Contrib. Plasma Phys. 42 (2002) 1, 55−60

Nonlinear Electrostatic Ion-Acoustic Wavesin the Solar Atmosphere

C.-V. Meistera), A. V. Volosevich

b)

a)Astrophysical Institute Potsdam, An der Sternwarte 16, 14482 Potsdam,Germany

b)Mogilev State University, Belaruse-mail: cvmeister@aip.de

Received 25 August 2000, in final form 4 February 2001

Abstract

Basing on recent solar models, the excitation of ion-acoustic turbulence in the weakly-collisional, fully and partially-ionized regions of the solar atmosphere is investigated. Withinthe frame of hydrodynamics, conditions are found under which the heating of the plasma byion-acoustic type waves is more effective than the Joule heating. Taking into account waveand Joule heating effects, a nonlinear differential equation is derived, which describes theevolution of nonlinear ion-acoustic waves in the collisional plasma.

1 Introduction

The solar atmosphere is a rather complicated inhomogeneous plasma structure wherethe values of the main parameters, e.g. of the temperature and densities of chargedand neutral particles, vary within orders of magnitude [1]. According to acceptednumerical models, one may conclude that the solar atmosphere consists of three dif-ferent layers: of the lower-laying weakly-ionized photospheric plasma with relativelystrong particle collisions, of the weakly-collisional chromospheric plasma, and, above avery thin transition region, of the fully-ionized high-temperature coronal plasma. Onemay take it for granted that in the solar atmosphere ion-acoustic waves, the so-calledp-modes, exist [2, 3]. In the present paper, hydrodynamic models of linear and non-linear ion-acoustic waves in the weakly-collisional solar atmosphere are developed. Inthis region, the electron density ne is about 1017 m−3, the neutral-particle density nn

is 1016 − 1020 m−3, the electron temperature equals Te ≈ 4200 − 6300 K, the elec-tron plasma frequency ωpe and the electron cyclotron frequency ωce are of the order of3 · 1010 Hz, electron Debye and electron cyclotron radius amout to 3 · 107 m3, and theratio of the electron-neutral collision frequency to the coulomb collision frequency νei

varies between 10−4 and 10−1.

2 Hydrodynamic model

The investigation of the ion-acoustic waves is done starting with the hydrodynamicsystem of equations for electrons and ions taking thermal conductivity effects intoaccount. The momentum and energy balances as well as continuity equations usedread

menedve

dt= −∇(neTe)− ene

E − νenmeneve + Re, (1)

c© WILEY-VCH Verlag Berlin GmbH, 13086 Berlin, 2002 0863-1042/02/01/01-0055 $ 17.50+.50/0

56 Contrib. Plasma Phys. 42 (2002) 1

minidvi

dt= −∇(niTi) + eni

E − νinminivi − Re + ηi∆vi, (2)

3

2nadTa

dt+ naTa∇va = −divqa +Qa, a = e, i (3)

∂na

∂t+∇(nava) = 0, a = e, i. (4)

Here the index a = e designates the electrons and a = i determines the ions, d/dt =∂/∂t + va∇. na, Ta, va are the density, temperature and velocity of the particleof kind a, respectively. Deriving Eq. (1, 2) the ion viscosity ηi = niTi/νi and thecollision frequencies of the charged particles with the neutrals νan were taken intoaccount. νi = νin + νie + νii. The friction force Re = −a1νeineme(ve − vi) − a2ne∇Te

consists of two parts. One part occurs because of the coulomb collisions between theelectrons and ions having a non-zero relative velocity. The second contribution is thethermoforce which is caused by the temperature gradient. The electron heat currentqe = a2neTe(ve−vi)−a3neTe∇Te/(meνei) is a consequence of the relative velocity of theelectrons with respect to the ions, and of the gradient of the electron temperature. Theion heat current reads qi = −ainiTi∇Ti/miνie. The amount of heat contained in theelectron system because of the electron-ion collisions and the existence of a temperaturegradient is Qe = a1meneνei(ve − vi)

2 + a2ne(ve − vi)∇Te − 3meneνei(Te − Ti)/mi, andthe heat of the ion component because of the collisions with the electrons may beestimated by Qi = 3meneνei(Te − Ti)/mi. The coefficients a1, a2, a3, and ai dependon the type of the collision integral chosen for the description of the plasma. Withinthe Braginsky model [4], the coefficients equal a1 = 0.51, a2 = 0.71, a3 = 3.16, andai = 3.9.

