Moist Potential Vorticity Generation in Extratropical Cyclones Z 1995.pdf · vorticity point of...

Moist Potential Vorticity Generation in Extratropical Cyclones

bv

Zuohao Cao

A thesis submitted in conformity with the requirements for the Degree o f Doctor o f

Philosophy in the Department o f Physics o f the University o f Toronto

© Copyright by Zuohao Cao 1995

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission

For my parents Mr. Jie Cao and Mrs. Sujuan Vang

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

Moist Potential Vorticity Generation in Extratropical Cyclones Zuohao Cao, Ph.D. 1995 Department of Physics, University of Toronto

Abstract

The mechanism o f moist potential vorticity (M PV) generation in a three-

dimensional moist adiabatic and frictionless flow is investigated. It is found that MPV

generation is governed by baroclinic vectors and moisture gradients. Negative (positive)

M PV can be generated in the region where baroclinic vectors have a component along

(against) the direction o f moisture gradients. Numerical simulations o f extratropical

cyclones with different moisture distributions show that at the different stages o f

cvclogenesis, negative MPV usually appears in the warm sector near the north part o f the

cold front, the bent-back warm front, the warm core and the cold front.

The effects o f the Boussinesq approximation on the distribution o f vorticity and

MPV are examined. The Boussinesq approximation neglects several components o f the

^olenoidal term in the vorticity equation. As a result, it underestimates the thermally direct

circulations in the cold front and the bent-back warm front by 25% to 30%. This effect is

more pronounced when latent heat release is taken into account. Consequently, the

influence o f the Boussinesq approximation on the MPV field is very significant.

Evidence o f the presence o f conditional symmetric instability in the negative MPV

region o f the extratropical cyclone is also presented.

i


Acknowledgments

Many individuals, who deserve special recognition, made the direct and

indirect contribution to this thesis. I would like to express my thanks to Professor

Man-Ru Cho. my supervisor, fo r his insights, encouragement, understanding and

guidance, to Professors G. YV. K. Moore and T. G. Shepherd fo r serving in my

supervisory committee, and to Professors M. K. Yau (the external examiner) and R.

List for their interest in this research.

Thanks are due to Professor D.-L. Zhang for providing the basic code o f

PSU/NCAR three-dimensional hydrostatic mesoscale model. I am indebted to Dr. M.

Medley for many enjoyable and helpful discussions on numerical methods. Thanks also

go to my colleagues in the department o f physics for their assistance and discussions,

especially Dr. John Koshyk. Murray Mackay and Paul Kushner.

I am grateful to my wife. Shu Chen, for her love and support, and my lovely

son. Eric Yixiao Cao, for his moral support. The specially important people are my

parents. Mr. Jie Cao and Mrs. Sujuan Yang. Their encouragement and support w ill be

remembered for ever.

My financial support by the department o f physics o f University o f Toronto

through various fellowships and by Professor Han-Ru Cho from his research grant is

highly appreciated. This research was supported by the NSERC and AES o f Canada.

ii


Contents

Abstract i

Acknowledgments ii

1. IN TR O D U C TIO N I

1.1 Extratropical cyclones 1

1.2 Dry and moist potential vorticity 3

1.3 Objectives and organization o f the thesis 8

2. G O V ER N IN G EQ UATION AND G ENERA TIO N OF M O IST

PO TE N TIA L V O R T IC IT Y 10

2 1 Governing equation o f moist potential vorticity 10

2.2 Formulation for the baroclinic generation o f moist potential

vorticity 1 -

3. N U M E R IC A L M O D EL 16

3.1 Model equations 16

3.2 Initial conditions 19

3.3 Experiment design 24

3.3.1 Control experiment 24

3.3.2 Sensitivity experiments 27

4. G EN E R A TIO N OF M O IST PO TE N TIA L V O R T IC IT Y IN

EX TR A TR O PIC A L CYCLONES 30

4.1 The control experiment 30

iii


4.1.1 Life cycle o f the extratropical cyclones and

M PV distribution 30

4 I 2 Baroclinic generation o f M PV 37

4 .1 3 Effects o f horizontal diffusion and rainwater evaporation 44

4.2 Sensitivity study 47

4.3 Summary 67

5. BOUSSINESQ A PPR O XIM ATIO N AND ITS IM P LIC A TIO N S 70

5.1 Vorticity dynamics 70

5.2 Moist potential vorticity dynamics 79

5.3 Summary 81

6. C O N D IT IO N A L SY M M E TR IC IN S TA B IL ITY (CSI) IN

EXTRATRO PICAL CYCLONES 82

6. 1 The criteria o f CSI used in this study 83

6.2 The scheme for taking two-dimensional cross sections o f equivalent

potential temperature and absolute momentum 84

6.3 CSI in extratropical cyclones 86

6.3.1 Evidence o f CSI in extratropical cyclones 86

6.3.2 Possible roles o f CSI in extratropical cyclones 92

6.3 .3 Effects o f moisture distribution on CSI in extratropical

cyclones 102

6.4 Summary 115

7. CONCLUSIONS AND SUGGESTIONS FOR FUTURE

RESEARCH 116

iv


APPENDIX A TH E ES TIM A TIO N OF FUNCTION A 119

APPENDIX B L IST OF SYMBOLS 120

REFERENCES 123

V


1

Chapter 1

INTRODUCTION

1.1 Extratropical cyclones

Extratropical cyclones have been recognized as an important class o f systems in

weather forecasting for at least 150 years because they frequently occur in the mid

latitudes. An example o f a marine extratropical cyclone is shown in Fig. 1.1. Many theories

have been developed to aid in the understanding o f cyclone behavior and structure. In the

19th century, the thermal theory o f cyclones based on Espv’s work claimed that the

decrease o f the surface pressure in storms is essentially related to the release o f latent heat

in the ascending air near the storm center. The ever-increasing knowledge o f dynamical

processes gave birth to the polar front theory (Bjerknes and Solberg 1922). According to

the polar front theory, the cyclone forms as a result o f an instability o f the polar front, a

surface o f discontinuity separating tropical and polar air masses. After the debates on the

relative importance o f dynamic and thermodynamic processes in extratropical storms in

the 1920s. the dynamic processes associated with low-level fronts and existence o f a

strong upper-level current were realized as important elements for the development o f

cyclones. However, the importance o f latent heat release from boundary layer and free

atmosphere was not ruled out. Another remarkable achievement o f cvclogenesis theory

was baroclinic instability theory introduced by Chamey (1947) and Eady (1949). This

theory emphasizes the instability o f the broad baroclinic westerlies rather than the frontal

discontinuities. The baroclinic theory' successfully predicts the structure o f the incipient

waves, realistic growth rates o f their development and their characteristic wavelengths.

The success o f the baroclinic theory and the failure o f earlier works on frontal instability


v rffii^s :

Fig. 1.1 Visible images of a storm occurred in the western Pacific Ocean at (a) 2100 UTC 10 April 1992, (b) 1800

UTC 11 April 1992, and (c) 2000 UTC 12 April 1992.


(Solberg 1928; Kotschin 1932; Bjerknes and Godske 1936) result in general acceptance o f

baroclinic instability as the fundamental cause o f cvclogenesis. These classical studies o f

cyciogenesis by Charney (1947) and Eadv (1949) can also be described from potential

vorticity point o f view (e.g., Chamev and Stem 1962).

Because in the middle latitudes precipitation is mainly caused by extratropical

cyclones, especially those associated with explosive deepening, the extratropical cyclones

have received much attention in recent years (e.g., Houze and Hobbs 1982). Although

precipitation processes cover temporal scales from 10'*s to I CPs and spatial scales from

meso-y (2-20 km in horizontal dimension) tc meso-a (200-2.000 km), most precipitation

in extratropical cyclones is organized at meso-p jcale (20-200 km) in the form o f

rainbands. According to Houze et al. (1976). Hobbs (1978) and Matejka et al. (1980), the

principal rainbands in mid-latitude cyclones are classified into six types ( see Fig. 1.2).

Several theories have been proposed to explain the formation o f these rainbands (see

Table 1.1 for details). Table 1.1 suggests that conditional symmetric instability is one o f

the successful candidates to elucidate the formation o f some rainbands in cyclones. The

observational studies by Bennetts and Ryder (1984), and Parsons and Hobbs (1983) also

show that the theory o f conditional symmetric instability can explain many o f the observed

features o f the warm-sector and wide cold-frontal rainbands. Because the value o f (moist)

potential vorticity is critical for the appearance o f (conditional) symmetric instability, the

importance o f (moist) potential vorticity is emphasized in the literature. Further

discussions and implications o f (moist) potential vorticity w ill be given in the next section.

1.2 Dry and moist potential vorticity

Although the importance o f potential vortic ity (PV) thinking has been

recognized only relatively recently, the application o f this concept to the study o f

extratropical cyclones can be traced back to Rossbv's work (Rossby 1940). As he


4

Open hexagonal ceils

Cirrus cloud boundary as seen from satellite

t

s y n o p t i c f e a t u r e s \ TYOCS CF MCSOSCAUE RAINBANOS

1 T | SURFACE LOW- i 1 L j 1 PRESSURE CENTER t WARM-FRONTAL

y j SURFACE COLO y* FRONT 1 2 . WARM-SECTOR

^ 1 SURFACE WARM ! y * I FRONT i

3 wiOE COLO- f r o n t a l

y * | SURFACE WARM 4 * 1 OCCLUOED FRONT I

4 NARROW c o l o - FRONTAL

S I COLO FRONT AL0 F7 I j 1

“ I 1 1 4 » o s t f » o n t a l1

Fin. | .2 Schematic depiction ot' iho tvpes o f rainbands ( numbers i -o i observed in extratropical cyclones I- rom I lobbs

i 19SD.


Table 1.1 The possible mechanisms fo r various rainbands

Tvpe o f rainbands Mechanisms

Wide cold-frontal bands The ducting o f gravity waves (Lindzen and

Tung 1976), and conditional symmetric

instability (Bennetts and Hoskins 1979).

Warm-sector rainbands The ducting o f internal gravity waves

(Lindzen and Tung 1976), wave-CISK

(Lindzen 1974; Raymond 1975),

conditional symmetric instability (Bennetts

and Hoskins 1979) and potential instability

(Kreitzberg and Perkev 1976, 1977).

Narrow cold-frontal rainbands Gravity-current and strong horizontal

shear (Matejka 1980; Hobbs and Persson

1982).

Prefrontal cold surge and postfrontal

rainbands

conditional symmetric instability (Bennetts

and Hoskins 1979) and Wave-CISK.


t)

pointed out. in a barotropic flow the absolute vortic ity divided by the depth o f a tluid

column remains constant fo llow ing the motion o f the tluid column. The hydrodynamic

generalization o f this concept was first stated two years later in the independent work

by Ertel (1942). Tn his paper, Ertel showed that potential vortic ity defined as

-C a -v e . (1.1)p

where Ca 1S absolute vortic ity vector and 0 is potential temperature, is a

conservative quantity in adiabatic and frictioniess flow. To distinguish it from the

concept o f moist potential vo rtic ity which includes the effects o f moisture, the

potential vortic ity defined by (1.1) is referred to as the Dry Potential Vortic ity or

DPV in this thesis.

Since Rossbv's time many applications o f PV thinking have been found in

atmospheric sciences. Because o f its conservative nature, Reed and Danielsen (1959)

considered DPV as a tracer to identify the stratospheric origin o f air in the tropopause

folds associated with upper-level frontogenesis. Kleinschmidt (1950, 1951, 1955, and

1957) used the notion o f DPV anomalies to explain observed cyclogenesis, indicating

the importance o f quasi-horizontal advection along isentropic surfaces from the

stratospheric reservoir o f high DPV. The well-known Petterssen Type-B cyclone

development (Petterssen and Smebve 1971), originally identified as an upper-level

positive vortic ity advection area moving over a low-level baroclinic zone, should

properly be considered as the development resulting from the advection o f an upper-

level DPV anomaly along an isentropic surface. Examples o f the applications o f the

DPV concept to the studies o f mid-latitude cyclones can be found in Hoskins et al.

(1985), Young et al. (1987), W hitaker et al. (1988), Davis and Emanuel (1991), and

Reed et al. (1992). The role o f upper-level DPV anomalies in rapidly deepening

cyclogenesis is still an active area o f research (e.g.. Hoskins and Berrisford 1988).

An important advance in the use o f PV thinking was the recognition by


7

Charnev and Stern (1962) that necessary conditions o f baroclinic instability, the origin

o f extratropical cyclones, can be formulated entirely in terms o f DPV. In an extensive

review paper by Hoskins et al (1985). the authors pointed out that most synoptic

scale processes in the mid-latitude such as baroclinic and barotropic instabilities can

be understood from the PV point o f view. Because o f the existence o f an invertib ility

principle in balanced systems, it is fair to say that the dynamics o f mid-latitude

cyclones as a first order approximation is the dynamics o f potential vortic ity.

Symmetric instability (SI) is an example o f using PV thinking in mesoscale

atmospheric processes. The possible importance o f SI in mesoscale phenomena has

been examined in a number o f studies (e.g., Bennetts and Hoskins 1979; Emanuel

1983). These studies have shown that when air is symmetrically unstable, it is possible

that air parcels are accelerated away from their equilibrium positions due to the

combined action o f buoyancy and Coriolis forces. The condition fo r SI is traditionally

stated as the Richardson number being smaller than unity, where the Richardson

number is defined by

u r‘6

cz

where 90 is a reference value o f potential temperature and v is velocity. Hoskins

(1974) is the first to show that this condition can also be conveniently stated as the

DPV being negative. This discovery has not only found an alternate way to identify

regions o f SI, but also raised a number o f questions concerning the instability itself.

One o f such questions is the likelihood o f SI to occur anywhere in the atmosphere.

Because o f the conservative nature o f DPV in an adiabatic inviscid flow, SI w ill not

occur in an adiabatic flow i f the initial DPV values are positive everywhere, which is

usually the case. A number o f studies have therefore been made to examine the non


8

conservation o f DPV due to latent heat release (Chan and Cho 1989; Cho and Chan

1991), and turbulence (Thorpe and Rotunno 1989; Cooper et al. 1992).

A more likely process to happen in the real atmosphere is the conditional

symmetric instability (CSI), a form o f SI made possible by the release o f latent heat o f

condensation. It is conditional on the saturation o f air parcels. The necessary criteria

for two-dimensional frictionless CSI in a geostrophic balanced flow are that the air be

saturated and that the moist potential vortic ity (MPV), defined as

—C~a • V 0C, (1.3)P-

where 0C is equivalent potential temperature, have a negative value. This kind o f

instability has extensively been studied in recent years since it was first proposed as a

possible mechanism for the formation o f some frontal rainbands (Bennetts and

Hoskins 1979) even though the mechanisms for the formation o f rainbands in cyclones

are not yet completely understood (Houze et al. 1976; Hobbs 1978; Matejka et al.

1980; Houze and Hobbs 1982). Reuter and Yau (1990) suggested that slantwise

convection due to CSI is likely to be ubiquitous in extratropical cyclones. The

possibility o f negative MPV generation in extratropical cyclones is therefore o f more

than pure academic interest.

1.3 Objectives and organization of the thesis

The question still remains to be answered o f the processes for MPV

generation. Unlike DPV which is not conserved when condensation occurs. MPV is

not conserved when the flow is three-dimensional and the air is at least partially

unsaturated. The M PV generation in an unsaturated three-dimensional flow is

examined in Chapter 2 after a b rie f discussion o f the governing equations. It w ill be

shown that M P V generation is due to the presence o f baroclinicity together with


9

gradients o f water vapour. The rate o f M PV generation can be expressed as the vector

product o f the baroclinic vector and the gradient o f moisture.

In order to examine the generation o f negative M PV in a typical m id-latitude

cyclone, a set o f numerical simulations are made using a three-dimensional regional

atmospheric model. The model is described in Chapter 3 together w ith the

experimental design and the initial conditions used in the simulations. The results o f

the main simulation and those o f the sensitivity experiments are presented in Chapter

4.

Chapter 5 discusses the implications o f using a Boussinesq approximation on

vorticity and MPV fields since the baroclinic vector is treated in different ways in the

Boussinesq and prim itive equation models. The Boussinesq approximation is

frequently used in theoretical studies, but it ignores some important part o f the

solenoidal term, and is therefore incapable o f properly describing the significant

development o f vortic ity and MPV. The differences between a Boussinesq flo w and a

flow described by the more complete set o f prim itive equations are compared and

discussed in that chapter.

As a result o f negative MPV generation in extratropical cyclones, CSI appears in

the bent-back warm front o f the extratropical cyclone. In chapter 6, evidences o f CSI are

shown and the relation between CSI. and the deepening o f surface low and the

precipitation is also presented.

Conclusions and suggestions for future research are made in chapter 7.


Chapter 2

10

GOVERNING EQUATION AND GENERATION OF MOIST POTENTIAL VORTICITY

2.1 Governing equation of moist potential vorticity

The moist potential vorticity equation can be derived from the governing equations

(2.1 ) - (2 .6 ).

cV - - - - 1 - F — + ( V - V ) V + 2Q:< V = — Vp h-G -f —c-t p p

(2.1)

^ £ ^ pv - v = odt

(2.2)

d0e = Q dt

(2.3)

0 e = Qexp( , q )I.

