"A simplified squall-line model revisited"
Robert G. Fovell and Pei-Hua Tan
Quarterly Journal of the Royal
Meteorological Society
(in press)
The discussion below is based upon, and further extends, the material covered in the Fovell and Tan (1998) paper, which should probably be reviewed first. This page presents a brief overview of the paper. Only a portion of the paper's content is covered below. Please direct comments, questions, and/or reprint requests to Robert Fovell.
Available here is a copy of the paper in Adobe Acrobat PDF format.
Links to sections on this page:
Motivation and Background
Fovell and Tan (1998; hereafter FT) studied the
generation, propagation,and dissipation
of the mature multicellular storm's
constituent convective cells. They showed that the typical new cell
first appears within the the cold pool's
forced updraft and quickly
becomes a positively buoyant free updraft.
The buoyancy induces a
local circulation that, at first, serves to intensify the cold pool
forced updraft, aiding the new cell's intensification. However, once the
cell begins moving rearward within the FTR flow, the buoyancy-induced
circulation starts actively suppressing the forced lifting, even as
it forces the entrainment of potentially cold air (from the cold pool
and/or the middle troposphere ahead of the storm) into the cell. This entrainment
causes the lower portion of the growing cell to become convectively
stable while destabilizing the forced updraft farther below.
The growing cell thereby becomes ``cut off'' from the forced updraft,
and continues propagating rearward within the FTR airflow. It was concluded
that the cellular transience was a
direct result of the convectively-generated, buoyancy-induced circulation.
To further
our understanding of the cut off and cell
regeneration processes,
we attempted to identify the simplest dynamical
framework capable of capturing this behavior.
First, the model was made
two-dimensional, as it has been demonstrated that multicellular
behavior can be dynamically similar in the two- and three-dimensional
frameworks (e.g., Rotunno et al. 1988; FT). Next, condensate
loading was deactivated and the complex
subgrid turbulence parameterization was
replaced by a simple eddy diffusion intended primarily to restrain
computational instability. These were found to have little dynamical
effect other than to permit the cells to become somewhat stronger and
longer-lived.
The explicit appearance of precipitation and its attendant
timescales were obviated by the adoption of a maintained, lower
tropospheric heat sink. Testing
revealed that the sink's artificially generated cold pool permitted
realistic, long-lived multicellular storms to form with relatively little
sensitivity attached to the cooling rate employed.
Finally, the chief role of water vapor is to release latent heat
to the surrounding air upon condensing. As the dominant (but not
sole) condensation production mechanism is lifting, very
simplified models have parameterized moisture by making
condensation warming proportional to ascent velocity. Our simplified
model follows the implementation discussed in
Garner and Thorpe (1992; hereafter GT).
Our study was specifically motivated by GT's reported failure to
find any multicell-type storm behavior in the experiment they
conducted. Based on FT, we believed that the parameterized moisture
(PM) model does contain the necessary physics to support sustained
multicellular behavior. Below, we show that when an initial sounding
with sufficient convective instability is employed, realistic
multicellular temporal behavior is obtained with the PM model.
Model design
In the PM model, the equations concerning moisture are eliminated and
the prognostic equation for potential temperature q is rewritten (neglecting mixing) as:
Next, condensation warming is made proportional to ascent velocity, but
only in a portion of the domain dubbed the unstable
region
.
This is justified in the following manner. The heating term may be
written as:
In a Lagrangian view, the first equivalence on the
right-hand side (RHS) illustrates the condensational heating is
generated owing to the change of qvs with time along a saturated
parcel's path. The second equivalence is based on the fact that
the motion affecting the qvs change is primarily in the vertical
direction. The third
expression shows that the path taken by the saturated parcel is
along a moist adiabat. qp is the potential temperature of
such a parcel, and its vertical gradient is the slope of a moist
adiabat in temperature-height space. The last term reveals that heat
release is proportional to ascent velocity (w).
The figure below depicts the model design:
The "unstable region" is where rising motion is presumed to generate
latent heating. The "cooling zone" is where evaporation cooling is
acting. (The paper explains what the other symbols mean.) In
practice, it is necessary to keep the cooling zone in sync with the
cold pool it produces lest the results become quite unphysical. Our
cooling zone is "storm-adaptive", meaning its location is chosen to
align with the subcloud cold pool, wherever it is and however it
happens to move.
Results
GT's sounding contained about 400 J/kg of convective available
potential energy, or CAPE. With this sounding, the model tends to
settle into steady state solutions. The "successful" steady solution
has a storm with a deep (but steady) updraft while the "unsuccessful"
steady solution has no deep convection, only a forced updraft
associated with the cold pool lifting. These two simulations differ
with respect to the strength of the low-level vertical wind shear.
The key point is that multicellular behavior was essentially absent.
The figure below shows an unsuccessful (left) and successful (right)
steady state, made using GT's sounding. These are vertical (x-z)
cross-sections across the
storm. (Only a portion of the simulation domain is shown, and the
horizontal axis is labeled relative to the left side.) The contoured
field is vertical motion, with solid (dashed) contours indicating
ascent (descent). The shaded field is perturbation potential
temperature.
