Both whitecaps on the surface and bubble plumes beneath them constitute sea foam. For passive remote sensing of the ocean surface, the foam skin depth dictates our interest in the floating foam layers, excluding deeper bubble plumes [25]. Whitecaps corresponding to the inception of wave breaking are usually thick and bright (visually and radiometrically) and are referred to as active whitecaps. Whitecaps in their decaying phase are thinner and dimmer and are referred to as residual whitecaps. Anguelova [24] gives an extended definition of the sea foam.

Sea foam in surface layers has a specific vertical structure comprising large thin-walled bubbles close to the air-foam interface characterized with high air content (dry foam) and smaller, thick-walled bubbles with high water content (wet foam) close to the seawater boundary [20]. A set of macroscopic quantities such as void fraction f_a (defined as the fraction of a unit volume of seawater that is occupied by air) and foam layer thickness t can describe this mechanical structure. A set of microscopic quantities such as bubble dimensions (radius a and wall thickness w) and concentration or size distribution N(a) are useful when one considers the characteristics of the bubbles forming the foam.

To describe oceanic whitecaps under various conditions and in different lifetime stages, the full range of void fractions f_a (from 0 to 1) needs to be considered [24]. It was shown that various functional forms could represent the shape of the void fraction profile in the foam depth [25]. A review of oceanographic data from laboratory and field experiments [25] established a plausible range of foam layer thicknesses encountered in the ocean, from 1 cm to more than 12 cm in active whitecaps and from 0.1 to 1 cm when the whitecaps decay.

Bubble size distributions that characterize bubble plumes at different depths and under various wind speeds have been published extensively; see summaries in [23,26,27]. Meanwhile, there are just a few reports for bubble size distributions within the foam layer. Militskii et al. (Figure 3 in [28]) report bubble diameters within the foam layer in the range from 0.2 mm to 2 mm with the most probable value being 0.3 mm to 0.5 mm. From analysis of 20 foam images, Guo et al. [29] report the mode of the bubble size distribution to be around 0.5 mm “in the bottom half of the foam layer”. Certainly, the range of bubble sizes is much wider in natural conditions under various wind speeds. To encompass as fully as possible this range of bubble sizes, we refer to recent laboratory and field measurements of bubble size distributions in bubble plumes close to the surface, about 0.5 m below the interface, and soon after the breaking event [27,30]. Thus, we consider bubbles with radii a of the air cavity from 0.05 mm to 10 mm with a broad peak around radii of 0.05 mm to 1 mm.

Basic physical relationships from the electromagnetic theory determine the dielectric properties of a medium. Table 1 lists the formulae used to obtain all foam dielectric properties from the complex permittivity of foam ε_f at microwave frequencies F from 1.4 to 37 GHz. The formulae used are valid for scatter free medium, but are applicable to sea foam because a survey of experimental and modeling observations has shown that at the frequencies of interest the scattering in foam is weak to negligible (Section 2.3 in [24]).

Discernible interfaces at the air-foam and foam-water boundaries point toward possible contribution to radiation attenuation due to surface scattering by foam. To investigate whether the foam interfaces with air and water are rough or smooth as seen by microwave radiation, we use two criteria appropriate at microwave frequencies ([32], p. 827). The Fraunhofer criterion defines a surface as smooth if σ < σ_ref, where σ is the standard deviation of the surface height (or RMS height) and the reference value is σ ref = λ 32   cos   θ, with λ being the wavelength and θ the incidence angle of the EM radiation. The other criterion expresses σ in electromagnetic units with the quantity kσ where k = 2π/λ is the radiation wavenumber.

Systematic investigation of various mixing rules yielded several possibilities for computing the dielectric constant of sea foam ε_f at microwave frequencies [24]. For consistency with the study on the foam skin depth [25], here we use ε_f obtained with the Polder-van Santen (PS) mixing rule (Equation (1) in Table 1). The calculations use a double Debye model for the complex dielectric constant of seawater ε [33]. Refer to [24] for the dependencies of ε_f obtained with the PS mixing rule on the frequency, sea surface temperature (SST), and salinity. All results are illustrated with an exponential profile for the foam void fraction f_a(z) with values at the air-foam and foam-water boundaries of 99% and 1%, respectively. Refer to [25] for comparison of exponential f_a(z) to other functional forms of the void fraction profile.

