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Author's personal copy

Marine Pollution Bulletin 77 (2013) 266–273

Contents lists available at ScienceDirect

Marine Pollution Bulletinjournal homepage: www.elsevier.com/locate/marpolbul

Analysis of a beach as a time-invariant linear input/output system
of marine litter
Tomoya Kataoka a,b, Hirofumi Hinata a,⇑, Shigeru Kato baCoastal Zone Systems Division, Coastal, Marine and Disaster Prevention Department, National Institute for Land and Infrastructure Management, 3-1-1 Nagase, Yokosuka,
Kanagawa 239-0826, Japanb
Department of Architecture and Civil Engineering, Toyohashi University of Technology, 1-1 Hibarigaoka, Tempaku, Toyohashi, Aichi 441-8580, Japana r t i c l e

i n f oKeywords:
Nii-jima Island
Linear system analysis
Unit impulse response
Residence time
Diffusion equation
Mark-recapture experimenta b s t r a c t
The exponential decay of the amount of new litter on Wadahama Beach, Nii-jima Island, Japan revealedby 20-month mark-recapture experiments demonstrates a linear response of the beach to the input oftarget items. Here we show the amplitude and phase characteristics of the beach as a time-invariant linear input/output system and discuss the hydrodynamic and geomorphological factors that would determine the characteristics with the aid of a diffusion equation. The characteristics are fully determined by
the residence time of the items (sr = 209 days) and can be described as functions of the ratio of sr to theperiod of input variability. The decay is reproduced well by the analytical solution of the equation with aconstant diffusion coefﬁcient (D), whose order was estimated by sr and the backshore width. Generally, D
would depend on hydrodynamical statistics and beach geomorphology as well as the dimensions anddensity of the items.
Ó 2013 Elsevier Ltd. All rights reserved.

1. Introduction
The exponential decay of the amount of new litter on Wadahama Beach, Nii-jima Island, Japan revealed by 20-month mark-recapture (MR) experiments demonstrates a linear response of thebeach to the input of litter. Here we show the amplitude and phasecharacteristics of the beach as a time-invariant linear input/output
system and discuss hydrodynamic and geomorphological factors
which would determine the system characteristics by analyzing a
diffusion equation.
A number of beach surveys on the abundance and categorization of beached litter and on its spatio-temporal variability havebeen conducted on beaches around the world (e.g., Walker et al.,
1997; Williams and Tudor, 2001; Kusui and Noda, 2003; Ivar do
Sul and Costa, 2007; Ryan et al., 2009; Ribic et al., 2012). These surveys have generally clariﬁed the present environmental conditionsof the beaches and, in some cases, their variability trends. On theother hand, when we assess the impacts of the litter on the beachenvironment and/or take measures for some scenarios of natural
disasters or run-off accidents, it becomes crucial to understandhow the beach responds to marine litter inputs from offshore.
More speciﬁcally, we need to answer the following questions: If
a beach receives time-invariant litter input, to what amount does
⇑ Corresponding author. Tel.: +81 46 844 5025; fax: +81 46 844 1145.
E-mail addresses: kataoka-t852a@ysk.nilim.go.jp (T. Kataoka), hinata-h92y2@
ysk.nilim.go.jp (H. Hinata), s-kato@ace.tut.ac.jp (S. Kato).
0025-326X/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved.
http://dx.doi.org/10.1016/j.marpolbul.2013.09.049

the beached litter ﬁnally approach? What is the relaxation time
to reach a steady state? And when the litter input varies with time,
what are the ampliﬁcation factor and the phase lag between the input and the abundance on the beach?
Mark-recapture (MR) experiments have been utilized for estimating population and other population parameters such as thesurvival, recruitment, and population growth rate of animals andﬁshes (e.g., Peterson and Cederholm, 1984; Smith et al., 1999).
The MR method has been applied to beached litter studies by someresearchers (e.g., Garrity and Levings, 1993; Bowman et al., 1998;
Williams and Tudor, 2001.). In these experiments, all of the target
items were sprayed with the same color in accordance with theexperimental dates or locations of strandlines where the items
were found. This method allows us to understand the population
decay and movement of each cohort, but not the movements ofindividual items.
Garrity and Levings (1993) found that debris tended to moveupshore of strandlines and then enter into upland areas, and thatthe movement of debris was seasonal with greater rates of disappearance out of transects in the dry season than the wet season.
They speculated that the average residence time on beaches is lessthan 1 year. From comparative analyses of the MR experiment results from the six Israeli Mediterranean beaches, Bowman et al.
(1998) concluded that beach geomorphology – beach width, ridge
and runnel morphology, and beach porosity – made the backshoreof beaches an efﬁcient trap for litter. Williams and Tudor (2001)
revealed that the residence time of litter on the beach surface

