User:Egm6322.s09.bit.sahin/final

 See my comments below. Egm6322.s09 14:29, 1 May 2009 (UTC)

=Modelling of Longshore Sediment Transport=

Introduction
Coasts are projected to be home at least 3 billion people by 2025. Not only by means of their aesthetical beauty in natural life, are coastal zones indispensable also because of their economical potential with eternal resources and expensive investments such as marinas, harbours, quays, etc.

Nearshore processes include several parameters such as waves, currents, tides and also movement of sediments, which affect dynamics of coastal zones significantly. Coastal sedimentation is one of the main concerns of coastal engineering profession since sediment transport cause movement of shoreline and change in the nearshore bathymetry of coastal zones. Wind wave induced sediment transport takes place in longshore (littoral drift) and cross-shore directions, first of which is taken as the governing pattern in long term shoreline changes and related one-line numerical models. Longshore sediment transport is transport of sediment in parallel to shoreline, which is assumed to be caused by waves breaking at an angle to the shore and wave induced nearshore current circulation in one line theory (Figure 1).

 It may be useful to point out to the general readers that Fig.1 is the top view of a beachline (longshore line) with the yellow region representing the beach, and the blue region water. Egm6322.s09 18:16, 1 May 2009 (UTC)



One-line Theory and Shoreline Change .
The one-line model is the simplest contour model, and it describes the time history of the shoreline position along a shoreline. The first one-line model was presented by Pelnard-Considère (1956), who examined the behaviour of groins on a beach. Since that time, the "diffusion" equation, which he developed, has been applied to many different situations.



The shoreline is shown in Figure 2, with the x-axis oriented alongshore and y-axis offshore. The equation for the shoreline position is

$$y=y\left ( x,t \right )$$

Owing to the irregularities in the shoreline, the shoreline normal does not point directly offshore at most locations. The local normal can be determined as

$$\mathbf{n}=\frac{-\frac{\partial y}{\partial x}\mathbf{i}+1\mathbf{j}}{\sqrt{1+\left ( \frac{\partial y}{\partial x} \right )^{2}}}$$

The vectors, i, j are the unit vectors in the $$x$$ and $$y$$ directions. The angle made by the local beach normal to the $$y$$- axis is, by definition of the dot product,

$$\gamma =cos^{-1}\left ( \boldsymbol{i}\cdot \boldsymbol{j} \right )$$

The first equation used in the development of the one-line model is the alongshore sediment transport formula

$$Q=\frac{K\rho H_{b}^{5/2}\sqrt{g/\kappa }sin2\left ( \delta _{b}-\gamma \right )}{16\left ( \rho _{s-\rho } \right )\left ( 1-p \right )}=C_{q}sin2\left ( \delta _{b}-\gamma  \right ) $$ .........(1),

where $$C_{q}$$ is defined here for covariance and $$\left ( \delta _{b}-\gamma \right ) $$ measures the angle of wave incidence relative to the shoreline normal. The breaking wave angle of incidence $$\delta _{b}$$ is measured from the fixed $$y$$-axis.

A wide range of expressions exists for the amplitude of the longshore sand transport rate, mainly based on empirical results. For example, the Shore Protection Manual (SPM) (1984) gives the following equation:

$$Q_{0}=\frac{\rho g}{16}H_{sb}^{2}c_{gb}\frac{K}{\left (\rho _{s}-\rho \right )\lambda  }$$

where $$\rho$$: density of water ($$kg/m^{3}$$), $$H_{sb}$$: significant breaking wave height (m), $$c_{gb}$$: wave group velocity at breaking point (m/sec), K: nondimensional empirical constant, $$\rho_{s}$$: density of sand ($$kg/m^{3}$$), $$\lambda$$ porosity of sand.

Kamphuis (1991) gives,

$$Q=7.3 H_{sb}^{2}T^{1.5}m_{b}^{0.75}D_{50}^{-0.25}sin^{0.6}\left ( 2\alpha _{bs} \right )$$

where $$m_{b}$$: beach slope at breaker point, $$D_{50}$$: median grain size diameter (m), $$\alpha_{bs}$$: effective wave breaking angle

