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Shell models as phenomenological models of turbulence

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The Seventh Israeli Applied and Computational Mathematics Mini-Workshop. ... Shell models serve as a very convenient ground for testing new ideas. ... – PowerPoint PPT presentation

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Title: 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

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), 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

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
    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.

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.
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