The finite Larmor radius regime for the Vlasov-Poisson equations. The three dimensional setting with non uniform magnetic field

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The finite Larmor radius regime for the Vlasov-Poisson equations. The three dimensional setting with non

uniform magnetic field

Mihai Bostan, Aurélie Finot

To cite this version:

Mihai Bostan, Aurélie Finot. The finite Larmor radius regime for the Vlasov-Poisson equations. The

three dimensional setting with non uniform magnetic field. 2018. �hal-01706309�

The finite Larmor radius regime for the

Vlasov-Poisson equations. The three dimensional setting with non uniform magnetic field

Miha¨ı BOSTAN

, Aur´ elie FINOT

Abstract

The subject matter of this paper concerns the mathematical analysis of toka- mak plasmas. We focus on the finite Larmor radius regime with general magnetic shapes. We concentrate on the asymptotic analysis for the Vlasov-Poisson equa- tions under strong external magnetic fields, by averaging with respect to the fast cyclotronic motion around the magnetic lines. The main properties of the limit model are emphasized : mass and energy balances, Hamiltonian structure.

Keywords: Vlasov-Poisson system, Hamiltonian formulation, Finite Larmor radius regime.

AMS classification: 78A35, 82D10.

1 Introduction

We consider a population of charged particles of mass m, charge q, whose presence density is denoted by f(t, x, v). Motivated by the study of tokamak plasmas, we analyse

∗Aix Marseille Universit´e, CNRS, Centrale Marseille, I2M, Marseille France, Centre de Math´ematiques et Informatique, UMR 7373, 39 rue Fr´ed´eric Joliot Curie, 13453 Marseille Cedex 13 France. E-mail : [email protected]

†Aix Marseille Universit´e, CNRS, Centrale Marseille, I2M, Marseille France, Centre de Math´ematiques et Informatique, UMR 7373, 39 rue Fr´ed´eric Joliot Curie, 13453 Marseille Cedex 13 France. E-mail : [email protected].

the dynamics of this population under the effect of a given magnetic field B(x) = B (x)e(x), with B > 0, |e| = 1. We use the Vlasov-Poisson equations, that is, the particles are transported under the action of the electro-magnetic force q( E(t, x) + B (x) v ∧ e(x) ), where E is the self-consistent electric field corresponding to the charge density ρ(t, x) = q R

R³

f(t, x, v) dv, (t, x) ∈ R

× R

. The evolution of the presence density f and the electric field E is given by the Vlasov-Poisson system :

∂

f + v · ∇

f + q

m ( E(t, x) + B(x) v ∧ e(x) ) · ∇

f = 0, (t, x, v) ∈ R

× R

× R

(1) E(t, x) = −∇

φ, −ε

∆

φ = q

Z

R³

f(t, x, v) dv, (t, x) ∈ R

× R

(2) supplemented by the initial condition

f(0, x, v) = f

(x, v) ≥ 0, (x, v) ∈ R

× R

. (3) Here ε

stands for the electric permittivity of the vacuum. By introducing the funda- mental solution of the Laplace operator in R

, x →

, x ∈ R

\ {0}, the Poisson equation (2) becomes

φ(t, x) = 1 4πε

Z

R³

ρ(t, y)

|x − y| dy = q ε

Z

R³

Z

R³

f(t, y, w) 4π|x − y| dwdy and therefore the electric field writes

E(t, x) = q 4πε

Z

R³

Z

R³

f (t, y, w) x − y

|x − y|

dwdy, (t, x) ∈ R

× R

.

The well posedness of the Vlasov-Poisson problem (1), (2), (3) is well understood.

Weak solutions have been constructed in [3], strong solutions have been investigated in [13]. For the propagation of the moments and regularity results we refer to [4, 15].

Clearly, as the field v · ∇

+

( E(t, x) + B(x) v ∧ e(x) )· ∇

is divergence free, we have the mass conservation

d dt

Z

R³

Z

R³

f(t, x, v) dvdx = 0, t ∈ R

.

The total energy is conserved as well. Indeed, the balance of the kinetic energy gives d

dt Z

R³

Z

R³

m |v |

2 f (t, x, v) dvdx − Z

R³

E(t, x) · j(t, x) dx = 0 (4)

where j(t, x) = q R

R³

f (t, x, v)v dv is the current density. Combining the continuity equation and the Poisson equation ε

div

E = ρ, we obtain

div

{ε

∂

E + j} = 0.

After multiplication by the potential φ and integration by parts one gets the balance for the electric energy

d dt

ε

2

Z

R³

|E(t, x)|

dx + Z

R³

E(t, x) · j(t, x) dx = 0. (5) The total energy conservation follows by (4), (5). We assume that the initial presence density has finite mass and total energy

Z

R³

Z

R³

f

(x, v) dvdx < +∞

Z

R³

Z

R³

m |v|

2 f

(x, v) dvdx + q

2ε

Z

R³

Z

R³

Z

R³

Z

R³

f

(x, v)f

(y, w)

4π|x − y| dwdydvdx < +∞.

