On the regularity of the Green-Naghdi equations for a rotating shallow fluid layer

The Green-Naghdi equations are an extension of the shallow-water equations that capture the effects of finite fluid depth at arbitrary order in the characteristic height to width aspect ratio H/L. The shallow-water equations capture these effects to first order only, resulting in a relatively simple two-dimensional fluid-dynamical model for the layer horizontal velocity and depth. The Green-Naghdi equations, like the shallow-water equations, are two-dimensional fluid equations expressing momentum and mass conservation. There are different 'levels' of the Green-Naghdi equations of rapidly increasing complexity. In the present paper we focus on the behaviour of the lowest-level Green-Naghdi equations for a rotating shallow fluid layer, paying close attention to the flow structure at small spatial scales. We compare directly with the shallow-water equations and study the differences arising in their solutions. By recasting the equations into a form which both explicitly conserves Rossby-Ertel potential vorticity and represents the leading-order departure from geostrophic-hydrostatic balance, we are able to accurately describe both the 'slow' predominantly sub-inertial balanced dynamics and the 'fast' residual imbalanced dynamics. This decomposition has proved fruitful in studies of shallow-water dynamics but appears not to have been used before in studies of Green-Naghdi dynamics. Importantly, we find that this decomposition exposes a fundamental inconsistency in the Green-Naghdi equations for horizontal scales less than the mean fluid depth, scales for which the Green-Naghdi equations are supposed to more accurately model. Such scales exhibit pronounced activity compared to the shallow-water equations, and in particular spectra for certain fields like the divergence are flat or rising at high wavenumbers. This indicates a lack of convergence at small scales, and is also consistent with the poor convergence of total energy with resolution compared to the shallow-water equations. We suggest a mathematical reformulation of the Green-Naghdi equations which may improve convergence at small scales.