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Shell models as phenomenological models of

turbulence

- The Seventh Israeli Applied and Computational

Mathematics Mini-Workshop. - Weizmann Institute of Science, June 14, 2007
- Boris Levant (Weizmann Institute of Science)
- Joint work with R. Benzi (Universita di Roma), P.

Constantin (University of Chicago), I. Procaccia

(Weizmann Institute of Science), and E. S. Titi

(University of California Irvine and Weizmann

Institute of Science)

Plan of the talk

- Introducing the shell models
- Existence and uniqueness of solutions
- Finite dimensionality of the long-time dynamics
- Anomalous scaling of the structure functions

Introduction

- Shell models are phenomenological model of

turbulence retaining certain features of the

original Navier-Stokes equations. - Shell models serve as a very convenient ground

for testing new ideas. - They are used to study energy cascade mechanism,

anomalous scaling, energy dissipation in the zero

viscosity limit and other phenomena of turbulence.

Navier-Stokes equations

- All information about turbulence is contained in

the dynamics of the Navier-Stokes equations - In the Fourier space variables it takes the form

The phenomenology of turbulence

- Let , where is a characteristic

length. - For small viscosity , there exist two scales

Kolmogorov , and viscous s.t. - Inertial range the

dynamics is governed by the Euler ( )

equation. - Dissipation range energy

from the inertial modes is absorbed and

dissipated - Viscous range the dynamics is

governed by the linear Stokes equation

Kolmogorovs hypothesis

- The central hypothesis of the Kolmogorovs theory

of homogeneous turbulence states that in the

inertial range, there is no interchange of energy

between the shell and the

shell if the shells

and are separated by

at least an order of magnitude. - One usually considers .

A drastic modification of the NSE

- The turbulent field in each octave of wave

numbers is replaced by a very

few representative variables. - The time evolution is governed by an infinite

system of coupled ODEs with quadratic

nonlinearities. - Each shell interacts with only few neighbors.

Different models

- The most studied model today is

Gledzer-Okhitani-Yamada (GOY) model. - Sabra model a modification of the GOY model

introduced by V. Lvov, etc. - Other examples include the dyadic model, Obukhov

model, Bell-Nelkin model etc.

Sabra shell model of turbulence

- The equation describe the evolution of complex

Fourier-like components , of the

velocity field . - with the boundary conditions .
- The scalar wave numbers satisfy

Quadratic invariants

- The inviscid ( ) and unforced (

) model has two quadratic invariants. - The energy
- Second quadratic invariant

The dimension of the shell model

- The 3-D regime . Forward

energy cascade from large to small scales. - W is associated with a helicity.
- The 2-D regime . The energy

flux is backward from the small to large

scales. - W is associated with an enstrophy.
- The value stands for the critical

dimension. It represents a point where the flux

of the energy changes its direction.

Typical spectrum in the 3-D regime

Existence and uniqueness of solutions

- P. Constantin, B. Levant, E. S. Titi, Analytic

study of the shell model of turbulence, Physica

D, 219 (2006), 120-141. - P. Constantin, B. Levant, E. S. Titi, A note on

the regularity of inviscid shell models of

turbulence, Phys. Rev. E, 75 (1) (2007).

Preliminaries sequence spaces

- Define a space to be a space of square

summable infinite sequences over , equipped

with an inner product and norm - Denote a sequence analog of the Sobolev spaces
- with an inner product and norm

Abstract formulation of the problem

- We write a Sabra shell model equation in a

functional form for - The linear operator is
- The bilinear operator is defined as
- where

Solutions of the viscous model

- The viscous ( ) shell model has a unique

global weak and strong solutions for any

. - Moreover, for , the solution of the

viscous shell model has an exponentially (in

) decaying spectrum - when the forcing applied to the finite number

of modes.

P. Constantin, B. Levant, E. S. Titi, Analytic

study of the shell model of turbulence, Physica

D, 219 (2006), 120-141.

Weak solutions of the inviscid model

- For the inviscid ( )

shell model has a global weak solution with

finite energy - for any .
- The solution is not necessarily unique.

