Introduction
Satellite imagery of cloud fields over the eastern edges of the oceanic
basins exhibits both closed and open cellular cloud patterns that have
captured the imagination of the atmospheric scientist and the layperson
alike. Interest in these cellular cloud modes has been spurred by both the
desire to understand these states and to evaluate their consequences for
shallow cloud reflectance and climate forcing. The closed cellular state is a
mostly cloudy state characterized by broad, weak updrafts in the opaque
cloudy cell center and stronger, narrower downdrafts around the cell edges.
The open-cell state is the “polar opposite” or “negative” in which
narrow, strong, cloudy updrafts surround broad, weak downdrafts in the
optically thin cell center. These states have been studied through
observation (Sharon et al., 2006; Stevens et al., 2005; Wood and Hartmann,
2006; Wood et al., 2011) and modeling (Savic-Jovcic and Stevens, 2008; Xue et
al., 2008; Wang and Feingold, 2009a, b; Kazil et al., 2011, 2014; Yamaguchi
and Feingold, 2015), with most efforts addressing the closed-to-open cell
transition. These studies have shown that rain is the likely initiator of the
closed-to-open cell transition, pointing to the importance of deepening of
the cloud (Mechem et al., 2012) and/or reduction in the cloud condensation
nucleus concentration. An interesting aspect of the precipitating open
cellular system is that strongly buoyant cloudy cells produce rain, which
imposes local negative buoyancy perturbations. The cloud–rain cycles thus
create an adaptive open-cell state that constantly rearranges itself as
clouds move through positive buoyancy (non-precipitating) and negative
buoyancy (precipitating) cycles (Feingold et al., 2010; Koren and Feingold,
2013). The closed-cell state has, in contrast, a more rigid structure that
maintains itself over many hours (Koren and Feingold, 2013).
A relatively under-studied aspect of the system is the two-way transition
from closed-to-open-to-closed cells, which will be the focus of the current
work. The results pertain to what have been termed “pockets of open cells”
(Stevens et al., 2005) or rifts (Sharon et al., 2006) in which open
cells periodically appear within a meteorological setting that promotes
closed cellular convection. This work does not address the broader question
of closed to open-cell transitions due to a warming sea surface temperature
as one moves westward from the stratocumulus-capped continental coastlines.
While some modeling work has addressed the two-way transition between states
(Wang and Feingold, 2009b; Berner et al., 2013) and there exists ample visual
evidence of ship tracks “filling in” cloudiness in open-cell fields (e.g.,
Goren and Rosenfeld, 2012), there remain open questions regarding the
relative ease of the two transitions and the extent to which aerosol
intrusions control this transition. For example, Wang and Feingold (2009b)
perturbed a cloud-resolving simulation of the open-cell state with a very
large aerosol perturbation, and while a thin layer of cloud did fill the open
cells, the aerosol was unable to convert the system to a closed state,
presumably because the cloud was too thin to generate sufficient radiative
cooling. The juxtaposition of these simulations and the observations suggests
that differences in meteorological conditions, aerosol perturbations, and the
timing within the diurnal cycle might matter (Wang et al., 2011). The latter
study explored other important factors such as the amount and distribution of
the aerosol perturbation (in the form of ship tracks).
To address this problem, we use a cloud-resolving atmospheric model that uses
a simple microphysical scheme with an ad hoc control over the drop
concentration and therefore, all else equal, the rain production. This is in
contrast to our earlier work (Kazil et al., 2011), in which the aerosol life cycle
was simulated from new particle formation through wet scavenging, and to more
recent two-dimensional, multi-day simulations of closed and open-cell systems
(Berner et al., 2013). The choice of a simple control over drop concentration
avoids a more direct assessment of the importance of the rates of aerosol
removal and replenishment. Supporting simulations are also performed using a
dynamical systems analogue to the aerosol–cloud–precipitation system in the
form of modified predator–prey coupled equations (Koren and Feingold, 2011;
Feingold and Koren, 2013; Jiang and Wang, 2014), which provides insight into
the essence of the system at minimal computational cost.
Model description
Cloud-system resolving model (CRM)
We use the System for Atmospheric Modeling (SAM) as described in
Khairoutdinov and Randall (2003) with a second-order centered scheme for
momentum advection and a monotonic fifth-order scheme for scalar advection
(Yamaguchi et al., 2011). SAM solves the anelastic Navier–Stokes equations
on an Eulerian spatial grid. Prognostic equations are solved for liquid water
static energy, mixing ratios of water vapor, cloud water, rain water, and
subgrid-scale turbulence kinetic energy (TKE). While our earlier work used
bin, or bin-emulating physics in large eddy simulation
and CRM (e.g., Feingold et al., 1996; Wang and
Feingold, 2009a, b), the Khairoutdinov and Kogan (2000) microphysics is
chosen here for expediency and because its level of complexity is
commensurate with the ad hoc specification of N(t).