(2.4)

e = T ( 1000)c (2.5)

p = pRT ( 1+a ;q). (2.6)

where F, Q. and q are frictional force, diabatic heating or cooling rate, and specific humidity

o f water vapour. T, is the temperature that an air parcel would have i f lifted adiabatically

to its condensation level, which is defined as


X I

T:. = — r — - a .In( r)T - a ; a .

where r is relative humidity. The constants a ,, a ; , a 3, and a 4 are equal to 2.675xl03 K.

0.61, 55.0 K and 2.84x 103 K, respectively.

The set o f equations (2. l)-(2.6) can be reduced to the following M PV equation

d ( ^ a -V0e ) E----------- = ( £a , V) j9 e _ V6e . ( VP.:-Z E ) + VQe ■ ( - V x - ) , (2.7)

dt p dt p ' p p

where F represents any frictional forces in the system, and the other variables have their

usual meaning. In this thesis we are primarily concerned with the second term on the right

hand side o f (2.7). The flow is therefore assumed frictionless, F=0. and moist adiabatic,

dt

A moist adiabatic process (saturated pseudoadiabat) can be described by the

following thermodynamic equation (p.57. Gill 1982; p.22. Rogers and Yau 1989).

L dq, ^ d 0 _ o (2.8)C rT ( i - q J 0

where Cp is the specific heat at constant pressure. L is the latent heat o f vaporization, qs is

saturated specific humidity, and 0 is potential temperature. I f the temperature change in

the process is not very large and qs is small, an approximate integral o f (2.8) yields the

expression o f equivalent potential temperature for saturated air

Lqs0c = 0 e x p ( ^ ) . (2.9)

P* . .

The quantity 0C defined in (2.9) is a function o f pressure p and temperature T only since it

refers only to saturation conditions. However, an equivalent potential temperature 0e,


which depends on specific humidity q. p. and T, can be defined for any parcel, whether

saturated or not. In other words. 9e is constant for a parcel provided that the changes are

adiabatic when the parcel is unsaturated and moist adiabatic when the parcel is saturated.

The more accurate empirical formula (2.4) o f 0e, suggested by Betts and Dugan (1973), is

used. Unless otherwise mentioned, "adiabatic" in the context o f the thesis is referred to as

d0drv adiabatic. — = 0 .

dt

The second term in (2.7), V 0e -(Vp .* V p ) / p ' \ is zero i f the How is either two-

dimensional or saturated. In the former case. 0e, p. and p are fiinctions o f only two o f the

three spatial variables (x.y.z). In the latter case. 0e is a function o f density p and pressure

p only. Therefore, the second term contributes to the rate o f change of M PV only when

the flow is three-dimensional and air is unsaturated. In this situation, it represents one o f

the MPV sources, and (2.7) can be written as:

d ( - - V 0 e ) 2. V 0e • ( - P v ? ) (2.10)

dt e p -'

2.2 Formulation for the baroclinic generation of moist potential vorticity

The moisture effect implicitly expressed by the right hand side o f (2.10) can not

directly be visualized. To be clear, we substitute (2.6) into the right-hand side ot (2.10)

and reduce it to:

V 0 , - [ 2 £ i S t l - — T - •! - ~ ■ Vp - g . ^ - V q . vp!p-' p “ T ( l - a ; q)

V0 V T R VpBecause — = ------------------- .

0 T C , p


13

V p ^ Vq x Vp}p- T (1 -r a : q)

= } - i - V 9 x Vp - Vq x V p }. (2.11)p - 9 ( i - a . q )

Substitute (2.4) and (2.5) into (2.11), make use o f the formula o f saturated specific

humidity qs,

0.662e<-qs = ----------

P - e s

where es is saturated vapour pressure.

es = b .lle x p (a 5- ^ - ) ,

in which a 5= 19.85, and a f,=5.41812x 10' K, expand and rearrange (2.11), it becomes:

d ( ^ • V6e ) --------- = A (V0 x Vp) • Vq, (2.12)

9 a ,T . a [a - ( T - a , ) l n ( r ) ]___________ I 4 _____whprp A — ^ t “

p- 0 T ( T - a J a - a , [ a - ( T - a J l n ( r ) ]1 .? 4 .> 4

a, ' a 4 [ ( T - a J 2 ( T . ( p - e s) + a , -<a6 q p ) - a 7 > a 4 T2q ( p - e s)]

[ a 4 T - a , ( T - a , )ln(r)]“ Tv ( p - e g)

in which Tv is virtual temperature. I f any other expression o f0 e such as 0e = 0 e x p ( ^ jL),C PT

instead o f (2.4), is used, one can obtain the same equation as (2.12) with a different

function A. The estimation o f function A is given in Appendix .A where it is shown that A

is usually negative under typical atmospheric conditions. The vector V0 x Vp is the

baroclinic vector and q is specific humidity. Formula (2.12) indicates that there w ill be a


14

-r- - -

VPv e \v

V Q

F ie .2 .1 Schematic diagram o f a baroclinic vector and moisture gradient in a frontal /one \ 0. \ p. and \ i| represent

tiie gradients o f potential temperature, pressure, and specific humidity. N . S. H and W stand for north, south, east,

and west.


15

decrease (increase) in the value o f M PV i f the moisture gradient has a component along

(against) the direction o f the baroclinic vector, i.e., the moisture gradient is parallel to the

lines o f intersection o f the constant p and constant 0 surfaces. This principle is general and

is applicable to any baroclinic systems. Fig.2.1 shows a schematic diagram o f a baroclinic

vector and moisture gradient in a frontal zone. As shown, the baroclinic vector points out

o f the paper, which has a component in the direction o f the moisture gradient. Hence,

negative MPV is expected to be generated in this frontal region.

The DPV equation in a frictionless flow is given by

d | C a .V0) - B = ( ^ a _ . v ) £ i (2.13)

dt p dt

It is clear from (2.12) and (2.13) that DPV is conserved in an unsaturated flow while MPV

in general is not. But in a saturated flow, because the right hand side o f (2.12) is zero,

MPV is conserved while DPV in general is not. A sufficient condition for PV invertibility

is that DPV > 0. Because o f the possibility o f CSI in saturated air fo r which MPV < 0, the

balance condition (relationship between the potential temperature and wind) may be

violated there. The invertibility condition should therefore be stated as DPV > 0

everywhere as well as MPV > 0 in saturated regions.


Id

Chapter 3

NUMERICAL MODEL

The PSU/NCAR three-dimensional hydrostatic mesoscale model (Anthes et al.,

1987) is used to examine the baroclinic generation o f MPV in mid-latitude cyclones. The

description o f model equations, initial conditions and experiment design is given here; the

results w ill be discussed in the next chapter.

3.1 Model equations

The model, originally developed by Anthes and Warner (1978) and further

documented by Anthes et al. (1987), has been applied to a wide variety o f problems

ranging from the synoptic scale to the small end o f the mesoscale (Anthes 1990). fo r

reference, the governing equations for the seven prognostic variables (momentum (u,v),

temperature T. pressure difference p* between the surface pressure (ps) and top pressure

(pt) o f the model, water vapour qv, cloud water qc, and rainwater qr) and three diagnostic• •

variables (geopotential height <|). vertical velocity co = p. and vertical velocity o ) are given

as follows;

f (p * u ) ,c (p *u u /m ) f (p *v u /m ), fp * u oa « + ,y to

+ fp*v - mp»( I S i V ; * l i , + F.,,. p*u ( 3 1 1 )p * - p t / a r/x r x


17

r(p*v) ,r(p*uv/m) c(p*w/m ), op vo— _— = - m: [— ------- ——z----------1 - zf.t ex cv CCS

c * * / R T " c p * . C $ _ y- tp*u - m p*( :-- -------— ) -r F.:n- p*vp * -rp, I g cy ry

f(p*T) ,r(up*T/m) c(vp*T/m), cp*Ta— .— = - m: [----- --------+ ] - ------

f t C X O ' CG

RT..O) . L ,.p *(P con - Prg) _

' Crm(a + pt / p * ) ' cpw

r(p*qv) _ , r(up*q.7m) r(vp*q,/m) cp*qvaet ex cy cg

- p^ p - p j ^Fhd- pX-

r(p*q.) _ r(up*q,7m) ^ r(vp*q,7m) rp*q.Gft fx cy cg

+ P*(Pa,n * Pr. “ Pr<> ~ Fhd P* Qc

f(p*qr) = , r(up*qf7m) ^ f(vp*q,7m) fp*qrart fx cy co

* P*(P„ + PK - p„) - S + F„„ p* qr

rp* nr(p*u/m) r(p*v/m), rp*C~z— = - m: [— ,------- + — _ ] - .—f t ex rv cg

1 a. .tp .„r(p*u/m) r(p*v/m), , , ,a = ~ r f H r- + m- [— , + —1 , ] dap* J ' et 1 ex rv s ’

rp* rp* rp*co = p*a + a [ ~r~ + m (u~^~ * v~z~~)

^ 1 et ex rv

(3.1.2)

(3.1.3)

(3.1.4)

(3.1.5)

(3.1.6)

(3.1.7)

(3.1.8)

(3.1.9)


18

(3.1.10)cln(cr + pt/p*)

where

Tv=T(l+0.608qv)

is the virtual temperature, and

Cpm=cr ( 1+a81clv)

is the specific heat at constant pressure for moist air. Pru, Pu„ Prtf, Pcim, and v, are the

tendency operators for the accretion rate o f cloud droplets by raindrops, the

autoconversion rate o f cloud droplets to raindrops, the evaporation rate o f raindrops, the

condensation rate o f water vapour or the evaporation rate o f cloud droplets and the mass-

weighted mean terminal velocity o f raindrops, respectively. FIID is an operator for

horizontal diffusion. The second order form is used for the row or column o f grid points

next to the lateral boundaries,

F . . D 2 = K „ V ; .

while the fourth order form is used in the interior.

IID4 = ^ n.

where K n is defined as

where A' is an amplitude factor, K [IO is a parameter dependent on the grid size As and time

increment At, and k and D are the von Karman constant (= 0.4) and the horizontal

deformation (Smagorinsky 1963), respectively. A ll other variables are assumed their usual

meteorological meaning.

The model has been modified to make it suitable for the simulation o f a baroclinic

channel flow with periodic boundary conditions at the eastern and western boundaries, and

K „ = (As)2K H,

and

K„ = A1 (K ho + t k W D ) .


19

rigid wall boundary conditions at the nonhem and southern boundaries. The map

projection in the original model is removed. The horizontal domain size is 4000 km x

8000 km with a horizontal grid spacing o f 50 km. The venical coordinate used is:

Ps Pt

where pt (= 300 mb) and ps are the top and the surface pressure o f the model,

respectively. The model contains 14 computational layers at a = 0.996, 0.986, 0.960.

0.920, 0.870. 0.805, 0.730, 0.645, 0.550, 0.450. 0.350. 0.250, 0.150 and 0.050.

To capture the major features o f synoptic scale cyclones, the physical processes

included in the simulations are: (1) explicit calculation o f cloud water and rain water as

time dependent variables (Hsie et al. 1984), and virtual temperature and water loading

effects; (2) horizontal diffusion, which is considered a part o f parameterization o f mixing

in a free atmosphere. Physical processes not included in the simulations are: ( I ) all o f

radiation; (2) ice-phase microphysics (i.e., freezing, melting, deposition, and sublimation);

(3) the cumulus parameterization schemes; (4) bounary-layer turbulence; (5) vertical

diffusion; (6) surface flux (e.g., heat and moisture). As a result, only three source terms

appear in the thermodynamics equation (3.1.3), condensation, rainwater evaporation, and

horizontal diffusion. Since 0C is conserved for a moist adiabatic process, the only source

terms for 0C are rainwater evaporation and horizontal diffusion. The effect o f rainwater

evaporation on MPV generation is extremely small (see the next chapter for details). The

experiment with and without horizontal diffusion shows little effect o f horizontal diffusion

on negative MPV distribution (discussed in the next chapter).

3.2 Initial Conditions

The initial conditions are generated analytically similar to those o f Fritsch et al.

(19S0) and Nuss and Anthes (1987). Compared with Nuss and Anthes' (1987) initiation,


20

two major changes have been made in this study: (1) DPV and M PV are initially positive

everywhere within the domain. (2) The initial velocity fields are determined by the

geostrophic wind relation rather than the nonlinear balance equation.

For reference, the six steps o f generating analytical initial conditions are given as

follows:

( I ) Specification o f a two-dimensional pressure field at a reference height o f 5.5

km:

P = P o + A P x + APy

where pu is a constant,

Apx = ax bp (x) Gp(y) sin + <D,(y) ] (3.2.2)N

in which

bp (x) = d, [ d, s in ( - y ^ ) ] (3.2.3)

introduces an east-west amplitude.asymmetry between the trough and ridge, and

Gp( y ) = s i n ( ^ ) (3.2.4)L v

forces the perturbation to vanish on the north and south boundaries.

APv = - a,, tanh[ y ~ y-g- ] - a,, t a n h [ ^ ^ ] (3.2.5)p ,F j ( x )dy P :dy

where

0 j r \

Fj(x> = 1 -b sin + <D,(y) ] (3.2.6)

produces a jet streak in the flow by introducing different isobar packing along the flow.

The constants ay,. ay,, ax. p,, p; , b, d,, d, and phase functions <P.(y), 0 2(y) are listed in

Table 3.1. The coordinate x, y and z are defined in km. yc and dv are the center of domain

and grid space, respectively.


21

(2) Specification o f the three-dimensional temperature structure:

T(x,y,z) = T0 + y(z) Az + ATX + ATy (3.2.7)

where

7 rrvATX = bxD(z)c-r (x) GT(y) sin — + <t>x(y ,z )], (3.2.8)

X

in which

D(z) = — - — tanh ( - —- ) (3.2.9)2 2 dz

2 _ , _ v 2and <t>T(y,z) = - [ * ° U ^ y ] + <&2(y) (3.2.10)

2 z5 “ (zr " zo )“

introduce vertical variation in the temperature wave amplitude and phase, where the

reference level zR is 5.5 km, the level o f maximum phase difference z0 is 7 km and dz is

vertical grid space o f 1 km, and

7 7T\c-r (x ) = d,, [ da + s in (-y“ - ) ] (3.2.11)

X

and GT(y )= s in (— ) (3.2.12)Lv

are similar to (3.2.3) and (3.2.4).

ATv = - bvl tanh[-— ^ - ] - bv, t a n h [ ^ £ - ] + FR (x,y,z) (3.2.13)Pbldy ‘ pb2dy

in which

Fr (x,y,z) = tfz) [ sin + it )+sin f 1 ) ] sin2 ( ^ - ) sin2 ( ^ ) (3.2.14)L x L y L x Ly

where f(z) = a.-[ — - — tanh(— —- ) ] (3.2.15)2 2 dz

produces a more intense low-level temperature gradient or front near the surface that

decreases w ith height in the light o f the function f(z). The constants T0, b y t , by2, b x , % , pbl,


Phi. du, da and phase function 0 :(y ) are listed in Table 3.1. The vertical temperature

variation y(z) Az is defined by the parabolic equation:

y(z) Az = (4 h k ) '2 - [ 4h(z h- k) ] ' : , (3.2.10)

where

h = s; k (3.2.17)

A T - ( n : } '

and k = ------ . (3.2.18)

zo - - ( 2 s }

in which s is the lapse rate (°C km '1) and AT is the temperature difference between the

mean surface and the top o f model.

(3) The three-dimensional relative humidity (RH) field is specified in the same way

as the temperature. Horizontal RH wave structure is similar to temperature with a

different amplitude (Fig. lc). and vertical distribution o f RH is adjusted in such a way that

initial M PV is positive everywhere. Mixing ratio is calculated w'hen RH and temperature

fields are specified.

(4) The pressure distribution except reference level is determined by integrating a

hydrostatic equation with the virtual temperature effect included.

(5) The three-dimensional winds are obtained from geostrophic relations:

RTv fp RT\ fpu = ------------ . and v = -------------. (u.2.19)

fp f y fp ex

(6) Initial fields are interpolated from z coordinate to c coordinate.


Table 3 .1 Constants to define the reference level pressure and the three-dimensional temperature

For Reference Level Pressure For 3-D Temperature Field

P., 495 mb T„ 273 K

as i 10 Lx,L> 4000 km, 8000 km

as. 18 by. 12

a\ 3 bv. 7

P. 18 b 5

24 t

b 0.0 Phi 22

d, 1/4 Ph2 16

d. 3.0 dt, 1/4

f 1.0 du 3.0

<t>i(v) (t>2(v)-5~/8=37t/8 d 1.0

<t>2(v) 71 02(V) K


24

3.3 Experiment Design

Ten experiments have been conducted to study the moisture effects on the

development and evolution o f MPV in extratropical cyclones. In each o f eight sensitivity

experiments, by changing the moisture distribution while holding all other model

conditions same as those in the Control, we are able to investigate the influences o f

different moisture distribution on MPV generation in the cyclones. A dry experiment is

performed for comparison.

The nine experiments with moisture can be classified into two categories.

