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1 Remote Sens. 2012, 4, ; doi: /rs Article OPEN ACCESS Remote Sensing ISSN Dielectric and Radiative Properties o Sea Foam at Microwave Frequencies: Conceptual Understanding o Foam Emissivity Magdalena D. Anguelova * and Peter W. Gaiser Remote Sensing Division, Naval Research Laboratory, Washington, DC 20375, USA; peter.gaiser@nrl.navy.mil * Author to whom correspondence should be addressed; maggie.anguelova@nrl.navy.mil; Tel.: ; Fax: Received: 14 March 2012; in revised orm: 16 April 2012 / Accepted: 18 April 2012 / Published: 27 April 2012 Abstract: Foam raction can be retrieved rom space-based microwave radiometric data at requencies rom 1 to 37 GHz. The retrievals require modeling o ocean surace emissivity ully covered with sea oam. To model oam emissivity well, knowledge o oam properties, both mechanical and dielectric, is necessary because these control the radiative processes in oam. We present a physical description o oam dielectric properties obtained rom the oam dielectric constant including oam skin depth; oam impedance; wavelength variations in oam thickness, roughness o oam layer interaces with air and seawater; and oam scattering parameters such as size parameter, and reraction index. Using these, we analyze the scattering, absorption, relection and transmission in oam and gain insights into why volume scattering in oam is weak; why the main absorption losses are conined to the wet portion o the oam; how the oam impedance matching provides the transmission o electromagnetic radiation in oam and maximizes the absorption; and what is the potential or surace scattering at the oam layers boundaries. We put all these elements together and oer a conceptual understanding or the high, black-body-like emissivity o oam loating on the sea surace. We also consider possible scattering regimes in oam. Keywords: passive remote sensing; whitecaps; sea oam; oam raction; whitecap raction; oam permittivity; dielectric constant; oam dielectric properties

2 Remote Sens. 2012, Introduction Oceanic whitecaps are a conspicuous expression o wave breaking with air entrainment. Creating rats o sea oam on and bubble plumes below the sea surace, breaking waves transer momentum, heat, and mass across the air-sea interace [1 4] and alter the optical ield o the water column, the surace roughness, and the ambient noise in the ocean [5 7]. One (among others) physical quantity that is used to evaluate these air-sea interaction processes is the whitecap raction, deined as the raction o the ocean surace covered by oam. Whitecap raction can be evaluated with a number o parameterizations developed or open ocean [8] and coastal zone [9] using photographic measurements [10 12] in the visible portion o the electromagnetic (EM) spectrum. Whitecaps can also be observed at inrared [13] and microwave wavelengths [14]. Recent eorts to retrieve whitecap raction rom space-based radiometric observations o the ocean surace at microwave requencies rom 1 to 37 GHz show promise [15 18]. The retrieval algorithm requires the modeling o the ocean surace emissivity ully covered with sea oam e. Laboratory and ield measurements have shown that e at microwave requencies is high, close to that o a blackbody [14,19 22]. To explain and model e, one needs to understand the radiative processes taking place in oam. Knowledge o the mechanical and dielectric properties o sea oam is necessary or such an understanding. While extensive oceanographic research acilitates the characterization o the oam mechanical properties (Chapters 4.4 and 4.5 in [23]), the knowledge o the oam dielectric properties is incomplete. Basis or the dielectric properties o a medium is its relative dielectric constant (permittivity). Permittivity determines such intrinsic characteristics o the medium as skin (and penetration) depth, impedance, reractive index, etc. These quantities control the attenuation due to absorption and scattering and the transmission and relection o the EM radiation in the medium. These radiative processes aect, in turn, the energy lost in and then emitted by the medium. Thereore, an understanding o the relative importance o all these characteristics and processes can assist the choice o an approach to model e, the ormulation o relevant assumptions and/or simpliications, and the interpretation o the model results. We address the need or more complete characterization o oam dielectric and radiative properties in a series o papers. Anguelova [24] analyzed the available inormation to determine suitable ormula to predict the complex permittivity o sea oam ε. Anguelova and Gaiser [25] ocused on the skin depth o oam layers with a vertical void raction proile. In the present paper, we consider the oam properties at microwave requencies, which control undamental radiative processes such as relection, scattering, and transmission in vertically structured oam layers. The parameters used or this consideration are oam impedance η, size parameter x, and reractive index m (Section 3). Analysis o these dielectric properties helps to identiy the unique traits, which the undamental radiative processes acquire when microwave radiation interacts with sea oam (Section 4). Finally, we put all our indings together to ormulate a concept o the high oam emissivity (Sections 5.1 and 5.2) and discuss scattering regimes in oam (Section 5.3). The inormation, indings, and generalizations in this series o papers orm the physical basis on which we have built our oam emissivity model (briely described in [16] and to be detailed in a orthcoming paper).

