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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

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Outline (#2) high-Rayleigh number horiz convection - observations instabilities and transitions in Ra-Pr inviscid model - turbulent plumes “filling-box” process steady “recycling-box” model compare solutions to experiments non-monotonic BC.s and 2 plumes (northern and southern hemispheres?) demo

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Experiment at large Ra heating cooling Box 1.25 m long x 0.2 m deep x 0.15 m wide, total flux 140 W Only one half of box is shown (Mullarney, Griffiths & Hughes, JFM 2004)

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Instabilities at large Ra ‘Synthetic schlieren’ image heated half of base 20cm x=0x=L/2=60cm

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Instabilities at large Ra heated base 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 stable outer BL convective instability shear instability eddying instability?

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Convective ‘mixed’ layer convective instability predicted for Ra F >10 12 fixed flux Assume mixed layer deepening through ‘encroachment’

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Instabilities at large Ra heated base, T h Cooled T c Applied temperature B.C.s Flow and instabilities are not sensitive to type of BC

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Infinite Pr - steady shallow intrusions momentum and thermal b.l.s have same thickness Chiu-Webster, Hinch & Lister, 2007 T

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Infinite Pr - steady shallow intrusions momentum 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 plume and 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) specific buoyancy flux F 0

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in the plume continuity momentum buoyancy z WpWp EW p R (Note: solution in terms of buoyancy flux F B = g Q cf. Baines & Turner 1969)

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in the interior continuity density wewe plume outflow EW P

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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 q c (cooling) q h (heating) zero net heating interior diffusion (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 q h = –q c wewe diffusion plume outflow

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Predicted temperature in sample experiment specific buoyancy flux F 0 = 7.1 x 10 -7 m 3 /s 3 diffusivity = 1.5 x 10 -7 m 2 /s (molecular) entrainment constant E z = 0.1 (Turner 1973) lab 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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Predicted downwelling in sample experiment specific buoyancy flux F 0 = 7.1 x 10 -7 m 3 /s 3 diffusivity = 1.5 x 10 -7 m 2 /s (molecular) entrainment constant E z = 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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Predicted buoyancy flux in sample experiment specific buoyancy flux F 0 = 7.1 x 10 -7 m 3 /s 3 diffusivity = 1.5 x 10 -7 m 2 /s (molecular) entrainment constant E z = 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) h WhLWhL specific buoyancy flux F 0 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) W H L specific buoyancy flux F 0 box length L depth H

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Model /lab /numerics comparisons Ra F dependence Model * LabNumerics* h/Lh/L= 3.39Ra F -1/6 2.652.87 UhL/*UhL/* = 0.33Ra F 1/3 0.460.40 Nu 0.75Ra F 1/6 0.820.62 Constants * constants evaluated for water at experimental conditions; Powers laws identical to viscous boundary layer scaling (Flux Rayleigh number Ra F ~ specific buoyancy flux F 0 )

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Mass transport pathways stabilising buoyancy source destabilising buoyancy source

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Buoyancy transport pathways destabilising buoyancy SINK destabilising buoyancy source

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Buoyancy transport pathways in the Munk & Wunsch diffusive ocean destabilising buoyancy SINK destabilising buoyancy source

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Non-monotonic B.C.s => two plumes effects on interior stratification? applied heat fluxapplied T c 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 RQ =

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Regime 3

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Interior Stratification

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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

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Schematic of the global thermohaline circulation

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Energetics ??????? My talk from AFMC

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Non-monotonic B.C.s => two plumes applied heat fluxapplied T c applied heat flux h = 0.2 m L = 1.25 m

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