Anguelova and Gaiser [25] obtained and analyzed in detail the skin depth d (Equation (2) in Table 1) of sea foam vertically stratified in layers with thickness t from 0.2 cm to 10 cm. For various frequencies F and the chosen f_a(z) profile (Section 3.1), d could vary from 0.17 cm to no more than 7 cm. Analysis of the relationship between d, t, and the f_a range (upper to lower values at the foam layer boundaries) showed that the thermal emission from foam-covered surfaces could be from the entire foam layer, from part of the foam layer, or from both the foam layer and the seawater.

For any frequency or any layer thickness, d is thicker than its corresponding penetration depth δ (Equation (3) in Table 1) by a factor of up to 2 [25]. Figure 1 shows δ as a function of frequency for three fixed f_a values. For f_a = 90%, δ ranges from a few centimeters to a few meters; for f_a < 60%, δ is mostly less than 1 cm.

The concept of impedance has been applied in different fields of physics ([38]; p. 282). In the EM theory, the intrinsic impedance of a medium follows from the relationship between the amplitudes of the electric and magnetic fields of EM radiation. It depends only on the medium properties and is important when the propagation of EM radiation through a medium is considered.

Figure 2(a) shows the magnitude Re 2 + Im 2 of a complex foam impedance η (Equation (4) in Table 1) for dry (constant vertical profile of f_a = 98%, dash-dot line) and wet (constant profile of f_a = 10%, dashed line) foam. The impedances of the seawater (solid line) and air (dotted line) are shown for comparison. Figure 2(b) depicts how the foam impedance changes as a function of f_a in a vertically structured foam layer for three frequencies. Over the range of considered frequencies, foam impedance varies from 1 to less than 0.2 as f_a decreases from 100% to 0%.

Vertical variations of the real part of the permittivity in foam thickness invoke changes in the radiation wavelength λ_f and propagation constant k_f as compared to that in air λ₀ (Equations (5) and (6) in Table 1). Table 2 shows values of λ_f in dry and wet foam represented with f_a = 98% and 10%, respectively, for all considered frequencies. The corresponding values in seawater λ are also included for reference.

Figure 3 depicts the change of λ_f from λ₀ to λ as it propagates through foam with varying void fraction. In the figure, for each frequency we discern three regions in the trend of λ_f based on the rate of change ∂λ_f/∂f_a. In dry foam, λ_f varies slowly remaining close to λ₀; ∂λ_f/∂f_a changes by a factor of 1.4 (1.1) at 1.4 (37) GHz as void fraction changes from 80% to 100%. For void fractions below 40%, λ_f is quite close to λ and also changes relatively slowly. The slope ∂λ_f/∂f_a changes by a factor of 2.6 on average over the range of frequencies when f_a varies from 40% to 10%. A steeper transition between the air-like and seawater-like behavior takes place in the void fraction range of 80% to 40% with ∂λ_f/∂f_a changing on average by a factor of 4.8. The lower the frequency the more clearly these three regions are seen.

On this basis we can broadly represent a vertically structured foam layer as a sequence of three relatively homogeneous sublayers, which stand for dry, intermediate wet, and wet foams. We identify these sublayers by their f_a values with vertical lines in Figure 3. While we base this conceptual representation on different rates of change of λ_f values, Raizer (Figure 1(a) in [39]) gives a similar schematic of a vertically stratified foam based on experimental observations [28].

Though this is a mechanical, not dielectric, property of the sea foam, we consider the roughness of the foam layer interfaces because it is useful in considering the radiative processes in foam (Section 4.2). Structurally, foam roughness comes about as concave and convex surfaces formed at the boundaries from the bubble caps protruding into the air and indenting into the water. Roughness due to bubbles at the foam boundaries has not been presented in the literature. Here we make an order-of-magnitude estimate using two smoothness criteria (Section 2.3).

