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VALIDATION OF THE CANADIAN REGIONAL
CLIMATE MODEL USING SPECTRAL ANALYSIS
by
Ravi Varma Mundakkara
A theses submitted to the Faculty of Graduate Studies and Research
in partial fulfillment of the requirements for the degree of
Master of Science
Department of Atmospheric and Oceanic Sciences
McGill University
Montréal, Quebec
Copyright O Ravi Vanna Mundakkara November 1998
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Le spectre de certains champs météorologiques sont calcul6s en utilisant des
transformations de Fourier deux-dimensionelles sur une région limitée du modele
de prévision à &lément fini régional (EFR) et du modèle canadien de climat
rbgional (MCCR). Les limites de soutirer les tendances linéaires des champs. et
ainsi la variance à grande échelle. sont discutées. Les spectres des champs maillés
sont calcul&, donnant une erreur de variance pour les differentes échelles.
Dans cette &tude, nous utilisons des methodes spectrales pour évaluer
I'habilete du MCCR de reproduire avec précision les systèmes à méso-échelle
durant ses intdgrations de courte durée, lorsque des conditions initiales et latérales
h faible résolution, telles que celles des MCGs. sont impostes. Deux cas. un dans
la région de Montréal et l'autre d;ms le bassin du fleuve Mackenzie (BFM), sont
étudiés. Il est ddterminé que la croissance de l'erreur relative de variance B la
plupart des echelles est moindre pour le BFM. possiblement associée aux effets
topographiques. Dans tous les cas et essais. l'erreur relative maximale de variance
est obtenue à une longueur d'onde d'environ 350-km.
Les &art-types et les 6cari-types nonnaiisés par le variance du champ simule
(RCMI) d'altitude geopotentielle pour les deux cas sont minimes et ne montrent
qu'un peu, sinon aucune, croissance lorsqu'une décomposition des &chelles est
effectuee. Cependant. l'erreur relative da variance. lorsqu'elle est exarninbe par
rapport à I'echelle, demontre des diffdrences consid6rables. Les erreun relatives B
differentes échelles dévoilent des taux de croissance et celles aux écheiles m6so-a
et synoptique demontrent une croissance avec le temps.
Spectra for various meteorologicai fields are computed using the two-
dimensional Fourier transfonn technique on a limited-area grid of the Regional
finite-element (WE) mode1 and the Canadian Regional Climate Modei (CRCM).
Limitations of removing the linear trend from the fields. and thereby removing the
large-scale variance. are discussed. The spectra of difference fields are calculated.
yielding the error variance for different scales.
Spectral methods are used widely in the evaiuation of global rnodels. In this
study. the same method is used for evaluating the CRCM in its ability to conectly
reproduce the mesoscale systems in short-term integrations. when low-resolution
GCM-like initial and lateral boundary conditions are provided. Two cases have
been chosen for this study, the first one over the Montréal region and second one
over the Mackenzie River Basin (MRB). It is found that the relative error variance
growth at most scdes. particularly the small scales. is less for the MRB region
possibly due to the topographie forcing. in both cases and ai1 experiments, the
maximum relative e w r variance is found to be at a wavelength of about 350-km.
Root mean square (mis) error and relative rms error for the geopotentid
height field for both cases are very small and show little or no growth, when scale
decompositions are not made. However, the relative error variance when
examined according to scaie, show considerable differences. The relative errors ai
different scales show different growth rates and that of the meso-a and synoptic
scales are found to be growing with time.
iii
Table of Contents
RÉsUMÉ
ABSTRACT
Table of Contents
List of Figures
Acknowledgrnents
Chapter 1 - Introduction
1 . 1 Introduction
1.2 issue of mesoscale predictability
1.3 Specval methods for mode1 evduations
1.4 Motivation and objective of this study
Chapter 2 - Mode1 Description
2.1 htroduc tion
2.2 The CRCM
2.2.A. Dynamics of the CRCM
2.2.B. Physical pararneterization
2.2.C. The diagnostic system
2.2.D. Experimental configuration
2.3 The RFE mode1
Chapter 3 - Spectra of fields on a lirnited-area grid
3.1 Introduction
3.2 Methodology
3.2.A. Removal of trends
3.2.B. Determination of spectra
3.2.C. Smoothing of r field
3.2.D. Vertically integrated spectral variance
xii
Chapter 4 - Mode1 Results
4.1 Case descriptions
4.2 Experimental design
4.3 Root mean square error
4.4 Relative error variance
4.5 Specua. of fields
4.6 Error growth for different scalcs
4.7 Effects of advection dirough the boundary
4.8 Other experiments
Chapter 5 - Summary and future recomrnendations
Chapter 6 - The '%lean" Run
6.1 Experimental design
6.2 Results
Appendix
A l .(a) Fourier senes
A 1 .(b) Higher dimensions
A 1 . (c) S pectra
A l .(d) Discrete Fourier transform
A 1 .(e) Fast Fourier vans form
A2. Some aspects of de-trending a field
References
List of Figures
Page
3.1 wavenumber k in the @,q) space, within the band A& 17
3 3 (a) u - component of 850-mb wind field at 48 h from CRCM 22
simulation. The domain is (4480 km)2 and is centered over Montréal.
(b) u - field after de-trendino. (c) trend only. (dl. ( e ) and ( f l are the
corresponding spectra computed for each field shown
3 3 (a) The 850-mb geopotential height field at 48 h from CRCM 23
simulation. The domain area is (4480 km)? over Montréal. (b) Sarne
field after de-trending. (c) trend only. (d). (e) and (f) are the
corresponding spectra computed for each field shown
3.4 An exarnple of smoothing of a field using spectral techniques. (a) the 24
u - wind field as shown in 3.2(a). (b) the smoothed wind field after
filtering out al1 the scales (wavelengths) below 450 km. (c) the
spectnim of the field in (a) and (d) the spectrum of the smoothed
field in (b)
3.5 (a) 850-mb u - wind field and (b) 850-mb v - wind field at 48 h from 25
CRCM simulation. (c) spectrum of the field in (a). (d) spectrum of the
field in (b). (e) kinetic energy (KE) spectrum ai 850-mb at 48 h and
(0 vertically integrated KE spectrum
4.1 Sea-Level pressure (PM) and the 1000-mb temperature (TT) for case4 43
at 00 Z November 8, 1996
4.2 Sea-level pressure (PM) and the 1000-mb temperature (TT) for case-2 44
at 00 Z January 10. 1996
4.3 Temporal change in rms error for RCMI (defined as the difference 45
between the control run and RFE analyses) of case-l (November 8-9,
1996). for (a) geopotential height (GZ, in dm), (b) u-wind (W. in ms'
'), (c) v-wind (W, in ms") and (d) temperature (TT. in OC) fields at
the 1000-hPa (thick solid). 700-hPa (light solid) and 500-hPa (dashed)
levels
4.4 Temporal change in nns enor as in fig.4.3, but for RCMl of case-2 46
(January 10-12. 1996)
4.5 Temporal change in relative nns error for RCMl (defined as the 47
difference between the control nin and RFE malyses, normalized by
the control run variance) of case-l (November 8-9, 1996). for (a)
geopotential height (GZ. in dm), (b) u-wind (UU, in ms-'), (c) v-wind
(W, in ms") and (d) temperature (TT, in OC) fields at the 1000-hPa
(thick solid), 700-hPa (light solid) and 500-hPa (dashed) levels
4.6 Temporal change in relative rms error as in fig. 4.5, but for RCM 1 of 48
case-2 (January 10- 12, 1996)
4.7 Temporal change in rms error for RCM2 (defined as the difference 49
between the RCM2 and control run) of case- 1 (Novernber 8-9, 1 W6),
for (a) geopotential height (GZ, in dm), (b) u-wind (UU, in ms-'), (c)
v-wind (W, in ms") and (d) temperature (TT, in OC) fields at the
1000-hPa (thick solid), 700-hPa (light solid) and 500-hPa (dashed)
levels
vii
4.8 Temporal change in rms error as in fig. 4.7, but for RCM2 of case-3 50
(January 10-12. 1996)
4.9 Temporal change in relative rms enor for RCM2 (defined as the 5 1
difference between the control run and RCM2, normalized by control
run variance) of case-1 (November 8-9. 1996). for (a) geopotential
height (GZ. in dm). (b) u-wind (UU. in ms"). ( c ) v-wind (VV, in mç")
and (d) temperature (TT, in OC) fields at the 1000-hPa (thick solid),
700-hPa (light solid) and 500-hPa (dashed) levels
4.10 Temporal change in relative rms error as in fig. 4.9. but for RCM2 of 52
case-2 (January 10- 12,1996)
4.11 Spectrum of the 500 mb geopotential height field of RCMl case-1 53
(initialized at 00 Z 6 November, 1996). (a) at the initial time, and after
(b) 12 h, (c) 24 h, (d) 48 h, (e) 60 h and (0 72 h of integration. The
wavelength plotted on x-axis gives the scale of motion. Solid line is
the spectmm of RFE analysis field, dashed line is that of the RCMl
(control nin) field and the dotted line is the spectnim of the difference
between the two
4.12 As in fig. 4.1 1, but for the vertically integrated Spectrum of GZ of 54
RCM 1 case- 1 (initialized at 00 Z 6 November. 19%)
4.13 As in fig. 4.1 1, but for the vertically integrated Spectrum of KE of 55
RCM 1 case- 1 (initialized at 00 Z 6 November, 1996)
4.14 As in fig. 4.11, but for the vertically integrated Spectrum of 56
temperature field (?T) of RCMl case4 (initiaiized at 00 Z 6
viii
November, 1996)
4.15 Vertically integrated relative error variance as a function of time for 57
different wavelengths (scales) of (a) GZ, (c) KE and (e) TT fields, for
RCM 1 case- 1 (November 8-9, 1996). The srndler the scale, the more
relative error is. Vertically integrated relative error variance as a
function of wavelength for different simulation times of (b) GZ. (d)
KE and (f) TT fields for the same case. As the scale decreases, error
increases for ail time. Here RCM 1 (control mn) is compared with the
RFE analyses
4.16 Sarne as in fig. 4.15, but for RCM 1 case-2 (January 10- 12, 1996) 58
4.17 Vertically integrated relative error variance as a function of time for 59
different wavelengths (scales) of (a) GZ. (c) KE and (e) TT fields. for
RCMZ case- 1 (Novernber 8-9, 1996). The smaller the scale, the more
relative error is. Vertically integrated relative error variance as a
function of wavelength for different simulation times of (b) GZ, (d)
KE and (f) Ti' fields for the same case. Here RCM2 is compared with
the control run
4.18 Same as in fig. 4.17, but for RCM2 case3 (January 10-1 2, 1996) 60
4.19 500-mb GZ field at initial time of RCMZ of case-2. (00 Z January 9, 61
1996). The domain is divided into four subdomains and the error
variance spectra for each subdomain are depicted in fig. 4.20
4.20 The relative error variance of GZ at 500 mb, for RCM2 of case-2 62
January 10-12, 1996), as a function of scale for different simulation
houn for (a) northwest (b) aonheast (c) southwest and (d) southeast
subdomains of the fig. 4.19 shown in the previous page. Here RCM2
is compared with the control nin
4.21 Sarne as in fig. 4.17, but for RCM3 case-l (November 8-9, 1996). 63
Here RCM3 is compared with the control run
4.22 Same as in Bg. 4.17, but for RCM4 case5 (January 10- 12, 1996). 64
Here RCM4 is compared with the control run
4.23 Same as in fig. 4.17, but for RCMS case-2 (January 10- 12, 1996). 65
Here RCMS is compared with the control run
4.24 The relative error variance as a function of wavelength for different 66
simulation times for the GZ field at (a) 1ûûû mb (b) 925 mb (c) 850
mb (d) 700 rnb (e) 500 mb and (0 300 mb for RCMZcase-2 (January
10-12, 1996).
6.1 Vertically integrated relative ermr variance as a function of time for 77
different wavelengths (scales) of (a) GZ, (c) KE and (e) TT fields, for
RCM2 case-2 (Januuy 10-12, 1996) in the new experiment. The
smder the scale, the more relative error is. Venically integrated
relative emr variance as a function of wavelength for different
simulation times of (b) GZ, (d) KE and (f) TT fields for the same
case. Here RCM2 is compared with the conirol run (RCMI 1).
6.2 The relative error variance as a function of wavelength for different 78
simulation times for the GZ field at (a) 100 mb (b) 925 rnb (c) 850
mb (d) 700 mb (e) 500 mb and ( f ) 3 0 mb for RCM2 case-2 (January
10- 12, 1996) in the new experiment.