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T. Kataoka et al. / Marine Pollution Bulletin 77 (2013) 266–273

depended on its size: items smaller than the substrate size had
shorter residence times mainly because they became buried in
the beach.
In terms of linear system analysis, the decrease in population ofeach cohort describes the unit impulse response (UIR) of a beachsystem. Once we obtain the UIR, we are able to understand thebeach response to the time-variant/invariant inputs and thus answer the questions. Here we consider Wadahama Beach as a linear
black box and calculate the UIR of target litter items based on MR
experiments to acquire an overall understanding of the beach response to marine litter input regardless of seasonality and positions of individual items. We also conducted a dynamic
examination of the beach system using a diffusion equation to deduce factors which determine the system characteristics. The target litter items of this study are three types of plastic ﬁshingﬂoats (see Section 2.1), which are comparatively small and hardly
moved by wind pressure on the beach. The UIR would depend on
litter items, that is, Wadahama Beach would have different systemcharacteristics for other litter items, such as plastic PET bottles,
plastic bags and expanded polystyrene ﬂoats, whose behavior on
the beach is greatly affected by the wind.

2. Materials and methods
2.1. Study site and target items
Wadahama Beach is located on the west coast of Nii-jima Island, about 150 km south of Tokyo (Fig. 1a). There are two isolated
mountains in the northern and southern parts of the island. When
the Kuroshio Current takes the nearshore nonlarge meander pathor the large meander path (e.g., Hinata et al., 2005), the current
strikes the west coast of the island. The beach is 900 m long and
30–50 m wide (Fig. 1b), with coarse sand with a mean sediment
diameter of d50 = 1.43 mm. The angle of the beach slope ranges
from about 5° to 10°. The coastal hinterland connects the beachwith an escarpment with an average angle of about 35° (Fig. 1b).
This steep escarpment blocks migration of the beached litter tothe hinterland except litter of lower density, such as plastic PET
bottles, plastic sheets and plastic bags. The beach is not a publicbathing site and thus generally there are few visitors to the beacheven in summer. Also, we have not found any people on the beachin the images taken every two hours by the webcam system(Kataoka et al., 2012) which has been installed on the northernhinterland since August 2011 (Fig. 1b).
There are no available measurements of long-term surface wave
or sea level data around the island. Instead, we used wind datafrom the Japan Meteorological Agency (JMA) weather station located in the central region of the island (Fig. 1a). The mountain behind the beach would create local characteristics of the wind ﬁeld
on and around the beach by blocking the easterly winds. Typhoons
passing by the island during the summer and autumn seasons will
episodically produce larger and longer sea surface waves. We also
analyzed the sea level record measured at Oshima Island, which issituated about 50 km north of Nii-jima Island, to understand theKuroshio Path variability (Fig. 1a).
The target litter items are three types of plastic ﬁshing ﬂoats(Fig. 1d). The dimensions and average weights of these ﬂoats measured in situ are: type 1: 13.0 cm  /max 2.4 cm, on average 38.8 g;
type 2: 13.1 cm  /max 7.8 cm, on average 116.9 g; type 3:
11.0 cm  /max 1.9 cm, on average 12.8 g. We have seen that theﬂoats on the beach are hardly moved even by the strong westerly
winter monsoon. We have found all three types of ﬂoats strandedon eight beaches distributed from the northernmost to the southernmost ends of the Japanese Archipelago, where we have installed
the webcam beached litter monitoring system (Kataoka et al.,2012). Thus, in the future, the system characteristics of Wadahama
Beach for the ﬂoats revealed in this study will be compared withthose of the beaches based on the MR experiments, which will
improve our understanding of the beach systems. In addition,
Nakashima et al. (2012) found that the type 1 plastic ﬂoats containa high concentration (13.5 ± 8.4 g/kg) of lead (Pb) which could
leach into surrounding water in beaches. Thus, it is important toestablish a method of calculating the residence time of ﬂoats on
actual beaches in order to estimate the total mass of Pb leaching
into the natural environment.
2.2. Mark-recapture experiments
Mark-recapture experiments have been conducted since September 2011 at one- to three-month intervals (Table 1). The ﬁrst
and second experiments were carried out in the northernmost
100 m- and 200 m-long areas, respectively (Fig. 1b), while theother experiments covered the entire beach. All the target itemson the beach were collected, numbered with a permanent marker
and replaced where they were found. The number consists of anexperiment number and a sequential number allocated to eachtype of ﬂoat (Fig. 1d). A thorough search for ﬂoats was conducteduntil no unnumbered target items could be found by repeatedobservations. The positions of all the items were recorded by ahandheld GPS receiver (GARMIN GPSMAP 60CSx) with a measurement error of about ±3 m (Fig. 1b). In addition, the topography ofthe beach was measured by a real-time kinematic GPS system(Trimble 5800 II, Trimble) with horizontal and vertical measurement errors of about ±5 mm (Fig. 1b). The distance between adjacent measurement points is about 10 m. Based on the experiments,
we calculated the numbers of ﬂoats that had beached (immigration) and backwashed (emigration) or been buried during the previous and present MR experiments and those of ﬂoats thatremained on the beach surface. Since we consider the beach as alinear black box, we used the positioning data of individual ﬂoats
only for setting the initial conditions of the one-dimensional diffusion equation given in Section 4.
2.3. Linear system analysis
Generally, a linear black box is described by a convolution integral. Speciﬁcally, in the time-domain, output y(t) is described by
the convolution integral of input x(t) and UIR h(t) as follows:yðtÞ ¼

