ON THE VLASOV-MAXWELL SYSTEM WITH A STRONG EXTERNAL MAGNETIC FIELD

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ON THE VLASOV-MAXWELL SYSTEM WITH A STRONG EXTERNAL MAGNETIC FIELD

Francis Filbet, Tao Xiong, Eric Sonnendr¨ucker

To cite this version:

Francis Filbet, Tao Xiong, Eric Sonnendr¨ucker. ON THE VLASOV-MAXWELL SYSTEM WITH A STRONG EXTERNAL MAGNETIC FIELD. SIAM Journal on Applied Mathematics, Society for Industrial and Applied Mathematics, 2018, 78 (2), pp.1030-1055. �hal-01437310�

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MAGNETIC FIELD

FRANCIS FILBET, TAO XIONG AND ERIC SONNENDR ¨UCKER

Abstract. This paper establishes the long time asymptotic limit of the 2d×3d Vlasov-Maxwell system with a strong external magnetic field. Hence, a guiding center approximation is obtained in the two dimensional case with a self-consistent electromagnetic field given by Poisson type equations.

Then, we propose a high order approximation of the asymptotic model and perform several numerical experiments which provide a solid validation of the method and illustrate the effect of the self-consistent magnetic field on the current density.

Keywords. Asymptotic limit; High order scheme; Vlasov-Maxwell system; Finite difference methods.

Contents

1. Introduction 1

2. Mathematical modeling 3

2.1. Rescaling of the Vlasov-Maxwell system 3

2.2. Asymptotic limit of the Vlasov-Maxwell system 5

2.3. Weak solutions for the asymptotic model 10

2.4. Guiding center model & linear instability 13

3. Numerical scheme 14

3.1. Hermite WENO finite difference scheme for the transport equation 14

3.2. Discretization of the elliptic equations 16

4. Numerical Examples 16

4.1. Linear equation 16

4.2. Diocotron instability 17

4.3. Kelvin-Helmholtz instability 20

5. Conclusion 23

Acknowledgement 24

References 25

1. Introduction

We consider a plasma confined by a strong external magnetic field, hence the charged gas evolves under its self-consistent electromagnetic field and the confining magnetic field. This configuration is typical of a tokamak plasma [3, 30], where the magnetic field is used to confine particles inside the core of the device.

We assume that on the time scale we consider, collisions can be neglected both for ions and electrons, hence collective effects are dominant and the plasma is entirely modelled with kinetic transport equations, where the unknown is the number density of particles f ≡ f(t,x,v) depending on time t≥0, position x∈D⊂R³ and velocityv∈R³.

Such a kinetic model provides an appropriate description of turbulent transport in a fairly general context, but it requires to solve a six dimensional problem which leads to a huge computational cost.

To reduce the cost of numerical simulations, it is classical to derive asymptotic models with a smaller number of variables than the kinetic description. Large magnetic fields usually lead to the so-called drift-kinetic limit [1, 8, 28, 27] and we refer to [4, 7, 19, 20, 14, 21] for recent mathematical results

1

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on this topic. In this regime, due to the large applied magnetic field, particles are confined along the magnetic field lines and their period of rotation around these lines (called the cyclotron period) becomes small. It corresponds to the finite Larmor radius scaling for the Vlasov-Poisson equation, which was introduced by Fr´enod and Sonnendr¨ucker in the mathematical literature [19, 20]. The two-dimensional version of the system (obtained when one restricts to the perpendicular dynamics) and the large magnetic field limit were studied in [14] and more recently in [4, 31, 24]. We also refer to the recent work [26] of Hauray and Nouri, dealing with the well-posedness theory with a diffusive version of a related two dimensional system. A version of the full three dimensional system describing ions with massless electrons was studied by Han-Kwan in [23, 25].

Here we formally derive a new asymptotic model under both assumptions of large magnetic field and large time asymptotic limit for the two dimensional in space and three dimensional in velocity (2d×3d) Vlasov-Maxwell system. An analogous problem for the Vlasov-Poisson system has already been carefully studied by F. Golse and L. Saint-Raymond in two dimension [21, 32, 22], and recently by P. Degond and F. Filbet in three dimension [12]. In this paper, we will follow [12] to introduce some main characteristic scales to rewrite the Vlasov-Maxwell system in a dimensionless form, and reformulate the Maxwell equations by defining two potential functions corresponding to the self-consistent electromagnetic field. We consider a small cyclotron period, where the plasma frequency is relatively small as compared to the cyclotron frequency, and study the long time behavior of the plasma. As- suming a constant strong external magnetic field and that the distribution function is homogeneous along the external magnetic field, an asymptotic kinetic model can be derived by performing Hilbert expansions and comparing the first three leading order terms in terms of the small cyclotron period, thanks to passing in the cylindrical coordinates. The new asymptotic model is composed of two two dimensional transport equations for the distribution functions of ions and electrons respectively, averaging in the velocity plane orthogonal to the external magnetic field, and a Poisson’s equation for determining the electric potential as well as an elliptic equation for the magnetic potential. It is incompressible with a divergence free transport velocity and shares several good features with the original Vlasov-Maxwell system, such as conservation of moments in velocity, total energy, as well as theL^p norm and physical bounds. The existence of weak solutions for the asymptotic model can also be obtained by following the lines of existence of weak solutions for the Vlasov-Poisson system [2, 13], with someL^p estimates on the charge density and current. Moreover, as the Mach number goes to 0 in the self-consistent magnetic field, we can recover the two dimensional guiding-center model, which is an asymptotic model for the Vlasov-Poisson system under the same scalings [21, 36, 29].

