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On the Parameterization of • Slantwise Convection in General Circulation Models

Liang Ma

Department of Atmospheric and Oceanic Sciences McGill University Montreal, Canada August, 2000

Submitted to the faculty of Graduate Studies and Research in partial fu]fillment of the requirements for the degree of Doctor of Philosophy

@Liang Ma 2000

i

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0-612-70086-0 •

Abstract

This study is concerned with the effect of slantwise convection in general circulation

models (GCl\JIs). The approach is through the development of a slantwise convective

parameterization scheme (SCPS) and its implementation into version 11 of the third

generation GCM of the Canadian Climate Centre for modelling and analysis (CCCma

gcm11).

"vVe first study the characteristics of conditional symmetric instability (CSI) in

an environment which is also unstable for conditional upright instability (CUI). The

results indicate features common to both upright and slantwise convection. This so

called slantwise buoyant instability (SBI) possesses two relevant time scales and its

horizontal scale can ranges from tens of km up to over one thousand km.

\Ve then analyze the 15-year ECMWF re-analysis (ERA) data to compute the

global distributions of convective available potential energy (CAPE) and slantwise

convective available energy eSCAPE). We show that the potential for CSI and CUI

indeed co-exists over most areas around the globe. Based on the results of the theo­

retical study and the data analysis, a parameterization for slantwise convection was • ü developed and implemented into gcml1. It was round that the scheme impacts sig­ • nificantly the simulated general circulation by the development of a direct meridional secondary circulation. The results of the 5-year simulations show that the scheme re­

duces SCAPE and SCAPE residual Ta over the mid-latitudes, leading to a weakening

of the thermal wind and the strength of the upper-level jets. The largest improve­

ment in the simulated dimate however lies in the reduced meridional transient eddy

transports of heat and zonal momentum. With the inclusion of the scheme, the eddy

transports agree much more favorably with the observational analysis.

• ili •

Résumé

Cette étude porte sur les effets de la convection oblique dans les modèles de circulation

générale (GCM). L'approche utilisée est la suivante. Un paramétrage de la convection

oblique (SCPS) a été développé et implanté dans la version Il du GCM (troisième

génération) du Centre Climatique Canadien pour la modélisation et l'analyse (CC­

Cma).

Les caractéristiques de l'instabilité conditionnelle symétrique (CSI) ont d'abord

été étudiées dans le contexte d'un environement propice au dévelopement de

l'instabilité conditionnelle verticale (CUI). Ceci a permis d'identifier certaines car­

actéristiques communes au.x deux types d'instabilité. L'instabilité résultante, qual­

ifiée de convective-oblique, possède deux échelles de temps caractéristiques. De plus,

nous avons déterminé que les échelles horizontales typiques associées à cette instabilité

varient entre quelques dizaines et plus d'un millier de kilomètres.

Les données provenant des ré-analyses de l'ECMWF ont été utilisées pour calculer

la distribution de l'énergie convective potentielle disponible (CAPE) et de l'énergie

convective oblique potentielle (SCAPE) sur la planète. Nous avons demontré qu'en

général, le CSI et le CUI co-existent. En se basant sur une étude théorique ainsi que • iv sur l'analyse de données, un paramétrage de la convection oblique a été développé • et mis en application dans le GCMIL Nous avons demontré que ce paramétrage a comme effet le développement d'une circulation méridienne directe, laquelle représente

un impact significatif sur la circulation générale simulée. Une série de simulations

de cinq ans a démontré que ce paramétrage a pour effet de réduire, aux latitudes

moyennes, le SCAPE ainsi que son résidu, rs' Ceci a pour effet de diminuer la force

des courants-jets et du vent thermique. Par contre, la réduction du transport de

la chaleur et de la quantité de mouvement zonale, par les tourbillons meridionau..'"<

transitoires, constitue l'amélioration du climat simulé la plus significative. Ainsi,

avec ce nouveau schéma, le transport par les tourbillons est mieu..x modélisé tel que

demontré par une comparaison avec des observations.

• v •

Statement of originality

The original aspects of this thesis are as follows:

1) An investigation of the instability in a 2D baroclinic flow with an unstable

stratification using a two-layer parcel model.

2) An analysis of the global distribution of CAPE and SCAPE using ECMWF

re-analysis and the finding that the conditions for upright convection and slantwise

convection generally co-exist in the atmosphere.

3) Oevelopment of a 20 parameterization scheme for slantwise buoyant instability

(SBI).

4) Oevelopment of a simplified ID version of the slantwise convective parameter­

ization scheme and its implementation in version Il of the third generation CCCma

GCM.

5) Sensitivity tests and 5 year simulations using the scheme leading to the findings

that the scheme

a) reduces SCAPE and SCAPE residual over the mid-latitudes.

b) significantly weakens the poleward transient eddy transports of temperature

and zonal momentum over the mid- and higher latitudes. • vi c) decreases the zonally averaged temperature gradient and high level jets over • the midlatitudes. d) generates a direct secondary meridional circulation.

• vii •

Acknowledgments

First of al1, 1 sincerely thank my supervisor Professor M. K. Yau. His advice and

suggestions help me to choose the research project. Over the years in my doctor­

ate program, he has been giving me countless and continuous encouragement and

instruction. Without his support and guidance, 1 would never have completed my

thesis work.

1 would also like to thank Dr. N. A. ~IcFarlane ofthe Canadian Climate Centre for

modelling and analysis (CCCma) for kindly introducing to me the CCCma GCM and

allowing me to use their computing facilities. Discussion with him greatly encouraged

me to continue my reseach. 1 am indebted to ail the people at CCCma who gave me

help and advice during and after my visit to implement my scheme into the CCCma

GCM. Special appreciation goes to Richard Harvey, Fouad ~Iajaess, Dr. Damin Liu,

Dr. Jiangnan Li, and Dr. Jian Sheng.

1 will oever forget the lectures and instructions 1 received from professors in the

Department of Atmospheric and Oceanic Sciences. 1 would aIso like to thank my

office mates Dr. Ning Bao, Dr. Nagarajan Batrinath, Stephen Dery, Dr. Yubao Liu,

Dr. Zhaomin Wang, Dr. F. Kong, and Yongsheng Chen. 1 was inspired from the • viii many useful discussions with them and received valuable help in model programming, • figure plotting, and thesis typesetting. 1 also thank Dr. Paul Vaillancourt for the translation of the thesis abtract ioto

French.

The finandaI support from grants awarded to Professor lVI. K. Yau are gratefully

acknowledged.

Finally, 1 would like ta thank my family, my wife Ye.xu Li for her support, my

aider son Fan for his understanding, and my little son ~lichel for the joy he gives me,

although he knows nothing about what 1 have been doing.

• ix •

Contents

List of Tables xiii

List of Figures xiv

1 Introduction 1

1.1 Motivation. 1

1.2 Review of the theory for symmetric instability 5

1.2.1 Critenon for conditional symmetric instability . . .. . 5

1.2.2 Charactenstic scales 9

1.2.3 Other characteristics Il

2 Slantwise Buoyant Instability 13

2.1 A 2-dimensional two-Iayer Lagrangian model .. 13

2.2 Governing equations 14

2.3 Analytical solutions . 20

2.3.1 Solution for the lower layer (z < hd ... 20

2.3.2 Solution for the upper layer (z > hd 23 • x 2.4 Numerical results ...... 24

• 3 Climatology of CAPE and SCAPE from ECMWF Re-Analysis 34 3.1 Calculation of CAPE and SCAPE ...... 35

3.2 The data and the thermodynamic process ..... 39

3.3 The global distribution of CAPE and SCAPE 40

3.4 Distributions of CAPE/SCAPE and precipitation 43

4 A Generalized Moist Convective Adjustment Scheme 53

4.1 Formulation . 53

4.2 Test under idealized 2-D conditions 60

5 A One-Dimensional Parameterization Scheme for Slantwise Convec-

tion 68

5.1 Formulation of the 1-D scheme. . 69

5.2 Slantwise precipitation ...... 74

5.3 Latitude fil ter and slantwise shift ...... 78

5.4 Triggering function and relaxation time ... 79

6 The Response of the Large-scale Mean Circulation to the Slantwise Convective Parameterization Scheme 81

6.1 The CCC GCM ...... 81

6.2 The three-month experiments ...... 83

6.3 Large-scale response ta slantwise momentum transport: a mechanism

via a secondary mean meridional circulation ...... 91 • xi 7 The Effects ofthe Slantwise Convective Parameterization Scheme in • a 5-Year Climate Simulation 101 7.1 Effect on reducing SCAPE in midlatitudes ...... 102

7.2 Effect on decreasing the strength of the upper-level jets in midlatitudes

and the cold temperature bias ...... 110

7.3 Effect on weakening the meridional transient eddy transports. 116

7.4 Other effects...... 123

7.5 Discussion... 128

8 The Representativeness of the 5-year Mean and Some Sensitivity

Experiments 135

8.1 The interannual variability of the 5-year simulation 135

8.2 Sorne sensitivity experiments...... 143

9 Summary and Conclusions 153

References 156

A Detailed Solution for 2.24 and 2.35 173

• xii •

List of Tables

6.1 Parameters and features of the seps used for the three month experiments (also

for the 5 year simulations in later chapters)...... 84

8.1 Summary of sensitivity test: Disable thermodynamic feedhack (~)alW and

( ~) in the SCPS for DJF and JJA of year 2 simulations (SLW42E~I-DJF2 .Lw and SLW42EM-JJA2). The comparisons are made with respect ta runs SLW42E-

DJF2 and SLW42E-JJA2, respectively. SR and NH denote Southern Hemisphere

and Northem Hemisphere respectively. 149

8.2 Summary of sensitivity test: Tum off the latitudinal filter in the SCPS for DJF of

the year 2 simulation (SLW42EF-DJF2). The comparisons are made with respect

to run SLW42E-DJF2...... • .. 149

8.3 Summary of sensitivity test: Change the relaxation time T from 3 ms ta 9 ms in

the SCPS for DJF of the year 2 simulation (SLW42ES-DJF2). The comparisons

are made with respect ta run SLW42E-DJF2...... 152

• xili •

List of Figures

2.1 A schematics ofthe distribution ofabsolute momentum ]\t! and entropy 8 or neutral

buoyancy surface defined in (2.16) for classic slantwise convection (a) and slantwise

buoyant convection (b). The arrows at point B indicate the acceleration an air tube

initially located at point A would experience if it is being displaced toward point B. 18

2.2 Time series of tube trajectory for a tube initially located at point A in an environ­

ment unstable ta slantwise buoyant convection with solid line as absolute mamen­

turo m (in mis) and dashed line as potential temperature 9 (in K). (a) analytic

5 3 1 trajectory for basic state parameters Nr = Ni = lx 10- S-2 t v; = 2 X 10- S- , '1 =

4 10- S-l and hl = 2.0 km; (h) numerical trajectory for the same environment as in

(a); (c) same as in (h) e.xcept that Rayleigh friction is added; and (d) same as (c)

but for the parameters Nr = 1 x 10-6 S-2 , Ni = 1 x 10-5 S-2 , v: = 2 X 10-3 s-1 , '1 =

10-4 8-1 and hl = 5.0 km...... 26

• xiv 2.3 Sensitivity to variation in 10g(Nr) along abscissa (in log(I/82» for (a) slope of • displaced tube from A to B (in degree), (b) the horizontal distance of the equilib- rium point C (in km), (c) the dimensionless time for the inertial mode scaled by

1 (/,.,)-1/2, (d) the dimensionless time fClr the buoyancy mode scaled by Nï , and

(e) the ratio V~:~. The basic state parameters are the same as those used in :s " Figure 2.2a...... 28

2.4 Same as Figure 2.3 but for variations in 10g(Ni) on abscissa (units log(l/s2). 29

2.5 Same as Figure 2.3 but for variations in vertical shear v= (units 10-3S-1) on abscissa. 30

2.6 Same as Figure 2.3 but for variations in absolute vorticity Tf (units S-1) on abscissa. 31

3.1 Schematics ofcalculating SCAPE from a vertical sounding. Adopted from Emanuel

1983b...... 36

3.2 The global distribution of CAPE for seasons (a) DJF and (b) JJA. The plots are

averaged over 14 years based on EMCF\V re-analysis data of 1979-1993. Values

are in units of 100 J /kg with a contour interval of 500 J /kg. 44

3.3 Same as Figure 3.2 but for SCAPE...... 45

3.4 The global distribution of SCAPE residual Til or SCAPE - CAPE for seasons (a)

DJF and (b) JJA. The plots are averaged over 14 years based on EMCFW re-

analysis data of 1979-1993. Values are in units of 100 J/kg with a contour interval

of 200 J/kg...... 46

3.5 The global distribution of SCAPE residual fraction f. or SCA{cEJufIPE for sea­

sons (a) DJF and (b) JJA. The plots are averaged over 14 years based on EMCFW

re-analysis data of 1979-1993. Values are in percent with a contour interval of 25% 47 • xv 3.6 The global distribution of gridded precipitation for seasons (a) DJF and (b) JJA.

The plots are averaged over 16 years based on a dataset of 1979-1995 constructed

• 1 by Xie and Arkin (1997). The units are in 10- m/s with a contour interval of

2 x 10-7 rn/s...... 50

3.7 The global distribution of CIN for seasons (a) DJF and (b) JJA. The plots are

averaged over 14 years based on El\'ICFW re-analysis data of 1979-1993. Values

are in units of 100 J /kg with a contour interval of 100 J /kg...... 51

4.1 Schematic illustration of the adjustment process for (top panel) pure conditional

slantwise instability or CSI and (bottom panel) slantwise buoyant instability or

SBI. (a) of top panel shows the configuration of momentum surface m, neutral

buoyancy surface s, and the reference momemtum surface mR in the physical two­

dimensional space. (b) of top panel is the profile of momentum and the reference

momentum along neutral buoyancy surface s, denoted as mand fflR respectively.

The tilde is used to represent values along the neutral buoyancy surface. ln the

bottom panel, (a) illustrates the configuration of the momentum surface m, the

neutral buoyancy surface s, the reference momemtum surface mR, and the reference

neutral buoyancy surface SR in the physica1 2-D space. (h) is the vertical profile of

s and SR. (c) shows the distribution of momentum and the reference momentum

along the neutral buoyancy surface S in the case of SBI...... 57

• xvi 4.2 Initial fields of (a) absolute momentum (soUd, in mis) and potential temperature • (dashed, in K) , (h) absolute momentum (solid) and equivalent potential tem­ perature (dasbed, in K), (e) absolute momentum (solid) and saturated equivalent

potential temperature (dashed, in K), and (d) equivalent potential temperature

(dashed, in K) and specifie humidity (solid, g/kg). The units for horizontal dis-

tance (x) and vertical height (z) are km. 62

4.3 Initial profiles (solid) and their reference states (dashed) for column 13 (a) equiva­

lent potential temperature (in K), (b) potential temperature (in K). and (c) specific

humidity (in g/kg). Column 13 is located at 200 km along the abscissa. The vertical

coordinate is height in km...... 63

4.4 The adjusted fields for (a) potential temperature (dashed) and absolute momentum

m (saUd), (b) equivalent potential temperature (dashed) and m (solid), and (c)

saturated equivalent potential temperature (dashed) and m (saUd) after 3 hours of

integration. The neutral buoyancy surface for column 13 is schematically shown by

the thick dashed line in (b). The units and coordinates are the same as in Figure 4.2 65

4.5 The initial (salid) and reference (dashed) absolute momentum profiles (in mis)

along the neutral buoyancy surface shawn in Figure 4.4b. The vertical coardinate

is in km...... 66

• xvii 5.1 Schematic illustration of the I-D slantwise convective parameterization scheme • (SepS) for three possible scenarios: (panel a) SBI, (panel b) CSI, and (panel c) stable condition. ~Tvg is the temperature equivalence of wind shear correction

defined in (3.9). ATv~ is the same as ATug but for the reference state defined by

(5.4). ~T.lw is the temperature equivalence ofthe slantwise momentum adjustment

as defined in (5.5)...... 75

5.2 Horizontal scale L and column shift. Considering the span of slantwise convection

(from column 0 through column -2 in this case), the calculated tendencies are

shifted from original Column 0 to Column -1 for better rapresentation...... 76

6.1 Zonally averaged slantwise tendencies of (a) local momentum (%t )61w, and (b) it's u

component and (c) v component produced by SCPS in experiment SLW42E-DJFl.

Plots are in units of 1 x IO-sm s2 with pressure (hPa) as ordinate and latitude

(degree) as abscissa...... •.. .. 85

6.2 Zonally averaged distributions of precipitation (uuits mm/day) produced by (a)

the slantwise convective parameterization scheme and (b) the total precipitation

vs. latitudes for run SLW42E-DJFl. 86

7 1 6.3 Zonallyaveraged (a) moisture tendency (units 10- gkg-1S- ) and (b) temperature

tendency (units 10-7 K/s) produced by the SCPS in experiment SLW42E-DJFl.

Coordinates are pressure (ordinate in bPa) and latitude (abscissa in degree). 87

6.4 Comparison of zonally averaged (a) SCAPE (in J/kg) and (b) SCAPE residual

r. (J/kg) between runs SLW42E-DJFl (solid) and NOSLW-DJFl (dashed). (c)

depicts the initial fractional SCAPE residual f. in percent from NOSLW-DJF1.. 88 • xvili 6.5 Zonally averaged diff'erences of (a) u (in mIs), (h) T(in 0.1 K), (c) v ( in mIs), and • (d) w (in 10-2 Pals) between SLW42E-DJFI and NOSLW-DJFl. The ordinate and abscissa are pressure (in hPa) and latitude (in degree), respectively...... 90

6.6 Zonally averaged (a) (~).lw from the artificial (~).lw 1 day experiment SLW40G­

IDAY (in 10-6 ms-2 ), and (b) the difference fields u (in mIs) and (c) T (in 0.1 K)

between SLW40G-IDAY and NOSLW-IDAY with the same ordinate and abscissa

as in Figure 6.1 and Figure 6.5...... 93

6.7 Zonally averaged difference fields of (a) v (in mIs) and (b) w (n 10-2 Pals) between

SLW40G-IDAY and NOSLW-1DAY with the same ordinate and abscissa as in

Figure 6.5...... 94

6.8 Zonally averaged differences of (a) u (in 0.1 mIs). (b) T (in 0.01 K), (c) v (in 0.1

mIs), and (d) w (in 0.01 Pals) between SLW42EM-2STP (two time-step, momen­

tum adjustment ooly) and NOSLW-2STP. The ordinate and abscissa are the same

as in Figure 6.5...... 96

6.9 Zonallyaveraged momentum tendency (~~)~lw (in 10-6 m s-2) from SLW42EM-

2STP with the same ordinate and abscissa as in Figure 6.1...... 97

6.10 Zonally averaged diff'erences of (a) u (in mIs), (b) T (in 0.1 K), (c) v (n mIs),

and (d) w (in 0.01 Pals) between SLW42El\JI-IDAY ( momentum tendency ooly 1

day integration) and NOSLW-1DAY. The ordinate and abscissa are the same as in

Figure 6.5...... 98

6.11 The same as Figure 6.5 but between runs SLW42EM-DJFl (the first DJF season

momentum tendency only) and NOSLW-DJFl. 99 • xix 7.1 5-year means of zonally averaged (a) (~).hu (in units of 10-5 m 5-2) and (h) • (qjf).,w (in units of 10-7 K/s) for DJF from SLW42E-5YR. The ordinate is height in hPa and the abscissa is latitude in degree. 102

7.2 The same as Figure 7.1 but for JJA. 103

7.3 The mean global distribution of column averaged ahsolute local momentum ten-

dency (k ~ 1 ~ 18 Vi 1.'10) in units of mis per 3 hours, where kt and kb t- b+ ~Ic at 1= " represent the levels of the cloud top and cloud base, respectively, and 3 hour is

the relaxation time T used in our simulation) for (a) DJF and (b) JJA from run

SLW42E-5YR...... 104

7.4 Zonally averaged mean SCAPE residual r. for (a) DJF and (b) JJA from runs

SLW42E-5YR (soUd line) and NOSLW-5YR (dashed line). The units in ordinate

are J/kg. . . 105

7.5 The global distribution of the difference in SCAPE residual r:l (in J/kg) between

SLW42E-5YR and NOSLW-5YR for (a) DJF and (b) JJA...... 107

7.6 The global distribution of SCAPE (in J/kg) for DJF from (a) NOSLW-5YR and

(b) SLW42E-5YR...... IDS

7.7 The same as Figure 7.6 but for JJA. 109

7.S The 5 DJF Mean ofzona1ly averaged u component ofwind from (a) SLW42E-5YR,

(h) NOSLW-5YR, and (c) NCEP re-analysis. The ordinate is in hPa and wind in

rn/s . 111

7.9 The same as Figure 7.8 but for JJA Mean.. 112 • xx 7.10 Zonally averaged difference fields of (a) u (units mIs) and (b) T (units O.lK) in • DJF between SLW42E-5YR and NOSLW-5YR runs. Also the differences of (c) u (units mIs) and (d) T (units O.lK) in DJF but between NOSLW-5YR and NCEP

lo-year re-analysis. The ordinate is in hPa and latitude along the abscissa in degree.114

