# Effects of microplastics and surfactants on surface roughness of water waves

Yukun Sun1  Christopher Ruf2  Thomas Bakker1  Yulin Pan1<EMAIL_ADDRESS>1Department of Naval Architecture and Marine Engineering, University of
Michigan, Ann Arbor MI 48105, USA 2Department of Climate and Space Sciences
and Engineering, University of Michigan, Ann Arbor MI 48105, USA

###### Abstract

In this paper, we study the flow physics underlying the recently developed
remote sensing capability of detecting oceanic microplastics, which is based
on the measurable surface roughness reduction induced by the presence of
microplastics on the ocean surface. In particular, we are interested in
whether this roughness reduction is caused by the microplastics as floating
particles, or by the surfactants which follow similar transport paths as
microplastics. For this purpose, we experimentally test the effects of
floating particles and surfactants on surface roughness, quantified by the
mean square slope (MSS), with waves generated by a mechanical wave maker or by
wind. For microplastics, we find that their effect on wave energy and MSS
critically depends on the surface area fraction of coverage, irrespective of
the particle sizes in the test range. The damping by particles is observed
only for fractions above $O(5-10\%)$, which is much higher than the realistic
ocean condition. For surfactants, their damping effect on mechanically
generated irregular waves generally increases with the concentration of
surfactants, but no optimal concentration corresponding to maximum damping is
observed, in contrast to previous studies based on monochromatic waves. In
wind-wave experiments, the presence of surfactants suppresses the wave
generation, due to the combined effects of reduced wind shear stress and
increased wave damping. For the same wind speed, the wind stress is identified
to depend on the concentration of surfactants with a power-law relation. The
implications of these findings to remote sensing are discussed.

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##  1 Introduction

Ocean plastics pollution is an urgent and global problem. An estimated eight
million tons of plastics trash enters the ocean each year, and most of it is
battered by sun and waves into microplastics. Information about the
distribution and volume of microplastics is vital to address the removal of
plastic pollution from the ocean environment. Recently the development of
global observational systems for microplastics has been under active
discussions among interdisciplinary communities (Maximenko et al., 2019;
Martínez-Vicente et al., 2019; King et al., 2021). Along this direction, a new
remote sensing technique (Ruf & Evans, 2020; Evans & Ruf, 2021) is introduced,
which enables the tracking of microplastics over time and across the globe.
The principle of the technique is to infer the microplastics concentration
through the ocean surface roughness anomaly (i.e., lower-than-expected
roughness) induced by microplastics, which can be accounted for by the
difference between the real-time measurement from a spaceborne radar and a
standard model of surface roughness.

The new technique has been applied to data from the NASA CYGNSS (Cyclone
Global Navigation Satellite System) (Ruf et al., 2013, 2016). Specifically,
CYGNSS measures through the GPS L1 signal the normalized bi-static radar cross
section (NBRCS), whose inverse provides the mean square slope (MSS) of the
ocean surface, defined as

$\textrm{MSS}=\int_{0}^{k_{c}}k^{2}S(k)dk,$ (1)

where $k$ is the wavenumber, $S(k)$ is the omni-directional spectrum, and
$k_{c}$ is the cut-off wavenumber depending on the incident angle and carrier
wave frequency in remote sensing. For CYGNSS, $k_{c}$ takes an average value
of $7.5$ rad m-1 (Wang et al., 2020). As defined by (1),
$\textrm{MSS}\rightarrow\overline{\nabla\eta\cdot\nabla\eta}$ (variance of
surface elevation gradient) for $k_{c}\rightarrow\infty$, and otherwise MSS
quantifies the surface roughness up to a finite scale $k_{c}$. In addition to
CYGNSS measurements, another source of MSS can be obtained by the standard
Katzberg model (Katzberg et al., 2006) which takes wind speeds from a NOAA
reanalysis model (Liu et al., 2018) as inputs. The MSS anomaly, defined as the
relative difference between the CYGNSS measurements and Katzberg model results
(normalized by the latter), is expected to account for the effect of
microplastics on surface roughness, among other factors (such as error of the
CYGNSS measurements and the influence of other physical processes). In Evans &
Ruf (2021), it is found that the MSS anomaly shows a favorable correlation
with the concentration of oceanic microplastics computed from a global
transport model, as shown in Figure 1.

Figure 1: Relation between the MSS anomaly (computed from CYGNSS data and
Katzberg model in the oceanic region within latitudes of $\pm 38^{\circ}$, and
averaged from June 1, 2017 to May 31, 2018) and number density of
microplasitics (computed by a global microplastic transport model (van Sebille
et al., 2015)). The error bars represent $95\%$ confidence intervals with each
one computed by data from the van Sebille model at locations with a given MSS
anomaly. The range with a correlation between the MSS anomaly and number
density is indicated ( ). The data for this plot is obtained from Evans & Ruf
(2021).

While this new technique shows promising applications, the underlying flow
physics, in particular regarding the correlation between MSS anomaly and
microplastics concentration, is not clear. In Ruf & Evans (2020), the authors
suggest that the correlation may be due to the enhanced damping of surface
waves by microplastics as floating particles, referring to the earlier
experimental results in Sutherland & Balmforth (2019). Indeed, in a sloshing
wave tank, Sutherland & Balmforth (2019) show that the presence of layers of
floating particles enhances the wave damping, with the enhancement increasing
with the increasing number of particle layers. However, the experimental setup
with the water surface fully covered by particles is not consistent with the
situation of microplastics that only covers a small fraction of the ocean
surface. To represent the latter situation, experiments are needed to study
the characteristics of surface waves in the presence of particles partially
covering the free surface. Such experiments are currently not available.

