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# Stochastic Modeling in Geophysical Fluid Dynamics

Long Li

► To cite this version:

Long Li. Stochastic Modeling in Geophysical Fluid Dynamics. 2024 - CALISTO Team Seminar, Oct 2024, Valbonne, France. http://dx.doi.org/10.4957/880

HAL Id: hal-04957880

https://hal.science/hal-04957880v1

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Distributed under a Creative Commons Attribution 4.0 International License
# Stochastic Modeling in Geophysical Fluid Dynamics
## CALISTO Team Seminar

Long Li

ERC STUOD Research Associate

ODYSSEY Team, INRIA, IRMAR, Rennes, France

October 1, 2024, Inria, Valbonne, France

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# 1. Introduction

## 2. Modeling

### 2.1 Stochastic Transport Formulation

### 2.2 Geophysical Flow Modeling

### 2.3 Numerical Modeling of Uncertainty

## 3. Applications

### 3.1 Large-Scale Atmosphere Dynamics

### 3.2 Mesoscale Ocean Dynamics

### 3.3 Submesoscale Ocean Dynamics

### 3.4 Upper Ocean Boundary Layer Dynamics

## 4. Conclusion and Perspectives
Multi-scale dynamics ⇒ Complex interactions ⇒ Numerical modeling challenges

<figure><img src="image_8.png" /></figure>
Motivations

* Rigorously model the impact of unresolved processes on resolved scales

* Enhance uncertainty representation to construct physically meaningful ensembles

* Strong interest in applying ensemble methods for forecasting and data assimilation

⇒ Transition from a deterministic, causal physics approach to a stochastic representation of geophysical fluid dynamics
# 1. Introduction

# 2. Modeling

## 2.1 Stochastic Transport Formulation

## 2.2 Geophysical Flow Modeling

## 2.3 Numerical Modeling of Uncertainty

# 3. Applications

## 3.1 Large-Scale Atmosphere Dynamics

## 3.2 Mesoscale Ocean Dynamics

## 3.3 Submesoscale Ocean Dynamics

## 3.4 Upper Ocean Boundary Layer Dynamics

# 4. Conclusion and Perspectives
# 1. Introduction

# 2. Modeling

## 2.1 Stochastic Transport Formulation

### 2.2 Geophysical Flow Modeling

### 2.3 Numerical Modeling of Uncertainty

# 3. Applications

## 3.1 Large-Scale Atmosphere Dynamics

### 3.2 Mesoscale Ocean Dynamics

### 3.3 Submesoscale Ocean Dynamics

### 3.4 Upper Ocean Boundary Layer Dynamics

# 4. Conclusion and Perspectives
* Stochastic Lagrangian flow:

$$d\mathbf{X}_t = \underbrace{\mathbf{v}(\mathbf{X}_t, t) dt}_{\text{resolved time-correlated}} + \underbrace{\sigma(\mathbf{X}_t, t) dB_t}_{\text{unresolved time-uncorrelated}}$$

X: Lagrangian displacement within bounded domain $S \subset \mathbb{R}^d(d = 2, 3)$, B: Cylindrical Brownian motion on $(\Omega, \mathcal{F}, (F_t)_t, \mathbb{P})$ taking values in the Hilbert space $\mathcal{H} := L^2(S)$

* Correlation operator:

$$\sigma(\mathbf{x}, t)f = \int_S \check{\sigma}(\mathbf{x}, \mathbf{y}, t)f(\mathbf{y}) d\mathbf{y}, \quad f \in \mathcal{H}, \quad \mathbf{x} \in S$$

$\check{\sigma}$: bounded symmetric integral kernel, $\sigma$: Hilbert-Schmidt operator

* Spectral decomposition:

$$\sigma(\mathbf{x}, t) dB_t = \sum_{n \in \mathbb{N}} \phi_n(\mathbf{x}, t) d\beta_t^n$$

($\phi_n$)$_n$: a family of eigenfunctions (scaled by eigenvalues) of $\sigma$, ($\beta_t^n$)$_n$: a sequence of independent standard Brownian motions
* Stochastic transport operator:

$$ \mathbb{D}q := dq + (\mathbf{v}^* dt + \sigma dB_t) \cdot \nabla q - \nabla \cdot (a \nabla q) dt = 0 $$

