Introduction
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 .
Since this skewness is also reproduced in upper levels, a cut cell model
produces reduced or greatly reduced errors in comparison models
with terrain-following coordinates .
Several techniques have been developed to overcome
these nonphysical errors associated with terrain-following grids,
especially when spatial scales of three-dimensional models become finer
(which leads to a steepening of the model orography).
introduced
a variable-step topography (VST) surface coordinate system within
a nonhydrostatic host model. Unlike the traditional discrete-step
approach, the depth of a grid box intersecting with a topographical
structure is adjusted to its height, which leads to straight cut
cells. Numerical tests show that this technique produces better
results than conventional approaches for different topography
(severe and smooth) types . In their cases, also
the computational costs with the VST approach are reduced because
there is no need of extra functional transform calculations due to
metric terms. derived approximations for
z coordinate nonhydrostatic atmospheric models by using the
shaved-element finite volume method. There, the dynamics are computed
in the cut cell system, whereas the physics computation remains in the
terrain-following system. Using a z-coordinate system
can also improve the prediction of meteorological parameters like clouds and
rainfall due to a better representation of the atmospheric flow near mountains
in a numerical weather prediction (NWP) model .
The cut cell method is also used in the
Ocean–Land–Atmosphere Model (OLAM) , which extends
the Regional Atmospheric Modeling System (RAMS) to a global model
domain.
In OLAM, the shaved-cell method is applied to an icosahedral
mesh .
simulated a two-dimensional flow over different
mountain slopes and compared the results of their cut cell model with
a model using terrain-following coordinates. Especially for steep slopes,
significant errors were reported in the terrain-following model.
A drawback is the generation of low-volume cells when a cut cell method is used.
To avoid instability problems around these small cells, the time
integration scheme has to be adapted. This can be achieved by using
semi-implicit or semi-Langrangian methods, for example.
In ASAM, a linear-implicit Rosenbrock time integration scheme is used .
Another option to handle the small cells problem is to merge small cut cells
with neighboring cells in either the horizontal or vertical direction .
However, this approach becomes more complicated when applying it to three spatial
dimensions, since a lot of special cases have to be considered.
To achieve reasonable vertical solutions near the ground, the usage of
local mesh refinement techniques becomes interesting for large-scale models .
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 O(1–100 m) can be performed with
respect to a sufficiently resolved terrain structure.
In previous studies, the model was used to demonstrate
the volume-of-fluid (VOF) method for nondissipative cloud transport
. ASAM also took part in an intercomparison study
of mountain-wave simulations for idealized and real terrain profiles,
where altogether 11 different nonhydrostatic numerical models were compared
. A partially implicit peer method is presented in
in order to overcome the small cell problem around orography when using cut cells.
The model was
recently used for a study of dynamic flow structures in a turbulent
urban environment of a building-resolving resolution
. There, the implementation
of a dynamic Smagorinsky subgrid-scale model is tested for a
convective atmospheric boundary layer and an inflow generation approach
that produces a turbulent flow field is presented.
A separately developed LES model at TROPOS is called ASAMgpu
. It includes some basic features of the ASAM code
and runs on graphics processing units (GPUs), which enables very
time-efficient computations and post-processing. However, this model
is not as adjustable as the original ASAM code and the inclusion of
three-dimensional orographical structures is not implemented so far.
ASAMgpu was applied for a study of heat island effects
on vertical mixing of aerosols by comparing the results of large eddy simulations
with wind and aerosol lidar observations .
Physical constants.
Symbol
Quantity
Value
p0
Reference pressure
105 Pa
Rd
Gas constant for dry air
287 Jkg-1K-1
Rv
Gas constant for water vapor
461 Jkg-1K-1
cpd
Specific heat capacity at constant pressure for dry air
1004 Jkg-1K-1
cpv
Specific heat capacity at constant pressure for water vapor
1885 Jkg-1K-1
cpl
Specific heat capacity at constant pressure for liquid water
4186 Jkg-1K-1
cvd
Specific heat capacity at constant volume for dry air
717 Jkg-1K-1
cvv
Specific heat capacity at constant volume for water vapor
1424 Jkg-1K-1
L00
Latent heat at 0 K
3.148×106 Jkg-1
g
Gravitational acceleration
9.81 ms-2
Cs
Smagorinsky coefficient
0.2
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 followed
by a flow past an idealized mountain ridge from .
Two other “classical” test examples have been chosen and modified so that cut cells
are included and interaction within these cells are guaranteed.
The first one is a falling cold bubble with a developing density current
but with a 1 km high mountain on the left part of the domain.
The second case is the moist bubble benchmark case reported by .
The bubble will rise and interact with a zeppelin-shaped cut area in the center of
the domain. For both test cases, energy conservation tests are carried out.
Another two-dimensional case by is presented to test the
accuracy of the cut cell method by advecting a smooth bump in a radial wind
field in an annulus.
For the last test case, a three-dimensional mountain overflow with subsequent
orographic cloud generation and precipitation is simulated .
Concluding remarks and future work are in the final section.
Description of the All Scale Atmospheric Model
Governing equations
The flux-form compressible Euler equations for the atmosphere
are
∂ρ∂t+∇⋅(ρv)=0,∂(ρv)∂t+∇⋅(ρvv)=-∇⋅τ-∇p-ρg-2Ω×(ρv),∂(ρϕ)∂t+∇⋅(ρvϕ)=-∇⋅qϕ+Sϕ,
where ρ is the total air density, v=(u,v,w)T
the three-dimensional velocity vector, p the air pressure, g
the gravitational acceleration, Ω the angular velocity
vector of the earth, ϕ a scalar quantity and Sϕ the sum of
its corresponding source terms.
The subgrid-scale terms are τ for
momentum and qϕ for a given scalar.
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. () is
represented by the (dry) potential temperature θ. In the
presence of water vapor and cloud water, this quantity is replaced by
the density potential temperature θρ as
a more generalized form of the virtual potential temperature
θv:
θρ=θ1+qvRvRd-1-qc,
where the equation of state can be expressed as follows:
p=ρRdθρpp0κm.
