Theory-Based Transport Modeling

The GA theory group has been a major contributor to the theory of stabilization of ITG modes by ExB velocity shear, in the contexts of both linear stability Staebler 91,Dominguez 93] and turbulence [Waltz 94,Waltz 95,Waltz 98].

Figure 1: GYRO scan (global with kinetic electrons, flat profiles) measuring the
equilibrium ExB shearing rate, , required to quench ion and electron transport.

GA was the first to develop transport models which demonstrated the existence of transport barriers due to the self-consistent feedback between the equilibrium electric field and the turbulent transport [Hinton 91,Hinton 93,Staebler 94] . The impact of ExB velocity shear on the ion thermal transport in both flux-tube gyrofluid simulations [Waltz 94,Waltz 95,Waltz 97] as well as 2-D global fluid simulations [Garbet 96,Garbet 98] was found to be well-modeled by a simple "quench rule". The quench rule states that the effective thermal diffusivity is reduced by the factor , where is the ExB shear rate (proportional to ) and is the maximum local growth rate from the linear gyrokinetic equations without ExB shear. When exceeds the turbulence is completely suppressed (quenched) so the effective thermal diffusivity due to ITG turbulence is zero. The quench rule was determined from ITG simulations with adiabatic electrons. Fig. 1 shows the stabilizing effect of ExB shear on ITG modes with kinetic electrons, as calculated with GYRO. With kinetic electrons the turbulence is found to be quenched at a higher level of ExB shear than for adiabatic electrons. The quench rule was incorporated into an otherwise gyroBohm transport model, GLF23 [Waltz 97]. The complex phenomenology of internal and edge transport barriers has been explored with this model [Kinsey 02], and with earlier simple models [Staebler 94], and found to be consistent in many cases with the quench rule.

In addition to being able to model transport barriers, where the turbulence is quenched, for sub-critical levels of ExB shear, the gyroBohm scaling is broken by the diamagnetic velocity shear contribution to the ExB shear. When the ExB rotation velocity is weak (at the diamagnetic level), the stabilization factor reduces to so that one finds gyroBohm scaling when and Bohm or worse as . We may therefore expect a transport scaling law which is not a simple power-law in ; for example, . GLF23 modeling of recent gyro-radius experiments in co- and counter-rotating DIII-D plasmas has demonstrated that the apparent scaling of the ion heat transport is sensitive to changes in ExB shear stabilization in H-mode plasmas. Transport modeling showed that the apparent scaling of the ion heat transport varied from gyro-Bohm (for co-injection) to Bohm-like for (counter-injection) in agreement with experimental analyses (see Fig. 2). The differences in the observed thermal transport scaling were attributed to differences in the ExB shear between the co- and counter-rotating plasmas. The electron thermal transport followed a gyro-Bohm scaling for both pairs. Although the quench rule gives a simple approximate model for ExB shear suppression it is not reliably predictive since the level of velocity shear required to quench the turbulence depends on parameters like magnetic shear. It may be that linear eigenmodes exist in the presence of ExB shear and that the linear stability point of these eigenmodes would give a calculation of the quench point which followed parametric dependencies. If the ExB shear could be included in the linear eigenmodes then the quasilinear weights for the transport used in GLF23 would include the ExB shear.

In sheared slab geometry this has been shown to be important [Staebler 91]. The parallel and perpendicular viscous stresses have also been computed for ITG modes in sheared slab geometry [Dominguez 93] and found to depend both on ExB and parallel velocity shear with strong off-diagonal terms. These off-diagonal terms were shown to be able to drive toroidal rotation in heated plasmas without momentum injection, a phenomenon which has been recently observed experimentally. These off-diagonal terms can only be computed quasilinearly if the eigenmodes include ExB shear. The turbulence-driven momentum transport has also been shown to be able to alter the poloidal flow [Staebler 04] from its neoclassical value. This is especially important near the separatrix and could have a large impact on the H-mode transition. Thus, there are a number of reasons to develop a theory of toroidal gyrokinetic eigenmodes with ExB shear.

