恒星湍对流过程研究以及在盾牌座δ型变星中的验证
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摘要
恒星内部结构和演化是天体物理研究中的根本问题之一，其中对流运动主宰了恒星对流区中的能量传递过程，对恒星结构和演化影响很大，而准确描述这一过程主要依赖于恒星对流理论的发展。恒星中的对流是一种湍流运动，一直以来都是用代数型方程的混合长理论来描述，属于局地理论，而这种理论对湍流运动仅仅是粗糙的描述，不能描述大部分湍流性质。20世纪70年代以来，人们逐步提出和发展了基于湍流二阶多方程模型的非局地理论，这种理论比混合长理论更为合理，对湍对流的描述基本达到了当今流体力学对湍流的认识。但是非局地理论在数值计算方面存在很多困难造成这种理论不能替代混合长，而可调参数模型在湍流模型中具有很宽的适用范围，有可能减少数值计算中的诸多问题，把非局地理论全面引入恒星结构和演化领域。必须指出半经验可调参数型非局地对流理论在恒星演化领域的地位，它是高阶湍流模型全面应用到恒星结构和演化领域必经之路，是二阶模型中的最佳实验模型，有可能解决恒星湍流最困难的问题-可算性。本文的工作主旨就是通过对可调参数模型的研究，逐步建立一个可以在恒星演化各个阶段都能被成功计算的非局地对流模型。
本论文的工作集中在半经验可调参数型的非局地湍对流理论的计算和应用上，通过研究湍流扩散、湍流耗散、热耗散、湍动能各向同性等过程，在不同恒星中建立合理参数范围，并通过对具体恒星的计算考察各个过程的合理性。
在第一章的引言开始部分，为了强调湍流在自然界的普遍性和湍流研究存在的困难，论文简要介绍了湍流研究发展历史，总结了湍流特点，湍流研究方法和研究现状，并专门介绍了可以应用到恒星环境的湍流模式理论，包括模式理论的历史和分类。由于人类对湍流的认识至今不清，所以我们接下去特意用一节阐述了我们对湍流现象的基本观点，并简单引出分析湍流所需要的一些无量纲数。第三节进入恒星对流这个主题。我们比较地球实验和太阳对流观测，并引入关于浮力和运热的无量纲数，通过雷诺数和Grashof数说明了恒星对流是一种多重尺度的湍流。之后介绍基于湍流处理方法的一些恒星对流理论，并详细演示二阶湍对流理论的推导过程以及不同湍流过程的模式化方法。
第二章简要介绍本文所采用的半经验可调参数模型，并在数值方法上对它进行了详细分析。第一节从雷诺输运方程出发，引入了六个可调参数模式封闭方程，经过坐标变换得到恒星结构和演化可用的湍对流方程。紧接着我们分析了湍对流理论的对流稳定性，对流稳定性分析表明非局地对流理论对于对流稳定性的刻画方式有别于局地理论，产生更合理的稳定性判据并预示了超射区是由多个部分组成。第三节我们在数值上对方程组进行详细的分析，包括数值方法、边界条件、估计值、限制条件、差分格式，最后指出所有问题将在非线性上耦合在一起。在此过程中，必须强调湍对流方程组和恒星结构方程组共同组成八方程十二阶微分方程组共同描述恒星，我们分析方程组后建立了合理有效的数值方案来解这组完整的恒星结构方程组。
第三章中我们结合1M_⊙恒星模型的计算结果研究了各种湍流过程，比较非局地对流理论和局地理论的区别。我们首先推导非局地对流模型的简化型模型，包括对应的局地模型和一些中间型模型，说明本文采用的模型可以通过调节参数逐步变换成各种其他类型的模型，这一特点兼顾了局地理论的可算性和非局地理论的合理性。第二节我们逐一分析了湍流过程，表明方程对各种过程的描述有以下合理性：梯度型的湍流扩散近似可以合理的描述湍流扩散过程和对流超射区，湍流量的径向分布具有复杂的特征；湍动能耗散的混合长近似可以基本描述耗散过程，引入的典型长度-耗散尺度可以从湍流能谱理论的耗散过程解释，相比标准混合长理论的混合长度更有意义；热耗散主要通过光子辐射耗散过程进行，一般情况下比湍动能耗散快得多；湍动能的各向同性化过程可以同时在不稳定区和稳定区符合湍流能量串级过程；非局地对流理论中的超射区简单可分为传热区和制冷区，而局地理论无法得到超射区，两者在对流不稳定区内的湍流量分布也不同。第三节专门讨论雷诺应力和湍流压，这是湍对流运动除传热以外的另一个重要作用，通过分析气体压和湍流压的不同，认识到计算湍流压的两种方法是两种不同程度的简化，其中单独计算湍流压的做法更合理。
