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A joint distribution of significant wave height and characteristic surf parameter
Coastal Engineering 57 (2010) 948–952
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Coastal Engineering j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / c o a s t a l e n g
Technical note
A joint distribution of significant wave height and characteristic surf parameter Dag Myrhaug a,⁎, Sébastien Fouques b a b
Department of Marine Technology, Norwegian University of Science and Technology, NO-7491 Trondheim, Norway Norwegian Marine Technology Research Institute, NO-7450 Trondheim, Norway
a r t i c l e
i n f o
Article history: Received 1 December 2009 Received in revised form 15 February 2010 Accepted 4 May 2010 Available online 12 June 2010 Keywords: Significant wave height Characteristic surf parameter Joint distribution Return period contour lines Surf zone processes Design of coastal structures
a b s t r a c t The paper provides a joint distribution of significant wave height and characteristic surf parameter. The characteristic surf parameter is given by the ratio between the slope of a beach or a structure and the square root of the characteristic wave steepness in deep water defined in terms of the significant wave height and the spectral peak period. The characteristic surf parameter is used to characterize surf zone processes and is relevant for e.g. wave run-up on beaches and coastal structures. The paper presents statistical properties of the wave parameters as well as an example of results corresponding to typical field conditions. © 2010 Elsevier B.V. All rights reserved.
1. Introduction The surf parameter, also often referred to as the surf similarity parameter or the Iribarren number, is used to characterize surf zone processes. It is given by the ratio between the slope of a beach or a structure and the square root of the wave steepness in deep water as introduced by Iribarren and Nogales (1949) and used later by Battjes (1974). Shallow water regions where waves break are referred to as the surf zone, and the different breakers on slopes are defined and classified in terms of the surf parameter. It also appears that the surf parameter enters in many empirical and theoretical models for wave-induced phenomena in the surf zone. The breaking of waves is associated with large loss of energy. Within the surf zone along beaches the wave energy flux from offshore is dissipated into turbulence and heat, and consequently the wave height decreases towards the shoreline. Wavebreaking also results in strong currents along the shoreline and thereby affects the nearshore circulation. The high intensity of turbulence caused by wave-breaking is also responsible for the intense sediment transport in the surf zone. Wave run-up on beaches and coastal structures such as e. g. breakwaters, seawalls and artificial reefs are characterized by using the surf parameter. The surf parameter is commonly defined in terms of individual wave parameters. However, a characteristic surf parameter defined in terms of sea state parameters is also used; see e.g. Herbich (1990) and Silvester and Hsu (1997). Recently Mase et al. (2004) used such a surf parameter to estimate the significant value of the wave run-up on coastal structures. ⁎ Corresponding author. E-mail addresses: [email protected] (D. Myrhaug), [email protected] (S. Fouques). 0378-3839/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.coastaleng.2010.05.001
The purpose of this study is to provide a simple analytical tool which can be used to give estimates of maximum run-up heights on beaches and coastal structures. This is obtained by providing a joint distribution of significant wave height and characteristic surf parameter by a transformation of a (HS, TP) joint distribution. As an example the bivariate distribution is obtained by using the joint distribution of HS and TP proposed by Moan et al. (2005). Statistical properties of the wave parameters are presented. An example of calculating the significant value of the wave runup on seawalls near shorelines corresponding to typical field conditions is also provided to demonstrate the application of the method. 2. Background The characteristic surf parameter based on sea state parameters is pffiffiffiffiffi defined as ξP = m = sP where m=tan α is the slope with an angle α with the horizontal, sP =2πHS/(gT2P) is the characteristic wave steepness of a sea state in deep water where g is the acceleration of gravity. Thus this surf parameter is defined in terms of the sea state parameters in deep water. Different parametric models of the joint probability density function (pdf) of HS and TP, or HS and TZ (=mean zero-crossing wave period), are given in the literature. Some examples are: Haver (1985), Moan et al. (2005) for HS and TP; Mathiesen and Bitner-Gregersen (1990), Bitner-Gregersen and Guedes Soares (2007) for HS and TZ. In this paper the joint distribution of HS and ξP is obtained by using the joint pdf of HS and TP proposed by Moan et al. (2005) (i.e. given in Eqs. (1)–(7)). This pdf was obtained as a best fit to data obtained from wave measurements made in the Northern North Sea during a 29 year period. The joint pdf was obtained as pðHS ; TP Þ = pðTP jHS ÞpðHS Þ
ð1Þ
D. Myrhaug, S. Fouques / Coastal Engineering 57 (2010) 948–952
949
where p(HS) is the marginal pdf of HS given by the following combined lognormal and Weibull distributions (this type of distribution was first suggested by Haver (1985))
pðHS Þ =
8 " # > ðlnHS −θÞ2 1 > > > pffiffiffiffiffiffi exp − ; HS ≤3:25m > > 2κ2 < 2πκHS β > β−1 > > HS HS > > exp − ; HS N 3:25m β > : ζβ ζ
ð2Þ
Here θ and κ2are the mean value and the variance, respectively, of lnHS, given by 2
θ = 0:801; κ = 0:371
ð3Þ
and ζ, β are the Weibull parameters given by ζ = 2:713;
β = 1:531:
Fig. 1. P(HS) versus HS in Weibull-scale.
