[{"content":" 原文信息\n出处：Numerical Simulations in Cosmology: Chapter 3 — Hydrodynamic methods and sub-resolution models for cosmological simulations\n作者：Milena Valentini (Universitá degli Studi di Trieste / INAF), Klaus Dolag (LMU München / MPI for Astrophysics)\narXiv：arXiv:2502.06954 页数：61 页，14 幅图\n翻译说明： 中文翻译，段落对照，上面是中文，下面是英文。Chinese translation, paragraph by paragraph, with Chinese on top and English below.\n虚线下划线 为个人添加的注释或评论。文献引用如 [1] 可点击复制完整引用信息。\n摘要 宇宙学模拟是结构形成研究中极为强有力的工具。Cosmological simulations are powerful tools in the context of structure formation.\n借助它们，我们得以探索暗物质（DM）晕的等级式并合与成团性，验证或排除各种可能的结构形成图景，并研究星系在宇宙时间尺度上的物理演化性质。They allow us to explore the hierarchical assembly of dark matter (DM) halos and their clustering, to validate or reject possible scenarios of structure formation, and to investigate the physical properties of evolving galaxies across cosmic time.\n宇宙学流体动力学模拟尤为关键：它研究的是形成中的星系内部，复杂的星际介质（ISM）如何响应星系演化中最剧烈的能量过程——例如超新星（SN）爆发驱动的恒星反馈，以及活动星系核（AGN）反馈。Cosmological hydrodynamical simulations are especially key to study how the complex interstellar medium (ISM) of forming galaxies responds to the most energetic processes during galaxy evolution, such as stellar feedback ensuing supernova (SN) explosions and feedback from AGN (active galactic nuclei).\n宇宙结构形成与演化涉及的天体物理过程跨越了极大的物理尺度动力学范围。有鉴于此，宇宙学模拟必须借助次网格模型，来捕捉那些发生在模拟分辨率极限之下的物理过程。Given the huge dynamical range of physical scales spanned by the astrophysical processes involved in cosmic structure formation and evolution, cosmological simulations resort to sub-resolution models to capture processes occurring below their resolution limit.\n针对同一物理过程，不同次网格方案给出的结果差异是惊人的——然而这一点却常常被忽视。The impact of different sub-grid prescriptions accounting for the same process is striking, though often overlooked.\n其中主要过程包括：气体冷却、恒星形成与反馈、恒星演化与化学增丰、黑洞（BH）增长及其反馈。Some among the main aforementioned processes include: hot gas cooling, star formation and stellar feedback, stellar evolution and chemical enrichment, black hole (BH) growth and their ensuing feedback.\n在大型计算体积中运行宇宙结构形成与星系演化的模拟，对于揭示宇宙第一批结构如何诞生及其后续演化的驱动机制，具有关键意义。Producing simulations of cosmic structure formation and galaxy evolution in large computational volumes is key to shed new light on what drives the formation of the first structures in the Universe, and their subsequent evolution.\n模拟预测不仅对于与当前和未来观测设备的数据进行比对至关重要，还能有效指导未来的观测计划。Not only are predictions from simulations crucial to compare with data from ongoing and upcoming observational instruments, but they can also effectively guide future observational campaigns.\n此外，我们业已迈入高性能计算时代。拥有这样的数值代码至关重要：不仅要尽可能完整地涵盖已实现的物理过程，还要在计算层面高效运行，并能在最先进的百亿亿次（exascale）基础设施上平稳扩展。Besides, since we have entered the era of high-performance computing, it is of paramount importance to have numerical codes which are not only as complete as possible as for the inclusion of physical processes implemented, but also very efficient from the computational point of view and able to smoothly scale on state-of-the-art exascale infrastructures.\n本章将回顾宇宙学模拟中采用的主要流体动力学方法，以及用于纳入驱动星系形成与演化的基本天体物理过程的最常用技术。In this chapter, we review the main hydrodynamic methods used in cosmological simulations and the most common techniques adopted to include the fundamental astrophysical processes which drive galaxy formation and evolution.\n系列导航\n→ 下一篇：§3.1 宇宙学流体动力学模拟\n进度记录\n2026-06-25：创建翻译框架，完成摘要翻译 2026-07-04：重构为按章节独立成篇 2026-07-06：迁移到 en.txt + en N 方案（英文分离为独立文件） 2026-07-24：迁移到 inline en 方案（英文直接写在 md 中，不再使用 en.txt） ","date":"2026-06-25","permalink":"/posts/2026-06-25-hydrodynamic-01/","series":["宇宙学模拟中的流体动力学方法与次网格模型"],"summary":" 原文信息\n出处：Numerical Simulations in Cosmology: Chapter 3 — Hydrodynamic methods and sub-resolution models for cosmological simulations\n作者：Milena Valentini (Universitá degli Studi di Trieste / INAF), Klaus Dolag (LMU München / MPI for Astrophysics)\narXiv：arXiv:2502.06954 页数：61 页，14 幅图\n翻译说明： 中文翻译，段落对照，上面是中文，下面是英文。Chinese translation, paragraph by paragraph, with Chinese on top and English below.\n虚线下划线 为个人添加的注释或评论。文献引用如 [1] 可点击复制完整引用信息。\n","tags":["翻译","astronomy:cosmology","astronomy:cosmology:simulation","numerical:hydrodynamics","numerical:hydrodynamics:SPH","numerical:hydrodynamics:AMR","numerical:subgrid"],"title":"宇宙学模拟中的流体动力学方法与次网格模型"},{"content":" 引力是驱动结构形成的基本力，因此大多数宇宙学模拟的核心构建模块便是 $N$ 体程序（关于替代方法的讨论，参见[1] ）。$N$ 体程序的目标是研究自引力、无碰撞系统的非线性动力学演化（综述见[2] ,[3] ）。Being gravity the force that drives structure formation, the building block of the majority of cosmological simulations is an N-body code (see e.g. [1] for a discussion on alternative approaches). The goal of an N-body code is to investigate the non-linear dynamical evolution of a self-gravitating, collisionless system (see [2] ,[3] , for reviews).\n无碰撞流体是这样一类系统：其组成元素在总质量分布所产生的集体引力场驱动下运动。在此类系统中，引力碰撞以及流体元素之间的二体相互作用均可忽略不计，不会影响组成元素自身的运动和一般性质。要定量判断一个系统是否可视为无碰撞，需将其弛豫时标与系统演化时标进行比较。考虑一个由 $N$ 个元素组成的系统，其元素以典型速度 $v$ 在尺度 $L$ 内运动，则特征穿越时间为 $t_{\\rm cross} = L/v$[4] 。与相邻粒子的可能碰撞会对每个粒子的速度产生二次方速度扰动 $(\\Delta v)^2$，这是因为每次碰撞贡献的 $\\Delta v$ 方向随机。弛豫时间定义为二次方速度扰动累积到大致与速度平方本身相当的时标：$t_{\\rm relax} = v^2/(\\Delta v)^2 \\times t_{\\rm cross}$。这里，$v^2/(\\Delta v)^2$ 量化了使速度改变达到与 $v^2$ 同量级所需的穿越次数，该量依赖于粒子数 $N$[4] 。因此，弛豫时间可写作 $t_{\\rm relax} = N/(8 \\ln N) \\times t_{\\rm cross}$。若一个系统的弛豫时间远超宇宙年龄 $\\sim 1/H_0$（$H_0$ 为哈勃常数的现今值），则可将该系统视为无碰撞系统。经过时标 $t_{\\rm relax}$ 之后，粒子运动会因与其他粒子的相互作用而发生偏转，系统便不再可被视为无碰撞。星系中的暗物质和恒星可表示为无碰撞的非相对论性粒子流体，而气体则属于有碰撞成分。