Towards closed and integral forms of the Harris dispersion relation

Towards closed and integral forms of the Harris dispersion relation

Daniel W. Crews May 3, 2021

1 Resonant denominators in integration over velocities

This note summarizes progress towards a simple integral form of the dispersion function for elec- trostatic plasma waves making a general angle θ with the magnetic field. It concludes with a few fascinating integrals which appear to generalize the well-known plasma dispersion function,

Z(ζ) ≡ 1

√π Z ∞

−∞

e^−z2

z − ζdz, Im(ζ) > 0 (1)

being the convolution of a Gaussian with a simple pole where ζ = ω/k is the phase velocity. The pole represents the wave-particle resonance in an un-magnetized plasma where the Doppler-shifted frequency is stationary, ω⁰ ≡ (ω − k_kv)/ωc → 0. In the magnetized case, it will be seen here that the integrals are instead convolutions with a cosecant kernel, csc(πω⁰). Thus poles occur at integer multiples of the cyclotron frequency ω − k_kv_k= nω_cwith n ∈ Z, resulting in Landau resonance at n = 0 and cyclotron-harmonic resonance for n 6= 0. Let’s now see how these integrals arise.

2 Linear modes of the Vlasov-Poisson system

The Vlasov-Poisson system with a background magnetic field B = B0 is linearized around a homogeneous equilibrium distribution f (x, v, t) = f0(v) + f1(x, v, t), resulting in the equations

(v × B0) · ∇vf0 = 0, (2)

∂tf1+ v · ∇xf1+ q

m(v × B0) · ∇vf1+ q

mE1· ∇_vf0 = 0. (3) The linearized first-order equation is converted to cylindrical coordinates in the velocity space relative to the magnetic field, (vx, vy, vz) → (v⊥, ϕ, vk), and Fourier-transformed in space and generalized-Fourier-transformed¹ in time for the spectral solution with azimuthal Fourier series [1],

f₁(k⊥, v⊥, ϕ, v_k, ω) = ₀iΦ(k) e

ω²_p ωc

e^{iβ sin ϕ}

∞

X

n=−∞

J_n(β)

n − ω⁰e^−inϕ k_k∂f₀

∂vk

+ nk⊥

1 β

∂f₀

∂v⊥



(4)

where the parameter β ≡ k⊥v⊥/ω_c and as before the cyclotron-normalized Doppler-shifted fre- quency is defined as ω⁰ ≡ (ω − k_kvk)/ωc. By solving for the particle density n(x) = R f₁d³v and

1That is, a one-sided transform is used in time, f (ω) =R∞

0 e^iωtf (t)dt valid for Im(ω) ≥ α > a for some α bounding the asymptotic behavior of the integrand, limt→∞f (t) → e^at, and the evolution is considered to be switched on at the beginning, i.e. f (t) = 0 for t < 0 (causality). The inverse transform is then f (t) = _2π¹ R∞+iα

−∞+iαe^−iωtF (ω)dω. For general distribution functions we can safely take α = 0⁺.

substituting into the Poisson equation, one arrives at the Harris dispersion relation [2],

(ω, k) = 1 +ω_p² k²

Z ∞

−∞

dvk

∞

X

n=−∞

Hn(vk) = 0 (5)

where the sum over H_n(v_k) is given by the series of polar velocity-space integrals

∞

X

n=−∞

H_n(v_k) = Z ∞

0

(2πv⊥dv⊥)h k⊥Υ1

β

∂f₀

∂v⊥

+ k_kΛ∂f₀

∂v_k i

(6)

by defining the two functions Υ and Λ as the infinite Bessel series, Υ =

∞

X

n=−∞

n

ω⁰− nJ_n²(β), (7)

Λ =

∞

X

n=−∞

1

ω⁰− nJ_n²(β). (8)

The sequence of simple poles at integer multiples of ωc is here split into an infinite series.