ð4Þ 2
p(TP|HS) is the conditional pdf of TP given HS, given by the following lognormal distribution " # 1 ðlnTP −μ Þ2 pðTP jHS Þ = pffiffiffiffiffiffi exp − 2σ 2 2πσTP
ð5Þ
where μ and σ2 are the mean value and the variance, respectively, of lnTP, given by a
μ = a1 + a2 HS 3 ða1 ; a2 ; a3 Þ = ð1:780; 0:288; 0:474Þ 2
b H
σ = b1 + b2 e 3 S ðb1 ; b2 ; b3 Þ = ð0:001; 0:097; −0:255Þ
ð6Þ
ð7Þ
σˆξ = σ
2
ð11Þ
where μ and σ2 are given in Eqs. (6) and (7), respectively. Fig. 1 shows the cumulative distribution function (cdf) of HS in Weibull-scale, corresponding to the results presented in Moan et al. (2005, Fig. 1) which also included the data upon which the model is based. Fig. 2 shows the cdf of ξ̂P in Weibull-scale. Fig. 3 shows the conditional cdf of ξ̂P given HS in Weibull-scale for HS = 3, 6, 9 and 12 m, by utilizing that " # lnξˆ P −μξˆ ˆ P ξP jHS = Φ σξˆ
The values of the parameters in Eqs. (3), (4), (6) and (7) have been provided by Gao (2007).
where Φ is the standard Gaussian cdf, i.e.
3. Joint distribution of HS and ξP
1 −t 2 = 2 dt ΦðνÞ = pffiffiffiffiffiffi ∫ e 2π −∞
ð12Þ
ν
ð13Þ
3.1. Joint distribution of HS and ξP̂ = ξP/m The joint pdf of HS and ξP̂ = ξP/m is obtained from Eq. (1) by a change of variables from (HS, Tp) to (HS, ξP̂ ), which takes the form p HS ; ξˆ P = p ξˆ P jHS pðHS Þ
ð8Þ
From Fig. 3 it is observed that for a given value of HS the probability of exceeding e.g. ξ̂P = 7 decreases as HS increases. Note that this is a typical value of ξ̂P, i.e. for m = 0.1, this gives ξP = 0.7, which is close to the value used in the example in Section 3.4.
where p(HS) is given in Eqs. (2)–(4). It should be noted that this change of variables only affects p(TP|HS) since TP = [HS/(g/2π)]1/2ξp̂ , yielding a lognormal pdf for ξp̂ given HS in the form (by using the Jacobian |dTp/dξp̂ |) 2 2 3 lnξˆ P −μ ξˆ 7 1 6 exp 4− p ξˆ P jHS = pffiffiffiffiffiffi 5 2 2πσξˆ ξˆ P 2σξˆ
ð9Þ
2
where μξ̂ and σξ̂ are the mean value and the variance, respectively, of ln ξP̂ , given by 0
1
B C B C 1 C μξˆ = ln B BsffiffiffiffiffiffiffiffiffiffiffiffiffiC + μ @ HS A g = 2π
ð10Þ Fig. 2. P(ξP̂ ) versus ξP̂ in Weibull-scale.
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Fig. 5. σξ̂|HS versus HS.
Fig. 3. P(ξP̂ |HS) versus ξ̂P in Weibull-scale for HS = 3 m(blue), 6 m(red), 9 m (green) and 12 m (black).
