‘ Horizontal convection’ 2 transitions solution for convection at large Ra two sinking regions

1. ‘Horizontal convection’ 2transitionssolution for convection at large Ratwo sinking regions Ross Griffiths Research School of Earth Sciences The Australian National University

2. 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

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

4. Instabilities at large Ra Applied heat flux cooled base heated base

5. Instabilities at large Ra Central region of heated base

6. Instabilities at large Ra eddying instability? shear instability stable outer BL convective instability end of heated base

7. Convective ‘mixed’ layer Assume mixed layer deepening through ‘encroachment’ fixed flux convective instability predicted for RaF >1012

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

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

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

11. 3 regimes?(almost unexplored!)

12. Entraining end-wall plume and interior eddies

13. 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 specific buoyancy flux F0 a la Baines & Turner (1969)

14. in the plume • continuity • momentum • buoyancy Wp EWp R z (Note: solution in terms of buoyancy flux FB = gQ cf. Baines & Turner 1969)

15. in the interior • continuity • density plume outflow we EWP

16. Asymptotic ‘filling box’ solution time Baines & Turner (1969)

17. interior diffusion (mixing?) 2. Steady, diffusive ‘recycling box’ qh (heating) qc (cooling) • localized destabilising flux (analytical convenience) • entrainment into plume (2D, 3D or geostrophic) • downwelling velocity varies with depth • zero net heating Killworth & Manins, JFM, 1980; Hughes, Griffiths, Mullarney & Peterson, JFM, 2007

18. plume equations as before, but add diffusion in the interior … • continuity • density • at base • heating = cooling qh = –qc plume outflow we diffusion

19. Predicted temperature in sample experiment • specific buoyancy flux F0 = 7.1 x 10-7 m3/s3 • diffusivity k = 1.5 x 10-7 m2/s (molecular) • entrainment constant Ez = 0.1 (Turner 1973) lab theory: a = 3.2 x 10-4 ºC-1 a = 1.5 x 10-4 ºC-1 (box 1.25 m long x 0.2 m depth)

20. Predicted downwelling in sample experiment • specific buoyancy flux F0 = 7.1 x 10-7 m3/s3 • diffusivity k = 1.5 x 10-7 m2/s (molecular) • entrainment constant Ez = 0.1 (Turner 1973) numerical theory: a = 3.2 x 10-4 ºC-1 a = 1.5 x 10-4 ºC-1 (box 1.25 m long x 0.2 m depth)

21. Asymptotic scalings for ‘recycling box’ (line plume) box length L • thermal boundary layer: • thickness • volume transport in boundary layer (per unit width) * specific buoyancy flux F0 WhL h

22. Asymptotic scalings for ‘recycling box’ (line plume) box length L depth H WH L • top-to-bottom density difference • overturning volume transport (per unit width) Dr specific buoyancy flux F0

23. Model /lab /numerics comparisons Constants RaF dependence *constants evaluated for water at experimental conditions; Powers laws identical to viscous boundary layer scaling (Flux Rayleigh number RaF ~ specific buoyancy flux F0 )

24. Non-monotonic B.C.s => two plumeseffects on interior stratification? L = 1.25 m h = 0.2 m applied heat flux applied Tc applied heat flux

25. Regime 1

26. Regime 2

27. Confluence Point RQ =

28. Regime 3

29. Interior Stratification

30. 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

31. 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

32. next lecture • rotation effects • thermohaline phenomena • responses to changed forcing