The initial and boundary conditions not only follow the Second Dynamics and
Chemistry of Marine Stratocumulus (DYCOMS-II RF02; Ackerman et al., 2009) but
also include a number of perturbations. The domain is
40 km × 40 km wide and 1.6 km deep with a grid spacing of 200 m
in the horizontal and 10 m in the vertical. Tests with finer horizontal grid
spacings (100 and 75 m) show that the key results are remarkably robust to
the model grid spacing (Appendix A). The lateral boundary conditions are
doubly periodic and the time step is 1 s. Our base case is the standard
Global Energy and Water Cycle Experiment (GEWEX) Cloud System Study (GCSS)
DYCOMS-II RF02 case with horizontal winds (u; v = 7.3;
-3.5 m s-1 at 1000 m; see Ackerman et al., 2009) and an interactive
surface model based on similarity theory; the large-scale subsidence is
computed based on the large-scale horizontal wind divergence of
3.75 × 10-6 s-1; long-wave radiative flux divergence is
calculated using either a simple liquid water path-dependent method (Ackerman
et al., 2009) or the coupled rapid radiative transfer model (RRTM; Mlawer et
al., 1997). Because of the shallow depth of the domain, a free tropospheric
sounding is patched above the domain top for the RRTM radiation calculations.
For the above-domain temperature sounding, we follow Cavallo et al. (2010).
The domain top value of water vapor mixing ratio is used as the above-domain
water vapor profile. Different domain-top values of water vapor mixing ratio
will be considered in Sect. 4.2. While in principle, the simple long-wave
radiation scheme could be tuned to mimic that of RRTM (e.g., Larson et al.,
2007), we have not done so. This has the salutary effect of providing
different responses of radiative cooling to liquid water path, which will
serve to elucidate sensitivity of transitions to long-wave radiative cooling.
Simulations are on the order of 18 h so that some include significant
periods of shortwave radiation. Perturbations to these initial and boundary
conditions are shown in Sect. 3.1.4.
Predator–prey model
The second model is an adaptation of the the predator–prey model (Koren and
Feingold, 2011). The model comprises three equations that describe the cloud
depth H, drop concentration N, and rain rate R for the cloud system:
dHdt=H0-Hτ1-αH2(t-T)c1N(t-T),dNdt=N0-Nτ2-c2N(t-T)RandR(t)=αH3(t-T)N(t-T),
where c1 is a temperature-dependent constant, and c2 and α are
constants based on theory. H0 is the cloud depth that would be reached
within a few timescales τ1 in the absence of rain-related losses. Thus
H0 represents “meteorological forcing” or, in population dynamics
nomenclature, the “carrying-capacity” of the system. Similarly, N0 is
the drop (or aerosol) concentration “carrying-capacity” that the system
would reach in a few τ2 in the absence of rain. The N loss term on
the right-hand side of Eq. (2) captures a physically based rate of removal.
The delay T represents the time required for cloud water to be converted
to rainwater by collision and coalescence between drops and introduces
significant complexity and nuanced response in the system of equations
(Feingold and Koren, 2013). Here we substitute Eq. (2) with a simple time-varying N similar to that imposed in the CRM simulations. Rain rate is
diagnosed from the prognostic variables H and N, again with delay T
(Eq. 3). While the system of Eqs. (1)–(3) is represented by five primary
parameters, H0, N0, τ1, τ2, and T, the use of a prescribed
N(t) instead of Eq. (2) reduces the free parameters to H0, τ1, and
T . In the current work, we will select values of these parameters that are
physically plausible and/or that help illustrate the key points.
Results
Cloud-resolving modeling
Time variation in N
A series of simulations with a prescribed evolution of drop concentration N
is applied to all simulations (Fig. 1). The time series starts with a steady
N = 90 mg-1 (equivalent to 90 cm-3 at an air density of
1 kg m-3), which for the current case generates closed-cell conditions
with minimal precipitation. It then mimics the rapid drop in N associated
with the runaway reduction in N in a developing open cell over the course
of 2 h (e.g., Feingold et al., 1996; Wang and Feingold, 2009a); a 4 h
period of steady, low N; and then an equally rapid (2 h) rise in N back
to pre-open-cell conditions. Four different values of low N are applied: 5,
15, 25, and 35 mg-1. The rapid rise back to 90 mg-1 is
unrealistic given earlier work that estimated a recovery time of
∼ 10 h (Berner et al., 2013) but, as will be shown, it provides a
(near) upper bound on replenishment of the drop concentration, and anything
less rapid serves to strengthen the arguments to be presented. The time
series of N, specifically the recovery to N = 90 mg-1, will be
varied in a number of sensitivity tests.
Imposed time series of drop concentration
N. The minimum N is varied between 5 and
35 mg-1.
Control Simulations
The control simulations use the GCSS specifications as described above, the
simple long-wave radiation scheme (no shortwave radiation) and surface latent
and sensible heat fluxes that respond to the local surface horizontal winds.
A series of snapshots of the cloud liquid water path (LWP) calculated from
the modeled cloud and rain water mixing ratios (Fig. 2) shows an initial
closed cellular state transitioning to an open-cell state (distorted by the
mean northwesterly flow), a filling in of cloud associated with the increase
in N, which gradually provides colloidal and dynamical stability to the
cloud and, finally, a more complete closed cellular cloud cover.
