so that $\lim_{k \to \infty} \mathbb{P}(l_k = \tau) = 1$. Hence, using the stochastic Grönwall lemma of [28] with $\eta < \zeta < l_k \wedge T$, for all $k$ there exists a $k$-dependent constant $C_k$, such that

$$ \mathcal{E}(0, l_k \wedge T) \leq C_k(1 + \|v_0\|_V^4). \qquad (6.47) $$

Furthermore, by Itô's lemma,

$$ \begin{aligned} d_t \|\partial_z v\|_{L^2}^4 + 4c \|\partial_z v\|_{L^2}^2 \|\nabla_3 \partial_z v\|_{L^2}^2 dt & \leq -4 \|\partial_z v\|_{L^2}^2 ((\partial_z v \cdot \nabla_H) v - (\nabla_H \cdot v) \partial_z v, \partial_z v)_{L^2} dt \\ & \quad + 4 \|\partial_z v\|_{L^2}^2 (g[\beta_T \nabla_H T + \beta_S \nabla_H S], \partial_z v)_{L^2} dt - 4 \|\partial_z v\|_{L^2}^2 (\partial_z \hat{F}_\sigma^v(U), \partial_z v)_{L^2} dt \\ & \quad + 4 \|\partial_z v\|_{L^2}^2 (\mathbf{P} \nabla_H [\nabla_3 \cdot (a_3^K \nabla_3(w(v) + w_s))] + K * [(u_s \cdot \nabla_3)(w(v) + w_s)]), \partial_z v)_{L^2} dt \\ & \quad + 2 \|\partial_z v\|_{L^2}^2 \|\partial_z \hat{G}_\sigma^v(U) - \partial_z \mathcal{M}^{U,\sigma}\|_{L_2(W,L^2)}^2 dt + 4 \|[(\partial_z \hat{G}_\sigma^v(U) - \partial_z \mathcal{M}^{U,\sigma})^* \partial_z v\|_{L_2(W,L^2)}^2 dt \\ & \quad + 4 \|\partial_z v\|_{L^2}^2 \sum_{k=0}^{\infty} \int_S \left[ \partial_z \hat{G}_\sigma^v(U)(\phi_k) - \partial_z \nabla_H \pi_k \right] \cdot \partial_z v \, d\beta_t^k. \end{aligned} \qquad (6.48) $$

Thus, reasoning as in subsection 6.2,

$$ \begin{aligned} \mathbb{E} \left[ \sup_{[\eta, \zeta]} \|\partial_z v\|_{L^2}^4 \right] + \mathbb{E} \left[ \int_{\eta}^{\zeta} \|\partial_z v\|_{L^2}^2 \|\nabla_3 \partial_z v\|_{L^2}^2 ds \right] & \leq C \mathbb{E} \left[ 1 + \|\partial_z v(\eta)\|_{L^2}^4 \right] \\ & + C \mathbb{E} \left[ \int_{\eta}^{\zeta} (1 + \|U\|_V^2 + \int_S |\nabla \tilde{v}|^2 |\tilde{v}|^2) (1 + \|\partial_z v\|_{L^2}^4) ds \right] \\ & + C \mathbb{E} \left[ \int_{\eta}^{\zeta} \|[(\partial_z \hat{G}_\sigma^v(U) - \partial_z \mathcal{M}^{U,\sigma})^* \partial_z v\|_{L_2(W,L^2)}^2 ds \right]. \end{aligned} \qquad (6.49) $$

It remains to estimate the last term on the RHS. Using the notations of subsection 6.2, notice that,

$$ \begin{aligned} \mathbb{E} \left[ \int_{\eta}^{\zeta} \|[(\partial_z \hat{G}_\sigma^v(U) - \partial_z \mathcal{M}^{U,\sigma})^* \partial_z v\|_{L_2(W,L^2)}^2 ds \right] & \leq C \mathbb{E} \left[ \int_{\eta}^{\zeta} \sum_{k=0}^{\infty} \left| (\partial_z \psi_k - \partial_z (\phi_k \cdot \nabla_3 v) - \mathbf{P} \nabla_H (\phi_k \cdot \nabla_3 w(v)), \partial_z v)_{L^2} \right|^2 ds \right] \\ & \leq C \mathbb{E} \left[ (1 + \sup_{[\eta, \zeta]} \|\partial_z v\|_{L^2}^4) \int_{\eta}^{\zeta} (1 + \|\nabla_3 \partial_z v\|_{L^2}^2) ds \right], \end{aligned} \qquad (6.50) $$

so that

$$ \begin{aligned} \mathbb{E} \left[ \sup_{[\eta, \zeta]} \|\partial_z v\|_{L^2}^4 \right] + \mathbb{E} \left[ \int_{\eta}^{\zeta} \|\partial_z v\|_{L^2}^2 \|\nabla_3 \partial_z v\|_{L^2}^2 ds \right] & \leq C \mathbb{E} \left[ 1 + \|\partial_z v(\eta)\|_{L^2}^4 \right] \\ & + C \mathbb{E} \left[ \int_{\eta}^{\zeta} (1 + \|U\|_V^2 + \int_S |\nabla \tilde{v}|^2 |\tilde{v}|^2 + \|\nabla_3 \partial_z v\|_{L^2}^2) (1 + \|\partial_z v\|_{L^2}^4) ds \right]. \end{aligned} \qquad (6.51) $$

Hence, the stochastic Grönwall lemma yields, with $\eta < \zeta < l_k \wedge T$ again,

$$ \mathbb{E} \left[ \sup_{[0, l_k \wedge T]} \|\partial_z v\|_{L^2}^4 \right] + \mathbb{E} \left[ \int_0^{l_k \wedge T} \|\partial_z v\|_{L^2}^2 \|\nabla_3 \partial_z v\|_{L^2}^2 ds \right] \leq C_k \mathbb{E} \left[ 1 + \|\partial_z v(0)\|_{L^2}^4 \right]. \qquad (6.52) $$

