The combined effect of wave–current interaction and mud-induced damping on nonlinear wave evolution

Ocean Modelling 41 (2012) 22–34

Contents lists available at SciVerse ScienceDirect

Ocean Modelling journal homepage: www.elsevier.com/locate/ocemod

The combined effect of wave–current interaction and mud-induced damping on nonlinear wave evolution James M. Kaihatu ⇑, Navid Tahvildari Zachry Department of Civil Engineering, Texas A&M University, College Station, TX 77843-3136, United States

a r t i c l e

i n f o

Article history: Received 4 February 2011 Received in revised form 21 September 2011 Accepted 22 October 2011 Available online 28 October 2011 Keywords: Nonlinear surface waves Wave–current interaction Bottom mud

a b s t r a c t The development of a phase-resolving nonlinear frequency-domain model with both wave–current interaction and viscous mud-induced energy dissipation is discussed. The model is compared to dissipation rates deduced from experimental data, with favorable results. The model is then run with cnoidal waves over a finite mud patch with both opposing and following currents. It is determined that wave height dissipation by mud is exacerbated by opposing currents and reduced by following currents, in agreement with previous work. It is shown that mud-induced damping affects the cnoidal wave shape; under significant damping, the resulting waveform resembles a sine wave, with some short-scale variability as phaselocking between the harmonics breaks down. In addition, the effect of uncompensated subharmonic interactions, a cause of high frequency damping over mud, is also evident with wave–current interaction. Finally, random wave spectra are used to initialize the model and allowed to evolve over a flat bottom with a mud patch, with and without co-flowing currents. As before, the dissipation of the random waves is enhanced by opposing currents and reduced by following currents. The degree of spectral broadening seen in wave–current interaction in non-dissipative environments is also seen here with mud-induced dissipation. High spectral frequencies strongly damped by bottom mud recover some energy (at the expense of low frequencies) in the lee of the finite mud patch. This recovery is evident even with substantial damping across the majority of the frequency range of the spectrum. Ó 2011 Elsevier Ltd. All rights reserved.

1. Introduction In shallow water, ocean surface waves are subject to strong evolution effects due to bathymetric variations, wave breaking, bottom characteristics (roughness and sediment size/type), ambient currents, and nonlinear (triad) wave–wave interactions. The general properties of the wave field can vary over the scale of O(10– 100 m) in these regions due to wave nonlinearity alone, as the shape of the waves transition toward flatter troughs, sharper crests, and finally a sawtooth-like shape during breaking. These effects can be seen in the evolving shape of the wave spectrum. As random waves shoal, harmonics of the spectral peak are amplified against the background energy level of the remainder of the spectrum. As this process continues and the surf zone is approached, energy in frequencies not at harmonics of the spectral peak also undergo evolution via nonlinear interactions, and the spectrum begins to flatten (Elgar et al., 1990). During the course of breaking, energy is preferentially lost from the high frequency tail of the spectrum, and asymptotically approaches a shape that is proportional to f2, the inverse of the square of the frequency, which is

⇑ Corresponding author. Tel.: +1 979 862 3511. E-mail address: [email protected] (J.M. Kaihatu). 1463-5003/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.ocemod.2011.10.004

in accordance with the sawtooth-like shape of waves observed in the surf zone (Kirby and Kaihatu, 1996). However, environmental factors external to the dynamics of wave propagation can also have a strong effect on spectral evolution. In many areas of the world (e.g. at the mouths of major rivers), significant currents and large deposits of bottom mud are both present, potentially leading to Doppler shifting of frequencies, strong damping of wave energy, and attendant effects on the nonlinear nature of the wave evolution. While there has been substantial work on the effect of each on wave evolution, there is little work on the combined effect. While wave–current interaction alters the linear properties of surface waves in a fundamental way (Bretherton and Garrett, 1968), the nonlinear characteristics of waves are also affected. Nonlinear effects were considered in studying large amplitude waves in adverse currents (Smith, 1976), waves near the blocking condition (Crapper, 1972; Chawla and Kirby, 2002), and the effect of currents on deep water (Turpin et al., 1983; Kirby, 1986) and shallow water (Chen et al., 1999; Kaihatu, 2009) waves. In particular, Chen et al. (1999) and Kaihatu (2009) investigated nearshore nonlinear energy exchange between wave frequencies, with the latter study extending this to irregular waves. The presence of cohesive bottoms can lead to large energy dissipation over very short distances; this was seen in both laboratory

23

J.M. Kaihatu, N. Tahvildari / Ocean Modelling 41 (2012) 22–34

(e.g. Gade, 1958) and field (e.g. Wells and Coleman, 1981; Mathew, 1992) experiments. Theoretical descriptions of mud-based wave damping have generally focused on the assignation of proxy behavior to the mud, and the resulting proxy studied for its dissipation characteristics. Mud has been variously represented as: viscous fluid (Gade, 1958; Dalrymple and Liu, 1978; Ng, 2000), poro-elastic solid (Yamamoto et al., 1978; Yamamoto and Takahashi, 1985), Bingham plastic (Mei and Liu, 1987), Voigt solid (Macpherson, 1980; Hsiao and Shemdin, 1980; Jiang and Mehta, 1995, 1996; Zhao et al., 2006), and various non-Newtonian materials (Chou et al., 1993; Foda et al., 1993). Most of these studies develop a description based on a well-defined mud state, though some (Chou et al., 1993; Zhao et al., 2006) prescribe some evolutionary behavior to the mud. In addition, the dissipation characteristics of mud depend on wave frequency; the presence of mud thus not only affects a spectrum of waves to different degrees, but also affects any process which transfers energy between spectral components. For example, Sheremet et al. (2005) and Kaihatu et al. (2007), while explaining a field observation by Sheremet and Stone (2003), showed that the presence of mud can indirectly drain energy from wave frequencies which are too high to be kinematically coupled to the muddy bottom; the damping of lower frequencies by the mud is mitigated by energy transfer from higher frequencies. In areas such as the Atchafalaya, Louisiana (USA) shelf, where the 10 m isobath may be as far as 50 km offshore, nonlinear energy exchange may play a major role in describing wave evolution in muddy regions. It was previously seen (Sheremet et al., 2005; Kaihatu et al., 2007; Kaihatu, 2009) how both wave–current interaction and mud-induced damping have individual impacts on the wave spectrum, and it would be interesting to examine how these would combine to alter spectral evolution in shallow water. In this study we describe a nonlinear, phase-resolving, frequency domain model capable of predicting changes in spectra based on triad interactions, to which both wave-current interaction and mud-induced damping have been added. Due to the deterministic aspect of the model, the effect of mud and currents on the nonlinear wave–wave interaction between frequency components can be simulated in a comprehensive manner. The model, therefore, does not make use of approximations to nonlinear processes to accommodate phaseaveraged models (e.g. Eldeberky and Battjes, 1995). In addition to describing the development and verification of this unique model, this study also demonstrates the robustness of nonlinear energy transfer even after significant damping from bottom mud.

cosh kn ðh þ zÞ cosh kh

fn ðzÞ ¼

ð2Þ

and a time-periodic velocity potential form which assumes a propagating wave:

~ n ¼  igAn ei / 2r n

R

kn dxxn t

þ c:c:

ð3Þ

where An is the amplitude of the free surface elevation g for the n-th frequency component and c.c. refers to the complex conjugate. Expanding the boundary value problem to second order in amplitude, using Green’s Second Identity (Smith and Sprinks, 1975) to derive a nonlinear mild-slope equation for wave propagation, limiting nonlinear interactions to resonant triads of frequency components (Phillips, 1980) and using the parabolic approximation (Radder, 1979) to reduce the equation to a computationally straightforward form results in a parabolic evolution equation for frequency domain wave propagation over varying depth and current (Kaihatu and Kirby, 1995). Further reducing the domain to one horizontal dimension results in:

