Ocean Dynamics DOI 10.1007/s10236-015-0869-z
Shoreline relaxation at pocket beaches Imen Turki 1,2 & Raul Medina 1 & Nabil Kakeh 1,3 & Mauricio González 1
Received: 8 February 2015 / Accepted: 14 August 2015 # Springer-Verlag Berlin Heidelberg 2015
Abstract A new physical concept of relaxation time is introduced in this research as the time required for the beach to dissipate its initial perturbation. This concept is investigated using a simple beach-evolution model of shoreline rotation at pocket beaches, based on the assumption that the instantaneous change of the shoreline plan-view shape depends on the long-term equilibrium plan-view shape. The expression of relaxation time is developed function of the energy conditions and the physical characteristics of the beach; it increases at longer beaches having coarse sediments and experiencing low-energy conditions. The relaxation time, calculated by the developed model, is validated by the shoreline observations extracted from video images at two artificially embayed beaches of Barcelona (NW Mediterranean) suffering from perturbations of sand movement and a nourishment project. This finding is promising to estimate the shoreline response and useful to improve our understanding of the dynamic of pocket beaches and their stability.
Keywords Equilibrium model . Shoreline rotation . Pocket beaches . Relaxation time Responsible Editor: Bruno Castelle * Imen Turki
[email protected] 1
Environmental Hydraulics Institute ‘IH Cantabria’, University of Cantabria, c/Isabel Torres 15, 39011 Santander, Spain
2
Present address: UMR CNRS 6143 Continental and Coastal Morphodynamics‘M2C’ University of Rouen, 76821 Mont-Saint-Aignan Cedex, France
3
Department of Applied Physics, Universitat Politècnica de Catalunya-Barcelona Tech, Barcelona, Spain
1 Introduction The better understanding of physical processes responsible for the beach responses to natural and human agents has become an urgent issue for scientists and engineers dealing with the coastline evolution. For example, beach nourishment or storm-driven waves are typical causes of beach changes and the time needed for the recovery from such events is a key variable useful to assess the coastal vulnerability and to make more informed decisions when dealing with coastal hazards and management. Many studies describe or attempt to predict shoreline erosion induced by natural events and/or human activities (e.g., Miller and Dean 2003; Ping et al. 2006; Callaghan et al. 2009; van Rijn 2009), but only a few studies have examined the time necessary for the beach to naturally return to its initial state and recover. The recovery of beaches has been investigated in previous research (e.g., Vousdoukas et al. 2011; Suanez et al. 2012). In terms of field observations, Wang et al. (2006) examined storm impact and post-storm recovery along a 200-km stretch of coast. They found that beach recovery began immediately after the storm and that within 90 days. More recently, Choowong et al. (2009) have shown that the coastline of Phang-nga requires 2 years to recover after the 2004 Indian Ocean tsunami and similar results were reported for the Khao Lak coast. Houser and Hamilton (2009) have used LIDAR data collected immediately following Hurricane Katrina in July 2006 (after almost 1 year of recovery) to quantify the dune recovery at the east of Pensacola Beach. They reported that beach recovery is related to a series of factors such as the offshore bathymetry and the pre-storm state of dunes. Similarly, the recovery of the Texan coast following a major hurricane has been extended up to 4 and 5 years (Morton et al. 1994). Gradual recovery of Barcelona beaches after abrupt modifications caused by storms has been investigated by Ojeda and Guillén (2008) and linked to episodes of beach
Ocean Dynamics
rotation. Corbella and Strech (2012) have studied 37 years of beach profile data on the east coast of South Africa and concluded that beach profiles need an average of 2 years to fully recover from major storms. Also, they indicate that recovery can be accelerated or retarded depending on the physical characteristics of the coastlines and the presence of beach structures. In the NW Mediterranean, the recovery from storm events and human activities of Barcelona embayed beaches was studied by Ojeda and Guillén (2008). They have demonstrated that storms are responsible for major changes in the configuration of Barcelona beaches, mainly a beach rotation caused by waves approaching obliquely to the coast. In terms of mathematical perspectives, some models have been developed to reproduce the beach recovery from storms and nourishment projects (e.g., Bramato et al. 2012; Rogers and Work 2006; Galofre et al. 1995). SBEACH (Larson and Kraus 1989) and XBeach (Roelvink et al. 2009) have been used to predict the erosive effect of storms and to reproduce beach recovery. A more successful approach is