On the interaction between ocean surface waves and seamounts
Jeison Sosa, Luigi Cavaleri & Jesús Portilla-Yandún
Ocean Dynamics Theoretical, Computational and Observational Oceanography ISSN 1616-7341 Ocean Dynamics DOI 10.1007/s10236-017-1107-7
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Author's personal copy Ocean Dynamics DOI 10.1007/s10236-017-1107-7
On the interaction between ocean surface waves and seamounts Jeison Sosa 1 & Luigi Cavaleri 2 & Jesús Portilla-Yandún 3
Received: 30 March 2017 / Accepted: 26 September 2017 # Springer-Verlag GmbH Germany 2017
Abstract Of the many topographic features, more specifically seamounts, that are ubiquitous in the ocean floor, we focus our attention on those with relatively shallow summits that can interact with wind-generated surface waves. Among these, especially relatively long waves crossing the oceans (swells) and stormy seas are able to affect the water column up to a considerable depth and therefore interact with these deep-sea features. We quantify this interaction through numerical experiments using a numerical wave model (SWAN), in which a simply shaped seamount is exposed to waves of different length. The results show a strong interaction that leads to significant changes in the wave field, creating wake zones and regions of large wave amplification. This is then exemplified in a practical case where we analyze the interaction of more realistic sea conditions with a very shallow rock in the Yellow Sea. Potentially important for navigation and erosion processes, mutatis mutandis, these results are also indicative of possible interactions with emerged islands and sand banks in shelf seas.
Keywords Surface waves . Seamounts . Wave model SWAN . Swell . Yellow Sea Responsible Editor: Alejandro Orfila * Jesús Portilla-Yandún
[email protected] 1
School of Geographical Sciences, University of Bristol, University Road, Clifton, Bristol BS8 1SS, UK
2
Institute of Marine Sciences, ISMAR-CNR, Arsenale – Tesa 104, Castello 2737/F, 30122 Venice, Italy
3
Mechanical Engineering Department, Escuela Politécnica Nacional, Ladrón de Guevara E11-253, Quito, Ecuador
1 Seamounts and waves Seamounts are a prominent feature of the oceans whose interest concerns both their origin (see, e.g., French and Romanowicz 2015) and their interaction with the ocean that surrounds them (Chen et al. 2015). On this second line of interest, we analyze how sea waves interact with the top of these mountains. This interaction affects the two parts, the wave field being perturbed with potentially large spatial gradients, while the bottom is subject to different dynamical forces and their consequent effects. These processes are fully determined by the characteristics of both the sea bottom (depth, dimensions, shape, material, among others) and those of the impinging waves (wave height, period, direction, and more in general the wave spectrum). Much attention has been given to sea-bottom interactions on the continental shelves and, more so, on the coastal zones, which are the center of attention of a continuous flow of research and literature (e.g., Babanin et al. 2005, Goda 2009; Nielsen 2009; DavidsonArnott 2009; Short and Woodroffe 2009). In this paper, the attention is focused on oceanic features, in particular seamounts whose top, still below the surface, is within reach of the wave effects. More specifically, we analyze how the surface wave field is modified by the presence of a shallow summit. Although very relevant for some practical applications such as navigation, and the endangerment of islands due to sea level rise and other geophysical processes (e.g., Weiss 2015; Wilson 1963; Whittaker 2007), these interactions have been very poorly studied, and there exist no dedicated in situ measurements to quantify these effects. This has been the main motivation in this study to explore the situation following a numerical approach. At present, numerical wave models, within their inherent limits, are capable to represent quite closely the main physical processes involved (Cavaleri et al.
