arXiv subjects

Jiyang He

Publications and source records attributed to Jiyang He.

6 recordsLinked to original sources

arXiv2026

Zonal flows driven by libration in rotating spherical shells: the case of periodic characteristic paths

This work investigates the weakly nonlinear dynamics of internal shear layers and the mean zonal flow induced by the longitudinal libration of an inner core within a spherical shell. Building on the work of He et al. (J. Fluid Mech., vol. 939, 2022, A3), which focused on linear dynamics, we adopt a similar setup to explore the nonlinear regime using both asymptotic theory and numerical computations, with Ekman numbers as low as $E=10^{-10}$. A specific forcing frequency of $\widehat{\omega}=\sqrt{2}\widehat{\Omega}$, where $\widehat{\Omega}$ denotes the rotation rate, is introduced to generate a closed rectangular path of characteristics for the inertial wave beam generated at the critical latitude. Our approach extends previous results by Le Diz\`es (J. Fluid Mech., vol. 899, 2020, A21) and reveals that nonlinear interactions are predominantly localized around regions where the wave beam reflects on the boundary. We derive specific scaling laws governing the nonlinear interactions: the width of the interaction region scales as $E^{1/3}$, and the amplitude of the resulting mean zonal flow scales as $E^{1/6}$ in general. However, near the rotation axis, where the singularity of the self-similar solution becomes more pronounced, the amplitude exhibits a scaling of $E^{-1/2}$. In addition, our study also examines the nonlinear interactions of beams which are governed by different scaling laws. Through comparison with numerical results, we validate the theoretical predictions of the asymptotic framework, observing good agreement as the Ekman number decreases.

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arXiv2025

Final states of two-dimensional turbulence above large-scale topography: stationary vortex solutions and barotropic stability

The final states of freely decaying two-dimensional (2D) topographic turbulence consist of a background flow and localized vortices. While the background flow satisfies a linear potential vorticity (PV)-streamfunction relation, the vortex structures remain poorly understood. To address this gap and ensure oceanic relevance, we examine quasi-stationary final states of 2D turbulence over a sinusoidal topography featuring a bump and a dip, where two oppositely signed vortices are locked to the topographic extrema. After subtracting the background flow, the vortices exhibit a "sinh"-like PV-streamfunction relation, as observed in flat-bottom turbulence. Motivated by Gaussian vortex profiles in flat-bottom turbulence, we propose an empirical model combining the background flow with Gaussian vortices centered at the topographic extrema. This model accurately reproduces quasi-stationary states and yields locally stationary solutions to the inviscid governing equation. We further test the model under complex topography and high-energy conditions, confirming that the "sinh"-like trend and Gaussian profiles are robust features of localized vortices. Linear stability analyses of these stationary vortex solutions reveal background flow-dependent stability: cyclone/elevation and anticyclone/depression configurations are stable at low background energy, while anticyclone/elevation and cyclone/depression configurations are stable at high background energy. These findings align with vortex-topography correlations observed in simulations across energy regimes. Our results provide explicit vortex solutions for quasi-stationary final states of 2D topographic turbulence and elucidate the mechanism underlying vortex-topography correlations through stability analyses of vortices embedded in topographic background flows.

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arXiv2024

Internal shear layers generated by a vertically oscillating cylinder in unbounded and bounded rotating fluids

