Introduction
Traditional cloud modeling methodologies apply a continuous medium approach for
all thermodynamic variables, that is, not only for the temperature and water
vapor, but also for all forms of cloud condensate and precipitation. Such
methodologies have been the workhorse of the cloud-scale modeling from its
early days
e.g.,,
but also in numerical weather prediction using global as well as limited-area
models and in climate simulation. Since the edge of an ice-free cloud
represents a sharp transition from droplet-laden air close to saturation to
unsaturated droplet-free air outside the cloud, numerical diffusion and
dispersion errors impose stringent constraints on numerical schemes suitable
for cloud modeling. For instance, since cloud and precipitation variables are
positive definite, any numerical scheme that introduces negative values to
the numerical solution (e.g., during advection in the physical space) is not
suitable for cloud simulation. Moreover, difficulties in representing sharp
cloud edge discontinuities in thermodynamic fields are well appreciated by
the cloud-scale modeling community, especially from the point of view of the
supersaturation field, the key variable for the formation and growth of water
and ice cloud particles e.g.,and references
therein.
The last couple of decades witnessed an increased interest in cloud-scale
computational approaches that limit the abovementioned problems and attempt
to better represent the truly multiphase nature of clouds. Among those, the
particle-based Lagrangian method, referred to as the Lagrangian cloud model
or the “super-droplet
method” , is of particular relevance see
alsoamong
others.
By representing formation and growth of natural cloud particles using a
subset of those particles (“super-particles”), many problems haunting
traditional Eulerian approaches are either eliminated or significantly
reduced. For instance, formation of cloud droplets through activation of
cloud condensation nuclei (CCN) can be formulated in a straightforward way,
and processing of CCN through collision–coalescence or chemical reactions
within droplets can be simulated from first principles. In the continuous
medium approach, however, these processes require either
extreme computational effort (i.e., multidimensional bin schemes) or are
simply impossible to consider without additional simplifications. In the
Lagrangian approach for warm (ice-free) clouds, each super-droplet (SD
hereafter) carries a set of attributes, such as the CCN size and composition,
wet particle mass and multiplicity parameter (the latter being the number of
real droplets each super-droplet represents), that allow the representation
of condensation and associated latent heat release as well as the development
of drizzle and rain. In previous applications of such a methodology, the
super-droplets outside clouds represent unactivated CCN (haze) particles that
become activated upon entering a cloud and can further grow through
diffusional and collisional processes. Since the information about the CCN is
available for each super-droplet, the methodology allows for detailed study
of not only effects of CCN on cloud microphysics and dynamics, but also CCN
processing by a cloud. However, when cloud processing is of no interest, the
Twomey activation can be used with super-droplets
forming when CCN is activated and no super-droplet existing outside a cloud
as often applied in Eulerian bin microphysics models
e.g.,. Since cloud volume is a small fraction
of the computational domain volume, the Twomey super-droplets allow
significant savings when compared to CCN-based Lagrangian methodology.
Moreover, significantly longer time steps can be used because modeling of CCN
deliquescence is avoided.
This paper discusses the development and testing of a novel Lagrangian
approach focusing on activation and diffusional growth of cloud
droplets. Our motivation is to use this methodology to study the
impact of turbulence and entrainment on the spectrum of cloud
droplets in shallow warm boundary layer clouds, such as tropical
or subtropical cumulus and subtropical stratocumulus see
idealized adiabatic parcel simulations discussed inhereafter
GA17. The key aspect, difficult if not
impossible to apply in the Eulerian approach, is the possibility
to formulate a subgrid-scale statistical scheme and apply it to
individual droplets taking advantage of a stochastic formulation
along the Lagrangian particle trajectory as in GA17. The developments
discussed here exclude collision–coalescence as only marginally
relevant to the spectral broadening problem. Collision–coalescence
can be included in a relatively straightforward way see a
review and tests of various approaches discussed
in and adding it to the model
described here will be pursued in the future.
The next section presents analytic formulation of the Twomey super-droplet
scheme and discusses its implementation in the Eulerian fluid flow model. The
specific aspects discussed in detail are the treatment of the activation on
the finite-difference fluid flow model grid, transport of super-droplets
across the Eulerian grid, and coupling between the super-droplets and
Eulerian thermodynamics. Section 3 presents examples of model simulations
where the Lagrangian thermodynamics is included in an anelastic small-scale
fluid flow model and applied in moist rising thermal simulations. A
traditional super-droplet scheme (i.e., following CCN particles and allowing
their activation and growth of resulting cloud droplets) is used to show
consistency between the two methods. Brief conclusions and the outlook are
presented in Sect. 4.
Formulation
Analytic formulation
Model equations describe evolution in space and time of the potential
temperature, water vapor mixing ratio, and a set of Lagrangian point
particles representing activated cloud droplets. The potential temperature
Θ and water vapor mixing ratio qv equations are as
follows:
DΘDt=LvcpΠCd,DqvDt=-Cd,
where D/Dt=∂/∂t+(u⋅∇) is the material (advective)
derivative, Cd is the condensation rate, Π=(p/p0)R/cp is the Exner function (p is the local pressure that in the
anelastic system comes from the environmental profiles and p0=1000 hPa), and Lv and cp are the latent heat of
vaporization and air specific heat at constant pressure, respectively. The
condensation rate Cd is defined as the rate of change of the mass
of cloud droplets. For the finite-difference model considered here, it can be
calculated from the rate of change of mass of all cloud droplets located
within a given grid cell:
Cd=ddt∑i=1N4ρw3ρaπri3Ni,
where ρw and ρa are the water and air density,
respectively, and ri and Ni are the radius and concentration of N cloud
droplet classes (bins) into which all droplets located within the grid cell
are grouped. Such a definition has some similarity to the way condensation
rate is calculated in Eulerian bin microphysics schemes, an analogy that will
be useful when droplet activation is discussed later in this section. Given
the supersaturation S=qv/qvs-1 (where qvs is
the saturated water vapor mixing ratio) the individual droplet growth
equation is as follows:
dridt=ASri+r0,whereA=qvsρaDvρw1+Lvcp∂qvs∂T,
r0=1.86 µm is a parameter that allows including kinetic
effects e.g.,, and Dv is
the diffusivity of water vapor in the air that depends on the temperature and
pressure. A convenient feature of Eq. (4) is that the rate of growth remains
bounded when ri approaches zero. The coefficient A used in Eq. (4) is an
approximate form of a more general formulation as given, for instance, by
Eq. (3) in . The approximate formulation (Eq. 4)
can be obtained by assuming that the thermal conductivity of air K is
approximately given by K=cpρaDv, that is,
replacing thermal diffusivity with the diffusivity of water vapor (this is
accurate to about 10–15 %). Note that and
GA17 applied a constant value A=0.9152×10-10 m2 s-1.