the one that uses long-term datasets to develop empirical models of beach response to storms and the subsequent recovery. For example, Davidson and Turner (2009) used a 6-year time series of shoreline position to calibrate a template model for seasonal to interannual beach profile evolution, in which the profile evolves toward a constantly changing template. The model has also simulated a recovery from a 10-day storm event with significant wave height up than 2.5 m between 20 and 30 days. Regarding nourishment projects, some models have been developed (Rogers and work 2006; Work and Dean 1995; Dean and Yoo 1992). Dean (1988) and Browder and Dean (2000) have also developed a simple analytical model under the assumption that an initially rectangular nourishment project alongshore spreading is essentially independent from wave direction and is symmetrical from the center of the shoreline. This model was applied and validated by Elko et al. (2005) using high-resolution video imagery to quantify the rapid evolution to the nourishment activity performed every 4–5 years at Upham beach on the west coast of Florida. They have found that 50 % of the nourished material was remaining after a period of 485 days compared to the 343 days obtained from an empirical estimation. Overall, the response of beaches to natural and anthropic perturbations depends on their physical parameters which should be considered to determine the response time scales. The present research focuses on the gradual response of pocket beaches and their relaxation after storms and maninduced activities. A new analytical formulation, derived from the equilibrium beach model (Turki et al. 2013b), is developed to estimate the time required for the beach to dissipate its initial perturbation. We define this time as the Brelaxation time.^ The paper presents five sections. Following the introduction, field observations of shoreline position and wave conditions at Barcelona beaches are described in Section 2. Then, the equilibrium beach evolution model is presented in its analytical form (Section 3). Section 4 develops the new
concept of relaxation time deriving the evolution model form. Finally, all findings are discussed and concluding remarks are presented in Sect. 5.
2 Data Two pocket beaches in the NW Mediterranean were considered in this work (Fig. 1): the longest beach Bogatell (600 m) with coarse sediments (D50=0.75 mm) and Somorrostro with a length of 400 m and a mean grain size of 0.45 mm. These beaches are continuously exposed to many humaninterventions such sand cleaning before the summer season and small-scale sand redistribution after storms. The shoreline changes are monitored using an Argus video system (Holman and Stanley 2007), located atop a building close to the beach at around 142 m high (Fig. 1a). The database of video images is provided by the Coastal Ocean Observatory (Institute of Marine Sciences, CSIC, Barcelona, Spain). The station comprises five cameras but only two of them (C1 and C5) were used in this study. Daily measurements of total shoreline movement and beach rotation were carried out at different cross-shore profiles (P1 to P10) of Bogatell and Somorrostro (Fig. 1b) during the periods of 2005–2007 with no human interventions (Turki et al. 2013a) and 2002–2004 marked by nourishment projects and sand movement (Ojeda and Guillén 2006). The timeaveraged images, the closest to the tidal level of +0.2 m extracted every 5 min from the REDMAR gauge located in the Barcelona harbor, were selected for the present analysis and used to detect manually the shoreline as the wet/dry interface. Shoreline rotation was studied at Barcelona beaches along a series of orthogonal profiles (from P1 to P10) spaced in time between 1 to 4 days depending on the availability of the video images (Fig. 1b). Under the assumption of the shoreline linear shape and the constant cross-shore profile, Turki et al. (2013a) have used the shoreline data to develop a simplified model which separates the overall shoreline movement into the contributions of rotation and translation. Hourly wave conditions (significant wave height Hs, peak period Tp, direction Dir) have been provided by a hindcast analysis for the period between 1991 and 2008 (Reguero et al. 2012). The wave rose in deep water (Fig. 2a) shows that the dominant directions range between ESE and WSW. Wave transformation has been modeled from the deep water to the breaking zone using the wave module (SP-Oluca) of the Coastal Management System (SMC) developed by the Environmental Hydraulics Institute IH Cantabria (University of Cantabria). SP-Oluca solves the parabolic approximation of the Mild Slope Equation and simulates random seas over irregular bottom bathymetry (Gonzalez and Medina 2007). The breaking wave conditions have been determined assuming that the breaking occurs when the ratio between wave height and water depth is 0.76 (Fig. 2b).
