Evaluation of an Explicit One-dimensional Time Dependent Tilting Cloud
Model: Sensitivity to Relative Humidity
Shu-Hua Chen12@ and Yang-Cheng Siao2
1
Department of Land, Air, and Water Resources, University of California, Davis, CA
2
Department of Atmospheric Sciences, National Central University, Chung-Li, Taiwan
Submitted to Journal of the Meteorological Society Japan
January, 2009
@
Corresponding author’s address: Shu-Hua Chen, Department of Land, Air, and Water
Resources, University of California, Davis, California 95616-8627; E-mail:
shachen@ucdavis.edu.
1
Abstract
The performance of the one-dimensional (1D) Explicit Time dependent Tilting cloud Model
(ETTM) was evaluated against three-dimensional (3D) cloud resolving model (CRM)
simulations in sheared environments. The Weather Research and Forecasting (WRF) model,
the cloud resolving model used in this study, was first applied to study 3D cloud properties
under two different environments, one from the Rain In Cumulus over the Ocean (RICO)
field experiment and the other from the International H2O Project (IHOP).
WRF
simulations were examined with different radii, from 1 km to 10 km, of thermal bubbles for
initiation.
Three-dimensional cloud features were quite different between RICO and IHOP
due to their environments, which were a sub-tropical maritime sounding and a mid-latitude
continental sounding, respectively. ETTM 1D cloud simulations corresponding to each of
the WRF simulations were conducted.
Simulated 1D clouds with the original thermal
bubble similar to those used in 3D cloud simulations were too weak.
The sensitivity of
model results to relative humidity was tested by imposing 88% (ER88) and 95% (ER95)
humidity into the thermal bubble in ETTM simulations.
Compared with the original
simulations, 1D results from ER85 and ER95 showed clear improvements, but were still
underestimated relative to 3D clouds and results from IHOP were slightly worse than those
from RICO.
The process of adding moisture into thermal bubbles for 1D cloud simulations,
in fact, is a reasonable approach. Results from the comparisons of mass, heat, and moisture
fluxes between WRF and ER95 were very encouraging and further improvements will be
made on ETTM, particularly for downdraft and cloud depth estimates, for use in cumulus
parameterization.
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1. Introduction
The purposes of cumulus parameterization are to estimate unresolved sub-grid scale
precipitation, to account for sub-grid scale latent heat release, and to vertically redistribute
heat, moisture, and momentum (Kain and Fritsch 1993).
Even though advances in
computational resources and techniques make it possible to run global cloud-resolving
models, they are mostly available at a few national centers, such as the Earth Simulator at
Japan Agency for Marine-Earth Science and Technology, and are often limited to short-time
forecasts and simulations. Therefore, cumulus parameterization in numerical models is and
will still be needed, in particular for long-term global climate integrations.
The importance
of the cumulus representation cannot be over-emphasized as errors in moist physics schemes
overwhelm those in dynamics.
In general, cumulus parameterization consists of three components: triggering
functions, which manage the timing and location of sub-grid scale convection; the vertical
redistribution of mass, momentum, heat, and moisture variables, which control the
modification of the grid-scale environment; and closure assumptions, which regulate the
strength of the convection. Arakawa (2004) gave a very comprehensive review of the
cumulus parameterization history.
Different cumulus parameterization, in particular for the
moist-convective adjustment scheme (Manabe et al. 1965), the Kuo scheme (Kuo 1974), and
the Arakawa scheme (Arakawa 1969), were reviewed.
The controversies, current trends,
and outstanding problems of cumulus parameterization were also discussed.
As the resolution of numerical models has increased, the approach of using 1D cloud
models to describe sub-grid scale effects on the vertical redistribution of those
aforementioned dynamics and thermodynamic variables (i.e., mass, momentum, heat, and
moisture) was proposed (Anthes 1977, Frank and Cohen 1985, Fritsch and Chappell 1980,
Grell 1993, Sun and Haines 1996). After computational resources were greatly improved, a
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2D cloud model was then used in each model grid column to directly resolve sub-grid scale
clouds (i.e., no need for the triggering function and the closure assumption) and is known as
the super parameterization scheme (Grabowski 2001).
Results from the super
parameterization approach were very promising, and the application has been expanded
(Khairoutdinov et al. 2005, Khairoutdinov et al. 2008, Tao et al. 2008). However, extensive
rainfall was found in the western Pacific during the Northern Hemisphere summer after the
use of the scheme (Khairoutdinov et al. 2005).
In super parameterization, as a 2D cloud
model is calculated in each model grid box during the entire model integration, it requires
about 2 orders of computational time than that of the conventional cumulus parameterization
approach (Khairoutdinov et al. 2005).
Therefore, the super parameterization approach is
difficult to be adopted widely, in particular for those with limited computer resources, and
cumulus parameterization with the use of a 1D cloud model might be more practical.
Most of 1D cloud models that were used in cumulus parameterization are either
simple plumes or bulk 1D cloud models (e.g., Anthes 1977, Fritsch and Chappell 1980, Frank
and Cohen 1985, Grell 1993, Hu 1997), which might not perform properly due to their
simplifications (Liu et al. 2001).
A slightly more sophisticated 1D entrainment/detrainment
plume model was then proposed by Kain and Fritsch (1990, 1993) and a more realistic 1D
model was developed by Haines and Sun (1994) for cumulus parameterization.
All these
1D models were time independent and most of them ignore the downdraft effects.
The
importance of the downdraft to cumulus parameterization has been revealed (Brown 1979;
Molinari and Corsetti 1985; Cheng 1989; Grell et al. 1991; Moncrieff 1992; Gray 2000; Liu
et al 2001).
Downdrafts, which have been often observed beside tilted updrafts, have
non-negligible effect on mass, moisture, and heat fluxes, in particular for low levels
(Moncrieff 1981, 1992).
As pointed out in Grell et al. (1991), a more realistic 1D model,
such as one that is time dependent and physically-based, might be needed in order to obtain a
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better estimate of the sub-grid-scale cloud thermodynamic properties in cumulus
parameterization.
Chen and Sun (2004) developed an Explicit one-dimensional Time dependent Tilting
cloud Model (ETTM) for potential use in cumulus parameterization.