A l Three sine waves which have the sarne k = -2 interpretation on an
eight point grid. The nodai values are denoted by the dark dots. Both
k = 6 and k = -10 waves are misinterpreted as a k = -2 wave on a
corne gnd. k is the wavenumber here
-42 (a) The geopotential field ( 2 ) ai 48 hours afler removing linear trend
along dl i only. (b) the z-field after trends along both directions are
removed. (c) trend alone. (d), (e) and (f) are the spectn of the fields
shown in (a). (b) and (c) respectively
A3 (a) The geopotential field (z) at 48 hours after removing linear trend
along al1 j only. (b) the z-field &ter trends along both directions are
rernoved. (c) trend done. (d), (e) and (f) are the spectn of the fields
shown in (a), (b) and (c) respectively
A4 The z - field in the North-West (upper left corner) subdomain of the
fig. 3.3(a) of chapter 3. The subdomain is (2240 km)'. (b) z - field
&ter trends are removed in both directions. (c) trend alone. (d), (e)
and (f') give the spectra of the fields in (a), (b) and (c) respectively
AS (a) The z - field in the upper middle subdomain of the fig. 3.3(a) of
chapter 3. The subdomain is (2240 km)2. (b) z - field after trends are
removed in both directions. (c) trend alone. (d), (e) and (f) give the
sDectra of the fields in (a). (b) and (c) resîxctivelv
Acknowledgements
1 would like to thank rny theses advisors, Prof. René Lapnse and Prof. isztar
Zawadzki, for giving me this opportunity and providing constant encouragement
and patience, as well as entrusting me with the academic freedom necessary to
undertake such an endeavour. My special thanks to the extemal examiner of this
thesis, Dr. George Boer of Canadian Centre for Climate Modelling and Andysis.
for his critical comments on the error analysis performed in the original version.
Many of my friends also spent countless hours discussing. explaining and
helping with the project. Michele Giguère h u ken always helpful to solve my
diffculties with running CRCM on various machines. Hélène Côte was very kind
to provide me with the scripts to deal with RFE analyses. Dominique Paquin
introduced me to the CRCM. Rick Danielson has ken a tnie fnend 1 could share
both my research ideas and also engage in discussions about a number of issues.
Thanks are also due to Badrinath Nagarajan, Jason Milbrandi, Louis-Philippe
Crevier, Marco Carrera and Stephan Ddry for their many help related to this
thesis.
Finally, 1 would like to thank my farnily back home for their continued
support and patience, without whom 1 could not have gotten to this point.
xii
Chapter 1 - Introduction
1.1, Introduct#on
General Circulation models (GCMs) are the main tools available today for
climate simulation and are run at resolutions which are too couse to adequately
descnbe mesoscale forcings and yield accurate regional climate details. The
increasing demand by the scientific community, policy makers, and the public for
realistic projections of possible regional impacts of future climate changes has
rendered the issue of regional climate simulations criticaily important.
The problem of projecting regional climate changes is essentially one of
representation of climatic forcings on two different spatial scales: the large scale,
defined as ranging from -1000 km to the earth's radius, and the mesoscale.
defined as ranging fmm a few kilometers to several hunâred kilometers (Giorgi
and Meams, 1991). Large-scale forcings, for example, those due to the Earth's
orbital characteristics or the abundance of atmospheric constituents, regulate the
generai circulation. This in tum determines the succession of weather events,
which characterize the climate regime of a given region. Mesoscale forcings, for
exarnple, those induced by complex topographical features and surface
characteristics, modify the structure of weather events and initiate local mesoscale
circulations. Embedded in the large-scale atmospheric systems, these circulations
contribute to regulate the regional distribution of climatic variables. Since the
resolution of current GCMs is not fine enougb to resolve small-scale atmospherîc
circulations, an alternative is to produce detded climate simulations for selected
regions by nesting a limited ana mode1 (LAM) within a GCM or within
observational analyses. Such nested models have corne to be termed Regional
Climate Models (RCMs) although the term could encompass variable-resolution
global GCMs (McGregor, 1997). Owing to their increased resolution. atmosphenc
forcings from the lower boundary, including details of land surface processes, can
be better represented in the RCMs. if the GCM simulation can produce reaiistic
intensities and frequencies for each type of major synoptic systems, then it can be
nested with an RCM to produce a realistic detailed climatology. at least for mid-
latitudes where the boundary forcing can determine the broad behaviour of the
regional systems. Many studies have been done in the field of regional ciimate
modelling (Giorgi et al. 1997; Giorgi et al. 1993; Giorgi and Bates 1989. Giorgi
1990; Chnstensen et al. 1997; Caya et al. 1995; iuid Laprise et al. 1998).
1.2. Issue of rne8oscaIe predctability
Classical predictability is an attribute of the atmosphere itself. Two
atmospheric states, which are initially close, are found to diverge as time proceeds
due to the natural operation of the physicd system. if one of these states is taken
to be that of the atmosphere and the other to be an observed state of the
atmosphere, then the difference between them represents error in the observations.
The divergence of these two states with time is interpreted as the growth in
forecast emr due to uncertainty in the initial conditions. There have been many
approaches in the study of atmospheric predictability in global numericd rnodels
(Cbarney et al., 1966; Lorenz, 1969; Leith and Kraichnan, 1972; Boer, 1984).
Several studies (Baumhefner, 1984; Shukla, 1984) have suggested that the
theoreticai limit of predictability decreases as the scale of the feature of interest
decreases in global models. The pmblem of error growth in regional numerical
models are different due to the effect of lateral boundary conditions specified
(Anthes. 1984 and 1986; Anthes et al., 1985). They are found to be insensitive to
small uncertainties in the initial conditions.
The idea of regional climate modelling has its roots in the work of Anthes et
d. (1985 and 1989) who proposed to apply the lirnited-nrea nested models io
environmental modelling problerns in which simulations are extended beyond the
deterministic predictability limit of forecasts (Laprise et al.. 1998). Anthes et ai.
(1985) studied the effect of uncertainties in initiai conditions and the effect of the
lateral boundary conditions (LX) on simulations using a regional-scale
numencal model. In contrast to previous large-scale studies using global models
(Lorenz, 1969; Shukla, 1984, etc.), the simulations showed little or no error
growth of differences in initial conditions over the time period 0-72 hours. In al1
the lirnited-area simulations with the same LBC, the rrns height differences ai 500
mb decreased with time. The rms differences in other variables, including
temperature, specific humidity and horizontal wind components, aiso showed that
little or no growth takes place over the penod of integration when only the initial
conditions are varied.
Two hypotheses were put fonvard by Anthes (1986) to explin these absences
of growth of initial emrs in the limited-area model. First, the same LBC may be
preventing different evolutions of the fiow in the intenor of the domain by
providing identicai large-scale information to the periphery of the pairs of
simulations. If the large-scaie flow, together with the forcing at the earth's surface
through topography and energy Buxes, is controlling the evolution of the
mesoscale as suggesied by Anthes ( 1984), then one would expect little sensitivity
of mesoscaie forecasts to variations in initial conditions. A second hypotheses is
that the synoptic weather type over the limited area was, by chance, more stable to
initial perturbations than typical global circulations. which always contain some
regions that are sensitive to initial perturbations.
Vukicevic and Errico (1990) found that the relative high degree of
predictability found in some limited-area simulations is due ro the use of observed
laterd boundary conditions. They also found that the presence of topographie
forcing also enhances model predictability for certain features. Weygiuidt and
Seaman (1994) suggest that the improvement of horizontal resolution will greatly
cnhance the predictability of the geographically related mesoscale features.
1.3. Spectral methoda for model evalustionr
Spectrai methods have been used to study the predictability and error in global
models (for example Boer, 1984 and 1994). In these studies, the growth of error
variance of 500 mb geopotential height with time have been studied, and has
permitted the study of error growth for different wavenumbers (scdes of motion).
These studies found that the ski11 of the mode1 is lost quickly for the smaller
scales (higher wavenumbers) and more slowly at larger scales (smaller
wavenumbers). We explore the possibility of such an andysis for the growth of
error in the case of regional models. in the case of global models, basic
atrnospheric flow variables may be expressed as a funciion of spatial scde, where
the variables (scalar fields) are expressed in terms of spherical harmonies (the
angular part of the solution of Laplace's equation in sphencal polar coordinates).
The order of the Legendre ploynodal, in the expansion of sphericd hmonics. is
the two-dimensional wavenumber on the sphere. which chmcterize the scales of
atmospheric flow (Boer, 1993). In the case of a regional lirnited-area rectangular
domain, a twodimensional Fourier transform is required and the spectnim
computed from Fourier coefficients. We address issues like aperiodicity of fields
and diasing of the spectmrn in the case of limited-area domains. Computing the
spectrum of a field present in a limited-ma domain based on Emco (1985) is
explined in chapter 3.
1.4. Mot~vation and objective of this study
The paradigm underlying the RCM is: (1) that the specification of the LBC
constrains the RCM's 'climate regime' broadly to that of the nesting GCM, but
(2) that redistic fine-grain structures will develop and evolve in RCM simulations
despite the fact that LBC only provide couse-grain information. The f i ~ t
postulate is generally verified for applications with modest regional domains. on
the order of one-tenth of the Earth's surface, at middle latitudes (Anthes et al.
1985; Caya et al. 1995, Laprise et al. 1998). The processes leading to the
emergence of fine-scale detnils in the high-resolution simulations nested with the
low-resolution LBC are still poorly understood.
The newly developed Canîdian Regional Climate Mode1 (CRCM) has been
used for many regional clirnate simulations (Caya et al 1995, Laprise et al. 1998).
CRCM is usuaily nested in the GCM simulated fields, which are of low-
resolution. A description of CRCM is given in chapter 2. The CRCM produces
rnany smali-scde features in the simulations, but it is not known whether the fine-
scale details emerging from the model simulations are realistic or not. The
objective of this study is to develop a methodology based on spectral analysis to
verify the smail scale details of CRCM sirnulated fields. We also attempt to
address the issue of the effect of LBCs on the simulated small-scale features. In
order to venfy this. the following approach is used: analyses from the Regional
Finite Element (RE) model, which are high-resolution fields provided by the
Canadian Meteorological Centre (CMC). are used to specify initial and lateral
boundary conditions for a similarly high-resolution CRCM simulation. This is
compared to another CRCM simulation of the sarne resolution but with low-
resolution information at the initial time and at lateral boundaries. permitting us to
eliminate the model erron from the total enor, while specifying GCM-like
information to drive CRCM. A spectral analysis will allow us to split the
simulated fields into different scales of motion and the growth of error at each
scde cm be determined seporately.
A description of the Canadian Regional Climate Mode1 (CRCM) is given in
chapter 2. The methodology for computing the spectra of atmospheric fields over
a limited-area domain is explûined in detail in chapter 3. Chapter 4 gives the case-
descriptions, details of the experiments performed using the CRCM and the
results of the simulations. Discussion and conclusions are given in chapter 5. An
appendix in which some basics of Fourier transformations and some limitations of
the method employed to detennine the spectra is included, followed by a
compnhensive list of the references consulted for this study.
Chapter 2 - Model description
This chapter explains the Canadian Regionai Climate Model (CRCM), and a
brief description of the Regionai Finite Element (RFE) model analysis-forecast
system. Simulation of regional ciimate with nested rnodeis empioys a high-
resolution LAM nested within a coarser resolution global driving model. The
initial condition and time-dependent lateral boundary conditions required by the
regional model are supplied by the global model. The newly developed Canadian
Regional Climate Model (CRCM) is a result of coupling the Cooperative Centre
for Research in Mesometeorology (CCRM) mesoscale non-hydrostatic
cornmunity rnodel dynamics (CCRM; Bergeron et al.. 1994) and the complete
physicai processes parameterization package of Canadian Centre for Climate
anaiysis and modelling (CCCma) GCMII (McFarlane et al., 1992). Use of the
same physical pûrameterizations in the CRCM and GCMII ensures consistency
between the two models and facilitates the transfer of information from the
driving to the nested rnodel (Caya et al., 1995; Caya and Laprise, 1998).
2.2. The CRCM
The CRCM system consists of t h e main components: the dyniimic kemel of
the model, its physical parameterization package and the diagnostic system, which
includes a number of software utilities to pre- and post-process model-simulated
(or observed) data. used to calculate and compare climatologies (Laprise et al.,
1998).
2.Z.A. Dynamics of the CRCM
The dynamics of the CRCM are based on the complete non-hydrostatic Euler
equations (Bergeron et al., 1994). Therefore, the dynamical formulation does not
restrict spatial sales at which the model c m be run. The integration of the
complete Euler equations is made affordable by the efficiency of the three
dimensional semi-Lagrangian semi-implicit scherne. The efficiency of the
integration scheme also permits longer timesteps compared to other numerical
schemes (Eulerian methods) for a non-hydrostatic model, at al1 resolutions.
Because of its numencal formulation, the CRCM is computationally less
expensive than other RCMs by a factor of 3 to 5, despite the fact that it is a more
general non-hydrostatic rnodel (Laprise et al., 1998).
