## Section: New Results

### Waterbag models: analysis and simulations

Participant : Nicolas Besse.

In paper [33] , we revisit the linear theory of kinetic flute-like modes such as ionic instabilities by using the exact geometric reduction of Vlasov equation yielded by waterbag invariants which are reminiscent to the geometric Liouville invariants. The waterbag representation of the statistical distribution function of particles can be viewed as a special class of exact weak solution of the Vlasov equation, allowing to reduce this latter into a set of hydrodynamic equations (with the complexity of a multi-fluid model) while keeping its kinetic features (Landau damping and resonant wave-particle interaction). For high toroidal-number-mode, from ballooning transformation and multi-scale WKB-type analysis, we demonstrate that one can construct eigenmode solutions of the two-dimensional integro-differential gyrowaterbag operator by solving a nested one-dimensional Fredholm-type integral equation. Qualitatively, the solution of the nested one-dimensional Fredholm-type equation is equivalent to first solving for the mode structure along the field lines locally in radius, and then constructing the two-dimensional global mode structure through a radially weighted superposition of local solutions. The radial weighted function is solution of a Schrödinger equation or a Riccati equation in the dual space. Solving the linear turning points problem and using connection formulas, the global dispersion relation arises from the WKB-type phase integral quantization condition which involves the local eigenfrequency. Finally we perform the spectral analysis of the nested one-dimensional Fredholm-type operator which constitutes a meromorphic family of compact operators and extend all the results proved for unstable eigenmodes to stable and damped ones by analytic continuation.

In paper [36] , we present two new codes devoted to the study of ion temperature gradient (ITG) driven plasma turbulence in cylindrical geometry using a drift-kinetic multi-water-bag model for ion dynamics. Both codes were developed to complement the Runge-Kutta semi-lagrangian multi-water-bag code GMWB3D-SLC described in [55] . The CYLGYR code is an eigenvalue solver performing linear stability analysis from given mean radial profiles. It features three resolution schemes and three parallel velocity response models (fluid, multi-water-bag, continuous Maxwellian). The QUALIMUWABA quasi-linear code is an initial value code allowing the study of zonal flow influence on drift-waves dynamics. Cross-validation test performed between the three codes show good agreement on both temporal and spatial characteristics of unstable modes in the linear growth phase.

In paper [32] , we first present the derivation of the anisotropic Lagrangian averaged gyrowaterbag continuum (LAGWBC) equations. The gyrowaterbag continuum can be viewed as a special class of exact weak solution of the Vlasov-gyrokinetic equation, allowing to reduce this latter into an infinite dimensional set of hydrodynamic equations (i.e. an infinite dimensional hyperbolic system of first-order conservation laws in several space dimensions with non-local fluxes) while keeping its kinetic features (Landau damping and nonlinear resonant wave-particle interaction). These models are very promising because they reveal to be very useful for analytical theory (such as the resolution of the eigenvalue problem for analytical description of various instabilities) and numerical simulations (when the continuum is converted into a small finite set of “fluid” or waterbag, the problem has the complexity of a multifluid model instead of a kinetic one) of laser-plasma and gyrokinetic physics (electrostatic turbulence problem). The gyrowaterbag waterbag continuum is derived from two phase-space variable reductions of the Vlasov equation through the existence of two underlying invariants. The first one, coming from physic properties of the dynamics (the fast gyromotion of particles around magnetic field lines) is adiabatic and called the magnetic moment. The second one, named "waterbag” and coming from geometric invariance property of the phase-space, is just the direct consequence of the Liouville Theorem and is reminiscent to the geometric Liouville invariant. In order to obtain the LAGWBC equations from the gyrowaterbag continuum we use an Eulerian variational principle and Lagrangian averaging techniques. Regarding to the original gyrowaterbag continuum, the LAGWBC equations show some additional properties and several advantages from the mathematical and physical viewpoints, which make this model a good candidate for describing accurately gyrokinetic turbulence in magnetically confined plasma. In the second part of this paper we prove local in time well-posedness of an approximated version of the anisotropic LAGWBC equations, that we call the “isotropic” LAGWBC equations, by using quasilinear PDE type methods and elliptic regularity estimates for several operators.