Plumes in turbulent convection

1. A short summary of convection

2. Clustering of convective plumes

3. The Prandtl problem

A. Provenzale, ISAC-CNR Torino and CIMA, Savona

Rayleigh-Benard convection:

Rayleigh-Benard convection:

Important parameters:

R = gD3T / = a = L D

Rayleigh-Benard convection:If R < Rcrit conduction

T(x,y,z,t)=Tcond(z)(u,v,w)=(0,0,0)

If R > Rcrit convectionT= Tcond +

(u,v,w) non zero

Rayleigh-Benard convection:

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Rayleigh-Benard convection:

flux fixed1,0

re temperatufixed10

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slipfree1,00z

u

z

u

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slipno1,00

conditionsBoundary

zCz

T

zT

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Rayleigh-Benard convection:

Linear stability analysis

Weakly nonlinear expansions

“Turbulent” convection

Convective patterns:

Photo by Hezi Yizhaq, Sede Boker, Negev desert

Turbulent convection (=0.71, a=2, R=107)

Turbulent convection (R=106):

Turbulent convection (=0.71, a=2, R=107)

Turbulent convection (=0.71, a=2, R=107)

Turbulent convection (=0.71, a=2, R=107)

Turbulent convection:

Statistical properties andtransition from soft to hard turbulence

Scaling of the heat transport:Nu vs R

Nu = 1 + w

Turbulent convection:

Dynamics of convective plumes

Turbulent convection:

Formation of large-scale structureand clustering of convective plumes

Turbulent convection:

Formation of large-scale structure and mean shear (wind)

Krishnamurti and Howard (1981)Massaguer, Spiegel and Zahn (1992)

Elperin, Kleeorin, Rogachevskii and Zilitinkevish (2003)

Hartlep, Tilgner and Busse (2003)

Parodi, von Hardenberg, Passoni, Spiegel, Provenzale (2003)

Clustering of convective plumes:

Clustering of convective plumes:

Clustering of convective plumes:

Clustering of convective plumes:

Clustering of convective plumes:

Clustering of convective plumes:

Turbulent RB convection undergoesa process of inverse energy cascade

from the scales of the linear instabilityto the largest scales (box size)

Once reached an approximate k-5/3 spectrum,the system becomes statistically stationary

(is there an upper scales where the cascade stops?)

It is not a mean shear ( k = 0 )but rather a circulation at the largest scales

Turbulent convection is either non-stationaryor dominated by finite-domain effects

The large-scale structures areclusters of individual plumes

What causes the clustering ?

Option 1: attraction of same-sign plumes

Option 2: the interplay of the lower and upper boundary layers

by the agency of plumes

Other view:The fixed-flux instabilityof a coarse-grained field

( with Reff << R )

Is RB convection a good model fornatural convective processes ?

Yes, as a first step (e.g. plumes)

No, for proper understanding

Most natural convective flowshave no up-down symmetry

Reasons:non-Boussinesq

non symmetric boundary conditions

Penetrative convection

Penetrative convection

Solar granulation

Tropical convective precipitation

GATE 1 data set. = 4 km, L=256 km, t=15 min

“True” dynamics:turbulent, moist, non-Boussinesq

precipitating convection

Can we find a simplified dynamical model ?

The Prandtl problem

Prandtl (1925)

The Prandtl problem

A Parodi, KA Emanuel, A Provenzale (2003)

The Prandtl problem

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The Prandtl problem

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Heat flux

The Prandtl problem

Average temperature profile

The Prandtl problem

The Prandtl problem

The Prandtl problem

The Prandtl problem

The Prandtl problem

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2

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tzyxHdzdttyxP

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P(x,y,t) is taken as a proxy for convective rainfall

The Prandtl problem

The Prandtl problem

still a long way to go,but the results are intriguing.

Linear stability, weakly nonlinear analysis,

properties of the turbulent plumes,particle transport.

And, then, addition of moisture.

The Prandtl problem

The Prandtl problem