Z txðsÞhðt  sÞds:

ð1Þ

1

Here in this study, x(t) is the immigration (litter input ﬂux) and y(t)is the total remnant on the beach. h(t) denotes the remnant functionof each cohort. From the MR experiments, we calculated x(t), y(t)and h(t).The Fourier transform of Eq. (1) yieldsYðxÞ ¼ HðxÞXðxÞ;ð2Þ

where x = 2p/T is the angular frequency and T the period of temporal variability of the input. Eq. (2) describes the system in the frequency domain. H(x), the Fourier transform of the UIR, is thefrequency response of the system:HðxÞ ¼

Z 1hðtÞ expðixtÞdt:ð3Þ

0

The amplitude and the phase response of the system becomeAðxÞ ¼ jHðxÞj;hðxÞ ¼ tan1ImðHðxÞÞ
:ReðHðxÞÞ

ð4Þ

Eq. (4) shows that the system characteristics are fully describedby the UIR, which can be determined by the MR experiments.

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T. Kataoka et al. / Marine Pollution Bulletin 77 (2013) 266–273

(a)

Tokyo

CoastalHinterland(b)Fore - Back shore shore
MHWL
MWL
MLWL
1st survey area
2nd survey area

North
Pacific

type 1(d)

Escarpment

North
Pacific

1-1

type 2

1-1
Hinterland

type 3

B

(c)

1-1

Fig. 1. (a) Location and enlarged map of Nii-jima Island. White triangle indicates the location of the tidal station at Oshima Island of the Japan Meteorological Agency (JMA).
In the enlarged map, the altitudes of Nii-jima Island (contour lines and gray-white gradation) and the meteorological observatory of the JMA (white circle) are shown. (b)Enlarged map of Wadahama Beach showing the altitudes (contour lines and gray-white gradation) and positions of target items newly washed ashore from the third to thesixth experiments (white circles), and the position of the webcam (white star). Note that the scale in the latitudinal direction is six times as large as that in the longitudinal
direction. The upper panel of (b) shows the cross-sectional beach topography along the dash-dotted line in the lower panel. MHWL, MWL and MLWL represent respectively
the mean high water level, the mean water level and the mean low water level calculated from the sea level records obtained during September 2011 to May 2013 at the JMA
tide station (see also Fig. 3a). (c) Cross-sectional distribution of the total abundance of immigration from the third to the sixth experiments. B represents the backshore width.
(d) Pictures of target items. The numbers on the items, which are used for identifying each item, consist of an experiment number and a sequential number allocated to eachtype.

Table 1
Cohort population and abundance of re-emergence (within parentheses) in each MR experiment. The abundance of re-emergence is added to the cohort population of theprevious experiments. The totals of remnant and re-emergence (within parentheses) in each experiment are shown in the last column.

*

Experiment number

Experiment date

1
2
3
4
5
6
7
8
9
10
11
12

2011/09/30
2011/10/27
2011/11/24
2011/11/26
2012/01/26
2012/03/23
2012/06/29
2012/08/21
2012/11/08
2012/12/27
2013/02/27
2013/05/08

Cohort no.

Remnant

1

2

3

4

5

6

7

8

9

10

47(0)
36 (0)*
34(1)
34(2)
28(2)
17(0)
12(3)
10(0)
6(3)
6(0)
6(1)
6(3)

–
76(0)
70(0)
70(2)
51(5)
37(2)
27(9)
22(1)
11(4)
9(3)
7(0)
5(0)

–
–
62(0)
59(0)
45(0)
35(6)
25(7)
21(0)
13(2)
11(3)
9(0)
6(4)

–
–
–
20(0)
15(0)
12(4)
6(1)
5(0)
4(1)
4(3)
3(0)
0(0)

–
–
–
–
18(0)
11(0)
8(3)
7(0)
4(0)
4(2)
2(0)
2(1)

–
–
–
–
–
18(0)
7(0)
6(0)
2(0)
2(1)
2(0)
0(0)

–
–
–
–
–
–
136(0)
116(0)
48(3)
45(2)
37(1)
33(23)

–
–
–
–
–
–
–
26(0)
8(0)
8(1)
6(0)
5(3)

–
–
–
–
–
–
–
–
57(0)
50(0)
44(4)
29(18)

–
–
–
–
–
–
–
–
–
20 (0)
16 (0)14(7)11

28 (0)13(0)12

191(0)47(0)112(0)
166(1)
183(4)
157(7)
130(12)221 (23)213(1)
153(13)159(15)160(6)
304(59)Population of cohort no. 1 in the second experiment is not used in Figs. 3 and 4 because the second experiment was carried out in the northernmost 200 m-long area.