We focus on the asymptotic regimes concerning the magnetic fusion : the charged particles are confined under the effect of strong magnetic fields. It is very instructive to consider first a uniform magnetic field B =

(0, 0, B), B > 0 and to neglect the electric field. In this case, the characteristic equations of (1), or the motion equations of the charged particles in the phase space writes

dX

dt = V (t; x, v), dV

dt = qB m

t

(V

(t; x, v), −V

(t; x, v), 0)

together with the conditions X (0; x, v) = x, V(0; x, v) = v. It is easily seen that

t

(V

(t; x, v), V

(t; x, v)) = R(−ωt)

(v

, v

), V

(t; x, v) = v

, X

(t; x, v) = x

+ tv

. The space coordinates in the orthogonal directions with respect to the magnetic field come by observing that

d dt

(X

, X

) + (V

, −V

) ω

= (0, 0), with ω = qB m . As p

(V

+ V

)(t; x, v) = p

v

+ v

, we deduce that the particle trajectories in the orthogonal plane are circles of center (x

, x

) +

and radius p

v

+ v

/|ω|. When

the magnetic field is high, this radius (called the Larmor radius) becomes small and

the plasma remains confined around the magnetic lines. Notice also that the dynam- ics in the orthogonal directions evolves on a smaller time scale than in the parallel direction. Indeed, X

, X

, V

, V

are periodic in time, of cyclotronic period T

=

, which is small when B is large. We are face to a multi-scale problem (see [1, 16]), and solving numerically for both slow and fast dynamics would require a huge amount of computation efforts. Alternatively, we can search for homogenization procedure : determine the effective particle trajectories, after averaging with respect to to the fast dynamics. Several mathematical analysis have been performed in the framework of uniform magnetic fields [11, 12, 14, 5, 8, 9].

In this paper we intend to investigate the setting of general magnetic fields, by taking into account the curvature of the magnetic lines. Our paper is organized as follows. In Section 2 we present the main results : the limit model and its conservations.

Section 3 is devoted to the homogenization process with respect to fast rotation. In Section 4 we extend the previous analysis to general transport operators. We appeal to the hamiltonian formalism. The finite Larmor radius regime is investigated in Section 5. We present formal arguments, by following the rigorous analysis in [9]. We refer to Appendix A for a classical result concerning the hamiltonian flows (the invariance of the symplectic structure).

2 Presentation of the main results

Based on the analysis in the case of uniform magnetic fields, we intend to extend the asymptotic study to non uniform magnetic fields. Therefore we have to average with respect to the fast motion in the orthogonal directions. This corresponds to a separation between the evolution time scales of the advections entering the vector field

v · ∇

+ q

m ( E(t, x) + B(x) v ∧ e(x) ) · ∇

. (6) Such a separation is obtained when splitting the kinetic energy of the Hamiltonian

H = m |v|

2 + qφ into the parallel and orthogonal contributions

H

= m (v · e(x))

2 + qφ, H

= m |v ∧ e(x)|

2 .

To this decomposition of H it corresponds the following decomposition of the vector field in (6)

a(t, x, v) · ∇

= (v · e(x))e(x) · ∇

+ h q

m E(t, x) − (v · e(x))

∂

e v i

· ∇

b(x, v) · ∇

= [v − (v · e(x))e(x)] · ∇

+

ω(x)v ∧ e(x) + (v · e(x))

∂

e v

· ∇

where ω(x) = qB(x)/m. The dynamics generated by the advection b · ∇

corresponds to the cyclotronic motion. It preserves the modulus of the perpendicular velocity and evolves at a time scale related to the cyclotronic period T

= 2π/ω. The dynamics generated by the advection a·∇

conserves the sum between the parallel kinetic energy and the potential energy. It contains oscillations in the parallel direction at the plasma frequency. We are interested on the asymptotic limit such that the dynamics generated by the advection a·∇

, characterizing the parallel motion, occurs at a much larger time scale with respect to the cyclotronic period. Therefore we are looking to the asymptotic behavior of the presence densities (f

)

solving the Vlasov-Poisson problems (1), (2), (3) when the ratio between the time scales characterizing the dynamics of b · ∇

and a · ∇

becomes small (the parameter ε refers to this ratio). We proceed by smoothing out the oscillations due to the fast cyclotronic motion. We search for a new family of presence densities (F

)

such that at any time t, f

(t) appears as the composition between F

(t) and the fast oscillating characteristic flow of b · ∇

f

(t, x, v) = F

(t, X (−t; x, v), V (−t; x, v)), (t, x, v) ∈ R

× R

× R

, ε > 0 where

dX

dt = [I

− e(X (t; x, v)) ⊗ e(X (t; x, v))] V (t; x, v) dV

dt = ω(X (t; x, v)) V (t; x, v) ∧ e(X (t; x, v)) + (V(t; x, v) · e(X (t; x, v)))

∂

e(X ) V(t; x, v) and X (0; x, v) = x, V(0; x, v) = v. We expect that the family (F

)

is stable, and determine the problem satisfied by the limit density F = lim

F

, giving us the asymptotic behavior for small ε

f

(t, x, v) = F (t, X (−t; x, v), V (−t; x, v)) + o(1), as ε & 0.

