In this work, the fully compressible, three-dimensional, nonhydrostatic atmospheric model called All Scale Atmospheric Model (ASAM) is presented. A cut cell approach is used to include obstacles and orography into the Cartesian grid. Discretization is realized by a mixture of finite differences and finite volumes and a state limiting is applied. Necessary shifting and interpolation techniques are outlined. The method can be generalized to any other orthogonal grids, e.g., a lat–long grid. A linear implicit Rosenbrock time integration scheme ensures numerical stability in the presence of fast sound waves and around small cells. Analyses of five two-dimensional benchmark test cases from the literature are carried out to show that the described method produces meaningful results with respect to conservation properties and model accuracy. The test cases are partly modified in a way that the flow field or scalars interact with cut cells. To make the model applicable for atmospheric problems, physical parameterizations like a Smagorinsky subgrid-scale model, a two-moment bulk microphysics scheme, and precipitation and surface fluxes using a sophisticated multi-layer soil model are implemented and described. Results of an idealized three-dimensional simulation are shown, where the flow field around an idealized mountain with subsequent gravity wave generation, latent heat release, orographic clouds and precipitation are modeled.

In this paper we present the numerical solver ASAM (All Scale
Atmospheric Model) that has been developed at the Leibniz Institute
for Tropospheric Research (TROPOS), Leipzig. ASAM was initially
designed for CFD (computational fluid dynamics) simulations around
buildings.
The model can also be used with spherical or
cylindrical grids. Stability problems with grid convergence in
special points (the pole problem) in both grids are handled through
the implicit time integration both for advection and the yet faster
gravity and acoustic waves.
For simulating the flow around obstacles, buildings or orography,
the cut (or shaved) cell approach is used. With this attempt one remains
within the Cartesian grid and the numerical pressure
derivative in the vicinity of a structure is zero
if the cut cell geometry is not taken into account,
which is not the case in terrain-following coordinate systems due to the
slope of the lowest cells

The here-presented model is a developing research code with
different options to choose from like different numerical methods
(e.g., split-explicit Runge–Kutta or partially implicit peer
schemes), number of prognostic
variables, physical parameterizations or the change to a spherical grid
type. Parallelization is realized by using the message
passing interface (MPI) and the domain decomposition method. The code
is easily portable between different platforms like Linux, IBM or Mac
OS. With these features, large eddy simulations (LES) with spatial
resolutions of

A separately developed LES model at TROPOS is called ASAMgpu

Physical constants.

This paper is structured as follows. The next section deals with
a general description of the model. It includes the basic equations
that are solved numerically and the used energy variable. Also, the
cut cell approach and spatial discretization as well as the time
integration scheme are described.
This approach can be extended to other orthogonal grids like the lat–long grid.
Section 3 deals with the model physics,
including a subgrid-scale model, a two-moment microphysics scheme
and surface flux parameterization.
Results of different idealized test
cases are shown in Sect. 4.
The first one is a dry rising heat bubble described by

The flux-form compressible Euler equations for the atmosphere
are

Possible configurations for cut cell intersection (cases 1–3) for different numbers of face intersection points (markers). The last two cases are excluded.

Stencil for third-order approximation.

Cut cell with face and volume area information (left) and arrangement of face- and cell-centered momentum (right).

The energy equation in the form of Eq. (

The number of additional equations like Eq. (

The spatial discretization is done on a Cartesian grid with grid
intervals of lengths

The spatial discretization is formulated in terms of the grid interval
length and the face and volume areas. The variables are arranged on
a staggered grid with momentum

The discretization of the advection operator is performed for
a generic cell-centered scalar variable

Assuming a positive flow in the

To solve the momentum equation, the nonlinear advection term is
needed on the face. This is achieved by a shifting technique
introduced by

Two neighbored cut cells with face and volume area information (left) and arrangement of face- and cell-centered tendency of momentum (right).

