Dynamics under location uncertainty and other energy-related stochastic subgrid schemes

(1)

HAL Id: hal-02103233

https://hal.archives-ouvertes.fr/hal-02103233

Submitted on 18 Apr 2019

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Dynamics under location uncertainty and other

energy-related stochastic subgrid schemes

Valentin Resseguier, Long Li, Gabriel Jouan, Pierre Dérian, Etienne Mémin,

Bertrand Chapron

To cite this version:

Valentin Resseguier, Long Li, Gabriel Jouan, Pierre Dérian, Etienne Mémin, et al.. Dynamics under

location uncertainty and other energy-related stochastic subgrid schemes. 2019 - Workshop

Conserva-tion Principles, Data and Uncertainty in Atmosphere-Ocean Modelling, Apr 2019, Potsdam, Germany.

pp.1-70. �hal-02103233�

(2)

Dynamics under location

uncertainty and  

other energy-related stochastic

subgrid schemes

V. Resseguier

L. Li, G. Jouan, P. Derian, 

E. Memin, B. Chapron

(3)

Motivations

• More rigorously identified sudgrid dynamics effects 

 

• Quantification of modeling errors (UQ) 

2

(4)

• Models under location uncertainty (LU)

• Some parameterization of the models under

location uncertainty

• A new energy-budget-based stochastic scheme: 

WaveHyperv

• Numerical comparisons

(5)

Part I

Models under location

uncertainty (LU)

(6)

v = w +

B

˙

LU :Adding

(7)

v = w +

B

˙

Resolved

large scales

LU :Adding

random velocity

5

(8)

v = w +

B

˙

Resolved

large scales

White-in-time

_{small scales}

LU :Adding

(9)

v = w +

B

˙

Resolved

large scales

White-in-time

_{small scales}

LU :Adding

random velocity

5 Large scales:

Small scales:

Variance

tensor:

w

˙

B

a = a(x, x) =

E{ dB ( dB)

T

}

dt

(10)

v = w +

B

˙

Resolved

large scales

White-in-time

_{small scales}

LU :Adding

random velocity

Large scales:

Small scales:

Variance

tensor:

w

˙

B

a = a(x, x) =

E{ dB ( dB)

T

}

dt

Mikulevicius &

Rozovskii, 2004

Flandoli, 2011

Holm, 2015

Holm and

Tyranowski, 2016

Arnaudon et al. 2017

Cotter and al. 2017

Memin, 2014

Resseguier et al. 2017 a, b, c

Cai et al. 2017

Chapron et al. 2018 

Yang & Memin 2019

Resseguier et al. 2019 a,b

Crisan et al., 2017

Gay-Balmaz & Holm 2017

Cotter and al. 2018 a, b

Cotter and al. 2019

(11)

LU

SALT

v = w +

B

˙

Resolved

large scales

White-in-time

_{small scales}

LU :Adding

random velocity

5 Large scales:

Small scales:

Variance

tensor:

w

˙

B

a = a(x, x) =

E{ dB ( dB)

T

}

dt

Mikulevicius &

Rozovskii, 2004

Flandoli, 2011

Holm, 2015

Holm and

Tyranowski, 2016

Arnaudon et al. 2017

Cotter and al. 2017

Memin, 2014

Resseguier et al. 2017 a, b, c

Cai et al. 2017

Chapron et al. 2018 

Yang & Memin 2019

Resseguier et al. 2019 a,b

Crisan et al., 2017

Gay-Balmaz & Holm 2017

Cotter and al. 2018 a, b

Cotter and al. 2019

(12)

Advection 

of tracer

Θ

D⇥

Dt

= 0

Large scales:

Small scales:

Variance

tensor:

w

˙

B

a = a(x, x) =

E{ dB ( dB)

T

}

dt

(13)

Advection 

of tracer

Θ

6 Large scales:

Small scales:

Variance

tensor:

w

˙

B

a = a(x, x) =

E{ dB ( dB)

T

}

dt

(14)

Advection 

of tracer

Θ

@

_t

⇥ + w

?

