# Numerical Simulation of Micro-Bubbles Dispersion by Surface Waves
## 5 Conclusions  
We have presented an algorithm for numerical modeling of microbubble dispersion in the near-surface water layer of the upper ocean, under the action of non-breaking, progressive surface waves. The algorithm is based on a Eulerian-Lagrangian approach where full, 3D Navier-Stokes equations for the carrier flow induced by a waved water surface are solved in a Eulerian frame, and the trajectories of individual bubbles are simultaneously tracked in a Lagrangian frame, taking into account the impact of the bubbles on the carrier flow and the impact of a wind-induced surface drift. The bubbles diameters are considered in the range from 200 to 400 microns (thus, micro-bubbles), and the effects related to the bubbles deformation and dissolution in water are neglected. The wave shape is prescribed and assumed to be unaffected by either bubbles or induced turbulent motions.  
The simulations results show that bubbles are capable of enhancing the carrier-flow turbulence, as compared to the bubble-free flow, and that the vertical water velocity fluctuations are mostly augmented, and increasingly so by larger bubbles. The results also show that the bubbles dynamics are governed by buoyancy, the surrounding fluid acceleration force, and the drag force whereas the impact of the lift force on the bubble dynamics remains negligible. On the basis of the simulation results, parameterizations for the void fraction fluxes have been obtained. The results show that the vertical component of the void-fraction flux remains unaffected by either the wave motion or wave-induced turbulence as compared to that in a quiescent water. The horizontal void-fraction flux is produced by a mean drift of the bubbles in the direction of the surface wave propagation and can be regarded as analogous to the Stokes drift of Lagrangian (non-inertial) particles in a 2D surface wave modified by the bubbles' ascent. The developed algorithm and parameterizations for void-fraction fluxes can be used for prediction of microbubble dispersion in the ocean upper layer and further employed in large-scale prognostic models.  
As a next step in the development of the algorithm, a more complicated problem is to be considered where the wave motion is fully resolved (i.e., not prescribed) thus allowing modification of the surface wave by either induced turbulence or bubbles. This however remains a subject for future research.  
**Author Contributions:** Algorithm development, performing DNS, data analysis, writing--original draft preparation, O.A.D.; methodology, formal analysis, writing--review and editing, W.-T.T. All authors have read and agreed to the published version of the manuscript.  
**Funding:** This work is supported by the Russian-Taiwanese Joint Research Project through the Russia RFBR Grant (No. 21-55-52005) and the Taiwan MOST Grant (MOST 110-2923-M-002-014 -MY3). ODis additionally supported by RFBR Grant (No. 20-05-00322,) and by the Ministry of Education and Science of the Russian Federation (Task No. 0030-2019-0020).  
**Institutional Review Board Statement:** Not applicable.  
**Informed Consent Statement:** Not applicable.  
**Data Availability Statement:** Not applicable.  
**Conflicts of Interest:** The authors declare no conflict of interest.