# Ams Bianka Model Set 146 __HOT__

We present a result on existence of solutions for a system of highly nonlinear partial differential equations related to a phase field model for non-isothermal solidification/melting processes in the case of two possible crystallization states and flow of the molten material.

## Ams Bianka Model Set 146

These results constitute a fundamental step in the proof of the existence of solutions of a complete model for solidification obtained by coupling the present equations with a singular Navier-Stokes system for the flow velocity. The analysis of this complete model is done in a forthcoming article.

All general circulation models (GCMs) analyzed and interpreted by Solomon et al. (2007) [Intergovernmental Panel on Climate Change (IPCC) Fourth Assessment Report, chapter 10] project an increasing global air temperature for the future. As one consequence, the number of days with heat stress will increase in many regions around the globe. The expected changes in strength and frequency of extreme daytime or nighttime temperatures will provoke enhanced discomfort and even an increase in mortality (Souch and Grimmond 2004).

This paper is structured as follows. The regional climate models used as input for the cuboid method and the basic features of the Microscale Urban Climate Model in 3 dimensions (MUKLIMO_3) are introduced in section 2. The cuboid method is described in section 3, and its application to Frankfurt is discussed in section 4. An evaluation of the cuboid method and its results is presented in section 5. In section 6 we discuss the results of the cuboid method. We close the paper with a summary and some conclusions in section 7.

Because of the coarse resolution of GCMs, climate-change signals projected by GCMs need to be scaled down to higher spatial resolutions before they can be used for climate impact studies. The first downscaling step is accomplished by regional climate models and statistical downscaling approaches. Based on these results, the second downscaling step uses MUKLIMO_3 at urban scale. In this section all models and simulations used are introduced.

In addition, we make use of the projections from the two statistical downscaling approaches Wetterlagen-basierte Regionalisierungsmethode (WETTREG) and Statistical Regional Model (STAR). The WETTREG model (Enke et al. 2005) is a circulation-pattern-based weather generator. The projection time series are generated by randomly rearranging the time slices of recent climate. However, the process of resampling is performed to conserve the previously derived frequency distribution of circulation patterns from the simulations of GCMs as closely as possible. This conditioning allows creation of new time series of characteristics that are significantly different from current climate. WETTREG data for all relevant climate parameters are available for 283 stations across Germany. In distinction from the numerical models, the WETTREG model only yields data points on the daily basis (e.g., daily mean temperature).

Similar to WETTREG, STAR model simulations (Orlowsky et al. 2008) are also generated by resampling observational data from weather stations. However, the coupling to global climate projections is more straightforward than within WETTREG. Here, temperature change projections for the region of interest are derived from the global models and are simply prescribed as linear trend for the recombination of weather episodes without regard to the simulated large-scale circulation patterns. STAR provides data for 2342 weather stations of Germany on a daily time resolution.

The current version of MUKLIMO_3 is intended to simulate the atmospheric temperature fields in urban environments. For this purpose, MUKLIMO_3 is augmented with prognostic equations for atmospheric temperature and humidity, balanced heat and moisture budgets in the soil, and a sophisticated vegetation model. Cloud processes and precipitation are not considered. MUKLIMO_3 simulates idealized atmospheric conditions with pronounced influence of local land-use properties. Typical integration times reach from several hours up to one day.

The consequences of the parameterization of the unresolved buildings for the model equations are as follows: In grid cells with unresolved buildings, 1) the volume of the atmosphere is reduced relative to that in the building-free grid cells, 2) heat and momentum are exchanged between the walls and roofs of the buildings and the atmosphere, and 3) subgrid-scale perturbations of the wind field modifying the turbulent exchange in the atmosphere are induced.

