## 92.60.-e Properties and dynamics of the atmosphere; meteorology (see also 92.40.Zg Hydrometeorology, hydroclimatology)

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- asymptotic analysis (2)
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- deep convective clouds (1)
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The thesis presents the analysis of a reduced model for modulation of internal gravity waves by deep convective clouds. The starting point for the derivation are conservation laws for mass, momentum and energy coupled with a bulk micro-physics model describing the evolution of mixing ratios of water vapor, cloud water and rain water. A reduced model for the identified scales of the regime is derived, using multi-scale asymptotics. The closure of the model employs conditional averaging over the horizontal scale of the convective clouds. The resulting reduced model is an extension of the anelastic equations, linearized around a constant background state, which are well-known from meteorology. The closure of the model is achieved purely by analytical means and involves no additional physically motivated assumptions. The essential new parameter arising from the coupling to a micro-physics model is the area fraction of saturated regions on the horizontal scale of the convective clouds. It turns out that this parameter is constant on the employed short timescale. Hence the clouds constitute a constant background, modulating the characteristics of propagation of internal waves. The model is then investigated by analytical as well as numerical means. Important results are, among others, that in the model moisture (i) inhibits propagation of internal waves by reducing the modulus of the group velocity, (ii) reduces the angle between the propagation direction of a wave-packet and the horizontal, (iii) causes critical layers and (iv) introduces a maximum horizontal wavelength beyond which waves are no longer propagating but become evanescent. The investigated examples of orographically generated gravity waves also feature a significant reduction of vertical momentum flux by moisture. The model is extended by assuming systematically small under-saturation, that is saturation at leading order. The closure is similar to the original case but requires additional assumptions. The saturated area fraction in the obtained model is no longer constant but now depends nonlinearly on vertical displacement and thus on vertical velocity.

By use of asymptotic analysis Carqué et al. [ZIB-Report 08-03] derived an asymptotic column model for deep convective clouds based on the three dimensional compressible flow equations and a bulk microphysics parameterization. In the present study we check the plausibility of the reduced model equations by comparing implications of the model for the scaling of various terms in the governing equations with those extracted from large eddy simulation data based on the computational model UCLA-LES1.1. This code solves an anelastic system of equations with complete droplet based microphysics and LES closures. We observe that the simulation data corroborate the basic assumptions of the asymptotic analysis and the main conclusions implied by the asymptotically reduced model. The code output reflects the scales of space and time: The deep convective clouds show an anisotropic structure where the horizontal scale is considerably narrower than the vertical scale; with a period of about 20 min, from emergence to breakup, the life cycle of one particular deep convective cloud corresponds exactly to the reference time of the reduced model. The characteristic properties of dynamics as predicted by the reduced model are also reflected in the simulation data: The horizontal flow is controlled by the pressure field; the vertical velocity develops freely independent of pressure over the depth of the convective column; the vertical velocity is directly determined by the buoyancy induced by the potential temperature deviation relative to the background stratification. With respect to grid resolution we observe that refining the spatial step size of the equidistant computational grid from 125 m to 62.5 m does not influence the results: Even with the coarser grid the relevant physical phenomena are sufficiently resolved. Somewhat surprisingly, the Coriolis term involving vertical velocity and acting on the horizontal (east-west) velocity component appears at leading order in the asymptotics. Accordingly, we expected to find a nontrivial impact of this Coriolis effect on the horizontal flow velocity components within columns of updrafts. However, switching the term on and off in subsequent simulations did not sizeably affect the results.

Asymptotic analyses of the three dimensional compressible flow equations coupled with transport equations for the mixing ratios of water vapour, cloud water and rain water are described. We obtain reduced systems of equations for two particular regimes of length and time scales: Models for the long time evolution of deep convective columns and for the short time evolution of shallow convective layers. The asymptotic deep convective column model is anelastic, yet the vertical motion is pressure free, i.e., it evolves freely in interaction with buoyancy while the horizontal divergence adjusts to fullfil the anelastic constraint. The perturbation pressure guaranteeing compliance with the horizontal divergence constraint obeys a Poisson-type equation. Surprisingly, the vertical velocity plays an important role in the horizontal dynamics through the Coriolis term. The vertical acceleration in a saturated column is directly determined by the buoyancy induced by potential temperature differences relative to the background stratification. This potential temperature deviation is a conserved quantity. Evaporation is the only important microphysical process in the undersaturated regime. The evaporation rate depends on the saturation deficit and the amount of rain water present and determines the (downward) vertical velocity and the distribution of water vapour. To connect the deep convective column solutions to top and bottom boundary conditions, a different flow regime needs to be accounted for. Within shallow layers whose depth is comparable to the column diameters, adjustment to physical boundary conditions can take place. This is the second regime considered in this report. The shallow convective layer regime is shown to be asymptotically described by Boussinesq-type equations. These equations are closed by evolution equations which show that, in the saturated regime, the distributions of potential temperature and cloud water are determined by a condensation rate that is directly proportional to the vertical velocity. In the undersaturated regime, the potential temperature distribution is determined by the amount of rain present, since the water vapour in this case is shown to be a conserved quantity. In both regimes the distribution of rain water depends on the rain water flux.