Fig. 4 shows the expected value of ξP̂ given HS, where (Bury, 1975) h i 1 2 E ξˆ P j HS = exp μ ˆ + σξˆ ξ 2
ð14Þ
It appears that E[ξP̂ |HS] decreases as HS increases. Fig. 5 shows the standard deviation of ξP̂ given HS, σξ ̂|HS, where (Bury, 1975) 2 σ 2 2 σξˆ j H = exp 2μξˆ + σˆξ e ˆξ −1 S
ð15Þ
It appears that σξ ̂|HS decreases as HS increases. Fig. 6 shows the isocontours of p(HS, ξ̂P). It appears that the peak of the joint pdf is located at HS ≈ 2 m and ξP̂ ≈ 7.5. More features of p(HS, ξP̂ ) will be discussed in the Section 3.3 where the return period contour lines are presented. 3.2. Joint distribution of HS and ξP The joint pdf of HS and ξP is obtained from Eq. (8) by a change of variables from (HS, ξP̂ ) to (HS, ξP), which takes the form pðHS ; ξP Þ = pðξP jHS ÞpðHS Þ
Fig. 4. E[ξP̂ /HS] versus HS.
ð16Þ
where p(HS) is given in Eqs. (2)–(4). This change of variables only affects p(ξ̂P|HS) in Eq. (8) since ξP̂ = ξP/m, yielding a lognormal pdf for ξP given HS in the form 2 2 3 ln ξP −μξ 7 1 6 exp 4− pðξP jHS Þ = pffiffiffiffiffiffi 5 2σξ2 2πσξ ξP
ð17Þ
where μξ and σ2ξ are the mean value and the variance, respectively, of ln ξP, given by 0
1
B C B m C C μξ = ln B BsffiffiffiffiffiffiffiffiffiffiffiffiffiC + μ = ln m + μξˆ @ HS A g = 2π 2
σξ = σ
2
ð18Þ
ð19Þ
where μ and σ2 are given in Eqs. (6) and (7), respectively. No additional figures will be presented for the statistics of ξP because they are similar to those shown for ξP̂ in Section 3.1. It follows that P(ξP) = P(ξ̂P), the latter is shown in Fig. 2, and that P(ξP|HS) = P(ξ̂P|HS), where P(ξ̂P|HS) is shown in Fig. 3. Moreover, E[ξP|HS] = mE[ξ̂P|HS], where E[ξ̂P|HS] is shown in Fig. 4, and σξ|HS = mσξ̂|HS, where σξ̂|HS is shown in Fig. 5.
Fig. 6. Isocontours of p(HS, ξP̂̂ ).
D. Myrhaug, S. Fouques / Coastal Engineering 57 (2010) 948–952
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3.3. Return period contour lines The joint distribution given in Section 3.1 can be used to determine the n-year return period contour lines. Here the contour lines are obtained using the so-called IFORM approach described in DNV (2007) by assuming a sea state duration of 3 hours. This approach is based on a transformation of the joint cdf of the concerned parameters to the standard Gaussian cdf, from which the return period contour lines are determined as circles. As a reference case, Fig. 7 shows the 1-, 10- and 100-year return period contour lines of HS and Tp. More results based on the same joint pdf model of HS and TP are given in Moan et al. (2005). Fig. 8 shows the 1-, 10- and 100-year return period contour lines of HS and ξP̂ . The contour plot shows two “extreme” regions; e.g. for the 100-year contour line, the first one for high values of HS(HS ≳ 10 m) and low values of ξ̂P(4 ≲ ξ̂P ≲ 8), and the second one for high values of ξ̂P(ξ̂P ≳ 24) and low values of HS(HS ≲ 2 m). It should be noted that the information concerning all the contour lines in these two regions pffiffiffiffiffiffi appears clearly from the (HS, TP) contour plot, since ξˆ P ∼TP = HS . 3.4. Example This example is included to demonstrate the application of the results. The significant value of the wave run-up on sea walls near shorelines, RS, according to the CIRIA/CUR manual (1991) (see e.g. Mase et al. (2004)) is given by RS = 1:35ξP HS
for
RS = 3:00−0:15ξP HS
0 <ξP ≤2
for
2<ξP < 12
ð20Þ
Fig. 8. 1-, 10- and 100-years contour lines of HS and ξ̂̂P; symbols as in Fig. 7.