Droplets are carried by the airflow (i.e., droplet sedimentation is
excluded), an assumption justifiable by the exclusion of droplet collisions,
the spatial scales considered (tens of meters and larger), and the length of
simulations (up to a few tens of minutes). Thus, the evolution of the ith
droplet position xi is calculated as
dxidt=u(xi,t),
where u is the air flow velocity predicted by the dynamical model.
Considering typical cloud droplet concentrations in natural clouds, from
several tens to a few thousands per cubic centimeter, it is computationally
impossible to follow all cloud droplets in the entire volume of even a very
small cloud. Thus, the Lagrangian methodology involves following only a
selected (typically relatively small) subset of cloud droplets, referred to
as super-droplets following . This is again in the
spirit of using a finite (and typically relatively small) number of classes
(bins) in the Eulerian bin microphysics scheme. As in
, , , and
, among others, the list of attributes for each
super-droplet includes the position xi, radius ri, and
multiplicity. The latter depicts the number of particles represented by a
single super-droplet. Other attributes can be added if needed, for instance,
the local supersaturation perturbation (on top of the grid-scale
supersaturation predicted by the flow model) that can affect super-droplet
growth in Eq. (4) as in GA17 or the subgrid-scale velocity perturbation that
can affect the motion of the super-droplet in Eq. (5).
Numerical implementation
As will be discussed in Sect. 3, the novel super-droplet scheme has been
included in the finite-difference anelastic model EULAG and its simplified
version referred to as babyEULAG. EULAG and babyEULAG apply
nonoscillatory-forward-in-time (NFT) integration scheme
e.g.,.
For the coupling with super-droplets, the NFT scheme for the potential
temperature (Eq. 1) and water vapor mixing ratio (Eq. 2) has been modified to
include the Euler-forward time integration, that is,
Ψ(t+Δt)=Ψ(t)+F(t)Δt0,
where Ψ is either Θ or qv, F represents the
right-hand-side of Eqs. (1) and (2), and subscript “0” depicts the
departure point of the fluid trajectory. This is the same as applied in the
bin microphysics versions of EULAG in Sect. 2.2
therein and babyEULAG in appendix
therein. Exploring the analogy between Lagrangian
(trajectory-wise) and Eulerian (control-volume-wise) description of the fluid
flow equations, Eq. (6) is solved using the flux-form monotone advection
scheme MPDATA e.g.,. Thus, the
second-order-in-space and centered-in-time advection scheme is combined with
the first-order-in-time (Euler forward) integration of the forcing term.
A similar approach is used for the super-droplets, where the super-droplet
transport is computed using the predictor–corrector scheme and droplet growth
is calculated using the first-order-in-time uncentered scheme. It should be
stressed the momentum equation in the host babyEULAG and EULAG models is
advanced applying the centered in time scheme.
Super-droplet initiation
The key element of the scheme presented here that makes it distinct from the
approach used in ,
, ,
, and others, is the way super-droplets are
created. The original implementations assume that super-droplets fill the
entire computational domain, and they initially represent deliquesced
(humidified) CCN in equilibrium with their local environment. These
unactivated super-droplets may become activated if environmental conditions
dictate so, for instance, when passing through the cloud base. When CCN dry
radius is one of the super-droplet attributes, the original approach allows
explicit representation of aerosol processing by a cloud when
collision–coalescence takes place in which case the dry CCN after
collision–coalescence combines dry CCN from colliding
droplets, or when chemical reactions are included (e.g.,
Anna Jaruga; PhD dissertation, University of Warsaw). However, if neither of
those processes is of interest, a significantly simpler approach can be used
based on the so-called Twomey activation as often
used in bin microphysics schemes e.g.,. The
Twomey approach links the number mixing ratio of activated CCN N to the
maximum supersaturation S experienced by the cloudy volume. We will refer
to the analytical or tabulated correspondence between N and S as the
N–S relationship. Cloud base
activation in the Eulerian bin microphysics scheme is simulated by
introducing cloud droplets into appropriate bins until the supersaturation
reaches its peak and activation is completed
see. Without collision–coalescence, the
local droplet number mixing ratio provides information about the maximum
supersaturation experienced by the volume in the past. With
collision–coalescence, an additional model variable, the number mixing ratio
of already activated CCN, needs to be used to control whether additional CCN
activation is required see Sect. 2c in. The
additional variable is also needed if a significant variability of the CCN
exists in the computational domain (e.g., in the vertical direction) or if
droplet sedimentation is included in the model physics.
The same approach can be used with super-droplets as already applied in GA17
in adiabatic parcel simulations. The key idea is that super-droplets are
created in supersaturated conditions when the local concentration of
activated droplets as given by the Twomey relationship is smaller than the
one dictated by the local supersaturation. When a complete evaporation of
cloud droplets occurs in subsaturated conditions, super-droplets are simply
removed from the super-droplet list. Hence, no super-droplets exist outside
of cloudy volumes, similarly to traditional Eulerian bin microphysics
schemes. It follows that super-droplets with Twomey activation provide
significant computational advantage over the traditional Lagrangian approach
because only a relatively small number of super-droplets has to be used. Note
that in the Eulerian bin scheme the computational expense of the droplet
transport in the physical space is independent of whether droplets fill a
small or a large fraction of the domain. This is because each bin needs to be
advected separately in the physical space and the computational effort is
independent of whether the entire domain or just its small fraction is filled
with droplets. It is worth pointing out that applying Twomey activation to
create cloud droplets in the Lagrangian warm-rain thermodynamics bears
similarities to the way ice particles are initiated in a particle-based
Lagrangian model targeting ice processes e.g.,.