A collisionless fluid is a system whose constituent elements move under the effect of the collective gravity field generated by the total mass distribution: in a collisionless system, gravitational collisions and interactions between fluid elements are negligible and do not influence the motion and general properties of the constituent elements themselves. In order to establish quantitatively whether a system can be considered collisionless or not, the relaxation timescale has to be contrasted to the timescale over which the system evolves. A system featuring N elements moving with typical velocity $v$ across its size $L$ is characterised by the crossing time $t_{\\rm cross} = L/v$ [4] . Possible collisions with nearby particles result in a quadratic velocity perturbation $(\\Delta v)^2$ of each particle's velocity, since each encounter contributes with a randomly oriented $\\Delta v$. The relaxation time is the timescale over which the quadratic velocity perturbation amounts to roughly the squared velocity itself $t_{\\rm relax} = v^2/(\\Delta v)^2 \\times t_{\\rm cross}$. Here, $v^2/(\\Delta v)^2$ quantifies the number of crossings required to have a velocity change of the same order as $v^2$ and depends on the number N of particles [4] . Therefore, the relaxation time can be cast as $t_{\\rm relax} = N/(8 \\ln N) \\times t_{\\rm cross}$. A system can be deemed as collisionless if its relaxation time exceeds by far the age of the Universe $\\sim 1/H_0$ ($H_0$ being the present-day value of the Hubble constant). After the timescale $t_{\\rm relax}$, deflections in a particle motion by interactions with other particles are expected, and the system cannot be considered collisionless anymore. DM and stars in galaxies can be represented as a collisionless, non-relativistic fluid made of particles, while gas is a collisional component.\n作为一级近似，宇宙结构的形成可通过 $N$ 体模拟来研究——这类模拟仅追踪无碰撞粒子在引力作用下的演化。此类模拟已在单个天体（如星系和星系团）以及极大尺度结构的研究中，以高分辨率得到广泛应用。关于在累积引力场中积分 $N$ 个粒子运动方程的各种数值方法和可行途径，在关于 $N$ 体的章节中有详细描述。To a first approximation, the formation of cosmic structures can be studied using N-body simulations, which only follow the evolution of collisionless particles under gravity. Such simulations have been performed with high resolution for individual objects, like galaxies and galaxy clusters, as well as for very large-scale structures. The numerical methods and different possible approaches used to tackle the problem of integrating the equations of motion of the N particles in the cumulative gravity field are described in N体章节.\n然而，除引力透镜这一可能的例外，观测所反映的主要是普通（重子）物质的状态。因此，要在宇宙结构演化框架下解释这些观测，就必须理解那些决定宇宙重子演化的复杂非引力物理过程。在等级式形成图景中，暗物质、气体和恒星如何共同演化——重子落入暗物质分布的势阱、冷却、最终凝聚形成恒星（见视频1，第xiii页）——这一过程决定了星系际介质（IGM）和星系团内介质（ICM）的状态与组成，并驱动着能量与金属的反馈、磁场的产生以及高能粒子的加速。取决于来源的不同，这些成分会通过喷流、星风或冲压效应被吹出，最终与周围的 IGM/ICM 混合。部分效应（如冲压效应）可在流体动力学模拟中被自然地追踪；而另一些（如恒星形成及其反馈、超新星对 ISM、IGM 和 ICM 的化学污染）则必须通过有效模型来纳入模拟。However, with the possible exception of gravitational lensing, observations mainly reflect the state of the ordinary (baryonic) matter. Therefore, their interpretation in the framework of cosmic structure evolution requires that we understand the complex, non-gravitational physical processes which determine the evolution of the cosmic baryons. How DM, gas and stars co-evolve – with baryons falling into the potential well of the underlying DM distribution, cooling, and finally condensing to form stars (see Video 1, page xiii) – within the hierarchical formation scenario contributes to the state and composition of the intergalactic and intracluster media (IGM and ICM, respectively), and is responsible for feedback in energy and metals, magnetic fields, and high-energy particles. Depending on their origin, these components will be blown out by jets, winds or ram pressure effects and finally mix with the surrounding IGM/ICM. While some of these effects will be naturally followed within hydrodynamic simulations (like ram pressure effects), others have to be included in simulations via effective models (like star formation and ensuing feedback, and chemical pollution of the ISM, IGM and ICM by SNe).\n得益于计算能力的提升和数值方法的进步，过去几十年中可用于此类模拟的分辨率单元¹数量急剧增长，如Fig. 1所示。值得注意的是，迄今为止最大规模的流体动力学模拟（如 Magneticum² Box0/mr[5] 和 FLAMINGO³ L2p8_m9[6] ）在整个宇宙演化历程中追踪了超过 $2 \\times 10^{11}$ 个粒子（包括 DM、气体、恒星和 BH 粒子）。 Thanks to the improved computing power and advancements in numerical methods, the number of resolution elements¹ which can be used in such simulations has increased dramatically over the last decades, as shown in Fig.\u0026nbsp;1. Note that the largest, hydrodynamical simulations up to date (e.g. Magneticum² Box0/mr[5] and FLAMINGO³ L2p8_m9[6] ) followed a total number of more than 2×10¹¹ particles (e.g., DM, gas, stars and BH particles) across the entire evolution of the Universe.\nFig.\u0026nbsp;1. 模拟中分辨率单元数量随时间的演化。黑线为1970年至今纯$N$体模拟标度关系的拟合。粉色标记表示额外包含了磁场和谱宇宙线电子与质子的模拟。 本章将回顾描述流体动力学的基本数值方法，并讨论在宇宙学流体动力学模拟中纳入基本天体物理过程的主要途径（见视频3，第xiii页）。磁场和高能粒子等进一步的成分需要对其注入过程和演化进行额外建模，因此必须与流体动力学自洽耦合——这些内容在关于星系团的章节中有更详细的描述。In this chapter, we will review the basic numerical methods which are used to describe the hydrodynamics, and discuss the main approaches adopted to incorporate the fundamental astrophysical processes in the context of cosmological hydrodynamical simulations (see Video 3, page xiii). Further components like magnetic fields and high-energy particles need additional modelling of their injection processes and evolution. Therefore, they must be self-consistently coupled with hydrodynamics and are described in more detail in 星系团章节.\n系列导航\n← 上一篇：Chapter 3 封面与摘要\n→ 下一篇：§3.2 流体动力学与数值方法\n","date":"2026-06-25","permalink":"/posts/2026-06-25-hydrodynamic-02/","series":["宇宙学模拟中的流体动力学方法与次网格模型"],"summary":" 引力是驱动结构形成的基本力，因此大多数宇宙学模拟的核心构建模块便是 $N$ 体程序（关于替代方法的讨论，参见[1] ）。$N$ 体程序的目标是研究自引力、无碰撞系统的非线性动力学演化（综述见[2] ,[3] ）。Being gravity the force that drives structure formation, the building block of the majority of cosmological simulations is an N-body code (see e.g. [1] for a discussion on alternative approaches). The goal of an N-body code is to investigate the non-linear dynamical evolution of a self-gravitating, collisionless system (see [2] ,[3] , for reviews).\n","tags":["翻译","astronomy:cosmology","astronomy:cosmology:simulation","numerical:hydrodynamics","numerical:N-body"],"title":"§3.1 宇宙学流体动力学模拟"},{"content":" 宇宙中引力成团具有高度非线性，这使数值模拟成为详细追踪星系和星系团等形成结构之演化的唯一手段。