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**Geoscientific Model Development**
An interactive open-access journal of the European Geosciences Union

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**Model description paper**
24 Aug 2018

**Model description paper** | 24 Aug 2018

The microscale obstacle-resolving meteorological model MITRAS v2.0: model theory

^{1}Meteorological Institute, CEN, University of Hamburg, Hamburg, Germany^{a}now at: Faculty of Energy Engineering, Aswan University, Aswan, Egypt^{b}now at: Finnish Meteorological Institute, Marine Research, Helsinki, Finland^{c}now at: Met Office, Exeter, UK

^{1}Meteorological Institute, CEN, University of Hamburg, Hamburg, Germany^{a}now at: Faculty of Energy Engineering, Aswan University, Aswan, Egypt^{b}now at: Finnish Meteorological Institute, Marine Research, Helsinki, Finland^{c}now at: Met Office, Exeter, UK

**Correspondence**: Mohamed Hefny Salim (mohamed.salim@uni-hamburg.de)

**Correspondence**: Mohamed Hefny Salim (mohamed.salim@uni-hamburg.de)

Abstract

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This paper describes the developing theory and underlying processes of the microscale obstacle-resolving model MITRAS version 2. MITRAS calculates wind, temperature, humidity, and precipitation fields, as well as transport within the obstacle layer using Reynolds averaging. It explicitly resolves obstacles, including buildings and overhanging obstacles, to consider their aerodynamic and thermodynamic effects. Buildings are represented by impermeable grid cells at the building positions so that the wind speed vanishes in these grid cells. Wall functions are used to calculate appropriate turbulent fluxes. Most exchange processes at the obstacle surfaces are considered in MITRAS, including turbulent and radiative processes, in order to obtain an accurate surface temperature. MITRAS is also able to simulate the effect of wind turbines. They are parameterized using the actuator-disk concept to account for the reduction in wind speed. The turbulence generation in the wake of a wind turbine is parameterized by adding an additional part to the turbulence mechanical production term in the turbulent kinetic energy equation. Effects of trees are considered explicitly, including the wind speed reduction, turbulence production, and dissipation due to drag forces from plant foliage elements, as well as the radiation absorption and shading. The paper provides not only documentation of the model dynamics and numerical framework but also a solid foundation for future microscale model extensions.

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Salim, M. H., Schlünzen, K. H., Grawe, D., Boettcher, M., Gierisch, A. M. U., and Fock, B. H.: The microscale obstacle-resolving meteorological model MITRAS v2.0: model theory, Geosci. Model Dev., 11, 3427–3445, https://doi.org/10.5194/gmd-11-3427-2018, 2018.

1 Introduction

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The urban boundary layer is considerably influenced by its surface characteristics. Within the canopy layer, atmospheric flow is disturbed by buildings and other obstacles located at the surface and hence all related atmospheric processes (Meng, 2015). This creates a complex three-dimensional, time-dependent flow, temperature, and humidity field. Studying the atmospheric boundary layer flow and its interactions with complex terrain in, e.g., urban areas is a complex problem for both the meteorological and engineering communities. Field experiments are one approach (e.g., Schafer et al., 2005); however, field measurements have a low spatial representativeness and largely depend on the turbulence structure within the urban area and the wind direction fluctuations. This implies that long averaging times are needed in order to obtain reasonable time-averaged values; on the other hand, long averaging times are not feasible because the atmospheric situation changes due to, e.g., the diurnal cycle or synoptic changes. Also, investigating future scenarios is not possible from field measurements. Subsequently, laboratory experiments in controlled conditions (wind tunnel) are used to overcome these problems after matching the major similarity parameters with the full-scale model. These physical models can be very detailed (Harms et al., 2011) and can provide comparison data for numerical models and field experiments (VDI, 2015). However, wind tunnel modeling is mostly limited to neutral atmospheric stratification due to the requirement of similarity to nature. Furthermore, it is sometimes difficult to satisfy the atmospheric boundary conditions and to resemble important features of the Earth system such as the Coriolis force effect. Alternatively, high-resolution numerical computer models are frequently used to simulate urban areas.

Numerical modeling of wind flow and pollutant dispersion in urban areas is a challenging task due to the geometrical variety of buildings. It inevitably involves impingement and separation regions, a multiple vortex system with building wakes, and jet effects in street canyons (Murakami et al., 1999). Furthermore, the neighbor buildings add their own impacts on the urban meteorology, resulting in interacting flow and dispersion patterns. Due to this complexity, explicit resolving of the buildings is necessary instead of only implicitly considering building effects by using surface roughness parameterizations. This gave rise to the development of obstacle-resolving microscale meteorological models such as PALM (Maronga et al., 2015), ASMUS (Gross, 2012), ENVI-met (Bruse and Fleer, 1998; Müller et al., 2014), MISKAM (Eichhorn, 1989; Eichhorn and Kniffka, 2010), MUKLIMO (Früh et al., 2011), MITRAS (Schlünzen et al., 2003; Salim et al., 2011), and OpenFOAM (Franke et al., 2012). These models are now widely used for environmental and engineering studies.

The microscale obstacle-resolving transport and stream model MITRAS is part of the M-SYS model system (Trukenmuller et al., 2004; Schatzmann et al., 2006). This model system is designed to investigate pollution transport, chemical reactions, and atmospheric phenomena in the atmospheric boundary layer. The obstacle-resolving MITRAS model calculates wind, temperature, humidity fields, cloud, and rainwater, as well as tracer transport within the obstacle layer. The model has been applied for more than 10 years; however, an overall description of the model theory has not been published in a refereed journal. This is timely because computers now allow for time-dependent long-term integration of the temperature and humidity equations in high resolution. In addition, MITRAS in its version 2 was extended and optimized for more realistic applications in urban areas (Salim et al., 2011; Röber et al., 2013). Specifically, more surface cover classes were added to better describe surface characteristics: fine tuning the code structure for maximum parallelization to make it faster and able to simulate larger domains and parameterizing the additional radiation, turbulence production and dissipation due to wind turbines, and urban vegetation.

Model validations of MITRAS-01 have been performed in comparison to wind tunnel data (Schlünzen et al., 2003; Grawe et al., 2013). MITRAS 2 is evaluated using the VDI guideline for obstacle-resolving microscale models (Grawe et al., 2015). The results will be presented in a separate paper.

Equations and the solution method will be described in Sect. 2, including the turbulence parameterization (Sect. 2.2) and numerical treatment (Sect. 2.3). Further sub-grid-scale processes need to be parameterized, even in a very highly resolving atmospheric model like MITRAS. This concerns cloud microphysical processes and radiation. Both are calculated with the same parameterizations as its sister model METRAS (Schlünzen, 2003; Schlünzen et al., 2018b). The boundary conditions, including surface, lateral, and top boundaries, are given in Sect. 3. The treatment of obstacle-induced effects is described in Sect. 4, including wind, shading, and heat transfer effects. MITRAS parameterizes momentum and heat fluxes on obstacle surfaces dependent on the local roughness length (Sect. 4.1, 4.2) and explicitly resolves obstacles such as buildings, including overhanging obstacles (e.g., bridges or overpasses), trees, and wind turbines, to account for its aerodynamics and thermodynamic effects. The handling of wind turbines in the model and their effects is described in Sect. 4.3. Vegetation effects, especially their effect on radiation, are given in Sect. 4.4.

2 Model theory

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MITRAS is based on the physical conservation equations, specifically the Navier–Stokes equations, the continuity equation, and the conservation equation for scalar quantities such as potential temperature and humidity. This set of equations is written in flux form, transformed in a terrain-following coordinate system and filtered before it is used in MITRAS.

The equations of MITRAS in flux form are transformed in a three-dimensional nonuniform terrain-following coordinate system ${\dot{x}}^{\mathrm{1}}$, ${\dot{x}}^{\mathrm{2}}$, ${\dot{x}}^{\mathrm{3}}$ so that the lower boundary conforms to the terrain. The vertical coordinate is defined by

$$\begin{array}{}\text{(1)}& {\dot{x}}^{\mathrm{3}}={z}_{t}{\displaystyle \frac{z-{z}_{s}\left(x,y\right)}{{z}_{t}-{z}_{s}\left(x,y\right)}}.\end{array}$$

Here *z*_{s}(*x*,*y*) is the orography height and *z*_{t} is the domain
height. The terrain-following coordinates ensure an easy specification of the
boundary conditions over orography and eases nesting of MITRAS in METRAS due
to the use of the same vertical coordinate transformation. The
transformation used allows for grid stretching in all three directions to keep a
high resolution in the focus area of the domain while allowing some distance
between the open boundaries (lateral and top) and the area of interest. In
addition, the coordinate system can be rotated against north in any desired
angle, allowing for additional flexibility. Figure 1 shows an example of a
vertical cross section in this terrain-following coordinate system.

Reynolds averaging is used to filter the equations after the coordinate
transformation: the atmospheric state variables are divided into a value
$\stackrel{\mathrm{\u203e}}{\mathit{\psi}}$, which is the average over a finite time, Δ*τ*,
and grid volume, $\mathrm{\Delta}x\cdot \mathrm{\Delta}y\cdot \mathrm{\Delta}z$, and its deviation
*ψ*^{′} (Pielke, 2002). The deviations are assumed to be zero when averaged
over Δ*τ*. This assumption is reflected in the choice of
parameterizations for sub-grid-scale (SGC) turbulent fluxes (Sect. 2.2).
Performing the filtering operation after the coordinate transformation
ensures that all possible fluctuations are included with their correct
contravariant and covariant components. The average $\stackrel{\mathrm{\u203e}}{\mathit{\psi}}$ is
further decomposed for temperature, humidity, concentration, pressure, and
density into a large-scale part *ψ*_{0}, the so-called basic state, and a
microscale deviation $\stackrel{\mathrm{\u0303}}{\mathit{\psi}}$. The basic state is intentionally chosen
to be steady state and fulfill basic concepts of meteorology. For example,
the base state pressure *p*_{0} is chosen so that it satisfies the
hydrostatic approximation and the vertical gradient is thus balancing
*g**ρ*_{0}. This ensures a higher order of accuracy when solving the
equations, especially for the vertical wind. *ψ*_{o} represents a domain
average value using the values at the same height above sea level:

$$\begin{array}{}\text{(2)}& {\mathit{\psi}}_{o}={\displaystyle \frac{\mathrm{1}}{\mathit{\delta}x\mathit{\delta}y}}\underset{{y}_{\text{a}}}{\overset{{y}_{\text{a}}+\mathit{\delta}y}{\int}}\underset{{x}_{\text{a}}}{\overset{{x}_{\text{a}}+\mathit{\delta}x}{\int}}\stackrel{\mathrm{\u203e}}{\mathit{\psi}}\text{d}x\text{d}y,\end{array}$$

where *y*_{a} and *x*_{a} are the coordinate of a corner
of the model and *δ**x**δ**y* the domain size in the *x* and
*y* direction.