We assume that half of a bubble projects out of or indents into a reference surface to form a vertical characteristics length scale. We can thus use bubble radii to represent the roughness height of foam at the air and water interfaces. Bubble radius data are obtained in laboratory and field experiments [7,30,31]. From the large-size end of the bubble data (a from 1 mm to 10 mm), foam rms height at the air-foam boundary is σ_af = 1.3 mm. At the foam-water interface, we estimate σ_fw = 0.2 mm using the small-size end of the bubble data, radii from 0.05 mm to 1 mm. For σ_ref and kσ (Section 2.3) we use λ₀ and λ_f at a void fraction of 10% (Table 2) for the air and water interfaces, respectively.

Figure 6 shows all these estimates together over the range of considered frequencies for θ = 53°. Panel (a) represents results for the Fraunhofer criterion (Section 2.3), while panel (b) shows results for the kσ criterion. At the air-foam interface, σ_ref (curved solid line) of the Fraunhofer criterion changes from about 10 mm at 1.4 GHz to 0.4 mm at 37 GHz. Compared to the rms height due to bubbles at this boundary, σ_af = 1.3 mm (horizontal solid line), the Fraunhofer criterion identifies dry foam as a smooth surface for frequencies below approximately 12 GHz (vertical solid line). For higher frequencies, the dry foam would be a rough surface. Approximately the same frequency limit of 12 GHz divides the wet foam into smooth and rough surfaces for the foam-water interface (dashed lines).

According to the kσ criterion, a surface is smooth if kσ < 0.2 and very rough if kσ > 1.0 ([32]; p. 828). Figure 6(b) shows how the kσ values for air-foam and foam-water boundaries (solid and dashed curves, respectively) compare to these limits (dotted lines). The smooth/rough separation for the boundaries of both dry and wet foam occurs at around 6.5 GHz. That is, according to the kσ criterion, dry and wet foam surfaces are rough for a wider range of frequencies than that established by the Fraunhofer criterion.

Only smooth boundary invokes a specular reflection as that illustrated in Figure 7(a). In real geophysical configurations, all boundaries are more or less rough. This results in a surface scattering pattern, which adds to or completely supersedes the specular reflection.

The strength of surface scattering is defined by the dielectric contrast of two media on both sides of an interface (as in Section 4.1) and the degree of roughness of this interface. In Section 4.1 we showed that, by the virtue of impedance matching, the foam boundaries are weakly to non-reflective surfaces. This relatively weak reflection might, however, be augmented if these boundaries are rough. In this section we address the question whether the roughness of the foam boundaries is large enough to affect the weak, dielectrically determined surface reflection of the sea foam.

Figure 6 showed that, according to the Fraunhofer criterion, both dry and wet foams appear rough to EM radiation above 12 GHz, while the kσ criterion places this limit to about 6.5 GHz. To put into prospective these results, we compare the foam rms height values σ_af and σ_fw used in Figure 6 to the rms height of a wind-roughened sea surface σ_u. We obtain σ_u from the variance of the sea surface displacement h about the mean surface at a given point. It is defined as the zeroth moment of the wave spectrum S(K), σ u ≡ m 0 = ∫ K low K up S ( K ) d K = 〈 h 2 〉 [43,44], where K = 2π/Λ is the wave number of an ocean wave with wavelength Λ. Note that σ_u is a measure of the geometric roughness of a wind-driven sea surface, not the aerodynamic roughness usually represented by the roughness length z₀ [45,46].