Another mechanism that can lead to the observed correlation, as hypothesized
in Evans & Ruf (2021), is the wave damping effect by surfactants which share
similar transport paths as microplastics (van Sebille et al., 2020). Compared
to a scarce number of studies on the effect of floating particles, there is a
much larger body of literature on the effect of surfactants to surface waves.
It has been observed in several field studies (Cox & Munk, 1954; Barger et
al., 1970; Hühnerfuss et al., 1981, 1983; Ermakov et al., 1986; Lombardini et
al., 1989; Bock et al., 1999) that the presence of surfactants on the ocean
surface results in some damping effect of surface waves. Physically, the
damping effect is believed to be caused by the Marangoni stresses (due to the
inhomogeneous adsorption of surfactants at the interface) that can act in
opposite directions of the wave motion (Manikantan & Squires, 2020). In
controlled experiments (Lapham et al., 2001) and numerical simulations
(Lucassen, 1968, 1982) where quantitative studies are possible, it is found
that an optimal surfactant level associated with maximum damping exists for a
given wave frequency. While these studies are conducted for different wave
frequencies (in the range of 4 to 200 Hz) and identify different optimal
surfactant levels, it is suggested by Manikantan & Squires (2020) that the
optimal level may correspond to the situation of the surfactant-induced
Marangoni wave having the same wavelength as the surface wave, leading to some
resonance-like effect. For irregular waves as in the ocean, however, the
effect of varying levels of surfactant is not sufficiently studied, and it is
not clear whether an optimal level for wave damping exists for a wave
spectrum. This question is also relevant to the correlation problem as such an
optimal level implies a non-monotonic relation between concentration and MSS
anomaly that somewhat contradicts the observed correlation.

For wind waves, experiments in wave tanks show that the presence of surfactant
significantly suppresses the wave generation (Mitsuyasu & Honda, 1986; Tang &
Wu, 1992; Wei & Wu, 1992; Savtchenko et al., 1997; Uz et al., 2002). This
phenomenon is usually explained through the reduction of wind shear stress by
surfactants. As shown in Uz et al. (2002), with a surfactant level of
$10^{-1}$ mol l-1, the shear stress can be reduced up to 30$\%$ compared to
clean water for the same wind speed of 9.6 m s-1. The minimum wind speed (and
shear stress) to excite waves is also found to significantly increase with the
presence of surfactant (Savtchenko et al., 1997; Tang & Wu, 1992). With
surfactant level of $9.0\times 10^{-4}$ mol l-1, Mitsuyasu & Honda (1986)
report that the minimum wind speed leading to a wave growth is between 10 and
12.5 m s-1. In spite of the scattered results, most studies focus on the wave
characteristics at one (or only a few) surfactant levels. As a result, there
are no systematic relations identified for the relevant parameters including
wind speed, shear stress, wave growth rate, surfactant level and the
associated surface tension. Uncovering these relations is not only helpful for
interpretation/application of the aforementioned CYGNSS data, but also
desirable in understanding wave generations on the ocean surface that is
inevitably contaminated by surfactants.

In the present study, we aim to understand the mechanism of the MSS anomalies
detected by CYGNSS as well as the general physics of wave damping by floating
particles and surfactants. For these purposes, we conduct wave-tank
experiments to test the effects of floating particles and surfactants to damp
the waves generated by either a mechanical wave maker or wind. For floating
particles, we use two sizes of particles with 0.5 cm and 0.8 cm (both in the
range of oceanic microplastics) and consider their partial coverage of the
free surface with varying area fraction. We find a critical dependence of the
damping effect of particles on the area fraction, irrespective of the size of
particles. The enhanced particle damping of MSS and energy of the surface
waves is only observed for fractions above $O(5\sim 10\%)$. For low fraction
of $O(0.1\%)$ that corresponds to real oceanic microplastics situations, the
wave energy is not affected and the MSS is slightly increased (probably due to
the diffraction by particles). For surfactants, results on mechanically
generated waves show that the presence of surfactant damps the wave energy and
MSS, but no optimal surfactant level (associated with maximum damping) can be
identified as in the previous cases with monochromatic waves. The surfactants
also suppress the wind-generated waves, leading to a higher critical wind
speed to excite waves and a lower growth rate (for excited waves). Physically,
we attribute this suppression to not only the previously argued effect of
reduced wind shear stress, but also the wave damping by surfactants.
Furthermore, at the same wind speed, we find that the wind shear stress
depends exponentially on the surface tension, or depends on the concentration
of surfactants in a power-law relation (with a negative exponent linearly
correlated with the wind speed). The presented results clearly identify the
presence of surfactant as the dominant contribution to the MSS anomaly
detected by CYGNSS.

The organization of the rest of this paper is as follows. The details of the
experimental setup and procedures, including methods of measurements and
choice of floating particles and surfactants, are described in §2. The
reference results with both mechanically and wind generated waves in clean
water are discussed in §3. The main results with the presence of microplastics
and surfactants are presented in §4 and §5, respectively. Finally, the
conclusion is provided in §6.

##  2 Experimental Setup and Procedures

###  2.1 Facility and Input Parameters

The experiments are conducted in the wind-wave tank facility in the Marine
Hydrodynamics Laboratory (MHL) at the University of Michigan, with a schematic
sketch of the tank shown in figure 2. The tank is 35 m long and 0.7 m wide
with a water depth of 0.68 m. Waves can be generated by either a mechanical
wave maker or wind through an open-loop tunnel, and are dissipated at one end
of the tank by a beach with slope of $5^{\circ}$.

The wave maker is wedge-shaped with an angle of $30^{\circ}$ to the vertical
direction, and spans the width of the wave tank. The motion of the wave maker
is numerically actuated by a servo motor manufactured by Kollmorgen® AKM2G.
The servo motor is feedback controlled with a proportional-integral-derivative
(PID) controller, according to an input frequency spectrum
$S_{\textrm{in}}(f)$ that describes the power spectral density of the surface
elevations $\eta(t)$. For an approximation of the real ocean scenario in this
study, we use the Bretschneider spectrum, a two-parameter wind-wave spectrum
empirically developed for fully-developed seas:

$S_{\textrm{in}}(f)=\frac{1.25}{8\pi}\frac{f_{p}^{4}}{f^{5}}H_{s}^{2}\mathrm{e}^{-1.25\left(f_{p}/f\right)^{4}},$
(2)

where $f_{p}$ is the frequency of the peak mode and $H_{s}$ the significant
wave height. In the current study, we use $f_{p}=1.25$ Hz, corresponding to
$\lambda_{p}=1$ m as deep-water waves, and $H_{s}=3.9$ cm, corresponding to a
moderate effective steepness $\epsilon=H_{s}k_{p}/2\approx 0.12$ with
$k_{p}\equiv 2\pi/\lambda_{p}$ the peak wavenumber. In each experiment, the
wave maker is actuated for 200 seconds, with acceleration and deceleration for
10 seconds at the start and the end of the actuation.