LWP at t = (a) 3 h (closed cell), (b) 5 h
(closed transitioning to open), (c) 7 h (open cell),
(d) 11 h (open transitioning to closed), (e) 13 h
(further recovery to closed), (f) 18 h (closed cell). Control
simulation with minimum N = 5 mg-1. The grey scale ranges from 0 to
450 g m-2.
Figure 3 shows time series of the domain mean cloud LWP, rain water path
(RWP), surface rain rate Rsfc, and the
mean surface rain rate conditionally sampled for
Rsfc ≥ 0.1 mm d-1 (Rcond). After the “spin
up” of turbulence, by t = 3 h the LWP is approximately steady at
110 g m-2 (although decreasing slowly). The reduction in N after
3 h results in rapid reduction in domain average LWP as rain ensues and
cloud cover decreases, a period of relatively steady LWP – particularly for
the low minimum N – and then a slow recovery after the increase in N at
t = 9 h. In spite of the symmetry in the ramping down and up of N,
there exists an asymmetry in LWP(t) commensurate with the minimum
imposed value of N. Asymmetry also exists in RWP(t); initially
strong RWP during the onset of drizzle (t ≈ 5 h) is followed by
a more steady but lower RWP (t = 6–9 h) and relatively steady
Rsfc. This period is characterized by a balance between dynamical
forcing that replenishes cloud liquid water and by drizzle losses. Note that
the start of the increase in N at 9 h does not put an immediate stop to
rain, as evidenced by the long tail of low RWP and Rsfc that
persists even after N = 90 mg -1 (t = 11 h). This is
because in these very clean conditions the increase in N initially helps to
boost LWP, which further boosts rain. (Recall that
R ∝ LWPα N-β; e.g., Pawlowska and Brenguier,
2003, with α approximately 3 × larger than β.)
Time series of (a) LWP, (b) rain water path (RWP),
(c) domain mean surface rain rate Rsfc, and
(d) surface rain rate conditionally sampled for
R ≥ 0.1 mm d-1 (Rcond) for the control case and for
the various minimum N as in Fig. 1. Recovery becomes progressively more
difficult with decreasing minimum N. The initial spike in surface
Rcond is related to the fact that during the first hour of
simulation, collision–coalescence and sedimentation are not simulated.
Figure 4 shows the mean cloud fraction fc (defined by a cloud
liquid water mixing ratio qc threshold of 0.01 g kg-1),
surface latent heat (LH) and sensible heat (SH) fluxes, inversion height
zi (based on the maximum gradient in liquid water potential
temperature θl), and cloud base/top height
(zb/zt; calculated based on a qc threshold of
0.01 g kg-1). (θl≈θ-qcLv/cpd; with Lv the latent heat of vaporization
and cpd the specific heat of dry air at constant pressure.) Cloud
fraction recovery is approximately symmetrical for high minimum N but
becomes increasingly more asymmetrical as the minimum N approaches
5 mg-1. Surface latent heat fluxes decrease, while sensible heat fluxes
increase during the open-cell period, consistent with the cooler and moister
surface outflows (see e.g., Kazil et al., 2014, for more detailed analysis of
the surface flux responses). The pre-open-cell rise in cloud base and top
height is suppressed during the raining period. Cloud bases for the different
N perturbations all tend to converge after full recovery of N, while
cloud tops for the stronger perturbations are up to ∼ 50 m lower. Note
that because the calculations of zt and zi apply
different criteria, the absolute values are somewhat different. The main
point, however, is to compare the response to different N(t).
Time series of (a) cloud fraction fc,
(b) surface latent and sensible heat fluxes (LH and SH,
respectively; LH > SH), (c) inversion height zi, and
(d) cloud top zt and cloud base zb for the
control case and the various minimum N as in Fig. 1. Note the suppression
of the deepening of the boundary layer associated with drizzle.
No aerosol perturbation
It is of interest to compare these perturbed simulations to one in which
there is no perturbation to N, i.e., N = 90 mg-1 for the entire
simulation. Figure 5 shows profiles of total liquid water, total water mixing
ratio qt, and θl for
the control case and a simulation without any N perturbation. In the
absence of a perturbation to N, the cloud does not produce substantial
drizzle; even though the boundary layer deepens steadily, it does not produce
enough liquid water to generate precipitation at N = 90 mg-1, and
it remains reasonably well mixed. In contrast, the control simulation with a
strong perturbation to N (5 mg-1) exhibits significant
drizzle-related reduction in cloud water and significant perturbation to the
well-mixed state (Fig. 5, left column). Notably, and in agreement with
earlier studies (e.g., Stevens et al., 2005), the cloud layer dries and thus
warms during the precipitating period while the surface cools and moistens.
For example, this can be deduced from the fact that θl is
steady while qc decreases, which means that θ must have
increased. By the end of the simulation, vertical mixing has increased;
θl is approximately constant with height. For qt,
vertical mixing also increases although a moister layer exists up to a depth
of 100 m. Overall, however, the morphological structure of the cloud field,
its flow structure (not shown) and the thermodynamic profiles at the end of
the simulation, are consistent with a closed cellular system.
