Geophysical Research Letters
RESEARCH LETTER
10.1002/2014GL061222
Key Points:
• Analyzed scaling of precipitation
extremes in radiativeconvective equilibrium
• Representation of ice- and
mixed-phase microphysics plays an
important role
• Response to warming depends
on accumulation period at
low temperatures
Supporting Information:
• Readme
• Sections S1 and S2 and Figures S1
to S5
Correspondence to:
M. S. Singh,
mssingh@MIT.EDU
Citation:
Singh, M. S., and P. A. O’Gorman
(2014), Influence of microphysics
on the scaling of precipitation
extremes with temperature,
Geophys. Res. Lett., 41, 6037–6044,
doi:10.1002/2014GL061222.
Received 16 JUL 2014
Accepted 4 AUG 2014
Accepted article online 8 AUG 2014
Published online 22 AUG 2014
Influence of microphysics on the scaling of precipitation
extremes with temperature
Martin S. Singh1 and Paul A. O’Gorman1
1 Department of Earth, Atmospheric and Planetary Sciences, Massachusetts Institute of Technology, Cambridge,
Massachusetts, USA
Abstract
Simulations of radiative-convective equilibrium with a cloud-system resolving model are
used to investigate the scaling of high percentiles of the precipitation distribution (precipitation extremes)
over a wide range of surface temperatures. At surface temperatures above roughly 295 K, precipitation
extremes increase with warming in proportion to the increase in surface moisture, following what is termed
Clausius-Clapeyron (CC) scaling. At lower temperatures, the rate of increase of precipitation extremes
depends on the choice of cloud and precipitation microphysics scheme and the accumulation period, and
it differs markedly from CC scaling in some cases. Precipitation extremes are found to be sensitive to the fall
speeds of hydrometeors, and this partly explains the different scaling results obtained with different
microphysics schemes. The results suggest that microphysics play an important role in determining
the response of convective precipitation extremes to warming, particularly when ice- and mixed-phase
processes are important.
1. Introduction
Increases in the intensity of precipitation extremes are seen in simulations of global warming [Kharin et al.,
2007; O’Gorman and Schneider, 2009] and have been identified in observational trends [Westra et al., 2013].
However, the rate at which precipitation extremes will strengthen as the climate warms remains uncertain
to the extent that they involve moist-convective processes that are difficult to represent in climate models
[Wilcox and Donner, 2007; O’Gorman, 2012; Kendon et al., 2014]. Observations of variability in the current
climate may also be used to derive relationships between precipitation extremes and the temperature at
which they occur [e.g., Allan et al., 2010], and analyses based on station observations have found that the
fractional increase in subdaily precipitation extremes with respect to temperature is up to twice that of daily
precipitation extremes [Lenderink and van Meijgaard, 2008; Lenderink et al., 2011; Berg et al., 2013]. However,
this dependence on timescale is not well understood, and it is unclear whether such relationships would
also hold under global warming.
Here we investigate the precipitation distribution in simulations with a cloud-system resolving model in the
idealized setting of radiative-convective equilibrium (RCE). Previous studies of the precipitation distribution in RCE have found that, at surface temperatures characteristic of Earth’s tropics, precipitation extremes
increase with warming following what is known as Clausius-Clapeyron (CC) scaling, increasing roughly in
proportion to the saturation specific humidity near the surface [Muller et al., 2011; Romps, 2011]. This conclusion holds even when the convection is organized [Muller, 2013]. CC scaling of precipitation extremes
with surface temperature is consistent with a simple view of the response of strong precipitation events to
warming in which the amount of converged water vapor increases due to the increased amount of moisture
near the surface. However, the close agreement with CC scaling found in studies of RCE may be coincidental
to some extent because the water vapor convergence does not occur only at the surface, and because the
strength and vertical profile of the updrafts change with warming [Muller et al., 2011; Romps, 2011].
In this study, we extend previous modeling results to a wider range of surface temperatures. Consistent with
previous work, precipitation extremes increase with warming at a rate similar to that implied by CC scaling
at surface temperatures above 295 K. At lower temperatures, however, the scaling of precipitation extremes
depends on the representation of cloud and precipitation microphysics used in the simulations as well as
the accumulation period considered. Instantaneous precipitation extremes increase with warming at up to
twice the rate implied by CC scaling for one of the microphysics schemes used.
SINGH AND O’GORMAN
©2014. American Geophysical Union. All Rights Reserved.
6037
Geophysical Research Letters
10.1002/2014GL061222
One factor contributing to such high rates of increase is a change in the mean fall speed of hydrometeors
in a warming atmosphere. The precipitation distribution is known to be sensitive to hydrometeor fall speed
[Parodi and Emanuel, 2009; Parodi et al., 2011]; here we find that as the atmosphere warms, the fraction of
the precipitating water in the column composed of frozen hydrometeors decreases, and the mean fall speed
of hydrometeors increases, amplifying the increase in precipitation extremes. But the degree to which the
fall speed increases with warming is sensitive to assumptions regarding the treatment of frozen hydrometeors, and hydrometeor fall speed is not the only microphysical property to have an influence on precipitation
extremes. As a result, different microphysics schemes lead to different scaling of precipitation extremes with
warming at surface temperatures for which ice- and mixed-phase microphysical processes are important,
even in the relatively simple case of RCE.
