i) \circ dB_{\epsilon t}^i.
\end{aligned} 
\quad (3.9) $$

Such a stochastic generalisation of Hamilton's canonical equations has been noted and discussed for wave hydrodynamics previously [20] for the classical water wave system.

### An Alternative, Energy-Conserving Approach to the Incorporation of Stochastic Noise

The second option of introducing stochasticity into the WMFI system is through the modification of mean flow and wave momentum to include different stochastically fluctuating components. The introduction of the stochastic momentum can be accomplished by making the deterministic Lie-Poisson bracket of the WMFI system
to include stochastic components. Following [11], the chosen modification is the addition of “frozen” Lie-Poisson bracket multiplying semi-martingales. The fixed (frozen) parameters in the frozen Lie-Poisson brackets are the spatially, possibly temporal varying noise “modes” which are transformed by the deterministic transport and phase velocities in the same way as the deterministic momentum. Let $\lambda_i$ and $\psi_i$ denote the stochastic fluctuations of the mean flow and wave momentum respectively, the stochastic Lie-Poisson equation can be written as

$$ 
\begin{aligned}
d \begin{bmatrix} m_j \\ D \\ \rho \\ p_j \\ N \end{bmatrix} = & - \begin{bmatrix} m_k \partial_{\epsilon j} + \partial_{\epsilon k} m_j & D \partial_{\epsilon j} - \rho_{,\epsilon j} p_k \partial_{\epsilon j} + \partial_{\epsilon k} p_j & N \partial_{\epsilon j} \\ \partial_{\epsilon k} D & 0 & 0 \\ \rho_{,\epsilon k} & 0 & 0 \\ p_k \partial_{\epsilon j} + \partial_{\epsilon k} p_j & 0 & 0 \\ \partial_k N & 0 & 0 \end{bmatrix} \\
& \times \begin{bmatrix} u^{Lk} \, d\epsilon t \\ -\pi \, d\epsilon t + d p_0 \\ D \, gz \, d\epsilon t \\ \mathbf{v}_G^k \, d\epsilon t \\ \alpha^2 \tilde{\omega} \, d\epsilon t \end{bmatrix} \\
& - \sum_i \begin{bmatrix} (\lambda_k^i \partial_{\epsilon j} + \partial_{\epsilon k} \lambda_j^i) \circ dW_{\epsilon t}^i & 0 & 0 & (\psi_k^i \partial_{\epsilon j} + \partial_{\epsilon k} \psi_j^i) \circ dB_{\epsilon t}^i & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ (\psi_k^i \partial_{\epsilon j} + \partial_{\epsilon k} \psi_j^i) \circ dB_{\epsilon t}^i & 0 & 0 & (\psi_k^i \partial_{\epsilon j} + \partial_{\epsilon k} \psi_j^i) \circ dB_{\epsilon t}^i & 0 \\ 0 & 0 & 0 & 0 & 0 \end{bmatrix} \\
& \times \begin{bmatrix} u^{Lk} \\ -\pi \, d\epsilon t + d p_0 \\ D \, gz \\ \mathbf{v}_G^k \\ \alpha^2 \tilde{\omega} \end{bmatrix}.
\end{aligned} 
\quad (3.10) $$

Here, the stochastic component of the pressure $dp$ is added as before following the semimartingale driven variational principle [21]. Similarly to the stochastic vector fields $\zeta^i$ and $\sigma^i$, we need to determine $\lambda^i$ and $\psi^i$ through calibration with existing data for each type of momentum. The influence of the stochasticity on the circulation dynamics of the mean flow and wave momentum is clear from the following modified Kelvin-circulation theorem
$$
\begin{align}
& d \oint_{c(\mathbf{u}^L)} (\mathbf{u}^L + \mathbf{\Omega} \times \mathbf{x}) \cdot d\mathbf{x} + \oint_{c(\mathbf{u}^L)} \frac{1}{\epsilon} \rho g \hat{z} \nonumber \\
& \qquad + \alpha^2 D^{-1} \left( N \nabla_{\epsilon} \tilde{\omega} + \mathbf{k} \operatorname{div}_{\epsilon} \mathbf{x} (N \mathbf{v}_G) \right) \cdot d\mathbf{x} d\epsilon t \nonumber \\
& \qquad + \sum_i \oint_{c(\mathbf{u}^L)} D^{-1} \left( \mathbf{u}^L \times \frac{\partial}{\partial \epsilon} \times \boldsymbol{\lambda}^i - \nabla_{\epsilon} \mathbf{u} (\mathbf{u}^L \cdot \boldsymbol{\lambda}^i) \right) \cdot d\mathbf{x} \circ dW_{\epsilon t}^i \nonumber \\
& \qquad + \sum_i \oint_{c(\mathbf{u}^L)} D^{-1} \left( \mathbf{v}_G \times \frac{\partial}{\partial \epsilon} \times \boldsymbol{\psi}^i - \nabla_{\epsilon} \mathbf{v} (\mathbf{v}_G \cdot \boldsymbol{\psi}^i) \right) \cdot d\mathbf{x} \circ dB_{\epsilon t}^i = 0, \tag{3.11}
\end{align}
$$

where the loop is moving with the *deterministic* velocity field. By construction, the Eq. (3.10) preserves the deterministic energy path-wise as the Poisson structure remain anti-symmetric and the variational derivative of the Hamiltonian is unchanged. However, the modification of the Poisson structure implies that the standard EB fluid Casimirs are no longer conserved.

## 4 Conclusion

In this paper we have derived a closed system of equations for the interaction of a GLM flow with the slowly varying envelope of a WKB field of internal gravity waves (IGW) by incorporating the two approximate descriptions into Hamilton's principle. Building on the work of Gjaja and Holm [5], we have shown that this approach boosts the canonical equations for the WKB IGW into the reference frame of the Lagrangian mean transport velocity, $\mathbf{u}^L$, satisfying the Euler-Boussinesq equations on the left-hand side of Eq. (2.26). Thus, GLM expresses WMFI as WKB wave motion boosted into the reference frame of the Euler-Boussinesq equations satisfied by the Lagrangian mean transport velocity, $\mathbf{u}^L$, and its corresponding pressure, $p_0$, and density, $\rho$. The dependence of the wave Lagrangian $\alpha^2 \bar{L}_W$ on the fluid variables $D$ and $\rho$ implies from its variation in $p_0$ that incompressibility of the Lagrangian mean transport velocity $\mathbf{u}^L$ does continue to hold for the order $O(\alpha^2)$ asymptotic expansion treated here.

