Numerical study on wave dynamics and wave-induced bed erosion characteristics in Potter Cove, Antarctica

Abstract

Wave generation, propagation, and transformation from deep ocean over complex bathymetric terrains to coastal waters around Potter Cove (King George Island, South Shetland Islands, Antarctica) have been simulated for an austral summer month using the Simulating Waves Nearshore (SWAN) wave model. This study aims to examine and understand the wave patterns, energy fluxes, and dissipations in Potter Cove. Bed shear stress due to waves is also calculated to provide a general insight on the bed sediment erosion characteristics in Potter Cove.A nesting approach has been implemented from an oceanic scale to a high-resolution coastal scale around Potter Cove. The results of the simulations were compared with buoy observations obtained from the National Data Buoy Center, the WAVEWATCH III model results, and GlobWave altimeter data. The quality of the modelling results has been assessed using two statistical parameters, namely the Willmott’s index of agreement D and the bias index. Under various wave conditions, the significant wave heights at the inner cove were found to be about 40–50 % smaller than the ones near the mouth of Potter Cove. The wave power in Potter Cove is generally low. The spatial distributions of the wave-induced bed shear stress and active energy dissipation were found to be following the pattern of the bathymetry, and waves were identified as a potential major driving force for bed sediment erosion in Potter Cove, especially in shallow water regions. This study also gives some results on global ocean applications of SWAN.

Appendices

Appendix 1: Governing equations of the wave model

1.1 Wind source terms

Wave growth by wind (SWAN Team 2011a) is described by

$$ S_{in}(\sigma,\theta) = A + BE(\sigma,\theta) $$

(11)

in which A describes linear growth and B exponential growth. The expression for the linear growth term A is given by Cavaleri and Malanotte-Rizzoli (1981):

$$ A = \frac{1.5\times 10^{-3}}{2\pi g^{2}}(U_{*}\:{\max}[0,{\cos}(\theta -\theta_{w})])^{4}H_{f} $$

(12)

where U _∗ is the friction velocity (Eq. 16), 𝜃 _w is the wind direction, and H _f is the filter (Eq. 13) to eliminate wave growth at frequencies lower than the Pierson–Moskowitz frequency. Equation 14 gives the peak frequency $\sigma _{\rm {PM}}^{*}$ of the fully developed sea state (Pierson and Moskowitz 1964).

$$ H_{f}={\text{exp}}\left\{-\left(\frac{\sigma }{\sigma_{\rm{PM}}^{*}}\right)^{-4}\right\} $$

(13)

$$ \sigma_{\rm{PM}}^{*} = \frac{0.13g}{28\:U_{*}}2\pi $$

(14)

For the exponential growth, the expression due to Komen et al. (1984) is used in this study (Eq. 15). The exponential growth term becomes significant when wave energy is present.

$$ B={\max}\left[0,0.25\frac{\rho_{\rm{a}}}{\rho_{\rm{w}}}\left(28\frac{U_{*}}{c_{\rm{ph}}}{\cos}(\theta -\theta_{w})-1\right)\right]\sigma $$

(15)

where ρ _a and ρ _w are the air and water densities, respectively, and c _ph is the phase speed of the wave.

While SWAN is driven by the wind speed at 10 m height U ₁₀, the friction velocity U _∗ is used for the computations of A and B and is obtained from the following equation:

$$ U_{*}^{2} = C_{\rm{d}}U_{10}^{2} $$

(16)

in which C _d is the wind drag coefficient after (Wu 1982):

$$ C_{\rm{d}}=\left\{\begin{array}{ll} 1.2875\, \times \, 10^{-3},\qquad \qquad\quad\text{for} \; U_{10} < 7.5 \; \mathrm{m/s}\\ ( 0.8\, + \, 0.065 \; U_{10}) \times \, 10^{-3},\;\text{for} \; U_{10} \geq 7.5 \;\mathrm{m/s} \end{array}\right. $$

(17)

1.2 Wave energy dissipation

The wave energy dissipation is controlled by three different mechanisms, as represented in the last three terms in Eq. 2, i.e., whitecap**, bottom friction, and depth-induced wave breaking. Wave spectral dissipation due to whitecap** is dominant in the oceanic waters and shelf seas. As the waves propagate towards intermediate and shallow waters, bottom friction becomes important. Eventually, depth-induced wave breaking contributes to a large portion of energy dissipation in the nearshore region.

