Equations for estimating the value of*σ*_{tgmax} and*σ*_{twmax} for stability analysis (Kogure & Matsukura, 2010b).

## 1. Introduction

The height of a tsunami as it surges into the coasts is altered by the presence of a coral reef. In particular, wave propagation over the reef crest induces wave breaking, and energy dissipation increases to 50-90% (Munk & Sargent, 1948; Kono & Tsukayama, 1980; Roberts & Suhayda, 1983). Shibayama et al. (2005) reported that the heights of tsunamis in Sri Lanka caused by theSumatra earthquake in 2004 were lower at coastlines with coral reefs than without. This reduction in the height of the tsunami is due to friction between the tsunami and coral reefs seaward of the coasts. Nott (1997) stated in contrast that tsunamis pass through the gaps in coral reefs; the Great Barrier Reef does not act as an effective barrier against tsunamis, since the coastline near Cairns in Australia has been impacted by tsunamis. Coral reefs are recognized as a breakwater against invading tsunamis, but details of the behavior of tsunamis around coral reefs are unknown.

Such “modern reefs” would form coastal cliffs if the reefs uplifted. Verticallimestone cliffsformed by the uplifted coral reefs are seen in tropical-subtropical areasaround the world (e.g. Tjia, 1985; Maekado, 1991). The limestone cliffs, i.e., “ancient reefs”, are subject to wave erosion while the “modern reefs” could control wave energy. Notches carved by the wave erosion often induce cliff collapses (Maekado, 1991; Kogure et al., 2006; Kogure and Matsukura, 2010a). Kogure et al. (2006) presented a mechanical equation for calculating the critical notch depth for cliff collapse. In addition to these cases, waves also break down or smash geomorphological environment of rocky coasts. Notched cliffs can be collapsed by waves as well as gravity (e.g. Kogure and Matsukura, 2010b). Here, one question has been raised about the relationship between the tsunami action affected by “modern” fringing reefs and cliff collapses induced by the tsunami: are there any relationships between the developments of “modern” and “ancient” fringing reefs?

We can see an interesting interaction between “modern” and “ancient” fringing reefs through tsunami wave action in Kuro-shima, a small island in the YaeyamaIslands. The YaeyamaIslands, in the south-western part of Japan, have been attacked by several large tsunamis in the last few thousand years (e.g. Nakata & Kawana, 1995). The largest tsunami on record is the 1771 Meiwa Tsunami caused by the Yaeyama Earthquake of April 24, 1771. The maximum run-up height of the tsunami was estimated by Nakata & Kawana (1995) to be more than 5 m on the east and south side of Kuro-shima. In Kuro-shima, a low-lying notched cliff of height 3–4 m, made of Ryukyu Limestone, currently has a flat top surface that is partly destroyed. Many blocks, apparently due to cliff collapses, are scattered on the reef flat. Based on the observed correspondence between geometries of these blocks and the scars on nearby cliffs, the blocks have been cleaved from the cliffs. In Kuro-shima, collapses of coastal cliffs have been induced not only by enlargement of the notch at the cliff base but also by the attack of extreme waves such as tsunamis or bores during a storm (Kogure & Matsukura, 2010a, 2010b).

The present study shows stability analysis models to evaluate the cliff collapses due to extreme waves and discusses the effects of wave heights on the cliff collapses. Finally, this paper presents the effect of coral reefs in reducing the height of extreme waves, especially tsunamis, by comparing the distribution of blocks produced by wave-induced collapses of coastal cliffs and the development of coral reefs around Kuro-shima.

## 2. Study area

Kuro-shima is a low-lying island made of uplifted coral limestone,located in the south-western area of the Ryukyu Islands (Fig. 1). Geomorphic, geologic and lithologic features are described below.

