A STUDY OF NON-ADIABATIC BAROCLINIC INSTABILITY
by
MAURIZIO FANTINI
SUBMITTED TO THE DEPARTMENT OF
EARTH ATMOSPHERIC AND PLANETARY SCIENCES
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS
FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY IN METEOROLOGY
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
September 1988
(n
Massachusetts Institute of Technology 1988
Signature of Author
-
Dept. of Eart , Atmospheric and Planetary Sciences
Certified b y
&-
Kery A.
Emanuel -
7
Thomas H.
Thesis Supervisor
Accepted by
/
Jordan
- Chairman
Departmental Committee on Graduate Students
Syrr
M
APR,,
A STUDY OF NON-ADIABATIC BAROCLINIC INSTABILITY
by
Maurizio Fantini
Submitted to the Department of Earth Atmospheric and Planetary Sciences
in partial fulfillment of the requirements for the degree of
Doctor of Philosophy in Meteorology
ABSTRACT
Observations of explosively deepening oceanic cyclones show the
coincidence of an essentially baroclinic structure, deep convective
activity and associated latent heat release, and air-sea exchanges of
sensible and latent heat.
In order to establish the role of these
effects in a possible explanation of the unusual features of those
storms a study is performed of the normal modes of non-adiabatic
baroclinic instability.
The effect of convection alone is considered first, by assuming
that a slantwise convective adjustment takes place on a short time
scale so as to 'prepare' a moist symmetrically neutral environment for
growing baroclinic waves. A hierarchy of two-dimensional models that
incorporate this assumption (i.e. a base state of zero equivalent
potential vorticity) is then used: a two-level semi-geostrophic, and a
two-level primitive equations model are solved analytically and a
multi-level non-hydrostatic PE numerical model is integrated in time to
examine the structure of the unstable perturbations to an Eady-like
base state.
An increase in growth rate of about 70% (in the more
realistic model) is found, with a most unstable wavelength about half
that of the dry case, and composed of a strong narrow updraft and a
large weak downdraft. Another important aspect of the solutions is
that frontal collapse occurs only at the surface and not at both top
and bottom boundaries as it is usual in dry models.
The increase in growth rate is not sufficient, however, to explain
explosive cyclones and sensible and latent heat fluxes from the bottom
boundary are then added to the models. For the 2-level this is done
with a linearized drag law for equivalent potential temperature, in
which the flux is proportional to an air-sea entropy difference in the
base state and to the perturbation meridional velocity. This is
applied to the updraft region only, under the assumption that only
there will the heat be transported effectively outside the boundary
layer. The results show a further increase of the growth rate,
linearly proportional to the air-sea temperature difference, but the
drastic simplifications involved in the formulation of the model leave
uncertainties as to the uniqueness of the solution.
- 2 -
The numerical model is run with heat and moisture fluxes and
momentum drag at the surface. In a realistic range of parameters the
most favorable cases reach a deepening rate of the surface low-pressure
center of 24 mb in 24 hrs, which is the conventional threshold for
shallow
The evolution often begins with a
explosive cyclones.
hurricane-like structure that gives way to a deep baroclinic mode after
The thermal structure in the presence
several hours (i.e. 10-20 hrs).
of heating from the bottom boundary displays a narrow warm core that
expands as the storm intensifies. The phase speed of the disturbance
is reduced as the shallow early state is advected by the low-level wind
and tends to stabilize when the 'moist baroclinic' mode takes over.
Some limitations related to the two-dimensionality of the models
used here need to be removed in the future, but it seems possible to
conclude that the most striking features of explosive cyclones can be
attributed to the presence of surface fluxes of sensible and latent
heat, redistributed upward by moist convective activity.
Thesis supervisor: Kerry A. Emanuel
Title: Professor of Meteorology
- 3 -
ACKNOWLEDGMENTS
I am grateful
interest
in
to
this work.
my
advisor
Kerry
82-14582
for
his
studies
by
to
NSF/g
grants
ATM, AF/ESD F19628-86-K-0001 and ONR/c N00014-87-K-0291.
numerical experiments were performed on the CRAY-X/MP at NCAR.
wish
constant
Rich Rotunno, of NCAR, provided the numerical
I was supported through my graduate
model.
Emanuel
thank
I
The
also
Diana Spiegel for help with the CMPO computer and Jane
McNabb for running the department.
-4-
Ben sai come nell'aere si raccoglie
quell'umido vapor, che in acqua riede
tosto che sale dove il freddo il coglie
Purgatorio,V, 109-111
-5-
Table of Contents
Abstract
...pag.2
Acknowledgments
...pag.4
Table of contents ...pag.6
1.
Introduction
1.1 Observational background ...pag.7
1.2 Theoretical background ...pag.15
2.Baroclinic instability in a saturated environment
2.1 A 2-level semi-geostophic model ...pag.21
2.2 A 2-level primitive-equation model ...pag.50
2.3 A primitive-equation numerical model ...pag.63
Appendixes to chapter 2
2.A Matching conditions for the linearized system ...pag.98
2.B Proof that
TZ is real for the 2L SG model ...pag.100
2.C QG and GM non-dimensional quantities
2.D
3.
G
for the 2L PE model ...pag.102
Air-sea interaction
3.1 A 2-level semi-geostophic model
...pag.104
3.2 A primitive-equation numerical model
4.
...pag.101
Conclusion ...pag.151
Bibliography ...pag.156
List of figures ...pag.165
List of tables ...pag.171
- 6 -
...pag.119
1.1
The role of heat sources in the atmosphere has traditionally
considered
significant
extratropical cyclones.
etc.)
and
the
but
not
dominant
in
The Norwegian cyclone
classical
theory
of
the
model
gravitational
potential
processes alone.
the
mean
of
(J.Bjerknes,1919
instability
planetary
scale
energy into cyclone scale energy by adiabatic
Non-adiabatic effects are obviously
responsible
for
meridional temperature gradient (differential solar heating)
that drives the general circulation,
systems
development
baroclinic
(Charney,1947;Eady,1949) emphasized the conversion of
been
and
for
small-scale
convective
(latent heat release), so that their influence on mid-latitude
synoptic scale motions is a natural subject of investigation.
The release of latent heat of condensation was first recognized in
early numerical models as crucial for a correct computation of vertical
velocities
(Smagorinsky,1956)
and
vorticity
and
pressure
fields
(Manabe,1956) and has been since then the subject of diagnostic studies
of
observed
Aubert,1957
or
numerically
and
Danard,1964.
simulated
cyclones,
with
Comparision of moist and dry-adiabatic
runs of Global Circulation Models (Manabe et al.,
shown,
beginning
1965; Gall,1976) have
among other features, a tendency of 'moist' cyclones to develop
faster and with a shorter horizontal scale.
For storms that develop in a maritime environment
source
is the flux of heat from the lower boundary.
of several cases of North Atlantic
- 7 -
cyclones
at
another
likely
After examination
different
stages
of
evolution
Pettersen
et
al.,1962
conclude that the inclusion of heat
fluxes and release of latent heat 'led to improvement' of the
computed
tendencies and state:
"It is evident, therefore, that the effect of heat
sources and sinks on cyclone development are essentially
complex and need not be small.
In the absence of direct
information it has been customary to assume that the effects
of heat and cold sources are unimportant except in regard to
changes over extended periods of time. Though this is true
for the general circulation of the atmosphere, it need not be
so for individual systems" (pag.261)
In this and later works (Pettersen and Smebye,1971)
types
of
cyclones were identified.
levels,
is
the
North
induced
the upper troposphere.
Type
baroclinic
zone
at
with little or no vorticity advection aloft; and a Type B
(prevalent over
cyclogenesis
distinct
A Type A (predominantly maritime)
in which the development is initiated in a highly
low
two
American
continent)
in
which
low-level
by a strong pre-existing vorticity center in
By geographic location and mode
of
initiation
A is obviously more sensitive to fluxes of heat and humidity from
the lower boundary in the
latent
heat
by
synoptic
early
stages,
and
subsequent
or convective scale updrafts.
hand Tracton,1973 identified the outbreak of convection
release
of
On the other
near
the
low
center as an alternative initiator of the development also for cyclones
over the continental U.S., east of the Rocky Mountains, in the presence
of moist low-level air from the Gulf of Mexico.
A parallel line of investigation has identified a different
of
marine
cyclones
class
that are characterized by small horizontal scale,
rapid growth, 'warm core' thermal structure,
- 8 -
and
active
presence
of
non-adiabatic
processes
- latent heat release and sensible and latent
heat fluxes from the lower boundary,
explosive
phase
of growth.
that
are
prevalent
relative
to
occurred
February
1950.
the
Gulf
of
Alaska
in
deepening coincided with an outbreak of extremely
ocean.
Diagnostic
the
The first detailed description of such an
event is probably that of Winston,1955,
in
during
a
cold
cyclone
that
The explosive
air
over
the
calculations show the vertical velocities obtained
in the assumption of adiabatic motion to be largely in error (including
the
wrong
low-level
sign
air
climatological
at
to
the
be
time
an
average.
of
order
maximum growth) and the heating of
of
Although
magnitude
this
larger
to be at least of the same magnitude.
latent
the
ice
edge,
are
termed
'polar
heat
Cyclones of this family,
that occur at high latitudes, north of the main baroclinic zone,
near
the
paper mentions only sensible
heat as the source, a later study (Pyke,1965) reveals the
fluxes
than
often
lows' and often compared to
tropical cyclones for the importance that non-adiabatic processes
in
their
development.
Some
works
essentially baroclinic structure
Browning,1969;
in
Mansfield,1974;
on polar lows have identified an
the
mature
Reed,1979)
stage
while
the
two
is
to
A
chose
and
to
combination
generally agreed upon, because the 'pure' mechanisms,
dry-baroclinic or moist processes with no
unable
(Harrold
others
emphasize the hurricane-like features (Rasmussen,1979).
of
have
explain
all
baroclinity,
of the observed features.
given in Rasmussen and Lystad,1987.
-9-
are
certainly
A recent review is
In contrast to the often insignificant
the
environment
at
the
early
stages
short
temporal
to develop in strong baroclinic zones, e.g.
near Japan (Matsumoto et al.,1970).
present
in
of development of polar lows,
mid-latitude events on a small spatial and
observed
baroclinicity
scale
are
the 'Baiu' front
Nitta and Ogura,1972 summarize the
observed features:
the
"...
warm-core type structure in the upper portion of
cyclone
...
The
particular
features of these
intermediate-scale cyclones in general, aside from their
smallness in size, are that these disturbances do not appear
to be associated with an upper tropospheric trough and their
kinetic energy is confined to the lower portions of the
troposphere
...
In many cases a low-level
jet
is
observed
(at a level around 700 mb), the air in the lower troposphere
is moist, the thermal stratification is less stable, and
heavy rainfall takes place ... The characteristic time scale
is short compared to that of a baroclinic wave.
These
cyclones also appear quite suddenly and are stationary or
move slowly. When an upper trough approaches, these cyclones
often appear to interact with it and develop into mature
extratropical cyclones."(pag. 1011,1012)
They attempt a numerical simulation with
evaporation
from
the
sea
a
moist
model
that
surface and in fact reproduces some of the
observed characteristics, including the small size and the
and
the
evolution
mature stage.
into
allows
a
more
warm
core,
clearly baroclinic structure in the
A 'dry' run of the model appears unable to maintain
the
'intermediate scale' cyclone.
The first systematic study of explosive cyclones is to be found in
Sanders
and
Gyakum,1980.
A
cut-off criterion was established - the
central pressure at low level falling by at least 1 mb per hour for
hrs at 60 N.
24
Since the time and space scales of baroclinic instability
are known to be proportional to f
-
,
10 -
this
effect
was
eliminated
by
allowing the cut-off deepening rate to vary with the sine of latitude in other words the selection was made in terms
normalized
with
local value of planetary vorticity.
the
among other results, that most of the deepest cyclones
middle
vorticity
relative
of
latitudes deepen explosively.
They found,
that
occur
They are predominantly maritime,
cloud
cold season events, with hurricane-like features in the wind and
The
fields.
geographic
western Atlantic and Pacific
of
currents
of
areas
Oceans,
most
in
frequent occurrence are the
association
with
the
actually occurs at the northern edge of these currents, where
is
gradient
maximum,
but
this
correlation
warm
The maximum frequency
Gulf Stream and the Kuroshio.
the
in
is
believed
the
SST
not to be
critical for the phenomenon since another relative maximum of frequency
in
occurs
the
northeastern Pacific where it can be associated with a
flow of polar air over warmer waters,
uniform.
but
where
the
SST
is
almost
The authors conclude that
"The circumstances point to the importance of both
large-scale horizontal temperature contrasts and transfer of
latent and sensible heat from the winter
ocean
into
relatively cold air... Explosive cyclones seem likely to be
mainly baroclinic events, strongly, and perhaps crucially,
aided by diabatic heating."(pag. 1595)
This conclusion is confirmed in successive works, at least for the
explosive
events
that occur off the East coast of the U.S., for which
both detailed individual case studies (Bosart,1981; Gyakum,1983a,b) and
average
properties
are available.
a
vorticity
over
a significant number of cases (Sanders,1986)
The phase of explosive growth is always associated with
maximum
at
500
mb,
which
is often independent of the
surface low in the early stages of development and
-
11 -
comes
to
interact
with
it
at the beginning of the explosive growth (Sanders,1986).
position
of
baroclinic
the
upper-level
is
consistent
with
deepening
waves, and positive vorticity advection at 500 mb is highly
correlated with the
Numerical
trough
The
simultaneous
rate
of
deepening
(Sanders,1986).
simulations of one of these storms (Anthes et al.,1983) have
shown that baroclinic instability was the primary
intensification
but
that
latent
heating
mechanism
played
a
of
early
greater role in
enhancing the development at later stages.
A further statistical study (Roebber,1984)
examines
the
maximum
24-hrs deepening rates of all Northern Hemisphere mid-latitude cyclones
during one year and shows deviations from a normal distribution at
largest
values,
beginning with a deepening rate that, when normalized
as in Sanders and Gyakum,1980, is of the order of 1 mb per
supporting
the
the
hour,
thus
choice of the cut-off value of Sanders and Gyakum,1980
and the conclusion that a physical mechanism
different
from
ordinary
baroclinic instability is at work.
Several mechanisms
have
been
suggested
that
may
enhance
the
development of oceanic cyclones:
The release of latent heat by
stronger
vertical
large-scale
velocities,
convergence-divergence pattern and
therefore
vorticity by the vortex stretching mechanism.
- 12 -
motion
generates
reinforces
the
generation
the
of
Cumulus convection may activate a self-sustaining large scale
circulation by means of a CISK mechanism.
Surface heating may act to increase baroclinicity and
static
reduce
in the lower troposphere, thus accounting for a
stability
stronger growth rate of baroclinic disturbances.
The effective static stability of upward saturated motions is
given
by
the gradient of equivalent potential temperature, which
is always less than the 'dry' static stability.
Replenishment of moisture from the ocean may feed any of
the
above mechanisms.
The relative smoothness of the sea surface compared
to
land
surface reduces the frictional dissipation of kinetic energy.
More recent diagnostic studies (Reed and Albright,1986;
al.,1987;
Liou
individual
and
cases
Elsberry,1987;
in
which
the
Wash
at
different
Chang
et
al.,1988)
depict other
ingredients
(large-scale
baroclinic forcing, high low-level baroclinicity, low static stability,
latent heat release, heat fluxes from the
their
relative
importance
and
time
sea)
are
all
of appearance vary.
present
but
As already
noted for the 'polar low' subset of explosive cyclones no single 'pure'
mechanism
can
account
for all of the observed features.
In much the
same way the time evolution from early to mature stage seems to
a
path
follow
in which elements of both of Pettersen's type A and type B are
present to a different degree.
- 13 -
oceanic
explosive
properties of the environment, as
the
a
to
opposed
from
insignificant
necessarily
features
observed
the
of
general and related to the stability
are
cyclones
of
much
how
In order to understand
point
of
(but
transient
not
of the weather)
view
case
behavior due to the peculiar initial state of each individual
it
seems necessary to identify the 'normal modes' of an atmosphere that is
baroclinically
non-adiabatic
at
and,
unstable
Of
influences.
the
of
latent
subject
we will
mechanisms
which
heat,
to
may
be
for ordinary land cyclones as well, and the air-sea exchange of
and
sensible
respectively.
a
is
time,
above-listed
the
consider only the effects of release
active
same
higher
latent
heat.
is
This
done
in
Chapters
2
and
3
In view of the observational evidence we mainly look for
growth
rate
than
dry
baroclinic
smaller
instability,
horizontal scale, 'warm core' thermal structure, and alterations in the
phase speed of the wave.
representation
of
the
The
we
way
effects
approach
the
problem
A
the
of condensation of water vapor on the
large-scale flow in a simple analytic model is explained
section.
of
in
the
next
full explanation of the phenomenon we consider, including
the role of the interaction with a pre-existing
mid-tropospheric
high
vorticity center, would require the study of an initial value problem this is not within the scope of this investigation.
model,
in
Chapters
2
and
3,
We use a numerical
only for the purpose of examining the
normal modes of the system in a less approximate context.
-
14 -
1.2
Attempts at a theoretical explanation of sub-synoptic scale,
included
have
cyclones
growing
either 'dry' or 'moist' models.
fast
The
'dry' models included non-geostrophic baroclinic instability (uniformly
Richardson number), for possible application to the disturbances
small
(e.g.,Tokioka,1970), where high wind shear and
on the 'Baiu' front
observed,
are
stability
static
to
or
waves'
'frontal
in general
of
(Orlanski,1968), but they were mostly indirect representations
effects
baroclinic
the
in
modifications
such
their
explains
the
inability
Pedlosky,'1964)
temperature
adequately
and
surface
maximum at the
troposphere,
upper
These are shallow modes, with
rapidly
decreasing
height.
with
of
either
or
profiles
represent
the
This
continuous
the
(Charney,1947;
their
behavior.
(Phillips,1951;
models
two-level
models
wind
and
Green,1960)
to
simple
with
Eady,1949;
In Staley and Gall's model the
when the low-level static stability is reduced.
can be useful for the early
mentioned
the
sensitivity to low-level environmental parameters, and
most unstable wavelength becomes shorter and its growth
higher
but
the geopotential and the temperature perturbation
both
of
show
lower troposphere produce instead a
destabilisation of the short waves.
amplitudes
to
only weakly affected by changes of static
are
waves
model
quasi-geostrophic
stability or vertical shear in the middle and
that
the
of surface heating through reduced low-level static stability.
Staley and Gall,1977 used a 4-level
that
low
'shallow
low'
phase
of
rate
modestly
This mechanism
the
phenomenon
in the previous section, but it seems inadequate to describe
-
15 -
the explosive
phase
of
growth.
Similar
results
are
obtained
by
Blumen,1981 in a two-layer model with different depth of the two layers
and different static
stability
in
each,
extended
by
Hyun,1981
to
include differences in both lapse rate and shear between the two layers
; and by Satyamurty et al.,1982 in
dimensional
a
high
vertical
resolution,
two
primitive equation model with 'curved' wind profiles (i.e.
non uniform shear).
The latter study also considered an
including
a
representation
mechanism and concludes that
generating
instability
in
of
explicit
latent
low-level
the
heat
effect
by
release via a wave-CISK
heating
sub-synoptic
heating
is
also
scales.
capable
of
This treatment
belongs to the 'moist' line of models seemingly initiated by Nitta,1964
(with
reference
to
Eliassen,1964 for
proportional
an
the
to
unpublished work by Ooyama and to Charney and
representation
condensational
heating
as
the frictionally induced vertical velocity at the top
of the boundary layer).
forms
of
(Mak,1982;
Sardie
The CISK models have been proposed in
various
and Warner,1983; Wang and Barcilon,1986) and
they generally conclude that the increase in maximum growth rate is not
large
enough
to
account
for
the
explosive
growth.
Wang
and
Barcilon,1986 point out however that:
"besides the destabilisation of increased moisture content,
there are a number of favorable factors for rapid development of
the disturbances, such as reduced static stability, increased
vertical shear, higher latitude, shallower convection, and deeper
moist convergence layer with its top above cloud base...
The
cooperative interaction between favorable factors listed earlier
may create large growth which substantially exceeds the linear
combination of their individual effects."(pag. 716)
-
16 -
Orlanski,1986 uses a numerical
model
to
study
the
development
of
meso-alpha cyclones and shows that localized (sensible) surface heating can
produce a more intense development of short baroclinic waves.
that
suggests
He
the weak stability of the moist atmosphere and release of latent heat
are the primary causes for the explosive growth but the instability of
dry,
pre-storm
the
environment can be responsible for the scale selection and
position of the storm.
He proposes a mechanism for the
rapid
development
of winter storms over ocean surfaces as follows:
"With the passage of a cold front or a land-sea contrast,
cold air can be advected over the rather warm ocean surface; the
heat fluxes from the ocean will then rapidly reduce the static
stability of the atmosphere in the first kilometer, increasing
the baroclinicity and thereby allowing meso-baroclinic waves to
develop. These waves are shallow, having a depth of the boundary
layer and horizontal scales of a few hundred kilometers.
They
can organize convergence of surface moisture in those scales.
With this addition of moisture, the wave will explosively develop
into an intense meso-alpha cyclone."(pag. 2882)
Lastly, Weng and Barcilon,1987
models
by
shift
the
attention
back
to
'dry'
sensible heating and Ekman dissipation into Blumen's
including
(1979) two-layer model and conclude that explosive cyclogenesis occurs in a
complex environment in which many conditions must be met.
