This paper reports a breakdown in linear stability theory under conditions of neutral stability that is deduced by an examination of exponential modes of the form |$h\approx{{e}^{i(kx-\omega t)}}$|⁠, where |$h$| is a response to a disturbance, |$k$| is a real wavenumber and |$\omega (k)$| is a wavelength-dependent complex frequency. In a previous paper, King et al. (2016, Stability of algebraically unstable dispersive flows. Phys. Rev. Fluids, 1, 073604) demonstrates that when Im|$[\omega (k)]=0$| for all |$k$|⁠, it is possible for a system response to grow or damp algebraically as |$h\approx{{t}^{s}}$| where |$s$| is a fractional power. The growth is deduced through an asymptotic analysis of the Fourier integral that inherently invokes the superposition of an infinite number of modes. In this paper, the more typical case associated with the transition from stability to instability is examined in which Im|$[\omega (k)]=0$| for a single mode (i.e. for one value of |$k$|⁠) at neutral stability. Two partial differential equation systems are examined, one that has been constructed to elucidate key features of the stability threshold, and a second that models the well-studied problem of rectilinear Newtonian flow down an inclined plane. In both cases, algebraic growth/decay is deduced at the neutral stability boundary, and the propagation features of the responses are examined.

中文翻译：

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关于经典中性流中波浪的稳定性

本文报告了在中性稳定性条件下线性稳定性理论的细目分类，该细目由检查形式| $ h \ approx {{e} ^ {i（kx- \ omega t）}} $ |⁠得出的指数模态得出，其中| $ h $ | 是对干扰的响应，| $ k $ | 是实波数，|| $ \ omega（k）$ | 是与波长有关的复数频率。在以前的论文中，金等人。（2016，代数不稳定色散流的稳定性。Phys。Rev . Fluids，1，073604）证明当Im | $ [\ omega（k）] = 0 $ | 对于所有| $ k $ |⁠，系统响应可能会随着| $ h \ approx {{t} ^ {s}} $ |的代数增长或衰减。其中| $ s $ | 是分数幂。增长是通过对傅立叶积分的渐近分析得出的，该分析固有地调用了无限多个模的叠加。在本文中，研究了从稳定到不稳定过渡的更典型情况，其中Im | $ [\ omega（k）] = 0 $ | 单一模式（即| $ k $ |⁠的一个值））保持中性。研究了两个偏微分方程系统，一个已建立以阐明稳定性阈值的关键特征，第二个已建模了经过充分研究的直线牛顿流向下倾斜的平面模型。在这两种情况下，都在中性稳定边界推导代数增长/衰变，并检查响应的传播特征。