2.2.8. Physical parameterization
The physicd parameterization package (imported from GCMII) takes into
consideration the following: vertical turbulent fluxes of momentum, mountain-
wave drag, radiation absorption and ernission (solu and terrestrial) by the
atrnosphere and by the surface, release (absorption) of latent heat when
condensation (evaporation) occurs, precipitation (solid and liquid), convection
(dry and moist), variation of the surface albedo (defined for the two spectral bands
and as a function of grwnd cover type, soil rnoisture, snow cover and snow age),
cloud cover (evaluated diagnostically from the prognostic water vapour and
temperature fields), surface energy budget, soi1 moisture ngime, vegetation and
soil characteristics. The physicai package of the GCMII (and therefore CRCM)
also includes an ocean mixed-layer model and a thermodynamic sea-ice rnodel.
For the experiments performed in this study, both of these models were tumed off
and climatological values of sea-surface temperature (SST) and ice-coverage were
used instead.
2.2.C.The Diagnostic system
The diagnostic system of the CRCM has to account for the polar-
stereographic (PS) projection used by the CRCM for its liorizontal representation
and for its scaled-height terrain-following vertical coordinate. The horizontal
resolution of the CRCM is uniform in the PS projection but the vertical resolution
is variable. The mode1 uses an Arakawa C-type staggered grid arrangement
(Bergeron et al., 1994) of its atmosphere. The GCMn physics and diagnostics
packages are modified to account for the different grids and projections of the
CRCM and GCMII.
2.2.D. Experimental configuration
The current configuration of the CRCM consists of a grid of 138 by 138
points covenng (4480 km12 area for experiment RCMl (see chapter 4) centred
over Montréal for case4 and over the Mackenzie River Basin (MRB) for case-2.
Case descriptions are also given in chapter 4. For al1 other experiments (RCMZ,
RCM3, RCM4 and RCMS), the CRCM consists of a 1 18 by 1 18 gnd point (41 30
km12 area ccntered over Montda1 for case-1 and the MRB for case-2. In al1
experiments the CRCM is run at 35-km resolution with a 450-second time step. in
the vertical, 19 unequally spaced terrain-following GalChen coordinate levels
(Bergeron et al., 1994), from the gound up to 32.6 km, are used. The following
fields at the initial time and at lateral boundaries for experiments RCMl and
RCM3 are denved from the Regional Finite Element (RE) operational analysis-
forecast model, provideci by the Canadian Meteorologicai Centre (CMC):
horizontai wind components, temperature. pressure and water vapour. They are
interpolated from the RFE archived data ont0 the CRCM laterai boundaries. A
sponge zone of ten grid points within the boundary of the nested domain is used
to gndually blend the CRCM fields and information received from the RFE. At
the lateral boundaries of the CRCM domain, the values of the driving model
variables are imposed, whereas throughout the rest of the grid within the sponge
zone (the free zone), variables of the regional model are not affected explicitly by
those of the global model. For al1 experiments other than RCMl and RCM3,
initial and lateral boundary conditions are obtained frorn the fields simulated by
experiment RCM 1.
2.3. The RF€ model
A brief description of the Regional Finite Elernent (RFE) mode1 system,
which is operational at CMC, from which the initial and lateral boundary
conditions for experiment RCMl are generated is given below. The RFE model,
which is used to produce the two 6-h forecasts during the 1 2 4 spinup period,
employs a semi-Lagrangian treatment of advection (Tanguay et al.. 1989;
Chouinard et al., 1994). It is a primitive equations rnodel defined on a polar-
stereographic projection true at 60' N using the sigma as vertical coordinate. It
has a variable resolution horizontal mesh with a maximum resolution of 35 km
over North Amerka. The use of trilinear finite elements permits the mesh to Vary
in aU three spatial dimensions and, in particular, to focus the horizontal resolution
over a uniform-resolution area of interest. The resolution varies smoothly away
from this area to the domain boundxies, which are nearly tangent to the equator.
Coupled to the dynamical model is a comprehensive set of pammeterizations of
physical processes (Benoit et al., 1989). These include a planetary boundary layer
based on turbulent kinetic energy, and a surface layer based on similarity theory,
solar and infrared radiation, large-scale precipitation, and moist convection. The
regional data assimilation system (Chouinard et ai.. 1994) is designed to provide
the RFE model with more detailed analyses in a dynamicdly consistent manner.
The large-scale flow is redefined every 12 h, thus eliminating the accumulation of
errors in tropical regions and their subsequent propagation to the middle-latitude
high-resolution region of interest.
The malysis component of the regional system produces analyses of the
di fferences between the observations and the triai field inrerpolated to the
observation locations of horizontai whd components. geopotential height.
temperature and dewpoint depression on isobaric surfaces. The analysis proceeds
from point to point of the RFE model's horizontai mesh, analyzing each of these
variables at the specified isobaric analysis levels. A vertical interpolation fmm
pressure to sigma levels is performed.
The model initialization consists of three iterations of an implicit nonlinear
normal-mode initialization, performed directly on RFE model's mesh (Temperton
and Roch, 1991).
Chapter 3 - Spectra of fields on a limited-area grid
3.1. lntroductbn
The anaiysis of the variance of a field as a function of horizontal scale is
fundamental to many theories conceming the dynamics of geophysical fluids.
This includes spectral analysis of data as observed or as simulated by a modei.
Spectral analysis is especially useful in comparing model results with theoretical
studies of geophysical turbulence or predictability. Transformation of the fields
from the physical space ont0 a wave-number space using Fourier transformation
is comrnonly used in meteorology. This study uses spectral analysis techniques to
separate different scaies of motion present in the Canadian Regional Climate
Modei (CRCM) described in chapter 2. and to study the growth of error variance
in these scales when a GCM-like initial and lateral boundary conditions are
specified.
Fast Fourier transform subroutines (Nobile and Robeno, 1986; Press et al.,
1992) are used in this study to compute the Fourier coefficients for each field and
the spectnim, which is the nom of the coefficient for each wavenumber. A
particular difficulty with trying to use spectral maiysis over a lirnited-area is that
the data is not naturally periodic, an underlying assumption of al1 discrete Fourier
msforms, hence some pre-pmcessing of the data is called for. Specific methods
for determining the spectrum of a field are described in detail below. Section 2(a)
explains the method to 'de-alias' the spectrum computed for a field by cemoving
Linear trends. The spectnim of wind and geopotential height fields, with and
without removing the linear trends, are presented to illustrate the signifcance of
'aliasing'. Detemination of spectra &ter removing the trends is explained in
section 2(b). We illustrate in section 2(c) how a field c m be smoothed by
removing information in al1 the scales smdler than a specific scale. Finally,
vertical integration of the spectra of meteorological fields are explained and
illustrated in the Iast section 2(d).
The methodology used for computing the spectmm of fields for a limited-area
grid is based on that of Emco (1985). The fields within the limited-area dornain
are generally aperiodic and the spectral analysis is not straightfonvard in this case
because of 'aliasing'. It is the misrepresentation of spectra of non-resolvable
scale waves by projecting them ont0 the spectra of resolvable scaie waves and
thereby contaminating them. Aliasing is explained in appendix A. In order to
make a field penodic, a linear trend is defined by the boundary values of the field
in the rectangular grid and is removed prior to the spectral analysis. This,
generally, will reduce misrepresentation of scales larger than the dornain.
3.2.A. Removal of trends
The aperiodic nature of the fields in the lirnited-area model (grid) domain
causes some major difficulties for spectral analysis. For atmospheric flows, the
variance peaks at large scales. For a chosen limited-area gnd, there are many
scales (for example zona1 wave numbers 1-8), larger than the model domain, that
cannot be resolved on a limited area. If these large scales are not rernoved from
the data they will alias onto smaiier scales, which are nsolved. The scdes larger
than the model domain appear as trends, resulting in significant aperiodicity for
the analysis region. Furthemore, these trends are not simple functions since they
represent a composite of planetary scales. When the analysis domain is treated as
periodic, as for the Fourier analysis, these trends project onto al1 spectral
components, analogous to the projection of a periodic saw-tooih function. This
will diston the spectnim computed from the limited-area grid. Due to large
variances associated with the trends, their rnisrepresentation as smaller scales
(resolved by the model grid) in this way may actually dominate the representation
of smaller scales, periodic perturbations within the domain. This problem is worse
for trends with greater variation within the analysis domain (Van Tuyl and Emco,
1989).
The method used to remove the large, unresolved scales from any field ois, is
to remove linear trends dong al1 points of constant i or j (corresponding to
domain's x and y directions). The linear trends are defined by the boundq values
of the field. Explicitly, the steps are described below:
1. First, for each j, determine the slope
2. Next, for each i, j remove the trend in the i direction to yield
Ni and Ni are the number of grid points in the domain dong i and j
respectively. In this study, a square domain is used and Ni is equal to NI. Steps 3
and 4 are to repeat steps 1 and 2 with the roles of i and j reversed, with a,,,
replacing a,, , , and obtain the new. de-trended field a , , . i.e.,
The a , , are actually independent of the order in which the trends are removed
(either i or j fint). The result is a periodic field where, u,, = a,,, for al1 i,
andaij = a i , , , , for dl j. A discussion on various aspects of removing linear
trends using this method is given in appendix A.
3.2.B. Deter rnination of spectra
The spectral coefficients c , , of any field are determined by the discrete two-
dimensional Fourier transforrn:
where p and q have discrete values
Ax is the distance between two adjacent grid points dong i (or]>. in our case Ax is
35 km, both dong i and j.
In (6) above, the quantities NR and NJ/2 should be tnincated to integers. in
practice, only half the coefficients need to be explicitly calculated since
- c , , -CI,.-, , where the asterisk denotes complex conjugation. This condition
results from the fact that fields in physicai space are red and not complex
numbers. For the velocity field, coefficients for the u and v components are
computed separateiy.
Values of c,, determine a spectrum in a two-dimensiond, discrete vector
wavenumber (p. q) space. They can also be used to determine a spectrum in a
one-dimensional (k) space. Here the spectrum S(k) will be determined by a
summation within discrete annuli in p. q space. and a set of k will be defined as
the central radii of those annuli such that,
The one-dimensional discrete k is shown in figure 1 below, given the
corresponding p values and q values in the @, q) space. Many combinations of p
and q are possible for the same vdue of k. The variance of the field for the one-
dimensional wavenumber k is calculated by adding up the variance for al1 the
points within the shaded annular region of width Ak in the figure. Note that p and
q can have negative values.
F i g . 1 Fig. 3.1: wavenumber k in the @,q) space, within the band M.
If NI # NJ, vaiues of Ak and k have k e n determined from the minimum of the
fundamental (1 = 1 ) values of p and q. This minimum value defines A&, the
wavenumber band over which summation (7) is made for computing S(k) for each
k. For a domain with 128 grid points in each x and y directions and 35-km spacing
between the grid points, the value of A& will be 1.4128 x 10" m-'. The values of k
are specified as
The values of k are truncated at Ak(Ni/2), since otherwise some combinations
of p. q would be missing from the summation in S(k) above, and the spectnim
would be distorted.
If the values of k were tnincated such that al1 c , , were inciuded in the above
surnmation, then
is the gnd-mean of the variance of the field aN . The spectnirn of kinetic energy
is given as:
where S.(k) and S,(k) are spectra of the variances of the u and v fields
respectively .
A 2-D fast Fourier transfomi subroutine was used to compute the Fourier
coefficients of the fields after de-trending them. A description of Fourier analysis
is given in the appendix A. Fast Fourier transform subroutines described by
Nobile and Robeno (1986) and Press et al. (1992) are used to compute the Fourier
coefficients. Figure 3.2 illustrates an example of the fields More and after de-
trending. The fields shown are the model-simulated horizontal wind dong x-
direction, u, at 48 hours. No geography is shown. Figure 3.2(a) shows the wind
field in x-direction, u, from the analysis at 48 hours. Figure 3.2(b) gives the de-
trended wind field at the same Ume and figure 3.2(c) shows the linear trend field.
The figures 3.2(d), 3.2(e) and 3.2(f) show the spectnim of the wind (u) field
before de-trending, f i e r de-trending and that of trend alone respectively. It is
evident from these figures that, if the linear trends are not removed, variance in
the smaller scales resolved by the grid are drasticaily misrepresented. Figure
3.2(f) is the spectnim of the trend done. which is the projection of dl the
variances at large scales, that are not resolved, ont0 those scales resolved by the
gnd. Figure 3.3 show corresponding figures for the geopotential height field (2) of
850 mb at the same time. It is to be noted that the linear trends are dependent on
the choice of the domain too because if the domain chosen were different, the
slopes of the field in either directions may be different and the corresponding
trends are also difierent consequently. Details of this aspect are discussed in
appendix A.
3.2.C Smoothing a field
The spectral rnethod con dso be used for the purpose of filtering out a desired
scde of motion from a given field. The spectral coefficients are computed using
FFî methods as in the previous sections. The coefficients corresponding to those
scdes of motion, which are to be removed from the field, are then set to zero.
Then an inverse fast Fourier transform is performed in order to retrieve the field
after filtenng. A field can thus be smoothed, by removing the variance in scales
smaller than a desired value and then performing inverse Fourier transformation.