· r⇥ +

B

˙

· r⇥ = r ·

✓

1

2 a

r⇥

◆ Large scales:

Small scales:

Variance

tensor:

w

˙

B

a = a(x, x) =

E{ dB ( dB)

T

}

dt

(15)

Advection

of tracer

Θ

@

_t

⇥ + 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

(16)

Advection

Diffusion

Advection 

of tracer

Θ

@

_t

⇥ + w

?

· r⇥ +

B

˙

· r⇥ = r ·

✓

1

2 a

r⇥

◆ Large scales:

Small scales:

Variance

tensor:

w

˙

B

a = a(x, x) =

E{ dB ( dB)

T

}

dt

(17)

Advection

Diffusion

Advection 

of tracer

Θ

Drift correction

@

_t

⇥ + 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

(18)

Advection

Diffusion

Advection 

of tracer

Θ

Drift correction

Multiplicative

random

forcing

@

_t

⇥ + w

?

· r⇥ +

B

˙

· r⇥ = r ·

✓

1

2 a

r⇥

◆ Large scales:

Small scales:

Variance

tensor:

w

˙

B

a = a(x, x) =

E{ dB ( dB)

T

}

dt

(19)

Advection

Diffusion

Advection 

of tracer

Θ

Drift correction

Multiplicative

random

forcing

@

_t

⇥ + 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

(20)

Advection

Diffusion

Advection 

of tracer

Θ

Drift correction

Multiplicative

random

forcing

@

_t

⇥ + w

?

· r⇥ +

B

˙

· r⇥ = r ·

✓

1

2 a

r⇥

◆ Large scales:

Small scales:

Variance

tensor:

w

˙

B

a = a(x, x) =

E{ dB ( dB)

T

}

dt

(21)

Advection

Diffusion

Advection 

of tracer

Θ

Drift correction

Multiplicative

random

forcing

Balanced

energy

exchanges

@

_t

⇥ + 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

(22)

Part II

Some parameterization 

of LU models

(23)

Part II

Some parameterization 

of LU models

7

=?

(24)

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

1 x 10

−3

10 -5

10 -4

10

2

10

4 w

Large scales:

Small scales:

Variance

tensor:

w

˙

B

a = a(x, x) =

E{ dB ( dB)

T

}

dt

Reference:

KE Spectrum

MU Spec

Spectral model 

(homogeneous and stationary )

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B

˙

B

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(25)

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

1 x 10

−3

10 -5

10 -4

10

2

10

4 w

Large scales:

Small scales:

Variance

tensor:

w

˙

B

a = a(x, x) =

E{ dB ( dB)

T

}

dt

8 Reference:

Resseguier, Memin & Chapron 2017b

KE Spectrum

Fixed spectrum

at small scales

MU Spec

Spectral model 

(homogeneous and stationary )

B

˙

B

(26)

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

1 x 10

−3

10 -5

10 -4

10

2

10

4 w

Large scales:

Small scales:

Variance

tensor:

w

˙

B

a = a(x, x) =

E{ dB ( dB)

T

}

dt

Reference:

KE Spectrum

Fixed spectrum

at small scales

MU Spec

Spectral model 

(homogeneous and stationary )

B

˙

B

(filter) (white noise)

_{B =}

˙

<latexit sha1_base64="1L0eoLDZQus/4Lt3o4EVUB1V/3g=">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</latexit><latexit sha1_base64="1L0eoLDZQus/4Lt3o4EVUB1V/3g=">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</latexit><latexit sha1_base64="1L0eoLDZQus/4Lt3o4EVUB1V/3g=">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</latexit><latexit sha1_base64="1L0eoLDZQus/4Lt3o4EVUB1V/3g=">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</latexit>

