• Abade, G. C., W. W. Grabowski, and H. Pawlowska, 2018: Broadening of cloud droplet spectra through eddy hopping: Turbulent entraining parcel simulations. J. Atmos. Sci., 75, 33653379, https://doi.org/10.1175/JAS-D-18-0078.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Ackerman, A. S., M. P. Kirkpatrick, D. E. Stevens, and O. B. Toon, 2004: The impact of humidity above stratiform clouds on indirect aerosol climate forcing. Nature, 432, 10141017, https://doi.org/10.1038/nature03174.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Ackerman, A. S., and Coauthors, 2009: Large-eddy simulations of a drizzling, stratocumulus-topped marine boundary layer. Mon. Wea. Rev., 137, 10831110, https://doi.org/10.1175/2008MWR2582.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Ahlm, L., A. Jones, C. W. Stjern, H. Muri, B. Kravitz, and J. E. Kristjánsson, 2017: Marine cloud brightening—As effective without clouds. Atmos. Chem. Phys., 17, 13 07113 087, https://doi.org/10.5194/acp-17-13071-2017.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Albrecht, B. A., 1989: Aerosols, cloud microphysics, and fractional cloudiness. Science, 245, 12271230, https://doi.org/10.1126/science.245.4923.1227.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Arrhenius, S., 1896: On the influence of carbonic acid in the air upon the temperature of the ground. Philos. Mag., 41, 237276, https://doi.org/10.1080/14786449608620846.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Betts, A. K., 2007: Coupling of water vapor convergence, clouds, precipitation, and land-surface processes. J. Geophys. Res., 112, D10108, https://doi.org/10.1029/2006JD008191.

    • Search Google Scholar
    • Export Citation
  • Bretherton, C., P. N. Blossey, and J. Uchida, 2007: Cloud droplet sedimentation, entrainment efficiency, and subtropical stratocumulus albedo. Geophys. Res. Lett., 34, L03813, https://doi.org/10.1029/2006GL027648.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Bronshtein, I. N., K. A. Semendyayev, G. Musiol, and H. Mühlig, 2007: Handbook of Mathematics. Springer Science & Business Media, 1159 pp.

  • Chandrakar, K. K., W. Cantrell, K. Chang, D. Ciochetto, D. Niedermeier, M. Ovchinnikov, R. A. Shaw, and F. Yang, 2016: Aerosol indirect effect from turbulence-induced broadening of cloud-droplet size distributions. Proc. Natl. Acad. Sci. USA, 113, 14 24314 248, https://doi.org/10.1073/pnas.1612686113.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Chen, J.-P., 1994: Theory of deliquescence and modified Köhler curves. J. Atmos. Sci., 51, 35053516, https://doi.org/10.1175/1520-0469(1994)051<3505:TODAMK>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Chen, Y.-C., M. W. Christensen, G. L. Stephens, and J. H. Seinfeld, 2014: Satellite-based estimate of global aerosol–cloud radiative forcing by marine warm clouds. Nat. Geosci., 7, 643646, https://doi.org/10.1038/ngeo2214.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Clark, T. L., 1973: Numerical modeling of the dynamics and microphysics of warm cumulus convection. J. Atmos. Sci., 30, 857878, https://doi.org/10.1175/1520-0469(1973)030<0857:NMOTDA>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Connolly, P. J., G. B. McFiggans, R. Wood, and A. Tsiamis, 2014: Factors determining the most efficient spray distribution for marine cloud brightening. Philos. Trans. Roy. Soc., A372, 20140056, https://doi.org/10.1098/rsta.2014.0056.

    • Search Google Scholar
    • Export Citation
  • Cooper, G., D. Johnston, J. Foster, L. Galbraith, A. Neukermans, R. Ormond, J. Rush, and Q. Wang, 2013: A review of some experimental spray methods for marine cloud brightening. Int. J. Geosci., 4, 7897, https://doi.org/10.4236/ijg.2013.41009.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Cooper, G., J. Foster, L. Galbraith, S. Jain, A. Neukermans, and B. Ormond, 2014: Preliminary results for salt aerosol production intended for marine cloud brightening, using effervescent spray atomization. Philos. Trans. Roy. Soc., A372, 20140055, https://doi.org/10.1098/rsta.2014.0055.

    • Search Google Scholar
    • Export Citation
  • Cui, Z., A. Gadian, A. Blyth, J. Crosier, and I. Crawford, 2014: Observations of the variation in aerosol and cloud microphysics along the 20°S transect on 13 November 2008 during VOCALS-REx. J. Atmos. Sci., 71, 29272943, https://doi.org/10.1175/JAS-D-13-0245.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Dziekan, P., and H. Pawlowska, 2017: Stochastic coalescence in Lagrangian cloud microphysics. Atmos. Chem. Phys., 17, 13 50913 520, https://doi.org/10.5194/acp-17-13509-2017.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Feingold, G., W. Cotton, B. Stevens, and A. Frisch, 1996: The relationship between drop in-cloud residence time and drizzle production in numerically simulated stratocumulus clouds. J. Atmos. Sci., 53, 11081122, https://doi.org/10.1175/1520-0469(1996)053<1108:TRBDIC>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Feingold, G., R. Boers, B. Stevens, and W. R. Cotton, 1997: A modeling study of the effect of drizzle on cloud optical depth and susceptibility. J. Geophys. Res., 102, 13 52713 534, https://doi.org/10.1029/97JD00963.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Feingold, G., W. R. Cotton, S. M. Kreidenweis, and J. T. Davis, 1999: The impact of giant cloud condensation nuclei on drizzle formation in stratocumulus: Implications for cloud radiative properties. J. Atmos. Sci., 56, 41004117, https://doi.org/10.1175/1520-0469(1999)056<4100:TIOGCC>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Feingold, G., I. Koren, T. Yamaguchi, and J. Kazil, 2015: On the reversibility of transitions between closed and open cellular convection. Atmos. Chem. Phys., 15, 73517367, https://doi.org/10.5194/acp-15-7351-2015.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Ghan, S. J., G. Guzman, and H. Abdul-Razzak, 1998: Competition between sea salt and sulfate particles as cloud condensation nuclei. J. Atmos. Sci., 55, 33403347, https://doi.org/10.1175/1520-0469(1998)055<3340:CBSSAS>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Gillespie, D. T., 1975: An exact method for numerically simulating the stochastic coalescence process in a cloud. J. Atmos. Sci., 32, 19771989, https://doi.org/10.1175/1520-0469(1975)032<1977:AEMFNS>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Glassmeier, F., F. Hoffmann, J. S. Johnson, T. Yamaguchi, K. S. Carslaw, and G. Feingold, 2021: Aerosol-cloud-climate cooling overestimated by ship-track data. Science, 371, 485489, https://doi.org/10.1126/science.abd3980.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Glenn, I. B., G. Feingold, J. J. Gristey, and T. Yamaguchi, 2020: Quantification of the radiative effect of aerosol–cloud interactions in shallow continental cumulus clouds. J. Atmos. Sci., 77, 29052920, https://doi.org/10.1175/JAS-D-19-0269.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Gryspeerdt, E., and Coauthors, 2019: Constraining the aerosol influence on cloud liquid water path. Atmos. Chem. Phys., 19, 53315347, https://doi.org/10.5194/acp-19-5331-2019.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Hartmann, D., and Coauthors, 2013: Observations: Atmosphere and surface. Climate Change 2013: The Physical Science Basis, T. F. Stocker et al., Eds., Cambridge University Press, 159–254.

  • Hoffmann, F., 2017: On the limits of Köhler activation theory: How do collision and coalescence affect the activation of aerosols? Atmos. Chem. Phys., 17, 83438356, https://doi.org/10.5194/acp-17-8343-2017.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Hoffmann, F., 2020: Effects of entrainment and mixing on the Wegener–Bergeron–Findeisen process. J. Atmos. Sci., 77, 22792296, https://doi.org/10.1175/JAS-D-19-0289.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Hoffmann, F., and G. Feingold, 2019: Entrainment and mixing in stratocumulus: Effects of a new explicit subgrid-scale scheme for large-eddy simulations with particle-based microphysics. J. Atmos. Sci., 76, 19551973, https://doi.org/10.1175/JAS-D-18-0318.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Hoffmann, F., S. Raasch, and Y. Noh, 2015: Entrainment of aerosols and their activation in a shallow cumulus cloud studied with a coupled LCM-LES approach. Atmos. Res., 156, 4357, https://doi.org/10.1016/j.atmosres.2014.12.008.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Hoffmann, F., Y. Noh, and S. Raasch, 2017: The route to raindrop formation in a shallow cumulus cloud simulated by a Lagrangian cloud model. J. Atmos. Sci., 74, 21252142, https://doi.org/10.1175/JAS-D-16-0220.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Hoffmann, F., T. Yamaguchi, and G. Feingold, 2019: Inhomogeneous mixing in Lagrangian cloud models: Effects on the production of precipitation embryos. J. Atmos. Sci., 76, 113133, https://doi.org/10.1175/JAS-D-18-0087.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Hoffmann, F., F. Glassmeier, T. Yamaguchi, and G. Feingold, 2020: Liquid water path steady states in stratocumulus: Insights from process-level emulation and mixed-layer theory. J. Atmos. Sci., 77, 22032215, https://doi.org/10.1175/JAS-D-19-0241.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Jenkins, A. K. L., P. M. Forster, and L. S. Jackson, 2013: The effects of timing and rate of marine cloud brightening aerosol injection on albedo changes during the diurnal cycle of marine stratocumulus clouds. Atmos. Chem. Phys., 13, 16591673, https://doi.org/10.5194/acp-13-1659-2013.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Jensen, J. B., and A. D. Nugent, 2017: Condensational growth of drops formed on giant sea-salt aerosol particles. J. Atmos. Sci., 74, 679697, https://doi.org/10.1175/JAS-D-15-0370.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Kazil, J., T. Yamaguchi, and G. Feingold, 2017: Mesoscale organization, entrainment, and the properties of a closed-cell stratocumulus cloud. J. Adv. Model. Earth Syst., 9, 22142229, https://doi.org/10.1002/2017MS001072.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Kerstein, A. R., 1988: A linear-eddy model of turbulent scalar transport and mixing. Combust. Sci. Technol., 60, 391421, https://doi.org/10.1080/00102208808923995.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Khairoutdinov, M. F., and Y. Kogan, 2000: A new cloud physics parameterization in a large-eddy simulation model of marine stratocumulus. Mon. Wea. Rev., 128, 229243, https://doi.org/10.1175/1520-0493(2000)128<0229:ANCPPI>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Khairoutdinov, M. F., and D. A. Randall, 2003: Cloud resolving modeling of the ARM summer 1997 IOP: Model formulation, results, uncertainties, and sensitivities. J. Atmos. Sci., 60, 607625, https://doi.org/10.1175/1520-0469(2003)060<0607:CRMOTA>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Khvorostyanov, V. I., and J. A. Curry, 1999: A simple analytical model of aerosol properties with account for hygroscopic growth: 1. Equilibrium size spectra and cloud condensation nuclei activity spectra. J. Geophys. Res., 104, 21752184, https://doi.org/10.1029/98JD02673.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Köhler, H., 1936: The nucleus in and the growth of hygroscopic droplets. Trans. Faraday Soc., 32, 11521161, https://doi.org/10.1039/TF9363201152.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Lamb, D., and J. Verlinde, 2011: Physics and Chemistry of Clouds. Cambridge University Press, 584 pp.

  • Lasher-Trapp, S. G., W. A. Cooper, and A. M. Blyth, 2005: Broadening of droplet size distributions from entrainment and mixing in a cumulus cloud. Quart. J. Roy. Meteor. Soc., 131, 195220, https://doi.org/10.1256/qj.03.199.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Latham, J., and M. Smith, 1990: Effect on global warming of wind-dependent aerosol generation at the ocean surface. Nature, 347, 372373, https://doi.org/10.1038/347372a0.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Latham, J., and Coauthors, 2012: Marine cloud brightening. Philos. Trans. Roy. Soc., A370, 42174262, https://doi.org/10.1098/rsta.2012.0086.