The second and the last equation is the conservation of sand equation. Consider a section of the coastline that is $$\Delta x$$ long, as shown in Figure 2. We will assume that the profile is in equilibrium in the cross-shore direction. This is a key assumption;regardless of the shoreline orientation, it is assumed that the equilibrium profile applies at all $$x$$ locations. Now, if the rate at which sand is entering the cross section owing to a longshore transport is greater than the rate at which it is leaving from the other side of the profile, there must be an accumulation of sand with time in the profile, causing the shoreline to advance offshore. The volume of sand necessary to move a profile seaward (shoreward) is the shoreline accretion (recession) times the height of the active profile, $$\left (h_{*}+B \right )$$, where $$h_{*}$$ is the depth of closure and B is the berm height. Restating this in mathematical terms,

$$\Delta t\left [ Q\left ( x \right )-Q\left ( x+\Delta x \right ) \right ]=\left [ y\left ( t+\Delta t \right ) -y\left ( t \right ) \right ]\left ( h_{*}+B \right )\Delta x $$

or, through the use of Taylor series and the argument that $$\Delta x$$ and $$\Delta y$$ become very small, we obtain

$$\frac{\partial y}{\partial t}+\frac{1}{\left ( h_{*}+B \right )}\frac{\partial Q}{\partial x}=0$$

Now, by substituting the expression for transport rate, Eq. (1), into this equation, we obtain the final equation. For computer calculation, this is sufficient. For analytical solutions, some further approximations are needed. Expanding the trigonometric terms in $$Q$$, we have

$$Q=C_{q}sin2\left ( \delta _{b}-\gamma \right )=C_{q}\left [ sin2\delta _{b}\left ( cos^{2}\gamma -sin^{2}\gamma  \right )-2cos 2\delta _{b}sin\gamma cos\gamma  \right ]$$

From the definition of the shoreline normal, $$\boldsymbol{n}$$, $$sin\gamma=-\boldsymbol{n}\cdot \boldsymbol{i}$$ and $$cos\gamma =\boldsymbol{n}\cdot \boldsymbol{j}$$. This leads us to

$$sin\gamma =\frac{\frac{\partial y}{\partial x}}{\sqrt{1+\left (\frac{\partial y}{\partial x} \right )^{2}}} $$, $$cos\gamma =\frac{1}{\sqrt{1+\left (\frac{\partial y}{\partial x}  \right )^{2}}} $$

and

$$tan\gamma =\frac{\partial y}{\partial x}$$

These definitions for $$sin\gamma $$ and $$cos\gamma $$ are substituted into the definition of $$Q$$, which yields, for small values of $$\frac{\partial y}{\partial x}$$,

$$Q=C_{q}sin2\delta _{b}-2C_{q}cos2\delta _{b}\frac{\partial y}{\partial x}=Q_{0}-G\left ( h_{*}+B \right )\frac{\partial y}{\partial x}$$,

where $$G=2C_{q}cos2\delta _{b}/\left ( h_{*}+B \right )$$. The term $$Q_{0}$$ is the background transport rate for a shoreline parallel to the $$x$$-axis, and the second term represents the transport induced by the alongshore shoreline slop due to the shoreline deviation from the $$x$$-axis. We can now carry out the derivatives required in Eq(3); again, we assume that $$\partial y/\partial x\ll 1$$ with the final expression

$$\frac{\partial Q}{\partial x}\simeq -G\left ( h_{*}+B \right )\frac{\partial ^{2}y}{\partial x^{2}}$$

Introducing this simplified form for the derivative of $$Q$$ into the sand conservation equation, we obtain the final version of the Pelnard-Considere equation,

$$\frac{\partial y}{\partial t}=G \frac{\partial ^{2}y}{\partial x^{2}}$$

This equation is the classical one-dimensional diffusion equation, which is well known in mathematical physics, and a variety of solutions exist for this equation. The parameter $$G$$ is referred to as the longshore diffusivity. For small angles of wave incidence with respect to the shore normal, $$\delta _{b}\ll 1$$ and $$G=2C_{q}/\left ( h_{*}+B \right )$$. The diffusivity parameter has units of length squared per unit of time.

Classification
We have the classical one-dimensional diffusion equation to solve:

$$\frac{\partial y}{\partial t}=G \frac{\partial ^{2}y}{\partial x^{2}}$$

Since $$ac-b^{2}=0$$, the PDE is clearly parabolic.

===General Formal Solution ===

An infinitely long beach is assumed to be exposed to waves of constant height and period with have crests parallel to the $$x$$-axis (parallel to the trend of the shoreline). The shoreline will adjust to reach an equilibrium state in which the longshore sand transport rate is equal at every point along the shoreline. Since the wave crests are parallel to the $$x$$-axis, the equilibrium sand transport rate is zero. An initially straight beach is thus the stable shoreline form in this case. If the shoreline shape at time $$t=0$$ is described by a function $$f(x)$$, the solution of the equation is given by the following integral :

$$y\left ( x,t \right )=\frac{1}{2\sqrt{\pi Gt}}\int_{-\infty }^{+\infty }f\left ( \xi \right )exp\left ( -\frac{\left ( x-\xi  \right )^{2}}{4Gt} \right )d\xi $$.........(2)

for $$t>0$$ and $$-\infty < x< \infty$$.