3 Remote Sens. 2012, Background 2.1. Sea Foam as a Medium Both whitecaps on the surace and bubble plumes beneath them constitute sea oam. For passive remote sensing o the ocean surace, the oam skin depth dictates our interest in the loating oam layers, excluding deeper bubble plumes [25]. Whitecaps corresponding to the inception o wave breaking are usually thick and bright (visually and radiometrically) and are reerred to as active whitecaps. Whitecaps in their decaying phase are thinner and dimmer and are reerred to as residual whitecaps. Anguelova [24] gives an extended deinition o the sea oam. Sea oam in surace layers has a speciic vertical structure comprising large thin-walled bubbles close to the air-oam interace characterized with high air content (dry oam) and smaller, thick-walled bubbles with high water content (wet oam) close to the seawater boundary [20]. A set o macroscopic quantities such as void raction a (deined as the raction o a unit volume o seawater that is occupied by air) and oam layer thickness t can describe this mechanical structure. A set o microscopic quantities such as bubble dimensions (radius a and wall thickness w) and concentration or size distribution N(a) are useul when one considers the characteristics o the bubbles orming the oam. To describe oceanic whitecaps under various conditions and in dierent lietime stages, the ull range o void ractions a (rom 0 to 1) needs to be considered [24]. It was shown that various unctional orms could represent the shape o the void raction proile in the oam depth [25]. A review o oceanographic data rom laboratory and ield experiments [25] established a plausible range o oam layer thicknesses encountered in the ocean, rom 1 cm to more than 12 cm in active whitecaps and rom 0.1 to 1 cm when the whitecaps decay Bubbles in a Foam Layer Bubble size distributions that characterize bubble plumes at dierent depths and under various wind speeds have been published extensively; see summaries in [23,26,27]. Meanwhile, there are just a ew reports or bubble size distributions within the oam layer. Militskii et al. (Figure 3 in [28]) report bubble diameters within the oam layer in the range rom 0.2 mm to 2 mm with the most probable value being 0.3 mm to 0.5 mm. From analysis o 20 oam images, Guo et al. [29] report the mode o the bubble size distribution to be around 0.5 mm in the bottom hal o the oam layer. Certainly, the range o bubble sizes is much wider in natural conditions under various wind speeds. To encompass as ully as possible this range o bubble sizes, we reer to recent laboratory and ield measurements o bubble size distributions in bubble plumes close to the surace, about 0.5 m below the interace, and soon ater the breaking event [27,30]. Thus, we consider bubbles with radii a o the air cavity rom 0.05 mm to 10 mm with a broad peak around radii o 0.05 mm to 1 mm Deinitions o Electromagnetic Properties Basic physical relationships rom the electromagnetic theory determine the dielectric properties o a medium. Table 1 lists the ormulae used to obtain all oam dielectric properties rom the complex permittivity o oam ε at microwave requencies F rom 1.4 to 37 GHz. The ormulae used are valid

4 Remote Sens. 2012, or scatter ree medium, but are applicable to sea oam because a survey o experimental and modeling observations has shown that at the requencies o interest the scattering in oam is weak to negligible (Section 2.3 in [24]). Discernible interaces at the air-oam and oam-water boundaries point toward possible contribution to radiation attenuation due to surace scattering by oam. To investigate whether the oam interaces with air and water are rough or smooth as seen by microwave radiation, we use two criteria appropriate at microwave requencies ([32], p. 827). The Fraunhoer criterion deines a surace as smooth i σ < σ re, where σ is the standard deviation o the surace height (or RMS height) and the reerence λ value is σ re =, with λ being the wavelength and θ the incidence angle o the EM radiation. 32cosθ The other criterion expresses σ in electromagnetic units with the quantity kσ where k = 2π/λ is the radiation wavenumber. 3. Foam Dielectric Properties 3.1. Sea Foam Permittivity Systematic investigation o various mixing rules yielded several possibilities or computing the dielectric constant o sea oam ε at microwave requencies [24]. For consistency with the study on the oam skin depth [25], here we use ε obtained with the Polder-van Santen (PS) mixing rule (Equation (1) in Table 1). The calculations use a double Debye model or the complex dielectric constant o seawater ε [33]. Reer to [24] or the dependencies o ε obtained with the PS mixing rule on the requency, sea surace temperature (SST), and salinity. All results are illustrated with an exponential proile or the oam void raction a (z) with values at the air-oam and oam-water boundaries o 99% and 1%, respectively. Reer to [25] or comparison o exponential a (z) to other unctional orms o the void raction proile Foam Skin and Penetration Depths Anguelova and Gaiser [25] obtained and analyzed in detail the skin depth d (Equation (2) in Table 1) o sea oam vertically stratiied in layers with thickness t rom 0.2 cm to 10 cm. For various requencies F and the chosen a (z) proile (Section 3.1), d could vary rom 0.17 cm to no more than 7 cm. Analysis o the relationship between d, t, and the a range (upper to lower values at the oam layer boundaries) showed that the thermal emission rom oam-covered suraces could be rom the entire oam layer, rom part o the oam layer, or rom both the oam layer and the seawater. For any requency or any layer thickness, d is thicker than its corresponding penetration depth δ (Equation (3) in Table 1) by a actor o up to 2 [25]. Figure 1 shows δ as a unction o requency or three ixed a values. For a = 90%, δ ranges rom a ew centimeters to a ew meters; or a < 60%, δ is mostly less than 1 cm.

5 Remote Sens Table 1. Basic relationships rom the electromagnetic theory used to obtain oam dielectric properties. Equation # Property Symbol (Units) Formula Reerence Notes Dielectric properties (1) Dielectric constant ε ε ε ε + 2ε ε = a ( ε ε ) 1+ 2ε + 2( ε ε ) d () z dz = 1 Equation (9.7) in [34] Seawater is environment with permittivity ε; Bubbles are inclusions with void raction a α α ield attenuation coeicient (2) Skin depth d (mm) 0 p. 847 in [32] F requency (Hz) 2π F α() z =. Im ε () c speed o light (cm s 1 ) z c (3) Penetration depth δ (mm) δ k a () z dz = 1 Scattering ignored 0 ditto Extinction Absorption δ = d ε () i.e., z = const k e k a = 2α (4) Intrinsic impedance η η = 1/ ε p. 226 in [35] Normalized (relative), complex (5) Wavelength in oam λ (cm) λ = λ 0 ε p in [36] λ 0 ree-space wavelength (6) Propagation constant (wave number) k (cm 1 ) k = 2π λ p. 116 in [37] Scattering parameters (7) Size parameter x x = a ( 2 π λ ) = k a p. 128 in [37] a bubble radius (8) Reraction index m m = ε = m im m and m real and imaginary p. 116 in [37] parts o m

