Stochastic modeling of the oceanic mesoscale eddies

HAL Id: hal-03140513

https://hal.inria.fr/hal-03140513

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Stochastic modeling of the oceanic mesoscale eddies

Long Li, Deremble Bruno, Noé Lahaye, Etienne Mémin

To cite this version:

Long Li, Deremble Bruno, Noé Lahaye, Etienne Mémin. Stochastic modeling of the oceanic mesoscale eddies. 6th Sandbox STUOD Sandbox Workshop, Feb 2021, London, United Kingdom. pp.1-24. �hal-03140513�

Long Li , Bruno Deremble , Noé Lahaye and Etienne Mémin

Inria Rennes Bretagne Atlantique, France

1

2

1

1

1

Stochastic modeling of the

oceanic mesoscale eddies

Motivations

I. Can LU model better represent the mesoscale eddies effect on the

large-scale circulation?

II. Can LU model improve the internal variability of coarse-resolution

ocean models?

Dynamical core

2000

3000

4000

5000

6000

7000

-4

-2

0

2

4

6

8

10

Z (km)

Y (km)

OCEAN LAYER 1

OCEAN MIXED LAYER

OCEAN LAYER 2

ATMOSPHERE

MIXED LAYER

ATMOSPHERE

LAYER 1

ATMOSPHERE

LAYER 2

INCOMING SOLAR

RADIATION

HEAT DISTRIBUTED

INTERNALLY BY

RADIATION AND

ENTRAINMENT

WIND STRESS

LEADS TO EKMAN

PUMPING

EKMAN PUMPING

QUASI-GEOSTROPHIC

DYNAMICS

QUASI-GEOSTROPHIC

DYNAMICS

ENTRAINMENT

Deterministic QG equations

•

Evolution of -th layer potential vorticity (PV)

k

@

_t

q

_k

=

r· (u

_k

q

_k

) +

A

2

f

₀

r

4

_p

k

A

₄

f

₀

r

6

_p

k

f

₀

H

_k

(w

k

w

k 1

)

u

_k

=

1

f

₀

r

?

_p

k

q

_k

=

1

f

₀

r

4

_p

k

+ (y

y

0

) +

f

₀

H

_k

(⌘

k

⌘

k 1

)

⌘

_k

=

p

k+1

p

k

g

_k

0

,

g

0

k

=

g(⇢

_k+1

⇢

_k

)

⇢

₀

•

Update -th layer pressure, height and velocity

k

u

_m

= u

₁

+

⌧

?

f

₀

H

_m

•

Evolution of mixed layer temperature (or SST)

@

_t

T

_m

=

r· (u

_m

T

_m

) + K

₂

_r

2

T

_m

K

₄

_r

4

T

_m

+ w

₀

T

1

+ T

m

2H

_m

F

m

w

₀

=

1

f

₀

r ⇥ ⌧

w

_N

=

ek

2f

₀

r

2

_p

N



w

₁

=

m

T

2

1

T

w

₀

Exp.1

Exp.2

Eddy-resolving simulations

Stochastic QG equations

Advection flux

Diffusion flux

Sources flux

Sinks flux

k

B

˙

t

•

k

-th layer eddy velocity noise (m/s)

a

_k

dt

=

E

h

k

B

˙

t

E[

k

B

˙

t

]

k

B

˙

t

E[

k

B

˙

t

]

T

i

•

k

-th layer variance tensor

(

m

2

/

s

2

)

F

_k

=

_k

B

˙

_t

1

2

r·a

k

q

k

+

1

2

a

k

rq

k

u

k

·r

?

k

B

˙

t

+a

k

rf +

1

2

X

i=1,2

@

_x

?

_i

a

_k

ru

i

_k

F

_m

=

₁

B

˙

_t

1

r· a

₁

T

_m

+

1

a

₁

rT

_m

•

Evolution of SST

@

_t

T

_m

=

r· (u

_m

T

_m

) + K

₂

_r

2

T

_m

K

₄

_r

4

T

_m

+ w

₀

T

1

+ T

m

2H

_m

F

m

+

r· F

m

•

Evolution of -th layer PV

k

@

t

q

k

=

r· (u

k

q

k

) +

A

₂

f

₀

r

4

_p

k

A

₄

f

₀

r

6

_p

k

f

₀

H

_k

(w

k

w

k 1

) +

r· F

k

h

u

?

_k

_{6= 0 if F}

2

_{6= F}

i

u

HR

k

u

HR

_k

=

Fu

HR

k

+ (1

F)u

HR

k

Fu

HR

_k

+ (1

F)u

HR

_k

Data-driven modeling

(i) LU-POD

= (u

LR

_k

u

LR

k

) + (1

F)u

HR

k

(1

F)u

HR

_k

{

k,n

}

n=1,...,N

t

POD

u

?

