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

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

Role of buoyancy? Potential temperature section 25ºW (Atlantic) – WOCE A16 65ºN – 55ºS N S 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?

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

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

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

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

Numerical solutions for thermal convection (linear variation of bottom temperature) (Rossby, Tellus 1998) Ra = 10 6 Ra = 10 8  T y x y x

Ra=10 3 Ra=10 4 Ra=10 5 Ra=10 6 Ra=10 7 Ra=10 8 Numerical solutions for thermal convection (linear variation of bottom temperature) (re-computing Rossby’s solutions, Tellus 1998)

Ra=10 3 Ra=10 4 Ra=10 5 Ra=10 6 Ra=10 7 Ra=10 8 Numerical solutions for thermal convection (linear variation of bottom temperature) (re-computing Rossby’s solutions, Tellus 1998)

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

back-step … to Sandström’s “theorem” (Sandström 1908, 1916) Sandström concluded that a thermally-driven circulation can exist only if the heat source is below the cold source “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’) 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)

Sandström experiments revisited 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 He reported: one large cell (maximum vel near source heights) approximately uniform temperature X X significant circulation two anticlockwise cells plume from each source reaches top or bottom X three anticlockwise cells plume from each source reaches nearest horizontal boundary X X C H H H C C

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

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

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

Boundary layer analysis for imposed  T (after Rossby 1965) Steady state balances: continuity + vertical advection-diffusion uh ~ wL ~  T L/h buoyancy - horizontal viscous stresses g  Th/L ~ u/h 2 conservation of heat FL ~  o c p  Tuh Nu ~ c 3 Ra 1/5 h ~ c 1 Ra –1/5 u ~ c 2 Ra 2/5 => u h TaTa TcTc THTH Ra = g  TL 3 / 

Solutions for infinite Pr Chiu-Webster, Hinch& Lister, 2007 Rossby scaling holds at Ra > 10 5 Linear T applied to top

Experiments at larger Ra, smaller D/L, applied  T or heat flux (Mullarney, Griffiths, Hughes, J. Fluid Mech. 2004) Parameters: Ra F = g  FL 4 /(  o c p  T 2  Pr = /  T A = D/L and define Nu = FL/(  o c p  T  T  = Ra F /Ra Room T a

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

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

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

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

Horizontal velocity

B. L. analysis for imposed heat flux (Mullarney et al. 2004) Steady state balances: continuity + vertical advection-diffusion uh ~ wL ~  T L/h buoyancy - horizontal viscous stresses g  Th/L ~ u/h 2 conservation of heat FL ~  o c p  Tuh Nu ~ b 0 -1 Ra F 1/6 h/L ~ b 1 Ra F –1/6 uL/  T ~ b 2 Ra F 1/3 => u h TaTa TcTc F  T/  T ~ b 0 Ra F -1/6 wL/  T ~ b 3 Ra F 1/6  T = FL/  o c p  T )

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

2D simulation Ra F = 1.75 x H/L = 0.16Pr = 5.18 (m/s) 0 Horizontal velocity

2D simulation (m/s) Ra F = 1.75 x H/L = 0.16Pr = vertical velocity Horizontal velocity

Snap-shot of solution at lab conditions  T Ra F = 1.75 x H/L = 0.16Pr = 5.18 Eddy travel times ~ min

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

Temperature distribution (K) Ra F = 1.75 x H/L = 0.16Pr = K contours

B.L. Scaling and experimental results Mullarney, Griffiths, Hughes, J Fluid Mech Circles - experiments; squares & triangle - numerical solutions After adjustment for different boundary conditions (Ra F = NuRa) these data lie at < Ra < Agreement also with Rossby experiments at Ra<10 8

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

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

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

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 Ra F ~ / Ra ~ circulation is robust to different types of surface thermal B.C.s, but sensitive to fluxes from other boundaries

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?)

Buoyancy-driven circulation? Potential temperature section 25ºW (Atlantic) – WOCE A16 65ºN – 55ºS heating cooling N S

Schematic of the Meridional Overturning Circulation Schmitz, 1996

North Atlantic sinking

Steady flow at small and moderate Ra flow across isotherms in B.L. warm plume rising in cooler surroundings and cooling with height Isotherm plot, Ra = 10 6, Pr = 5 Mullarney et al (cf. Rossby 1998) cold hotApplied temp gradient