Domain average profiles of (a) and (b) liquid
water mixing ratio qc, (c) and (d) total water
mixing ratio qt, and (e) and (f) liquid water
potential temperature θl. Left column: control case with
minimum N = 5 mg-1; right column: N = 90 mg-1
throughout the simulation. Drizzle results in a drying of the cloud layer and
a moistening of the surface (c). The drizzling period is
characterized by poor average vertical mixing. The drizzling system
eventually recovers to a well-mixed state, although surface moisture
persists.
Tests of robustness of LWP recovery. (a) Control case but
with fixed surface fluxes and no winds. Line types as in Fig. 1;
(b) RRTM (shortwave and long wave) and start time of 21:00 LT. Line
types as in Fig. 1. The arrow points to sunrise at 06:00 LT
(t = 30 h); (c) as in (b) but with start times
staggered by 1 h between 20:00 and 23:00 LT; (d) control case
(solid line); dry air aloft (dashed line), dry air aloft and divergence
increased to 5 × 10-6 s-1 (dotted line), and dry air
aloft but divergence decreased to 1 × 10-6 s-1
(dash-dotted line). The drier air aloft is calculated according to
qv=qv,0-3 g kg-1 [1-exp((795-z)/500)] with qv,0 = 3 g kg-1 rather than
qv,0 = 5 g kg-1 as in the
control case. (Terms in the exponent are in meters.) The precipitable water
at t = 0 is 10.0 mm compared to 11.7 mm for the standard profile. In
(c) and (d) the minimum N = 5 mg-1.
Sensitivity tests
A number of sensitivity tests and perturbations to the initial and boundary
conditions were performed to gauge robustness in the response to the N(t)
perturbation. These include (i) fixed surface fluxes
(SH = 15 W m-2 and LH = 93 W m-2); (ii) simulation of
both shortwave and long-wave radiation using RRTM; (iii) changing start times
in the diurnal cycle; and (iv) varying free tropospheric humidity and
large-scale subsidence. This is just a subset of the various tests that could
be performed. Figure 6 shows time series plots of LWP for these various
tests. In all cases the asymmetry in LWP(t) in response to N(t) is
clear. Of interest is that RRTM tends to generate stronger long-wave radiative
cooling and therefore even in the presence of shortwave radiation, LWP
recovery after the open-cell period is much more effective (cf. Figs. 3a and
6b; see further discussions in Sect. 4.2). Delays in the start time of the
simulation slow the LWP recovery (progressively weaker slopes with increasing
delay in Fig. 6c) because of shortwave absorption, but once N has returned
to 90 mg-1 the simulations converge. Other significant changes to the
simulations are in response to changes in subsidence and free tropospheric
humidity (Fig. 6d). A drier free troposphere (see figure caption for details)
reduces LWP during the first 4 h of simulation before the onset of drizzle.
This reduction in LWP is magnified in the case of stronger subsidence, but in
the case of weaker subsidence the loss in LWP is countered by the ability of
the boundary layer to generate a deeper cloud. As might be expected, recovery
to the closed-cell state is slowest in the case of a dry free troposphere in
combination with strong subsidence. Thus meteorological conditions that
influence cloudiness itself set the stage for the rate of recovery after the
drizzling period.
Analysis of RRTM simulation and minimum N = 5 mg-1
(solid line, Fig. 6b). Time series of (a) domain mean LWP and
(b) TKE averaged horizontally, over the boundary-layer depth (solid
line), and CAPE (dashed line); (c) a phase diagram of (a)
vs. (b). Colored arrows indicate stages of evolution of the system.
Vertical dashed lines are included to focus on temporal phase lags between
LWP and TKE. Red arrow: LWP falls rapidly while TKE continues to increase;
green arrow: both LWP and TKE decrease; black arrow: LWP begins to recover
while TKE still decreases; blue arrow: LWP and TKE increase in unison as the
closed-cell state recovers. For t>24 h, subtract 24 to get local time.
Relationship between recovery, turbulence kinetic energy, and convective available potential energy
Stratocumulus cloud water provides a source of long-wave radiative cooling,
which generates negative buoyancy and turbulence. Surface precipitation
removes liquid water from the cloud layer and deposits it to the surface.
Surface precipitation therefore reduces the amount of cooling associated with
the evaporation of cloud water in the cloud layer and warms the cloud layer.
Near-surface evaporation of precipitation cools the surface layer. Thus
surface precipitation serves to stabilize the boundary layer (e.g., Stevens
et al., 1998). We therefore expect rain processes to manifest in TKE and
convective available potential energy (CAPE). We analyze an illustrative case
that includes a diurnal cycle (start time 21:00 LT) and a cycle of N from
90 to 5 to 90 mg-1 (Fig. 6b solid line). TKE is the grid-resolved
component averaged over the boundary layer depth. Sunrise is at
approximately 06:00 LT, i.e., about the time of the beginning of N
recovery. A time series of LWP, TKE, and CAPE reveals that during the initial
closed-cell phase (prior to t = 24 h), LWP drives production of TKE
(Fig. 7a, b), which in turn drives higher LWP. The prescribed drop in N
results in precipitation and a loss of LWP. TKE also drops but with a delay
of approximately 1 h. This delay is associated with the surge in surface TKE
on transition to the open-cell state, associated with the surface outflows
(Fig. 8a, t ≈ 27 h). The surface TKE slowly wanes as the
surface rain rate and outflows weaken. (The peak transitions back to cloud
top upon recovery of the closed-cell state at t > 32 h.) TKE continues
to decrease during the open-cell drizzling phase and only begins to rebound
approximately 1 h after the introduction of N and the LWP recovery
(Fig. 7b). Later, LWP and TKE increase in unison and eventually peak
simultaneously at maximum cloud recovery. During the last 4 h of the
simulation, absorption of shortwave radiation results in a decrease in LWP.