To increase the numerical accuracy further, the pressure deviation,
$\stackrel{\mathrm{\u0303}}{p}=p-{p}_{\mathrm{0}}-{p}^{\prime}$, is decomposed into a quasi-hydrostatic pressure
*p*_{1} (in balance with $g\stackrel{\mathrm{\u0303}}{\mathit{\rho}}$) and a more or less dynamically
impacted part *p*_{2}, i.e., $\stackrel{\mathrm{\u0303}}{p}={p}_{\mathrm{2}}+{p}_{\mathrm{1}}$. So the pressure can be
expressed as $p=\stackrel{\mathrm{\u0303}}{p}={p}_{\mathrm{0}}+{p}_{\mathrm{1}}+{p}_{\mathrm{2}}+{p}^{\prime}$. Here ${p}^{\prime}\ll \stackrel{\mathrm{\u0303}}{p}$ so
*p*^{′} is neglected.

The Boussinesq approximation is used, and thus density variations in the
Navier–Stokes equations are neglected except for the buoyancy term. *p*_{2}
is determined from an elliptic differential equation, which is derived from
the anelastic approximation of the continuity equation, resulting in a
decoupling of pressure and density in the model and hence no sound wave
propagation. In addition to the above equations, the ideal gas law and the
equation for the potential temperature are solved diagnostically to couple
the thermodynamic and aerodynamics of the model.

The solved prognostic equations in MITRAS are as follows.

$$\begin{array}{ll}\text{(3)}& {\displaystyle \frac{\partial {\mathit{\rho}}_{\mathrm{0}}{\mathit{\alpha}}^{*}\stackrel{\mathrm{\u203e}}{u}}{\partial t}}=& {\displaystyle}-{\displaystyle \frac{\partial}{\partial {\dot{x}}^{\mathrm{1}}}}\left(\stackrel{\mathrm{\u203e}}{u}{\dot{x}}_{x}^{\mathrm{1}}{\mathit{\rho}}_{\mathrm{0}}{\mathit{\alpha}}^{*}\stackrel{\mathrm{\u203e}}{u}\right)-{\displaystyle \frac{\partial}{\partial {\dot{x}}^{\mathrm{2}}}}\left(\stackrel{\mathrm{\u203e}}{v}{\dot{x}}_{y}^{\mathrm{2}}{\mathit{\rho}}_{\mathrm{0}}{\mathit{\alpha}}^{*}\stackrel{\mathrm{\u203e}}{u}\right){\displaystyle}& {\displaystyle}-{\displaystyle \frac{\partial}{\partial {\dot{x}}^{\mathrm{3}}}}\left(\stackrel{\mathrm{\u203e}}{{\dot{u}}^{\mathrm{3}}}{\mathit{\rho}}_{\mathrm{0}}{\mathit{\alpha}}^{*}\stackrel{\mathrm{\u203e}}{u}\right)-{\mathit{\alpha}}^{*}{\dot{x}}_{x}^{\mathrm{1}}\left({\displaystyle \frac{\partial {p}_{\mathrm{1}}}{\partial {\dot{x}}^{\mathrm{1}}}}+{\displaystyle \frac{\partial {p}_{\mathrm{2}}}{\partial {\dot{x}}^{\mathrm{1}}}}\right)\\ {\displaystyle}& {\displaystyle}-{\mathit{\alpha}}^{*}{\dot{x}}_{x}^{\mathrm{3}}{\displaystyle \frac{\partial {p}_{\mathrm{2}}}{\partial {\dot{x}}^{\mathrm{3}}}}+\stackrel{\mathrm{\u0303}}{\mathit{\rho}}{\mathit{\alpha}}^{*}g{\dot{x}}_{x}^{\mathrm{3}}{\displaystyle \frac{\partial z}{\partial {\dot{x}}^{\mathrm{3}}}}+f{\mathit{\rho}}_{\mathrm{0}}{\mathit{\alpha}}^{*}\left(\stackrel{\mathrm{\u203e}}{v}-{V}_{\text{g}}\right)\\ {\displaystyle}& {\displaystyle}-{f}^{\prime}{d}^{\prime}{\mathit{\rho}}_{\mathrm{0}}{\mathit{\alpha}}^{*}\stackrel{\mathrm{\u203e}}{w}-{\stackrel{\mathrm{\u203e}}{F}}_{\mathrm{1}}\end{array}$$

$$\begin{array}{ll}\text{(4)}& {\displaystyle \frac{\partial {\mathit{\rho}}_{\mathrm{0}}{\mathit{\alpha}}^{*}\stackrel{\mathrm{\u203e}}{v}}{\partial t}}=& {\displaystyle}-{\displaystyle \frac{\partial}{\partial {\dot{x}}^{\mathrm{1}}}}\left(\stackrel{\mathrm{\u203e}}{u}{\dot{x}}_{x}^{\mathrm{1}}{\mathit{\rho}}_{\mathrm{0}}{\mathit{\alpha}}^{*}\stackrel{\mathrm{\u203e}}{v}\right)-{\displaystyle \frac{\partial}{\partial {\dot{x}}^{\mathrm{2}}}}\left(\stackrel{\mathrm{\u203e}}{v}{\dot{x}}_{y}^{\mathrm{2}}{\mathit{\rho}}_{\mathrm{0}}{\mathit{\alpha}}^{*}\stackrel{\mathrm{\u203e}}{v}\right){\displaystyle}& {\displaystyle}-{\displaystyle \frac{\partial}{\partial {\dot{x}}^{\mathrm{3}}}}\left(\stackrel{\mathrm{\u203e}}{{\dot{u}}^{\mathrm{3}}}{\mathit{\rho}}_{\mathrm{0}}{\mathit{\alpha}}^{*}\stackrel{\mathrm{\u203e}}{v}\right)-{\mathit{\alpha}}^{*}{\dot{x}}_{y}^{\mathrm{2}}\left({\displaystyle \frac{\partial {p}_{\mathrm{1}}}{\partial {\dot{x}}^{\mathrm{2}}}}+{\displaystyle \frac{\partial {p}_{\mathrm{2}}}{\partial {\dot{x}}^{\mathrm{2}}}}\right)\\ {\displaystyle}& {\displaystyle}-{\mathit{\alpha}}^{*}{\dot{x}}_{y}^{\mathrm{3}}{\displaystyle \frac{\partial {p}_{\mathrm{2}}}{\partial {\dot{x}}^{\mathrm{3}}}}+\stackrel{\mathrm{\u0303}}{\mathit{\rho}}{\mathit{\alpha}}^{*}g{\dot{x}}_{y}^{\mathrm{3}}{\displaystyle \frac{\partial z}{\partial {\dot{x}}^{\mathrm{3}}}}-f{\mathit{\rho}}_{\mathrm{0}}{\mathit{\alpha}}^{*}\left(\stackrel{\mathrm{\u203e}}{u}-{U}_{\text{g}}\right)\\ {\displaystyle}& {\displaystyle}+{f}^{\prime}d{\mathit{\rho}}_{\mathrm{0}}{\mathit{\alpha}}^{*}\stackrel{\mathrm{\u203e}}{w}-{\stackrel{\mathrm{\u203e}}{F}}_{\mathrm{2}}\end{array}$$

$$\begin{array}{ll}\text{(5)}& {\displaystyle \frac{\partial {\mathit{\rho}}_{\mathrm{0}}{\mathit{\alpha}}^{*}\stackrel{\mathrm{\u203e}}{w}}{\partial t}}=& {\displaystyle}-{\displaystyle \frac{\partial}{\partial {\dot{x}}^{\mathrm{1}}}}\left(\stackrel{\mathrm{\u203e}}{u}{\dot{x}}_{x}^{\mathrm{1}}{\mathit{\rho}}_{\mathrm{0}}{\mathit{\alpha}}^{*}\stackrel{\mathrm{\u203e}}{w}\right)-{\displaystyle \frac{\partial}{\partial {\dot{x}}^{\mathrm{2}}}}\left(\stackrel{\mathrm{\u203e}}{v}{\dot{x}}_{y}^{\mathrm{2}}{\mathit{\rho}}_{\mathrm{0}}{\mathit{\alpha}}^{*}\stackrel{\mathrm{\u203e}}{w}\right){\displaystyle}& {\displaystyle}-{\displaystyle \frac{\partial}{\partial {\dot{x}}^{\mathrm{3}}}}\left(\stackrel{\mathrm{\u203e}}{{\dot{u}}^{\mathrm{3}}}{\mathit{\rho}}_{\mathrm{0}}{\mathit{\alpha}}^{*}\stackrel{\mathrm{\u203e}}{w}\right)-{\mathit{\alpha}}^{*}{\dot{x}}_{z}^{\mathrm{3}}{\displaystyle \frac{\partial {p}_{\mathrm{2}}}{\partial {\dot{x}}^{\mathrm{3}}}}\\ {\displaystyle}& {\displaystyle}+{f}^{\prime}{\mathit{\rho}}_{\mathrm{0}}{\mathit{\alpha}}^{*}\left(\stackrel{\mathrm{\u203e}}{u}{d}^{\prime}-\stackrel{\mathrm{\u203e}}{v}d\right)-{\stackrel{\mathrm{\u203e}}{F}}_{\mathrm{3}}\end{array}$$

$$\begin{array}{ll}\text{(6)}& {\displaystyle \frac{\partial {\mathit{\rho}}_{\mathrm{0}}{\mathit{\alpha}}^{*}\stackrel{\mathrm{\u203e}}{\mathit{\chi}}}{\partial t}}=& {\displaystyle}-{\displaystyle \frac{\partial}{\partial {\dot{x}}^{\mathrm{1}}}}\left(\stackrel{\mathrm{\u203e}}{u}{\dot{x}}_{x}^{\mathrm{1}}{\mathit{\rho}}_{\mathrm{0}}{\mathit{\alpha}}^{*}\stackrel{\mathrm{\u203e}}{\mathit{\chi}}\right)-{\displaystyle \frac{\partial}{\partial {\dot{x}}^{\mathrm{2}}}}\left(\stackrel{\mathrm{\u203e}}{v}{\dot{x}}_{y}^{\mathrm{2}}{\mathit{\rho}}_{\mathrm{0}}{\mathit{\alpha}}^{*}\stackrel{\mathrm{\u203e}}{\mathit{\chi}}\right){\displaystyle}& {\displaystyle}-{\displaystyle \frac{\partial}{\partial {\dot{x}}^{\mathrm{3}}}}\left(\stackrel{\mathrm{\u203e}}{{\dot{u}}^{\mathrm{3}}}{\mathit{\rho}}_{\mathrm{0}}{\mathit{\alpha}}^{*}\stackrel{\mathrm{\u203e}}{\mathit{\chi}}\right)+{\mathit{\rho}}_{\mathrm{0}}{\mathit{\alpha}}^{*}{\stackrel{\mathrm{\u203e}}{Q}}_{\stackrel{\mathrm{\u203e}}{\mathit{\chi}}}-{\stackrel{\mathrm{\u203e}}{F}}_{\stackrel{\mathrm{\u203e}}{\mathit{\chi}}}\end{array}$$

