One of the main features of astrophysical shocks is their ability to accelerate particles to extremely high energies. The leading acceleration mechanism, the diffusive shock acceleration, is reviewed. It is demonstrated that its efficiency critically depends on the injection of thermal plasma into acceleration which takes place at the subshock of the collisionless shock structure that, in turn, can be significantly smoothed by energetic particles. Furthermore, their inhomogeneous distribution provides free energy for magnetohydrodynamic (MHD) turbulence regulating the subshock strength and injection rate. Moreover, the MHD turbulence confines particles to the shock front controlling their maximum energy and bootstrapping acceleration. Therefore, the study of the MHD turbulence in a compressive plasma flow near a shock is a key to the understanding of the entire process. The calculation of the injection rate became part of the collisionless shock theory. It is argued that the further progress in diffusive shock acceleration theory is impossible without a significant advance in these two areas of plasma physics.

A two-field model of potential vorticity (PV) staircase structure and dynamics relevant to both beta-plane and drift-wave plasma turbulence is studied numerically and analytically. The model evolves averaged PV whose flux is both driven by and regulates, a potential enstrophy field, $\varepsilon$. The model employs a closure using a mixing length model. Its link to bistability, vital to staircase generation, is analyzed and verified by integrating the equations numerically. Long-time staircase evolution consistently manifests a pattern of meta-stable quasi-periodic configurations, lasting for hundreds of time units, yet interspersed with abrupt ($\Delta t\ll1$) mergers of adjacent steps in the staircase. The mergers occur at the staircase lattice defects where the pattern has not completely relaxed to a strictly periodic solution that can be obtained analytically. Other types of stationary solutions are solitons and kinks in the PV gradient and $\varepsilon$ - profiles. The waiting time between mergers increases strongly as the number of steps in the staircase decreases. This is because of an exponential decrease in inter-step coupling strength with growing spacing. The long-time staircase dynamics is shown numerically be determined by local interaction with adjacent steps. Mergers reveal themselves through the explosive growth of the turbulent PV-flux which, however, abruptly drops to its global constant value once the merger is completed.

An acoustic instability of a shock precursor driven by the pressure gradient of accelerated particles is studied in the nonlinear regime. The nonlinearity steepens unstable acoustic waves and turns them into shocks. The shocks form a "shocktrain" but they may merge into each other. Traveling wave solutions are obtained analytically in two different cases. In the first case, only acoustic instability is included and the characteristic scale (distance between the shocks) is limited only by the system size (while shocks merge). In the second case, the instability develops out of cyclotron unstable seed magnetohydrodynamic (MHD) waves. The spatial distance between the MHD wave packets sets the scale of the acoustic shocktrain. The internal structure of the individual shocks is presumably determined by the ion skin depth c/ωpi and by the relaxation length of slightly superthermal particle distributions near these shocks. The shocks are assumed to be arbitrarily thin compared with the distance between them which is ensured by a small viscous term. Both types of solutions are dynamically verified by numerical calculations. The hydromagnetic flow in the shock precursor emerging from the acoustic instability is crucial for two recently suggested phenomena in diffusive shock acceleration. One phenomenon is the enhancement of the acceleration rate well above its standard (Bohm) value due to the narrowing of the shock precursor. The second phenomenon is the amplification of the long-scale magnetic field by an inverse cascade of Alfvén waves generated by accelerated particles and scattered in k-space on the acoustic perturbations. © 2009. The American Astronomical Society. All rights reserved.

We suggest a physical mechanism whereby the acceleration time of cosmic rays (CRs) by shock waves can be significantly reduced. This creates the possibility of particle acceleration beyond the knee energy at ∼ 10 15 eV. The acceleration results from a nonlinear modification of the flow ahead of the shock supported by particles already accelerated to the knee momentum at p ∼ p *. The particles gain energy by bouncing off converging magnetic irregularities frozen into the flow in the shock precursor and not so much by recrossing the shock itself. The acceleration rate is thus determined by the gradient of the flow velocity and turns out to be formally independent of the particle mean free path (mfp). The velocity gradient is, in turn, set by the knee particles at p ∼ p* as having the dominant contribution to the CR pressure. Since it is independent of the mfp, the acceleration rate of particles above the knee does not decrease with energy, unlike in the linear acceleration regime. The reason for the knee formation at p ∼ p * is that particles with p > p* are effectively confined to the shock precursor only while they are within limited domains in the momentum space, while other particles fall into "loss islands," similar to the "loss cone" of magnetic traps. This structure of the momentum space is due to the character of the scattering magnetic irregularities. They are formed by a train of shock waves that naturally emerge either from unstably growing and steepening magnetosonic waves or as a result of acoustic instability of the CR precursor (CRP). These losses steepen the spectrum above the knee, which also prevents the shock width from increasing with the maximum particle energy. © 2006. The American Astronomical Society. All rights reserved.

