ELSEVIER

25 July 1994

Physics Letters A 190 (1994) 309-316

PHYSICS LETTERS A

A kinetic derivation of a generalized Zakharov-Kuznetsov equation for ion acoustic turbulence in a magnetized plasma

A.M. Hamza Physics Department, University of Western Ontario, London, Ontario, Canada N6A 3K7

Received 24 January 1994; revised manuscript received 1 April 1994; accepted for publication 12 April 1994 Communicated by M. Porkolab

Abstract

The Zakharov-Kuznetsov equation has been derived by a number of authors in the context of a fluid description as well as a kinetic description in a nonmagnetized plasma. We propose, in this short paper, to discuss the derivation of the equation governing the evolution of ion acoustic waves in a magnetized plasma using kinetic theory. This allows us to introduce Landau damping more specifically, as well as other purely nonlinear kinetic effects. We first investigate the effects of Landau damping on the conservation laws, and therefore on the solitary solution known to exist in the fluid regime. We also derive a conservation law very much like the one-dimensional case.

1. Introduction

In this paper we shall derive the equation governing the development and evolution of nonlinear ion acoustic waves in a strongly magnetized plasma. A similar work was done by Ott and Sudan [ 1 ] and Taniuti [2] in the case of a nonmagnetized one-dimensional plasma. Along the same lines a kinetic derivation of a "modified Korteweg-de Vries" (MKdV) equation was proposed by Karpman [3]. The magnetized case was first studied by Zakharov and Kuznetsov [4] using a fluid description of the plasma. Shivamoggi [5] rederived the Zakharov-Kuznetsov equation in the context of fluid theory and discussed the recurrence and Lagrange stability of solutions to this equation. In a less recent paper Qian et al. [6] derived the Zakharov- Kuznetsov equation based on a kinetic description. However they used phase space density conservation along particle orbits to solve the Vlasov equation, and then substituting into the Poisson equation they were able to recover the Zakharov-Kuznetsov equation with an extra term due to the reflected electron population. This method requires the evaluation of the orbits exactly. By including only the reflected electron species they have automatically made a drastic assumption; that is to say that the electrostatic potential of the structure that develops is rarefactive. Dupree [7 ] was able to investigate thoroughly the effects of nonlinear resonant particle effects on the evolution of a time dependent ion hole; his investigation included trapped and reflected particles, i.e., a complete analysis of the problem. The equation derived by Qian et al. [6 ] does not include the effects of electron Landau damping even though their final equation was obtained in the limit of 7] -~ 0, and so did Shivamoggi [ 5 ]. In the fluid derivation, the electron inertia is neglected, and an adiabatic behavior of

0375-9601/94/$07.00 (~ 1994 Elsevier Science B.V. All rights reserved SSDI

310 A.M. Hamza / Physics Letters A 190 (1994) 309-316

the electrons is assumed, however when T~ --, 0 Landau damping is due to electrons as pointed out by Ott and Sudan [1 ]. We shall in this paper remedy both problems, namely derive the equation with the inclusion of electron Landau damping, as well as other nonlinear effects. As opposed to Qian et al. [6] we will not consider particle orbits but rather use the classical approach of Fourier and Laplace transforms of the Vlasov-Poisson system to derive the evolution equation as is done by Galeev and Sagdeev [8]. The derivation we propose includes nonlinear resonant particle effects such as trapping. However, we will not investigate the effects of the nonlinear resonant particle effects thoroughly. We will rather reproduce the Zakharov-Kuznetsov solitary solution in the absence of electron Landau damping, and then investigate the decaying behavior of the soliton due to Landau damping. In the case where the reflected electrons are included, a localized solution of the BGK type does not exist as was shown by Hamza in the one-dimensional nonmagnetized case [9] and in the magnetized case [ 10 ]. Because of the complexity of this problem we shall not enter into the details of trapping effects, these effects will be addressed in a subsequent paper.

We shall first derive the equation governing the evolution of the ion acoustic fluctuations using the Vlasov- Poisson system in the limit of low frequency long wavelength. We will briefly describe the solitary solution in the absence of electron Landau damping, and then discuss the effects of electron Landau damping on the stationary solution. We finally conclude.

