View source for Stellar convection simulations

==What is convection?==

Convection is an instability occurring in a stratified fluid. It is a mechanism for transporting thermal energy by means of the bulk displacement of a fluid. What is hot goes up due to buoyancy, what is cold comes down. A common instance is that of a pan of water set on a heat source (electric heating element, gas ring, etc.). The water heated at the bottom of the pan is lighter, and rises to the surface, where it cools, sinks back, is warmed again, rises once more, and so on. Such convective motion tends to reduce the difference in temperature between the bottom of the pan and the surface. In the case of a star, convective motions is one of three ways to carry away the nuclear energy generated in its core. The other two are radiation where photons (or light) carries the energy, and conduction (direct contact between a hot and a cold body) which is only important in white dwarfs and neutron stars. The location of convection zones is strongly dependent on the star’s mass. Cool and low-mass stars are fully convective, going towards hotter and more massive stars a radiative zone forms and expands from the core. Stars slightly more massive and warmer than the Sun, also form a convective core which expands towards the hot stars while the convective envelope shrinks and disappears as the radiative envelope reaches the surface. The Sun has a convective envelope stretching from the surface and about 1/3 towards the centre. When there is steep variation in density, as in the solar convection zone, the fluid’s entropy turns out to be the natural variable for the characterization of convective efficiency; highly efficient convection maintains a nearly adiabatic stratification such that the integrated heat flux through the convection zone is small relative to the thermal energy of the plasma.

===Schwarzschild & Ledoux criteria for convective stability===

In the inviscid limit (<math>\nu=\kappa=0</math>), the criteria to not trigger convection is very simple. Let’s displace a fluid element by a small radial distance <math>dr</math> in a stratified media (s=surrounding & e=element), we get:

<math>\rho_{\rm e}=\rho_{\rm e}^* + dr \left(\frac{d\rho}{dr}\right)_{\rm e}</math> (eq 1) ; 
<math>\rho_{\rm s}=\rho_{\rm s}^* + dr \left(\frac{d\rho}{dr}\right)_{\rm s}</math> (eq 2)

[[Image:StellarConvection_BrunMiesch_Fig1.jpg|thumb|100px|right|F1|vertical displacement of a fluid element in a stratified media]]

Where <math>\rho_{\rm e}^*\ ,</math> <math>\rho_{\rm e}</math> are respectively, the density of the element before and after the displacement dr, and <math>\rho_{\rm s}^*\ ,</math> <math>\rho_{\rm s}</math> the density of the surrounding media (and the same convention is used for the pressure <math>P</math> and temperature <math>T</math>). Let’s take the difference between equations (1) & (2):

<math>\rho_{\rm e} - \rho_{\rm s} = dr \left[\left(\frac{d\rho}{dr}\right)_{\rm e} - \left(\frac{d\rho}{dr}\right)_{\rm s}\right] > 0</math> (eq 3), if positive then the stratification is STABLE against convection

or equivalently <math>\left(\frac{d\rho}{dr}\right)_{\rm e} - \left(\frac{d\rho}{dr}\right)_{\rm s} > 0</math> (eq 4)

To proceed, we need an equation of state (EOS), <math>\rho=\rho(P,T,\mu)\ ,</math> to provide the relations between the various thermodynamic variables (with <math>\mu</math> the mean molecular weight). An EOS can be stated generally as

<math>\frac{d\rho}{\rho}=\alpha\frac{dP}{P}-\delta\frac{dT}{T}+\phi\frac{d\mu}{\mu}</math> (eq 5)

Note that for a perfect gas <math>P= R \rho T / \mu\ ,</math> with <math>R</math> the gas constant, the coefficients  <math>\alpha=\delta=\phi=1\ .</math>
 
where <math>\alpha, \delta, \phi</math> are thermodynamic coefficients. Substituting equation (5) in equation (4) leads to:

<math>\left(\frac{\alpha}{P}\frac{dP}{dr}\right)_e - \left(\frac{\delta}{T}\frac{dT}{dr}\right)_e - \left(\frac{\alpha}{P}\frac{dP}{dr}\right)_s + \left(\frac{\delta}{T}\frac{dT}{dr}\right)_s - \left(\frac{\phi}{\mu}\frac{d\mu}{dr}\right)_s > 0 </math> (eq 6)

