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.
Published Version
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