第四章我们计算了几组1.85M_⊙左右恒星的主序演化，并针对一颗盾牌座δ型变星，FG Vir，进行了模型计算和振动分析。我们首先介绍了盾牌座δ型变星的观测特点，FG Vir的研究现状，这是一颗观测频率最多但很多频率都没有模式证认的典型盾牌座δ型变星。本章重点集中在第二节，包括湍流研究和频率分析两个部分。前一部分研究表明对于这一类星的主序，非局地和局地两类对流理论在恒星结构上是相似的，但外部对流区的湍流量却相差近十倍。仔细研究非局地湍流的每一过程，进一步证实了在这类星中非局地对流的各种过程是合理的，并发现湍流扩散是这类星外层对流不稳定区中不可忽视的一个过程。后一部分的频率分析中，我们对两类模型进行了绝热和非绝热振动分析，发现非局地和局地理论在振动频率大小上比较相近，但其他方面存在很大差别。最重要的区别就是两者具有不同的稳定性：非局地对流恒星模型的p模式径向和非径向振动的低阶谐频都是不稳定的，g模的低阶谐频也可能是不稳定的；而局地模型所有观测范围的频率都是振动稳定的。此外，这类星中的“avoidedcrossing”现象会影响p模式振动的低阶谐频频率大小，并且该现象敏感地依赖于绝热或非绝热，局地对流或非局地对流。在结构和振动的分析基础上，我们用非局地对流理论模型构造FG Vir的恒星结构，分析了非绝热振动频率，并且进一步计算自转频率分裂，较为合理地从理论上验证了观测所证认的一些频率。最后指出，湍流扩散、”avoided crossing”、非绝热、较差自转、和频差频、l=3的非径向振动频率，这些因素不同程度地耦合在一起，使该类星的理论频率证认还需要更多的努力。
第五章我们总结全文并展望未来的工作。我们得出总的结论：半经验可调参数型非局地对流理论不仅可以合理描述1M_⊙恒星的外部单层厚对流，还正确构造出1.85M_⊙恒星的双层薄不稳定区的对流运动特征和振动特征，两方面的工作都证实该理论不仅比混合长理论更合理，而且该理论可以适用于不同恒星的湍对流环境。
最后再概述一下本论文的主要结构框架：首先介绍了湍流的特点和模式理论，突出强调恒星中的对流过程和湍对流理论(第一章)，然后集中讨论了半经验可调参数型非局地对流理论的基本概况和数值分析(第二章)。接下来主要阐述自己的研究工作，在不同类型的恒星对流区中考察非局地对流的合理性，包括1M_⊙恒星的对流过程(第三章)，1.85M_⊙恒星对流和振动分析(第四章)。最后一章总结所做的工作的主要内容和结果，并对未来的理论工作作了展望。
需要指出本文的工作背景是湍流研究，工作基础是湍流模式理论，由于湍流研究中很多问题都存在争议，因此本文中对湍流运动分析过程中不可避免地存在一些矛盾，解决这些问题有可能需要人类对湍流本质更深的认识。
Turbulence is ubiquitous in astrophysics, ranging from cosmology, interstellar medium to stars, accretion disks, etc. Large scales and small viscosities combine to form turbulence with extreme large Reynolds numbers which can not be gotten in the laboratories. Stars exist rely on turbulence convection to transport heat when radiate process is insufficient. But common usage of phenomenolog-ical turbulence expressions in stellar convection, just like mixing-length theory and other kind of local theories, makes turbulent convection models perennially unpredictive. At the same time, Reynolds stress models (RSM) are deduced by many researchers, but it is hard to apply them in the stellar structure and evolution, mainly because the calculation of the differential turbulence convection equations is the main obstacle on the way to the better theories. The semi-empirical theory with tunable parameters is flexible to some extent and can be used widely in turbulence, so we deduced this kind of theories and resolved corresponding four equations. It has possibility to be the first non-local theory extensively applied to the stellar structure and evolution.