We assume the same CCN characteristics as in GA17 and
. CCN characteristics include the chemical
composition, the number mixing ratio of activated CCN for a given
supersaturation (the Twomey relationship) and the activation radius. CCN are
assumed to be composed of sodium chloride (sea salt; NaCl). Idealized CCN
distribution, the same as in , is represented
by a sum of two lognormal distributions with number mixing ratio, mean radii,
and geometric standard deviations (unitless) 57.33 and 38.22 m g-1
(i.e., per cubic centimeter for
the air density of 1 kg m-3; these values come from converting the 60
and 40 cm-3 concentrations to the number mixing ratio using air density
at the bottom of the computational domain in simulations discussed in
Sect. 3), 20 and 75 nm, and 1.4 and 1.6, respectively. The N–S
relationship is tabulated and the table is used as input to the super-droplet
scheme. Once activated, the initial radius corresponding to the activation
radius is assigned for each super-droplet. The latter is approximated as 8×10-10/Sact (m) as in GA17, where Sact is
the activation supersaturation; see Eq. (6) and Fig. 2 in
. In addition to the droplet radius, the model
keeps track of the super-droplet multiplicity parameter (or attribute), that
is, the number of droplets the super-droplet represents,
. A newly created super-droplet is placed randomly
within a given grid cell and added to the super-droplet list.
Thick line: number mixing ratio of activated CCN as a function of
the supersaturation, the Twomey relationship, used in simulations described
herein. Thin dashed lines illustrate numerical implementation of the CCN
activation scheme. See text for details.
Figure , adopted from GA17, shows the N–S relationship and
illustrates the way super-droplets are created. First, the maximum
supersaturation Smax is selected. Smax has to exceed
the maximum supersaturation anticipated in the simulation. Smax
equal to 4 % is used here as shown in Fig. . The corresponding
maximum number mixing ratio of activated droplets Nmax is divided
by the number of droplet classes to be used in the simulations. The example
in Fig. assumes 10 classes whereas simulations typically
apply several tens to several thousands of classes. New super-droplets are
introduced to a given grid cell when the supersaturation predicted for that
grid cell exceeds the supersaturation corresponding to the activation
supersaturation of super-droplets already present in the grid cell. The
approach illustrated in Fig. ensures that the multiplicity
parameter is the same for all super-droplets. This is beneficial because
equal multiplicity minimizes statistical fluctuations of derived cloud
quantities (such as the droplet concentration or liquid water content) when
super-droplets are advected from one grid cell to another. The approach
adopted here was suggested by simple one-dimensional advection tests
completed during early stages of the scheme development. However, equal
multiplicity is possibly the worst choice when collision–coalescence is added
to the scheme physics as pointed out by .
We note in passing that discuss various
methodologies for introducing Lagrangian particles, including stochastic
particle initiation as well as particle merging and splitting, that all aid
computational efficiency of the Lagrangian cloud model. These need to be
considered while expanding the scheme to include collision–coalescence.
Illustration of the activation as represented on the fluid flow
grid. Panel (a) shows locations of CCN activated at a given model time
step. Panel (b) shows the situation at the next time step when activated
CCN are advected away from the grid cell and activation of new CCN is
required.
When applied in a multidimensional fluid flow model, there is an additional
issue with the proposed scheme that needs to be addressed.
Figure shows a single two-dimensional grid cell at which
formation of new super-droplets takes place at time t. At the next time
step, t+Δt, super-droplets are advected upwards by the updraft, and a
droplet-free volume is advected into the grid cell. Assuming that the
supersaturation within the grid cell does not change, there is a need to
activate new super-droplets as some of those present within the grid cell at
the previous time step moved upwards. The new super-droplets should be
introduced into the droplet-free volume (i.e., in the lower part of the grid
cell in Fig. b) because unactivated CCN
would be there. However, keeping track of volumes void of super-droplets
during activation followed by advection is cumbersome. At the same time,
adding new super-droplets randomly into the entire grid cell leads to the
situation where super-droplets are not randomly distributed (i.e., more
super-droplets is present in the upper part of the grid cell in
Fig. ). A simple approach adopted here is that all
super-droplets are always randomly repositioned within a given grid cell once
additional activation within that cell takes place.
Transport of super-droplets in the physical space
Super-droplets are advected in the physical space applying a
predictor–corrector scheme to solve Eq. (5). The predictor step estimates the
n+1 time level position from n time level velocity as follows:
xpn+1=xn+un(xn)Δt,
where the subscript “p” depicts the predictor solution. The corrector step
(subscript “c”) is subsequently applied as follows:
xcn+1=xn+un+1(xpn+1)+un(xn)Δt2.
The predictor–corrector scheme ensures the second-order accuracy for the time
integration of the super-droplet transport. However, to increase accuracy,
the corrector step can be repeated by replacing xp by the already-calculated xc in the un+1 velocity on the right-hand side of Eq. (7). Note that velocity needs to be interpolated to the
super-droplet position and repeating the corrector step increases the overall
computational cost. We will test the benefit of the second correction step in
the droplet advection procedure later in this section. It also needs to be
pointed out that the super-droplet transport requires knowledge of the flow
velocity at the n+1 time level in Eq. (8). Similarly to the case of EULAG`s
and babyEULAG's advection of the temperature and water vapor mixing ratio
where advecting velocities need to be known at n+1/2 time level, the n+1
time level velocities in Eq. (8) are extrapolated from velocities available
at n-1 and n time levels.
Velocity interpolation to calculate super-droplet transport is the key
element of the Lagrangian scheme. Since the EULAG model applies unstaggered
grid (i.e., all variables are located at the same position), one possibility
is to consider a grid cell whose four corners in two dimensions (eight vertices in three dimensions) form a
rectangular (cuboid-shaped in three dimensions) grid cell. For a super-droplet located in
such a grid cell, flow velocity at the droplet position can be interpolated
from the velocity values at the corners and vertices. Arguably the simplest
possibility is to apply a bilinear (trilinear in three dimensions) interpolation scheme,
but a more advanced scheme may be considered as well. However, the bilinear
interpolation (and likely more advanced interpolation schemes) does not lead
to physically consistent results as documented below.
Illustration of the interpolation scheme used in the super-droplet
transport scheme referred to as “simple” in the text. The rectangular box
represents a single grid cell with u and w depicting horizontal and
vertical velocities perpendicular to grid cell boundaries used in the
advection scheme of the Eulerian model. The large dot represents droplet
position.
