Chapter 2: Fractal Antennae

• Fractal Antennae and Coherence
• Radiation and Simple Fractal Models
  • Gain Due to Tortuosity
  • Fractal Tortuous Walk
  • Branching and Spatial Structure
• Modeling Lightning as a Fractal Antenna
  • Fractal Lightning: Stochastic Model
  • Computing the Fields from the Fractal Structure
  • How does the Fractal Dimension Affect the Field Pattern?
  • Fields from the Stochastic Model
  • Scaling of the fields

Fractal Antennae and Coherence

For the purposes of this work, we assume that a fractal antenna can be formed as an array of ''small'' line elements having a fractal distribution in space. Such description is consistent with our understanding of fractal discharges and lightning observations as discussed by LeVine and Meneghini [1978], Niemeyer et al. [1984], Sander [1986], Williams [1988], and Lyons [1994]. Appendix A develops the theory for the calculation of the fields produced by a fractal antenna composed of small line elements and for the calculation of the array factor in the far field of the fractal.

Fractals are characterized by their dimension. It is the key structural parameter describing the fractal and is defined by partitioning the volume where the fractal lies into boxes of side $\varepsilon$. We hope that over a few decades in $\varepsilon$, the number of boxes that contain at least one of the discharge elements will scale as $N(\varepsilon )\sim \varepsilon ^{-D}$. It is easy to verify that a point will have D=0, a line will have D=1 and a compact surface will have D=2 . The box counting dimension [Ott, 1993] is then defined by  

$$D\simeq \frac{\ln N(\varepsilon )}{\ln (\frac 1\varepsilon )}$$ (1)

For a real discharge there is only a finite range over which the above scaling law will apply. If $\varepsilon$ is too small, then the elements of the discharge will look like one-dimensional line elements. Similarly, if $\varepsilon$ is too large, then the discharge will appear as a single point. It is, therefore, important to compute D only in the scaling range, which is hopefully over a few decades in $\varepsilon$. The fractal dimension will be an important parametrization for the fractal discharge models that we will explore later, and will impact significantly the intensity and spatial structure of the radiated pattern.

To understand the concept of a fractal antenna, we start with a long line antenna which radiates as a dipole (top of Fig. 2.1). Then we make the antenna path tortuous (middle of Fig. 2.1), essentially generating a two dimensional phased array. The resulting tortuous (fractal) antennae can then be considered as a non uniform distribution of radiating elements (bottom of Fig. 2.1). The radiation pattern will exhibit an antenna gain if the radiating elements radiate, at least partially, coherently.

  

Figure 2.1: A long line antenna (top). A tortuous antenna connecting the same two endpoints (middle). Such descriptions arises naturally fractal discharges like lightnin. A spatially nonuniform distribution of radiators (bottom).

Each of the elements contributes to the total radiated power density at a given point with a vectorial amplitude and phase, i.e.  

$${\bf E}\cdot {\bf E}^{*}\sim (\sum_{n=1}^N{\bf A}_ne^{i\phi ... ...=\sum_{n,m}({\bf A}_n\cdot {\bf A}_m^{*})e^{i(\phi _n-\phi _m)}$$ (2)

The vector amplitudes A_n represent the strength and orientation of each of the individual elements, while the phases $\phi _n$ are in general related to the spatial distribution of the individual elements over the fractal, e.g. for an oscillating current of the form e$^{i\omega t}$ the phases vary as $\phi \sim kr$ where $k=\frac \omega c$ and r is the distance to the detector position.

In the sense of statistical optics, we can consider the ensemble average of Eq. (2.2), using an ergodic principle, over the spatial distribution $P(\phi _1,\phi _{2,}\phi _{3,}...,{\bf A}_1,{\bf A}_2, {\bf A}_3,...)$ of the fractal elements [Goodman, 1985]. For simplicity we assume that the distributions for each of the elements are independent, and also the same, hence

$$G=\sum_{n,m}^N\left\langle ({\bf A}_n\cdot {\bf A}_m^{*})e^{... ...left\vert \left\langle e^{i\phi }\right\rangle \right\vert ^2)$$

By requiring that $\left\langle \left\vert {\bf A}\right\vert ^2\right\rangle =\left\vert \left\langle {\bf A}\right\rangle \right\vert ^2=1$ we obtain that the ensemble average is

$$\left\langle {\bf E}\cdot {\bf E}^{*}\right\rangle \sim G=N^... ...left\vert \left\langle e^{i\phi }\right\rangle \right\vert ^2)$$

If the distribution of the phases is uniform (e.g. random) then $<e^{i\phi }\gt=0$ and G=1/N. On the other hand, if there is perfect coherence we have $<e^{i\phi }\gt=1$ and G=1. In general, a fractal antenna will display a power law distribution in the phases $P_\alpha (\phi )\sim \phi ^{-\alpha }$(multiplied by the factor $1-e^{-\phi ^{-\alpha }}$ so it is finite at the origin), where $\alpha =0$ corresponds to the uniform distribution case and $\alpha \rightarrow \infty$ corresponds to perfect coherence. Figure 2.2 shows the plot of $\left\vert \left\langle e^{i\phi }\right\rangle \right\vert$ as a function of $\alpha .$ It can be seen that a power law distribution of phases, or similarly a power law in the spatial structure, gives rise to partial coherence.

