Efficient Dynamical Downscaling of General Circulation Models Using
Continuous Data Assimilation
Srinivas Desamsetti1, Hari Prasad Dasari1, Sabique Langodan1, Edriss S. Titi2, Omar
Knio1 and Ibrahim Hoteit1,*
1. King Abdullah University of Science and Technology (KAUST), Physical Sciences
and Engineering Division, Thuwal, Saudi Arabia
2. Department of Mathematics, Texas A&M University, College Station, USA
*Corresponding author: Prof. Ibrahim Hoteit,
King Abdullah University of Science and Technology (KAUST),
Physical Science and Engineering Division,
Thuwal 23955-6900, Saudi Arabia.
E-mail: ibrahim.hoteit@kaust.edu.sa
1
Abstract
Continuous data assimilation (CDA) is successfully implemented for the first time for
efficient dynamical downscaling of a global atmospheric reanalysis. A comparison of the
performance of CDA with the standard grid and spectral nudging techniques for
representing long- and short-scale features in the downscaled fields using the Weather
Research and Forecast (WRF) model is further presented and analyzed. The WRF model
is configured at 0.25° ´ 0.25° horizontal resolution and is driven by 2.5° ´ 2.5° initial and
boundary conditions from NCEP/NCAR reanalysis fields. Downscaling experiments are
performed over a one-month period in January, 2016.
The similarity metric is used to evaluate the performance of the downscaling
methods for large (2000 km) and small (300 km) scales. Similarity results are compared
for the outputs of the WRF model with different downscaling techniques, NCEP/NCAR
reanalysis, and NCEP Final Analysis (FNL, available at 0.25° ´ 0.25° horizontal
resolution). Both spectral nudging and CDA describe better the small-scale features
compared to grid nudging. The choice of the wave number is critical in spectral nudging;
increasing the number of retained frequencies generally produced better small-scale
features, but only up to a certain threshold after which its solution gradually became closer
to grid nudging. CDA maintains the balance of the large- and small-scale features similar
to that of the best simulation achieved by the best spectral nudging configuration, without
the need of a spectral decomposition. The different downscaled atmospheric variables,
including rainfall distribution, with CDA is most consistent with the observations. The
Brier skill score values further indicate that the added value of CDA is distributed over the
entire model domain. The overall results clearly suggest that CDA provides an efficient
new approach for dynamical downscaling by maintaining better balance between the global
model and the downscaled fields.
Key words: Dynamical downscaling, WRF, Nudging, Spectral Nudging, Continuous Data
Assimilation, Similarity index.
2
1. Introduction
Dynamical downscaling using Regional Atmospheric Models (RAMs) is a broadly
recognized approach for resolving high resolution regional atmospheric features, e.g.,
Dickinson et al., (1989); Giorgi, (1990); Jacob and Podzun, (1997); Giorgi and Mearns,
(1999); Meehl et al., (2007); Rinke and Dethloff, (2000); Dasari et al., (2010, 2014),
Srinivas et al., (2015); Yesubabu et al., (2016), to cite but a few. It is widely implemented
in wide-range of applications, such as local weather forecasts, regional climate change
projections, air quality studies, energy applications, and numerous industrial applications
(Jacob and Podzun, 1997; Meehl et al., 2007; Langodan et al., 2014, 2016).
Although dynamical downscaling has been proven efficient for producing high
resolution information, the resulting outputs may still hold systematic and transient errors
(von Storch et al., 2000; Leung and Gustafson, 2005; Steiner et al., 2006). Generally, the
main sources of errors in dynamical downscaling result from either imperfect model
physics, and from the dynamical downscaling method itself (Dickinson et al.,1989; Giorgi,
1990). The uncertainties related to the model, such as the size of the domain, vertical and
horizontal resolutions, spin-up period, topography, and physical parameterizations, have
been investigated in several previous studies, e.g., Davies, (1983); Giorgi and Mearns,
(1999); Denis et al., (2002); Vincent and Hahmann, (2015).
To simulate fine-scale solutions, a balance between the RAM and the global down
scaled fields should be maintained during the downscaling simulations by retaining the
global large-scales and evolving the RAM to generate its own small-scale features; this has
been considered as one of the most challenging problems in dynamical downscaling
(Rockel et al., 2008; Vincent and Hahmann, 2015). To address this issue, the lateral
3
boundary relaxation, or nudging, technique was introduced by Davis (1976). It basically
consists of adding a nudging term to the predictive equation of the variable to be nudged.
In this study, we investigate an innovative dynamical downscaling technique for RAMs,
called Continuous Data Assimilation (CDA), and compare its performance with the stateof-the-art methods, the grid and spectral nudging techniques.
To capture the features of the driving large-scale fields throughout the domain, the
grid (Stauffer and Seaman, 1990) and spectral (Waldron et al., 1996; von Storch, 1995; von
Storch et al., 2000) nudging techniques have been proposed. Grid nudging is performed at
every grid point of the domain, constraining with equal weights the whole spectrum of the
atmospheric phenomena to the global fields (Stauffer and Seaman 1990). Spectral nudging
aims at better maintaining the balance of the RAMs by only constraining the large-scale
features while allowing the RAM to develop its local variability (Miguez-Macho et al.,
2005). This technique allows the model to better represent the small-scale effects due to
topography, land-sea contrast, and land-use distribution, and their interactions with the
large-scale fields (Feser, 2006; Feser and von Storch, 2005; Rockel et al., 2008; Winterfeldt
and Weisse, 2009; Vincent and Hahmann, 2015). However, the performance of spectral
nudging strongly depends on the choice of the cut-off wave number, with no systematic
way to set the value of this threshold other than conducting trial-and-error runs (von Storch
et al., 2000; Liu et al., 2012). Closely related methods have been also proposed, replacing
the large-scale fields of the RAM with the corresponding large-scale fields of the GCM at
specified time intervals as in Kida et al. (1991) and Sasaki et al. (1995), or adding finerscale perturbations to the large-scale GCM solution (Juang and Kanamitsu, 1994; Juang et
al., 1997) within the spirit of ‘‘anomaly models’’ (Navarra and Miyakoda, 1988).
4
Continuous Data Assimilation (CDA) methods (Charney et al., 1969; Daley, 1991)
can assimilate atmospheric observations into dynamical models during the integration
time. Recently, significant progress has been made in the CDA approach by introducing a
nudging term to the model equations to directly assimilate the observations (Henshaw et
al., 2003; Korn, 2009; Olson and Titi, 2009; Hayden-, et al., 2011; Azouani et al., 2014;
Bessaih et al., 2014; Altaf et al., 2017). The introduced nudging term constrains the model
large-scale variability to available information, which is computed as a misfit between
interpolants of the assimilated coarse grid information and fine grid model predictions.
This new CDA method has been designed, implemented, and tested for different physical
dynamical systems, including the Navier-Stokes equations, Rayleigh-Benard convection
model and planetary geostrophic ocean circulation model (Azouani et al., 2014; Farhat et
al., 2015, 2016a, b, c; Gesho et al., 2016; Altaf et al., 2017).
This study presents the first successful implementation of CDA for dynamical
downscaling with a three-dimensional, non-hydrostatic regional circulation atmospheric
model. We use the Advanced Research Weather Research and Forecasting (WRF-ARW;
Skamarock et al., 2008) model version 3.9 developed by NCEP/NCAR for this purpose.
We further evaluate and compare the results of the WRF simulations with grid nudging,
spectral nudging, and CDA for resolving the large- and small-scale atmospheric features
of a regional domain covering most of the African continent and the Middle East. Section
2 describes the model and the different downscaling techniques. Section 3 presents the
evaluation method used for the analysis of the downscaled fields. The results are discussed
in Section 4. Section 5 summarizes the main results of the study.
