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Weak and Strong Turbulence in the Complex Ginzburg Landau Equation

  • J. D. Gibbon
Conference paper
Part of the The IMA Volumes in Mathematics and its Applications book series (IMA, volume 55)

Abstract

We present analytical methods whereby weak and strong turbulence are predicted in the D dimensional complex Ginzburg Landau (CGL) equation \(A_{t}=RA+(1+i\nu)\Delta A-(1+i\mu)A\vert A\vert^{2q}\) on a periodic domain [0,1]. Strong (hard) turbulence is characterised by large fluctuations away from space & time averages while no such fluctuations occur in weak (soft) turbulence. In the Δ-ν plane, there are different areas where weak & strong behaviour can occur. In the strong case (Δ, ν → ±∞, ‡∞), the corresponding areas go out to the inviscid limit where the CGL equation becomes the NLS equation in which a finite time singularity occurs when qD ≥ 2. A new infinite set of differential inequalities for the “lattice” of functionals \(F_{n,m}=\int\left[{\vert\nabla^{n-1}A\vert}^{2m}+\alpha_{n,m}\vert A\vert^{2m\left[q(n-1)+1\right]}\right]d\underline{x}\) enables us to construct large time upper bounds on the F n,m . The occurrence of strong spiky turbulence is predicted for qD = 2 by showing that exponents of R in the upper bounds of F n,m & ‖A in the strong regions are dependent on the quantity |ν| which gets large in the inviscid limit. The critical value qD = 2 plays an important role: when qD > 2 the CGL equation has some similarities with the 3D Navier equations. A comparison is made between the two & the possibility of having a 1D system which mimics some limited features of the Navier Stokes equations is discussed.

Keywords

Navier Stokes Equation Differential Inequality Interpolation Inequality Inertial Subrange Incompressible Navier Stokes Equation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag New York, Inc. 1993

Authors and Affiliations

  • J. D. Gibbon
    • 1
  1. 1.Department of MathematicsImperial CollegeLondonUK

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