Waves under location uncertainty

(1)

HAL Id: hal-01891185

https://hal.archives-ouvertes.fr/hal-01891185

Submitted on 9 Oct 2018

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Waves under location uncertainty

Valentin Resseguier, Bertrand Chapron, Fabrice Collard

To cite this version:

Valentin Resseguier, Bertrand Chapron, Fabrice Collard. Waves under location uncertainty. Ocean

Remote Sensing Synergy, Sep 2018, Brest, France. �hal-01891185�

(2)

Waves under location

uncertainty

Valentin Resseguier,

(3)

Motivations

• Mimic interaction wave / (small-scale) current 

Small-scale currents are hardly observed or simulated 

Waves see them during very short time 

• Quantification of associated modelling errors 

 

• Improve the Scalian’s wave simulator 

• Study influence of swell / current interactions onto satellite altimetry

2

(4)

Part I

(5)

CEN « Simulation » (~ 70 people)

R&D and engineering

Expertise:

• Radar, optronics, sonar

• Geophysical fluid dyn.

• Mechanical and 

thermal

Business:

• Scientific softwares

• Simulations, HPC

• VR & AR

L@b 

(~ 15 peoples)

Research, R&T, R&D

Expertise:

• Geophysical fluid dyn.

• Signal, data assimilation

• Machine Learning

• Multi-agents systems

• Drones

Other Business Units

~ 2400 people

(6)

SeaMotion-radar

simulator

• Simple (mainly linear dynamics) and fast

simulation of surface gravity waves 

• Coupled with radar simulation 

• Lot of physical phenomena taking into

account 

• Used to study satellite altimetry

(7)

Part II

Waves LU

(8)

v = w +

B

˙

LU :Adding

(9)

v = w +

B

˙

Resolved

large scales

LU :Adding

random velocity

7

(10)

v = w +

B

˙

Resolved

large scales

White-in-time

_{small scales}

LU :Adding

(11)

v = w +

B

˙

Resolved

large scales

White-in-time

_{small scales}

LU :Adding

random velocity

7 Large scales:

Small scales:

Variance

tensor:

w

˙

B

a = a(x, x) =

E{ dB ( dB)

T

}

dt

(12)

v = w +

B

˙

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 & Holm 2017

Cotter and al 2018 a, b

References

:

LU :Adding

random velocity

Large scales:

Small scales:

Variance

tensor:

w

˙

B

a = a(x, x) =

E{ dB ( dB)

T

}

dt

(13)

Dispersion ratio

At the first order 

in steepness :

8 Large scales:

Small scales:

Variance

tensor:

Wave:

Doppler

frequency:

w

˙

B

a = a(x, x) =

E{ dB ( dB)

T

}

dt

ae

✏

i

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!

₀

=

p

g

_kkk

(14)

Dispersion ratio

At the first order 

in steepness :

D

Dt

= !

0 (k) =

p

g

kkk

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Large scales:

Small scales:

Variance

tensor:

Wave:

Doppler

frequency:

w

˙

B

a = a(x, x) =

E{ dB ( dB)

T

}

dt

ae

✏

i

!

₀

=

p

g

_kkk

(15)

Stochastic

transport

Dispersion ratio

At the first order 

in steepness :

8 D

Dt

= !

0 (k) =

p

g

kkk

Large scales:

Small scales:

Variance

tensor:

Wave:

Doppler

frequency:

w

˙

B

a = a(x, x) =

E{ dB ( dB)

T

}

dt

ae

✏

i

!

₀

=

p

g

_kkk

(16)

Frequency without currents

Stochastic

transport

Dispersion ratio

At the first order 

in steepness :

D

Dt

= !

0 (k) =

p

g

kkk

Large scales:

Small scales:

Variance

tensor:

Wave:

Doppler

frequency:

w

˙

B

a = a(x, x) =

E{ dB ( dB)

T

}

dt

ae

✏

i

!

₀

=

p

g

_kkk

(17)

Conservation of

action

Case 0:

Stationary deterministic velocity

Stretching of phase

9 cst. = A =

a

2 !

₀

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˙k = rw

T

_k

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White & Fornberg 1997

Random ray

˙

X

_R

= u

_g

(18)

Conservation of

action

Case 0:

Stationary deterministic velocity

Stretching of phase

cst. = A =

a

2 !

₀

˙k = rw

T

_k

White & Fornberg 1997

Random ray

˙

X

_R

= u

_g

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Large scale

group velocity:

Wave:

Doppler

frequency:

w

ae

✏

i

!

₀

=

p

g

_kkk

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(19)

Ratio

effective rotation / strain rate

Simple linear case 1:

stationary deterministic, 

divergence-free and 

linear in x large-scale velocity

Stretching of phase

10 ˙k = rw

T

_k

r =

!

w

D⇣

D⌧

= r

cos(⇣)

Elliptic

Hyperbolic

D

D⌧

(ln (

kr⇥k)) =

sin(⇣)

0

20

40

60

80 −0.6

−0.4

−0.2

0

0.2

0.4 r = 0. 9

ζ

τ

0

20

40

60

80 −10

0

10

20

30

40 ln

(∥∇

Θ

∥

)

τ

0

2

4

6

8

10 −4

−2

0

2

4 r = 2

ζ

τ

0

2

4

6

8

10 −1.2

−1

−0.8

−0.6

−0.4

−0.2

0 ln

(∥∇

Θ

∥

)

τ

|r| < 1

|r| > 1

ln (

_kkk)

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ln (

_kkk)

ln (

_kkk)

Large scale

group velocity:

Wave:

Doppler

frequency:

w

ae

✏

i

!

₀

=

p

g

_kkk

(20)

Conservation of action

Log-normal amplitude

Simple linear case 2:

no large-scale current 

+ homogeneous and divergence-free 

small-scale velocity

Wave with

Brownian phase

Stochastic stretching of phase

Log-normal wave- number

˙k = r( ˙B)

T

_k

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(x)e

i(!

0 t+k

· B

t

)

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cst. = A =

a

2 !

₀

Large scale

group velocity:

Small scale

group velocity:

Variance

tensor:

Wave:

Doppler

frequency:

w

˙

B

a = a(x, x) =

E{ dB ( dB)

T

}

dt

ae

✏

i

!

₀

=

p

g

_kkk

(21)

Simple linear case 3:

divergence-free and linear-in-x large-scale current 

+ homogeneous and divergence-free 

small-scale velocity

Stochastic stretching

of phase

˙k = r(w + ˙B)

T

_k

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12 Action

cst. =

a

2 !

₀

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