The derivation of the equation satisfied by the limit density F relies on the averaging

techniques for hamiltonian vector fields cf. Proposition 4.2. The key point is to appeal

to the classical result saying that any hamiltonian vector field preserves the symplectic structure, see Proposition A.1.

Theorem 2.1 Let f

be a non negative smooth enough presence density, with finite mass and total (kinetic and electric) energy

Z

R³

Z

R³

f

(x, v) dvdx < +∞

Z

R³

Z

R³

m |v|

2 f

(x, v) dvdx + q

2ε

Z

R³

Z

R³

Z

R³

Z

R³

f

(x, v)f

(y, w)

4π|x − y| dwdydvdx < +∞.

We denote by (f

, φ

)

the solutions of the Vlasov-Poisson problems (1), (2), (3).

Let (X , V) be the characteristic flow of the vector field b(x, v) · ∇

= [v − (v · e(x))e(x)] · ∇

+

ω(x)v ∧ e(x) + (v · e(x))

∂

e v

· ∇

and E the function given by E (X, V, Y, W ) = lim

T→+∞

1 T

Z

dt

4π|X (t; X, V ) − X (t; Y, W )| , (X, V, Y, W ) ∈ R

. We denote by F the solution of the problem

∂

F + ( F (t), H[F (t)] ) = 0, (t, X, V ) ∈ R

× R

× R

(7) F (0, X, V ) = f

(X, V ), (X, V ) ∈ R

× R

(8) where

H[F (t)](X, V ) = m h(v · e)

i

2 (X, V ) + q

ε

Z

R³

Z

R³

E (X, V, Y, W )F (t, Y, W ) dW dY (9) and the Poisson bracket (·, ·) is given by

m( F, H[F ] ) = ∇

H[F ] · ∇

F − ∇

H[F ] · ∇

F + (∇

F ∧ ∇

H[F ]) · qB

m . (10) Then we have the asymptotic behavior

f

(t, x, v) = F (t, X (−t; x, v), V(−t; x, v)) + o(1), as ε & 0.

The notation h·i stands for the average along the flow (X , V), see Proposition 4.2.

The above limit model conserves the mass and the total energy. Actually, the perpen-

dicular kinetic energy is preserved as well.

Theorem 2.2 Assume that the hypotheses in Theorem 2.1 hold true. Let us denote by F the solution of the problem (7), (8), (9), (10). Then we have

d dt

Z

R³

Z

R³

F (t, X, V ) dV dX = 0, t ∈ R

d

dt Z

R³

Z

R³

m |V ∧ e(X)|

2 F (t, X, V ) dV dX = 0, t ∈ R

d

dt Z

R³

Z

R³

F (t, X, V )

m h(v · e)

i 2 + q

2ε

Z

R³

Z

R³

E (X, V, Y, W )F (t, Y, W ) dW dY

dV dX = 0.

3 Effective transport under fast rotation

Before analyzing the non linear Vlasov-Poisson system, it is very instructive to consider the linear transport problem

∂

g + a(t, z) · ∇

g + ω

z · ∇

g = 0, (t, z) ∈ R

× R

(11)

g(0, z) = g

(z), z ∈ R

(12)

where a(t, z) is a given smooth, divergence free vector field of amplitude a := kak

, ω > 0 and

z = (z

, −z

), for any z = (z

, z

) ∈ R

. We assume that the variations of the initial condition g

occurs on a typical length L, that is

∇zgⁱⁿ gⁱⁿ

∼

, and that L

a >> 1

ω . (13)

The above condition says that the dynamics generated by the advection a · ∇

evolves on a much longer time scale than the dynamics generated by the advection ω

z · ∇

, whose period is 2π/ω. In this case it is legitimate to approximate the global dynamics, by averaging with respect to the fast rotation corresponding to the transport operator ω

z · ∇

, see also [9, 7]. The idea is to filter out the fast rotation and after that to pass to the limit when ε =

becomes small. More exactly, let us introduce the flow

dZ

dt = ω

Z (t; z), Z (0; z) = z and the new unknowns (G

)

given by

g

(t, z) = G

(t, Z), Z = Z(−t; z) = R(ωt)z

where (g

)

solve (11), (12) with

= ε, and R(s) stands for the rotation of angle s ∈ R . The family (G

)

satisfy

∂

G

+ ˜ ϕ(ωt)a(t) · ∇

G

= 0, (t, Z ) ∈ R

× R

(14) G

(0, Z) = g

(Z), Z ∈ R

where for any vector field c = c(z), the notation ˜ ϕ(s)c stands for the vector field ( ˜ ϕ(s)c)(Z) = R(s)c(R(−s)Z), Z ∈ R

, s ∈ R .

