electromagnetic waves in magnetized plasma the …despres/chrome/documents/...electromagnetic waves...

Electromagnetic waves in magnetized plasmaThe dispersion relation

Bruno Despres (LJLL-UPMC)

Electromagnetic waves in magnetized plasma The dispersion relation p. 1 / 32

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Dispersionrelation forone speciesAllcoefficientsconstant

Dispersionrelation formultispecies

Groupvelocity

Vlasov-Maxwell

Vectors are in bold.

For each species s = 1, . . . ,, solve

∂t fs + v · ∇x fs +qsms

(E + v ∧ B) · ∇v fs = 0

for the kinetic unknown fs = fs(t, x , v), and couple with theMaxwell’s equations

− 1c2∂tE +∇∧ B = µ0J, ∇ · E = ρ

ε0,

∂tB +∇∧ E = 0, ∇ · B = 0,J =

∑s qs

∫vfsdv ,

ρ =∑

s qs∫fsdv .

Expensive.


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Groupvelocity

Euler-Maxwell

Simplify

For s = 1, . . . , solve for ns = ns(t, x) and ue = ue(t, x){∂tns +∇ · (nsus) = 0,

∂t(nsus) +∇ · (nsus ⊗ us + ∇psms

) = qsns (E + us ∧ B) ,

coupled with the Maxwell’s equations− 1

c2∂tE +∇∧ B = µ0J, ∇ · E = ρ

ε0,

∂tB +∇∧ E = 0, ∇ · B = 0,J =

∑s qsnsus ,

ρ =∑

s qsns .

In the cold plasma approximation the pressure ps is negligible

∇psms≈ 0.

This is a closed system. Still expensive.


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Groupvelocity

Newton-Maxwell

Simplify further

For each species s = 1, . . . ,, solve

ms (∂tus + us · ∇us) = qs (E + us ∧ B) ,

coupled with the Maxwell’s equations− 1

c2∂tE +∇∧ B = µ0J,

∂tB +∇∧ E = 0, ∇ · B = 0,J =

∑s qsnsus .

Notice that ns has to be provided to close the system.


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Groupvelocity

Geometry and constant coefficients

x

y

z

E.M. wavek

E0 = 0

B0 = B0 ez 6= 0

B0 = B0ez and k = (sin θ cosϕ, sin θ sinϕ, cos θ) .

Invariance by rotation around ez : ϕ = 0 without restriction.

k = (sin θ, 0, cos θ) .


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Groupvelocity

Linearization

One linearizes E = 0 +E1 + . . . ,B = B0 +B1 + . . . ,us = 0 +us

1 + . . . .

One obtainsms∂tu

s1 = qs (E1 + us

1 ∧ B0) ,

coupled with − 1

c2∂tE1 +∇∧ B1 = µ0J1,

∂tB1 +∇∧ E1 = 0,J1 =

∑s qsnsu

s1.

Here the densities ns are given.The system is closed.


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Groupvelocity

One species (e−)

− 1

c2∂tE1 +∇∧ B1 = µ0qeNe(x)ue

1,∂tB1 +∇∧ E1 = 0,ms∂tue

1 = qe (E1 + ue1 ∧ B0) .

Stix : A general analysis of this model is able to provide asurprisingly comprehensive view of plasma waves.

If you discuss this topic with a plasma physicist, just keep inmind that it is a model.With this respect, the situation is very different from Maxwell’sequation in the vaccum.


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Groupvelocity

Dispersion relation

Consider simple solutions

E1 = E(x)e−iωt with E(x) = e e i(k,x), and e ∈ C3, . . .

Propagative properties can be locally understood checkingwether

ω ∈ R for k ∈ R3.

The phase velocity of a wave e i((k,x−ωt)) is

vϕ =ω

|k|.

Definition The accessible domain (in the phase space) is

A(ω) ={

(Ne , |B0|) , ∃ k ∈ R3 and a non trivial sol.}.