Experiments (a)-(f) belong to the first type. The initial moisture gradients in this type o f

experiment are almost perpendicular to the gradients o f potential temperature. In the

second type o f experiment, the initial moisture gradients with various magnitudes are

parallel to the potential temperature gradients, such as the control experiment, experimen*

(g) and (h).

3.3.1 Control experiment

Because baroclinicitv, suggested by Charnev (1947) and Eadv (1949), is one o f the

key ingredients for cyclogenesis. the initial conditions in our experiments are highly

favorable for baroclinic instability. Most o f surface meridional temperature gradients

shown in Fig.S.la are localized in a 1250 km baroclinic zone. Unlike the initialization

method o f a very small perturbation imposed on the basic flow (Hoskins and West. 1979;

Peltier et al.. 1990; Palavarapu and Peltier. 1990), a perturbation with moderate amplitude

is used, which represent the early phase o f a typical mid-latitude cyclogenesis (Fig.3. lb).

The relative humidity field varies from 44% to 77% at low levels (Fig.3. lc) and decreases

upward (Fig. Id). The combination o f a moist surface and dry upper-level flow is favorable


25

I'ig .VI (a) Initial surface temperature field. The contour interval is 4°C. (bf Initial surface pressure Held with a

contour interval of 4 mb. tci Initial surface relative humidity. Hie contour interval is 11%. (d) Vertical cross section

of the initial relative humidity taken at X = 200(1 km. The contour interval is 11%. (e) Horizontal cross section of

MPV field at the initial lime taken at W mb. The contour interval is 0.1 PVU. For laH c) and (e), the abscissa is in

the X direction and the ordinate is m the Y direetion. N t here denotes the north. For (d). the abscissa is in the Y

direction and the ordinate is upward. The horizontal domain size is 40ut) km s SOOO km and the distance between

two ticks is lot) km.


66


27

environment for explosive cyclogenesis (Barry and Chorley 1982). The initial M PV is

positive everywhere in the domain (Fig.3.Id) and the atmosphere is therefore conditionally

symmetrically stable.

3,3,2 Sensitivity experiments

The description o f moisture distribution in experiments (a)-(h) is given as follows,

(a) A high RH band is located at the surface low (Fig.3.2a). This RH distribution is similar

to the observation by Bennetts and Ryder (1984) (see their Figs. 1 and 2 for details), (b)

High RH bands are distributed on both sides o f the surface low (Fig.3.2b). Some

observational evidence can be found in Figs.3 and 5 o f Locatelli et a l.'s (1989) paper, (c)

A high RH band positions behind the surface low (Fig.3,2c). (d) Contrast to the

experiment (c), a high RH band is ahead o f the surface low (Fig.3.2d). In the experiments

(a)-(d). only one band o f high moisture content is specified in the entire domain. While in

the experiments (e)-(f), multiple bands o f high RH are specified, (e) Two bands o f high

RH are located on both sides o f the surface low (Fig.3.2e). (0 Two bands o f high RH are

ahead o f the surface low (Fig.3.2f). The next two experiments are almost same as the

Control except the different range o f RH variation. In the experiment (g), surface RH

varies from 66% to 88% (Fig.3.2g). In the experiment (h), surface RH changes from 33%

to 55% (Fig.3.2h).


I'iu .3.2 Initial surface relative Imnndilv fields lor the experiments (a H in . ‘Ihe contour interval is 11 %


30

Chapter 4

GENERATION OF MOIST POTENTIAL VORTICITY IN EXTRATROPICAL CYCLONES

In this chapter, the generation processes for negative MPV in typical mid-latitude

cyclones are examined using the three-dimensional model described in the previous

chapter. The results o f the control experiment are presented in section 4 .1 followed by a

set o f sensitive experiments in section 4.2.

4.1 The control experiment

The initial and boundary conditions o f the control experiment are described in

chapter 3. The simulation is started with a moderate perturbation and integrated for 8.5

days o f model time.

4 .1.1 Life cycle o f the extratropical cyclone and MPV distribution

Fig.4.1 shows the evolution and the structure o f the surface temperature field at

different stages o f the cvclone development. By 30 hours, the warm conveyor belt,

oriented northeast-southwest, has formed ahead o f a strong baroclinic zone (Fig.4. la). At

48 hours, a bent-back warm front and a T-bone structure (Shapiro and Keyser 1990; Kuo

et al. 1991) have appeared (Fig.4. lb). The warm core enclosed by the 0°C isotherm is

completely cut off'at 66 hours (Fig.4.1c) and then split into two parts at 72 hours


31

(a) N T (b) (C)

ivK\ * : w ' f W /IWvA *— 1 1 Mi' '\\* * % * * * 1 1 CO/ .w i\» * * • • • 1 h i S 1 \ \ * * % * • • I M r/ \\ * \ *—• ' ' /H i*/ l \ \V ^ __

\ : \ : } cl rS 1 /// V * *i J&y 1 1 *E * / r 1 \ VV\ \ » *

+ * "■s

/ $ F o & » T'* . .y i\ lffi\ f\V» *.*»*' • »•% * • . v5\ t )}•• » 1 ,* « »\ i;s\t-'y'* 'i1'1 \ ■ .- >

is_

(d) (e) r*(0 '.............................

( S b ,

f , :

**/v. J

7 »»#■*.• |!/' i ;**%'* ui * * • i » t /#*•,»» i

\^ 4 v k S > ft,;*-...* % ' i * t"*V •«/■'\ \ ’ * \; ...... :i V “• J :5\ \! '• / J R •% % tlwt. » 1• ' --'rtfcvsS&sS

%%% ** *S .

: **:*?«*.V ' » \ A * l» *i'— i s * *'*. Vi- , .. «N-

.V —X __ 5

-i”\ t'** #/# 'V

1 'm.4 I Surface temperature field al hours (at 30. ibt 4S. (ct 0 6 . (d) 72. (e) 84 and (0 204. The contour interval is

4°C


32

(Fig.4. Id). One o f them is lifted o ff the ground by 84 hours (Fig.4. le). At the end o f the

simulation (8.5 days), the remaining warm core is mostly lifted oft* the ground (Fig.4.10.

The surface temperature simulation presented here is similar to Shapiro and Keyser's

(1990) version o f the life cycle o f marine cyclones, and agrees well with recent

observational studies by Neiman and Shapiro (1993). and Neiman et al. (1993). It is

someu'hat different fror.i the life cycle simulated by Schultz and Mass (1993) in which the

occlusion process in the sense o f a cold front catching up to a warm front, as explained in

the classical Norwegian model, is observed in their simulation o f a mid-latitude cyclone

over land.

Fig.4.2 shows the evolution o f MPV on the 871 mb pressure surface at different

stages o f cyclogenesis. and Fig.4.3 the corresponding distributions o f relative humidity

Negative MPV' has not yet appeared during the development stage (Fig.4.2a) At the

mature stage (Fig.4.2b). an area o f negative MPV appears to the south o f the bent-back

warm front, where the air is unsaturated with respect to water vapor (Fig 4.3b). After the

mature stage, the negative MPV moves into the part o f the warm core (Fig.4.2c) where

relative humidity is about 90% (Fig.4 3c). At the end o f simulation, the MPV is positive

everywhere.

Figs.4.4 and 4.5 give the distributions o f MPV and relative humidity on the 757 mb

surface. As can be seen from Fig.4.4. the negative MPV first appears in the warm sector

near the north end o f the cold frontal zone (Fig.4.4a) close to the position o f the surface

low where the air is unsaturated (Fig.4 .5a). The observ ations shown in Fig. 2 o f Zhang and

Cho (1992) supports our MPV simulation at this stage. The negative MPV in their Fig.2

occurs in the warm sector ahead o f the cold frontal zone indicating that the generation

mechanism is due to strong baroclinicity as suggested in chapter 2. Parsons and Hobbs

(1983) also found a negative "moist symmetric stability parameter", proportional to the

geostrophic M PV (Thorpe and Clough 1991). in the warm sector o f the cyclone. At the

mature stage o f the cyclone, the area o f negative MPV (Fig.4.4b) moves into an


33

u 11111111111 ii it 11 u ii i n i u 111 u n nr:

T mi mu 111 m 11 n mu IIIII m n i i i r

j 11111.1111111111 u 111111 n 11 n rrn 111

n T

o

■*tnt i t t i n 111 n i I I I I I I I I I i I I I I i I H I 111

J IIIIII11111111’ 11111IIII111111111111ITE

O

■ m n ii in iu iL in t i im i i i i i t tn im tr

L i i i i i i m i i i i i i i i i i i i i i i i m i i i i i i i i . m

i i i i i u u i i i i i i i i i n n i i i i i i i im i i i i i f

Fig.4 2 I [on/onlal cross section of M I’V taken at the ST I mb at hours ta) 30. (b) 48. (c) 6 6 . and (d) 204. rite contour

interval is (I 1 I’Vt i for 40 hours and 0.5 PVU for the rest. The shadings indicate regions of negative MPV.


l'ig.4.3 Horizontal cross section of relative luimiditv taken at the S7I nth al hours in ) 30. (b) 4X. (c) <rf>. and (d)

Hie contour interval is 3.'%.


35

I a 'j imiMTtMtmiinmMinmmmrij

T I I 1 I I I I I I I I I I I I I I I I It I t I I I I I t1 I I I I I I I

u in m 111 i n i i i 11 m rr m 11111 i n i rma

N t

T i i i i i i i i i i i i i i i i i i i i im im i i i im i in

Ta

j 11 i 11 i j i i i i i j i n 11 i i h j i I i i j i i ri f c

h 11 it 11 n 11111111111 n m i n n i m hilt

tmmmmmnrqTTT

ii im n n in iiiii it iim;iniiiiittii_d

l'isi.4 4 1 lori/ontal cross section of MPV taken at the 757 mb at hours (a ) 30. (b i 48. (e) 6 6 . and (d) 204. The contour

interval is 0 I PVU for 30 hours and 0.5 PVIJ tor the rest.


36

l . v

n T

t a

I'iii.4.5 Horizontal cross section of relative inuiiidttv taken at the 757 mb at hours iai 30. (bj -IS. t c»*>*». and (di 204

Hie contour interval is 33".7


37

unsaturated region south o f the bent-back warm front (Fig.4.5b); meanwhile negative

values begin to appear along the cold front (Fig.4.4b). At 66 hours (Fig.4.4c) the former

feature progresses toward the unsaturated part o f the warm core (Fig.4.5c) while the latter

further intensifies and becomes a major negative MPV region in an unsaturated

environment. At the end o f the model simulation o f 8.5 days, all regions o f negative MPV

at this level have disappeared.

The vertical cross section o f the wave structure at 48 hour model time along the

line AA' marked on Figs. 4.4 and 4.5 are shown in Fig.4.6. An upper-level westerly jet

with maximum speed o f 60 m/s (Fig.4.6b) is associated with the cold front (Fig.4.6a). At

the low-level, an easterly jet at the speed o f 50 m/s is linked with the bent-back warm

front. The generated negative MPV area in the vicinity o f the bent-back warm front has

moved into upper levels due to the induced secondary circulation. Based on the

calculation o f geostrophic MPV using the Fronts 87 dataset, Thorpe and Clough (1991)

found that negative geostrophic MPV is partiy distributed in the statically stable warm

sector aloft, perhaps due to the reasons just explained. Since the tropopause in the form o f

a PV inversion is not included in the model, the potential vorticity intrusion from the

stratosphere into the troposphere do not appear in the simulation.

4.1.2 Baroclinic generation o f MPV

Qualitative information about negative MPV generation can be obtained by

analyzing baroclinic vectors and moisture gradients on isobaric surfaces, while quantitative

information can only be obtained by evaluating (2.12). On a constant pressure surface, the

right hand side o f (2.12) can be written as

A (V^l) v — nW^q. (4.1)cn

where V^ is the horizontal gradient operator on a constant pressure surface, and n is a


38

0.050

0.150

0.250

0.350

0.450

0.550

0.645

cr 0.730

0.805

0.870

0.920

0.960

0.986

0.996

Fia.4 <> Vortical cross section taken at the X = 400 kni at 48 hours ol'/ai equivalent potential temperature at an

interval of 4 K; (hi u component of veloeiiv at an interval of 10 m/s; (c) MPV field at an interval ot 0 I P V If


0.050

0.150

0.250

0.645

0.730

0.870

0.920

0.996

0 .150

0.250

0.450

0 .730

0 .805

0 .870

0.920

0.960

0.986

0 .996

Fig.4.0 (b) and(c)


40

unit vector normal to the pressure surface. Since the baroclinic vector is parallel to the

lines o f intersection o f isentropic and isobaric surfaces, the potential temperature contours

on a constant pressure surface give the direction o f the baroclinic vectors. The direction o f

the gradient o f specific humidity on the surface can be determined from the specific

humidity contours. These maps give therefore a qualitative indication where the MPV

generations may be significant. The only drawback o f such an approach is that the strength

o f pressure gradient cannot be determined from one pressure surface map alone. A similar

analysis can also be carried out on a constant potential temperature surface.

Fig.4.7 shows the potential temperature and specific humidity contours on the 757

mb surface during the evolution o f the cyclone. The distributions o f the function A in

(2.12) at 30 hours on the 757 mb and the 406 mb surfaces are shown in Fig.4.8. They are

negative everywhere. The directions o f the potential temperature gradients and specific

humidity become substantially different at the boundary o f the condensation region

(Fig.4.7a). It is not difficult to see. therefore, that negative MPV on the 757 mb surface

first forms in the warm sector near the northern end o f the cold frontal zone, where

condensation first takes place. This effect can also be seen at 48 hours (Fig.4 .7b) and 66

hours (Fig.4.7c) when negative MPV along the cold front occurs in regions next to the

areas o f condensation. At the end o f the simulation (Fig.4.7d). the potential temperature

contours are almost parallel to the specific humidity contours particularly in the warm core

and near the fronts, and the source term in (2.12) is thus nearly zero at this time. The

above analysis can directly be visualized in a three-dimensional configuration (Fig.4.9), in

the warm core and the cold front where the orientations o f the baroclinic vectors,

represented by the line o f intersections between constant 0 and p surfaces, and the

moisture gradients, indicated by the normal o f a constant specific humidity surface, are in

favour o f negative MPV generation.


41

M u r« m m n m i m i 11m * n h i n ij

nmmiiLL

m m ii rii m m 1111iiii,

Fiu.4.7 Distributions of specific humidity (solid lines! and potential temperature (dashed lines) on the 757 mb

pressure surface at hours (a) 30. (b) 48. fc) 66. and (d) 204 The contour intervals of specific humidity and potential

temperature are 2 g/kg and 6 1C. respectively.


42

mil i n nmnmmi m i 11 u m hi imj

(a) N t |

ji ii ii. i iii ii ii ii iu 1111

l'in.4.8 Horizontal cross section of the function A at 30 hours taken at (a) the 757 mb and (b) the 406 mb. Hie

contour interval is 1 m(’kn'-.


43

l-'ig.4.9 The distribution of three (kids at 72 hours, the constant specific humidity surface (2 g/kg) indicated by the

green surface, the constant potential temperature surface (296 K) indicated by the red surface and the constant

pressure surface (757 mb) denoted bv the purple surface. In the figure, the origin is at the right bottom comer close to

us and the x-axis points into the page, y-axis to the letl, and z-axis upward. The three-dimensional perspectives arc

made using F.xplorer on a Silicon Graphics computer.


44

4.1.3 Effects o f horizontal diffusion and rainwater evaporation

Horizontal diffusion and rainwater evaporation are two source terms o f MPV

which have not been discussed so far. In order to examine relative significance o f their

effects on MPV generation, two sensitivity experiments are carried out. The sensitivity

experiments performed have a domain size 4000 \ 4000 km.

(1) Horizontal diffusion

Orlanski and Katzfey (1987) found that horizontal diffusion begins to affect the

intensity o f model cyclones when horizontal diffusion coefficient K n has the values from

1.x 1(P to 1.x 106 m ^s'1. In the simulation presented in the previous sections, the value o f

horizontal diffusion coefficient K H ranges from 3 .95*104 to 5 .75xl04 m V 1. Fig.4.10

shows the results o f the experiments with and without horizontal diffusion (Figs.4 .10a and

4.10b, respectively) at 24 hours o f model time. Except for some expected noise

(Fig.4.10b), the two M PV fields are quite similar.

(2) Rainwater evaporation

Rainwater evaporation has little effect on MPV generation since MPV fields are

almost the same in two experiments w ith and without rainwater evaporation (Figs.4.1 la

and 4.11b). MPV generation due to diabatic cooling o f rainwater evaporation often

happens in a substantial unsaturated downdraught. This effect is very small because

rainwater evaporation spreads in a deep column compared w ith snow evaporation (Clough

and Franks 1991) and other phase changes like melting, and only produces a weak

gradient o f diabatic cooling.


45

10 Horizontal cross section of MPV taken at the 757 mb at 24 hours of ta'i the experiment with horizontal

diffusion and (hi the experiment without horizontal diffusion. Hie contour interval is 0.1 PVU.


46

Fig.4.11 Horizontal cross section of MPV taken at the 757 mb at 42 hours of (a) the experiment with rainwater

evaporation and(b) the experiment without rainwater evaporation. The contour interval is 0.2 PVTJ.