5
2. Model and nudging methods
The Advanced Research WRF (WRF-ARW) model version 3.9 developed by
NCEP/NCAR (Skamarock et al., 2008) is used to conduct the dynamical downscaling
experiments. The model configuration consists of a single domain covering the Africa
continent and the Middle East with 25 km x 25 km (~0.25° ´ 0.25°) horizontal resolution
and 35 vertical levels. Terrain elevation, land-use, and soil types were obtained from the
United States Geological Survey (USGS) data available at arc 2’ resolution. The
simulations are performed over a one-month period starting from January 1, 2016. The
model initial and 6- hourly boundary conditions are extracted from the NCEP/NCAR
reanalysis data available at 2.5° ´ 2.5°.
Different model free runs were conducted without nudging (control run), and with
the three downscaling methods: grid and spectral nudging, and CDA. All experiments were
performed with the same WRF configuration. The grid and spectral nudging methods are
implemented as in WRF-3.1.1 following Liu et al. (2012). CDA is implemented under the
same conditions as spectral nudging. Nudging is performed every 6 hours over the entire
simulation period and at every model grid point, with a nudging coefficient of 0.0003 s-1
for all nudged variables (Stauffer and Seaman, 1990; Gomez and Miguez, 2017). The
default spline interpolation operator in WRF model is used as the interpolant for the
downscaling experiments with CDA. We also performed several sensitivity experiments
using spectral nudging with different cut-off wave numbers (three, five, seventh, nine,
eleven and thirteen) in both the x- and y-directions to investigate its sensitivity to the choice
of this threshold; these are referred to as S33, S55, S77, S99, S1111, and S1313,
respectively.
6
2.1 Downscaling techniques
Nudging or Newtonian relaxation is commonly used in RAMs to maintain consistency with
the large-scale forcing fields while allowing mesoscale features to develop their own
variability in the regional simulations (von Storch, 1995; von Storch et al., 2000; Hogrefe
et al., 2004; Leung and Gustafson, 2005; Steiner et al., 2006). In this method, the model
state is relaxed toward a reference (could be an observation or an analysis) state by adding
an artificial tendency term, which is computed based on the difference between the
observed and model predicted states. This section briefly summarizes the general
formulations, and the differences, of the three tested downscaling methods. WRF is used
in this study, and since its nudging implementation is based on Stauffer and Seaman (1990),
we follow the same notations, in which nudging is defined as:
!"
!#
= F(α, *x⃗, t) + G" W(x*⃗, t)ε(x*⃗)(α
23 − α),
------------ (1)
where the term F(α, x*⃗, t) is the tendency predicted by the atmospheric model, x*⃗ represents
the spatial variables (x, y, z), α(x*⃗, t) is a particular atmospheric state variable (to be
nudged), and α
23 is the value toward which the state variable is nudged. G" is the nudging
factor that determines the relative magnitude of the tendency term in relation to the rest of
the model processes included in F(α, *x⃗, t). In WRF, its spatial and temporal variations are
set by a time-dependent (four-dimensional) weighting function W(x*⃗, t) = W67 W8 W9 , in
which W67 and W8 are respectively the horizontal and vertical weighting functions defined
based on a radius of influence and the distance from the observation. Similarly, the time
weighting function :9 depends on the model-relative time and the time of the observation
(Stauffer and Seaman, 1990). The analysis quality factor ε(x*⃗), typically varying between
0 and 1, depends on the distribution and quality of the nudging data. The nudged variables,
7
α, include the zonal and meridional wind components, and the potential temperature (or
the water vapor mixing ratio).
2.1.1 Grid nudging
Grid nudging in the WRF modeling system does not consider the quality of the analysis
ε(x*⃗) but includes a vertical weighting factor V(z) with values ranging between 0 and 1.
V(z) is included to remove the impact of nudging near the surface, so that to allow the
downscaling model to develop its own physics in the lower levels while nudging the
circulation in the upper levels to the reference data (Stauffer and Seaman, 1990). The
nudging equation in the WRF model is then expressed as
!"
!#
= F(α, *x⃗, t) + G" W(x*⃗, t)V(z)(α
23 − α),
----------- (2)
The model prediction is therefore nudged towards a reference field, typically a coarse
resolution global analysis, after it was interpolated to the RAM’s grid. Eq. (2) is then
applied assuming a perfect observation at every grid point.
2.1.2 Spectral nudging
Spectral nudging is implemented in a similar way to grid nudging, after applying a spectral
filtering to the tendency term (α
23 − α), first in the x-direction and then in the y-direction.
No filtering is applied in the vertical direction. The spectral nudging equation is then
expressed as:
!"
!#
= F(α, x*⃗, t) + G" W(x*⃗, t)V(z)Filt ?@ [(α
23 − α)],
------------- (3)
where Filt ?@ represents a spectral filtering above a certain cut-off wave number. Filtering
is then performed in three steps, as follows:
i)
A Fast Fourier Transform (FFT) algorithm is first applied on each row of the
tendency term (α
23 − α) to transform it to the spectral space.
8
ii) All wave numbers above a certain cut-off wavenumber in x-direction are set to
zero.
iii) Then, using the inverse FFT, the remaining Fourier coefficients are transformed
back to the spatial space.
The same procedure is then applied to each column of (α
23 − α) in the y-direction. The
spectral filtering removes all spatial frequencies higher than the selected cut-off wave
number, ensuring that nudging is only applied on the low wavelengths. Due to the
orthogonality of the functions of the Fourier expansion, only the same spectral components
of the physical space term F(α, *x⃗, t) in Eq. (3) are affected by nudging.
2.1.3 Continuous Data Assimilation (CDA)
CDA exploits the fact that instabilities in turbulent flows occur at spatial large-scales and
that spatial small-scales are stabilized by the viscous dissipation term in the Navier-Stokes
equations (Currie, 1974). A rigorous mathematical framework was then developed
showing that indeed the asymptotic, in time, behavior of the spatial large-scale of any
solution to the Navier-Stokes equations determines in a unique fashion the asymptotic
behavior of the full solution (Foias and Prodi, 1967; Jones and Titi, 1993; Cockburn et al.,
1997). As such, the potential issue of solution multiplicity in numerical weather models
(Weisse et al., 2000) would not manifest itself unless an extended setup is considered,
involving
for
e.g.
ensembles
of
uncertain
trajectories,
and/or
stochastic
parameterizations. Such a stochastic framework is yet to be covered by the theory of CDA.
To fix ideas, let us present some examples of the above results. Let us divide the
physical domain Ω into disjoint subdomains ΩD , E = 1, ⋯ , H, such that IJKLMΩD N ≤ ℎ,
and let us choose randomly points *x⃗D ∈ ΩD for E = 1, ⋯ , H. Suppose R(S⃗, T) and U(S⃗, T)
9
are two different solutions of the Navier-Stokes equations. If VRMS⃗D , TN − UMS⃗D , TNW → 0 as
T → ∞ for E = 1, ⋯ , H , then MR(S⃗, T) − U(S⃗, T)N → 0 as T → ∞ for all S⃗ ∈ Ω, provided
ℎ ≤ ℎ3 , where ℎ3 is a length scale that depends only on the Reynolds number, but is
independent of the specific solutions. This means that the nodal values of the solutions at
a spatial coarse scale of size ℎ determine the solution on the whole domain, asymptotically
^
in time. Similarly, let us set [\D ≡ _` _ ∫` [(S⃗, T)IS , the local volume average of the
a
a
function [ , for E = 1, ⋯ , H . If VR\D (T) − UD̅ (T)W → 0 as T → ∞ for E = 1, ⋯ , H , then
MR(S⃗, T) − U(S⃗, T)N → 0 as T → ∞ for all S⃗ ∈ Ω, provided ℎ ≤ ℎ3 as before. This means
that the spatial coarse scale local volume averages at the domain of size ℎ determine the
solution in a unique fashion, asymptotically in time.