2. Simulations of Radiative-Convective Equilibrium
We analyze simulations of RCE in a doubly periodic domain conducted using version 16 of the Bryan Cloud
Model [Bryan and Fritsch, 2002]. Singh and O’Gorman [2013] used a similar set of simulations to investigate
increases in convective available potential energy with warming. The model is compressible and nonhydrostatic, and it uses sixth-order spatial differencing coupled with a sixth-order diffusion scheme for numerical
stability and a split-explicit time-stepping scheme following Wicker and Skamarock [2002]. Surface fluxes
are calculated using bulk aerodynamic formulae, while subgrid motions are parameterized through a
Smagorinsky turbulence scheme. Interactive radiation is included, but there is no diurnal cycle; all simulations are performed with a constant solar flux of 390 W m−2 incident at a zenith angle of 43◦ .
Our primary focus is on simulations conducted using a six-species, one-moment microphysics scheme
based on Lin et al. [1983] as modified by Braun and Tao [2000] in which there are three hydrometeor species
(rain, snow, and hail), each with a different fall speed that depends on the mixing ratio. We refer to this as
the Lin-hail scheme. Additionally, we consider simulations with an alternate form of the Lin et al. [1983]
scheme in which the hail category is replaced by a graupel category, which we refer to as the Lin-graupel
scheme. The major difference between these two schemes is that the density, mean size, and fall speed of
hail are taken to be considerably larger than those of graupel. Finally, we consider simulations conducted
using a third microphysics scheme based on that of Thompson et al. [2008]. This scheme also includes six
water species, but it additionally includes prognostic variables representing the number concentration of
cloud ice and (in the implementation used here) rain water, effectively being a two-moment scheme for
these species. Further details of the microphysics schemes used in this study are given in section S1 in the
supporting information.
Simulations are conducted with different imposed CO2 concentrations in the range 1–640 ppmv and at
three different horizontal resolutions. First, a set of low-resolution simulations (2 km horizontal grid spacing, 80 × 80 km domain) are run to equilibrium over a slab ocean using the Lin-hail microphysics scheme.
The slab has a depth of 1 m and a horizontally uniform temperature that responds to domain-integrated
energy fluxes at the surface. The wide range of CO2 concentrations used allows the slab ocean simulations
to achieve equilibrium sea surface temperatures (SSTs) in the range 281–311 K. While the varying CO2 concentration also affects radiative cooling rates in the simulations, previous work suggests that changes in the
magnitude of tropospheric radiative cooling may have only a weak effect on precipitation intensity in RCE
[see Figure 3 of Parodi and Emanuel, 2009]. Nevertheless, the low CO2 concentration used in the coldest simulations and the lack of large-scale forcing in RCE must be taken into account when comparing our results
to observations of convective precipitation extremes at similar surface temperatures on Earth.
A set of intermediate-resolution simulations (1 km horizontal grid spacing, 84 × 84 km domain) and a set
of high-resolution simulations (0.5 km horizontal grid spacing, 160 × 160 km domain) are then conducted
using the Lin-hail scheme over the same range of CO2 concentrations and using the equilibrium SSTs of
the corresponding slab ocean simulations as a fixed lower boundary condition. Finally, a subset of the
intermediate-resolution simulations are repeated using the alternate microphysics schemes (Lin-graupel
and Thompson). All simulations include 64 vertical levels, and the intermediate-resolution (high-resolution)
simulations are each run for 40 (30) days with statistics collected at hourly intervals after 20 days of simulation. While the behavior of precipitation extremes is broadly similar across resolutions, we primarily show
results from the intermediate-resolution simulations in order to directly compare the different microphysics
schemes. Results for the high-resolution, Lin-hail simulations are shown in Figure S1.
SINGH AND O’GORMAN
©2014. American Geophysical Union. All Rights Reserved.
6038
precipitation extremes (mm/h)
Geophysical Research Letters
a
10.1002/2014GL061222
3. Scaling of Precipitation
Extremes
102
effective fall speed (m/s)
Figure 1a shows precipitation extremes
as measured by the 99.99th percentile
of instantaneous gridbox precipitation
rates as a function of the mean temperature at the lowest model level (Ts ; the
lowest level is at 50 m). Here instanta101
neous precipitation rates are measured
by the flux of hydrometeors through the
lower boundary at a given time step,
and zero precipitation rates are included
in the calculation of percentiles. The
b
99.99th percentile is taken to be rep10
resentative of the extremes in these
simulations; higher percentiles behave
similarly, but lower percentiles begin to
behave more like the mean precipitation
intensity (the mean precipitation rate in
precipitating gridboxes), and this is left
for future study. For surface temperaLin−hail
tures above 295 K, the rate of increase
1
Lin−graupel
of precipitation extremes with warming
Thompson
is similar to that implied by CC scaling
(gray lines) for all microphysics schemes
280
290
300
310 considered, although it is slightly sub-CC
in the Thompson simulations. At lower
Ts (K)
temperatures, however, the scaling
of precipitation extremes is strongly
Figure 1. (a) The 99.99th percentile of the instantaneous precipitation
rate and (b) the effective hydrometeor fall speed conditioned on the
dependent on the microphysics scheme
precipitation exceeding its 99.99th percentile (vf ). Simulations using
used. For simulations with the Lin-hail
the Lin-hail (black), Lin-graupel (thick gray), and Thompson (dashed)
scheme, instantaneous precipitation
microphysics schemes are shown as a function of the mean temperextremes increase rapidly with temperature at the lowest model level (Ts ). In Figure 1a, thin gray lines are
contours proportional to the surface saturation specific humidity, and
ature, exceeding the rate of increase
red dashed lines are proportional to the square of the surface saturagiven by CC scaling (6–7% K−1 ) and
tion specific humidity; each successive line corresponds to a factor of
approaching twice that rate (red dashed
2 increase.