  

Figure 2.2: A plot of $\left\vert \left\langle e^{i\phi }\right\rangle \right\vert ^2$as a function of $\alpha$.

If the distribution of the vector amplitudes does not satisfy the above relations, e.g. the radiators are oriented in arbitrary directions, then the power density will be less coherent due to $\left\langle \left\vert {\bf A}\right\vert ^2\right\rangle \geq \left\vert \left\langle {\bf A}\right\rangle \right\vert ^2$. A similar result can be achieved by having a power law distribution in the amplitudes. In conclusion, the radiation field from a power law distribution of phases will have a point where the phases from the radiators will add up almost (partially) coherently showing a significant gain over a random distribution of phases.

The parameter $\alpha$ is related to the fractal dimension D [Ott, 1993] of the antenna. We conjecture that the most relevant structural parameter in determining the radiation pattern from a fractal antenna is its fractal dimension. The fractal dimension will be an important parametrization for the fractal models that we will explore later, and will impact significantly on the intensity and spatial structure of the radiated pattern. The partial coherence of the radiators depends on the spatial power law distribution.

Such a power law distribution of phases can be visualized with the help of Cantor sets [Ott, 1993]. A family of Cantor sets is constructed by successively removing the middle $\eta <1$ fraction from an interval, taken as [0,1], and repeating the procedure to the remaining intervals (see Fig. 2.3). At the n^th step, a radiator is placed at the mid-point of each of the remaining intervals.

  

Figure 2.3: The construction of the fractal distribution of the radiators from the $\eta -$Cantor set.

Note that for $\eta =0$ we obtain a uniform distribution of elements, but for $\eta \neq 0$ the radiators are non-uniformly distributed, and in fact the spatial distribution follows a power law that can be described by its fractal dimension. Suppose that for $\varepsilon$ we require $N(\varepsilon )$ intervals to cover the fractal, then it is clear that with $\varepsilon ^{^{\prime }}\rightarrow \frac \varepsilon 2(1-\eta )$ we would require $2N(\varepsilon ^{^{\prime }})$ intervals to cover the fractal. But the fractal is the same, therefore, $N(\varepsilon )=2N(\varepsilon ^{^{\prime }})$. From the scaling $N(\varepsilon )\sim \varepsilon ^{-D}$ we obtain that the dimension is given by

$$D=-\frac{\ln 2}{\ln (\frac{1-\eta }2)}$$

We can go further, and write a formula for the radiation field due to the $\eta -$Cantor set of radiators. Note that if at the n^th step we have the radiators placed at the sequence of points $S_n=\{x_i\vert i=1,...,2^{n-1}\}$then at the n^th+1 step each radiator at x_i will be replaced by two radiators at $x_i\pm \frac 1{2^{n+1}}(1-\eta )^{n-1}(1+\eta )$ generating the sequence $S_{n+1}=\{x_i\vert i=1,...,2^n\}$. Since we start with $S_1=\{\frac 12\}$ the sequences S_n at the n^th step are trivially constructed. The radiation field (see Eq. (2.1)) from this $\eta -$Cantor set at the n^th step can then be written as  

$$E=\sum_{m=1}^n(-1)^m\mu _me^{ikLa\;x_m+i\phi _m}$$ (3)

where $k=\frac \omega c$, L is the spatial extent of the fractal, $a= \widehat{{\bf x}}\cdot \widehat{{\bf r}}$ is the angular position of the detector, and $\phi _m$ (taken as zero) is the phase of the m^th element. The radiators are given a strength proportional to the measure $\mu _i$ (or length) of the segment which defines it.

The space dependence of the radiation fields is plotted in Fig. (2.4)a-b for $\eta =1/3$ (D=0.63) and $\eta =0$ (D=1) respectively, where the sets have been taken to the 5^th level. The most relevant issue for our purposes is the fact that there is a direction at which phases add coherently (partially) for $\eta =1/3$ while this does not happen for the homogeneous case $\eta =0$.

  

Figure 2.4: The spatial dependence of the radiation fields for (a) $\eta =1/3$, D=0.63 and (b) $\eta =0$, D=1.

Therefore, partial coherence occurs naturally in systems that have power-law spatial distributions. We are now ready to turn to the properties of fractal antennae with propagating currents. Specifically, how tortuosity and branching can increase the radiated field intensity in some locations as compared with single dipole antennae.

Radiation and Simple Fractal Models

To illustrate the properties of fractal antennae compared to those of simple dipole radiators, we take the fractal antenna as composed of small line elements and compute its far field radiation pattern. For an oscillating current $I(t)=I_oe^{-i\omega t}$ that propagates with speed $\beta=$v/c along the antenna, the contribution from each line element to the total radiation field is (from Eq. (2.1))  

$$\frac{cr{\bf E}_n({\bf r},t)}{I_o}=e^{-i\omega t}e^{ikr}\fra... ..._n}e^{i\frac{ks_n}\beta }(e^{ikL_na_n}e^{i\frac{kL_n}\beta }-1)$$ (4)

where $a_n=\widehat{{\bf L}}_n\cdot \widehat{{\bf r}}$, ${\bf p}_n$ is the position of the beginning of the line element from the origin, and $b_n=\widehat{{\bf p}}_n\cdot \widehat{{\bf r}}$. Radiation occurs when there is a change in the direction of the propagating current. Also note that mathematically we can describe a radiator with a nonpropagating current in the non-physical limit $\beta \rightarrow \infty$.