Here $\stackrel{\mathrm{\u203e}}{u}$, $\stackrel{\mathrm{\u203e}}{v}$, and $\stackrel{\mathrm{\u203e}}{w}$ are
the wind velocity components in the Cartesian coordinates,
$\stackrel{\mathrm{\u203e}}{{\dot{u}}^{\mathrm{3}}}$ is the contravariant vertical component of the wind
vector, *α*^{∗} denotes a grid volume, (${\dot{x}}^{\mathrm{1}}$,
${\dot{x}}^{\mathrm{2}}$, ${\dot{x}}^{\mathrm{3}})$, and (*x*, *y* , *z*) are the coordinates of
the terrain-following coordinate system and of the Cartesian system,
respectively. *U*_{g}, *V*_{g} denote the horizontal components of
the geostrophic wind, $\stackrel{\mathrm{\u203e}}{{Q}_{\mathit{\chi}}}$ sources and sinks of
$\stackrel{\mathrm{\u203e}}{\mathit{\chi}}$. The Coriolis parameters *f* and $\dot{f}$ are calculated
according to the local geographic latitude *ϕ* and the angular velocity
of the Earth's rotation Ω as *f*=2Ωsin*ϕ* and $\dot{f}=\mathrm{2}\mathrm{\Omega}\mathrm{cos}\mathit{\varphi}$. The variables *d*=sin*ξ* and ${d}^{\prime}=\mathrm{cos}\mathit{\xi}$ account
for the rotation of the coordinate system by an angle *ξ* against north.
The terms ${\stackrel{\mathrm{\u203e}}{F}}_{\mathrm{1}}$, ${\stackrel{\mathrm{\u203e}}{F}}_{\mathrm{2}}$, ${\stackrel{\mathrm{\u203e}}{F}}_{\mathrm{2}}$, and
${\stackrel{\mathrm{\u203e}}{F}}_{\stackrel{\mathrm{\u203e}}{\mathit{\chi}}}$ are the sub-grid-scale (SGS) turbulent
momentum and scalar fluxes emerging from filtering the equations. Subscript
notation for the coordinate transformation terms is used.

Due to the filtering, sub-grid-scale (SGS) fluxes arise. The three SGS
turbulent fluxes in the momentum equations (*j*=1, 2, 3) are

$$\begin{array}{ll}\text{(7)}& {\displaystyle}{\stackrel{\mathrm{\u203e}}{F}}_{j}=& {\displaystyle \frac{\partial}{\partial {\dot{x}}^{\mathrm{1}}}}\left({\mathit{\rho}}_{\mathrm{0}}{\mathit{\alpha}}^{*}\stackrel{\mathrm{\u203e}}{{u}^{\prime}}\stackrel{\mathrm{\u203e}}{{{u}_{j}}^{\prime}}{\dot{x}}_{x}^{\mathrm{1}}\right)+{\displaystyle \frac{\partial}{\partial {\dot{x}}^{\mathrm{2}}}}\left({\mathit{\rho}}_{\mathrm{0}}{\mathit{\alpha}}^{*}\stackrel{\mathrm{\u203e}}{{v}^{\prime}}\stackrel{\mathrm{\u203e}}{{{u}_{j}}^{\prime}}{\dot{x}}_{y}^{\mathrm{2}}\right){\displaystyle}& {\displaystyle}+{\displaystyle \frac{\partial}{\partial {\dot{x}}^{\mathrm{3}}}}{\mathit{\rho}}_{\mathrm{0}}{\mathit{\alpha}}^{*}\left(\stackrel{\mathrm{\u203e}}{{u}^{\prime}}\stackrel{\mathrm{\u203e}}{{{u}_{j}}^{\prime}}{\dot{x}}_{x}^{\mathrm{3}}+\stackrel{\mathrm{\u203e}}{{v}^{\prime}}\stackrel{\mathrm{\u203e}}{{{u}_{j}}^{\prime}}{\dot{x}}_{y}^{\mathrm{3}}+\stackrel{\mathrm{\u203e}}{{w}^{\prime}}\stackrel{\mathrm{\u203e}}{{{u}_{j}}^{\prime}}{\dot{x}}_{z}^{\mathrm{3}}\right).\end{array}$$

The SGS fluxes can be expressed in terms of the Reynolds stress tensor
*τ*_{ij}, which is related to the deformation tensor through the turbulent
mixing coefficient (Lilly, 1962; Smagorinsky, 1963).

At the lowest model layer, the validity of Monin–Obukhov surface layer
similarity theory (Monin and Obukhov, 1954; Foken, 2006) is assumed. The grid-box-averaged values of *u*_{∗}, *θ*_{∗}, *q*_{v∗} are
calculated using stability functions of Dyer (1974) with von Karman constant
*κ*=0.4.

Above the lowermost model layer the SGS turbulent fluxes are derived from a first-order closure (Detering, 1985; Etling, 1987).

$$\begin{array}{ll}\text{(8)}& {\displaystyle}{\mathit{\tau}}_{ij}=& {\displaystyle}-{\mathit{\rho}}_{\mathrm{0}}\stackrel{\mathrm{\u203e}}{{{u}_{i}}^{\prime}}\stackrel{\mathrm{\u203e}}{{{u}_{j}}^{\prime}}{\displaystyle}=& {\displaystyle}{\mathit{\rho}}_{\mathrm{0}}{K}_{ij}\left({\displaystyle \frac{\partial \stackrel{\mathrm{\u203e}}{{u}_{i}}}{\partial {\dot{x}}^{k}}}{\displaystyle \frac{\partial {\dot{x}}^{k}}{\partial {x}_{j}}}+{\displaystyle \frac{\partial \stackrel{\mathrm{\u203e}}{{u}_{j}}}{\partial {\dot{x}}^{k}}}{\displaystyle \frac{\partial {\dot{x}}^{k}}{\partial {x}_{i}}}\right)-{\displaystyle \frac{\mathrm{2}}{\mathrm{3}}}{\mathit{\rho}}_{\mathrm{0}}{\mathit{\delta}}_{ij}\stackrel{\mathrm{\u203e}}{E}\end{array}$$

*K*_{ij} is the turbulent exchange coefficient for momentum in the
*x*_{j} direction. The last term in Eq. (8) represents the reduction of the
diagonal fluxes due to pressure. Since this term is small compared to the
deformation tensor term, it is neglected in MITRAS. Due to the symmetry of
*τ*_{ij}, the actually calculated exchange coefficients are only a
horizontal exchange coefficient (*K*_{hor}) and a vertical exchange
coefficient (*K*_{vert}).

The turbulent fluxes for scalar quantities, e.g., potential temperature, are expressed as

$$\begin{array}{ll}\text{(9)}& {\displaystyle}{\stackrel{\mathrm{\u203e}}{F}}_{\mathit{\chi}}=& {\displaystyle \frac{\partial}{\partial {\dot{x}}^{\mathrm{1}}}}\left({\mathit{\rho}}_{\mathrm{0}}{\mathit{\alpha}}^{*}\stackrel{\mathrm{\u203e}}{{u}^{\prime}{\mathit{\chi}}^{\prime}}{\dot{x}}_{x}^{\mathrm{1}}\right)+{\displaystyle \frac{\partial}{\partial {\dot{x}}^{\mathrm{2}}}}\left({\mathit{\rho}}_{\mathrm{0}}{\mathit{\alpha}}^{*}\stackrel{\mathrm{\u203e}}{{v}^{\prime}\mathit{\chi}}{\dot{x}}_{y}^{\mathrm{2}}\right){\displaystyle}& {\displaystyle}+{\displaystyle \frac{\partial}{\partial {\dot{x}}^{\mathrm{3}}}}{\mathit{\rho}}_{\mathrm{0}}{\mathit{\alpha}}^{*}\left(\stackrel{\mathrm{\u203e}}{{u}^{\prime}{\mathit{\chi}}^{\prime}}{\dot{x}}_{x}^{\mathrm{3}}+\stackrel{\mathrm{\u203e}}{{v}^{\prime}\mathit{\chi}}{\dot{x}}_{y}^{\mathrm{3}}+\stackrel{\mathrm{\u203e}}{{w}^{\prime}\mathit{\chi}}{\dot{x}}_{z}^{\mathrm{3}}\right).\end{array}$$

These fluxes are also parameterized by a first-order closure.

$$\begin{array}{}\text{(10)}& {\displaystyle}-{\mathit{\rho}}_{\mathrm{0}}\stackrel{\mathrm{\u203e}}{{{u}_{i}}^{\prime}{\mathit{\chi}}^{\prime}}={\mathit{\rho}}_{\mathrm{0}}{K}_{i\mathit{\chi}}\left({\displaystyle \frac{\partial \stackrel{\mathrm{\u203e}}{\mathit{\chi}}}{\partial {\dot{x}}^{k}}}{\displaystyle \frac{\partial {\dot{x}}^{k}}{\partial {x}_{j}}}\right)\end{array}$$

*K*_{iχ} is the corresponding mixing coefficient in the
*x*_{i} direction, which is related to *K*_{ij}.

The exchange coefficients in MITRAS are either calculated using the
Prandtl–Kolmogorov closure (Sect. 2.2.3, first subsection) or the
*E*–*ε* turbulence closure (Sect. 2.2.3, second subsection). In
both turbulence closures the exchange coefficients are calculated as a
function of the SGS turbulent kinetic energy *E*, for which an additional
prognostic equation similar to Eq. (6) is solved. For the *E*–*ε*
turbulence closure the dissipation *ε* is additionally calculated
from a prognostic equation (López et al., 2005).

The exchange coefficients are calculated as follows.

$$\begin{array}{}\text{(11)}& {\displaystyle}{K}_{\text{vert}}={c}_{\mathrm{1}}l{\stackrel{\mathrm{\u203e}}{E}}^{\mathrm{1}/\mathrm{2}}f\left(Ri\right)\end{array}$$

*c*_{1} is a free constant determined by matching the theoretical model
against experimental values. It has the value 0.55 in MITRAS (López,
2002). The Richardson number *Ri*-dependent term is defined as

$$\begin{array}{}\text{(12)}& {\displaystyle}f\left(\mathit{Ri}\right)=\left\{\begin{array}{ll}\mathrm{1}-\mathrm{5}\phantom{\rule{0.125em}{0ex}}\mathit{Ri}& \mathrm{for}\phantom{\rule{0.25em}{0ex}}\mathit{Ri}>\mathrm{0}\\ (\mathrm{1}-\mathrm{16}\phantom{\rule{0.125em}{0ex}}\mathit{Ri}{)}^{\mathrm{1}/\mathrm{4}}& \mathrm{for}\phantom{\rule{0.25em}{0ex}}\mathit{Ri}\le \mathrm{0},\end{array}\right.\end{array}$$

and in accordance with the stability functions used in the surface layer. For
the SGS turbulent kinetic energy, *E*, a prognostic equation is solved. The
dissipation rate *ϵ* is calculated diagnostically as

$$\begin{array}{}\text{(13)}& {\displaystyle}\mathit{\u03f5}={c}_{\mathit{\epsilon}}{\displaystyle \frac{{\stackrel{\mathrm{\u203e}}{E}}^{\mathrm{3}/\mathrm{2}}}{l}}.\end{array}$$

*c*_{ε} is a constant set to 0.166 (López, 2002). The local
mixing length *l* is related to the stability function *φ*_{m} and
the distance to the nearest solid surface *z*_{w}, which can be the ground
surface or a surface of a resolved obstacle. The equation of *l* reads

$$\begin{array}{}\text{(14)}& {\displaystyle}l={\displaystyle \frac{\mathit{\kappa}{z}_{w}}{\mathrm{1}+\mathit{\kappa}\frac{{z}_{w}}{\mathit{\lambda}}}}{\mathit{\varphi}}_{\text{m}}.\end{array}$$

*λ* is the maximum eddy size (a value of 100 m is used).