### 2.1. Geomorphological setting

The Pleistocene coral limestone is known as RyukyuLimestone. The circumference of the island is about 10 km, and its highest point is about 10 m above mean sea level. Fringing reefs have developedaround the island, except for the northern part; the coralreefs that are developing at the north of the island meet a large lagoon. Coastal areas in the south-east and south-westareas of the island face the Pacific Ocean. The maximum andminimum distance from the coast to the offshore reef edge isabout 850 m in the south-east and 200 m in the south-west (Fig. 2). Waves break at the reef edge at low tide, and reach the coastalcliffs at high tide. A fault running NW–SE has developed in thenorth-eastern part of the island, and development of the coral reefis discontinuous along an extension of the fault (Fig. 2). Arange of coastal cliffs, 6 km long andless than 5 m high (Fig.3a), has developed in the area, except at the northernsandy beaches (Fig. 2). A reef flat has developed in front ofthe coastal cliff. The seaward side of the reef flat is a lagoonal environment, with depth of water 1–3 m (Fig.3b).

There is no visiblevegetation on the top surface of the coastal cliff, certainly not near the shore. The cliff has a notch at its base; this notched cliff has a visor extending seaward(Fig.3b). The depth of the notch, defined as the horizontaldistance of the visor from the retreat point of the notch to theseaward end of the visor, is 3–4 m at most sites. The elevationof the retreat point of the notch is almost exactly equal to themaximum sea level (HSL) at spring tide, which is about 1 mhigher than mean sea level (MSL); see Fig.3b.

Vertical joints with cracks having width between ten andseveral tens of centimetres have developed on coastal cliffs or reef flats. These joints appear to develop systematically rather than randomly, so that they are believed to have come about by geological processes. The depth of a joint is usually largeenough to incise both cliffs and reef flats (Fig. 4a). Thesejoints intersect coastal cliff faces at various angles, perpendicular, oblique (Fig. 4b) or parallel to the cliff. A cliffbounded by vertical joints therefore appears to be separatedfrom the main cliff. The degree of development of a horizontalbedding plane on a cliff face varies from area to area; cliffswith and without bedding planes are distributed along the rocky coasts of this island with no apparent pattern.

### 2.2. Cliff collapses and blocks

Many angular blocks are scattered at cliff bases, and haveapparently been produced by failures of notched cliffs (Fig. 3a). Along the 6 km coastal cliffs, more than 150 blocks were found having one side ofdimension exceeding 1 m. Theblocks are composed of Ryukyu Limestone identical to thecliffs behind the blocks. Vegetation or joints are rarely seenon the surfaces of the blocks. Some blocks can be clearlyidentified as having originated from the cliffs immediatelybehind them (Fig.5a, 5b); the place of origin of other blocks cannot be identified because they appear to have moved.Many identifiable blocks are inclined seaward, and have atriangular or quadrangular flat surface which appears to coincidewith the top surface of the cliff (Fig.5a, 5b).Although the lower part of these blocks is inundated at high tide, the flat surface displays no development of a notch, as shownon the cliff. A few blocks have a breadth of 10 m in the directionfollowing the shoreline; in most cases the breadth isapproximately 5 m, as shown in Fig.5 a and 5b, and themaximum thickness of the blocks is about 3 m.

Some cliffs behind a fallen block show a failure scar. Theshape of the scar on the cliffs corresponds to that of the block,having a triangular or quadrangular flat surface (Fig. 5c and 5d). Many scars have similar features, including: (1) the cliffabove the retreat point of a notch has detached from the maincliff (Fig. 5b); (2) the scar consists of a nearly horizontalsurface and bounding vertical surfaces (Fig. 5d), and has nosliding or shearing striation; (3) the height of the horizontalsurface is almost equal to that of the retreat point of the notch;and (4) the vertical surfaces intersect the horizontal surface,with the vertical extension of the joints reaching the reef flat(Fig. 5b and 5d). In view of the resulting visible joints onthe top surface of a cliff (Fig. 4b), Kogure & Matsukura (2010a) inferred that the verticalsurfaces on the scar are joint surfaces, and that the horizontalsurface is a failure surface at which cliff collapse has occurred.The cliff, which is separated from the main cliff by verticaljoints, will collapse when the notch extends sufficiently farinward from the sea.

