Reversal of the Parallel Drift Frequency in Anomalous Transport of Impurity Ions.

It is found that, in the studies of heavy ion transport with gyrokinetic simulations, the ion parallel drift frequency can reverse sign in velocity space when the amplitude variation of the electrostatic potential fluctuation is strong along the magnetic field line. As a result, the particle transport related to the parallel dynamics is strongly enhanced. It is noted that, while parallel gradient of the fluctuation amplitude can be instigated by a large magnetic shear or safety factor in a tokamak, the generic mechanism is independent of its cause, which suggests broader applications to kinetic plasma problems. Some relevant topics are briefly addressed in the end.

Transition from linear Landau damping to nonlinear Bernstein-Greene-Kruskal modes via phase synchronization.

Dynamics of the transition from a linear plasma wave to a nonlinear state characterized by the Bernstein-Greene-Kruskal mode is studied within the framework of the Vlasov-Poisson system. In the linear stage, the plasma distribution function (f) develops finer and finer structures in velocity space through a series of "mixing" processes leading to the Landau damping of the plasma wave. These mixing processes inevitably result in strong phase irregularities in velocity space. Using numerical simulations, it was observed that starting from the wave-particle resonance region, this irregular phase pattern gets "smoothed out" through a process of spreading of phase synchronization, which tends to reduce Landau damping, facilitating the formation of the nonlinear plasma wave as a fully synchronized final state. It is also found that there exists a residual damping for the quasisteady nonlinear wave when the phases of the particles are not fully synchronized.

Spiral chain models of two-dimensional turbulence.

Reduced models, mirroring self-similar, fractal nature of two-dimensional turbulence, are proposed, using logarithmic spiral chains, which provide a natural generalization of shell models to two dimensions. In a turbulent cascade, where each step can be represented by a rotation and a scaling of the interacting triad, the use of a spiral chain whose nodes can be obtained by scaling and rotating an original wave vector provides an interesting perspective. A family of such spiral chain models depending on the distance of interactions can be obtained by imposing a logarithmic spiral grid with a constant divergence angle and a constant scaling factor and imposing the condition of exact triadic interactions. Scaling factors in such sequences are given by the square roots of known ratios such as the plastic ratio, the super-golden ratio, or some small Pisot numbers. While spiral chains can represent monofractal models of a self-similar cascade, which can span a large range of wave numbers and have good angular coverage, it is also possible that spiral chains or chains of consecutive triads play an important role in the cascade. As numerical models, the spiral chain models based on decimated Fourier coefficients have the usual problems of shell models of two-dimensional turbulence such as the dual cascade being overwhelmed by statistical chain equipartition due to an almost stochastic evolution of the complex phases. A generic spiral chain model based on evolution of energy is proposed, which is shown to recover the dual cascade behavior in two-dimensional turbulence.

Nested polyhedra model of isotropic magnetohydrodynamic turbulence.

A nested polyhedra model has been developed for magnetohydrodynamic turbulence. Driving only the velocity field at large scales with random, divergence-free forcing results in a clear, stationary k^{-5/3} spectrum for both kinetic and magnetic energies. Since the model naturally effaces disparate scale interactions, does not have a guide field, and avoids injecting any sign of helicity by random forcing, the resulting three-dimensional k spectrum is statistically isotropic. The strengths and weaknesses of the model are demonstrated by considering large or small magnetic Prandtl numbers. It was also observed that the timescale for the equipartition offset with those of the smallest scales shows a k^{-1/2} scaling.

Nested polyhedra model of turbulence.

A discretization of the wave-number space is proposed, using nested polyhedra, in the form of alternating dodecahedra and icosahedra that are self-similarly scaled. This particular choice allows the possibility of forming triangles using only discretized wave vectors when the scaling between two consecutive dodecahedra is equal to the golden ratio and the icosahedron between the two dodecahedra is the dual of the inner dodecahedron. Alternatively, the same discretization can be described as a logarithmically spaced (with a scaling equal to the golden ratio), nested dodecahedron-icosahedron compounds. A wave vector which points from the origin to a vertex of such a mesh, can always find two other discretized wave vectors that are also on the vertices of the mesh (which is not true for an arbitrary mesh). Thus, the nested polyhedra grid can be thought of as a reduction (or decimation) of the Fourier space using a particular set of self-similar triads arranged approximately in a spherical form. For each vertex (i.e., discretized wave vector) in this space, there are either 9 or 15 pairs of vertices (i.e., wave vectors) with which the initial vertex can interact to form a triangle. This allows the reduction of the convolution integral in the Navier-Stokes equation to a sum over 9 or 15 interaction pairs, transforming the equation in Fourier space to a network of "interacting" nodes that can be constructed as a numerical model, which evolves each component of the velocity vector on each node of the network. This model gives the usual Kolmogorov spectrum of k^{-5/3}. Since the scaling is logarithmic, and the number of nodes for each scale is constant, a very large inertial range (i.e., a very high Reynolds number) with a much lower number of degrees of freedom can be considered. Incidentally, by assuming isotropy and a certain relation between the phases, the model can be used to systematically derive shell models.

