# Numerical Simulation of Micro-Bubbles Dispersion by Surface Waves
## 1 Introduction  
Understanding the dynamics of small-scale processes occurring in the vicinity of the sea surface is important for modeling the exchange of mass, heat and momentum between the atmosphere and the ocean [1, 2]. In many practical situations, clouds of bubbles, produced mainly by breaking surface waves, may also affect the state of the near-surface water layer [3, 4]. Laboratory and field observations [3, 5, 6, 7] as well as recent direct numerical simulations of breaking waves [8] indicate that whereas comparatively large bubbles (with diameters \(d>1\) mm) quickly rise to the surface and burst, smaller (or micro-) bubbles (\(d\)\(\sim\)100 \(\upmu\)m) are suspended in water for a considerably longer times and thus contribute mostly to the void fraction observed in the near-surface oceanic layer.  
According to field observations the concentration of microbubbles (and thus the void fraction) in the upper ocean layer can be considerable even at relatively low winds and affects gas exchange between the air and water [9, 10], production of sea-salt aerosols [11], and the propagation of sound in the upper ocean [12, 13]. Therefore, modeling microbubble dynamics is important for many practical applications.  
Numerical modeling of the dispersion of microbubbles by surface waves and currents taking into account their impact on the near-surface turbulence represents a challenging problem, and various models are employed in numerical investigations of bubbly flows [14]. Known numerical studies of microbubble dispersion are mainly restricted to either bubble-laden isotropic turbulence (with periodic boundary conditions) [15], or boundary-layerflows in the vicinity of a fixed, flat boundary [16]. However, performing numerical simulation of a flow in the vicinity of a waved boundary involves additional efforts required to resolve a strong geometric nonlinearity. There are mainly two methods employed to cope with this problem in numerical studies. One approach is related to using the Volume of Fluid method where the air and water phases are directly resolved (cf., e.g., [8]). However, in order to model a sufficiently high void fraction (on the order of \(10^{-5}\)[12]), the number of microbubbles to be considered in the framework of a fully-resolved numerical experiment with an adaptive mesh refinement may become prohibitively large. Another approach (also used and discussed in the present study) employs a mapping of the physical coordinates onto the curvilinear coordinates where the waved surface is reduced to a flat surface.  
The objective of the present paper is to present a numerical algorithm for evaluation of the dispersion of micro-bubbles by progressive, non-breaking surface (Stokes) waves. The wave shape is prescribed and assumed to be stationary, and unaffected by either bubbles or induced turbulent motions. Thus it is assumed that for typical void fractions observed in the upper ocean layer (on the order of O(\(10^{-5}\)) or less), the impact of bubbles on the carrier surface wave flow remains negligible. Results of numerical experiments also reveal that typical amplitudes of turbulent wave-induced fluctuations are by orders of magnitude smaller as compared to the energy-containing mother wave [17]. However, in the present study, the impact of bubbles on the induced turbulence, although not so significant at the considered void fractions, is accounted for. The algorithm also allows to investigate how wind-induced surface drift affects bubbles dynamics employing a parameterization of the drift velocity at the water surface in terms of the air friction velocity (determined by the velocity scale tied to the surface wave celerity) [18]. The numerical method is based on the algorithms previously developed for modeling atmospheric boundary layer over progressive surface waves under various conditions (stable air-stratification, parasitic capillaries, droplets) [19; 20; 21; 22; 23] and adapted here for modeling a bubble-laden near-surface water layer.