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ATMOSPHERIC SCIENCE LETTERS
Atmos. Sci. Let. 7: 2–8 (2006)
Published online in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/asl.121
An idealized numerical simulation of mammatus-like
clouds
Katharine M. Kanak1 and Jerry M. Straka2 *
1 Cooperative Institute for Mesoscale Meteorological Studies (CIMMS), University
2 School of Meteorology, University of Oklahoma, Norman, Oklahoma, USA
*Correspondence to:
Jerry M. Straka, School of
Meteorology, University of
Oklahoma, 100 East Boyd
Street, Room 1310, Norman,
Oklahoma, 73019, USA.
E-mail: jstraka@ou.edu
of Oklahoma, Norman, Oklahoma, USA
Abstract
A three-dimensional numerical simulation of mammatus-like clouds is presented. A portion
of a cirrus outflow anvil cloud is simulated including cloud ice and snow microphysical
representations. The simulated mammatus clouds appear in a cellular pattern and are
compared with the few available previously published physical observations of mammatus.
Copyright  2006 Royal Meteorological Society
Keywords:
mammatus; numerical simulation; ice microphysics; cirrus outflow anvils
Received: 15 August 2005
Revised: 11 October 2005
Accepted: 4 January 2006
1. Introduction
Mammatus clouds are defined as ‘hanging protuberances, like pouches, on the under surface of
a cloud’ (Huschke, 1959). Mammatus can appear
in clusters of unequal sized lobes (Ludlum and
Scorer, 1953), cellular type patterns (Agee, 1975;
Schaefer and Day, 1981), linear patterns (e.g.
Imai, 1957; Warner, 1973), or wave type patterns (e.g.
Clarke, 1962; Winstead et al., 2001). Mammatus have
also been documented in association with many types
of cloud morphologies including in association with
high level cirrus not associated with any thunderstorms
(Hlad, 1944; Schultz et al. 2006). In this article, only
the case of mammatus associated with a cirrus outflow
anvil cloud, associated with a deep convective thunderstorm, will be considered, and in this case, it is a
cellular type formation.
A picture of a typical field of mammatus clouds
(Figure 1(a)) shows protuberances (or lobes) hanging
from the underside of a parent cumulonimbus cirrus
outflow anvil cloud that occurred on 20 May 2001 at
2251 UTC. The mammatus clouds appear smooth and
are approximately equally spaced, with some evidence
of smaller lobe structures within them (Figure 1(a)).
The mammatus continued to be present at this location
for over an hour following the time of the photograph. A National Weather Service (NWS) sounding
was released ∼5 km from the site of the photograph,
valid for 0000 UTC 21 May 2001, and is shown
in Figure 1(b). Since most soundings are released
at about 2315 UTC, we expect this sounding balloon may possibly have penetrated the mammatus
field or at least the parent anvil cloud. It can be
inferred from Figure 1(b) that mammatus might be
located at about 400 mb, at a temperature of about
Copyright  2006 Royal Meteorological Society
−25 ◦ C, and in the vicinity of a shallow inversion as
labeled on Figure 1(b). Thus, the mammatus shown
in Figure 1(a), likely comprise ice crystals and snow
aggregates. The region below the assumed mammatus cloud base is characterized by a much drier layer,
which would support sublimation of any particles that
fell below cloud base (Figure 1(b)).
Although there have been many mammatus formation mechanisms proposed, we examine the three
mechanisms proposed by Scorer (1958): subsidence
of a cloud layer, hydrometeor fallout, and evaporation sublimation of hydrometeors below cloud base.
In particular, the specific case of fallout of frozen
snow aggregates from the parent cirrus anvil cloud is
emphasized and is examined through the use of idealized numerical simulation. The simulation is designed
to include only a portion of the cirrus anvil cloud in
which mammatus-like clouds are manifest. The entire
simulated cloud layer subsides, but at a rate different
from the terminal velocity of the snow aggregate particles, and sublimation of the snow aggregates occurs at,
or just below cloud base. Thus, although the focus of
the simulation is to mimic the fallout of hydrometeors,
it may be possible that all three of Scorer’s mechanisms may be occurring. This article is intended to
present work in progress and its purpose is to present
preliminary results. To the authors’ knowledge this
may be one of the first numerical simulations of mammatus.
2. Methodology
2.1. Model description
The numerical model used for the current experiment is a three-dimensional, fully compressible, nonhydrostatic model. The model equations are given
Numerical simulation of mammatus-like clouds
(a)
3
simulation include sublimation and deposition of ice
and snow aggregates, along with aggregation of ice
crystals into snow aggregates. As there is little or no
available cloud water in the simulation domain, there
is no riming and thus, no graupel or other types of ice
particles form.