6 Remote Sens. 2011, Figure 1. Foam penetration depth as a unction o requency δ (F) obtained with Polder-van Santen mixing rule with exponential void raction proile ranging rom 99% at the air-oam interace and 1% at the oam-seawater boundary at ixed seawater temperature (T s = 20 C) and salinity (S = 34 psu) at three values o oam void raction a Foam Impedance The concept o impedance has been applied in dierent ields o physics ([38]; p. 282). In the EM theory, the intrinsic impedance o a medium ollows rom the relationship between the amplitudes o the electric and magnetic ields o EM radiation. It depends only on the medium properties and is important when the propagation o EM radiation through a medium is considered. Figure 2. Normalized intrinsic impedance o sea oam obtained rom oam permitivity (Equation (4) in Table 1) at ixed seawater temperature (T s = 20 C) and salinity (S = 34 psu): (a) as a unction o requency or seawater, air, wet oam ( a = 10%), and dry oam ( a = 98%); (b) as a unction o oam void raction at three requencies. Figure 2(a) shows the magnitude 2 2 Re + Im o a complex oam impedance η (Equation (4) in Table 1) or dry (constant vertical proile o a = 98%, dash-dot line) and wet (constant proile o a = 10%, dashed line) oam. The impedances o the seawater (solid line) and air (dotted line) are

7 Remote Sens. 2012, shown or comparison. Figure 2(b) depicts how the oam impedance changes as a unction o a in a vertically structured oam layer or three requencies. Over the range o considered requencies, oam impedance varies rom 1 to less than 0.2 as a decreases rom 100% to 0% Wavelength Changes in Foam Layers Vertical variations o the real part o the permittivity in oam thickness invoke changes in the radiation wavelength λ and propagation constant k as compared to that in air λ 0 (Equations (5) and (6) in Table 1). Table 2 shows values o λ in dry and wet oam represented with a = 98% and 10%, respectively, or all considered requencies. The corresponding values in seawater λ are also included or reerence. Table 2. Wavelengths at ixed SST (20 C) and salinity (34 psu) in air (λ 0 ), in seawater (λ), in dry and wet oam (λ ). F GHz λ 0 cm (in air) λ cm λ cm (in seawater) a = 98% a = 10% Figure 3. Wavelength in oam λ (cm) (Equation (5) in Table 1) as a unction o oam void raction at ixed seawater temperature (T s = 20 C) and salinity (S = 34 psu) and dierent requencies. Vertical lines divide the a range into three regions whose values can be associated with bubble diamaters predominant or each region. Figure 3 depicts the change o λ rom λ 0 to λ as it propagates through oam with varying void raction. In the igure, or each requency we discern three regions in the trend o λ based on the rate o change λ / a. In dry oam, λ varies slowly remaining close to λ 0 ; λ / a changes by a actor o 1.4 (1.1) at 1.4 (37) GHz as void raction changes rom 80% to 100%. For void ractions below 40%,

8 Remote Sens. 2012, λ is quite close to λ and also changes relatively slowly. The slope λ / a changes by a actor o 2.6 on average over the range o requencies when a varies rom 40% to 10%. A steeper transition between the air-like and seawater-like behavior takes place in the void raction range o 80% to 40% with λ / a changing on average by a actor o 4.8. The lower the requency the more clearly these three regions are seen. On this basis we can broadly represent a vertically structured oam layer as a sequence o three relatively homogeneous sublayers, which stand or dry, intermediate wet, and wet oams. We identiy these sublayers by their a values with vertical lines in Figure 3. While we base this conceptual representation on dierent rates o change o λ values, Raizer (Figure 1(a) in [39]) gives a similar schematic o a vertically stratiied oam based on experimental observations [28] Scattering Parameters o Foam Characteristic Bubbles or Each Foam Sublayer The known scattering rules o individual particles [37] can be applied to oam mixture because the oam macroscopic parameters (e.g., a and λ ) are related to those o a separate bubble (e.g., b and λ b ). Qualitatively, such a relationship is possible because the vertical structure o a oam layer is associated with segregation by size o the bubbles orming the oam (see [24] and Section 2.1). Indeed, we expect that dry oam consist predominantly o large bubbles with an upper limiting radius o 10 mm (Section 2.2). Meanwhile, most o the smaller bubbles with a lower limiting radius o 0.05 mm would concentrate preerentially in the lower, wet part o the oam layer. The intermediately wet oam between these two limiting portions would comprise bubbles with wider size variation, with most typical sizes around the peak o the bubble size distribution (Section 2.2). Thus, the three sublayers o vertically structured oam associated with speciic a values (Section 3.4) could be alternatively dierentiated by expected speciic, predominant bubble radius a (Figure 3). Quantitatively, this relationship is based on the act that oam void raction can be obtained rom the bubble size distribution [7]. The void raction a o a collection o N bubbles with a ixed outer radius a and a wall (shell) o seawater w = a qa, where qa is the inner radius o a bubble, is 1 air volume in a bubble 3 a b = N = N total volume o a bubble (1) For example, i a = 10 mm and w = 100 µm (i.e., q = 1 w/a = 0.99), then b 97%. Since a bubble with such dimensions (large a, thin wall w) is characteristic or the dry oam sublayer represented with a = 98%, b can characterize the mixture s a. Similarly, the void ractions o smaller bubbles, e.g., a = 0.3 mm and 0.1 mm, having walls with a thickness w o 0.05 mm (i.e., q = 0.83 and 0.5 respectively) are representative o oam with a b = q 3 60% and 12%, respectively Size Parameter in a Foam Layer The size parameter x (or diraction parameter [40]) describes the relationship between the radiation wavelength in a mixture and the sizes o the scatterers in it. In the case o oam, x quantiies how this relation changes in oam layer thickness as both λ and the dominant bubble size a change. q