We have further demonstrated how stochasticity in the fluid can permeate through both the wave and mean flow dynamics, and that such terms can be included through the variational structure. Moreover, this paper has identified the nested semidirect-product Lie-Poisson structure possessed by the Hamiltonian formulation of the GLM WMFI equations. The continued preservation of the fundamental Lie algebraic structure for the Hamiltonian formulation of the stochastic GLM WMFI system implies that its data calibration and uncertainty quantification can still be treated systematically using the stochastic advection by Lie transport (SALT) approach [8]. Future work will focus next on deriving a 2D vertical slice model for these 3D GLM WMFI equations and developing data calibration methods for
the 2D vertical slice model, in order to investigate the inclusion of mean internal gravity wave effects on the responses of the stochastic Eady problem.

**Acknowledgments** We are grateful to our friends, colleagues and collaborators for their advice and encouragement in the matters treated in this paper. DH especially thanks C. Cotter, F. Gay-Balmaz, I. Gjaja, J.C. McWilliams, T. S. Ratiu and C. Tronci for many insightful discussions of corresponding results similar to the ones derived here for WMFI, and in earlier work together in deriving hybrid models of complex fluids, turbulence, plasma dynamics, vertical slice models and the quantum-classical hydrodynamic description of molecules. DH and RH were partially supported during the present work by Office of Naval Research (ONR) grant award N00014-22-1-2082, “Stochastic Parameterization of Ocean Turbulence for Observational Networks”. DH and OS were partially supported during the present work by European Research Council (ERC) Synergy grant “Stochastic Transport in Upper Ocean Dynamics” (STUOD)—DLV-856408.

## Appendix: Asymptotic Expansion

This appendix fills in details of the derivations of the approximations discussed in Sect. 2. Namely, the displacement of a fluid element from its mean trajectory is represented by

$$ \mathbf{X}_t = \mathbf{x}_t + \alpha \xi(\mathbf{x}_t, t), \qquad (A.1) $$

and the associated velocity is given by

$$ \mathbf{U}_t(\mathbf{X}_t) = \mathbf{u}^L(\mathbf{x}_t, t) + \alpha \left( \partial_t \xi(\mathbf{x}_t, t) + \mathbf{u}^L \cdot \nabla_{\mathbf{x}_t} \xi(\mathbf{x}_t, t) \right) =: \mathbf{u}^L(\mathbf{x}_t, t) + \alpha \frac{d}{dt} \xi(\mathbf{x}_t, t). \qquad (A.2) $$

The fluctuating terms are assumed to have a WKB structure, lending the pressure an associated slow/fast decomposition

$$ \xi(\mathbf{x}, t) = \mathbf{a}(\epsilon\mathbf{x}, \epsilon t)e^{i\phi(\epsilon\mathbf{x}, \epsilon t)/\epsilon} + \mathbf{a}^*(\epsilon\mathbf{x}, \epsilon t)e^{-i\phi(\epsilon\mathbf{x}, \epsilon t)/\epsilon}, \qquad (A.3) $$

$$ p(\mathbf{X}, t) = p_0(\mathbf{X}, t) + \sum_{j \ge 1} \alpha^j \left( b_j(\epsilon \mathbf{X}, \epsilon t) e^{ij\phi(\epsilon \mathbf{X}, \epsilon t)/\epsilon} + b_j^*(\epsilon \mathbf{X}, \epsilon t) e^{-ij\phi(\epsilon \mathbf{X}, \epsilon t)/\epsilon} \right). \qquad (A.4) $$

Making these approximations within a fluid governed by the Euler-Boussinesq equations may be performed by substituting them into Hamilton's principle, asymptotically expanding, and truncating to leave only the leading order terms. The relevant variational principle in this case is as follows

$$ 0 = \delta \int_{t_0}^{t_1} \int_{\mathcal{M}} \mathcal{D} \left( \frac{1}{2} |\mathbf{U}|^2 + \mathbf{U} \cdot \mathbf{\Omega} \times \mathbf{X} - g\varrho Z \right) + p(1 - \mathcal{D}) d^3 X dt. \qquad (2.6 \text{ revisited}) $$
We first note that the volume form must be written in terms of the mean basis, as

$$ \mathcal{D}(\mathbf{X})d^3X = \mathcal{D}^\xi(\mathbf{x})d^3X = \mathcal{D}^\xi(\mathbf{x})\mathcal{J}d^3x := Dd^3x, \quad (A.5) $$

where

$$ \mathcal{J} = \det \left( \delta_{ij} + \alpha \frac{\partial \xi^i}{\partial x^j} \right). $$