Many fallen blocks have a quadrangularor triangular surface. In many cases, the cliffs behind the blocks have a matching failure surface. Kogure&Matsukura (2010a) denoted collapseshaving a quadrangularor triangular failure surface asH_{q}-type or H_{t}-type collapse, respectively (Fig. 6). H_{q}-type collapse takes place at a cliffwhere three joints have intersected each other to form a quadrangularshape (Fig. 5a and 5c); two parallel joints developperpendicular to a cliff face, and one joint runs almost parallelto the cliff face.Fig. 5b and 5d show aH_{t}-type collapse. Two vertical joints that weredeveloping oblique to a cliff face intersected each other a fewmeters inland from the cliff face. The cliff bounded by the twovertical joints has collapsed at the horizontal surface (Fig. 5d). Cliff collapses in Kuro-shima therefore display two shapes of failure surface, quadrangular and triangular,according to the number of joints and the angle between thejoints and the cliff face.Blocks of both H_{q}- andH_{t}-type are distributed widely around the coasts of Kuro-shima.The distributions of H_{q}- and H_{t}-type collapses appear tobe controlled by the angle between the vertical joints and thecliff face (Kogure&Matsukura, 2011).

### 2.3. Tsunamis

The most significant tectonic processes in the area of the Ryukyu Islands are the rifting of the Okinawa Trough and subductionof the Philippine Sea Plate. As a result of theseprocesses the Ryukyu Arc is characterized by intense seismicand tectonic activity, leading to raised and faulted Pleistoceneand Holocene coral reef tracts. Seismic activity often induces tsunamis. Coastal cliffs in the Ryukyu Islands havesuffered damage from large tsunamis several times in history(e.g. Imamura, 1938; Kato & Kimura, 1983; Nakata &Kawana, 1995).

The largest tsunami on record is the 1771 Meiwa Tsunamicaused by the Yaeyama Earthquake (M 7.4) on April 24, 1771.The maximum run-up height of the tsunami was estimated byNakata &Kawana (1995) to be more than 30 m on thesouth-eastern side of Ishigaki Island, and 5 m on the east andsouth side of Kuro-shima. Kato & Matsuo (1998) have indicated the possibility of coastal cliff collapses due to historical tsunamis, especially the 1771 Meiwa Tsunami,in Kuro-shima. Nakata &Kawana (1995) inferred the actionof other large tsunamis prior to the Meiwa Tsunami, based onradiocarbon dating analysis. Many tsunami blocks that arecomposed of Holocene fossil corals are much older than the Meiwa period; in a few islands around Ishigaki Island,some are at much higher locations than the run-up height ofthe 1771 Meiwa Tsunami. According to Nakata &Kawana(1995), such large tsunamis recur on timescale of hundreds orthousands of years in the southern Ryukyu Islands. Theyasserted that the southern Ryukyu Islands were inundated by large tsunamis about 600, 1000 and 2000 years before the present (yr BP); the tsunami at 2000 yr BP was much higher thanthe Meiwa Tsunami.

The Ryukyu Islands are in a subtropical area, and in thesummer and autumn they often experience typhoons.These give rise to bores that manifest as an extreme wave inthe Ryukyu Islands. Severe damage to the coastal environment as a result of these bores has been reported from the late 1980s (e.g.Nakaza et al., 1988). Also in recent years, and especially in July2007, extreme waves generated by a strong typhoon destroyedsome social infrastructures in the southern part of Okinawa Island. Despite studies of the wave heightand estimated wave pressure of extreme offshore waves duringtyphoons around the Ryukyu Islands, there is little information about these parameters for bores in coral reef lagoons which permits us to estimate the wave pressure. Below, we discuss the possibility of cliff collapses due to tsunamis.

## 3.Methods

### 3.1. Stability analysis models

According to Kogure&Matsukura (2010b), wave-induced collapse proceeds as follows: (1) a developing notch forms a visor-like cantilever beam; (2) tensile stress is generated, from its own weight, on the horizontal plane that is level with the retreat point of the notch; (3) collapse occurs due to stress from wave impact, resulting in an ‘upset’ mode of collapse. The cantilever beam model (Timoshenko & Gere, 1978, pp.108-110) is suitable for estimating the critical notch depth. To obtain the critical failure stresses in H_{q}- and H_{t}-type collapse, equations were derived based on this model.

