Spatial kinetic energy spectra in the
convection-permitting limited-area NWP model
COSMO-DE
Lotte Bierdel∗ , Petra Friederichs, Sabrina Bentzien
Lotte Bierdel
Meteorological Institute, University of Bonn
Auf dem Hügel 20, 53121 Bonn, Germany
phone: +49 228 735108
e-mail: [email protected]
1
Abstract
2
Kinetic energy spectra derived from commercial aircraft observations of horizontal wind
3
velocities exhibit a k −5/3 wavenumber dependence on the mesoscale that merges into a
4
k −3 dependence on the macroscale. In this study, spectral analysis is applied to evalu-
5
ate the mesoscale ensemble prediction system using the convection-permitting NWP model
6
COSMO-DE (COSMO-DE-EPS). One-dimensional wavenumber spectra of the kinetic en-
7
ergy are derived from zonal and meridional wind velocities, as well as from vertical velocities.
8
Besides a general evaluation, the model spectra reveal important information about spin-up
9
effects and effective resolution.
10
The COSMO-DE-EPS well reproduces the spectral k −5/3 dependence of the mesoscale
11
horizontal kinetic energy spectrum. Due to the assimilation of high-resolution meteorologi-
12
cal observations (mainly rain radar), there is no significant spin-up of the model simulations
13
within the first few hours after initialization. COSMO-DE-EPS features an effective reso-
14
lution of a factor of about 4 to 5 of the horizontal grid spacing. This is slightly higher in
15
comparison to other limited area models. Kinetic energy spectra derived from vertical ve-
16
locities exhibit a much flatter wavenumber dependence leading to relatively large spectral
17
energy on smaller scales. This is in good agreement with similar models and also suggested
18
by observations of temporal variance spectra of the vertical velocity.
2
19
Zusammenfassung
20
Aus Flugzeugmessungen der horizontalen Windkomponenten abgeleitete Spektren kine-
21
tischer Energie weisen eine Wellenzahl-Abhängigkeit von k −5/3 auf der Mesoskala auf.
22
Auf der Makroskala geht diese in eine k −3 -Abhängigkeit über. In dieser Arbeit wird mit
23
Hilfe der Spektralanalyse das Ensemble-Vorhersagesystem des konvektionserlaubenden Wet-
24
tervorhersagemodells COSMO-DE (COSMO-DE-EPS) untersucht. Eindimensionale, hori-
25
zontale Wellenzahlspektren der kinetischen Energie werden aus horizontalen und vertikalen
26
Windgeschwindigkeiten abgeleitet. Neben der allgemeinen Auswertung ermöglichen die
27
Modellspektren eine Ableitung wichtiger Modelleigenschaften wie der Spin-Up Zeit und der
28
effektiven Auflösung.
29
Das COSMO-DE-EPS reproduziert die k −5/3 -Abhängigkeit des Spektrums der kinetis-
30
chen Energie auf der Mesoskala. Durch die Assimilation hochaufgelöster Radardaten zur
31
Initialisierung des Modells gibt es keinen signifikanten Spin-Up während der ersten Mod-
32
ellstunden. Die effektive Auflösung des COSMO-DE-EPS beträgt das vier- bis fünffache des
33
horizontalen Gitterabstands. Im Vergleich zu anderen Modellen mit begrenztem Modellgebiet
34
ist die effektive Auflösung damit höher. Aus Vertikalgeschwindigkeiten abgeleitete Spektren
35
der kinetischen Energie weisen eine flache Wellenlängenabhängigkeit auf. Dies führt zu einer
36
relativ hohen spektralen Energie auf kleineren Skalen, was mit Beobachtungen zeitlicher Var-
37
ianzspektren der Vertikalgeschwindigkeit übereinstimmt.
3
38
1
Introduction
39
The spectral variance of variables such as the kinetic energy as a function of horizontal scale
40
is fundamental in many theoretical studies of geophysical fluids and in turbulence theory. A
41
general observation when analyzing atmospheric horizontal kinetic energy spectra is a k −3 slope
42
on larger scales and a k −5/3 slope on the mesoscale, in observational as well as model data
43
(L ARSEN et al., 1982; NASTROM and G AGE, 1985; L INDBORG, 1999; C HO and L INDBORG,
44
2001; S KAMAROCK, 2004), where k is the wavenumber.
45
NASTROM et al. (1984) analysed commercial aircraft measurements of wind velocity under
46
the assumption of frozen turbulence, where a Taylor transformation is applied to convert fre-
47
quency spectra into wavenumber spectra. In their study two possible interpretations of the em-
48
pirically found k −5/3 powerlaw dependence below scales of about 500 km are given: an inverse
49
energy cascade (from high to low wavenumbers) in quasi two-dimensional (2D) turbulence or a
50
forward energy cascade (from low to high wavenumbers) associated with a spectrum of internal
51
gravity waves.
52
Since then, the theoretical interpretation of the mesoscale kinetic energy spectrum is under
53
vital debate (L INDBORG, 2005, 2007; T ULLOCH and S MITH, 2009; L OVEJOY et al., 2010).
