- Email: [email protected]
The mystery of boulders moved by tsunamis and storms
Marine Geology 295-298 (2012) 28–33
Contents lists available at SciVerse ScienceDirect
Marine Geology journal homepage: www.elsevier.com/locate/margeo
The mystery of boulders moved by tsunamis and storms Robert Weiss ⁎ Department of Geosciences, Virginia Tech, Blacksburg, VA 24061, USA
a r t i c l e
i n f o
Article history: Received 30 March 2011 Received in revised form 29 November 2011 Accepted 2 December 2011 Available online 16 December 2011 Communicated by J.T. Wells Keywords: tsunami storms boulders theory
a b s t r a c t Boulders are moved differently in storm and tsunami and produce different characteristics of the boulder deposits. This contribution is motivated by two observations. One by Bourgeois and MacInnes (2010), which described that boulders were moved selectively due to different bed roughness during 15 November 2006 tsunami on the island of Matua. The second topic is motivated by the boulder lines on Ishigaki Island by Goto et al. (2010). Both topics are approached with linear wave theory and stability analysis. From both, the safety factor is derived for a spherical boulder with bed roughness and exposure as moment arms, with which we are able to quantify the influence of bed roughness on the incipient motion of boulders. For constant forces, a bed roughness of about 30% of the boulder radius will prevent boulder transport. Furthermore, the comparison between storm and tsunami waves in terms of the amplitude necessary to move boulders revealed that amplitude of storm waves is smaller than tsunami, which we ascribe to the contribution of both velocity components to lift forces. The comparison of total energy and number of waves revealed that storms have a larger total energy and a much larger number of waves, which lead us to the conclusion that tsunamis produce unorganized boulder deposits; whereas, storms are capable of organizing boulders along lines and in clusters. © 2012 Elsevier B.V. All rights reserved.
1. Introduction 1.1. Motivation and background The motivation of this paper comes from two recent observations regarding boulders moved by storms and tsunamis. The observations on the island of Matua by Bourgeois and MacInnes (2010) of the 15 November 2006 central Kuril Island tsunami revealed that boulders were moved somewhat selectively. The tsunami waves exceeded 12 m of runup with a maximum of about 20 m with inundations between 40 and 90 m. Both values demonstrate the steepness of the terrain. The main observation was that some boulders were moved, indicated by recently deceased attached intertidal fauna, and some were not moved even though the flow depth inferred from the measurement was sufficient. Bourgeois and MacInnes (2010) speculated that roughness might be the key to understanding this phenomenon. The second publication that inspired this study is from Goto et al. (2010) on the observed boulder distributions on Ishigaki Island, Japan. The key observation by Goto et al. (2010) is that boulders moved by tsunami are distributed erratically near-shore and onshore, whereas boulders moved by a storm formed a line on the reef crest, but did not travel into the moat and onto the beach.
⁎ Tel.: + 1 540 231 2334. E-mail address: [email protected]. 0025-3227/$ – see front matter © 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.margeo.2011.12.001
Inversion methods have been designed to help distinguishing boulders moved by tsunamis from those moved by storms. For the boulders under consideration, wave heights of storm and tsunami waves needed for incipient motion were estimated, and then argued that, often times, storm waves have unrealistically large values and must therefore be ruled out as the causative process. In this regard, Costa (1983) derived a set of equations to infer the flow power needed to transport boulders during the flash flood peaks in the Colorado Front Range. Bryant et al. (1997) employed these equations to infer tsunami height from boulders along the Australian coast. Furthermore, Nott (2003) related the incipient motion of boulders to the causative waves and derived for different scenarios of boulder emplacement a set of equations that is widely known as Nott's equations. These equations have been extended in Nandasena et al. (2011). Alternative expressions were proposed by Benner et al. (2010) and Buckley et al. (2011). Nott's equations and its enhancements have been employed to reconstruct wave heights of the 2004 Sumatra tsunami along the coasts of Indonesia (Paris et al., 2010) and Thailand (Kelletat et al., 2007) and in the Mediterranean (Mastronuzzi and Sanso, 2004; Barbano et al., 2010). Furthermore, Nott's equations have also been used in the very controversial Australian Megatsunami Hypothesis (Bryant et al., 1997; Dominey-Howes et al., 2006). The approaches to consider incipient boulder movement mentioned above rely on the quadratic form of the forces, in which the free stream horizontal velocity is squared. Furthermore, these approaches have a drag coefficient that needs to be approximated because for arbitrary geometries, the drag coefficient is unknown. The
R. Weiss / Marine Geology 295-298 (2012) 28–33
quadratic form of lift and drag force only includes the onedimensional, horizontal free-stream velocity. The flow field under waves is two dimensional, and this was taken into account to estimate the forces exerted by the flow field. To overcome the limitations of the quadratic form of drag and lift forces, the Kelvin–Helmholtz circulation theorem was applied to compute the forces from the twodimensional fluid motion. Furthermore, incipient motion of boulders is approached with linear theory for both the water-related forcing and stability of a boulder. It is also assumed that the boulder is circular in cross section and that is of cylindrical shape for which the lateral extent can be neglected. With this assumption, even a linear theory can provide quantitative information on the influence of bed roughness, for comparing tsunami and storm waves, and for inferences of the boulder clustering by storms and tsunamis. However, linear theory does not allow for a derivation of equations to infer wave amplitude for arbitrary boulder geometries. The derivation of Nott's equations also reveals that certain geometric characteristics need to be present. It should be noted that such an attempt in the same framework as presented herein requires the employment of nonlinear wave theory with an arbitrary geometry of the integration contour used in the Kelvin–Helmholtz circulation theorem. 1.2. Boulder and sand transport In geology and sedimentology, boulders are defined to represent grain sizes larger than 256 mm in diameter (ϕ b −8). Sand has ϕ values between −1 and 4. In terms of size, there are three to five orders of magnitude between sand and boulders, which leads to the intuitive conclusion that sand is transported differently by a storm or tsunami waves than boulders. Furthermore, the magnitude of storm and tsunami waves can vary over several orders of magnitude. Hence, sediment transport by tsunami and storm waves is a multiscale problem in terms of grain sizes involved as well as causative power, which may be the reason for the challenge that respective deposits pose for inferences of the causative process and respective magnitudes. Boulder and sand deposits are expected to be different, which can be ascribed to the very different fashion of sand and boulder transport. The obvious difference between sand and boulder transport is that sand grains are easily lifted and will spend significant amount of time in the water column before they touch the ground again. Boulders on the other hand, probably will rotate or be pushed very close to the bed with very short or even neglectable lift periods compared to the time it takes to travel one diameter. A fluid body that exhibits circulation exerts drag and lift forces, which can be quantified with the help of the vorticity. Circulation and vorticity are an expression of the turbulence. From the stochastic nature of turbulence, it follows that the eddies created by circulation have a size distribution. For a boulder under consideration, we can assume that the mode of the eddy distribution M(le) is M(le) ≤ lb, in which lb is a boulder length scale. A boulder length scale is the radius of a boulder or a length of its horizontal extent. From this condition for the mode of the eddy distribution, it follows that drag and lift forces exhibit large gradients along the boulder, and it is possible that drag and lift forces do not exceed the threshold of motion on one side of the boulder. However, the lift force on the front side causes a reduction in resistance. If the drag force is smaller than the lift force, the boulder will rotate; if the drag force is similar, the boulder will be pushed. For sand grains, we define that the mode of the eddy distribution is M(le) > > ls, in which ls is a sand-grain length scale, such as the grain diameter. Therefore, the force gradient along the sand grain can be neglected, resulting in a lift of the entire grain. Once lifted, these particles experience the delicate and complex balance between turbulent fluctuations of the flow and gravity. If the influence of turbulence is larger than the influence of gravity, the particle pathways become
29
completely random due to the stochastic nature of turbulence and the relatively weak influence of gravity. This transport mode is commonly known as suspended load. If gravity has a larger influence on the particle under consideration than turbulence, the pathway of the particle in the fluid will obey a ballistic pathway (with stabilizing Magnus effect). This transport mode is commonly known as bed load. The duration of the lift phase of these grains in case of bed load depends on the magnitude of the turbulence and respective stresses compared to gravity. However if the duration is much longer than it takes for a grain to travel the distance of its size, then different grain sizes are sorted and are able to form sedimentary structures that are larger than the grain size. 1.3. Setting In order to be able to compare transport of a boulder by storms and tsunamis, it is assumed that boulder transport takes place in some water depth and not directly at the wave front of the tsunami. However, transport of boulders by the wave front may also be possible for tsunamis. The effects of a storm consist of a surge and the storm waves. The surge may flood low-lying land with storm waves bringing destructive energy to the inundated areas. Boulder characteristics are described by the height of the boulder, d, and its length, lb. For theoretical analysis, a cylindrical boulder shape is employed for which only the circular cross section of radius is considered. Then the boulder is defined by the radius rb = lb/2= d/2. For wave theory, it is assumed that A is the wave amplitude, L is the wavelength, and d is the water depth. The two-dimensional velocity (u = (u, w)) field is determined with the help of linear wave theory, in which ω is the angular frequency (ω = (2π)/T; T-wave period) and k is the wave number (k = (2π)/L). Linear wave theory defines the framework for storm waves because the storm surge causes an increase in water depth. Furthermore, depth-limited waves are employed, which means that the height of the waves is limited to 0.8 of the water depth. Even though it is assumed that boulder transport in a tsunami takes place well behind the moving wave front, from the length-scale to water-depth ratio of a tsunami it can be concluded that the horizontal velocity u computed with linear theory represents a gross underestimation of the actual velocity. The computation of the vertical velocity w with linear theory is thought to be robust even for tsunamis. In this setting, it can be assumed that the velocity is related to the depth of the flow. The scaling constant in the relationship between the flow velocity and depth of flow is the Froude number Fr. Spiske et al. (2008), Jaffe and Gelfenbaum (2007), and Matsutomi et al. (2001) reported maximum Froude numbers of 2. However, Lynett (2007) found Froude numbers as high as 5 for the largest waves employed in the study, but using a reference height. Employing the entire water column, Fr = 5 decreases to about Fr = 3. Hence, we assume that the Froude number can vary from 0.5 to 3. 2. Boulder transport model 2.1. Drag and lift forces Usually, drag and lift forces are expressed in a quadratic form of the velocity, which works well for one-dimensional flows. For example this approximation is used in Nott (2003). However, a wave can only be described by a two-dimensional flow field u = (u, w). Hence, drag and lift forces are exerted by the two-dimensional wave motion. As an assumption, the circulation of the flow under a wave can be calculated with the Kelvin–Helmholtz circulation theorem to the first order.
