Self Organization in Beach Cusp Formation
What is Beach cusp?
Beach cusps are cuspate patterns with concave seaward, are usually formed at the shoreline, and have longshore wavelengths varying from less than 1 m until 700 m. They enumerate some traditional names given cuspate features on ocean shorelines. The classification are Typical Beach Cusps which wavelength 8-25 m, Storms Cusps which wavelength 70-120 m and Giant Cusps which wavelength 700-1500 m (Guza et al 1975).
Why beach cusp appears?
Many studies have been conducted in case to know how beach cusp formed. Early studies said that beach cusp formed by the action of nearshore current system. The generation mechanisms are classified into two types: those due to a longshore variation in the external force and those due to hydrodynamic instability (Tanikawa et al, 2004). The former one includes theories in which the wave field loses its uniformity due because of external causes. Such as edge waves and cross wave (Guza 1973) and the other one means that rhythmic formation is because fluid dynamic instability breaks the uniformity of radiation stress even without an external cause (Hino 1973).
How to explain their formation?
Beach cusp research has been studied to approach their formation in several ways
1. Field Experiment (Gerhard Masselink, 2003)
2. Laboratory (Guza,1973)
3. Numerical Program (Coco, et al, 2003)
4. Mathematic Analysis (Hino, 1975; Izumi, 2004)
Which one is the most powerful to approach beach cusp formation?
Not clear though. When doing any research, we didn’t know which one is the most powerful. In my case, I want to show their formation by mathematic analysis.
Why use mathematic analysis?
Beach cusp is more like a perturbation on linear shoreline. Their pattern can reminder us to sinusoidal function. They have a constant wavelength in each formation. In mathematic analysis, we would find any solution to represent the case. This solution would be real and/or unreal. If the solution real, the base state in linear stability analysis become stable, it means beach cusp don’t occur on linear shoreline. Otherwise, we would find beach cusp when we have unreal solution.
Numerical analysis can’t solve the unreal solution. There will be error or we can not continue our programming.
Therefore, we need mathematic analysis.
What is perturbation equation?
An equation which perturbated the linear base state. We can assume anything in this equation.
For example, we assume that:
a) Beach cusp is a function of beach slope
b) Function of grain size sediment
c) Function of wave height
d) Swash action
e) Breaking zone width
f) Radiation stress
g) etc
From this assumption, we can make any equation which have a function of those. In mathematic analysis, if we have a complex function, we solve one by one. Then, we make a graphic. From this graphic, later on, we could know, which one of those functions, which closely approach the formation of beach cusp. When there is a function which have less contribution to beach cusp, then this function will be denied and become an error.
What is Base State?
Base state is initial condition in linear stability analysis. A system equation which have some function like in the perturbation equation, but we make the linear function.
Conclusion?
So, with mathematic analysis, there will be many ways to develop the early model since numeric analysis can not solve this.
This analysis will provide the scientist to explore all his hipotesa to base state and perturbation equation.
More simple and free.
Some mathematic tricks
Non-dimensional analysis needed to get simpler.
Often, the equations we want to solve in fluid problems can not solved with all the terms present. By using this analysis, we often gain an insight into what physical processes.
Still need to study though !!!
Beach cusps are cuspate patterns with concave seaward, are usually formed at the shoreline, and have longshore wavelengths varying from less than 1 m until 700 m. They enumerate some traditional names given cuspate features on ocean shorelines. The classification are Typical Beach Cusps which wavelength 8-25 m, Storms Cusps which wavelength 70-120 m and Giant Cusps which wavelength 700-1500 m (Guza et al 1975).
Why beach cusp appears?
Many studies have been conducted in case to know how beach cusp formed. Early studies said that beach cusp formed by the action of nearshore current system. The generation mechanisms are classified into two types: those due to a longshore variation in the external force and those due to hydrodynamic instability (Tanikawa et al, 2004). The former one includes theories in which the wave field loses its uniformity due because of external causes. Such as edge waves and cross wave (Guza 1973) and the other one means that rhythmic formation is because fluid dynamic instability breaks the uniformity of radiation stress even without an external cause (Hino 1973).
How to explain their formation?
Beach cusp research has been studied to approach their formation in several ways
1. Field Experiment (Gerhard Masselink, 2003)
2. Laboratory (Guza,1973)
3. Numerical Program (Coco, et al, 2003)
4. Mathematic Analysis (Hino, 1975; Izumi, 2004)
Which one is the most powerful to approach beach cusp formation?
Not clear though. When doing any research, we didn’t know which one is the most powerful. In my case, I want to show their formation by mathematic analysis.
Why use mathematic analysis?
Beach cusp is more like a perturbation on linear shoreline. Their pattern can reminder us to sinusoidal function. They have a constant wavelength in each formation. In mathematic analysis, we would find any solution to represent the case. This solution would be real and/or unreal. If the solution real, the base state in linear stability analysis become stable, it means beach cusp don’t occur on linear shoreline. Otherwise, we would find beach cusp when we have unreal solution.
Numerical analysis can’t solve the unreal solution. There will be error or we can not continue our programming.
Therefore, we need mathematic analysis.
What is perturbation equation?
An equation which perturbated the linear base state. We can assume anything in this equation.
For example, we assume that:
a) Beach cusp is a function of beach slope
b) Function of grain size sediment
c) Function of wave height
d) Swash action
e) Breaking zone width
f) Radiation stress
g) etc
From this assumption, we can make any equation which have a function of those. In mathematic analysis, if we have a complex function, we solve one by one. Then, we make a graphic. From this graphic, later on, we could know, which one of those functions, which closely approach the formation of beach cusp. When there is a function which have less contribution to beach cusp, then this function will be denied and become an error.
What is Base State?
Base state is initial condition in linear stability analysis. A system equation which have some function like in the perturbation equation, but we make the linear function.
Conclusion?
So, with mathematic analysis, there will be many ways to develop the early model since numeric analysis can not solve this.
This analysis will provide the scientist to explore all his hipotesa to base state and perturbation equation.
More simple and free.
Some mathematic tricks
Non-dimensional analysis needed to get simpler.
Often, the equations we want to solve in fluid problems can not solved with all the terms present. By using this analysis, we often gain an insight into what physical processes.
Still need to study though !!!
posted by yessi | Friday, March 11, 2005
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