□

## 6.5 Globality-in-time of the solution

Our final argument for showing the existence of a global pathwise solution is inspired by what was proposed in [2] (see the arguments p.38-39), yet different. In the following, we use a result that is similar to Proposition 4.1 of [2] [Part 4], which we prove in the appendix (Appendix A): let $T > 0$,
and define two stopping times $\eta, \zeta < \tau \wedge T$, such that $\eta < \zeta$. Then the following holds,

$$ \mathbb{E}\left[\sup_{s \in [\eta, \zeta]} \|v\|_V^2\right] + \mathbb{E}\left[\int_{\eta}^{\zeta} \|v\|_{\mathcal{D}(A)}^2 ds\right] \le C \mathbb{E}\left[1 + \|v(\eta)\|_V^2 + \int_{\eta}^{\zeta} \|v \cdot \nabla v\|_{L^2}^2 ds + \int_{\eta}^{\zeta} \|w(v) \partial_z v\|_{L^2}^2 ds\right]. \quad (6.53) $$

Moreover, remark that by Hölder inequality and Sobolev embeddings,

$$ \mathbb{E}\left[\int_{\eta}^{\zeta} \|v \cdot \nabla v\|_{L^2}^2 ds\right] \le C_{\xi} \mathbb{E}\left[\int_{\eta}^{\zeta} (1 + \|v\|_V^2) ds\right] + \xi \mathbb{E}\left[\int_{\eta}^{\zeta} \|v\|_{\mathcal{D}(A)}^2 ds\right]. \quad (6.54) $$

Furthermore,

$$ \begin{aligned} \|w(v) \partial_z v\|_{L^2(S)}^2 &\le \int_{-h}^{0} \|w(v)\|_{L^4(S_H)}^2 \|\partial_z v\|_{L^4(S_H)}^2 dz \\ &\le \left( \sup_{[-h,0]} \|w(v)\|_{L^4(S_H)}^2 \right) \left( \int_{-h}^{0} \|\partial_z v\|_{L^4(S_H)}^2 dz \right) \\ &\le C \left( \int_{-h}^{0} \|\nabla \cdot v\|_{L^4(S_H)}^2 dz \right) \left( \int_{-h}^{0} \|\partial_z v\|_{L^4(S_H)}^2 dz \right) \\ &\le C \left( \int_{-h}^{0} \|v\|_{H^1(S_H)} \|v\|_{H^2(S_H)} dz \right) \left( \int_{-h}^{0} \|\partial_z v\|_{L^2(S_H)} \|\partial_z v\|_{H^1(S_H)} dz \right) \\ &\le C \|v\|_{H^1} \|v\|_{H^2} \|\partial_z v\|_{L^2} \|\partial_z v\|_{H^1} \le C_{\xi} \|v\|_V^2 \|\partial_z v\|_{L^2}^2 \|\partial_z v\|_{H^1}^2 + \xi \|v\|_{\mathcal{D}(A)}^2. \end{aligned} \quad (6.55) $$

Thus,

$$ \mathbb{E}\left[\int_{\eta}^{\zeta} \|w(v) \partial_z v\|_{L^2(S)}^2 ds\right] \le C_{\xi} \mathbb{E}\left[\int_{\eta}^{\zeta} \|v\|_V^2 \|\partial_z v\|_{L^2}^2 \|\partial_z v\|_{H^1}^2 ds\right] + \xi \mathbb{E}\left[\int_{\eta}^{\zeta} \|v\|_{\mathcal{D}(A)}^2 ds\right]. \quad (6.56) $$

Using those estimates in equation (6.53), we reach

$$ \begin{aligned} \mathbb{E}\left[\sup_{[\eta, \zeta]} \|v\|_V^2\right] + \mathbb{E}\left[\int_{\eta}^{\zeta} \|v\|_{\mathcal{D}(A)}^2 ds\right] \le C \mathbb{E}\left[1 + \|v(\eta)\|_V^2\right] \\ + C \mathbb{E}\left[\sup_{[\eta, \zeta]} \|v\|_V^2 \int_{\eta}^{\zeta} (1 + \|\partial_z v\|_{L^2}^2 \|\partial_z v\|_{H^1}^2) ds\right]. \end{aligned} \quad (6.57) $$

Therefore, by the stochastic Grönwall lemma of [1] (rather than the one of [28]), the following holds a.s.,

$$ \sup_{[0, l_k \wedge T]} \|v\|_V^2 + \int_0^{l_k \wedge T} \|v\|_{\mathcal{D}(A)}^2 ds < \infty. \quad (6.58) $$

Since $\lim_{k \to \infty} \mathbb{P}(l_k = \tau) = 1$, we deduce that, almost surely,

$$ \sup_{[0, \tau \wedge T]} \|v\|_V^2 + \int_0^{\tau \wedge T} \|v\|_{\mathcal{D}(A)}^2 ds < \infty. \quad (6.59) $$

Thus, $\tau = \infty$ a.s. The aforementioned solution to the problem $(\hat{P}_K)$ is hence global-in-time, with values almost surely in the space

$$ L_{\text{loc}}^2([0, +\infty), \mathcal{D}(A) \cap C([0, +\infty), V). $$

□

## 6.6 Continuity in probability with respect to the initial data and the noise operator

This subsection concerns the continuity of the solutions of $(\hat{P}_K)$ with respect to initial data and to the noise. We prove the second point of Theorem 2 below.
Let $U_0^n, U_0, \sigma^n, \sigma, U^n, U$ and $R^n$ as in the second point of Theorem 2, and let $T > 0$. Remark that,

$$R^n \in L^2([0, T], \mathcal{D}(A)) \cap C([0, T], V), \quad \forall T > 0 \text{ a.s.} \qquad (6.60)$$