9 Remote Sens. 2012, Figure 4 shows a amily o curves or all considered requencies representing x as a unction o a o a oam mixture comprising bubbles with a ixed radius a = 0.3 mm (a size within the peak o the bubble size distribution, Section 2.2). Changes o a in the depth o oam comprising bubbles with the ixed radius a are realized via variations o bubble wall thickness w which yield q 3 = a (Section 3.5.1). Figure 4. Size parameter in oam x (Equation (7) in Table 1) as a unction o oam void raction at various requencies and bubble radius a = 0.3 mm at the peak o the bubble size distribution obtained rom oceanographic measuremetns. The igure shows that x increases as oam changes rom dry to wet. This change o x in oam depth is stronger or lower requencies in accord to the variations o the λ values shown in Table 2. For instance, at 37 GHz the value o x at a = 98% changes by a actor o 3.8 compared to the values o x at a = 10%, while at 10.7 GHz this change is by a actor o 6.6. Table 3 summarizes the ranges o variation o x in oam thickness using the values o the predominant bubble size or each sublayer (Columns 1 and 2 in Table 3). Table 3. Ranges o variation o size parameter (x), and oam reractive index (Re{m} and Im{m} ) in oam depth. Foam Mixture x (Figure 4) Re{m} (Figure 5(a)) Im{m} (Figure 5(b)) a = 98% a = 10 mm a = 60% a = 0.3 mm a = 10% a = 0.05 mm Reractive Index o Foam The reractive index o oam m is obtained with Equation (8) in Table 1. Figure 5 presents the real and imaginary parts o m (m and m in panels a and b, respectively) as a unction o requency at void ractions representative or each o the three sublayers 98%, 60%, and 10%. At a ixed void raction value, the dependence on requency is noticeable. But the variations due to changes o the void

10 Remote Sens. 2012, raction are stronger, especially or the imaginary part. As expected, in dry oam ( a = 98%) m 1 and m eatures the lowest values o O(10 3 ). As the seawater content increases in oam depth, so does the reraction index (Table 3): at a = 60%, m = O(3) and m = O(1); at a = 10% m = O(5 to 9) and m = O(3). Figure 5. Foam reractive index m (Equation (8) in Table 1) as a unction o requency at oam void ractions representing dry oam ( a = 98%), wet oam ( a = 10%) and intermediate stage oam ( a = 60%) at ixed seawater temperature (T s = 20 C) and salinity (S = 34 psu): (a) Real part, Re{m}; (b) Imaginary part, Im{m} Roughness o Foam Layer Interaces Though this is a mechanical, not dielectric, property o the sea oam, we consider the roughness o the oam layer interaces because it is useul in considering the radiative processes in oam (Section 4.2). Structurally, oam roughness comes about as concave and convex suraces ormed at the boundaries rom the bubble caps protruding into the air and indenting into the water. Roughness due to bubbles at the oam boundaries has not been presented in the literature. Here we make an order-o-magnitude estimate using two smoothness criteria (Section 2.3). We assume that hal o a bubble projects out o or indents into a reerence surace to orm a vertical characteristics length scale. We can thus use bubble radii to represent the roughness height o oam at the air and water interaces. Bubble radius data are obtained in laboratory and ield experiments [7,30,31].

11 Remote Sens. 2012, From the large-size end o the bubble data (a rom 1 mm to 10 mm), oam rms height at the air-oam boundary is σ a = 1.3 mm. At the oam-water interace, we estimate σ w = 0.2 mm using the small-size end o the bubble data, radii rom 0.05 mm to 1 mm. For σ re and kσ (Section 2.3) we use λ 0 and λ at a void raction o 10% (Table 2) or the air and water interaces, respectively. Figure 6 shows all these estimates together over the range o considered requencies or θ = 53. Panel (a) represents results or the Fraunhoer criterion (Section 2.3), while panel (b) shows results or the kσ criterion. At the air-oam interace, σ re (curved solid line) o the Fraunhoer criterion changes rom 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 a = 1.3 mm (horizontal solid line), the Fraunhoer criterion identiies dry oam as a smooth surace or requencies below approximately 12 GHz (vertical solid line). For higher requencies, the dry oam would be a rough surace. Approximately the same requency limit o 12 GHz divides the wet oam into smooth and rough suraces or the oam-water interace (dashed lines). Figure 6. Criteria or surace smoothness (Section 2.3) at the air-oam boundary (solid lines) and oam-water boundary (dashed lines): (a) Fraunhoer criterion; (b) kσ criterion. Fraunhoer criterion, σ re (mm) a) Dry oam Smooth Rough Wet oam Smooth Rough σ re at boundary: Air-oam Foam-water θ = 53 ο σ a = 1.3 mm σ w = 0.2 mm b) Criterion kσ 10-1 Smooth Rough kσ = 0.2 kσ at boundary: Air-oam Foam-water Frequency, F (GHz) According to the kσ criterion, a surace is smooth i kσ < 0.2 and very rough i kσ > 1.0 ([32]; p. 828). Figure 6(b) shows how the kσ values or air-oam and oam-water boundaries (solid and dashed curves, respectively) compare to these limits (dotted lines). The smooth/rough separation or the boundaries o both dry and wet oam occurs at around 6.5 GHz. That is, according to the kσ criterion, dry and wet oam suraces are rough or a wider range o requencies than that established by the Fraunhoer criterion.