In general, the radiation pattern of an antenna can be effectively excited, only by certain frequencies corresponding to the characteristic length scales of the antenna, e.g. $kL\sim 1$ (see Eq. (2.4)). Therefore, if there is no characteristic size, as in the case of a power law structure, then the antenna will generate an effective radiation pattern for a whole range of frequencies controlled by the smaller and largest spatial scale. Such antenna is called a broad band antenna, and that is why fractal antennae are so important in many applications.

By spatially superposing these line radiators we can study the properties of simple fractal antennae. Of special interest, to our high altitude lightning work, is to compare the radiation pattern of these fractal models with a simple (meaning one line element) dipole antenna.

Gain Due to Tortuosity

The first element in understanding fractal antennae is the concept of tortuosity in which the path length between two points is increased by requiring that the small line elements are no longer colinear. A simple tortuous model is displayed in Fig. 2.5, where the parameter $\varepsilon$ represents the variation from the simple dipole model (line radiator), i.e. the dipole is recovered as $\varepsilon \rightarrow 0$.

  

Figure 2.5: A simple tortuous variation of a line radiator. Note that the antenna will radiate every time there is a change in direction.

Except for the propagation effect, we can observe that this antenna (Fig. 2.5) can be considered as the contribution from a long line element (a dipole) plus the contribution from a Cantor set of radiators as described in the previous section (see Eq. (2.3)). Therefore, the tortuosity naturally increases the radiation field intensity, at least in some direction, as compared with the single dipole element.

The field can be written for the structure of 2.5, with the help of Eq. (2.4), as the superposition of the 2N line elements, and is given by the normalized field  

$${\bf E}(\varepsilon )=\frac{\beta (e^{ikla_x}e^{i\frac{kl}\b... ...epsilon kl}\beta }-1)}{(1-\beta a_y)} \widehat{{\bf y}}+\ldots$$ (5)

where $l=\frac LN$ is the length of the small segments composing the tortuous path, and $a_x=\widehat{{\bf x}}\cdot \widehat{{\bf r}}$ and $a_y=\widehat{{\bf y}}\cdot \widehat{{\bf r}}$. It is clear that in the limit $\varepsilon \rightarrow 0$ we recover the single dipole radiation pattern. The effect of the tortuosity can now be posed as the behavior of the normalized ${\rm P}(\varepsilon )={\bf E}(\varepsilon )\cdot {\bf E}^{*}(\varepsilon )$ for $\varepsilon \neq 0$. In general the analysis can be simplified in the limit for small $\varepsilon$, i.e. ${\rm P}(\varepsilon )\simeq {\rm P}(0)+{\rm P}^{\prime }(0)\varepsilon +\ldots$. Of course P(0) is the dipole contribution, and ${\rm P}^{\prime }(0)\varepsilon$ is the change in the radiated power density due to tortuosity. The dipole has a maximum in the radiated power density ${\rm P}(0)\simeq \frac{4\beta ^2}{(1-\beta a_x)^2}$, while the tortuous contribution goes as ${\rm P}^{\prime }(0)\varepsilon \simeq \frac{4\beta kL\varepsilon }{(1-\beta a_x)^2}{\rm f}(a_x,a_y,kL,\beta ,N)$ . The function f depends on the given parameters, but its maximum is of the order ${\rm f}\sim 1$ with clear regions in (a_x,a_y) where it is positive.

For our purposes, the most important contribution comes from the fact that ${\rm P}^{\prime }(0)\varepsilon$ is essentially independent of N and it scales as $\Delta {\rm P}\sim \beta k\Delta s=\beta kL\varepsilon ,$ which corresponds to the increase in the path length of the antenna due to the tortuosity. Such technique can be applied to other geometries, giving essentially the same scaling $\Delta {\rm P}\sim \beta k\Delta s$ result. This fact will be extremely relevant in our analysis since lightning has naturally a tortuous path.

Fractal Tortuous Walk

More generally, a fractal tortuous path can also be constructed in terms of a random walk between two endpoints [Vecchi, et al., 1994]. We start with a straight line of length L, to which the midpoint is displaced using a Gaussian random generator with zero average and deviation $\sigma$(usually $\sigma =0.5L_i$). The procedure is then repeated to each of the straight segments N times. There is a clear repetition in successive halving of the structure as we go to smaller scales, making this antenna broad band. Figure 2.6a shows a typical tortuous fractal where the division has been taken to the N=8 level and in which the pathlength s has increased 5 times, i.e. s=5L. We can estimate the fractal dimension by realizing that the total length should go as $L_{tot}\sim L(\frac s{\left\langle \ell \right\rangle })^{D-1}$, where $\left\langle \ell \right\rangle$ is the average segment size. This formulation is completely equivalent to Eq. (2.1).