Let $\eta < \zeta$ two positive stopping times. Reasoning as in the proof for uniqueness in subsection 5.2, we infer that

$$\mathbb{E} \left[ \sup_{[\eta, \zeta]} \|R^n\|_V^2 + \int_{\eta}^{\zeta} \|R^n\|_{\mathcal{D}(A)}^2 ds \right] \le \mathbb{E} \left[ \|R^n(0)\|_V^2 \right] + C \mathbb{E} \left[ \int_{\eta}^{\zeta} \|R\|_V^2 (1 + \|U\|_{\mathcal{D}(A)}^2 + \|U^n\|_{\mathcal{D}(A)}^2) ds \right] \\ + C \mathbb{E} \left[ \int_{\eta}^{\zeta} \delta_{\sigma}^n (1 + \|U\|_{\mathcal{D}(A)}^2 + \|U^n\|_{\mathcal{D}(A)}^2 + (\beta_{\sigma}^n)^2 + (\delta_{\sigma}^n)^2 (\beta_{\sigma}^n + 1)) ds \right], \quad (6.61)$$

where

$$\delta_{\sigma}^{n} := \sup_{[0,T]} \|\sigma^{n} - \sigma\|_{\mathcal{L}_{2}(W, H^{4})}^{2} + \|a_{3} - a_{3}^{n}\|_{H^{1}([0,T], H^{3})}, \qquad (6.62)$$

$$\beta_{\sigma}^{n} := \sup_{[0,T]} (\|\sigma\|_{\mathcal{L}_{2}(W,H^{4})}^{2} + \|\sigma^{n}\|_{\mathcal{L}_{2}(W,H^{4})}^{2}) + \|a_{3}\|_{H^{1}([0,T],H^{3})} + \|a_{3}^{n}\|_{H^{1}([0,T],H^{3})}.$$

In addition, we define

$$\mathcal{N}(\eta, \zeta; X) := \sup_{[\eta, \zeta]} \|X\|_V^2 + \int_{\eta}^{\zeta} \|X\|_{\mathcal{D}(A)}^2 ds, \qquad (6.63)$$

and denote by

$$\tau_{k,n}^* := \inf\{t > 0 \mid \mathcal{N}(0, t; U) + \mathcal{N}(0, t; U^n) > k\}. \qquad (6.64)$$

Using equation (6.61) with $(U^1, U^2, \tau') = (U, U^n, \tau_{k,n}^*)$, by the stochastic Grönwall lemma of [1], we infer that there exists a constant $C_0$ such that, for all $\epsilon, K > 0$, for all integer $k > 0$,

$$\mathbb{P}(\mathcal{N}(0, \tau_{k,n}^* \wedge T; R^n) \ge \epsilon) \le \frac{C_0}{\epsilon} e^{C_0 K} (\|R_0^n\|_V^2 + \delta_{\sigma}^n) \\ + \mathbb{P} \left( \int_0^{\tau_{k,n}^* \wedge T} (\|U\|_{\mathcal{D}(A)}^2 + \|U^n\|_{\mathcal{D}(A)}^2) ds + \chi_n \ge K \right), \quad (6.65)$$

where $\chi_n := T\delta_{\sigma}^n(1 + \beta_{\sigma}^n) + T(1 + (\beta_{\sigma}^n)^2)$. Since $\delta_{\sigma}^n$ and $\beta_{\sigma}^n$ are bounded, so is $\chi_n$. Then, denote by $\hat{\chi} = \sup_n \chi_n$, and set $K = k + 1 + \hat{\chi}$, so that

$$\mathbb{P}(\mathcal{N}(0, \tau_{k,n}^* \wedge T; R^n) \ge \epsilon) \le \frac{C_0}{\epsilon} e^{C_0(k+1+\hat{\chi})} (\|R_0^n\|_V^2 + \delta_{\sigma}^n). \qquad (6.66)$$

Hence, by choosing $k_n = \left\lfloor \frac{-\ln(\|R_0^n\|_V^2 + \delta_{\sigma}^n)}{2C_0} \right\rfloor \to \infty$ for a large enough $n$, we find that

$$\mathbb{P}(\mathcal{N}(0, \tau_{k,n}^* \wedge T; R^n) \ge \epsilon) \le \frac{C_0}{\epsilon} e^{C_0(1+\hat{\chi})} \sqrt{\|R_0^n\|_V^2 + \delta_n} \to 0. \qquad (6.67)$$

As the solutions $U$ and $U^n$ are global-in-time a.s., $\tau_{k,n}^* \to \infty$ in probability. Therefore,

$$\mathbb{P}(\mathcal{N}(0, T; R^n) \ge \epsilon) \to 0, \qquad (6.68)$$

that is to say $U^n \to U$ in probability, in the space $C([0, T], V) \cap L^2([0, T], \mathcal{D}(A))$.

□

## A Appendix 1: Proof of equation (6.53) – An estimate on the velocity and the temperature

We show equation (6.53), which is similar to Proposition 4.1 of [2], and which we use in the final globality argument. The result is the following lemma,

**Lemma A.1** Let $T > 0$ a real number. Let $U = (v, T, S)^T$ the solution of either $(\mathcal{P}_K)$ or $(\hat{\mathcal{P}}_K)$ – equations (3.16) and (3.22) – given by the second point of Theorem 1, and $\tau_0$ its associated a
stopping time. In addition, let two stopping times $\eta, \zeta < \tau_0 \wedge T$, such that $\eta < \zeta$. Then,

$$ \mathbb{E}\left[\sup_{s \in [\eta, \zeta]} \|v\|_V^2\right] + \mathbb{E}\left[\int_{\eta}^{\zeta} \|v\|_{D(A)}^2 ds\right] \le C \mathbb{E}\left[1 + \|v(\eta)\|_V^2 + \int_{\eta}^{\zeta} \|v \cdot \nabla v\|_{L^2}^2 ds + \int_{\eta}^{\zeta} \|w(v) \partial_z v\|_{L^2}^2 ds\right]. \quad (A.1) $$