7.11 The same as Figure 7.10 but for JJA...... 115

7.12 l\'Iean zonally averaged differences of (a) v (units mis) and (h) w (units 0.01 Pals)

between SLW42E-5YR and NOSLW-SYR for DJF mean. The same differences of

(c) v and (d) w but for JJA...... •...... •...•. .. 117

7.13 The 5 DJF mean of zonally averaged transient eddy transports (a) u'v' (units

m2s-2 ) and (h) viT' (units mKs-L) from fun NOSLW42-SYR, and the diŒerences

of (c) u'v' and (d) v'T' between NOSLW-5YR and NCEP re-analysis. . 118

7.14 The same as Figure 7.13, but for the 5 JJA mean...... 119

7.15 The 5 DJF mean of zonally averaged transient eddy transports (a) u'v' (units

m2s-2 ) and (h) viT' (units Km S-2) from run SLW42E-5YR, and the differences

of (c) u'v' and (d) viT' between SLW42E-SYR and NOSLW-5YR. 121

7.16 The same as Figure 7.15, but for 5 JJA seasons...... 122

7.17 The global distribution of mean sea level pressure in DJF from (a) NOSLW-5YR

and (h) NCEP re-analysis. The UDits are hPa...... 124

7.18 The differences of mean sea level pressure in DJF (a) between NOSLW-SYR and

NCEP re-analysis and (h) between SLW42E-SYR and NOSLW-5YR. The units are

hPa...... 125

7.19 The same as Figure 7.17 but for JJA. 126 • xxi 7.20 The same as Figure 7.18 but for JJA...... 127 • 7.21 The global distribution of precipitation for DJF from (a) NOSLW-SYR and (b) NCEP re-analysis. The units are mm/day...... 129

7.22 The differences of precipitation for DJF Ca) between NOSLW-5YR and NCEP re­

analysis and (b) between SLW42E-5YR and NOSLW-5YR. The units are mm/day. 130

7.23 The same as Figure 7.21 but for JJA. 131

7.24 The same as Figure 7.22 but for JJA...... 132

8.1 Zonally averaged momentum tendency (~~).lw (in units of 10-5 ms-2) for (a) the

first DJF, (b) the second, (c) the third, (d) the fourth, (e) the fifth, and (f) the

5-year DJF mean from run SLW42E-5YR. The ordinate is in hPa and the latitudes

along the abscissa in degree...... 137

8.2 The same as Figure 8.1, but for seasons of JJA. 138

8.3 Zonallyaveraged u diff'erences (in mis) between SLW42E-SYR and NOSLW-5YR

for (a) the first DJF, (h) the second, (c) the third, (d) the fourth, (e) the fifth, and

(f) the 5-year DJF mean. The ordinate and the abscissa are in units of hPa and

degree. 139

8.4 The same as Figure 8.3, but for JJAs. . 140

8.5 Zonally averaged v'T' differences (units K m S-1) between SLW42E-5YR and

NOSLW-5YR for (a) the first DJF, (b) the second, (c) the third, (d) the fourth,

(e) the fifth, and (f) the 5-year DJF mean...... 141

8.6 The same as Figure 8.5, but for JJAs. ... 142 • xxii 8.7 Difference fields of zonally averaged (a) T (in 0.1 K), (b) u (in mis), (c) v (in

2 1 2 • mis), (d) w (in 10- hPa/s), (e) viT' (in Kms- ) and (f) u'v' (in m s-2) between SLW42E-DJF2 and NOSLW-DJF2. SLW42E-DJF2 and NOSLW-DJF2 denote re-

spectively the second DJF season in run SLW42E-5YR and NOSLW-5YR..... 144

8.8 Difference fields of zonally averaged (a) T (in 0.1 K), (b) u (in mis), (c) v (in

mis), (d) w (in 10-2 hPa/s), (e) 'v'T' (in Kms-1) and (f) u'v' (in m2 s-2) between

SLW42E-JJA2 and NOSLW-JJA2. SLW42E-JJA2 and NOSLW-JJA2 denote re-

spectively the second JJA season in ron SLW42E-5YR and NOSLW-5YR..... 145

8.9 Difference fields of zonally averaged (a) T (in 0.1 K), (b) u (in mis), (c) v (in

mis), (d) w (in 10-2 hPa/s), (e) viT' (in K ms-1) and (f) u'v' (in m2 S-2) between

SLW42EM-DJF2 and NOSLW-DJF2...... 147

8.10 Same as in Figure 8.9, but between test SLW42El\il-JJA2 and run NOSLW-JJA2. 148

8.11 Zonally averaged differences of (a) T (in O.lK), (h) u (in mis), (c) v ( in mis),

(d) w (iD 10-2 hPa/s), (e) v'T' (in Km s-I), and (f) u'v' (in m:.! s-2) between test

SLW42EF-DJF2 and run NOSLW-DJF2...... 150

8.12 Zonally averaged differences of (a) T (in O.lK), (b) u (in mis), (c) v ( in mis), (d)

w (in 10-2 hPa/s), (e) v'T' (in Km s-I), and (f) v'T' (in m2 S-2) between test

SLW42ES-DJF2 and ron NOSLW-DJF2...... 151

• xxiii •

Chapter 1

Introduction

1.1 Motivation

In atmospheric modeling, the atmosphere is described by discrete time and space

intervals. Physical processes on scales smaller than these intervals, the so called

subgrid-scale processes, cannot be resolved explicitly and are generally represented

as Reynolds 8uxes in the goveming equations. To close the set of equations, it is

necessary to fonnulate the collective effects of the subgrid-scale processes in terms of

the prognostic variables on the resolved scale. This is the problem ofparameterization.

In operational numerical weather prediction (NWP) and general circulation mod­

els (GCM), one of the important subgrid-scale processes is cumulus convection. Its

importance on the larger scale 80w through the release of latent heat and the ver­

tical transports of heat, moisture and momentum was first recognized in studies in

the tropics (Riehl and Ma1kus 1957; Ooyama 1964; Charney and Eliassen 1964). In • 1 mid-latitudes, the effect of cumulus convection on cyclogenesis was documented by • Tracton (1973), Gyakum (1983) and Anthes et al. (1983). Various parameterizations have been proposed and implemented in coarse-grid models. They include the moist

adjustment schemes (Manabe et al. 1965; Kurihara 1973; Betts and ~Iiller 1986), the

Kuo scheme (Kuo 1965, 1974; Anthes 1977), the Arakawa- Schubert scheme (Arakawa

and Schubert 1974), the mass flux scheme (Tiedtke 1989), and the Emanuel (1991)

scheme. Recently new schemes have been developed for mesoscale modeling, such

as the Fritsch-Chappell (Fritsch and Chappe111980) and the Kain-Fritsch (Kain and

Fritsch 1990) schemes. In spite ofrapid progress in representing convection, important

questions remained unanswered. One of which is the parameterization of mesoscale

convection in coarse-grid models.

For operational GC~Is and NWP models, the grid size is usually a few tens of

kilometers to a few hundred kilometers. However, many mesoscale circulation sys­

tems are in the mesO-,6 « 200 km) and meSO-i « 20 km) scales (Orlanski 1975).

Diagnostic studies ofconvective weather systems reveal large residuals in the momen­

tum and vorticity budgets, which cannat be explained fully by simple convective-scale

transports (Stevens and Shapiro 1977; Shapiro 1978; Cho et al. 1979; Reeves et al.

1979). It appears that at least some of the apparent sourcefsink terms result from

the effects of mesoscale convective circulations.

Existing cumulus parameterizations do not deal with mesoscale convection because

they focus on convection on a scale smaller than 10 km (Ooyama 1982). Although

sorne studies seem to suggest that moist convective processes, including those on the • 2 mesoscale, are basically parameterizable (Arakawa and Chen 1987; Chen 1989; Xu • 1991), schemes suitable for the representation of large, quasi-balanced circulations are still in a state of evolution due largely to uncertainties on the nature and importance

of mesoscale circulations (Frank 1983).

It is weil known that the traditional conditional instability (hereafter referred to

as Conditional Upright Instability or CtH) plays a fundamental role in theoretical

studies and the parameterization of convective-scale phenomena. On the mesoscale,

there are mounting evidence to suggest that conditional symmetric instability or 1 conditional slantwise instability (CSI) is responsible for sorne rnesoscale features like

rainbands. The theoretical basis for CSI is given in Bennetts and Hoskins (1979) and

Emanuel (1979). Studies on the importance of csr in the formation of rainbands and

snowbands, explosive marine cyclogenesis, and mid-latitude squalllines can he found

in Wolfsberg et al. (1986), Parsons and Hobbs (1983), Bennetts and Sharp (1982),

Bennetts and Ryder (1984), Seltzer et al. (1985), Gyakum (1987), Reuter and Yau

(1990; 1993), Balasubramanian and Yau (1994a, b, and 1995), and Zhang and Cho

(1992). Emanuel (1983a) showed from a Lagrangian parcel model that the unstable

displacement for CSI is along an isentropic surface with a typicallength scale of about

100 km. The structure, evolution and energetics of CSI from numerical simulations

with idealized initial conditions are reported in Ducrocq (1993), Thorpe and Rotunno

(1989), Huang (1991) and Innocentini and Neto (1992). The relation between CSI

and baroclinic instability ( Miller and Antar 1986; Emanuel and Thorpe 1987) and

convective instability (Seman 1994) have been investigated. Progress has also been • 3 made to understand the three-dimensional nature of the classical two-dimensional • symmetric instability problem using semi-geostrophic theory (Shutts and CuHen 1987) and non-linear simulations (Jones and Thorpe 1992).

In short, increasing attention is being directed toward CSI because it is a lead­

ing candidate for sorne mesoscale cloud and precipitation bands and because it plays

an important role in larger scale systems like extratropical cyclones and midlatitude

squall tines. However, CSI or slantwise convection is distinctIy mesoscale and is Dot

resolved in current coarse-grid models like GCl'.[s. At present, only a few param­

eterizations for slantwise convection exist, inciuding the ooes by Nordeng ( 1987,

1993), Chou and Thorpe (1993 for dry CSI), and Balasubramanian and Yau (1993a

for a two layer mode!). The latter two redistribute the absolute momentum along

a neutral buoyancy surface using a mass flu." approach while the former employs a

Kuo-type approach to adjust the thermodynamic field along an absolute momentum

surface. Only Nordeng's scheme was actually implemented in a mesoscale model for

real case studies. However, the adjustment of the thermodynamic field along an ab­

solute momentum surface implies mainly a thermodynamic eddy transport, which is

not consistent with the theoretical study of Emanuel (1983a) which suggests that in

CS!, the momentum eddy transport plays a predominant role.

The purpose of this thesis is to parameterize slantwise convection in a general cir­

culation model. Alter the description of the motivation of the study, we will review

relevant literature on the theory of slantwise convection in this chapter. Chapter

2 presents the analytic solutions of a two-dimensional two-Iayer Lagrangian parcel • 4 model and introduces the concept of slantwise buoyant instability (SBI). In Chapter • 3, the global climatology of convective available potentiaI energy (CAPE) and slant­ wise convective available potential energy (SCAPE) are computed from 15 years of

ECMWF re-analysis data. Based on the results of the theoretical study and data

analysis, a generalized 2-0 moist convective parameterization scheme and its simpli­

fied 1-0 version are developed and tested respectively in Chapters 4 and Chapter 5.

Then the 1-0 scheme is implemented into the Canadian Climate Centre for modelling

and analysis (CCCma) GCM. Five year simulations as weIl as various shorter-term

experiments were performed and the results are presented in Chapters 6, 7, and 8.

Chapter 9 presents the summary and conclusions.

1.2 Review of the theory for symmetric instability

Ta represent the collective effect of CS! on the larger-scale circulation, it is important

to understand the dynamicaI and thermodynamical properties ofslantwise convection.

Here, we shall review the instability criterioD, the characteristic time and length

scales, the unstable circulation and the structure of the eddy flu.xes.

1.2.1 Criterion for conditional symmetric instability

The concept of CS! represents an extension of dry symmetric instability (SI) which is

a type of inertial instability arising from an unstable balance of the pressure gradient

and inertial (e.g. centrifugai or Coriolis) forces. SI was first investigated by Rayleigh

(1916) for a homogeneous, incompressible and inviscid circular vortex. Solberg (1933) • 5 extended the analysis to an inviscid, Boussinesq circular vortex with constant vertical • shear and static stability. The stability criterion for axisymmetric disturbances is that the square of the angular momentum of the basic circular flow increases with

radius in the horizontal direction for a homogeneous flow or increases with radius

along isentropic surfaces in the presence of baroclinicity. A rigorous derivation of the

instability criterion on the basis of the variational method is given by Fj0rtoft (1950).

He obtained a symmetric stability tensor from a certain quadratic fonn related to

the second variation of an effective potential energy for two-dimensional meridional

motion. If the stability tensor is positive definite, then the second variation of energy

is positive. Thus the equilibrium balance becomes a minimum energy state and is

stable.

For a simplified case in which the curvature effects of the circular motion are ne-

glected and the coefficients in the perturbation equation are assumed to be constants,

the stability tensor may be approximately given by a matrbc

(1.1)

where

F2 = 2nsin .p(2n sin.p + ~: ),

B 2 = ! 89 = 20 . ,1. av 9 ex Slntp 8z '

and

• 6 Here the domain is considered to have local Cartesian coordinates with the x- • axis directed horizontally southward and the z-axis vertically upward. The symbol il denotes the mean absolute angular velocity of the vortex. V is the relative zonal

velocity and ~ the mean latitude of the domaine

The two diagonal elements F2 and fV2 represent measures for pure inertial stability

and static stability respectively. The two identical off-diagonal elements B2 give

a measure of baroclinicity. A suflicient condition for the stability of a baroc1inic

vortex with respect to axially symmetric disturbances is that both the trace and the

determinant of the tensor are positive everywhere in the flow.

A further analysis by Ooyama (1966) indicated that the above sufficient condition

is also a necessary condition. Thus the necessary and sufficient condition for SI is

that the determinant or the trace (or possibly both) of the stability tensor Q becomes

negative in at least some part of the vortex. An interesting interpretation of the

eriterion can he obtained by noting that

av~ q = F2N2 _ B4 = yI [(1 + av) aB _ aB] , (1.2) B 8x az az ax

is proportional to the Ertel potential vorticity of the basic fiow (Chamey 1973) with

the Coriolis parameter 1 given as

1 = 2ilsin cP.

Therefore, if negative potential vorticity occurs in even a part of the flow, it is a

suflicient condition for symmetric baroclinic instability. It should be mentioned that

F2/B 2 is the slope of the constant absolute momentum surfaces 1\1/, defined by M = • 7 / x + V. The ratio B2/ N2 is the slope of the constant potential temperature surfaces • in the basic f1ow. Thus if N2 and F2 are positive, q < 0 implies that the potential temperature surfaces are steeper than that of the absolute momentum surfaces and

the ftow is unstable. Altematively the instability criterion can be written in terms of

the Richardson number as B.ï < J, where ''1 is the absolute vorticity and Tl 9 il! R; = (J (t~r (1.3)

If the horizontal shear (or relative vorticity) of the basic flow is neglected relative to

/, we recover the instability eriterion ~ < 1 given by Stone (1966).

In the simplified case, perturbations do not have to be perfectly axially symmetric

to realize symmetric instability. Ifthe dimension of the domain under consideration is

much smaller than the average distance from the rotation axis, curvature effects can

be neglected and the axially symmetric motion becomes unidirectional. Displacement

of a circular ring then becomes the displacement of a two-dimensional tube elongated

in the direction of the flow (or the direction of the thermal wind). The disturbance

is still symmetric because it is independent of one directional coordinate just as an

axi-symmetric disturbance is iodependent of the azimuthal coordinate.

The criterion for dry symmetric instability is rarely satisfied 00 a scale of 100

km in the real atmosphere (Sawyer 1949; Bennetts and Hoskins 1979). Studies are

therefore directed toward the inclusion of latent heat release in SI. Theoretical results

show that, in a saturated atmosphere, (1.1) is replaced by a moist stability tensor

(1.4) • 8 where N2 = iL 88e e 8 8z • 0 and (Je is the equivalent potential temperature (or wet-bulb potential temperature

used in sorne papers). The necessary condition for CSI is that the equivalent Ertel

potential vorticity qe, defined by qe = N; F2 - B4, is negative but the dry potential

vorticity is positive ( q > 0 ) (Bennetts and Hoskins 1979; Xu 1986). Equivalently, the

constant M surfaces must be more vertical than (J surfaces but less vertical than (Je

surfaces. In terms of the moist and dry Richardson numbers, the instability criterion

is given by ~ f R;. = ~ (~)2 < 1/' (1.5)

with the dry Richardson number Hï > f. It should be mentioned that the stability 11 tensor (1.4) is obtained by assuming that the basic moisture field is horizontally

uniform at least within each moist updraft region (Xu 1986).

1.2.2 Characteristic seales

From a normal mode analysis based on an Eady liked basic state without horizontal

wind shear, Stone (1966) showed that the largest length scale for SI is on the order of

1 U (1-RY f2 • With llï less than unity, the maximum value of this scale is under 100 km.

The characteristic horizontal scale can also be estimated by the ratio of the depth of

the unstable domain and the slope of the isentropic surfaces. For typical atmospheric

conditions, one obtains a length scale between lOB and 100H, where H is the depth

of the unstable layer (Emanuel 1979, 1983a). By changing the basic state parameters • 9 and the viscosity in a linear model, Xu (1986) showed that the preferred CSI mode • can vary from the small scale ta the subsynoptic scale (or from meso-'Y scale ta meso-a scale).

Like the length scale, the time scale for CSI also spans a considerable range.

Slantwise convection in the context of a Lagrangian parcel model suggests an inertial

e- folding time scale proportional ta (!Vz )-l/2, which amounts ta under 1 hour at mid­

latitudes and is generally larger at low latitudes where the magnitudes of the Coriolis

parameter ! and the wind shear li; are on the average smaller. Linear analysis by Xu

(1986) revealed that the time scale for CSI is a function of H, J.ty2, N;, B2, and F2 as

weIl as the horizontal time scale L. By defining To as the time scale (NF)- L/2 with a typical value around 0.3 h, Xu (1986) concluded that

1) For small scale CSI (L < O.5H), the time scale T varies from 0.5To to 3.0Ta, and it is very sensitive ta variations in N;, especially when N; < o.

2) For large-scale CSI (L > 5.0H) t T is around 5.0Ta "J 10.0TOt and it depends on

the inertial stability pl.

3) For medium-scale CSI (0.5H < L < 5.0H), T is in a range somewhere between

case (i) and case (H), and it depends strongly on baroclinicity.

Except at the low end of case (i), which is the situation for conditional upright

instability, T is on the order of 1 "J 3 hours. Simple numerical experiments by

Bennetts and Roskins (1979) indicated an approximate e-folding time scale from 3.7

"J 5.7 hauts, while their corresponding theoretical analysis suggested an e-folding • 10 time of 2.8 "-J 3.7 hours. Although the complete life cycle of SI/CSI obtained by • more sophisticated numerical experiments ( Thorpe and Rotunno 1989; Huang 1991; Innocentini and Neto 1992; Ducrocq 1993; Persson and Wamer 1991) seem to yield

longer times, limited observational evidence tends to support a time scale of less

than 3 hours. In an analysis of precipitation bands observed during the Canadian

Atlantic Storms Program, Reuter and Yau (1990) show that the condition of slantwise

instability can be removed by slantwise convection within three hours.

1.2.3 Other characteristics

Despite the haroclinic basic state, SI/CSI develops at the expense of the mean kinetic

energy of the unperturbed zonal flow. In contrast, conventional baroclinic instability

draws its energy from the mean available potential energy of the unperturbecl zonal

flow. This property of energy transfer in SI/CSI is associated with the statically sta­

ble condition assumed in most SI/CSI studies. The unstable circulations are nearly

aligned along isentropic surfaces for SI or along saturated equivalent potential tem­

perature surfaces for CS!. Evidence for the alignment can be found in normal mode

analysis (Stone 1966), linear variational analysis (Emanuel 1979), Lagrangian parcel

dynamics analysis (Emanuel 1983a) and numerical simulations (Thorpe and Rotunno

1989; Huang 1991; Ducrocq 1993). The implication is that slantwise convection in a

statically stable atmosphere must transport momentum clown gradient along neutral

buoyancy surfaces, while the transport ofavailable potential energy may be negligible. • Il In closing this chapter, we mention that most studies of SI/CSI assume the condi­

tion for pure slantwise convection where the initial state is statically stable in the ver­

• 2 2 tical (N > 0 or Ni > 0 ifsaturated) and inertially stable in the horizontal (F > 0).

Because the requirement of positive definiteness of the stability tensor breaks down

once N2 < 0 (or Ni < 0), the behaviour ofslantwise convection in a background Bow

unstable to upright convection has been examined only by a few authors. As men­

tioned earlier, Xu (1986) predicted a meso-"Y horizontal scale but a nearly convective

time scale for CSI with J.V; < o. Analytical analysis and numerical experiments by

Seman (1994) show that in a convectively unstable environment, slant,vise convective

updrafts are more vertical than that found in conventional CSI theory. In addition

the slantwise updrafts are no longer aligned along neutral buoyancy surfaces. Large

vertical heat fluxes occurred in Seman's numerical simulation, in contrast to classical

csr where the heat Buxes are negligible. \Vhile the horizontal momentum transports

occurred on an inertial time scale, the vertical flu..xes are governed by a convective

time scale.

It is clear from the above discussion that the picture for classical slantwise convec­

tion can be considerably modified in an atmosphere conditionally unstable to upright

convection. In the next chapter, we will construct a 2-D two-layer Lagrangian parcel

model to explore the effect of the presence of CUI on CS!. That is, we will extend

CUI ta a 2-D flow in the presence of.strong vertical wind shear.

12 • •

Chapter 2

Slantwise Buoyant Instability

2.1 A 2-dimensional two-Iayer Lagrangian model

In studies of the classical CUI, a horizontally homogeneous atmosphere is generally

assumed. This one-dimensional assumption is justifiable because the time and length

scales ofCUI is much smaller than those of the slowly evolved large-scale environment.

To investigate CSI in the context of CUI, the one-dimensional treatment has to be

extended to two dimensions with the addition of one more degree of freedom. We will

therefore explore a combination of upright and slantwise instabilities, and we will call

it slantwise buoyant convection or slantwise buoyant instability (SBI). It should be

noted that the criteria for CSI and CUI are not always separable, since the condition

for CUI (N; < 0) is also the condition for symmetric instability as the symmetric

stability tensor given in (lA) is no longer positive definite when N; < O. • 13 We will study this problem using a Lagrangian parcel model in a manner follow- • ing Emanuel (1983a; 1994). The results would he useful in developing a generalized convective adjustment scheme including both upright and slantwise convective insta-

bilities in later chapters.

