[Paper Review] Nonlinear waves in stratified Taylor--Couette flow. Part 1. Layer formation

Colin Leclercq, Jamie Partridge+4 more|arXiv (Cornell University)|Sep 9, 2016

Oceanographic and Atmospheric Processes10 references3 citations

TL;DR

This study reveals that non-axisymmetric helical instabilities—specifically mixed-ribbons and cross-spirals—drive nonlinear coherent structures in axially stratified Taylor–Couette flow, leading to dynamic layering and influencing the formation of density interfaces. Despite axisymmetric modes being linearly critical, these non-axisymmetric waves dominate layer depth selection, especially at high Schmidt numbers, where static staircase-like profiles emerge.

ABSTRACT

This paper is the first part of a two-fold study of mixing, i.e. the formation of layers and upwelling of buoyancy, in axially stratified Taylor--Couette flow, with fixed outer cylinder. Using linear analysis and direct numerical simulation, we show the critical role played by non-axisymmetric instability modes, despite the fact that the flow is centrifugally unstable in the sense of Rayleigh's criterion. Interactions between helical modes of opposite handedness leads to the formation of nonlinear coherent structures: (mixed)-ribbons and (mixed)-cross-spirals. These give birth to complex density interface patterns, seemingly appearing and disappearing periodically as the coherent structure slowly rotates around the annulus. These coherent structures seem to be responsible for the formation of layers reported in a recent experiment by Oglethorpe et al. (2013). We distinguish `dynamic layering', instantaneous, localized and caused by the vortical motions, from `static layering' corresponding to the formation of a `staircase profile' in the adiabatically sorted background density. The latter only occurs at large enough Schmidt number, revealing the significant impact of the Schmidt number in the layering process.

Motivation & Objective

• To investigate the role of non-axisymmetric instabilities in layer formation within axially stratified Taylor–Couette flow, challenging prior assumptions that axisymmetric modes dominate.
• To resolve the discrepancy between early experiments (Boubnov et al., 1995) and later simulations (Hua et al., 1997; Caton et al., 1999) regarding the primary instability type.
• To examine how coherent structures such as ribbons and cross-spirals govern the depth and dynamics of density layers, especially at large Schmidt numbers.
• To assess the impact of the Schmidt number on the transition from dynamic to static layering, and on the robustness of layer depth selection mechanisms.
• To reconcile experimental observations (Oglethorpe et al., 2013) with numerical simulations, particularly regarding probe-averaged measurements and unobserved non-axisymmetric dynamics.

Proposed method

• Employed linear stability analysis to identify the thresholds and growth rates of non-axisymmetric helical modes, including m = 1, m = 3, and mixed modes.
• Conducted large-gap, high-Reynolds-number direct numerical simulations (DNS) up to Re = 10⁴ and Sc = 730 to resolve nonlinear interactions and coherent structure formation.
• Used viscous linear stability analysis to confirm the supercritical nature of bifurcations for ribbon and cross-spiral structures, resolving prior contradictions in the literature.
• Revisited numerical and experimental data from Boubnov et al. (1995), Hua et al. (1997), and Caton et al. (1999, 2000) in a larger domain with impulsive start conditions to assess sensitivity to initial conditions.
• Applied shadowgraphy and 3D density field reconstruction to visualize non-axisymmetric coherent structures and their role in layering.
• Analyzed buoyancy flux and density variance to distinguish dynamic layering (instantaneous vortical overturning) from static layering (adiabatically sorted staircase profiles).

Experimental results

Research questions

• RQ1 Why do experimental observations of layer formation in STC (Oglethorpe et al., 2013) differ from earlier simulations that assumed axisymmetric instabilities?
• RQ2 What is the role of non-axisymmetric helical modes in the formation of coherent structures such as ribbons and cross-spirals in stratified Taylor–Couette flow?
• RQ3 How does the Schmidt number influence the transition from dynamic layering to static, staircase-like density profiles?
• RQ4 Why do axisymmetric simulations and experiments fail to capture the true dynamics of layer formation when non-axisymmetric modes are present?
• RQ5 Can the depth of well-mixed layers be predicted by coherent structures rather than by Ozmidov or buoyancy length scales?

Key findings

• Non-axisymmetric helical modes of opposite handedness interact nonlinearly to form coherent structures such as mixed-ribbons and mixed-cross-spirals, which are responsible for the observed periodic appearance and disappearance of density interfaces.
• Despite Rayleigh’s criterion indicating axisymmetric centrifugal instability, the primary instability is non-axisymmetric, with linear growth rates of non-axisymmetric modes closely matching those of the axisymmetric mode, suggesting near-simultaneous bifurcation.
• The bifurcation to ribbon and cross-spiral structures is supercritical, as confirmed by DNS, resolving the long-standing discrepancy between early experiments (Boubnov et al., 1995) and later simulations (Hua et al., 1997).
• Dynamic layering—caused by vortical overturning of the density field—is dominant at low Schmidt numbers, while static layering (staircase profiles) only emerges at high Schmidt numbers (Sc ≥ 16), indicating a strong dependence on molecular diffusion.
• The Ozmidov length scale significantly underestimates the actual layer depth, which is instead governed by the underlying coherent structure dynamics rather than turbulent mixing scales.
• Experimental layer depth measurements in Oglethorpe et al. (2013) likely reflect axisymmetrically averaged values due to probe filtering, suggesting that non-axisymmetric coherent structures were not resolved without shadowgraphy.