然而，这需要捕获巨大的空间和时间动力学范围。例如，结构等级式并合跨越的物理尺度从星系内部的亚 kpc 尺度，一直延伸到数百兆秒差距——后者是宇宙中最大的相干尺度。在这方面，基于 $N$ 体和流体动力学技术的现代宇宙学模拟程序，最适合在结构等级式形成过程中精确追踪暗物质和气体在其全部复杂性下的联合动力学。As a result of the high non-linearity of the gravitational clustering in the Universe, numerical simulations are the only method to follow the evolution of the forming structures like galaxies and galaxy clusters in detail. However, a huge dynamic range in space and time has to be captured. For instance, the range of physical scales over which the hierarchical assembly of structures develops spans from sub-kpc scales in galaxies up to several hundreds of megaparsecs, the latter being the largest coherent scale in the Universe. Here, modern cosmological simulation codes based on N-body and hydrodynamics techniques are best suited to accurately follow the joint dynamics of DM and gas in their full complexity, during the hierarchical build-up of these structures.\n系列导航\n← 上一篇：§3.1 宇宙学流体动力学模拟\n→ 下一篇：§3.3 基本方程与技术\n","date":"2026-06-25","permalink":"/posts/2026-06-25-hydrodynamic-03/","series":["宇宙学模拟中的流体动力学方法与次网格模型"],"summary":" 宇宙中引力成团具有高度非线性，这使数值模拟成为详细追踪星系和星系团等形成结构之演化的唯一手段。然而，这需要捕获巨大的空间和时间动力学范围。例如，结构等级式并合跨越的物理尺度从星系内部的亚 kpc 尺度，一直延伸到数百兆秒差距——后者是宇宙中最大的相干尺度。在这方面，基于 $N$ 体和流体动力学技术的现代宇宙学模拟程序，最适合在结构等级式形成过程中精确追踪暗物质和气体在其全部复杂性下的联合动力学。As a result of the high non-linearity of the gravitational clustering in the Universe, numerical simulations are the only method to follow the evolution of the forming structures like galaxies and galaxy clusters in detail. However, a huge dynamic range in space and time has to be captured. For instance, the range of physical scales over which the hierarchical assembly of structures develops spans from sub-kpc scales in galaxies up to several hundreds of megaparsecs, the latter being the largest coherent scale in the Universe. Here, modern cosmological simulation codes based on N-body and hydrodynamics techniques are best suited to accurately follow the joint dynamics of DM and gas in their full complexity, during the hierarchical build-up of these structures.\n","tags":["翻译","astronomy:cosmology","astronomy:cosmology:simulation","numerical:hydrodynamics"],"title":"§3.2 流体动力学与数值方法"},{"content":" 过去几十年中，人们发展了多种数值方案来求解描述宇宙重子成分与无碰撞暗物质的耦合方程组。绝大多数重子（即气体）可被描述为理想流体，其演化由一组方程——即欧拉方程——所支配。积分上述方程的流体求解器可归为两大类，如Fig. 2所示：粒子方法（离散化质量，见[47] 及其参考文献）和网格方法（离散化计算域，见[48] 及其参考文献）。近年来又涌现出新的求解器：它们融合了两种方法的特征（见[49] 及其参考文献），将在后续章节中详细讨论。A variety of numerical schemes has been developed in the past decades to solve the coupled system of equations describing the baryonic content of the Universe and the collisionless DM. The majority of the baryons (i.e. gas) can be described as an ideal fluid, whose evolution is ruled by a set of equations, namely, the Euler equations. The hydro solvers which integrate the aforementioned equations fall into two main categories, that are summarized in Fig.\u0026nbsp;2: particle methods, which discretize mass (see [47] and references therein), and grid-based methods, which discretize the computational domain (see [48] and references therein). Recently, additional solvers have been developed: they combine characteristics of both methods (see [49] and references therein) and will be discussed in detail in the following sections.\n拉格朗日描述 欧拉描述 核密度估计 网格划分 Fig.\u0026nbsp;2. 所有求解器均需通过状态方程来封闭，该方程将气体压强 $P$ 与（单位质量）内能 $u$ 及密度 $\\rho$ 联系起来。假设为理想单原子气体，则有 $P = (\\gamma - 1) \\rho u$，多方指数 $\\gamma = 5/3$。All of them have to be closed by an equation of state, relating the gas pressure P to the internal energy (per unit mass) u and the density ρ. Assuming an ideal, monoatomic gas, this will be P = (γ−1) ρ u with the polytropic index γ = 5/3.\n将这些方程应用于宇宙学结构形成问题时，与在非膨胀背景下的纯流体动力学模拟相比，会浮现出几个新的特征。首先，通常被忽略的自引力（表现为 $\\nabla \\Phi$ 项）必须纳入考量（其贡献可按 $N$ 体章节中描述的方法求解）。When applying these equations to the problem of cosmological structure formation, there are several features emerging in comparison to purely hydrodynamic simulations in a non-expanding background. First, the otherwise often neglected self-gravity, emerging as the ∇Φ term, has to be accounted for (its contribution can be solved following the methods described in N体章节).\n其次，辐射损失——通过冷却函数 $\\Lambda(u, \\rho)$ 量化，其中 $n^2 \\Lambda$（$n^2 \\Lambda / \\rho$）表示单位体积（质量）的冷却率，$n$ 为气体数密度——对重子成分的演化起着关键作用（亦见 §3.11）。冷却对描述宇宙恒星形成历史尤为关键，如 §3.14 所述。Second, radiative losses, quantified through the cooling function Λ(u,ρ), where n²Λ (n²Λ/ρ) represents the cooling rate per unit volume (mass), with n being the gas number density, play a key role in influencing the evolution of the baryonic component (see also Section 11). Cooling is especially key to describe the history of star formation of the Universe, as outlined in Section 14.\n此外，方程必须适配宇宙学背景（即计入宇宙的膨胀历史）。在几乎所有情形中，这最后一步都是通过切换到所谓的共动坐标并变换时间变量来实现的。这一改变通常不改变前述方程的一般结构，只是方程中会出现额外的、依赖于宇宙学时间的因子（关于严格推导，建议读者参阅[50] 第5章）。Additionally, the equations have to be adapted to the cosmological background (to take into account the expansion history of the Universe). In almost all cases, the latter step is achieved by switching to the so called comoving coordinates and through a transformation of the time variable. This change does not typically alter the general structure of the equations above, except for additional, cosmological, time dependent pre-factors which appear in the equations (we refer the reader to chapter 5 in [50] for a rigorous derivation).\n系列导航\n← 上一篇：§3.2 流体动力学与数值方法\n→ 下一篇：§3.4 欧拉网格方法\n","date":"2026-06-25","permalink":"/posts/2026-06-25-hydrodynamic-04/","series":["宇宙学模拟中的流体动力学方法与次网格模型"],"summary":" 过去几十年中，人们发展了多种数值方案来求解描述宇宙重子成分与无碰撞暗物质的耦合方程组。绝大多数重子（即气体）可被描述为理想流体，其演化由一组方程——即欧拉方程——所支配。积分上述方程的流体求解器可归为两大类，如Fig. 2所示：粒子方法（离散化质量，见[47] 及其参考文献）和网格方法（离散化计算域，见[48] 及其参考文献）。近年来又涌现出新的求解器：它们融合了两种方法的特征（见[49] 及其参考文献），将在后续章节中详细讨论。