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Convection driven by differential buoyancy fluxes on a horizontal boundary

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Convection driven by differential buoyancy fluxeson a horizontal boundary

‘Horizontal convection’

Ross Griffiths

Research School of Earth Sciences The Australian National University

#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

- 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

convection in a rotating, rectangular basin

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

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

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

(Rossby, Deep-Sea Res. 1965)

10 cm

24.5 cm

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)

Ra=103

Ra=104

Ra=105

Ra=106

Ra=107

Ra=108

Linear T applied to top

Solutions for infinite Pr

Chiu-Webster, Hinch & Lister, 2007

“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)

- • 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

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

- 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

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

Rayleigh-Benard

low T

FB

higher T

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

Ta

h

u

Tc

TH

Steady state balances:

• continuity +vertical advection-diffusion

uh ~ wL ~ TL/h

• buoyancy - horizontal viscous stresses

gTh/L ~ u/h2

• conservation of heat

FL~ ocpTuh

h ~ c1Ra–1/5

=>

u ~ c2Ra2/5

Nu ~ c3Ra1/5

Ra = gTL3/

Linear T applied to top

Solutions for infinite Pr

Rossby scaling holds at Ra > 105

Chiu-Webster, Hinch& Lister, 2007

Room

Ta

Parameters:

RaF = gFL4/(ocpT2

Pr = /T

A = D/L

and define Nu = FL/(ocpTT= RaF/Ra

Recent experimentslarger Ra, smaller D/L(Mullarney, Griffiths, Hughes, J. Fluid Mech. 2004)

Movie - whole tank

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

Movie - whole tank

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

x=0

x=L/2=60cm

20cm

imposed heat flux

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

gTh/L ~ u/h2

• conservation of heat

FL~ ocpTuh

h/L ~ b1RaF–1/6

uL/T ~ b2RaF1/3

=>

wL/T ~ b3RaF1/6

Nu ~ b0-1RaF1/6

T = FL/ocpT)

temperature profiles

Above heated base (fixed F)

Above cooled base (fixed T)

0

RaF = 1.75 x 1014

H/L = 0.16

Pr = 5.18

Horizontal velocity

(m/s)

0

RaF = 1.75 x 1014

H/L = 0.16

Pr = 5.18

Horizontal velocity

vertical velocity

(m/s)

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

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

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

- 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)

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

Buoyancy fluxes from opposite boundary(eg. geothermal heat input to ocean)

Mullarney, Griffiths, Hughes, Geophys. Res. Lett. 2006

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

• 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?)