Squall lines are coherent turbulent traveling waves on scales of order 100 km in
the atmosphere that emerge in a few hours from the interaction of strong vertical shear and moist
deep convection on scales of order 10 km. They are canonical coherent structures in the tropics and
middle latitudes reflecting upscale conversion of energy from moist buoyant sources to horizontal
kinetic energy on larger scales. Here squall lines are introduced through high resolution numerical
simulations which reveal a new self-similarity with respect to the shear amplitude. A new multi-scale
model on mesoscales which allows for large vertical shears, appropriate for squall lines, is developed
here through systematic multi-scale asymptotics. Mathematical and numerical formulations of the
new multi-scale equations are utilized to illustrate both new mathematical and physical phenomena
captured by these new models. In particular, non-hydrostatic Taylor-Goldstein equations govern
the upscale transports of momentum and temperature from the order 10 km microscales to the
order 100 km mesoscales; surprisingly, upright single mode convective heating without tilts can
lead to significant upscale convective momentum transport from the microscales to the mesoscales
due to the strong shear. The multi-scale models developed here should be especially useful for
dynamic parameterizations of upscale transports as well as for new theory in three-dimensions with
a transverse shear component, where contemporary theoretical understanding is meager.