1.2.1 Whitecap**

Whitecap** on the ocean surface is an important phenomenon for ocean wave evolution. In deep water, considerable wave energy is lost through whitecap** and transferred to surface currents and turbulence. Whitecap** is mainly controlled by the wave steepness and the formulation as adapted by the WAMDI Group (1988) reads as follows:

$$ S_{ds,\rm{w}}( \sigma ,\theta) = -\Gamma \tilde{\sigma}\frac{k}{\tilde{k}}E( \sigma ,\theta) $$

(18)

where $\tilde {\sigma }$, k, and $\tilde {k}$ denote the mean frequency, wave number, and mean wave number, respectively. Γ is a steepness-dependent coefficient and is given by the following:

$$ \Gamma = C_{ds}\left((1-\delta)+\delta \frac{k}{\tilde{k}}\right) \left ( \frac{\tilde{s}}{\tilde{s}_{\rm{PM}}} \right )^{p} $$

(19)

in which the coefficients C _{d s} , δ and p are tunable coefficients, $\tilde {s}$is the overall wave steepness, $\tilde {s}_{\rm {PM}} = \sqrt {3.02\times 10^{-3}}$ is the value of the wave steepness for the Pierson–Moskowitz spectrum. For the wind input of Komen et al. (1984), which is used in the present study, the default values are C _{d s} = 2. 36 × 10^{≤ 5}, δ = 0 and p = 4. The overall wave steepness $\tilde {s}$is defined as follows:

$$ \tilde{s}=\tilde{k}\sqrt{E_{\rm{tot}}} $$

(20)

where E _{t o t} is the total wave energy:

$$ E_{\rm{tot}}=\int_{0}^{2\pi }\int_{0}^{\infty }E(\sigma ,\theta)d\sigma d\theta $$

(21)

1.2.2 Bottom friction

Wave–bottom interaction also contributes to the wave evolution. In water of finite depth, wave energy is attenuated by bottom friction, which involves the turbulent boundary layer at the sea bottom. The empirical model of JONSWAP (Hasselmann et al. 1973) for bottom friction is applied in this study, and the formulation can be expressed as follows:

$$ S_{ds,\rm{b}} = -C_{b}\frac{\sigma^{2}}{g^{2}\sinh^{2}(kh)}E( \sigma ,\theta ) $$

(22)

where g is the gravitational acceleration, h is the water depth, and C _b is the bottom friction coefficient that generally depends on the bottom orbital motion U _{r m s} :

$$ U_{\rm{rms}}=\sqrt{\int_{0}^{2\pi }\int_{0}^{\infty }\frac{\sigma^{2}}{\sinh^{2}(kh)}E( \sigma ,\theta )d\sigma d\theta } $$

(23)

A constant friction coefficient, i.e., C _b = 0. 067 m ²s^{≤ 3}, by default in the SWAN model, is adopted in this study.

1.2.3 Depth-induced wave breaking

In very shallow waters (surf zone), energy dissipation due to depth-induced wave breaking becomes dominant, and waves begin to break when the ratio of wave height to water depth reaches its limiting value. In SWAN, the dissipation is computed with

$$ S_{ds,\rm{br}}(\sigma ,\theta )=-D_{\rm{tot}}\frac{E(\sigma ,\theta )}{E_{\rm{tot}}} $$

(24)

where D _{t o t} is given by

$$ D_{\rm{tot}}=\frac{1}{4}\alpha Q_{b}\left ( \frac{\tilde{\sigma }}{2\pi } \right )H_{\max}^{2} $$

(25)

in which α = 1 in SWAN, and H _max is the maximum individual wave height that can exist at a given depth. H _max can be calculated from H _max = γ h, where γ is the breaker parameter. In this study, γ = 0. 73 is used. Q _b is the fraction of breaking waves calculated in SWAN with:

$$ Q_{b}=\left\{\begin{array}{lll} 0, \qquad \qquad \qquad\qquad\quad\;\;\,\text{for} \;\; \beta \leq 0.2\\ Q_{0} - \beta^{2} \frac{Q_{0}-\text{exp}(Q_{0}-1)/\beta^{2}}{\beta^{2}-\text{exp}(Q_{0}-1)/\beta^{2}}, \text{for} \;\; 0.2 < \beta < 1\\ 1\qquad \qquad \qquad \qquad \qquad\;\text{for} \;\; \beta \geq 1 \end{array}\right. $$

(26)

where β = H _{r m s} / H _max and H _{r m s} is the root mean square wave height. In addition, for β ≤ 0. 5, Q ₀ = 0 and for 0. 5 < β ≤ 1, Q ₀ = (2β ≤ 1)².

Appendix 2: Wave-induced bed shear stress and threshold of sediment motion

1.1 Wave-induced bed shear stress

Wave skin friction bed shear stress τ _{w a v e} can be computed from the wave orbital velocity near the seabed U _w and the wave friction factor f _w (Soulsby 1997):

$$ \tau_{\rm{wave}} = \frac{1}{2}\rho f_{w}U_{w}^{2} $$

(27)

where ρ = 1, 027 kg/m ³ is the density of the seawater. It is noteworthy that only the skin friction (or grain-related) bed shear stress acts directly on the sediment grains, and therefore only this parameter is used to calculate the threshold of sediment motion (Soulsby 1997; Reeve et al. 2004).

