Twodimensional Finite Larmor Radius approximation in canonical gyrokinetic coordinates
Abstract
In this paper, we present some new results about the approximation of the VlasovPoisson system with a strong external magnetic field by the 2D finite Larmor radius model. The proofs within the present work are built by using twoscale convergence tools, and can be viewed as a new slant on previous works of Frénod & Sonnendrücker and Bostan on the 2D finite Larmor Radius model. In a first part, we recall the physical and mathematical contexts. We also recall two main results from previous papers of Frénod & Sonnendrücker and Bostan. Then, we introduce a set of variables which are socalled canonical gyrokinetic coordinates, and we write the Vlasov equation in these new variables. Then, we establish some twoscale convergence and weak* convergence results.
1 Introduction
Nowadays, domestic energy production by using magnetic confinement fusion (MCF) techniques is a huge technological and human challenge, as it is illustrated by the international scientific collaboration around ITER which is under construction in Cadarache (France).
Since magnetic confinement, needed to reach nuclear fusion reaction, is a very complex physical phenomenon, the mathematical models which are linked with this plasma physics subject need to be rigorously studied from theoretical and numerical points of view. Such a work programme based on rigorous mathematical studies and high precision numerical simulations can bring some additional informations about the behavior of the studied plasma before the launch of real experiments.
The present paper can be viewed as a part of the recent work programme about the mathematical justification of the mathematical models which are used for numerical simulations of MCF experiments. Indeed, the first tokamak plasma models have been proposed by Littlejohn, Lee et al., Dubin et al. or Brizard et al. (see [21], [19, 20], [8], [6, 7]) nevertheless most of these models were established by using formal assumptions. For ten years, many mathematicians have been working on mathematical justification of these models, especially the gyrokinetic approaches like guidingcenter approximations and finite Larmor radius approximations: many results in this research field are due to Frénod & Sonnendrücker et al. [10, 12, 13, 14], Golse & SaintRaymond [15, 16], Bostan [5] or, more recently, HanKwan [18]. These mathematical results mostly rely on twoscale convergence theory (see Allaire [3], Nguetseng [25]) or compactness arguments.
In this paper, we are focused on the 2D finite Larmor radius model and its mathematical justification: more precisely, the goal is to make a synthesis of previous mathematical proofs of the convergence of , where
(1.1) 
and where is the solution of the following 2D VlasovPoisson system
(1.2) 
towards the couple which is the solution of the 2D finite Larmor radius model given in [5] (see also Theorem 2 below). The main results on this model are due to Sonnendrücker, Frénod and Bostan, and indicate that somehow weak* converges to the solution of the finite Larmor radius model. However the proofs within these articles are based on various assumptions and use various tools.
The first part of the present paper is devoted to a stateoftheart about the twodimensional finite Larmor radius approximation. Firstly, we recall the procedure which allows us to obtain the dimensionless model (1.2) from the complete VlasovPoisson model by considering specific assumptions. Then we recall the twoscale convergence theorem of Frénod & Sonnendrücker [14] on the one hand, and the weak* convergence theorem of Bostan [5] on the other hand.
In a second part, we introduce a set of variables which are socalled canonical gyrokinetic coordinates and we reformulate the VlasovPoisson system (1.2) in these new variables. Then, we establish a twoscale convergence theorem which only relies on Frénod & Sonnendrücker’s assumptions. Finally, we deduce directly from this result Bostan’s Finite Larmor radius model (see [5]) by adding a somehow nonphysical assumption on the electric field which consists in considering a strong convergence of the sequence .
2 Stateoftheart
2.1 Scaling of the VlasovPoisson model
This paragraph is devoted to the scaling of the following VlasovPoisson model:
(2.3) 
where is the position variable, is the velocity variable, is the time variable, is the ion distribution function, is the electron density, is the selfconsistent electric field generated by the ions and the electrons, is the magnetic field which is applied on the considered plasma, is the electric potential linked with , is the elementary charge and is the elementary mass of an ion.
In this model, the external magnetic field is assumed to be uniform and carried by the unit vector . We also assume that the electron density is given for any . Following the same approach as in Bostan [5], Frénod et al. [10, 14], Golse et al. [15, 16] and HanKwan [18], we add the following assumptions:

