Parametrizing the Energy Dissipation Rate in Stably Stratified Flows
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Parametrizing the Energy Dissipation Rate in Stably Stratified Flows Sukanta Basu1
· Ping He2 · Adam W. DeMarco3
Received: 8 January 2020 / Accepted: 4 August 2020 © The Author(s) 2020
Abstract We use a database of direct numerical simulations to evaluate parametrizations for energy dissipation rate in stably stratified flows. We show that shear-based formulations are more appropriate for stable boundary layers than commonly used buoyancy-based formulations. As part of the derivations, we explore several length scales of turbulence and investigate their dependence on local stability. Keywords Buoyancy length scale · Integral length scale · Outer length scale · Ozmidov scale · Stable boundary layer
1 Introduction Energy dissipation rate is a key variable for characterizing turbulence (Vassilicos 2015). It is a sink term in the prognostic equation of turbulence kinetic energy (TKE; e) ∂e (1) + ADV = BNC + SHR + TRP + PRC − ε, ∂t where, ε is the mean energy dissipation rate. The terms ADV , B N C, S H R, T R P, and P RC refer to advection, buoyancy production (or destruction), shear production, transport, and pressure correlation terms, respectively. Energy dissipation rate also appears in the celebrated “− 5/3 law” of Kolmogorov (1941) and Obukhov (1941a, b) E(κ) ≈ ε2/3 κ −5/3 ,
B
(2)
Sukanta Basu [email protected] Ping He [email protected] Adam W. DeMarco [email protected]
1
Faculty of Civil Engineering and Geosciences, Delft University of Technology, Delft, The Netherlands
2
Department of Aerospace Engineering, University of Michigan, Ann Arbor, USA
3
United States Air Force, Washington, D.C., USA
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where, E(κ) and κ denote the energy spectrum and wavenumber, respectively. In field campaigns or laboratory experiments, direct estimation of ε has always been a challenging task as it involves measurements of nine components of the strain rate tensor. Thus, several approximations (e.g., isotropy, Taylor’s hypothesis) have been utilized and a number of indirect measurement techniques (e.g., scintillometers, lidars) have been developed over the years. In parallel, a significant effort has been made to correlate ε with easily measurable meteorological variables. For example, several flux-based and gradient-based similarity hypotheses have been proposed (e.g., Wyngaard and Coté 1971; Wyngaard et al. 1971; Thiermann and Grassl 1992; Hartogensis and de Bruin 2005). In addition, a handful of papers also attempted to establish relationships between ε and either the vertical velocity variance (σw2 ) or TKE (e). One of the first relationships was proposed by Chen (1974). By utilizing the Kolmogorov–Obukhov spectrum (i.e., Eq. 2) with certain assumptions, he derived ε ∝ σw3 ,
(3)
where, the proportionality constant is not dimensionless. Since this derivation is only valid in the inertial range of turbulence, a band-pass filtering of vertical velocity measurements was recommended prior to computing σw . A few years later, Weinstock (1981) revisited the work of Chen (1974) and again made use of
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