lines). (Slightly higher rates of increase
are found if the SST is used instead of Ts as a measure of the surface temperature.) Scaling of precipitation
extremes above the CC rate is also found for the Thompson simulations, but only over a much narrower
range of surface temperatures than for the Lin-hail simulations. For the Lin-graupel simulations, the increase
in precipitation extremes with warming varies somewhat with temperature but remains relatively close to
the CC rate.
Figure 2 shows precipitation extremes for 1, 3, and 6 h accumulation periods in addition to the instantaneous precipitation extremes shown earlier (daily precipitation extremes are also shown for the
high-resolution, Lin-hail simulations in Figure S1). As with the instantaneous precipitation extremes, the
scaling of the accumulated precipitation extremes is relatively robust at high surface temperatures, and it
is similar to CC scaling for all microphysics schemes. At lower temperatures, however, the rate of increase of
accumulated precipitation extremes with warming varies with temperature and the microphysics scheme
used. For example, for surface temperatures below 295 K, the 6-hourly precipitation extremes increase
with warming at close to the CC rate in the Lin-hail simulations, whereas in the Thompson simulations they
increase weakly or decrease with warming. Both the Thompson simulations and the Lin-hail simulations
(but not the Lin-graupel simulations) have much greater fractional increases in instantaneous precipitation
extremes compared with 6-hourly precipitation extremes for temperatures below 295 K.
SINGH AND O’GORMAN
©2014. American Geophysical Union. All Rights Reserved.
6039
precipitation extremes (mm/h)
Geophysical Research Letters
Lin−hail
a
10.1002/2014GL061222
Lin−graupel
b
Thompson
c
inst
1h
3h
101
6h
100
280
290
300
310
280
Ts (K)
290
300
310
280
290
Ts (K)
300
310
Ts (K)
Figure 2. The 99.99th percentile of the instantaneous precipitation rate (inst) and precipitation rate averaged over 1 h,
3 h, and 6 h in simulations with (a) Lin-hail, (b) Lin-graupel, and (c) Thompson microphysics schemes (black). Gray lines
are contours proportional to the surface saturation specific humidity with each successive line corresponding to a factor
of 2 increase.
In the following sections we seek to understand which aspects of the different microphysics schemes lead
to the differences in the scaling of precipitation extremes and the deviations from CC scaling documented
above. While accumulated precipitation rates also vary across the microphysics schemes considered, we
focus on instantaneous precipitation extremes because they are simpler to analyze.
4. Scaling of Condensation Extremes
We first consider extremes of the instantaneous column net-condensation rate, which we define as the
vertical integral over the column of the net microphysical sink of water vapor at a given time step. In
contrast to the scaling of instantaneous precipitation extremes, net-condensation extremes roughly follow
CC scaling at all temperatures (Figure 3a) as well as across the three microphysics schemes (Figure S2). The
roughly CC scaling of net-condensation extremes occurs despite the peak vertical velocity conditioned on
net-condensation extremes increasing substantially with warming (Figure S3). As explained by Muller et al.
[2011] and discussed in detail in section S2, the net-condensation rate is particularly sensitive to the vertical velocity at low levels. In our simulations, the vertical velocity conditioned on net-condensation extremes
condensation extremes (Ce; mm/h)
0
2
10
4
fall speed (m/s)
8
102
8 m/s
4 m/s
2 m/s
101
101
1 m/s
280
290
300
310
Ts (K)
280
290
300
precipitation extremes (Pe; mm/h)
b
a
310
Ts (K)
Figure 3. The 99.99th percentile of instantaneous (a) column net-condensation rate and (b) surface precipitation rate
as a function of the mean temperature of the lowest model level (Ts ). Black lines correspond to Lin-hail simulations and
colored lines correspond to altered Lin-hail simulations in which the fall speeds of all hydrometeors are set to constant
values of 1 (blue), 2 (cyan), 4 (green), and 8 (maroon) m s−1 . Marker colors in Figure 3b correspond to the effective fall
speed (vf ) of hydrometeors in the Lin-hail simulations (see text). Thin gray lines are contours proportional to the surface
saturation specific humidity, with each successive line corresponding to a factor of 2 increase.
SINGH AND O’GORMAN
©2014. American Geophysical Union. All Rights Reserved.
