Turbulent Transport In Tokamaks Research Articles

Quasilinear gyrokinetic theory: a derivation of QuaLiKiz

In order to predict and analyse turbulent transport in tokamaks, it is important to model transport that arises from microinstabilities. For this task, quasilinear codes have been developed that seek to calculate particle, angular momentum and heat fluxes, both quickly and accurately. In this tutorial, we present a derivation of one such code known as QuaLiKiz, a quasilinear gyrokinetic transport code. The goal of this derivation is to provide a self-contained and complete description of the underlying physics and mathematics of QuaLiKiz from first principles. This work serves both as a comprehensive overview of QuaLiKiz specifically as well as an illustration for deriving quasilinear models in general.

Gyrokinetic particle simulation of electrostatic microturbulence with impurity ions

Impurity is an important factor that can affect significantly turbulent transport in tokamaks. In order to study the impurity physics, we implement a new impurity module in the gyrokinetic particle simulation code GTC (Gyrokinetic Toroidal Code). With an improved numerical scheme, we expand the validity of gyrokinetic Poisson equation in the GTC to the short wavelength region, for both non-zonal and zonal parts of the perturbed Poisson equation. Verifications of this new scheme are carried out on the linear instability and zonal flow response. The linear simulation of the ion temperature gradient (ITG) instability including the impurity ions shows that the new Poisson solver can obtain the correct linear growth rate and frequency at the thermal ion gyro-radius scale. The residual zonal flow with impurities obtained via the new zonal flow solver is consistent with the numerical and analytical predictions in the large aspect-ratio limit. The nonlinear simulation of the ITG turbulence shows that the turbulent transport is significantly reduced by the impurity ions through decreasing the linear growth rate of the instability.

Review of mixing length estimates and effects of toroidicity in a fluid model for turbulent transport in tokamaks

Basic aspects of turbulent transport in toroidal magnetized plasmas are discussed. In particular the fluid closure has strong effects on zonal flows which are needed to create an absorbing boundary for long wave lengths and also to obtain the Dimits nonlinear upshift. The fluid resonance in the energy equation is found to be instrumental for generating the L–H transition, the spin-up of poloidal rotation in internal transport barriers, as well as the nonlinear Dimits upshift. The difference between the linearly fastest growing mode number and the corresponding longer nonlinear correlation length is also addressed. It is found that the Kadomtsev mixing length result is consistent with the non-Markovian diagonal limit of the transport at the nonlinearly obtained correlation length.

Interaction between neoclassical effects and ion temperature gradient turbulence in gradient- and flux-driven gyrokinetic simulations

Neoclassical and turbulent transport in tokamaks has been studied extensively over the past decades, but their possible interaction remains largely an open question. The two are only truly independent if the length scales governing each of them are sufficiently separate, i.e., if the ratio ρ* between ion gyroradius and the pressure gradient scale length is small. This is not the case in particularly interesting regions such as transport barriers. Global simulations of a collisional ion-temperature-gradient-driven microturbulence performed with the nonlinear global gyrokinetic code Gene are presented. In particular, comparisons are made between systems with and without neoclassical effects. In fixed-gradient simulations, the modified radial electric field is shown to alter the zonal flow pattern such that a significant increase in turbulent transport is observed for ρ*≳1/300. Furthermore, the dependency of the flux on the collisionality changes. In simulations with fixed power input, we find that the presence of neoclassical effects decreases the frequency and amplitude of intermittent turbulent transport bursts (avalanches) and thus plays an important role for the self-organisation behaviour.

Review of Mixing Length Estimates and Effects of Toroidicity in a Fluid Model for Turbulent Transport in Tokamaks

Review of Mixing Length Estimates and Effects of Toroidicity in a Fluid Model for Turbulent Transport in Tokamaks

Gyrokinetic particle simulation of microturbulence for general magnetic geometry and experimental profiles

Developments in gyrokinetic particle simulation enable the gyrokinetic toroidal code (GTC) to simulate turbulent transport in tokamaks with realistic equilibrium profiles and plasma geometry, which is a critical step in the code–experiment validation process. These new developments include numerical equilibrium representation using B-splines, a new Poisson solver based on finite difference using field-aligned mesh and magnetic flux coordinates, a new zonal flow solver for general geometry, and improvements on the conventional four-point gyroaverage with nonuniform background marker loading. The gyrokinetic Poisson equation is solved in the perpendicular plane instead of the poloidal plane. Exploiting these new features, GTC is able to simulate a typical DIII-D discharge with experimental magnetic geometry and profiles. The simulated turbulent heat diffusivity and its radial profile show good agreement with other gyrokinetic codes. The newly developed nonuniform loading method provides a modified radial transport profile to that of the conventional uniform loading method.

Transport Bifurcation in a Rotating Tokamak Plasma

The effect of flow shear on turbulent transport in tokamaks is studied numerically in the experimentally relevant limit of zero magnetic shear. It is found that the plasma is linearly stable for all nonzero flow shear values, but that subcritical turbulence can be sustained nonlinearly at a wide range of temperature gradients. Flow shear increases the nonlinear temperature gradient threshold for turbulence but also increases the sensitivity of the heat flux to changes in the temperature gradient, except over a small range near the threshold where the sensitivity is decreased. A bifurcation in the equilibrium gradients is found: for a given input of heat, it is possible, by varying the applied torque, to trigger a transition to significantly higher temperature and flow gradients.

