1. 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)

2. 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

3. 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.

4. 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

5. 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

6. Kolmogorov’s hypothesis
The central hypothesis of the Kolmogorov’s 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

7. 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.

8. Different models
The most studied model today is Gledzer-Okhitani-Yamada (GOY) model.
Sabra model – a modification of the GOY model introduced by V. L’vov, etc.
Other examples include the dyadic model, Obukhov model, Bell-Nelkin model etc.

9. 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

10. Quadratic invariants
The inviscid ( ) and unforced ( ) model has two quadratic invariants.
The energy
Second quadratic invariant

11. 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.

12. Typical spectrum in the 3-D regime

13. Existence and uniqueness of solutions
P. Constantin, B. Levant, E. S. Titi, “Analytic study of the shell model of turbulence”, Physica D, 219 (2006),
P. Constantin, B. Levant, E. S. Titi, “A note on the regularity of inviscid shell models of turbulence”, Phys. Rev. E, 75 (1) (2007).

14. 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

15. 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

16. 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),

17. 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).

18. 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).

19. 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).

20. 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

21. 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.

22. 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

23. 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),
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),

24. 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 d=2 it was shown that the fractal dimension of the global attractor of NSE satisfies

25. 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

26. 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

27. 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’’

28. 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.

29. 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.

30. 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

31. 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.

32. 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.

33. 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.

34. Stability of a single mode solution
Bifurcation diagram of the single mode stationary solution vs
3-D D

35. 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

36. 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

37. 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),

38. 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

39. 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

40. 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.

41. 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.

42. 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

43. 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.

44. 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.

45. Numerical evidence
The ``compensated’’ sixth order structure function for different values of

46. 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

47. 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.

48. 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.