 It is possible to relate the above solution in Eq.(2) to what was done in class. Egm6322.s09 18:16, 1 May 2009 (UTC)

The shoreline position is denoted by $$y$$ and is a function of $$x$$ and $$t$$. The quantity $$\xi$$ is a dummy integration variable. Consequently, the change in both natural and manipulated beach forms can be determined if equation!!(last) is evaluated. Equation (2) may be interpreted as a superposition of an infinite number of plane sources instantaneously released at $$t=0$$. The source located at point $$\xi$$ contributes an amount $$f\left (\xi \right ) d\xi$$ to the system. Infinitely far away from such a single source no effect on the shoreline position is assumed (boundary condition).

Initial and Boundary Conditions
A groyne (groin in the United States) is a rigid hydraulic structure built from an ocean shore (in coastal engineering) or from a bank (in rivers) that interrupts water flow and limits the movement of sediment. The problem is to determine analytically the deposition of sand on the updrift side of a groin and the subsequent (and antisymmetric) erosion of sand on the downdrift side of a groin from a one-line contour model (Figure 3). The analytical solution for the beach change at a groin or any thin shore-normal structure which blocks alongshore sand transport was fist obtained by Pelnard-considere (1956). Initially, the beach is in equilibrium (parallel to the $$x$$-axis) with the same breaking wave angle existing everywhere, thus leading to a uniform sand transport rate along the beach. At time t=0 a thin groin is instantenaously placed at x=0, blocking all transport. Mathematically, this boundary condition can be formulated as



$$\frac{\partial y\left ( 0,t \right )}{\partial x}=y_{x}\left ( 0,t \right )=-tan\delta _{b}$$

This equation states that the shoreline at the groin is at every instant parallel to the wave crests. The wave crests make an angle $$\delta _{b}$$ with the $$x$$-axis according to Figure 3, giving rise to longshore sand transport in the negative $$x$$-direction.

A groin interrupts the transport of sand alongshore, causing an accumulation at the updrift side and erosion at the downdrift side. The solution describing the accumulation part is

$$y\left ( x,t \right )=2tan\delta _{b} \sqrt{Gt}\mathrm{ierfc}\left ( \frac{x}{2\sqrt{Gt}} \right )$$

for $$t>0$$ and $$x\geq 0$$.

The solution can also be written as follows:

$$y\left ( x,t \right )=2tan\delta _{b}\left [ \sqrt{\frac{Gt}{\pi }}e^{-x^{2}/4Gt}-\frac{x}{2}erfc\left ( \frac{x}{2\sqrt{Gt}} \right ) \right ]$$

where the complementary eroor function, $$\mathrm{erfc}\left ( \right )=1-\mathrm{erf}\left (  \right )$$.

Some Analytical Results
Four cases have been considered and shoreline change with time, for different wave angles and different wave heights have been discussed. In each case, these parameters haven't been changed:

wave period, $$T=8s$$

beach slope, $$m_{b}=0.05$$

median sediment diameter, $$D_{50}=1.5 mm$$

The cases are:

Case 1) Deep water significant wave hight, $$H_{0}=1 m$$ and deep water wave angle, $$\alpha_{0}=5^{o} $$

Case 2) Deep water significant wave hight, $$H_{0}=1 m$$ and deep water wave angle, $$\alpha_{0}=2^{o} $$

Case 3) Deep water significant wave hight, $$H_{0}=2 m$$ and deep water wave angle, $$\alpha_{0}=5^{o} $$

Case 4) Deep water significant wave hight, $$H_{0}=2 m$$ and deep water wave angle, $$\alpha_{0}=2^{o} $$

Matlab code that is used in this study

Result 1 Shoreline change with time


Shoreline change at up-drift of a groin located at x(alongshore ccordinate)=0 on a sandy beach of 1.25 mm median grain size due to waves with T=8 sec. period and varying wave heights and approach angles. It is easily observed in all three cases that rate of shoreline change decreases with time. Results for $$H_{s}=1m$$ and $$\alpha_{0}=5^{o}$$ can be shown in Figure 4.

Result 2: Effect of wave height on shoreline change


Due to increasing wave height ($$H_{0}$$), wave forcing (deep water approach angle is 5 degrees) and long-shore sediment transport rate increase, which increase trapped sand volume and shoreline change at up-drift a groin (Figure 5). Figure 5 shows the shoreline change after 15 days.

