Dynamics under location uncertainty: model derivation, modified transport and uncertainty quantification

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Dynamics under location uncertainty: model derivation, modified transport and uncertainty quantification

Valentin Resseguier, Baylor Fox-Kemper, Etienne Mémin, Bertrand Chapron

To cite this version:

Valentin Resseguier, Baylor Fox-Kemper, Etienne Mémin, Bertrand Chapron. Dynamics under loca- tion uncertainty: model derivation, modified transport and uncertainty quantification. AGU 2017 - American Geophysical Union, Dec 2017, New Orleans, United States. pp.1-47. �hal-01891163�

Dynamics under location uncertainty:

model derivation, modified transport

and uncertainty quantification

Valentin Resseguier, Baylor Fox-Kemper Etienne Memin, Bertrand Chapron

1

• Rigorously identified subgrid dynamics effects


• Injecting likely small-scale dynamics


• Predict extreme events


• Quantification of modeling errors


• Studying different likely scenarios and attractors


2

Climate projections

Ensemble forecasts and data assimilation

Motivations

Contents

I. Location uncertainty

II. SQG under moderate uncertainty

3

Part I

Location uncertainty

4

v = w + B ˙

5

Adding random velocity

v = w + B ˙

5

Resolved large scales

Adding random velocity

v = w + B ˙

5

Resolved

large scales White-in-time

small scales

Adding random velocity

v = w + B ˙

5

Large scales:

Small scales:

Variance tensor:

w

B˙

a = a(x, x) =

E{ dB ( dB)^T } dt

Resolved

large scales White-in-time

small scales

Adding random velocity

v = w + B ˙

5

Large scales:

Small scales:

Variance tensor:

w

B˙

a = a(x, x) =

E{ dB ( dB)^T } dt

Resolved

large scales White-in-time

small scales

Mikulevicius and Rozovskii, 2004 Flandoli, 2011

Holm, 2015

Holm and Tyranowski, 2016 Arnaudon et al., 2017

Memin, 2014

Resseguier et al. 2017 a, b, c Chapron et al. 2017

Cai et al. 2017

Cotter and al 2017 Crisan et al., 2017

Gay-Balmaz and Holm 2017

References :

Adding random velocity

Advection of tracer Θ

D ⇥

Dt = 0

6

Large scales:

Small scales:

Variance tensor:

w

B˙

a = a(x, x) =

E{ dB ( dB)^T } dt

Advection of tracer Θ

6

Large scales:

Small scales:

Variance tensor:

w

B˙

a = a(x, x) =

E{ dB ( dB)^T } dt

Advection of tracer Θ

@

⇥ + w

· r ⇥ + B ˙ · r ⇥ = r ·

✓ 1

2 a r ⇥

◆

6

Large scales:

Small scales:

Variance tensor:

w

B˙

a = a(x, x) =

E{ dB ( dB)^T } dt

Advection

Advection of tracer Θ

@

⇥ + w

· r ⇥ + B ˙ · r ⇥ = r ·

✓ 1

2 a r ⇥

◆

6

Large scales:

Small scales:

Variance tensor:

w

B˙

a = a(x, x) =

E{ dB ( dB)^T } dt

Advection

Diffusion

Advection of tracer Θ

@

⇥ + w

· r ⇥ + B ˙ · r ⇥ = r ·

✓ 1

2 a r ⇥

◆

6

Large scales:

Small scales:

Variance tensor:

w

B˙

a = a(x, x) =

E{ dB ( dB)^T } dt

Advection

Diffusion

Advection of tracer Θ

Drift

correction

@

⇥ + w

· r ⇥ + B ˙ · r ⇥ = r ·

✓ 1

2 a r ⇥

◆

6

Large scales:

Small scales:

Variance tensor:

w

B˙

a = a(x, x) =

E{ dB ( dB)^T } dt

Advection

Diffusion

Advection of tracer Θ

Drift

correction

Multiplicative random

forcing

@

⇥ + w

· r ⇥ + B ˙ · r ⇥ = r ·

✓ 1

2 a r ⇥

◆

6

Large scales:

Small scales:

Variance tensor:

w

B˙

a = a(x, x) =

E{ dB ( dB)^T } dt

Advection

Diffusion

Advection of tracer Θ

Drift

correction

Multiplicative random

forcing

@

⇥ + w

· r ⇥ + B ˙ · r ⇥ = r ·

✓ 1

2 a r ⇥

◆

6

Large scales:

Small scales:

Variance tensor:

w

B˙

a = a(x, x) =

E{ dB ( dB)^T } dt

Advection

Diffusion

Advection of tracer Θ

Drift

correction

Multiplicative random

forcing

@

⇥ + w

· r ⇥ + B ˙ · r ⇥ = r ·

✓ 1

2 a r ⇥

◆

6

Large scales:

Small scales:

Variance tensor:

w

B˙

a = a(x, x) =

E{ dB ( dB)^T } dt

Advection

Diffusion

Advection of tracer Θ

Drift

correction

Multiplicative random

forcing

Balanced energy exchanges

@

⇥ + w

· r ⇥ + B ˙ · r ⇥ = r ·

✓ 1

2 a r ⇥

◆

6

Large scales:

Small scales:

Variance tensor:

w

B˙

a = a(x, x) =

E{ dB ( dB)^T } dt

Navier-Stokes

Derived random models

7

D Dt

Large scales:

Small scales:

Variance tensor:

w

B˙

a = a(x, x) =

E{ dB ( dB)^T } dt

Conservations (mass, linear momentum, …)

Navier-Stokes

Derived random models

7

D Dt

Large scales:

Small scales:

Variance tensor:

w

B˙

a = a(x, x) =

E{ dB ( dB)^T } dt

Conservations (mass, linear momentum, …)

Simplifications

Reduced Order Model Data

Navier-Stokes

Derived random models

7

D Dt

Large scales:

Small scales:

Variance tensor:

w

B˙

a = a(x, x) =

E{ dB ( dB)^T } dt

Conservations (mass, linear momentum, …)

Simplifications

Reduced Order Model Data

Boussinesq

Navier-Stokes

Derived random models

7

D Dt

Large scales:

Small scales:

Variance tensor:

w

B˙

a = a(x, x) =

E{ dB ( dB)^T } dt

Conservations (mass, linear momentum, …)

Simplifications

Reduced Order Model Data

Lorenz 63

Boussinesq

Navier-Stokes

Derived random models

7

D Dt

Large scales:

Small scales:

Variance tensor:

w

B˙

a = a(x, x) =

E{ dB ( dB)^T } dt

Conservations (mass, linear momentum, …)

Simplifications

Reduced Order Model Data

QG

Lorenz 63

Boussinesq

Navier-Stokes

Derived random models

7

D Dt

Large scales:

Small scales:

Variance tensor:

w

B˙

a = a(x, x) =

E{ dB ( dB)^T } dt

Conservations (mass, linear momentum, …)

Simplifications

Reduced Order Model Data

QG

Lorenz 63

Boussinesq

Navier-Stokes

Derived random models

7

D Dt

Uncertainty

a/2 U L

Large scales:

Small scales:

Variance tensor:

w

B˙

a = a(x, x) =

E{ dB ( dB)^T } dt

Conservations (mass, linear momentum, …)

Simplifications

Reduced Order Model Data

QG

QG MU Lorenz 63

Boussinesq

Navier-Stokes

Derived random models

7

D Dt

Uncertainty

a/2 U L

Large scales:

Small scales:

Variance tensor:

w

B˙

a = a(x, x) =

E{ dB ( dB)^T } dt

Conservations (mass, linear momentum, …)

Simplifications

Reduced Order Model Data

QG

SQG MU

QG MU Lorenz 63

Boussinesq

Navier-Stokes

Derived random models

7

D Dt

Uncertainty

a/2 U L

Large scales:

Small scales:

Variance tensor:

w

B˙

a = a(x, x) =

E{ dB ( dB)^T } dt

Conservations (mass, linear momentum, …)

Simplifications

Reduced Order Model Data

QG

SQG MU

QG MU

SQG SU Lorenz 63

Boussinesq

Navier-Stokes

Derived random models

7

D Dt

Uncertainty

a/2 U L

Large scales:

Small scales:

Variance tensor:

w

B˙

a = a(x, x) =

E{ dB ( dB)^T } dt

Conservations (mass, linear momentum, …)

Simplifications

Reduced Order Model Data

QG

SQG MU

QG MU

SQG SU Lorenz 63

Boussinesq

Navier-Stokes

Derived random models

7

D Dt

Uncertainty

a/2 U L

Large scales:

Small scales:

Variance tensor:

w

B˙

a = a(x, x) =

E{ dB ( dB)^T } dt

Conservations (mass, linear momentum, …)

