We use a full-wave approach to find the field of monochromatic whistler waves, which are excited and propagating in the low nighttime ionosphere. The source current is located in the horizontal plane and can have arbitrary finite distribution over horizontal coordinates. The ground-based horizontal magnetic field and electric field at

ELF/VLF waves, which propagate in the ionosphere in whistler mode, are an important part of the ionosphere dynamics. Such waves can be emitted by various natural phenomena such as atmospheric lightning discharges and volcanic eruptions, magnetospheric chorus and hiss. Artificial ELF/VLF waves have been produced by ground-based transmitters and by modulated high-frequency (HF) heating of the ionosphere current system responsible for

Several numerical methods have been developed for calculating whistler wave fields in the Earth's ionosphere (Pitteway, 1965; Wait, 1970; Bossy, 1979; Nygre'n, 1982; Budden, 1985; Nagano et al., 1994; Yagitani et al., 1994; Shalashov and Gospodchikov, 2011). One of the main difficulties is numerical instabilities caused by a large imaginary part of the vertical wave number. General full-wave analysis, including the problem of numerical “swamping” of the evanescent wave solutions, was performed, for example, by Nygre'n (1982), Nagano et al. (1994), and Budden (1985). A traditional approach in full-wave analysis is to divide a stratified ionosphere into a number of thin horizontal and homogeneous slabs and then connect the solutions in each slab by applying the boundary conditions. Such a technique has been used by Yagitani et al. (1994) to study ELF/VLF propagation from an infinitesimal dipole source located in the lower ionosphere. The idea of recursive calculation of mode amplitudes was developed and used for an arbitrary configuration of the radiating sources by Lehtinen and Inan (2008). Nevertheless, finding fields created by both natural and artificial ELF/VLF radiating sources is still very relevant.

In this paper, we use numerical methods to find the field of ELF/VLF waves, which have been produced in a low nighttime ionosphere. On the one hand, significant inhomogeneity of plasma parameters, strong wave mode attenuation and effect of wave mode transformation (for example, whistler to vacuum electromagnetic) in the low-altitude nighttime ionosphere make the problem considered difficult enough and fundamentally important. On the other hand, it has practical significance, as an example, for interpretation of numerous experimental results on HF heating which modulate natural ionospheric currents at altitudes of 60–

In calculations, we use a technique known as the two-point boundary-value problem for ordinary differential equations (Kierzenka and Shampine, 2001). Using this technique in early work (Bespalov and Mizonova, 2017; Bespalov et al., 2018) has provided numerically stable solutions of a complete system of wave equations for arbitrary altitude profiles of plasma parameters and in the stratified ionosphere for arbitrary angles of wave incidence from above. Here, we find a wave field created by a monochromatic source current located in the low night ionosphere. We examine the influence of peculiarities of current distribution on the proportion in which source energy supplies the Earth–ionosphere waveguide or flows upward. As an example of calculations, we use current distributions similar to those simulated by HF heating of the auroral electrojet (Payne et al., 2007). The obtained results are important for analysis of the ELF/VLF emission phenomena observed both in the ground-based observatories and onboard satellites.

We consider a whistler wave which is excited and propagating in the layer

To solve the system in Eq. (5), in Eq. (6) we define four boundary conditions. We write two of them on the plane

We take into account that out of the plane

At first we find two linearly independent solutions

In the calculations, we use the altitude profiles of the plasma density shown in Fig. 1a and the collision frequencies between charged and neutral particles shown in Fig. 1b. The plasma density data are taken from International Reference Ionosphere (Bilitza and Reinisch, 2007), available from CCMC at

Nighttime ionosphere model:

We assume that currents occupy a volume which has a shape of a horizontal pancake and use for calculations a Gaussian distribution over

Fields in the

Altitude dependences of the amplitudes of horizontal electric and magnetic fields.

We use a full-wave approach to find the field of monochromatic whistler waves which are excited and propagating at night in the strongly inhomogeneous low ionosphere. A source current is assumed to be located in the horizontal plane and to have in this plane an arbitrary finite-space distribution. At first, we consider a plane wave with the horizontal components of the refractive index

Source currents and field space distributions:

Current density (red line) and field space distributions at

As an example, we calculate the fields created by varying at a frequency of

At altitudes of source and above, the two ELF/VLF wave modes Eq. (8) are weakly damped and have right-hand polarization, and another two modes are evanescent and have left-hand polarization. At altitudes

In the second case of ratio

If the horizontal size of radiating currents is small enough,

The current distributions used as an example in our calculation and presented in Fig. 4 can be similar to electrojet currents modulated in the D region by the HAARP HF heating facility (Keskinen and Rowland, 2006; Payne et al., 2007). For example, according to data collected during an experimental campaign run in April 2003 and results of numerical simulations (Payne et al., 2007; Lehtinen and Inan, 2008), the maximum change in modulated conductivity occupies approximately

We find a field of monochromatic whistler waves which are excited and propagating in the low nighttime ionosphere. Using a MATLAB boundary-value problem solver enables us to find numerically stable solutions of a full set of the wave equations applying to conditions of an inhomogeneous ionosphere at altitudes below

The paper is theoretical, and no new experimental data are used. The data are taken from the International Reference Ionosphere model (Bilitza and Reinisch, 2007) (

VGM produced the calculations, analyzed results, and wrote the paper. PAB proposed the problem, discussed results, and wrote the paper.

The authors declare that they have no conflict of interest.

The work (Sects. 2–4) is supported by RFBR grant no. 20-02-00206A. The work of Peter A. Bespalov (Sects. 1, 5, and 6) is supported by RSF grant no. 20-12-00268. The numerical calculations were performed as part of the State Assignment of the Institute of Applied Physics RAS, project no. 0030-2021-0002.

This research has been supported by the RSF (grant no. 20-12-00268).

This paper was edited by Dalia Buresova and reviewed by three anonymous referees.