A variety of numerical schemes has been developed in the past decades to solve the coupled system of equations describing the baryonic content of the Universe and the collisionless DM. The majority of the baryons (i.e. gas) can be described as an ideal fluid, whose evolution is ruled by a set of equations, namely, the Euler equations. The hydro solvers which integrate the aforementioned equations fall into two main categories, that are summarized in Fig. 2: particle methods, which discretize mass (see [47] and references therein), and grid-based methods, which discretize the computational domain (see [48] and references therein). Recently, additional solvers have been developed: they combine characteristics of both methods (see [49] and references therein) and will be discussed in detail in the following sections.\n","tags":["翻译","astronomy:cosmology","astronomy:cosmology:simulation","numerical:hydrodynamics","numerical:hydrodynamics:SPH","numerical:hydrodynamics:AMR"],"title":"§3.3 基本方程与技术"},{"content":"Eulerian (grid) methods 网格方法在结构或非结构化网格上求解欧拉方程（参见Fig. 2），用以表示流体。其中，原始变量描述流体的热力学性质（如 $\\rho$、${\\vec v}$ 或 $P$），而守恒变量则定义守恒定律（如 $\\rho$、$\\rho{\\vec v}$ 或 $\\rho u$）。早期研究者尝试使用中心差分格式：流体仅以网格中心的数值表征，导数则通过有限差分表示求得（参见例如 [51] ）。这些方法采用人工黏性来处理激波（类似于§3.6中描述的平滑粒子流体动力学方法），否则在出现间断的区域会失效。此外，从构造上讲，它们仅具有一阶精度。Grid-based methods solve the Euler equations (see Fig.\u0026nbsp;2) based on structured or unstructured grids, representing the fluid. While primitive variables encode the thermodynamic properties of the fluid (e.g., $\\rho$, ${\\vec v}$, or $P$), conservative variables define the conservation laws (e.g., $\\rho$, $\\rho{\\vec v}$, or $\\rho u$). Early attempts were made using a central difference scheme, where fluid is only represented by the centered cell values and derivatives are obtained by the finite-difference representation (see, for example, [51] ). These methods use artificial viscosity to handle schocks (similar to the smoothed particle hydrodynamics method described in §3.6), as they would otherwise break down in regimes where discontinuities appear. Also, by construction, they are only first-order accurate.\n经典方法则采用重构格式，根据其阶数，利用数个相邻网格（即所谓的模板点）来重构任意流体力学变量（$u_n$）的场，从而在网格界面（$u_{\\pm0.5}^{l/r}$）处逐步提高精度阶数。典型的格式包括分段常数方法（PCM）、分段线性方法（PLM；如 [52] ）和分段抛物线方法（PPM；[53] ），如Fig. 2右下部分所示。然后，利用重构函数的形状来计算某个物理量在网格单元上的总积分除以每个网格的体积（即网格平均值，${\\bar u}_n$），而非在网格中心处做逐点近似（即中心变量，$u_n$）。Classical approaches use instead reconstruction schemes which, depending on their order, take several neighboring cells (so-called stencils) into account to reconstruct the field of any hydrodynamical variable ($u_n$), with increasing order of accuracy at the interfaces of the cells ($u_{\\pm0.5}^{l/r}$). Typical schemes are the piecewise constant method (PCM), the piecewise linear method (PLM; e.g., [52] ), and the piecewise parabolic method (PPM; [53] ), as illustrated in the lower right part of Fig.\u0026nbsp;2. The shape of the reconstruction function is then used to calculate the total integral of a quantity over the grid cell, divided by the volume of each cell (e.g., cell average, ${\\bar u}_n$), rather than pointwise approximations at the grid centers (e.g., central variables, $u_n$).\n现代高精度格式通常基于至少五个网格点构成的模板，并采用本质无振荡（ENO；[54] ）或所谓的加权本质无振荡（WENO；[55] ）方案进行重构，从而保持高阶精度（近期综述参见例如 [56] ）。Modern, high-order schemes usually have stencils based on at least five grid points and implement essentially nonoscillatory (ENO; [54] ) or the so-called weighted essentially nonoscillatory (WENO; [55] ) schemes for reconstruction, which maintain high-order accuracy (see e.g., [56] for a recent review).\n重构后的物理量用于计算网格边界左侧和右侧的值。这些值随后用作初始条件来求解所谓的Riemann问题（见Fig. 3右图），其解给出穿过网格边界的各种物理量（如质量、能量等）的通量。此外，还需引入额外约束以抑制重构中的振荡（例如避免出现新的极值），即所谓的斜率限制器，用于估计重构所允许的最大斜率。一种方法是要求界面间的总变差不随时间增加。这些所谓的总变差递减格式（TVD；[57] ）如今提供了多种不同的斜率限制器，由不同作者提出。The reconstructed quantities are used to calculate the left- and right-hand side values at the cell boundaries. These values are then exploited as initial conditions to solve the so-called Riemann problem (see right panel of Fig.\u0026nbsp;3), whose solution provides the fluxes of various quantities (e.g., mass, energy, etc.) across the cell borders. Additional constraints are included to avoid oscillations (e.g., the development of new extrema) in these reconstructions, such as the so-called slope limiters, which estimate the maximum slope allowed for the reconstruction. One way is to demand that the total variation among the interfaces does not increase with time. These so-called total variation diminishing schemes (TVD; [57] ) nowadays provide various different slope limiters, as suggested by different authors.\n初始状态 最终状态 Fig.\u0026nbsp;3. 如何求解一般Riemann问题——即初始分隔两种状态的间断的演化——可参见教科书（如 [58] ）。这里，我们仅以激波管为例做简要描述。这对应于一个初始时刻两侧都处于静止的系统。Fig. 3的左侧部分显示了初始（上图）和演化后（下图）的系统。后者可划分为五个区域。区域1和5的值与初始构型相同。区域2是一个稀疏波，由区域1和区域3的状态决定。求解此问题可借助一般的Rankine\u0026ndash;Hugoniot条件，该条件描述了间断处的跳跃关系。它们列在右上部分，其中我们假设了一个随激波速度 $v_s$ 运动的坐标系。结合这些条件并定义初始密度比 $\\lambda=\\rho_1/\\rho_5$，可得到关于压力比 $P=P_{3,4}/P_5$ 的非线性代数方程，如Fig. 3右下部分所示。一旦求解该方程得到 $P_{3,4}$，其余未知量便可从四个条件中逐步导出。How to solve the general Riemann problem, i.e. the evolution of a discontinuity initially separating two states, can be found in textbooks (e.g., [58] ). Here, we only want to give a brief description of the solution of a shock tube as an example. This corresponds to a system where both sides are initially at rest. The left part of Fig.\u0026nbsp;3 shows in the initial (upper panel) and the evolved (lower panel) system. The latter can be divided into five regions. The values for regions 1 and 5 are identical to the initial configuration. Region 2 is a rarefaction wave which is determined by the states in regions 1 and 3. The solution can be obtained by invoking the general Rankine--Hugoniot conditions, describing the jump conditions at a discontinuity. They are listed in the right upper part, where we have assumed a coordinate system which moves with the shock velocity $v_s$. By combining such conditions and defining the initial density ratio $\\lambda=\\rho_1/\\rho_5$, it is possible to get the nonlinear, algebraic equation for the pressure ratio $P=P_{3,4}/P_5$, as displayed in the lower right part of Fig.