ay ay ay. Nope.
It is true that temperature gradients account for jet streaks (or vise versa), but thunderstorms don't care about what a temperature gradient is on the large scale (100-1000s of km).
Thnderstorms derive their energy from latent heat... which is dependent on the temperature of the air (warmer = better). As T goes up, the air can support more water vapor. When water vapor is then transported aloft and cooled... it can then condensate to form water droplets. This transformation from gas->liquid causes a monster relase of latent heat which in turns heats the air. Air with more water vapor is lighter in density.... increase the temperature and that also decreases the density... parcel of air becomes more bouyant and as long as there is no cap, off you go to a thunderstorm.
Now you can have something called baroclinic generation of voriticity along the cold pool and inflow of a supercell. This can result in tubes of horizontal vorticity aligned with the density gradient. THis gradient is due to the drastic difference in air temp/dewpoint. IF the inflow to the storm is aligned along this horizontal rotating tube of air... you can easily ingest it into the storm's updraft.. and wham you can enhance or get the mesocyclone cranking.
Aaron
