The convective approximations

(1)

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The convective approximations

Raphaël Raynaud, Ludovic Petitdemange, Emmanuel Dormy

To cite this version:

Raphaël Raynaud, Ludovic Petitdemange, Emmanuel Dormy. The convective approximations. Euro-pean GdR Dynamo, Jul 2013, Ascona, Switzerland. �hal-02428472�

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The Convective Approximations

Raphaël Raynaud, Ludovic Petitdemange, Emmanuel Dormy

raphael.raynaud@ens.fr

LRA,Département de Physique, École normale supérieure

I

NTRODUCTION

Convection can occur in a shallow layer of fluid with a small temperature contrast across the layer. Thus, it is natural to hope for a simpler descrip-tion of buoyantly driven flows via an expansion of the fully compressible equations. The Boussinesq approximation performs well in modeling con-vection in laboratory experiments, but it is however unsatisfactory for large stratified systems, the lower part of which is compressed by the overly-ing material. More general approximations are then required to describe convection in natural systems like oceans, stars or planetary cores.

C

ONSERVATION LAWS

•

Mass conservation

∂

_t

ρ + ∇ · (ρV) = 0

•

Momentum conservation

ρD

_t

V = −∇P + ρg + F

ν

with

g = −∇U

and

F

ν

= ∇ · (τ )

and

τ

_ij

= 2µ

1

2 (∂

i

v

j

+ ∂

j

v

i

) −

1

3 ∂

k

v

k

δ

ij

where

µ

is the dynamic viscosity of the fluid

•

Energy conservation

∂

_t

u

tot

+ ∇ · I

tot

= 0

where

u

tot

= ρ E

k

+ E

I

+ U

(total energy)

I

tot

= ρV E

k

+ E

H

+ U

+ I

q

+ I

ν (total energy flux) with

E

H

= E

I

+ P/ρ

(specific enthalpy)

I

q

= −K∇T

(heat flux)

I

_iν

= −V

_j

τ

_ji (viscous energy flux)

•

Conservation laws and the first principle of thermodynamics

d E

I

= −P d ρ

−1

+ T d S

lead to the heat transfert equation

ρT D

_t

S + ∇ · (I

q

) = Q

ν

with the viscous heating

Q

ν

= τ

_ij

∇

_j

V

_i

T

HE REFERENCE STATE

Decomposition of the thermodynamics variables into the sum of a steady variable corresponding to the reference atmosphere and a convective dis-turbance

f = f

_a

+ f

_c

The reference state must be in quasiequilibrium

•

mechanical quasiequilibrium: hydrostatic balance

−∇P

_a

+ ρ

_a

g = 0

•

thermal quasiequilibrium: “well mixed” isentropic reference state

∇S

_a

= 0

This defines the adiabatic gradient

∇T

_a

=

∂T

∂P

S

∇P

_a

= α

s

g =

αT

a

C

_p

g

∇T

_a

=

g

C

_p (in the case of a perfect gas)

T

HE

B

OUSSINESQ APPROXIMATION

•

“thin layer approximation”: a dimensionless parameter must tend to

zero when the size of the convective region tends to zero

•

resulting set of equations

∇ · (V) = 0

D

_t

V = −∇Π − αT g +

µ

ρ

₀

∇

2

V

D

_t

T = κ∇

2

T

HE ANELASTIC APPROXIMATION

•

non-dimensionalizing the equations: a dimensionless parameter

must tend to zero when the temperature contrast goes to zero

•

resulting set of equations

∇ · (ρ

_a

V) = 0

D

_t

V = −∇

P

c

ρ

_a

− α

S

_c

g + F

ν

/ρ

_a

D

_t

(T

_a

S

_c

) + ∇ · (I

q_a

+ I

q_c

) − ρ

_a

α

S

_c

V · g = Q

ν

N

UMERICAL IMPLEMENTATION

•

mean field approach for the heat transfert equation

split the variables into a mean and fluctuating part,

f = hf i + f

t

so that

hhf ii = hf i

and

hf

t

i = 0

This creates a turbulent entropy flux term

I

St

= ρ

_a

hS

t

V

t

i ∝ ∇Sc

Hypothesis: the turbulent diffusion of entropy dominates over the molecular diffusion of temperature in the heat transfert equation

•

assume a polytropic reference state (

P ∝ ρ

γ) for a perfect gas Two additional non-Boussinesq control parameters:

– the polytropic index

n = 1/(γ − 1)

– the degree of stratification

N

_ρ

= ln

ρa(bottom)

ρ_a(top)

C

ONCLUSIONS

•

practical interest: filter out fast processes (acoustic/seismic

pro-cesses) and retain slow processes (convection)

=⇒

low Mach approximation

=⇒

faster numerical integration

•

method: from a thermodynamical point of view, consider the

con-vective state as a small departure from a steady reference state References:

– Braginsky & Roberts, GAFD, 1995

– The anelastic dynamo benchmark: Jones et al., Icarus, 2011 – Brown et al., APJ, 2012

I

SSUES

•

use of a non steady reference state

•

consistency and domain of validity of the different variants of the

anelastic approximation

•

averaging the heat transfert equation

The convective approximations