Following the above notations, once can introduce interpolants approximation
operators based on the model values or the local volume averages, respectively.
Specifically, one can introduce, for example, the interpolant approximation operators by
step functions as follows:
gh (φ)(S⃗) = ∑l
⃗D Nk`a (S⃗)
Dm^ φMS
or
gh (φ)(S⃗) = ∑l
\D k`a (S⃗)
Dm^ [
for the case of nodal values or the case of local value averages, respectively. Here,
k`a (S⃗) = 1 whenever S⃗ ∈ ΩD ; and k`a (S⃗) = 0 whenever S⃗ ∉ ΩD the characteristic
function of the subdomain ΩD . The above interpolant operators gh (φ) are approximation
functions of φ at the spatial scale of size ℎ.
Capitalizing on these results, Azouani et al. (2014) introduced a new approach for
CDA that uses as nudging function the difference between coarse-scale interpolant of the
spatial downscaled data and a coarse-scale interpolant of the model outputs. Note that
10
gh (φ) are functions, and hence the nudging is not only at the grid points, but it is done
adequately on a full neighborhood diameter size ℎ at each grid point. This sets constraints
on the spatial large-scale flow of the model, which in turn forces the solution of the model
to behave like the unknown reference solution that corresponds to the observational coarse
mesh data. In particular, Azouani et al. (2014) were also able to show that the method is
not sensitive to the model initial conditions, i.e., no matter how one initializes the model,
its solution always converges, at an exponential rate in time, to the same unique reference
solution that corresponds to the given coarse mesh data, provided the observational grid is
fine enough depending on the Reynolds number. The CDA equation can be expressed as
!"
!#
= F(α, x*⃗, t) + G" W(x*⃗, t)V(z)[gh (α
23 ) − gh (α)],
------------- (4)
where gh is linear, coarse-mesh interpolation operator (Azouani et al., 2014). Specifically,
gh is used to define a smooth function that (i) interpolates the data provided at the nodes of
the coarser spatial mesh, and (ii) varies linearly between neighboring nodes.
Furthermore, stability analysis has enabled Mondaini and Titi (2018) and Ibdah et
al., (2018) to establish uniform in time error estimates for the spatial discretization and the
full discretization of the model, respectively, which makes its computational
implementation reliable. Notably, it has also been observed that this CDA approach is
equally applicable to other relevant dissipative systems, and that for certain systems it is
sufficient to collect coarse mesh measurements of only part of the state variables (Farhat
et al., 2016a,b,c; 2017). In this context, it has been shown rigorously (Farhat et al., 2016c)
that for planetary scale geostrophic circulation models, the coarse mesh observations of the
temperature are sufficient for determining and recovering the full reference solution, i.e.,
both the velocity and the temperature, as it has been asserted by Charney et al. (1969). In
11
addition, this CDA approach has also been extended by Foias et al. (2016) to incorporate
fully-discrete, in time and space, observations and to extract from these measurements
statistical information concerning the corresponding unknown reference solution. Biswas
et al. (2018) have recently extended this approach for recovering the probability
distribution of the reference solutions from probability distribution of the observed
measurements.
3. Evaluation method
To evaluate the results of the different downscaling techniques at different scales,
we used the similarity concept proposed by von Storch et al. (2000) and implemented, for
example, in Liu et al. (2012). The similarity index is measured based on a metric, q(T, r),
defined as
q(T, r) = 1 −
〈[t(9,u)vt∗ (9,u)]x 〉
〈t(9,u)x 〉
,
----------- (5)
where t is the model simulation time, L the length scale of interest, z(T, r) the input field
(NCEP/NCAR data in this study) to the RAM, z ∗ (T, r) the RAM (WRF) output field,
〈 〉 denotes the 2D spatial-average over the domain. Every 6-hours, the similarity at
different scales of interest are computed after obtaining z(T, r) and z ∗ (T, r), based on
which the performances of the downscaling techniques are evaluated for both the largeand small-scales in opposing ways. A high similarity is desired for large-scales, as it
suggests consistency between the large-scale features of the downscaled fields and the
input fields (von Storch et al., 2000; Liu et al., 2012). In contrast, lower similarities are
expected for the small-scale features in the RAM simulation as it develops its own
variability that is not present in the global fields.
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Following Liu et al. (2012), we compare the large- and small-scale features from
NCEP and the RAM at the horizontal scales 2000 km and 300 km, respectively. To
compute the similarity at these scales, we first interpolate NCEP/NCAR data (the input to
WRF) to the same grid as the RAM (WRF, 25 km). The 25 km resolution grid cells in the
modeling domain are next re-divided according to the scale of interest, in which the new
cell includes several original grid cells. The representative values of the input, z(T, r) and
output, z ∗ (T, r), fields are then computed for each new cell.
To evaluate the small-scale features in the downscaled WRF fields we introduce an
independent dataset, NCEP Final Analysis (FNL) available at 0.25° ´ 0.25° (which is of
same grid resolution as our downscaled fields) as suggested by Liu et al. (2012). The idea
is to compute the similarities between NCEP /NCAR and FNL for both large- and smallscales and use these as the reference to evaluate the similarities between the downscaled
WRF and NCEP/NCAR fields at the small-scales. If FNL and NCEP/NCAR are consistent
(i.e. have high similarity) at the large-scale, then the similarity between FNL and
NCEP/NCAR at the small-scales could be used as criteria for a reasonable similarity of the
WRF and NCEP/NCAR small-scale results. If not, the latter cannot be used directly and
instead the difference of similarity between the large- and small-scales would be used to
assess whether the change in similarity between the input and downscaled fields is
reasonable (Liu et al., 2012).
To further assess quantitatively the solution of the nudging methods, we evaluated
the Added Value (AV) of the downscaled fields with respect to the FNL observations
compared to the model solution without nudging using the modified Brier Skill Score
(BSS, von Storch and Zwiers, 1999; Feser et al., 2011; Li et al., 2016), defined as:
13
} x ( a ~ ,a )
, if Å Ç (aÉ , aÑ ) < Å Ç (aÜ , aÑ ),
----------- (6)
{|| = }x (a Ä ,a ) − 1, if Å Ç (aÉ , aÑ ) > Å Ç (aÜ , aÑ ),
----------- (7)
{|| = 1 −
} x ( a Ä ,a )
} x (a ,a )
~
where Å is the root-mean-square-error (rmse) between two atmospheric variable solutions.
aÑ , aÜ , KàI aÉ denote the FNL observations, the model control run without nudging, and
the model solutions with the different nudging techniques (grid, spectral, CDA),
respectively. Based on this definition, BSS varies between −1 and 1, with negative BSS
suggesting the control run is more in agreement with the observations, and positive BSS
indicating WRF with nudging has generated AV over the control run.
4. Results
This section assesses the downscaling performance of the CDA method and compares its
results to those of the control, grid nudging, and spectral nudging solutions. The evaluation
is performed based on surface pressure (PS), temperature (T850 and T500), the zonal (U)
and meridional (V) wind components, and kinetic energy (KE850 and KE500), as a
surrogate for wind speed, at the 850 and 500 hPa levels over the simulation period January
1–31, 2016. The similarity metrics q(T, r) for these variables at large- and small-scales are
first computed between the model and NCEP/NCAR, as described in Section 3, and their
temporal mean and standard deviation are outlined in Table 1 and Table 2, respectively.
The time evolutions of the similarity for large- and small-scales for temperature (T850 and
T500), and kinetic energy (KE850 and KE500) are shown in Figures 1 and 2.