Similarly, $\rho$ also transforms as

$$ \rho(\mathbf{X}) = \rho^\xi(\mathbf{x}) =: \rho. $$

Before calculating the terms featuring $\mathbf{U}$, note that

$$ \begin{aligned} \partial_t \xi(\mathbf{x}_t, t) + \mathbf{u}^L \cdot \nabla_{\mathbf{x}_t} \xi(\mathbf{x}_t, t) &= \epsilon \frac{\partial \mathbf{a}}{\partial \epsilon t} e^{i\phi/\epsilon} + a i \frac{\partial \phi}{\partial \epsilon t} e^{i\phi/\epsilon} \\ &\quad + \epsilon \frac{\partial \mathbf{a}^*}{\partial \epsilon t} e^{-i\phi/\epsilon} - \mathbf{a}^* i \frac{\partial \phi}{\partial \epsilon t} e^{-i\phi/\epsilon} \\ &\quad + \epsilon e^{i\phi/\epsilon} \mathbf{u}^L \cdot \nabla_{\epsilon x} \mathbf{a} + i a e^{i\phi/\epsilon} \mathbf{u}^L \cdot \nabla_{\epsilon x} \phi \\ &\quad + \epsilon e^{-i\phi/\epsilon} \mathbf{u}^L \cdot \nabla_{\epsilon x} \mathbf{a}^* - i \mathbf{a}^* e^{-i\phi/\epsilon} \mathbf{u}^L \cdot \nabla_{\epsilon x} \phi \end{aligned} \quad (A.6) \\ = ia e^{i\phi/\epsilon} \left( \frac{\partial \phi}{\partial \epsilon t} + \mathbf{u}^L \cdot \nabla_{\epsilon x} \phi \right) + ia^* e^{-i\phi/\epsilon} \left( -i \frac{\partial \phi}{\partial \epsilon t} - \mathbf{u}^L \cdot \nabla_{\epsilon x} \phi \right) \\ + \epsilon e^{i\phi/\epsilon} \left( \frac{\partial \mathbf{a}}{\partial \epsilon t} + \mathbf{u}^L \cdot \nabla_{\epsilon x} \mathbf{a} \right) + \epsilon e^{-i\phi/\epsilon} \left( \frac{\partial \mathbf{a}^*}{\partial \epsilon t} + \mathbf{u}^L \cdot \nabla_{\epsilon x} \mathbf{a}^* \right) \\ = -i \tilde{\omega} a e^{i\phi/\epsilon} + i \tilde{\omega} \mathbf{a}^* e^{-i\phi/\epsilon} + \mathcal{O}(\epsilon), \end{aligned} $$

where we define $\tilde{\omega} := -\frac{d}{d\epsilon t}\phi = -\frac{\partial}{\partial\epsilon t}\phi + \mathbf{u}^L \cdot \nabla_{\epsilon x}\phi$ and $\mathbf{k} := \nabla_{\epsilon x}\phi$ as in (2.9) and (2.8). The may now calculate the energy terms, beginning with kinetic energy, making use of the above relation and taking the mean.<fnref n="3" /> Note that the following relations are true within the Lagrangian, but are expressed here in isolation.

$$ \begin{aligned} \frac{1}{2}|\mathbf{U}|^2 &= \frac{1}{2}|\mathbf{u}^L + \alpha(\partial_t \xi(\mathbf{x}_t, t) + \mathbf{u}^L \cdot \nabla_{\mathbf{x}_t} \xi(\mathbf{x}_t, t))|^2 \\ &= \frac{1}{2}|\mathbf{u}^L|^2 + \alpha^2|\partial_t \xi(\mathbf{x}_t, t) + \mathbf{u}^L \cdot \nabla_{\mathbf{x}_t} \xi(\mathbf{x}_t, t)|^2 \\ &= \frac{1}{2}|\mathbf{u}^L|^2 + 2\alpha^2 \tilde{\omega}^2 \mathbf{a} \cdot \mathbf{a}^* + \mathcal{O}(\alpha^2 \epsilon). \end{aligned} $$

<footnote n="3">³ In taking the mean within the action integral, we discard the terms multiplied by rapid oscillations $\exp(\pm i\phi/\epsilon)$ and $\exp(\pm 2i\phi/\epsilon)$. These non-resonant terms are assumed to oscillate to zero under the time integral.</footnote>
The rotation term and potential energy are

$$
\begin{align*}
\mathbf{U} \cdot \mathbf{\Omega} \times \mathbf{X} &= (\mathbf{u}^L + \alpha(\partial_t \xi(\mathbf{x}_t, t) + \mathbf{u}^L \cdot \nabla_{\mathbf{x}_t} \xi(\mathbf{x}_t, t))) \cdot \mathbf{\Omega} \times (\mathbf{x} + \alpha \xi(\mathbf{x}, t)) \\
&= \mathbf{u}^L \cdot \mathbf{\Omega} \times \mathbf{x} + \mathbf{u}^L \cdot \mathbf{\Omega} \times (\alpha \xi) + \alpha(\partial_t \xi + \mathbf{u}^L \cdot \nabla_{\mathbf{x}_t} \xi) \cdot \mathbf{\Omega} \times \mathbf{x} \\
&\quad + \alpha(\partial_t \xi + \mathbf{u}^L \cdot \nabla_{\mathbf{x}_t} \xi) \cdot \mathbf{\Omega} \times (\alpha \xi) \\
&= \mathbf{u}^L \cdot \mathbf{\Omega} \times \mathbf{x} + \alpha^2(\partial_t \xi + \mathbf{u}^L \cdot \nabla_{\mathbf{x}_t} \xi) \cdot \mathbf{\Omega} \times \xi \\
&= \mathbf{u}^L \cdot \mathbf{\Omega} \times \mathbf{x} + 2i\alpha^2 \tilde{\omega} \mathbf{\Omega} \cdot (\mathbf{a} \times \mathbf{a}^*) + \mathcal{O}(\alpha^2 \epsilon), \\
g\varrho Z &= g\rho(z + \alpha\xi_3) = g\rho z.
\end{align*}
$$

Within the pressure term, we need to take care of expansion in both $p^\xi$ and $\mathcal{J}$. We
have

$$
\overline{(1 - \mathcal{D}^\xi) p^\xi d^3 \overline{X}} = \overline{(\mathcal{J} - D) p^\xi d^3 \overline{x}} = \overline{(\mathcal{J} p^\xi - D p^\xi)} d^3 \overline{x}
$$

