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The radial structure of tokamak profiles determined by anomalous transport is elucidated by studying the statistical mechanics of a sand pile automaton for which the toppling conditions depend on local gradient, alone. In this representation, the sand pile dynamics is Markovian, and spatial profiles may be obtained from calculated expectation values of the local gradient. The Markovian structure of the dynamics is exploited to analytically derive a local gradient probability distribution function from a generalized kinetic equation. For homogeneous, weak noise, the calculated expectation value of the gradient is well below the marginally stable state. In the over‐driven limit (i.e., strong homogeneous noise), a region of super‐critical gradient is shown to form near the bottom of the pile. For the case of localized noise, the mean self‐organized profile is always sub‐critical. These results are consistent with numerical studies of simple automata. Their relevance to and implications for tokamak confinement are discussed.

The ideal magnetohydrodynamic ballooning stability of a poloidally rotating plasma in a sheared radial electric field is investigated for an axisymmetric toroidal configuration. An analytical criterion for the ballooning instability of such a system is derived. Electric field shearing is the dominant effect. The sign of the electric field gradient enters the stability criterion via the angular momentum gradient a la Taylor–Couette flow. Thus, Er’

The existence of a critical gradient is an important feature of many transport models. [M. Kotschenreuther et al., Phys. Plasmas 2, 2381 (1995)]. However, fundamental differences exist in the dynamics near marginal stability, depending on whether the transport phenomena are controlled by strict (linear) marginal stability or by a self-organized criticality. One of the most striking differences is in the stiffness of the profiles. In this paper, a sand pile model is used to gain some basic understanding of the stiffness of the profile under different dynamics.

The secondary instability of large scale flows, such as zonal flows and streamers, in toroidal ion temperature gradient turbulence is investigated. It is shown that diamagnetic effects (finite pressure fluctuations), such as those for toroidal ion temperature gradient modes, significantly increase the total Reynolds force. In general, pressure fluctuations have a finite phase shift relative to the electrostatic potential. It is shown that both parts (in-phase and out-of-phase) contribute to the diamagnetic Reynolds stress tensor. Also, the out-of-phase component of the pressure fluctuations is responsible for anomalous energy flux. It is shown here, that a specific contribution of the out-of-phase component to the Reynolds stress tensor allows us to decouple zonal flow dynamics from slow evolution of the ion pressure profile. General equations for the zonal flow and streamer instabilities in toroidal ion temperature gradient turbulence are derived and the growth rates are determined.