Shell models as phenomenological models of turbulence

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
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 .

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. Lvov, 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

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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.

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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), 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).

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), 120-141.
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), 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.

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 d2 it was shown that the fractal dimension
  of the global attractor of NSE satisfies

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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

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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

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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 .

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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.

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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.

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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 .

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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.

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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.

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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.

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Stability of a single mode solution

• Bifurcation diagram of the single mode stationary
  solution vs. .
• 3-D
  2-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),
  1431-1441.

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.

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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.

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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.