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Geophysical Research Letters
10.1002/2014GL061222
decreases with warming at levels below 800 hPa. Consistent with Muller et al. [2011], low-level changes in
vertical velocity, therefore, have a negative influence on net-condensation extremes, explaining how the
peak updraft increases, but the net-condensation rate roughly follows CC scaling.
The simulations using the Thompson scheme experience a slightly larger increase in net-condensation
extremes (Figure S2) and peak updraft strength (Figure S3) with warming compared to the Lin-hail and
Lin-graupel simulations. These dynamical differences, however, are relatively small, and differences in condensation extremes are not sufficient to explain the different scaling of precipitation extremes across the
different microphysics schemes used.
5. The Effect of Hydrometeor Fall Speed on Instantaneous Precipitation Extremes
It is useful to represent the relationship between instantaneous net-condensation extremes and instantaneous precipitation extremes by an efficiency 𝜖P such that
Pe = 𝜖P Ce ,
(1)
where Pe and Ce are the 99.99th percentiles of the instantaneous precipitation rate and instantaneous
column net-condensation rate, respectively. Since we consider net condensation and since the occurrence of precipitation extremes may not be exactly collocated in space and time with the occurrence of
net-condensation extremes, 𝜖P is not a conventional precipitation efficiency. Nevertheless, 𝜖P represents
the efficiency by which large net-condensation events are translated into large precipitation events as the
condensate falls to the surface.
The behavior of 𝜖P differs across simulations using different microphysics schemes. For simulations with
the Lin-hail scheme, the fractional rate of increase of instantaneous precipitation extremes with warming is larger than that of instantaneous net-condensation extremes at low temperatures, indicating that 𝜖P
increases with warming (Figure 3). For the Lin-graupel and Thompson simulations, 𝜖P varies nonmonotonically with surface temperature, with an overall slight increase with warming occurring in the Lin-graupel
simulations and a slight decrease in the Thompson simulations (Figure S4).
In order to understand the deviations of instantaneous precipitation extremes from CC scaling, we need
to understand the causes of variations in 𝜖P with temperature. One factor influencing the value of 𝜖P is the
fall speed of hydrometeors. A low fall speed results in a longer time between condensation and precipitation, and it increases the probability that a given hydrometeor might evaporate before it reaches the surface
(Figure 4a). Additionally, an increased hydrometeor lifetime allows more time for the precipitation event
to be smeared out spatially relative to the condensation event by turbulence or the cloud-scale circulation
(Figure 4b). While simple advection of the column does not affect 𝜖P (since precipitation extremes and condensation extremes need not be collocated), a horizontal smearing of precipitation over a larger area may
contribute to a reduction in 𝜖P . Finally, a low hydrometeor fall speed may also affect 𝜖P by smearing out the
precipitation in time. Since condensation occurs at a range of heights, precipitation will reach the ground
at different times, even if the condensation were to occur at a single instant. This reduces the magnitude of
the maximum instantaneous precipitation rate relative to the maximum column condensation rate, even
if it does not alter the total precipitation from a given convective event (Figure 4c). The effect of this temporal smearing on precipitation extremes is thus largest for instantaneous precipitation extremes, while
precipitation extremes accumulated over time periods longer than a convective event (∼ 1 hour) may be
relatively unaffected.
The hydrometeor fall speed is sensitive to temperature changes; frozen hydrometeors can have significantly
different fall speeds compared to that of rain [see, e.g., Pruppacher and Klett, 1997]. To investigate the effect
that these fall speed changes may have on the response of precipitation extremes to warming, we examine additional intermediate-resolution simulations in which the Lin-hail microphysics scheme is altered such
that the fall speeds of all hydrometeors are fixed to a constant value. Sets of simulations are conducted with
fall speeds in the range 1–8 m s−1 ; for each fall speed, simulations are run with different SST boundary conditions corresponding to a subset of the SSTs used in the simulations described in section 2. Apart from the
values of the fall speeds of snow, rain, and hail, the model used for each of these simulations is identical to
the model used for the corresponding Lin-hail simulation at the same SST (see also section S1). We focus
here on the Lin-hail simulations because they exhibit the largest variations in 𝜖P , but changes in fall speeds
in the other schemes are also discussed later.
SINGH AND O’GORMAN
©2014. American Geophysical Union. All Rights Reserved.
6041
Geophysical Research Letters
evaporation/sublimation
b
spatial smearing
c
precip. or cond. rate
a
10.1002/2014GL061222
temporal smearing
decreasing
fall speed
time
Figure 4. Schematic showing mechanisms affecting the value of the precipitation efficiency, 𝜖P . (a) Evaporation and
sublimation of precipitation, (b) spatial smearing of precipitation via removal of liquid and solid water from the column
by turbulence (both resolved and subgrid) and the cloud-scale circulation, and (c) smearing of the precipitation event in
time as the hydrometeor fall speed decreases (maroon to green to blue). Dashed black line in Figure 4c represents the
column net-condensation rate.
