‘ horizontal convection’ 2 transitions solution for convection at large ra two sinking regions
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‘ Horizontal convection’ 2 transitions 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 - PowerPoint PPT PresentationTRANSCRIPT
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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
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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
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Instabilities at large Ra
‘Synthetic schlieren’ image
heated half of base
20cm
x=0 x=L/2=60cm
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Instabilities at large Ra
heated base
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cooled base
Applied heat flux
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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?
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Convective ‘mixed’ layer
convective instability predicted for RaF
>1012
fixed flux
Assume mixed layer deepening through ‘encroachment’
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Instabilities at large Ra
heated base, ThCooled Tc
Applied temperature B.C.s
Flow and instabilities are not sensitive to type of BC
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Infinite Pr - steady shallow intrusionsmomentum and thermal b.l.s have same thickness
Chiu-Webster, Hinch & Lister, 2007
T
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Infinite Pr - steady shallow intrusionsmomentum and thermal b.l.s have same thickness
Chiu-Webster, Hinch & Lister, 2007
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3 regimes?(almost unexplored!)
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Entraining end-wall plumeand interior eddies
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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
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in the plume
• continuity
• momentum
• buoyancyz
Wp
EWp
R
(Note: solution in terms of buoyancy flux FB = gQ cf. Baines & Turner 1969)
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in the interior
• continuity
• densitywe
plumeoutflow
EWP
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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)
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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
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plume equations as before, but add diffusion in the interior …
• continuity
• density
• at base– heating = cooling
qh = –qc
we
diffusion
plumeoutflow
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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
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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
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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
*
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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
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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 )
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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
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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 =
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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
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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
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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
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next lecture
• rotation effects
• thermohaline phenomena
• responses to changed forcing