   @An rn @ C gn þ U An þ D n An þ @x 2ðC gn þ UÞ @x rn ¼

n1 Nn X X i RAl Anl eiHl;nl þ 2 SAl Anþl eiHnþl;l 8ðC gn þ UÞrn l¼1 l¼1

! ð4Þ

where the nonlinear coupling coefficients are:

 R¼

 

g



r2 k k þ ðkl þ knþl Þðrnl kl þ rl knl Þ rl rnl n l nl r2   n r2l þ rl rnl þ r2nl g

S¼



 

g

r2 k k þ ðknþl  kl Þðrnþl kl þ rl knþl Þ rl rnþl n l nþl r2   n r2l  rl rnþl þ r2nþl g

ð5Þ

 ð6Þ

and the phase mismatches Hl,nl and Hn+l,l are:

Hl;nl ¼

Z

Hnþl;l ¼

kl þ knl  kn dx

Z

knþl  kl  kn dx

ð7Þ ð8Þ

The intrinsic frequency r in (4) is defined relative to a coordinate system moving at the current velocity U. It is related to the absolute frequency x (defined with respect to a stationary coordinate system) by:

2. Nonlinear wave model 2.1. Frequency domain wave–current interaction model The model used as a basis for development is the nonlinear wave–current interaction model of Kaihatu and Kirby (1995), with the second-order correction described by Kaihatu (2009). The model was used previously in a study of the effect of wave–current interaction on the recurrence phenomenon in random waves (Kaihatu, 2009). A full account of the derivation of the model can be found in Kaihatu and Kirby (1995); herein we restrict explanation of the model to salient details. The ambient current field is assumed to be constant in the vertical (z) coordinate but is allowed to vary in the horizontal (x, y) directions; this makes it consistent with a slowly-varying depth h(x, y) and irrotationality. The current field U = rh/0 is assumed to be O(1) (the subscript 0 represents the zeroth-order mean flow). The assumed form of the velocity potential / is:

~ n ðkn ; xn ; x; y; tÞ / ¼ /0 þ fn ðkn ; h; zÞ/

with:

ð1Þ

x ¼ r þ kU

ð9Þ

where:

r2 ¼ gk tanh kh

ð10Þ

The term DnAn in (4) represents the effect of energy dissipation and is written generically. This dissipation can be the result of breaking, damping by mud, or any other damping mechanism. The form of Dn for wave breaking has been discussed by Kirby and Kaihatu (1996), Eldeberky and Battjes (1996), and Chen et al. (1997), among others. The form of Dn for energy dampening by bottom viscous mud has been detailed by Kaihatu et al. (2007), and will be discussed in more detail in the next section. In the spirit of numerical models which include wave breaking effects, the mud-induced dissipation term is added to the wave evolution equation rather than derived via a suitable bottom boundary condition. A disadvantage of this approach is that higher

24

J.M. Kaihatu, N. Tahvildari / Ocean Modelling 41 (2012) 22–34

order phenomena such as surface–interface wave interactions (Hill and Foda, 1998; Tahvildari and Jamali, 2009) are not included. However, the primary focus in this study is on the behavior of the surface wave, for which this approach is reasonable. Eldeberky and Madsen (1999) detailed a deficiency of the basic model of Kaihatu and Kirby (1995); this was remedied by Kaihatu (2001), who formulated a correction to the amplitude An which accounts for the second order terms in the dynamic free surface boundary condition. Kaihatu (2009) rederived this correction for the case of wave–current interaction. This correction is:

" # n1 Nn X 1 X Bn ¼ An þ IAl Anl eiHl;nl þ 2 JAl Anþl eiHnþl;l 4g l¼1 l¼1

ð11Þ

where Bn is the complete second-order amplitude of the free surface and the interaction coefficients are:

r3nl xl þ r3l xnl  g 2 kl knl þ r2l r2nl rl rnl r3nþl xl þ r3l xnþl  g 2 kl knþl  r2l r2nþl J¼ rl rnl

I¼

while the real part of k2 contributes the effect of mud to the overall wavenumber:

k ¼ k1 þ Reðk2 Þ ¼ k1 

ReðBÞk1 sinh k1 h cosh k1 h þ k1 h

where B is a complex coefficient:

k1 dm ðB1  B2 Þ þ ck1 dm 2B3 k1 dm ðB1 þ B2 Þ ImðBÞ ¼ 2B3

ReðBÞ ¼

~ þ sinh2 dÞ ~ ~ cosh d ~  c2 fðcosh2 d B1 ¼ cð2c2 þ 2c  1  f2 Þ sinh d 2~ ~ þ sinh2 d ~ sin2 dÞ ~  ðc2  1Þ2 fðcosh d cos2 d

~ þ c sinh dÞ ~ cos d; ~  2cð1  cÞðf cosh d

ð13Þ

~ cos d ~ B2 ¼ cð2c2 þ 2c  1 þ f2 Þ sin d ~ þ c cosh dÞ ~  sin d; ~  2cð1  cÞðf sinh d

ð21Þ

~ þ ðf sinh d ~ þ c cosh dÞ ~ ~ þ c sinh dÞ ~ 2 cos2 d ~ 2 sin2 d: B3 ¼ ðf cosh d The complex wavenumber K is a function of several nondimensional parameters which serve to specify the mud characteristics:

qw qm rffiffiffiffiffiffi mm f¼ mw c¼

ð23Þ

~¼ d d dm where the subscript w refers to water. The ease of implementation is evident, as all mud effects are at second order and all explicit in terms of the nondissipative wavenumber (Eqs. (9) and (10)). Kaihatu et al. (2007) incorporated this mud dissipation mechanism into the nonlinear transformation model of Kaihatu and Kirby (1995) and compared the resulting model to the data of de Wit (1995). Despite the thin-layer assumption of the Ng (2000) dissipation mechanism, the comparison to data was favorable even though the mud layer depth of the experiment of de Wit (1995) would not generally be considered thin relative to the water depth. The consequences of the thin layer assumption of Ng (2000) are primarily apparent in the wavelength of the damped wave (real part of the complex wavenumber); Tahvildari and Kaihatu (2011) describe this in the context of a comparison to the unapproximated viscous mud layer model of Dalrymple and Liu (1978).

ð14Þ

where k = Re(K), is the wave amplitude, dm is the mud layer thickpaffiffiffiffiffiffiffiffiffiffiffiffi ness, and d ¼ 2m=x is the Stokes boundary layer thickness, in which m is the kinematic viscosity of the bottom mud layer. The subscript m denotes mud. Using this scaling to characterize the lower mud layer and develop the dissipation mechanism, Ng (2000) obtained the following form for the complex wavenumber:

ð15Þ

The leading order wavenumber k1 is real and is a solution of the nondissipative linear dispersion relation. The second order wavenumber is complex and contains the effect of fluid mud. The imaginary part of k2 is the damping rate:

ImðBÞk1 sinh k1 h cosh k1 h þ k1 h

ð20Þ

ð22Þ

In this section we outline the form of the dissipation function due to viscous bottom mud. As stated in a previous section, there are several different proxy behaviors for representing the effect of mud on the overlying surface waves; their ease of use depends on the resulting dispersion relation from the theory, which ties properties of the mud proxy with wave frequency x, wavenumber k, and water depth h. Linear dissipative systems generally have complex wavenumbers (Whitham, 1974), the real and imaginary parts of which govern wavelength and dissipation rate, respectively. In general, search for wavenumber roots must be undertaken in the complex plane (e.g. Dalrymple and Liu, 1978), with the attendant laborious calculation and ambiguity of multiple roots (Mendez and Losada, 2004). The problem of nonuniqueness of roots would make implementation into a predictive wave model difficult unless pre-calculation of the relevant roots was undertaken (Ng and Chiu, 2009). As a straightforward alternative, the thin-layer viscous mud dissipation theory of Ng (2000) was used. This development is essentially a boundary layer reduction of the model of Dalrymple and Liu (1978) which assumes that the effect of mud on surface waves can be considered at a higher order of approximation relative to the lower order nondissipative propagation. This was done by assuming a scale for the mud layer thickness that is equivalent to its Stokes boundary layer:

