ACPAtmospheric Chemistry and PhysicsACPAtmos. Chem. Phys.1680-7324Copernicus PublicationsGöttingen, Germany10.5194/acp-16-12127-2016Turbulence effects on warm-rain formation in precipitating
shallow convection revisitedSeifertAxelaxel.seifert@dwd.dehttps://orcid.org/0000-0001-9760-3550OnishiRyohttps://orcid.org/0000-0001-9250-0712Deutscher Wetterdienst, Offenbach, GermanyCenter for Earth Information Science and Technology, Japan Agency for Marine-Earth
Science and Technology, Yokohama Kanagawa, JapanAxel Seifert (axel.seifert@dwd.de)28September20161618121271214117May201624May20161September20169September2016This work is licensed under a Creative Commons Attribution 3.0 Unported License. To view a copy of this license, visit http://creativecommons.org/licenses/by/3.0/This article is available from https://acp.copernicus.org/articles/16/12127/2016/acp-16-12127-2016.htmlThe full text article is available as a PDF file from https://acp.copernicus.org/articles/16/12127/2016/acp-16-12127-2016.pdf
Two different collection kernels which include turbulence effects on the
collision rate of liquid droplets are used as a basis to develop a
parameterization of the warm-rain processes autoconversion, accretion, and
self-collection. The new parameterization is tested and validated with the help
of a 1-D bin microphysics model. Large-eddy simulations of the rain formation
in shallow cumulus clouds confirm previous results that turbulence effects
can significantly enhance the development of rainwater in clouds and the
occurrence and amount of surface precipitation. The detailed behavior differs
significantly for the two turbulence models, revealing a considerable
uncertainty in our understanding of such effects. In addition, the
large-eddy simulations show a pronounced sensitivity to grid resolution, which
suggests that besides the effect of sub-grid small-scale isotropic
turbulence which is parameterized as part of the collection kernel also the
larger turbulent eddies play an important role for the formation of rain in
shallow clouds.
Introduction
The formation of rain in warm liquid clouds is a result of the condensational
growth on cloud condensation nuclei and the subsequent growth of these
droplets by binary collisions . Especially in strongly turbulent clouds,
like cumulus convection, the in-cloud turbulence can potentially increase the
frequency of such binary collisions and thereby enhance rain formation
(; ). This problem has attracted
considerable attention over the last 2 decades, culminating in the
formulation of the semi-empirical collision–coalescence kernel of Ayala and
Wang . This collection kernel
attempts to provide a complete and quantitative description of the collision
processes in turbulent (warm) clouds. Subsequently, have
applied this kernel and formulated a two-moment bulk microphysical model that
takes into account the turbulence effects on autoconversion and accretion as
predicted by the Ayala–Wang kernel. In large-eddy simulations (LESs) of trade
wind cumulus convection have shown a significant impact
of the turbulence effect on in-cloud rain formation and surface rain
rates. These results, which were based on a two-moment bulk scheme, have later
been largely confirmed by using a bin microphysics
model in an LES.
The semi-empirical collision–coalescence kernel of Ayala and Wang is to a
large extent based on the results of direct numerical simulation (DNS) which
are necessary to quantify the turbulence effects on the collision statistics
in terms of, e.g., the radial distribution function to describe the
preferential concentration effect. As the DNS results are obtained at fairly
low Reynolds number, much lower than observed within clouds, the formulation
of the collection kernel includes an extrapolation to large Reynolds numbers.
An alternative collection kernel recently proposed by
yields similar results at low Reynolds numbers where DNS data are available
but differs significantly in the Reynolds number dependency and the predicted
values at high Reynolds numbers .
In the following we revisit the results of and repeat
most of their study, but now we apply the Onishi kernel and an updated
version of the Ayala–Wang kernel. First, we derive and validate the
corresponding two-moment bulk schemes, which already allows us some insights
into the differences between the two kernels. Next, we apply the two-moment bulk
scheme in a large-eddy simulation study to test whether the differences
between the two kernels matter in LESs of trade wind cumulus clouds.
The structure of this paper very much follows in the steps of the
study. After a short review of the basic relations the
two collection kernels are presented in Sect. 2. In Sect. 3 we use a box
model to derive the enhancement factor for autoconversion. In Sect. 4 the
two-moment scheme is applied and validated in a 1-D kinematic model. The
large-eddy simulations are presented and discussed in Sect. 5, followed by the
Conclusions.
Parameterizations of the turbulence effects in the
collision–coalescence kernel
For pure gravitational collisions the collection kernel can be written as
see, e.g.,Kgrav(r1,r2)=π[r1+r2]2|v(r1)-v(r2)|Ecoll,
where r1 and r2 are the radii of the two droplets, v(r) is the
terminal fall velocity of droplets, and Ecoll is the collision
efficiency. For a turbulent flow the more general definition of the
collision–coalescence kernel
K(r1,r2)=2π[r1+r2]2wrg12EcollηE,
has to be used. Here wr is the radial relative velocity at contact
. The radial distribution
function g12 quantifies the effect of preferential concentration on the
pair number density statistics, and ηE represents an enhancement
factor due to a modification of the collision efficiency by the turbulent
flow. For further details and explanations of the basic concepts we refer to
the recent reviews by and .