Fusion, space and solar plasmas as complex systems

Complex systems science seeks to identify simple universal models that capture the key physics of extended macroscopic systems, whose behaviour is governed by multiple nonlinear coupled processes that operate across a wide range of spatiotemporal scales. In such systems, it is often the case that energy release occurs intermittently, in bursty events, and the phenomenology can exhibit scaling, that is a significant degree of self-similarity. Within plasma physics, such systems include Earth's magnetosphere, the solar corona and toroidal magnetic confinement experiments. Guided by broad understanding of the dominant plasma processes—for example, turbulent transport in tokamaks or reconnection in some space and solar contexts—one may construct minimalist complex systems models that yield relevant global behaviour. Examples considered here include the sandpile approach to tokamaks and the magnetosphere and a multiple loops model for the solar coronal magnetic carpet. Such models can address questions that are inaccessible to analytical treatment and are too demanding for contemporary computational resources; thus they potentially yield new insights, but risk being simplistic. Central to the utility of these models is their capacity to replicate distinctive aspects of observed global phenomenology, often strongly nonlinear, or of event statistics, for which no explanation can be obtained from first principles considerations such as the underlying equations. For example, a sandpile model, which embodies critical-gradient-triggered avalanching transport associated with nearest-neighbour mode coupling and simple boundary conditions (and little else), can be used to generate some of the distinctive observed elements of tokamak confinement phenomenology such as ELMing and edge pedestals. The same sandpile model can also generate distributions of energy-release events whose distinctive statistics resemble those observed in the auroral zone. Similarly, a multiple loops model, which embodies random footpoint motion combined with reconnection of intersecting loops (and little else), can generate global magnetic field structure resembling the solar coronal magnetic carpet, with power law distributions for energy-release events also similar to those observed in the solar corona. These reduced models thus focus on identifying the key physical ingredients that are necessary and sufficient to generate the observed phenomenology.

GLF23 Modeling of Turbulent Transport in DIII-D

During the past decade, there has been significant progress made in our predictive understanding of turbulent transport in tokamaks. Theoretical advances have led to the development of comprehensive theoretical transport models based on drift wave physics. This paper summarizes the development of the GLF23 drift wave transport model, its application to modeling of DIII-D experiments, and burning plasma projections. The model predicts the transport due to ion temperature gradient, trapped electron, and electron temperature gradient modes and includes the effects of E × B shear flow and Shafranov shift stabilization. GLF23 has been successful in predicting the core profiles in a wide variety of discharges. Examples of published results are given along with a discussion of some outstanding physics issues.

Edge transport and the low-to-high transition in tokamaks with D-shaped magnetic flux surfaces

Nonlinear, three-dimensional numerical simulations of finite-β drift-ballooning turbulence in the edge of tokamaks with D-shaped magnetic flux surfaces are presented. The simulations are based on the reduced Braginskii equations. The turbulent transport in tokamaks with D-shaped magnetic flux surfaces is directly compared to that in tokamaks with circular magnetic flux surfaces. The energy flux can be extremely different depending on the shape of the magnetic flux surfaces. While a tokamak with circular flux surfaces exhibits exceedingly poor confinement, an otherwise identical tokamak with D-shaped flux surfaces is in the high (H) mode where the transport rate is small. The magnitude of the energy flux in these two cases differs by a multiplicative factor larger than 1000.

Description of turbulent transport in tokamaks by invariants

In general, turbulent transport drives a plasma toward a state of turbulent equipartition, in which Lagrangian invariants are uniformly distributed. Different invariants decay with different rates, and in tokamaks the frozen-in law of particles in the poloidal magnetic field survives longer than the corresponding law for the toroidal field, assuming that the trapped particles dominate the turbulent transport. Therefore, the plasma profiles depend on the safety factor q(r), and the condition for convection of trapped particles is that the shear dq/dr is positive. There are two ways to suppress this convection and thereby enhance confinement. The first one is to reverse the magnetic shear. The energy of typical trapped particles then increases outward instead of inward, which suppresses instabilities. The second method is to eliminate the trapped ions by poloidal rotation, and thereby create a transport barrier.

Turbulent equipartition and up-gradient transport

Turbulent transport in tokamaks is often observed to be directed up-gradient. It is proposed that such up-gradient fluxes represent a tendency to approach turbulent equipartition (TEP). At TEP the phase space fluid is well mixed along surfaces defined by those invariants that are not destroyed by the turbulence. This implies that the distribution function f depends only of these invariants, since, according to Liouville's theorem, f is a Lagrangian invariant of the phase space flow. The nature of the TEP depends entirely on what the invariants are. Assuming that the magnetic moment of the particles is conserved, the general form of the transport equations is found for a two-dimensional electrostatic plasma model. These equations describe the relaxation to TEP, and show that in the presence of turbulence there may be fluxes without gradients in the thermodynamic variables. For tokamaks the two first adiabatic invariants of the single-particle motion are assumed to be conserved in the presence of the turbulence. This leads to the density profile n ~ 1/q of trapped particles, which has a maximum in the centre. The proposed model implies that trapped particle modes dominate turbulent transport in tokamaks. Since negative shear stabilizes these modes, it is predicted that negative shear improves confinement significantly.

Edge gradient and safety-factor effects on electrostatic turbulent transport in tokamaks.

Electrostatic turbulence and transport has been measured in the tokamak edge as the safety factor (and, concomitantly, the energy-confinement time) and the edge equilibrium gradients are varied substantially. The turbulence is largely independent of safety factor, but strongly sensitive to the local-density gradient, which itself depends upon the limiter configuration.