Fig.7 shows the relation between wave height, *H*_{w}, and cliff height, *H*^{*}. When *H* and *h* are as shown in Fig.6, and *H*’ denotes the height from the reef flat to the retreat point of the cliff, *H*^{*} is equal to the sum of *H*’,*H* and *h*. The relation can be divided into four cases (Kogure&Matsukura, 2010b): (1) *H*_{w}≤*H*’ (Case 1), in which that wave pressure does not operate on the notched roof; (2) *H*’ <*H*_{w} ≤*H*’ + *H* (Case 2), in which horizontal stress acts only on the notched roof; (3) *H*’ + *H*<*H*_{w}≤*H*^{*} (Case 3), in which horizontal stress acts on the notched roof and part of the vertical cliff surface; and (4) *H*^{*}<*H*_{w} (Case 4), in which horizontal stress acts on the notched roof and the vertical cliff surface. In Case 1, waves have no effect on collapse because no stress acts on the notched roof. Stability analysis models are derived below for Cases 2, 3 and 4.

Three forces are involved in cliff instability: (1) downward stress due to gravity, *σ*_{tgmax}, (2) upward stress due to waves, *σ*_{twmax}, and (3) the strength of the cliff material, i.e., the tensile strength, *S*_{t}. The following equation gives the condition for wave-induced collapse:

For a H_{q}-type cliff having a cliff height above the retreat point (referred tohenceforth as the cliff height), *H* + *h*, the value of *σ*_{tgmax} operating on the retreat point of the notch (bold E in Fig.8a) is shown as the sum of gravity-induced stress, *σ*_{A}, and downward loading due to gravity, *σ*_{B}. (Both are shown in bold arrows in Fig. 8a). This relation is

The values of *σ*_{A} and *σ*_{B} are:

where*ρ*is the rock density and *g* is the acceleration due to gravity. The value of *σ*_{tgmax} is given by substituting Eqs. (3) and (4) into Eq. (2), as:

Wave pressure acts on a notched roof when waves fill up a notch; a cliff experiences both vertical and horizontal components of wave pressure through the whole of the notched roof (Fig. 9). Cliffs are also subject to buoyancy according to the underwater volumes of the cliffs. For a H_{q}-type cliff, in Case 2 (Fig. 7b), having cliff height *H* + *h*, with distance*L* between the cliff face and a vertical joint parallel to the cliff face, notch depth *βL* (0 ≤ *β*< 1) and angle *γ*, the maximum value of *σ*_{twmax} at E in Fig. 9a is given by:

where*P* is the wave pressure and*ρ*_{sw} is the density of sea water. In Cases 3 and 4, the value of *σ*_{twmax} is given respectively by:

For aH_{t}-type cliff (Figs. 8b and 9b), the values of *σ*_{tgmax} and *σ*_{twmax} can be obtained as above for a H_{q}-type cliff. The equations obtained for aH_{t}-type cliff are set out in Table 1, numbered from Eq. (9) to (12).

H_{q}-type | H_{t}-type |

σ_{tgmax} | |

Equation (5): | Equation (9): |

σ_{twmax} | |

Case 2 | |

Equation (6): | Equation (10): |

Case 3 | |

Equation (7): | Equation (11): |

Case 4 | |

Equation (8): | Equation (12): |

### 3.2. Parameters for stability analysis

The stability analysis requires information on the dimensions ofcliffs and blocks, *H’*, *H*, *h*, *L*, *βL* and *γ*, the physical and mechanicalproperties *S*_{t} and *ρ* of the cliff material, i.e. Ryukyu limestone,and the wave pressure, *P*. These parameters are characterized asfollows.

