Rotating shallow water flow under location uncertainty

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Submitted on 23 Sep 2020

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Rotating shallow water flow under location uncertainty

Rüdiger Brecht, Long Li, Werner Bauer, Etienne Mémin

To cite this version:

Rüdiger Brecht, Long Li, Werner Bauer, Etienne Mémin. Rotating shallow water flow under location

uncertainty: Part II: some numerical results. Seminar STUDO, Sep 2020, Rennes, France. �hal-

02946835�

Rotating shallow water flow under location uncertainty Part II: some numerical results

Rüdiger Brecht

Long Li

Werner Bauer

Etienne Mémin

1Department of Mathematics and Statistics Memorial University of Newfoundland, Canada

2Fluminance Group

Inria Rennes - Bretagne Atlantique, France

3Department of Mathematics Imperial College London, UK

Governing equations

Conservation of momentum

D

u + f × u dt = −g∇h dt (1)

Conservation of mass

D

h + h ∇· u dt = 0 (2) Incompressible constraints

∇·σdB

= 0, ∇· ∇· a = 0 (3) Conservation of energy

d

Z

Ω

ρ

2 h|u|

+ gh

dx = 0 (4)

Stochastic transport operator: Dt[•] =

dt+ (u−∇·¹₂a)dt+σdBt

·∇[•]−∇·

1 2a∇[•]

Governing equations

Conservation of momentum

D

u + f × u dt = −g∇h dt (1) Conservation of mass

D

h + h ∇· u dt = 0 (2)

Incompressible constraints

∇·σdB

= 0, ∇· ∇· a = 0 (3) Conservation of energy

d

Z

Ω

ρ

2 h|u|

+ gh

dx = 0 (4)

Stochastic transport operator: Dt[•] =

dt+ (u−∇·¹₂a)dt+σdBt

·∇[•]−∇·

1 2a∇[•]

Governing equations

Conservation of momentum

D

u + f × u dt = −g∇h dt (1) Conservation of mass

D

h + h ∇· u dt = 0 (2) Incompressible constraints

∇·σdB

= 0, ∇· ∇· a = 0 (3)

Conservation of energy d

Z

Ω

ρ

2 h|u|

+ gh

dx = 0 (4)

Stochastic transport operator: Dt[•] =

dt+ (u−∇·¹₂a)dt+σdBt

·∇[•]−∇·

1 2a∇[•]

Governing equations

Conservation of momentum

D

u + f × u dt = −g∇h dt (1) Conservation of mass

D

h + h ∇· u dt = 0 (2) Incompressible constraints

∇·σdB

= 0, ∇· ∇· a = 0 (3) Conservation of energy

d

Z

Ω

ρ

2 h|u|

+ gh

dx = 0 (4)

Stochastic transport operator: Dt[•] =

dt+ (u−∇·¹₂a)dt+σdBt

·∇[•]−∇·

1 2a∇[•]

Energy diagnosis

Figure: Convergence in temporal resolution of LU ensemble energy to the reference (at

spatial resolution

).

Energy diagnosis

0 0.5 1 1.5 2

days 0

0.5 1 1.5 2 2.5 3 3.5 4

relative error

#10^-8

Figure: Relative errors of LU ensemble mean energy compared to the reference (at

spatial resolution

).

Barotropic instability

Video: Evolution of vorticity fields.

Barotropic instability

Reference LES

-3 -2 -1 0 1

0.4 0.6 0.8 1 1.2

-3 -2 -1 0 1

0.4 0.6 0.8 1 1.2

LU online LU offline

-3 -2 -1 0 1

0.4 0.6 0.8 1 1.2

-3 -2 -1 0 1

0.4 0.6 0.8 1 1.2

Figure: Contour snapshots of vorticity field at day

with CI

.

Barotropic instability

10⁰ 10¹ 10²

Wavenumber 10⁰

10² 10⁴ 10⁶ 10⁸

Kinetic Energy

10⁰ 10¹ 10²

Wavenumber 10^-4

10^-2 10⁰ 10²

Enstrophy

Figure: Spectrums of kinetic energy (left) and enstrophy (right) at day

lines are power laws of slope

(left) and

(right).

Ensemble forecast

LU offline PIC

LU online PIC

Video: Evolution of rank histograms from day

to day

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