Dynamics of magnetic field-aligned small-scale irregularities in the electron concentration, existing in the F-layer ionospheric plasma, is investigated with the help of a mathematical model. The plasma is assumed to be a rarefied compound consisting of electrons and positive ions and being in a strong, external magnetic field. In the applied model, kinetic processes in the plasma are simulated by using the Vlasov-Poisson system of equations. The system of equations is numerically solved applying a macroparticle method. The time evolution of a plasma irregularity, having initial cross-section dimension commensurable with a Debye length, is simulated during the period sufficient for the irregularity to decay completely. The results of simulation indicate that the small-scale irregularity, created initially in the F-region ionosphere, decays accomplishing periodic damped vibrations, with the process being collisionless.

Occurrence of electron density irregularities is a natural phenomenon in the Earth’s ionosphere. These irregularities have a wide range of spatial scales, ranging from a few Debye lengths to thousands of kilometers. The electron density increases and depletions inside irregularities can lie in the range from a few portions to some tens of percentages. The well-known equatorial anomaly is the example of large-scale irregularities in the ionospheric F layer. Another example is the main ionospheric trough observed at subauroral latitudes [

It is known that ionospheric irregularities may be originated not only by natural processes but also as a result of active experiments in the ionospheric plasma, in particular, as a result of release of various chemically active substances into the ionosphere. Moreover, ionospheric irregularities may be formed by high-power high-frequency radio waves, pumped into the ionosphere by ground-based ionospheric heaters. These waves cause a variety of physical processes in the ionospheric plasma. Some such processes result in the formation of both large-scale electron temperature and density irregularities and small-scale, geomagnetic field-aligned irregularities in the ionosphere. Mathematical modeling of the large-scale F-layer modification by powerful high frequency waves was performed in some studies, in particular, in the papers by Meltz and LeLevier [

Just small-scale, geomagnetic field-aligned ionospheric irregularities are investigated in the present study utilizing a mathematical model developed recently in the Polar Geophysical Institute (PGI).

The ionospheric plasma at F-layer altitudes is supposed to be a rarefied compound consisting of electrons and positive ions in the presence of a strong, external, uniform magnetic field. The studied irregularities are assumed to be geomagnetic field-aligned, with their cross-sections being circular. The initial cross-section diameters of the irregularities are supposed to be commensurable with the Debye length. At F-layer levels, the mean free path of particles (electrons and ions) between successive collisions is much more than the cross-section diameters of the considered irregularities. Therefore, the plasma is assumed to be collisionless. Kinetic processes in such plasma are described by the Vlasov-Poisson system of equations which has been considered, for example, in the studies by Hockney and Eastwood [

As pointed out previously, the investigated irregularities are geomagnetic field-aligned. Their longitudinal sizes are much more than the cross-section diameters. Gradients of the plasma parameters in the longitudinal direction are much less than those in a plane perpendicular to a magnetic field in the vicinity of the irregularity. Therefore, plasma parameters in the vicinity of the irregularity may be considered as independent on the longitudinal coordinate. This simplification allows us to consider a two-dimensional flow of plasma in a plane perpendicular to a magnetic field line.

Not long ago, in the PGI, the two-dimensional mathematical model has been developed which is intended to simulate dynamics of the near-earth rarefied plasma [

In this work, the simulation region lays in the plane perpendicular to the magnetic field line. The simulation region is a square and its side length is equal to 96 Debye lengths of plasma. The dimension of the simulation region is consistent with recommendations of the applied macroparticle method using a discrete Vlasov-Poisson system of equations. According to these recommendations, for the adequate representation of the real plasma, the dimension of the simulation region should be no less than 60–100 Debye length of the plasma [

The grid width is equal to one eighth of the Debye length of the plasma. The quantity of the grid cells is ^{13}.