Both large- (Table 1) and small-scales (Table 2) of PS, T850, and T500 show a
similar magnitude (> 0.9999) of similarity between CDA, grid, spectral, and no-nudging.
In all experiments and over the whole downscaling period, the high similarity of T850, and
T500 for the large-scale (Fig. 1a, d) suggest that all the downscaling techniques
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successfully capture the features of the driving large-scale fields. Although the differences
in the magnitudes are relatively small in all the experiments, the similarity for temperature
with NCEP/NCAR is relatively higher at small-scale for CDA and spectral nudging (Fig.
1b, e). Small temporal variations in the similarity for PS at both large- and small-scales
indicate (not shown) a slight improvement in similarity at both scales with the spectral
nudging and CDA compared to control run and grid nudging.
The similarity between the model and NCEP/NCAR at large-scale for KE850
and KE500 (Table 1) show (Figure 2) significant differences between the downscaling
methods. All evaluated downscaling methods, grid, spectral, and CDA, suggest much
higher means in similarity (> 0.9) at the large-scale than the control run without nudging
(<0.47), outlining the ability of the downscaling techniques to retain the features of the
global downscaled fields. In accordance with the discussion in Section 3, the similarity
values (Table 2) are significantly lower for the small-scale for all three downscaled
simulations. These lower similarities are expected due to the added variability produced by
the RAMs (Liu et al., 2012).
Important differences were observed in the sensitivity runs of spectral nudging
with different cut-off wave number. For instance, an increase in the retained spectral wave
frequencies from S33 to S99 increased the similarity between grid and spectral nudging
solution at both large- and small-scales. The differences between S77 and S99 were not
significant, while the results of S1111 and S1313 quickly deteriorated (not shown). This
confirms that spectral nudging should be implemented with a suitable number of waves,
large enough to constrain the large-scales, but not too large to avoid the damping of the
small-scales, as in grid nudging.
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The high similarity between the NCEP/NCAR and FNL fields at the large-scales
(Table 1) confirms the consistency between these two products. As discussed in Section 3,
we further computed the similarity between NCEP/NCAR and FNL at small-scale (Table
2) to determine the consistency of lower similarities of the different downscaled fields at
these scales. Although all the downscaling techniques provide a lower similarity at the
small-scale (Fig. 1c, Fig. 1f, Fig. 2c, Fig. 2f) compared to the large-scale, CDA and S99
are consistent in producing comparable decrease in similarity to that of FNL. The close
results of CDA to those obtained with the best simulation achieved using spectral nudging
(S99) at both large- and small-scales indicate the advantage of CDA in producing robust
downscaled fields without the need of any spectral decomposition.
As outlined in Table 2, on average and for both large and small scales, the
differences in the similarity indices of the different downscaling techniques are in the order
of 10-6, while their standard deviations are in the order of 10-7. To assess the significance
of these small differences in the similarity indexes in term of the final downscaled solution,
we compared the distributions of the differences between the WRF outputs (with the
different downscaling techniques) and NCEP/NCAR at the large- and small- scales with
those of the differences between FNL and NCEP/NCAR (Tables 3 and 4). The width of
the distribution represents the variability added by the different downscaling techniques to
global coarse fields (Liu et al., 2012). A small width reflects an over-nudging of the
downscaled fields towards the driving global fields, while a larger width emphasizes the
capability of the downscaling model to internally develop its small-scale variability that is
missing in the driving coarse fields. The differences in the distribution width of the largeand small-scales for temperature and wind components at different levels using grid
16
nudging are quite small compared to those of S99, CDA, and FNL. This suggests that grid
nudging over fits the RAM small-scale features to the driving global fields. CDA on the
other hand produces a distribution comparable to that of S99 at both large- and small-scales
that is more consistent with the FNL fields.
To further assess the significance of the larger variability in the similarity index
resulting from spectral nudging and CDA, we performed a linear regression analysis
between the differences of WRF outputs and NCEP/NCAR fields, and those of FNL to
NCEP/NCAR fields. The corresponding scatter plots are presented in Figures 3 and 4 for
T500 (Fig. 3) and KE500 (Fig. 4). CDA and spectral nudging improve the correlation and
slope with FNL compared to grid nudging at both large- and small-scales. Similar results
are obtained at 850 hPa (Fig. 5 and 6). Overall, the differences distributions analysis results
are consistent with those obtained from the analysis of the similarity index above,
indicating the relevance of the small differences in the similarity index in evaluating the
performances of the downscaling techniques, as has been already suggested by Lui et al.
(2012).
The differences in KE similarities computed between NCEP/NCAR and WRF
using spectral nudging and CDA may reflect significant differences in the surface sea level
pressure (SLP), relative humidity (RH) at 850 hPa and 500 hPa, clouds, and precipitation
patterns of the downscaled solutions, which are important parameters for regional climate,
air quality, and hydrological modeling studies. We therefore analyzed the mean SLP (Fig.
7) and RH at 850 hPa (Fig. 8) and 500 hPa (Fig. 9) as simulated by the different
downscaling methods along with their corresponding NCEP/NCAR fields. The dominant
pressure patterns observed in the NCEP/NCAR (Fig. 7a) are a relatively low pressure about
17
1010 hPa to 1015 hPa between 15N to 15S and a high pressure above 1020 hPa north of
15N. Similar patterns are well produced by the downscaling simulations, but not the control
run (Fig. 7c), which shows a larger bias of about 20-25 hPa over the oceanic regions and
of about 20 hPa over the Arabian Peninsula. The spatial patterns of RH at 850 hPa from
NCEP/NCAR clearly indicate (Fig. 8) the dry conditions over the desert regions (around
15N to 25N) and the coastal regions of the Arabian Sea between the Somali coast and
northwestern India. These dry and humid conditions are well simulated by CDA and
spectral nudging. The control and grid-nudging simulations (Fig. 8c and 8d) show higher
humidity over the dry regions and mountains compared to NCEP/NCAR. Similar to 850
hPa, the RH patterns (Fig. 9) at 500 hPa level are also well simulated by all experiments,
except the control run. Low humid values north of the equator extending up to 30N, and
high humid values south of the equator extending up to 20S are dominant features in the
NCEP/NCAR (Fig. 9a). Again, these are well reproduced by all simulations except the
control run (Fig. 9c), which exhibits low humid values around the equator from 20S to
30N. The small isolated regions around 10S with higher humidity in CDA (Fig. 9b) and
spectral nudging (Fig. 9e, f) are associated with convective activities in the Intertropical
Convergence Zone. The changes in the MSLP and RH between the different simulations
can modulate the cloud formation and associated rainfall. To confirm this, we compared
the total cloud coverage in the different simulations to the Modern-Era Retrospective
analysis for Research and Applications data (MERRA, Rienecker et al., 2011) in Figure
10. The mean total cloud coverage from MERRA shows (Fig.10a) a higher coverage of
about 70 - 80% between 15S and the equator compared to other regions where the cloud
coverage is relatively low (about 10-40%). These observed mean cloud features are well
18
produced by the CDA (Fig.10b) and the spectral nudging simulations (Fig. 10e and f). The
control (Fig. 10c) and grid-nudging (Fig. 10d) simulations failed to reproduce the observed
cloud patterns as observed by MERRA, with higher cloud coverage over the ocean regions,
and over land around the equator. The CDA and S99 solutions are in good agreement with
MERRA not only in terms of spatial distribution, but also in terms of magnitude. S99 shows
(Fig. 10e) better mean cloud patterns compared to S33, confirming the sensitivity of
spectral nudging to the choice of the cut-off wave number.