**Proof:** We will prove this lemma for the problem $(\mathcal{P}_K)$, the proof for $(\hat{\mathcal{P}}_K)$ being similar. Remind first the definition of $(\mathcal{P}_K)$,

$$ \partial_t U^* + [AU^* + B(U^*) + \Gamma U^* + \frac{1}{\rho_0} \mathbf{P} \nabla_H p + F_\sigma(U^*)] dt = G_\sigma(U^*) dW_t - \frac{1}{\rho_0} \mathbf{P} \nabla_H dp_t^\sigma. \quad (A.2) $$

Then, let $\theta = (T, S)^{\mathrm{T}}$, so that

$$ \partial_t v^* + [Av^* + B(v^*) + \Gamma v^* + \frac{1}{\rho_0} \mathbf{P} \nabla_H p + F_\sigma^v(v^*)] dt = G_\sigma^v(v^*) dW_t - \frac{1}{\rho_0} \mathbf{P} \nabla_H dp_t^\sigma, \quad (A.3) $$

$$ \partial_t \theta + [A\theta + B(v^*, \theta) + F_\sigma^\theta(\theta)] dt = G_\sigma^\theta(\theta) dW_t, \quad (A.4) $$

where $F_\sigma := (F_\sigma^v, F_\sigma^\theta)^{\mathrm{T}}$ and $G_\sigma := (G_\sigma^v, G_\sigma^\theta)^{\mathrm{T}}$. Let $t > 0$, and apply Itô's lemma with $\|\cdot\|_V^2$ and $\|\cdot\|_H^2$,

$$ d_t \|v_n^*\|_V^2 = -2(Av_n^*, Av_n^* + B(v_n^*) + \Gamma v_n^* + \frac{1}{\rho_0} \mathbf{P} \nabla_H p + F_\sigma^v(v_n^*))_H dt \quad (A.5) $$

$$ + 2(Av_n^*, G_\sigma^v(v_n^*) dW_t - \frac{1}{\rho_0} \mathbf{P} \nabla_H dp_t^\sigma)_H + \|G_\sigma^v(v_n^*) - \mathcal{M}^{U,\sigma}\|_{L_2(W,V)}^2 dt, $$

$$ d_t \|\theta_n\|_H^2 = -2(\theta_n, A\theta_n + B(v_n^*, \theta_n) + F_\sigma^\theta(\theta_n))_H dt + 2(\theta_n, G_\sigma^\theta(\theta_n) dW_t)_H + \|G_\sigma^\theta(\theta_n)\|_{L_2(W,H)}^2 dt. \quad (A.6) $$

Also, remind that the regularising properties of $K$ lead to

$$ \frac{1}{2} \sum_{k=0}^{\infty} \| \mathbf{P} \nabla_H \pi_k \|_H^2 \le C(1 + \|U\|_H^2), \quad \frac{1}{2} \sum_{k=0}^{\infty} \| \mathbf{P} \nabla_H \pi_k \|_V^2 \le C(1 + \|U\|_V^2). \quad (A.7) $$

**Estimate of the bounded variation terms** For the advection term we have

$$ (Av_n^*, B^n(v_n^*))_H \le \|Av_n^*\|_H \|B^n(v_n^*)\|_H \le \xi \|AU_n^*\|_H^2 + C_\xi \left( \|v_n^* \cdot \nabla v_n^*\|_{L^2}^2 + \|w_n^* \partial_z v_n^*\|_{L^2}^2 \right). \quad (A.8) $$

Moreover, for all $\xi > 0$,

$$ |(Av_n^*, \Gamma^n(v_n^*))_H| + |(Av_n^*, (\frac{1}{\rho_0} \mathbf{P} \nabla_H p)^n)_H| \le C \|v_n^*\|_{D(A)} \|U_n^*\|_V \le C_\xi (1 + \|U_n^*\|_V^2) + \xi \|v_n^*\|_{D(A)}^2. \quad (A.9) $$

For $F_\sigma^v$, by Young's inequality,

$$ (Av_n^*, F_\sigma^v(v_n^*))_H \le M_\sigma^1 + M_\sigma^2 + C \|Av_n^*\|_H \le M_\sigma^1 + M_\sigma^2 + C_\xi + \xi \|Av_n^*\|_H, \quad (A.10) $$

with

$$ |M_\sigma^1| \le |b(v_n^*, v_s, Av_n^*)| \le C \|v_n^*\|_V^{1/2} \|v_n^*\|_{D(A)}^{3/2} \le C_\xi \|v_n^*\|_V^2 + \xi \|v_n^*\|_{D(A)}^2, \quad (A.11) $$

$$ \begin{aligned} M_\sigma^2 &= -\frac{1}{2}(D_{\mu,\nu}\nabla_3 \cdot a_3 \nabla_3 v_n^*, Av_n^*)_{L^2} = -\frac{1}{2}(D_{\mu,\nu}\partial_i(a_{ij}\partial_j v_{n,k}^*), -\mu_l \partial_{ll} v_{n,k}^*)_{L^2} \\ &= -\frac{1}{2}(D_{\mu,\nu}\partial_l(a_{ij}\partial_j v_{n,k}^*), -\mu_l \partial_{il} v_{n,k}^*)_{L^2} \\ &= -\frac{1}{2}(D_{\mu,\nu}(a_{3,ij}\partial_{jl}v_{n,k}^*), -\mu_l \partial_{il} v_{n,k}^*)_{L^2} + C_\xi \|U_n^*\|_V^2 + \xi \|U_n^*\|_{D(A)}^2, \end{aligned} \quad (A.12) $$

where $D_{\mu,\nu} = \mathrm{Diag}(\mu, \mu, \nu)$, and using Einstein's notation convention of summation over repeated indices, with $i, j, l \in \{x, y, z\}$ and $k \in \{x, y\}$. We also used the notation $\mu_x = \mu_y = \mu$ and $\mu_z = \nu$.
**Estimate of the martingale terms** Let $\psi_k = \mathbf{P}[\phi_k \cdot \nabla_3 v + \phi_k \cdot \nabla_3 v_s] + A\phi_k^H + C\phi_k^H$ and
$\psi_k^w = \mathbf{P}K * [\phi_k \cdot \nabla_3 w_s] + A\phi_k^z$, so that