All the models that we labeled 'moist', above, are
hypothesis,
that
is
they
environment the convective
(either
through
Ekman
assume
activity
that
in
initiated
a
by
based
on
conditionally
the
a
CISK
unstable
large-scale
flow
pumping or by the secondary circulation associated
with the baroclinic wave itself) can have a 'collective' positive feed-back
effect on the large-scale vorticity.
and Fichtl,1983:
The one exception is the work by Tang
since the static stability for upward
-
17 -
saturated
motions
'dry'
temperature, and for downward motion by the
potential
equivalent
temperature,
potential
upward and downward branches of the secondary
the
separately
treat
they
of
rate
lapse
environmental
the
by
given
is
stability
circulation of a baroclinic wave, using a different value of the
in
parameter
each
Then the effect of release of latent heat is
region.
represented by the use of a smaller static stability for upward compared to
The normal modes of the problem, which have to amplify at
downward motion.
wavelength
the same rate everywhere to be so called, have then a different
in
the two regions, and have to be matched at an intermediate point.
solve
procedure is similar to the way we
some
and
chapter,
following
the
in
on the technical differences are made there.
comments
The physical approach is different, however.
a
problem
the
This
Tang and Fichtl,1983 consider
stable saturated atmosphere - no consideration is given to the effect of
place,
that
Instead we assume
convection.
continues
and
place,
take
to
conditionally unstable, on a time
as
so
motion,
to
large scale flow.
symmetric
instability
basis
(e.g.
scale
whenever
the
faster
much
has
adjustment
taken
atmosphere becomes
the
than
synoptic
a conditionally neutral environment for the
'prepare'
On the
convective
a
of
theoretical
results
on
conditional
Bennetts and Hoskins,1979) and observational
evidence from recent field experiments (Emanuel,1985a;1988) we assume
a
slantwise
vertical
convection,
upright
adjustment occurs, i.e., after the lapse rate has
convective
been reduced to
that
neutrality
the
(zero
atmosphere
is
moist
still
static
unstable
stability)
to
by
slantwise
convection (along absolute angular momentum surfaces), which is then active
until
the
equivalent
potential
vorticity
(
, which is the stability
parameter for this form of instability) is reduced to zero.
- 18 -
The
value
of
the
'vertical'
static
in
stability
state of conditional symmetric
the
neutrality is determined by other environmental factors, but it
anyhow
is
The features of the large-scale flow are determined by the 'dry'
positive.
environmental parameters where there is downward motion, and by the 'moist'
parameters,
which
updraft region.
of
equivalent
vorticity, in the
potential
by Emanuel, 1985b and Thorpe and Emanuel,1985, in
vorticity
the
study
baroclinic
of
instability.
lead to
of a breakdown of the approximation in the limit
The
reformulate the model from the primitive equations, in sect.2.2.
section
that
plays in that approximation (see Hoskins,1975) and it
is extended here (in sect.2.1) to
Suspicions
role
the
association with the semi-geostophic equations, because of
potential
study
the
This assumption has been previously applied to
circulations
frontal
zero
include
last
chapter 2 presents simulations with a multi-level PE numerical
of
model aimed at a more detailed look at the solutions found with the 2-level
analytic models of sect.s 2.1 and 2.2.
As we noted in the previous section, there
is
similarity
a
between
some of the observed features of mid-latitude explosive events and those of
tropical cyclones.
latent
heat)
has
The role
been
of
shown
exchanges
air-sea
to
be
of
heat
(especially
crucial for the intensification of
tropical cyclones (e.g. Ooyama,1969), and even suggested (Emanuel,1986) to
be
their
sole
cause.
A
recent
study (Davis and Emanuel,1988) finds a
strong correlation between the potential for
air-sea
temperature
atmospheric
heating
due
to
and moisture contrast and the amount of pressure fall
in mid-latitude explosive oceanic cyclones.
In Chapter 3 we introduce this
source of heat into the models of chapter 2 and examine its consequences on
-
19 -
the development of 'moist' baroclinic waves.
- 20 -
2.1
We begin the investigation of this
problem
with
model formulated in the semi-geostrophic approximation.
Eady-like
model
approximation;
vertical:
(uniform shear
f-plane)
the
instability.
minimal
that
and
static
SG
Boussinesq
we will solve only on two levels in the
discretisation
that
is
suggested
by
retains
except
the
baroclinic
for
the
basic
form
of
The use
convective
chosen, because potential vorticity appears naturally
in the equations in place of static stability.
finite
We consider an
gradient that balances the vertical shear.
approximation
parameterisation
of
2-level
The perturbations are assumed to have infinite meridional
temperature
of the
2-D
stability;
scale, so that all the y-derivatives are zero,
state
a
amplitude
effects
(for
It also allows
a
2-D perturbations only) since the
equations transformed in 'geostrophic space' become linear without
need
of
any
'small
amplitude'
study
the
assumption (with one exception to be
mentioned later).
We then start with
the
equations
in
the
approximation, that we write
+ %X
U-O L+
~~t3
77-If~y
e,~t
N
(3\
'f
q/.t8
- 21 -
~~u
4L) 0
-
OCr
geostrophic
momentum
where the symbols have their usual meteorological meaning
wind
has
been
used
in
place
of
and
thermal
the hydrostatic relation to avoid
explicit reference to pressure.
Consider a
0OU.--
/e*)
steady
'
W
solution
and
The
)
satisfying
to this mean
perturbations
y-independent
o=0
).
state (which implies (
9CC,
equations
for
the
perturbations
are:
-lz0
We here derive the linearized equations first, and then
that
the
coordinate
same result.
we
will
show
transformation to geostrophic space leads to the
If we linearize at this point we get
)
LLe
CZ .3)
0
for the momentum equation, and
for the thermodynamics.
: 0-ee+
Using the first of (2.2) this can be written
+%W(
- 22 -
)zo
'25=
(
or, recalling the definition of potential vorticity
fl'
-*
t&
~,t
qd:
lo
A)
1C
L& :is
so that
Q
er8
the potential vorticity of the
mean
state
C2.4)
o-°
From the continuity equation we can define
secondary
circulation
,
such
that
a
streamfunction
Wi --J
and
0
for
-Z
the
;
system (2.2) reduces to
CO~C -(C
which
A-
is
a
(L /o@
closed
system
for
the
unknowns
,
with
as a diagnostic relation for the ageostrophic component
of meridional velocity.
On the other hand we could apply to (2.2) the transformation
X-,, Iz
)q)
23 -
-23 -
I
3x-o
I
which immediately gives
It is easy to see that these equations imply
so we identify the quantity
coincides
as the potential vorticity; this definition
(see
o(Ro)
the
GM
McWilliams
To
recover
the
physical
Equations (2.9) are
depends
conserved following
relation for "f
the
on
individual
assumes the form
~
-p8XI
- 24 -
I,
solution
itself,
of
fluid).
parcels
to
d is conserved, if q1
in particular they are linear, because
t(X'tT)
identical
formally
is chosen to be uniform at some initial time (if
then
the
quantities
inverse transformation from 'geostrophic' to physical space
shall have to be applied.
(2.8);
to
and Gent, 1980) and it is then consistent with
approximation.
non-linear
(2.6)
with
is
not
although
The
uniform
it
is
diagnostic
We derived the above set of equations for an adiabatic system, in which
the
dry
(cB4
entropy
-
individual
parcels
Boussinesq
approximation)
of
is
conserved
fluid: (t/Jt)
(/W(d
moist air as long as no changes of
substitute
GV
realistic
ratio).
When
=
AJ*
0
O
..
;
phase
occur
the
or
this
is
motion of
(within the
still true of
(strictly
we
should
but the difference is of order one percent for
atmospheric
values
of
potential
temperature
and
mixing
condensation occurs the source term that appears in the
thermodynamic
defining
6
for
following
equation
-
(*)
can
be
an equivalent potential temperature
accounted
-
-
for
by
(*) that is
conserved for all motions in which the only heat source is the
release
of latent heat of condensation; for all ascending motions in our model,
that we assume saturated, we then use the equation of
:9/dt
'O
instead of Cdd/
:0.
conservation
of
Therefore the thermodynamic
equation in regions of ascending motion is written:
where
so that
7
temperature
is
has
the
absolute
with
vorticity.
The
equivalent
potential
the moist entropy the same relation that
(*) Both these definitions use the approximation in which
replaced by a constant-.
0
- 25 -
has
i
is
with dry entropy; and there exists a relation
between the differentials at constant pressure of dry and moist entropy
for a saturated parcel of air (see Emanuel,1986).
The above is exactly
true at constant pressure - we assume that it holds at constant
and
't
write
'OM .
adiabatic lapse rate and fr
/ C--
Here
the moist adiabatic lapse
then rewrite the equation of conservation of
e
height
is the dry
rate.
We
can
(2.11) as
or
As the quantity
(equivalent potential vorticity) is conserved by the flow this equation
is
entirely
similar to the thermodynamic equation in (2.9) and we can
write the two of them as
+
wn
updraft
regions of descending motion.
- 26 -
(13)
regions
and
/
o)
in
Two observations have to be
treatment
of
saturated
and
made
unsaturated
adiabatic lapse rate is in general
variables,
and
so
a
thermodynamic
conserved
where
only
position,
similarity
of
First, the moist
of
the
even
at
mid-level,
and give it a constant value.
thermodynamic
though
the main
so
we
ignore
Second, 7d
all
is
not
condensation occurs - the effect of release of latent
heat is to create an anomaly of
region
motions.
function
of
the
In the two-level model below we will solve
equation
variability of r
a
function
variability is with height.
the
concerning
d
that is of no concern in the updraft
itself, because there 10 is the dynamically relevant parameter;
however this anomaly of
fd
ascending
into
motion
and
can
be
the
advected
downdraft,
out
of
where
the
region
it is dynamically
significant.
We ignore this effect as small in the following - we
regard
approximation
this
as
a
of
can
linearization of the term
in
13
(2.gI, and this is then the only linearization that we
make
in
order
to
are
derive a linear system of equations.
forced
to
On the other
hand, by writing (2.12) as
and deriving the potential vorticity equation from here, we
get
(e.g.
Thorpe and Emanuel, 1985)
Now, again, in the 2-level discretisation the thermodynamic equation is
solved only at mid-level;
is
; the z-dependence of
is initially uniform by assumption, and so
i
has
4
- 27 -
already
been
neglected
and
its
contribution
anyhow
of (2.14
the r.h.s.
and
is
is
small as
~
; the only remaining term on
--O0
7
, but if
is
taken equal to zero
at
top
bottom boundaries (where the physical boundary condition is w = 0,
i.e.
= 0 orT = const.) then T
= 0 at mid- level (*)
is
value consistent with the two level discretisation, and ?
the
only
at mid-level
is conserved even in the presence of condensation, because the
use
of
only one level as representative of the thermodynamics is equivalent to
the assumption that heating is vertically uniform does not change potential vorticity.
consider the
solutions
2-level
in
solutions
geostrophic
condition
that
For all purposes we can therefore
derived
space.
a
below
as
finite
amplitude
We get exponential growth at finite
amplitude because the infinite meridional temperature gradient provides
an infinite reservoir of APE and the perturbations do not interact with
the basic state:
(**).
In
any
they are added to the mean flow and do not modify
it
system with a finite meridional scale the amplitude of
the perturbation would of course be bounded as
t->c>
and
not
grow
indefinitely.
Taking the basic wind to
be
zero
at
mid
level,
(*) Note that this is true in geostrophic space.
transformation
derivative of T
not
necessarily
extrema,
is
applied
vertical
lines
become
so
that
the
When the inverse
M
lines,
so the
along M surfaces will have a maximum at mid-level, but
so
where both
the
and
vertical
,
or
in both coordinate systems.
- 28 -
derivative.
and
'
However the absolute
are zero are
at
mid-level
unstable
Eady
waves
2-level (shown in fig.
are
stationary,
we write the equations for the
2.1) as
Ax- +_(
where it has been assumed that the time dependence of the
IrT
S,
the
boundary
been applied,
condition
_
(.-
solution
is
= 0 at top and bottom boundaries has
is the thermal wind relation for the
base
state, and
1
-
(**) Notice however that this is so
physical
space
in
geostrophic
space
only.
In
the self-advection of the perturbation, represented by
the transformation of coordinates, introduces an algebraically
growing
term that leads to the frontal collapse, after which the transformation
is no longer valid (and the GM approximation itself breaks down
this occurs)
- 29 -
before
We will solve the system for
the
unknowns qllJ\
regions of ascending and descending motion:
I
r
separately
in each of them the system
is linear with constant coefficients, but one of the coefficients (
has
different
values
in the two regions.
in
)
The solutions will then be
matched at the interfaces to insure continuity of the relevant physical
quantities.
The use of a zero (or small)
motion constitutes the parameterization of
convection
on
? in
the region of ascending
the
effects
of
the large scale flow that we discussed in Ch.l.
(2.1 ) can be written in non-dimensional form as
where (denoting dimensional variables with an *)
I
slantwise
fi
-e40
- 30 -
*
System
//-0/I
1)------------------
h
2)
----------
Vg=V1
Vg= /2
-U-
=0/
Fig.2.1 - Structure of the 2-level model , and
levels where variables are defined.
7
Region 1
Region 2
t
I
-L 1
1
0
L
1
-L2
2
Fig. 2.2 - Subdivision of the model in a 'moist'
and a 'dry' region . Horizontal coordinate X is
defined separately in each region. Solutions are
matched at points A=B and C.
-
31
-
Note that the horizontal length
Rossby
scale
( A/
is
or
,
the
radius of deformation in the SG approximation, where the static
stability
has
dimensional
been
quantity
replaced
by
potential
vorticity.
The
non
q is the ratio of the stability parameter in the
region of ascending motion to the dry potential vorticity.
Its
value
is
From (2.16) an equation for
/
can be derived
and the other variables are given in terms of
Equation (2.17) is valid everywhere,
assume
that
the
for
a
by
general
) .
We
now
solution is periodic over a length 2L and divide the
domain in two regions as in fig.
2.2, and we require the
solution
to
have positive vertical velocity in region 1 and negative in region 2.We
then specialize (2.11F) to q = const.
+(Z6,)2.,15)
X
to get
-
- 32 -
that is valid in the interior of the two regions, with q = 1 in
2
and q = r in region 1.
and A (=B) of fig.
region
The solutions have to be matched at points C
The appropriate matching conditions will
2.2.
now
be derived.
The derivation of the matching conditions is slightly different in
physical
space, for the linearized system, or in geostrophic space for
the full system, although the resulting conditions are
show
here
the
we
linearized
version).
study
in
this
First
section
we
(see
require
continuous, which implies ~Ir continuous, or
Since
we
will
*-)t
that
Appendix
the
2.A
f
the
to
be
and J4 continuous in the
'
and
for
transformation
solve the equation (2.13) for t
write those conditions in terms of
gives
We
argument for the finite amplitude model in geostrophic
space that
2-level.
identical.
its
we want to
(2 .11)
derivatives:
have to be continuous at the
ad
and
condition
in
interface.
The usual approach to the kinematic matching
this
of problems is to require the displacement of the interface
kind
This is right if the interface is a
to be the same in the two regions.
material
surface,
so
that
the
fluid
cannot
go
(e.g.,Lamb,1932), but this is not the case in our system.
different
not
so
We
argument based on the equation of continuity, i.e.
velocity normal to the interface has to be continuous the
it
through
if
adopt
a
that the
this
were
requirement of mass conservation would generate a jump of
the tangential velocity along the original interface, and so generate a
new
surface
of
discontinuity,
orthogonal
to
interior of the regions where the solutions are
- 33 -
the first one, in the
supposed
to
be
well
behaved.
Since we are thinking of a vertical interface in geostrophic
space, it will be an M line in physical space; we can
write
the
unit
vector normal to an M line as
in physical coordinates, so that
--
W
(2.2o)
in geostrophic coordinates.
N7
We have already required the continuity of
in geostrophic space, and we note that the continuity of a function
across a vertical line implies that all its derivatives along that line
are
continuous too.
1
the continuity of
pressures
at
the
-----;_
(2.16) by
(-jf
Therefore A5
, or t
two
is continuous and (2. 0) reduces to
itself in the 2-level.
sides
Finally we need the
of the interface to be the same:
( in non-dimensional form) we multiply
the
to get
and integrate across the discontinuity in geostrophic space
Now we make use again of
jT -7
or
-+r
-
-c
34 -
to write
since
first
of
Therefore the continuity
StIX
o ,
-C
because
of
now
is
equivalent
we have already imposed k
same result is obtained for
can
PI
p
to
the
requirement
to be continuous.
from the second of eq.s
express this condition in terms of T
The
(2.1() (*).
We
and its derivatives by
integration of 2.14 across the interface:
-t
'C
The term in the first square bracket is zero because
i2(fJ-)
(see
eq.
continuity of q
I II
.
of
to
is
equal
and so we are left with the requirement of
(2.21)
C+ I 0 X
2
consider
that
we
have
a
requirement
on
interface conditions (2.21).
equation
(2.13)
or
(*) note that while we always write 'continuity', what we
the
value
of
the
function
interface is equal to the value on
guarantee
that
the
the
We assume that w has a definite sign when
we choose to apply the 'moist' thermodynamic equation to region
that
the
the solution, namely that w > 0 in region 1 and w < 0 in
region 2, that is not included in either the
is
to
To summarize, the four matching conditions are on
T)( I C V)
The next step is
structure
2.11)
it
the
evaluated
other
on
side.
1
really
and
mean
one side of the
This
does
not
function does not have a singularity right at the
interface, like a 8 -function, that is not
detected
itself but results in a finite jump of its primitive.
have physical reasons to require the primitive
have to impose it as an added condition.
- 35 -
to
be
by
the
function
Therefore, if we
continuous,
we
one to region 2, but if we solve the problem in the form it
'dry'
the
requirement.
this
satisfy
general
not
do
has been posed up to this point we will get solutions that
in
We would have then to look at the
that
structure of the solutions and retain only those
are
acceptable
from this point of view
.
%
4
0 with
O
for
solutions
, n=1,2,...
WAiV$
(*).We
interfaces
the
0
the
that
at
=
then
positive;
is continuous
o
0
too, so it has to be zero at the interface - again
so
modes
change sign at the
interface condition guarantees that
second
the
to
want
unwanted
fX has to change sign as well, because q is
interfaces, so
but
We
argument.
following
the
with
the
We can however filter out some of
replace
the
everywhere
condition
When this condition is imposed we
.
r=l,
i.e.
know
the 'dry' 2-level model, will be
( 4L being the wavelength and Ll=L2 in fig 2.2).
The only solution that we consider acceptable is the one with n=l, that
does not have any zeroes in the interior of each region, but
at
the
between the two regions.
interface
one
only
We will then proceed from
this solution at r=l with small decrements of r, using the solution
each
an
as
step
close to
zero.
solution
is
that
initial guess for the next one, down to values of r
Another
observation
both eq.
odd or even derivatives of
be continuous the jump in t
that
helps
in
the
numerical
2.11 and the conditions 2.21 imply either
the
(*) in other words the jump in I
is zero.
at
streamfunction,
but
not
both
kinds
9
to
conserves the sign, then for
must conserve the sign as well, unless it
And we discard the former.
- 36 -
together.
In
this
situation,
and
systems in the two regions as shown
separate
problems
for
the
in
2
two
fig
symmetric (
part of the solution, defined as
A
using
different coordinate
2.2,
4')
we
can
write
two
and antisymmetric ( A
(If{)g$
'Q.)-
'-)
purely symmetric solution will have a zero of w in the interior, and
so cannot be used.
We cannot exclude solutions with a small (not large
enough
the
to
change
sign
of
w)
antisymmetric one but we note that t
dispersion
relations:
combined
and
will
of
points
the two dispersion relations intersect.
L
following only
:
have
two
different
solutions are then possible (if they
exist at all) only at a discrete set
where
symmetric part superposed on the
this will
give
in
parameter
space,
We will consider in the
all
the
eigenvalues;
the
possible degeneration in the form of the eigenfunctions for some values
of the parameters would probably have
to
be
resolved
invoking
some
other constraint not included in this model.
We are now ready to proceed from a general solution (*)
in the form
(lit)
()
0(
=
CL+
(Z t)
X
L%,
X
t
j
(&1
(*) There are a few degenerate cases in
general
solution.
The
relation
While the solution
.01
.,
e.
4t,
2
which
this
is
only case of practical interest is
which is the maximum growth rate of the dry (r=l)
dispersion
(2.1l)
Ila
+
+ k
to
model,
not
2:2-J-
because
the
,
the
found numerically takes this value at two points.
, exists anyway at these two points, we did not
examine the consequences of adding the extra terms k
- 37 -
t4.(x) .
q
Here we allow the wave numbers
shown
to
be
real
antisymmetry reduces
-
(
and
(see
Appendix
"
to
-
X..
to be complex, while T
2.B).
Now
the
can be
requirement
.(tL
of
I
and the conditions w = 0 give
i%(,,
L
4C' )L t
so that
_ u,) ILI
e ,4iCtjz)L
d~(~'~)L2.
so that
and the remaining matching conditions on
has
definite
symmetry
j'
fl'9',9|
properties,
need
to
be
(that,
because
applied at one
interface only) can be written as
31 chIL,
a cha Lisha Ll+shlL1
P2 chP 2L 2
a 2 cho2L2 sha2L22+sh 2L2
02 ch
q alchaL
l1chiL,
1
2
q(
a L
2
X2chX2L2 sha2
2h
achOl 1"lchcxLl+plch o
L22
Ll
a2cha2L2 c
2
2L3-3
2s p2L2
b2 = 0
h2L2
This system has non-trivial solutions if the matrix has rank one, which
requires
that
two
second
order determinants be zero.
- 38 -
Having proved
that 1
is real the determinants are real or pure imaginary.
In either
case
we
6 X
.L L,'
. Two of them are then free: we choose L
and '
L2.
we treat as independent variables, and solve for the eigenvalues
that
6 and
have two real conditions that relate the four real parameters
.
We begin the presentation of the results with
(i.e.
q=l
everywhere).
is
the
classic
explicitly
result
for
an
the
rather than
has
to
r=l
2-level
.
model
dimensional
(e.g.
quantities
This explicit solution was used to test the
performance of the numerical procedure.
wavenumber
case
eigenvalue I-
f-plane
Pedlosky,1979 - eq.(7.11.13) ) except that the
involve
dry
In this case (2.18) can be solved for plane
waves of wavenumber k, giving
This
the
be
real
-
Note that
in
this
case
the
if q is uniform there are no vertical
boundaries and the solution has to be bounded at infinity.
