We make a physical–mathematical analysis of the implications that the presence of a large number of tiny bubbles may have, when present, on the thin upper layer of the sea. In our oceanographic example, the bubbles are due to intense rain. It was found that the bubbles increase momentum dissipation in the near surface and affect the surface tension force. For short waves, the implications of increased vorticity are momentum exchanges between wave and mean flow and modifications to the wave dispersion relation. For the direct effect we have analyzed, the implications are estimated to be non-significant when compared to other processes of the ocean. However, we hint at the possibility that our analysis may be useful in other areas of research or practical application.

Our interest is in the consequences of intense rain on the small-scale roughness of the ocean surface. More specifically, we analyze how the presence of tiny, rain-induced air bubbles in the thin upper layer of the ocean affects the high-frequency tail of the gravity wave spectrum, including capillary waves. Although tiny, both in length and height, with respect to the majestic motion of the surface,
these waves are important for both physical and practical reasons. Physically, they control a large part of the exchanges between ocean and atmosphere, and they act as roughness elements for atmospheric turbulence and modulate the interaction between longer waves, currents, and wind

The disappearance of capillary sea waves with rain is not new. Known to mariners since their early attempts (reduced small-scale roughness leads also to reduced breaking), the subject has also been studied in recent times by, e.g.,

On this basis, we make a physical–mathematical analysis of the ensuing processes. In Sect.

To determine the impact of air bubbles on the thin upper level of the ocean, we perform homogenization on the Navier–Stokes equations. Homogenization is a well-established technique in modeling transport in complex media, endowed with statistical homogeneity in the media, as viewed at large scales (see

We assume that air bubbles are distributed uniformly in the transverse direction within the vicinity of the ocean surface. For our specific example, we explore how the presence of bubbles affects the effective viscosity of the fluid averaged over a cell

Velocity and position are denoted by

Let

In the following, we derive the fluid mechanics equations for averaged dynamic quantities (see the related work of

Collecting by orders in

Since the above derivation is not explicitly connected to rain, one can imagine that the same approach applies to similar situations where, for whichever reason, a large number of bubbles is distributed in the enclosing medium. We will touch on this point further in the final discussion.

The tensor

With a fixed ratio

Relative effective momentum dissipation

If we adopt the specific form of

In this section, we assume a statistical distribution for raindrops and derive the consequent density of air bubbles in the ocean surface layer.

The density of raindrops is assumed to follow the Marshall–Palmer distribution (see

To complete our quantitative example, we need to relate the number of bubbles in the upper layer of the ocean to rain. Falling
raindrops generate subsurface air bubbles in the neighborhood of the sea surface (see

The production of air bubbles by rain can be measured by acoustic means (see

Raindrops with a radius greater than 1.1 mm produce type II air resonant bubbles of varying radius. The relationship between the raindrop radius

The relation between the entrained air bubble radius

Small raindrops (

Probability of air bubble creation by large raindrops as a function of raindrop radius

Tying back to Sect.

In doing so, we will make some approximations. First, bubbles are injected at a very limited depth by raindrops, and this depth may vary with bubble size (see

The bubble distribution

Solutions of the advection–diffusion equation necessitate the specification of the velocity and the dispersion, generating non-stationary descriptions that might also include mixing due to wave turbulence (see

In the following, the advection–diffusion equation is simplified to estimate the air bubble concentration in a thin layer close to the surface, similar to those in

Referring to the above approximations, the factor

Making use of Eq. (

The subject of mathematical models for wave dissipation that reproduce the observed dissipation rates observed in laboratory experiments is considered in the study by

In what follows, we will invoke a far less general, but more consistent, way of addressing the question on how the presence of rain bubbles affects water waves.
The derivation of infinitesimal amplitude gravity waves dynamics, starting from Navier–Stokes, appears in

By combining the boundary and non-boundary solutions, the approximate solution of the system, assuming a vanishingly small upper layer thickness and vanishing velocity at depth, is as follows:

The dispersion relation for the waves is as follows:

The effective wave dissipation

The dependence of the effective wave dissipation on the rain rate, using Eqs. (

In both cases, the increase in effective wave damping due to the injection of air, at least as estimated by simple considerations, is small compared to the values reported in the experiments in

Figure

Relative effective wave dissipation

Stimulated by an oceanographic problem, we have developed a physical–mathematical approach to analyze how the presence of a large number of small bubbles affects the characteristics of the containing liquid. Given the general method and having specified the necessary conditions for its application, we focused on a specific example, namely the presence of air bubbles in the thin upper layer of the ocean, and their impact on ocean surface waves. We first derived the (upscaled) equations for mass and momentum conservation. The upscaled velocity and pressure are averages over sub-wave scales over which the material with heterogeneities assumes a stationary distribution. These equations revealed that the effective (upscaled) momentum dissipation is increased when bubbles are present with respect to the no-bubble case.