Γ ¼ ∮C u⋅dl
ð1Þ
30
R. Weiss / Marine Geology 295-298 (2012) 28–33
in which Γ is circulation. We assume that the maximum length scale of the eddies created by flow field underneath a wave is the height of the boulder under consideration. For simplicity, the integration contour, C, in Eq. (1) is defined to be of circular shape of radius rc = 0.5db, where db is the boulder height. The derivation of the drag and lift forces is similar to the one presented independently by Kutta and Jousowski and uses the wellknown Kutta–Jousowski formula (Batchelor, 1967): F ¼ ρu eΓ, in which e represents the unit vector. In Kutta and Jousowski's derivation of forces, the velocity is defined as the free-stream velocity, which is possible for the cases Kutta and Jousowski considered. However, in a two-dimensional flow situation, the derivation of drag and lift forces from circulation with the help of the Kutta–Jousowski formula is more difficult and bears some mathematical challenges, as described in Crighton (1985). Intuitively, the drag force and lift force receive contributions from the horizontal velocity. However, the lift force also is influenced by the vertical flow component. As an approximation for the drag force computed from circulation, the approximation FD = ρu|Γ| is used and for the lift force FL = ρ(u − w)|Γ|. Please note that subtracting of the vertical velocity from the horizontal still is a positive contribution because the vertical velocity bears a negative sign (see Eq. (2)). For the contour integration, the equations for the horizontal and vertical velocities from the linear wave theory are simplified, assuming the shallow-water approximation, with χ = L/d ≥ 20, in which L is the wavelength and d the water depth. With ω = (2π)/T, k = (2π)/L, pffiffiffiffiffiffi and T ¼ gd=ð20dÞ, it is assumed that 2πðz þ dÞ cosh χd ≈1 2π sinh χ 2π ðz þ dÞ sinh z χd ≈ 1þ : 2π d sinh χ
2.2. Force balance and incipient motion In the last section, it was derived how wave characteristics influence the drag and lift forces. In this section, the safety factor is employed to boulder stability and incipient motion. The stability of a boulder is a function of flow velocity and the slope, but is also a function of how the boulder is locked by surrounding boulders and particles. In classic sedimentology and engineering, the work by Stevens and Simons (1971) was ground-breaking and resulted in the well-established linear stability analysis based on moments. Here, the theory developed by Stevens and Simons (1971) is employed to compute the force equilibrium between the weight, lift and drag forces. Fig. 1a depicts the stability with the help of moment arms with regard to a rotation at point 0 where incipient motion will take place. From the figure, the following equilibrium condition can be derived: l2 W s cos α ¼ l1 W s sin α þ l3 F D þ l4 F L :
ð5Þ
Eq. (5) is the original form of the equation in Simons and Sentürk (1992), with the assumption that the flow is parallel to the sloping bed, for which the angle is α. The moment arms l1, l2, l3, and l4 need to be determined and may have a significant influence on stability. From Fig. 1a, it can be assumed that l2 ≡ l4. Furthermore, it is assumed that the boulder is of circular shape and that the onset point for the weight vector is the center of the circle. Then by defining l2 = r − ξx, the relation l1 + l2 = r and l1 = r − l2 = ξx can be obtained, following the general equation for circles. The parameter r denotes the radius of the circular boulder. The momentum arm l3 can be approximated by l3 = l1 + r − ξz with the final formula for l3: l3 = r + ξx − ξz. Substituting these expressions of the momentum arms into Eq. (5) yields: ðr−ξx ÞW s cos α ¼ ξx W s sin α þ ðr þ ξx −ξz ÞF D þ ðr−ξx ÞF L :
ð6Þ
To evaluate the stability of a setting, the safety factor Π is defined as the ratio of the left-hand and right-hand side of Eq. (6):
With the two substitutions, the velocity components simplify to: pffiffiffiffiffiffi gdA cos ðωt−kxÞ χdpffiffiffiffiffiffi 2π gdA z 1 þ sin ðωt−kxÞ: wðx; z; t Þ ¼ − χd d
uðx; z; t Þ ¼
Π¼
2π
ð7Þ
ð2Þ
Eq. (2) is now used for contour integration to find the circulation. The contour is defined as a circle with the radius rc, which is defined as half of the boulder height. The absolute value of circulation |Γ| then becomes: jΓ j ¼ ωAr c sin ðωt−krc Þ:
ðr−ξx ÞW s cos α : ξx W s sin α þ ðr þ ξx −ξz ÞF D þ ðr−ξx ÞF L
ð3Þ
Eqs. (2) and (3) can now be used to compute the drag and lift force. The expression for the drag and lift forces: pffiffiffiffiffiffi 2πρrc A2 gd sin ðωt−krc Þcos ðωt−kxÞ FD ¼ χdpffiffiffiffiffiffi 2 h i 2πρr c A gd z 1 þ sin ðωt−kxÞ−cos ðωt−kxÞ sin ðωt−kr c Þ: FL ¼ d χd ð4Þ As a reminder, both equations are based on linear wave theory, which is suitable for computing the magnitude of forces. However, the computation of instantaneous forces, for example to estimate how far a boulder is moved by a wave, requires the application of the non-linear wave theory, which is not considered here.
From Eq. (7), it becomes obvious that for Π ≥ 1 the setting of a boulder is stable; whereas for Π b 1, the setting is unstable and results in boulder movement. A discussion on the safety factor of a boulder setting, also known as riprap stability, can be found in Julien (1998) and also in Simons and Sentürk (1992). The definition of the safety factor is part of the standard procedures in engineering geology and geotechnical engineering to guarantee slope stability. 2.3. Limitation of the approach As mentioned earlier, this contribution introduces an alternative to the approaches for incipient boulder movement that relies on the quadratic form for the forces. This approach is based on the Kelvin– Helmholtz circulation theorem and the two-dimensional Kutta– Jousowski formula. In order to introduce this approach, both are coupled with linear wave theory and a simplified boulder geometry. The Kelvin–Helmholtz circulation theorem in Eq. (1) requires the integration along a given contour line. In this regard, the combination of the Kelvin–Helmholtz circulation theorem and Kutta–Jousowski formula with the linear wave theory and simplified boulder geometry ensures a good balance between mathematical and physical sophistication. Even with these simplifying assumptions pivotal general conclusions can be drawn on the initiation of boulder movement during storms and storm waves, as well as tsunamis.