ETTM is more
realistic compared with those simple plume or bulk 1D cloud models used in cumulus
parameterization (e.g., Anthes 1977, Fritsch and Chappell 1980, Frank and Cohen 1985, Grell
1993, Sun and Haines 1996, Hu 1997).
2.2.
A brief introduction to the model is given in Section
Before applying ETTM to a cumulus parameterization scheme, it is important to
evaluate the model performance under realistic environments.
Cloud resolving models
(CRMs) have been used to study three-dimensional (3D) or two-dimensional (2D) cloud
properties (Lin 1999, Akihiko and Mitsuru 2005, Gao and Li 2008, Luo et al. 2008).
CRMs
were also applied to validate the performance of cumulus parameterization schemes (Liu et al.
2001).
In this study, the weather research and forecasting (WRF) model was used as a CRM
to study 3D cloud properties under different shear and convective available potential energy
(CAPE) environments.
The results from WRF CRM were then used to evaluate and
improve ETTM’s performance.
This paper is organized as follows.
Both models, 3D WRF CRM and 1D ETTM,
and initial soundings are introduced in Section 2. WRF model experiment designs and
simulation results are given in Section 3, while those of ETTM simulations and comparisons
between two model results are present in Section 4. Brief concluding remarks are given at
the end.
2. Brief Description of Models and Initial Soundings
2.1 Three-dimensional cloud-resolving Weather Research and Forecasting (WRF) model
WRF is a community model recently developed by several organizations.
5
There are
two dynamical cores available for public: the Advanced Research WRF (ARW) and the
Nonhydrostatic Mesoscale Model (NMM). The ARW version 2.2 (Skamarock et al. 2005)
was adopted for this study.
ARW is a fully compressible, nonhydrostatic model, and the
governing equations are written in the flux form to conserve mass, dry entropy, and scalars.
The Runge-Kutta third-order time scheme was employed and fifth- and third-order advection
schemes were chosen for the horizontal and vertical directions, respectively.
WRF was
designed to help researchers develop and study advanced physics and data assimilation
systems, and it can be used for both idealized and real case studies. Therefore, in this study
WRF was chosen as the cloud-resolving model (CRM) for idealized 3D cloud simulations.
More information about the WRF model can be found on the WRF website at
http://www.wrf-model.org.
2.2 Explicit one-dimensional Time dependent Tilting cloud Model (ETTM)
ETTM was developed by Chen and Sun (2004) for potential use in cumulus
parameterization.
ETTM includes several important features: the model is anelastic,
non-hydrostatic and time dependent; the cloud can tilt; an updraft and a downdraft can
coexist; and a sophisticated cloud microphysics is used. In addition, non-hydrostatic pressure,
entrainment/detrainment, lateral eddy mixing and vertical eddy mixing are considered; the
entrainment/detrainment layers are physically determined and the entrainment/detrainment
rates for temperature, moisture, and hydrometeors are physically calculated using the
continuity equation.
Its cloud microphysics scheme (Chen and Sun 2002) is one of the
microphysics schemes available in the WRF model (i.e., Purdue Lin microphysics scheme).
Therefore, WRF is one of the most suitable 3D CRM models for comparing ETTM results
with 3D cloud simulations.
The equations (i.e., Navier Stokes equations) that governed both updraft and
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downdraft in ETTM were mapped from Cartesian coordinates onto tilting cylindrical
coordinates. All fields within the cloud were assumed to be horizontally symmetric with
respect to the tilting axis.
Therefore, the tangential wind is zero for ETTM at all times.
Given the assumption of the pressure perturbation distribution, which followed the
zeroth-order Bessel function of the first kind, the governing equations were, therefore,
reduced from 3D to 1D after horizontal integration.
The lateral eddy exchange, entrainment,
vertical flux, and pressure perturbation gradient force were solved using the
Forward-Backward scheme in time (Sun 1984) and the central difference in space.
parameters were introduced, the cloud radius (R) and the tilting angle (  o ).
Two
The tilting
angle, which was assumed to be the same for both updraft and downdraft, was defined as the
angle from the vertical axis in erect cylindrical coordinates to the tilting axis in tilting
cylindrical coordinates.
The cloud radius of the downdraft was assumed to be 40% of that
for the updraft, as in Chen and Sun (2002), which was based on Lemone and Zipser (1980).
Although the cloud radius and tilting angle are functions of time, for simplicity they were
assumed to be constant in the ETTM model.
Both parameters were estimated from
three-dimensional clouds in this study and will be parameterized in the near future.
In addition to better representing clouds in nature, the tilting effect of the cloud can
naturally activate and enhance a downdraft in the model.
When an updraft develops and
produces precipitation, a portion of precipitation will separate from the tilting updraft due to
the gravitational force.
The evaporative cooling and the drag force from separated
precipitation can trigger the downdraft as shown in Fig. 1 and enhance it later on.
Moreover,
hydrometeors that are detrained from the upper part of the updraft can induce and enhance
the downdraft as well, through the same mechanisms (i.e., evaporative cooling and the drag
force). Therefore, the downdraft occurs after the updraft develops.
Note that in ETTM all
downdrafts, either due to separated precipitation or detrained hydrometeors, are considered
7
together.
2.3 Soundings
Both WRF and ETTM were initialized with the same vertical soundings.
it was assumed that the initial fields were horizontally homogeneous.
In WRF,
Two 1D
environmental soundings (Fig. 2 and Table 1) were used for this idealized study. One is
from the Rain In Cumulus over the Ocean (RICO) field experiment and the other is from the
International H2O Project (IHOP). The sounding from RICO was observed at 0000 UTC
November 18, 2004 from San Juan, Puerto Rico, a sub-tropical environment where the low
levels were moist (Fig. 2a). The vertical wind shear was about 3 to 4 km deep and winds
changed from easterly-southeasterly near the surface to westerly-northwesterly as the height
increased (Fig. 2b), giving approximately unidirectional shear with height. The other one
from IHOP was collected at 2100 UTC June 4, 2002 and is a sounding from the Great Plains
in the US, in the continental middle latitude, where the low levels were relatively dry (Fig.
2c). For IOP, in addition to a drier lower atmosphere, the convective available potential
energy (CAPE) was larger (2450 J kg-1) than that from RICO (1400 J kg-1) and the wind shear
direction changed with height, i.e., multi-directional shear (Fig. 2b vs. 2d).