2.2 Governing equations

We start with the inviscid equations of motion on an 1 plane with the forro

du ap dt = -oax + Iv, (2.1) dv éJp dt = -afJy - lu, (2.2) dw éJp di = -aaz -g, (2.3)

where 11., 'V, and w are the eastward, northward, and upward components of velocity,

a is the specifie volume and f is the Coriolis parameter. Let us consider a flow so

that tbere exists a direction, called y', such that the pressure gradient force does not

vary with height. Then u~, the geostrophic wind in the corresponding x' direction, is

also constant. We now rotate the coordinate system about the z axis from (x, y, z) to

(x', y', Zl) and then translate the rotated coordinate system (x', y', Zl) with a constant

speed u g in the x' direction. In this new reference frame (x", y", Zl') , we have

x" = x' - u~t, y" = y', z" = z' (2.4)

u" = 11.' - u~, v" = v', w" = W' (2.5) • 14 The equations of motion (2.1) to (2.3) become

du" ôp (2.6) • dt = -0ôx" + f v," d ~" -fu" (2.7) dt = ' dw" ôp ~ = -oôz" - 9, (2.8)

From the thermal wind relation, u~ = constant (or u; = 0 in the (x", y", z") sys­

tem) implies that the component of thermal wind in the x' or x" directions is zero.

Hence the horizontal isotherms must lie along the yi or y" direction, and the x' or

x" axes lie along the temperature gradient, pointing toward warmer air. For conve-

nience, we drop the double prime notation with the understanding that henceforth

our coordinate system will be (31' , y", z") .

By noting that 'U = dx/dt, (2.7) can be written as

diV! = 0 where AI 'U + fx. (2.9) dt ' =

(2.9) is a statement of the conservation of absolute momentum N! in an inviscid flow

on an f plane where the thermal wind is unidirectional. Now we assume a symmetric

disturbance independent of y being displaced in the x direction. By neglecting the

effects of the displacement on the basic state flow, the goveming equation for the

displaced tube of air in the x direction becomes

du ôp dt = -Oax +fv

=f(v-vg ) (2.10)

= f(Mt - lvIg ), • 15 where Mt is the tube's absolute momentum, and Mg = vg + f x is the geostrophic • absolute momentum. (Hereafter, we will use subscript t to represent the tube and q the geostrophically balanced environment.) The vertical motion of the tube is govemed

by

dw 0t -Qg (2.11) dt = 9 Qg ,

where 0t is the specifie volume of the tube and Og is that of its environment. From

the ideal gas law and the definition of virtual potential temperature, (2.11) may be

written as

dw dt (2.12)

The last relation is obtained by assuming that the pressure of the tube adjusts in-

stantaneously to that of its environment, a reasonable assumption provided that the

velocity of displacement is much less than the speed of sound.

Following Emanuel (1994), (2.11) can also be written in tenns of a conserved

variable as follows. Assume that 0t - Og is small relative to Og and that 0t - Og

undergoes an isobaric process, then 0t - Og can be expanded as function of entropy S

(Ot - Qg)p ::= (~:) (St - Sg) + O(other), (2.13) p,other

where S is entropy and other denotes other dependent variables such as the water

vapor content. Neglecting second order terms and applying one of ~Iaxwell's thermo-

dynamic relations, (2.13) becomes

(a, - ( 9 )p ~ (':'). (s, - 59)· (2.14) • 16 Using the hydrostatic approximation, (2.14) becomes dw - ~ r(St - s ) (2.15) • dt 9 , where

dry adiabatic lapse rate when the environment is unsaturated,

moist adiabatic lapse rate when the environment is saturated,

Cp ln 8 for an unsaturated tube, , (2.16) ( S = Cp ln 8;, for a saturated tube, and 8; is the saturated equivalent potential temperature. Since 8 is conserved in a

dry adiabatic displacement and 8; is conserved in a moist adiabatic displacement,

St is always conserved following the displacement of the tube, provided that the

displacement is reversible.

Let us assume a typical environment with the saturated equivalent potential tem-

perature decreasing with height (Figure 2.1). For a saturated tube originating at

point A, any displacement in region B between the constant NI and 8; surfaces will

produce an upward acceleration and a horizontal acceleration in the -x direction if

vertical wind shear is present (see (2.10) and (2.15)). Convection is therefore slant-

wise, except in the case where the NI surfaces are nearly vertical or the environment

barotropic. When the vertical wiDd shear is strong, the slope of the trajectory of the

displaced tube is more horizontal, but in this case the c1assical condition for esr that

the M surfaces be less vertical than the equivalent potential temperature surfaces is

not required. This kind of convection will he known as slantwise huoyant instahility

(SBI). • 17 •

Classic Slantwise Convection

a

Slantwise Buoyancy Convection

b

Figure 2.1: A schematics of the distribution of absolute momentum M and entropy s or neutral buoyancy surface defined in (2.16) for classic slantwise convection (a) and slantwise buoyant con­ vection Cb). The arrows at point B indicate the acce1eration an air tube initial1y located at point A would experience ifit is being displaced toward point B.

• 18 To understand the dynamical behaviour ofSBI, we consider a two-layer dry atmo- • sphere \Vith constant vertical and horizontal wind shears. The lower layer is statically unstable with a constant Brunt-VaisalHi frequency -lYf. The upper layer is stable

\Vith Brunt-VâisâlUi frequency lVi. Here lVl and lV2 are both positive. The absolute

momentum J."v[g and the entropy 8 g of the basic flow are given by

1.\1[9 = VzZ + Tjx, (2.17)

C ~T·) - 80 - ~lvt-Z S:z;X, g + 8 9 = (2.18) ( 8t + :J?.lV;z + s:z;X, 9 -

where hl is the height separating the two layers and So is the value of 8 g at x = o. z = o.

St can he written in terms of other parameters as

(2.19)

For this stationary basic state, aH parameters in (2.17) and (2.18) are considered

constants, and the thermal \vind relation is

(2.20)

At the original point (x = 0, Z = 0),

St - 8 g (0,0) = SO,

and they are conserved variables in the symmetric system. Substituting (2.17) and

(2.18) into the tube equations (2.10) and (2.15) and using the thermal wind relation • 19 (2.20), we obtain the governing equations for a tube initially located at x = z = 0

du • dt = - f(vzz + Tjx), (2.21) z - fvzx, dw {Nr (2.22) di = (Nf + fq)h L - Niz - fvzx,

We can eliminate the variable z from (2.21) and (2.22) by noting

du cPx dw cPz dt = dt2 ' di = dt2 '

Thus we obtain the following homogeneous linear differential equations with con-

stant coefficients

(2.23)

and

(z > hd. (2.24)

2.3 Analytical solutions

2.3.1 Solution for the lower layer (z < hl)

..I\ssume that initially a tube of air is displaced from x = 0, z = 0 to x = Xo, z = Zo.

Solving (2.23) yields the position of x at time t below hl as

(2.25) • 20 • where

and 0'1. and 0'2 are the eigenvalues

(72 -(fr; - Nf) +----:V -I(Nf + fr;)2 + 4f2Vz_ 2 1. - 2 (2.26) 2 (fr; - Nf) + J~(m-1.-+-fr;-)-2+-4f-2-v-::2 (72 = 2 . (2.27)

Substituting x into (2.21), z can easily be obtained by rewriting (2.21) as

~ + fr;x Z=-~--- (2.28) fv: (2.29)

For an unstable displacement, the right hand side of (2.26) must be positive or

Nf f ->---2 _t (2.30) V;: 1]

which is always satisfied for 1] > o.

In the case of unstable motion, the situation is similar to that found in classical

SI, the tube Dot only accelerates upward but also horizontally. Since A2 ,...., ~, and

3 2 the magnitude of which is 10- - 10- , the disturbed tube will experience a slanted

trajectory toward the negative x axis as long as the slope of the displacement is not • 21 less than A2 • The cosine terms in (2.25) and (2.29) imply that the tube will oscillate • during its exponential growth. For sufficiently long time, the asymptotic siope can be obtained as

a = Hm ( ~) ~ -Al t-..inf X

2 = - 2;V: ([fi + N~ + ..;(Nf + f'fj)2 + 4j2v: )• (2.31)

Nonnally we have

and aiso

thus

(2.32)

The asymptotic slope is directed toward the two-dimensional vector with compo-

nents (1, a). The slope of the neutral buoyancy surfaces in this case is

or (l, -lia) in vector fonn. It is obvious that the asymptotic slope of the trajectory

of the tube is perpendicular to that of the surfaces of neutral buoyancy. We should

aiso remark that the limiting slopes of w and u aIso approach -Al, meaning that the

asymptotic velocity is aIso perpendicular to the surfaces of neutral buoyancy.

For the case of a stable stratification, the buoyant frequency in the lower layer

becomes Nf instead of -Nf. The characteristic frequencies in (2.26) and (2.27) • 22 become

<7~ = -(Jfj+ Nf) + .j(/fi- Nf)2 + 4J2V;2, (2.33) • 2 _2 _ (Ifi + Nf) + J(/fi - Nf)2 + 4/2v;2 02 - 2 ' (2.34)

where 0'1 bas positive real root only when ur > 0, or N2 1 Rï = _1 <-. v;'l fi

\Ve therefore recover the classical criterion for SI.

2.3.2 Solution for the upper layer (z > hl)

For z > hl , (2.24) can be rewritten as a homogeneous equation

(2.35)

under the following coordinate translation

I(Nf + 1V:;)V;h 1 Xl = X + 1:\.12- 12- 2 ' (2.36) i.Vi'TJ - V:z:

The detailed solution for (2.35) is given in Appendix A. Here we ooly show the

eigenvalues of (2.35), which are of the form

2 -(Ir; + Ni) + J(/fi - l'ViF + 4f2vz? .À 1 = 2 ' (2.37) (fTi _/(/fi - Ni)2 4f2 ;:2 \2 _ + Ni) + V + v ""2 - 2 (2.38)

The displacement of the tube is still unstable if .ÀI > 0 when B.ï < f /Tl , where B.ï is defined by using Ni. Otherwise, the tube will oscillate vertically and horizontally • 23 around its equilibrium point defined by

- f(vzz + Tjx) - 0, • (2.39) (Nf + Ni)ht - Ni - Jv::x - 0,

at

(Nf + Ni)hl 11; x= (2.40) Tf - JF;l and

(Nf + Ni)h1Ti (2.41) Z = TiNi - fvr? . In the latter case, the upper layer acts as a cap to the unstable lower layer and

prevents the lifted tube from running away indefinitely.

In the next section, we will present some quantitative numerical results under

typical atmospheric conditions.

2.4 Numerical results

Figure 2.2a shows the analytic trajectory for an idealized atmosphere with Nf ­

Ni = 1 X 10-58-2 ,v:: = 2 x 10-38-1,77 = 10-48-1, and hl = 2.0 km. Point A

represents the initial position of the air tube. \Vhen the tube reaches point B the

potential temperature of the tube is the same as that of the environment. Point B

therefore represents an equilibrium point in the traditional theory of upright buoyant

instability. However, the trajectory ofthe air sample does Dot end at point B. Instead,

the tube is heading toward point C. At point C (given by (2.40) and (2.41)), both

the absolute momentum and the potential temperature of the tube are the same as • 24 those of its environment. Thus point C is an equilibrium point for slantwise buoyant • instability. Without friction, the tube does not stop at C and will continue to reach point D before it starts to move back toward the equilibrium point.

An inspection of Figure 2.2a indicates that the characteristics of the trajectory

from A to B is different from those from B to C or D. The trajectory from A to

B is more or less vertical and the surfaces of neutral buoyancy are quasi-horizontal

(note that the vertical scale is much stretched relative to the horizontal scale in these

figures). The slope from A to B is -50.07 or 88.9° from the negative x axis, and the

slope of the entropy surface s below z = hl is 0.02 or Ll° degree from the positive

x direction. This result is consistent with our asymptotic analysis presented earlier

where the trajectory from A to B is shawn to be perpendicular to the surfaces of

neutral buoyancy in the lower layer.

The trajectory from B to C or D represents a complicated oscillation around

the equilibrium point C. If we ignore the high frequency oscillation in the vertical

direction, then it is clear that the tube is displaced in a slantwise direction along

the neutral buoyancy surfaces at the upper layer. The tube overshoots point C and

reacbes point 0 before it reverses its direction and moves towards B. It should also

be pointed out that the trajectory from A to B and from B to C are characterized

by two different time scales. From A to B, the e-folding time is arouod ~l which cao

be obtained from (2.27) by noting that the parameter J.V'f dominates over the other

terms for typical atmospheric conditions. The characteristic time for motion from B

to C is IÀ11-l (see Appendix A and the discussion on Figure 2.3e to 2.6e below), which • 25 l''4. '1. '2.

Il. t. 1. ..7 • ~. & J. 2.

-.... -'... 1•.' JN.' ,...

Figure 2.2: Time series oftube trajectory for a tube initially located at point A in an environment unstable to slantwise buoyant convection with solid line as absolute momentum m (in mIs) and dashed line as potential temperature 9 (in K). (a) analytic trajectory for basic state parameters N'f = Ni = 1X 10-5S-2 , v: = 2x10-3S-1 , 11 = 10-4 S-1 and hl = 2.0 km; (h) numerical trajectory for the sameenvironment as in (a); (c) same as in (h) except that Rayleighfriction is added; and (d) same as Cc) but for the parameters N'f = 1 X 10-6 S-2, Ni = 1 x 10-5 S-2.v: = 2 X 10-3 S-l. 11 = 10-4 $-1 and hl = 5.0 km. • 26 is related ta the inertial time scale (fTI) -1/2. Thus we may characterize the motion • from A to B as the buoyancy mode and motion from B to C the inertial mode. Then SBr can be considered as a combination of these two modes.

The tube in Figure 2.2a will osciUate around C forever because no frictional forces

are considered. To relax this assumption, we calculated sorne tube trajectories by

including Rayleigh friction. Adding friction however renders the governing differential

equations 4-th arder, and analytic solutions were not possible. We therefore resort to

numerical solutions. The results of these calculations are plotted in Figure 2.2c and

2.2d. Figure 2.2c shows that with proper friction or mLxing, the oscillation decays

and the tube eventually stops close to the equilibrium point C. In Figure 2.2d, the

amplitude of the oscillation becomes smaller as the magnitude of Nf decreases. It

should be pointed out that the numerical solutions are quite accurate as indicated

by a comparison of the analytic and numerical solutions under the same conditions

displayed in Fig. 2.2a and Fig. 2.2b.

To determine how the trajectory would change with variations in environmental

conditions, sensitivity tests were performed and the results are displayed in Figures 2.3

through 2.6. In these figures, the parameters varied along the abscissa are NI (Figure

2.3), N2 (Figure 2.4), 'U: (Figure 2.5) and TJ (Figure 2.6). The vertical coordinates in

the different panels are the slope of the trajectory from A ta B (panel a), the location

of the equilibrium point (panel c), a measure of the time scale in the upper layer « (f~:/2 ,panel c), a measure of the time scale in the lower layer (~, panel d), and the ratio {3 =J~:4 (panel e, note that a large value of /3 means that for the upper • 27 • e;ra.3

87.4

(a) 84.5

81 .6

7B.7

416.7

326.7

(b) 236.7

t46.7

'5b.7 1.36 ["".. 'o. ,... 1.32

(c) 1.2e;

, .26

, .23 1

,. Bill

I.BIlI

(d) IlI.qq

IlI.qq

1ll.Q8

5'576.

45e;1lI.

(e) J6"4.

26t8.

1632. -b.llIe; -5.11 -S.34 -4.C;b -4.SC; -4.21

Figure 2.3: Sensitivity ta variation in log(N;) along abscissa (in logel/82»for (a) slopeofdisplaced tube from A ta B (in degree), Cb) the horizontal distance of the equilibrium point C (in km), (c) the dimensionless time for the inertial mode scaled by (/'1)-1/2, (d) the dimensionless time for the

1 buoyancy mode scaled by Nï t and Ce) the ratio ~:~. The basic state parameters are the same J3 4 as those used in Figure 2.2a. • 28 •

Q3.4 .Ia • '" Section 1 t4. r •• ~

Q1.1

8b.6

94.3

137.9

113.7

(b) 8Q.6

05.5

41.4

1.31

1.23

(c) 1. lb

1.08

1.0' 1.05 ~." ..._..., 1 1."3 -l

1 (d) 1.1lI"

0.Q7 ,L....- -l.- --l.. L....- j 1ll.QS 35b21. f" 'd.,....,."l" .,.,~•." 2689b.

(e) 18171 . 9446. ~ 721. 1 -5.a5 -4.83 -4.61

Figure 2.4: Same as Figure 2.3 but for variations in log(Ni) on abscissa (units log(1/s2).

• 29 •

8q.S

BQ.2

(a) 88.Q

88.S

88.2

1258.Q -a 0'

q4".~

(b) 622.2

3"3.Q

-14.S

3.27 •.••1. tar- .""1

2.bQ

(c) 2.11

1.53

a.qS

1.'15 •.•••• 'a.- 6. hl

1 _"3 (d) 1 . "" 0.Q7

0.qS

• ,.tlet ·:I.e2·2l'. ,.t le]·2• .,.·21 1468".

1124" .

(e) 7B"". 435Q.

q,eL 0.Qe 1.34 1.78 2.22 2.6b 3.10

3 1 Figure 2.5: Same as Figure 2.3 but for variations in vertical shear v= (unit! 10- S- ) on abscissa.

• 30 •

QJ.4 [OP. ", S.CtIOft Il Id.e.....

Ql.l (a) 8B.Q ~ 86.6 ~ 84.3 1

207.7

lb6.3

(b) 125.0

83.7

42.4

•. , ••1. 'a~ L,Io'" 1.43 i 1.35 j (c) 1.27 ~ 1.18 j 1.10 1 1.05 ~...... , 1 1.03

(d) 1."" l ".97 j ".cas !

3b4e. et.. 1'/.1 1 2 1H5. -i 1 (e) 21Q". 14bS. j 74fi1 . Ill. 74E-1lI4 a.lfi1E-"J 3.13E-a3 a.15E-aJ a.1BE-a3 0.21E-fiI3

Figure 2.6: Same as Figure 2.3 but for variations in absolute vorticity TI (units S-l) on abscissa.

• 31 layer, the amplitudes of the terms with time scale given by lTl in (A.2) is much larger • than those with the time scale given by lT2). An inspection of these figures show that 1) The slope of the trajectory of the tube for the buoyancy mode is quasi-upright

because the isentropic surfaces are quasi-horizontal. The slopes ranged from

800 ta 900 degrees for the values of the parameters varied (panel a of Figures

2.3 to 2.6).

2) The horizontal scale, measured by the location of the equilibrium point C,

varies from tens of km ta almost 1000 km, which is in agreement with result

of Xu (1986) who showed that CS! can range from small scales ta subsynoptic

scales in the presence ofconditional unstable stratification. The horizontal scale

is strongly affected by the vertical wind shear and absolute vorticity in the

environment. It increases with wind shear and anti-cyclonic vorticity (Figures

2.5b and 2.6b). SBI degenerates into the classical buoyant instability when the

effects of wind shear and vorticity can be neglected.

3) There is a clear separation of time scales for the buoyancy mode and the inertial

mode. In the solution in the lower layer, (2.25) and (2.29), the characteristic

time scale is l/lTl because the cash terms predominate. For the solution for x in

(A.2) when z > hl, the terms associated with lTL dominate over those associated

with 0"2 since the ratio V~:~ is unquestionably large in all conditions shown in panel e of Figures 2.3 through 2.6. Hence, l/Ul in the upper layer cao be

considered the characteristic time scale for the horizontal motion. In Figures

2.3 through 2.6, the value for 1/lT1 ranges from 0.981ViL ~ 1.0Ni1 for motion • 32 from A to B (panel d) and from 0.94(/1])-1/2 "V 2.2(f1])-1/2 for motion up to • point C or 0 (panel c). This suggests that the buoyancy time scale is NïL and the inertial time scale (/T/)-L/2. These time scales are fairly separable under

typical conditions in the atmosphere.

• 33 •

Chapter 3

Climatology of CAPE and SCAPE

from ECMWF Re-Analysis

In the previous chapter, we investigated sorne of the characteristics of conditional

syrnmetric instability and slantwise buoyant instability. Before attempting ta param­

eterize them in GCMs, it is important to determine the potential for occurrence of

these instabilities over the globe. To this end, we will analyze the global distribution

of CAPE (convective available potential energy) and SCAPE (slantwise convective

available potential energy) using the 15-year ECMWF (European Centre for l\t"Iedium­

Range Weather Forecasts) re-analysis (ERA). Although Shutts (1990) has studied the

distribution of SCAPE for individual cases of severe storms, the analysis of SCAPE

over extended areas and long time span has not appeared in the literature.

• 34 3.1 Calculation of CAPE and SCAPE

• It is weil known that CAPE is the positive area in the tephigram ofa vertical sounding. For the calculation of SCAPE, we followed closely Emanuel (1983b) who defined

SCAPE as the maximum potential energy a symmetrically displaced tube of air may

realize in a slantwise conditional unstable atmosphere. Using the Lagrangian parcel

equations (2.10) and (2.12), SCAPE can be written in terms of the work done in

moving the tube from an initial equilibrium point or level of free convection (LFC)

to an upper equilibrium point (EP)

EP SCAPE - kFe F . dl EP - lLFCr [ f(Atlt - _Mg)'. + 89 (8vt - (Jvg)k] • dl, (3.1) 0

where the subscripts t and 9 refer respectively to the lifted tube of air and the en-

vironment which is in exact geostrophic and hydrostatic balance. The subscript v

indicates that virtual potential temperature is used. By taking the cud of F, noting

that Mt is conserved, and neglecting the variations ofBvt in the x direction, we obtain

v x F = (_/8Mg _iL88v9 )i. (3.2) 8z 8vo 8x

The cud of Fis zero since the right hand side of (3.2) vanishes under the assumption

that the environmental flow is in thermal wind balance. Thus the integral (3.1) is

independent of path and SCAPE can be cast into a form similar ta CAPE by choosing

the path of integration along a constant absolute momentum surface M

SCAPE = r (J9 «(Jvt - (Jv9)dz, (3.3) lM vO • 35 or in pressure coordinates

• (3.4)

where R is the gas constant for dry air. Note that both Bvg and Tuy are environmental

quantities along a constant 1\1/ surface.

