# Presentation: 002689

## Phenomenology of Rayleigh-Taylor Turbulence

- _Misha Chertkov_   (T-13, Theory Division, Los Alamos)

- Phys.Rev.Lett **91,** 115001 (2003)

- Thanks:   David Sharp (LANL)
-                 Brad Plohr (LANL)
-                 Vladimir Lebedev (Landau Inst)
-                 Tim Clark  (LANL)
-                 Ray Ristorcelli (LANL)

## Condition:  

- The developed (mixing) regime of
-  Rayleigh-Taylor instability (turbulence)

_*Question:*_
- Explain/understand hierarchy of spatial/temporal scales of velocity and density (temperature) fluctuations, deep inside the mixing zone.

## Menu

- _Cascade picture_ (phenomenology)

_*Menu*_

- _Navier-Stokes__ Turbulence_   (Kolmogorov,Obukhov’41)
- _Passive Scalar Turbulence_     (Obukhov’48, Corrsin’51)

- _Rayleigh-Taylor turbulence phenomenology_

- _3d_

- _2d_

- _Small Atwood number, Boussinesq_

- _2d_
- vs
- _3d_

-  Miscible, incompressible
-  Immiscible, incompressible
-  Richtmyer-Meshkov + decay turb.

-  Anisotropy
-  Intermittency
-  Mixing
-  Chemical reactions  *

- vs     Rayleigh-Benard

_*Plans:*_

*  **Lagrangian *
*phenomenology*

_***_

*  **Anomalous *
*     **scaling*

_***_

## Navier-Stokes Turbulence  (steady 3d)

- _Menu_

- kinetic energy flux
-  scale independent !!!
-   time independent !!!

- typical velocity fluctuation on scale  “r”

- Kolmogorov ’41
- Obukhov ‘41

## Passive scalar turbulence    (steady)

- _Menu_

-       scalar flux
-  scale independent !!!
-   time independent !!!

- typical temperature fluctuations on scale  “r”

- Obukhov ’48
- Corrsin ‘51

## Low Atwood number, Boussinesq approximation

- e.g. Landau-Lifshitz
- ``Hydrodynamics”

_*Free convection*_* **(one fluid)*

_*Navier-Stokes*_

- Oberbeck 1879
- Lord Rayleigh  1883
- J. Boussinesq 1903

- _unstable_

- G.I. Taylor ‘1950
- Chandrasekhar ‘1961
-   ...

- _Menu_

- Rayleigh-Taylor  vs Rayleigh-Benard
-  (different initial/boundary conditions)

## Rayleigh-Taylor turbulence        3d

- L(t)  ~ turbulent (mixing) zone width
-            also energy-containing scale

- Adiabatic picture:
-        decreases with r

- _Menu_

- Sharp-Wheeler ’61 +
- Review:  Sharp ‘84

## Rayleigh-Taylor turbulence  3d   

-                                         smallish scales

- velocity is smooth
- passive scalar adveciton is
- ``Batchelor”

- _Menu_

_*viscous scale*_

_*dissipative scale*_

## Rayleigh-Taylor turbulence        2d

- L(t)  ~ turbulent (mixing) zone width
-            also energy-containing scale

- _Menu_

- _so far the same as in 3d_

- _Two false attempts_

## Rayleigh-Taylor turbulence     2d  active scalar regime

-  Bolgiano ‘59-Obukhov ‘59
- (Rayleigh-Bernard turb scenario)

- _Menu_

*consistent with RB numerics*
*     **Celani,Matsumoto,*
*    **Mazzino,Vergassola ‘02*

## Miscible case

- _Menu_

- compatibility
-   condition

## 2003 Dirac Medal

- On the occasion of the birthday of P.A.M. Dirac
- the Dirac Medal Selection Committee takes pleasure in announcing that
- the 2003 Dirac Medal and Prize
- will be awarded to:
**Robert H. Kraichnan**
- (Santa Fe, New Mexico)
- and
**Vladimir E. Zakharov**
- (Landau Institute for Theoretical Physics)
*The 2003 Dirac Medal and Prize is awarded to Robert H. Kraichnan and Vladimir E. Zakharov for their distinct contributions to the theory of turbulence, particularly the exact results and the prediction of inverse cascades, and for identifying classes of turbulence problems for which in-depth understanding has been achieved.*
* *
*Kraichnan’s most profound contribution has been his pioneering work on field-theoretic approaches to turbulence and other non-equilibrium systems; one of his profound physical ideas is that of ****the inverse cascade for two-dimensional turbulence****.  Zakharov’s achievements have consisted of putting the theory of wave turbulence on a firm mathematical ground by finding turbulence spectra as exact solutions and solving the stability problem, and in introducing the notion of ****inverse**** and dual cascades in wave turbulence.*
* *
* *
* *
* *
- 8 August 2003

- _Menu_ *

*2d **

## Lagrangian phenomenology of Turbulence

-    velocity gradient tensor
- coarse-grained over the blob

- tensor of inertia of the blob

- _Stochastic minimal model_  verified against DNS

- Chertkov, Pumir, Shraiman
- Phys.Fluids. 99, Phys.Rev.Lett. 02

*Steady, isotropic Navier-Stokes turbulence*

_*Challenge !!!*_      To extend the Lagrangian phenomenology  (capable of describing small
-                                      scale anisotropy and intermittency) to non-stationary world, e.g. of
-                                                          Rayleigh-Taylor Turbulence

- _Menu_ *

## Intermittency (anomalous scaling) of density fluctuations

- Small scale fluctuations of passive scalar shows intermittency
- (and anisotropy) even in a self-similar velocity field  !!!!
_*Kraichnan model:*_
-  1/d-expansion                          Chertkov, Falkovich, Kolokolov,Lebedev  ‘95
-  ``almost diffusive” limit              Gawedzki, Kupianen   ‘95
-  ``almost smooth” limit                 Shraiman, Siggia         ’95
-  exponent saturation (large n)       Chertkov  ’97;   Balkovsky, Lebedev ‘98

_*Challenge !!!*_*     *
*                      *To extend the passive scalar steady turbulence intermittency description
-                               to non-stationary cases, e.g.  to explain density filed intermittency
-                               and mixing in the 3d Rayleigh-Taylor Turbulence.

- _Menu_ *

## Acceleration of chemical reactions by turbulence

_*Acceleration of chemical reactions by turbulence*_

- Chertkov, Lebedev  Phys. Rev. Lett. 90, 034501 (2003)
-                                  Phys. Rev. Lett. 90, 134501 (2003)

- Explanation of the chemical concentration
-   (and its fluctuations) decay in time in
-      stationary (isotropic or bounded)
-                chaotic/turbulent   flow

_*Challenge !!!*_*   *
*                      *To extend the chemical reaction description  to the case of non-stationary
-                               advection, e.g. by  Rayleigh-Taylor Turbulence.

- _Menu_ *