Ocean Dynamics
Fig. 1 a Study area; Barcelona city beaches, Spain. b Oblique video images from ARGUS station: Somorrostro and Bogatell are filmed by camera 1 and camera 5, respectively
3 Beach evolution model
The proposed methodology uses the beach evolution model, developed by Turki et al. (2013b), which assumes that the variation of the shoreline response is proportional to the difference between the instantaneous shoreline response R(t) and its final or equilibrium form R∞:
2013b). The instantaneous shoreline response R(t) increases exponentially with time and reaches 64 % of the equilibrium response R∞ at t=Ts. The main hypotheses of the model are the following: (1) the beach plan-form is linear and rotates around a pivotal point; (2) the alongshore variability in wave characteristics and sediment grain size are nonsignificant. The model determines the instantaneous shoreline response R(t) from Eq. 1 which is solved numerically for each sea state (Hb, Dirb, Tp) as shown by Eq. 2.
dR 1 ¼ ⋅ðR∞ −Rðt ÞÞ dT Ts
−1 = Rðt Þ ¼ R∞ ⋅ 1−e T s ðtÞ
3.1 Overview of the theoretical development
ð1Þ
The relation (Eq. 1) is controlled by the characteristic time scale Ts (Kriebel 1986; Miller and Dean 2003; Turki et al.
ð2Þ
The instantaneous R(t) and the equilibrium R∞ shoreline response were defined geometrically as a function of the
Ocean Dynamics
Fig. 2 Wave conditions at Barcelona coast. a Wave rose in deep water (hindcast database). b Modeled wave conditions at Bogatell and Somorrostro
beach length l, the instantaneous beach angle α(t), and the equilibrium angle β (Fig. 2a): Rðt Þ ¼ R∞ ¼
l ⋅tanðαðt ÞÞ 2
l ⋅tanðβÞ 2
ð3Þ ð4Þ
The instantaneous angle α(t) is defined as the angle between the initial shoreline position and the instantaneous one, produced under a specific set of wave conditions. β represents the angle between the initial shoreline position (continuous
black line in Fig. 3a) and the equilibrium one (which would imply a shoreline parallel to the direction of the wave crests, dashed black). The characteristic time scale of the system Ts was derived by Turki et al. (2013b) as a function of the wave energy and the physical characteristics of the beach (length and sediment grain size) as illustrated in Eq. 5. T s ðt Þ ¼ with
l 2 ⋅h* ⋅ðtanβ−tanαðt ÞÞ ^ F r ⋅χðβ; Rðt ÞÞ 4⋅K⋅E
ð5Þ
Ocean Dynamics Fig. 3 Synopsis of the beach plan-form evolution. a Relation between the instantaneous R and the equilibrium R∞ shoreline response produced above a reference forcing. b Changes in beach rotation under two different wave conditions produced at t1 and t2
χðβ; Rðt ÞÞ ¼ sinð2⋅βÞ−
4⋅Rðt Þ ⋅cosð2⋅β Þ l
ð6Þ
where the magnitude of the energy flux required to move sediments, EFr (Eq. 7), is a function of the breaking wave height (Hb), the critical wave height (Hcr), the wave celerity (Cb) at breaking and a coefficient relating the group celerity Cg to the celerity C (nb): E Fr ¼
1 ⋅ρ ⋅g⋅ðH b −H cr Þ2 ⋅C b ⋅nb 8 w
The critical wave height Hcr is given as 1 2⋅π⋅h ^ pffiffiffi ⋅U cr ⋅ 2 H cr ¼ ⋅T P ⋅sin π L
ð7Þ
1 ^ cr ¼ 0:014⋅T P ⋅ðs−1Þ2 ⋅g2 ⋅D50 3 U
ð9Þ
Going back to Eq. 5, h* is the closure depth which is a function of the offshore wave height Hs. The coefficient Cc (Copobianco et al. 1997) is a constant that differs from one beach to another and usually varies between 2.4 and 3.4 (here, a value of 2.8 was used). h* ¼ C c :ðH s Þ0:67
^¼ K
k ðρs −ρw Þ⋅g⋅a0
ð10Þ
ð11Þ
ð8Þ
L and h are the wave length and depth at breaking, respectively. Ûcr is the critical depth-averaged speed defined as (Van Rijin et al. 2003):
where ρs is the sediment density, ρw is the water density, g is gravity, a’ is a function of the sediment porosity P (a’=1−P), and k represents a dimensionless proportionality coefficient which depends on the mean grain size of the sediment.
Ocean Dynamics
Following Valle et al. (1993), this coefficient is defined as k= 1.4e−2.5.D50. At each time step, Ts was calculated numerically using Eq. 2 to determine R(t). A synopsis of beach changes in shown in Fig. 3b. β1/R∞1 (dotted black line) and β2/R∞2 (dotted grey line) are the equilibrium response of the beach plan-form, angle/distance, at t1 and t2, respectively. α1/R1 (dashed black line) and α2/R2 (dashed gray line) are the instantaneous response of the beach plan-form, angle/distance, at t1 and t2, respectively. The difference between the equilibrium β1/β2 and the instantaneous angles α1/α2 is described by Ωb1/Ωb2. θwc1 and θwc2 are the angles between the initial shoreline position (solid black line) and the equilibrium positions R∞1 and R∞2 produced at t1 and t2, respectively.