SCAPE as defined in (3.4) can be estimated from a vertical sounding as follows.

Consider a hypothetical sounding located at x = 0 and an absolute momentum surface

1\;/1 which intersects the ground at the same point (Figure 3.1). Suppose a tube of air

at x = 0 at the surface is reversibly lifted to point 1 while conserving its 1\;/(= 1\1/d.

Let ~J.V/ he the difference between the 1\;/ of the tube and that of its environment

Figure 3.1: Schematics of calculating SCAPE from a vertical sounding. Adopted from Emanuel 1983b.

at point 1 (.LltI2 ). Assuming that the variation of the absolute vorticity TJ(p) with x in

small in the region between the lifted tube and the NIL surface, the horizontal distance • 36 L between the tube and the constant momentum surface 1\11 can he approximated as

~kl ~M L= (3.5) • aM/ax = - TJ(p)·

Furthermore , by neglecting the variation ofthe thermal wind in the x direction along

L, the virtual temperature of the environment along the Al surface can he estimated

from the first order Taylor expansion as

aTug TU9 1M - Tug+L âx f av - Tuy + Râ(-lnp) (3.6)

The latter expression results from the use of the thermal wincl relation of the fonn

(3.7)

Substituting (3.5) and (3.6) into (3.4), and considering /.ll\tf = v - Vo in a vertical

column, where Vo is the velocity at the surface, the SCAPE up to point 1 can be

estimated in a local column as

_ fl [ 1 f d(v - '00)2] SCAPE(Tvt ,Tvg,p) Jo R Tut - Tuy + 2 RT/(p) de -lnp) d( -lnp)

- f R(T.t - T.g')d(-lnp), (3.8)

where

Tuy' - TU9 - ~Tu9 1 f d(v -vO)2 (3.9) - Tuy - 2 RTJ(P) de-lnp) . • 37 (3.8) shows that SCAPE can be calculated in a manner similar ta the traditional • CAPE by replacing the virtual temperature of the environment by a new temperature T~g which cantains the contribution from the vertical \Vind shear. ~Tvg in (3.9) can

be viewed as the temperature equivalence of the effect of wind shear in the calculation

of SCAPE. It can also he regarded as the contribution of the inertial part to the total

SBL

The total SCAPE in a column between the level of free convection (LFC) and the

top equilihrium point (EP) can then bewritten as ~ 1/~) dd~_~:~2] SCAPE(Tut •Tuy) - l:: [R(Tut - Tuy) + d( -lnp)

- CAPE(Tvt , Tug ) + T s , (3.10)

where

EP T s - R.!lTugd( -lnp) lLFC rE? 1 f d(v - '00)2 (3.11) - lLFc2 11(P) d(-lnp) d(-lnp)

is the residual of SCAPE over CAPE. Ts can be viewed as a measure of the contri-

bution of the inertial mode to SBI When CAPE > O~ SCAPE is composed of the

positive contributions from CAPE and T s • "Vhen CAPE $ 0, SBI degenerates into

the traditional Conditional Symmetric Instability (CSI) if SCAPE remains positive.

Because C.A.PE can be negative in this case, for the environment flow ta he condi-

tionally slannvise uustahle, T s must he large enough to result in a positive SCAPE.

This means that CSI is only possible in areas where stratification is weak and the

vertical wind shear is strong, a condition weIl described by the Richardson Nurober • 38 • in (1.3). 3.2 The data and the thermodynamic process

We will use the lS-year global ERA data at 6-hour intervals at DO, 06, 12, and 18

UTC. The horizontal resolution is 2.5 x 2.So on a regular grid. Besides 14 surface

variables, there are 7 fields at 17 upper air levels (1000, 92S, 8S0, 77S, 700, 600, SOO,

400, 300, 250, 200, ISO, 100, 70, 50, 30, and 10 hPa). Interested readers are referred

to Gibson et al. (1997), Kallberg et al. (1997), and Uppala et al. (1997) for further

details on the ERA project.

As described in section 3.1, SCAPE is estimated by modifying the temperature

buoyancy with a wind shear correction factor. However, there are three possible ther­

modynamic processes ta calculate the buoyancy of a lifted parcel: 1) the standard

irreversible or pseudoadiabatic process, 2) the reversible process including the water

loarling, and 3) the reversible process including both the liquid and the ice phases. In

the pseudoadiabatic process, all liquid condensate is assumed to be removed instan­

taneously from the air parcel. Therefore, water loading and the heat capacity of the

condensate are not considered in the calculation ofparcel buoyancy. For the reversible

process, it is assumed that the condensed water is carried upward with the parcel.

Because the effects of water loading and the heat capacity of the condensate are in­

cluded, the buoyancy is significantly reduced relative to the pseudoadiabatic process

(Xu and Emanuel 1989). However, if the reversible process is further expanded ta

include the ice phase, the significant positive effect of the latent heat of fusion tends • 39 to compensate the negative effect of the loading of condensate. Williams and Renno • (1993), using a large numbers of soundings from GATE (the Global Atlantic Trop­ ical Experiment) and AMEX (the Australian Monsoon Experiment), demonstrated

that the inclusion of the ice phase in the reversible process results in values of CAPE

similar to those obtained in a pseudoadiabatic process. Based on this finding, we will

calculate the parcel buoyancy by lifting a well-mixed parcel between the surface and

the lowest 60 hPa along a pseudoadiabat.

3.3 The global distribution of CAPE and SCAPE

Figure 3.2 shows the distribution of mean CAPE over the globe for the two seasoos

DJF (December, January, and February) and JJA (June, July, and August). AI­

though the ERA data extend from January 1979 to December 1993, ooly data from

June 1979 to August 1993 are used in the calculatioos with 14 samples in each sea­

son. In DJF, the most significant areas for CAPE (contours larger than 500 J /kg) lies

between 10° N and 30° S. Loogitudinally, it extends over the central and western Pa­

cifie Ocean, .Australia, the Indian Ocean, the southern part ofAfrica, central America,

the east and central part of South America, and the western Atlantic Ocean. In the

eastern Pacifie and Atlantic Oceans, oarrow bands only occur close to the equator.

In the Northem Hemisphere summer (JJA.), the pattern shifts northward. The 500

J/kg contour in generallies north of 20° S. The main areas of signfficant CAPE are

located in the lower latitudes and in the tropics with sorne patches extending to the

midlatitudes in North America and the Eurasian Continent. Again the bands over • 40 the tropical east Pacific and the Atlantic Oceans are much narrower than at other • longitudes. This feature is attributable to the relatively cold sea surface temperatures over the two regions.

Fig. 3.3 displays the global distribution of mean SCAPE for the 14 seasons. The

pattern of SCAPE is similar to CAPE but its north-south extent is wider in many

areas. Note that in the sub-tropical region of the west Pacifie, a new 500 J/kg branch

appears in DJF. This larger area of SCAPE over CAPE is not too surprising because

according to (3.10), SCAPE can be decomposed into the traditional CAPE plus the

effect of the vertical wind shear, which is usually positive in the troposphere. As a

result, SCAPE is larger than CAPE in most cases.

To reveal the inertial part of SCAPE (Le. the wind shear effect in (3.10)), we

calculated the SCAPE residual r s = SCAPE - CAPE and displayed the result in

Figure 3.4. In general, T s is smaIl relative to SCAPE or CAPE and is located mostly

within areas with positive CAPE. This result suggests that the potential for slantwise

convection exists in regions with upright convection and SBI is more likely to occur

than the pure CSI. In DJF, the residuals are largely found around the sub-tropical

and midlatitude regions with the most significant patch over the middle and lower

latitudes of the western Pacific. In these areas, the vertical wind shear is stronger

than that over the equatorial region but the stratification is still relatively weak.

Near the equator, the residuals are small with the exception of two patches over the

eastern Pacific and the Atlantic. In JJA, the patch of significant SCAPE residual

over the ocean is clearly shifted ta the Southem Hemisphere. However, the strongest • 41 residuals are located over the sub-continent in southern Asia and its eastern vicinity. • The relatively large values are related to the strong vertical wind shear associated with the monsoon season over the region.

Another parameter which can be used to gauge the relative importance of the

inertial mode and buoyancy modes is the fractional SCAPE residual Is, defined as

r" SCAPE - CAPE III = SCAPE - SCAPE (3.12)

In general the value of Is ranges from 0 to 1. When SCAPE = CAPE, f" = 0 which is the condition favorable for pure CUI. When CAPE=O, III = l, and the situation is

favorable for pure CSI. Therefore in regions where Is is small, upright convection is

likely, and in regions where III is close ta one, there is a high potential for slantwise

convection. The SBI type of convection is most favored \Vith intermediate values for

Fig. 3.5 shows that for the l4-year mean, the fractional SCAPE residual is less

than 1. However, at a fixed time, there are occasional instances when 1" = 1 (not

shown). Comparison of Fig. 3.4b and Fig. 3.5b indicates that in JJA, the most

likely location for slantwise convection lies in the sub-tropical and the midlatitude

belts of the southem Indian Ocean and the western and central portions of the South

Pacifie Ocean. Note that the large values of SCAPE residual r" over India become

much smaller in terms of 1" because CAPE is very large during the monsoon season.

For DJF, slantwise convection is most favored in the midlatitudes especially over the

western and central portions of the North Pacifie ocean. In general, CSI and SBI

is expected to play an important role over the midlatitude oceans during the winter • 42 • hemisphere, where buoyancy is weaker than that during the summer hemisphere. 3.4 Distributions ofCAPE/SCAPE and precipita-

tion

It is ofinterest to ask whether the distributions ofCAPE and SCAPE represent actual

convective activities in the atmosphere. Before attempting to answer this question,

it is important to hear in rnind three limitations in our calculation of CAPE and

SCAPE. First, our calculation assumed the ascent of an undiluted air parce!. In

reality, an air parcel cannot he entirely separated from its environment. The CAPE

and SCAPE calculated only represent the maximum amount of energy that may

he released. Second, as was mentioned previously, the calculation depends on the

choice of a thermodynamic process, we assume that the parcel ascent is along a

pseudoadiabat. Third, even though SCAPE or CAPE is present, work still bas to be

expanded to bring the air parcel to saturation herore the energy cao he released. In

otber words, we need to overcome an energy harrier generally known as the convective

inhibition energy (CIN) defined hy

LFC R CIN = -(Tut - Tvg)dp, (3.13) fs5FC p

where Tvg and Tut represent respectively the virtual temperatures of the environment

and the lifted parcel (or tube in the case of unidirectional fiow). Graphically, CIN is

the negative area below the LFC on a tephigram.

CIN is usually small and bardly exceeds a few hundreds of J kg-l, or an order • 43 •

(a)

(b)

Figure 3.2: The global distribution of CAPE for seasons (a) DJF and (b) JJA. The plots are averaged over 14 years based on E~[CFW re-analysis data of 1979-1993. Values are in units of 100 J /kg with a contour interval of 500 J/kg. • 44 •

Ca)

(b)

Figure 3.3: Same as Figure 3.2 but for SCAPE.

• 45 •

(a)

(h)

Figure 3.4: The global distribution ofSCAPE residual r. or SCAPE - CAPE for seasons (a) DJF and (h) JJA. The plots are averaged over 14 years based on EMCFW re-analysis data of 1979-1993. Values are in units of 100 J /kg with a contour interval of 200 J/kg. • 46 ·'

(a)

Ch)

Figure 3.5: The global distribution of SC~P_E residual fraction f. or SC·'Vè\l~i~PE for seasons (a) DJF and (b) JJA. The plots are averaged over 14 years based on EMCFW re-analysis data of 1979-1993. Values are in percent with a contour interval of 25% • 47 of magnitude smaller than the ma..ximum values for CAPE or SCAPE. However, ta • overcome CIN, the parcel still requires substantiaI initial vertical velocity which can be estimated from

\!to = v'2CIN. (3.14)

For a CIN of 50 J kg-l, Via turns out to he about 10 m S-l. Such a requirement

for disturbance velocity can inhibit the release of CAPE or SCAPE. vVe will exam­

ine this barrier effect by comparing the global distributions of CAPE/SCAPE and

precipitation.

Fig. 3.6 presents the global mean precipitation over a 17-year period from 1979

ta 1995. The data source is the gridded global monthly precipitation dataset con­

structed by Xie and Arkin (1997). The dataset includes gauge observations, a variety

of satellite observations! and the NCEP reanalysis. For DJF. the distribution of pre­

cipitation (Fig. 3.6a) compares weIl with the distribution of CAPE (Fig. 3.2a) over

the tropics and most regions in the Southern Hemisphere! lending support to the

predominant role of buoyancy instability in deep convection in the tropics. However t

the distribution of CAPE fails ta explain two major patches of precipitation over the

midlatitude oceans in the Northem Hemisphere. On the other hand, a comparison of

Fig. 3.3a, Fig. 3.5a, and Fig. 3.6a clearly suggests that slant,,;se convection or the

inertial mode in SBI plays an important role in affecting precipitation in these areas.

It should aIso be borne in mind that during Northem Hemisphere winter, larger scale

baroclinic systems must aIso contribute to the distribution of precipitation in the

midlatitudes. • 48 Fig. 3.7 shows the distribution of CIN. Comparison of Fig. 3.7a, Fig. 3.3a, and • Figure 3.6a reveal possible inhibition effects over Australia, the eastern Pacifie Ocean, and the western part of the south Atlantic Ocean.

Similar features also appear in JJA. The precipitation over the tropics is largely

correlated with CAPE. The location ofthe precipitation over the oceans in the South­

ern Hemisphere agrees with the appearance of significant fractional SCAPE residual

fs (Figure 3.6b and Figure 3.5a). The inhibiting effect of CIN is mainly evident over

Northeastem Africa and the area of the Middle East. Some trace of this effect can

also he detected over the central Pacific Ocean. The patch of precipitation over the

west coast of Canada is not correlated with CAPE or SCAPE and is likely the result

of the semi-permanent Aleutian low and the orographie etrect of the high mountains.

The main conclusions from this chapter are as follows:

1) In our 14-year mean global analysis, SCAPE appears mostly in areas where

the CA.PE is also positive. This result suggests that the potential for slantwise

convection exists in regions with upright convection and SBI is more likely to

occur than the pure CS!.

2) While CAPE dominates over the tropics, especially over the warmer western

oceans, slantwise convection or the inertial mode of SBI is likely ta play a

significant raIe over the midlatitude oceans during the winter hemisphere.

• 49 •

(a)

(b)

Figure 3.6: The global distribution of gridded precipitation for seasoos (a) DJF and (b) JJA. The plots are averaged over 16 years based on a dataset of 1979-1995 coostructed by Xie and Arkin (1997). The units are in 10-7 mis with a contour interval of 2 x 10-7 mis. • 50 •

(a)

(h)

Figure 3.7: The global distribution ofCIN for seasons (a) DJF and (b) JJA. The plots are averaged over 14 years based on EMCFW re-analysis data of 1979-1993. Values are in units of 100 J/kg with a contour interval of 100 J/kg. • 51 3) A comparison of the distribution of precipitation, CAPE, SCAPE, and frac­ • tional SCAPE residual indicated that there is a good correlation between ar­ eas of precipitation, and areas with significant CAPE, SCAPE, and fractional

SCAPE residual.

• 52 •

Chapter 4

A Generalized Moist Convective

Adjustment Scheme

4.1 Formulation

Based on the results of theoretical studies and data analysis presented in previous

chapters, we propose here a generalized moist convective adjustment scheme. The

main consideration is that CAPE is frequently associated with SCAPE. Indeed over

the ocean, the houndary layer is often potentially unstable for upright and slantwise

convection. As such, the most common type of instability is likely to he slantwise

buoyant instahility (SBI) with its two time scales associated respectively with the

buoyancy mode and the inertial mode.

Our scheme follows the approach ofBetts and Miller (Betts 1986; Betts and Miller

1986), but differs from it in important ways: • 53 1} Not only does the scheme adjust the atmospheric temperature and moisture • structures toward a reference quasi-equilibrium thermodynamic structure in the presence of large-scale forcing, the momentum field would also be adjusted to-

ward a corresponding reference state.

2} The reference states are neutral to upright convection as weIl as slantwise con-

vection.

3} Instead of one adjustment time scale, two adjustment time scales for the buoy-

aney mode and the inertial mode are specified. The convective eddy fluxes of

heat, moisture and momentum are parameterized as

88) _ Ils - 8 R" - 8 ( - - + , (4.1) ôt conv Ti Tb

where R and 8 are 3-D vectors denoting the temperature, specifie humidity

and momentum of the resolved scale and the reference quasi-equilibrium states,

respectively. The subseribes i and b represents respectively the inertial and the

buoyancy mode. Ti and Tb are the inertial and buoyaney adjustment time scales.

The adjustment begins by constructing a local 2-D plane perpendicular to the

local thermal wind direction for each vertical column of the 3-D grid. We will adjust

the temperature, moisture, and momentum along these local 2-D planes. The local

two dimensionality is justifiable from numerical experiments by Jones and Thorpe

(1992) who showed that for a 2-D basic state, 3-D perturbations are still more or less

two dimensional with the axis of the perturbation aligned within 6° of the direction

of the thermal wind. • 54 To clarify the formulation of our adjustment scheme, we first discuss the simpler • case when pure slantwise convection is present (Le. CAPE = 0 and SCAPE>O). In this ease, a displaced tube of air will move more or less along a neutral buoyancy

surface. There are negligible heat fluxes but the absolute momentum will be adjusted

towards a reference state neutral to CSI characterized by a constant value of absolute

momentum along a neutral buoyancy surface.

For each vertical column there is an associated neutral buoyancy surface for a

tube of air lifted from the ground. Along this surface, the potential temperature of

the tube when lifted reversibly is always equal to that of the environment. For an

unsaturated tube initially located at the lowest grid point, the x and P coordinates

of the associated neutral buoyancy surface can be obtained by assuming that the air

tube is first lifted along a constant 8 surface of the environment while conserving its

own 8 until the lifting condensation level (LCL) is reached, so that

(4.2)

where the over-tilde refers to the environmental quantity along the neutral buoyancy

surface and the subscript t to that of the tube. The coordinates Xo and Po represent

the initial coordinates of the tube.

ACter saturation is reached, the temperature of the tube follows that of a moist

adiabat and the neutral buoyancy surface is charaeterized by a constant value of

saturated equivalent potential temperature 8; , snch that

(4.3)

where Xc and Pc are the eoordinates of the LCL and qo denotes the initial specifie • 55 humidity ofthe tube. The right hand sides of (4.2) and (4.3) are constants, determined • by the initial conditions of the air tube. The horizontal coordinate x ofthe neutral huoyancy surface at each model vertical

level p can he estimated by a Taylor series expansion

if p > Pc, (4.4)

(4.5)

where the overbar refers to grid point values. In practice Xc in (4.5) can he replaced

by Xo as the difference between these two values is small.

Having ohtained the neutral buoyancy surface, we cao then estimate the distrihu-

tion of ahsolute momentum, in, along the surface from

_() ôm(xo,p) ( ) m=m Xo,P + 8x X-Xo· (4.6)

We can also define two neutral points, PB and PT ,above the LCL. They represent

respectively the level of the cloud base and the cloud top (Figure 4.1, top panel). The

counterpart of the lower equilibrium point PB in CUI is the level of free convection

(LFe).

The last step is to construct a reference absolute momentum profile, characterized

by a constant value mR along the neutral buoyancy surface. This can be obtained by

using the constraint of conservation of absolute momentum, in the form

lpo[PT (mR - m)dp =0, (4.7)

where the path of integration is along the neutral buoyancy surface s. • 56 Classic Slantwise Convection • (a) (h) m

PB -'-----.lIIr-~r_-----

Slantwise Buoyant Convection

m

Figure 4.1: Schematic illustration of the adjustment process for (top panel) pure conditional slantwise instability or CSI and (bottom panel) slantwise buoyant instabilityor SBI. (a) oftop panel shows the configuration of momentum surface m, neutral buoyancy surface s, and the reference momemtum surface m R in the physical two-dimensional space. Ch) of top panel is the profile of momentum and the reference momentum along neutral buoyancy surface s, denoted asm and mR respectively. The tilde is used to represent values along the neutral buoyancy surface. In the bottom panel, (a) illustrates the configuration of the momentum surface m, the neutral buoyancy surface s, the reference momemtum surface mR, and the reference neutral buoyancy surface SR in the physical 2-D space. (h) is the vertical profile of s and SR. (c) shows the distribution of momentum and the reference momentum along the neutral buoyaÎlcy surface S in the case of SBI. • 57 The momentum flux along s can then be calculated, according to the adjustment • approach (4.1), as (4.8)

where Ti is the inertial adjustment time scale to be specified. The momentum tendency

computed from (4.8) will be interpolated to the two nearest grid points on the local

2-D plane, and finally interpolated ûnto the 3-D model grid.

The effect of the adjustment along an s surface on a local 2-D plane is illustrated

schematically in Figure 4.1 (top pïWèlj.

For the more general case when both CAPE and SCAPE are present, we have to

account for both the buoyancy mode and the inertial mode. Since the buoyancy time

scale is much shorter than the inertial time scale, we will first construct a thermody-

namic reference state TR and qR and then construct a momentum reference state mR

along the neutral buoyancy surface defined by TH. and qR.

