Lightning 1: Mechanics of Lightning

In the first instalment of the Lightning Series, I will take you through the mechanics of lightning. You will appreciate the spectacle of lightning more once you learn the physics behind it.

Introduction

Lightning is a massive spark of electricity observed in the atmosphere between two electrically charged areas, such as within a cloud, between clouds or between a cloud and the ground. Lightning is such a spectacle and also causes thunder which sends an explosive sonic shock wave to your ears. Unsurprisingly, people in the Middle Ages regarded lightning as supernatural interventions.

Even today, little may be known to many about the physics behind lightning. Although we learn at school that air is an insulator, electricity observed in lightning clearly travels in air. Lightning appears unpredictable and its geometry is beautiful and awe-striking.

In this article, I walk the reader through the mechanics of lightning and simulate lightning as a nonlinear dynamical system. To grasp the gist of the discussion, the reader can skip anything technical herein should they find somewhat overwhelming.

Basic data for lightning

Before we delve into the nitty gritty of lightning, shown below are basic data for lightning.

· On average lightning travels 4km in length.

· The air around lightning can heat up to 30,000 degrees Celsius, which is five times as hot as the surface of the Sun.

· The amount of energy per lightning bolt is around 5,000,000,000 joules which is enough to power an average household for a month.

· There are about 1,800 thunderstorms in the world. The amount of energy per average thunderstorm amounts to 10^15 joules, which is equivalent to the amount of energy produced by an atomic bomb.

· There are 44 lightning strikes per second.

· Thunder can be heard up to 40km away from the lightning discharge. The audible frequency changes with distance, since higher frequencies tend to be more quickly absorbed by the air.

· The total estimated cost of damage caused by lightning per year is in the order of billion dollars in the world.

As we know, lightning can be disastrous to our life. However, lightning is also a potential source of alternative energy, which may provide a colossal amount of electricity without any artificial generator. Unfortunately, we currently waste all of it. Why? We just don’t have any technology available to catch and store sudden bursts of tremendous energy produced by lightning. I will come back to this point in more detail in a sequel.

Mechanics of lightning

Lightning is most commonly observed in a thunderstorm. A thunderstorm occurs in cumulonimbus clouds (see [CC]). A cumulonimbus cloud floats at around 500m high up to 10,000m high in altitude and has an overshooting top. The formation of a cumulonimbus cloud requires a moist unstable air mass with a rapid lifting force due to strong atmospheric convection [AC]. That is, warmer less dense air rises to the top while cooler denser air sinks to the bottom. Consequently, a cumulonimbus cloud grows vertically.

As warm moist air rapidly rises, water droplets and ice crystals within the cloud violently collide with each other. These collisions result in enormous friction, which consequently creates static electricity. Lighter positively charged ice crystals rise to the top of the cloud, while heavier, negatively charged particles accumulate at the bottom. This creates a charge imbalance. See Figure 1.

Figure 1. Charge distribution in a cumulonimbus cloud

When this charge difference becomes too great, the air’s insulating properties are overcome, and a sudden, visible flash of lightning occurs as the negative charges rush to balance with positive charges on the ground or within the cloud. This phenomenon is called dielectric breakdown [DI].

It is very counter-intuitive and weird that electricity can travel in an insulator, such as air. To understand dielectric breakdown a little more in detail, recall that air mainly consists of oxygen and nitrogen molecules. Both oxygen and nitrogen molecules are stable, i.e., having no freely moving electrons, hence, air molecules are an insulator.

There are about 2.6910^25 air molecules per cubic metre in a typical cumulonimbus cloud. Each air molecule is about 0.5 nanometre wide in size. On average an air molecule travels 66 nano metres without colliding with others. This distance is known as the mean free path* in physics [ME]. The average speed of air molecule is about 470m/sec. This means that each air molecule collides with others approximately seven billion times per second.

On a cloud scale, the total number of such collisions is uncountably large so that these collisions knock electrons from the stable air molecules, creating positively and negatively charged particles. The ejected electrons hit air molecules and subsequently push more electrons out of the molecules, creating a series of wide-scale chain reactions. This phenomenon is known as electron avalanche. See Figure 2.

