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Graham Feingold, Shalvn Tzivion (Tzitzvashvili), and Zev Leviv

Abstract

We present a solution to the stochastic collection/breakup equation (SCE/SBE) using our recently developed method of moments and the Low and List fragment distribution function. We prove that the collisional breakup equation conserves overall mass irrespective of the degree to which the fragment distribution function conserves mass for individual collisions. The method is compared with analytical solutions to the steady state collection/breakup equation, as well as with the breakup equation, for simple kernels. The proposed method produces better approximations to the analytical solutions for simple kernels, than those obtained using a single moment method. In writing the breakup terms, an approximation to the breakup kernel and fragment distribution function in discrete categories is used. This approach allows one to write the algorithm without prescribing the shape of the distribution function within the category. The approximation to the fragment distribution function in a discrete category is validated using existing single moment methods. In cases of breakup dominated evolution, one and two-moment solutions to the collecion/breakup equation are shown to differ little, though the solutions to collection dominated evolution are expected to differ appreciably. The method of moments thus represents a more universal, accurate approach to solving mass transfer equations.

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Graham Feingold, Zev Levin, and Shalva Tzivion (Tzitzvashvili)

Abstract

The evolution of raindrop spectra below cloud base in subsaturated atmospheres is traced with the aid of an axisymmetrical rainshaft model which includes the detailed warm microphysical treatment presented in parts I and II of this series. As input to the model, a stationary cloud provides rainfall with a predetermined drop spectrum. Mass loading and evaporative cooling generate downdrafts below cloud base. For near-adiabatic lapse rates and moderate mass loading, microbursts develop. For a given liquid water content, the magnitude of these downdrafts depends primarily on the lapse rate of temperature, but also on the drop spectrum injected at cloud base. For a given liquid water content, spectra comprising a relatively large number of small drops tend to generate significantly stronger downdrafts than spectra with a greater component of large drops. It is shown that drop collection and breakup may also affect the magnitude of the generated downdrafts significantly. When spectra comprising mainly small drops evolve to create larger drops, or when spectra comprising mainly large drops evolve to create smaller drops, neglect of collection and breakup can modify the downdrafts by up to about 50%. It is shown that in a steady state situation the drop spectra evolve toward bi- or trimodal spectra as predicted by simple rainshaft models with fixed dynamics.

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Paquita Zuidema, Huiwen Xue, and Graham Feingold

Abstract

The net shortwave radiative impact of aerosol on simulations of two shallow marine cloud cases is investigated using a Monte Carlo radiative transfer model. For a shallow cumulus case, increased aerosol concentrations are associated not only with smaller droplet sizes but also reduced cloud fractions and cloud dimensions, a result of evaporation-induced mixing and a lack of precipitation. Three-dimensional radiative transfer (3DRT) effects alter the fluxes by 10%–20% from values calculated using the independent column approximation for these simulations. The first (Twomey) aerosol indirect effect is dominant but the decreased cloud fraction reduces the magnitude of the shortwave cloud forcing substantially. The 3DRT effects slightly decrease the sensitivity of the cloud albedo to changes in droplet size under an overhead sun for the two ranges of cloud liquid water paths examined, but not strongly so. A popular two-stream radiative transfer approximation to the cloud susceptibility overestimates the more directly calculated values for the low liquid-water-path clouds within pristine aerosol conditions by a factor of 2 despite performing well otherwise, suggesting caution in its application to the cloud albedos within broken cloud fields. An evaluation of the influence of cloud susceptibility and cloud fraction changes to a “domain” area-weighted cloud susceptibility found that the domain cloud albedo is more likely to increase under aerosol loading at intermediate aerosol concentrations than under the most pristine conditions, contrary to traditional expectations.

The second simulation (cumulus penetrating into stratus) is characterized by higher cloud fractions and more precipitation. This case has two regimes: a clean, precipitating regime where cloud fraction increases with increasing aerosol, and a more polluted regime where cloud fraction decreases with increasing aerosol. For this case the domain-mean cloud albedo increases steadily with aerosol loading under clean conditions, but increases only slightly after the cloud coverage decreases. Three-dimensional radiative transfer effects are mostly negligible for this case. Both sets of simulations suggest that aerosol-induced cloud fraction changes must be considered in tandem with the Twomey effect for clouds of small dimensions when assessing the net radiative impact, because both effects are drop size dependent and radiatively significant.

