Oceanic fluid dynamics under location uncertainty Part.I : Towards a stochastic modeling for the Quasi-Geostrophic system

HAL Id: hal-02710890

https://hal.inria.fr/hal-02710890

Submitted on 1 Jun 2020

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Oceanic fluid dynamics under location uncertainty Part.I : Towards a stochastic modeling for the

Quasi-Geostrophic system

Long Li, Etienne Mémin, Bruno Deremble

To cite this version:

Long Li, Etienne Mémin, Bruno Deremble. Oceanic fluid dynamics under location uncertainty Part.I :

Towards a stochastic modeling for the Quasi-Geostrophic system. Seminar of Mutiple-Scale Ocean

Model (MSOM) project, Jun 2018, Brest, France. �hal-02710890�

Oceanic fluid dynamics under location uncertainty

Part.I : Towards a stochastic modeling for the Quasi-Geostrophic system

Long Li

,

Étienne Mémin

, Bruno Deremble

1

FLUMINANCE Inria/Irmar Rennes,

LMD ENS Paris

Brest, June 12, 2018

Introduction

Objectives

better small-scale representation

identify regously subgrid effects correct false numerical dissipation better ensemble spreading

Plan

fluid flows under location uncertainty stochastic QG equations

multi-layer stochastic shallow water system

Introduction

Objectives

better small-scale representation identify regously subgrid effects

correct false numerical dissipation better ensemble spreading

Plan

fluid flows under location uncertainty stochastic QG equations

multi-layer stochastic shallow water system

Introduction

Objectives

better small-scale representation identify regously subgrid effects correct false numerical dissipation

better ensemble spreading

Plan

fluid flows under location uncertainty stochastic QG equations

multi-layer stochastic shallow water system

Introduction

Objectives

better small-scale representation identify regously subgrid effects correct false numerical dissipation better ensemble spreading

Plan

fluid flows under location uncertainty stochastic QG equations

multi-layer stochastic shallow water system

Introduction

Objectives

better small-scale representation identify regously subgrid effects correct false numerical dissipation better ensemble spreading

Plan

fluid flows under location uncertainty stochastic QG equations

multi-layer stochastic shallow water system

Introduction

Objectives

better small-scale representation identify regously subgrid effects correct false numerical dissipation better ensemble spreading

Plan

fluid flows under location uncertainty

stochastic QG equations

multi-layer stochastic shallow water system

Introduction

Objectives

better small-scale representation identify regously subgrid effects correct false numerical dissipation better ensemble spreading

Plan

fluid flows under location uncertainty stochastic QG equations

multi-layer stochastic shallow water system

Introduction

Objectives

better small-scale representation identify regously subgrid effects correct false numerical dissipation better ensemble spreading

Plan

fluid flows under location uncertainty stochastic QG equations

multi-layer stochastic shallow water system

Uncertainty formulation (Mémin, 2014)

Principes

Lagrangian displacement :

dX ( x, t ) = w ( X ( x, t ) , t ) dt + σ ( X ( x, t ) , t ) dB

, ∀( x, t ) ∈ D × R

, D ⊂ R

X ( x, 0 ) = x, ∀ x ∈ D

Eulerian velocity :

U ( x, t ) = w ( x, t )

´¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¶

large-scale

+ σ ( x, t ) B ˙

´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶

small-scale

, ∀ ( x, t ) ∈ D × R

Multiplicative noise

( B

)

is a cylindrical Wiener process in L

( D, R

) Spatial correlation :

σ ( x, t ) dB

= ∫

σ ˜ ( x, y, t ) dB

( y ) dy, ∀ ( x, t ) ∈ D × R

Subgrid tensor :

a = σσ

= 1

dt E[( σdB

)( σdB

)

]

Uncertainty formulation (Mémin, 2014)

Principes

Lagrangian displacement :

dX ( x, t ) = w ( X ( x, t ) , t ) dt + σ ( X ( x, t ) , t ) dB

, ∀( x, t ) ∈ D × R

, D ⊂ R

X ( x, 0 ) = x, ∀ x ∈ D

Eulerian velocity :

U ( x, t ) = w ( x, t )

´¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¶

large-scale

+ σ ( x, t ) B ˙

´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶

small-scale

, ∀ ( x, t ) ∈ D × R

Multiplicative noise

( B

)

is a cylindrical Wiener process in L

( D, R

) Spatial correlation :

σ ( x, t ) dB

= ∫

σ ˜ ( x, y, t ) dB

( y ) dy, ∀ ( x, t ) ∈ D × R

Subgrid tensor :

a = σσ

= 1

dt E[( σdB

)( σdB

)

]

Uncertainty formulation (Mémin, 2014)

Principes

Lagrangian displacement :

dX ( x, t ) = w ( X ( x, t ) , t ) dt + σ ( X ( x, t ) , t ) dB

, ∀( x, t ) ∈ D × R

, D ⊂ R

X ( x, 0 ) = x, ∀ x ∈ D

Eulerian velocity :

U ( x, t ) = w ( x, t )

´¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¶

large-scale

+ σ ( x, t ) B ˙

´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶

small-scale

, ∀ ( x, t ) ∈ D × R

Multiplicative noise

( B

)

is a cylindrical Wiener process in L

( D, R

) Spatial correlation :

σ ( x, t ) dB

= ∫

σ ˜ ( x, y, t ) dB

( y ) dy, ∀ ( x, t ) ∈ D × R

Subgrid tensor :

a = σσ

= 1

dt E[( σdB

)( σdB

)

]

Uncertainty formulation (Mémin, 2014)

Principes

Lagrangian displacement :

dX ( x, t ) = w ( X ( x, t ) , t ) dt + σ ( X ( x, t ) , t ) dB

, ∀( x, t ) ∈ D × R

, D ⊂ R

X ( x, 0 ) = x, ∀ x ∈ D

Eulerian velocity :

U ( x, t ) = w ( x, t )

´¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¶

large-scale

+ σ ( x, t ) B ˙

´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶

small-scale

, ∀ ( x, t ) ∈ D × R

Multiplicative noise

( B

)

is a cylindrical Wiener process in L

( D, R

)

Spatial correlation :

