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We study numerically the mutual propagation of two fractional topological charge Laguerre-Gaussian (LG) light beams in the moving atmosphere with the thermal nonlinearity. The fractional charge beam is constructed as a weighted superposition of a finite number of the standard LG modes with integer charges. Numerical simulations demonstrate an enhanced stability of the cross-section intensity distribution of the fractional charge LG beams against the thermal blooming, as compared with the case of integer charge beams. The dominant mechanism of such a stability results from the multimodal structure of the fractional charge beams.