Thus the corresponding standard deviation of the wave run-up is σR = 1.35 · 0.072 · 7.5 m = 0.73 m. An estimate of the extreme wave run-up can be obtained by using the results in the contour plots shown in Figs. 7 and 8. For example, the extreme wave run-up corresponding to the 100-year contour line is then obtained by searching along the (HS, ξP̂ ) contour in Fig. 8 for the condition giving maximum wave run-up according to Eqs. (20) and (21). It should be noted that use of the (HS, TP) contours will give the same result, since the transformations involved are one-to-one. However, these aspects will not be elaborated further in this example.
ð21Þ
The given flow conditions are: • Significant wave height in deep water, HS = 7.5 m • Slope of seawall, m = 0.1 Here an estimate of RS is exemplified by using the expected value of ξP given HS, i.e. by using E[ξP|HS] = mE[ξP̂ |HS] where E[ξ̂P|HS] is given in Eq. (14). Now it follows by substituting Eqs. (10), (11), (6) and (7) in Eq. (14) that E[ξP|HS = 7.5 m] = 0.576, which corresponds to the value obtained from the curve shown in Fig. 4. Thus the wave run-up is obtained from Eq. (20) as RS = 1.35 · 0.576 · 7.5 m = 5.83 m. Moreover, from Eq. (15) it follows that σξ|HS = 7.5 m = 0.072, which corresponds to the value obtained from the curve shown in Fig. 5.
3.5. Comments Finally some comments are given on the present method versus common practice in coastal engineering. For assessment of e.g. maximum run-up height on beaches and coastal structures common practice would be to start from available data on joint statistics of significant wave height and characteristic wave period (TP or TZ) within directional sectors at a nearby offshore location; then to transform these by using a wave simulation model to obtain joint statistics of significant wave height and characteristic wave period at the relevant near-shore location; and to use this information as input for the assessment of wave run-up height. Alternatively, the present method provides a simple analytical tool which might serve to give first estimates of maximum run-up height on beaches and coastal structures. Such estimates are useful for comparison and verification of more complete computational methods, as well as in situations when time and access to computational resources are limited (for example under field conditions). Moreover, it also serves as a first inexpensive estimate of the quantities of interest before eventually applying more work-intensive computational tools. Although the present results are valid for the specifically chosen joint distribution of HS and TP, it gives an analytically based method which can be used for other joint distributions of HS and TP (or TZ). 4. Summary
Fig. 7. 1-, 10- and 100-years contour lines of HS and TP; 1-year (blue), 10-year (green), 100-year (red).
A joint distribution of significant wave height and characteristic surf parameter is provided by a transformation of a (HS, TP) joint distribution. As an example this is demonstrated by using the joint pdf model of HS and TP given by Moan et al. (2005) representing 29 year of wave data in the Northern North Sea. Statistical properties of the wave parameters including return period contour lines, as well as an example of results corresponding to typical field conditions are presented.
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References Battjes, J.A., 1974. Surf similarity. Proceedings 14th Int. Conf. on Coastal Engineering, Vol. 1. ASCE, New York, pp. 466–479. Bitner-Gregersen, E., Guedes Soares, C., 2007. Uncertainty of average wave steepness prediction from global wave databases. Proc. of MARSTRUCT, Glasgow, UK, pp. 3–10. Bury, K.V., 1975. Statistical Models in Applied Science. John Wiley & Sons, New York. CIRIA/CUR, 1991. Manual on the use of rock in coastal and shoreline engineering. : Special publication, 83. Construction Industry Research and Information Association, London (607 pp). DNV, 2007. Recommended Practice DNV-RP-C205 Environmental Conditions and Environmental Loads. (April 2007). Gao, Z. 2007. Private communication.
Haver, S., 1985. Wave climate off northern Norway. Applied Ocean Research 7 (2), 85–92. Herbich, J.B., 1990. Handbook of Coastal and Ocean Engineering. Volume 1. Wave Phenomena and Coastal Structures. Gulf Publishing Co, Houston, Texas. Iribarren, C.R., Nogales, C., 1949. Protection des ports. Sect. 2. Comm. 4, 17th Int. Nav. Cong. Lisbon, pp. 31–80. Mase, H., Miyahira, A., Hedges, T.S., 2004. Random wave runup on seawalls near shorelines with and without artificial reefs. Coastal Engineering Journal, JSCE 46 (3), 247–268. Mathiesen, J., Bitner-Gregersen, E., 1990. Joint distributions for significant wave height and wave zero-up-crossing period. Applied Ocean Research 12 (2), 93–103. Moan, T., Gao, Z., Ayala-Uraga, E., 2005. Uncertainty of wave-induced response of marine structures due to long-term variation of extratropical wave conditions. Marin Structures 18 (4), 359–382. Silvester, R., Hsu, J.R.C., 1997. Coastal Stabilization. World Scientific, Singapore.
