Multiple states of two-dimensional turbulence above topography

TL;DR

This paper investigates how initial length scales influence the long-term states of two-dimensional turbulence over topography, revealing dependencies on initial conditions that affect vortex behavior, enstrophy, and potential vorticity homogenization.

Contribution

It extends previous work by demonstrating the impact of initial length scale on turbulence states, linking them to minimum enstrophy and PV homogenization.

Findings

Long-term states depend on initial length scale and enstrophy.

Large initial scales lead to near-minimum enstrophy states.

Small initial scales produce more vortices and PV homogenization.

Abstract

The recent work of Siegelman \& Young (PNAS, vol. 120(44), 2023, pp. e2308018120) revealed two extreme states reached by the evolution of unforced and weakly-damped two-dimensional turbulence above random rough topography, separated by a critical kinetic energy $E_\#$. The low- and high-energy solutions correspond to topographically-locked and roaming vortices, surrounded by non-uniform and homogeneous background potential vorticity (PV), respectively. However, we found that these phenomena are restricted to the particular intermediate length scale where the energy was initially injected into the system. Through simulations initialised by injecting the energy at larger and smaller length scales, we found that the long-term state of topographic turbulence is also dependent on the initial length scale and thus the initial enstrophy. If the initial length scale is comparable with the domain size, the long-term flow field resembles the minimum-enstrophy state proposed by Bretherton \& Haidvogel (J. Fluid Mech., vol.78(1), 1976, pp. 129-154), with very few topographically-locked vortices; the long-term enstrophy is quite close to the minimum value, especially when the energy is no larger than $E_\#$. As the initial length scale becomes smaller, more vortices nucleate and become more mobile; the long-term enstrophy increasingly deviates from the minimum value. Simultaneously, the background PV tends to homogenization, even if the energy is below $E_\#$. These results complement the phenomenology of topographic turbulence documented by Siegelman \& Young, by showing that the minimum-enstrophy and background PV homogenization states can be adequately approached by large- and small-scale initial fields, respectively, with relatively arbitrary energy.