Changes in hydrometeor fall speed have a large effect on precipitation extremes (Figure 3b). For example,
an increase in fall speed from 1 to 8 m s−1 results in an increase in instantaneous precipitation extremes by
more than a factor of 5 at a surface temperature of Ts = 298 K. That fall speeds influence precipitation statistics is consistent with the results of Parodi et al. [2011]. Parodi and Emanuel [2009] argue that hydrometeor
fall speed also influences updraft velocities because a lower fall speed results in the lofting of a greater quantity of condensed water that then reduces updraft buoyancy through water loading. But in our simulations,
net-condensation extremes (Figure 3a), as well as the vertical velocity conditioned on net-condensation
extremes (not shown), only weakly depend on fall speed. A stronger dependence of the updraft strength
on hydrometeor fall speed is found if only warm-rain microphysical processes are allowed as in Parodi and
Emanuel [2009], but in general, the effect of fall speed on 𝜖P is a much larger factor in determining the
intensity of precipitation extremes.
To quantify the variations in fall speed more generally, we define the effective hydrometeor fall speed, vf , as
the hydrometeor-mass-weighted mean of the fall speeds of all hydrometeors in the column conditioned on
the precipitation rate exceeding its 99.99th percentile. In the fixed-fall speed simulations, this is simply equal
to the imposed hydrometeor fall speed. In the simulations with the Lin-hail microphysics scheme, vf ranges
from less than 1 m s−1 in the coldest simulation to more than 8 m s−1 in the warmest simulation (Figure 1b).
This is primarily a result of the increasing fraction of rain compared to snow in the column as the atmosphere
warms, but the increase in rain mixing ratios with warming also increases the effective fall speed of rain itself
(Figures S5a and S5d). Hail has a greater fall speed than both snow and rain, but in the Lin-hail simulations,
it contributes only a small fraction of the total hydrometeor loading of the atmosphere.
Based on vf , and by comparison with the fixed-fall speed simulations, the increase in precipitation extremes
with warming in the Lin-hail simulations may be seen to consist of a component related to the increase
in surface temperature at fixed fall speed and a component due to the effect of increasing fall speed. The
precipitation extremes in a given Lin-hail simulation are roughly consistent with those in a fixed-fall speed
simulation at the same surface temperature and with the same effective fall speed (compare marker colors
and line colors in Figure 3b), confirming the utility of considering the effects of increasing hydrometeor fall
speed separately to other effects of warming.
At fixed fall speed, the increase of precipitation extremes with warming is somewhat below the CC rate and
somewhat smaller than the increase in condensation extremes, implying a decrease in 𝜖P with warming
(Figure 3b). Part of this reduction in 𝜖P may relate to an increase in hydrometeor lifetime caused by
an increase in the mean height over which precipitation falls, since the typical formation height of
hydrometeors rises with warming. In the Lin-hail simulations, the increase in effective fall speed vf amplifies
the increase in precipitation extremes with warming. This results in super-CC scaling of instantaneous
precipitation extremes at low temperatures (for which the fractional rate of increase in vf with warming
is largest), and it results in roughly CC scaling at temperatures greater than ∼ 295 K (for which the
hydrometeor distribution is dominated by rain, and the fractional increase in vf with temperature is smaller).
SINGH AND O’GORMAN
©2014. American Geophysical Union. All Rights Reserved.
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Geophysical Research Letters
10.1002/2014GL061222
In simulations using the Lin-graupel and Thompson microphysics schemes, the fractional increase in vf
with warming is smaller than in the Lin-hail case (Figure 1b), and the fractional rates of increase of instantaneous precipitation extremes are also smaller (Figure 1a). The difference in behavior of the fall speed may
be attributed to the larger abundance of rimed ice (i.e., graupel or hail) during heavy precipitation events
in simulations using these alternate microphysics schemes, which, because of its faster fall speed compared
to that of snow, increases the mean hydrometeor fall speed at low temperatures (Figure S5). The smaller
increase in vf with warming in the Lin-graupel and Thompson simulations at least partially accounts for
the lower fractional rate of increase of precipitation extremes found with these schemes when compared
to the Lin-hail simulations. Other aspects of the microphysical schemes used and the different scaling of
condensation extremes in the Thompson simulations may also be expected to contribute to the disagreement of instantaneous precipitation extremes across the different microphysics schemes.
6. Conclusions
Our results show that precipitation extremes in radiative-convective equilibrium increase with warming at
a rate roughly consistent with Clausius-Clapeyron scaling at surface temperatures above 295 K. At lower
temperatures, uncertainty regarding the representation of ice- and mixed-phase microphysics has a large
effect on simulated precipitation extremes, and a variety of behaviors occur.
The fall speed of hydrometeors is found to be one factor influencing the intensity of precipitation extremes.
The precipitation rate increases as the hydrometer fall speed is increased because of changes in the efficiency with which large net-condensation events are translated into large precipitation events. The mean
hydrometeor fall speed in the simulations increases with warming as the fraction of hydrometeors in the
column consisting of frozen species decreases. This amplifies the increase of precipitation extremes relative
to the case in which hydrometeor fall speeds are held fixed, particularly for low temperatures at which the
fractional change in fall speed with warming is largest.