R. Weiss / Marine Geology 295-298 (2012) 28–33
31
Fig. 1. (a) depicts the classic riprap stability problem as defined in Simons and Sentürk (1992). The level arms for the different forces are given for the case where the flow is parallel to the slope θ. (b) simplification of the stability problem in (a) to account for bed roughness and exposure. The relationship between the level arms in (a), and exposure and bed roughness is given in the text.
3. Results 3.1. Variable moment arms and constant forces With the help of Eq. (6), the influence of the moment arms on the stability of a boulder for constant lift and drag forces and constant boulder characteristics can be studied. The left- and right-hand sides of Eq. (6) are employed to compute the safety factor Π (Eq. (7)). The moment arms are r − ξx and r − ξz. The physical interpretation of r − ξx is trivial as the magnitude of the bed roughness (Fig. 1b), which can be a characteristic of the surface on which the boulder in question rests, or is generated by adjacent boulders. The moment arm r − ξz can be seen as the exposure of the boulder (Fig. 1b) to the flow, which is a complicated and non-linear function. The magnitude of the exposure not only depends on the geometry of the boulder, but also on the spatio-temporal flow dynamics around the boulder. The latter is impossible to infer reliably in hindsight. Fig. (2) plots the safety factor Π as a function of the bed roughness on the left panel and as a function of the exposure on the right panel. Again, if Π ≥ 1 the boulder sits in stable condition (gray area in both panels), and for Π b 1 the boulder will move. On the left-hand side of Fig. 2, the envelope of the dark-gray area is defined as the minimum (ξz = r) and maximum exposure (ξz = 0). Intuitively, if the moment arm, r − ξx, is small, which also means
that the bed roughness is small. From the figure it is evident that with increasing bed roughness, the safety factor increases as well. For maximum exposure, the safety factor exceeds one at a normalized bed roughness of (r − ξx)/r ≈ 0.6. That means if the roughness is larger than 30% of the total boulder height, the boulder will not move given the force remains constant in the system. If the boulder radius is assumed to be 1 m, the safety factor is Π = 0.06 for roughness length of 5 cm, and Π = 1 for a roughness length of 30 cm. The right panel of Fig. 2 depicts the safety factor as a function of normalized exposure ([r − ξz]/r). The dark-gray area represents the envelope created by the minimum (ξx = r) and maximum (ξx = 0) bed roughness. Interestingly only for low roughness, the safety factors cross the threshold value of one. Only for safety factors that are already close to the threshold due to respective bed roughness, travel of the safety factor from the unstable to a stable condition is possible. We can also observe that when the exposure length is low, the difference in the safety factors for the maximum and minimum bed roughness is the largest, reaching asymptotic characteristics. The left panel of Fig. 2 directly demonstrates the exceptional relevance that bed roughness has for the stability of a boulder. This fact is supported also by the right side of Fig. 2, due to the almost asymptotic difference in safety factors as a function of exposure length between minimum and maximum bed roughness. 3.2. Variable forces and constant moment arms
Fig. 2. The left panel shows the safety factor Π as the function of the normalized moment arm that is related to bed roughness, the right panel depicts Π as the function of the normalized exposure.
With the help of the safety factor from Eq. (7) and the equations to compute lift and drag forces, the difference in boulder movement between storm and tsunami waves can be studied. Tsunamis and storm waves have very different periods. Therefore, parameter χ of Eq. (4), which is the wavelength to water-depth ratio, is chosen for both types based on a reference for storm and tsunami waves. The reference for storm waves is based on the reference water depth ds∗, the respective length-to-depth ratio is χs∗. Respectively, theses reference parameters are dt∗ and χt∗ for tsunamis. Then, the local χ is computed with χ = (χ ∗d ∗)/d, in which the subscripts are dropped and d is the local water depth. As reference depth for storm waves a water depth of ds∗ = 50 m is used and for tsunamis dt∗ = 5000 m. The refer~ is normalized ence χ ∗'s are χs∗ = χt∗ = 20. The mass of the boulders m by the mass of the unit boulder, which has a radius of 1 m and the ~ are normalunit density of ρ = 1800 kg/m 3 = ρ ∗. The amplitudes A ized by amplitude A ∗ which represents the amplitude needed to move the unit boulder. Fig. 3a,b shows the threshold amplitude for boulder motion as a function of the boulder mass for tsunamis (Fig. 3a) and storm (Fig. 3b) waves. As a reminder, a Froudenumber scaling is employed for the horizontal velocity, u, with pffiffiffiffiffi ffi u ¼ Fr gd, to account for the very different fluid dynamics of storm
32
R. Weiss / Marine Geology 295-298 (2012) 28–33
~ t ¼ At =A ) as the function of normalized mass (m ~ s ¼ As =A ) as ~ ¼ m=m ). (b) gives the normalized storm-wave amplitude (A Fig. 3. (a) gives the normalized tsunami amplitude (A t s ~ (c) depicts the amplitude ration of storm and tsunami waves (At/As) as a function of m ~ for three different Froude numbers (F1 : Fr = 1.0, F2 : Fr = 2.0, F3 : Fr = 3.0). the function of m.
and tsunami waves in shallower water. For Fig. 3a, a Froude number of Fr = 1.5 is used. Comparing tsunami and storm waves, the envelope spanning from 0.5ρ ∗ to 2ρ ∗ has the same funnel-shaped geometry. However, the amplitude of tsunamis to move boulders of larger mass is about five times larger compared to the unit boulder. For similar boulder masses, the threshold amplitude is only about three times larger than unity. This difference is ascribed to an increased, perhaps underestimated, relevance of the vertical velocity, v, for boulders ~ ¼ 1 and transported in tsunamis and by storm waves. Near unity (m ~ ¼ 1), the normalized amplitudes for storm and tsunami waves are A ~ ¼ 1, a boulder of density 0.5ρ ∗ very similar. For a boulder mass of m ~ requires wave amplitude of A≈0:7, and if the boulder density is 2ρ ∗ ~ the required threshold amplitude is A≈1:3. From the lower to the upper bound of the envelope depicted in Fig. 3a,b, the density qua~ druples, but the amplitude only doubles. If a constant amplitude A of one is assumed then the boulder mass that is possible to move ~ ¼ 0:25 and for 2ρ ∗ the mass is m ~ ¼ 3:9. For the quadrufor 0.5ρ ∗ is m pled boulder density, the mass increases 15 times. Fig. 3c shows the amplitude ratio (At/As) of storm and tsunami waves as the function of normalized boulder mass. The different lines denoted with F1, F2, and F3 indicate different Froude numbers (F1 : Fr = 1.0, F2 : Fr = 2.0, F3 : Fr = 3.0) for the tsunami. For small Froude numbers the amplitude ratio At/As is large, which means that the amplitude of tsunamis required to move a boulder is larger than the amplitude of a storm wave to move the same boulder. As the Froude number increases the amplitude ratio approaches unity. Only for relative light boulders and large Froude numbers, the amplitude of tsunamis is smaller than storm waves to transport the same boulder. It should be noted that the understanding of amplitude is no longer trivial. In shallow water, the amplitude of a tsunami is related to the water depth and is understood as such in the Froudenumber-scaled horizontal velocity. For storm waves, the common
understanding of amplitude is valid; however, a certain water depth must be present to support wave propagation. Because a finitedepth breaking criterion is assumed, the surge underneath the waves can be computed from the wave amplitude. The depth limit ~ s is is set to 2A = H = 0.8d and coincides where the amplitude A found. If the water depth during the storm is only due to the storm surge, for example on coastal plains very close to the still-water level, the surge level can be estimated with d = 1.25H. Furthermore, it would be possible to derive an explicit expression to invert amplitudes of tsunami and storm waves from boulder masses with the equations employed in this section. However, the linearity of the applied theories only allows for general conclusions, and the nonlinearity of the quantification process prohibits the derivation of an explicit expression. 3.3. Storms versus tsunamis Storms and tsunamis are very different processes from a hydrodynamic point of view. Tsunamis normally consist of a succession of two to four large waves of different wave amplitudes and wavelengths for the periods are between 10 and 30 min. The entire energy available for transport of particles of any size arrives in consecutive waves. The effects of storms include a surge and storm waves. The storm surge may submerge boulders in the surf zone or even on land. The storm waves then may cause the motion of boulders. The number of storm waves is at least 3 orders of magnitude larger than in tsunamis. The total energy per wavelength is the sum of the kinetic and potential energy. It is common to write that Et ¼ Ek þ Ep ¼ ρ∫Ω 12 v2 þ gz dΩ ¼ 12 ρgA. The integration along the wave is performed over a wavelength. The total amount of energy then is computed by multiplying Et by the number of waves. As example, data of the 2004 Sumatra at Port La Rue and of a tropical storm in Bisbane,