D ¼ Imðk2 Þ ¼ 

ð19Þ

where:

2.2. Model for mud–induced dissipation

K ¼ k1 þ k2 ; jk1 j  jk2 j

ð18Þ

ð12Þ

We use the full model with the corrections applied in this study.

ka  kdm  kdm  1

ð17Þ

ð16Þ

3. Model results In this section we show several results from the completed model. We first perform some comparisons to laboratory experiments in which waves, current and mud were present, and show that the model can recapture the measured damping rates. We then describe some results of wave–mud–current interaction in an effort to determine how the incidence of following or adverse currents affects the overall damping experienced by the waves. 3.1. Dissipation rates from laboratory data Ideally, comparisons to field data would have been performed in order to assess the accuracy of the model. However, at this stage of model development, we are interested in testing the model in

25

J.M. Kaihatu, N. Tahvildari / Ocean Modelling 41 (2012) 22–34

conditions where the uncertainties of wave forcing, current velocities and mud properties are minimized. This is particularly true of the measurement of mud properties, which may have substantial variability in field conditions. We therefore use laboratory data of wave, current and mud damping to validate the model. While several laboratory data sets for waves propagating over expanses of bottom mud have been published, wave–mud interaction experiments in the presence of currents are quite rare in the literature. The only two data sets readily available are by An and Shibayama (1995) and Zhao et al. (2006). Both studies also describe the development of viscoelastic mud dissipation models, and use the data to calibrate and validate the models. The published measurements for both studies consisted of damping rates, to which we compare our model results. The viscoelastic representation of the bottom mud layer employed by both An and Shibayama (1995) and Zhao et al. (2006) essentially lead to the following decomposition of the effect of mud on the surface waves:

me ¼ m þ

iG

ð24Þ

rq

where me is the ‘‘comprehensive dynamic viscosity’’ as denoted by An and Shibayama (1995), G is the elastic pffiffiffiffiffiffiffi modulus, r the wave frequency, q the mud density, and i ¼ 1. The real part of me is the viscosity m. In theory, therefore, it should be straightforward to obtain direct measures of viscosity from the viscoelastic characteristics of the mud. However, in a viscoelastic medium, the viscosity m

Estimates of D from An and Shibayama (1995) and Zhao et al. (2006) using Ng (2000) − Deduction of ν 0.14

0.12

is dependent on the strain-rate intensity, which is proportional to the square of the gradients of the horizontal and vertical velocities at the mud surface (Zhao et al., 2006). It is thus not clear whether the required information can be gleaned at all from the published data. We instead adopt a simpler approach and deduced from the data the kinematic viscosity m required in order to provide a comparable amount of damping as measured. Fig. 1 shows a comparison between the measured damping rate and that best-matched via the viscous mud mechanism of Ng (2000), both as a function of deduced (termed ‘‘equivalent’’) mud viscosity. It is apparent that the dissipation rate D appears to climb with increase in viscosity as estimated by Ng (2000). It is noted that equivalent mud viscosities were not deducible for all of the experiments of An and Shibayama (1995) and Zhao et al. (2006); this may be due to the limitations of the thin-layer model of Ng (2000) used in (4). However, equivalent viscosities were obtained for the majority of the experimental cases. We note further that the deduction of equivalent viscosities was done using the Ng (2000) mechanism independent of the nonlinear model. The resulting kinematic viscosities were then input to the model (4). The reported incident wave conditions were converted into nonlinear permanent form solutions for the model as developed by Kaihatu (2009). The dissipation rate was then derived from the results and compared to measurements. This was done by assuming that the wave height can be expressed as:

HðxÞ ¼ HO eDx

ð25Þ

where HO is the initial wave height at x = 0. The dissipation rate D was calculated at several points along the mud patch and averaged along the extent of the mud. It could be argued that this is not necessarily a validation of the model, since the choice of the appropriate equivalent viscosity may have insured that the dissipation rate would compare well with the measurements. However, as mentioned above, the viscosities were selected based on linear estimates of mud-induced damping from Ng (2000), while the model being verified (Eq. (4)) is nonlinear. The test, therefore, would be to determine if the nonlinear nature of the model affects the dissipation comparisons. Prior work of

0.1

Comparison of Dissipation Rates

0.14

D (1/m)

0.08

0.12 0.06

meas

(1/m)

0.1

D

0.04 Zhao et al. (2006) estimated using Ng (2000)

0.08

Measurement of Zhao et al. (2006) An and Shibayama (1995) estimated using Ng (2000) Measurement of An and Shibayama (1995)

0.06

0.02

0.04 0

0

0.002

0.004

0.006 ν (m 2/s)

0.008

0.01

0.012

Fig. 1. Comparison of dissipation rates: experiment of An and Shibayama (1995) and Zhao et al. (2006). Open circles: measured dissipation rates of Zhao et al. (2006). Asterisks: dissipation rate estimated using Ng (2000) and viscosities deduced from experiment of Zhao et al. (2006). Plus signs: measured dissipation rates of An and Shibayama (1995). X-signs: dissipation rate estimated using Ng (2000) and viscosities deduced from experiment of An and Shibayama (1995).

0.02 0.02

0.04

0.06

0.08 0.1 D (1/m)

0.12

0.14

mod

Fig. 2. Comparison of measured dissipation rates from An and Shibayama (1995, plus signs) and Zhao et al. (2006, - open circles) to those from the nonlinear model (4). Slope of best-fit line is 1.052.

26

J.M. Kaihatu, N. Tahvildari / Ocean Modelling 41 (2012) 22–34

ζ=100 T=10s h=1m Ho=0.1m

−1

10

Waveheight (m)

U=−0.15m/s D=−0.0085/m U=0m/s D=−0.0082/m U=+0.15m/s D=−0.0079/m

−2

10

−3

10

300

350

400

450

500

550 x (m)

600

650

700

750

800

Fig. 3. Wave height decay from (4), cnoidal wave propagation over a muddy bottom in water of constant depth. Solid line: Current opposing wave propagation. Dash-dot line: No current. Dashed line: Current and wave propagation in same direction.

U=+0.15 m/s

0

−1

−1

−2

−1

10

−2

−2

10

−3

10

−3

10

−3

10

10

−4

Amplitude (m)

10

−5

10

−6

10

−4

10

Amplitude (m)

−4

Amplitude (m)

10

10

10

−5

10

−6

10

−7

−8

−8

−9

−8

10

−9

−9

10

−10

10

−10

0

1 f (Hz)

2

10

−6

10

10

10

−5

10

−7

10

10

10

10

−7

10

U=−0.15 m/s

0

10

10

10

U=0.0 m/s

0

10

−10

0

1 f (Hz)

2

10

0

1 f (Hz)

2

Fig. 4. Amplitude spectra of cnoidal waves over mud. Left: U = 0.15 m/s. Center: U = 0. Right: U = 0.15 m/s. Dashed line: amplitude spectrum at x = 0. Solid line: amplitude spectrum at x = 800 m, the downwave end of the mud patch, for f = 10, dm = 0.02 m. Dash-dot line: Amplitude spectrum at x = 800 m, with f = 100, dm = 0.2 m.