σ ( x, t ) dB

= ∫

σ ˜ ( x, y, t ) dB

( y ) dy, ∀ ( x, t ) ∈ D × R

Subgrid tensor :

a = σσ

= 1

dt E[( σdB

)( σdB

)

]

Uncertainty formulation (Mémin, 2014)

Principes

Lagrangian displacement :

dX ( x, t ) = w ( X ( x, t ) , t ) dt + σ ( X ( x, t ) , t ) dB

, ∀( x, t ) ∈ D × R

, D ⊂ R

X ( x, 0 ) = x, ∀ x ∈ D

Eulerian velocity :

U ( x, t ) = w ( x, t )

´¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¶

large-scale

+ σ ( x, t ) B ˙

´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶

small-scale

, ∀ ( x, t ) ∈ D × R

Multiplicative noise

( B

)

is a cylindrical Wiener process in L

( D, R

) Spatial correlation :

σ ( x, t ) dB

= ∫

σ ˜ ( x, y, t ) dB

( y ) dy, ∀ ( x, t ) ∈ D × R

Subgrid tensor :

a = σσ

= 1

dt E[( σdB

)( σdB

)

]

Uncertainty formulation (Mémin, 2014)

Principes

Lagrangian displacement :

dX ( x, t ) = w ( X ( x, t ) , t ) dt + σ ( X ( x, t ) , t ) dB

, ∀( x, t ) ∈ D × R

, D ⊂ R

X ( x, 0 ) = x, ∀ x ∈ D

Eulerian velocity :

U ( x, t ) = w ( x, t )

´¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¶

large-scale

+ σ ( x, t ) B ˙

´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶

small-scale

, ∀ ( x, t ) ∈ D × R

Multiplicative noise

( B

)

is a cylindrical Wiener process in L

( D, R

) Spatial correlation :

σ ( x, t ) dB

= ∫

σ ˜ ( x, y, t ) dB

( y ) dy, ∀ ( x, t ) ∈ D × R

Subgrid tensor :

a = σσ

= 1

dt E[(σdB

)(σdB

)

]

Stochastic Reynolds Transport Theorem (SRRT)

SRRT for solenoidal turbulence

Volumetric rate of change of a scalar for 0 = ∇ ⋅ σdB

= ∇ ⋅ σ :

d ∫

q ( x, t ) dx = ∫

[ d

q + ( ∇ ⋅ ( qw ) − 1 2

d

∑

i,j=1

∂

∂x

∂x

( qa

)) dt + ∇q ⋅ σdB

] dx

Conservation of extensive scalar :

d

q + w

dt ⋅ ∇q + σdB

⋅ ∇q − ∇ ⋅ 1

2 ( a∇q ) dt = − q∇ ⋅ w

dt advection

forcing diffusion

energy balance

Effective drift :

w

= w − 1

2 ∇ ⋅ a Stokes drift ?

Stochastic Reynolds Transport Theorem (SRRT)

SRRT for solenoidal turbulence

Volumetric rate of change of a scalar for 0 = ∇ ⋅ σdB

= ∇ ⋅ σ :

d ∫

q ( x, t ) dx = ∫

[ d

q + ( ∇ ⋅ ( qw ) − 1 2

d

∑

i,j=1

∂

∂x

∂x

( qa

)) dt + ∇q ⋅ σdB

] dx

Conservation of extensive scalar :

d

q + w

dt ⋅ ∇q + σdB

⋅ ∇q − ∇ ⋅ 1

2 ( a∇q ) dt = − q∇ ⋅ w

dt advection

forcing diffusion

energy balance

Effective drift :

w

= w − 1

2 ∇ ⋅ a Stokes drift ?

Stochastic Reynolds Transport Theorem (SRRT)

SRRT for solenoidal turbulence

Volumetric rate of change of a scalar for 0 = ∇ ⋅ σdB

= ∇ ⋅ σ :

d ∫

q ( x, t ) dx = ∫

[ d

q + ( ∇ ⋅ ( qw ) − 1 2

d

∑

i,j=1

∂

∂x

∂x

( qa

)) dt + ∇q ⋅ σdB

] dx

Conservation of extensive scalar :

d

q + w

dt ⋅ ∇q + σdB

⋅ ∇q − ∇ ⋅ 1

2 ( a∇q ) dt = − q∇ ⋅ w

dt advection

forcing diffusion

energy balance

Effective drift :

w

= w − 1

2 ∇ ⋅ a Stokes drift ?

Stochastic conservation laws

Navier Stokes equations under location uncertainty

Momentum equation :

d

w + ( w

dt + σdB

) ⋅ ∇w − 1 ρ ∇ ⋅ ( 1

2 ρa∇w ) dt + f × ( wdt + σdB

)

= − 1

ρ ∇ ( pdt + dp

) − ρkdt + ν∇

( wdt + σdB

) ,

where ν =

is the kinematic viscosity and dp

is a centered random process.

Mass equation :

d

ρ + ( w

dt + σdB

) ⋅ ∇ρ − ∇ ⋅ ( 1

2 a∇ρ ) dt = − ρ∇ ⋅ w

dt

∇ ⋅ σdB

= 0

Continuity equation (a sufficient constraint) :

0 = ∇ ⋅ w = ∇ ⋅ ( ∇ ⋅ a ) = ∇ ⋅ σ

Stochastic conservation laws

Navier Stokes equations under location uncertainty

Momentum equation :

d

w + ( w

dt + σdB

) ⋅ ∇w − 1 ρ ∇ ⋅ ( 1

2 ρa∇w ) dt + f × ( wdt + σdB

)

= − 1

ρ ∇ ( pdt + dp

) − ρkdt + ν∇

( wdt + σdB

) ,

where ν =

is the kinematic viscosity and dp

is a centered random process.

