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# A Stochastic Formulation of the Ekman-Stokes Model

Long Li

► To cite this version:

Long Li. A Stochastic Formulation of the Ekman-Stokes Model. 2024 - Atelier sur la représentation des fines échelles océaniques dans les simulations numériques, Oct 2024, Plouzané, France. hal-04957875

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https://hal.science/hal-04957875v1

Submitted on 20 Feb 2025

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Distributed under a Creative Commons Attribution 4.0 International License
# A Stochastic Formulation of the Ekman-Stokes Model

## Atelier sur la représentation des fines échelles océaniques dans les simulations numériques

Long Li*,¹, Etienne Mémin¹, Bertrand Chapron¹,²

* ERC STUOD Research Associate

¹ Odyssey Team, Inria Rennes, France

² LOPS, Ifremer Plouzané, France

October 11, 2024, Ifremer, Plouzané, France

Inria

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Ifremer

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erc

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STUOD
# 1. Introduction

2. Stochastic Formulation

3. Uncertainty Representation

4. Statistical Analyses

5. Conclusion
Classical Ekman surface boundary layer [Ekman, 1905]

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:

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

* A well-established theory, but challenging to observe due to measurement limitations and unaccounted phenomena like variable eddy viscosity, waves, stratification, and convection.

<figure><img src="image_6.png" /></figure>
* **Motivation:** Develop a stochastic model to capture wind, wave, and current interactions in the upper ocean boundary layer, incorporating uncertainties from smaller scales.

* **Approach:** Derive the model through the stochastic Craik-Leibovich equations and constrain uncertainties with established physical parameterizations.

* **Reference:** Li, L., Mémin, E., and Chapron, B. (2024). A generalized stochastic formulation of the Ekman-Stokes model with statistical analyses. *Preprint*. https://hal.science/hal-04672188.
1. Introduction

2. Stochastic Formulation

3. Uncertainty Representation

4. Statistical Analyses

5. Conclusion
Stochastic Boussinesq momentum equations

$$
\begin{gather*}
d\mathbf{v} + (\mathbf{v}^* \cdot \nabla \mathbf{v} - \nabla \cdot (\boldsymbol{a}\nabla \mathbf{v}) + \hat{f}\boldsymbol{z} \times \mathbf{v}) dt + (\boldsymbol{\sigma} \cdot \nabla \mathbf{v} + \hat{f}\boldsymbol{z} \times \boldsymbol{\sigma}) dB_t = (b\hat{z} - \nabla p) dt - \nabla dP_t \\
\boldsymbol{a} = \frac{1}{2}\boldsymbol{\sigma}\boldsymbol{\sigma}^T, \quad \mathbf{v}^* = \mathbf{v} - \mathbf{v}_s, \quad \mathbf{v}_s = \nabla \cdot \boldsymbol{a}
\end{gather*}
$$

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

$\sigma dB_t$: unresolved flow component, $\boldsymbol{a}$: diffusion coefficient, $\mathbf{v}^*$: effective advection velocity, $\mathbf{v}_s$: Itô-Stokes drift, $dP_t$: unresolved pressure component

Applying a change of variable from **v** to **v**\* and a classical vector calculus identity, and
assuming a quasi-stationary **v**<sub>s</sub> ⇒

Generalized stochastic Craik-Leibovich momentum equations

$$
\begin{align*}
d\mathbf{v}^* &+ ((\mathbf{v}^* - \mathbf{v}_s) \cdot \nabla \mathbf{v}^* - \nabla \cdot (\boldsymbol{a} \nabla \mathbf{v}^*) + \hat{f}\mathbf{z} \times \mathbf{v}^* - b\hat{z} + \nabla p) dt + (\boldsymbol{\sigma} \cdot \nabla \mathbf{v}^* + \hat{f}\mathbf{z} \times \boldsymbol{\sigma}) dB_t + \nabla dP_t \\
&= \underbrace{(-\hat{f}\mathbf{z} \times \mathbf{v}_s - \nabla (\mathbf{v}^* \cdot \mathbf{v}_s))}_{\text{Coriolis Stokes}} + \underbrace{\mathbf{v}_s \times (\nabla \times \mathbf{v}^*)}_{\text{pressure correction}} - \underbrace{(\nabla \times \mathbf{v}_s) \times \mathbf{v}^*}_{\text{Craik-Leibovich vortex force}} + \underbrace{\nabla \cdot (\boldsymbol{a} \nabla \mathbf{v}_s)}_{\text{Itô-Stokes force}} dt - \underbrace{\sigma \cdot \nabla \mathbf{v}_s \, dB_t}_{\text{Itô-Stokes transport}}
\end{align*}
$$
# Scaling for the Ekman Boundary Layer

In this study, we do not account for stratification effects within the boundary layer ($b = 0$).

