The diffusion term is given in the terrain-following coordinate system as 1F‾ϕ‾=∂∂x˙1α*ρ0u′ϕ′‾∂x˙1∂x+∂∂x˙2α*ρ0v′ϕ′‾∂x˙2∂y+∂∂x˙3α*ρ0u′ϕ′‾∂x˙3∂x+α*ρ0v′ϕ′‾∂x˙3∂y+α*ρ0w′ϕ′‾∂x˙3∂z with the grid volume α*, the basic state atmospheric density ρ0, the wind velocity components u, v, w, and Cartesian coordinates x, y, z. The fluxes are parameterised using a first-order closure 2-ρ0u′ϕ′‾=ρ0Khor∂ϕ‾∂x˙1∂x˙1∂x+∂ϕ‾∂x˙3∂x˙3∂x3-ρ0v′ϕ′‾=ρ0Khor∂ϕ‾∂x˙2∂x˙2∂y+∂ϕ‾∂x˙3∂x˙3∂y4-ρ0w′ϕ′‾=ρ0Kver∂ϕ′‾∂x˙3∂x˙3∂z

with the horizontal and vertical exchange coefficients Khor and Kver, respectively . Inserting the expressions for the subgrid-scale turbulent fluxes (Eqs. –) into the diffusion term (Eq. ) leads to the equations as used in the prior model version MITRAS v3.0 .

In the M-SYS model system , a Kessler-type parameterisation is applied . It is a three-category (water vapour q11, cloud water q12c, and rain water q12r) bulk water-continuity model designed for warm clouds . In Sect.  Fig. , the processes included in the warm scheme are shown in grey and black. Liquid water drops in the atmosphere are distinguished by their droplet size. Drops with a mean drop radius of about 10µm are considered cloud water, whereas drops with a mean radius of about 100µm are defined as rain water. The separation radius is 40µm . For rain drops, the Marshall–Palmer size distribution is assumed. A terminal velocity is the mass weighted mean of the individual sedimentation speeds. The following expression for the terminal velocity of rain, vTR, is used 5vTR=68.81⋅10-3⋅ρ0kg m-3⋅q12rkg kg-10.1905 which is taken from , recalculated to SI-units. To take the smaller densities at higher altitudes into account, a correction factor 6F=ρrefρ0 with the reference density ρref=1.29kg m-3 is included. The terminal velocity is thus larger for larger altitudes. The coagulation of cloud water drops leads to new rain water drops, which is called autoconversion. For the autoconversion process to start, enough cloud drops that are big enough to allow coagulation have to be present . The critical value is taken as q1,cri2c=10-3kg kg-1. Above the critical value, rain water production depends linearly on the cloud water content with the inverse autoconversion interval kwarmr=10-3s-1. Consequently, the autoconversion rate for the warm rain scheme is 7Bauw=max0,kwarmr⋅q12c-q1,cri2c .

Accretion is the growth of rain drops by collecting cloud drops. The parameterisation of this process is based on the continuous model for droplet growth. It assumes a uniform and continuous distribution of cloud drops, as well as that their radii are much smaller than the rain drop radii and that the cloud drop sedimentation speed is zero . This leads to the accretion equation 8Baccw=934.63⋅q12ckg kg-1⋅10-3⋅ρ0kg m-3⋅q12rkg kg-10.875 which is the original equation by converted to SI units.

The calculation of condensation and evaporation to and from cloud droplets is based on the method of saturation adjustment . It assumes that within a cloud, saturation is achieved. The sedimentation flux of cloud droplets is neglected. If the humidity exceeds the saturation specific humidity q1,sat1, condensation is 9Bcond=q11-q1,sat1αcond with the condensation parameter 10αcond=1+L21⋅q1,sat1⋅4028Kcp⋅T⋅p0pSRcp-38.33K2 with the latent heat of vapourisation L21, the specific heat for dry air cp, temperature T, basic state pressure p0, the ground surface pressure pS, and the gas constant for dry air R .