Any physical model of wr, g12, and ηE should be
formulated in the dimensionless numbers that characterize the system. These
are first of all the two Stokes numbers of the two colliding particles with
the Stokes number being defined by
St=τpτk,
where τp is the particle relaxation timescale and
τk is the Kolmogorov timescale. The particle relaxation timescale is given by
τp=29ρpρar2νa,
with the material density of the particle ρp (here liquid water
with ρp=103 kg m-3), the air density ρa,
and the kinematic viscosity of air νa. The Kolmogorov timescale
τk is related to the Kolmogorov length scale ℓk and
the turbulent dissipation rate ϵ by
τk=ℓk2νa=νaϵ.
Due to the r2 dependency of τp, the Stokes number increases
with droplet size. Typical cloud droplets with radii smaller than
20 µm have a Stokes number below 0.2, large cloud droplets and small
rain drops are close to St=1, and larger raindrops have a large
Stokes number St≫1. Preferential concentration effects, i.e.,
large values of g12, occur for St≈1. Smaller droplets
with smaller Stokes numbers simply follow the flow and show no clustering,
while drops with St≫1 do not feel the small-scale turbulence
due to their inertia, and their trajectories are, in addition, largely
determined by their significant terminal fall velocity. Therefore large cloud
droplets and small raindrops with radii between 20 and 100 µm are
most strongly affected by turbulence effects.
A turbulent flow is not yet fully characterized by τk (or
ϵ) alone. To quantify the root mean square of the turbulent velocity
fluctuations, urms, we introduce the Taylor-microscale Reynolds
number defined by
Enhancement factor of the collision–coalescence kernel for a
dissipation rate of ϵ=1000 cm2 s-3. Shown are
(a) the Ayala–Wang kernel for a Taylor-microscale Reynolds number of
1000, (b) the Ayala–Wang kernel for
Reλ=20 000, and (c) the ratio of the Ayala–Wang kernel at Reλ=20 000 and
Reλ=1000. The second row shows the same plot for the Onishi kernel at
ϵ=1000 cm2 s-3 and (d)Reλ=1000 and (e)Reλ=20 000, and (f) the ratio
between the kernels at those two Reynolds numbers.
Reλ=urmsλTνa=15νaϵurms2νa.
The Taylor-microscale Reynolds number is important for the collision
statistics as it is closely related to the two-point correlation and the
autocorrelation functions of turbulent flows. In general, turbulence has
three independent length scales: the Kolmogorov scale, ℓk; the
Taylor microscale, λT; and a large-eddy or integral length
scale . Therefore we will throughout most of this paper
treat ϵ and Reλ as two independent variables.
Only later when we apply the collision–coalescence model in LES will we
parameterize Reλ as a function of ϵ.
Various models have been suggested to parameterize wr, g12, and
ηE in terms of St and Reλ. Here we
focus on the models of Wang and Ayala
and Onishi
. A detailed discussion of these two models
has recently been given by . We refer to those
papers for the relevant parameterization equations.
Figure shows the enhancement factor of the collision
kernel due to turbulence effects, i.e., the ratio
K(r1,r2;ϵ,Reλ)/Kgrav(r1,r2), for the
Ayala–Wang and the Onishi model at ϵ=1000 cm2 s-3 for
two different values of Reλ.
The Ayala–Wang model shows a significant increase of the collection kernel
for high Reynolds numbers for droplets smaller than 80 µm radius,
roughly a factor-of-2 increase from Reλ=1000 to
Reλ=20 000 (Fig. a, b, c). For
the Onishi kernel the Reλ dependency is more subtle and
can be characterized as a shift of the maximum of the enhancement from
smaller to larger droplets; i.e., the kernel decreases for small droplets (r<40µm) but increases for larger droplets (r>40µm)
as the Reynolds number increases. For an in-depth discussion of the Reynolds
number dependencies we refer again to .
Parameterization of turbulence effects on autoconversion
The evolution of the drop size distribution f(x) as a function of drop mass
x, where f(x)dx is the number of drops per unit volume in the size
range [x,x+dx], is governed by the kinetic equation also known as
the Smoluchowski coagulation equation
, which in its continuous form,
∂f(x)∂tkoag=12∫0xf(x-x′)f(x′)K(x-x′,x′)dx′-∫0∞f(x)f(x′)K(x,x′)dx′,
was first derived by . A detailed discussion of this
equation and its mathematical properties is given in the classic review by
and more recently by . Another
classic but still-interesting contribution on the interpretation of the
continuous form of the Smoluchowski equation is the paper by
. Although various numerical methods are available to
solve Eq. () directly
e.g.,, this
is most often seen as computationally too expensive in three-dimensional
atmospheric models. Therefore bulk parameterizations are used which predict
only a limited number of (partial) moments of the drop size distribution.
Following and motivated by the emergence of bi-modal
mass distributions as a consequence of the colloidal instability, the size
distribution is decomposed into two parts. Drops smaller than some threshold
x* are called cloud droplets; drops larger than x* are called rain
drops. The value of x*=2.6×10-10 kg, which corresponds to a radius
of 40 µm, is not arbitrary but should be chosen as the local minimum
of the bi-modal mass distribution function g(x)=xf(x) during the colloidal
instability . This minimum exists due to
the properties of the (gravitational) coagulation kernel K(x,y) which
becomes less steep for x>x*. Having defined the two drop
categories, we can identify the following bulk microphysical processes:
autoconversion is the formation of rain drops due to collisions between cloud
droplets, and accretion is the growth of rain drops due to the collection of
cloud droplet by rain drops. The change of the number density within one
category due to coagulation within this drop category is called
self-collection. For a more detailed review of the basic ideas of warm-rain
parameterizations we refer to the review of . The increase
in rainwater content Lr due to autoconversion and accretion is
given by the integrals ∂Lr∂tau=12∫x′=0x*∫x′′=x*-x′x*f(x′)f(x′′)K(x′,x′′)x′dx′′dx′,∂Lr∂tac=∫x′=x*∞∫x′′=0x*f(x′)f(x′′)K(x′,x′′)x′dx′′dx′.
