On the Reversibility of Transitions between Closed and Open Cellular Convection

. The two-way transition between closed and open cellular convection is addressed in an idealized cloud resolving modeling framework. A series of cloud resolving simulations shows that the transition between closed and open cellular states is asymmetrical, and characterized by a rapid (“runaway”) transition from the closed- to the open-cell state, but slower recovery to the closed-cell state. Given that precipitation initiates the closed-open cell transition, and that the recovery requires 5 a suppression of the precipitation, we apply an ad hoc time-varying drop concentration to initiate and suppress precipitation. We show that the asymmetry in the two-way transition occurs even for very rapid drop concentration replenishment. The primary barrier to recovery is the loss in turbulence kinetic energy (TKE) associated with the loss in cloud water (and associated radiative cooling), and the vertical stratiﬁcation of the boundary layer during the open-cell period. In transitioning from 10 the open to the closed state, the system faces the task of replenishing cloud water fast enough to counter precipitation losses, such that it can generate radiative cooling and TKE. It is hampered by a stable layer below cloud base that has to be overcome before water vapor can be transported more efﬁciently into the cloud layer. Recovery to the closed cell state is slower when radiative cooling is inefﬁcient such as in the presence of free tropospheric clouds, or after sunrise, when it is hampered 15 by the absorption of shortwave radiation. Tests suggest that recovery to the closed-cell state is

other important factors such as the amount and distribution of the aerosol perturbation (in the form of shiptracks). 60 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) where the aerosol lifecycle 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 to the essence of the system, at minimal computational cost. We use the System for Atmospheric Modeling (SAM) as described in Khairoutdinov and Randall (2003) with a 2nd order centered scheme for momentum advection and a monotonic 5th order scheme for scalar advection (Yamaguchi et al. 2011). SAM solves the anelastic Navier-Stokes 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). 95 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 Section 4.2. While in principle, the simple longwave 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 100 longwave 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 Section 3.1.4.
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: and, where c 1 is a temperature-dependent constant, and c 2 and α are constants based on theory. H 0 is the cloud depth that would be reached within a few timescales τ 1 in the absence of rain-related 105 losses. Thus H 0 represents "meteorological forcing", or in population dynamics nomenclature, the "carrying-capacity" of the system. Similarly, N 0 is the drop (or aerosol) concentration "carryingcapacity" that the system would reach in a few τ 2 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, 110 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 Equations (1 -3) is represented by five primary parameters, H 0 , N 0 , τ 1 , τ 2 and T , the use of a prescribed N (t) instead of Eq.
(2) reduces the free parameters 115 to H 0 , τ 1 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. specifically the recovery to N = 90 mg −1 , will be varied in a number of sensitivity tests.

Control Simulations
The control simulations use the GCSS specifications as described above, the simple longwave 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 135 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. Figure 3 shows time series of the domain mean cloud LWP, rain water path (RWP) and surface rain 140 rate R sf c , and the mean surface rain rate conditionally sampled for R sf c ≥ 0.1 mm d −1 (R cond ).
After the "spin up" of turbulence, by t = 3h 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 145 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 R sf c . 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 150 an immediate stop to rain, as evidenced by the long tail of low RWP and R sf c 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 ∝ LWP α N −β ; e.g., Pawlowska and Brenguier 2003, with α approximately 3 × larger than β.)

No Aerosol Perturbation
It is of interest to compare these perturbed simulations to one in which there is no perturbation 170 to N , i.e., N = 90 mg −1 for the entire simulation. Figure 5 shows profiles of total liquid water, total water mixing ratio q t , and liquid water potential temperature θ l 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.

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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 θ l is steady while q c decreases, which means that θ 180 must have increased. By the end of the simulation, vertical mixing has increased; θ l is approximately constant with height. For q t , 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.

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 longwave radiation using RRTM; (iii) changing start times in the diurnal cycle; and (iv) varying free tropospheric hu-190 midity and largescale subsidence. This is just a subset of the various tests that could be performed. tion 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.

Relationship between Recovery, Turbulence Kinetic Energy and Convective Available Potential Energy
Stratocumulus cloud water provides a source of longwave 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 210 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 Further analysis of the recovery shows that recovery is hampered by below cloud buoyancy consumption of TKE (  of N is clearly an important controlling factor for recovery, even immediate replenishment does not erase the asymmetry in the LWP and TKE recovery.

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 var-255 ious ways that this can be explored in modeling world but 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 section 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 260 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 Fig. 11 shows results for the meteorological forcing timescale τ 1 =3 h (Eq. 1) associated with "recharge" of liquid water (or cloud 285 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 290 in the H transitions is readily apparent, with larger reductions in N exhibiting stronger asymmetry, much as in the CRM.
The meteorological timescale τ 1 is now increased to 6 h (right column of Fig. 11), representing a slower rise to H 0 and is akin to a weaker external meteorological forcing. Weaker forcing can also be achieved by decreasing H 0 itself, but this can generate values of R that are unrealistic for 295 stratocumulus and it is therefore more desirable to tune τ 1 for these exercises. For τ 1 = 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 Eqns. (1) and (3) 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).

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A comparison between results based on the simple radiation scheme (Stevens 2005) as opposed to RRTM shows a much stronger recovery in the RRTM simulation (c.f. Fig. 3a and 6b). As noted earlier, RRTM generates stronger longwave 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.

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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 320 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 325 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.
For perspective, Fig. 12 also includes comparison with the control simulation (simple longwave 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 330 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 z i is larger and rebounds more rapidly in the Control case (simple longwave radiation) than in the RRTM-based simulations. Closer inspection shows that for the same N perturbation, the Control simulation generates less surface precipitation than does the RRTM simulation (Fig. 13). The weaker thermodynamic stabilization in the Control The influence of absorption of solar radiation on cloud recovery ( Fig. 6c and Fig. 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.

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This point has also been raised in other modeling (Wang and Feingold, 2009b;Wang et al. 2011) and observational studies (Burleyson and Yuter, 2014).

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 350 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 is one of many factors determining cloud amount. The delayed N perturbation simulation generates a higher z i 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 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 ( 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 longwave cooling and delay recovery (Fig. 12).

Sensitivity to Grid Spacing
The standard simulations are all performed on a relatively coarse grid spacing of ∆x × ∆y × ∆z = 200 m × 200 m × 10 m (aspect ratio of 20:1). Before embarking on the more extensive simulations presented here, system response was explored for finer grids and smaller aspect ratios: ∆x × ∆y × 435 ∆z = 100 m × 100 m × 10 m (aspect ratio of 10:1); and 75 m × 75 m × 10 m (aspect ratio of 7.5:1).
All simulations were performed on the same domain size (40 km). Figure  to perform all simulations at higher resolution and smaller aspect ratios, Fig. A1 suggests that the key aspects of the system response are robust to grid spacing.          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. 3ab. Smaller values of N perturbation exhibit faster H recovery after reintroduction of N . Recovery is also faster in the case of smaller τ1.    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. Figure A1. Sensitivity of results to grid spacing (meters).