Open Access Article

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DOI: 10.1039/D4EA00073K
(Paper)
Environ. Sci.: Atmos., 2024, Advance Article

J. I. Katz*

Department of Physics and McDonnell Center for the Space Sciences, Washington University, St. Louis, Mo, USA. E-mail: katz@wuphys.wustl.edu; Fax: +1-314-935-6219; Tel: +1-314-935-6202

Received
30th May 2024
, Accepted 14th August 2024

First published on 23rd August 2024

Increasing marine haze and clouds has been considered as a possible means of increasing the Earth's albedo. This would reduce solar heating and global warming, counteracting the effects of the anthropogenic increase in greenhouse gases. One proposed method of doing so would inject small droplets of seawater or condensation nuclei into the marine boundary layer, creating artificial haze and cloud. The equilibrium size of such droplets is described by the Köhler equation that includes the vapor pressure reduction attributable to the solute according to Raoult's law and the vapor pressure increase of a small droplet as a result of surface tension according to Kelvin. Here we apply this classic result to small droplets in the marine boundary layer, where the partial pressure of water vapor is less than the equilibrium vapor pressure because it is in equilibrium with the saline ocean. We calculate the equilibrium size of a droplet containing dissolved ions and find that the radius of a droplet of seawater shrinks greatly before it achieves equilibrium.

## Environmental significanceIncreasing the Earth's albedo by nucleating clouds or injecting seawater aerosols into the atmosphere has been proposed to counteract global warming produced by anthropogenic greenhouse gases. The equilibrium sizes of these aerosols are determined by evaporation and condensation of atmospheric water vapor. |

This work is only concerned with the region very close to the ocean surface, not with subsequent enhancement of marine clouds at higher altitudes.^{5–9} While the general physical principles have long been known,^{10,11} here I express their interaction in simple and transparent analytic results. I do not consider the effects of possible surfactants^{12} because they are poorly characterized and vary greatly from place to place. In application to albedo modification, water could be drawn from below the ocean surface and would have little surfactant content.

(1) |

p_{i} = p_{0}x_{i}
| (2) |

(3) |

(4) |

Application of Raoult's law (eqn (2)) to the vapor pressure p(∞) of a flat surface of seawater indicates that p(∞) = 0.9796p_{0}(∞); we define the undersaturation ε = 1 − p(∞)/p_{0}(∞) = 1–0.9796 = 0.0204.

(5) |

Although the formalism indicates that for some values of the parameters there is a meaningful minimum N that permits formation of a stable droplet, for droplets large enough to justify the use of continuum theory this minimum (eqn (5)) is so small as to be irrelevant. It is meaningful to solve eqn (4) for r(N) or N(r) when r ≫ 0.1 nm and N ≫ 2.

Fig. 1 Relation between equilibrium droplet radius r and number of solute ions N, evaluated in the marine atmospheric equilibrium boundary layer. The air is slightly undersaturated in water vapor compared to air over pure water because of ocean salinity. At smaller N and r the quadratic (surface tension) term in eqn (4) dominates and r ∝ N^{1/2}, while at larger N and r the cubic (undersaturation) term dominates and r ∝ N^{1/3}. The cross-over between these regimes occurs for r ≈ r_{AB} ≈ 50 nm and N ≈ 5 × 10^{5} (eqn (6)). r_{BC} is identical to r(N) for fresh water. |

There are three cross-overs in eqn (4), values of r when two terms are equal. The cubic term (in r) equals the quadratic term when

(6) |

The cubic term in r in eqn (4) equals the negative term when

(7) |

The term quadratic in r equals the negative term when

(8) |

Very small droplets (those with r ≪ 100 nm) are ineffective scatterers of light, so we now consider the regime in which the quadratic term in eqn (4) may be neglected in comparison to the cubic term (r ≫ r_{AB} ≈ 50 nm). Neglecting the quadratic term, we find

(9) |

Then the number of water molecules in the droplet

(10) |

Condensation of seawater multiplies the volume of the aerosol by a factor and multiplies its scattering by another large factor. This latter factor is proportional to the square of the volume in the Rayleigh scattering regime applicable to very small droplets (r ≲ 100 nm), and to the 2/3 power of the volume in the geometrical optics regime applicable to larger droplets (r ≳ 1000 nm)^{14}.