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T. Kataoka et al. / Marine Pollution Bulletin 77 (2013) 266–273

Fig. 2. Time series of the total numbers of immigration, remnant and emigration
revealed by the MR experiments. Dashed lines with solid symbols show the timeseries of the remnant and the emigration calculated from the observed immigration
using Eqs. (1) and (5). Black arrows show the date of each experiment.

3. Results
3.1. System characteristics of Wadahama Beach
The time series of the total immigration (x(t)), remnant (y(t))
and emigration (z(t)) of all types are shown in Fig. 2. The total
immigration has local maximums in October 2011 and June
2012, whereas the total emigration takes local maximums in January and November 2012. The resultant total remnant has local
maximums (minimums) in November 2011 and June 2012 (inMarch and November 2012).
Generally, the time series (x(t), y(t), z(t)) of each ﬂoat type (notshown here) has local maximums and minimums in the corresponding months of the time series of the totals shown in Fig. 2.
The similar temporal variation pattern of immigration suggests

269

that all types have similar spatial distribution and drifting characteristics in the offshore, while the similar emigration time series
would be the result of similar movements of all the types on thebeach. Thus, here we derived the UIR regardless of the ﬂoat type.
The population decay of each cohort is tabulated in Table 1. The
remnants include the number of re-emerged ﬂoats which were notrecovered in the previous surveys but were found in the present
survey. However, the ratio of re-emerged ﬂoats to the remnants
is very small (6% on average). Natural or anthropogenic large debris, such as logs and lumbers, was not common on the beach: it did
not form any strandline, but was stranded separately and sparsely.
In the experiments, we searched for the target items behind andbeneath such debris. Presumably, re-emerged ﬂoats had been buried in the sand. Williams and Tudor (2001) found that litter items
larger than the surrounding substrate accumulated more readilyon the beach surface, since these items would be exhumed byenergetic wind waves. Thus, we assume that the emigration of
the ﬂoats is mainly driven by backwashing to the sea.
Fig. 3 shows the time series of the 30-day moving averages ofadjusted sea level (Fig. 3a), monthly wind velocity and direction(Fig. 3b), and the population decay of all the cohorts normalized
by the initial input values (Fig. 3c). The positive sea level anomaly
corresponds well with the Kuroshio Path variation: the Kuroshio
took the nearshore nonlarge meander path and would hit Nii-jimaIsland in September 2011, May 2012, from September to November 2012 and in April 2013 (http://www1.kaiho.mlit.go.jp/KANKYO/KAIYO/qboc/index_E.html). Although the sea level changes
due to tide (by about ±40 cm) around Nii-jima Island, we are unable to discuss the relation between the population decay revealedby bimonthly MR experiments and tide-induced sea level changes.
For the wind ﬁelds, weaker easterly winds dominated in summer
whereas stronger westerly winds prevailed from late autumn toearly spring.
In general, the populations decrease exponentially with lower(higher) decreasing rates from June to August 2012 and from

Fig. 3. Time series of (a) 30-day moving average of adjusted sea level anomaly at the JMA tidal station (see white triangle in Fig. 1a), (b) monthly average wind speed (solidline with white square) and direction in degrees clockwise from north (black circle) observed at the JMA wind station (white circle in Fig. 1a) and (c) the population decay ofeach cohort. Black arrows in (c) show the date of each experiment.

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270

T. Kataoka et al. / Marine Pollution Bulletin 77 (2013) 266–273

Fig. 5. System characteristics of Wadahama Beach as functions of sr/T.

Fig. 4. Unit impulse response functions revealed by all the MR experiments of
Wadahama Beach (solid line) and from the MR experiments by Garrity and Levings(1993) (dash-dotted line) which was calculated by reading the value of the cohort
population from their Fig. 3. The dashed line represents the normalized population
decay obtained by solving the diffusion Eq. (12) with the diffusion coefﬁcient set to
0.17 cm2 s1.