An example of smoothing by this method is shown in figure 3.4. The same wind
field, as in fig. 3.2 to be smoothed is shown in 3.4(a). The corresponding
specuum of the field is shown in fig. 3.4(c). The spectnim with coefficients of al1
wavenumkn below 9 x lob m" set to zero is shown in fig. 3.4(d). The filtering
is actually done in @, q ) space. For a particular k, al1 possible p a d q and
determined and the corresponding coefficients c,,,, in @, q) space, are set to zem.
This is done for all k, which are to have zero variance in order to be removed. The
inverse transformation of the remaining coefficients will give a smoothed wind
field as shown in fig. 3.4(b). The linear trend field, which was removed from the
(wind) field before performing the filtering of smaller scales is added back to the
smoothed field, &ter the inverse Fourier transfoni, so that the variance of al1 the
large scales present before filtering are also present after smoothing the field.
3.2.D Vertically integrated spectral variance
The spectrum of any field, a, (for example z. u and v fields), at any pressure
level in the model (or RFE analysis) can be computed using the method explained
above. It is sornetimes useful to integnte the variance of the fields through the
vertical column of atmosphere. The CRCM data, in our case, consists of sixteen
vertical (pressure) levels which are 1000 mb, 925 mb, 850 mb, 700 mb. 500 mb,
400 mb, 300 mb, 250 mb, 200 mb, 150 mb, 100 mb, 70 mb, 50 mb, 30 mb, 20 mb
and 10 mb. The model output is interpolated onto these (arbitrary) pressure levels
from the model terrain-following sigma coordinate, which has nineteen levels. An
integrated spectrum, Si,, of the field a, can be caiculated using the following
me thod.
Let pi (where i goes from O to 15) be the pressure levels present in the model
(or analysis). Then, the average between any two adjacent pressure levels is given
as*
We can then define Mi, difference between the two consecutive avenged pressure
levels as
Here po, corresponds to 1000 mb and pl5 corresponds to 10 mb. The integrated
spectrum of the field is then given by,
where Si is the spectnim of the field a, at the pressure level pi.
An example of the integrated kinetic energy spectrum is given in fig. 3.5. The
wind fields u and v at 48 houn is shown in figs. 3.5(a) and 3.5(b). The kinetic
energy (KE) specuum for 850 mb pressure level over which the wind fields are
shown is given in fig. 3.5(e). The KE spectrurn integnted verticaily over al1 levels
for the same time is shown in fig. 330. The KE specva for each level are
calculated following equation (10) and then integrated vertically using the
surnrnation (1 1).
The details of the cases examined and the mode1 simulations in each
expriment are explained in the following chapter.
Fig. 3.2: (a) u - component of 850-mb
(d) sp.ctrum of u in (O) 1
(a) sprnrum of u (no trrnd) ln (b)
'"$
wind field at 48 h from CRCM simulation.
The domain is (4480 km)' and is centered over Montréal. (b) u - field after de-
trending. (c) trend only. (d), (e) and (f) are the corresponding spectra computed
for each field shown.
(r) rpeclrum of z (no Irmd) in (b) 7 1
(1) rpetrurn ot t (trend cnv) in (c) I 1
FLg. 3.3: (a) The 850-mb geopotential height field at 48 h from CRCM
simulation. The domain area is (4480 km)2 over Montréal. (b) Same field after de-
trending. (c) trend only. (d), (e) and ( f ) are the corresponding spectra computed
for each field shown.
Fig. 3.4: An exampie of smoothing of a field using spectral techniques. (a) the u -
wind field as shown in 3.2(a). (b) the smoothed wind field after filtering out dl
the scdes (wavelergths) below 450 km. (c ) the spectmm of the field in (a) and
(d) the spectrum of the smoothed field in (b).
(e) KE rpsctnim of total iind firM 1 1
10% (d) spectrum of v in (b)
1 t
Fig. 3.5: (a) 850-mb u - wind field and (b) 850-mb v - wind field at 48 h from
CRCM simulation. (c) spectmm of the field in (a). (d) spectrum of the field in (b).
(e) kinetic energy (KE) spectrum at 850-mb ai 48 h and (0 vertically integrated
KE spectrum.
Chapter 4 - Mode1 results
Two cases have been chosen for this study. The cases were fairly stratifonn
events which have occurred in late fa11 or earlier winter. These were chosen to
avoid the deep convective events of smaller scales, which might not be well
parameterized in the model.
4.1. Case Descriptions
The fint case was the passage of a cold front over the region of Montréal in
eastern Canada on 8-9 November, 1996. This cold front stayed over the same
region for a long tirne and heavy precipitation also fell in the region. The sea-level
pressure and temperature fields at 002 on Nov. 8 are shown in figure 4.1. A
tonguc of warm air in tbe warm sector of the cold front stretched north-eastward
dong the St. Lawrence river, associated with strong warm advection. Heavy rain
was produced by the passage of this cold front, which resulted in a flash flood
south of Montréal (Yu et al. 1998).
The second case chosen was the passage of a cyclone during 10-12 January,
1996 over the Mackenzie River Basin (MW) east of the Canadian Rocky
mountains in western Canada. Our interest is to assess the ability of the CRCM to
simulate the realistic regional features of a representative wintertime MRB event.
This is important to understand because the climatology of the wintertime
precipitation simulated by an RCM nested within a GCM is useful in the study of
the water cycle of the MRB region. The amount of precipitation in tum is
dependent on the cyclogenesis and moisture transport from the northem pacific.
We chose to simulate a representative wintertime lee cyclogenesis event that
occurred during 10-12 January. 1996 (Lackmann et al., 1998). Much of the spring
runoff from the MRB to the Arctic Oceui in the north cornes from melting of the
snow pack that accumulates during cold-season precipitation episodes such as this
one. A lee cyclone forms over the southem MRB during this type of event and is
accornpanied by heavy precipitation. The sea-level pressure and tempenture ai
002 on January 10, 1996 for this case is shown in figure 4.2.
Two expenments (model runs) were performed using the Canadian Regional
Climate Mode1 (CRCM) for each of the two cases studied. A description of
CRCM and Regional Finite Elernent ( R E ) models c m be found in chapter 2.
The initial and laterai boundq conditions were generated for the first model
run from the Canadian Meteorological Centre (CMC) RFE model analysis. which
is updated every 12 hours and archived every 6 hours. The horizonid resolution
of this analysis is 35 km. We perform the CRCM runs also ai 35 km resolution.
The domain for the fmt run, which makes use of the RFE analysis for initial and
lateral boundary conditions, is of 138 grid points in each x and y directions. The
m a of the domain is (4830 km)'. The integration penod is 96 hours.
The model-simulated fields are cornpared to those of the driving analysis
fields in the fmt nin. The spectra of fields of both the mode1 simulation and RFE
analyses are then computed, as well as spectra of the difference fields. The
difference field is the difference between the RFE analysis and the model
simulation, for a particular field, at the same time and level. The spectnim of a
difference field gives the variance of the error of the simulated field with respect
to the RFE analysis. We note that the variance of the difference of a field is not
the same as the difference of the variances, since variance is a quadratic quantity.
Since both the model simulation and the driving anaiysis are of high resolution.
we take this as Our control run.
Our objective is to study to what extent srnall-scale featiires of a CRCM
simulation are dictated and controlled by the large-scale forcing at the lateral
boundaries. if large-scale forcing dictates the smaller scde features, then we
should be able to simulate the fields as in the first run. employing low-resolution
information at the initial time and at the laterai boundaries. To achieve this, the
simulated fields from the first (control) run are degraded in resolution by
removing dl the scales (wavelengths) smailer than approximately 450 km by
Fourier decomposition. This smoothing is done as explined in section 3.2.C. of
chapter 3. Our choice of a cutoff wavelength of 450 km is based on two
considerations: The average separation between upper air sounding stations for
rneteorological observations is approximately of this distance in Nonh Arnenca,
and the present resolution of General Circulation Models (GCM) üpproaches this
value. For instance. from sampling theonm one cm argue ihat the GCMs with
400 km by 400 km horizontal grid increments would have a resolution of no
better than 800 km (Pielke, 1991). Laprise (1992) analyzed the definition of
resolution for global spectral models and suggested that a T3 1 triangular tnincated
spectral model, with the rnost optimistic view, will have an effective horizontal
resolution of 426 km. This estimate is made by considering the average spacing
between Gaussian latitudes of the transfon grid to define the horizontal
resolution. The RCM is usuaily nested in a GCM field and run for months or
seasons to study the details of climatology over a region of interest. Hence, it is
important to know whether the RCM is capable of sirnulating the smailer-scaie
(mesoscale) features correctiy from the low-resolution boundary conditions
provided by the GCM.
The resulting smoothed fields are then used to generate the initial conditions
and laterai boundary conditions to nin the CRCM again in a slightly smaller
domain. From the initial model domain used for fint run, ten grid points are
removed from near the boundary to account for the sponge zone. This is the
reason for a smdler domain in the second experiment, which consists of 1 18 grid
points in each x and y direction, and has an area of (4130 kmf. Results of the
second mn is then compared to the simulations of the control mn. The growth of
error at smdler scales is again determined by obtaining a time series of the spectm
of difference fields at various scales.
Though these are the two main experiments performed on both cases, an
experiment was performed on case4 in which the initial and laterai boundary
conditions are obtained from the smoothed field of RFE analysis instead of from
the simulated control run. Since this procedure did not give a very different result
from that of the second run, it was not used in case-2. instead of choosing a single
cut-off wavelength as 450 km, two case-2 model runs were performed with
different cut-off wavelengths of 600 km and 320 km. The results of the model
simulations are explained in the following sections.
4.3. Root mean square error
The CRCM was run for case4 from 00 2, November 6 for a period of 96 h.
The integration for case3 began at 00 2. January 9. again for a penod of 96 h.
The fiat integration of CRCM, which makes use of the high resolution CMC
analysis for initial and lateral boundary conditions. is dcsignated hereafter as
RCM 1. For the second xun of the CRCM over the same penod (RCM?) the initiai
and laterai boundary conditions w e n obtained from the smoothed fields simulated
by RCM 1.
The root mean square error (rms error) for RCM 1 for case- 1 is shown in fig.
4.3. The RCMl fields are compared to the RFE andysis fields. First, the
difference field in physical space is computed and rms error is calculated for the
mode1 domain. Fig. 4.3(a) gives the rrns error for geopotentid height (GZ) at
100 mb, 700 mb and 500 mb. The corresponding figure for case3 is shown in
fig. 4.4. It may be appropriate to look at relative error (ratio of the m i s error to
the rms variance of the RCMl field at the same level) in order to get an idea of
the predictability of the field. Figure 4.5(a) gives the relative error variance of the
GZ field for RCMl, when compared to RFE analyses field. The 500-rnb and 700-
mb level relative erroa are lower than 0.01. The 1 0 - m b relative error is large
because of the small variance associated with this field. Figure 4.5(b) and 4.5(c)
gives the comsponding relative enon for the u and v wind fields (UU and VV in
figures). respective1 y. The corresponding figure for temperature (Ti') is depicted
in fig. 4.5(d). The relative errors for the corresponding fields for case-2 are shown
in fig. 4.6. The rms emr for the geopotential height field is not seen to increase
with tirne. and in fact the relative error for the GZ field, say at 500 mb, is very
low at al1 times. This feature corresponds very well to other results of regional
limiteci-area models. as explûined by Anthes (1986). in which rms error is not
found to grow with time.
4.4. Relative error variance
The error between the model simulation and the control run cm be written as
e = y - x. where y is the model-simulated (forecast) field and x is that of the
control run (following Boer. 1993). Then the error variance is,
where S: and S: are the calculated variances of model simulated field and that
of the control run respectively and r, is the correlation between them. f l is a
srnwth function of these variances. The overbar represents the area averaging in
the analysis domain. Al1 terms in the above equation are functions of simulation
(forecast) tirne. If the variances of the model simulated fields are approximately
equal to that of the control nin, then /3 will be close to one and the error variance
can be written as,
- e2 = 2~:(1-r,) (2)
One can define a relative error variance with respect to the variance of the control
run from the equation (1) as,
When S: = S; , then eq. (3) becomes,
if there is no correlaiion between the model-simulated fields and thrit of the
control run. then r, will be zero and the relative error variance will becorne 2.
This is the limiting relative error variance value when the variance of the model
and that of the control run are nearly equal. When the correlation is negative. then
from eq. (3) we obtain,
In the worst case the correlation is -1 so that the limiting relative error variance
becomes,
When S: = S: , this quantity is 4, which is the maximum value the relative error
variance can assume when forecast is anti-correlated with the control run.