Our study is focused on the semi-empirical theory, especially in different turbulence processes, including the turbulent diffusion, the dissipation of turbulent energy, the radiation dissipation, the "return to isotropic" and the effect of the buoyancy. We apply the theory into two kind of stars during their Main Sequence, which are 1M_⊙star and 1.85M_⊙star.
My thesis is organized as follows:
In Chapter I, the importance of turbulence is recalled with the development process. The historical and characters of turbulence are introduced, including the difficulty in front of the turbulence research. Then we showed our understanding to turbulence in one section since the turbulence is an unknown and controversial question up to now. At the last section, we compared the laboratory phenomenon and the solar convection observation to confirm that the stellar convection is turbulent flow. We reviewed different stellar convection theories, especially introduce typical one of them.
In Chapter II, the numerical method of turbulent convection theory is carefully constructed. We firstly deduce the semi-empirical formula from RSM. Then we analyzed the convection stability and compressibility in this theory since they are two features in the turbulence modeling. Based on these preparations, we analyzed the detail during solving the equations in some important sides, which are boundary condition, estimated values, special condition and nonlinear, and finally constructed an effective numerical method to solve the equations.
In Chapter III and Chapter IV, we focus on how much our convection theory can describe turbulence in two kinds of stars. We firstly deduced the medial and local theory from the non-local formula, and then mainly compared the non-local with the local theories. In solar-like star, all the description of turbulence in the models, namely diffusion, dissipation, isotropic and buoyancy, were agreed well with the characters of turbulence. In theδScuti type stars, the present conclusion were reconfirmed and we found the turbulent value in the non-local models are greatly lager than those in the local models. Furthermore, we made models of one of theδScuti type stars, named FG Vir. It is found that the non-local theory shows some exciting frequencies in the nonadiabatic oscillation, but the local one shows all damping. Therefore, the frame rotating frequencies are estimated to identify the observation oscillation frequencies of FG Vir. About nine frequencies are fitted and many others can not be corresponding with the calculation. The oscillation of the FG Vir is far away from being clearly understood.
In the last Chapter, conclusion and discussion are made. All of this work showed the semi-empirical convection theory is better than the mixing-length theory.
本论文的工作集中在半经验可调参数型的非局地湍对流理论的计算和应用上，通过研究湍流扩散、湍流耗散、热耗散、湍动能各向同性等过程，在不同恒星中建立合理参数范围，并通过对具体恒星的计算考察各个过程的合理性。
在第一章的引言开始部分，为了强调湍流在自然界的普遍性和湍流研究存在的困难，论文简要介绍了湍流研究发展历史，总结了湍流特点，湍流研究方法和研究现状，并专门介绍了可以应用到恒星环境的湍流模式理论，包括模式理论的历史和分类。由于人类对湍流的认识至今不清，所以我们接下去特意用一节阐述了我们对湍流现象的基本观点，并简单引出分析湍流所需要的一些无量纲数。第三节进入恒星对流这个主题。我们比较地球实验和太阳对流观测，并引入关于浮力和运热的无量纲数，通过雷诺数和Grashof数说明了恒星对流是一种多重尺度的湍流。之后介绍基于湍流处理方法的一些恒星对流理论，并详细演示二阶湍对流理论的推导过程以及不同湍流过程的模式化方法。