The temperature
profiles were close to dry adiabatic at low levels and became toward to a moist adiabatic
structure in the middle levels for both soundings.
atmosphere was much deeper in IHOP than in RICO.
However, the dry adiabatic-like lower
As a result, the temperature for IHOP
dropped about 20 oC from the surface to 700 hPa, which was more than that for RICO (about
16 oC).
3. WRF Simulations
3.1 Experiment design
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The idealized supercell case in the WRF model was adopted for idealized 3D
simulations.
As mentioned earlier, the initial fields were assumed horizontally
homogeneous and two case simulations, RICO and IHOP, were conducted using the
soundings in Fig. 2.
One domain with 321 x 321 x 51 grid points in the east-west,
north-south, and vertical directions, respectively, was configured for all WRF simulations.
The horizontal resolution was 250 m, which gave a horizontal domain size of 80 km x 80 km.
The vertical grids were stretched from a resolution of about 200 m close to the surface to
1000 m close to the model top at approximate 16-km height.
A weakly-damped 5-km
Rayleigh sponge layer was placed at the top of the domain to absorb reflected waves.
Because of the high spatial resolution, a sub-grid eddy diffusion scheme, the Smagorinsky,
was used instead of a boundary layer parameterization.
The Purdue Lin microphysics
scheme (Chen and Sun 2002) was chosen, as mentioned earlier.
A free slip boundary lower
condition was used.
The 3D cloud simulation was initiated using a thermal bubble. For both RICO and
IHOP, simulations with different horizontal radii of bubbles were conducted to examine cloud
properties with different cloud sizes. A maximum potential temperature perturbation (  'max )
of 3 K was assigned at the center of the bubble.
The formula of the thermal bubble ( ' )
was:

'   'max  cos 2  r / 2

(1)
 x  x o   y  yo   z  zo 
where r  
 
 
 .
 R   R   1500 
2
2
2
The center of the bubble, (xo, yo, zo), was located at 1.5 km height over the center of the
model domain.
The bubble radius, R, varied from 2 km to 10 km for RICO and 3 km to 10
km for IHOP with an interval of 1 km. Simulations with smaller bubble radii (i.e., 1 km for
9
RICO and 1-2 km for IHOP) were also examined but not included because their simulated
cloud diameters were less than 5-grid spacings (i.e., less than 1.25 km).
integrated for one hour with a time step of 1 second.
The model was
Note that in order to keep the
simulated storms close to the center of the domain, mean winds of (u,v) = (-5.9 m s-1, -2.5 m
s-1) for RICO and (u,v) = (-1.17 m s-1, 5.76 m s-1) for IHOP were deducted from the initial
soundings.
The mean wind was estimated from the storm’s propagation speed during the
first 20-min integration using the original wind sounding with the experiment of R=10 km for
each case.
3.2 WRF simulation results
Figure 3 shows the time variation of the maximum vertical velocity from simulated
3D clouds.
When a bigger thermal bubble was applied, a convective cloud took longer to
develop and reach its maximum vertical velocity for both RICO and IHOP.
With the same
size of bubble, RICO, which was moister in the lower atmosphere, took a slightly shorter
time to develop.
The small peak for each cloud was due to the imposed thermal bubble and
it was weaker as the bubble became larger.
It is interesting to see that the peak value of the
maximum vertical velocity kept increasing for IHOP (Fig. 3b) but quickly flattened out for
RICO (Fig. 3a) as the bubble size increased.
The maximum vertical velocity from RICO
was stronger than that from IHOP with a small thermal bubble, while that from IHOP was
stronger when the radius of the thermal bubble was larger than 5 km and this is probably due
to a larger CAPE for the IHOP sounding.
Moreover, when a larger thermal bubble was used,
the maximum vertical velocity remained at a high value after passing its peak value for RICO,
and this was due to the development of multi-cells, one after another.
multicell clouds in the middle latitude.
This is similar to
For all IHOP simulations, only one major cloud
developed.
10
The simulated cloud bases were independent of bubble size and were located at
approximately the same level (Fig. 4), the lifting condensation level, for both cases. The
cloud base for IHOP was slightly higher than that for RICO because a dryer boundary layer
was present in the former sounding (Fig. 2a versus 2b).
In contrast to the cloud base, the
simulated cloud top strongly depended on the initial bubble and it reached higher (i.e., a
greater cloud depth) when the bubble size increased.
With the same size of the thermal
bubble, the cloud top for IHOP was higher than for RICO when the bubble radius was greater
than 5 km.
Cloud depth is an essential output of cloud parameterization schemes.
The
simulated depth is important to the vertical redistribution of those dynamic and
thermodynamic properties, which is critical to the adjustment of the atmospheric instability.
It is important that a 1D cloud model can at least approximately reproduce a 3D cloud depth
when it is considered in a cumulus parameterization scheme.
It is worth mentioning that the simulated cloud tops were lower than the equilibrium
level (EL), in particular for small clouds. Therefore, the use of EL as the top of the sub-grid
scale vertical adjustment will overestimate the depth of the adjustment.
Further more, for
any given environment sounding with no negative CAPE between the level of free
convection and EL, such as the two soundings in the study, a cloud will consume more
environment energy (i.e., CAPE) when the cloud top reaches higher. As a result in theory, a
stronger maximum vertical velocity, which is proportional to the squire root of the consumed
CAPE, should be expected when the cloud top reaches higher (i.e., larger clouds here). This
is shown for the IHOP case but not for RICO (Figs. 3 and 4), and the reason behind the RICO
case is worth for further study.
Figures 5a and 5b show the horizontal cross sections of the vertical velocity at three
different levels with the bubble radius of 10 km after a 23-min integration, at the time when
the downward motion development due to detrainment, for RICO and IHOP, respectively.
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The simulated 3D cloud from RICO tilted from the west-northwest to the east-southeast
direction, which is close to the vertical shear direction in Fig. 2c.
Due to the
multi-directional shear in IHOP, the tilting direction of the simulated 3D cloud changed with
height and it also approximately followed the variation of the vertical shear (Fig. 2d).
It is
well known that the multi-directional shear can sometimes dynamically help support a longer
life cycle of a convective cloud, such as supercell (Klemp and Wilhelmson 1978).
It is
expected that ETTM will have difficulties presenting the influence of the multi-directional
shear effect on sub-grid cloud development.