R. Weiss / Marine Geology 295-298 (2012) 28–33
Australia are used (see Weiss and Bahlburg (2006)). The storm data consist of about ns = 33000 waves and the number of waves is nt = 120 in case of the tsunami. Then the total amount of wave energy for all waves in the storm is in the order of nsEt ≈ 10 11J and in the tsunami it is in the order of ntEt ≈ 10 9J. For boulder transport by storms and tsunamis, it is assumed that the pre-transport boulder distribution is unorganized. In a storm, the energy per wave is relatively low, and only a certain fraction of all storm waves are capable of transporting boulders. However, the storm surge changes in order of hours, allowing large number of waves to move boulders. With slow changing storm surge and storm-wave periods in the order of 20 s, we can assume that depthlimited breaking occurs in a certain area with respective water depth. This fact leads us to the conclusion that in the area of the wave breaking, boulders are deposited to form lines or even clusters. On the other hand, the energy per wave in a tsunami is large. The tsunami will pick up a boulder and transport it to a certain location. The transport distance is a sensitive function of boulder characteristics and wave forcing. However, subsequent waves may not be able to move certain boulders under consideration. It is therefore very unlikely that boulders will be organized to clusters or lines during the transport process. However, clustering is possible if the transport distance is short, and pre-transport clusters existed. 4. Conclusions The scope of this study was to explore boulder transport during storms and tsunami waves. Different processes were explored not only to better establish the limitations that our understanding of boulder transport bears but also on the opportunities that simple analysis can offer. Due to the linearity of the employed wave theory, it is not legitimate to derive a different set of equations, such the equations derived in Nott (2003), to infer the amplitude of storm and tsunami waves from the moved boulder characteristics. Even though the theories employed herein are linear, the results point to very interesting conclusions. For example, bed roughness seems to be a major factor, which has not been incorporated in inversion methods previously. Bourgeois and MacInnes (2010) speculated based on the relevance of bed roughness, but Fig. 2 offers a quantitative view point. It appears that leaving the forces constant, the existence of bed roughness with length scales in the order of ten percent of the boulder height will prevent a boulder from being moved. Smaller roughness length will let the boulder be moved. This suggests that for field studies, the bed roughness of the potential pre-transport area is important to survey along with the boulder characteristics. However, it should be noted that smaller boulders and cobbles that were in the vicinity of the boulder in question could have created bed roughness but have been moved as well. The comparison of amplitudes between storm and tsunami waves required to move a boulder revealed that both waves can move boulders of similar sizes with similar amplitudes. It is even possible that the amplitude of storm waves necessary to move a boulder is smaller than the amplitude of a tsunami required to move the same boulder. With this fact, it is possible to draw the conclusion that it may be impossible to discriminate those boulders moved by storms from those moved by tsunamis easily without taking also the periods of storm and tsunami waves into account. However, it is assumed that the mechanics of boulder movement is different in a storm compared to a tsunami. If the boulders are distributed unorganized in the pre-transport area, boulders can also be distributed in an unorganized fashion in case of a tsunami. However for storms, it is possible that boulders can be organized in clusters or along lines due to the depth-finite character of the storm waves and their relation to the surge level, such as observed by Goto et al. (2010). Assuming depth-limited storm waves, a boulder line or cluster should form at around 2.25 of the amplitude needed to move boulders of the same size. This helps to establish the
33
magnitude of the storm-surge levels for historical or paleo-storm events if boulder deposits are present. From a general point of view and even with the linear concepts and theories employed, this study revealed the complexity of boulder transport, which might not be fully understood until three-dimensional flow simulation around boulders with arbitrary geometries is carried out to study the three dimensional stress distribution. Furthermore, measuring the sizes and densities of boulders in the field is not sufficient. Quantifications of the bed roughness and careful documentation of boulder arrangements are pivotal to distinguish storm-moved from tsunamimoved boulders. There is a critical need for more field observations and measurements, but theoretical analysis is equally important to resolve the mystery of boulders moved by storm and tsunami waves. Acknowledgements The author would like thank Bruce Jaffe and an anonymous reviewer for their helpful comments. This research was partially funded by the grant NSF-EAR-1136534 to the author. References Barbano, M., Pirrotta, C., Gerardini, F., 2010. Large boulders along the south-eastern Ionian coast of Sicily: storm or tsunami deposits? Marine Geology 275 (1–4), 140–154. Batchelor, G.K., 1967. An Introduction to Fluid Dynamics. Cambridge Univ. Press. Benner, R., Browne, T., Brückner, H., Kelletat, D., Scheffers, A., 2010. Boulder transport by waves: progress in physical modelling. Zeitschrift für Geomorphologie 54, 127–146. Bourgeois, J., MacInnes, B., 2010. Tsunami boulder transport and other dramatic effects of the 15 November 2006 central Kuril island tsunami on the island of Mantua. Zeitschrift für Geomorphologie 54 (3), 175–195. Bryant, E., Young, R., Price, A., Wheeler, D., Pease, M., 1997. The impact of tsunami on the coastline of Jervis Bay southeastern Australia. Physical Geography 18 (5), 440–459. Buckley, M., Jaffe, B.E., Wei, Y., Watt, S., 2011. Estimated velocities and inferred cause of overwash that emplaced inland fields of cobbles and boulders at Anegada, British Virgin Islands. Natural Hazards. doi:10.1007/S11069-011-9725-8. Costa, J., 1983. Paleohydraulic reconstruction of the flash-flood peaks from boulder deposits in the Colorado Front Range. Geological Society of America Bulletin 94, 986–1004. Crighton, D., 1985. The Kutta condition for unsteady flow. Annual Review of Fluid Mechanics 17, 411–445. Dominey-Howes, D., Humphreys, G., Hesse, P.P., 2006. Tsunami and paleotsunami depositional signatures and their potential value in understanding the late-Holocene tsunami record. The Holocene 16 (8), 1095–1107. Goto, K., Miyage, K., Kawamato, H., Imamura, F., 2010. Discrimination of boulders by tsunamis and storm waves at Ishigaki Island, Japan. Marine Geology 269 (1–4), 34–45. Jaffe, B., Gelfenbaum, G., 2007. A simple model for calculating tsunami flow speed from tsunami deposits. Sedimentary Geology 200, 347–361. Julien, P., 1998. Erosion and Sedimentation. Cambridge Univ. Press. Kelletat, D., Scheffers, S., Scheffers, A., 2007. Field signatures of the SE-Asia mega-tsunami along the west coast of Thailand compared to the Holocene paleo-tsunami from the Atlantic region. Pure and Applied Geophysics 164 (2–3), 413–431. Lynett, P., 2007. The effects of shallow water obstruction on long wave runup and overland flow velocity. Journal of Waterway, Port, Coastal, and Ocean Engineering (ASCE) 133 (6), 455–462. Mastronuzzi, G., Sanso, P., 2004. Large boulder accumulations by extreme waves along the Ionian coast of Apulia (Southern Italy). Quaternary International 120, 173–184. Matsutomi, H., Shuto, N., Imamura, F., Takhasi, T., 2001. Field surveys of the 1996 Inian Jaya earthquake tsunami in Brak Island. Natural Hazard 24, 199–212. Nandasena, N.A.K., Paris, R., Tanaka, N., 2011. Reassessment of hydrodynamic equations to initiate boulder transport by high-energy events (storms, tsunamis). Marine Geology 281, 70–84. Nott, J., 2003. Waves, coastal boulders and the importance of pre-transport setting. Earth and Planetary Science Letters 210, 269–276. Paris, R., Fourier, J., Poizot, E., Etienne, S., Morin, J., 2010. Boulder and fine sediment transport and deposition by the 2004 tsunami in Lhok Nga (western Banda Aceh, Sumatra, Indonesia): a coupled offshore–onshore model. Marine Geology 268 (43–54). Simons, D., Sentürk, F., 1992. Sediment Transport Technology. Water Resources Publications. Spiske, M., Weiss, R., Roskosch, J., Bahlburg, H., 2008. Inversion of flow depth and speed from tsunami deposits using TsuSedMod. Eos Trans. AGU, Fall Meet. Suppl., 89 (52), pp. OS53A–1288. Stevens, M., Simons, D., 1971. River Mechanics. Stability Analysis for Coarse Granular Material on Slopes. Shen Publisher, Ch. Weiss, R., Bahlburg, H., 2006. A note on the preservation potential of offshore tsunami deposits. Journal of Sedimentary Research 76 (12), 1267–1273.
