We have investigated the heterogeneously heated free Convective Boundary Layer (CBL) by means of dimensional analysis and results from Large-Eddy Simulations (LES) and Direct Numerical Simulations (DNS). We have created scaling laws that can express the state of the system as a function of the variables defining the atmospheric flow and the parameters that describe the properties of the surface heterogeneity.
The flow over a heterogeneously heated surface can manifest itself in three regimes, which can be classified based on their blending height, i.e., the height at which the influence of heterogeneity vanishes (Mahrt, 2000): in the micro-scale regime, heterogeneities have a blending height within the surface layer and their overlying CBL scales as if it were a horizontally homogeneous one. Heterogeneities in the macro-scale regime have a blending height above the boundary layer, i.e., their horizontal extent is so large that a heterogeneous CBL develops that scales with its corresponding local surface properties. In between those, we find the meso-scale regime that has a blending height within the mixed layer. Under these conditions, strong secondary circulations can form that lead to a peak in kinetic energy, which we call the optimal state.
One of the particular aims of our scaling laws is to describe the properties of the optimal state and the transition from the meso- to the micro-scale regime. In order to do this, we have created an idealized system that consists of a stably stratified atmosphere heated from below by square patches with high surface buoyancy fluxes. On this system we have applied dimensional analysis in order to derive parametrizations of relevant properties. By varying the non-dimensional parameters related to the size and distance of the patches and that related to the surface buoyancy fluxes of the patch and its surroundings, a wide range of surface conditions can be created, such as chessboard-like patterns, but also isolated plumes over each patch.
Within the non-dimensional system, there are three parameters that describe the surface heterogeneity; First, there is the heterogeneity size XH , which is the distance between two patches. Second, there is the patch size XP , which is the width of a patch. Third, there is the heterogeneity amplitude, which is the degree to which the patches stand out of their environment. This is expressed in the ratio B0P/B0, where B0P is the buoyancy flux of the patch and B0 the area-averaged surface flux. The three length scales have been non-dimensionalized with the reference Ozmidov length L0 ≡ (B0/N^3)^(1/2), which is a scaling height for the initial size of the plumes that form.
2 Experiments
Our simulations consist of a combination of LES and DNS. LES has been used because it opens the possibility of studying a large parameter space and it has been proven useful for studying the heterogeneously heated CBL. For simulations with very small patches and relatively high fluxes, we have chosen for DNS, because the usage of an LES surface model is questionable under such high variability at the surface.
Within each performed simulations, we travel through a range of heterogeneity regimes, such that it contains the formation of a peak in kinetic energy, corresponding to the optimal state with the strongest secondary circulations, and the subsequent transition from the meso- to the micro-scale regime. We do this by starting the simulation under the condition that the heterogeneity regime classifies as either macro- or meso-scale and subsequently we let the CBL grow until the transition is finished.
Following this simulation approach, we have conducted two experiments. The first experiment is used to study the importance of heterogeneity size and amplitude, based on a surface buoyancy flux pattern with equal area coverage for patch and non-patch regions. We have performed a sensitivity study on the heterogeneity size for three different amplitudes. In the second experiment we have investigated the influence of the patch size We have done a series of simulations in which, for a given reference CBL height, the heterogeneity size and the mean surface buoyancy flux is constant, but with a decreasing patch size and increasing patch heat flux for the different simulations.
3 Results
The main findings are:
The optimal state and the transition to a horizontally homogeneous CBL do not occur at a fixed ratio of the heterogeneity size and the CBL height h, but occur at larger ratios of XH /h for larger heterogeneity sizes. This means that larger heterogeneity sizes reach their optimal state and transition relatively earlier. This is related to the development of structures in the downward moving air that grow faster than the CBL itself and eventually break down the organization imposed by the heterogeneous heating.
The state of the system can be expressed in a non-dimensional variable (h/L0)(XR/XH)(XH/L0)^(2/3). This variable relates the CBL height h to the heterogeneity size XH and the patch size XR using the reference Ozmidov length of the CBL L0 as a scaling variable. This scaling takes into account the heterogeneity size XH properly, such that the scaled time instance of optimal state and transition is unmodified if the heterogeneity size is changed.
An increase in the heterogeneity amplitude at the same mean surface buoyancy flux results in earlier optimal states and delays the transition.
A reduction of the patch size XR combined with an increase of the patch heat flux, such that the total energy input remains the same, results in later transitions from the meso- to the micro-scale regime and a lower integrated kinetic energy in the meso-scale regime.
Small heterogeneity sizes, which are of the same sizes as the initial height of convective plumes, tends to reduce the entrainment flux of buoyancy, whereas larger heterogeneity sizes, that exceed the initial size of the plumes appear to enhance entrainment.
To conclude, our findings improve our understanding of a large part of the parameter space that defines the simplest case of heterogeneous heating: the one without a large scale forcing.
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