## 1. Introduction

[2] Modeling very low frequency (VLF, 3–30 kHz) wave propagation in the Earth-ionosphere waveguide remains an important yet computationally difficult problem. In free space, VLF waves have typical wavelengths in the range of tens of kilometers. However, the situation rapidly changes in the partially ionized lower ionosphere. Near the VLF reflection height (approximately 85 km at night) for a typical midlatitude ionosphere at 24 kHz, the wavelength can drop to less than a quarter of its free-space value for propagation parallel to the background magnetic field or even zero for non-parallel propagation [*Stix*, 1962]. There have been many approaches to modeling propagation and scattering of VLF waves in the Earth-ionosphere waveguide. Some full-wave models solve a series of one-dimensional reflection problems at the boundaries between uniform, stratified layers and then use these reflection coefficients to find the solution to arbitrary problems as the sum of waves in *k*-space. Such techniques have been used to successfully model propagation of VLF waves over very large distances [*Pappert and Ferguson*, 1986; *Lehtinen and Inan*, 2008]. Inhomogeneities can be handled in a number of different ways. Modal techniques such as those used by the long-wave propagation capability (LWPC) [*Pappert and Ferguson*, 1986] solve for propagation within “slabs,” each of which is assumed to be composed of horizontally stratified layers, infinite in extent. Mode-coupling calculations are used to propagate the fields in each slab into adjacent slabs. Weak, localized disturbances can be handled as done by *Lehtinen et al.* [2010], who uses the Born approximation to model the inhomogeneity as a set of equivalent currents. This approach is powerful but is limited to relatively mild density perturbations. Recently, generic field solvers like finite difference time domain (FDTD) [*Taflove and Hagness*, 2005; *Yee*, 1966] have gained in popularity and have been successfully applied to VLF propagation and scattering off ionospheric disturbances [*Cummer*, 2000; *Peter et al.*, 2006; *Marshall and Inan*, 2010]. These simulations are typically bound by the number of unknowns required to accurately discretize the space. Typical rule-of-thumb guidelines state that on the order of 10 cells per wavelength are required to accurately represent a solution using FDTD [*Taflove and Hagness*, 2005, p. 34]. While this is not necessarily prohibitive for moderately-sized problems (on the order of a few hundred kilometers per spatial dimension) within the Earth-ionosphere waveguide, the situation rapidly degrades above the VLF reflection height, where the wavelength can easily drop to less than a quarter of its free space value. For reasonable accuracy on a fixed FDTD grid, then, the grid spacing must be chosen on the basis of the smallest wavelength of interest in the solution, and thus is enormously prohibitive. While FDTD can be used on semi-structured (plaid) and even fully unstructured grids, doing so typically reduces the formal order of accuracy of the scheme from second to first order, and also significantly complicates the derivation of current update equations for magnetized plasmas or other anisotropic, dispersive materials. High-order FDTD techniques, by contrast, suffer from non-locality of the finite difference operators, which leads to failure of the scheme in the presence of sharp gradients or material discontinuities.

[3] In this paper, we discuss the application of a relatively new class of techniques known as nodal discontinuous Galerkin (DG) methods to the problem of wave propagation in the Earth-ionosphere waveguide. Since it is straightforward to use the DG method on unstructured grids and doing so has no impact on the formal order of accuracy, we can manage the scale problem by sampling more finely only where it is needed, i.e., near the VLF reflection height, while using a coarser grid elsewhere. This approach allows accurate and flexible simulation of scattering off ionospheric disturbances in the VLF regime with fewer restrictions on the type or magnitudes of the plasma density perturbations that can be modeled.

### 1.1. Background

[4] Lightning discharges can modify the ionosphere via a number of different mechanisms. Because of the impulsive, energetic nature of a lightning return stroke, a large electromagnetic pulse (EMP) is radiated from a lightning channel. The fields associated with this EMP are intense enough to cause significant heating and ionization of the ionosphere above the discharge [*Inan et al.*, 1991; *Cheng et al.*, 2007; *Marshall et al.*, 2008, 2010]. A number of secondary effects are also present; for instance, the amount of charge removed by a lightning return stroke creates a large quasi-electrostatic (QE) field above a lightning discharge, which can further modify the levels of ionization [*Pasko et al.*, 1995, 1998]. In either the QE case or the EMP case, the magnitude of the change in the ambient electron density can be on the order of a few percent to over a hundred percent, peaking in an altitude range of approximately 80 to 100 km and extending over radial distances of over 100 km [*Cheng et al.*, 2007; *Marshall et al.*, 2010]. Because of the abrupt onset of such disturbances, their altitudes near the VLF reflection height, and their relatively wide spatial extent, they are expected to modify the properties of VLF waves propagating within the Earth-ionosphere waveguide. This can be observed by a ground-based receiver as a phase or magnitude perturbation on a received narrowband signal observed shortly after a causative lightning discharge. Indeed, this connection was observed quite early [*Armstrong*, 1983], but the exact nature of the disturbances and their effects on received VLF signals have not yet been fully resolved.