第二章简要介绍本文所采用的半经验可调参数模型，并在数值方法上对它进行了详细分析。第一节从雷诺输运方程出发，引入了六个可调参数模式封闭方程，经过坐标变换得到恒星结构和演化可用的湍对流方程。紧接着我们分析了湍对流理论的对流稳定性，对流稳定性分析表明非局地对流理论对于对流稳定性的刻画方式有别于局地理论，产生更合理的稳定性判据并预示了超射区是由多个部分组成。第三节我们在数值上对方程组进行详细的分析，包括数值方法、边界条件、估计值、限制条件、差分格式，最后指出所有问题将在非线性上耦合在一起。在此过程中，必须强调湍对流方程组和恒星结构方程组共同组成八方程十二阶微分方程组共同描述恒星，我们分析方程组后建立了合理有效的数值方案来解这组完整的恒星结构方程组。
第三章中我们结合1M_⊙恒星模型的计算结果研究了各种湍流过程，比较非局地对流理论和局地理论的区别。我们首先推导非局地对流模型的简化型模型，包括对应的局地模型和一些中间型模型，说明本文采用的模型可以通过调节参数逐步变换成各种其他类型的模型，这一特点兼顾了局地理论的可算性和非局地理论的合理性。第二节我们逐一分析了湍流过程，表明方程对各种过程的描述有以下合理性：梯度型的湍流扩散近似可以合理的描述湍流扩散过程和对流超射区，湍流量的径向分布具有复杂的特征；湍动能耗散的混合长近似可以基本描述耗散过程，引入的典型长度-耗散尺度可以从湍流能谱理论的耗散过程解释，相比标准混合长理论的混合长度更有意义；热耗散主要通过光子辐射耗散过程进行，一般情况下比湍动能耗散快得多；湍动能的各向同性化过程可以同时在不稳定区和稳定区符合湍流能量串级过程；非局地对流理论中的超射区简单可分为传热区和制冷区，而局地理论无法得到超射区，两者在对流不稳定区内的湍流量分布也不同。第三节专门讨论雷诺应力和湍流压，这是湍对流运动除传热以外的另一个重要作用，通过分析气体压和湍流压的不同，认识到计算湍流压的两种方法是两种不同程度的简化，其中单独计算湍流压的做法更合理。
第四章我们计算了几组1.85M_⊙左右恒星的主序演化，并针对一颗盾牌座δ型变星，FG Vir，进行了模型计算和振动分析。我们首先介绍了盾牌座δ型变星的观测特点，FG Vir的研究现状，这是一颗观测频率最多但很多频率都没有模式证认的典型盾牌座δ型变星。本章重点集中在第二节，包括湍流研究和频率分析两个部分。前一部分研究表明对于这一类星的主序，非局地和局地两类对流理论在恒星结构上是相似的，但外部对流区的湍流量却相差近十倍。仔细研究非局地湍流的每一过程，进一步证实了在这类星中非局地对流的各种过程是合理的，并发现湍流扩散是这类星外层对流不稳定区中不可忽视的一个过程。后一部分的频率分析中，我们对两类模型进行了绝热和非绝热振动分析，发现非局地和局地理论在振动频率大小上比较相近，但其他方面存在很大差别。最重要的区别就是两者具有不同的稳定性：非局地对流恒星模型的p模式径向和非径向振动的低阶谐频都是不稳定的，g模的低阶谐频也可能是不稳定的；而局地模型所有观测范围的频率都是振动稳定的。此外，这类星中的“avoidedcrossing”现象会影响p模式振动的低阶谐频频率大小，并且该现象敏感地依赖于绝热或非绝热，局地对流或非局地对流。在结构和振动的分析基础上，我们用非局地对流理论模型构造FG Vir的恒星结构，分析了非绝热振动频率，并且进一步计算自转频率分裂，较为合理地从理论上验证了观测所证认的一些频率。最后指出，湍流扩散、”avoided crossing”、非绝热、较差自转、和频差频、l=3的非径向振动频率，这些因素不同程度地耦合在一起，使该类星的理论频率证认还需要更多的努力。
第五章我们总结全文并展望未来的工作。我们得出总的结论：半经验可调参数型非局地对流理论不仅可以合理描述1M_⊙恒星的外部单层厚对流，还正确构造出1.85M_⊙恒星的双层薄不稳定区的对流运动特征和振动特征，两方面的工作都证实该理论不仅比混合长理论更合理，而且该理论可以适用于不同恒星的湍对流环境。
最后再概述一下本论文的主要结构框架：首先介绍了湍流的特点和模式理论，突出强调恒星中的对流过程和湍对流理论(第一章)，然后集中讨论了半经验可调参数型非局地对流理论的基本概况和数值分析(第二章)。接下来主要阐述自己的研究工作，在不同类型的恒星对流区中考察非局地对流的合理性，包括1M_⊙恒星的对流过程(第三章)，1.85M_⊙恒星对流和振动分析(第四章)。最后一章总结所做的工作的主要内容和结果，并对未来的理论工作作了展望。
需要指出本文的工作背景是湍流研究，工作基础是湍流模式理论，由于湍流研究中很多问题都存在争议，因此本文中对湍流运动分析过程中不可避免地存在一些矛盾，解决这些问题有可能需要人类对湍流本质更深的认识。
Turbulence is ubiquitous in astrophysics, ranging from cosmology, interstellar medium to stars, accretion disks, etc. Large scales and small viscosities combine to form turbulence with extreme large Reynolds numbers which can not be gotten in the laboratories. Stars exist rely on turbulence convection to transport heat when radiate process is insufficient. But common usage of phenomenolog-ical turbulence expressions in stellar convection, just like mixing-length theory and other kind of local theories, makes turbulent convection models perennially unpredictive. At the same time, Reynolds stress models (RSM) are deduced by many researchers, but it is hard to apply them in the stellar structure and evolution, mainly because the calculation of the differential turbulence convection equations is the main obstacle on the way to the better theories. The semi-empirical theory with tunable parameters is flexible to some extent and can be used widely in turbulence, so we deduced this kind of theories and resolved corresponding four equations. It has possibility to be the first non-local theory extensively applied to the stellar structure and evolution.