Contents lists available at ScienceDirect
Coastal Engineering j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / c o a s t a l e n g
Technical note
A joint distribution of significant wave height and characteristic surf parameter Dag Myrhaug a,⁎, Sébastien Fouques b a b
Department of Marine Technology, Norwegian University of Science and Technology, NO-7491 Trondheim, Norway Norwegian Marine Technology Research Institute, NO-7450 Trondheim, Norway
a r t i c l e
i n f o
Article history: Received 1 December 2009 Received in revised form 15 February 2010 Accepted 4 May 2010 Available online 12 June 2010 Keywords: Significant wave height Characteristic surf parameter Joint distribution Return period contour lines Surf zone processes Design of coastal structures
a b s t r a c t The paper provides a joint distribution of significant wave height and characteristic surf parameter. The characteristic surf parameter is given by the ratio between the slope of a beach or a structure and the square root of the characteristic wave steepness in deep water defined in terms of the significant wave height and the spectral peak period. The characteristic surf parameter is used to characterize surf zone processes and is relevant for e.g. wave run-up on beaches and coastal structures. The paper presents statistical properties of the wave parameters as well as an example of results corresponding to typical field conditions. © 2010 Elsevier B.V. All rights reserved.
1. Introduction The surf parameter, also often referred to as the surf similarity parameter or the Iribarren number, is used to characterize surf zone processes. It is given by the ratio between the slope of a beach or a structure and the square root of the wave steepness in deep water as introduced by Iribarren and Nogales (1949) and used later by Battjes (1974). Shallow water regions where waves break are referred to as the surf zone, and the different breakers on slopes are defined and classified in terms of the surf parameter. It also appears that the surf parameter enters in many empirical and theoretical models for wave-induced phenomena in the surf zone. The breaking of waves is associated with large loss of energy. Within the surf zone along beaches the wave energy flux from offshore is dissipated into turbulence and heat, and consequently the wave height decreases towards the shoreline. Wavebreaking also results in strong currents along the shoreline and thereby affects the nearshore circulation. The high intensity of turbulence caused by wave-breaking is also responsible for the intense sediment transport in the surf zone. Wave run-up on beaches and coastal structures such as e. g. breakwaters, seawalls and artificial reefs are characterized by using the surf parameter. The surf parameter is commonly defined in terms of individual wave parameters. However, a characteristic surf parameter defined in terms of sea state parameters is also used; see e.g. Herbich (1990) and Silvester and Hsu (1997). Recently Mase et al. (2004) used such a surf parameter to estimate the significant value of the wave run-up on coastal structures. ⁎ Corresponding author. E-mail addresses: [email protected] (D. Myrhaug), [email protected] (S. Fouques). 0378-3839/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.coastaleng.2010.05.001
The purpose of this study is to provide a simple analytical tool which can be used to give estimates of maximum run-up heights on beaches and coastal structures. This is obtained by providing a joint distribution of significant wave height and characteristic surf parameter by a transformation of a (HS, TP) joint distribution. As an example the bivariate distribution is obtained by using the joint distribution of HS and TP proposed by Moan et al. (2005). Statistical properties of the wave parameters are presented. An example of calculating the significant value of the wave runup on seawalls near shorelines corresponding to typical field conditions is also provided to demonstrate the application of the method. 2. Background The characteristic surf parameter based on sea state parameters is pffiffiffiffiffi defined as ξP = m = sP where m=tan α is the slope with an angle α with the horizontal, sP =2πHS/(gT2P) is the characteristic wave steepness of a sea state in deep water where g is the acceleration of gravity. Thus this surf parameter is defined in terms of the sea state parameters in deep water. Different parametric models of the joint probability density function (pdf) of HS and TP, or HS and TZ (=mean zero-crossing wave period), are given in the literature. Some examples are: Haver (1985), Moan et al. (2005) for HS and TP; Mathiesen and Bitner-Gregersen (1990), Bitner-Gregersen and Guedes Soares (2007) for HS and TZ. In this paper the joint distribution of HS and ξP is obtained by using the joint pdf of HS and TP proposed by Moan et al. (2005) (i.e. given in Eqs. (1)–(7)). This pdf was obtained as a best fit to data obtained from wave measurements made in the Northern North Sea during a 29 year period. The joint pdf was obtained as pðHS ; TP Þ = pðTP jHS ÞpðHS Þ
ð1Þ
D. Myrhaug, S. Fouques / Coastal Engineering 57 (2010) 948–952
949
where p(HS) is the marginal pdf of HS given by the following combined lognormal and Weibull distributions (this type of distribution was first suggested by Haver (1985))
pðHS Þ =
8 " # > ðlnHS −θÞ2 1 > > > pffiffiffiffiffiffi exp − ; HS ≤3:25m > > 2κ2 < 2πκHS β > β−1 > > HS HS > > exp − ; HS N 3:25m β > : ζβ ζ
ð2Þ
Here θ and κ2are the mean value and the variance, respectively, of lnHS, given by 2
θ = 0:801; κ = 0:371
ð3Þ
and ζ, β are the Weibull parameters given by ζ = 2:713;
β = 1:531:
Fig. 1. P(HS) versus HS in Weibull-scale.
