A theoretical approach suggests that the surface heterogeneity on a scale of tens of kilometres can generate mesoscale motions that are not in a quasi-stationary state. The starting point of the theoretical approach is the equations of horizontal velocity and potential temperature that are low-pass filtered with a mesoscale cut-off wavelength. The transition of the generated mesoscale motions from a quasi-stationary state to a non-stationary state occurs when horizontal advection is strong enough to level out the potential temperature gradient on the surface heterogeneity scale. Large-eddy simulations (LES) suggest that the convective boundary layer (CBL) changes to a non-stationary state when forced by a surface heat-flux variation of amplitude of 100W m⁻² or higher and a wavelength of the order of 10 km. Spectral analysis of the LES reveals that when the mesoscale motions are in a quasi-stationary state, the energy provided by the surface heat-flux variation remains in organized mesoscale motions on the scale of the surface variation itself. However, in a non-stationary state, the energy cascades to smaller scales, with the cascade extending down into the turbulence scale when the wavelength of the surface heat-flux variation is on a scale smaller than 100 times the CBL height. The energy transfer from the generated mesoscale motions to the CBL turbulence results in the absence of a spectral gap between the two scales. The absence of an obvious spectral gap between the generated mesoscale motions and the turbulence raises questions about the applicability of mesoscale models for studies on the effect of high-amplitude surface heterogeneity on a scale of tens of kilometres.