54
Mainly two hypothesis have been put forward. The first hypothesis (G AGE, 1979; G AGE and
55
NASTROM, 1986) suggests an interpretation of the mesoscale kinetic energy spectrum in terms
56
of geostrophic turbulence (C HARNEY, 1971) which is in turn based on quasi 2D turbulence
57
theory (K RAICHNAN, 1967). The energy spectrum of 2D-turbulence is characterised by both
58
the conservation of energy and enstrophy. F JO/ RTOFT (1953) and C HARNEY (1971) based their
59
analysis on triadic transfer in the wavenumber space. They obtain a k −3 forward enstrophy
60
cascade with zero energy flux and a k −5/3 inverse energy cascade with zero enstrophy flux. It is
61
pointed out by M ERILEES and WARN (1975) that the latter proof is flawed, and G KIOULEKAS
62
and T UNG (2007) give a more general proof considering net fluxes. They conclude that enstrophy
4
63
is predominantly transferred downscale and energy is transferred upscale in forced-dissipative
64
2D-turbulence under statistical equilibrium. Furthermore they emphasise that a downscale energy
65
transfer is not prohibited although it is shown that the net energy flux is directed from small to
66
large scales.
67
The interpretation of atmospheric motions in terms of 2D-turbulence theory has been ques-
68
tioned (L INDBORG, 1999) mainly for two reasons: Firstly, 2D-turbulence theory requires a small-
69
scale source for the formation of an upscale energy flux on the mesoscales. However, this source
70
would be located on scales which are dominated by three-dimensional structures. Therefore, a
71
downscale energy cascade as predicted by 3D-turbulence theory should evolve rather than an
72
upscale cascade characteristic for 2D-turbulence. Secondly, there is observational evidence that
73
the k −5/3 spectral slope associated to the energy cascade appears on smaller scales than the
74
k −3 spectral slope of the enstrophy cascade. This is ’paradoxical’ (F RISCH (2004), p. 242) since
75
K RAICHNAN (1967) predicted the opposite in 2D-turbulence theory. The mesoscale portion of
76
the spectrum can still be explained as in 2D-turbulence by the formation of an inverse energy
77
cascade (G AGE, 1979; L ILLY, 1983; M ÉTAIS et al., 1996) which requires a small-scale energy
78
source. L ILLY (1983) suggested that this source can be generated by decaying convective clouds
79
and thunderstorm anvil outflows.
80
The second hypothesis bases on small wavenumber gravity waves which break down con-
81
tinuously to shorter waves providing a downscale (from large to small scales) energy flux (D E -
82
WAN , 1979, 1997; VAN
83
cascade of fully-developed, isotropic 3D-turbulence derived by Kolmogorov (see for example
84
F RISCH (2004), p.72 ff.). Recent interest has focused on the interpretation of the k −5/3 spec-
85
tral slope in terms of 3D stratified turbulence (L INDBORG, 2005; R ILEY and L INDBORG, 2008)
86
where further evidence for the forward cascade hypothesis is given. L INDBORG (2006) gives an
87
analysis of highly stratified flows with Froude numbers much smaller than one. He argues that
Z ANDT, 1982). The resulting forward cascade resembles the well-known
5
88
two-dimensional vertically oriented vortices split into thin layers through the influence of strong
89
stratification. These layers, in turn, split into thinner layers and so on. As the vertical extend of
90
the layers becomes smaller, they show sensitivity to Kelvin-Helmholtz instabilities which cause
91
a horizontal break-up of the layers. Through this mechanism total energy is transferred within
92
a highly anisotropic forward cascade (L INDBORG, 2005). Advocating the ’wave-hypothesis’,
93
L INDBORG (2006) argues, that the word ’wave’ should be replaced by ’layer’ in order to avoid
94
confusion with linear waves and to take account of the strong directional dependence of highly
95
stratified motion simultaneously.
96
In addition to theoretical studies the observed spectral slope of the mesoscale portion of
97
the spectrum is reproduced by direct numerical simulations (see for example R ILEY and DE -
98
B RUYN KOPS (2003); L INDBORG and B RETHOUWER (2007); T UNG and O RLANDO (2002))
99
and numerical weather prediction (NWP) models. KOSHYK and H AMILTON (2001) performed
100
general circulation model (GCM) simulations with the hydrostatic SKYHI GCM of the Geo-
101
physical Fluid Dynamics Laboratory with a horizontal grid spacing of 35 km of the highest
102
resolution version. Their simulations reproduced both the k −3 spectral slope on large scales of
103
about 5000 km to 500 km and the −5/3 power-law dependence below 500 km. S KAMAROCK
104
(2004) computed kinetic energy spectra using NWP forecasts from the non-hydrostatic Weather
105
Research and Forecast (WRF) model. He found that the k −5/3 -range is reproduced and actually
106
develops in simulations which were initialized with large-scale fields lacking mesoscale variabil-
107
ity.
108
The k −5/3 slope on the mesoscale has significant implications for the predictability when
109
using high-resolution mesoscale (NWP) models, since error growth on small scales may be much
110
faster for mesoscale dynamics compared to synoptic scale motions (S KAMAROCK, 2004). It is
111
thus important to evaluate NWP models with respect to their spectral characteristics. In this study,
112
spectral analysis is applied to evaluate the mesoscale ensemble prediction system (EPS) using
6
113
the convection-permitting NWP model COSMO-DE (BALDAUF et al., 2011). Horizontal, one-
114
dimensional wavenumber spectra of the kinetic energy are derived from zonal and meridional
115
wind velocities as well as from vertical velocity. The spectra not only represent the spatial
116
characteristics of mesoscale fields in COSMO-DE, but also give information about the scale
117
of the dynamics that is effectively resolved by the model. Spin-up effects and the development
118
of mesoscale disturbances during the forecast as well as the effects of data assimilation are
119
investigated.
120
121
122
123
124
125
The spectral behavior of the kinetic energy components is assessed with special emphasis on
the following questions:
• Does COSMO-DE reproduce the k −5/3 spectral slope for the horizontal velocity components
on the mesoscale?
