Bubble size distribution in surface wave breaking entraining process

From the similarity theorem,an expression of bubble population is derived as a function of the air entrainment rate,the turbulent kinetic energy (TKE) spectrum density and the surface tension.The bubble size spectrum that we obtain has a dependence of a-2.5+nd on the bubble radius,in which nd is positive and dependent on the form of TKE spectrum within the viscous dissipation range.To relate the bubble population with wave parameters,an expression about the air entrainment rate is deduced by introducing two statistical relations to wave breaking.The bubble population vertical distribution is also derived,based on two assumptions from two typical observation results.