The adjustment scheme is illustrated in Figure 4.1 (bottom panel). Similar to

the case in the top panel, the LCL is obtained in a like manner. Different from the

case of CUI, the LCL here is the level of free convection (LFC) since the air tube

is lifted along an isentropic surface instead of in the upright direction. The cloud

base is then given by the LFC. There are two equilibrium points above the LFC. The buoyancy equilibrium point 14 is determined by assuming that the air is lifted vertically until it reaches the thermodynamic neutral level. The inertial equilibrium point 14- represents the intersection ofthe neutral buoyancy surface s with the absolute

momentum surface originating from the initial location of the air tube (Figure 4.1, • 58 hottom panel). • The reference thermodynamic profile can he obtained following the procedure of Betts and Miller (1986). The first guess reference temperature profile is obtained

from

Tlt(P) = T(PB) + r m(PB - p) (4.9)

for p > p}, where r m = dT/ dp is the moist pseudoadiabatic lapse rate. The reference

specific humidity profile is in tum given by

(4.10)

where q~ is the saturated specific humidity and x(P) is a specified relative humidity

profile with X(p) E [O,l}. The first guess profiles of TA and qh are then modified iteratively until they satisfy the total enthalpy constraint

P!j. - [ep(TR - T) + L(qR - q)Jdp = o. (4.11) lPO

Once the reference thermodynamic profile is established, the neutral buoyancy

surface for the inertial convection mode can be defined following the same approach

as in (4.1), but with the newly obtained reference temperature state used in the

calculation. Similarly the absolute momentum mofthe environment along the neutral

buoyancy surface is determined from (4.6). The constant reference momentum state

ma on s is obtained by the constraint of the conservation of absolute momentum as

(4.12) • 59 where the integral is along the neutral buoyancy surface s and Ti is the inertial ad- • justment time scale. The flux tendencies of temperature, specifie humidity and mo- mentum caused by slantwise buoyant convection are estimated as

TR-T - (4.13) (~) con" Tb qR-q - (4.14) (Z) con" Tb mR-'m - (4.15) (~) con" Ti

by specifying the adjustment time scales Tb and Ti. The momentum Oux obtained in

(4.15) is still on s and must at first be interpolated onto the 2-D local plane. Then the

temperature, humidity, and momentum fluxes on the 2-D local plane are interpolated

to the 3-D grid.

We remark that in (4.13) and (4.14) the adjustment will he assumed to be in

the vertical direction since the most unstahle direction of the buoyancy mode is

perpendicular to the initial neutral buoyancy surface which is more or less horizontal.

For the absolute momentum flux (4.15), the adjustment would be along the neutral

buoyancy surface defined hy TR and qR.

4.2 Test under idealized 2-D conditions

The parameterization introduced in Section 4.1 for SBI is tested in an idealized two-

dimensional environment. The initial conditions are specified as foUows

3 1 -V;: =25 . x 10- s,- • 60 • 5 2 Nl2 -1 X 10- S,-

5 2 N.22 -X- 5 10- S,-

The horizontal and vertical grid sizes ~x and ~z are 100 km and 500 ID, respectively,

and hl = ~z. There are a total of 21 vertical columns.

The potential temperature 8, absolute momentum m, and saturated equivalent

potential temperature 8; of the initial state are shown in Figure 4.2a and c. The

initial specifie humidity field is depicted in Figure 4.2d. The resulting equivalent

potential temperature field 8e is displayed in Figure 4.2b. It is clear that the initial

environment is conditionally unstable to both the traditional upright convection and

the newly defined slantwise buoyant convection (Fig. 4.2c).

In this simple test, we simply lifted a sample of air from the lowest level at each

column. If the LFC for this air sample is found, the slantwise buoyant convection

scheme is activated. The cloud base is set at the LFC, and the buoyancy cloud top

14 is given by the buoyancy equilibrium point above the LFC. The thermodynamic

reference state for this vertical column is then computed iteratively following the

procedure described by (4.9) through (4.11). The in-cloud relative humidity profile X

is assumed constant with a value ofunity (Le. the cloud column is assumed saturated).

Figure 4.3 shows an example of the reference temperature profile and the reference·

humidity profile for column 13 at x = 200 km. It can be seen that the buoyancy mode

tends to warm the column at the lower part of the cloud and to cool it at the upper • 61 ''5.

'2.

'1 •

o. • J ...... -2.... 2.... 0 ...... (e)

Figure 4.2: Initial fields of (a) absolute momentum (solid, in mis) and potential temperature (dashed, in K) , (b) ab501ute momentum (50lid) and equivalent potential temperature (dashed, in K), (c) absolute momentum (solid) and saturated equivalent potential temperature (dashed, in K), and (d) equivalent potential temperature (dashed, in K) and specifie humidity (solid, g/kg). The units for horizontal distance (x) and vertical height (z) are km. • 62 •

50 1 j d. in i t the solid. init tneta 50 1 id 1 in i t q d.5hed· ~.f/adid tne dashed. ~ef/adjd tneta dashed. ~.f/adid q un i ts. 1( un i ts. 1( un i ts. g/kg 1'5. ,....-----r------..,..--,1'5. 1'5. il H

12. 12. 12.

.IJ / J ,/ t' 1 1 1 1 1 Il 1 1 o. 1 lb. [ i 1 ri~ ~ 1 1 1 1 1

3. ~ ~ ~ 1 J J

1 1 / J 1 1 ~---JI ..[ '-- ....l..- ...... l QI. L..--/ L..-- 1 iL l 324. 3'53. 27Q. 316. 3'54. -0.4 4 . 4 Q.3 0.0 hOU~5 Column 13 (a) (b) (c)

Figure 4.3: Initial profiles (soUd) and their referenee states (dashed) for column 13 (a) equivalent potential temperature (in K), (b) potential temperature (in K), and (e) specifie humidity (in g/kg). Column 13 is located at 200 km along the abscissa. The vertical eoordinate is height in km.

• 63 levels and in the subcloud layers (Figure 4.3b). Moisture is transported upward • from the subcloud layers (Figure 4.3c). The equivalent potential temperature for the reference state plotted in Figure 4.3a shows that the reference thermodynamic state

is fairly neutral to upright buoyant instability. It should be mentioned that a three-

point moving average operator has been applied to the reference temperature profile

to reduce the discontinuity around the buoyancy cloud top level at 14-. The reference absolute momentum profile for this column is then constructed by

first interpolating the absolute momentum values at the grid points onto the reference

neutral buoyancy surface shown in Figure 4.4. The neutral buoyancy surface is, in

general, slantwise and not upright. Then the reference absolute momentum, which is

constant along the neutral buoyancy surface, is calculated as

(4.16)

by using the constraint (4.12). Both m and mR for column 13 along the neutral buoyancy surface given in Figure 4.4 are shown in Figure 4.5.

We chose a value of 0.5 hours for Tb and a value of 3 hours for Ti' The tendencies

for T and q for each vertical column where slantwise buoyant convection is active

are computed from (4.13) to (4.14) at each vertical grid point of the column. The

momentum tendencies are computed along the neutral buoyancy surface defined by

TH. and qR using (4.15). The absolute momentum tendencies calculated at a location

between two grid points at the level p are interpolated to the two grid points by

applying an interpolation scheme. The absolute momentum and equivalent potential

temperature fields after 3 hours of adjustment are shown in Figure 4.4. For the • 64 •

..

D.

J . .. -1.... ·0.... -2••.• 21•.• e••• ,.... (a) 1'.

12 . ..

.. -'.... 1 -0.... ·2.... 2.... e.... ,.... (b) ".

'2 ..

D •

.. -II•.' -e.... -2.... 211.' e.... ln•. (c)

Figure 4.4: The adjusted fields for (a) potential temperature (dashed) and absolute momentum m (solid), (h) equivalent potential temperature (dashed) and m (solid), and (c) saturated equivalent potential temperature (dashed) and m (solid) after 3 hours ofintegration. The neutral buoyancy sur­ face for column 13 is schematically shown by the thick dashed line in Cb). The units and coordinates are the same as in Figure 4.2 • 65 •

501 id. in j t mom­ d4shed. ~.f/4djd mom~ uni ts. mis

-70. -12. 4b.

Figure 4.5: The initial (saUd) and reference (dashed) absolute momentum profiles (in mis) along the neutral buoyancy surface shown in Figure 4.4b. The vertical coordinate is in km. • 66 specified initial condition, slantwise buoyant convection takes place for columns 11­ • 14, Iocated from x = 0 km ta x = 300 km. It can be seen that not onlyare the (saturated) equivalent potential temperature surfaces within the convective region

more or less neutraI to upright buoyant convection, but the adjusted momentum

surfaces are parallel to the constant Be surfaces, implying that the adjusted state is

neutrai to both upright and slantwise convection.

• 67 •

Chapter 5

A One-Dimensional

Parameterization Scheme for

Slantwise Convection

A. major goal ofthis study is to investigate the effect of the parameterization ofCS! in

general circulation models. We presented in Chapter 4 a generalized parameterization

scheme for SBr in two dimensions. However, this scheme is rather expensive to ron in

a GCM. Furthermore, aIl GCMs contain an existing cumulus parameterization scheme

(CPS) for upright convection. Therefore, for practical consideration, we will simplify

the 2-D scheme ta a 1-D version by making use of the assumptions in the calculation

of SCAPE in a vertical column presented in Chapter 3. A subset of this 1-0 version,

which includes only the slantwise convective parameterization scheme (SepS), would

then he implemented into the cce (Canadian Clïmate Centre) GCM (Zhang and • 68 MeFarlane, 1995a). In sa doing, we are assuming that the buoyaney mode or vertical • convection is taken care of by the existing CPS, and the SCPS is invoked after the CPS has released the vertical instability.

5.1 Formulation of the I-D scheme

The basic assumption of the 1-D scheme is that in the release of SBI, the atmosphere

adjusts toward a referenee state characterized by the vanishing ofSCAPE. Because the

existing CPS is used to represent the effect of upright convection, the temperature

and specifie humidity adjustment by the buoyancy mode in (4.13) and (4.14) are

effected by the CPS. For the adjustment of momentum in (4.15), the theory of SBI

discussed in Chapter 4 indicates that the adjustment should be carried out not only

for the case of classical slantwise convection (Fig. 4.1, top panel) but also for the case

of slantwise buoyant instability (Fig. 4.1, bottom panel).

Reeall that (4.15) applies to a neutral buoyancy surface s for CSI, or to a reference

neutral buoyancy surface SR. for SBI. Furthermore, min (4.15) denotes the absolute

momentum along S or SR in a local 2-D plane normal ta the local thermal wind vector.

Ta simplify the scheme ta one dimension, we made the same assumption as in Chapter

3 in the ealculation of SCAPE.

We now consider the expression for the SCAPE of the reference state in bath SBI

and CS!. In the case of SBI, the referenee state is characterized by the reference mo­

mentum m R , temperature TR and specifie humidity qR. The SCAPE of the referenee • 69 R state, denoted as SCAPE , vanishes such that • (5.1) where Tvt is the virtual temperature of the air tube, and Tv~' a function of T R and

qR, is the virtual temperature of the reference state. (5.1) is simply the statement

that the reference momentum surface m R is parallel ta the reference neutral buoyancy

surface sR (Figure 4.1). Any air tube lifted along an m R surface is equivalent to being

lifted along an sR surface and will not affect the TR and qR of the reference state.

Because classical csr is a special case of SBI, (5.1) is still valid. In this case the

buoyancy mode is absent and there is no adjustment for temperature and humidity.

The reference state for T R, qR, and sR is simply given by that of the environment, or

and in the case of pure CS!.

Since m R is constructed ta he parallel ta the neutral buoyancy surface s (Figure 4.1),

the SCAPE of the reference state would again vanish.

By neglecting the horizontal variation of the thermal wind and the absalute vor-

ticity TI, (5.1) can he transformed from an integral along a slanted direction to an

integral along the vertical direction as

(5.2)

where

(5.3) • 70 and

RIf d (R R)2 (5.4) • t!iTvg = 2 RTl d(-lnp) V - ~ . Here t!iT;: is the buoyancy of the air tube relative to the virtual temperature of

the reference state T~ at each height. ~Tv~ is the temperature equivalence of the

correction due to the presence of vertical wind shear in the reference state.

Note that in the case of SBI, T R and qR are now deflned by a vertical neutral

buoyancy surface and the virtual temperature of the air tube lifted vertically is the

same as Tv~. Thus, ~T:: equals 0 for SBI. In the case of pure CSI, the environment

(Tg, qg) is stable for an upright displacement. A tube lifted vertically with respect ta

(Tg, qg) will experience a negative buoyancy. Therefore

for SBI,

for CS!.

R (5.2) gives an integral constraint for V , the component of the wind parallel to the

thermal wind vector in the reference state. By definition, m R = f X + V R , with X

increasing horizontally in a direction normal to the local thermal wind. Since at the

grid point of each vertical column, )( = 0, and 'm R = VR, we will reCer to v R as

the reference wind component or the reference momentum. ~R is simply VR at the

surface or the lowest model level. For simplicity, the local thermal wind is given by

the mean thermal wind through the troposphere in each vertical column.

Even though the reference momentum is gjven implicitly in (5.2), a unique solution

for V R cannot be obtained without additional assumptions. The simplest approach

is to apply the constraint (5.2) at each of the N vertical level, from n=O to N-2, that • 71 is

fp a(Vn~l - VOR)2 = -Rf).TR , n = D,···,N - 2. (5.5) • 21J ap ut Solving for VR, we obtain

RR 2R1JÔT!i~p (V: R _ TfR)2 Vn + 1 = va + f p + n Vo , n = D,·· ·,N - 2. (5.6)

Since there are N - 1 equations with N unknown VOR, ViR, ...,VU_L' an additional

equation is required and will be given by the constraint of the conservation of mo-

mentum over the column, with the fonn

N-l N-l L VnR~p = L Vn~P· (5.7) n=O n=O R Mathematically, VO ,ViR, ... ,v:-1 can be solved by combining the N -1 equations from (5.6) with (5.7). In practice, if we let VaR = VO, the first guess of VIR' is easily

obtained from

(5.8)

aod the rest of the VR'S cao be calculated iteratively as follows

R' Vn+1 = \10 + n = 0, ... , J.V - 2. (5.9)

Then, the final VRS can be re-calculated by subtracting out a constant given by the

arithmetic mean of the difference between the first guess VnR' and the environment

profile Vn as v:R ' _ E(v;.R' - ~)~p = v:R (5.10) n n EÂp'

The VnR calculated in this way automatically satisfies the momentum conservation

(5.7). • 72 By specifying the relaxation time T of the inertial mode, the momentum tendency • along the direction of the thermal wincl can he calculated as (5.11)

The momentum tendencies for the x and y components, (:) and (:) ,can then slw slw he obtained by projecting (~) to the longitudinal and the meridional directions slw for each column.

To illustrate the working of the SCPS, we first define

(5.12)

as the difference between the temperature equivalence of the reference state and the

temperature equivalence ofthe environment due to the presence ofvertical wind shear.

~Tslw can be thought as the temperature equivalence of the momentum adjustment

to the reference state mR, or V R. Fig. 5.1 depicts schematically three possibilities.

For the general case of SBI (panel a), the atmosphere is conditionally unstable to

both upright and slantwise convection (panel al). An existing CPS is assumed to

first adjust the T and q of the environment to TR and qR (panel a2) so that as far

as the SCPS is concemed, ~Tv~ = o. From (5.2) and the assumption made in (5.5),

it follows that 6.Tvg = 0 and hence ~Tslw = -tlTt!g. This result is equivalent to

constructing a reference momentum profile so that the temperature equivalence of

the correction due to vertical wind shear in the environmental sounding is completely

eliminated (panel a3).

For the case of pure CS!, the atmosphere is stable to vertical convection so that

~Tvg(see Fig. 5.1bl) > ~Tv~(see Fig. 5.1b2) > 0, and I6.Tslwl < ~Tvg. It means • 73 that the effect of the vertical wind shear is only partially removed and the sounding • is brought back to astate that is neutral to pure slantwise convection (Figure 5.1 panel b).

The third possibility is that the environment is stable to SBI, i.e. both CAPE

and SCAPE are non-positive. In this case, no adjustment is required, or ~T81w = 0

(Figure 5.lc).

5.2 Slantwise precipitation

Although the focus of the SCPS is the momentum transport by the inertial mode

of SBI along a slantwise slope, the transport does have a vertical component. We

would calculate the amount of precipitation produced by the scheme by estimating

the vertical component of the slanted updraft.

Similar to the parcel theory of vertical convection, the maximum slantwise up­

draft a 2-D tube may realize is V2SCAPE. Since the slantwise cloud covers only a

fraction of the grid box, we introduce the slantwise cloud fraction (j which represents

the percentage area occupied by the slantwise cloud in one grid box. Denoting the

characteristic slope of the slantwise path by J.L, the vertical component of the slantwise

updraft is approximated as

Udw = (jf8J.LV2SCAPE (5.13)

where the fractional SCAPE residual gives a measure of the relative strength of the

inertial mode in SBI. • 74 a) SBI case • Wind shear effec:t is complelely removed:

I1TslMI =-I1T'Ig ....bySCPS I1rvx = 0

al a2 a3

, b) CSI case Wind shear effec:t only panialy removed: II1TJlMlI < I1T'Ix ~ I1r'lK~O by SCPS

bl b2

Sv S , \ \ c) Stable ta bath vertical and slantwise lifting \I1T~ ~ No adjustmenl al ail: 11TsiMI = 0 11r 'IX = ATvg \ \ \ \ \ \ __ S \ \ Dry/moisture adiabats \ \ Original soundings \\ ---- Sv Vinual soundings with wind shear effeet \\ ~

c

Figure 5.1: Schematic illustration ofthe 1-D slantwise convective parameterizationscheme (SepS) for three possible scenarios: (panel a) SBI, (panel h) CSI, and (panel c) stable condition. ~ug is the temperature equivalence of wind shear correction defined in (3.9). ~T~ is the same as ~Tug but for the reference state defined by (5.4). ~T.'1lJ is the temperature equivalence of the slantwise momentum adjustment as defined in (5.5). • 75 Note that in (5.13), when fs = 0, SCAPE = CAPE, and Uslw = O. This is just the • case for pure buoyant instability. \Vhen fs = 1, CA.PE = 0, we recover the situation for the classical slantwise convection. There is no buoyancy mode in this case and the

SCAPE may be Cully released by the inertial mode. For simplicity, the cloud fraction

u is assumed constant with a value of 0.5. The slope Il- cau be estimated from ~ in

Figure 5.2, where H is the height of the slantwise cloud top and L is the horizontal

Col-2 Col·1 Col 0

Figure 5.2: Horizontal scale L and column shift. Considering the span of slantwise convection (from column 0 through column -2 in this case), the calculated tendencies are shifted from original Column 0 to Column -1 for better representation.

scale of the convection. L can, in turn, he obtained by

~m ml-ma L = - am = fi ' (5.14) ax where ml and mo are respectively the momentum at the top of cloud and the lowest

level. fi is the averaged absolute vorticity over that layer. • 76 Assuming that the air in the slantwise cloud is saturated and that any supersat­ • uration results in condensation, the condensation rate ~~ produced by the ascent is given by dx. Qi dt = Q2 Us1w , (5.15)

where Qi and Q2 are thermodynamic functions given in Rogers and Yau (1989, see

their (7.23) and (7.24)). The precipitation rate Rs1w is the integral of ~~ over the

cloud layer,

Pb,ue Qi dp Rs1w = - -QUs1w -, (5.16) lPtoJ' 2 9 if ail condensate precipitates out immediately.

The total moisture lost to the precipitation due to slantwise convection is sub-

tracted out from the cloud and subcloud layers as follows

Rslw if in cloud layer 8q Ci ~h ' _- cloud (5.17) ( 8tL. -{ (1 _ a) R.1w , if in subcloud layer ~hsub where ~hcloud and ~hsub denote the depths of the cloud and subcloud layers, respec­

tively, and a is a constant E [0,1]. Care was taken ta ensure that (:) calculated slw in (5.17) would not result in negative specifie humidity when it is applied to the total

moisture tendency equation. The latent heat produced by the condensate is released

uniformly in the cloud and subcloud layers as [top L(8q ) = PaIr .dw dp ffI') G:' lit (5.18) (8t {Ptop slw ln dp PaIr where L and Cp are the latent heat of condensation and the specifie heat at constant

pressure, respectively. • 77 5.3 Latitude BIter and slantwise shift

• The momentum tendency in our scheme is mainly determined by the SCAPE residual Ts in the case of SBI (Fig. 5.1a), or by a fraction of the SCAPE residual in the case

of CSI (Fig. 5.1b). From the analysis of Chapter 3, we showed that ra tends to be

relatively large over the lower latitudes relative to la because both CAPE and SCAPE

are large over low latitudes and tropical areas. Thus, simply using Ts without regard to Is may result in an overestimation of (a;;) in the lower latitudes and an alw underestimation in the middle and high latitudes. To correct this bias, la together

with the Coriolis parameter 1 are used to constructed a latitudinal filter Fi in the

Conn

(5.19)

where 10 is dimensionless given by the ratio of1 and the angular velocity of the earth

2f!, L is a constant, and 'Y is a normalization constant because both Is and 10 are less

than 1. The introduction of 10 in (5.19) reflects the emphasis for slantwise instability

in higher latitudes.

Recall that the 1-D SCPS is based on the assumption ofsmall horizontal variations

of the thermal wind and absolute vorticity. The 1-D approach would perform similar

to the 2-D scheme if the flow is exactly uniform in the horizontal direction. In

principle, the momentum tendencies in a local column should represent the average

over a slantwise path. If the slanted path is weil within a grid box, the momentum

tendency should be applied to the same grid box. However, if the slanted path is

close ta or even larger than the scale oC a grid box, it is more appropriate to apply the • 78 momentum tendency to a neighboring column to better represent the average effect • of the convective eddy transport. Ta detennine the exact grid box to apply the momentum tendency, we first cal-

culate the horizontal scale of the slantwise convection between the cloud hase and

the cloud top using (5.14). Strictly speaking, this length scale is ooly appropriate in

a local plane perpendicular to the direction of the thermal wind. In practice, this

plane is more or less parallel to the meridional direction in a GCM. Let L y denote

the meridional component of the horizontal scale L. Then the numher of grid point

the momentum tendency will he shifted is determined by

y À y ) Jshi/t..= lnt (L~y + ILylL ' (5.20)

where f)"y is the meridional grid scale and À is a constant parameter which controls

where the shift starts. For example, if À = 0, then iShi/t increases by one only if L y

exceeds one grid length (Figure 5.2). When À = 0.5, one grid point is shifted if L y

exceeds half a grid length. When À = 1, any slantwise length scale larger than zero

will result in shifting the tendency ta the next grid point. Note that since the shift

criterion is based on the horizontal scale of the slantwise path at each grid column,

the number of grid point shifted is different for different grid points.