Advection of the potential temperature and water vapor mixing ratio (as well
as the velocity components) in EULAG is performed on the C grid (i.e., with
the horizontal–vertical velocities at the vertical–horizontal grid cell
boundaries). Advective velocities come from interpolating velocity components
predicted on the unstaggered grid into the C grid. Advective velocities
satisfy the anelastic incompressibility condition ∇⋅(ρu)=0, where ρ(z) is the anelastic density profile. In two dimensions, the
divergence of advecting velocities can be written in the finite-difference
form as follows (see Fig. ):
ui+12,k-ui-12,kΔx+wi,k+12-wi,k-12Δz=-wρ∂ρ∂z,
where the term on the right-hand side of Eq. (9) representing the change of
the anelastic density with height is left in the analytic form as it is irrelevant
to the discussion. With a single super-droplet located in the grid cell (see
Fig. ), the horizontal and vertical velocities can be
interpolated using a simple scheme similar to that used in
:
u=αui+12,k+(1-α)ui-12,k,w=γwi,k+12+(1-γ)wi,k-12,
where α and γ are nondimensional distances of the super-droplet
position to the cell boundary as shown in Fig. . As documented
in the Appendix A, such a definition ensures that the incompressibility
condition (Eq. 9) is maintained on the subgrid scale of the grid cell. This,
however, ensures that a deformation of the initially rectangular
grid cell, as represented by passive advection of all passive particles
initially located inside the cell, preserves the cell area (volume in three dimensions). We
refer to the interpolation scheme (Eq. 10) as “simple” in the following
discussion in contrast to the bilinear (or trilinear in three dimensions) interpolation
scheme introduced previously.
To investigate the accuracy of the super-droplet transport scheme, a
relatively simple test problem was designed. In the test, two-dimensional
rising moist thermal simulations driven by the Eulerian bulk condensation
scheme were used, applying the
same simulation setup as in the super-droplet simulations (see Sect. 3.1).
The predicted rising thermal flow (similar to the one shown later in the
paper applying super-droplets) was applied to advect a large number of
passive particles introduced to a fraction of the computational domain
including the thermal and its immediate environment at the onset of the
simulation. The number of passive particles varied from several tens to a few
thousands per grid cell in various tests. In the rising thermal flow
simulated by the model, one should expect the average number of particles per
grid volume to slightly decrease because of the density decreasing with
height. Moreover, the number should show statistical fluctuations due to
advection of particles from one grid cell to another. The fluctuation
amplitude should vary approximately as an inverse of the square root of the
initial number of particles per grid cell. These assumptions provide the
basis for evaluating the accuracy of the super-droplet transport.
Evolution of the maximum and the minimum number of passive particles
per grid cell in the bulk thermal simulations applying (a, b) the
bilinear advection scheme and (c, d) the “simple” scheme
(Fig. ). Panels (a) and (c) show results applying
100 passive particle per grid cell, and (b) and (d) show
results with 1000 passive particles per grid cell. Both schemes apply either
the predictor step or the predictor–corrector with either one or two
corrective iterations. The vertical bar length in (c)
and (d) corresponds to six standard deviations of the expected
number of particles per grid cell. See text for details.
Figure shows evolution of the minimum and maximum number of
passive particles per grid cell advected using the predictor–corrector
scheme, with the upper and lower panels showing results from the bilinear and simple
flow velocity interpolation, respectively. The extrema are calculated using
only grid cells with the cloud water mixing larger than 0.01 g kg-1
(i.e., cloudy cells). Results are shown from simulations applying the
predictor-only scheme (Eq. 7), the predictor–corrector scheme (Eqs. 7 and 8),
and the predictor–corrector scheme with additional iteration of Eq. (8). The
initial number of passive particles is either 100 (panels a and c) or 1000
(panels b and d) per grid cell. The standard deviation of the number of
particles per grid cell after advection should be close to the square root of
the initial number, that is, close to 10 and 30 particles (i.e., close to 10
and 3 % of the particle number per grid cell) in the left and right panels,
respectively. It follows that the difference between the maximum and minimum
number of passive super-droplets per grid cell should be not significantly
larger than a few standard deviations. As the upper panels of
Fig. show, the bilinear interpolation scheme leads to a
larger difference starting around minute 2 of the simulations, and the
difference increases with time. This is clearly unphysical as argued above.
In contrast, the simple scheme with a single corrective iteration provides
physically consistent results, that is, the difference between the maximum
and minimum is several times the standard deviation of the initial particle
number per grid cell, and such a difference is maintained throughout the
10 min long simulations. Only an insignificant improvement is simulated with
the additional iteration of the corrective step (Eq. 8). Finally, a slight
reduction of the mean concentration (located somewhere between the maximum
and minimum symbols) is apparent in bottom panels of Fig. .
This is because of the reduction of the droplet concentration due to the
decrease of the air density with height (i.e., the term on the right-hand
side of Eq. 9). It should be also pointed out that the difference between the
predictor-only and the predictor–corrector schemes should decrease if a
significantly shorter time step is used (e.g., 0.1 s instead of 1 s used in
Fig. simulations).
This simple example, together with similar simulations using different
numbers of passive particles not shown here as well as results of the
super-droplet approach available at the University of Warsaw (Arabas et al.,
2015), suggests that the simple scheme (Eq. 10) (and its extension into a
three-dimensional
framework) should be used in the Lagrangian microphysics. Hence, such a
scheme is used in all super-droplet simulations presented in this paper.
Coupling thermodynamic Eulerian and Lagrangian fields
The overall strategy for the time integration of the coupled Eularian and
Lagrangian components of the model thermodynamics is to advance the
temperature and moisture fields using Eq. (6) first, then to transport
Lagrangian super-droplets using Eqs. (7) and (8), and finally to calculate
condensation or evaporation of cloud droplets according to Eq. (4), with the
condensation or evaporation providing temperature and moisture tendencies
calculated from Eq. (3) in each grid cell. These tendencies are applied in
the next model time step. Condensation or evaporation of individual
super-droplets require knowledge of the supersaturation that needs to be
calculated from updated temperature and water vapor fields. The flow-resolved
supersaturation field can be supplemented with the subgrid-scale fluctuations
as in GA17. By the same token, the resolved flow used to transport
super-droplets through the predictor–corrector scheme can be supplemented
with the subgrid-scale velocity fluctuations estimated from the predicted
subgrid-scale turbulent kinetic energy. These additions are not included in
the initial formulation and testing of the Twomey super-droplets discussed in
this paper, but will form an important component of the model application in
the future.