The family (G

)

contains no fast rotation, and therefore we expect stability as ε & 0. In order to identify the limit model, we appeal to the following two-scale Hilbert development

G

(t, Z) = G(t, s = ωt, Z) + G

(t, s = ωt, Z) + G

(t, s = ωt, Z) + ... (15) where G

(t, s, Z ) ∼ εG(t, s, Z), G

(t, s, Z ) ∼ ε

G(t, s, Z ).... Plugging the Ansatz (15) in (14) we obtain

∂

G(t, ωt, Z) + ω∂

G(t, ωt, Z) + ∂

G

(t, ωt, Z) + ω∂

G

(t, ωt, Z) + ...

+ ˜ ϕ(ωt)a(t) · (∇

G(t, ωt, Z) + ∇

G

(t, ωt, Z ) + ...) = 0.

Identifying the contributions of the same orders, one gets

∂

G(t, s, Z) = 0, (t, s, Z ) ∈ R

× R

× R

∂

G(t, s, z)+ω∂

G

(t, s, Z )+ ˜ ϕ(s)a(t)·∇

G(t, s, Z ) = 0, (t, s, Z ) ∈ R

× R

× R

. (16) Therefore G depends only on t and Z and its evolution comes by (16), after eliminating G

. As ˜ ϕ(s) is 2π-periodic, we expect that all dependencies with respect to s are 2π- periodic and therefore, averaging for s ∈ [0, 2π] leads to the limit problem

∂

G(t, Z) + 1 2π

Z

˜

ϕ(s)a(t) ds · ∇

G(t, Z) = 0, (t, Z ) ∈ R

× R

(17)

G(0, Z) = g

(Z), Z ∈ R

. (18)

We point out that in the non periodic case, it is possible to replace the mean over one period by the ergodic mean. Indeed, when S → +∞, we have by (13)

1 S

Z

ω∂

G

ds = ωG

S

0

∼

ωεG S

∼ 1 S

h a L G i

0

<< 1 S

Z

˜

ϕ(s)a(t) ds · ∇

G

and therefore in the general case (not necessarily periodic), the equation (17) becomes

∂

G + lim

S→+∞

1 S

Z

˜

ϕ(s)a(t) ds · ∇

G = 0, (t, Z) ∈ R

× R

provided that the family (

R

0

ϕ(s)a(t)ds) ˜

admits a limit, when S → +∞, for any t ∈ R

. The leading term in the development (15) satisfies a transport equation, whose advection vector field at any time t appears as the average of a(t) along the characteristic flow of ω

z · ∇

. We call it the effective advection vector field under fast rotation

ha(t)i := lim

S→+∞

1 S

Z

˜

ϕ(s)a(t) ds.

At this stage, the argument are completely formal. Nevertheless we will prove that the family (G

)

converges strongly in L

( R

, L

( R

)) toward G when ε & 0, under suitable hypotheses. For doing that, let us analyse the family of transformations ( ˜ ϕ(s))

in L

( R

)

.

Proposition 3.1 The family of linear transformations c → ϕ(s)c ˜ = R(s)c(R(−s)·), s ∈ R is a C

-group of unitary operators on L

( R

)

. If c ∈ L

( R

)

is divergence free, then so is ϕ(s)c, for any ˜ s ∈ R

Proof. We only check that div

ϕ(s)c ˜ = 0 for any divergence free vector field c ∈ L

( R

)

. Let c ∈ L

( R

)

be a vector field such that div

c = 0 in D

( R

). For any test function Ψ(Z) ∈ C

( R

) and any s ∈ R we have

Z

R²

˜

ϕ(s)c · ∇

Ψ dZ = Z

R²

R(s)c(R(−s)Z ) · ∇

Ψ(Z) dZ

= Z

R²

c(R(−s)Z) · (∇

(Ψ ◦ R(s)))(R(−s)Z ) dZ

= Z

R²

c(z) · (∇

(Ψ ◦ R(s)))(z) dz = 0 saying that div

ϕ(s)c ˜ = 0 in D

( R

).

We denote by L the infinitesimal generator of ( ˜ ϕ(s))

L : domL ⊂ L

( R

)

→ L

( R

)

, domL =

c ∈ L

( R

)

: ∃ lim

s→0

˜

ϕ(s)c − c

s in L

( R

)

and

Lc = lim

s→0

˜

ϕ(s)c − c

s =

z · ∇

c −

c, c ∈ domL.

The properties of the operator L are summarized below.

Proposition 3.2

1. The domain of L is dense in L

( R

)

and L is closed.

2. The operator L is skew-adjoint.

3. The average operator c ∈ L

( R

)

→ hci :=

R

0

ϕ(s)c ˜ ds coincides with the orthogonal projection on ker L = {c ∈ L

( R

)

: ˜ ϕ(s)c = c, s ∈ R }. If the vector field c is divergence free, then so is the vector field hci.

4. We have the Poincar´ e inequality

kck

≤ π

2 kLck

for any c ∈ domL ∩ (ker L)

and RangeL = (ker L)

. Proof.

1. The operator L is the infinitesimal generator of a C

-group, and therefore domL is dense in L

( R

)

and L is closed.

2. The skew-adjointness of L is a consequence of the fact that ( ˜ ϕ(s))

is a C

-group of unitary operators.

