[2] The scientific community pays (especially in the recent years) a great attention to studies of the phenomena related to thunderstorm activity. It became clear relatively recently that many of these phenomena are accompanied by local changes in the properties of the lower ionosphere (electron concentration and collision frequency). These changes should influence the propagation of VLF signals [Dowden et al., 1994; Inan et al., 1991, 1995]. Because of that, the "Trimpi" effect may be interpreted as a result of the scatter of the electromagnetic field propagating in the wavequide Earth-ionosphere at a local irregularity in the lower ionosphere. The goal of this paper is a mathematical simulation of the Trimpi effect which is a short-time variation in the amplitude and phase of VLF signal caused by appearance of a local three-dimensional perturbation in the lower ionosphere. The experimental recordings of the amplitude and phase of VLF signals published by Nunn [1996] show that sudden changes in the amplitude and phase may reach 6 dB and 10o, respectively. Such phenomena as heating of the ionosphere by electromagnetic pulse of the lightning [Inan et al., 1991], energetic electron precipitation [Rycroft, 1973], and "cloud-ionosphere discharges" [Dowden et al., 1994] may be causes of such changes. The Trimpi effect related to sprites (what can be also considered as three-dimensional irregularities of the propagation medium) attracts increased interest especially recently [Rodger, 2003]. Sprites are rather rare phenomena and it is still impossible to reproduce them in laboratory, so for their studies all possible methods should be studied including the VLF remote sounding method observing and studying variations in the fields of permanently operating VLF transmitters.
[3] Numerical simulation of the process of low-frequency electromagnetic wave propagation in the Earth-ionosphere waveguide in the presence of a local three-dimensional irregularity (the scattering properties of which should correspond to the currently available information on the plasma formations in the lower ionosphere) may help in choosing between this or that theory proposed to describe the causes and development of the above noted phenomena. The available observations do not provide complete information on the nature of such phenomena, and their principal parameters are known fairly approximately. One may state definitely that such irregularity is located in the region ~50-90 km over the Earth's surface, that is, in the vicinity of the lower boundary of the ionosphere where its least ionized layers are located. So, if one identifies the appearance of the Trimpi effect with a sudden appearance of a three-dimensional local irregularity in the lower ionosphere, the variations in the amplitude and phase of the received signal at a particular radio wave propagation path may be explained by the scatter of radio waves at such irregularity. In this case one has to assume that the irregularity is characterized by increased (relative to the surrounding medium) charged particle concentration and probably by higher temperature (which fact may be interpreted as an increase in the effective collision frequency of electrons).
[4] Two types of the Trimpi effect are described in publications. The first one is a classical Trimpi effect [Helliwell et al., 1973]. It is assumed to be a result of whistlers caused by energetic electron precipitations [Rycroft, 1973]. Part of the electromagnetic pulse energy the source of which was a lightning discharge may propagate with small losses along the geomagnetic field lines reflecting from the ionospheric boundary in the magnetically conjugated points and interacting to energetic electrons at the geomagnetic equator. The latter process may lead to the transfer of part of the energy from electrons to the electromagnetic wave. The electrons that have lost part of their energy are precipitating into the lower ionosphere altering its properties [Rycroft, 1973]. The classical Trimpi effect is characterized by some delay after the formation of a spherical (~0.6 s) and rather long relaxation time (up to 100 s) [Rodger, 2003].
[5] The so-called "Early Trimpi" has much shorter delay (less than 100 m s) between the lightning discharge and registration of the disturbance [Inan et al., 1988]. Dowden et al. [1996] and Rodger [2003] supposed that the ionospheric disturbances related to sprites and elves are manifested in the form of Early Trimpi. The Trimpi effect caused by sprites has a typical logarithmic decrease in time [Dowden and Rodger, 1997].