In a
model
where q = const.< 1 everywhere we would get
for the dispersion relation.
of
both
'axis by a factor
In a flk plane this is just a
.
If we insist on real wavenumbers in a
model like the one presented here, with one region of q=l1
q=r,
each
one
will
have
appropriate scale factor.
the
the
stretching
dispersion
relation
and
one
(2.23) with the
We would probably be unable to
satisfy
matching conditions (2.2t), but the minimal requirement that I
the same in both regions will give solutions
growth
rate
of
the
dry
model, i.e.
- 39 -
that
have
of
at
most
all
be
the
the least unstable part of the
whole
domain will determine the properties of the
Tang
instance
and
We
Fichtl,1983).
able
are
to
for
(see
solution
this
overcome
-
limitation because we are in fact allowed to use complex wavenumbers
when the model is not uniform in x the normal modes are not necessarily
plane waves.
In this model we solve (2.13) on a bounded domain and
we
do not have to worry about the behavior at infinity.
Before going to the actual numerical results we can
priori
considerations
on
the
effect
of
reducing
classical baroclinic instability theory that
scales
with
the
Rossby
each
region
to
be
).
most
a
We know from
unstable
mode
We can expect the horizontal structure
defined
by
own
its
Moreover the growth rate scales with
2.23).
r.
some
radius of deformation (that in this model is
defined with q in place of i
in
the
make
q, i.e.
~~:/,.
when q is uniform
(see
also
We could use as a preliminary guess for the growth rate in this
model a linear combination of
d
at q=l and
$
at q=r, weighted
with
the relative area of the two regions, i.e.:
Lt C t LZ.4
2
This crude estimate turns out to be a rather good approximation to
behavior
of the actual solutions as long as r >.l, while for smaller r
the growth rates obtained numerically ( shown in fig.
than
that
(table
(2.22)
for
2.3) are
larger
2.1 gives comparision of solutions of 2.22 and 2.24
for one value of L2 ).
solve
the
Although (2.13) is singular for
D-90 we can
very small r - the curve labelled r=O in fig 2.3 is
obtained asymptotically, and confirmed by
-
40 -
numerical
investigation
of
Table 2 .1
L2 = 1.8 ; G1 = .59
2
Or
.5
.3
.2
.1
.01
0.
.63
.71
.69
.69
.46
.36
.24
.07
.55
.45
.78
.84
.94
1.24
.76
.82
.90
1.07
1.47
1.18
.32
.10
0.
-
41 -
1.6
1.4
r= 0.
1.2
r = .01
1.0
r = .1
(O
.8
.6
r-.5
.4
r=.
.2
0
0
.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
L2
Fig.2.3 - Growth rate 6 versus half-width of the downdraft L2
for different values of r; r=1 is the dry model; r=0O is obtained
(see text)
asymptotically and confirmed numerically for rAk.1
-
42 -
5.0
1.6
L 2 =2.
1
1.4 -
=3. I =4.
=1.5
-
r =0.
1.2 -
.01
1.0 -
0
\
2
4
\
6
\
8
10
12
14
16
18
LTOT
Fig.2.4 - As 2.3 but d' vs.
total wavelength LT. TrZLz4J+A);
dashed
lines join points of constant La , i.e. points that lay on vertical
lines in fig.2.3, thus showing the reduction of total wavelength due to
the narrow updraft
-
43 -
20
(2.22) for r as small as
\O
.The approximate
dispersion
relation
at
r=0 is
(see
Emanuel
et
al.,1987,
Appendix
C
for
the
derivation).
An
approximate solution of this is
2 L Z,
2 Ls,.v,
L
valid when the second term in parenthesis is <<1, and so
L2 > 1.5.
This
illustrates
one
of
fig.2.3 from (2.24) - the growth rate
L-' 0
and
not
to
the
to
L2 > 3.
of
the oscillation around
i£
finite
value
for
r
Lz'(f/8t2j )2T but
the
does not exceed .01 for any
The absolute maximum is the first one,
1.95, which differs from
a
The oscillatory part gives a
sequence of relative maxima of growth rate for
amplitude
for
qualitative differences of
tends
zero as d' does.
anywhere
and
occurs
at
L2
=
by less that .02.
8
The abscissa in fig 2.3 is L1,
which
is
1/4th
%
--.
Thus
of
the
total
the half-width of the w<0
wavelength
in general, as the width of
the
most
unstable
mode,
the
at
r=l, but
updraft
which
has
I
decreases
an
LZ
increasing as r decreases has a shorter total wavelength, as
fig.2.4 where the same curves are redrawn as functions of
-
44 -
region,
Y
with
slightly
shown
LT*T .
in
Fig 2.5 shows the horizontal structure of the solution for
and
The relative amplitude of the various fields are shown,
=1.8.
L2
r=0.08
A definite amplitude must be
with an arbitrary normalisation.
assumed
in order to apply the inverse transformation to physical space shown in
so
the
is
evident,
and
due to release of latent heat.
The
in
the
The narrowness and strength of the updraft are
fig.2.6.
net
heating
average
effect of vortex stretching
stronger
positive
already
is
in
evident
fig.2.5
vorticity at low level compared to the upper level,
is
that feeds back on itself when the inverse transformation
Since
applied.
in the inverse transformation from geostrophic to physical space
areas of positive
vorticity
are
vorticity
stretched
are
compressed
and
(horizontally) this
frontal collapse occurring at the rear of the
areas
of
asymmetry
updraft
at
negative
results
low
in
level,
while the upper frontal zone, situated in a weak subsidence region, and
thus less reinforced by stretching, is still only
shear
zone.
a
broad
horizontal
The positive vorticity at the rear of the updraft and the
negative vorticity ahead of it translate in a very sharp gradient of
w
at the western edge of the updraft (see fig.2.6 where a large amplitude
of
the
solution
transformation).
is
assumed,
While
these
to
emphasize
results
the
effect
of
the
are reasonably satisfying, in
that they show more adherence to observed structure and deepening rate,
in
contrast
to
dry
baroclinic modes, there are a few remarks on the
validity of the approximation that should be taken into
departure
of
the computed 4
r
-
The
from the estimate (2.24) based on length
scales suggests that the scaling itself may be
small
account.
inappropriate
at
very
indeed the model predicts an updraft of zero width at r=O,
-
45 -
and
Thorpe
(Emanuel,1985;
that the semi-geostrophic approximation breaks
conclude
Emanuel,1985)
even
frontogenesis in an
of
studies
model
this
to
similar
environment
Previous
longer.
any
hydrostatic,
not
and
and before this occurs the system will not be balanced,
down on the approach to (slantwise convective) neutrality and derive
lower
from eq.s (2.1).
made here, e.g.
term
is
A similar argument can be
on the admissible values of q.
bound
region
in
o(1)
in
2;
Since this is non-dimensional
region
a
1
X A.
however
each
so the
following relations apply:
The ratio of advection of ageostrophic motion to
motion,
of
that
geostrophic
which has to be small for the validity of the GM approximation
is (in a linearized sense):
the latter relation being derived from 2.10 and
0
2
the
and
Since
breaks
approximation
down
t
when
ve
This relation
.
means that the Rossby number of the updraft region, defined either with
its
width
the
inertia
neglected.
with its own stability parameter, becomes o(1), so that
or
of
For
the
circulation
secondary
these
reasons,
-
and
46 -
cannot
be
consistently
encouraged by the fact that the
a smaller growth rate than predicted here when
show
In
a
we will
PE
dynamics.
PE model we expect the growth rate of baroclinic modes to become
Ro
smaller as
be
must
Stone,1972)
(e.g.
increases
energy
potential
wind:
indeed
P--v0
include
in the next section extend the present model to
do
2.3
numerical experiments that will be discussed in sect.
available
the
because
partly used to accelerate the ageostrophic
the
the kinetic energy, that for a balanced model includes only
geostrophic
terms will have all the non-geostrophic component as well.
In particular we will have a prognostic equation
for
the
zonal
wind
that is missing here.
As a final observation, this model could have been written, in the
version, from the quasi-geostrophic approximation, using tj
linearized
instead of q (e.g. Tang
justification
that
indicate
vertically
for
Fichtl,
and
does
q-IO
convective
adjustment
occurs
for N
on
M
:
physical
observations
surfaces,
not
X as scaled here is a Rossby number for
indeed
the perturbation, and when
work
the
However
The two theories should approach each
(e.g.Emanuel,1988).
other asymptotically:
not
1983).
the M surfaces become vertical.
-~
-
47 -
S-.4
/
I'
/
2
..- 4.4
e
\
/
/
-I
-2
/
/0
I
2
3
/"
-.4
x
r=0.08.
Bottom:
same but
for
meridional
r-
-
velocity
v
and
-.-
-4
-t
Fig.2.
-2
3 Top:
Dimensionless
-r
modified
O
and
I
as
2
X -
function V
of
the
3
Fig.2.3- Top: Dimensionless 4
and 9' as function of the
geostrophic coordinate X, for the most rapidly growing mode when
r=0.08. Bottom: same but for meridional velocity v and modified
geopotential +i/C+vr/t
at upper and lower level
-
48 -
N
3 7
I/
/
I
I
I
2-
\
w
W
-
-
-
0-1
-
II
I
i
X-
SI
2
V
-2
- 1
0
Fig.2.( - Same as 2.5 but
low-level vorticity is 10f#
-
1
in
49 -
physical
2
space
when
the
maximum
2.2
In
this
section
primitive equations.
we
reformulate
there
is
2-level
model
from
the
The motivations for this step have been discussed
at the end of the previous
because
the
no
section.
The
transformation
coordinates' for the PE (*).
model
is
comparable
now
linearized,
to the 'geostrophic
The equations for small 2D
perturbations
to an Eady basic state are
0.
00
(22and)
Q'9t
as
W
in
the
'.fJ-
-
preceding
The
CIaC
. /I
and
C,a K4-/t.
Therefore the x-momentum equation becomes
%
J'r Z 0
(*)
section
X="
=-X/to
transformation
2.z C
l~_
e2~~+
)
/
-
would
take care of the y-momentum and thermodynamic equations in the same way
as (2.R ) does in the GM
x-momentum
equations
approximation.
however
assume
transformation.
- 50 -
The
a
thermal
wind
and
the
very complex form under this
that we solve for I~4,
then
are the counterparts of (2.3) and (2.5)
written on the r.h.s.
respectively,
and
terms
the
are those missing in the GM approximation.
From these two equations we can derive, for the continuous
model,
an equation for
The corresponding GM equation is
(compare Eady,1949).
(4 -.
in which
)
appears as a stability parameter in place of
The two-level discretisation is done as in the preceding
this
implies Vo
at mid-level.
assumes the same form as in GM.
in
the
~~{~
this
stability
"
'
parameter
section:
Therefore the thermodynamic equation
However, the cancellation of
comes
not
from
in the momentum equation and
we
the IA
this term but from the
are
then
preserving
charachteristic of the continuous PE system in the 2-level, as we
- 51 -
will see from (2.32) below.
we have
Using again
2
I\s;+
Cc+ i~ D,)
~t~ ( ca,~L
"
0,.
eolt
-
h
0 4;
set
Bo qy~
g,
( a
-
I,
where
-
(230o)
a.
-q,
=0
=O
(JZ_-
t)
With the same scaling as in sect 2.1, and defining
C6 +k) NJ
Q
2X
S+
.i
2-
and an equation for
q
(2.31)
can be derived as
(2.3z)
wind field given by
with the %4:
i
IC-Yr)
+ Iq+ aOL) ?
2a rti)
C+~f\yu+
+~~\~
+ 2
2
- 52 -
60+1
t2
.2 .33)
Ot~s
The term
+
;:Q
N Z.
We
-
in (2.32) is the non-dimensionalisation of
use q explicitly for comparision with the results of
the preceding section, while a
would
be
more
appropriate
formulation
to
compare
with
with
other
non-geostrophic
baroclinic instability
studies
formulation
have the Richardson numbers in the two regions,
say
we
L4 and
=C'
|
would
,
/
as
(e.g.
Stone,1966).
non-dimensional
-
and
6:
I2
/
parameters;
"
(see Appendix 2.C).
longer:
can
When ,L
O
the limit
the
here
The -1
.
allows us to recover the GM approximation by letting
r
In
use
0o -
O
at fixed
t%-O is not singular any
(Z%>O.
We
examine the behavior of baroclinic waves in a saturated
environment even when the lapse
rate
unstable,
not
as
we
in the radicand
in fact we can find solutions for allt>-Ui.e for
therefore
latter
long
as
it
is
is
conditionally
symmetrically
vertically unstable.
We are not,
however, studying symmetric instability; we will use a finite but small
Ro,
so that the dry region is always stable to symmetric perturbations
- also note that the definition of Ro won't allow
only
(4
41 ;
moreover
we
consider perturbations of infinite meridional extent, so, even in
region 1, where symmetric instability is
possible,
we
only
consider
modes on the 'baroclinic axis' in the terminology of Stone,1966.
The same arguments used in Appendix
conditions
=0
2.A
lead
to
the
interface
and the quantities
being the same at the two sides of
- 53 -
the
interfaces.
Everything
then
proceeds
2.1 with the exception that we cannot prove that
as in sect.
6 is real.
is
It
to
found
however
real
be
the
by
numerical
calculations (see Appendix 2.D).
Table 2.2 shows the behavior of I'
values
of
r
sect.
2.1.
while
the
and Ro.
, :1.i
and X at
for
various
Recall that Ro = 0 is the GM model presented in
It is apparent that the growth
updraft becomes larger.
rate
decreases
with
This behavior can be understood if
we consider that the stability parameter here is again tJ , which
appears
both
in
the
time
scale
Ro,
and
then
the length scale for the most
unstable mode, and increasing the Rossby number (i.e.
the
shear)
at
fixed q, actually increases the BV frequency.
When going into the negative r region the g.r.
seen
in
fig
obvious.
system
2.7.
increases again as
The relevance of the 'negative r' solutions is not
The hypothesis on which this work rests is that the time
spends
in a
<0
synoptic scale motions.
the
state is much shorter than the time scale of
4 -D
While this is certainly true of
states,
the typical doubling time of symmetric instability is of the order of a
few hours (e.g.
'much
shorter
Bennetts and Hoskins,1979) and this
than'
the
time
it
not
order
of
twelve
hours.
is conceivable that the developing cyclone will, at some
time during its growth, have to deal
instability
strictly
scale of explosive cyclones which, in
extreme cases may double in intensity on the
Therefore
is
to
slantwise
with
convection,
an
environment
of
slight
especially since the stability
parameter q is itself a conserved quantity
that
can
be
modified
diabatic or frictional processes only (in a saturated environment).
- 54 -
by
TABLE
r
: -. 08
-. 05
---
2.2
-. 03
-. 01
---
---
.02
.3
1.14
r : -. 08
1.00
-.05
.07
.01
.1
(1.41)
1.24
1.26
.5
1.
.94
.69
.58
1.23
.94
.69
.58
1.23
1.15
.92
.69
.58
.87
.67
.57
.80
.64
.56
.5
1.
1.19
1.08
1.04
1.01
.95
.92
.90
.89
-.03
-. 01
.02
.3
0.
0.
.01
(0.0)
.07
.24
.63
.06
.07
.24
.63
.07
.10
.25
.64
---
.0
.12
.14
.16
.29
.65
.14
.18
.21
.22
.24
.34
.67
- 55 -
This model does not allow the base state to vary with time on the scale
of
the
perturbation
or shorter and so the solutions at negative r do
not necessarily represent the behavior of
adjustment is taking place.
the
system
our
instance.
hope
that
a
'slow'
However, the simple fact that the trend to
higher growth rates and narrower updrafts continues
strengthen
when
beyond
r = 0 may
no major qualitative changes occur in that
The numerical simulations presented in the next section will
confirm this assumption.
The sensitivity of the system is greater near the singular
so
that
a
change
in
Ro
at
r = 0 produces dramatic changes in the
eigenvalues, while a similar change at r = .1 has
(see fig.
2.8).
so
little
Note that the stability parameter is
and Ro appear in different combinations in the
(2.32),
point,
other
9+
consequences
P.o
but q
coefficients
of
the PE solutions are not simply a translation in parameter
space.
The cut-off wavelength appears to be independent of r, as
in
GM
(see
fig.2.3) but it depends on Ro.
expression for this from the r = 1 case.
and
setting
L
- =
(I
60=
z
in
.
(2.32)
we
it
was
We can derive an analytic
Assuming plane wave solutions
get
k
I+/tg o
Table 2.3 gives the values of the
)
or
cut-off
L2
This further, although weak destabilisation of the short waves
by
for the Rossby numbers shown in fig.
2.8
reducing Ro can also be seen in fig.2.8.
The Rossby number at constant
q is proportional to the vertical wind shear so
- 56 -
that
an
increase
in
non-dimensional
C
for
scale factor for time.
a
longer waves is easily offset by the smaller
Only the shortest waves, whose
6
increases by
large factor (especially those that go from 6d.0 to a finite value)
actually show a larger dimensional growth rate when the
Rossby
number
is reduced
TABLE
Ro
2.3
cut-off L2
0.
1.1107
.1
1.1163
.2
1.1327
.3
1.1596
This effect is entirely due to keeping
shear,
which
dimensional
implies
cut-off
qd constant
and
changing
the
a change in the Brunt - Vaisala frequency.
The
L
is
and
doef not
depend
explicitly on the shear, except in the sense that the BV frequency of a
base state with constant potential vorticity increases with
vertical shear
(*).
As in the GM
destabilisation
increasing
case
of
sect.
2.1
there
is
however
a
'true'
of short waves when the width of the updraft decreases
as we move closer to the singularity.
-
57 -
Table 2.4
shows
LTor. for
the
marginally stable wave at two values of r.
From the preceding
discussion
it
natural
scales
with q.
In this model we insist on
because
we
to
want
use
to
is
fairly
obvious
by
its
N
with the PE are those defined with
study
the
the
use
behavior
of
of
potential
equivalent
potential
vorticity
the
and not
vorticity
baroclinic
conditionally symmetrically neutral environment, which is
identified
that
waves in a
most
easily
rather than by a
(*) To accompany this there is a slight shift of the most unstable wave
to larger L 2
as Ro increases - both these effects can be understood in
terms of the simplest heuristic model of baroclinic
slope
instability.
of the isentropes in the meridional plane is 1,a/
JP/
The
and
the slope of the trajectories of the perturbations (at small enough Ro)
is, vi
L
.
When
the
Rossby
number
(i.e.
the
increased at fixed N the secondary circulation
same
amount
as
is
vertical
shear) is
reinforced
the slope of the isentropes, because
L
N
by
,
so that
the wedge of instability is unchanged from the 'parcel' point of
Here
we
increase
Ro
trajectories is still
but
that
of
fixed
-
in
Therefore the wedge
as Ro increases.
region.
It
view.
this case the slope of the
ov?@L*(with the appropriate definition of Ro)
, which is always less
of
instability
is
effectively
This takes some of the shortest waves, whose
trajectories are closer to
unstable
q
the isentropes goes as
than the former.
reduced
at
the
also
the
maximum
reduces
corresponds then to longer waves.
- 58 -
the
allowed
slope,
most
unstable
outside
the
slope, that
combination of static stability and vertical shear.
TABLE
2.4
LTOT
r=O.
r=. 1
0.
1.12
1.12
.1
1.12
.1
1.23
.2
1.14
.19
1.36
.3
1.16
.29
1.50
0.
1.12
.31
1.47
.1
1.12
.33
1.49
.2
1.14
.37
1.56
.3
1.16
.42
1.65
We now proceed to the next section where we examine how
two-level
model
results
compare
multi-level PE numerical model.
- 59 -
with
a
(still
two
well
the
dimensional)
.32
.
30
1.2 1.1 1.0
.9
.28
.26
.24
.22
.20 -
singular
.18
Ro .16
.14
.12
.10
-
.08
.06
.04
.02
-.10
L2
-
-.05
0
.05
Fig.2.7 - Contours of constant
1.8
- 60 -
.10
r
.15
" on the r - Ro
.20
plane
for
.25
fixed
.30
r =0.
1.6
1.4
Ro=0.
1.2
Ro=.1
1.0
Ro=.2
or,X
.8
.6
.4
.2
-
--
Ro=.2
0
0
.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
L2
Fig 2.8 a) - Growth rate
d (solid) and ratio of updraft to
downdraft width A
(dashed) for various values of Ro at fixed r = 0.
Ro = 0 is the GM approximation
- 61 -
5.0
r =.1
1.6
1.4
1.2
1.0
o',X
.Ro=O.
*Ro=.1
-Ro=.2
.8
Ro=.3
.6
.4
Ro=.l
Ro=O.
0
0
.5
1.0
1.5
2.0
2.5
3 .0
L2
Fig.2.8 b) - same as a) but for r = .1
- 62 -
3.5
4.0
4.5
5.0
2.3
In this section we describe the results of
performed
with
a
of
testing
experiments
two-dimensional non-hydrostatic
primitive-equation
model for the purpose
numerical
the
conclusions
of
the
previous
The model itself was written by R.Rotunno at NCAR and it is
sections.
described in Rotunno and Emanuel,1987, where an axisymmetric version of
it
was used to study tropical cyclones.
Only the main characteristics
of the model are described here.
The model integrates the PE in a 2-D domain, with a leap-frog time
stepping.
The compressible terms are integrated separately with a very
short time step (1.
waves,
while
a
sec.
due
here) to avoid instabilities
longer (20.
).
All
sound
seconds) time step is used for all other
terms (this 'splitting technique' is described in detail in
Wilhelmson,1978
to
Klemp
and
experiments are run here in a 10.