\u0026nbsp;3. Once $P_{3,4}$ is known by solving this equation, the remaining unknowns can be inferred step by step from the four conditions.\n在流体力学代码中精确求解Riemann问题可能代价高昂，并严重影响代码性能。因此，有多种近似方法可用于求解Riemann问题，包括所谓的ROE方法（如 [59] ）、HLL/HLLE方法（如参见 [60] 、[61] 、[62] ）以及HLLC（如参见 [63] ）。对所有方法的详细描述超出了本综述的范围，请读者参阅所引文献或教科书（如 [64] ）。Solving the full Riemann problem in a hydrodynamics code can be expensive and it can severely affect the code performance. Therefore, there are various approximate methods to solve the Riemann problem, including the so-called ROE method (e.g., [59] ), the HLL/HLLE method (e.g., see [60] ,[61] ,[62] ), and HLLC (e.g., see [63] ). A description of all these methods is outside the scope of this review, so we refer the reader to the references given or to textbooks (e.g., [64] ).\n宇宙学应用中采用了多种网格代码，包括基于TVD的代码（如 [65] 和 CosmoMHD [66] ）以及基于PLM的代码（如 ART [67] 、[68] 和 RAMSES [69] ）。基于PPM的代码包括 Zeus [70] 、ENZO [71] 、COSMOS [72] 和 FLASH [73] 。值得指出基于WENO的代码 [74] 。A wide variety of grid codes are used for cosmological applications, including TVD-based codes (e.g., [65] and CosmoMHD [66] ) and PLM-based codes (like ART [67] ,[68] and RAMSES [69] ). PPM-based codes include Zeus [70] , ENZO [71] , COSMOS [72] , and FLASH [73] . It is worth mentioning the WENO-based code by [74] .\n系列导航 ← 上一篇：§3.3 基本方程与技术 → 下一篇：§3.5\n","date":"2026-06-25","permalink":"/posts/2026-06-25-hydrodynamic-05/","series":["宇宙学模拟中的流体动力学方法与次网格模型"],"summary":"Eulerian (grid) methods 网格方法在结构或非结构化网格上求解欧拉方程（参见Fig. 2），用以表示流体。其中，原始变量描述流体的热力学性质（如 $\\rho$、${\\vec v}$ 或 $P$），而守恒变量则定义守恒定律（如 $\\rho$、$\\rho{\\vec v}$ 或 $\\rho u$）。早期研究者尝试使用中心差分格式：流体仅以网格中心的数值表征，导数则通过有限差分表示求得（参见例如 [51] ）。这些方法采用人工黏性来处理激波（类似于§3.6中描述的平滑粒子流体动力学方法），否则在出现间断的区域会失效。此外，从构造上讲，它们仅具有一阶精度。Grid-based methods solve the Euler equations (see Fig. 2) based on structured or unstructured grids, representing the fluid. While primitive variables encode the thermodynamic properties of the fluid (e.g., $\\rho$, ${\\vec v}$, or $P$), conservative variables define the conservation laws (e.g., $\\rho$, $\\rho{\\vec v}$, or $\\rho u$). Early attempts were made using a central difference scheme, where fluid is only represented by the centered cell values and derivatives are obtained by the finite-difference representation (see, for example, [51] ). These methods use artificial viscosity to handle schocks (similar to the smoothed particle hydrodynamics method described in §3.6), as they would otherwise break down in regimes where discontinuities appear. Also, by construction, they are only first-order accurate.\n","tags":["翻译","astronomy:cosmology","astronomy:cosmology:simulation","numerical:hydrodynamics"],"title":"§3.4 欧拉（网格）方法"},{"content":"Adaptive mesh refinement 为拓宽数值格式的动力学范围，多种网格程序已采用网格细化策略（例如 ART、RAMSES、ENZO 和 FLASH）。多数情况下采用简单的密度（即每个网格的质量）判据：若某网格内的质量超过阈值 $m \\equiv \\rho , \\Delta x^3 \u0026gt; m_{min}$， 则将该网格分割为多个（例如八个）子网格，并将内部属性从原网格插值到新子网格上。这确保了引力质量（即引力源）在计算域内均匀分布。如此一来，底层网格便以准拉格朗日方式跟随质量流演化，如 Fig. 4 左图所示，该图展示了宇宙学模拟中细化网格的典型结构。To widen the dynamical range of the numerical schemes, mesh refinement strategies have been applied in several grid codes (e.g., ART, RAMSES, ENZO, and FLASH). In most of the cases, a simple density (e.g., mass per cell) criterion is used. If the mass within one cell exceeds a certain threshold, $m \\equiv \\rho \\, \\Delta x^3 \u003e m_{min}$, the cell is divided in multiple (e.g., eight) sub-cells and the internal properties are interpolated from the original cell onto the new sub-cells. This ensures that the gravitational mass (e.g., the source of gravity) is homogeneously distributed within the computational domain. In this way, the underlying grid evolves in a quasi-Lagrangian fashion following the mass flow, as illustrated in the left panel of Fig.\u0026nbsp;4, which shows the typical structure of the refinement grid in a cosmological simulation.\n基于速度判据的其他细化策略也常被采用 [75] ,[76] ，用于研究星系团中的激波与湍流。 将前述判据扩展，额外对速度跳变进行细化，便可在整个宇宙结构中以前所未有的高空间分辨率追踪湍流和激波的形成，如 Fig. 4 右图所示。 为高精度、高准确度地追踪湍流级串，可额外使用次尺度湍流模型来初始化细化网格上的速度：这可防止湍流级串受到抑制 [77] 。Other refinement strategies based on velocity criteria are often used [75] ,[76] to study shocks and turbulence in galaxy clusters. By extending the aforementioned criteria to additionally refining on velocity jumps, the formation of turbulence and shocks can be followed with unprecedented high spatial resolution throughout the cosmic structures, as shown in the right part of Fig.\u0026nbsp;4. In order to follow the turbulent cascade with high precision and accuracy, sub-scale turbulence models can additionally be used to initialize the velocities on the refined cells: this prevents the turbulent cascade from being suppressed [77] .\nFig.\u0026nbsp;4. 系列导航 ← 上一篇：§3.4 → 下一篇：§3.6\n","date":"2026-06-25","permalink":"/posts/2026-06-25-hydrodynamic-06/","series":["宇宙学模拟中的流体动力学方法与次网格模型"],"summary":"Adaptive mesh refinement 为拓宽数值格式的动力学范围，多种网格程序已采用网格细化策略（例如 ART、RAMSES、ENZO 和 FLASH）。多数情况下采用简单的密度（即每个网格的质量）判据：若某网格内的质量超过阈值 $m \\equiv \\rho , \\Delta x^3 \u003e m_{min}$， 则将该网格分割为多个（例如八个）子网格，并将内部属性从原网格插值到新子网格上。这确保了引力质量（即引力源）在计算域内均匀分布。如此一来，底层网格便以准拉格朗日方式跟随质量流演化，如 Fig. 4 左图所示，该图展示了宇宙学模拟中细化网格的典型结构。To widen the dynamical range of the numerical schemes, mesh refinement strategies have been applied in several grid codes (e.g., ART, RAMSES, ENZO, and FLASH). In most of the cases, a simple density (e.g., mass per cell) criterion is used. If the mass within one cell exceeds a certain threshold, $m \\equiv \\rho \\, \\Delta x^3 \u003e m_{min}$, the cell is divided in multiple (e.g., eight) sub-cells and the internal properties are interpolated from the original cell onto the new sub-cells. This ensures that the gravitational mass (e.g., the source of gravity) is homogeneously distributed within the computational domain. In this way, the underlying grid evolves in a quasi-Lagrangian fashion following the mass flow, as illustrated in the left panel of Fig. 4, which shows the typical structure of the refinement grid in a cosmological simulation.