# 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
The situation is different in a lightning storm (right plots of Figure 5): (i) the outer wall retains (bottom right of Figure 5) the same atmospheric value (78), and the intense Joule heating within the thin region of skin effect appears as a higher initial temperature \(T_{\rm j}\) in Figure 5 (top right); (ii) the highest temperature \(T_{\rm j}\) is for titanium, which is a heat-resistant metal, with clearly lower values for aluminium, copper and steel, in decreasing order, as shown in Table 6; (iii) Table 6 indicates the peak temperature \(T_{\rm j}\) due to the Joule heating in the skin effect and the depth \(x_{\rm j}\) at which it occurs for each metal, showing that the penetration depth is largest for titanium and smallest for steel, with copper and aluminium in between; (iv) outside the skin effect \(x_{\rm j}\leq x\leq L\), the Joule heating is negligible and the temperature varies linearly from the \"peak\" value \(T_{\rm j}\) at \(x_{\rm j}\) in Table 6 to the fixed value (79) at the inner wall; (v) for all metals except titanium, the \"peak\" temperature \(T_{\rm j}\) due to Joule heating at \(x=x_{\rm j}\) is lower than the inner wall temperature \(T_{\rm 1}>T_{\rm j}\), and thus, beyond the \"peak\" \(x_{\rm j}\leq x\leq L\), the temperature rises almost linearly from \(T_{\rm j}\) at \(x_{\rm j}\) to \(T_{\rm 1}\) at \(x=L\). For titanium, the peak temperature \(T_{\rm j}\) at \(x_{\rm j}\) is larger than the inner wall temperature \(T_{\rm j}>T_{\rm 1}\), leading to a linear decay to the same temperature (79) at the inner wall; (vi) in the full range plot (top plots of Figure 5) the temperature always ends at the inner wall value (79); (vii) the temperature always starts at the outer wall temperature (78), as is apparent for fair weather (left plots of Figure 5); (viii) for a lightning storm, the temperature also starts (right bottom plots of Figure 5) at the outer wall value (78), but, since the Joule heating region is so thin (Table 6), the temperature appears to start at \(T_{\rm j}\
eq T_{\rm 0}\) in the full thickness plot (top-right plots of Figure 5); (ix) the values of \(T_{\rm j}\) are noticeably higher for titanium and close for aluminium, copper and steel (Table 6); (x) thus, over the full thickness of the slab in a lightning storm (top-right plots of Figure 5), the temperature for titanium appears higher than for fair weather, whereas, for aluminium, copper and steel, there is not much difference between fair weather and the lightning storm because the Joule heating  
Figure 4: Magnetic field along the initial part of the slab using (68) with penetration depth \(l\) from Table 2 and dielectric permittivity \(\varepsilon\) and magnetic susceptibility \(\mu\) from Table 1, with surface magnetic field for (**left**) fair weather; (**right**) a lightning storm. is not too significant in either case. Thus the hierarchy is different in detail in a thin skin heating region  
\[T:\quad 0\leq x\leq x_{t}:\quad\overline{T}>\overline{A}>\overline{C}>\overline{S} =S=C=A=T,\] (91a) and, overall, over the whole thickness of the slab, \[T:\quad 0\leq x_{t}\leq L:\quad\overline{T}>\overline{A}\sim\overline{C}\sim \overline{S}=S=C=A=T. \tag{91b}\]  
The hierarchy for temperature in the Joule heating region (91a) is the same as (88) for the electric and magnetic fields in a lightning storm, whereas, in fair weather, the hierarchy in (88) becomes negligible in (91a). The hierarchy over the full thickness of the slab (91b) in the top plots of Figure 5 (i) also applies to the weak Joule heating in fair weather in the skin region (bottom-left plots of Figure 5); (ii) does not show the detail (bottom-right plots of Figure 5) of more intense Joule heating in the skin region in a lightning storm (Table 6) that corresponds to the hierarchy (91b). The heat flux (Figure 6) would be (69a)-(69b) constant (dotted line),  
\[G_{0}=k(T_{1}-T_{0}), \tag{92}\]  
in the absence of Joule heating.