Figure 2: Schematic sketch of the side view of the wave tank, including a wave
maker, a wind tunnel, six ultrasonic sensors, a hot-wire anemometer, and a
beach. The spacings between the sensors are not perfectly scaled, with
detailed information provided in Table 1.

Winds of different speeds are generated in an open-loop wind tunnel, which is
powered by a 40-hp fan and controlled by the percentage of its maximum output
power. The bottom of the wind tunnel outlet is 10 cm above the water level.
The wave tank is well sealed to prevent air leakage that may cause pressure
fluctuations and therefore unsteady freestream wind speeds. A hot-wire
anemometer manufactured by Extech® is placed at 13 m downstream of the wind
tunnel outlet to measure the wind speed. Operated at a sampling rate of 1 Hz,
the anemometer is mounted on a vertical traverse that allows one-degree-of-
freedom movements in the $z-$axis. Wind profiles are measured by adjusting the
location of the traverse, with all measurements taken over the stationary
state of the wind. It is found that the maximum wind speed along the vertical
axis (hereafter the reference wind speed) satisfies a linear relation with the
fan power, as shown in Figure 3(a). In the current study, we use three
reference wind speeds of 4.29, 6.59, and 9.09 m s-1 at fan powers of $20$,
$30$, and $40\%$, with measured wind profiles plotted in figure 3(b).

The data acquisition system consists of 6 Senix ToughSonic® model-14
ultrasonic sensors mounted on the top of the wave tank. The intervals between
the sensors are non-uniform, with distances from each sensor to the wave maker
and the wind tunnel outlet listed in Table 1. The measurement of surface
elevation $\eta(t)$ is converted from the voltage signals delivered by the
sensors to a National Instruments® data acquisition board. The sampling
frequency of the sensors is fixed at 100 Hz (which is sufficient for the
current study) although the maximum sampling frequency is up to 2000 Hz.

Sensor | 1 | 2 | 3 | 4 | 5 | 6
---|---|---|---|---|---|---
Distance to wave maker (m) | 9.35 | 11.38 | 14.23 | 18.66 | 22.76 | 28.45
Distance to wind outlet (m) | 2.03 | 4.06 | 6.91 | 11.34 | 15.44 | 21.13
Table 1: Distances from the 6 wave sensors to the wave maker and to the wind
outlet

Figure 3: (a) Reference wind speeds $u_{0}$ for different fan powers
($\diamond$), with the linear fit ( ). (b) Wind profiles with fan powers of
$20\%$ ($\circ$), $30\%$ ($\square$), and $40\%$ ($\triangle$), correspondig
to $u_{0}=4.29$, 6.59, and 9.09 m s-1, respectively.

###  2.2 Testing Procedures

####  2.2.1 Experiments with microplastics

The major materials of floating plastic debris in real ocean are polyethylene
and polypropylene (Cózar et al., 2014). In this study, we use two types of
particles both made of polypropylene (density $\rho_{p}=0.92$ kgm-3): one as
the PolyFil PolyPellets® microbeads with irregular shape and average diameter
$D_{p}\approx 0.5$ cm, and the other as McMaster Carr® plastic balls with
regular shape and diameter $D_{p}=0.8$ cm (both within the size range of the
oceanic microplasitics (Cózar et al., 2014)). To quantitatively test the wave
damping by particles, we focus on mechanically generated waves, since the area
fraction of particles (defined in (3)) is very difficult to control in wind
waves (due to the drift by wind).

Each experiment with particles in mechanically generated waves is conducted
with the following procedures. We first place the particles in the wave tank
at 8.53 m from the wave maker in calm water. The particles then spread out
both up- and downstream due to particle-particle and particle-surface
interactions until a balance is reached with a recorded length of $L_{i}$ (see
Figure 4(a)). As waves pass through the particles, they further spread out and
drift downstream due to the wave effect (e.g., the Stokes drift as discussed
in Stokes (1847); van Sebille et al. (2020)), with the final spreading length
(after waves pass through) recorded as $L_{f}$ (see Figure 4(b)). To quantify
the concentration of particles with varying spreading length, we define an
(average) area fraction

$C=\frac{N_{p}S_{p}}{W\overline{L}},$ (3)

where $N_{p}$ is the number of particles, $S_{p}=\pi D_{p}^{2}/4$ is the
planform area of one particle, $W$ is the width of the tank and
$\overline{L}=(L_{i}+L_{f})/2$ is the average length of the spreading. In
practice, we choose $N_{p}$ such that the value of $C$ ranges from 0.1$\%$ to
20$\%$, with the lowest fraction close to the oceanic microplastics
concentration. A total of 9 values of $C$ in this range are tested (5 and 4
for smaller and larger particles), with each experiment repeated for 3 times
to quantify the uncertainty level.

To test the repeatability of using (3) to measure the concentration, we
perform 3 repetitions for each value of $N_{p}$ for both types of particles,
with the results shown in figure 4 including the error bars that represent one
standard deviation on each side. Along with very small error bars (which
illustrates sufficient repeatability), we see a power-law relation between $C$
and $N_{p}$ for both types of particles and that the value of $C$ is
independent of the particle sizes for large $N_{p}$. While these behaviors
themselves are intriguing and may imply deeper physics, we leave the
investigation of them to future work and focus on the effect of particles on
surface waves in this study.

Figure 4: Schematic sketch of location and spread of particles at the (a)
beginning and (b) the end of each experiment. For simplicity, only the
locations of sensors 1 and 2 represented by ($\bullet$) are shown. The
particle-covered regions are marked by green color with particles illustrated
by ($\bullet$).

Finally, due to the drift of particles, the final length $L_{f}$ (for larger
$N_{p}$) may cover the surface area beneath sensors 1 and 2. In order to
robustly measure the effect of particles on waves (i.e., to avoid the acoustic
reflection from particles and consistently consider waves after passing
through all particles), we focus on sensor measurements downstream of the
particle patch $L_{f}$. In addition, we exclude data from sensor 6 since it is
relatively far away from the particles (so that the wave properties may be
modified too much by nonlinear interactions instead of particle damping).
Therefore, in §4 we present results based on sensors 3, 4 and 5 for floating
particle experiments.

Figure 5: Values of $C$ for particles with $D_{p}$=0.5 cm ( ) and $D_{p}$=0.8
cm ( ) at different $N_{p}$, with error bars representing one standard
deviation on each side calculated from three repetitions.