12 Remote Sens. 2012, Radiative Processes in Foam Following the Kircho s law or thermal radiation [41] the emissivity o a medium at thermal equilibrium equals its absorptivity we know that in order to model the medium emissivity we need to evaluate the absorption losses in it. We recognize two possible pathways o absorption in oam layers (Section 4.5 in [25]). We call medium-intrinsic the absorption o the radiation propagating along the line o observation due to the presence o seawater, a lossy medium. We call scattering-driven the absorption caused by a lengthened path o propagation o the radiation along the line o observation due to the scattering among densely packed bubbles. The scattering-driven absorption enhances the medium-intrinsic absorption. Note that we consider the attenuation caused by scattering-driven absorption dierent rom the attenuation caused by loss o radiation scattered out o the line o observation. Both o these scattering eects cause attenuation in the medium, but one o them has a positive consequence or our considerations because it enhances the absorption and thus contributes to the emission o the medium. Our descriptive analysis does not allow evaluation o the relative magnitude o these two scattering eects. However, heeding the act that sea oam has high thermal emission, which necessarily derives rom strong absorption losses, we assume that the lost o radiation available or absorption via scattering is small in oam. The processes, which deliver the EM radiation available or absorption in the oam are relection and surace scattering at the air-oam and oam-water boundaries, scattering within the oam, and transmission through the oam. In the ollowing, we analyze the oam dielectric properties presented in Section 3 and gain physical understanding o how these undamental radiative processes acquire unique eatures because o the presence o vertical void raction proile in a oam layer Foam Relection and Transmission Foam as Impedance Matching In absence o oam, the air water interace is highly relective. The reason or this is the large dierence between the dielectric constants o air (ε 0 = 1 i.0) and seawater, which leads to very dierent intrinsic impedances (Figure 2(a), solid and dotted lines). Because the impedance dierence between two media determines the relectivity at their interace, r Δη (e.g., p. 228 in [35]), the relectivity at the air water boundary is large. Thus, only a small portion o down-welling atmospheric radiation impinging on the air water interace would pass and be available or absorption within the oam-seawater system. As Kircho s law holds, a oam ree water surace has low emission. The cartoon in Figure 7(a) illustrates this. The situation changes when oam is present at the water air interace because the oam acts as matching impedance. Dry oam (e.g., a 98%) at the top o the oam layer has dielectric properties close to those o air, while wet oam ( a 10%) at the bottom o the oam layer has dielectric properties similar to those o seawater [25]. Accordingly, the intrinsic impedances or wet and dry oam are close to those o seawater and air, respectively, Figure 2(a) (dashed and dash-dotted lines). Thereore, as the oam void raction gradually changes rom the top o the oam layer to its bottom, it provides a seamless matching o the dielectric properties o air and seawater, Figure 2(b). This action o a oam layer on the water

13 Remote Sens. 2012, surace is the natural counterpart o the technology used to reduce relection in optical systems ([42], Chapter 2, 16). That is, the oam layer acts as an antirelective coating on the sea surace allowing eective transmission o down-welling atmospheric radiation into the oam-seawater system. As more o the transmitted radiation is absorbed in the oam-water system, more is emitted, ultimately resulting in high emissivity rom oam-covered water suraces. The cartoon in Figure 7(b) illustrates this situation. Figure 7. Schematic representation o relection and transmission o incident (let) and emitted (right) radiation under dierent conditions at air-sea interace: (a) oam-ree interace; (b) oam-covered interace; (c) bubbly mixture in the water without oam on the surace. Sr c Sr Sr c (a) (b) (c) Eectiveness o Impedance Matching Williams [19] suggested that the most eective impedance matching o sea oam would necessarily involve a surace layer. Figure 2(b) conirms this suggestion by showing that the impedance dierence between seawater and air is best minimized only when the ull range o void raction is present, rom 100% to 0%. This could be ulilled only when oam loats on the surace. Strong emission would not be observed when bubble plumes linger in the water column but oam is not present at the air-water interace above them (Figure 7(c)). The reason is that the lower impedance o the aerated seawater in the bubble plume is shielded rom the surace or microwaves ([19], p. 223) and thus its matching role is obviated. Besides presence or lack o oam on the surace, another possibility that might inluence the eectiveness o the impedance matching is that the vertical proiles in oam layers with dierent thicknesses may cover limited instead o the ull range o values. For instance, active whitecaps are thick thus they are anticipated to have impedance corresponding to wider range o void ractions, rom an upper value o % down to about 10%. Meanwhile, the oam layers o residual whitecaps are thinner and may have a lower impedance associated with a narrower a range shited toward lower values, e.g., a (<60% to 0%) ([25], Section 4.4). Figure 2(b) suggests, however, that even oam with a limited a range could provide impedance matching. It may not be the most eective, but still it could mediate some decrease in the impedance mismatch between air and seawater. The most important requirement thereore or the impedance matching is the oam to loat on the surace.