* Diffusion coefficient tensor:

$$ \mathbf{a}(\mathbf{x}, t) = \frac{1}{2} \int_S \mathbf{\tilde{\sigma}}(\mathbf{x}, \mathbf{y}, t) \mathbf{\tilde{\sigma}}^T(\mathbf{y}, \mathbf{x}, t) \mathbf{d}\mathbf{y} = \frac{1}{2} \sum_{n \in \mathbb{N}} (\phi_n \phi_n^T)(\mathbf{x}, t) $$

* Effective advection drift:

$$ \mathbf{v}^* := \mathbf{v} - \underbrace{\nabla \cdot \mathbf{a}}_{\text{noise}} + \underbrace{\sigma(\nabla \cdot \mathbf{\sigma})}_{\text{noise compressibility}} $$

* Stochastic Reynolds Transport Theorem:

$$ d\left(\int_{V(t)} q(\mathbf{x}, t) \mathrm{d}\mathbf{x}\right) = \int_{V(t)} \left(\mathbb{D}q + q \nabla \cdot \left(\mathbf{v}^* dt + \sigma dB_t\right)\right) \mathrm{d}\mathbf{x} $$

⇒ Passage from Lagrangian to Eulerian coordinates
Under incompressible flow ($\nabla \cdot \mathbf{v}^* = \nabla \cdot \sigma = 0$) and adequate boundary conditions (e.g., torus):

* Energy conservation:

$$ 
\begin{aligned}
\frac{1}{2} d (\|q\|_{\mathcal{H}}^2) &= \int_S (q \, dq + \frac{1}{2} d\langle q \rangle_t) \, dx \\
&= \underbrace{(\int_S q \nabla \cdot (a \nabla q) \, dx)}_{\text{energy loss by diffusion}} + \underbrace{\int_S (\nabla q)^T a \nabla q \, dx}_{\text{energy intake by noise}}
\end{aligned}
$$
$$ dt = 0 $$

⇒ Fluctuation-dissipation theorem encoded directly by stochastic transport

* High-order conservation ($p > 1$):

$$ \mathbb{D} \left( \frac{1}{\rho} q^p \right) = q^{p-1} \mathbb{D} q = 0, \quad d \left( \int_S \frac{1}{\rho} q^p \, dx \right) = 0 $$

⇒ It shares same conservation properties as deterministic framework
* Stratonovich representation [Bauer et al., 2020]:

$$Dq = dq + (\mathbf{v}^* dt + \sigma \circ dB_t) \cdot \nabla q$$

* Girsanov transformation [Li et al., 2023a, Li et al., 2023b]:

$$W_t = B_t + \int_0^t \Gamma_s \, ds, \quad t \in [0, T]$$

$$\frac{dQ}{dP} = \exp \left( \int_0^T \left\langle \Gamma_t, dB_t \right\rangle_H - \frac{1}{2} \int_0^T \| \Gamma_t \|_H^2 dt \right)$$

$$Dq = dq + ((\mathbf{v}^* - \sigma \Gamma_t) dt + \sigma dW_t) \cdot \nabla q - \nabla \cdot (a \nabla q) dt \\ = dq + ((\mathbf{v}^* - \sigma \Gamma_t) dt + \sigma \circ dW_t) \cdot \nabla q$$

⇒ Incorporates slow components of subgrid scales into the initial formulation, and is useful for nudging techniques in data assimilation
# 1. Introduction

# 2. Modeling

## 2.1 Stochastic Transport Formulation

## 2.2 Geophysical Flow Modeling

## 2.3 Numerical Modeling of Uncertainty

# 3. Applications

## 3.1 Large-Scale Atmosphere Dynamics

## 3.2 Mesoscale Ocean Dynamics

## 3.3 Submesoscale Ocean Dynamics

## 3.4 Upper Ocean Boundary Layer Dynamics

# 4. Conclusion and Perspectives
**Quasi-Geostr.**
(u, v, η)

geostrophic
approximation

**Shallow Water**
(u, v, η)

isentropic/isopycnal
layers

**Primitive**
(u, v, η, T, S)

hydrostatic
approximation

**Boussinesq**
(u, v, w, η, T, S)

small
density variation

## A hierarchy of equations sets

Complexification of the physics

Simplifications

* Obtained from power series expansion in small number and scaling

* Same strategy and proper scaling of the noise term
  [Bauer et al., 2020, Brecht et al., 2021, Li et al., 2023a]
## Governing equations