2. Derivation of the evolution equation: the model

Let us consider a strongly magnetized plasma (low fl << 1), with Te >> Ti, where Te and Ti represent the electron and ion temperatures, respectively. The equations that describe the magnetized system under consideration are the Vlasov and Poisson equations written in the following forms,

Ofj(x ,v , t ) Ot + v . V f j ( x , v , t ) + v × l g j . V ~ f j ( x , v , t ) = q J ~ v ~ . V ~ f j ( x , v , t ) , (1)

my

V2cI) = -4rr E qj f dv f j (x , v, t), (2) J J

where the subscript j in this case represents the plasma species, and DJ represents the gyrofrequency of the species j pointing along the magnetic field. We note that we have placed the nonlinear term in the Vlasov equation on the right hand side. The left hand side is nothing but the total time derivative of the distribution function along the particle orbits defined as follows,

dx dv dt - v, dt - vxg- / j . (3)

Therefore one can write the Vlasov equation in the following form,

Df j ( x , v , t ) _ q_Lv ~ . V~f j (x ,v , t ) . (4) Dt rnj

One can rewrite this partial differential equation in the form of an integral along particle orbits (see Ref. [8] or the work of Dupree)

t

f y (x ,v , t ) = Cl1 f d r V ~ ( x ( z ) , z ) . Vv f j ( x ( z ) , v ( z ) , z ) . (5) mj J

- o o

Following the classic procedure of solving this equation by iteration, see, for example, Ref. [8] or Ref. [11 ], one can express the Fourier transform of the distribution function in the following form,

A.M. Hamza / Physics Letters A 190 (1994) 309-316 311

]~(k,v,~o) = Zfj(")(k,v, co), (6) n

t

fj(n)(k,v,o~)exp [ i ( k . x - t o t ) ] = i qj Z f dzO(ki,oJ1)exp{i[ki .x(z) -- OJIT]} mj /'l +i2=/' -oo

X k l " O f J (n - l ) (k2 'v ' ° )2 ) Ov exp {i [k2. x (z) - w2z]}. (7)

Substituting expression (7) into the Poisson equation and using a perturbation scheme to second order in amplitude leads to

k2E( I ) ( ° ) )* (k ' ( ° ) ~ Z Ck(~)k2 ( ( ° l ' ° ' )2 )* (k l ' ( ' ° l )* (k2 ' ° )2 ) = 0, (8)

il +/'2=i

where ]¢ - (k, to), and therefore the sum is constrained to momentum and energy conservation. The next step consists of evaluating et °) (co) and e (2) (~01,072) for the case of ion acoustic waves that develop in a strongly kl, k2 magnetized plasma. The case of Langmuir waves was treated by Galeev and Sagdeev [8]. Eq. (8) can be expanded around the ion acoustic eigenfrequency, i.e, we can write

(1) (C0 t°=t°k (I). OekRx ) ' - °) (oJk), (9) e21)(to) "~ ekR tOOk) + (CO-- COk) Oco +lekI

where -(i) and -(1) represent the real and imaginary parts of the dielectric function, respectively. We then CkR eki easily recognize the quasilinear expression for the growth rate,

")(oJk) ¢~kI (10) ~k = ") (oJ)/OoJIo~=o,k" OekR

At this stage the equation for the Fourier transform of the electrostatic potential can be written as

( o J - o ~ t - i y k ) ¢ ( k , oJ) + y]~ t k 2 0 ~ o j _ m , *(kl,O)l)CJ)(k2,0)2) = O. (11) /q +/,2=i

The linear dielectric function is given by

o9~j n~ / dv J~(kxv-d£21) (k 0 ~k[l'((.O) = 1 + Z - ~ " o~-- k~lvtl---n~ j Ilo-~ll +

j n~-oo nK2j 0 ~f02. (12) V.a. OV.a_ ]

We now look for a solution corresponding to the ion acoustic wave, in the long wavelength regime, and in the frequency domain such that

vt~i << IoJ/kiil << Vth~, eO << -Qi, (13)

where vt~j is the thermal velocity of species j , ions and electrons, respectively, and g2i is the ion gyrofrequency. Then the dielectric function can be simplified and written in a more explicit form, see, for example, Ref. [ 12],

