We continue to investigate the possibility that interstellar turbulence is caused by nonlinear interactions among shear Alfven waves. Here, as in Paper I, we restrict attention to the symmetric case where the oppositely directed waves carry equal energy fluxes. This precludes application to the solar wind in which the outward flux significantly exceeds the ingoing one. All our detailed calculations are carried out for an incompressible magnetized fluid. In incompressible magnetohydrodynamics (MHD), nonlinear interactions only occur between oppositely direct waves. Paper I contains a detailed derivation of the inertial range spectrum for the weak turbulence of shear Alfven waves. As energy cascades to high perpendicular wavenumbers, interactions become so strong that the assumption of weakness is no longer valid. Here, we present a theory for the strong turbulence of shear Alfven waves. It has the following main characteristics. (1) The inertial-range energy spectrum exhibits a critical balance beween linear wave periods and nonlinear turnover timescales. (2) The "eddies" are elongated in the direction of the field on small spatial scales; the parallel and perpendicular components of the wave vector, kz and k1., are related by kz ~ ki_ 13 L -113 , where Lis the outer scale of the turbulence. (3) The "one-dimensional" energy spectrum is proportional to kl. 5 1 3 -an anisotropic Kolmogorov energy spectrum. Shear Alfvenic turbulence mixes specific entropy as a passive contaminant. This gives rise to an electron density power spectrum whose form mimics the energy spectrum of the turbulence. Radio, wave scattering by these electron density fluctuations produces anisotropic scatter-broadened images. Damping by ion-neutral collisions restricts Alfvenic turbulence to highly ionized regions of the interstellar medium. We expect negligible generation of compressive MHD waves by shear Alfven waves belonging to the critically balanced cascade. Viscous and collisionless damping are also unimportant in the interstellar medium (ISM). Our calculations support the general picture of interstellar turbulence advanced by Higdon.

In 1965, Kraichnan proposed that MHD turbulence occurs as a result of collisions between oppositely directed Alfve n wave packets. Recent work has generated some controversy over the nature of nonlinear couplings between colliding Alfve n waves. We Ðnd that the resolution to much of the confusion lies in the existence of a new type of turbulence, intermediate turbulence, in which the cascade of energy in the inertial range exhibits properties intermediate between those of weak and strong turbulent cascades. Some properties of intermediate MHD turbulence are the following : (1) in common with weak turbulent cascades, wave packets belonging to the inertial range are long-lived ; (2) however, components of the strain tensor are so large that, similar to the situation in strong turbulence, perturbation theory is not applicable ; (3) the breakdown of perturbation theory results from the divergence of neighboring Ðeld lines due to wave packets whose perturbations in velocity and magnetic Ðelds are localized, but whose perturbations in displacement are not ; (4) three-wave interactions dominate individual collisions between wave packets, but interactions of all orders n º 3 make comparable contributions to the intermediate turbulent energy cascade ; (5) successive collisions are correlated since wave packets are distorted as they follow diverging Ðeld lines ; (6) in common with the weak MHD cascade, there is no parallel cascade of energy, and the cascade to small perpendicular scales strengthens as it reaches higher wavenumbers ; (7) for an appropriate weak excitation, there is a natural progression from a weak, through an intermediate, to a strong cascade.

We study weak Alfvenic turbulence of an incompressible, magnetized fluid in some detail, with a view to developing a firm theoretical basis for the dynamics of small-scale turbulence in the interstellar medium. We prove that resonant 3-wave interactions are absent. We also show that the Iroshnikov-Kraichnan theory of incompressible, magnetohydrodynamic turbulence-which is widely accepted-describes weak 3-wave turbulence; consequently, it is incorrect. Physical arguments, as well as detailed calculations of the coupling coefficients are used to demonstrate that these interactions are empty. We then examine resonant 4-wave interactions, and show that the resonance relations forbid energy transport to small spatial scales along the direction of the mean magnetic field, for both the shear Alfven wave and the pseudo Alfven wave. The threedimensional inertial-range energy spectrum of 4-wave shear Alfven turbulence guessed from physical arguments reads E(kz, kJ ~ VA vLL-113 kJ: 1013 , where VA is the Alfven speed, and vL is the velocity difference across the outer scale L. Given this spectrum, the velocity difference across A1_ ~ kJ: 1 is V;.~ ~ vL(A.1_jL) 213 • We derive a kinetic equation, and prove that this energy spectrum is a stationary solution and that it implies a positive flux of energy in k-space, along directions perpendicular to the mean magnetic field. Using this energy spectrum, we deduce that 4-wave interactions strengthen as the energy cascades to small, perpendicular spatial scales; beyond an upper bound in perpendicular wavenumber, k1_ L ~ (V AfvL) 312 , weak turbulence theory ceases to be valid. Energy excitation amplitudes must be very small for the 4-wave inertial-range to be substantial. When the excitation is strong, the width of the 4-wave inertial-range shrinks to zero. This seems likely to be the case in the interstellar medium. The physics of strong turbulence is explored in Paper II.

