NEW APPROACH TO COASTAL ZONE VULNERABILITY
CLASSIFICATIONMargarita Shtremel, P.P. Shirshov Institute of Oceanology RAS, Russia
6805919@mail.ru
New approach of assessment of coastal zone vulnerability to wave action, based on
nonlinear properties of wave transformation, is presented on example of Anapa bay-bar
coasts.
Key words: wave transformation, relief deformation, coastal zone
I. ABSTRACT
While approaching the coastline, waves reach intermediate depth where they transformdue to shoaling and nonlinear near-resonant triad interactions between wave components. The
last process causes periodical in space variations of wave symmetry that leads to occurrence ofcross-shore sediment transport gradients, that results in corresponding relief changes. In thispaper method of coastal zone classification depending on its vulnerability to wave impact isdeveloped based on analysis of nonlinear wave transformation effects. Mean bottom slope,sediment size and density, and wave height and period are the main parameters that allow
predicting whether coastal zone will be stable or the erosion will occur. The method is based on
assessment of balance between onshore component of sediment transport caused by wave
asymmetry and offshore component driven by undertow. This approach was applied to theAnapa coasts.
II. INTRODUCTION
Problem of influence of nonlinear waves and undertow on sediment transport is not new.
For example, Abreu with colleagues [1] developed new bed shear stress predictor based on index
of skewness and waveform parameter that characterize wave skewness and asymmetry. If
incorporated in formula for sediment transport rate this predictor provides better prediction of
sediment transport in laboratory experiments than original method of Nielsen. But in their work
process of periodic in space changes of asymmetry due to energy exchange between wave
harmonics is not considered. This work, on the other hand, is focused on manifestations of triad
interactions in cross-shore sediment transport patterns.
In this work coastal zones with mild and sandy bottom slopes are studied. Such coasts arecharacterized by presence of sandbars, which indicates realization of scenarios of wave
transformation with periodic exchange of energy between wave harmonics due to near-resonant
triad interactions, as was shown in [2]. Such scenarios are realized when relatively steep waves
propagate above mild bottom slope and can be separated from scenarios with small amplitudesof second harmonics using Iribarren number (𝐼𝐼𝐼𝐼 =during analysis of field experiment data in work [6].

tan 𝛼𝛼�𝐻𝐻�𝐿𝐿) as a criterion. This feature was foundIII. METHOD DESCRIPTION
To calculate cross-shore sediment transport caused by waves Bailard's formula based on
Bagnold's energetic approach was used

1εεq = 2 fw ρ �tanbφ �������u|u|2 + Ws �������u|u|3 �(1)swhere 𝑓𝑓𝑤𝑤 = 0.01 – bottom friction coefficient, 𝜌𝜌- sand density, 𝜀𝜀𝑏𝑏 = 0.1 and 𝜀𝜀𝑠𝑠 = 0.01 – turbulent
viscosity and turbulent diffusion coefficients respectively, tan 𝜑𝜑 = 0.5 – internal particle friction�������2 and �������
coefficient, 𝑊𝑊𝑠𝑠 - hydraulic size of sediment particles, u|u|u|u|3 - second and third
statistical moments of horizontal component of orbital velocities, respectively. Following
Leont’yev [3], consider the horizontal velocity of wave motion as sum of short wave component
and component caused by undertow:𝑢𝑢 = 𝑢𝑢𝑠𝑠 + 𝑢𝑢� = 𝑢𝑢1 𝑐𝑐𝑐𝑐𝑐𝑐 𝜔𝜔𝜔𝜔 + 𝑢𝑢2 𝑐𝑐𝑐𝑐𝑐𝑐(2𝜔𝜔𝜔𝜔 + 𝜑𝜑) + 𝑢𝑢�
1𝐻𝐻𝐻𝐻(2)where 𝑢𝑢𝑚𝑚 = 2 sinh(𝑘𝑘ℎ) – amplitudes of first and second harmonics of wave velocity, φ – phase lag
1 𝐻𝐻 2

between them (biphase), 𝑢𝑢� = − 8 ℎ2 �𝑔𝑔ℎ – undertow velocity.