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2007) allowing us to analyze and eventually quantify the disturbance of the wave field. Naturally, the verification of these results remains relevant, especially in the light of the large magnitude of the disturbances derived from the model. However, such verification is out the reach of the present possibilities considering that there are no suitable data for the purpose. The dimensions and heights of seamounts cover a very wide range. Hydrographic surveys and shipping experience have revealed a plethora of seamounts whose top are, in marine terms, very close to the surface (e.g., Keating et al. 1987; Sinton 1989; Kearey et al. 2009; Wessel et al. 2010; Yesson et al. 2011). As a few examples, we quote the Bowie and Cobb seamounts in the gulf of Alaska (with summit depth at 24 and 34 m, respectively), the Muirfield seamount in the Indian Ocean (18 m), the Kawachi volcano (20 m) in the Solomon islands, and the Socotra Rock, also called Ieodo or Parangdo (4.6 m below the surface in the Yellow Sea). In turn, windgenerated waves are the most ubiquitous phenomenon on the sea surface. Under severe storms, waves may grow large and then propagate on the ocean surface for long distances in the form of swell, which is a permanent feature of the oceans, in particular of the tropical zone (see, among others, PortillaYandún et al. 2016). Shorter waves tend to disappear more rapidly with distance. Given the orbital vertical motion of the water particles associated to waves, their action is present on a limited vertical range just below the sea surface (following Airy theory, this depth corresponds to about half the wavelength). Therefore, for an effective interaction with the local bottom, waves need to be long enough, typically the double of the local depth or more. It is then natural to focus on waves that hold swell characteristics up to 20-s period and about 600m wavelength (see Section 3). Considering these wave characteristics, there is a large set of geological features whose peak is just within their range of action. To evaluate the variety of effects and implications of a seamount on the surface wave field, we have first considered a simple-shaped bottom feature under also relatively simple, nearly sinusoidal wave conditions (Section 2). The analysis of these results (Section 3) provides a general idea of the possibly large wave height enhancements that, e.g., a passing vessel may experience on the surface. For a more realistic case like in stormy sea conditions, the wave energy distribution will change from sinusoidal to a more general and broad wave spectrum. This implies different patterns of interaction with the seamount, the result being the superposition of these different modifications, with a consequent smoothing of the peak values. This is done in Section 4 where we explore the interaction of a hindcast storm in the Yellow Sea with the mentioned Socotra Rock. In Section 5, we critically analyze our results, especially in view of the approximations present in wave modeling. Granted the different situations, hence results, we point out how most of the reported results can provide
information also in the case of islands or shallow banks. We also point out that this level of technical knowledge has been for centuries part of the know-how that allowed Pacific islanders to travel from island to island across vast spaces of the ocean (e.g., Lewis 1994). We conclude in Section 6 itemizing our basic results.
2 Evaluating the effect of a submerged seamount on the wave field 2.1 The numerical approach In general terms, wave conditions in a certain area and at a given time are described by the wave spectrum (see Holthuijsen 2007, henceforth H07, for an exhaustive, but simple, description—we refer the user to this reference for all the basic definitions concerning wind waves). In practice, we consider the sea surface as the superposition (Pierson and Marks 1952) of a large number of sinusoidal waves, each characterized by its height, direction, and period (hence wavelength and frequency). For our purposes, we focus on a simpler and effective situation, consisting on a nearly monochromatic (i.e., single sinusoidal) wave train (soon to be detailed). This simple configuration is important to understand the primary wavebottom and wave-wave interactions. In the presence of a submerged obstruction, the wave field gets disturbed, giving origin to subharmonic components that in turn may interact among themselves. In reality (see Portilla et al. 2009), the incoming wave field can be composed by several different wave trains moving into different directions, so the interaction gets more complex. Nevertheless, real swell trains do not depart much from the monochromatic description (see H07), and given their relatively large periods, together with heavy storm waves, they are the ones most effectively interacting with submerged bathymetry features. Given a wave with height H and period T, there is an associated orbital motion at the surface whose circular velocity (in deep water) is u = πH/T. This motion attenuates rapidly with depth z (exponentially in deep water, see H07 for a full documentation). At a depth equal to half the wavelength, a wave and its orbital motion are attenuated to 4% of their surface value. However, the wavelengths present in the ocean (see Table 1) and the proximity of many seamounts to the surface clearly point to possible strong interactions. When waves are long enough, they are able to Bfeel^ the bottom through the orbital motion of the water particles. Relevant for our purpose Table 1
Deep-water wavelength for waves of different period
Period (s) Wavelength (m)
8 100
10 156
12 225
14 306
16 400
18 506
20 625