⇤

<latexit sha1_base64="v//LS0nZAWkEUMFOPunsLTcwR08=">AAAB6XicdVDJSgNBEK2JW4xb1KOXxiCIh6EnRDO5Bb14jGIWSIbQ0+lJmvQsdPcIYcgfePGgiFf/yJt/Y2cRVPRBweO9Kqrq+YngSmP8YeVWVtfWN/Kbha3tnd294v5BS8WppKxJYxHLjk8UEzxiTc21YJ1EMhL6grX98dXMb98zqXgc3elJwryQDCMecEq0kW7PUL9Ywrbr1pyKi7Dt4FoVVwzBtfJF9Rw5Np6jBEs0+sX33iCmacgiTQVRquvgRHsZkZpTwaaFXqpYQuiYDFnX0IiETHnZ/NIpOjHKAAWxNBVpNFe/T2QkVGoS+qYzJHqkfnsz8S+vm+rA9TIeJalmEV0sClKBdIxmb6MBl4xqMTGEUMnNrYiOiCRUm3AKJoSvT9H/pFW2HcNvKqX65TKOPBzBMZyCA1WowzU0oAkUAniAJ3i2xtaj9WK9Llpz1nLmEH7AevsERUWNMA==</latexit><latexit sha1_base64="v//LS0nZAWkEUMFOPunsLTcwR08=">AAAB6XicdVDJSgNBEK2JW4xb1KOXxiCIh6EnRDO5Bb14jGIWSIbQ0+lJmvQsdPcIYcgfePGgiFf/yJt/Y2cRVPRBweO9Kqrq+YngSmP8YeVWVtfWN/Kbha3tnd294v5BS8WppKxJYxHLjk8UEzxiTc21YJ1EMhL6grX98dXMb98zqXgc3elJwryQDCMecEq0kW7PUL9Ywrbr1pyKi7Dt4FoVVwzBtfJF9Rw5Np6jBEs0+sX33iCmacgiTQVRquvgRHsZkZpTwaaFXqpYQuiYDFnX0IiETHnZ/NIpOjHKAAWxNBVpNFe/T2QkVGoS+qYzJHqkfnsz8S+vm+rA9TIeJalmEV0sClKBdIxmb6MBl4xqMTGEUMnNrYiOiCRUm3AKJoSvT9H/pFW2HcNvKqX65TKOPBzBMZyCA1WowzU0oAkUAniAJ3i2xtaj9WK9Llpz1nLmEH7AevsERUWNMA==</latexit><latexit sha1_base64="v//LS0nZAWkEUMFOPunsLTcwR08=">AAAB6XicdVDJSgNBEK2JW4xb1KOXxiCIh6EnRDO5Bb14jGIWSIbQ0+lJmvQsdPcIYcgfePGgiFf/yJt/Y2cRVPRBweO9Kqrq+YngSmP8YeVWVtfWN/Kbha3tnd294v5BS8WppKxJYxHLjk8UEzxiTc21YJ1EMhL6grX98dXMb98zqXgc3elJwryQDCMecEq0kW7PUL9Ywrbr1pyKi7Dt4FoVVwzBtfJF9Rw5Np6jBEs0+sX33iCmacgiTQVRquvgRHsZkZpTwaaFXqpYQuiYDFnX0IiETHnZ/NIpOjHKAAWxNBVpNFe/T2QkVGoS+qYzJHqkfnsz8S+vm+rA9TIeJalmEV0sClKBdIxmb6MBl4xqMTGEUMnNrYiOiCRUm3AKJoSvT9H/pFW2HcNvKqX65TKOPBzBMZyCA1WowzU0oAkUAniAJ3i2xtaj9WK9Llpz1nLmEH7AevsERUWNMA==</latexit><latexit sha1_base64="v//LS0nZAWkEUMFOPunsLTcwR08=">AAAB6XicdVDJSgNBEK2JW4xb1KOXxiCIh6EnRDO5Bb14jGIWSIbQ0+lJmvQsdPcIYcgfePGgiFf/yJt/Y2cRVPRBweO9Kqrq+YngSmP8YeVWVtfWN/Kbha3tnd294v5BS8WppKxJYxHLjk8UEzxiTc21YJ1EMhL6grX98dXMb98zqXgc3elJwryQDCMecEq0kW7PUL9Ywrbr1pyKi7Dt4FoVVwzBtfJF9Rw5Np6jBEs0+sX33iCmacgiTQVRquvgRHsZkZpTwaaFXqpYQuiYDFnX0IiETHnZ/NIpOjHKAAWxNBVpNFe/T2QkVGoS+qYzJHqkfnsz8S+vm+rA9TIeJalmEV0sClKBdIxmb6MBl4xqMTGEUMnNrYiOiCRUm3AKJoSvT9H/pFW2HcNvKqX65TKOPBzBMZyCA1WowzU0oAkUAniAJ3i2xtaj9WK9Llpz1nLmEH7AevsERUWNMA==</latexit>