Mass equation :

d

ρ + ( w

dt + σdB

) ⋅ ∇ρ − ∇ ⋅ ( 1

2 a∇ρ ) dt = − ρ∇ ⋅ w

dt

∇ ⋅ σdB

= 0

Continuity equation (a sufficient constraint) :

0 = ∇ ⋅ w = ∇ ⋅ ( ∇ ⋅ a ) = ∇ ⋅ σ

Stochastic conservation laws

Navier Stokes equations under location uncertainty

Momentum equation :

d

w + ( w

dt + σdB

) ⋅ ∇w − 1 ρ ∇ ⋅ ( 1

2 ρa∇w ) dt + f × ( wdt + σdB

)

= − 1

ρ ∇ ( pdt + dp

) − ρkdt + ν∇

( wdt + σdB

) ,

where ν =

is the kinematic viscosity and dp

is a centered random process.

Mass equation :

d

ρ + ( w

dt + σdB

) ⋅ ∇ρ − ∇ ⋅ ( 1

2 a∇ρ ) dt = − ρ∇ ⋅ w

dt

∇ ⋅ σdB

= 0

Continuity equation (a sufficient constraint) :

0 = ∇ ⋅ w = ∇ ⋅ ( ∇ ⋅ a ) = ∇ ⋅ σ

Stochastic governing equations for stratified ocean

Simple Boussinesq equations under location uncertainty

Boussinesq approximations :

ρ ( x, t ) = ρ

+ δρ ( x, t ) , with ∣ δρ ∣ ≪ ρ

p ( x, t ) = p

( z ) + δp ( x, t ) , with ∣ δp ∣ ≪ ∣ p

∣ Hydrostatic balance :

∂p

∂z ( z ) = − gρ

Boussinesq momentum equation :

d

w + ( w

dt + σdB

) ⋅ ∇w − ∇ ⋅ ( 1

2 a∇w ) dt + fk × ( udt + ( σdB

)

)

= − ∇ ( pdt ˜ + d˜ p

) + bkdt + ν∇

( wdt + σdB

) , b = − g δρ ρ

Boussinesq thermodynamic equation :

▷ Applies SRRT with ∇ ⋅ σdB

= 0 for b ( x, t ) = b

( z ) + b

( x, t ) : d

b

+ ( w

dt + σdB

) ⋅ ∇b

+ N

( w

dt + ( σdB

)

) = ∇ ⋅ ( 1

2 a∇b

) dt + ∇ ⋅ ( 1

2 a

N

) dt

Stochastic governing equations for stratified ocean

Simple Boussinesq equations under location uncertainty

Boussinesq approximations :

ρ ( x, t ) = ρ

+ δρ ( x, t ) , with ∣ δρ ∣ ≪ ρ

p ( x, t ) = p

( z ) + δp ( x, t ) , with ∣ δp ∣ ≪ ∣ p

∣ Hydrostatic balance :

∂p

∂z ( z ) = − gρ

Boussinesq momentum equation :

d

w + ( w

dt + σdB

) ⋅ ∇w − ∇ ⋅ ( 1

2 a∇w ) dt + fk × ( udt + ( σdB

)

)

= − ∇ (˜ pdt + d˜ p

) + bkdt + ν∇

(wdt + σdB

), b = −g δρ ρ

Boussinesq thermodynamic equation :

▷ Applies SRRT with ∇ ⋅ σdB

= 0 for b ( x, t ) = b

( z ) + b

( x, t ) : d

b

+ ( w

dt + σdB

) ⋅ ∇b

+ N

( w

dt + ( σdB

)

) = ∇ ⋅ ( 1

2 a∇b

) dt + ∇ ⋅ ( 1

2 a

N

) dt

Stochastic governing equations for stratified ocean

Simple Boussinesq equations under location uncertainty

Boussinesq approximations :

ρ ( x, t ) = ρ

+ δρ ( x, t ) , with ∣ δρ ∣ ≪ ρ

p ( x, t ) = p

( z ) + δp ( x, t ) , with ∣ δp ∣ ≪ ∣ p

∣ Hydrostatic balance :

∂p

∂z ( z ) = − gρ

Boussinesq momentum equation :

d

w + ( w

dt + σdB

) ⋅ ∇w − ∇ ⋅ ( 1

2 a∇w ) dt + fk × ( udt + ( σdB

)

)

= − ∇ (˜ pdt + d˜ p

) + bkdt + ν∇

(wdt + σdB

), b = −g δρ ρ

Boussinesq thermodynamic equation :

▷ Applies SRRT with ∇ ⋅ σdB

= 0 for b ( x, t ) = b

( z ) + b

( x, t ) :

Continuously stratified QG system

Geostrophic scaling assumptions

1. Classical scalings :

A small Rossby number : Ro ≪ 1 A small variation of f : ∣βL∣ ≪ f

The scale of motion is not significantly larger than the deformation scale :

Ro

Bu = O( Ro ) ⇒ ∂b

∂z ≪ N

Continuously stratified QG system

Geostrophic scaling assumptions

1. Classical scalings :

A small Rossby number : Ro ≪ 1

A small variation of f : ∣βL∣ ≪ f

The scale of motion is not significantly larger than the deformation scale :

Ro

Bu = O( Ro ) ⇒ ∂b

∂z ≪ N

Continuously stratified QG system

Geostrophic scaling assumptions

1. Classical scalings :

A small Rossby number : Ro ≪ 1 A small variation of f : ∣βL∣ ≪ f

The scale of motion is not significantly larger than the deformation scale :

Ro

Bu = O( Ro ) ⇒ ∂b

∂z ≪ N

Continuously stratified QG system

Geostrophic scaling assumptions

1. Classical scalings :

A small Rossby number : Ro ≪ 1 A small variation of f : ∣βL∣ ≪ f

The scale of motion is not significantly larger than the deformation scale :

Ro

Bu = O( Ro ) ⇒ ∂b

∂z ≪ N

Continuously stratified QG system

Geostrophic scaling assumptions

2. Uncertainties scalings :

The vertical uncertainty is small compared with the horizontal uncertainties : ( σdB

)

∥( σdB

)

∥ ∼ Ro

Bu D, D = h L ≪ 1

A moderate uncertainty such that the energy dissipated by the horizontal small-scale flow is the same order than the large-scale kinetic energy :

a

∼ U L ⇐ M KE ∼ U

, T KE ∼ A

/ T

Results :

( σdB

)

∂

∂z = O( Ro Bu ) a

a

∼ Ro

Bu D, a

a

∼ ( Ro Bu )

