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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Anthes, R. A.. 1986: The General Question of Predictability. In "MesosCale

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