Electromagnetic Radiation Spectrum of a Composite System

In many physics fields, the radio emission of a composite system composed of a large number of randomly occurring but similar emission sources is encountered. In general, the composite system lasts longer than each individual component and individual source currents vary much more rapidly. This Letter presents a theory to understand the electromagnetic radiation spectrum of such a system. If the temporal distribution of the random occurrence of the component and the distribution to describe the relevant emission source properties are known, the spectrum of the composite system can be readily found from this theory. There are two main terms that define the spectrum: one term results from the coherent summation of the contributions from individual sources and is proportional to the square of the total number of the components in the system; the other term results from an incoherent summation and is proportional to the first power of that number. This can lead to drastically different spectral magnitudes in different spectral regions, typically with the spectral magnitude in the lower frequency region many orders of magnitude stronger than that in the higher frequency region.

Figure 1

(a) A sprite, composed of many streamers, above a thunderstorm. (b) Monte Carlo simulation of a relativistic runaway electron avalanche inside a strong thundercloud field [45]. (c) The energy spectral density of a thunderstorm narrow bipolar event, which is thought to be caused by a system of streamers. Figure adapted from [29]. (d) The energy spectral density of the radio emissions associated with a terrestrial gamma ray flash. Figure adapted from [49].

Figure 2

Simulation results of an ensemble of ${10}^7$ identical streamers. The ensemble current moment (a) and its time derivative (b). (c) The ESD of a single streamer ensemble. (d) The average ESD of ten streamer ensembles. The red dashed line shows the spectrum obtained by using Eq. (2). The two horizontal dashed lines give the ESD levels at the VLF and HF bands, respectively.

Figure 3

(a) Probability for the streamer current rise time. The rise time for each bin is set at its mid value. (b) The streamer current pulses for the shortest and longest rise times. The inset shows an enlarged view of the fast rise of the current, and the thick black curve shows the average current pulse. (c) The spectral magnitudes of the current pulses in (b). The thick black curve shows the spectral magnitude of the average current pulse and the dashed curve shows the mean of the spectral magnitude.

Figure 4

Modeling results of an ensemble of ${10}^7$ nonidentical streamers. (a) The ESD of a single streamer ensemble and (b) the average ESD of ten streamer ensembles, where the streamer current rise time follows a nominal Gaussian distribution with a mean of 1 ns and a standard deviation of 10 ns (Fig. 3). The average ESD for a reduced deviation of the streamer current rise time: (c) 1 ns and (d) 0.1 ns. The yellow and red dashed lines show the ESD from Eqs. (1) and (2), respectively, where the spectrum of the average streamer current pulse is used [thick black lines in Fig. 3 for (a) and (b)].