\n","tags":["翻译","astronomy:cosmology","astronomy:cosmology:simulation","numerical:hydrodynamics"],"title":"§3.5 自适应网格细化"},{"content":"Lagrangian methods and Smoothed Particle Hydrodynamics 经典的拉格朗日方法是所谓的光滑粒子流体动力学方法（SPH；[78] ,[79] ），它求解Euler方程的拉格朗日形式（参见Fig. 2），并可在高密度区域获得良好的空间分辨率。The classical Lagrangian method is the so-called Smoothed Particle Hydrodynamics method (SPH; [78] ,[79] ), which solves the Lagrangian form of the Euler equations (see Fig.\u0026nbsp;2) and can achieve good spatial resolution in high-density regions.\nSPH的基本思想是用质量元（即粒子）而非像Euler方法那样用体积元来离散流体（参见Fig. 2）。显然，已坍缩天体中的平均粒子间距小于低密度区域。因此，该方案通过保持质量分辨率固定，实现了空间分辨率的自适应。关于SPH格式的全面综述及其严格推导，参见[47] 。The basic idea of SPH is to discretize the fluid by mass elements (i.e. particles), rather than by volume elements as in Eulerian methods (see Fig.\u0026nbsp;2). It is evident that the mean interparticle distance in collapsed objects is smaller than in underdense regions. The scheme will thus be adaptive in spatial resolution by keeping the mass resolution fixed. For a comprehensive review with rigorous derivations of the SPH formalism, see [47] .\nFig. 5总结了SPH的主要特征。核函数光滑方法的一般定义，是为任意变量$X$构建连续流体量的第一步。其中，核函数仅依赖于距离的模，同时还要求是单调且可微的（参见Fig. 5上部）。在离散化形式中，我们将积分的体积元$d {\\vec x} = d^3 x$替换为粒子的质量与密度之比$m_j/\\rho_j$。虽然该方程对空间中的任意位置${\\vec x}$都成立，但这里我们只关心原始粒子位置${\\vec x}_i$处的流体表示，这些位置是后面唯一需要流体表示的地方。需要注意的是，对于具有紧支撑的核函数（即当$|{\\vec x}|\u0026gt;h$时$W({\\vec x},h)=0$），求和无需对所有粒子进行，而只需对半径为$h$的球体内的粒子（参见Fig. 2）——即所考虑粒子$i$周围的邻居粒子——进行求和。最初，最常用的核函数是$B_2$样条，但现代方法采用了全新的核函数族，如HOCT核函数[80] 或所谓的Wendland核函数[81] ，它们表现出更好的稳定性和更高的精度（参见Fig. 5右上图和[82] 的最新综述及其参考文献）。Fig.\u0026nbsp;5 summarizes the main characteristics of SPH. The general definition of a kernel smoothing method is the first step to build continuous fluid quantities for an arbitrary variable $X$. Here, the kernel depends on the distance modulus only, and is in addition required to be monotonic and differentiable (see upper part of Fig.\u0026nbsp;5). In the discretized version, we replace the volume element of the integration, $d {\\vec x} = d^3 x$, with the ratio of the mass and density $m_j/\\rho_j$ of the particles. Although this equation holds for any position ${\\vec x}$ in space, we are only interested here in the fluid representation at the original particle positions ${\\vec x}_i$, which are the only locations where we will need the fluid representation later on. It is important to note that for kernels with compact support (i.e., $W({\\vec x},h)=0$ for $|{\\vec x}|\u003eh$), the summation does not have to be done over all the particles but only over the particles within the sphere of radius $h$ (see Fig.\u0026nbsp;2), namely, the neighbors around the particle $i$ under consideration. Originally, the most frequently used kernel is the $B_2$-Spline, but modern schemes invoke an entire new family of kernels like the HOCT kernels [80] or the so-called Wendland kernels [81] which show better stability and higher accuracy (see upper right panel of Fig.\u0026nbsp;5 and also recent review by [82] and references therein).\nFig.\u0026nbsp;5. 导数可利用核函数的解析导数来计算。当使用成对对称公式时，守恒定律在数值上可得到更好的满足：这可通过在利用$\\rho X$或$X/\\rho$的导数时引入一些恒等式来实现（参见Fig. 5左中部）。为利用SPH方法的自适应特性，通常允许每个粒子$i$具有不同的光滑长度$h_i$，并通过在核函数内包含固定数量的邻居粒子或固定质量来确定。然而，上述独立光滑长度常常带来两方面的复杂性：首先，成对对称公式需要为每对粒子定义一个平均核函数，历史上已提出过各种构建平均值的方法。其二，核函数的导数——因光滑长度在空间上变化——伴随着修正项（$\\partial h_i/\\partial \\rho_i$），这些修正项最初总被忽略，因其无法直接计算。从熵表述出发，[83] 首次从拉格朗日形式导出了包含变化光滑长度的适当修正项的SPH公式，这些公式列在Fig. 5的右中部。注意，该形式同时也规定了粒子对平均核函数的构造方式。Derivatives can be calculated using the analytically known derivatives of the kernel. Conservation laws are numerically better achieved when pairwise symmetric formulations are used: this can be obtained by exploiting some identities when using derivatives of $\\rho X$ or $X/\\rho$ (see left middle part of Fig.\u0026nbsp;5). To profit from the adaptive nature of the SPH method, the smoothing length $h_i$ is typically allowed to vary for each individual particle $i$ and is determined by encompassing either a fixed number of neighbours or a fixed mass within the kernel. However, two complications often stem from the abovementioned individual smoothing lengths: first, a pairwise symmetric formulation needs to define an averaged kernel for each particle pair, and historically various ways to build averages have been discussed. The other is that the derivatives of the kernel -- being the smoothing length spatial dependent -- come with correction terms ($\\partial h_i/\\partial \\rho_i$), which originally have been always ignored because they can't be computed directly. Starting from an entropy formulation, [83] derived for the first time an SPH formulation including the proper correction terms for the varying smoothing length from a Lagrangian formalism, which are listed in the middle right part of Fig.\u0026nbsp;5. Note that this formalism then also specifies the way in which the averaged kernel for a particle pair has to be constructed.\n这还可进一步推广，如[84] 所示。在该工作中，从$x$加权的体积平均 $$ \\bar{y} = y_i = \\sum_j x_j W_{ij}(h_i), $$ （$x$加权的）体积元$\\Delta\\nu_i\\equiv x_i/y_i$以及压强$P_i$、单位质量内能$u_i$和熵函数$A_i$之间的广义关系 $$ P_i = (\\gamma-1)u_i\\frac{m_i}{\\Delta\\nu_i} = A_i\\left(\\frac{m_i}{\\Delta\\nu_i}\\right)^\\gamma $$ 出发，得到了这一广义公式。由此导出广义SPH方程组 $$ m_i \\frac{d {\\vec v}i}{d t} = - \\sum_j x_i x_j \\left(f{ij}\\frac{P_j}{y_j^2}{\\vec \\nabla}i W{ij}(h_i) + f_{ji}\\frac{P_i}{y_i^2}{\\vec \\nabla}i W{ij}(h_j) \\right)!, $$ 和 $$ f_{ij} \\equiv 1 - \\frac{\\tilde{x_i}}{x_j} \\left(\\frac{h_i}{3\\tilde{y_i}} \\frac{\\partial y_i}{\\partial h_i}\\right) \\left[1 + \\frac{h_i}{3\\tilde{y_i}}\\frac{\\partial \\tilde{y_i}}{\\partial h_i}\\right]^{-1}. $$ 当取$x_i=\\tilde{x_i}=m_i$时，有$y_i=\\tilde{y_i}=\\bar{\\rho_i}$和$\\Delta\\nu_i=m_i/\\rho_i$，并遵循熵$A_i$（即$P_i=A_i\\bar{\\rho_i}^\\gamma$），这组方程将给出[83] 中提出的熵守恒SPH公式。