“Can the multi-directional shear effect be
parameterized in a 1D cloud model” is an interesting question for study.
Figures 5a and 5b
also show that during the early mature stage of cloud life cycles, downward motions occurred
approximately outside the cloud edges, which can be seen in the vertical cross sections in
Figs. 6a and 6b as well. The downward motions resulted from the evaporative cooling and the
drag force of hydrometeors after the effects of detrainment and advection on those
hydrometeors (D in Figs. 6a and 6b).
For IHOP, the locations of downward motions with
respect to the cloud at different levels reflected the change of the wind direction with height.
Three minutes later (i.e., a 27-min integration), the downward motions developed underneath
the clouds, but leaned toward the tilting side of the clouds, becoming more evident for both
cases (alphabet P in Figs. 5c, 5d, 6c and 6d).
These downdrafts were triggered by
precipitation that separated from upper-level updrafts, as they started at the levels that were
below the maximum vertical velocity (i.e., below the detrainment layers).
With a larger
CAPE and a stronger maximum vertical velocity in IHOP for R=10 km, the overshooting
above the cloud top was more pronounced (i.e., the top of the updraft is higher than that of
the cloud field).
Figure 7 shows the vertical soundings passing the maximum vertical velocity of the
convective clouds at different times for the cases with a 10-km bubble radius. At the earlier
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time of the cloud development, the storm presented a moist adiabatic thermodynamic
structure after the convective adjustment occurred for both cases (Figs. 7a and 7b).
A few
minutes later, the intrusion of the environmental dry air into the cloud, which was induced by
dynamical mechanisms, resulted in the evaporative cooling and made the inner cloud depart
from moist adiabatic (i.e., 700 hPa to 850 hPa for RICO in Fig. 7c and 480 hPa to 550 hPa for
IHOP in Fig. 7d).
In the later stage of the cloud life cycle, mixing with the environment
further destroyed the moist adiabatic properties of the clouds (Figs. 7e and 7f), which became
unsaturated later during the dissipation stage (figure not shown) when downward motion
became dominant.
cloud.
The intrusion of dry air into a cloud can suppress the development of the
It is also expected that ETTM will have difficulties representing this effect.
However, the entrainment of the dry air from upper levels of clouds during the dissipation
state is included in ETTM (the latter is through the mass continuity equation).
Usually, the cloud radius and the tilting angle vary with time and height.
As 1D
ETTM simulations need inputs for the tilting angle and cloud radius, their approximations
from 3D clouds were roughly estimated here. At each level, an equivalent cloud radius ( R e )
was defined assuming that R e2 was equivalent to the sum of areas where the upward
motion was greater than 2 m s-1.
In general, the maximum equivalent cloud radius increased
when the initial thermal bubble size increased (Fig. 8a).
It is worth pointing out that the
maximum equivalent cloud radius was smaller than the radius of the initial thermal bubble, in
particular that they were much smaller for larger clouds.
With the same initial perturbation
(i.e., the same size of the thermal bubble), the maximum equivalent cloud radius from RICO
was comparable or slightly bigger than that from IHOP.
Figure 9 shows the time evolution
of the vertical profiles of the equivalent cloud radius for the bubble radii of 5 and 10 km from
the initial time to the time when the maximum equivalent cloud radius was reached.
Results
from the later time were not plotted since downward motion developed, which may lead to
13
underestimation of the equivalent cloud radius.
A small updraft was developed before the
primary one in Figs. 9a, 9c, and 9d; this was because of the imposed thermal bubble.
Note
that no such a small updraft occurred in the experiment of the 10-km radius bubble from
RICO (i.e., Fig. 9b) as the upward motion due to the thermal bubble did not reach the
minimum requirement of 2 m s-1 (see Fig. 3a).
The maximum equivalent cloud radius
located at the middle to upper levels of the cloud and was shifted upward as the cloud became
stronger.
This differs from the assumption of a constant cloud radius made in ETTM.
Future study is needed to determine the importance of variation of the 1D cloud model radius.
Instead of estimating the variation of the tilting angle with respect to time and
height, an averaged angle from the updraft was estimated at the time when the maximum
vertical velocity was reached and this was used for ETTM simulations.
To estimate the
angle, the equivalent centers at two selected levels within the updraft were first identified.
The upper level was about 1/4 of the cloud depth from the cloud top, and the lower level was
about the same depth from the cloud base.
The equivalent center point, (x, y) at each level,
was the average of the locations for grid points with vertical velocity greater than 2 m s-1,
which is similar to the criteria used for the equivalent cloud radius.
The ratio of the
horizontal displacement (ds) to the vertical displacement (dz) between these two points was
used to estimate the cloud tilting angle ( o ) using the formula, αo  tan 1(ds / dz) .
Results
in Fig. 8b show that the tilting angle was almost constant (about 10o) with respect to different
bubble sizes for the RICO case. However, for the IHOP case, the averaged tilting angle
decreased when the bubble size increased and values ranged from about 20o to 40o.
4. ETTM Simulation Results and Comparison with WRF Simulations
4.1 Experiment design
The same initial soundings as the WRF CRM simulations (Fig. 2) were used in
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ETTM, except that horizontal winds were set to zero.
The stretched resolution and grid size
in the vertical were the same as those at the initial time in WRF simulations.
ETTM
experiments with various cloud radii and tilting angles were conducted to mirror each WRF
simulation (i.e., 9 bubble size experiments for RICO and 8 for IHOP).
For convenience,
ETTM experiments were named following the convention Exxkm, where E indicates the
ETTM model and “xx” indicates the radius of the thermal bubble in the WRF simulation.
As mentioned earlier, the tilting angle and the cloud radius in ETTM were assumed constant
during the model integration, and they were input parameters.
Before parameterizing the
tilting angle and cloud radius using environment or grid-scale soundings, we assessed how
ETTM performs when the best of both parameters are provided.
In ETTM simulations
conducted here, both numbers were directly estimated from simulated 3D clouds from WRF,
and the estimated maximum equivalent cloud radius and the tilting angle shown in Fig. 8
were used.
Results from ETTM were then compared with 3D cloud simulations to evaluate
ETTM’s performance.
For each experiment, the thermal bubbles used in ETTM simulations
were also estimated from horizontally averaged potential temperature perturbations in WRF
simulations (within the area of the estimated 3D updraft).