[6] Cloud-ionosphere discharges [Winckler, 1995] was one of the initially used terms for red sprites discovered in 1989 [Franz et al., 1990]. Sprites are manifested in the form of light flashes observed over the thunderstorm clouds. It is interesting that for a human eye these flashes look red only in 37.5% of the cases (in other cases they look rose, orange, or even green) [Lyons, 1996]. These picturesque phenomena [Dowden et al., 1997a; Lyons, 1994; Rairden and Mende, 1995; Sentman and Wescott, 1995; Winckler et al., 1996] immediately attracted common attention. However, their study appeared to be a rather complicated problem due to their low optical brightness [Stenbaek-Nilsen et al., 2000] and short lifetime equal to tens of milliseconds [Lyons, 1996]. Probably that is why their observations are registered only at night. Theoretically, it is possible to observe sprites by a naked eye but only in exceptional cases. So to study these phenomena special optical systems are used [Rairden and Mende, 1995; Sentman and Wescott, 1995]. By their appearance sprites may be subdivided to the following types: "similar to a carrot", "having the shape of a column or "jelefish" [Rodger, 1999]. Their occurrence is usually related to strong ( > 50 kA) lightning discharges between a cloud and the surface of positive polarity ( + CG) [Boccippio et al., 1995; Winckler, 1995]. Sprites are generated at a height of about 50 km, that is, approximately by 30 km above the thunderstorm cloud. Their upper boundary is located at a height of about 90 km over the Earth's surface. The horizontal diameter of sprites (in the case when there are not less than two "columns") is 25-50 km [Rodger, 1999]. The frequency of sprite generation is rather low. Over the entire globe, lightning discharges occur 50-100 times per minute, but only a few are accompanied by sprites. The causes of this are, first, the fact that the lightning discharges of the positive polarity occur much more seldom than the discharges of negative polarity (10% "+CG" and 90% "-CG") [Uman, 1987]. Second, all sprites are related to strong positive discharges, whereas only in rare cases strong positive discharges are accompanied by these phenomena. Appearance of sprites has been detected in various climatic conditions both over the land and seas [Boeck et al., 1995; Vaughan et al., 1992].
[7] There exist various theories of the sprite generation mechanism. It is assumed that the electromagnetic pulse accompanying a lightning discharge (EMP), or the quasi-electrostatic field (QE) may be the cause of their generation [Pasko et al., 1997; Rowland et al., 1995; Valdivia et al., 1997].
[8] Besides red sprites, such events as blue jets and elves are observed over thunderstorm clouds. Blue jets appear directly over the center of a thunderstorm cloud and are a narrow upward cone. Being born at the height of the upper boundary of the cloud (about 20 km), they reach a height of about 40-50 km over the Earth's surface spreading away in their upper part to the dimension of about 5 km with the cone aperture angle of about 15o. There exists a theory assuming that the blue jets initial stage is blue starters. They also present cones covering the region from approximately 17 km (the upper boundary of the cloud) to 25 km over the Earth's surface and in their upper part have a width of about 2 km. It was detected that they are generated in the vicinity of negative charges [Wescott et al., 1996].
[9] The phenomena called elves are observed at altitudes of about 75-105 km after strong lightning discharges. [Fukunishi et al., 1996a, 1996b]. It is assumed that these formations having the diameter of 100-300 km occur due to the heating of the electrons in the lower ionosphere by the electromagnetic pulse of the lightning discharge [Inan et al., 1996].
[10] Many authors paid attention to studies of the Trimpi effect. A fairly detailed review of these studies was presented by Soloviev and Hayakawa [2002], who analyzed various methods of solution of the three-dimensional problems in the radio wave propagation theory. As far as we know from the publications, currently nothing cardinally new can be added to the list presented earlier. This paper presents a development of the studies begun by Soloviev and Hayakawa [2002]. Unlike in the latter paper, in this paper the influence of the geomagnetic field in the problem of diffraction at a three-dimensional irregularity is considered. This influence in the considered frequency range is most strongly manifested at night when all sprites described in publications were optically observed.
[11] In this paper to solve the problem on the field of a vertical electric dipole in the Earth-ionosphere waveguide having a local irregularity at the upper wall, an original method based on the integral equations theory [Soloviev and Hayakawa, 1997] is applied. The irregularity is chosen in the form of a finite by height cylinder without any limitations on the shape and dimensions of its cross section. Though in publications there are indications to a presence of a fine structure of the sprites, this model does not take into account such structure, but models a sprite as a scattering volume. It is evident that using VLF electromagnetic field with a wavelength of the order of 15 km (for a frequency of 20 kHz) one cannot resolve the fine structure of a sprite. The scatter from the system of close-located "columns" would not differ from the scatter at the whole cylinder.
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