Given the considerable differences in surface pressure, relative humidity, cloud
coverage, and KE similarities with the different nudging methods, investigating the
associated changes in the simulation of precipitation is of interest. We therefore compared
the spatial patterns of the total rainfall from the different WRF runs with and without
nudging to the Global Precipitation Measurements (GPM, Huffman, 2016). Daily GPM
observations are extracted for the studied domain from half-hourly data at 10 ´ 10 km. The
GPM observations suggest (Fig. 11a) isolated rainfall patterns over the AP and adjoining
regions; heavy rainfall over the northern parts; small localized mountain regions with
isolated heavy rainfall between the equator to south of the AP; and a continuous rain belt
south of the equator. The control (Fig. 11c) and grid nudging (Fig. 11d) simulations fail to
predict both rainfall intensity and spatial distribution over the study region. The observed
large-scale features and mesoscale variations are much better reproduced by CDA (Fig.
11b) and spectral nudging (Fig. 11e, f). Experiments with different wave numbers suggests
that the higher wave numbers (S99) generate a comparable precipitation patterns to those
of GMP, primarily over northern Saudi Arabia, which are comparable to those simulated
by CDA.
19
The qualitative analysis of the spatial distribution of the different atmospheric
variables suggest that both CDA and S99 are in good agreement with the observations. For
a quantitative analysis, and to assess the added value (AV) from the different nudging
methods with respect to the FNL observations compared to the control run, we computed
the modified Brier skill score (BSS) for T850 and KE850 as representatives of the
thermodynamic and dynamic fields and plotted the results in Fig. 12 and 13. CDA and
spectral nudging result in important improvement for both T850 (Fig. 12) and K850 (Fig.
13) over the entire model domain compared to the control run, except for a small
mountainous region of northeast Africa, southwestern Red Sea, and northeastern Arabian
Peninsula. The BSS results are overall consistent with those of the analysis of similarity.
5. Summary
Nudging techniques enforce a dynamical balance between Regional Atmospheric Models
(RAMs) and global atmospheric fields play an important role in producing reliable regional
simulations. In this study, we successfully implemented and tested a new method,
continuous data assimilation (CDA), for dynamical downscaling of a three-dimensional
general circulation atmospheric model. We also evaluated its performance in reproducing
the large- and small-scale features of the downscaled region. We further compared the
results of CDA against those obtained using the standard grid and spectral nudging
methods. As one of the most widely used atmospheric models for downscaling, we used
the Weather Research and Forecast (WRF) model in our downscaling experiments, which
offers advanced packages for downscaling with grid and spectral nudging.
WRF model simulations at 25 km x 25 km (~0.25° ´ 0.25°) horizontal resolution
were carried out over a period of 31 days starting on January 1, 2016 with no, grid, spectral,
20
and CDA nudging. We also performed three more simulations with spectral nudging using
different wave cut-off numbers (3, 5, and 9) to assess its sensitivity to the choice of retained
wave number. The initial conditions and boundary forcing (updated every six hours) were
obtained from the NCEP/NCAR reanalysis available at 2.5° ´ 2.5°. Another independent
source of data, the NCEP Final Analysis (FNL) available at 0.25° ´ 0.25° resolution, was
used for validation. The evaluation of the downscaled fields was performed based on the
similarity metric suggested by Liu et al. (2012) for two different scales 2000 km and 300
km representing the large- and small-scale, respectively. We analyzed different parameters
such as temperature, pressure, winds and kinetic energy to assess the performance of the
different downscaling methods. The Brier skill score (BSS) index for temperature and
kinetic energy at 850 hPa was also computed to assess the added value (AV) of different
nudging methods with respect to FNL.
Overall, the downscaling experiments well captured the features of the driving
large-scale fields. At small-scales, spectral nudging with well-tuned cut-off wave number
and CDA produced stronger spatial and temporal variability than grid nudging, suggesting
the ability of these methods to enable more small-scale features in the model simulations,
which may be constrained by coarse downscaled fields in grid nudging. Spectral nudging
and CDA performances were also significantly closer to FNL than grid and no-nudging for
surface pressure (PS), temperature, and kinetic energy. The choice of the cut-off wave
number was very important in spectral nudging: an increase in the number of waves
produced better small-scale features, up to a certain wave number after which its results
became close to grid nudging. The results of the newly implemented CDA method were
21
consistent with the FNL and its performance is comparable to the best spectral nudging
simulation in producing both large- and small-scale features.
Our initial results suggest that the CDA method is a promising approach for
dynamical downscaling and enables to efficiently retain the balance between the RAMs
and GCMs by constraining the RAM large-scale features to the GCM fields and allowing
to develop its own small-scale features without the need of a spectral decomposition, which
also means saving in terms of computational time. More tests and experiments will be
conducted in future studies to further assess the performance of CDA with different grid
resolution, other driving global fields, different interpolation operators, and different
physical processes for predicting various regional atmospheric phenomena.
Acknowledgments
The research was supported by the office of Sponsor Research (OSR) at King Abdullah
University of Science and Technology (KAUST) under the Virtual Red Sea Initiative
(Grant #REP/1/3268-01-01) and the Saudi ARAMCO-KAUST Marine Environmental
Observatory (SAKMEO). This research made use of the Supercomputing Laboratory
resources at KAUST. The work of Edriss S. Titi was supported in part by ONR grant
N00014-15-1-2333, the Einstein Stiftung/Foundation - Berlin, through the Einstein
Visiting Fellow Program, and by the John Simon Guggenheim Memorial Foundation.
References
Altaf, M.U., Titi, E.S., Gebrael, T., Knio, O., Zhao, Z., McCabe, M.F., Hoteit, I., 2017.
Downscaling the 2D Benard Convection Equations using continuous data
assimilation. Computational Geosciences, 21:393.
Azouani, A., Olson, E., Titi, E.S., 2014. Continuous data assimilation using general
interpolant observables. J. Nonlinear Sci., 24, 277-304.
22
Bessaih, H., Olson, E., Titi, E. S., 2015. Continuous assimilation of data with stochastic
noise. Nonlinearity, 28, 729–753.
Biswas, A., Foias, C., Mondaini, C., Titi, E.S., 2018. Downscaling data assimilation
algorithm with applications to statistical solutions of the Navier--Stokes equations,
Annales de l'Institut Henri Poincare (C) Analyse Non-Lineaire, 39, 295-326.
https://doi.org/10.1016/j.anihpc.2018.05.004.
Charney, J., Halem, J., Kastrow, M., 1969. Use of incomplete historical data to infer the
present state of the atmosphere, Journal of Atmospheric Science, 26, 1160-1163.
Cockburn, B., Jones, D., Titi, E.S., 1997. Estimating the number of asymptotic degrees of
freedom for nonlinear dissipative systems, Mathematics of Computation, 66, 10731087.
Currie, I.G., 1974. Fundamental Mechanics of Fluids, McGraw-Hill, ISBN 0-07-0150001.
Daley, R., 1991. Atmospheric Data Analysis, Cambridge Atmospheric and Space Science
series, Cambridge University Press.
Dasari, H., Challa, VS., 2015. A study of precipitation climatology and its variability over
Europe using an Advanced advanced Regional Model (WRF). American Journal of
Climate Change, Vol.4, No.1.
Dasari, H., Challa, VS., BhaskarRao, D.V., Anjaneyulu, Y., 2011. Simulation of Indian
Monsoon extreme rainfall events during the decadal period of 2000-2009 using a
high-resolution mesoscale model. Advances in Geo Sciences, Volume 22 (AS-22),
31-48. World Scientific Publishing Co. Pte. Ltd.
Dasari, H., Salgado, R., Perdigao, J., Challa, V.S., 2014. A regional climate simulation
study using WRF-ARW model over Europe and evaluation for extreme temperature
weather events. International Journal of Atmospheric Sciences, 704079, 22.
http://dx.doi.org/10.1155/2014/704079.