Dealing with the terms separately, we have the expanded expression for $p^\xi$

$$
\begin{align*}
p^{\xi}(\mathbf{x}) &= p_0(\mathbf{x}, t) + \alpha \frac{\partial p_0}{\partial x_i} \xi_i + \frac{\alpha^2}{2} \frac{\partial^2 p_0}{\partial x_i \partial x_j} \xi_i \xi_j + \mathcal{O}(\alpha^3) \\
&\quad + \sum_{j \le 1} \alpha^j \left( b_j (\epsilon \mathbf{x}, \epsilon t) + \alpha \epsilon \frac{\partial b_j}{\partial \epsilon x_i} \xi_i + \frac{\alpha^2 \epsilon^2}{2} \frac{\partial^2 b_j}{\partial \epsilon x_i \partial \epsilon x_k} \xi_i \xi_k + \mathcal{O}(\alpha^3) \right) \\
&\quad \exp \left( \frac{ij}{\epsilon} \left( \phi(\epsilon \mathbf{x}, \epsilon t) + \alpha \epsilon \frac{\partial \phi}{\partial \epsilon x_i} \xi_i + \frac{\alpha^2 \epsilon^2}{2} \frac{\partial^2 \phi}{\partial \epsilon x_i \partial \epsilon x_k} \xi_i \xi_k + \mathcal{O}(\alpha^3) \right) \right) + \text{c.c.} \\
&= p_0 + \alpha \frac{\partial p_0}{\partial x_i} \xi_i + \frac{\alpha^2}{2} \frac{\partial^2 p_0}{\partial x_i \partial x_j} \xi_i \xi_j + \mathcal{O}(\alpha^3) \\
&\quad + \sum_{j \le 1} \alpha^j \left( b_j + \alpha \epsilon \frac{\partial b_j}{\partial \epsilon x_i} \xi_i + \mathcal{O}(\alpha^2) \right) \\
&\quad \times \exp \left( \frac{ij\phi}{\epsilon} \right) \left( 1 + ij\alpha \frac{\partial \phi}{\partial \epsilon x_i} \xi_i + \mathcal{O}(\alpha^2) \right) + \text{c.c.} \\
&= p_0 + \alpha \frac{\partial p_0}{\partial x_i} \xi_i + \frac{\alpha^2}{2} \frac{\partial^2 p_0}{\partial x_i \partial x_j} \xi_i \xi_j \\
&\quad + \exp \left( \frac{i\phi}{\epsilon} \right) \left( \alpha b_1 + \alpha^2 \epsilon \frac{\partial b_1}{\partial \epsilon x_i} \xi_i + b_1 i \alpha^2 \frac{\partial \phi}{\partial \epsilon x_i} \xi_i \right) \\
&\quad + \exp \left( \frac{2i\phi}{\epsilon} \right) \alpha^2 b_2 + \text{c.c.} + \mathcal{O}(\alpha^3),
\end{align*}
$$
which gives the phase averaged expression

$$ \overline{p^{\xi}} = p_0 + \frac{\alpha^2}{2} \frac{\partial^2 p_0}{\partial x_i \partial x_j} (a_i a_j^* + a_i^* a_j) + \alpha^2 \left( \epsilon a_i^* \frac{\partial b_1}{\partial \epsilon x_i} + i b_1 a_i^* \frac{\partial \phi}{\partial \epsilon x_i} + \epsilon a_i \frac{\partial b_1^*}{\partial \epsilon x_i} - i b_1 a_i^* \frac{\partial \phi}{\partial \epsilon x_i} \right) + \mathcal{O}(\alpha^3). $$

Note that

$$ \begin{aligned} \mathcal{J} &= \det \left( \delta_{ij} + \alpha \frac{\partial \xi_i}{\partial x_j} \right) = 1 + \alpha \frac{\partial \xi_i}{\partial x_i} + \alpha^2 (2\delta_{ij} - 1) \frac{\partial \xi_i}{\partial x_j} \frac{\partial \xi_j}{\partial x_i} + \mathcal{O}(\alpha^3) \\ &= 1 + \alpha \frac{\partial \xi_i}{\partial x_i} + \alpha^2 \left( \frac{\partial \xi_1}{\partial x_1} \frac{\partial \xi_2}{\partial x_2} + \frac{\partial \xi_3}{\partial x_3} \frac{\partial \xi_1}{\partial x_1} + \frac{\partial \xi_3}{\partial x_3} \frac{\partial \xi_2}{\partial x_2} - \frac{\partial \xi_1}{\partial x_2} \frac{\partial \xi_1}{\partial x_2} \right. \\ & \qquad \left. - \frac{\partial \xi_1}{\partial x_3} \frac{\partial \xi_3}{\partial x_1} - \frac{\partial \xi_2}{\partial x_3} \frac{\partial \xi_3}{\partial x_2} \right) + \mathcal{O}(\alpha^3) \\ &= 1 + \alpha \left( \left( \frac{\partial a_i}{\partial x_i} + \frac{i}{\epsilon} a_i \frac{\partial \phi}{\partial x_i} \right) \exp(i\phi/\epsilon) + \left( \frac{\partial a_i^*}{\partial x_i} - \frac{i}{\epsilon} a_i^* \frac{\partial \phi}{\partial x_i} \right) \exp(-i\phi/\epsilon) \right) \\ & \qquad + \alpha^2 (2\delta_{ij} - 1) \left( \left( \frac{\partial a_i}{\partial x_j} + \frac{i}{\epsilon} a_i \frac{\partial \phi}{\partial x_j} \right) \exp(i\phi/\epsilon) + \text{c.c.} \right) \\ & \qquad \times \left( \left( \frac{\partial a_j}{\partial x_i} + \frac{i}{\epsilon} a_j \frac{\partial \phi}{\partial x_i} \right) \exp(i\phi/\epsilon) + \text{c.c.} \right) + \mathcal{O}(\alpha^3). \end{aligned} $$

