TY - JOUR
T1 - On the possibility of soft and hard turbulence in the complex Ginzburg-Landau equation
AU - Bartuccelli, Michele
AU - Constantin, Peter
AU - Doering, Charles R.
AU - Gibbon, John D.
AU - Gisselfält, Magnus
N1 - Funding Information:
The authors would especially like to thank Darryl Holm of the Theory Division and the CNLS at Los Alamos, with whom we have had many interesting conversations on this topic and who hosted three of us (PC, CRD and JDG) when some of this work was done. We are also grateful to C.D. Levermore, N. Ercolani, J.M. Hyman, Roy Jacobs, D.W. McLaughlin and H. Rose for discussions. Petre Constantin is supported on NSF Grant DMS-860-2031 and is also a Sloan Research Fellow. Charles Doering is supported on NSF Grant No. PHY-8907755 and NSF Grant No. PHY-8958506 and is a Presidential Young Investigator. Michele Bartuccelli would like to thank the UK SERC Nonlinear Systems Panel for a postdoctoral research assistantship.
PY - 1990/9/1
Y1 - 1990/9/1
N2 - We present analytical methods which predict the occurrence of both soft (weak) and hard (strong) turbulence in the complex Ginzburg-Landau (CGL) equation: At=RA+(1+iν)δA-(1+iμ)A|A|2 on a periodic domain [0,1] D in D spatial dimensions. Hard turbulence is characterised by large fluctuations away from spatial and temporal averages with a cascade of energy to small scales. This form of hard turbulence appears to occur not in 1D but only in 2D and 3D in parameter regions which are bounded by hyperbolic curves in the second and fourth quadrants of the μ-ν planes where the system is modulationally unstable (ε{lunate}=1+μν<0). This region goes out to the dissipationless limit (μ,ν→±∞,∓∞) where the CGL equation becomes the NLS equation. When D≥2 this latter equation blows up in finite time and it is clear from our results that this finite time singularity is fundamental in causing strong turbulent behaviour. The CGL equation has an attractor consisting of C∞ functions for all finite μ and ν when D=1 and 2. When D=3 we have the same regularity in part of the μ-ν plane, which covers some of the predicted hard turbulent area. The CGL equation also possesses inertial manifolds when D=1 and 2. Our results are based on a new method where we consider an infinity of Lyapunov functionals of rank 2n: Fn=∫(|∇n-1A|2+αn|A|2n)dx, for αn > 0. For large times and large R, upper bounds exist for the infinite set of Fn's, constructed from the hierarchy of differential inequalities Fn≤(2nR+cn{norm of matrix}A{norm of matrix}2∞)F n-bnF2n/Fn-1, for cn, bn > 0 (F0≡1). Estimates for the "bottom rung" F2 give upper bounds for the whole ladder. Long time upper bounds on F2 and {norm of matrix}A{norm of matrix}2∞ (and hence all Fn) are well controlled in the soft region but become much larger in the hard region, whereas spati al and temporal averages remain comparatively small. When the nonlinearity is A|A|2q, the critical case qD=2 gives parallel results.
AB - We present analytical methods which predict the occurrence of both soft (weak) and hard (strong) turbulence in the complex Ginzburg-Landau (CGL) equation: At=RA+(1+iν)δA-(1+iμ)A|A|2 on a periodic domain [0,1] D in D spatial dimensions. Hard turbulence is characterised by large fluctuations away from spatial and temporal averages with a cascade of energy to small scales. This form of hard turbulence appears to occur not in 1D but only in 2D and 3D in parameter regions which are bounded by hyperbolic curves in the second and fourth quadrants of the μ-ν planes where the system is modulationally unstable (ε{lunate}=1+μν<0). This region goes out to the dissipationless limit (μ,ν→±∞,∓∞) where the CGL equation becomes the NLS equation. When D≥2 this latter equation blows up in finite time and it is clear from our results that this finite time singularity is fundamental in causing strong turbulent behaviour. The CGL equation has an attractor consisting of C∞ functions for all finite μ and ν when D=1 and 2. When D=3 we have the same regularity in part of the μ-ν plane, which covers some of the predicted hard turbulent area. The CGL equation also possesses inertial manifolds when D=1 and 2. Our results are based on a new method where we consider an infinity of Lyapunov functionals of rank 2n: Fn=∫(|∇n-1A|2+αn|A|2n)dx, for αn > 0. For large times and large R, upper bounds exist for the infinite set of Fn's, constructed from the hierarchy of differential inequalities Fn≤(2nR+cn{norm of matrix}A{norm of matrix}2∞)F n-bnF2n/Fn-1, for cn, bn > 0 (F0≡1). Estimates for the "bottom rung" F2 give upper bounds for the whole ladder. Long time upper bounds on F2 and {norm of matrix}A{norm of matrix}2∞ (and hence all Fn) are well controlled in the soft region but become much larger in the hard region, whereas spati al and temporal averages remain comparatively small. When the nonlinearity is A|A|2q, the critical case qD=2 gives parallel results.
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U2 - 10.1016/0167-2789(90)90156-J
DO - 10.1016/0167-2789(90)90156-J
M3 - Article
AN - SCOPUS:0000922548
VL - 44
SP - 421
EP - 444
JO - Physica D: Nonlinear Phenomena
JF - Physica D: Nonlinear Phenomena
SN - 0167-2789
IS - 3
ER -