3. It is easily seen that for any c ∈ L

( R

)

, the average hci is left invariant by the group ( ˜ ϕ(s))

. Indeed, by periodicity, we have for any h ∈ R

˜

ϕ(h) hci = ˜ ϕ(h) 1 2π

Z

˜

ϕ(s)c ds = 1 2π

Z

˜

ϕ(s + h)c ds = 1 2π

Z

˜

ϕ(s)c ds = hci and therefore hci ∈ ker L. For any a ∈ ker L we have

Z

R²

a · c dz = Z

R²

˜

ϕ(s)a · ϕ(s)c ˜ dz = Z

R²

a · ϕ(s)c ˜ dz, s ∈ R . Averaging with respect to s over one period, one gets

Z

R²

a · c dz = Z

R²

a · hci dz

saying that c−hci ⊥ ker L. Therefore we have the equality hci = Proj

c, c ∈ L

( R

)

. Assume that c ∈ L

( R

)

is a divergence free vector field. By Proposition 3.1 we know that ˜ ϕ(s)c is divergence free for any s ∈ R . For any test function Ψ ∈ C

( R

) we have

Z

R²

hci · ∇

Ψ dZ = Z

R²

1 2π

Z

˜

ϕ(s)c ds · ∇

Ψ dZ = 1 2π

Z

Z

R²

˜

ϕ(s)c · ∇

Ψ dZ ds = 0

saying that div

hci = 0.

4. Let a ∈ L

( R

)

be a vector field such that hai = 0. We consider the vector field c = 1

2π Z

0

s ϕ(s)a ˜ ds.

We have

˜

ϕ(h)c = 1 2π

Z

s ϕ(s ˜ + h)a ds = 1 2π

Z

(σ − h) ˜ ϕ(σ)a dσ = 1 2π

Z

σ ϕ(σ)a ˜ dσ and we deduce that lim

( ˜ ϕ(h)c − c)/h = a, saying that c ∈ domL and Lc = a.

Observe also that c has zero average 4π

hci =

Z

Z

s ϕ(s ˜ + h)a dsdh = Z

0

s Z

0

˜

ϕ(s + h)a dh

| {z }

2πhai=0

ds = 0.

Moreover, we have the inequality kck

=

1 2π

Z

(s − π) ˜ ϕ(s)a ds

≤ kak

2π

Z

|s − π| ds = π

2 kak

. The range of L is closed, since we have

(ker L)

⊂ RangeL ⊂ RangeL = (ker L

)

= (ker L)

.

Remark 3.1

1. The group ( ˜ ϕ(s))

acts also on L

( R

)

. More exactly, if c ∈ L

( R

)

, then

˜

ϕ(s)c ∈ L

( R

)

and k ϕ(s)ck ˜

= kck

, for any s ∈ R .

2. We define the average operator on L

( R

)

by the same formula hci = 1

2π Z

0

˜

ϕ(s)c ds, c ∈ L

( R

)

and we have k hci k

≤ kck

for any c ∈ L

( R

)

.

3. For any a ∈ L

( R

)

∩L

( R

)

∩(ker L)

, there is a unique c ∈ domL∩L

( R

)

∩ (ker L)

such that Lc = a. Moreover we have

kck

≤ π

2 kak

.

In order to check the asymptotic behavior G

−G = O(ε) in L

( R

, L

( R

)), as ε & 0, we need to introduce a corrector. Thanks to the fourth statement of Proposition 3.2, for any t ∈ R

we consider c(t) ∈ domL ∩ (ker L)

such that a(t) − ha(t)i = Lc(t).

Observe that

∂

G + ˜ ϕ(s)a(t) · ∇

G = ˜ ϕ(s)a(t) · ∇

G − ha(t)i · ∇

G

= ˜ ϕ(s)(a(t) − ha(t)i) · ∇

G

= ˜ ϕ(s)Lc(t) · ∇

G

= d

ds { ϕ(s)c(t) ˜ · ∇

G}

and therefore (16) is equivalent to

ωG

(t, s, Z ) + ˜ ϕ(s)c(t) · ∇

G = ωG

(t, 0, Z) + c(t) · ∇

G.

The choice G

(t, 0, Z) = 0, (t, Z ) ∈ R

× R

, leads to the corrector

ωG

(t, s, Z ) = (c(t) − ϕ(s)c(t)) ˜ · ∇

G. (19) If the initial condition g

is smooth enough, we prove that G remains smooth and thanks to the inequalities

kc(t) − ϕ(s)c(t)k ˜

≤ 2kc(t)k

≤ πkLc(t)k

≤ 2πka(t)k

we expect that

sup

s∈R+

kG

(t)k

≤ 2πa ω

k∇

G(t)k

kG(t)k

kG(t)k

∼ 2πa ω

k∇

g

k

kg

k

kG(t)k

∼ εkG(t)k

.

