论文标题
二维磁性雷利 - 纳德问题的分叉分析
Bifurcation analysis of a two-dimensional magnetic Rayleigh-Bénard problem
论文作者
论文摘要
我们使用称为Deflated Continature的数值技术对二维磁性雷利 - 贝纳德问题进行分叉分析。我们的目的是研究磁场对分叉图的影响，因为Chandrasekhar Number $ Q $增加并将其与标准（非磁性）Rayleigh-Bénard问题进行比较。我们以高chandrasekhar的稳定状态计算$ q = 10^3 $的稳定状态在雷利号$ 0 \ leq \ ra \ leq 10^5 $上。这些解决方案是通过将通气与低chandrasekhar数量的稳态延续相结合而获得的，这使我们能够探索磁场强度的影响，因为$ q $从低耦合增加而增加，在低耦合中，磁效应几乎可以忽略不计，到$ q = 10^3 $的强耦合。我们发现了大量具有丰富动力学的状态，并观察到具有几个干草叉，霍夫夫和马鞍节分叉的复杂分叉结构。我们的数值模拟表明，当$ Q $增加时，问题中分叉的开始延迟，而具有流体速度模式的解决方案与背景垂直磁场对齐的流体速度模式具有特权。此外，我们报告了一个在高磁耦合处稳定的状态分支，这表明人们可以利用磁场来区分溶液。
We perform a bifurcation analysis of a two-dimensional magnetic Rayleigh--Bénard problem using a numerical technique called deflated continuation. Our aim is to study the influence of the magnetic field on the bifurcation diagram as the Chandrasekhar number $Q$ increases and compare it to the standard (non-magnetic) Rayleigh--Bénard problem. We compute steady states at a high Chandrasekhar number of $Q=10^3$ over a range of the Rayleigh number $0\leq \Ra\leq 10^5$. These solutions are obtained by combining deflation with a continuation of steady states at low Chandrasekhar number, which allows us to explore the influence of the strength of the magnetic field as $Q$ increases from low coupling, where the magnetic effect is almost negligible, to strong coupling at $Q=10^3$. We discover a large profusion of states with rich dynamics and observe a complex bifurcation structure with several pitchfork, Hopf, and saddle-node bifurcations. Our numerical simulations show that the onset of bifurcations in the problem is delayed when $Q$ increases, while solutions with fluid velocity patterns aligning with the background vertical magnetic field are privileged. Additionally, we report a branch of states that stabilizes at high magnetic coupling, suggesting that one may take advantage of the magnetic field to discriminate solutions.