A high order numerical scheme will be proposed for solving the new asymptotic model, which is an extension of the one developed by C. Yang and F. Filbet [36] for the two dimensional guiding center model. Some other recent numerical methods for the Vlasov-Poisson system or the two dimensional guiding-center model can be referred to [15, 34, 10, 9, 18, 16, 35] and reference therein. Here a Hermite weighted essentially non-oscillatory (HWENO) scheme is adopted for the two dimensional transport equation, as well as the fast Fourier transform (FFT) or a 5-point central difference scheme for the Poisson equation of the electric potential and the 5-point central difference scheme for the elliptic equation of the magnetic potential. We will compare the asymptotic kinetic model with the two dimensional guiding-center model. With some special initial datum as designed in the numerical examples, we will show that under these settings, the two dimensional guiding-center model stays steady or nearly steady, while the asymptotic model can create some instabilities with a small initial nonzero current for the self-magnetic field. These instabilities are similar to some classical instabilities, such as Kelvin-Helmholtz instability [18], diocotron instability [36] for the two dimensional guiding- center model with some other perturbed initial conditions, which can validate some good properties of our new asymptotic model.

The rest of the paper is organized as follows. In Section 2, the dimensionless Vlasov-Maxwell system under some characteristic scales and the derivation of an asymptotic model will be presented. The verification of preservation for some good features as well as the existence of weak solutions for the asymptotic model will also be given. The numerical scheme will be briefly described in Section 3 and followed by some numerical examples in Section 4. Conclusions and our future work are in Section 5.

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2. Mathematical modeling

In this paper, we start from the Vlasov equation for each species of ions and electrons, (2.1) ∂_tf_s + v· ∇_xf_s + qs

m_s

E + v×(B+B_ext)

· ∇_vf_s = 0, s=i, e,

where fs ≡fs(t,x,v) is the distribution function, ms and qs are the mass and charge, with s=i, e for the ions and electrons respectively. Here we assume that the ions have an opposite charge to the electrons qi = e = −q_e and consider a given large magnetic field Bext, as well as self-consistent electromagnetic fieldsE and B, which satisfy the Maxwell equations

(2.2)











∇_x×E = −∂_tB,

∇_x×B = 1

c²∂tE + µ0J,

∇_x·E = ρ ε0

,

∇_x·B = 0,

where c is the speed of light, µ₀ is the vacuum permeability, ε₀ is the vacuum permittivity and µ0ε0 = 1/c². The density ns, average velocityus are related to the distribution functionfs by

n_s = Z

R³

f_sdv, n_su_s = Z

R³

f_svdv,

hence we define the total charge density ρ and total current density J as ρ = e(n_i−n_e) and J = e(n_iu_i−n_eu_e).

2.1. Rescaling of the Vlasov-Maxwell system. In the following we will derive an appropriate dimensionless scaling for (2.1) and (2.2) by introducing a set of characteristic scales.

We assume that the plasma is such that the characteristic density and temperature of ions and electrons are of the same order, that is,

(2.3) n := ni = ne, T := Ti = Te.

We choose to perform a scaling with respect to the ions. On the one hand, we set the characteristic velocity scalev as the thermal velocity corresponding to ions,

v :=

κBT m_i

^1/2 ,

whereκB is the Boltzmann constant. Then we define the characteristic length scale ofx given by the Debye length, which is the same for ions and electrons

x := λ_D =

ε₀κBT ne²

^1/2 .

It allows to define a first time scale corresponding to the plasma frequency of ionsωp := v/x.

Finally, the characteristic magnitude of the electric fieldE can be expressed fromnand x by E := e n x

ε₀ ,

hence the characteristic magnitude of the self-consistent magnetic field B, which is denoted by B, is related to the scale of the electric field byE =v B.

On the other hand, by denotingBextthe characteristic magnitude of the given magnetic fieldBext, we define the cyclotron frequency corresponding to ions by

ωc := eBext

m_i

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and ω⁻¹_c corresponds to a second time scale.

With the above introduced scales, we define the scaled variables as v⁰ = v

v, x⁰ = x

x, t⁰ = t t, and the electromagnetic field as

E⁰(t⁰,x⁰) = E(t,x)

E , B(t⁰,x⁰) = B(t,x)

B , B⁰_ext(t⁰,x⁰) = Bext(t,x) Bext

.

Furthermore, for each species, we define the characteristic velocity and subsequently, by letting f = n/v³,

f_s⁰(t⁰,x⁰,v⁰) = f_s(t,x,v)

f , s = i, e.

Inserting all these new variables into (2.1), dividing by ωp and dropping the primes for clarity, we obtain the following dimensionless Vlasov equation

(2.4)









 1

ωpt ∂_tf_i + v· ∇_xf_i +

E + v×B + ω_c

ω_p v×B_ext

· ∇_vf_i = 0, 1

ω_pt ∂tfe + v· ∇_xfe − mi

me

E + v×B + ωc

ωp

v×Bext

· ∇_vfe = 0,

while the dimensionless Maxwell equations (2.2) are scaled according to the plasma frequency of ions,

(2.5)











∇_x×E = − 1 ωpt ∂_tB,

∇_x×B = Ma² 1

ω_pt ∂tE + J

,

∇_x·E = ρ,

∇_x·B = 0, where Ma =v/cis the Mach number and

(2.6) ρ = n_i−n_e, J = n_iu_i − n_eu_e.