1 1 -- A0( f l i ) k[~ 2. = ~ ~ A 0 ( f l i ) ek(l)(OJ) 1 + k2X~ + k2X2 k2

+i [o o 1 • ~-TZ,2 , (14) Ikl[ Ivth¢ k22~ + [k, ]Vthi k2~2Di exp 2k, Vth i

312 A.M. Hamza / Physics Letters A 190 (1994) 309-316

where Ao is the modified Bessel function and fli I b 2 .2 = I~±~'i" In the long wavelength limit this leads to the following eigenfrequency and growth rate for the ion acoustic waves in the magnetized plasma,

tok = ]ktllcs(1 - 1 / .232 l i t 2 . 2 . ~ . !_~. f ~ - i me I n" 'ODe-- ~ n ' l P s 1, Yk = -va-VmilklllCs. (15)

We can also write explicitly the partial derivative of the dielectric function in the long wavelength, low frequency limit,

-- (1) t°=t°k 0)2 i 1 k2aeks (to) Oto ~ 2 ~ Ik, lcs" (16)

The final step of the process, and probably the most tedious, is to evaluate the coupling coefficent e (2). In order to do so we need to evaluate fj(2)(k, v, to). Using expression (7) we obtain

t

_ ( q j ~ 2 ~ / dr~(k, , to ,)exp(i{k.[x-x(z)]-og(t-r)})

o / oj;j x k l . - ~ Z dz'~(k3,to3)exp(i{k2.[x-x(z')]-to(z-z')})k3. O-'v-" (17)

Using the trajectories of a particle in a magnetic field one can easily evaluate the time integrals, and rewrite expression (17) in terms of Bessel functions to obtain

4rtq} f ~ exp[i(n-m)(qb-O)]Jn(k±v±/12j)Jm(k±v±/12j) £(2) - - dv ,l.k:(tol,C02) = Z m 2 t o - kllVll - nf2j

3 n , m = - o o

x k , . a-~ t . ,=-~ exp [i(l -p)(~b -~o_2_~11_~11_/-~JO2)]Jl(k2±vx/12J)JP(k2xvx/12J) \v-~Ov---~(lI2J 0 + k211a_~l I 3~j, (18)

where 4) represents the angle between the perpendicular component of the velocity v± and a fixed axis in the plane perpendicular to the magnetic field, while 0j is the angle between k j± and a fixed axis in the plane perpendicular to the magnetic field. Because of the long wavelength, low frequency regime that we are considering only the zeroth order Bessel functions contribute to the sums, which leaves us with the following approximation for the coupling coefficient,

4nq 3 f kill 0 kll - kill tgJ~j e (2) (tol,to) = Z ~ j dv (19) k~,k J mj to - kllvll Ovll 09 - to! - (kll - k~ll)vll Ovll"

At this stage we shall introduce a slow time scale characterized by the frequency to' = t o - IklllCs. Noting that co'/Ik u ]cs << 1 one can further approximate expression (19) after neglecting the resonant interaction. In other words we shall take into account only the principal value associated with the velocity integral and finally write

art@ f kill 1 0 1 OFoj = ~ __4re@ f kill 7 9 OFoj E(2) ( t o l ' t o ) = L - ' ¢ ~ m~ J kll Cs vllOvllc~ vii Ovll L_~ m ~ J dvll )30vl[ t ~ , k - - d V l l - -

J J kll (cs - v i i

(20)

where ? stands for principal value, while Foj represents the distribution function integrated over the perpen- dicular component of the velocity.

A.M. Hamza / Physics Letters A 190 (1994) 309-316 313

Substituting expressions (20), (16) into Eq. (11 ) we obtain the following equation governing the evolution of the electrostatic potential,

I k 2 2 2 2 ~ ~ {09' -I- ~l IllCs[kllADe -k k . ( 2 ~ + p~)]}O(k, oJ) + i IktllCsO(k, og)

+ Z A(cs)klll~(kl'°gl)~(k2'°92) = 0, (21)