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Shalva Tzivion (Tzitzvashvili), Graham Feingold, and Zev Levin

Abstract

A new, accurate, efficient method for solving the stochastic collection equation (SCE) is proposed. The SCE is converted to a set of moment equations in categories using a new analytical form of Bleck&'s approach. The equations are written in a form amenable to solution and to a category-by-category analysis of drop formation and removal. This method is unique in that closure of the equations is achieved using an expression relating high-order moments to any two lower order moments, thereby restricting the need for approximation of the category distribution function only to integrals over incomplete categories. Moments in categories are then expressed in terms of complete moments with the aid of linear or cubic polynomials. The method is checked for the case of the constant kernel and a linear polynomial kernel. Results show that excellent approximation to the analytical solutions for these kernels are obtained. This is achieved without the use of weighting functions and with modest computing time requirements. The method conserves two or more moments of the spectrum (as required) and successfully alleviates the artificial enhancement of the collection process which is a feature of many schemes.

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Hailong Wang, William C. Skamarock, and Graham Feingold

Abstract

In the Advanced Research Weather Research and Forecasting Model (ARW), versions 3.0 and earlier, advection of scalars was performed using the Runge–Kutta time-integration scheme with an option of using a positive-definite (PD) flux limiter. Large-eddy simulations of aerosol–cloud interactions using the ARW model are performed to evaluate the advection schemes. The basic Runge–Kutta scheme alone produces spurious oscillations and negative values in scalar mixing ratios because of numerical dispersion errors. The PD flux limiter assures positive definiteness but retains the oscillations with an amplification of local maxima by up to 20% in the tests. These numerical dispersion errors contaminate active scalars directly through the advection process and indirectly through physical and dynamical feedbacks, leading to a misrepresentation of cloud physical and dynamical processes. A monotonic flux limiter is introduced to correct the generally accurate but dispersive solutions given by high-order Runge–Kutta scheme. The monotonic limiter effectively minimizes the dispersion errors with little significant enhancement of numerical diffusion errors. The improvement in scalar advection using the monotonic limiter is discussed in the context of how the different advection schemes impact the quantification of aerosol–cloud interactions. The PD limiter results in 20% (10%) fewer cloud droplets and 22% (5%) smaller cloud albedo than the monotonic limiter under clean (polluted) conditions. Underprediction of cloud droplet number concentration by the PD limiter tends to trigger the early formation of precipitation in the clean case, leading to a potentially large impact on cloud albedo change.

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Fabian Hoffmann, Bernhard Mayer, and Graham Feingold

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Marine cloud brightening (MCB) is a geoengineering approach to counteract climate change by the deliberate seeding of sea salt aerosol particles that, once they activated to cloud droplets, directly increase cloud reflectance and hence global albedo. However, a large fraction of the seeded aerosol may remain interstitial, i.e., unactivated particles among cloud droplets. Because the consideration of interstitial aerosol optical properties usually requires computationally expensive simulations of the entire particle spectrum and direct Mie calculations, we develop a simple parameterization to be used with computationally efficient bulk and even bin cloud microphysical schemes that do not treat the unactivated aerosol explicitly. Using parcel and largeeddy simulations with highly detailed Lagrangian cloud microphysics and direct Mie calculations as a reference, we show that the parameterization captures the variability in the interstitial aerosol extinction successfully. By applying the parameterization to typical MCB cases, we find that the consideration of interstitial aerosol extinction is important for the assessment of MCB in shallow clouds with weak updrafts, in which only a small fraction of aerosol particles is activated to cloud droplets.