* Noise scalings derived from eddy viscosity:

$$ 
\begin{aligned}
\boldsymbol{a}_{xx} &= \frac{1}{2} \sigma_x \sigma_x^T \sim A_h, & a_{zz} &= \frac{1}{2} \sigma_z^2 \sim A_v \\
\Rightarrow \sigma_x &\sim \sqrt{2} A_h^{1/2}, \quad \sigma_z \sim \sqrt{2} A_v^{1/2} & \boldsymbol{a}_{xz} &\sim A_h^{1/2} A_v^{1/2}
\end{aligned}
$$

* Assumptions on dimensionless numbers:

$$ 
\begin{gathered}
\mathrm{Ro} = \frac{U}{f_0 L} \ll 1, \quad \mathrm{E}_h = \frac{A_h}{f_0 L^2} \sim \mathrm{Ro}^2, \quad \mathrm{E}_v = \frac{A_v}{f_0 H^2} \sim \delta_e^2, \quad \delta = \frac{H}{L} \ll 1 \\
(E_h / \mathrm{Ro} = 1 / \mathrm{Re} \sim \mathrm{Ro})
\end{gathered}
$$

## Nondimensional momentum equations

$$ 
\begin{aligned}
\mathrm{Ro} \, \mathrm{d}u &= \left( -f(\mathbf{u} + \partial_z \boldsymbol{a}_{xz})^{\perp} - \nabla p + \partial_z (\boldsymbol{a}_{zz} \partial_z (\mathbf{u} + \partial_z \boldsymbol{a}_{xz}) + \mathcal{O}(\mathrm{Ro})) \right) \mathrm{d}t \\
&\quad - \sqrt{2} \mathrm{Ro}^{1/2} \left( \left( \partial_z \partial_z (\mathbf{u} + \partial_z \boldsymbol{a}_{xz}) + f \boldsymbol{\sigma}_x^{\perp} \right) \mathrm{d}B_t + \partial_x \mathrm{d}P_t \right) + \mathcal{O}(\mathrm{Ro}) \mathrm{d}B_t \\
\delta^2 \mathrm{Ro} \, \mathrm{d}w &= \left( -\frac{1}{\delta_e^2} \partial_z p + \mathcal{O}(\delta^2) \right) \mathrm{d}t - \sqrt{2} \frac{\mathrm{Ro}^{1/2}}{\delta_e^2} \partial_z \mathrm{d}P_t + \mathcal{O}(\delta^2) \mathrm{d}B_t
\end{aligned}
$$
# Generalized Steady Ekman-Stokes Model

* An intermediate time scale $t = O(1)$ gives the prognostic equations of O(1):

$$f(\mathbf{u}^* + \mathbf{u}_s)^\perp = -\nabla p + \partial_z(a_{zz}\partial_z(\mathbf{u}^* + \mathbf{u}_s)), \quad \mathbf{u}_s = \partial_z\mathbf{a}_{xz}, \quad \partial_z p = 0$$

* Separating the geostrophic and ageostrophic from $\mathbf{u}^* = \mathbf{u}_g + \mathbf{u}_e$, restoring dimensions, applying boundary conditions ⇒ Ekman-Stokes model in complex notation:

$$\begin{cases} \partial_z(a_{zz}\partial_z\mathbf{u}_e) - if\mathbf{u}_e = if\mathbf{u}_s - \partial_z(a_{zz}\partial_z\mathbf{u}_s) \\ \rho_o a_{zz}\partial_z\mathbf{u}_e|_{z=0} = \tau_w, \quad \rho_o a_{zz}\partial_z\mathbf{u}_e|_{z=-H} = 0 \end{cases}$$