However, the size of the increase in hydrometeor fall speed is sensitive to the fraction of frozen precipitation existing as rimed ice as well as the fall speed of the rimed ice species itself. Differences in
the treatment of frozen hydrometeors at least partially explain the different responses of precipitation
extremes to warming in simulations with different microphysics schemes. Indeed, the fall speed characteristics of rimed ice have been found previously to be important in determining the precipitation and
radar reflectivities in modeling case studies of supercell [Morrison and Milbrandt, 2011] and squall line
[Bryan and Morrison, 2012] convection.
Other mechanisms not considered in this paper, such as those relating to the formation of precipitating
hydrometeors, may also be important for the scaling of precipitation extremes. The scaling of accumulated
precipitation extremes does not appear to be related to hydrometeor fall speeds in a simple way, and other
dynamical and microphysical factors may play a role in giving the varied behavior of accumulated precipitation extremes found here. Thus, while hydrometeor fall speeds are clearly important in determining
the intensity of convective precipitation extremes for short accumulation periods, further work is required
to fully understand the mechanisms by which microphysical processes may influence the precipitation
distribution in RCE.
Our results may have implications for the behavior of precipitation extremes under climate change or in
observed variability in the current climate. For instance, a change in hydrometeor fall speed is one factor
that potentially contributes to the super-CC scaling of subdaily precipitation extremes found in some
high temporal resolution station observations when stratified by surface temperature [e.g., Lenderink and
van Meijgaard, 2008; Lenderink et al., 2011; Berg et al., 2013]. Dynamical effects have also been argued to
be relevant in explaining the potential for super-CC scaling of precipitation extremes with warming [Loriaux
et al., 2013], and relationships between temperature and specific dynamical regimes or moisture availability
[Hardwick Jones et al., 2010] complicate the interpretation of precipitation and temperature covariability in
observations. This study highlights the role of cloud and precipitation microphysics in helping to determine
the response of convective precipitation extremes to warming, and it emphasizes the need for continued
research to better constrain the modeling of ice- and mixed-phase precipitation processes.
SINGH AND O’GORMAN
©2014. American Geophysical Union. All Rights Reserved.
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Geophysical Research Letters
Acknowledgments
We thank Richard Allan and an
anonymous reviewer for helpful comments. Model results were obtained
using the numerical code CM1,
which is maintained by George
Bryan and is available at http://
www.mmm.ucar.edu/people/bryan/
cm1/. High-performance computing support from Yellowstone
(ark:/85065/d7wd3xhc) was provided
by NCAR’s Computational and Information Systems Laboratory, sponsored
by the NSF. We acknowledge support from NSF grant AGS-1148594 and
NASA ROSES grant 09-IDS09-0049.
The Editor thanks Richard Allan and
an anonymous reviewer for their
assistance in evaluating this paper.
SINGH AND O’GORMAN
10.1002/2014GL061222
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©2014. American Geophysical Union. All Rights Reserved.
6044
Influence of microphysics on the scaling of precipitation
extremes with temperature
Martin S. Singh & Paul A. O’Gorman
S1
Details of microphysics schemes
Here we describe in further detail the different microphysics schemes used in this study.
Lin-hail
The Lin-hail microphysics scheme is based on Lin et al. (1983) as modified by the Goddard Cumulus
Ensemble Modeling Group (Tao and Simpson, 1993; Braun and Tao, 2000). It includes six classes
of water species representing water vapor, cloud water, cloud ice, rain, snow and hail. The size
distributions of the precipitating species (rain, snow and hail) are assumed to follow an exponential
distribution such that the concentration of particles of species i with diameter Di per unit size interval
is given by
n(Di ) = n0i exp (−λi Di ) .
(S1)
Here n0i is the intercept parameter and λi is the slope parameter, defined by
λr =
λs =
λh =
�
�
�
�14
πρr n0r
ρa r r
�14
πρr n0s
ρa r s
πρh n0h
ρa r h
,
,
�14
,
where ρ is the density, r is the mixing ratio and the subscripts refer to dry air (a), rain (r), snow (s)
and hail (h). Following Potter (1991), the density of liquid water ρr is used rather than the density of
snow when calculating λs . Some of the values of the particle densities and intercept parameters differ
from those used originally by Lin et al. (1983) and are given in table S1.
Based on the size distributions given above and an assumed form for the fall speed of each particle,
the mass-weighted fall speeds of each hydrometeor species Ui may be written,
Ur =
ar Γ(4 + br )
6λbrr
�
ρ0
ρa
�12
,
� �1
as Γ(4 + bs ) ρ0 2
,
ρa
6λbss
�
�1
Γ(4.5)
4gρh 2
Uh =
.
1
3CD ρa
6λ 2
Us =
h
Here, Γ(·) is the gamma function, ρ0 is the dry air density at the surface, and the constants are defined
in table S1. The microphysical source and sink terms of each water species are calculated following
Lin et al. (1983) as modified by Braun and Tao (2000), and a saturation adjustment similar to Tao
et al. (1989) is applied to prevent supersaturation. Cloud water is assumed to have zero fall speed
relative to air, while cloud ice is assumed to fall at a uniform velocity of 0.2 m s−1 .