####  2.2.2 Experiments with surfactants

In the current study, we use a soluble surfactant, Triton X-100 (molecular
weight $625$ g mol-1), that has been commonly used in many previous studies
(e.g. Lapham et al., 2001; Uz et al., 2002; Liu & Duncan, 2006). We consider
nine concentrations $\Gamma=\Gamma_{0}-\Gamma_{8}$ (in terms of mol l-1) as
listed in Table 2, with $\Gamma_{0}$, $\Gamma_{1}$, $\Gamma_{2}$,
$\Gamma_{6}$, $\Gamma_{7}$, and $\Gamma_{8}$ tested for mechanically-generated
waves, and $\Gamma_{0}\sim\Gamma_{7}$ tested for wind waves. The surface
tension $\sigma$ for each concentration is measured from water samples by an
Mxbaoheng® BZY-101 surface tensiometer (figure 6(a)), which uses the Wilhelmy
plate method based on the balancing of surface tension, gravitational, and
buoyant forces on a platinum plate (e.g. Lapham et al., 1999). The measured
surface tensions (with an error within $\pm 0.3$ mN/m for this device) are
listed in Table 2 and plotted in figure 6(b), showing a logarithmic relation
with the value of $\Gamma$. All tested values of $\Gamma$ are below the
critical micelle concentration (CMC) limit for Triton X-100, which is
$\Gamma=23\times 10^{-5}$ mol l-1 corresponding to a saturated surface tension
$\sigma=30.6$ mN m-1.

In each day, we perform a total of (maximum) 12 experiments related to one
concentration $\Gamma$, that include three cases with different wind speeds
and one case with the wave maker, with each repeated for three times. Between
each two experiments we wait for 30 minutes, which allow the surfactants to
reach a uniform distribution over the tank. At the end of each day more
surfactants are added to the tank until the next level of desired
concentration $\Gamma$ is reached. A circulation pump is turned on overnight
to well mix the surfactant and water before the starting of experiments on the
next day.

Figure 6: (a) The surface tensiometer BZY-101. (b) The measured surface tension $\sigma$ ($\square$) depending on the concentration $\Gamma$, with a logarithmic fit (– – –) to show the relation $\sigma\sim\log(\Gamma)$. Symbol | $\Gamma_{0}$ | $\Gamma_{1}$ | $\Gamma_{2}$ | $\Gamma_{3}$ | $\Gamma_{4}$ | $\Gamma_{5}$ | $\Gamma_{6}$ | $\Gamma_{7}$ | $\Gamma_{8}$
---|---|---|---|---|---|---|---|---|---
$\Gamma\times 10^{5}$ (mol l-1) | 0 | 0.05 | 0.20 | 0.31 | 0.61 | 0.81 | 1.53 | 3.11 | 5.09
$\sigma$ (mN m-1) | 72.0 | 69.1 | 60.1 | 57.5 | 54.4 | 53.5 | 49.9 | 45.5 | 43.5
Table 2: Surfactant concentrations $\Gamma_{0}-\Gamma_{8}$ used in the
experiments, with the corresponding values of surface tension $\sigma$.

Finally, we summarize all experiments conducted for microplastics and
surfactants in Table 3.

Particles/Surfactants | Concentration | Wave generation | Repetitions
---|---|---|---
$D_{p}\approx 0.5$ cm |  $C=0.12$, $7.07$, $12.48$, and $18.69\%$ | Mechanical | 3
$D_{p}=0.8$ cm |  $C=0.30$, $0.60$, $6.44$, $12.88$, and $23.55\%$ | Mechanical | 3
TX-100 |  $\Gamma_{0}$ to $\Gamma_{2}$ and $\Gamma_{6}$ to $\Gamma_{8}$ | Mechanical | 3
TX-100 |  $\Gamma_{0}$ to $\Gamma_{7}$ | Wind | 3
Table 3: Summary of setups for experiments with microplastics and surfactants.

###  2.3 Quantities of interest

Frequency spectra of surface elevations are calculated from the time series
measured by the sensors in a time interval of 100 s of stationary state (in
time). We use a standard way (Adak, 1998) to compute the spectra as an
ensemble average over spectra of 9 segments of the time series with $50\%$
overlapping of each two segments (see figure 7). For each segment ($\Delta
t=20$ s with 2000 data points), we use a Tukey window function to taper the
tails of the segment, and evaluate the spectrum as

$S_{f}(f)=(\Delta t/2)\left|\hat{\eta}(f)\right|^{2},$ (4)

with $\hat{\eta}(f)$ the coefficient of cosine Fourier series of the tapered
segment.

Given $S_{f}(f)$, we are interested in two quantities of the mean square slope
(MSS) and the wave energy ($E$), defined respectively by (1) and

$E=\int_{0}^{k_{c}}S(k)\,dk,$ (5)

with $S(k)=gS_{f}(f)/(8\pi^{2}f)$. While the evaluation by (5) is not
sensitive to the cutoff wavenumber $k_{c}$ for $k_{c}$ above 7.5 rad m-1
(correponding to CYGNSS applications), the value of MSS evaluated by (1)
depends more significantly on $k_{c}$ up to $O(1000)$ rad m-1. This dependence
will also be discussed in §4 and §5 in the context of the implications of the
experimental results to CYGNSS applications.

Figure 7: A typical time series $\eta(t)$ ( ) and 9 segments ( ), with data
taken from sensor 1 in clean water test with mechanically generated waves.
Each segment is vertically shifted for better visualization with the mean
water level indicated by (– – –).

##  3 Reference results in clean water

In this section, we present the reference results in clean water, with the
purpose of characterizing the properties, especially quantifying the
uncertainty levels, of the results.

###  3.1 Wave maker experiments

Figure 8: (a) MSS and (c) $E$ for varying $k_{c}$ at sensor 1 ( ), 2 ( ), 3 (
), 4 ( ), 5 ( ), and 6 ( ) in a typical clean-water experiment. (b) MSS and
(d) $E$ with $k_{c}=1000$ rad m-1 at 6 sensors, including the mean values ( )
and error bars as one standard deviation on both sides, calculated over 10
repetitions in one day.