The choice of integration limits K_low and K_up above determines the wavelength scales contributing to σ_u. But these length scales, determined mostly by geometric consideration, also need to be tied to length scales important for the aerodynamic surface roughness because this is the variable used for air-sea interaction studies. Our reasoning in choosing K_low and K_up is thus as follows. First, note that remote sensing of the sea surface is effective when the probing EM wavelengths λ₀ are comparable to the length scales of the ocean waves. For example, for scatterometers, ocean length scales ranging from about 40 cm to less than 2 cm correspond to λ₀ from 25 cm to 3 cm at incidence angles of 20° and 60° ([36], p. 1705). On this premiss, one can expect the probing wavelengths λ₀ for the range of frequencies investigated here (Table 2) to be most relevant to ocean lengths scales from about 40 cm to a few millimeters. These length scales represent short gravity and capillary water waves. Next, note that analysis of the mean square slope of wave number spectrum, resulting from oceanographic surface wave measurements, has shown that intermediate-scale waves with Λ between 6 m and 2 cm are the dominant contributors to the aerodynamic surface roughness in oceans [47]. Combining these two considerations, we restrict our estimates to ocean length scales from 40 cm to 2 cm. The corresponding wavenumbers K range from 3.14 rad·cm⁻¹ to 0.16 rad·cm⁻¹. To cover this range of K values, we obtained m₀ for three portions of the wave spectrum: S₂(K), S₃(K), and S₄(K) as defined in [43]. In the estimates we used a friction velocity u_* of 24 cm·s⁻¹ (U₁₀ ≈ 7 m·s⁻¹), a value close to the globally averaged wind.

For the considered range of length scales for ocean waves, we find that the wind-roughened rms height σ_u ranges from about 1 mm for the short end of water waves (Λ of 2 cm) to 5.4 mm for longer water waves (Λ of 40 cm). We obtain similar estimates for σ_u if an empirically formulated wave spectrum in terms of mean square slope is used [47] instead of S(K).

A comparison of these estimates to σ_af = 1.3 mm implies that there could be cases in which the roughness of the dry foam could be comparable to the roughness created by the shortest capillary waves. But the dielectric contrast between dry foam and air is almost none existing (Figure 2 and Section 4.1). This leads ultimately to lack of surface scattering at the air-foam boundary.

For the wet foam boundary, its roughness (σ_fw = 0.2 mm) is an order of magnitude below that of the shortest capillary waves (1 mm). This implies that wet foam, whether at the surface or as a sublayer in the foam thickness, is usually smoother than the sea surface roughness. We thus deem the wet foam boundary as incapable of providing significant surface scattering as well.

Our conclusion of negligible surface scattering by any of the foam layer boundaries is consistent with the general observation that the reflectivity of a foam-covered surface is not comparable to that of foam-free rough sea. In fact, microwave models (e.g., [48]) usually assume that “foam tends to decrease the reflectivity” ([49], p. 24). Note that the conclusions in this section result from the combined effect of the geometric roughness and the dielectric contrasts at the boundaries of a foam layer with well-stratified void fraction. Deviations from these conclusions are possible when the void fraction profile is limited (Section 5.2).

While we recognize the presence of scattering-driven pathway of attenuation in foam and expect it to play an important role in a structure conducive to significant scattering (namely, the close packing of bubbles), a previous survey ([24], Section 2.3) has shown that at the considered frequencies the scattering within the sea foam is weak. Here, we explain this counterintuitive conclusion.

Generally, one conjectures on the presence, or lack of, volume scattering in a medium by considering its effective penetration depth and the spatial and dielectric irregularities in its volume ([32], Chapter 11–4). The spatial and dielectric irregularities are usually quantified with their spatial distribution and dielectric contrast. The size parameter and the refractive index of a medium express these two quantities. Thus, in the following we analyze δ, x and m for clues regarding the volume scattering in foam.

Our reasoning proceeds in two steps. First, employing general scattering rules, we establish the expected scattering behavior for various foam structures. Second, we show how this expected behavior is altered (or not followed) in natural, vertically stratified foam layers.

Section 4.3 provides a physical explanation of why, of the two pathways for absorption losses in foam, the scattering-driven one, though present, is not a major contributor. This thus qualitatively confirms the conclusion Anguelova and Gaiser ([25]; Section 4.5) reached that the medium-intrinsic absorption is the defining controlling factor for high foam absoprtivity. Anguelova and Gaiser [25] also showed that the thin water-laden wet foam could provide a strong emission from foam-covered sea surfaces. Here we give further evidence for this conclusion using the results for x and m.

