Volume 142, number 6,7 PHYSICS LETTERS A 18 December 1989 HARD TURBULENCE IN A FINITE DIMENSIONAL DYNAMICAL SYSTEM? Michele BARTUCCELLI a Peter CONSTANTIN b Charles R. DOERING John D. GIBBON a and Magnus GISSELFALT a,I Department of Mathematics, Imperial College, London SW7 2BZ, UK b Department of Mathematics, University of Chicago, Chicago, IL 60637, USA 0 Department of Physics and institutefor Nonlinear Studies, Clarkson University, Potsdam, NY 136 76, USA Received 11 October 1989; acceptedfor publication 17 October 1989 Communicated by D.D. HoIm We present new time-averaged and time-asymptotic bounds on various norms of solutions to the 2-d complex Ginzburg—Lan- dau (CGL) equation, 8A/t9t=RA + (1 + iv) /~A — (I + ~u) IAI 2A. These bounds establish the existence of a finite dimensional global attractor and inertial manifolds, so that the dynamics on the attractor are those of a finite dimensional dynamical system. The CGL equation is known to have chaotic solutions characterized by relatively long length-scale, low-modal dynamics, also known as sofi turbulence. Our new estimates suggest that near the nonlinear Schrodinger (NLS) limit (I v I, I ii I —c~, ifand only if v~ <0), solutions of the CGL equation are dominated by the remnants of the blow-up solutions of the NLS equation, resulting in asymptotic dynamics marked by large intermittent spikes in space and time, i.e., hard turbulence. Experimental and numerical studies of turbulence the Zakharov equations ~ occurs when a self-focus- generally distinguish between soft or weak turbu- ing instability intermittently drives the electric field lence, on the one hand, and hard or strong turbu- into spatially localized spikes which finally burn out lence on the other. The salient feature of weak tur- due to dissipation, only to be re-ignited elsewhere. bulence is that of low dimensional temporal, and low Here, the inertial range in the power spectrum is modal spatial dynamics, with no major excursions identified with the scaling of the self-similar form of from space and time averages. In contrast, strong the spikes as the system tries to blow up. turbulence is characterized by intermittent large de- Finite dimensional dynamical system methods and viations from space or time averages, controlled only concepts have been widely used to analyze weak tur- by dissipation at small scales. The experiments on bulence, with the greatest success near the onset of convection in gaseous helium by Heslot, Castaing and instabilities of laminar flows [3]. This approach has Libchaber [1] show soft and hard regions of tur- been rigorously justified in recent years with the de- bulence as a function of the Rayleigh number. There velopment of techniques for bounding the global at- is a definite change of scales between the regions; tractor dimension of continuum systems [41 and, small-scale turbulent boundary layers are associated perhaps more to be point, with the development of with the hard turbulence region. In general, hard tur- the concept of inertial manifolds [5] which directly bulence in fluid dynamical systems can be defined establish the connection between continuum dynam- by the appearance of an inertial range in the power ics, described by partial differential equations spectrum, and a clear separation of scales between (PDEs), and finite systems of ordinary differential long wavelength driving forces and short scale dis- equations (ODEs). An inertial manifold provides sipation. Strong turbulence in plasmas described by the “slaving function” through which high wave- number modes are explicitly controlled by low wave- number modes. Although inertial manifolds have Also at Department of Theoretical Physics, Chalmers Univer- sity of Technology, GOteborg S-412 96, Sweden. ~ For a review, see ref. [21. 0375-9601/89/s 03.50 © Elsevier Science Publishers B.V. (North-Holland) 349