Dasari, H., Wibig, J., Repaz, M., 2010: Numerical Modeling of the Severe Cold Weather
event over Central Europe (January 2006). Advances in Meteorology, 619478.
Davies, H.C., 1976: A lateral boundary formulation for multi-level prediction models.
Quart. J. Roy. Meteor. Soc., 102, 405–418.
23
Davies, H.C., 1983. Limitations of some common lateral boundary schemes used in
regional NWP models. Mon. Wea. Rev.,111, 1002–1012.
Denis, B., Laprise, J., Cote, J., Caya, D., 2002. Downscaling ability of one-way nested
regional models: The big-brother experiment. Clim. Dyn., 18, 627–646.
Dickinson, R. E., Errico, R. M., Giorgi, F., Bates, G. T., 1999. A regional climate model
for the western United States. Cim. Change, 15, 383–422
Farhat, A., Jolly, M.S., Titi, E.S., 2015. Continuous data assimilation for 2D Benard
convection through velocity measurements alone. Phys. D, 303, 59-66
Farhat, A., Lunasin, E., Titi, E.S., 2016a. Abridged continuous data assimilation for the 2D
Navier-Stokes equations utilizing measurements of only one component of the
velocity field. J. Math. Fluid Mech., 18, 1-23.
Farhat, A., Lunasin, E., Titi, E.S., 2016b. Data assimilation algorithm for 3D Bernard
convection in porous media employing only temperature measurements, Journal of
Mathematical Analysis and Applications, 438, 492-506.
Farhat, A., Lunasin, E., Titi, E.S., 2016c. On the Charney conjecture of data assimilation
employing temperature measurements alone: the paradigm of 3D planetary
geostrophic model. Mathematics of Climate and Weather Forecasting, 2, 61-74.
Farhat, A., Lunasin, E., Titi, E.S., 2017. Continuous data assimilation algorithm for a 2D
Bénard convection through horizontal velocity measurements alone. J. Nonlinear
Sci., 27, 1065-1087.
Feser, F., 2006. Enhanced detectability of added value in limited- area model results
separated into different spatial scales. Mon. Wea. Rev., 134, 2180–2190,
doi:10.1175/MWR3183.1.
Feser, F., von Storch, H., 2008: A dynamical downscaling case study for typhoons in
Southeast Asia using a regional climate model. Mon. Weather Rev., 136(5), 1806–
1815.
Feser, F., von Storch, H., 2005. A spatial two-dimensional discrete filter for limited area
model evaluation purposes. Mon. Weather Rev., 133, 1774–1786.
Feser, F., Rockel, B., von Storch, H., Winterfeldt, J., and Zahn, M., 2011. Regional climate
models add value. B. Am. Meteor. Soc., 92, 1181–1192.
24
Foias, C., Mondaini, C., Titi, E.S., 2016. A discrete data assimilation scheme for the
solutions of the 2D Navier-Stokes equations and their statistics. SIAM Journal of
Applied Dynamical Systems (SIADS), 15, 2109-2142.
Foias, C., Prodi, G., 1967. Sur le comportement global des solutions non stationnaires des
équations de Navier-Stokes en dimension 2. Rend. Sem. Mat. Univ. Padova, 39, 134.
Gesho, M., Olson, E., Titi, E.S., 2016. A computational study of a data assimilation
algorithm for the two-dimensional Navier-Stokes equations. Communications in
Computational Physics, 19, 1094-1110.
Giorgi, F., 1990. On the simulation of regional climate using a limited area model nested
in a general circulation model. J. Climate, 3, 941–963.
Giorgi, F., Marinucci, M. R., Bates, G. T., 1993. Development of second-generation
regional climate model (RegCM2). Part II: Convective processes and assimilation of
lateral boundary conditions. Mon. Wea. Rev., 121, 2814–2832.
Giorgi, F., Mearns, L. O., 1999. Introduction to special section: regional climate modeling
revisited. J. Geophys. Res., 104, doi:10.1029/98JD02072, 6335–6352.
Hayden, K., Olson, E., Titi, E.S., 2011. Discrete data assimilation in the Lorenz and 2D
Navier–Stokes equations. Physica D, 240, 1416–1425.
Henshaw, W.D., Kreiss, H.O., Ystrom, J., 2003. Numerical experiments on the interaction
between the large- and small-scale motion of the Navier-Stokes Equations. SIAM J.
Multiscale Modeling and Simulation, 1, 119-149.
Hogrefe, C., and co-authors., 2004. Simulation changes in regional air pollution over the
eastern United States due to changes in global and regional climate and emissions. J.
Geophys. Res., 109, D22301, doi:10.1029/2004JD004690.
Huffman, G., 2016. GPM IMERG Late Precipitation L3 1 day 0.1degree x 0.1degree V04,
Greenbelt, MD, Goddard Earth Sciences Data and Information Services.
https://disc.gsfc.nasa.gov/datacollection/GPM_3IMERGDL_04.html.
Ibdah, H.A., Mondaini, C., Titi, E.S., 2018. Uniform in time error estimates for fully
discrete
numerical
schemes
of
arXiv:1805.01595[math.NA]
25
a
data
assimilation
algorithm.
Jacob, D., Podzun, R., 1997. Sensitivity studies with the Regional Model REMO. Meteor.
Atmos. Phys., 63, 119–129.
Jones, D., Titi, E.S., 1993. Upper bounds on the number of determining modes, nodes, and
volume elements for the Navier-Stokes equations. Indiana University Math. Jour.,
42, 875-887
Juang, H.-M. H., and M. Kanamitsu, 1994: The NMC Regional Spec- tral Model. Mon.
Wea. Rev., 122, 3–26.
Juang, S.-Y. Hong, and M. Kanamitsu, 1997: The NCEP Regional Spectral Model: An
Update. Bull. Amer. Meteor. Soc., 78, 2125– 2143.
Kida, H., T. Koide, H. Sasaki, and M. Chiba, 1991: A new approach to coupling a limited
area model with a GCM for regional climate simulation. J. Meteor. Soc. Japan, 69,
723–728.
Korn, P., 2009. Data assimilation for the Navier-Stokes-α equations. Physica D., 238,
1957–1974.
Langodan, L., Cavaleri, L., Yesubabu, V., Hoteit, I., 2014. The Red Sea: a natural
laboratory for wind and wave modeling. Journal of Physical Oceanography., 44 (12),
3139-3159
Langodan, L., Yesubabu, V., Hariprasad, D., Knio, O., Hoteit, I., 2016. A high-resolution
assessment of wind and wave energy potentials in the Red Sea. Applied Energy, 181,
244-255
Lee, J.‐W., Hong, S.‐Y., 2014: Potential for added value to downscaled climate extremes
over Korea by increased resolution of a regional climate model. Theor. Appl.
Climatol., 117(3–4), 667–677.
Leung, L. R., ustafson, W. I., 2005. Potential regional climate change and implications to
US air quality. Geophys. Res. Lett., 32, L16711, doi:10.1029/2005GL022911.
Li, D., 2016: Added value of high-resolution regional climate model: selected cases
over the Bohai Sea and Yellow Sea areas. Int. J. Climatol., 37, 1, 169-179.
Liu, P., Tsimpidi, A. P., Hu, Y., Stone, B., Russell, A. G., and Nenes, A., 2012. Differences
between downscaling with spectral and grid nudging using WRF. Atmos. Chem.
Phys., 12, 3601–3610.
26
Meehl, G. A., and co-authors., 2007. Global climate projections, in Climate Change 2007:
The Physical Science Basis Contribution of Working Group to the Fourth Assessment
Report of the Inter-Governmental Panel on Climate Change. Cambridge Univ. Press,
Cambridge, UK, 747– 845.