The temporal evolution of the small-scale irregularity, created initially in the F-layer ionospheric plasma, is numerically studied during a period sufficient for the irregularity to decay completely. The 2D2V variant of the mathematical self-consistent model of dynamics of the near-earth rarefied plasma is utilized. More specific details of the utilized mathematical model and peculiarities of the applied numerical method may be found in the study of Mingalev et al. [

The utilized mathematical model can describe the behavior of the near-earth plasma under various conditions. The results of calculations to be presented in this paper were obtained using the input parameters of the model typical for the nocturnal ionospheric plasma at the level of 300 km. In particular, the value of the nondisturbed electron concentration (equal to the positive ion concentration) is 10^{11} m^{−3}. The electron and ion temperatures are supposed to be equal to 1213 K and 930 K, respectively. The bulk flow velocities of electrons and positive ions are assumed to be zero. The value of the magnetic field, ^{−5} T.

The above pointed out values yield the following quantities of some physically significant parameters. The electron thermal velocity, ^{7} s^{−1}. The Debye length of the plasma,

Taking the input parameters of the mathematical model typical for the nocturnal ionosphere at the level of 300 km, we have calculated the time evolution of the distribution functions of charged particles as well as self-consisting electric field for two distinct on principle situations. In these situations, the initial distributions of electric charge density have been different. The first situation corresponds to homogeneous spatial distributions of the electron and positive ion concentrations at the initial moment inside the simulation region, with the plasma being electrically neutral and the electric charge density being equal to zero.

The second situation corresponds to homogeneous spatial distribution of the positive ion concentration only. The spatial distribution of the electron concentration, at the initial moment, contains a circular irregularity at the center of the simulation region. Inside the irregularity, the electric neutrality of the plasma is broken whereas, beyond it, the plasma is electrically neutral at the initial moment.

Simulation results, obtained for the first situation when the process started from the completely electrically neutral state, indicate that the spatial distributions of the electron and positive ion concentrations tend to retain a homogeneity and electrical neutrality of the plasma. However, short-scale nonregular fluctuations of the calculated parameters of the plasma arise near their initial values. In particular, the electric charge density and electric field fluctuate, with amplitudes of the fluctuations being very little. We calculate two orthogonal components of the electric field,

It is of interest to consider a decrease of electron concentration, ^{−5} at all grid cells of the simulation region (Figure

The time variations of maximal (top curve) and minimal (bottom curve) values of the relative decrease of the electron concentration, computed at all grid cells of the simulation region. Also, the time variation of the relative decrease of the electron concentration, calculated at the center of the simulation region (middle curve). The results were obtained for the first situation with the start from the completely electrically neutral state. The values are given in units of 10^{−6}. The normalized time,

One of the physically significant parameters of the plasma, filling in a volume

Results of simulation, obtained for the first situation with the start from the completely electrically neutral state, indicate that the normalized potential energy of the plasma fluctuates (Figure ^{−11} (Figure

The time variation of the normalized potential energy of the plasma filling up all simulation region, ^{−11}. The normalized time,

The presence of the short-scale nonregular fluctuations of the calculated parameters of the plasma, referred to as a discrete noise, is due to the specific character of the applied macroparticle method, with the fluctuation amplitudes being conditioned by the number of macro-particles used in the calculations. It can be noticed that, in the present calculations, we use the discrete 2D2V Maxwell distribution simulated by 2^{11} macro-particles with 16 levels of energy and 128 points for azimuth angle of velocity, which were uniformly located in space in a square having the side length of

It should be emphasized that the amplitudes of the fluctuations, obtained for the first situation when process started from the completely electrically neutral state, characterize the accuracy of the applied numerical method that cannot be exceeded in following calculations. The determination of this accuracy is one of the goals of performing calculations for the first situation corresponding to homogeneous spatial distributions of electrons and positive ions inside the simulation region at the initial moment.