3 Linear ion-acoustic waves

In the following the changes of the electron and ion temperatures because of the waveand Joule heating by collisions will be considered. It is assumed that the equilibriumtemperatures of the electrons and ions equal Teo and Tio respectively. Further it issuggested that in the plasma weak disturbances of the density na1, velocity va1 andtemperature Ta1 of the charged particles exist. The disturbances are approximatelyproportional exp(−iωt + ikr), and the quasineutrality condition nio = neo = no issatisfied. Then, from the continuity equation Eq. (4) follows

ne1 =kve1

ω∗e

, ni1 =kvi1

ω∗i

, ω∗e = ω − kveo, ω∗

i = ω − kvio. (5)

voe and voi are constant drift velocities of the electrons and ions, which may be causedby constant electric fields or by the existence of beams of charged particles. ¿From theenergy balances Eqs. (3) and Eq. (5), it follows in first order of the wave disturbances(voi = 0, vte-thermal velocity of particle a)

Te1

To=ne1ω

∗e

noωe

(1 + i

[2a1

νekvo

k2v2te

+ 3meνeω

∗i

miω∗e ωi

])(1 + 9

m2eν

2e

m2i ωiωe

)−1

, (6)

Ti1

To=

ω

dωi

ni1ω

∗i

noω+

3imeνene1Ωe

miωωeno− 3imeνeni1

miωeno

(a2 +

2ia1νekvo

k2v2te

), (7)

C.-V. Meister, A.V. Volosevich, Ion-Acoustic Waves in the Solar Atmosphere 57

Ωe = ω∗e +a2ω+ i2a1νekvo/(k

2v2te)(ω−kvo/2), d = 1+9m2

eν2e/(m

2i ωiωe), ωe = 3ω∗

e/2+ia3k

2v2te/νe + 3imeνe/mi, ωi = 3ω∗

i /2 + iaik2v2

ti/νe + 3imeνe/mi. Under the condition|d−1| 1, the heating of the plasma by the waves is stronger than the Joule heating.In the following, it is assumed that the plasma is almost fully ionized and consists ofelectrons and heavy ions. Ion viscosity plays the main role in the wave damping, andthe electron temperature gradients cause an electron heat current. From Eqs. (1, 2)then follows (me mi, v ≈ vi, and ne = ni = n)

∂vi

∂t+ (vi∇)vi = − 1

nmi∇(Te + Ti)n +

ηi

mi∆vi − νinvi. (8)

Linearizing Eq. (8), one may find for the ion velocity

vi =kc2s

ω∗∗i + ik2η

(n

no+

1

1 + τ

[τTe1

Teo+Ti1

Tio

]), Te = Teo + Te1, Ti = Tio + Ti1, (9)

ω∗∗i = ω − kvio + iνin, c2s =

(Teo + Tio)kB

mi, τ =

Teo

Tio. (10)

In the case | d− 1 | 1, from Eqs. (5-7) the dispersion equation

ω∗i ω

∗∗i +ik2ηω∗

i = k2c2s(1+a+ib), a = τRere+Reri, b = τ Imre+Imri, re =Ωe

ωe, ri =

ω∗i

ωi

(11)may be found. Introducing as usual the real ωr and imaginary γ parts of the wavefrequency, ω = ωr + iγ, one has

ω2r = k2c2s(1 + a), γ =

1

2

[bkcs −

(νin +

k2v2ti

νi

)]. (12)

