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

NS

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?

Previewconvection in a rotating, rectangular basin

heated over 1/2 of the base, cooled over 1/2 of the base

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Stommel’s meridional overturning:the “smallness of sinking regions”

higher templow temp

Highlatitudes

lowlatitudes

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

Highlatitudes

lowlatitudes

Imposed surface temperature gradient

Thermocline + small region of sinking maximal downward diffusion of heat

thermocline

Stommel, Proc. N.A.S. 1962

abyssal flow?

Early experiments: thermal convection with a linear variation of bottom temperature

(Rossby, Deep-Sea Res. 1965)

24.5 cm

10 cm

Ra=103 Ra=104 Ra=105

Ra=106 Ra=107 Ra=108

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

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

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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

HC

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.

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Comparison of three classes of (steady-state) convection

Rayleigh-Benard

higher T

low T

FB

HigherT

Side-wall heating and cooling

LowTFB

Horizontal convection

higher temp lower temp

FB

In Boussinesq case, zero net buoy flux through any level•

•heating

cooling

Horizontal convection

Ocean orientation

higher templower temp

FB

Zero net buoy flux through any level

higher temp lower temp

FB

laboratory orientation

Destabilizing buoyancy forces deep circulation

Boundary layer analysisfor imposed T (after Rossby 1965)

Steady state balances:

• continuity + vertical advection-diffusionuh ~ wL ~ T L/h

• buoyancy - horizontal viscous stressesgTh/L ~ u/h2

• conservation of heat FL ~ ocpTuh

Nu ~ c3Ra1/5

h ~ c1Ra–1/5

u ~ c2Ra2/5=>

u h

Ta

TcTH

Ra = gTL3/

Solutions for infinite Pr

Chiu-Webster, Hinch& Lister, 2007Rossby scaling holds at Ra > 105

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:

RaF = gFL4/(ocpT2

Pr = /T

A = D/L and define Nu = FL/(ocpTT= RaF/Ra

RoomTa

Movie - whole tank

Recent experimentslarger Ra, smaller D/L

(Mullarney, Griffiths, Hughes, J. Fluid Mech. 2004)

Movie - whole tank

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

Recent experimentslarger Ra, smaller aspect ratio, applied heat flux

(Mullarney, Griffiths, Hughes, J. Fluid Mech. 2004)

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imposed heat flux

20cm

x=0 x=L/2=60cm

‘Synthetic schlieren’ image showing vertical density gradients (above heated end)

B. L. analysis for imposed heat flux(Mullarney et al. 2004)

Steady state balances:

• continuity + vertical advection-diffusionuh ~ wL ~ T L/h

• buoyancy - horizontal viscous stressesgTh/L ~ u/h2

• conservation of heat FL ~ ocpTuh

Nu ~ b0-1RaF

1/6

h/L ~ b1RaF–1/6

uL/T ~ b2RaF1/3=>

u h

Ta

Tc

F

T/T ~ b0RaF-1/6

wL/T ~ b3RaF1/6

T = FL/ocpT)

temperature profiles

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

2D simulation

RaF = 1.75 x 1014 H/L = 0.16 Pr = 5.18

(m/s)

0

Horizontal velocity

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2D simulation

(m/s)

RaF = 1.75 x 1014 H/L = 0.16 Pr = 5.18

0

vertical velocity

Horizontal velocity

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Snap-shot of solution at lab conditions

T

RaF = 1.75 x 1014 H/L = 0.16 Pr = 5.18

Eddy travel times ~ 20 - 40 min

Time-averaged solutions for larger Ra

T

RaF = 1.75 x 1014 H/L = 0.16 Pr = 5.18

Horizontal velocity reversal ~ mid-depthTime-averaged downward advection over most of the box

B.L. Scaling and experimental results

Mullarney, Griffiths, Hughes, J Fluid Mech. 2004

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

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. 2006

T

T

T

Differential forcing at top only (applied flux and applied T)

Add 10% heatinput at base

Or add 10% heatloss 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 RaF ~ 1012 / Ra ~ 1010

• 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?)