Communications in Mathematical Sciences

Volume 8 (2010)

Number 1

Special Issue on the Occasion of Andrew Majda’s Sixtieth Birthday: Part I

New multi-scale models on mesoscales and squall lines

Pages: 113 – 134



Andrew J. Majda

Yulong Xing


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.


Squall lines; multi-scale asymptotics; Taylor-Goldstein equations; upscale convective momentum transport

2010 Mathematics Subject Classification

65T50, 76F55, 93A30

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