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 of extratropical
cyclones with different moisture distributions show that at the different stages of
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 of vorticity and
M PV 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 of the extratropical cyclone is also presented.
i
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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, for his insights, encouragement, understanding and
guidance, to Professors G. YV. K. Moore and T. G. Shepherd for 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 of
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 of physics for their assistance and discussions,
especially Dr. John Koshyk. M urray 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 will be
remembered for ever.
My financial support by the department of physics of University of 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
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Abstract i
Acknowledgments ii
1. INTRODUCTION 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. GOVERNING EQUATION AND GENERATION OF MOIST
POTENTIAL VORTICITY 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 E L 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. GENERATION OF MOIST POTENTIAL VORTICITY IN
EXTRATROPICAL CYCLONES 30
4.1 The control experiment 30
iii
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4.1.1 Life cycle o f the extratropical cyclones and
MPV distribution 30
4 I 2 Baroclinic generation o f MPV 37
4 .1 3 Effects o f horizontal diffusion and rainwater evaporation 44
4.2 Sensitivity study 47
4.3 Summary 67
5. BOUSSINESQ APPROXIMATION AND ITS IMPLICATIONS 70
5.1 Vorticity dynamics 70
5.2 Moist potential vorticity dynamics 79
5.3 Summary 81
6. CONDITIONAL SYMMETRIC INSTABILITY (CSI) IN
EXTRATROPICAL 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 of moisture distribution on CSI in extratropical
cyclones 102
6.4 Summary 115
7. CONCLUSIONS AND SUGGESTIONS FOR FUTURE
RESEARCH 116
iv
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APPENDIX B LIST OF SYMBOLS 120
REFERENCES 123
V
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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 of
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
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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.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (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 of 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 of potential vorticity (PV) thinking has been
recognized only relatively recently, the application of this concept to the study of
extratropical cyclones can be traced back to Rossbv's work (Rossby 1940). As he
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Open hexagonal ceils
t
Cirrus cloud boundary as seen from satellite
synoptic features \ TYOCS CF MCSOSCAUE RAINBANOS 1 T | SURFACE LOW- i 1 L j 1 PRESSURE CENTER t WARM-FRONTAL
y j SURFACE COLO 2 . WARM-SECTOR y* FRONT 1 ^ 1 SURFACE WARM ! 3 wiOE COLO- y * I FRONT i f r o n t a l y * | SURFACE WARM 4 NARROW c o l o - 4 * 1 OCCLUOED FRONT I FRONTAL
S I COLO FRONT AL0 F7 I j 1
“ I 1 1 4 »ostf»ontal 1
Fin. | .2 Schematic depiction ot' iho tvpes o f rainbands ( numbers i -o i observed in extratropical cyclones I- rom I lobbs
i 19SD.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Table 1.1 The possible mechanisms for 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 conditional symmetric instability (Bennetts
rainbands and Hoskins 1979) and Wave-CISK.
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pointed out. in a barotropic flow the absolute vorticity divided by the depth of a tluid
column remains constant following the motion of the tluid column. The hydrodynamic
generalization of this concept was first stated two years later in the independent work
by Ertel (1942). Tn his paper, Ertel showed that potential vorticity defined as
-Ca-ve. (1.1) p
where Ca 1S absolute vorticity vector and 0 is potential temperature, is a
conservative quantity in adiabatic and frictioniess flow. To distinguish it from the
concept of moist potential vorticity which includes the effects of moisture, the
potential vorticity defined by (1.1) is referred to as the Dry Potential Vorticity 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 of 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 of quasi-horizontal advection along isentropic surfaces from the
stratospheric reservoir of high DPV. The well-known Petterssen Type-B cyclone
development (Petterssen and Smebve 1971), originally identified as an upper-level
positive vorticity advection area moving over a low-level baroclinic zone, should
properly be considered as the development resulting from the advection of an upper-
level DPV anomaly along an isentropic surface. Examples o f the applications o f the
DPV concept to the studies of 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 of PV thinking was the recognition by
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Charnev and Stern (1962) that necessary conditions o f baroclinic instability, the origin
of extratropical cyclones, can be formulated entirely in terms of 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 invertibility
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 of potential vorticity.