/q +/~2=~

where the coefficient A (cs) is given by the following expression,

c~ ~-~ 4nqf f kql 79 )3 OFOYovll. (22) A(c ) = dr , mj kll (cs - vii

We now introduce a slow time scale • defined as follows,

dto' . . . . _ _ ~ -~--~-- exp lltK .x - og'z) ], (23)

k

operating on Eq. (21) with the inverse Fourier transform operator leading to a partial differential equation for the potential ~ ,

o f,2 o2 +

+ aT' ? dz' O~(x ±,z', z) 1 Oz' z - z' = O, (24)

- - O O

where a = ~/me/8r~mi. This easily reduces to the modified Korteweg-de Vries equation in the one-dimensional nonmagnetized

plasma with electron Landau damping derived by Ott and Sudan [ 1 ]. We should point out that the electron Landau damping term is recovered only for times much shorter than the electron and ion trapping times as was shown by Lotko [13] and Hamza [9]. For times longer than the electron and ion trapping times the contribution of the resonant electrons and ions has to be modified. Qian et al. [6] and Hamza [9] have discussed the case of reflected electrons off a rarefactive structure and an ion acoustic wave packet, respectively. In the absence of Landau damping, in a fluid description, the nonlocal term disappears and one recovers the Zakharov-Kuznetsov equation [4]. One can easily generalize the results of the one-dimensional modified KdV to the three-dimensional case. One can easily show that for any localized square integrable potential disturbance with • (x±, [z I ~ oo, z) ~ 0 we obtain the following result after integrating over the z axis, using

P i dz' ,~ _ Z------ 7 = 0,

~ O O

(25)

(26)

and consequently taking the first moment leads to

314 A.M. Hamza I Physics Letters A 190 (1994) 309-316

" i i s i 2 dT d r± d z ~ 2 ( x , r ) = -2rt2c~ dr± dkl l lk l l l l~(x±,k l i , z )12~ O.

- - (3O - - ~

(27)

This, in other words, shows that a steady state solution does not exist when condition (27) is satisfied by the initial condition. However, it was shown by Hamza [9], that steady state solutions for which Eq. (27) is not satisfied, such as shocks or double layers, can exist. Double layers can form and evolve on time scales longer than the electron trapping time.

3. The stationary solitary solution

In this section we shall argue that if one assumes the same boundary conditions imposed on the Zakharov- Kuznetsov equation, that allow the formation of a soliton, then the evolution equation does not have a stationary solution in our case. However, if one imposes boundary conditions that allow a double layer structure, or shock, to form then it is possible to show analytically that a stationary solution can be obtained. First we start by looking for a stationary solution by using a moving frame defined by

( - 2~'z ~ _ ~ ' x ± t ---* Ogvit, (28)

then Eq. (14) becomes

0 ~ 0 ~2 + ~ ( V 2 ~ + ) = 0, (29) O-7

where • = • ((, ~, t) and n eR((, ~, t). We now look for a stationary solution such that • = • ( ~ - At, ~),

0 0 - - A b e . (30) Ot

This therefore leads to rewriting Eq. (29) in the following form,

0 t~2 0--((V2~ - 2 ~ + ) = 0. (31)

By imposing the following boundary conditions,

Ill ~ o~, • ---, 0, 4~¢ ~ 0, ¢,¢¢ ~ 0,

Ill ~ c~, • --. 0, ~e ~ 0, ~¢e ~ 0, (32)

Zakharov and Kuznetsov were able to find a stationary solitary solution to a generalized Korteweg-de Vries (KdV) equation in a magnetized plasma. The equation obtained included the quadratic nonlinearity due to the ions, but they did not include any of the electron nonlinearities, and no electron Landau damping,

V2~ - ( A - ~ ) ~ = 0. (33)

They were then able to find a spherically symmetric solution; a three-dimensional soliton by assuming the following property,

r/ = ~ + u~ -At . (34)

Eq. (33) reduces to a one-dimensional equation and can be solved using the pseudo-potential approach to lead to the solitary solution presented by Shivamoggi [ 5 ].

A.M. Hamza / Physics Letters A 190 (1994) 309-316 315

Shivamoggi suggested that the Zakharov-Kuznetsov soliton seems to be in good agreement with the exper- imental observations of Raychaudhuri et al. [ 14], namely that the solitary wave tends to undergo focussing as the magnetic field increases; this is physically plausible, because the scale length of the soliton in the per- pendicular direction depends on ps, that is the gyroradius of the ions evaluated with the electron temperature; ps decreases with an increasing magnetic field. Moreover, the dispersion term in the perpendicular direction becomes smaller, i.e., the spreading of momentum and energy in the perpendicular direction becomes smaller and therefore the quadratic nonlinearity plays a more significant role in the focussing of the solitary solution.