Taking the phase average gives

$$ \begin{aligned} \overline{\mathcal{J}} &= 1 + \alpha^2 (2\delta_{ij} - 1) \left( \left( \frac{\partial a_i}{\partial x_j} + \frac{i}{\epsilon} a_i \frac{\partial \phi}{\partial x_j} \right) \left( \frac{\partial a_j^*}{\partial x_i} - \frac{i}{\epsilon} a_j^* \frac{\partial \phi}{\partial x_i} \right) + \text{c.c.} \right) + \mathcal{O}(\alpha^3) \\ &= 1 + i\alpha^2 \frac{\partial \phi}{\partial \epsilon \mathbf{x}} \cdot \frac{\partial}{\partial \mathbf{x}} \times (\mathbf{a} \times \mathbf{a}^*) + \mathcal{O}(\alpha^3), \end{aligned} $$

where the last equality uses the fact that we are operating under a spatial integral and integration by parts applies. Then, we have

$$ \begin{aligned} p^{\xi} \mathcal{J} = p_0 &+ \alpha \frac{\partial p_0}{\partial x_i} \xi_i + \frac{\alpha^2}{2} \frac{\partial^2 p_0}{\partial x_i \partial x_j} \xi_i \xi_j \\ &+ \exp\left(\frac{i\phi}{\epsilon}\right) \left(\alpha b_1 + \alpha^2 \epsilon \frac{\partial b_1}{\partial \epsilon x_i} \xi_i + b_1 i \alpha^2 \frac{\partial \phi}{\partial \epsilon x_i} \xi_i\right) \\ &+ \exp\left(\frac{2i\phi}{\epsilon}\right) \alpha^2 b_2 + \text{c.c.} \end{aligned} $$
$$ 
\begin{aligned}
&+ p_0 \alpha \frac{\partial \xi_i}{\partial x_i} + \alpha^2 \frac{\partial \xi_i}{\partial x_i} \frac{\partial p_0}{\partial x_j} \xi_j + \alpha^2 \exp\left(\frac{i\phi}{\epsilon}\right) b_1 \frac{\partial \xi_i}{\partial x_i} \\
&+ p_0 \alpha^2 (2\delta_{ij} - 1) \frac{\partial \xi_i}{\partial x_j} \frac{\partial \xi_j}{\partial x_i} + \mathcal{C} \mathcal{C} + \mathcal{O}(\alpha^3).
\end{aligned}
$$

Applying phase averaging gives

$$ 
\begin{aligned}
\overline{p^{\xi} \mathcal{J}} = p_0 &+ \frac{\alpha^2}{2} \frac{\partial^2 p_0}{\partial x_i \partial x_j} (a_i a_j^* + a_i^* a_j) \\
&+ \alpha^2 \left( \epsilon a_i^* \frac{\partial b_1}{\partial \epsilon x_i} + i b_1 a_i^* \frac{\partial \phi}{\partial \epsilon x_i} + \epsilon a_i \frac{\partial b_1^*}{\partial \epsilon x_i} - i b_1^* a_i \frac{\partial \phi}{\partial \epsilon x_i} \right) \\
&+ \alpha^2 \left( \left( \frac{\partial a_i}{\partial x_i} + i \frac{\partial \phi}{\partial \epsilon x_i} a_i \right) \left( a_j^* \frac{\partial p_0}{\partial x_j} + b_1^* \right) \right. \\
&\left. + \left( \frac{\partial a_i^*}{\partial x_i} - i \frac{\partial \phi}{\partial \epsilon x_i} a_i^* \right) \left( a_j \frac{\partial p_0}{\partial x_j} + b_1 \right) \right) \\
&+ p_0 i \alpha^2 \frac{\partial \phi}{\partial \epsilon \mathbf{x}} \cdot \frac{\partial}{\partial \mathbf{x}} \times (\mathbf{a} \times \mathbf{a}^*) + \mathcal{O}(\alpha^3).
\end{aligned}
$$

We may assemble these statements into the following action integral, which may be regarded as an approximation of (2.6).

$$ 
\begin{aligned}
S = & \int_{t_0}^{t_1} \int_{\mathcal{M}} D \left( \frac{1}{2} |\mathbf{u}^L|^2 + \alpha^2 \tilde{\omega}^2 |\mathbf{a}|^2 - \rho g z + \mathbf{u}^L \cdot \mathbf{\Omega} \times \mathbf{x} + 2i \alpha^2 \tilde{\omega} \mathbf{\Omega} \cdot (\mathbf{a} \times \mathbf{a}^*) \right) \\
& + \left( \frac{\alpha^2}{2} \frac{\partial^2 p_0}{\partial x_i \partial x_j} (a_i a_j^* + a_i^* a_j) \right. \\
& \left. + \alpha^2 \left( \epsilon a_i^* \frac{\partial b_1}{\partial \epsilon x_i} + i b_1 a_i^* \frac{\partial \phi}{\partial \epsilon x_i} + \epsilon a_i \frac{\partial b_1^*}{\partial \epsilon x_i} - i b_1^* a_i \frac{\partial \phi}{\partial \epsilon x_i} \right) \right) (1-D) \\
& + \alpha^2 \left( \left( \frac{\partial a_i}{\partial x_i} + i \frac{\partial \phi}{\partial \epsilon x_i} a_i \right) \left( a_j^* \frac{\partial p_0}{\partial x_j} + b_1^* \right) \right. \\
& \left. + \left( \frac{\partial a_i^*}{\partial x_i} - i \frac{\partial \phi}{\partial \epsilon x_i} a_i^* \right) \left( a_j \frac{\partial p_0}{\partial x_j} + b_1 \right) \right) \\
& + p_0 (1-D) + i \alpha^2 p_0 \frac{\partial \phi}{\partial \epsilon \mathbf{x}} \cdot \frac{\partial}{\partial \mathbf{x}} \times (\mathbf{a} \times \mathbf{a}^*) d^3 x dt.
\end{aligned}
\tag{A.7}
$$
We now seek to simplify this integral. Firstly, we note that the following relation-
ships hold for the last four terms on the second row of Eq. (A.7)

$$
\begin{align*}
& \alpha^2 \left( i b_1 a_i^* \frac{\partial \phi}{\partial \epsilon x_i} - i b_1^* a_i \frac{\partial \phi}{\partial \epsilon x_i} \right) (1 - D) + \alpha^2 \left( i \frac{\partial \phi}{\partial \epsilon x_i} a_i b_1^* - i \frac{\partial \phi}{\partial \epsilon x_i} a_i^* b_1 \right) \\
& = -\alpha^2 i D \left( b_1 \mathbf{k} \cdot \mathbf{a}^* - b_1^* \mathbf{k} \cdot \mathbf{a} \right),
\end{align*}
$$