There is a steady decrease in CAPE over the course of the simulation, which
is also indicative of the inability of the system to rebound.
The asymmetry of the closed–open–closed transition cycle is nicely
demonstrated as a plot in LWP-TKE phase space (Fig. 7c). During the delay in
TKE recovery upon reintroduction of N(t=30–31 h), turbulence does not
reinforce the LWP increase. Thus LWP recovery following the introduction of
N is hampered by the inability of the system to generate turbulence via
radiative cooling – itself a function of LWP.
Further analysis of the recovery shows that recovery is hampered by below
cloud buoyancy consumption of TKE (Fig. 8b) at t ≈ 32 h and a
height of ≈ 250 m (marked by a white minus sign on the figure).
Analysis of other cases shows that this is a robust feature during the
recovery stage, although it varies in magnitude and extent. Horizontal
x–y slices through this region reveal that the buoyancy consumption of
TKE is related to rising of cold air and sinking of warm air (e.g., Moeng,
1987). (Figure not shown.) After the disappearance of this region of buoyancy
consumption of TKE, the total water flux (vapor plus cloud water) into the
cloud increases significantly (Fig. 8c, t > 33 h).
Time–height cross sections associated with Fig. 7:
(a) resolved TKE (m2 s-2), (b) buoyancy production
of TKE (m2 s-3), and (c) qt flux (W m-2).
Note how in (a), the peak TKE transitions from the cloud layer to
the surface during closed-to-open cell transition (t ≈ 26 h) and
back to the cloud layer upon recovery. In (b), an area of negative
buoyancy production of TKE (consumption) is indicated by a white minus sign
(t ≈ 33 h). After the disappearance of this region of buoyancy
consumption of TKE, stronger qt flux into the cloud is evident
(panel c, t ≈ 33 h).
Influence of rate of N replenishment on recovery
Given the simplicity of the N representation, it is useful to consider
whether recovery is limited by the rate of recovery of N at the end of the
open-cell phase. Two variations on the control simulations (Figs. 1, 3) are
repeated. The first ramps N up from 5 to 90 mg-1 within 5 min (as
opposed to 2 h); the second ramps N up to 300 mg-1, also within
5 min (Fig. 9); both are highly unrealistic, considering the aerosol
replenishment rates via new particle formation, mechanical surface
production, and entrainment (Kazil et al., 2011). It is clear that even these
unrealistically high N recharge rates make little difference in terms of
the rate of increase in LWP and TKE. Small enhancements in recovery in
fc and deepening of the boundary layer are, however, evident. A
more realistic N recovery rate of t = 10 h further delays recovery.
Thus while the rate of replenishment of N is clearly an important
controlling factor for recovery, even immediate replenishment does not erase
the asymmetry in the LWP and TKE recovery.
Simulations testing the importance of N for recovery of the closed-cell state for control case set up with variations. Simulations prior to
t = 9 h are the same (slight differences are due to different machine
compilers and processors). After t = 9 h, N increases to
90 mg -1 within 2 h as in Fig. 1 (solid line, control); N recovers
to 90 mg-1 within 5 min (dashed line); N recovers to 300 mg-1
within 5 min (dotted line); and N recovers to 90 mg-1 within 10 h
(dash-dotted line).
Influence of meteorological forcing on recovery
Given the close relationship between LWP and TKE – albeit with delay – we
hypothesize that an appropriately placed influx of energy and water into the
system should help recovery. There are various ways that this can be explored
in modeling world; one straightforward way is
by increasing surface sensible and latent heat fluxes which generate
surface-driven buoyancy and moisture (e.g., Xue and Feingold, 2006). Another
is through stronger cloud top cooling, which is explored in Sect. 4.2. The
control simulation is repeated, but this time the interactively calculated
values of SH and LH are both increased by a factor of 2, coincident with and
following the beginning of the ramp up of N at 9 h. This is an ad hoc
simulation and is not meant to be tied to a specific scenario. As shown in
Fig. 10, recovery is significantly stronger. This simulation also exhibits a
layer of buoyancy consumption of TKE centered on ∼ 300 m during the
recovery stage (t > 10 h for this case), as in Fig. 8b. However it is
significantly weaker and diminished in size compared to the control case with
standard interactive fluxes (figures not shown). Moreover, the qt
flux into the cloud layer is stronger and starts earlier in the recovery
stage than for the control case. Thus the increased SH and LH help to reduce
the strength, extent, and duration of this layer of buoyancy consumption of
TKE, thereby accelerating recovery. An additional case in which only LH was
doubled and SH was kept the same produced similar results vis-à-vis
recovery in LWP during the open-to-closed cell transition, although there was
a proliferation of shallow cumulus with low cloud base during this stage
resulting from the lower lifting condensation level. Meteorological forcing
that generates thicker cloud appears to be important for increasing the rate
of LWP and TKE recovery and transition back to the open-cell state.
