A process-based evaluation of the Intermediate Complexity Atmospheric Research Model (ICAR) 1.0.1
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Geosci. Model Dev., 14, 1657–1680, 2021
https://doi.org/10.5194/gmd-14-1657-2021
© Author(s) 2021. This work is distributed under
the Creative Commons Attribution 4.0 License.
A process-based evaluation of the Intermediate Complexity
Atmospheric Research Model (ICAR) 1.0.1
Johannes Horak1 , Marlis Hofer1 , Ethan Gutmann2 , Alexander Gohm1 , and Mathias W. Rotach1
1 Department of Atmospheric and Cryospheric Sciences, Universität Innsbruck, Innsbruck, Austria
2 Research Applications Laboratory, National Center for Atmospheric Research, Boulder, Colorado, USA
Correspondence: Johannes Horak (johannes.horak@uibk.ac.at)
Received: 22 September 2020 – Discussion started: 19 October 2020
Revised: 28 January 2021 – Accepted: 8 February 2021 – Published: 23 March 2021
Abstract. The evaluation of models in general is a nontriv- different to the standard zero-gradient boundary condition is
ial task and can, due to epistemological and practical rea- shown to reduce errors in the potential temperature and water
sons, never be considered complete. Due to this incomplete- vapor fields. Furthermore, the results show that there is a low-
ness, a model may yield correct results for the wrong rea- est possible model top elevation that should not be undercut
sons, i.e., via a different chain of processes than found in to avoid influences of the model top on cloud and precipita-
observations. While guidelines and strategies exist in the at- tion processes within the domain. The method to determine
mospheric sciences to maximize the chances that models the lowest model top elevation is applied to both the idealized
are correct for the right reasons, these are mostly applica- simulations and the real terrain case study. Notable differ-
ble to full physics models, such as numerical weather pre- ences between the ICAR and WRF simulations are observed
diction models. The Intermediate Complexity Atmospheric across all investigated quantities such as the wind field, wa-
Research (ICAR) model is an atmospheric model employing ter vapor and hydrometeor distributions, and the distribution
linear mountain wave theory to represent the wind field. In of precipitation. The case study indicates that the precipita-
this wind field, atmospheric quantities such as temperature tion maximum calculated by the ICAR simulation employing
and moisture are advected and a microphysics scheme is ap- the developed recommendations is spatially shifted upwind
plied to represent the formation of clouds and precipitation. in comparison to an unmodified version of ICAR. The cause
This study conducts an in-depth process-based evaluation of for the shift is found in influences of the model top on cloud
ICAR, employing idealized simulations to increase the un- formation and precipitation processes in the ICAR simula-
derstanding of the model and develop recommendations to tions. Furthermore, the results show that when model skill
maximize the probability that its results are correct for the is evaluated from statistical metrics based on comparisons
right reasons. To contrast the obtained results from the linear- to surface observations only, such an analysis may not reflect
theory-based ICAR model to a full physics model, ideal- the skill of the model in capturing atmospheric processes like
ized simulations with the Weather Research and Forecasting gravity waves and cloud formation.
(WRF) model are conducted. The impact of the developed
recommendations is then demonstrated with a case study for
the South Island of New Zealand. The results of this inves-
tigation suggest three modifications to improve different as- 1 Introduction
pects of ICAR simulations. The representation of the wind
field within the domain improves when the dry and the moist All numerical models of natural systems are approximations
Brunt–Väisälä frequencies are calculated in accordance with to reality. They generate predictions that may further the un-
linear mountain wave theory from the unperturbed base state derstanding of natural processes and allow the model to be
rather than from the time-dependent perturbed atmosphere. tested against measurements. However, the complete verifi-
Imposing boundary conditions at the upper boundary that are cation or demonstration of the truth of such a model is im-
possible for epistemological and practical reasons (Popper,
Published by Copernicus Publications on behalf of the European Geosciences Union.
1658 J. Horak et al.: A process-based evaluation of ICAR
1935; Oreskes et al., 1994). While the correct prediction of considered by ICAR but may be accounted for with other
an observation increases trust in a model, it does not ver- methods (e.g., Jarosch et al., 2012; Horak et al., 2019). For
ify the model; e.g., correct predictions for one situation do such cases Schlünzen (1997) advises that a model has to be
not imply that the model works in other situations or even assessed with respect to its limit of application. Therefore,
that the model arrived at the prediction through what would a direct comparison to a full physics-based model is gen-
be considered the correct chain of events according to scien- erally not sufficient for an evaluation of ICAR since ICAR
tific consensus. In contrast, a model prediction that disagrees is not intended to provide a full representation of atmo-
with a measurement falsifies the model, thereby indicating, spheric physics. Furthermore, whether the results obtained
for instance, issues with the underlying assumptions. From from ICAR simulations are correct for the right reasons can-
a practical point of view, the incompleteness and scarcity of not be inferred from, for instance, precipitation measure-
data, as well as the imperfections of observing systems place ments alone. Similar spatial distributions of precipitation
further limits on the verifiability of models. The same limita- may result from a variety of different atmospheric states.
tions apply to model evaluation as well. However, evaluation Therefore, the modeled processes yielding the investigated
focuses on establishing the reliability of a model rather than result need to be considered as well.
its truth. However, in the literature the evaluation efforts for ICAR
These propositions include models employed in the Earth have so far focused mainly on comparisons to precipitation
sciences, such as coupled atmosphere–ocean general circu- measurements or WRF output. Gutmann et al. (2016) com-
lation models, numerical weather prediction models and re- pared monthly precipitation fields for Colorado, USA, ob-
gional climate models. Those models approximate and sim- tained from ICAR to WRF output and an observation-based
plify the world and processes in it by discretizing the gov- gridded data set. While Gutmann et al. (2016) additionally
erning equations in time and space and by modeling subgrid- performed idealized hill experiments, these focused on the
scale processes with adequate parameterizations (e.g., Sten- qualitative comparison of the vertical wind field and the dis-
srud, 2009). The applied simplifications are often the re- tribution of precipitation between ICAR and WRF. Bernhardt
sult of a trade-off between physical fidelity of the modeled et al. (2018) applied ICAR to study changes in precipita-
processes and the associated computational demand. How- tion patterns in the European Alps that are dependent on the
ever, even with a firm basis in natural laws, such models chosen microphysics scheme. Horak et al. (2019) evaluated
may generate results that match measured data but arrive ICAR for the South Island of New Zealand based on multi-
at them through a causal chain differing from that inferred year precipitation time series from weather station data and
from observations (“right, but for the wrong reason”; e.g., diagnosed the model performance with respect to season, at-
Zhang et al., 2013). Additionally, the reason for a matching mospheric background state, synoptic weather patterns and
result may even be found in unphysical artifacts introduced the location of the model top. By comparing to measure-
by the numerical methods of these models (e.g., Goswami ments, Horak et al. (2019) observed a strong dependence of
and O’Connor, 2010). In acknowledgment of the fundamen- the performance of ICAR on the location of the model top,
tal limitation of verification, models are evaluated rather than finding an optimal setting of 4.0 km above topography that
verified, and best practices and strategies have been outlined minimized the mean squared errors calculated at all weather
to maximize the probability that the results obtained from stations. However, the analysis of cross sections revealed nu-
a model are correct for the right reasons (e.g., Schlünzen, merical artifacts in the topmost vertical levels, suggesting
1997; Warner, 2011). Most of these criteria, however, apply these to be responsible for the high model skill, thus ren-
to full physics-based models, such as regional climate mod- dering the model right for the wrong reason.