Horizontal momentum

$$ \mathbb{D}u + f(\mathbf{u} \, dt + \sigma \, dB_t)^\perp = -\nabla(\rho \, dt + dP_t) $$

Vertical momentum

$$ \mathbb{D}w = -\partial_z(\rho \, dt + dP_t) + b \, dt $$

Temperature and salinity

$$ \mathbb{D}T = \kappa_t \Delta T \, dt, \quad \mathbb{D}S = \kappa_s \Delta S \, dt $$

Equation of state

$$ b = b(T, S, z) $$

Incompressibility

$$ \nabla \cdot \mathbf{v}^* = 0, \quad \nabla \cdot \sigma \, dB_t = 0 $$

* Hydrostatic approximation:

$$ \left\{ \begin{array}{l} \partial_z p = b + (\nabla \cdot \mathbf{a}) \cdot \nabla w + \nabla \cdot (\mathbf{a} \nabla w) \\ \partial_z dP_t = \sigma \, dB_t \cdot \nabla w \end{array} \right. $$

⇒ **Stochastic Primitive Model**

* Mathematical convergence results [Debussche et al., 2023]
## Governing equations

$$ egin{cases} 
\mathbb{D}\mathbf{u} + \beta y(\mathbf{u} \, \mathrm{d}t + \sigma \, \mathrm{d}B_t)^\perp = -f_0 \mathbf{u}_a^\perp \, \mathrm{d}t \\ 
\mathbb{D}b = -N^2 w_a \, \mathrm{d}t \\ 
\mathbf{u} = \nabla^\perp \psi, \quad b = f_0 \partial_z \psi, \quad \nabla \cdot \mathbf{u}_a + w_a = 0 \\ 
\nabla \cdot \sigma = \nabla \cdot \nabla \cdot \mathbf{a} = 0 
\end{cases} $$

u: geostrophic velocity, (u_a, w_a): ageostrophic velocity, b: buoyancy, ψ: stream function, f = f₀ + βy: Coriolis parameter, N: buoyancy frequency

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* Conservation of total energy:

$$ d(\|\mathbf{u}\|^2 + \|b\|^2/N^2) = 0 $$

* Conversion of ensemble energy:

$$ egin{cases} 
\mathbf{u} = \bar{\mathbf{u}} + \mathbf{u}', \quad b = \bar{b} + \mathbf{b}' \\ 
\mathrm{MKE} = \|\bar{\mathbf{u}}\|^2, \quad \mathrm{MPE} = \|\bar{b}\|^2/N^2 \\ 
\mathrm{EKE} = \|\mathbf{u}'\|^2, \quad \mathrm{EPE} = \|\mathbf{b}'\|^2/N^2 
\end{cases} 
\begin{array}{l} 
- \langle (\nabla \bar{\mathbf{u}}, (\mathbf{u}^*)' \mathbf{u}')_\mathrm{P} \rangle + \frac{1}{2} \langle (\nabla \bar{\mathbf{u}}, \bar{a} \nabla \bar{\mathbf{u}} + \bar{a}' \nabla \mathbf{u}')_\mathrm{P} \rangle \\ 
+ \langle (\nabla \bar{\mathbf{u}}, (\mathbf{u}^*)' \mathbf{u}')_\mathrm{P} \rangle - \frac{1}{2} \langle (\nabla \bar{\mathbf{u}}, \bar{a} \nabla \bar{\mathbf{u}} + \bar{a}' \nabla \mathbf{u}')_\mathrm{P} \rangle \\ 
\langle \frac{1}{N^2} \mathbf{u}^2 \rangle + \langle \frac{1}{N^2} \mathbf{b}^2 \rangle + \langle \frac{1}{N^2} (\nabla \bar{\mathbf{u}})^2 \rangle \\ 
- \langle (\nabla \bar{b}/N^2) \cdot (\mathbf{u}^*)' \mathbf{b}' \rangle + \frac{1}{2} \langle (\nabla \bar{b}/N^2) \cdot (\bar{a} \nabla \bar{b} + \bar{a}' \nabla \mathbf{b}') \rangle \\ 
+ \langle (\nabla \bar{b}/N^2) \cdot (\mathbf{u}^*)' \mathbf{b} \rangle - \frac{1}{2} \langle (\nabla \bar{b}/N^2) \cdot (\bar{a} \nabla \bar{b} + \bar{a}' \nabla \mathbf{b}') \rangle 
\end{array} 
\begin{array}{l} 
\mathrm{MKE} \\ 
\mathrm{MPE} \\ 
\mathrm{EPE} 
\end{array} 
$$