3.2 Analytical form of the model In this part, an alternative formulation of the model has been developed to define an analytical solution. Substituting the expression of Ts (Eq. 5) into Eq. 1 gives ^ E F r ðt Þ dRðt Þ 4:K ¼ þ * þ sinð2:Ωb ðt ÞÞ dt l h ðt Þ
ð12Þ
The angle Ωb is a function of the initial equilibrium angle θwc and the instantaneous angle α(t) as
As shown in Eq. 3, the angle α(t) depends on the instantaneous shoreline response R(t): 2⋅Rðt Þ αðt Þ ¼ tan ð14Þ l This angle takes a negative (positive) sign for shoreline retreat (accretion). It is relatively small and can be assumed to be less than 15°, in which case, based on a paraxial approximation, Eq. 12 can be written as αðtÞ∼
RðtÞ l
ð15Þ
Obviously, the beach length l, is larger than the instantaneous shoreline response R(t). Then, the term RðltÞ is small and 2 its square value RðltÞ is close to zero. Using these simplifications, sin(2.Ωb(t)) can be approximated as sinð2:bðt ÞÞ∼sinβð2:θwc ðt ÞÞ þ
4:Rðt Þ :cosðθwc ðt ÞÞ l
ð16Þ
Substituting Eq. 16 in Eq. 12, we obtain a linear ordinary differential equation (ODE) of first order dR þ Pðt Þ⋅Rðt Þ ¼ Qðt Þ dt
ð17Þ
where P(t) and Q(t) are continuous functions defined as b ðt Þ
¼ θwc ðt Þ−αðt Þ
ð13Þ
where the initial equilibrium angle θwc(t) is measured between the initial shoreline position, before simulations and the equilibrium one, associated to simulated sea state. For a better understanding of the different angles, we use Fig. 2b where the beach rotation under two different wave conditions produced at t1 and t2 is shown. At time t1, the initial shoreline moves and, if the same wave conditions continued indefinitely, would reach the equilibrium state defined by β1/R∞1 (dotted black line). In reality, wave conditions are not definitively applied, and then, the shoreline movement is limited to α1/R1 (dashed black line). At time t2, β2/R∞2 (dotted gray line) and α2/R2 (dashed gray line) are the equilibrium and the instantaneous responses, respectively. The angles Ωb1 and Ωb2 represent the difference between the equilibrium angles β1/β2 and the instantaneous angles α1/α2. θwc1 and θwc2 are measured between the initial shoreline positions (solid black and gray lines) and the equilibrium ones R∞1 and R∞2 (dotted black and gray lines) produced at t1 and t2, respectively. Note that this angle is different from the equilibrium angle β(t). In fact, θwc(t) is the angle between the wave crest and the shoreline position observed at the initial time (t0) before the sea state simulations while β(t) is the angle between the wave crest and the shoreline position observed at the beginning of the current sea state.
PðtÞ ¼
^ 16:K : E F r ðt Þ:cosð2:θwc ðt ÞÞ 2 l
ð18Þ
QðtÞ ¼
^ 4:K : E F r ðt Þ: sinð2: θwc ðt ÞÞ l
ð19Þ
By substituting and solving, R(t) results in Z
t
−
R ðt Þ ¼ e
t0
2 pðt Þ:dt 6 :6 4 R ðt 0 Þ þ
Z
Z t t0
QðtÞ:e
τ¼t τ¼t 0
3 Pðτ Þ:dτ
7 :dt 7 5ð20Þ
According to Eq. 20, the instantaneous shoreline response can be expressed as a function depending on its initial position R(t0) and the wave energy conditions, produced between t0 and t, described by the terms P(t) and Q(t). 3.3 Shoreline response Using the analytical form of the model, changes in the shoreline position at Bogatell and Somorrostro were computed between April 2005 and July 2006 and compared to results obtained by the numerical form (Turki et al. 2013b). Figure 4a, b shows the beach rotation at the northern side (P10). The initial value of the shoreline response R0 on 1 April 2005 was
Ocean Dynamics
Fig. 4 Shoreline response R computed using numerical (solid black line) and analytical (solid gray line) forms of the beach-evolution model over 15 months (April 2005–July 2006) at P10: a Bogatell and b Smorrostro.