D

∀ i ∈ H, a

∂

∂x

∂z dt = O( Ro Bu ) , a

∂

∂z

dt = O(( Ro

Bu )

)

Continuously stratified QG system

Geostrophic scaling assumptions

2. Uncertainties scalings :

The vertical uncertainty is small compared with the horizontal uncertainties : ( σdB

)

∥( σdB

)

∥ ∼ Ro

Bu D, D = h L ≪ 1

A moderate uncertainty such that the energy dissipated by the horizontal small-scale flow is the same order than the large-scale kinetic energy :

a

∼ U L ⇐ M KE ∼ U

, T KE ∼ A

/ T

Results :

( σdB

)

∂

∂z = O( Ro Bu ) a

a

∼ Ro

Bu D, a

a

∼ ( Ro Bu )

D

∀ i ∈ H, a

∂

∂x

∂z dt = O( Ro Bu ) , a

∂

∂z

dt = O(( Ro

Bu )

)

Continuously stratified QG system

Geostrophic scaling assumptions

2. Uncertainties scalings :

The vertical uncertainty is small compared with the horizontal uncertainties : ( σdB

)

∥( σdB

)

∥ ∼ Ro

Bu D, D = h L ≪ 1

A moderate uncertainty such that the energy dissipated by the horizontal small-scale flow is the same order than the large-scale kinetic energy :

a

∼ U L ⇐ M KE ∼ U

, T KE ∼ A

/ T

Results :

( σdB

)

∂

∂z = O( Ro Bu ) a

a

∼ Ro

Bu D, a

a

∼ ( Ro Bu )

D

∀ i ∈ H, a

∂

∂x

∂z dt = O( Ro Bu ) , a

∂

∂z

dt = O(( Ro

Bu )

)

Continuously stratified QG system

Geostrophic scaling assumptions

2. Uncertainties scalings :

The vertical uncertainty is small compared with the horizontal uncertainties : ( σdB

)

∥( σdB

)

∥ ∼ Ro

Bu D, D = h L ≪ 1

A moderate uncertainty such that the energy dissipated by the horizontal small-scale flow is the same order than the large-scale kinetic energy :

a

∼ U L ⇐ M KE ∼ U

, T KE ∼ A

/ T

Results :

( σdB

)

∂

∂z = O( Ro Bu ) a

a

∼ Ro

Bu D, a

a

∼ ( Ro

Bu )

D

Continuously stratified QG system

Non-dimensional primitive equations under location uncertainty

Momentum :

Ro[d_tu+ (u^∗dt+ (σdB_t)_H)⋅ ∇_Hu−∇_H⋅( 1

2a_H∇_Hu)dt+O(

Ro Bu)]

+(f0+Roβ(y−y0))k× (udt+ (σdBt)_H) = −∇_H(pdt+˜ d˜p^′_t)

Hydrostasy :

b^′dt+O(RoD²)=

∂

∂z(˜pdt+d˜p^′_t) Continuity :

∇_H⋅u+

∂w

∂z =0

∇_H⋅(∇_H⋅a_H) + Ro Bu

∂

∂z(∇_H⋅a_Hz) =∇_H⋅(σdBt)_H+ Ro Bu

∂(σdBt)_z

∂z =0 (1)

Thermodynamic :

Ro[dtb^′+ (u^∗dt+ (σdBt)_H)⋅ ∇_Hb^′−∇_H⋅( 1

2a_H∇_Hb^′)dt+

∂b^′

∂zwdt] +Bu wdt+Ro[(σdBt)_z−

1

2∇_H⋅a_Hzdt+O( Ro Bu)] =0

Continuously stratified QG system

Non-dimensional primitive equations under location uncertainty

Momentum :

Ro[d_tu+ (u^∗dt+ (σdB_t)_H)⋅ ∇_Hu−∇_H⋅( 1

2a_H∇_Hu)dt+O(

Ro Bu)]

+(f0+Roβ(y−y0))k× (udt+ (σdBt)_H) = −∇_H(pdt+˜ d˜p^′_t)

Hydrostasy :

b^′dt+O(RoD²)=

∂

∂z(˜pdt+d˜p^′_t)

Continuity :

∇_H⋅u+

∂w

∂z =0

∇_H⋅(∇_H⋅a_H) + Ro Bu

∂

∂z(∇_H⋅a_Hz) =∇_H⋅(σdBt)_H+ Ro Bu

∂(σdBt)_z

∂z =0 (1)

Thermodynamic :

Ro[dtb^′+ (u^∗dt+ (σdBt)_H)⋅ ∇_Hb^′−∇_H⋅( 1

2a_H∇_Hb^′)dt+

∂b^′

∂zwdt] +Bu wdt+Ro[(σdBt)_z−

1

2∇_H⋅a_Hzdt+O( Ro Bu)] =0

Continuously stratified QG system

Non-dimensional primitive equations under location uncertainty

Momentum :

Ro[d_tu+ (u^∗dt+ (σdB_t)_H)⋅ ∇_Hu−∇_H⋅( 1

2a_H∇_Hu)dt+O(

Ro Bu)]

+(f0+Roβ(y−y0))k× (udt+ (σdBt)_H) = −∇_H(pdt+˜ d˜p^′_t)

Hydrostasy :

b^′dt+O(RoD²)=

∂

∂z(˜pdt+d˜p^′_t) Continuity :

∇_H⋅u+

∂w

∂z =0

∇_H⋅(∇_H⋅a_H) + Ro Bu

∂

∂z(∇_H⋅a_Hz) =∇_H⋅(σdBt)_H+ Ro Bu

∂(σdBt)_z

∂z =0 (1)

Thermodynamic :

Ro[dtb^′+ (u^∗dt+ (σdBt)_H)⋅ ∇_Hb^′−∇_H⋅( 1

2a_H∇_Hb^′)dt+

∂b^′

∂zwdt] +Bu wdt+Ro[(σdBt)_z−

1

2∇_H⋅a_Hzdt+O( Ro Bu)] =0

Continuously stratified QG system

Non-dimensional primitive equations under location uncertainty

Momentum :

