# Stochastic closure for partially-resolved turbulence

Valentin Resseguier, Chloé Caville, Merveille Talla, Antoine Moneyron, Guillaume Lepape, Agustin Martin Picard, Matheus Ladvig, Erwan Hascoët, Laurence Wallian, Pierre Jacquet, et al.

<figure><img src="image_1.png" /></figure>

## To cite this version:

Valentin Resseguier, Chloé Caville, Merveille Talla, Antoine Moneyron, Guillaume Lepape, et al. Stochastic closure for partially-resolved turbulence. Houches 2024 - Complex Systems, Statistical Mechanics and Machine Learning Crossover (in Memory of Giovanni Paladin), Mar 2024, Les Houches, France. pp.1-1, 2024. hal-04549011

HAL Id: hal-04549011

https://hal.science/hal-04549011

Submitted on 16 Apr 2024

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entific research documents, whether they are pub-
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abroad, or from public or private research centers.

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

Distributed under a Creative Commons Attribution 4.0 International License
# Stochastic closure for partially-resolved turbulence

Valentin Resseguier, Chloé Caville, Merveille Talla, Antoine Moneyron, Guillaume Lepape, Agustin Martin Picard, Matheus Ladvig, Erwan Hascoët, Laurence Wallian, Pierre Jacquet, et al.

► To cite this version:

Valentin Resseguier, Chloé Caville, Merveille Talla, Antoine Moneyron, Guillaume Lepape, et al. Stochastic closure for partially-resolved turbulence. Houches 2024 - Complex Systems, Statistical Mechanics and Machine Learning Crossover (in Memory of Giovanni Paladin), Mar 2024, Les Houches, France. pp.1-1, 2024. hal-04549011

HAL Id: hal-04549011

https://hal.science/hal-04549011v1

Submitted on 16 Apr 2024

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_1.png" /></figure>
# STOCHASTIC CLOSURE FOR PARTIALLY-RESOLVED TURBULENCE

Resseguier V., Caville C., Talla M., Moneyron A., Lepape G., Picard A., Ladvig M., Hascoët E., Wallian L., Jacquet P., Stabile G., Mémin E., Heitz D., Chapron B.

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

## Abstract

To better constraint and explain data-based algorithms (e.g., data assimilation, Bayesian inference, machine learning), we tie priors to physics. Although statistical physics propose stochastic processes to be used as priors, many real systems hardly accommodate with some usual statistical simplifications (e.g., stationarity, delta-correlation). Therefore, our framework stands halfway between analytic stochastic descriptions and heavy numerical simulations. We rely on turbulence multiscale decomposition and stochastic transport. This framework is applied to wave-turbulence interactions and model errors quantification for data assimilation in geophysical fluid mechanics and reduced order models.

## Multi-scale velocity decomposition

<figure><img src="image_4.png" /></figure>

## Data assimilation / Bayesian inference

<figure><img src="image_5.png" /></figure>

**Notations**
Velocity wave-number
$κ = |κ|$

## Stochastic subgrid parametrization

### Stochastic transport

We consider $v' = \sigma \dot{B}_t$, a divergence-free (conditionally-)Gaussian process, $\delta$-correlated in time & correlated in space

to be injected in physics equation, typically through stochastic transport of any $Q$ along the flow $\frac{dx}{dt} = \bar{v} + \sigma \dot{B}_t$

$$ (1) \quad \frac{DQ}{Dt} = \partial_t Q + \bar{v}^* \cdot \nabla Q + \left( \sigma \dot{B}_t \cdot \nabla Q - \nabla \cdot \left( \frac{a}{2} \nabla Q \right) \right) = S_Q \quad (\text{sources, e.g. forces}) \quad \begin{cases} a = \mathbb{E}\{\sigma dB_t (\sigma dB_t)^T\}/dt \\ \bar{v}^* = \bar{v} - \nabla \cdot a/2 \end{cases} $$

Some references:

* Kunita (1990); Piterbarg & Ostrovskii (1997); Mikulevicius & Rozovskii (2004). *SIAM J Math Anal* 35.5: 1250-1310.

* Location Uncertainty (LU): Mémin (2014). *Geophys. Astro. Fluid* 108.2: 119-146; Resseguier et al. (2017). *Geophys. Astro. Fluid*. hal-01391420

* Stochastic Advection by Lie transport (SALT): Holm (2015). *Proc. Royal Society A.* 471.2176 (2015): 20140963.