Besides small skin depth ([25]; Section 4.1), another quantitative measure for the high absorptivity of foam as a medium is the fact that the refractive index (and permittivity ε_f) of foam is a complex number (m = m′ – im″, Section 3.5.3). Media with even very small m″ values have been called strongly absorbing. On the basis of m″ ≠ 0 and m – 1 << 1, which means that a negligible fraction of the radiation is reflected, one could conjecture that even dry foam (Figure 5, solid lines) would be absorptive. In fact, a combination of refractive index values similar to that of dry foam with large size parameter (e.g., x ≥ 100) could represent a medium whose behavior approaches that of a black body ([37], p. 269).

However, such a combination of m and x is not observed in natural dry foam because x does not exceed 10 (Figure 8(a)). The combination of a narrow range of x (x ≤ 10) with m″ << 1 thus excludes the possibility that dry foam contributes to medium-intrinsic absorption losses. Meanwhile, m′ and m″ values for wet foam are of the same order (Figure 5, dashed lines) signifying that this is an intrinsically absorptive medium. However little the volume scattering in wet foam (Section 4.3), it adds to the medium-intrinsic absorption, making the wet foam a more effective absorber than the dry foam.

Various suggestions can be found in the literature explaining the high emissivity of foam-covered surfaces. Williams [50] was the first to suggest the importance of the impedance matching that the sea foam provides between the air and seawater. Consequent experiments led to the tentative conclusion that roughness expressed as “distortion of the water surface at the interstices between the bubbles at the bottom of the foam layer” ([51], p. 3097) together with the dielectric constant of foam “produce a zone of radically tapered dielectric constant” ([19], p. 223; cited also by [21]; [36], p. 1454; [48]). The Russian literature on the subject approaches the problem from a microscopic point of view and promotes the explanation that the bubbles within the foam “represent some kind of small black bodies, which intensively absorb the EM energy owing to inter-bubble diffraction” ([52], p. 511). Describing the sea foam as “a nonreflecting … absorbing but partially transparent medium”, Wilheit ([48], p. 246) touched upon the effective transmission of EM radiation through foam. Williams [19] measurements involving foam on different underlying substrates, water or metal, suggest that not just the foam itself, but also the system of foam floating on seawater is the one that could lead to high emissivity.

This brief account of proposed explanations demonstrates that one or another element of the physical system that a foam-covered sea surface represents has been identified and used to interpret the observed high emissivity. None of these explanations, however, combines these elements into a consistent whole. Meanwhile, the analysis in this study of the radiative processes in foam (Sections 4.1–4.4) shows that none of these separate elements alone is enough to account for the high emissivity e_f of sea foam. Rather, all foam properties and all radiative processes engage in an emergent behavior (i.e., the whole is greater than the sum of its parts [53]) and contribute in some way to bring about the overall effect of a high e_f. Moreover, we argue that what makes the sea foam an exceptional emitter is the combination of these properties and processes in a vertically structured foam layer floating on seawater surfaces. The following is a summary of our conceptual understanding of how this happens.

First, two components ensure the high emissivity of foam-covered ocean surfaces: (1) High absorption losses in a coupled foam-seawater system; and (2) effective transmission of EM radiation through the foam. If one of these components were missing, a high e_f could not be guaranteed. The potential of the foam-seawater system for strong absorption and emission cannot be realized without effective transmission and propagation of radiation across layer boundaries into the foam; and vice versa—effective transfer across the layer boundaries is not enough if a highly absorptive agent is not in place to carry on high absorption and emission.

Second, absorption losses in foam are provided by medium-intrinsic and scattering-driven absorption. Absorption in wet foam is relatively more important than absorption in dry foam. Furthermore, though strong, the absorption of the foam layer itself might remain somewhat limited. The seawater below the foam layer is a major absorptive agent whose emission adds to that of the foam layer to yield the highest absorption losses in a coupled foam-seawater system.

Third, with its intrinsic impedance, foam on the ocean surface, in any stage of its lifetime, provides impedance matching between the contrasting dielectric properties of air and seawater. This decreases reflections at the air-foam and foam-seawater boundaries and assists effective entry of EM radiation into the foam-seawater system. Existing, yet weak surface scattering at these boundaries and weak volume scattering within the foam provide the maximal portion of the propagating radiation to be delivered to and absorbed by the wet foam and seawater.