Km - high
boundary
condition
troposphere with a rigid lid at the top
(*);
the
w = 0 is applied at top and bottom, while periodicity is imposed in the
x-direction:
sub-harmonics
therefore
only
waves
of
can grow in the model.
the
imposed
length
The grid has a 1.
with
a
50.
Km
horizontal resolution.
Fig.
its
Km spacing in
the vertical; most of the experiments shown in this section
run
or
have
been
2.9 (adapted from
(*) The option of a sponge layer at the top boundary for the absorption
of
gravity
waves
was
available
bfcause it would also deform the
in
base
the model but we did not use it
state
wind
profile
and
comparision with the 2-level and the Eady models more difficult.
- 63 -
make
Rotunno and Emanuel,1987) shows the arrangement of the variables on the
grid.
The
variables
thermodynamic
and v are calculated at the same
there are N=10 points in the vertical, the first one being
points:
The vertical velocity w is
m height and the last one at 9,500 m.
500.
at
known at N+1 points displaced half a grid length in the vertical first
and
zero.
The zonal velocity u is known at points at the same height as v,
but
last
displaced
integrated
(IT
of
these
the
are on the boundaries and are set equal to
horizontally
seven
half a grid length.
equations
for
the
Originally the model
u,v,w, %7&
variables
but we did not use the equation for liquid water - when
=Exner)
q
is above saturation condensation occurs and latent heat is added to the
9
equation,
re-evaporation.
but
we
do
not
keep
track
This is done, once again,
of
to
1.
and do not allow
comparision
make
with
analytic models more direct.
A linearized saturation law for water vapor
following
reason.
When
there
the
thermal
wind,
and
'
it
perturbations remain y-independent if they
time.
The equation for
9
,
a
:
is
we
highly
its
non-linear
value at t=0 for all times.
are
at
such
linearly
behavior
compute
the
.
initial
The use of
varying
potential
%.
Since we
of
we are
forced
64 -
to
It is easy to see how this leads
rapidly to the advection of very high values of moisture in the
-
the
in space, so the
uniform
cannot keep track of the meridional variation of
use
for
easily
can
, however, includes a term
any realistic saturation law gives, for a
temperature
assumed
is no moisture the only variable that
depends on the meridional coordinate is
from
is
region
of southerly wind, and of negative values of
1%
in the northerly wind.
To delay as much as possible the occurrence of this unphysical event we
use
a
linearized
1
+
,
chosen to give a reasonable value of
where
the two coefficients are
at the surface (25.
g/Kg
at
300 K) and a small enough meridional gradient that no negative moisture
is
advected
integration
during
with
the
give
a
We finally choose
of approximately -1.8-to
slightly decreasing with height.
course
Since
we
always
start
imply a y-dependence of
o= -.035
M
,
that we ignore.
b= .~.Q ,
at the surface, and
The dependence of
1
and
on z
would
of
The values of the
thermodynamic variables for a typical run are given in table
is
the
a very small meridional wind perturbation this value
is not exceedingly small.
which
integration.
2.5.
obviously too high in the upper troposphere, but its absolute value
is of little importance - we always run
saturated
the
model
for
an
initially
atmosphere, so that supersaturation occurs anywhere there is
ascent, and only the derivatives count in the advection terms.
When a grid point is supersaturated after advective and
diffusive
terms have been computed condensation occurs, and heat and moisture are
recalculated isobarically from
The model of Rotunno and Emanuel,1987 used a more sophisticated
(Soong
and
Ogura,1973)
but
our
saturation law is already idealized
enough that we do not worry about this order of inaccuracy.
- 65 -
scheme
Turbulence representation is as in Mason and Sykes,1982.
diffusion
Vertical
is switched on only when the Richardson number goes below 1.
and eddy viscosity is then
where S
m.
is a vertical mixing length that is here taken equal
to
200.
Horizontal eddy viscosity is
We experimented with various values of
fields
to produce reasonably smooth
=6x,
without altering the overall picture and finally chose
the horizontal grid spacing.
Table 2.6 gives a list of the
values
of
the model parameters.
The base state is designed to have
vorticity.
We
uniform
equivalent
potential
choose a constant shear and a surface temperature; the
Boussinesq hydrostatic equation is then integrated (from 7,,=
to give pressure at all levels.
Finally
6
is obtained from
which gives
-
+ _
tt
with
C
-
66 -
)
Table 2.7 summarizes the major characteristics of the
experiments
All the fields are displayed as seen from a
presented in this section.
reference frame that moves with the
base
wind
at
mid-level.
w
is
averaged vertically, and u horizontally, to be shown on the same points
Consequently w does not appear to go
as the other variables.
at
the
to
zero
top and bottom of the graphs because those are not the top and
bottom of the model.
Fig.
This
is
a
dry
run
8
5
2.10 shows
that
at various times for experiment
we show for comparision with the saturated
The dimensional wavelength is 4,000 Km
experiments.
AO.
and
the
average
potential vorticity
o
is .1407 ,W
which
(*) We design the basic state
"s
means
d
that
for
uniform
q
,
is not uniform, because the gradients of Y. vary
-,
The value at the top is .17-.o
with height.
surface
.
.12.
The
domain
average
and
the
value
the
at
is also the value of
d
at
mid-level that we use for scaling purposes to compare with the 2-level.
With
this
q
the
non
dimensional length of the channel is 6.81, or
L =1.70 in the notation of the preceding sections.
the
2-L
SG eigenvalue is
(*) The model uses
section
variables.
L2
; the vertical shear
6=.586 or .074,t0
dimensional
For this r and
All
quantities
in
this
are dimensional, unless otherwise explicitly noted, and in MKS
units.
- 67 -
is
.3-0O
-3 -4
Cr =.566
rate
5 , which gives a Ro = .255; then the
or
.073 .o
evaluated
perturbation,
from
every
5_
.
the
domain-integrated
hour,
PE
2-L
eigenvalue
is
Fig 2.11 shows the time series of growth
during
kinetic
energy
the time integration.
of
the
The initial
condition for this run is
Nr.^O
.Z & Z To
71.r+ ;LF
i.e.
a wave with a westward tilt of W
top.
'
is
radians
m/s and pressure and
1.
D
between
and
fields are geostrophically
balanced to this v, while w and u' are equal to zero.
potential vorticity perturbation of order 10
of magnitude smaller than the base state.
surface
This
m S , i.e.
As it can be
induces
a
three orders
seen
in
fig.
2.11 the system takes a few hours to readjust to the most unstable Eady
mode and then settles to a very steady behavior, with growth rate
is
not
far
from
the
that
one predicted by the 2-L PE model of sect 2.2,
indicated by the lower straight horizontal line (the
other
being
the
2-L SG growth rate from sect.2.1), the difference being very likely due
in ltrge part to the presence of dissipation in
The average growth rate of AO between 100.
of
which corresponds to an e-folding time
the
numerical
model.
and 150.
hours is .069,l
"
approx.
40.
We
hours.
initialize the run with a very small perturbation and by the end of the
integration only a slight decline
Examination
of
Fig.2.10
shows
of
the
that
growth
the
rate
is
observed.
structure
of the solution
closely resembles the most unstable Eady wave with the
frontogenetical
effects
included
in
semi
geostrophic
theory.
The small asymmetry
between top and bottom boundaries is due to non-uniform VI , while
-
68 -
the
asymmetry between northerly and southerly winds at either boundary is a
effect
proper
personal
(S.Garner,
advection
ageostrophic
of
communication).
Exp.
Al is a replica of AO in a saturated atmosphere.
state has r=0., i.e.
it is neutral to slantwise convection.
parameters are as in AO.
normal
mode
growth rate undergoes
r=0.
25.
oscillations
hrs.
as
at 50.
0
hrs).
the
cold
front
the
case
for
AO
Lro =6.81,
=.92 corresponding
to
a
time
of
X =.11 or a dimensional width of
The SG values would be C =1.38 and
the symmetry of
anomaly
was
Km, which is, within the grid resolution, the same
course greatly off mark.
surface
moist
During this time the
, corresponding to an e-folding
.112-iO
The second eigenvalue is
observed.
the
which is a little higher than the actual
=.118 .Q4o
hrs (
base
All other
to
For the non dimensional
and Ro=.255 the 2-L PE eigenvalue is G
the updraft of 396.
as
large
to a dry normal mode.
dimensional
6
Fig 2.12 shows the adjustment
taking place in the first 50.
readjusting
The
As observed in
fig.2.12
=0.
at
which are of
this
time
the
is already stronger than the upper-level one, but
updraft
generates
an
equally
strong
negative
of relative vorticity at the top boundary in correspondence to
a warm frontal zone.
This phase of
the
evolution
stops
before
the
absolute vorticity becomes negative.
As seen in Fig.2.13 the growth rate declines slowly from this time
on
and
the
updraft becomes larger, although the form of the solution
remains essentially the same
estimate
of
growth
rate
until
from the
- 69 -
100.
L
hrs
(see
fig.2.14).
observed at 100.
The
hrs is still
good ( L=2.72 -
=.743-
=.95 *\o
) but
=.33
is
larger
than
observed.
The potential vorticity anomaly, in fig.2.15 shows
the
expected
by
this
dipole, but very little change at mid level.
I
time
itself
is not exactly conserved (*) but fig.2.15 shows that its change at
mid
of
point
that determine the solution in the 2-level
evolution
successive
Since the parameters
updraft is very nearly zero.
the
of
the
model
are
unchanged,
the
the disturbance must be attributed to finite
amplitude effects setting in.
Indeed the fields between 100.
hrs.
to resemble the SG solutions in physical
(see
fig.2.14)
begin
space (i.e.
of finite amplitude - see sect.2.1) rather than
geostrophic
space
(i.e.
of
small
amplitude).
The
reduced
in
those
The updraft becomes
larger, but with a maximum at the western edge, at a lower
before.
and 150.
level
than
at the top of the updraft makes the
divergence
upper anticyclonic region become progressively weaker than the cyclonic
one,
and an upper level cold front begins to develop, well to the rear
of the surface front.
Finally we note that the
9e
anomaly has become fairly strong by
(*) The considerations in Rotunno and Klemp,1985
but
5
,
(Appendix A)
does not remain small in this 2-D model because the base
state is allowed to have meridional gradients of a
perturbation
apply,
cannot
Jacobian small.
adjust
its
meridional
With no baroclinicity 0:
not arise.
-
70 -
and IV,
structure
zO
to
while
the
keep
the
and the problem
would
150.
hrs, increasing r in the updraft region, and there is a negative
anomaly on the western edge of the updraft, corresponding to
that
would
be
slantwise
convectively
unstable (also see
fields and recall that there is a meridional
unstable
symmetric
a
f
region
4
gradient, so the
and M
most
mode is not on the x-z plane) if it were saturated
(which is not, see the relative humidity in fig.2.15).
As an additional
solution
check
on
the
'normal
mode'
nature
of
this
we run a similar experiment (A2) with the same base state and
random initial condition.
A random field of v with
values
between
+
and -.01 is generated, and used as initial condition for the model run.
The initial amplitude is very
small
so
perturbation
state
potential
to
the
base
as
not
to
create
vorticity.
integration is then required to reach amplitudes similar
a
A
to
large
longer
Al.
The
structure of the expected most unstable mode is already present at 200.
hrs.
(not shown) although superimposed
that
make
pace
As the moist baroclinic
(see
growth
Fig.2.17
shows
mode
keeps
the
very
similar
oscillations
growing
at
a
shape
to
Al
end of the integration of A2, when the
amplitude is comparable to that of Al in fig.2.14.
in
scale
rate in fig.2.16) the small scale noise is
dissipated away, until only a solution of
remains.
small
it difficult to estimate the true width of the main updraft
from the w field.
regular
to
Apart from a
shift
the horizontal location due to the randomness of the initialization
this is virtually indistinguishable from Al.
-
71 -
An experiment similar to Al but with momentum drag at the
(labeled A3)
was
performed
to
dissipation on the growth rate.
see
the
effect
A momentum
flux
of
is
surface
this
form
applied
at
of
the
lower boundary with the form
where C =.004 (constant) and the zonal wind is only
field,
i.e.
it is assumed that
is
shown
in
fig.2.18
perturbation
A=0 at the surface but we are looking
from a reference frame that moves with the
rate
the
together
mid-level
wind.The
growth
with the growth rate of Al for
comparision, and selected fields are in fig.2.19.
Almost
happens
the growth begins to
until
50.
hrs.
In the next 50.
hrs.
slow down but the structure of the solution is
After
100.
hrs.
essentially
shown
new
unchanged.
the tendency of the meridional wind to be reduced to
zero at the ground can be seen in the graphs (recall
point
nothing
is at 500.
m height) and the
e
that
the
lowest
surfaces are flattened.
The updraft becomes larger and more complex than in Al at the same time
and
the maximum of w is at an higher level than before.
The effect of
surface drag on the structure of the low-level front is similar to that
described in previous (dry) studies, e.g.
Keyser and Anthes,1982.
Fig.2.20 shows the growth rate for exp.
order
B, which has
In
to get this value of r we have to change the vertical profile of
9 , and consequently % , which is here .155'1.-8
given
r=.ll.
Precise numbers are
in Table 2.7, and fig.2.20 shows the SG and PE estimates for the
growth rate.
- 72 -
Experiment C is back to r=0.
Al.
but at a lower
The vertical shear is reduced to 10.
from 30. mnw
.085.
as it was in Al.
Fig.2.21
S
number
the
Rossby
of
the
constant.
gives
as
the
amplitude
is
still
the
parameter
values
As seen in fig.2.21
presence
of
horizontal
6
6~ is higher than the
Both occurrences can
probably
diffusion
and
hrs.
6 =1.18 or
of C,
it had been for the previous experiments,and
than expected.
by
X
The
shows
=.504.O'
numerical
the
highest curve.
observed is larger
be
attributed
to
the
wavelength.
lowest
The negative value corresponds to a vertically
growth
values at 30.
the parameter value for
rates
compare
fairly
which
After
this
growth to slow down.
numerically
are
well
the
For all
with
the
hrs., that is the time when the adjustment is
completed and the solution is the linear normal mode
sections.
Al.
Fig.2.23
2-L PE model is singular as discussed in the previous section.
numerical
the
Km wave at high Ro number for three
neutral saturated atmosphere, i.e.
three cases the 2-L PE
and
value,
different values of r, namely r=.11, r=0., and r=-.07, from the
to
PE
- even so the updraft appears much
reducing
the growth rate of a 3,000.
the
2-L
narrower in the low-Rossby-number environment than it was in exp.
Several runs have been made
to
small and the growth rate
Fig.2.22 shows v and w at 160.,180.,200.
for
=.03.
run
number
Since the time scale is three
times what it was in Al the evolution is that much slower
end
than
between top and bottom,
This reduces
shows the growth rate.
Rossby
time
the
previous
the updraft begins to spread out and the
All during this phase the
consistent
of
with
- 73 -
a
J
solution
and
of
evaluated
the
2-L
PE
of
approximately the right
average
ij
in
integration.
the
wavelength
updraft
and
region
increasing
r.
However
the
does not change much during the
The observed solution seems to reach the non-linear stage
sooner than at longer wavelength.
At even shorter wavelengths the 'destabilisation' of
mentioned in sect.2.2 can be seen when Ro is decreased.
the growth rate of exp.H and fig.s 2.25 and 2.26
exp.s G and H.
the shear.
The
latter
is
unstable
neutral.
Its
Ro
fields
but differ for
number
(.085).
initial
condition
The
wave
has
a
in
exp.
G
(compare Farrell,1982) reaching in 20.
and
the
An
exhibits
wave
a
dissipated away.
which
is
however,
tips over to the other side, looses coherence and
experiment
only
hours
the same amplitude that H
hrs.; at this stage the vertical tilt is reduced,
soon
decays.
hrs.
is
westward tilt with
height, so we observe a fast 'non-modal' growth in the first few
has at 50.
for
and quickly reaches the normal mode form and
follows the usual behavior (see fig.2.24).
instead
low
waves
Fig 2.24 shows
selected
Both have a 2,000.Km-long wave and r=0.
G has high Ro number (.255) and H
short
initialized
neutral
wave
with
after
random
noise
(not
shown)
the small scale noise has been
Note that for these two experiments the 2-L SG model,
insensitive to the Ro number, would give the same, unstable,
solution.
-
74 -
The model results presented in this section show,
characteristic,
as
a
the asymmetry between the upward and downward branches
of the secondary circulation, and the effects it induces on
fields.
This
behavior,
the
other
obtained in a model that explicitly resolves
convection, is fairly well described by the analytic models
2.1
dominant
of
sect.s
and 2.2, that used the slantwise convective adjustment hypothesis.
This suggests that this form of parameterization might be
used
in
other
cyclones.
that
models
Previous
simulations of moist baroclinic
attempt
results
from
parameterization schemes (e.g.
models
of
the
that
circulation
updrafts)
(e.g.
fail
Nehrkorn,1985
structure of Eady waves modified by CISK - most of
only discuss the eigenvalues).
use
alternative
CISK, or a latent heat release given by
a combination of convective and stable
asymmetry
successfully
the
to
show
who
shows
other
this
the
studies
It is however possible that this aspect
of the analytic solution is more a product of the separate treatment of
'dry' and 'moist' regions in the same wave, rather than of the specific
parameterisation chosen.
works
that
treat
To the author's knowledge the
influence
on
previous
the updraft and downdraft regions separately are by
Lilly,1960, who was only concerned with convection 'per
its
only
a
se',
and
not
baroclinic flow, and Tang and Fichtl, 1983.
The
latter model has similarities to the one presented here, that have been
discussed
in
the
Introduction,
and produces two distinct asymmetric
modes - one with a narrow updraft and one with
The
lack
of
selectivity
between
different matching conditions.
-
74a -
the
two
a
large
updraft
(*).
modes is probably due to
The moist baroclinic modes found up to this
have
a
growth
in
still
do
not
rate high enough to compare with the observed 'bombs'.
Even more important, the spatial structure
observed
point
both
of
the
eigensolutions
is
land and sea storms, bombs and non-bombs, and so it
does not provide a mean to separate the two classes of phenomena.
(*) This result can be understood if we consider that that
used
real wavenumbers.
same value of
Fig.2.3).
see curve
the
r=l
in
a given wavelength in the 'dry' region, and so a given
growth rate, there are two wavelength that correspond to the
in
only
The dispersion relation in that case gives the
G- at two different wavelength (e.g.
For
model
dispersion
relation
for
same
-
the 'moist' region, one of which is
shorter and the other longer than the
'dry'
wavelength
(recall
that
when only real wavenumbers are allowed the dispersion relation 6(k) is
just stretched by a factor
both axis, i.e.
,
z
k
N
, or (
)
- 74b -
depending
on
the
scaling,
in
TABLE
2.5
9r (k)
(k )
z (km)
p (mb)
10.
252.0
.675
345.0
11.6
232.8
387.8
9.5
273.9
.691
342.2
12.3
236.4
386.6
8.5
321.8
.723
336.7
13.7
243.6
384.1
7.5
375.5
.756
331.5
15.1
250.6
381.5
6.5
435.2
.789
326.6
16.5
257.5
378.9
5.5
501.3
.821
322.0
17.9
264.3
376.4
4.5
574.4
.854
317.5
19.2
271.0
373.8
3.5
654.8
.886
313.3
20.5
277.6
371.2
2.5
743.0
.919
309.3
21.8
284.1
368.7
1.5
839.3
.952
305.5
23.1
290.6
366.2
.5
944.2
.984
301.8
24.4
296.9
363.7
0.
1000.0
1.00
300.0
25.0
300.0
362.5
- 75 -
8
T (K)
(
TABLE
2.6
model parameters
200. m
50. km
10. km
variable
4000. km in exp. Al
S1.
km
variable
50. km in exp Al
Lt
20. sec
(long)
1. sec
10-4
- 76 -
5.-
(short)
TABLE
EXP.
r
1.
Ro
.255
wavel.
2.7
C(2-L)
A(2-L)
comments
(Km)
(s-
4,000
.073E-4
1.
dry reference run
.118E-4
.11
satur. reference run
1"
)
same as Al except with random initial condition
same as Al except with surface drag
.11
.243
4,000
.099E-4
.23
as Al but higher 34
0.
.086
"
.051E-4
.03
as Al but lower vertical
shear
0.
.255
3,000
.11
.243
-. 07
.124E-4
.16
shorter wavelength
"
.101E-4
.32
as D but higher
.265
"
.138E-4
.10
as D but negative
0.
.255
2,000
0.
.086
"
1.
.053E-4
.06
4
neutral normal mode
as G but lower vertical
shear
- 77 -
4
w
H
0
w
I
w
w
w
w
w
w
Fig.2.9 - Domain and arrangement of dependent variables on the
staggered grid covering the domain. All the thermodynamic variables
are located at v points (adapted from Rotunno and Emanuel,1987)
- 78 -
Time= 170. hrs
Time = 150. hrs
Time = 100. hrs
8.00
u
.0
.
-- 0.
-14.4
_
MA.-- 12 _-
MAX
MW -14.2408
14.2081
MI -17.8190
17.82Z4
MAX
IW -18.1817
16.0113
V
(rn/3)
MAX
IN -13016
MAZ 12.2981
-:
.
.
\,\ t
-,, \',,,tt
,cm/)
MAX 0.285777
.
.
\,,i,
Ki 9' ",'.
= , 1.39435
. I-
]t
MiII -. 282484
(
(K)
..-- :
I1
.1,0
" '7..','..-
'~ ~~1-' '"'~
-
-
MI -1.38887
\ %
,,
0
%
-_..-_,,
, ,_._-
, ,,
\ I
"-
-4.85082
1"
MAX, 4.85577
,
% \
... .
,,..
%
_,
( K).