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Groupvelocity

Derivation

Plugging the dependancy with respect to ω in the cold plasmamodel, one gets the linear system

iωc2

E +∇∧ B = µ0qeNeue ,−iωB +∇∧ E = 0,−iωmeue = qe (E + ue ∧ B0)

Elimination of the magnetic field yields (µ0ε0c2 = 1)

∇∧∇ ∧ E− ω2

c2E = i

ω

c2ε0qeNeue .

It remains to compute the velocity ue with respect to theelectric field.


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Groupvelocity

First simple case : B0 = 0

Then ue = − qeiωme E. One gets

∇∧∇ ∧ E− ω2

c2E = − q2eNe

c2ε0meE

which is conveniently rewritten as

∇∧∇ ∧ E− ω2

c2

(1−

ω2p

ω2

)E = 0

where the plasma frequency is

ω2p =

q2eNe

c2ε0me.


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Groupvelocity

Spatial dependancy

Remind the Ansatz

E(x) = e e i(k,x), B(x) = b e i(k,x), e,b ∈ C3,

where a priori k ∈ R3.

A possibility is to writek = kn

where k > 0 andn ∈ R3, |n| = 1.

is the direction in space of the wave.


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Groupvelocity

With this Ansatz

∇∧∇ ∧ E− ω2

c2

(1−

ω2p

ω2

)E = 0

becomes

−k2n ∧ n ∧ e− ω2

c2

(1−

ω2p

ω2

)e = 0

or also

−n ∧ n ∧ e =ω2

k2c2

(1−

ω2p

ω2

)e.


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Groupvelocity

Interpretation

Notice that −n ∧ n ∧ e = M(n)e where the matrix is symetricnon negative

M(n) = −n ∧ n∧ = I − n⊗ n = M(n)t ≥ 0.

The whole problem reduces to

M(n)e = λe, λ =ω2

k2c2

(1−

ω2p

ω2

)=ω2 − ω2

p

k2c2.

This is an classical eigenproblem for asymetric real matrix :the eigenvector is e ;the eigenvalue is λ.


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Groupvelocity

Discussion

The characteristic polynomial is

det (M(n)− λI ) = (λ− 1)2λ = 0.

The eigenspace k⊥ is associated to the same eigenvalueλ = 1 with multiplicity 2. One has the relation

k2c2 = ω2 − ω2p.

The eigenvector e = k is associated to λ = 0 withmultiplicity 1. One gets

0 = ω2 − ω2p.

Warnig : the classical discussion systematically disregard thezero eigenvalue.


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Groupvelocity

Consequence for propagation

Corollary : Propagation (ω, n ∈ R) is possible iff

ω2 ≥ ω2p =

q2eNe

c2ε0me.

The phase velocity of the wave e ik((n,x)−ωkt) is

vϕ =ω

k= c

√1 +

ω2p

k2c2︸︷︷︸vϕ(k)

= c1√

1− ω2p

ω2︸︷︷︸=vϕ(ω)

.


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Groupvelocity

Accessibility in physical space

Consider a problem where Ne = Ne(x).

Definition : The accessible domain in physical space is

B0(ω) ={

x ∈ Rd ;Ne(x) ∈ A0(ω)}⊂ Rd

An antenna with given frequency ω is facing a fusion plasmawith increasing electronic density d

dωNc > 0 :”the accessible domain” = ”the prop. region” is near the wall.

?

Accessible Non propagative= Non accessible

Nc(ω)

Ne < Nc

Nc < Ne


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Groupvelocity

Principal case : B0 6= 0

The linear system writes

−iωme

1 iωcω 0

−iωcω 1 0

0 0 1

uxuyuz

= qe

Ex

Ey

Ez

where the cycloton frequency is ωc = |qe |B0

me.