els or numerical weather prediction models, that are expected This study aims to improve the understanding of the
to model atmospheric processes comprehensively. ICAR model and develop recommendations that maximize
The Intermediate Complexity Atmospheric Research the probability that the results of ICAR simulations, such
model (ICAR; Gutmann et al., 2016) employed in this study as the spatial distribution of precipitation, are correct and
is intended to be a simplified representation of atmospheric caused by the physical processes modeled by ICAR (cor-
dynamics and physics over mountainous terrain. With a ba- rect for the right reasons) and not by numerical artifacts or
sis in linear mountain wave theory, it is a computationally any influence of the model top. For a given initial state, a
efficient alternative to full physics regional climate models, correct representation of the fields of wind, temperature and
such as the Weather Researching and Forecasting (WRF; moisture, as well as of the microphysical processes, are a
Skamarock et al., 2019) model. Compared to simpler linear- necessity to obtain the correct distribution of precipitation.
theory-based models of orographic precipitation (e.g., Smith Therefore, simulations of an idealized mountain ridge are
and Barstad, 2004), ICAR allows for a spatially and tempo- employed to investigate and evaluate the respective fields and
rally variable background flow and a detailed vertical struc- processes in ICAR. This study first quantitatively and qual-
ture of the atmosphere and employs a complex microphysics itatively analyzes how closely the ICAR wind and potential
scheme. However, for instance, precipitation induced by con- temperature fields match the analytical solution for the ideal
vection or enhanced by nonlinearities in the wind field is not ridge and contrasts them with a WRF simulation to infer the
Geosci. Model Dev., 14, 1657–1680, 2021 https://doi.org/10.5194/gmd-14-1657-2021
J. Horak et al.: A process-based evaluation of ICAR 1659
aspects not captured by linear theory (Sect. 4.1). In a sec- the intrinsic frequency σ , and the fluid displacement η̂ are
ond step the influence of the height of the model top and given by
the upper boundary conditions on the microphysical cloud
formation processes are quantified with a sensitivity study η̂ = ĥeimz , (3)
(Sects. 4.2–4.4). Thirdly, the differences in the hydrometeor N2 − σ2
and precipitation distribution due to nonlinearities and other m2 = (k 2 + l 2 ), (4)
σ2 −f 2
processes not represented by linear theory are investigated in
a comparison of ICAR to WRF (Sect. 4.5). Finally, the im- σ = U k + V l. (5)
pact of recommendations derived from the preceding steps
on a real case are demonstrated (Sect. 4.6). The case study Here ĥ denotes the Fourier transform of the topography
is conducted for the South Island of New Zealand and con- h(x, y), z the elevation and N the Brunt–Väisälä frequency.
trasted to the results of Horak et al. (2019). All findings are Note that depending on whether a grid cell is saturated or
discussed in Sect. 5, and the conclusions, including the rec- not, either the moist, Nm (Emanuel, 1994), or dry, Nd , Brunt–
ommendations, are summarized in Sect. 6. Väisälä frequency is employed in Eq. (4) and calculated as
d ln θ
Nd2 = g , (6)
2 ICAR model dz
2 1 d
2.1 Model description Nm = 0m (cp + cl qw ) ln θe
1 + qw dz
dqw
ICAR is an atmospheric model based on linear mountain − cl 0m ln T + g , (7)
wave theory (Gutmann et al., 2016). The input data sets re- dz
quired by ICAR are a digital elevation model supplying the with the acceleration due to gravity g, the temperature T , the
high-resolution topography h(x, y) and forcing data, i.e., a potential temperature θ , the equivalent potential temperature
set of 4-D atmospheric variables as supplied by atmospheric θe , the saturated adiabatic lapse rate 0m , the saturation mix-
reanalysis such as ERA5 or coupled atmosphere–ocean gen- ing ratio qs , the cloud water mixing ratio qc , and the total
eral circulation models. The forcing data set represents the water content qw = qs + qc and the specific heats at constant
background state of the atmosphere and must comprise the pressure of dry air and liquid water cp and cl . Note that ICAR
horizontal wind components (U ,V ), pressure p, potential employs quantities from the perturbed state of the domain to
temperature 2 and water vapor mixing ratio qv0 . calculate N even though in linear mountain wave theory N
ICAR stores all dependent variables on a 3-D staggered is a property of the background state (e.g., Durran, 2015).
Arakawa C-grid (Arakawa and Lamb, 1977, pp. 180–181) Statically unstable atmospheric conditions (i.e., N 2 < 0) in
and employs a terrain-following coordinate system with con- the forcing data are avoided by enforcing a minimum Brunt–
stant grid cell height. In particular, mass-based quantities Väisälä frequency of Nmin = 3.2 × 10−4 s−1 throughout the
such as water vapor are stored at the grid cell center, while the domain.
horizontal wind components u and v are stored at the centers The vertical wind speed perturbation w = w0 is calculated
of the west–east or south–north faces of the grid cells, and from the divergence of the horizontal winds u and v, where
the vertical wind component w is stored at the center of the u = U + u0 and v = V + v 0 , as follows:
top–bottom faces of each grid cell.
In contrast to dynamical downscaling models, ICAR ∂w ∂u ∂v
− = + . (8)
avoids solving the Navier–Stokes equations of motion explic- ∂z ∂x ∂y
itly. Instead, ICAR calculates the perturbations to the hori-
zontal background winds analytically for a given time step ICAR does solve the Eqs. (1)–(8) for every grid cell in the
by employing linearized Boussinesq-approximated govern- ICAR domain separately and for every forcing time step as
ing equations that are solved in frequency space with the to allow for a spatially and temporally variable background
Fourier transformation (Barstad and Grønås, 2006). With the state. To make this task computationally viable, ICAR em-
Fourier transform of, for example, the east–west wind per- ploys a lookup table; see Gutmann et al. (2016) for details.
turbation u0 denoted as û the perturbations to the horizontal ICAR allows for the selection of different microphysics
wind field are (MP) schemes. In this study an updated version of the
Thompson MP scheme is employed (Thompson et al., 2008).