There are two issues that need to be considered for the coupling between
Eulerian and Lagrangian model components. The first one concerns spurious
supersaturation fluctuations near cloud edges seeand references
therein. This problem is particularly serious when the
Twomey activation is used as illustrated later in the paper because of the
direct link between the local
supersaturation and the concentration of activated cloud droplets.
Specifically, numerical overshoots of the supersaturation lead to an
immediate activation of new cloud droplets. In contrast, when deliquescence
and droplet activation are explicitly considered in the traditional
super-droplet method, these transient overshoots may have a smaller impact on
the droplet activation. This is one of the conclusions of the
study, also confirmed by simulations discussed in
this paper. developed a relatively simple method
to cope with this problem for the case of a double-moment Eulerian
microphysics scheme and suggested how it can be extended to the bin
microphysics. We apply the methodology to the
super-droplet simulations as discussed below.
The second issue concerns the interpolation of the thermodynamic fields to
the super-droplet position. (see Sect. 5.1.2),
(Sect. 2.2.3), and Miroslaw Andrejczuk (personal communication, 2017) interpolate the potential
temperature and water vapor mixing ratio and then derive the local
supersaturation. Such an approach is not appropriate due to the nonlinear
relationship between the supersaturation and the potential temperature.
Interpolating the supersaturation would be more appropriate. However,
supersaturation interpolation brings conceptual issues similar to those
concerning super-droplet transport: if a single super-droplet represents a
large ensemble of real cloud droplets, should growth of the ensemble be
represented using the grid-averaged conditions? Moreover, one-dimensional
tests with a stationary cloud–environment interface show that the
supersaturation interpolation results in a gradual erosion of the cloud edge.
This is because supersaturation interpolation between a cloudy grid cell near
the cloud edge and a subsaturated cell outside the cloud results in
subsaturated conditions for super-droplets located near the cell boundary
leading to their evaporation. In contrast, applying the mean supersaturation
maintains the steady conditions near the motionless cloud–environment
interface. Moreover, applying the methodology to
cope with the spurious cloud-edge supersaturation discussed below becomes
cumbersome (if not impossible) when the supersaturation interpolation to the
super-droplet position is used. Overall, our tests, similar to those discussed
in the next section, suggest that the impact of supersaturation interpolation
in a rising thermal simulations is small, and thus we decided to proceed with
the simpler and computationally more efficient method of applying the
grid-cell supersaturation to growth and evaporation of all super-droplets within
a given grid cell.
Avoiding spurious cloud-edge supersaturations
The key aspect of the (GM08 hereinafter) method
is to rely on the prediction of the absolute supersaturation (the difference
between the water vapor mixing ratio and its saturated value) and to locally
adjust the water vapor, cloud water, and temperature to maintain the
predicted absolute supersaturation. This is in the spirit of
, who used the temperature and supersaturation
as main model variables and diagnosed the water vapor mixing ratio. Such a
method results in a physically consistent supersaturation field but does not
conserve water. GM08 circumvent this problem and apply the approach to the
Eulerian double-moment cloud microphysics (i.e., predicting number and mass
mixing ratios of the cloud water field). They also suggest how this approach
can be used in the bin scheme (see Sect. 4 therein). Here we explain how this
method is used with Twomey super-droplets.
The crux of the method is to calculate the amount of cloud water ϵ
that needs to condense or evaporate to ensure that the predicted potential
temperature and water vapor mixing ratio fields give the absolute
supersaturation that agrees with the predicted one. Thus, in addition to the
prediction of the potential temperature and water vapor mixing ratio, the
scheme predicts the evolution of the absolute supersaturation (see Eq. A8 in
, and Eq. 4 in GM08). Once the amount of cloud
water involved in the adjustment is calculated as in Eq. (7) of GM08, one
needs to decide how that amount is distributed among super-droplets present
within a given grid cell. Following GM08, the amount of cloud water
ϵ that needs to be distributed among N super-droplets from a given
cell is calculated as
ϵ=∑i=1Nϵi,ϵi=ϵβτi,whereβ=∑k=1N1τk,
where τi=(4πDvniri)-1 is the phase
relaxation timescale for the ith super-droplet (ni is the concentration
of droplets ith super-droplet represents); see Eq. (A5) in .
Knowing ϵi, the radius of each super-droplet within a given
grid cell is subsequently modified, keeping the multiplicity unchanged.
Example of application: two-dimensional moist thermal simulations
The scheme described above has been merged with the EULAG model
e.g.,www2.mmm.ucar.edu/eulag/ and its
simplified version referred to as babyEULAG
. Here we present
results from the babyEULAG model as it is simpler and thus more convenient
for the scheme testing and improvement. The University of Warsaw Lagrangian
Cloud Model (UWLCM) briefly described in the next section is used in the
comparison. The Lagrangian approach applied in the UWLCM is referred to as
the traditional super-droplet method in the discussion below. Both the
babyEULAG model and the UWLCM apply the implicit large-eddy simulation
approach, that is, without modeling of the unresolved subgrid-scale transport
see references to other studies applying this method
in.
The University of Warsaw Lagrangian Cloud Model, UWLCM
The UWLCM is an open-source software for two-dimensional and three-dimensional modeling of clouds
with super-droplet or bulk microphysics. Advection of the Eulerian
fields is done using the libmpdata++
implementation of the MPDATA algorithm .
Cloud microphysics is modeled using the libcloudph++ library .
Coupling between Eulerian and Lagrangian model
components is done in the same way as in the Twomey model. Potential
temperature and water vapor mixing ratio are not interpolated to
the position of a super-droplet, but the same value is used for all
droplets within a cell. The procedure for limiting spurious
supersaturation, which was described in Sect. 2.6, is not used. The
super-droplets are advected with a predictor–corrector method with
velocities interpolated to super-droplet position using the “simple”
scheme defined in Sect. 2.4. In the MPDATA algorithm used in
simulations presented in this paper, variable-sign fields were
handled using the “abs” option of the libmpdata++ library see
Sect. 3.1.5 and 3.4.1 in. Advection of
Eulerian fields and of super-droplets is done with a Δt=1 s time step. Water condensation is done using 10 substeps per
time step of advection, resulting in a Δt=0.1 s time step
for condensation. Details of the substepping procedure are discussed
in .