  • Lewis, E., and S. E. Schwartz, 2004: Sea Salt Aerosol Production: Mechanisms, Methods, Measurements, and Models. Geophys. Monogr., Vol. 152, Amer. Geophys. Union, 413 pp.

    • Crossref
    • Export Citation
  • Maahn, M., F. Hoffmann, M. D. Shupe, G. Boer, S. Y. Matrosov, and E. P. Luke, 2019: Can liquid cloud microphysical processes be used for vertically pointing cloud radar calibration? Atmos. Meas. Tech., 12, 31513171, https://doi.org/10.5194/amt-12-3151-2019.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Mitchell, D. L., 2000: Parameterization of the Mie extinction and absorption coefficients for water clouds. J. Atmos. Sci., 57, 13111326, https://doi.org/10.1175/1520-0469(2000)057<1311:POTMEA>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Mordy, W., 1959: Computations of the growth by condensation of a population of cloud droplets. Tellus, 11, 1644, https://doi.org/10.1111/j.2153-3490.1959.tb00003.x.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Pinsky, M., and A. Khain, 2002: Effects of in-cloud nucleation and turbulence on droplet spectrum formation in cumulus clouds. Quart. J. Roy. Meteor. Soc., 128, 501533, https://doi.org/10.1256/003590002321042072.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Platnick, S., and S. Twomey, 1994: Determining the susceptibility of cloud albedo to changes in droplet concentration with the advanced very high resolution radiometer. J. Appl. Meteor. Climatol., 33, 334347, https://doi.org/10.1175/1520-0450(1994)033<0334:DTSOCA>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Pruppacher, H. R., and J. D. Klett, 1997: Microphysics of Clouds and Precipitation. 2nd ed. Kluwer Academic Publishers, 954 pp.

  • Qian, L., and J. Lin, 2011: Modeling on effervescent atomization: A review. Sci. China Phys. Mech. Astron., 54, 21092129, https://doi.org/10.1007/s11433-011-4536-1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Schwenkel, J., F. Hoffmann, and S. Raasch, 2018: Improving collisional growth in Lagrangian cloud models: Development and verification of a new splitting algorithm. Geosci. Model Dev., 11, 39293944, https://doi.org/10.5194/gmd-11-3929-2018.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Sedunov, Y. S., 1974: Physics of Drop Formation in the Atmosphere. John Wiley and Sons, 234 pp.

  • Seinfeld, J. H., and S. N. Pandis, 2016: Atmospheric Chemistry and Physics: From Air Pollution to Climate Change. John Wiley and Sons, 1120 pp.

  • Shima, S.-I., K. Kusano, A. Kawano, T. Sugiyama, and S. Kawahara, 2009: The super-droplet method for the numerical simulation of clouds and precipitation: A particle-based and probabilistic microphysics model coupled with a non-hydrostatic model. Quart. J. Roy. Meteor. Soc., 135, 13071320, https://doi.org/10.1002/qj.441.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Stevens, B., and G. Feingold, 2009: Untangling aerosol effects on clouds and precipitation in a buffered system. Nature, 461, 607613, https://doi.org/10.1038/nature08281.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Twomey, S., 1959: The nuclei of natural cloud formation. Part II: The supersaturation in natural clouds and the variation of cloud droplet concentration. Pure Appl. Geophys., 43, 243249, https://doi.org/10.1007/BF01993560.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Twomey, S., 1974: Pollution and the planetary albedo. Atmos. Environ., 8, 12511256, https://doi.org/10.1016/0004-6981(74)90004-3.

  • Twomey, S., 1977: The influence of pollution on the shortwave albedo of clouds. J. Atmos. Sci., 34, 11491152, https://doi.org/10.1175/1520-0469(1977)034<1149:TIOPOT>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Unterstrasser, S., F. Hoffmann, and M. Lerch, 2017: Collection/aggregation algorithms in Lagrangian cloud microphysical models: Rigorous evaluation in box model simulations. Geosci. Model Dev., 10, 15211548, https://doi.org/10.5194/gmd-10-1521-2017.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Unterstrasser, S., F. Hoffmann, and M. Lerch, 2020: Collisional growth in a particle-based cloud microphysical model: Insights from column model simulations using LCM1D (v1. 0). Geosci. Model Dev., 13, 51195145, https://doi.org/10.5194/gmd-13-5119-2020.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Vaughan, N. E., and T. M. Lenton, 2011: A review of climate geoengineering proposals. Climatic Change, 109, 745790, https://doi.org/10.1007/s10584-011-0027-7.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Victor, D. G., D. Zhou, E. H. M. Ahmed, P. K. Dadhich, J. G. J. Olivier, H.-H. Rogner, K. Sheikho, and M. Yamaguchi, 2014: Introductory chapter. Climate Change 2014: Mitigation of Climate Change, O. Edenhofer et al., Ed., Cambridge University Press, 111–150, https://www.ipcc.ch/site/assets/uploads/2018/02/ipcc_wg3_ar5_chapter1.pdf.

  • Wang, H., P. J. Rasch, and G. Feingold, 2011: Manipulating marine stratocumulus cloud amount and albedo: A process-modelling study of aerosol-cloud-precipitation interactions in response to injection of cloud condensation nuclei. Atmos. Chem. Phys., 11, 42374249, https://doi.org/10.5194/acp-11-4237-2011.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Wang, S., Q. Wang, and G. Feingold, 2003: Turbulence, condensation, and liquid water transport in numerically simulated nonprecipitating stratocumulus clouds. J. Atmos. Sci., 60, 262278, https://doi.org/10.1175/1520-0469(2003)060<0262:TCALWT>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Warner, J., 1973: The microstructure of cumulus cloud: Part IV. The effect on the droplet spectrum of mixing between cloud and environment. J. Atmos. Sci., 30, 256261, https://doi.org/10.1175/1520-0469(1973)030<0256:TMOCCP>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Wood, R., 2012: Stratocumulus clouds. Mon. Wea. Rev., 140, 23732423, https://doi.org/10.1175/MWR-D-11-00121.1.

  • Wood, R., 2021: Assessing the potential efficacy of marine cloud brightening for cooling Earth using a simple heuristic model. Atmos. Chem. Phys., https://doi.org/10.5194/acp-2021-327, in press.

    • Search Google Scholar
    • Export Citation
  • View in gallery
    Fig. 1.

    (a) The (dry) aerosol radius as a function of the spray droplet radius and (b) the equilibrium radius as a function of the spray droplet radius for a supersaturation ratio of S = −20% and a temperature of T = 10°C. Both quantities are plotted for different salinities (colored lines). The black 1:1 line has been added to guide the reader’s eye.

  • View in gallery
    Fig. 2.

    The aerosol concentration as a function of the geometric mean aerosol radius and the geometric standard deviation for a spray liquid water mixing ratio of 0.01 g kg−1 and a salinity of 3.5%. The black dots show the seeded cases analyzed in this study. The black line indicates the concentration of the background aerosol.

  • View in gallery
    Fig. 3.

    The initial aerosol size distributions analyzed for this study. The following color code will be used throughout the study to indicate the different aerosol cases: Green lines indicate geometric mean radii of 0.1 μm, orange lines 0.3 μm, and red lines 0.9 μm. Continuous lines indicate a geometric standard deviation of 1.5 μm, long-dashed lines 2.0 μm, and short-dashed lines 2.5 μm. The black line marks the background aerosol.

  • View in gallery
    Fig. 4.

    The droplet size distribution in the nonseeded case as a function of the droplet radius and the distance from cloud base in the parcel simulation. The black line indicates pure diffusional growth. The blue letters A to E mark certain microphysical processes further detailed in the main text. The line segment in the lower-right corner indicates the color used to depict this nonseeded case in subsequent plots.

  • View in gallery
    Fig. 5.

    Continuation of Fig. 4. Droplet size distributions for all analyzed seeded cases as a function of the droplet radius and the distance from cloud base in the parcel simulations. The black line indicates pure diffusional growth in the nonseeded case (cf. Fig. 4). The line segment in the lower-right corner indicates the color used to depict this case in other plots of this study (e.g., Fig. 3).

  • View in gallery
    Fig. 6.

    Vertical profiles of (a) supersaturation ratio, (b) LWP, (c) RWP, (d) cloud-layer-averaged droplet number concentration, (e) cloud optical thickness, (f) cloud albedo, (g) rCRE susceptibility to aerosol seeding caused by microphysical processes, and (h) total rCRE susceptibility to aerosol seeding as a function of the distance from cloud base for all seeded and the nonseeded case of the parcel model simulations. The gray line indicates the average cloud-top height of the LES results presented in Fig. 7.

  • View in gallery
    Fig. 7.

    Time series of (a) LWP, (b) RWP, (c) cloud-layer-averaged droplet concentration, (d) cloud fraction, (e) entrainment velocity, (f) cloud optical thickness, (g) cloud albedo, and (h) total rCRE susceptibility to aerosol seeding for all seeded and the nonseeded case for the LES model.

  • View in gallery
    Fig. A1.

    The equilibrium radius of a wetted aerosol as a function of the (dry) aerosol radius determined using the new analytical solution (A3) (short-dashed black line), a numerical solver (green line), and the approximate solutions by Khvorostyanov and Curry (1999) and Chen (1994) (blue and red lines, respectively). The four panels show the equilibrium radius for different supersaturation ratios.

  • View in gallery
    Fig. B1.

    The seeded aerosol mass mixing qaero in a developing plume as a function of the distance from the sprayer for different seeded aerosol lifetimes (τaero, colored lines). The black dashed line shows the average value of all qaero curves, determined over 1000 km, which is the typical length of a plume.

All Time Past Year Past 30 Days
Abstract Views 940 574 0
Full Text Views 478 371 52
PDF Downloads 533 383 44

Cloud Microphysical Implications for Marine Cloud Brightening: The Importance of the Seeded Particle Size Distribution

Fabian HoffmannaLudwig-Maximilans-Universität München, Meteorologisches Institut, Munich, Germany
bCooperative Institute for Research in Environmental Sciences, University of Colorado, Boulder, Colorado
cChemical Sciences Laboratory, NOAA/Earth System Research Laboratories, Boulder, Colorado

Search for other papers by Fabian Hoffmann in
Current site
Google Scholar
PubMed
Close
https://orcid.org/0000-0001-5136-0653
and
Graham FeingoldcChemical Sciences Laboratory, NOAA/Earth System Research Laboratories, Boulder, Colorado

Search for other papers by Graham Feingold in
Current site
Google Scholar
PubMed
Close
Full access

Abstract

Marine cloud brightening (MCB) has been proposed as a viable way to counteract global warming by artificially increasing the albedo and lifetime of clouds via deliberate seeding of aerosol particles. Stratocumulus decks, which cover wide swaths of Earth’s surface, are considered the primary target for this geoengineering approach. The macroscale properties of this cloud type exhibit a high sensitivity to cloud microphysics, exposing the potential for undesired changes in cloud optical properties in response to MCB. In this study, we apply a highly detailed Lagrangian cloud model, coupled to an idealized parcel model as well as a full three-dimensional large-eddy simulation model, to show that the choice of seeded particle size distribution is crucial to the success of MCB, and that its efficacy can be significantly reduced by undesirable microphysical processes. The presence of even a small number of large particles in the seeded size spectrum may trigger significant precipitation, which will reduce cloud water and may even break up the cloud deck, reducing the scene albedo and hence counteracting MCB. On the other hand, a seeded spectrum comprising a large number of small particles reduces the fraction of activated cloud droplets and increases entrainment and evaporation of cloud water, which also reduces the efficiency of MCB. In between, there may exist an aerosol size distribution that minimizes undesirable microphysical processes and enables optimal MCB. This optimal size distribution is expected to be case dependent.