Symmetric instability (SI) is an example of 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 for 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 of 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 of such questions is the likelihood of SI to occur anywhere in the atmosphere.
Because of the conservative nature of DPV in an adiabatic inviscid flow, SI will not
occur in an adiabatic flow if 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
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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 of 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 vorticity (MPV), defined as
—C~a • V 0C, (1.3) P-
where 0C is equivalent potential temperature, have a negative value. This kind of
instability has extensively been studied in recent years since it was first proposed as a
possible mechanism for the formation of 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 of 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 MPV generation in an unsaturated three-dimensional flow is
examined in Chapter 2 after a brief discussion of the governing equations. It will be
shown that M P V generation is due to the presence o f baroclinicity together with
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gradients o f water vapour. The rate o f MPV 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 of negative MPV in a typical mid-latitude
cyclone, a set of numerical simulations are made using a three-dimensional regional
atmospheric model. The model is described in Chapter 3 together with the
experimental design and the initial conditions used in the simulations. The results of
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 M PV fields since the baroclinic vector is treated in different ways in the
Boussinesq and primitive 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 of properly describing the significant
development o f vorticity and MPV. The differences between a Boussinesq flow and a
flow described by the more complete set of primitive 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 of surface low and the
precipitation is also presented.
Conclusions and suggestions for future research are made in chapter 7.
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Chapter 2
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 ).
c V - - - - 1 - F — + (V-V)V + 2Q:< V = — Vp h-G -f — (2.1) c-t p p
^ £ ^ pv - v = o (2.2) dt
d0e = Q (2.3) dt
0 e = Qexp( ,q) (2.4)
I.
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 if lifted adiabatically
to its condensation level, which is defined as
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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) j9e _ V6e . (VP.:-ZE) + 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, ^d0_o (2.8) CrT ( i- q J 0
where C p 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
Lqs 0c=0exp(^). (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,
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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
d0 drv adiabatic. — = 0 . dt
The second term in (2.7), V 0 e -(Vp .* Vp)/p'\ is zero if 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. 0 e 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:
V0,-[2£iStl- — T^- •! - ~ ■ Vp - g.^ - V q . vp! p-' p“ T ( l - a ;q)
V0 VT R Vp Because — = ------. 0 T C , p
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Vp ^ Vq x Vp} p- T (1 -r a : q)
= } -i-V 9 x Vp - Vq x Vp}. (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-es
where es is saturated vapour pressure.
es = b.llexp(a5-^ -),
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 , )ln(r)] whprp A — ^ t “ ______I 4 _____ p- 0 T (T-aJa -a,[a -(T-aJln(r)] 1 .? 4 .> 4
a, ' a 4[(T -a J 2(T.(p-es) + a, -
[a4 T-a ,(T -a , )ln(r)]“ Tv(p-eg)
in which Tv is virtual temperature. If any other expression of0e 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
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-r- - -
VP v e \v VQ
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.
