The Critical Richardson Number and Limits of Applicability of Local Similarity Theory in the Stable Boundary Layer

TL;DR

This study identifies a critical Richardson number (~0.20-0.25) beyond which local similarity theory in the stable boundary layer fails due to the breakdown of turbulence energy cascades, limiting its applicability.

Contribution

It establishes a quantitative threshold for the Richardson number that delineates the limits of Monin-Obukhov similarity theory in stable atmospheric conditions.

Findings

The inertial subrange disappears when Ri and Rf exceed 0.20-0.25.

Turbulence becomes non-Kolmogorov and decays rapidly beyond the critical Richardson number.

Classical similarity theory applies only when Ri and Rf are below the critical thresholds.

Abstract

Measurements of atmospheric turbulence made over the Arctic pack ice during the Surface Heat Budget of the Arctic Ocean experiment (SHEBA) are used to determine the limits of applicability of Monin-Obukhov similarity theory (in the local scaling formulation) in the stable atmospheric boundary layer. Based on the spectral analysis of wind velocity and air temperature fluctuations, it is shown that, when both of the gradient Richardson number, Ri, and the flux Richardson number, Rf, exceed a 'critical value' of about 0.20 - 0.25, the inertial subrange associated with the Richardson-Kolmogorov cascade dies out and vertical turbulent fluxes become small. Some small-scale turbulence survives even in this supercritical regime, but this is non-Kolmogorov turbulence, and it decays rapidly with further increasing stability. Similarity theory is based on the turbulent fluxes in the high-frequency part of the spectra that are associated with energy-containing/flux-carrying eddies. Spectral densities in this high-frequency band diminish as the Richardson-Kolmogorov energy cascade weakens; therefore, the applicability of local Monin-Obukhov similarity theory in stable conditions is limited by the inequalities Ri < Ri_cr and Rf < Rf_cr. However, it is found that Rf_cr = 0.20 - 0.25 is a primary threshold for applicability. Applying this prerequisite shows that the data follow classical Monin-Obukhov local z-less predictions after the irrelevant cases (turbulence without the Richardson-Kolmogorov cascade) have been filtered out.