$$
\begin{align*}
\frac{1}{2} \|G_{\sigma}^{n}(v_{n}^{*}) - \mathcal{M}^{U,\sigma}\|_{\mathcal{L}_{2}(W,V)}^{2} &= \frac{1}{2} \sum_{k=0}^{\infty} \|\psi_{k} + \psi_{k}^{w} - \mathbf{P}[\phi_{k} \cdot \nabla_{3} v_{n}^{*}] - \mathbf{P}\nabla_{H} \pi_{k}\|_{V}^{2} \\
&\leq \frac{1}{2} \sum_{k=0}^{\infty} \|\psi_{k} + \psi_{k}^{w}\|_{V}^{2} + \|\mathbf{P}[\phi_{k} \cdot \nabla_{3} v_{n}^{*}]\|_{V}^{2} + 2C \underbrace{|(\nabla_{3}(\phi_{k} \cdot \nabla_{3} v_{n}^{*}), \nabla_{3}(\psi_{k} + \psi_{k}^{w}))_{L^{2}}|}_{\leq C|b(\phi_{k}, v_{n}^{*}, \Delta_{3}(\psi_{k} + \psi_{k}^{w}))|} \\
&\quad + \|\mathbf{P}\nabla_{H} \pi_{k}\|_{V}^{2} + C \underbrace{|(\nabla_{3}\mathbf{P}\nabla_{H} \pi_{k}, \nabla_{3}(\psi_{k} + \psi_{k}^{w}))_{H}|}_{\leq C\|\mathbf{P}\nabla_{H} \pi_{k}\|_{H}\|\psi_{k} + \psi_{k}^{w}\|_{H}^{2}} + C|(\mathbf{P}[\phi_{k} \cdot \nabla_{3} v_{n}^{*}], \mathbf{P}\nabla_{H} \pi_{k})_{V}|. \quad \text{(A.13)}
\end{align*}
$$

Additionally, remark that

$$
\sum_{k=0}^{\infty} (\mathbf{P}[\phi_k \cdot \nabla_3 v_n^*], \mathbf{P}\nabla_H \pi_k)_V \le C_\xi \sum_{k=0}^{\infty} \|\mathbf{P}\nabla_H \pi_k\|_V^2 + \xi \|v_n^*\|_{D(A)}^2.
$$

Moreover, for all $\xi > 0$,

$$
\begin{align*}
\sum_{k=0}^{\infty} \|\mathbf{P}[\phi_k \cdot \nabla_3 v_n^*]\|_V^2 &\le \sum_{k=0}^{\infty} \|D_\mu^{1/2} \nabla_3 (\phi_k \cdot \nabla_3 v_n^*)\|_{L^2(S, \mathbb{R}^3)}^2 \\
&\le C_\xi \sum_{k=0}^{\infty} \|(\nabla_3 \phi_k)^\mathrm{T} \cdot \nabla_3 v_n^*\|_{L^2(S, \mathbb{R}^{3 \times 2})}^2 + (1+\xi) \sum_{k=0}^{\infty} \|D_\mu^{1/2} \phi_k \cdot \nabla_3 (\nabla_3 v_n^*)\|_{L^2(S, \mathbb{R}^{3 \times 2})}^2 \\
&\le C_\xi \|v_n^*\|_V^2 + (1+\xi) (D_{\mu,\nu} a_{3,ij} \partial_{il} v_{n,k}^*, \mu_l \partial_{il} v_{n,k}^*)_H,
\end{align*}
$$

where we used Einstein's notation with i, j, l ∈ {x, y, z} and k ∈ {x, y} again. Hence, for all ξ > 0,

$$
\frac{1}{2} \|G_{\sigma}^{m}(v_{n}^{*}) - \mathcal{M}^{U_{n},\sigma}\|_{\mathcal{L}_{2}(W,V)}^{2} \leq -M_{\sigma}^{2} + C_{\xi}(1 + \|v_{n}^{*}\|_{V}^{2}) + \xi \|U_{n}^{*}\|_{\mathcal{D}(A)}^{2}. \quad (A.14)
$$

Thus, using the equation (A.5), we reach

$$
d_t \|v_n^*\|_V^2 + \|v_n^*\|_{D(A)}^2 dt \le C \left( \|v_n^* \cdot \nabla v_n^*\|_H^2 + \|w_n^* \partial_z v_n^*\|_H^2 \right) \\
\qquad + 2(v_n^*, G_\sigma^v(v_n^*) dW_t)_H + C(1 + \|U_n^*\|_V^2) dt. \quad (A.15)
$$

For all $T > 0$ and all stopping times $0 \le \eta < \zeta \le T$, by integrating, then taking the supremum
and the expectation, we have

$$
\begin{align}
\mathbb{E}[\sup_{[\eta, \zeta]} \|v_n^*\|_V^2] + \mathbb{E}[\int_{\eta}^{\zeta} \|v_n^*\|_{D(A)}^2 dr] &\le \mathbb{E}[\|v_n^*(\eta)\|_V^2] + C \mathbb{E} \left[ \int_{\eta}^{\zeta} \|v_n^* \cdot \nabla v_n^*\|_{L^2}^2 + \|w_n^* \partial_z v_n^*\|_{L^2}^2 dr \right] \tag{A.16} \\
&\quad + C \mathbb{E} \left[ \int_{\eta}^{\zeta} (1 + \|U_n^*\|_V^2) dr \right] + \mathbb{E} \left[ \sup_{[\eta, \zeta]} \left| \int_{\eta}^{s} \sum_{k=0}^{\infty} I_7^k d\beta_s^k \right| \right]. \nonumber
\end{align}
$$