We let an oscillating current, e.g. $I_oe^{i\omega t},$ propagate along the fractal, but in real applications we can imagine the oscillating current lasting for only a finite time $1/\alpha$. In order to have a finite current pulse propagating through the fractal random walk, we let $I(t)=I_o(e^{-\alpha t}-e^{-\gamma t})(1+\cos (\omega t))\theta (t)$ with $\omega =2\pi \alpha n_f$ and $\theta (t)$ as the step function. Here n_f represent the number of oscillations during the decay time scale 1/$\alpha$. We chose the decay parameters as $\alpha =10^3$ s^-1 and $\gamma =2\times 10^5$ s^-1, hence $\gamma /\alpha =200$, which correspond to realistic parameters for lightning [Uman, 1987]. The radiated power density is then computed using Eq. (A.4) and is shown in Fig. 2.6b for n_f=5 and $\beta =0.1$ at the position a_x=0, a_y=0, r=60 km. The dipole equivalent is given by the dashed lines in all 3 panels. The peak in the radiated power density is about 10 times larger than for the dipole case, which agrees well with the results $\frac{{\rm P} ^{\prime }(0)\varepsilon }{{\rm P}(0)}\sim$ $\frac{2\pi \alpha \Delta s}{ c\beta }$n$_f\sim 10$ even though the effect from the tortuosity is not small. The larger path length of the tortuous discharge produces an increase in the radiation as compared with a dipole radiator. Of course there is a limit due to energy conservation, but in practical applications we are well under it. The increase in the high frequency components of the radiated field power spectrum (Fig. 2.6c), as compared with the dipole antenna, will be responsible for the spatially structured radiation pattern.

  

Figure 2.6: The fractal random walk (a) and its instantaneous radiated power density (b) as well as its power spectrum (c). The dashed lines represent the behavior of the single dipole.

The far field array factor $R=\alpha \int dt{\bf E}^2$ (defined in Appendix A) and the peak power density depend on the path length, or equivalently on the number N of divisions of the fractal. Figure 2.7b shows the array factor as a function of the path length for the fractal shown in Fig. 2.7a. Here n_f=5 so that the peak of the array factor is at a_x=0 and a_y=0. There is a clear increase in the array factor from the tortuous fractal as compared with the single dipole.

  

Figure 2.7: (a) The tortuous discharge. (b) The array factor dependence, normalized to the dipole, on the pathlength.

Therefore, the effect of tortuosity can increase the radiated power density at certain locations as compared to a single dipole antenna.

Another important concept related to fractal antennae is the spatial structure of the radiation field. We can see from the array factor, Eq. (A.5), that for large n_f $f[\tau ,\zeta ]\simeq e^{-\tau }(2+\cos (\zeta \tau ))$. The spatial dependence of the array factor will be determined by the factor $\frac{\alpha \zeta \Delta r}c$ over the fractal. Consequently, the radiation pattern will have spatial structure when $\frac{ \alpha \zeta \Delta r}c\gt 2\pi ,$ which translate into n_f>50. Figure 2.8b shows the array factor at the height h=60 km for the discharge structure shown in Fig. 2.8a with n_f=200. Therefore, such a tortuous fractal can also display a spatial structure in the radiation pattern. But it is more natural for the spatial structure to be generated through a branching process as we will see in the next section.

  

Figure 2.8: The fractal structure (a) and its array factor(b) showing clear spatial structure in the radiation pattern.

There is an energy constraint that limits the degree of tortuosity of a fractal lightning discharge since we cannot radiate more energy than what is initially stored as separated charge. Also, if the line elements of the antenna given by Fig. 2.5 get too close together, then their contribution to the radiated field will tend to cancel each other. Therefore, there is an optimal number of elements forming an antenna, and this optimal number translates into an optimal dimension of the fractal, more on this later.

Branching and Spatial Structure

Another element in understanding fractal antennae is the concept of branching. Take the simple branching element shown in Fig. 2.9 where the current is divided between the two branching elements. We can compute the radiation field, for a propagating current I$_oe^{i\omega t}$,as  

$${\bf E}(\varepsilon )=\widehat{{\bf x}}\frac{\beta (e^{ik\el... ...\varepsilon a_y}e^{i\frac{\varepsilon k\ell }\beta }-1)\}\ldots$$ (6)

where $\ell =L/2$ and $\varepsilon$ is the variation from the single dipole, i.e. we recover the dipole as $\varepsilon \rightarrow 0$.

  

Figure 2.9: A simple branching situation in which we distribute the current among the branching elements.

Again, the analysis can be simplified in the limit for small $\varepsilon$,i.e. ${\rm P}(\varepsilon )\simeq {\rm P}(0)+{\rm P}^{\prime }(0)\varepsilon +\ldots$ . Of course P(0) is the dipole contribution, and ${\rm P}^{\prime }(0)\varepsilon$ is the change in the radiated power density due to the line branching. The dipole has a maximum in the radiated power density ${\rm P}(0)\simeq \frac{4\beta ^2}{(1-\beta a_x)^2}$, while the branching contribution goes as ${\rm P}^{\prime }(0)\varepsilon \simeq \frac{\beta kL\varepsilon }{(1-\beta a_x)^2}{\rm f}(a_x,a_y,kL,\beta ,N)$. The function f depends on the given parameters, but its maximum is of the order ${\rm f}\sim 1$ with clear regions in (a_x,a_y) where it is positive. Therefore, the branching process can give rise to an increase in the radiated power density at certain position. Of course this increase is due to the increase in the path length. This effect will saturate as $\varepsilon$ is increased passed one, since then the strongest contribution will come from the dipole radiator given by 2$\varepsilon L$.

