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Waterbags and gyrokinetic-waterbags

Figure 1: Density perturbation of the plasma obtained as a sum over all the contours. This test case deals with the development of an ion-temperature-gradient instability driving the plasma to a turbulent state. From Coulette & Besse (2013b).

We have developed semi-lagrangian and discontinuous-Galerkin schemes for reduced models of waterbag-type (gyrowaterbag, waterbag-{Poisson, Maxwell, quasineutral}) which are multi-fluid models of one to three space dimensions and are similar to hyperbolic systems of conservation laws (Besse & Bertrand 2009; Coulette & Besse 2013a, 2013b). To do this, we have used efficient parallel numerical schemes to run the codes on a large number of processors, which is essential because of the high dimensionality of kinetic models.

Figure 1 provides an example of a simulation of the 3D gyrowaterbag model in cylindrical geometry [GMWB3D-SLCS code (Besse & Bertrand, 2009; Coulette & Besse 2013a, 2013b) with 12 3D-contours of the 4D phase-space]. It illustrates the development of an ion-temperature-gradient instability in cylindrical geometry driving the plasma to a turbulent state and depicts the density perturbation of the plasma which is obtained as a sum over all the contours. As an additional example, Figure 2 and 3 below show respectively, in toroidal geometry, the initial conditions of a simulation and the expected eigenmodes of the gyrowaterbag model.

We next plan to develop a Lagrangian gyrokinetic-waterbag code for the toroidal geometric configuration, based on the tesselation approach currently developped in 6D for cold dark matter dynamics. The method would consist in following the lagrangian evolution of 3D-manifolds in a 4D dimensional phase-space.


  1. N. Besse, P. Bertrand, Gyro-water-bag approach in nonlinear gyrokinetic turbulence, J. Comput. Phys., 228 (2009), 3973-3995
  2. N. Besse, D. Coulette, Asymptotic and spectral analysis of the gyrokinetic-waterbag integro-differential operator in toroidal geometry, J. Math. Phys. 57 (2016), 081518
  3. D. Coulette, N. Besse, Numerical comparisons of gyrokinetic multi-water-bag models, J. Comput. Phys., 248 (2013a), 1-32.
  4. D. Coulette, N. Besse, Multi-water-bag models of ion temperature gradient instability in cylindrical geometry, Phys. Plasmas, 20 (2013b), 052107.
  5. D. Coulette, N. Besse, Numerical resolution of the global-eigenvalue problem for the gyrokinetic-waterbag model in toroidal geometry, J. Plasma Phys., 83 (2017), 905830207

Figure 2: Gyrokinetic-waterbag equilibrium contours in toroidal geometry for the CYCLONE test case. On the left-top (a) using moment equivalence principle, on the right-top (b) following the level line of the continuous Maxwellian distribution. Poloidal view of a closed contour [left-bottom, (c)] and open contour [right-bottom (d)]. From Coulette & Besse (2017).

Figure 3: Poloidal section of eigenmodes of gyrokinetic-waterbag model in toroidal geometry for the CYCLONE test case. On the left computed by the asymptotic method (Besse & Coulette 2016, Coulette & Besse 2017), on the right computed by a quasilinear code (without the approximation induced by the asymptotic methods, Coulette & Besse 2017), at the top the real part of the electrical potential, at the bottom the modulus of the electrical potential.