‘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