In the previous two equations θ=T(p0/p)κm is the
potential temperature, qv=ρv/ρ is the
mass ratio of water vapor in the air (specific humidity),
qc=ρc/ρ is the mass ratio of cloud water
in the air, p0 a reference pressure and κm=(qdRd+qvRv)/(qdcpd+qvcpv+qccpl) the
Poisson constant for the air mixture (dry air, water vapor, cloud
water) with qd=ρd/ρ. Rd and
Rv are the gas constants for dry air and water vapor,
respectively.
The number of additional equations like Eq. ()
depends on the complexity of the microphysical
scheme used. Furthermore, tracer variables can also be included. The
values of all relevant physical constants are listed in
Table .
Cut cells and spatial discretization
Definition of cut cells
The spatial discretization is done on a Cartesian grid with grid
intervals of lengths Δxi,Δyj,Δzk and can
easily be extended to any orthogonal, logically rectangular structured grid
(i.e., it has the same logical structure as a regular Cartesian grid)
like spherical or cylindrical coordinates.
First, it is described for the Cartesian case.
Generalizations are discussed afterwards.
Orography and other obstacles like buildings are presented by cut cells, which
are the result of the intersection of the obstacle with the underlying
Cartesian grid. In Fig. different possible and excluded
configurations are shown for the three-dimensional case.
For the spatial discretization only the six partial face areas and
the partial cell volume and the grid sizes of the underlying Cartesian mesh
are used. For a proper representation the orography is smoothed in such a way
that the intersection of a grid cell and the orography can be described by
a single possible nonplanar polygon. Or in other words, a Cartesian cell is
divided in at most two parts, a free part and a solid part.
For each Cartesian cell, the free face area of the six faces and the free
volume area of the cell are stored, which is the part outside
of the obstacle. These values are denoted for the grid cell i,j,k
by FUi-1/2,j,k,
FUi+1/2,j,k,
FVi,j-1/2,k,
FVi,j+1/2,k,
FWi,j,k-1/2,
FWi,j,k+1/2 and
V i,j,k,
respectively. In the following, the relative notations
FUL and FUR are used, e.g., as shown in
Fig. .
Spatial discretization
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
V=(U,V,W)=(ρu,ρv,ρw)
at the cell faces and all other variables at the cell center.
The discretization is a mixture of finite volumes and finite differences.
In the finite volume context the main task is the reconstruction of values
and gradients at cell faces from cell-centered values.
The discretization of the advection operator is performed for
a generic cell-centered scalar variable ϕ. In the context of
a finite volume discretization, point values of the scalar value ϕ
are needed at the faces of this grid cell. Knowing these face values,
the advection operator in the x direction is discretized by
(FURUFRϕR-FULUFLϕL)/VC where UF is the discretized momentum
at the corresponding faces. To approximate these values at the
faces, a biased upwind third-order procedure with additional limiting
is used .
Assuming a positive flow in the x direction, the third-order
approximation at xi+1/2 is obtained by quadratic interpolation
from the three values as shown in Fig. .
The interpolation condition is that the three cell-averaged values are
fitted:
ϕFR=ϕC+hC(hL+hC)(hC+hR)(hL+hC+hR)(ϕR-ϕC)+hChR(hL+hC)(hL+hC+hR)(ϕC-ϕL)=ϕC+α1(ϕR-ϕC)+α2(ϕC-ϕL).
To achieve positivity in Eq. (), we apply state
limiting. For this task, Eq. () is rewritten in slope-ratio
formulation:
ϕFR=ϕC+K(ϕC-ϕL),
where
K=α1ϕR-ϕCϕC-ϕL+α2.
Then K is replaced by limiter function Ψ and Eq. () is rewritten as
ϕFR=ϕC+ΨϕR-ϕCϕC-ϕL(ϕC-ϕL),
Ψ(r)=max0,minr,min(δ,α1r+α2),δ=2,
as proposed by . This limiter has the property that
the unlimited higher-order scheme (Eq. ) is used as much as
possible and is utilized only then when it is needed. In the case
of Ψ=0, the scheme degenerates to the simple first-order upwind
scheme. The coefficients α1 and α2 can be
computed in advance to minimize the overhead for a nonuniform grid.
In the case of a uniform grid the coefficients are constant, i.e., they
are equal to 1/3 and 1/6. For a detailed discussion of the
benefits of this approach and numerical experiments also see
.
This procedure is applied in all three grid directions, where the
virtual grid sizes h are defined as
hL=VL/FL,hC=0.5VC/(FL+FR),hR=VR/FR.
Momentum
To solve the momentum equation, the nonlinear advection term is
needed on the face. This is achieved by a shifting technique
introduced by for the incompressible
Navier–Stokes equation. For each cell, two cell-centered values of
each of the three components of the Cartesian velocity vector are
computed and transported with the above advection scheme for
a cell-centered scalar value.
The obtained tendencies are then interpolated back to the faces.
This approach avoids separate advection routines
for the momentum components.
For a normal cell the shifted values are obtained from the six momentum face values,
whereas for a cut cell the shift operation takes into account the weights of
the faces of the two opposite sides, compare with Fig. for the used notation.
ULC=UFLifFUL≥FUR(UFLFUL+UFR(FUR-FUL))/FURelse.
The interpolation of the cell tendencies TULC and
TURC back to a face tendency TUF is obtained by
the arithmetic mean of the two tendencies of the two shifted cell
components originating from the same face. For a cut face the
interpolation takes the form (see Fig. )
TUF=TURCVLFUL+FUC+TULCVRFUR+FUC/VLFUL+FUC+VRFUR+FUC.
The pressure gradient and the Buoyancy term are computed for all faces
with standard difference and interpolation formulas with the grid
sizes taken from the underlying Cartesian grid.