2.2. Experiment design
(b)
Figure 1. (a) Photograph from Norman, Oklahoma, USA, on 20
May 2001 at 2251 UTC. (b) National Weather Service (NWS)
sounding from Norman, Oklahoma, USA, on 21 May 2001 at
0000 UTC, courtesy of University of Wyoming online sounding
archives. Bold lines denote the upper and lower boundaries of
the numerical simulation domain
in Carpenter et al. (1998). The velocity fields are
advected with a sixth-order local spectral scheme
(Straka and Anderson, 1993). Divergence and the
pressure gradient are solved using sixth-order centered
spatial differences. Finally, the scalars are advected
with a sixth-order Crowley flux scheme (Tremback
et al., 1987) with a flux limiter (Leonard, 1991). The
subgrid turbulence closure is a 1.5-order scheme
(Deardorff, 1980). The integration of the fast and slow
solution modes are split from each other with the
centered-in-time leapfrog scheme used for the slow
modes, and an explicit forward–backward solver for
the pressure gradient in the velocity equations and
mass flux divergence in the pressure equation.
The microphysics of the model is described by
Straka and Mansell (2005). Water vapor and ice crystal mixing ratios are specified in the initial cirrus anvil
cloud using an inverse exponential distribution and
snow aggregates form from the ice crystals. There are
numerous other larger ice categories available in the
model, but the processes that activate them are not
occurring in this simulation. The microphysical processes that are represented for the conditions of this
Copyright  2006 Royal Meteorological Society
The model domain is intended to represent a roughly
rectangular portion of a thunderstorm cirrus outflow
anvil at some distance downwind of the main thunderstorm updraft, as mammatus are often observed to be
located at great distances from the main thunderstorm
(e.g. Stith (1995) reports mammatus located 70 km
from the main storm updraft (see Stith’s Figure 1(b)).
The model domain is 4000 × 4000 × 6000 m with
a 50 m grid spacing in all directions. The domain
extends from 3500 m above ground level (AGL) to
9500 m AGL in the vertical. A time step of 1 s can
be used for the slow modes, while the small time step
0.04 s is used for the fast modes to ensure numerical
stability. Periodic lateral boundary conditions are used.
At the upper and lower boundaries, the vertical velocity is equal to zero and the boundaries are specified as
free-slip.
The initial conditions are horizontally homogeneous
and mean potential temperature and mean dewpoint
are specified using the observed sounding thermodynamic data (Figure 1(b)). No mean winds are prescribed in this simulation. The model is initialized by
specifying a constant ice crystal mixing ratio, between
8500 and 9500 m, of 1.25 × 10−3 Kg Kg−1 (consistent with the values observed by Heymsfield and
Knollenberg (1972) and Stith (1995). Then random
perturbations of ice crystal mixing ratio of ±0.25 ×
10−3 Kg Kg−1 are added at the initial time step in
this layer. While it is not known if the amplitude of
these perturbations is reasonable, a reasonable snow
aggregate field is generated by this cloud ice field.
These fields serve the purpose of providing an initial
microphysical state field that falls into the layers below
owing to gravitational settling and loading, which initiate an initial vertical motion field.
3. Results
Figure 2(a) shows a three-dimensional isosurface
movie of the 1 × 10−5 m snow aggregate diameter
surface for the full simulation domain (except the top
20 grid points have been excluded so that a time label
could be included) at times t = 10 min to t = 30 min.
This diameter is the characteristic diameter, which is
the inverse of the slope of the size distribution and
the mean diameter for an inverse exponential distribution. Hereafter, the characteristic diameter will just be
referred to as ‘diameter’. The entire layer descends,
but certain snow aggregates do so at a faster rate and
fall out below the base of the main cloud. This is similar to Scorer’s fallout mechanism for the formation
Atmos. Sci. Let. 7: 2–8 (2006)
4
(a)
(b)
Figure 2. (a) Isosurface movie of snow aggregate diameter
value of 1 × 10−5 mm as sampled at 1 min intervals from
t = 10 min to t = 30 min is available from the supplementary
materials page. The full three-dimensional simulation domain is
shown. (b) XZ cross-section of the full domain at t = 26 min
and y = 3425 m of snow aggregate diameter contours in m
of mammatus clouds. Near the end of the simulation,
e.g. t = 26 min, there are smooth, lobe-like protuberances, with rounded lower boundaries, in a cellular
pattern extending from the parent anvil cloud that
show a qualitative resemblance to mammatus clouds.
However, it cannot be proven that the features simulated are actually mammatus clouds, or just something
similar to them. Nevertheless, for simplicity they will
simply be referred to as ‘mammatus.’