Figure 2: Comparison of measured ion temperature profiles (circles) with the GLF23 transport model with ExB velocity
shear (solid lines) and without (dashed lines) for (a) low- co-rotation, (b) high- co-rotation, (c) low-
counter-rotation, and (d) high- counter-rotation[Petty 02].

A major upgrade to the theory-based transport model GLF23 [Waltz 97] was begun in the last grant period. A shifted-circle electrostatic version is approaching completion. The motivation for the upgrade is provided by the extensive testing against experiment which has identified both the strengths and weaknesses of the theory. GLF23 was constructed with a theory-based philosophy: a model fit to linear and non-linear turbulence theory, rather than directly to experiment. For L-modes and the core of H-mode plasmas, without internal transport barriers, the predicted temperature profiles from GLF23 are in good agreement with tokamak data [Kinsey 03]. This tests the ITG and trapped electron mode (TEM) sectors of the model. The model computes approximate linear eigenmodes by solving a set of gyro-Landau fluid equations using a trial wavefunction, and then uses these eigenmodes to compute the quasilinear energy and particle fluxes. The trial wavefunctions use free parameters which are fit by matching to gyrokinetic growth rates. A saturation rule, which is fit to nonlinear simulations, is used to model the fluctuation amplitude. The temperature profile prediction is more sensitive to the linear threshold than it is to the saturation rule. The ExB velocity shear is included using the quench rule. The broad goals of the next generation of GLF23 are as follows. First, to extend the range of validity of the model to strong shear and reversed shear. This has been accomplished by replacing the trial wavefunction with a Hermite basis function solution which is valid for the full range of plasma parameters. Second, to test the hypothesis that coupling between electron temperature gradient (ETG) and ITG modes produces the unexplained anomalous particle and momentum transport observed within transport barriers where the ITG/TEM modes are quenched. A new set of gyro-Landau fluid equations is under development in order to achieve this goal. The new equations extend the GLF23 equations by including full circulating electron dynamics rather than isothermal passing electrons assumed in GLF23 [Staebler 03].

Figure 3: Growth rates computed using new gyro-Landau fluid
equations. Note that GKS [Kotschenreuther 95] is the linear
forerunner of GS2.

Ions and electrons are coupled so the ion transport due to electron modes can be computed quasilinearly, whereas in GLF23, ETG and ITG modes are uncoupled. The first two goals will be completed in the shifted circle version of the new model. The third goal is to include shaped geometry and finite- electromagnetic modes. These effects are linked, because without the shaped geometry the threshold for the kinetic ballooning mode is not realistic. Completion of the latter goal will require a database of non-linear turbulence simulations in shaped geometry at finite-. These will be used to define a saturation rule. The fourth goal is to include ExB velocity shear into the eigenmodes directly rather than use the quench rule. This is a fundamental theoretical challenge. Progress towards this goal has begun with the derivation of a linearized Vlasov equation with sheared poloidal and toroidal flows in the Vlasov equilibrium [transportbiblio#Staebler 04]. The main obstacle is that it is known that conventional ballooning mode representation cannot be used to reduce the linear stability problem to one dimension. Either a new reduction scheme or a 2-D eigenmode solution needs to be found. GYRO can be used to search for such a 2-D linear eigenmodes with ExB shear. The first step is to use GYRO to establish the existence of local eigenmodes (rather than Floquet modes). If eigenmodes do exist, a method of finding them in a gyro-Landau fluid system of moment equations needs to be found. Once this is done, the quasilinear toroidal and parallel viscous stress components can be computed. This will finally complete the transport matrix required for the evolution of density, energy, toroidal and parallel momentum. With the full transport matrix the theory for spontaneous toroidal rotation in heated tokamaks can be tested. The role of the velocity shear layer at the separatrix in the H-mode transition can also be explored.