To consider an asymptotic limit, we introduce a dimensionless cyclotron period of ions ε := ω_p

ωc

,

whereε is a small parameter and study the long time asymptotic, that is,ε= 1/(ωpt)1. We also denote byα the mass ratio between electrons and ions

α := me

mi

.

Under these two scalings, the Vlasov equation (2.4) takes the form

(2.7)











ε ∂_tf_i + v· ∇_xf_i +

E + v×B + 1

εv×B_ext

· ∇_vf_i = 0,

ε ∂tfe + v· ∇_xfe − 1 α

E + v×B + 1

εv×Bext

· ∇_vfe = 0

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and the Maxwell equations (2.5) are

(2.8)











∇_x×E = −ε ∂_tB,

∇_x×B = Ma² (ε ∂_tE + J),

∇_x·E = ρ,

∇_x·B = 0, withρ and Jgiven by (2.6).

2.2. Asymptotic limit of the Vlasov-Maxwell system. To derive an asymptotic model from (2.7)-(2.8), let us set our assumptions

Assumption 2.1.ConsiderΩ⊂R² and D= Ω×[0, L_z], the external magnetic filed only applies in thez-direction

B_ext= (0,0,1)^t.

For simplicity we consider here periodic boundary conditions in space for the distribution function and the electromagnetic field.

Assumption 2.2.The plasma is homogeneous in the direction parallel to the applied magnetic field.

Hence, the distribution functionsfi and fe do not depend on z.

For any x = (x, y, z)^t ∈ R³, we decompose it as x = x⊥ +x_k according to the orthogonal and parallel directions to the external magnetic field B_ext, that is, x⊥ = (x, y,0)^t and x_k = (0,0, z). In the same manner, the velocity is v=v⊥+v_k ∈R³ with v⊥= (vx, vy,0)^t and v_k = (0,0, vz). Under these assumptions and notations, the Vlasov equation (2.7) can be written in the following form,

(2.9)











ε ∂_tf_i + v· ∇_xf_i + (E+v×B)· ∇_vf_i + v^⊥

ε · ∇_vf_i = 0, ε ∂_tf_e + v· ∇_xf_e − 1

α(E+v×B)· ∇_vf_e − v^⊥

ε α · ∇_vf_e = 0, wherev^⊥= (v_y,−v_x,0) for anyv∈R³.

Now we reformulate the Maxwell equations using Assumption 2.2. Here and after, we will drop the subindexxfor spatial derivatives of macroscopic quantities which do not depend on v, such asEand B and their related quantities, for clarity. On the one hand, from the divergence free condition of (2.8), we can writeB=∇_x×A, whereA is a magnetic potential verifying the Coulomb’s gauge

∇_x·A = 0.

On the other hand, the electric field E is split into a longitudinal part and a transversal part E = EL+ET, with







∇_x×EL= 0,

∇_x·ET = 0.

From (2.8) it is easy to see that E_L = −∇_xΦ, where the electrical potential Φ is a solution to the Poisson’s equation,

(2.10) −∆_xΦ =ρ.

Then, from (2.8) we get that

∇_x×ET = −∂_tB = −ε∇_x×(∂tA),

hence using the uniqueness of the decomposition for given boundary conditions, we necessarily have, assuming periodic boundary conditions, that ET =−ε∂_tA and the electric field Eis given by

(2.11) E = −∇_xΦ − ε∂_tA.

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Furthermore, the second equation in (2.8) gives the equation satisfied by the potentialA, that is, (2.12) (εMa)²∂_tt²A − ∆_xA = Ma² (J − ε ∂_t∇_xΦ).

Gathering (2.10)-(2.12), we finally have E = −∇_xΦ − ε ∂_tAand B = ∇_x×A, (2.13)







(εMa)²∂_tt²A − ∆_xA = Ma² (J − ε ∂_t∇_xΦ),

−∆_xΦ = ρ.

Now we remind the basic properties of the solution to (2.9) and (2.13)

Proposition 2.3.We consider that Assumptions 2.1 and 2.2 are verified and (f_i^ε, f_e^ε,Φ^ε,A^ε)ε>0 is a solution to (2.9) and (2.13). Then we have for all t≥0,

kf_s^ε(t)k_L^p =kf_s^ε(0)k_L^p, s=i, e.

Moreover we define the total energy at timet≥0, as E^ε(t) :=

Z

T²×R³

[f_i^ε(t) + α f_e^ε(t)] |v|²

2 dx⊥dv + 1 2

Z

T²

|∇_xΦ|² + ε|∂_tA|² + 1

Ma²|∇_x×A|²

dx⊥, which is conserved for all timet≥0, E^ε(t) =E^ε(0).

We now derive the asymptotic limit of (2.9) and (2.13) by lettingε→0. We denote the solutions to the above equations (2.9) and (2.13) as (f_i^ε, f_e^ε,A^ε,Φ^ε), and perform Hilbert expansions fors=i, e

(2.14)











f_s^ε=fs,0+εfs,1+ε²fs,2+· · · , A^ε = A₀ + εA₁ +· · · , Φ^ε = Φ₀ + εΦ₁ +· · · , correspondingly

E^ε = E₀ + εE₁ + · · ·, B^ε = B₀ + εB₁ + · · ·. We prove the following asymptotic limit

Theorem 2.4(Formal expansion). Consider that Assumptions 2.1 and 2.2 are satisfied. Let(f_i^ε, f_e^ε,A^ε,Φ^ε) be a nonnegative solution to the Vlasov-Maxwell system (2.9) and (2.13) satisfying (2.14). Then, the leading term (fi,0, fe,0,Φ0,A0) is such that