It is however important to note that if one were to include the electron Landau damping term, one would find that the soliton would decay in time as was shown by Ott and Sudan [ 1 ] in the case of a one-dimensional ion acoustic KdV soliton.

4. Summary and conclusion

The problem of ion acoustic turbulence in both laboratory and space plasma physics has been debated for a long time, the leading theoretical models suggesting that turbulent fluctuations of soliton type can explain the different observations.

It has been shown theoretically [9,6] as well as numerically, that rarefactive ion acoustic solitons cannot develop in a one-dimensional unmagnetized two-species plasma. Compressive solitons, on the other hand, are the natural solution to the KdV equation.

A formalism based on the classical KdV or the Zakharov-Kuznetsov equations alone cannot explain the one-dimensional results of several numerical solutions as well as satellite observations. In a magnetized plasma, it was shown by Hamza [9,10] that when neglecting the ion quadratic nonlinearity and including the source of free energy in the electron, and imposing proper boundary conditions, a stationary shock solution can form.

In this paper we have presented a kinetic derivation of a generalized Zakharov-Kuznetsov equation including the electron Landau damping which becomes the only damping rate when / ] ~ 0. The electron Landau damping automatically introduces a wave-particle resonant time scale. Karpman [3] has discussed in detail, in the case of an unmagnetized two-species plasma, the domain of validity of a solitary solution, and the effects of resonant particles on its evolution. It is clear that similar arguments can be brought up in the magnetized case.

We have not discussed the effects of trapped particles since the calculations presented are only valid for times less than the trapping times of both the electrons and ions, respectively. This would indeed lead to a reevaluation of the steady state solutions in the presence of resonant particles. Hamza [10] has argued that the equation governing the evolution of ion acoustic fluctuations in a magnetized plasma, subject to boundary conditions similar to those imposed on the classical Zakharov-Kuznetsov equation does not admit a solitary solution. However, the possibility of shock solutions, and double layers is discussed [ I0]. It would be interesting and challenging at the same time to investigate the trapped particle effects. This will be the subject of a forthcoming paper.

Acknowledgement

The author would like to thank the referee for very useful comments. Funding for this research has been provided by the Canadian Network of Space Science (CNSR), one of the fifteen Networks of centres of excellence supported by the government of Canada.

References

[1] E. Ott and R.N. Sudan, Phys. Fluids 12 (1969) 2388.

316 A.M. Hamza / Physics Letters A 190 (1994) 309-316

[2] T. Taniuti, Prog. Theor. Phys. Suppl. 55 (1974). [3] V.I. Karpman, Soy. Phys. JETP 50 (1979) 695. [4] V.E. Zakharov and E.A. Kuznetsov, Soy. Phys. JETP 39 (1974) 285. [5] B.K. Shivamoggi, J. Plasma Phys. 41 (1989) 83. [6] S. Qian, W. Lotko and M. Hudson, Phys. Fluids 31 (1988) 2190. [7] T.H. Dupree, Phys. Fluids 26 (1983) 2460. [8] A.A. Galeev and R.Z. Sagdeev, Wave-particle-wave interaction, in: Handbook of plasma physics, Vol. 1 (1983) p.

699. [9] A.M. Hamza, in: Development of intermittent, spatially localized ion fluctuations in a two-species plasma, Ph.D Thesis,

M.I.T. (1988/89). [10] A.M. Hamza, Phys. Rev. E 48 (1993) 2055. [ 11] R.C. Davidson, in: Methods in nonlinear plasma theory (Academic Press, New York, 1972). [ 12] S. Ichimaru, in: Basic principles of plasma physics. A statistical approach (Benjamin/Cummings, Menlo Park, 1973). [13] W. Lotko, Phys. Fluids 26 (1983) 1771. [14] S. Raychaudhuri, J. Hill, P. Forshing, H.Y. Chang, S. Sukarto, C. Lien and K.E. Lonngren, Phys. Scr. 36 (1987) 508.