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is also the speed with which waves advance, which also decreases by the interaction with the bottom. This implies that, coming across a sloping surface, different parts of the wave front will move on different depths, hence with different speeds, and therefore the wave will tend to turn (H07). Because of different speeds, the interaction with the bottom implies also an increase of the wave height (at first interaction there can be a decrease, but this is irrelevant for our purposes). The above provides the essential information for properly interpreting the later results. All these effects, to some extent intuitive for an expert wave modeler, can be properly evaluated and quantified in their whole complexity with the aid of a numerical model. For the practical tests, we have used SWAN, an advanced wave model amply cited in the literature (see, among others, Booij et al. 1996 and again H07). Wave models can be used in two different and opposite ways. On the one hand, we can aim at improving the model, both its physics and numerics. This is typically done by first formulating a conceptual approach to describe a physical process, then verifying the results versus measured data, which are an essential part of the procedure. Alternatively, a welltested model can be used as a tool to obtain information of a particular situation otherwise not available. This is typically the case when, granted a reliable input, e.g., the meteorological fields, we need to infer something where measurements are not available. For the purpose of this paper, we are in the second, albeit simpler, situation. We want to explore how submerged mountains that are within reach of wind waves can affect their surface field. For this, we use a spectral model from the well-established phase averaged, third-generation family such as WW3 (Tolman et al. 2013), WAM (Janssen 2004), and SWAN (Ris et al. 1999; Zijlema et al. 2012). The choice has been for SWAN because it is better suited for the present purpose. SWAN has also been extensively used in different, but somehow similar, tests. These (see, among others, Hope et al. 2013 and Huang et al. 2013) dealt with the interaction of laboratory waves with a bottom bump. Although carried out in a wave channel, but possibly more demanding because of the very shallow water conditions, these tests have shown the good performance of SWAN in representing these processes. SWAN is used here with its standard formulation (see, e.g., The SWAN team 2017), which includes most of the relevant processes. Dealing with a very directionally narrow swell, the basic processes at work are refraction and shoaling. Since we focus our attention on the seamount-wave interactions, wave generation by wind is not considered. Consequently, nonlinear interactions and whitecapping breaking are also excluded. We have also purposely worked with relatively low input wave height to avoid whitecapping, although this appears to become relevant in the test with large wave heights. However, all these processes have been set at work in the more realistic Socotra Rock case. From our initial tests, and also to be expected,
bottom friction is not relevant for the final results because the interaction time with the bottom is very short. Diffraction is also not considered, because it is still without a rigorous numerical solution within the SWAN model (see website of SWAN manual and, among others, Lin 2013). In any case, the resulting wave height distributions lack the sharp transversal gradients that make diffraction a relevant process of the field. 2.2 Tests setup Having framed the general idea, the testing approach could be to select one storm in an area where a seamount is present. There is an infinite variety of the seamount geometry and wave conditions that can be found in the oceans. However, our main purpose here is not to provide exact figures as the result of a specific, possibly complicated, situation for which we would in any case lack the necessary information. Moreover, in such a situation, interpreting the effects of the single processes at work would not be straightforward due to the complexity of both the local bathymetry and the wave conditions. Rather, we follow the principle of simplifying the problem to be able to group the essential results and to explain why they are so. Therefore, we first focus on a simplified solution suitable for better understanding the interaction and the related physical aspects, and allowing for a better interpretation of the latter more real case. In this section, we give a brief overview of the modeling characteristics relevant to this interaction. After these, we carry out a Breal^ case exploring the interaction of a storm in the Yellow Sea with the Socotra Rock. This final test is described in Section 4. Because in this case we will be dealing with an active storm, all the physical terms of the SWAM model equations are Bturned on,^ including, in contrast to the idealized cases, the wind input and the fourth order nonlinear interactions. For 2D simulations, the input wave field needs to correspond to a 2D wave spectrum considering a large number of frequencies, typically 30 or 36, from 0.03 to 0.5 Hz, and a large number of different directions. Therefore, rather than with a single sinusoidal wave, we have worked with a spectrum with a narrow spectral Gaussian shape centered on the selected frequency (i.e., period). For the practical computation, the wave direction is from west to east, also narrowly distributed around the main direction, with spectral resolution at 2° interval. The overall energy corresponds to 1 m significant wave height. For different tests, we consider seven different periods from 8 to 20 s at 2-s interval (see Table 1 providing also the corresponding wavelengths in deep-water conditions). The narrow distribution in frequency makes peak and mean period, Tp and Tm, practically identical. We indicate them as T. The grid for the numerical simulations is 40 km wide (north to south) and 80 km long (west to east) at 100-m interval. The
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wave input conditions are uniform at the west border (see Section 3 for the general grid used for computation). The seamount is centered at position (0., 0.). In this first set of tests, only swell-like conditions are considered; hence, no wind input and fourth-order nonlinear interactions have been taken into consideration. Finally, we look only for stationary states, i.e., for equilibrium conditions; hence, no evolution in time is considered. All this holds for the idealized cases.
Table 2 Seamount bathymetry parameters
Case
A
λ (×103)
zo (m)
1
700
13.2
20
2 3
700 700
15.2 17.8
30 40
4
700
21.6
50
5
700
27.2
60
2.3 Bathymetry The overall bathymetry consists of a flat deep bottom where the seamount has been imposed. We have parameterized the seamount as a 2D Gaussian function submerged at different levels. The function describing the bathymetry is given by: r2 zðrÞ ¼ A e− λ −1 −zo ð1Þ where z(r) is the water depth as a function of the distance r from the center of the seamount (r2 = x2 + y2). A is a constant that defines the vertical dimension of the seamount (700 m). The parameter λ is introduced to control the horizontal configuration of the seamount potentially exposed to waves. In this case, a constant diameter of 8 km is set at 100-m water depth in all cases. zo specifies the depth d of the seamount summit which ranges from 20 to 60 m (see Fig. 1 and Table 2).