This closure is based on the standard *E*–*ε* model and the Kato
and Launder (1993) modifications, which eliminate the excessive turbulent
kinetic energy produced by the standard *E*–*ε* model
in stagnation regions (López et al., 2005). The mechanical production
term *P*_{M} in the *E* equation can be derived according to the Kato
and Launder (1993) modifications as

$$\begin{array}{}\text{(15)}& {\displaystyle}{P}_{\text{M}}={c}_{\mathit{\mu}}\mathit{\epsilon}S\mathrm{\Omega}.\end{array}$$

*S* is the dimensionless strain rate.

$$\begin{array}{}\text{(16)}& {\displaystyle}S={\displaystyle \frac{\stackrel{\mathrm{\u203e}}{E}}{\mathit{\epsilon}}}\sqrt{{\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}{\left({\displaystyle \frac{\partial \stackrel{\mathrm{\u203e}}{{u}_{i}}}{\partial {x}_{j}}}+{\displaystyle \frac{\partial \stackrel{\mathrm{\u203e}}{{u}_{j}}}{\partial {x}_{i}}}\right)}^{\mathrm{2}}}\end{array}$$

Ω is the rotation rate:

$$\begin{array}{}\text{(17)}& {\displaystyle}\mathrm{\Omega}={\displaystyle \frac{\stackrel{\mathrm{\u203e}}{E}}{\mathit{\epsilon}}}\sqrt{{\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}{\left({\displaystyle \frac{\partial \stackrel{\mathrm{\u203e}}{{u}_{i}}}{\partial {x}_{j}}}-{\displaystyle \frac{\partial \stackrel{\mathrm{\u203e}}{{u}_{j}}}{\partial {x}_{i}}}\right)}^{\mathrm{2}}},\end{array}$$

which goes to zero near the stagnation point, so *P*_{M} is
significantly reduced.

It has been shown by Schlünzen et al. (2003) that using Kato and
Launder (1993) modifications for both the turbulent kinetic energy equation
and the dissipation equation in MITRAS leads to overestimation of the
momentum fluxes at the stagnation point. To overcome this drawback, they
suggested limiting the Kato and Launder reformulation to the energy equation
only. So, for the *ε* equation, the *P*_{M} term is calculated
the same way as in the standard *E*–*ε* model.

$$\begin{array}{}\text{(18)}& {\displaystyle}{P}_{\text{M}}={c}_{\mathit{\mu}}\mathit{\epsilon}{S}^{\mathrm{2}}\end{array}$$

The buoyancy term is calculated in the same way as in the
Prandtl–Kolmogorov closure. The values for the constants *c*_{1}, *c*_{2},
*c*_{μ} are taken as suggested by López (2005) as *c*_{μ}=0.09,
*c*_{1}=1.44, *c*_{2}=1.92.

The model equations are solved using finite-volume methods on a staggered
Arakawa C grid (Arakawa and Lamb, 1977). On this grid, scalar variables are
defined at the cell center, while the velocity components are defined on their
respective normal cell faces (Fig. 2). The obstacles faces are set where the
corresponding wall normal velocity components are defined. Since *u*, *v*,
*w*, and the scalar variables are defined at different locations in space,
four index arrays are needed to describe the obstacle in the discretized 3-D
model domain. The discretization allows for grid stretching in both the vertical and
horizontal directions to keep focus on the inner part in the domain. The
scalar points are always in the center of a grid cell, while the wind
components might have different distances to the next-neighbor scalar grid
point.

The advection and diffusion terms in the momentum equations are solved in
MITRAS using the Adam–Bashforth scheme in time and centered differences in
space. The vertical diffusion terms are determined using the Crank–Nicholson
implicit scheme in order to increase the time step for vertical exchange
processes. All other terms in the momentum equations except dynamic pressure
*p*_{2} are solved forward in time and centered in space.

The dynamic pressure *p*_{2} is determined iteratively from a Poisson
equation to satisfy the anelastic approximation. MITRAS allows two
user-selected options for the iterative procedures, the iterative IGCG scheme
(idealized generalized conjugate gradient; Kapitza and Eppel, 1987) and the
preconditioned BiCGStab method (biconjugate gradient stabilized; Van der
Vorst, 1992).

To avoid numerical artifacts that might appear due to nonlinear interactions and result in shortwave energy accumulation, artificial diffusivity is added.

The advection of scalar quantities is solved forward in time and using the
upstream scheme for advection. Even though the upstream method is known for being
diffusive with its implicit diffusivity *K*_{num} in, e.g., the *x* direction
given by ${K}_{\text{num}}=\mathrm{0.5}\cdot \left|U\right|\cdot \mathrm{\Delta}x\cdot \left(\mathrm{1}-\mathit{Co}\right)$ (Roache, 1982), the artificial diffusivity is
acceptable in the target area of the domain for two reasons. (a) The
isotropic grid and building impacts ensure advection and diffusion in
the horizontal and vertical direction being of comparable size, and (b) advection
is often larger than diffusion terms since the SGS turbulence is small within
the canopy layer. *U* is the local wind speed, Δ*x* is the local grid
width, and *Co* is the Courant number.

Since Eq. (13) implies that dissipation rate and sub-grid-scale turbulent
kinetic energy are directly coupled, the dissipation cannot be calculated
with very large time steps. Equation (13) is solved by determining an
analytic solution (Appendix A) as suggested by Fock (2015). This avoids
unphysical values of *ε*, which might even be negative for large
time steps Δ*t*.

3 Boundary conditions

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In MITRAS, several types of boundary conditions can be used in physically consistent combinations to allow for different kinds of simulations. At the ground surface (lower boundary) and obstacle faces (Sect. 4) material interfaces are given, while the lateral boundaries and the upper boundary are artificial due to the use of a limited area model.

For the horizontal wind velocity components, a no-slip boundary condition
($\stackrel{\mathrm{\u203e}}{{u}_{j}}=\mathrm{0}$, *j*=1, 2) is used at all vertical surfaces. With
this the vertical wind defined in the staggered grid at the surface also
results in zero. Since *u* and *v* are defined at the lowest level at *k*=0.5
(Fig. 2) on the staggered grid, a Neumann boundary condition is used with a
constant gradient using the zero surface values. With this the no-slip
condition is achieved at the surface. Additional details for buildings are
given in Sect. 4.1.

The boundary condition for the pressure *p*_{2} is formulated by considering
the wind velocity boundary condition, the grid staggering, the position of
the domain boundaries, and the dynamic pressure equation. Consistent with the
no-slip boundary condition, the boundary condition used for *p*_{2} at the
wall is

$$\begin{array}{ll}\text{(19)}& {\displaystyle}& {\displaystyle \frac{\partial {\dot{x}}^{\mathrm{1}}}{\partial x}}{\displaystyle \frac{\partial {\dot{x}}^{\mathrm{3}}}{\partial x}}{\displaystyle \frac{\partial {p}_{\mathrm{2}}}{\partial {\dot{x}}^{\mathrm{1}}}}+{\displaystyle \frac{\partial {\dot{x}}^{\mathrm{2}}}{\partial y}}{\displaystyle \frac{\partial {\dot{x}}^{\mathrm{3}}}{\partial y}}{\displaystyle \frac{\partial {p}_{\mathrm{2}}}{\partial {\dot{x}}^{\mathrm{2}}}}{\displaystyle}& {\displaystyle}+\left({\left({\displaystyle \frac{\partial {\dot{x}}^{\mathrm{3}}}{\partial x}}\right)}^{\mathrm{2}}+{\left({\displaystyle \frac{\partial {\dot{x}}^{\mathrm{3}}}{\partial y}}\right)}^{\mathrm{2}}+{\left({\displaystyle \frac{\partial {\dot{x}}^{\mathrm{3}}}{\partial z}}\right)}^{\mathrm{2}}\right){\displaystyle \frac{\partial {p}_{\mathrm{2}}}{\partial {\dot{x}}^{\mathrm{3}}}}=\mathrm{0}.\end{array}$$

Due to the terrain-following coordinate system (Eq. 1) the vertical gradient
of the dynamic pressure *p*_{2} needs to be zero perpendicular to the
surface.

The calculated temperature values of all physical boundaries (ground and obstacles surfaces, i.e., wall and roof) are used at the lower boundary and at the obstacle surfaces. The necessary additions for buildings are provided in Sect. 4.2. These temperature values are calculated using the force-restore method for the ground soil heat flux. Following Tiedtke and Geleyn (1975) and Deardorff (1978), the temperature at the surface, $\stackrel{\mathrm{\u203e}}{{T}_{\text{s}}}$, is calculated as

$$\begin{array}{}\text{(20)}& {\displaystyle \frac{\partial \stackrel{\mathrm{\u203e}}{{T}_{\text{s}}}}{\partial t}}={\displaystyle \frac{\mathrm{2}\sqrt{\mathit{\pi}}{\mathit{\kappa}}_{\text{s}}}{{\mathit{\upsilon}}_{\text{s}}{h}_{\mathit{\theta}}}}\left(H-\sqrt{\mathit{\pi}}{\mathit{\upsilon}}_{\text{s}}{\displaystyle \frac{\stackrel{\mathrm{\u203e}}{T}-\stackrel{\mathrm{\u203e}}{T\left(-{h}_{\mathit{\theta}}\right)}}{{h}_{\mathit{\theta}}}}\right).\end{array}$$

*H* is the force term of the fluxes of the surface energy budget: the net
shortwave (*R*_{SW,net}) and longwave radiation flux
(*R*_{LW,net}), the sensible (*Q*_{S}) and latent heat flux
(*Q*_{L}), and the anthropogenic heat emission flux (*Q*_{a}).
$\stackrel{\mathrm{\u203e}}{T(-{h}_{\mathit{\theta}})}$ is the deep soil temperature, *h*_{θ}
represents the depth of the daily temperature wave in the restore term, and
*κ*_{s} is the thermal diffusivity of the surface cover material.

Both *R*_{SW,net} and *R*_{LW,net} are calculated using the
atmospheric radiation scheme of the MITRAS sister model METRAS (Schlünzen
et al., 2018b) and the surface characteristics, i.e., the albedo value. The
influence of vegetation and the shading from the obstacles is taken into
account in the calculation of radiation (Sect. 4.4). The fluxes *Q*_{S}
and *Q*_{L} are calculated with respect to the thermal stratification
using the friction velocity *u*_{∗} and the scaling values for heat,
*θ*_{∗}, and water vapor, *q*_{∗}.