#### 3.2.1. Dimensions of blocks and cliffs and rock properties of Ryukyu limestone

Of the blocks scattered around the coastal cliff, blocksof clearly identifiable type with identifiable place of origin on the cliff face were chosen for the present analysis. In addition to the blocks, the dimensions of two cliffs identified as H_{t}-type, including that in Fig. 4b, were measured by a laser finder and a measuring stick. Table 2 shows the data for the heights (*H* and *h*) andlengths (*L*) of the blocks and cliffs. The values of *H*, *h*, *L* and *βL* are0.9-1.5 m, 0–1.7 m, 2.0–7.0 m, and 0.7–4.8 m for H_{q}-type blocks,1.3–1.5 m, 0–1.7 m, 2.4–6.1 m and 0.9–3.6 m forH_{t}-typeblocks, and 1.4 m, 0 m, 3.4–4.7 m and 2.0–2.7 m forH_{t}-type cliffs.

Kogure&Matsukura (2010b) measured the values of *γ* and *H’*. The value of*γ*is 20° for every block, and thevalue of *H’* is the same for every cliff in Kuro-shima, i.e. 1.5 m.The value of *S*_{t} for Ryukyu limestone can be estimated usingthe scaling equation proposed by Kogure et al. (2006); *S*_{t} is given in terms of the length of the horizontalfailure surface, (1−*β*)*L*, as:

where the units of *S*_{t} and *L* are kPa and meters. The physicalproperties of Ryukyu limestone in Kuro-shima have been given by Kogure&Matsukura (2010a), and*ρ* = 2.35 Mg/m^{3}.

Height | Length | Notch depth | ||||||

No. | H (m) | h (m) | H + h(m) | L (m) | βL (m) | |||

H_{q}-1H _{q}-2H _{q}-3H _{q}-4H _{q}-5H _{q}-6H _{q}-7H _{q}-8H _{q}-9H _{q}-10H _{q}-11H _{q}-12H _{q}-13H _{q}-14H _{q}-15H _{q}-16H _{q}-17H _{q}-18H _{q}-19H _{q}-20H _{q}-21H _{q}-22H _{q}-23H _{q}-24H _{q}-25H _{q}-26H _{q}-27H _{q}-28H _{q}-29H _{q}-30H _{q}-31H _{q}-32H _{q}-33H _{q}-34H _{q}-35H _{t}-1H _{t}-2H _{t}-3H _{t}-4H _{t}-5H _{t}-6H _{t}-7H _{t}-8H _{t}-9H _{t}-10H _{t}-11H _{t}-12H _{t}-13H _{t}-14H _{t}-15H _{t}-16H _{t}-17H _{t}-18H _{t}-19H _{t}-C1^{*}H _{t}-C2^{*} | 1.5 1.5 1.5 1.5 1.5 1.5 1.0 1.5 1.2 1.5 1.5 1.5 1.0 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.4 1.4 0.9 1.0 1.2 1.5 1.5 1.3 1.5 1.5 1.3 1.5 1.5 1.5 1.4 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.3 1.5 1.5 1.3 1.5 1.4 1.4 1.4 | 1.1 0.5 1.5 0.4 1.5 1.2 0 0.3 0 0.7 1.0 0.1 0 0.8 0.2 1.1 0.9 0.3 0.9 0.6 0.9 1.1 0.8 1.0 0 0 0 0 0 0.5 1.7 0 0.4 0.3 0 0.8 0.5 0 0 0.8 0.8 0.7 1.7 0.9 0.5 0.9 0.6 0.5 0 0.2 0.8 0 0.6 0 0 0 | 2.6 2.0 3.0 1.9 3.0 2.7 1.0 1.8 1.2 2.2 2.5 1.6 1.0 2.3 1.7 2.6 2.4 1.8 2.4 2.1 2.4 2.6 2.3 2.5 1.4 1.4 0.9 1.0 1.2 2.0 3.2 1.3 1.9 1.8 1.3 2.3 2.0 1.5 1.4 2.3 2.3 2.2 3.2 2.4 2.0 2.4 2.1 2.0 1.3 1.7 2.3 1.3 2.1 1.4 1.5 1.5 | 7.0 3.4 5.0 2.8 3.0 3.8 3.6 4.6 2.3 3.9 3.2 5.4 3.5 6.8 2.5 3.6 4.2 5.0 3.2 2.5 3.6 2.5 3.9 3.7 2.0 2.6 3.7 4.0 2.5 4.4 3.4 2.0 3.9 4.1 2.2 3.8 2.6 3.3 2.4 5.0 4.7 3.1 6.1 5.1 4.3 2.7 2.8 4.5 3.3 3.3 3.4 3.4 3.7 2.8 3.4 4.7 | 4.8 2.3 3.6 2.2 2.4 3.2 2.0 2.2 1.9 2.0 1.4 3.5 0.7 3.6 2.1 2.7 2.7 3.9 2.6 1.9 3.0 1.9 2.5 3.1 1.6 1.5 3.1 2.0 1.4 3.6 2.6 1.3 2.9 2.0 1.6 2.1 0.9 1.9 1.4 3.2 2.2 1.7 3.6 3.1 2.6 1.5 1.8 2.8 2.1 1.9 1.8 1.0 1.9 1.5 2.0 2.7 |