Several possible calibrations:

* Brute-force data-driven: PCA from residues $v'$

$$ v'(x, t) = \sum_{i=1}^{n} \xi_i(x) \dot{W}_t^{(i)} $$

* Physics-driven: e.g., using scale symmetry

$$ v'(x, t) = \nabla \times \psi_{\sigma} * \dot{B} \quad \text{with} \quad |\dot{\psi}_{\sigma}(\kappa)|^2 \sim A\kappa^{-\mu} $$

* PCA of the multiplicative noise

$$ (v' \cdot \nabla) g(x, t) = \sum_{i=1}^{n} F_i[g](x, t) \dot{W}_t^{(i)} $$

* More in the review: Resseguier et al. (2021). *Arch Comp Eng*. hal-02558016.

<figure><img src="image_6.png" /></figure>

## Geophysical Fluid Dynamics
application
for forecastings & reanalyzes

The example of randomized SQG model for $\frac{l_{v'}}{l_v} \ll 1$

(SQG: Surface Quasi-Geostrophic)

* Stochastic transport of buoyancy $b$
  $\frac{Db}{Dt} = S_b$ (forcing & dissipation)

with $\bar{v} = \text{cst.} \nabla^\perp \Delta^{-\frac{1}{2}} b$

with initial condition:

<figure><img src="image_7.png" /></figure>

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

From one realization,
better small-scale representation.

The stochastic transport triggers instabilities.

<figure><img src="image_9.png" /></figure>

Low-resolution
deterministic
simulation

High-resolution
deterministic
simulation

Low-resolution
stochastic
simulation

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

From an ensemble of simulations,
accurate uncertainty quantification (“covariance inflation”)
to eventually improve data assimilation.

<figure><img src="image_11.png" /></figure>

At a fixed point :

<figure><img src="image_12.png" /></figure>

### Trigger and track bifurcations

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

### References

* Resseguier et al. (2017). *Geophys. Astro. Fluid*. hal-01391476v3

* Resseguier et al. (2020). *Nonlin. Proc. Geo*. hal-02371789v3

* Resseguier et al. (2021). *Arch Comp Eng*. hal-02558016

Further reading with LU/SALT assimilation :

* Cotter et al. *SIAM-ASA J Uncertain.* 8.4 (2020) 1446-1492.

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

## Surrogate application
for system
monitoring & control

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

Stochastic reduced order model (SROM) from
randomized Navier-Stokes & subsampled synthetic data
for real-time data assimilation
PCA/POD modes

* Stochastic transport of $\bar{v}(x, t) = \sum_{i=0}^{n} b_i(t) \Phi_i(x)$
  $\frac{D\bar{v}}{Dt} = S_{\bar{v}} = \text{forces}$, constrained in a small space $\text{Span}((\Phi_i)_i)$

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

POD-Galerkin and statistical estimations
with projected additive & multiplicative noises
$\int_{\Omega} \Phi_i \cdot (\text{Randomized Navier-Stokes})$ leads to :

$$ M \frac{d}{dt} b(t) = \left( \mathbf{H} \left( \tilde{b}(t) \right) + \frac{n \times (n+1) \times (n+1)}{M \times 1} \tilde{b}(t) \right) \quad \begin{array}{l} \text{Full order: } M \sim 10^7 \\ \text{Reduced order: } n \sim 10 \end{array} $$

2nd order polynomial
with coefficients given by :

$$ \frac{D\bar{v}}{Dt} = S_{\bar{v}} $$

$$ (\Phi_j)_j $$

$$ a \approx \Delta t \bar{v}'(v')^T $$

Estimator for the noise covariance

<figure><img src="image_17.png" /></figure>

### Flow prediction far outside the learning period

<figure><img src="image_18.png" /></figure>

DNS, Re 300,
$n = 8$ modes
1 measur. point

LES, Re 3900,
$n = 4$ modes
9 measur. points

### References

* ITHACA-FV: Stabile & Rozza (2018). *Comp. Fluids* arXiv:1710.11580

* Resseguier et al. (2021). *SIAM-ASA J Uncertain*. hal-03169957

* Resseguier et al. (2022). *J Comp.Phys*. hal-03445455

## Wave-turbulence interaction
application
for better observation analyses

Ray tracing for fast waves : $c_g^0/v \gg l_{v'}/l_v$

For a wave $h = He^{i\phi(\epsilon t, \epsilon x)}$

the (Stratonovich) dispersion ratio reads: $\omega - v \cdot k = \omega_0(k)$

with $\omega = -\frac{\partial \phi}{\partial t}$, $k = \nabla \phi$, and $\omega_0 \propto k^\alpha$.