Logarithmic discretization and systematic derivation of shell models in two-dimensional turbulence.

A detailed systematic derivation of a logarithmically discretized model for two-dimensional turbulence is given, starting from the basic fluid equations and proceeding with a particular form of discretization of the wave-number space. We show that it is possible to keep all or a subset of the interactions, either local or disparate scale, and recover various limiting forms of shell models used in plasma and geophysical turbulence studies. The method makes no use of the conservation laws even though it respects the underlying conservation properties of the fluid equations. It gives a family of models ranging from shell models with nonlocal interactions to anisotropic shell models depending on the way the shells are constructed. Numerical integration of the model shows that energy and enstrophy equipartition seem to dominate over the dual cascade, which is a common problem of two-dimensional shell models.

Wave-number spectrum of dissipative drift waves and a transition scale.

We study the steady state spectrum of the Hasegawa-Wakatani (HW) equations that describe drift wave turbulence. Beyond a critical scale k_{c}, which appears as a balance between the nonlinear time and the parallel conduction time, the adiabatic electron response breaks down nonlinearly and an internal energy density spectrum of the form F(k_{⊥})∝k_{⊥}^{-3}, associated with the background gradient, is established. More generally a dual power law spectrum, approximately of the form F(k_{⊥})∝k_{⊥}^{-3}(k_{c}^{-2}+k_{⊥}^{-2}) is obtained, which captures this transition. Using dimensional analysis, an expression of the form k_{c}∝C/κ is derived for the transition scale, where C and κ are normalized parameters of the HW equations signifying the electron adiabaticity and the density gradient, respectively. The results are numerically confirmed using a shell model developed and used for the Hasegawa-Wakatani system.

Finding the elusive E×B staircase in magnetized plasmas.

Turbulence in hot magnetized plasmas is shown to generate permeable localized transport barriers that globally organize into the so-called "ExB staircase" [G. Dif-Pradalier et al., Phys. Rev. E, 82, 025401(R) (2010)]. Its domain of existence and dependence with key plasma parameters is discussed theoretically. Based on these predictions, staircases are observed experimentally in the Tore Supra tokamak by means of high-resolution fast-sweeping X-mode reflectometry. This observation strongly emphasizes the critical role of mesoscale self-organization in plasma turbulence and may have far-reaching consequences for turbulent transport models and their validation.

Elasticity in drift-wave-zonal-flow turbulence.

We present a theory of turbulent elasticity, a property of drift-wave-zonal-flow (DW-ZF) turbulence, which follows from the time delay in the response of DWs to ZF shears. An emergent dimensionless parameter |〈v〉'|/Δωk is found to be a measure of the degree of Fickian flux-gradient relation breaking, where |〈v〉'| is the ZF shearing rate and Δωk is the turbulence decorrelation rate. For |〈v〉'|/Δωk>1, we show that the ZF evolution equation is converted from a diffusion equation, usually assumed, to a telegraph equation, i.e., the turbulent momentum transport changes from a diffusive process to wavelike propagation. This scenario corresponds to a state very close to the marginal instability of the DW-ZF system, e.g., the Dimits shift regime. The frequency of the ZF wave is ΩZF=±Î³d1/2γmodu1/2, where γd is the ZF friction coefficient and γmodu is the net ZF growth rate for the case of the Fickian flux-gradient relation. This insight provides a natural framework for understanding temporally periodic ZF structures in the Dimits shift regime and in the transition from low confined mode to high confined mode in confined plasmas.

Physics of stimulated L→H transitions.

We report on model studies of stimulated L→H transitions. These studies use a novel reduced mesoscale model. Studies reveal that L→H transitions can be triggered by particle injection into a subcritical state (i.e., P

How the propagation of heat-flux modulations triggers E × B flow pattern formation.

We propose a novel mechanism to describe E×B flow pattern formation based upon the dynamics of propagation of heat-flux modulations. The E × B flows of interest are staircases, which are quasiregular patterns of strong, localized shear layers and profile corrugations interspersed between regions of avalanching. An analogy of staircase formation to jam formation in traffic flow is used to develop an extended model of heat avalanche dynamics. The extension includes a flux response time, during which the instantaneous heat flux relaxes to the mean heat flux, determined by symmetry constraints. The response time introduced here is the counterpart of the drivers' response time in traffic, during which drivers adjust their speed to match the background traffic flow. The finite response time causes the growth of mesoscale temperature perturbations, which evolve to form profile corrugations. The length scale associated with the maximum growth rate scales as Δ(2) ~ (v(thi)/λT(i))ρ(i)sqrt[χ(neo)τ], where λT(i) is a typical heat pulse speed, χ(neo) is the neoclassical thermal diffusivity, and τ is the response time of the heat flux. The connection between the scale length Δ(2) and the staircase interstep scale is discussed.

Effect of sheared flow on the growth rate and turbulence decorrelation.