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Bjorn Stevens, Robert L. Walko, William R. Cotton, and Graham Feingold

Abstract

The production of anomalous supersaturations at cloud edges other than cloud base has presented a vexing challenge for modelers attempting to represent the evolution of a droplet spectrum across an Eulerian grid. Although the problem manifests itself most dramatically for models that explicitly predict on the supersaturation field, it is also present in models with bulk condensation schemes in which condensation happens implicitly. Although the problem has been discussed in the context of truncation errors associated with finite difference approximations to advection, this note demonstrates more generally that the cloud-edge supersaturation problem is a fundamental problem associated with the ubiquitous assumption that the forcings on the droplet spectra are well represented by the mean thermodynamic fields. In certain respects, this assumption is equivalent to failing to represent fractional cloudiness within a grid. Although well-known consequences of this problem are the underprediction of temperature and the erroneous representation of the mean buoyancy flux within a grid box, we also demonstrate that the spurious production of droplets can arise in response to the spurious production of supersaturations in models with detailed microphysical representations.

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Jerry Y. Harrington, Graham Feingold, and William R. Cotton

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The impact of solar heating and infrared cooling on the growth of a population of drops is studied with two numerical modeling frameworks. An eddy-resolving model (ERM) simulation of Arctic stratus clouds is used to generate a dataset of 500 parcel trajectories that follow the mean dynamic motions of the simulated cloud. The 500-parcel dataset is used to drive a trajectory ensemble model (TEM) coupled to an explicit microphysical model that includes the radiative term in the vapor growth equation. The second framework is that of the ERM itself.

Results from the TEM show that the production of drizzle-sized drops is strongly dependent upon parcel cloud-top residence time for both radiative- and nonradiative-influenced growth. Drizzle-sized drops can be produced between 20 and 50 min earlier through the inclusion of the radiative term, which corroborates earlier results. The radiative effect may also cause drops with r < 10 μm to evaporate, producing a bimodal size spectrum. Parcel cloud-top residence times as short as 12 min can initiate this bimodal spectrum. TEM results show that the radiative effect increases drizzle drop mass predominately in parcels that tend to contribute to drizzle even in the absence of the radiative term. Activation of large cloud condensation nuclei appears to have a larger effect on drizzle production than does the radiative term. ERM simulations show a weak overall influence of the radiative term. Drizzle onset occurs earlier when the radiative term is included (about 20 min), but there is no strong change in the overall structure or evolution of the cloud.

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Graham Feingold, W. R. Cotton, Bjorn Stevens, and A. S. Frisch

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This paper considers the production of drizzle in statocumulus clouds in relation to the boundary-layer turbulent kinetic energy and in-cloud residence times. It is shown that drizzle production in statocumulus of the order of 400 m in depth is intimately related to the vertical velocity structure of the cloud eddies. In a series of two dimensional numerical experiments with fixed cloud condensation nucleus concentrations, the effect on drizzle production of enhanced or diminished vertical velocities is simulated. Rather than do this by simulating clouds exhibiting more or less energy, we modify drop terminal velocities in a manner that conserves the fall velocity relative to the air motions and allows droplet growth to occur in a similar dynamical environment. The results suggest that more vigorous clouds produce more drizzle because they enable longer in-cloud dwell times and therefore prolonged collision-coalescence. In weaker clouds, droplets tend to fall out of the cloud before they have achieved significant size, resulting in smaller amounts of drizzle. In another series of experiments, we investigate the effects of the feedback of drizzle on the boundary-layer dynamics. Results show that when significant amounts of drizzle reach the surface, the subcloud layer is stabilized, circulations are weaker, and the boundary layer is not well mixed. When only small amounts of drizzle are produced, cooling tends to be confined to the region just below cloud base, resulting in destabilization, more vigorous circulations, and a better mixed boundary layer. The results strongly suggest that a characteristic time associated with collision-coalescence be incorporated into drizzle parameterizations.

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Zev Levin, Graham Feingold, Shalva Tzivion, and Albert Waldvogel

Abstract

A comparison is made between the evolution of raindrop spectra as measured at stations in the Swiss Alps separated by vertical distances of the order of 600 m, with that modeled in an axisymmetrical model including detailed microphysics. Results show that under steady rain, weak advective conditions, and rain rates greater than 2 mm h−1, the model satisfactorily reproduces the features of the observed drop spectrum. Results deteriorate for low rain rates (of the order of 1 mm h−1) since drop collisions are too few to modify the spectrum significantly. The general agreement between modeled and observed spectra suggests that further considerations of this kind are justified.

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