© 2021 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Fabian Hoffmann, fa.hoffmann@lmu.de

Abstract

Marine cloud brightening (MCB) has been proposed as a viable way to counteract global warming by artificially increasing the albedo and lifetime of clouds via deliberate seeding of aerosol particles. Stratocumulus decks, which cover wide swaths of Earth’s surface, are considered the primary target for this geoengineering approach. The macroscale properties of this cloud type exhibit a high sensitivity to cloud microphysics, exposing the potential for undesired changes in cloud optical properties in response to MCB. In this study, we apply a highly detailed Lagrangian cloud model, coupled to an idealized parcel model as well as a full three-dimensional large-eddy simulation model, to show that the choice of seeded particle size distribution is crucial to the success of MCB, and that its efficacy can be significantly reduced by undesirable microphysical processes. The presence of even a small number of large particles in the seeded size spectrum may trigger significant precipitation, which will reduce cloud water and may even break up the cloud deck, reducing the scene albedo and hence counteracting MCB. On the other hand, a seeded spectrum comprising a large number of small particles reduces the fraction of activated cloud droplets and increases entrainment and evaporation of cloud water, which also reduces the efficiency of MCB. In between, there may exist an aerosol size distribution that minimizes undesirable microphysical processes and enables optimal MCB. This optimal size distribution is expected to be case dependent.

© 2021 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Fabian Hoffmann, fa.hoffmann@lmu.de

1. Introduction

Earth’s climate is warming as a result of anthropogenic emissions of greenhouse gases (Arrhenius 1896; Hartmann et al. 2013). To avoid the perilous impacts on human societies and the environment, significant reductions in these emissions are required (e.g., Victor et al. 2014). While this approach will likely require decades to be effective, geoengineering might offer more immediate means by either removing carbon dioxide from the atmosphere or by enhancing the amount of radiant energy reflected by the planet (e.g., Vaughan and Lenton 2011).

Marine cloud brightening (MCB) is a geoengineering approach to increasing the reflection of incident shortwave radiation by enhancing the albedo and lifetime of clouds, via the seeding of additional particles into these clouds (Latham and Smith 1990; Latham et al. 2012). Marine stratocumulus are the primary target for such an approach since they cover consistently wide swaths of the subtropical oceans, providing ample potential for MCB (e.g., Wood 2012). Macroscopically, MCB can be understood as an attempt to increase the relative cloud radiative effect (rCRE), that is, the fraction of shortwave radiation reflected by clouds (Betts 2007). The rCRE can be approximated as
rCREfcAc,
where fc is the fraction of surface covered by clouds, referred to as the cloud fraction, and the cloud albedo Ac, which can be approximated as
Ac=τγ+τ,
where γ = 13.3 depends on the degree of forward scattering for overhead sun (e.g., Glenn et al. 2020). Accordingly, Ac is primarily determined by the cloud optical thickness
τ=0[0Qe(r,λ)πr2n(r,z)dr]dz,
that is, the vertically (z) integrated product of the droplet size distribution n(r, z) and the attenuation cross section Qe(r, λ)πr2, where Qe is the extinction efficiency, which depends on the wavelength λ of the radiation and the droplet radius r. For a given cloudy boundary layer, this definition of τ indicates that the pathway for MCB to change macroscale cloud properties is paved with cloud microphysical processes that shape the droplet size distribution.
To better understand the microphysical processes shaping τ, it is commonly approximated as
τϕN1/3LWP5/6,
where ϕ is a parameter that depends on temperature and (weakly) on pressure, N is the average concentration of cloud and rain droplets, and LWP is the liquid water path, that is, the vertically integrated liquid water content. Using this definition of τ, and differentiating the rCRE with respect to the aerosol concentration, Naero, yields
dln(rCRE)dln(Naero)=13dln(N)dln(Naero)+56dln(LWP)dln(Naero)+dln(fc)dln(Naero)dln(1+τ/γ)dln(Naero),=dln(rCRE)dln(Naero)|processesdln(rCRE)dln(Naero)|saturation,
which is the susceptibility of the rCRE to aerosol seeding, that is, a measure of the efficiency of MCB (cf. Platnick and Twomey 1994). While the first three terms on the right-hand side represent potential microphysical impacts on MCB, which are of primary importance for this study, the last term represents the saturation of Ac for τγ, which indicates that additional brightening of already very reflective clouds is ineffective. Note that this term is usually negligible for shallow clouds like the marine stratocumulus targeted in MCB, for which Ac < 0.5 (e.g., Wood 2012). [Formally, this is equivalent to the approximation Acτ/γ, due to which the dln(rCRE)/dln(Naero)|saturation term vanishes].

The first term on the right-hand side of (5) represents the change in τ resulting from the addition of new cloud droplets due to seeding, typically causing a stronger reflection of shortwave radiation as desired for MCB (Twomey 1974, 1977). Although dln(N)/dln(Naero) is usually >0, the increase in N is not necessarily proportional to Naero since their activation is buffered by the more efficient condensation in aerosol-laden clouds (Twomey 1959; Stevens and Feingold 2009). The second and third term on the right-hand side of (5) represent changes in LWP and fc. Albrecht (1989) argued that by reducing the average droplet size, a higher N prevents collisional growth and hence precipitation, which maintains a larger LWP, cloud lifetime, and hence fc. The opposite effect has been found for even larger increases in N, that is, a reduction in LWP due to stronger entrainment caused by accelerated evaporation of droplets (Wang et al. 2003; Ackerman et al. 2004; Bretherton et al. 2007). Accordingly, not only the magnitude but also the sign of dln(LWP)/dln(Naero) may vary depending on the MCB seeding scenario (Glassmeier et al. 2021). However, it is not only the number of seeded aerosol particles that matters. Large particles in the seeded aerosol size spectrum may act as precipitation embryos that initiate or enhance collisional growth, and hence reduce LWP and fc accordingly (Feingold et al. 1999; Jensen and Nugent 2017). In fact, the aforementioned process might also result in a reduction of N due to the coalescence of cloud droplets as well as precipitation scavenging (Feingold et al. 1997), decreasing the efficiency of MCB even further.

This short overview of extant studies already indicates that MCB may result in unwanted microphysical processes vis-à-vis MCB depending on the seeded aerosol size spectrum. Connolly et al. (2014) considered the most efficient aerosol size distribution for seeding based on the energy required to spray seawater into the atmosphere, where it breaks up into droplets that subsequently evaporate and act as cloud condensation nuclei (e.g., Cooper et al. 2013, 2014). Connolly et al. (2014), however, based their calculations on a simple parcel model, which is not able to represent the entire range of microphysical interactions outlined above. Accordingly, for this study, we use a highly detailed Lagrangian cloud model (LCM) that covers the entire range of relevant hydrometeors and associated microphysical processes in the required detail (e.g., Shima et al. 2009; Hoffmann et al. 2015), coupled with full three-dimensional large-eddy simulations (LESs) necessary to represent processes that cannot be included in a parcel model, for example, evaporation–entrainment feedbacks. Nonetheless, idealized parcel model simulations will complement the study to gain a deeper microphysical understanding of MCB.

The paper is organized as follows: In section 2, we will investigate how the seeded aerosol spectrum is formed, that is, how it relates to the spectrum of the spray droplets. Then, we will briefly summarize the modeling approach, before results on the interactions of MCB and cloud microphysics are presented using both the parcel model and LES in section 3. The study is summarized in section 4. Concluding remarks are given in section 5. In appendix A, an analytical expression for the equilibrium radius of wetted aerosol particles in a subsaturated environment is presented, continuing the discussions of section 2. In appendix B, the temporal and spatial variation of seeded aerosol mass in an MCB spraying plume is addressed.

2. The seeded aerosol spectrum

To understand how the seeded aerosol affects clouds during MCB, it is first necessary to illuminate how the seeded aerosol spectrum is formed. For this, we need to understand the relationship between the seawater spray droplets and the dissolved aerosol. The salinity of seawater is defined as
fsalinity=msaltmseawater,
where msalt is the mass of sea salt dissolved per mass unit of seawater, mseawater. Accordingly, the radius of a dry aerosol particle, raero, is related to the radius of its corresponding spray droplet, rspray, via
raero=rspray(fsalinityρsprayρaero)1/3,
assuming a spherical shape for both particles. ρspray and ρaero are the mass densities of the spray droplet and the dry aerosol, respectively. For dilute mixtures, as is typical for seawater,
ρsprayρwater(1+fsalinity)
yields a good approximation, using the density of pure water ρwater. Figure 1a presents raero as a function of rspray. As expected, raero is smaller than rspray, and increases in proportion to fsalinity.
Fig. 1.
Fig. 1.

(a) The (dry) aerosol radius as a function of the spray droplet radius and (b) the equilibrium radius as a function of the spray droplet radius for a supersaturation ratio of S = −20% and a temperature of T = 10°C. Both quantities are plotted for different salinities (colored lines). The black 1:1 line has been added to guide the reader’s eye.

Citation: Journal of the Atmospheric Sciences 78, 10; 10.1175/JAS-D-21-0077.1

Based on Köhler (1936), an equilibrium radius, requi, can be assigned to every particle situated in a subsaturated environment. This is the size a wetted aerosol will attain to balance the effects of curvature and dissolved aerosol mass on its saturation pressure. Using the analytical relationship for requi, (A3), derived in appendix A, Fig. 1b shows that rspray is typically larger than requi. Only for unusually high salinities or in almost saturated conditions (not shown), can rspray be smaller than requi. Thus, a typical spray droplet needs to evaporate to attain its equilibrium size. This might have important implications since it is well known that large, dry particles with a commensurately large equilibrium radius—commonly referred to as giant cloud condensation nuclei (GCCN)—constitute effective precipitation embryos and can initiate collision and coalescence in clouds (e.g., Feingold et al. 1999; Jensen and Nugent 2017). While Mordy (1959) argued that it may take a prohibitively long time span for large particles to grow to their equilibrium sizes in a subsaturated environment, the opposite situation is encountered here: seeded droplets will take time to evaporate to their equilibrium sizes, with commensurate implications for the colloidal stability of clouds during MCB.

The relationship (7) allows one to construct the seeded aerosol size distribution based on the distribution of the spray droplets. Since measurements of seeded aerosol particles produced by potential MCB apparatus indicate a lognormal distribution (Cooper et al. 2014), the spray droplets must also follow this distribution. This can be shown by a simple transformation operation (e.g., Lamb and Verlinde 2011, chapter 2.1.4), which relates the concentration density distributions of seeded aerosol particles to the distribution of spray droplets as
n(rspray;Nspray,rm,spray,σspray)=n(raero;Naero,rm,aero,σaero)draerodrspray=n(raero;Naero,rm,aero,σaero)(fsalinityρsprayρaero)1/3,
where the radii of spray particles and aerosols relate as rspray = raero(fsalinityρspray/ρaero)−1/3, and the parameters of the lognormal distributions as Nspray = Naero, rm,spray = rm,aero(fsalinityρspray/ρaero)−1/3, and σspray = σaero, which are the respective concentrations, geometric mean radii, and geometric standard deviations.

For this study, we construct 9 different seeded aerosol distributions using the aforementioned lognormal size distribution (9) with systematically varying geometric mean radii, rm,aero = 0.1, 0.3, or 0.9 μm, and geometric standard deviations,σaero = 1.5, 2.0, or 2.5. We choose these parameters following Ahlm et al. (2017), who varied rm,aero between 0.1 and 0.44 μm and σaero between 1.5 and 2.0. We also include larger values that may result from collision and coalescence of spray droplets in the direct vicinity of the spray nozzle, a process favored under very high droplet concentrations (Qian and Lin 2011). Note, however, that the optimal value of rm,aero is strongly debated, with Connolly et al. (2014) and Wood (2021), suggesting rm,aero to be in the range of 0.015 to 0.05 μm.