5.4 Triggering function and relaxation time

The triggering function used in this scheme is the combination of SCAPE residual

r, defined as SCAPE - CAPE and the fractional SCAPE residual Is. The scheme is • 79 activated whenever

T. > 100 J/kg Ta> 0 J/kg • or (5.21) { !& > 10% { !& > 30%

are satisfied. The use of rais because of the emphasis on the inertial mode in the

scheme. Since over the tropical areas, r & is usually a small residual ofthe large SCA.PE

and CAPE, !& is also added in the triggering function as the measure of the relative

importance of the inertial mode relative to the buoyancy mode. The first condition

in the triggering function is designed to suppress convective activities in areas where

Ta is relative large but still a small residual between SCAPE and CAPE, particularly

in the tropics and the low latitudes. The second condition reflect an emphasis to

those areas where rs is small but the fractional SCAPE residual is becoming more

important, like over the middle and higher latitudes.

Ta close the scheme, we need ta specify the convection relaxation time 'j in (5.11).

The results of the Lagrangian model in Chapter 2 indicated that the time scale of the

4 1 inertial mode for SBI ranges from 0.94 f"oJ 2.2(/,,,)-1/2. Taking 10- 8- as the typical

value for f and TI for midlatitudes, the relaxation time turns out to he 3 f"oJ 6 hours.

Other studies Ce.g. Bennetts and Hoskins 1979, Thorpe and Rotunno 1989, Xu 1986,

and Reuter and Yau 1990), show the time scale for CSI ranges from 1 f"oJ 5 hours, or

longer. In our experiments with this scheme, the relaxation time is set at 3 hours.

Sorne sensitivity tests on varying the relaxation time will he reported in Chapter 8.

• 80 •

Chapter 6

The Response of the Large-scale

Mean Circulation to the Slantwise

Convective Parameterization

Scheme

6.1 The CCC GCM

The first atmospheric general circulation model developed at the Canadian Clïmate

Centre (CCC GC~U) was reported in Boer et al. (1984a). Since then, CCC, DOW CC­

Cma (the Canadian Centre for Clïmate Modelling and Analysis), has made numerous

improvement in the model for better simulation of climate. The second-generation • 81 CCC general circulation model (GC~III, ~IcFarlane et al. 1992) employed a trian­ • gular spectral truncation of 32 longitudinal waves (T32), a resolution substantially higher than the standard T20 of GClVII. Vertically, it uses a hybrid coordinate TI

which is terrain following in the lower troposphere, but becomes more like a pressure

coordinate further away from the surface (Laprise and Girard 1990; NlcFarlane et al.

1992). GCMII is also coupled to a simple mbced layer ocean model and a sea ice

model. The moist convective adjustment and the large-scale condensation schemes

are similar to those outlined in Daley et al. (1976). Interested readers are referred to

Boer et al. (1984a) and ~IcFarlane et al. (1992) for further details.

In this study, the implementation of the slantwise convective parameterization

scheme (SepS) and subsequent experiments are based on version Il of the third­

generation CCC GCJ\iI (gcm11). Among other changes from GClVIlI, gcm11 replaces

the moist convective adjustment scheme with a penetrative mass flu..x CPS (Zhang and

wlcFarlane 1995a) based on the plume ensemble concept of Arakawa and Schubert

(1974) but simplified in three aspects. First, the updraft ensemble is assumed to

comprise only of plumes which are sufficiently buoyant to penetrate through a local

conditionally unstable layer. Second, aIl such plumes are equally likely and they have

the same upward mass fltL"<: at the base ofthe convective layer. Third, moist convection

occurs only when CAPE is present and CAPE is removed according to an exponential

rate with a specified adjustment time scale. The first two assumptions result in

an analytical expression for the ensemble cloud mass flux. The third assumption

yields a closure condition similar to the work function of Arakawa and Schubert. • 82 Another improvement in gcm11 is the higher resolution both in the vertical and the • horizontal directions. In all the experiments presented in this study, 22 vertical levels and 47 longitudinal truncation waves are used. The model also has a 96 x 48, or

3.75° x 3.75° grid mesh on which ail physical processes, including the CPS and SCPS,

are calculated.

Although gcmll simulates the current climate quite weil, some deficiencies still

exist. One of which is that the strength of the zonally averaged mid-latitude jets are

overestimated, especially for the Southem Hemisphere (lvlcFarlane et al. 1992). An­

other weakness is that the model produces much stronger mean meridional transient

eddy transports of temperature and momentum (v'T' and v'u') relative to the climate

of the real atmosphere. We will show that implementation of the seps cao alleviate

sorne of these problems.

6.2 The three-month experiments

Before undertaking a long-term simulation with the SCPS in the next chapter, it

is important to understand how the SCPS affects the general circulation in gcrn Il.

We therefore carried out a set of three month experiments, with the goal of diag­

nosing the response of the model to the inclusion of the 1-D slantwise convective

parameterization.

We initiated gcml1 from a restart data file at the end of a November. The restart

file was obtained using the NCEP (National Center for Environmental Prediction)

analysis as the initial condition in gcm11 which is then integrated for more than a • 83 year. The experiments are mn for three months to cover a full Northem Hemisphere • winter season (DJF). Table 6.1lists the parameters and features of the seps used in the experiments. They are described in detail in Chapter 5.

Parameter/Feature Values/Status Equation Momentum tendency Enabled (5.11) Slantwise precipitation Enabled (5.16) Moisture tendency Enabled (5.17) Temperature tendency Enabled (5.18) T > T OJ/kg Triggering function s 100J/kg or { s > (5.21) { fs > 10% fs > 30% Cloud fraction (u) 0.5 (5.13) Moisture distribution factor (Ct) o (5.17) Filter power factor (1.) 2 (5.19) Filter compensation factor ('Y) 32 (5.19) Relaxation time (T) 3 hours (5.11)

Table 6.1: Parameters and features of the seps used for the three month experiments (a1so for the 5 year simulations in later chapters).

Before describing the results, we comment brieflyon the notation in labeling the

experiments. Each label begins with either SLW or NOSLW, denoting whether the

SCPs is included or excluded. For the experiments with the inclusion of slantwise

convection, SLW is followed by a seria! number nn and then one or two characters

xx. This is then followed by a time marker. For example, SLW42E-DJFl represents

an experiment including slantwise convection with a seria! numher 42E. The time

period is the first DJF since the start of the integration.

Figure 6.1 shows the zonally averaged momentum tendencies produced in

SLW42E-DJFl. AlI figures are time averaged over the three month integration

period. 1t is clear that the u-component momentum tendency (::) is similar ta 5lw • 84 •

Figure 6.1: Zonally averaged slantwise tendencies of (a) local momentum (~).,Ut, and Cb) it's u component and (c) v component produced by seps in experiment SLW42E.DJFl. Plots are in units of 1 x IÜ-5m 52 with pressure (hPa) as ordinate and latitude (degree) as ab5cissa. • 85 the momentum tendency (a: ).lw in the direction of the local thermal wind. In other • words, the thermal wind is on average roughly aligned along the longitudinal direc- tion, in agreement with the mostlyeast-west alignment of the large-scale isotherms

in the midlatitudes. As a result, the meridional momentum tendency (:;).lw is rela­

tively small (Figure 6.1e). Another striking feature is that the momentum tendencies

are mainly located over the mid-latitudes, with negative values at upper levels and

positive values below. This eharaeteristic reflects the design of the SCPS which tends

to reduce the vertical wind shear in regions with sufficient SCAPE residual Ts and

fractional SCAPE residual fs.

Figure 6.2 presents the zonally averaged total precipitation and the precipitation produced by the seps. The moisture and temperature tendencies resulting from

c. ] os

1 •

...

~. J' 1 -JI l.HlrJQt,Il'

Figure 6.2: Zonallyaveraged distributions of precipitation (units mm/day) produced by (a) the slantwise convective parameterization scheme and (b) the total precipitation vs. latitudes for run SLW42E-DJFL

the precipitation produeed by slantwise convection are shawn in Figure 6.3. The peak • 86 ...IZ, ..••1 • u. Q~ 1.' 112 lU Z5Z Z52 .ul ~ .ul u. ~.

512 l SlZ

700 100 r ,~, /\ " ... ,,~ .... _,'\ ... "' ' ~ v .. ' , .,. ,"':,'::;.:.":.\"'-:::.~ .,. ..2 "Z /+,. tn tU tG 10 JIl 0 -JIl -10 -tO 10 10 JO 0 -JO -10 -10 (.1 1.1

Figure 6.3: Zonally averaged (a) moisture tendency (units 10-1 g kg-1S-1) and (b) temperature tendency (units 10-1 K/s) produced by the SCPS in experiment SLW42E-DJFl. Coordinates are pressure (ordinate in hPa) and latitude (abscissa in degree).

values oftotal precipitation over the mid-latitudes ranges from 2 to 3 mmlday (Figure

6.2a). Slantwise convection contributes about 20-25% to the total precipitation rate

over the mid-latitudes. The moisture deficit is mainly located at the lower model

layers because the moisture distribution weight factor a in (5.17) is set ta 0, which

effectively assumes that aIl condensed liquid water is taken out from the subcloud

layer. The temperature tendency profile in Figure 6.3b is the result of the even

distribution of latent heating from the subcloud layer up to the cloud top. We will

show later that the contribution ta the large scale circulation from the thermodynamic

part of the scheme is not significant but it serves to ensure a proper energy balance.

Fig. 6.4 shows the effect ofthe SCPS on the zonally averaged SCAPE and SCAPE

residual Ts • Consistent with the EClVlWF analysis presented in Chapter 3, the most

significant region of SCAPE ( > 1000 J/kg ) is over the low and tropical latitudes • 87 15N

• liN

4511

~IN

15••

1'" 'J"

1 'li' D' -DI 11' 48. 40' 4.'n. U. II'

~o. IC' 1.' Iii

~. e' .11

J'

• Je "'. LA .='"IlŒ• '" •• ~.

45 ••

1'5 1. :~

·li 0.

Figure 6.4: Comparison of zonally averaged (a) SCAPE (in J/kg) and (h) SCAPE residual r. (J/kg) hetween runs SLW42E-DJFl (solid) and NOSLW-DJFl (dashed). (c) depiets the initial fractional SCAPE residual f. in percent from NOSLW-DJFL • 88 (from 15° N to 25° S for this DJF). However, this is also where CAPE co-exists • with SCAPE. Furthermore, in this region, CAPE is almost always the predominant part in SCAPE, meaning that the buoyancy mode dominates the inertial mode. The

zonally averaged fractional SCAPE residual fa, which is a measure of the relative

importance of inertial mode in SBI, indicates that the inertial mode is relatively

important ( > 10% ) over the latitude belts 30 - 50° N and 30 - 60° S (Figure 6.4c). Note that the seps produces most of the reduction in SCAPE and ra over the same

regjons (Figures 6.4a and b) in agreement with the location of the largest momentum

tendencies depicted in Figure 6.1. The reduction in ra peaks around 40° N and 50°

S where r.s is reduced respectively to one fourth and one third of the original values

in the NOSLW-DJFl ron. The latitudes for maximum reduction also coincide with

the location of peak fractional SCAPE residual fa (30 ....- 40° N and 50° S) shown in

Figure 6.4c.

Figure 6.5a gives the difference field (SLW42E-DJFl minus NOSLW-DJF1) of

zonally averaged u between the two rons. As expected, u is reduced near the

centers where the SCPS is active, particularly around 35° N and 45° S. However,

the latitudinal extent of zonal wind reduction is narrower than the latitudinal extent

with substantial momentum tendencies (Fig. 6.1a). Furthermore, within the regions

marked bya negative momentum tendency above 700 hPa, positive zonal u difference

patches aIso occur. Fig. 6.5b depicts two major patches of positive temperature

difference below 250 hPa over 35 - 60° N and 45 - 60° S, and patches of negative

temperature diff'erence below 250 hPa over 20 - 30° N and 74 - 90° N. As a result, • 89 •

Figure 6.5: Zonally averaged differences of (a) u (in mIs), (b) T(in 0.1 K), (c) v ( in mIs), and (d) w (in 10-2 Pals) between SLW42E-DJFI and NOSLW-DJFl. The ordinate and abscissa are pressure (in hPa) and latitude (in degree), respectively.

• 90 the meridional temperature gradient in SLW42E-DJF1 decreases in the midlatitudes • but increases in the lowand high latitudes relative to NOSLW-DJFl in the Northem Hemisphere. A similar picture can also be seen in the Southem Hemisphere. By

comparing Fig. 6.5a and Fig. 6.5b, it is clear that the regions with negative (positive)

zonal wind difference correspond to regions with a decreasing (increasing) meridional

temperature gradient in SLW42E-DJFl. This observation strongly suggests a thermal

wind balance between the difference fields in u and T.

While the thermal wind relation can explain why there are positive patches of

zonal u difference within a region with overwhelmingly negative momentum fluxes in

the upper levels, it is still not clear how the temperature difference arises. Before one

attributes too hastily the warming (Figure 6.5b) to the direct effect ofdiabatic heating

produced by the precipitation in the seps, we should note that Figures 6.5c and 6.5d

display a strong signal in the response of the mean meridional circulation, implying

that another process may contribute to forming the warm temperature differences.

We will design a few special experiments in the next section to clariCy this issue.

6.3 Large-scale response ta slantwise momentum

transport: a mechanism via a secondary mean

meridional circulation

Momentum transport is the major characteristic in this slantwise parameterization

scheme. To isolate the response to the momentum transport, an artificial momentum • 91 tendency CC:)OI.. only, Fig. 6.6a) is specified over the latitude belt of 25 - 55° • in both hemispheres. The tendency is distributed uniformly over latitude circles with an amplitude found typically in our 3 month experiments. The tendency is

heId fbced during a one-day integration and the experiment will be referred to as

SLW40G-1DAY. The temperature and moisture tendencies in (5.17) and (5.18) are

disabled. The large-scale response, indicated by the differences bet,veen SL\V40G-

IDAY and NOSLW-1DAY exhibits features similar to those ofSLW42E-DJFl (Figure

6.6b and c). There are patches of negative differences in 'U over the latitudes where

C:)'I" is active, with positive wave-like patches on both sides. Significant positive

temperature differences dominate over regions where the SCPS is active, with smaller

patches of negative temperature difference over the lower latitudes. The difference

plots for the zonal averaged 'U and w clearly indicate two meridional ceUs surrounding

the two regions with active momentum transport in both hemispheres (Figure 6.7).

These ceUs are called secondary meridional circulations because they are induced by

the momentum tendency of the scheme and are anomalous circulations relative to

run NOSL\V-1DAY. Since there is no direct temperature tendency from the SCPS

in this experiment, the temperature difference in Figure 6.6c must be related to the

secondary meridional circulation. But how does the u momentum tendency generates

the meridional ceUs in the first place'?

In two classical papers, Eliassen (1952) and Kuo (1956) came to the following

conclusion using a linear, zonally averaged, 2-D steady state model including diabatic

heating, friction, and eddy heat and momentum flu.xes (the sO-called Kuo-Eliassen • 92 •

••Il 1•• III lU ' .. UI

'-J. .;;~~~:~ t,------;1 •1"___-----, S'2 , 1 \ -----_..-' 700 j ... Il. '-U tU 10 -10 -10

150 1 tU

,aool-ol__...... """"':l,I-_""""'....._....Io.Io...... _ ....._ ..~_.IoI..~__~ 10

100

100

Figure 6.6: Zonally averaged (a) (~)."u from the artificial (~)dlU 1 day experiment SLW40G­ IDAY (in lO-6ms-2), and (b) the difrerence fields u (in mIs) and (c) T (in 0.1 K) between SLW40G­ 1DAY and NOSLW-IDAY with the same ordinate and abscissa as in Figure 6.1 and Figure 6.5. • 93 10 la 50 I:~U ,.. 50 • lOG ) ... lOG 150 150 20G \.. 200 ~ L 250 ---"""=-, ( 250 JOG Q ~ L5!'-~):~) JOG <\:,:::,~,:~/ 400, .. .00 .0 . . ~ ~J ~.... ---.".' loa

70G \ 700 ~,"-" JI ,,-- ..., U 150 1 ---.'~ 150 J : ,,", __ ":~\.', GA Â tU l' \...... , n~ \'~ 100G " 111111: 1 ta 10 JO -JO -10 -10

Figure 6.7: Zonally averaged düference fields of (a) v (in mis) and (h) w (n 10-2 Pals) between SLW40G-IDAY and NOSLW-IDAY with the same ordinate and abscissa as in Figure 6.5.

equation):

A source of westerly momentum near the ground and a sink at higher levels will

induce a direct circulation.

Note that the word "direct" is a terminology used in their papers. It refers to rising

motion in the lower and warmer latitudes and descending motion in the higher and

cooler latitudes. Because a linear model is used, their conclusion can he applied to any

separate eddy forcing, including the slantwise convective momentum flux in our case.

Recognizing that the (::).Iw tendency from our seps represents a source ofwe'lterly

u at lower levels and a sink of u above, we can explain the secondary meridional

circulation presented in Figure 6.7, as weil as Figures 6.5c and d by the Eliassen-Kuo

theory. Note that the areas of "warming" coincide with the descending branches ofthe

"direct" cells, while the areas of "cooling" are associated with the ascending branches. • 94 Thus the temperature difference displayed in Figure 6.6c is expected to result from • adiabatic heating generated by the secondary "direct" meridional circulation. Once the temperature changes occur, the zonal wind adjusts according to the temperature

anomalies so that the large-scale flow remains in thermal wind balance. As a result,

we observe negative (positive) u difference over the regions with weakened (enhanced)

temperature gradients.

Finally, we configured the model with the same set of parameters as in run

SLW42E-DJF1 except that we disabled the slantwise precipitation and its thermo­

dynamic effects (q, T tendencies). This momentum flux only run is denoted as

SLW42EM-DJFl. Fig. 6.8 shows that at a very short time of just 2 time steps

(denoted as SLW42EM-2STP and NOSL\V-2STP in Fig. 6.8), the secondary merid­

ionaI circulation has yet to be set up in the v and w fields. Therefore the T anomaly

is negligible (Fig. 6.8b) and the differential u is similar to the tendency field in Figure

6.9.

After one day of integration, the secondary meridional cells are emerging in the

v and w fields (Figure 6.10c and cl). Then the corresponding adiabatic warming or

cooling anomalies appear in the differentiaI T plot (Figure 6.10b). As a result, the

differential u field starts to adjust to the thermal wind balance, especially in the upper

level from 500 hPa to 100 hPa. For example, in the Southem Hemisphere, there is

a patch of negative u difference between 30° S and 45° S and a patch of positive u

difference poleward of the negative patch in u centered around 300 hPa between 45°

S and 65° S above the level of 500 hPa. The location of the negative (positive) patch • 95 •

Figure 6.8: Zonal1y averaged differences of (a) u (in 0.1 mis), (h) T (in 0.01 K), (c) v (in 0.1 mis), and (d) w (in 0.01 Pals) between SLW42EM-2STP (two time-step, momentum adjustment ooly) and NOSLW·2STP. The ordinate and abscissa are the same as in Figure 6.5.

• 96 •

-10

6 Figure 6.9: Zonally averaged momentum tendency c%T ).'w (in 10- ms-2 ) from SLW42EM-2STP with the same ordinate and abscissa as in Figure 6.1.

well matched the location of the negative (positive) temperature gradient difference.

In the Northem Hemisphere, the field of difference in 'u exhibits finer structure but

operates in a like manner. For instance, there are two patches of negative u difference

at around 35° N and 55° N above 500 hPa with a positive patch in between. This

positive patch is located over a region with a positive temperature gradient difference.

As expected, the response toward a thermal wind balance is not clear at the lower

layers because of the presence of the boundary layer and the lower surface.

Fig. 6.11 shows the difference between SLW42EM-DJFl and NOSLW-DJFl av-

eraged over the same three month periode When compared to the difference field

between SLW42E-DJFl and NOSLW-DJFl in Figure 6.5, it is evident that without

the latent heating from the slantwise precipitation the magnitude of the temperature

difference is about 5 degree smaller over the midlatitudes. However, the differential

mean meridional circulation represented by v and w (Figures 6.l1c and d) exhibits • 97 •

Figure 6.10: Zonally averaged difl'erences of (a) u (in mis), (h) T (in 0.1 K), (c) v (n mis), and (d) w (in 0.01 Pals) between SLW42EM-1DAY ( momentum tendency only 1 day integration) and NOSLW-IDAY. The ordinate and abscissa are the same as in Figure 6.5.

• 98 •

ICI 1 SO SO 1CIO 1CIO I~CI ISO ZCIO ZCIO Z5C1 Z5Cl 5110 5C1O eoo eoo

5C1O 5C1O

'CIO ICIO

7C1O 7C1O

150 ,1.5

10 10 10 -10 -10 -10

Figure 6.11: The same as Figure 6.5 but between runs SLW42E~I-DJFl (the first DJF season momentum tendency only) and NOSLW-DJFl.

• 99 almost the same pattern as in Fig. 6.5, but with a slightly stronger amplitude. This • result affirms that the secondary meridional circulation is induced mainly by the ma.. mentum flux in the SCPS. It is aIso useful ta point out that the shape orthe difference

field in u is similar between the two experiments because away from the boundary

layer, the u field responds mainly to the gradient of the temperature difference rather

than the absolute value of the temperature difference. This is not surprising since

momentum flux is the primary forcing in the SCPS while the role of the temperature

flux is only secondary but acts to ensure the conservation of energy.