Φ₀ ≡Φ(t,x),

A₀ ≡(0,0, A(t,x))^t. Furthermore, we define(Fi, Fe) as

Fi(t,x, pz) = 1 2π

Z

R²

fi,0(t,x,v)dvxdvy, Fe(t,x, qz) = α⁻¹ 1 2π

Z

R²

fe,0(t,x,v)dvxdvy

where pz =vz+A(t,x) and qz =α vz−A(t,x), and the two Hamiltonians H_i = Φ +1

2 (A−pz)² and H_e = Φ − 1

2α (qz+A)², where (Fi, Fe,Φ, A) is a solution to the following system

(2.15)











∂tFi − ∇^⊥_xH_i· ∇_xFi = 0,

∂_tF_e − ∇^⊥_xH_e· ∇_xF_e = 0,

−∆_xΦ = ρ,

−∆_xA + Ma²

n_i+n_e α

A = Ma²J_z,

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and the density ni and ne are given by

(2.16) ns =

Z

R

Fs(t,x, rz)drz, s=i, e,

hence the charge density is ρ=ni−ne and the current density corresponds to

(2.17) J_z =

Z

R

r_z

F_i(t,x, r_z) − 1

αF_e(t,x, r_z)

dr_z, where the Mach number Ma=v/c.

Remark 2.5.Observe that the drift velocity in (2.15) called E×B =∇^⊥_xΦ is the same for the two species, since it does not depend on the charge of the particle.

Proof. We first start with the self-consistent electromagnetic fields, we can easily find from (2.13) that E₀ = −∇_xΦ₀ and B₀ = ∇_x×A₀ with

(2.18)







−∆_xΦ0 = ρ0,

−∆_xA₀ = Ma²J₀,

and at the next orderE1=−∇_xΦ1−∂tA0 and B1 =∇_x×A1, with (2.19)







−∆_xΦ₁ = ρ₁,

−∆_xA1 = Ma² (J1 − ∂t∇_xΦ0), where fork= 0,1,

ρ_k = Z

R³

[f_i,k − f_e,k]dv, J_k = Z

R³

v [f_i,k −f_e,k]dv.

Substituting the Hilbert expansions into (2.9), and comparing the orders of ε, such as ε⁻¹, ε⁰ and ε, we obtain the following three equations for ions:

(2.20)











v^⊥· ∇_vfi,0 = 0,

v· ∇_xf_i,0+ (E₀+v×B₀)· ∇_vf_i,0 = −v^⊥· ∇_vf_i,1,

∂tfi,0+v· ∇_xfi,1+ (E0+v×B0)· ∇_vfi,1+ (E1+v×B1)· ∇_vfi,0 = −v^⊥· ∇_vfi,2

and for electrons:

(2.21)











v^⊥· ∇_vf_e,0 = 0,

αv· ∇_xfe,0−(E0+v×B0)· ∇_vfe,0 = v^⊥· ∇_vfe,1,

α (∂tfe,0+v· ∇_xfe,1)−(E0+v×B0)· ∇_vfe,1−(E1+v×B1)· ∇_vfe,0 = v^⊥· ∇_vfe,2. We now pass in cylindrical coordinates in velocityv=v⊥+vk, with

v⊥ = ωeω, where we have setω =|v⊥|and

eω =



 cosθ sinθ 0



, eθ=





−sinθ cosθ

0



.

Using these notations, we now derive the asymptotic limit according to the orders ofεin (2.20)-(2.21).

First the leading order term in (2.20)-(2.21) written in cylindrical coordinates becomes

−∂_θfs,0 = 0, s = i, e,

which means that f_s,0 does not depend on θ, hence from Assumption 2.2, it yields that f_s,0 ≡ f_s,0(t,x_⊥, ω, v_z).

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As a consequence, the current density is such that (nsus,0)⊥ :=

Z

R³

v⊥fs,0dv = Z

R

Z ∞ 0

fs,0

Z 2π 0

eωdθ

ω²dω dvz = 0,

which implies that only the third component of the total current density J0 might be nonzero and therefore only the third component of A₀ in (2.18) might be nonzero, that is, A₀ = (0,0, A₀) is a solution to the Poisson’s equation with the source termJ₀ = (0,0, j_z)

−∆_xA₀ = Ma²j_z,

hence fromB₀ =∇_x×A₀, it yields thatB₀=∇^⊥_xA₀ and particularlyB_0,z = 0.

Finally, since the electric field E0 =−∇_xΦ0 and from Assumption 2.2, we also have that E0,z = 0.

Now we treat the zeroth order term in (2.20)-(2.21) and use the cylindrical coordinates in the velocity variable, it gives

(2.22) ∂_θf_i,1 = e_ω · G_i,0, ∂_θf_e,1 = e_ω · G_e,0, with

(2.23)







G_i,0 = + ω∇_xf_i,0 − (∇_xΦ₀ − v_z∇_xA₀) ∂_ωf_i,0 − ω∇_xA₀∂_v_zf_i,0 , G_e,0 = − α ω∇_xf_e,0 + (∇_xΦ₀ − v_z∇_xA₀) ∂_ωf_e,0 + ω∇_xA₀∂_v_zf_e,0

.

