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\settitle{Electrostatic Charging \\*of Clouds -- Thunderstorms}
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\setabstract{The weather phenomenon of lightning is an everyday example of a naturally appearing electrostatic effect. While nowadays it is commonly known that the discharge is driven by enormous electric potentials in clouds, the actual mechanisms effectuating the charge separation are still subject to active research. This essay shall give an overview of the  physical effects and mechanisms leading to lightning in thunderstorms.}
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\section{Introduction}
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Probably everyone has witnessed a thunderstorm and lightning in her life. About 50--100 discharges per second are registered globally. However, despite the omnipresence of this weather phenomenon, the actual mechanisms behind it are only slowly being recognized. The reason for this impasse is twofold. Firstly, the number of effects leading to charge separation, transport, accumulation, and recombination in the atmosphere is vast~\cite{NRC:1986,Harrison:2003}. Secondly, actually available data is scarce and mostly limited to a small set of local environment variables. Furthermore, up to the late 1960s there have been no accurate measurements of the charge and wind distributions within electrostatically charged clouds and the observation from the ground could only give very limited and unreliable data. In order to obtain more systematic, and time resolved data, measurements had to be conducted within the clouds of thunderstorms. This, however, seemed unfeasible for a long time since aircraft and balloons would be struck by lightning and the contact resistance between different parts of the vehicle would lead to sparking and, hence, the generation of heat. That, in turn, caused small explosions and structural damage - both of which would down planes or balloons. Only after the development of modern aircraft hull structures that are electrically connected throughout so that a hit can be diffused without structural damage, allowed meteorologists to enter thunderstorms and to acquire data about the internal distribution of wind, charge, and other parameters. Nowadays, the most important processes involved in the creation of lightning are understood but still there remain open questions. This article attempts to give a simple but accurate description of the subject.
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\section{Basic charging mechanisms}
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In order to generate discharges in the atmosphere, local differences in the space charge $\rho(x)$ (respectively electric potential $\phi(x)$) are required. These are created by two effects at different length scales. First, the generation of ionized particles, and second their charge-dependent transport in different directions.\\
Ionization is mainly caused by high-energetic cosmic radiation and terrestrial radioactivity, but also by human activity\footnote{The last point is interesting, as the exhaust of industry, cars, etc. and electric fields of mobile communication and power distribution increase the local ion concentration. In addition, also the temperature in concrete deserts on a sunny day is higher than in natural landscapes. This leads to the so-called `urban heat island' effect that cities create their own micro-climate and it can be observed that thunderstorms often build up right above densely populated areas.~\cite{Li:2012,Villarini:2013}}. However, ion generation and recombination are influenced by a large number of chemical and physical effects (for a review see~\cite{Harrison:2003}). For example, in fair weather the charge carrier concentration near ground is about $n_e=60\,e_c$/cm$^3=9.6\,$pC/m$^3$ (with the electron charge $e_c$), while during precipitation of wet snow it can reach $5500\,e_c$/cm$^3=880\,$pC/m$^{3}$. 
According to Poisson's law, $\partial E/\partial z=\rho(z)/\varepsilon_0$, the charge density (assumed to vary only in vertical direction $z$) creates an electric field $E$. Initially, the net space charge vanishes (i.e. $\int\!{\rm d}V\,\rho(x)=0$ in a volume $V$), as approximately equal numbers of positive and negative ions are produced ($n=n_+-n_-=0$). Then, however, charges are transported by several mechanisms that depend on the metrological situation.\\
The basic origin of charge separation is the Earth's own electric field $E_\oplus$. To estimate $E_\oplus$, we write Gau{\ss}'s law as the integration over a spherical shell $S$ of radius $r$ with area element ${\rm d}\mathbf{S}=r^2\sin\vt {\rm d}\vt\,{\rm d}\vp$,\footnote{A note on notation: Vectors $\mathbf{x}$ are printed bold and have an absolute value (norm) $x\equiv |\mathbf{x}|$. Their components $x_i$ are always indexed.}
\begin{align}
 \label{eq:gauss_spherical}
 \phi(\mathbf{x})=\int\limits_{S}\!{\rm d}\mathbf{S}\,\mathbf{E}_\oplus(\mathbf{x})=\frac{Q}{\varepsilon_0}\,,
\end{align}