The idea will be to estimate G

(t, Z) − G(t, Z) − G

(t, ωt, Z) and to combine with a triangular inequality, by taking into account that G

∼ εG. Notice that

d

dt {G(t, Z) + G

(t, ωt, Z )} + ˜ ϕ(ωt)a(t) · ∇

{G(t, Z ) + G

(t, ωt, Z)} = ∂

G(t, Z)

+ ∂

G

(t, ωt, Z ) + ω∂

G

(t, ωt, Z ) + ˜ ϕ(ωt)a(t) · ∇

{G + G

}

= ∂

G

(t, ωt, Z ) + ˜ ϕ(ωt)a(t) · ∇

G

(t, ωt, Z ) which together with (14), gives

d

dt {G

(t, Z ) − G(t, Z ) − G

(t, ωt, Z)} + ˜ ϕ(ωt)a(t) · ∇

{G

(t, Z) − G(t, Z ) − G

(t, ωt, Z)}

= −∂

G

(t, ωt, Z ) − ϕ(ωt)a(t) ˜ · ∇

G

(t, ωt, Z ).

As at any time t, the vector field a(t) is divergence free, we know by Proposition 3.1 that ˜ ϕ(ωt)a(t) is also divergence free, and by standard computations one gets

d

dt kG

(t) − G(t) − G

(t, ωt)k

≤ sup

s∈R+

k∂

G

(t, s)k

+ sup

s∈R+

k ϕ(s)a(t) ˜ · ∇

G

(t, s)k

. Integrating with respect to t and taking into account that G

(0) − G(0) − G

(0, 0) = g

− g

− 0 = 0, we obtain

kG

(t) − G(t)k

≤ sup

s∈R+

kG

(t, s)k

(20)

+ Z

0

sup

s∈R+

k∂

G

(t, s)k

+ sup

s∈R+

k ϕ(s)a(t) ˜ · ∇

G

(t, s)k

ds, t ∈ [0, T ].

It remains to estimate the L

norms of G

(t, s), ∂

G

(t, s), ϕ(s)a(t) ˜ · ∇

G

(t, s) locally in t and uniformly in s ∈ R

. This is a little bit tedious, but can be done under suitable smoothness assumptions. It comes by appealing to the explicit expression (19) of the corrector G

and the regularity of G, the solution of the limit model (17), (18). The estimates are uniform with respect to s thanks to the fact that ( ˜ ϕ(s))

is a C

-group of unitary transformations in L

( R

). For details we refer to [10]. Finally, under the condition (13), we obtain

sup

(t,s)∈[0,T]×R+

kG

(t, s)k

+ k∂

G

(t, s)k

+ k ϕ(s)a(t) ˜ · ∇

G

(t, s)k

≤ C(T, G)ε and (20) becomes

sup

t∈[0,T],ε>0

kG

(t) − G(t)k

≤ C(T, G)ε ˜

for some constants C, C ˜ depending on T and G. Coming back to the unknowns g

(t, z) = G

(t, R(ωt)z), we deduce that

g

(t, z ) − G(t, R(ωt)z) = O(ε) in L

( R

, L

( R

)), as ε & 0.

4 Effective transport under fast advection

We generalize the previous analysis to the case of two vector fields a(t, z) · ∇

, b(z) · ∇

whose associated phase flows evolve on different time scales T

, T

. We only indicate

the main lines, following the arguments developed in Section 3. For a complete rigorous

study we refer to [10]. We investigate the asymptotic behavior of the solutions (g

)

solving

∂

g

+ a(t, z ) · ∇

g

+ b(z) · ∇

g

= 0, (t, z) ∈ R

× R

(21)

g

(0, z) = g

(z), z ∈ R

(22)

when ε = T

/T

becomes small. We suppose that the initial density belongs to L

( R

) and that the total advection vector field is divergence free

div

{a(t) + b} = 0, t ∈ R

which guarantees the L

norm conservation Z

R^m

(g

(t, z))

dz = Z

R^m

(g

(z))

dz, t ∈ R

, ε > 0.

If the vector field b(z) · ∇

is smooth, with growth at most linear at infinity, it possesses a smooth global flow Z (t; z)

dZ

dt = b(Z(t; z)), Z(0; z) = z, (t, z) ∈ R × R

.

Filtering out the fast motion along b, and taking into account the mass constraint, we are led to the new unknown given by

g

(t, z) dz = G

(t, Z) dZ, Z = Z(−t; z) that is

G

(t, Z ) = g

(t, z)J(t, Z), z = Z (t; Z), J(t, Z ) = det ∂Z

∂Z (t; Z).

Recall that the Jacobian determinant satisfies J(0, ·) = 1 and

∂

J(t, Z ) = J(t, Z )(div

b)(Z(t; Z )) = −J(t, Z)(div

a(t))(Z (t; Z)), (t, Z ) ∈ R × R

. In the next proposition we identify the transport problems satisfied by the densities (G

)

.

Proposition 4.1 The new densities (G

)

given by G

(t, Z)dZ = g

(t, z)dz, Z = Z (−t; z) verify

∂

G

+ div

{G

ϕ(t)a(t)} = 0, (t, Z) ∈ R

× R

(23)

G

(0, Z ) = g

(Z), Z ∈ R

(24) where for any vector field c = c(z), the notation ϕ(t)c stands for the vector field

(ϕ(t)c)(Z ) = ∂Z (−t; Z(t; Z ))c(Z(t; Z)), (t, Z) ∈ R × R

.