For obstacles, the calculated surface temperature (Sect. 4.2) of the obstacle surfaces is used at the corresponding grid cells.

The following budget equation, introduced by Deardorff (1978), is used to calculate the humidity at the lower boundary (${\stackrel{\mathrm{\u203e}}{q}}_{\text{1s}})$.

$$\begin{array}{}\text{(21)}& {\displaystyle}{\stackrel{\mathrm{\u203e}}{q}}_{\text{1s}}={\mathit{\alpha}}_{q}{\stackrel{\mathrm{\u203e}}{q}}_{\mathrm{1}\mathrm{sat}}\left({\stackrel{\mathrm{\u203e}}{T}}_{\text{s}}\right)+\left(\mathrm{1}-{\mathit{\alpha}}_{q}\right){\stackrel{\mathrm{\u203e}}{q}}_{\mathrm{1}}\end{array}$$

*α*_{q} is the soil water availability,
${\stackrel{\mathrm{\u203e}}{q}}_{\mathrm{1}\mathrm{sat}}\left({\stackrel{\mathrm{\u203e}}{T}}_{\text{s}}\right)$ is the saturated
humidity at ${\stackrel{\mathrm{\u203e}}{T}}_{\text{s}}$, and ${\stackrel{\mathrm{\u203e}}{q}}_{\mathrm{1}}$ is the specific
humidity at the first model level. The initial value of *α*_{q} is given
for each surface cover class (Sect. 5.2). A prognostic equation is used to
calculate *α*_{q} in the time-dependent model integration.

$$\begin{array}{}\text{(22)}& {\displaystyle \frac{\partial {\mathit{\alpha}}_{q}}{\partial t}}={\displaystyle \frac{{Q}_{\text{E}}/l+P}{{\mathit{\rho}}_{\text{w}}{W}_{k}}}\end{array}$$

*Q*_{E} is the turbulent humidity flux at the surface (calculated in
consistency with *Q*_{L}), $l=\mathrm{2.5}\times {\mathrm{10}}^{\mathrm{6}}$ J kg^{−1} is the
latent heat of vaporization of water, *P* is the precipitation (if any),
*ρ*_{w} water density, and *W*_{k} is the saturation value for water
content. This is prescribed for each surface cover class (Sect. 5.2).

At the ground surface and at the obstacle surface $\stackrel{\mathrm{\u203e}}{E}$ and
*ε* are calculated as a function of local friction velocity, as
follows.

$$\begin{array}{}\text{(23)}& {\displaystyle}{\stackrel{\mathrm{\u203e}}{E}}_{z\phantom{\rule{0.125em}{0ex}}=\phantom{\rule{0.125em}{0ex}}\mathrm{0}}={\displaystyle \frac{{u}_{*}^{\mathrm{2}}}{{c}_{\mathrm{1}}^{\mathrm{2}}}},\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}{\mathit{\epsilon}}_{z\phantom{\rule{0.125em}{0ex}}=\phantom{\rule{0.125em}{0ex}}\mathrm{0}}={\displaystyle \frac{{u}_{*}^{\mathrm{3}}}{\mathit{\kappa}{z}_{\mathrm{0}}}}.\end{array}$$

The empirical constant *c*_{1} is set to the same value as in Eq. (11). This
helps together with Eq. (12) to obtain continuous fluxes at the top of the
lowest model cell, which employs the surface layer scheme (Lopez, 2002; Fock,
2015).

Dirichlet boundary conditions are used in two different formulations. They can be used in MITRAS in arbitrary combinations to describe the lateral boundaries of the domain: open boundary (radiative) and fixed boundaries. The appropriate combination of boundary value calculations depends on the application. For instance, a realistic application with comparison to field data in mind needs open boundaries. In these the boundary normal wind components are calculated as far as possible from the prognostic equations. The boundary normal advection is treated with the use of the Orlanski condition at inflow boundaries and by the upstream scheme at outflow boundaries. For the boundary parallel velocity components a zero-flux condition is assumed (Schlünzen, 1990).

When comparison with wind tunnel measurements (e.g., Grawe et al., 2013b) is performed, fixed boundaries are advantageous. In these the wind profiles are to be imposed at the inlet boundary and kept fixed at the initial values, while at the outflow the wind velocity is treated as an open boundary.

The normal gradients of pressure *p*_{2}, temperature, humidity, and
concentrations are set to zero at the lateral boundaries.

For the vertical wind, which is defined at the model top, the Dirichlet condition is used, prescribing it to initial values (mostly vertical wind zero). For all other variables a Neumann boundary condition is employed for which the gradients normal to the boundary are zero. In order to avoid reflections of vertically propagating waves at the upper model boundary, Rayleigh damping layers (absorbing layers) are used in MITRAS. The Rayleigh damping terms, which are added to the flow equations (Eqs. 3–6), are written here.

$$\begin{array}{}\text{(24)}& {\displaystyle}& {\displaystyle}{R}_{\mathrm{1}}=-{\mathit{\rho}}_{\mathrm{0}}{\mathit{\alpha}}^{*}\left(\stackrel{\mathrm{\u203e}}{u}-{U}_{\text{g}}\right){\mathit{\nu}}_{\text{R}}\text{(25)}& {\displaystyle}& {\displaystyle}{R}_{\mathrm{2}}=-{\mathit{\rho}}_{\mathrm{0}}{\mathit{\alpha}}^{*}\left(\stackrel{\mathrm{\u203e}}{v}-{V}_{\text{g}}\right){\mathit{\nu}}_{\text{R}}\text{(26)}& {\displaystyle}& {\displaystyle}{R}_{\mathrm{3}}=-{\mathit{\rho}}_{\mathrm{0}}{\mathit{\alpha}}^{*}\stackrel{\mathrm{\u203e}}{w}{\mathit{\nu}}_{\text{R}}\end{array}$$

*U*_{g} and *V*_{g} are the geostrophic wind velocity components
and *v*_{R} is the relaxation coefficient, defined as

$$\begin{array}{}\text{(27)}& {\displaystyle}{\displaystyle}{v}_{\text{R}}=\left\{\begin{array}{ll}\mathrm{0}& \text{for}\phantom{\rule{0.25em}{0ex}}k<{k}_{D}\\ {\mathit{\delta}}^{\left({k}_{t}-k\right)}& \text{for}\phantom{\rule{0.25em}{0ex}}k\ge {k}_{D}.\end{array}\right.\end{array}$$

*k* denotes the vertical grid point index, *k*_{t} the index of the highest
grid point at the upper boundary, and *k*_{D} the index of the first absorbing
layer. The coefficient *δ* is set to 0.2 and, based on our experience,
five
damping layers are sufficient.

The normal pressure gradient, temperature gradient, turbulent momentum fluxes, and their gradients are all set to zero at the upper boundary. This assumption results in zero vertical heat and moisture fluxes as well as zero momentum fluxes at the upper model boundary.

4 Treatment of obstacles

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The concept of the mask method (Briscolini and Santangelo, 1989) is employed in MITRAS to explicitly resolve the buildings within the 3-D model domain. This method is based on the immersed boundary method (Mittal and Iaccarino, 2005), which allows for flow simulation in the vicinity of complex geometries that do not conform on Cartesian grids. Impermeable grid cells at the building positions are defined using 3-D fields of weighting factors, vol($x,y,z)$, defined at each grid cell to give information about whether a grid cell is an atmospheric cell or a building cell (which lies mostly or completely inside a building). Any faces of a grid volume that are a wall or roof of a building are marked. This means that the grid cells in a domain are divided into three groups: grid cells in the free atmosphere surrounded by atmosphere without any adjacent building, grid cells next to building surfaces, and grid cells within buildings, as shown in Fig. 3. This separation facilitates the model coding and economizes the computational requirements.

A building mask containing these data is prepared by the preprocessor GRIMASK (Sect. 5.3). In the model, e.g., the wind velocity components vanish at the building boundaries by multiplying the fluxes with the face markers (impermeable walls). Additional wall functions are included to address friction effects properly.

Building surfaces influence the ambient air temperature. Their effect is
taken into account by simulating the sensible heat flux. In grid cells that
are adjacent to building surfaces, the term *Q*_{θ} is added to the
turbulent fluxes of heat (Eq. 9).

$$\begin{array}{}\text{(28)}& {\displaystyle}{Q}_{\mathit{\theta}}=-{u}_{\text{b}*}{\mathit{\theta}}_{\text{b}*}\end{array}$$

represents the temperature flux, which is calculated from the friction
velocity at buildings, ${u}_{{\text{b}}_{*}}$, and the scaling value for
potential temperature, *θ*_{b*}. ${u}_{{\text{b}}_{*}}$ is calculated
following the approach of Lopez (2002) as

$$\begin{array}{}\text{(29)}& {\displaystyle}{u}_{\text{b}*}=\mathit{\kappa}{\displaystyle \frac{\left|{\mathit{v}}_{\text{b}}\right|}{\mathrm{ln}\left(\frac{{d}_{\text{b}}}{{z}_{\text{b},\mathrm{0}}}\right)}},\end{array}$$

assuming a logarithmic wind profile with neutral stratification over the
building surface. |*v*_{b}| is the wind speed
parallel to the building surface at the first scalar grid cell next to the
building surface, i.e., in the distance *d*_{b}. *z*_{b,0} is the
roughness length of the building surface. The scaling value for potential
temperature at buildings is calculated as

$$\begin{array}{}\text{(30)}& {\displaystyle}{\mathit{\theta}}_{\text{b}*}=\mathit{\kappa}{\displaystyle \frac{\left(\stackrel{\mathrm{\u203e}}{{\mathit{\theta}}_{\text{d,b}}}-\stackrel{\mathrm{\u203e}}{{\mathit{\theta}}_{\text{b}}}\right)}{\mathrm{ln}\left(\frac{{d}_{\text{b}}}{{z}_{\text{b},\mathrm{0},\mathit{\theta}}}\right)}}\end{array}$$

from the difference of the building surface temperature, *θ*_{b},
and the temperature at the first grid cell next to the building,
*θ*_{d,b}. The roughness length for temperature at the building
(${z}_{\text{b},\mathrm{0},\mathit{\theta}})$ depends on the Reynolds number, *Re*.
Following Brutsaert (1975), the roughness length ratio is calculated as

$$\begin{array}{}\text{(31)}& {\displaystyle \frac{{z}_{\text{b},\mathrm{0}}}{{z}_{\text{b},\mathrm{0},\mathit{\theta}}}}=\mathrm{exp}\left(\mathit{\kappa}\left(\mathrm{7.3}{\mathit{Re}}^{\mathrm{1}/\mathrm{4}}\sqrt{\mathit{Pr}}-\mathrm{5}\right)\right),\end{array}$$

with the Prandtl number (*Pr*) set to 0.71.

