Comparing vorticity and curvature Rossby numbers

In ocean dynamics, there is often a need to measure the point-by-point significance of the nonlinear term compared with the Coriolis term in the momentum equations. The bulk Rossby number (i.e., $\frac{U}{fL}$) does not meet this need, which necessitates the proposal for the pointwise Rossby number. Conventionally, two different formulations are used to represent the pointwise Rossby number approximately. One is the vorticity Rossby number defined as the ratio of the relative vorticity to the planetary vorticity (i.e., $\frac{\zeta }{f}$), and the other is the curvature Rossby number formulated as the ratio of the curvature vorticity to the planetary vorticity (i.e., $\frac{{\zeta }_{curv}}{f}$). It remains unknown which approximate representation is more accurate. Here we compare their accuracies on the basis of theoretical and data analysis. The vorticity Rossby number is found to overestimate the pointwise nonlinearity of oceanic flows due to its neglect of the spatial variation of the kinetic energy. The curvature Rossby number is shown to intrinsically consider the kinetic energy term; thus, it is more accurate and useful for diagnosing and understanding the nonlinearity of ocean circulation.