Kaihatu et al. (2007) indicates that the effect of mud dissipation on wave nonlinearity can be substantial; in this sense the viscosities so deduced by a linear model might not be the most appropriate for a nonlinear model, so good model-data agreement is not a foregone conclusion. The comparison between calculated and measured dissipation rates is shown in Fig. 2. It is apparent that the model (4) is able to replicate the dissipation rates of the experiment; the slope of

the best-fit line through the points shown is 1.052. However, it is noteworthy that the nonlinear nature of the model does affect the damping, particularly at high damping rates; this is evident in the larger deviation of the modeled dissipation rates from the measured rates at high D, the equivalent viscosity values of which were calculated from the Ng (2000) model without nonlinearity. A likely explanation for this lack of agreement at high damping is that the dissipation mechanism of Ng (2000) varies with fre-

27

J.M. Kaihatu, N. Tahvildari / Ocean Modelling 41 (2012) 22–34

0.04

1.5

0.03

1

0.02

0.5

0.01

−0.5

−0.01

−1

−0.02

−1.5

1020

1040 1060 x (m)

1080

x 10

0

0

−0.03 1000

ζ=100

−3

2

eta (m)

eta (m)

ζ=10 0.05

−2 1000

1100

1020

1040 1060 x (m)

1080

1100

Fig. 5. Free surface elevations from model (4) forced by cnoidal waves. Left: f = 10, dm = 0.02 m. Right: f = 100, dm = 0.2 m. Solid line: U = 0.15 m/s. Dashed line: U = 0. Dashdot line: U =  0.15 m/s.

ζ=100 U=+0.15 m/s

0

−1

−1

−2

−1

10

−2

−2

10

−3

10

−3

10

−3

10

10

−4

Amplitude (m)

10

−5

10

−6

10

−4

10

Amplitude (m)

−4

Amplitude (m)

10

10

10

−5

10

−6

10

−7

−8

−8

−9

−8

10

−9

−9

10

−10

10

−10

0

1 f (Hz)

2

10

−6

10

10

10

−5

10

−7

10

10

10

10

−7

10

ζ=100 U=−0.15 m/s

0

10

10

10

ζ=100 U=0.0 m/s

0

10

−10

0

1 f (Hz)

2

10

0

1 f (Hz)

2

Fig. 6. Amplitude spectra of cnoidal waves over mud (f = 100, dm = 0.2 m); comparison of full nonlinear model (4), linear model, and nonlinear model with subharmonic interactions deactivated. For all spectra: x = 800 m. Left: U = 0.15 m/s. Center: U = 0. Right: U =  0.15 m/s. Solid line: nonlinear model (4). Dashed line: model (4) with nonlinear terms deactivated. Dash-dot line: model (4) with subharmonic terms deactivated.

quency. As will be further explored in a later section, this variation affects the nonlinear energy transfer within the wave train, which would, in turn, affect the overall wave height. To provide some context for the differences in damping rates: the largest damping rate error depicted in Fig. 2 will lead to an error between measured and modeled wave height of around 145%.

3.2. Permanent form waves over a muddy bottom: effect of currents We now use the model to investigate the effect of currents on the damping rate caused by the viscous bottom mud. We do this for both monochromatic (cnoidal) and random waves; in this section we discuss cnoidal wave damping. While these solutions of

28

J.M. Kaihatu, N. Tahvildari / Ocean Modelling 41 (2012) 22–34

Table 1 Wave conditions, random wave recurrence tests. Ur

Hrms (m)

fp (Hz)

h (m)

0.78 2.08

0.09 0.24

0.0625 0.0625

2 2

Table 2 Current velocities used in random wave recurrence tests. U (m/s)

Fr

1.10 0 1.10

0.25 0 0.25

permanent form are not generally indicative of ocean waves under field conditions, there are two advantages to their use. First, they are a canonical wave form, so deviations from this form due to interaction with mud and currents can be easily discerned. In addition, because of the frequency-domain form of the model (4), the cnoidal wave form is represented as component amplitudes which are harmonics of the base frequency (the inverse of the specified period of the cnoidal wave). Only a small number of harmonics are thus needed in order to span a wide range of frequencies and (as a result) a large range of dissipative behavior from the mud. The overall length of the domain is 1000 m, with mud placed between x = 300 m and x = 800 m. The (constant) water depth over the domain is h = 1 m, and the values used for the current pffiffiffiffiffiffiwere ±0.15 m/ s, which corresponds to a Froude number F r ¼ U= gh ¼ 0:05. A case with zero current was also examined. The incident wave is a cnoidaltype permanent form solution of (4), as developed in Kaihatu (2009), with a period T = 10 s and wave height H = 0.1 m. Fifteen harmonics of the base frequency f = 0.1 Hz were used to resolve the cnoidal wave form, leading to a frequency range 0.1 Hz 6 f 6 1.5 Hz. We note that the combination of water depth and incident wave condition was chosen to illuminate certain aspects of nonlinear evolution

over the finite patch of mud. However, this combination is comparable to many areas which have long shallow shelves and extensive areas of bottom mud; one example is the shelf near Marsh Island, Louisiana, USA, which has a depth of less than 2 m over a distance of more than 7 km (Sheremet, personal communication). The resolution used for the model was Dx = 0.025 m. We specified two different sets of parameters for mud dissipation, with each set leading to very different damping conditions. One set of conditions yields a low damping rate; for this case we use a viscosity m = 0.00013 m2/s and a mud layer depth dm = 0.02 m, resulting in a value of the viscosity parameter f = 10 (assuming that the kinematic viscosity of water is 1.3  106 m2/s). Conversely, the other set of conditions results in very high mud-induced damping; for this case we use m = 0.013 m2/s and a mud layer depth dm = 0.2 m, leading to f = 100. These conditions are also the extremes of the viscosity values recovered from the data of Zhao et al. (2006) mentioned in the previous section. Fig. 2 of Kaihatu et al. (2007) shows how the dissipation D changes with kh; at low kh (long waves) the boundary layer of the viscous mud will be greater than the actual thickness of the mud layer and damping is reduced, while at high kh (short waves) the waves are kinematically uncoupled with the bottom. One of the conclusions of An and Shibayama (1995) and Zhao et al. (2006) was that following currents tended to reduce the amount of mud-induced dissipation, while opposing currents enhanced it. Fig. 3 shows the wave heights from the model over the mud patch for the f = 100 case using following, zero, and opposing currents. It is clear that the opposing current, even as mild as used here, enhances the mud-induced damping, while the following currents reduce the damping. The damping rate D was calculated using (25). It was found that D = 0.0079 m1 for the following current and D = 0.0085 m1 for the opposing current, thus confirming that the model (4) replicates the findings of An and Shibayama (1995) and Zhao et al. (2006). One possible reason for this occurrence in the context of the present viscous mud formulation is that following currents increase the near-bottom orbital velocity (by lengthening the wave), and thus increase the

ζ=100 U =2.08 r

0

10

F = −0.25 D= −0.00044/m r

F = 0 D= −0.00039/m r

H

rms

(x)/H

rms

(1000 m)

Fr= +0.25 D= −0.00037/m

1000

1050

1100

1150

1200

1250 x (m)

1300

1350

1400

1450

1500

Fig. 7. Decay of normalized Hrms over mud patch in water of constant depth: random wave tests with opposing and following currents, f = 100, dm = 0.2 m. Solid line: Fr =  0.15. Dashed line: Fr = 0. Dash-dot line: Fr = 0.15.