4.2 ETTM simulation results and comparison with WRF CRM results
Figures 10 and 11 show the updrafts and downdrafts from E10km runs for RICO
and IHOP, respectively.
The simulated updrafts from ETTM (i.e., Figs. 10a and 11a)
developed right after the model integration (i.e., no time lag), while it took about 10 to 15
min for a 3D cloud to develop (see Figs. 9b and 9d).
This implies that a time delay might be
needed for sub-grid cloud development if ETTM is used in a time dependent cumulus
parameterization scheme and that the delay will depend on the size of the convective cloud.
Downdrafts quickly developed before updrafts reached their mature stage (Figs. 10b and 11b).
15
For RICO, the maximum downdraft developed at the same height as the upper half updraft
during the early stage and shifted toward the surface in the dissipation stage.
The 1D cloud
life cycle was shorter and both updraft and downdraft were weaker for the IHOP case than
those for RICO, which is the opposite of the 3D cloud results.
The maximum vertical velocity from ETTM and WRF CRM with respect to
different maximum equivalent cloud radii, Re (i.e., different sizes of bubbles), are plotted in
Fig. 12.
For both cases, the maximum vertical velocities from 1D clouds (i.e., ETTM in Fig.
12) were much weaker than those from 3D clouds (WRF-wmax; i.e., the maximum point
values) as expected; however, most of them were also weaker than the maximum averaged
3D vertical velocity (WRF-avg; i.e., the maximum of the averaged values within the 3D
updraft). The averaged vertical velocity of WRF simulations was the horizontal average of
vertical velocities from those grids in which values were greater than 2 m s-1.
For the IHOP
case the maximum averaged value from WRF CRM increased when the maximum equivalent
cloud radius (i.e., bubble size) increased, while that from ETTM remained almost constant or
slightly decreased when the cloud radius increased.
The simulated cloud tops and bases from ETTM were also examined since they are
important to the vertical adjustment of mass, momentum, heat, and moisture variables in
cumulus parameterization.
Note that no cloud developed for the E02km run from IHOP and,
therefore, no results are plotted for this experiment.
The simulated 3D cloud bases, which
were close to the lifting condensation levels, were well reproduced by ETTM with different
equivalent cloud radii for both RICO and IHOP (Fig. 13).
However, the 1D cloud tops were
significantly underestimated, in particular, those from IHOP were capped at about 3-4 km
height.
Moreover, the cloud life cycle became shorter for IHOP ETTM simulations when
the cloud radius became bigger, which is the opposite of its counterpart from the 3D cloud
(figure not shown), indicating that ETTM results got worse when the cloud radius increased
16
for the IHOP case.
Overall speaking, the simulated 1D cloud results for IHOP were worse
than those for RICO.
The underestimation of the cloud strength from ETTM was expected
since the 1D cloud presents an average of a 3D cloud and therefore, the penetration effect of
the 3D updraft core was under-represented. Moreover, some mechanisms in a 3D cloud,
such as the multi-directional shear effect, can not be explicitly included and some
simplifications were made in 1D cloud.
improved?
Can these simplified, simulated 1D clouds be
Some sensitivity tests were conducted in next section.
4.3 ETTM sensitivity tests
To increase the intensity of the 1D clouds, a sensitivity test that doubled the number
of the vertical levels (i.e., double the vertical resolution) was carried out.
The depth and
strength of the 1D cloud were only slightly improved (figures not shown).
During time taken for a 3D cloud to develop with a thermal bubble (Fig. 3), the
boundary layer over the updraft region was moistened due to low-level convergence before
the cloud developed.
This is consistent with the difference in time delay for 3D cloud
development between RICO and IHOP.
lower atmosphere was moister.
A shorter lag time was required for RICO since its
In ETTM, the convergence/divergence effect was taken into
account (i.e., through the continuity equation).
the model integration (Figs. 10 and 11).
However, the 1D cloud developed right after
Therefore, no moistening in the lower atmosphere
would occur before the 1D cloud developed. To account for this process, for each ETTM
simulation two sensitivity runs with different moistening assumptions within the thermal
bubble were conducted (i.e., use of a moistened thermal bubble).
(RH) within the thermal bubble was specified as follows:
17
The Relative Humidity
max RH o , RH c ,
RH  
RH o ,
z  3km
z  3km
,
where RH o is the original RH in the initial sounding, and RH c is the lower bound of the
relative humidity within the bubble.
Two different RH c values, 88% and 95%, were
examined and sensitivity trials were named ER88 and ER95, respectively, for convenience.
With a fixed cloud radius, the intensity of the simulated 1D cloud was clearly
improved when the low-level moisture was increased.
The cloud developed deeper and the
maximum vertical velocities became stronger for both updraft and downdraft (Figs. 10-13), in
particular for IHOP that larger clouds were able to penetrate the cap at 3-4 km height that
occurred in the original bubble simulations in Section 4.1.
More moisture was able to
further enhance simulated 1D cloud intensity (i.e., ER88 vs. ER95) for both cases.
Different
values of relative humidity ( RH c ) might be needed for different sizes of 1D cloud
simulations and this requires further study.
The maximum vertical velocity from ER88 was
close to the maximum average value from WRF CRM for RICO.
For IHOP, with either
ER88 or ER95, the maximum vertical velocity was stronger than the maximum averaged
value from WRF CRM simulations and again remained close to constant but increased
slightly when the cloud radius increased.
Although the simulated cloud depths were
improved for both cases, they were still underestimated when compared with 3D clouds (Fig.
13).
The time variation of the vertical profiles of integrated mass, heat, and moisture
fluxes from WRF 3D (i.e., horizontal integration) and ER95 1D (i.e., summation of updraft
and downdraft) clouds were calculated and results with the bubble radii of 5 km and 10 km
for RICO (Fig.14) and IHOP (Fig. 15), respectively, are presented. Since the simulated
cloud depth is important to the vertical adjustment of variables and that from ER95
18
reproduced the 3D cloud depth better than those from ER85 and the original bubble
simulations, only results from ER95 are presented.
The depth of the vertical distribution of fluxes was shallower from ER95 as
expected since the 1D cloud top did not reach as high as that of the 3D cloud.
For the RICO
case, ETTM erroneously produced negative mass flux near the surface, which was
contributed from the downdraft at the dissipation stage.