Contents lists available at SciVerse ScienceDirect
Marine Geology journal homepage: www.elsevier.com/locate/margeo
The mystery of boulders moved by tsunamis and storms Robert Weiss ⁎ Department of Geosciences, Virginia Tech, Blacksburg, VA 24061, USA
a r t i c l e
i n f o
Article history: Received 30 March 2011 Received in revised form 29 November 2011 Accepted 2 December 2011 Available online 16 December 2011 Communicated by J.T. Wells Keywords: tsunami storms boulders theory
a b s t r a c t Boulders are moved differently in storm and tsunami and produce different characteristics of the boulder deposits. This contribution is motivated by two observations. One by Bourgeois and MacInnes (2010), which described that boulders were moved selectively due to different bed roughness during 15 November 2006 tsunami on the island of Matua. The second topic is motivated by the boulder lines on Ishigaki Island by Goto et al. (2010). Both topics are approached with linear wave theory and stability analysis. From both, the safety factor is derived for a spherical boulder with bed roughness and exposure as moment arms, with which we are able to quantify the influence of bed roughness on the incipient motion of boulders. For constant forces, a bed roughness of about 30% of the boulder radius will prevent boulder transport. Furthermore, the comparison between storm and tsunami waves in terms of the amplitude necessary to move boulders revealed that amplitude of storm waves is smaller than tsunami, which we ascribe to the contribution of both velocity components to lift forces. The comparison of total energy and number of waves revealed that storms have a larger total energy and a much larger number of waves, which lead us to the conclusion that tsunamis produce unorganized boulder deposits; whereas, storms are capable of organizing boulders along lines and in clusters. © 2012 Elsevier B.V. All rights reserved.
1. Introduction 1.1. Motivation and background The motivation of this paper comes from two recent observations regarding boulders moved by storms and tsunamis. The observations on the island of Matua by Bourgeois and MacInnes (2010) of the 15 November 2006 central Kuril Island tsunami revealed that boulders were moved somewhat selectively. The tsunami waves exceeded 12 m of runup with a maximum of about 20 m with inundations between 40 and 90 m. Both values demonstrate the steepness of the terrain. The main observation was that some boulders were moved, indicated by recently deceased attached intertidal fauna, and some were not moved even though the flow depth inferred from the measurement was sufficient. Bourgeois and MacInnes (2010) speculated that roughness might be the key to understanding this phenomenon. The second publication that inspired this study is from Goto et al. (2010) on the observed boulder distributions on Ishigaki Island, Japan. The key observation by Goto et al. (2010) is that boulders moved by tsunami are distributed erratically near-shore and onshore, whereas boulders moved by a storm formed a line on the reef crest, but did not travel into the moat and onto the beach.
⁎ Tel.: + 1 540 231 2334. E-mail address: [email protected]. 0025-3227/$ – see front matter © 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.margeo.2011.12.001
Inversion methods have been designed to help distinguishing boulders moved by tsunamis from those moved by storms. For the boulders under consideration, wave heights of storm and tsunami waves needed for incipient motion were estimated, and then argued that, often times, storm waves have unrealistically large values and must therefore be ruled out as the causative process. In this regard, Costa (1983) derived a set of equations to infer the flow power needed to transport boulders during the flash flood peaks in the Colorado Front Range. Bryant et al. (1997) employed these equations to infer tsunami height from boulders along the Australian coast. Furthermore, Nott (2003) related the incipient motion of boulders to the causative waves and derived for different scenarios of boulder emplacement a set of equations that is widely known as Nott's equations. These equations have been extended in Nandasena et al. (2011). Alternative expressions were proposed by Benner et al. (2010) and Buckley et al. (2011). Nott's equations and its enhancements have been employed to reconstruct wave heights of the 2004 Sumatra tsunami along the coasts of Indonesia (Paris et al., 2010) and Thailand (Kelletat et al., 2007) and in the Mediterranean (Mastronuzzi and Sanso, 2004; Barbano et al., 2010). Furthermore, Nott's equations have also been used in the very controversial Australian Megatsunami Hypothesis (Bryant et al., 1997; Dominey-Howes et al., 2006). The approaches to consider incipient boulder movement mentioned above rely on the quadratic form of the forces, in which the free stream horizontal velocity is squared. Furthermore, these approaches have a drag coefficient that needs to be approximated because for arbitrary geometries, the drag coefficient is unknown. The
R. Weiss / Marine Geology 295-298 (2012) 28–33
quadratic form of lift and drag force only includes the onedimensional, horizontal free-stream velocity. The flow field under waves is two dimensional, and this was taken into account to estimate the forces exerted by the flow field. To overcome the limitations of the quadratic form of drag and lift forces, the Kelvin–Helmholtz circulation theorem was applied to compute the forces from the twodimensional fluid motion. Furthermore, incipient motion of boulders is approached with linear theory for both the water-related forcing and stability of a boulder. It is also assumed that the boulder is circular in cross section and that is of cylindrical shape for which the lateral extent can be neglected. With this assumption, even a linear theory can provide quantitative information on the influence of bed roughness, for comparing tsunami and storm waves, and for inferences of the boulder clustering by storms and tsunamis. However, linear theory does not allow for a derivation of equations to infer wave amplitude for arbitrary boulder geometries. The derivation of Nott's equations also reveals that certain geometric characteristics need to be present. It should be noted that such an attempt in the same framework as presented herein requires the employment of nonlinear wave theory with an arbitrary geometry of the integration contour used in the Kelvin–Helmholtz circulation theorem. 1.2. Boulder and sand transport In geology and sedimentology, boulders are defined to represent grain sizes larger than 256 mm in diameter (ϕ b −8). Sand has ϕ values between −1 and 4. In terms of size, there are three to five orders of magnitude between sand and boulders, which leads to the intuitive conclusion that sand is transported differently by a storm or tsunami waves than boulders. Furthermore, the magnitude of storm and tsunami waves can vary over several orders of magnitude. Hence, sediment transport by tsunami and storm waves is a multiscale problem in terms of grain sizes involved as well as causative power, which may be the reason for the challenge that respective deposits pose for inferences of the causative process and respective magnitudes. Boulder and sand deposits are expected to be different, which can be ascribed to the very different fashion of sand and boulder transport. The obvious difference between sand and boulder transport is that sand grains are easily lifted and will spend significant amount of time in the water column before they touch the ground again. Boulders on the other hand, probably will rotate or be pushed very close to the bed with very short or even neglectable lift periods compared to the time it takes to travel one diameter. A fluid body that exhibits circulation exerts drag and lift forces, which can be quantified with the help of the vorticity. Circulation and vorticity are an expression of the turbulence. From the stochastic nature of turbulence, it follows that the eddies created by circulation have a size distribution. For a boulder under consideration, we can assume that the mode of the eddy distribution M(le) is M(le) ≤ lb, in which lb is a boulder length scale. A boulder length scale is the radius of a boulder or a length of its horizontal extent. From this condition for the mode of the eddy distribution, it follows that drag and lift forces exhibit large gradients along the boulder, and it is possible that drag and lift forces do not exceed the threshold of motion on one side of the boulder. However, the lift force on the front side causes a reduction in resistance. If the drag force is smaller than the lift force, the boulder will rotate; if the drag force is similar, the boulder will be pushed. For sand grains, we define that the mode of the eddy distribution is M(le) > > ls, in which ls is a sand-grain length scale, such as the grain diameter. Therefore, the force gradient along the sand grain can be neglected, resulting in a lift of the entire grain. Once lifted, these particles experience the delicate and complex balance between turbulent fluctuations of the flow and gravity. If the influence of turbulence is larger than the influence of gravity, the particle pathways become
29
completely random due to the stochastic nature of turbulence and the relatively weak influence of gravity. This transport mode is commonly known as suspended load. If gravity has a larger influence on the particle under consideration than turbulence, the pathway of the particle in the fluid will obey a ballistic pathway (with stabilizing Magnus effect). This transport mode is commonly known as bed load. The duration of the lift phase of these grains in case of bed load depends on the magnitude of the turbulence and respective stresses compared to gravity. However if the duration is much longer than it takes for a grain to travel the distance of its size, then different grain sizes are sorted and are able to form sedimentary structures that are larger than the grain size. 1.3. Setting In order to be able to compare transport of a boulder by storms and tsunamis, it is assumed that boulder transport takes place in some water depth and not directly at the wave front of the tsunami. However, transport of boulders by the wave front may also be possible for tsunamis. The effects of a storm consist of a surge and the storm waves. The surge may flood low-lying land with storm waves bringing destructive energy to the inundated areas. Boulder characteristics are described by the height of the boulder, d, and its length, lb. For theoretical analysis, a cylindrical boulder shape is employed for which only the circular cross section of radius is considered. Then the boulder is defined by the radius rb = lb/2= d/2. For wave theory, it is assumed that A is the wave amplitude, L is the wavelength, and d is the water depth. The two-dimensional velocity (u = (u, w)) field is determined with the help of linear wave theory, in which ω is the angular frequency (ω = (2π)/T; T-wave period) and k is the wave number (k = (2π)/L). Linear wave theory defines the framework for storm waves because the storm surge causes an increase in water depth. Furthermore, depth-limited waves are employed, which means that the height of the waves is limited to 0.8 of the water depth. Even though it is assumed that boulder transport in a tsunami takes place well behind the moving wave front, from the length-scale to water-depth ratio of a tsunami it can be concluded that the horizontal velocity u computed with linear theory represents a gross underestimation of the actual velocity. The computation of the vertical velocity w with linear theory is thought to be robust even for tsunamis. In this setting, it can be assumed that the velocity is related to the depth of the flow. The scaling constant in the relationship between the flow velocity and depth of flow is the Froude number Fr. Spiske et al. (2008), Jaffe and Gelfenbaum (2007), and Matsutomi et al. (2001) reported maximum Froude numbers of 2. However, Lynett (2007) found Froude numbers as high as 5 for the largest waves employed in the study, but using a reference height. Employing the entire water column, Fr = 5 decreases to about Fr = 3. Hence, we assume that the Froude number can vary from 0.5 to 3. 2. Boulder transport model 2.1. Drag and lift forces Usually, drag and lift forces are expressed in a quadratic form of the velocity, which works well for one-dimensional flows. For example this approximation is used in Nott (2003). However, a wave can only be described by a two-dimensional flow field u = (u, w). Hence, drag and lift forces are exerted by the two-dimensional wave motion. As an assumption, the circulation of the flow under a wave can be calculated with the Kelvin–Helmholtz circulation theorem to the first order.