Thus, under the condition bkcs > νin + k2v2ti/νi, an instability occurs. If in

the plasma the collisions between charged and neutral particles are negligible andthe coulomb collisions play the main role, this condition may be represented byvo/cs > k2v2

te/(2a1ν2ei(1 + τ )) = βcr. Obtaining the last inequality, it was used that

b ≈ 2a1νeikvo/k

2v2te. Thus, for the solar atmosphere at altitudes of h = 500 km, with

Te ≈ 4.2 · 103 K, vte ≈ 2.5 · 105m/s and νe ≈ 5 · 107 Hz, one finds βcr ≈ 5 · 10−3 · k2.At altitudes h = 1000 km, one has Te ≈ 6 · 103 K, vte ≈ 3 · 105m/s and νei ≈ 2.2 · 106

Hz, βcr ≈ 2 · 10−2 · k2. Then it follows from k <√vo/csνe/vte, that k < 200 m−1

at h = 500 km, and k < 7.3 m−1 at h = 1000 km. At the temperature minimumwith Tmin = 4.2 · 103 K (h = 515 km), vte ≈ 2.5 · 105m/s and νe ≈ 8 · 106 Hz,one has wave numbers below 32 m−1 and wave lengths above 19 cm. Thus the con-dition for the plasma heating by wave-particle collisions | d − 1 | 1 is satisfied,kcs 3 · 5 · 10−4 · νe = 1.2 · 10−2, and thus k 2 · 10−6 m−3 as cs ≈ 5.9 · 103

m/s. ¿From the above estimates follows, that in the case 3meνe/mi a2k2v2

te/νe theplasma of the solar atmosphere is essentially heated by the waves. If there would notbe any heating by the waves, the considered oscillations would be ion-acoustic ones.If a 1 is satisfied, the heating process modifies the wave dispersion only a little.The amplitudes of the linear ion-acoustic waves increase (γ > 0) if the heating by thewaves (first term in the brackets of Eq. (12)) is more effective than the damping of thewaves by the collisions of the ions with the neutral particles and by the ion viscosity.The waves increase because of the (assumed to be) constant electron drift velocity,which may be a consequence of almost constant electric fields or of the appearance ofelectron beams.

58 Contrib. Plasma Phys. 42 (2002) 1

4 Nonlinear ion-acoustic waves

As follows from the linear theory, under supercritical conditions in the solar atmo-sphere occur ion-acoustic waves with increasing amplitudes. Thus here the system ofequations Eqs. (1-4) will be solved again, but this time the wave disturbances willnot be considered as small ones. Taking into account the main terms describing theincrease of the wave amplitudes (heat currents because of temperature gradients) andtheir saturation (ion viscosity), Eqs. (1-4) may be transformed into

∂vi

∂t+ (vi∇)vi = −c2s(∇n+ 0.5∇Te) + η∆vi − νinvi, (13)

∂

∂tlnn+ vi∇n+ divvi = 0, (14)

nTed

dtln

(T

3/2e

n

)= divκe∇Te + nTe∇ lnn. (15)

For simplicity, in Eqs. (13, 15) it was assumed Teo = Tio = To, c2s = 2To/mi and

κe = a2nev2te/νe, and the ion heating was neglected. The physical sense of the energy

balance Eq. (15) consists in the fact, that the changes of the entropy (S = T3/2e /N

is the entropy transferred from the system by one electron, N = neV represents thenumber of electrons) are caused by dissipative processes which are the result of thegradients of the temperature and the density of the charged particles. In this case,Eq. (15) may be approximately represented by

κo∆Te + (vo∇) lnN = 0, κo = a2v2te/νe. (16)

In the case of weak disturbances, Eq. (16) gives a relation between the electron temper-ature disturbances and the density disturbances, Te1/To ∼ ikvoνe/(a2k

2v2te) = bne1/no.