In the sub-saturated areas below the cloud, evaporation of rain water may occur. The evaporation is given as 11Bevap=Atkg m-3kg kg-1s-2⋅ρ0kg m-3×10-3⋅q12rkg kg-1⋅Fv⋅S⋅ρ0kg m-3×10-3-1 with saturation S and the parameter for the rain droplet spectrum 12At=2.623×10-3m3kg s2⋅ρ0×10-3⋅q1,sat11+1.282×1010m3K2kg⋅ρ0×10-3⋅q1,sat1θ-2 and the ventilation factor 13Fv=0.78+80.73⋅ρ0kg m-3×10-3⋅q12rkg kg-10.225 . The potential temperature is given as θ.

The influence of liquid water on radiation is included in the radiation parameterisation in MITRAS . There are two radiation schemes implemented: the two-stream approach and the vertically integrated approach, which does not consider clouds. Thus, only the two-stream approach can be applied when atmospheric liquid water is present. For the long wave radiation, cloud and rain water is included in the calculation of the absorption coefficient. For the short wave radiation, only small droplets like cloud water are taken into account when deriving scattering and absorption by liquid water .

The horizontal subgrid-scale fluxes depend on both the grid surface parallel gradient of ϕ (first terms in brackets on the right hand side in Eqs.  and ), and on the vertical gradients (second terms in brackets). The latter result from the transformation into the terrain-following coordinate system. However, the terrain-following coordinates create some numerical problems at obstacle walls. To explain these, the calculation of the diffusion in x-direction of e.g. liquid water content in a terrain-following grid cell will serve as an example (Fig. ). An obstacle cell (shaded grey) is assumed below and in x-direction of the grid cell for which the diffusion is calculated (thick black boundary). The diffusion in x-direction following Eqs. ()–() results in 14F‾ϕ‾,x˙1=-∂∂x˙1α*ρ0Khor∂ϕ‾∂x˙1∂x˙1∂x∂x˙1∂x+α*ρ0Khor∂ϕ‾∂x˙3∂x˙3∂x∂x˙1∂x.

The diffusion is the grid following gradient ∂/∂x˙1 of the fluxes through the vertical grid cell boundaries (purple and grey arrow in Fig. ). With a building at the right hand side of the grid cell, the flux is not calculated following Eq. (). Instead, a specific building surface flux is applied (Sect. ).

For the calculation of the fluxes between atmospheric grid cells, the grid following gradient ∂/∂x˙1 and vertical gradients of ϕ are required. The first term in brackets is represented as the violet arrow and the second as the blue arrow in Fig. . The calculation of the grid following ∂/∂x˙1 gradient of ϕ is straight forward using the values of ϕ located at the grid cell centres left and right from the grid cell boundary (crosses in the central and middle left grid cell in Fig. ). For the calculation of the vertical gradient, however, values of ϕ from all six grid cells that surround the grid cell boundary are used (two left columns of grid cells). This includes in the example provided in Fig.  two building grid cells (grey), which give no physically useful values. When the horizontal diffusion is calculated, only fluxes through vertical walls are taken into account.

In general, the orography in MITRAS with the resolution of a few metres is relatively flat. Therefore, the second term in brackets in Eq. (), which includes the slope of the terrain as ∂x˙3/∂x, is small. By neglecting the influence of terrain steepness in the horizontal flux, Eq. () simplifies to only the first term. Similar simplifications can be done for the diffusion in the y-direction. This leads to the following expressions for the horizontal turbulent fluxes: 15-ρ0u′ϕ′‾=ρ0Khor∂ϕ‾∂x˙1∂x˙1∂x16-ρ0v′ϕ′‾=ρ0Khor∂ϕ‾∂x˙2∂x˙2∂y.

Boundary fluxes had been defined before for all scalar quantities at obstacle surfaces, except for cloud, rain and snow water content. For MITRAS, those quantities are added and the treatment of building surface fluxes at obstacle surfaces is adapted for all scalar quantities.

Previously, building surface fluxes for ρ0u′ϕ′‾, ρ0v′ϕ′‾, and ρ0w′ϕ′‾ were defined as a three-dimensional variable located at scalar grid points. Wall orientation was taken into account in the calculation of the building surface fluxes, but when more than one building surface was present, only the value that has been calculated last was stored in the variable. For the example of a building edge as in Fig. , the same building surface flux was used for the vertical wall (grey arrow) as for the roof below (bottom boundary of central grid cell). Therefore, the structure of the building surface flux variables has been adjusted for all scalar quantities. Like other building surface variables (e.g. building surface temperature), a value is defined for each obstacle adjacent atmospheric grid cell and for each wall orientation. For the calculation of the pre-existing building surface fluxes, boundary conditions as described in and are applied.