Γ ¼ ∮C u⋅dl
ð1Þ
30
R. Weiss / Marine Geology 295-298 (2012) 28–33
in which Γ is circulation. We assume that the maximum length scale of the eddies created by flow field underneath a wave is the height of the boulder under consideration. For simplicity, the integration contour, C, in Eq. (1) is defined to be of circular shape of radius rc = 0.5db, where db is the boulder height. The derivation of the drag and lift forces is similar to the one presented independently by Kutta and Jousowski and uses the wellknown Kutta–Jousowski formula (Batchelor, 1967): F ¼ ρu eΓ, in which e represents the unit vector. In Kutta and Jousowski's derivation of forces, the velocity is defined as the free-stream velocity, which is possible for the cases Kutta and Jousowski considered. However, in a two-dimensional flow situation, the derivation of drag and lift forces from circulation with the help of the Kutta–Jousowski formula is more difficult and bears some mathematical challenges, as described in Crighton (1985). Intuitively, the drag force and lift force receive contributions from the horizontal velocity. However, the lift force also is influenced by the vertical flow component. As an approximation for the drag force computed from circulation, the approximation FD = ρu|Γ| is used and for the lift force FL = ρ(u − w)|Γ|. Please note that subtracting of the vertical velocity from the horizontal still is a positive contribution because the vertical velocity bears a negative sign (see Eq. (2)). For the contour integration, the equations for the horizontal and vertical velocities from the linear wave theory are simplified, assuming the shallow-water approximation, with χ = L/d ≥ 20, in which L is the wavelength and d the water depth. With ω = (2π)/T, k = (2π)/L, pffiffiffiffiffiffi and T ¼ gd=ð20dÞ, it is assumed that 2πðz þ dÞ cosh χd ≈1 2π sinh χ 2π ðz þ dÞ sinh z χd ≈ 1þ : 2π d sinh χ
2.2. Force balance and incipient motion In the last section, it was derived how wave characteristics influence the drag and lift forces. In this section, the safety factor is employed to boulder stability and incipient motion. The stability of a boulder is a function of flow velocity and the slope, but is also a function of how the boulder is locked by surrounding boulders and particles. In classic sedimentology and engineering, the work by Stevens and Simons (1971) was ground-breaking and resulted in the well-established linear stability analysis based on moments. Here, the theory developed by Stevens and Simons (1971) is employed to compute the force equilibrium between the weight, lift and drag forces. Fig. 1a depicts the stability with the help of moment arms with regard to a rotation at point 0 where incipient motion will take place. From the figure, the following equilibrium condition can be derived: l2 W s cos α ¼ l1 W s sin α þ l3 F D þ l4 F L :
ð5Þ
Eq. (5) is the original form of the equation in Simons and Sentürk (1992), with the assumption that the flow is parallel to the sloping bed, for which the angle is α. The moment arms l1, l2, l3, and l4 need to be determined and may have a significant influence on stability. From Fig. 1a, it can be assumed that l2 ≡ l4. Furthermore, it is assumed that the boulder is of circular shape and that the onset point for the weight vector is the center of the circle. Then by defining l2 = r − ξx, the relation l1 + l2 = r and l1 = r − l2 = ξx can be obtained, following the general equation for circles. The parameter r denotes the radius of the circular boulder. The momentum arm l3 can be approximated by l3 = l1 + r − ξz with the final formula for l3: l3 = r + ξx − ξz. Substituting these expressions of the momentum arms into Eq. (5) yields: ðr−ξx ÞW s cos α ¼ ξx W s sin α þ ðr þ ξx −ξz ÞF D þ ðr−ξx ÞF L :
ð6Þ
To evaluate the stability of a setting, the safety factor Π is defined as the ratio of the left-hand and right-hand side of Eq. (6):
With the two substitutions, the velocity components simplify to: pffiffiffiffiffiffi gdA cos ðωt−kxÞ χdpffiffiffiffiffiffi 2π gdA z 1 þ sin ðωt−kxÞ: wðx; z; t Þ ¼ − χd d
uðx; z; t Þ ¼
Π¼
2π
ð7Þ
ð2Þ
Eq. (2) is now used for contour integration to find the circulation. The contour is defined as a circle with the radius rc, which is defined as half of the boulder height. The absolute value of circulation |Γ| then becomes: jΓ j ¼ ωAr c sin ðωt−krc Þ:
ðr−ξx ÞW s cos α : ξx W s sin α þ ðr þ ξx −ξz ÞF D þ ðr−ξx ÞF L
ð3Þ
Eqs. (2) and (3) can now be used to compute the drag and lift force. The expression for the drag and lift forces: pffiffiffiffiffiffi 2πρrc A2 gd sin ðωt−krc Þcos ðωt−kxÞ FD ¼ χdpffiffiffiffiffiffi 2 h i 2πρr c A gd z 1 þ sin ðωt−kxÞ−cos ðωt−kxÞ sin ðωt−kr c Þ: FL ¼ d χd ð4Þ As a reminder, both equations are based on linear wave theory, which is suitable for computing the magnitude of forces. However, the computation of instantaneous forces, for example to estimate how far a boulder is moved by a wave, requires the application of the non-linear wave theory, which is not considered here.
From Eq. (7), it becomes obvious that for Π ≥ 1 the setting of a boulder is stable; whereas for Π b 1, the setting is unstable and results in boulder movement. A discussion on the safety factor of a boulder setting, also known as riprap stability, can be found in Julien (1998) and also in Simons and Sentürk (1992). The definition of the safety factor is part of the standard procedures in engineering geology and geotechnical engineering to guarantee slope stability. 2.3. Limitation of the approach As mentioned earlier, this contribution introduces an alternative to the approaches for incipient boulder movement that relies on the quadratic form for the forces. This approach is based on the Kelvin– Helmholtz circulation theorem and the two-dimensional Kutta– Jousowski formula. In order to introduce this approach, both are coupled with linear wave theory and a simplified boulder geometry. The Kelvin–Helmholtz circulation theorem in Eq. (1) requires the integration along a given contour line. In this regard, the combination of the Kelvin–Helmholtz circulation theorem and Kutta–Jousowski formula with the linear wave theory and simplified boulder geometry ensures a good balance between mathematical and physical sophistication. Even with these simplifying assumptions pivotal general conclusions can be drawn on the initiation of boulder movement during storms and storm waves, as well as tsunamis.