ð4Þ 2
p(TP|HS) is the conditional pdf of TP given HS, given by the following lognormal distribution " # 1 ðlnTP −μ Þ2 pðTP jHS Þ = pffiffiffiffiffiffi exp − 2σ 2 2πσTP
ð5Þ
where μ and σ2 are the mean value and the variance, respectively, of lnTP, given by a
μ = a1 + a2 HS 3 ða1 ; a2 ; a3 Þ = ð1:780; 0:288; 0:474Þ 2
b H
σ = b1 + b2 e 3 S ðb1 ; b2 ; b3 Þ = ð0:001; 0:097; −0:255Þ
ð6Þ
ð7Þ
σˆξ = σ
2
ð11Þ
where μ and σ2 are given in Eqs. (6) and (7), respectively. Fig. 1 shows the cumulative distribution function (cdf) of HS in Weibull-scale, corresponding to the results presented in Moan et al. (2005, Fig. 1) which also included the data upon which the model is based. Fig. 2 shows the cdf of ξ̂P in Weibull-scale. Fig. 3 shows the conditional cdf of ξ̂P given HS in Weibull-scale for HS = 3, 6, 9 and 12 m, by utilizing that " # lnξˆ P −μξˆ ˆ P ξP jHS = Φ σξˆ
The values of the parameters in Eqs. (3), (4), (6) and (7) have been provided by Gao (2007).
where Φ is the standard Gaussian cdf, i.e.
3. Joint distribution of HS and ξP
1 −t 2 = 2 dt ΦðνÞ = pffiffiffiffiffiffi ∫ e 2π −∞
ð12Þ
ν
ð13Þ
3.1. Joint distribution of HS and ξP̂ = ξP/m The joint pdf of HS and ξP̂ = ξP/m is obtained from Eq. (1) by a change of variables from (HS, Tp) to (HS, ξP̂ ), which takes the form p HS ; ξˆ P = p ξˆ P jHS pðHS Þ
ð8Þ
From Fig. 3 it is observed that for a given value of HS the probability of exceeding e.g. ξ̂P = 7 decreases as HS increases. Note that this is a typical value of ξ̂P, i.e. for m = 0.1, this gives ξP = 0.7, which is close to the value used in the example in Section 3.4.
where p(HS) is given in Eqs. (2)–(4). It should be noted that this change of variables only affects p(TP|HS) since TP = [HS/(g/2π)]1/2ξp̂ , yielding a lognormal pdf for ξp̂ given HS in the form (by using the Jacobian |dTp/dξp̂ |) 2 2 3 lnξˆ P −μ ξˆ 7 1 6 exp 4− p ξˆ P jHS = pffiffiffiffiffiffi 5 2 2πσξˆ ξˆ P 2σξˆ
ð9Þ
2
where μξ̂ and σξ̂ are the mean value and the variance, respectively, of ln ξP̂ , given by 0
1
B C B C 1 C μξˆ = ln B BsffiffiffiffiffiffiffiffiffiffiffiffiffiC + μ @ HS A g = 2π
ð10Þ Fig. 2. P(ξP̂ ) versus ξP̂ in Weibull-scale.