-7.98404
J
MAX 7.90982
10,
%
.;.,:.-
,,,
'0
('b)
(mb)
,\1
,
()
A
%,
\0"
MAX
2.43828
MI -2.88898
I
MAX 8.19191
-92i
MAX 13.2772
I
-15.2884
0
times
at
AO
,p'
'p fields for exp.
u,v,w, 5 ,
Note that contour spacing is not the
190,200 hrs.
same for perturbation fields at different times
Fig.2.10 100,150,170,180,
- 79
-
Time= 200. hrs
Time= 190. hrs
Time= 180. hrs
U
-
lp
- -
I
---
MAX
MI
18.7778
MAX 20.2890
-19.0222
ZZ Z-
.
-I
MAI 22.1810
MIN -20zm~
----
I'
5
MIX -22.4258
\ \\\%.
(m/s)
.*I X
;\\
l/
%
\\\
-
.
\\
V-
(m/s)
\\
r
o
.
t
MAX 25.8357
(Cm/u)
(=,
(K)
MAX 350.737
MAX
MUN 291.565
363.182
xU
288.892
I
q,[ ..
',"]
I
(IC ;t/
,I
I.
,
j
"'
/
II
-
MIN -10.2281
MAX 12.5852
(K)
p
A'N
MAX 10.0520
1
MN -13.0912
P(
P'
(nb)
(mb)
MAX 21.0270
MI -25.3213
Fig. 2.10 (cont.)
- 80 -
MAX 25.1244
MIN -32.2987
Exp. AO
.24
.22
.20
.18
.16
.14
U)
C
.12
V-I
.10
.08
2-L SG
2-L PE
.06
.02
0
0
100
20
120
160
140
hrs
Fig.2.11 - Growth rate C vs. time for exp.
AO
two-level an ytic models with the same parameters as AO
- 81 -
and
for
the
Time= 10. hrs
~:
\\\\
MAX 0.210201
U
MAL 0.316159
MN
MA
U
MAX 0.128772
Time= 20. hrs
0.517868-01
MDI -. 23 088E-01
"to
Time= 30. hrs
MAX0.584414
Time= 40. hrs
S--
MAI 0.357489
Time=
50.
hrs
MIY -. 2982018-01
-
MAX 1.44519
MDIN
-
Fig.2.12 - v and w fields for exp.
- 82 -
Al from 10 to 50 hrs
MIN -. 427120E-01
Exp. Al
.24
.22
.20
.18
2-L SG
.16
I
.14
0
b
2-L PE
.10
I
,
I
,
I
i
,
l
~I ~I
100
hrs
Fig.2.13 - same as 2.11 but for exp.
- 83 -
I
,
I
Al
I
I
120
I
I
140
I
160
Time= 100.
lrs
Time= 110. hrs
Time= 120. hrs
U
7-:
---
aOO
--
--
,,
-
-
---
..-
-
--
,v
M"
18.7643
IN -15.133
MAX
20.2251
~
V
%
MW -18.989
\ ......-
\%\
V
(m/s)
II
MAX 12.9983
M=
8.9035
-T-.,
%
'1
w
\~\\\\\\\
///
(M/
MN -11.4106
101 -. 29135I
(cm/a)
iX2.318756
I
0(-.428190
MAX 3.48275
( -. 818544
~- ~---~--0
MAX
346.893
hMI 299.053
SIl
()
MAX
5.64392
I
-3.91886
(mK)
Pr
--
..
t
MA
7.70790
RI -7.79125
MAX
10.8998
.i
'.
(mb'
'
•
MIN -11.0550
Fig.2.14 - same as 2.10 but for exp.
- 84 -
Al from 100 to 150 hrs
Time= 130. hrs
Time= 140. hrs
Time= 150. hrs
U
(u)
MhA 22.4992
1IN -18.4862
MAX 25.7406
MIN -20.5211
MAX 38.5340
IN -34.5315
\
I
,
wC/8
(cm/u)
MAX 7.44485
YN -1.31544
MAX 11.6578
()
M
'o
a'
363.520
XI -1.94378
M4D41
N -3.10802
YI
MAX
14.
11IN-.00
IN 298.879
I
k
(K)
MAX
I''
i,,,x,,\,/1
i"
\;,
.,-.;" ,! ,/
11.3884
Ill
MIN -7.90430
|
[
P'
(mb)
MAX 21.7289
MIN -22.2814
MAX 41.1488
Fig. 2.14 (cont.)
- 85 -
MDN-41.7968
Time= 100. hrs
Time= 150. hrs
q;
r.h.
r.h.
MAX
1.00000
MI 0.698537
4e
.1
0.
(K)
(x)
I
L
MAX 422.172
I .II
M
I,( III
11
1
I
11111£~O:I(
I
MIN 337.175
*'i";"'""I
i
I
M
til
. 1
MA
I0.783
III)in,
191.803
1(11((1%
;
/
i I , .
rI
I'"
'"'".
111
11
I
MD -. 24
MR -204.784
MAX
,
,I
I/.
I
177.284
!
I .
I
(I
I
I
MI -848.388
I
i..
1iJ~7roff
(1o-y/r)
(to-yms)
WA
0.378535
MIN -. 424036
.170
.X
MI -1.30295
[j"f.,//
."/Il.II -. .
Il''I , 'I
qe
(o-yms)
(1o-/s)
MAX 0.190047
Fig.2.15 -
qe
e
1a* ' 6,i,
MIN -. 22748
t
- 86 -
for exp.
MAI 1.85177
M I -1.57959
Al at 100 and 150 hrs
Exp. A2
.24
.22
.20
.18
2-L SG
.16
.14
cn
'-
.12
2-L PE
.10
.08
.06
.02
I
140
I
160
I
I
180
I
I
200
,
I
,
I
II
*
220
240
hrs
Fig.2.16 - same as 2.11 but for exp.
- 87 -
,
A2
I
I
260
•
280
I
300
U
I
V
Time= 300. hrs
Timne= 290. hrs
Time= 280. hrs
U
''
-*-..-_
-
,,--
--
Ma
22.4692
MU
.W
A
*.
MAX
--
a\
0
I-.7
N II• i .- "X\\
.
\6
MX -23.730
-.',."%
%
-
V
-
(m/w
MD( -6.2747
..- I.
0
%
Ma
I >-;.:%•r
W
\
%-.
(M9
w
.
\
--
-,
'
'a\\
28296
2
%%
%',
%. -•
\
V
-- -.
-- -
, ,-
= --
MAX 31.4678
MNl -21.2387
MAX 28.827
MiX -18.445
-
..
'55.8w
0
'a\\ MW -41
9.5709
w
%
(c,/u)
MAX
9.40101
MAX
MN -1.49919
14.4774
NN -3.55272
(K)
MA
353.887
MIN 293.197
r
MAX 369.593
\\
MI
90.223
XA" 387.701
1IN 284.888
(K)
p'
MA
17.4306
-11.860
(b)
Pg
P'
(mb
(rnb)
MAX
22.5186
MIN -23.208
Fig.2.17 - same as 2.10 but for exp.
88 -
A2 from 280 to 300 hrs
Exp.s A1,A3
.24
.22
.20
.18
.16
.14
*
0
.12
b
.10
.08
.06
.02
0
100
120
hrs
Fig.2.18 - Growth rate
d
vs.
- 89 -
time for exp.s Al and A3
140
160
Time= 140. hrs
Time= 130. hrs
U
Time= 150. hrs
U
............
-. 0
0-----~' -- 2) )1.
.-.
:--. -- : -- _----
q' ---
-.
2:.
.5-
I
MA
MXa 20.1787
I
-19.9752
MAX 18.8289
XIN -1&5372
Osl
I
MAX 21.4814
MAX 25.0491
MI
MAX 11.87
iP
51612
MAX8.
'
.. JII
-
258
- -
IN -24.204
(,,/3)
IX -242043
INI -21.8688
N -25.1191
w
(cm/-)
S3.97704
-1.15899
'mdII
~
M -7.975M
II
IXN-1.57078
Is
II
S
I
ViAi
38.71
29
1,
(r)
(K)
MA
35.817
IN 296.6
n'
'f
(K)
(K)
P1
(m b)
MAX
21.7988
MIN -18.3156
Fig.2.19 - same as 2.10 but for exp.
- 90 -
MA
27.3311
RN -19.4406
A3 from 130 to 150 hrs
Exp. B
.24 .22
.20
.18
.16
S.14
.02
0
0
20
40
60
80
100
hrs
Fig.2.20 -
same as 2.11 but for exp.
-
91 -
B
120
140
160
Exp. C
.24
.22
.20
.18
.16
.14
o
.12
4
.10
.08
.06
III
2-L SG
A
!1
-II
A
2-L PE
.02
I
0
I
a
I
,
I
I
•
!
•
20
!
100
hrs
Fig 2.21 - same as 2.11 but for exp.
- 92 -
C
.
o
120
m
140
160
(cm/s
w
I0
1
A
Time= 160. hrs
MAX 0.812903
s
\\%
MAX:0.81213
IN -. 3179148-01
\
MflI -. 17913-01
'
11%
Time= 180. hrs
.N
a
-.
436462Z-01
Time= 200. hrs
MAX 3.2354
MN -3.848683
Fig.2.22 - same as 2.12 but for exp.
- 93 -
MAX
1.41976
mIN
-. 5716281-01
C at 160,180,200 hrs
Exp.s D,E,F
.24
.22
.20
2-L PE
I
.14
r--.07
'\
;1-
~I .12
numer.
b
.10
F : r=-.07
D : r=O.
.08
E : r=.11
.06
.04
.02
S
I
i
I
I
I
II
II
I
I
I
I
I
100
I
I
I
,
120
I
I
I
- 94 -
and
for
II
160
140
hrs
Fig.2.23 - Growth rate S6 vs. time for exp.s D,E,F
two-level PE analytic model with the same parameters
I
the
Exp.
.22
.20
.18
.16
.14
Y
0 *4
b
.12
.10
.08
.06
2-L PE
.04
.02
0
0
20
40
60
100
80
hrs
Fig.2.24 - same as 2.11 but
for exp.
- 95 -
H
120
140
160
Fig.2.25 - same as 2.12 but for exp.
- 96 -
G at 20 and 50 hrs
Time= 20. hrs
V
r/s
. v' ,
M
\
" 01
w
|II
S
cm/s
0
_ II'
II
, lI '!
l I!,
l
iI
i
',
I
'I
S•
.
0
iI ,
I5
kI
MDI -. 171540
MAX 0.19357
MAX 0.58
lZ-01 ,
III
I,
1DIN-.31832
I
•
I
,-01
•
I|
I•
•
Time= 50. hrs
MIN
MAX 0.6033
Time= 100. hrs
"'_,,,,:''.
MAX
1.07297
MIN -1.00737
Fig.2.26 - same as 2.12 but for exp.
- 97 -
,
MAX 0.450453
,
-. 318328E-01
MY'"
1,'
'.
MIN -. j50810E-01
H at 20,50 and 100 hrs
Appendix 2.A - Matching conditions for the linearized system
We
present
linearized
for
here
the
reference
argument
on
of 4t and
ir!
the
transformation
from some other source.
we
have
for
the
When we do
that will be used in the next sections.
system
not have a requirement
continuity
future
to
get
the
On the other hand the
interfaces are vertical in physical space, so the 'kinematic' condition
on
the
normal
velocity
is
-
just
=\
or
0 in
the
2-level.
The 'dynamic'
pressure.
In
is
condition
the
requirement
of
continuity
of
order to derive this condition in a generalized fashion
we rewriteSystem (2.14) in a frame of reference that moves at
a
speed
'A in the positive x-direction
Now we integrate the momentum equations across the interface
then + e me
O
-then
requirements
the
then the requirements
c
?A\+j
?A0
and subtraction as
0o
- 98 -
can be expressed, by
sun 1
Since the
continuity
reference
we
have
of
to
pressure
cannot
require
depend
-
the
and
separately in order to satisfy the first of the
any
on
U 0 . The second then becomes just
above
.
-6
by integrating (2.1Y), or, more simply,
get
an
frame
of
-0
conditions
for
We can express this
equivalent
condition
from the thermodynamic equation above:
%O )- -"IL)t,
which gives
1
7
j t) "
't"
C*',
_0
SWe
and either
'
)
I
XJ t
therefore
k
or
f+-
- 99 -
apply
four
conditions
on
a
6
Appendix 2.B - Proof that
Multiply (2.1.
by
is real for the SG two-level model
(the complex conjugate of t
) and integrate
over a wavelength (end points are immaterial and will not be noted)
For the 2nd and 3rd term we know that neither factor in
has
a
S-like
behavior
at
the
interface
conditions imposed ( see footnote pag.
3
).
because
the
of
integrand
the matching
Integrate by
parts
use the periodicity of the solution to get
Since
z(t- tl
but because
(~C)- 6 L-A))
CC
the r.h.s is
9
this
t(A=4 is zero and therefore
-
100 -
'Z is real.
and
Appendix 2.C - QG and GM non-dimensional quantities
Let
z
zL
then the non-dimensional form of (2.29) is
Dtt
the limit for
-- 00
write
-
of this is QG, as is
known.
well
However,
and rescale with
L
QAO-I
LL
then the non-dimensional form of (2.29) is
I
+-oT.+
Z
r
9-VtjG~
YH
-r-i
0
the limit of this for
words
= O
is now the GM approximation.
we recover QG by neglecting
GM by neglecting
either P_
-o o
1
compared to
or
compared to
t
N
- h
moves
the
Since
; and we recover
NE
- 1A
for
, either form is correct in this limit.
o
In the opposite limit the shift in the
number
.
N
In other
singularity
-
101
-
definition
of
from M=0
the
(014
Richardson
o
)
to
Appendix 2.D -
A
for the PE two-level model
With the same procedure described in App.
2.9 we find
Let
Since the other two
coefficients
behavior
in
of
C
terms
of
are
A
positive
(but
we
since
can
A
discuss
depends
eigensolution itself we cannot say a priori whether any of the
the
on
the
classes
of solutions that follow are not empty).
A40,
For
which is always the
2
case
depending
on
the
values of
0
are real, one being positive
CX- 0
6
the
form
of
@ (0
when
,
and
the eigensolution, for some
and
one
>0O, the two
When
negative root disappears, so in fact the modes of positive
positive root has become 6
0.
sign of the discriminant is.
4.CA
At 4>o
traveling
Assume there is a
A>Q
o we get two real negative 4
modes.
However,
At
A = 0
for which
, i.e.
complex
102 -
-0
;
four neutral,
S
and
since this condition means
large and positive, it is doubtful that this class of solutions
-
the
we cannot say a priori what the
traveling solutions; for all A)A there would be
unstable
occur,
negative.
with Ac.O are the continuation of the GM solutions.
then for
may
so
N
exists
-
more likely the eigensolutions in this range of
is never larger than
Ao .
- 103 -
are such that A
3.1
In this chapter we examine the effect of adding a heat
the lower boundary in the models of ch.2.
the behavior of developing baroclinic
environment
flux
from
This is intended to simulate
waves
over
the
ocean,
in
an
favourable for the explosive growth observed and described
in the studies mentioned in the Introduction.
We
will
use
for
this
purpose the numerical model of sect.2.3, but first, in this section, we
extend the SG model of sect 2.1..
problem
as
It turns
possible
of the updraft, and phase speed.
range
internal
diffusion
would
of
growth
rate,
suspect
that
a
model
the
interfaces,
and
the
thus selected would not necessarily be the most unstable of those
found here.
to
eigenvalue
resolve the ambiguity by securing the
continuity of all physical variables across
mode
the
Although it can be argued that
the most unstable of these will be favoured we
with
that
it is formulated here does not have a unique solution, but
rather that we can only determine a
width
out
We think however that a better way to accomplish
this
is
simply use the numerical model of sect 2.3 - the investment of time
and computation needed to include internal diffusion in the
would
not
be
adequate
to
the
sole
result
of
alterations of an already very idealized situation.
model,
we
rate,
and
model
small quantitative
As for the 2-L
PE
already know from sect.2.2 what consequences the use of the
primitive equations has on the moist baroclinic modes - it
sensitive
2-L
to
makes
them
the Rossby number of the base state, reducing the growth
allows
us
to
find
solutions
in
a
slantwise
environment, as long as it is not vertically unstable.
-
104 -
unstable
The simplifying
assumptions that we have to make in order to be able
level
model
solve
a
two
with heat fluxes at the bottom boundary easily offset the
better accuracy of the PE response in this situation.
still
to
wish
to
present
the
2-L
SG
In
summary,
we
for logical continuity with the
previous chapter and then we will focus, in the next
section,
on
the
numerical simulations.
We now proceed to introduce a heat flux from the lower boundary as
an
explicit diabatic heating term in the thermodynamic equation of the
2-L SG model of sect.2.1.
The 1st principle of thermodynamics
can
be
written
cSdT- ±j + L4
d
Applying this law to adiabatic reversible processes the conservation of
equivalent
potential temperature
6
is deduced.
The transfer of heat
from the sea surface to the air in contact with it has two
CyT
and
L1,
respectively.
,
that
components:
are referred to as sensible and latent heat,
The fluxes are usually parameterized
(see
Jacobs,1942;
Malkus,1962) with a drag law of the form
where Q. is
C(
or LV ; the subscript 'sea' refers to saturated air at
the temperature of the sea surface, and 'air' to the air at some height
(a few meters) above the surface.
are
If the empirical drag coefficients C
the same for the two quantities the two laws can be combined in an
expression
for
c
*TL+
L
,
-
or,equivalently,
105 -
c % .
Following
Ooyama,1969
we
write
the sensible plus latent heat flux from tha sea
surface as
The heat exchanged with the ocean
will
remain
in
the
boundary
layer unless some mechanism more efficient than turbulent diffusion can
transport it upward - this role is played by convective activity in the
moist
region and we here assume that there is no such mechanism in the
dry region (*).
point
located
moist
region
heating.
Since we know the potential temperature
in
is
We
the
thermodynamic
equation
further
assume,
consistently
heat
received
in the troposphere, i.e.
from the value
FO
at
one
troposphere we assume that only in the
the
discretisation, that the
distributed
middle
only
from
affected
the
with
sea
by
diabatic
the
2-level
is
uniformly
that the flux decreases linearly
at z=0 to zero at z=H
, so that
the
thermodynamic
equation is written
To deal with this term it is necessary to consider small amplitude
perturbations
and
linearized
equations;
we now assume a basic state
(*) In the real world, and in the numerical model of the next
it
section,
is of course possible to have convection, dry or moist, anywhere if
the latent and sensible heat input at
enough to create instability.
-
106 -
the
bottom
boundary
is
large
temperature
6C
t.
(
.
of Q is then
form
perturbation
The
For convenience we will later write the equations
coordinate
moving
9
difference
frame,
thermal
the
forcing
in
a
it is understood that in the system in
but
zero
is
which the sea is at rest the basic state surface velocity
that
equivalent
air-sea
constant
a
with zero velocity at the surface and
so
not enter the equation for the basic
does
state.
the base state is balanced for any LZro
by
to consider a heat flux given
temperature
with
the
&
surface
therefore
is
unlikely
)G\ .:
wind -c
and its
&1
is positive and viceversa.
that
this
term
it is
gradient
Then it reduces
advection.
temperature
both the amplitude of the thermal wave and the
It
then
perturbation
the
of
apparent that this heating always acts to reduce
- it is negative where
0
In this case it is possible
coupling
the
state
basic
.
MY
if
A different linearisation of (3.1) is possible:
could be responsible for
increased instability. )On the other hand the heating term that we
reduces
(see
for
eq.(3.2) below)
,
which
is
v,
thermodynamic
meridional advection term in the
r~e'-l)
positive
equivalent
to
an
to
t"%
,
equation
increase
a
so that the
if
becomes
of the basic state
(*) Note that this argument is valid for the 2-level only.
with
use
In a
model
higher vertical resolution we cannot a priori exclude that the
wave modifyies its structure in such a way as to take adventage of this
form
of heating (e.g. by changing phase with height more rapidly than
an ordinary baroclinic wave).
-
107 -
baroclinicity (**).
if
exclusive
two
The
chapter.
The use of both terms would imply that both
non-zero
-
therefore
mutually
are
to use the same basic state as in the previous
want
we
.
and
terms
CO
heat flux in the form
a
aF
iand
are
&
would
appear in the zero-order equation and modify the base state.
The term IW
that
component
is not
it
unless
tractable
easily
has
does not change sign, in which case it becomes linear.
model
RLkare
both
Moreover their values at z=0 are not explicitly known in
the
proportional
being
present.
is
AX
The perturbation velocity at z=0 has two components (U\t )J):
zero,
one
just
-
w (see eq.2.2) but Q-e
to
and
they can either be assumed to be equal to the values at level
2, or extrapolated from the two known levels to the surface.
Since
we
are at this point more interested in the qualitative changes induced in
the solutions by this form of diabatic heating,
use
only
"\.
we
surface.
We found in sect.2.1 that AF)>
that
flux.
accuracy
of
everywhere in region 1 and
we
this will not change, at least for small values of the heat
This of course simplifies things, in that we can
setup of the model as in 2.1
and
not
acts only in the updraft, which is order .1;
so that, in a domain average, %-'
(see fig 3.1).
the
use
same
(as opposed to having to further subdivide
(**) This is only a qualitative argument,
rate
in
as the perturbation velocity at the
details,
hope
will
than
P
correct:
entirely
itself is order 1.
is only 10% larger than
A posteriori we can say that
the
increase
in
S,
growth
observed cannot be solely attributed to increased meridional heat
flux.
-
108 -
the updraft region according to the sign of V
) (*).
We finally write the thermodynamic equation in the moist region as
From sect.2.1 we know that
for
Jt
9
is not in phase with
and
at
least
small heating this character should be preserved (the limit 9-
is not singular).
homogeneo~p
equations
We then have a thermal forcing out of phase with the
solution
the same phase
and we cannot expect the forced solution to have
speed.
For
is
mathematical
convenience
we
write
the
in a frame of reference that moves with a velocity UL+4
in
the positive x-direction.
62
o
real
and
sect.2.1) and A
o
We assume that such a system exists in which
solve
the
eigenvalue
problem
for
)
A
(as in
.