−m(σ k + ilf )i η̂
û(k, l) = , (1) It predicts mixing ratios for water vapor qv , cloud water qc ,
k2 + l2 cloud ice qi , rain qr , snow qs and graupel qg , from here on
−m(σ l + ikf )i η̂ referred to as microphysics species, as well as the number
v̂(k, l) = , (2)
k2 + l2 concentrations for cloud ice ni and rain nr . The Thompson
with the horizontal wavenumbers k and l, the Coriolis term MP scheme is a double-moment scheme in cloud ice and rain
f , and the imaginary number i. The vertical wavenumber m, and a single-moment scheme for the remaining quantities.
https://doi.org/10.5194/gmd-14-1657-2021 Geosci. Model Dev., 14, 1657–1680, 2021
1660 J. Horak et al.: A process-based evaluation of ICAR
The microphysics species, ni , nr and θ are advected with 2.2.1 Calculation of the Brunt–Väisälä frequency
the calculated wind field according to the advection equation
(Gutmann et al., 2016): From the initial state of θ and the microphysics species fields
at t0 (see Eq. 12), ICAR calculates the (moist or dry; see
∂ψ ∂(uψ) ∂(vψ) ∂(wψ)
=− + + , (9) Eqs. 6 and 7, respectively) Brunt–Väisälä frequency N for all
∂t ∂x ∂y ∂z model times tm smaller than the first forcing time tf1 . During
where ψ denotes any of the advected quantities. At the lateral each model time step, the θ and microphysics species fields
domain boundaries L located at nx = 0, nx = Nx , ny = 0, in the ICAR domain are modified by advection and micro-
and ny = Ny , where Nx and Ny are the number of grid points physical processes. Therefore, for model times tm > t0 , θ and
along the x and y direction, the value of ψ is given by the all the microphysics species, q represents the perturbed state
forcing data set and specified by a Dirichlet boundary condi- of the respective fields, denoted as
tion as
θ = 2 + θ 0 and (13)
ψ(x, y, z, t) = ψF (x, y, z, t), (10) 0
(x,y)∈L q = q0 + q . (14)
with ψF as the respective quantity in the forcing data set tem-
Note that in this notation, the perturbed water vapor field
porally and spatially interpolated to the ICAR grid and model
is denoted as qv , the background state water vapor field as
time. At the upper boundary T where nz = Nz and Nz as the
qv0 and the perturbation field as qv0 . Consequently, during
grid points along the z direction, a zero-gradient Neumann
all intervals tfn ≤ tm < tfn+1 , where tfi are subsequent forc-
boundary condition is imposed:
ing time steps, N is based on the perturbed states of poten-
∂ψ(x, y, z, t) tial temperature and the microphysics species at tfn . More
= 0. (11)
∂z z∈T specifically, all atmospheric variables ICAR uses for the cal-
The initial conditions at t0 for the 3-D fields of all atmo- culation of N with Eqs. (6) and (7) are represented by the
spheric quantities 9 in the ICAR domain are prescribed by perturbed fields.
linearly interpolating the corresponding field in the forcing However, in linear mountain wave theory N is a property
data set 9F to the high-resolution ICAR domain: of the unperturbed background state (e.g., Durran, 2015), an
assumption that is not satisfied by the calculation method em-
9(x, y, z, t0 ) = 9F (x, y, z, t0 ). (12) ployed by the standard version of ICAR. This study there-
Note that capital 9 denotes not only the advected quantities fore employs a modified version of ICAR that, in accordance
ψ but also p. with linear mountain wave theory, calculates N from the state
In linear mountain wave theory, the wind field is entirely of the atmosphere given by the forcing data set if the corre-
determined by the topography and the background state of sponding option is activated. In the following, the modifica-
the atmosphere (Sawyer, 1962; Smith, 1979) and, for a hori- tion of ICAR basing the calculation of N on the background
zontally and vertically homogeneous background state, given state is referred to as ICAR-N, while the unmodified version,
by a set of analytical equations (e.g., Barstad and Grønås, which bases the calculation on the perturbed state of the at-
2006). This formal simplicity is achieved by a number of mosphere, is referred to as the original version (ICAR-O). If
simplifications, for instance neglecting the interaction of properties applying to both versions are discussed, the term
waves with waves; waves with turbulence or nonlinear ef- ICAR is chosen.
fects such as gravity wave breaking, time-varying wave am-
plitudes, or low-level blocking and flow splitting. Discus- 2.2.2 Treatment of the upper boundary in the
sions of the limitations of linear theory resulting from this advection numerics
reduction of complexity can be found in the literature (e.g.,
Dörnbrack and Nappo, 1997; Nappo, 2012). ICAR imposes a zero-gradient boundary condition (ZG BC)
Note that since ICAR is based on the equations derived at the upper boundary on all quantities subject to numerical
by Barstad and Grønås (2006), it currently neglects the re- advection; see Eq. (11). This section details how, specifically
flection of waves at the interface of atmospheric layers with for the microphysics species, a ZG BC has the potential to
different Brunt–Väisälä frequencies. Furthermore, it neglects cause problems by, e.g., triggering influx of additional water
the vertical increase of the amplitude of the wind field pertur- vapor into the domain. Due to its conceptual simplicity, the
bations with decreasing density. A full description of ICAR issue is illustrated for the upwind advection scheme, which
is given by Gutmann et al. (2016). is the standard advection scheme employed by ICAR.
In the following, the mass levels are indexed from 1 to Nz
2.2 Modifications to ICAR and the half levels bounding the kth mass level are denoted as
k − 1/2 and k + 1/2. Note that the vertical wind components
The investigations described in this study were conducted are calculated at half levels with Eq. (8) and that, in particu-
with a modified version of ICAR 1.0.1. All modifications are lar, no boundary condition is required to determine w at the
publicly available for download (Gutmann et al., 2020). model top.