Simplifications

Part II

SQG under Moderate Uncertainty

SQG MU

Code available online

8

9

SQG

Hyper- viscosity

Reference flow:

deterministic SQG

1024 x 1024 Db

Dt = ↵_HV ⁴b

u =

✓

cst.r^{? 12}

◆ b

9

SQG

Hyper- viscosity

Reference flow:

deterministic SQG

1024 x 1024 Db

Dt = ↵_HV ⁴b

u =

✓

cst.r^{? 12}

◆ b

One realization : Stochastic destabilization

Location Uncertainty 128 x128

x ( m)

y(m)

t= 17 d ay s

0 2 4 6 8 10

x 10⁵ 0

2 4 6 8

x 10⁵

10⁻⁵ 10⁻⁴

10⁻⁶ 10⁻⁴ 10⁻²

|ˆb(κ)|2

κ!

r a d . m⁻¹"

t= 17 d ay s

10

Deterministic 1024 x 1024 Deterministic 128 x 128

One realization : Stochastic destabilization

Location Uncertainty 128 x128

x ( m)

y(m)

t= 17 d ay s

0 2 4 6 8 10

x 10⁵ 0

2 4 6 8

x 10⁵

10⁻⁵ 10⁻⁴

10⁻⁶ 10⁻⁴ 10⁻²

|ˆb(κ)|2

κ!

r a d . m⁻¹"

t= 17 d ay s

Spectrum

10

Deterministic 1024 x 1024 Deterministic 128 x 128

One realization : Stochastic destabilization

Location Uncertainty 128 x128

x ( m)

y(m)

t= 17 d ay s

0 2 4 6 8 10

x 10⁵ 0

2 4 6 8

x 10⁵

10⁻⁵ 10⁻⁴

10⁻⁶ 10⁻⁴ 10⁻²

|ˆb(κ)|2

κ!

r a d . m⁻¹"

t= 17 d ay s

Spectrum

10

Deterministic 1024 x 1024 Deterministic 128 x 128

One realization : Stochastic destabilization

Location Uncertainty 128 x128

x ( m)

y(m)

t= 17 d ay s

0 2 4 6 8 10

x 10⁵ 0

2 4 6 8

x 10⁵

10⁻⁵ 10⁻⁴

10⁻⁶ 10⁻⁴ 10⁻²

|ˆb(κ)|2

κ!

r a d . m⁻¹"

t= 17 d ay s

Spectrum

10

Deterministic 1024 x 1024 Deterministic 128 x 128

One realization : Stochastic destabilization

Location Uncertainty 128 x128

x ( m)

y(m)

t= 17 d ay s

0 2 4 6 8 10

x 10⁵ 0

2 4 6 8

x 10⁵

10⁻⁵ 10⁻⁴

10⁻⁶ 10⁻⁴ 10⁻²

|ˆb(κ)|2

κ!

r a d . m⁻¹"

t= 17 d ay s

10

Deterministic 1024 x 1024 Deterministic 128 x 128

One realization : Stochastic destabilization

Location Uncertainty 128 x128

x ( m)

y(m)

t= 17 d ay s

0 2 4 6 8 10

x 10⁵ 0

2 4 6 8

x 10⁵

10⁻⁵ 10⁻⁴

10⁻⁶ 10⁻⁴ 10⁻²

|ˆb(κ)|2

κ!

r a d . m⁻¹"

t= 17 d ay s

10

Deterministic 1024 x 1024 Deterministic 128 x 128

One realization : Stochastic destabilization

Location Uncertainty 128 x128

x ( m)

y(m)

t= 17 d ay s

0 2 4 6 8 10

x 10⁵ 0

2 4 6 8

x 10⁵

10⁻⁵ 10⁻⁴

10⁻⁶ 10⁻⁴ 10⁻²

|ˆb(κ)|2

κ!

r a d . m⁻¹"

t= 17 d ay s

10

Deterministic 1024 x 1024 Deterministic 128 x 128

One realization : Stochastic destabilization

Location Uncertainty 128 x128

x ( m)

y(m)

t= 17 d ay s

0 2 4 6 8 10

x 10⁵ 0

2 4 6 8

x 10⁵

10⁻⁵ 10⁻⁴

10⁻⁶ 10⁻⁴ 10⁻²

|ˆb(κ)|2

κ!

r a d . m⁻¹"

t= 17 d ay s

10

Deterministic 1024 x 1024 Deterministic 128 x 128

x ( m)

y(m)

t= 17 d ay s

0 2 4 6 8 10

x 10⁵ 0

2 4 6 8

x 10⁵

10⁻⁵ 10⁻⁴

10⁻⁶ 10⁻⁴ 10⁻²

|ˆb(κ)|2

κ!