当取$x_i=\\tilde{x_i}=(\\gamma-1)m_i u_i$时，有$y_i=\\bar{P_i}$和$\\Delta\\nu_i=(\\gamma-1)m_i u_i/P_i$，这将给出[85] 中提出的压强-能量SPH公式。由于此时压强是核函数加权的量，接触间断面便得到了正确处理。第三种可能性是取$x_i=m_iA_i^{1/\\gamma}$，它将导出压强-熵公式。更多细节见[84] 。This can be further generalized, as shown in [84] . There, such a generalized formulation was obtained starting from an x-weighted volume average $$ \\bar{y} = y_i = \\sum_j x_j W_{ij}(h_i), $$ the (x-weighted) volume element $\\Delta\\nu_i\\equiv x_i/y_i$ and the generalized relation $$ P_i = (\\gamma-1)u_i\\frac{m_i}{\\Delta\\nu_i} = A_i\\left(\\frac{m_i}{\\Delta\\nu_i}\\right)^\\gamma $$ between the pressure $P_i$, the internal energy per unit mass $u_i$ and the entropic function $A_i$. This leads to the set of generalized SPH equations $$ m_i \\frac{d {\\vec v}_i}{d t} = - \\sum_j x_i x_j \\left(f_{ij}\\frac{P_j}{y_j^2}{\\vec \\nabla}_i W_{ij}(h_i) + f_{ji}\\frac{P_i}{y_i^2}{\\vec \\nabla}_i W_{ij}(h_j) \\right)\\!, $$ and $$ f_{ij} \\equiv 1 - \\frac{\\tilde{x_i}}{x_j} \\left(\\frac{h_i}{3\\tilde{y_i}} \\frac{\\partial y_i}{\\partial h_i}\\right) \\left[1 + \\frac{h_i}{3\\tilde{y_i}}\\frac{\\partial \\tilde{y_i}}{\\partial h_i}\\right]^{-1}. $$ For the choice of $x_i=\\tilde{x_i}=m_i$, which implies $y_i=\\tilde{y_i}=\\bar{\\rho_i}$ and $\\Delta\\nu_i=m_i/\\rho_i$, and following the entropy $A_i$ (e.g., $P_i=A_i\\bar{\\rho_i}^\\gamma$), this set of equations will result in the entropy-conserving formulation of SPH as presented in [83] . For the choice of $x_i=\\tilde{x_i}=(\\gamma-1)m_i u_i$, implying $y_i=\\bar{P_i}$ and $\\Delta\\nu_i=(\\gamma-1)m_i u_i/P_i$, this results in a pressure--energy formulation of SPH, as presented in [85] . As now pressure is a kernel weighted quantity, contact discontinuities are properly treated. A third possibility is to choose $x_i=m_iA_i^{1/\\gamma}$ which leads to a pressure--entropy formulation. For more details, see [84] .\n然而，SPH方法在低密度区域的表现不如高密度区域。由于引入了可观的人工黏性，它在激波区域的分辨率也会下降。在其经典实现中，离散化误差会在密度梯度陡峭的区域——特别是在接触间断面附近——对粒子引入虚假压力。这导致了一个尺度为SPH光滑核函数半径的边界间隙，该间隙内的相互作用受到严重阻尼。标准实现通常不包含显式的混合项来补偿这种效应。因此，经典实现无法分辨和处理多相流体相互作用中的动力学不稳定性，如Kelvin-Helmholtz不稳定性或Rayleigh-Taylor不稳定性。这两个缺点均可通过本节末尾描述的现代实现来克服。此外，在宇宙学背景下，SPH方法的自适应特性、其与引力的简单耦合方式以及可使用独立时间步长的能力，常常弥补了这些不足，从而使SPH仍为数值流体宇宙学中最常用的方法之一。The SPH method, however, is not performing as well in low-density regions as in the higher density ones. It also suffers from degraded resolution in shocked regions due to the introduction of a sizeable artificial viscosity. In its classical implementation, discretization errors introduce spurious pressure forces on particles in regions with steep density gradients in particular near contact discontinuities. This results in a boundary gap of the size of an SPH smoothing kernel radius over which interactions are severely damped. The standard implementation typically does not involve an explicit mixing term, which can compensate this effect. Therefore, the classical implementation does not resolve and treat dynamical instabilities in the interaction of multi-phase fluids, such as Kelvin--Helmholtz or Rayleigh--Taylor instabilities. Both these shortcomings can be overcome by a modern implementation as described at the end of this section. In addition, in the cosmological context, the adaptive nature of the SPH method, its simple way to couple to gravity and the possibility to have individual time steps often compensate for such shortcomings, thus making SPH still one of the most commonly used methods in numerical hydrodynamical cosmology.\n如前所述，还需添加所谓的人工黏性$\\Pi_{ij}$，使得熵守恒SPH公式的最终方程为 $$ \\frac{d {\\vec v}_i}{d t} = - \\sum_j m_j \\left(f_j\\frac{P_j}{\\rho_j^2}{\\vec \\nabla}_i W_{ij}(h_j)+f_i\\frac{P_i}{\\rho_i^2}{\\vec \\nabla}_i W_{ij}(h_i) + \\Pi_{ij} {\\vec \\nabla}_i {\\bar W}_{ij} \\right), $$ 以及 $$ \\frac{d A_i}{d t} = \\frac{1}{2} \\frac{\\gamma-1}{\\rho_i^{\\gamma-1}} \\sum_j m_j \\Pi_{ij} \\left({\\vec v}_j - {\\vec v}_i\\right){\\vec \\nabla}_i {\\bar W}_{ij}. $$\n这一描述人工黏性的项通常是捕捉激波所必需的，其构造方式与其他流体动力学格式类似。流行的公式有Monaghan和Gingold[86] 以及Balsara[87] 提出的形式，这些公式包含体黏性项和von Neumann-Richtmeyer黏性项，并辅以限制器以减少低粒子数下剪切流中的角动量输运[88] 。现代方法采用[89] 提出的人工黏性形式，它基于可压缩气体动力学Riemann解的类比。为减少至少无激波流动区域中的人工黏性，一种可行途径是遵循Morris和Monaghan[90] 提出的思想：每个粒子携带自己的人工黏性，该黏性在激波区域之外会逐渐衰减。关于这种实现对ICM的影响的详细研究可参见[91] 。在高阶人工耗散项的实现方面还有各种进一步的改进[92] ,[93] 。现代SPH公式还利用了速度梯度的高阶计算方案，参见Fig. 5下部以及[94] ,[95] 中的相关讨论。通过采用[92] 并辅以这种速度梯度的高阶计算方案，可更好地抑制人工黏性，如[96] 所示。This term describing an artificial viscosity is usually needed to capture shocks and its construction is similar to other hydrodynamical schemes. Popular formulations are those proposed by Monaghan and Gingold [86] and Balsara [87] , which includes a bulk viscosity and a von Neumann--Richtmeyer viscosity term, supplemented by a limiter reducing angular momentum transport in the presence of shear flows at low particle numbers [88] . Modern schemes implement a form of the artificial viscosity as proposed by [89] , based on an analogy with Riemann solutions of compressible gas dynamics. To reduce this artificial viscosity, at least in those parts of the flows where there are no shocks, a posibility is to follow the idea proposed by Morris and Monaghan [90] : every particle carries its own artificial viscosity, which eventually decays outside the regions which undergo shocks. A detailed study of the implications on the ICM of such an implementation can be found in [91] . There are various further improvements on the implementation a higher order artificial dissipation term [92] ,[93] . Modern SPH formulations also make use of higher order calculation schemes for velocity gradients, see lower part of Fig.