47

4.2 Sensitivity study

Eight sensitivity experiments are conducted, each with a different initial moisture

field as explained in chapter 3. The results o f these experiments are summarized in this

section.

(1) Experiment (a). Although the warm core enclosed by 0°C isotherm is not entirely

lifting o f f the ground by the end o f the simulation (8.5 days), the evolution and structure

o f the surface temperature is generally similar to the control experiment. Figs.4.13 and

4.14 shows the evolution o f MPV and relative humidity on the 871 mb pressure surface at

different stages o f cyclogenesis. Negative MPV first appears in the unsaturated area to the

south o f the bent-back warm front at the mature stage, and then moves to the warm core

after the mature stage. Additional negative MPV takes place along the cold front. While in

the control experiment, there is no net generation o f negative MPV on the 871 mb surface

along the cold front. By the end o f the simulation, all negative MPV become positive. The

contours o f the potential temperature and specific humidity at different model time are

shown in Figs. 15. The regions o f large differences in direction between the gradients o f

potential temperature and specific humidity are located in the area where MPV

generations are significant.

(2) Experiment (b). Because the initial band o f high RH in this experiment is distributed on

both sides o f the surface low. the first appearance o f negative M PV on the 871 mb

pressure surface is no longer in the warm sector near the northern end o f the cold frontal

zone but at the warm side o f the warm front (Fig.4.16a). The negative MPV is westwards

tilted in vertical at 24 hours as shown in Fig.4.16a and Fig.4.17a. The negative M PV area

on both the 871 mb and the 757 mb pressure surfaces, located in unsaturated area

(Figs.4.18 and 4.19) then moves into the bent-back warm front and intensifies there. The


48

u 11111111111 rrr rr1111111ri 111111111111

•! i m i n m i i n t n 1 1 1 1 i i 1 1 1 1 1 1 1 1 1 1 1 1 1 1 n

LM11II111 TiTrTTTTnTinm nm

T i l l " n l i m i 11 m u II i l l M i i i i u i i i i r

1111111 11111111111111

TTTX l l l t l l l l l l t l U H H f l K K I ITTTfl

XITI'11111 I t 11 U I ! 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ! | L

-iLLLLLiiittitntnnitiiittttnninitr

I- ig.4. 13 I lonzontal cross section of MPV for the experiment ta) taken at the 871 mb at hours (a) 24, (b) 48. (c) 72,

and (d) 204. The contour interval is 0.5 P VI). 'Hie shadings indicate regions ot negative MPV.


49

l-'ia.-4.14 Horizontal cross section of relative humidity tor the experiment (a) taken at the 871 mb at hours (a) 24. (b)

48. (c) 72. and Id) 204. The contour interval is 33%.


50

111111m11111111111ii111111 rnit

... nn .......... n ...... m .. i. ..... in

m

Tin if»nil iiimmiMimiirnmiir

..m m .ii.................

Fig. 15 Distribution of specific humidity (solid lines) and potential temperature (dashed lines) on the 871 mb pressure

surface for the experiment (a i at hours (a) 2-4. (b) 48. (c) 72. (d) 96 and fe) 204 The contour intervals of specific

humidity and potential temperature are 2 g/kg and 6 K. respectively.


51

Li ii n i i n ri n m m i rm t vrrrvr r m n rz

-r 11 n i n ii 111 ii titiLLLU m iiiim iilifi

i i i i i i m i i i i i i i i i i i i i n n i i i m i ' i '

t11 i i n1111111n11i i m inimum

U1:111II111M 111M 11111 n U1111111! 11! LI

7 n 111 t i i n n 111 m n i r r i 1 I I I H 1 t r u i i f

■ii i i m i i m m m Tn-r m n in i t im i iu

7 n 11 n 111 m i n 11 n 1111 n 111 n 11 n 1111

Fiir.4 16 Horizontal cross section of MPV for the experiment (b) taken at the 871 mb at hours (a) 24, (b) 48, (c) 72

and (d) 204. The contour interval is 0.5 PVU. Hie shadings indicate regions of negative MPV.


52

j iu m iim i TTTTTTTTJ

»i i i n 11111111 m n i n 1111111 n 11111 m

1111111 III Tr riTTnrni im iunm

( b )

iLLitii m m i i i m m i n 11 n i r

m i n i u m i i . t m i i i i i i i i n i n n n i iH

t i i ii i ii n i ii n i m 111 ii 11 uni 11 in i ii i

(d)

h jllim n m in m in im um i nn.tr

Ha.4.17 Horizontal cross section of MPV for the experiment (hi taken at the 7?7 mb at hours (a) 2-1. (hi 4K. (c) 72

and (d) 204. Hie contour interval is 0.5 PVU. The shadings indicate regions of negative MPV.


53

I j i / r ^

i'ig.4. IS Horizontal cross section of relative humidity for the experiment (b) taken at the 871 mb at hours (a) 24, (b)

48, (e) 72 and (d) 2tt4. The contour interval is 33 %.


54

Fig.4.19 Horizontal cross section of relative humidity for the experiment (b) taken at the 757 mb at hours (a j 24. (b)

48. (c) 72 and (d) 2t)4. Hie contour interval is 35 %.


55

part o f them finally progresses to the saturated warm core. Different from the control

experiment, the negative MPV stays in the unsaturated warm core until the end o f the

simulation. Another significant feature is no net generation o f negative MPV in the cold

frontal zone because there is no condensation at the low or middle levels, and therefore

the gradients o f potential temperature and specific humidity are either parallel to each

other at the low level (Fig.4.20) or weak at the middle level (Fig.4.21).

(3) Experiment (c). The initial high moisture content in experiment (c) is located behind

the surface low. Fig.4.22 and Fig.4.23 show the generation and evolution o f M PV through

the life cycle o f the cyclone, which occur in unsaturated environment (Figs.4.24 and 4.25).

On the 871 mb pressure surface (Fig.4.22), negative MPV at the first place appears along

the warm front at the development stage and then in the bent-back warm front at the

mature stage. After the cyclone matures, the negative MPV moves into the warm core.

Meanwhile, negative values start to form in the cold front (Fig.4.22c) on the 871 mb

surface, which does not happen in the control experiment. A t the end o f the simulation,

the negative MPV become positive. In contrast to the control experiment, no negative

MPV is found on the 757 mb pressure surface at the development stage. The negative

MPV is first produced in the bent-back warm front, which is very much weak in strength

compared with the control experiment, and it disappears when the negative M PV

generates in the cold front. This cold-front negative MPV then moves to the warm core

and eventually disappears. While in the control experiment, the significant feature is that

negative M PV exits in the bent-back warm front/the warm core and the cold front

simultaneously.

(4) Experiment (d). Because initial high RH band is positioned ahead o f the surface low

and moisture is therefore advected along the warm conveyor belt approximately parallel to

the cold front, the negative M PV generation on the 871 mb pressure surface first takes


56

TTTTJ

\ \

t . mi., n .......... ,t ..... .

Fiti.4.20 Distribution of specific humidity isolid lines) and potential temperature (dashed lines) on the 871 mb

pressure surface for the experiment (b> at hours (a) 24. tbi 48. ic'i 72 and (d) 204. Hie contour intervals ot specific

humidity and potential temperature are 2 g'tcg and 6 K. respectively.


57

iu iiu i i i i u m i ii'i1 1 1 1m i ii n 11111it

. m i n . n i m . i i IIM H I I

l:ig.4.21 Distribution of specific humidity (solid lines) and potential temperature (dashed lines) on the 757 mb

pressure surface tor the experiment (b) at hours (a) 24. (b) 48. (c) 72 and (d) 204. The contour intervals of specific

humiditv and potential temperature are 2 g/kg and 6 1C. respectively.


58

1111 rm r rm i m u it t i m > r r r r g r g

= (a)

■ i i i i i m n i i i i i i m i i i i i i i i i i i i i i i i i n -i 1111 ii 1111111 i i i TTTTTTTnTTTTTTTTTTTTT

t i i i i l n m in i m i i 11 m t u n ) n in iLnn i ) 11 n 1111 n n i ? m n l n n 111 n i t 1 1 1L

31111111 m u i i i l l m i '1111! 113

cm in n m m m i t i im ii i i i i in n n in ii in in m m iiiiiin iin im in iiiir

Fig.4.22 Horizontal cross section of MPV for the experiment (c) taken at the X7! nib at hours (a) 24. (h) 4X. (o 72.

(d) ‘76 and (o) 204. Hie contour intervals arc 0 1 PVIJ for la) and 0 5 PVU for the rest. The shadings indicate regions

of negative MPV.


59

;! i I'm i m i i i h i i h d ri i ru m vm m u

\ <*> I

I Lin n ILI H « n i » 11 n 111111 n 11 n 11 n

11 I 11 111 I l l|i i 111 n m 1111m 'it

(b)

-Mum inniinmniinnninnn m3 h_t 111 i i luj in iin iiinn im iiiiiiiiL

.nn m rn 'in 11 m m i'i 11 m 11 i t h i m u

1‘ig.4.23 I lori/.ontal cross section of MPV for the experiment (c) taken at the 757 mb at hours (a) 24. (b) 48, (c) 72,

til) % anil (el 204. lhe contour intervals are 0.15 PVU for (hi and 0.5 PVU for the rest, rhe shadings indicate

regions of negative MPV.


60

n » M I I r T T M I ! I I ' l l I T I I I I ! M 11' 111111111'1

Tin.4.24 Horizontal cross section of relative humidity for the experiment (ci taken at the X7! inb at hours t a ; 24. (b)

48. (c i 72. (d) 96 and ( e ) 204 The contour mten'al is 33 %.


61

O

/

l'ivt.4.25 I lonzonial cross section of relative humidity for the experiment (cl taken at the 757 mb at hours (a) 24, (b)

48. (c) 72. (d) % and (e) 204. The contour interval is 33 %.


62

place in the warm sector at the north part o f the cold front, i.e., at the intersection o f the

cold and the warm fronts. Then, this negative MPV area moves into and intensifies at the

bent-back warm front and the warm core (Fig.4.26). Note that all negative MPV

generations happen in the neighborhood o f condensation (Fig.4.27) where big differences

in direction exist between the gradients o f potential temperature and specific humidity

(Fig.4.28). The development and evolution o f negative MPV on the 757 mb pressure

surface are similar to that on the S71 mb pressure surface. Different from the control

experiment, at/after the mature stage no negative MPV appears in the cold front because

the gradients o f potential temperature and specific humidity are either near parallel to each

other in direction or weak in strength. In addition, in the cold front region no condensation

takes place to change these gradients.

(5) Experiments (e)-(f). At the development stage, M PV generation in the cyclones is

mainly affected by the initial moisture distribution. Fig.4.29 shows the MPV evolution on

the 871 mb pressure surface for the experiment (f). Because o f the band-shaped moisture

structure (Fig.4.30), the negative MPV at the development stage is well organized in the

manner o f the bands. They are located in the warm side o f the warm front and the north

end o f the cold front, respectively (Fig.4.29a). Then, these bands o f the negative MPV

emergence into the bent-back warm front and move to the warm core (Figs.4.29b and

4.29c). They stay there till the end o f the simulation (Fig.4.29d). The structure o f negative

M PV on the 757 mb pressure surface is similar to that on the 871 mb pressure surface, but

the former is much weaker in strength than the latter. For the experiment (e), the structure

o f the generated negative MPV on the 871 mb and the 757 mb pressure surfaces at the

different stage o f cvclogenesis is almost identical to that o f the experiment (f). It seems

that initial moisture bands ahead o f the surface low rather than behind the surface low are

the most important in determining the generation and evolution o f negative MPV. The

slight deviation o f the experiment (e) from the experiment (f) is that an extra weak band-


63

i n m . i m im m i f i im m n im m u

hum m i I I I m um m ini 111 miniH

u immmmi ium im nim inn iiu

t m n i i n u 111». i u i i n 1 1 i m i n n i m r t

mriTtTmm m m inmi Tq

n t u litLLi 1111 m m iiilli ; 11 n n i i n n

u i i im i i i n i i m i i m i i i i m u r irm i lj

t i im im i m i t < i u~i immi niiTh111 r

Ftn.4 2<> I lorizonlal cross section of MPV for the experiment (<i) taken at the S7I mb at hours (a) 24. (b) 48, (c) 72,

and (d) 20-4. flic contour interval is 0.5 PVU. The shadings indicate regions ot'negative MPV.


64

i

l'i".4.27 Uorizonlal cross section of relative humidity for the experiment <d; taken at the X7! mb at hours taj 24. (b)

4X. (ci 72 and (d) 204. The contour interval is 33%.


65

\ _ /

o

I-ig.-4.2S Distribution of specific hiuniiiity (solid lines) and potential temperature (dashed lines) on the 871 mb

pressure surlaee for the experiment (d) at hours (a) 24. (h) 48. (e) 72. and (d) 204. The contour intervals of specific

humidity and potential temperature are 2 g/kg and 6 K. respectively.


66

LirnrrmtTi liim iiiimiiim iiiim

■t n n 11111111111 n i n i n i ii n 111 n 1111 rt

j n i n 11111 n n in i iTTTTt i ii i n 1111 n i

h n 111 »i iii n m nILLLUii inn 11m 11 r

j T k i if i ¥ Y f i f i Y U T i ' i i u 1111111111 m i m u

i ii m 11111 n t n i»111 n 11 n ii t n i ii n i

J iiii i im m riiiii i

hn ii i ii 1111111111 ii r n 111 ii i ii i ii i ii i r

Fia.4.29 Horizontal cross section of MPV for the experiment (0 taken at the X7I inb at hours (a) 24. (b) 4X. (c ) 72.

and (dl 204. The contour interval is 0.5 PVU. Tlie shadings indicate regions of negative MPV


67

shaped negative MPV is produced on the 871 mb pressure surface probably due to the

initial bands o f high moisture content at the rear o f the surface low.

(6) Experiment (g) and (h). The initial moisture content specified in the experiment (g)

ranges from 66 % to 88 % while in the experiment (h) from 33 % to 55 %. In the

experiment (g), the warm core enclosed by 0°C isotherm is completely lifted o tf the

ground by the end o f 6 days, 2.5 days earlier than in the control experiment. This stage in

the experiment (h), however, has never arrived till the end o f the simulation. Although the

experiments (g) and (h) are mainly similar to the control experiment, some differences are

found among them: ( I) The negative MPV in the experiment (g) does not last longer

period and disappears after the cyclone matures because most o f condensation take place

before the mature stage o f cyclogenesis. and the contours o f potential temperature and

specific humidity are parallel to each other. (2) In the experiment (h), there is no net

generation o f negative MPV along the cold front on the 757 mb surface presumably due to

lack o f moisture.

4.3 Summary

In this Chapter, the mechanism o f MPV generation in a three-dimensional

frictionless and moist adiabatic flow has been applied to mid-latitude cyclones with

different moisture distribution. Because in the frictionless and unsaturated atmosphere,

MPV generation is governed by baroclinic vectors and moisture gradients, the effects o f

different moisture distribution are investigated by designing nine numerical experiments. In

these experiments, the moisture gradients are required either perpendicular to (the first

type o f experiments) or approximately parallel to (the second type o f experiments) the

potential temperature gradients. It has been shown that MPV generation in extratropical

cyclones is sensitive to moisture distribution particularly at the development stage o f


Fig.4.30 Horizontal cross section of relative humidity lor the experiment (0 taken at the K71 mb at hours (a) 24

48. (c) 72. and (d) 204. The contour interval is 33 %.


59

cyclogenesis. The main results o f the control experiment and sensitivity experiments are

summarized as follows:

(1) In extratropical cyclones, negative M PV generation usually takes place in the

warm sector near the north end o f the cold front, the bent-back warm front, the warm core

and the cold front. The most favourable places for negative MPV generation are the bent-

back warm front and the warm core. The warm-front negative MPV simulated in this

study is in an excellent agreement with the observations.

(2) After the cyclone matures, the negative MPV moves into the warm core. It w ill

stay there till the end o f the simulations i f air is not saturated. Otherwise, negative M PV

becomes positive.

(3) The development o f negative MPV along the cold front is, to large extent,

dependent on the location and strength o f high moisture content. The bands o f initial high

moisture content w ith the wave length - 2000 km located at and just behind the surface

low is accounted for a favorable condition for low-level (871 mb) negative M PV

generation along the cold front.

(4) The initial multiple bands o f high moisture content normally affect negative

MPV generation at the development stage o f the cyclones, and the bands ahead o f the

surface low are favorable for negative MPV generation.


70

Chapter 5

BOUSSINESQ APPROXIMATION AND ITS IMPLICATIONS

It is well known that the Boussinesq approximation retains the density variations in

the buoyancy term while it ignores density fluctuations in the inertial terms. Since this

approximation drastically simplifies the solenoidal term in the vorticity equation, it is often

used in theoretical studies o f for example, baroclinic instability and frontogenesis.

Because the solenoidal term is one o f the main source terms for vorticity and it is

proportional to the MPV source term o f (2.12). it would be interesting to examine the

effects o f this approximation on the vorticity and MPV dynamics. A complete comparison

o f the differences and similarities between a primitive equation model and a Boussinesq

model would be based on the integrations o f these two models. Since simulations using

the Boussinesq approximation have not been made in this study, we will make the

comparisons in the following two ways.

