{"title":"理想MHD湍流向平衡过渡和相干结构,第2部分","authors":"John V. Shebalin","doi":"10.3390/fluids8060181","DOIUrl":null,"url":null,"abstract":"We continue our study of the transition of ideal, homogeneous, incompressible, magnetohydrodynamic (MHD) turbulence from non-equilibrium initial conditions to equilibrium using long-time numerical simulations on a 1283 periodic grid. A Fourier spectral transform method is used to numerically integrate the dynamical equations forward in time. The six runs that previously went to near equilibrium are here extended into equilibrium. As before, we neglect dissipation as we are primarily concerned with behavior at the largest scale where this behavior has been shown to be essentially the same for ideal and real (forced and dissipative) MHD turbulence. These six runs have various combinations of imposed rotation and mean magnetic field and represent the five cases of ideal, homogeneous, incompressible, and MHD turbulence: Case I (Run 1), with no rotation or mean field; Case II (Runs 2a and 2b), where only rotation is imposed; Case III (Run 3), which has only a mean magnetic field; Case IV (Run 4), where rotation vector and mean magnetic field direction are aligned; and Case V (Run 5), which has non-aligned rotation vector and mean field directions. Statistical mechanics predicts that dynamic Fourier coefficients are zero-mean random variables, but largest-scale coherent magnetic structures emerge and manifest themselves as Fourier coefficients with very large, quasi-steady, mean values compared to their standard deviations, i.e., there is ‘broken ergodicity.’ These magnetic coherent structures appeared in all cases during transition to near equilibrium. Here, we report that, as the runs were continued, these coherent structures remained quasi-steady and energetic only in Cases I and II, while Case IV maintained its coherent structure but at comparatively low energy. The coherent structures that appeared in transition in Cases III and V were seen to collapse as their associated runs extended into equilibrium. The creation of largest-scale, coherent magnetic structure appears to be a dynamo process inherent in ideal MHD turbulence, particularly in Cases I and II, i.e., those cases most pertinent to planets and stars. Furthermore, the statistical theory of ideal MHD turbulence has proven to apply at the largest scale, even when dissipation and forcing are included. This, along with the discovery and explanation of dynamically broken ergodicity, is essentially a solution to the ‘dynamo problem’.","PeriodicalId":12397,"journal":{"name":"Fluids","volume":"77 1","pages":"0"},"PeriodicalIF":1.8000,"publicationDate":"2023-06-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Transition to Equilibrium and Coherent Structure in Ideal MHD Turbulence, Part 2\",\"authors\":\"John V. Shebalin\",\"doi\":\"10.3390/fluids8060181\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"We continue our study of the transition of ideal, homogeneous, incompressible, magnetohydrodynamic (MHD) turbulence from non-equilibrium initial conditions to equilibrium using long-time numerical simulations on a 1283 periodic grid. A Fourier spectral transform method is used to numerically integrate the dynamical equations forward in time. The six runs that previously went to near equilibrium are here extended into equilibrium. As before, we neglect dissipation as we are primarily concerned with behavior at the largest scale where this behavior has been shown to be essentially the same for ideal and real (forced and dissipative) MHD turbulence. These six runs have various combinations of imposed rotation and mean magnetic field and represent the five cases of ideal, homogeneous, incompressible, and MHD turbulence: Case I (Run 1), with no rotation or mean field; Case II (Runs 2a and 2b), where only rotation is imposed; Case III (Run 3), which has only a mean magnetic field; Case IV (Run 4), where rotation vector and mean magnetic field direction are aligned; and Case V (Run 5), which has non-aligned rotation vector and mean field directions. Statistical mechanics predicts that dynamic Fourier coefficients are zero-mean random variables, but largest-scale coherent magnetic structures emerge and manifest themselves as Fourier coefficients with very large, quasi-steady, mean values compared to their standard deviations, i.e., there is ‘broken ergodicity.’ These magnetic coherent structures appeared in all cases during transition to near equilibrium. Here, we report that, as the runs were continued, these coherent structures remained quasi-steady and energetic only in Cases I and II, while Case IV maintained its coherent structure but at comparatively low energy. The coherent structures that appeared in transition in Cases III and V were seen to collapse as their associated runs extended into equilibrium. The creation of largest-scale, coherent magnetic structure appears to be a dynamo process inherent in ideal MHD turbulence, particularly in Cases I and II, i.e., those cases most pertinent to planets and stars. Furthermore, the statistical theory of ideal MHD turbulence has proven to apply at the largest scale, even when dissipation and forcing are included. This, along with the discovery and explanation of dynamically broken ergodicity, is essentially a solution to the ‘dynamo problem’.\",\"PeriodicalId\":12397,\"journal\":{\"name\":\"Fluids\",\"volume\":\"77 1\",\"pages\":\"0\"},\"PeriodicalIF\":1.8000,\"publicationDate\":\"2023-06-14\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Fluids\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.3390/fluids8060181\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"MECHANICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Fluids","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.3390/fluids8060181","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MECHANICS","Score":null,"Total":0}
引用次数: 0
摘要
我们继续研究理想的、均匀的、不可压缩的、磁流体动力学(MHD)湍流从非平衡初始条件到平衡的转变,使用1283周期网格上的长时间数值模拟。采用傅里叶谱变换方法对动力学方程进行时间正演数值积分。之前接近平衡的六次运行在这里延伸到平衡状态。和以前一样,我们忽略了耗散,因为我们主要关注的是在最大尺度上的行为,而这种行为已经被证明在理想和实际(强迫和耗散)MHD湍流中基本上是相同的。这六次运行具有施加旋转和平均磁场的不同组合,代表了理想、均匀、不可压缩和MHD湍流的五种情况:情况1(运行1),没有旋转或平均场;情况II(运行2a和2b),其中只施加旋转;情形III(运行3),只有平均磁场;情形IV (Run 4),旋转矢量与平均磁场方向对齐;Case V (Run 5),它具有不对齐的旋转矢量和平均场方向。统计力学预测,动态傅里叶系数是零均值随机变量,但最大规模的相干磁结构出现并表现为傅里叶系数,其平均值与其标准差相比非常大,准稳定,即存在“破遍历性”。这些磁相干结构在过渡到接近平衡状态的所有情况下都出现了。在这里,我们报告说,随着运行的继续,这些相干结构只在情形1和情形2中保持准稳定和能量,而情形4保持其相干结构,但能量相对较低。在案例III和案例V中,在过渡时期出现的连贯结构随着其相关运行扩展到平衡状态而崩溃。最大规模的相干磁结构的产生似乎是理想MHD湍流所固有的一个发电机过程,特别是在情况I和II中,即与行星和恒星最相关的那些情况。此外,理想MHD湍流的统计理论已被证明适用于最大尺度,即使包括耗散和强迫。这与动态破缺遍历性的发现和解释一起,本质上是“发电机问题”的解决方案。
Transition to Equilibrium and Coherent Structure in Ideal MHD Turbulence, Part 2
We continue our study of the transition of ideal, homogeneous, incompressible, magnetohydrodynamic (MHD) turbulence from non-equilibrium initial conditions to equilibrium using long-time numerical simulations on a 1283 periodic grid. A Fourier spectral transform method is used to numerically integrate the dynamical equations forward in time. The six runs that previously went to near equilibrium are here extended into equilibrium. As before, we neglect dissipation as we are primarily concerned with behavior at the largest scale where this behavior has been shown to be essentially the same for ideal and real (forced and dissipative) MHD turbulence. These six runs have various combinations of imposed rotation and mean magnetic field and represent the five cases of ideal, homogeneous, incompressible, and MHD turbulence: Case I (Run 1), with no rotation or mean field; Case II (Runs 2a and 2b), where only rotation is imposed; Case III (Run 3), which has only a mean magnetic field; Case IV (Run 4), where rotation vector and mean magnetic field direction are aligned; and Case V (Run 5), which has non-aligned rotation vector and mean field directions. Statistical mechanics predicts that dynamic Fourier coefficients are zero-mean random variables, but largest-scale coherent magnetic structures emerge and manifest themselves as Fourier coefficients with very large, quasi-steady, mean values compared to their standard deviations, i.e., there is ‘broken ergodicity.’ These magnetic coherent structures appeared in all cases during transition to near equilibrium. Here, we report that, as the runs were continued, these coherent structures remained quasi-steady and energetic only in Cases I and II, while Case IV maintained its coherent structure but at comparatively low energy. The coherent structures that appeared in transition in Cases III and V were seen to collapse as their associated runs extended into equilibrium. The creation of largest-scale, coherent magnetic structure appears to be a dynamo process inherent in ideal MHD turbulence, particularly in Cases I and II, i.e., those cases most pertinent to planets and stars. Furthermore, the statistical theory of ideal MHD turbulence has proven to apply at the largest scale, even when dissipation and forcing are included. This, along with the discovery and explanation of dynamically broken ergodicity, is essentially a solution to the ‘dynamo problem’.