A interesting and manageable broadband antenna can be described in terms of the Weierstrass functions [Werner and Werner, 1995]. We take successive branching elements, as shown in Fig. 2.10a, where we distribute the current at each branching point so that the branching element keeps a fraction $\alpha$ of the current. The n^th branching element is displaced by a factor $\varepsilon ^n$ with respect to the origin. If we concentrate only on the contribution from the last branching set, as shown in Fig. 2.10a, we can write the field as

$${\rm E}_x\sim \sum_{n=1}^N\varepsilon ^{n(d_o-2)}\cos (kl\varepsilon ^na_y+\phi _n(\beta ))$$

$$\phi _n(\beta )=\frac{kl}\beta \varepsilon ^{n-1}(2+\varepsilon )$$

where we have redefined $\alpha =\varepsilon ^{d_o-2}$ and a$_y=\cos \theta$. In the limit $\beta \rightarrow \infty$ and N$\rightarrow \infty$ we obtain the Weierstrass function that is continuous but not differentiable, i.e. is a fractal, and furthermore, its dimension in the sense given by Eq. ( 2.1) is d_o. For the purpose of illustration we truncate the above sum to N=8. In Fig. 2.10b, we show the dependence of the field as a function of $a_y\;\epsilon \;[-1,1]$ with a_x=0 for $\beta \rightarrow \infty$. The parameters values are shown in the figure caption. Figure 2.10c shows the gain factor given by

$$G=\frac{\max \left\vert E\right\vert ^2}{\frac 12\int da_y\left\vert E\right\vert ^2}$$

as a function of the dimension $d_o\;\epsilon \;[0,1]$. We chose this range since the fractal already has a dimension 1 in the perpendicular directions, i.e. D=1+d_o.

  

Figure 2.10: The branching process to produce a Weierstrass radiation pattern. (a) The brunching process with the branching length increasing as $\varepsilon ^nL$ and the current decreasing as $\alpha ^nI_o$. (b) the radiation pattern with $\beta \rightarrow \infty$ given perfect coherence. (c) The gain vs the dimension. It also contains the parameters used in all 3 figures. (d) Patial coherence for $\beta =0.1$.

Note the increase in the gain as a function of dimension. In general, there is an optimal value of D that generates the highest power density and that does not necessarily has to be for D=2. In Fig. 2.10b all the elements from the antenna add up coherently at a_y=0, hence providing perfect coherence. For a finite $\beta <1$ the propagation brings a different phase shift at each element. Figure 2.10d shows the effect for $\beta =0.1$ as a function of $\theta$. Note that at no point there is perfect coherence, but there is clear partial coherence. The peak value of E² is actually sensitive to $\beta$.

Even though fractal antennae naturally lead to the concept of an increase in the peak radiated power, it also has a second important consequence due to branching. As we have seen in the case of the Wiertrauss function, fractal antennae naturally result in the generation of a spatial structure in the radiated power density. This interplay between the spatial structure and the increase in the peak radiated power are the essential ingredients of fractal antennae and why they are so important. A clear example can be illustrated in Fig. 2.10d where there are multiple relevant peaks of the radiated power in space.

Fractal Antenna/Field Gain simple fractal antenna examples gain due to tortuosity + branching

Modeling Lightning as a Fractal Antenna

The hypothesis of this work is that the structure of the red sprites can be attributed to the fact that the power density generated by lightning does not have the smooth characteristics expected from the dipole model of Eq. 2.7, but the structured form expected from a fractal antenna [ Kim and Jaggard, 1986; Werner et al., 1995]. Previous studies of lightning assumed that the RF fields causing the atmospheric heating and emissions, were produced by an horizontal dipole cloud discharge moment M that generates an electric field at the height z, given by  

$$E=\frac M{4\pi \epsilon _oz^3}+\frac 1{4\pi \epsilon _ocz^2}\frac{dM}{dt}+\frac 1{4\pi \epsilon _oc^2z}\frac{d^2M}{dt^2}$$ (7)

where c is the velocity of light and $\epsilon _o$ is the permittivity of free space. It is important to realize that a lightning discharge must be horizontal, as in intracloud lightning, to project the energy upwards into the lower ionosphere. A vertical discharge, as in cloud-to-ground lightning, will radiate its energy horizontally as a vertical antenna.

It is obvious that such a horizontal dipole results in electric fields that vary smoothly with distance. However, it is well known that lightning discharges follow a tortuous path [LeVine and Meneghini, 1978]. It was shown [ Williams, 1988] that intracloud discharges resemble the well known Lichtenberg patterns observed in dielectric breakdown. In fact a time-integrated photograph of a surface leader discharge is illustrated by Figure 2.11 . These patterns have been recently identified as fractal structures of the Diffusion Limited Aggregate (DLA) type with a fractal dimension $D\approx 1.6$ [Sander, 1986; Niemeyer et al., 1984].