Notice

∣∣∣∣ 1 iωcω

−iωcω 1

∣∣∣∣ = 1− ω2cω2 = ω2−ω2

cω2 and

uxuyuz

= −qeiω

ω2

ω2−ω2c

i ωωcω2−ω2

c0

−i ωωcω2−ω2

c

ω2

ω2−ω2c

0

0 0 1

Ex

Ey

Ez

(1)


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Groupvelocity

The elimination of the velocity yields the generalizedeigenvalue problem

M(n)e =ω2

k2c2

I −ω2p

ω2

ω2

ω2−ω2c

i ωωcω2−ω2

c0

−i ωωcω2−ω2

c

ω2

ω2−ω2c

0

0 0 1

︸︷︷︸=M=M?

e.

If B0 = 0, then ωc = 0 and M = I .

The parallel direction ez is not modified by ωc .

Singular for ω = ωc .


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Groupvelocity

Matrix M

One writes usually

M =

S −iD 0iD S 00 0 P

with S = 1

2 (R + L), D = 12 (R − L) and

R = 1− ω2p

ω2ω

ω−ωc

L = 1− ω2p

ω2ω

ω+ωc

P = 1− ω2p

ω2 .


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Groupvelocity

Discussion a la Stix

As before the eigenproblem that one consider writes

det (M(n)− λM) = 0, λ =ω2

k2c2

Following Stix we study instead

det (M− µM(n)) = 0, µ =1

λ=

k2c2

ω2.

The dispersion relation reduces to the second order polynomialequation

Aµ2 − Bµ+ C = 0

with

A = S sin2 θ + P cos2 θ, B = RL sin2 θ + PS(1 + cos2 θ)

andC = det(M) = PRL.


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Groupvelocity

Expansion of the dispersion relation

The dispersion relation Aµ2 − Bµ+ C = 0 can be rewritten as

c4

ω4

(S(ω)k21 + P(ω)k23

) (k21 + k33

)− c2

ω2

(R(ω)L(ω)k21 + P(ω)S(ω)

(k21 + 2k23

))+ C (ω) = 0

where S = 12 (R + L), C = PRL and

R = 1− ω2p

ω2ω

ω−ωc

L = 1− ω2p

ω2ω

ω+ωc

P = R = 1− ω2p

ω2 .

Notice that B0 = 0 turns into ωc = 0 and S(ω) = P(ω).


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Groupvelocity

Cutoff

Definition : Cutoff= {k = 0}. x

t

That is PRL = 0.

P = 0 is the previous cutoff : ω = ±ωp.

R = 0 yieldsω3 − ω2ωc − ωω2

p = 0

that is ω =ωc±√ω2c+4ω2

p

2 .

L = 0 yieldsω3 + ω2ωc − ωω2

p = 0

that is ω =−ωc±

√ω2c+4ω2

p

2 .


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Groupvelocity

Cutoff and accessibility (B0 6= 0)

Assume ω and ωc are given. Boundaries of the accesibilitydomain are given by

ω2p = ω2 + ηωωc , η = −1, 0, 1.

The antenna with given frequency ω still facing the fusionplasma with increasing electronic density : the prop. region isnear the wall ( d

dωNc > 0). The cyclotron frequency is assumedconstant.

?

O mode X 2 mode

X 1 mode


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Groupvelocity

Resonance : B0 6= 0 necessary

Definition : Resonance= {k =∞}. x

t

Therefore A = 0 and tan2 θ = −PS . Two major cases of interest

follow.

If θ = 0, it yields S =∞ which is true for ω2 = ω2c . This

is the cyclotron resonnance. Clearly the velocity (ofelectrons) goes to infinity, see (1).

θ = π2 , it yields

S = 0.

This is the more subtle hybrid resonance. We will seethat the velocity (of electrons) also goes to infinity.


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Groupvelocity

The resonant cone Kr

Definition : it is Kr ⊂ R2 the set of (k1, k3) such that

S(ω)k21 + P(ω)k23 = 0.