The bottom orbital velocity U _w can either be extracted directly from the wave model output (i.e., $U_{\rm {bot}}=\sqrt {2}U_{\rm {rms}}$), which is the case in the present study, or approximated by linear wave theory assuming a monochromatic wave with significant wave height H _s and peak period T _p in water of depth h:

$$ U_{w} = \frac{\pi H_{s}}{T_{p}\sinh(kh)} $$

(28)

where the wave number k = 2Π / L, and L is the wavelength. The wave spectral properties ( H _s , T _p , and L) were also computed by the wave model.

Depending on the types of flow (laminar, smooth turbulent, or rough turbulent), the wave friction factor f _w can be determined based on the formulations and procedures described in Soulsby (1997) and Whitehouse et al. (2000) as follows:

$$ f_{w}=\max\{f_{wr},f_{ws}\} $$

(29)

The rough bed friction factor f _{w r} can be calculated from:

$$ f_{wr}=1.39\left ( \frac{A}{z_{0}} \right )^{-0.52} $$

(30)

where A = U _w T / 2Π is the semi-orbital excursion, T is the wave period ( T _p is used), z ₀ = d ₅₀ / 12 is the bed roughness length. The median grain diameter d ₅₀ is set to 63 μm, which is the borderline between coarse and fine-grained soils according to a soil classification (BS EN ISO 14688-1 2002).

The smooth bed friction factor f _{w s} can be approximated by

$$ f_{ws}=BR_{w}^{-N} $$

(31)

Depending on the wave Reynolds number R _w (Eq. 32), the coefficients B and N are given in Table 7.

$$ R_{w} = \frac{U_{w}A}{\nu} $$

(32)

where ν = 1. 787 × 10 ^{≤ 6} m ²/s is the kinematic viscosity. The flow is said to be rough turbulent when f _{w r} > f _{w s} .

Table 7 The coefficients B and N for the determination of $f_{ws}$ (Soulsby 1997; Whitehouse et al. 2000)

A similar approach was also used in the investigations on the wave impacts on bed shear stress, e.g., in the East Frisian Wadden Sea by Stanev et al. (2006), Lettmann et al. (2009), and in the Black Sea by Stanev and Kandilarov (2012). In general, both waves and currents are important drivers for bed sediment dynamics. The skin friction bed shear stress component due to currents τ _{c u r r e n t} is a quantity that represents the current flow-induced frictional force acting on sediment grains on the seabed and has not been considered in this investigation.

1.2 Threshold of motion

The threshold of motion of sediments at the seabed can be given in terms of the threshold orbital velocity U _{w c r} or critical (or threshold) bed shear stress τ _{c r} .

1.2.1 Threshold orbital velocity

Based on the equations of Komar and Miller (1974), the threshold orbital velocity U _{w c r} can be approximated as follows:

$$ U_{\rm{wcr}} = [0.118g(s-1)]^{2/3}d_{50}^{1/3}T^{1/3} \;\;\text{for}\;\;d_{50} < 0.5\;\text{mm} $$

(33a)

$$U_{\rm{wcr}} = [1.09g(s-1)]^{4/7}d_{50}^{3/7}T^{1/7} \;\;\text{for}\;\;d_{50} > 0.5\;\text{mm} $$

(33b)

where s = ρ _s / ρ is the ratio of densities of sediment grain and sea water. Bed sediments begin to move when U _w > U _{w c r} .

1.2.2 Critical bed shear stress

The incipient motion of a sediment grain in response to waves occurs when the wave force exerted by the skin friction bed shear stress acting on the grain exceeds the submerged weight of the grain counteracting it (Soulsby 1997). The threshold of movement appears if the Shields parameter reaches a critical value, and this can be represented by the threshold Shields parameter 𝜃 _{c r} :

$$ \theta_{\rm{cr}} = \frac{\tau_{\rm{cr}}}{g(\rho_{s}-\rho )d_{50}} $$

(34)

where τ _{c r} is the critical bed shear stress. The study by Monien et al. (2011) suggested that the geochemical composition of the sediments in Maxwell Bay is characterized by tholeiitic basaltic andesite bed material, mostly determined by the lithogenic background of Barton Peninsula and adjacent areas. Thus, it is assumed that the bottom sediment of Potter Cove consists of similar bed material, and therefore the sediment density is set to ρ _s = 2, 650 kg/m ³.

The erosional behavior of noncohesive sediments has been determined empirically and described in Soulsby (1997) as the dimensionless threshold Shields parameter 𝜃 _{c r} , which is given as an algebraic expression as follows:

$$ \theta_{\rm{cr}} = \frac{0.30}{1+1.2D_{*}} + 0.055 [ 1-\mathrm{exp}(-0.020D_{*})] $$

(35)

where the dimensionless grain size D _∗ is defined as follows:

$$ D_{*} = \left [ \frac{g(s-1)}{\nu^{2}} \right ]^{1/3}d_{50} $$

(36)

Therefore, based on Eq. 34, the critical bed shear stress τ _{c r} is calculated to be 0.13 N/m ². In this paper, the bed sediment erosion is quantified and assessed based on the critical bed shear stress.