The wave functions of Sinai-billiard-shaped microwave cavities are experimentally studied. Some of the general features observed are parity breaking in the lowest eigenstates, "bouncing-ball" states, and states with quasirectangular or quasicircular symmetry. The above features are associated with nonisolated periodic orbits. Some states are observed which can be associated with isolated periodic orbits, leading to scars. This work represents the first direct experimental observation of scarred eigenfunctions. At high frequencies the eigenfunctions are very complex, and are yet to be classified in a general scheme.

We present a phenomenological model of imbalanced MHD turbulence in an incompressible magnetofluid. The steady state cascades, of waves traveling in opposite directions along the mean magnetic field, carry unequal energy fluxes to small length scales, where they decay as a result of viscous and resistive dissipation. The inertial range scalings are well understood when both cascades are weak. We study the case in which both cascades are, in a sense, strong. The inertial range of this imbalanced cascade has the following properties: (1) The ratio of the rms Elsässer amplitudes is independent of scale and is equal to the ratio of the corresponding energy fluxes. (2) In common with the balanced strong cascade, the energy spectra of both Elsässer waves are of the anisotropic Kolmogorov form, with their parallel correlation lengths equal to each other on all scales, and proportional to the two-thirds power of the transverse correlation length. (3) The equality of cascade time and wave period (critical balance) that characterizes the strong balanced cascade does not apply to the Elsässer field with the larger amplitude. Instead, the more general criterion that always applies to both Elsässer fields is that the cascade time is equal to the correlation time of the straining imposed by oppositely directed waves. (4) In the limit of equal energy fluxes, the turbulence corresponds to the balanced strong cascade. Our results are particularly relevant for turbulence in the solar wind. Spacecraft measurements have established that in the inertial range of solar wind turbulence, waves traveling away from the Sun have higher amplitudes than those traveling toward it. Result 1 allows us to infer the turbulent flux ratios from the amplitude ratios, thus providing insight into the origin of the turbulence.

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We present a conjecture relating the density of quantum resonances for an open chaotic system to the fractal dimension of the associated classical repeller. Mathematical arguments justifying this conjecture are discussed. Numerical evidence based on computation of resonances of systems of n disks on a plane are presented supporting this conjecture. The result generalizes the Weyl law for the density of states of a closed system to chaotic open systems.PACS numbers: 05.45. Mt, 03.65.Sq, 05.45.Ac, 31.15.Gy, 95.10.Fh The celebrated Weyl law concerning the density of eigenvalues of bound states is a central result in the spectroscopy of quantum systems [1]. The Weyl formula states that the asymptotic level number N (k), defined as the number of levels with k n < k (where k → ∞) is given after smoothing by.., for a quantum system bounded in a region R of D-dimensional space whose volume is V . For closed systems with smooth boundaries, the Weyl formula is well-established, and although primarily valid in the semi-classical limit, nevertheless can be applied with astonishing accuracy to very low energies extending almost down to the ground state of integrable and chaotic closed systems. Generalizations of the Weyl law to other situations have long been sought. A notable example is the conjecture by Berry [2] for the density of states of closed systems with fractal boundaries, i.e. "fractal drums".Open systems are characterized by resonances defined by complex wavevectork n = Re(k n ) + iIm(k n ), corresponding to states with finite life times arising from escape to infinity. Open chaotic systems, which occur in a variety of physical situations, are generically characterized by a classical phase space repeller that is fractal. In this letter we present a conjecture relating the density of resonances for an open chaotic system to the fractal dimension of the associated classical repeller. It can be stated as:where d H is the partial Hausdorff dimension of the repeller [3, §4.4]. This relation generalizes the Weyl law for the density of states of a closed system to chaotic open systems.In this letter, we will provide a heuristic argument for the validity of this conjecture, and present new computations that confirm its validity.The repeller in a scattering problem is defined as the set of points in phase space which do not escape to infinity at both positive or negative times. The Hausdorff dimension of the repeller is given by D H = 2d H +2, where we did not restrict ourselves to an energy surface. For closed two dimensional systems, such as compact surfaces of constant negative curvature, we have real zeros only and N (k) = #{k n : k n ≤ k} ∼ k 2 , which is consistent with (1) as d H = 1 then everything is trapped.Our motivation comes from rigorous work on quantum resonances and in particular from the work of Sjöstrand Here we consider a different but related problem. Suppose that Z(k) is the semi-classical Selberg-Ruelle zeta function, with k the wave number. In some situations the zeros of its meromorphic continua...

Wavefunctions in chaotic and disordered quantum billiards are studied experimentally using thin microwave cavities. The chaotic wavefunctions display universal density distributions and density auto-correlations in agreement with expressions derived from a 0-D nonlinear σ-model of supersymmetry, which coincides with Random Matrix Theory. In contrast, disordered wavefunctions show deviations from this universal behavior due to Anderson localization. A systematic behavior of the distribution function is studied as a function of the localization length, and can be understood in the framework of a 1-D version of the nonlinear σ-model. PACS number(s): 05.45.+b, 03.65.G, 71.55.J Typeset using REVT E X