If ωc = 0 or is small enough, S(ω)P(ω) > 0. SoKr = (0, 0).

If |B0| is large enough, so S(ω)P(ω) < 0. So Kr is theunion of 2 straight lines : at infinity one gets a solution ofthe dispersion relation.

The case SP = 0 is described before.


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Groupvelocity

A short summary

The dispersion relation of the cold plasma theory exhibitsmany features of waves in magnetic plasmas, withsurprising accuracy (Stix).The algebra is tricky and needs time to be understood.

Main features

Accessible domaincut-offresonanceresonant conedependance with respect to the principal plasmaparameters which are ωc and ωp


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Groupvelocity

Additivity principle

The equations are linear. Additivity principle yields

M =

S −iD 0iD S 00 0 P

with S = 1

2 (R + L), D = 12 (R − L) and

R = 1−∑

s(ωs

p)2

ω2ω

ω−εsωsc

L = 1−∑

s(ωs

p)2

ω2ω

ω+εsωsc

P = R = 1−∑

s(ωs

p)2

ω2 ,

where εs = ±1 depending on the charge.Different combinations yields an enormous number of differentmodes.This feature is the main justification of the method used.


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Groupvelocity

Definition

Phase velocity (Reminder) : By definition vϕ = ωk .

Group velocity : It is defined by

vg = ∇kω ∈ R3.

It is the velocity of wave packets : a priori vg 6= |vϕ|.

vg

vϕ

If B0 = 0, it is easy to check that in 1D dωdk = c√

1+ω2p

c2k2

.


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Groupvelocity

Symetrization

Rewrite the eigenvalue problem without elimination 0 −ic2k∧ µ0c2qeNe I

ik∧ 0 0− qe

meI 0 ωcez∧

ebue

= iω

ebue

Symetrize 0 ck∧ −iωpI

−ck∧ 0 0iωpI 0 −iωcez∧

ecbωpue

= ω

ecbωpue

That is A(k)U = ωU where A(k) = A(k)?.

Consider k = k0 + αd : A(α) = A(k0 + αd) is linear in α.By theorem (Kato) : all eigenvalues ωj(α) are real analyticwith respect to α. Orthonormal eigenvectors are analytic.


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Groupvelocity

A simple consequence

∀d such that |d| = 1 ∣∣ω′j(0)∣∣ ≤ c , ∀j .

Proof :1) A(α)uj(α) = ωj(α)uj(α)

2) A′(α)uj(α) + A(α)u′j(α) = ω′j(α)uj(α) + ωj(α)u′j(α)

3)(A′(α)uj(α), uj(α)

)= ω′j(α) (uj(α), uj(α)) = ω′j(α)

Here A′(α) is easy to compute. And it is clear that‖A′(0)|| ≤ c. Therefore

Prop. : The group velocity is bounded by c .


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Groupvelocity

Sign of k · vg

Take d = k. So

A′(0) = A(k)−

0 0 −iωpI0 0 0

iωpI 0 −iωcez∧

︸︷︷︸

H

ecbωpue

Therefore : (A′(0)uj(0), uj(0)) = ((ωI− H)uj(0), uj(0)).

Prop : Assume ω2p < ω2 − ωωc , then ωI− H > 0. Therefore

the group velocity is in the direction of k, that is

ω′j(0) = k · ∇kωj > 0.


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Groupvelocity

Some references

Stix : The theory of plasma waves McGraw-Hill, 1962.

Stix : Waves in plasmas, AIP, 1992.

Swanson : Plasma Waves, Academic Press, 1989.

Miyamoto : Plasma physics and controled nuclear fusion,Springer, 2005.

Brambilla : Kinetic Theory of Plasma Waves-HomogeneousPlasmas, Clarendon, 1998

Freidberg : Plasma physics and fusion energy, CambridgeUniversity Press, 2007.

Kato : Perturbation theory of linear operators, Springer, 1980.


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