# On Anisothermal Electromagnetic Elastic Deformations in Flight in Fair Weather and Lightning Storms
## 4 Metal Slab in Flight in Fair and Stormy Weather
### Comparison of Flight in Fair Weather and Lightning Storms  
Figure 2 shows that the electric field (68) decays exponentially with depth from a surface value that is much higher \(\overline{E}_{0}\) in a lightning storm than \(E_{0}<\overline{E}_{0}\) in fair weather. In both the fair weather (left) and lightning storm (right) cases in Figure 2, the electric field is highest for titanium and lowest for steel, with aluminium and copper in between. In all cases, the values in a lightning storm are higher than in fair weather, as represented by the scheme  
\[E:\quad\overline{T}>\overline{A}>\overline{C}>\overline{S}>T>A>C>S. \tag{88}\]  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Quantity** & **Symbol** & **Equation** & \multicol  
In Figure 3, the exponential decay of the electric current with depth (68) is the same as for the electric field in Figure 2, but the starting values are all distinct due to the Ohmic electrical conductivity and much higher in a lightning storm than in fair weather. For the electric current in Figure 3, the hierarchy is not as clear as for the electric field in Figure 2, with some curves for different metals crossing at intermediate depth instead of being separated over the whole thickness of the slab. Both for the fair weather case (left) and for the lightning storm case (right) in Figure 3, the electric field starts highest for copper, but then crosses with aluminium and titanium, and remains always larger only relative to steel. In Figure 4, the magnetic field decays exponentially with depth (68) like the electric field in Figure 2 and the electric current in Figure 3, with starting values that are (a) much higher in a lightning storm than in fair weather and (b) in both cases different for distinct metals, namely, maximum for titanium, and decreasing for aluminium and copper and minimum for steel, thus leading to the same hierarchy (88) as for the electric field in Figure 2.  
In all three figures, specifically, Figure 2 for the electric field, Figure 3 for the electric current and Figure 4 for the magnetic field, the exponential decay in (68) is very rapid because the penetration depth is very small. In order to make the \"skin effect\" visible, and avoid the curves collapsing to the \(x\) axis, only a small depth below the surface,  
\[0\leq x\leq 10^{-9}\,\mathrm{m}\ll L \tag{89}\]  
is shown in Figures 2-4. The very fast decay with the depth of the electric current in Figure 2 implies that the Joule heating is hardly visible on (76) the full length scale  
\[0\leq x\leq L=2\times 10^{-3}\,\mathrm{m} \tag{90}\]  
used in Figure 5 for the temperature, in Figure 6 for the heat flux, in Figure 7 for the displacement, in Figure 8 for the strain and in Figure 9 for the stress. Figure 3: Electric current along the initial part of the slab using (68) with penetration depth \(l\) from Table 2 and Ohmic electrical conductivity \(\chi\) in Table 1, with surface electric current for (**left**) fair weather; (**right**) lightning storm. The temperature profile (49) would be linear (dotted line) between the fixed boundary values (Figure 5) in the absence of the Joule heat source due to the resistive dissipation of electric currents shown in Figure 3. The Joule heating in fair weather (left plots of Figure 5) is small enough not to affect the linear temperature profile between the temperatures (78) at the outer and (79) at the inner wall, with no discernible difference between different metals.