The value of rm,aero is crucial since it determines the number concentration of seeded aerosol particles Naero that is produced per mass unit of seawater sprayed in the atmosphere. Thus, a smaller rm,aero is beneficial since it reduces the energy required for spraying (Connolly et al. 2014). For this study, we prescribe a spray liquid water mixing ratio of q1,spray = 0.01 g kg−1, that is, 0.01 g seawater spray per 1 kg dry air. Assuming a typical seawater salinity of 3.5%, the corresponding total seeded dry aerosol mass mixing ratio is qaero = 350 μg kg−1. The corresponding seeded aerosol concentrations1 Naero vary between 1.2 mg−1 and 18 374.9 mg−1 and are indicated by black dots in Fig. 2. Note that Connolly et al. (2014) and Wood (2021) suggest qaero to be around 1 μg kg−1, which is expected from using a smaller rm,aero value, as discussed above. Assuming that 1 μg kg−1 is indeed a target seeded aerosol mass mixing ratio, accounting for plume dilution and other processes that diminish the aerosol, requires that one inject much higher masses to obtain a mean mass mixing ratio of 1 μg kg−1 over an extended plume. Appendix B shows that our value of 350 μg kg−1 averages to 1 μg kg−1 over a distance of 1000 km, and that these higher concentrations are relevant for the first hours or first tens of kilometers after injection.

Fig. 2.
Fig. 2.

The aerosol concentration as a function of the geometric mean aerosol radius and the geometric standard deviation for a spray liquid water mixing ratio of 0.01 g kg−1 and a salinity of 3.5%. The black dots show the seeded cases analyzed in this study. The black line indicates the concentration of the background aerosol.

Citation: Journal of the Atmospheric Sciences 78, 10; 10.1175/JAS-D-21-0077.1

The background aerosol consists of three lognormal distributions with rm,aero = (9.0, 19.5, 77.0) nm, σaero = (1.416, 1.425, 1.592), and Naero = (11.7, 38.4, 41.7) mg−1. These parameters correspond to Connolly et al. (2014), with only the aerosol concentrations divided by a factor of 4 to create a constant LWP in the LESs of the nonseeded case, in which weak drizzle balances cloud replenishing processes (cf. Fig. 7). Note that the resultant droplet concentration in the nonseeded case (50–70 mg−1) is well within the range of observed values for maritime stratocumulus (e.g., Wood 2012). With this setup, we will be able to investigate a wide range of potential microphysical MCB responses, from enhanced precipitation to stronger entrainment caused by accelerated evaporation. All investigated aerosol size distributions are depicted in Fig. 3. For initializing the particle spectra in the simulation results presented below, the background aerosol particles are adjusted to their equilibrium radii (A3), considering the current saturation ratio and temperature, while the injected spray droplets are initialized using (7), that is, at superequilibrium size, which creates the potential for precipitation embryos, as discussed above.

Fig. 3.
Fig. 3.

The initial aerosol size distributions analyzed for this study. The following color code will be used throughout the study to indicate the different aerosol cases: Green lines indicate geometric mean radii of 0.1 μm, orange lines 0.3 μm, and red lines 0.9 μm. Continuous lines indicate a geometric standard deviation of 1.5 μm, long-dashed lines 2.0 μm, and short-dashed lines 2.5 μm. The black line marks the background aerosol.

Citation: Journal of the Atmospheric Sciences 78, 10; 10.1175/JAS-D-21-0077.1

3. Numerical results

The previously described aerosol size distributions are used to initialize the LCM, which has been coupled to a simple parcel model and a three-dimensional LES model for this study. Before advancing to the results, a quick overview of the modeling framework will be given; the reader is referred to the referenced literature for more details.

a. Modeling framework

For all simulations, the LCM is used for representing cloud microphysics. In this framework, all hydrometeors are represented by computational particles, each representing an ensemble of real particles. This approach allows a wetted aerosol particle to grow continuously to a cloud droplet and to a raindrop without artificial categorization. The transition between these hydrometeor categories is explicitly modeled in the LCM and not parameterized, as is frequently done in other cloud microphysical models. For example, the activation of a wetted aerosol to a cloud droplet is considered explicitly by including the effects of curvature and solute aerosol mass within the diffusional growth equation (Hoffmann et al. 2015). Collisional growth is based on a statistical method that approaches the exact method by Gillespie (1975) once the number of hydrometeors represented by each computational particle, called the weighting factor, approaches unity (Dziekan and Pawlowska 2017; Unterstrasser et al. 2017; Hoffmann et al. 2017). This representation of collisional growth also considers the redistribution of aerosol mass correctly (Hoffmann 2017).

Note that we distinguish between wetted aerosols, cloud droplets, and rain drops for analysis purposes nonetheless. A wetted aerosol is considered a cloud droplet when its radius exceeds 1 μm. Köhler theory is not applied for differentiation since it can be misleading when the dry aerosol mass is large (e.g., Hoffmann et al. 2017). A cloud droplet is considered a rain drop when its radius exceeds 25 μm. This radius has been chosen in analogy to many simple cloud microphysical models that need to distinguish between cloud and rain particles for parameterization purposes (e.g., Khairoutdinov and Kogan 2000).

The first results presented below are gained from the LCM coupled to a simple parcel model, in which a volume of air is cooled by adiabatic lifting (e.g., Pruppacher and Klett 1997; Maahn et al. 2019; Hoffmann 2020). The second set of simulations are obtained from the LES model system for atmospheric modeling (SAM; Khairoutdinov and Randall 2003) coupled with the LCM (Hoffmann et al. 2019; Hoffmann and Feingold 2019). The parcel model and the LES are initialized using the intercomparison case derived from the second research flight of the DYCOMS II measurement campaign (Ackerman et al. 2009), representing a nocturnal, weakly drizzling stratocumulus layer. The case is useful in that it produces clouds whose ability to produce drizzle is quite sensitive to the aerosol representation, and given the range of aerosol injections, is potentially sensitive to evaporation–entrainment feedbacks.

While the parcel model does not require prescription of spatial dimensions, the LES model follows the setup originally proposed in Ackerman et al. (2009): 128 × 128 grid boxes in the horizontal with a grid size of 50 m, and 256 grid boxes in the vertical with a grid size of 5 m. Note that this rather coarse LES grid is necessary due to the massive computational requirements of the LCM. While the LES is run for 6 h, the parcel model is executed for 4500 s, with a typical updraft velocity of 0.3 m s−1, allowing assessment of a wide range of cloud depths of up to 950 m. (In contrast, the cloud depth of the LESs is around 350 m, depending on the seeded aerosol distribution.) For all processes, a model time step of 0.1 s is used in the parcel simulations. For the LESs, the time step is increased to 0.5 s, however, combined with the semianalytical method of Clark (1973) to accommodate supersaturation changes during the slightly longer time step.

To represent the hydrometeors, several computational particles are placed in the single volume of the parcel model and about three million grid boxes in the 200 lowermost layers of the LES. For the parcel model (LES model), 100 (20) computational particles are assigned to each mode of the aforementioned background aerosol distribution, that is, 300 (60) in total, as well as additional 100 (20) computational particles for the seeded aerosol distribution, which is sufficient for an accurate representation of the microphysical processes that are encountered (Unterstrasser et al. 2017, 2020; Schwenkel et al. 2018). A constant number of computational particles for the seeded aerosol has the advantage that the seeded and the background particles are always adequately represented, and not over or undersampled as the seeded aerosol concentration varies by four orders of magnitude. A random generator following each aerosol distribution individually is used to prescribe the aerosol mass for each computational particle. The initial weighting factors are calculated from the number of particles in each distribution and distributed by a random generator to create a broader spectrum of weighting factors, which can be beneficial to the representation of collisional growth in Lagrangian cloud models (Unterstrasser et al. 2017). The wet radii of the background aerosol particles are initialized by their respective equilibrium radii (A3), while the wet radii of the seeded aerosol particles are determined by (7), that is, the size of the spray droplets, which are typically larger than their equilibrium radius.

To derive Ac, τ is calculated using (3). The extinction coefficients Qe are calculated for each hydrometeor at a wavelength of 500 nm, being representative for the bulk of solar radiation, using a Mie theory parameterization by Mitchell (2000). As pointed out by Connolly et al. (2014), this procedure is important since small unactivated particles, frequently neglected in the determination of Ac, can reflect a significant fraction of shortwave radiation, that is, contribute significantly to MCB.

b. Parcel simulations

A parcel model can represent clouds reasonably well as long as the effects of entrainment are negligible and the droplets are sufficiently small that the assumption of particles not leaving the parcel is valid. During collisional growth, however, some droplets may grow to sizes with appreciable fall velocities. Since there is no representation of sedimentation, these droplets will not leave the parcel, and collisional growth could be significantly overestimated. To mitigate this problem, droplets with a substantial (virtual) sedimentation velocity are randomly removed from the parcel, effectively limiting droplet sizes to radii below 200 μm. (The probability of a droplet leaving the parcel is prescribed as p = wsΔt/100 m, where ws is the size-dependent sedimentation velocity and Δt the model time step.) Note that due to this artificial loss of large particles, the parcel model should not be considered adiabatic.

The hydrometeors in the parcel model also experience individual stochastic supersaturation fluctuations, added to the mean parcel supersaturation, using the linear eddy model (Kerstein 1988; Hoffmann 2020). For this, an integral turbulence length scale of 100 m and a relatively low but still typical kinetic energy dissipation rate of 1 cm2 s−3 are prescribed. This has been done to mimic the natural broadening of the droplet size distribution in clouds, which is essential for the onset of collisional growth, while diffusional growth alone would result in an unrealistically narrow distribution (e.g., Warner 1973; Chandrakar et al. 2016). To reduce statistical fluctuations, each parcel simulation is repeated 30 times with a different set of random numbers. Thus, the following parcel model results constitute ensemble averages.

1) Nonseeded case

The general development within the parcel model is exemplified in Fig. 4, which shows the droplet size distribution as a function of height for the nonseeded case. Below cloud base (marked by the letter A), the background aerosol particles have adapted to their thermodynamical environment, that is, they attain their respective equilibrium radii, which increase slightly toward cloud base due to the increasing saturation ratio [see Eq. (A3) in appendix A]. At cloud base, a fraction of these particles activates and grows by diffusion (B). Particles that are too small remain unactivated and persist as interstitial aerosol (C). At about 400 m above cloud base, collisional growth begins (D), which increases the drop size substantially compared to the expected diffusional growth alone (black line). The commensurate decrease in cloud droplet number and therefore integral droplet surface area limits the uptake of excess water vapor by condensation. Accordingly, increasing supersaturations develop that enable the in-cloud activation of interstitial aerosol (E). Figure 5 is a continuation of Fig. 4 for seeded cases.

Fig. 4.
Fig. 4.

The droplet size distribution in the nonseeded case as a function of the droplet radius and the distance from cloud base in the parcel simulation. The black line indicates pure diffusional growth. The blue letters A to E mark certain microphysical processes further detailed in the main text. The line segment in the lower-right corner indicates the color used to depict this nonseeded case in subsequent plots.

Citation: Journal of the Atmospheric Sciences 78, 10; 10.1175/JAS-D-21-0077.1

Fig. 5.
Fig. 5.

Continuation of Fig. 4. Droplet size distributions for all analyzed seeded cases as a function of the droplet radius and the distance from cloud base in the parcel simulations. The black line indicates pure diffusional growth in the nonseeded case (cf. Fig. 4). The line segment in the lower-right corner indicates the color used to depict this case in other plots of this study (e.g., Fig. 3).