We first investigate the results for mechanically generated waves, with figure
8(a) and (c) showing typical results of MSS and $E$ as functions of $k_{c}$
measured by the six sensors in one of the runs. It can be seen that the values
of MSS and $E$ converge for $k_{c}$ above $\mathcal{O}(10^{3})$ rad m-1 and
$\mathcal{O}(10)$ rad m-1, respectively. The main results of the paper will be
presented with $k_{c}$=1000 rad m-1 for both quantities (a lower value of
$k_{c}$ results in inaccurate calculation in terms of (1) while a higher value
results in higher uncertainty among repetitions of the experiments). Figure
8(b) and (d) show the mean values and error bars (as one standard deviation on
both sides) of MSS and $E$ with $k_{c}$=1000 rad m-1, computed from 10
repetitions of each experiment within one day. For this value of $k_{c}$, the
errors of MSS and $E$ are both acceptable (with this $k_{c}$ for MSS
reflecting a balance between the accuracy of (1) and uncertainties in
repetitions), although the uncertainty levels of MSS are generally larger
especially for sensor 2 in this case. This larger uncertainty of MSS is mainly
caused by the noise in the high-frequency motion of the wave maker (that is
not precisely controlled due to the relatively low-frequency implementation of
the feedback control), which is amplified due to the $k^{2}$ dependence in the
MSS calculation (1). We note that both MSS and $E$ can increase toward
downstream over two adjacent sensors (say MSS for sensors 4 and 5, $E$ for
sensors 1 and 2). These are due to the nonlinear wave effect which may
transfer energy to high wavenumbers (leading to an increased MSS) or exchange
energy among linear and nonlinear parts (leading to an increased $E$ as the
linear part of the energy).

We also remark that the MSS measured over different days has a higher
uncertainty level compared to that shown in 8(b) (which is quantified within
one day). This is probably due to the different levels of impurities (e.g.,
organic materials) that are flushed into the tank during the cleaning
procedure at the start of each day (which is difficult to avoid for tank of
this size), as well as the background noise (e.g., the wind-wave tank is
located nearby a 109.7 m long towing tank which is under different operations
each day). These factors may have larger influence to the higher wavenumber
component of the spectrum resulting in higher uncertainty in MSS. In order to
avoid such uncertainty introduced by cross-day experiments, we compare our
results with and without microplastics (with multiple repetitions for each)
within the same day to quantify the effect of microplastics, as will be
discussed in §4.

###  3.2 Wind wave experiments

Figure 9: (a) MSS and (b) $E$ with $k_{c}=1000$ rad m-1 at 6 sensors,
generated by reference wind speeds of 4.29 ( ), 6.59 ( ), and 9.09 ( ) m s-1,
including the mean values and error bars as one standard deviation on both
sides, calculated over 10 repetitions in one day.

The MSS and energy $E$ measured in wind wave experiments at three different
reference wind speeds are shown in figure 9(a) and (b), as functions of the
fetch (measured by the locations of sensors 1-6). The error bars are
reasonably small for both quantities, indicating the robustness in the systems
of wind-wave generation and measurement. For energy $E$, the uncertainty is
higher than the cases with mechanically generated waves mainly because of the
variation close to the wind-generated peak mode located at the relatively low
frequency range of the spectrum (so it mainly adds uncertainty to $E$ but not
MSS). It is also clear that both MSS and $E$ increase with the fetch and wind
speeds, consistent with previous results (Cox & Munk, 1954; Charnock, 1955;
Hay, 1955; Ellison, 1956; Phillips, 1958). The condition of fully-developed
sea cannot be achieved in the current 35-m tank for the three operational wind
speeds.

##  4 Results of experiments with floating particles

The values of MSS and $E$ at sensors 3 to 5 with small ($D_{p}\approx 0.5$ cm)
and large ($D_{p}=0.8$ cm) particles are presented respectively in figures 10
and 11, along with the clean-water results. The mean values and error bars in
each sub-figure are computed from 3 repetitions of experiments with particles
(with corresponding size and area fraction $C$), as well as 10 repetitions of
clean-water experiments within the same day for reference (or comparison). In
general, we find that the damping effects for MSS and $E$ only become obvious
for sufficiently large value of $C$ for both particle sizes, e.g., in figures
10(d), 10(h), 11(e), and 11(j), where we see smaller values of the two
quantities in experiments with particles relative to the clean-water
experiments. For small values of $C$, the MSS and energy $E$ are barely
affected for either particle size, in terms of the average values over the
three sensors compared to the clean-water results (cf. figures 10(b), 10(f),
11(b), and 11(g)). The only exception to this general behavior is the MSS at
small area fraction $C$ especially for the small particle (figure 10(a)),
which shows higher values at all sensors compared to the clean-water results.
This is probably caused by the diffraction of short waves by the particles (or
a patch of particles) that tends to increase the MSS especially when the
particles are smaller.

Since the results measured by a single sensor may be subject to higher
uncertainty and influenced by other physical processes (e.g., different
nonlinear effects introduced after the modification of spectra by the presence
of particles) in addition to the particle damping, we further quantify the
effects of particles by examining the average quantities:

$\overline{\textrm{MSS}}_{ij}=\frac{1}{j-i+1}\sum_{n=i}^{j}\textrm{MSS}_{n},\quad\overline{E}_{ij}=\frac{1}{j-i+1}\sum_{n=i}^{j}E_{n},$
(6)

where $\textrm{MSS}_{n}$ and $E_{n}$ are MSS and $E$ measured at sensor $n$.
For results in this section, we use $i=3$ and $j=5$ to obtain average values
of $\overline{\textrm{MSS}}_{35}$ and $\overline{E}_{35}$. We further define
the ratio $\alpha$ to quantify the effect of particles relative to the clean-
water results, as

$\alpha_{\phi}=\overline{\phi}_{35}^{p}/\overline{\phi}_{35}^{c},$ (7)

where $\phi$ represents MSS or $E$, and the superscripts $p$ and $c$ denote
results with the presence of particles and in clean water respectively. The
quantities $\alpha_{\textrm{MSS}}$ and $\alpha_{E}$ are presented in figure 12
as functions of the area fraction $C$ for both particle sizes. We observe
that, when measured by the area fraction, the results obtained with the two
particle sizes almost collapse to a single curve, indicating that the effects
of particles on surface waves are irrespective of the particle size in the
test range. The damping effects for both quantities become evident for values
of $C$ above $C^{*}\sim O(5-10\%)$. The effect of particles on MSS is much
larger than their effect on $E$ (e.g., at the highest value of $C$,
$\alpha_{\textrm{MSS}}=0.90$ but $\alpha_{E}=0.97$), suggesting that the
damping effect of the tested particles focuses more on the short waves.
Finally, for higher values of $C$, the damping effect for MSS seems to exhibit
a logarithmic relation with $\alpha_{\textrm{MSS}}-1\sim-\log(C/C^{*})$ over
half a decade indicated by the data in figure 12(a).

