Convection driven by differential buoyancy fluxes on a horizontal boundary Ross Griffiths Research...
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Transcript of Convection driven by differential buoyancy fluxes on a horizontal boundary Ross Griffiths Research...
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?)