Turbulent Mechanical Energy Budget in Stably Stratified Baroclinic Flows over Sloping Terrain

Abstract

Analysis of second-moment budget equations in a slope-oriented coordinate frame exhibits the pathways of exchange between the potential energy of mean flow and the total turbulent mechanical energy. It is shown that this process is controlled by the inclination of the potential temperature gradient. Hence, this parameter should be considered in studies of turbulence in slope flows as well as the slope inclination. The concept of turbulent potential energy is generalized to include baroclinicity, and is used to explain the role of along-slope turbulent heat flux in energy conversions. A generalization of static stability criteria for baroclinic conditions is also proposed. In addition, the presence of feedback between the turbulent heat flux and the temperature variance in stably-stratified flows is identified, which implies the existence of oscillatory modes characterized by the Brunt-Väisäla frequency.

Scalar-Flux Similarity in the Layer Near the Surface Over Mountainous Terrain

“… Coordinate system (i.e., the frame of reference for the projection of the temperature and momentum fluxes): while it is customary to use a terrain-following coordinate system over sloped surfaces, so that the temperature and momentum fluxes are normal to the surface, this introduces some difficulty because even if the local (perturbation) isentropes are parallel to the slope, the dominant direction of heat fluxes at some distance away from the surface is vertical (Stiperski and Rotach 2016 ; Oldroyd et al. 2016a , b ; Lobocki 2017 ). We have, therefore, tested the hypothesis that the vertical (rather than the normal) heat fluxes constitute the appropriate scaling variable by comparing calculated with slope-normal and vertical temperature fluxes for the four i-Box sites with sloping terrain (Figs.…”

Boundary-Layer Flow Over Complex Topography

“…• In contrast to the shear production term for TKE, the gradient of mean virtual potential temperature and the slope-normal temperature flux are both large at the jet peak so the dominant gradient production term in the rate equation for the variance of virtual potential temperature, , remains large and profiles of exhibit a local maximum near the jet peak (Denby 1999;Grachev et al 2016). Compared to TKE, and its relation to turbulent potential energy (Zilitinkevich et al 2009;Łobocki 2017) has received much less attention with the exception of variance similarity scaling efforts as discussed next.…”

Prediction of gradient‐based similarity functions from the Mellor–Yamada model

“…Indeed, as 𝜑 x appears in the denominators of fractions in this scaling (Table 1, last column), and 𝜑 x = 0 when Ri 𝜑 300 = 0.9 𝜑 250 , there is a singularity at Ri ≈ 0.686 followed by a change of sign. As follows from the definition in Equation 15, U 2 𝜃 is the turbulent potential energy (TPE: Dalaudier and Sidi, 1987;Łobocki, 2017), hence the function presented in the second plot (center) is the TKE/TPE ratio. As the TPE is null at neutral equilibrium, this ratio tends to infinity when Ri → 0.…”

“…Note that the mechanism of turbulence collapse described by the MSHF theory does not preclude the existence of turbulence beneath the maximum location and does not take into account restoration mechanisms such as generation by gravity waves or other submeso motions or conversion between turbulent kinetic energy (TKE) and turbulent potential energy (TPE). The last concept was also introduced in the last quarter of a century and earned due attention in the development of similarity theory (Zilitinkevich and Esau, 2007) and modeling (Zilitinkevich et al ., 2007; Li et al ., 2016; Łobocki, 2017). The absence of such mechanisms in the parametrizations used in numerical weather prediction and climate models is a likely reason for a frequent problem of models—a spurious decoupling of flow from the underlying surface associated with unrealistically rapid and intense surface cooling (e.g., Derbyshire, 1999; Cuxart et al ., 2005), commonly termed “runaway cooling”.…”

“…All versions of the model predict a decrease of this share with growing stability. The neutral equilibrium value ranges between MY82 and all the MYNN versions due to different model constants.As follows from the definition in Equation15, U 2 𝜃 is the turbulent potential energy (TPE:Dalaudier and Sidi, 1987;Łobocki, 2017), hence the function presented in the second plot (center) is the TKE/TPE ratio. As the TPE is null at neutral equilibrium, this ratio tends to infinity when Ri → 0.…”