Interacting drift wave–zonal flow turbulence is examined at the spectral level of description using an extended “predator–prey” model. Analytic solutions that describe both the linear scaling of transport with ion–ion collisionality as well as the saturation regime are obtained for a simple model of drift wave turbulence. A theory of self-regulation in this system is presented. The possibility of bifurcation to a state with higher turbulence level and transport is demonstrated. This bifurcation is associated with the appearance of a condensate solution at the largest scales. The possible relevance of this phenomenon to the bursting events of turbulence and transport recently observed in gyrokinetic simulations of ITG instability is discussed.

A simple one-field L-H transition model is studied in detail, analytically and numerically. The dynamical system consists of three equations coupling the drift wave turbulence level, zonal flow speed, and the pressure gradient. The fourth component, i.e., the mean shear velocity, is slaved to the pressure gradient. Bursting behavior, characteristic for predator-prey models of the drift wave - zonal flow interaction, is recovered near the transition to the quiescent H-mode (QH) and occurs as strongly nonlinear relaxation oscillations. The latter, in turn, arise as a result of Hopf bifurcation (limit cycle) of an intermediate fixed point (between the L- and H-modes). The system is shown to remain at the QH-mode fixed point even after the heating rate is decreased below the bifurcation point (i.e., hysteresis, subcritical bifurcation), but the basin of attraction of the QH-mode shrinks rapidly with decreasing power. This suggests that the hysteresis in the H-L transition may be less than that expected from S-curve models. Nevertheless, it is demonstrated that by shaping the heating rate temporal profile, one can reduce the average power required for the transition to the QH-mode. © 2009 American Institute of Physics.

The two-field (pressure/density) model for the L→H transition is extended and analyzed qualitatively. In its original form the model is ambiguous as to the location of the transition within the range of bistability of particle and thermal fluxes. Here, the model is regularized by including (i) hyperdiffusion, (ii) time dependence, and (iii) curvature of the pressure profile. The regularizations (i)-(ii) agree and indicate that the Maxwell rule for the forward and back transition applies, as opposed to the maximum flux forward and minimum flux backward transition rules (which yields hysteresis) as suggested previously. Regarding (i)-(ii), simple models suggest that for a pressure gradient driven electric field shear bifurcation, the basic scale of the pedestal is inexorably tied to the particle fueling depth, which normally is the neutral penetration depth. There is no hysteresis predicted by the local model of transport suppression. However, the effect of pressure profile curvature (iii) changes these results substantially. When it dominates, the curvature effect reduces the transition threshold to the lower end of the range of heating power, which falls within the phase coexistence region for both forward and back transitions. This softens the transition threshold requirements. In this limit, the model with pressure curvature also predicts transitions which occur in regimes of flat density and driven exclusively by the temperature gradient. This allows the pedestal to extend beyond the fueling depth, and also allows some decoupling of density and pressure profiles. In a parameter range where the pressure curvature is less important the transition occurs somewhere between the above two limits. © 2008 American Institute of Physics.

We present a theory for the generation of mesoscale (krg ≪ 1, where rg is the cosmic-ray gyroradius) magnetic fields during diffusive shock acceleration. The decay or modulational instability of resonantly excited Alfvén waves scattering off ambient density perturbations in the shock environment naturally generates larger scale fields. For a broad spectrum of perturbations, the physical mechanism of energy transfer is random refraction, represented by the diffusion of Alfvén wave packets in k-space. The scattering field can be produced directly by the decay instability or by the Drury instability, a hydrodynamic instability driven by the cosmic-ray pressure gradient. This process is of interest to acceleration since it generates waves of longer wavelength, and so enables the confinement and acceleration of higher energy particles. This process also limits the intensity of resonantly generated turbulent magnetic fields on rg scales. © 2007. The American Astronomical Society. All rights reserved.

A simple criterion that allows one to determine whether or not a given wave spectrum will generate zonal flows, is derived and analyzed. In the context of a coupled drift wave–zonal turbulence, the results are pertinent to the limit of small zonal flow damping, γd→0, in which previous analyses found that the turbulence vanishes. However, the practically important issue of the drift wave amplitude threshold for zonal flow excitation was not resolved. In its formal mathematical appearance, the criterion obtained is similar to the well-known Penrose criterion that is used for stability analysis of stellar distributions and particle distributions in plasmas. By contrast, the derived criterion, being applied to wave quanta rather than to particle distribution, shows that even “normal” (wave density decaying with wave number) distributions with an intensity above the threshold should generate zonal flows. This clearly points at the ubiquity of the latter.

The predictions of the extended predator-prey model of the coupled spectral dynamics of drift wave-zonal flow turbulence are presented. The model exhibits three possible types of time-dependent solutions, depending on system parameters, which are: (1) quasiperiodic bursting of the transport and turbulence intensity levels; (2) oscillatory relaxation to a stationary state, and in the collisionless limit; (3) an intensity pulse followed by saturation of zonal flow. These solutions are consistent with the time dependent behavior recently observed in the global gyrokinetic simulations. © 2001 American Institute of Physics.