#### 3.2.2. Estimation of wave pressure

The dynamic pressure of waves at a cliff, *P*, used in Eqs.6–8 and 10–12 (Table 1), is given as a simple product ofparameters such as *ρ*_{sw}, *g*, and wave height, *H*_{w} (e.g. Hom-maand Horikawa, 1964) as:

where *k* is termed the coefficient of wave pressure. Eq.14 provides the maximum dynamic pressure exerted on avertical seawall in the horizontal direction, derived from experimental data. Use of Eq. 14 in this study should take into account the shape of the target object (verticalseawall and notched cliff) and the direction of wave pressure(horizontal and obliquely upward). In spite of detailed research into coastal engineering, no equations have been derived for the wave pressure in the present situation. Extreme waves compacted into a notch are believed to generate the same pressure in all directions, because the water conforms to Pascal’s principle during such extreme events. If a significant notch developed on the cliff, it would experience high pressure on its face. We therefore useEq. 14 to calculate the wave pressure under theassumption that large waves filling up the entire notch exert thesame pressure on the notched face perpendicularly and in the horizontaldirection.

Some studies have been made of the value of *k*for tsunamis. Ikeno et al. (1998) considered the *k*-value of a tsunami in which the wave crest is dividedinto two or more parts in propagation at a shallow beach, and the crests collapse just before reaching a coast having cliffs and breakwaters. This type of tsunami causes severe damageto the coastal environment. Ikeno et al. (1998) proposed a*k*-value of 3.5, and Kogure&Matsukura (2010b) used this value. We do the same.

## 4. Results and discussion

### 4.1. Calculation of critical wave height

To determine the effect of wave impact on the cliffs, we perform a stability analysis for the blocks and the cliffs listed in Table 2.The critical condition for wave-induced collapse is equality of the left- and right-hand side of Eq. 1. The right-hand side is given by Eqs.5 and 13 for H_{q}-type, and by Eqs. 9 and 13 for H_{t}-type:

To estimate the wave pressure in Eq. 14, Kogure&Matsukura (2010b) took *k* =3.5 and *ρ*_{sw} = 1.02 Mg/m^{3}; this is the density of sea water at26.3 °C, which was the annual sea surface temperatureobserved in 1998 by Ishigaki-jima Local MeteorologicalObservatory. Also,*g* = 9.81 m/s^{2}.