* WKB approximation with variation of parameters for Ekman current stress $\tau_e = \rho_o a_{zz} \partial_z \mathbf{u}_e$ [Wenegrat and McPhaden, 2016]:

$$\tau_e(z) = \tau_w \left( \frac{a_{zz}(z)}{a_{zz}(0)} \right)^{1/4} \frac{\sinh(\theta(z))}{\sinh(\theta(0))} + \int_{-H}^{0} G(z, \zeta) (if\tau_s - a_{zz}\partial_{zz}^2\tau_s)(\zeta) d\zeta$$

$$\tau_s = \rho_o a_{zz} \partial_z \mathbf{u}_s, \quad \theta(z) = \sqrt{if} \int_{-H}^{z} a_{zz}^{-1/2}(\zeta) d\zeta, \quad G: \text{a symmetric Green's function}$$

* Modified integral properties ($T = \int \mathbf{u} \, dz, w = \int \nabla \cdot \mathbf{u} \, dz$):

$$\mathbf{T}_e = -\frac{i}{\rho_o f}(\tau_w + \tau_s^0) - \mathbf{T}_s, \quad w_e(-H) = \frac{1}{\rho_o f}\nabla \times (\tau_w + \tau_s^0) - \nabla \cdot \mathbf{T}_s$$
Time-Dependent Stochastic Ekman-Stokes Model

* Considering a fast time scale by a change of time coordinates [Crowe and Taylor, 2018], $\tau = t/R_o \Rightarrow B_\tau = B_t/R_o^{1/2}$, separating geostrophic and ageostrophic $\Rightarrow$ SPDE:

$$ \begin{cases} \mathrm{d}\mathbf{u}_e = \left(-\mathrm{if}(\mathbf{u}_e + \mathbf{u}_s) + \partial_z(a_{zz}\partial_z(\mathbf{u}_e + \mathbf{u}_s))\right)\mathrm{d}t - (\mathrm{if}\sigma_x + \sigma_z\partial_z(\mathbf{u}_e + \mathbf{u}_s))\mathrm{d}B_t \\ \rho_o a_{zz}\partial_z\mathbf{u}_e|_{z=0} = \tau_w, \quad \rho_o a_{zz}\partial_z\mathbf{u}_e|_{z=-H} = 0 \end{cases} $$

* Statistical properties of the SPDE were investigated numerically

* Conditional mean transport ($\mathbb{E}^\sigma[\bullet] := \mathbb{E}[\bullet|\sigma]$, $\sigma$ is random and independent to $B_t$):

$$ \begin{gathered} \partial_t \mathbb{E}^\sigma[\mathbf{u}_e] = -\mathrm{if}(\mathbb{E}^\sigma[\mathbf{u}_e] + \mathbf{u}_s) + \partial_z(a_{zz}\partial_z(\mathbb{E}^\sigma[\mathbf{u}_e] + \mathbf{u}_s)) \\ \Rightarrow \mathbb{E}^\sigma[\mathbf{T}_e](t) = e^{-ift}\mathbb{E}^\sigma[\mathbf{T}_e](0) + \frac{1}{\rho_o} \int_0^t e^{-if(t-r)}(\tau_w + \tau_s^0)(r) \mathrm{d}r - \mathbf{T}_s(1 - e^{-ift}) \end{gathered} $$
1. Introduction

2. Stochastic Formulation

**3. Uncertainty Representation**

4. Statistical Analyses

5. Conclusion
Physical Parameterizations

* K-profile parameterization (KPP) for vertical eddy viscosity [McWilliams and Huckle, 2006] :

$$a_{zz}(z) = c_1 u_* h G(\zeta), \quad \zeta = -z/h, \quad h = c_2 u_* / f$$

$$G(\zeta) = \zeta(1-\zeta)^2 + H(\zeta_0 - \zeta) \frac{(\zeta - \zeta_0)^2}{2\zeta_0}$$

$u_*$: friction velocity, $h$: boundary layer depth, $H$: Heaviside step function

* Stokes drift corresponding to a monochromatic deep-water wave [Phillips, 1977] :

$$\eta = \alpha \cos(kx - \omega t), \quad \omega = (gk)^{1/2}$$

$$\mathbf{u}_s(z) = U_0 e^{2kz} e^{i\theta_s}, \quad U_0 = \omega k \alpha^2$$

$\eta$: surface elevation, $\alpha$: wave amplitude, $k$: horizontal wavenumber, $\omega$: angular frequency, $U_0$: Stokes magnitude, $\theta_s$: wave propagation direction