For the parameterization of autoconversion we follow SB2001
hereafter. For a cloud droplet distribution which
initially obeys a gamma distribution in particle mass xf(x)=Axνe-Bx,
SB2001 derived the autoconversion parameterization
∂Lr∂tau=kcc20x*(ν+2)(ν+4)(ν+1)2Lc2x¯c21+Φau(τ)(1-τ)2.
Here Lc is the cloud water content;
x¯c=Lc/Nc is the mean cloud droplet mass, with
Nc being the cloud droplet number density; and x* is again the
separating mass between cloud and rain drops. The dimensionless ratio
τ=Lr/(Lc+Lr) with the rainwater content
Lr acts as an internal timescale and modulates the autoconversion
rate due to the universal function Φau(τ) given by
Φau(τ)=600τ0.68(1-τ0.68)3.
In the case of purely gravitational collection the kernel parameter for
autoconversion is given by kcc=kcc,0=9.44×109 s-1 kg-2 and originates from a piecewise polynomial
approximation of the collection kernel .
Following we extend this autoconversion parameterization
to include turbulence effects by making kcc a function of
ϵ, Reλ, and r¯c. The
third
dependency is necessary, because the turbulence effects are different for
droplets of different size. have shown that the
Ayala–Wang kernel can be approximated with the following ansatz:
kcc(r¯c,ν,ϵ,Reλ)=kcc,0{1+ϵReλpαcc(ν)exp-r¯c-rcc(ν)σcc(ν)2+βcc,
where
Enhancement factor of the autoconversion rate
for the Ayala–Wang kernel (upper row) and the Onishi kernel (lower row) at
Reλ=20 000 (a, c), the Reynolds number dependency
of the enhancement factor at ϵ=600 cm2 s-3(b, d), and
the dependency on dissipation rate for Reλ=20 000 (c, f). Data
points (dots) are based on numerical solutions of the stochastic
collection equation (SCE); the parameterization shown (dashed lines) is
Eq. (10) with the coefficients as given in Table 1. All plots are shown
for ν=1. Note the different scaling of the y axis for both kernels.
αcc(ν)=a1+a2ν1+a3νrcc(ν)=b1+b2ν1+b3νσcc(ν)=c1+c2ν1+c3ν
are functions of the shape parameter ν only. Here we use the same ansatz
for the updated Ayala–Wang kernel and for the Onishi kernel. The 11
coefficients of this model have been determined by a nonlinear least-squares
fit using a database of numerical solutions of the stochastic collection
equation (SCE). The parameter space covered by the SCE simulations is
ϵ∈[0,1000] cm2 s-3, Reλ∈[1000,25 000], Lc∈[0.2,2] g m-3, r¯c∈[8,20]µm, and ν∈[0,4]. Note that in contrast to
we have extended the range for ϵ to values up
to 1000 cm2 s-3 to allow for the higher dissipation rates that
occur, for example, in cumulus congestus. The resulting coefficients for both
turbulence kernels are given in Table 1.
Coefficients as a result of the nonlinear regression for
kcc as given by
Eqs. ()–().
The most notable difference between the two kernels is that for the
Ayala–Wang kernel the autoconversion rate increases with
Reλ, resulting in p=1/4, whereas autoconversion
decreases slowly with increasing Reλ for the Onishi
kernel with a power law exponent p=-1/8.
The different autoconversion enhancement factors for the two kernels and the
quality of the fits are shown by Fig. , in which the Reynolds
number dependency is also shown in more detail. The results for the
Ayala–Wang kernel show somewhat higher enhancement factors compared to
, mostly due to the improved treatment of the collision
efficiency cf.. The Onishi kernel shows much
lower enhancement factors, and the maximum is shifted to larger (mean) droplet
radii compared to the Ayala–Wang kernel. The
Reλ dependency reveals that especially for the Onishi
kernel the value of the exponent, p=-1/8, is not actually constant, but the
slope has significant dependencies on r¯c and
Reλ. This more complicated behavior is consistent with
the analysis presented by , who showed that the
Reynolds number dependency of the kernel varies with Stokes number
(e.g., their Fig. 2). For the Ayala–Wang kernel the numerical data show a
slightly steeper increase with Reλ compared to the
parameterization. This is mostly because we kept the exponent at p=1/4 as in
, although the extended range of the dissipation rate in
the current study would call for a slightly higher exponent. The dependency
on dissipation rate is assumed to be linear in Eq. (), and
this is confirmed for the Onishi kernel, but for the Ayala–Wang kernel the
ϵ dependency becomes slightly weaker for high dissipation rates.
