convection convection matt penrice astronomy 501 university of victoria
TRANSCRIPT
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
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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
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∇' ≡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
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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