Figure 2. Electron avalanche

An electron avalanche is a rapid multiplication of free electrons in a medium, which is triggered by a strong electric field that accelerates the electrons. As a result, the electrons collide with other electrons and ionize neutral atoms. This process leads to an exponential increase in the number of electrons and ions, which can result in a sudden, significant current or plasma formation. This leads to the polarisation where the lighter, positively charged ice crystals are carried to the top of the cloud, while the heavier, negatively charged particles fall to the bottom. An overpowering polarised electric field due to an electron avalanche brings about a massive electrical discharge as lightning.

As a matter of fact, every one of us utilises an electron avalanche every day in many different industrial applications, such as semiconductors, combustion engines or gas lighters where sparks are used to ignite the fuel. As mentioned earlier, there are three main forms of lightning.

Cloud-to-ground lightning

The negative charge at the base of the cloud repels negative charges in the ground, leaving the ground’s surface positively charged. A downward-moving stepped leader, a stream of negatively charged air, begins to form from the cloud. As the stepped leader gets closer to the ground, a positively charged upward leader rises from the ground to meet it. When these two leaders connect, a conductive path is formed. Then, a massive electrical discharge or lightning flows rapidly down to the ground.

Intra-cloud lightning

The most common type of lightning is intra-cloud lightning, where the electrical discharge occurs within a single cloud. This is because the electrical charge imbalances within a single thunderstorm cloud are much closer together and are more frequently established than other forms of lightning where the distance is farther apart, for instance, the distance between the cloud base and the ground.

Cloud-to-cloud lightning

Cloud-to-cloud lightning occurs when the electrical discharge travels between different clouds. This is also a common form of lightning due to relatively closer distances of charge imbalances between two separate nearby thunderstorm clouds. Cloud-to-cloud lightning is often portrayed as sheet lightning or dancing in the sky as if it so appeared to our eyes.

Lightning as a fractal

“Clouds are not spheres, mountains are not cones, and lightening does not travel in a straight line”, said Mandelbrot [MA]. Many emergent phenomena observed in nature do not follow Euclidian geometry but more complex patterns or fractal geometry. Lightning also exhibits fractal characteristics.

What is a fractal?

Lightning is a branching electric discharge which is a Lichtenberg figure [LI]. Consider a lightning strike from a cloud to the ground. A negatively charged channel of a lightning strike, also known as a stepped leader, appears as a series of short bursts or steps. Each step lasts about a millisecond. As it moves along, it branches out in many directions that appear unpredictable. The shape of lightning is self-similar in the sense that on both macroscopic and microscopic scales it resembles itself, regardless of the degree of magnification. See Figure 3.

Figure 3. Fractal characteristics of lightning

If we try to measure the channel length of a lightning strike L by dividing the entire length into N steps (measurement intervals) where each step length is set to a constant epsilon, we can express the length L as

The channel length L depends on its regularity and the unit length epsilon. For lightning, L increases as the scale of measurement epsilon gets smaller. This is because we can more accurately include the lengths of irregularities in the entire length L, as the unit length epsilon gets smaller. The irregularities in the geometry of lightning are so complex and intricate that no linear yardstick can appear to accurately capture the entire length.

It is known that the number of steps N has a power law dependence on the unit step length epsilon such that

where gamma is a parameter which represents the fractal dimension. Fractal characteristics are encapsulated into this power law relationship. Substituting Equation (2) into Equation (1), we have

where F is a proportionality constant. We can empirically estimate the fractal dimension gamma by linearly regressing data for L on data for epsilon such that

Why does lightning strike have a fractal geometry?

The reason why a lightning strike exhibits fractal characteristics is due to the fact that the stepped leader searches for the easiest path to the ground. At each step, every leader’s tip is not sensing the actual ground but rather positive charges around the tip induced on the ground by the leader’s electric field. A positive charge at each electrode determines how likely the stepped leader reaches the electrode. Therefore, the subsequent step of a lightning strike is stochastically determined based on positively charged electrodes neighbouring around the tips of the stepped leader. The charge distribution, which has the power law dependence, results in the fractal structure of lightning.

Simulating lightning

We consider simulating cloud-to-ground lightning. In theory, there are two ways to describe the dynamics of a cloud-to-ground stepped leader: an exact method and a heuristic approach for phenomenological modelling. The former relies on exactly solving a macroscopic system of lightning, whilst the latter approach focuses on explaining the emergent phenomenon of lightning commonly adopted in statistical physics. In this article, we exploit a heuristic approach known as the dielectric breakdown model [NPW].