29

J.M. Kaihatu, N. Tahvildari / Ocean Modelling 41 (2012) 22–34

U =0.78 ζ=10

(a)

U =2.08 ζ=10

r

0

−2

−2

S(f) (m2s)

10

2

S(f) (m s)

10

−4

10

−4

10

−6

10

r

(b) 100

10

−6

0

0.1

0.2

10

0.3

0

f (Hz)

(c)

Ur=0.78 ζ=10 Fr=−0.25

0

10

−1

−2

2 S(f) (m s)

2

S(f) (m s)

10

10

−3

10

−4

−2

10

−3

10

−4

10

10

0

20 40 60 Number of wavelengths of spectral peak Ur=2.08 ζ=10 Fr=−0.25

0

10

0

(f)

−1

20 40 60 Number of wavelengths of spectral peak Ur=2.08 ζ=10 Fr=+0.25

0

10

−1

10

2

S(f) (m2s)

10 S(f) (m s)

0.3

Ur=0.78 ζ=10 Fr=+0.25

(d) 100

−1

−2

10

−3

−2

10

−3

10

10

−4

10

0.2 f (Hz)

10

(e)

0.1

−4

0

20 40 60 Number of wavelengths of spectral peak

10

0

20 40 60 Number of wavelengths of spectral peak

Fig. 8. Wave spectra and evolution of harmonics of the spectral peak from nonlinear model (4); f = 10 and dm = 0.02 m. Top row: wave spectra for (a) Ur = 0.78 and (b) Ur = 2.08. For (a) and (b), solid line is spectra at x = 0, dashed line is spectra at x = 21Lp (downwave edge of mud patch, where Lp is the wavelength of the spectral peak) with Fr =  0.25, and dash-dot line is spectra at x = 21Lp with Fr = 0.25. Middle and bottom rows: Spectral energy density at several frequencies evolving over distance for: (c) Ur = 0.78, Fr =  0.25; (d) Ur = 0.78, Fr = 0.25; (e) Ur = 2.08, Fr =  0.25; and (f) Ur = 2.08, Fr = 0.25. For (c) through (f): Energy density at spectral peak of fp = 0.0625 Hz (solid line); second harmonic of peak (dashed line); third harmonic of peak (dash-dot line); and subharmonic of peak (fp/2; dash-x).

boundary layer thickness beyond that of the mud layer thickness. This would reduce the mud-induced dissipation. Opposing currents, in contrast, reduce the wavelength and would thus have the opposite effect on the dissipation. Since the model (4) evolves individual spectral amplitudes of the cnoidal waves, it is possible to determine how the dissipation affects different frequency components of the wave field. For the cnoidal wave train used here, kh ranges from 0.2 (for f = 0.1 Hz) to 9 (for f = 1.5 Hz), which would lead to a wide range of mud dissipation over this range. Fig. 4 shows amplitude spectra for following, zero and opposing currents for both f = 10 and f = 100. While the f = 10 case clearly undergoes less overall damping than the f = 100 case, Kaihatu et al. (2007) showed that both cases had very similar distributions of dissipation over a range of kh. The difference in dissipation is clearly displayed in Fig. 4. The f = 10 case exhibits a change in the slope of the amplitude spectrum caused

by the mud, while the f = 100 case results in a very different amplitude spectrum. The frequency experiencing the greatest dissipation (f = 0.4 Hz) corresponds to kh  0.9, which is close to that for the maximum dissipation for the present case shown by Kaihatu et al. (2007). Fig. 5 shows the corresponding free surface elevations downwave of the mud patch for both f = 10 and f = 100 and for Fr = 0.15, Fr = 0 and Fr = +0.15. While the distinctive shape of the cnoidal wave is evident for f = 10, the wave shape is quite different for f = 100. Rather than cnoidal, the shape is quite sinusoidal due to the strong dampening of energy at frequencies responsible for maintaining the flat troughs and sharp crests endemic of cnoidal waves. Additionally, the waveforms for the Fr = 0.15 (opposing current) case exhibit some short-wave variability relative to the other cases; it appears that damping of wave energy at particular frequencies by the mud, and the increase of the wavenumber by

30

J.M. Kaihatu, N. Tahvildari / Ocean Modelling 41 (2012) 22–34

(a)

U =0.78 ζ=100 r

0

(b)

10

−2

10 S(f) (m2s)

S(f) (m2s)

r

−2

10

−4

10

−4

10

−6

10

U =2.08 ζ=100 0

10

−6

0

0.1

0.2

10

0.3

0

f (Hz)

(c)

Ur=0.78 ζ=100 Fr=−0.25

0

10

(d)

−1

−2

S(f) (m2s)

S(f) (m2s)

Ur=0.78 ζ=100 Fr=+0.25

0

10

−1

−3

10

−4

−2

10

−3

10

−4

10

10

0

20 40 60 Number of wavelengths of spectral peak Ur=2.08 ζ=100 Fr=−0.25

0

10

0

(f)

−1

20 40 60 Number of wavelengths of spectral peak Ur=2.08 ζ=100 Fr=+0.25

0

10

−1

10 S(f) (m2s)

10 S(f) (m2s)

0.3

10

10

−2

10

−3

−2

10

−3

10

10

−4

10

0.2 f (Hz)

10

(e)

0.1

−4

0

20 40 60 Number of wavelengths of spectral peak

10

0

20 40 60 Number of wavelengths of spectral peak

Fig. 9. Wave spectra and evolution of harmonics of the spectral peak from nonlinear model (4); f = 100 and dm = 0.2 m. Top row: wave spectra for (a) Ur = 0.78 and (b) Ur = 2.08. For (a) and (b), solid line is spectra at x = 0, dashed line is spectra at x = 21Lp (downwave edge of mud patch, where Lp is the wavelength of the spectral peak) with Fr =  0.25, and dash-dot line is spectra at x = 21Lp with Fr = 0.25. Middle and bottom rows: Spectral energy density at several frequencies evolving over distance for: (c) Ur = 0.78, Fr = 0.25; (d) Ur = 0.78, Fr = 0.25; (e) Ur = 2.08, Fr = 0.25; and (f) Ur = 2.08, Fr = 0.25. For (c) through (f): Energy density at spectral peak of fp = 0.0625 Hz (solid line); second harmonic of peak (dashed line); third harmonic of peak (dash-dot line); and subharmonic of peak (fp/2; dash-x).

both the mud and the adverse current, is reducing the phase locking between the wave frequencies required for the wave to propagate as a permanent-form wave. It can be seen in Fig. 4 that the high frequencies in the amplitude spectrum in the cnoidal wave train undergoes damping. As noted above, the relative water depth kh = 9 at the highest frequency. Kaihatu et al. (2007) showed that the energy loss at these frequencies were primarily due to subharmonic interactions with highlydamped lower frequencies; the lack of compensated energy transfer from low frequencies due to their direct damping caused loss of energy in the high frequencies. To illustrate this, as well as show how a co-flowing current might affect the results, we alter the model (4) in two ways. First, we deactivate all nonlinear summations in (4), creating a completely linear model. We then deactivate only the first summation of (4), allowing only superharmonic nonlinear interactions. Kaihatu et al. (2007) showed that these two altered models gave equivalent results with no high frequency

damping, confirming the role of subharmonic interactions in damping energy at high frequencies. Fig. 6 shows the resulting amplitude spectra just beyond the mud patch for following, zero and opposing currents. As in Kaihatu et al. (2007), the high frequency components in the altered models (linear and reduced nonlinear) show little damping over the mud layer for all three cases. This indicates that the presence of a co-flowing current does not affect the essential conclusion linked to the role of subharmonic interactions in dampening out wave energy in very high relative water depths. 3.3. Random waves, currents and mud In this section, we investigate how the presence of mud affects the evolution of random waves with and without a current. We use the same specification for the mud as in the previous section (f = 10, dm = 0.02 m for low damping, f = 100, dm = 0.2 m for high

31

J.M. Kaihatu, N. Tahvildari / Ocean Modelling 41 (2012) 22–34

F =−0.25 x r

0

=1000m−1500m

F =−0.25 x

mud

−1

−1

10

−2

−2

10 S(f) (m2s)

S(f) (m2s)

10

−3

10

−4

10

−5

−5

10

10

−6

−6

0

0.1

0.2 f (Hz)

10

0.3

Fr=+0.25 xmud=1000m−1500m

0

0

0.1

0.2 f (Hz)