For liquid water at the ground surface, the model allows for three surface boundary conditions: zero gradient, prescribed fixed value, and flux at the boundary equal to flux in the atmosphere above. For obstacle surfaces, the latter is chosen. Water and snow reaching a building close to the wall is considered to be absorbed by the adjacent surface. Considering again the configuration in Fig. : In order to get the building surface flux at the obstacle wall in positive x-direction of the atmospheric grid cell (grey arrow), the flux at the opposite grid cell boundary is used, calculated following Eqs. () and () . The same approach is applied for other wall directions and for cloud, rain and snow water content.

Without snow, the change of temperature at ground surface, TS, with time t is calculated following Eq. () in MITRAS considering net short and long wave radiation SW_net and LW_net, sensible and latent heat fluxes HS and LS, heat flux to and from the soil at the surface Gsoil (Eq. ), using thermal diffusivity ksoil, thermal conductivity νsoil, and deep soil temperature Th,soil at the depth hsoil. 17∂TS∂t=B*SWnet+LWnet+HS+LS+Gsoil with 18B*=2πksoilνsoilhsoil

The soil heat flux Gsoil (Eq. ), is expressed using the force-restore method . The deep soil temperature can be assumed constant for shorter time ranges (< 3 d). 19Gsoil=-πhsoilνsoil-1TS(t)-Th,soil

In case of snow on ground, an additional snow layer is assumed, which impacts the temperature on and near the ground surface as described for METRAS in . The treatment of the snow layer is based on . The thermal diffusivity of snow ksnow is given in Eq. () using the snow volumetric heat capacity cv,snow (Eq. ). The snow thermal conductivity νsnow (Eq. ) and volumetric heat capacity cv,snow both depend on the density of the snow pack ρsnow (Eq. ). The snow thermal conductivity, the snow volumetric heat capacity, and the depth of the temperature wave into the snow are given in Eqs. ()–() using the specific heat capacity of ice cice, the density of ice ρice, the thermal conductivity of ice νice, and the period of the temperature wave τ=86400s=1d. 20ksnow=νsnowcv,snow21cv,snow=cice⋅ρsnowρice22νsnow=νice⋅ρsnowρw1.8823hsnow=τ⋅νsnowcv,snow

The snow depth zsnow (Eq. ) is calculated using the snow water equivalent SWE (Eq. ) and the density of water ρw. 24zsnow=SWE⋅ρwρsnow

Two cases form limit value situations which are to be treated as follows: In case of a shallow snow cover, meaning, the depth of the temperature wave into the snow hsnow (Eq. ) exceeds the snow depth zsnow (Eq. ), the heat conduction of snow cover and snow soil heat flux can be expressed with Eqs. () and () using Eq. (). 25B*=2π1cv,snowzsnow26Gsoil=-πzsnowνsnow+hsoilνsoil-1⋅TS(t)-Th,soil.

In case of a very thick snow cover, the heat wave does not reach below the snow and hsoil becomes zero leading to a soil heat flux of Eq. () . The heat conduction B* is calculated following Eq. () but using the temperature depth in snow hsnow instead of the snow depth zsnow. 27Gsoil=-πhsnowνsnow-1TS(t)-Th,soil

As a snow pack ages, its density (Eq. ) increases. assumes an asymptotic solution with time from a minimum density ρmin to a maximum density ρmax with the empirical parameters τf and τ1 and time step Δt following , , and . 28ρsnow(t+Δt)=ρsnow(t)-ρmax⋅exp-τfΔtτ1+ρmax

The parameters were chosen according to with ρmin=100kg m-3, ρmax=300kg m-3, τf=0.24 and τ1=86400s.

For the snow albedo (Sect. ), the parameters suggested by were chosen in the implementation of the obstacle-resolving microscale model over those suggested by and , as they fit observations in an urban area better by considering anthropogenic pollution. For the snow density, however, we decided to keep the parameters suggested by . The parameterisation in MITRAS is supposed to represent a snow event in a city like Hamburg (Germany). The parameters suggested by fit well for snow climate cities like Montreal or Helsinki, but they do not necessarily fit equally well for Hamburg, where larger snow packs are rare.

