Buoyancy fluxes in stratified flows: observations and parameterizations

\textsc{We present a synthesis of observations of turbulent buoyancy fluxes, }\textsc{\textit{B}}\textsc{, made at five sites where flows and turbulence are primarily associated with internal waves, both breaking and non-breaking. In four cases, }\textsc{\textit{B}}\textsc{ was calculated from the covariance of velocity and density whereas in the fifth case, it was inferred from the rate of temperature variance dissipation,}$\chi $\textsc{. Overall, we find that the flux Richardson number, }\textsc{\textit{Ri}}$_{f}$\textsc{, depends on the Gibson number, }\textsc{\textit{Gi}}\textsc{ }$=$\textsc{ }$\varepsilon $\textsc{/}$\nu $\textsc{N}$^{\mathrm{2}}$\textsc{: when }\textsc{\textit{Gi}}\textsc{ \textless 100, }\textsc{\textit{Ri}}$_{f}$\textsc{ }$\approx $\textsc{ 0.27, and when }\textsc{\textit{Gi}}\textsc{ \textgreater 100 }\textsc{\textit{Ri}}$_{f}$\textsc{ }$\approx $\textsc{ 2.7 }\textsc{\textit{Gi}}$^{\mathrm{-0.5}}$\textsc{, in agreement with the functional relationship found originally using direct numerical simulation (DNS). Our observations do not match well other DNS-derived models that parameterize }\textsc{\textit{Ri}}$_{f}_{\mathrm{\thinspace }}$\textsc{in terms of the gradient Richardson number, }\textsc{\textit{Ri,}}\textsc{ or the turbulence Froude numbers, }\textsc{\textit{Fr}}$_{k}$\textsc{ and }\textsc{\textit{Fr}}$_{t}$\textsc{. Similarly, }\textsc{\textit{Ri}}$_{f}$\textsc{(}\textsc{\textit{Gi}}\textsc{) is found to be the same for all the covariance data sets, despite the fact that these 4 flows produce turbulence that falls in different regimes defined by several pairs chosen from the 5 non-dimensional numbers that the Buckingham }$\Pi $\textsc{ theorem shows may affect }\textsc{\textit{Ri}}$_{f}$\textsc{ . }