To approximate the pressure gradient at the interface of two grid cells with only the
pressure values of the two grid cells there is some freedom in choosing the grid size.
Whereas in the grid size is chosen to preserve energy in their model,
we follow and do not take in to account the cut cell structure.
Both versions are implemented in the ASAM code and it became apparent that the second
one is more suitable to simulate flows in hydrostatic balance.
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 Qsurf is distributed
within the dotted area.
Terminal fall velocity of raindrops after
Eq. ().
Boundary flux distribution near cut cells
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 x and z directions.
An example configuration is displayed in Fig. .
The common partial face area of two neighboring cells is greater than the opposite face
(i.e., left vs. right in x and y directions, bottom vs. top), e.g., in our case
FUL>FUR and FWT>FWB.
For the flux distribution we first define a virtual volume
over the cut cell face FC through
Vvirt=FChvirt,
where
hvirt=Δx|nx|+Δz|nz|.
Here, (nx,nz)T is the normal unit vector of the cut cell face FC.
Then the flux fraction with a weight of VC/Vvirt is added to the cut cell
(cell 2 in Fig. ).
The remaining part that has to be distributed is weighted by the face values
|FUL-FUR|/Fsurf⋅VC/(ΔxΔyΔz) to the neighbored cells in x direction
(cell 1 in the example) and
|FWB-FWT|/Fsurf⋅VC/(ΔxΔyΔz) in z direction (cell 4 in the example), where
Fsurf=|FUL-FUR|+|FWB-FWT|.
With this approach, only the available information of the considered cut cell
(volume and common face areas with neighboring cells) and not of its surrounding cells is needed.
The extension of this method to the third spatial dimension is done analogously.
Coefficient table for ROS2.
0
2/3
-5/43/4‾
-4/3
12+163
A-Matrix
Γ-Matrix
γ
Coefficient table for ROSRK3.
0
0
1/3
-11/27
1
11/54
1/2
17/27
-11/4
-17/2711/41‾
A-Matrix
Γ-Matrix
γ
Time integration
After spatial discretization, an ordinary differential equation
y′(t)=F(y(t))
is obtained that has to be integrated in time (method of lines). To
tackle the small time step problem connected with tiny cut cells,
linear implicit Rosenbrock W methods are used
.
A Rosenbrock method has the form
I-τγJki=τFyn+∑j=1i-1αijuj+∑j=1i-1γijkj,i=1,…,s,yn+1=yn+∑j=1sαs+1jkj,
where yn is a given approximation at y(t) at time tn and
subsequently yn+1 at time tn+1=tn+τ. In addition, J is
an approximation to the Jacobian matrix ∂F/∂y.
A Rosenbrock method is therefore fully described by the two matrices
A=(αij), Γ=(γij) and the parameter
γ.
Among the available methods are a second-order two-stage method after
.
Sk1=τF(yn),Sk2=τFyn+23k1-43k1,yn+1=yn+54k1+34k2,S=I-γτJ,J≈F′(yn),
with γ=12+163 or in matrix form in
Table .
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 or in the Consortium for Small-scale Modeling
(COSMO) model .
Its coefficients are given in Table .
The previously described Rosenbrock W methods allow a simplified solution
of the linear systems without losing the order. When
J=JA+JB the matrix S can be
replaced by S=(I-γτJA)(I-γτJB). Further simplification can be
reached by omitting some parts of the Jacobian or by replacing of the
derivatives by the same derivatives of a simplified operator
F̃(wn). For instance, higher-order interpolation formulae are
replaced by the first-order upwind method. The structure of the
Jacobian is
J=∂Fρ∂ρ∂Fρ∂V0∂FV∂ρ∂FV∂V∂FV∂Θ0∂FΘ∂V∂FΘ∂Θ.
A 0 block indicates that this block is not included in the
Jacobian or is absent. The derivative with respect to ρ is only
taken for the buoyancy term in the vertical momentum equation. Note
that this type of approximation is the standard approach in the
derivation of the Boussinesq approximation starting from the
compressible Euler equations. The matrix J can be
decomposed as
J=JT+JP=∂Fρ∂ρ00∂FV∂ρ∂FV∂V000∂FΘ∂Θ+0∂Fρ∂V000∂FV∂Θ0∂FΘ∂V0
or
J=JT+JP=∂Fρ∂ρ000∂FV∂V000∂FΘ∂Θ+0∂Fρ∂V0∂FV∂ρ0∂FV∂Θ0∂FΘ∂V0.
The first part of the splitting JT is called the
transport/source part and contains the advection, diffusion and source
terms like Coriolis, curvature, buoyancy, latent heat, and so on. The
second matrix JP is called the
pressure part and involves the derivatives of the pressure
gradient, with respect to the density-weighted potential temperature, and of the divergence, with respect to momentum
of the density and potential temperature equation. The difference
between the two splitting approaches is the insertion of the
derivative of the gravity term in the transport or pressure
matrix. The first splitting (Eq. ) damps sound waves.
For this splitting the second linear system with the pressure part of the
Jacobian can be reduced to a Poisson-like equation.
The second splitting (Eq. ) damps sound and gravity waves
but the dimension of the pressure system is doubled.
Both systems are solved by
preconditioned conjugate-gradient (CG)-like methods .
For the
transport/source system
the Jacobian can be further split into
JT=JAD+JS,
where the matrix JAD is the derivative of the
advection and diffusion operator where the unknowns are coupled
between grid cells. The matrix JS assembles the
source terms. Here the coupling is between the unknowns of different
components in each grid cell.
With this additional splitting the linear equation
(I-γτJAD-γτJS)Δw=R
is preconditioned from the right with the matrix
Pr=(I-γτJAD)-1
and from the left with the matrix
Pl=(I-γτJS)-1.
The matrix
Pl(I-γτJAD-γτJS)Pr
can be written in the following form by using the Eisenstat trick :
(I-γτPlJAD)Pr=(I+Pl((I-γτJAD)+I))Pr.