(●): spatial integration, [●, ●]ₚ: Frobenius inner product
## Governing equations

$$ egin{cases} 
\mathbb{D}\mathbf{u} + f\mathbf{u}^{\perp} dt = -g\nabla\eta dt \\ 
\mathbb{D}h + h\nabla \cdot \mathbf{u} dt = 0 \\ 
\nabla \cdot \mathbf{\sigma} = \nabla \cdot \nabla \cdot \mathbf{a} = 0 
\end{cases} 
$$

<figure><img src="image_10.png" /></figure>

* Conservation of total energy (KE + PE):

$$ d\left(\int_S \frac{1}{2}(h|\mathbf{u}|^2 + g\mathbf{h}^2) \, d\mathbf{x}\right) = 0 $$

* Mathematically well-posed [Lang et al., 2023]
# 1. Introduction

# 2. Modeling

## 2.1 Stochastic Transport Formulation

## 2.2 Geophysical Flow Modeling

## 2.3 Numerical Modeling of Uncertainty

# 3. Applications

## 3.1 Large-Scale Atmosphere Dynamics

## 3.2 Mesoscale Ocean Dynamics

## 3.3 Submesoscale Ocean Dynamics

## 3.4 Upper Ocean Boundary Layer Dynamics

# 4. Conclusion and Perspectives
# Proper Orthogonal Decomposition (POD)

[Brecht et al., 2021, Li et al., 2023a] Linear procedure that provides a basis of the “most probable” realizations of the input data ⇒ Data-driven stationary noise model

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Girsanov drift can be build by $\sigma(\mathbf{x})\Gamma = \sum_{k=1}^{N} \langle \mathbf{u}_{LR}^t, \mathbf{\Phi}_k \rangle \mathbf{\Phi}_k(\mathbf{x})$

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# Dynamic Mode Decomposition (DMD)

[Li et al., 2023b, Tucciarone et al., 2024] Builds a linear dynamical system from data and provides expansion in terms of linear recurrent solution

Filtered Velocity
time series

<figure><img src="image_13.png" /></figure>

$$v_{LR} = [\mathcal{B}_1 v_{HR}]_{LR} \downarrow$$

DMD complex-conjugate
modes

<figure><img src="image_14.png" /></figure>

$$\mathcal{M} = \{ (\varphi_m, \omega_m) \}_{m=1}^N$$

Time-scale separation of $\mathcal{M}$

$$\mathcal{M}^u = \{ m \in [1, N] : |\omega_m| > \pi/\tau_c \}$$

$$\mathcal{M}^c = \{ m \in [1, N] : |\omega_m| < \pi/\tau_c \}$$

<figure><img src="image_15.png" /></figure>

Fast modes ($\tau < \tau_c$)

<figure><img src="image_16.png" /></figure>

Slow modes ($\tau > \tau_c$)

$$\sigma\Gamma_t = \sum_{m \in \mathcal{M}^c} \exp(i\omega_m t)\varphi_m(\mathbf{x}),$$

$$\sigma dB_t = \sum_{m \in \mathcal{M}^u} \exp(i\omega_m t)\varphi_m(\mathbf{x}) d\beta_t^m$$
[Bauer et al., 2020, Brecht et al., 2021] Build a set of random observations (pseudo-observations) and apply POD online (at each step)

<figure><img src="image_17.png" /></figure>
[Tuciarone et al., 2024] Create a horizontal convolution noise with a vertical profile imposed connect the noise to the current state of the flow

