Diapycnal mixing in layered stratified plane Couette flow quantified in a tracer-based coordinate

Abstract

The mixing properties of statically stable density interfaces subject to imposed vertical shear are studied using direct numerical simulations of stratified plane Couette flow. The simulations are designed to investigate possible self-maintaining mechanisms of sharp density interfaces motivated by Phillips’ argument (Deep-Sea Res., vol. 19, 1972, pp. 79–81) by which layers and interfaces can spontaneously form due to vertical variations of diapycnal flux. At the start of each simulation, a sharp density interface with the same initial thickness is introduced at the midplane between two flat, horizontal walls counter-moving at velocities$\pm U_{w}$. Particular attention is paid to the effects of varying Prandtl number$\mathit{Pr}\equiv \unicode[STIX]{x1D708}/\unicode[STIX]{x1D705}$, where$\unicode[STIX]{x1D708}$and$\unicode[STIX]{x1D705}$are the molecular kinematic viscosity and diffusivity respectively, over two orders of magnitude from 0.7, 7 and 70. Varying$\mathit{Pr}$enables the system to access a considerable range of characteristic turbulent Péclet numbers$\mathit{Pe}_{\ast }\equiv {\mathcal{U}}_{\ast }{\mathcal{L}}_{\ast }/\unicode[STIX]{x1D705}$, where${\mathcal{U}}_{\ast }$and${\mathcal{L}}_{\ast }$are characteristic velocity and length scales, respectively, of the motion which acts to ‘scour’ the density interface. The dynamics of the interface varies with the stability of the interface which is characterised by a bulk Richardson number$\mathit{Ri}\,\equiv \,b_{0}h/U_{w}^{2}$, where$b_{0}$is half the initial buoyancy difference across the interface and$h$is the half-height of the channel. Shear-induced turbulence occurs at small$\mathit{Ri}$, whereas internal waves propagating on the interface dominate at large$\mathit{Ri}$. For a highly stable (i.e. large$\mathit{Ri}$) interface at sufficiently large$\mathit{Pe}_{\ast }$, the complex interfacial dynamics allows the interface to remain sharp. This ‘self-sharpening’ is due to the combined effects of the ‘scouring’ induced by the turbulence external to the interface and comparatively weak molecular diffusion across the core region of the interface. The effective diapycnal diffusivity and irreversible buoyancy flux are quantified in the tracer-based reference coordinate proposed by Winters & D’Asaro (J. Fluid Mech., vol. 317, 1996, pp. 179–193) and Nakamura (J. Atmos. Sci., vol. 53, 1996, pp. 1524–1537), which enables a detailed investigation of the self-sharpening process by analysing the local budget of buoyancy gradient in the reference coordinate. We further discuss the dependence of the effective diffusivity and overall mixing efficiency on the characteristic parameters of the flow, such as the buoyancy Reynolds number and the local gradient Richardson number, and highlight the possible role of the molecular properties of fluids on diapycnal mixing.

Stratified shear instability in a field of pre-existing turbulence

“…Thorpe et al (2013) and Li et al (2015) showed that turbulence can act to either stabilize or destabilize a flow, depending on its distribution and the flow parameters. Furthermore, recent numerical studies have shown that external turbulence can modify the evolution of other stratified flows in a variety of ways, including sharpening an existing shear layer at high Pr (Zhou et al 2017) and increasing the rate of decay of a stratified wake (Pal & Sarkar 2015). As such, understanding the impact of different distributions of pre-existing turbulence on the fully nonlinear evolution of stratified shear flows such as the one we have considered here is an important avenue of future research.…”

Mixing and Formation of Layers by Internal Wave Forcing

“…They can be explained by the vertical heterogeneity of the normal mode shear stress in the present experiments. A precise evaluation of the different terms in the stratification budget with a similar approach as in Zhou et al () is needed to quantitatively describe the relation between internal wave induced mixing and the generation of layers. Of particular interest is the link between the formation of a forbidden zone for internal wave propagation and the emergence of layers.…”

“…In fact, the largest values of are measured in the case where the triadic resonant generation is favored, leading to the oblique propagation of secondary waves with spatial scale smaller than the forcing scale. These waves participate in the direct energy cascade toward the scales of mixing and contribute to enhance , by comparison with the case of weak TRI (Dossmann et al, ) However, while a fraction of the secondary waves energy participates to irreversible mixing, some of this energy is involved in reversible transfers between the potential and kinetic energy compartments of the flow (e.g., Hughes et al, ; Venayagamoorthy & Koseff, ; Zhou et al, ), such as wave‐induced stirring. These reversible transfers occur at spacetime scales that are different from the forcing scales over which the buoyancy/velocity correlations are averaged and are visibly accounted for in .…”

“…Thorpe () offers a review of the various generation mechanisms for layer generation, arguing that the Phillips‐Posmentier mechanism can sustain layers forced by other processes. The conditions for formation and stability of a layer have been recently refined relying on numerical studies in a stratified shear flow (Taylor & Zhou, ; Zhou et al, ). Precise stratification budgets have allowed for the identification of nondimensional parameters for layer formation.…”

“…While parameterizations commonly rely on Reynolds‐averaged buoyancy flux as a prognostic value for irreversible mixing properties, recent studies argue that reversible processes such as large‐scale stirring may also contribute to the buoyancy flux (Venayagamoorthy & Koseff, ; Zhou et al, ). To evaluate the irreversible component of the buoyancy flux that eventually contributes to raise the center of mass of the flow, it is important to provide a diagnostic of mixing processes based on the evolution of the stratification between subsequent mixing events.…”