Geosci. Model Dev., 14, 1657–1680, 2021 https://doi.org/10.5194/gmd-14-1657-2021
J. Horak et al.: A process-based evaluation of ICAR 1661
To arrive at the discrete equations of the upwind advection, model (Skamarock et al., 2019). These models place the lo-
the flux divergences ∂(uψ)/∂x, ∂(vψ)/∂y and ∂(wψ)/∂z cation of the upper boundary at elevations high enough that
on the right-hand side of Eq. (9) are discretized as, e.g., in moisture fluxes across the boundary are negligible. While
Patankar (1980). The vertical flux gradient φz across mass applying either treatment to ICAR is, in general, an op-
level k at time step t due to downdrafts (wk+1/2 t < 0 and tion, it is undesirable since both necessarily result in higher
t
wk−1/2 < 0) is then approximated by model tops and therefore would severely increase the compu-
tational cost of ICAR simulations. Hence, this study investi-
∂(wψ) 1 t t
gates whether the application of computationally cheaper al-
φz = ≈ ψk+1 wk+1/2 − ψkt wk−1/2
t
, (15) ternative boundary conditions is able to reduce errors caused
∂z 1z
by, e.g., the unphysical mass influx and loss described above.
with 1z as the vertical grid spacing. The resulting value of ψ To this end, additional boundary conditions are added to the
at mass level k at time step t +1 is calculated with an explicit ICAR code with the option to apply different boundary con-
first-order Euler forward scheme as ditions to different quantities ψ. Furthermore, this study as-
sesses whether the lowest possible model top elevation nec-
1t t
ψkt+1 = ψkt − t
ψk+1 wk+1/2 − ψkt wk−1/2
t
, (16) essary to avoid the model top’s impact on the results can be
1z chosen substantially below that of full physics models with-
where 1t denotes the length of the time step. At the upper out sacrificing the physical fidelity of the results.
boundary, where k = Nz , with Nz being the number of verti-
cal levels, by default ICAR applies a zero-gradient boundary
condition to ψ by setting ψNz +1 = ψNz . In case of down- 3 Methods
drafts, ψNz > 0 and vertical convergence in the wind field
across the topmost vertical mass level (wNz +1/2 < wNz −1/2 ), To investigate ICAR with respect to the influence of the
this results in a negative vertical flux gradient and an associ- elevation of the upper boundary and the boundary condi-
ated increase in ψ (see Eq. 16). If wNz +1/2 < wNz −1/2 per- tions applied to it, idealized numerical simulations and a real
sists for more than one time step, the concentration of the case study are conducted. Simulations are run with ICAR-O,
quantity in the topmost vertical level will continue to increase ICAR-N and WRF in order to assess to what degree ICAR
until it is redistributed within the domain via advection or simulations approximate the results of the analytical solution
conversion into other microphysics species. As observed by and a full physics model. In addition, WRF is employed to
Horak et al. (2019), this influx of additional water therefore infer differences due to nonlinearities.
may cause numerical artifacts such as the formation of spu-
rious clouds. 3.1 Simulation setup
While the effect described above is related to downdrafts
at the model top, note that updrafts, on the other hand, may Simulations in this study are conducted with version 1.0.1
cause moisture to be transported out of the domain, lead- of ICAR (ICAR-O) and version 4.1.1 of WRF. Additionally,
ing to a mass loss. However, for k = Nz and wN t >0 a modification of ICAR-O, referred to as ICAR-N, where
z +1/2
t the Brunt–Väisälä frequency N is calculated from the back-
and wNz −1/2 > 0, the discretization of the vertical flux diver-
ground state given by the forcing data set is employed. Note
gence in Eq. (9) yields
that ICAR-O, on the other hand, calculates N from the per-
∂(wψ) 1 t turbed state of the atmosphere predicted by the ICAR-O. In
t t t
≈ ψNz −1 wN z −1/2 − ψ N z
w N z +1/2 . (17) the idealized simulations the forcing data set is represented
∂z 1z
by an idealized sounding while for the real case it is the
Therefore, this issue cannot be addressed by applying differ- ERA-Interim reanalysis. For idealized simulations, a period
ent boundary conditions, since Eq. (17) does not depend on of 18 h is used for spinup and the model output from t = 19 h
ψNz +1 . to t = 30 h, with an interval of 1 h, is evaluated. The ICAR
A solution to address both issues would potentially be to setup for the real case is described in Horak et al. (2019).
include a relaxation layer directly beneath the model top The ideal case consists of an infinite ridge extending along
(see, e.g., Skamarock et al., 2019). Within this relaxation the south–north direction in the domain and westerly flow.
layer, vertical wind speeds would tend towards zero with The horizontal grid spacings of ICAR and WRF are chosen
decreasing distance to the model top and perturbed quanti- as 1x = 1y = 2 km with 404 grid points along the west–east
ties would be relaxed towards their value in the background axis and open boundary conditions at the western and eastern
state. Another potential solution is employed by full physics boundaries. Since ICAR does not currently support periodic
models such as the Integrated Forecasting System (IFS) boundary conditions, 104 grid points are employed along the
of the European Center for Medium-Range Weather Fore- south–north axis to minimize the influence of the bound-
casts (ECMWF, 2018), the COSMO model (Doms and Bal- aries on the domain center. For ICAR, open boundary con-
dauf, 2018) or the Weather Research and Forecasting (WRF) ditions are imposed at the southern and northern boundaries.
https://doi.org/10.5194/gmd-14-1657-2021 Geosci. Model Dev., 14, 1657–1680, 2021
1662 J. Horak et al.: A process-based evaluation of ICAR
WRF, on the other hand, just uses three grid points along chosen as 1013 hPa. For the comparison of the ICAR and
the south–north axis and periodic boundary conditions. The WRF wind fields to an analytical solution, dry conditions
vertical spacing in ICAR simulations is set to 1z = 200 m, with RH = 0 % are employed, while otherwise saturated con-
while the 26 km high WRF domain is subdivided in 130 grid ditions with RH = 100 % are prescribed throughout the ver-
cells, resulting in an average vertical spacing of approxi- tical column at all heights. The sensitivity to the base state is
mately 200 m. At the lower boundary ICAR and WRF em- investigated by either varying U between 5 and 40 m s−1 in
ploy a free-slip boundary condition. An implicit Rayleigh steps of 5 m s−1 or varying N between 0.005 and 0.015 s−1
dampening layer (Klemp et al., 2008) is applied to the up- with a step size of 0.0025 s−1 for the 1 km high and 20 km
permost 16 km of the WRF domain, with a dampening coef- wide ridge. An overview of the parameter space covered by
ficient of 0.3 s−1 . The model time step of ICAR is automati- the simulations is given in Table 1. A specific combination of
cally calculated by ICAR to satisfy the Courant–Friedrichs– topography and sounding is referred to as scenario.