0.3

Fr=+0.25 xmud=500m−1000m

0

10

10

−1

−1

10

10

−2

−2

10 S(f) (m2s)

10 S(f) (m2s)

−3

10

−4

10

−3

10

−4

−3

10

−4

10

10

−5

−5

10

10

−6

10

=500m−1000m

mud

10

10

10

r

0

10

−6

0

0.1

0.2 f (Hz)

0.3

10

0

0.1

0.2 f (Hz)

0.3

Fig. 10. Wave spectra from nonlinear model (4): f = 100 and dm = 0.2 m for all simulations shown. Top row: Fr = 0.25. Bottom row: Fr = +0.25. Left column: 1000 m 6 xmud 6 1500 m. Right column: 500 m 6 xmud 6 1000 m. Solid line: spectrum at beginning of mud patch. Dashed line: spectrum at downwave end of mud patch.

damping). Our domain for this study consists of a constant water depth h = 2 m and an overall length of 4900 m or 70 times the wavelength of the spectral peak (Lp). This overall domain length was chosen to allow the spectra to reach a flat equilibrium shape; in the absence of mud, no significant evolution of the spectrum occurred after this point (Kaihatu, 2009). The mud patch in this section extends from x = 1000 m (about 14Lp) to x = 1500 m (about 21Lp), which allows the spectral peak of the wave spectra to undergo some degree of quasi-recurrent cycling in energy before encountering the mud patch. The wave spectra uses here is of the TMA form (Bouws et al., 1985). Specification of the spectral parameters for the wave field are displayed in Table 1. We use wave conditions with different Ursell numbers (Ur), where:

Ur ¼

d

l2

ð26Þ

where:

d¼

Hrms 2h

and:

ð27Þ

l ¼ kh

ð28Þ

The characteristics of the spectral evolution are quite dependent on the Ursell number. Kaihatu (2009) showed that, for this domain and a Ur = 0.78, the spectrum would undergo some degree of cyclic evolution; quasi-recurrent effects between the frequency peak and its harmonics would be evident before the spectrum flattens into a ‘‘whitened’’ shape. In contrast, Kaihatu (2009) also showed that a spectrum with Ur  2 would experience broad cross-spectral energy exchange immediately, with little recurrent cycling. The current velocities used here are listed in Table 2, and the conditions are referenced by their associated Froude numbers Fr. Both the wave and current conditions detailed here were also used in Kaihatu (2009). Fig. 7 shows mud-induced dissipation rates from the models, as calculated from (25) with Hs substituted for H and using f = 100 and dm = 0.2 m. As with the cnoidal wave case, the dissipation rate D under opposing currents is higher than under following currents. Results from the model are shown in Fig. 8 (f = 10, dm = 0.02 m) and Fig. 9 (f = 100, dm = 0.2 m) for Fr = ±0.25. For the case of f = 10, dm = 0.02 m (Fig. 8), which exhibits little apparent damping, the evolution of the spectrum follows the trends mentioned earlier: the spectral maxima at the peak frequency and its harmonics are

32

J.M. Kaihatu, N. Tahvildari / Ocean Modelling 41 (2012) 22–34

ζ=100 x

ζ=100 x

=1000m−2000m

mud

0

−5

2

10

−10

=1000m=3000m

mud

(b) 100

10

S(f) (m2s)

S(f) (m s)

(a)

−5

10

−10

10

10

0

0.1

0.2

0.3

0

0.1

0.2

f (Hz)

(c)

Fr=−0.25 xmud=1000m−2000m

0

10

−2

−2

10 2

S(f) (m s)

2 S(f) (m s)

Fr=+0.25 xmud=1000m−2000m

(d) 100

10

−4

10

−4

10

−6

−6

10

10 0

20 40 60 Number of wavelengths of spectral peak F =−0.25 x

(e)

r

0

0

=1000m−3000m

mud

10

−2

r

0

=1000m−3000m

mud

10

−2

10 S(f) (m2s)

−4

10

−6

10

−8

−4

10

−6

10

−8

10

10

−10

10

20 40 60 Number of wavelengths of spectral peak F =+0.25 x

(f)

10 S(f) (m2s)

0.3

f (Hz)

−10

0

20 40 60 Number of wavelengths of spectral peak

10

0

20 40 60 Number of wavelengths of spectral peak

Fig. 11. Wave spectra and evolution of harmonics of the spectral peak from nonlinear model (4); f = 100, Ur = 0.78 and dm = 0.2 m. Top row: wave spectra for (a) xmud = 1000 m–2000 m and (b) xmud = 1000 m–3000 m. For (a) and (b), solid line is spectra at x = 0, dashed line is spectra at downwave edge of mud patch with Fr = 0.25, and dash-dot line is spectra at downwave edge of mud patch with Fr = 0.25. Middle and bottom rows: Spectral energy density at several frequencies evolving over distance for: (c) Fr = 0.25, xmud = 1000 m–2000 m; (d) Fr = 0.25, xmud = 1000 m–2000 m; (e) Fr = 0.25, xmud = 1000 m–3000 m; and (f) Fr = 0.25, xmud = 1000 m–3000 m. For (c) through (f): Energy density at spectral peak of fp = 0.0625 Hz (solid line); second harmonic of peak (dashed line); third harmonic of peak (dash-dot line); and subharmonic of peak (fp/2; dash-x).

evident in the evolution of moderate Ur conditions, while relatively high Ur conditions broaden rapidly in frequency with no discernible peaks in the spectrum. However, it is clear that opposing currents (Fr = 0.25) enhance the dissipation caused by the mud patch, while following currents (Fr = +0.25) reduce this damping. This is particularly evident in the case where Ur = 0.78 (Fig. 8a), but can also be seem in the Ur = 2.08 case (Fig. 8b), especially at the higher frequencies (f > 0.1 Hz). Fig. 8c through f show the evolution of the energy density at the frequency peak, the second and third harmonics, and the subharmonic (fp/2). Energy density at these harmonic components undergo some evolution before reaching a state of apparent equilibrium. There is more disparity in energy level between harmonics in the case of Ur = 0.78 (Fig. 8c and d) than for Ur = 2.08 (Fig. 8e and f). Similar to the cnoidal wave case of the previous section, there is more dissipation of high frequency components for opposing currents and Ur = 0.78 than for following

currents. In contrast, dissipation is not as apparent for Ur = 2.08 for either following or opposing currents, though the scale of evolution toward an apparent equilibrium state is more rapid for opposing currents (Fig. 8e) than following currents (Fig. 8f); this was also seen for wave–current interaction alone by Kaihatu (2009). For the case of f = 100, dm = 0.2 m (Fig. 9), the mud-induced damping is far more apparent, as is the effect of the orientation of the current, relative to f = 10. This is seen in the wave spectra shown for Ur = 0.78 (Fig. 9a) and Ur = 2.08 (Fig. 9(b). As with the f = 10 case, evolution of the spectral amplitudes at selected harmonics of the peak frequency are shown (Fig. 9c through f). What is interesting in this case, however, is that the energy in the frequencies beyond the spectral peak appear to rebound in energy beyond the leeward end of the mud patch (x P 21Lp). This is true for all cases of Ur, with this rebound being quite small for Ur = 0.78 and Fr = 0.25 (Fig. 9e). The increase in high frequency energy comes at the expense of low

