Likely chaotic transitions of large-scale fluid flows using a stochastic transport model

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Submitted on 9 Nov 2016

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Likely chaotic transitions of large-scale fluid flows using a stochastic transport model

Valentin Resseguier, Etienne Mémin, Bertrand Chapron

To cite this version:

Valentin Resseguier, Etienne Mémin, Bertrand Chapron. Likely chaotic transitions of large-scale fluid flows using a stochastic transport model. 9th Chaotic Modeling and Simulation International Conference (CHAOS2016), May 2016, Londres, United Kingdom. �hal-01377702�

Likely chaotic transitions of large-scale fluid flows

using a stochastic transport model

Valentin Resseguier, Etienne Mémin, Bertrand Chapron

1

Motivations for deriving

random fluid dynamics models

• Rigorously identified sudgrid dynamics effects

• Injecting likely small-scale dynamics

• Quantification of modeling errors:

especially for Data assimilation: ensemble forecasts

• Predicting possible distinct scenarios

2

• Chaotic transitions

• Dynamics under location uncertainty

• Ensemble of simulations

Contents

3

Chaotic transitions

4

in SQG dynamics

Reference flow: deterministic SQG 512² Initial condition 1 Scenario 1

5

512² 512²

Reference flow: deterministic SQG 512² Initial condition 1 Scenario 1

5

512² 512²

6

Reference flow: deterministic SQG 512² Initial condition 2 Scenario 2

512² 512²

6

Reference flow: deterministic SQG 512² Initial condition 2 Scenario 2

512² 512²

7

Reference flow:

deterministic SQG

512² versus 128² Initial condition 1

?

512² 512²

128² 128²

7

Reference flow:

deterministic SQG

512² versus 128² Initial condition 1

?

512² 512²

128² 128²

Dynamics under location uncertainty

8

Random equations

• Random initial conditions

• Arbitrary Gaussian forcing

• Averaging, homogenization

• Adding white random velocity

Underdispersive + need large ensemble

Adding energy + wrong phase

Assumptions and energy issues

v = w + B˙

9

Advection of tracer Θ

D ⇥

Dt = 0

10

Advection of tracer Θ

10

Advection of tracer Θ

@_t⇥ + w^? · r⇥ + B˙ · r⇥ = r ·

✓ 1

2 ar⇥

◆

10

Advection

Advection of tracer Θ

@_t⇥ + w^? · r⇥ + B˙ · r⇥ = r ·

✓ 1

2 ar⇥

◆

10

Advection

Diffusion

Advection of tracer Θ

@_t⇥ + w^? · r⇥ + B˙ · r⇥ = r ·

✓ 1

2 ar⇥

◆

10

Advection

Diffusion

Advection of tracer Θ

@_t⇥ + w^? · r⇥ + B˙ · r⇥ = r ·

✓ 1

2 ar⇥

◆

10

Drift

correction

Advection

Diffusion

Advection of tracer Θ

Multiplicative random

forcing

@_t⇥ + w^? · r⇥ + B˙ · r⇥ = r ·

✓ 1

2 ar⇥

◆

10

Drift

correction

Advection

Diffusion

Advection of tracer Θ

Multiplicative random

forcing

@_t⇥ + w^? · r⇥ + B˙ · r⇥ = r ·

✓ 1

2 ar⇥

◆

10

Drift

correction

Advection

Diffusion

Advection of tracer Θ

Multiplicative random

forcing

@_t⇥ + w^? · r⇥ + B˙ · r⇥ = r ·

✓ 1

2 ar⇥

◆

10

Drift

correction

Advection

Diffusion

Advection of tracer Θ

Multiplicative random

forcing

Balanced energy exchanges

@_t⇥ + w^? · r⇥ + B˙ · r⇥ = r ·

✓ 1

2 ar⇥

◆

10

Drift

correction

Ensemble of simulations

11

with SQG MU

Code available online

12

examples of realizations:

Spectrum mean²/variance:

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

128² 128²

12

examples of realizations:

Spectrum mean²/variance:

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

128² 128²

Visualization of the ensemble

At a fixed time t,

Principal Component Analysis (EOF)

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

⇥⁽ⁱ⁾(x, t) ⇡ Eˆ(⇥)(x, t) +

NX_EOF

n=1

c⁽ⁱ⁾_n (t) _n(x, t)

Visualization of the ensemble

At a fixed time t,

Principal Component Analysis (EOF)

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

⇥⁽ⁱ⁾(x, t) ⇡ Eˆ(⇥)(x, t) +

NX_EOF

n=1

c⁽ⁱ⁾_n (t) _n(x, t)

i^th realization

Visualization of the ensemble

At a fixed time t,

Principal Component Analysis (EOF)

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

⇥⁽ⁱ⁾(x, t) ⇡ Eˆ(⇥)(x, t) +

NX_EOF

n=1

c⁽ⁱ⁾_n (t) _n(x, t)

Mean i^th realization

Visualization of the ensemble

At a fixed time t,

Principal Component Analysis (EOF)

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

(random) EOF coefficient

⇥⁽ⁱ⁾(x, t) ⇡ Eˆ(⇥)(x, t) +

NX_EOF

n=1

c⁽ⁱ⁾_n (t) _n(x, t)

Mean i^th realization

Random initial

conditions

Under location uncertainty

Visualization of the ensemble (200 realizations)

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

Random initial

conditions

Under location uncertainty

Visualization of the ensemble (200 realizations)

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

Random initial

conditions

Under location

uncertainty

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

pdf of the 1^st PCA coefficient along time

20 30 40 50 60 70 80

Time (day) -4

-2 0 2 4

1st PCAcoefficient

×10^-4

0 2000 4000 6000 8000 10000 pdf of the 1^st PCA coefficient along time

35 40 45 50 55 60 65 70 75 80

Time (day) -4

-2 0 2 4

1st PCAcoefficient

×10^-4

0 2000 4000 6000 8000 10000

Mean of scenario A

0 5 10

x ×10⁵

0 2 4 6 8

y

×10⁵

-1 -0.5 0 0.5

×101^-3 Mean of scenario B

0 5 10

x ×10⁵

0 2 4 6 8

y

×10⁵

-1 -0.5 0 0.5

×101^-3

Mean of scenario A

0 5 10

x ×10⁵

0 2 4 6 8

y

×10⁵

-1 -0.5 0 0.5

×101 ^-3 Mean of scenario B

0 5 10

x ×10⁵

0 2 4 6 8

y

×10⁵

-1 -0.5 0 0.5

×101^-3

Conclusion

16

Conclusion

• Random transport applicable to any dynamics

• Link (inhomogeneous) diffusion, drift correction and multiplicative noise

• Predict likely scenarios with few realizations

• Other results:

Better small scales

Estimate positions and amplitudes of errors Extreme events

Additional physical information

17

Code SQG MU:

link from Fluminance website - V. Resseguier

Thank you for your attention

18

Drift correction

19

Drift correction

20

Drift correction

w^? = w 1

2 (r · a)^T

20

SQG under Moderate Uncertainty

SQG MU

Code available online

21

Reference flow:

deterministic SQG

512 x 512

22

Reference flow:

deterministic SQG

512 x 512

22