This concept allows for a consideration of not only surface-mounted buildings but also overhanging obstacles such as bridges and overpasses or pathways to courtyards. They can all be considered in complex urban geometries.

In order to obtain an accurate surface temperature of the buildings (obstacles), most exchange processes at the building surfaces are considered in MITRAS, including turbulent and radiative processes (Gierisch, 2011). Thus, the physical properties of the façade and wall materials are to be introduced as model inputs. These properties include reflectivity, emissivity, heat transfer coefficient, and specific heat capacity.

The surface temperature of a building surface, *T*_{b}, is calculated
from the energy budget of the infinitely thin outermost slab of the building
façade. The slab is heated or cooled from outside by a heat source *H*
and supported from inside by the rest of the façade that is connected to
the building interior. The latter is assumed to be maintained at a
temperature *T*_{in}.

The rate of temperature change of the slab is governed by the imbalance
between the forcing term *H* and a restoring term. The prognostic equation for
the surface temperature reads

$$\begin{array}{}\text{(32)}& {\displaystyle}{c}_{\text{wall}}D{\displaystyle \frac{\partial \stackrel{\mathrm{\u203e}}{{T}_{\text{b}}}}{\partial t}}=H-{\displaystyle \frac{\mathit{\lambda}}{D}}\left(\stackrel{\mathrm{\u203e}}{{T}_{\text{b}}}-\stackrel{\mathrm{\u203e}}{{T}_{\text{in}}}\right).\end{array}$$

Here the forcing term *H* is calculated from

$$\begin{array}{}\text{(33)}& {\displaystyle}H={R}_{\text{SW,abs}}+{R}_{\text{LW,abs}}-\mathit{\epsilon}\mathit{\sigma}{\stackrel{\mathrm{\u203e}}{{T}_{\text{b}}}}^{\mathrm{4}}+{Q}_{\text{S}}+{Q}_{\text{L}}.\end{array}$$

*R*_{SW,abs} and *R*_{LW,abs} are the absorbed incoming shortwave
and longwave radiation, *Q*_{S} and *Q*_{L} are the sensible and
latent heat fluxes at the surface, which are calculated from the local
friction velocity and the local scaling values for temperature and humidity
(Gierisch, 2011), *λ* is the thermal conductivity, *D* is the wall
thickness, and *c*_{wall} is the wall volumetric heat capacity.

The surface energy balance for the inside wall surface can be written as

$$\begin{array}{}\text{(34)}& {\displaystyle}{Q}_{\text{C}}-{h}_{\text{i}}\left(\stackrel{\mathrm{\u203e}}{{T}_{\text{in}}}-\stackrel{\mathrm{\u203e}}{{T}_{\text{room}}}\right)=\mathrm{0}.\end{array}$$

*h*_{i} is the heat transfer coefficient for the internal wall,
*Q*_{C} the heat conduction flux through the wall calculated as
${Q}_{\text{C}}=\frac{\mathit{\lambda}}{D}\left(\stackrel{\mathrm{\u203e}}{{T}_{\text{b}}}-\stackrel{\mathrm{\u203e}}{{T}_{\text{in}}}\right)$, and
$\stackrel{\mathrm{\u203e}}{{T}_{\text{room}}}$ is the room temperature.

From Eq. (34), the relation between $\stackrel{\mathrm{\u203e}}{{T}_{\text{in}}}$ and $\stackrel{\mathrm{\u203e}}{{T}_{\text{b}}}$ is

$$\begin{array}{}\text{(35)}& {\displaystyle}\stackrel{\mathrm{\u203e}}{{T}_{\text{in}}}={\displaystyle \frac{{h}_{\text{i}}}{{h}_{\text{i}}+\frac{\mathit{\lambda}}{D}}}\stackrel{\mathrm{\u203e}}{{T}_{\text{room}}}+{\displaystyle \frac{\frac{\mathit{\lambda}}{D}}{{h}_{\text{i}}+\frac{\mathit{\lambda}}{D}}}\stackrel{\mathrm{\u203e}}{{T}_{\text{b}}}.\end{array}$$

Substituting for *H* and $\stackrel{\mathrm{\u203e}}{{T}_{\text{in}}}$ in Eq. (32) yields

$$\begin{array}{}\text{(36)}& {\displaystyle \frac{\partial \stackrel{\mathrm{\u203e}}{{T}_{\text{b}}}}{\partial t}}={\displaystyle \frac{\mathrm{1}}{{c}_{\text{wall}}D}}\left[H-C\left(\stackrel{\mathrm{\u203e}}{{T}_{\text{b}}}-\stackrel{\mathrm{\u203e}}{{T}_{\text{room}}}\right)\right].\end{array}$$

The right-hand side of Eq. (36) is a function of $\stackrel{\mathrm{\u203e}}{{T}_{\text{b}}}$. Thus

$$\begin{array}{}\text{(37)}& {\displaystyle \frac{\partial \stackrel{\mathrm{\u203e}}{{T}_{\text{b}}}}{\partial t}}=F\left(\stackrel{\mathrm{\u203e}}{{T}_{\text{b}}}\right).\end{array}$$

Solving Eq.(37) numerically,

$$\begin{array}{}\text{(38)}& {\displaystyle}F\left({\stackrel{\mathrm{\u203e}}{{T}_{\text{b}}}}^{t+\mathrm{\Delta}t}\right)=F\left({\stackrel{\mathrm{\u203e}}{{T}_{\text{b}}}}^{t}\right)+{\left.{\displaystyle \frac{\partial F}{\partial \stackrel{\mathrm{\u203e}}{{T}_{\text{b}}}}}\right|}_{{\stackrel{\mathrm{\u203e}}{{T}_{\text{b}}}}^{t}}\left({\stackrel{\mathrm{\u203e}}{{T}_{\text{b}}}}^{t+\mathrm{\Delta}t}-{\stackrel{\mathrm{\u203e}}{{T}_{\text{b}}}}^{t}\right),\end{array}$$

and solving for ${\stackrel{\mathrm{\u203e}}{T}}_{\text{b}}^{t+\mathrm{\Delta}t}$ gives the time-dependent equation for the surface temperature:

$$\begin{array}{}\text{(39)}& {\displaystyle}{\stackrel{\mathrm{\u203e}}{{T}_{\text{b}}}}^{t+\mathrm{\Delta}t}={\stackrel{\mathrm{\u203e}}{{T}_{\text{b}}}}^{t}+{\displaystyle \frac{F\left({\stackrel{\mathrm{\u203e}}{{T}_{\text{b}}}}^{t}\right)}{\frac{\mathrm{1}}{\mathrm{\Delta}t}-{\left.\frac{\partial F}{\partial \stackrel{\mathrm{\u203e}}{{T}_{\text{b}}}}\right|}_{{\stackrel{\mathrm{\u203e}}{{T}_{\text{b}}}}^{t}}}}.\end{array}$$

Wind turbines are represented in MITRAS by impermeable grid cells at the position of the tower and the nacelle, similar to other buildings (i.e., vanishing wind speed and zero turbulent kinetic energy are assumed at grid points within the tower and nacelle). The orientation of the nacelle changes in relation to the wind direction during the model simulation. The wind turbine rotor is parameterized by using the actuator-disk concept (Molly, 1978; Mikkelsen, 2003; El Kasmi and Masson, 2008). In this concept the rotor is replaced by an imaginary permeable disk subjected to a distribution of forces that acts upon the incoming flow at a rate defined by the period-averaged kinetic energy that the rotor extracts from the atmosphere.

According to the actuator-disk model, the reduction of the wind speed is
caused by the rotor thrust, *T*, which is formulated as

$$\begin{array}{}\text{(40)}& {\displaystyle}T={\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}{c}_{\text{T}}\mathit{\rho}A{V}_{\mathrm{1}}^{\mathrm{2}}.\end{array}$$

*V*_{1} is the speed of the approaching flow at wind turbine level, *A* is
the rotor area, *ρ* is the air density, and *c*_{T} is the
nondimensional thrust coefficient for the corresponding wind speed. The wind
speed deficit is limited to those cells located at the actual rotor position.
The speed of the approaching flow is calculated with respect to the
orientation of the rotor, which depends on wind direction and changes
direction during the simulation. The thrust coefficient depends on wind speed
and therefore the wind turbine automatically switches on and off in MITRAS at
the cut-in and cutoff wind speed, respectively.

This wind turbine rotor blades create wake vortices of the wind turbines,
which are associated with increased turbulence intensity. The turbulence
generation in the wake is parameterized in MITRAS by adding an additional
term, *Q*_{wt}, to the turbulence mechanical production, *P*_{M},
in the turbulent kinetic energy equation to account for the turbulence
generation at the rotor position. This term is formulated as

$$\begin{array}{}\text{(41)}& {\displaystyle}{Q}_{\text{wt}}={\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}{c}_{\text{wt}}{u}_{\text{wt}}^{\mathrm{2}}.\end{array}$$

*u*_{wt}
is a scale velocity used to characterize the turbulence. The factor
*c*_{wt} (s^{−1}) includes the wind turbine characteristics that
govern the amount of produced turbulence, namely the rotor size, the number
of blades, and the rotational speed.

In MITRAS, the tangential velocity *v*_{θ}, which is frequently used to
model the vortex developing behind an aircraft, is used as the scale
velocity. The Rankine vortex model (Gerz et al., 2001) is applied here to
calculate *v*_{θ}.

$$\begin{array}{}\text{(42)}& {\displaystyle}{v}_{\mathit{\theta}}={\displaystyle \frac{{\mathrm{\Gamma}}_{O}}{\mathrm{2}\mathit{\pi}{r}_{\text{c}}}}\end{array}$$

*r*_{c} is the vortex core radius and Γ_{O} is the rotor
circulation, which is related to the rotor rotational speed, lift
coefficient, and aspect ratio. More details about modeling the wind turbines
in MITRAS are given in Linde (2011).

There are two modes of vegetation treatment in MITRAS: the implicit mode and the explicit mode. In the implicit mode, the effect of the vegetation (grass, bushes, trees, etc.) is implicitly considered in the surface parameterization using the roughness length. This is done by allocating the vegetation surface cover class for the corresponding surface grid cells and using the corresponding input parameters (e.g., roughness length, soil water, content, etc.; Sect. 5.2).

In explicit mode, vegetation effects are explicitly resolved. These effects include wind speed reduction (Schlüter, 2006), turbulence dissipation due to drag forces from plant foliage–atmosphere interaction (Salim et al., 2015), and radiation absorption and shading.

The wind speed reduction is parameterized by introducing a local
three-dimensional sink term, ${S}_{{u}_{i}}$, with *i*=1, 2, and 3 for the *u*,
*v*, and
*w* component. ${S}_{{u}_{i}}$ is added to the momentum equations (Eqs. 3–5).
Following Liu (1996), the sink term is calculated as

$$\begin{array}{}\text{(43)}& {\displaystyle}{S}_{{u}_{i}}=-{c}_{\text{d}}\text{LAD}\left({\dot{x}}^{\mathrm{3}}\right)\cdot U\cdot \stackrel{\mathrm{\u203e}}{{u}_{i}}.\end{array}$$

Here *c*_{d} is a drag coefficient, *U* is the mean wind speed at height
${\dot{x}}^{\mathrm{3}}$, and LAD(${\dot{x}}^{\mathrm{3}})$ is the equivalent leaf area density
of the plant at height ${\dot{x}}^{\mathrm{3}}$. The value of *c*_{d}=0.2
determined by Katul (1998) is chosen here.