where $Q$ is the total charge of the Earth. Assuming for simplicity perfect spherical symmetry, $\mathbf{E}_\oplus$ must be a constant on $S$, pointing in radial direction. The surface normal vector ${\rm d}\mathbf{S}$ of a sphere points outward and is therefore (anti-)parallel to $\mathbf{E}_\oplus$. This allows us to pull $\mathbf{E}_\oplus\equiv E_\oplus(r)$ out of the integral in \eqnref{eq:gauss_spherical}, leading to
\begin{align}
 \phi(\mathbf{x})&=E_\oplus(r) r^2\int\limits_0^\pi\!\sin\vt\,{\rm d}\vt\int\limits_0^{2\pi}{\rm d}\vp=4\pi r^2 E_\oplus(r)=\frac{Q}{\varepsilon_0}\nonumber\\
 \Rightarrow E_\oplus(r)&=\frac{Q}{4\pi\varepsilon_0\,r^2}\,.\label{eq:E_solution}
\end{align}
Therefore, the electric field of the Earth is strongest at the surface $r=r_\oplus\approx 6371\,$km, and falls off with the square of the radius. From measurements of electric fields at different altitude $Q\approx -6\times10^5\,$C~\cite{Demtroeder:1999} could be obtained. Earth is charged negatively at all times, resulting in $E_\oplus\approx100$--$300\,$V/m near the ground.\\
As mentioned above, there are always ions of charge $q$, which drift due to the Coulomb force $qE_\oplus$, to counter $E_\oplus$. Therefore, positive charges accumulate near the ground, while negative charges move to higher altitudes, and the effective electric field $E_\text{eff}$ falls off exponentially with the altitude instead of $r^{-2}$.\\
Air always contains small droplets (particles) of water and hydrocarbons of only a few \textmu{}m diameter. Within these particles, ions are trapped. In the presence of $E_\text{eff}$, positive ions move to the bottom of the particle and vice versa for negative ions, thereby creating an electrical dipole. This dipole-character caused by $E_\text{eff}$ is the root cause of all cloud charging. We consider the ion current $j_i$ ($i=+,-$), created by wind speeds $u$, and $E_\text{eff}$ in dependence on the so-called `mobility' $M_i$ of ions\footnote{Turbulence (eddy) effects are not considered here.}
\begin{align}
\mathbf{j}_i=\rho_i \mathbf{u}+\rho_iM_i\mathbf{E}_\text{eff}\label{eq:current}\,.
\end{align}
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\begin{figure}[!ht]
 \centering
 \includegraphics{fig_clouds.eps}
 \caption{Cloud stages. a) Small cumulus. Wind speeds are low ($u<v$) and charging is driven by drift of ions in the electric fields. The internal charge distribution leads to a current imbalance (first term in \eqnref{eq:current} such that $j_+$ is stronger on the top, where particles acquire more positive charge, and $j_-$ stronger on the bottom, leading to negatively charged particles there (see \figref{fig:drift}a). This creates $E_\text{eff}>E_\oplus$ in the cloud. b) Large cumulus with convection currents $u>10\,{\rm m/s}>v$. The first term in \eqnref{eq:current} leads to stronger currents and charge collection due to particle movement (see \figref{fig:drift}b), which in turn augments $E_\text{eff}$, and increases the positive feedback mechanism. $E_\text{eff}$ can reach several kV/m.\label{fig:clounds}}
\end{figure}
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\begin{figure}[!ht]
 \centering
 \includegraphics{fig_drift.eps}
 \caption{Selective drift charge collection. a) Slow particle case. Ions move in the vertical $E_\text{eff}$. They are attracted by the polarization charge of particles. If $j_+\neq j_-$ the particle acquires net charge. b) Gravity or wind move particles at speeds $u$ larger than the ion drift velocity $v$. On the `front' side of the particles, ions of opposite charge to the polarization exhibited by the particle in the direction of $u$ are collected, leading to net charging of the particle. This effect occurs in larger clouds (\figref{fig:clounds}b) and enhances the drift charging due to ion currents.\label{fig:drift}}
\end{figure}
In the earliest cloud stage (see \figref{fig:clounds}a) of small scattered cumulus with $u<1\, {\rm m/s}$ one finds predominantly small droplets ($\leq 10\,$\textmu{}m), which will simply float. Ions drift freely (at velocity $v$, in a mostly vertical current $j_i$, seeking to counterbalance $E_\oplus$ (second term in \eqnref{eq:current}). As shown in \figref{fig:drift}a, particles will selectively capture ions, depending on the local strength of the currents $j_i$. In the lower part of the cloud, $j_->j_+$, leading to negative charging of droplets there, and vice versa on the top. The resulting charge distribution is shown in \figref{fig:clounds}a. Note that particle charging increases $E_\text{eff}$, while the pure ion currents aim to counterbalance $E_\text{eff}$. The stronger the electric field the stronger the ion current and the stronger particle charging. This is a positive feedback circuit.\\
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\begin{figure}[!ht]
 \mbox{}\hspace*{-3ex}\includegraphics[width=86mm]{fig_collision.eps}
 \caption{Charge collection by collisions. a) Liquid particles collide and split again if the relative speeds are high enough. The process leaves an unequal amount of charges in each particle. b) Ice particles collide and obtain charge by induction.\label{fig:collision}}
