Victor S. L'vov

The saga of YIG: Spectra, thermodynamics, interaction and relaxation of magnons in a complex magnet

Abstract A review of magnon properties of yttrium-iron garnet (YIG), a classical object for experimental studies in magnetism, is presented. Both experimental and theoretical results concerned with thermodynamics and kinetics of YIG are described. The main purposes of the review are to introduce a new method of approximate calculation of the magnon spectra in magnets with large unit cell and to obtain by means of this method some basic properties of YIG. In particular, it is shown that the problem of calculating the frequencies of all the 20 magnon branches over the entire Brillouin zone contains two small parameters. First, because of the large number of magnetic atoms in the unit cell the distance between the nearest interacting magnetic atoms is small in comparison with the lattice constant and, accordingly, with the wavelength of a spin wave. An effective long-wavelength character thus arises in the problem. Second, there are a large number of wave-vector directions along which many elements of the Hamiltonian matrix vanish by symmetry in the basis which diagonalizes this matrix for k = 0. These matrix elements thus have an additional, angular smallness for arbitrary directions of k. These matrix elements can be taken into account using perturbation theory. As a result, the large elements of the Hamiltonian matrix are few in number, and they can be eliminated by several two-dimensional rotations. Approximate expressions, differing from the computer calculations by ⪅ 10%, are thus obtained for the frequencies. Neutron scattering data are used to find the values of the exchange integrals in YIG and to obtain the magnon spectra. It is shown that in the energy range T ⪅ 260 K only magnons of the lower branch are excited; the spectrum of these “ferromagnons” is quadratic in the wave vector only up to 40 K and becomes linear in the region ωk ⪆ 40 K. For temperatures up to 400 K the temperature dependence of the magnetization is calculated in the spin-wave approximation and good agreement with experimental data is found. A brief review of experimental data on magnon relaxation in YIG is presented. The magnon-magnon interactions which cause the magnon relaxation are described. The amplitude of the four-magnon exchange interaction is determined, and the temperature correction to the frequency is evaluated. This temperature correction is positive, in contrast to the case of simple cubic ferromagnet with nearest-neighbour interaction. The exchange relaxation rate is calculated for normal and umklapp processes. It is shown that the magnetic dipole interaction is important only for the ferromagnons; the amplitude of this interaction and the corresponding relaxation rate are determined. Three-magnon scattering processes are allowed only for wave vectors larger than a certain k/; at k = k/ there is a discontinuity in the wave-vector dependence of the damping. A calculation is given for the nonvanishing contribution to the relaxation at k = 0 on account of scattering processes involving optical magnons; this contribution is due to the local uniaxial anisotropy. The relative role of each of the investigated relaxation mechanisms is discussed, and the correspondence of the present results with the experimental data is examined.

Clustering instability of the spatial distribution of inertial particles in turbulent flows

A theory of clustering of inertial particles advected by a turbulent velocity field caused by an instability of their spatial distribution is suggested. The reason for the clustering instability is a combined effect of the particles inertia and a finite correlation time of the velocity field. The crucial parameter for the clustering instability is a size of the particles. The critical size is estimated for a strong clustering (with a finite fraction of particles in clusters) associated with the growth of the mean absolute value of the particles number density and for a weak clustering associated with the growth of the second and higher moments. A new concept of compressibility of the turbulent diffusion tensor caused by a finite correlation time of an incompressible velocity field is introduced. In this model of the velocity field, the field of Lagrangian trajectories is not divergence-free. A mechanism of saturation of the clustering instability associated with the particles collisions in the clusters is suggested. Applications of the analyzed effects to the dynamics of droplets in the turbulent atmosphere are discussed. An estimated nonlinear level of the saturation of the droplets number density in clouds exceeds by the orders of magnitude their mean number density. The critical size of cloud droplets required for clusters formation is more than

Quantum turbulence in a propagating superfluid vortex front

20 \mu

Anomalous scaling in a model of passive scalar advection: Exact results

m.

Model of intraseasonal oscillations in earth's atmosphere.

We present experimental, numerical, and theoretical studies of a vortex front propagating into a region of vortex-free flow of rotating superfluid 3He-B. We show that the nature of the front changes from laminar through quasiclassical turbulent to quantum turbulent with decreasing temperature. Our experiment provides the first direct measurement of the dissipation rate in turbulent vortex dynamics of 3He-B and demonstrates that the dissipation becomes mutual-friction independent with decreasing temperature, and it is strongly suppressed when the Kelvin-wave cascade on vortex lines is predicted to be involved in the turbulent energy transfer to smaller length scales.