Therefore, only the LU-decomposition of the matrix (I-γτJS) has to be stored. The matrix
(I-γτJAD) is inverted by
a fixed number of Gauss–Seidel iterations.
The second matrix of the splitting approach is written in case of the
first splitting (Eq. ) as follows:
(I-γτJP)=IγτGRADDΘγτDIVDVI,
where DV and DΘ are diagonal matrices.
GRAD and DIV are matrix representations of the discrete
gradient and divergence operator. The entries of the matrix DV are
the potential temperature at cell faces and the entries of the matrix DΘ
are the derivative of the pressure in the cell center with respect to the
density-weighted potential temperature.
Elimination of the
momentum part gives a Helmholtz equation for the increment of the
potential temperature. This equation is solved by a CG-method with
a multi-grid as a preconditioner. For the second splitting
(Eq. ), the resulting matrix is doubled in dimension and
not symmetric anymore.
As basic iterative solvers, BiCGStab is applied for the transport/source system
and GMRES for the pressure part .
The number of iterations for the two iterative methods are problem dependent. They increase
with increasing time step and are usually in the range of 2–5 iterations.
Physical parameterizations
Smagorinsky subgrid-scale model
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. ()–() numerically with
a grid size which is larger than the size of the smallest turbulent scales,
the equations have to be filtered. Large eddy simulation employs
a spatial filter to separate the large-scale motion from the small
scales. Large eddies are resolved explicitly by the prognostic Euler
equations down to a predefined filter-scale Δ, while smaller
scales have to be modeled. Due to the filtering operation, additional
terms that cannot be derived trivially occur in the set of Euler
equations.
Nevertheless, to solve the filtered set of equations, it is necessary
to parameterize the additional subgrid-scale stress terms τij=uiuj‾-u‾iu‾j for momentum and
qij=uiqj‾-u‾iq‾j for
potential. Note that τij expresses the effect of
subgrid-scale motion on the resolved large scales and is often
represented as an additional viscosity νt with the
following formulation:
τij=-2νtS‾ij,
where S‾ij=12∂u‾i∂xj+∂u‾j∂xi is the strain rate tensor and
νt the turbulent eddy viscosity. To determine the
additional eddy viscosity, the standard Smagorinsky subgrid-scale
model is used:
νt=(CsΔ‾)2|S‾|,
where Δ‾ is a length scale, Cs the
Smagorinsky coefficient, and using the Einstein summation notation for
standardization |S‾|=2S‾ijS‾ij. The grid spacing is used as
a measure for the length scale. This standard Smagorinsky
subgrid-scale model is widely used in atmospheric and engineering
applications. The Smagorinsky coefficient Cs has
a theoretical value of about 0.2, as estimated by .
Applying this value to a turbulence-driven flow with thermal
convection fields results in a good agreement with observations as
shown by .
To take stratification effects into account, modified
the standard Smagorinsky formulation by changing the eddy viscosity to
νt=(CsΔ‾)2max0,|S‾|21-RiPr1/2
with
Ri=gθρ∂θρ∂z|S‾|2.
Here, Ri is the Richardson number and Pr is the turbulent Prandtl
number. In a stable boundary layer the vertical gradient of the
potential temperature is greater than zero (positive), which leads to
a positive Richardson number and, thus, the additional term
Ri/Pr reduces the square of the strain rate tensor
and decreases the turbulent eddy viscosity. Therefore, less turbulent
vertical mixing takes place.
The implementation in the ASAM code is accomplished in the main
diffusion routine of the model. It develops the whole term of
∂/∂xjρDSij for every time
step. The coefficient D represents Dmom for the
momentum and Dpot for the potential subgrid-scale stress.
Further routines describe the computation of Dmom and
Dpot in the following way:
Dmom=(CsΔ‾)2|S‾|.
The potential subgrid-scale stress is related to the Prandtl
similarity and can be developed by dividing the subgrid-scale stress
tensor for momentum by the turbulent Prandtl number Pr that typically
has a value of 1/3 . The length scale
Δ‾ in the standard Smagorinsky formulation is set to
the value of grid spacing. However, the cut cell approach makes it
difficult because of tiny and/or anisotropic cells. To overcome this
deficit the value is defined after :
Δ‾=(Δ1Δ2Δ3)1/3f(a1,a2).
Δ is the grid spacing in orthogonal directions, and
a correction function f is applied as follows:
f(a1,a2)=cosh427ln2a1-lna1lna2+ln2a21/2witha1=Δ1Δ3,a2=Δ2Δ3.
Here, a1 and a2 are the ratios of grid spacing in different
directions with the assumption that Δ1≤Δ2≤Δ3. For an isotropic grid f=1.
Two-moment warm cloud microphysics scheme
The implemented microphysics scheme is based on the work of
. This scheme explicitly represents two moments
(mass and number density) of the hydrometeor classes cloud droplets
and rain drops. Ice phase hydrometeors are currently not implemented
in the model. Altogether, seven microphysical processes are included:
condensation/evaporation (COND), cloud condensation nuclei
(CCN) activation to cloud droplets at
supersaturated conditions (ACT), autoconversion (AUTO),
self-collection of cloud droplets (SCC), self-collection of rain
drops (SCR), accretion (ACC) and evaporation of rain
(EVAP):
∂(ρqv)∂t+∇⋅(ρvqv)=-SCOND-SACT+SEVAP,∂(ρqc)∂t+∇⋅(ρvqc)=+SCOND+SACT-SAUTO-SACC,∂(ρqr)∂t+∇⋅(ρvqr)=+SAUTO+SACC-SEVAP,∂NCCN∂t+∇⋅(vNCCN)=-SCONDN-SACTN+SEVAPN,∂Nc∂t+∇⋅(vNc)=+SCONDN+SACTN-SAUTON-SACCN-SSCC,∂Nr∂t+∇⋅(vNr)=+SAUTON+SACCN-SEVAPN-SSCR.
Details on the conversion rates can be found in .