3 Results and analysis 3.1 Physics and details of the interactions To provide a better idea of the overall results and of the various reasons behind them, we first focus on a single case, detailing the implied physics. The most prominent case is the one with the shallowest seamount (top at – 20-m depth)
depth (m)
0 −50 −100 -700 −5
0 distance (km)
5
Fig. 1 Seamount configuration. Zero depth indicates the water surface. The different lines indicate the different bathymetry scenarios used for the summit depth
and the longest waves (T = 20 s). Figure 2 shows the corresponding results for the significant wave height Hs (top panels, (a) and (c)) and the mean period (lower panels, (b) and (d)). Panels (c) and (d) cover the whole computational grid, 80 km in x (east) direction (the one of propagation of the incoming waves), 40 km in the y (north) direction. The top of the seamount is at position (0., 0.). Panels (a) and (b) focus around this point (see the spatial scale, km, for reference). Note the 100-m isobath to better relate the surface changes to the local bathymetry. In panel (a), we see the large increase of Hs just behind (1 km eastwards of) the seamount summit. Note that the maximum enhancement is much larger, 3.27 m, than what visible in the plot, 2.4 m, because the corresponding area is very small (there are very large spatial gradients). On a general perspective, we find a growing wave height while waves approach the seamount (from the west). This is due to shoaling occurring when the swells interact with the submerged obstacle on the west side of the seamount. A characteristic feature is given by the two lobes of large wave height, on the east side of the seamount, that protrude respectively in the north and south directions. The explanation is given in the 3D representation of Fig. 3. Granted that the dimensions are stretched for a better grasping of the situation, we see that, while shoaling during their approach to the seamount, the waves, e.g., on the south side, i.e., to the right, of the submerged peak (negative y in panel Fig. 2a) feel the bottom on their left side. This leads to their drastic refraction towards the left (counterclockwise) resulting in a north-east going wave component being developed (the northward lobe seen in the same panel). Clearly, there is a symmetry with respect to the y = 0 x-axis, so we find also a clockwise turning of the wave approaching from the region y > 0, leading to the development of the southward lobe (wave system). It is the combination of these two systems, crossing each other downstream the submerged seamount that leads to the peak of the significant wave height (3.27 m). The overall pattern is given by the combination of shoaling, peaking on top of the seamount, and of the refraction Baround it^ of the waves converging from the right and left of the peak. We point out that, considered or not, nonlinear interactions have no role in this evolution because of the very short time scale involved and the rapidly varying spatial conditions. Shown also in Fig. 2a, we see this better in Fig. 3.
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1
c 20
1
5 1
a
1
1. 2
.7 21.4
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This double drastic deviation of wave energy off the main direction is not without consequences. In fact, we see in Fig. 2a, more extensively in Fig. 2b, and also in Fig. 3 the large extended area of reduced wave height on the lee (eastward) side of the seamount position. We generally expect lower wave heights on the lee of an island because this creates a physical block. However, in the present case, the lower Hs behind the seamount are due to the refraction induced by the shallow local bathymetry and the consequent deviation of the flowing energy off the original direction. Note also that the shadowed area extends well beyond the 40 km considered in the computational grid. A first-hand estimate suggests it may extend, with a progressively decreasing effect, for hundreds of kilometers. This is due to the swell conditions (narrow directional spectrum—see Section 2) we have selected for the tests.
10
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20 20 20 20 20 20 20 202020 19.5
20
−10
19.5
21
0
30
19.5 21
22 21
22
19.5
x(km)
21
20
19.5
10
20
x (km)
20
21
0
20
0
10
d 20
1
x(km)
5
20
b
0 y(km)
y (km)
20
−5 −5 −5
−5 −5 −5
Fig. 3 An idealized 3D view of the waves impinging on a submerged seamount. Note how behind the island the incoming wave system turns into different directions
0.8
0. 8
1. 5
0
x(km)
1.4
1
0
1
10
1
x (km)
1.2
1.2
Fig. 2 Significant wave height (top panels) and mean wave period (lower panels) fields around the seamount for a homogeneous input wave field (from left to right) of Hs = 1 m and T = 20 s. The seamount summit water depth is 20 m. Red colors indicate enhancement; blue colors indicate decrease of the wave field parameters compared to the input at the boundary. The two curved arrows in panels a and b show how the input waves tend to rotate around the submerged peak. The red line circles indicate the seamount isobaths
−30
−20
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0
19.5
10
20
30
40
y(km)
A wind sea, under active generation, with its wider spectrum both in frequency and direction, would lead to a much shorter wake because the lateral, off the main direction, components would soon advect energy in the area behind the seamount. Using narrow spectrum as input wave conditions highlights well, in Fig. 2b, d, the different refraction effects on the waves with different period, hence wavelength. This is easily recognized in Fig. 2b showing the distribution of the dominant wave period in the various areas of the grid. We recognize at once the two pronounced lobes, pointing, respectively, to north and south, with a longer period, 21–22 s, respect to the central one of the impinging waves. The reason is that the various wave components in the considered spectrum have different frequencies, hence wavelengths. This leads to a selective refraction effect, more intense for the longer (lower
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frequency, longer period) waves. Of course this implies that, at least for a while, we find higher frequency (lower period) waves in the lee of the seamount.