<figure><img src="image_18.png" /></figure>
# 1. Introduction

# 2. Modeling

## 2.1 Stochastic Transport Formulation

## 2.2 Geophysical Flow Modeling

## 2.3 Numerical Modeling of Uncertainty

# 3. Applications

## 3.1 Large-Scale Atmosphere Dynamics

## 3.2 Mesoscale Ocean Dynamics

## 3.3 Submesoscale Ocean Dynamics

## 3.4 Upper Ocean Boundary Layer Dynamics

# 4. Conclusion and Perspectives
# 1. Introduction

# 2. Modeling

## 2.1 Stochastic Transport Formulation

## 2.2 Geophysical Flow Modeling

## 2.3 Numerical Modeling of Uncertainty

# 3. Applications

## 3.1 Large-Scale Atmosphere Dynamics

## 3.2 Mesoscale Ocean Dynamics

## 3.3 Submesoscale Ocean Dynamics

## 3.4 Upper Ocean Boundary Layer Dynamics

# 4. Conclusion and Perspectives
Comparison of Barotropic Instabilities [Brecht et al., 2021]

Mid-latitudinal unstable jet, initial value problem, RSW model with biharmonic eddy viscosity, structure-preserving discretization on sphere

<figure><img src="image_19.png" /></figure>

• STO triggers better instability than DET
Comparison of Ensemble Spread

**Figure:** Evolution of vorticity at different latitudes. Black lines: observed trajectories. Blue shaded area: ensemble spread of deterministic model with perturbed initial conditions (PIC). Red shaded area: ensemble spread of stochastic model using POD noise.

<figure><img src="image_20.png" /></figure>

• STO produces more efficient spread containing observations than PIC
For a reliable ensemble forecast, the mean squared error (MSE) of the ensemble matches the mean intra-ensemble variance (MEV):

$$ \underbrace{\frac{1}{N} \sum_{n=1}^{N} (\hat{\mathbb{E}}[q_n] - q_n^0)^2}_{\text{MSE}} \approx \underbrace{\frac{1}{N} \sum_{n=1}^{N} \hat{\mathbb{V}}[q_n]}_{\text{MEV}} $$

$q_n^0$: n-th observation, $\hat{\mathbb{E}}[q_n]$ and $\hat{\mathbb{V}}[q_n]$: empirical ensemble mean and variance of n-th forecast

<figure><img src="image_21.png" /></figure>

* STO provides a better trade-off between model error representation and ensemble spread than PIC

* Data-driven forecasts are more reliable than model-driven forecasts
# 1. Introduction

# 2. Modeling

## 2.1 Stochastic Transport Formulation

## 2.2 Geophysical Flow Modeling

## 2.3 Numerical Modeling of Uncertainty

# 3. Applications

## 3.1 Large-Scale Atmosphere Dynamics

### 3.2 Mesoscale Ocean Dynamics

## 3.3 Submesoscale Ocean Dynamics

## 3.4 Upper Ocean Boundary Layer Dynamics

# 4. Conclusion and Perspectives
Comparison of Non-Eddy-Resolving Simulations [Li et al., 2023a]

* Western boundary current, doublegyre configuration with steady symmetric wind forcing and biharmonic eddy viscosity, 3 layers QG model, unresolved baroclinic instability

* STO-POD-P: projection of noise and drift on iso-surfaces of layer thickness anomaly ($\sigma dB_t \perp \nabla h'$)

REF (5km)

<figure><img src="image_22.png" /></figure>

DET (40km)

<figure><img src="image_23.png" /></figure>

STO-POD (40km)

<figure><img src="image_24.png" /></figure>

STO-POD-P (40km)

<figure><img src="image_25.png" /></figure>
Comparison of Time Averages

<figure><img src="image_26.png" /><figcaption>Top: Barotropic stream function, Bottom: 1st Baroclinic stream function</figcaption></figure>

* The coarse resolution stochastic models enable to reproduce the eastwards jet
Comparison of Energy Spectrum

<figure><img src="image_27.png" /><figcaption>**Figure:** Temporal mean of vertically integrated kinetic energy (KE) spectrum</figcaption></figure>

* STO-EOF-P brings higher KE backscattering than STO-EOF at both resolutions
Comparison of Temporal Variability