Citation: Journal of the Atmospheric Sciences 78, 10; 10.1175/JAS-D-21-0077.1

This behavior is in agreement with the profiles displayed in Fig. 6 (thick black line). The supersaturation ratio (Fig. 6a) is negative below cloud base and increases toward cloud base, where it reaches its maximum. Thereafter, the supersaturation decreases up to a height of about 400 m by the condensation of excess water vapor to liquid water, as shown by the increasing LWP (Fig. 6b). At higher levels, the supersaturation increases again. This is caused by the decelerated uptake of excess water vapor due to the decreasing integral surface area once collisional growth transfers cloud water into rain, as shown by the increasing rainwater path (RWP), which is defined in analogy to LWP but restricted to droplets with r > 25 μm (Fig. 6c). Consequently, the cloud-layer-averaged N also decreases (Fig. 6d), which indicates that the in-cloud activation clearly visible in Fig. 4 is not able to counteract the reduction in N by collision and coalescence. Regarding the in-cloud activation, it is, however, interesting to note that although the supersaturation increases at higher levels above cloud base, it does not exceed the maximum at cloud base, which is usually required to activate aerosols aloft (Pinsky and Khain 2002). Accordingly, the in-cloud activations here are most likely caused by supersaturation fluctuations, either in the ensemble of parcel simulations or the aforementioned stochastic supersaturation fluctuations in each simulation, which although they do not affect the mean supersaturation, enable localized supersaturations to exceed the maximum at cloud base (Abade et al. 2018).

Fig. 6.
Fig. 6.

Vertical profiles of (a) supersaturation ratio, (b) LWP, (c) RWP, (d) cloud-layer-averaged droplet number concentration, (e) cloud optical thickness, (f) cloud albedo, (g) rCRE susceptibility to aerosol seeding caused by microphysical processes, and (h) total rCRE susceptibility to aerosol seeding as a function of the distance from cloud base for all seeded and the nonseeded case of the parcel model simulations. The gray line indicates the average cloud-top height of the LES results presented in Fig. 7.

Citation: Journal of the Atmospheric Sciences 78, 10; 10.1175/JAS-D-21-0077.1

Finally, Figs. 6e and 6f show τ and Ac, respectively. Generally, τ and Ac increase throughout the entire analyzed cloud depth due to the increasing LWP (Fig. 6b). However, the growth rate of τ decreases when the production of rain begins (about 400 m above cloud base, Fig. 6c). This already indicates that inhibiting collisional growth can be an effective strategy to amplify MCB.

2) Seeded cases

The simplest approach to MCB is, however, to increase τ and hence Ac by a larger N due to the seeding of additional aerosol. Indeed, most analyzed seeded cases exhibit a τ that is larger than in the nonseeded case (5). This is especially the case for all aerosol-laden simulations with rm,aero = 0.1 μm and the narrowest distribution of rm,aero = 0.3 μm with σaero = 1.5, which all result in an increase in an N ≥ 500 mg−1 (green lines, continuous orange line; Figs. 6d,e and 5a,d,g,h). (To ease comparison of Fig. 5 with other figures, a short line segment is added to the lower-right corner of each panel, the color of which corresponds to the specific seeded aerosol case.)

These aerosol-laden cases also inhibit or significantly reduce collisional growth. For all seeded cases in Fig. 5, a black line shows the droplet size expected from the nonseeded diffusional growth alone (cf. Fig. 4). By comparing this line to the changes in the droplet size distribution, it is easy to see that a large number of seeded aerosol particles limits the production of precipitation-sized embryos and hence subsequent collisional growth by the increased competition for water vapor. If, however, the seeded aerosol distribution becomes wider or contains larger particles, collisional growth can be enhanced, which occurs for all simulations with rm,aero = 0.9 μm and the widest distribution of rm,aero = 0.3 μm with σaero = 2.5 (red lines, short-dashed orange line; Figs. 6b,c and 5b,c,f,i). The reason for this is large aerosol particles with commensurately large equilibrium radii in the seeded aerosol spectrum—the aforementioned GCCN—that initiate the collision process without requiring substantial diffusional growth within the cloud (e.g., Feingold et al. 1999; Jensen and Nugent 2017). In fact, some seeded aerosol particles are so large that they are considered rain drops (r > 25 μm) even at cloud base (Fig. 5). Therefore, all seeded cases that contain substantial numbers of GCCN result in smaller τ and Ac than in the nonseeded case (Figs. 6e,f).

The seeded case with rm,aero = 0.3 μm and σaero = 2.0 can be considered a “chimera” (long-dashed orange line; Fig. 5f). Although the added aerosol particles increase N and hence τ substantially (Figs. 6d.e), the seeded aerosol spectrum comprises GCCN that trigger significant losses due to collisional growth at about 800 m above the cloud base (Fig. 6c). While stratocumulus rarely reach these depths (e.g., Wood 2012), the presence of GCCN in shallower clouds must be taken into account for the long-term colloidal stability of seeded clouds, as addressed in the next subsection.

Finally, we need to consider that although seeded aerosol particles increase N (Fig. 6d), not all seeded particles activate. Figure 5 shows that a large fraction of seeded aerosol particles does not participate in significant diffusional growth above cloud base, that is, they remain interstitial aerosol. In fact, it is hard to discern a distinct mode for diffusional growth in the two cases with the highest numbers of seeded aerosol (Figs. 5d,e). The reason for this is the reduced supersaturation inside the cloud (Fig. 6a), which buffers the activation of the seeded aerosol as a result of the more efficient condensation in the aerosol-laden cases (Stevens and Feingold 2009).

While a large fraction of particles may remain unactivated in the seeded cases, Connolly et al. (2014) showed that these interstitial particles can contribute significantly to τ. While this primarily happens inside the cloud, it also takes place below cloud base where the seeded particles may contribute to τ and hence Ac, in contrast to the negligible values below cloud base in the nonseeded case (Figs. 6e,f). Note that calculations of the extinction coefficient Qe for these particles must consider Mie theory, which yields typically smaller values than the extinction coefficient in the limit of geometric scattering, which is only valid for (larger) cloud droplets (r ≫ 0.1 μm). The commensurate implications for τ are illustrated nicely by comparing the seeded case with rm,aero = 0.3 μm and σaero = 1.5 (continuous orange line) to the case with rm,aero = 0.1 μm and σaero = 2.5 (short-dashed green line) (Fig. 6e). Although fewer particles are seeded in the first case (Naero = 680.6 mg−1 vs Naero = 880.4 mg−1), it exhibits a larger τ and hence Ac. This is another aspect that should be considered regarding optimization of aerosol size distributions to be used for MCB.

Note, however, that N might be biased to larger values due to the differentiation between wetted aerosol and cloud droplets by a fixed radius of 1 μm. Interstitial aerosol particles with dry aerosol radii exceeding 0.1 μm exhibit equilibrium radii of more than 1 μm in saturated conditions. Thus, these particles are considered cloud droplets although curvature effects still impede their growth.

3) The efficiency of seeding

The rCRE susceptibility, neglecting the Ac saturation, is shown in Fig. 6g. The total susceptibility, including the Ac saturation, is presented in Fig. 6h. The susceptibilities exhibit very large positive values for all seeded cases at the cloud base. These values are indicative of the additional liquid water transported into the cloud by the seeded particles as discussed above (section 2). Their contribution to the rCRE is, however, soon exceeded by the liquid water produced by the condensation inside the cloud. Accordingly, the associated increase in the absolute rCRE is commensurately weak at the cloud base. For much deeper clouds (≥500 m), the Ac saturation, included in the total rCRE susceptibility (Fig. 6h), also renders MCB increasingly inefficient.

Subtropical stratocumulus, with typical cloud depths between 200 and 400 m (Wood 2012), are generally located between these extremes, but their rCRE susceptibility is significantly modulated by microphysical processes as shown in Figs. 6g,h. Just above cloud base, the cases with a substantial number of GCCN (red lines) begin to exhibit negative rCRE susceptibilities (Figs. 6g,h), indicating that seeded GCCN enhance collisional growth and thus decrease LWP and N even in very shallow clouds (e.g., Feingold et al. 1997). If one neglects the Ac saturation (Fig. 6g), the cases without significant collisional growth attain almost stationary rCRE susceptibilities from 200 m above the cloud base upward. These values may be used to identify the potential for the most efficient increase in the rCRE due to seeding, which is found for the case with rm,aero = 0.3 μm and σaero = 2.0 (long-dashed orange line) with a susceptibility of about 0.3. Since the LWP of this case is very similar to the nonseeded case (Fig. 6b), liquid water adjustment might be negligible [dln(LWP)/dln(Naero) ≈ 0]. Accordingly, this case could be indicative of a very efficient conversion of seeded aerosol into cloud droplets [dln(N)/dln(Naero) ≈ 1]. The lower susceptibilities found for the other seeded cases are therefore caused by either limited activation of seeded particles into active cloud droplets (mostly green lines) or GCCN-enhanced collisional growth (mostly red lines).

c. Three-dimensional LES modeling

The parcel model gives important insights into cloud microphysical processes, albeit in an idealized framework. The parcel model lacks, however, several processes that are included in LES, for example, evaporation–entrainment feedbacks, the correct representation of sedimentation, and the fractional breakup of the cloud deck. Results for LESs that encompass a much more complete set of physical interactions are explored below.

As described earlier, the setup for the LES is based on Ackerman et al. (2009). A constant aerosol distribution is initialized throughout the model domain up to a height of 1000 m, assuming that the seeded and the background aerosol are well mixed at the beginning of the simulation. This is a valid assumption for particles with a wet radius smaller than 25 μm (Lewis and Schwartz 2004) but might overestimate the presence of very large aerosol particles in the upper levels of the domain. The unintended effect of these particles on precipitation is prevented by allowing collision and coalescence only after the first hour of simulation time.

If seeded aerosol mass is lost by precipitation, it is reinjected at a height of 50 m above the surface, which is suggested to be a realistic height for the injection of spray droplets during MCB (Cooper et al. 2014). The size of the reinjected particles is determined by the same randomized procedure as outlined for the initialization of aerosol spectra detailed above, including the adjustment of their initial size according to (7), that is, the superequilibrium size of sprayed droplets discussed in section 2. This is done to maintain the same seeded aerosol mass throughout the simulation. This procedure has not been applied to the background aerosol. (Since the coalescence of droplets might combine background and seeded aerosol mass, it was necessary to trace the seeded aerosol mass as an additional parameter in each computational particle).

Figure 7 shows time series of various parameters derived from the LESs. In these cases, the stratocumulus layer is approximately 350 m deep. This height is indicated by a horizontal gray line in Fig. 6 to facilitate comparison with the parcel model result.

Fig. 7.
Fig. 7.

Time series of (a) LWP, (b) RWP, (c) cloud-layer-averaged droplet concentration, (d) cloud fraction, (e) entrainment velocity, (f) cloud optical thickness, (g) cloud albedo, and (h) total rCRE susceptibility to aerosol seeding for all seeded and the nonseeded case for the LES model.

Citation: Journal of the Atmospheric Sciences 78, 10; 10.1175/JAS-D-21-0077.1

Figure 7 a shows the LWP time series for all cases. Interestingly, in all seeded cases, the LWP is smaller than in the nonseeded case, and reduces throughout the simulation, while the nonseeded case is in an almost stationary state. For the cases with a substantial number of GCCN (red lines, short-dashed orange line), this loss of liquid water can be attributed to precipitation (e.g., Cui et al. 2014), as clearly indicated by the higher RWP (Fig. 7b). Moreover, the cloud-layer-averaged N decreases due to precipitation scavenging in the cases with the strongest collisional growth (short- and long-dashed red lines), indicating that a significant number of droplets grown on background aerosol particles are involved in collisions and lost to the surface (Fig. 7c). Additionally, the enhanced precipitation also initiates the breakup of the cloud deck, which reduces fc, here defined as the fraction of columns exceeding a τ of 5, further counteracting the MCB effort (Fig. 7d). Overall, enhanced collisional growth results in a smaller τ and Ac than in the nonseeded case (Figs. 7f,g). Note that this decrease is also observed for the seeded case with σaero = 2.5 and rm,aero = 0.3 μm (short-dashed orange line) for which the parcel model required a cloud depth of about750 m, while the LES cloud exhibits a depth of about 350 m only. This difference is likely associated with processes that widen the droplet distribution and therefore accelerate the collision process, which are absent in the parcel model: for example, droplet recirculation that results in longer droplet residence times and more variable droplet growth histories (e.g., Feingold et al. 1996; Lasher-Trapp et al. 2005; Hoffmann et al. 2017).

