Finite Prandtl Number Convection

Catherine A. Hier-MajumderDepartment of Terrestrial Magnetism

Carnegie Institution of Washington

Collaborators

• University of Minnesota, Minneapolis:David A. Yuen

• Université de Montréal:Alain P. Vincent

Purpose of Study

• Many fluids with large, but finite Pr numbers (Pr~104) in geophysical systems (mushy ice, Mg-magmas)

• Usually approximated as infinite Pr, same as mantle (Pr~1025)

• Pr~104 fluids still have significant inertial behaviors missed by infinite Pr approximation

Outline

• Equations of Finite Prandtl Convection

• 2-D Finite Prandtl Plumes

• Infinite vs. finite Prandtl plumes

• 3-D Finite Prandtl Plumes

• Conclusions

Finite Prandtl Convection Equations

0=•∇ v

vev 21

∇++−∇= zRaTpDtD

Pr

TDt

DT 2∇=

Nondimensionalized by

Finite Prandtl Convection Equations

0=•∇ v

vev 2∇++−∇=

RaPr

TpDtD

z

TRaPrDt

DT 2∇=

zgLTU )(by onalizedNondimensi Δ≡ α

Ra = 107,Pr = 7Ra = 107,Pr = 7

Nondimensional Temperature

Red is hot T > 1Blue is cold T ~ 0

Ra = 107,Pr = 7

Nondimensional Temperature

Red is hot T > 1Blue is cold T ~ 0

Ra = 108,Pr = 2 x 104

Nondimensional Temperature

Red is hot T > 1Blue is cold T ~ 0

Ra = 108,Pr = ∞

Nondimensional Temperature

Red is hot T > 1Blue is cold T ~ 0

Ra = 108,Pr = 2 x 104

t = t1

Ra = 108,Pr = infinityt = 2t1

Implications for Plume Heat Flux

• Heat flux = vz * T • Real heat fluxes are likely to

be at least 2X larger than those calculated using infinite Pr approximation

Conclusions

• Prandtl number 104 plumes have significantly different behaviors from infinite Prandtl number plumes:

o Hotter

o Grow faster

• Differences increase with Rayleigh number

• This may have significant effects on heat fluxes in icy satellites, very hot bodies like Io, and magma ocean phases of the early terrestrial planets and moons

Ra = 6.57 x 105, Pr = 10

Questions?