There are numerous applications in physics, statistics and electrical circuit simulation where it is required to bound entries and the trace of the inverse and the determinant of a large sparse matrix. All these computational tasks are related to the central mathematical problem studied in this paper, namely, bounding the bilinear form uXf(A)v for a given matrix A and vectors u and v, wherefis a given smooth function and is defined on the spectrum of A. We will study a practical numerical algorithm for bounding the bilinear form, where the matrix A is only referenced through matrix-vector multiplications. A Monte Carlo method is also presented to efficiently estimate the trace of the inverse and the determinant of a large sparse matrix.

Tokamak transport modeling scenarios, including ITER [ITER Physics Basis Editors, Nucl. Fusion 39, 2137 (1999)] performance predictions, are based exclusively on reduced models for core thermal and particle transport. The reason for this is simple: computational cost. A typical modeling scenario may require the evaluation of thousands of individual transport fluxes (local transport models calculate the energy and particle fluxes across a specified flux surface given fixed profiles). Despite continuous advances in direct gyrokinetic simulation, the cost of an individual simulation remains so high that direct gyrokinetic transport calculations have been avoided. By developing a steady-state iteration scheme suitable for direct gyrokinetic and neoclassical simulations, we can now compute steady-state temperature profiles for DIII-D [J. L. Luxon, Nucl. Fusion 42, 614 (2002)] plasmas given known plasma sources. The new code, TGYRO, encapsulates the GYRO [J. Candy and R. E. Waltz, J. Comput. Phys. 186, 545 (2003)] code, for turbulent transport, and the NEO [E. A. Belli and J. Candy, Plasma Phys. Controlled Fusion 50, 095010 (2008)] code, for kinetic neoclassical transport. Results for DIII-D L-mode discharge 128913 are given, with computational and experimental results consistent in the region 0≤r/a≤0.8.

Electron temperature gradient (ETG) transport is conventionally defined as the electron energy transport at high wave number (high-k) where ions are adiabatic and there can be no ion energy or plasma transport. Previous gyrokinetic simulations have assumed adiabatic ions (ETG-ai) and work on the small electron gyroradius scale. However such ETG-ai simulations with trapped electrons often do not have well behaved nonlinear saturation unless fully kinetic ions (ki) and proper ion scale zonal flow modes are included. Electron energy transport is separated into ETG-ki at high-k and ion temperature gradient-trapped electron mode (ITG/TEM) at low-k. Expensive (more computer-intensive), high-resolution, large-ion-scale flux-tube simulations coupling ITG/TEM and ETG-ki turbulence are presented. These require a high effective Reynolds number R≡[k(max)∕k(min)]2=μ2, where μ=[ρsi∕ρsi] is the ratio of ion to electron gyroradii. Compute times scale faster than μ3. By comparing the coupled expensive simulations with (1) much cheaper (less compute-intensive), uncoupled, high-resolution, small, flux-tube ETG-ki and with (2) uncoupled low-resolution, large, flux-tube ITG/TEM simulations, and also by artificially turning “off” the low-k or high-k drives, it appears that ITG/TEM and ETG-ki transport are not strongly coupled so long as ETG-ki can access some nonadiabatic ion scale zonal flows and both high-k and low-k are linearly unstable. However expensive coupled simulations are required for physically accurate k-spectra of the transport and turbulence. Simulations with μ⩾30 appear to represent the physical range μ>40. ETG-ki transport measured in ion gyro-Bohm units is weakly dependent on μ. For the mid-radius core tokamak plasma parameters studied, ETG-ki is about 10% of the electron energy transport, which in turn is about 30% of the total energy transport (with negligible E×B shear). However at large E×B shear sufficient to quench the low-k ITG/TEM transport, the high-k tail of the ETG-ki transport survives. Decreasing the trapping to minimize the TEM opens a stability gap between ITG and ETG. High-k ETG transport driven by low-k ITG instability in an ETG linearly stable plasma is demonstrated.

This work reports on numerical studies of small-scale electrontemperature-gradient (ETG) turbulence embedded in large-scale turbulence driven by both ion-temperature-gradient (ITG) modes and trapped-electron modes (TEM). To begin with, we find that the simplified adiabatic-ion model of ETG turbulence does not always saturate nonlinearly, suggesting that corrections to the purely adiabatic ion response are required for robust saturation. Our results also qualitatively confirm a prediction of Holland and Diamond that the back-reaction of ETG on ITG turbulence is insignificant. For the parameters studied, we find that ETG turbulence levels are reduced as ion driving gradients are increased. This result is at least partially explained by linear physics. An important practical result of this work is the finding that most of the electron energy transport arises from ion scales (k θ ρ i < 1) in cases for which ionscale instabilities are not suppressed. Specifically, for the Cyclone base case parameters, only 16% of the energy diffusivity, χ e , arises from the spectral region k θ ρ i > 1.0.

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