Turbulent Diffusion in Magnetized Parametrically Unstable Plasma
V. N. Pavlenko and V. G. Panchenko
Institute for Nuclear Research of the National Academy of Sciences of Ukraine,
Prospekt Nauki, 47, Kyiv, 03680, Ukraine
Abstract. In the present report the processes of the turbulent diffusion in a magnetized plasma with lower-hybrid pump
wave are investigated. Under parametric instability conditions we found the magnitude of the diffusion coefficient and its
dependence on various plasma parameters. Our results can be of interest for the investigation of anomalous transport
processes in space and laboratory plasma with a small value of β and also can take place in a plasma with ion-temperature
Keywords: turbulent diffusion, parametrically unstable plasma, relaxation, upper-hybrid pump, magnetized plasma.
It is known that important macroscopic plasma properties are governed by particle collisions. So their calculation
calls for the solution of a kinetic equation with a collision integral. The determination of the collision integral and,
consequently, the associated diffusion of particles are important aspects of plasma nonlinear theory.
The collision integral of charged particles in the plasma may be presented in the form 
where α indicates the kind of particles. In (1) δ fα (ω, k, p)
is the fluctuation of the distribution function and δ E
is the fluctuating electric field. We shall study fluctuations in an electron-ion magnetized plasma (with the constant
magnetic field B0
directed along the z-axis) under the influence of radio-frequency pump-wave field
E0 (t) = E0 y cosω0t
, which is taken in the dipole approximation.
Now we are interested in the case when the frequency of the pump wave 0
ω is the lower-hybrid frequency
are the frequency and the damping rate of lower-hybrid wave.
Consider the parametric decay of the pump wave into a daughter lower-hybrid wave and modified convective cells:
ω0 =ωLH +ωc . (3)
Here ωc = (mi /me )1/ 2Oi ⋅cosθ is the frequency of the modified convective cell and γ c is the damping rate,
c 2 ei γ ≈ − v ( vei is the electron-ion collision frequency). Note that these long-wavelength (kρ⊥i <1) convective
modes arise in a magnetized plasma with a small value of β (β is the ratio of the plasma pressure to the magnetic
pressure) and can occur in the ionospheric plasma.
Let us investigate the case when the external electric field exceeds the threshold for parametric instability and
fluctuations resulting from this instability are well developed. We assume that the instability saturates due to the
stabilization mechanism associated with the scattering of charged particles by turbulent fluctuations of the electric
field. We will characterize this scattering by the effective collision frequency veff which determines the rate of
plasma heating by HF waves and is connected with the coefficient of turbulent damping D⊥ by the expression 
veff ≈ k⊥D⊥ (4)
To calculate veff we use the equation for the plasma energy balance (5)
where Iα ( p)
is the collision integral determined by (1).
In the region above the instability threshold, from equation (5), we obtain the following expression for the
Numerical calculations demonstrate that the diffusion term due to the pump wave is dominated in comparison with
the diffusion which takes place in the plasma in the presence of the thermal fluctuations only.
Now we will study the parametric interaction of the lower-hybrid wave with the harmonics of the ion-cyclotron
frequency (ω ≈ nOi ) . The frequency and the damping rate of such oscillations for the case n = 1 are determined
by the known relations :
In Eqs (7) and (8), ( ) ( ) i
An i In i e β β −β⊥
⊥ = ⊥ , where In is the modified Bessel function, ( )2 1 β⊥i = kρ⊥i > .
Moreover, Eqs (5) and (6) were obtained at v i ω / kz v e
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