However, this limiting value is not valid when the above approximation, viz., the
variance of model forecast equal to that of control run, does not hold. An example
of this is present in fig. 4.15 for case-1 (fig. 4.16 for case-2) where the RCMl
(control run in the subsequent experiments) is compared with the RFE. In the case
of GZ field, emr variances at scales srnalier than around LOO km are much higher
thon the limiting value discussed above. Here the variance from the RFE and
subsequently the emr variance is much higher (an order of magnitude) than that
of the RCM l at those scales and is evident from the spectra plotted in fig. 4.12
(Details of spectral malysis is given in section 4.5 below). We point out that since
in al1 subsequent experiments the error variance at each scale is normalized by the
variance, of RCMf at that scale to obtain relative error variance, we used RCMl
as x and RFE as y to obtain fig. 4.15 and fig. 4.16. One cÿn nonnalize the enor
variance with the sum of the variances of both mode1 and the control run in which
case the limit to the relative error variance will always be limited to unity when
correlation is positive or two when correlation is negative. In this study, we used
the control run variance to nomaiize the error.
If we consider the error variance as a measure of predictability and if a field
with relative error variance less than two is considered predictable up to the
simulation time when it reaches two, then the geopotential field is predictable
throughout the simulation period for both the cases. Error variance in both
components of the wind field are larger compared to the geopotential and
temperature fields, in both cases.
The enor for the second run (RCM2) is different from that of the RCMl in
both case-1 and case-2. The temporal variation of the rms emor for various fields
for case-l is shown in fig. 4.7 and for case3 in fig. 4.8. The geopotential emr
variance increases with height in both cases, though they tend not to grow at the
lower lcvels. The temperature emr variance is found to increase with time and is
more obvious in case-2. The RCM2 relative error variance for the case-l is showa
in fig. 4.9 and the corresponding figure for case-:! is depicted in fig. 4.10. In fact,
the relative errors of ai1 fields give aimost a Bat curve, implying no growth,
though that of temperature grows slightly. However, the relative errors in wind
components and 1000 mb geopotential height field in both the cases are close to
one from the very beginning. It is also noticeable that the relative errors are
greater at lower levels in both the cases. If again the relative error variance is
considered as a measure of predictability. then the simulation of al1 fields (except
for v wind field after 24 houn) are somewhat predictable (or usefui) at al1 times
and levels. Looking ai the sirnulated fields, most major large-scale structures are
k i n g reproduced.
4.5. Spct i , of field.
The spectrum of a field will give the variance of the field at al1 scales present
in the atmosphere within the domain of study. Horizontal spectra of the wind
field. which gives kinetic energy, have been studied using rnodels and from
observations (Lilly and Peterson, 1983; Leith. 1971). It will be useful to look at
the spectra of fields md the spectra of difference fields instead of the rms or
relative error in the physical space. as the latter will be the sum of errors at al1
scales present. Hence the mis error will represent the erroa at those scales where
variance is the highest. It is well known that the spectra of atmosphenc fields
have negative dopes, with variances ai the scale of the order of 1000 km king
several orders of magnitude higher than those of small scales of a few hundred
kilometers. Errors at smUer scales may not even be noticed while taking the rms
errors into consideration. Therefore it is worthwhile employing the spectral
techniques to study the growth of errors at different scales present in the lirnited-
area domain.
Figure 4.1 1 gives the spectra of the 500-mb geopotential height field (GZ
hereafter) for case-1, caiculated by the method of chapter 3. The solid line gives
the spectrum of the RFE analysis field, the dashed line gives the spectmm of the
CRCM simulated field for the first nin (RCM1) and the dotted line is the
spectrum of the difference field. The largest scale shown in the plot is 2250 km
and the smallest shown is 70 km. At the initial time, the RFE fields and the
CRCM fields are the same and the difference field is zero. At my later time, the
model-simulated field is different from that of the analysis field and hence there
will be variance in the difference field. The spectra shown here are very similar to
the 500-mb geopotential spectra shown by Errico and Baurnhefner (1987). We
note that there is a maximum in the spectnim of RFE-analysed GZ field at around
100 km, which is absent in the mode1 simulated field except at the initial time.
This may be due to some smail scale features (wiggles in the analysis field)
present in the analysis field which are not realistic. This also contributes to the
error variance at small scales when comparing RCMl to the RFE analysis. This
small spectral maximum, however, is not present in the spectra of wind fields (or
kinetic energy fields) and in the temperature field spectrum. Fig. 4.12 shows the
corresponding vertically integrated GZ spectra, computed using the method
explained in chapter 3. Since the vertically integrated specwm of the difference
field wiii take into account erroa at ail vertical pressure levels, it is convenient to
analyse the verticdy integrated spectra rather than spectra at each level. Fig. 4.13
shows the vertically integrated spectra of horizontal kinetic energy (KE),
calculated using the method explained in the previous chapter, from the horizontal
wind fields. The horizontal KE energy spectnim is obtained by adding the spectra
of the horizontal wind fields ai the s m e level. Figure 4.14 shows the
corresponding temperature spectra. The variance in difference field in each case is
expected to be lower than that of the model simulated field during the initial
hours. This is found to be tme for scales larger than approximately 300 km.
In a 3-D turbulent fluid with a - 9 3 power law, energy rnovcs toward both
higher and lower wave numbers. at a rate greater than the transfers in a fluid with
a -3 power law. Leith and Kraichnan (1972) found that if errors are introduced in
large wave numbers for a 3-D turbulent fluid, then initiaily. total error decreased
as information propagated from lower to higher wave numbers. But later the
errors in the small scale rnoved toward larger scales and the total error increased.
If the error propagates from smailer to larger scales. the spectrum of the
difference field (which is the error variance) is expected to cross that of the model
field at a lower scale and the point of intersection moves towards the larger scales.
The point of intersection is where the error variance is the same as that of model
field variance. However, this could not be observed since the model varinces at
small scales are lost very quickly and also, srnall unrealistic features in the
analysis are not reproduced by the CRCM.
The corresponding spectn of fields for case-2 are very similar to the ones for
the case- 1 and are not shown here.
4.6. Enor growth for different 8cales
It is important to know the growth of error variance at each scale with time.
Fig. 4.15a shows the growth of vertically integrated relative error variance with
time for case-1, nonnalized by the RCMl field variance, for the GZ field.
Relative error variance equd to two for a particular scde is when the error
variancc at that scale is twice ihnt of the mode1 simulated field. and predictability
at that scale is then considered to be lost. The error variances üt luge scales, from
fig. 4.15(a), are very low and do not seem to be growing. Error increases as the
scale becomes smailer and the relative error variance at scales smdler than 450
km are close to 2 even at six hours of simulation. These scales seem to be
unpredictable even after providing high-resolution initial and lateral boundary
conditions, Growth of error at most scales, with relative error less than 2, show
the property of not growing much, similar to the RMS error growth described by
Anthes ( 1986).
Growth of errors with scale is depicted in fig. 4.15(b) for various simulation
times. The difference in variance at smdl d e s in the RFE analysis fields and the
CRCM field is the major reason for the large errors at srnaller scales. The relative
error variance at small scaies, particularly for GZ field, is high because of the
large variance of RFE at those scaies compared to RCMI as explained in the
section 4.4 above. Also, both the rnodels have different physical
parameterizaiions and resolution of topographie fields. This may contribute to the
differences at smaller scales.
The error in KE is due to errors in both u and v wind components. The growth
of relative error in KE with time for different scales is depicted in fig. 4.15(c) for
the RCM 1 for case- 1. The relative error growth with scale for different times is
shown in fig. 4.15(d). The error in KE is growing faster because of the greater
variations of u and v within the domain. Sirnilar figures for the temperature field
for case- 1 RCM 1 are depicted in figures 4.15(e) and 4.15(0. Error variances of
the largest scales (2250 km) are higher for temperature and KE fields compared to
geopotential height.
The corresponding figures of RCMl for case-2 are shown in fig. 4.16.
Although they are similar to that of case-1, the smdler scaies (450 - 1200 km)
have relative enor variance reaching two fater than in case-1. Remarkably, the
error variance for the temperature field at dl scales up to 60 h lies below the 2
line. Both cases show large relative error variance at small scales, especially for
the GZ field, when compared to the R F ' analysis. These may be due to smail-
scale features present in the RFE fields, which are responsible for the small
distortion of the specinim of GZ (fig. 4.12). Since these small-scde structures are
not present in the RCMl field, the erroa at around LOO km are much larger. These
e m r s may be due to the difference between the CRCM and RFE models, such as
physical parameterization, resolution of topography etc.
in order to avoid the problem of differences in the two models, a second mn
of CRCM (RCMZ) is made with lower resolution initial and boundary conditions.
This is then compared to the RCMl simulations, which is taken as the control run
assuming that RCMl is the best possible simulation on account of the high-
resolution initial and lateral boundary conditions provided.
The growth of errors for various scales and times are depicted in fig. 4.17 for
case- 1 and fig. 4.18 for case-2. From fig. 4.16(a), it can be seen that the largest
scales (2250 km) have relative error variance less than two throughout the
simulation period. However. al1 scales smaller than 700 km have relative error
variance growing very quickly reaching close to 2. This is the case for the KE and
temperature fields as c m be seen from figs. 4.17(c) and 4.17(e). There is a drop of
relative error variance at amund 42 hours. which is when the low-pressure system
enten the domain. As the cyclone enten the domain. the error starts to decrease.
However, after the cyclone is well within the domitin, the rrror once again starts
growing. This shows that the forcing at the boundary is important and when the
forcing of winds and geopotential heights are strong, the error is found to
decrease.
We note that the relative error for GZ grows up to a scale of around 350 km in
both the cases according io figures 4.17(b) and 4.18(b). After that, the error is
found to decrease with scales up to around 165 km. This is five times the grid
spacing. The scaies below this may not be well represented because at least 5 grid
points are needed for representing a wave properly. Hence, the growth of relative
error below 165 km scale may not have great significance. We recdl that only
discrete wavelengths are present in the spectrum, which are calculaied from the
discrete wave numbea separated by increments of dk as explained in the previous
chapter.
Although both cases give similar trends for the growth of relative errors with
scale, there is significant difference between two cases as far as predictability is
concemed. For case-2. the GZ and TT simulations seem to be very useful
particularly at scales smaller than 350 km. The topography might be playing a
significant role in forcing these scales. However. the wind fields (from the KE
plots) are completely useless for scales below 700 km for case-1 and 500 km for
case3 even ai 12 houn of simulation. We note that the scales larger than - i 500
km are predictable for al1 the fields at al1 times. This is probably due to the lateral
boundary forcing where the energy in the largest scales is specified accurately and
the model is forced to satisfy that condition. The evolution of the field in the
interior of the domain will affect al1 other scdes and hence the laterai boundary
will have less impact on controlling the growth of error at these smdler scales. If
only the mis error growth in physical is considered, it will be dominated by errors
at large scales and will always be less than one if normalized by the RCM 1 field.
4.7. Effects of advection through the boundary
The lateral boundary condition of a regional model gives continuous updating
of information at regular time intervals at the boundary. This controls the systems
moving in and out of the model domain. The systems are advected into the
domain through the upsmam boundary, whereas the model-simulated (evolved)
systems are moving outward through the downstream boundiuy. Hence. the
information advected into the downstream portion of the regional model domain
is contaminated with errors induced by the evolution in the upstream portion of
the domain. This may enhance the error downstream and hence the total error of
the simulation. It is interesting to h o w whether this advection h a any significant
effect on the emr growth in different parts of the mode1 domain. The analysis
below has been performed to study the effect of the advection through the
boundary on simulated fields.
We took the simulated 500-mb GZ fields from RUN2 of case-2. As above,
error is defined relative to that of the control run of the same case. However, the
domain within which the cornparisons are made was split into four as shown in
fig. 4.19. This figure shows the GZ field from RCM2 at 00 Z on Jmuary 9, 1996
when the CRCM was initialised. Let us denote the northwest corner of the domain
as (a), northeast corner as (b), southwest corner üs (c) and the southeast corner as
(d). The corresponding error analyses are shown in fig. 4.20. Since each
subdomain contains only 54 grid points. the discrete wave numbers (and
corresponding wave lengths) and the discrete wave number band (dk) are different
from that of our previous analyses. The largest wavelength shown on the plots in
fig. 4.20 is 560 km (ccrresponding to the wave nurnber 3.38) and smallest
wavelength is 70 km. We note that the wont error growth occurs in the southeast
subdomain (d) whereas in the northwest subdomain (a) the relative error remains
below one throughout the simulation period. The low-pressure system entering
the northwest subdomain (a) from the West (after 12 hours in the simulation) and
the low-pressure system entering the northeast subdomiiin (b) from the north
together strongly advect the information into the northwest subdomain (a) from
both the boundaries. Hence, this subdomain is constantly being well updated until
the cyclone from the West dissipates. The low-pressure system in the northeast
subdomain (b) entenng from north moves south and later south-east and helps the
updating of northeast subdornain (b) in the initiai times, up to when the flow at the
boundary becomes weak. Although the growth of error in the southwest
subdomain (c) was large in the beginning, it reduces to a large extent when a
high-pressure system shows up from the south. Once the low-pressure system in
the southeast subdornain (d) moves oui, the error there shows large growths even
for the largest scde shown. The advection into this part of the total domain is
from the other subdomains and not at al1 from the laterd boundary. Hence it
shows the worst growth of emr. We provided a couse resolution initiai and
laterai boundary conditions in the above experirnent, in which al1 scales below
450 km were removed.