Result 3: Effect of deep water aproach angle on shoreline change


Similarly, waves (H=1 m.) approaching with larger angles cause higher long-shore sediment transport rates and shoreline changes at up-drift of a groin (Figure 6). Figure 6 shows the shoreline change after 15 days.

Numerical Solution


In the structure of one-line numerical model, instead of analytical solution, sediment continuity equation is converted to an explicit finite difference scheme depending on longshore distance, $$y(x,t)$$ and longshore sediment transport rate, $$Q(x,t)$$ (Figure 7). In this scheme, gradual change of longshore sediment transport rate in alongshore direction is calculated by the following expression:

$$\frac{\partial Q_{i}}{\partial x}=\frac{Q_{i+1}-Q_{i}}{\Delta x}$$

Consequently, the expression given below is derived to solve sediment continuity equation explicitly:

$$y_{i}^{'}=y_{i}+\frac{\Delta t}{\left ( h_{*}+B \right )\Delta x}\left ( Q_{i}-Q_{i+1} \right )$$

where

$$\Delta t$$: time increment

$$\Delta x$$: longshore distance increment

subscript "i+1" indicates the next increment in alongshore direction and prime (') indicates values at next time step. Explicit solution of sediment continuity using finite difference scheme is described as fixing sediment transport rates to compute shoreline displacements at the next time step

Appropriate boundary and possibly internal conditions must be posed to complete problem formulation. Various types of boundary conditions are possible and are implemented fairly directly for the explicit model. The simplest boundary condition might be to fix the displacement of the shoreline at the ends of a represented shoreline segment, say, $$y(0)$$ and $$y(l)$$, where l is the length of the shoreline segment represented by the model. With these values fixed, the adjacent $$Q_{s}$$ are computed using these specified values and the adjacent $$y$$ values.

In explicit scheme, since shoreline coordinates $$(y)$$ at $$t=t_{1}+\Delta t$$ depend on shoreline coordinates and longshore sediment transport rates at $$t=t_{1}$$, stability comes out to be an important parameter. At every time step and longshore distance icrement, stability is checked by the following expression:

$$R_{s}=\frac{Q}{\delta _{b}\left ( h_{*}+B \right )}\frac{\Delta t}{\Delta x^{2}}\leq \frac{1}{2}$$

In the model, $$\Delta x$$ and $$\Delta t$$ values are assigned by the user. Taking greater time increments and/ or smaller longshore distance increments to solve sediment continuity equation decrease stability of the solution and may cause oscillations in resulting shoreline

An example for Comparison between Analytical and Numerical Solutions




Ari et al.(2006) used one-line model to determine the shoreline changes. They investigated a sedimentation problem at the entrance of Karaburun fishery harbor.

The case study field is Karaburun coastal village located at the south west coast of the Black Sea which is at northwest of Istanbul City, Turkey. Karaburun has morphological dynamic shoreline in 4 km length (Figure 8)

The harbor operations are affected by the sedimentation problem because of considerable rate of westward sediment transport towards the harbor entrance, thus the water depth shallows and the navigation to and from the harbor is prevented. The goal of the study is to examine the hydrodynamic parameters effecting shoreline changes and sedimentation near fishery port and discuss their effects on the sedimentation.

They used a numerical model to determine shoreline change based on one-line theory. They validated the model by benchmarking with analytical solution for a simple idealized case. A comparison of the results of the analytical and numerical solutions for shoreline change after one year is shown in Figure 9. For more detail about the study, see Ari et al. (2006).

Conclusion
One dimensional coastal morphology model is the simplest of all coastal numerical models. At the same time it is the most important, since any study involving coastal change normally involves a 1-D computation somewhere. A 1-D model solves two simple 1-D simultaneous equations: conservation of sand (mass) and a bulk sediment transport rate formula. The assumptions of one-line theory are:

-Beach profile moves parallel to itself

-Long term shoreline changes are due to long-shore sediment transport variations

-Long-shore sediment transport is proportional to wave breaking parameters

For analytical solution, it is also assumed that angle between breaking wave crests & shoreline and shoreline & x-axis are small. Results of analytical solution show that longshore sediment transport rate increases with increasing wave height and wave approach angle. For practical problems, the equations, the input wave conditions and the boundary conditions cannot normally be simplified sufficiently for analytical solutions to be valid. In that case numerical solution is needed. Obviously, due to the fact that each coastal zone and sedimentation problem are totally different, the results of the model needs to be compared with field measurements at site and physical model studies.

Signature
Egm6322.s09.bit.sahin 15:37, 30 April 2009 (UTC)