Because UWLCM explicitly represents CCN deliquescence, a more
detailed droplet growth equation is used see Sect. 5.1.3
in. The κ–Köhler parametrization
of aerosol hygroscopicity is used. We assume κ=1.28 for
the sea salt aerosol used in this paper see Table 1
in. Super-droplets in the UWLCM typically
have different multiplicities and only the initial number of
super-droplets per grid cell is prescribed. The super-droplet
initialization scheme is the same as in (the
“constant SD” type of simulation described in Sect. 2 therein).
Setup of moist thermal simulations
Rising moist thermal simulations follow with small modifications. The
environmental profiles are taken as constant stability dlnΘv/dz=1.3×10-5 m-1 for
the temperature (Θv is the virtual potential temperature;
the potential-temperature-based stability was used in
) and constant relative humidity of 20 %. Surface temperature
and pressure are taken as 283 K and 850 hPa. Note that the Θv-based
stability profile requires an iterative procedure when moving upwards
from the surface because the temperature, moisture, and pressure
(the latter resulting from the hydrostatic balance) have all to be
adjusted to give stability and relative humidity profiles exactly
as specified above. The circular moisture perturbation is introduced
in the middle of the 3.6 km horizontal domain, with the center
located at the 800 m height. The vertical extent of the domain is
2.4 km. The air inside the 250 m perturbation radius (200 m was
used in ) is assumed to be saturated, and the relative
humidity decreases to the environmental value as cosine squared
over the 100 m radial distance. Uniform horizontal and vertical
grid length of 20 m is used. A 1 s time step is used in simulations
applying the babyEULAG model.
The average number of super-droplets per grid cell affects the
amplitude of statistical fluctuations due to the transport of
super-droplets across the Eulerian grid. Because of different
formulations of CCN activation, a direct match of the super-droplet
number per grid cell between UWLCM and the Twomey scheme is impossible.
In the Twomey scheme simulations, the number of Smax
divisions considered was 50, 200, 1000, and 4000. The corresponding
number of super-droplets per cloudy grid cell was around 40 for 50
divisions, ∼ 150 for 200 divisions, ∼ 700 for 1000 divisions,
and ∼ 2700 for 4000 Smax divisions. The UWLCM
simulations used in the comparison with the Twomey scheme simulations
discussed in the next section applied 200 and 4000 super-droplets
per grid cell. Only some of them became activated, and
the averaged fraction of activated super-droplets (i.e., those with
the radius larger than the critical radius) was around 45 % for
both cases.
Comparison between UWLCM and the Twomey super-droplets
When comparing results from the two models, one needs to keep in mind that
microphysical schemes differ in some additional details. In particular, the
UWLCM applies the κ–Köhler parametrization
to prescribe CCN activation characteristics whereas
the Twomey scheme applies the N–S relationship derived from activation
calculations applying CCN chemical composition information i.e., as
in. Our tests with the adiabatic parcel model
applying either the κ–Köhler parametrization used in UWLCM or the
approach based on the CCN chemical composition show that difference of a couple of percentage points between droplet concentration predicted by the two methods for the
same supersaturation is not unusual. Moreover, different droplet growth
equations are used in the two schemes, although this factor does not affect
the favorable comparison presented in . Finally,
the number of SDs representing activated CCN used in both models is not
exactly the same as explained before.
Distribution of (a) the water vapor and (b) the
cloud water mixing ratios at 2, 6 and 10 min for the Twomey super-droplet
scheme in the rising thermal simulation.
As Fig. , but for the UWLCM model.
Figures and show spatial
distributions of the water vapor and cloud water mixing ratios for the two
simulations, that is, using either the Twomey super-droplets with the
babyEULAG model (Fig. ) or the traditional
super-droplets with UWLCM model (Fig. ). Both
simulations apply a similar number of super-droplets per grid cell (4000
aerosol sizes in UWLCM and 4000 divisions in the Twomey simulation). Overall,
the transition of the initial circular perturbation to a cloudy rising vortex
pair proceeds similarly in the two models. The most obvious difference comes
from the development of instabilities near the thermal top. These
instabilities are forced by fluctuations of thermodynamic fields (and thus
cloud buoyancy) that result from a finite number of super-droplets in each
grid cell. As discussed in ,
these cloud–environment instabilities represent a combination of
Rayleigh–Taylor and Kelvin–Helmholtz instabilities occurring in a complex
geometry near the thermal leading edge. The spatial scale of the instability
depends on the depth of the shear that develops near the cloud–environment
interface as the thermal pushes upwards see Sect. 4b
in. The specific realization of the instability
pattern changes with the number of super-droplets used in the simulation, and
with the selection of random numbers applied during positioning
super-droplets
on the Eulerian grid during activation. It follows that the direct comparison
between the simulations is possible only before the development of the
instabilities, say, up to the 6th minute of the simulation (i.e., middle
panels in Figs. and ).
A more detailed comparison between the two simulations is facilitated
by applying two different statistical measures. The first one involves
conditional sampling of various fields across the thermal, including points
with the cloud water mixing ratio exceeding a threshold of 0.1 g kg-1.
A smaller threshold allows incorporation of more significant fraction of
points from the thermal edge that are affected by the Eulerian model
numerics. The statistics include the mean values of conditionally sampled
fields and the standard deviations of the spatial variability of a given
field across the thermal. The second measure is the time evolution of various
quantities at the center of mass of the cloud water field, that is, at the
height of zcm=∫zqcds/∫qcds (where qc depicts the cloud water mixing ratio
and the integral is over the entire computational domain) and a similar
expression for the horizontal position xcm.
Comparison of the mean supersaturation averaged over the cloudy
points for UWLCM (a, d) and the Twomey super-droplet scheme
without (b, e) and with (c, f) the adjustment to avoid
unphysical cloud-edge supersaturation fluctuations. The blue and brown lines
represent the mean and the mean plus the standard deviation of the spatial
distribution. Panels (a–c) and (d–f) are for simulations
with 200 and 4000 super-droplets (UWLCM) or number of divisions (Twomey
scheme). Data are plotted at every time step of the fluid flow model.