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* Markovian wind process [McWilliams and Huckle, 2006] :

$$ \mathbf{u}_a = \bar{\mathbf{u}}_a + \mathbf{u}'_a, \quad d\mathbf{u}'_a = -\frac{1}{T_a}\mathbf{u}'_a dt + \sqrt{\frac{2}{T_a}} \Sigma_a d\mathbf{W}_t $$

* Random wind stress (upper boundary) via bulk formula :

$$ \tau_w = C_D \rho_a |\mathbf{u}_a| \mathbf{u}_a = \rho_o u_*^2 e^{i\theta_*} \\ \Rightarrow u_* \text{ random} \Rightarrow a_{zz} \text{ random (via KPP)} $$

* Random surface wave stress (derived in SPDE):

$$ \tau_s^0 = a_{zz} \partial_z \mathbf{u}_s |_{z=0} = a_{zz}(0) U_0 e^{i\theta_s}, \quad \theta_s \sim \mathcal{N}(\Theta_s, \Sigma_s^2) $$

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# Noise Representation

**Objective:** Determine noise terms ($\tilde{\sigma}_x$ dBt, $\tilde{\sigma}_z$ dBt) such that the associated diffusion coefficient $\tilde{a}_{zz}$ and Itô-Stokes drift $\partial_z \tilde{a}_{xz}$ match or approximate the given parameterizations ($a_{zz}$, $u_s$).

* Typically, this involves solving optimization problems for weighted coefficients with a predefined set of basis functions

* Here, we consider a particular solution projected onto a set of localized basis functions {$e_n$} in $L^2([-H, 0], \mathbb{R})$ (e.g., regularized Dirac functions, wavelets):

$$ \tilde{\sigma}_z(z) \, dB_t = \sqrt{2} \sum_n \langle a_{zz}^{1/2}, e_n \rangle e_n(z) \, d\beta_t^n $$

$$ \tilde{\sigma}_x(z) \, dB_t = \sqrt{2} \sum_n \langle a_{zz}^{-1/2} u_s, e_n \rangle e_n(z) \, d\beta_t^n $$

$u_s = \int_{-H}^2 u_s(\zeta) \, d\zeta = u_s/(2k)$, $\beta_t^n$: standard Brownian motions (independent to $W_t$)

* Stronger winds correspond to increased vertical noise, while smaller waves enhance horizontal noise at the surface

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1. Introduction

2. Stochastic Formulation

3. Uncertainty Representation

4. Statistical Analyses

5. Conclusion
* Parametrized PDE (p-PDE) [McWilliams et al., 1997]:

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

* Parametrized stochastic PDE (p-SPDE):

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

* With the same type of boundary condition and initial condition:

$$ \rho_0 a_{zz} \partial_z \mathbf{u}_e|_{z=0} = \tau_w, \quad \rho_0 a_{zz} \partial_z \mathbf{u}_e|_{z=-H} = 0, \quad \mathbf{u}_e(t=0) = 0 $$

<table><thead><tr><th>Parameters</th><th>Value</th><th>Description</th></tr></thead><tbody><tr><td><i>f</i></td><td>0.73 × 10<sup>-4</sup> s<sup>-1</sup></td><td>Coriolis freq.</td></tr><tr><td><b>u&#773;<sub>a</sub></b></td><td>(5, 0) m s<sup>-1</sup></td><td>Mean wind</td></tr><tr><td>Σ<sub>a</sub></td><td>5 m s<sup>-1</sup></td><td>SD of transient wind</td></tr><tr><td><i>T</i><sub>a</sub></td><td>1 day</td><td>Memory time of wind</td></tr><tr><td>α</td><td>0.8 m</td><td>Surface wave amp.</td></tr><tr><td>λ = 2π/k</td><td>60 m</td><td>Stokes wavelength</td></tr><tr><td>Θ<sub>s</sub></td><td>0°</td><td>Stokes mean angle</td></tr><tr><td>Σ<sub>s</sub></td><td>5°</td><td>SD of Stokes angles</td></tr></tbody></table>
* Pseudo-spectral Chebyshev method in space and linear-implicit Euler scheme in time for the SPDE:

$$ \mathbf{u}_j^{n+1} = (I_{j,k} - A_{j,k}\Delta t)^{-1}(\mathbf{u}_k^n + F_k^n(\mathbf{u}_s)\Delta t + G_k^n(\mathbf{u}^n, \mathbf{u}_s)\Delta B) $$

$\mathbf{u}_j^n := \mathbf{u}(z_j, t_n)$, $A = -ifI + DaD$, $D$: Chebyshev differentiation matrix, $F, G$: RHS operators.