and

$$
\begin{align*}
& \alpha^2 \int_M \left( \epsilon a_i^* \frac{\partial b_1}{\partial \epsilon x_i} + \epsilon a_i \frac{\partial b_1^*}{\partial \epsilon x_i} \right) (1-D) + \frac{\partial a_i}{\partial x_i} b_1^* + \frac{\partial a_i^*}{\partial x_i} b_1 d^3 x \\
&= -\alpha^2 \epsilon \int_M D \left( a_i^* \frac{\partial b_1}{\partial \epsilon x_i} + a_i \frac{\partial b_1^*}{\partial \epsilon x_i} \right) d^3 x = \mathcal{O}(\alpha^2 \epsilon),
\end{align*}
$$

after integration by parts. We have thus far involved several of the order $\alpha^2$ terms on
the third line of (A.7). The remainder of these are handled as follows

$$
\begin{align*}
& \alpha^2 i \int_M \frac{\partial \phi}{\partial \epsilon x_i} a_i a_j^* \frac{\partial p_0}{\partial x_j} - \frac{\partial \phi}{\partial \epsilon x_i} a_i^* a_j \frac{\partial p_0}{\partial x_j} \, d^3 x \\
&= \alpha^2 i \int_M -p_0 \frac{\partial}{\partial x_j} \left( \frac{\partial \phi}{\partial \epsilon x_i} a_i a_j^* \right) + p_0 \frac{\partial}{\partial x_j} \left( \frac{\partial \phi}{\partial \epsilon x_i} a_i^* a_j \right) \, d^3 x \\
&= i \int_M -\alpha^2 p_0 \frac{\partial \phi}{\partial \epsilon x_i} \frac{\partial}{\partial x_j} \left( a_i a_j^* \right) + \alpha^2 p_0 \frac{\partial \phi}{\partial \epsilon x_i} \frac{\partial}{\partial x_j} \left( a_i^* a_j \right) \, d^3 x \\
&= - \int_M i \alpha^2 p_0 \frac{\partial \phi}{\partial \epsilon \mathbf{x}} \cdot \left( \mathbf{a}(\nabla \cdot \mathbf{a}^*) - \mathbf{a}^*(\nabla \cdot \mathbf{a}) + (\mathbf{a}^* \cdot \nabla)\mathbf{a} - (\mathbf{a} \cdot \nabla)\mathbf{a}^* \right) \, d^3 x \\
&= - \int_M i \alpha^2 p_0 \frac{\partial \phi}{\partial \epsilon \mathbf{x}} \cdot \frac{\partial}{\partial \mathbf{x}} \times (\mathbf{a} \times \mathbf{a}^*) \, d^3 x,
\end{align*}
$$

and

$$
\begin{align*}
& \int_M \alpha^2 \epsilon \left( \frac{\partial a_i}{\partial \epsilon x_i} a_j^* \frac{\partial p_0}{\partial x_j} + \frac{\partial a_i^*}{\partial \epsilon x_i} a_j \frac{\partial p_0}{\partial x_j} \right) d^3 x \\
&= \int_M \alpha^2 \epsilon \left( \frac{\partial a_i}{\partial \epsilon x_i} a_j^* \frac{\partial p_0}{\partial x_j} - a_i^* \frac{\partial}{\partial \epsilon x_i} \left( a_j \frac{\partial p_0}{\partial x_j} \right) \right) d^3 x \\
&= - \int_M \alpha^2 a_i^* a_j \frac{\partial^2 p_0}{\partial x_i \partial x_j} d^3 x = - \int_M \frac{\alpha^2}{2} \left( a_i a_j^* + a_i^* a_j \right) \frac{\partial^2 p_0}{\partial x_i \partial x_j} d^3 x.
\end{align*}
$$
Assembling this back into the action integral (A.7) finally yields the expression for S in (2.7),

$$
\begin{equation}
\begin{aligned}
S = \int_{t_0}^{t_1} \int_{\mathcal{M}} D \left[ \frac{1}{2} |\mathbf{u}^L|^2 + \alpha^2 \tilde{\omega}^2 |\mathbf{a}|^2 - \rho g z + \mathbf{u}^L \cdot \mathbf{\Omega} \times \mathbf{x} + 2i \alpha^2 \tilde{\omega} \mathbf{\Omega} \cdot (\mathbf{a} \times \mathbf{a}^*) \right. \\
\left. - \alpha^2 i (b \mathbf{k} \cdot \mathbf{a}^* - b^* \mathbf{k} \cdot \mathbf{a}) - \alpha^2 a_i^* a_j \frac{\partial^2 p_0}{\partial x_i \partial x_j} \right] \\
+ (1-D)p_0 + \mathcal{O}(\alpha^2 \epsilon) d^3 x \, dt.
\end{aligned}
\tag{A.8}
\end{equation}
$$

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Quentin Jamet, Etienne Mémin, Franck Dumas, Long Li, and Pierre Garreau

**Abstract** Current climate models are known to systematically overestimate the rate of deep water formation at high latitudes in response to too deep and too frequent deep convection events. We propose in this study to investigate a misrepresentation of deep convection in Hydrostatic Primitive Equation (HPE) ocean and climate models due to the lack of constraints on vertical dynamics. We discuss the potential of the Location Uncertainty (LU) stochastic representation of geophysical flow dynamics to help in the process of re-introducing some degree of non-hydrostatic physics in HPE models through a pressure correction method. We then test our ideas with idealized Large Eddy Simulations (LES) of buoyancy driven free convection with the CROCO modeling platform. Preliminary results at LES resolution exhibit a solution obtained with our Quasi-nonhydrostatic (Q-NH) model that tends toward the reference non-hydrostatic (NH) model. As compared to a pure hydrostatic setting, our Q-NH solution exhibits vertical convective plumes with larger horizontal structure, a better spatial organization and a reduced intensity of their associated vertical velocities. The simulated Mixed Layer Depth (MLD) deepening rate is however too slow in our Q-NH experiment as compared to the reference NH, a behaviour that opposes to that of hydrostatic experiments of producing too fast MLD deepening rate. These preliminary results are encouraging, and support future efforts in the direction of enriching coarse resolution, hydrostatic ocean and climate models with a stochastic representation of non-hydrostatic physics.