J.M. Kaihatu, N. Tahvildari / Ocean Modelling 41 (2012) 22–34

frequency energy, most clearly seen for Ur = 2.08 and Fr = 0.25 (Fig. 9f). It can also be seen that the Ur = 2.08 case still undergoes faster spectral broadening than the Ur = 0.78 case, despite the high degree of mud-induced dissipation. For almost all cases, the harmonic spectral amplitudes (Fig. 8c through f; Fig. 9c through f) reach some degree of apparent equilibrium by Lp = 50. 3.3.1. Effect of mud patch location and extent Ideally, it would be useful to run more cases using different wave conditions to broaden some of the observations discussed here. However, the wave conditions cover a wide range of Ursell number, and given the model’s tendency to quickly broaden at Ur = 2.08, more insight might not be gained with higher wave heights (higher Ur). Instead, to add some more generality to the conclusions, we alter the spatial extent of the mud for two different cases. For both cases we specify the high damping mud parameters (f = 100, dm = 0.2 m) and use the wave condition where Ur = 0.78; the latter was used since the wave spectra appeared to show more harmonic structure over a longer spatial extent than those of a higher Ursell number. The location of the mud patch in the previous simulations was determined by gauging the length needed for the spectral peak to undergo some degree of cyclic quasi-recurrent evolution (an indication of developing nonlinear wave–wave interaction) prior to damping. To gauge the effect of the location of the mud patch on the evolution of the wave spectrum, the mud patch was relocated to 500 m 6 x 6 1000 m, intercepting the wave an earlier stage in nonlinear interaction development. Fig. 10 shows the resulting spectra. In the case of opposing current, there is little harmonic structure in the spectra in either the original or modified mud placements, and the primary difference appears to be the presence of a discernable peak in the spectrum at the end of the mud patch of the modified mud placement case. In the case of following current, however, the harmonic structure of the spectra in both the original and modified mud placement cases are evident. This is likely due to the strengthening of nonlinear interactions caused by following currents (Chen et al., 1999; Kaihatu, 2009). Additionally, it was generally noted in the prior sets of simulations that the higher frequencies of the wave spectra appeared to gain in energy (at the expense of the lower frequencies) some distance beyond the lee of the mud patch. This is due to nonlinear interactions moving energy from low to high frequencies. It is of interest to examine how robust this mechanism is if the extent of the mud is increased (and, consequently, more damping takes place). We extended the mud patch first from 1000 m 6 x 6 2000 m, then from 1000 m 6 x 6 3000 m. The results are shown in Fig. 11; spectra are shown in Fig. 11a and b, and spectral amplitudes for selected harmonics of the peak frequency are shown in Fig. 11c through Fig. 11f. For all cases, spectra are output at the beginning of the domain at the leeward edge of the mud patch (x = 28Lp and x = 43Lp for the first and second cases of an extended mud patch, respectively). For all simulations, the longer mud patches show very strong damping at the higher frequencies of the spectrum with some gain in energy in these frequencies beyond the mud patches; these are seen in Fig. 11c through f. As with most of the previous simulations, the subharmonic frequency appears to be little-affected by direct mud-induced dissipation regardless of the extent of the mud; reduction in energy in these frequencies are generally due to the transfer of energy to highly-damped higher frequencies. This has implications for long wave generation and potential low-frequency surge; similar phenomena has been observed in experiments of wave breaking on fringing reefs (Nwogu and Demirbilek, 2010). In addition, the increase in high frequency energy beyond the mud patch has an effect on the near-bottom velocity statistics, which may impact sediment transport mechanisms which use this information (Bailard, 1981). At present, the degree of spectral evolution seen in these

33

results cannot be simulated at present by phase-averaged nearshore wave models.

4. Conclusion A model which combines the effect of wave nonlinearity, mudinduced dissipation and wave–current interaction is described. The model is based on the frequency-domain nonlinear wave model of Kaihatu and Kirby (1995), using separate extensions to wave–mud interactions (Kaihatu et al., 2007) and wave–current interaction (Kaihatu, 2009). For dissipation by mud, we used the thin-layer viscous mud mechanism of Ng (2000), due primarily to its ease of implementation, though we note that Kaihatu et al. (2007) showed that this damping mechanism was relatively skillful even when outside the ostensible thin-layer limitation. The model was first compared to measured dissipation rates from laboratory experiments (An and Shibayama, 1995; Zhao et al., 2006). Because that study treated the mud as a viscoelastic (rather than viscous) material, the viscous damping mechanism of Ng (2000) was used (independent of the nonlinear model) to deduce the viscosity needed to provide the measured dissipation rates. With these viscosity values, the nonlinear model (4) with both mud and wave–current interaction was used to replicate the measured dissipation rates. For the most part, the dissipation rates were well recovered, though somewhat less so at high damping rate. In this range of damping rate, the dissipation strongly affects the nonlinear energy transfer and, consequently, the estimation of the wave height. The model was then used to study the propagation of nonlinear permanent form cnoidal waves over a muddy bottom and through opposing and following currents. It was first shown that the model predicts trends in dissipation rates that are in line with the conclusions of previous studies; namely, that following currents reduce the dissipation due to a muddy bottom, while opposing currents increase this dissipation. In the context of the thin layer viscous formulation of Ng (2000) used in the model, it is thought that these trends occur due to the relationship between the boundary layer of the fluid and the thickness of the mud layer. The increase in near bottom wave orbital velocity due to a following current increases the boundary layer thickness of the fluid relative to the mud depth, and would thus decrease the dissipation possible with a given mud thickness and viscosity compared with no ambient current; the opposite is true for opposing currents. Amplitude spectra from the model showed strong frequency dependence of mud-induced dissipation for the case of deep, highly viscous mud, as well as the influence of currents on this dissipation. One of the primary conclusions of the prior work of Kaihatu et al. (2007) was that uncompensated subharmonic energy transfer to low frequencies undergoing strong damping was responsible for draining energy from frequencies too high to be otherwise affected by the muddy bottom. The addition of both opposing and following currents did not affect this basic conclusion. We then applied the model to the problem of random wave propagation, using the same spatial domain, wave and current conditions as that of Kaihatu (2009), with the exception of the addition of a finite patch of mud. Again, the calculated damping rate from the model for both following and opposing currents revealed that the damping rate for following currents is less than that for opposing currents, as was the case for cnoidal waves. Kaihatu (2009) determined that random wave propagation over a flat bottom would generally lead to slow spectral broadening for moderate Ursell numbers and relatively rapid broadening for higher Ursell numbers; the presence of currents would also affect the spatial scale of this broadening. For the case of light damping by mud, these trends are evident in the results, with the additional feature