This is mainly because it is infeasible to provide the exact solution for lightning with a vast number of particles involved with their initial conditions and thermodynamical conditions. Even if solvable, it would also contain too much irrelevant detail, such as the precise position and momentum of every single particle. For this purpose, a simple heuristic approach best describes the most probable collective behaviours of the complex lightning system.

Dielectric breakdown model

We use a two-dimensional stochastic dielectric breakdown model to simulate the fractal properties for the propagation of stepped leader from a cloud to the ground. Clearly, lightning in the real world occurs in three dimensions. This simplified model is designed to elucidate the fractal characteristics of lightning and can be easily extend to higher dimensions.

First, we create a 2-dimensional N-by-N square lattice. See Figure 4 (a). The lattice represents the atmosphere. Cells on the lattice represent areas (electrodes) in the atmosphere with some electric potential which the stepped leader may potentially connect to. The height and the width of each square is assumed to be tens of metres due to the fact that a lightning strike covers a long range per step in the sky. The top row is assumed to be a strongly negatively charged thundercloud, whilst the bottom row represents the ground with induced positive charges.

Figure 4. N-by-N square lattice

Here are the basic rules for the growth of the discharge pattern.

· Grid initialisation and boundary conditions: The electric potential for cells in the top row (cloud) is set to zero (negative potential), whilst those of the cells in the bottom row (ground) is set to one (positive potential). The stepped leader first branches out from the middle cell in the top row.

· Determination of electric potential: Given the boundary conditions and the negative potential leader branches, the electric potential phi is defined for all cells adjacent to each existing leader branch by iteratively solving the discrete analogue of Laplace’s equation at each step. See Figure 4 (b). A realised discharge pattern in the stepped leader is represented by shaded cells. See Figure 4 (c).

· Stepwise random growth pattern: The “stepped” nature of the leader is random. At each step, one new bond (a new conductive path) is randomly chosen and added to the stepped leader. It might branch off. The core of the dielectric breakdown model is the probabilistic rule for growth. The direction of the step favours paths with a stronger electric field which maximises the potential difference but does not always exclusively follow them. The local discharge probability for each candidate cell is proportional to the local electric field raised to the power eta such that

The probability for the cell (i,j) is given by

The stronger the local electric field, the higher the probability for the cell to be added to the stepped leader. The parameter eta relates to the power law dependence which controls the strength how likely cells with stronger electric potential are chosen over the others in the local electric field.

· Iterations and termination. Repeat recalculating the electric potential for a new iteration to provide a new path until either the stepped leader reaches the ground or the maximum number of iterations is reached.

Electric field represented by Laplace’s equation

Laplace’s equation is given by

Laplace’s equation describes a steady state of a system. Similar to preferential attachment probabilities in [HD], Laplace’s equation relates to the local discharge probabilities in our context which characterises the fractal structure of lightning.

· The discrete form of Laplace’s equation reveals that the electric potential at a particular cell is an average of the electric potentials at its neighbours adjacent to the particular cell,

In our simulation, we numerically solve the finite-difference version of Laplace’s equation (7) by using the successive overrelaxation (SOR) method.

· The reason why the above specification can explain the fractal phenomenon observed in lightning is that it incorporates the growth propensity associated with strong electric fields. For instance, if we purposefully assign a priori larger probabilities for the growth of tips of the stepped leader to side branching (e.g., eta = 0), this simplification completely neglects the growth nature of the electric field. As a result, its global dimensionality remains as the Euclidean dimension (gamma = 2), which does not result in the fractal nature of lightning.

Results

In our simulation, the grid size is set to 100 times 100 (N=100) and the maximum number of iterations is set to 3,000, as a stopping criterion in case the stepped leader does not reach the ground within 3,000 steps.

Figure 5 shows simulation results under various levels of eta. The higher the value of eta, the less concentrated the geometry around the initial discharge. Compare one extreme case with eta = 0 with another extreme with eta = 10. For eta = 0, even after 3,000 steps, the discharge pattern does not reach the ground level and heavily concentrates around the origin of the stepped leader, indiscriminately branching out into all directions. On the other hand, for eta = 10, the stepped leader attained the ground level only in 272 steps. The stepped leader hardly branches out. The discharge pattern shows a strong tendency to follow strongest electric charges. The shape looks too simplistic to resemble any real-world lightning strike.