No. | H_{*} (m) | H_{wc} (m) | Case | No. | H^{*} (m) | H_{wc} (m) | Case | |

H_{q}-1H _{q}-2H _{q}-3H _{q}-4H _{q}-5H _{q}-6H _{q}-7H _{q}-8H _{q}-9H _{q}-10H _{q}-11H _{q}-12H _{q}-13H _{q}-14H _{q}-15H _{q}-16H _{q}-17H _{q}-18H _{q}-19H _{q}-20H _{q}-21H _{q}-22H _{q}-23H _{q}-24H _{q}-25H _{q}-26H _{q}-27H _{q}-28 | 4.1 3.5 4.5 3.4 4.5 4.2 2.5 3.3 2.7 3.7 4.0 3.1 2.5 3.8 3.2 4.1 3.9 3.3 3.9 3.6 3.9 4.1 3.8 4.0 2.9 2.9 2.4 2.5 | 2.8 2.4 2.6 1.1 1.7 1.3 5.4 7.6 0.6 4.6 4.4 2.7 76.0 5.5 0.7 1.9 2.8 1.1 1.3 1.4 1.2 1.7 2.6 1.2 0.8 4.2 0.5 8.1 | 2 2 2 1 2 1 4 4 1 4 4 2 4 4 1 2 2 1 1 1 1 2 2 1 1 4 1 4 | H_{q}-29H _{q}-30H _{q}-31H _{q}-32H _{q}-33H _{q}-34H _{q}-35H _{t}-1H _{t}-2H _{t}-3H _{t}-4H _{t}-5H _{t}-6H _{t}-7H _{t}-8H _{t}-9H _{t}-10H _{t}-11H _{t}-12H _{t}-13H _{t}-14H _{t}-15H _{t}-16H _{t}-17H _{t}-18H _{t}-19H _{t}-C1^{*}H _{t}-C2^{*} | 2.7 3.5 4.7 2.8 3.4 3.3 2.8 3.8 3.5 3.0 2.9 3.8 3.8 3.7 4.7 3.9 3.5 3.9 3.6 3.5 2.8 3.2 3.8 2.8 3.6 2.9 4.2 4.1 | 4.9 1.0 2.2 2.4 1.5 6.8 1.4 2.1 3.6 1.3 1.2 1.4 3.4 2.0 2.4 1.7 1.5 2.0 1.1 1.3 0.9 1.5 2.4 12.0 2.6 1.8 2.0 2.2 | 4 1 2 2 1 4 1 2 4 1 1 1 3 2 2 2 1 2 1 1 1 1 2 4 2 2 2 2 |

Kogure&Matsukura (2010b) calculated the critical wave height *H*_{wc} at which wave-inducedcollapse occurs, as follows: (1) the value of *σ*_{tgmax} + *S*_{t} was calculated by substituting the data for each block and cliff into Eq. 15 for H_{q}-type, or Eq. 16 for H_{t}-type; (2) Eq. 14 giving the wave pressure was substituted into Eqs. 6 or 7 or 8 (H_{q}-type), or Eqs. 10 or 11 or 12 (H_{t}-type), to calculate the value of *σ*_{twmax}; (3) once the data for each block and cliff except *H*_{w} had been substituted into those equations, *H*_{wc} was determined as the critical wave height, so that the value of *σ*_{twmax} matched *σ*_{tgmax} + *S*_{t}. The value of *H*_{wc} shows the minimum wave height required to induce collapse for each block-and-cliff combination. Table 3 shows the calculated values of *H*_{wc}, with reference to the relationship between wave height and cliff height (Fig. 7). The resulting values of *H*_{wc} are 0.5–76.0 m. Kogure & Matsukura (2010b) argued that *H*_{wc} = 76.0 m for H_{q}-13 is not a realistic value, and the analysis therefore excludes H_{q}-13.

### 4.2. Distribution of wave-provided blocks

Todistinguish between blocks produced by gravitational or wave-induced collapse,the blocks and the cliffs were classified into Cases 1–4 in Fig.7 according to the relationshipbetween *H*^{*} and *H*_{wc} (Table 3). Blocks classified as Case 1 appear to have beenproduced by gravitational collapse, because the waves cannotreach a notched roof in this case (Fig.7a). Blocks classified into Cases 2–4 may involve collapse due to waves (Fig. 7b, 7c and 7d). Especially, blocks classified as Case 4derive from wave-induced collapse. The cliffs H_{t}-C1 and C2 were classified as Case 2.

