论文标题
在沙滩石的椭圆形形状上
On the Oval Shapes of Beach Stones
论文作者
论文摘要
本文介绍了一种新的地球物理理论，即以单个简单的局部整数差异方程的形式，以解释在平面海滩上单独的石头摩擦磨损如何导致经验观察到的椭圆形形状。该理论中的基本思想是直观的观察，即石头表面的一个点的消融速率与那时石头曲率的产物成正比，并且该点可能与海滩接触的频率很可能与海滩接触。具体而言，这种新模型中的关键作用是由随机波过程和石头的全局（非本地）形状扮演的，即其形状远离与海滩的接触点。该过程的基本物理机制是能量从波过程转化为石材的势能。即使在二维设置中，也没有封闭形式甚至渐近解决方案，即使在二维设置中，但使用标准曲线缩短算法和使用Monte Carlo Simulation的随机离散时间多面部性设置的确定性连续时间设置也呈现了基本的数值解决方案。
This article introduces a new geophysical theory, in the form of a single simple partial integro-differential equation, to explain how frictional abrasion alone of a stone on a planar beach can lead to the oval shapes observed empirically. The underlying idea in this theory is the intuitive observation that the rate of ablation at a point on the surface of the stone is proportional to the product of the curvature of the stone at that point and how often the stone is likely to be in contact with the beach at that point. Specifically, key roles in this new model are played by both the random wave process and the global (non-local) shape of the stone, i.e., its shape away from the point of contact with the beach. The underlying physical mechanism for this process is the conversion of energy from the wave process into potential energy of the stone. No closed-form or even asymptotic solution is known for the basic equation, even in a 2-dimensional setting, but basic numerical solutions are presented in both the deterministic continuous-time setting using standard curve-shortening algorithms, and a stochastic discrete-time polyhedral-slicing setting using Monte Carlo simulation.