Lewy criterion (Courant et al., 1928; Gutmann et al., 2016) For the default scenario with the 1 km high and 20 km
and is approximately 40 s, while for WRF it is set to 2 s. wide ridge and a background state with U = 20 m s−1 , N =
Idealized ICAR simulations are run for different model 0.01 s−1 and RH = 100 %, the vertical wavelength of hy-
top elevations. The elevation of the upper boundary of the drostatic mountain waves is λz = 2π U/Nd = 12.6 km and
domain, referred to as model top elevation ztop , is increased the non-dimensional mountain height is = hm Nd /U = 0.5.
by adding additional vertical levels while keeping the verti- While the listed values for λz and are valid only for dry
cal spacing constant. The lowest model top is set at 4.4 km conditions, they are employed to summarize the basic char-
while the highest is located at 14.4 km, with steps of 1 km acteristics of the background state. For the witch of Agnesi
in between. The lower end of the model top range reflects ridge, the critical value for the onset of wave breaking in a
the lowest settings employed in preceding studies, such as dry (unsaturated) atmosphere is c = 0.85 (Miles and Hup-
Horak et al. (2019), where the optimal setting was deter- pert, 1969). Note that while a saturated atmosphere has been
mined at 4.0 km, or Gutmann et al. (2016), who set the top shown to increase the values of and c (Jiang, 2003), wave
of the ICAR domain to 5.64 km. An additional simulation breaking does not occur due to < c . Nonetheless, other
with ztop = 20.4 km is conducted to serve as a reference sim- nonlinear effects, such as wave amplification, cannot be com-
ulation where the cloud processes within the troposphere are pletely neglected. The combination of this sounding and to-
not affected by the model top. The Thompson microphysics pography is therefore suitable as an indicator of how well
scheme as described in Sect. 2 is employed in all models. The the ICAR solution approximates scenarios in which nonlin-
ICAR implementation of the Thompson MP was forked from earities occur, a situation ICAR is very likely to encounter
WRF version 3.4. Preliminary tests were conducted, showing in real-world applications. To this end an ICAR-N simula-
that WRF 3.4 and WRF 4.1.1 yielded the same results for the tion is compared to a WRF simulation employing the same
default scenario, with only negligible differences. Addition- topography and sounding.
ally, the code of the Thompson MP implementation in ICAR
and WRF 4.1.1 was reviewed and tested to ensure that dif- 3.3 Analytical solution
ferences between the implementations did not affect the re-
sults. All input files and model configurations are available ICAR calculates the perturbations to the horizontal back-
for download (Horak, 2020). ground wind with Eq. (1) and Eq. (2), while the vertical
wind speed is calculated according to Eq. (8). Perturbations
3.2 Topographies and initial soundings to the potential temperature and microphysics species fields,
on the other hand, result from advection and microphysi-
The topography is given by a witch of Agnesi ridge defined cal processes calculated with numerical methods. In ICAR-O
by h(x) = hm a 2 /(x 2 + a 2 ) with a height of hm = 1 km this introduces a time dependency for N and, in turn, for the
at the domain center at x = 0 km and a half width at half wind field perturbations that depend on N as input variable.
maximum of a = 20 km. Along the y axis the ridge extends Furthermore, ICAR assembles the wind field with an algo-
through the entire domain. To investigate the influence of rithm that allows for a spatially variable background state
the topography, additional ICAR simulations for ridge con- (Gutmann et al., 2016). It is therefore necessary to ascertain
figurations with a = 20 km and heights of 0.5, 2 and 3 km how well the exact analytical perturbations are reproduced
are conducted, as well as 1 km high ridges with a = 10 km, by ICAR. This cannot be inferred from a direct comparison
a = 15 km, a = 30 km and a = 40 km, respectively. to WRF since the wind field of the latter is influenced by non-
The vertical potential temperature profile of the base state linear processes not modeled by ICAR. For the topography
2(z) is characterized by a potential temperature at the sur- given in Sect. 3.2 linear-theory-based analytical expressions
face of 270 K, a constant Brunt–Väisälä frequency, and N = for the resulting perturbations to a horizontally and vertically
0.01 s−1 and is calculated by solving Eq. (6) for θ. The hor- uniform background state have been derived as follows (e.g.,
izontal wind components of the base state are chosen as Smith, 1979):
U = 20 m s−1 and V = 0 m s−1 , and the surface pressure is
Geosci. Model Dev., 14, 1657–1680, 2021 https://doi.org/10.5194/gmd-14-1657-2021
J. Horak et al.: A process-based evaluation of ICAR 1663
Table 1. Overview of the combinations of topographies and soundings (scenarios) used to initialize the idealized ICAR simulations. Here
hm denotes the ridge height, a the half width at half maximum of the ridge, U the west–east wind component of the base state, RH the
relative humidity, Nd the dry Brunt–Väisälä frequency of the base state, λz the vertical wavelength of the hydrostatic mountain waves for dry
conditions and the non-dimensional mountain height for dry conditions. The default scenario used for the comparison of ICAR to WRF is
highlighted in bold.
hm (km) 0.5 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 2.0 3.0
a (km) 20 10 15 20 20 20 20 20 20 20 20 20 20 20 20 20 20 30 40 20 20
U (m s−1 ) 20 20 20 20 5 10 15 20 20 20 20 20 20 25 30 35 40 20 20 20 20
RH (%) 100 100 100 100 100 100 100 100 100 0 100 100 100 100 100 100 100 100 100 100 100
Nd (s−1 ) 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.005 0.0075 0.01 0.01 0.0125 0.015 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01
λz (km) 12.6 12.6 12.6 12.6 3.1 6.3 9.4 25.1 16.8 12.6 12.6 10.1 8.4 15.7 18.8 22 25.1 12.6 12.6 12.6 12.6
(1) 0.25 0.5 0.5 0.5 2 1 0.67 0.25 0.38 0.5 0.5 0.63 0.75 0.4 0.33 0.29 0.25 0.5 0.5 1 1.5
representing different physical situations. The respective dis-
cretizations of the equations given in Table 2 then determine
a sin(lz) + x cos(lz) the value of ψNz +1 .
u0 (x, z) = A(z)N , (18)
a2 + x 2 For this study, options are added to the ICAR code that
(x 2 − a 2 ) sin(lz) − 2ax cos(lz) allow the application of different BCs to water vapor, poten-
w0 (x, z) = A(z)U , (19) tial temperature and the hydrometeors (cloud water, ice, rain,
(a 2 + x 2 )2
N 2 a cos(lz) − x sin(lz) snow and graupel), hereafter referred to as a set of bound-
θ 0 (x, z) = −A(z) 2, (20) ary conditions. To indicate which BCs were applied to what
g a2 + x 2
group in a specific model run, the runs are labeled with a
three-digit code; see Table 3. The first digit indicates the BC
with u0 as the perturbation to the horizontal background
imposed on θ , the second digit the BC imposed on qv and
wind U , w0 the perturbation to the vertical wind speed, θ 0
the third digit the BC imposed on qhyd , which encompass
the perturbation to the background potential temperature 2,
all remaining MP species (qc , qi , qr , qs and qg ). The num-
g = 9.81 m s−2 as the gravitational acceleration, l the Scorer
ber ID associated with each BC is listed in Table 2. In this
parameter defined as l = N/U and A(z) as the elevation-
notation, for instance, 014 denotes a simulation imposing a
dependent amplitude of the perturbations. A(z) is given by
zero-gradient BC to θ , a constant-gradient BC to qv , and a
p constant flux gradient BC to the hydrometeors qhyd .