${S}_{{u}_{i}}$ represents a source of turbulence due to the extraction of mean kinetic energy, $\stackrel{\mathrm{\u203e}}{E}$, from the flow. However, the typical effect of vegetation is to reduce the overall turbulence by enhancing the dissipation of turbulence. To parameterize the additional turbulence creation and dissipation, additional source terms are added to the turbulent kinetic energy and dissipation equations. Following Wilson (1988) and Liu et al. (1996) these terms read as follows.

$$\begin{array}{}\text{(44)}& {\displaystyle}& {\displaystyle}{Q}_{\text{veg},\stackrel{\mathrm{\u203e}}{E}}={c}_{\text{d}}\text{LAD}\left({\dot{x}}^{\mathrm{3}}\right)\cdot {U}^{\mathrm{3}}-\mathrm{4}{c}_{\text{d}}\text{LAD}\left({\dot{x}}^{\mathrm{3}}\right)\cdot \left|U\right|\cdot \stackrel{\mathrm{\u203e}}{E}\text{(45)}& {\displaystyle}& {\displaystyle}{Q}_{\text{veg},\mathit{\epsilon}}=\mathrm{1.5}{c}_{\text{d}}\text{LAD}\left({\dot{x}}^{\mathrm{3}}\right)\cdot {U}^{\mathrm{3}}-\mathrm{6}{c}_{\text{d}}\text{LAD}\left({\dot{x}}^{\mathrm{3}}\right)\cdot \left|U\right|\cdot \mathit{\epsilon}\end{array}$$

The reduction of the shortwave radiation flux is considered by including local reduction coefficients (ranging from 1 to 0) according to the vegetation characteristics. The reduction coefficients are described in terms of the vertical leaf area index, LAI, of the plant (see Sect. 5.4).

$$\begin{array}{}\text{(46)}& {\displaystyle}{\mathit{\sigma}}_{\text{SW}}\left({\dot{x}}^{\mathrm{3}}\right)=\mathrm{exp}\left(F\cdot \text{LAI}\left({\dot{x}}^{\mathrm{3}}\right)\right)\end{array}$$

5 Model input

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Several model inputs are required to run MITRAS to accurately simulate a domain for, e.g., an urban area (Fig. 4). These include, for instance, the orography heights of the domain, the surface cover types, the building data (dimensions, shape, and position), and the vegetation data for such a domain. Integrating these inputs to the computational domain of MITRAS is done in a separate preprocessor called GRIMASK (Salim, 2014). A complete description of this preprocessor is outside the scope of this paper, but the required input data and how they are in general achieved is outlined here (Sect. 5.1–5.4). Moreover, the representative meteorological conditions for the domain are required as inputs to run MITRAS; they are provided in consistency with the model physics and numeric using a preprocessor.

Urban domains might include elevated terrain. To better describe the domain
terrain, the orographic effects of the domain are considered in MITRAS by
virtue of the terrain-following coordinate system (Sect. 2.1). Both the
aerodynamic and the radiative (shading) effects of the slopes are
considered in MITRAS. For realistic applications, the orography data (terrain
height above sea level) of the domain are introduced to GRIMASK in the standard
ASCII grid format of a geographic information system (GIS). Usually these
data are in much finer resolution (less than 0.25 m) compared to the
computational domain horizontal resolution (∼1 m). GRIMASK then
aggregates these data to the surface grid cells to calculate the average
orography height for each surface grid cell. This is done by splitting
each grid cell into *n* sub-grids and calculating the orography height of
each sub-grid. The eventual orography height, *z*_{s}, of a grid cell
(*i*,*j*) is calculated from

$$\begin{array}{}\text{(47)}& {\displaystyle}{z}_{\text{s}}(i,j)={\displaystyle \frac{\mathrm{1}}{n}}\sum _{\mathrm{1}}^{n}{z}_{\text{sub}}\left(x,y\right).\end{array}$$

*z*_{sub}(*x*,*y*) is the orography height of a sub-grid.

For idealized studies and test cases, GRIMASK can generate artificial orography heights according to the objective of the test case, e.g., a bell mouth hill or a Gaussian hill.

It is essential to define the surface cover characteristics of the urban
domain because they govern the surface energy budget (Eq. 19) and all surface-dependent fluxes. In realistic cases, the urban domain usually contains
several surface cover types (water, sealed surfaces, vegetation, sand, ice,
etc.) and it becomes imperative to define an appropriate data structure to
encode information on surface cover characteristics. To this purpose, the
surface cover data are first introduced to GRIMASK in the GIS standard ASCII
grid format. GRIMASK then integrates these data into the computational grid
cells at the surface. Each grid cell is composed of at least one surface
cover class, but more SGS surface covers are allowed. The preprocessor
calculates how many surface cover classes exist in the domain and the
portion of each surface cover class in each grid cell. This is done following
the approach used to calculate the orography heights. Each grid cell at the
surface is divided into sub-grids and the surface cover class of each
sub-grid is defined. The data structure of the surface cover consists of two
datasets: (a) the portion of each surface cover class in each grid cell and
(b) a list of surface cover classes existing in the domain. Several classes
are also prepared in the surface cover class database for the different
vegetation types (coniferous trees, deciduous trees, bushes, etc.). A
database of several surface cover classes with attributed physical
parameters is available in the 1-D MITRAS model. The physical parameters
given per surface cover class include albedo (*A*), roughness length
(*z*_{0}), thermal diffusivity (*K*), thermal conductivity (*ν*_{s}),
soil water availability (*α*_{q}), and saturation value for water
content (*W*_{k}). The physical parameters are given in Table 1 for selected
surface cover classes.

For buildings the explicit treatment is normally chosen. If the implicit
consideration of obstacles is chosen, i.e., they are not explicitly resolved
in the model grid, a much larger roughness length would be required, which
conflicts with a high vertical grid resolution. This is similarly true for
trees. The roughness length for water is modified during the model
calculations with dependence on wind speed (Fischereit et al., 2016). The
roughness length of scalar quantities over water, *z*_{0a}, is
calculated dependent on the roughness length for momentum, *z*_{0}
(Brutsaert, 1975, 2013),

$$\begin{array}{}\text{(48)}& {\displaystyle \frac{{z}_{\mathrm{0}}(x,y)}{{z}_{\text{0a}}(x,y)}}=\mathrm{exp}\left(\mathit{\kappa}\left(\mathrm{7.3}{\mathit{Re}}^{\mathrm{1}/\mathrm{4}}\sqrt{{C}_{\text{a}}}-\mathrm{5}\right)\right),\end{array}$$

for temperature and humidity by substituting *C*_{a} with the Prandtl
number *Pr*=0.76 and the Schmidt number *Sc*=0.6,
respectively.

To distinguish surface cover classes, water, buildings, and sea ice, identifiers are incorporated for each surface cover class. These act as the Kronecker delta function to mark the particular class.

In order to generate the building mask used to provide the building data to the model, detailed information about the buildings in the domain is required. For instance, the building dimensions, shape, and location are needed for each building located in the domain to calculate the 3-D array volume and the building wall-based markers discussed in Sect. 4.1. This process is done in the preprocessor, GRIMASK, which allocates the buildings to the computational grid.

Since in the current version of MITRAS the 3-D field volume can be either 0 (building cell) or 1 (atmosphere cell), buildings are approximated to fit into the grid. For grid cells that are partially filled with buildings, the determination of whether these cells are building or atmosphere cells depends on how much volume of the cell is filled with building. A grid cell is considered a building cell if at least 50 % of its volume contains building. Otherwise it is counted as an atmosphere cell. This approximation is computationally efficient to consider the effect of buildings since the model equations only need to be multiplied by the 3-D field volume.

For realistic applications, complex urban building geometry can be
provided to GRIMASK in either the raster digital elevation model (DEM)
format, which is commonly used due to the advances in remote sensing
technologies, or in the ASCII 3-D computer-aided design (CAD) format. GRIMASK
integrates the high-resolution DEM data, which is a grid of squares
representing the elevation of each small grid, to the computational grid and
calculates how much volume of the building is contained in each grid cell.
When the building data are provided in the ASCII CAD format, GRIMASK uses an
approach similar to *z* buffering to integrate the building surfaces (usually
triangles) to the computational grid and calculates the array volume and the
face markers.

The vegetation input to MITRAS depends on the selected vegetation treatment in the model. In the implicit mode, the vegetation is defined as a surface cover class (Sect. 5.2). In explicit mode, however, vegetation inputs are the 3-D arrays LAD and LAI prepared by GRIMASK. Two approaches are available in GRIMASK in order to calculate these arrays based on the available plant data: the measurement approach and the analytic approach. In the first approach, the following data for each plant in the model area are processed in GRIMASK: the measured 1-D vertical leaf area index profile LAI(${\dot{x}}^{\mathrm{3}})$, the plant height, and the plant location. The following relation is used to relate LAD and LAI.

$$\begin{array}{}\text{(49)}& {\displaystyle}\text{LAI}\left({\dot{x}}^{\mathrm{3}}+\mathrm{\Delta}z\right)=\underset{z}{\overset{z+\mathrm{\Delta}z}{\int}}\text{LAD}\left({\dot{x}}^{\mathrm{3}}\right)\text{d}z\end{array}$$

In the analytical approach, GRIMASK uses the following empirical relation proposed by Lalic and Mihailovic (2004) to describe LAD profile from plant parameters.

$$\begin{array}{ll}\text{(50)}& {\displaystyle}& {\displaystyle}\text{LAD}\left({\dot{x}}^{\mathrm{3}}\right)={\displaystyle}& {\displaystyle}{\text{LAD}}_{\text{m}}{\left({\displaystyle \frac{h-{z}_{\text{m}}}{h-{\dot{x}}^{\mathrm{3}}}}\right)}^{n}\mathrm{exp}\left[n\left(\mathrm{1}-{\displaystyle \frac{h-{z}_{\text{m}}}{h-{\dot{x}}^{\mathrm{3}}}}\right)\right]\end{array}$$

LAD_{m} is the maximum LAD, *h* is the plant height above
*z*_{s}, *z*_{m} is the corresponding height above *z*_{s},
and

$$\begin{array}{}\text{(51)}& {\displaystyle}n=\left\{\begin{array}{ll}\mathrm{6}& \mathrm{0}\le {\dot{x}}^{\mathrm{3}}<{z}_{\text{m}}\\ {\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}& {z}_{\text{m}}\le {\dot{x}}^{\mathrm{3}}<h.\end{array}\right.\end{array}$$

The plant parameters used in these equations can be obtained from the forest phenology calendar.