Figures 2(b), 3, and 4 show XZ vertical crosssections at y = 3425 m. Figure 2(b) shows the snow
Copyright  2006 Royal Meteorological Society
K. M. Kanak and J. M. Straka
aggregate diameter contours for the full domain. It
should be noted that the visible portion of the cloud
would be the lower edge or smallest nonzero contour
in plots of the snow diameter. In Figure 2(b), the
bottom edge of the snow aggregate diameter field
represents the ‘fallout front’ as named by Scorer
(1958). Figures 3 and 4 display expanded views of
the region enclosed by the black box shown in
Figure 2(b).
Figure 3 shows the wind vectors along with contours of snow aggregate diameter in meters at four
times: (a) t = 22 s, (b) t = 24 min, (c) t = 26 min,
and (d) t = 28 min. At t = 22 min, the first descending elements are beginning to form. There are two
dominant elements (labeled A and B on Figures 3
and 4) and about four other smaller elements (not
labeled), all of which can be identified at each of
the four times shown. Between t = 22 min and t =
24 min there is not much change in the magnitude
of the maximum downward velocities (about 4 m s−1
for both times), but the dominance of lobes A and
B is more prominent at t = 24 min. By t = 26 min,
the maximum downward velocities are on the order
of 6.68 m s−1 , and the two dominant lobes are starting to exhibit a slight horizontal spreading near the
bottom edges (denoted by a white arrow), particularly
for lobe A. This lateral spreading was also documented
by Winstead et al. (2001) and Kollias et al. (2005) and
predicted by Scorer (1958, his Figure 4). In association
with this structure, the velocity vectors show horizontal divergence and some evidence of recirculation
upward on the peripheries of the mammatus elements.
By t = 28 min, the other initially smaller mammatus
elements are now interacting more closely with the
originally dominant A and B elements. The shapes of
the leading edges of the elements have become much
more rounded. Further lateral spreading is evident,
especially for the A and B elements.
Closer examination of t = 26 min, (Figure 3(c))
reveals that the A and B mammatus elements extend a
few 100 m below the mean cloud base and have widths
of ∼500 m. The other smaller elements have depths
and widths of about 200 m or so. This would suggest
an aspect ratio of 1 : 1 or 1 : 1.5 for a given mammatus element which appears to hold at t = 28 min as
well. Such widths are consistent with those reported
by Clarke (1962), Warner (1973), and Kollias et al.
(2005), but are smaller than those reported by Stith
(1995), Martner (1995), and Winstead et al. (2001).
The decreasing values of snow aggregate diameter
(and amounts of particles) toward the bottoms of simulated lobes (Figure 3) is consistent with the often
observed translucence of mammatus at their edges
(e.g. Hlad, 1944; Stith, 1995; Emanuel, 1981). There
are some regions in which large values of snow aggregate diameters, and thus larger numbers of snow aggregates, are caught up in the upward return flows on
the peripheries of the mammatus lobes. Therefore,
one interpretation might be that the visible mammatus
Atmos. Sci. Let. 7: 2–8 (2006)
Numerical simulation of mammatus-like clouds
5
(a)
(b)
(c)
(d)
Figure 3. XZ cross-sections of the partial domain at y = 3425 m of snow aggregate diameter contours with interval of 0.0006 (m)
with velocity vectors (m s−1 ) overlain in black. A and B denote the two more dominant mammatus elements. The white arrow
denotes a region of lateral spreading. (a) t = 22 min, maximum vector length corresponds to 4.07 m s−1 ; (b) t = 24 min, maximum
vector length corresponds to 4.0 m s−1 . (c) t = 26 min, maximum vector length corresponds to 6.68 m s−1 ; (d) t = 28 min,
maximum vector length corresponds to 7.0 m s−1
cloud comprises both upward and downward transports of snow aggregates.
Figure 4 shows the total buoyancy and its components from one time, t = 26 min. The total buoyancy
term (Figure 4(a)) is calculated as,
Total Buoyancy = g
θ −θ
θ
+ 0.608(qv − q v )
−(qice + qsnow )
(1)
where g is gravity, qv is the mixing ratio of water
vapor, qice , the mixing ratio of ice crystals, and qsnow
the mixing ratio of snow aggregates. The potential
temperature is θ , and the overbar denotes the mean
value of a variable. The first term on the right hand
Copyright  2006 Royal Meteorological Society
side is due to latent heat release with sublimation
(Figure 4(b)), the second term is the effect of water
vapor on buoyancy (Figure 4(c)), and the third term
(Figure 4(d)) is due to loading by cloud ice and snow
aggregates. The magnitude of sublimation contributes
more strongly to the downward motion within the
mammatus lobes than does loading or water vapor for
this simulation. It is apparent that the largest sublimation values are just above the leading edge of the fallout front (Figure 4(b)). The strongest effects of loading
are well above the fallout front (Figure 4(d)). Thus, it
may be said that for the parameters of this simulation,
sublimation, similar to Scorer’s third mechanism of
evaporation, appears to be the significant factor in driving the downward motion of the mammatus elements.