With the same scaling as in sect.2.1 we
get
the
non-dimensional
equations as
(*) This is a model of a continuous fluid resolved only at
in
the
of fluid.
Accordingly
it
would
v at the surface.
be
consistent
cases
however
use
and no major
the extrapolated ir
region 1 when zero or low heating is
obtained is inconsistent.
to
a
linearly
We checked the results for some values
of the parameters with this choice of t"
some
levels
vertical, and not a model of two superposed homogeneous layers
extrapolated
In
two
applied,
change
occurs.
becomes negative inside
so
that
the
solution
In all cases an increased heat flux forces a
positive v at the surface.
-
109 -
(. - C +t.)
- t,
CC 6'
--
Where
P
)
-
o
%
-
in
The equation for t
in
IL
:
region 2 and
region 1;
derived from here is
and the meridional geostrophic velocities are given by
;S
Z(It,-)Co.) i, -
The matching conditions are derived as in Appendix.2.A.
to
require
of pressure.
continuity of
t ;* ~ W
Expressed in terms of
-
110 -
We
have
and these imply the continuity
and
its
derivatives,
and
of
, they are
1~a
(recall that we use " -":,,
general '.
These
).
but
these
expressions
expressions are valid unless
are
valid
for
a
L-j[ . When this
happens eq.3.3 becomes 3rd order and the momentum equations in 3.2 give
r-
/6"
required.
or
Tjt//
so that only three matching conditions are
We will use 3.6 above and check a posteriori that U,
Condition ii) does not imply w=0O
sect.2.1
but
we
still
at
A,B
and
want w to change sign there.
impose conditions more restrictive than ii) that
some of the unwanted modes(*).
help
it
was
in
is useful to
filter
out
Let
I
at the interface.
C
.
I
'
Then condition ii) gives
'LA3
So we want
-T >U
that
Tt4wI
I0r
-I.
10
O
*.
.
We theintroduce a factor
,-
nz c
(*) see sect.2.2 for analogous discussion in the no-flux case
- 111 -
6
such
When
D
=(
:O
then J
then
and *J
C> and
J.
has a
\A.
has
negative
a
non-zero
positive
value.
non-zero
When
value.
The
discontinuity of w at the interface is given by
A
We cannot say a priori what its value is, because
but
eigensolution,
O< r
note that for
the
part
I
, the term in
O
06
parenthesis is always positive, so the vertical
of
is
velocity
will
always
have a discontinuity at the interface.
We replace condition ii) in 3.6 with the two separate conditions
A)
(where a
and O
is allowed to have different values at
AIc
)
and
we
then
conditions at the two interfaces.
solve
properties
linear
as
in
system
eighth-order
that
the
eq(3.4) and
two
interfaces
apply
the
Since there are no obvious
will
determinants
given L 2LI.
yield
are
non-trivial
set
The
arbitrary (as long as they are
presented
6A
g
next
the
symmetry
sect 2.1 we are left with the full 10x8 homogeneous
equal
solutions
to
zero.
when
most
last
between
unstable
two
0
parameters
and
solution
and the least unstable the one with
A C
- 112 -
1).
is
6
three
The 'dispersion
relation' then will determine three eigenvalues - we solve for
for
five
IU.o
are perfectly
In
the
results
always the one with
C~
I
.
Since the
width
of
the
updraft changes by changing
these parameters
matching
is
as
arbitrarily
A and
choosing
the
C we can think of
point
at
which
the
performed within an interval of allowed widths determined
by the other (physical) conditions (*).
Fig.3.1 shows the behavior of the eigenvalues when the heating VO
is
increased at fixed r and L
is shown.
.
i
The spread due to the choice of
The more unstable (at fixed
o ) has a larger updraft
C
(which
is understandable, since heating acts only in the updraft,its effect is
larger when it acts on a larger portion of the wave).
The phase
is also larger for the more unstable modes.
The value of
realistic:
F'nf.
In
(A
=
for average atmospheric values
Fig
3.2
)
Co
the
eigenvalues
for
L2 are shown.
versus
the
speed
o shown are
*
most
unstable
case
Notice, above all, the further
destabilisation of the short waves, and the large
increase
rate.
., but we will see in
The
phase speed is also increasing with
in
growth
the next section that this is an artifact of the 2-level model (**).
The form of the eigensolution
displayed
in
differences with the l
presence
of
This
fig.3.3.
mode
has
G2I.5,
=..
r oI-2
A= *I
is
,=.
.
Major
o case are only the discontinuity of w and
the
a double minimum of vertical velocity at the edges of the
(*)In a work by Lilly,1960
upright
for L2 t.8 1
convection
and
the
a
similar
technique
same problem arises.
is
applied
The growth rate is
there found after arbitrarily picking the width of the updraft.
- 113 -
to
updraft, like a return flow that is not spread
out
uniformly
in
the
downdraft region.
3.4 is the growth rate as function of
Finally fig.
O
and
for a fixed wavelength, showing a remarkably linear behavior with
O
(**) As already noted the vertical structure is rigidly determined in a
2-level.
The heating is proportional to
ahead of the thermal wave (see fig.3.3)
forward motion.
1& , so it is maximum slightly
and
This does not change if S
therefore
induces
it
is linearly extrapolated.
It does not seem impossible that a heating proportional, say,
which
is
maximum
sligthly
behind
the
a
to %F5 ,
thermal wave would produce a
backward motion, but of course there is no justification for doing this
in
the
present
model.
This effect, that is in fact observed in the
numerical model of the next section
vertical structure of the wave.
-
114 -
is
accomplished
by
a
different
1.6
1.4
1.2
1.0
.8
.6
ge
.4
.2
0
.4
.6
.8
1.0
1.2
1.4
1.6
1.8
2.0
2.2
1o
Fig.3.1 - Eigenvalues 6,
w
Ul,
of the SG 2-level model with heat
flux from the lower boundary vs. heating parameter to for r=.l,
L2=1.8.
Solid:most unstable 6A.=
o
, dashed:
least unstable
6.A
=
- 115 -
2.4
1.6
r-.1,ro -2.
1.4
, =1.
1.2
1.0
.8
.6
rm1.,Po =. - dry
z----
.2S
.
-
-
-
- -
--
- ---
---"-
-
.-.
0
0
.5
1.0
1.5
2.0
2.5
3.0
Fig.3.2 - Eigenvalues T ,
L.o vs. L2 for various values of the
heating parameter r o for the most unstable case 6 A Z 9c = O (see
text).
Solid: growth rate G- ; long dash: ratio of the width of the
updraft to downdraft X ; short dash: relative phase speed Ut .
- 116 -
3.5
10
8
/1/
4
/
\
/
-2
-
$.
0
.2
-
.4
.6
.8
1.0
1.2
1.4
1.6
1.8
2.0
Fig.3.3 - Form of the eigensolution for L2=1.8, r=.1, [o =2.,
0 . Eigenvalues for this mode are c- -1.5,
=.17, U =.23.
is known only from it derivative v, so that an arbitrary
can be added
to t both pl and
t
I p2.
I
II
can be added to both p1 and
p2.
SA
-c :
Pressure
-8 constant
constant
- 117 -
2.2
0.5 0.4 0.3
r
0.2
0.1
0.0
0.0
0.5
1.0
2.0
1.5
Po
Fig.3.4 - Contours of constant
S,:
0 , on the f'o,r plane.
- 118 -
growth
rate
r
for
L2=1.8,1A-
3.2
The numerical model of sect.2.3 is used again in this section with
added heat and moisture fluxes from the lower boundary in the form used
in Rotunno and Emanuel,1987, i.e.:
where
The latter quantities are functions of x and t through the surface
pressure
perturbation
I
.
The
relative
temperature difference in the base state
space.
Both
these
parameters
humidity
&b
are
contribute to 4
humidity is fixed at 100% and only &
is varied.
and the air-sea
chosen
uniform
in
- here the relative
The
limitations
of
the linearized saturation law that we use become evident at this point.
In the model's thermodynamics
good
approximation
at the surface (
world
(using
1, At
6Ir
always, and this is not a
of the real thermodynamics.
For example a
=300 K ;p=1000.
-=25.2
mb) gives
total
A
K in the real
the value given in the Smithsonian meteorological tables
for saturation mixing ratio) but only 7.5 K in the model.
the
c=
r
is
relevant
we
-
can
119 -
As
long
as
use arbitrarily high values of
6G(model) to reach a 4b,
However,
the
corresponding to a much lower
we
heat
mechanism
for
future
reinforces
development
have
an
impact
on
the
are trying to model - a larger part of the entropy flux
goes into warming of the boundary
latent
(world).
different proportion of sensible and latent heat flux in
the model and in the atmosphere is likely to
phenomenon
L
than
the
occurs
layer
release
updraft
in
in
we
instead
the
updraft.
may
reality,
of
expect
especially
being
saved
as
Since the latter
here
if
a
the
weaker
air
is
sub-saturated at sea level, so that the entropy flux has an even larger
component of latent heat.
The drag coefficient C
coefficient
is here assumed equal to the momentum drag
CD that appears in the momentum flux:
and
A summary of the experiments presented in this chapter is in Table
3.1.
The first experiment that we show (labeled I) is run with CE =0 ,
so as to provide a control experiment for the rest of the section.
(*) This form is appropriate for marine environment.
one
experiment
constant
with
drag shown in sect.
Recall
that
the
2.3 (exp.A3) was run with a
C =.004; the drag coefficient that we use here increases
the perturbation and does not reach that value until
- 120 -
It
'A~
with
70 m/sec.
has
3,000 km wavelength and r=0, i.e.
it is similar to exp.D of sect.
2.3, except that it has frictional drag at
specified
50.Km.
50.
above,
and
The
surface
in
the horizontal resolution is 30.
The horizontal 'mixing length'
Km.
the
initial
Q
perturbation
the
form
Km instead of
defined in sect
2.3
remains
is the same as in sect.2.3 (see
pag.68)
Fig.
the
3.5 shows the evolution of the growth rate
perturbation
kinetic
energy
with fig 2.23 will show that no
calculated
as a function of time.
difference
appears
from
Comparision
until
50.
hrs,
after which time a slight decrease of the growth rate due to frictional
dissipation takes place.
There are no surprises in
the
evolution
the disturbance, that is shown at selected times in fig.
Fig 3.6 shows the growth rate vs.
the
same
as
time for exp.
3.7.
FI2,
The evolution follows that of exp.
time faster growth takes place.
At 50.
field at the
thermal
and 60.
surface,
gradients.
occurring.
As
base
70.
hrs
state.
shown
in
hours there is only a slightly stronger wind
70.
due
hrs
the
to
the
increased
strong
increase
that
horizontal
in
w
and
convection
we assumed in the previous section removal of heat
from the boundary layer occurs in the updraft and we can begin
at
is
hrs, after which
The form of the solution is
presumably
At
I until about 65.
one-grid-point oscillations in the wind field suggest
is
which
I except that heat and moisture fluxes are active and an
air-sea temperature difference 1e=10 K is imposed in the
fig.3.8.
of
that this process is taking place.
to
In the following 20.
hrs distinct narrow updrafts are evident, embedded in a larger area
- 121 -
see
of
weak
w,
positive
associated
and,
with
distinct
them,
At 100 hrs
vorticity and potential temperature are observed.
have
updrafts
into
collapsed
of
maxima
the
all
one, which is strong and narrow enough
that its effect is clearly seen on the zonal wind field as well
as
in
the vorticity maximum at the surface and in the thermal maximum, all of
which coincide with the minimum surface
middle
(see
troposphere
also fig.
pressure
3.9).
and
extend
to
the
Note that at this time the
310 K isentropic (*) surface is into the ground, so
the
sign
the
of
surface flux at the center of the perturbation is reversed and only the
low-level convergence feeds heat into the updraft.
hrs
the
The vertical velocities
were
at
actually
but the updraft becomes wider.
hrs
100.
are
structure is suggestive of a tropical storm and the wind
can
30.
next
the inner warm core of the cyclone extends to higher levels but it
also spreads out.
they
In
weaker
than
The thermal
itself
field
be looked at as the superposition of one of the 'moist baroclinic'
modes of Ch.
2 and an entity resembling a shallow hurricane,
as
seen
in fig.3.10 where the difference of meridional wind fields of exp.
and I at the same hours are
shallow
shown.
This
'2-D
Fig.3.11 shows the location
(*) The flux is determined by the moist entropy
at
being
310
K
in
the
S
of
the
humidity
as the
B,
is
0.
which for saturated
model's thermodynamics is 377.5 K.
Since the
updraft is saturated (see relative humidity in fig.3.14) and the
relative
a
feature, is advected by the low-level wind, and the whole wave
appears to be moving with it.
air
hurricane',
FI2
'sea'
100% the argument can be made with either 9- or
field in fig.3.9 confirms.
- 122 -
m/s
isotach
of
meridional wind at the lowest level versus time.
largest westward displacement,
hours
that
occurs
on the western side, and 20.
between
100.
slower
than
the
mid-level
qualitative disagreement with
section.
Given
zonal
the
wind.
two-level
This
model
Km/hr Cf
is
of
reality.
(Sanders,1986)
growing
a
point of
the
previous
that
the
more
rapidly
Observations
show
bombs tend to travel
faster, although this is not a general feature, while in the
model
an
heat flux seems to induce a slower phase speed (the westward
displacement relative to the mid-level wind is larger
de ).
4.2
the more realistic features of the numerical model we
would expect its results to be closer to
increased
110.
hours later at the eastern edge of
the southerly winds, corresponds to a phase speed of 15.
m/s
and
The
The
with
increasing
only point in which the numerical model is weaker than the
two level is the relative amount of sensible
to
latent
heat
in
the
entropy flux, as we mentioned earlier, and this point should be further
explored, possibly with a 3-D model using
(recall
a
realistic
thermodynamics
that we are forced to use a linearized saturation law in order
to be consistent with a constant meridional gradient of
mixing
ratio,
which is all a 2-D model can do)
The phase of rapid growth seems coincident with the occurrence
convection, vertical or slantwise, in the model.
Initially the wind is
only zonal and the model has zero resolution on the
so
there
cannot
be any slantwise convection.
increases, the surface
projection
on
the
x-z
normal
to
the
wind
of
meridional
plane,
As the meridional wind
acquires
a
non-trivial
plane but still only very long wavelengths of
- 123 -
symmetric modes can be resolved, while it is known
are
the
most
unstable.
Fig.
3.14
shows
that
that
relative
humidity)
CL S
vorticity
a
the
is
buoyancy
positive,
for
while
in
field
(and
in
the
equivalent
potential
is negative, in the interior and they are both negative at
9
fig.
this
effect on exp.
Later on
(see
90.
3.14) a region of negative vertical buoyancy is formed
in both I and FI2 where the
Neither
hrs, when
saturated vertical motion
the lowest level, where direct heating is applied.
hrs.
shortest
at 60.
individual updrafts can already be identified in the w
the
the
nor
the
91
perturbation
interior
I or exp.
FI2.
negative
decreases
with
height.
ie region seem to have any
Ten hours later a region
of
unstable
air (both vertically and slantwise) extends continuously over the depth
of the troposphere.
and
The main updraft is located in the
same
position
the inner core of the perturbation reaches the upper levels of the
model.
At this same time, which is the beginning of the phase of
fast
growth (see fig.3.6), the domain integrated 'symmetric' conversion term
U
i
L-.v
end
in the KE equation becomes positive and remains so
until
the
of the integration, with values an order of magnitude smaller than
the baroclinic terms.
become
broad,
and
In the later stages, when
the
the
inner
core
has
growth rate is beginning to decline, both the
vertical and the slantwise stability are positive again where
the
air
is saturated (not shown).
Fig.
3.12 shows that
an
increase
of
the
air-sea
temperature
difference changes the maximum growth rate reached and also the time of
onset of the phase of rapid growth.
- 124 -
The same effect is observed in fig
3.13
where
the
different he .
deeper
for
minimum
stronger
effect
pressure
is
plotted for runs with
In the first stage the low pressure center is
heating
significantly altered.
the
surface
and
the
KE of the perturbation is not
The fields in fig.
3.8 show that in this phase
of surface heating is confined to the lowest levels of the
atmosphere, that are directly subjected to the heat flux.
very
rapid
time,
and
potential
modestly
growth
occurs,
hurricane-like
vorticity.
The
with
Later
on
deepening approximately linear with
features
in
pressure
the
drop
temperature,
during
wind
and
this phase reaches
values above the conventional threshold for exposive cyclones, that
c 19.5
b
mb/24
a
is
hrs at the latitude (450) at which the model is run, for
15 K.
The environmental stability influences
both
the
maximum
growth
rate achieved and the time of onset of the 'explosive' phase of growth,
as shown by the sequence of experiments L,FL1,FL2,FL3 (see table
whose
growth-rate is shown in Fig.
in Fig.
a
3.16, versus time.
negative
displacement.
r,
3.1),
3.15, and minimum surface pressure
These are runs in which the base state has
corresponding
to
neutrality
for
saturated vertical
Comparision with 3.12,3.13 shows that each
of
them
is
faster and deeper than its counterpart at r=O.
The evolution of the perturbation in exp.
hrs
in
fig.
3.17
and
FL2 is shown
model.
state
o0
30.
it follows the same pattern as FI2, with the
exception that the phase of explosive growth begins sooner.
base
from
With
this
everywhere initially, which is allowed within a 2-D
In a 3-D world we would expect widespread slantwise
- 125 -
convection
to restore a condition of symmetric neutrality in a short time and then
the evolution to follow
slantwise
and
that
of
FI2.
only
relative
effect
most
unstable
has
symmetric
grows
that
mb
whether
a direct causal effect on the onset of the 'explosive' stage
3.13
and
the change of pace from the initial 'exponential' growth to
the later 'linear' growth seems to take place in
10.
modes.
-
faster
we cannot say at this point, but it may be observed in fig.s
3.16
of
of the reduced stability is that, as already seen in
the 2-level model, the moist baroclinic wave
this
importance
vertical convection in the evolution cannot be assessed
in a model that does not resolve the
The
The
in
all
but
the
weakest
cases
amplitude dependence for this transition.
the
shown,
A
neighborhood
of
thus suggesting an
possible
explanation
of
this behavior is that in the first stage only the vertical advection by
the large-scale flow can transport heat upward of the lowest levels
the atmosphere, and weaker disturbances take longer to do so.
latent heat acquired in the boundary layer is released, and
of
Once the
begins
to
modify the thermal structure in the updraft, convection can take place.
Its relative importance for the vertical heat transport at
is
uncertain,
but
more
stage
its kinematic consequences on the structure of the
solution are evident in the subsequent evolution.
narrower,
this
intense
and
The updraft
becomes
more effective in the transport of heat.
Notice that the collapse of the updraft region
in
this
stage
is
in
direct contrast with the 2-level model - there we directly applied heat
to the middle of the troposphere, here the system has to
own
means to do so.
organize
its
This result should be compared with the result of
Davis and Emanuel,1988 that show that the correlation between deepening
- 126 -
rate
and
heat
flux undergoes a transition from
deepening rate goes above 1.2 bergerons
our
experiments
with
the
same
(*).
.5 to -. 8 when the
Since we
amplitude
of
the
initialize
perturbation
growth-rate threshold is equivalent to an amplitude threshold
experiments.
On
the
other
all
for
a
our
hand Rotunno and Emanuel,1987 have shown
that tropical hurricanes need a finite amplitude initiator to begin the
development
and
the
behavior of our '2-D hurricane' seems consistent
,with this conclusion.
Finally Fig 3.18 shows the wavelength
rate.
The
curves
are
similar to FI2 (i.e.
domain,
the
r=0 ,=9:10
at the same (30.
levelling off at 3600.
pressure
K),
dependence
drop vs.
but for
Km) resolution.
of
the
growth
time for experiments
varying
length
of
the
The growth rate seems to be
Km, but we could not explore further
for
lack
of computer time.
(*) 1 bergeron is the threshold for 'bombs' as defined in
Gyakum,1980.
Its
Sanders
and
value varies with the sine of latitude and is equal
to a pressure drop of 24.
mb in 24.
-
127 -
hrs at 600 N
TABLE
wavel.
EXP
0.
3,000 Km
3.1
comments
--
slantwise neutral - as in D plus
momentum drag
FIO
0 K
FIl
5 K
FI2
10 K
FI3
15 K
FI4
20 K
-. 07
--
as I plus heat flux at the bottom
vertically neutral - as in F plus
momentum drag
FL1
-.07
5 K
FL2
-.07
10 K
FL3
-.07
15 K
- 128 -
as L plus heat flux at the bottom
Exp. I
.22
.20
.18
.16
.14
ca
.12
.10
.08
.06
.02
0
I
I
I
II
I
I
I
I
Ii
I
I
1I
1I
100
I
120
hrs
Fig.3.5 - growth rate
T
vs.
time for exp.
- 129 -
I
I
140
160
Exp. FI2
.24
.22 .20
.18
.16
0
S.12
b .10
.08
.06
.04
.02
0
20
40
60
80
100
hrs
Fig.3.6 - same as fig.3.5 but for exp.
-
130 -
FI2
120
140
160
Time= 50. hrs
Time= 60. hrs
Time= 70. hrs
.00
.00---------
U
U
0.0
-- - -- - -- - -- - -- - -- - -- --
---
--
-
-
-
--
-
(m/s)
-
|
MAX
V
(m/s)
!
*
-
MAX
14.1804
UH -13.9174
MAX
14.4803
MN -14.0901
14.8428
MAX
MIX -14.3529
.
'
\
\\
'
;
%
.
-.--
%'
.--:
V
--
I
\
(m/s)
'
."
1.75522
I
mN-1.80406
MX 3M.8206
NN -8.10142
MAX
MN -. 328798
W
(0=/S)
MAX 0.886238
MIN -. 143958
MAT 0.943864
m -2z8
.2,
-
-0.
1.19440
336.
".
•
"320
.. -----.
.
-.zo.3.