r a d . m⁻¹"

t= 17 d ay s x ( m)

y(m)

t= 17 d ay s

0 2 4 6 8 10

x 10⁵ 0

2 4 6 8

x 10⁵

10⁻⁵ 10⁻⁴

10⁻⁶ 10⁻⁴ 10⁻²

|ˆb(κ)|2

κ!

r a d . m⁻¹"

t= 17 d ay s x ( m)

y(m)

t= 17 d ay s

0 2 4 6 8 10

x 10⁵ 0

2 4 6 8

x 10⁵

10⁻⁵ 10⁻⁴

10⁻⁶ 10⁻⁴ 10⁻²

|ˆb(κ)|2

κ!

r a d . m⁻¹"

t= 17 d ay s x ( m)

y(m)

t= 17 d ay s

0 2 4 6 8 10

x 10⁵ 0

2 4 6 8

x 10⁵

10⁻⁵ 10⁻⁴

10⁻⁶ 10⁻⁴ 10⁻²

|ˆb(κ)|2

κ!

r a d . m⁻¹"

t= 17 d ay s x ( m)

y(m)

t= 17 d ay s

0 2 4 6 8 10

x 10⁵ 0

2 4 6 8

x 10⁵

10⁻⁵ 10⁻⁴

10⁻⁶ 10⁻⁴ 10⁻²

|ˆb(κ)|2

κ!

r a d . m⁻¹"

t= 17 d ay s x ( m)

y(m)

t= 17 d ay s

0 2 4 6 8 10

x 10⁵ 0

2 4 6 8

x 10⁵

10⁻⁵ 10⁻⁴

10⁻⁶ 10⁻⁴ 10⁻²

|ˆb(κ)|2

κ!

r a d . m⁻¹"

t= 17 d ay s

Ensemble:

random coherent structures

11

x ( m)

y(m)

t= 17 d ay s

0 2 4 6 8 10

x 10⁵ 0

2 4 6 8

x 10⁵

10⁻⁵ 10⁻⁴

10⁻⁶ 10⁻⁴ 10⁻²

|ˆb(κ)|2

κ!

r a d . m⁻¹"

t= 17 d ay s x ( m)

y(m)

t= 17 d ay s

0 2 4 6 8 10

x 10⁵ 0

2 4 6 8

x 10⁵

10⁻⁵ 10⁻⁴

10⁻⁶ 10⁻⁴ 10⁻²

|ˆb(κ)|2

κ!

r a d . m⁻¹"

t= 17 d ay s x ( m)

y(m)

t= 17 d ay s

0 2 4 6 8 10

x 10⁵ 0

2 4 6 8

x 10⁵

10⁻⁵ 10⁻⁴

10⁻⁶ 10⁻⁴ 10⁻²

|ˆb(κ)|2

κ!

r a d . m⁻¹"

t= 17 d ay s x ( m)

y(m)

t= 17 d ay s

0 2 4 6 8 10

x 10⁵ 0

2 4 6 8

x 10⁵

10⁻⁵ 10⁻⁴

10⁻⁶ 10⁻⁴ 10⁻²

|ˆb(κ)|2

κ!

r a d . m⁻¹"

t= 17 d ay s x ( m)

y(m)

t= 17 d ay s

0 2 4 6 8 10

x 10⁵ 0

2 4 6 8

x 10⁵

10⁻⁵ 10⁻⁴

10⁻⁶ 10⁻⁴ 10⁻²

|ˆb(κ)|2

κ!

r a d . m⁻¹"

t= 17 d ay s x ( m)

y(m)

t= 17 d ay s

0 2 4 6 8 10

x 10⁵ 0

2 4 6 8

x 10⁵

10⁻⁵ 10⁻⁴

10⁻⁶ 10⁻⁴ 10⁻²

|ˆb(κ)|2

κ!

r a d . m⁻¹"

t= 17 d ay s

Ensemble:

random coherent structures

11

12

Ensemble : uncertainty quantification

x ( m)

y(m)

t= 17 d ay s

0 2 4 6 8 10

x 10⁵ 0

2 4 6 8

x 10⁵

10⁻⁵ 10⁻⁴

10⁻⁶ 10⁻⁴ 10⁻²

|ˆb(κ)|2

κ!

r a d . m⁻¹"

t= 17 d ay s x ( m)

y(m)

t= 17 d ay s

0 2 4 6 8 10

x 10⁵ 0

2 4 6 8

x 10⁵

10⁻⁵ 10⁻⁴

10⁻⁶ 10⁻⁴ 10⁻²

|ˆb(κ)|2

κ!