(11) |

(12) |

r^{3} + R_{0}r^{2} − r_{0}^{3}
| (13) |

In the relevant regime r ≪ R_{0} (larger droplets fall rapidly as rain) and the cubic term may be neglected, leading to the approximation

(14) |

The solution of eqn (4) and is shown in Fig. 2. In fact, shrinkage is limited by the breakdown of Raoult's law when the solution becomes concentrated. r/r_{0} cannot be less than ; the result is a particle of moist salt or highly concentrated brine containing all the originally dissolved salt but little or none of the original water. Shrinkage stops because the increase in vapor pressure as a result of reduced radius is balanced by the (more rapidly growing) decrease as a result of growing salinity.

Fig. 2 Equilibrium radius r of seawater drop of original radius r_{0} in oceanic equilibrium boundary layer according to eqn (13). Dashed line (r = 0.27r_{0}) shows where Raoult's law breaks down because the droplet becomes concentrated brine or moist salt; values of r lying below this line are unphysical. This may be dealt with a modified Raoult's law,^{15} but without the simple result shown here. |

Using atomized seawater as cloud condensation nuclei differs from the mechanism producing ship tracks from dry products of combustion, whose physical chemistry must differ from and be more complicated than that of salt. While seawater droplets do have well-defined “dry radii” (the radius of a dry salt particle containing their salt content), they can never become that small because the decrease of their vapor pressure as they shrink stops evaporation at the equilibrium radii given by eqn (4) and (13) and shown in the figures. A significant consequency of the evaporative shrinkage of an injected seawater droplet is that its gravitational sink rate is reduced, facilitating its entrainment by turbulent flow in the marine boundary layer, and therefore its transport to altitudes at which supersaturation may make it grow sufficiently to become an effective scatterer of sunlight.

Saline water droplets may be produced in great numbers by flow of bulk seawater through a nozzle or a wire grid. They may be subsequently lofted in the turbulent marine boundary layer to altitudes at which adiabatic cooling has rendered the air supersaturated. Further growth may make them effective scatterers of sunlight.^{5–9,15} This resembles the mechanism by which cloud seeding may stimulate rainfall from clouds supersaturated by evaporation of small droplets, although in the marine case supersaturation results from adiabatic cooling of rising air rather than by evaporation of smaller droplets (Ostwald ripening). This further growth involves turbulent mixing in the marine boundary layer, a complex process beyond the scope of this paper.

- J. Latham, Nature, 1990, 347, 339–340 CrossRef .
- J. Latham, Atmos. Sci. Lett., 2002, 3, 52–58 CrossRef .
- J. Latham, K. Bower, T. Choularton, H. Coe, P. Connolly, G. Cooper, T. Craft, J. Foster, A. Gadian, L. Galbraith, H. Iacovides, D. Johnston, B. Lawader, B. Leslie, J. Meyer, A. Naukermans, B. Ormond, B. Parkes, P. Rasch, J. Rush, S. Salter, T. Stevenson, H. Wang, Q. Wang and R. Wood, Philos. Trans. R. Soc., A, 2012, 370, 4217–4262 CrossRef PubMed .
- F. J. Millero, R. Feistel, D. G. Wright and T. J. McDougall, Deep Sea Res., Part I, 2008, 55, 50–72 CrossRef .
- M. K. Yau and R. R. Rogers, A Short Course in Cloud Physics, Elsevier, Amsterdam, edn. 3rd, 1989 Search PubMed .
- K. C. Young, Microphysical Processes in Clouds, Oxford U. Press, Oxford, 1993 Search PubMed .
- J. M. Wallace and P. V. Hobbs, Atmospheric Science: an Introductory Survey, Elsevier, Amsterdam, edn. 2nd, 2006 Search PubMed .
- R. A. Houze, Int. Geophys., 2014, 104, 47–76 Search PubMed .
- K. N. Fossum, J. Ovadnevaite, D. Ceburnis, J. Preiß, J. R. Snifer, R.-J. Huang, A. Zuend and C. O'Dowd, npj Clim. Atmos. Sci., 2020, 3, 14 CrossRef CAS .
- L. D. Landau and E. M. Lifshitz, Statistical Physics, Pergamon, London, 1958, p. 279 Search PubMed .
- A. B. Pippard, The Elements of Chemical Thermodynamics, Cambridge U. Press, Cambridge, 1966, p. 111 Search PubMed .
- W. D. Garrett, J. Geophys. Res., 1968, 73, 5145–5150 CrossRef CAS .
- M. H. P. Ambaum, Thermal Physics of the Atmosphere, Wiley-Blackwell, Chichester, West Sussex, UK, 2010 Search PubMed .
- H. C. van de Hulst, Light Scattering by Small Particles, Dover, New York, 1981 Search PubMed .
- M. D. Petters and S. M. Kreidenweis, Atmos. Chem. Phys., 2007, 7, 1961–1971 CrossRef CAS .

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