November 2012 to March 2013 (August to November 2012). There
was no signiﬁcant relationship between the temporal variability ofsea level record and that of the decreasing rates. On the other hand,
local maximums of the immigration in June 2012 and May 2013(see Fig. 2) were preceded by the nearshore nonlarge meander
path, which suggests that the Kuroshio plays an important rolein the transportation of the target items. Unexpectedly, a signiﬁcant relationship was not found between the temporal change inthe rates and that in the monthly wind velocity. In actuality, theﬂoats on the beach may be backwashed intermittently to the sea
by storm events which are not represented by the monthly averagewind velocities; however, we will not examine here the mechanisms of the temporal change in the decreasing rates.
The plot of the normalized population made from all the experiment data setting the experiment dates equal to 0 (Fig. 4) provides
a much clearer exponential decay. We approximated the plots as
an exponential function (exp[kt]) with 95% conﬁdence level(R2 = 0.867, P < 0.01) and obtained the UIR function of the beachfor the target items as follows:hðtÞ ¼texpð 209Þ;ðt  0Þ;

0;ðt < 0Þ;

ð5Þ

where t is elapsed time in days. From the integral of this function
from t = 0 to inﬁnity, the residence time of the ﬂoats on the beachis estimated as sr = 1/k = 209 days, that is, about seven months.
From Eqs. (3)–(5), we obtain the Wadahama Beach system characteristics as functions of sr/T (Fig. 5):

A

s 
r

T

s 
 s
sr
r
r
¼ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ; h¼  tan1 2p
:
2
T
T
sr
1 þ ð2p T Þ

ð6ÞThe ampliﬁcation factor A/sr and the phase lag (not time lag)respectively approach 1 and 0 as sr/T ? 0. For sr/T  1, A/sr  0
and h  p/2. Practically, time-variant ﬂoat inputs with a much
shorter period than the residence time, such as T < 20 days (sr/T > 10), would have almost no impact on the beach environment.
We calculated the remnants (y(ti)) from the observed immigrations (x(t1), x(t2), . . ., x(ti)) by using Eqs. (1) and (5), where ti is thedate of the i-th experiment. Then the emigrations were calculated
as z(ti) = (y(ti)  y(ti1))  x(ti). Fig. 2 shows that the calculated

remnants and emigrations agree well with the corresponding
observation results except in November 2012. Although the discrepancy appears to arise from the exclusion of the temporalchange in the decreasing rate, this overall agreement conﬁrms thatthe beach responds linearly to the ﬂoat inputs. At present it is difﬁcult to perform precise frequency analyses on the time series of
the immigrations and remnants and on their relations based on
twelve MR experiments. In the future, much longer MR experiments or experiments with shorter time intervals will allow us
to conduct, for instance, Fourier transformation and correlation
analysis of the time series.
Despite the inclusion of the data from the ﬁrst and second
experiments in which the survey areas were limited to the northern part of the beach, the calculated remnants and emigrations correspond well with the observed ones (Fig. 2). The temporal decayof the ﬁrst and second cohorts’ population indicates the residence
time of the ﬂoats, which had washed ashore in the northern part ofthe beach, on the whole Wadahama Beach. From the third to thesixth experiment, 19 (50)% of the immigrations were found within
the northernmost 100 (200) m-long area. Thus, the decay exhibitedby the ﬁrst and second experiments would describe the overallsystem characteristics of the whole beach.
3.2. Beach response to idealized inputs
To answer the question raised in Section 1, here we calculate
the beach response to idealized inputs. The time evolution of theabundance of the remnants on the beach is calculated by Eq. (1),whereas Eq. (6) gives the beach response in the steady-state. First,
assuming that Wadahama Beach receives a time-invariant litter input ﬂux of x0 (ind./day) after a total clearance of beach litter, thatis, y(0) = 0, the time evolution becomesyðtÞ ¼

Z t
0






ts
t
ds ¼ sr x0 1  exp :
x0 exp 

sr

sr

ð7Þ

The second term of the right-hand side is a decay term. The
abundance ﬁnally approaches y1 = srx0 = 209x0 as t ? 1. The
abundance reaches 90% of y1 at t90 = -sr  ln(0.1) = 2.3sr = 481 days. The relaxation time t90 is proportional to the residence timeof the beach, not depending on the litter input ﬂux. If a beach thathas an exponential decay type of the UIR and much longer residence time receives the constant litter input of x0, the litter slowly
but greatly accumulates on the beach.
Second, the beach response to a sinusoidal input with an amplitude of x0 (ind./day) and with a period of T0 (day) is calculated.
Although a negative litter input is not realistic, this does not cause
any problem when analyzing the basic characteristics of the beachsystem. The response becomesAuthor's personal copy