First notice thatGe,0 and Gi,0 do not depend onθ∈(0,2π) since f0 does not depend on θand Z 2π

0

eωdθ = 0,

then the solvability condition of (2.22) is automatically satisfied and after integration inθ, we obtain f1 as,

(2.24)







fi,1(t,x⊥, ω, θ, vz) = −e_θ·Gi,0(t,x⊥, ω, vz) + hi(t,x⊥, ω, vz), fe,1(t,x⊥, ω, θ, vz) = −e_θ·Ge,0(t,x⊥, ω, vz) + he(t,x⊥, ω, vz), whereh_i and h_e are arbitrary functions which do not depend on θ.

Now we focus on the first order with respect to ε in (2.20)-(2.21). Similarly, from the periodic boundary condition inθ∈(0,2π), we have the following solvability condition

1 2π

Z 2π 0

∂θfs,2dθ = 0, s = i, e.

Therefore, we have (2.25)











∂_tf_i,0+ 1 2π

Z 2π 0

v· ∇_xf_i,1+ (E₀ + v×B₀)· ∇_vf_i,1+ (E₁+v×B₁)· ∇_vf_i,0

dθ= 0,

α∂tfe,0+ 1 2π

Z 2π 0

αv· ∇_xfe,1−(E0 + v×B0)· ∇_vfe,1−(E1+v×B1)· ∇_vfe,0

dθ= 0.

Each integral term can be explicitly calculated by substituting f_i,1 and f_e,1 from (2.24). On the one hand, observing that







∂ωfs,1 =−e_θ·∂ωGs,0+∂ωhs, s = i, e ,

∂θfs,1 =eω·Gs,0, s = i, e , it yields fors=i, e,

(2.26) 1

2π Z 2π

0

v· ∇_xfs,1dθ = −ω

2∇_x·G^⊥_s,0.

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On the other hand, the same kind of computation leads to for s=i, e, (2.27) 1

2π Z 2π

0

(E0+v×B0)· ∇_vfs,1dθ=−1 2

(E0+vz∇_xA0)

ω ·∂ω

ωG^⊥_s,0

−ω∇_xA0·∂vzG^⊥_s,0

. Finally since f_s,0 does not depend on θ ∈ (0,2π) and the electric field does not depend on z, the last term in (2.25) only gives

(2.28) 1

2π Z 2π

0

(E1+v×B1)· ∇_vfs,0dθ=−∂_tA0∂vzfs,0, s = i , e.

Gathering (2.26)-(2.28), and recalling thatE₀=−∇_xΦ₀, we get for the distribution function f_i,0,

∂_tf_i,0 − ω

2∇_x·G^⊥_i,0 + 1 2

∇_x(Φ₀−v_zA₀)

ω ·∂_ω

ωG^⊥_i,0

+ω∇_xA₀·∂_v_zG^⊥_i,0

− ∂tA0∂vzfi,0 = 0.

and for the distribution functionf_e,0, α

∂_tf_e,0 − ω

2∇_x·G^⊥_e,0

− 1 2

∇_x(Φ₀−v_zA₀)

ω ·∂_ω(ωG^⊥_e,0) +ω∇_xA₀·∂_v_zG^⊥_e,0

+ ∂_tA₀∂_v_zf_e,0 = 0.

Using the definition ofGs,0 fors=i, ein (2.23) and after some calculations, it finally yields that

(2.29)











∂tfi,0− ∇^⊥_x(Φ0−vzA0)· ∇_xfi,0 −

∇_xΦ0· ∇^⊥_xA0+∂tA0

∂vzfi,0 = 0, α

∂tfe,0− ∇^⊥_x(Φ0−vzA0)· ∇_xfe,0

+

∇_xΦ0· ∇^⊥_xA0+∂tA0

∂vzfe,0 = 0.

Observing that this equation does not explicitly depend onω, we define F_s,0(t,x⊥, v_z) := 1

2π Z

R²

f_s,0(t,x⊥,v)dv_xdv_y, s = i, e . Multiplying (2.29) byω and integrating with respect to ω, we get

(2.30)











∂_tF_i,0 − ∇^⊥_x(Φ₀−v_zA₀)· ∇_xF_i,0 −

∇_xΦ₀· ∇^⊥_xA₀+∂_tA₀

∂_v_zF_i,0 = 0, α

∂_tF_e,0 − ∇^⊥_x (Φ₀−v_zA₀)· ∇_xF_e,0 +

∇_xΦ₀· ∇^⊥_xA₀+∂_tA₀

∂_v_zF_e,0 = 0.

This last equation can be reformulated to remove the time derivative ofA₀ in the velocity field. To this aim, we introduce a new variable forp_z = v_z + A₀(t,x) in F_i,0 andq_z = α v_z − A₀(t,x) in F_e,0 and perform a change of variable in velocity

F_i(t,x⊥, p_z) = F_i,0(t,x⊥, v_z), F_e(t,x⊥, q_z) = α⁻¹F_e,0(t,x⊥, v_z).

From now on, we will use Φ(t,x) and A(t,x) in short of Φ0(t,x) and A0(t,x) respectively. Hence (2.30) now becomes







∂_tF_i − ∇^⊥_xH_i· ∇_xF_i = 0,

∂_tF_e − ∇^⊥_xH_e· ∇_xF_e = 0, with

H_i = Φ + 1

2(A−p_z)² and H_e = Φ − 1

2α(A+q_z)²,

where the charge density is always given byρ=ni−ne, whereas the current density is now given by j_z = J_z −

n_i+n_e α

A,

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where (ni, ne) and J_z are respectively defined in (2.16) and (2.17). Finally, the potentials (Φ, A) are now solutions to







−∆_xΦ = ρ,

−∆_xA + Ma²

ni+ne

α

A = Ma²J_z,

where Ma =v/cis the Mach number.

