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PHYSICS
A. A. GALEEV, V. I. KARPMAN, R. Z. SAGDEEV
ON ONE SOLVABLE PROBLEM IN THE THEORY OF PLASMA TURBULENCE
(Presented by Academician M. A. Leontovich on 28 II 1964)
In recent years substantial progress has been made in a number of works on the theory of plasma turbulence. It has turned out that in those cases where the spectrum of turbulent plasma pulsations is a collection of plasma waves, the picture of turbulence can be described by a closed system of integro-differential equations for the averaged distribution function of the plasma particles (the so-called quasilinear kinetic equation ($^{1,2}$)) and for the spectral energy densities of various plasma waves (kinetic equations for interacting waves ($^3$)). This closed system is readily obtained in passing from the dynamical description of the plasma and of the self-consistent electromagnetic fields to a statistical description, if one uses the random-phase hypothesis and perturbation theory—an expansion in the small parameter $\gamma/\omega$ ($\gamma,\omega$ are the imaginary and real parts of the frequency). In essence, such an expansion is an expansion in the smallness of the ratio of the energy of interaction of the waves to their total energy. The most important application of the theory of turbulence obtained in this way is the construction of a scheme of transport phenomena in a turbulent plasma. Here the most difficult part of the problem proves to be the determination of the spectrum of turbulent pulsations. The difficulty arises because of the nonlinearity of the integral kinetic equation for the spectral energy density of the waves. All attempts made so far ($^{4-9}$) have amounted only to rather rough estimates.
In this connection, it seems important to us to investigate particular classes of problems that admit an analytic solution. It turns out that one of the solvable problems of this type is the problem of the nonlinear time evolution of the spectrum of Langmuir electron oscillations in a homogeneous plasma without a magnetic field. As will be seen below, this problem is very instructive for another reason as well: in the nonlinear relaxation of electron oscillations the main role is played by ions, which at first sight might seem paradoxical.
Since the derivation of the initial integral equations is sufficiently simple, we shall reproduce it in full. In doing so we shall use the method of asymptotic perturbation theory ($^{1,2}$) (the cumbersome nature of other methods ($^{10,11}$) does not seem to us sufficiently justified).
Thus, we shall start from the kinetic equations for the distribution functions of ions and electrons and from Poisson’s equation for the potential $\varphi$ of the electric field of the oscillations:
\[ \frac{\partial f_j}{\partial t}+\mathbf{v}\nabla f_j-\frac{e_j}{m_j}\nabla\varphi\,\frac{\partial f_j}{\partial \mathbf{v}}=0,\qquad \Delta\varphi=-4\pi\sum_j e_j\int f_j(\mathbf{v})\,d\mathbf{v}. \tag{1} \]
We decompose the particle distribution function $f_j(\mathbf{v},t)$ into a slowly varying $f_{0j}(\mathbf{v},t)$ and rapidly oscillating parts $f_j^{(1)}$ ($^{1,2}$). The electric potential—
we represent the oscillations of the electric field in the form
\[ \varphi=\sum_{\mathbf{k},\omega}\varphi_{\mathbf{k}\omega}e^{-i(\omega t-\mathbf{k},\mathbf{r})}, \qquad \varphi_{\mathbf{k}\omega}=\varphi_{\mathbf{k}}\delta_{\omega,\omega_k}+\varphi'_{\mathbf{k}\omega}, \tag{2} \]
where \(\varphi_{\mathbf{k}}\) is the amplitude of the natural oscillation, the real part of whose frequency \(\omega_k\) is related to the wave vector \(\mathbf{k}\) by the dispersion relation; \(\varphi'_{\mathbf{k}\omega}\) is the amplitude of the “forced” oscillation.