14 Remote Sens. 2012, Surace Scattering o Foam Only smooth boundary invokes a specular relection as that illustrated in Figure 7(a). In real geophysical conigurations, all boundaries are more or less rough. This results in a surace scattering pattern, which adds to or completely supersedes the specular relection. The strength o surace scattering is deined by the dielectric contrast o two media on both sides o an interace (as in Section 4.1) and the degree o roughness o this interace. In Section 4.1 we showed that, by the virtue o impedance matching, the oam boundaries are weakly to non-relective suraces. This relatively weak relection might, however, be augmented i these boundaries are rough. In this section we address the question whether the roughness o the oam boundaries is large enough to aect the weak, dielectrically determined surace relection o the sea oam. Figure 6 showed that, according to the Fraunhoer criterion, both dry and wet oams 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 oam rms height values σ a and σ w used in Figure 6 to the rms height o a wind-roughened sea surace σ u. We obtain σ u rom the variance o the sea surace displacement h about the mean surace at a given point. It is deined as the zeroth moment o the wave K up 2 spectrum S(K), σ m = S( K dk = h [43,44], where K = 2π/Λ is the wave number o an ocean u 0 ) K low wave with wavelength Λ. Note that σ u is a measure o the geometric roughness o a wind-driven sea surace, not the aerodynamic roughness usually represented by the roughness length z 0 [45,46]. The choice o 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 or the aerodynamic surace roughness because this is the variable used or air-sea interaction studies. Our reasoning in choosing K low and K up is thus as ollows. First, note that remote sensing o the sea surace is eective when the probing EM wavelengths λ 0 are comparable to the length scales o the ocean waves. For example, or scatterometers, ocean length scales ranging rom about 40 cm to less than 2 cm correspond to λ 0 rom 25 cm to 3 cm at incidence angles o 20 and 60 ([36], p. 1705). On this premiss, one can expect the probing wavelengths λ 0 or the range o requencies investigated here (Table 2) to be most relevant to ocean lengths scales rom about 40 cm to a ew millimeters. These length scales represent short gravity and capillary water waves. Next, note that analysis o the mean square slope o wave number spectrum, resulting rom oceanographic surace wave measurements, has shown that intermediate-scale waves with Λ between 6 m and 2 cm are the dominant contributors to the aerodynamic surace roughness in oceans [47]. Combining these two considerations, we restrict our estimates to ocean length scales rom 40 cm to 2 cm. The corresponding wavenumbers K range rom 3.14 rad cm 1 to 0.16 rad cm 1. To cover this range o K values, we obtained m 0 or three portions o the wave spectrum: S 2 (K), S 3 (K), and S 4 (K) as deined in [43]. In the estimates we used a riction velocity u * o 24 cm s 1 (U 10 7 m s 1 ), a value close to the globally averaged wind. For the considered range o length scales or ocean waves, we ind that the wind-roughened rms height σ u ranges rom about 1 mm or the short end o water waves (Λ o 2 cm) to 5.4 mm or longer water waves (Λ o 40 cm). We obtain similar estimates or σ u i an empirically ormulated wave spectrum in terms o mean square slope is used [47] instead o S(K).

15 Remote Sens. 2012, A comparison o these estimates to σ a = 1.3 mm implies that there could be cases in which the roughness o the dry oam could be comparable to the roughness created by the shortest capillary waves. But the dielectric contrast between dry oam and air is almost none existing (Figure 2 and Section 4.1). This leads ultimately to lack o surace scattering at the air-oam boundary. For the wet oam boundary, its roughness (σ w = 0.2 mm) is an order o magnitude below that o the shortest capillary waves (1 mm). This implies that wet oam, whether at the surace or as a sublayer in the oam thickness, is usually smoother than the sea surace roughness. We thus deem the wet oam boundary as incapable o providing signiicant surace scattering as well. Our conclusion o negligible surace scattering by any o the oam layer boundaries is consistent with the general observation that the relectivity o a oam-covered surace is not comparable to that o oam-ree rough sea. In act, microwave models (e.g., [48]) usually assume that oam tends to decrease the relectivity ([49], p. 24). Note that the conclusions in this section result rom the combined eect o the geometric roughness and the dielectric contrasts at the boundaries o a oam layer with well-stratiied void raction. Deviations rom these conclusions are possible when the void raction proile is limited (Section 5.2) Weak Volume Scattering Throughout a Foam Layer While we recognize the presence o scattering-driven pathway o attenuation in oam and expect it to play an important role in a structure conducive to signiicant scattering (namely, the close packing o bubbles), a previous survey ([24], Section 2.3) has shown that at the considered requencies the scattering within the sea oam is weak. Here, we explain this counterintuitive conclusion. Generally, one conjectures on the presence, or lack o, volume scattering in a medium by considering its eective penetration depth and the spatial and dielectric irregularities in its volume ([32], Chapter 11 4). The spatial and dielectric irregularities are usually quantiied with their spatial distribution and dielectric contrast. The size parameter and the reractive index o a medium express these two quantities. Thus, in the ollowing we analyze δ, x and m or clues regarding the volume scattering in oam. Our reasoning proceeds in two steps. First, employing general scattering rules, we establish the expected scattering behavior or various oam structures. Second, we show how this expected behavior is altered (or not ollowed) in natural, vertically stratiied oam layers Expected Scattering in Foam Layers Using δ, we can identiy the structure that could contribute to volume scattering the most. Because compared to ixed a values the integral eect o a vertical a proile is to limit d [25], we iner rom Figure 1 that signiicant volume scattering could be expected in active whitecaps as they involve more dry oam and thus have the largest penetration depth. Because the penetration depth o intermediate-wet and wet oam ( a 60%, e.g., residual whitecaps) is much smaller, the inhomogeneities within could be assumed to contribute little to volume scattering. Using x and m, we can urther assess the strength and relative importance o the volume scattering in dry and wet oam. For this purpose, we employ two basic rules established or scattering by particles. Namely, (i) the larger the particle compared to the wavelength (λ << a or x >> 1) the more eective the scattering ([37], p. 68); and (ii) the larger the dielectric contrast between the reractive indices o the