Next, we consider simulations made using a more favorable sounding,
one with about 1800 J/kg. This value is more typical of springtime
severe storm events according to Bluestein and Jain's (1985)
climatology.
When a more convectively favorable (but
also more realistic) sounding is adopted, multicellular behavior is
ubiquitous. Below shows the results of one case at four times during
a single "repeat cycle". The sequence of pictures is quite similar to
that seen in the more physically realistic cloud model and analyzed by
FT. Further similarities, and some caveats, are discussed in the paper.
Deficiencies of the parameterized moisture model
The PM model accomplishes some substantial simplifications, and is
quite economical to run. Unfortunately, our model solutions present
some problems that point to inherent deficiencies in the model. Most
importantly, the degree to which the storm modifies its own upstream
environment as it organizes appears to be quite excessive. This is
important because the upstream environment (into which the storm is
propagating) directly feeds back upon the storm.
The figure above shows vertical profiles of the horizontal wind taken
at a location ahead of the storm during the mature phase. Shown are:
the initial profile; the profile from the PM model simulation
shown in Fig. 2; a
modified PM model simulation (see below); and two simulations from "full physics"
cloud models. The latter indicate little alteration from the initial
profile occurs in the important lower layer. In contrast, the PM
model simulation evinces substantial differences, especially at the low
levels.
The excessive upstream modification issues from a fundamental
inconsistency in the PM model. In reality, as convective cells
retreat towards the storm's rear, they "run out" of water vapor and
thus condensation heating eventually ceases. In the PM model,
however, rising motion creates warming, which serves to maintain the
rising motion that creates the warming. Thus, as the cells age and
retreat, parameterized condensation warming never ceases.
As a result, excessive warming accumulates in the storm's trailing
region (see x < 20 km in Fig. 2). This warming:
As a test of our explanation, we attempted a crude "fix" by
incorporating a "convective sponge" into the model. This sponge,
located within the unstable zone,
removed warming from the trailing region at a time scale long
compared to the evolution and movement of the convective cells but
short relative to the trailing region's accumulation of warming.
Figure 5, above, shows x-z cross-sections from the original
and sponged simulations. Comparing the simulations, it is seen that:
Conclusions
GT employed a PM model to elucidate the role of vertical wind shear in
organizing squall-line storms. During maturity, a typical storm
possesses a large-scale background circulation in which sequentially
generated,
short-lived convective cells are embedded. GT chose the simplified
model because, in their view, it inherently "suppressed cell decay and
redevelopment", permitting them to focus on the much steadier background
circulation. Indeed, their model
yielded "extraordinarily steady" solutions that were either
successful (consisting of a single, persistent deep updraft) or
unsuccessful (devoid of deep convection) during maturity.
"Multicellular" storms were ostensibly absent.
Based on FT's analysis, however, we suspected the PM model should support cell
redevelopment. Our examination showed that GT's initial environment
lacked sufficient convective instability to drive the
episodic entrainment process critical to cellular transience. Given a
commonly available amount of instability, even the PM model was found to
support realistic-looking multicellular storms. Steady unsuccessful and
successful storms still existed, but in between, the parameter space
region occupied by multicell storms was substantially larger.
Less realistic was the degree to which PM model storms influenced
their upstream environments. Analysis suggested the exaggerated
impact could be traced to a fundamental shortcoming of the simplified
model: that convective cells, once formed, did not dissipate, even as
they propagated far away from the storm's leading edge. This
permitted an excess of parameterized condensation warming to build up
in the storm's trailing region, with a substantial impact on the
storm's overall circulation.
GT's study helped demonstrate that both shear intensity and shear
layer depth combine to influence the mature squall-line storm's
background, quasi-steady
circulation. Indeed, by examining an environment with small convective
instability, they showed the transition between the successful and
unsuccessful cases can be very
narrow, thereby emphasizing the contrast between these two extreme states
without the distraction posed by multicellular transience. Our
contribution is to show that the PM model also supports reasonable
multicellular behavior when more realistic environmental conditions
are employed. However, our analysis also points to deficiencies in the PM
framework that serve to limit its general applicability.
Acknowledgement
This work was supported by the National Science Foundation (NSF) under
grant ATM94-21847.
References
Bluestein, H. B., and M. H. Jain, 1985: Formation of mesoscale
lines of precipitation: severe squall lines in Oklahoma during the
spring. Journal of the
Atmospheric Sciences, 42,1711-1732.
Page created September, 1999, by
Robert Fovell
Fovell, R. G., and P.-H. Tan, 1998: The temporal behavior of numerically simulated multicell-type
storms, Part II: The convective cell life cycle and cell
regeneration. Monthly Weather Review, 126, 551-577.
Garner, S. T., and A. J. Thorpe, 1992: The development of organized
convection in a simplified squall-line model. Quarterly Journal
of the Royal Meteorological Society, 118, 101-124.
Rotunno, R., J. B. Klemp, and M. L. Weisman, 1988:
A theory for strong, long lived squall lines. Journal of the
Atmospheric Sciences, 45, 463-485.