Miguez-Macho, G., 2004. Spectral nudging to eliminate the effects of domain position and
geometry in regional climate model simulations. J. Geophys. Res.-Atmos., 109,
D13104, doi:10.1029/2003JD004495.
Miguez-Macho, G., 2005. Regional climate simulations over North America: Interaction
of local processes with improved large- scale flow, J. Climate, 18, 1227–1246.
Mondaini, C., Titi, E.S., 2018. Uniform in time error estimates for the postprocessing
Galerkin method applied to a data assimilation algorithm, SIAM Jour. Numerical
Analysis, 56, 78-110.
Navarra, A., and K. Miyakoda, 1988: Anomaly general circulation models. J. Atmos. Sci.,
45, 1509–1530.
Olson, E., Titi, E.S., 2009. Determining modes and Grashoff number in 2D turbulence.
Theoretical and Computational Fluid Dynamics, 22, 799-840.
Rienecker, M.M. et al. 2011. MERRA: NASA’s Modern-Era Retrospective Analysis for
Research and Applications. Journal of Climate, 24, 3624–3648.
Rinke, A., Dethloff, K., 2000. The influence of initial and boundary conditions on the
climate of the Arctic in a regional climate model. Climate Res., 14, 101–113.
Rockel, B., Castro, C.L., Pielke Sr., R. A., von Storch, H., Leoncini, G., 2008. Dynamical
downscaling: Assessment of model system dependent retained and added variability
for two different regional climate models. J. Geophys. Res., 113, D21107,
doi:10.1029/2007JD009461.
Sasaki, H., J. Kida, T. Koide, and M. Chiba, 1995: The performance of long term
integrations of a limited area model with the spectral boundary coupling method. J.
Meteor. Soc. Japan, 73, 165–181.
Skamarock, W.C., and co-authors., 2008. A Description of the Advanced Research WRF
Version 3, NCAR Technical note NCARTN-475+STR.
27
Srinivas, C.V., Hari Prasad, D., Bhaskar Rao, D.V., Bhaskaran, R., Venkatraman, B., 2015.
Simulation of the Indian summer monsoon onset-phase rainfall using a regional
model. Ann. Geophys., 33, 1097-1115, doi:10.5194/angeo-33-1097-2015.
Stauffer, D.R., Seaman, N.L., 1990. Use of four-dimensional data assimilation in a limitedarea mesoscale model, Part I: Experiments with synoptic-scale data, Mon. Wea. Rev.,
118, 1250–1277.
Steiner, A. L., Tonse, S., Cohen, R. C., Goldstein, A. H., Harley, R. A., 2006. Influence of
future climate and emissions on regional air quality in California. J. Geophys. Res.,
111, D18303, doi:10.1029/2005JD006935.
Vincent, C.L., Hahmann, A.N., 2015. The Impact of Grid and Spectral Nudging on the
Variance of the Near-Surface Wind Speed. Journal of Applied Meteorology And
Climatology, 54(5): 1021–1038, 10.1175/JAMC-D-14-0047.1.
von Storch H, Zwiers FW. 1999. Statistical Analysis in Climate Research. Cambridge
University Press: Cambridge, UK.
von Storch, H., 1995: Inconsistencies at the interface of climate impact studies and global
climate research. Meteor. Z., 4, 72–80.
von Storch, H., Langenberg, H., Feser, F., 2000. A spectral nudging technique for
dynamical downscaling purposes. Mon. Wea. Rev., 128, 3664 – 3673.
Weisse, R., Heyen, H., von Storch, H., 2000. Sensitivity of a regional atmospheric model
to a sea state dependent roughness and the need of ensemble calculations. Mon. Wea.
Rev. 128: 3631-3642.
Waldron, K.M., Peagle, J., Horel, J. D., 1996. Sensitivity of a spectrally filtered and nudged
limited area model to outer model options. Mon. Wea. Rev., 124, 529–547.
Winterfeldt, J., Geyer, B., Weisse, R., 2011. Using QuikSCAT in the added value
assessment of dynamically downscaled wind speed. Int. J. Climatol., 31 (7), 1028–
1039
Winterfeldt, J., Weisse, R., 2009. Assessment of value added for sur- face marine wind
speed obtained from two regional climate models. Mon. Wea. Rev., 137, 2955–2965.
28
Yesubabu, V., Hari Prasad, D., Langodan, S., Srinivas, C.V., Hoteit, I., 2016. Climatic
features of the Red Sea from a regional assimilative Model. International Journal of
Climatology, DOI: 10.1002/joc.4865
29
Table 1. Mean and standard deviation of the similarity between the RAM and NCEP/NCAR at large-scales
Mean of the similarity for large-scale waves (2000 km)
PS
T850
U850
V850
KE850
T500
U500
V500
KE500
CDA
0.99998605
0.99999714
0.97167140
0.97152543
0.96723545
0.99999964
0.99854946
0.99049723
0.99777579
Grid
0.99998581
0.99999744
0.98094010
0.97748172
0.99035347
0.99999976
0.99912435
0.99274999
0.99860662
No-Nudging
0.99986231
0.99972951
0.05710086
0.33376178
0.46544221
0.99996454
0.84965920
0.30325863
0.87648469
S33
0.99998665
0.99999714
0.94595301
0.91221452
0.91652679
0.99999917
0.99386102
0.96675873
0.99145013
S55
0.99998653
0.99999756
0.97086024
0.96998733
0.94466907
0.99999964
0.99846143
0.98823738
0.99735242
S99
0.99998629
0.99999785
0.97609609
0.97662634
0.98445141
0.99999964
0.99895734
0.99211472
0.99839503
FNL
0.99999946
0.99999064
0.84402716
0.82412124
0.90915436
0.99999577
Standard deviation of the similarity for large-scale waves (2000 km)
0.97440130
0.85004336
0.98543674
PS
T850
U850
V850
KE850
T500
U500
V500
KE500
CDA
0.000005
0.000001
0.014733
0.016004
0.030686
0.000000
0.000770
0.007707
0.001503
Grid
0.000005
0.000001
0.010876
0.013207
0.005458
0.000000
0.000378
0.006230
0.000723
No-Nudging
0.000042
0.000087
0.466062
0.303761
0.425571
0.000020
0.072125
0.577311
0.061159
S33
0.000005
0.000001
0.028365
0.041847
0.060352
0.000000
0.003972
0.029892
0.004708
S55
0.000005
0.000001
0.014842
0.016339
0.047274
0.000000
0.000767
0.009983
0.001695
S99
0.000005
0.000001
0.013099
0.012828
0.010660
0.000000
0.000456
0.007248
0.000999
FNL
0.000000
0.000003
0.086345
0.100384
0.056032
0.000002
0.014739
0.128554
0.008931
PS: surface pressure; T850 and T500: Temperature at 850 and 500 hPa levels; U850 and U500: Zonal wind components at 850 and 500 hPa levels; V850 and
V500: Meredional wind components at 850 and 500 hPa levels; KE850 and KE500: Kinetic Energy at 850 and 500 hPa levels.
