We study the nonlinear mode competition of various spiral instabilities in magnetised Taylor-Couette flow. The resulting finite-amplitude mixed-mode solution branches are tracked using the annular-parallelogram periodic domain approach developed by Deguchi & Altmeyer (2013). Mode competition phenomena are studied in both the anti-cyclonic and cyclonic Rayleigh-stable regimes. In the anti-cyclonic sub-rotation regime, with the inner cylinder rotating faster than the outer, Hollerbach, Teeluck & Rudiger (2010) found competing axisymmetric and non-axisymmetric magneto-rotational linearly unstable modes within the parameter range where experimental investigation is feasible. Here we confirm the existence of mode competition and compute the nonlinear mixed-mode solutions that result from it. In the cyclonic super-rotating regime, with the inner cylinder rotating slower than the outer, Deguchi (2017) recently found a non-axisymmetric purely hydrodynamic linear instability that coexists with the non-axisymmetric magneto-rotational instability discovered a little earlier by Rudiger, Schultz, Gellert & Stefani (2016). We show that nonlinear interactions of these instabilities give rise to rich pattern-formation phenomena leading to drastic angular momentum transport enhancement/reduction.

中文翻译：

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磁化 Taylor-Couette 流中的完全非线性模式竞争

我们研究了磁化 Taylor-Couette 流中各种螺旋不稳定性的非线性模式竞争。使用 Deguchi & Altmeyer (2013) 开发的环形平行四边形周期域方法跟踪得到的有限幅度混合模式解分支。模式竞争现象在反气旋和气旋瑞利稳定机制中都得到了研究。在反气旋子旋转状态下，内圆柱比外圆柱旋转得更快，Hollerbach、Teeluck & Rudiger (2010) 在实验研究可行的参数范围内发现了相互竞争的轴对称和非轴对称磁旋转线性不稳定模式. 在这里，我们确认模式竞争的存在并计算由此产生的非线性混合模式解决方案。在气旋超旋转状态下，由于内圆柱的旋转速度比外圆柱慢，Deguchi (2017) 最近发现了一种非轴对称的纯流体动力学线性不稳定性，它与 Rudiger、Schultz、Gellert 和 Stefani（2016）早先发现的非轴对称磁旋转不稳定性共存. 我们表明，这些不稳定性的非线性相互作用会产生丰富的模式形成现象，从而导致角动量传输的急剧增强/减少。