First, using the simulation results o f the primitive equation model, we w ill compare

ail terms that affect the evolution o f the vorticity and MPV fields, and identify those terms

that are kept in the Boussinesq approximation. Based on these comparisons, we will

consider next the similarities between a Boussinesq and a primitive equation model o f

circulation patterns due to vorticity and MPV distributions. We will start with the vorticity

field.

5.1 Vorticity dynamics


71

The difference in the vorticity dynamics between non-Boussinesq and Boussinesq

flow is in the way they treat the density in the governing equations. In the Boussinesq

approximation (Spiegel and Veronis 1960), one ignores the density variations in the mass

continuity equation and the horizontal momentum equation, but retains the density

variation in the buoyancy force where the fractional density variation is assumed to be

equal to the fractional temperature variation, and the pressure fluctuation is ignored.

Neglecting the frictional force, one can write the vorticity equation in the primitive

equation model as follows:

^ tl - « a - V ) V = - CaV . V +- L ( V |ip . V hp + V hp x ^ k + a k x V hp). (5.1)

where the solenoidal term, ( V p < V p ) / p : , has been split into three components

represented by the three terms in parentheses on the right hand side o f the equation. In the

Boussinesq fluid, the vorticity equation is simplified to:

^ - - ( L . V ) V = - - ^ k * V0. (5.2)dt a 0 o

Note that the term £ aV - V is ignored in (5.2) because o f the incompressibility

assumption. In the Boussinesq model the solenoidal term is only in the horizontal direction

while in the primitive equation model, the solenoidal term has both horizontal and vertical

components.

To examine the differences in vorticity dynamics between (5.1) and (5.2), we have

evaluated the terms on the right hand side o f these two equations using the output o f the

dry experiment, which is identical to the control experiment except the relative humidity

being set to zero everywhere. The results at 48 hour simulation are shown in Fig.5.1.

Panels (a) and (b) shows the values o f the horizontal and vertical components o f

the vector (C • V ) V , with maunitudes o f the order 10*7 s-2 and 10'9 s-2. Since these terms a

appear in both (5.1) and (5.2). they serve as useful references. Panels (c) and (d) show the


horizontal and vertical components o f the compressible term Q V ■ V . As expected, thetl

vertical component o f this term is not negligible while the horizontal component o f this

term is small as compared to the other terms in equation (5.1). Similar results were

obtained by Hoskins (1972).

The three solenoidal terms o f (5.1) are shown in panel (e), (f) and (g), while the

right hand term o f (5.2) is shown in panel (h). The vertical component o f the solenoidal

term, 1 / p : (V p » V p ), which is absent in the Boussinesq approximation, is small as can

be seen from panel (e) o f Fig.5.1. We note that the second term in parentheses on the right

hand side o f (5.1) can be reduced by straightforward manipulations to the simplified

solenoidal term in the Boussinesq form o f the equation (5.2) i f one assumes thei i t

hydrostatic approximation and makes use o f the approximation 0 /9 = - p /p (0 and

0 are perturbed and reference potential temperature). These two terms are shown in

panel (g) and (h) in Fig.5.1. It can be seen by comparing them that two panels are quite

similar in both magnitude and in pattern, but the simplified solenoidal term in the

Boussinesq model underestimates the corresponding part o f the solenoidal term in a

primitive equation model. The horizontal component o f the solenoidal term

l / p : ( c p / c z k x V p). however, is completely neglected in the Boussinesq

approximation. As shown in panel (f) o f Fig. 5.1, this term is smaller than but has the same

magnitude as the other horizontal term in the solenoidal term. These comparisons are

quite representative o f the results at other levels and other times even though we have

shown the results only at a particular time and a particular level. In Fig.5.2 we show the

same comparisons for the control experiment. Similar conclusions can be drawn from the

control experiment w ith moisture included.

In Fig.5.3 the sum o f the two horizontal components o f the solenoidal term in a

primitive equation model at levels 2.4 km and 6.9 km is shown in panel (b) and (e), while

the simplified solenoidal term in the Boussinesq approximation is shown in panel (a) and


7.3

■o—

1'iii.5.1 I lon/ontal cross sections lor the ilrv cxpennienmiken at 48 hours at 2.4 km above tlie ground ot la) the horizontal component

of tire vector I C ■ V i V with a contour interval ot 5. * Ilf ' s'-, (b) tlie vertical componcit ot the vector (C, ■ \ )V with a contour a a

interval oi' I . >■ It)''* s'- mid tlie direction of the corres xii iding vector (arrows lie) same as (a) except tlie horizontal component ot vector

" \ ' ■ V with a contour interval of 3 ' It) s'-. id) same as tb) except tor the vertical conuxment ot' vector^ V • V with a contour a a

intenal of I v it)'0 s'-.


74

;ls i b ) except Ibr llic vector \ ' p x V p / p “ with a contour interval o f 2. * |ir*°s “ (I) icune as (a) except lor tlie

vector t-p / (7.1c x \ ' p / p 2 with a contour interval of5. x K r7 s°: (g) same as Cl) except for tlie vector V ^ p z / T p / (7.k / p“

with a contour interval ot'5.* I l l 's 0 : (li) samcas(a) except fortlie vector -e nnk - \n with a contour interval ol 5 a Id ' s2


Fie.5 2iaH J> Same as Fie.5.1 except for the control experiment.


76

Fi;;.5.2 (CM")


77

•........ jN t j! (a)

i -T iN v c \

Vs —

■\Y\:NJi

r - I I t 1 1 t l !_LI_1_L1_L1 .1 I 1 ■ 1 i I I I 1 1 L H .U 1 I 1 1 I I ■ !

niiiniiii

;.um jm uum i mimmu

M i i n r i i i i i i i i M n T

O’

TTTTTTTTTTTTTPTT'PrTTTTTTTTTTTTTTTTTTTTT

( d ) .........................

11. H I , . . . . . . . . . .

I'm.5 3 Horizontal cross sccuons for the dry experiment taken at 48 hours at 2.4 km above the ground ot (a) -

2g 0 o li x VO with a contour interval of 5. * It)*7 s'2’, (bl same as (a) except for tlie vector (c p / c?zk x V^p / p“ +

p x fp / f /k / p~); ic) same as la) except tor the difference between (b) and (a) with a contour interval ofh

2.5 x 10*' s'2: (dHt3 are same as (aH d except die cross section taken at 6.9 km above the ground.


78

> <

c>

Fig. 5.4 Same as Fig.5.3 except for the control experiment.


79

(d), respectively. The differences between panel (a) and (b) {(d) and (e)} are presented in

panel (c) {(f)!- These differences between the horizontal components o f two versions o f

the solenoidal terms are generally smaller than the individual components themselves. The

results show that the Boussinesq approximation generally underestimates the thermally

direct circulations near the cold front and the bent-back warm front by 25% to 30%.

These differences are more pronounced in the control experiment where the effect o f

condensation o f water vapor is included (Fig.5.4).

5.2 Moist potential vorticity dynamics

The vorticity differences due to the Boussinesq approximation can cause the

differences o f MPV dynamics between a Boussinesq fluid and the fluid described by the

primitive equations (PE).

For a moist adiabatic and inviscid fluid, the equation o f MPV in the PE model can

be written as (same term as r. h. s. o f 2 . 1 2 ):

d ( ^ a -V0 )— ^-------— = V 6 e - ( V-- V, V- ) . (5.3)

dt p -

while the equation o f MPV under the Boussinesq approximation is expressed as:

d ( ^ a -V0 )— B--------- - = i - k . ( V 0 e v V 9 > ( 5 -4 )

dt p0 . h e h

where 0 O is a reference potential temperature, and denotes the horizontal gradient

operator.

Fig.5.5 shows the MPV source terms on the right hand side o f (5.3) and (5.4) in

panel (a) and (b), respectively. These source terms are multiplied arbitrarily by At = 3

hours in order to express them in the PV unit. It can be seen that the different treatments


90

o

I iu-5.5 Horaontal cross section ot' the source terms ot' MPV tor tlie control experiment taken at 2 4 km above the

eround at 48 hours ot'ui) A tV U A pi \'u .\t. and (b) e / p / 0oli i VO • VO) \t. Hie contour interval is 0 I I’VtJ. \t

= hours


81

o f the solenoidal terms have significant effects on the MPV generation. The difference is

mainly due to the horizontal component o f the solenoidal term neglected in the Boussinesq

approximation. The Boussinesq approximation overestimates the negative MPV

generation in the bent-back warm front region while underestimates the source term along

the cold front. These results are quite similar at other levels and other times o f the

simulation. From these results it is clear that the Boussinesq approximation w ill give

wrong features o f the MPV in the most intense development region ot the cvclone.

5.3 Summary

The results o f these comparisons suggest that the Boussinesq approximation: (1)

underestimates the thermally direct circulations in the cold front and in the bent-back

warm front by 25% to 30%, which is more pronounced i f the effect o f latent heat release

is taken into account; and (2) w ill not give the correct MPV features in the intensive

development portions o f the cyclone.


82

Chapter 6

CONDITIONAL SYMMETRIC INSTABILITY IN EXTRATROPICAL CYCLONES

Conditional symmetric instability (CSI) has been extensively investigated since it

was first proposed as a possible mechanism for the formation o f frontal rainbands by

Bennetts and Hoskins (1979), and Emanuel (1979, 1983). Bennetts and Sharp (1982)

showed that the necessary condition o f CSI was satisfied in several cases o f observed

extratropical cyclones. Using a fine-mesh model, Shutts (1990) revealed substantial

amounts o f available potential energy for CSI prior to cases o f explosive cyclone

development. Other observations (Emanuel 1988; Reuter and Yau 1990) showed that

extratropical cyclone regions are typically in a state o f neutrality to CSI. Simulations by

Kuo and Reed (1988) o f an explosively deepening cyclone in the eastern Pacific have

shown similar results. To understand CSI and its possible roles in extratropical cyclones,

three-dimensional effects must be taken into account because extratropical cyclones often

demonstrate the small scale horizontal structure and associated strongly curved flows.

The three-dimensional criterion for symmetric instability (SI) is an open question,

and it is difficult to derive analytically. Instead o f directly attacking this problem, Jones

and Thorpe (1992) investigated the basic dynamics o f a three-dimensional motion in a

region o f negative DPV, where negative DPV is a sufficient condition for two-dimensional

frictionless SI (Hoskins 1974; Bennetts and Hoskins 1979). With the release o f latent heat,

the sufficient condition for two-dimensional frictionless CSI becomes M PV having a

negative value. Hence, the development o f negative MPV in extratropical cyclones


83

indicates the possibility o f exciting two-dimensional structure o f CSI. Reuter and Yau

11990) suggested that slantwise convection due to CSI is likely to be ubiquitous in

extratropicai cyclones. In the real atmosphere, CSI is a more likely process than SI

because it is easier for M PV to be negative than DPV. Nevertheless, SI may occur more

frequently than we think (Thorpe and Clough 1991). The development o f negative MPV

discussed in the previous chapters suggests the possibility o f CSI. In this chapter, we will,

first o f all, present our criterion for CSI and technique for taking two-dimensional cross

sections o f momentum and equivalent potential temperature in section o .l and 6 .2 ,

respectively. The possible roles o f CSI in extratropical cyclones and the sensitivity study

o f different moisture distributions are discussed in section 6.3. The summary is then given

in section 6.4.

6.1 The criteria of conditional symmetric instability used in

this study

The CSI or SI falls into two categories, i.e., layer instability (Bennetts and Hoskins

1979) and parcel instability (E nanuel 1983). In general, layer instability is a necessary but

insufficient condition for parcel instability. The criteria for layer SI are subject to various

constraints. For example, the criteria for layer SI are based on a normal mode method

which discomposes the flow into two parts, then estimates the instability with respect to

the basic state. Therefore, the perturbation imposed on the basic state is arbitrary. This

method, strictly speaking, is only suitable for linear analysis, and incapable o f describing

the nonlinear instability. Usually, this method gives a necessary condition fo r the

instability. In this section, we will take the parcel instability approach. By adapting the

criterion o f parcel instability and combining the criterion o f layer instability, we use the

following criteria as the rule for the appearance o f CSI: (1) The slopes o f equivalent

potential temperature surfaces are steeper than those o f absolute momentum surfaces; (2 )


84

The air in the region satisfied w ith the condition ( I ) is saturated; (3) The parcel is moving

slantwisely in this area; (4) M PV is negative. Because no three-dimensional criterion o f

CSI exits, our goal is therefore to find the regions satisfied with above conditions in three-

dimensional extratropical cyclones. It goes without saying that two-dimensional cross

sections o f equivalent potential temperature and absolute momentum are important to

determine whether those criteria are satisfied or not. Usually, the cross section is taken

along either the x or y direction, and the corresponding absolute momentum surface is

calculated using either M = v -r f \ or M = f y -u. which is suitable for some special case

and not consistent with the original assumptions o f CSI theory in a three-dimensional llow

in general. The more general and consistent scheme for taking cross sections o f equivalent

potential temperature and absolute momentum will be given in the next section.

6.2 The scheme for taking two-dimensional cross sections of equivalent potential temperature and absolute

momentum

The CSI problem is usually treated as a two-dimensional problem where one

component o f horizontal derivatives o f dynamical fields is neglected. In the study o f CSI

associated with two-dimensional fronts, the fronts are usually parallel to either y or x

direction, thus the cross sections with vanishing cp / cy or fp / rx are taken along the x or

y direction. The corresponding conservative absolute momentum is defined as M = v + fx

or M = f y - u. In a three-dimensional situation, however, the fronts are not necessary to

be oarallel to v or x direction, and the cross sections are in general required along the

horizontal potential temperature gradient in order to have symmetry and conservation

properties. Suppose we take the cross section in the direction o f a three-dimensional

potential temperature gradient, a general expression o f constant absolute momentum M

(Nordeng 1987) in this plane can be written as


85

M = fs + V n, (6.2.1)

where s is the horizontal distance along the direction is,

Ts - - V0/|V9|, (6.2.2)

and V „ is the wind component normal to the direction is:

V n = v i / I s. (6.2.3)

The horizontal distance s from the grid point to the point where the M surface crosses the

level k (see Fig.6 . 1) is determined by

f s(7C) + V n(7t,s) = f • 0 - Vn( - O,0 ), (6.2.4)

which is equivalent to

s(7t) = f '^ V ^ T t^O ) - V n(-.s)]. (6.2.5)

R

where k - ( — ■—) p . Because coordinates relative to the srid point (i, j) are:1000

s0 x - ldx = S U- 1 = ------ ;---- ~r •.";'-r h ,( 0 ; + 6 ? )1'-

- - S0V _ land dy = sts- j = ------ 5-V m r *1 -

( 0 - -© v )■'-

where h. 0 V, and 0 X are the grid length, the gradient o f potential temperature in x and y

direction, respectively, we can express s as:

s = “ ( 0- ;+ 8 ; ) l/2. (6.2.6)

dvh 2 n 2 J / 2or s = - - r — (0 * +0y ) (6.2.7)

Similarly, V,, - v k x is = — ,— * (u9v - v0x ). (6.2.8)(0 X _r0y ) •

Instead o f directly finding M surfaces using (6.2.4) and (6.2.5) as suggested by Nordeng

(1087). we can use formula (6.2.6). (6.2.7) and (6.2.8) to calculate the value o f M at each


86

grid point at which the cross section is parallel to potential temperature gradients. In

general, this scheme is to make a cross section perpendicular to potential temperature

contours no matter how one takes the cross section. It is clear that absolute momentum

defined as M = v + tx or M = fy - u is a special case o f M = ts + V „ when 0 V or 0 N is

zero. In the following section, we will use this technique to examine the appearance and

the possible roles o f CSI in extratropical cyclones.

6.3 CSI in extratropical cyclones

6.3.1 Evidence o f CSI in extratropical cyclones

The CSI criteria in section 6.2 are used to diagnose the structure o f CSI in three-

dimensional extratropical cyclones. Figs.6 .2a and 6.2b show the horizontal cross sections

o f negative MPV and cloud field taken on the 477 mb pressure surface. Negative MPV

occurred in the saturated region (Fig.6 .2a) shows a clear three-dimensional structure with

a length about 750 km (in y direction) and a width range from 200 to 500 km (in x

direction).

The negative MPV generated aloft can be understood by considering the jet

structure. Fig.6.3 shows the vertical cross sections in meridional direction through the line

BB' marked on Fig.6.2. An upper-level westerly jet with maximum speed o f 30 m/s

(Fig.6.3b) is associated with the cold front (Fig.6.3a). At low-levels. an easterly jet with

maximum speed o f 40 m/s (Fig.6.3b) is linked with the bent-back warm front (Fig . 6 3a).

Because o f the induced secondary circulation (Fig.6.3c), the negative MPV adjacent to the

bent-back warm front has moved into saturated areas at upper levels, which is in

agreement with the observation by Thorpe and Clough (1991) They found that negative

geostrophic MPV is partly distributed in the statically stable warm sector aloft.