• Does the COSMO-DE ensemble prediction system (COSMO-DE-EPS) exhibit significant
spin-up effects with respect to the spectral variance?
126
• What is the effective resolution of COSMO-DE with a grid spacing of 2.8 km?
127
• What are the spatial characteristics of the vertical velocity component?
128
Since COSMO-DE-EPS is aimed at forecasting high-impact weather conditions, we concen-
129
trate on cases that bear potential for hazardous impacts. The most dominant mesoscale weather-
130
related impacts in western Europe are related to strong mean winds, severe gusts, and heavy
131
precipitation (C RAIG et al., 2010). Particularly during summer, these weather situations are of-
132
ten related to deep moist convection.
133
Only very few observations exist of spatial variance spectra of vertical velocities on the atmo-
134
spheric mesoscale. BACMEISTER et al. (1996) present horizontal wavenumber spectra of the ver-
135
tical velocity derived from stratospheric NASA aircraft campaigns. Time-dependent spectra have
136
been measured amongst others by radiosondes, aircrafts or mesosphere-stratosphere-troposphere
7
137
(MST) radar. Temporal variance spectra of horizontal velocities follow a k −5/3 power law on
138
temporal scales between 2 and 50 hours (L ARSEN et al., 1982), whereas variance spectra of ver-
139
tical velocities show a much flatter scale dependence (DAREN et al., 1987). Model spectra seem
140
to reproduce the flatter spectrum for vertical wind velocities (S KAMAROCK and B RYAN, 2006;
141
F IORI et al., 2010). However, S KAMAROCK and B RYAN (2006) point out that the spectral char-
142
acteristics of the vertical velocity change with the grid spacing indicating that convergence of the
143
numerical solution is not yet achieved.
144
Spectral decomposition of fields from limited-area models by simple Fourier transformation
145
leads to largely biased estimates of the spectrum. Fourier transformation requires periodic
146
boundary conditions. Otherwise, unresolved large scale changes alias to smaller scales and cause
147
artificially high energy on small scales (E RRICO, 1985; D ENIS et al., 2002). It is thus necessary to
148
remove the aperiodic changes within the horizontal fields. In this study we follow E RRICO (1985)
149
and remove a linear two-dimensional trend. We furthermore assume horizontal isotropy, and
150
average the two-dimensional spectra over a torus defined by the length of the two-dimensional
151
wavenumber vector. In order to reduce the sampling noise, the individual spectra from each of the
152
20 members of the EPS are averaged. The spread over the ensemble accounts for the uncertainty
153
of the estimates.
154
In this article we proceed as follows: Section 2 shortly introduces the COSMO-DE model
155
and the ensemble prediction system setup. The weather situation during the selected cases is
156
represented in section 3. This is followed by a description of how the energy spectra are derived
157
in section 4. In section 5 we investigate the horizontal and vertical energy spectra in the COSMO-
158
DE-EPS for different forecast times and during four weather situations. The conclusions are
159
given in Section 6.
8
160
2
Data
161
The Consortium for Small-scale Modeling (COSMO) model is a non-hydrostatic limited-area
162
atmospheric model (www.cosmo-model.org) which is part of the operational model chain of
163
the German Meteorological Service (Deutscher Wetterdienst, DWD). COSMO-DE is the high-
164
resolution version with a horizontal grid spacing of about 2.8 km. It is nested into the COSMO-
165
EU with a horizontal grid spacing of 7 km which in turn receives the lateral boundary conditions
166
from the global NWP model GME. The model domain is centered over Germany and covers
167
parts of its neighboring countries and the Alpine region with 421 × 461 grid points on a rotated
168
regular longitude-latitude grid. The vertical discretization of the non-hydrostatic model uses 50
169
layers in orography-following σ-coordinates. The operational forecasts at DWD are issued for
170
21 hours and start every 3 hours.
171
On account of the relatively small horizontal grid spacing, COSMO-DE is able to explicitely
172
simulate deep convection without parameterization (D OMS et al., 2005). An important feature
173
of the operational COSMO-DE prediction system is the assimilation of surface precipitation
174
rates derived from the German radar network using latent heat nudging (S TEPHAN et al., 2008,
175
LHN). A cloud microphysics scheme accounts for precipitation in form of graupel, snow and rain
176
(R EINHARDT and S EIFERT, 2006). Sub-grid scale turbulence is parameterized according to the
177
level 2.5-model of M ELLOR and YAMADA (1982) with a turbulence closure of order 1.5 (D OMS
178
et al., 2005).
179
The σ-levels reach from 21500 m above ground down to 10 m. The dynamics on lower
180
levels are highly affected by orography, whereas the upper few levels are strongly damped by
181
Rayleigh damping (D OMS et al., 2005). Horizontal energy spectra of horizontal and vertical wind
182
velocities are investigated on the σ-levels in order to avoid artificial signals from interpolation. As
183
pointed out by S KAMAROCK (2004) computing spectra on constant pressure or height surfaces
184
only leads to little significant differences in the results.
9
185
The present study investigates forecasts from an experimental version of the ensemble pre-
186
diction system COSMO-DE-EPS, which is being developed at DWD. Under optimal conditions,
187
COSMO-DE-EPS consists of 20 forecasts using the operational version of COSMO-DE. The
188
different forecasts are driven by five perturbations of model physics and four different lateral
189
boundary conditions. No perturbation of initial conditions is applied in this version of the EPS.