P. Constantin, B. Levant, E. S. Titi, A note on

the regularity of inviscid shell models of

turbulence, Phys. Rev. E, 75 (1) (2007).

Weak solutions uniqueness

- The solution is unique

up to time T if - The solution conserves

the energy as long as . In other

words, if - The last statement is an analog of Onsager

conjecture for the solutions of Euler equation.

P. Constantin, B. Levant, E. S. Titi, A note on

the regularity of inviscid shell models of

turbulence, Phys. Rev. E, 75 (1) (2007).

Solutions of the inviscid model

- For there exists T 0, such that

the inviscid ( ) shell model has a unique

solution - In the 2-D parameter regime

there exists a such that the

norm of the solution is conserved. Using this and

the Beale-Kato-Majda type criterion for the

blow-up of solutions of the shell model, we show

that in this 2-D regime the solution exists

globally in time.

P. Constantin, B. Levant, E. S. Titi, A note on

the regularity of inviscid shell models of

turbulence, Phys. Rev. E, 75 (1) (2007).

Looking for the blow-up

- The goal is to show that the norm of the

initially smooth strong solution becomes infinite

in finite time for some initial data. - This will allow to address the problem of

viscosity anomaly. Namely, that the mean rate of

the energy dissipation in the 3-D flow - is bounded away from zero when .

Dyadic model of turbulence

- For the following inviscid dyadic shell model
- one can show that for any smooth initial data

the norm of the solution becomes infinite

in finite time. - This was proved in the series of papers by N.

Pavlovich, N. Katz, S. Friedlander, A. Cheskidov,

and others.

Damped inviscid equation

- Consider the inviscid equation with damping
- for some .
- For any which are supported on the finite

number of modes, the solution of the damped

equation exists globally in time for any

.

Finite dimensionality of the long-time dynamics

- P. Constantin, B. Levant, E. S. Titi, Analytic

study of the shell model of turbulence, Physica

D, 219 (2006), 120-141. - P. Constantin, B. Levant, E. S. Titi, Sharp

lower bounds for the dimension of the global

attractor of the Sabra shell model of

turbulence, J. Stat. Phys., 127 (2007),

1173-1192.

Degrees of freedom of turbulent flow

- Classical theory of turbulence asserts that

turbulent flow has a finite number of degrees of

freedom. In the dimension d 2,3 - For d2 it was shown that the fractal dimension

of the global attractor of NSE satisfies

Finite dimensionality of the attractor

- The shell model has a finite-dimensional global

attractor. - The fractal and Hausdorff dimensions of the

global attractor satisfy - Moreover, we get an estimate in terms of the

generalized Grashoff number

Attractor dimension in 2-D

- In the 2-D parameter regime

there exists a

such that the norm of the solution is

conserved. - Assume that the forcing is

applied to the finite number of modes

for . Then the fractal and Hausdorff

dimensions of the global attractor satisfy

Around the critical dimension 2-D

- Note that as .
- Therefore, the number of degrees of freedom of

the model tends to as we approach the

critical dimension .

Inertial manifold

- An inertial manifold is a finite dimensional

Lipschitz, globally invariant manifold which

attracts all solutions of the equation in the

exponential rate. Consequently, it contains the

global attractor. - The concept was introduced by Foias, Sell and

Temam in 1988. - The existence of an inertial manifold for the

Navier-Stokes equations is an open problem.

Dimension of the inertial manifold

- Let the forcing satisfy

for . Then the shell model has an

inertial manifold of dimension - This bound matches the upper bound for the

fractal dimension of the global attractor. - The estimate takes into account the structure of

the forcing if the equation is forced only at

the high modes, the attractor is small.

Reduction of the long-time dynamics

- For such an denote a projection of

onto the first modes, and .

There exists a function

whose graph is an inertial manifold. - The long-time dynamics of the model can be

exactly reduced to the finite system of ODEs - for .

How big the attractor can be?

- The bounds obtained until now predict that the

global attractor is finite-dimensional for any

force. But are those bounds tight? - In the 2-D regime of parameters

for the forcing concentrated

on the first mode the stationary solution

is globally stable. - Our goal is to construct the forcing for which

the upper bound for the dimension of the global

attractor are realized.