Time t10 that is needed to convert 10 %
of the initial cloud water to rainwater: (a)t10 as a function of
dissipation rate ϵ for various ν (and r¯c=15µm,
Reλ=10 000), (b)t10 as a function of mean cloud droplet radius
r¯c for various values of dissipation rate ϵ (and ν=2,
Reλ=10 000), and (c)t10 as a function of the initial cloud liquid
water content for various values of dissipation rate ϵ (and
r¯c=14, ν=2, Reλ=10 000). Data points are numerical solution of the
SCE; dashed lines represent the solutions of the two-moment bulk scheme
with the enhancement factor for autoconversion based on the Onishi kernel
as given by Eq. (10) and the
coefficients of Table 1.
A first test of the autoconversion parameterization is obtained by
simulations of exactly the same kind as used as training data, i.e., SCE
simulations with an initial condition following a gamma distribution. As a
metric for evaluation we use the timescale t10, which is defined as the
time needed to convert 10 % of the initial liquid water to rainwater.
Figure shows the dependencies of t10 on dissipation
rate ϵ, initial mean drop radius r¯c, and initial
cloud water content Lc. This confirms that the fit is reasonable
and that the autoconversion parameterization captures those dependencies
correctly.
Turbulence effects in a 1-D kinematic model
As in we use the 1-D kinematic model of
as a slightly more complete test problem for the
warm-rain scheme. The 1-D kinematic model is especially useful as it
describes the various stages of the warm-rain formation in an isolated
cumulus cloud. This is necessary to test and validate our assumptions
regarding accretion and self-collection of raindrops. Those two processes
depend strongly on drop sedimentation and the resulting drop size
distribution and can therefore hardly be tested in pure SCE simulations.
Although the 1-D model provides a reasonable idealized framework for such a
test, we would recommend using a kinematic 2-D model
e.g., in future studies,
because the 1-D framework might not be sensitive enough to differences in the
treatment of sedimentation which are more relevant in a more complex flow
field. Here we apply the simpler 1-D model for consistency with
.
The accretion rate and self-collection of rain are parameterized as
∂Lr∂tac=kcrLcLrΦac(τ)ηac,withΦac=ττ+5×10-44,
and
∂Nr∂tsc=-krrNrLrηsc,
with kcr=5.78 m3 kg-1 s-1,
krr=4.33 m3 kg-1 s-1, and turbulent enhancement
factors ηac and ηsc. In the case of the Ayala–Wang
kernel we use the same enhancement factors as in :
ηac=ηsc=1+c^rϵ14,
with c^r=0.05 cm-1/2 s3/4. For the Onishi kernel
we apply a stronger enhancement which is linear in the dissipation
rate ϵ:
Accumulated surface precipitation of the 1-D
kinematic model as a function of the assumed in-cloud turbulent
dissipation rate ϵ (other parameters are temperature gradient
Γ0=1.5 K km-1, the maximum updraft speed w0=2 m s-1, and the
updraft timescale τw=40 min). Shown are results from the Ayala–Wang
model at Reλ=1000(a) and Reλ=20 000 (b), as well as the Onishi model
at those two Reynolds numbers (c, d). Results of the spectral bin reference
model are depicted with solid lines, and the results of the two-moment
parameterization with dashed lines.
ηac=ηsc=1+cˇrϵx*x¯r23,
with cˇr=0.8×10-3 cm-2 s3. For a
dissipation rate of 1000 cm2 s-3 this corresponds to an increase in
accretion of 28 % in the case of the Ayala–Wang kernel and 80 % for the
Onishi kernel. For the Onishi kernel we have included an additional
dependency on x¯r=Lr/Nr to suppress the turbulent
enhancement for very large (mean) raindrop sizes that do not feel the effect
of small-scale turbulence. The enhancement factors for accretion and
self-collection cannot be directly derived from the collection kernel alone.
The turbulent enhancement of the collision rate leads also to changes in the
drop size distribution; i.e., the increase in accretion and self-collection is
attributed, first, to the direct increase in the collision rates by the local
turbulence and, second, to a modification of the drop size distribution by
the turbulence effect. The latter constitutes a memory effect and makes it
also difficult to discuss the turbulence effects on accretion and
self-collection separately, because these two processes are strongly linked.
In the following we always mean the combined action of self-collection of rain
and accretion when we discuss effects of turbulence on the droplet growth by
accretion.
Extensive tests with the 1-D kinematic model have shown that the
parameterization compares reasonably well with the bin microphysics solution
for both collection kernels. The most important metric to evaluate the
warm-rain scheme in the 1-D kinematic model is the precipitation amount at the
surface. One could argue that the timing is almost as relevant as the
precipitation amount, but as shown by the
precipitation efficiency in the 1-D cloud model depends mostly on the timescales
of dynamics and microphysics, or rather their ratio, the
Damköhler number. Therefore we discuss here only the precipitation amounts
which are presented in Fig. as a function of dissipation
rate (which is assumed as homogeneous within the cloud) for two different
Reynolds numbers and various aerosol number concentrations Na. For
further details, e.g., on the treatment of activation we refer to
. For the Ayala–Wang kernel we find a significant
increase in surface precipitation; for example, we find an increase by a
factor of 2 for low Na (clean conditions) when ϵ is as
large as 1000 cm2 s-3 compared to pure gravitational kernel
(ϵ=0). For high Na the cloud does not produce any rain
without the effect of turbulence on the collision rate (ϵ=0) but
yields significant rain when turbulence can contribute to rain formation. For
the Onishi kernel we find qualitatively the same behavior, but the rain
amounts are significantly lower especially for low dissipation rates
ϵ. The different Reynolds number dependencies of both kernels are
also visible in these surface rain amounts. For the Ayala–Wang kernel the
rain amounts increase significantly for higher Reynolds numbers. In the case of
the Onishi kernel a slight decrease is observed for high Na when
increasing Reλ from 1000 to 20 000. For
Na=50 cm-3 a slight increase with Reλ is
visible for the spectral model but not for the two-moment scheme. This can
be attributed to the increase in the accretion rate in the Onishi kernel for
high Reλ, and this effect we have neglected in the bulk
scheme (mostly because the Re dependency is quite weak and in addition the
low Reλ case is not important for cloud physics
applications). Nevertheless, the 1-D kinematic model suggests that the
turbulence effect on accretion is significant, and even more so in the case of
the Onishi kernel. Especially for low Na, when autoconversion is
quite efficient, accretion can become the limiting process for droplet
growth, and an increase in accretion due to turbulence effects can significantly
affect surface rain amount. This will be further investigated using
large-eddy simulations in the following section.