190
Model physics are perturbed within the parameterizations of COSMO-DE by constant perturba-
191
tions. The entrainment rate for shallow convection is changed from 0.0003 (default) to 0.002, the
192
scaling factor for the thickness of the laminar boundary layer for heat from 1. (default) to 10 and
193
0.1, respectively, the critical value for normalized over-saturation from 4 (default) to 1.6, and the
194
maximal turbulent length scale from 500 m (default) to 150 m. A description of these parameters
195
and their default values is given in (S CH ÄTTLER et al., 2009). The lateral boundary conditions
196
are taken from four members of the COSMO-SREPS ensemble prediction system (M ARSIGLI
197
et al. (2008), ARPA-SIM, Bologna) which is a nested EPS with a grid spacing of 10 km. The
198
four selected members of COSMO-SREPS a driven by the global models IFS, GME, GFS and
199
UM.
200
The experimental 24h-forecasts were initialized daily at 00 UTC and performed for a period
201
from May to October 2009. Technical problems as well as an unsteady data flow from COSMO-
202
SREPS lead to large gaps within the data set of the experiment, particularly with regard to the
203
number of available members. Only 35 days of the above mentioned period consist of the entire
204
20 member forecasts.
205
3
206
We investigate COSMO-DE-EPS forecasts for four days during the period from May to October
207
2009 that exhibit potentially high-impact weather conditions. During all days chosen, heavy
208
precipitation occurred over parts of the model domain. All weather situations are prone for deep
Weather situations
10
209
convection, either within a synoptic scale frontal system or related to a large scale convectively
210
instable vertical stratification. Another criterion choosing these days is the availability of an EPS
211
with at least 15 members. The EPS is complete for May 20th, July 2nd, and August 21th 2009,
212
whereas for July 17th 2009 only 15 members are available.
213
During May 20th 2009, a cold front crosses the model domain from west to east. This leads to
214
a bisection of weather conditions over the model domain with a postfrontal north-western and a
215
prefrontal south-eastern region. The postfrontal part gets under the influence of cold polar air that
216
infiltrates in higher altitudes and leads to a destabilization of the atmosphere. The passage of the
217
cold front involves showery precipitation events in north-western Germany. During the course
218
of the day, the cold front becomes stationary and vanishes. The prefrontal part of the model
219
domain is dominated by subtropical, unstable air masses. After a clear morning, due to solar
220
irradiation cumulus clouds arise very locally in the south and south-east and lead to precipitation
221
and thunderstorms.
222
On July 2nd 2009, the weather condition over Germany is dominated by a high-pressure
223
system with weak horizontal pressure gradients. While it is fair in the morning, cumulus clouds
224
form over the central and sout-eastern part of the model domain in the late afternoon. The
225
development of intense and small-scale deep convection is a result of solar irradiation which
226
heats moist and warm air located over the model domain. Thunderstorms with hail and heavy
227
squalls arise locally, but no synoptic-scale lifting occurs during this day.
228
During July 17th 2009, a cold front of a low centered over England crosses the model domain
229
from south-west to north-east and infiltrates polar maritime air behind the front line. The front
230
is preceded by a convergence line with squall-line character lying in the warm-sector of the low,
231
in which course showers and thunderstorm cells form and intensify around noon. The synoptic-
232
scale induced storms are accompanied by hail and storm rainfall.
233
August 21th 2009 follows the hitherto warmest day of the year in Germany. It is characterized
11
234
by the passage of a cold front which is preceded by a convergence line. The prefrontal infiltration
235
of cold air leads to strong temperature differences between the eastern and western part of the
236
model domain. Along the convergence line thunderstorms and strong precipitation occur.
237
4
238
Generally, spectral analysis uses eigenfunctions of the Laplace operator within the respective
239
domain. The solutions of the two dimensional Laplace operator on a global domain are spherical
240
harmonics. On a limited area with regular grid spacing and periodic boundary conditions,
241
these solutions are sinusoidal functions. However, fields from limited-area NWP models have
242
non-periodic boundaries and are often formulated on a non-Cartesian horizontal grid. Spectral
243
analysis is thus not straightforward.
Methods
244
Fortunately, the horizontal grid of COSMO-DE is a rotated latitude-longitude grid where the
245
south pole is located at 40◦ S and 10◦ E. Consequently, the equator is located near the center of
246
the model domain, and the convergence of the meridians is minimized and the deviation from a
247
regular grid is negligible.
248
COSMO-DE-EPS is nested into members of coarser-scale COSMO-SREPS simulations (Sec.
249
2). The nesting uses a relaxation scheme. The influence of the lateral boundary condition
250
diminishes with distance to the boundary. However, the dynamics near the lateral boundaries
251
are strongly smoothed. For that reason, we removed 50 grid points (about 140 km) along each
252
edge of the model domain, so that it is reduced to 321 × 361 horizontal grid points.
253
Due to the lateral forcing, the boundary conditions are not periodic. If these aperiodicities are
254
not removed, the variance spectra are largely distorted due to the aliasing of the large, unresolved
255
changes onto smaller scales. In this study, they are removed by subtracting appropriate linear
256
trends following E RRICO (1985). Another methodology to account for the aperiodicities is the
257
discrete cosine transform (DCT, D ENIS et al., 2002), which takes a mirror image of the original
12
258
field prior to the Fourier transformation, and therefore makes the analyzed field periodic.
259
The removal of trends will be shortly described in the following. We are given a field ai,j
260
with i = 1, ..., Nx and j = 1, ..., Ny on the approximately regular reduced grid of COSMO-DE.
261
The linear trend in y-direction is defined as
sj =
262
aNx ,j − a1,j
,
Nx − 1
and removed from ai,j by
0
1
ai,j = ai,j − (2i − Nx − 1) · sj .