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D. Myrhaug, S. Fouques / Coastal Engineering 57 (2010) 948–952
Fig. 5. σξ̂|HS versus HS.
Fig. 3. P(ξP̂ |HS) versus ξ̂P in Weibull-scale for HS = 3 m(blue), 6 m(red), 9 m (green) and 12 m (black).
Fig. 4 shows the expected value of ξP̂ given HS, where (Bury, 1975) h i 1 2 E ξˆ P j HS = exp μ ˆ + σξˆ ξ 2
ð14Þ
It appears that E[ξP̂ |HS] decreases as HS increases. Fig. 5 shows the standard deviation of ξP̂ given HS, σξ ̂|HS, where (Bury, 1975) 2 σ 2 2 σξˆ j H = exp 2μξˆ + σˆξ e ˆξ −1 S
ð15Þ
It appears that σξ ̂|HS decreases as HS increases. Fig. 6 shows the isocontours of p(HS, ξ̂P). It appears that the peak of the joint pdf is located at HS ≈ 2 m and ξP̂ ≈ 7.5. More features of p(HS, ξP̂ ) will be discussed in the Section 3.3 where the return period contour lines are presented. 3.2. Joint distribution of HS and ξP The joint pdf of HS and ξP is obtained from Eq. (8) by a change of variables from (HS, ξP̂ ) to (HS, ξP), which takes the form pðHS ; ξP Þ = pðξP jHS ÞpðHS Þ
Fig. 4. E[ξP̂ /HS] versus HS.
ð16Þ
where p(HS) is given in Eqs. (2)–(4). This change of variables only affects p(ξ̂P|HS) in Eq. (8) since ξP̂ = ξP/m, yielding a lognormal pdf for ξP given HS in the form 2 2 3 ln ξP −μξ 7 1 6 exp 4− pðξP jHS Þ = pffiffiffiffiffiffi 5 2σξ2 2πσξ ξP
ð17Þ
where μξ and σ2ξ are the mean value and the variance, respectively, of ln ξP, given by 0
1
B C B m C C μξ = ln B BsffiffiffiffiffiffiffiffiffiffiffiffiffiC + μ = ln m + μξˆ @ HS A g = 2π 2
σξ = σ
2
ð18Þ
ð19Þ
where μ and σ2 are given in Eqs. (6) and (7), respectively. No additional figures will be presented for the statistics of ξP because they are similar to those shown for ξP̂ in Section 3.1. It follows that P(ξP) = P(ξ̂P), the latter is shown in Fig. 2, and that P(ξP|HS) = P(ξ̂P|HS), where P(ξ̂P|HS) is shown in Fig. 3. Moreover, E[ξP|HS] = mE[ξ̂P|HS], where E[ξ̂P|HS] is shown in Fig. 4, and σξ|HS = mσξ̂|HS, where σξ̂|HS is shown in Fig. 5.
Fig. 6. Isocontours of p(HS, ξP̂̂ ).
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3.3. Return period contour lines The joint distribution given in Section 3.1 can be used to determine the n-year return period contour lines. Here the contour lines are obtained using the so-called IFORM approach described in DNV (2007) by assuming a sea state duration of 3 hours. This approach is based on a transformation of the joint cdf of the concerned parameters to the standard Gaussian cdf, from which the return period contour lines are determined as circles. As a reference case, Fig. 7 shows the 1-, 10- and 100-year return period contour lines of HS and Tp. More results based on the same joint pdf model of HS and TP are given in Moan et al. (2005). Fig. 8 shows the 1-, 10- and 100-year return period contour lines of HS and ξP̂ . The contour plot shows two “extreme” regions; e.g. for the 100-year contour line, the first one for high values of HS(HS ≳ 10 m) and low values of ξ̂P(4 ≲ ξ̂P ≲ 8), and the second one for high values of ξ̂P(ξ̂P ≳ 24) and low values of HS(HS ≲ 2 m). It should be noted that the information concerning all the contour lines in these two regions pffiffiffiffiffiffi appears clearly from the (HS, TP) contour plot, since ξˆ P ∼TP = HS . 3.4. Example This example is included to demonstrate the application of the results. The significant value of the wave run-up on sea walls near shorelines, RS, according to the CIRIA/CUR manual (1991) (see e.g. Mase et al. (2004)) is given by RS = 1:35ξP HS
for
RS = 3:00−0:15ξP HS
0 <ξP ≤2
for
2<ξP < 12
ð20Þ
Fig. 8. 1-, 10- and 100-years contour lines of HS and ξ̂̂P; symbols as in Fig. 7.