• 100 •

Chapter 7

The Effects of the Slantwise

Convective Parameterization

Scheme in a 5-Year Climate

Simulation

Ta examine the effects of the slantwise convective parameterization on the simulated

climate, a 5-year run (SLW42E-5YR) using the same settings as in Table 6.1 has been

performed. The simulation begins on December lst of the first year, and terminates

at the end of August in the sixth year. The time averages of various quantities over

the 5 DJF and 5 JJ.~ seasons were then compared to those from a 5-year run with no

slantwise convection (NOSLW-5YR). Comparison is aiso made with well-established

long-term observational analyses. For SCAPE and its derived variables (rs and !s), • 101 our analysis is based on the 15-year ERA data. For other climatic fields, the 10- • year NCEP re-analysis (NRA) available at CCC is used because it is the traditional benchmark to gauge the CCC GC~I simulation results.

7.1 Effect on reducing SCAPE in midlatitudes

The 5-year mean tendencies of zonally averaged momentum (u component) and tem-

perature for DJF in SLW42E-5YR (Figure 7.1) are similar to those from SLW42E-

DJFl (Figure 6.1b and 6.3b). The momentum tendency peaks around 40° in both

hemispheres. The latitudinal extent and the magnitudes are however larger in the

tZr------, Q. ,.z'~--=:=c?~~ __ ~ z'z , " ... ' ", J,SI , ' • '. , 1..• r .J" ,,- ... :,' ~.: r \.~: " l' 1 ., , '&2 1 l ,,1 # ~. ,,,...../', .,-'' 100 ,'li.. ..,2 JO ,.,0 -tel ·10 .0 ·10

5 Figure 7.1: 5-year means ofzonally averaged (a) (~).ltu (in units of 10- ms-2 ) and (b) (~).ltu (in units of 10-7 K/s) for DJF from SLW42E-5YR. The ordinate is height in hPa and the abscissa is latitude in degree.

Northem Hemisphere. The situation reverses itself in JJA (Figure 7.2), confirming

the fact that slantwise convection is more active in winter seasons because of the

stronger vertical wind shear in accordance with the theory of slantwise instability. • 102 However, the seps generates a more asymmetric distribution in the zonallyaver- • aged tendencies for u and T in JJA than in DJF. A comparison of the local tendency

Figure 7.2: The same as Figure 7.1 but for JJA.

(a;; ).,., with (::).,., and (:).1., (not shown) demonstrates the dominance of the zonal component and supports the conclusion of the last chapter that the local planes

are atigned close to the meridional direction.

To reveal the geographic distribution of the slantwise convective activity, we

plotted the mean column averaged absolute tendency of the local momentum V 1 kt al!; (k _ kIL 1 ôt' I.slw, where kt and kb represent the levels of the cloud top and t b + î=kb cloud base, respectively) for both seasons in Figure 7.3. The main areas of activity

are midlatitude oceans in the winter hemispheres. The stronger zonally averaged

momentum tendency in the winter season of the Southem Hemisphere is associated

with the mostly uninterrupted midiatitude oceans.

The time mean of the zonally averaged SCAPE residual r s over the 5 years is • 103 •

(a)

(b)

Figure 7.3: The Mean global distribution of column averaged absolute local momentum tendency (k _ ~ 1 t 1a; I.,cu) in units of mis per 3 hours, where kt and k6 represent the levels of the t 6 + i=t. cloud top and cloud base, respectively, and 3 hour is the relaxation time T used in our simulation) for (a) DJF and (h) JJA frOID run SLW42E-SYR. • 104 sunHar to the one season run (Figure 7.4a and Figure 6.4b). For JJA (Figure 7.4b), • the difference between the runs with and without slantwise convection is larger over the Southem Hemisphere. Sînce the SCAPE residual is generally smaller in SLW42E-

SYR, the differences in the SCAPE residual between SLW42E-SYR and NOSLW-SYR

will he called the reduction of r s for experiment SLW42E-5YR.

Fig. 7.S displays the global distribution of the reduction. In DJF, there are two

major bands in the midlatitudes over the North Atlantic and the North Pacific (Fig.

Jn JII' 28. 2111 20. .1>1 '411 U. 22. '2' 21. Jill 'B' ,aa '1>tI lI.a '4. ,., ,-, , \ -,. '21 , \ ...... ,- \ ·'1 , ,., \ , \ 1111 1 \ , Il' \ o. 1 O. \ 4. 1 41 \ 1" \ / 'II / 21 \ .... \ Il , ~I Il' J' • JI 1111 'II 'III JI • J' ~H:~ ~Hl f'JO[ lM ,',

Figure 7.4: Zonallyaveraged Mean SCAPE residual r. for (a) DJF and (h) JJA from runs SLW42E­ 5YR (solid line) and NOSLW-SYR (dashed Une). The units in ordinate are J/kg.

7.Sa) . In JJA, a continuous band extends around the whole longitude circle in the

midlatitudes of the Southem Hemisphere (Fig. 7.5b). Maxima are found around the

eastern South Pacific, western South Atlantic, and the western Indian Ocean. Unlike

DJF, there are significant reduction in r 5 even in the summer hemisphere, particularly

over the midlatitude continents ofNorth America, Europe and Northeast Asia. While

the reduction over the ocean waters in the winter hemispheres is associated with the • lOS strong large-scale baroclinic systems, the reduction over the continents in summer • may be associated with midlatitude mesoscale convective systems. Figure 7.6 depicts the distribution of the mean SCAPE for the two experiments.

In general, the distribution in SLW42E-5YR agrees better with the ERA (Figure 3.3).

Specifically, in DJF, the ron without slantwise convection produces too much SeA.PE

over the North Atlantic and the Northeast Pacifie. Furthermore, the magnitude for

SCAPE is also too large over tropical areas in NOSLW-5YR. In contrast, the seps in

SLW42E-5YR reduces the excess SCAPE over the North Atlantic and the Northeast

Pacifie. However, the reduction appears to be a bit excessive over the western and

central parts of the North Pacifie. The large SCAPE over the tropical areas is not

reduced because the SCPS represents only the effect of the inertial mode in SBI, and

has little effect at lowand tropical latitudes.

A comparison of Figure 7.7 with Figure 3.3 shows that in JJA, the distribution of

SCAPE in SLW42E-5YR is significantly better than in NOSL\V-5YR over the mid­

latitudes of the Southern Hemisphere, particularly in areas over the South Pacifie,

South Atlantic, and the Indian Ocean. Over the continents in the Northern Hemi­

sphere, the scheme over-reduced SCAPE over eastem Europe, but under-reduced it

over Northeast Asia and Northwest America.

• 106 •

(a)

-100.0 -50.00 O.OOOOE 50.00 100.0 .liliiliillIDD0~

(b)

Figure 7.5: The global distribution of the difference in SCAPE residual T. (in J /kg) between SLW42E-5YR and NOSLW-5YR for (a) DJF and Cb) JJA. • 107 •

(a)

100.0 ~OO.O 1000. 3000. 5000. DlilllliillItzd~B.

(b)

Figure 7.6: The global distribution of SCAPE (in J/kg) for DJF from (a) NOSLW-SYR and (h) SLW42E-SYR. • 108 •

(a)

100.0 500.0 1000. 3000. 5000. DliillillillrzJ~~.

(b)

Figure 7.7: The same as Figure 7.6 but for JJA. • 109 7.2 Effect on decreasing the strength ofthe upper­ • level jets in midlatitudes and the cold temper­ ature bias

As mentioned earlier, one of the deficiencies in gcmll is that the upper level jets

over the Southem Hemisphere is too strong. Figures 7.8 and 7.9 depict a comparison

of the mean zonal wind for SLW42E-5YR, NOSLW-5YR, and the 10-year NCEP

re-analysis (NRA). The locations of the jet maximum in NRA for the summer and

winter seasons are at 30 and 45 degrees respectively (Figures 7.8c and 7.9c). Without

slantwise convection (Figs. 7.8b and 7.9b), the upper level jet maximum is about 10

mis stronger than the analysis in the Southern Hemisphere in bath DJF and JJA.

Although the magnitude of the jet maximum in the Northem Hemisphere is quite

weIl simulated in DJF, it is again tao strang in JJA.

When the SCPS is included, the zonal jet is reduced by about 5 mis in the South­

em Hemisphere for DJF (compare Figures 7.8a and 7.8b) and is in closer agreement

with the NRA (Figure 7.8c). The jet maximum over the Northem Hemisphere re­

mains simHar to experiment NOSLW-5YR and agrees weil with the NRA. For JJA,

the location of the jet maximum relative ta the NRA in the Southem Hemisphere is

better in SLW42E-5YR. Over the Northem Hemisphere, the magnitude of the jet in

SLW42E-5YR is a bit weaker than the NRA.

Ta clarify the effect of the SCPS on the upper level jet, we plotted the difference

fields of the mean zonal wind and the mean temperature between NOSLW-5YR and • 110 •

100

700

ISO lU laoo~_-.--~~""""'_"""""'__~=--'--"'-~ 10

Figure 7.8: The 5 DJF Mean of zonally averaged u component of wind from (a) SLW42E-5YR, Ch) NOSLW-5YR, and (c) NCEP re-analysis. The ordinate is in bPa and wind in mIs. • 111 •

5010r:;==~~~~~~~ 100 150 200 no 100

150 .n 1000 tG to

la 50 100 150 200 250 JOO

.00

500

.00

700

150 .25 1000 tG

1000 0 ta IcI

Figure 7.9: The same as Figure 1.8 but for JJA mean. • 112 NRA as well as between SLW42E-5YR and NOSLW-5YR in Figures 7.10 and 7.1l. • The first thing to note is that the zonal wind difference between NOSLW-SYR and NRA can be largely explained by the temperature bias. A strong and deep cold bias

relative to the NRA appears poleward of the jet axes over both hemispheres in DJF

(Figure 7.10d). However, a warm bias extends from SSO to 2S0 hPa equatorward

of the jet axis in the Southern Hemisphere. In JJA (Figure 7.11d), there is a cold

bias poleward of the jet axis in the Southem Hemisphere, a warm bias north of the

Northem Hemisphere jet below around 450 hPa and a cold bias aloft. The region

equatorward of the jet axes is marked by a cold bias below and a warm bias aloft.

The temperature differences result in a thermal wind adjustment which is reflected

in the zonal wind differences.

The effect of the seps can now' be assessed from the difference fields between

SLW42E-SYR and NOSLW-SYR. Figures 7.10b and 7.11b show that the temperature

in the run with slantwise convection is much warmer than the ron without poleward of

the jet axes in both hemispheres in both seasons below about 200 hPa. In particular,

the wanner temperatures in SLW42E-5YR compensate the cold bias in NOSLW-5YR

over the Northem Hemisphere and the Southem Hemisphere in DJF (Figure 7.10d)

and over the Southern Hemisphere in JJA (Figure 7.11d), thus bringing the tempera­

ture distribution in SLW42E-5YR doser- to the NRA. An exception however occurs in

the summer over the Northern Hemisphere, where the temperature in SLW42E-SYR

would be even wanner than the too warm temperature in NOSLW-5YR poleward of

the jet axis below about 250 hPa. • 113 •

400 soo

lOG

700

ua tU 1000 -10 to 10 Il -JC1 Ici

Figure 7.10: Zonally averaged diff'erence fields of (a) u (units mis) and (b) T (units O.lK) in DJF between SLW42E-5YR and NOSLW-5YR runs. AIso the difFerences of (c) u (units mis) and (d) T (units O.lK) in DJF but between NOSLW-SYR and NCEP lo-year re-analysis. The ordinate is in hPa and latitude along the abscissa in degree.

• 114 •

F·19ure 7.11: The same as F·19ure 7.10 but for JJA.

• 115 Similar to the mechanism in the one season experiment discussed in Chapter 6, • the WarIn temperature anomaly in SLW42E-5YR results from the direct secondary meridional circulation in response to the momentum transport by slantwise convection

(Figure 7.12). Through the thermal wind relation, the strength of the zonal jet is

altered. Comparison of Figure 7.11a and Figure 7.11c indicates that in JJA, the

zonal wind difference between SLW42E-5YR and NOSLW-5YR is in general opposite

to that between NOSLW-5YR and NHA. This opposite phase relation also appears in

DJF over the Southem Hemisphere (compare Figure 7.10a and Figure 7.10c) and in

the Northem Hemisphere between 30N and SON. However, poleward of SON, the ron

with slantwise convection tends ta overestimate the already tao strong zonal wind

during Northern Hemisphere winter. Our results therefore suggests that the seps

acts in the right direction in reducing the strength of the too strong jet and the cold

temperature bias in gcmll, particularly for the Southem Hemisphere.

1.3 Effect on weakening the meridional transient

eddy transports

The seps affects strongly the mean meridional transient eddy transports and alters

their maximum values by as much as SO% in some cases. Figure 7.13 shows the mean

meridional transient eddy transports of momentum u'v' and heat v'T' in NOSLW­

5YR and the corresponding difference fields between NOSLW-SYR and NRA. in DJF.

Without slantwise convection, very large poleward transports ofu momentum (Figure • 116 •

IO'..-.....-..,...,.-...... - ...... -.,..-.....--~,...... '=",...... "..-...---. 3CI 100 ua zoo m 100

-10

Figure 7.12: Mean zonally averaged differences of (a) v (units mis) and (h) w (units 0.01 Pals) hetween SLW42E-5YR and NOSLW-5YR for DJF mean. The same differences of (c) v and (d) w but for JJA.

• 117 •

,,

• : ~ 1 " 1 1 l, • 1 ,,/ / ;:1 1 , ·10 '1 ., ,"f,. :, '...... __ ,' 1_,1

-tll -tll

2 2 Figure 7.13: The 5 DJF mean ofzonally averaged transient eddy transports (a) u'v' (units m s- ) and (h) v'T' (units m K S-1) from run NOSLW42-5YR, and the differences of (c) u'v' and (d) v'T' between NOSLW-SYR and NCEP re-analysis.

• 118 7.13a) and temperature (Figure 7.13b) at the midlatitudes are evident. Relative • to the NRA, gcmll over-predicted the poleward transports especially at the upper levels in midlatitudes (Figures 7.13c and 7.13d). In JJA, NOSLW-5YR depicts in

the 80uthern Hemisphere a strong convergence of u'v' between 50°8 - 60°8 (Figure

'O·I:":""""-----=-=~=""-=::r-....".:!"":"::---:-="'--:l ,: ;> ,~ 200 ua JOO

100

1100

100 100

150 ISO us lU 'OOOta~~~=--~~~a~--.:--~JO...... :-~&O~-...... ~-to laoo __~ ...... _~...... _~~---..~ .....-..;,:::..o-~ ta -&0 -tG 1"1

Figure 7.14: The same ?s Figure 7.13, but for the 5 JJA mean.

7.14a) accompanied by a strong poleward transport of heat (Figure 7.14b). Relative

to the NRA, gcmll over-estimates the transient eddy momentum flux convergence

in the 80uthem Hemisphere but under-estimates the poleward u transport in the • 119 Northem Hemisphere (Figure 7.14c). Similar to DJF, the poleward transport viT' is • again over-estimated in the Southem Hemisphere (Figure 7.14d). Figures 7.15 and 7.16 depict the transient eddy transports in SLW42E-5YR and

the corresponding difference fields between the mns with and without slantwise con­ vection. It is evident that the seps exerts a strong influence on u'v' and v'T'. Com­

parison of Figures 7.13a,b and 7.15a,b indicates that the transient eddy transports in

SLW42E-5YR are significantly weaker in DJF. Indeed, the overestimated poleward

transports (Figure 7.13c,d) in the run without slantwise convection are significantly

compensated by the difference fields between SLW42E-5YR and NOSLW-5YR (Fig­

ure 7.15c,d). The much better agreement between SL\V42E-5YR and NRA in terms

of the transient eddy transports ofzonal momentum and heat can be attributed to the

change in the mean meridional temperature gradient as a result of the seps. Specifi­

calIy, as discussed in the previous chapter, the momentum eddy flu..x generated by the

slantwise convective parameterization induces a direct secondary meridional circula­

tion which in tum leads to adiabatic cooling (warming) in the ascending (descending)

branch in the lower (higher) latitude. As a resuIt, the meridional temperature gradi­

ent decreases between the ascending and descending branches and the baroclinicity

in the midlatitude weakens. Since baroclinic weather systems sucb as troughs and

cyclones are the main mechanism for the poleward transient eddy transports (Peixoto

and Oort, 1984), the magnitude ofthe transports therefore decreases. A similar mech­

anism is also responsible for the increase baroclinicity and the stronger poleward u'v'

around 50° N in Figure 7.15c. • 120 •

Figure 7.15: The 5 DJF mean ofzonally averaged transient eddy transports (a) u'v' (units m2s-2 ) and (h) v'T' (units Km s-2) from run SLW42E-5YR, and the difFerences of (c) u'v' and (d) viT' between SLW42E-5YR and NOSLW-5YR.

• 121 •

la :MI 100 :J I:MI ZOO Z50 500

'00

500

telO

700

ua tU 1000 to 10 Ja •••0 ·50 ·tel ·90 -tel -90

la la :MI 5a 100 loa 150 I:MI ZOO ZOO 250 ua JOO JOO

tao 500 -500 100 100

700 700

ua ua n, 1000 10 50 0 -50 -10 -90 90 -90 1••

Figure 7.16: The same as Figure 7.15, but for 5 JJA seasons.

• 122 For JJA, the picture of the transient eddy momentum and heat transports in • the Southem Hemisphere is similar to that in DJF, with u'v' and viT' being smaller in the case with slantwise convection than in the case without (compare Figures

7.14a,b and Figures 7.16a,b). The difference fields of (SLW42E-5YR - NOSLW-5YR)

exhibit similar patterns as those from (NOSLW-5YR - NRA) but are opposite in sign,

indicating that SLW42E-5YR would agree better with NRA in terms of the eddy

transports. However, the picture is different in the Northem Hemisphere. Although

SLW42E-5YR still results in a weaker poleward momentum transport relative to

NOSLW-5YR (Figure 7.16c), the error relative to NRA actually increases because the

northward momentum transport is aIready too weak in the midlatitudes in NOSLW­

5YR (Figure 7.14c).

7.4 Other effects

We also investigate the effect of the SCPS on the mean sea level pressure (MSLP)

and the surface precipitation. Even though the distribution of MSLP in NOSLW­

5YR agrees weIl with NRA. (Figures 7.17a, b and 7.19a, b), there is a tendency for

the subtropical highs to he too strong over the oceans during the summer (Figures

7.18a and 7.20a). The incorporation of the SCPS partially offsets the positive MSLP

anomalies over the subtropics in summer, especially in the Southern Hemisphere

(compare Figures 7.18a and 7.18b).

The situation is more complicated during the winter. Over the Northem Hemi­

sphere, positive MSLP anomalies appear over the regions of the Aleutian Low, and • 123 •

(a)

'15.0 '15.0 1005. 1015. 1025. CIlDDDD~

(b)

Figure 7.17: The global distribution ofmean sea level pressure in DJF from (a) NOSLW-5YR and (b) NCEP re-analysis. The units are hPa. • 124 •

(a)

-1.000 - ••000 O.OOOOE 6.000 a.aoo .IililliillIDD0.

(b)

Figure 7.18: The diff'erences of mean sea leve1 pressure in DJF Ca) between NOSLW-SYR and NCEP re-analysis and (b) between SLW42E-SYR and NOSLW-5YR. The units are hPa. • 125 •

(II)

185.0 915.0 1005. 1015. 1025. DDDDD~

(b)

Figure 7.19: The same as Figure 7.11 but for JJA. • 126 •

(II)

-11.000 -4.000 O.OOOOE 4.000 Il.000 .IillillllilDD0.

(b)

Figure 7.20: The same as Figure 7.18 but for JJA. • 127 the high pressure areas of North Arrica and East Asia (Figure 7.18a). In addition, a • stronger Polar Lowextends from North America, crossing the North Atlantic Ocean to reach Northeast Asia. In the Southern Hemisphere, the MSLP anomalies exhibit

a wave pattern over the midlatitude oceans (Figure 7.20a). In general, the inclusion

of the SCPS improves the situation over the Southern Hemisphere (compare Figures

7.20a and 7.20b) but worsens the situation over most regions of Northem America.

Figures 7.21 through 7.24 show the global distribution of precipitation. In general,

the agreement between NOSLW-5YR and NRA is fairly good (Figures 7.21a, b and

7.23a, b). The inclusion of precipitation from slantwise convection did not change

significantly the precipitation distribution in NOSLW-5YR (Figures 7.22b and 7.24b).

Even though we showed in Chapter 6 that the precipitation associated with slantwise

convection can account for 20-25% ofthe total precipitation in midlatitudes, the SCPS

does not change significantly the total amount of precipitation relative to the NOSLW­

5YR ron. The reason is that the increase in precipitation associated with slantwise

convection in the midlatitudes is offset by the decrease in large scale precipitation in

the subsiding branch of the direct secondary meridional circulation.

7.5 Discussion

We now summarize the response of gcml1 to the incorporation of the slantwise con­

vective parameterization scheme as follows:

1) The momentum tendency (source at low levels and sink at high levels) generated

by the seps induces a direct secondary mean meridional circulation over the • 128 •

(Il)

2.000 ••000 &.000 8.000 12.00 20.00 DlS."Sl~.~œ.

(b)

Figure 7.21: The global distribution of precipitation for DJF from (a) NOSLW-5YR and (b) NCEP re-analysis. The units are mm/day. • 129 •

(a)

-1.000 -2.000 a.DOOO[ 2.000 1.000 .lilililiillDDrzJ.