T. Kataoka et al. / Marine Pollution Bulletin 77 (2013) 266–273







Rt
s
yðtÞ ¼ 0 x0 sin 2Tp0 t exp  t
sr ds
h



i
¼ Ax0 sin 2Tp0 t þ h  sin h  exp  str ;4. Discussion

ð8Þ

where A and h are the ampliﬁcation factor and phase lag in Eq. (6),respectively. Generally, the decay term in Eq. (8) is much smaller
than that in Eq. (7). If the input varies seasonally with a period of
one year, the maximum of the remnants on the beach appears at
about 75 days after the input maximum. If the beach receives a
combined input, which is a more realistic input, the abundance
approaches.

yðtÞ ! x0

sr þ A sin


2p
tþh
T0

; ðt ! 1Þ;

271

ð9Þ

as predicted by Eq. (6). In this case, the relaxation time is almost
determined by the decay term for the constant input. Speciﬁc examples of the response of Wadahama Beach with x0 = 1(ind./day) andT0 (=2p/x0) = 180 days are shown in Fig. 6.
If the inputs of the ﬂoats (x(t)) to Wadahama Beach are given
according to some scenarios based on numerical models (e.g.,
Kubota, 1994; Kako et al., 2011; Potemra, 2012; Maximenko
et al., 2012), we can estimate the time evolution of the abundanceon Wadahama Beach (y(t)). Conversely, when y(t) is measured by
the webcam system or beach surveys by humans, x(t) can be estimated by the inverse Fourier transform of X(x) = Y(x)/H(x) andthen compared with the model predictions, which is useful for
improving the numerical models. We note that existing numericalmodels calculate the abundance of litter on the sea surface but not
on beaches, and that the linear system analysis serves as a mediator between numerical models and beach monitoring.

Naturally, the following two questions arise from the above linear system analysis: What are the factors determining sr? Do allbeaches have an exponential decay type of unit impulse response?
If a beach has an exponential decay type of unit impulse response(exp[t/sr]), the system characteristics of the beach are fully determined by sr. However, it is not realistic to conduct long-term MR
experiments of all beaches involving human effort. If we canunderstand the factors and develop a mathematical model of sr,
for instance, as a function of hydro- and morphodynamic statistics,
we will be able to produce a residence time map for a region ofinterest with much less effort. The map would allow us to assess
the long-term impacts of marine litter caused by natural disasters
and/or loss of ﬂow accidents on beaches and to take measures tominimize the overall damage in the region.
4.1. Governing equation of population decay
Before discussing the questions, dynamic investigations of thesystem will be helpful. From Eq. (5), the population of each cohort
obeys the following equation:dyc ðtÞ yc ðtÞ
þ
¼0
dt
sr

ð10Þ

The outﬂow ﬂux from the beach to the sea is proportional to the
population itself. The solution of Eq. (10) becomestyc ðtÞ ¼ yc ð0Þ exp ¼ yc ð0Þct ;

sr

ð11Þ

where c = exp[1/sr] is the mean unit-time survival rate on thebeach. This solution denotes that randomly selected ﬂoats ofyc(1  c) backwash to the sea in every unit time. The residence timeof 209 days of the ﬂoats on Wadahama Beach corresponds to a daily
survival rate of 99.5%, that is, a yearly survival rate of 17.4%. Thisimplies that the backwash process can be described by a random
diffusion process, in which case the governing equation isconsidered to be:@ qðx; tÞ@ 2 qðx; tÞ
¼D
@t
@x2

ð12Þ

where the x-axis points positive in the offshore direction with theorigin at the landward edge of the backshore (see Fig. 7), D is atime-invariant diffusion coefﬁcient and q(x, t) is the number ofﬂoats per unit length. Since we have not found any ﬂoats on the hinterland or foreshore of the beach, the boundary conditions are(

D @@xq ¼ 0;

ðx ¼ 0Þ;

q ¼ 0;

ðx P BÞ;where B denotes the backshore width. Integrating Eq. (12) fromx = 0 to x = B with a boundary condition of no net ﬂux at x = 0 gives

dyc ðtÞ
@q
¼D
:
dt
@x B

ð14Þ

The order of D can be estimated by comparing Eq. (14) with Eq.
(10):

y
@q
y =B
y
 c¼D
 D c ¼ D 2c :
B
sr
@x L
B
)D
Fig. 6. The response of Wadahama Beach to idealized inputs. The responses toconstant litter input ﬂux (a), sinusoidal litter input ﬂux (b) and combined litter
input ﬂux (c), respectively. The legends of the lines are shown in (a).