The snow water equivalent SWE (Eq. ) with the unit of metres represents the mass of snow using an equivalent water height. In contrast, the height of the snow pack is given in Eq. (). The snow water equivalent is reduced by evaporation E and melting Msnow (Eq. ). The rate of snowfall Pr_snow adds to it . 29∂SWE∂t=Prsnow-E-Msnow

In , no precipitating snow is calculated. Rain reaching the ground is assumed to be snow, if the surface temperature is below the freezing point T0. In MITRAS, precipitating snow is calculated (Sect. ) and considered in the rate of snowfall. For simplicity, rain is assumed to be snow on ground for surface temperatures below the freezing point. Similarly, precipitating snow reaching the ground for temperatures above freezing point, is assumed to be rain in the calculation of the soil water content. Diffusive fluxes into the ground are only possible in the absence of a snow cover. As a consequence, for air temperatures above freezing point, with both snow and rain falling, the snow water equivalent might be overestimated, if the surface temperatures are below the freezing point. Snow melt (Eq. ) is calculated following using the latent heat of fusion L32. 30Msnow=νsnowρwL32zsnowTS-T0

The roughness length of snow-covered areas is reduced compared to snow-free areas as a snow pack smooths a surface. The roughness length z0 under the influence of snow (Eq. ) is calculated using the snow roughness length z0snow=10-3m, the roughness length of areas without snow cover z0ini, and the snow cover fraction psnowz0 (Eq. ) following . 31z0=(1-psnowz0)⋅z0ini+psnowz0⋅z0snow

The parameterisation of the snow cover fraction (Eq. ) is based on with the empirical factor β=0.408. 32psnowz0=SWESWE+β⋅z0ini

For now, the influence of snow cover on the roughness length of roofs is neglected. For obstacle surfaces including roofs, the roughness length for concrete (z0=10-3m) is assumed regardless of snow cover.

If there is already a snow pack present, the albedo α of the ground surface (Eq. ) is increased to a maximum albedo αmax after one hour of snowfall with the magnitude 0.01m h-1, or an equivalent value is used, e.g. with a higher magnitude for 0.01 m of snow in a shorter time . The amount of snowfall is represented by the change of the snow water equivalent ΔSWE. 33α(t+Δt)=α(t)+min1,ΔSWE⋅3600sΔt⋅0.01m⋅αmax-α(t)

found, that a minimum snow depth of 5cm is required to completely mask the albedo of the underlying soil. The effect of the surface covers shining through the snow surface is included in MITRAS. However, in a city like Hamburg, which aims at black roads in winter, snow rarely remains untouched because of winter services and traffic, which means the albedo of the underlying soil (αini) shines through for snow depths greater than 5cm. These effects of direct human activities are included by using a basic approach. The underlying albedo is considered until a snow depth of 0.5m is reached, then a snow cover of fresh snow is assumed. This is represented by a linear relation (Eq. ) using the critical snow water equivalent SWEcrit=0.05m, which corresponds a snow depth of 0.5m (Eq. ). In MITRAS, Eq. () is used for snow albedo in case of snowfall with snow water equivalent higher than SWE_crit. Equation () is used for values below SWE_crit with and without snowfall because the impact of the underlying soil is assumed to be larger than the impact of aging of the snow pack. 34α=αini+min1,SWESWEcrit⋅αmax-αini

Without snowfall and a snow water equivalent greater than SWE_crit, the albedo is simultaneously decreased due to the aging of the snow pack. For temperatures below the freezing point, a linear decrease of the albedo to the minimum albedo αmin is assumed (Eq. ) and if it is warmer, an exponential decrease (Eq. ) is assumed using the empirical factors τα and τf,α following . 35α(t+Δt)=α(t)-τα⋅Δtτ1for TS<273.16K36α(t+Δt)=α(t)-αmin⋅exp-τf,α⋅Δtτ1+αminfor TS>273.16K