  

Figure 2.11: Time-integrated photograph of a surface leader discharge (Lichtenberg pattern) [Niemeyer et al., 1984]

As noted previously, the tortuous path increase the effective dipole moment, since now the pathlength along the discharge is longer that the Euclidean distance. To understand this analogy, we construct a tortuous walk between two points separated by a distance R as shown in Fig 2.12. Take the tortuous path as N small steps of averaged step length $L_o\ll R$, then the total path length S along the tortuous discharge is

$$S\sim NL_o\sim (\frac R{L_o})^{D-1}R$$

where the number of small steps is $N\sim (\frac R{L_o})^D$ with D as the box counting dimension [Ott, 1993]. As we have seen before, the change in the path length increases the radiated power density as E$^2=E_o^2+\beta k(S-R)$ where E$_o^2\sim \beta ^2$ Therefore, for $R\sim 10$ km (typical for an intracloud discharge), $L_o\sim 50$ m, $\beta =0.1$, and $D\sim 1.6$,with obtain $\frac{E^2}{E_o^2}\sim 1+\frac{kR}\beta ((\frac R{L_o})^{D-1}-1)\sim 1+5f(kHz)$ where f is the frequency of the current.

  

Figure 2.12: Tortuous path between two point.

This is only an analogy, but it gives us good intuition that a fractal lightning discharge will produce an increase in the radiated field intensity, at least locally, as compared with a dipole model and a spatially structured radiation pattern. A fractal dielectric discharge of size R can be modeled as a set of non-uniformly distributed small current line elements [ Niemeyer et al., 1984] that represent the steps of the discharge breakdown as it propagates during an intracloud lightning discharge. The size of the elementary current steps is about $L_o\sim 50$ m [Uman, 1987]. As a current pulse propagates along this horizontal fractal discharge pattern it radiates energy upwards (see Appendix A on how the fields are calculated) as well as downward.

To determine the extent over which the non-uniformity of the lightning discharge current affects the power density structure projected in the lower ionosphere, we will now construct a simple fractal model of the lightning discharge that will yield a spatio-temporal radiation pattern at the relevant heights.

Fractal Lightning: Stochastic Model  

We want to generate a fractal model that can be parametrized by its fractal dimension. For this purpose, we follow Niemeyer et al. [1984] who proposed a two dimensional stochastic dielectric discharge model that naturally leads to fractal structures. In this model the fractal dimension D can be easily parametrized by a parameter $\eta$. Femia et al. [1993] found experimentally that the propagating stochastic Lichtenberg pattern is approximately an equipotential. Then, the idea is to create a discrete discharge pattern that grows stepwise by adding an adjacent grid point to the discharge pattern generating a new bond. The new grid point, being part of the discharge structure, will have the same potential as the discharge pattern. Such local change will affect the global potential configuration, see Fig. 2.13.

  

Figure 2.13: Diagram of the discrete discharge model.

The potential for the points not on the discharge structure is calculated by iterating the discrete two dimensional Laplace's equation

$$\begin{array} {c} \nabla ^2\phi =0 \ \phi _{i,j}=\frac 14... ... _{i+1,j}+\phi _{i-1,j}+\phi _{i,j+1}+\phi _{i,j-1})\end{array}$$

until it converges. This method reproduces the global influence of a given discharge pattern as it expands. The discharge pattern evolves by adding an adjacent grid point. The main assumption here is that an adjacent grid point denoted by (l,m) has a probability of becoming part of the discharge pattern proportional to the $\eta$ power of the local electric field, which translates to

$$p(i,j)=\frac{\phi _{i,j}^\eta }{\sum_{l,m}\phi _{l,m}^\eta }$$

in terms of the local potential. Here we have assumed that the potential at the discharge is zero. The structure generated for $\eta =1$, corresponding to a Lichtenberg pattern, is shown in Fig. 2.14.

      

Figure 2.15: The plot of $lnN(\epsilon )$ vs $\frac 1\epsilon$ for $\eta =1$.Figure 2.14: Fractal discharge generated with $\eta =1$.

The color coding corresponds to the potential. Figure 2.15 shows a plot of $N(\varepsilon )$ vs. $Log(\varepsilon )$ for the fractal discharge of Fig. 2.14, i.e. $\eta =1.0$. Again the scaling behavior only occurs over a few decades, but it is very clear. The dimension of this structure is $D\simeq 1.6$.

  

Figure 2.16: The dimension of the stochastic model as a function of $\eta$ with the estimated error bars.

Note that this model, and also the dimension of the discharge, is parametrized by $\eta$. Intuitively we expect that when $\eta =0$ the discharge will have the same probability of propagating in any direction, therefore, the discharge will be a compact structure with a dimension D=2. If $\eta \rightarrow \infty$ then the discharge will go in only one direction, hence D=1. Between these two limits, the dimension will be the function $D(\eta )$ shown in Fig 2.16. As an example the corresponding structure generated for $\eta =3$ (Fig 2.17) has a dimension of D=1.2.