Additionally, a limiter function is used to ensure numerical stability
and avoid nonphysical negative values . Since there
is no saturation adjustment technique in ASAM, the condensation
process is taken as an example to demonstrate the physical meaning of
the limiter functions. Considering the available water vapor density
ρv and the cloud water density ρc, the
process of condensation (or evaporation of cloud water, respectively)
is forced by the water vapor density deficit and limited by the
available cloud water.
FOR=ρv-(pvsT/Rv)LIM=ρcSCOND=FOR-LIM+(FOR2+LIM2)1/2τCOND
Here, pvs is the saturation vapor pressure and the
relaxation time is set to τCOND=1 s. The
numerator term is called the Fischer–Burmeister function and has
originally been used in the optimization of complementary problems
cf.. A simple model after is
applied to determine the corresponding changes in the number
concentrations and to ensure a reduction of the cloud droplet number
density to zero if there is no cloud water present. This means that
Nc decreases when droplets are getting too small,
SCONDN=min0,Cρcxmin-Nc,
and increases when droplets are getting too large,
SCONDN=max0,Cρcxmax-Nc,
where xmin and xmax are limiting parameters for cloud water.
This ensures that the cloud droplet number concentration is within a certain range
defined by distribution parameters in if condensate is present.
A timescale factor of C=0.01 s-1 controls the speed of this correction
and appears to be reasonable for this particular process.
Precipitation
The sedimentation velocity of raindrops is derived as in the
operationally used COSMO model from the German Weather Service
. There, the following assumptions are made. The
precipitation particles are exponentially distributed with respect to
their drop diameter (Marshall–Palmer distribution):
fr(D)=N0rexp-λrD.
Here, λr is the slope parameter of the distribution
function and N0r=8×106m-4 is an
empirically determined distribution parameter. The terminal fall
velocity of raindrops is then assumed to be uniquely related to drop
size, which is expressed by the following empirical function:
Wf(D)=crD1/2
with cr=130m1/2s-1. Finally, the
precipitation flux of rainwater can be calculated as
Pr=ρrWf(ρr)=∫0∞m(D)Wf(D)fr(D)dD
with the raindrop mass
m(D)=πρWD3/6,
where ρW=1000kgm-3 is the mass density
of water. This leads to an expression for the terminal fall velocity
of raindrops in dependence on their density:
Wf(ρr)=-crΓ(4,5)6ρrπρWN0r1/8.
This takes place at the tendency equation for the rain water density:
∂(ρqr)∂t+∇h⋅(ρvhqr)+∂∂zρqrw+Wf=Sqr.
Surface fluxes
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. ), the source term
for this quantity has to be calculated:
∂(ρθρ)∂t+∂∂xj(ρθρuj)=ρ∂θρ∂t+θρ∂ρ∂t+θρ∂ρuj∂xj+ρuj∂θρ∂xj=ρ∂θρ∂t+uj∂θρ∂xj+θρ∂ρ∂t+∂ρuj∂xj=ρdθρdt+θρSv.
Sv is the source term of water vapor (in kgm-3s-1). Considering
Eq. (), adding the sensible heat flux and
neglecting phase changes leads to
∂(ρθρ)∂t+∂∂xj(ρθρuj)=Sθρ
with
Sθρ=ρθρShT+SvρdRvRm-lnπRvRm-cpvcpml,
where Sh is the heat source (in Ks-1), Rm=Rd+rvRv and cpml=cpd+rvcpv+rlcpl are the gas constant and the
specific heat capacity for the air mixture, respectively. The
corresponding surface fluxes (in Wm-2) are
Ssens=ShρdcpmlρA,Slat=SvLv(T)VA.
Here, Lv=L00+(cpv-cpl)T is the
latent heat of vaporization, A is the cell surface at the bottom
boundary and V the cell volume.
Soil model
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:
τzx=ρCm|vh|u(h),-ρcpw′θ′‾=ρcpCh|vh|θ(h)-θ(z0T),-ρLw′q′‾=ρLCq|vh|q(h)-q(z0q).
Cm, Ch and Cq are the bulk
transfer coefficients and it is considered that
Ch=Cq.
In , the bulk transfer coefficients are defined as
follows:
Cm, h=k2ΨMΨM, H
with
ΨM ,H=lnz+z0z0-ϕm, hz+z0L+ϕm, hz0L
and ϕm, h representing the integrated similarity
functions. L stands for the Obukhov length and k is the
von Kármán constant. In neutral to highly stable conditions
ϕm, h follows and in unstable
situations the ϕ-functions follow . For
further details concerning limitations and restrictions see
.
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:
∂Wsoil,k∂t=∂∂zDiff∂Wsoil,k∂z+κsoil,k
with the diffusion coefficient
Diff=κsoil,k∂Ψsoil,k∂Wsoil,k.
Wsoil,k is the volumetric water content in the kth
soil layer. Ψsoil stands for the matric potential and
κsoil is the hydraulic conductivity. Ψsoil
and κsoil are parameterized based on
:
κsoil=κsatWeff1-1-Weff1mm2′Ψsoil=ΨsatWeff-1m-11n.
Weff describes the effective soil wetness, which takes
a residual water content Wres into account, restricting the
soil from complete desiccation. κsat and
Ψsat are the hydraulic conductivity and the matric
potential at saturated conditions, respectively. The parameters m
and n describe the pore distribution with
m=1-1/n (also see Tables and ).
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
evaporation of bare soil is adjusted to the parameterization proposed
by .
The variation of the soil temperature is a result of heat conductivity
depending on the soil texture and the soil water content of the
respective soil layer:
∂Tsoil∂t=1ρc∂∂zλ∂Tsoil∂z+EqρwcwT‾soil.