The magnetic field is supposed to be strong,

The finite Larmor radius effects are taken into account,

The ion gyroperiod is supposed to be small.
We define the dimensionless variables and unknowns , , , , and by
(2.4) 
In these definitions, is the characteristic length in the direction perpendicular to the magnetic field, is the characteristic length in the direction of the magnetic field, is the characteristic velocity and is the characteristic time. We also rescale the electron density as follows:
(2.5) 
Following the assumptions on the magnetic field , we set as being such that
(2.6) 
Then, we set as the size of the physical device in game. We also link , and with the characteristic Debye length by
(2.7) 
and we take as the characteristic length in the direction perpendicular to the magnetic field, i.e.
(2.8) 
Since we want to take into account the smallness of the gyroperiod and the finite Larmor radius effects, we define the characteristic gyrofrequency and the characteristic Larmor radius as
(2.9) 
With these notations, the VlasovPoisson system is rescaled as follows:
(2.10) 
Taking into account the finite Larmor radius effects consists in considering a regime in which the Larmor radius is of the order of the Debye length. This implies
(2.11) 
Since the magnetic field is assumed to be strong, the Larmor radius is small when compared with the size of the physical domain. Then it is natural to take
(2.12) 
where is small.
Assumption (iii) can be translated in terms of characteristic scales by
(2.13) 
Assumption (i) means that the magnetic force is much stronger than the electric force, so we consider
(2.14) 
Then, removing the primes and adding in subscript, the rescaled VlasovPoisson model writes
(2.15) 
which is the model studied in previous works of Frénod & Sonnendrücker [14], Golse & SaintRaymond [15, 16], and Bostan [5].
2.2 Previous results
In this paragraph, we recall two main results about the asymptotic behavior of the sequences and when goes to 0. The first one is based on the use of twoscale convergence and homogenization techniques developed by Allaire [3] and Nguetseng [25], and was established by Frénod and Sonnendrücker in [14]. The second one relies on compactness arguments and was proved by Bostan in [5]. After recalling these two results, we discuss the main differences between them. These differences are the source of the motivation of the present paper.
In order to simplify, we consider that the whole model (2.15) does not depend on nor , and we assume that for all . Then, it is reduced to a singularly perturbed 2D VlasovPoisson model of the form
(2.16) 
where and .
The following theorem can be attributed to Frénod and Sonnendrücker [14] (see Theorem 1.5) even if the setting of this paper is a charged particle beam VlasovPoisson model not involving any electron density. Nonetheless, the proof of [14] works again in the setting of model (2.16) which involves an electron density.
Theorem 1 (Frénod & Sonnendrücker [14]).
We assume that, for a fixed , is in , is positive everywhere and such that
(2.17) 
We also assume that does not depend on , is in and satisfies
(2.18) 
Then, the sequence is bounded independently of . Furthermore, by extracting some subsequences,
(2.19) 
Moreover, is linked with by the relation
(2.20) 
and is the solution of
(2.21) 
where
(2.22) 
In this theorem, stands for the space of functions being in and periodic with respect to .
As a consequence of this theorem, extracting some subsequences, we have
(2.23) 
where
(2.24) 
By using the relation between and , we can easily remark that is solution of
(2.25) 
We notice that these equations still involve and . In order to make these dependencies disappear, Bostan has proposed in [5] a reformulation of the Vlasov equation in guidingcenter coordinates. Before presenting it, we introduce the sequence defined by
(2.26) 
and, in the same spirit, we define the initial guidingcenter distribution by
(2.27) 
Theorem 2 (Bostan [5]).
We assume that , and that is periodic in and , is positive everywhere and satisfies
(2.28) 
We also assume that there exists such that
(2.29) 
We also assume that admits a strong limit denoted with in . Then, up to a subsequence, weakly* converges to a function in verifying
(2.30) 
where is the solution of
(2.31) 
In this theorem, stands for the space of functions being in and periodic with respect to and .
This last result introduces a mathematical justification of the approximation of the VlasovPoisson model (2.16) by the finite Larmor radius model which is exactly (2.30)(2.31). However, in order to prove this convergence result, Bostan considered stronger assumptions on and than needed to get existence of the weak* limit from Theorem 1: the initial distribution is supposed to be periodic in and and is supposed to admit a strong limit in some Banach space.
3 Convergence result in canonical gyrokinetic coordinates
This section is devoted to a twoscale convergence result for model (2.16) written in a new set of variables which are socalled canonical gyrokinetic coordinates. Then, by adding a nonphysical hypothesis for the electric field , we obtain straightforwardly Bostan’s weak* convergence result.
3.1 Reformulation of Vlasov equation
Following the ideas of Littlejohn [21], Lee [19, 20], and Brizard et al. [6, 7], we define the variables by linking them with by
(3.1) 
This set of variables is socalled canonical gyrokinetic coordinates: indeed, if we define the characteristics linked with by
(3.2) 
where are the characteristics associated with the Vlasov equation (2.16.a), i.e. satisfying
(3.3) 
we have
(3.4) 
where the hamiltonian function is defined by
(3.5) 
and is linked with by the relation
(3.6) 
Then it is straightforward to see that, in the gyrokinetic canonical coordinates, the VlasovPoisson system (2.16) has the following shape:
(3.7) 
where and , and where and are linked with and by
(3.8) 
3.2 Twoscale convergence
We set the following notations
(3.9) 
and we consider the following Banach spaces, involving periodicity with respect to :
and we can state the following theorem.