The LWP also decreases in the aerosol-laden cases (green lines, continuous orange line), albeit at a significantly reduced RWP compared to the nonseeded case (Figs. 7a,b). Accordingly, the loss of liquid water must be attributed to a different process than enhanced collisions. Figure 7e shows the entrainment velocity we. Clearly, we is substantially larger in the aerosol-laden cases, for which two explanations exist in the literature: Precipitating clouds exhibit reduced entrainment velocities due to the enhanced removal of liquid water from the cloud top by sedimentation, thereby reducing the potential for evaporative cooling and hence entrainment (Ackerman et al. 2004; Bretherton et al. 2007). Aerosol-laden clouds, on the other hand, tend to evaporate faster, which accelerates the evaporative cooling at cloud top and hence intensifies the mixing of the cloud with the free troposphere aloft (Wang et al. 2003). Irrespective of their relative importance, both processes increase entrainment at higher N and hence contribute to the reduction in LWP (Hoffmann et al. 2020). From an MCB perspective, this loss in LWP is outweighed by the effect of the much higher N, with the result that τ and Ac are higher for the aerosol-laden cases (Figs. 7f,g), at least over the duration of these simulations. On longer time scales, the absorption of shortwave radiation during daytime may amplify the reduction in LWP during MCB. And the overall loss of LWP could reduce the emission of longwave radiative cooling and hence the kinetic energy driving the stratocumulus dynamics, accelerating the decreasing trend in LWP and potentially fc even further (e.g., Wang et al. 2011).

The entrainment feedbacks also affect the rCRE susceptibility (Fig. 7h), which is up to a factor of 2 smaller than in the parcel model at corresponding cloud depths (green lines, continuous orange line), indicating stronger evaporation of liquid water [dln(LWP)/dln(Naero) < 0]. Interestingly, the same cases as in the parcel model exhibit negative susceptibilities although the cloud is shallower (red lines, short-dashed orange line), indicating that the collisional processes are generally accelerated in the LES as outlined above, resulting in losses of LWP [dln(LWP)/dln(Naero) < 0], N [dln(N)/dln(Naero) < 1], and fc [dln(fc)/dln(Naero) < 0]. As in the parcel model, the case with σaero = 2.0 and rm,aero = 0.3 μm (long-dashed orange line) exhibits the highest susceptibility, presumably due to the very efficient activation of particles to cloud droplets for this case (see also Fig. 7c).

4. Summary and discussion

In this study, we analyzed the impact of different seeded aerosol size distribution on MCB from a cloud microphysical perspective. To this end, we used an LCM that provides a highly detailed representation of cloud microphysical processes typically unavailable in other microphysical schemes. By coupling the LCM to a simple parcel model as well as an LES model, we identified basic microphysical implications for MCB, first in the idealized parcel setup, and then in LESs of a stratocumulus layer, with the aim of determining the best aerosol size distribution for MCB from a cloud microphysical and macrophysical perspective.

Overall, nine cases with seeded aerosol and one case without seeded aerosol have been investigated. In the seeded cases, the mass of seawater spray, or equivalently the mass of aerosol, has been kept constant. However, the geometric mean radius (rm,aero = 0.1, 0.3, 0.9 μm) and the geometric standard deviation (σaero = 1.5, 2.0, or 2.5) have been varied systematically to produce aerosol concentrations between 1.2 and 18 374.9 mg−1 to represent a wide range of potential seeded aerosol concentrations and sizes. To further understand the relationship between the spray droplets and the seeded aerosol, the study started with a theoretical assessment of their relationship, revealing that the spray droplets are usually larger than their equilibrium radius, and indicating the potential for precipitation embryos if the dissolved aerosol mass is sufficiently large.

Other studies such as Connolly et al. (2014) and Wood (2021) have considered seeding scenarios in which the mass of injected aerosol is lower or the size of the seeded aerosol particles smaller than those applied here. From the perspective of simple albedo brightening, it seems plausible that as long as the number concentration and size of cloud droplets resulting from different seeding approaches agree, our results are relevant to other seeding scenarios. However, as discussed in this paper, and summarized below, nuanced responses, only represented by our more detailed and coupled modeling, need to be considered.

a. The risks of precipitation embryos

Parcel and LES modeling confirm the deleterious effect of precipitation embryos for all seeded cases with rm,aero = 0.9 μm and the broadest distribution with rm,aero = 0.3 μm, showing that enhanced collisional growth can result in a cloud albedo that is smaller than in the nonseeded case, hampering MCB efforts. While the potential for these so-called GCCN is well known (e.g., Feingold et al. 1999; Jensen and Nugent 2017), it is important to reiterate their effect for MCB since some spraying techniques may produce sufficiently large droplets and hence particles that will trigger these unhelpful (for MCB) collision and coalescence responses (Qian and Lin 2011; Cooper et al. 2013, 2014). We emphasize that parcel models do not reflect the full extent of the risks posed by precipitation. First, LES droplet residence times in the cloud layer can be significantly longer than in the parcel framework, which increases the probability for precipitation (e.g., Feingold et al. 1996). This effect is enhanced by the consideration of entrainment and mixing in the LES, which allows droplets to experience different saturation histories, resulting in a broader droplet size distribution that is also more beneficial to collision and coalescence (e.g., Lasher-Trapp et al. 2005; Hoffmann et al. 2017). Overall, the collisional growth is stronger in the LES, which manifests itself in more rainwater at lower cloud depths compared to the parcel model. Additionally, the LESs also show reduced cloud fractions in precipitating conditions, which amplifies the negative effect on MCB even further.

b. The risks of evaporation–entrainment

To avoid the risks of precipitation, the obvious alternative is to seed with smaller and hence more numerous aerosol particles (mainly produced for rm,aero = 0.1 μm and the narrowest distribution for rm,aero = 0.3 μm). These cases are generally favorable for MCB since the higher number of cloud droplets results in a significant increase in the cloud albedo. Parcel and LES modeling show, however, that the production of new cloud droplets in the aerosol-laden cases is not as effective as in the other scenarios. Many seeded aerosol particles, as well as a large fraction of the background aerosol, do not activate (e.g., Twomey 1959; Ghan et al. 1998). While these particles still contribute to the overall albedo (Connolly et al. 2014), their extinction coefficient can be significantly smaller [cf. Eq. (3)], making MCB less efficient. More importantly, the increased entrainment of free tropospheric air into the cloud layer simulated by LES for the more aerosol-laden cases, results in a stronger reduction of liquid water than in the nonseeded case (Wang et al. 2003; Ackerman et al. 2004; Bretherton et al. 2007). While the albedo is still higher than in the nonseeded case, the ongoing loss of liquid water due to enhanced entrainment may restrict the overall lifetime and spatial extent of seeded stratocumulus decks, with commensurate limitations to the effectiveness of MCB. This negative LWP adjustment must be considered an imminent threat to long-term MCB efforts (Glassmeier et al. 2021). To mitigate this effect in future MCB efforts, it might be beneficial to consider spraying smaller numbers of (smaller) aerosol particles. Since the resultant increase in cloud albedo would decrease, MCB would need to be applied to a larger fraction of the globe.

5. Concluding remarks

This study must be viewed as a case study, and the microphysical implications discussed here might not be relevant to all potential MCB scenarios. Given the sensitivity of both precipitation and evaporation–entrainment processes to the cloud microphysical structure, the optimal size distribution will likely need to be matched to the target cloud. For instance, the results indicate that shallower stratocumulus might be a better target for MCB. In these clouds, droplets are typically smaller and hence less likely to collide, limiting the influence of seeded GCCN. Additionally, the effect of enhanced entrainment rates due to seeding might also be mitigated since shallower stratocumulus do not entrain as readily as deeper ones (Hoffmann et al. 2020). The above-cloud moisture has been found to be an important factor to regulate aerosol effects on the entrainment rate (Chen et al. 2014; Gryspeerdt et al. 2019), suggesting that a more humid free troposphere bears a greater potential for MCB, independent of cloud depth.

Unfortunately, the large computational resources required for the LCM limit further investigations here. As mentioned above, the long-term behavior, including the diurnal heating by the absorption of shortwave radiation, must be studied in more detail to understand and potentially avoid the dissipation of stratocumulus decks in response to aerosol injection (e.g., Feingold et al. 2015). For example, Wang et al. (2011) and Jenkins et al. (2013) showed how preventing early morning drizzle is key to maintaining a stronger cloud radiative effect into the late morning and afternoon hours. Furthermore, larger model domains are required to represent changes in the mesoscale organization of stratocumulus, including their effects on the cloud fraction and the potential for precipitation (e.g., Kazil et al. 2017). These factors may also have a bearing on the optimal time and location of seeding. Nevertheless, the detailed microphysical insights gained from the LCM are not possible using other modeling schemes. Therefore, we regard the results of this study as motivation for a general MCB assessment strategy: A hierarchy of models containing microphysically simpler but computationally faster approaches that allow one to investigate the large-scale, long-term behavior of stratocumulus, complemented by the highly detailed microphysical modeling presented here. Only in combination, meaningful conclusions on MCB can be made.

Acknowledgments

FH appreciates support from the Cooperative Institute for Research in Environmental Sciences (CIRES) Visiting Fellowship of the University of Colorado Boulder and the NOAA/Chemical Sciences Laboratory, as well as the Emmy Noether program of the German Research Foundation (DFG) under Grant HO 6588/1-1. GF and FH also acknowledge support from an Earth’s Radiation Budget grant, NOAA CPO Climate and CI 03-01-07-001. Marat Khairoutdinov graciously provided the SAM model.

Data availability statement.

The data that support the findings of this study are available from the corresponding author upon request.

APPENDIX A

An Analytical Expression for the Equilibrium Radius

Current literature provides many approximations for the equilibrium radius of wetted aerosol particles in a subsaturated environment (e.g., Chen 1994; Khvorostyanov and Curry 1999). Initially unfamiliar with the work of Sedunov (1974), we derived a complete analytical expression for the equilibrium radius requi with Eqs. (6)–(26b) in Pruppacher and Klett (1997):
ln(S+1)=ArequiBrequi3raero3,
where S is the supersaturation ratio, and raero the (dry) radius of the aerosol. The parameters A and B=braero3 comprise several parameters relevant to the curvature and solute effects, which determine the equilibrium size of the wetted aerosol (Köhler 1936). We assume a fully soluble aerosol with raero3requi3 and ln(S + 1) ≈ S, such that
S=ArequiBrequi3.
Note that the simplification ln(S + 1) ≈ S is not necessary for the following derivation, that is, S can be replaced by ln(S + 1) without changing the validity of the results.
Following chapter 1.6.2.3 in Bronshtein et al. (2007), we rewrite (A2) as a reduced cubic, which generates the solutions
requi=A3S{{12cos[arccos(σ)/3]}for1σ1,{12cosh[acosh(σ)/3]}forσ>1,
with
σ=2S2Scrit21,
and the critical supersaturation for activation Scrit=4A3/27B. Note that this solution equals Eq. (3.3.25) in the textbook of Sedunov (1974) once appropriate substitutions are applied. Nonetheless, we believe it is worthwhile to represent this solution here due to the fading availability of older, undigitized literature.