87

QJ

distance

Fig.6.1 A possible geometric configuration for an absolute momentum surface M and the model grid. The upper part

shows a vertical section along the temperature gradient (s increases downgradient). Tlie M-surface is locally

perpendicular to this direction and crosses a pressure surface f re,) along tlie thick line in the lower part of the ligure

which is a horizontal section. Coordinates relative to the actual grid point for the intersection between the M-surliice

and the constant pressure surface (it.) along the s-direction is shown in the lower part of the figure (Nordeng 19X7).


88

The most important evidence for the appearance o f CSI is the relative position o f

equivalent potential temperature and absolute momentum surfaces, and parcel motion with

respect to these surfaces. F'g.6.4 shows a meridional cross section taken along the line

BB'. At the upper-level o f the bent-back warm front, the slope o f equivalent potential

temperature is substantially steeper than the slope o f absolute momentum (Fig.6 .4a). The

parcel is moving between two surfaces, which is in favour o f CSI (Fig.6 .4a). By taking

more meridional cross sections (not presented), it can be shown that the areas favourable

for CSI have a length from 300 to 600 km (in v direction) and a width up to 500 km (in ::

direction). The combined restoring forces o f buoyancy and Coriolis accelerate the parcel

moving in the regions where an equivalent potential temperature surface is more vertical

than a absolute momentum surface. The absolute momentum surface here is calculated

using the formula (6.2.6) and (6.2.8) given in section 6.3. For simplicity, the expression o f

absolute momentum defined as M = v -r (x or M = fy - u is used in three-dimensional

numerical simulations (e.g., Kuo and Reed 1988: Shutts 1990; Lindstrom and Nordeng

1991). In an idealized two-dimensional front, potential temperature contours are parallel

to the y or x axis, and the absolute momentum is therefore conservative in the y or x

direction. However, in the three-dimensional case the potential temperature in general is

not parallel to the y o r x axis especially at and after the development stages o f cyclones.

Therefore, the absolute momentum defined as M = v + tx or M = fy - u is not conserved in

the three-dimensional configuration, inconsistent with the original assumption o f

momentum conservation. Equation (6.2.6) and (6.2.8) used in this study are not only more

accurate but also more consistent. M (= fs + V n) defined bv (6.2.6) and (6.2.8) is in

general conservative in a three-dimensionai situation.

As shown in Figs.6.4b and 6.4c. CSI is released in the region where the air is

saturated, and the equivalent potential temperature surface is more vertical than the

absolute momentum surface. In agreement with the simulation using observational data by

Shutts (1990), CSI takes place at the height about the 477 mb. The prediction o f CSI is


S9

vB'

i

CO—>

(a) N T j

14

1 II 1 M I I I I 1 111 MM' ..........11 III III 1 Ml i

(b) N t

f!

^ •

Fig.6.2 Horizontal cross section taken 011 the 477 mb pressure surface at 36 hours of (a) saturated negative MPV with

a contour interval (J. 1 PVU: (b) cloud tields with a contour interval of 5x I0"~ g/kg.


90

Ha)

329

B B'

I'tu.tv.' Vertical cross section taken along the line BB' tX - 200 kin) at .'6 hours ot (a) equivalent potential

temperature at an interval of 4 K.. (b) u component of velocity at an interval of 10 m/s: (c) MPV field at an interval ot

o.l PVU.


91

0.0500.150

0.250

0.450

0.550

0.645

0.730

0.805

0.870

0.920

0.960

0.986

0.996

B B’

a

0.050

0.150

0.250

0.350

0.450

0.550

0.645

0.730

0 .805

0.870

0 .920

0.960

0 .986

0 .996

B B'

Fig.6.3 (b) and (c).


92

important and difficult because in most cases CSI cannot be forecast by conventional

convective instability analysis because CSI frequently occurs in a convectivelv stable

environment (e.g., Emanuel 1983). Furthermore, the observations (e.g., Emanuel 1988)

and numerical simulations (e.g., Kuo and Reed 1988) show that CSi is often neutral in

extratropical cyclones. Based on fine-resolution observations. Thorpe and Clough (1991)

found that the near-neutral condition is representative o f the mid to upper frontal zone and

that there is evidence o f active CSI. Instead o f reasoning CSI happening from the previous

neutral state o f CSI, in this study CSI is explicitly simulated in the mid- to upper-level o f

the bent-back warm front. In the next several figures, we will show that this active CSI is

adjusted toward its neutrality.

By 42 hours, the negative MPV in saturated areas becomes weak (Figs.6.5a and

6.5b). A meridional cross section (Fig.6 .6 ) taken along the line CC' marked on Fig.6.5

shows that the atmosphere is in a state o f slightly conditionally symmetric instability or

neutral conditionally symmetric stability in the bent-back warm front (Figs.6 .6 a and 6 .6 b).

By 48 hours. CSI is adjusted toward the neutrality (not shown). The nature o f CSI

adjustment is pooriv understood. Thorpe and Rotunno (1989) addressed this problem for

a two-dimensional dry case. The SI circulations themselves cannot stabilize or even

neutralize the flow due to conservation o f DPV in an inviscid two-dimensional case. By

adding diffusion in the model, they found that a parameterization o f subgrid scale

turbulence does not lead to down-gradient DPV flux.

6.3.2 Possible roles o f CSI in extratropical cyclones

The development rates o f extratropical cyclones are significantly enhanced by the

interaction o f diabatic and dynamic processes, especially those related to the release o f

latent heat (e.g.. Kuo and Reed 1988). In extratropical cyclones, the frontal zones are


93

0.0500.150

0.250

0 .350

0.450

0.550

0.645

0.730

0.805

0 .870

0.920

0.960 I »0 .986

0.996 nuim m inihnuu

-SM

I -18 m/i

B B'

0 .050

0 .150

0 .250

0 .350

0.450

0 .550

0 .645

0.730

0 .805

0 .870

0 .920

0.960

0 .986

0.996

(b)

. . . . . . . . . . . . . . . . . . . . . . . . . .

B B'

Fig.6.4 Vertical cross section taken along the line BB’ (X = 200 km) at 36 hours of fa) equivalent potential

temperature (dashed lines) at an interval o f -4 fv. absolute momentum (solid lines) at an interval of 100 in/s and

velocity field indicated by arrows: (b) cloud fields with a contour interval of 5x 10' g/lcg.


94

I /-MvC

tc

'C '» M M l 11 M l » T I

....

Fig.o 5 1 lorizontal cross section taken on the 477 tub pressure surface at 42 hours of (a) saturated negative MPV with

a contour interval oft). I I’VU. ib) cloud fields with a contour interval of 5s 10"- g/kg.


95

0.050 f t T M I 1 | T f f T T M f l 1 I ! I ! »JJ j n im m n w ntii

£ 1 '0.150

0.250

0.350

• m . ^ ^ ^ *jjiji *0 .450

0.550

■ujfu*'*

0.730

0.805

0.870

0.920

0.960

0.986

0.996 _ 56 m/s>C

0.050

0.150

0.250

0.350

0.450

0.550

0.645

a 0.730

0.805

0 .870

0.920

0.960

0.986

0.996

C c*Fig.6.6 Vertical cross section taken along the line CC." (X = 400 kin) at 42 hours ol fa) equivalent potential

temperature (solid lines', at an interval of 4 K. absolute momentum (dashed lines) at an interval ol 100 in/s and

velocity lield indicated by arrows: (b) cloud fields with a contour interval of 5x 10"- g/kg.


96

0.050

0.150

0.250

0.350

0.450

0.550

0.645

ct 0 .730

0.805

0.870

0.920

0.960

0.986

0.996C C’

Kig.o.6 (c) MPV field at an interval of 0.1 PVU.


97

ideal environments for the feedback between latent heat release and cyclogenesis or

frontogenesis. In addition, the parcel displacements favourable for CSI riong the sloping

front are exactly those produced by the cross-frontal circulation (Thorpe 1994). However,

observations (Emanuel 1988; Reuter and Yau 1990) indicate that frontal zones in

extratropical cyclones are often neutral to CSI. and CSI would be quickly neutralized by

slantwise convection i f present. Although it may be questioned whether a deep

conditionally symmetrically unstable layer exists, fme-mesh numerical simulations using

real data as initial conditions (Shutts 1990) show that CSI appears through a deep layer

prior to and during the early stage o f explosive development (see his Figs. 8 and 10). The

simulation presented here based on idealized initial conditions is in excellent agreement

w ith Shutts’ simulations (Shutts 1990). Therefore, it is interesting to pursue the question

o f how CSI affects the structure and evolution o f extratropical cyclones.

As shown in Fig.6.4, CSI happens in an environment stable to upright convection,

in agreement with Lindstrom and Nordeng’s (1992) simulations. A narrow sloping sheet o f

rapidly ascending air indicated by the velocity vectors is positioned both within and ahead

o f the surface bent-back warm frontal zone. The maximum upward motions are

approximately 5.2 ub/s within the region o f slantwise circulations. A similar result is also

reported by Thorpe and Emanuel (1985), and Xu (1986). In Kuo and Reed's (1988)

simulation, most o f the regions in the warm front (their Fig. 11) are symmetrically neutral

or slightly symmetrically unstable, and they are located in the lower and middle portions o f

the frontal cloud band. Kuo and Reed (1988) claimed that the neutral conditional

symmetric stability or slight conditional symmetric instability occurred in their study were

responsible for the intensification and the creation o f the vortex (see their Fig.9c and Table

2 ) because low-level upward motion produces strong vertical stretching near the boundary

layer and an associated strong spinup o f low-level vorticity. Our simulations, by contrast,

show that CSI mostly appears in the middle or upper troposphere, consistent with the

observational study by Thorpe and Clough (1991). In our simulations, the vertical


98

component o f surface vorticity is substantially intensified (Fig.6.7a) and the upper-level

flow is divergent (Fig.6 .7b) in and adjacent to the region o f CSI. This can clearly be

observed in Fig.6 . 8 by comparing the upper-level MPV and divergence with the surface

vorticity and pressure fields at the position o f y = 4800 km (96x50 km) where CSI takes

place aloft.

The surface vorticity is continuously intensified when the atmosphere is in a state

o f neutral symmetric stability or slight CSI (Figs.6.9a and 6.9b). Similar results were

obtained by Kuo and Reed (1988). One question is why there should be an enhanced

response in the region o f slantwise convective adjustment since responses to atmospheric

forcings decrease as the stability increases (e.g.. Sutcliffe 1947), or as the M PV increases.

Two possibilities have been hypothesized by Lindstrom and Nordeng (1992), but the

mechanism remains elusive. Because CSI is often in a neutral state through an adjustment

process. Bennetts and Hoskins (1979) suggested that the adjustment o f CSI leads to

distorting absolute momentum and equivalent potential temperature surfaces, and

produces local regions o f purely inertial and convective instabilities. I t seems in this study

that convective instability cannot be generated by CSI adjustment because the atmosphere

is convective stable throughout the simulation.

Fig.6 . 10 shows six-hour accumulated precipitation from 30 hours to 48 hours. It is

clear that the rainband in the bent-back warm frontal zone is associated w ith CSI (cf.

Fig.6 .2a and Fig.6 .10b), in agreement with the observation by Parsons and Hobbs (1983).

It is CSI that is likely responsible for the band precipitation in the bent-back warm front.

As CSI takes place, the equivalent potential temperature surfaces are steeper than absolute

momentum surfaces. This directly results in a steeper frontal surface since the frontal

surface can be represented by an equivalent potential temperature surface in a moist

adiabatic process. Because the frontal surface can affect the precipitation in such a way

that the steeper the siope o f the cold-frontal surface the stronger the precipitation


99

0.050

0.150

0.250

0.350

0.450

0.550

0.645

ct 0 .730

0 .805

0.870

0.920

0.960

0.986

0 .996

B B*

0.050

0.150

0 .250

0.350

0.450

0 .550

0 .645

cr 0 .730

0 .805

0 .870

0 .920

0 .960

0 .986

0 .996B B*

l'ig.6.7 Vertical cross section taken along the line BB' ( X = 2U0 km) at 36 hours ot (a) vertical component ol relative

vorticity with a contour interval ot" I * 10~* s' _ (hi two-dimensional divergence with a contour interval ot I / I O'3 s *


100

1020

-1015o

O> -1010ov3 A + -1005

3C/3

"33

-1000

cfy -9900.5O33

y>

"5

>a<S

-985

-980

-975

■9700 112 128 144 16016 80 9664

B Y (x50 km) B’■ 3 - surface vorticity —i— divergence aloft - 3 - upper-level M P V surface pressure

I'ig.i'.S 1 lori/ontal distributions along the line 13B’ i X =200 kin I at 36 hours of (a ) MPV Hold at 477 mb surface with

a 0 I PVIJ: (b) divergence Held at 477 mb surface .vuh an unit of 1x10° s'*: toi surface pressure at an unit of mb:

id) surface relative vorticity at an inut of I \ It)- * s '*.


Surfa

ce

pres

sure

(m

b)

101

(7

0.050 f r i i i n i m i i i m i t i m i h u n i i i i n i n i J T T H T r r r t ! I I 1 1 1 1 1 1 1 I I I ! M I M

0.150

0.250

\0.350

0.450

0.550

0.645

0.305

0.870

0.920

0.960

0 .986

0.996

11ii111ii n i i i i in i i ij n i in i i i n ? i in rv x /fT T rr IIIITTJ0 .050

0 .150

0.250

0.350

0.450

0.645

0.730

0 .805

0.870

0 .920

0.960

0 .986

0.996

Fig.6.9 Vortical cross section taken along the line CC (X = 400 km) at 42 hours of fa) vertical component of relative

vorticitv with a contour interval of I * 10~* s' . (bj two-dimensional divergence with a contour interval ol 1 * 10 s .


102

(Locatelli et al. 1994). This conclusion, we believe, can be extended to the bent-back

warm front case since the bent-back warm front can be considered as a 'cold' front in the

cold air stream behind the conventional cold front. Hence, CSI can enhance the

precipitation in a band form at the bent-back warm front.

6.3.3 Effects o f moisture distribution on CSI in extratropical cyclones

To explore the effects o f the spatial distribution o f relative humidity on CSI, a

series o f sensitivity experiments are performed. Initial conditions in each simulation are

identical to the control experiment except for moisture distribution. The details o f the

numerical experiment design have been given in chapter 3.

(a) Comparison befiveen experiment (d) and the control experiment

The initial moisture distribution o f experiment (d) is almost the same as that o f the

control experiment. The regions o f high moisture content are located ahead o f the surface

low for both experiments. However, the control experiment has a more continuos broad

moisture zone than experiment (d). As shown in Figs.6 .1 la and 6 .1 lb, the negative M PV

and CSI in experiment (d) occurred in the same saturated areas (Figs.6.12a-c). Note that

at the upper level o f the warm sector the absolute momentum surface is more horizontal

than the equivalent potential temperature surface (Fig.6 .12a). The slope o f the absolute

momentum surface in that region is flat, and the value o f the absolute momentum is more

or less uniform. The latter means that the restoring force due to momentum conservation

is very small. The negative MPV and cloud fields shown in this region are consistent with

the CSI criterion. Similarly to the control experiment, below the region o f CSI the surface

vorticity is intensified (Fig.6.13a) and above this region the flow is divergent (Fig.6.13b),

which favour surface cyclone deepening. This comparison indicates that the high moisture


103

m M m I 1 I I I I I | n T fTT

(c)

a

Fig.6.10 The surface precipitation increment produced (a) between 30 and 36 hours: (b) between 36 and 42 hours: (c)

between 42 and 48 hours. The contour interval is 2 min.


104

content ahead o f the surface low is very important in negative MPV and CSI generation

aloft and surface cyclone intensification.

(b ) Comparison benveen experiments (d). (e) and (f)

This intercomparison is helpful to understand whether multiple bands o f high

moisture content ahead o f surface low have a significant impact on CSI generation.

Figs.6.14 and 6.15 show the absolute momentum, equivalent potential temperature, cloud

field, and M PV for experiment (e) and (f), respectively. Neither experiment identifies the

appearance o f CSI due to equivalent potential temperature surfaces being more horizontal

than absolute momentum surfaces in the saturated area. In addition, the saturated region is

not exactly matched by negative MPV Hence, the synoptic scale, rather than mesoscale.

high moisture bands ahead o f the surface low favour the generation o f CSI, and the

intensification o f the surface low. It is clear that the high moisture bands initially located

behind the surface low have little effect on the deepening o f surface low (cf. Figs.6.16a

and 6 .16b).

(c) Comparisons among experiments (d). (a), (c) and (b)

In contrast with the experiment (d), the high moisture bands are located at and

behind the surface low, and on either side o f the low in experiment (a), (c), and (b),

respectively. None o f these experiments shows CSI. However, the surface low at 24 hour

simulation o f experiment (a) is deeper than that o f experiment (b) and (c) (Figs.6.17a and

6 .17b). Therefore, the drier air ahead o f the surface low inhibits cyclone development (cf.

Fig.0 .17b).


105

11>1 ■ ■ ■ ■' ■»11 ■ ■ ■1 ■ ■1 ■111

?D

4 d ’i i 11 u i i i i i i ii i i i i i i i i i i r rvi'i i i i i i i rn

(b)

Sfi

E ■ ■, ■ ■ ■.... ■ i in I ii 11 n i in.. ■ i

T d

Fig.6.11 Horizontal cross section for the experiment! d) taken on the 618 mb pressure surface at 24 hours of (a) MPV

with a contour interval oi'0.1 PVU: (b) cloud fields with a contour interval of 5* 10'“ g/kg.