<figure><img src="image_28.png" /><figcaption>Figure: Comparison of variability measures across different coarse models. Lower RMSE and Gaussian Relative Entropy (GRE) values and higher Pattern Correlation (PC) values indicate better model performance.</figcaption></figure>

* STO-EOF-P provide the best results for all the metrics of variability at different resolutions ⇒ importance of the non-stationary characteristic of the noise
# 1. Introduction

# 2. Modeling

## 2.1 Stochastic Transport Formulation

## 2.2 Geophysical Flow Modeling

## 2.3 Numerical Modeling of Uncertainty

# 3. Applications

## 3.1 Large-Scale Atmosphere Dynamics

## 3.2 Mesoscale Ocean Dynamics

### 3.3 Submesoscale Ocean Dynamics

## 3.4 Upper Ocean Boundary Layer Dynamics

# 4. Conclusion and Perspectives
# NEMO Configuration [Tucciarone et al., 2024]

NEMO is a state-of-the-art ocean primitive model, doublegyre circulation driven by seasonal wind stress and surface heat fluxes, 31 vertical layers

<figure><img src="image_29.png" /></figure>
Comparison of Eddy-Permitting Simulations (1/3°)

* DMD (η): DMD performed on the geostrophic velocity reconstructed from the sea surface elevation η → Towards real data !

* Data-driven noise models have been projected along the isopycnals

<figure><img src="image_30.png" /></figure>
Comparison of Eddy-Resolving Simulations (1/9°)

R27d

<figure><img src="image_31.png" /></figure>

R9d

<figure><img src="image_32.png" /></figure>

POD'((u)ₜ)

<figure><img src="image_33.png" /></figure>

G' * dBₜ

<figure><img src="image_34.png" /></figure>

DMD(η) + G' * dBₜ

<figure><img src="image_35.png" /></figure>
Comparison of Enstrophy Spectrum

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

* Performance in eddy-permitting regime: STO-DMDs > STO-PODs > DET, best: DMD ($\eta$)

* Performance in eddy-resolving regime: Model-driven (VPP) > Data-driven (POD,DMD)
# 1. Introduction

# 2. Modeling

## 2.1 Stochastic Transport Formulation

## 2.2 Geophysical Flow Modeling

## 2.3 Numerical Modeling of Uncertainty

# 3. Applications

## 3.1 Large-Scale Atmosphere Dynamics

## 3.2 Mesoscale Ocean Dynamics

## 3.3 Submesoscale Ocean Dynamics

# 3.4 Upper Ocean Boundary Layer Dynamics

# 4. Conclusion and Perspectives
Classical Ekman Surface Boundary Layer

* For steady, linear flow, uniform density and viscosity, driven by surface wind stress:

$$f\mathbf{u}_e = a\partial_{zz}^2\mathbf{u}_e, \quad \rho_o a\partial_z\mathbf{u}_e|_{z=0} = \tau, \quad \mathbf{u}_e|_{z \to -\infty} = 0$$

* Explicit solution - Ekman spiral:

$$\left\{ \begin{aligned} u_e &= \frac{\sqrt{2}}{\rho_o f h_e} e^{z/h_e} \left( \tau^x \cos\left(\frac{z}{h_e} - \frac{\pi}{4}\right) - \tau^y \sin\left(\frac{z}{h_e} - \frac{\pi}{4}\right) \right) \\ v_e &= \frac{\sqrt{2}}{\rho_o f h_e} e^{z/h_e} \left( \tau^x \sin\left(\frac{z}{h_e} - \frac{\pi}{4}\right) + \tau^y \cos\left(\frac{z}{h_e} - \frac{\pi}{4}\right) \right) \end{aligned} \right. \quad (\text{Ekman layer depth: } h_e = \sqrt{2a/f})$$

* Ekman transport and Ekman vertical velocity:

$$\mathbf{T}_e = \int \mathbf{u}_e \, dz = \frac{1}{\rho_o f} \tau^{\perp}, \quad w_e = \int \nabla \cdot \mathbf{u}_e \, dz = \frac{1}{\rho_o f} \nabla \times \tau$$

* A beautiful theory, but generally difficult to observe due to measurement challenges and the influence of phenomena not accounted for, such as nonuniform eddy viscosity, inertial and gravity waves, stratification, and convection.

