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Bolgiano-Obukhov spectrum and mixing efficiency in stably stratified turbulence

In this paper, using a shell model, we simulate highly turbulent stably stratified flow for weak to moderate stratification at unitary Prandtl number. We investigate the energy spectra and fluxes of velocity and density fields. We observe that for moderate stratification, in the inertial range, the kinetic energy spectrum E_{u}(k) and the potential energy spectrum E_{b}(k) show dual scaling-Bolgiano-Obukhov scaling [E_{u}(k)∼k^{-11/5} and E_{b}(k)∼k^{-7/5}] for k<k_{B}, where k_{B} is the Bolgiano wave number, and Kolmogorov scaling (∼k^{-5/3}) for k>k_{B}. In addition, we find that the mixing efficiency η_{mix} varies as η_{mix}∼Ri for weak stratification, whereas η_{mix}∼Ri^{1/3} for moderate stratification, where Ri is the Richardson number.

 

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In the referenced paper, the authors utilize a shell model to conduct simulations of highly turbulent stably stratified flow. The focus of their investigation is on weak to moderate stratification at a unitary Prandtl number. They examine the energy spectra and fluxes of both the velocity and density fields. Here are the key findings described in the paper:

1. Dual Scaling in the Inertial Range: For moderate stratification, within the inertial range of scales, the kinetic energy spectrum E_{u}(k) and the potential energy spectrum E_{b}(k) exhibit dual scaling patterns. Specifically, they follow Bolgiano-Obukhov scaling for wavenumbers k less than a critical value k_{B}, and Kolmogorov scaling for k greater than k_{B}. The scaling exponents are E_{u}(k)∼k^{-11/5} and E_{b}(k)∼k^{-7/5} for k<k_{B}, while they adhere to the familiar Kolmogorov scaling (∼k^{-5/3}) for k>k_{B}.

2. Mixing Efficiency: The authors also investigate the mixing efficiency η_{mix}, which is a measure of how effectively the turbulence mixes the fluid in the presence of stratification. They find that the mixing efficiency varies depending on the strength of stratification. Specifically:
   - For weak stratification, the mixing efficiency η_{mix} scales linearly with the Richardson number Ri.
   - For moderate stratification, the mixing efficiency η_{mix} scales as η_{mix}∼Ri^{1/3} with respect to the Richardson number Ri.

It's important to note that these findings are specific to the model and assumptions used in the paper. The authors have employed a shell model, which is a simplified representation of turbulence, and have made assumptions about the flow properties and parameters. Therefore, the results should be interpreted within the context of the model and its limitations.

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