Weak turbulence theory and simulation of the gyro-water-bag model

The thermal confinement time of a magnetized fusion plasma is essentially determined by turbulent heat conduction across the equilibrium magnetic field. To achieve the study of turbulent thermal diffusivities, Vlasov gyrokinetic description of the magnetically confined plasmas is now commonly adopted, and offers the advantage over fluid models (MHD, gyrofluid) to take into account nonlinear resonant wave-particle interactions which may impact significantly the predicted turbulent transport. Nevertheless kinetic codes require a huge amount of computer resources and this constitutes the main drawback of this approach. A unifying approach is to consider the water-bag representation of the statistical distribution function because it allows us to keep the underlying kinetic features of the problem, while reducing the Vlasov kinetic model into a set of hydrodynamic equations, resulting in a numerical cost comparable to that needed for solving multifluid models. The present paper addresses the gyro-water-bag model derived as a water-bag-like weak solution of the Vlasov gyrokinetic models. We propose a quasilinear analysis of this model to retrieve transport coefficients allowing us to estimate turbulent thermal diffusivities without computing the full fluctuations. We next derive another self-consistent quasilinear model, suitable for numerical simulation, that we approximate by means of discontinuous Galerkin methods.

Figure 1

Gyro-water-bag: Phase space plot for a three-bag model.

Figure 2

Gyro-water-bag: Distribution function for a three-bag model.

Figure 3

(Color online) Initial ion density profile.

Figure 4

(Color online) Initial ion temperature profile.

Figure 5

(Color online) Initial $\eta$ profile.

Figure 6

(Color online) $L^2$-norm at $r=r_0$ of the electrical potential, ${\kappa }_n=-0.009$, ${\kappa }_T=-0.067$.

Figure 7

(Color online) $L^2$-norm at $r=r_0$ of the electrical potential, ${\kappa }_n=-0.01$, ${\kappa }_T=-0.1$.

Figure 8

(Color online) Normalized mean heat flux at $r=r_0$, ${\kappa }_n=-0.009$, ${\kappa }_T=-0.067$.

Figure 9

(Color online) Normalized mean heat flux at $r=r_0$, ${\kappa }_n=-0.01$, ${\kappa }_T=-0.1$.

Figure 10

(Color online) Relative error on the $L^1$-norm (or mass) of $f$, ${\kappa }_n=-0.009$, ${\kappa }_T=-0.067$.

Figure 11

(Color online) Relative error on the $L^1$-norm (or mass) of $f$, ${\kappa }_n=-0.01$, ${\kappa }_T=-0.1$.

Figure 12

(Color online) Relative error on the $L^2$-norm of $f$, ${\kappa }_n=-0.009$, ${\kappa }_T=-0.067$.

Figure 13

(Color online) Relative error on the $L^2$-norm of $f$, ${\kappa }_n=-0.01$, ${\kappa }_T=-0.1$.

Figure 14

(Color online) Relative error on the entropy of $f$, ${\kappa }_n=-0.009$, ${\kappa }_T=-0.067$.

Figure 15

(Color online) Relative error on the entropy of $f$, ${\kappa }_n=-0.01$, ${\kappa }_T=-0.1$.

Figure 16

(Color online) Relative error on the total energy, ${\kappa }_n=-0.009$, ${\kappa }_T=-0.067$.

Figure 17

(Color online) Relative error on the total energy, ${\kappa }_n=-0.01$, ${\kappa }_T=-0.1$.