Reply to comment by Rajeev Thottappillil and Vladimir A. Rakov on “Radio frequency radiation beam pattern of return strokes: A revisit to theoretical analysis”


1. Introduction

[1] Thottappillil and Rakov [2005] argued that equation (10) given by Shao et al. [2004] (hereinafter referred to as SJF) is not general and cannot be used to derive complete solutions for return stroke models. This is not in conflict with the analysis of SJF, since it was only used to demonstrate the generality of our equation (7) under special conditions stated by SJF. To avoid possible future confusion, this reply gives complete solutions for all three models brought up by Thottappillil and Rakov [2005]. We further show that the F factor is fundamental for the radiation beam pattern as long as a moving current is concerned, although it may take different forms for different models. We also show that by traveling along with the propagating current pulse the analyses can be significantly simplified, a clear contrast to previous work. In this reply, we give a general radiation solution for the modified transmission line (MTLE) model that has not been reported before.

[2] SJF derived the F factor (1 − v cos θ/c)−1 for a moving current pulse through a simple differential transformation between the retarded time and stationary time/space and claimed that this factor is fundamental for the radiation beam pattern as long as the current pulse is in motion relative to an observer. To demonstrate its generality, we applied the result to two simple return stroke models: the transmission line (TL) and the traveling current source (TCS) models. We appreciate Thottappillil and Rakov's [2005] comments (hereinafter referred to as TR) on the paper and would like to take this opportunity to address their comments and to further discuss and clarify the analysis presented by SJF.

2. Discussion

[3] First, as stated by SJF, equations (6) and (7) (equation (7) is reproduced as equation (1) by TR) are general results for far-field radiation from a moving current pulse. Without repeating the discussion of SJF we emphasize here that these equations clearly show the physical nature behind the radiation intensity (∝∂i(z′, t′)/∂t′) and the beam pattern (∝ sin θ(1 − v cos θ/c)−1), which are not immediately clear in previous reports.

[4] In their comments, TR agreed that equation (7) is general and argued that equation (10) is not. We agree with these comments; however, they are not in conflict with the SJF discussion! In the SJF paper, derivations of equations (10)(12) were merely used for demonstration and were not meant to give comprehensive answers for all of the different discharge components, as we stated in paragraph 23 following equation (12). TR should have been aware of the special conditions that led to equation (10), especially the first one that says “v is constant.” In addition to the explicitly assumed conditions of SJF, an implicit but well-understood assumption of continuous, lossless current propagation was also imposed in order to carry out the integral (from (2a) to (2b), as given by TR). TR are correct that our equation (10) is only valid for the TL type of current models, as it was intended to be in the SJF paper under the special conditions.

[5] Having said the above, we seem to have answered most of TR's comments toward SJF. However, it appears necessary to some readers to give a more complete treatment for each of the three suggested models, namely, TL, TCS, and MTLE, based on equation (7) of SJF. Using the framework laid out by SJF, one can express the change of current at (z′, t′) as

equation image

From equation (1) one can get

equation image

Here v = dz′/dt′ is the speed of the current pulse, di(z′, t′) is the net current change that could be due to current injection (source) or current attenuation (sink), and ∂i(z′, t′) is the current gradient along the channel.

[6] For the simplest model of continuous, lossless current propagation, i.e., the TL model and the part of the downward current propagation in the TCS model, di(z′, t′) = 0 over [0, L′). If the process is viewed from a stationary frame, equation (2) becomes

equation image

From this and equation (10) of SJF one gets

equation image

Note that we missed the minus sign in the SJF paper, as correctly pointed out by TR. From this equation we obtained the analytic results for the TL (equation (11) given by SJF should have an opposite sign) and TCS models, regarding the continuous propagation portion of the models. For the purpose of simple demonstration we stopped here in the SJF paper. To address TR's comments on the complete solutions for the TL and TCS models, we need to extend the analysis a bit further.

[7] In addition to the continuous current propagation, at any t′, the channel extends instantaneously at the front (z′ = L′). This extension adds extra current and is equivalent to a current source at the top of the channel. From equation (2) we have, for the TL model,

equation image

Adding this to the part of continuous current propagation, one gets the total radiation

equation image

The current source di(L′, t′) in equation (5) is similar to a current discontinuity discussed by others [e.g., Rubinstein and Uman, 1990; Thottappillil et al., 1998]. However, in the analysis here, no special treatment is necessary, for a current source of either nonzero (TL with a so-called “turn-on” term) or zero.

[8] For the TCS model the channel extends at a rate u, and one gets the same equation as (5) for the current source after replacing v with u. Putting this together with the part of continuous propagation (equation (12) of SJF), we can readily obtain equation (3) of TR. Notice that for the downward propagation in TCS the polar angle between v (∣v∣ = −c) and r is π − θ, and for the upward extension (u) the polar angle is θ.