Under homogenization conditions, the model yields a dynamic equation that has a higher effective diffusion constant with bubbles than without bubbles. This translates into an increased kinematic viscosity of the air–water mixture, which is directly related to an increase in the damping of surface waves. The source for the tiny uniformly distributed bubbles is, in our case, rain, whose small drops generate the bubble distribution (large drops lead to splashing and turbulence in the upper layer, which is a different process). For typical rain rates, we found that the resulting wave damping due to bubbles is not significant with respect to other damping sources, such as turbulence-induced dissipation due to the impact of large raindrops. We also found that the presence of bubbles affects the capillary waves' dispersion relation through a change in the effective surface tension felt by the waves.

The increase in wave damping (related to kinematic viscosity) when bubbles are present may appear counterintuitive at first sight, but it results from kinematic viscosity being the ratio of the dynamic viscosity and the density of the fluid of interest. As bubbles are injected into water, the homogenized kinematic viscosity decreases, but so does the homogenized fluid density, with the overall effect being an increase in the homogenized kinematic viscosity. Physically, the presence of small bubbles at the typically small void ratios of non-breaking waves enhances the momentum transfer between the waves and the mean flow. The enhanced wave rotational component results in higher viscous effects. Moreover, since the density enters explicitly in the definition of the surface tension force, when the volume fraction of air to water is not zero, then the average density of the medium decreases, and, thus, with this decrease comes an increase in surface tension forces. Finally, the viscous effects and changes in the surface tension also lead to changes in the dispersion relation and, as a consequence, in the wave group velocity as well.

Many simplifications were made in the formulation of the rain/wave model. In the momentum equation, we used the most simplistic model for the tensor

A last simplification is related to the fact that waves occur in a salty ocean. The densities of the fresh water due to rain and the salt water of the ocean are different. Both fresh water and air bubbles are, thus, affecting the composition of the ocean in the near surface and, thus, the local density and the tensor

There are several processes in which the mixture of a liquid containing a large number of bubbles or droplets has different characteristics. Examples are a frothy ocean, whatever the content, or cavitating flows, as it happens in ship propellers or in pressurized flows. An example we came across while preparing this article describes how foams are used to ameliorate unwanted ship motion due to sloshing of their holding tank contents (see

We itemize our main conclusions here, as follows:

A general methodology has been developed to quantify the effect of small-scale bubbles on the properties of the containing liquid. Averaged dynamic quantities can be defined, and for these, averaged momentum and mass conservation equations of general applicability can be derived.

A model was derived that permits an estimate of the amount of air injected by rain.

The presence of tiny air bubbles, which we assume are uniformly distributed, changes the physical characteristics of the containing liquid, affecting the momentum balances – the momentum dissipation of irrotational motions is enhanced due to an increase in the rotational component of the flow.

We have also specifically addressed the impact of the presence of air bubbles on gravity waves. We found that air bubbles increase the effective wave damping and, hence, that rain has a damping effect on waves. Changes in the density near the free surface will also affect the surface tension. Since the generation of bubbles by rain is small in the natural setting, the damping effect is small, akin to the damping of waves due to surface contamination. An increase in wave damping has implications for momentum exchanges between the waves and the mean flow. Wave damping changes and surface tension changes impact the wave dispersion relation and the group velocity. These effects are primarily relevant to the dynamics of small and capillary waves.

We have cited a number of situations where, granted certain conditions, our approach can be applied for a general estimate of the overall increased dissipation.

All data are synthetic and generated by the expressions in the paper.

LC brought the unresolved problem of the mechanism, whereby rain calms rough seas, to the attention of the other authors. All authors developed the homogenization model for momentum exchanges near the ocean surface when impurities (in this case air) were present. All authors participated in the model for the connection between rain rates and the presence of resulting air bubbles in the upper ocean. All authors participated in equal measure in the preparation of the paper.

The authors declare that they have no conflict of interest.

The authors wish to thank the Kavli Institute for Theoretical Physics (KITP), at the University of California, Santa Barbara, for their hospitality and for supporting this research project. The KITP is supported, in part, by the National Science Foundation (NSF). Alex Ayet acknowledges the support from the DGA and the French Brittany Regional Council. Juan M. Restrepo acknowledges the support from the NSF/OCE grant. Alex Ayet would like to thank Bertrand Chapron for the insightful comments. We also acknowledge the comments and suggestions from the reviewers, which helped improve the clarity of the paper.

This research has been supported by the National Science Foundation Directorate for Geosciences (NSF/OCE; grant no. 1434198), the NSF (grant no. PHY-1748958) and the DGA (grant no. D0456JE075).

This paper was edited by Joachim Peinke and reviewed by two anonymous referees.