Turbulence effects in large-eddy simulations of trade wind cumuliModel setup
To investigate the effect of in-cloud turbulence on rain formation in trade
wind cumulus clouds, we perform large-eddy simulations of the Rain In Cumulus
over the Ocean (RICO) case as described by . We use the
standard RICO case and not the moister initial condition as in
. We apply the UCLA-LES model
on a domain of 51.2 km × 51.2 km
with doubly-periodic boundary conditions, a simulation time of at least
30 h, and a horizontal mesh size of 50 m with additional simulations at finer and
coarser grid spacing. The model time step is variable with a maximum Courant
number below 0.5. The time step is mostly dominated by the vertical grid
spacing and velocity and approximately 1 s. The cloud microphysical
parameterization follows SB2001 and with the
modifications described in the previous sections. For the shape parameter of
the cloud droplet size distribution we use ν=1 in all simulations. The
sub-grid-scale (SGS) turbulence model is a Smagorinsky–Lilly closure
including a proper treatment of anisotropic grids
. As described in detail in
, the SGS model provides the local (grid point) turbulent
dissipation rate ϵ which is needed for the turbulence effect on
cloud microphysics. Additional assumptions are necessary for the Reynolds
number Reλ as the SGS model provides neither
Reλ nor urms. Here we follow
and parameterize Reλ as a
function of ϵ alone. Consistent with homogeneous isotropic
turbulence, we use the scaling relation Reλ=Re0(ϵ/ϵ0)1/6, with Re0=10 000 and
ϵ0=100 cm2 s-3.
Turbulence effect on rain formation
Figure shows time series from a first set of
simulations with grid spacing Δx=50 m. After some initial spin-up
the cloud liquid water path (CWP) increases slowly with time, corresponding to a
slowly deepening cloud layer. Rainwater develops after a few hours, and
surface precipitation is observed subsequently. The rainwater path (RWP), surface
rain rate, and timing of the rain formation differ strongly between the
various simulations. The control simulation which uses the purely
gravitational kernel develops only marginal rain and surface precipitation
within the 30 h period. In contrast, the simulation which applies the
Ayala–Wang kernel develops rain much earlier, and the rain rate reaches
1 mm day-1 after about 20 h with some fluctuations later on.
Using the Onishi kernel leads to faster rain formation compared to the
control simulations, but slower than for the Ayala–Wang kernel. At the end of
the simulation period the Onishi kernel yields similar rain rates to the
Ayala–Wang kernel; i.e., in the last hours both turbulence kernels increase
the surface rain rate by a factor of 7 relative to the control run. Especially
for the Onishi kernel the enhancement of the rain formation is due to the
combined action of the increased autoconversion and accretion. This is
illustrated by an additional simulation which uses only the enhancement for
autoconversion, ignoring the effect on accretion. The resulting time
series are much closer to the control run and show only a significant
increase in rain rate at the very end of the simulation period. This
underpins our results of the previous section that the rain formation in
shallow cumulus clouds is limited not only by autoconversion but also by
accretion. Although accretion increases more strongly in the Onishi kernel
than in the Ayala–Wang kernel, the LES results show that this cannot
compensate for the weaker increase in autoconversion resulting in a reduced
turbulence effect on rain formation. The main feedback of the different
microphysical developments on the dynamics and the evolution of the boundary
layer as a whole is that rain formation arrests the growth of the cloud layer
as can be seen in the time series of the inversion height in
Fig. ; i.e., the Ayala–Wang kernel leads to a much
shallower cloud layer in the precipitating regime. A similar behavior for
different cloud droplet number densities was shown by
and . For the RICO case the
boundary layer deepens and supports successively deeper clouds until moisture
is efficiently removed by precipitation. Eventually the precipitating regime
reaches a quasi-stationary state, the subsiding radiative-convective
equilibrium . This is also consistent with the finding
that the enhancement of the warm-rain process by taking into account
turbulence effects on collisions has a very similar effect on cloud patterns,
cloud fields, vertical profiles, etc. to a change in the cloud droplet number.
Time series of the cloud liquid water
path, rainwater path, the surface rain rate, and the inversion height for
four simulations using the three different collection kernels. The
simulation marked “au only” applies the turbulent enhancement only to
autoconversion, ignoring the effect on accretion. We have applied a
running average to all time series with an averaging window of 120 min for
the surface rain rate and 30 min for RWP, CWP, and inversion height.