Before concluding the section on floating particles, we briefly discuss the
implication of the results to the remote sensing (e.g., CYGNSS) applications.
Figure 13 shows a typical result of MSS as a function of $k_{c}$ measured by
sensor 3 for both sizes of particles at their highest area fraction $C$, along
with the reference results in clean water. It is clear from the plots that for
$k_{c}=7.5$ rad m-1 (corresponding to CYGNSS application as discussed in §1),
the effect of particles on MSS is negligible even for the highest
concentration (let alone the oceanic concentration which is much lower).
Therefore, the results are sufficient for us to conclude that the MSS
anomalies observed by CYGNSS are not caused by the effects of microplastics as
floating particles.

Figure 10: The MSS (— $\square$ —, top row) and energy $E$ (— $\square$ —,
bottom row) measured in experiments with particles of size $D_{p}\approx 0.5$
cm by sensors 3 to 5, with area fractions of (a)(e) $0.12\%$, (b)(f) $7.07\%$,
(c)(g) $12.48\%$, and (d)(h) $18.69\%$, together with the clean-water results
(— $\circ$ —) (obtained in the same day as particle experiments) for
reference. The error bars of all results correspond to one standard deviations
on both sides of the mean values. Figure 11: The MSS (— $\square$ —, top row)
and energy $E$ (— $\square$ —, bottom row) measured in experiments with
particles of size $D_{p}=0.8$ cm by sensors 3 to 5, with area fractions of
(a)(f) $0.30\%$, (b)(g) $0.60\%$, (c)(h) $6.44\%$, and (d)(i) $12.88\%$, and
(e)(j) $23.35\%$, together with the clean-water results (— $\circ$ —)
(obtained in the same day as particle experiments) for reference. The error
bars of all results correspond to one standard deviations on both sides of the
mean values. Figure 12: (a) $\alpha_{\textrm{MSS}}$ and (b) $\alpha_{E}$ as
functions of area fraction $C$ for particle sizes $D_{p}\approx 0.5$ cm
($\circ$) and $D_{p}=0.8$ cm ($\circ$). The reference value of
$\alpha_{\textrm{MSS}}=1$ and $\alpha_{E}=1$ (i.e., no effect from particles)
are indicated (– – –). A logarithmic fit to $\alpha_{\textrm{MSS}}$ at large
values of $C$ is shown (\- $\cdot$ -).

Figure 13: MSS computed with different $k_{c}$ with (a) particles of
$D_{p}\approx 0.5$ cm, $C=18.69\%$ ( ) and (b) particles of $D_{p}=0.8$ cm,
$C=23.35\%$ ( ). The reference clean-water results ( ), as well as the
indication of $k_{c}=7.5$ rad m-1 (– – –), are shown in both (a) and (b).

##  5 Results of experiments with surfactants

###  5.1 Results for mechanically generated waves

Figure 14: (a) MSS and (b) $E$ measured by different sensors in experiments of
mechanically generated waves with $\Gamma=\Gamma_{0}$ and $\sigma=72.0$ mN m-1
( ), $\Gamma=\Gamma_{1}$ and $\sigma=69.1$ mN m-1 ( ), $\Gamma=\Gamma_{7}$ and
$\sigma=45.5$ mN m-1 ( ), and $\Gamma=\Gamma_{8}$ and $\sigma=43.5$ mN m-1 (
). The error bars represent one standard deviation on each side of the mean
value computed from 3 repetitions.

Figure 14 shows values of MSS and $E$ measured by all six sensors at different
surfactant concentrations $\Gamma$ (leading to different surface tensions
$\sigma$). We observe significant damping of both MSS and $E$ with the
presence of surfactants, compared to the results in clean water (i.e.,
$\Gamma=\Gamma_{0}$ and $\sigma=72.0$ mN m-1). Another interesting phenomenon
seen in figure 14 is that the major surfactant-induced damping effect of MSS
and energy $E$ occur in a short distance from the wave maker, namely before
sensor 2 (15.47 m) for MSS and before sensor 1 (13.00 m) for $E$. After this
short distance, the surfactant-induced damping effect becomes insignificant,
i.e., the values of MSS and $E$ do not further deviate from the results in
clean water. This phenomenon suggests a hypothetical mechanism that for each
surfactant concentration, there might exist a critical level of wave amplitude
below which there is no enhanced damping by surfactants. This mechanism,
although yet to be verified, may be related to the Marangoni wave due to the
compression/expansion of the surface, that is only strong enough with
sufficiently high surface waves to result in a significant Marangoni damping.

We are also interested in the dependence of surfactant-induced damping on the
concentration levels. For this purpose, we plot $\overline{\textrm{MSS}}_{16}$
and $\overline{E}_{16}$ as functions of the concentration $\Gamma$. These
results are remarkably different from the previous experiments/simulations
based on a monochromatic wave (with frequencies both larger and smaller than
the peak frequency in (2)) (Lucassen, 1968, 1982; Lapham et al., 2001), where
an optimal concentration level exists corresponding to the maximum reduction
rate for each wave frequency. Nevertheless, for irregular waves as in our
study, the relation between the damping effect and the concentration level is
not monotonic, with $\overline{\textrm{MSS}}_{16}$ showing a local maximum at
$\Gamma=\Gamma_{7}$ and $\overline{E}_{16}$ at $\Gamma=\Gamma_{2}$. These
complicated behaviors indicate that to further quantitatively understand the
enhanced damping effect of surfactant to irregular ocean waves, one needs to
consider the Marangoni damping in the presence of nonlinear wave interactions
among a range of wave frequencies.

Figure 15: (a) $\overline{\textrm{MSS}}_{16}$ and (b) $\overline{E}_{16}$ at
different concentration levels of the surfactants. The results for
$\Gamma=\Gamma_{0}$ (i.e., clean water) are indicated by (– – –) in both sub-
figures.

