Oceanic Dynamics under Location Uncertainty: Towards a consistent stochastic modeling

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Oceanic Dynamics under Location Uncertainty: Towards a consistent stochastic modeling

Long Li, Werner Bauer, Etienne Mémin

To cite this version:

Long Li, Werner Bauer, Etienne Mémin. Oceanic Dynamics under Location Uncertainty: To-

wards a consistent stochastic modeling. Workshop Conservation Principles, Data and Uncertainty

in Atmosphere-Ocean Modelling, Apr 2019, Potsdam, Germany. �hal-02708028�

Oceanic Dynamics under Location Uncertainty

Towards a consistent stochastic modelling

Fluminance Group

Inria Rennes France & Universit´e Rennes 1 France

Long Li Werner Bauer Etienne Mémin

Objective

Why stochastic modeling ?

Why consistent model ?

Take into account small-scale / unresolved processes

Uncertainty Quantification, Ensemble Forecasts

Both scales must respect appreciate physical laws

Losing consistency may provide wrong statistics

Outline

3 Summary

4

2 Experimental Evaluations

Structure—preserving of Rossby wave

Ensemble forecasting verification of SQG _MU

Time—statistics of wind—driven circulation

1 Governing Equations

Location Uncertainty Principles

Barotropic Quasi—Geostrophic Model under LU

homogeneous / heterogeneous

Location Uncertainty Principles

small-scale unresolved large-scale

resolved

dX _t = u(X _t , t)dt + (X _t , t)dB _t

Stochastic flow :

a = ^{4 T} = E ⇥

dB _t ( dB _t ) ^T ⇤

/dt

Variance tensor :

space-time white noise symmetric

kernel correlated in space

uncorrelated in time

Functional process :

(x, t)dB _t = Z

D

˘ (x, y, t)dB _t (y)dy

multiplicative noise

subgrid diffusion corrected drift

u 1

2 r · a

Balanced Energy

d _t ✓ + u ^? · r ✓dt + dB _t · r ✓ 1

2 r · (a r ✓)dt = 0 D _t ✓ = ⁴

Transport of a random tracer : ( r · = 0 )

d dt

Z

D

1

2 ✓ ² = 0

( r · u ^? = 0 )

( Mémin 2014; Resseguier et al. 2017 )

Barotropic Quasi–Geostrophic Model under LU ( Resseguier et al. 2017; Li et al. 2019 )

Evolution of the potential vorticity with source processes : D _t q = S ₁ ( r u)dt + S ₂ ( r u)dB _t

Evolution of the stream function :

q = /L ² _R + f

Strong incompressible constraints :

u = r ^? , dB _t = r ^? 'dB _t , r · u ^? = 0 Conservation of total energy : ^d

dt Z

D

1 2

h kr k ² + ( /L ² _R ) ² i

= 0

Transport of a prognostic passive tracer : D _t q ⁰ = 0

Conservation of tracer energy : ^d

dt Z

D

1

2 (q ⁰ ) ² = 0

h Sketch : Z

D

dB _t · r (q f ) = Z

D

S ₂ ( r u) dB _t i

Outline

3 Summary

4

2 Experimental Evaluations

Structure—preserving of Rossby wave

Ensemble forecasting verification of SQG _MU

Time—statistics of wind—driven circulation

1 Governing Equations

Location Uncertainty Principles

Barotropic Quasi—Geostrophic Model under LU

Structure–preserving of Rossby wave

Flowchart : q

( Potential Vorticity )

q′

( Passive Tracer )

Illustration of conservations :

Passive Tracer Tracer Energy Total Energy

ψ

( Stream Function )

Ensemble forecasting verification of SQG _MU

Time

Initial uncertainty Forecast uncertainty Model

PIC

LU

Illustration of ensemble prediction systems :

Ensemble forecasting verification of SQG _MU

Reliability — Ensemble Spread match MSE of the Ensemble Mean :

( Resseguier et al. 2019 )

Ensemble forecasting verification of SQG _MU

LU

homogeneous

PIC

homogeneous

LU

heterogeneous

PIC

heterogeneous

Day 10 Day 20 Day 30 Reliability

Model

Reliability — Ranked Histogram of Ensemble Members :

( Resseguier et al. 2019 )

Time–statistics of wind–driven circulation

DNS 256 ⇥ 512 with Re = 450 and Ro = 0.0016

Time–statistics of wind–driven circulation

DNS 256 ⇥ 512 LU 32 ⇥ 64 LES _LU 32 ⇥ 64

Truth

Outline

3 Summary

4

2 Experimental Evaluations

Structure—preserving of Rossby wave

Ensemble forecasting verification of SQG _MU

Time—statistics of wind—driven circulation

1 Governing Equations

Location Uncertainty Principles

Barotropic Quasi—Geostrophic Model under LU

Summary

A consistent stochastic model :

Physical conservation laws satisfied Better ensemble spread represented

Time-averaged profil well described on coarse mesh

Future works :

Multi-layers QG model under LU

( Q-GCM Projects : Hogg et al. 2003 )

Data Assimilation with Particle Filters

( Cotter et al. 2018 )

Thanks for Your Attention !

Ensemble of random stream functions

Homogeneous parameterization

low-pass band filter ^? white noise

Heterogeneous parameterization

y( m )

x(m)

t = 0 d ay s

0 2 4 6 8

x 10 ⁵ 0

5

10 x 10 ⁵

1 0.5 0

0.5 x 10 1 ³

Neighbor Centered neighbor

Random selection

y( m )

x(m) t = 0 d ay s

0 2 4 6 8

x 10

0

5 10 x 10

1 0.5 0 0.5 x 10 1

y( m )

x(m) t = 0 d ay s

0 2 4 6 8

x 10

0

5 10 x 10

1 0.5 0 0.5 x 10 1

y( m )

x(m) t = 0 d ay s

0 2 4 6 8

x 10

0

5 10 x 10

1 0.5 0 0.5 x 10 1

y( m )

x(m) t = 0 d ay s

0 2 4 6 8

x 10

0

5 10 x 10

1 0.5 0 0.5 x 10 1

y( m )

x(m) t = 0 d ay s

0 2 4 6 8

5

0 5 10 x 10

1 0.5 0 0.5 x 10 1

Global ensemble

Local ensemble :

SVD

Noise