In what follows it will be convenient for us to represent the slow time dependence of the amplitude \(\varphi_{\mathbf{k}\omega}\) in exponential form and to include it in the frequency \(\omega\). For oscillations of small amplitude, equation (1) for the rapidly oscillating part of the distribution function \(f^{(1)}_j(\mathbf{v},t)\), represented in the form (2), will be solved by the method of successive approximations. Then, taking into account terms of third order in the amplitudes \(\varphi_{\mathbf{k}\omega}\), we have
\[ \begin{aligned} f^j_{\mathbf{k},\omega} &=\mu^j_{\mathbf{k}\omega}(\mathbf{v})\varphi_{\mathbf{k}\omega} +\sum_{\omega',\omega'',\,\mathbf{k}'+\mathbf{k}''=\mathbf{k}} \mu^{kj}_{\mathbf{k}'\omega',\,\mathbf{k}''\omega''} \varphi_{\mathbf{k}'\omega'}\varphi_{\mathbf{k}''\omega''} e^{-i(\omega'+\omega''-\omega)t} \\ &\quad+ \sum_{\omega',\omega'',\omega''',\,\mathbf{k}'+\mathbf{k}''+\mathbf{k}'''=\mathbf{k}} \mu^{kj}_{\mathbf{k}'\omega',\,\mathbf{k}''\omega'',\,\mathbf{k}'''\omega'''} \varphi_{\mathbf{k}'\omega'}\varphi_{\mathbf{k}''\omega''}\varphi_{\mathbf{k}'''\omega'''} e^{-i(\omega'+\omega''+\omega'''-\omega)t}, \tag{3} \end{aligned} \]
where
\[ \mu^j_{\mathbf{k}\omega}(\mathbf{v}) = -\frac{e_j}{m_j}\, \frac{\mathbf{k}\,df_{0j}/d\mathbf{v}}{\omega-\mathbf{k}\mathbf{v}+i0}. \]
\[ \begin{aligned} \mu^{kj}_{\mathbf{k}'\omega',\,\mathbf{k}''\omega''} &= \frac{e_j^2}{2m_j^2}\, \frac{1}{\omega'+\omega''-\mathbf{k}\mathbf{v}+i0} \left[ \mathbf{k}'\frac{\partial}{\partial \mathbf{v}} \frac{1}{\omega''+\mathbf{k}''\mathbf{v}+i0} \mathbf{k}''\frac{\partial}{\partial \mathbf{v}} \right. \\ &\qquad\left. +\mathbf{k}''\frac{\partial}{\partial \mathbf{v}} \frac{1}{\omega'-\mathbf{k}'\mathbf{v}+i0} \mathbf{k}'\frac{\partial}{\partial \mathbf{v}} \right]f_{0j}(\mathbf{v}), \\[1ex] \mu^{kj}_{\mathbf{k}'\omega',\,\mathbf{k}''\omega'',\,\mathbf{k}'''\omega'''} &= -\frac{e_j^3}{2m_j^3}\, \frac{1}{\omega'+\omega''+\omega'''-\mathbf{k}\mathbf{v}+i0} \mathbf{k}'\frac{\partial}{\partial \mathbf{v}}\, \frac{1}{\omega''+\omega'''-(\mathbf{k}''+\mathbf{k}''',\mathbf{v})+i0} \\ &\qquad\times \left[ \mathbf{k}''\frac{\partial}{\partial \mathbf{v}} \frac{1}{\omega'''-\mathbf{k}'''\mathbf{v}+i0} \mathbf{k}'''\frac{\partial}{\partial \mathbf{v}} + \mathbf{k}'''\frac{\partial}{\partial \mathbf{v}} \frac{1}{\omega''-\mathbf{k}''\mathbf{v}+i0} \mathbf{k}''\frac{\partial}{\partial \mathbf{v}} \right]f_{0j}(\mathbf{v}). \tag{4} \end{aligned} \]
Assuming the distribution of the phases of the oscillations to be random, one can average over them in Poisson’s equation, where the charge density is calculated according to (3). As a result, for the amplitude of the oscillations we obtain the equation:
\[ \begin{aligned} \operatorname{Im}\Biggl\{ \varepsilon^{(1)}(\omega,\mathbf{k}) &- \sum_{\mathbf{k}',\omega'} \frac{ \varepsilon^{(2)}_{\mathbf{k}'\mathbf{k}''}(\omega_{\mathbf{k}},\omega_{\mathbf{k}}-\omega_{\mathbf{k}'}) \varepsilon^{(2)}_{\mathbf{k},-\mathbf{k}'}(\omega_{\mathbf{k}},-\omega_{\mathbf{k}}) }{ \varepsilon^{(1)}(\omega_{\mathbf{k}}-\omega_{\mathbf{k}'},\mathbf{k}-\mathbf{k}') } |\varphi_{\mathbf{k}'}|^2 \\ &\quad+ \sum_{\mathbf{k}'} \varepsilon^{(3)}_{\mathbf{k}',\mathbf{k},-\mathbf{k}'} (\omega',\omega,-\omega') |\varphi_{\mathbf{k}'}|^2 \Biggr\} |\varphi_{\mathbf{k}}|^2 \\ &\quad -\operatorname{Im} \sum_{\mathbf{k}'+\mathbf{k}''=\mathbf{k}} \frac{ \left|\varepsilon^{(2)}_{\mathbf{k}'\mathbf{k}''}(\omega_{\mathbf{k}'},\omega_{\mathbf{k}''})\right|^2 }{ \varepsilon^{(1)}(\omega_{\mathbf{k}'}+\omega_{\mathbf{k}''},\mathbf{k}) } |\varphi_{\mathbf{k}'}|^2|\varphi_{\mathbf{k}''}|^2 =0. \tag{5} \end{aligned} \]