Example configuration for surface flux distribution around a cut cell.
Green shading represents the solid part of the cells, whereas blue shading is the
“virtual volume” normal to the cut cell face. The total surface flux

Terminal fall velocity of raindrops after
Eq. (

Due to small cell volumes around cut cells, boundary fluxes such as sensible and latent
surface heat fluxes (cf. Sect. 3.4) have to be distributed to the surrounding cells
to avoid instability problems because of sharp gradients in the respective scalar fields.
For simplicity we consider a two-dimensional case in

Coefficient table for ROS2.

Coefficient table for ROSRK3.

After spatial discretization, an ordinary differential equation

A Rosenbrock method has the form

Among the available methods are a second-order two-stage method after

Moreover, a new Rosenbrock method was constructed from a low storage three-stage
second-order Runge–Kutta method, which is used in split-explicit time
integration methods in the Weather Research and Forecasting (WRF)
Model

The previously described Rosenbrock W methods allow a simplified solution
of the linear systems without losing the order. When

For the
transport/source system
the Jacobian can be further split into

The second matrix of the splitting approach is written in case of the
first splitting (Eq.

As basic iterative solvers, BiCGStab is applied for the transport/source system
and GMRES for the pressure part

The set of coupled differential equations can be solved for a given
flow problem. For simulating turbulent
flows with large eddy simulations, the Euler equations mentioned at the beginning
have to be modified.
Within the technique of LES it is necessary to characterize the unresolved motion.
By solving
Eqs. (

Nevertheless, to solve the filtered set of equations, it is necessary
to parameterize the additional subgrid-scale stress terms

To take stratification effects into account,

The implementation in the ASAM code is accomplished in the main
diffusion routine of the model. It develops the whole term of

The implemented microphysics scheme is based on the work of

The sedimentation velocity of raindrops is derived as in the
operationally used COSMO model from the German Weather Service

This takes place at the tendency equation for the rain water density:

A simple way to parameterize surface heat fluxes is the usage of
a constant flux layer. There, the energy flux is directly given and
does not depend on other variables. With the density potential
temperature formulation (Eq.

In order to account for the interaction between land and atmosphere and the high diurnal variability of the meteorological variables in the surface layer, a soil model has been implemented into ASAM. In contrast to the constant flux layer model, the computation of the heat and moisture fluxes are now dependent on radiation, evaporation and the transpiration of vegetated area. Phase changes are not covered yet and intercepted water is only considered in liquid state.

The implemented surface flux scheme follows the description of Jiménez et al. (2012), which is the revised flux scheme used in the WRF model.
The surface fluxes of momentum, heat and moisture
are parameterized in the following way, respectively:

The transport of the soil water as a result of hydraulic pressure due
to diffusion and gravity within the soil layers is described by
Richard's equation:

Further addition/extraction of soil water is controlled by the
percolation of intercepted water into the ground and the evaporation
and transpiration of water from bare soil and vegetation. The
mechanisms implemented are based on the Multi-Layer Soil and
Vegetation Model TERRA_ML as described in

The topmost layer is exposed to the incoming radiation and thus has
the strongest variation in temperature in comparison to the other soil
layers within the ground. The temperature equation of the first layer
is, in addition to the incoming radiation, determined by the latent
and sensible heat flux.

Results for the dry bubble simulation after

In this section, we present six example test cases.
In five of these cases, orography
or obstacles are included to test conservation properties and model accuracy.
The first test case is the rising heat bubble prescribed in

A two-dimensional simulation of a rising thermal
is presented in

Steady-state solution for the simulation of the

Time series of total energy error for the density current test case with and without
the hill. The error is expressed as

Potential temperature field at

Equivalent potential temperature field for the moist rising bubble test case with
background wind of

In this test case, a flow over a mountain ridge is simulated

A nonlinear test problem is the density current simulation
study documented in

Another analysis is carried out by switching on the physical viscosity
of

Convergence study for the density current test case with a 1

Convergence study for the annulus advection test.