Along the 4-dimensional stochastic flow,

$$ \left\{ \begin{aligned} \frac{dx_r}{dt} &= c_g^0 + \bar{v} + \sigma \dot{B}_t & \quad \text{(eikonal equations)} \\ \frac{dk}{dt} &= -\nabla(\bar{v} + \sigma \dot{B}_t)^T k \end{aligned} \right. $$

Group velocity

$$ c_g^0 = \nabla_k \omega_0 = \frac{c_g^0}{k} k $$

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

Acoustic
$\omega_0 = ck$

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

Swell in
deep ocean
$\omega_0 = \sqrt{gk}$

* the wave action spectrum $N$ is transported

$$ \frac{DN}{Dt} = 0 $$

and energy adds up: $H^2(x, t) = \int \omega_0(k) N(x, k, t) dk$

Single-ray statistics

$$ \begin{cases} \frac{d}{dt} \log k = -\bar{\sigma} \sin \zeta(\theta_k) + \gamma_0 + \sqrt{\gamma_0 \dot{B}_t^{(3)}} \\ \frac{d}{dt} \theta_k = -\frac{\partial V}{\partial \theta_k} (\zeta(\theta_k)) + \sqrt{3\gamma_0 \dot{B}_t^{(4)}} \end{cases} \quad \begin{array}{l} \text{wave-number log-normality} \\ \text{stochastic shortening} \end{array} $$
$\nabla \bar{v}^T = \bar{\sigma} S_{\bar{v}} + \bar{\omega} J$
Wave wave-vector $k$
$k = k \begin{pmatrix} \cos \theta_k \\ \sin \theta_k \end{pmatrix}$

with $V = \frac{\bar{\sigma}}{4} (\sin \zeta - \frac{\bar{\omega}}{\bar{\sigma}} \zeta)$, $\zeta = 2(\theta_k + \bar{\psi})$, and $\gamma_0 = \frac{1}{8} \mathbb{E} \left\| \nabla (\sigma \dot{B}_t)^T \right\|^2 / dt$.

Then, $\mathbb{E}N(x, k, t) = \iint dx_r^0 dk^0 N(x_r^0, k^0) p(x, k | x_r^0, k^0, t)$

### Stationary solution for linearized flow ($\bar{\sigma}, \bar{\omega}, \bar{\psi}$ cst.)

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

Wave groups trapped in an adversarial turbulent jet

* Our fast wave approximation still valid for strong current if $l_{v'}/l_v$ small enough (steep $v$ spectrum or fine resolution).

* For $\bar{u} \approx \bar{U}_0 - \frac{1}{2}\bar{\beta}y^2$ & $\bar{v} \approx 0$, $\bar{U}_0, \bar{\beta} < 0$, $\frac{d^2}{dt^2}y_r + |c_g^0\bar{\beta}|y_r = c_g^0\sqrt{3\gamma_0}\dot{B}_t^{(4)}$

with analytic oscillatory GP solution
slowly diffusing along $y$ (jet trap)

$(\sigma_{y_r}^2 \propto t$ whereas the usual branching regime ($\bar{v} = 0$) predicts $\sigma_{y_r}^2 \propto t^3)$

### Reference

* Resseguier, Hascoët & Chapron (2024). Submitted to J Fluid Dyn.
# STOCHASTIC CLOSURE FOR PARTIALLY-RESOLVED TURBULENCE

Resseguier V., Caville C., Talla M., Moneyron A., Lepape G., Picard A., Ladvig M., Hascoët E., Wallian L., Jacquet P., Stabile G., Mémin E., Heitz D., Chapron B.

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

## Abstract

To better constraint and explain data-based algorithms (e.g., data assimilation, Bayesian inference, machine learning), we tie priors to physics. Although statistical physics propose stochastic processes to be used as priors, many real systems hardly accommodate with some usual statistical simplifications (e.g., stationarity, delta-correlation). Therefore, our framework stands halfway between analytic stochastic descriptions and heavy numerical simulations. We rely on turbulence multiscale decomposition and stochastic transport. This framework is applied to wave-turbulence interactions and model errors quantification for data assimilation in geophysical fluid mechanics and reduced order models.