The blocks broken off by tsunamis were plotted on an air photograph of Kuro-shima, together with blocks fallen due to gravity (Fig. 10). The tsunami-induced blocks are concentrated in the south-eastern and south-western coasts, whereas the gravity-caused blocks are distributed evenly along the coastlines.

The distance from the coast to the offshore reef edge is about 800 m in the south-eastern area, the farthest offshore in Kuro-shima and indicating that the fringing reefs are highly developed. These fringing reefs are cut by a fault running NW-SE which makes them discontinuous. Tsunamis must therefore have invaded the south-eastern coastal area through this gap, giving rise to the cliff collapses. In the south-western area, the distance between the coastline and reef edge is about 200 m, the smallest value in Kuro-shima. Additionally, the reef crest is poorly developed and its width is the narrowest in the island. Reduction of the height of tsunamis by these fringing reefs must have been inadequate to prevent the collapses.

### 4.3. Effects of fringing reefs in reducing the height of tsunamis

Fig. 10 indicates that there are two factors affecting the height of tsunamis invading these coasts. One factor is the distance between the coastlines and reef edge; the other is continuity of the fringing reefs and their development.

Comparison of the distribution of the blocks between the southern and south-western areas clearly shows the effect of distance from the coastlines to the reef edge. For all blocks classified as Case 4 (H_{q}-7,8,10,11,14,26,28,29,34,H_{t}-2 and 17), the average value of *H*_{wc} is 6.6 m. It follows that tsunamis having height approximately 6.6 m must have attacked the south-eastern and south-western coasts, causing collapse of the cliffs. In contrast, the average value of *H*_{wc} is 2.1 m for the cliffs (H_{t}-C1 and C2) on the south coast where the distance between the coastline and reef edge is about 700 m (see the white arrow in Fig. 10). This implies that H_{t}-C1 and C2 have never experienced tsunami(s) higher than 2.1 m, and have not yet fallen even though the heights of tsunamis are almost the same everywhere before the reef crest around Kuro-shima. We infer that the heights of tsunamis reaching the coasts decrease with increasing distance between the coastlines and the edge of the offshore reefs. The height of the reef crest may also affect the tsunami height, but we do not consider this further here.

Fringing reefs do not necessarily act as an effective barrier against invading tsunamis if fringing reefs develop discontinuously. Some wave-induced blocks are seen at south-eastern coasts although the distance between the coastline and offshore reef edge is the greatest in Kuro-shima. This appears to be due to the cutting of the fringing reefs by the fault running in the NW-SE direction. Surging tsunamis from the Pacific ocean could be concentrated at the gap. This phenomenon is often referred to as the “channelling effect”. Overall, reefs will always absorb some of the wave energy, but, by channelling the water masses, the concentrated impact could be greater on coastal stretches with nearby reefs (Cochard et al., 2008). Our observation is consistent with records by Fernando et al. (2005) and Marris (2005), who reported the effects of channelling on the heights of tsunamis in the 2004 tsunami in Sri Lanka. Also, the height of tsunamis would be amplified, not reduced, during propagation through narrow paths or patchy reefs in the Great Barrier Reef (Nott, 1997).

## 5. Conclusion

This study showed stability analysis models to evaluate the cliff collapses due to extreme waves. We determined the distribution of the blocks caused by cliff collapses due to tsunamis, by means of stability analysis. Comparison of the distributions of blocks caused by tsunamis and by gravitational processes shows that two factors influence the height of tsunamis that reach the coasts. One factor is the distance between the coastline and reef edge. The heights of tsunamis reaching coasts decrease with increasing distance between the coastline and offshore reef edge. The other factor is continuity in the fringing reefs and their development. The height of tsunamis invading through the coral reef must have been constant or amplified by deep and narrow gaps, known as the channelling effect. Tsunamis which passed through the gap were able to reach the coasts without any loss of height and induce collapses of coastal cliffs. Therefore, broad and continuous “modern” fringing reefs may act as an effective barrier against tsunamis to collapse coastal cliffs, i.e., “ancient” fringing reefs.