16 Remote Sens. 2012, particle and the surrounding medium the stronger the scattering. That is, in the case o particles in air, there is no scattering or m = 1 and scattering is small or m near 1 ([37], p. 172). We apply these rules or particles to oam on the basis o the connection between oam layers and bubbles (Section 3.5.1). We start with the analysis o x values. Figures 8a-c show x as a unction o requency in dry, intermediate-wet, and wet oams or bubble sizes characteristic or each sublayer (Figure 3 and Table 3). In each panel, the size parameter associated with the characteristic bubble size, which is the least likely or the depicted void raction is plotted in gray. We observe two trends rom panels (a) to (c) in Figure 8. One trend is that or most requencies, oam structures comprising large bubbles (a = 10 mm) have x > 1 or x >> 1 (dashed lines), whatever the void raction those bubbles orm by having various wall thicknesses w (recall Section 3.5.1). According to rule (i) above, this means that oam containing larger bubbles would have strong scattering. Another trend observed in Figure 8 is that x >> 1 in wetter oam (panels (b) and (c)) than in dry oam (panel (a)). From these two trends, we surmise that signiicant and eective volume scattering would be present in wet oam comprising large, thick-walled bubbles; Figure 8(c) (dashed line) exempliies this generalization Altered Scattering in Foam Layers The expected scattering behavior established in Section is not readily applicable or natural oams in which bubbles sort themselves by size in the oam depth (Section 2.1). Because o the bubble size stratiication, large bubbles are the least likely to occur or non-existing in wetter sublayers; Figure 8(c) shows this (the dashed line is gray). Thus, eective volume scattering should be conined to mostly dry oam. However, dry oam has the lowest x values (Figure 8(a)) compared to x o other oam structures (Figure 8(b,c)). Analysis o x thus reveals the irst limitation or achieving strong volume scattering in natural oam layers. Namely, the volume scattering is restricted to the least eective scattering structure, the dry oam. Analysis o m reveals another limitation or eective volume scattering in natural oam layers. From Figure 5 and rule (ii) (Section 4.3.1), we see that the x limited eectiveness o the dry oam as a scattering medium is urther diminished by its small dielectric contrast. Having m 1 (solid line in Figure 5), the reractive index o the dry oam does not dier much rom that o the air. Thus, the scattering in dry oam is negligible. Meanwhile, as m increases in oam depth, the perceived ineectiveness o the wet oam as a scattering medium based on x << 1 could be somewhat strengthened. This is especially true when wet oam is adjacent to air (residual whitecaps might exempliy such a case). But when wet oam is part o a vertically structured oam layer (as in active whitecaps), the increased values o m would still lead to modest enhancement o the volume scattering because the immediately adjacent sublayers have close m values. Overall, we iner that, in a oam mixture comprising a distribution o bubble sizes sorted in oam depth, the simultaneous decrease o λ and the number o bubbles with radii eective or scattering keeps the x values relatively low and conined in a narrow range. As a result, the scattering in oam remains less eective. This is compounded with the eect o the dielectric contrast, which is expressed via m increasing in oam depth yet staying adjacent to sublayers with similar m values. The scattering in wet oam is somewhat more eective than the scattering in dry oam, however the thickness, thus the

17 Remote Sens. 2012, penetration depth, o wet oam restricts this urther. The overall outcome is weak scattering throughout the oam. Figure 8. Size parameter in oam x (Equation (7) in Table 1) as a unction o requency and oam void raction representing: (a) dry oam ( a = 98%); (b) intermediate-wet oam ( a = 60%) and (c) wet oam ( a = 10%). For each a region the size parameters or three possible bubble radii are shown. The radii least likely or each type o oam are in gray. Fixed seawater temperature (T s = 20 C) and salinity (S = 34 psu).

18 Remote Sens. 2012, Strong Absorption by Wet Foam Revisited Section 4.3 provides a physical explanation o why, o the two pathways or absorption losses in oam, the scattering-driven one, though present, is not a major contributor. This thus qualitatively conirms the conclusion Anguelova and Gaiser ([25]; Section 4.5) reached that the medium-intrinsic absorption is the deining controlling actor or high oam absoprtivity. Anguelova and Gaiser [25] also showed that the thin water-laden wet oam could provide a strong emission rom oam-covered sea suraces. Here we give urther evidence or this conclusion using the results or x and m. Besides small skin depth ([25]; Section 4.1), another quantitative measure or the high absorptivity o oam as a medium is the act that the reractive index (and permittivity ε ) o oam 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 o m 0 and m 1 << 1, which means that a negligible raction o the radiation is relected, one could conjecture that even dry oam (Figure 5, solid lines) would be absorptive. In act, a combination o reractive index values similar to that o dry oam with large size parameter (e.g., x 100) could represent a medium whose behavior approaches that o a black body ([37], p. 269). However, such a combination o m and x is not observed in natural dry oam because x does not exceed 10 (Figure 8(a)). The combination o a narrow range o x (x 10) with m << 1 thus excludes the possibility that dry oam contributes to medium-intrinsic absorption losses. Meanwhile, m and m values or wet oam are o the same order (Figure 5, dashed lines) signiying that this is an intrinsically absorptive medium. However little the volume scattering in wet oam (Section 4.3), it adds to the medium-intrinsic absorption, making the wet oam a more eective absorber than the dry oam. 5. Discussion 5.1. Concept or the High Foam Emissivity Various suggestions can be ound in the literature explaining the high emissivity o oam-covered suraces. Williams [50] was the irst to suggest the importance o the impedance matching that the sea oam provides between the air and seawater. Consequent experiments led to the tentative conclusion that roughness expressed as distortion o the water surace at the interstices between the bubbles at the bottom o the oam layer ([51], p. 3097) together with the dielectric constant o oam produce a zone o radically tapered dielectric constant ([19], p. 223; cited also by [21]; [36], p. 1454; [48]). The Russian literature on the subject approaches the problem rom a microscopic point o view and promotes the explanation that the bubbles within the oam represent some kind o small black bodies, which intensively absorb the EM energy owing to inter-bubble diraction ([52], p. 511). Describing the sea oam as a nonrelecting absorbing but partially transparent medium, Wilheit ([48], p. 246) touched upon the eective transmission o EM radiation through oam. Williams [19] measurements involving oam on dierent underlying substrates, water or metal, suggest that not just the oam itsel, but also the system o oam loating on seawater is the one that could lead to high emissivity. This brie account o proposed explanations demonstrates that one or another element o the physical system that a oam-covered sea surace represents has been identiied and used to interpret the observed high emissivity. None o these explanations, however, combines these elements into a consistent whole.