4.8. Other experimentr
Apart from the two experhents above, some other expenments have also
been perfonned as explained below .
For the second nin of CRCM above, we have used the control-run simulated
fields, after smoothing, for generating the initial and lateral boundary conditions.
We have performed another CRCM run (RCM3), in which the initial and laterd
boundary conditions came from smoothed RFE fields rather than from the
simulated control nin. The results are shown in fig. 4.21 in which the temporal
change of the relative error variance as well as the growths of the relative error
variance with scale for various fields are plotted. Again the cornparisons are made
with respect to the control run (RCM1). This experiment did not give a better
nsult when compared to RCM2. In fact, the growths of relative error are worse
even as the tendency for the same with scale and time are sirnilu, for ail fields
considered. This experiment was not perfomed in case-2.
For the RCMZ in both case-1 and case3 was perfomed after smoothing the
control-nin simulated field such that ail scaies below 450 km w;is rernoved. Two
experiments were performed in case-2 in which this cut-off wavelengths (450 km
for RCMZ) are different. In the fourth experiment (RCM4), a cutoff wavelength
of 630 km is chosen. The CRCM, without any changes, was run with initial and
boundary conditions derived frorn this smoothed field in which al1 scales below
630 km were absent. The results are shown in fig. 4.22. This result when
compared to fig. 4.18 (for RCM2) shows no woaening of the relative error
variance even for the wind field. From the left panels of the fig. 4.22 it cm be
seen that the relative error variance of 450 km scale, which was absent at the
initial time, decreases initially before starts growing. This result indicates that the
large-scale forcing is capable of controlling to a certain extent the evolution of the
srnail scales within the domain.
In order to test the usefulness of providing a higher resolution initial and
lateral boundary conditions, a fifth experirnent (RCMS) was performed again in
case-2. In this experiment a cut-off wavelength of 3 15 km was used. The results
of this experiment are shown in fig. 4.23. Again the tendencies for the growth of
relative error variance remain sirnilar to RCM2. However, the relative error
variance for KE even at 12 h of iniegration is found to be higher (almost double)
at largest scales and grows quicker with scale when compared to that of RCM2.
in fact, for RCMS, the relative emx variance of temperature field, after 12 h.
reaches one only at a scale around 150 km. For RCMS, it happens at around 350
km at which scale a i i the fields in al1 expenments of case-2 show a maximum
relative error variance.
As a final note, we have considered in al1 of the above experiments the
relative error variance at different scales integrated vertically. The vertical
integrarion, as exploined in chapter 3. takes c m of the errors at d l levels.
However, the relative error and its growth vary with height for most scales. Fig.
4.24 shows the relative error variance for 1000-mb, 925-mb, 850-mb, 700-mb,
500-mb and 300-mb GZ field. The relative errors at smdl scales, smalIer than
around 350 km (wavelength), show considenble predictability at d l tirnes in the
lowest levels (up to 850 mb). As the height increases, the relative error in the
srnall scaies increases. Whereas the large scales, larger than about 1000 km
(wavelength), show the opposite trend. The relative error at these scales grows
rapidly in the lower levels. The larger the scale, the smaller the relative error
growth with time, for scales larger than 1 0 km, as we go to higher levels. The
increase in relative error variance with time is smaller as the scale decreases at
lower levels whereas it is higher at higher levels, although some sporadic changes
appear in the intermediate scales between 100 km and 350 km. In short, larger
scales seem to be better predictable at higher levels whereas smdler scales are
better predictable at lower levels. Surface inhomogeneities like topography may
be the reason for this increased predictability of smaller scales at lower levels
whereas its influence is less at higher levels.
Fig. 4.1: Sea-level pressure (PM) and the 1000-mb temperature (TT) for
case-1 at 00 Z November 8,1996.
PM 002 10 JAN 96 RFE
Fig. 43: Sea-level pressure (PM) and the 1000-mb temperature (Ti') for
case92 at 00 Z January 10, 1996.
(a) RCMl CASE-1 GZ
-, 1000 rnb - 700 mb ......... 500rnb
(c) RCMl CASE-1 W
, , 1000 mb - 700 rnb ......... 500 mb
O 20 40 60 80 100 time (hrs)
(b) R C M l CASE-1 UU ' 5 L
(d) RCM1 CASE-1 TT 1 5 ~ ~ ~ " " " ~ ' ~ " ~ " " ~ l
L 10- : L al V) 2
O 20 40 60 80 100 time (hrs)
. . , 1000 rnb
. - 700 mb
.......... 500 mb
.
Fig. 4.3: Temporal change in rrns error for RCMl (defined as the difference
between the control nin and RFE analyses) of case- l (November 8-9, 1996). for
(a) geopotential height (GZ, in dm), (b) u-wind (UU, in md), ( c ) v-wind (W. in
ms*') and (d) temperature (TT, in OC) fields at the 1000.hPa (mck solid). 700-hPa
(iight solid) and 500-hPa (dashed) levels.
" 51
(a) RCM1 CASE-2 GZ
. 1000 mb - 700 mb ......... 500 mb
(c) R C M l CASE-2 W 1 5 ~ ' ' ~ ' ~ ~ ' ' ' ~ ' ' ' ' ~ ' 7 ~ ~ ~
, - 1000 mb , - 700 rnb +
.......... 500 rnb
L 10- L 0)
O 20 40 60 80 100 time (nrs)
(d) RCMl CASE-2 TT 1 5 1 ~ ~ ' ' ' ' ~ " ' ' ' . " " ' \
(b) R C M l CASE-2 UU 1 5 ~ ~ ~ ~ ' ' ~ ' ' ' ~ ' ' ~ ~ ~ " ~ ' ~
O 20 40 60 80 100 time (hrs)
L 10- 2 01
Fig. 4.4: Temporal change in rms error as in fig. 4.3, but for RCMl of case-2
(January 10-12, 1996).
. . 1000 mb , - 700 mb . . . . . . . . . . 500mb
(CI) R C M l CASE-1 GZ z
......... 500mb - 700 rnb
. 1000 mb
O 20 40 60 80 100 time (hrs)
(b) RCM1 CASE- 1 UU 10.00
......... 500 mb - 700 mb . , 1000 mb
(d) R C M l CASE-1 TT 10.00
500mb ......... - 700 mb , 1000 mb
O 20 40 60 80 100 time (hrs)
Fig. 4.5: Temporal change in relative rms error for RCMl (defmed as the
difference between the control run and RFE analyses, normalized by the control
nin variance) of case- 1 (Novernber 8-9, 1996), for (a) geopotential height (GZ. in
dm), (b) u-wind (UU, in ms-'), (c) v-wind (W, in ms") and (d) temperature (TT,
in OC) fields at the 1000-hPa (thick solid), 70-hPa (light solid) and 500-hPa
(dashed) leveb.
(b) RCMl CASE-2 UU 10.00
......... 500mb
. 700 rnb
. 1000 mb
(c) RCMl CASE-2 W 10.00
......... 500mb - 700 mb -, 1000 mb
O 20 40 60 80 100 time (hrs)
(d) R C M l CASE-2 TT
. 700 mb
. IO00 mb
5 1.00- 8 E al
i . - 4
O - _ 1
0.10 - ......... ..........
1 ..............
0.01 1 - I . I - - - l - - - . . O 20 40 60 80 100
time (hrs)
Fig. 4.6: Temporal change in relative nns error as in fig. 4.5, but for RCMl of
case-2 (January 10-12, 1996).
(a) RCM2 CASE-1 GZ l ~ ~ ~ r - v - l ~ - ~ ~ ' ~ '
, , 1000 mb . 700 mb ......... 500mb
?
..........
(c) RCM2 CASE-1 W
- 700 mb ......... 500 mb :
(b) RCM2 CASE-1 UU
I
0 1 . . t . . .
O 20 40 60 80 IO0 timeihrs)
(d) RCM2 CASE- 1 TT I
0 20 40 60 80 100 time (hrs) time (hrs) . ,
Fig. 4.7: Temporal change in rms error for RCM2 (defined as the difference
between the RCM2 and controi run) of case-1 (November 8-9. 1996). for (a)
geopotential height (GZ. in dm), (b) u-wind (UU. in ms"). (c) v-wind (VV, in
m d ) and (d) temperature (TT. in OC) fields at the 1000-hPa (thick solid), 700-hPa
(light solid) and 500-Ma (dashed) levels.
(a) RCM2 CASE-2 GZ 1 5 r r ' ' " ' . ' ' . v ' ' ' ' 1 ' - ' '
._.-.. , , 1000 rnb -..-..
, (c), ,RCM2 ,CASE-,21 W .
1000 mb - 700 mb
O 20 40 60 80 100 time (hrs)
(d) RCM2 CASE-2 TT 1 5 1
O 20 40 60 80 100 tirne (hrs)
Fig. 4.8: Temporal change in rms enor as in fig. 4.7, but for RCM2 of case-:!
(Jmuary 10- 12, 1996).
(a) RCM2 CASE-1 GZ 1 0 . 0 0 ~ - - - ' - ~ - ' - ~ - ' ~ ~ ~ ' ~ ~ 7 ~
(c) RCM2 CASE- 1 W
0.01 1 O 20 40 60 80 100
time (hrs)
(b) RCMZ CASE-] UV 10.00
......... 500 rnb
. 700 mb , 1000 mb
(d) RCM2 CASE-1 T l 1 0 . 0 0 ~ 7 ~ . ' - - . ' - - ' [ - - . ' . - . j
. . . . . . . . 500 mb - 700 rnb . , 1000 mb
0.01 1 O 20 40 60 80 100
time (tirs)
Fig. 4.9: Temporal change in relative rms error for RCMZ (defined as the
difference between the control run and RCMZ, nomalized by control run
variance) of case4 (November 8-9, 1996). for (a) geopotential height (GZ, in
dm), (b) u-wind (W. in md), (c) v-wind (VV, in ms") and (d) temperature (TT,
in OC) fields at the 1000-hPa (thick solid), 700-hPa (light solid) and 500-hPa
(dashed) levels.
(c) RCMZ CASE-2 W 1 0 . 0 0 ~ - - - ' - ~ ~ ' - ' - " - ' ' - ' ' !
time (hrs)
(b) RCM2 CASE-2 UU 10.00~ . . ' - - ' - - - ' - - . ' . - . l
(d ) RCMZ CASE-2 TT 10.OOL . . v - - 7 r - - - 1 - - - 8 - . 7 i
O 20 40 60 80 100 tirne (hrs)
Fig. 4.10: Temporal change in relative rms error as in fig. 4.9, but for RCM2 of
case-2 (January 10- 12,1996).
(b) 500 mb CZ SpKtnim 12 h n - - - - . - - . I V - . - - - . -
- RFE 1
(c) 5Cül mb CZ 50.ctrum 24 h n - - - RFE 1
(O) 5W mb GZ SprcVum 80 h n 1 " - . - . . - - - . . - - -
RFF 1
I ......... DtFF 1
(d) 500 mb CZ Speclnim 48 h n I O ' " - - - - - - r n - - - - - - - 'I - RFE .
, , , . RCM1,
(I) MO mC CZ Spsctrum 72 h n 10'
- - - . RCM1, \.... . . DIFF ,
- . - *<.
Fig. 4.11: Spectrum of the 500 mb geopotential height field of RCMl case-1
(initialized at 00 Z 6 November, 1996). (a) at the initial time, and after (b) 12 h,
(c) 24 h, (d) 48 h, (e) 60 h and ( f ) 72 h of integration. The wavelength plotted on
x-axis gives the scale of motion. Solid line is the spectrum of RFE analysis field,
dashed line is that of the RCMl (control run) field and the dotted iine is the
spectrum of the difference between the two.
ru- - - - . - . ,.-.. . . .
i 7 - - - - - -
- RFE 1
(b) int GZ Spectrum 12 k n 1 0 * - - ~ - - . . - . . - . . - . -
- RFE . -,,. RCM1,
\ - - m . \ : \
-
RFE . RCM1, DIFF .
10-2
Fig. 4.12: As in fig. 4.11. but for the vertically integrated Spectmm of GZ of
RCM 1 case- 1 (initidized at 00 Z 6 November, 1996).
ION ta 10 t##0 lm 100 1 O ( h l rronunpi (W
(c) Int CZ Spactrum 24 hm (d) ln1 GZ Spechm 48 h n 1 0 6 ~ - - . - - * - - - - - ' - - - - - - - - 1 I O ' + - - - - , - - - - . - - - -
L 1 - RFE 1 1 - RF€ 1
- 10-2 \ - \ -
-
(a) Int KE Spectnm initia4 time 1 0 ~ 6 1
(c) int KE Speehm 24 hm . . - . - ' I V - - - - - - - l - . - - - - ' .
- RFE RCM 1 - DiFF .
.
(d) In1 KE Spectnim 46 h n io6--+.- - - - -
- RFE .