Q. Jamet (✉)

INRIA, ODYSSEY Group, Plouzané, France

e-mail: quentin.jamet@inria.fr

E. Mémin · L. Li

INRIA, ODYSSEY Group, Rennes, France

F. Dumas

SHOM, Brest, France

P. Garreau

Laboratoire d'Océanographie Physique et Spatiale, IFREMER, Plouzané, France
# 1 Introduction

Deep ocean convection is a crucial mechanism for large scale ocean circulation and climate. It controls the rate of deep ocean water masses formation, sequestrating atmospheric properties such as heat and carbon in the abyssal ocean. In the North Atlantic basin, deep ocean convection in the Labrador Sea and the Nordic Seas is part of the large scale Atlantic Meridional Overturning Circulation (AMOC), an oceanic metric with many climate implications (Zhang et al. 2019). Coarse resolution (i.e. $\Delta x \sim O(100)$ km) climate models are known to overestimate the rate of deep water formation at high latitudes in response to too deep and too frequent convective events (Heuzé 2017; 2021), a bias that is expected to worsen with next generation climate models with ocean components at higher resolution (Masson-Delmotte et al. 2021). Among other possibilities (e.g. preconditioning, air-sea interactions), we explore in this paper the possible misrepresentation of deep ocean convection in current climate models in response to their hydrostatic formulation.

Ocean modulus of current climate models solve the Hydrostatic Primitive Equations (HPE), a simplified version of the full Navier-Stokes equations (NS). Geophysical fluids have specific characteristics that allow some approximations from the general NS, leading to drastic simplifications in their numerical implementation which, in turn, allow us to model the global ocean at climate scales (i.e. for several decades/centuries) with the currently available computational resources. Among those approximations is the hydrostatic balance which arises from the relatively thin thickness of the ocean ($H \sim O(1)$ km) as compared to the horizontal extension of its large scale dynamics ($L \sim O(1000)$ km for gyres and $L \sim O(10 - 100)$ km for ocean mesoscale eddies). The aspect ratio $\delta = H/L$ is thus orders of magnitude smaller than unity. Scaling the vertical velocity $\bar{W} = \delta U$, with $U$ the typical horizontal velocity of the flow, leads to small contribution of vertical acceleration as compared to horizontal components. For a regime satisfying such a scaling, only vertical pressure gradients are able to balance gravitational acceleration in the vertical component of the NS equations, and the system can be simplified as:

$$ \partial_t u + \nabla \cdot (uu) - fv = -\frac{1}{\rho_0} \partial_x p + \mathcal{F}_u + \mathcal{D}_u, \quad (1a) $$

$$ \partial_t v + \nabla \cdot (uv) + fu = -\frac{1}{\rho_0} \partial_y p + \mathcal{F}_v + \mathcal{D}_v, \quad (1b) $$

$$ 0 = -\frac{1}{\rho_0} \partial_z p - b, \quad (1c) $$

where $u = (u, v, w)$ is the three-dimensional velocity field, $f = 2\Omega\sin(\theta)$ is the traditional Coriolis pseudo-force, $p$ is pressure, $b = \frac{\rho-\rho_0}{\rho_0}g$ is the buoyancy defined for Boussinesq fluids (i.e. when density $\rho$ is replaced by its constant value $\rho_0$,
unless multiplied by gravity in which case it is expressed as density anomaly $\frac{\rho-\rho_0}{\rho_0}$), and $\mathcal{F}$ and $\mathcal{D}$ are forcing and dissipative processes, respectively. Equations (1a)–(1c) are the HPE momentum equations used in current climate models. From a numerical viewpoint, using HPE instead of general NS or other Non-Hydrostatic (NH) sets of equation greatly simplifies the procedure as only (1a) and (1b) have to be stepped forward in time for each discretized ocean layers, while (1c) is used (diagnostically) to obtain the pressure field through vertical integration of density variations subject to gravitational acceleration. In HPE models, convection is part of the parameterized (i.e. unresolved) three-dimensional turbulence and mixing processes which are encapsulated in $D_{u,v}$. Usually, these operators are formulated with a down-gradient approach, where the vertical fluxes of a scalar $\theta$ are parameterized as $\overline{w'\theta'} = -K_{\theta}\partial_z\overline{\theta}$, with $\overline{\theta}$ the local, resolved field. Several models can be used to estimate the dissipation coefficient $K_{\theta}$ (e.g. TKE (Gaspar et al. 1990), GLS (Umlauf and Burchard 2003), KPP (Large et al. 1994)), but in case of convection, this coefficient is usually set to an unrealistically large value (0.1 to $10^2 \text{ m}^2\text{s}^{-1}$) to quickly restore static instabilities associated with convective processes and avoid model instabilities. More recently, Giordani et al. (2020) proposed an oceanic application of the eddy-diffusivity mass-flux formulation initially derived by the atmospheric community (e.g., Hourdin et al. 2006, Suselj et al. 2019), which allows a better representation of vertical advective fluxes associated with convection. The approximations leading to HPE are likely to be satisfied in most of the ocean where vertical velocities are small and their spatial patterns are of small scales. However, for the case of deep ocean convection where vertical velocities can reach $W \sim \mathcal{O}(10 \text{ cm s}^{-1})$ and over horizontal scales of $\mathcal{L} \sim \mathcal{O}(1 \text{ km})$, such approximations become questionable. In case such approximations turn out to be violated, it becomes necessary to find ways of re-introducing some form of non-hydrostasy within HPE.