19 Remote Sens. 2012, Meanwhile, the analysis in this study o the radiative processes in oam (Sections ) shows that none o these separate elements alone is enough to account or the high emissivity e o sea oam. Rather, all oam properties and all radiative processes engage in an emergent behavior (i.e., the whole is greater than the sum o its parts [53]) and contribute in some way to bring about the overall eect o a high e. Moreover, we argue that what makes the sea oam an exceptional emitter is the combination o these properties and processes in a vertically structured oam layer loating on seawater suraces. The ollowing is a summary o our conceptual understanding o how this happens. First, two components ensure the high emissivity o oam-covered ocean suraces: (1) High absorption losses in a coupled oam-seawater system; and (2) eective transmission o EM radiation through the oam. I one o these components were missing, a high e could not be guaranteed. The potential o the oam-seawater system or strong absorption and emission cannot be realized without eective transmission and propagation o radiation across layer boundaries into the oam; and vice versa eective transer across the layer boundaries is not enough i a highly absorptive agent is not in place to carry on high absorption and emission. Second, absorption losses in oam are provided by medium-intrinsic and scattering-driven absorption. Absorption in wet oam is relatively more important than absorption in dry oam. Furthermore, though strong, the absorption o the oam layer itsel might remain somewhat limited. The seawater below the oam layer is a major absorptive agent whose emission adds to that o the oam layer to yield the highest absorption losses in a coupled oam-seawater system. Third, with its intrinsic impedance, oam on the ocean surace, in any stage o its lietime, provides impedance matching between the contrasting dielectric properties o air and seawater. This decreases relections at the air-oam and oam-seawater boundaries and assists eective entry o EM radiation into the oam-seawater system. Existing, yet weak surace scattering at these boundaries and weak volume scattering within the oam provide the maximal portion o the propagating radiation to be delivered to and absorbed by the wet oam and seawater. Forth, the oam-seawater system reaches its maximal eectiveness and becomes a black-body-like emitter when a vertical proile o properties in the depth o a oam layer controls the relative importance and contribution o the radiative processes. Only gradual increase o seawater content and gradual change o spatial inhomogeneities minimize radiation attenuation in dry oam and maximize the absorption losses in wet oam and underlying seawater. Only a gradual change o the oam intrinsic impedance over the widest possible range o values, rom that o the air to that o the seawater, provides weakly relective to non-relective oam boundaries and maximizes the entry o EM radiation. In short, only in vertically structured oam layers every property and process conspires to bring maximum EM radiation to the most eective absorptive agent and thus producing maximum absorption and emission Foam as a Dynamic System Our discussion o the sub-surace bubble plume and the overlying oam layer generated is presented up to this point as a static, time-independent system. The impact that the dynamic o the system may have on the oam structure and its emissivity was alluded to in our considerations o the eectiveness

20 Remote Sens. 2012, o the impedance matching provided by whitecaps in their active and residual lietime stages (Section 4.1.2). Here we address urther the temporal variations o the system. The rational or these considerations is that the void raction in the bubble clouds ormed ater wave breaking varies by roughly two orders o magnitude between initial entrainment and the loss o large bubbles, an interval o roughly a second or so. Indeed, in a laboratory study, Anguelova and Huq [59] observed disintegration o the initial compact orm o a whitecap into winding streaks o oam ater only 100 ms. Such a time scale is comparable to the time period o 90 ms reported by Deane and Stokes [60] or the dispersion o a compact mass o bubbles into individual ones in a sur zone. For about one wave period (about 1 s), Anguelova and Huq [59] documented a decrease o the void raction rom near unity to about 20%. They also observed that the evolution o the bubble plume, and the oam layer above, could persist or many wave periods. The question is: What is the impact o such dynamical changes on a ield o whitecaps in the open ocean? Statistical analysis o void raction values representative or the initial air entrainment and or the subsequent decaying stage suggest the presence o both thick oam layers with high void ractions o 70% 90%, and thinner, wetter oam layer with lower void raction values (<80%) (e.g., Figures 12 and 13 in Anguelova and Huq [59]). Because evolving bubble clouds keep replenishing the oam layers on the surace, their statistics reveal that the temporal variations lead to a high variability o the oam layer properties. That is, in the open ocean at any moment oam layers with a distribution o void raction values at their boundaries could exist. These would include combinations o dry oam next to a oam-water boundary or wet oam next to air. Thereore, some o the various oam structures may exhibit a distinct impedance discontinuity at the mean ocean surace; others may represent stratiied oam well. This in turn has important implications or the relection and transmission o microwave energy. The next logical question is: How can we model such a variability o the oam layer properties? One simple approach is to assume that the oam structure analyzed in the present study represents a likely scenario in the early stage o oam development. A signiicant improvement upon such a simple approach would be to develop an elaborate physical model or the time-varying properties o sub-surace bubble clouds, about which relatively little is known, or the surace oam they generate, about which even less is known. A starting point could be to generate a void raction proile likely to be representative o the bubble plume/oam system during whitecap decay. Anguelova and Gaiser ([25], Section 4.4) discussed the modeling o void raction proiles by noting that one needs to consider the proile shape, proile range, and the proile variations with oam thickness, and how these choices aect the oam emissivity Modeling Scattering in Foam Foam emissivity can be modeled ollowing two approaches [36,52,54]. The intensity approach employs the radiative transer theory to calculate the microwave radiance o a medium. The wave approach employs solutions o Maxwell equations to compute the scattering losses in a medium and use those to obtain emissivity. The PS ormula represents the most rigorous starting point when obtaining the scattering o sea oam is pursued. The reason is that the PS ormula is identical to the low requency limit orm o the eective permittivity obtained with the strong permittivity luctuations theory or scattering rom a random medium ([54], pp ). According to Tsang et al. ([54], p. 429), the applicability