106

0.050'""m T I .....IUJ,,n ' I1”0.150

0.250

0.450

0.550

0.645

0.730

0 .805

0.870 14 4 / t t f f

0.920/ / / / / /

1 I I0.960

0.986

0 .996 m i i i i i m i n u

r~"i

. 22.5 m/s

i'ig.o.12 Vertical crass section tor the experiment (d) taken along die line DD’ (X = 300 km) at 24 hours ot (a)

equivalent potential temperature (dashed lines) at an interval ot'4 K. absolute momentum (solid lines) at an interval

of 100 m/s and velocity tield indicated by arrows: (b) cloud fields with a contour interval of 5x 10'- g/kg: (c) MPV

field at an interval of 0.1 PVU.


107

0.050

0 .150

0.250

0 .350

0 .450

0 .550

0 .645

0 .730

0 .805

0 .870

0 .920

0 .960

0 .986

0 .996

iiniiiiin iTTrrm r r rrr

(b)

H . I I ■ I , , I . I H .1 . ■ 1 I . ■ I .

N

D D’

O.ObO0 .150

0 .250

0 .350

0 .450

0 .645

0 .730

0 .805

0 .870

0.920

0 .960

0 .986

0 .996

Fig. 6.12 (bi ami (c).


108

0.050

0.150

0.250

0.350

0.450

0.550

0.645

a 0 .730

0.805

0.870

0.920

0.960

0.986

0.996

0.050

0.150

0.250

0.350

0.450

0.550

0.645

ct 0 .730

0.805

0.870

0.920

0.960

0.986

0.996

I'ig.6.13 Vertical cross section for the experiment (dl taken along the line DD' (X = 300 km) at 24 hours of (a)

vertical component ot'relative vorticity with a contour interval of 1x10“* s'*: vb) two-dimensional divergence with a

contour interval of 1 x It)'3 s'*.

D D*

D'


109

0 .050

0 .150

0 .250

0 .350

0 .450

0 .550

0 .645

G 0 .730

0 .805

0 .870

0 .920

0.960

0 .986

0 .996

Fig.6.14 Vertical cross section lor the experiment !e) taken along the line X = 300 km at 24 hours of fa) equivalent

potential temperature (dashed lines) at an interval of 4 K. absolute momentum (solid lines) at an interval of 100 m/s

and velocity Held indicated by arrows: (b) cloud fields with a contour interval of 5x10*2 g/kg; (c) MPV field at an

interval of 0 .1 PVU.


0 .050

0 .150

0 .250

0 .350

0 .450

0 .550

0 .645

a 0 .730

0 .805

0 .870

0.920

0 .960

0 .986

0 .996

0 .050

0 .150

0 .250

0 .350

0 .450

0 .550

0 .645

a 0 .730

0 .805

0 .870

0 .920

0 .960

0 .986

0 .996

Fig.6.14 (b)and (c).


I l l

O’

0.050

0.150

0 .250

0.350

0.450

0.550

0.645

0 .730

0.805

0.870

0.920

0.960

0 .986

0.996

i pi m m ttmm i eiiwi tu v

23.4 m/s

Fig.6.15 Vertical cross section for the experiment (cl) taken along the line X = 350 km at 24 hours ol (a) equivalent

potential temperature (dashed lines) at an interval of 4 K, absolute momentum (solid lines) at an interval of 100 m/s

and velocity field indicated by arrows', (b) cloud fields with a contour interval of 5x10"^ g/kg. (c) MPV Held at an

interval of 0.1 I’Vli.


112

0 .050

0 .150

0 .250

0 .350

0 .450

0 .550

0 .645

ct 0 .730

0 .805

0 .870

0 .920

0 .960

0 .986

0 .996

0 .050

0 .150

0 .250

0 .350

0 .450

0 .550

0 .645

ct 0 .730

0 .805

0 .870

0 .920

0 .960

0 .986

0 .996

I-ig.6.15 (b) and (c).

tT I T l I T U I I I 1'M ‘TTTH M r f t T H I 1 1 111 11 M 1 1 1 1 11) H 1 1 1 111 H f 1 1 I I 1 1 1 11 ! 1 1 H I I T


1 13

Hg.fi. 16 Surface pressure fields at 2-1 hours for (a) the experiment (e)'. (h) the experiment (f) Hie contour interval is

4 mb.


114

Fig.h. 17 Surface pressure fields at 24 hours for (a) the experiment (b)'. (b) the experiment (c). The contour interval is

4 mb.


115

6.4 Summary

Three-dimensional CSI and its possible roles in extratropical cyclones have been

examined in this chapter. It is suggested that CSI appears when the parcel is moving in the

region where the slope o f equivalent potential temperature surfaces is steeper than

absolute momentum surfaces, implying the possibility o f negative MPV generation. To be

consistent with the assumption o f CSI theory, a scheme is proposed for taking two-

dimensional cross sections o f equivalent potential temperature and absolute momentum,

which ensures that the cross section is parallel to potential temperature gradients. Based

on these criteria and the scheme, CSI in extratropical cyclones is found at the upper level

o f the bent-back warm front, and is then adjusted towards a neutral state. The simulations

agree with observations. With the release o f CSI, the slantwise updraught is enhanced and

the upper-level flow is divergent, which in turns intensifies the surface low. The sensitivity

experiments conducted show' that synoptic scale high moisture bands ahead o f the surface

low have a significant impact on the generation o f CSI and on the deepening o f the surface

low. No CSI has been found when initial high moisture bands are distributed behind and at

the surface low' or on either side o f the low. In fact, the drier air ahead o f the surface low

prevents the cyclones from deepening.


116

Chapter 7

CONCLUSIONS AND SUGGESTIONS FOR FUTURE RESEARCH

We have found in this thesis that M PV generation in a three-dimensional moist

adiabatic and frictionless flow is governed by baroclinic vectors and moisture gradients.

Negative (positive) MPV can be generated in the region where baroclinic vectors have a

component along (against) the direction o f moisture gradients. This criterion o f MPV

generation has been evaluated in numerical simulations o f mid-latitude cyclones with

several dill'erent moisture distributions. The moisture gradients in these simulations are

designed to be either almost perpendicular to, or approximately parallel to the potential

temperature gradients. It turns out that the MPV generation in extratropical cyclones is

particularly sensitive to moisture distribution at the development stage o f cyclogenesis.

The main results o f numerical simulations are summarized as follows:

( 1) Negative M PV usually appears in the warm sector near the north part o f the

cold front, the bent-back warm front, the warm core and the cold front. The bent-back

warm front and the warm core are the most favourable places for negative MPV

appearance.

(2) After the cyclone matures, the negative MPV moves into the warm core. It w ill

stay there until the end o f the simulation in the unsaturated areas. The negative M PV tends

to become positive in saturated regions.

(3) The development o f negative M PV along the cold front is, to a large extent,

dependent on the location o f high moisture content. High moisture content bands with

wave length - 2 0 0 0 km located at and just behind the surface low is favorable for low-


117

level negative MPV generation along the cold front.

(4) The initial multiple bands o f high moisture content affect negative MPV

generation at the development stage o f the cyclones, and the bands ahead o f the surface

low are favourable regions for negative M PV generation.

The importance o f the baroclinic contribution to vorticity and MPV generation is

diagnosed in the Boussinesq and the PE models. The neglect o f part o f the horizontal

components in the solenoidal term by the Boussinesq approximation leads to an

underestimation o f the thermally direct circulations in the cold front and in the bent-back

warm front by 25% to 30%. This effect is more significant when latent heat release is

considered. Furthermore, the simplifications made in the Boussinesq model result in the

incorrect representation o f MPV distributions in the intensive regions o f mid-latitude

cyclones.

The CSI and its possible roles in extratropical cyclones have also been investigated

in this thesis. It is suggested that CSI takes place when the MPV is negative and the How

is moving in a direction between the equivalent potential temperature surfaces and

absolute momentum surfaces. A scheme is used in this study for taking two-dimensional

cross sections o f equivalent potential temperature and absolute momentum, which is

consistent with the assumption o f CSI theory and ensures that the cross section is parallel

to potential temperature gradients. Based on these criteria, CSI is found at the upper-level

o f the bent-back warm front at the development stage, and is then adjusted towards a

neutral state. Sensitivity experiments show that the presence o f synoptic-scale high

moisture bands ahead o f the surface low favour CSI generation and surface low

deepening.

Although the idealized model initialization presented in this thesis is similar to real

pre-storm environments, further research using real data initialization would be o f interest

to understand the roles o f the generated MPV in extratropical cyclones. Diabatic effects,

such as radiation and ice-phase microphysics, may also be important for the generation o f


118

MPV and CSI in extratropical cyclones, should be investigated.


APPENDIX A

119

THE ESTIMATION OF FUNCTION A

The function A is negative in most cases. I f typical atmospheric values are chosen

as T = 300.0 K, T v = 300.0 K, q = 5.Ox 10‘ 3 g/g, r = 0.7, p = 1000.0 mb, es = 40.0 mb, the

order o f magnitude o f each term in the braces o f the function A can be evaluated as

follows:

1 Ge ° S Tv a , [ a 4 “ ( T - a 3 ) ln( r ) ]

p2 0 ' T ( T - a J a 4 l a ^ a ^ - ( T - a ^ ) ln( r ) |

1 0 - ' 1 0 1

a I x a 4 [ ( T - a , ) 2 (Tv ( p - e s ) + a 9 * a f iq p ) - a 2 x a 4 T 2 q(p - eg)]

[ a 4 T- cx 3 ( T - a 3 ) l n ( r ) ] 2 7 v ( p - e s )

10 - '

where the constants a,, a 2, a ,, a A and a f)are equal to 2.675x10* K, 0.61, 55.0 K, 2.84x

10* K, and 5.41812x 103 K, respectively. The third term in the braces o f the function A is

not sensitive to the variation o f pressure. For example, i f p=250.0 mb is used, the order o f

magnitude o f this term is same as one estimated above.


APPENDIX B

LIST OF SYMBOLS

120

Symbol Unit Description

Cp J K-'kg - 1 specific heat o f dry air at constant pressure

Cpni J K-'kg ’ 1 specific heat o f moist air at constant pressure

D s-' horizontal deformation

e, Pa saturation vapour pressure

f s-1 Coriolis parameter

F N n r ' frictional force per unit volume

FMI) ___ an operator for horizontal diffusion (defined in section 3.1)

G m s’ 2 gravitational acceleration

h m grid length

KM m2 s-' coefficient o f horizontal diffusion

Kmo m2 s-' background value o f K,,

Lv J kg-' iatent heat o f water vapour condensation

m ------ map scale factor

M m s-' absolute momentum

p Pa air pressure

p* Pa difference between the top pressure (pt) and surface

pressure (ps) o f the model

Peon kg kg* 1 s-' condensation o f water vapour or evaporation o f cloud drops

Pra kg kg-' s-' accretion o f cloud drops by rain drops

Prc kg kg-' s' 1 autoconversion o f cloud drops to rain drops


121

Pre kg kg- 1 s-

q kg kg- 1

q kg kg - 1

qr kg kg-'

qs kg kg- 1

qv kg kg - 1

Q K s - 1

r

R J K-'kg - 1

S in

T K

T v K

u m s' 1

V 111 S' 1

V m s’ 1

Vn 111 S' 1

v. m s-1

4> 1112 s- 2

K

K

e K

K

e' K

CD * CD K n r 1

evaporation o f rain drops

specific humidity

mixing ratio o f cloud water

mixing ratio o f rain water

saturation specific humidity

mixing ratio o f water vapor

diabatic heating or cooling rate

relative humidity

gas constant for dry air

horizontal distance from the grid point to the point where

the M surface crosses the level u

temperature

virtual temperature

x-component o f velocity

y-component o f velocity

three-dimensional velocity vector

wind component normal to the direction o f potential

temperature gradient

mass-weighted mean terminal velocity o f rain drops

geopotential

Von Karman constant (“ 0.4)

defined as (p /1 0 0 0 )Rci’

potential temperature

reference value o f potential temperature

potential temperature perturbation

gradient o f potential temperature in x and y direction,

respectively


122

0 c K equivalent potential temperature

0 * K equivalent potential temperature fo r saturated air

p kg nr3 air density

p., kg nr3 reference value o f air density

p kg nv3 air density perturbation

a nondimensional vertical coordinate o f the model*

a S '1 vertical velocity in a o-coordinate

(!) Ps S '1 vertical velocity in an isobaric coordinate

£ 2 S' 1 angular velocity o f the Earth

Cm S '1 absolute vorticity


REFERENCES

123

Anthes, R. A., and T. T. Warner, 1978: The development o f mesoscale models suitable for

air pollution and other meteorological studies. Mon. iVea. Iiev., 106, 1045-1078.

Anthes, R. A.. E.-Y. Hsie and Y.-H. Kuo, 1987: Description o f the Penn Sate / NCAR

Mesoscale Model Version 4 (MM4). NCAR Technical Note, NCAR / TN-282, 6 6 pp.

Anthes, R. A., 1990: Recent applications o f the Penn State /NCAR mesoscale model to

synoptic, mesoscale, and climate studies. Hull. Anter. Meteor. Sac., 71, 1610-1629.

Asselin, R., 1972: Frequency filter for time integrations. Mon. l i ’ea. Rev., 100, 487-490.

Barry, R. G. and R. J. Chorley, 1982: Atmosphere, weather am i climate. Methuen & Co.

Ltd.

Benard, P., J.-P. Lafore, and J.-L. Redelsperger, 1992: Nonhydrostatic simulation o f

frontogenesis in a moist atmosphere. Part II: Moist potential vorticity budget and

wide rainbands../. Atmos. Set., 49, 2218-2235.

Bennetts, D. A., and B. J. Hoskins, 1979: Conditional symmetric instability - a possible

explanation for frontal rainbands. 0 . J. R. Met. Soc., 105, 945-962.

Bennetts, D. A., and J. C. Sharp, 1982: The relevance o f conditional symmetric instability

to the prediction o f mesoscale frontal rainbands. Quart. .!. Roy. Meteor. Soc., 108,

595-602.

Bennetts, D. A., and P. Ryder, 1984: A study o f mesoscale convective bands behind cold

fronts. Part I: Mesoscale organization. Quart. J. Roy. Meteor. Soc., 110, 121-145.

Betts, A. K., and F. J. Dugan, 1973: Empirical formula fo r saturation pseudoadiabats and

saturation equivalent potential temperature../. Appl. Meteor., 1 2 , 731-732.


124

Bjerknes, J., and C. Godske, 1936: On the theory o f cyclone formation o f extratropical

fronts. Astrophys. Norv., 1, No. 6 , 199-235.

Bjerknes, J., and H. Solberg, 1922: Life cycle o f cyclones and the polar front theory o f

atmospheric circulation. Geofys. Pttbl., 3, N o .l, 1-18.

Bolton, D., 1980: The computation o f equivalent potential temperature. Mon. Weo. Rev.,

108, 1046-1053.

Bosart, L. P., 1981: The President's Day snowstorm o f 18-19 February 1979: A

subsynoptic scale event, Mon. IVea. Rev., 109, 1542-1566.

Browning, K. A., 1990: Organization o f clouds and precipitation in extratropical cyclones.

I'lxtra irop ica l ( yclones. The Erik Palmen Memorial Volume. C. VV. Newton and E.

Nolopainen, Eds., Amer. Meteor. Soc., 132.

Cao, 7 , and H.-R. Clio, 1993: Numerical study o f the structure and the complete

occlusion proccs" o f explosive cyclogenesis. The IS th General Assembly o f European

Geophysical Society, Annales Geophysicae Supplement II to Vol. 11, 202.

Cao, Z.. and H.-R. Clio, 1994: Generation o f moist potential vorticity in extratropical

cyclones. Submitted to./. Atmos. Sci.

Carlson, T. N„ 1980: A irflow through midlatitude cyclones and the comma cloud pattern.

Mon. IVea. Rev., 108, 1498-1509.

Chan, D. S. T., and H.-R. Clio, 1989: Meso-P-scale potential vorticity anomalies and

rainbands Part 1: Adiabatic dynamics o f potential vorticity anomalies../. Atmos. Sci.,

46, 1713-1723.

Charney. J.G., 1947: The dynamics o f long waves in a baroclinic westerly current. ./.

Meteor., 4, 135-162.

Charney, J. G., and M. E. Stern, 1962: On the stability o f internal baroclinic jets in a


125

rotating atmosphere../. .4tmos. Sci., 19, 159-172.

Cho, H.-R., and D. S. T. Chan, 1991: Meso-P-scale potential vorticity anomalies and

rainbands. Part II: Moist model simulations../. Atmos. Sci., 48, 331-341.

Clough, S. A., N. S. Grahame and A. O'neill, 1985: Potential vorticity in the stratosphere

derived using data from satellites. 0 . J. Ii. Met. Soc., I l l , 335-358.

Clough, S. A., and R. A. A. Franks, 1991: The evaporation o f frontal and other stratiform

precipitation. Quart. ./. Roy. Meteor. Soc., 117, 1057-1080.