Sensitivity of LES results to variations in the
cloud droplet number density. Shown are the rainwater path, surface rain
rate, inversion height, and accretion / autoconversion ratio for the
three different collection kernels of the control simulations
using the purely gravitational kernel (bullets, grey shading), the
Ayala–Wang kernel (squares, blue shading), and the Onishi kernel
(diamonds, red shading). The shaded area indicates the standard error at a
95 % confidence level.
The strong turbulence effect of both kernels suggested by
Fig. is consistent with and
, but two important aspects have to be considered.
First, this behavior is transient; i.e., even the purely gravitational case
would develop significant rain of order 1 mm day-1 after some time.
Extending the simulation further shows that this happens after about 35 h.
Second, Fig. shows only simulations for a specific
intermediate value of the cloud droplet number density. A lower value will
make rain formation easier and more efficient also for the gravitational
kernel and lead to smaller differences; a higher droplet number may suppress
precipitation even for the collection kernels that include turbulence
effects. To get a more complete picture, we have to discuss both effects.
Statistics for the large-eddy simulations assuming different
collection kernel. Nx is the number of grid points in the horizontal, and
Δx and Δz are the horizontal and vertical grid spacing.
Listed variables are the timescales t1 and t2, which characterize the
transition to precipitating shallow convection (0.1 mm day-1 as rain
rate threshold for t1, 0.8 mm day-1 for t2), the area-averaged
cloud cover C, the inversion height zi, cloud liquid water path (CWP) in
g m-2, rainwater path
(RWP) in g m-2, surface rain rate R in W m-2 (29 W m-2
corresponds to mm day-1). The ratio of accretion over autoconversion,
AC / AU, and the rain efficiency,
RE=1-EV/(AU+AC) (both evaluated over the whole
column). Time averages are from 24 to 30 h. The simulations shown in
Fig. are indicated by bold font. Simulations with identical model configuration (kernel,
Nx, Δx, Δz, Nc) differ only by the random seed
of the initial condition.
nKernelNxΔxΔzNct1t2CziCWPRWPRAC / AURE1No turb.1024502535.07.521.616.7223812.617.343.05.2746.42No turb.1024502535.011.718.716.4224512.815.937.84.9842.73No turb.1024502550.019.631.916.0237015.03.75.23.4024.14No turb.1024502550.018.932.415.2237514.94.15.93.4223.25No turb.1024502550.021.334.915.6237514.83.85.83.3824.26No turb.1024502570.034.445.715.2238815.41.11.42.9818.77No turb.1024502570.028.543.715.4238515.31.32.23.4425.58No turb.1024502570.029.237.615.5238515.61.42.13.4923.39No turb.10245025105.046.050.515.2239215.60.20.32.8520.210Onishi1024502535.08.420.213.9221310.718.041.85.1143.511Onishi1024502535.06.217.714.4218010.415.843.95.8351.712Onishi1024502550.016.829.016.7235115.09.417.04.4532.413Onishi1024502550.013.025.718.1231714.612.529.35.6437.514Onishi1024502550.013.627.216.9233715.413.430.15.3440.815Onishi1024502550.012.825.119.4230815.914.033.15.8544.116Onishi1024502550.014.224.917.8229515.016.342.56.3146.816*Onishi, au only1024502550.016.828.816.3236215.08.715.23.8931.617Onishi1024502570.019.236.415.2237014.83.04.64.0024.918Onishi1024502570.021.738.215.3237715.12.84.43.8624.419Onishi1024502570.021.436.715.5237715.22.94.43.8825.420Onishi1024502570.024.033.515.8237815.43.25.04.0125.521Onishi10245025105.030.743.115.3239215.50.91.64.3727.722Ayala–Wang1024502535.04.813.610.520166.411.534.25.4753.523Ayala–Wang1024502535.04.413.712.719017.614.146.26.6862.624Ayala–Wang1024502550.05.617.813.621239.914.341.26.1851.625Ayala–Wang1024502550.06.415.814.120919.715.248.37.8261.326Ayala–Wang1024502550.06.118.015.0214310.515.847.56.5555.627Ayala–Wang1024502550.07.218.214.0215110.415.341.85.8248.328Ayala–Wang1024502570.013.726.216.4230914.012.730.25.5441.829Ayala–Wang1024502570.09.722.017.8226513.515.342.76.6349.630Ayala–Wang1024502570.010.621.417.5224413.214.642.06.6550.331Ayala–Wang10245025105.019.335.215.9236415.14.79.54.9533.9Sensitivity to cloud droplet number
We have performed a larger set of large-eddy simulations for different cloud
droplet number densities. In addition, simulations have been repeated with
different random seeds to sample the stochastic uncertainty of the system and
to reduce the standard error in the statistical evaluation.
Table summarizes the results in terms of domain-mean
statistical quantities like cloud cover, inversion height, and rainwater path.