2
(4.1)
263
Linear trend determination and subtraction are then repeated with interchanged i and j. The
264
resulting field is independent of the order of detrending in x- and y-direction and has periodic
265
boundaries. Trend estimation and subtraction are separately performed for each model variable.
266
The estimation of the one-dimensional variance spectrum again follows E RRICO (1985).
267
First, the two-dimensional spectrum is estimated via fast Fourier transformation (FFT). Assum-
268
ing horizontal isotropy, the one-dimensional wavenumber spectrum I(κ) is derived by summa-
269
tion of the squared absolute value of the two-dimensional Fourier coefficients C(p, q) over a
270
discrete annulus Aκ in wavenumber space
I(κ) =
X
∗
Cp,q · Cp,q
,
p,q∈Aκ
∗
denotes the complex conjugate. p and q are the discrete wavenumbers in x and y-
271
where
272
direction and given by
p
=
q
=
1
2πlx
,
∆ Nx − 1
2πly
1
,
∆ Ny − 1
Nx
,
2
Ny
ly = 0, ±1, ..., ±
,
2
lx = 0, ±1, ..., ±
p
273
where ∆ is the horizontal grid spacing. Aκ denotes the annulus given by κ− 21 ∆κ <
274
κ + 21 ∆κ . Furthermore, ∆κ is defined as the minimum of the fundamental values of p and q, and
13
p2 + q 2 <
275
since in our case Ny > Nx , we use
∆κ =
2π
∆(Ny − 1)
276
to define the size of the annuli. The central radii of the annuli are defined as κ = l · ∆κ with
277
l = 0, 1, ...,
278
spectra are independently estimated for the u- and v-wind component and then averaged.
Ny
2 .
For the kinetic energy spectrum of the horizontal wind, the one-dimensional
279
In order to display the ensemble mean and spread of the spectra for the EPS simulations, we
280
first calculated the spectra for the wind fields of each simulation, separately. The ensemble mean
281
spectrum is then derived as the mean over the single spectra, whereas the spread indicates the
282
range of the spectral estimates.
283
5
284
5.1
285
Results
Height-dependence of the mass specific, horizontal kinetic energy
spectrum of the horizontal wind velocity
286
We start by investigating the height-dependence of the horizontal kinetic energy spectrum in
287
the COSMO-DE-EPS. In the lower troposphere orography has a large impact on the spatial
288
characteristics of the atmospheric dynamics, whereas the upper tropospheric flow is dominated
289
by large scale structures. The upper most levels are strongly damped and exhibit very smooth
290
horizontal wind fields. This is due to a damping layer, which is included in NWP models in
291
order to absorb upward moving wave disturbances and prevent reflection of gravity waves from
292
the upper model boundary (D OMS et al., 2005). In COSMO-DE the damping layer starts in an
293
elevation of about 11 km (σ-levels < 10).
294
Fig. 1 illustrated the effect height-dependence of the horizontal kinetic energy spectra of
295
the horizontal wind components. The spectra are presented for the 1h forecast time on May
296
20th 2009. The unphysical properties of the damping layer are reflected in the associated kinetic
14
297
energy spectra and are not considered (Figs. 1 a,b). The energy spectrum in σ-level 50 mainly
298
represents the influence of the orography with large energy particularly on smaller scales. Except
299
for the uppermost and the lowest levels, the k −5/3 wavenumber dependence is well reproduced
300
throughout the entire troposphere. In order to minimize the influence of the upper damping layer
301
as well as the lower boundary, we concentrate on σ-level 20 for the investigation of the horizontal
302
kinetic energy spectra.
303
5.2
Mass specific, horizontal kinetic energy spectra of the horizontal wind
velocity
304
305
In the following, spectra of the mass specific, horizontal kinetic energy for the four days described
306
in Sec. 3 are presented. Only σ-level 20 is considered here, which corresponds to a height of about
307
7 km.
308
Figs. 2 to 5 show energy spectra as well as the associated horizontal wind fields for the 1h,
309
10h, and 20h forecast times, respectively. The wind fields are composed of the absolute value of
310
the detrended fields of the wind components in m/s and the arrows which indicate the direction
311
and strength of the original fields of the wind components. We refrain from displaying the spectra
312
and wind fields for 0h forecast time (initialization). The variance spectra are almost identical to
313
those of 1h forecast time, except that as a result of the LHN (Sec. 2) the ensemble spread is
314
almost zero at initialization. The similarity of the spectra at 0h and 1h forecast times indicates
315
that no significant spin-up or adjustment takes place during the first forecast hours.
316
In Figs. 2 to 5 we display the ensemble mean of the energy spectrum as well as the spread
317
within the ensemble. Linear trends are fitted to the ensemble mean spectra on the log-log scale
318
over the range of the meso-β scale (20 to 200 km) indicated by vertical gray lines. The slope
319
estimate β is given in the caption together with a 95% confidence interval. Note however, that
320
the underlying regression model assumes independent Gaussian errors. Although most slope
15
321
estimates are very close to k −5/3 , there are significant differences.
322
We start with May 20th 2009. The horizontal wind in a height of about 7 km is dominated
323
by a strong south-westerly component that weakens during the course of the day (right panels
324
of Fig. 2). The weak maximum of wind speed in the east of the model domain indicates the
325
anticyclonic arm of the jetstream. Due to LHN (Sec. 2), atmospheric fields in COSMO-DE-
326
EPS already contain small scale disturbances at the initialization of the forecasts (not shown).