B2

sr

:

ð15Þ

Substituting B = 30 m and sr = 209 days into Eq. (15), D is of theorder of 101 cm2 s1. The solution of Eq. (12) subject to theboundary conditions of Eq. (13) isAuthor's personal copy

272

T. Kataoka et al. / Marine Pollution Bulletin 77 (2013) 266–273

4.2. Factors determining sr

Nearshore zone
Foreshore
Backshore (B 30 m)

Offshore zone
Breaker zone

x

B

float

0

Wadahama Beach

Hinterland

Windage effect

Wave run-up dominated regime

Ocean current
Ocean current anddominated regime wave-driven current
dominated regime
Fig. 7. Schematic image of current regimes and the behavior of a target ﬂoat (redlines) on and around Wadahama Beach with the coordinate used for onedimensional diffusion calculation. B represents the location of the boundary
between the foreshore and backshore. Residence time (sr) corresponds to the time
taken for the ﬂoat to move along the bold red line. However, the time taken fordrifting in the nearshore region is considered to be much shorter than the duration
for which the ﬂoat stays on the beach. (For interpretation of the references to colourin this ﬁgure legend, the reader is referred to the web version of this article.)

8
N
X
>
>
< qðx; tÞ ¼
an expðk2n DtÞ cosðkn xÞ;
n¼1

>
RB
>
:a ¼
4kn
qðx; 0Þ cosðkn xÞdx;nsinð2kn BÞþ2kn B 0

ð16Þ

where kn (=(2n-1)p/2B) is the eigenvalue of the n-th mode and N the
highest mode number. The population of the cohort is then obtained by integrating the density over the backshore width:yc ðtÞ ¼

Z B

qðx; tÞdx:

ð17Þ

0

From the individual ﬂoat positioning survey results (Fig. 1c), we
assumed a sinusoidal initial distribution of the ﬂoats as follows:qðx; 0Þ ¼

p

2B

0;

sinðpBxÞ;ð0 6 x < BÞ;
ðx < 0; x P BÞ:

ð18Þ

The population decay obtained by substituting Eqs. (16)(N = 10) and (18) into Eq. (17) and setting D equal to 0.17 cm2 s1
reproduce well the UIR function from the MR experiments (Fig. 4).
Although the value of D was actually adjusted so that the population decay agrees well with the UIR from the MR experiments, it isimportant to note that the order of D is in accordance with Eq. (15).
This demonstrates that the long-term ﬂoat population dynamics
on Wadahama Beach are governed by a diffusion equation and thatthe order of the diffusion coefﬁcient can be estimated from the residence time and the backshore width.

From the individual ﬂoat positions recorded by the handheld
GPS receiver, it was found that the population of each cohort decreased exponentially as the ﬂoats changed position laterally andlongitudinally (not shown here). Considering that the wind has little effect on the movement of ﬂoats on the beach, this implies thatthe exponential decay results from iteration of the following
sequential dynamic processes: washing ashore the beach – backwashing to the sea – drifting in the nearshore region – washingashore again/drifting far offshore (Fig. 7). The drifting of the ﬂoats
on the sea surface is affected by the wind. Therefore, the resultant
residence time sr will depend on the statistics of wind-waves,wind- and/or wind-wave-driven coastal currents and winds inthe nearshore region.
Eq. (15) shows that the determination of sr can be replaced by aderivation of the diffusion coefﬁcient D, which more directly explains the hydrodynamics. Let us consider two beaches withwidths of B and 2B, having the same residence time of sr. The diffusion coefﬁcient of the wider beach is about four times as large asthat of the narrower beach, which would be the result of moreenergetic wave-current systems. Conversely, when the two beaches have the same hydrodynamic statistics, Eq. (15) suggests thatthe wider beach would have even longer residence time, that is, amuch higher abundance of litter.
From comparisons of the MR experiment results on six Israeli
Mediterranean beaches, Bowman et al. (1998) concluded thatswash energy to beach width and beach morphology are the dominant factors controlling the quantity of litter on the beaches. They
found that the two widest beaches had the most litter. Tsouk et al.
(1985) reached the same conclusion in their study on oil-polluted
Israeli beaches. Their widest beach ranked lowest in self-cleaning
performance. Bowman et al. (1998) also reported that high beachporosity is the most dominant physical factor, because it increases
the subsurface beach drainage and diminishes the backwash of litter, thus preventing self-cleaning and encouraging litter accumulation. Williams and Tudor (2001) revealed that the residence time oflitter on the beach surface depended on litter size: items that were
smaller than the substrate had a shorter residence time mainly due
to burial of items in the beach.
When we consider the residence time of litter with lower densities such as PET bottles and plastic bags, wind pressure force acting on the beached litter surface seems to be a crucial factor, since
we frequently found that, for instance, plastic bags were moved
down to the foreshore or up to the escarpment from the backshoreof the beach by wind pressure, which appears to reduce the residence time of such litter on the beach.
From the above discussions, generally, sr of litter on a beach isconsidered to be given by

sr 

B2
;
DðS; W; d50 ; ql ; lÞ

ð19Þ

where S is wave-current ﬁeld statistics, W the statistics of windﬁeld, d50 the mean sediment diameter, ql the density of the litterand l the representative length of the litter. The density is included
to represent litter behavior on a beach and in its offshore. For thetarget ﬂoats, the wind statistics affect sr via the wave-current ﬁelds
and wind effects in the nearshore.
4.3. The UIR function of the beaches on the Caribbean Coast
Garrity and Levings (1993) identiﬁed the temporal decay of ﬁfty
representative items in a 1  50 m transect at four sites along the
Caribbean Coast of Panama. In general, the number of each itemin the four transects exponentially decreased with respect to time,Author's personal copy

