Convection driven by differential buoyancy fluxes on a horizontal boundary

1. Convection driven by differential buoyancy fluxeson a horizontal boundary ‘Horizontal convection’ Ross Griffiths Research School of Earth Sciences The Australian National University

2. Overview #1 • What is ‘horizontal convection’? • Some history and oceanographic motivation • experiments, numerical solutions • controversy about “Sandstrom’s theorem” • how it works #2 • instabilities and transitions • solution for convection at large Rayleigh number • two sinking regions #3 • Coriolis effects • adjustment to changing boundary conditions • thermohaline effects

3. Role of buoyancy? • Surface buoyancy fluxes --> deep convection • dense overflows, slope plumes (main sinking branches of MOC). Can sinking persist? How is density removed from abyssal waters? Does the deep ocean matter? N S Potential temperature section 25ºW (Atlantic) – WOCE A16 65ºN – 55ºS

4. Preview convection in a rotating, rectangular basin heated over 1/2 of the base, cooled over 1/2 of the base

5. Stommel’s meridional overturning:the “smallness of sinking regions” Imposed surface temperature gradient low temp higher temp High latitudes low latitudes Solution: Down flow in only one pipe ! Stommel, Proc. N.A.S. 1962

6. Stommel’s meridional overturning:the “smallness of sinking regions” Imposed surface temperature gradient low temp higher temp thermocline High latitudes low latitudes abyssal flow? • Thermocline + small region of sinking • maximal downward diffusion of heat Stommel, Proc. N.A.S. 1962

7. Early experiments: thermal convection with a linear variation of bottom temperature (Rossby, Deep-Sea Res. 1965) 10 cm 24.5 cm

8. Numerical solutions for thermal convection (linear variation of bottom temperature) (re-computing Rossby’s solutions, Tellus 1998) Ra=103 Ra=104 Ra=105 Ra=106 Ra=107 Ra=108

9. Numerical solutions for thermal convection (linear variation of bottom temperature) (re-computing Rossby’s solutions, Tellus 1998) Ra=103 Ra=104 Ra=105 Ra=106 Ra=107 Ra=108

10. Linear T applied to top Solutions for infinite Pr Chiu-Webster, Hinch & Lister, 2007

11. back-step … to Sandström’s “theorem” (Sandström 1908, 1916) “a closed steady circulation can only be maintained in the ocean if the heat source is situated at a lower level than the cold source” (Defant 1961; become known as ‘Sandstrom’s theorem’) • Sandström concluded that a thermally-driven circulation can exist only if the heat source is below the cold source Surface heat fluxes … “cannot produce the vigorous flow we observe in the deep oceans. There cannot be a primarily convectively driven circulation of any significance” (Wunsch 2000)

12. • one large cell (maximum vel near source heights) • approximately uniform temperature X X X • significant circulation • two anticlockwise cells • plume from each source reaches top or bottom X X • three anticlockwise cells • plume from each source reaches nearest horizontal boundary Sandström experiments revisited He reported: C H I: Heating below cooling • still upper and lower layer, circulating middle layer • three layers of different temperature II: Heating/cooling at same level • circulation ceases III: Heating above cooling • water remains still throughout • upper (lower) layer temperature equal to hot (cold) source, stable gradient between C H H C

13. Sources at same level • diffusion (Jeffreys, 1925) • heating at levels below the cooling source • cooling at levels above the heating source • horizontal density gradient • drives overturning circulation throughout fluid • physically and thermodynamically consistent view of Sandström’s experiment and horizontal convection • no grounds to justify the conclusion of no motion when heating and cooling applied are at the same level. C H

14. Side-wall heating and cooling Higher T Low T FB Horizontal convection In Boussinesq case, zero net buoy flux through any level FB heating • • higher temp lower temp cooling Comparison of three classes of (steady-state) convection Rayleigh-Benard low T FB higher T

15. Horizontal convection Ocean orientation lower temp higher temp Destabilizing buoyancy forces deep circulation Zero net buoy flux through any level FB laboratory orientation FB higher temp lower temp

16. Boundary layer analysisfor imposed T(after Rossby 1965) Ta h u Tc TH Steady state balances: • continuity +vertical advection-diffusion uh ~ wL ~ TL/h • buoyancy - horizontal viscous stresses gTh/L ~ u/h2 • conservation of heat FL~ ocpTuh h ~ c1Ra–1/5 => u ~ c2Ra2/5 Nu ~ c3Ra1/5 Ra = gTL3/