<figure><img src="image_37.png" /></figure>
* A novel formulation encodes the interactions between wind, waves, and currents through the introduction of the uncertainty of unresolved motions (p-SPDE):

$$d\mathbf{u}_e = (-if(\mathbf{u}_e + \mathbf{u}_s) + \partial_z(a_{zz}\partial_z(\mathbf{u}_e + \mathbf{u}_s)))dt - (if\sigma_x + \sigma_z\partial_z(\mathbf{u}_e + \mathbf{u}_s))dB_t$$

$$\rho_o a_{zz} \partial_z \mathbf{u}_e |_{z=0} = \tau_w, \quad \rho_o a_{zz} \partial_z \mathbf{u}_e |_{z=-H} = 0$$

u_e: Ekman current, u_s: Stokes drift (wave), τ_w: wind stress

* Comparison with a benchmark model (p-PDE):

$$\partial_t \mathbf{u}_e + if(\mathbf{u}_e + \mathbf{u}_s) = \partial_z(a_{zz}\partial_z \mathbf{u}_e)$$

with the same boundary conditions.

* Both models incorporate established physical parameterizations: Markovian wind process, random wave direction, and K-Profile vertical mixing

<figure><img src="image_38.png" /></figure>
Comparison of Ensemble Statistics

<figure><img src="image_39.png" /></figure>

* The p-SPDE yields smoother mean profiles, higher uncertainty, increased energy, and stronger correlations than the p-PDE.
Comparison of Higher-Order Statistics

<figure><img src="image_40.png" /></figure>

* The p-SPDE exhibits more skewed distributions and frequent extreme values than the p-PDE, especially near the surface.
Comparison of Evolution After Wind Stops

**Figure:** The wind is stopped ($\tau_W$ = 0) after 30 days of simulation. The mean Ekman spirals at different days in (a, b) are represented by various colors. In each spiral, a darker color indicates a deeper position with a depth step of 5 meters.

<figure><img src="image_41.png" /></figure>

* The p-SPDE maintains its mean spiral and exhibits higher residual uncertainties than the p-PDE after wind stoppage.
# 1. Introduction

# 2. Modeling

## 2.1 Stochastic Transport Formulation

## 2.2 Geophysical Flow Modeling

## 2.3 Numerical Modeling of Uncertainty

# 3. Applications

## 3.1 Large-Scale Atmosphere Dynamics

## 3.2 Mesoscale Ocean Dynamics

## 3.3 Submesoscale Ocean Dynamics

## 3.4 Upper Ocean Boundary Layer Dynamics

# 4. Conclusion and Perspectives
* The stochastic transport formulation preserves key physical properties, such as energy conservation.

* This stochastic scheme provides a good trade-off between model error representation and ensemble spread.

* The random models allow us to accurately reproduce the statistical distribution of large-scale flow on a coarse mesh.

* Temporal and spatial characteristics of the unresolved flow component have strong influences on the numerical results.

* By incorporating uncertainty from smaller scales, the approach helps explain interactions between intermittent winds, waves, and currents.
# Mesoscale Air-Sea Interaction

<figure><img src="image_42.png" /></figure>

<figure><img src="image_43.png" /></figure>
<figure><img src="image_44.png" /></figure>

<figure><img src="image_45.png" /></figure>

* Introducing transport noises into the coupled system reduces both the eddy wind work and the eddy buoyancy work
## Generalized stochastic Craik-Leibovich momentum equations