## Multi-scale velocity decomposition

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

## Data assimilation / Bayesian inference

<figure><img src="image_4.png" /></figure>

Notations
Velocity wave-number
$κ = |κ|$

## Stochastic subgrid parametrization

### Stochastic transport

We consider $v' = \sigma \dot{B}_t$, a divergence-free (conditionally-)Gaussian process, $\delta$-correlated in time & correlated in space

to be injected in physics equation, typically through stochastic transport of any $Q$ along the flow $\frac{dx}{dt} = \bar{v} + \sigma \dot{B}_t$

$$ (1) \quad \frac{DQ}{Dt} = \partial_t Q + \bar{v}^* \cdot \nabla Q + \left[ \sigma \dot{B}_t \cdot \nabla Q - \nabla \cdot \left( \frac{a}{2} \nabla Q \right) \right] = S_Q \quad (\text{sources, e.g. forces}) \quad \left\{ \begin{array}{l} a = \mathbb{E}\{\sigma dB_t (\sigma dB_t)^T\}/dt \\ \bar{v}^* = \bar{v} - \nabla \cdot a/2 \end{array} \right. $$

Some references:

* Kunita (1990); Piterbarg & Ostrovskii (1997); Mikulevicius & Rozovskii (2004). *SIAM J Math Anal* 35.5: 1250-1310.

* Location Uncertainty (LU): Mémin (2014). *Geophys. Astro. Fluid* 108.2: 119-146; Resseguier et al. (2017). *Geophys. Astro. Fluid*. hal-01391420

* Stochastic Advection by Lie transport (SALT): Holm (2015). *Proc. Royal Society A.* 471.2176 (2015): 20140963.

Several possible calibrations:

* Brute-force data-driven: PCA from residues $v'$

$$ v'(x, t) = \sum_{i=1}^{n} \xi_i(x) \dot{W}_t^{(i)} $$

* Physics-driven: e.g., using scale symmetry

$$ v'(x, t) = \nabla \times \psi_{\sigma} * \dot{B} \quad \text{with} \quad |\dot{\psi}_{\sigma}(\kappa)|^2 \sim A\kappa^{-\mu} $$

* PCA of the multiplicative noise

$$ (v' \cdot \nabla) g(x, t) = \sum_{i=1}^{n} F_i[g](x, t) \dot{W}_t^{(i)} $$

* More in the review: Resseguier et al. (2021). *Arch Comp Eng.* hal-02558016.

<figure><img src="image_5.png" /></figure>

## Geophysical Fluid Dynamics application for forecastings & reanalyzes

The example of randomized SQG model for $\frac{l_{v'}}{l_v} \ll 1$

(SQG: Surface Quasi-Geostrophic)

* Stochastic transport of buoyancy $b$

$\frac{Db}{Dt} = S_b$ (forcing & dissipation)

with $\bar{v} = \text{cst.} \nabla^\perp \Delta^{-\frac{1}{2}} b$

with initial condition:

<figure><img src="image_6.png" /></figure>

<figure><img src="image_7.png" /></figure>

From one realization,

better small-scale representation.

The stochastic transport triggers instabilities.

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

Low-resolution
deterministic
simulation

High-resolution
deterministic
simulation

Low-resolution
stochastic
simulation

<figure><img src="image_9.png" /></figure>

From an ensemble of simulations, accurate uncertainty quantification (“covariance inflation”) to eventually improve data assimilation.

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

At a fixed point :

<figure><img src="image_11.png" /></figure>

### Trigger and track bifurcations

<figure><img src="image_12.png" /></figure>

### References

* Resseguier et al. (2017). *Geophys. Astro. Fluid*. hal-01391476v3

* Resseguier et al. (2020). *Nonlin. Proc. Geo*. hal-02371789v3

* Resseguier et al. (2021). *Arch Comp Eng.* hal-02558016

Further reading with LU/SALT assimilation :

* Cotter et al. *SIAM-ASA J Uncertain.* 8.4 (2020) 1446-1492.

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

## Surrogate application for system monitoring & control

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

Stochastic reduced order model (SROM) from randomized Navier-Stokes & subsampled synthetic data for real-time data assimilation

PCA/POD modes

* Stochastic transport of $\bar{v}(x, t) = \sum_{i=0}^{n} b_i(t) \Phi_i(x)$

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

POD-Galerkin and statistical estimations with projected additive & multiplicative noises $\int_{\Omega} \Phi_i \cdot (\text{Randomized Navier-Stokes})$ leads to:

$$ M \frac{d}{dt} b(t) = \left[ \mathbf{H}(\tilde{b}(t)) + \begin{pmatrix} n \times (n+1) & (n+1) \times 1 \\ M \times 1 & M \times 1 \end{pmatrix} \tilde{b}(t) \right] \quad \begin{array}{l} \text{Full order: } M \sim 10^7 \\ \text{Reduced order: } n \sim 10 \end{array} $$