Until now we have included the slow change of the oscillation amplitude in the imaginary part of the frequency. It is now convenient to pass to real frequencies and to regard the amplitudes \(\varphi_{\mathbf{k}}\) as slowly dependent on time. This transition is easily carried out by means of the replacement
\[ \operatorname{Im}\varepsilon^{(1)}(\omega_k,\mathbf{k})|\varphi_k|^2 \to -\frac{\partial\varepsilon^{(1)}(\omega_k,\mathbf{k})}{\partial i\omega_k} \frac{\partial|\varphi_k|^2}{\partial t} + \operatorname{Im}\varepsilon^{(1)}(\omega_k,\mathbf{k})|\varphi_k|^2. \tag{6} \]
In doing this it should be remembered that the poles in the integrals determining the dielectric permittivity are bypassed according to the rules indicated in (4). The connection of \(\varepsilon^{(1)}(\omega,\mathbf{k})\), \(\varepsilon^{(2)}_{\mathbf{k}'\mathbf{k}''},\ldots\) with \(\mu^j_{\mathbf{k}\omega}\), \(\mu^{kj}_{\mathbf{k}'\mathbf{k}''},\ldots\) is obvious.
For the slowly varying averaged distribution function \(f_{0j}(\mathbf{v}, t)\) under the action of the oscillations, following the general rules \((^{1,2})\), we obtain, taking into account the interaction of the oscillations,
\[ \frac{\partial f_{0j}}{\partial t} = \frac{e_j^2}{m_j^2} \sum_{\mathbf{k}} |\varphi_{\mathbf{k}}|^2 \mathbf{k} \frac{\partial}{\partial \mathbf{v}} \frac{\nu_{\mathbf{k}}}{(\omega_{\mathbf{k}}-\mathbf{k}\mathbf{v})^2+\nu_{\mathbf{k}}^2} \mathbf{k}\frac{d f_{0j}}{d\mathbf{v}} - \]
\[ - \operatorname{Im}\frac{e_j}{m_j} \sum_{\mathbf{k}}\mathbf{k}\frac{\partial}{\partial\mathbf{v}} \left\{ \sum_{\mathbf{k}'} \mu^{kj}_{\mathbf{k}'\omega',\,\mathbf{k}-\mathbf{k}',\,\omega-\omega'} \frac{\varepsilon^{(2)}_{\mathbf{k},-\mathbf{k}'}(\omega_{\mathbf{k}},-\omega_{\mathbf{k}'})} {\varepsilon^{(1)}(\omega_{\mathbf{k}}-\omega_{\mathbf{k}'},\,\mathbf{k}-\mathbf{k}')} |\varphi_{\mathbf{k}}|^2|\varphi_{\mathbf{k}'}|^2 +\right. \]
\[ \left. + \sum_{\mathbf{k}'+\mathbf{k}''=\mathbf{k}} \frac{ \mu^{kj*}_{\mathbf{k}'\omega',\,\mathbf{k}''\omega''} \varepsilon^{(2)}_{\mathbf{k}',\,\mathbf{k}''}(\omega_{\mathbf{k}'},\omega_{\mathbf{k}''}) } {\varepsilon^{(1)}(\omega_{\mathbf{k}'}+\omega_{\mathbf{k}''},\,\mathbf{k})} |\varphi_{\mathbf{k}'}|^2|\varphi_{\mathbf{k}''}|^2 - \right. \]
\[ \left. - \sum_{\mathbf{k}'} \mu^{kj}_{\mathbf{k}'\omega',\,\mathbf{k}'\omega'',\,-\mathbf{k}''',\,-\omega'''}(\mathbf{v}) |\varphi_{\mathbf{k}'}|^2|\varphi_{\mathbf{k}}|^2 \right\}. \tag{7} \]
Equations (5), (7) constitute a complete system of equations describing the turbulent kinetics of a rarefied plasma without a magnetic field, accurate up to terms quadratic in the energy of the oscillations. We now turn directly to the question of the nonlinear relaxation of electron plasma oscillations.