10' - -,-. RCM1, ......... DlFF .
102 -
1 0 0 - 4
Fig. 4.13: As in fig. 4.11. but for the vertically integrated Spectrum of KE of
RCM 1 case- 1 (initiaiized at 00 Z 6 November, 1996).
(a) int NE Sprctrum 60 hm ( 1 ) lnt KE 5pectrum 72 hrs - - - . . . - - - II+-* cc6-- - - - l . . . . . -
- RFE .
10' - , -, . RCM1, ,,4, -, ,. RCM1- ......... DlFi .
; id- e P '
18 -
10-2
ioOQ1 lm 1W tm lm ia !O lm
IIPiiYiSlh (hi) Mr*npth (hi)
- 10-2 - ......... 1 - - . - -
Fig. 4J4: As in fig. 4.1 1, but for the vertically integrated Spectrum of
temperature field (TI') of RCM 1 case- 1 (initialized at 00 Z 6 Novernber, 1996).
(cl RCMl CASE-1 KE
0.011 1 - O
1 20 (O 00 M 100
unuomaww (01 RCMl CASE-1 TT
r - I
id) RCMl CASE-1 KE
Fig. 4.15: Verticdly integrated relative error variance as a function of time for
different wavelengths (scales) of (a) GZ, (c) KE and (e) TT fields, for RCMl
case-1 (November 8-9, 1996). The smaller the scde, the more relative error is.
Vertically integrated relative error variance as a function of wavelength for
different simulation times of (b) GZ (d) ?CE and (f) TT fields for the same case.
As the scale decreases, error increases for ail time. Here RCMl (control mn) is
compllred with the RFE analyses.
(c) RCMI CM€-2 KE r - - I - . - r - - - i - - i - . ' l
aoi l I l
(CI RCM t CASE2 TT
Flg. 4.16: Sarne as in fig. 4.15, but for RCM 1 case3 (January 10-12, 1996).
(b) RCMZ CASE1 02 I .
(d) RCM2 CASE-1 KE
! : i
1 2 hm 0.8, . .
Fig. 4.17: Vertically integroted relative error variance as a function of time for
different wavelengths (scales) of (a) GZ, (c) KE and (e) TT fields, for RCMZ
case4 (November 8-9, 1996). The smaller the scale. the mon relative error is.
Vertically integrated relative error variance as a function of wavelength for
different simulation times of (b) GZ, (d) KE and (0 TT fields for the same case.
Here RCM2 is compared with the control mn.
(b) RCM2 CASE-2 Gt 1 . . - I
id) RCMZ CASEZ KE r - 1
Fig. 4.18: Same as in fig. 4.17, but for RCMZ case5 (January 10- 12, 1996).
RCM2 500 00 GZ 09 JAN 96
Fig. 4.19: 500-mb GZ field at initial tirne of RCM2 of case-2. (00 Z January 9,
1996). The domain is divided into four subdomains and the error variance spectra
for each subdomain are depicted in fig. 4.20.
(a} AûV CASE-2 GZ
, , , , 60 hrs .......... 48hrs . 24 hrs
E
(c) ADV CASE-2 (d) ADV CASE-2 CZ
' O . O r - - - - - l
Fig. 4.20: The relative error variance of GZ at 500 mb, for RCM2 of case-2
Ianuary 10-12. 1996). as a function of scale for different simulation hours for
(a) northwest (b) northeast (c) southwest and (d) southeast subdomains of the
fig. 19 shown in the previous page. Here RCM2 is compared with the control
run.
(a) RCM3 CASE-1 û2 - . - - : . . 1
(b) RCM3 CASE-1 Gt 1 1
1 2 itn
(d) K M 3 CASE-1 KE
1 - -
Fig. 4.21: Same as in fig. 4.17, but for RCM3 case- l (November 8-9, 1996). Here
RCM3 is compared with the control run.
Fig. 4.22: Same as in fig. 4.17, but for RCM4 case-2 (January 10-12. 1996). Here
RCM4 is compared with the control nin.
(a) CICM5 CASE-2 GZ
. . . - .
(a) RCMS CASE-2 TT T t
(b) RCMS CASE-2 û Z
1 - 1
(c) RCMS CASE.2 TT 1 - - 1
Fig. 4.23: Same as in fig. 4.17, but for RCMS case-2 (January 10- 12, 1996). Here
RCMS is cornpmd with the contrd run.
(a) RCM2 CASE-2 OZ 1M)O mb 1 . . - 1
F i t 4.24: The relative error variance as a function of wavelength for different
simulation times for the GZ field at (a) 1000 mb (b) 925 mb (c ) 850 mb (d) 700
mb (e) 500 mb and ( f ) 300 mb for RCMZcase-2 (January 10-12, 1996).
Chapter 5 - Summary and future recommendations
Spectra of different fields for a limited-area domain have been cornputed
using the Fourier transform technique following Emco (1985). The Fourier
spectral analysis of a meteorological field on a limited-area grid is demonstrated
to be an appropriate scale analysis tool. Linear trends have been removed before
performing the two-dimensional Fourier transformation on a field, which permits
the representation of the field in the wavenumber domain.
The spectra of the geopotential height field. both horizontal components of the
wind field and the temperature field of both the regional finite-element (RE)
model and the Canadian regional climate model (CRCM) are examined üt
different simulation times. The spectra of the difference field. which gives the
error variance at various scales, are also studied. The spectra of fields displayed in
chapter 4 are similar to the spectra obtained from previous studies. Large variance
is at the larger scales and the spectra have a negative dope. The srna11 peak in the
spectrum of geopotential field from RFE model ai small scales around 1 0 km
(wavelength) is due io the interpolation between the pressure levels and the model
Gal-Chen levels and back to pressure levels in the CRCM grid. Also. the
geopotential field is not a model variable and is calculated from the surface
pressure and temperature fields. Vertically integrated spectral variance has also
been examined. The vertical integration of spectra of a difference Field accounts
for errors at d l vertical pressure levels.
There are a few limitations of the spectral technique used in this study. For
exarnple, any singularly large value within a field will project onto al1 scales. The
spectrum of such a field may be misunderstood as that of a field characterized by
mûny scales of motion over the whole domain, rather than by very locülized or
isolated behaviour. Precipitation patterns may be a good example of this. For this
reüson we have not attempted to Fourier decompose the precipitation fields.
Another limitation of this technique is the possible removal of the wrong linear
trends while there are significant variations in the field at the boundaries. If a field
were constant nearly everywhere except ai the boundaries, detrending may create
significant variations in the interior. These variations then would project ont0 al1
spectral components. These scenarios for the limited-area domain for the RFE or
CRCM fields studied here occur when the centre of highs and lows of the fields
move in or out of the domain. Once the highs and lows of the field are well within
or out of the limited-are domain, this enor would not anse. An example of this
type of misrepresentation in the case of geopotential field is given in the
appendix.
The newly developed CRCM is usually nested within a GCM to study the
climatology of a region of interest. The spectral analysis method h a been used to
evaluate the ability of CRCM to simulate the mesoscale feanires correctly given
GCM-like initial and lateral boundary conditions. We have attempted to
investigate the short-term simulations of individual cases by the CRCM in this
study, though the long-term monthly integrations and the climatology are planned
for future studies. Two cases have been chosen. The first one is a passage of a
cold front over the Montréal region in late fall and the other is the passage of a
low-pressure system over the Mackenzie River Basin (MRB) in early winter. The
experiment RCMl, in which the initial and lateral boundary conditions are
obtained from high resolution RFE model, is considered as the control run. Al1
scales below 450 km (wavelength) are removed by smoothing the RCMl fields.
For RCMZ, the initial and laterd boundary conditions are GCM-likc ficlds
defined by the smoothed RCMl fields. The root mean square (RMS) error and
relative rms error for RCM2 when compared with RCMl show little or no error
growth. especially for the geopotential field. The relative m i s error in geopotential
height at most levels are on the order of 0.01 (1 % of the variance of the RCMl
field at the same level), indicating a high predictability in CRCM simulation.
This, however, is very different if the relative errors are examined according to
scale. The relative error for the largest scales of al1 fields is on the order of O. 1 at
al1 times. Though most of the variance is at the Iargest scales. the relative error at
these scales is an order of magnitude greater when compared to the relative rms
error without scale decomposition. This is because the linecir trend was removed
from the fields before spectral decomposition and this trend represents scales
larger than the limited-are? grid ûssociated with quite large variance. Besides, the
error variance at each scale is obtûined by normdizing the error variance by the
corresponding RCM 1 variance at the same scale.
Error growth is less in case-2 especially at small scales for the geopotential
and wind fields. This could be attributed to the forcing by topography. The Rocky
Mountains rnay have a profound influence on the systems produced over MRB
region at these scales. However, the relative errors for the meso-a and synoptic
scales grow with time and are more visible in case-2. In case-1, a low-pressure
system enters the model domain at around 36-h of the simulation and helps reduce
errors at synoptic scales at around this tirne. The relative error, however, grows
again once the system is completely inside the dornain.
Two more experirnents were performed in case-2 with different cut-off
wavelengths for smoothing. The experiment RCM4, in which al1 scales
(wavelengths) below 630 km were rernoved. gave results similar to RCMZ. The
experiment RCMS, in which al1 scales (wavelengths) below 315 km were
removed also gave results similar to RCMZ.
Effects of advection through the lateral boundary on the simulations have
been exmined by dividing the domain of case2 into four subdomains and
studying the relative enor growth at 500-mb in each of these subdomüins
separately. it is found that the advection of information from outside the boundary
has a profound influence on development of systems in each of the subdornains.
The northwest subdomain has the least error growth at al1 scales because of the
flow into this subdomain from both boundaries for most of the time.
In short, the spectral method employed here is demonstrated to be a useful
tool to verify the scale-dependant error growth in a regional-scale model. The rms
error or relative rms error for the entire domain without taking different scales
into consideration may often provide an overestimate of the predictability of
meteorological fields. However, when the relative error growth of different scales
are considered separately, the emrs at srnall and intermediate (meso) scales are
found to grow with time. The relative error variance at each scaie is level-
dependant. Large scales have small growth of relative error variance at higher
Ievels whereas the opposite is true for the smaller scales.
As possible future work, the CRCM can be used for a range of temporal
integrations. It would be interesting io study the monthly. seasonai and yearly
deviations of the CRCM simulations from a corresponding climatology. A r t u d ~
of a January simulation is currently undenvay. Spectral methods have been used
in the p s t (for exümple Boer 1993) for studying global models systematic- and
random-error variances from the mem (climate) in the extended-range
forecasting.
Chapter 6 - The "Clean" Run
We have perfomed additional mode1 runs after the initiai thesis submission
was made. Some of the results of the new experiments are explained below.
6.1. Experimental design
There are huge differences in the control run (RCMl) and the RCM1,
especially at small scales, as explained in the previous chapters. One reûson for
this could be the influence of initial condition and latenl boundary conditions in
the RCM 1, which comes from the RFE. Small scdes present at the initial time
and at the boundary evolve and have profound influence in the small-scde
feeatures simulnted by RCMI. Since these were not present at the initial time and
at the boundary in RCM2, the small-scale features simulated are not likely to be
influenced by the RFE small-scale features. Thus the bias in RCMl would
contribute to the errors in the evaluation of RCM2 and subsequent runs.
To avoid this we have perfomed an additional CRCM mn to improve the
control run. The experiments were perfonned on case-2 (Jan 10- 12, 1996) as
follows. The first run of CRCM (RCMI) is driven by the initial and lateral
boundary conditions from RFE as explained in chapter 4. The only difference is
that the CRCM simulation started at 002 on Jan 8, 1996 instead of 00Z on Jan 9,
1996 as in chapter 4. This new RCMl was not taken as control mn. We make
another CRCM run (hereafter RCM 1 I ) , which stiirts 24 hours later (002 on Jan 9,
1996) with initial and laterd boundary conditions coming frorn RCM 1. This new
run (RCMI 1) is taken as the control mn. The RCMZ run was made the same way
as before. Initial and lateral boundary conditions for RCMZ come from the
smoothed fields of RCM1. Again the RCMI, as RCM 1 1, begins at 002 on Jan 9,
1996. Now the control run and the RCM2 are over the same domain with same
resolution. The initial and boundary conditions come from the same source. The
diffcrcncc is in the resolution of initial and boundary conditions. RCM? is nin
with a course resolution initial and lateral boundary conditions in which al1 the
scales below 450 km are absent.
6.2. Results
Results of this experiment show considerable differences compared to the
previous experirnents. Relative errors for al1 the fields at a11 scales are
considerably decreased which show the bias in RCM 1 simulations due to the
influence of RFE small-sale features in the initial time. Fig. 6.1 gives the
integrated relative error variance for RCM2 when compared to the control run
(RCMI 1) for GZ, KE and 'TT fields. This figure corresponds to f ig . 4.18 in
chapter 4. Although the trends in the error growth with time for various scales
remain more or less the same, the magnitude of the relative error variance have
drastically decreased and stays below around O. I for GZ and TT and 0.5 for KE.