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* Exact scheme for the Markovian wind process:

$$ \mathbf{u}_a^{n+1} = \bar{\mathbf{u}}_a + (\mathbf{u}_a^n - \bar{\mathbf{u}}_a)e^{-\Delta t/T_a} + \sum_a \sqrt{1 - e^{-2\Delta t/T_a}} \xi^n $$

$\bar{\mathbf{u}}_a$: mean wind, $T_a$: memory time, $\xi$: standard normal variables.

* Code: https://github.com/matlong/SGESM

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Comparison of Ensemble Statistics

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* The p-SPDE produces smoother mean profiles, greater spread, increased energy, and stronger correlations compared to the p-PDE.
Comparison of Higher-Order Statistics

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

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* After wind stoppage, the p-SPDE maintains its mean spiral and exhibits higher residual variability compared to the p-PDE.
1. Introduction

2. Stochastic Formulation

3. Uncertainty Representation

4. Statistical Analyses

# 5. Conclusion
* This novel stochastic formulation is derived to capture the interplay between random motions induced by wind, waves, and currents with enhanced uncertainty representation.

* Numerical results indicate higher kinetic energy, stronger correlations, and more skewed distributions with increased extreme events near the surface compared to traditional models.
* Validate the model using reference data from high-quality numerical simulations [McWilliams et al., 2012]

* Analyze wave feedback mechanisms in a 1D fully coupled atmosphere-ocean boundary layer model [Lewis and Belcher, 2004]

* Quantify the influence of wave mixing on the evolution of thermal fronts in a 2D (x, z) stratified Ekman model [Crowe and Taylor, 2018]

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# Questions?
<figure><img src="image_19.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_20.png" /></figure>

Crowe, M. N. and Taylor, J. R. (2018).

The evolution of a front in turbulent thermal wind balance. Part 1. theory.

J. Fluid Mech., 850:179-211.

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

Ekman, V. W. (1905).

On the influence of the earth's rotation on ocean-currents.

Ark. Mat. Astr. Fys., 2(11):1-52.

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

Lewis, D. M. and Belcher, S. E. (2004).

Time-dependent, coupled, Ekman boundary layer solutions incorporating Stokes drift.

Dyn. Atmos. Oceans, 37(4):313-351.

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

McWilliams, J. C. and Huckle, E. (2006).

Ekman layer rectification.

J. Phys. Oceanogr., 36(8):1646-1659.
<figure><img src="image_24.png" /></figure>

McWilliams, J. C., Huckle, E., Liang, J.-H., and Sullivan, P. P. (2012).

The wavy Ekman layer: Langmuir circulations, breaking waves, and Reynolds stress.

J. Phys. Oceanogr., 42(11):1793-1816.

<figure><img src="image_25.png" /></figure>

McWilliams, J. C., Sullivan, P. P., and Moeng, C.-H. (1997).

Langmuir turbulence in the ocean.

J. Fluid Mech., 334:1-30.

<figure><img src="image_26.png" /></figure>

Phillips, O. M. (1977).

The Dynamics of the Upper Ocean.

Cambridge University Press.

<figure><img src="image_27.png" /></figure>

Wenegrat, J. O. and McPhaden, M. J. (2016).

Wind, waves, and fronts: Frictional effects in a generalized Ekman model.

J. Phys. Oceanogr., 46(2):371-394.
Comparison across Transient Wind Levels

<figure><img src="image_28.png" /></figure>

* Higher transient winds lead to increased global energy, deeper boundary layers, and stronger Ekman transport with greater angular uncertainty.

<figure><img src="image_29.png" /></figure>

* Higher wind fluctuations, and thus stronger noise, lead to an increase in extreme events over time, particularly near the surface.
Comparison across Stokes Parameters

<figure><img src="image_30.png" /></figure>

* Increasing wavelength reduces global energy, transport magnitude, and Ekman layer depth, with smaller wavelengths enhancing wave mixing effects.

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

* Global energy and transport magnitude peak when waves align with wind, decreasing symmetrically with rotation; transport angle closely follows wave direction changes.