The case eta = 0 means that growth probabilities are homogeneous and totally independent from the local electric field, analogous to trajectories of Brownian particles. This specification is similar to Eden model for cancer growth [EGM]. As mentioned before, eta = 0 corresponds to the fractal dimension eta = 2.

In summary, the parameter eta controls the nonlinearity of local discharge probabilities as follows:

· For eta < 1, the model introduces more randomness. It results in more branched bushy tree-like structures. This effect is similar to that of diffusion-limited aggregation [DLA]. The propagation time with respect to the number of steps taken to reach the ground is much larger for smaller values of eta.

or the shape tends towards a 2D plane.

· For eta = 1, a discharge probability for each candidate cell is proportional to the local electric field. Therefore, the model favours growth in the direction of the strongest electric field. This is more realistic than cases with much smaller values of eta. The resultant geometry looks more self-similar like a fractal.

· For eta > 1, the model increases the likelihood of breakdown at cells with higher electric fields. This leads to more directional and less branched growth. The propagation time is much smaller for larger values of eta. The resultant geometry looks more sparse, more skeletal and more simplistic.

or the shape tends towards a line.

Clearly, neither extremely high nor extremely low eta is appropriate for the expected fractal structure to emerge. As discussed in [NPW] and [TS], the empirically estimated fractal dimension of lightning is

To restore this fractal dimension, the value of eta should be around one. Indeed, the discharge patterns around eta = 1 below look realistic with a fair amount of branching and self-similar geometric structure.

Figure 5. Simulation results under various values of parameter eta, ranging from eta = 0 to eta = 10

Concluding remarks

· Fractal structures are ubiquitously found around us. The fractal structure observed in lightning is no coincidence to other phenomena, such as viscous fingering or coastal shore lines. See [HD].

· It is possible to refine the 2D simulation by including more realistic situations, such as an attachment process. An attachment process is related to an upward-moving discharge from the ground to meet the downward-moving stepped leader. As the downward stepped leader nears the ground or a tall object, the intense local electric field causes an upward-moving discharge or a streamer to form. A strike occurs when the downward leader connects with an upward streamer. With some tweaks to the current approach, this extension is straightforward.

· We can also make the fractal structure more intricate and realistic. This can be achieved by adopting an adaptive mesh, for instance, applying a finer mesh to cells near tips of the stepped leader or a coarser mesh to cells in far fields. For faster convergence, a multigrid extension is also naturally compatible with the current approach.

· A 3D extension of the simulation is straightforward in principle. The main challenge will be to develop an efficient fast convergent numerical solution. The simple numerical method adopted in this article will likely severely suffer from the curse of dimensionality.

· In the sequel, I will discuss how we can practically predict potential locations where lightning discharges will occur.

References

· [AC] Atmospheric convection, Wikipedia, https://en.wikipedia.org/wiki/Atmospheric_convection

· [CC] Cumulonimbus cloud, Wikipedia, https://en.wikipedia.org/wiki/Cumulonimbus_cloud

· [DI] Electrical breakdown, Wikipedia, https://en.wikipedia.org/wiki/Electrical_breakdown

· [DLA] Diffusion-limited aggregation, Wikipedia, https://en.wikipedia.org/wiki/Diffusion-limited_aggregation

· [EGM] Eden growth model, Wikipedia, https://en.wikipedia.org/wiki/Eden_growth_model

· [HD] Harmonic Dynamics, Mechanics of meme propagation, (2025), Medium, https://medium.com/@harmonic.dynamics.00/mechanics-of-meme-propagation-1384d8bf1cb9

· [LF] Lichtenberg figure, Wikipedia, https://en.wikipedia.org/wiki/Lichtenberg_figure.

· [MA] Mandelbrot, B. The Fractal Geometry of Nature, (1982), W. H. Freeman & Co Ltd

· [ME] Mean free path, Wikipedia, https://en.wikipedia.org/wiki/Mean_free_path

· [NPW] Niemeyer L., Pietronero L., & Wiesmann H.J. (1984). Fractal dimension of dielectric breakdown. Physical Review Letters 52 (12): 1033–1036.

· [SOR] Successive over-relaxation, Wikipedia, https://en.wikipedia.org/wiki/Successive_over-relaxation

· [TS] Tsonis A.A., A fractal study of dielectric breakdown in the atmosphere. (1991) In Scaling, Fractals and Non Linear Variability in Geophysics (Eds. D. Schertzer and S. Lovejoy). pp167–174 Kluwer.