For IHOP, the negative flux in
ER95 was too low in altitude and, again, this was caused by the downdraft.
the negative flux will require improvements in the ETTM downdraft.
The problem of
Moreover, the WRF
vertical fluxes for IHOP were sustained over a longer time than from the 1D cloud because
the 1D cloud life cycle was too short.
Although there were discrepancies between WRF CRM and ER95 simulations, the
flux patterns from both models showed some similarities, such as the shapes of those fluxes.
For RICO, WRF produced negative mass flux (Fig. 14a) at the upper half cloud height and
this was reproduced by ER95 (Fig. 14b); moreover, the negative heat fluxes due to overshoot
cooling above the cloud top and at the middle to upper level height of the cloud during the
later stage of the cloud life cycle were presented in both models. These positive results
show that ETTM has promise for use in cumulus parameterization, given further
improvements in performance, in particular for downdraft and cloud depth.
It was noticed that the magnitudes of fluxes from 1D cloud were larger than those
from 3D cloud after adding moisture into the thermal bubble (Figs. 14 and 15). This might
be due to a larger maximum vertical velocity compared with the maximum averaged value
from the 3D cloud (Fig. 12), and it can be potentially overcome using a partial cloud
approach (i.e., less than one full cloud effect) in cumulus parameterization.
5. Concluding Remarks
19
The cloud-resolving Weather Research and Forecasting (WRF) model and the
Explicit Time dependent Tilting cloud Model (ETTM) were used to simulate
three-dimensional (3D) and one-dimensional cloud properties, respectively, under two
different atmospheric environments.
One is a sub-tropical maritime sounding from the Rain
In Cumulus over the Ocean (RICO) experiment and the other is a mid-latitude continental
sounding from the International H2O Project (IHOP). For the IHOP case, the convective
available potential energy (CAPE) was higher (2450 J kg-1 for IHOP and 1400 J kg-1 for
RICO) and the lower atmosphere was drier than those from RICO. The vertical wind shear
direction varied with height for IHOP, while it was almost uniformly directed for RICO.
The 1D cloud model examined, ETTM, consists of an updraft and a downdraft and
was developed for potential use in cumulus parameterization (Chen and Sun 2004). The
updraft was initiated by a thermal bubble, while the downdraft was triggered by the drag
force and evaporative cooling of detrained hydrometeors and departed precipitation from the
updraft (Fig. 1).
Both WRF and ETTM are non-hydrostatic model and the same
microphysics scheme was used in both models for this study.
For both RICO and IHOP, 3D cloud simulations were conducted and initialized with
different radii of thermal bubbles. There was a time lag between the model initial time and
the 3D cloud development time, and the delay increased as the bubble radius increased.
Moreover, the delay can be longer if the lower atmosphere is originally drier (i.e., the IHOP
case). The tilting direction of the 3D cloud strongly correlated to its environment shear, and
the tilting angle from RICO was smaller than that from IHOP, which is consistent with
observed clouds in tropics versus in mid-latitude. For RICO, WRF simulations produced
multi-cells, one cloud after another one, when the bubble size got larger and the maximum
vertical velocity was capped after the bubble radius was larger than 3 km.
For IHOP, each
simulation produced one convective cloud only and the maximum vertical velocity and the
20
cloud life cycle increased when the bubble size increased (Fig. 3b). The simulated 3D cloud
base was almost independent of the bubble size, while the 3D cloud top developed higher
when a larger bubble was used for both cases.
ETTM 1D cloud simulations corresponding to each of the WRF CRM simulations
were conducted. The cloud radius and the tilting angle required in ETTM were estimated
from WRF 3D cloud simulations and will be parameterized using grid-scale information in
the future. Compared with 3D clouds, results from ETTM were too weak with the original
thermal bubbles for all cloud parameters:
depth was too shallow, etc.
the vertical velocity was too weak; the cloud
While the 3D cloud took time to develop, the 1D cloud
developed right after ETTM integration. Therefore, there was a moistening process in the
low levels before 3D clouds developed, and this process was missing in 1D cloud
development.
To compensate for the missing moistening process, two sensitivities with relative
humidities of 88% (ER85) and 95% (ER95) imposed into the thermal bubbles were carried
out.
The simulated 1D cloud became stronger and deeper, in particular for IHOP that larger
clouds were able to penetrate the cap at 3-4 km height that occurred in the original bubble
simulations, though the depth was still underestimated. For ER95, when compared with 3D
cloud results, the depth of the vertical mass, heat, and moisture flux distribution was too
shallow; the erroneous negative mass fluxes were caused by ETTM downdraft; and the life
expansion of 1D cloud from IHOP was too short.
Overall speaking, the simulated 1D cloud
results for IHOP were slightly worse than those for RICO, and this might be because the
environment sounding from IHOP was more sophisticated than that from RICO.
It is very
unlikely that 1D ETTM can fully reproduce 3D cloud features. The addition of moisture
into the thermal bubble, which is a reasonable approach, clearly improved ETTM
performance. Although there were discrepancies between ER95 and WRF simulation, the
21
overall patterns of mass, heat, and moisture fluxes from both models were somewhat similar.
This is quite promising when considering the use of ETTM in cumulus parameterization in
the future.
It is understood that more improvements are required to enhance ETTM
performance, such as to increase cloud depth, to address the upward shift of the downdraft
near the surface, etc.
ACKNOWLEDGEMENTS: The authors would like to acknowledge the WRF model
development teams for their efforts on the model development.
National Science Council grant # 95-2111-M-008-040 in Taiwan.
22
This work was supported by
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26
Table Captions:
Table 1: Information for two soundings that were used in this study.
Figure Captions:
Fig.1:
A schematic diagram showing the initiation of downdraft from a tilted updraft.
Fig. 2: The soundings used for (a) RICO and (b) IHOP simulations.
(c) and (d) are
hodographs of (a) and (b), respectively.
Fig. 3: Time evolution of the simulated maximum vertical velocity with different initial
thermal bubble radii for (a) RICO and (b) IHOP.
Fig. 4: Height of the simulated maximum cloud top (solid lines) and minimum cloud base
(dashed lines) with respect to different thermal bubbles during 1-h model integrations
for RICO (black lines) and IHOP (gray lines).