Furthermore,

$$
\begin{align*}
\mathbb{E} \left[ \sup_{[\eta, \zeta]} \left| \int_{\eta}^{s} \sum_{k=0}^{\infty} I_{\tau}^{k} d\beta_{\tau}^{k} \right| \right] &\le C \mathbb{E} \left[ \left( \int_{\eta}^{\zeta} \sum_{k=0}^{\infty} \underbrace{\left| (\psi_k + \psi_k^w - \mathbf{P}[\phi_k \cdot \nabla_3 v_n^*] - \mathbf{P}\nabla_H \pi_k, A v_n^*)_H \right|^2}_{\le C(\|\psi_k\|_H^2 + \|\phi_k\|_{L^\infty}^2 (1+\|v_n^*\|_V^2) + \|\mathbf{P}\nabla_H \pi_k\|_H^2)} ds \right)^{1/2} \right] \\
&\le C \mathbb{E} \left[ \left( \sup_{[\eta, \zeta]} (1 + \|v_n^*\|_V^2) \right)^{1/2} \left( \int_{\eta}^{\zeta} \|A v_n^*\|_H^2 ds \right)^{1/2} \right] \\
&\le C_{\xi} \mathbb{E} \left[ \sup_{[\eta, \zeta]} (1 + \|v_n^*\|_V^2) \right] + \xi \mathbb{E} \left[ \int_{\eta}^{\zeta} \|U_n^*\|_{D(A)}^2 ds \right]. \tag{A.17}
\end{align*}
$$
Hence, using Lemma 4.1,

$$
\begin{equation}
\begin{aligned}
\mathbb{E}[\sup_{[\eta, \zeta]} \|v_n^*\|_V^2] + \mathbb{E}[\int_{\eta}^{\zeta} \|v_n^*\|_{D(A)}^2 dr] &\le C \left(1 + \mathbb{E}[\|v_n^*(\eta)\|_V^2]\right) \\
&\quad + C \mathbb{E}\left[\int_{\eta}^{\zeta} \|v_n^* \cdot \nabla v_n^*\|_{L^2}^2 + \|w_n^* \partial_z v_n^*\|_{L^2}^2 dr\right].
\end{aligned}
\tag{A.18}
\end{equation}
$$

That is the result we seek.

<figure><img src="image_2.png" /></figure>

B Appendix 2: Energy estimates for Theorems 3 & 4

Now we give a sketch of the proof for the energy estimates for the problems ($\mathcal{P}_{EB}^{\gamma_r}$) and ($\mathcal{P}_{EV}^{\gamma_r}$) –
equations (3.32) and (3.31). These are used in the proof of Theorem 2 and 3, which run similarly
as for Theorem 1. Therefore, the other steps, including the straightforward existence of Galerkin
solutions, are omitted.

B.1 Energy estimate for the problem ($\mathcal{P}_{EB}^{\gamma_r}$) (Theorem 3)

Assume that $\gamma_r > 1$. For the energy estimates, we only need to check for the pressure term, the noise terms and the additional correction terms $\frac{1}{2}\mathbf{P}\nabla \cdot \hat{a}[\nabla w]$ and $\mathbf{P}\mathcal{C}_\sigma$.

Covariation correction term

$$
\frac{1}{2}(\mathbf{P}\nabla \cdot \hat{a}[\nabla w(v_n^*)], v_n^*)_H = -\frac{1}{2}(\hat{a}[\nabla w(v_n^*)], \nabla v_n^*)_L^2
$$

$$
\text{and } \|\mathbf{P}\mathcal{C}_{\sigma}\|_{H} \le \sum_{k=0}^{\infty} \|\phi_k\|_{L^{\infty}} \|\phi_k\|_{H^4} \le (\sum_{k=0}^{\infty} \|\phi_k\|_{L^{\infty}}^2)^{1/2} (\sum_{k=0}^{\infty} \|\phi_k\|_{H^4}^2)^{1/2} \le C.
$$

Bounded variation pressure term

$$
\begin{align*}
\left| \left(U_n^*, \frac{1}{\rho_0} \mathbf{P} \nabla_H p\right)_H \right| &\le \left| \left(g \nabla_H \int_z^0 (\beta_T T_n + \beta_S S_n) dz', v_n^*\right)_{L^2} \right| \\
&\quad + \left| \left(\nabla_H \int_z^0 \left[v_s \cdot \nabla_H w(v_n^*) + w_s \partial_z w(v_n^*)\right] dz', v_n^*\right)_{L^2} \right| \\
&\quad + \left( \nabla_H \int_z^0 \left[\frac{1}{2} \nabla \cdot \hat{a}[\nabla v_n^*] + \frac{1}{2} \nabla \cdot \hat{a}[\nabla w(v_n^*)] - \alpha(-\Delta)^{\gamma_r} w(v_n^*)\right] dz', v_n^* \right)_{L^2} \\
&\quad + \left| \left(\nabla_H \int_z^0 \left[v_s \cdot \nabla_H w_s + w_s \partial_z w_s - \left[\frac{1}{2} \nabla \cdot \hat{a}[\nabla v_s] + \frac{1}{2} \nabla \cdot \hat{a}[\nabla w_s]\right]\right] dz', v_n^* \right)_{L^2} \right| \\
&\quad + \left| \left(\nabla_H \int_z^0 \left[\frac{1}{2} \sum_k \phi_k \cdot \nabla \partial_z \pi_k + \frac{1}{2} \sum_k (\phi_k \cdot \nabla) \Delta \phi_k^z dz'\right], v_n^* \right)_{L^2} \right| \\
&\le \left| \left(v_s \cdot \nabla_H w(v_n^*) + w_s \partial_z w(v_n^*), w(v_n^*)\right)_{L^2} \right| + \left( \frac{1}{2} \left[ \nabla \cdot \hat{a}[\nabla v_n^*] + \nabla \cdot \hat{a}[\nabla w(v_n^*)] \right], w(v_n^*) \right)_{L^2} \\
&\quad - (\alpha(-\Delta)^{\gamma_r} w(v_n^*), w(v_n^*))_{L^2} + C \|U_n^*\|_H \|U_n^*\|_V + C \|U_n^*\|_V + \frac{1}{2} \left| \left( \sum_k \phi_k \cdot \nabla \partial_z \pi_k, w(v_n^*) \right)_{L^2} \right| \\
&\le -\frac{1}{2} \left( \hat{a}[\nabla v_n^*] + \hat{a}[\nabla w(v_n^*)], \nabla_3 w(v_n^*) \right)_{L^2} - \alpha \|w(v_n^*)\|_{H^{\gamma_r}}^2 + C \|U_n^*\|_H \|U_n^*\|_V + C \|U_n^*\|_V \\
&\quad + C \left( \sum_k \|\phi_k\|_{L^\infty} \right) \underbrace{\|\nabla w(v_n^*)\|_{L^2} \|\partial_z \pi_k\|_{L^2}}_{\le C(1+\|w(v_n^*)\|_{H^1}^2)}
\end{align*}
$$