Ro[d_tu+ (u^∗dt+ (σdB_t)_H)⋅ ∇_Hu−∇_H⋅( 1

2a_H∇_Hu)dt+O(

Ro Bu)]

+(f0+Roβ(y−y0))k× (udt+ (σdBt)_H) = −∇_H(pdt+˜ d˜p^′_t)

Hydrostasy :

b^′dt+O(RoD²)=

∂

∂z(˜pdt+d˜p^′_t) Continuity :

∇_H⋅u+

∂w

∂z =0

∇_H⋅(∇_H⋅a_H) + Ro Bu

∂

∂z(∇_H⋅a_Hz) =∇_H⋅(σdBt)_H+ Ro Bu

∂(σdBt)_z

∂z =0 (1)

Thermodynamic :

Ro[dtb^′+ (u^∗dt+ (σdBt)_H)⋅ ∇_Hb^′−∇_H⋅(

1a_H∇_Hb^′)dt+

∂b^′ wdt]

Continuously stratified QG system

Zeroth order relations

Pressure balances rotation :

f₀k× (u₀dt+ (σdBt)_H) = −∇_H(p₀dt+d˜p^′_t)⇔

⎧⎪

⎪⎪

⎪⎪

⎪⎪

⎨

⎪⎪

⎪⎪

⎪⎪

⎪⎩ f₀v₀=

∂p₀

∂x, f₀u₀= −

∂p₀

∂y f₀(σdBt)_y=

∂d˜p^′_t

∂x , f₀(σdBt)_x= −

∂d˜p^′_t

∂y

Horizontal incompressibilities :

0=∇_H⋅u₀=∇_H⋅(σdB_t)_H (1) Ð→

∂(σdB_t)_z

∂z

≈0, ∂

∂z(∇_H⋅a_Hz) ≈0 (2)

Relative vorticity :

∂v₀

∂x −

∂u₀

∂y =

∇²_Hp0 f₀ Hydrostrasy :

∂p₀

∂z =b₀, ∂p˜^′_t

∂z = O(RoD²) Thermal wind balance :

∂u₀

∂z ⋅ ∇_Hb₀=0, ∂(σdB_t)_H

∂z = O(RoD²), ∂a_H

∂z = O(Ro²D⁴) (3)

Continuously stratified QG system

Zeroth order relations

Pressure balances rotation :

f₀k× (u₀dt+ (σdBt)_H) = −∇_H(p₀dt+d˜p^′_t)⇔

⎧⎪

⎪⎪

⎪⎪

⎪⎪

⎨

⎪⎪

⎪⎪

⎪⎪

⎪⎩ f₀v₀=

∂p₀

∂x, f₀u₀= −

∂p₀

∂y f₀(σdBt)_y=

∂d˜p^′_t

∂x , f₀(σdBt)_x= −

∂d˜p^′_t

∂y Horizontal incompressibilities :

0=∇_H⋅u₀=∇_H⋅(σdB_t)_H Ð→⁽¹⁾

∂(σdB_t)_z

∂z

≈0, ∂

∂z(∇_H⋅a_Hz) ≈0 (2)

Relative vorticity :

∂v₀

∂x −

∂u₀

∂y =

∇²_Hp0 f₀ Hydrostrasy :

∂p₀

∂z =b₀, ∂p˜^′_t

∂z = O(RoD²) Thermal wind balance :

∂u₀

∂z ⋅ ∇_Hb₀=0, ∂(σdB_t)_H

∂z = O(RoD²), ∂a_H

∂z = O(Ro²D⁴) (3)

Continuously stratified QG system

Zeroth order relations

Pressure balances rotation :

f₀k× (u₀dt+ (σdBt)_H) = −∇_H(p₀dt+d˜p^′_t)⇔

⎧⎪

⎪⎪

⎪⎪

⎪⎪

⎨

⎪⎪

⎪⎪

⎪⎪

⎪⎩ f₀v₀=

∂p₀

∂x, f₀u₀= −

∂p₀

∂y f₀(σdBt)_y=

∂d˜p^′_t

∂x , f₀(σdBt)_x= −

∂d˜p^′_t

∂y Horizontal incompressibilities :

0=∇_H⋅u₀=∇_H⋅(σdB_t)_H Ð→⁽¹⁾

∂(σdB_t)_z

∂z

≈0, ∂

∂z(∇_H⋅a_Hz) ≈0 (2)

Relative vorticity :

∂v₀

∂x −

∂u₀

∂y =

∇²_Hp0 f₀

Hydrostrasy :

∂p₀

∂z =b₀, ∂p˜^′_t

∂z = O(RoD²) Thermal wind balance :

∂u₀

∂z ⋅ ∇_Hb₀=0, ∂(σdB_t)_H

∂z = O(RoD²), ∂a_H

∂z = O(Ro²D⁴) (3)

Continuously stratified QG system

Zeroth order relations

Pressure balances rotation :

f₀k× (u₀dt+ (σdBt)_H) = −∇_H(p₀dt+d˜p^′_t)⇔

⎧⎪

⎪⎪

⎪⎪

⎪⎪

⎨

⎪⎪

⎪⎪

⎪⎪

⎪⎩ f₀v₀=

∂p₀

∂x, f₀u₀= −

∂p₀

∂y f₀(σdBt)_y=

∂d˜p^′_t

∂x , f₀(σdBt)_x= −

∂d˜p^′_t

∂y Horizontal incompressibilities :

0=∇_H⋅u₀=∇_H⋅(σdB_t)_H Ð→⁽¹⁾

∂(σdB_t)_z

∂z

≈0, ∂

∂z(∇_H⋅a_Hz) ≈0 (2)

Relative vorticity :

∂v₀

∂x −

∂u₀

∂y =

∇²_Hp0 f₀ Hydrostrasy :

∂p₀

∂z

=b₀, ∂p˜^′_t

∂z = O(RoD²)

Thermal wind balance :

∂u₀

∂z ⋅ ∇_Hb₀=0, ∂(σdB_t)_H

∂z = O(RoD²), ∂a_H

∂z = O(Ro²D⁴) (3)