[5] We discuss one particular class of VLF signal disturbances, termed “early” VLF events. Early VLF events are abrupt changes in the amplitude of a received narrowband VLF signal following a lightning discharge. They are characterized by their abrupt onset (< 20 ms) after a causative lightning discharge (hence the term “early”) and their relatively slow recovery time (10–100 s). Early/fast VLF events are also characterized by 15 dB beam widths of less than 20°–30° [*Johnson et al.*, 1999; *Moore et al.*, 2003]. A number of different causative mechanisms have been postulated [*Inan et al.*, 1991; *Pasko et al.*, 1998; *Moore et al.*, 2003]. More recent modeling efforts have focused on EMP-induced ionospheric disturbances and shown that they are not inconsistent with such events [*Cheng and Cummer*, 2005; *Cheng et al.*, 2007; *Marshall and Inan*, 2010]. A recent survey paper [*Inan et al.*, 2010] summarizes what is known (as of 2010) of these lightning-ionosphere interactions, among others. Recent modeling efforts by *Cheng and Cummer* [2005], *Cheng et al.* [2007], and *Marshall and Inan* [2010] have focused on scattering from lightning EMP-induced ionospheric disturbances, which we investigate further in this paper.

[6] We compute the narrowband VLF fields scattered from such lightning EMP-induced density perturbations. Previous work by *Marshall and Inan* [2010] used a 2-D finite difference model to show that the EMP from very intense lighting discharges, with *E*_{100} (defined by *Uman and McLain* [1970] as the magnitude of the electric field as observed on the ground at 100 km from a discharge) in the range of 7–40 V/m, can perturb the ambient electron density enough to cause measurable perturbations on the amplitude of a received narrowband VLF signal. Later work by *Lehtinen et al.* [2010] used the Born approximation to approximate the scattered field in a medium of homogeneous, horizontally stratified layers. This work revealed the full three-dimensional structure of the scattered field from a lighting EMP-induced density perturbation for a range of incidence angles. The technique used by *Lehtinen et al.* [2010], however, assumes that the scattered field is much smaller than the incident field and as such is unsuitable for the whole range of density perturbations considered by *Marshall and Inan* [2010]. In addition, the technique cannot account for multiple scattering or modes propagating exactly parallel to the stratified layers, necessitating the use of a more general solution method for strong perturbations. The magnitude, angular extent, and shape of the scattered field and amount of transverse variation for intense, spatially complicated (e.g., ring-shaped) disturbances were previously unknown; this work seeks to address this.

[7] We extend the work of *Lehtinen et al.* [2010] and *Marshall and Inan* [2010], using a fully three-dimensional continuum electromagnetic DG solver to solve for the VLF scattered fields from an EMP-induced ionospheric disturbance. The DG method has an arbitrary order of accuracy but is also highly local and parallelizable, combining desirable features from the finite element method and the finite volume method. Similar to low-order finite volume methods, the only data that must be communicated between CPUs are the field values at the faces between elements. In contrast to traditional finite element methods, the mass matrices are strictly local to each element; thus, no large, sparse system of equations need be solved. In addition, the polynomial order (and thus the accuracy) can be increased as much as desired without increasing the physical width of the stencil or destroying the locality of the scheme. As a continuum, time domain method, backscattering and multiple scattering are implicitly handled; the only significant limitation, similar to those in FDTD techniques, is that the wavelengths of interest must be sufficiently sampled. For traditional staggered-grid FDTD, a minimum 2 cells per wavelength are required for convergence, but typical rule-of-thumb guidelines dictate that on the order of 10 cells per wavelength are required for acceptable accuracy. The DG method has similar restrictions: at minimum, on the order of *π* unknowns per wavelength are required for convergence. However, in contrast to FDTD, the order of accuracy can be increased without restrictions on the material parameters or the amount of grid inhomogeneity. This means that for a given domain, the error in a high-order DG simulation will decrease faster than that of an FDTD simulation for a given reduction in the grid size. Further, the DG technique is easily adapted for use on completely unstructured, strongly inhomogeneous meshes, which allows us to use smaller cells only where they are needed, e.g., in the ionosphere where the VLF whistler wavelengths are much shorter than their free-space equivalents for a given frequency. In the context of this simulation work, the free-space wavelength is equal to 12.5 km, but in the ionosphere drops to less than 2 km for propagation parallel to the magnetic field.

### 1.2. Outline and Summary

[8] We begin by discussing the computational model in section 2. We show how to incorporate linear, anisotropic dispersive materials, applying the technique to the cold plasma model used in this paper and to the perfectly matched layer (PML), an absorbing boundary first introduced by *Berenger* [1994], we use to truncate part of the domain for our simulations. We validate the technique in section 3, showing that the scheme is consistent with analytical results. We discuss the ambient model, scattering model, computational domain, and source conditions in section 4. In section 5, we discuss the results, showing the scattered field amplitudes on the ground for a variety of incident source conditions.

[9] Our three-dimensional simulation results show that the scattered fields from lightning EMP-induced ionospheric disturbances can have complicated structure with strong transverse variation, particularly in the region near a disturbance produced by a vertical lightning discharge. Our model shows that while not all features of “early” VLF events can be reproduced by a single, large vertical lighting EMP-induced disturbance, the scattered field amplitudes are nonetheless significant, with measurable amplitude changes greater than 0.2 dB possible under smooth ambient conditions. As an extreme case, we show that the disturbance induced by 60 repeated, intense intercloud lightning discharges can result in strong perturbations on the order of 0.2 to 0.5 dB under smooth ambient conditions, with the near-field values showing strong spatial variation. We also show that while the bulk of the wave energy from such large (with respect to the wavelength) disturbances is scattered in the forward direction, there are nonetheless non-negligible scattered fields in the transverse direction. We also show the phase response from the same simulations, showing that it may also be a useful diagnostic measure when considering narrowband signal perturbations from ionospheric scatterers.