Then, substituting (2) into (1) and simplifying the results owing to negligible items
according to [3] for typical conditions of wave and coastal zone parameters, the equations will beas follows:33�������u|u|2 ≈ 4 u1 2 u2 cos φ + 2 u1 2 u�and equation (1) converts to13 ε1616�������u|u|3 ≈ 5π u1 3 u2 cos φ + 3π u1 3 u�

16u5q ≅ 2 fw ρu1 2 �4 tanbφ (u2 cos φ + 2u� ) + 5π εs W1 (u2 cos φ + 3 u� )�
s(3)

(4)

Thus, to calculate sediment transport, the following variables should be determined:horizontal amplitudes of wave motion velocity of first and second harmonics (𝑢𝑢1 , 𝑢𝑢2 ), phaseshift between them (biphase, 𝜑𝜑) and undertow velocity (𝑢𝑢�)When reaching intermediate depth, waves start to transform due to near-resonant triad
interactions that manifest in growth of higher wave harmonics. Typical for mild sloping beaches
wave regimes produce scenarios of wave transformation that are characterized by periodical
spatial exchange of energy between first and higher harmonics of wave motion. This process isaccompanied with corresponding changes of biphase and determines spatial structure of onshore
component of sediment transport that is caused by asymmetry of waves, as well as offshore
component provided by undertow. As was shown in [4], when second harmonic starts to grow,biphase is negative and approximately equals –pi/2, when second harmonic reaches its
maximum, biphase equals zero, and when it decreases, biphase becomes positive and tends topi/2 (Fig. 1).

Fig. 1. Spatial evolution of first and second harmonic amplitudes of free surface elevation(upper plot) and corresponding biphase values (lower plot) in monochromatic waves
propagating above mild bottom slope (0.01) (input wave height – 1.8 m, period – 8 s). Data ofnumeric modeling with SWASH.
It is possible to predict value of second harmonic maximum using the following empiricalformula constructed based on data of laboratory experiments [4]:a2

a1 +a2=0,014H�L(5)

To calculate onshore component of sediment transport with Bailard’s formula (4) it isneeded to know amplitudes of harmonics, phase shift between them and water depth. Propertiesof sediments are considered constant across the profile. Spatial evolution of determinative
parameters of sediment transport from example in Fig. 1 produce following sediment transport
pattern (Fig. 2).

Fig. 2. Onshore component of sediment transport caused by wave asymmetry calculatedusing Bailard’s formula. Initial data obtained with SWASH numeric model (monochromaticwaves, bottom slope 0.01, input wave height – 1.8 m, period – 8 s).
Offshore component of sediment transport is caused by action of undertow that can be
estimated based on principle of mass conservation. Stokes drift leads to onshore transfer of waterand undertow compensates it. Following formula represents depth-averaged value of undertow
velocity [3]:1 𝐻𝐻 2𝑢𝑢� = − 8 ℎ2 �𝑔𝑔ℎ(6)where H – significant wave height, h – water depth. Wave height grows with decreasing ofdepth, so undertow velocity has the following spatial distribution (Fig. 3).

Fig. 3. Spatial evolution of undertow velocities (monochromatic waves propagating
above mild bottom slope (0.01) (input wave height – 1.8 m, period – 8 s)). Data of numericmodeling with SWASH.
Fig. 4 demonstrates spatial changes in offshore component of sediment transport causedby action of undertow.

Fig. 4. Offshore component of sediment transport caused by undertow, calculated using
Bailard’s formula. Initial data obtained with SWASH numeric model (monochromatic waves,bottom slope 0.01, input wave height – 1.8 m, period – 8 s).
Undertow causes offshore sediment transport, wave asymmetry – onshore transport, sumof these two components determines weather onshore or offshore sediment transport and,
consequently, accumulation or erosion would prevail in each point of cross-shore profile.

Periodical in space distribution of harmonic amplitudes and biphase lead to occurrence ofmaximum onshore sediment transport in the position of second harmonic amplitude maximum(Fig. 5). Balance of sediment transport in said point is used as measure of coastal vulnerability towave action: if total sediment transport is directed to the shore in said point, coastal zone isconsidered resistant to this wave regime (this situation is shown in the example (Fig. 1-5)), if
total sediment transport is directed seaward, than this coastal zone is vulnerable to the wave
regime.