As a measure for the temporal, i.e., transient, behavior we have
calculated two timescales that characterize the rain formation by the
exceedance of thresholds for the domain-averaged rain rate, t1 for a
threshold of 0.1 mm day-1 and t2 for 0.8 mm day-1. While t1
measures the first occurrence of rain at the surface, the larger threshold
value of t2 characterizes the transition to organized precipitation
shallow convection . The most important results are
summarized in Fig. , which illustrates the turbulence effects
on the rain formation for different values of the cloud droplet number
density. Shown are domain-mean quantities from 24 to 30 h of the
simulations, and standard error is depicted by shaded areas. The standard error is
estimated as σx/nx, where σx is the standard deviation of
that variable and nx is its effective sample size. For each simulation we
estimate the effective sample size during the sampling period of 6 h as nx=n0(1-r1)/(1+r1), where r1 is the lag-1 autocorrelation and n0 is
the number of samples in the time series. This simple formulation gives
almost the same results as a more sophisticated implementation following
. As shown in Fig. , rainwater
path and surface rain rate increase with decreasing cloud droplet number;
a pronounced impact of turbulence-induced collisions is also shown. For
Nc=50 cm-3, i.e., the simulations which are also shown in
Fig. , both the Ayala–Wang kernel and the Onishi
kernel lead to a strong increase in RWP and rain rate. For the lower value of
Nc=35 cm-3 the purely gravitational kernel used in the
control simulations is sufficient to produce similar values of RWP and rain
rate, and the differences between the three kernels are no longer
statistically significant. For an increase in droplet number the rain
formation gets suppressed. Already for Nc=70 cm-3 the rain
rate and RWP for the Onishi kernel drops to values which are hardly different
from the purely gravitational case, while the Ayala–Wang kernel still shows a
strong enhancement leading to rain rates of order 1 mm day-1 during the
30 h period. Finally, for Nc=105 cm-3 the rain formation
starts to get suppressed even for the Ayala–Wang kernel, and for droplet
number exceeding that value all three collection kernels would only yield
marginal precipitation within the 30 h period.
Transition timescales t1 (dashed, grey
symbols) and t2 (solid, black symbols), defined as the
time when the domain-averaged rain rate exceeds 0.1 and 0.8 mm day-1,
respectively, for the first time. The transition times are averaged over
multiple simulations with different random seeds.
For low cloud droplet numbers we do not find a significant difference for the
rainwater path and the surface rain rate between the three different kernel
during the 24 to 30 h sampling period, because all three simulations develop
a rain rate that is close to the quasi-equilibrium rainwater flux.
Nevertheless, the transient behavior is different between the three kernels
for all droplet number densities as, e.g., seen from the timescales t1
and t2 in Fig. . The Ayala–Wang kernel leads to an
acceleration of the rain formation by more than 10 h for high drop number
and still several hours for low droplet numbers. The acceleration caused by
the Onishi kernel is less strong and becomes smaller for t2 for low drop
numbers, while the difference in t1 to the control run remains also for low
drop numbers. This difference in the transient behavior leaves an imprint in
the structure of the boundary layer even for long simulation times in the
sense that the Ayala–Wang kernel, which develops rain most easily, arrests the
growth of the boundary layer much earlier, leading to the lowest inversion
height in the precipitating regime (Fig. c). For the Onishi
kernel this cloud macroscopic effect of the microphysical processes is much
weaker. That the cloud droplet number and the microphysical efficiency of the
cumulus clouds modulate the inversion height is consistent with the results
of and .
As Fig. but showing the
dependency of the results in the sampling period 24 to 30 h on grid
spacing for a cloud droplet number density of Nc=50 cm-3.
The turbulence effects on the collision rate, as postulated by the two
different turbulence models, lead to a strong increase of the autoconversion
rate and a moderate increase of accretion. This is true for both kernels,
although the Onishi model has a weaker enhancement of autoconversion and a
stronger increase in accretion, especially at high Reynolds numbers. It is
therefore interesting to check whether a significant shift in the importance
of those two warm-rain processes can be observed in the large-eddy
simulations. Figure d shows the ratio of accretion over
autoconversion, AC / AU, for the sampling period of 24 to 30 h. For all
simulations accretion is the dominant process, and total accretion exceeds
autoconversion by a factor of 3 or more. Interestingly, the simulations which
take into account turbulence effects show a higher AC / AU ratio compared
to the control simulations, which is counterintuitive as the enhancement
mostly affects autoconversion. This behavior can be understood from the
relation between autoconversion and accretion. A higher autoconversion rate
will most likely lead to a subsequent increase in accretion, because more
small rain drops become available for accretional growth. Therefore an
increase in the autoconversion rate, as caused by the turbulence effects, has
little effect on the AC / AU ratio. In fact, the higher rain rate regimes
of the simulations with the turbulence kernels favor accretion over
autoconversion. Therefore the observed AC / AU ratio is not directly
linked to the turbulent enhancement factors of the process rates.
Sensitivity to grid resolution
Previous studies, e.g., by and
, have emphasized that especially the precipitating RICO case exhibits a strong
sensitivity to the grid spacing used in large-eddy simulations. We have
therefore performed another set of simulations to test the sensitivity to
grid spacing using 100, 50, and 25 m horizontal mesh size for the three
different collection kernels. The vertical grid spacing for all simulations
is fixed at 25 m. Figure summarizes the main results of
the resolution study. The detailed statistics of the individual simulations
are given in Table . For cloud liquid water path
hardly any sensitivity to grid spacing is found, but the simulations with the
Ayala–Wang kernel lead in general to a reduced CWP. This can be explained by
the more rapid conversion of cloud water to rain and by the shallower cloud
layer in the precipitating regime. For rainwater path and surface rain rate
we find a strong increase with increasing resolution for the Onishi kernel
and the control simulations. At 25 m grid spacing all three models give
similar RWP and surface rain rate, and differences are not statistically
significant for those two variables. This is a similar behavior to that for the
reduced cloud droplet number. A small grid spacing in the LES makes the rain
formation more rapid, and the differences between the kernels becomes smaller
when they all reach the precipitating regime before the chosen sampling
period. This is confirmed by Fig. , which shows that the
timescale t2 decreases with resolution and at 25 m grid spacing all
three kernels have a t2 smaller than 20 h; i.e., the sampling period of
24 to 30 h is in the precipitating regime for all three collision kernels.