1
Lin-graupel
The Lin-graupel scheme is similar to the Lin-hail scheme, except that it includes graupel as one of the
frozen hydrometeor species instead of hail. The size distribution of graupel is defined by (S1), but it is
assumed to have a smaller density, ρg , and a larger intercept parameter, n0g , compared to that of hail
(table S1) and a different form for the mass-weighted fall speed given by (Rutledge and Hobbs, 1984),
� �1
ag Γ(4 + bg ) ρ0 2
,
(S2)
Ug =
b
ρ
6λgg
where the slope parameter for graupel is defined,
�
�1
πρg n0g 4
λg =
.
ρa r g
The functional form of the microphysical source and sink terms for graupel are identical to those of
hail in the Lin-hail scheme, but, because of the different density, intercept parameter and fall speed of
graupel, the magnitude of these tendency terms differ, even for identical environmental conditions.
Fixed-fall speed
The fixed-fall speed simulations described in section 5 use an altered form of the Lin-hail scheme in
which the fall speed of all hydrometeors are fixed to given values for the purposes of the precipitation
fallout calculation. However, the fall speed of cloud ice remains set to 0.2 m s−1 , and the microphysical
source and sink terms are calculated based on the fall speeds as given by the original Lin-hail scheme.
Thompson
The microphysics scheme referred to as the Thompson scheme in this study is based on that of Thompson et al. (2008). As with the Lin-based schemes, it includes six water species (vapor, cloud water,
cloud ice, rain, snow and graupel). However, unlike the Lin-based schemes, the Thompson scheme assumes the size distributions of all hydrometeors except snow follow a generalized gamma distribution,
and it includes as prognostic variables the number concentration of cloud ice particles and rain drops
in addition to the mixing ratio of each water species. Thus, the Thompson scheme is a two-moment
scheme for cloud ice and rain water, and a one-moment scheme for all other condensate species1 . In
addition, Thompson et al. (2008) allows for the distribution parameters of the hydrometeor species
to depend on their mixing ratio in an attempt to emulate the behavior of more comprehensive, fully
two-moment schemes (e.g., Morrison et al., 2005). In the simulations used in this study, we assume a
fixed number concentration of cloud droplets characteristic of maritime conditions of 100 cm−3 .
S2
Condensation extremes in RCE
Here we analyze the scaling of instantaneous net-condensation extremes2 in the simulations in further
detail. We consider a decomposition of the net condensation similar to the scaling used by Muller
et al. (2011):
� zt ∗
∂q
Ce = −�C
ρwe dz.
(S3)
∂z
0
1 In Thompson et al. (2008) only cloud ice is treated as two-moment, but in the implementation used in this study
rain is also included as a two-moment species.
2 Considering condensation rather than net condensation (i.e., neglecting the contribution of grid boxes in which the
microphysical tendency of water vapor is positive) results in values of condensation extremes that differ by less than
three percent.
2
Table S1: Parameters used in the Lin-hail and Lin-graupel microphysics schemes.
Symbol
n0r
ρr
ar
br
n0s
as
bs
n0h
ρh
CD
n0g
ρg
ag
bg
Description
Intercept parameter for rain
Density of rain water
Rain fall speed constant
Rain fall speed exponent
Intercept parameter for snow
Snow fall speed constant
Snow fall speed exponent
Intercept parameter for hail
Density of hail stones
Drag co-efficient
Intercept parameter for graupel
Density of graupel particles
Graupel fall speed constant
Graupel fall speed exponent
Value
8 × 106
1000
842
0.8
1 × 108
12.4
0.42
2 × 104
900
0.6
4 × 106
400
19.3
0.37
Units
m−4
kg m−3
m1−br s−1
m−4
m1−bs s−1
m−4
kg m−3
m−4
kg m−3
m1−bg s−1
Here Ce is the 99.99th percentile of column net condensation, ρ is the air density, q ∗ is the saturation
specific humidity, the over-bar represents a time and domain mean and zt is the height of the tropopause
(taken to be the level at which the lapse rate equals 2 K km−1 ). The vertical velocity profile we (z) is
calculated as the mean vertical velocity for points at which the column net-condensation rate exceeds
its 99.99th percentile. The integral in (S3) represents an estimate of the column net-condensation rate
derived from the dry static energy budget (see Muller et al., 2011). The derivation neglects storage,
horizontal advection and deviations from moist adiabatic lapse rates; the inaccuracies in the estimate
associated with these approximations are collected into the condensation efficiency �C so that (S3) is
exactly satisfied.
According to (S3), for a simple mass-flux profile with convergence near the surface and divergence
near the tropopause, net-condensation extremes follow CC scaling if the mass-flux profile and the
efficiency �C are invariant under warming. Deviations of net-condensation extremes from CC scaling
result from a more realistic mass-flux profile, or from changes in the mass-flux profile or condensation
efficiency with temperature. As discussed in Muller et al. (2011), the expression for the condensation
rate (S3) includes the vertical velocity within an integral weighted by the vertical gradient in saturation
specific humidity. This vertical gradient maximizes at the surface and decreases exponentially above,
which gives the low-level vertical velocity more weight in determining the condensation rate compared
to the vertical velocity at higher levels. As pointed out in section 4, the negative low-level changes in
we are thus an important negative influence on net-condensation extremes in our simulations.