327
Consequently, Fig. 2 a) exhibits an energy spectrum that is very close to k −5/3 over the range
328
of the meso-β scale. The spectrum slightly subsides at scales below about 12 km. The general
329
character of the horizontal kinetic energy spectra is conserved throughout the simulation and a
330
spin-up effect in which small scale disturbances evolve is not detected. However, at forecast time
331
10h, the energy at wavelengths between 12 km to 30 km is significantly increased with respect to
332
a k −5/3 slope. It should be noted, that spectra at forecast times 10h and 20h feature an overall loss
333
of energy on the mesoscale compared to the first hours of simulation. This is due to a weakening
334
and northward shift of the maximum wind speed as indicated in the right panels of Fig. 2.
335
The ensemble spread due to perturbed model physics and different boundary conditions
336
(compare Sec. 2) increases over time. In order to investigate the influence of the lateral boundary
337
conditions and the perturbed model physics, the spectra of each group of simulation setup are
338
compared. However, no systematic differences are found between the groups and all members
339
equally contribute to the spread of the ensemble (not shown).
340
The formulation of dissipation, the grid size and the utilized integration scheme influence the
341
models capability to resolve processes acting on scales near the resolution-limits of the model.
342
Thus, the spectra reveal important information about the models ability to resolve small scale
343
dynamics. The wavelength at which a model spectrum decays relative to a simulation with a finer
344
grid spacing is referred to as the effective resolution (S KAMAROCK, 2004). Here, the effective
345
resolution is roughly estimated as the wavelength where the spectrum decays significantly below
16
346
the k −5/3 slope. The rationale behind this is that the k −5/3 slope is expected to extend further
347
down to smaller wavelengths. If the spectrum drops more rapidly then those modes are damped
348
by numeric dissipation and hence are dynamically inaccurate.
349
With regard to the spectra on May 20th 2009 (Fig. 2), the effective resolution is estimated to
350
about 15 km since on smaller scales the spectral slope decays significantly faster than k −5/3 .
351
S KAMAROCK (2004) investigates the effective resolution of the WRF model with different
352
grid spacing. He observes that the effective resolution is a constant multiple of the horizontal
353
grid spacing ∆, which is generally around 7∆. In our case, an effective resolution of 15 km
354
corresponds to 5∆, which is higher than in S KAMAROCK (2004).
355
On July 2nd 2009, as described previously (Sec. 3), the model domain is under the influence
356
of a high pressure system and small-scale convection arises during the course of the day.
357
Consistently, the absolute values of the horizontal wind components (right panel of Fig. 3)
358
are very low and large scale dynamics are less pronounced. In contrast to May 20th 2009, no
359
jetstream passes the model domain. The k −5/3 slope on the mesoscale is again well reproduced
360
(Fig. 3) and the ensemble spread evolves similarly to that on May 20th. One major difference is
361
the less pronounced decay of the spectrum at higher wavenumbers indicating a relatively high
362
portion of small-scale variations of the horizontal kinetic energy. This is consistent with a weather
363
situation where the dynamics are dominated by local thermal convection rather than synoptic-
364
scale processes. The weak transition from resolved to unresolved scales induces difficulties in
365
determining the effective resolution. An approximate estimation yields an effective resolution of
366
about 11 km corresponding to 4∆. At forecast hour 20, a significant increase is observed with
367
respect to a k −5/3 slope in the kinetic energy on scales below 30 km.
368
Fig. 4 represents the kinetic energy spectra and the wind fields for July 17th 2009. The
369
temporal evolution of the wind field is characterized by a jetstream entering the model domain.
370
Again, the observed k −5/3 slope is reproduced and maintained during the simulation. Compared
17
371
to the other spectra, the decay at the effective resolution of about 15 km is more pronounced. The
372
ensemble spread grows faster than during the previously discussed days. At forecasts hour 10 the
373
jetstream is located in the model domain. This coincides with an increase of small-scale energy
374
at the forecast hours 10 and 20, which in turn correlates well with the associated horizontal wind
375
fields, where small-scale variability enhances across the entire model domain. A striking feature
376
in Fig. 4 is the bulge across the wavelength range of about 120 km to 250 km in the spectrum
377
of the 20th forecast hour. This local increase of energy over a narrow range of wavelengths
378
accompanied by an enlarged ensemble spread probably represents the jetstream.
379
The horizontal kinetic energy spectra and the wind fields for August 21th 2009 are displayed
380
in Fig. 5. The temporal evolution of the horizontal wind field shows the weakening and north-
381
eastward displacement of the cyclonic arm of the jetstream. Again, the k −5/3 slope is well
382
reproduced, and the strong decay of the spectra at small scales reveals an effective resolution of
383
about 13 km. As in the previous cases the ensemble spread grows with time and some enhanced
384
small scale energy is observed at forecast hour 20.
385
Figs. 2 to 5 indicate the spread within the ensemble in terms of the spectral kinetic energy.
386
It represents the spread in an average over the squared amplitudes of the respective harmonics.
387
Note, that the spectra are largely smoothed, firstly by averaging over all coefficients with similar
388
absolute value of the wavenumber vector, and secondly by averaging over the meridional and
389
zonal wind components. Another component of the ensemble spread is related to the phase shift,
390
which is associated to the displacement errors in an ensemble and is not investigated in this study.
391
Consequently, Figs. 2 to 5 only represent one smoothed component of the spread in the forecast,
392
and further studies are needed in order to assess the spread of the forecasts of the horizontal
393
kinetic energy spectra in the COSMO-DE-EPS with respect to observations.