  

Figure 2.17: Fractal discharge generated for $\eta =3$ .

To compute the radiated fields, we must describe the current along each of the segments of the fractal discharge. We start with a charge Q_o at the center of the discharge. The current is then discharged along each of the dendritic arms. At each branching point we chose to ensure conservation of current, but intuitively we know that a larger fraction of the current will propagate along the longest arm. Suppose that a current I_o arrives at a branching point, and if L_i is the longest distance along the i^th branching arm, we intuitively expect that the current on the i^th arm should be proportional to $L_i^\eta$. Therefore, we satisfy charge (or current) conservation if the current along the i^th branching arm is

$$I_i=\frac{L_i^\eta }{\sum_{\,j}L_j^\eta }$$

Computing the Fields from the Fractal Structure

A current pulse propagates along the horizontal (in the x-y plane) 2 dimensional fractal discharge structure, e.g. $I(x,t)=I(t-\frac s{{\rm v} })$generating radiation fields. The radiation field is the superposition, with the respective phases, of the small line current elements. The intracloud current pulse is taken as a series of train pulses that propagate along the arms of the antenna

$$I(t)=I_o(e^{-\alpha t}-e^{-\gamma t})(1+\cos (\omega t))\theta (t)$$

with $\omega =2\pi \alpha n_f$ and $\theta (t)$ as the step function. Here n_f represent the number of oscillations during the decay time scale 1/$\alpha$. We chose the decay parameters as $\alpha =10^3$ s^-1 and $\gamma =2\times 10^5$ s^-1, hence $\gamma /\alpha =200$, which correspond to realistic parameters for lightning [Uman, 1987]. The total charge discharged is then $Q=I_o/\alpha$, which for $I_o=100\;kA$gives $Q\sim 100\;C$. As we have seen before, we require $n_f\sim 100$ to create the spatial structures so that the exponential decay $e^{-\alpha t}$can be considered as the envelope of the oscillating part.

On a given position the time dependence of the field intensity E² has a fractal structure, as it is shown in Fig. 2.18a for the stochastic discharge model with $\eta =3$. The frequency spectrum of the electric field is shown in Fig. 2.18b. It is very important to realize that the relevant frequencies are below a few hundred kHz. By restricting the field frequencies to below a few hundred kHz, the analysis is greatly simplified, since then the conductivity and dielectric tensors can be considered as independent of time in the lower ionosphere (see Appendix B).

The large conductivity of the ground at these frequencies can be included by assuming to first order an image discharge of opposite current below a perfectly conductive plane. The primary discharge is taken to be at z_o=5 km above the ground. This parameter is not very relevant, since we are interested in the field at heights of about $h\sim 80$ km, therefore, moving the discharge from 5 to 10 km will only change the field strength by a marginal 10%.

  

Figure 2.18: (a) The field power density due to the stochastic discharge model at a given position as a function time (b) and the frequency spectrum of the field.

How does the Fractal Dimension Affect the Field Pattern?

For a 2-dimensional fractal structure, we expect that the strength of the radiated power density depends on the fractal structure, i.e. its fractal dimension. If the strength of the k Fourier component is A_k then the field in the far field [ Jackson 1975] at r along the axis of the fractal will be given by

$${\bf E\sim }\int dkA_k\int_\mu d\mu {\bf R}(\mu )\sin (k\rho )$$

where d$\mu$ (the fractal measure) is the contribution of the fractal from a given polar position ($\rho$,$\phi$). ${\bf R}(\mu )$ is the phase contribution from the elements of the fractal at position ($\rho$,$\phi$)and in the far field should be proportional to the direction of the local current. Note that a radially propagating uniform 2 dimensional current structure will generate no field at the axis since contributions to ${\bf R}(\mu )$ from different parts of the fractal will cancel each other.

The cross section of the fractal at a given radius $\rho$ will resemble a Cantor set in $\phi \;\epsilon \;[0,2\pi ]$, and the phase contribution will be given by ${\bf S}(\rho )=\int_\mu d\phi {\bf R}(\rho ,\phi )$ which will be finite for an asymmetrical fractal. The integration can be carried as a Lebesgue integral or as a Riemann-Stieltjes integral over this pseudo-Cantor set [Royden, 1963]. Note that if the fractal is uniformly distributed along $\phi$, corresponding to D=2, then ${\bf S} (\rho )=0$. Similarly, for a delta function at $\phi =\phi _o$ corresponding to D=1, ${\bf S}(\rho )$ gives a positive contribution. ${\bf S}(\rho )$ is a very complicated function that depends on the details of the current distribution along the fractal. In an average sense we can suppose that $S(\rho )\sim f(D)$ where f(D=1)=1 and f(D=2)=0 but f can be greater than one for other values as has been investigated in previous sections when branching and propagation occurs. Therefore,