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decrease (increase) in the value o f M PV if 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|Ca.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 for 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.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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*uu/m) f(p*vu/m), fp*uo a « + ,y to
+ fp*v - mp»( I S i V ; * l i , + F.,,. p*u (311) p * - p t /a r/x rx
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r(p*v) ,r(p*uv/m) c (p*w/m ), op v o — _— = - m: [— ------——z------1 - z f.t ex cv CCS
- ctp*u * - * / m p*( R T " c : p- *------. — C $ ) -r F.:n-_ p*v y (3.1.2) p * -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) _ (3.1.3) ' Crm(a + pt / p*) ' cpw
r(p*qv) _ , r(up*q.7m) r(vp*q,/m) cp*qva et ex cy cg
- p^ p^ - p j ^Fhd- pX- (3.1.4)
r(p*q.) _ r(up*q,7m) ^ r(vp*q,7m) rp*q.G ft fx cy cg
+ P*(Pa,n * Pr. “ Pr<> ~ Fhd P* Qc (3.1.5)
f(p*qr) = , r(up*qf7m) ^ f(vp*q,7m) fp*qra rt fx cy co
* P*(P„ + PK - p„) - S + F„„ p* qr (3.1.6)
rp* nr(p*u/m) r(p*v/m), rp*C ~z— = - m: [— ,------+ — _ ] - .— (3.1.7) f t ex rv cg
1 a. .tp .„r(p*u/m) r(p*v/m), , , , a = ~ r f Hr- + m- [— , + —1, ] da (3.1.8) p* J ' et 1 ex rv s ’
rp* rp* rp* co = p*a + a [ ~r~ + m (u~^~ *v~z~~) (3.1.9) ^ 1 et ex rv
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(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 of 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..D2 = K „ V ;.
while the fourth order form is used in the interior.
^IID4 = ^ n.
where K n is defined as
K „ = (As)2KH,
and
K„ = A1 (K ho + t k W D ) .
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. All 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
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 of 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 of horizontal diffusion
on negative MPV distribution (discussed in the next chapter).
3.2 Initial Conditions
The initial conditions are generated analytically similar to those of Fritsch et al.
(19S0) and Nuss and Anthes (1987). Compared with Nuss and Anthes' (1987) initiation,
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 = Po+APx + APy
where pu is a constant,
Apx = ax bp (x) Gp(y) sin + 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,, tanh[^^] (3.2.5) p,Fj(x)dy P:dy where 0 j r \ Fj(x> = 1 -b sin + 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 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. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 rrv ATX = bxD(z)c-r(x) GT(y) sin — + in which D(z) = — - — tanh ( - —- ) (3.2.9) 2 2 dz 2 _, _ v 2 and 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 of 1 km, and 7 7T\ c-r (x) = d,, [ da + sin(-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,tanh[^£-] + FR (x,y,z) (3.2.13) Pbldy ‘ pb2dy in which Fr (x,y,z) = tfz) [ sin + it )+sin f1^ ) ] 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, byt, by2, bx, % , pbl, Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 = (4hk)'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\ fp u = ------. and v = ------. (u.2.19) fp fy fp ex (6) Initial fields are interpolated from z coordinate to c coordinate. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Table 3 .1 Constants to define the reference level pressure and the three-dimensional temperature For Reference Level Pressure For 3-D Tem perature 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 24 3.3 Experiment Design Ten experiments have been conducted to study the moisture effects on the development and evolution o f M PV 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 of 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 66 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 of 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). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. I'iu.3.2 Initial surface relative Imnndilv fields lor the experiments (aH in. ‘Ihe contour interval is 11 % Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 31 (a) N T (b) (C) + * "■s / $Fo&» T'*..y i • \. lffi\ v5\ t f\V» )}•• *.*»*' » 1 • ,* »•%« * : w 'f W »\ i;s\t-'y'* 'i1'1 \ ■ .- > ivK\ * / \ : \ : } IWvA\\* * % *— 1 * 1* Mi'* 1 1 CO/ ' . cl rS 1 /// V * * • • • 1 h i S w1 \ i\» \ * * *% * • • I Mr / \\ * \ *—• ' ' / H i* i J&y 1 1 ^ * / l\\V ^ __ E * /r 1 \ VV\ \ » * is_ (d) (e) r*(0 '...... %% %* * * 7 »»#■*. ( S b , i• *|!/' * • i » i t /#*•,»» ;**%'* u i S. : **:*?«*. V ' » \ A * l» * f , : \^ 4 v k S > ft,;*-... i'— i s * *'*. Vi- * % ' i * t , .. «N- \ \ "*V •«/■' ’ * \ **/v. J .V — ; ...... :i V “ X __ 5 • J :5\ \ !