(b)

Figure 7.22: The difl'erences of precipitation for DJF (a) between NOSLW-5YR and NCEP re­ analysis and (b) between SLW42E-5YR and NOSLW-5YR. The units are mm/day. • 130 •

:LOOO ••000 '.000 1.000 t:LOO 20.00 DlSSl~.OO ••

(b)

Figure 7.23: The same as Figure 7.21 but for JJA.

131

•i •

(3)

-1.000 -1.000 O.oooot 2.000 1.000 • Iiliiliill 0 0 rzl •

(b)

Figure 7.24: The same as Figure 7.22 but for JJA. • 132 midlatitudes. • 2) Adiabatic warming in the descending branch (at mid- and high latitudes) of the secondary circulation and adiabatic cooling in the ascending branch (over

lower latitudes) decrease the mean temperature gradient in the region between

the two branches over the midlatitudes while increasing the mean temperature

gradient over the higher and lower latitudes.

3) To restore the thermal wind balance, the zonally averaged vertical wind shear

weakens in the midlatitudes where the temperature gradient decreases but

strengths at higher and lower latitudes where the temperature gradient in-

creases.

4) The weaker temperature gradient in the midlatitudes decreases the baroclinicity

and baroclinic activities resulting in weaker poleward transient eddy transports

for both u and T in midlatitudes.

The above summary is somehow idealized. The actual response depends on the

season and the hemisphere. In general, the effect of the seps is stronger in winter

than in summer. In terms of improving the simulated climate, we showed that inclu­

sion of the SCPS general1y improves the SCAPE, U, T, u'v', v'T', and MSLP fields

in the Southem Hemisphere in DJF and JJA. The improvement over the Northem

Hemisphere is however mixed, and the simulated climate can become worse in sorne

cases. The reason may be due to the fact that gcm11 is tuned to yield a good cli­

mate simulation in the Northem Hemisphere and readjustment ofother parameters is • 133 needed when a new process is introduced. The other possibility is the more complex • distribution of land and sea in the Northem Hemisphere. Despite these short comings, our SCPS significantly improves the simulated cli­

mate in the following ways:

1) It decreases the excess SCAPE over the open waters in the midlatitudes.

2) It reduces the overly strong zonally averaged jet in the midlatitudes of the

Southem Hemisphere.

3) It decreases the cold bias in the mid- and high latitudes.

4) It weakens the overly strong poleward transient eddy transports of u and T.

5) It yields better simulation of the ~ISLP over the midlatitudes of the South­

ern Hemisphere, especially in the strengths of the subtropical highs during.the

summer.

• 134 •

Chapter 8

The Representativeness of the

5-year Mean and Some Sensitivity

Experiments

8.1 The interannual variability of the 5-year sim­

ulation

Because of the heavy computationalload, we only performed 5-year simulations. It is

of interest to ask whether this time period is sufficiently long to al10w meaningful test

of the effect of the slantwise convective parameterization scheme on the simulated

climate. To answer this question, we analyzed the interannual variability of the 5­

year simulations. Sorne sensitivity experiments are also performed to determine the • 135 variation of the results ta certain parameters. âu • Figures 8.1 and 8.2 show the zonally averaged momentum tendency or flux (ôt )dw produced by the SCPS for each of the 5 DJF and J JA seasons in SLW42E-5YR. The

profiles are similar to each other and to the 5-year mean (Figures 8.lf and 8.2f) and

their variation from year to year is small. Examination ofother fluxes like temperature

and moisture also indicates no significant interannual variability.

The large scale response for an individual year is again consistent with the 5-

year mean. Figures 8.3 and 8.4 depict the differences in the zonally averaged jet (u

component) between the runs with and without slantwise convection. Although the

maximum values and the location ofthe maximum values undergo small changes from

year to year, the basic pattern in the difference fields remains the same. For each

DJF season, the jet in SLW42E-5YR is weaker than the jet in NOSLW-SYR around

40° N and 50° S. In each JJA, the strength of the jet is reduced around 40° Sand

50° N in the run with slantwise convection.

The small year-tO-year variability of the key features in the 5 years is also found

in other fields like temperature, SCAPE, SCAPE residual (r.5) , and the meridional

transient eddy transports of zonal momentum and heat. As an example, Figures 8.5

and 8.6 show the difference in transient eddy transport of heat between the run with

the SCPS and the mn without. In DJF (Figure 8.5), there is a major weakening of

the transport around the midlatitudes in Narthem Hemisphere for each ofthe years in

SLW42E-5YR. On the other hand, the decreased transport occurs in the midlatitudes

of the Soutbem Hemisphere in each JJA (Figure 8.6). • 136 • .,IZ,-..------, .,u~r_------:I U •• 1" ... ln l'Z zn lSZ

o -10 -10 la'

IZ,....------:I U•• ".IIZ lSZ

UI

-10

u,....------...., ...::~-~~-----, IIZ l'Z UI u.

SlZ 1

100

o -10 -10 ln

5 2 Figure 8.1: Zonally averaged momentum tendency (~).'w (in units of 10- ms- ) for (a) the first DJF, (h) the second, (c) the third, (d) the fourth, Ce) the fifth, and (f) the 5-year DJF mean from run SLW42E-SYR. The ordinate is in hPa and the latitudes aIong the abscissa in degree. • 137 •

12:~------'""] ::~----....., 1.i112 2U UI

III Il.,.2 tI~ji.,;;,. __~;':""lo-4-_:"""''''';::::;::t..._--.-.,....J...... ICI 110

...,1Zr------~ l4i ln 252 r UI Ur-

U%

100 71lll

'11 III ... 1.. 142 142 "~~~.,.,...... ~;:;:;:~'-_ ...... _---'- lU lCl

12:r------, .,12 ::~----..., .. .- UI 1.1 III III 15% Z52 UI UI

4oS4

~12

1CO '1' 1•• 142 tl5 o -.sa -10 lCl (st

Figure 8.2: The same as Figure 8.1, but for seasons of JJA. • 138 •

;,

",1 "f

100

150 12' laool-l--...... -...... ------...... - ...... eo ---.-1tG

400

500

100

100

~ ~. 'Z5 1000 tG -JQ -10 -tG 10 10 JQ cl -JQ -10 -10

Figure 8.3: Zonally averaged u differences (in mIs) between SLW42E-SYR and NOSLW-SYR for (a) the first DJF, (b) the second, (c) the third, (d) the fourtb, (e) the fifth, and Cf) the 5-year DJF mean. The ordinate and the abscissa are in units of hPa and degree. • 139 •

:.g- 100 ISO ZOO zsa 100

100

100

lsa tU laoal-...... J:.....~...,.I ...... -__-l-...... J....,.--II,-L~u...-..,1.~_..- ...... ICI

Figure 8.4: The same as Figure 8.3, but for JJA5. • 140 •

-10

.ao

~o lao lao ",, no : lU 1cao 10

10 !ICI tao t!ICI zao 150 ~ao

6llll

500

lOG Ica

'!ICI 1 tU tCao .. 10

Figure 8.5: Zonally averaged v'T' differences (units Kms-L) between SLW42E-5YR and NOSLW­ 5YR for (a) the first DJF! (h) the second, (c) the tbird, (d) the fourth! (e) the fifth, and (f) the 5-year DJF mean. • 141 •

.00 ,

~oo }

.00 IGCI

100 700 } '--, 1 ~ 150 l5G " ~ ' ... : .~ - .n --, .000 'GOG tel 10 tel 10 )Cl .2. -)Cl -10 -H

.aa

:lOG

100

100

l5G ns 'GOG ta -10

Figure 8.6: The same as Figure 8.5, but for JJAs. • 142 Based on the results of the analysis for each individual season, we conclude that • the year-to-year variation in the simulation is small and the time mean over 5 years ean yield a sufficiently stable climate.

8.2 Sorne sensitivity experiments

A few sensitivity tests are performed in the context of one or two season simulations

because ofeconomy and the fact that the results from individual seasons are consistent

with the 5-year simulations. For convenience, the experiments are carried out for DJF

and JJA of the second year.

Three sets of experiments are performed. The first set is designed to test the

importance of the thermodynamie forcing in the seps. Specifically, the tendencies

( :) and (:) are switched off and ooly the momentum tendencies (a;) slw dw .5lw and (:) are allowed. The two experiments are designated as SLW42EM-DJF2 dw and SLW42ElVI-JJA2 for the DJF and the JJA seasons respectively.

The second set is directed toward the sensitivity of the results on the latitude

filter Fi. In experiment SLW42EF, we set Fi = 1 (no latitude filter) for a DJF season

simulation. The third set foeuses on the relaxation time Î by increasing its value to

9 hours in run SLW42ES-DJF2.

Figures 8.7 and 8.8 show the difference fields between the run with the full seps

and the mn with no SCPS for DJF2 and JJA2 respectively. Figures 8.9 and 8.10 depict

the difference fields between the run with the thermodynamic tendencies suppressed

in the SCPS and the no SCPS run for the same seasons. Consistent with the results • 143 •

IOE7 LtI:~~~;-'~;::::~-...,~::.r.:::o-t"'\'\"";""':-;--;:-:~::=:.., ,: \ _1 ,,,,,__,.','~'

1'0 =,ZOO eoo •

100

150 lU IOOO~_-.J""',""",__""''''''''''_...... I,'''''''__'''''''''''''''''''''''''''''~-__--l ~ -~

Figure 8.7: Difference fields of zonally averaged (a) T (in 0.1 K), (h) u (in mis), (c) v (in mis), (d) w (in 10-2 bPa/s), (e) viT' (in Km S-1) and (0 "'V' (in m2 s-2) between SLW42E-DJF2 and NOSLW-DJF2. SLW42E-DJF2 and NOSLW-DJF2 denote respectively the second DJF season in run SLW42E-SYR and NOSLW-SYR.

• 144 •

Figure 8.8: Difference fields of zonally averaged (a) T (in 0.1 K), (b) u (in mis), (c) v (in mis), (d) w (in 10-2 hPa/s), (e) v'T' (in Km s-1) and (f) u'v' (in m2 5-2 ) between SLW42E-JJA2 and NOSLW-JJA2. SLW42E-JJA2 and NOSLW-JJA2 denote respectively the second JJA season in run SLW42E-5YR and NOSLW-SYR.

• 145 in Chapter 6, the characteristics of the direct secondary meridional circulation remain • intact when the temperature and moisture tendencies in the SCPS are turned off for both the DJF (compare Figure 8.9 and Figure 8.7) and JJA (compare Figures 8.10 and

Figure 8.8) seasons. The transient eddy transports of heat and momentum also did

not undergo drastic changes. In terms of the difference in temperature from the run

without slantwise convection, SL\lV42El\tI-JJA2 (Figure 8.l0a) indicates smaller values

but rather similar patterns as in SLW42E-.JJA2 (Figure 8.8a). The same also applies

to the difference in 'U for the same season (Figure 8.lDb and 8.8b). However, in DJF,

the gradient of the difference in T from (SL\.y42E~I-DJF2- NOSLW-DJF2) is weaker

than from (SLW42E-DJF2 - NOSL\V-DJF2) in the Northem Hemisphere. As a result,

the gradient in the difference in 'u is also weaker for the ron with no temperature and

moisture tendencies in the seps (compare Figures 8.9a,b and 8.7a,b). Table 8.1 gives

a summary of the results.

When the latitudinal filter is turned off. the direct meridional secondary circulation

becomes stronger in the lower latitudes (Figures 8.1lc,d). For DJF, the activity for

sIantwise convection dominates in the Northern Hemisphere. For JJA (not shown),

the effect of turning off the latitudinal filter is qualitatively similar except that the

major effect now appears in the Southern Hemisphere. The results are summarized

in Table 8.2.

With a larger rela.xation time T of 9 hours~ the response in SL\lV42ES-DJF2 are

weaker (Figure 8.12). Table 8.3 summarizes the results.

146 • •

10 sa .~ \ \ 100 \\ , \ :' ISO ~ '.l ZOO Z5G Joo ) ~ 1 ~oo ... , 1 500 1 • ~ 100 , . 150 . .". IZ~ ," ~. 10Q(l 10 10 JO .l'1) -JO -10 .IQ

Figure 8.9: Difference fields of zonally averaged (a) T (in 0.1 K), (h) u (in mIs), Cc) v (in mIs), (d) w (in 10-2 hPa/s), (e) v'T' (in Kms-l ) and Cf) u'v' (in m2 5-2) between SLW42EM-DJF2 and NOSLW-DJF2. • 147 •

~·~,~="'=::;~---C""'-;~""'I!l;;---O:::::::--~-~~--' IQQ ç I~ lOG ~ !QQ

Ill!

laoo~...-.L.". -';:"'_...... I.-,....L...,..l.~_'&'-t.-~_~ ....04 III .e'Q -10 -10

1:IQIr:'7:::O=:::::>"-OC::~~;T!:::-:~""';---:;=-=~-""'I':=--:7""'-":>-. I~ / lOG Z~ JOO

Figure 8.10: Same as in Figure 8.9, but between test SLW42E~I-JJA2and run NOSLW-JJA2. • 148 Zonally A.veraged SLW42EM-DJF2 SLW42EM-JJA2 • Difference in (m Tendency Only) (m Tendency Ooly) T Patterns are similar with better Patterns are rather similar for agreement in SH. Smaller maxi­ both NH and SH with smaller mum values maximum values u Stronger reduction in SH. The Similar patterns in bath NH and area of reduction in NH nearly SH disappeared v Same structure Same structure w Almost same response Almost same response viT' Similar patterns with NH's reduc­ Similar reduction patterns in tion extending to higher latitudes both NH and SH Similar patterns with a little Similar reduction patterns in stronger reduction both NH and SH

Table 8.1: Summary of sensitivity test: Disable thermodynamic feedback (~) 1 and (~) • w .lw in the seps for DJF and JJA of year 2 simulations (SLW42EM-DJF2 and SLW42EM-JJA2). The comparisons are made with respect to runs SLW42E-DJF2 and SLW42E-JJA2, respectively. SR and NR denote Southem Hemisphere and Northem Hemisphere respectively.

Zonally Averaged SLW42EF-DJF2 Difference in (Fl=l) T Similar patterns, larger warming in NH u Similar patterns but stronger in NH v Stronger cells, especially at lower latitudes in NH Direct cell extends toward Equator Similar patterns in both NH and SH Similar reduction patterns but extending fur­ ther southward in NH

Table 8.2: Summary of sensitivity test: Tum off the latitudinal filter in the seps for DJF of the year 2 mnulation (SLW42EF-DJF2). The comparisons are made with respect to run SLW42E-DJF2. • 149 • la sa u 100 r" lsa ZOO lSO Joo

400 $00 -JOO 100 100

100 100

ISO ISO

tz~

1000 10

la 10 sa sa 100 100 ISO lsa ZOO ZOO ZSO zso Joo JOO 4QO 1 400 $00 500

100 100 : ,~, , 100 l /00 , , " ': -,' 1 " 1 .. " ~4 , " 1 . ' , ," t ISO l , ISO , . "0," 1 ., , , , ~ , , tz~ Uli . , , , '.' ..."-:,' . ., 1000 1000 10 ·10 10 10 JO 0 (e,

la sa 100 lsa ZOO ZSO JOO

4QO

500

100

100

ISO lU 1000 la

Figure 8.11: Zonally averaged differences of (a) T (in O.IK), (b) u (in mis), Cc) v ( in mis), (d) w 2 (in 10- hPa/s), (e) v'T' (in Km 5-1), and Cf) u'v' (in m2 5-2) between test SLW42EF-DJF2 and run NOSLW-DJF2. • 150 •

la SIl ', , 100 , . 'SIl . . 100 : ~ lS1l ·. 100 · . 1 · , 400 .aa soo , ~aa .00 , .aa 100 1aa n ' . . ' V 'SIl '.' 125 0 100Q IQQQ tG III JO a tG III JO 0 -JO .IQ .,. 1'1

Figure 8.12: Zonally averaged differences of (a) T (in O.1K), (h) u (in mis), (c) v ( in mis), (d) w (in 10-2 hPa/s), (e) viT' (in K ms-I), and (f) viT' (in m2 s-2) hetween test SLW42ES-DJF2 and run NOSLW-DJF2. • 151 •

Zonally Averaged SLW42ES-DJF2 Difference in (r = 9 hours) T Weaker warming, especially in SH u Weaker reduction pattern v The cells almost disappeared w Weaker circulation v'T' Weaker reduction especially over SH u'v' Weaker reductions

Table 8.3: SUJnIDary of sensitivity test: Change the relaxation time T from 3 hrs to 9 hrs in the seps for DJF of the year 2 simulation (SLW42ES-DJF2). The comparisons are made with respect to run SLW42E-DJF2.

• 152 •

Chapter 9

Summary and Conclusions

This study is concemed with the effect of slantwise convection in general circulation

models (GCMs). The approach is through the development of a slantwise convective

parameterization scheme (SCPS) and its implementation into version Il of the third

generation GCM ofthe Canadian Clîmate Centre for modelling and analysis (CCCma

gcm11). The study is carried out in four major steps.

First, we study the characteristics of conditional symmetric instability (CSI) in

an environment which is also unstable for upright convection using a 20 2-layer La­

grangian parcel mode!. The results show that the convection which develops exhibits

features of both conditional upright instability (CUI) and conditional symmetric În­

stability and is therefore known as slantwise buoyant instability (SBI). Analysis of

the analytical and numerical solutions from the model reveals that SBI can he decom­

posed into a buoyancy mode in the vertical and an inertial mode in the horizontal. • 153 The buoyancy mode has a time scale given by the traditional Brunt-VaisalHi fre­ • quency while the inertial mode occurs on a much slower inertial time scale. The horizontal scale of SBI ranges from tens of km up to over one thousand km, and is

strongly affected by the vertical wind shear as weIl as the absolute vorticity. As a

result, momentum transport as weIl as temperature and moisture transports are aIl

important in SBI.

In the second part, the 15-year ECM\VF re-analysis (ERA) is used to compute

the global distributions of CAPE and SCAPE. We show that the potential for CSI

and CBI indeed co-exists over most areas around the globe. Over the tropical regions,

the buoyancy mode predominates while over a significant portion of the subtropics

and the midlatitudes, the effect of the inertial mode is likely to he significant. The

condition for the pure form ofCSI occurs only infrequently. The analysis also indicates

that the buoyancy mode of SBI tends to he more active in the summer hemisphere,

and the inertial mode more active over the winter hemisphere.

Third, a 20 parameterization based on the concept of SBI was constructed and

tested under idealized environmental conditions. Then a simplified ID version, treat­

ing mainly the inertial mode of SBI, was developed and implemented into gcmll. Ta

understand the physical mechanism 'underlying the impact of the slantwise convective

parameterization scheme (SCPS), some short-term experiments and sensitivity tests

were carried out. It was found that the SCPS produces a source of westerly ma­

mentum near the ground and a sink at higher levels. A direct meridional secondary

circulation develops according to the manner described by Eliassen (1952) and Kuo • 154 (1956). Adiabatic warming in the descending branch and adiahatic cooling in the • ascending hranch of the circulation reduce the mean meridional temperature gradient in the midlatitudes to weaken the thermal wind and the upper-level jets. A weaker

temperature gradient at midlatitudes results in decreased baroclinicity and baroclinic

activities, which in tum reduces the poleward transient eddy transports of heat and

zonal momentum.

In the last part, 5-year simulations including the SCPS were performed. It was

found that the scheme reduces SCAPE and SCAPE residual r s over the midlatitudes.

It also helps to reduce the thermal wind and the strength of the upper-level jets

over the region. The largest improvement in the simulated climate however lies in

the effect of the SCPS on the mean meridional transient eddy transports. With­

out slantwise convection, gcml1 significantly over-estimates the meridional transient

eddy transports of temperature and U over the midlatitudes of both hemispheres.

Implementation of the SCPS led to substantial reduction of the over-predicted eddy

transports.

This study represents a first attempt to test a SCPS in the CCCma GCM. The

results are highly encouraging but there are rooms for improvement. For example,

the scheme can lead to a worse simulated climate in the Northern Hemisphere during

JJA. There is aIso sensitivity of the results to the values of the latitude filter Fi and

the relaxation time T. Future work should he directed toward determining an optimal

interphase of the seps with the cumulus parameterization scheme and the testing of the generalized SBI scheme in a GCM. • 155 •

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• 172 •

Appendix A

Detailed Solution for 2.24 and 2.35

For z > hL, (2.24) can be reduced to the homogeneous form (2.35) by the coordinate

transform (2.36). The eigenvalue equation of (2.35) can he shown to be

(A.l)

The roots À L and À2 of (A.l) were given in (2.37) and (2.38). ,\I is negative when .1Vi f 2 > =, V z Tl while À~ is always negative.

Assume that

and

the solution of (2.35) is given by

(A.2)

or the solution of (2.24) is

(A.3) • 173 where

LI = f(Nf + Ni)V;hl . • fNiTï - f2V ; The vertical coordinate of the trajectory can be obtained as

Z = -fI (fi + ffix) 1 (A.4) v;:

where

Let (XlO l ZLO = hd and (UlO l WLO) denote the initial position and velocity com-

ponents, respectively, which are determined from (2.25) and (2.28) as weil as their

derivatives at z = hl. The coefficients Cl through C4 thus can be written as functions

of the initial conditions and the environmental parameters

d2 Cl - - dl, (•.\..5) Bi -B2 B2da C2 - (A.6) CFl (BI - 8 2 ), Bld - d ca - l 2 , (A.7) 8 1 -82 Bida - d4 C4 - (•.\..8) 0"2(B 1 - B2)

where

dl - xlO + LI!

d2 - fV;:ZIO - ffjLt,

da - UIO,

d4 - fVzWIO, • 174 BI - O'r - ffj, • B2 - O'i - ffj· As shown in Figures 2.3e through 2.6e, et+~ » ci+~ for a range ofenvironmental

conditions. This means that the characteristic terms with frequency 0'1 dominates in

the inertial mode of the slantwise buoyant instability, thus the characteristic time

for inertial mode can be given by O'lL. O'r May he estimated as fTJ if Ni » fTJ and

Ni » fvz , a condition generally satisfied under typical atmospheric conditions.

• 175