# 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
For fair weather (left plot of Figure 6), the Joule heating is very small or negligible, leading to a horizontal line for the heat flux. For a lightning storm, the Joule heating is also small outside the skin area, leading to nearly the same horizontal line as for fair weather if the full range of depth (90) is used. In order to show the effect of Joule heating in the case of a lightning storm (right plot of Figure 6), the heat flux is plotted \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Metal** & **T** & **A** & **C** & **S** & **Unit** \\ \hline Peak temperature \(T_{1}\) & 333.57 & 223.28 & 219.07 & 216.50 & \(K\) \\ \hline
**Location \(x_{2}\)** & \(8.32\times 10^{3}\) & \(2.72\times 10^{2}\) & \(1.52\times 10^{2}\) & \(3.94\) & \(\times 10^{-12}\)m \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Peak temperature due to the Joule heating in the skin effect for a flight in lightning storms. Figure 5: Temperature along the entire slab with length (90) (**top** plots) and along the initial part of the slab (**bottom** plots) in fair weather (**left**) and lightning storm (**right**) cases using (49) and (47b) with outer \(T_{0}\) and inner \(T_{1}\) boundary temperatures from Table 3, penetration depth \(l\) from Table 2 and heating function at the surface \(Q_{0}\) from Table 4.  
in a subset \(10^{-10}\,\mathrm{m}\) with a smaller depth within the thin skin region (89). The modulus of the heat flux is highest for copper and lowest for steel with titanium and aluminium in between in fair weather (left plot of Figure 6) when the heat flux is constant. In a lightning storm (right plot of Figure 6), in the thin region of the skin effect, the intense Joule heating leads to an increase in heat flux that is fastest for titanium and slowest for steel, with copper and aluminium in between. Thus the hierarchy for heat flux is different in a lightning storm to in fair weather,  
\[G:\quad\overline{T}>\overline{A}>\overline{C}>\overline{S}\gg C>A>T>S, \tag{93}\]  
with much larger values in the former case. The hierarchy for the heat flux (93) is broadly similar to those for the temperature (91a)-(91b) and electric and magnetic fields (88). Figure 7 shows that the displacement is zero at the inner boundary and negative at the outer boundary, with larger values in modulus depending more on the metal than on the weather. The displacement is (59), the sum of a quadratic function and a negative exponential of position. The displacement in Figure 7 is close for fair weather (left) and a lightning storm (right) for all metals except titanium. In both cases, the displacement in the modulus is largest for aluminium and decreases for copper and steel. For fair weather, the smallest displacement in the modulus is for titanium:  
\[|u|:\quad\overline{A}>A>\overline{C}>C>\overline{S}>S>T. \tag{94}\]  
For a lightning storm, the curve for titanium corresponds to a smaller displacement in modulus than for aluminium, and crosses with the curve for copper close to the outer wall, and also crosses with the curve for steel close to the inner wall. The hierarchy (94) for the displacement differs from (88) for the electric and magnetic fields (93) and from (91a)-(91b) for the temperature and (93) for the heat flux because the ordering of metals is distinct. Figure 6: Heat flux along the entire slab with thickness (90), using (50b) and (47b), with the same values of \(l\), \(T_{0}\), \(T_{1}\) and \(Q_{0}\) as in Figure 5.  
Figure 8 shows that the strain (60) does not vanish at either boundary and is (60) a linear function (dotted line) plus an exponential. The exponential effect away from the surface is small and the strain is almost linear, with the magnitude and slope depending more on the metal than on the weather. Although not zero, the strain is very small at the outer wall in fair weather (Figure 8, left) and increases towards the inner wall. At the inner wall, the strain in a lighting storm (Figure 8, right top) is not too different from in fair weather, starting at the outer wall with small values.