This relatively simple structure allowed the development of simplified mixed-layer models to evaluate the CBL growth, based on the daily evolution of turbulent heat flux at ground level. These models typically assume a vertically constant potential temperature profile throughout the mixed layer, and reproduce the upper inversion either as a sharp discontinuity (zero-order jumps) or as a finite-depth constant-gradient stable layer (first-order jumps).

By introducing this structure in the energy conservation equations, and applying the appropriate initial/boundary conditions, suitable evolution equations can be obtained for the CBL height, the potential temperature, and the inversion strength.

Such a simplified scheme is hardly applicable to reproduce the CBL growth in mountain valleys, where different structures are known to occur, as shown by both field measurements and model simulations. Indeed the convective boundary layer growing during daytime in mountain valleys typically displays a shallower mixed-layer, whereas the stably stratified capping inversion layer is usually deeper. These effects have been explained as a consequence of the core valley atmosphere subsidence, compensating the heat and mass advection associated with thermally driven up-slope flows along the sunlit valley sidewalls. Such processes are not parameterised in the above simplified models, valid for flat terrain.

In the present work a simplified model of CBL growth is proposed, which extends previously introduced concepts to cover processes specifically occurring in mountain valleys. Assuming a simple trapezoidal valley cross-section, a first order mixed layer scheme is adopted to reproduce the thermal structure of the atmosphere above the valley floor. The up-slope mass and heat flux is estimated by means of the Defant's (1949) extension of Prandtl (1942) theory, reproducing a time-varying up-slope flow.

Accordingly the effect of subsidence warming is evaluated by estimating the compensating downward motion from mass conservation and assuming a linearly stratified free atmosphere.

By combining all the governing equations and related constraints, arising from continuity and consistency conditions, an evolution equation for the mixed layer height is obtained, which incorporates all the effects and the physical and geometrical parameters concurring to control the diurnal evolution of the layer.

As a result, the model allows to evaluate how the air in the valley mixed-layer gets warmed more than over an adjacent plain, as estimated by a classical CBL model, and how various factors concur to determine this enhanced heating.