Figure compares evolution of the supersaturation
field conditionally sampled over the rising thermal for UWLCM with 200 and
4000 super-droplets per grid cell and the Twomey approach with 200 and 4000
Smax divisions. The Twomey results either include or exclude (marked adjust and
noadjust in the figure) temperature and moisture adjustment as described in
Sect. 2.6. The figure shows that application of the adjustment scheme is
critical for maintaining a physically consistent supersaturation field as the
supersaturations are significantly higher without the adjustment. The
physical consistency is measured by comparing the supersaturation predicted
locally with the quasi-equilibrium supersaturation, that is, the
supersaturation resulting from a balance between production due to the
updraft and removal due to condensation
. Except for the initial
first minute when droplets are small, the supersaturation predicted by the
model agrees well with the quasi-equilibrium supersaturation (not shown). An
important point is that simulations with 200 divisions (upper panels) differ
little from simulations applying 4000 divisions (lower panels) until
different flow realizations in the final few minutes cause the divergence.
The agreement suggests that about a hundred super-droplet per grid cell is
sufficient to obtain statistics that change little with further increase of
the super-droplet number. The reduction of both the standard deviation and
the amplitude of the fluctuations when the adjustment scheme is applied is
also apparent. The corresponding results from UWLCM simulations show that the
mean supersaturation evolution is similar to those for the Twomey simulations
without adjustment as one might expect.
As in Fig. , but for the droplet
concentration Nact.
Figure shows statistics of the droplet concentration in the
format similar to Fig. . As expected, the mean
concentration is higher when adjustment is not used in the Twomey approach.
The mean concentration slowly decreases in time because of the air expansion
due to rise of the thermal. For Twomey simulations with 200 Smax
divisions, the mean concentration at minute 2 is around 77 cm-3 and the
standard deviation of the spatial distribution is around 5 cm-3 in
simulation with the adjustment versus 85 cm-3 and higher standard
deviation without the adjustment. The mean concentration at 2 min decreases
to around 72 cm-3 for the 4000 division simulations. For the UWLCM, the
mean concentration at minute 2 is around 75 cm-3 and the standard
deviation of the spatial distribution is around 9 cm-3. The similarity
of the mean concentration between the Twomey super-droplets with adjustment
and UWLCM documents the limited impact of the cloud-edge supersaturation
fluctuations when details of the CCN activation are resolved in the original
super-droplet approach. Larger standard deviation of the spatial distribution
is likely because of the variable multiplicity attribute among original
super-droplets in the UWLCM model.
Evolution of various parameters at the center of mass of the cloud
water in the rising thermal simulations applying 200 super-droplets (UWLCM;
purple lines) or 200 divisions (Twomey scheme with adjustment; green lines).
The panels show (a) droplet concentration, (b) droplet
mean radius, (c) spectral width of the droplet size distribution,
(d) height of the center of mass, (e) supersaturation, and
(f) vertical velocity.
As Fig. , but for 4000 super-droplets (UWLCM)
or 4000 divisions (Twomey scheme).
Figures and compare various
statistics between the Twomey scheme and UWLCM at the center of mass of the
cloud water field with 200 and 4000 super-droplets (UWLCM) or
Smax divisions (Twomey scheme), 200 in
Fig. , and 4000 in Fig. . The
data are plotted at every model time step. Microphysical properties such as the
droplet concentration, mean radius, and the spectral width show oscillations
that are reduced with the increased number of super-droplets per grid cell.
It is important to note that center of mass is calculated on the Eulerian
grid, that is, it jumps from one grid box to another as the thermal moves
upwards. The period of the oscillations in Figs.
and , about 10 s, matches approximately the
propagation of the center of mass over the grid as the updraft velocity is
about 2 m s-1 and the grid length is 20 m. The amplitude of the
oscillations can be estimated by comparing the original evolution (i.e., the
one shown in Figs. and ) and
the evolution sufficiently smoothed in time so the oscillations are removed.
In the case of the droplet concentration for the Twomey simulations, the
amplitude decreases from 6.0, 2.0, 1.1, and 0.6 cm-3 for the number of
divisions increasing from 50, 200, 1000, and 4000. This is roughly the expected
scaling, that is, along the square root of the number of super-droplets that
increases from around 40 to 2700 for the number of divisions increasing from
50 to 4000. As mentioned before, the direct comparison between various
simulations is only possible up to about the 6th minute as different flow
evolutions make results impossible to compare at later times. Except for the
oscillation amplitude, the results for different number of Smax
divisions compare well for the Twomey super-droplets.
The differences between Figs.
and are consistent with the differences between
conditionally averaged statistics. For instance, droplet concentrations
fluctuate between 60 and 80 cm-3 for Twomey super-droplets with
adjustment and 200 Smax divisions and UWLCM with 200
super-droplets per grid cell. The evolution of the center of mass height is
very similar in Twomey and UWLCM simulations. The mean radius is close to
15 µm at minute 6 for both the Twomey and UWLCM. Droplet
concentration fluctuations are larger in UWLCM arguably because of the way
CCN is sampled when super-droplets are created, that is, with different
multiplicity parameters that increase the oscillation amplitude. The
evolution of the vertical velocity at the cloud water center of mass
increases in both simulations up to about 3.5 min and decreases thereafter
(the evolutions after minute 6 cannot be compared due to different flow
realizations as already explained). The vertical velocity maxima around
3.5 min are similar.
Evolution of the maximum supersaturation in the domain for UWLCM
simulations (purple line) and Twomey scheme with (blue line) and without
(green line) adjustment for unphysical supersaturation fluctuations. Data
are plotted at every model time step with 200 super-droplets in UWLCM and 200 division in the model with the Twomey activation.