2.3. Weak solutions for the asymptotic model. First notice that the asymptotic model (2.15) is now two dimensional in space since we assume that the plasma is homogeneous in the parallel direction to the external magnetic field and one dimensional in moment since we have averaged in the orthogonal direction to the external magnetic field.

To simplify the presentation, from now onx represents the orthogonal part of x⊥= (x, y,0) with (x, y)∈Ω.

For the sake of simplicity in the analysis we have only considered periodic boundary conditions in space, forx∈Ω := (0, Lx)×(0, Ly),

(2.31)











Φ(t, x+L_x, y) = Φ(t, x, y), Φ(t, x, y+L_y) = Φ(t, x, y), A(t, x+L_x, y) =A(t, x, y), A(t, x, y+L_y) =A(t, x, y),

F_i(t, x+L_x, y, p_z) =F_i(t, x, y, p_z), F_i(t, x, y+L_y, p_z) =F_i(t, x, y, p_z), p_z ∈R, F_e(t, x+L_x, y, q_z) =F_e(t, x, y, q_z), F_e(t, x, y+L_y, q_z) =F_e(t, x, y, q_z), q_z ∈R.

But other kinds of boundary conditions may be treated for the asymptotic model as homogeneous Dirichlet boundary conditions for the potential Φ and A

(2.32) Φ(t,x) = 0, A(t,x) = 0, x∈∂Ω.

Then let us review the main features of the asymptotic model (2.15), which make this mathematical model consistent with the initial Vlasov-Maxwell model (2.9) and (2.13).

Proposition 2.6.Consider a solution to the asymptotic model (2.15) with the boundary conditions (2.31), or (2.32), or a combination of both, then it satisfies

• the flow remains incompressible ;

• for any m >1, we have conservation of moments in velocity, for any time t≥0, (2.33)

Z

Ω×R

|r_z|^mF_s(t,x, r_z)dr_zdx= Z

Ω×R

|r_z|^mF_s(0,x, r_z)dr_zdx, s = i, e;

• for any continuous function φ:R7→R, we have for any timet≥0, (2.34)

Z

Ω

Z

R

φ(Fs(t,x, rz))dxdrz = Z

Ω

Z

R

φ(Fs(0,x, rz))dxdrz, s = i, e;

• the total energy defined by (2.35) E(t) :=

Z

R

Z

Ω

|r_z−A|²

2 F_i + |r_z+A|²

2α F_edxdr_z + 1 2

Z

Ω

|∇_xΦ|²+ 1

Ma² |∇_xA|² dx, is conserved for all timet≥0.

Proof. The velocity field in (2.15) can be written as

U_s(t,x, p_z) =−∇^⊥_xH_s, s=e, i,

hence∇_x·Us= 0 is automatically satisfied and the flow is incompressible.

Then observing that the variabler_z ∈Ronly appears as a parameter in the equation, we prove the conservation of moments with respect to r_z : for anym >1 we have fors=i, e,

Z

Ω×R

|r_z|^mF_s(t,x, r_z)dr_zdx= Z

Ω×R

|r_z|^mF(0,x, r_z)dr_zdx.

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For a given smooth function φ:R7→R and s=i, e, if we multiply the first equation in (2.15) by φ⁰(F_s), it becomes

∂tφ(Fs) +∇_x·(Usφ(Fs)) = 0.

Integrating the above equation in space Ωwe obtain

∂

∂t Z

Ω

φ(Fs)dx=− Z

∂Ω

φ(Fs)Us(t,x, rz)·νxdσx,

where νx is the outward normal to ∂Ω at x. Now for periodic boundary conditions (2.31), the right hand side is obviously zero, and for homogeneous Dirichlet boundary conditions (2.32), we observe that the tangential derivatives verify∇_xΦ·τx=∇_xA·τx= 0, whereτxis the tangential vector to∂Ω atx. Hence since

U_s·ν_x= 0, onx∈∂Ω,

the right hand side is also zero in that case. Finally a further integration onrz shows that

(2.36) d

dt Z

Ω

Z

rz

φ(F_s)dr_zdx= 0 or

Z

Ω

Z

pz

φ(F_s(t))dr_zdx= Z

Ω

Z

R

φ(F_s(0))dr_zdx, t≥0.

Notice that this result still holds true when φ is only continuous. Taking φ(F) = F, it ensures the conservation of mass, φ(Fs) = max(0, Fs) gives the non-negativity of the distribution function for nonnegative initial datum, whileφ(F_s) =Fs^p for 1≤p <∞, it yields the conservation of L^p norm.

Now let us show the conservation of total energy. On the one hand, we multiply the equation onFi

byH_i and the one on F_e by H_e, it gives after a simple integration by part and using the appropriate boundary conditions (2.31) or (2.32),

Z

Ω×R

H_i∂tFi + H_e∂tFedxdrz = 0.

or (2.37)

Z

Ω×R

(A−r_z)²

2 ∂_tF_i + (A+r_z)²

2α ∂_tF_edxdr_z + Z

Ω×R

∂_t(n_i−n_e) Φdx = 0.

The first and second terms in the latter equality can be written as









 I₁ :=

Z

Ω×R

(A−r_z)²

2 ∂_tF_idxdr_z= d dt

Z

Ω×R

(A−r_z)²

2 F_idxdr_z − Z

Ω

(n_iA− n_iu_i)∂_tA dx

I₂ :=

Z

Ω×R

(A+rz)²

2α ∂_tF_edxdr_z = d dt

Z

Ω×R

(A+rz)²

2α F_edxdr_z − 1 α

Z

Ω

(n_eA+ n_eu_e)∂_tA dx, which yields using the equation onA in (2.15),

I₁+I₂= d dt

Z

Ω×R

"

(A−r_z)²

2 F_i + (A+r_z)² 2α F_e

#

dxdr_z + 1 2 Ma²

d dt

Z

Ω

|∇_xA|²dx.