Fig. 5: The horizontal cross sections of vertical velocity at 1.8 km (shading), 2.7 km (gray
lines), and 3.5 km (black lines) after (a) 23 min and (c) 27 min simulations for RICO
for the experiment with a bubble radius of 10 km.
(b) and (d) are the same
information as (a) and (c), respectively, for IHOP at 1.8 km (shading), 3.0 km (gray
lines), and 4.2 km (black lines). For each case, negative values at the first plotted
level (i.e., 1.8 km) were not plotted, while they were plotted as dashed lines at the
other two levels (i.e., 2.7 km and 3.5 km for RICO and 3.0 km and 4.2 km for IHOP).
Fig. 6: The vertical cross sections of vertical velocity for (a) RICO at 23 min, (b) IHOP at 23
min, (c) RICO at 27 min, and (d) IHOP at 27 min for simulations with an initial
bubble radius of 10 km.
The locations of vertical cross sections in (a)-(d) are
indicated in Fig. 5a-d, respectively. The solid black line is the contour line of 1 g
kg-1 for the total hydrometeors (i.e., cloud edge).
The gray lines are the total
precipitation (i.e., rain, snow and graupel) with an interval of 1 g kg-1.
27
Fig. 7: The vertical sounding within the 3D cloud passing the maximum vertical velocity
after (a) 20-min (c) 24-min, and (e) 33-min simulations for RICO with an initial
bubble radius of 10 km.
(b), (d), and (f) present the same information but for IHOP
after 20-min, 27-min, and 30-min, respectively.
Fig. 8: Estimated (a) maximum equivalent cloud radius (R; km) and (b) tilting angle (o) for
RICO (black lines) and IHOP (gray lines) at the time when the maximum vertical
velocity occurred.
Fig. 9: Time evolution of the vertical profiles of the equivalent cloud radius (km) from initial
time to the approximate time when the maximum equivalent cloud radius was reached
for the experiments with bubble radii of (a) 5 km and (b) 10 km from RICO. (c) and
(d) are the same as (a) and (b), except from IHOP.
Fig. 10: Time evolution of vertical velocity (ms-1) from the (a) updraft and (b) downdraft
from E10km for the RICO case. (c) and (d) are the same as (a) and (b), respectively,
except from the ER88 experiment and (e) and (f) from the ER95 experiment.
Fig. 11: Same as Fig. 10, except for the IHOP case.
Fig. 12: The WRF simulated maximum vertical velocity (WRF-wmax), the WRF maximum
horizontally averaged vertical velocity, and the ETTM maximum vertical velocity
with the original thermal bubble (ETTM) and with different relative humidities within
the thermal bubble (i.e., ER88 for the 88% humidity run and ER95 for the 95%
humidity run) with respect to different ETTM cloud radii (or 3D bubble radii) for (a)
RICO and (b) IHOP.
Fig. 13: The simulated cloud top and base with respect to different cloud radii (or 3D bubble
radii) from WRF and ETTM with sensitivity tests of different low level relative
humidities from (a) RICO and (b) IHOP. Note that the base lines overlap from
different runs and therefore not easy to see. No cloud developed in the ETTM
28
simulation for the cloud radius of 0.73 km with the original thermal bubble for the
IHOP sounding.
Fig. 14: The simulated (a) mass (×108 kg s-1), (d) heat (X1012 J s-1), and (e) moisture (×106kg
s-1) fluxes from WRF simulations for the RICO case when the bubble size is 5 km.
(b), (d), and (f) are the same as (a), (c), (e), respectively, except from the
corresponding 1D cloud simulation from ER95.
Fig. 15: Same as Fig. 14, except for the IHOP ER95 run with the bubble size of 10 km.
29
Table 1: Information for two soundings that were used in this study.
Sounding 1
Sounding 2
Field Experiment
RICO
IHOP
Station
SJU San Juan, Puerto Rico
OUN Norman, Oklahoma,
USA
Latitude/Longitude
-66.0o / 18.4 o
-97.40 o / 35.20 o
Observed time
0000 UTC December 18, 2004
2100 UTC June 4, 2002
Convective Available
Potential Energy
~ 1400 J kg-1
~ 2450 J kg-1
Deducted mean wind
U (m s-1), v (m s-1)
-5.9,
-2.5
30
-1.17,
5.76
Z
R
Detrainment
Detrainment
Downdraft
Downdraft
Downdraft
Entrainment
Entrainment
Fig.1: A schematic diagram showing the initiation of downdraft from a tilted updraft.
31
(a)
(b)
(c)
(d)
Fig. 2: The soundings used for (a) RICO and (b) IHOP simulations.
hodographs of (a) and (b), respectively.
32
(c) and (d) are
(a)
25
RICO-02km
RICO-03km
20
Wmax(m/s)
RICO-04km
15
RICO-05km
RICO-06km
10
RICO-07km
RICO-08km
5
RICO-09km
RICO-10km
0
0
5
10
15
20
25
30
35
40
Time (min)
(b)
35
IHOP-03km
Wmax (m/s)
30
IHOP-04km
25
IHOP-05km
20
IHOP-06km
15
IHOP-07km
IHOP-08km
5
10
IHOP-09km
5
IHOP-10km
0
0
5
10
15
20
25
Time (min)
30
35
40
Fig. 3: Time evolution of the simulated maximum vertical velocity with different initial
thermal bubble radii for (a) RICO and (b) IHOP.
33
14
RICO-cloud base
Height (km)
12
RICO-cloud top
10
IHOP-cloud base
8
IHOP-coud top
6
4
2
0
1
2
3
4
5
6
7
8
9
10
Bubble radius (km)
Fig. 4: Height of the simulated maximum cloud top (solid lines) and minimum cloud base
(dashed lines) with respect to different thermal bubbles during 1-h model integrations
for RICO (black lines) and IHOP (gray lines).
34
(a) RICO 23 min
(b) IHOP 23 min
(c) RICO 27 min
(d) IHOP 27 min
P
P
Fig. 5: The horizontal cross sections of vertical velocity (cm s-1) and wind vectors at 1.8 km
(shading), 2.7 km (gray lines), and 3.5 km (black lines) after (a) 23 min and (c) 27
min simulations for RICO for the experiment with a bubble radius of 10 km. (b) and
(d) are the same information as (a) and (c), respectively, for IHOP at 1.8 km (shading),
3.0 km (gray lines), and 4.2 km (black lines). For each case, negative values at the
first plotted level (i.e., 1.8 km) were not plotted, while they were plotted as dashed
lines at the other two levels (i.e., 2.7 km and 3.5 km for RICO and 3.0 km and 4.2 km
for IHOP).