As a final element of the comparison, we show in Fig.
evolutions of the maximum supersaturation in the computational domain for
simulations with the Twomey super-droplets with and without the adjustment to
limit unphysical cloud-edge supersaturations and for the UWLCM. The
simulations apply similar number of super-droplets per grid cell (200
divisions in the Twomey super-droplet simulations and 200 samples of the CCN
distribution in the UWLCM simulation). The maximum supersaturations occur at
different spatial locations near the cloud edge as the cloud–environment
interface moves across the Eulerian grid. This is why large fluctuations in
the UWLCM simulation impact the mean droplet concentration to a smaller
degree than in the Twomey approach. In a nutshell, CCN has no time to respond
to these fluctuations when deliquescence is explicitly calculated by the
model. In contrast, Twomey activation immediately adds new droplets when
supersaturation fluctuations take place. These additional droplets can
evaporate in subsequent time steps, but some survive and lead to the
increased mean droplet concentration as documented in Fig. .
It follows that the adjustment is the key element of the Twomey
super-droplets, but is less significant for the traditional super-droplet
approach; see .
In summary, we believe that simple tests presented in this section document
the efficacy of the super-droplet approach with the Twomey activation.
Unfortunately, we cannot provide a direct comparison of the computational
effort between the two approaches because the two models run on different
computer systems. However, since the cloud covers about 2.5 % of the
two-dimensional computational domain, the Twomey scheme requires roughly 40
times less computational effort for simulations presented here (this estimate
excludes the difference in the time steps used by both models). However, for
a hypothetical three-dimensional
simulation with a domain extending 3.6 km in the second horizontal
direction, the volume of the initial spherical bubble with the same radius
would only constitute about 0.1 % of the computational domain volume.
Thus, the computational effort in similar three-dimensional simulations would
be about 3 orders of magnitude larger in UWLCM than in the babyEULAG with
Twomey super-droplets. UWLCM makes up a lot of this difference by applying
modern software engineering techniques including parallel processing and
application of graphics processing units; see .
It is important to note, however, that parallelization of the numerical model
applying Twomey super-droplets requires a different strategy than the domain
decomposition approach typically applied in finite-difference numerical
models. This is to avoid significant load imbalances between subdomains
featuring cloudy grid cells (i.e., with super-droplets) and those that are
cloud-free. It remains to be seen whether such a different parallelization
will lead to problems due to an increased communication.
Conclusions
This paper discusses technical details of a novel Lagrangian condensation
scheme to model nonprecipitating warm (ice-free) clouds. The idea is to use
Lagrangian point particles (“super-droplets” following the nomenclature
introduced by ), rather than continuous medium
variables such as number or mass mixing ratios, to represent condensed cloud
water. Previous studies applying such methodology
e.g.,
demonstrate significant advantages of the super-droplet method, such as
reduced numerical diffusion, formulation of the governing equations from
first principles, and straightforward application of suitable statistical
techniques to represent unresolved subgrid-scale variability as in GA17.
However, in previous applications of the Lagrangian microphysics, the
super-droplets outside clouds represent unactivated CCN that become
activated upon entering a cloud and can further grow through diffusional and
collisional processes. Thus, the super-droplets fill the entire computational
domain and need to be transported even if they exist far away from a cloud
and do not affect cloud processes. The original methodology allows for the detailed study of not only effects of CCN on cloud microphysics and dynamics, but
also CCN processing by a cloud. When applying the super-droplet method to
problems where CCN processing is of secondary importance (e.g., the impact of
entrainment on the spectrum of cloud droplets), a simpler and
more computationally efficient approach can be used. The idea is to create super-droplets
only when CCN is activated and to remove them when a complete evaporation
(i.e., CCN de-activation) takes place. Thus, no super-droplet exists outside
a cloud and a significantly smaller number of super-droplets need to be
followed in space and time when compared to the traditional super-droplet
scheme with the same number of super-droplets per grid cell. The new
super-droplet approach is possible by applying the Twomey activation method
where the local supersaturation dictates the concentration of cloud droplets
(and thus the number of the super-droplets) that need to be present inside a
cloudy volume. Twomey activation excludes details of the CCN deliquescence
and activation, and super-droplets simply disappear when a complete
evaporation of cloud droplets occurs. Twomey activation is often used in
Eulerian bulk e.g., and
bin microphysics schemes
e.g.,. Moreover,
simulation of the CCN deliquescence requires short time steps and avoiding it
with the Twomey activation provides additional computational advantage. As
mentioned previously, applying Twomey activation to create cloud droplets
bears similarities to the way ice particles are initiated in a Lagrangian
model targeting ice processes e.g.,.
We apply the traditional Lagrangian super-droplet model, the University of
Warsaw Lagrangian Cloud Model
UWLCM; and compare
results from UWLCM and the novel Twomey super-droplet method. The simulations
apply an idealized setup of a moist thermal rising in a stratified
environment . Overall, the
comparison demonstrates the efficacy of the new approach as simulation
results differ little between UWLCM and the new scheme. This is consistent
with adiabatic parcel results discussed in that
– away from the cloud base – show good agreement between cloud properties
simulated applying a scheme with Twomey activation and a scheme where details
of the CCN deliquescence are modeled explicitly. The results presented here
show that avoiding spurious cloud-edge supersaturation fluctuations is
essential with the Twomey activation. This is because these fluctuations
immediately translate into unphysical droplet concentrations that affect
subsequent evolution of cloud microphysical properties. In contrast, these
fluctuations that are highly transient in space and time seem to have small impact on
simulations using the original super-droplet method, in agreement with
results discussed in .
As noted in , , and
, modeling cloud base activation in the
Eulerian cloud model requires high vertical resolution to resolve cloud base
supersaturation maximum, say, of the order of 10 m. The same is true for the
Lagrangian super-droplets. In the case of lower vertical resolution (i.e.,
when the cloud base supersaturation maximum is poorly resolved), an
activation parameterization can be used, for instance, linking the
concentration of activated CCN to the strength of the updraft velocity
e.g.,among
others. Such a
parameterization can also be used with the methodology presented in this
paper, for instance, in simulations of deep convection that only allow low
vertical resolution. As deep convection requires incorporation of ice physics
into the Lagrangian methodology, the possibility of applying an even simpler
representation of super-droplet formation through the activation
parameterization is appealing. Such a methodology will pave the way for
applications of the Lagrangian methodology beyond high-spatial-resolution
large-eddy simulation today to the cloud-resolving (convection-permitting)
weather and climate simulation of the future. We plan to include such
developments to the Twomey super-droplet scheme presented here, together with
the inclusion of the collision–coalescence that will be the focus of future
scheme expansions. These developments will be reported in forthcoming
publications.