On the other hand, from the equation on Φ in (2.15), we get I₃:=

Z

Ω×R

∂_t(n_i−n_e) Φdx= 1 2

d dt

Z

Ω

|∇_xΦ|²dx.

Finally, using thatI₁+I₂+I₃= 0 in (2.37), we obtain the energy conservation (2.35).

From the conservation of moments (Proposition 2.6), we get L^p estimates [5] on the macroscopic quantities

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Lemma 2.7.If F ∈L¹∩L^∞(Ω×R) and |r_z|^mF ∈L¹(Ω×R) with 0≤ m <∞, then we define n_F =

Z

R

F dr_z, nu_F = Z

R

F r_zdr_z, e_F = Z

R

F|r_z|²dr_z and there exists C >0 such that

kn_Fk_L1+m ≤ CkFk^m/(m+1)_L∞

Z

Ω×R

|r_z|^m|F|dr_zdx

1/(m+1)

and











knu_Fk_L(1+m)/2 ≤ CkFk(m−1)/(m+1) L^∞

Z

Ω×R

|r_z|^m|F|dr_zdx

2/(m+1)

,

ke_Fk_L(1+m)/3 ≤ CkFk(m−2)/(m+1) L^∞

Z

Ω×R

|r_z|^m|F|dr_zdx

3/(m+1)

.

From Proposition 2.6 and Lemma 2.7 we can prove the existence of weak solutions to (2.15) Theorem 2.8 (Existence of weak solutions). Assume that the nonnegative initial condition Fs,in ∈ L¹∩L^∞(Ω×R) for s=i, e and for any m >5

(2.38)

Z

Ω×R

|r_z|^mF_s(0,x, r_z)dr_zdx<∞.

Then, there exists a weak solution (Fi, Fe,Φ, A) to (2.15), with Fi, Fe ∈ L^∞(R⁺, L¹ ∩L^∞(Ω×R)), and Φ, A∈L^∞(R⁺, W₀^1,p(Ω)), for anyp >1.

Proof. The proof follows the lines of the existence of weak solutions for the Vlasov-Poisson system [2, 13]. The main point here is to get enough compactness on the potential A since its equation is nonlinear

−∆_xA + Ma²

n_i+n_e α

A = Ma²J_z.

From (2.38) and Proposition 2.6, we first get the conservation of moments for anyl ∈(0, m] and s=i, e

Z

Ω×R

|r_z|^lF_s(t)dr_zdx= Z

Ω×R

|r_z|^lF_s,indr_zdx <∞,

hence applying Lemma 2.7, it yields that for anyr∈[1, m+ 1] andq ∈[1,(m+ 1)/2]

ρ=n_i−n_e∈L^∞(R⁺, L^r(Ω)), J_z ∈L^∞(R⁺, L^q(Ω)).

Thus, from the elliptic equations in (2.15) forA and Φ,







−∆_xΦ = ρ,

−∆_xA + Ma²

ni+ne

α

A = Ma²J_z, it yields

∇_xΦ ∈ L^∞(R⁺, W₀^1,r(Ω)), ∇_xA ∈ L^∞(R⁺, W₀^1,q(Ω)).

Since we can chooser and q >2, using classical Sobolev inequalities, we have in particular that both

∇_xΦ and∇_xA are uniformly bounded in L^∞(R⁺×Ω).

Furthermore, we obtain an estimate on the time derivative ∂_t∇_xΦ and ∂_t∇_xA by differentiating with respect to the two Poisson equations in (2.15)











−∆_x∂tΦ =∂tρ,

−∆_x∂tA+ Ma²

ni+ne

α

∂tA= Ma²∂tJ_z−Ma²

∂tni+∂tne

α

A.

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Then using the evolution equation satisfied byρ and J_z











∂_tρ=∇_x·

ρ∇^⊥_xΦ +

n_i+ne

α

∇^⊥_xA²

2 − ∇^⊥_xAJ_z

,

∂tJ_z=∇_x·

J_z∇^⊥_xΦ +

niui+neue

α²

∇^⊥_xA²

2 − ∇^⊥_xA

ei− ee

α²

, wheree_s corresponds to the second order moment in r_z,

e_s(t,x) = Z

R

F_s(t)|r_z|²dr_z, fors=i, e

and applying Lemma 2.7, we have thate_i, e_e∈L^∞(R⁺, L²(Ω)), hence both terms∂_t∇_xA and ∂_t∇_xΦ are uniformly boundedL^∞(R⁺, L²(Ω)).

From these estimates, we get strong compactness on the electromagnetic field E = −∇_xΦ and B = ∇_x ×A in L² and weak compactness in L² allowing to treat the nonlinear terms and prove

existence of weak solutions for (2.15).

Remark 2.9.Observing that starting from (2.15), and taking the limit Ma → 0, it gives from the Poisson’s equation thatA= 0. Then we integrate (2.15) inrz ∈Rand we recover the two dimensional guiding-center model [21, 36, 29]

(2.39)







∂tρ+∇_x·(Uρ) = 0,

−∆_xΦ =ρ, with the divergence free velocityU=−∇^⊥_xΦ.