Simulations considering the influence of surface forcing on
recovery. Solid line: control simulation; dashed line: latent and sensible
heat fluxes are double their interactively calculated values after
t = 9 h, i.e., concurrent with the increase in N.
Discussion
The predator–prey model
We explore the ability of the predator–prey model to capture key responses of
the system to changes in N emanating from the CRM simulations. To do so, we
replace Eq. (2) with a time series much like that in Fig. 1 with high and low
N values of 90 cm -3 and 5, 15, 25, or 35 cm -3, respectively.
The transition times and the duration of the low N state are the same as in
Fig. 1 except the time axis is shifted by 4 h to allow the model to spin
up. (The low N state is reached at 9 h in these predator–prey calculations
rather than 5 h as in Fig. 1.) Thus, in keeping with the CRM simulations we
prescribe N(t), which essentially overrides the replenishment time τ2
and prescribes two “carrying capacity” values N0 = 90 cm -3 or
in the low N state, N0 = 5, 15, 25, or 35 cm -3. Other system
parameters are H0 = 650 m and microphysical delay T = 20 min.
The left column of Fig. 11 shows results for the meteorological forcing
timescale τ1 = 3 h (Eq. 1) associated with “recharge” of liquid
water (or cloud depth H). It is apparent that the larger the imposed
reduction in N, the larger is the decrease in H and the associated
increase in R at the onset of heavy rain, much as in Fig. 3b. Thereafter,
differences in R during the low H period are relatively small, again in
agreement with Fig. 3b. However, we do note that for the 90 to 5 cm-3
simulation, R behavior is anomalous because of the overshoot to very low
H (50 m) upon transition to very low N caused by the large loss term in
Eq. (1). The asymmetry in the H transitions is readily apparent, with
larger reductions in N exhibiting stronger asymmetry much as in the CRM.
Predator–prey analog to the cloud system (Eqs. 1 and 3). N(t) is
prescribed as in Fig. 1 but the timing is 4 h later to allow for
predator–prey model spin-up. Model parameters are H0 = 650 m,
T = 20 min. Left column: H and R solutions for H recovery time
τ1 = 3 h; Right column: H and R solutions for
τ1 = 6 h. Line types represent the various N time series (as in
Fig. 1, but with N in cm-3). In both cases the concurrence of rapid
increase in R during the rapid reduction in LWP is simulated as in Fig. 3a,
b. Smaller values of N perturbation exhibit faster H recovery after
reintroduction of N. Recovery is also faster in the case of smaller
τ1.
The meteorological timescale τ1 is now increased to 6 h (right column
of Fig. 11), representing a slower rise to H0, and is akin to a weaker
external meteorological forcing. Weaker forcing can also be achieved by
decreasing H0 itself, but this can generate values of R that are
unrealistic for stratocumulus and it is therefore more desirable to tune
τ1 for these exercises. For τ1 = 6 h, the initial build-up
in H prior to the N perturbation is slower and the rain rates are weaker.
The asymmetry in the transitions is even more pronounced. This is in broad
agreement with Fig. 10, where it was shown that stronger forcing (in the form
of higher surface fluxes) had a significant effect on recovery.
Note that because of the existence of delay terms in Eqs. (1) and (3), one
might a priori anticipate asymmetry in transitions for the imposed N(t).
For example, when transitioning from high H and high N (analogous to the
closed state) to low H and low N (analogous to the open state), the
source term for H is relatively small (because H is closer to H0) and
the loss term (H2(t-T)/N(t-T)) is large. This explains the very rapid
closed-to-open cell transition. When transitioning from low H and low N
to high H and high N, the source term is relatively large but the loss
term is also relatively large, particularly when the imposed N(t-T) is
small. This helps explain the slower recovery as well as the dependence of
the recovery time on the imposed minimum value of N.
Finally, as with the CRM results in Fig. 9, solution to the predator–prey
equations with instantaneous replenishment in N also fails to produce rapid
LWP recovery (figures not shown).
Influence of radiation
A comparison between results based on the simple radiation scheme (Stevens et
al., 2005) as opposed to RRTM shows a much stronger recovery in the RRTM
simulation (cf. Figs. 3a and 6b). As noted earlier, RRTM generates stronger
long-wave radiative cooling than the (untuned) simple calculation, which
serves to support the contention that the slow recovery is related to the
delay in TKE production by cloud radiative cooling.