Since <math>P_{\rm e}=P_{\rm s}</math> and <math>P_{\rm e}^*=P_{\rm s}^*</math> (pressure equilibrium) and <math>d\mu</math> is null for the element (the element is fully mixed), we get after multiplying by the pressure scale height <math>H_p = -1/(d \ln P/dr)\ ,</math> the following stability criteria:

<math>\left(\frac{d\ln T}{d\ln P}\right)_{\rm s} < \left(\frac{d\ln T}{d\ln P}\right)_{\rm e} + \left(\frac{\phi}{\delta}\frac{d\ln\mu}{d\ln P}\right)_{\rm s}</math>  (eq 7); STABLE against convection

or using stellar physics classical gradient notation: <math>\nabla < \nabla_{\rm e} + \frac{\phi}{\delta}\nabla_{\mu}</math> (eq 8)

In an atmosphere in which the energy is transported only per radiation, then <math>\nabla = \nabla_{\rm rad}\ .</math> Let’s test the stability of this atmosphere by considering the ''adiabatic'' displacement of an element: <math>\nabla_{\rm e} = \nabla_{\rm ad}</math> 

The atmosphere is convectively stable if:
Ledoux criterion: <math>\nabla_{\rm rad} < \nabla_{\rm ad} + \frac{\phi}{\delta}\nabla_{\mu}</math> (eq 9)

Schwarzschild criteria (if no variation of composition or ionization is assumed): <math>\nabla_{\rm rad} < \nabla_{\rm ad}</math> (eq 10)

Remark: The gradient of the specific entropy per unit mass <math>S</math> can easily be related to the difference between <math>\nabla</math> and <math>\nabla_{\rm ad}</math> such that (in the case where we neglect variation of composition or ionization):

<math>\frac{dS}{dr}= - \frac{c_p}{H_p}\left(\nabla - \nabla_{\rm ad}\right)</math> (eq 11)
 
with <math>c_p</math> the specific heat at constant pressure. This leads to a simple criteria for inviscid convection: There is convection when <math>dS/dr</math> is negative or conversely; the medium is convectively stable for a positive <math>dS/dr\ .</math>

===Rayleigh number===

In reality even though atomic viscosity can be very small in a stellar plasma, the threshold to trigger convection will be higher than in the inviscid limit since diffusion effects will suppress the convection instability. The instability criteria for a horizontal layer heated from below is described in detail by Chandrasekhar (1961) and subsequent works have considered rotating spherical shells (Roberts 1968, Busse 1970; Gilman 1975). For illustration, we briefly list the instability criteria for the case of a convective layer heated from below assuming either no slip or stress free boundary conditions for the velocity field. The Rayleigh number is defined as:

<math>Ra = \frac{g \alpha \Delta T d^3}{\kappa\nu}</math> in Boussinesq (i.e nearly incompressible) convection (with <math>g</math> the gravity and <math>d</math> the convection zone thickness).

For stress free conditions, top and bottom, the critical Rayleigh number is : <math>Ra_{\rm c}</math> = 658
For stress free and no slip : <math>Ra_{\rm c}</math> = 1100
For no slip top and bottom: <math>Ra_{\rm c}</math> = 1708

The Rayleigh number has to be above this threshold for convection to start. In stellar convection zones the Rayleigh number can exceed <math>10^{15}</math> so such instability conditions are easily realized.

The presence of an imposed field generally raises the instability threshold; magnetic fields have a stabilizing influence. For stress-free velocity boundary conditions and a purely radial magnetic field at the boundaries, <math>Ra_{\rm c}</math> depends on the Hartman number, <math>Ha\ ,</math> such that <math>Ra_{\rm c} = \pi^2(Ha)^2</math> for <math>Ha >> 1\ .</math>
The Hartman number is defined as:  <math>Ha^2 = \frac{\sigma B_0^2 d^2}{\rho \nu}\ ,</math> with <math>\sigma</math> the electrical conductivity and <math>B_0</math> the imposed vertical field.

So the stronger the imposed magnetic field the harder it is to trigger convection (check Cattaneo, Emonet & Weiss 2003 for a recent discussion on dynamo and magnetoconvection in an unstable slab with an externally imposed field).