Let us consider the results of simulation, obtained for the second situation when, at the initial moment, the spatial distribution of the electron concentration contains a circular irregularity at the center of the simulation region, with the spatial distribution of the positive ion concentration being homogeneous in all simulation region. Calculations were made for the case in which the initially created irregularity has the cross-section diameter of

The calculated spatial distributions of the relative decrease of the electron concentration,

The time evolution of the initially created irregularity was numerically simulated using the mathematical model described above. Simulation results indicate that, after initial moment, the spatial distribution of the electron concentration changes essentially while the positive ion concentration is retained practically invariable. It turns out that the initially created irregularity vanishes completely during a short period, with the plasma becoming electrically neutral in all simulation region at the moment near to the equilibrium period of Langmuir oscillations of the electrons

It is of interest to note that, in the process of evolution, additional almost symmetrical alternate rings with an excess of charge of different sign begin to appear around the initial irregularity situated in the center of the simulation region. Such rings are seen in Figure

Simulation results, obtained for the second situation when the temporal evolution of the initially created irregularity was studied, indicate that the calculated parameters fluctuate at separate points of the simulation region. Examples of fluctuating parameters of the plasma are presented in Figure ^{−5} V/m while, in the second situation, they can exceed a value of 0.2 V/m (Figure

The time variations of one component of the electric field, namely, ^{−6} V/m in (a) and in V/m in the (b). The normalized time,

Results of simulation, obtained for the second situation, indicate that the normalized potential energy of the plasma can fluctuate. From Figure

The time variation of the normalized potential energy of the plasma filling up all simulation region,

As was noted earlier, the applied mathematical model has been utilized for numerical simulation of the behavior of small-scale irregularities of the electron density which can exist in the magnetospheric plasma [

In this paper, the time evolution of the small-scale irregularities in the F-layer ionospheric plasma was numerically simulated using the two-dimensional mathematical model, developed recently in the PGI. The model is based on numerical solving the Vlasov-Poisson system of equations, with the Vlasov equations describing the distributions functions of charged particles and the Poisson equation governing the self-consistent electric field. The full implicit variant of the macro-particle method is applied for numerical solving of the system of equations, with the real charge-mass ratio for electrons having been used.

The investigated irregularities are supposed to be geomagnetic field-aligned, with their cross-section being circular. The cross-section diameters of the irregularities are much less than their longitudinal dimensions. Gradients of the plasma parameters in a plane perpendicular to a magnetic field are much more than those in the longitudinal direction in the vicinity of the irregularity. Consequently, the latter gradient may be omitted and the flow of plasma may be considered as two-dimensional in the plane perpendicular to the magnetic field.

Calculations were made of the time evolution of the distribution functions of charged particles as well as the self-consisting electric field for conditions typical for the nocturnal ionospheric plasma at the level of 300 km. Firstly, to determine the accuracy of the mathematical model, calculations were performed for the situation corresponding to homogeneous spatial distributions of electrons and positive ions inside the simulation region at the initial moment, that is, when the process started from the completely electrically neutral state. As a consequence, the amplitudes of the so-called discrete noise of the numerical model were determined.

Secondly, the temporal evolution of the initially created irregularity was numerically simulated which has the parameters typical for small-scale irregularities in the F-layer ionospheric plasma. Simulation results indicated that, in the course of time, the irregularity decays accomplishing periodic damped vibrations.

The irregularity vanishes and recovers periodically, with its parameters fluctuating. In particular, the normalized potential energy of the plasma demonstrates periodic damped vibrations. The period of the vibrations with the maximal amplitudes is close to the period of cyclotron oscillations of electrons which is approximately a factor of 2.3 larger than the equilibrium period of Langmuir oscillations of electrons. During the process of evolution, around the initial irregularity, additional almost symmetrical alternate rings with an excess of charge of different sign began to appear. In the course of time, these additional rings filled up all simulation region.

The time interval of about 15 vibration periods of the irregularity is sufficient for the irregularity to lose almost completely its initial structure, to be diffused, and to decay. This time interval is much less than the mean free time of electron between successive collisions with other particles. Therefore, the process of destroying the initially created irregularity is substantially collisionless.

This study was partially supported by the Division of Physical Sciences of the Russian Academy of Sciences through the program “Plasma processes in the solar system” and by the RFBR Grant no. 10-01-00451.