Klingbeil and Burchard (2013) have proposed a direct implementation of full non-hydrostatic effects into an HPE model through a pressure correction method. Instead of solving the full three-dimensional velocity field equations

$$ \partial_t u + \nabla \cdot (uu) - fv + \tilde{f}w = -\frac{1}{\rho_0} \partial_x p + \mathcal{F}_u + \mathcal{D}_u, \quad (2a) $$

$$ \partial_t v + \nabla \cdot (uv) + fu = -\frac{1}{\rho_0} \partial_y p + \mathcal{F}_v + \mathcal{D}_v, \quad (2b) $$

$$ \partial_t w + \underline{\nabla \cdot (uw)} - \tilde{f}u = -\frac{1}{\rho_0} \partial_z p - b + \mathcal{F}_w + \underline{\mathcal{D}_w}, \quad (2c) $$

where $\tilde{f} = 2\Omega\cos(\theta)$ is the non-traditional Coriolis pseudo-force (shown for consistency but not considered in the following), and non-hydrostatic contributions are shown in blue. To avoid the complexity of solving a three dimensional Poisson equation to recover the non-hydrostatic pressure (as usually done in NH pressure correction methods, e.g. Marshall et al. 1997) Klingbeil and Burchard (2013)
proposed to account for non-hydrostatic pressure correction through a vertical integration of a so-called *non-hydrostatic buoyancy*, i.e. following the strategy of HPE models. This strategy offers a general implementation of NH physics in HPE, but still suffers from numerical instabilities in the case of strongly non-hydrostatic dynamics. For the case of deep ocean convection, it can be shown that further simplifications can be made by only accounting for the horizontal viscosity acting on the vertical velocities in the computation of the NH pressure correction (through vertical integration; Pierre Garreau, personal communication). As will be shown later through the analysis of different idealized Large Eddy Simulations (LES), HPE models tend to produce convective plumes near the grid size of the model, leading to unstructured (on the horizontal) convective cells. From one grid point to the next, vertical velocities could be of opposite sign leading to intense horizontal gradients. Including a horizontal viscous operator on the HPE vertical velocities (we recall here that in HPE vertical velocities are diagnosed from the horizontal velocity field though continuity) leads to a broadening of the convective plumes and a more realistic horizontal organization. In other words, when convective plumes start to form, they 'entrain' the neighboring points thus communicating horizontally their vertical momentum. Such a process can be seen as a simplified entrainment/detrainment mechanism discussed by Giordani et al. (2020) for the case of eddy-diffusivity mass-flux parameterization. In the present study, we consider the approach of Klingbeil and Burchard (2013) as a starting point and discuss a strategy to extend this idea in the context of a stochastic parameterization. The results presented here are all obtained at LES resolution, such that a clear connection with climate scale regimes is still lacking. However, these results provide a first step toward the development of robust stochastic parameterization for climate models, which will be the subject of dedicated studies.

The paper is organized as follows. In Sect. 2 we briefly recall the Location Uncertainty (LU; Mémin 2014, Bauer et al. 2020, Resseguier et al. 2017) framework used to represent the inertial and dissipative effects on vertical momentum (underlined terms in (2c)) as a result of a strong noise regime or for application to flow dynamics where the hydrostatic approximation becomes questionable. Section 3 is dedicated to the numerical implementation of the stochastic, non-hydrostatic pressure correction into the terrain-following Coastal and Regional Ocean Community (CROCO), along with the description of the simulations we have conducted. Preliminary results are described and discussed in Sect. 4. We summarize our paper and provide some perspectives for further work in Sect. 5.

## 2 Stochastic Formulation of Direct Non-hydrostatic Pressure Correction

Following Mémin (2014), the stochastic version of the horizontal momentum equation (in vector notation) reads:
$$ \mathbb{D}_t u_h + f k \times (u_h dt + \sigma d\mathbf{B}_t^H) = -\frac{1}{\rho_0} \nabla_H (pdt + dp_t^\sigma), \quad (3) $$

with $u_h = (u, v, 0)$, $\mathbb{D}_t$ the stochastic transport operator defined as:

$$ \mathbb{D}_t u_h = d_t u_h + (u^* dt + \sigma d\mathbf{B}_t) \cdot \nabla u_h - \frac{1}{2} \nabla \cdot (a \nabla u_h) dt, \quad (4) $$

with $u^*$ the incompressible (i.e. $\nabla \cdot \sigma d\mathbf{B}_t = 0$) modified advection defined as:

$$ u^* = u - \frac{1}{2} \nabla \cdot a \quad (5) $$

where $u = (u_h, w)$ is the three dimensional velocity field, $\sigma d\mathbf{B}_t$ represents the stochastic flow and $a$ its associated variance tensor. The term $\frac{1}{2} \nabla \cdot a$ can be interpreted as an equivalent of the Stokes drift for an inhomogeneous random fast component $\sigma d\mathbf{B}_t$ (Bauer et al. 2020).

The introduction of the stochastic pressure $dp_t^\sigma$ in (3) requires some discussion. This stochastic pressure is associated with the small scale velocity component encoded through the noise. Following Resseguier et al. (2017), for smooth-in-time momentum equation subject to a classical deterministic large scale momentum equation, its (three dimensional) gradient can be expressed as:

$$ -\frac{1}{\rho_0} \nabla p_t^\sigma = (\sigma d\mathbf{B}_t) \cdot \nabla u + f \times \sigma d\mathbf{B}_t \quad (6) $$

such that its interpretation (and scalling) should be related to the processes the stochastic formulation aims at representing. In the context of large scale modelling parameterization such as Tucciarone et al. (2023), the stochastic Primitive Equations they derived is meant to represent the effects of meso (and potentially submeso) scale eddies onto the large scale gyre circulation. The usual hydrostatic arguments are thus used, such that the vertical gradient of the stochastic pressure is identically zero (i.e. $\partial_z d p_t^\sigma = 0$) and its horizontal gradient is strictly balanced by the stochastic Coriolis pseudo-force. Here, we are interested in relaxing the hydrostatic approximation on the noise structure, but retaining it for the smooth-in-time, resolved flow, and derive the stochastic equation for the vertical momentum. As a first step in this direction, we will not include the contribution of the non-traditional Coriolis pseudo-force. After some manipulations, we obtain the following equation for the vertical momentum:

$$ \left(-\frac{1}{2} \nabla \cdot a\right) \cdot \nabla w dt - \frac{1}{2} \nabla \cdot (a \nabla w) dt + \sigma d\mathbf{B}_t \cdot \nabla w = \left(-\frac{1}{\rho_0} \partial_z p - b\right) dt - \frac{1}{\rho_0} \partial_z d p_t^\sigma, \quad (7) $$

Black terms in (7) are associated with hydrostatic physics and terms in blue are the different stochastic contributions that emerge when applying non-hydrostatic