Drift wave turbulence and anomalous plasma diffusion in lower hybrid cavities observed in the ionosphere

In plasma of the Earth’s upper ionosphere, regions with a depleted plasma density and an increased level of oscillations with a lower hybrid frequency compared to the environment were found. It was established that such plasma density cavities have cylindrical symmetry and are elongated along the geomagnetic field, so that the longitudinal dimensions significantly exceed the transverse ones. Such structures, called lower hybrid cavities, are quite stable, so that during the passage through them spacecraft do not observe significant changes in the parameters of the cavities. Thus, the nature of the change in cavities over time remains unclear. In this paper, we theoretically investigate the temporal evolution of a cavity in plasma of ionosphere. Since depletion of the plasma density is a cylindrically symmetric region, it creates a radial inhomogeneity in the plasma. In turn, the inhomogeneity of plasma leads to the development of low-frequency drift instability and a turbulent state of plasma. The anomalous plasma diffusion across the geomagnetic field resulting from the development of turbulence of drift waves of inhomogeneous plasma is considered as a mechanism for changing the cavity. In this paper the equation of plasma diffusion in cavity is solved, where the initial radial distribution of plasma density is the upside-down Gaussian. Plasma diffusion occurs radially towards the center, since the plasma density increases with increasing radial coordinate. Obtained solution of the diffusion equation gives the rate of decrease in the depth of cavity. In addition to reducing the depth of the cavity, its expansion also occurs, however, the expansion of the cavity is slower than the decrease in depth. The paper gives plots of the plasma density distribution over the radius for several time values, which show the temporal evolution of the cavity. These dependences show that in a time of the order of 1 second the cavity changes significantly, but does not completely disappear.


Introduction
Studies of ionosphere by satellites and sounding rockets have established that in the plasma of the ionosphere there are regions with a higher level of lower hybrid oscillations compared to the surrounding plasma, which correlate with a depletion of plasma density. These regions are elongated along the geomagnetic field, so that the longitudinal dimensions much exceed the transverse ones were called lower hybrid solitary structures (LHSS), or lower hybrid cavities (LHC). Since of the high velocities of the spacecraft and small transverse dimensions of LHC, which are of the order of tens of meters, the measurement time is no more than tens of milliseconds. However, during this time, the structure has not changed much, that is the LHC is a relatively stable formation. Despite that there is a number of works to explain this phenomenon, the references to which are given in the review [1], the mechanisms of the appearance of cavities, as well as their stability, are not entirely clear. There are also no explanations for their disappearance and estimates of their lifetime.
This article discusses the problem of the time evolution and disappearance of LHC due to drift turbulence and anomalous transport of inhomogeneous plasma across the magnetic field. Drift turbulence develops due to the growth of unstable drift waves in inhomogeneous magnetized plasma, the frequency of which is much lower than the ion cyclotron frequency. This turbulence has long been a serious problem in plasma confinement in studies of controlled thermonuclear fusion, since it led to anomalous plasma diffusion across the magnetic field. We consider the same turbulence as a candidate for the role of the mechanism leading to the diffusion of plasma in the cavity and filling it with plasma.
Measurements showed that the cavities have cylindrical symmetry, and therefore, an analysis of turbulence, as well as diffusion processes in the LHC plasma, should be carried out using the model of cylindrical waves. Such a model developed in our works [2][3][4][5] was used in [6] for LHC conditions in plasma. In [6], we considered both linear and nonlinear stages of drift instability for cavities, found the level of turbulence in the LHC, and previously investigated the temporal evolution of plasma density in the cavity due to anomalous transport through the magnetic field caused by drift turbulence.
In this work, the anomalous plasma diffusion caused by drift turbulence is studied in detail. We give here the procedure for solving the diffusion equation for the characteristic distribution of the plasma density over the radius and give a graphic illustration of the evolution of the distribution of the plasma density of the cavity.

Drift wave turbulence in lower hybrid cavities
We first briefly consider the conditions in LHC under which anomalous plasma diffusion across the magnetic field occurs. Observations showed that the plasma density distribution in LHC has the form of an upside-down Gaussian distribution [1], which can be described by the equation Evaluation shows [6] that at altitudes of up to 1000 km where the main ion component is the singly ionized oxygen, the frequency of the drift oscillations in LHC is of the order of 3-5 Hz, whereas at the altitudes of 1500-2000 km, where the main ion component is protons, the frequency is of the order of 7-9 Hz. The growth time of instability, which is equal to   Anomalous diffusion of plasma in lower hybrid cavities Due to drift turbulence in the plasma of LHC, the distribution of the density of the plasma components along the radius changes, and is governed by the diffusion equation where  D is the diffusion coefficient across the magnetic field. Note that the diffusion coefficients of electrons and ions are equal,     ei D D D , so that the diffusion is ambipolar. For the level of turbulence of (3) we have [6]    00 es cT D eB r .
The equation (4) gives diffusion of plasma in the direction of axis of LHC, since the plasma density increases from the center.
To solve the equation (4) we introduce the notation   4DD and rewrite them as 2 22 nn rD t rr Take the Laplace transform of (6) To solve the partial differential equation (11), we use the method of characteristics. First integral of the eq. (11) is found from the characteristic equation 2 1 dt dp Dp The solution of eq. (12), i.e. the first integral of eq. (11) is 1 ut Dp To obtain the second integral of (11) rewrite it as Then the second characteristic equation for (11) is dp dN pN Thus the second integral is v Np  .
First and second integrals yield a general solution of (11): Now we take into account the initial dependence (i. e. at 0 t  ) of plasma density on the radius (1). Solving the diffusion equation (11), we consider only second term in (1), since the first does not depend on either time or coordinate. Thus we study the evolution of the distribution   Thus, if the depth of the cavity decreases by a factor of k , then its transverse dimension increases by a factor of k , so that anomalous diffusion leads not only to a decrease in the depth of the cavity, but also to its expansion, however at a lower rate. Figure 1 shows the temporal evolution of the plasma density distribution over the radius, where dependence (25 5 10 D cm s . Figure 1 shows that the cavity disappears in a time longer than 1 second.

Conclusions
Radial inhomogeneity of the plasma density and temperature in the plasma of LHC leads to the development of the drift instability and drift turbulence of plasma in the cavity. In turn, drift turbulence causes an anomalous diffusion of the plasma across the magnetic field, which leads to the disappearance of the cavity, and the cavity disappearance time exceeds 1 s.