Continuously stratified QG system

Zeroth order relations

Pressure balances rotation :

f₀k× (u₀dt+ (σdBt)_H) = −∇_H(p₀dt+d˜p^′_t)⇔

⎧⎪

⎪⎪

⎪⎪

⎪⎪

⎨

⎪⎪

⎪⎪

⎪⎪

⎪⎩ f₀v₀=

∂p₀

∂x, f₀u₀= −

∂p₀

∂y f₀(σdBt)_y=

∂d˜p^′_t

∂x , f₀(σdBt)_x= −

∂d˜p^′_t

∂y Horizontal incompressibilities :

0=∇_H⋅u₀=∇_H⋅(σdB_t)_H Ð→⁽¹⁾

∂(σdB_t)_z

∂z

≈0, ∂

∂z(∇_H⋅a_Hz) ≈0 (2)

Relative vorticity :

∂v₀

∂x −

∂u₀

∂y =

∇²_Hp0 f₀ Hydrostrasy :

∂p₀

∂z

=b₀, ∂p˜^′_t

∂z = O(RoD²) Thermal wind balance :

∂u₀

∂z ⋅ ∇_Hb₀=0, ∂(σdB_t)_H

∂z = O(RoD²), ∂a_H

∂z = O(Ro²D⁴) (3)

Continuously stratified QG system

First order equations

Momentum equations :

d

u

+ ( u

dt + ( σdB

)

) ⋅ ∇

u

− ∇

⋅ ( a

2 ∇

u

) dt − f

v

dt

− β ( y − y

)( v

dt + ( σdB

)

) = − ∂p

∂x dt − A

∇

u

dt (4) d

v

+ ( u

dt + ( σdB

)

) ⋅ ∇

v

− ∇

⋅ ( a

2 ∇

v

) dt + f

u

dt + β ( y − y

)( u

dt + ( σdB

)

) = − ∂p

∂y dt − A

∇

v

dt (5)

Continuity equation :

∂u

∂x + ∂v

∂y + ∂w

∂z = 0 (6)

Cross-differentiating (4)-(5) combining with (6) : d

[ ∇

p

f

+ β(y − y

)] + (u

dt + (σdB

)

) ⋅ ∇

[ ∇

p

f

+ β(y − y

)]

− ∇

⋅ ( a

2 ∇

[ ∇

p

f

+ β(y − y

)])dt = (f

∂w

∂z − A

f

∇

p

)dt + R

Continuously stratified QG system

First order equations

Momentum equations :

d

u

+ ( u

dt + ( σdB

)

) ⋅ ∇

u

− ∇

⋅ ( a

2 ∇

u

) dt − f

v

dt

− β ( y − y

)( v

dt + ( σdB

)

) = − ∂p

∂x dt − A

∇

u

dt (4) d

v

+ ( u

dt + ( σdB

)

) ⋅ ∇

v

− ∇

⋅ ( a

2 ∇

v

) dt + f

u

dt + β ( y − y

)( u

dt + ( σdB

)

) = − ∂p

∂y dt − A

∇

v

dt (5) Continuity equation :

∂u

∂x + ∂v

∂y + ∂w

∂z = 0 (6)

Cross-differentiating (4)-(5) combining with (6) : d

[ ∇

p

f

+ β(y − y

)] + (u

dt + (σdB

)

) ⋅ ∇

[ ∇

p

f

+ β(y − y

)]

− ∇

⋅ ( a

2 ∇

[ ∇

p

f

+ β(y − y

)])dt = (f

∂w

∂z − A

f

∇

p

)dt + R

Continuously stratified QG system

First order equations

Momentum equations :

d

u

+ ( u

dt + ( σdB

)

) ⋅ ∇

u

− ∇

⋅ ( a

2 ∇

u

) dt − f

v

dt

− β ( y − y

)( v

dt + ( σdB

)

) = − ∂p

∂x dt − A

∇

u

dt (4) d

v

+ ( u

dt + ( σdB

)

) ⋅ ∇

v

− ∇

⋅ ( a

2 ∇

v

) dt + f

u

dt + β ( y − y

)( u

dt + ( σdB

)

) = − ∂p

∂y dt − A

∇

v

dt (5) Continuity equation :

∂u

∂x + ∂v

∂y + ∂w

∂z = 0 (6)

Cross-differentiating (4)-(5) combining with (6) : d

[ ∇

p

f

+ β(y − y

)] + (u

dt + (σdB

)

) ⋅ ∇

[ ∇

p

f

+ β(y − y

)]

− ( a

[ ∇

p

+ ( − )]) = ( ∂w

− A

) +

Continuously stratified QG system

First order equations

Nonlinear source-sink terms :

R

= ( − ∂u

∂x

∂v

∂x + ∂u

∂y

∂u

∂x − ∂v

∂x

∂v

∂y + ∂v

∂y

∂u

∂y ) dt

= − tr (D[ u ] J D[− ∇

⋅ a

2 ]) dt R

= − ∂ ( σdB

)

∂x

∂v

∂x + ∂ ( σdB

)

∂y

∂u

∂x − ∂ ( σdB

)

∂x

∂v

∂y + ∂ ( σdB

)

∂y

∂u

∂y

= − tr (D[ u ] J D[( σdB

)

]) R

= ∇

⋅ ( 1

2

∂a

∂x ∇

v

) dt − ∇

⋅ ( 1 2

∂a

∂y ∇

u

) dt R

= − β∇

⋅ a

dt

▷ a homogeneous ⇒ R = R

, E[ R ] = 0

N.B. D[ u ] =

( ∇

u + ∇

u ) : deformation tensor, J = ( 0 − 1

1 0 ) : 90° rotation matrix

Continuously stratified QG system

First order equations

Nonlinear source-sink terms :

R

= ( − ∂u

∂x

∂v

∂x + ∂u

∂y

∂u

∂x − ∂v

∂x

∂v

∂y + ∂v

∂y

∂u

∂y ) dt

= − tr (D[ u ] J D[− ∇

⋅ a

2 ]) dt R

= − ∂ ( σdB

)

∂x

∂v

∂x + ∂ ( σdB

)

∂y

∂u

∂x − ∂ ( σdB

)

∂x

∂v

∂y + ∂ ( σdB

)