Our study is focused on the semi-empirical theory, especially in different turbulence processes, including the turbulent diffusion, the dissipation of turbulent energy, the radiation dissipation, the "return to isotropic" and the effect of the buoyancy. We apply the theory into two kind of stars during their Main Sequence, which are 1M_⊙star and 1.85M_⊙star.
My thesis is organized as follows:
In Chapter I, the importance of turbulence is recalled with the development process. The historical and characters of turbulence are introduced, including the difficulty in front of the turbulence research. Then we showed our understanding to turbulence in one section since the turbulence is an unknown and controversial question up to now. At the last section, we compared the laboratory phenomenon and the solar convection observation to confirm that the stellar convection is turbulent flow. We reviewed different stellar convection theories, especially introduce typical one of them.
In Chapter II, the numerical method of turbulent convection theory is carefully constructed. We firstly deduce the semi-empirical formula from RSM. Then we analyzed the convection stability and compressibility in this theory since they are two features in the turbulence modeling. Based on these preparations, we analyzed the detail during solving the equations in some important sides, which are boundary condition, estimated values, special condition and nonlinear, and finally constructed an effective numerical method to solve the equations.
In Chapter III and Chapter IV, we focus on how much our convection theory can describe turbulence in two kinds of stars. We firstly deduced the medial and local theory from the non-local formula, and then mainly compared the non-local with the local theories. In solar-like star, all the description of turbulence in the models, namely diffusion, dissipation, isotropic and buoyancy, were agreed well with the characters of turbulence. In theδScuti type stars, the present conclusion were reconfirmed and we found the turbulent value in the non-local models are greatly lager than those in the local models. Furthermore, we made models of one of theδScuti type stars, named FG Vir. It is found that the non-local theory shows some exciting frequencies in the nonadiabatic oscillation, but the local one shows all damping. Therefore, the frame rotating frequencies are estimated to identify the observation oscillation frequencies of FG Vir. About nine frequencies are fitted and many others can not be corresponding with the calculation. The oscillation of the FG Vir is far away from being clearly understood.
In the last Chapter, conclusion and discussion are made. All of this work showed the semi-empirical convection theory is better than the mixing-length theory.
引文
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[3] 欣茨[荷兰]．湍流(上册)．科学出版社，书号：13031-3575，北京，1987．
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[5] 忻孝康，刘儒勋，蒋伯诚．计算流体动力学．国防科技大学出版社，书号：7-81024-065-x，1989．
[6] 李焱．博士论文：双系统恒星振动理论．中国科学院，云南天文台，1992．
[7] 是勋刚．湍流．天津大学出版社，书号：7-5618-0566-7，天津，1994．
[8] 章梓雄，董曾南．粘性流体力学．清华大学出版社，书号：7-302-02852-4，北京，1998．
[9] 黄润乾．恒星物理．科学出版社，书号：7-03-005836-4，北京，1998．
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