T. Kataoka et al. / Marine Pollution Bulletin 77 (2013) 266–273

as shown in their Fig. 3. They did not show speciﬁc data on the
population decay. Thus, by reading the population decays fromthe ﬁgure and combining all the data from the four sites, weapproximated the time decay as an exponential function(exp[kt]) with 95% conﬁdence level (R2 = 0.769, P < 0.01) as follows (dash-dotted line in Fig. 4):(
hðtÞ ¼t
exp  104;

ðt P 0Þ;

0;ðt < 0Þ:

ð20Þ

From this equation, the residence time is found to be 104 days,
which is half the residence time of Wadahama Beach probably due
to the following two reasons: (1) The transect area (1  50 m) is
much smaller than that of Wadahama Beach (30–50  900 m);
and (2) the representative items included litter with lower density
such as Styrofoam, which was the second most abundant categories on the transect and is likely to be moved by wind.
This suggests that other beach systems could have exponential
decay type of UIR, and also that the residence time depends on thedimensions of the particular site and the litter items. This ﬁnding
supports the above discussion on the residence time and diffusion
coefﬁcient based on our MR experiments.

5. Conclusion
The exponential decay of the amount of new ﬁshing ﬂoats on
Wadahama Beach, Nii-jima Island, Japan revealed by 20-monthmark-recapture (MR) experiments demonstrated a linear responseof the beach to the input of litter. By approximating the decay as an
exponential function, we obtained the unit impulse response (UIR)of the linear beach system. Also, we obtained the system characteristics from the Fourier transform of the UIR.
The characteristics are fully determined by the residence time(sr) of the ﬂoats on the beach and can be described as functionsof the ratio of sr to periods of the ﬂoat input variability (T). The
ampliﬁcation factor A/sr and the phase lag (not time lag) respectively approach 1 and 0 as sr/T ? 0. For sr/T  1, A/sr  0 andh  p/2. Practically, time-variant ﬂoat inputs with a much shorter
period than the residence time would have almost no impact on
the beach environment.
Assuming idealized time-invariant and/or variant litter input
ﬂuxes, time evolutions of the abundance on Wadahama Beachwere examined by using the convolution integral of the inputs
and the UIR. When the beach receives the combined litter ﬂux,
the abundance reaches a dynamic steady-state in about twice theresidence time, where the relaxation time is determined by thetime-invariant component of the input.
Combinations of existing ocean models and linear system analysis would allow us to predict the abundance of litter on beaches
by the convolution integral of the model-derived input and theUIR of the beaches. Conversely, it is possible to estimate the inputs
from the abundance of litter on beaches measured by webcam systems or human beach surveys, then the calculated inputs could becompared with the results of existing ocean models and used to
improve the models.
The exponential decay was well reproduced by a diffusion
equation with a constant diffusion coefﬁcient, whose order of magnitude was found to be roughly estimated by the backshore width
and the residence time of the ﬂoats. This demonstrates that
determination of sr can be replaced by a derivation of the diffusion
coefﬁcient, which more directly explains the hydrodynamics. From
our MR and individual ﬂoat positioning experiments as well as theresults of previous studies, the coefﬁcient would depend on273

wave-current and wind-ﬁeld statistics and beach geomorphology
as well as the density of the litter.
This study identiﬁed the characteristics of the Wadahama Beachsystem for three types of ﬂoats. To understand more comprehensively the mechanisms that determine sr or its equivalent, the diffusion coefﬁcient, long-term MR and individual ﬂoat positioning
experiments are required for various debris items on different beaches which have different speciﬁc morphologies and wave-current
and wind-ﬁeld statistics.
Acknowledgements
We sincerely thank Dr. Atsuhiko Isobe of Ehime University, Dr.
Masahisa Kubota of Tokai University, Dr. Hideshige Takada ofTokyo University of Agriculture and Technology, Dr. Shin’ichi Aoki
of Osaka University and Drs. Takanobu Inoue and Takumi Okabe of
Toyohashi University of Technology for their valuable comments.
The authors also sincerely thank Mr. Ryotaro Fuji of Kokusai Kogyo
Co., Ltd. for assistance with the MR experiments. This research was
supported by the Environment Research and Technology
Development Fund (B-1007) of the Ministry of the Environment,
Japan and by JSPS KAKENHI Grant Numbers 23656309, 25820234.
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