To explore the influence of radiation further, we consider the representation
in RRTM of the effective free tropospheric air above the domain top. In the
RRTM simulations thus far (Fig. 6b, c), the upper tropospheric humidity is
maintained constant at the value at the domain top
(≈ 2 g kg-1). An additional simulation in which the effective
free tropospheric humidity is reduced to 0.01 g kg-1 is repeated. This
would indicate a more efficient cooling of the system, e.g., in the absence
of free tropospheric clouds. Note that this value only pertains to the
effective radiative layer above the model top and does not directly
affect the thermodynamics within the model domain. (Simulations with varying
modeled free tropospheric air are shown in Fig. 6d.) As shown in Fig. 12, the
more efficient cooling associated with this drier effective free troposphere
generates significantly stronger turbulence and a more rapid recovery to the
closed-cell state. Towards the end of the simulation this recovery of LWP is
modulated to some extent by the stronger entrainment associated with the
higher TKE – compounded by the solar absorption – so that LWP increases are
small.
Perspective of different parameters controlling recovery. Solid
line: control simulation with time shift such that N(t) time series
coincide; dashed line: RRTM simulation as in Fig. 6b; dotted line: same as
dashed line but with N recovering to 90 mg-1 over 10 h; dash-dotted
line: RRTM simulation as in Fig. 6b but with drier free troposphere imposed
above model domain top. Minimum N = 5 mg-1 in all cases. Slow
replenishment of N retards cloud recovery while stronger radiative forcing
enhances recovery.
For perspective, Fig. 12 also includes comparison with the control simulation
(simple long-wave radiation and standard N replenishment timescale of 2 h,
with the time axis shifted so that the perturbations to N coincide) and the
standard RRTM simulation but with a replenishment timescale of 10 h. We note
that the rate of recovery of N is clearly an important factor in recovery
of the LWP and the closed-cell state. Also of interest is that zi
is larger and rebounds more rapidly in the control case (simple long-wave
radiation) than in the RRTM-based simulations. Closer inspection shows that
for the same N perturbation, the control simulation generates less surface
precipitation than the RRTM simulation does (Fig. 13). The weaker
thermodynamic stabilization in the control simulation allows for a deepening
boundary layer. Nevertheless, the deeper boundary layer by itself is not able
to sustain a deeper cloud during the recovery because the weaker radiative
cooling limits the regeneration of condensate. The RRTM simulation is
characterized by significantly more positive vertical velocity skewness and
by
stronger qt flux. Thus while the boundary layer is on average
poorly mixed, the stronger updrafts supply moisture to the top of the
boundary layer, which helps to boost fc and LWP.
Domain average profiles of (a) and (b) rain water
content qr, (c) and (d) qt flux, and
(e) and (f) vertical velocity skewness. Left column:
control case with minimum N = 5 mg-1; right column: RRTM
simulation (dashed line in Fig. 12). Stronger drizzle in the RRTM simulation
generates stronger positive skewness and higher qt flux during the
open-cell period, which help maintain higher fc and LWP (Fig. 12).
Contour intervals: rain water content: 0.01, 0.02, 0.04, 0.06, 0.08, 0.12,
0.18 g kg-1 ; qt flux: 20 to 160 W m-2 in increments
of 20; skewness: -1.5 to 1.5 in increments of 0.25. Color scales are
identical for left and right columns.
The influence of absorption of solar radiation on cloud recovery (Figs. 6c
and 12) is clearly manifested in both the initial stages after introduction
of particles and towards the end of the simulations when LWP decreases
markedly. The timing of the reintroduction of particles to the system
relative to the diurnal cycle must therefore be considered to be a
fundamental aspect of recovery. This point has also been raised in other
modeling (Wang and Feingold, 2009b; Wang et al., 2011) and observational
studies (Burleyson and Yuter, 2015).
Investigation of importance of timing of N perturbation. Solid
line: control case; dashed line: control case but with N perturbation
delayed by 8 h during which time the boundary layer has deepened (time axis
shifted so N perturbation coincides). In both cases LWP and fc
recovery are similar.
Influence of boundary layer depth
The DYCOMS-II RF02 boundary layer has a tendency to deepen steadily over the
course of the simulation (Fig. 5). We now consider possible influence of the
boundary layer depth on open–closed cell recovery. To address this, we
simulate the system described in Fig. 3 but delay the application of the
perturbation in N(t) until 11 h, i.e., 8 h later than the standard
simulation when the boundary layer is deeper. In all other respects the
perturbation is the same. This result is shown as the dashed curve in
Fig. 14, shifted by t-8 h. Clearly, recovery to the closed-cell state is
very similar to that for the shallower boundary layer. However, this result
cannot be generalized since boundary layer depth is one of many factors
determining cloud amount. The delayed N perturbation simulation generates a
higher zi throughout the simulation commensurate with the higher
TKE. However, curiously LWP is lower prior to the precipitation and very
close to the control simulation thereafter, so that radiative cooling is
similar during the recovery stage. We have argued that recovery is closely
related to the ability of the system to regenerate cloud water and radiative
cooling. Why is recovery in the delayed perturbation case so similar to the
control case when the same LWP has to drive circulations over a deeper
boundary layer? Analysis shows that the delayed perturbation simulation
produces less surface rain R (both in rate and areal cover) and less
thermodynamic stabilization, thus allowing the system to recover more readily
(figures not shown). A more rigorous evaluation of recovery in deeper
boundary layers such as those observed in the Southeast Pacific (Wood et al.,
2011) is left to later study.