The effect of a large scale flow shear on a linearly unstable turbulent system is considered. A cubic equation describing the effective growth rate is obtained, which is shown to reduce to well-known forms in weak and strong shear limits. A shear suppression rule is derived which corresponds to the point where the effective growth rate becomes negative. The effect of flow shear on nonlinear mode coupling of drift or Rossby waves is also considered, and it is shown that the resonance manifold shrinks and weakens as the vortices are sheared. This leads to a reduction of the efficiency of three-wave interactions. Tilted eddies can then only couple to the large scale sheared flows, because the resonance condition for that interaction is trivially satisfied. It is argued that this leads to absorbtion of the sheared vortices by large scale flow structures. Studying the form of the effective growth rate for weak shear, it was shown that in addition to reducing the overall growth rate, a weak flow shear also reduces the wave number where the fluctuations are most unstable.

Edge temperature gradient as intrinsic rotation drive in Alcator C-Mod tokamak plasmas.

Intrinsic rotation has been observed in I-mode plasmas from the C-Mod tokamak, and is found to be similar to that in H mode, both in its edge origin and in the scaling with global pressure. Since both plasmas have similar edge ∇T, but completely different edge ∇n, it may be concluded that the drive of the intrinsic rotation is the edge ∇T rather than ∇P. Evidence suggests that the connection between gradients and rotation is the residual stress, and a scaling for the rotation from conversion of free energy to macroscopic flow is calculated.

Anisotropic shell model of turbulence.

An anisotropic shell model has been proposed for two-dimensional (2D) turbulence. It is similar to the 2D version of the Gledzer-Ohkitani-Yamada model but with the angular variable in wave-number space divided into three distinct directions representing structures elongated in different directions. In the case when the drive is isotropic the usual isotropic solution is recovered as the fixed point of this model. The Hasegawa-Mima limit of the model is considered in particular due to its relevance for 2D anisotropic systems such a quasigeostrophic and plasma turbulence. It is observed from this simple model that the anisotropy diminishes as a function of scale during the cascade process, and the maximum of the energy is not at the node that has maximum drive, but at a nearby node that is directly coupled to that one.

Wave-number spectrum of drift-wave turbulence.

A simple model for the evolution of turbulence fluctuation spectra, which includes neighboring interactions leading to the usual dual cascade as well as disparate scale interactions corresponding to refraction by large scale structures, is derived. The model recovers the usual Kraichnan-Kolmogorov picture in the case of exclusively local interactions and midrange drive. On the other hand, when disparate scale interactions are dominant, a simple spectrum for the density fluctuations of the form |nk|2 proportional to k(-3)/(1+k2)2 is obtained. This simple prediction is then compared to, and found to be in fair agreement with, Tore Supra CO2 laser scattering data.

Turbulence in the TORE SUPRA tokamak: measurements and validation of nonlinear simulations.

Turbulence measurements in TORE SUPRA tokamak plasmas have been quantitatively compared to predictions by nonlinear gyrokinetic simulations. For the first time, numerical results simultaneously match within experimental uncertainty (a) the magnitude of effective heat diffusivity, (b) rms values of density fluctuations, and (c) wave-number spectra in both the directions perpendicular to the magnetic field. Moreover, the nonlinear simulations help to revise as an instrumental effect the apparent experimental evidence of strong turbulence anisotropy at spatial scales of the order of ion-sound Larmor radius.

Toroidal rotation driven by the polarization drift.

Starting from a phase space conserving gyrokinetic formulation, a systematic derivation of parallel momentum conservation uncovers a novel mechanism by which microturbulence may drive intrinsic rotation. This mechanism, which appears in the gyrokinetic formulation through the parallel nonlinearity, emerges due to charge separation induced by the polarization drift. The derivation and physical discussion of this mechanism will be pursued throughout this Letter.

Turbulent equipartition and homogenization of plasma angular momentum.

A physical model of turbulent equipartition (TEP) of plasma angular momentum is developed. We show that using a simple, model insensitive ansatz of conservation of total angular momentum, a TEP pinch of angular momentum can be obtained. We note that this term corresponds to a part of the pinch velocity previously calculated using quasilinear gyrokinetic theory. We observe that the nondiffusive TEP flux is inward, and therefore may explain the peakedness of the rotation profiles observed in certain experiments. Similar expressions for linear toroidal momentum and flow are computed and it is noted that there is an additional effect due the radial profile of moment of inertia density.

Nonlinear triad interactions and the mechanism of spreading in drift-wave turbulence.

We present the results of a derivation of the fluctuation energy transport matrix for the two-field Hasegawa-Wakatani model of drift wave turbulence. The energy transport matrix is derived from a two-scale direct interaction approximation assuming weak turbulence. We examine different classes of triad interactions and show that radially extended eddies, as occurs in penetrative convection, are the most effective in turbulence spreading. We show that in the near-adiabatic limit internal energy spreads faster than the kinetic energy. Previous theories of spreading results are discussed in the context of weak turbulence theory.