What happens when Ra grows large:Turbulent convection

Statistical approach

Dynamics of individual (coherent) plumes

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

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

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

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

Turbulent convection:

Statistical properties andtransition from soft to hard turbulence

Scaling of the heat transport:Nu vs Ra

Nu =Qtotal / Qconduction= 1 + < w >

Turbulent convection

Turbulent convection

Turbulent convection

Turbulent convective plumes

Clustering of plumes andformation of large-scale order

Large-scale wind and generation of mean shear (rectification process, k=0):

Krishnamurti and Howard (1981)Howard and Krishnamurti (1986)

Massaguer, Spiegel and Zahn (1992)

Large-scale wind leads to plume clustering: Heslot et al (1987)

Kadanoff (2001)

Instability of the long-wave modes inturbulent convection: Elperin et al (2003)

Numerical simulation with periodic b.c.:No k=0 mode

Hartlep, Tilgner and Busse (2003)Parodi, von Hardenberg, Passoni, Provenzale, Spiegel, PRL (2004), PLA (2008)

Coarsening of the plume pattern:

Coarsening of the plume pattern:

Coarsening of the plume pattern:

Coarsening of the plume pattern:

The coarsening is due toclustering of convective plumes:

Turbulent RB convection undergoesa process of energy transfer

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

At later times,the system becomes statistically stationary.

It is not a mean shear ( k = 0 )but rather a circulation at the largest scales (k=1)

The large-scale structures areclusters of individual plumes

The large-scale wind in closed containeris probably due to the same process,

with the energy piling up at the k=1/2 mode

It is not the large-scale circulation that generates plume clustering but viceversa

Is there an upper scale where the clustering is arrested ?

How does the clustering depend on theRayleigh number ?

Is there an upper scale where the clustering is arrested ?

Is there an upper scale where the clustering is arrested ?

Dependence on the Rayleigh number

What causes the clustering ?

Option 1: attraction of same-sign plumes

Option 2: the interaction of the lower and upper

boundary layers by the agency of plumes

Other view:The fixed-flux instability of a coarse-grained field

( with Reff << R )

In the fully turbulent regime, the system recovers a “statistical” roll pattern

Convection has still a lot to teach:

Effect of rotation

Transport

Predictability

Simplified models of moist convection

A summary of NS and fully-developed turbulence

(incompressible, homogeneous flow)

€

∇⋅r

u = 0

∂

∂t+

r u ⋅∇

⎛

⎝ ⎜

⎞

⎠ ⎟r u = −∇p + ν 0∇

2 r u

Non dimensional version

€

∇⋅r

u = 0

∂

∂t+

r u ⋅∇

⎛

⎝ ⎜

⎞

⎠ ⎟r u = −∇p +

1

Re∇ 2 r

u

Reynolds number : Re =UL

ν 0

A summary of NS and fully-developed turbulence

(incompressible, homogeneous flow)

In the limit :

formation of boundary layers

€

Re → ∞

Far from boundary layers:

inertial range(Kolmogorov 1941,

or K41)

€

ετ

=u3

l= constant → u ≈ l1/ 3

E(k)dk ≈ u2 ≈ l2 / 3

k ≈1/ l

E(k) ≈ k−5 / 3

log k

log E(k)

k-5/3

E

Direct energy cascadefrom large

to small scales