In the Lin-hail simulations, the condensation efficiency �C decreases between 0.84 and 0.75 across
the intermediate-resolution simulations (Fig. S4), with the condensation efficiency behaving similarly
for simulations with the other microphysics schemes. However, the changes in �C depend on the precise
form of (S3) used. Muller (2013), used a slightly different approach in which the net-condensation
rate was estimated based on the vertical gradient of dry static energy rather than saturation specific
humidity. With this approach, the condensation efficiency �C does not decrease monotonically with
warming in the Lin-hail simulations. Regardless of the precise form of (S3) used, however, the fractional changes in �C with warming are generally considerably smaller than fractional changes in the
precipitation efficiency �P defined in section 5.
3
References
Braun, S. A., and W.-K. Tao (2000), Sensitivity of high-resolution simulations of hurricane Bob (1991)
to planetary boundary layer parameterizations, Mon. Wea. Rev., 128, 3941–3961.
Lin, Y.-L., R. D. Farley, and H. D. Orville (1983), Bulk parameterization of the snow field in a cloud
model, J. Clim. Appl. Meteorol., 22, 1065–1092.
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parameterization for application in cloud and climate models. Part I: Description, J. Atm. Sci., 62,
1665–1677.
Muller, C. (2013), Impact of convective organization on the response of tropical precipitation extremes
to warming, J. Climate, 26, 5028–5043.
Muller, C. J., P. A. O’Gorman, and L. E. Back (2011), Intensification of precipitation extremes with
warming in a cloud-resolving model, J. Climate, 24, 2784–2800.
Potter, B. E. (1991), Improvements to a commonly used cloud microphysical bulk parameterization,
J. Appl. Meteorol., 30, 1040–1042.
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clouds and precipitation in midlatitude cyclones. XII: A diagnostic modeling study of precipitation
development in narrow cold-frontal rainbands, J. Atm. Sci., 41, 2949–2972.
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Rev., 117, 231–235.
Thompson, G., P. R. Field, R. M. Rasmussen, and W. D. Hall (2008), Explicit forecasts of winter
precipitation using an improved bulk microphysics scheme. Part II: Implementation of a new snow
parameterization, Mon. Wea. Rev., 136, 5095–5115.
4
precipitation extremes (mm/hr)
2
10
inst
1 hr
3 hr
6 hr
1
10
24 hr
0
10
280
290
Ts (K)
300
310
condensation extremes (mm/hr)
Figure S1: As in Fig. 2 but for high-resolution simulations using the Lin-hail microphysics scheme.
The larger sample size at this resolution allows daily accumulations (24 hr) to be also included.
2
10
Lin−hail
Lin−graupel
Thompson
1
10
280
290
Ts (K)
300
310
Figure S2: The 99.99th percentile of instantaneous column net-condensation rate as a function of
the mean temperature of the lowest model level (Ts ). Simulations with Lin-hail (black), Lin-graupel
(thick-gray) and Thompson (dashed) microphysics schemes are shown. Thin gray lines are contours
proportional to the surface saturation specific humidity, and red-dashed lines are proportional to the
square of the surface saturation specific humidity; each successive line corresponds to a factor of two
increase.
5
Lin−hail
a
Lin−graupel
b
Thompson
c
pressure (hPa)
200
400
600
800
1000
0
5 10
w e (m/s)
15
0
5 10
w e (m/s)
15
0
5 10
w e (m/s)
15
Figure S3: Mean vertical velocity profiles for columns in which the instantaneous column netcondensation rate exceeds its 99.99th percentile. (a) Lin-hail simulations with mean temperatures
of the lowest model level (Ts ) of 277 (black), 292, 298 and 309 K (orange). (b) Lin-graupel simulations
with Ts equal to 277 (black), 292, 298 and 308 K (orange). (c) Thompson simulations with Ts equal
to 277 (black), 292, 298 and 308 K (orange).
1
ε
Efficiency
C
0.1
Lin−hail
Lin−graupel
Thompson
280
290
Ts (K)
300
ε
P
310
Figure S4: Condensation efficiencies [�C ; defined in (S3)] and precipitation efficiencies [�P ; defined in
(1)] as a function of the mean temperature of the lowest model level (Ts ). Results for simulations using
the Lin-hail (black), Lin-graupel (gray) and Thompson (dashed) microphysics schemes are shown.
6
fraction of hydrometeor mass
effective fall speed (m/s)
Lin−hail
a
Lin−graupel
b
Thompson
c
10
1
snow
e
d
rain
hail/graupel
f
0.75
0.5
0.25
0
280
290 300
Ts (K)
310
280
290 300
Ts (K)
310
280
290 300
Ts (K)
310
Figure S5: (a,b,c) Effective hydrometeor fall speed, vf (black), and effective fall speed of snow (blue),
rain (green) and hail/graupel (maroon) defined analogously to vf but only considering the specific
hydrometeor. (d,e,f) Fraction of total column hydrometeor mass comprised of each hydrometeor species
for columns in which the instantaneous precipitation rate exceeds its 99.99th percentile [colors same
as in (a)]. Lin-hail (left), Lin-graupel (center) and Thompson (right) simulations are shown.
7