_k

=

_F(1

_F)u

HR

k

a

_k

dt

=

M1

X

n=M0

k,n

T

k,n

k

B

˙

t

=

M1

X

n=M0

k,n

⇠

n

+ u

?

k

⇠

_n

_{⇠ N (0, 1)}

(M

₀

< M

₁

< N

_t

)

P

_k

= I

r✓

k

(

r✓

k

)

T

kr✓

k

k

2

,

✓

_k

=

f

0

H

_k

(⌘

k

⌘

k 1

)

(ii) LU-POD-P

Data-driven modeling

e

k

B

˙

t

= P

k

k

B

˙

t

,

a

e

k

= P

k

a

k

P

T

_k

LU-POD

Coarse-resolution simulations

Δx = 80

L

= [39,22]

LR

LU-POD

LU-POD-P

0

50

100

150

0

0.02

0.04

0.06

0.08

0.1

0.12

Coarse-resolution simulations

LR

LU-POD

LU-POD-P

Upper-layer PV (

Δx = 80

km,

L

_d

= [39,22]

km)

of Exp.2 (with SST)

0

50

100

150

0

0.05

0.1

0.15

0.2

0.25

Statistical prediction

1st

Baroclinic

mode

Barotropic

mode

Comparison of

mean (60-180

yrs) contour of

pressure for

coarse-model

(120 km)

simulations

(Exp.1)

Statistical prediction

Barotropic

mode

Comparison of

mean (60-180

yrs) contour of

pressure for

coarse-model

(120 km)

simulations

(Exp.2)

1st

Baroclinic

mode

Z Z

Measures of statistics

Dispersion =

1

|A|

Z Z ⇣

2

REF

2

MOD

1

log

2

REF

2

MOD

⌘

dA

PC =

k

REF

MOD

k

2

A

k

2

REF

k

A

k

MOD

2

k

A

GRE =

1

2

p

_REF

p

_MOD

REF

2

A

+

1

2

Dispersion

RMSE of p

_MOD

=

_kp

_MOD

p

_REF

_k

_A

RMSE of

MOD

=

k

MOD

REF

k

A

,

= (p

p)

2

Measures

Values Accuracy

+

+

+

+

+

0

0.2

0.4

0.6

0.8

1

40

80

120

0

100

200

300

400

500

40

80

120

0

50

100

150

200

250

300

40

80

120

Long-term

statistics

0

0.1

0.2

0.3

0.4

0.5

40

80

120

0

0.2

0.4

0.6

0.8

1

1.2

40

80

120

Comparison of statistical

measures (integrated

vertically) for pressures

provided by different coarse

models (Exp.2)

10

0

10

2

10

1

10

2

10

3

Exp.2

(with SST)

10

-6

10

-5

10

0

10

1

10

2

10

-6

10

-5

10

1

10

2

Exp.1

(without SST)

Comparison of

KE spectral

density

averaged in

time (60-75 yrs)

and in layers

for different

coarse models

Energy analysis

EKE

_k

=

⇢

0

H

k

2

ku

k

k

2

A

+

⇢

₀

H

_k

2

ku

0

k

k

2

A

•

KE decomposition

Standing

EKE

Transient

EKE

0

50

100

150

0

200

400

600

800

1000

0

50

100

150

0

500

1000

1500

Comparison of EKE

(integrated over

layers) for different

coarse models

(Solid lines —

standing EKE,

Dashed lines —

transient EKE)

Low-pass filtering

Energy analysis

u

_k

= u

_k

+ u

0

_k

+⇢

₀

_hu

₁

, ⌧

_i

_A

⇢

0

f

0 ek

2

ku

N

k

2

A

+

N

X

k=1

⇢

₀

H

_k

_hu

_k

, A

₂

_r

2

u

_k

A

₄

_r

4

u

_k

_i

_A

@

_t

N

X

k=1

⇢

₀

H

_k

2

ku

k

k

2

A

=

@

t

N

X

1

k=1

⇢

₀

g

_k

0

2

k⌘

k

k

2

A

N

X

1

k=1

⇢

₀

g

_k

0

_h⌘

_k

, w

_k

_i

_A

Z Z

•

KE contributions

KE

PE

Buoyancy

forcing

( = 0 in Exp.1)

Dissipation

Wind

forcing

Bottom

drag

Energy analysis

0

50

100

150

0

5

10

15

10

-5

0

50

100

150

0

5

10

15

10

-5

0

50

100

150

0

5

10

15

10

-5

0

50

100

150

0

5

10

15

10

-5

0

50

100

150

0

5

10

15

10

-5

0

50

100

150

0

5

10

15

10

-5

Comparison of contributions to the rate of KE for different coarse models (Exp.1)

0

50

100

150

-5

0

5

10

15

20

10

-4

0

50

100

150

-5

0

5

10

15

20

10

-4

0

50

100

150

-5

0

5

10

15

20

10

-4

0

50

100

150

-5

0

5

10

15

10

-4

0

50

100

150

-5

0

5

10

15

10

-4

0

50

100

150

-5

0

5

10

15

10

-4

Energy analysis

Conclusion

•

The correction drift is found to be important in reproducing the meandering

jet on coarse mesh.

•

The non-stationary noise based on the projection method enables us to

Working on …

1

2

3

4

1

3

4

Coupled model