**Abstract**

Fractal dimension, which is a measure for the degree of complexity or that of fractals, is given for the erosion and sedimentation of fluvial beds. An alternative to fractal dimension is ht-index, which quantifies complexity in a unique way while sediment particles begin to move if a situation is eventually reached when the hydrodynamic force exceeds a certain critical value. Back to question, the physical meaning of fractal dimension is that many natural and social phenomena are nonlinear rather than linear, and are fractal rather than Euclidean. We need a new paradigm for studying our surrounding phenomena, not Newtonian physics for simple systems, but complexity theory for complex systems, not linear mathematics such as calculus, Gaussian statistics, and Euclidean geometry, but online mathematics including fractal geometry, chaos theory, and complexity science in general.

**Keywords:** natural dimensions, nonlinearity, fractals, meanders, plume