2nd order polynomial with coefficients given by:

$$ \cdot \frac{D\bar{v}}{Dt} = S\bar{v} $$

$$ \cdot (\Phi_j)_j $$

$$ \cdot a \approx \Delta t \bar{v}'(v')^T $$

Estimator for the noise covariance

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

### Flow prediction far outside the learning period

<figure><img src="image_17.png" /></figure>

DNS, Re 300,
$n = 8$ modes
1 measur. point

LES, Re 3900,
$n = 4$ modes
9 measur. points

### References

* ITHACA-FV: Stabile & Rozza (2018). *Comp. Fluids* arXiv:1710.11580

* Resseguier et al. (2021). *SIAM-ASA J Uncertain*. hal-03169957

* Resseguier et al. (2022). *J Comp.Phys*. hal-03445455

## Wave-turbulence interaction application for better observation analyses

Ray tracing for fast waves : $c_g^0/v \gg l_{v'}/l_v$

Group velocity

For a wave $h = He^{i\phi(\epsilon t, \epsilon x)}$

$$ c_g^0 = \nabla_k \omega_0 = \frac{c_g^0}{k} k $$

the (Stratonovich) dispersion ratio reads: $\omega - \mathbf{v} \cdot \mathbf{k} = \omega_0(k)$

with $\omega = -\frac{\partial \phi}{\partial t}$, $\mathbf{k} = \nabla \phi$, and $\omega_0 \propto k^\alpha$.

<figure><img src="image_18.png" /></figure>

Acoustic
$\omega_0 = ck$

Along the 4-dimensional stochastic flow,

$$ \left\{ \begin{aligned} \frac{dx_r}{dt} &= c_g^0 + \bar{v} + \sigma \dot{B}_t & \quad \text{(eikonal equations)} \\ \frac{dk}{dt} &= -\nabla(\bar{v} + \sigma \dot{B}_t)^T \mathbf{k} \end{aligned} \right. $$

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

Swell in
deep ocean
$\omega_0 = \sqrt{gk}$

* the wave action spectrum $N$ is transported

$$ \frac{DN}{Dt} = 0 $$

and energy adds up: $H^2(x, t) = \int \omega_0(k) N(x, k, t) dk$

Single-ray statistics

$$ \begin{cases} \frac{d}{dt} \log k = -\bar{\sigma} \sin \zeta(\theta_k) + \gamma_0 + \sqrt{\gamma_0 \dot{B}_t^{(3)}} \\ \frac{d}{dt} \theta_k = -\frac{\partial V}{\partial \theta_k} (\zeta(\theta_k)) + \sqrt{3\gamma_0 \dot{B}_t^{(4)}} \end{cases} \quad \begin{array}{l} \text{wave-number log-normality} \\ \text{stochastic shortening} \end{array} $$ $$ \text{Refraction} \quad \text{Scattering} $$ $$ \nabla \bar{v}^T = \bar{\sigma} S \bar{\psi} + \bar{\omega} J $$ $$ \text{Wave wave-vector } \mathbf{k} $$ $$ \mathbf{k} = k \begin{pmatrix} \cos \theta_k \\ \sin \theta_k \end{pmatrix} $$ $$ \text{with } V = \frac{\bar{\sigma}}{4} \left( \sin \zeta - \frac{\bar{\omega}}{\bar{\sigma}} \zeta \right), \quad \zeta = 2(\theta_k + \bar{\psi}), \quad \text{and } \gamma_0 = \frac{1}{8} \mathbb{E} \left\| \nabla (\sigma \dot{B}_t)^T \right\|^2 / dt. $$ $$ \text{Then, } \mathbb{E}(x, k, t) = \iint dx_r^0 dk^0 N(x_r^0, k^0) p(x, k | x_r^0, k^0, t) $$ $$ \text{Stationary solution for linearized flow } (\bar{\sigma}, \bar{\omega}, \bar{\psi}) \text{ stet.} $$ $$ \text{Wave groups trapped in an adversarial turbulent jet} $$

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

Our fast wave approximation still valid for strong current if $l_{v'}/l_v$ small enough (steep $v$ spectrum or fine resolution).

* For $\bar{u} \approx \bar{U}_0 - \frac{1}{2} \bar{\beta} y^2$ & $\bar{v} \approx 0$, $\bar{U}_0, \bar{\beta} < 0$

$$ \frac{d^2}{dt^2} y_r + |c_g^0 \bar{\beta}| y_r = c_g^0 \sqrt{3\gamma_0 \dot{B}_t^{(4)}} $$

with analytic oscillatory GP solution slowly diffusing along $y$ (jet trap) ($\sigma_{y_r}^2 \propto t$ whereas the usual branching regime ($\bar{v} = 0$) predicts $\sigma_{y_r}^2 \propto t^3$)

### Reference

* Resseguier, Hascoët & Chapron (2024). Submitted to J Fluid Dyn.