(K)
()
MAX 342.78
MIN
301.377
MAX
-
3493.85
-.-
MIX
300.930
I
,'
(K)
MAX 0.832588
M
-.40539
MAX0.94401
RIN -. 188220
,36
I"
M 3
7I
I
.
p'
X0
P
7-
I,
(mb)
(Mb)
I\ #
I
MAX
128128
MN -128798
MAX 1.8078
I
MN -1.882
MA
r
2.51983
MM -2.52742
Fig.3.7 - u,v,w,%&',p fields for exp.I, shown every 10.
hours
from 50.
hrs to 130.
hrs. Contour interval is not constant with
time.
- 131 -
-.
u
Time= 100. hrs
Time= 90. hrs
Time= 80. hrs
00
.0.
MAX 15.4150
%
%-
(/)U
(rn/u)
Mm -15.2787
16.2385
MAX
Mm -14.7459
---
MAX 17.2299
.\
%
%
MIN -15.8828
V
rn/u)
N\
(m/u)
MI -6.72893
MAX 8.84857
MMN-8.09017
MAX 9.12928
I
w
.2-
Ma
NN -.818840
2.11021
-. 851124
MAX 2.48519
/8
-.36.
(1)
)
--20,
(K)
t
MAX 344.88
Mme
300 8 ,
"M
300.28W
MX
348.198
In 299.8s
'
r-:~/'/;h~'"//////0//
(K)
;
MAX~~~~=
MAX 3. 615
M
3.858
()
1.6
xx - 1.459
pI
P
(Mb)
(mb)
MAX
3.50935
MN -3.47828
Fig.3.7
MIN -4.76382
AX 4.91411
(cont)
-
132 -
Time= 110. hrs
-----
u
Time= 120. hrs
Time= 130. hrs
-----------__--------- -
U)
MAX
1851
EN -16.6701
~
V
.1
*I
i(
~*..s"\ ~()
/
MN -18.880
-
MIN -15.1473
4.53170
_~338.
L ~~----
(K)
20.8728
I
I'
'
MIN -20.0942
-2.09805
-
,m.L
-
---
'
~CC:
i--4~------~L
27~
299.248
MAX
MAX 8.88019
WN-1.59910
----
MD
\\\o\
- %...
MAX
348.239
21.2411
,
\\
MAX 15.7242
MA
MAX
%
--
.,
(r/i)
MDN-17.5493
MAX 19.5724
I
MAX 347.808
MIN 298.528
MAX
349.880
MW 297.720
I
-
,r
MAX
6.02313
RN -2.58886
/I
MAX 6.88997
r
(')
MN -3.83193
(K)
(Kb)
MAX 9.00519
MW -8.20301
Fig.3.7
MAX 15.2483
(cont)
- 133 -
M
-12.4911
Time= 50. hrs
0.0
8.00
U
Time= 70. hrs
Time= 60. hrs
~~ 0.0~---------.
00
-
~
-
U
S0.0
, ------
- -T- -- - T-- ----8.00
MAX 14.1688
MA
M -13.9808
MIN
14 .43
-14.128
- -
MAX
--
-
--
-
14.7892
-8.00- --
-
--
-
--
-
V
%I
'
(r/,)
-- ----- - MI -14.4222
-
'
(m/s)
"'
\
(=v
'I,'\
MA
1.88481
MIW-1.89437
IN -2.30068
2.50M
w
MAX
M .- 3
MIN
1.35594
-A29602
MAX
2.53480
36
MfN -1.32125
20.
(K)
,at
(K)
MAX 342.763
MIN 302.118
MA
342.96
.
MN
308.448
MX 343.253
MIN 303.025
,0'
'0b
Sr"
MAX 2.50848
MN -. 548861
MAX 3.56258
MIN
-. 709198
P'
p'
(Kb)
(mb)
MAX
1.29651
0N -1.5088
MAX
1.81901
MN -2.14011
Fig.3.8 - same as fig.3.7 but for exp.
- 134 -
MA
FI2
2.52928
MI -2.99616
Time= 80. hrs
Time= 90. hrs
.00
Time= 100. hrs
00
U
U
(u)
-- .-
-.
. . . .
.
0---
--
-... -
.7p
MAX
MI -15.0606
15.2818
MAX 15.9184
MDI -16.1838
MIN
MAX 18.4433
V
iO
(n/a)
I
-18.6313
V
"v
rr
A\
MAX 7.80896
MW -6.98077
MAz 11.5486
'
I:
')
--
M
'
~AI
'.
(m/s
-8.66744
'
"
w
"W
Cm/
MAX 4.99396
,
MIN -1.77710
--
36
-- ,
,
MAX
10.4832
MN -2.74850
MAX 31.1050
MIN -3.12270
.
--
3
.
,'l
20.
(K)
()
I
MA
343.552
MDI 303.622
MA
1
344.137
301422
MX 344.528
DIN 304.897
I
Mi)
n3,I
.60
(K)
(K)
MAX
4.96809
MIN -.915710
MAX
8.97956
MIN -11155
MAX
9.3091
1.N -1.51771
PI
. .,
, (nib)
(Mb)
MAX 4.81786
Fig.3.8
MIN -L79632
(cont)
- 135 -
Time= 120. hrs
Time= 110. hrs
Time= 130. hrs
u)
-
".
,...
,.---.
""
,
i.
"
IMA 21.2876
"-- -
. . .,-
"q -- ,
" ..-- ---
- .
Mh -18.8893
. ... . . :
(m
-- ,I
MI 23.8281
IN -20.7946
.A.2471.:;
M...
: -. 889
A
MIN -238590
2.2471
(m/s)
-
%
v
A'> .7!
%,
MAX
18.4277
mN-13.9754
~MAXZ
~I
MAX 28.0534
MAX 32.6489
M4 -22.3409
MAX
351.889
MDI 30.472
MAX 22.1484
IN -27.1956
IN
-24110
N.4
I -.
41
(cm/8)
MA
34887MI
308.87
YM 347.742
Mfi
306.273
9'
MAX11.88
MaUS5
Mm567
1698
MAZ
11."21
MN -l.96895
MAX
9.82797
M -15.4371
C7
x=a
p'
(mb)
MAX 14.754
-21.8943
Fig.3.8 (cont)
-
136 -
(il
Time= 130. hrs
Time= 100. hrs
(K)
().
MAX 390.120
(io-,)
I. VI.
-..
MAX 0.290067
_ _,
MI-1.81880
MIN
MAX 401.132
MIW 364.307
-
'-,
MX
3.00200
.0
3.189
. .
(1o-/ms)
IX -2.11930
Fig.3.9 - equivalent potential temperature
4- and perturbation
potential vorticity Q' for exp. FI2 at 100. hrs and 130. hrs
- 137 -
Vdiff
m/s)
I,
'
Time= 90. hrs
>
M"
2.97131
/
,
I,
r
r
P
S
.
.
IN -1.35881
o>
Time= 100. hrs
00
Time= 110. hrs
Ma
14.9823
MIN -11.3721
Fig.3.10 - difference in meridional wind field v between exp.
and exp. I at 90,100,110 hrs. The graph shows FI2 minus I
- 138 -
FI2
300
280
260
240 220
200
180
C 160
140
120
100
80
60 -
40
20
0
l
200
400
I
600
1
IIiI
11
1f
il
l
1
800 1000 1200 1400 1600 1800 2000 2200 2400 2600 2800 3000
x km)
Fig.3.11 - Location of the two O.m/s isotach$ of meridional wind,
at the
lowest resolved level (500. m height), for exp. FI2, versus
time, showing the westward displacement of the disturbance in the frame
of reference in which the zonal basic wind is zero at mid-level
- 139 -
FI4 (20 K)
.24
.22
.20
.18
FI3 (1i K)
.16
A F12 (10
4.
I
0
b
K)
.12
.10
.08
.06
.02
0
I
,
I
I
I
,
I
I
•
I
100
I
•
I
I
120
.
.
"
I
140
hrs
Fig.3.12 - growth rate
r
vs.
-
time for exp.s I,FI2,FI3,FI4
140 -
1I
160
80 -
70
A8=20 K
FI4
60
A=15 K
50 -
FI3
40
Ae=10 K
-
FI2
30
Ae=5 K
FI1
20
-
Ae=0
K
CE =0
10
'10
40
0 20
60I
60
100
80
120
160
140
hrs
FIO
Fig.3.13 - minimum surface pressure vs.
(5 =0 K), FI1,FI2,FI3,FI4
-
141 -
time for exp.s I
(C =0),
Time= 60. hrs
Time= 90. hrs
Time= 100. hrs
I
r.h.
r.h.
&
,.r
MA 1.00000
I
MI 0.904081
r
aO
qe
a.
10"/ms
,
.
r
'
- o
xa
'lKl- iese
or-4i8
MAX0.382828
M I6~
4)
Bsat
p
MAX 0.21392E-04
and
exp.
MIN -. 475359-05
MX 0.407790-04
MI -. 29097E-04
MAX 0.550453Z-04
MY -. 4213623-04
Fig.3.14a - relative humidity, equivalent potential vorticity g,
buoyancy for saturated vertical motions 8S-(t
I e'
,) for
r
I at 60.,90.,100.
hrs
'
-
142 -
y"
10o-m
Time= 100. hrs
Time= 90. hrs
Time= 60. hrs
r.h.
r.h.
N.Y.ea
MAX 1.00000
qe
MIN 0.900006
41
*
10-/ms,
J
AIN,
10-i/ms
~O
MAX 0.879108E-01
MIN -1.04004
MI -2.19364
MAX 0.257404
Bsat
Bsat
(.-2)
MAX 0.3414022-04
IN-.123654-03
Fig.3.14b - same as a) but for exp.
- 143 -
FI2
MAX 0.423828E-04
UIN -. 124337E-03
.22
FL3 (15 K)
.20
.18
.16
FL2 (10 K)
r
I
I
0
b
.12
.10
.08
FLI (5 K)
.06
.02
0
I
i
_
I
I
I
40
I
•
.
60
t
100
120
hrs
Fig.3.15 - same as fig.3.12 but for exp.s L,FL1,FL2,FL3
-
144 -
140
160
80 -
70
60
Ae=15 K
FL3
50
'
-
A8=10 K
FL2
40 -
Ae=5 K
30 -
FLi
CE =0
20-
L
10
20
40
60
100
80
120
hrs
Fig.3.16 - same as fig.3.13 but for exp.s L,FL1,FL2,FL3
-
145 -
140
160
Time= 30. hrs
Time= 40. hrs
8.00
U
3.00
0.M0
0
-
'- - -- - -- - -8.00 -- - -- - -- MAX
139200
HN -13.7150
0.0
(
r- - -- - -- - -- - -- - -- - -- - rn/,)
.,
- -- - --- -8.00 -- - -- - -MA
V
14.1048
N -13.829
'
%
(
(
(m/s)
MAX 1.57488
v
MIN -1.32815
(
V
w"
(cm/8)
MAX 0.94885680
MIN -. 969975.-01
Ma
M
1.84818
-. 388187
332.
324.
,'
316.
(K)
16.
JOB
(K)
-50.
MAX 39.707
MIN 301.700
MAX
339.873
MIN 301.508
(K)
(K)
MAX 0.782410
MIN -. 184255
MAX 1.31912
MIN -. 278387
,'. .
.
p I
(nib)
(nib)
MAX 0.840774
130.
MIN -. 708788
Fig.3.17 - same as fig.3.7 but for exp.
hrs
-
146 -
MIN -L1S090
MAX 1.02227
FL2 between
30
hrs
and
Time= 70. hrs
Time= 60. hrs
Time= 50. hrs
8.00
U
u
.0
--
MAX
13
MaZ 14C3701
M-I 14
1
IN -14.0148
MAX
MN -14.2588
14.8013
-
--
'---
-
--
-
--- -
MAY 15.3900
-
--.
----
MIN -14.7378
2\\:
%
V
V
\\\\
--'-- %\
(,-,/8)
(m/s)
(,, /,
r.
MAX
2.03407
2l66M
MM
3
-2.80244
V4
W -. 98=5
MX 2.4487
MN -. 269482
(m/s)
MAX 4.70024
1)
MN -1.89820
--
. --
(K)
302.373
MAX 340.01
MIN 301.9
MAX 340.380
M
MAX 2.00797
MN -4A258M
MAX 2.92407
MN -. 880847
MAX
340.799
MAX 4.28088
MIW 303.073
MIN -. 900116
pm'
P'
(K)
-
MAX 1.52310
Fig.3.17
MN -2.54748
MAX 223454
MN -1.78828
-\ -
-
(cont)
-
147 -
MAX 3.18783
.0
MN -3.9679
Time= 80. hrs
"--
U
(u)
(M/8)
Time= 90. hrs
. ,o00
Time= 100. hrs
.
-
_
7=.--_.----" "" "
. I
18.2271
MAX
I
L"
5,
MA--. 118
J
2AX
MN1-15.7951
3. 716
- MI -4856-- -= " "
121-
0. 54
%\
%
V
-\
5\\
\
MAX 7.06539
MN -6.8841
MAX
UN -3.28008
w
(cm/u)
4
10.3288
-336.
Ma
19.0728
MN -236898
MA 286587
MN -3.03880
-
20
8.
1)
(K)r
,a
(K)
AX 341.411
MIN 303.743
M=Z 342.042
MIN 304.377
(K)
(K)
6.14502
M -
p
P
t
(nib)
]
I
,
MAX
PI
--
-
~
453483
l
\
I
O
(nib)
MIN -6.11993
Fig.3.17
MAX 10.2722
(cont)
-
148 -
MIN
-15.8976
Time= 110. hrs
u
(/,)
Time= 120. hrs
..
-.. ..-a. s .
"__' - a a,"--
-o,
X
s7.39
MA
1.28
6
MAX 30.7675
u
?u
.0
,-- - --, ,, (rn/u
.0x ,,---_-
IF
f
b
Time= 130. hrs
..-.
M1
-X
N,I -29M8
IN -2.47050
( )(
.33
Ma189.361a
MAX 2596-0
V
w
()
I
.
N
3.3
-6.799L
NM-. =N0
X
3.2
MAX
22.0.48
6
MAX 28338
o
0.8
D -82.96
-M9.8Z40
(K)
(K)
x"Z 347.339
WN
305.871
(mb L,
WU 17.1219
w
352.339
MN
306.M4
X
369.201
-IN 306500
it
-22.3MA
Fig.3.17
IIN 25.3
MWN-2"131
(cont)
- 149 -
MAX 37.086
UN -
80
70
60
50
3600.Km
3300.Km
3000.Km
P12
40
30
2700.Km
20
2400.Km
10
0
40
60
100
80
120
140
hrs
F9g.3.18 - minimum surface pressure vs.
time for experiments
similar to FI2 but with varying horizontal length of the domain. Exp.
FI2 has a 3000 Km wavelength
- 150 -
160
4.1
This work consisted of two parts.
studied
In the
2)
we
the behavior of 2-D Eady waves in a saturated environment.
We
assumed that a slantwise
convective
adjustment
reduce the lapse rate to symmetric neutrality.
base state with uniform vertical shear and
vorticity.
first
(chapter
had
taken
place
to
Therefore we designed a
zero
equivalent
potential
The release of latent heat was accounted for explicitly in
the numerical model (section 2.3) and
implicitly,
by
conserving
the
equivalent potential temperature in the updraft, in the analytic models
(this representation assumes that all upward motion is saturated).
numerical
experiments
The
confirmed the finite amplitude semi-geostrophic
results as far as the structure of the eigenfunctions is concerned, but
revealed
that
the
non-geostrophic
contributions
ignored
in
semi-geostrophic approximation have a large impact on the growth
-0
in the limit
, where the SG system is singular.
the
rates
Specifically the
non-dimensional growth rates increase as the Rossby number of the
flow
decreases,
as
was
expected
from
previous
studies
non-geostrophic baroclinic instability, and in a larger measure as
mean
of
the
system gets closer to conditional symmetric neutrality.
The linearized
2-level PE model
rates
numerically
waves
simulated
exhibit
reproduced
reproduces
in
some
dry
fairly
well
the
growth
waves at small amplitude.
features
models,
that
are
of
the
The moist baroclinic
commonly
observed
and
not
such as an updraft much narrower than the
downdraft, a sharp gradient of vertical velocity across the cold front,
and
frontal
collapse
occurring
-
151 -
at
low
level
first,
and
not
symmetrically at top and bottom boundaries.
The
asymmetry
of
the
w
field accounts for a much shorter wavelength of the most unstable mode,
that is about 60% of the dry most unstable wavelength,
rate
about
as
twice
conversion
terms
'baroclinic'
large.
reveals
conversion
A
that
cursory
during
mean
from
most
of
the
to eddy potential
energy
being
provided
by
the
the
release
a
growth
of the energy
examination
than the eddy potential to eddy kinetic term w'l ,
potential
with
evolution
V''
the
is smaller
missing
eddy
of latent heat of
condensation.
Although this first part gives results in the
both
the
compare
growth
rate
favourably
cyclones.
As
we
the
direction,
and the structure of the eigensolutions do not
with
observed
the
mentioned
normal modes.
cases
of
explosive
marine
in the Introduction some of the observed
features may however be due to
studying
desired
non-modal
growth
while
we
are
only
In Chapter 3 we included a representation
of heat and moisture fluxes at the lower boundary in the models of
previous
chapter.
A
heating
term
proportional
velocity of the perturbation and to an air-sea
linearization
of
entropy
region
the meridional
difference
only,
having
SG
model,
most
unstable
the
aloft.
modes thus found have a growth rate that increases
almost linearly with the air-sea temperature difference.
simulations,
in
assumed that only there can latent heat
flux from the lower boundary be realized as sensible heat input
The
(a
a drag law for equivalent potential temperature) was
added to the thermodynamic equation of the 2-level
updraft
to
the
performed
with
the
The numerical
model of section 2.3 with fluxes of
- 152 -
heat in the form of drag laws for potential temperature
confirm
result,
this
respect to
the
structure,
which
addition
in
mid-level
is
base
very
narrow
at
and
to
a
the
beginning of the phase of
'explosive' growth and becomes broader later on.
in
two
distinct
phases
boundary accumulates
low-pressure
increased,
confined
in
-
warm-core
occurs
in the first one the heat from the bottom
lowest
the
levels,
the
deepening
of
the
center and the growth rate of kinetic energy are modestly
but
to
the
the
differences
low
levels.
with
the
no-flux
simulations
are
In a second stage heat is efficiently
develops,
a hurricane-like circulation superimposed on the moist baroclinic
wave, the growth rate
values
thermal
The evolution
transported upward in the updraft region, a narrow warm core
with
moisture,
a retrograde phase speed with
to
wind,
and
two
to
perturbation
kinetic
energy
reaches
peak
three times higher than in the no-flux experiments and
the deepening rate
conventional
of
of
the
central
surface
pressure
threshold for explosive cyclones.
is
above
the
When compared with the
numerical simulations we observe that the two-level model qualitatively
reproduces
the
increase
in
growth-rate
in the 'explosive' phase of
growth but it misses the 'shallow' early phase because of poor vertical
resolution.
Moreover,
a complex vertical structure is created by the
more realistic model in order to transport heat upward,
while
in
the
two-level heat was applied directly at mid-level.
The models also give
conflicting results as to the phase speed of
explosive
the
cyclones,
that appear to be faster than the mean wind in the two-level and slower
than the mean wind in the numerical model.
The
observations
of
real
cases leave uncertainties on this point so we cannot decide which model
- 153 -
of
performs better from this point
view.
limitations
The
both
of
models (especially with regard to the linearized saturation law used in
the numerical simulations)
sections.
deepening
The
have
been
discussed
appropriate
the
in
rate of the pressure perturbation increases
almost linearly with the air-sea temperature difference, and it must be
in
kept
that the same entropy flux can be achieved with smaller
mind
temperature differences if the air in contact with the ocean surface is
not
as
saturated,
seems to be the case, so that a larger latent heat
flux can take place.
Our goal at the outset had been to provide a
for
developments
explosive
maritime
influences on baroclinic instability.
study
seems
to
indicate
by
mechanism
plausible
non-adiabatic
studying
Specifically,
(1984)
Roebber's
that the distribution of deepening rates of
mid-latitude cyclones is biased toward the strongest events, suggesting
that a distinct physical mechanism is active.
The results of Chapter 2
well
show features that are commonly observed in land cyclones as
therefore
do
not offer the means to discriminate between ordinary and
explosive developments in the way implied
They
are
relevant
for
the
description
by
of
anywhere moisture is present, presumably with
features
and
depending
on
the
relative
climatological
baroclinic
humidity,
improvement over those earlier studies mentioned
and
in
development
less
or
more
studies.
enhanced
they offer some
the
Introduction
that do not consider the ability of the marine environment to provide a
sustained source of moist entropy.
are
more
frequent
It is likely
that
'moist'
events
over sea and 'dry' events over land, but when land
-
154 -
and sea cyclones are considered together there is no 'a priori'
to
expect
reason
non-gaussian distribution of events from this mechanism.
a
On the other hand the experiments of Chapter 3, in addition to a better
adherence
to
the
observed
suggest that only those
amplitude
(or
events
normal
that
of
explosive cyclones, seem to
reach
above
a
'threshold'
of
growth rate) are further enhanced by the effect of heat
fluxes from the ocean.
a
behavior
It is then conceivable that the deviations from
distribution,
if
they
are
confirmed
extensive samples of events, are
caused
'feedback'
disturbances
mechanism
in
those
by
the
by
studies of more
activation
that
are
of
this
already
the
strongest beforehand.
We have only been studying this problem
point
from
the
'normal
mode'
of view, and the common assumption in this regard is that if the
development lasts long enough the most unstable
observed,
after
the transients are dissipated away.
the work by Farrell,1982, 1984,1985
initial
normal
has
shown
that
mode
an
be
Recently however
the
'non-modal'
growth may be more important than previously thought.
model, specifically, the suggested presenceof
will
amplitude
In this
threshold
makes it possible that a favorable initial condition (as well as any of
the observed features that we have ignored, like the interaction with a
pre-existing
reach
trough in the upper level flow) may help the perturbation
sooner
efficiently
the
use
stage,
the
or
energy
the
organization,
in
which
it
can
provided by the heat fluxes at the lower
boundary.