For longitudinal electron oscillations it is impossible to satisfy the “decay” conditions \((\omega_{\mathbf{k}}=\omega_{\mathbf{k}'}+\omega_{\mathbf{k}''})\); therefore resonant transfer of energy only between waves is impossible, and the interaction between two waves occurs only with the participation of a part of the “background.” The latter can interact intensively only with forced oscillations whose frequency is equal to the difference of the natural frequencies of the oscillations, \(\omega=\omega_{\mathbf{k}}-\omega_{\mathbf{k}'}\).
As Drummond and Pines showed for the example of a one-dimensional spectrum \((^2)\), in the first nonvanishing order with respect to \(r_D/\lambda\) \(\bigl(r_D \simeq \sqrt{T/4\pi e^2 n}\) is the Debye radius, \(\lambda\) is the wavelength of the oscillations\()\), the only effect of the interaction of oscillations is the transfer of energy from short-wavelength scales to long-wavelength ones. The damping effects of a one-dimensional packet, appearing in higher orders with respect to \(r_D/\lambda\), were considered by V. I. Karpman \((^7)\).* However, in these works the influence of ions was neglected, which, generally speaking, in a number of cases becomes substantial.
As an example, let us consider an isotropic (three-dimensional) wave packet. It turns out that if its width satisfies the condition
\[ r_D\Delta k \lesssim \left(\frac{1}{k r_D}\frac{V_{Ti}}{V_{Te}}\right)^{2/3}, \tag{8} \]
where \(V_{Ti,e}=\sqrt{2T_{i,e}/m_{i,e}}\) are the thermal velocities of the particles, then in (5) one should take into account the interaction of forced oscillations only with the ions. Keeping the terms leading with respect to \(r_D/\lambda\), both in the energy transfer and in the damping of the oscillations, one can write (5), with allowance for (6), in the form
\[ \frac{\partial|\varphi_{\mathbf{k}}|^2}{\partial t} = \omega|\varphi_{\mathbf{k}}|^2 \int d\mathbf{k}'\, \frac{(\mathbf{k}\cdot\mathbf{k}')^2}{k^2} |\varphi_{\mathbf{k}'}|^2/16\pi n_0 T \times \]
\[ \times\operatorname{Im} \frac{1+4(\omega_{\mathbf{k}}-\omega_{\mathbf{k}'})/\omega} { 2-i\sqrt{\pi}\, \frac{\omega_{\mathbf{k}}-\omega_{\mathbf{k}'}}{|\mathbf{k}-\mathbf{k}'|V_{Ti}}\, w\!\left( \frac{\omega_{\mathbf{k}}-\omega_{\mathbf{k}'}} {|\mathbf{k}-\mathbf{k}'|V_{Ti}} \right) }, \tag{9} \]
where
\[ w(z)=\frac{i}{\pi}\int_{-\infty}^{+\infty}\frac{e^{-t^2}}{z-t}\,dt, \qquad T_i=T_e. \]
It follows from this expression that, as in the one-dimensional case, the damping effects are smaller than the transfer of ener-
* The magnitude of the damping given in that article turned out to be numerically overestimated because of an excessive truncation made in the calculations.
