Convection

Matt Penrice Astronomy 501 University of Victoria

Outline

Convection overviewMixing Length Theory (MLT)

Issues with MLTImprovements on MLTConclusion

Conditions for convectionRadiation temperature gradient

Convective temperature gradient

Convection condition

€

∂T

∂r rad

= −3

16πac

κρL

r2T 3

€

∂T

∂r ad

= 1 −1

γ

⎛

⎝ ⎜

⎞

⎠ ⎟T

P

dP

dr

€

∂T

∂r rad

>∂T

∂r ad

Mixing length theory (MLT)

Assume groups of convective elements which have same properties at given r

Each element travels on average a distance know as the mixing length before mixing with the surrounding matter

The are assumed to have the same size and velocity at a given r, respectively

€

Λ

€

Λ,v

MLT 11Assume complete pressure

equilibriumAssume an average temperature

T(r) which is the average of all elements at a given r at an instant in time

Therefore elements hotter then T will be less dense and rise because of the assumed pressure equilibrium and vice versa for cooler elements

MLT 111What we are really interested in is the

convective flux

Cp=Specific heat at constant pressure =Mixing length =The distance over which the pressure

changes by an appreciable fraction of itself

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Fc =1

2ρvcPT

Λ

λ P

∇ −∇'( )

€

λP =P

ρg€

λP

€

Λ

MLT 1V =The average temperature

gradient of all matter at a given radius

=The temperature gradient of the falling or rising convective elements

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∇≡d lnT

d lnP

€

∇' ≡d lnT '

d ln P'

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∇

€

∇'

Issues with MLTNeglecting turbulent pressure

Neglecting asymmetries in the flow

Clear definition of a mixing length

Failure to describe the boundaries

Improvements to MLTArnett, Meakin, Young (2009)

Convective Algorithms Based on Simulations (CABS)

Creates a simple physical model based on fully 3D time-dependant turbulent stellar convection simulation

Kinetic Energy EquationMLT does not deal with KE loss

due to turbulenceTurbulent energy will cascade

down from large scales to small (large scale being the size of the largest eddy)

Energy is dissipated through viscosity at small scales (Kolmogorov micro scales)

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∂t ρEk +∇⋅ ρEkuo = −∇⋅ Fp + Fk + p'∇⋅ u' + ρ 'g⋅ u' −ε k

FluxesPressure perturbation (sound

waves)

Convective Turbulent motions

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FP = p'u'

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Fk = ρEku'

Boundaries Redefine a convective zone as a region in

which the stratification of the medium is unstable to turbulent mixing

Defined using the Bulk Richardson number

The Bulk Richardson number is the ratio of thermally produced turbulence and turbulence produced by vertical shear

u is the rms velocity of the fluid involved with the shear and l is the scale length of the turbulence

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RiB =Δbl

u2

Boundaries 11 = The change in buoyancy

across a layer of thickness

N=The Burnt-Vaisala frequency or the frequency at which a vertically displaced parcel will oscillate in a statistically stable environment

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N 2 =g

ρ o

∂ρ

∂r

€

Δb = N 2drΔr

∫€

Δr

€

Δb

Arnett, Meakin &Young 2009

Arnett, Meakin &Young 2009

ConclusionMixing Length Theory provides a

simple description of convection but has numerous draw backs

CABS introduces a way to take into account loss of KE due to turbulence as well as a dynamic definition of the boundary layers