For the mesoscale model METRAS, the parameters provided by are applied with αmin,V91=0.5, τα,V91=0.008, and τf,α,V91=0.24 . However, the albedo in urban areas is generally lower than in rural areas mainly due to pollution. assessed and evaluated parameters in a snow scheme for two cold climate cities (Helsinki and Montreal) and suggested the parameters: αmin,J14=0.18, τα,J14=0.018, and τf,α,J14=0.11. Both and assume a maximum snow albedo of 0.85. In Fig. , the decreasing albedo as described with Eqs. () (blue lines) and () (black lines) is shown for the parameters used in METRAS after (solid lines) and the parameters based on in MITRAS (dashed lines). According to , their suggested parameters fit well with observations in an urban area, which is why their parameters were chosen for MITRAS as well.

The terminal velocity for snow of 42vTS=4.82⋅10-3⋅ρ0kg m-3⋅q13skg kg-10.075 is introduced. Smaller densities at higher altitudes again need to be taken into account (compare with Eq. ). The terminal velocity of snow is lower than that of rain with a maximum value of 2.2m s-1 (blue line in Fig. ).

In general, precipitation rates and amounts are only given at ground surface in atmospheric models. In MITRAS, however, these precipitation quantities should be given on roofs as well. Precipitation quantities on roofs are calculated similar to the precipitation variables on ground . For the sedimentation fluxes (first terms in Eqs. and ) in the atmosphere, the fluxes of rain and snow water content through the grid cell boundaries are calculated using the terminal velocity defined at vector points and rain and snow water content defined at grid cell centres. For the precipitation quantities at ground, a terminal velocity for the scalar point just above ground is calculated. The precipitation quantities are then derived from the flux of rain and snow above surface. In case of a roof, the terminal velocity is calculated similarly, thus using the scalar value of the terminal velocity just above the roof.

The development of snow from cloud water by subsequent diffusional growth is named nucleation. For the one-category ice scheme in MITRAS v3.3, a temperature dependence is considered: 43ϵ(T)=0if T≥T00.51+sinπ⋅0.5T0+T2-TT0-T2if T2<T<T01if T≤T2.

According to , it is based on observations of the frequency distribution of water and ice in mixed-phase clouds. ϵ(T) is one below the minimum temperature T2=235.16K and zero above the freezing point (T0). Note, that the conversion coefficient for Kelvin in MITRAS is 273.16 and not 273.15 as in .

The conversion rates are given as 44Bauc=max(0,kcoldr⋅1-ϵ(T)⋅q12c-q1,cri2c45Bnuc=max(0,kcolds⋅ϵ(T)⋅q12c-q1,cri2c with the autoconversion interval for rain kcoldr=10-4s-1 and for snow kcolds=10-3s-1 . Note, that unlike in , a nonzero value for q1,cri2c is chosen in MITRAS. In Fig.  the autoconversion rates for warm clouds (Eq. , red) and mixed phase clouds (Eq. , blue) are shown as well as the nucleation rates (Eq. , black). All processes increase with increasing cloud water contents. While the autoconversion in the warm rain scheme only depends on the cloud water content, the other processes additionally depend on temperature. When mixed-phase clouds are assumed, less rain water is created by autoconversion than in warm clouds for the same cloud water content. Below T2=235.16K the nucleation of snow is highest and the autoconversion is zero. The nucleation gradually decreases and the autoconversion increases with higher temperatures until the freezing point is reached.

For riming, the continuous model for particle growth by collection is applied. Snow particles have no spherical shapes. Instead, they are assumed to be rimed aggregates of crystals and they have the form of thin circular plates. The temperature dependent mass-size relation of snow is defined as 46am(T)=amc-amv1+cos2πT-0.5(T0+T1)T0-T1if T0>T>T1amcelse with the constant parameters amc=0.08kg m-2 and amv=0.02kg m-2 and the temperature T1=253.16K .

For the accretion term (Eq. ) in the one-category ice scheme, the temperature dependency (Eq. ) is considered: 47Baccc=(1-ϵ(T))⋅934.63⋅q12ckg kg-1⋅ρ0kg m-3×10-3⋅q12rkg kg-10.87548Brim,MITRAS=am(T)kg m-2-1⋅3307.24⋅q12ckg kg-1⋅ρ0kg m-3×10-3⋅q13skg kg-11.075if T<T00ifT≥T0.