- 155 -
In addition to the previously discussed linearized saturation law,
at
one
least
other
problem related to the two-dimensionality of the
model appears in the results - the onset
of
the
explosive
phase
is
related to a 'hurricane' type of organization, and while 2-D baroclinic
we
know
that
the natural structure of hurricanes is axisymmetric, so
that the 3rd dimension, in a cartesian space, is as
other
two.
significantly
extent,
meridional
waves are qualitatively similar to waves of finite
Whether
or
alter
our
not
the
conclusion
addition
that
as
important
the
of a 3rd dimension would
the
most
distinguishing
features of mid-latitude explosive cyclones are due to air-sea exchange
of latent and sensible heat transported upward by
convective
activity
cannot be determined at this point, and only a 3-D model can answer the
question.
The presence of a 'hurricane' phase at the beginning of
the
exposive growth in real cases could however be detected by observation,
since its duration is, in the model, of the order of 20.
- 155a -
hours.
BIBLIOGRAPHY
Anthes.R.A.,Y.-H.
Kuo and J.R.Gyakum,1983 - Numerical simulations
of a case of explosive marine cyclogenesis.
Mon.Wea.Rev.,111,1174-1188
Aubert,E.J.,1957 - On the release of latent heat as
large-scale atmospheric motions.
Bennetts,D.A.
instability
-
and
a
in
J.Meteor.,14,527-542
B.J.Hoskins,1979
possible
factor
a
-
Conditional
explanation
symmetric
for
frontal
rainbands.
of
moving
cyclones.
Quart.J.Roy.Meteor.Soc,105, 945-962
-
Bjerknes,J.,1919
On
structure
the
Geofys.Publ.,1,1-8
Blumen,W.,1979
-
On
short
wave
baroclinic
instability.
J.Atmos.Sci.,37, 1925-1933
Bosart,L.F.,1981 - The President's Day snowstorm of 18-19 February
1979:
a subsynoptic-scale event.
Chang,C.B.,D.J.Perkey
and
Mon.Wea.Rev.,109,1542-1566
W.-D.Chen,1987
structure bf an intense oceanic cyclone.
Charney,J.G.
depression.
Observed
dynamic
Mon.Wea.Rev.,115,1127-1139
Charney,J.G.,1947 - The dynamics of long
westerly current.
-
waves
in
a
baroclinic
J.Meteor.,2,136-163
and A.Eliassen,1964 - On the growth of the hurricane
J.Atmos.Sci.,21,68-75
- 156 -
Danard,M.B.,1964 - On the influence of
cyclone development.
Davis,C.A.
influence
of
released
latent
heat
on
J.Appl.Meteor.,3,27-37
and K.A.Emanuel,1988 - Observational evidence for
surface heat fluxes on maritime cyclogenesis.
the
Submitted
to Mon.Wea.Rev.
Eady,E.T.,1949 - Long waves and cyclone waves.
Tellus,1,33-52
Emanuel,K.A.,1985 - Frontal circulation in the presence
moist symmetric stability.
Part I:
small
J.Atmos.Sci, 42,1062-1071
Emanuel,K.A.,1986 - An air-sea
cyclones.
of
interaction
Steady-state maintenance.
theory
for
tropical
J.Atmos.Sci,43,585-604
Emanuel,K.A.,1988 - Observational evidence of slantwise convective
adjustment.
Submitted to Mon.Wea.Rev.
Emanuel,K.A.,M.Fantini
instability
convection.
in
an
Part
and
environment
I:
A.J.Thorpe,1987
-
Baroclinic
of small stability to slantwise moist
Two-dimensional
models.
J.Atmos.Sci.,
44,
1559-1573
Farrell,B.F.,1982 baroclinic flow.
The
initial
growth
of
disturbances
in
a
J.Atmos.Sci., 39, 1663-1686
Farrell,B.F.,1984
-
Modal
and
J.Atmos.Sci., 41, 668-673
-
157 -
nonmodal
baroclinic
waves.
Farrell,B.F.,1985 - Transient growth of damped
baroclinic
waves.
J.Atmos.Sci., 42, 2718-2727
Gall,R.L.,1976 - The effect of released
baroclinic waves.
latent
heat
in
growing
J.Atmos.Sci,33,1686-1701
-
Green,J.S.A.,1960
A
problem
in
baroclinic
stability.
Quart.J.Roy.Meteor.Soc., 86,237-251
Gyakum,J.R.,1983 synoptic aspects.
On
the
evolution
the
QEII
storm.
I:
Mon.Wea.Rev.,111,1137-1155
Gyakum,J.R.,1983b - On the
evolution
dynamic and thermodynamic structure.
Harrold,T.W.
of
and
the
QEII
storm.
II:
Mon.Wea.Rev.,111,1156,1173
K.A.Browning,1969
baroclinic disturbance.
of
-
The
polar
low
as
a
Quart.J.Roy.Meteor.Soc.,95,710-723
Hoskins,B.J.,1975 - The geostrophic momentum approximation and the
semi-geostrophic equations.
J.Atmos.Sci.,32,233-242
Hyun,J.M.,1981 - Separate and combined effects of static stability
and
shear
current.
variation
on
the
baroclinic
instability
of a two-layer
J.Atmos.Sci,38, 322-333
Jacobs,W.C.,1942
atmosphere.
-
On
the
energy
J.Mar.Res.,5,37-66
- 158 -
exchange
between
sea
and
Keiser,D.
and
R.A.Anthes,1982
-
The
influence
of
planetary
boundary layer physics on frontal structure in the Hoskins - Bretherton
horizontal shear model.
Klemp,J.B.
and
J.Atmos.Sci., 39, 1783-1802
R.B.Wilhelmson,1978
-
three-dimensional convective storm dynamics.
The
simulation
of
J.Atmos.Sci.,35,1070-1096
Lilly,D.K.,1960 - On the theory of disturbances in a conditionally
unstable atmosphere.
Liou,C.-S.
forecasts
Mon.Wea.Rev.,88,1-17
and
of
R.L.Elsberry,1987
an
explosively
-
Heat
deepening
budget
analyses
maritime
and
cyclone.
Mon.Wea.Rev.,115,1809-1824
Mak,M.K.,1982 - On moist quasi-geostrophic baroclinic instability.
J.Atmos.Sci.,39,2028-2037
Malkus,J.S.,1962 - Large scale interactions.
The Sea, ch.4.
John
Wiley and Sons
Manabe,S.,1956
condensation
-
to
On
the
the
contribution
change
in
of
heat
released
pressure
by
pattern.
J.Meteor.Soc.Japan,34,12-24
Manabe,S.,J.Smagorinsky
climatology
of
and
R.F.Strickler,1965
-
Simulated
a general circulation model with a hydrological cycle.
Mon.Wea.Rev.,93, 769-798
- 159 -
Mansfield,D.A.,1974 - Polar Lows:
disturbances
in
the development
cold
of
baroclinic
air
outbreaks.
Quart.J.Roy.Meteor.Soc.,100,541-554
Mason,P.J.
of
and R.I.Sykes,1982 - A two dimensional numerical study
horizontal
layer.
roll vortices in an inversion capped planetary boundary
Quart.J.Roy.Meteor.Soc.,108,801-823
Matsumoto,S.,S.Yoshizumi and M.Takeuchi,1970 - On the structure of
the
'Baiu front' and the associated intermediate-scale disturbances in
the lower atmosphere.
J.Meteor.Soc.Japan,48,479-491
McWilliams,J.C.
planetary
and
P.R.Gent,1980
circulations
in
the
-
Intermediate
atmosphere
models
of
and
ocean.
state.
Ph.D.
J.Atmos.Sci.,37,1657-1678
Nehrkorn,T.,1985 - Wave-CISK in a baroclinic basic
dissertation.
Massachusetts Institute of Technology
Nitta,T.,1964 - On
cyclone
due
to
the
the
development
release
of
of
latent
relatively
heat
by
small
scale
condensation.
J.Meteor.Soc.Japan,42,260-267
Nitta,T.
development
and
of
the
Y.Ogura,1972
-
Numerical
intermediate-scale
atmosphere - J.Atmos.Sci,29, 1011-1024
- 160 -
cyclone
simulation
in
a
of
moist
the
model
-
Ooyama,K.,1969
tropical cyclones.
Numerical
simulation
of
the
life
cycle
of
J.Atmos.Sci.,26,3-40
Orlansky,I.,1968 - Instability of
frontal
waves.
J.Atmos.Sci.,
25, 178-200
-
Orlansky,I.,1986
meso-alpha cyclones.
Localised
baroclinicity:
a
source
for
J.Atmos.Sci., 43, 2857-2885
Pedlosky,J.,1964 - The stability of currents in the atmosphere and
ocean:
Part I.
J.Atmos.Sci.,21,201-219
Pedlosky,J.,1979 - Geophysical Fluid Dynamics.
Pettersen,S.,D.L.Bradbury
cyclone
models
in
and
Springer Verlag
-
K.Pedersen,1962
relation
to
heat
and
The
Norwegian
cold
sources.
Geofys.Publ.,24,243-280
Pettersen,S.
and
-
S.Smebye,1971
extratropical cyclones.
On
the
simple
three-dimensional
study of large-scale extratropical flow patterns.
On
development of cyclones.
Rasmussen,E.,1979
disturbance.
of
Quart.J.Roy.Meteor.Soc., 97, 457-482
Phillips,N.A.,1951 - A
Pyke,C.B.,1965 -
development
the
role
of
air-sea
model
for
the
J.Meteor.,8,381-394
interaction
in
the
Bull.Amer.Meteor.Soc.,46,4-15
-
The
polar
low
as
an
Quart.J.Roy.Meteor.Soc.,105,531-550
-
161 -
extratropical
CISK
Rasmussen,E.
a
summary
of
the
1986, Oslo, Norway.
and M.Lystad,1987 - The Norwegian Polar Low project:
international
conference on Polar Lows, 20-23 May
Bull.Amer.Meteor.Soc.,68,810-816
Reed,R.J.,1979
-
Cyclogenesis
in
polar
air
streams.
Mon.Wea.Rev.,107,38-52
Reed,R.J.
and M.D.Albright,1986
cyclogenesis in the Eastern Pacific.
-
A
case
Rotunno,R.
for
tropical
of
explosive
Mon.Wea.Rev.,114,2297-2319
Roebber,P.J.,1984 - Statistical analysis and
of explosive cyclones.
study
updated
climatology
Mon.Wea.Rev.,112,1577-1589
and K.A.Emanuel,1987 - An air-sea
cyclones.
Part
II:
interaction
Evolutionary
nonhydrostatic axisymmetric numerical model.
using
a
J.Atmos.Sci.,44,542-561
Rotunno,R, and J.Klemp,1985 - On the rotation and
simulated supercell thunderstorms.
study
theory
propagation
of
J.Atmos.Sci, 42, 271-292
Sanders,F.,1986 - Explosive cyclogenesis in the west-central North
Atlantic
Ocean
behavior.
Part
I:
composite
structure
and mean
Mon.Wea.Rev., 114,1781-1794
Sanders,F.
the 'bomb'.
1981-1984.
and J.R.Gyakum,1980 - Synoptic-dynamic climatology
Mon.Wea.Rev.,108,1589-1606
- 162 -
of
Sardie,J.M.
and
T.T.Warner,1983
development of polar lows.
Satyamurty,P.,
in
the
mechanism
for
the
and
A.D.Moura,1982
-
Subsynoptic-scale
J.Atmos.Sci.,39,1052-1061
Smagorinsky,J.,1956
processes
On
J.Atmos.Sci.,40,869-881
V.B.Rao
baroclinic instability.
-
-
numerical
On
the
prediction
inclusion
models.
of
moist
Bericht
adiabatic
des
Deutchen
Wetterdienst,38,82-90
Soong,S.-T.
and Y.Ogura,1972 - Numerical simulation of warm
development in an axisymmetric cloud model.
Staley,D.O.
and R.L.Gall,1977 -
baroclinic instability.
Stone,P.H.,1966
-
On
rain
J.Atmos.Sci.,31,1262-1285
the
wavelength
of
maximum
J.Atmos.Sci.,34,1679-1688
On
non-geostrophic
baroclinic
stability.
J.Atmos.Sci., 23, 390-400
Stone,P.H.,1972 - Comments on 'Ageostrophic effects on
instability'.
baroclinic
J.Atmos.Sci.,29,986-987
TangC.-M.
and G.H.Fichtl,1983 - The role of latent heat
in baroclinic waves - without beta-effect.
Thorpe,A.J.
J.Atmos.Sci.,40,53-72
and K.A.Emanuel,1985 - Frontogenesis in the
of small stability to slantwise convection.
- 163 -
release
presence
J.Atmos.Sci,42,1809-1824
Tokioka,T.,1970 - Non-geostrophic and non-hydrostatic stability of
a baroclinic fluid.
J.Meteor.Soc.Japan,48,503-520
-
Tracton,M.S.,1973
The
role
of
development of extratropical cyclones.
Wang,B.
zonal
with
convection
in
the
Mon.Wea.Rev.,101,573-593
and A.Barcilon,1986 - Moist
flow
cumulus
conditionally
stability
of
unstable
a
baroclinic
staratification.
J.Atmos.Sci.,43,705-719
Wash,C.H.,J.E.Peak,W.E.Calland
and
W.A.Cook,1988
-
Diagnostic
study of explosive cyclogenesis during FGGE - Mon.Wea.Rev.,116,431-451
Weng,H.Y.
explosive
and
A.Barcilon, 1987
cyclogenesis
in
a
-
Favorable
modified
environments
two-layer
Eady
for
model.
Tellus,39A,202-214
Winston,J.S.,1955 - Physical aspects of rapid cyclogenesis in
Gulf of Alaska.
Tellus,7,481-500
- 164 -
the
LIST OF FIGURES
-
Fig.2.1
Structure
of
the
2-level
model
and
levels
where
and
'dry'
variables are defined ...pag.31
Fig.2.2 - Subdivision of the
region.
model
in
a
'moist'
a
Horizontal coordinate X is defined separately in each region.
Soutions are matched at points A=B and C ...pag.31
Fig.2.3 - Growth rate 6' versus half-width of
for
different
values
of
r;
r=l
is
lines
join
points
of constant
downdraft
L
the dry model; r=O is obtained
asymptotically and confirmed numerically for rawto
Fig.2.4 - As 2.3 but (P vs.
the
.6
...pag.42
total wavelength L
LL , i.e.
2Lf~\+ );
dashed
points that lay on vertical
lines in fig.2.3, thus showing the reduction of total wavelength due to
the narrow updraft ...pag.43
Fig.2.5 - Top:
geostrophic
r=0.08.
Dimensionless W
coordinate
Bottom:
geopotential fI t
X,
same but
for
for
and
the
most
meridional
a
as
rapidly
function
of
the
velocity
growing mode when
v
and
modified
'F' at upper and lower level ...pag.48
Fig.2.6 - Same as 2.5 but
in
physical
space
when
the
maximum
low-level vorticity is 10f,...pag.49
Fig.2.7 - Contours of constant
6" on the r - Ro
L2 = 1.8 ...pag.60
- 165 -
plane
for
fixed
Fig 2.8 a) - Growth rate
downdraft
)
width
Sw
(solid) and
ratio
of
updraft
to
(dashed) for various values of Ro at fixed r = 0.
Ro = 0 is the GM approximation ...pag.61
Fig.2.8 b) - same as a) but for r = .1 ...pag.62
Fig.2.9 - Domain and arrangement of
staggered
grid
are located
at
covering
v
the domain.
points
dependent
variables
on
the
All the thermodynamic variables
(adapted from
Rotunno
and
Emanuel,1987)
...pag.78
Fig.2.10
100,150,170,180,
u,v,w,-
p
190,200
hrs.
fields
Note
for
exp.
AO
at
times
that contour spacing is not the
same for perturbation fields at different times ...pag.79
Fig.2.11 - Growth rate
C vs.
time
for
exp.
AO
and
for
the
two-level analytic models with the same parameters as AO ...pag.81
Fig.2.12 - v and w fields for exp.
Al from 10 to 50 hrs
Fig.2.13 - same as 2.11 but for exp.
Fig.2'.14 - same as 2.10 but for exp.
...pag.82
Al ...pag.83
Al
from
100
to
150
hrs
and
150
hrs
...pag.84
Fig.2.15 - q
1
2,
1,!f ,
...pag.86
- 166 -
for exp.
Al at 100
Fig.2.16 -
same as 2.11 but for exp.
Fig.2.17 - same as 2.10 but for exp.
A2 ...pag.87
A2
from
280
to
300
hrs
...pag.88
Fig.2.18 - Growth rate
C
vs.
time for exp.s Al and A3
Fig.2.19 - same as 2.10 but for exp.
A3
from
...pag.89
150
hrs
160,180,200
hrs
130
to
...pag.90
Fig.2.20 - same as 2.11 but for exp.
B ...pag.91
Fig 2.21 - same as 2.11 but for exp.
C ...pag.92
Fig.2.22 - same as
2.12
but
for
exp.
C
at
...pag.93
Fig.2.23 - Growth rate T* vs.
time for exp.s D,E,F
and
for
the
two-level PE analytic model with the same parameters ...pag.94
Fig.2.24 - same as 2.11 but for exp.
H ...pag.95
Fig.2.25 - same as 2.12 but for exp.
G at 20 and 50 hrs
Fig.2.26 - same as 2.12 but for exp.
H
at
20,50
and
..,.pag.96
100
hrs
...pag.97
Fig.3.1 -
flux
from
L2=1.8.
Eigenvalues
the
lower
boundary
Solid:most unstable
S=
I
.
6 )i
U.
O
vs.
A
...pag.115
- 167 -
of the SG 2-level model with heat
heating parameter
-CO
, dashed:
F'
least
for r=.l,
unstable
Fig.3.2 - Eigenvalues
heating
text).
parameter
Solid:
updraft
to
Po
G xA
for
N
the
vs.
L2 for various values of the
;
4
most unstable case 9-
6 ; long dash:
growth rate
downdraft
40
short
dash:
O
(see
ratio of the width of
the
relative phase speed
0 "
...pag.116
Fig.3.3 - Form of
S
=O
.
the
eigensolution
derivative
constant can be added to both pl and p2.
g=o
Contours
tp
, on the
of
constant
,r plane.
Fig.3.5 - growth rate
q
v,
from
time.
50.
hrs
to
130.
so
r=.l,
=.17,
that
an
0=.23.
arbitrary
growth
rate
C
for
L2=1.8,
...pag.118
vs.
time for exp.
FI2
I ...pag.129
...pag.130
l,p'fields for exp.I, shown every
hrs.
a =2.,
...pag.117
Fig.3.6 - same as fig.3.5 but for exp.
Fig.3.7 - u,v,w,&
L2=1.8,
C =1.5,
Eigenvalues for this mode are
Pressure is known only from it
Fig.3.4 -
for
10.
hours
Contour interval is not constant with
...pag.131
Fig.3.8 -
same as fig.3.7 but for exp.
FI2
Fig.3.9 - equivalent potential temperature
potential
vorticity
for
exp.
...pag.137
- 168 -
FI2
at
...pag.134
r
100.
and
perturbation
hrs and 130.
hrs
Fig.3.10 - difference in meridional wind field v between exp.
and exp.
I at 90,100,110 hrs.
The graph shows FI2 minus I ...pag.138
Fig.3.11 - Location of the two O.m/s isotach of
at
the
lowest
FI2
resolved level (500.
meridional
m height), for exp.
wind,
FI2, versus
time, showing the westward displacement of the disturbance in the frame
of
reference
in
which
the
zonal
basic
wind
is zero at mid-level
...pag.139
Fig.3.12 - growth rate
1
vs.
time
for
exp.s
I,FI2,FI3,FI4
...pag.140
Fig.3.13 - minimum surface pressure vs.
FIO
and
exp.
(&W=0 K),
(C =O),
FI1,FI2,FI3,FI4 ...pag.141
Fig.3.14a - relative humidity, equivalent potential vorticity
q
buoyancy
for
for
saturated
I at 60.,90.,100.
Fig.3.14b -
vertical motions
* +Bt
hrs ...pag.142
same as a) but for exp.
FI2
...pag.143
Fig.3.15 -
same as fig.3.12 but for exp.s L,FL1,FL2,FL3 ...pag.144
Fig.3.16 -
same as fig.3.13 but for exp.s L,FL1,FL2,FL3 ...pag.145
Fig.3.17 - same as fig.3.7 but for exp.
130.
time for exp.s I
hrs
..pag.146
- 169 -
FL2 between
30
hrs
and
pressure
Fjg 3.18 - minimum surface
similar
vs.
time
for
experiments
to FI2 but with varying horizontal length of the domain.
FI2 has a 3000 Km wavelength ...pag.150
-
170 -
Exp.
LIST OF TABLES
Table 2.1 - comparision of exact and
approximate
eigenvalues
of
the 2-level SG model ...pag.41
Table 2.2 - eigenvalues of 2L
PE
model
for
various
r
and
Ro
...pag.55
Table 2.3 - cut-off L2 for various Ro in the 2L PE model ...pag.57
Table 2.4 - total wavelength of eigensolutions of 2L PE model
for
various Ro ...pag.59
Table 2.5 -
base
state
thermodynamic
variables
for
exp.
Al
...pag.75
Table 2.6 - parameters of the numerical model ...pag.76
Table 2.7 - summary of experiments presented in sect.2.3 ...pag.77
Table
3.1
-
summary
of
experiments
...pag.128
-
171 -
presented
in
sect.3.2
E non pur io qui
anzi n'e' questo
che tante lingue
a dicer millibar
piango bolognese:
loco tanto pieno
non son ora apprese
tra Savena e Reno
adapted from:
Inferno,XVIII,58-61
- 172 -