Proof. Let us pick a compactly supported smooth test function Ψ(t, Z ) in R

× R

. Using the weak formulation of (21), (22) with ψ(t, z) = Ψ(t, Z(−t; z)), one gets 0 =

Z

R^m

g

(z)ψ(0, z) dz + Z

R+

Z

R^m

(∂

ψ + a(t, z) · ∇

ψ + b(z) · ∇

ψ)g

(t, z) dzdt

= Z

R^m

g

(Z)Ψ(0, Z) dZ + Z

R+

Z

R^m

a(t, Z(t; Z)) ·

∂ Z(−t; Z(t; Z ))∇

Ψ(t, Z)G

(t, Z) dZdt +

Z

R+

Z

R^m

{(∂

ψ )(t, Z (t; Z)) + b(Z(t; Z)) · (∇

ψ)(t, Z(t; Z ))}G

(t, Z) dZdt

= Z

R^m

g

(Z)Ψ(0, Z) dZ + Z

R+

Z

R^m

{∂

Ψ(t, Z ) + ϕ(t)a(t) · ∇

Ψ}G

(t, Z ) dZdt and therefore the density G

satisfies (23), (24).

Remark 4.1 If g

∈ L

( R

), 1 ≤ p < +∞, then we have Z

R^m

|g

(t, z)|

dz = Z

R^m

|g

(z)|

dz, t ∈ R

, ε > 0 implying that

Z

R^m

|G

(t, Z)|

J

(t, Z ) dZ =

Z

R^m

|g

(Z)|

dZ, t ∈ R

, ε > 0.

Remark 4.2 For any vector field c, we have the formula

div

{J (t, Z )ϕ(t)c} = J (t, Z)(div

c)(Z (t; Z)). (25) Indeed, let us pick a compactly supported smooth test function Ψ(Z ) and consider ψ(z) = Ψ(Z), z = Z(t; Z). By the chain rule, we obtain

hdiv

{J(t, Z )ϕ(t)c}, Ψi

= − Z

R^m

J(t, Z )ϕ(t)c · ∇

Ψ dZ

= − Z

R^m

∂Z (−t; z)c(z) · (∇

Ψ)(Z(−t; z)) dz

= − Z

R^m

c(z) ·

∂Z(−t; z)(∇

Ψ)(Z (−t; z)) dz

= − Z

R^m

c(z) · ∇

(Ψ ◦ Z(−t; ·)) dz

= − Z

R^m

c(z) · ∇

ψ(z) dz

= hdiv

c, ψi

saying that (25) holds true. In particular, if the vector field b is divergence free, we have J = 1, and in that case we obtain

div

ϕ(t)c = (div

c) ◦ Z(t; ·).

Moreover, if the vector field c is divergence free, then so is the vector field ϕ(t)c, see also Proposition 3.1.

Motivated by the example in Section 3, we introduce the average operator hci = lim

T→+∞

1 T

Z

ϕ(t)c dt = lim

T→+∞

1 T

Z

∂Z (−t; Z(t; ·))c(Z (t; ·)) dt and we claim that G = lim

G

solves the problem

∂

G + div

(G ha(t)i) = 0, (t, Z ) ∈ R

× R

G(0, Z) = g

(Z ), Z ∈ R

.

Moreover, under suitable hypotheses, we expect the asymptotic behavior g

(t, ·) = G(t, Z(−t; ·))

J (t, Z(−t; ·)) + O(ε) in L

( R

, L

( R

)), as ε & 0.

The average operator behaves nicely with respect to hamiltonian vector fields. Consider a symplectic structure on R

(with m an even integer), that is, let σ be a differential 2-form on R

, which is non degenerate and closed. We denote by Σ the matrix field corresponding to the differential 2-form σ.

Proposition 4.2 Let b and c be two hamiltonian vector fields on the symplectic man- ifold ( R

, σ), corresponding to the Hamiltonians H

, H

respectively. Then the average vector field

hci = lim

T→+∞

1 T

Z

∂ Z(−t; Z (t; ·))c(Z(t; ·)) dt is hamiltonian, and corresponds to the average Hamiltonian

hH

i := lim

T→+∞

1 T

Z

H

(Z(t; ·)) dt.

Proof. The differential 2-form is left invariant by any hamiltonian flow cf. [2] pp. 204, see also Appendix A. In particular we have (here Z is the flow of b)

Z (t; ·)

σ = σ, t ∈ R

or equivalently

t

∂Z(t; ·)Σ(Z(t; ·))∂Z(t; ·) = Σ(·), t ∈ R . Taking into account that

∂ Z(−t; Z (t; ·))∂Z (t; ·) = I

we obtain

∂Z(−t; Z (t; ·))Σ

(Z(t; ·))

∂ Z(−t; Z(t; ·)) = Σ

(·), t ∈ R . As the vector field c is hamiltonian, we have c = −Σ

∇H

and therefore

ϕ(t)c = ∂Z(−t; Z (t; ·))c(Z(t; ·))

= −∂Z(−t; Z (t; ·))Σ

(Z(t; ·))(∇H

)(Z (t; ·))

= −Σ

(·)

∂ Z(t; ·)(∇H

)(Z (t; ·))

= −Σ

(·)∇(H

◦ Z(t; ·)).

