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Register a new userChaotic diffusion in multi-scale turbulence
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Turbulence plays a very important role in determining the transport of energy and particles in tokamaks. This work is devoted to studying the chaotic diffusion in multi-scale turbulence in the context of the nonlinear wave-particle interaction. Turbulent waves with different scales of characteristic wavelengths can interact with the same group of charged particles when their phase velocity is close to the velocities of the charged particles. A multi-wavenumber standard mapping is developed to model the chaotic diffusion in multi-scale turbulence. The diffusion coefficient is obtained by calculating the correlation functions analytically. It is found that the contribution of the largest scale turbulence dominates the deviation from the quasi-linear diffusion coefficient. Increasing the overlap parameters of the smaller scale turbulence by just the increasing the wavenumber cannot make the diffusion coefficient to be the quasi-linear diffusion coefficient for a finite wave amplitude. Especially, in two-scale turbulence, the diffusion coefficient is mostly over the quasi-linear diffusion coefficient in the large wavenumber (of the smaller scale turbulence) limit. As more scales of components are added in the turbulence, the diffusion coefficient approaches the quasi-linear diffusion coefficient. The results can also be applied to other resonance-induced multi-scale turbulence in Hamiltonian systems with 1.5 or 2 degrees of freedom.
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We investigate the two-point correlations in the band spectra of spatially periodic systems that exhibit chaotic diffusion in the classical limit. By including level pairs pertaining to non-identical quasimomenta, we define form factors with the winding number as a spatial argument. For times smaller than the Heisenberg time, they are related to the full space-time dependence of the classical diffusion propagator. They approach constant asymptotes via a regime, reflecting quantal ballistic motion, where they decay by a factor proportional to the number of unit cells. We derive a universal scaling function for the long-time behaviour. Our results are substantiated by a numerical study of the kicked rotor on a torus and a quasi-one-dimensional billiard chain.
Particle transport, acceleration and energisation are phenomena of major importance for both space and laboratory plasmas. Despite years of study, an accurate theoretical description of these effects is still lacking. Validating models with self-consistent, kinetic simulations represents today a new challenge for the description of weakly-collisional, turbulent plasmas. We perform two-dimensional (2D) hybrid-PIC simulations of steady-state turbulence to study the processes of diffusion and acceleration. The chosen plasma parameters allow to span different systems, going from the solar corona to the solar wind, from the Earths magnetosheath to confinement devices. To describe the ion diffusion, we adapted the Nonlinear Guiding Center (NLGC) theory to the 2D case. Finally, we investigated the local influence of coherent structures on particle energisation and acceleration: current sheets play an important role if the ions Larmor radii are on the order of the current sheets size. This resonance-like process leads to the violation of the magnetic moment conservation, eventually enhancing the velocity-space diffusion.
Differential rotation is known to suppress linear instabilities in fusion plasmas. However, even in the absence of growing eigenmodes, subcritical fluctuations that grow transiently can lead to sustained turbulence. Here transient growth of electrostatic fluctuations driven by the parallel velocity gradient (PVG) and the ion temperature gradient (ITG) in the presence of a perpendicular ExB velocity shear is considered. The maximally simplified case of zero magnetic shear is treated in the framework of a local shearing box. There are no linearly growing eigenmodes, so all excitations are transient. The maximal amplification factor of initial perturbations and the corresponding wavenumbers are calculated as functions of q/epsilon (=safety factor/aspect ratio), temperature gradient and velocity shear. Analytical results are corroborated and supplemented by linear gyrokinetic numerical tests. For sufficiently low values of q/epsilon (<7 in our model), regimes with fully suppressed ion-scale turbulence are possible. For cases when turbulence is not suppressed, an elementary heuristic theory of subcritical PVG turbulence leading to a scaling of the associated ion heat flux with q, epsilon, velocity shear and temperature gradient is proposed; it is argued that the transport is much less stiff than in the ITG regime.
The chaotic diffusion for particles moving in a time dependent potential well is described by using two different procedures: (i) via direct evolution of the mapping describing the dynamics and ; (ii) by the solution of the diffusion equation. The dynamic of the diffusing particles is made by the use of a two dimensional, nonlinear area preserving map for the variables energy and time. The phase space of the system is mixed containing both chaos, periodic regions and invariant spanning curves limiting the diffusion of the chaotic particles. The chaotic evolution for an ensemble of particles is treated as random particles motion and hence described by the diffusion equation. The boundary conditions impose that the particles can not cross the invariant spanning curves, serving as upper boundary for the diffusion, nor the lowest energy domain that is the energy the particles escape from the time moving potential well. The diffusion coefficient is determined via the equation of the mapping while the analytical solution of the diffusion equation gives the probability to find a given particle with a certain energy at a specific time. The momenta of the probability describe qualitatively the behavior of the average energy obtained by numerical simulation, which is investigated either as a function of the time as well as some of the control parameters of the problem.
We consider the statistical description of steady state fully developed incompressible fluid turbulence at the inertial range of scales in any number of spatial dimensions. We show that turbulence statistics is scale but not conformally covariant, with the only possible exception being the direct enstrophy cascade in two space dimensions. We argue that the same conclusions hold for compressible non-relativistic turbulence as well as for relativistic turbulence. We discuss the modification of our conclusions in the presence of vacuum expectation values of negative dimension operators. We consider the issue of non-locality of the stress-energy tensor of inertial range turbulence field theory.
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