∂y

∂u

∂y

= − tr (D[ u ] J D[( σdB

)

]) R

= ∇

⋅ ( 1

2

∂a

∂x ∇

v

) dt − ∇

⋅ ( 1 2

∂a

∂y ∇

u

) dt R

= − β∇

⋅ a

dt

▷ a homogeneous ⇒ R = R

, E[ R ] = 0

Continuously stratified QG system

First order equations

Thermodynamic equation :

d

b

+( u

dt +( σdB

)

) ⋅∇

b

− ∇

⋅ ( a

2 ∇

b

) dt + Bu [( w

− ∇

⋅ a

2 ) dt +( σdB

)

] = 0 (7)

Derivating (7) along z : d

∂b

∂z + ( u

dt + ( σdB

)

) ⋅ ∇

∂b

∂z − ∇

⋅ ( a

2 ∇

∂b

∂z ) dt + Bu [( ∂w

∂z − ∂

∂z ∇

⋅ a

´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶ 2

0⇐(2)

) dt + ∂ ( σdB

)

´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶ ∂z

0⇐(2)

] + S = 0

S = [( ∂u

∂z − 1

2 ∇

⋅ ∂a

∂z ) dt + ∂ ( σdB

)

∂z ] ⋅ ∇

b

− ∇

⋅ ( 1 2

∂a

∂z ∇

b

) dt = 0 ⇐ (3) Results :

− ∂w

∂z = 1 Bu [ d

∂b

∂z + ( u

dt + ( σdB

)

) ⋅ ∇

∂b

∂z − ∇

⋅ ( a

2 ∇

∂b

∂z ) dt ]

Continuously stratified QG system

First order equations

Thermodynamic equation :

d

b

+( u

dt +( σdB

)

) ⋅∇

b

− ∇

⋅ ( a

2 ∇

b

) dt + Bu [( w

− ∇

⋅ a

2 ) dt +( σdB

)

] = 0 Derivating (7) along z : (7)

d

∂b

∂z + ( u

dt + ( σdB

)

) ⋅ ∇

∂b

∂z − ∇

⋅ ( a

2 ∇

∂b

∂z ) dt + Bu [( ∂w

∂z − ∂

∂z ∇

⋅ a

´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶ 2

0⇐(2)

) dt + ∂ ( σdB

)

´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶ ∂z

0⇐(2)

] + S = 0

S = [( ∂u

∂z − 1

2 ∇

⋅ ∂a

∂z ) dt + ∂ ( σdB

)

∂z ] ⋅ ∇

b

− ∇

⋅ ( 1 2

∂a

∂z ∇

b

) dt = 0 ⇐ (3) Results :

− ∂w

∂z = 1 Bu [ d

∂b

∂z + ( u

dt + ( σdB

)

) ⋅ ∇

∂b

∂z − ∇

⋅ ( a

2 ∇

∂b

∂z ) dt ]

Continuously stratified QG system

First order equations

Thermodynamic equation :

d

b

+( u

dt +( σdB

)

) ⋅∇

b

− ∇

⋅ ( a

2 ∇

b

) dt + Bu [( w

− ∇

⋅ a

2 ) dt +( σdB

)

] = 0 Derivating (7) along z : (7)

d

∂b

∂z + ( u

dt + ( σdB

)

) ⋅ ∇

∂b

∂z − ∇

⋅ ( a

2 ∇

∂b

∂z ) dt + Bu [( ∂w

∂z − ∂

∂z ∇

⋅ a

´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶ 2

0⇐(2)

) dt + ∂ ( σdB

)

´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶ ∂z

0⇐(2)

] + S = 0

S = [( ∂u

∂z − 1

2 ∇

⋅ ∂a

∂z ) dt + ∂ ( σdB

)

∂z ] ⋅ ∇

b

− ∇

⋅ ( 1 2

∂a

∂z ∇

b

) dt = 0 ⇐ (3)

Results :

− ∂w

∂z = 1 Bu [ d

∂b

∂z + ( u

dt + ( σdB

)

) ⋅ ∇

∂b

∂z − ∇

⋅ ( a

2 ∇

∂b

∂z ) dt ]

Continuously stratified QG system

First order equations

Thermodynamic equation :

d

b

+( u

dt +( σdB

)

) ⋅∇

b

− ∇

⋅ ( a

2 ∇

b

) dt + Bu [( w

− ∇

⋅ a

2 ) dt +( σdB

)

] = 0 Derivating (7) along z : (7)

d

∂b

∂z + ( u

dt + ( σdB

)

) ⋅ ∇

∂b

∂z − ∇

⋅ ( a

2 ∇

∂b

∂z ) dt + Bu [( ∂w

∂z − ∂

∂z ∇

⋅ a

´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶ 2

0⇐(2)

) dt + ∂ ( σdB

)

´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶ ∂z

0⇐(2)

] + S = 0

S = [( ∂u

∂z − 1

2 ∇

⋅ ∂a

∂z ) dt + ∂ ( σdB

)

∂z ] ⋅ ∇

b

− ∇

⋅ ( 1 2

∂a

∂z ∇

b

) dt = 0 ⇐ (3) Results :

− ∂w

= 1

[ d ∂b

+ ( u

dt + ( σdB ) ) ∂b

− ( a

∂b

) dt ]

Continuously stratified QG system

QG equations under location uncertainty

Potential vorticity (PV) : q

=

∇

p

f

+ β(y − y

) +

∂

∂z ( f

Bu b

), b

=

∂p

∂z

Evolution of PV :

d

q

+ (u

dt + (σdB

)

) ⋅ ∇

q

− ∇

⋅ ( a

2 ∇

q

)dt = − A

f

∇

p

dt + R

Dimensional version : q =

∇

p ˜ f

+ f + ∂

∂z ( f

N

∂ p ˜

∂z ) d

q + (u

dt + (σdB

)

) ⋅ ∇

q − ∇

⋅ (

a

2 ∇

q)dt = − A

f

∇

pdt ˜ + R

Continuously stratified QG system

QG equations under location uncertainty

Potential vorticity (PV) :

q

=

∇

p

f

+ β(y − y

) +

∂

∂z ( f

Bu b

), b

=

∂p

∂z Evolution of PV :

d

q

+ (u

dt + (σdB

)