The general procedure

- The global attractor contains all the steady

solutions together with their unstable manifolds.

- The plan is construct a specific forcing, find

a corresponding stationary solution and estimate

the dimension of its unstable manifold. - This method has been used by Meshalkin-Sinai,

Babin-Vishik, and Liu to estimate the lower bound

for the dimension of the global attractor for the

2-D NSE.

Single mode stationary solution

- The natural candidate forcing concentrated on

the single mode and the corresponding

stationary solution - This is an analog of the Kolmogorov flow for the

NSE, used by Babin-Vishik and others. - However, in our case, because of the locality of

the nonlinear interactions, the dimension of the

unstable manifold is at most 3.

Stability of a single mode solution

- Bifurcation diagram of the single mode stationary

solution vs. . - 3-D

2-D

Construction of the large attractor

- The conclusion for any and for

small enough viscosity, there exists such

that is stable for all and

unstable for all . - To build a large attractor, we consider the

following lacunary forcing and the corresponding

stationary solution

Lower bound for the dimension

- The solution has a large unstable

manifold. Counting its dimension we conclude that

the Sabra shell model at has a large

global attractor of dimension satisfying - Therefore, the upper-bounds for the fractal

dimension of the global attractor are sharp. - The constant depends only on and

tends to as .

Anomalous scaling of the structure functions

- R. Benzi, B. Levant, I. Procaccia, E. S. Titi,

Statistical properties of nonlinear shell models

of turbulence from linear advection models

rigorous results, Nonlinearity, 20 (2007),

1431-1441.

Structure functions

- The n-th order structure function of the velocity

field is defined as - where denotes the ensemble or time

average. - Assuming that the turbulence is homogeneous and

isotropic, one concludes that the structure

functions depend only on .

Kolmogorov scaling

- Under various assumptions on the flow, and in

particular, assuming that the mean energy

dissipation rate is bounded

away from zero when viscosity tends to 0,

Kolmogorov derived the 4/5 law - Applying dimensional arguments he conjectured

Anomalous scaling

- Recent experiments, both numerical and

laboratory, predict that the structure functions

are indeed universal and for each there

exist scaling exponents' , such that for

large Reynolds number - Moreover, , as predicted by the 4/5 law,

but the rest of the exponents are anomalous,

different from the prediction n/3.

Application of the shell model

- Shell models of turbulence serve a useful

purpose in studying the statistical properties of

turbulent fields due to their relative ease of

simulation. - In particular, shell models allowed accurate

direct numerical calculation of the scaling

exponents of their associated structure

functions, including convincing evidence for

their universality. - In contrast, simulations of the Navier-Stokes

equations much harder, and one still does not

know whether these equations in 3-D are well

posed.

Structure functions of the shell model

- We define the structure functions
- For sufficiently small viscosity, and a large

forcing, there exists an inertial range of

-s for which the structure functions follow a

universal power-law behavior - All the exponents are anomalous

except for the .

Linear problem

- In the recent years a major breakthrough has been

made in understanding the mechanism of anomalous

scaling in the linear models of passive scalar

advection. - The linear shell model reads
- where is a solution of the nonlinear

problem.

Connection to the nonlinear case

- Let be real, and consider the system
- Observe, that for any the two equations

exchange roles under the change . - This leads to the assumption that if the scaling

exponents of the two field exist they must be the

same for any . - Angheluta, Benzi, Biferale, Procaccia, Toschi

(2006), Phys. Rev. Lett. 87.

Numerical evidence

- The compensated sixth order structure

function for different values of

Rigorous result

- For and

the solution of the coupled

system exists globally in time. - For any , the solutions converge

uniformly, as to

the corresponding solutions of the system with

Conclusions

- If the scaling exponents of the fields

are equal for any they will be the

same for . - For the is a solution of the

nonlinear equation, while is a solution of

the linear equation advected by . - This result is valid for the large but finite

time interval.

Summary

- Shell models are useful in studying different

aspects of the real world turbulence, by being

much easier to compute than the original NSE. - Further analytic study of the models may shed

light on the long standing conjectures in the

phenomenological theory of turbulence.