35
(a) RICO 23 min
(b) IHOP 23 min
D
D
D
D
(c) RICO 27 min
(d) IHOP 27 min
P
P
P
Fig. 6: The vertical cross sections of vertical velocity (cm s-1) for (a) RICO at 23 min, (b)
IHOP at 23 min, (c) RICO at 27 min, and (d) IHOP at 27 min for simulations with an
initial bubble radius of 10 km. The locations of vertical cross sections in (a)-(d) are
indicated in Fig. 5a-d, respectively. The solid black line is the contour line of 1 g
kg-1 for the total hydrometeors (i.e., cloud edge). The gray lines are the total
precipitation (i.e., rain, snow and graupel) with an interval of 1 g kg-1.
36
(a) RICO-10km-20min
(b) IHOP-10km -23min
(c) RICO-10km -24min
(d) IHOP-10km-27min
(e) RICO-10km -33min
(f) IHOP-10km-37min
Fig. 7: The vertical sounding within the 3D cloud passing the maximum vertical velocity
after (a) 20-min (c) 24-min, and (e) 33-min simulations for RICO with an initial
bubble radius of 10 km. (b), (d), and (f) present the same information but for IHOP
after 23-min, 27-min, and 37-min, respectively.
37
(a)
3
Max Re (km)
RICO
IHOP
2
1
0
1
50
2
3
4
5
6
7
Bubble radius (km)
9
10
(b)
RICO
Titlting angle.
8
IHOP
40
30
20
10
0
1
2
3
4
5
6
7
Bubble radius (km)
8
9
10
Fig. 8: Estimated (a) maximum equivalent cloud radius (R; km) and (b) tilting angle (o) for
RICO (black lines) and IHOP (gray lines) at the time when the maximum vertical
velocity occurred.
38
Height (km)
(b)
Height (km)
(a)
(d)
Height (km)
(c)
Time (min)
Time (min)
Fig. 9: Time evolution of the vertical profiles of the equivalent cloud radius (km) from initial
time to the approximate time when the maximum equivalent cloud radius was reached
for the experiments with bubble radii of (a) 5 km and (b) 10 km from RICO. (c) and
(d) are the same as (a) and (b), except from IHOP.
39
Fig. 10: Time evolution of vertical velocity (ms-1) from the (a) updraft and (b) downdraft
from E10km for the RICO case. (c) and (d) are the same as (a) and (b), respectively,
except from the ER88 experiment and (e) and (f) from the ER95 experiment.
40
Fig. 11: Same as Fig. 10, except for the IHOP case.
41
(a)
WRF-wmax
ER88
ETTM
-1
Vertical velocity (m s )
20
WRF-avg
ER95
15
10
5
0
0.75/2
0.75 0.98/3
0.98 1.22/4
1.22 1.41/5
1.41 1.66/6
1.66 1.82/7
1.82 2.02/8
2.02 2.17/9
2.17 2.36/10
2.36
Max cloud
equivalent
radiusradius
(km)(km)
ETTM
radiuscloud
/ 3D bubble
(b)
-1
Vertical velocity (m s )
30
WRF-wmax
ER88
WRF-avg
ER95
ETTM
25
20
15
10
5
0
0.73/3
0.73
1.22 1.37/6
1.37 1.72/7
1.72 1.93/8
1.93 2.18/9
2.18
cloud
radius
(km)
ETTMMax
cloudequvalent
radius / 3D
bubble
radius
(km)
0.96/4
0.96
1.22/5
2.40/10
2.40
Fig. 12: The WRF simulated maximum vertical velocity (WRF-wmax), the WRF maximum
horizontally averaged vertical velocity, and the ETTM maximum vertical velocity
with the original thermal bubble (ETTM) and with different relative humidities within
the thermal bubble (i.e., ER88 for the 88% humidity run and ER95 for the 95%
humidity run) with respect to different ETTM cloud radii (or 3D bubble radii) for (a)
RICO and (b) IHOP.
42
(a)
12
WRF-cbase
WRF-ctop
Height (km)
9
ETTM-cbase
ETTM-ctop
6
ER88-cbase
ER88-ctop
3
ER95-cbase
ER95-ctop
0
0.75/2
0.75 0.98/3
0.98 1.22/4
1.22 1.41/5
1.41 1.66/6
1.66 1.82/7
1.82 2.02/8
2.02 2.17/9
2.17 2.36/10
2.36
Max equivalent cloud radius (km)
ETTM cloud radius / 3D bubble radius (km)
(b)
12
WRF-cbase
WRF-ctop
Height (km)
9
ETTM-cbase
ETTM-ctop
6
ER88-cbase
ER88-ctop
3
ER95-cbase
ER95-ctop
0
0.73/3
0.73
0.96/4 1.22
1.22/5 1.37
1.37/6 1.72
1.72/7 1.93/8
0.96
1.93 2.18/9
2.18 2.40/10
2.40
ETTM
radius /cloud
3D bubble
radius
Maxcloud
equivalent
radius
(km)(km)
Fig. 13: The simulated cloud top and base with respect to different cloud radii (or 3D bubble
radii) from WRF and ETTM with sensitivity tests of different low level relative
humidities from (a) RICO and (b) IHOP. Note that the base lines overlap from
different runs and therefore not easy to see. No cloud developed in the ETTM
simulation for the cloud radius of 0.73 km with the original thermal bubble for the
IHOP sounding.
43
(a)
(b)
(c)
(d)
(e)
(f)
Fig. 14: The simulated (a) mass (×108 kg s-1), (d) heat (X1012 J s-1), and (e) moisture (×106kg
s-1) fluxes from WRF simulations for the RICO case when the bubble size is 5 km.
(b), (d), and (f) are the same as (a), (c), (e), respectively, except from the
corresponding 1D cloud simulation from ER95.
44
(a)
(b)
(c)
(d)
(e)
(f)
Fig. 15: Same as Fig. 14, except for the IHOP ER95 run with the bubble size of 10 km.
45