Finally the average of c is given by hci = lim

T→+∞

1 T

Z

ϕ(t)c dt = −Σ

∇ lim

T→+∞

1 T

Z

H

(Z(t; ·)) dt = −Σ

∇ hH

i saying that the average field hci is hamiltonian, and corresponds to the average Hamil- tonian hH

i (see [6] for more details on function averages).

5 The finite Larmor radius regime

Let us come back to the Vlasov-Poisson equations (1), (2), (3). In order to apply the previous formalism, we need to identify the advection vector fields acting on different time scales. Clearly, when the magnetic field is uniform, that is B =

(0, 0, B), the advection vector field in the Vlasov equation writes (a(t, x, v) + b(x, v)) · ∇

, where

a(t, x, v) · ∇

= v

∂

+ q

m E(t, x) · ∇

(26)

b(x, v) · ∇

= v

∂

+ v

∂

+ ω(v

∂

− v

∂

), ω = qB

m . (27)

Notice that when the magnetic field is strong, b · ∇

generates a periodic group, whose

period 2π/ω is much smaller than the time scale on which evolves the group generated

by a · ∇

. Therefore it is legitimate to average a · ∇

with respect to b · ∇

. In order to better understand the decomposition of the particle dynamics into slow and fast motions, let us recall that the charged particle flows under electro-magnetic fields are hamiltonians. More exactly, if φ, A are the scalar and vector potentials of the electro-magnetic field (E, B), that is

E = −∇

φ, B = ∇

∧ A

let us consider the differential 2-form θ on R

, whose matrix field is Θ =





qM [B] mI

−mI

O



 .

Here, for any vector u ∈ R

, the notation M [u] stands for the matrix of the endomor- phism v ∈ R

→ u ∧ v ∈ R

, that is

M [u] =











0 −u

u

u

0 −u

−u

u

0











, u ∈ R

.

By straightforward computations we obtain the formula

θ = d[(qA

+ mv

)dx

+ (qA

+ mv

)dx

+ (qA

+ mv

)dx

]

and therefore the differential 2-form θ is closed, i.e., dθ = 0. Notice also that θ is non degenerate

Θ

=





O

−

I3

m q

m²

M [B]





and therefore θ is a symplectic structure on R

. Observe that the vector field v · ∇

+

q

m

(E + v ∧ B) · ∇

is hamiltonian, and corresponds to the Hamiltonian H = m |v |

2 + qφ.

Indeed, we have

∇

H = −Θ

v, q

m (E + v ∧ B) or equivalently

dH(η) = θ η,

v, q

m (E + v ∧ B)

, η ∈ R

.

The Poisson bracket of the functions f and H, that is, the derivative of the function f in the direction of the characteristic flow with Hamiltonian H, is given by

(f, H) = ∇

f ·

v, q

m (E + v ∧ B)

= −∇

f · Θ

∇

H

= 1 m

∇

H · ∇

f − ∇

H · ∇

f + (∇

f ∧ ∇

H) · qB m

. Therefore, the Vlasov-Poisson equations (1), (2) write

∂

f + (f(t), H [f(t)]) = 0, (t, x, v) ∈ R

× R

× R

H[f (t)](x, v) = m |v|

2 + q

ε

Z

R³

Z

R³

f(t, y, w)

4π|x − y| dwdy, (x, v) ∈ R

× R

.

We distinguish between the parallel and orthogonal kinetic energy, leading to the de- composition

H = H

+ H

, H

= m (v · e(x))

2 + qφ, H

= m |v ∧ e(x)|

2 .

To these Hamiltonians it corresponds the vector fields a · ∇

and b · ∇

, given by

∇

H

= −Θa, ∇

H

= −Θb.

Obviously, these hamiltonian vector fields represent a decomposition of v · ∇

+

(E + v ∧ B) · ∇

−Θ

v, q

m (E + v ∧ B)

= ∇

H = ∇

(H

+ H

) = −Θ(a + b).

This decomposition writes

a(t, x, v) · ∇

= (v · e(x))e(x) · ∇

+ h q

m E(t, x) − (v · e(x))

∂

e v i

· ∇

(28) b(x, v) · ∇

= [v − (v · e(x))e(x)] · ∇

+

ω(x)v ∧ e(x) + (v · e(x))

∂

e v

· ∇

(29) where ω(x) = qB(x)/m. Notice that both a · ∇

, b · ∇

are divergence free

div

a = div

[(v · e)e] + div

h q

m E − (v · e)

∂

e v i

= (v · e)div

e + e(x) ·

∂

e v − e(x) ·

∂

e v − (v · e) trace(

∂

e) = 0

div

b = div

(a + b) − div

a = 0.