Cooper, I. M., A. J. Thorpe, and C. II. Bishop, 1992: The role o f dillusivc ell'ects on

potential vorticity in fronts. Quart. .J. Roy. Meteor. Soc., 118, 629-047.

Daniclsen, IT F., and R. S. 1 lipskinct, 1980: Stratospheric-tropospheric exchange at polar

latitudes in summer../. (ieophys. Res., 85(C1), 393-400.

Daniclsen, 11. F., R. S. Hipskind, S. E. Gaines, G. W. Sachse, G. E. Gregory and G. F.

Hill, 1987: Three-dimensional analysis o f potential vorticity associated with

tropopause folds and observed variations o f ozone and carbon monoxide../. (ieophys.

Res., 92(1)2), 2103-21I I .

Davis, C. A., and K. A. Emanuel, 1991: Potential vorticity diagnostics o f cyclogenesis.

Mon. IVea. Rev., 119, 1929-1953.

Eady, E. T., 1949: Long waves and cyclone waves. Tellus, 1(3), 33-52.

Egger. J., 1990: Some aspects o f potential vorticity inversion. ./. Atmos. Sci., 47, 1269-

1275.

Emanuel, K. A., 1979: Inertial instability and mesoscale convective systems. Part I: linear

theory o f inertial instability in rotating viscous fluids../. Atmos. Sci., 36, 2425-2449.

Emanuel, K. A., 1982: A variational theorem for circulation integrals applied to inviscid

symmetric flows with variable stability and shear. Quart. ./. Roy. Meteor. Soc., 108,


126

825-832.

Emanuel, K. A., 1983: The lagrangian parcel dynamics o f moist symmetric instability. ./.

Atmos. Sci., 40, 2368-2376.

Emanuel, K. A., 1988: Observational evidence o f slantwise convective adjustment. Mon.

Wea. Rev., 116, 1805-1816.

Ertel, H., 1942: Ein Neuer hydrodynamischer Wirbelsatz. Met. Z , 59, 271-281.

Fritsch, J. M., E. L. Magaziner and C. F. Chappell, 1980: Analytical initialization for

three-dimensional models../. Appl. Meteor., 19, 809-818.

Gill, A. E , 1982: Atmosphere-ocean dynamics. Academic press.

Medley, M. and M. K. Yau, 1991: Anclastic modeling o f explosive cyclogenesis../. Atmos.

Soc., 48, 71 1-727.

I lertenstein, R. F. A., and W. M. Schubert, 1991: Potential vorticity anomalies associated

with squall lines. Mon. IVea. Rev., 119, 1663-1672.

Hobbs, P. V., 1978: Organization and structure o f clouds and precipitation on the

mesoscale and microscale in cyclonic storms. Rev. Geophys. Space Rhys., 16, 741 -

755.

Holton, J. R., i 992: An introduction to dynamics meteorology. Academic press. pp509.

Hoskins, B. J ., 1972: Non-Boussinesq effects and further development in a model o f upper

tropospheric frontogenesis. 0. J. R. Met. Soc., 98, 532-541.

Hoskins, B. J., and F. P. Bretherton, 1972: Atmospheric frontogenesis models:

mathematical formulation and solution../. Atmos. Sci., 29, 1 1-37.

Hoskins. B. J., 1974: The role o f potential vorticity in symmetric stability and instability,

G. .A R. Mel. Soc., 100, 480-482.


127

Hoskins, B. J., and N. V. West, 1979: Baroclinic waves and frontogenesis. Part II:

Uniform potential vorticity jet flows — cold and warm fronts. ./. Atmos. Sci., 36,

1663-1680.

Hoskins, B. J., M. E. McIntyre and A. W. Robertson, 1985: On the use and significance o f

isentropic potential vorticity maps. O . ./. I t Met. Soc., I l l , 877-946.

Hoskins, B., and P. Berrisford, 1988: A potential vorticity perspective o f the storm o f 15-

16 October 1987. Weather, 43, 122-129.

Houze, R. A., and P. V., Hobbs, K. R., Biswas, and W M., Davis, 1976: Mesoscale

rainbands in extratropical cyclones. Mon. IVea. Rev., 104, 868-878

Houze, R. A., and P. V., Hobbs, 1982: Precipitating cloud systems Ai/vances in

(.ieophysics. Vol. 24, Academic Press Inc., New York.

Ilsie, E.-Y., R. A. Anthes and D. Keyser, 1984: Numerical simulation o f frontogenesis in a

moist atmosphere../. Atmos. Sci., 41, 2581-2594.

Innocentini, V.. and E. D. S. C. Neto, 1992: A numerical study o f the role o f humidity in

the updrafl driven by moist slantwise convection../. Atmos. Sci., 49, 1092-1 106

Joly, A., and A. J. Thorpe, 1990a: Frontal instability generated by tropospheric potential

vorticity anomalies. (J ../. I t Met. Soc., 116, 525-560.

Joly, A., and A. J. Thorpe, 1990b: The stability o f a steady horizontal shear front with

uniform potential vortic ity ../. Atmos. Sci., 47, 2612-2622.

Jones, S. ('., and A. J., Thorpe. 1992: The three-dimensional nature o f ’symmetric’

instability../. Roy. Meteor. Soc., 118, 227-258.

Keyser, D., and R. Rotunno, 1990: On the formation o f potential vorticity anomalies in

upper-level je t-front systems. Mon. Wea. Rev., 118, 1914-1921.

Kleinschmidt, E., 1950a: Uber Aufbau und Entstehung von Zyklonen ( 1 . Teil). Met.


128

Ruuds., 3, 1-6.

Kleinschmidt, E., 1950b: Uber Aufbau und Entstehung von Zyklonen (2. Teil). Met.

Ruuds., 3, 54-61.

Kleinschmidt. E , 1951: Uber Aufbau und Entstehung von Zyklonen (3. Teil). Met.

Ruuds., 4. 89-96.

Kleinschmidt, E., 1955: Die Entstehung einer Hohenzyklone uber Nordamerika. Te/Ius, 7,

96-110.

Kleirisehmidt, E . 1957: In 'dynamic meteorology' by Eliassen, A. and Kleinschmidt, E„

Handbuch der Physik, 48, 1 12-129.

Knight, D. J , and P. V. I Iobbs. 1988: The mesoscale and microscale structure and

organization o f clouds and precipitation in midlatitude cyclones. Part XV: A

numerical modeling study o f frontogenesis and cold-frontal rainbands. ./. Atmos. Sci.,

45, 915-930.

Kotschin, N.. 1932: Uber die stabilitat van Mangulesschen Diskontinuitats-flachen Beitr.

P/tys. Atmos., 18, 121-164.

Kuo, Y.-H., and R. J. Reed, 1988: Numerical simulation o f an explosively deepening

cyclone in the eastern Pacific. Mon. IVea. Rev., 116, 2081-2105.

Kuo, Y.-H.. M. A. Shapiro and E. G. Donall, 1991: The interaction between baroclinic

and diabalic processes in a numerical simulation o f a rapidly intensifying extratropical

marine cyclone. Mon. IVea. Rev., 119, 368-384.

Kuo, Y.-H.. and R. J. Reed, 1988: Numerical simulation o f an explosively deepening

cyclone in Eastern Pacific. Mon. IVea. Rev., 116, 2081-2105.

I.emaitre. V. K. Y „ and M. Petitdidier, 1994: Dynamics o f a low-level je t observed during

the Front 87 experiment. O . ./. R. Met. Soc., 120, 277-303.


129

Lindstrom, S. S., and T. E. Nordeng, 1992: Parameterized slantwise convection in a

numerical model. Mon. IVea. Rev.. 1 2 0 , 742-756.

Locatelli, J. D., J. M. Sienkiewicz, and P. V. Hobbs, 1989: Organization and structure o f

clouds and precipitation on the Mid-Atlantic Coast o f the United States. Part I:

Synoptic evolution o f a frontal system from the Rockies to the Atlantic Coast. ./.

Atmos. Sci., 46, 13"_7-1348.

Locatelli, J. D., J. E. Martin, and P. V. Hobbs, 1994: A wide cold-frontal rainbands and its

relationship to frontal topography. (J. ./. R. Met. Soc., 120, 259-275.

Matejka, T. J., R. A., I lou/.e, and P. V., Hobbs, 1980: Microphysics and dynamics o f

clouds associated with mesoscale rainbands in extratropical cyclones ./. Roy. Meteor.

Soc., 106, 29-56.

McIntyre. M. E., 1970: Diffusive destabilization o f the baroclinic circular vortex.

(Ieophys. l- 'lu h il\v n ., I, 19-57.

Montgomery, M. 'I'., and B. F. Farrell, 1991: Moist surface frontogenesis associated with

interior potential vorticity anomalies in a semigeostrophic model. ./. Atmos. Sci., 48,

343-367.

Montgomery. M. T., and B. F. Farrell, 1992: Polar low dynamics. ./. Atmos. Sci., 49,

2484-2505.

Montgomery, M. T., and B. F. Farrell, 1993: Tropical cyclone formation. ./. Atmos. Sci.,

50, 285-3 10.

Neiman. P. J., and M. A. Shapiro, 1993: The life cycle o f an extratropical marine cyclone.

Part I: Frontal-cyclone evolution and thermodynamic air-sea interaction. Mon. Wen.

Rev., 121, 2153-2176.

Neiman, P. J.. M. A. Shapiro, and L. S. Fedor, 1993: The life cycle o f an extratropical


130

marine cyclone. Part II: Mesoscale structure and diagnostics. Mon. Wea. Rev., 121,

2177-2199.

Nordeng, T. E., 1987: The effect o f vertical and slantwise convection on the simulation o f

polar lows. Te/lns, 39A, 354-375.

Nuss, W. A., and R. A. Anthes, 1987: A numerical investigation o f low-level processes in

rapid cyclogenesis. Mon. IVea. Rev., 115, 2728-2743.

Orlanski, I., and J. J. Katzfey, 1987: Sensitivity o f model simulations for a coastal cyclone.

Mon. IVea. Rev., 115, 2792-2821.

Parsons, D. B , and P. V. Hobbs, 1983: The mesosclae and microscale structure and

organization o f clouds and precipitation in mid-latitude cyclones. XI: Comparisons

between observational and theoretical aspects o f rainbands../. Atmos. Sci., 40, 2377-

2397.

Persson, P. O. (5., and T. T. Warner, 1993: Nonlinear hydrostatic conditional symmetric

instability implications for numerical weather prediction. Mon. Wea. Rev., 121, 1821-

1833.

Pettersscn, S., and S. J. Smebye, 1971: On the development o f extratropical storms. O . ./.

R. Met. Soc., 97, 457-482.

Petterssen, S.. 1956: Weather analysis and forecasting. Vol. 1, Motion and motion

systems, 2nd edition, McGraw-Hill, 428pp.

Peltier. W. R., G. W. K. Moore, and S. M. Polavarapu, 1990: Cyclogenesis and

frontogenesis. Te/lns, 42A, 3-13.

Polavarapu, S. M., and W. R. Peltier, 1990: The structure and nonlinear evolution o f

synoptic scale cyclones: Life cycle simulations with a cloud-scale model. J. Atmos.

Sci., 47, 2645-2672.


131

Reed, R. J., and E. F. Danielsen, 1959. Fronts in the vicinity o f the tropopause. Arch. Met.

Geophys. B io k i, A l l , 1-17.

Reed, R. J., M. T. Stoelinga, and Y.-H., Kuo, 1992: A model-aided study o f the origin

and evolution o f the anomalously high potential vorticity in the inner region o f a

rapidly deepening marine cyclone. Mon. IVea. Rev., 120, 893-913.

Reuter, G. W., and M. K. Yau, 1990: Observations o f slantwise convective instability in

w inter cyclones. Mon. IVea. Rev., 118, 447-458.

Robinson, VV. A., 1989: On the structure o f potential vorticity in baroclinic instability.

'letIns, 41 A, 275-284.

Rogers, R. II., and M. K. Yau, 1989; A .short course in cloud physics. Pergamon press

Rossby, C. G., 1940: Planetary How patterns in the atmosphere. O. ./. R. Met. Soc., 6 6 ,

Suppl., 68-87.

Ryan, B. F., G. J. Tripoli, and W. R. Cotton, 1990: Convection in high stratiform cloud

bands. Some numerical experiments. O. .1. R. Met. Soc., 116, 943-964.

Sanders, F., 1987: Skill o f NMC operational models in precipitation o f explosive

cyclogenesis. Weather and forecasting , 2 , 322-336.

Schar, C., and H. C. Davies, 1990: An instability o f mature cold fronts../. Atmos. Sci., 47,

929-950.

Schubert, W. H „ and B. T. Alworth, 1987: Evolution o f potential vorticity in tropica!

cyclones. O . ./. R. Met. Soc., 113, 147-162.

Schultz, D. M., and C. F. Mass, 1993: The occlusion process in a mid-latitude cyclone

over land. Mon. Wea. Rev., 121, 918-940.

Shapiro, M. A., 1974: A multiple structured frontal zone-jet stream system as revealed by

meteorologically instrumented aircraft. Mon. Wea. Rev., 102, 244-253.


132

Shapiro, M. A., 1976: The role o f turbulent heat flux in the generation o f potential

vorticity in the vicinity o f upper-level je t stream systems. Mon. Wea. Rev., 104, 892-

906.

Shapiro, M. A., 1978: Further evidence o f the mesoscale and turbulent structure o f upper-

level je t stream-frontal zone systems. Mon. IVea. Rev., 106, 1100-1 111.

Shapiro, M. A., and D. Keyser, 1990: Fronts, jet streams, and the tropopause.

lix tra tro p ica l cyclones. The Erik Palmen Memorial Volume. C. W. Newton and E.

Holopainen, Eds., Ainer. Meteor. Soc., 161-191.

Shutts, G. J ., 1990: Dynamical aspects o f the October storm 1987: A study o f a successful

linc-mesh simulation. 0 . ./. 11. Met. Soc., 116, 1315-1347.

Smagorinsky, J , 1963: General circulation experiments with the primitive equations. I:

The basic experiment. Mon. Wea. Rev., 91, 99-165.

Solberg, II., 1928: Integration der almospharischem storungsgleichungen. ( ieofys. Pul., 5,

No. 9, 1-120.

Spiegel, E. A., and G. Veronis, I960: On the Boussinesq approximation for a

compressible fluid. Astrophysical Journa l, 131, 442-447.

Stein, U., and P. Alpert, 1993: Factor separation in numerical simulations.J. Atmos. Sci.,

50, 2107-2115.

Sun, W .-Y., 1984: Rainbands and symmetric instability../. Atmos. Sci., 41, 3412-3426.

Sutcliffe, R. C , 1947: A contribution o f the problem o f development. J. Roy. Meteor.

Soc.. 73, 370-383.

Thorpe, A. J., 1985: Diagnosis o f balanced vortex structure using potential vorticity.

Atmos. Sci., 42, 397-406.

Thorpe, A. J., and K. Emanmuel, 1988: Frontogenesis in the presence o f small stability to


133

slantwise convection../. Atmos. Sci., 42, 1809-1824.

Thorpe, A. J., and R. Rotunno, 1989: Nonlinear aspects o f symmetric instability. J. Atmos.

Sci., 46, 1285-1299.

Thorpe, A. J., 1990: Frontogenesis at the boundary between air-masses o f different

potential vorticity. 0 . J. Ii. Met. Soc., 116, 561-572.

Thorpe, A. J., and S. A. Clough, 1991: Mesoscale dynamics o f cold fronts: Structures

described by dropsoundings in FRONTS 87. Q. ./. R. Met. Soc., 117, 903-941.

T horpe. A. J., 1994: Dynamics o f mesoscale substructure at fronts. In The International

Symposium o f Life Cycles o f R xtra lrop ica l Cyclones, Bergen, Norway. Vol. 1, 220-

228.

Weldom, R. B., 1979: Satellite training course notes. Part IV Cloud patterns and upper

wind field. United States A ir Force AWS/TR-79/003.

Whitaker, J. S., L. W. Uccellini and K. F. Brill, 1988: A model-based diagnostic study o f

the rapid development phase o f the Presidents' Day cyclone. Mon. IVea. Rev., 116,

2337-2365.

Xu, Q., and X. P. Zhou, 1982: Baroclinic inertial instability in the nonstatic equilibrium

atmosphere. Scientia Atmospherica. Sinica. 6 , 355-367.

Xu, Q., 1986: Conditional symmetric instability and mesoscale rainbands. Quart. ./. Roy.

Meteor. Soc., 1 1 2 , 315-334.

Xu, Q., 1992: Formation and evolution o f frontal rainbands and geostrophic potential

vorticity anomalies../. Atmos. Sci., 49, 629-648.

Young, M. V., G. A. Monk and K. A. Browning, 1^87: Interpretation o f satellite imagery

o f a rapidly deepening cyclone. Quart. J. Roy. Meteor. Soc., 113, 1089-1115.

Zhang, D.-L., and H.-R. Cho, 1992: The development o f negative moist potential vorticity


134

in the stratiform region o f a simulated squall line. Mon. Wea. Rev., 120. 1322-1341.

Zhang, D.-L., and H.-R. Clio, 1994: Three-dimensional structure and evolution o f frontal

rainbands and conditional symmetric instability in the Eady-Wave model. Tellus, In

press


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