17. Linear T applied to top Solutions for infinite Pr Rossby scaling holds at Ra > 105 Chiu-Webster, Hinch& Lister, 2007

18. Experiments at larger Ra, smaller D/L, applied T or heat flux(Mullarney, Griffiths, Hughes, J. Fluid Mech. 2004) Room Ta Parameters: RaF = gFL4/(ocpT2 Pr = /T A = D/L and define Nu = FL/(ocpTT= RaF/Ra

19. Recent experimentslarger Ra, smaller D/L(Mullarney, Griffiths, Hughes, J. Fluid Mech. 2004) Movie - whole tank

20. RaF = 1.75 x 1014H/L = 0.16Pr = 5.18 (Mullarney, Griffiths, Hughes, J. Fluid Mech. 2004) Movie - whole tank

21. Recent experimentslarger Ra, smaller aspect ratio, applied heat flux(Mullarney, Griffiths, Hughes, J. Fluid Mech. 2004)

22. ‘Synthetic schlieren’ image showing vertical density gradients (above heated end) x=0 x=L/2=60cm 20cm imposed heat flux

23. B. L. analysis for imposed heat flux(Mullarney et al. 2004) Ta h u Tc F T/T ~ b0RaF-1/6 Steady state balances: • continuity +vertical advection-diffusion uh ~ wL ~ TL/h • buoyancy - horizontal viscous stresses gTh/L ~ u/h2 • conservation of heat FL~ ocpTuh h/L ~ b1RaF–1/6 uL/T ~ b2RaF1/3 => wL/T ~ b3RaF1/6 Nu ~ b0-1RaF1/6 T = FL/ocpT)

24. temperature profiles Above heated base (fixed F) Above cooled base (fixed T)

25. 0 RaF = 1.75 x 1014 H/L = 0.16 Pr = 5.18 2D simulation Horizontal velocity (m/s)

26. 0 RaF = 1.75 x 1014 H/L = 0.16 Pr = 5.18 2D simulation Horizontal velocity vertical velocity (m/s)

27. RaF = 1.75 x 1014 H/L = 0.16 Pr = 5.18 Snap-shot of solution at lab conditions  T Eddy travel times ~ 20 - 40 min

28. RaF = 1.75 x 1014 H/L = 0.16 Pr = 5.18 Time-averaged solutions for larger Ra  T Horizontal velocity reversal ~ mid-depth Time-averaged downward advection over most of the box

29. B.L. Scaling and experimental results Circles - experiments; squares & triangle - numerical solutions After adjustment for different boundary conditions (RaF = NuRa) these data lie at 1011 < Ra < 1013. Agreement also with Rossby experiments at Ra<108 Mullarney, Griffiths, Hughes, J Fluid Mech. 2004

30. Asymmetry and sensitivity • Large asymmetry (small region of sinking) • maximal downward diffusion of heat • suppression of convective instability (at moderate Ra) by advection of stably-stratified BL • interior temperature is close to the highest temperature in the box • A delicate balance in which convection breaks through the stably-stratified BL only at the end wall • maximal horiz P gradient, maximal overturn strength, and a state of minimal potential energy (compared with less asymmetric flows - from a GCM, Winton 1995) => sensitivity to changes of BC’s and to fluxes through other boundaries

31. Buoyancy fluxes from opposite boundary(eg. geothermal heat input to ocean) Differential forcing at top only (applied flux and applied T)  T  Add 10% heat input at base T  Or add 10% heat loss at base T Mullarney, Griffiths, Hughes, Geophys. Res. Lett. 2006

32. Buoyancy fluxes from opposite boundary(eg. geothermal heat input to ocean) Mullarney, Griffiths, Hughes, Geophys. Res. Lett. 2006

33. Summary Experiments with ‘horizontal’ thermal convection show • convective circulation through the full depth in steady state, but a very small interior density gradient at large Ra • tightly confined plume at one end of the box • interior temperature close to the extreme in the box (10-15% from the extremum at end of B.L.) • stable boundary layer in region of stabilizing flux, consistent with vertical advective-diffusive balance • suppression of instability up to moderate Ra by horizontal advection of the stable ‘thermocline’, but onset of instability at RaF ~ 1012 / Ra ~ 1010 • circulation is robust to different types of surface thermal B.C.s, but sensitive to fluxes from other boundaries

34. Next time: • instabilities, transitions in Ra-Pr plane • inviscid model for large Ra and comparison with measurements • sensitivity to unsteady B.C.s, temporal adjustment, and transitions between full- and partial-depth overturning (shutdown of sinking?)