Further, combining Eqs. (13, 14, 16) one finds

∂2vi

∂t2− c2s∆vi = − c2s

2κovo∇vi + ηo

∂

∂t∆vi − νin

∂vi

∂t− ∂

∂t(vi∇)vi. (17)

Taking the relation (14) between the density and the velocity of the charged particlesinto account, an analogeous equation may also be derived for N . The nonlinear partialdifferential equation describes the nonlinear evolution of ion-acoustic waves. The leftside of Eq. (17) determines the propagation velocity of a nonlinear structure (of thenonlinear wave). The first term on the right side of Eq. (17) describes the excitationof the wave by the current (caused by a constant electric field), the second term isresponsible for the high-frequency harmonics because of the ion viscosity, and the thirdcontribution describes the damping by the collisions with the neutral particles. Thelast term takes the nonlinear effect into account.It is extremely difficult to find an analytical solution of the nonlinear differential equa-tion Eq. (17). Thus, here the evolution of one-dimensional nonlinear structures willbe investigated. The one-dimensional form of Eq. (17) reads

∂2vi

∂t2− c2s

∂2vi

∂x2= − c2s

2κovo∂vi

∂x+ ηo

∂3

∂t∂x2vi − νin

∂vi

∂t− 1

2

∂2

∂t∂xv2

i . (18)

C.-V. Meister, A.V. Volosevich, Ion-Acoustic Waves in the Solar Atmosphere 59

To look for possible stationary nonlinear structures in the plasma, in Eq. (18) a newspatial coordinate s = x−vtwill be introduced, where v is the velocity of the stationarystructure. Thus, one obtains the equation

βo∂3vi

∂s3+ (M2 − 1)

∂2vi

∂s2+ α

∂vi

∂s− σ

∂2v2i

∂s2= 0. (19)

M = v/cs equals the propagation velocity of the structure which is normalized by thesound velocity, β = viηo/c

2s, α = vo/(2κo) −Mνi/cs, and σ = 1/c2s . Integrating Eq.

(19) once over s, one observes (vi, ∂vi/∂s and ∂2vi/∂s2 vanish at s→ ∞)

βo∂2vi

∂s2+ (M2 − 1)

∂vi

∂s+ αvi − σ

∂v2i

∂s= C. (20)

Solving Eq. (20) in first order with respect to | M2 −1 | (using the Krylov-Bogoljubovmethod [5]), one gets for the ion velocity disturbances

vi = r(s) cos ϕ(s) = ro exp

(−(M2 − 1)s

2βo

)cos

(ϕo +

√α

βos

), (21)

where ro = r(s = 0) and ϕo = ϕ(s = 0) are the amplitude and the phase of thenonlinear wave at s = 0.

5 Conclusions

• The excitation of ion-acoustic turbulence in the weakly-collisional, fully and partiallyionized regions of the solar atmosphere is studied taking into account temperature gra-dients.• Therefore the variations of the electron and ion temperatures because of effectiveparticle collisions - including heating by the waves - are considered.• Nonlinear structures in the solar atmosphere are studied using the hydrodynamicsystem of equations for electrons and ions taking thermal conductivity effects intoaccount.• A nonlinear partial differential equation is derived describing the behaviour of non-linear solar ion-acoustic waves.

Acknowledgements

A.V. Volosevich would like to thank the organization Deutsche Forschungsgemeinschaftfor financing her stay at the Astrophysical Institute Potsdam to work on the giventopic within the frame of the contract WER17/2/00. The both authors gratefullyacknowledge financial support by the project 24-04/055-2000 of the Ministerium furWissenschaft, Forschung und Kultur des Landes Brandenburg.

References

[1] Stix M., The Sun. An Introduction, Springer-Verlag, Berlin (1991).[2] Roberts B., Advances in Solar System Magnetohydrodynamics, eds. E.R. Priest and

A.W. Hood, University Press, Cambridge (1991), chap. 6, 105

60 Contrib. Plasma Phys. 42 (2002) 1

[3] Sturrock P.A., Plasma Physics. An Introduction to the Theory of Astrophysical,Geophysical and Laboratory Plasmas, University Press, Cambridge (1994)

[4] Braginsky S.I., Problems of Plasma Theory, Gozatomizdat, Moscow (1963), 183[5] Korn G.A., Korn T.M., Mathematical Handbook, McGraw-Hill Book Company, New

York (1968), 296