On advanced fluid modelling of drift wave turbulence

The Dupree-Weinstock renormalization is used to prove that a reactive closure exists for drift wave turbulence in magnetized plasmas. The result is used to explain recent results in gyrokinetic simulations and is also related to the Mattor-Parker closure. The level of closure is found in terms of applied external sources. A major complication in the description of drift-wave turbulence in magnetic confinement systems is that phase velocities of the turbulent perturbations are comparable to thermal velocities of the plasma. Although this is true both parallel and perpendicularly to the magnetic field, the perpendicular (magnetic drift) resonance has turned out to play a particularly important role in particular in H-mode plasmas with flat density profiles. Although the development of nonlinear gyrokinetic s has been rapid, it is still not feasible to use gyrokinetic codes as transport codes because of the computation time needed. Thus for predictive transport simulations we still need fluid models. In order to deal with the wave particle resonances, advanced fluid models were developed in the end of the 1980’s and 1990’s. We will here define advanced fluid models as fluid models which give a rule for treating the wave particle resonances in such a way that the model can be used near these resonances also in collisionless plasmas. The first model designed to do this was the advanced reactive fluid model5 and later the gyro fluid models in the US followed. The improvement of the description of tokamak transport was dramatic8. The most significant feature was that a new regime, the ‘flat density regime’, which typically persists in the major part of L-mode plasmas and all the way out to the edge pedestal in H-mode plasmas, could now be described and the problem with the radial variation of drift wave transport coefficients was resolved. Thus a general feature of all advanced fluid models is that the description of experimental results is much better than with previous standard fluid models. The main difference between the reactive model and gyrofluid models is that gyro-fluid models add linear dissipative kinetic resonances thus getting a dissipative closure. Such models originally got too large transport as compared to gyro-kinetic models. The main reference is here from the Cyclone project. In comparisons with gyro-kinetic models the question of which is the better closure obviously depends on to which extent nonlinearities in velocity space will modify or remove the dissipative linear kinetic resonances. Here, clearly, it is essential to keep velocity space nonlinearities4. These are often neglected in nonlinear Vlasov codes with the argument that they are small. When these are ignored the difference to quasilinear codes is usually rather small. A simple example of when linear resonances are completely removed due to flattening in velocity space is the bump in tail instability. In this case the bump is removed completely due to simple quasilinear diffusion in velocity space. It seems unlikely that simple quasilinear diffusion could give complete flattening of the bulk distribution since the whole distribution can not be flattened and since a local flattening would give steeper gradients for neighboring velocities. It has recently been pointed out, however, that a complete flattening is not needed to remove dissipative resonances which are due only to moments higher than those included self-consistently. This is because the full kinetic resonance involves all fluid resonances while in advanced fluid models the lowest order fluid resonances are kept unexpanded and treated self consistently. A significant effect, in kinetic theory, is, however played by the nonlinear frequency shift. It may shift the mode frequency between regions with negative and positive wave energy in such a way that dissipative wave particle resonances change sign. Thus the dissipative kinetic resonance can be averaged out in a way very similar to particle trapping without any change in the distribution function. This was actually the reason for the very strong difference between the linear and nonlinear closures in the Mattor-Parker work. The Mattor-Parker system v f ∂ ∂ ∇ φ was later generalized to include effects of background turbulence through a diffusion term. In the present paper we will generalize the Mattor-Parker work to include the kinetic nonlinearities and to show that wave-particle interaction can be seen as a collision between wave and particle where both change their velocity so that overall momentum is conserved. The change in wave phase velocity is due to a nonlinear frequency shift. We will also show that a renormalization leads to a situation where a reactive closure is valid, In such a state energy is clearly conserved. In order to have conservation of energy, a kinetic code must include the nonlinear response to linear wave-particle resonant interaction. This has been generally done in Particle In Cell (PIC) codes but not always in Vlasov codes. Since Finite Larmor Radius (FLR) effects are not an essential part of the following discussion we will here start from the drift-kinetic equation.