A(z) = hm a ρ(0)/ρ(z), (21)
The 10 combinations of BCs tested in the sensitivity study
are listed in Table 3. While a much larger set of combinations
where ρ is the height-dependent air density of the back-
of BCs exists, physically not meaningful BC combinations,
ground state. However, since the underlying equations em-
such as a zero-value BC imposed on potential temperature,
ployed by ICAR neglect the effect of wave amplification
√ due
were ruled out beforehand. Additionally, to reduce the pa-
to decreasing density with height, the term ρ(0)/ρ(z) in
rameter space further, a preliminary study was conducted to
Eq. (21) is set to unity in the following.
exclude sets of BCs that yielded results with distinctly higher
3.4 Boundary conditions at the model top errors than the standard zero-gradient BC.
In this study the effect of the boundary conditions (BCs) im- 3.5 Evaluation
posed by ICAR at the upper boundary of the simulation do-
main is investigated. To this end several alternative BCs to All evaluations conducted in this study focus on cross sec-
the existing zero-gradient boundary condition are added to tions along the west–east axis of the domain, oriented par-
the ICAR code, their abbreviations, mathematical formula- allel to the background flow. Since ICAR does not currently
tion and their numerical implementation are summarized in support periodic boundary conditions, the ICAR domain is
Table 2. All BCs constitute Neumann BCs except for the extended along the south–north axis to minimize influences
zero-value Dirichlet BC. Per default ICAR imposes a ZG from the boundaries (see Sect. 3.1). Additionally, for ICAR
BC at the model top to all quantities, corresponding to the the four centermost west–east cross sections from the south–
assumption that, e.g., the mixing ratio of hydrometeors qhyd north axis in the domain are averaged, and the average is
above the domain is the same as in the topmost vertical level. found to be representative of the domain center in prelimi-
A zero-value (ZV) BC imposed on, e.g., qhyd avoids any ad- nary tests (not shown). In WRF the central west–east cross
vection from outside of the domain into it. The constant- section from the south–north axis is used.
gradient (CG), constant-flux (CF) and constant-flux-gradient The effect of the Brunt–Väisälä frequency calculation
(CFG) BCs assume that either the gradient, flux or flux gra- method is investigated with a comparison of the u0 and w 0
dient of ψ, respectively, remains constant at the model top, fields obtained from ICAR-N and ICAR-O simulations to
https://doi.org/10.5194/gmd-14-1657-2021 Geosci. Model Dev., 14, 1657–1680, 2021
1664 J. Horak et al.: A process-based evaluation of ICAR
Table 2. Overview of all types of boundary conditions that were imposed at the model top of ICAR in the sensitivity study. The table lists
the ID number, the abbreviation used in this study, the full name and equation of the BC evaluated at z = ztop , and the resulting equation for
ψNz +1 required to calculate the flux at the top boundary of the domain in Eq. (16). Note that the zero-gradient BC is a special case of the
constant-gradient BC and that the constant c is chosen as ψNz − ψNz −1 . Due to the upwind advection scheme each BC is only applied if
wNz < 0.
ID Abbreviation Boundary condition ψNz +1
∂ψ
0 ZG Zero gradient ∂z = 0 ψNz
∂ψ
1 CG Constant gradient ∂z = c max(0, 2ψNz − ψNz −1 )
2 ZV Zero value ψ =0 0
∂(wψ) wNz −1
3 CF Constant flux ∂z = 0 wN ψNz
z
∂ 2 (wψ) 1
4 CFG Constant flux gradient =0 wNz (2ψNz −1 wNz −1 − ψNz −2 wNz −2 )
∂z2
Table 3. Combinations of BCs tested in the sensitivity study with idealized simulations. Each column represents a combination of three
BCs used in a specific simulation. Each digit of the three-digit code refers to the ID number of a specific BC listed in Table 2 that was
applied to one of the three quantities listed in the rows below. For all combinations of BCs, simulations for all of the topographic settings
and background conditions listed in Table 1 were performed.
Quantity BC combination
Code 000 011 111 114 113 014 044 141 142 133
θ ZG ZG CG CG CG ZG ZG CG CG CG
qv ZG CG CG CG CG CG CFG CFG CFG CF
qhyd ZG CG CG CFG CF CFG CFG CG ZV CF
the fields given by the analytical expressions in Eqs. (18) mass of the MP species in the cross section and the spa-
and (19). Nonlinear effects on the wind field are investigated tial distribution of potential temperature, the MP species and
by a comparison of ICAR to WRF. Differences between the the 12 h accumulated precipitation P12 h . Except for P12 h , all
models’ and the analytical solution are quantified with the quantities are averaged over the 12 h period after a spinup of
bias (B) and the mean absolute error (MAE, Wilks, 2011b, 18 h when an approximately steady state is reached. P12 h is
chap. 8). Since WRF uses a different model grid than ICAR, the precipitation accumulated over the same period.