2.1 Exploring closed and integral forms of the dispersion function Each of the series are known in closed form as cases of the Newberger sum rule,

Υ = πω⁰

sin(πω⁰)J_ω⁰(β)J−ω⁰(β) − 1, (9)

Λ = π

sin(πω⁰)J_ω⁰(β)J_−ω⁰(β), (10)

in terms of the product of Bessel functions J_zJ−z of complex-order.

2.1.1 Closed forms using hypergeometrics

Direct integration with these closed forms is not yet straight-forward. For example, in the case of a Maxwellian plasma one may be able to evaluate the polar velocity-space integrations in terms of hyper-geometric functions by evaluation of the integral [3, p. 699]

√ 2 2πα²

Z ∞ 0

e^−v2/2α2 πµ

sin(πµ)J_µ(qv)J−µ(qv)dv =₂F₂

 ₁

2, ¹₂ 1 + µ, 1 − µ



− 2α²q²

(11) and its more general cousin for use with loss-cone-like ring distributions,

1 2ⁿ⁻¹α²ⁿΓ(n)

Z ∞ 0

v²ⁿ⁻¹e^−v2/2α2 πµ

sin(πµ)J_µ(qv)J−µ(qv)dv =₂F₂

 ₁

2, n 1 + µ, 1 − µ



− 2α²q² . (12) Yet to evaluate the resulting integrals over the parallel velocity remains a difficult problem as the integration is with respect to the index as a singular parameter. That is, by rearranging the defining power series of the hyper-geometric with Euler’s reflection formula Γ(x)Γ(1 − x) = π csc(πx) to

2F2

 ₁

2, n 1 + µ, 1 − µ



(z) = πµ sin(πµ)

∞

X

m=0

(1/2)m(n)m

Γ(1 + m + µ)Γ(1 + m − µ) z^m

m! (13)

one can see that because the reciprocal Gamma function is entire (i.e. has no poles), the singularity in the parallel-velocity integral (here related to µ) will remain entirely in the cosecant kernel.

2.1.2 Integral forms using Neumann’s theorem

One may also write the J_n² series as trigonometric integrals over a zero-order Bessel function. That is, forms for Eqs.9-10 follow from Neumann’s integral

πJα(β)J−α(β) = Z π

0

J0(2β cos(θ/2)) cos(αθ)dθ (14) as the following two integral expressions

Υ = −β

sin(πω⁰) Z π

0

J1(2β cos(θ/2)) sin(θ/2) sin(ω⁰θ)dθ, (15)

Λ = 1

sin(πω⁰) Z π

0

J0(2β cos(θ/2)) cos(ω⁰θ)dθ. (16)

The expression for Υ is arrived at by integrating by parts and using J₀⁰ = −J₁. A useful expression for integration by parts with the distribution function is

∂Υ

∂β = − β sin(πω⁰)

Z π 0

J0(2β cos(θ/2)) sin(θ) sin(ω⁰θ)dθ. (17) Now consider the distribution function to be separable, of the form f0(v) = f⊥(v⊥)fk(vk). For example, we will take f⊥(v⊥) as a ring distribution and f_k(v_k) as a Maxwellian for a representative loss-cone distribution. Then upon substitution of the integral forms of these series into the Harris dispersion relation, the velocity space integration can be written in two parts. The first of these is, after integration by parts using the above expression for ∂_βΥ,

I₁ ≡ k_⊥ Z π

0

dθ sin(θ) Z ∞

−∞

dv_ksin(θω⁰) sin(πω⁰)f_k(v_k)

Z ∞ 0

(2πv⊥dv⊥)f⊥(v⊥)J₀(2β cos(θ/2)) (18) and the second is similarly (without integration by parts)

I2 ≡ k_k Z π

0

dθ Z ∞

−∞

dvk

cos(θω⁰) sin(πω⁰)

∂f_k(v_k)

∂vk

Z ∞ 0

(2πv⊥dv⊥)f⊥(v⊥)J0(2β cos(θ/2)) (19) where recall β = k⊥v⊥/ω_c and ω⁰ = (ω − k_kv_k)/ω_c. The inner most integral is a zero-order Hankel transform and relatively easily evaluated. Denoting it as

