23F04
Equilibrium velocity distributions in Alfvenic turbulenceand “apparent” temperature
アルフェン乱流中の平衡分布と「見かけの」温度
Yasuhiro Nariyuki
成行 泰裕
Kochi National College of Technology, Nankoku City, Kochi 783-8508, Japan
高知工業高等専門学校 〒783-8508 高知県南国市
In this presentation, the equilibrium velocity distribution in parallel propagating Alfvenic turbulence ispresented. The “apparent temperature” due to the non-resonant ion heating is explained on the basis of theequilibrium velocity distribution.

1. Introduction and Result
In this presentation, we report our recent
results on “Alfvenic” equilibrium velocity
distributions in collisionless plasmas[1-2]. In thepresent proceeding, a simple equilibrium
distribution
relatedtothenonresonant“pseudo-“ heating is presented.
We briefly discuss the Sonnerup-Su solution,
which is an exact Alfven wave solution of theVlasov-Maxwell system[3]. We here assume that
all the physical quantities depend only on one
spatial coordinate (x). The velocity distributionfunctions (f) obey the Vlasov equation

df ∂f
∂f dv
=+ vx
+ ⋅∇v f = 0,∂x dt
dt ∂t(1)where v=(vx, vy, vz),

dv q
= (e + v × b),
dt m

(2)

e=(ex, ey, ez), b=(bx, by, bz). We here assume ex=0,and bx=B0=const. When a monochromatic wave
( b = b0 exp(-iφk ) , where φk = ωt - kx , b0, ω ,and k=const.) is given, there are two constants ofmotion[1-4]
mv 2  q
 vφ mvx - x  + v ⋅ b ≡ H ,2  km((vx − vφ ) 2 + v y2 + vz2 ) ≡ E.
2(3)
(4)

The Sonnerup-Su solution f(x,v,t)=f (αH + βE ) isan exact solution of (1), where α , β are theconstants which satisfy the conditions[1-3].
Remark that the solution is exact even in plasmas
with finite temperature. In the entropy-maximizedform, the Sonnerup-Su can be written as abi-Maxwellian distribution with the oscillating
bulk velocity[1]. When we consider thenondispersive waves in the low-frequency limit,
the Sonnerup-Su solutions can be extended tothose of incompressible Alfvenic turbulence withthe broadband spectrum[2], v x2 + (v y − cby ) 2 + (v z − cbz ) 2 f = A exp −, (5)
vth2

whereA = n0 /( π 3 vth2 ),velocity,

vth is the thermalb = ∑ bk exp(−iφk ),c = −vφ / B0 ,respectively. We here assume ux=0.
The distribution (5) directly gives the “apparenttemperature”[2]TA ⊥ = ∫m(v y2 + v z2 )2

 vφ2 | b |2 fdv = n0 k BT 1 + 2 2 , (6) v B th0 which is equivalent to the past work by Wang etal.[5], in which the nonresonant ion heating in
Alfvenic turbulence was firstly demonstrated. The
result merely shows that the nonresonant ion
heating in Alfvenic turbulence is a relaxation
process to “Alfvenic” equilibrium state. Remarkthat the distribution is a mechanical equilibrium andthe ionizations and collisions in the solar
atmosphere, the chromospheric and coronalplasmas may play an important role in increasing
the physical entropy.
Acknowledgments
This work was supported by Grant-in-Aid forYoung Scientists (B) No.22740324 from MEXT ofJapan.
References[1] Y. Nariyuki: Phys. Plasmas, 18 (2011), 052112 .
[2] Y. Nariyuki: Phys. Plasmas, 18 (2011), 092118 .
[3] B. U. O. Sonnerup, and S. Y-. Su, Phys. Fluids, 10(1967), 462.
[4] D. Verscharen, and E. Marsch, Ann. Geophys., 29(2011), 909.
[5] C. B. Wang, C. S. Wu, and P. H. Yoon, Phys. Rev.
Lett., 96 (2006) 125001.