[9] In the discussions above, the integrals were carried out in the stationary frame in an attempt to relate our analysis to previous works [e.g., Thottappillil et al., 1998]. This is fine as long as the same physical channel and correct discharge processes are considered. However, the analysis can be further simplified if one moves along with the traveling current pulse (dz′). For the TL model, if one travels at the same speed v, one would see that ∂i(z′, t′)/∂t′ = 0 everywhere along (0, L′] since the current waveshape is assumed to be fixed after it emerges from the base of the channel. This is analogous to a smoothly moving train on which the driver sees no changes anywhere along the train if all passengers sit in their seats (all travel at the same constant speed) and no one jumps on or off the train (lossless), even though an observer on the ground may see changes of passenger density while different sections of the train pass through his fixed field of view (e.g., equation (3)). In the moving frame the channel has an instantaneous extension (current source) at the back end (z′ = 0), instead of at the front, as viewed from the stationary frame. Under these considerations the only physical process that contributes to the radiation is the current source at (z′ = 0, t′). So for TL, one can simply get

equation image

A similar but brief discussion like the above is given by SJF (paragraph 21, below equation (11)). It should be emphasized here again that a moving (at constant speed) but unchanging current wave, like the propagation portion of the TL, will not produce any radiation, as discussed previously by SJF.

[10] For TCS, there are two current sources at the front end (z′ = L′); one travels at speed c with a polar angle π − θ, and the other travels at speed u with a polar angle θ. The downward current is absorbed at the channel base (a current sink) at speed c and is viewed at π − θ. Like the TL, along (0, L′) the current propagates losslessly at a constant speed c and will not contribute to the radiation. Putting these together, we have

equation image

From this it is straightforward to arrive at equation (3) of TR. It should be noticed here that for each source/sink the integral is along current movement and the corresponding velocity is then always positive.

[11] In the modified transmission line model the current decays exponentially with height [Nucci et al., 1988]. Without the simplified step function assumption [Wait, 1998] the current can be described as

equation image

where λ is the decay length measured in the stationary frame. In a moving frame (the situation here) the apparent length is λ′ = λ(1 − v cos θ/c) for λ ≪ r, as indicated by Le Vine and Willett [1992] in their equation (10c). Similar to the TL model, there is a current source at z′ = 0. However, along the moving (0, L′], ∂i(z′, t′)/∂t′ ≠ 0 because of the current decay. In equation (9), i(0, t′ − z′/v) is the same as in the TL model, and it has no temporal change as one moves along with the current. The factor ez′/λ′ (which replaces λ with λ′ for the moving frame), on the other hand, has an apparent temporal change viewed from the moving frame. Therefore

equation image

So the total field for the MTLE model can be readily expressed as

equation image

This is the general solution for the MTLE model that we believe has not been reported before! The second term is effectively due to the current sinks along the channel. As λ′ → ∞, equation (11) becomes (7) for the TL model, as one would expect.

[12] If the current is simplified to i(0, t′) = I0u(t′), where u(t′) is the step function at t′ = 0, as assumed by Wait [1998] in his analysis, equation (11) is simplified to

equation image

Note that t′ − z′/v ≥ 0 and t′ = tr/c. This is exactly the same as equation (4) of TR, who apparently reproduced the result from Wait [1998]. It is not clear, though, why TR used “≈” rather than “=” in their equation. From equation (11), readers should be able to get the final analytical solutions for some other simple current waveshapes.

3. Summary

[13] We agree with TR that equation (10) of SJF is not general, since it was not meant to be. We should have made it more obvious. Nevertheless, we found it necessary for some readers to extend the SJF analysis. Current discontinuity at the wave front was a major concern of TR, and they have in the past treated it very carefully with thorough mathematic analyses [e.g., Thottappillil et al., 1998]. In this reply, by using the fundamental relation of equation (7) given by SJF, the discontinuity needs no special treatment. However, it might be reasonable to argue that the concept of current source/sink is analogous to the discontinuity.

[14] This reply further shows that the F factor is a fundamental radiation beam factor for moving current, which may, however, appear in different forms in different models. It is also shown that if one moves along with the current pulse, the analysis can be significantly simplified with only a few steps to arrive at the solution, in contrast to lengthy mathematical derivations in previous reports.

[15] After all, the derivation of equation (7) of SJF is nothing new, and similar treatment between moving and stationary frames can be found in almost all the classical electrodynamics textbooks. Our only contribution is introducing it to the lightning research community.


[16] This work was performed at the Los Alamos National Laboratory under the auspices of the U.S. Department of Energy.