The transport equations for the heat and the moisture in the soil are derived from Sievers et al. (1983). The treatment of vegetation in the canopy model is based on Siebert et al. (1992). The latter is ameliorated using three vertical levels for the vegetation. The topmost level contains the trees with the mean tree height ht, the leaf area density for the tree top ρt, and the fraction of tree cover σt. The stem room differs from that by a lower leaf area density ρs. All lower-level vegetation is characterized by the height of the canopy layer hc, the vegetation cover (συ not including σt), and the leaf area index for the canopy layer (LAIc). LAIc is defined as the ratio of the upper leaf surface divided by the surface area on which the vegetation grows. The impact of leaves in the model is threefold. 1) Leaves act as obstacles for the airflow. They are sources and sinks of 2) radiative energy as well as 3) water vapor. The values for these parameters are listed for different land-use classes in Table 3.

Radiation above the buildings is computed separately for short and long wavelengths. For the shortwave radiation, we compute the direct and the diffuse irradiance using an empirical approach [Verein Deutscher Ingenieure (VDI) 1994]. The parameterization for the longwave net radiative flux is based on the approach of Möller (1954) and Zdunkowski et al. (1975) with an additional parameterization to consider the effect of clouds. In layers with buildings, shortwave radiation is reflected and absorbed by their walls and roofs depending on the building density and height as well as by the soil. The model also includes a scheme for the emission and absorption of longwave radiation by the buildings and the soil.

Since MUKLIMO_3 simulates a limited area of the atmosphere, the model requires initial and boundary specifications. From a given set of initial air temperature T, relative humidity rh, and horizontal wind vector υ characterizing the environmental conditions, the model first computes a one-dimensional profile up to 1100-m altitude. This profile represents the background meteorological conditions with low influence from elevation and land use. It is, in the beginning, stamped to the complete three-dimensional model domain. In the course of the three-dimensional integration the one-dimensional profile values at 750-m altitude are transferred to the top of the three-dimensional domain and taken as time-dependent upper boundary values. This transfer is done for the wind velocity and the air temperature as well as the exchange coefficients.

The promising results of the cuboid method for the heat load in Frankfurt indicate that this method might represent a useful downscaling technique for a wide range of applications. In our study, three meteorological parameters (regional air temperature, humidity, and wind speed) were identified as dominating factors for possible urban-heat-load conditions. This required 23 MUKLIMO_3 simulations and a trilinear interpolation procedure. If for a different application only two dominating factors are sufficient to describe most of the expected variability (e.g., wind speed and mixing layer depth for estimating urban ventilation), then only 22 MUKLIMO_3 simulations and a bilinear interpolation procedure are required; that is, the cuboid method could be reduced to an even faster rectangle method. By analogy, 24 MUKLIMO_3 simulations and a four-dimensional linear interpolation are necessary if four dominating meteorological parameters can be identified, and so on. The approximate validity of linear interpolation has to be evaluated for all applications. Furthermore, for nonurban climate-change impact studies the urban climate model can be replaced by other computationally expensive impact models, such as air-quality models or hydrological models.

In case of a moderate to strong southerly large-scale flow component, the Inn Valley may be affected by south foehn which strongly influences the MoBL conditions and interacts with the thermally driven circulations. To assess the likelihood of foehn in the investigation area, we used the foehn probability at Steinach in the Wipp Valley and Innsbruck in the Inn Valley (Fig. 1a) which is based on the method of Plavcan et al. (2014) (Fig. 5c and Table 4). The method diagnoses foehn automatically and probabilistically using a statistical mixture model. The variables considered are wind direction and speed at the stations in the valleys and the potential temperature difference between the valley stations and a station higher up. For Innsbruck relative humidity is additionally taken into account. During four IOPs (IOPs 2b, 3, 5, and 10), foehn likely prevailed for several hours judging from the probability at Steinach. Foehn at Innsbruck in the Inn Valley occurs less often than at Steinach in the Wipp Valley. This indicates that foehn more frequently breaks through to the valley floor at Steinach than at Innsbruck. The latter is about 500 m lower than the station elevation at Steinach and often affected by a cold air pool (e.g., Haid et al. 2020).

Thermally driven flows regularly developed in the Inn Valley during all IOPs. However, their strength, depth, and onset time varied considerably just like the large-scale wind speed and direction, c