Tsoil is the absolute temperature in the kth soil layer (in K), T‾soil is the mean soil temperature of two
neighboring soil layers. The change in internal energy due to changes
in moisture by the inner soil water flux, evapotranspiration and
evaporation from the upper soil layer and the interception reservoir
is treated by the second term in brackets. The heat
conductivity λ and the volumetric heat capacity ρc are
variables that depend on the soil texture. The heat capacity of the
soil ρc formulated by is the sum of the heat
capacity of dry soil (ρ0c0, see Tables
and ), the heat capacity of wet soil (ρwcw) and the heat capacity of the air within the soil pores
(ρaca).
ρc=Wsoilρwcw+1-Wpvρ0c0+Wpv-Wsoilρaca
with Wpv corresponding to the soil pores,
ρwcw=4.18×106 Jm-3K-1 and ρaca=1298 Jm-3K-1. The heat conductivity
λ is defined after :
λ=418exp-Ψlog-2.7ifΨlog≤5.10.172ifΨlog>5.1
with Ψlog=log10100Ψsoil.
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.
∂Tsoil,1∂t=1ρc∂∂zλ∂Tsoil,1∂z+ΔQ
with
ΔQ=Qdir+Qdif-σTsfc4-cpQSH-LvQLH.
Here, QLH is the latent heat flux, describing the moisture flux
between soil and atmosphere as the sum of evaporation and
transpiration and QSH is the sensible heat
flux. Qdir and Qdif represent the direct and
diffusive irradiation, respectively.
Results for the dry bubble simulation after at t=1000 s:
contours of perturbation potential temperature (left panel) and
vertical velocity (right panel).
Solid (dashed) lines indicate positive (negative) values.
Contour intervals are 0.25 K (zero contour omitted) and
2 ms-1, respectively.
Test cases
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 .
The bubble is initially defined by a radial temperature perturbation,
which leads to rising motion due to buoyant forces.
Results of a two-dimensional gravity wave simulation
are shown in the second test case. There, the gravity waves are induced by an idealized
mountain ridge. Two subcases with different atmospheric stabilities
and reference temperatures are examined.
The third case is a sinking cold bubble in a dry
environment, from which a density current develops .
A 1 km tall hill is added at the left side of the domain so that
the resulting current overflows over the mountain.
Considering moisture effects and phase changes, the moist bubble case by
with the addition of a mid-air zeppelin
is performed. Besides analyzing the flow field in the vicinity of the
obstacles, conservation studies regarding total energy are performed for
both cases.
Another idealized benchmark case is carried out to analyze the accuracy
of the presented discretization method for cut cells. There, a scalar field
is advected by a radial wind field in an annulus . This
is also a suitable test for convergence studies by calculating L1
and L∞ error norms since an analytical solution can be used for comparison.
The last test case is a three-dimensional simulation study
regarding flow dynamics around an idealized mountain
and orographic precipitation by .
Dry bubble
A two-dimensional simulation of a rising thermal
is presented in .
This test case is also used as a dry reference case for the moist bubble simulation
in .
The domain is 20 km in the horizontal direction
and 10 km in the vertical direction with a uniform grid spacing of 125 m.
A mean flow of U=20 ms-1 is applied. After 1000 s the
bubble has been transported through the lateral periodic boundaries and is again
located at the center of the domain.
The perturbation field takes the following form:
θ′=2cos2πL2
with
L=x-xcxr2+z-zczr2≤1 .
The parameters xc=10 km, zc=2 km and xr=zr=2 km determine
the position and radius of the heat bubble.
The atmosphere is in hydrostatic balance and neutrally stable with
a surface pressure p0=1000 hPa and
a constant potential temperature θ=300 K.
Results of this simulation are displayed in Fig. .
The overall shape is reproduced and comparable to the reference solution.
Slight asymmetries are observed due to lateral transport.
However,
it becomes apparent that third-order Runge–Kutta time integration
together with a fifth-order advection scheme produces
better results in terms of minimum/maximum values and
symmetry cf. than the third-order advection scheme that is used here.
Steady-state solution for the simulation of the test case:
contours of vertical velocity for (a) N=0.01 s-1 and (b) N=0.01871 s-1.
Solid (dashed) lines indicate positive (negative) values.
Contour intervals are (a) 0.05 ms-1 and (b) 0.09355 ms-1, respectively.
Time series of total energy error for the density current test case with and without
the hill. The error is expressed as 10-4 % of the total energy at the beginning
of the simulation.
Potential temperature field at t=900 s for the density current test case
with an Agnesi hill on the left side of the domain and for different grid spacings
Δx=Δz=200, 100, 50, and 25 m (top to bottom).
Equivalent potential temperature field for the moist rising bubble test case with
background wind of U=20 ms-1. Snapshot taken at t=1000 s simulation time.
2-D mountain gravity waves
In this test case, a flow over a mountain ridge is simulated
.
A dry stable atmosphere is defined by a constant
Brunt–Väisälä frequency N, upstream surface temperature T0,
surface pressure p0 and a uniform inflow velocity U.
The domain extends
200 km horizontally and 19.5 km vertically with grid
spacings of Δx=500 m and Δz=300 m.
The structure of the mountain ridge is represented by a bell curve
shape with superposed variations:
h(x)=h0exp-[x/a]2cos2πx/λ,
where h0=250 m, a=5 km and λ=4 km.
The simulation result for the steady state is shown in
Fig. . There are no nonphysical distorted wave
patterns and the result agrees well with the analytical and reference solutions
shown in .
Cold bubble with orography interaction
A nonlinear test problem is the density current simulation
study documented in . In this case, the
computational domain extends from -18 to 18 km in
the horizontal direction and from 0 to 6.4 km in the vertical
direction with isotropic grid spacing of Δx=Δz=100 m.
Boundary conditions are periodic in the x direction and the free-slip
condition is applied for the top and bottom model boundaries.
The total integration time is t=1800 s. The
initial atmosphere is in a dry and hydrostatically balanced state
and there is a horizontally homogeneous environment with
θ‾=300 K (i.e., neutrally stratified).