) ⋅ ∇

q

− ∇

⋅ ( a

2 ∇

q

)dt = − A

f

∇

p

dt + R

Dimensional version : q =

∇

p ˜ f

+ f + ∂

∂z ( f

N

∂ p ˜

∂z ) d

q + (u

dt + (σdB

)

) ⋅ ∇

q − ∇

⋅ (

a

2 ∇

q)dt = − A

f

∇

pdt ˜ + R

Continuously stratified QG system

QG equations under location uncertainty

Potential vorticity (PV) :

q

=

∇

p

f

+ β(y − y

) +

∂

∂z ( f

Bu b

), b

=

∂p

∂z Evolution of PV :

d

q

+ (u

dt + (σdB

)

) ⋅ ∇

q

− ∇

⋅ ( a

2 ∇

q

)dt = − A

f

∇

p

dt + R

Dimensional version : q =

∇

p ˜ f

+ f + ∂

∂z ( f

N

∂ p ˜

∂z ) d

q + (u

dt + (σdB

)

) ⋅ ∇

q − ∇

⋅ (

a

2 ∇

q)dt = − A

f

∇

pdt ˜ + R

Parametrisation of noise

Some existing approaches

Homogeneous and isotropic turbulence model :

a is diagonal and constant

▷ Through a pass-band spectral cutoff :

In 2D, ( σ ( x ) dB

)

= ∇

ψ

⋆ dB

, ψ ˆ

( κ ) = A1

∣ κ ∣

Reduced order model :

▷ POD approach using observations from velocity field

a is stationnary

Parametrisation of noise

Some existing approaches

Homogeneous and isotropic turbulence model :

a is diagonal and constant

▷ Through a pass-band spectral cutoff :

In 2D, ( σ ( x ) dB

)

= ∇

ψ

⋆ dB

, ψ ˆ

( κ ) = A1

∣ κ ∣

Reduced order model :

▷ POD approach using observations from velocity field

a is stationnary

Parametrisation of noise

A new proposition

A type of uncertainty living on the iso-surface of buoyancy, i.e. σdB

⋅ ∇b = 0 :

▷ Iso-surface projector :

P

= ⎛

⎜ ⎝

1 0 α

0 1 α

α

α

∣ α ∣

⎞ ⎟

⎠

α = ( α

, α

)

= − ∇

b

∂b / ∂z = − ∇

b

/ N

1 + O( Ro ) = − ∇

∂p / ∂z N

▷ Divergence-free projector :

( σdB

)

= ( I

− ∆

∇

∇

´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶

δ_ij−^κiκj

∣κ∣2 in Fourier

)( αξ

) ,

where ξ

is the 3rd component of an original noise ξ

, which may be may be the homogeneous and isotropic one, or Kraichnan turbulent model.

a anisotropic, inhomogeneous and non-staionnary

N.B. In a layered model, the derivative along z will be approximated by a finite difference

between layer-averaged quantities.

Parametrisation of noise

A new proposition

A type of uncertainty living on the iso-surface of buoyancy, i.e. σdB

⋅ ∇b = 0 :

▷ Iso-surface projector :

P

= ⎛

⎜ ⎝

1 0 α

0 1 α

α

α

∣ α ∣

⎞ ⎟

⎠

α = ( α

, α

)

= − ∇

b

∂b / ∂z = − ∇

b

/ N

1 + O( Ro ) = − ∇

∂p / ∂z N

▷ Divergence-free projector :

( σdB

)

= ( I

− ∆

∇

∇

´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶

δ_ij−^κiκj

∣κ∣2 in Fourier

)( αξ

) ,

where ξ

is the 3rd component of an original noise ξ

, which may be may be the homogeneous and isotropic one, or Kraichnan turbulent model.

a anisotropic, inhomogeneous and non-staionnary

N.B. In a layered model, the derivative along z will be approximated by a finite difference

between layer-averaged quantities.

Parametrisation of noise

A new proposition

A type of uncertainty living on the iso-surface of buoyancy, i.e. σdB

⋅ ∇b = 0 :

▷ Iso-surface projector :

P

= ⎛

⎜ ⎝

1 0 α

0 1 α

α

α

∣ α ∣

⎞ ⎟

⎠

α = ( α

, α

)

= − ∇

b

∂b / ∂z = − ∇

b

/ N

1 + O( Ro ) = − ∇

∂p / ∂z N

▷ Divergence-free projector :

( σdB

)

= ( I

− ∆

∇

∇

´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶

δ_ij−^κiκj

∣κ∣2 in Fourier

)( αξ

) ,

where ξ

is the 3rd component of an original noise ξ

, which may be may be the homogeneous and isotropic one, or Kraichnan turbulent model.

a anisotropic, inhomogeneous and non-staionnary

A multi-layer model

Work in progress

(Hogg et al., 2003) - A QG coupled model

▷ only taken the ocean case in a double gyre basin, without heat flux

Figure: A multi-layer Shallow Water QG system

A multi-layer model

Work in progress

(Hogg et al., 2003) - A QG coupled model

▷ only taken the ocean case in a double gyre basin, without heat flux

A multi-layer model

An N-layers QG shallow water system

Evolution of q

, k = 1, . . . , N : d

q

+ 1

f

J ( p

, q

) dt − ( ∇

⋅ a

2 ) ⋅ ∇

q

dt + ( σdB

)

⋅ ∇

q

− ∇

⋅ ( a

2 ∇

q

) dt = − A

f

∇

p

dt + R

+ f

H

w

δ

´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶

surface pumping

− h

2f

∇

p

δ

´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶

bottom drag

Evolution of p

, k = 1, . . . , N : q

= ∇

p

f

+ β ( y − y

) + f

H

( η

− η

) Perturbation interface height :

η

= 0; η

= p

− p

g

, k = 1, . . . , N − 1; η

= D ( x, y ) g

= g

ρ

( ρ

− ρ

) , k = 1, . . . , N − 1 Horizonal uncertainties :

(σdB

)

= (I

− ∆

∇

∇

)(( ∇

η

− ∇

η

)ξ

), k = 1, . . . , N