Turbulent Convection in the Laboratory

K.R. SreenivasanNew York University

September 5, 2014

Gänseliesel

• Interior convection is an important ingredient of solar physics

• I have been working on laboratory convection for many years

• And have always thought controlled laboratory experiments might shed some light particularly on interior convection

• Although you are all experts on the subject, I will explain some laboratory experiments and computer simulations which may have some bearing on your expertise.

Flu id

T+ T

T

D

H

Basic Notation

Q = vertical heat fluxk = thermal conductivity of the fluid

Rag TH

3

Rayleigh number:

Prandtl number:

Pr

Aspect ratio:H

D

S ~ detailed shape ??

Nu depends on…

Nusselt numberNu = Q/(k T/H)

Niemela, Skrbek, KRS & Donnelly, Nature 404, 837 (2000) Slightly revised: Niemela & KRS, J. Low Temp. Phys. 143, 163 (2006)

[Pioneers: Threlfall (Cambridge); Libchaber, Kadanoff and coworkers (Chicago)]

(exponent close to 1/3)

1010

Ra=102424

106

108

Nu ≈ 5106

Nu ≈ 2.9109

Nu ≈ 2.61010

(almost the same as the extrapolated value)

Kraichnan (1962)“ultimate state”

Plasting & Kerswell (2003)“upperbound”

Seems consistent with Hanasoge, Duvall and KRS (2012)

Convective processes are far from being optimally efficient.

Rag TH

3

Nu = Q/(k T/H)

Urban et al. (2014)

See also: Roche et al. (2010) Chillá & Schumacher (2012)

It is disappointing that we still don’t know with confidence the heat transport law at

high Rayleigh numbers even in the simple case of Rayleigh-Bènard convection

Data on Rotating Convection

our data

Sun

(from Cheng et al. (2014), modified by me)

0.4 0.5 0.6 0.7 0.8 0.9 1 20.960

0.965

0.970

0.975

0.980

0.985

0.990

0.9954.23x1015<Ra<4.31x1015

Nu(0) adjusted accodring to local slopefor calculating ratio

log-log fit:

Nu=1.019Nucorr

(0)Ro0.024

Nu

/Nu co

rr(0

)

convective Rossby number

Heat transport decreases only modestly with rotation,and this appears true for the conditions of the Sun

exponent: 0.024

Rotating Convection

Nu/Nu0

Rossby number

“Giant Convection Cells Found on the Sun”---title of a Science paper

“Large-scale toroidal cells a challenge to theories of the Sun”---a website declares

Large scale circulation

(wind)

the container

large-scale

circulation (“mean wind”)

The “mean wind” breaks symmetry, with its own

consequencesThe mean wind

For convection in

a round cylinder, the mean wind precesses

freely.

For convection in a cubic box, the mean wind is constrained

along a diagonal.

0 2000 4000 6000 8000 10000

-10

0

10

VM

VM

V(t

), c

m/s

t, sec

Glatzmaier, Coe, Hongre & Roberts, Nature 401, 885-890 (1999)

Geomagnetic polarity reversals

The mean wind…with occasional reversals(KRS, Bershadskii & Niemela, PRE 65, 056306, 2000; Niemela et al. JFM, 2001)

Segment of continuous 120-hour record;

The reversals become more frequent with increasing Ra.

50 cm

J.J. Niemela and KRSJ. Fluid Mech. 557, 411-422 (2006).

Ra = 1.9 x 109 cm

12.5

Aspect ratio effect

Summary remarks

High-Rayleigh-number convection experiments tantalize us with quantitative connections to the convection processes in the Sun: heat transport law, large-scale convection cells, rotation, etc.

Alas, the connections seem to become weaker upon scrutiny, but there are reasons to be optimistic.