r a d . m⁻¹"

t= 17 d ay s x ( m)

y(m)

t= 17 d ay s

0 2 4 6 8 10

x 10⁵ 0

2 4 6 8

x 10⁵

10⁻⁵ 10⁻⁴

10⁻⁶ 10⁻⁴ 10⁻²

|ˆb(κ)|2

κ!

r a d . m⁻¹"

t= 17 d ay s x ( m)

y(m)

t= 17 d ay s

0 2 4 6 8 10

x 10⁵ 0

2 4 6 8

x 10⁵

10⁻⁵ 10⁻⁴

10⁻⁶ 10⁻⁴ 10⁻²

|ˆb(κ)|2

κ!

r a d . m⁻¹"

t= 17 d ay s x ( m)

y(m)

t= 17 d ay s

0 2 4 6 8 10

x 10⁵ 0

2 4 6 8

x 10⁵

10⁻⁵ 10⁻⁴

10⁻⁶ 10⁻⁴ 10⁻²

|ˆb(κ)|2

κ!

r a d . m⁻¹"

t= 17 d ay s x ( m)

y(m)

t= 17 d ay s

0 2 4 6 8 10

x 10⁵ 0

2 4 6 8

x 10⁵

10⁻⁵ 10⁻⁴

10⁻⁶ 10⁻⁴ 10⁻²

|ˆb(κ)|2

κ!

r a d . m⁻¹"

t= 17 d ay s

12

Ensemble : uncertainty quantification

x ( m)

y(m)

t= 17 d ay s

0 2 4 6 8 10

x 10⁵ 0

2 4 6 8

x 10⁵

10⁻⁵ 10⁻⁴

10⁻⁶ 10⁻⁴ 10⁻²

|ˆb(κ)|2

κ!

r a d . m⁻¹"

t= 17 d ay s x ( m)

y(m)

t= 17 d ay s

0 2 4 6 8 10

x 10⁵ 0

2 4 6 8

x 10⁵

10⁻⁵ 10⁻⁴

10⁻⁶ 10⁻⁴ 10⁻²

|ˆb(κ)|2

κ!

r a d . m⁻¹"

t= 17 d ay s x ( m)

y(m)

t= 17 d ay s

0 2 4 6 8 10

x 10⁵ 0

2 4 6 8

x 10⁵

10⁻⁵ 10⁻⁴

10⁻⁶ 10⁻⁴ 10⁻²

|ˆb(κ)|2

κ!

r a d . m⁻¹"

t= 17 d ay s x ( m)

y(m)

t= 17 d ay s

0 2 4 6 8 10

x 10⁵ 0

2 4 6 8

x 10⁵

10⁻⁵ 10⁻⁴

10⁻⁶ 10⁻⁴ 10⁻²

|ˆb(κ)|2

κ!

r a d . m⁻¹"

t= 17 d ay s x ( m)

y(m)

t= 17 d ay s

0 2 4 6 8 10

x 10⁵ 0

2 4 6 8

x 10⁵

10⁻⁵ 10⁻⁴

10⁻⁶ 10⁻⁴ 10⁻²

|ˆb(κ)|2

κ!

r a d . m⁻¹"

t= 17 d ay s x ( m)

y(m)

t= 17 d ay s

0 2 4 6 8 10

x 10⁵ 0

2 4 6 8

x 10⁵

10⁻⁵ 10⁻⁴

10⁻⁶ 10⁻⁴ 10⁻²

|ˆb(κ)|2

κ!

r a d . m⁻¹"

t= 17 d ay s

Conclusion

Models under location uncertainty blindly describe unresolved triades

•

Conserve energy

•

Model derivation

•

Instabilities triggered, 


possibly followed by extreme events

•

Uncertainty quantification to address filter divergence

13 V. Resseguier - [email protected]

Related works and outlooks

• Bifurcations (SQG) and attractor (Lorenz 63) exploration

• Stabilization / destabilization of Reduced Order Model

• Comparisons with data-driven and Stochastic Lie Derivative approaches (Holm and coauthors)

• Parametrization and tests based on higher-order statistics
 (curvature, energy flux through scales, bispectrum, …)

• Mimic barotropization

• Girsanov theorem for MLE and Bayesian estimations with satellite images

• Learning on SWOT data

• Filtering / smoothing with (reduced) models under location uncertainty

14 V. Resseguier - [email protected]