We study structure formation in two-dimensional turbulence driven by an external force, interpolating between linear instability forcing and random stirring, subject to nonlinear damping. Using extensive direct numerical simulations, we uncover a rich parameter space featuring four distinct branches of stationary solutions: large-scale vortices, hybrid states with embedded shielded vortices (SVs) of either sign, and two states composed of many similar SVs. Of the latter, the first is a dense vortex gas where all SVs have the same sign and diffuse across the domain. The second is a hexagonal vortex crystal forming from this gas when the linear instability is sufficiently weak. These solutions coexist stably over a wide parameter range. The late-time evolution of the system from small-amplitude initial conditions is nearly self-similar, involving three phases: initial inverse cascade, random nucleation of SVs from turbulence and, once a critical number of vortices is reached, a phase of explosive nucleation of SVs, leading to a statistically stationary state. The vortex gas is continued in the forcing parameter, revealing a sharp transition towards the crystal state as the forcing strength decreases. This transition is analysed in terms of the diffusivity of individual vortices using ideas from statistical physics. The crystal can also decay via an inverse cascade resulting from the breakdown of shielding or insufficient nonlinear damping acting on SVs. Our study highlights the importance of the forcing details in two-dimensional turbulence and reveals the presence of non-trivial SV states in this system, specifically the emergence and melting of a vortex crystal.

中文翻译：

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不稳定驱动的二维湍流中从涡旋气体到涡旋晶体

我们研究由外力驱动的二维湍流中的结构形成，在线性不稳定强迫和随机搅拌之间进行插值，并受到非线性阻尼的影响。通过广泛的直接数值模拟，我们发现了一个丰富的参数空间，具有四个不同的稳态解分支：大规模涡流、具有任一符号的嵌入屏蔽涡流 (SV) 的混合态以及由许多相似 SV 组成的两个态。后者中，第一个是稠密涡旋气体，其中所有 SV 具有相同的符号并在域中扩散。第二种是当线性不稳定性足够弱时，由这种气体形成的六方涡旋晶体。这些解决方案在很宽的参数范围内稳定共存。系统从小振幅初始条件的后期演化几乎是自相似的，涉及三个阶段：初始逆级联、湍流中SV的随机成核，以及一旦达到临界数量的涡流，爆炸成核阶段SV，导致统计静止状态。涡流气体在强迫参数中继续存在，随着强迫强度的降低，显示出向晶态的急剧转变。利用统计物理学的思想，根据各个涡旋的扩散性来分析这种转变。由于屏蔽击穿或作用于 SV 的非线性阻尼不足，晶体还会通过反向级联而衰减。我们的研究强调了二维湍流中强迫细节的重要性，并揭示了该系统中非平凡的 SV 态的存在，特别是涡旋晶体的出现和熔化。