Thus the corresponding standard deviation of the wave run-up is σR = 1.35 · 0.072 · 7.5 m = 0.73 m. An estimate of the extreme wave run-up can be obtained by using the results in the contour plots shown in Figs. 7 and 8. For example, the extreme wave run-up corresponding to the 100-year contour line is then obtained by searching along the (HS, ξP̂ ) contour in Fig. 8 for the condition giving maximum wave run-up according to Eqs. (20) and (21). It should be noted that use of the (HS, TP) contours will give the same result, since the transformations involved are one-to-one. However, these aspects will not be elaborated further in this example.
ð21Þ
The given flow conditions are: • Significant wave height in deep water, HS = 7.5 m • Slope of seawall, m = 0.1 Here an estimate of RS is exemplified by using the expected value of ξP given HS, i.e. by using E[ξP|HS] = mE[ξP̂ |HS] where E[ξ̂P|HS] is given in Eq. (14). Now it follows by substituting Eqs. (10), (11), (6) and (7) in Eq. (14) that E[ξP|HS = 7.5 m] = 0.576, which corresponds to the value obtained from the curve shown in Fig. 4. Thus the wave run-up is obtained from Eq. (20) as RS = 1.35 · 0.576 · 7.5 m = 5.83 m. Moreover, from Eq. (15) it follows that σξ|HS = 7.5 m = 0.072, which corresponds to the value obtained from the curve shown in Fig. 5.
3.5. Comments Finally some comments are given on the present method versus common practice in coastal engineering. For assessment of e.g. maximum run-up height on beaches and coastal structures common practice would be to start from available data on joint statistics of significant wave height and characteristic wave period (TP or TZ) within directional sectors at a nearby offshore location; then to transform these by using a wave simulation model to obtain joint statistics of significant wave height and characteristic wave period at the relevant near-shore location; and to use this information as input for the assessment of wave run-up height. Alternatively, the present method provides a simple analytical tool which might serve to give first estimates of maximum run-up height on beaches and coastal structures. Such estimates are useful for comparison and verification of more complete computational methods, as well as in situations when time and access to computational resources are limited (for example under field conditions). Moreover, it also serves as a first inexpensive estimate of the quantities of interest before eventually applying more work-intensive computational tools. Although the present results are valid for the specifically chosen joint distribution of HS and TP, it gives an analytically based method which can be used for other joint distributions of HS and TP (or TZ). 4. Summary
Fig. 7. 1-, 10- and 100-years contour lines of HS and TP; 1-year (blue), 10-year (green), 100-year (red).
A joint distribution of significant wave height and characteristic surf parameter is provided by a transformation of a (HS, TP) joint distribution. As an example this is demonstrated by using the joint pdf model of HS and TP given by Moan et al. (2005) representing 29 year of wave data in the Northern North Sea. Statistical properties of the wave parameters including return period contour lines, as well as an example of results corresponding to typical field conditions are presented.
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Haver, S., 1985. Wave climate off northern Norway. Applied Ocean Research 7 (2), 85–92. Herbich, J.B., 1990. Handbook of Coastal and Ocean Engineering. Volume 1. Wave Phenomena and Coastal Structures. Gulf Publishing Co, Houston, Texas. Iribarren, C.R., Nogales, C., 1949. Protection des ports. Sect. 2. Comm. 4, 17th Int. Nav. Cong. Lisbon, pp. 31–80. Mase, H., Miyahira, A., Hedges, T.S., 2004. Random wave runup on seawalls near shorelines with and without artificial reefs. Coastal Engineering Journal, JSCE 46 (3), 247–268. Mathiesen, J., Bitner-Gregersen, E., 1990. Joint distributions for significant wave height and wave zero-up-crossing period. Applied Ocean Research 12 (2), 93–103. Moan, T., Gao, Z., Ayala-Uraga, E., 2005. Uncertainty of wave-induced response of marine structures due to long-term variation of extratropical wave conditions. Marin Structures 18 (4), 359–382. Silvester, R., Hsu, J.R.C., 1997. Coastal Stabilization. World Scientific, Singapore.