Equivalent potential temperature field for the moist rising bubble test including
a zeppelin-shaped cut area in the center of the domain. Snapshot taken at

Same as in Fig.

Computational meshes and difference scalar fields of

Convergence study for the annulus advection test.
L1 (red) and

Computational grid around the mountain for an

Horizontal cross section of horizontal wind vectors at

Vertical cross section (

Vertical cross section (

The moist bubble benchmark case after

The test problem reported in

In this section, a test case described in

A detailed description of the three-dimensional, fully compressible, nonhydrostatic All Scale Atmospheric Model (ASAM) was presented. Main focus of this work was the description of the cut cell method within a Cartesian grid structure. With this method there is no accuracy loss near steep slopes, which can occur around mountains using a high spatial resolution or when obstacles or buildings are embedded. The concepts of the spatial discretization of the advection operator and a nonlinear term in the momentum equation were outlined. A technique to distribute surface fluxes around cut cells was described. An implicit Rosenbrock time integration scheme with two splitting approaches of the Jacobian were presented, which is particularly useful to bypass the small cell problem. With the described scheme, relatively large time steps are possible. Physical parameterizations (Smagorinsky subgrid-scale model, two-moment warm microphysics scheme, multi-layer soil model) which are necessary for performing particular atmospheric simulations, e.g., large eddy simulations of marine boundary layers, are implemented in ASAM. The model produces good results when comparing scalar and velocity fields for typical benchmark test cases from the literature. It was shown that energy conservation is not affected when it comes to interaction with the flow in the vicinity of cut cells. However, perfect energy conservation cannot be expected by design. Accuracy tests show that the EOC is almost second-order for the annulus advection test.

Other model features that could not be presented in the framework of this paper are local mesh refinement and parallel usage of the model. They will be part of future studies. There, performance tests for highly parallel computing with a large number of processors will be conducted. Furthermore, high-frequency output is desired for statistical data analysis. For this reason, efficient techniques like adaption of the output on modern parallel visualization software will be developed. There are no operator splitting techniques used, which leads to a consistent treatment of new processes with respect to time and to a simple programming style for the most part.

Another focus on future model development lies in the model physics,
which includes further testing of current implementations as well as
adding new parameterizations, e.g., an ice microphysics scheme.
For the description of turbulence, other (dynamic)
Smagorinsky models

ASAM already was and will further be applied for large eddy simulations of urban and marine boundary layers. Another ongoing study deals with island effects on boundary layer modification in the trade wind area exemplified by the island of Barbados, where the island topography plays a significant role and can be well described by the cut cell method.

In this section, a straightforward derivation of the density potential
temperature tendency equation is given to get the necessary source
terms for microphysics, surface fluxes and precipitation. Therefore,
phase changes are allowed and a water vapor source term

A prognostic equation for the internal energy

A prognostic equation for the (moist) potential temperature is derived
here. This is necessary because it appears in the density potential
temperature equation later on. Quantities that contain water vapor
and liquid water are marked with a tilde to distinguish them from
their dry equivalents (e.g., dry potential temperature

The moist potential temperature is

With the definition of the density potential temperature

Varying ratios of silt, clay and sand significantly change the
properties of soil and thus determine the heat and moisture fluxes of
the surface. Accordingly, these different ratios are referred to
specifically defined soil types. In Tables

Soil parameters from

Soil parameters as used in

The ASAM code is managed with Git, a distributed revision control and
source code management (SCM) system. To get access to the source code
and additional scripts for pre- and postprocessing, registration at
the TROPOS Git hosting website

As visualization tool, the free and open source software VisIt
(

This work is internally funded by TROPOS. The authors like to thank Luca Bonaventura and the anonymous reviewer for their constructive comments to improve the quality of the paper. We are also grateful to our technical employees Sabine Reutgen and Birgit Heinrich for developing and maintaining the grid generator as well as data maintenance and converting, respectively. Edited by: H. Weller