A significant difference in this case is that the relative error variance is not found
decreasing with scale below 400 km, which was the case before. The enor for GZ
and TT tend to get saturated around 0.1 for dl the scales below 400 km. The
relative emr variance in KE grows with decreasing scale. Also, the largest rate of
error gmwth is at around 1 0 0 km as in the case before.
The growth of relative error with height for GZ is depicted in fig. 6.2. The
corresponding figure in the runs before is 4.24. Again while the genenl trend is
preservrd, the magnitude of relative error is almost an order of magnitude less in
the new experiment for most scales. In al1 future works suggested in chapter 5 , the
control run should be made as done in this experiment so that the difference
between RCMl and the control run would be minimal and not influenced by RFE.
(c) RCMZ CASE-2 KE l . - - , - . , . - , . . - ' - . . l
O 20 40 dD y#) ammon mm (hm)
(e) RCMZ CASE-2 TT 1
. . . . . 48 hrs - 24 hm
~~ km)
[cl RCMZ CASE2 n
Fig. 6.1: Verticaily integrated relative error variance as a function of time for
different wavelengths (scales) of (a) GZ, (c) KE and (e) TT fields, for RCM2
case-2 (January 10- 12, 1996) in the new experiment. The smaller the scale, the
more relative error is. Vertically integrated relative error vdance as a function of
wavelength for different simulation times of (b) GZ. (d) KE and (f) TT fields for
the same case. Here RCM2 is compared with the control mn (RCM 1 1)
(a) RCM2 CASE-2 GZ 1OOO mb 1 - - ' 1
(b) ACM2 CASE-2 GZ 925 rnb 1 - - - 1
(el FICM2 CASE.2 GZ 500 mb (1 ) RCM2 CASE-2 GZ 300 rnb 1 - - - 1
Fig. 6.2: The relative error variance as a function of wavelength for different
simulation times for the GZ field at (a) 1000 mb (b) 925 rnb (c) 850 mb (d) 7 0
mb (e) 500 mb and (f) 300 mb for RCM2 case-2 (January 10-12, 1996) in the new
experiment
Appendix
Al(a). Fourier Series
A Fourier series is an expansion of a periodic functionJx), of period 2x. in
the interval O 5 x 5 2x in an infinite series of the forrn
where the Fourier coefficients, c,, off are defined by,
Equations (a.\) and (a.2) are together called the direct and inverse Fowirr
trnnsform respectively. Since f determines c, and vice versa, the trünsforrns
contain the same information and so represent wavenumber and real domain
descriptions of the function.
Fourier series can be written in a number of equivalent ways. Euler's relation
for e" gives,
f (x) = 5 3 + (a, c o r n + basinm)
where a, = c, + c-, and ba = i(c, - c, ) or
% = ffl(x)cogn dr and 1 5 0
For real f the conjugate of (a.2) gives c, = c,, where the astetisk denotes
complex conjugation. This is the r e n l i ~ condition.
The Parseval relation (energy relation) states that
Al(b). Higher dimensions
If f is a function of x and v on a square with sides of length 2n. then a Fourier
series in x gives (a. 1 ) except f and c, now also depend on v . Expanding c&) in a
second Fourier series in y then gives the two dimensional transform pair
and
where the sum is taken over al1 integer cornponents of a. Here D denotes the
square domain with sides of length 2n and d'x indicütes integration over both
variables.
Al(c). Spectra
The quantity I I f II' is called the energy, power, or variance of f (the
teminology varies depending on the application and physical dimensions off ).
According to Parseval's relation (a.3), i t is the superposition of contributions from
individual cornponents. The nue-sided spectnun off,
describes the contribution as a function of a as does the one-sided spectrum
albeit in a more compressed form. Since Ical = Ic.,l from the reality condition, for
realf S, is an even function of a while Su = 2 1 ~ ' . Since boih spectra contain the
same information, the one-sided spectrum is used.
Al@). Discrete Fourier transform
When f is given at J unifomly spaced points x, = %j/J on the interval [0,2x],
then the Fourier senes evaluated at x, leads to the J equations
in the infinite number of unknowns c,. While it is clearly impossible to determine
the coefficients from such limited information, the trîgonometric functions are no
longer independent on the grid since
eita+p ")
im, = e
for y = f 1, k2, . . . In other words wavenumbers a+yJ and a are associated with
the same spatial structure on the gnd and so are indistinguishable. Higher
harmonies with this property are called aliases of a. Some examples of which are
illustrated in Fig. Al. If a is restricted to the range O 5 a 5 J-l and if each a and
its aliases are grouped into a single term then (a.4) reduces to a system of J
equations in J unknowns
(a. 5)
where
Since #a (x, ) = e'"' are orthogonal vectors (of J dimensions), we get from (a.6).
the expression for a, as
(a.5) is calied the discrete Foiirirr series ofJ and the a, are the discrere Fortrier
coefficients. In fact (a.5) is unchanged when any J consecutive values are used or
equivaiently when the upper and lower lirnits are shifted by a constant. Thus for J
even (which turns out to be desirable for computational reasons), (a.5) can be
written as
112-1 im,
fi = Case a=-' l 2
(a.7) and (a.8) together constitute a discrete Fourier transform pair.
Only a finite set of wavenumbers 412. .. .., JI2 - 1 can be detected from J
evenly spaced measurements. The lowest detectable wavenumber is 1
corresponding to a waveiength 2x (the size of the interval) while the highest or
Nyquist wavenumber is JI2 corresponding to the wavelength,
2R/(J/2) = 4RIJ = 2&
which is the shortest that can be detected from discrete data at resolution hir.
Fig. Al: Three sine waves which have the same k = -2 interpretation on
an eight point grid. The nodal values are denoted by the dark dots. Both
k = 6 and k = - 10 waves are misinterpreted as a k = -2 wave on a couse
grid. k is the wavenumber here.
Since c,,z is aliased into c . 1 ~ and c,~+, is aliased into c-,c+! and so on. and
since -a is mapped onto a in the one-sided spectrum. energy at wavenumber JI2 +
fl outside the resolved range is aliased to -JE + and then to JI2 - P in the
discrete one-sided spectmm. In other words, wavenumbers in the range JI2 + 1 , J
- 1 are folded about the Nyquist frequency (112Ar) into the resolved range.
Higher wüvenumbers are similarly mapped into 1. 2, . . . ,512.
A l (e). Fast Fourier Transform
It is clear from equations (a.7) and (a.8) that the direct evaluation off; or ci,
requires O(J) operations per wavenumber and o(J') operations overall.
Remarkably, when J is a power of 7 or. more generally. when it has ü
factorizat ion
J = 2P3q4r5s6' where p. q, . . . . are integers,
then bothA and its transform can be evaluated in O(J1og.l) operations using a F c ~
Fourier Transform (FFï). The algorithms use the fact that transforms of length J
can be written as sums of shoner transfonns. For example, a transform of length J
can be constructed from two transforms of length J!2 so that when J = 2" a
transform of length J can be constmcted recursively beginning with transforms of
length one. The savings permit large data sets to be treated that would otherwise
impossible to deal with. Calculations that would require weeks of CPU time by
direct methods can be done in a matter of minutes using the FFT.
A2. Some aspects of de-trending a field
In chapter 3, we have seen the method used to remove a linear trend from a
field, over a limited area domain. before using the FFT to cornpute the spectrum.
It is worthwhile looking into the attendant assumptions used in this procedure.
The method assumes that removing the linear trend first dong i and then
dong j is the same as removing the linear twnd first dong j and tlieii dong i.
Rernoving the trend is in fact nullifying the slope of a linear surface in üny
particular direction. This depends on the end points, or in other words the
boundary values of the field. If the linear trend is removed dong any particulür
direction, the boundary values to be used to define the slope in the other direction
will be altered. Thus, during the process of removing linear trend almg i = 1 and i
= Ni, the value of the field at al1 j dong these boundary lines are changed thereby
changing the slope dong each j. Hence, it is possible that the trend removed
dong each j, once the trends dong al1 i are removed may be different from that if
computed before perfonning the same dong i. However, the trends we obtain by
both methods are effectively the same for the fields tested. An example is shown
in figures A2 and A3. The geopotential height field (z) of 850 hPa at 48 hours of
the simulation is shown in fig. 3(a), of Chapter 3, and the field after removing the
linear trend is shown in fig. 3(b). The corresponding spectra are depicted in
figures 3(d) and 3(e). Figure M ( a ) shows the same field z, with trends removed
dong al1 i only and the conesponding spectrum is shown in fig. A2(d). Figure
A3(a) shows the field z, with the trends removed dong j only and the
comsponding spectrum is shown in fig. A3(d). It is to be noted that the final
fields, after removing trends in both directions. are the same irrespective of the
choice of direction to remove the trend fint. The figures A3(b) is the same as in
A2(b) and their corresponding spectra. The trend alone and its spectrum are also
shown in both situations.
The idea of removing a linear trend is to make the field penodic so that
variance iii the luge-scda unresolvabla wavas will iiut be projejecied on to [lie
resolvable scales. Figure 3, of chapter 3 illustrates this 'aliasing'; i.e.. the
contamination of variance at the resolvable scales by folding of the spectrum of
the non-resolvable scales. The rnethod assumes that removal of the linear trend
wil
al 1
poi
1 remove the variance in the large-scale waves. However. the combination of
the non-resolvable scale waves need not have the same slope rit different grid
nts inside the domain. Hence, the choice of domain, which contains the field,
will greatly influence the linear trends removed. To illustrate this the following
experiment is done. Frorn Fig. 3(a) of chapter 3, which is the field at 48 hours,
two different subdomains are chosen and the linear trends removed from them.
The first subdomain is of (2240 km)' in area and is in the nonhwest (upper leA)
corner of the domain. It is shown in Fig. A4(a). A low-pressure system is well
within the domain. The spectnim of this field is shown in Fig. A4(d). Frorn the
figure we can see that the slope is very small in either direction. Hence, removing
the linear trend is expected to give a de-trended field, which is not much different
from the original one, as is shown in fig. A4(b), with the corresponding spectrum
in Fig. A4(e). The trend alone and corresponding spectra are depicted in Figs.
A4(c) and A4(9 respectively. Now, consider another subdomain having the same
area but is different from the one described above in that the center of the low-
pressure system is on the left boundary of the subdomain. This significantly
changes the slope of the non-resolvable waves in this subdomain when compared
to the previous one. Figure A5(a) shows this domain and the corresponding
spectmm in Fig. A5(d). The de-trended field in this domain is shown in Fig.
A5tb). We note thar the fields before and after de-rrending are significantly
different from each other as are the corresponding spectra. The trend alone. shown
in Fig. AS(c), obviously shows large variance because of the large slope found in
the original field. This is depicted in Figs. A5(c) and A5(f). This suggests that the
choice of the domain is very important if the linear trends are to be properly
removed and spectral analysis is to be done. It may be more representative if lows
and highs or any kind of sharp variations in the fields can be avoided at the
boundary of the limited aren domain.
(O) z - j trend rmnovld
- (b) 2 - bolh trend ramuwd
(d) rpectrum of r in (a) 102 j 1 'I
(a) specirum of 1 (no Irena) .n (b) 10;: 1
( f ) tpsctrum o l 2 (Irmnd oniy) ln (c) 1 1
Fig. Al: (a) The geopotential field (2 ) at 48 hours after removing linear trend
along al1 i only. (b) the z-field after trends along both directions are removed. (c)
trend alone. (d), (e) and (f) are the spectra of the fields shown in (a), (b) and ( c )
respective1 y.
ld ( 8 ) l pe~ t rum of ; I (no trend) in (5)
1
Fig. A3: (a) The geopotential field ( z ) at 48 hours after removing linear trend
dong al1 j only. (b) the z-field after trends dong both directions are rernoved. (c)
trend alone. (d), (e) and (f) are the spectra of the fields shown in (a), (b) and (c)
respective1 y.
(b) r - nwc ken0 removed m 10
W o n nummr I la+ (m")
(e) spactrurn of z (no trend) in (b) 102 r
l
Fig. A4: (a) The z - field in the northwest (upper left corner) subdomain of the
fig. 3a of chapter 3. The subdomain is (2240 km)'. (b) t - field after trends are
removed in both directions. (c) trend alone. (d), (e) and (f) give the specira of the
fields in (a), (b) and (c) respectively.
( O ) rpeclrum al z in (a) I 1
(O) spectrum a i z (no trend) on (b) 102 (
1s 1 ( f ) rpectrum of r (trend oniy) in (c)
1 1 i
C
1 0 4 1 IO la
#on numm X IO* (rn")
Fig. AS: (a) The z - field in the upper middle subdornain of the Ag. 3a of chapter
3. The subdomain is (2240 km)? (b) z - field after trends are removed in both
directions. (c) trend alone. (cl), (e) and (f) give the spectra of the fields in (a), (b)
and (c) respec tivel y.
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