H₀[f⊥](λ) ≡ Z ∞

0

(2πv⊥dv⊥)f⊥(v⊥)J0(2β cos(θ/2)) (20) with λ = λ(θ) the Harris dispersion relation may be written in the following form,

(ω, k) = 1 +ω²_p k²

Z π 0

H₀[f⊥](λ)h

k⊥sin(θ)V1+ k_kV2

i

dθ (21)

having defined the two parallel-velocity singular integrals V1 and V2 as

V1 = Z ∞

−∞

sin

_θk

k

ωc (z − v)

 sin

_πk

k

ωc (z − v)

 fk(v)dv, (22)

V2 = Z ∞

−∞

cos_θk

k

ωc (z − v) sin_πk

k

ωc (z − v)

∂f_k(v)

∂v dv (23)

where z = ω/k_k is the parallel phase velocity, and defined for Im(z) > 0 according to the con- vention of the chosen one-sided time transform. Generally the parallel background distribution will be Maxwellian, though one can examine other distributions. In this case, the integrand of Eqs. 22 & 23 will be Gaussian. These integrals clearly generalize the singular integral of the plasma dispersion function. In the limit k⊥ → 0 the usual dispersion integral is recovered as Rπ

0 cos(αθ)dθ = α⁻¹sin(πα), and similarly in kk → 0 the perpendicular dispersion relation in angular integral form. So, it is of great interest for cyclotron wave theory to investigate the integral

Z ∞

−∞

e^−v2/2

sin(π(z − v))dv, Im(z) > 0 (24)

as an extension of the plasma dispersion function to the cyclotron harmonics of all orders. This same integral kernel will arise even if the entirely closed form approach involving the hypergeometric function₂F₂(z) is pursued, as Eq.13 shows its poles to be of the cosecant form.

2.2 Closed forms for perpendicular propagation using hypergeometrics

The following notes use the integral transforms in Eq. 12 to express the dispersion function for perpendicular propagation as entirely closed forms involving generalized hypergeometric functions of type ₂F₂. Research along these lines has an old history. For example, in D.G. Swanson’s book “Plasma Waves,” results involving generalized hypergeometrics of type2F3 and 1F2 are used extensively, with an accompanying mathematical appendix B.5 [4].

2.2.1 The base case: Maxwellian-distributed plasma

In the case of purely perpendicular propagation there is no singular integral and the math simplifies considerably! Consider the Fourier mode form of the zeroth moment of f₁ for k_k = 0,

en = ω_p²₀Φe e

Z ∞ 0

(2πdv)∂f₀

∂v



1 − πω

sin(πω)J_ω(kv/ω_c)J−ω(kv/ω_c)

, (25)

for the case of a purely Maxwellian plasma,

∂f₀

∂v = − v

2πα⁴e^−v2/2α2. (26)

Each integral may be done analytically, α⁻⁴

Z ∞ 0

ve^−v2/2α2 πω

sin(πω)J_ω(kv/ω_c)J−ω(kv/ω_c)dv = α⁻²₂F₂

 ₁

2, 1 1 + ω, 1 − ω



(−2(kr_L)²) (27) α⁻⁴

Z ∞ 0

ve^−v2/2α2dv = α⁻² (28)

giving a zeroth moment,

en = ω²_p α²

₀Φe e



2F₂

 ₁

2, 1 1 + ω, 1 − ω



(−2(kr_L)²) − 1

(29) which substitutes into the dispersion relation to give

(ω, k) = 1 + 1 (kλD)²



2F₂

 ₁

2, 1 1 + ω, 1 − ω



γ
− 1

(30)

where γ ≡ −2(kr_L)² as an entirely closed form solution! That is, if you consider generalized hypergeometric functions with complex parameters to be a “closed form”. Now for purposes of calculation it’s more useful to extract the poles from the Pochhammer symbol coefficients of the hypergeometric. That is, manipulate the power series as follows