WRF fields are linearly interpolated to the ICAR grid for this Differences in the spatial distribution of time-averaged
comparison. quantities ψ̄, P12 h and time-averaged total mass of the MP
For the evaluation in this study the mixing ratios of the mi- species Q̄ with respect to the reference simulation are quan-
crophysics species are assigned to three groups: water vapor tified with the sum of squared errors (SSE). The SSE is cal-
qv , suspended hydrometeors qsus = qc + qi and precipitating culated between ICAR simulations with different values of
hydrometeors qprc = qr +qs +qg . The total mass of water va- ztop and the reference simulation employing the default zero-
por Qv , suspended hydrometeors Qsus and precipitating hy- gradient BCs at the upper boundary where ztop is zmax =
drometeors Qprc is calculated as 20.4 km. This model top is high enough that cloud processes
within the troposphere are not affected by the model top. The
Nz
Nx X
X SSE is calculated over all vertical levels, defined in both sim-
Q(t) = V ρij (t) qij (t), (22) ulations as
i=0 j =0
Nz
Nx X
X 2
SSE(ψ, ztop , BCs) = ψ̄ij (ztop , BCs) − ψ̄ij (zmax ) . (23)
where Nx and Nz are the horizontal and vertical number i=0 j =0
of grid cells, respectively, V the grid cell volume, qij (t)
the mixing ratio of the respective hydrometeor species, and Here ψ̄ij (ztop , BCs) is the time-averaged value of a quantity
ρij (t) is the density of dry air within the grid cell. Note that ψ in an ICAR simulation at grid point (i, j ) with the model
in contrast to WRF the grid cell volume in ICAR is constant top at ztop and the set of upper BCs, and ψ̄ij (zmax ) is the
and all vertical levels have the same height 1z. value of a quantity at the same location in the reference sim-
The sensitivity of the physical processes simulated by ulation with ztop = zmax . For 12 h accumulated precipitation
ICAR-N to the elevation of the upper boundary and the im- a one-dimensional version of equation (23), with the sum-
posed boundary conditions (BCs) is inferred from the total mation only along the x axis, is employed while for total
Geosci. Model Dev., 14, 1657–1680, 2021 https://doi.org/10.5194/gmd-14-1657-2021
J. Horak et al.: A process-based evaluation of ICAR 1665
mass no summation is necessary and only the squared dif- 3.6 Case study
ference (Q̄(ztop , BCs) − Q̄(zmax ))2 is calculated. The SSE is
preferred over the mean squared error (MSE) since differ- To investigate the effects of the suggested modifications to
ent model top settings result in different domain sizes, po- ICAR on the distribution of precipitation for a real-world ap-
tentially favoring simulations with higher model tops due to plication, a case study is conducted for the Southern Alps on
the larger area that the errors are averaged over. While the the South Island of New Zealand located in the southwest-
SSE conversely tends to favor smaller domains, lower SSEs ern Pacific Ocean. Furthermore, the procedure to identify the
obtained for simulations with higher model tops are then a lowest possible model top elevation Zmin , as described in
stronger indicator that increasing the model top effectively Sect. 3.5, is applied to this real case scenario and the result
reduces errors. compared to the optimal model top elevation of 4 km found
To quantify the improvement of one simulation, with a set by Horak et al. (2019) for this region. In their study the model
of boundary conditions BCs and model top ztop , over another top elevation was chosen as the elevation that led to the low-
by choosing a different set of boundary conditions, BCs0 , at est MSEs between simulated and measured 24 h accumulated
the upper boundary or another model top elevation ztop 0 , the precipitation for 11 sites in the Southern Alps. Section 4.6
reduction of error (RE) measure is employed (Wilks, 2011a, additionally investigates whether this seemingly optimal re-
chap. 8). It is given by sult, as suggested by the lowest MSEs, was achieved due to
the low model top potentially influencing the microphysical
0 , BCs0 )
SSE(ψ, ztop processes within the domain and the calculation of N being
RE(ψ) = 1 − . (24)
SSE(ψ, ztop , BCs) based on the perturbed fields. To this end, the hydrometeor
and precipitation distribution along cross sections through
This way, RE can be interpreted as a percentage improve- the Southern Alps are compared.
ment due to the alternative choice of ztop 0 or BCs0 over the To maintain comparability to Horak et al. (2019), the
original settings ztop and BCs, with RE = 0 corresponding ICAR simulations for ICAR-O and ICAR-N are forced with
to no improvement and RE = 1 corresponding to a complete the ERA-Interim reanalysis (ERAI, Dee et al., 2011) instead
removal of errors. of the more recent ERA5 reanalysis. For the ICAR-O sim-
To characterize the effect of increasing the model top ele- ulation the model top is set to 4 km, the elevation that was
vation on the SSE while keeping the set of boundary condi- identified as seemingly optimal in Horak et al. (2019) and ZG
tions unchanged, RE is evaluated for increasing values of ztop0
BCs are applied to θ and all microphysics species (BC code
between 4.4 and 14.4 km with ztop = 4.4 km and BCs = BCs0 000). For the ICAR-N simulation Zmin is determined for the
in Eq. (24). The resulting RE values then are equivalent to the day of the case study as described in Sect. 3.5 by conducting
percentage change of the SSEs achieved by increasing ztop in multiple simulations with model tops between 5–20 km. A
comparison to the lowest tested model top setting. Similarly, ZG BC is imposed on the potential temperature field to avoid
to investigate the effect of an alternative set of boundary con- numerical instabilities arising for a CG BC due to strongly
0 and BCs 6 = BCs0 . Here
ditions, RE is evaluated for ztop = ztop stratified atmospheric layers and a CG BC is imposed on the
the resulting RE values quantify the percentage improvement microphysics species (BC code 011). The remaining setup
of the SSEs achieved by changing the imposed boundary for ICAR-O and ICAR-N, such as the forcing data set and
conditions at the upper boundary while leaving the model the model domain, have been described in detail in Horak
top elevation unchanged. et al. (2019).
The quantity zmin (ψ, BCs) is introduced, which defines The case study focuses on 6 May 2015 LT (local time),
the model top elevation for a given set of boundary con- a day with stably stratified large-scale northwesterly flow
ditions BCs and parameter ψ for which RE exceeds 95 % throughout the troposphere impinging on the Southern Alps
for the first time and remains above that threshold for ztop ≥ over a 24 h period. Upstream of the South Island, ERAI ex-
zmin . In preliminary studies, the 95 % threshold value was hibits a 24 h averaged relative humidity of more than 80 % in
found as a suitable indicator for reaching a saturation in error the lowest 2 km of the atmosphere, an averaged moist Brunt–
reduction (not shown). The lowest possible model top eleva- Väisälä frequency of 0.012 s−1 , a mean near-surface temper-
tion Zmin is then calculated as the maximum of zmin (ψ, BCs) ature of 16.5 o C and a mean specific humidity at the surface
for all quantities ψ and a specific combination of bound- of 11 g kg−1 .
