Or in other words, does it increase the amount of heat+moisture advection from the Gulf of Mexico into the Central United States? And is this increase associated with increased CAPE/tornado/thunderstorm activity?
According to the NOAA document A Comparison of Rawinsonde Data from the Southeastern United States during El Niño, La Niña, and Neutral Winters (Sankovich et al.) report that
Schaefer and Tatom (1998) considered the number of tornadoes per year in the United States and sea surface temperatures (SST) in various portions of the Pacific Ocean to try to discern an impact of ENSO on the occurrence of tornadoes. Their statistical inquiries showed no major influence, but their data does show a signal that more tornadoes tend to occur in the mideastern states during the La Niña phase.
However, another study cited by Agee and Zurn-Birkhimer (1998) state
that tornado occurrences do not favor one ENSO phase but rather exhibit a shift in geographic location. For example, their results suggest that more tornadoes occur in the lower Midwest, Ohio Valley, Tennessee Valley, and mid-Atlantic region during the La Niña phase than in any other phase.
This is further concluded by Bove (1998) concluded that
increased tornado activity over the Ohio and Tennessee River Valleys during the cold La Niña phase.
Sankovich et al. conclude that
Like all climatological signals, they [ENSO] represent long-term trends. They do not provide a tool for making seasonal or other extended forecasts; rather, they show that other things being equal, the atmospheric stratification over the Southeastern United States is typically less compatible for the occurrence of severe thunderstorms (3/4” or larger hail, thunderstorm gusts of 58 mph or faster, tornados) during a La Niña winter than during Neutral or El Niño winters.
In the paper A seasonal-scale climatological analysis correlating spring tornadic activity with antecedent fall–winter drought in the southeastern United States (Shepherd et al. 2009), state that to
remove the uncertainty associated with the ENSO debate, we consider a new approach by focusing on antecedent drought conditions for a region.
Whereby, they conclude that the relationship between drought and incidences of tornadic and severe thunderstorm behaviour to possess a
statistically significant tendency for fall–winter drought conditions to be correlated with below-normal tornado days the following spring in north Georgia (i.e. 93% of the years) and other regions of the Southeast. Non-drought years had nearly twice as many tornado days in the study area as drought years and were also five to six times more likely to have multiple tornado days
Schaefer, J. T., and F. B. Tatom, 1998: The relationship between El Niño, La Niña, and United States tornadoes, Preprints, 19th Conf. on Severe Local Storms, Minneapolis, MN, Amer. Meteor. Soc., 416-419.
Agee, E., and S. Zurn-Birkhimer, 1998: Variations in USA tornado occurrences during El Niño and La Niña, Preprints, 19th Conf. on Severe Local Storms, Minneapolis, MN, Amer. Meteor. Soc., 287-290.
Bove, M.C., 1998: Impacts of ENSO on United States tornadic activity, Preprints, 19th Conf. on Severe Local Storms, Minneapolis, MN, Amer. Meteor. Soc., 313-316.