Figures and reveal that the LES is not
yet converged even at 25 m grid spacing. Unfortunately, higher resolution
than the 25 m grid becomes very expensive and cannot be tested here.
Differences in inversion height remain present even at the highest
resolution; especially the Ayala–Wang kernel leads to much shallower cloud
layers. A hint towards the causes of the strong resolution dependency
may be given by the AC / AU ratio, which increases strongly for higher
resolution. Especially the control run exhibits a significant increase from
below 4 at 50 m grid spacing to almost 8 at 25 m. The rain efficiency –
defined as the ratio of evaporation of rain over the sum of autoconversion
and accretion, 1-EV/(AU+AC) – shows a behavior very
similar to the AC / AU ratio and suggests that the growth by accretion
leads to large raindrops which are less susceptible to evaporation and thus more
rain reaching the ground. The strong sensitivity of the rain formation to
grid spacing may be surprising at first as individual precipitating cumulus
clouds have horizontal scales of at least 1000 m and should be well resolved
by the LES already at 50 m grid spacing. We suggest two possible mechanisms
to explain the observed sensitivity. First, due to the strong nonlinearity of
the autoconversion rate, small-scale fluctuations in cloud water may trigger
autoconversion earlier and more often and initiate the rain formation more
effectively at high resolution. Second, the in-cloud circulations which are
better resolved at higher resolution increase the in-cloud residence time of
the rain drops and therefore their overall growth by self-collection and
accretion. The latter effect has recently been emphasized as an important
growth mechanism for raindrops in shallow cumulus clouds
. Although it remains questionable whether a
two-moment bulk scheme can represent recirculation properly, the strong
increase of accretion observed in Fig. d would favor the
second explanation. Whatever the detailed mechanism is, the strong
sensitivity to grid spacing suggests that the larger modes of turbulence –
like turbulent entraining eddies, which are resolved by high-resolution LES
– play an important role in enhancing the rain formation. This provides a
second mechanism in addition to the effect of the small-scale isotropic
turbulence on collision rates which is parameterized by the Ayala–Wang or
Onishi kernel and sub-grid for any LES model.
As previous table but for the
simulations to investigate the resolution dependency at Nc=50 cm-3.
We have derived a warm-rain bulk two-moment scheme which incorporates the
effects of small-scale isotropic turbulence on the collision rate following
the two alternative models of Ayala–Wang and Onishi. The two collision
kernels differ mostly in their Reynolds number dependency. While the
Ayala–Wang model postulates an increase of autoconversion with Reynolds
number, the Onishi model predicts a decrease of autoconversion but an
increase in accretion for high Reynolds number. The two newly derived
variants of the Seifert–Beheng warm-rain scheme have been tested and
validated in 1-D simulations and compare favorably with the bin microphysics
model that acts as a reference.
The new bulk scheme has been applied in large-eddy simulations of
precipitating shallow convection to investigate the impact of the different
collision kernels. Both turbulence kernels lead to a significant enhancement
of the rain formation in shallow convective clouds, but the turbulence effect
is much weaker for the Onishi kernel. Especially for intermediate cloud
droplet numbers – in our simulations 50 cm-3, but this might differ from
case to case – the turbulence enhancement can lead to a strong increase in
rainwater path and surface rain rate compared to a purely gravitational
collection kernel. For the Ayala–Wang kernel we find a significant reduction
of the height of the trade wind inversion, because the rapid rain formation
arrests the growth of the cloud layer. This effect is not significant for the
Onishi kernel. Overall, we found that the enhancement of the warm-rain
process by taking into account turbulence effects on collisions has a very
similar effect on the evolution of the cloud field, the cloud patterns, and
the vertical profiles, etc. to a corresponding change in the cloud droplet number.
As Fig. but
showing the dependency of the rain formation timescales t1 and t2
on horizontal grid spacing for a cloud droplet number density of 50 cm-3.
The large-eddy simulations show a strong sensitivity to horizontal grid
spacing with a more rapid rain formation at higher resolution. This suggests
that the larger turbulent eddies like in-cloud circulations, which are
resolved by high-resolution LES, can play an important role for the growth of
rain drops. It is hypothesized that rain drops with large Stokes numbers,
St> 1, can interact with these large turbulent eddies. For
example, in the two-moment bulk scheme used in the present study such effects
are not yet accurately parameterized, and they need to be investigated in more
detail in future studies.
Our results show that the differences between the Ayala–Wang model and the
Onishi models are significant, and it needs to be clarified either by
observations or by additional DNS studies which collision kernel is more
realistic at high Reynolds numbers.
Data availability
The UCLA-LES model is distributed under GNU General Public
License and can easily be downloaded from https://github.com/uclales.
Model code and input files necessary to reproduce the specific experiments
of this study are available from the corresponding author upon
request.
Acknowledgements
We thank the computing center of ECWMF, where all simulations were performed
using resources provided through DWD. We thank Ann Kristin Naumann for
helpful comments on the manuscript. We thank the editor Graham Feingold and
three anonymous reviewers for their comments that helped to improve the
manuscript. Edited by: G. Feingold Reviewed by: three
anonymous referees
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