where we used that $\phi_k$ is divergence-free, and that $\partial_z \pi_k = -\phi_k \cdot \nabla w + \Delta \phi_k^z$. Then, we may conclude
by Sobolev interpolation and Young's inequality.
First term of the total covariation noise contribution

$$
\begin{aligned}
\| [G_{\sigma}^{n}(U_{n}^{*}) - \mathcal{M}^{\sigma, U_{n}^{*}}]^{*} U_{n}^{*} \|_{H}^{2} &= \sum_{k=0}^{\infty} \left[ \left\langle (w(v_{n}^{*}), \phi_{k} \cdot \nabla_{3} w(v_{n}^{*}))_{L^{2}} + (w(v_{n}^{*}), \phi_{k} \cdot \nabla_{3} w_{s})_{L^{2}} \right\rangle \right. \\
&\quad \left. + \sum_{k=0}^{\infty} \left[ (w(v_{n}^{*}), A\phi_{k}^{z})_{L^{2}} - (v_{n}^{*}, [\phi_{k} \cdot \nabla_{3} v_{n}^{*}])_{L^{2}} - (v_{n}^{*}, \phi_{k} \cdot \nabla_{3} v_{s} + A\phi_{k}^{H} + \Gamma\phi_{k}^{H})_{L^{2}} \right] \right]^{2} \\
&\leq \sum_{k=0}^{\infty} C \|\phi_{k}\|_{H^{3}}^{2} (\|v_{n}^{*}\|_{H}^{2} + \|w(v_{n}^{*})\|_{L^{2}}^{2}) \leq C (\|U_{n}^{*}\|_{H}^{2} + \|w(v_{n}^{*})\|_{L^{2}}^{2}).
\end{aligned}
$$

Second term of the total covariation noise contribution

$$
\begin{aligned}
\frac{1}{2} \| G_{\sigma}^{n}(U_{n}^{*}) - \mathcal{M}^{\sigma, U_{n}^{*}} \|_{\mathcal{L}_{2}(W, H)}^{2} &= \frac{1}{2} \sum_{k=0}^{\infty} \| \mathbf{P}[\phi_{k} \cdot \nabla_{3} v_{n}^{*} + \phi_{k} \cdot \nabla_{3} v_{s}] + A\phi_{k}^{H} + \Gamma\phi_{k}^{H} \|_{H}^{2} \\
&\quad + \frac{1}{2} \sum_{k=0}^{\infty} \| \mathbf{P}\nabla_{H}\pi_{k} \|_{H}^{2} + \sum_{k=0}^{\infty} (\mathbf{P}[\phi_{k} \cdot \nabla_{3}(v_{n}^{*} + v_{s})] + A\phi_{k}^{H} + \Gamma\phi_{k}^{H}, \mathbf{P}\nabla_{H}\pi_{k})_{L^{2}} \\
&\leq \frac{1}{2} (a_{3}\nabla_{3}v_{n}^{*}, \nabla_{3}v_{n}^{*})_{L^{2}} + C_{\xi}(1 + \|U_{n}^{*}\|_{H}^{2}) + \xi(\|U_{n}^{*}\|_{V}^{2} + \|w(v_{n}^{*})\|_{H^{\gamma r}}^{2}) \\
&\quad + (\hat{a}[\nabla_{3}v_{n}^{*}] + \frac{1}{2}\hat{a}[\nabla_{3}w(v_{n}^{*})], \nabla_{3}w(v_{n}^{*}))_{L^{2}}.
\end{aligned}
$$

Using those inequalities, we conclude by similar arguments as in subsection 4.2 (Step 3). Notice
that the “new” covariation compensation terms lead to the following w-dependent energy term
($\hat{a}[\nabla_3 v_n^*] + \frac{1}{2}\hat{a}[\nabla_3 w(v_n^*)]$, $\nabla_3 w(v_n^*)$)$_{L^2}$, arising from the computation of the bounded variation pres-
sure associated energy $(U_n^*, \frac{1}{\rho_0} \mathbf{P} \nabla_H p)_H$. As expected, it balances exactly the quadratic covaration
term arising from the martingale component $\frac{1}{2} \|G_\sigma^n(U_n^*) - \mathcal{M}^{\sigma, U_n^*}\|_{L_2(W,H)}^2$. Furthermore, the
term $\xi \|w(v_n^*)\|_{H^{\gamma r}}^2$ is absorbed by the additional viscosity term $\alpha \|w(v_n^*)\|_{H^{\gamma r}}^2 \ge \xi \|w(v_n^*)\|_{H^1}^2$. In
particular, for all $T > 0$, there exists a constant $C$ such that, for all $n \in \mathbb{N}$,

$$
\int_0^T \|w(v_n^*)\|_{H^{\gamma r}}^2 \le C.
$$

□

B.2 Energy estimate for the problem ($\mathcal{P}_{EV}^{\gamma_r}$) (Theorem 4)