2.4. Guiding center model & linear instability. To study the growth rate of the linear instability for our asymptotic model (2.15), we follow the classical linearization procedure: consider an equilibrium solution (Fi,0, Fe,0,Φ0, A0) to (2.15) and assume that

(2.40)

Z

R

r_zF_i,0dr_z = Z

R

r_zF_e,0dr_z = 0.

Therefore the potential A₀ satisfies a linear Poisson equation with a null source term together with periodic boundary condition or zero Dirichlet boundary conditions, which means thatA0 ≡0.

Now we consider (Fi, Fe,Φ, A) a solution to the nonlinear system ((2.15)) and decompose it as the sum of the equilibrium (F_i,0, F_e,0,Φ₀,0) and a perturbation (F_i⁰, F_e⁰,Φ⁰, A⁰) ,

F_i=F_i,0+F_i⁰, F_e=F_e,0+F_e⁰, ρ=ρ₀+ρ⁰, Φ = Φ₀+ Φ⁰, A=A⁰.

Then we substitute them into (2.15) and drop the high order small perturbation terms, a linearized system is obtained as follows:

(2.41)











∂tF_i⁰ − ∇^⊥_xΦ0· ∇_xF_i⁰− ∇^⊥_x Φ⁰−pzA⁰

· ∇_xFi,0 = 0,

∂tF_e⁰ − ∇^⊥_xΦ0· ∇_xF_e⁰− ∇^⊥_x

Φ⁰−qz

αA⁰

· ∇_xFe,0 = 0,

−∆_xΦ⁰ = ρ⁰,

−∆_xA⁰ + Ma²

n_i,0+n_e,0 α

A⁰ = Ma²J_z⁰:= Ma² Z

R

r_z

F_i⁰−F_e⁰ α

dr_z.

Now we integrate the first equation inpz ∈Rand the second one in qz ∈Rand using (2.40), we get a linearized system for the perturbed charge density

(2.42)







∂tρ⁰ − ∇^⊥_xΦ0· ∇_xρ⁰− ∇^⊥_xΦ⁰· ∇_xρ0 = 0,

−∆_xΦ⁰ = ρ⁰,

which is exactly the linearized system for the two dimensional guiding-center model (2.39).

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Therefore, from an equilibrium (ρ0,Φ0) for the guiding-center model (2.39), we can easily construct an equilibrium for (2.15) by choosingF_s,0 such that it satisfies (2.40) and

(2.43)

Z

R

F_s,0dr_z = n_s,0, fors=i, e.

wherens,0 is the equilibrium density satisfying ρ0 =ni,0−ne,0. For instance, we can choose F_s,0 = ns,0

√2π exp

−r_z² 2

.

In terms of the electric charge density ρ and potential Φ, our asymptotic model has the same mechanism for generating instabilities as the two dimensional guiding-center model, so that the growth rate of instabilities for the electric field will be the same. We can refer to [33, 29, 11] for the analytical and numerical studies of the two dimensional guiding-center model. In the next section, we will numerically verify that the linear growth rates of instabilities for the electric potential of the two models are the same.

From this point, we observe that by choosing a nonzero initial potential A, that is a small current density J_z, we can initiate an instability on the asymptotic model (2.15), whereas the purely electrostatic guiding center model remains stationary.

Remark 2.10.We would notice that for the distribution function F_i orF_e, due to the extra term of

∇^⊥_x(pzA⁰)· ∇_xFi,0 and∇^⊥_x(qzA⁰/α)· ∇_xFe,0 in the first two equations of (2.41), some other instabilities might also happen toF_i⁰ orF_e⁰, which is much more complicated to analyze.

3. Numerical scheme

In this section, we will describe a high order numerical scheme for solving the asymptotic kinetic model (2.15), which depends on (x, p_z) or (x, q_z) and time t. We will apply a conservative finite difference scheme with Hermite weighted essentially non-oscillatory (WENO) reconstruction for solving the conservative transport equations. The Poisson’s equation for the electric potential function Φ will be solved by a 5-point central finite difference discretization for Dirichlet boundary conditions, or by the fast Fourier transform (FFT) for periodic boundary conditions on a rectangular domain. The elliptic equation for the magnetic potentialAis solved by a 5-point central finite difference discretization. The methods described here are natural extensions of those proposed in [36] for solving the two dimensional guiding-center model (2.39), which we will also adopt here for comparison in the numerical tests. In the following, we will briefly review these methods and explicitly mention the main differences, we refer to [36] for more details. Besides, since some instability problems and the linear growth rates will be numerically studied in the next section, at the end of this section, we will linearize the asymptotic kinetic model (2.15) and build a link to the two dimensional guiding-center model (2.39) as is analyzed in [33].

3.1. Hermite WENO finite difference scheme for the transport equation. We consider the transport equation for ions in a conservative form

∂_tF_i+∇_x·(U_iF_i) = 0,

with Fi = Fi(t,x, pz) and Ui = Ui(t,x, pz), first in the moment pz direction we consider a cut-off domain of Ω_c = [−V_c, V_c], where V_c is large enough so that F_i outside it is nearly zero. We take a uniform discretization withNc grid points on Ωc. Since pz inFi is a dummy argument, on each fixed p_z the transport equation can be taken as a 2d problem on x. A finite difference scheme for the two dimensional transport equation is applied dimension by dimension, similarly for the transport equation of the electrons. In the following we will only need to describe a Hermite WENO reconstruction for a 1dtransport equation.

Let us consider a prototype 1dconservative transport equation

(3.1) f_t+ (uf)_x= 0,