ary conditions BCs. However, θ is excluded since this study
focuses mainly on hydrometeors. Nonetheless any relevant
error in θ influences the MP fields and the distribution of 4 Results
precipitation, thereby directly affecting Zmin . In this context
Zmin can then be interpreted as the lowest possible model 4.1 Comparison to the analytical solution
top elevation such that the cloud and precipitation processes
in the domain are sufficiently independent from influences of Figure 1 shows the horizontal and vertical perturbations to
the model top. the background state, as well as the isentropes of the per-
https://doi.org/10.5194/gmd-14-1657-2021 Geosci. Model Dev., 14, 1657–1680, 2021
1666 J. Horak et al.: A process-based evaluation of ICAR turbed potential temperature field as calculated with the an- gree ICAR is able to capture the results obtained with a full alytical solution based on linear theory and simulated with physics model. As expected, the WRF simulation shows a ICAR-N, ICAR-O and WRF up to an elevation of 15 km. larger deviation from the analytical wind field (cf. Fig. 1a, ICAR-N and ICAR-O simulations were run with ztop = e with Fig. 1d, h). The amplitudes in the perturbation fields 20.4 km and zero-gradient boundary conditions (BC code in WRF are larger and exhibit the elevation dependence in- 000). The simulations are conducted for a 2-D ridge and the dicated by Eq. (21). For w0 , for instance, the amplitude in- default scenario with the modification that RH = 0 % (see creases by 0.7 m s−1 from 4 to 10 km, resulting in an in- Sect. 3.2). creased orographic lift compared to ICAR. The range of ob- Generally, the horizontal west–east and vertical perturba- served values for u0 is −14.8 to 14.6 m s−1 and values of tions to the background state calculated by ICAR-N repro- w 0 lie between −1.7 and 2.4 m s−1 . These larger maximum duce those obtained from the analytical expressions well (cf. values in comparison to the analytical solution can mainly Fig. 1a–b and e–f). The range of values of u0 in ICAR-N be attributed to the amplification of the perturbations due to is −8.4 to 8.2 m s−1 compared to the −10.0 to 10.0 m s−1 the exponential decrease in density with height. For instance, derived from the analytical expression. Whereas for the at the elevation of the w0 maximum (Fig. 1h), the pressure north–south perturbations, the analytical solution yields v 0 = has dropped to about one-third of the surface pressure. Ac- 0 m s−1 , and ICAR-N calculates an average magnitude of cording to the pressure amplification term in Eq. (21) this 0.02 m s−1 . The minimum and maximum of v 0 are −1.6 and increases the amplitude by a factor of 1.7. The remaining dif- 1.5 m s−1 respectively, localized in close proximity to the ference of 0.7 m s−1 is most likely caused by wave amplifica- western and eastern domain boundaries. Along the domain tion due to nonlinearities and wave reflections at the damping center, v 0 lies between −0.5 and 0.5 m s−1 . For w0 , values layer. However, the general characteristics of the perturbation obtained with ICAR-N lie between ±1.1 m s−1 as opposed fields, such as the periodicity of the perturbations with eleva- to ±1.0 m s−1 for the analytical solution. The mean abso- tion and the approximate location of the positive and nega- lute error (MAE) in relation to the analytical solution of u0 tive perturbations, are similar to that of their corresponding is 0.9 m s−1 , which corresponds to 11 % of the absolute per- analytical counterparts. The increase in the amplitude of the turbation maximum. For w0 the MAE is 0.027 m s−1 or 2 % of perturbations due to the exponential decrease in density with the absolute perturbation maximum. This indicates a smaller height continues up until approximately 15 km (not shown), error in the w 0 field in ICAR-N in contrast to the u0 field. above which the dampening effects of the damping layer be- In comparison to the analytical fields (Fig. 1a) the u0 field come increasingly noticeable. in ICAR-N exhibits slightly lower values of u0 , particularly visible in the region where u0 < 0 m s−1 from approximately 4.2 Sensitivity to the set of upper boundary conditions 8 km upward, resulting in higher horizontal wind speeds in this region (Fig. 1b). The isentropes in ICAR-N are overall Figure 2a–e show the reduction of error (RE) achieved for very similar to those calculated analytically (see Fig. 1a–b), ICAR-N simulations for a given model top elevation ztop by yielding an MAE of 0.26 K. applying different upper boundary conditions than the ICAR The wind and potential temperature fields simulated by default (BC code 000). RE values are largest when a CG ICAR-O (Fig. 1c, g) exhibit clear differences to the analyt- BC is chosen for θ (Fig. 2a), more dependent on ztop for qv ical solution, especially above an elevation of about 6 km. (Fig. 2b) and smallest for the remaining quantities (Fig. 2c– The deterioration increases with elevation and is clearly visi- e) with similar results for all tested topographies and the re- ble from approximately z = 8 km upward, particularly for w0 spective time-averaged total masses Qv , Qsus and Qprc (not (Fig. 1g) but is still well pronounced for u0 and the isentropes shown). Most tested BC combinations reduce the error in (Fig. 1c). This is reflected in slightly elevated MAEs in com- at least one of the investigated quantities, but generally not parison to ICAR-N with 1.0 m s−1 in u0 , 0.034 m s−1 in w0 for all, with the exception of the combinations 141 and 142. and 0.32 K in θ . The reason for the relatively small differ- However, in the case of qsus , qprc , and P12 h no improvements ence to the MAEs of ICAR-N is that the MAE calculation for any BC combination are observed once ztop > 4.4 km across the entire cross section averages out the large devia- (Fig. 2c–e). The water vapor field shows improvements for tions in the small spatial area around the topographical ridge all BCs except for a CF BC, with the largest REs found for at the center. a CG BC imposed on qv . For the hydrometeors and P12 h the WRF is not expected to perfectly reproduce the analyti- improvement at the lowest model top setting of 4.4 km is only cal solution due to the occurrence of nonlinearities for the found if a CFG BC is applied to water vapor and either a CG, chosen non-dimensional mountain height of = 0.5 and the ZV or CFG to qhyd , otherwise the RE is approximately zero. amplification of perturbations due to the decrease in density The choice of an alternative BC over the standard ZG BC with height. Furthermore, the occurrence of partial wave re- has the largest potential for a reduction of error when (i) the flections from the model top is not entirely mitigated despite grid cells of the uppermost vertical level coincide with re- the careful selection of a damping layer (see Sect. 3.1). How- gions of vertical convergence where w < 0 and dw/dz < 0 ever, the WRF simulation serves as an indicator to what de- and (ii) the vertical flux gradients φz in these regions are Geosci. Model Dev., 14, 1657–1680, 2021 https://doi.org/10.5194/gmd-14-1657-2021
J. Horak et al.: A process-based evaluation of ICAR 1667 Figure 1. Vertical cross sections of the horizontal perturbation wind component u0 (top row) and vertical perturbation wind component w0 (bottom row) calculated analytically (left column) and calculated by ICAR-N (second column), ICAR-O (third column), and WRF (right column). The dashed–dotted horizontal line shows the vertical wavelength of the two-dimensional hydrostatic mountain wave λz , the dotted curve shows the 0 m s−1 contour line and the solid black contour lines show the isentropes. For panel (a) and (e), where the perturbation field is evaluated on constant height levels starting at z = 0 m, the topography is indicated by the dashed curve as to not obscure the perturbation field. All simulations are conducted for a 2-D ridge with hm = 1 km and a = 20 km and a background state with U = 20 m s−1 , Nd = 0.01 s−1 and RH = 0 %. Figure 2. The reduction of error (RE), dependent on the chosen combination of boundary conditions (x axis; see Table 3 for the key to the BC combination code), for (a) potential temperature θ, (b) water vapor q v , (c) suspended hydrometeors q sus , (d) precipitating hydrometeors q prc and (e) the 12 h precipitation sum P12 h . Note that overbars denote the temporal average of the respective quantity over 12 h following 18 h of model spinup. REs were calculated between an ICAR-N simulation with an alternative set of boundary conditions imposed at the upper boundary and an ICAR-N simulation employing the standard zero-gradient boundary condition (BC code 000), both run with the same model top elevation ztop (indicated by line color). All simulations are conducted for the default scenario. https://doi.org/10.5194/gmd-14-1657-2021 Geosci. Model Dev., 14, 1657–1680, 2021
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