Field observations of beach rotation are also plotted; they are represented by gray cross forms
extracted from video images at P10; R0 is 0.5 m at Bogatell and −2 m at Somorrostro. Fifteen months of beach rotation data from video images were used to validate modeled results. Strong similarities between the analytical and the numerical forms were observed (Fig. 4). Despite being based on a series of approximations, the analytical form of the model evaluates the explicit solution and provides more detailed information on the interaction between the shoreline response and the different variables involved (initial conditions, wave energy, and beach characteristics). Results illustrating the sensitivity of the shoreline response to the initial position are shown in Figs. 4a and 5a where the shoreline response R (black line) was computed over 15 months at Bogatell and Somorrostro, respectively. The initial condition R0 (extracted from observations) was perturbed to R’0 by a distance X and to R^0 by a distance 2X (X was taken as 3 m). Using these new initial conditions, computed shoreline response R’ (dashed black line) and R^ (dotted black line) display a temporal evolution similar to the nonperturbed response R with a displacement of ΔR’. The amplitude of these perturbations (ΔR’=R’-R and ΔR^=R^-R) decreases from April 2005 (ΔR’0=X and ΔR^0 =2X) to July 2006 when both trajectories are neighboring and converge to the nonperturbed response R (solid black line). The evolution of perturbations
ΔR’ and ΔR^ varies exponentially in time (Figs. 4b and 5b) and depends on the term P(t) which depends of the wave energy and the physical characteristics of the beach (length and sediment grain size). Note that the evolution is faster at Somorrostro (Fig. 6) where the beach length is shorter and the grain size is finer than at Bogatell. Initially, the difference between R’(t) (dashed black line) and R^(t) (dotted black line) is 3 m at both beaches; it reaches 0.5 m at Bogatell and 0.1 m at Somorrostro on 23 July 2006 representing 17 and 3 % of the initial perturbation, respectively.
4 Relaxation time 4.1 Theoretical development The analytical form of the model (Eq. 20) shows that the dependence of the shoreline response R(t) on its initial condition R(t0) diminishes with time. Figure 7 shows a synopsis of two shoreline responses R’(t) (dotted black line) and R^(t) (solid black line) for a same beach system. Let us assume an initial position R’(t0) to compute R’(t) and then a new shoreline response R^(t) caused by a perturbation ΔR0. Both responses converge over time, their difference is measured as ΔR, and reach approximately the same value (ΔR=~0) after a
Ocean Dynamics
Fig. 5 a Shoreline responses, R, R’, and R^ at Bogatell (northern side, P10) between April 2005 and July 2006 using different initial conditions R0 (Fig. 3a), R’0 (R’0 =R0 +X) and R^0 (R^0 =R0 +2.X), respectively (X=3m). b The evolution of the perturbations between R and R’/R^
Fig. 6 a Shoreline responses, R, R’, and R^ at Somorrostro (northern side, P10) between April 2005 and July 2006 using different initial conditions R0, R’0 (R’0 =R0 +X) and R^0 (R^0 =R0 +2.X), respectively (X=3m). b The evolution of the perturbations between R and R’/R^
Ocean Dynamics
Fig. 7 Synopsis describing the evolution of the perturbed shoreline responses R’ and R^ (vertical axis) with time (horizontal axis). The initial perturbation ΔR0 defined at t0 decreases and reaches small value
at t=t0 +PdT. Pd is the threshold value of ΔR which the difference between the perturbed and the nonperturbed shoreline response is negligible
period of time PdT (Fig. 7). We define this period as the relaxation time. The perturbation of the shoreline response at t and t0 is expressed as Eqs. 21 and 22, respectively
considered the same and that would imply the same shoreline position is predicted. Here, we assumed that the tolerance Pd represents 1 % of the initial perturbation:
0
00
ΔRðt Þ ¼ R ðt Þ−R ðt Þ
ð21Þ
ΔR0 ¼ R0 0 −R00 0
ð22Þ
As shown in Fig. 6, the perturbation ΔR(t), at a time t>t0, can be expressed as a function of its initial value ΔR0 produced at t0. Z t − Pðt Þ⋅dt t0 ΔRðt Þ ¼ ΔR0 ⋅e ð23Þ The sensitivity of the shoreline response to the initial conditions can be quantified using robust methods derived from dynamical systems theory. The definition of the exponential stagnation point or stable and unstable manifolds comes originally from the Lyapunov exponent theory (Steven and Strogatz 2001; Lyapunov 1992). This quantity characterizes the rate of separation of infinitesimally close trajectories. The difference between both trajectories can be defined as a function of the Lyapunov exponent λ which is given by ΔRðtÞ ¼ ΔR0 :eðt−t0 Þ
ð24Þ
Quantitatively, two trajectories in phase space with initial separation ΔR0 diverge or converge depending on the sign of λ (Steven and Strogatz 2001). When λ