Forth, the foam-seawater system reaches its maximal effectiveness and becomes a black-body-like emitter when a vertical profile of properties in the depth of a foam layer controls the relative importance and contribution of the radiative processes. Only gradual increase of seawater content and gradual change of spatial inhomogeneities minimize radiation attenuation in dry foam and maximize the absorption losses in wet foam and underlying seawater. Only a gradual change of the foam intrinsic impedance over the widest possible range of values, from that of the air to that of the seawater, provides weakly reflective to non-reflective foam boundaries and maximizes the entry of EM radiation.

In short, only in vertically structured foam layers every property and process conspires to bring maximum EM radiation to the most effective absorptive agent and thus producing maximum absorption and emission.

Our discussion of the sub-surface bubble plume and the overlying foam layer generated is presented up to this point as a static, time-independent system. The impact that the dynamic of the system may have on the foam structure and its emissivity was alluded to in our considerations of the effectiveness of the impedance matching provided by whitecaps in their active and residual lifetime stages (Section 4.1.2). Here we address further the temporal variations of the system.

The rational for these considerations is that the void fraction in the bubble clouds formed after wave breaking varies by roughly two orders of magnitude between initial entrainment and the loss of large bubbles, an interval of roughly a second or so. Indeed, in a laboratory study, Anguelova and Huq [59] observed disintegration of the initial compact form of a whitecap into winding streaks of foam after only 100 ms. Such a time scale is comparable to the time period of 90 ms reported by Deane and Stokes [60] for the dispersion of a compact mass of bubbles into individual ones in a surf zone. For about one wave period (about 1 s), Anguelova and Huq [59] documented a decrease of the void fraction from near unity to about 20%. They also observed that the evolution of the bubble plume, and the foam layer above, could persist for many wave periods.

The question is: What is the impact of such dynamical changes on a field of whitecaps in the open ocean? Statistical analysis of void fraction values representative for the initial air entrainment and for the subsequent decaying stage suggest the presence of both thick foam layers with high void fractions of 70%–90%, and thinner, wetter foam layer with lower void fraction values (<80%) (e.g., Figures 12 and 13 in Anguelova and Huq [59]). Because evolving bubble clouds keep replenishing the foam layers on the surface, their statistics reveal that the temporal variations lead to a high variability of the foam layer properties. That is, in the open ocean at any moment foam layers with a distribution of void fraction values at their boundaries could exist. These would include combinations of dry foam next to a foam-water boundary or wet foam next to air. Therefore, some of the various foam structures may exhibit a distinct impedance discontinuity at the mean ocean surface; others may represent stratified foam well. This in turn has important implications for the reflection and transmission of microwave energy.

The next logical question is: How can we model such a variability of the foam layer properties? One simple approach is to assume that the foam structure analyzed in the present study represents a likely scenario in the early stage of foam development. A significant improvement upon such a simple approach would be to develop an elaborate physical model for the time-varying properties of sub-surface bubble clouds, about which relatively little is known, or the surface foam they generate, about which even less is known. A starting point could be to generate a void fraction profile likely to be representative of the bubble plume/foam system during whitecap decay. Anguelova and Gaiser ([25], Section 4.4) discussed the modeling of void fraction profiles by noting that one needs to consider the profile shape, profile range, and the profile variations with foam thickness, and how these choices affect the foam emissivity.

Foam emissivity can be modeled following two approaches [36,52,54]. The intensity approach employs the radiative transfer theory to calculate the microwave radiance of a medium. The wave approach employs solutions of Maxwell equations to compute the scattering losses in a medium and use those to obtain emissivity. The PS formula represents the most rigorous starting point when obtaining the scattering of sea foam is pursued. The reason is that the PS formula is identical to the low frequency limit form of the effective permittivity obtained with the strong permittivity fluctuations theory for scattering from a random medium ([54], pp. 378–379). According to Tsang et al. ([54], p. 429), “the applicability of the mixing formula to a particular dense medium in the very low frequency limit will help in deciding the multiple scattering theory to be employed at higher frequencies.” Because we use the PS mixing rule in this study (Section 3.1), we are in good position to qualitatively discuss which scattering theory would be appropriate when the wave approach is chosen to obtain sea foam emissivity.