The perturbation (cold bubble with negative buoyancy) is defined by
a temperature perturbation of
T′=0.0∘CifL>1.0-15.0∘C(cos[πL]+1.0)/2ifL≤1.0,
where
L=(x-xc)xr-12+(z-zc)zr-120.5
and xc=0.0 km, xr=4.0 km,
zc=3.0 km and zr=2.0 km.
At first, there is no fixed physical viscosity turned on as in the original test case
(with ν=75 m2s-1) since a conservation test regarding total energy is carried out.
For this test, two simulation runs are performed with
(a) the above described standard setup and
(b) a modified setup where a mountain is added at the left part of the domain.
The mountain follows the “Witch of Agnesi” curve:
h(x)=H/(1+[(x-xM)/a1]2)ifx<xMH/(1+[(x-xM)/a2]2)ifx≥xM,
with half-width lengths a1=a2=1 km, mountain peak center position
xM=-6 km
and mountain height H=1 km.
Figure shows the temporal evolution of the total energy error for both simulations.
In a dry atmosphere, the total energy is
Ed=ρ(qdcvdT+gz+0.5|v|2).
Since exact energy conservation is not expected due to the model design, there is some kind of
energy loss for both simulations in the order of 10-3 % at the end of the integration time.
However, this is still acceptable due to the fact that in the test case there are very sharp gradients
in potential temperature and wind speeds. Also, the difference of the total energy error between
the two cases is very small (10-4 %). This means that, in this case,
cut cells do not affect the conservation properties in the model at all.
Total mass is always conserved within machine precision.
Another analysis is carried out by switching on the physical viscosity
of ν=75 m2s-1 as in . Four simulations
are performed with different isotropic grid spacings of 200, 100, 50 and 25 m,
respectively.
The potential temperature field after 900 s of integration time for these spatial resolutions
is shown in Fig. . Table shows minimum/maximum values
of horizontal wind speed and potential temperature at this time.
pointed out that their solutions
show convergence at the 50 m spacing for this test case (without hill)
with a fully compressible nonhydrostatic model.
The same behavior can be observed with ASAM simulations (not shown here), which does
also not change when the mountain is added to the domain.
Despite that there is a slight change in maximum wind speed, the potential temperature
field for the 25 and 50 m resolutions are nearly identical.
Some notable differences in the field can be observed for the 100 m resolution,
which are even more pronounced for the 200 m simulation.
Convergence study for the density current test case with a 1 km tall hill.
Minimum/maximum values of horizontal velocity and potential temperature for different grid spacings.
Δx
umin (ms-1)
umax (ms-1)
θmin (K)
θmax (K)
200
-25.93
35.64
291.89
300.01
100
-28.87
38.52
290.85
300.01
50
-28.90
38.31
290.71
300.00
25
-28.91
37.89
290.70
300.00
Convergence study for the annulus advection test.
L1 error norm (full domain), experimental order of convergence (EOC),
L∞ error norm, minimum and maximum tracer values for different meshes.
N
Domain L1 error
EOC
L∞ error
ϕmin
ϕmax
50
1.6377×10-2
–
1.9176×10-1
-2.0022×10-3
1.00048
100
4.9439×10-3
1.73
1.1860×10-1
-8.3884×10-4
1.00066
200
1.3653×10-3
1.86
6.3112×10-2
-8.5424×10-13
0.99977
400
3.7196×10-4
1.88
3.1822×10-2
-2.6209×10-16
0.99977
800
9.7302×10-5
1.93
1.9877×10-2
-1.0274×10-12
0.99978
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
t=1250 s
simulation time.
Same as in Fig. , but for the zeppelin and the lateral transported
moist bubble test cases.
Computational meshes and difference scalar fields of ϕ for
(a) N=50, (b) N=100, (c) N=200, (d) N=400, (e) N=800,
and (f) the scalar field for N=400 after one rotation.
Convergence study for the annulus advection test.
L1 (red) and L∞ (green) error norms for the full domain and
reference line (blue dotted) for “perfect” second-order convergence.
Computational grid around the mountain for an x–z cut plane
at y=1.38 km (cell center).
Horizontal cross section of horizontal wind vectors at
z=200 m height for the RH95 case (black) and the RH50 case (grey).
Surface grid cells around the mountain in green, circle lines represent
200 m orography intervals.
Vertical cross section (x–z plane) of vertical wind speed
for the RH95 case (black) and the RH50 case (grey). Updrafts in solid lines
(0.2 ms-1 contour interval, zero line included), downdrafts in dashed lines
(0.2 ms-1 contour interval, zero line excluded).
Vertical cross section (x–z plane) of microphysical properties
for the RH95 case.
Liquid water content (shaded),
contours of specific cloud water content qc (red–yellow) and
specific rain water content qr (blue).
Moist bubble with mid-air zeppelin
The moist bubble benchmark case after is based on
its dry counterpart described in . There,
a hydrostatic and neutrally balanced initial state is realized by
a constant potential temperature. A warm perturbation in the center of
the domain leads to the rising thermal. For the present test case,
a moist neutral state can be expressed with the equivalent potential
temperature θe and two assumptions: the total water
mixing ratio rt=rv+rl remains
constant and phase changes between water vapor and liquid water are
exactly reversible.
The perturbation field is identical to the dry bubble test case
(Eqs. and ).
The domain is
20 km long in the x direction and the vertical extent is
10 km. Grid spacing is again isotropic with Δx=Δz=100 m.
Periodic boundary conditions are applied in the lateral
direction, whereas free-slip conditions are used for the top and bottom
boundaries.
Again, a total energy test is performed by comparing two modifications
of the present test case: (a) a uniform horizontal wind speed of
U=20 ms-1 is applied. With that, the center of the bubble
is again located at x=0 m at t=1000 s after passing through the
periodic boundaries. (b) In the center of the domain, a zeppelin-shaped
region is cut out and acts as an obstacle for the rising bubble. A similar
test like this was already introduced in and .
However, their tests were carried out with the dry bubble, which was also shifted
1 km
to the left.
The result for the first case is shown in Fig. .