$$ 
\begin{aligned}
d\mathbf{v}^* + ((\mathbf{v}^* - \mathbf{v}_s) \cdot \nabla \mathbf{v}^* - \nabla \cdot (\boldsymbol{a}\nabla \mathbf{v}^*) + \boldsymbol{f}\ddot{\mathbf{z}} \times \mathbf{v}^* - b\dot{\mathbf{z}} + \nabla p) dt + (\boldsymbol{\sigma} \cdot \nabla \mathbf{v}^* + \boldsymbol{f}\ddot{\mathbf{z}} \times \boldsymbol{\sigma}) dB_t + \nabla dP_t \\
= \underbrace{(-\boldsymbol{f}\ddot{\mathbf{z}} \times \mathbf{v}_s - \nabla (\mathbf{v}^* \cdot \mathbf{v}_s))}_{\text{Coriolis Stokes}} + \underbrace{\mathbf{v}_s \times (\nabla \times \mathbf{v}^*)}_{\text{pressure correction}} - \underbrace{(\nabla \times \mathbf{v}_s) \times \mathbf{v}^*}_{\text{Craik-Leibovich vortex force}} + \underbrace{\nabla \cdot (\boldsymbol{a}\nabla \mathbf{v}_s)}_{\text{Itô-Stokes force}} dt - \underbrace{\boldsymbol{\sigma} \cdot \nabla \mathbf{v}_s \, dB_t}_{\text{Itô-Stokes diffusive mixing}}
\end{aligned}
$$

<figure><img src="image_46.png" /></figure>

<figure><img src="image_47.png" /></figure>

<figure><img src="image_48.png" /></figure>
<figure><img src="image_49.png" /></figure>

Surface curl (040 yrs 005 days)

<figure><img src="image_50.png" /></figure>

QG assimilated with high-quality SSH and SST data ⇒ Short-term forecast
after DA ⇒ Compared to NEMO reanalysed data (Copernicus Marine)
Questions?
<figure><img src="image_51.png" /></figure>

Bauer, W., Chandramouli, P., Chapron, B., Li, L., and Mémin, E. (2020).

Deciphering the role of small-scale inhomogeneity on geophysical flow structuration: a stochastic approach.

J. Phys. Oceanogr., 50(4):983-1003.

<figure><img src="image_52.png" /></figure>

Brecht, R., Li, L., Bauer, W., and Mémin, E. (2021).

Rotating shallow water flow under location uncertainty with a structure-preserving discretization.

J. Adv. Model. Earth Syst., 13(12):e2021MS002492.

<figure><img src="image_53.png" /></figure>

Debussche, A., Hug, B., and Mémin, E. (2023).

A consistent stochastic large-scale representation of the Navier-Stokes equations.

Journal of Mathematical Fluid Mechanics, 25(1):19.

<figure><img src="image_54.png" /></figure>

Lang, O., Crisan, D., and Mémin, E. (2023).

Analytical properties for a stochastic rotating shallow water model under location uncertainty.

Journal of Mathematical Fluid Mechanics, 25(2):29.

<figure><img src="image_55.png" /></figure>

Li, L. (2021).

Stochastic modeling and numerical simulation of ocean dynamics.

PhD thesis, Université de Rennes 1.

https://hal.science/tel-03207741/file/PhD-thesis-Li2021.pdf.
<figure><img src="image_56.png" /></figure>

Li, L., Deremble, B., Lahaye, N., and Mémin, E. (2023a).

Stochastic data-driven parameterization of unresolved eddy effects in a baroclinic quasi-geostrophic model.

J. Adv. Model. Earth Syst., 15(2):e2022MS003297.

<figure><img src="image_57.png" /></figure>

Li, L., Mémin, E., and Chapron, B. (2024).

A generalized stochastic formulation of the ekman-stokes model with statistical analyses.

https://hal.science/hal-04672188.

<figure><img src="image_58.png" /></figure>

Li, L., Mémin, E., and Tissot, G. (2023b).

Stochastic parameterization with dynamic mode decomposition.

In *Stochastic Transport in Upper Ocean Dynamics*, volume 10 of *Mathematics of Planet Earth*, pages 179–193. Springer.

<figure><img src="image_59.png" /></figure>

Mémin, E. (2014).

Fluid flow dynamics under location uncertainty.

Geophys. Astrophys. Fluid Dyn., 108(2):119–146.

<figure><img src="image_60.png" /></figure>

Thiry, L., Li, L., Mémin, E., and Roullet, G. (2024).

A unified formulation of quasi-geostrophic and shallow water equations via projection.

J. Adv. Model. Earth Syst., 16:e2024MS004510.
<figure><img src="image_61.png" /></figure>

Tucciarone, F., Li, L., and Mémin, E. (2024).

Derivation and numerical assessment of a stochastic large-scale hydrostatic primitive model.

In press JAMES.