R. Weiss / Marine Geology 295-298 (2012) 28–33
31
Fig. 1. (a) depicts the classic riprap stability problem as defined in Simons and Sentürk (1992). The level arms for the different forces are given for the case where the flow is parallel to the slope θ. (b) simplification of the stability problem in (a) to account for bed roughness and exposure. The relationship between the level arms in (a), and exposure and bed roughness is given in the text.
3. Results 3.1. Variable moment arms and constant forces With the help of Eq. (6), the influence of the moment arms on the stability of a boulder for constant lift and drag forces and constant boulder characteristics can be studied. The left- and right-hand sides of Eq. (6) are employed to compute the safety factor Π (Eq. (7)). The moment arms are r − ξx and r − ξz. The physical interpretation of r − ξx is trivial as the magnitude of the bed roughness (Fig. 1b), which can be a characteristic of the surface on which the boulder in question rests, or is generated by adjacent boulders. The moment arm r − ξz can be seen as the exposure of the boulder (Fig. 1b) to the flow, which is a complicated and non-linear function. The magnitude of the exposure not only depends on the geometry of the boulder, but also on the spatio-temporal flow dynamics around the boulder. The latter is impossible to infer reliably in hindsight. Fig. (2) plots the safety factor Π as a function of the bed roughness on the left panel and as a function of the exposure on the right panel. Again, if Π ≥ 1 the boulder sits in stable condition (gray area in both panels), and for Π b 1 the boulder will move. On the left-hand side of Fig. 2, the envelope of the dark-gray area is defined as the minimum (ξz = r) and maximum exposure (ξz = 0). Intuitively, if the moment arm, r − ξx, is small, which also means
that the bed roughness is small. From the figure it is evident that with increasing bed roughness, the safety factor increases as well. For maximum exposure, the safety factor exceeds one at a normalized bed roughness of (r − ξx)/r ≈ 0.6. That means if the roughness is larger than 30% of the total boulder height, the boulder will not move given the force remains constant in the system. If the boulder radius is assumed to be 1 m, the safety factor is Π = 0.06 for roughness length of 5 cm, and Π = 1 for a roughness length of 30 cm. The right panel of Fig. 2 depicts the safety factor as a function of normalized exposure ([r − ξz]/r). The dark-gray area represents the envelope created by the minimum (ξx = r) and maximum (ξx = 0) bed roughness. Interestingly only for low roughness, the safety factors cross the threshold value of one. Only for safety factors that are already close to the threshold due to respective bed roughness, travel of the safety factor from the unstable to a stable condition is possible. We can also observe that when the exposure length is low, the difference in the safety factors for the maximum and minimum bed roughness is the largest, reaching asymptotic characteristics. The left panel of Fig. 2 directly demonstrates the exceptional relevance that bed roughness has for the stability of a boulder. This fact is supported also by the right side of Fig. 2, due to the almost asymptotic difference in safety factors as a function of exposure length between minimum and maximum bed roughness. 3.2. Variable forces and constant moment arms
Fig. 2. The left panel shows the safety factor Π as the function of the normalized moment arm that is related to bed roughness, the right panel depicts Π as the function of the normalized exposure.
With the help of the safety factor from Eq. (7) and the equations to compute lift and drag forces, the difference in boulder movement between storm and tsunami waves can be studied. Tsunamis and storm waves have very different periods. Therefore, parameter χ of Eq. (4), which is the wavelength to water-depth ratio, is chosen for both types based on a reference for storm and tsunami waves. The reference for storm waves is based on the reference water depth ds∗, the respective length-to-depth ratio is χs∗. Respectively, theses reference parameters are dt∗ and χt∗ for tsunamis. Then, the local χ is computed with χ = (χ ∗d ∗)/d, in which the subscripts are dropped and d is the local water depth. As reference depth for storm waves a water depth of ds∗ = 50 m is used and for tsunamis dt∗ = 5000 m. The refer~ is normalized ence χ ∗'s are χs∗ = χt∗ = 20. The mass of the boulders m by the mass of the unit boulder, which has a radius of 1 m and the ~ are normalunit density of ρ = 1800 kg/m 3 = ρ ∗. The amplitudes A ized by amplitude A ∗ which represents the amplitude needed to move the unit boulder. Fig. 3a,b shows the threshold amplitude for boulder motion as a function of the boulder mass for tsunamis (Fig. 3a) and storm (Fig. 3b) waves. As a reminder, a Froudenumber scaling is employed for the horizontal velocity, u, with pffiffiffiffiffi ffi u ¼ Fr gd, to account for the very different fluid dynamics of storm
32
R. Weiss / Marine Geology 295-298 (2012) 28–33
~ t ¼ At =A ) as the function of normalized mass (m ~ s ¼ As =A ) as ~ ¼ m=m ). (b) gives the normalized storm-wave amplitude (A Fig. 3. (a) gives the normalized tsunami amplitude (A t s ~ (c) depicts the amplitude ration of storm and tsunami waves (At/As) as a function of m ~ for three different Froude numbers (F1 : Fr = 1.0, F2 : Fr = 2.0, F3 : Fr = 3.0). the function of m.
and tsunami waves in shallower water. For Fig. 3a, a Froude number of Fr = 1.5 is used. Comparing tsunami and storm waves, the envelope spanning from 0.5ρ ∗ to 2ρ ∗ has the same funnel-shaped geometry. However, the amplitude of tsunamis to move boulders of larger mass is about five times larger compared to the unit boulder. For similar boulder masses, the threshold amplitude is only about three times larger than unity. This difference is ascribed to an increased, perhaps underestimated, relevance of the vertical velocity, v, for boulders ~ ¼ 1 and transported in tsunamis and by storm waves. Near unity (m ~ ¼ 1), the normalized amplitudes for storm and tsunami waves are A ~ ¼ 1, a boulder of density 0.5ρ ∗ very similar. For a boulder mass of m ~ requires wave amplitude of A≈0:7, and if the boulder density is 2ρ ∗ ~ the required threshold amplitude is A≈1:3. From the lower to the upper bound of the envelope depicted in Fig. 3a,b, the density qua~ druples, but the amplitude only doubles. If a constant amplitude A of one is assumed then the boulder mass that is possible to move ~ ¼ 0:25 and for 2ρ ∗ the mass is m ~ ¼ 3:9. For the quadrufor 0.5ρ ∗ is m pled boulder density, the mass increases 15 times. Fig. 3c shows the amplitude ratio (At/As) of storm and tsunami waves as the function of normalized boulder mass. The different lines denoted with F1, F2, and F3 indicate different Froude numbers (F1 : Fr = 1.0, F2 : Fr = 2.0, F3 : Fr = 3.0) for the tsunami. For small Froude numbers the amplitude ratio At/As is large, which means that the amplitude of tsunamis required to move a boulder is larger than the amplitude of a storm wave to move the same boulder. As the Froude number increases the amplitude ratio approaches unity. Only for relative light boulders and large Froude numbers, the amplitude of tsunamis is smaller than storm waves to transport the same boulder. It should be noted that the understanding of amplitude is no longer trivial. In shallow water, the amplitude of a tsunami is related to the water depth and is understood as such in the Froudenumber-scaled horizontal velocity. For storm waves, the common
understanding of amplitude is valid; however, a certain water depth must be present to support wave propagation. Because a finitedepth breaking criterion is assumed, the surge underneath the waves can be computed from the wave amplitude. The depth limit ~ s is is set to 2A = H = 0.8d and coincides where the amplitude A found. If the water depth during the storm is only due to the storm surge, for example on coastal plains very close to the still-water level, the surge level can be estimated with d = 1.25H. Furthermore, it would be possible to derive an explicit expression to invert amplitudes of tsunami and storm waves from boulder masses with the equations employed in this section. However, the linearity of the applied theories only allows for general conclusions, and the nonlinearity of the quantification process prohibits the derivation of an explicit expression. 3.3. Storms versus tsunamis Storms and tsunamis are very different processes from a hydrodynamic point of view. Tsunamis normally consist of a succession of two to four large waves of different wave amplitudes and wavelengths for the periods are between 10 and 30 min. The entire energy available for transport of particles of any size arrives in consecutive waves. The effects of storms include a surge and storm waves. The storm surge may submerge boulders in the surf zone or even on land. The storm waves then may cause the motion of boulders. The number of storm waves is at least 3 orders of magnitude larger than in tsunamis. The total energy per wavelength is the sum of the kinetic and potential energy. It is common to write that Et ¼ Ek þ Ep ¼ ρ∫Ω 12 v2 þ gz dΩ ¼ 12 ρgA. The integration along the wave is performed over a wavelength. The total amount of energy then is computed by multiplying Et by the number of waves. As example, data of the 2004 Sumatra at Port La Rue and of a tropical storm in Bisbane,