$$E{\bf \sim }f(D)\int dkA_k\int_\mu dm(\rho )\;I(\rho )\sin (k\rho )$$

where dm($\rho$) represents the amount of the fractal between $\rho$ and $\rho +d\rho$ and I($\rho$) is the averaged current over $\phi$ at radius $\rho$. A fractal will have a mass up to a radius $\rho$ given by $m(\rho )=(\frac \rho {L_o})^D$ by noting that a 2 dimensional antenna will have more elements than a one dimensional fractal. In general, due to the branching process, some of the current does not reach the radius R. But for simplicity, if all of the current reaches the end of the fractal at radius R, then $dm(\rho )\;I(\rho )=d\rho$. In such case, the above integral gives

$$E{\bf \sim }f(D)\int dk\;k^{-1}A_k(1-\cos (kR))$$

Note that $\varepsilon =(\frac \ell R)$ in some sense selects the Fourier component $k=\frac{2\pi \beta }{\varepsilon R}$ which has a strength $A_k\sim N(\varepsilon )\sim \varepsilon ^{-D}$. The integral over k gives $g(D)\simeq (\frac{Lo}R)^{D-1},$ therefore, the field is given by  

$$E{\bf \sim }f(D)g(D)$$ (8)

which shows that the field has a maximum value at a specific value of $D\;\epsilon \;[1,2]$ since f(D) decreases and g(D) increases with D respectively.

On the other hand, the Rayleigh length, the distance beyond which the field start decaying to their far field values, behaves as $R_L(\varepsilon )\sim \frac{\varepsilon R}{2\pi \beta }$ for a given $\varepsilon$. Red sprites occur at a height $z\sim 80$ km, therefore, for $z\gt R_L(\varepsilon ),$elements with sizes smaller than $\varepsilon$ do not contribute to the field, i.e. as we increase z we wash out the information of increasingly larger spatial scales of the fractal. It is the power law dependence, as specified by the fractal dimension, that determine the field pattern.

Even though, the radiation pattern will depend on the details of the fractal structure, we expect that the most relevant parameter in determining the radiation pattern will be the fractal dimension, as found by Myers et al. [1990] for simple fractals. There is an interplay between the dimension and the spatial structure of the radiation pattern. For a dimension close to $D\sim 1$or $D\sim 2$, there will be no significant spatial structure. While an intermediate dimension can produce a significant spatial structure.

Fields from the Stochastic Model

First we start by computing the array factor based on the far field approximation (see Appendix A). We take n$_f\sim 200$ and $\beta =0.1$ and compute the array factor at a height z=60 km. Figure 2.19 shows the array factor for the discharge structure shown in Fig. 2.17 with $\eta =3$.

  

Figure 2.19: The array factor for $\eta =3$.

The length of the elementary current elements is about 100 m. The array factor shows clear structure. A cross-section of the normalized array factor are shown in Fig. 2.20 and Fig. 2.21 for $\eta =1$ and $\eta =2$ respectively.

    

Figure 2.21: Cross-section of the array factor for $\eta =2.$Figure 2.20: Cross-section of the array factor for $\eta =1.$

Similarly, the array factor at x=10 km, y=10 km, z=60 km is shown as a function of the fractal dimension of the discharges for $\beta =0.05$ and $\beta =0.025$ in Fig. 2.22a for n_f=0 and Fig. 2.22b for n_f=200.

  

Figure 2.22: The array factor (a) For n_f=1 and (b) for n_f=200. The graph has been interpolated for the purpose of illustration.

The fractal dimension dependence of the array factor is very intriguing, but is of clear significance for our lightning studies. What about the time dependence of the radiations fields? Figure 2.23 shows the time dependence of the radiation fields for $\eta =1,2,3,5,10$ with n_f=200,50,1 where each figure is carefully labeled.

  

Figure 2.23: The time dependence of the radiation fields for the fractal models. See explanation in text.

Again the relevance of the $\eta =3$ case is very striking. Each column of graphs represent the time dependence for n_f=200,50,1 respectively, where the rows represent the case for $\eta =1,2,3,5,10$. The amplitude of the field has been multiplied by the factor displayed next to the graph.

We take the case for $\eta =3$ and we study the dependence of the array factor as a function of the current frequency as parametrized by n_f. Figure 2.24 shows the frequency dependence of the array factor at this location x=10 km, y=10 km, z=60 km. Initially the array factor increases linearly with n_f as expected but then it starts to oscillate as the spatial variation of the field pattern becomes relevant.

  

Figure 2.24: The array factor as a function of n_f for $\eta =3.$

In conclusion, the fractal nature of the discharges, being a simple random walk or a stochastic discharge model, leads naturally to an increase in the peak power density as compared with the dipole model. This increase is related to the increase in the antenna path length, or tortuosity, and on the branching process. It will be shown later that this gain in peak power density leads to a significant reductions in the discharge properties (e.g. charge, peak current) required to produce the observed sprite emissions.

Scaling of the fields

In conclusion certain fractals can radiate more effectively than others, but in general this problem is very complicated. The power density, and thus the emission pattern and intensity, scales as $S(W/m^2)\sim \beta ^2I_o^2\,f(D,L,\omega )\,g(\theta ,D,\beta ,\omega )$, where $\,f(D,L,\omega )$ is the efficiency function of the different fractals. $g(\theta ,D,\beta ,\omega )$ represent the spatial structure of the radiation pattern with $\theta$ as the angular position.