\end{figure}
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When the clouds grow to a height of several km, the wind speeds of convection currents are generally $>10\,$m/s (see \figref{fig:clounds}b), such that even heavier particles circulate in the cloud. This leads to the outside of the cloud being more positively charged than the inside, as the charged particles are now transported with the winds. When particles larger than $1\,$mm (rain drops) collide, then collision charge separation can take place (see \figref{fig:collision}a). At this stage, there are also several other effects (such as the induction charging in \figref{fig:collision}b) that typically scale with the radius of the water or ice particles (see the review in Ref.~\cite{NRC:1986}), and contribute to the total charge buildup within the clouds. Measurements have shown that hail pellets of a few mm size can acquire charges of up to $100\,$pC (i.e. $6\times 10^{8}\,e_c$).
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\section{Discharges}
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As soon as the field strength reaches $10^5$--$10^6\,$V/m, the air gap between two regions of different space charge `breaks down', resulting in avalanche ionization leading to a discharge.
Since the electric fields between different regions of a thundercloud are typically larger than between the cloud and the ground, statistically only a small fraction of all discharges occurs towards the ground. In any case, the initial break-down occurs in steps. Originating in an area of $E_\text{eff}\geq 10^6\,$V/m, a so-called `leader channel' is formed. This low-resistance highly ionized path effectively short circuits its end points. Leader channels have a length of $30$--$90\,$m and take roughly $15$--$100\,\m$s to form. Different channels originating from separated volumes can connect, forming the typical `stepped leader' branch-structures of a discharge. Finally, when one of the channels arrives at a distance of 30 to $100\,$m above the ground, the electric field near objects on the surface is increased to several kV/m. This leads to the formation of a reverse break-down channel from the ground up. The reverse channel normally extends from the locally most `pointy' object, such as trees, towers, or rocks, as the electric field strength is largest there. As soon as a forward and a return channel connect, the actual stroke starts. At 1/3 of the speed of light the discharge current of typically $40\,$kA propagates in the leader channel, emitting $10^8\,$W per meter length, and heating the channel to $30\,$kK in typically $100\,$ns. In intervals of 40--$80\,$ms two to four subsequent return strokes are formed, for which a discharge appears as flickering to the human eye.
\begin{figure}[!ht]
\centering
 \includegraphics{fig_discharges.eps}
 \caption{Discharge development. 1) Most leaders originate in a zone of $-10$ -- $-15^\circ$C, which is found in summer storms at elevations of 4--$10\,$km. They propagate side wards and down in steps. 2) As soon as the ionized leader channel short circuits the cloud to $\sim100\,$m above ground, a return channel is created. 3) When the channels connect, the discharge is initiated.\label{fig:discharge}}
\end{figure}

The detection of atmospheric discharges is interesting not only from the meteorological point of view but also has practical applications, such as the detection of possible wood fires, and proof evidence for insurance companies. Starting in the 1970s, a global system of ground-based sensors detects discharge activity~\cite{Virts:2013} by triangulation of received broadband electric pulses. In addition, dedicated weather satellites are tracking atmospheric discharges. Modern computers allow us to analyze the enormous amount of data collected during the past few decades, leading to new insight via extensive use of statistics on the correlation between the numerous parameters involved in the formation of lightning. Despite all these efforts, details of the initiation of channels, various particle charging mechanisms, the influence of environmental parameters,  as well as the processes leading to radio frequency emission in discharges are still subject to active research.
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\section{Conclusion}
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Despite the omnipresence of lightning on our planet, the details about physical processes leading to this phenomenon are only slowly being understood. The Earth's electric field causes ion drift, leading to positive charge at high altitude. Condensation drops capture floating ions and become polarized. When these particles move through ionized air, they collect negative charges when moving downwards and positive ones when moving upwards, which enhances the field continuously. When the resulting electric potential between different regions causes ionization, a leader channel is formed, which then acts as conductor for the main discharge.
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