Fig. 5. Balance between sediment transport components calculated using Bailard’s
formula. Initial data obtained with SWASH numeric model (monochromatic waves, bottom slope
0.01, input wave height – 1.8 m, period – 8 s).
As was shown in [4], distance from the shoreline to nearest to the shore second harmonic
maximum position that corresponds to the nearest zero value of biphase can be found using
empirical formulaℎ𝑚𝑚𝑚𝑚𝑚𝑚
𝑡𝑡𝑡𝑡𝑡𝑡= 𝐿𝐿𝑏𝑏 ,(7)2𝜋𝜋where tan 𝛼𝛼 - bottom slope, 𝐿𝐿𝑏𝑏 = 𝑘𝑘 −2𝑘𝑘 - beat length, 𝑘𝑘1 and 𝑘𝑘2 - first and second harmonic
21wavenumbers.
This method is applicable not only for non-breaking, but for breaking waves as well, as it
was shown in [5] that breaking does not interrupt processes of nonlinear wave-wave interactions
and does not change significantly wave spectrum. In this work only non-breaking region of wave
transformation will be discussed, leaving breaking zone to further detailed studies. In theexample (Fig. 1-5) breaking starts in 370 m to the shore (Fig. 6).

Fig. 6. Changes in wave height along the cross-shore profile. A drop in wave height on 370 m
from the shore indicates beginning of breaking zone.
IV. APPLICATION TO ANAPA COASTS
Anapa bay-bar is extensive accumulative body, constructed by sandy deposits. It is
located on Northwest part of Russian Black Sea coast (Fig. 7).
Continental shelf in this region is represented by a mildly sloping plain covered by sandydeposits. Bottom slope of Anapa bay-bar is characterized by presence of several (up to 3)
sandbars [7].

Fig. 7. Anapa bay-bar.

During number of expeditions employees of the Southern branch of P.P.Shirshov Instituteof oceanology of Russian Academy of Sciences have carried out the complex coastal research of
Anapa bay-bar and obtained large amount of data concerning in particular bathymetry and grain
size of bottom sediments.
It was found out that mean grain size of sediments that construct sandbar crest is 0.2 mm[8].
Mean bottom slope values fluctuate around 0.01.
Waves propagating perpendicular to the shore (in North-East direction) were chosen foranalysis. According to calculations provided in [9], mean wave height of 25 yr repeatability is3.4 m.
Anapa bay-bar coasts are characterized by presence of multiple sandbars that indicate
dominance of wave regimes with periodical exchange of energy between harmonics [2]. The
range of periods that will be analysed in this case is determined by two following concepts:
1) to provide realization of scenarios of wave transformation with periodic exchange ofenergy between harmonics and significant amplitudes of second harmonics wave
steepness should exceed 0.013 for mean bottom slope 0.01 (see Iribarren numbercriterion) – that will give upper limit of wave periods (Tmax=13 s);2) the lower limit is chosen so that waves should not break in the nearest to the shore pointwith maximum of second harmonic (see (5)) (Tmin=11 s)That gives the following distribution of sediment transport in dependence on wave period(Fig. 8)Fig. 8. Sediment transport (sediment discharge) in point with maximum onshore
component in dependence on wave period, wave height H=3.4 m

As it is shown in the plot (Fig. 8), sediment transport in point with maximum onshore
component is positive in narrow range of wave periods (from 11.8 to 13 s) for waves with H=3.4m. Waves of such regimes will produce sediment transport that will change its direction from
offshore to onshore and backwards along the cross-shore profile, which will construct sandbars
and prevent coast from erosion. The rest of the wave regimes are more hazardous for thebeaches, as offshore component of sediment transport caused by action of undertow will not becompensated by onshore component caused by wave asymmetry and bottom sediments will beremoved from the coastal zone and beaches will erode.
V. CONCLUSION
On the example of Anapa bay-bar coasts new method of coastal zone vulnerability
prediction was tested. The interesting point, discovered in the course of this study, is that not
only steep waves can be hazardous to the coasts and cause erosion, but also very mild ones that
do not grow higher harmonics. Such waves will only cause offshore sediment transport due toundertow that will not be compensated by onshore component induced by wave asymmetry.
V. ACKNOWLEDGEMENT
This study was supported by Russian Science Foundation project 14-17-00547
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