3.2 Overall results The maximum significant wave heights reached for the single combinations of wave period and depth of seamount top are provided in Table 3. Remember that for all the cases, the input wave height is 1 m. As expected, the enhancement decreases with decreasing wave period (hence wavelength) and with an increasing depth of the seamount. However, it is still remarkable (more or less doubling the wave energy ~ Hs2) for waves of 10–12 s (a relatively short period in the oceans) and 50– 60 m depth. In Fig. 4, we try to give a general and more precise view of the trends with period and depth, and of the position of the seamount at which we find the maximum wave height. For each combination of period and depth, we plot in Fig. 4 the maximum Hs obtained over the whole computational grid and its position (kilometers off the center). The continuous lines show how, for a given depth, the results vary with the wave period. The dotted lines refer to a given period and show how the results change with depth. Note how in this case the lines are asymptotic to the unitary x-axis while the seamount effects decrease for increasing depth. This happens at a different rate depending on the period, till disappearing when the submerged top is out of the range of action of the surface waves. A quick estimate based on the respective wavelengths suggests that the minimum 1.02 value in Fig. 4 is reached for an 8-s wave when the seamount top is at about 60-m depth.
3.5 20m 20s
3
depth mean wave period
18s 30m
Hs [m]
16s
2.5
40m 50m
14s
60m 12s
2
10s 8s
1.5
1 0
5
10 x [Km]
15
20
Fig. 4 Maximum significant wave height over the computational grid for the different scenarios (see Table 3)
Table 3 Maximum significant wave height (meters) over the computational grid for different summit depths and wave conditions (Tm-1,0) Tm-1,0
20 m
30 m
40 m
50 m
60 m
8s
1.75
1.38
1.16
1.06
1.02
10 s 12 s
2.03 2.21
1.82 2.06
1.55 1.89
1.35 1.70
1.21 1.53
14 s
2.46
2.22
2.09
1.96
1.82
16 s 18 s
2.74 3.03
2.44 2.67
2.25 2.44
2.13 2.28
2.02 2.17
20 s
3.27
2.87
2.61
2.43
2.30
We complement the results in Figs. 3 and 4 and Table 3 showing in Fig. 5a, b, c the distribution of the wave heights along the x-axis, from − 10 till 20 km before and after the summit, for three different depths (20, 30, 50 m) and all the considered periods. We will come back later to panel (d). Note that the same vertical and horizontal scales have been used for the three panels. We recognize at once as, for each depth, the enhancement decreases moving to shorter (lower period) waves. Also the spatial scale, in x-direction, of the changes is longer. As explained in the previous subsection, longer waves have a stronger refraction; hence, they focus at a position in the shadow zone closer to the top of the mountain. For a similar reason, but related to a lower refraction for deeper seamounts, lower enhancements, also moving progressively to higher x-positions, are found passing from (a) to (b), and then to (c). From a more utilitarian point of view, it has been suggested for wave energy harvesting or simply for coastal surfing or protection (e.g., Jackson et al. 2005; Langhamer et al. 2010; Hanley et al. 2013) to take advantage of a natural, or to build an artificial, dune on the bottom to focus, exactly for the described processes, the wave energy towards a specific position. A suitable machine there located would produce up to an order of magnitude more energy than if facing the undisturbed field. Apart of the difficulty of dealing with wave systems refracted in completely different directions, a substantial problem would be the drastic change of the maximum energy position as a function of the dominant period of the incoming waves. Of course we could plan on the base of the most common conditions (or the most energetic ones), but the overall efficiency would be highly degraded. From this perspective, we show in Fig. 6 the wave power flux P (kW/m, from west to east) across the ± 20 km of the yaxis (see Fig. 2) at different positions along the x-axis. P = ρg.∫[E(f). cg(f)]df, where ρ is the water density, g is gravity, E is the wave energy density, f is frequency, and cg the wave group velocity. The case considered is the d = 20 m, T = 20 s. Starting from the uniform distribution before feeling the slopes of the seamount (x = − 10 km), we find above the
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b
Depth=20m
Depth=30m
3.5
3.5
3
3
2.5
2.5
2
2
Hs (m)
Hs (m)
a
1.5
1.5
1
1
0.5
0.5
0 −10
−5
0
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20
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0
x (km)
c
d
Depth=50m 3.5
5 x (km)
10
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10
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Depth=10m 9 8
3
7 6
2
Hs (m)
Hs (m)
2.5
1.5
5 4 3
1
2 0.5 0 −10
1 −5
0
5 x (km)
10
15
20
0 −10
−5
0
5 x (km)
Fig. 5 Significant wave height profile along the x-axis (y = 0) for period from 8 to 20 s, for different depths 20, 30, and 50 m in panels (a) (b), and (c), respectively. Panel (d) refers to the special case of 10-m summit depth
and a 4-m significant wave height impinging on the seamount. The dash line indicates the magnitude of the input background field
summit (x = 0.) the strong peak due to both shoaling and refraction with on the sides a limited decrease due to the
refracting components (hence not moving parallel to the xaxis). Moving to east to progressive x-positions, we come across the extended shadow, or wake, zone. Note the slight increase towards the larger (absolute) y-values, associated to the refracted waves, skewed with respect to, but with still a component in, the x-direction. Having provided a physical description of the possible situations and of the reasons behind, we can better interpret the 2D wave spectra shown in Fig. 7. In Fig. 7a, we see the approaching spectrum, representing the assigned wave conditions. The spectrum is centered at 0.05 Hz and flowing in the x-direction (angle θ = 0°). At x = − 0.5 km, Fig. 7b, we already perceive a slight widening of the spectrum, with a consequent limited decrease of its peak value, now 0.36 m2 s. When centered on the seamount (Fig. 7c), the peak value has grown because of shoaling while refraction is beginning to widen the directional distribution. The two refracted systems are well evident in Fig. 7d, 1.2 km after the summit. Note how distributing the overall available energy on a wider (f-θ) zone leads