Interaction of Kelvin waves and nonlocality of energy transfer in superfluids

Kraichnan’s model of passive scalar advection in which the driving velocity field has fast temporal decorrelation is studied as a case model for understanding the appearance of anomalous scaling in turbulent systems. We demonstrate how the techniques of renormalized perturbation theory lead (after exact resummations) to equations for the statistical quantities that reveal also non perturbative effects. It is shown that ultraviolet divergences in the diagrammatic expansion translate into anomalous scaling with the inner length acting as the renormalization scale. In this paper we compute analytically the infinite set of anomalous exponents that stem from the ultraviolet divergences. Notwithstanding, non-perturbative effects furnish a possibility of anomalous scaling based on the outer renormalization scale. The mechanism for this intricate behavior is examined and explained in detail. We show that in the language of L’vov, Procaccia and Fairhall [Phys. Rev. E 50, 4684 (1994)] the problem is “critical” i.e. the anomalous exponent of the scalar primary field � = �c. This is precisely the condition

Maximum drag reduction asymptotes and the cross-over to the Newtonian plug

We suggest a way of rationalizing intraseasonal oscillations of Earths atmospheric flow as four meteorologically relevant triads of interacting planetary waves, isolated from the system of all of the rest of the planetary waves. Our model is independent of the topography (mountains, etc.) and gives a natural explanation of intraseasonal oscillations in both the Northern and the Southern Hemispheres. Spherical planetary waves are an example of a wave mesoscopic system obeying discrete resonances that also appears in other areas of physics.

Drag reduction by a linear viscosity profile.

We argue that the physics of interacting Kelvin Waves (KWs) is highly nontrivial and cannot be understood on the basis of pure dimensional reasoning. A consistent theory of KW turbulence in superfluids should be based upon explicit knowledge of their interactions. To achieve this, we present a detailed calculation and comprehensive analysis of the interaction coefficients for KW turbuelence, thereby, resolving previous mistakes stemming from unaccounted contributions. As a first application of this analysis, we derive a local nonlinear (partial differential) equation. This equation is much simpler for analysis and numerical simulations of KWs than the Biot-Savart equation, and in contrast to the completely integrable local induction approximation (in which the energy exchange between KWs is absent), describes the nonlinear dynamics of KWs. Second, we show that the previously suggested Kozik-Svistunov energy spectrum for KWs, which has often been used in the analysis of experimental and numerical data in superfluid turbulence, is irrelevant, because it is based upon an erroneous assumption of the locality of the energy transfer through scales. Moreover, we demonstrate the weak nonlocality of the inverse cascade spectrum with a constant particle-number flux and find resulting logarithmic corrections to this spectrum.

Scale invariant theory of fully developed hydrodynamic turbulence-Hamiltonian approach

We employ the full FEN E-P model of the hydrodynamics of a dilute polymer solution to derive a theoretical approach to drag reduction in wall-bounded turbulence. We recapture the results of a recent simplified theory which derived the universal maximum drag reduction (MDR) asymptote, and complement that theory with a discussion of the cross-over from the MDR to the Newtonian plug when the drag reduction saturates. The FENE-P model gives rise to a rather complex theory due to the interaction of the velocity field with the polymeric conformation tensor, making analytic estimates quite taxing. To overcome this we develop the theory in a computer-assisted manner, checking at each point the analytic estimates by direct numerical simulations (DNS) of viscoelastic turbulence in a channel.

Universal Model of Finite Reynolds Number Turbulent Flow in Channels and Pipes

Drag reduction by polymers in turbulent flows raises an apparent contradiction: the stretching of the polymers must increase the viscosity, so why is the drag reduced? A recent theory proposed that drag reduction, in agreement with experiments, is consistent with the effective viscosity growing linearly with the distance from the wall. With this self-consistent solution the reduction in the Reynolds stress overwhelms the increase in viscous drag. In this Rapid Communication we show, using direct numerical simulations, that a linear viscosity profile indeed reduces the drag in agreement with the theory and in close correspondence with direct simulations of the FENE-P model at the same flow conditions.