Assume that $\gamma_r > 2$. For the energy estimates, we only need to check for the bounded variation pressure and noise terms. Furthermore, notice that

$$
\| \mathbf{P} \nabla_H \pi_k \|_H^2 \le C(1 + \|w(v_n^*)\|_{H^2}^2) \le C_\xi(1 + \|U_n^*\|_H^2) + \xi \|w(v_n^*)\|_{H^{\gamma_r}}^2.
$$
## Bounded variation pressure term

$$ \begin{aligned} \left| \left( U_n^*, \frac{1}{\rho_0} \mathbf{P} \nabla_H p \right)_H \right| & \le \left| \left( g \nabla_H \int_z^0 (\beta_T T_n + \beta_S S_n) dz', v_n^* \right)_{L^2} \right| \\ & \quad + \left| \left( \nabla_H \int_z^0 \left[ (v_s \cdot \nabla_H + w_s \partial_z) w(v_n^*) \right] dz', v_n^* \right)_{L^2} \right| \\ & \quad + \left| \left( \nabla_H \int_z^0 \left[ \frac{1}{2} \nabla \cdot (a \nabla v_n^*) - \alpha(-\Delta)^{\gamma_r} w(v_n^*) \right] dz', v_n^* \right)_{L^2} \right| \\ & \quad + \left| \left( \nabla_H \int_z^0 \left[ (v_s \cdot \nabla_H + w_s \partial_z) w_s - \frac{1}{2} \nabla \cdot (a \nabla v_s) \right] dz', v_n^* \right)_{L^2} \right| \\ & \le \left| \left( v_s \cdot \nabla_H w(v_n^*) + w_s \partial_z w(v_n^*), w(v_n^*) \right)_{L^2} \right| + \left( \frac{1}{2} \nabla \cdot (a \nabla v_n^*), w(v_n^*) \right)_{L^2} \\ & \quad - \left( \alpha(-\Delta)^{\gamma_r} w(v_n^*), w(v_n^*) \right)_{L^2} + C (\|U_n^*\|_H + 1) \|U_n^*\|_V \\ & \le \frac{1}{2} \left| \left( a \nabla v_n^*, \nabla w(v_n^*) \right)_{L^2} \right| - \alpha \|w(v_n^*)\|_{H^{\gamma_r}}^2 + C_{\xi} (\|U_n^*\|_H^2 + 1) + \xi \|U_n^*\|_V^2. \end{aligned} $$

## First term of the total covariation noise contribution

$$ \begin{aligned} \left\| \left[ G_{\sigma}^{n}(U_{n}^{*}) - \mathcal{M}^{\sigma, U_{n}^{*}} \right]^{*} U_{n}^{*} \right\|_{H}^{2} &= \sum_{k=0}^{\infty} \left[ \left( (w(v_{n}^{*}), \phi_{k} \cdot \nabla_{3} w(v_{n}^{*}))_{L^{2}} + (w(v_{n}^{*}), \phi_{k} \cdot \nabla_{3} w_{s})_{L^{2}} \right) \right. \\ & \qquad \left. + \sum_{k=0}^{\infty} \left[ (w(v_{n}^{*}), A\phi_{k}^{z})_{L^{2}} - \left( \underbrace{(v_{n}^{*}, [\phi_{k} \cdot \nabla_{3} v_{n}^{*}])}_{\text{under}} \right)_{L^{2}} - (v_{n}^{*}, \phi_{k} \cdot \nabla_{3} v_{s} + A\phi_{k}^{H} + \Gamma\phi_{k}^{H})_{L^{2}} \right] \right]^{2} \\ & \le \sum_{k=0}^{\infty} C \|\phi_{k}\|_{H^{3}}^{2} \|v_{n}^{*}\|_{H}^{2} \le C \|U_{n}^{*}\|_{H}^{2}. \end{aligned} $$

## Second term of the total covariation noise contribution

$$ \begin{aligned} \frac{1}{2} \left\| G_{\sigma}^{n}(U_{n}^{*}) - \mathcal{M}^{\sigma, U_{n}^{*}} \right\|_{L_{2}(W, H)}^{2} &= \frac{1}{2} \sum_{k=0}^{\infty} \left\| \mathbf{P}[\phi_{k} \cdot \nabla_{3} v_{n}^{*} + \phi_{k} \cdot \nabla_{3} v_{s}] + A\phi_{k}^{H} + \Gamma\phi_{k}^{H} \right\|_{H}^{2} + \left\| \mathbf{P} \nabla_{H} \pi_{k} \right\|_{H}^{2} \\ & \quad + \sum_{k=0}^{\infty} (\phi_{k} \cdot \nabla_{3}(v_{n}^{*} + v_{s}) + A\phi_{k}^{H} + \Gamma\phi_{k}^{H}, \mathbf{P} \nabla_{H} \pi_{k})_{L^{2}} \\ & \le \frac{1}{2} (a_{3} \nabla_{3} v_{n}^{*}, \nabla_{3} v_{n}^{*})_{L^{2}} + C_{\xi} (1 + \|U_{n}^{*}\|_{H}^{2}) + \xi (\|U_{n}^{*}\|_{V}^{2} + \|w(v_{n}^{*})\|_{H^{\gamma_{r}}}^{2}) + \underbrace{\frac{1}{2} \left( a \nabla v_{n}^{*}, \nabla w(v_{n}^{*}) \right)_{L^{2}}}_{\le C \|U_{n}^{*}\|_{H} \|w(v_{n}^{*})\|_{H^{2}}} . \end{aligned} $$

Again, we conclude by using similar arguments as in subsection 4.2 (Step 3).

<figure><img src="image_3.png" /></figure>

**Acknowledgements:** The authors acknowledge the support of the ERC EU project 856408-STUOD and benefit from the support of the French government "Investissements d'Avenir" program ANR-11-LABX-0020-01. AD acknowledges the support of the ANR project ADA.

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