34

J.M. Kaihatu, N. Tahvildari / Ocean Modelling 41 (2012) 22–34

of damping of the high frequency portion of the spectrum. This damping is less for following currents and enhanced for opposing currents. In contrast, the case of significant damping by mud shows very strong damping of the spectral frequencies above the peak; as before, this damping is exacerbated by opposing currents. However, well into the leeward side of the mud patch, it could be seen that these high frequencies have recovered some energy at the expense of the lower frequencies. This recovery appears to be robust, as it occurs even after overwhelming damping of the high frequencies from long extents of mud. 5. Future work This work is part of a larger effort to improve and extend the capabilities of coastal wave models, both phase resolved and phase averaged. The integration of these models into a larger overall framework is the focus of future work. In addition, a method of validation with field data which takes into account the uncertainty of the environment will be developed. Recently Tahvildari and Kaihatu (2011) developed a method for inverting free-surface information in conjunction with the model of Kaihatu et al. (2007) for mud parameter information, revealing many details of the sensitivity of predicted mud-induced damping on the specified parameters. In combination with field data, this can provide a means of validation while accounting for uncertainty. Furthermore, it is felt that the model can be used as a tool for aiding the planning of field campaigns. Acknowledgments This work was supported by the Office of Naval Research through the National Ocean Partnership Program (Award N00014–10-0389). Discussions with Dr. Alex Sheremet, University of Florida, proved useful for this paper. Anonymous reviewers provided suggestions which substantially improved this work. References An, N.N., Shibayama, T., 1995. Wave–current interaction with mud bed. In: Proc. 24th Intl. Conf. Coast. Eng., 2913–2927. Bailard, J.A., 1981. An energetics total load sediment model for a plane sloping beach. J. Geophys. Res. 86, 10938–10954. Bouws, E., Gunther, H., Rosenthal, W., Vincent, C.L., 1985. Similarity of the wind wave spectrum in finite depth water: 1. Spectral form. J. Geophys. Res. 90, 975– 986. Bretherton, F.P., Garrett, C.J., 1968. Wavetrains in inhomogeneous moving media. Proc. Roy. Soc. Ser. A 302, 529–554. Chawla, A., Kirby, J.T., 2002. Monochromatic and random wave breaking at blocking points. J. Geophys. Res. 107. doi:10.1029/2001 JC001042. Chen, Q., Madsen, P.A., Basco, D.R., 1999. Current effects on nonlinear interactions of shallow-water waves. J. Wtrway. Port Coast. Oc. Eng. 125, 176–186. Chen, Y., Guza, R.T., Elgar, S., 1997. Modeling specra of breaking surface waves in shallow water. J. Geophys. Res. 102, 25035–25046. Chou, H.T., Foda, M.A., Hunt, J.R., 1993. Rheological response of cohesive sediments to oscillatory forcing. In: Mehta, A.J. (Ed.), Nearshore and Estuarine Cohesive Sediment Transport, vol. 42, AGU, pp. 126–148. Crapper, G.D., 1972. Nonlinear gravity waves on steady non-uniform currents. J. Fluid Mech. 52, 713–724. Dalrymple, R.A., Liu, P.L.-F., 1978. Waves over soft muds: a two-layer fluid model. J. Phys. Oc. 8, 1121–1131. de Wit, P.J., 1995. Liquifaction of cohesive sediment by waves. Ph.D. dissertation, Delft University of Technology. Eldeberky, Y., Battjes, J.A., 1995. Parameterization of triad interactions in wave energy models. Proc. Coast. Dyn. ’95, 140–148.

Eldeberky, Y., Battjes, J.A., 1996. Spectral modeling of wave breaking: Application to Boussinesq equations. J. Geophys. Res. 101, 1253–1264. Eldeberky, Y., Madsen, P.A., 1999. Deterministic and stochastic evolution equations for fully dspersive and weakly nonlinear waves. Coast. Eng. 38, 1–41. Elgar, S., Freilich, M.H., Guza, R.T., 1990. Recurrence in truncated Boussinesq models for nonlinear waves in shallow water. J. Geophys. Res. 95, 11547–11556. Foda, M.A., Hunt, J.R., Chou, H.T., 1993. A nonlinear model for the fluidization of marine mud by waves. J. Geophys. Res. 98, 7039–7047. Gade, H.G., 1958. Effects of a nonrigid, impermeable bottom on plane waves in shallow water. J. Mar. Res. 16, 61–82. Hill, D.F., Foda, M.A., 1998. Subharmonic resonance of oblique interfacial waves by a progressive surface wave. Proc. Royal Soc. London A 454, 1129–1144. Hsiao, S.V., Shemdin, O.H., 1980. Interaction of ocean waves with a soft bottom. J. Phys. Oc. 10, 605–610. Jiang, F., Mehta, A.J., 1995. Mudbanks of the southwest coast of India IV: mud viscoelastic properties. J. Coast. Res. 11, 918–926. Jiang, F., Mehta, A.J., 1996. Mudbanks of the southwest coast of India V: wave attenuation. J. Coast. Res. 12, 890–897. Kaihatu, J.M., 2001. Improvement of parabolic nonlinear dispersive wave model. J. Wtrway. Port Coast. Oc. Eng. 127, 113–121. Kaihatu, J.M., 2009. Application of a nonlinear frequency domain wave–current interaction model to shallow water recurrence in random waves. Oc. Mod. 26, 190–205. Kaihatu, J.M., Kirby, J.T., 1995. Nonlinear transformation of waves in finite water depth. Phys. Fluids 7, 1903–1914. Kaihatu, J.M., Sheremet, A., Holland, K.T., 2007. A model for the propagation of nonlinear surface waves over viscous muds. Coast. Eng. 54, 752–764. Kirby, J.T., 1986. Higher-order approximations in the parabolic equation method for water waves. J. Geophys. Res. 91, 933–952. Kirby, J.T., Kaihatu, J.M., 1996. Structure of frequency domain models for random wave breaking. In: Proc. 25th Intl. Conf. Coast. Eng., 1144–1155. Macpherson, H., 1980. The attenuation of water waves over a non-rigid bed. J. Fluid Mech. 97, 721–742. Mathew, J., 1992. Wave–mud interaction in mud banks. Ph.D. dissertation, Cochin University of Schience and Technology, Cochin, India. Mendez, F.J., Losada, I.J., 2004. A perturbation method to solve dispersion equations for water waves over dissipative media. Coast. Eng. 51, 81–89. Mei, C.C., Liu, K.F., 1987. A Bingham-plastic model for a muddy seabed under long waves. J. Geophys. Res. 92, 14581–14594. Ng, C.-O., 2000. Water waves over a muddy bed: a two-layer Stokes’ boundary layer model. Coast. Eng. 40, 21–242. Ng, C.-O., Chiu, H.-S., 2009. Use of Mathcad as a calculation tool for water waves over a stratified muddy bed. Coast. Eng. J. 51, 69–79. Nwogu, O., Demirbilek, Z., 2010. Infragravity wave motions and runup over shallow fringing reefs. J. Wtrwy. Port Coast. Oc. Eng. ASCE 136 (6), 295–305. Phillips, O.M., 1980. The Dynamics of the Upper Ocean. Cambridge University Press, New York, p. 336. Radder, A.C., 1979. On the parabolic equation method for water–wave propagation. J. Fluid Mech. 95, 159–176. Sheremet, A., Stone, G.W., 2003. Observations of nearshore wave dissipation over muddy sea beds. J. Geophys. Res. 108, 3357–3368. Sheremet, A., Mehta, A.J., Kaihatu, J.M., 2005. Wave–sediment interaction on a muddy shelf. In: Proc. 5th Intl. Symp. Oc. Wave Meas. An., Madrid, Spain. Smith, R., 1976. Giant waves. J. Fluid Mech. 77, 417–431. Smith, R., Sprinks, T., 1975. Scattering of surface waves by a conical island. J. Fluid Mech. 72, 373–384. Tahvildari, N., Jamali, M., 2009. Analytical cubic solution to weakly nonlinear interactions between surface and interfacial waves. In: Proc. 28th Intl. Conf. Oc. Off. Arc. Eng. ASME, Honolulu, HI. doi:10.1115/OMAE2009-80120. Tahvildari, N., Kaihatu, J.M., 2011. Optimized determination of viscous mud properties using a nonlinear wave–mud interaction model. J. Atmos. Ocean Tech. doi:10.1175/JTECH-D-11-00025.1. Turpin, F.-M., Benmoussa, C., Mei, C.C., 1983. Effects of slowly varying depth and current on the evolution of a Stokes wavepacket. J. Fluid. Mech. 132, 1–23. Wells, J.T., Coleman, J.M., 1981. Physical processes and fine-grained sediment dynamics, coast of Surinam, South America. J. Sed. Petro. 15, 1053–1068. Whitham, G.B., 1974. Linear and Nonlinear Waves. Wiley-Interscience, New York. Yamamoto, T., Takahashi, S., 1985. Wave damping by soil motion. J. Wtrway. Port Coast. Oc. Eng. (ASCE) 111, 62–77. Yamamoto, T., Koning, H.L., Sellmeiher, H., van Hijum, E.V., 1978. On the response of a poro-elastic bed to water waves. J. Fluid Mech. 87, 193–206. Zhao, Z.-D., Lian, J.-J., Shi, J.Z., 2006. Interactions among waves, current, and mud: numerical and laboratory studies. Adv. Water Res. 29, 1731–1744.