The equivalent potential temperature field is very close to the benchmark
simulation, despite that the maximum value of θe is a little bit lower
in our case compared to the literature values and that there is a slight asymmetry
at the top of the thermal due to lateral transport.
The position of the rising thermal for the zeppelin case after
t=1250 s
is shown in Fig. . Because of the centered obstacle,
the bubble is split into two parts and deformed but, still, two
typical rotors are formed by each bubble and the result remains
symmetric.
When moisture and liquid water are present, the total energy takes the form
Et=ρqdcvd+qvcvv+qlcplT+qvL00+gz+0.5|v|2.
Again, energy is not fully conserved but the total relative
energy error after 2000 s of simulation time (there, in both cases,
the bubbles reach the top boundary resulting in zonal divergence)
stays in an acceptable range of 10-4 % (Fig. ), which is
1 order of magnitude smaller than in the cold bubble test case.
The difference of the error in total energy between the zeppelin and the classical case
is again very small. So, even with very small cut cells (≈1 % of full cell volume)
and microphysical conversions there is no indication that conservation properties are deteriorated.
For all cases, total mass is conserved within the numerical accuracy.
After , both mass and energy conservation are required to obtain the benchmark result.
Annulus advection test
The test problem reported in describes
the advection of a smooth bump by a radial wind field in an annulus.
It is described by the radius of the inner circle R1=0.75 and the
radius of the outer circle R2=1.25 within a rectangular domain [-1.5, 1.5]×[-1.5, 1.5].
The initial scalar field takes the following form:
ϕ=0.5erf5ϑ-π/3+erf52π/3-ϑ,
where ϑ=arctan(y/x).
Deriving the velocity field from the stream function
ψ(x,y)=π(R22-r2)/5 with r=(x2+y2)1/2, one full rotation is reached at t=5 s.
Figure a–e shows the difference fields between the analytical and numerical
solutions (Δϕ) for different mesh sizes, where N is the amount of grid cells in each spatial direction.
Figure f shows the final field after 5 s of integration time for N=400.
With greater N, the order of magnitude of the error decreases for the inner and outer
boundaries and the intermediate part of the annulus is less affected.
Table shows the results of the convergence study,
including error norms, experimental orders of convergence (EOC) as well as minimum and maximum values
of the tracer field for different mesh sizes.
For a fixed time step (0.01, 0.005, 0.0025 and 0.00125 s, respectively)
the advection scheme used in ASAM together with the Koren limiter
shows almost second-order convergence in the L1 norm, whereas
the L∞ norm is nearly first-order accurate (see Fig. ).
3-D mountain flow in a moist atmosphere
In this section, a test case described in is chosen.
It includes forced lifting around a 1 km high mountain (see Fig. ),
latent heat release and orographic precipitation.
Compared to the first three test cases, this case is now three-dimensional and uses a more
realistic initial profile, which mimics atmospheric conditions when it comes to
orographically dominated precipitation in the mountainous area of southwestern Germany.
In their work, they used the three-dimensional, nonhydrostatic weather prediction
model COSMO with terrain-following coordinates to describe the orography of the idealized mountain.
The model setup for the ASAM simulations is as follows: the domain extends
553 km × 553 km with a horizontal grid spacing of
2.765 km
and 70 vertical layers with uniform spacing of Δz=200 m.
A bell-shaped mountain is located at the center of the domain:
h(x,y)=Hx2+y2a2+11.5
with the mountain peak height H=1 km and the half-width length a=11 km.
Inflow and outflow boundary conditions are set according to the initial conditions.
A Rayleigh damping layer above 11 km is applied to suppress gravity wave reflections
from the top boundary. Surface heat fluxes and Coriolis force are turned off.
For turbulence parameterization, the standard Smagorinsky subgrid-scale model is used.
Microphysics are parameterized by the warm (i.e., no ice phase present) two-moment scheme
described in Sect. 3.2.
Initial profiles are obtained by assuming hydrostatic equilibrium,
a near-surface temperature Ts=283.15 K,
a constant mean flow U=10 ms-1,
a constant dry static stability Nd=11×10-3 s-1 and
a relative humidity profile, which is constant up to
zm=5 km
and rapidly decreases above this level according to
RH(z)=RHS0.5+π-1arctanz-zm500
with the near-surface humidity RHS=95 %
(RH95 case). To compare the results with its dry counterpart, another simulation with
RHS=50 % is performed (RH50 case).
Figure shows the wind field at 200 m height around
the mountain for both cases.
In the nearly saturated atmosphere, there is a more direct overflow over the mountain,
which is caused by the reduced stability due to high moisture.
These different flow characteristics also affect gravity wave structure (Fig. ).
The resulting waves are steeper and have a greater wavelength, which is
in agreement with gravity wave theory and the results from .
Most notable differences in the numerical results are discrepancies in vertical
wind strength in the lowest model layer at the windward side the mountain
(w≈0.6 ms-1 in ASAM vs. w≈0.2 ms-1 in COSMO),
which can be explained by the different surface coordinate systems of the models
(Cartesian grid with cut cells in ASAM and generalized terrain-following coordinates in COSMO).
Overall, the amplitude of vertical wind is higher for the resulting gravity waves.
Typical patterns of orographic clouds (one cloud upstream of the mountain and a larger cloud
with a high amount of liquid water content (LWC) and precipitation that reaches the ground
in the lee of the mountain) are also reproduced (Fig. ).
In this particular case the resulting patterns as well as the cloud and rain water contents
are comparable to the literature results, despite using different coordinate systems
and cloud microphysical schemes.
Conclusions and future work
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 e.g., might be
better suited for particular simulations compared to the present model
version. Also, a so-called implicit LES will be tested and
verified. There, no turbulence model is used and the numerics of the
discretization generate unresolved turbulent motions themselves. In
this type of LES, the sensitivity of the thermodynamical formulation
(especially in the energy equation) on the resulting motions has to be
analyzed.
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.