Fig. 6 Distribution of the wave power flux P (kW/m) from west to east (see Fig. 2) across transects at different positions along the x-axis. Input conditions Hs 1 m, T 20 s, depth of the mountain top 20 m. See text for the related definition
Author's personal copy Ocean Dynamics Hs= 1.0m, (x=−40Km, y=0Km) 90
0.15
90
60
120
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Hs= 0.9m, (x=−0.5Km, y=0Km) 0.4
60
0.35
0.1
a
30
0.3
0.35
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0.1 150
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0.25
0.05 0.25
0.2 180
0.2 180 0.15 0.15 330
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270 Hs= 3.3m, (x=1.2Km, y=1Km)
90
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30 0.05
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300
Hs= 2.0m, (x=0Km, y=0Km) 120
150
0.1
330
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0.05 300
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Hs= 1.5m, (x=2.4Km, y=1Km)
Hs= 1.5m, (x=2.4Km, y=−1Km)
90
0.15
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Hs= 0.7m, (x=10Km, y=0Km)
Hs= 0.6m, (x=30Km, y=0Km) 90
0.15 60
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Fig. 7
Wave spectra at different points of the simulation domain. The radial axis goes from 0. (center) till 0.15 Hz, the flow directions are shown with a Cartesian convention, i.e., with 0° to east (consistent with the undisturbed wave flow direction) with the angle increasing counterclockwise. Input conditions Hs = 1 m, Tm = 20, summit depth 20 m. All the spectra are scaled on the single peak value, provided for each spectrum by the vertical scale. Color bar units are m2 × s
interpretation of the results. Indeed, in the 4-m case, there is significant breaking on the summit of the seamount and at the convergence of the two systems. This breaking limits the maximum wave height reachable at and just behind the summit position.
to a lower spectral peak notwithstanding its increased integral value (Hs = 3.3 m). Figure 7e, f shows the spectra at two symmetrical (y = ±1 km) positions, 2.4 km after the summit. We recognize the two refracting systems, each one now not symmetrical because off the x-axis. Finally, in Fig. 7g, h, we show the spectra far in the wake zone. The further we go, the more the system converges towards the x-direction because of the more close-to-x rays pointing from the seamount position towards the considered point. As a final test on this series, we have considered the case of a much more energetic wave system with 4 m high, approaching the seamount, and assuming for this a more shallow summit, at 10-m depth. The results for the different wave periods are given in Fig. 5d. We note first the double Hs peak, the first one associated to shoaling, the second higher one (almost 9 m high) due to the two refracted converging systems. The first peak is located on the summit of the seamount after which it rapidly decreases following the increasing depth. However, shortly after we find the convergence zone due to the waves coming from the side of the mountain peak (see Fig. 3). Therefore, after an intermediate minimum, the significant wave height grows rapidly again to a much higher value. Note the extremely difficult conditions a vessel should stand in this area, with energy basically spread over almost 180° and two large wave systems moving at cross angles. Note that ignoring bottom-induced breaking and bottom friction would have qualitatively reproduced, but with a much higher peak, the distributions in panel Fig. 5a. A direct inspection of the source terms in the energy balance equation shows that both bottom friction and bottom-induced breaking are at work at the summit, the dominant and limiting factor being breaking. The peak Hs values decrease with wave period, but not as much as in the previous cases. For 20 and 8 s, the maximum Hs are close to 9 and 7 m, respectively, while in Fig. 4a, we find corresponding 3.3 and 1.75 m. The 7 m Hs with 8-s period deserves an explanation. A thumb rule of wind sea is that Hs is about 4% of the dominant wavelength, in this case 100 m. However, the value of 4 m Hs would correspond to the dynamical equilibrium between wind input and whitecapping (see H07). In the present case, in the absence of wind and whitecapping, the two wave systems come suddenly together so there is physically no time for whitecapping to dissipate wave energy before reaching these (7 m) extreme conditions. The selection of 1 m Hs for the input waves in the general tests was done to avoid too large wave heights that would lead to breaking that could impede a clean