R. Weiss / Marine Geology 295-298 (2012) 28–33
Australia are used (see Weiss and Bahlburg (2006)). The storm data consist of about ns = 33000 waves and the number of waves is nt = 120 in case of the tsunami. Then the total amount of wave energy for all waves in the storm is in the order of nsEt ≈ 10 11J and in the tsunami it is in the order of ntEt ≈ 10 9J. For boulder transport by storms and tsunamis, it is assumed that the pre-transport boulder distribution is unorganized. In a storm, the energy per wave is relatively low, and only a certain fraction of all storm waves are capable of transporting boulders. However, the storm surge changes in order of hours, allowing large number of waves to move boulders. With slow changing storm surge and storm-wave periods in the order of 20 s, we can assume that depthlimited breaking occurs in a certain area with respective water depth. This fact leads us to the conclusion that in the area of the wave breaking, boulders are deposited to form lines or even clusters. On the other hand, the energy per wave in a tsunami is large. The tsunami will pick up a boulder and transport it to a certain location. The transport distance is a sensitive function of boulder characteristics and wave forcing. However, subsequent waves may not be able to move certain boulders under consideration. It is therefore very unlikely that boulders will be organized to clusters or lines during the transport process. However, clustering is possible if the transport distance is short, and pre-transport clusters existed. 4. Conclusions The scope of this study was to explore boulder transport during storms and tsunami waves. Different processes were explored not only to better establish the limitations that our understanding of boulder transport bears but also on the opportunities that simple analysis can offer. Due to the linearity of the employed wave theory, it is not legitimate to derive a different set of equations, such the equations derived in Nott (2003), to infer the amplitude of storm and tsunami waves from the moved boulder characteristics. Even though the theories employed herein are linear, the results point to very interesting conclusions. For example, bed roughness seems to be a major factor, which has not been incorporated in inversion methods previously. Bourgeois and MacInnes (2010) speculated based on the relevance of bed roughness, but Fig. 2 offers a quantitative view point. It appears that leaving the forces constant, the existence of bed roughness with length scales in the order of ten percent of the boulder height will prevent a boulder from being moved. Smaller roughness length will let the boulder be moved. This suggests that for field studies, the bed roughness of the potential pre-transport area is important to survey along with the boulder characteristics. However, it should be noted that smaller boulders and cobbles that were in the vicinity of the boulder in question could have created bed roughness but have been moved as well. The comparison of amplitudes between storm and tsunami waves required to move a boulder revealed that both waves can move boulders of similar sizes with similar amplitudes. It is even possible that the amplitude of storm waves necessary to move a boulder is smaller than the amplitude of a tsunami required to move the same boulder. With this fact, it is possible to draw the conclusion that it may be impossible to discriminate those boulders moved by storms from those moved by tsunamis easily without taking also the periods of storm and tsunami waves into account. However, it is assumed that the mechanics of boulder movement is different in a storm compared to a tsunami. If the boulders are distributed unorganized in the pre-transport area, boulders can also be distributed in an unorganized fashion in case of a tsunami. However for storms, it is possible that boulders can be organized in clusters or along lines due to the depth-finite character of the storm waves and their relation to the surge level, such as observed by Goto et al. (2010). Assuming depth-limited storm waves, a boulder line or cluster should form at around 2.25 of the amplitude needed to move boulders of the same size. This helps to establish the
33
magnitude of the storm-surge levels for historical or paleo-storm events if boulder deposits are present. From a general point of view and even with the linear concepts and theories employed, this study revealed the complexity of boulder transport, which might not be fully understood until three-dimensional flow simulation around boulders with arbitrary geometries is carried out to study the three dimensional stress distribution. Furthermore, measuring the sizes and densities of boulders in the field is not sufficient. Quantifications of the bed roughness and careful documentation of boulder arrangements are pivotal to distinguish storm-moved from tsunamimoved boulders. There is a critical need for more field observations and measurements, but theoretical analysis is equally important to resolve the mystery of boulders moved by storm and tsunami waves. Acknowledgements The author would like thank Bruce Jaffe and an anonymous reviewer for their helpful comments. This research was partially funded by the grant NSF-EAR-1136534 to the author. References Barbano, M., Pirrotta, C., Gerardini, F., 2010. Large boulders along the south-eastern Ionian coast of Sicily: storm or tsunami deposits? Marine Geology 275 (1–4), 140–154. Batchelor, G.K., 1967. An Introduction to Fluid Dynamics. Cambridge Univ. Press. Benner, R., Browne, T., Brückner, H., Kelletat, D., Scheffers, A., 2010. Boulder transport by waves: progress in physical modelling. Zeitschrift für Geomorphologie 54, 127–146. Bourgeois, J., MacInnes, B., 2010. Tsunami boulder transport and other dramatic effects of the 15 November 2006 central Kuril island tsunami on the island of Mantua. Zeitschrift für Geomorphologie 54 (3), 175–195. Bryant, E., Young, R., Price, A., Wheeler, D., Pease, M., 1997. The impact of tsunami on the coastline of Jervis Bay southeastern Australia. Physical Geography 18 (5), 440–459. Buckley, M., Jaffe, B.E., Wei, Y., Watt, S., 2011. Estimated velocities and inferred cause of overwash that emplaced inland fields of cobbles and boulders at Anegada, British Virgin Islands. Natural Hazards. doi:10.1007/S11069-011-9725-8. Costa, J., 1983. Paleohydraulic reconstruction of the flash-flood peaks from boulder deposits in the Colorado Front Range. Geological Society of America Bulletin 94, 986–1004. Crighton, D., 1985. The Kutta condition for unsteady flow. Annual Review of Fluid Mechanics 17, 411–445. Dominey-Howes, D., Humphreys, G., Hesse, P.P., 2006. Tsunami and paleotsunami depositional signatures and their potential value in understanding the late-Holocene tsunami record. The Holocene 16 (8), 1095–1107. Goto, K., Miyage, K., Kawamato, H., Imamura, F., 2010. Discrimination of boulders by tsunamis and storm waves at Ishigaki Island, Japan. Marine Geology 269 (1–4), 34–45. Jaffe, B., Gelfenbaum, G., 2007. A simple model for calculating tsunami flow speed from tsunami deposits. Sedimentary Geology 200, 347–361. Julien, P., 1998. Erosion and Sedimentation. Cambridge Univ. Press. Kelletat, D., Scheffers, S., Scheffers, A., 2007. Field signatures of the SE-Asia mega-tsunami along the west coast of Thailand compared to the Holocene paleo-tsunami from the Atlantic region. Pure and Applied Geophysics 164 (2–3), 413–431. Lynett, P., 2007. The effects of shallow water obstruction on long wave runup and overland flow velocity. Journal of Waterway, Port, Coastal, and Ocean Engineering (ASCE) 133 (6), 455–462. Mastronuzzi, G., Sanso, P., 2004. Large boulder accumulations by extreme waves along the Ionian coast of Apulia (Southern Italy). Quaternary International 120, 173–184. Matsutomi, H., Shuto, N., Imamura, F., Takhasi, T., 2001. Field surveys of the 1996 Inian Jaya earthquake tsunami in Brak Island. Natural Hazard 24, 199–212. Nandasena, N.A.K., Paris, R., Tanaka, N., 2011. Reassessment of hydrodynamic equations to initiate boulder transport by high-energy events (storms, tsunamis). Marine Geology 281, 70–84. Nott, J., 2003. Waves, coastal boulders and the importance of pre-transport setting. Earth and Planetary Science Letters 210, 269–276. Paris, R., Fourier, J., Poizot, E., Etienne, S., Morin, J., 2010. Boulder and fine sediment transport and deposition by the 2004 tsunami in Lhok Nga (western Banda Aceh, Sumatra, Indonesia): a coupled offshore–onshore model. Marine Geology 268 (43–54). Simons, D., Sentürk, F., 1992. Sediment Transport Technology. Water Resources Publications. Spiske, M., Weiss, R., Roskosch, J., Bahlburg, H., 2008. Inversion of flow depth and speed from tsunami deposits using TsuSedMod. Eos Trans. AGU, Fall Meet. Suppl., 89 (52), pp. OS53A–1288. Stevens, M., Simons, D., 1971. River Mechanics. Stability Analysis for Coarse Granular Material on Slopes. Shen Publisher, Ch. Weiss, R., Bahlburg, H., 2006. A note on the preservation potential of offshore tsunami deposits. Journal of Sedimentary Research 76 (12), 1267–1273.