Otherwise it remains the same as in MITRAS v3.0. Note, that uses different parameters for the accretion.

In Fig. , the accretion rates for the warm rain parameterisation used in MITRAS v3.0 (grey) and the mixed-phase cloud parameterisation as used in MITRAS v4.0 (light blue) is shown as well as the riming rates (dark blue). Accretion increases for rain for temperatures above T2 (Eq. ) reaching the same value as warm clouds at the freezing point T0. The riming rate is highest at -10 °C, as the largest snowflakes can be found at -10 °C (Eq. ). Snow production is higher than rain production for the same initial snow respectively rain amount. At temperatures below the freezing point T0, snow is produced by riming. If it is warmer, rain water content is produced by shedding. The equation is the same as riming (Eq. ), but it produces rain water and not snow.

With the inclusion of snow, depositional growth (diffusion growth of snow particles) and sublimation occur, which is according to given as 49Bdep=αdepam(T)0.51+βdepam(T)0.25ρ0q13s0.225q11-q1,sat,ice1ρ0q13s0.625 with the factors 50αdep=1.09×10-3-3.34×10-5T-T0 and βdep=13.0. q1,sat,ice1 denotes the saturation specific humidity over ice. For better readability, the units of αdep and βdep are not provided here. They can be found in .

The melting rate is derived similarly to the evaporation and deposition rates leading to 51Bmelt=αmeltamc0.51+βmeltamc0.25ρ0q13s0.225T-T0ρ0q13s0.625 with the factors αmelt=7.2×10-6 and βmelt=13.0 .

Rain drops can be activated as ice nuclei due to various drop impurities (immersion freezing), this is represented in the model as 52Bifrz=αifeαif(T0-T)-1ρ0q12r1.75 with the parameter αif=9.95×10-5 .

The process of falling rain drops collecting ice nuclei (contact freezing nucleation) is represented as 53Bcfrz=αcfEcfNcf,0(270.17-T)1.3(ρ0q12r)1.625if T<270.17K0if T≥270.17K

with the parameter αcf=1.55×10-3, the collection efficiency Ecf=5.0×10-3, and the concentration of natural contact ice nuclei active at -4 °C at sea level Ncf,0=2.0×105 . Again, the units can be found in .

All simulations are performed for the same model domain (Sect. ). For the plausibility tests in Sect. , simulations with three different initial surface level temperatures (“C” for cold, “W” for warm, “H” for hot in Table ) are performed. In Fig. , the initial temperature profiles are provided in black. For the profile, a potential temperature gradient of 0.001K m-1 is assumed. For all simulations except Winit and Wwr_noprecip, nonzero initial cloud water contents (blue solid line in Fig. ) are prescribed which lead to heavy precipitation. The initial surface level pressure is 990hPa, the initial wind at 150m a.g.l. is from west (2m s-1). Due to Coriolis force effects, the wind direction at 10m height is south west with a friction reduced wind speed of 1.2m s-1. The initial relative humidity is set to 60% at ground and reduces to 30% at the top of the model domain (5km) yielding the specific humidity profiles in Fig.  (dash-dotted, dashed, dotted blue lines). At the lateral boundaries and at the upper boundary a gradient of zero is assumed except for boundary normal winds, where boundary values are either calculated (lateral) or large-scale values are prescribed (top). At the bottom boundary for wind fixed values are prescribed. For temperature and specific humidity, the surface energy budget is calculated (Sect. ) and for cloud, rain, and snow water content, the flux at the boundary equals the flux in the model (Sect. ).

The plausibility tests are performed using hit rates (Sect. ). To determine them, allowed uncertainty ranges have to be found for the meteorological variables. This is based on published methods to asses obstacle-resolving model performance and is done for variables by comparing model results with and without parallelisation (Wwr and Wwr_np in Table ).