394
Preliminary investigations of precipitation, surface temperature and wind gusts show that
395
COSMO-DE-EPS is underdispersive. This underdispersiveness is also observed in the AROME
18
396
mesoscale EPS from Méteo-France (V I É et al., 2011). Note that the EPS we investigate here
397
is started from unperturbed initial conditions. A perturbation of the initial conditions increases
398
the spread during the first 6 forecasts hours (P ERALTA and B UCHHOLD, 2011). However, the
399
ensemble remains underdispersive (personal communication S. Theis, 2011).
400
5.3
401
Mass specific, horizontal kinetic energy spectra of the vertical wind
velocity
402
This section discusses the variance spectra of the mass specific vertical velocity (Fig. 6) on σ-
403
level 20. We restrict the presentation to the 20h forecast time of the four days described in Sec. 3.
404
The computation is described in Sec. 4. The spectral behavior of the vertical velocity exhibits a
405
much flatter wavenumber dependence compared to the horizontal velocity components. A k −5/3
406
dependence, and therefore an inertial subrange, is not existent. In contrast, all the spectra show
407
almost no wavenumber dependence on the mesoscale.
408
This is consistent with horizontal power spectra of the stratospheric vertical velocities derived
409
from aircraft observations by BACMEISTER et al. (1996). Similarly to the simulated spectra,
410
their spectral slope is flat on wavelengths above about 15 km, and drops rapidly towards small
411
wavelengths below 15 km. The flat shape of the spectrum is also observed in model studies by
412
S KAMAROCK and B RYAN (2006) and F IORI et al. (2010), and consistent with MST-observations
413
of temporal variance spectra of the vertical velocity by L ARSEN et al. (1982) and DAREN et al.
414
(1987). The spectral characteristics of the horizontal and vertical components of the kinetic
415
energy are very different. The large scale variations are dominated by the horizontal kinetic
416
energy component, whereas the contribution of the vertical component becomes significant for
417
small-scale variations.
418
The general characteristics of the vertical velocity spectra for the other forecasts times are
419
very similar, and therefore not displayed. As for the horizontal energy spectra, the ensemble
19
420
spread is very small during initialization, but the ensemble spread grows faster for the vertical
421
compared to the horizontal velocity components (not shown).
422
The decay of the spectra at very high wavenumbers strongly varies between the four days. On
423
July 2nd 2009 (Fig. 6 b)) the damping of the spectrum is much less pronounced than on the other
424
days. A similar behavior was indeed observed on July 2nd 2009 for the horizontal velocity spectra
425
in Figs. 2 to 5. The estimation of an effective resolution on the basis of vertical velocity spectra
426
is critical since no hypothesis exists about the behavior on smaller scales. However, the spectra
427
drop significantly at wavelengths of about 4∆ to 5∆ probably due to numerical dissipation. This
428
is consistent with the estimates of the effective resolution for the horizontal wind velocities.
429
6
430
Mass specific horizontal kinetic energy spectra have been computed from horizontal fields of the
431
horizontal and vertical wind velocity as simulated by the COSMO-DE-EPS. The energy spectra
432
are estimated following E RRICO (1985). The aim of the study is to assess the spectral behavior of
433
the kinetic energy components in COSMO-DE, i.e. whether COSMO-DE reproduces the k −5/3
434
spectral slope for the horizontal velocity components on the mesoscale. Furthermore, we assess
435
the effective resolution of COSMO-DE with a grid spacing of 2.8 km and investigate the spatial
436
characteristics of the vertical velocity component.
Conclusions
437
To that end, 21 hour forecasts for four days in Summer 2009 are investigated during weather
438
situations with a potential for high-impact weather. We discuss the height dependence of the
439
spectral behaviour and variance spectra for velocities in the upper troposphere are presented.
440
Despite some significant deviations, the k −5/3 slope characteristic of the mesoscale as suggested
441
by flight observations by NASTROM and G AGE (1985) is generally reproduced in the COSMO-
442
DE-EPS. Albeit significant, the deviations are small and represent specific weather conditions
20
443
such as squall-lines or small convective cells. A consolidated view gives confidence in the high-
444
resolution COSMO-DE in the sense that its dynamical representation is energetically correct.
445
Observations of the horizontal variance spectra of the vertical velocity are rare. The spectra
446
of the vertical velocity component of COSMO-DE are very flat and show almost no scale
447
dependence on the mesoscale. This is not inconsistent with observations of temporal variance
448
spectra (L ARSEN et al., 1982; DAREN et al., 1987) and confirms the findings by F IORI et al.
449
(2010). It’s to be assumed, that the convergence of the numerical solution with respect to the
450
spectral characteristics of the vertical velocity component is not yet achieved, as pointed out by
451
S KAMAROCK and B RYAN (2006). Moreover, the spatial characteristics of the vertical velocity
452
might strongly depend on the respective turbulence parameterization, and further studies are
453
needed to assess the reasons of the flat vertical velocity spectra and how representative they are.
454
In all cases, the ensemble spread is very small at initialization. The horizontal wind fields
455
exhibit a k −5/3 energy spectrum already at initialization, and no spin-up effect or adjustment is
456
observed. This is attributable to the LHN which assimilates high-resolution radar observations.
457
With increasing forecast time the ensemble spread grows. The error growth as represented by an
458
increased spread of the energy spectra depends on the weather situation. It is particularly large
459
on July 17th 2009, when the jetstream enters the model domain. Similar results are obtained for
460
the vertical velocity component. No spin-up is detected, but the error growth within the ensemble
461
is larger.
462
The effective resolution of COSMO-DE is about 11 km to 15 km which corresponds to a
463
multiple of 4 to 5 of the horizontal grid spacing. S KAMAROCK (2004) concluded, that with
464
an effective resolution of 7∆ they are close to the limit of the resolving capabilities of finite-
465
difference models that use explicit methods. This suggests, that the tuning of the dissipation or
466
the formulation of the integration scheme in COSMO-DE optimizes the dynamical resolution of
467
the model.
