Stochastic parameterization of geophysical flows through modelling under location uncertainty

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Submitted on 9 Nov 2016

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Stochastic parameterization of geophysical flows through modelling under location uncertainty

Valentin Resseguier, Etienne Mémin, Bertrand Chapron, Pierre Dérian

To cite this version:

Valentin Resseguier, Etienne Mémin, Bertrand Chapron, Pierre Dérian. Stochastic parameterization of geophysical flows through modelling under location uncertainty. Data Analysis and Modeling in Earth Sciences (DAMES) 2016, Sep 2016, Hambourg, Germany. �hal-01377719�

Stochastic parametrization of geophysical flows

through modeling

under location uncertainty

Valentin Resseguier, Pierre Dérian, Etienne Mémin, Bertrand Chapron

Motivations

• Rigorously identified sudgrid dynamics effects


• Injecting likely small-scale dynamics


• Studying bifurcations and attractors


• Quantification of modeling errors


Climate projections

Ensemble forecasts and data assimilation

• Randomized dynamics

• SQG under Moderate Uncertainty

• Lorenz under location uncertainty

Contents

Randomized

dynamics

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˙

Advection of tracer Θ

D ⇥

Dt = 0

Advection of tracer Θ

Advection of tracer Θ

@_t⇥ + w^? · r⇥ + B˙ · r⇥ = r ·

✓ 1

2 ar⇥

◆

Advection

Advection of tracer Θ

@_t⇥ + w^? · r⇥ + B˙ · r⇥ = r ·

✓ 1

2 ar⇥

◆

Advection

Diffusion

Advection of tracer Θ

@_t⇥ + w^? · r⇥ + B˙ · r⇥ = r ·

✓ 1

2 ar⇥

◆

Advection

Diffusion

Advection of tracer Θ

Drift

correction

@_t⇥ + w^? · r⇥ + B˙ · r⇥ = r ·

✓ 1

2 ar⇥

◆

Advection

Diffusion

Advection of tracer Θ

Drift

correction

Multiplicative random

forcing

@_t⇥ + w^? · r⇥ + B˙ · r⇥ = r ·

✓ 1

2 ar⇥

◆

Advection

Diffusion

Advection of tracer Θ

Drift

correction

Multiplicative random

forcing

@_t⇥ + w^? · r⇥ + B˙ · r⇥ = r ·

✓ 1

2 ar⇥

◆

Advection

Diffusion

Advection of tracer Θ

Drift

correction

Multiplicative random

forcing

@_t⇥ + w^? · r⇥ + B˙ · r⇥ = r ·

✓ 1

2 ar⇥

◆

Advection

Diffusion

Advection of tracer Θ

Drift

correction

Multiplicative random

forcing

Balanced energy exchanges

@_t⇥ + w^? · r⇥ + B˙ · r⇥ = r ·

✓ 1

2 ar⇥

◆

Derived random models

Conservations (mass, linear momentum, …)

D Dt

Navier-Stokes

Boussinesq

Lorenz

Derived random models

Conservations (mass, linear momentum, …)

D Dt

Navier-Stokes

Boussinesq

Lorenz

Uncertainty

QG MU

SQG MU SQG SU

Derived random models

Conservations (mass, linear momentum, …)

D Dt

Navier-Stokes

Boussinesq

Lorenz

Uncertainty

QG MU

SQG MU SQG SU

Derived random models

Conservations (mass, linear momentum, …)

D Dt

Navier-Stokes

Boussinesq

Lorenz

Uncertainty

QG MU

SQG MU SQG SU

Derived random models

Conservations (mass, linear momentum, …)

D Dt

Navier-Stokes

Boussinesq

SQG under Moderate Uncertainty

SQG MU

Code available online

Reference flow:

deterministic SQG

512 x 512

Reference flow:

deterministic SQG

512 x 512

One realization

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

One realization

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 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

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

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

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

Summary of QG models

• Better small scales

• Estimate position and amplitude of errors

• Extreme events

• Bifurcations

• under Strong Uncertainty:


Simple 2D description of frontolysis/frontogenesis

pdf of the 1^stPCA coefficient along time

20 30 40 50 60 70 80

Time (day) -4

-2 0 2 4

1stPCAcoefficient

×10^-4

0 2000 4000 6000 8000 10000

y(m)

x(m) t= 0 d ay s

02468

x 10⁵ 0 2 4 6 8 x 10⁵

−1

−0.5 0 0.5 1 x 10⁻³

y(m)

x(m) t= 70 d ay s

0 5 10

x 10⁵ 0 2 4 6 8 x 10⁵

−1

−0.5 0 0.5 x 101⁻³

y(m)

x(m) t= 70 d ay s

02468

x 10⁵ 0 2 4 6 8 x 10⁵

−1

−0.5 0 0.5 x 101⁻³

Code SQG MU:

Lorenz model under location

uncertainty

Do large-scale (diffusive) models lead to over-representing "stable"-states 


in ensemble simulations?

Lorenz model(s)

• (usual) deterministic model ~ DNS, accurate but impossible to compute

• (deterministic) diffusive model ~ LES

• stochastic model under location uncertainty

dX

dt = Pr (Y X) 4

2⌥X dY = h

X(⇢ Z) Y 4

2⌥Y i

dt + ⇢ Z

⌥^1/2 dB_t dZ = h

XY bZ 8

2⌥bZi

dt + Y

⌥^1/2 dB_t

time error

distribution over many ensembles mean

particle reference

closest particle

bias

min distance

RMS distance

ensemble

Short time behavior

Comparison ensemble ↔ reference

3 metrics: minimum distance, bias, RMS distance

time error

distribution over many ensembles mean

particle reference

closest particle

bias

min distance

RMS distance

ensemble

Short time behavior

Comparison ensemble ↔ reference

3 metrics: minimum distance, bias, RMS distance

Long time

Long time

Long time

Attractor

visit rates

Attractor

visit rates

Conclusion

Conclusion

• Random transport applicable to any dynamics

• Better small scales

• Efficient spreading of the ensemble

• Likely scenarios

• Exploration of the attractor

Code SQG MU:

link from Fluminance website - V. Resseguier

Thank you for your attention

Drift correction

Drift correction

Drift correction

Drift correction

w^? = w 1

2 (r · a)^T

Bifurcations in SQG

tracked by SQG MU

Reference flow:

deterministic SQG

512² versus 128² Initial condition 1

?

512² 512²

128² 128²

Reference flow:

deterministic SQG

512² versus 128² Initial condition 1

?

512² 512²

128² 128²

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

-2 0 2 4

1st PCAcoefficient

×10^-4

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

×10^{5 ×10}1

-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

SQG under Strong Uncertainty

SQG SU

Mesoscale divergence

Horizontal Diffusion Geostrophic balance

f ⇥ u = 1

⇢_b rp⁰ + a

2 u

r · u / r

· u

Filtering of model outputs:

Gula, Jonathan, M. Jeroen Molemaker, and James C. McWilliams

"Gulf Stream dynamics along the southeastern US seaboard.”

Journal of Physical Oceanography 45.3 (2015): 690-715.

Filtering of model outputs:

Gula, Jonathan, M. Jeroen Molemaker, and James C. McWilliams

"Gulf Stream dynamics along the southeastern US seaboard.”

Journal of Physical Oceanography 45.3 (2015): 690-715.

Spatial test

Spatial test

10⁻⁴ 10⁻²

10⁻¹ 10⁰ 10¹

|ˆf1(κ)|/|ˆf2(κ)|

κ!

r a d . m⁻¹"

Me an sp e c t r u m r at i o 10⁻⁴

10⁻⁴ 10⁻² 10⁰ 10²

|ˆf(κ)|2

κ!

r a d . m⁻¹"

Nor mal i z e d me an sp e c t r u m of t h e i r r ot at i on al v e l o c i ty an d of i t s e st i mat i on

0.5 1 1.5

θ

Me an p h ase sh i f t

Spectral

test

Long time:

weak noise

Long time:

weak noise

Long time:

weak noise

Attractor

visit rates

Attractor

visit rates

Computing the visit rate

• Discrete covering of the Lorenz attractor (GAIO)


→ 611,550 cubic boxes of radius=0.15625

• For each ensemble, the visit rate:

⌧ (T ) = #{ unique boxes visited by ensemble over [0; T ]} total # of boxes