Figure 16: MSS and $E$ with $\Gamma=\Gamma_{0}$ and $\sigma=72.0$ mN m-1 (—
$\circ$ —), $\Gamma=\Gamma_{1}$ and $\sigma=69.1$ mN m-1 (— $\circ$ —),
$\Gamma=\Gamma_{2}$ and $\sigma=60.1$ mN m-1 (— $\circ$ —),
$\Gamma=\Gamma_{3}$ and $\sigma=57.5$ mN m-1 (— $\circ$ —),
$\Gamma=\Gamma_{4}$ and $\sigma=54.4$ mN m-1 (— $\circ$ —), and
$\Gamma=\Gamma_{5}$ and $\sigma=53.5$ mN m-1 (— $\circ$ —) at fetches $X$
corresponding to sensors 1 to 6. The top and middle rows plot the MSS and $E$
respectively in a linear-linear scale, and the bottom row plots $E$ in a
linear-logarithmic scale. The columns from left to right are for reference
wind speeds of 4.29, 6.59, and 9.09 m s-1, respectively.

###  5.2 Results for wind waves

Figure 16 plots the MSS and energy $E$ as functions of the fetch $X$ for
different wind speeds and different concentration levels of surfactant. We
first note that for cases with sufficiently low wind speed and/or high
surfactant concentration, there exist sensor measurements at a similar signal
level to the measurements in calm water. These data points (with a criterion
of $E$ less than 1.5 times that of noises measured in calm water) are
physically considered as cases of no wave generation and are practically
excluded in figure 16. It is also noted that in figure 16 there are some
curves which decrease with the increase of $X$ as well as “broken” lines (say
in (a), (c) and (g)), indicating wave growth at shorter fetch but then decay
at longer fetch. This phenomenon is due to the convection of surfactant
downstream by wind, which results in a higher concentration downstream
associated with stronger damping effect.

Figure 17: (a) Linear growth rate $\beta_{\textrm{MSS}}$ for MSS (with MSS
$\sim\beta_{\textrm{MSS}}X$) and (b) exponential growth rate $\beta_{E}$ for
$E$ (with $E\sim\exp(\beta_{E}X)$) for varying surfactant concentrations
$\Gamma$ at reference wind speeds $u_{0}=4.29$ (— $\circ$ —), $6.59$ (—
$\square$ —), and $9.09$ m s -1 (— $\triangle$ —). Only cases with wave
excitation are included, and negative values indicate wave decay from sensor 1
to 6. The levels of $\beta_{\textrm{MSS}}=0$ and $\beta_{E}=0$ are marked (– –
–).

From figure 16 we find that MSS and energy $E$ show respectively a linear
growth (cf. (a)-(c)) and an exponential growth (cf. (d)-(i)) with fetch $X$,
especially for the initial stage (i.e., short fetch) far before the fully-
developed condition (i.e., dissipation much weaker than growth). While the
exponential growth of energy $E$ is consistent with previous results
(Mitsuyasu & Honda, 1986; Lin et al., 2008; Wu & Deike, 2021), we are not
aware of any previous finding regarding the linear growth of MSS. These
results indicate that in a wind sea, the peak mode grows much faster than the
high-wavenumber portion of the spectrum in the initial stage of development.
The linear growth rate $\beta_{\textrm{MSS}}$ for MSS (with MSS
$\sim\beta_{\textrm{MSS}}X$) and exponential growth rate $\beta_{E}$ for $E$
(with $E\sim\exp(\beta_{E}X)$) are fitted from data and summarized in figure
17, both as functions of $\Gamma$ for each reference wind speed (only cases
with wave excitation are shown). We observe that higher surfactant
concentrations consistently reduce the growth rates $\beta_{\textrm{MSS}}$ and
$\beta_{E}$.

Figure 18: Wind profiles $u(z)/u_{0}$ with surfactant concentrations
$\Gamma_{0}$ ($\circ$) and $\Gamma_{4}$ ($\circ$), with linear fittings ( )
and ( ), at reference wind speeds (a) $u_{0}=4.29$, (b) 6.59, and (c) 9.09 m
s-1.

To understand the mechanisms underlying the effect of surfactants on MSS and
$E$, we first investigate the wind shear stress under different levels of
concentration. For this purpose, we follow the procedure of Liu (2016) to
extract the friction velocity $u_{*}$ (and thus shear stress $\tau=\rho
u_{*}^{2}$) from the logarithmic layer of the wind profile, defined as

$u(z)=\frac{u_{*}}{\kappa}\ln{\frac{z}{z_{0}}},$ (8)

where $\kappa=0.41$ is the von Kármán constant and $z_{0}$ is a roughness
parameter (characterizing the viscous length scale or thickness of viscous
sub-layer). Figure 18 shows some typical wind profiles as well as the fitting
using $u(z)=A\ln{z}+B$ from which the friction velocity $u_{*}=\kappa A$ and
wind shear stress $\tau$ can be calculated.

The wind (shear) stress $\tau$ for varying surfactant concentrations (thus
varying surface tensions) is plotted in figure 19 for all three reference wind
speeds $u_{0}$. For each wind speed we mark the critical point
($\Gamma_{c},\sigma_{c},\tau_{c}$) corresponding to the boundary below which
no waves are excited. It is clear that for cases both above and below the
critical points, the wind stress shows an exponential dependence on surface
tension ($\tau\sim\exp(a_{1}\sigma)$) and a power-law dependence (which can be
expected from figure 6(b)) on the surfactant concentration level
($\tau\sim\Gamma^{-a_{2}}$) at each same wind speed. The inset of figure 19(b)
plots the values of $a_{2}$ for different reference wind speeds $u_{0}$, which
suggests a linear relation of $a_{2}\sim u_{0}$ implying also $a_{1}\sim
u_{0}$ (although more data at other wind speeds may be needed to further
consolidate this relation).

In addition, the critical points ($\Gamma_{c},\sigma_{c},\tau_{c}$) in figure
19 reveal interesting physics on wind-wave generation. While higher wind
stress implies stronger tendency for wave generation, it is possible, as
suggested by figure 19, that waves are excited at lower $\tau$ (say
$u_{0}=4.29$ m s-1 and $\sigma$ around 70 mN m-1) but suppressed at higher
$\tau$ (say $u_{0}=6.59$ m s-1 and $\sigma$ around 50 mN m-1). This fact is
only possible if the latter is subject to higher damping effect (which
overcomes the higher wind stress) due to higher concentration of surfactants.
Therefore, to understand wave generation in the presence of surfactants, it is
important to consider the surfactant-induced damping as an additional physical
factor. For cases in clean water with a monochromatic wave profile (for which
damping can be clearly quantified), it has been demonstrated by numerical
simulations (Wu & Deike, 2021) that the wave growth rate uniquely depends on
wind stress above some balanced value (which characterizes the critical state
of growth from wind stress balancing the damping). The situation for our case,
however, is significantly more complicated, as the damping effect depends on
both the surfactant concentration level and wind speed (which affect the
wavelength of the peak mode to be excited that in turn changes the damping by
surfactants). These dependence relations need to be further investigated in
order to fully understand the wind-wave generation in the presence of
surfactants.