(27)

Absolute Diffusivity 

Spectral Density

(

homogeneous but non-stationary and tuning-free )

w

Large scales:

Small scales:

Variance

tensor:

w

˙

B

a = a(x, x) =

E{ dB ( dB)

T

}

dt

9 MU ADSD

Reference:

Resseguier, Pan & Fox-Kemper 2019a

˙

B

(28)

Absolute Diffusivity 

Spectral Density

(

homogeneous but non-stationary and tuning-free )

w

Large scales:

Small scales:

Variance

tensor:

w

˙

B

a = a(x, x) =

E{ dB ( dB)

T

}

dt

Absolute Diffusivity 

Spectral Density

A() = E()⌧ ()

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MU ADSD

Reference:

˙

B

(29)

Absolute Diffusivity 

Spectral Density

(

homogeneous but non-stationary and tuning-free )

w

Large scales:

Small scales:

Variance

tensor:

w

˙

B

a = a(x, x) =

E{ dB ( dB)

T

}

dt

Absolute Diffusivity 

Spectral Density

A() = E()⌧ ()

9 MU ADSD

Reference:

Resseguier, Pan & Fox-Kemper 2019a

˙

B

(30)

Absolute Diffusivity 

Spectral Density

(

homogeneous but non-stationary and tuning-free )

w

Large scales:

Small scales:

Variance

tensor:

w

˙

B

a = a(x, x) =

E{ dB ( dB)

T

}

dt

Absolute Diffusivity 

Spectral Density

A() = E()⌧ ()

MU ADSD

Reference:

˙

B

(31)

Absolute Diffusivity 

Spectral Density

(

homogeneous but non-stationary and tuning-free )

w

Large scales:

Small scales:

Variance

tensor:

w

˙

B

a = a(x, x) =

E{ dB ( dB)

T

}

dt

Absolute Diffusivity 

Spectral Density

A() = E()⌧ ()

9 MU ADSD

Reference:

Resseguier, Pan & Fox-Kemper 2019a

˙

B

(32)

Absolute Diffusivity 

Spectral Density

(

homogeneous but non-stationary and tuning-free )

w

Large scales:

Small scales:

Variance

tensor:

w

˙

B

a = a(x, x) =

E{ dB ( dB)

T

}

dt

Absolute Diffusivity 

Spectral Density

A() = E()⌧ ()

MU ADSD

Reference:

Residual

non-stationary

ADSD

˙

B

(33)

Absolute Diffusivity 

Spectral Density

(

homogeneous but non-stationary and tuning-free )

w

Large scales:

Small scales:

Variance

tensor:

w

˙

B

a = a(x, x) =

E{ dB ( dB)

T

}

dt

Absolute Diffusivity 

Spectral Density

A() = E()⌧ ()

9 MU ADSD

Reference:

Resseguier, Pan & Fox-Kemper 2019a

(filter) (white noise)

_{B =}

˙

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⇤

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