in the spectrum; moreover, because of the narrowness of the Maxwellian distribution for the ions, only waves with very close values of the modulus of the wave vector interact with one another, in the interval \(\delta k r_D < \sqrt{m_e/m_i} \ll \Delta k r_D\). The latter circumstance makes it possible to use an expansion of the subintegral expression in a series in the small difference \((k'^2 - k^2)r^2\), as a result of which the integro-differential equation (9) reduces to a nonlinear partial differential equation:
\[ \frac{\partial E_k}{\partial \tau} - E_k \frac{\partial E_k}{\partial \chi} = -\gamma E_k^2, \tag{10} \]
where
\[ E_k = \frac{4\pi k^3}{3}\, \frac{|\varphi_k|^2}{4\pi nT}, \qquad \tau = \frac{\pi m_e}{6\eta m_i}\,\omega t, \qquad \chi = \frac{k^2}{\Delta k\cdot k_0}, \qquad \frac{\gamma}{6} = \bigl(\sqrt{\Delta k k_0}\,r_D\bigr)^2 \lesssim \sqrt{\frac{m_e}{m_i}}, \]
\(\Delta k\) is the characteristic width of the packet, and \(k_0\) is the mean wave number of the wave packet.
The solution of this equation has the form
\[ E_k = e^{\gamma\chi} f\!\left( \frac{1-e^{-\gamma\chi}+\tau\gamma e^{-\gamma\chi}E_k}{\gamma} \right), \tag{11} \]
where \(f((1-e^{-\gamma\chi})/\gamma)\) is the energy distribution in the packet at the initial instant of time. It follows from this that the main effect of the time evolution of the packet is its narrowing. However, equation (10) itself is valid only under the condition that the spread of phase velocities in the wave packet is considerably larger than the thermal spread of the ion velocities \((\Delta k\cdot r_D \gg \sqrt{m_e/m_i})\).
As soon as the packet has become sufficiently narrow, we cannot describe its evolution analytically. However, the physical picture remains clear. The packet continues to narrow until four-plasmon interaction becomes significant.* The time of broadening of the packet due to four-plasmon interaction is easily estimated, knowing \(\varepsilon^{(2)}\), \(\varepsilon^{(3)}\). In order of magnitude it is given by the expression
\[ \tau \sim \omega^{-1}(\Delta k r_D)^2(W/nT)^{-2}, \tag{12} \]
where \(W=\sum_k k^2\varphi_k^2\) is the total energy of the wave packet. Comparing it with the characteristic narrowing time, one can find the established quasistationary width \(\Delta k\):
\[ \Delta k r_D \sim (W/nT)^{1/3}(m_e/m_i)^{1/6} \ll \sqrt{m_e/m_i}. \tag{13} \]
The establishment of the narrow quasistationary packet occurs so rapidly that the damping (or growth, for the other sign of \(\left.\dfrac{df}{dv}\right|_{v=\Delta\omega/\Delta k}\)) during the establishment process could be disregarded.
Novosibirsk State
University
Received
20 II 1964
CITED LITERATURE
- A. A. Vedenov, E. P. Velikhov, R. Z. Sagdeev, Nuclear Fusion, 1, 82 (1961).
- W. Drummond, D. Pines, Report No. 134, presented at the conference on plasma physics and controlled thermonuclear reactions, Salzburg, September 1961.
- M. Camac et al., Nuclear Fusion, Supplement 1962, book 2, 423 (1962).
- A. A. Galeev, V. I. Karpman, ZhETF, 44, 592 (1963).
- A. A. Galeev, S. S. Moiseev, R. Z. Sagdeev, Preprint of the Institute of Nuclear Physics, Siberian Branch, USSR Academy of Sciences, 1963.
- A. A. Galeev, L. I. Rudakov, ZhETF, 45, 547 (1963).
- V. I. Karpman, DAN, 152, 587 (1963).
- B. B. Kadomtsev, ZhETF, 45, 1230 (1963).
- V. I. Karpman, Applied Mathematics and Technical Physics, No. 6, 34 (1963).
- B. B. Kadomtsev, V. I. Petviashvili, ZhETF, 43, 2234 (1962).
- V. P. Silin, Journal of Applied Mathematics and Technical Physics, No. 1, 31 (1964).
* With the aid of the quasilinear equations (7) it is easy to show that the time of establishment of the “plateau” on the distribution function is always considerably greater than (12).