Figure 19: Wind shear stress $\tau$ as a function of (a) $\sigma$ and (b)
$\Gamma$ at $u_{0}=4.29$ ($\circ$), 6.59 ($\circ$), and 9.09 m s-1 ($\circ$),
with exponential and power-law fits marked by (– – –) for each $u_{0}$.
Critical points of $\sigma_{c}$, $\Gamma_{c}$, $\tau_{c}$ corresponding to
threshold of wave excitation are marked by $\times$ for each $u_{0}$. Values
of $a_{2}$ ($\triangle$) at different $u_{0}$ with linear fit (\- $\cdot$ -)
are shown in the inset of sub-figure (b). Figure 20: (a) MSS as a function of
cut-off wavenumber $k_{c}$ for mechanically generated waves with surfactant
concentrations $\Gamma_{0}$ ( ) and $\Gamma_{8}$ ( ); (b)
$\overline{\textrm{MSS}}_{16}$ computed with $k_{c}=7.5$ rad m-1 for different
concentration levels $\Gamma$ for mechanically generated waves, with result
from $\Gamma_{0}$ marked (\- $\cdot$ -); (c-e) MSS as a function of $k_{c}$ at
reference wind speeds $u_{0}=4.29$, 6.59, and 9.09 m s-1 respectively, with
surfactant concentrations $\Gamma_{0}$ ( ), $\Gamma_{1}$ ( ), $\Gamma_{2}$ (
), $\Gamma_{3}$ ( ), $\Gamma_{4}$ ( ), and $\Gamma_{5}$ ( ). The locations of
$k_{c}=7.5$ rad m-1 are indicated by vertical dashed lines in (a) and (c-e).

Finally, we discuss the implication of the results in surfactant experiments
to the MSS anomalies in CYGNSS remote sensing. Figure 20(a) shows the MSS at
sensor 6 as a function of $k_{c}$ for mechanically generated waves in clean
water and with the highest concentration $\Gamma_{8}$. At CYGNSS cut-off
wavenumber $k_{c}=7.5$ rad m-1 indicated in the figure, we see a difference of
the two values of MSS which is not significant but larger than the difference
in cases of floating particles (figure 13). A more quantitative picture is
shown in figure 20(b), which plots the counterpart of figure 15(a) but for
$k_{c}=7.5$ rad m-1. Comparing the clean-water result and the result with
$\Gamma_{8}$, we see a reduction of MSS by $O(10\%)$. For wind-wave cases,
figure 20(c)-(e) show the MSS at sensor 6 as a function of $k_{c}$ for
different concentration levels (for which waves are excited). In these figures
we see clear difference of MSS even at $k_{c}=7.5$ rad m-1. Quantitatively,
with surfactant levels $\Gamma_{2}$, $\Gamma_{3}$ and $\Gamma_{5}$, the MSS is
reduced by 8$\%$, 17$\%$ and 50$\%$ relative to the clean-water results for
$u_{0}$=4.29, 6.59 and 9.09 m s-1, respectively. Considering the CYGNSS MSS
anomaly which is $O(20\%)$ (in terms of reduction from the Katzberg model
results), it is clear that the presence of surfactants in wind seas is the
most influential factor for this remote sensing application.

##  6 Conclusions

In this paper, we investigate the physical mechanisms underlying the MSS
anomaly measured by CYGNSS which has been used to track oceanic microplastics.
For this purpose, we experimentally study the effect of floating particles and
surfactants on surface waves (in terms of the energy and surface roughness)
generated by a mechanical wave maker and/or wind. The floating particles are
tested with two sizes of 0.5 cm and 0.8 cm, and our results show that their
damping effect on surface waves critically depends on the area fraction of
coverage, irrespective of the sizes. Damping effects on both energy and MSS
are only observed for fractions above $O(5\sim 10\%)$, which are much higher
than the oceanic microplastics situation of $O(0.1\%)$. For surfactants,
experiments with mechanically generated irregular waves show that they
generally result in enhanced damping for both the energy and MSS. However, the
“optimal” concentration level corresponding to the maximum damping for
monochromatic waves cannot be identified for irregular waves tested in our
study. In the experiments of waves generated by wind, we find that the
presence of surfactants significantly suppresses the wave generation, due to
its combined effect of reducing the wind stress and increasing the surfactant-
induced damping. The wind stress, obtained from the measured wind profile, is
further found to follow an exponential dependence on the surface tension, and
a power-law dependence on the concentration level of the surfactant (with the
power-law exponent linearly related to the wind speed).

The key results of MSS in this paper are also presented for different cut-off
wavenumber $k_{c}$. When considering the CYGNSS cut-off wavenumber $k_{c}=7.5$
rad m-1, we find that the effect of floating particles on MSS is negligible
even at the highest fraction above $10\%$ tested in the current work. The
mechanically generated waves with surfactants of highest concentration
$\Gamma_{8}$ result in a $O(10\%)$ reduction relative to the MSS in clean
water. In comparison, the wind-waves generated with moderate surfactant
concentration of $\Gamma_{3}$ leads to a $O(17\%)$ reduction from the MSS in
clean water, for a wind speed about 9 m s-1. Considering the MSS anomalies
observed by CYGNSS, which corresponds to CYGNSS MSS $O(20\%)$ lower than the
results from the standard Katzberg model, we conclude that the effect of
surfactants in a wind sea is the most influential factor for this remote
sensing application.

## Declaration of Interests

The authors report no conflict of interest.

## Acknowledgement

The authors would like to thank Professor James Duncan and An Wang for useful
discussions, and Mr. Jason Bundoff, Mr. Alexander Flick and Mr. Jim Smith for
technical help at the Marine Hydrodynamics Laboratory. This work was supported
in part by NASA Science Mission Directorate contract NNL13AQ00C with the
University of Michigan.

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