A large-scale surface-friction-free meteorological situation is required as input to MITRAS to calculate the microclimate of a certain domain. This input is prepared by a one-dimensional model without explicit consideration of buildings but with inclusion of all relevant turbulence processes (including surface friction and Coriolis force) to provide the required meteorology data needed for the initialization of all the variables in the three-dimensional model. Among the inputs for the one-dimensional model are the large-scale speed wind components, which are taken to be the geostrophic wind and should not include any frictional effects or wind rotation with height that will both be imposed by the 1-D model, the large-scale potential temperature gradient or temperature profile, the large-scale relative humidity profile, the deep soil temperature, and the number of days without precipitation (dry days) prior to the simulation. The one-dimensional model calculates the initial values, the wind inflow profile if fixed boundary values are used, and the values at the top boundary. Since the one-dimensional model calculates the average mesoscale conditions, large-scale phenomena can be integrated into the model by controlling the inlet boundary condition using the time-slice approach for nesting (Schlünzen et al., 1990). For some applications (e.g., comparisons with wind tunnel data) it is essential to fix the inflow profiles as described in Sect. 3.2.

6 Examples of model applications

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This section provides examples of some simulations recently performed using MITRAS. The intent is not to provide model validations or verifications, as these will be done in a separate paper with a focus on this aspect, but rather to give the readers some impression about the model capacities and potential.

MITRAS results have been frequently compared to measurements of physical
models. For instance, Grawe et al. (2013) compared MITRAS results based on an
earlier model version to quality-ensured wind tunnel data for both idealized
(flow around quasi-two-dimensional beam, single cubic obstacle, and array of
cubic obstacles) and realistic (1×1 km^{2} urban domain around
Göttinger Straße in Hannover, Germany) test cases using standard
evaluation procedures (VDI, 2005). The model results show a very good
agreement with the measurements of the wind tunnel. Also, the model results
based on the current version have been compared to the wind tunnel
measurements of the Michelstadt test case (an idealized model of a Central
European city, which is publicly available in the CEDVAL-LES database:
http://www.mi.uni-hamburg.de/cedval-les). This was done during the
model validation using an updated evaluation guideline for prognostic
microscale wind field models (Grawe et al., 2015).

MITRAS has been used to simulate the wind flow field in the city center of
Hamburg in Germany. The selected domain has a size of 2×2 km^{2}
and represents a typical urban area with many features of urban complexity
such as complex building geometries, different street configurations, and
diverse surface covers (including water bodies, street pavements, and open
spaces). All available building data (shapes, dimensions, locations, etc.),
orography heights, and the surface cover characteristics of the domain are
utilized to generate the required input data for MITRAS. The meteorological
conditions used in this simulation are set so that the wind speed is
3 m s^{−1} at 200 m above the ground and the wind direction is
230^{∘}, which is a typical wind direction in Hamburg. The results are
shown in Fig. 5 in which wind speed and vertical wind are plotted at the
pedestrian height (vertical) level. The results describe the effects of
buildings on the flow field and show several flow features such as wind speed
acceleration in open areas, deceleration within dense building
configurations, and updrafts and downdrafts around the buildings. Moreover,
the figure shows that buildings alter the wind flow field and, additionally,
the orography and surface characteristics modify the wind.

MITRAS simulations with different parameterizations of urban vegetation (especially trees) were recently performed by Salim et al. (2015) to study the effect of the inclusion of trees when simulating the wind flow in different urban complexities. This study showed a significant effect of trees on the wind field and thus highlights the importance of the explicit representation of urban trees in microscale simulations. A snapshot of an animation created from the simulations performed in this study is shown in Fig. 6 and displays the wind field when the trees are considered in the simulation together with the tree sizes and locations. More details about this study can be found in Salim et al. (2015).

To show the thermal effect of a building on its surroundings, several simulations have been done using MITRAS. The building surface temperature is calculated as described in Sect. 4.2 to simulate a single high-rise building located in an urban area in Hamburg on 1 February at 13:30 (GMT + 2; Gierisch, 2011). Figure 7 shows the thermal effect of such building by comparing the air temperature field with a reference case, in which the thermal energy flux from building façades is neglected.

The model MITRAS, with embedded wind turbine parameterizations (see Sect. 4.3), has been used to produce simulation data for model validations (Linde, 2011). The selected case in this study involved the Nibe wind turbines in Nibe, Denmark. This case is selected because there is a meteorological measurement dataset for these wind turbines (Taylor, 1990). Figure 8 gives one example of the model results from this study. Comparisons of model results with the measurements showed a good agreement.

7 Summary and outlook

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The model theory of the obstacle-resolving microscale meteorological model
MITRAS version 2 has been described in this paper. Detailed descriptions of
the model equations and their formulations and approximations are presented.
The sub-grid-scale turbulence parameterization used in MITRAS is outlined
showing the Prandtl–Kolmogorov closure and the 1.5-order *E*–*ε*
turbulence closure. Also, detailed parameterizations of obstacles such as
wind turbines and vegetation (trees) are introduced. The different
boundary conditions implemented in MITRAS and the model inputs are
also outlined. The model dynamics and numerical framework of MITRAS
are established to provide a solid foundation for future model extensions.

Verification experiments of MITRAS version 2 with the simulation of urban areas with explicitly resolved obstacles (including buildings, wind turbines, and trees) will be presented in a separate paper. A recent application of MITRAS version 2 to vegetation effects in urban areas can be found in Salim et al. (2015).

Code availability

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Code availability.

Currently the MITRAS source code is distributed upon request under the terms of a user agreement with the Mesoscale and Microscale Modeling (MeMi) working group at the Meteorological Institute, University of Hamburg (https://www.mi.uni-hamburg.de/memi). A copy of the user agreement is available upon request. Due to current copyright restrictions, users are requested to contact the corresponding authors to obtain access to the code free of charge for research purposes under a collaboration agreement (metras@uni-hamburg.de).

Documentation for the M-SYS model system (Schlünzen et al., 2018a, b), in which MITRAS is included, is available online at https://www.mi.uni-hamburg.de/memi under “Numerical Models”.

Appendix A: Implicit method for dissipation of TKE

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The numerical method for calculating the dissipation term in the SGS TKE
equation is based on splitting the SGS TKE prognostic equation into two
parts. In the first part all processes except dissipation are integrated
within a time step Δ*t* to get a preliminary value of TKE, $\widehat{E}$.
In the second part TKE at the end of the complete time step ${\widehat{E}}^{n+\mathrm{1}}$
is used to integrate the dissipation term.

$$\begin{array}{}\text{(A1)}& {\displaystyle \frac{\partial {\widehat{E}}^{n+\mathrm{1}}}{\partial t}}=-\mathit{\epsilon}\end{array}$$

The dissipation is parameterized according to Eq. (13), which can be simplified as

$$\begin{array}{}\text{(A2)}& {\displaystyle}\mathit{\epsilon}=C{\displaystyle \frac{{\stackrel{\mathrm{\u203e}}{E}}^{\mathrm{3}/\mathrm{2}}}{l\left(\mathit{Ri}\right)}},\end{array}$$

where *C* is a constant.
Overall, the following equation is solved in the second step.

$$\begin{array}{}\text{(A3)}& {\displaystyle \frac{\partial {\mathit{\rho}}_{\mathrm{0}}{\mathit{\alpha}}^{*}{\widehat{E}}^{n+\mathrm{1}}}{\partial t}}=-{\mathit{\rho}}_{\mathrm{0}}{\mathit{\alpha}}^{*}C{\displaystyle \frac{{\widehat{E}}^{\mathrm{3}/\mathrm{2}}}{l\left(\mathit{Ri}\right)}}\end{array}$$

By integrating Eq. (A3), assuming *ρ*_{0}, *α*^{∗}, and
*l*(*Ri*) to be constant within one time step, the analytic solution
for the TKE dissipation is calculated as follows.

$$\begin{array}{}\text{(A4)}& {\displaystyle}& {\displaystyle}\underset{{\widehat{E}}^{n+\mathrm{1}}}{\overset{{\stackrel{\mathrm{\u203e}}{E}}^{n+\mathrm{1}}}{\int}}{\left({\widehat{E}}^{n+\mathrm{1}}\right)}^{-\mathrm{3}/\mathrm{2}}\text{d}{\widehat{E}}^{n+\mathrm{1}}=-{\displaystyle \frac{C}{l\left(\mathit{Ri}\right)}}\underset{t}{\overset{t+\mathrm{\Delta}t}{\int}}\text{d}t\text{(A5)}& {\displaystyle}& {\displaystyle}{\stackrel{\mathrm{\u203e}}{E}}^{n+\mathrm{1}}={\widehat{E}}^{n+\mathrm{1}}{\left({\displaystyle \frac{{C}^{\mathrm{2}}}{\mathrm{4}l{\left(\mathit{Ri}\right)}^{\mathrm{2}}}}\mathrm{\Delta}{t}^{\mathrm{2}}{\widehat{E}}^{n+\mathrm{1}}+{\displaystyle \frac{C}{l\left(\mathit{Ri}\right)}}\mathrm{\Delta}t\sqrt{{\widehat{E}}^{n+\mathrm{1}}}+\mathrm{1}\right)}^{-\mathrm{1}}\end{array}$$

A similar solution for the dissipation term in the TKE equation of the SGS model suggested by Deardorff (1980), suitable for large eddy simulation, is presented in Fock (2015).

Author contributions

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Author contributions.

MS organized the paper and collected the contributions.
He also developed the preprocessor GRIMASK and is responsible for the ideas
behind it (Sect. 4). He included different vegetation treatments in MITRAS
(Sect. 4.4) and provided model results on this (Sect. 6.3) as well as a
realistic application (Sect. 6.2). HS coordinated the model development since
the beginning and is overall responsible for the model and its documentation.
She provided a number of comments on the paper, as did DG, who is
responsible for model evaluation (Sect. 6.1) and aspects concerning model quality
assurance and code provision. MB contributed the wind turbine development to
MITRAS (Sect. 4.3) and corresponding results (Sect. 6.5), and AG implemented the
calculation of building surface temperatures (Sect. 4.3) and provided results
on this (Sect. 6.4). BF derived the analytic solution for the *E*−*ϵ*
relation (Eq. 13, Appendix A).

Competing interests

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Competing interests.

The authors declare that they have no conflict of interest.

Acknowledgements

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Acknowledgements.

This research is supported through the Cluster of Excellence “CliSAP”
(EXC177) and the research project UrbMod funded by the state of Hamburg,
Germany. The authors would like to thank the two anonymous reviewers and
the topical editor David Ham for their helpful and constructive comments and
support during the review process.

Edited by:
David Ham

Reviewed by: two anonymous referees

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Short summary

This paper gives a detailed description of the model theory of the obstacle-resolving microscale meteorological model MITRAS version 2. Detailed descriptions of the model equations and their formulations and approximations are presented. Also, detailed parameterizations of buildings, wind turbines, and vegetation in the model are introduced. Some example applications of the model are shown to demonstrate the model capacities and potential.

This paper gives a detailed description of the model theory of the obstacle-resolving microscale...

Geoscientific Model Development

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