  Severe thunderstorms cause substantial economic and human losses in the
United States. Simultaneous high values of convective available potential
energy (CAPE) and storm relative helicity (SRH) are favorable to severe
weather, and both they and the composite variable
$\mathrm{PROD}=\sqrt{\mathrm{CAPE}} \times \mathrm{SRH}$ can be used as
indicators of severe thunderstorm activity. Their extremal spatial dependence
exhibits temporal non-stationarity due to seasonality and large-scale
atmospheric signals such as El Ni\~no-Southern Oscillation (ENSO). In order to
investigate this, we introduce a space-time model based on a max-stable,
Brown--Resnick, field whose range depends on ENSO and on time through a tensor
product spline. We also propose a max-stability test based on empirical
likelihood and the bootstrap. The marginal and dependence parameters must be
estimated separately owing to the complexity of the model, and we develop a
bootstrap-based model selection criterion that accounts for the marginal
uncertainty when choosing the dependence model. In the case study, the
out-sample performance of our model is good. We find that extremes of PROD,
CAPE and SRH are generally more localized in summer and, in some regions, less
localized during El Ni\~no and La Ni\~na events, and give meteorological
interpretations of these phenomena.