4 A more realistic case Having illustrated the physics involved in the interaction of surface waves with an idealized seamount, we now consider a more realistic case to have a more complete feeling of the implications. For this, we consider the Socotra Rock (see Section 1) in the Yellow Sea (32.12° N, 125.18° E) whose summit is just 4.6 m below the sea surface. Figure 8 provides a view of the location and the bathymetry derived from the General Bathymetric Chart of the Oceans (http://www.gebco. net/). While on the west side, depth increases rapidly, and on the east side, there is an extended plateau at about 40 m depth 2 km long (panels Fig. 8e, h). At this location, the Korean Ocean Research and Development Institute (now KIOST, Korean Institute of Ocean Science and Technology) installed in 2003 the so-called Ieodo meteo-oceanographic tower (Shim et al. 2004). Based on the wave climate at the location, we have considered a typical storm with average wave height, period, and direction (date and time are irrelevant for the present purpose). The estimated spectrum, derived from the ERA-Interim reanalysis (Dee et al. 2011), is shown in Fig. 8b. The input significant wave height is 3.2 m, with the peak period at 15 s (Tm = 11.4 s), and 180° mean flow direction (moving to south). The modeled conditions involve an input spectrum with waves in a broad range of frequencies and directions moving on a real, hence more complicated, bathymetry, leading to more complex interactions. On the one hand, the wide, in frequency and direction, wave spectrum tends to smear the clean patterns we have seen in the idealized tests (see Figs. 2 and 5). The reason is that wave components with different frequency have different interactions and different spatial scales for their distributions. Also waves from different directions, as in a real wind wave spectrum, react in a different way. Notwithstanding these differences, the impact of the rock on the wave system is clear. From the undisturbed 3.1 m Hs, we reach at the summit position a 4.6 m Hs value (Fig. 8g). However, contrarily to the previous tests, the increased Hs is here due mainly to shoaling. This is evident by the position of the maximum Hs at the summit of the rock. However, refraction is present, as evident again by the asymmetrical Hs distribution in (Fig. 8h), with respect to the incoming wave direction. Note that the high wave conditions at the summit are such that, theoretically at least, the rock can be exposed in the deepest troughs. This can be the case under typhoon conditions, with potentially much higher wave conditions. As a side
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Fig. 8
The Socotra Rock test (South Korea). (a) Geographical location (circle), (b) input spectrum at the north boundary, (c), (d), and (e) bathymetry, Hs, and Tm in the simulation domain. (f), (g), and (h) same as (c), (d), and (e), but zoomed in the seamount area. Tm is defined as Tm-1,0 (see H07)
note, monitoring typhoon conditions has been the main reason to place the oceanographic tower, Ieodo, on the Socotra Rock. In addition, sea level variations, including tides, will also have an effect on the magnitude of the interactions. However, these effects have not been considered in this assessment. Focusing again on the simulated storm, the shadowing effect downstream the rock, partly due to dissipation, but mostly to refraction, is clearly visible in (Fig. 8d) with the wave heights in the most southern part of the plot lower than the input one north of the rock.
5 Discussion The results reported in the previous sections, although basic in conception, are quite robust and cover a wide range of practical possibilities, providing a solid reference for the wave heights and their distribution on the sea surface. These results are based on sound physics and on a well-tested tool (the SWAN model). Taking into account that 1-m wave height is a low average wave height in the oceans, it is clear that, wherever shallow seamounts are present, the wave conditions can indeed be very peculiar. As we have seen for the Socotra Rock, with realistic wave spectra and bathymetry, any real case will have its own characteristics, with enhancements and wake zones of different amplitude and extension. Truly enough, our general tests and to a certain extent also that of the Socotra Rock are simplified examples. It is also clear that the validity of the results remains within the limits of representation of the physical processes included in the model and their simplifications. For instance, in the open ocean, there are also currents, which are not necessarily uniform with depth, especially when interacting with a submerged nonsmall obstacle. For light currents, the Taylor column or T-P theorem (Proudman 1916; Taylor 1917) is a possibility. Temperature gradients in the vertical may affect the specific interaction with the system. Radiation stresses, associated to the changes waves experience interacting with the seamount, and more so to the possible whitecapping and bottom-induced breaking, lead to currents that in turn affect the local wave field, a classical coupled system. Apart from missing in the real cases most of, if not all, the necessary information, a quick order of magnitude estimate shows that, at least for the purpose of this paper, their consideration would not change appreciably the main results and, most of all, the main physics behind our test. Our purpose has not been to provide with great potential accuracy the results for a specific case. For all the practical
purposes having 3.27-m peak value (as in the test of Fig. 2) or a possibly Bmore exact^ one between 3.20 and 3.40 m would not affect our conclusions. Of course all this holds also for the Socotra Rock. For the benefit of an analytical test, we could have chosen an analytical shape suitable for a mathematical solution. Although an interesting exercise (but failing as soon as nonlinear dissipative processes as whitecapping are introduced), our aim was to have a numerical tool to be promptly applied to any bathymetry (e.g., the Socotra Rock). In this respect, the practical difficulty is to have a sufficiently detailed and correct bathymetry. The same comment applies to more sophisticated numerical approaches, as e.g., the Boussinesq equation. Although we have purposely been dealing with seamounts to stress the effects they can have on the surface wave fields, to a large extent, partly at least, our results provide information also for the interaction of ocean waves with islands of limited dimensions and with a suitable surrounding bathymetry. The wake of an island or even an archipelago is a known phenomenon, regularly considered in operational wave modeling (see, e.g., Chawla and Tolman 2008 and Stopa et al. 2011) whose results strictly depend on how one models the situation. In the early time of the WAM wave model (WAMDI Group 1988), we came across a case where the wake of the Hawaii islands was unrealistically extending indefinitely south of the archipelago. It was a large swell perfectly from the north direction. The bug, soon corrected, was the lack of energy advection from the sides towards the shadowed zone due to the coarse, at the time, model directional resolution. However, the classical case we have in mind is the typhoon Krosa, 6 October 2007, during which the world record significant and maximum single wave heights were recorded, 23.9 and 32.3 m, respectively, see Liu et al. (2008). What is unexpected in reading the paper is that these exceptional data were recorded by a buoy on the lee of an island hit by the maximum strength of the typhoon. The peak significant wave height in front of the island was modeled at about 12 m (personal information from the authors; see also Babanin et al. 2011). A possible explanation is given, as in our examples in Section 3, by the convergence of the two strongly refracted wave systems from both sides of the island. We stress that verification of the above results is out of the scope of the present paper and practically not possible simply because no suitable in situ data exist for the purpose (this is the reason to use a reliable model tool instead). Satellite data, typically altimeter, are neither an option because they are concentrated on narrow tracks with a discontinuous monitoring at a fixed site, while the processes involved in the interaction of surface waves and submerged seamounts are very local, and found in sparse areas of the oceans. Besides, the spatial Hs gradients close to the seamount can be very large, while the altimeter Hs values are derived from a several kilometers diameter area. All this makes large scale validation very
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difficult. A possibility worthwhile to mention is the use of stereo video wave recording systems (see Benetazzo et al. 2012), whose installation is under consideration at the Ieodo tower. This system can provide the wave field on a high spatial resolution (e.g., a few meter interval) and with a relatively large spatial coverage. We close with a less technical, but perhaps more interesting, note. When we, modern and technological people, dig into a natural subject and come across unexpected results, we may feel a sense of satisfaction, and even some complacency with ourselves for Bdiscovering^ the secrets of nature. This is not the case here, but, for the subject we have been dealing with, it is interesting to explore a bit the ancient art of navigation in the Pacific islands (Polynesia, Micronesia, Solomon, etc.). A good reference book is We, the navigators by David Lewis (1994). See also Genz et al. (2009). The local people used to navigate hundreds of kilometers from island to island, these mostly visible only when close to them, without any modern instrument, relying only on reading the stars in the sky and Bliterally reading^ the waves. Obviously with a previous knowledge of the general local (but not very local) conditions, they were able to derive with amazing accuracy their position, looking, often not by eye, but by direct feeling, the locally present wave systems. Two similar wave systems, coming at a certain angle, could identify the presence of an island in the up-swell direction, its distance suggested (remember our Fig. 7g, h) by the angle between the two converging swells. No doubt the modern technology has brought to us a wealth of fantastic tools. However, we feel that at the same time, the capability of solving (some of) the problems of nature with only our inner and learned knowledge should not be lost.
6 Summary We itemize here the main results of our tests. 1. A seamount sufficiently close to the surface (order of some tens of meters or less) may affect heavily the impinging wave fields 2. Three main zones can be distinguished for this interaction: (a) a direct shoaling over the summit of the seamount, (b) a convergence peak just behind the seamount, (c) a long distance wake extending for tens of kilometers or more 3. The shoaling depends on the wavelength and the depth and extension of the submerged summit 4. Refraction to the right and left of the summit leads to a potentially strong convergence zone where wave heights can be amplified. An increase factor of three times or more is possible 5. The refracted components are turned away from the original direction. The overall level of refraction varies, for a
given seamount, with the wavelength, hence its period. This leads to a large directional dispersion of wave energy behind the seamount 6. The results behind the seamount is a wake, an area of lower than original wave heights that extends for tens of kilometers. Some quick tests suggest the extension can reach 100 or 200 km 7. Interpreted as applied to islands, result (4) may help to explain documented cases of very or even extremely large wave conditions on the opposite side of where the typhoon was hitting the island 8. The pattern of wave systems behind an island have been used for centuries by the Pacific islanders to find their target on the vast expanse of the ocean. Acknowledgements We acknowledge funding from Escuela Politécnica Nacional (Projects PIMI1402 and PIJ1503) and Senescyt Scholarship AR2Q-000105-2015. Luigi Cavaleri was partially supported by the EU contract 730030 (call H2020-EO-2016, BCEASELESS^). We are warmly indebted to our good friend Gerbrant van Vledder for leading us to the discovery of the art of the Polynesian navigators. We acknowledge the useful suggestions by the anonymous reviewers that helped to straighten the paper and make more clear the general results.
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