30
Table 2. Mean and standard deviation of the similarity between the RAM and NCEP/NCAR at small-scales
Mean of the similarity for small-scale waves (300 km)
PS
T850
U850
V850
KE850
T500
U500
V500
KE500
CDA
0.99998337
0.99999058
0.91240192
0.89633244
0.89494669
0.99999863
0.98933464
0.97649366
0.99176407
Grid
0.99998373
0.99999225
0.95990425
0.94997531
0.97067863
0.99999911
0.99773675
0.98945946
0.99676031
No-Nudging
0.99985540
0.99964535
0.07106254
0.08139410
0.99993056
0.74900043
0.15007074
0.74443442
S33
0.99998409
0.99998194
0.68842328
0.63695365
0.63966984
0.99999440
0.94511408
0.82956451
0.92996812
S55
0.99998403
0.99998814
0.83709604
0.78103155
0.79188532
0.99999708
0.97539669
0.92701846
0.96858078
S99
0.99998385
0.99999207
0.92231774
0.90175653
0.92826903
0.99999875
0.99454635
0.97957581
0.99282628
FNL
0.99999863
0.99997193
0.65930980
0.60649222
0.70462215
0.99998736
Standard deviation of the similarity for small-scale waves (300 km)
0.92336237
0.72812873
0.92406327
PS
T850
-0.0519144
U850
V850
KE850
T500
U500
V500
KE500
CDA
0.000006
0.000003
0.027247
0.034035
0.074811
0.000000
0.004850
0.013146
0.004275
Grid
0.000005
0.000003
0.011793
0.017309
0.010606
0.000000
0.000569
0.005710
0.001229
No-Nudging
0.000043
0.000104
0.413987
0.270213
0.461699
0.000032
0.096030
0.493374
0.107139
S33
0.000005
0.000004
0.120302
0.105091
0.173989
0.000001
0.015521
0.097885
0.029512
S55
0.000005
0.000003
0.040479
0.058794
0.075941
0.000001
0.007464
0.040535
0.013222
S99
0.000006
0.000003
0.018289
0.026641
0.021350
0.000000
0.001449
0.011168
0.003153
FNL
0.000000
0.000005
0.101990
0.104339
0.087572
0.000003
0.024534
0.139838
0.026483
PS: surface pressure; T850 and T500: Temperature at 850 and 500 hPa levels; U850 and U500: Zonal wind components at 850 and 500 hPa levels; V850 and
V500: Meredional wind components at 850 and 500 hPa levels; KE850 and KE500: Kinetic Energy at 850 and 500 hPa levels.
31
Table 3. Mean and standard deviation of the distribution between the RAM and NCEP at large-scales
FNL
CDA
Grid
S99
T850
0.332
-0.204
-0.208
-0.164
FNL
CDA
Grid
S99
T850
0.225
0.079
0.068
0.075
Mean of large-scale waves (2000 km)
U850
V850
T500
0.058
-0.196
-0.019
-0.100
0.058
-0.054
-0.110
0.085
-0.066
-0.101
0.051
-0.054
Standard deviation of large-scale waves (2000 km)
U850
V850
T500
0.394
0.281
0.161
0.140
0.107
0.061
0.116
0.090
0.053
0.129
0.090
0.059
U500
-0.009
0.027
0.027
0.042
V500
0.317
0.045
0.058
0.054
U500
0.507
0.100
0.089
0.092
V500
0.474
0.101
0.086
0.091
T850 and T500: Temperature at 850 and 500 hPa levels; U850 and U500: Zonal wind components at 850 and 500 hPa levels; V850 and V500: Meredional wind
components at 850 and 500 hPa levels.
32
Table 4. Mean and standard deviation of the distribution between the RAM and NCEP at small-scales
FNL
CDA
Grid
S99
T850
0.357
-0.194
-0.206
-0.155
FNL
CDA
Grid
S99
T850
0.250
0.080
0.075
0.075
Mean of small-scale waves (300 km)
U850
V850
T500
0.060
-0.004
0.093
-0.046
-0.043
0.042
-0.088
-0.055
0.028
-0.064
0.029
-0.044
Standard deviation of small-scale waves (300 km)
U850
V850
T500
0.390
0.281
0.153
0.130
0.107
0.060
0.108
0.090
0.052
0.117
0.086
0.058
U500
-0.004
-0.043
-0.055
0.045
V500
0.260
0.024
0.040
0.036
U500
0.524
0.097
0.093
0.098
V500
0.462
0.090
0.078
0.082
T850 and T500: Temperature at 850 and 500 hPa levels; U850 and U500: Zonal wind components at 850 and 500 hPa levels; V850 and V500: Meredional wind
components at 850 and 500 hPa levels.
33
Figure 1. Time series of similarity in temperature at 850 (top panel) and 500 hPa (bottom panel) between NCEP and different
experiments (a, b, d, and e) with the RAM model, and between NCEP and FNL (c and f) at large- and small-scale waves.
34
Figure 2. Time series of similarity in kinetic energy at 850 (top panel) and 500 hPa (bottom panel) between NCEP and different
experiments (a, b, d, and e) with the RAM model, and between NCEP and FNL (c and f) at large- and small-scale waves.
35
Figure 3. Correlations between the temperature anomalies from NCEP/NCAR at 500 hPa
for different WRF model experiments of with a) Grid nudging, b) CDA, c) Spectral
nudging with three waves, and d) Spectral nudging with nine waves in both X and Ydirections for large- and small-scale waves.
36
Figure 4. Correlations between kinetic energy anomalies from NCEP/NCAR at 500 hPa
for different WRF model experiments of with a) Grid nudging, b) CDA, c) Spectral
nudging with three waves, and d) Spectral nudging with nine waves in both X- and Ydirections for large- and small-scale waves.
37
Figure 5. Correlations between the temperature anomalies from NCEP/NCAR at 850 hPa
for different WRF model experiments of with a) Grid nudging, b) CDA, c) Spectral
nudging with three waves, and d) Spectral nudging with nine waves in both X and Ydirections for large- and small-scale waves.
38
Figure 6. Correlations between kinetic energy anomalies from NCEP/NCAR at 850 hPa
for different WRF model experiments of with a) Grid nudging, b) CDA, c) Spectral
nudging with three waves, and d) Spectral nudging with nine waves in both X- and Ydirections for large- and small-scale waves.
39
Figure 7. Spatial distribution of mean surface pressure (hPa) from a) NCEP/NCAR and
different WRF model simulations of with b) CDA, c) No-nudging (NONDG), d) Gridnudging (GRID), e) Spectral nudging with three waves (S33), and f) Spectral nudging ,with
nine waves (S99) in both X and Y- directions for January 1–31, 2016.
40
Figure 8. Spatial distribution of Relative humidity (%) at 850 hPa from a) NCEP/NCAR
and different WRF model simulations of with b) CDA, c) No-nudging (NONDG), d) Gridnudging (GRID), e) Spectral nudging with three waves (S33), and f) Spectral nudging with
nine waves (S99) in both X and Y- directions for January 1–31, 2016.
41
Figure 9. Spatial distribution of Relative humidity (%) at 500 hPa from a) NCEP/NCAR
and different WRF model simulations of with b) CDA, c) No-nudging (NONDG), d) Gridnudging (GRID), e) Spectral nudging with three waves (S33), and f) Spectral nudging with
nine waves (S99) in both X and Y- directions for January 1–31, 2016.
42
Figure 10. Spatial distribution of total cloud cover (%) from a) MERRA observations and
different WRF model simulations of with b) CDA, c) No-nudging (NONDG), d) Gridnudging (GRID), e) Spectral nudging with three waves (S33), and f) Spectral nudging with
nine waves (S99) in both X and Y- directions for January 1–31, 2016.
43
Figure 11. Spatial distribution of total precipitation (mm) from a) GPM observations (OBS)
and different WRF model simulations of with b) CDA, c) No-nudging (NONDG), d) Gridnudging (GRID), e) Spectral nudging with three waves (S33), and f) Spectral nudging with
nine waves (S99) in both X and Y- directions for January 1–31, 2016.
44
Figure 12. Brier skill score distribution of the different nudging experiments relative to the
control simulations with FNL temperature (C) at 850 hPa level as a reference (true) field.
45
Figure 13. Brier skill score distribution of the different nudging experiments relative to the
control simulations with FNL kinetic energy (Joule) at 850 hPa level as a reference (true)
field.
46