Winstead et al. (2001) demonstrate the dynamic
circulation within a mammatus as observed using
airborne radar (their Figure 6 reproduced here as
Atmos. Sci. Let. 7: 2–8 (2006)
6
K. M. Kanak and J. M. Straka
(a)
(b)
(c)
(d)
Figure 4. XZ cross-sections of the partial domain at t = 26 min and y = 3425 m. Velocity vectors (m s−1 ) overlain in black. A
and B denote the two more dominant mammatus elements. The maximum vector length corresponds to 6.68 m s−1 . (a) Total
buoyancy term which is the left hand side of (1) in (m s−2 ). (b) First term on the right hand side of (1) which owes to sublimation
(m s−2 ). (c) Second term of equation (1), which is the effect of water vapor on buoyancy (m s−2 ); (d) Third term of equation (1)
which is the loading of snow aggregates and ice crystals (m s−2 )
Figure 5(a,b)). The general shape of the observed
lobes (Figure 5(a)) compares reasonably well with the
simulated flow at t = 24 or 26 min (Figure 3(c,d)).
Figure 5(c), at t = 26 min, is provided for clarity
to demonstrate the return flow (marked by bold
arrows) on the edges of the mammatus-like structures. There are downward vertical motions of magnitude up to −6.5 m s−1 within the mammatus lobes,
while there are upward vertical motions of +0.5
to +3.2 m s−1 between the mammatus lobes. These
vertical motions for the simulated mammatus lobes
are larger than those documented by Stith (1995),
Martner (1995), and Winstead et al. (2001) who
observed values of −3 to +1.0 m s−1 , but are on the
order of those reported by Kollias et al. (2005) who
observed −5 to +0.2 m s−1 . However, the vertical
Copyright  2006 Royal Meteorological Society
velocity amplitudes may also be a function of the
stage of development of a mammatus element since
an individual cell may last only 10 min (Warner,
1973).
4. Summary
An idealized numerical simulation of mammatus-like
clouds has been conducted. The experiment is intended
to simulate a portion of a cirrus outflow anvil cloud
that is far away from the parent cumulonimbus updraft.
Thus, a rectangular horizontally periodic domain is
used in which initial water vapor and ice crystal
mixing ratios are specified. An observed sounding, that
was released within an hour of a mammatus observation and within 5 km of the event, was used to
Atmos. Sci. Let. 7: 2–8 (2006)
Numerical simulation of mammatus-like clouds
7
Figure 5. From Winstead et al. (2001), Monthly Weather Review, American Meteorological Society (AMS). Data from airborne
Doppler radar. Pseudo-RHI plot of (a) reflectivity and (b) radial velocity for a representative mammatus element. This mammatus
cloud is almost directly above the aircraft flight track. (c) Velocity (m s−1 ) vectors from the current simulation at time t = 26 min
and y = 3425 m
initialize the simulation. Small-scale random perturbations of the ice crystal mixing ratios were imposed
to initiate the formation of snow aggregates and initial
motion fields.
The results show that the snow aggregate diameter
field qualitatively resembles mammatus and has physical dimensions that are roughly consistent with prior
observations. The buoyancy field and the terms that
contribute to the buoyancy indicate that sublimation
is the main driving force for the downward motions
in association with the simulated mammatus. Vertical
velocities are slightly larger than those reported from
most previous observational studies, except for Kollias et al. (2005). Wind vectors show a return flow
circulation on the peripheries of the downward lobes
as observed by Winstead et al. (2001) and others and
as predicted by Scorer (1958).
Limitations of this study include the highly idealized
nature of the simulation domain and the uncertainty
inherent in the specification of the initial conditions.
Furthermore, limited observations make it difficult to
state with any certainty that the methods used in this
study represent the mechanisms by which mammatus
actually form.
As this article is intended to present work in
progress, future work will include a series of numerical
simulation experiments of mammatus designed to
determine the formation conditions, variables that
control the size and shape, and the influence of
hydrometeor fallout and size sorting.
Copyright  2006 Royal Meteorological Society
Acknowledgements
The authors gratefully acknowledge Dr Kerry Emanuel, Dr
David Schultz, the editor, and two anonymous reviewers for helpful comments and discussions on this work.
K. Kanak is supported by the Cooperative Institute for
Mesoscale Meteorological Studies (CIMMS), provided by
NOAA/OAR/NSSL under NOAA–OU Cooperative Agreement
NA17RJ1227. Funding was also provided for K. Kanak by NSF
grant ATM-0135510 and for J. Straka by NSF grant ATM0340639.
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Atmos. Sci. Let. 7: 2–8 (2006)
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