Figure A1 shows that the solution (A3) (dashed black line) is identical to the numerical solution (Newton’s method, green line), which can approximate requi to arbitrary accuracy. The solutions of Chen (1994) [their Eq. (11)] and Khvorostyanov and Curry (1999) [their Eq. (21) with β = 0.5] deviate substantially for small particles and supersaturations approaching 0, while (A3) does not show any deviations.

Fig. A1.
Fig. A1.

The equilibrium radius of a wetted aerosol as a function of the (dry) aerosol radius determined using the new analytical solution (A3) (short-dashed black line), a numerical solver (green line), and the approximate solutions by Khvorostyanov and Curry (1999) and Chen (1994) (blue and red lines, respectively). The four panels show the equilibrium radius for different supersaturation ratios.

Citation: Journal of the Atmospheric Sciences 78, 10; 10.1175/JAS-D-21-0077.1

For the limits raero → 0 and raero → ∞, Taylor series expansion of (A3) yields
requi={raero3/2(bA)1/2forraero0,raero(bS)1/3forraero.
The exponents of raero are indicated by slopes in Fig. A1d. The behavior of the analytical solution for raero → ∞ is in good agreement with Chen (1994) and Khvorostyanov and Curry (1999). The behavior of the analytical solution for raero → 0 is not considered in Chen (1994). The solution of Khvorostyanov and Curry (1999), however, also agrees with the analytical solution for raero → 0, but only for large subsaturations (Fig. A1a). For almost saturated conditions (Fig. A1d), the exponent of Khvorostyanov and Curry (1999) is overestimated. Unfortunately, the unapproximated form (A3) shrouds these dependences entirely.

APPENDIX B

The Seeded Aerosol Mass in a Developing Plume

The seeded aerosol mass is an important constraint on the energy required to implement a successful MCB program. Recent studies estimated a seeded aerosol mass mixing ratio of about qaero = 1 μg kg−1 to be sufficient to cause the necessary increase in cloud albedo to counteract global warming (Connolly et al. 2014; Wood 2021). Since an MCB spraying apparatus emits aerosol in plumes that widen and lose mass via various processes (dry or wet deposition, the mixing with the typically cleaner free troposphere aloft) as they are advected with the mean wind, the same qaero cannot be applied for any location within the plume.

We build upon a simple plume model recently presented by Wood (2021). This model assumes a linear increase in the width of the plume and represents aerosol loss by an exponential decay. Mixing across the width of the plume and the boundary layer height is assumed instantaneous. Based on these assumptions, one obtains
qaero(t,x)=Qaerouh×(Kx)exp(tτaero)/ρ,
where x is the distance from the spraying apparatus with the plume propagating along x, and t the time an air parcel traveled inside the plume. The dimensions x and t relate via the horizontal wind speed u, such that t = x/u.

As done in various plume models, the plume width is parameterized by the distance from the source, x, multiplied by a growth factor K ≈ 0.1, which is representative for neutral stability (e.g., Seinfeld and Pandis 2016, chapter 18). The boundary layer height is assumed to be h = 1000 m; ρ = 1 kg m−3 is the average air density within the boundary layer, and u = 5 m s−1 a typical horizontal wind speed for marine stratocumulus-topped boundary layers. The seeded aerosol lifetime τaero is varied between 0.5 and 2.0 days. While 2.0 days might be a more typical lifetime (Wood 2021), the smaller values reflect to some extent the substantial microphysical effects on the cloud discussed in this study (wet deposition of GCCN, stronger entrainment in aerosol-laden conditions).

The source strength of the spraying apparatus Qaero is determined from
qaero¯=0lplume[Kx/2Kx/2qaero(t,x)dy]dx/0lplumeKx/2Kx/2dydx,
where the plume width extends in the y direction, which is perpendicular to x. We set qaero¯=1μgkg1, which is the average value suggested by Connolly et al. (2014) and Wood (2021). We set lplume = 1000 km, which is a typical plume length (Wood 2021).

The development of qaero for different τaero is shown in Fig. B1 (colored lines). Although Qaero is determined such that all curves yield the same average qaero¯ (dashed black line), it is clear that qaero can be two orders of magnitude larger than the plume average in the first tens of kilometers downstream of the spraying apparatus. Depending on the wind speed u, these conditions can prevail for a few hours. Thus, if qaero¯=1μgkg1 is really the optimal seeded aerosol mass, the results of this study, which have been calculated with qaero = 350 μg kg−1, will be representative for the initial period of plume development, and the microphysical consequences discussed in this study must be considered in the development of the plume farther downstream.

Fig. B1.
Fig. B1.

The seeded aerosol mass mixing qaero in a developing plume as a function of the distance from the sprayer for different seeded aerosol lifetimes (τaero, colored lines). The black dashed line shows the average value of all qaero curves, determined over 1000 km, which is the typical length of a plume.

Citation: Journal of the Atmospheric Sciences 78, 10; 10.1175/JAS-D-21-0077.1

REFERENCES

  • Abade, G. C., W. W. Grabowski, and H. Pawlowska, 2018: Broadening of cloud droplet spectra through eddy hopping: Turbulent entraining parcel simulations. J. Atmos. Sci., 75, 33653379, https://doi.org/10.1175/JAS-D-18-0078.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Ackerman, A. S., M. P. Kirkpatrick, D. E. Stevens, and O. B. Toon, 2004: The impact of humidity above stratiform clouds on indirect aerosol climate forcing. Nature, 432, 10141017, https://doi.org/10.1038/nature03174.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Ackerman, A. S., and Coauthors, 2009: Large-eddy simulations of a drizzling, stratocumulus-topped marine boundary layer. Mon. Wea. Rev., 137, 10831110, https://doi.org/10.1175/2008MWR2582.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Ahlm, L., A. Jones, C. W. Stjern, H. Muri, B. Kravitz, and J. E. Kristjánsson, 2017: Marine cloud brightening—As effective without clouds. Atmos. Chem. Phys., 17, 13 07113 087, https://doi.org/10.5194/acp-17-13071-2017.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Albrecht, B. A., 1989: Aerosols, cloud microphysics, and fractional cloudiness. Science, 245, 12271230, https://doi.org/10.1126/science.245.4923.1227.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Arrhenius, S., 1896: On the influence of carbonic acid in the air upon the temperature of the ground. Philos. Mag., 41, 237276, https://doi.org/10.1080/14786449608620846.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Betts, A. K., 2007: Coupling of water vapor convergence, clouds, precipitation, and land-surface processes. J. Geophys. Res., 112, D10108, https://doi.org/10.1029/2006JD008191.

    • Search Google Scholar
    • Export Citation
  • Bretherton, C., P. N. Blossey, and J. Uchida, 2007: Cloud droplet sedimentation, entrainment efficiency, and subtropical stratocumulus albedo. Geophys. Res. Lett., 34, L03813, https://doi.org/10.1029/2006GL027648.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Bronshtein, I. N., K. A. Semendyayev, G. Musiol, and H. Mühlig, 2007: Handbook of Mathematics. Springer Science & Business Media, 1159 pp.

  • Chandrakar, K. K., W. Cantrell, K. Chang, D. Ciochetto, D. Niedermeier, M. Ovchinnikov, R. A. Shaw, and F. Yang, 2016: Aerosol indirect effect from turbulence-induced broadening of cloud-droplet size distributions. Proc. Natl. Acad. Sci. USA, 113, 14 24314 248, https://doi.org/10.1073/pnas.1612686113.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Chen, J.-P., 1994: Theory of deliquescence and modified Köhler curves. J. Atmos. Sci., 51, 35053516, https://doi.org/10.1175/1520-0469(1994)051<3505:TODAMK>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Chen, Y.-C., M. W. Christensen, G. L. Stephens, and J. H. Seinfeld, 2014: Satellite-based estimate of global aerosol–cloud radiative forcing by marine warm clouds. Nat. Geosci., 7, 643646, https://doi.org/10.1038/ngeo2214.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Clark, T. L., 1973: Numerical modeling of the dynamics and microphysics of warm cumulus convection. J. Atmos. Sci., 30, 857878, https://doi.org/10.1175/1520-0469(1973)030<0857:NMOTDA>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Connolly, P. J., G. B. McFiggans, R. Wood, and A. Tsiamis, 2014: Factors determining the most efficient spray distribution for marine cloud brightening. Philos. Trans. Roy. Soc., A372, 20140056, https://doi.org/10.1098/rsta.2014.0056.

    • Search Google Scholar
    • Export Citation
  • Cooper, G., D. Johnston, J. Foster, L. Galbraith, A. Neukermans, R. Ormond, J. Rush, and Q. Wang, 2013: A review of some experimental spray methods for marine cloud brightening. Int. J. Geosci., 4, 7897, https://doi.org/10.4236/ijg.2013.41009.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Cooper, G., J. Foster, L. Galbraith, S. Jain, A. Neukermans, and B. Ormond, 2014: Preliminary results for salt aerosol production intended for marine cloud brightening, using effervescent spray atomization. Philos. Trans. Roy. Soc., A372, 20140055, https://doi.org/10.1098/rsta.2014.0055.

    • Search Google Scholar
    • Export Citation
  • Cui, Z., A. Gadian, A. Blyth, J. Crosier, and I. Crawford, 2014: Observations of the variation in aerosol and cloud microphysics along the 20°S transect on 13 November 2008 during VOCALS-REx. J. Atmos. Sci., 71, 29272943, https://doi.org/10.1175/JAS-D-13-0245.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Dziekan, P., and H. Pawlowska, 2017: Stochastic coalescence in Lagrangian cloud microphysics. Atmos. Chem. Phys., 17, 13 50913 520, https://doi.org/10.5194/acp-17-13509-2017.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Feingold, G., W. Cotton, B. Stevens, and A. Frisch, 1996: The relationship between drop in-cloud residence time and drizzle production in numerically simulated stratocumulus clouds. J. Atmos. Sci., 53, 11081122, https://doi.org/10.1175/1520-0469(1996)053<1108:TRBDIC>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Feingold, G., R. Boers, B. Stevens, and W. R. Cotton, 1997: A modeling study of the effect of drizzle on cloud optical depth and susceptibility. J. Geophys. Res., 102, 13 52713 534, https://doi.org/10.1029/97JD00963.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Feingold, G., W. R. Cotton, S. M. Kreidenweis, and J. T. Davis, 1999: The impact of giant cloud condensation nuclei on drizzle formation in stratocumulus: Implications for cloud radiative properties. J. Atmos. Sci., 56, 41004117, https://doi.org/10.1175/1520-0469(1999)056<4100:TIOGCC>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Feingold, G., I. Koren, T. Yamaguchi, and J. Kazil, 2015: On the reversibility of transitions between closed and open cellular convection. Atmos. Chem. Phys., 15, 73517367, https://doi.org/10.5194/acp-15-7351-2015.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Ghan, S. J., G. Guzman, and H. Abdul-Razzak, 1998: Competition between sea salt and sulfate particles as cloud condensation nuclei. J. Atmos. Sci., 55, 33403347, https://doi.org/10.1175/1520-0469(1998)055<3340:CBSSAS>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Gillespie, D. T., 1975: An exact method for numerically simulating the stochastic coalescence process in a cloud. J. Atmos. Sci., 32, 19771989, https://doi.org/10.1175/1520-0469(1975)032<1977:AEMFNS>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Glassmeier, F., F. Hoffmann, J. S. Johnson, T. Yamaguchi, K. S. Carslaw, and G. Feingold, 2021: Aerosol-cloud-climate cooling overestimated by ship-track data. Science, 371, 485489, https://doi.org/10.1126/science.abd3980.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Glenn, I. B., G. Feingold, J. J. Gristey, and T. Yamaguchi, 2020: Quantification of the radiative effect of aerosol–cloud interactions in shallow continental cumulus clouds. J. Atmos. Sci., 77, 29052920, https://doi.org/10.1175/JAS-D-19-0269.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Gryspeerdt, E., and Coauthors, 2019: Constraining the aerosol influence on cloud liquid water path. Atmos. Chem. Phys., 19, 53315347, https://doi.org/10.5194/acp-19-5331-2019.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Hartmann, D., and Coauthors, 2013: Observations: Atmosphere and surface. Climate Change 2013: The Physical Science Basis, T. F. Stocker et al., Eds., Cambridge University Press, 159–254.