Mean vs. standard deviation of LWP phase diagrams
While the asymmetry in the closed–open–closed cell transitions shows up
clearly in the LWP and TKE time series, the system also displays asymmetry in
other temporal evolution aspects. Considering parameterization applications,
Yamaguchi and Feingold (2015) examined the domain mean LWP
(μ(LWP)) vs. domain standard deviation of LWP
(σ(LWP)) and showed that for the same case (and model)
described here, the closed- to open-cell system follows a fairly predictable
path from high μ(LWP) and low σ(LWP) in the closed-cell
state towards lower μ(LWP) and high σ(LWP) in the
open-cell state. Similar analysis
is repeated here for the closed–open–closed transition for a number of
different simulations with and without a diurnal solar cycle and with
variations in the subsidence and free tropospheric humidity. One illustrative
example associated with Fig. 7 is shown in Fig. 15 but all exhibit similar
features. First, the simulations all show similar phase paths as in Yamaguchi
and Feingold (2015) for the closed–open transition. Of note is that for a
given μ(LWP) the open–closed transition is characterized in all
cases by higher σ(LWP) than for the closed–open transition.
The higher σ(LWP) on the open–closed path is an expression of
the slow recovery; i.e., the low cloudiness (high variance) state attempting
to achieve a more cloudy (lower variance) state (see also Fig. 2).
Phase diagram for the relative dispersion of LWP(σ(LWP)/μ(LWP)) vs. the mean LWP (μ(LWP)) for the
simulation in Fig. 7. Colors indicate simulation time. Note that the recovery
from open- to closed-cell state is characterized by higher σ/μ for
given μ.
Summary
This work is motivated by the radiative impacts of the large difference in
the amount of solar radiation absorbed at the Earth's surface in open vs.
closed cellular convection and a desire to (i) understand the propensity of
cellular systems to transition back and forth between states and
(ii) elucidate key processes controlling the transitions. Satellite imagery
often shows ship track effluent closing open cells, and yet cloud-resolving
models that include different levels of complexity in the representation of
the aerosol life cycle produce more ambiguous results regarding the ability of
aerosol perturbations to fill in open cells (Wang and Feingold, 2009b; Wang
et al., 2011; Berner et al., 2013). Rather than include detailed
representation of aerosol processes as in Kazil et al. (2014), we have
elected to prescribe a simple symmetrical time series of the drop
concentration evolution N(t). Even this symmetrical N(t) does not produce
a symmetrical LWP(t), suggesting that some underlying system behavior
is responsible for the relatively slow recovery. The key results of this
study can be recapitulated as follows.
In stratocumulus clouds driven by cloud-top radiative cooling, changes in
LWP precede changes in TKE. Once in the open-cell state, the recovery of the
system depends on regeneration of LWP and attendant radiative cooling; thus
the lag in TKE build-up represents a barrier to recovery. Although injection
of aerosol into the system helps suppress precipitation and generate LWP,
until N is large enough, the increasing LWP also helps generate
precipitation, representing further barrier to recovery. Thus while a
recharge of N is a necessary condition for recovery from the open-cell
state, it cannot explain the basic asymmetry in the recovery.
The relatively slow open–closed transition is related to the stabilization
caused by the rain during the low N open-cell state and the relatively long
time it takes for a build-up in the TKE after the reintroduction of N. The
recovery is slower when long-wave cooling is countered by shortwave absorption
(Fig. 6c), for large imposed reductions in N (Figs. 3, 4, 6), and when the
rate of reintroduction of N is slow (Fig. 12) or the amount too small (Wang
et al., 2011). Cloud layers within the free troposphere would also reduce the
effectiveness of long-wave cooling and delay recovery (Fig. 12).
A region of sub-cloud buoyancy consumption of TKE during the recovery
from open-to-closed cells has been identified (Fig. 8b). Examination of a
sample of the simulations presented herein show that the extent, magnitude,
and persistence of this area is proportional to the amount of rain generated
during the open-cell phase. Recovery to the closed-cell state proceeds once
this barrier has been removed and surface moisture can be transported more
effectively to the cloud (Fig. 8c). For example, recovery is more rapid when
stronger surface latent and sensible heat fluxes are coincident with the
replenishment of N (Fig. 10). In this case, the region of buoyancy
consumption of TKE is significantly reduced and surface vapor can reach the
cloud layer more readily.
In the predator–prey model, asymmetry is a fundamental property of the
equations because they include delay terms. It is shown that the degree of
the asymmetry is controlled by the timescale for replenishment of H, i.e.,
τ1 (or alternatively H0; Eq. 1) after recovery from the open-cell
state. Simple tests with either small τ1 or large H0, i.e., strong
meteorological forcing accompanying the injection of aerosol, result in more
symmetric transitions.
These results shed light on why the transition from open to closed cellular
state can be significantly more difficult than the reverse and point to the
need to understand the meteorological, radiative, and surface flux
environment in which these transitions occur. Transitions from the open to
the closed cellular state are expected to be slower during the daytime, when
the free troposphere is cloudier (Fig. 12), and when aerosol
perturbation/replenishment is slow (Fig. 12). Aspects of this hypothesis can
be tested with satellite observations and reanalysis.