21
468
This study did not examine spectra with respect to the different theories described in the
469
introduction. In order to confront COSMO-DE with theoretical aspects a range of hypothesis
470
related to the different theoretical models for the k −5/3 spectral slope on the mesoscales might
471
be tested. Much insight could be gained for example through the analysis of third order structure
472
functions and the division of the spectra into divergent and rotational modes.
473
Acknowledgments
474
The authors would like to thank the German Meteorological Service (DWD) for kindly providing
475
the COSMO-DE ensemble simulations, and Susanne Theis (DWD) for her support. Special
476
thanks are due to Andreas Hense, Andreas Bott, Matthieu Masbou, Werner Schneider and the
477
two reviewers for their thoughtful comments on the manuscript.
22
478
479
List of Figures
1
Kinetic energy spectra of horizontal wind velocities for August 21th 2009 and
480
1h forecast time on σ-level a) 1 in 21.500 m, b) 10 in 13.620 m, c) 20 in 7.280
481
m, d) 30 in 3.136 m, e) 40 in 832 m, and f) 50 in 10 m. The thin line indicates the
482
ensemble mean of the spectra, and the filled area is limited by the range of the
483
ensemble spread. The dashed line corresponds to a −5/3 slope. The wavenumber
484
is given in rad/m (lower x-axis) and the wavelength in km (upper x-axis). . . . . 30
485
2
Kinetic energy spectra and wind fields in about 7000 m for May 20th 2009 for
486
a) 1h, b) 10h, and c) 20h forecast time are displayed. The thin line indicates
487
the ensemble mean of the spectra (left panels), and the filled area is limited
488
by the range of the ensemble spread. The dashed line corresponds to a −5/3
489
slope. The slope coefficients as displayed by dotted lines are estimated over
490
the spectral range between the vertical gray lines. They amount to about a)
491
−1.78 ± 0.1, b) −1.59 ± 0.12, and c) −1.78 ± 0.09, respectively. The shading
492
of the corresponding horizontal wind components (right panels) represent the
493
absolute value of the detrended fields in m/s, and the arrows indicate the wind of
494
the original fields. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
495
3
b) −1.84 ± 0.1, and c) −1.25 ± 0.1. . . . . . . . . . . . . . . . . . . . . . . . . 32
496
497
4
500
Same as Fig. 2 but for July 17th 2009 (15 member ensemble) with slope
coefficients a) −1.58 ± 0.1, b) −1.67 ± 0.07, and c) −1.73 ± 0.08. . . . . . . . 33
498
499
Same as Fig. 2 but for July 2nd 2009 with the slope coefficients a) −1.78 ± 0.1,
5
Same as Fig. 2 but for August 21th 2009 with slope coefficients a) −1.46 ± 0.08,
b) −1.55 ± 0.08, and c) −1.46 ± 0.11. . . . . . . . . . . . . . . . . . . . . . . . 34
23
501
6
Kinetic energy spectra of the vertical velocity in about 7000 m for forecast time
502
20h for a) May 20th 2009, b) July 2nd 2009, c) July 17th 2009, and d) August
503
21th 2009. The thin line indicates the ensemble mean. The filled area is limited
504
by the range of the ensemble spread. . . . . . . . . . . . . . . . . . . . . . . . . 35
24
505
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Figure 1: Kinetic energy spectra of horizontal wind velocities for August 21th 2009 and 1h forecast time on σ-level a)
1 in 21.500 m, b) 10 in 13.620 m, c) 20 in 7.280 m, d) 30 in 3.136 m, e) 40 in 832 m, and f) 50 in 10 m. The thin line
indicates the ensemble mean of the spectra, and the filled area is limited by the range of the ensemble spread. The dashed
line corresponds to a −5/3 slope. The wavenumber is given in rad/m (lower x-axis) and the wavelength in km (upper
x-axis).
30
Figure 2: Kinetic energy spectra and wind fields in about 7000 m for May 20th 2009 for a) 1h, b) 10h, and c) 20h forecast
time are displayed. The thin line indicates the ensemble mean of the spectra (left panels), and the filled area is limited by
the range of the ensemble spread. The dashed line corresponds to a −5/3 slope. The slope coefficients as displayed by
dotted lines are estimated over the spectral range between the vertical gray lines. They amount to about a) −1.78 ± 0.1,
b) −1.59 ± 0.12, and c) −1.78 ± 0.09, respectively. The shading of the corresponding horizontal wind components
(right panels) represent the absolute value of the detrended fields in m/s, and the arrows indicate the wind of the original
fields.
31
Figure 3: Same as Fig. 2 but for July 2nd 2009 with the slope coefficients a) −1.78 ± 0.1, b) −1.84 ± 0.1, and c)
−1.25 ± 0.1.
32
Figure 4: Same as Fig. 2 but for July 17th 2009 (15 member ensemble) with slope coefficients a) −1.58 ± 0.1, b)
−1.67 ± 0.07, and c) −1.73 ± 0.08.
33
Figure 5: Same as Fig. 2 but for August 21th 2009 with slope coefficients a) −1.46 ± 0.08, b) −1.55 ± 0.08, and c)
−1.46 ± 0.11.
34
Figure 6: Kinetic energy spectra of the vertical velocity in about 7000 m for forecast time 20h for a) May 20th 2009, b)
July 2nd 2009, c) July 17th 2009, and d) August 21th 2009. The thin line indicates the ensemble mean. The filled area is
limited by the range of the ensemble spread.
35