  • Hoffmann, F., 2017: On the limits of Köhler activation theory: How do collision and coalescence affect the activation of aerosols? Atmos. Chem. Phys., 17, 83438356, https://doi.org/10.5194/acp-17-8343-2017.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Hoffmann, F., 2020: Effects of entrainment and mixing on the Wegener–Bergeron–Findeisen process. J. Atmos. Sci., 77, 22792296, https://doi.org/10.1175/JAS-D-19-0289.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Hoffmann, F., and G. Feingold, 2019: Entrainment and mixing in stratocumulus: Effects of a new explicit subgrid-scale scheme for large-eddy simulations with particle-based microphysics. J. Atmos. Sci., 76, 19551973, https://doi.org/10.1175/JAS-D-18-0318.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Hoffmann, F., S. Raasch, and Y. Noh, 2015: Entrainment of aerosols and their activation in a shallow cumulus cloud studied with a coupled LCM-LES approach. Atmos. Res., 156, 4357, https://doi.org/10.1016/j.atmosres.2014.12.008.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Hoffmann, F., Y. Noh, and S. Raasch, 2017: The route to raindrop formation in a shallow cumulus cloud simulated by a Lagrangian cloud model. J. Atmos. Sci., 74, 21252142, https://doi.org/10.1175/JAS-D-16-0220.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Hoffmann, F., T. Yamaguchi, and G. Feingold, 2019: Inhomogeneous mixing in Lagrangian cloud models: Effects on the production of precipitation embryos. J. Atmos. Sci., 76, 113133, https://doi.org/10.1175/JAS-D-18-0087.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Hoffmann, F., F. Glassmeier, T. Yamaguchi, and G. Feingold, 2020: Liquid water path steady states in stratocumulus: Insights from process-level emulation and mixed-layer theory. J. Atmos. Sci., 77, 22032215, https://doi.org/10.1175/JAS-D-19-0241.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Jenkins, A. K. L., P. M. Forster, and L. S. Jackson, 2013: The effects of timing and rate of marine cloud brightening aerosol injection on albedo changes during the diurnal cycle of marine stratocumulus clouds. Atmos. Chem. Phys., 13, 16591673, https://doi.org/10.5194/acp-13-1659-2013.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Jensen, J. B., and A. D. Nugent, 2017: Condensational growth of drops formed on giant sea-salt aerosol particles. J. Atmos. Sci., 74, 679697, https://doi.org/10.1175/JAS-D-15-0370.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Kazil, J., T. Yamaguchi, and G. Feingold, 2017: Mesoscale organization, entrainment, and the properties of a closed-cell stratocumulus cloud. J. Adv. Model. Earth Syst., 9, 22142229, https://doi.org/10.1002/2017MS001072.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Kerstein, A. R., 1988: A linear-eddy model of turbulent scalar transport and mixing. Combust. Sci. Technol., 60, 391421, https://doi.org/10.1080/00102208808923995.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Khairoutdinov, M. F., and Y. Kogan, 2000: A new cloud physics parameterization in a large-eddy simulation model of marine stratocumulus. Mon. Wea. Rev., 128, 229243, https://doi.org/10.1175/1520-0493(2000)128<0229:ANCPPI>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Khairoutdinov, M. F., and D. A. Randall, 2003: Cloud resolving modeling of the ARM summer 1997 IOP: Model formulation, results, uncertainties, and sensitivities. J. Atmos. Sci., 60, 607625, https://doi.org/10.1175/1520-0469(2003)060<0607:CRMOTA>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Khvorostyanov, V. I., and J. A. Curry, 1999: A simple analytical model of aerosol properties with account for hygroscopic growth: 1. Equilibrium size spectra and cloud condensation nuclei activity spectra. J. Geophys. Res., 104, 21752184, https://doi.org/10.1029/98JD02673.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Köhler, H., 1936: The nucleus in and the growth of hygroscopic droplets. Trans. Faraday Soc., 32, 11521161, https://doi.org/10.1039/TF9363201152.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Lamb, D., and J. Verlinde, 2011: Physics and Chemistry of Clouds. Cambridge University Press, 584 pp.

  • Lasher-Trapp, S. G., W. A. Cooper, and A. M. Blyth, 2005: Broadening of droplet size distributions from entrainment and mixing in a cumulus cloud. Quart. J. Roy. Meteor. Soc., 131, 195220, https://doi.org/10.1256/qj.03.199.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Latham, J., and M. Smith, 1990: Effect on global warming of wind-dependent aerosol generation at the ocean surface. Nature, 347, 372373, https://doi.org/10.1038/347372a0.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Latham, J., and Coauthors, 2012: Marine cloud brightening. Philos. Trans. Roy. Soc., A370, 42174262, https://doi.org/10.1098/rsta.2012.0086.

  • Lewis, E., and S. E. Schwartz, 2004: Sea Salt Aerosol Production: Mechanisms, Methods, Measurements, and Models. Geophys. Monogr., Vol. 152, Amer. Geophys. Union, 413 pp.

    • Crossref
    • Export Citation
  • Maahn, M., F. Hoffmann, M. D. Shupe, G. Boer, S. Y. Matrosov, and E. P. Luke, 2019: Can liquid cloud microphysical processes be used for vertically pointing cloud radar calibration? Atmos. Meas. Tech., 12, 31513171, https://doi.org/10.5194/amt-12-3151-2019.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Mitchell, D. L., 2000: Parameterization of the Mie extinction and absorption coefficients for water clouds. J. Atmos. Sci., 57, 13111326, https://doi.org/10.1175/1520-0469(2000)057<1311:POTMEA>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Mordy, W., 1959: Computations of the growth by condensation of a population of cloud droplets. Tellus, 11, 1644, https://doi.org/10.1111/j.2153-3490.1959.tb00003.x.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Pinsky, M., and A. Khain, 2002: Effects of in-cloud nucleation and turbulence on droplet spectrum formation in cumulus clouds. Quart. J. Roy. Meteor. Soc., 128, 501533, https://doi.org/10.1256/003590002321042072.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Platnick, S., and S. Twomey, 1994: Determining the susceptibility of cloud albedo to changes in droplet concentration with the advanced very high resolution radiometer. J. Appl. Meteor. Climatol., 33, 334347, https://doi.org/10.1175/1520-0450(1994)033<0334:DTSOCA>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Pruppacher, H. R., and J. D. Klett, 1997: Microphysics of Clouds and Precipitation. 2nd ed. Kluwer Academic Publishers, 954 pp.

  • Qian, L., and J. Lin, 2011: Modeling on effervescent atomization: A review. Sci. China Phys. Mech. Astron., 54, 21092129, https://doi.org/10.1007/s11433-011-4536-1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Schwenkel, J., F. Hoffmann, and S. Raasch, 2018: Improving collisional growth in Lagrangian cloud models: Development and verification of a new splitting algorithm. Geosci. Model Dev., 11, 39293944, https://doi.org/10.5194/gmd-11-3929-2018.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Sedunov, Y. S., 1974: Physics of Drop Formation in the Atmosphere. John Wiley and Sons, 234 pp.

  • Seinfeld, J. H., and S. N. Pandis, 2016: Atmospheric Chemistry and Physics: From Air Pollution to Climate Change. John Wiley and Sons, 1120 pp.

  • Shima, S.-I., K. Kusano, A. Kawano, T. Sugiyama, and S. Kawahara, 2009: The super-droplet method for the numerical simulation of clouds and precipitation: A particle-based and probabilistic microphysics model coupled with a non-hydrostatic model. Quart. J. Roy. Meteor. Soc., 135, 13071320, https://doi.org/10.1002/qj.441.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Stevens, B., and G. Feingold, 2009: Untangling aerosol effects on clouds and precipitation in a buffered system. Nature, 461, 607613, https://doi.org/10.1038/nature08281.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Twomey, S., 1959: The nuclei of natural cloud formation. Part II: The supersaturation in natural clouds and the variation of cloud droplet concentration. Pure Appl. Geophys., 43, 243249, https://doi.org/10.1007/BF01993560.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Twomey, S., 1974: Pollution and the planetary albedo. Atmos. Environ., 8, 12511256, https://doi.org/10.1016/0004-6981(74)90004-3.

  • Twomey, S., 1977: The influence of pollution on the shortwave albedo of clouds. J. Atmos. Sci., 34, 11491152, https://doi.org/10.1175/1520-0469(1977)034<1149:TIOPOT>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Unterstrasser, S., F. Hoffmann, and M. Lerch, 2017: Collection/aggregation algorithms in Lagrangian cloud microphysical models: Rigorous evaluation in box model simulations. Geosci. Model Dev., 10, 15211548, https://doi.org/10.5194/gmd-10-1521-2017.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Unterstrasser, S., F. Hoffmann, and M. Lerch, 2020: Collisional growth in a particle-based cloud microphysical model: Insights from column model simulations using LCM1D (v1. 0). Geosci. Model Dev., 13, 51195145, https://doi.org/10.5194/gmd-13-5119-2020.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Vaughan, N. E., and T. M. Lenton, 2011: A review of climate geoengineering proposals. Climatic Change, 109, 745790, https://doi.org/10.1007/s10584-011-0027-7.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Victor, D. G., D. Zhou, E. H. M. Ahmed, P. K. Dadhich, J. G. J. Olivier, H.-H. Rogner, K. Sheikho, and M. Yamaguchi, 2014: Introductory chapter. Climate Change 2014: Mitigation of Climate Change, O. Edenhofer et al., Ed., Cambridge University Press, 111–150, https://www.ipcc.ch/site/assets/uploads/2018/02/ipcc_wg3_ar5_chapter1.pdf.

  • Wang, H., P. J. Rasch, and G. Feingold, 2011: Manipulating marine stratocumulus cloud amount and albedo: A process-modelling study of aerosol-cloud-precipitation interactions in response to injection of cloud condensation nuclei. Atmos. Chem. Phys., 11, 42374249, https://doi.org/10.5194/acp-11-4237-2011.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Wang, S., Q. Wang, and G. Feingold, 2003: Turbulence, condensation, and liquid water transport in numerically simulated nonprecipitating stratocumulus clouds. J. Atmos. Sci., 60, 262278, https://doi.org/10.1175/1520-0469(2003)060<0262:TCALWT>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Warner, J., 1973: The microstructure of cumulus cloud: Part IV. The effect on the droplet spectrum of mixing between cloud and environment. J. Atmos. Sci., 30, 256261, https://doi.org/10.1175/1520-0469(1973)030<0256:TMOCCP>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Wood, R., 2012: Stratocumulus clouds. Mon. Wea. Rev., 140, 23732423, https://doi.org/10.1175/MWR-D-11-00121.1.

  • Wood, R., 2021: Assessing the potential efficacy of marine cloud brightening for cooling Earth using a simple heuristic model. Atmos. Chem. Phys., https://doi.org/10.5194/acp-2021-327, in press.

    • Search Google Scholar
    • Export Citation
1

Note that we express number concentrations as mixing ratios, i.e., the number of particles per mass unit of air, while the number of particles per volume unit of air has been used historically. The advantage of using mixing ratios is that this quantity is conserved under vertical motion. For shallow boundary layer clouds, both expressions yield approximately the same numerical value when expressed in units of mg−1 and cm−3, respectively.

Save