In rotating fluids, the viscous smoothing of inviscid singular inertial waves leads to the formation of internal shear layers. In previous works, we analysed the internal shear layers excited by a viscous forcing (longitudinal libration) in a spherical shell geometry (He \textit{et al.}, \textit{J. Fluid Mech.} {\bf 939}, A3, 2022; {\bf 974}, A3, 2023). We now consider the stronger inviscid forcing corresponding to the vertical oscillation of the inner boundary. We limit our analysis to two-dimensional geometries but examine three different configurations: freely-propagating wave beams in an unbounded domain and two wave patterns (a periodic orbit and an attractor) in a cylindrical shell geometry. The asymptotic structures of the internal shear layers are assumed to follow the similarity solution of Moore \& Saffman (\textit{Phil. Trans. R. Soc. Lond. A}, 264 (1156), 1969, 597-634) in the small viscous limit. The two undefined parameters of the similarity solution (singularity strength and amplitude) are derived by asymptotically matching the similarity solution with the inviscid solution. For each case, the derivation of the latter is achieved either through separation of variables combined with analytical continuation or the method of characteristics. Global inviscid solutions, when obtained, closely match numerical solutions for small Ekman numbers far from the critical lines, while viscous asymptotic solutions show excellent performance near those lines. The amplitude scalings of the internal shear layers excited by an inviscid forcing are found to be divergent as the Ekman number $E$ decreases, specifically $O(E^{-1/6})$ for the critical point singularity and $O(E^{-1/3})$ for attractors, in contrast to the convergent scalings found for a viscous forcing.

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arXiv2024

Multiple states of two-dimensional turbulence above topography

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.

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arXiv2023

Internal shear layers in librating spherical shells: the case of attractors

Following our previous work on periodic ray paths (He et al, 2022), we study asymptotically and numerically the structure of internal shear layers for very small Ekman numbers in a three-dimensional (3D) spherical shell and in a two-dimensional (2D) cylindrical annulus when the rays converge towards an attractor. We first show that the asymptotic solution obtained by propagating the self-similar solution generated at the critical latitude on the librating inner core describes the main features of the numerical solution. The internal shear layer structure and the scaling for its width and velocity amplitude in $E^{1/3}$ and $E^{1/12}$ respectively are recovered. The amplitude of the asymptotic solution is shown to decrease to $E^{1/6}$ when it reaches the attractor, as it is also observed numerically. However, some discrepancies are observed close to the particular attractors along which the phase of the wave beam remains constant. Another asymptotic solution close to those attractors is then constructed using the model of Ogilvie (2005). The solution obtained for the velocity has an $O(E^{1/6})$ amplitude, but a different self-similar structure than the critical-latitude solution. It also depends on the Ekman pumping at the contact points of the attractor with the boundaries. We demonstrate that it reproduces correctly the numerical solution. Surprisingly, the solution close to an attractor with phase shift (that is an attractor that touches the axis in 3D or in 2D with a symmetric forcing) is found to be much weaker.

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arXiv2018

Effects of Distributed Roughness on Crossflow Instability through Generalized Resonance Mechanisms

Experiments have shown that micron-sized distributed surface roughness can significantly promote transition in a three-dimensional boundary layer dominated by crossflow insta- bility. This sensitive effect has not yet been fully explained physically and mathematically. Unlike past researches focusing on the receptivity of the boundary layer to surface roughness, or on the local stability of the modified mean flow, this paper seeks possible inherent mechanisms by investigating the effects of distributed surface roughness on crossflow instability through resonant interactions with eigenmodes. A key observation is that the perturbation induced by roughness with specific wavenumbers can interact with two eigenmodes (travelling and stationary vortices) through triadic resonance, or interact with one eigenmode (stationary vortices) through Bragg scattering. Unlike the usual triadic resonance of neutral, or nearly neutral, eigenmodes, the present triadic resonance can take place among modes with O(1) growth rates, provided that these are equal; unlike the usual Bragg scattering involving neutral waves, crossflow stationary vortices can also be unstable. For these amplifying waves, the generalized triadic resonance and Bragg scattering are put forward, and the resulting corrections to the growth rates are derived by a multiple-scale method. The analysis is extended to the case where up to four crossflow vortices interact with each other in the presence of suitable roughness components. The numerical results for Falkner-Skan-Cooke boundary layers show that roughness with a small height (a few percent of the local boundary-layer thickness) can change growth rates substantially (by a more-or-less O(1) amount)....

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