% '• % tlwt./JR• » 1 • ' --'rtfcvsS&sS -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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 of 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 M PV 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 M PV 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 M PV (Fig.4.4b) moves into an Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 33 j 11111.1111111111 u 111111 n 11 n rrn 111 u 11111111111 ii it 11 u ii i n i u 111 u n nr: n T o ■*tntit tin 111 n i II III IIII i IIII i IHI 111 T mi mu 111 m 11 n mu IIIII mniiir Liiiiiimiiiiiiiiiiiiiiiimiiiiiiii.m J IIIIII11111111’ 11111IIII111111111111ITE O iiiiiuuiiiiiiiiinniiiiiiiimiiiiif ■mniiiniuiLintiimiiiiittnimtr 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. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. l'ig.4.3 Horizontal cross section of relative luimiditv taken at the S7I nth al hours in ) 30. (b) 4X. (c) Hie contour interval is 3.'%. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 35 I a ' u in m 111 in i i i 11 m rr m 11111 i n i rma j imiMTtMtmiinmMinmmmrij N t Tiiiiiiiiiiiiiiiiiiiiim im iiiimiin 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 Ta TTT tmmmmmnrq 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 iiim nniniiiiiitiim;iniiiiittii_d h 11 it 11 n 11111111111 n m i n n i m hilt 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. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 of 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 of MPV Qualitative information about negative M PV 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 of (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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission 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 of 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 M PV generation. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 41 M u r« m m n mim i11m *n h i n ij m m iirii m m 1111iiii, nmmiiLL 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. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 42 mil i n nmnmmi mi 11 u m hi imj (a) N t | jiiiii.iiiiiiiiiiiu 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'-. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 44 4.1.3 Effects of 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 with and without rainwater evaporation (Figs.4.1 la and 4.11b). MPV generation due to diabatic cooling of rainwater evaporation often happens in a substantial unsaturated downdraught. This effect is very small because rainwater evaporation spreads in a deep column compared with snow evaporation (Clough and Franks 1991) and other phase changes like melting, and only produces a weak gradient o f diabatic cooling. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 ff the ground by the end of 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 M PV 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 of 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 of large differences in direction between the gradients o f potential temperature and specific humidity are located in the area where M PV 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 48 u 11111111111 rrr rr1111111ri 111111111111 LM11II111 TiTrTTTTnTi n m n m •! i m i n m i i n t n 1111i i 11111111111111n 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 XITI'11111 I t 11 U I ! 1111111111111111111! | L TTTX l l l t l l l l l l t l U H H f l K K I ITTTfl -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. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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%. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 50 111111m11111111111ii111111 rnit Tin if»nil iiimmiMimiirnmiir m ... nn...... n ...... m .. i...... in ..mm.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. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 51 iiiiiim iiiiiiiiiiiiin n iiim i'i' Li ii n ii n ri n m m i rm t vrrrvrrm nrz -r 11 n i n ii 111 ii titiLLLU m iiiim iilifi t11 ii n1111111n11i i m inim um ■iiiim iim m m Tn-rm ninitim iiu U1:111II111M 111M 11111 n U 1111111! 11! LI 7 n 11 n 111 m i n 11 n 1111 n 111 n 11 n 1111 7 n 111 ti i n n 111 m n ir r i 1IIIH 1t r u i i f 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. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 52 jiu m iim i TTTTTTTTJ 1111111 III TrriTTnrniim iunm (b) »i i i n 11111111 m n i n 1111111 n 11111 m iLLitii m m i ii m m i n 11 n i r t i i ii i ii n i ii n i m 111 ii 11 uni 11 in i ii i (d) m i n i u m i i . t m i i i i i i i i n i n n n iiH h jllim n m in m inim u m 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. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 %. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 %. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 M PV 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 57 iuiiuiiiiumiii'i1111m iii n11111it . 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. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 58 1111 rm r rm i m u it t i m > rr rr g r g ■ iiiiim n iiiiiim iiiiiiiiiiiiiiiiin -i 1111ii 1111111 ii i TTTTTTTnTTTTTTTTTTTTT = (a) n i ) 11 n 1111 n n i ? m n l n n 111 n it 111L t i i i i l n m in i m ii 11 m t u n ) n in iLn 31111111 m u ii i l l m i '1111! 113 cm inn mmmi tiim iiiiiiinnnin iiininmmiiiiiiniiniminiiiir 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. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 59 ;! i I'm i m iiih iih d ri i ru m vm m u i i 111 n m 1111m 'it 11 I 11 111 I l l| \ <*> I (b) I Lin n ILI H « n i»11 n 111111 n 11 n 11 n -Muminniinmniinnninnn m3 h_t 111 i i luj iniiniiinnim iiiiiiiiL .nn m rn'in 11 m m i'i 11 m 11 it 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. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 60 n » M I I r T T M I ! I I ' l l IT 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 %. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 %. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 62 place in the warm sector at the north part of 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 of 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 M PV 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 M PV 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- Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 63 uimmmmiiumimniminniiu inm.im im m ifiimmnimmu t m n i i n u 111». iu iin 11imi n n im rt hum m i I II m um m ini 111 miniH mriTtTmm mm inmi Tq uiiim iiin iim iim iiiim u rirm i lj n t u litLLi 1111 m m iiilli ; 11 n n i i n n tiim im i mi t < i u~i immi niiTh111 r Ftn.4 2<> I lorizonlal cross section of MPV for the experiment ( and (d) 20-4. flic contour interval is 0.5 PVU. The shadings indicate regions ot'negative MPV. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 64 i l'i".4.27 Uorizonlal cross section of relative humidity for the experiment 4X. (ci 72 and (d) 204. The contour interval is 33%. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 66 LirnrrmtTi liim iiiimiiimiiiim j n i n 11111 n n in i iTTTTt i ii i n 1111 n i ■t n n 11111111111 n i n i n i ii n 111 n 1111 rt h n 111 »i iii n m nILLLUii inn 11m 11 r jTk i if i ¥ Y f i f i Y U T i'i i u 1111111111 m i m u Jiiiiiim m riiiiii i ii m 11111 n t n i»111 n 11 n ii t n i ii n 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 67 shaped negative M PV 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 of 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 %. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 of 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 M PV 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 will stay there till the end o f the simulations if 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 with 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 of 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. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 will 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 M PV distributions. We will start with the vorticity field. 5.1 Vorticity dynamics Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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+ Vhp x ^ k +a k x V hp). (5.1) where the solenoidal term, (Vp 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 of 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 of 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. horizontal and vertical components o f the compressible term Q V ■ V . As expected, the tl 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) if one assumes the i i t hydrostatic approximation and makes use of 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 : (cp/czk 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 with 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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'-. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 74 ;ls i b ) except Ibr llic vector \ ' p x V p / p “ with a contour interval of 2. * |ir*°s “ (I) icune as (a) except lor tlie vector t-p / (7.1c x \ ' p/p2 with a contour interval of5. x K r7 s°: (g) same as Cl) except for tlie vector V^pz/Tp/ (7.k / p“ with a contour interval ot'5.* Ill's 0 : (li) samcas(a) except fortlie vector -e nnk - \n with a contour interval ol 5 a Id ' s2 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Fie.5 2iaH J> Same as Fie.5.1 except for the control experiment. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 76 Fi;;.5.2 (CM") Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 77 •...... j niiiniiii M iinriiiiiiiiM nT ! (a) N t j i -TiN v c \ ■\Y\ :NJi Vs — O’ 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 ■ ! ;.umjmuumi mimmu TTTTTTTTTTTTTPTT'PrTTTTTTTTTTTTTTTTTTTTT ( d ) ...... 11. HI, ...... I'm.5 3 Horizontal cross sccuons for the dry experiment taken at 48 hours at 2.4 km above the ground ot (a) - 2 g 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 of h 2.5 x 10*' s'2: (dHt3 are same as (a H d except die cross section taken at 6.9 km above the ground. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 78 > < c> Fig. 5.4 Same as Fig.5.3 except for the control experiment. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 M PV under the Boussinesq approximation is expressed as: d ( ^ a -V0 ) —B------= i-k.(V 0 evV9> ( 5 -4 ) dt p 0 . 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 if the effect o f latent heat release is taken into account; and (2) will not give the correct MPV features in the intensive development portions of the cyclone. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 of 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 for the instability. In this section, we will take the parcel instability approach. By adapting the criterion of 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 ) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 84 The air in the region satisfied with 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 85 M = fs + Vn, (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: Vn= 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^Tt^O) - Vn(-.s)]. (6.2.5) R where k - ( — ■—) p. Because coordinates relative to the srid point (i, j) are: 1000 s0 x - l dx = S U -1 = ------;---- ~r •.";'-r h , ( 0 ; + 6 ? ) 1'- - - S0V _l and 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 /2 or 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 of 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 of CSI in extratropical cyclones The CSI criteria in section 6.2 are used to diagnose the structure of 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. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 or 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. S9 — CO vB' > 1 II 1 M I I I I 1 111 MM' i ...... 11 III III 1 Ml i (a) N T j (b) N t 41 ! 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. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 91 0.050 0.150 0.250 0.450 0.550 0.645 0.730 0.805 0.870 0.920 0.960 0.986 0.996 B B’ 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 B B' Fig.6.3 (b) and (c). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 of 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 93 0.050 0.150 0.250 0.350 0.450 0.550 0.645 0.730 0.805 0.870 0.920 -S 0.960 M I » 0.986 0.996 nuimminihnuu I -18 m/i B B' 0.050 0 .150 (b) 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.6.4 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. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 94 vCI /-M 'C' » M M l 11 M l »TI .... tc 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. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 95 0.050 f t T M I 1 |Tf fT T M fl 1 I ! I ! »JJ j nimmnwntii £ 1 ' 0.150 0.250 0.350 0.450 • m.^^^*jjiji *^ 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. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 0.996 C C’ Kig.o.6 (c) MPV field at an interval of 0.1 PVU. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 if 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 P V 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. It 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 with 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 .996 B 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 * Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 100 1020 o -1015 O > -1010 o v A + 3 -1005 3 C/3 -1000 "3 3 cf y -990 O 0.5 33 -985 y > Surface pressure (mb) "5 -980 > a< -975 S ■970 0 16 80 9664 112 128 144 160 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'*. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 101 0.050 f r i i i n i m i i i m it i m i h u n i i i in in iJTTH Trrrt! II 1111111 II I!M IM 0.150 0.250 0.350 \ 0.450 0.550 0.645 (7 0.305 0.870 0.920 0.960 0 .986 0.996 11ii111ii n i ii i in i i ij n iin i i i n ? i in rv x /fT T rr IIIITTJ 0.050 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 ^. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 of 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 103 m M m I 1 I I I I I | n T f TT (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. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 of 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). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 105 4 d ’ i i 11 u i i i i i ii i i i i i i i i i i r rvi'i iiiiiirn (b) Sfi 11>1 ■ ■ ■ ■' ■»11 ■ ■ ■1 ■ ■1 ■111 E ■ ■, ■ ■ ■.... ■ i in I ii 11 n i in.. ■ i ?D Td 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. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 106 0.050 '""m T I ..... 0.150 IUJ,,n ^ ' I1” 0.250 0.450 0.550 0.645 0.730 0.805 0.870 14 4 / t t f f 0.920 ////// 0.960 r~ 1 I I " i 0.986 0.996 m i i ii im in u . 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. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 107 0.050 iiniiiiin iTTrrm r r rrr 0 .150 (b) 0.250 0 .350 0.450 0 .550 0 .645 0 .730 0 .805 0.870 0.920 0 .960 N 0 .986 0.996 H . I I ■ I , , I . I H .1 . ■ 1 I . ■ I . D D’ O.ObO 0 .150 0 .250 0 .350 0 .450 0 .645 0.730 0 .805 0.870 0.920 0 .960 0 .986 0 .99 6 Fig. 6.12 (bi ami (c). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 D' 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 D D* 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'*. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 .73 0 0 .80 5 0 .870 0 .92 0 0 .96 0 0 .98 6 0 .99 6 Fig.6.14 (b)and (c). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Ill i pi m m ttmm i eiiwi tu v 0.050 0.150 0.250 0.350 0.450 0.550 0.645 O’ 0.730 0.805 0.870 0.920 0.960 0.986 0.996 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. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 112 tT IT l I TU I II 1'M‘TTTH M r ft T H I 1111111 M 111111) H 111 111 H f 11 I I1 1 111! 11 HI I T 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 .05 0 0 .15 0 0 .25 0 0 .35 0 0 .45 0 0 .55 0 0 .6 4 5 ct 0 .7 3 0 0 .8 0 5 0 .8 7 0 0 .92 0 0 .96 0 0 .98 6 0 .9 9 6 I-ig.6.15 (b) and (c). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1 1 6 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) M PV 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 M PV 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 M PV appearance. (2) After the cyclone matures, the negative MPV moves into the warm core. It will 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- Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 of 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 M PV 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 118 MPV and CSI in extratropical cyclones, should be investigated. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 119 APPENDIX A 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 ,[a4 “ ( T - a 3 )ln(r)] p2 0 ' T (T-a Ja 4 l a ^ a ^ - (T-a^ )ln(r)| 10-' 101 a I xa 4 [(T -a ,) 2 (Tv(p -e s)+ a 9 * a fiq p ) - a 2 x a 4 T2 q(p - eg)] [a 4 T-cx3 (T -a 3 )ln(r)] 2 7v (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 10 3 K, respectively. The third term in the braces o f the function A is not sensitive to the variation o f pressure. For example, if p=250.0mb is used, the order of magnitude o f this term is same as one estimated above. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 120 APPENDIX B LIST OF SYMBOLS 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 nr' 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 of horizontal diffusion Kmo m2 s-' background value of 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 of cloud drops to rain drops Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 121 Pre kg kg- 1 s- evaporation o f rain drops q kg kg- 1 specific humidity q^ kg kg- 1 mixing ratio of cloud water qr kg kg-' mixing ratio of rain water qs kg kg- 1 saturation specific humidity qv kg kg- 1 mixing ratio of water vapor Q K s - 1 diabatic heating or cooling rate r relative humidity R J K-'kg- 1 gas constant for dry air S in horizontal distance from the grid point to the point where the M surface crosses the level u T K temperature T v K virtual temperature u m s' 1 x-component of velocity V 111 S' 1 y-component of velocity V m s’ 1 three-dimensional velocity vector V n 111 S' 1 wind component normal to the direction o f potential temperature gradient v. m s-1 mass-weighted mean terminal velocity o f rain drops 4> 1112 s- 2 geopotential K Von Karman constant (“ 0.4) K defined as (p / 1 0 0 0 )Rci’ e K potential temperature K reference value o f potential temperature e' K potential temperature perturbation gradient o f potential temperature in x and y direction, CD CD K n r 1 * respectively Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 122 0 c K equivalent potential temperature 0 * K equivalent potential temperature for 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 123 REFERENCES 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 of the Penn Sate / NCAR Mesoscale Model Version 4 (MM4). NCAR Technical Note, NCAR / TN-282, 6 6 pp. Anthes, R. 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