‘horizontal convection’ 2 transitions solution for convection at large ra two sinking regions...
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‘Horizontal convection’ 2transitions
solution for convection at large Ra
two sinking regions
Ross Griffiths
Research School of Earth Sciences The Australian National University
Outline (#2)
• high-Rayleigh number horiz convection - observations• instabilities and transitions in Ra-Pr• inviscid model -
turbulent plumes“filling-box” processsteady “recycling-box” model
• compare solutions to experiments• non-monotonic BC.s and 2 plumes (northern and southern hemispheres?)
demo
Instabilities at large Ra
‘Synthetic schlieren’ image
heated half of base
20cm
x=0 x=L/2=60cm
Instabilities at large Ra
heated base
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cooled base
Applied heat flux
Instabilities at large Ra
Central region of heated base
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Instabilities at large Ra
end of heated base
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stable outer BL
convective instability
shear instability
eddying instability?
Convective ‘mixed’ layer
convective instability predicted for RaF
>1012
fixed flux
Assume mixed layer deepening through ‘encroachment’
Instabilities at large Ra
heated base, ThCooled Tc
Applied temperature B.C.s
Flow and instabilities are not sensitive to type of BC
Infinite Pr - steady shallow intrusionsmomentum and thermal b.l.s have same thickness
Chiu-Webster, Hinch & Lister, 2007
T
Infinite Pr - steady shallow intrusionsmomentum and thermal b.l.s have same thickness
Chiu-Webster, Hinch & Lister, 2007
3 regimes?(almost unexplored!)
Entraining end-wall plumeand interior eddies
Toward a model for flow at large Ra 1. the ‘filling box’ process
• closed volume
• localized buoyancy source– turbulent plume– entrainment of ambient fluid– upwelling velocity varies with
height– asymptotically steady flow and
shape of density profile – unsteady density– no diffusion
a la Baines & Turner (1969)
specificbuoyancyflux F0
in the plume
• continuity
• momentum
• buoyancyz
Wp
EWp
R
(Note: solution in terms of buoyancy flux FB = gQ cf. Baines & Turner 1969)
in the interior
• continuity
• densitywe
plumeoutflow
EWP
0
0.2
0.4
0.6
0.8
1
-30 -25 -20 -15 -10 -5 0
0 0.1 0.2 0.3 0.4 0.5 0.6
Dimensionless density anomaly e(ζ)−e(1)
Dimensionless upwelling velocity we and entrainment flux rwe
Asymptotic ‘filling box’ solution
time
Baines & Turner (1969)
2. Steady, diffusive ‘recycling box’
• localized destabilising flux (analytical convenience)
• entrainment into plume (2D, 3D or geostrophic)
• downwelling velocity varies with depth
qc (cooling)qh (heating)
• zero net heating
interiordiffusion(mixing?)
Killworth & Manins, JFM, 1980; Hughes, Griffiths, Mullarney & Peterson, JFM, 2007
plume equations as before, but add diffusion in the interior …
• continuity
• density
• at base– heating = cooling
qh = –qc
we
diffusion
plumeoutflow
Predicted temperature in sample experiment
• specific buoyancy flux F0 = 7.1 x 10-7 m3/s3
• diffusivity = 1.5 x 10-7 m2/s (molecular)
• entrainment constant Ez = 0.1 (Turner 1973)
lab
theory:
(box 1.25 m long x 0.2 m depth)
10-3
10-2
10-1
15 20 25 30 35
T (oC)
= 3.2 x 10-4 ºC-1
= 1.5 x 10-4 ºC-1
0
0.05
0.1
0.15
0.2
0 100 5 10-5 1 10-4 1.5 10-4
We (m/s)
Predicted downwelling in sample experiment
• specific buoyancy flux F0 = 7.1 x 10-7 m3/s3
• diffusivity = 1.5 x 10-7 m2/s (molecular)
• entrainment constant Ez = 0.1 (Turner 1973)
numerical
theory:
(box 1.25 m long x 0.2 m depth)
= 3.2 x 10-4 ºC-1
= 1.5 x 10-4 ºC-1
Asymptotic scalings for ‘recycling box’ (line plume)
• thermal boundary layer:
– thickness
– volume transport in boundary layer (per unit width)
hWhL
specificbuoyancyflux F0
box length L
*
Asymptotic scalings for ‘recycling box’ (line plume)
• top-to-bottom density difference
• overturning volume transport (per unit width)
WH L
specificbuoyancyflux F0
box length Ldepth H
Model /lab /numerics comparisons
RaF dependence
Model* Lab Numerics*
h/L = 3.39RaF-1/6 2.65 2.87
UhL/* = 0.33RaF1/3 0.46 0.40
Nu 0.75RaF1/6 0.82 0.62
Constants
*constants evaluated for water at experimental conditions;Powers laws identical to viscous boundary layer scaling(Flux Rayleigh number RaF ~ specific buoyancy flux F0 )
Non-monotonic B.C.s => two plumeseffects on interior stratification?
applied heat flux
applied Tc applied heat flux
h =
0.2
m
L = 1.25 m
Regime 1
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Regime 2
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Confluence Point
0.0
0.2
0.4
0.6
0.8
1.0
-0.2 -0.1 0.0 0.1 0.2RQ
xc/L
Regime 3
Regime 3Regime 2
Regime 2
Regime 1
RQ =
Regime 3
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Interior Stratification
0.0
2.0
4.0
6.0
8.0
10.0
-1.0 -0.5 0.0 0.5 1.0
RQ
normalised gradient (x 10
-3)
two plumes 280W
one plume 271W
Julia 140Waverage T3,4Julia 374W
Julia 69W
one plume 140W
Conclusions• Flow regimes are barely explored• Both convective and shear instabilities
occur at large Ra --> partially turbulent box
• inviscid model of a diffusive ‘filling box’-like process with zero net buoyancy input gives:– B.L. properties and Nu(Ra) in agreement with
viscous B.L. scaling, laboratory and numerical results
– downwelling velocity is depth-dependent – A residual advection–diffusion balance in the
interior is essential for steady state– Stratification (or vertical diffusivity required to
maintain a given stratification) is reduced by greater entrainment into the plume
Conclusions
• Circulation with two sinking regions is very sensitive to the difference in buoyancy fluxes
• Unequal plumes can increase the interior stratification by ~ 2
• The stronger plume sets the interior stratification
next lecture
• rotation effects
• thermohaline phenomena
• responses to changed forcing