The model domain is small to ensure a fast integration. However, it includes orography, slanted roofs, obstacle corners and different surface cover classes to assess, if the model can represent the different features for a realistic urban area. The domain extends 240m×210m horizontally and 6400m vertically with an equidistant area of 3m resolution in the middle (Fig. ). The grid increases towards the lateral boundaries to a maximum horizontal resolution of 10m. The maximum vertical resolution of 500m is achieved at 3km a.g.l. Typically, domains of obstacle-resolving models extend only few hundred metres vertically . However, with interest in cloud and precipitation development in the influence of a building, the upper level is chosen high enough to ensure a vertical extension of clouds is not hindered by a too low model top. Grid sizes increase in horizontal as well as vertical direction by a factor of 1.175. The equidistant area ranges from -28 to 26 m in the x-direction and -16 to 18m in the y-direction and from the ground up to 32m.

A single building with roof heights at 15 and 10m, and a size of 45m by 45m (Fig. ) including edges and slanted surfaces is placed on top a small mound. This setup resembles a terp, which is a North European form of housing, where an artificial mound protects the building from flooding. At the boundaries is cropland. The pavement and the streets are made of asphalt. At obstacles, the surface cover class is concrete (Fig. ). The simulations start at 7:30 and finish at 8:32. Within this time span, a precipitation event takes place and the sun rises. The focus area of the analyses lies in the centre of the domain (dashed lines in Figs.  and ).

The assessment, whether model results are identical or different is based on a method described in which uses hit rates. Hit rates have been applied in the past for assessing microscale models e.g.. as well as for other model scales (e.g. weather forecast , mesoscale air quality and meteorology models ). The hit rate q is defined as: 54q=1N∑i=1Nni with N being the total number of compared values and ni equal 1 (hit) or 0 (fail) depending on the deviation of model results and comparison data: 55ni=1if Pdi-OiOi≤D or Pdi-Oi≤W0else.

For every atmospheric grid cell at location i, the wind speed is assessed per component (u, v, w). The comparison results in a hit (ni=1), if the relative deviation does not exceed the threshold value, D, or the absolute deviation remains below the corresponding threshold W. When comparing model results to observational data, Pd_i denotes the predicted and Oi the observed values. In the present study, results of different model versions are compared. Pd_i denotes the newer model version and Oi the older version. Following and , a hit rate of q≥95% between two simulation results are considered similar. There, the threshold values for absolute and relative deviations, W and D, for the wind speed components speeds are WVDI=0.01m s-1 and DVDI=0.05. Wind speed components are normalised with the reference wind speed Uref for the test cases. In the current study, the initial wind speed of 2m s-1 is used as reference wind speed. As non-normalised values are compared, W has to be adjusted to W=WVDI⋅Uref=0.02m s-1 (Table ). For the relative deviation, DVDI is assumed for D for each of the wind speed components (Table ). It should be noted that these values are about a factor of 10 smaller than the required measurement uncertainties for wind speed given in .

In this study, not only the results for the components of the wind vector but also the results for temperature, long and short wave radiation, and precipitation amount on ground P are assessed. The W and D values for these meteorological variables are derived by comparing the results of MITRAS v3.0 with one running in parallel processing mode and the other one with single processor use (with and without parallelisation; Wwr and Wwr_np) after one hour of simulation. This measure is taken, since the same model might yield different results depending on compilers, installed packages, and hardware. Therefore, hit rates below 100% can occur (e.g. 98% for v, Table ). As a hit rate above 95 % means the results lie within the required accuracy, Wwr and Wwr_np can be considered identical (Table ).

With computational accuracy a strict criterion is applied for temperature, radiation, and precipitation. The resulting W and D values are consistent with the allowed uncertainty values for the wind speed components, as they also are about a factor of 10 smaller than given in . For example, the achievable uncertainty for temperature suggested by is 0.2K and here 0.05K is chosen. For long and short wave radiation 1/10th of the values for direct solar radiation 4–6 W m⁻² is used. The allowed absolute deviation of 0.001mm for precipitation is well below the accuracy of commonly used rain gauges 0.1 mm,. This strict value is chosen because precipitation is the target value of our model extensions. Low hit rates give the impression of larger differences. It should be noted that the dependency of the hit rate on the allowed deviations is a shortcoming of this validation metric as discussed in . We use this dependency to our advantage. The comparison of results of different model versions (e.g. with/without parallelisation, code extensions used with values of zero for the variables) should lead to very similar results. In contrast, extensions of a model with new or changed process descriptions and nonzero values for variables should provide different model results and thus small hit rates.