2F₂

 ₁

2, 1 1 + ω, 1 − ω



− 2(kr_L)²
− 1 =

∞

X

s=0

(1/2)_s(1)_s (1 + ω)s(1 − ω)s

(−2(kr_L)²)^s

s! − 1 (31)

= (−2(krL)²) πω sin(πω)

∞

X

s=0

(1/2)s+1(−2(krL)²)^s

Γ(2 + s + ω)Γ(2 + s − ω) (32) by first subtracting off the first term s = 0 which is equal to one, then second shifting the index back to the origin, and lastly expanding the denominators according to the two functional relations

(a)n≡ Γ(a + n)

Γ(a) , (33)

Γ(1 + ω)Γ(1 − ω) = πω csc(πω), (34)

with the second relation a consequence of Euler’s reflection formula. Substitution into the dispersion relation and rearranging the cosecant yields the formula

sin(πω) πω + 2

ωp

ω_c

2 ∞

X

s=0

(1/2)s+1(−2)^s

Γ(2 + s + ω)Γ(2 + s − ω)(krL)^2s= 0. (35) which is observed to produce the correct dispersion relation for a fifty term partial sum. Unfortu- nately, ₂F₂ functions are not rapidly convergent.

Figure 1: Zero contours of the power series expression with a 50-term partial sum of power series.

2.2.2 Closed form for perpendicular propagation in a loss-cone distribution

Similar expressions may be found for loss-cone-type distributions with ring parameter j. That is, for a general ring distribution with gradient,

f₀(v) = 1 2πα²Γ(j + 1)

− v² 2α²

j

exp

− v² 2α²



, (36)

∂f₀

∂v = j − v²

2α²

1

vf₀, (37)

then asR∞

0 ∂vf0dv = 0 for a ring distribution, the integration evaluates to

(ω, k) = 1 + 1 (kλ_D)²



2F2

 ₁

2, j + 1 1 + ω, 1 − ω



γ
−₂F2

 ₁

2, j 1 + ω, 1 − ω

 γ


= 0 (38)

where again γ ≡ −2(krL)². Two possible approaches to calculation are to either directly compute the difference of hypergeometric functions by combining their defining series and extracting the simple poles,

2F2

 1

2, j + 1 1 + ω, 1 − ω



γ
−₂F2

 1

2, j 1 + ω, 1 − ω



γ
= πω sin(πω)

∞

X

s=0

(1/2)s (j + 1)s− (j)_s
Γ(s + 1 + ω)Γ(s + 1 − ω)

γ^s s! (39) or to apply a differentiation formula to express the dispersion equation in the following form

ω_p ωc

2 d dγ

n

2F2

 ₁

2, j 1 + ω, 1 − ω

 γ
o

= j

2 (40)

in which case it may be possible to express the eigenvalues of perpendicular propagation in a ring-distributed plasma as the solution to the single transcendental equation

ωp

ω_c

2 2F₂

 ₁

2, j 1 + ω, 1 − ω



γ
= j

2γ. (41)

This remains to be verified with numerical experiments. Nevertheless, the results are important for the general oblique-propagation case by suggesting that the perpendicular-velocity integrals may be done in exactly this way with cosecant-type poles. The parallel velocity-space integration will then involve the singular integral given in Eq. 24!

References

[1] D.A. Gurnett and A. Bhattacharjee. Introduction to plasma physics. Cambridge University Press, Second edition, 2017.

[2] J.A. Tataronis and F.W. Crawford. Cyclotron harmonic wave propagation and instabilities: II.

Oblique propagation. J. Plasma Phys., 4, 1970.

[3] I.S. Gradshteyn and I.M. Ryzhik. Table of Integrals, Series, and Products. Academic Press, Seventh edition, 2015.

[4] D.G. Swanson. Plasma Waves. Institute of Physics, First edition, 2003.