\u0026nbsp;5 as well as related discussion in [94] ,[95] . Even better suppression of the artificial viscosity can be reached by following [92] in combination with such higher order calculation schemes for velocity gradients, as shown in [96] .\n过去十年中，研究者付出了相当大的努力来结合Lagrange方法和Euler方法的优势。一方面，将Godunov方法引入SPH形式取得了显著进展，参见[97] ,[98] ,[99] ,[100] ，这些工作最近促成了所谓的无网格方法的诞生。另一方面，在将Euler方法推广到移动网格方法方面也付出了巨大努力。这两种新方法将在接下来的两个小节中简要介绍。Considerable effort has been made in the last decade to combine the advantages of Langrangian and Eulerian methods. On one hand, significant progress has been achieved in involving Godunov methods into the SPH formalism, see [97] ,[98] ,[99] ,[100] , which recently led to the so called meshless methods. On the other hand, large effort has been made to extend Eulerian methods to moving mesh methods. Both these new methods are briefely described in the next two sub-sections.\n系列导航 ← 上一篇：§3.5 → 下一篇：§3.7\n","date":"2026-06-25","permalink":"/posts/2026-06-25-hydrodynamic-07/","series":["宇宙学模拟中的流体动力学方法与次网格模型"],"summary":"Lagrangian methods and Smoothed Particle Hydrodynamics 经典的拉格朗日方法是所谓的光滑粒子流体动力学方法（SPH；[78] ,[79] ），它求解Euler方程的拉格朗日形式（参见Fig. 2），并可在高密度区域获得良好的空间分辨率。The classical Lagrangian method is the so-called Smoothed Particle Hydrodynamics method (SPH; [78] ,[79] ), which solves the Lagrangian form of the Euler equations (see Fig. 2) and can achieve good spatial resolution in high-density regions.\n","tags":["翻译","astronomy:cosmology","astronomy:cosmology:simulation","numerical:hydrodynamics"],"title":"§3.6 拉格朗日方法与光滑粒子流体动力学"},{"content":"Moving mesh methods 研究者投入了大量精力，将§3.4中描述的Euler方法重新表述为Lagrange网格方法。移动网格流体动力学在宇宙学中应用的更多细节，可参见[101] 的开创性工作及其参考文献。这种早期方法从规则网格出发，然后通过跟随流体流动使网格变形。Euler方程通过计算网格边界上的通量来演化。宇宙学模拟中得到的网格示例见Fig. 6左图。该技术在实际应用中的一个缺点（或挑战）是单个网格单元可能严重变形和拉伸。现代方案通过基于Voronoi或Delaunay镶嵌构建非结构网格，规避了这一问题（参见[102] 及其参考文献）。基于网格生成点$\\vec{r}_i$和$\\vec{r}_j$的网格单元相关几何结构如Fig. 6右图所示。然后必须在界面的质心处计算通量（注意，该质心不一定位于两个网格生成点之间的直线上，如虚线所示） $$ \\mathbf{Q}i^{(n+1)} = \\mathbf{Q}i^{(n)} - \\Delta t \\sum_j A{ij}\\hat{\\mathbf{F}}{ij}^{(n+1/2)}. $$ 该界面的运动$\\vec{w}$由速度$\\vec{w}_i$和$\\vec{w}_j$唯一定义，并且必须在旋转坐标系($x\u0026rsquo;,y\u0026rsquo;$)中使用Riemann求解器计算通量。该技术的详细描述及其在测试问题中的表现，可参见[102] 。Substantial effort has gone into reformulating Eulerian methods as described in §3.4 into Lagrangian mesh approaches. More details on the idea of hydrodynamics on moving mesh for cosmological application can be found in the pioneering work by [101] and references therein. This early approach started from a regular mesh which then, by following the flow of the fluid, was deformed. The Euler equations were evolved by calculating the fluxes across the cell borders. An example of the resulting mesh for a cosmological simulation can be seen in the left part of Fig.\u0026nbsp;6. One disadvantage (or challenge) of this technique in practical applications is that individual cells can be extensively deformed and stretched. Modern schemes circumvent this problem by constructing an unstructured mesh based on a Voronoi or Delaunay tessellation (see [102] and references therein). The relevant geometry of the cells, based on the mesh generating points $\\vec{r}_i$ and $\\vec{r}_j$ is illustrated in the right part of Fig.\u0026nbsp;6. The fluxes then have to be calculated at the centroid of the interface (note that this is not necessarily on the straight line between the two mesh generating points, as indicated by the dotted line) $$ \\mathbf{Q}_i^{(n+1)} = \\mathbf{Q}_i^{(n)} - \\Delta t \\sum_j A_{ij}\\hat{\\mathbf{F}}_{ij}^{(n+1/2)}. $$ The motion $\\vec{w}$ of this interface is uniquely defined by the velocities $\\vec{w}_i$ and $\\vec{w}_j$, and the fluxes have to be calculated with the Riemann solver in the rotated frame ($x',y'$). A detailed description of this technique along with its performance in test problems can be found in [102] .\nFig.\u0026nbsp;6. 系列导航 ← 上一篇：§3.6 → 下一篇：§3.8\n","date":"2026-06-25","permalink":"/posts/2026-06-25-hydrodynamic-08/","series":["宇宙学模拟中的流体动力学方法与次网格模型"],"summary":"Moving mesh methods 研究者投入了大量精力，将§3.4中描述的Euler方法重新表述为Lagrange网格方法。移动网格流体动力学在宇宙学中应用的更多细节，可参见[101] 的开创性工作及其参考文献。这种早期方法从规则网格出发，然后通过跟随流体流动使网格变形。Euler方程通过计算网格边界上的通量来演化。宇宙学模拟中得到的网格示例见Fig. 6左图。该技术在实际应用中的一个缺点（或挑战）是单个网格单元可能严重变形和拉伸。现代方案通过基于Voronoi或Delaunay镶嵌构建非结构网格，规避了这一问题（参见[102] 及其参考文献）。基于网格生成点$\\vec{r}_i$和$\\vec{r}_j$的网格单元相关几何结构如Fig. 6右图所示。然后必须在界面的质心处计算通量（注意，该质心不一定位于两个网格生成点之间的直线上，如虚线所示） $$ \\mathbf{Q}i^{(n+1)} = \\mathbf{Q}i^{(n)} - \\Delta t \\sum_j A{ij}\\hat{\\mathbf{F}}{ij}^{(n+1/2)}. $$ 该界面的运动$\\vec{w}$由速度$\\vec{w}_i$和$\\vec{w}_j$唯一定义，并且必须在旋转坐标系($x’,y’$)中使用Riemann求解器计算通量。该技术的详细描述及其在测试问题中的表现，可参见[102] 。Substantial effort has gone into reformulating Eulerian methods as described in §3.4 into Lagrangian mesh approaches. More details on the idea of hydrodynamics on moving mesh for cosmological application can be found in the pioneering work by [101] and references therein. This early approach started from a regular mesh which then, by following the flow of the fluid, was deformed. The Euler equations were evolved by calculating the fluxes across the cell borders. An example of the resulting mesh for a cosmological simulation can be seen in the left part of Fig. 6. One disadvantage (or challenge) of this technique in practical applications is that individual cells can be extensively deformed and stretched. Modern schemes circumvent this problem by constructing an unstructured mesh based on a Voronoi or Delaunay tessellation (see [102] and references therein). The relevant geometry of the cells, based on the mesh generating points $\\vec{r}_i$ and $\\vec{r}_j$ is illustrated in the right part of Fig. 6. The fluxes then have to be calculated at the centroid of the interface (note that this is not necessarily on the straight line between the two mesh generating points, as indicated by the dotted line) $$ \\mathbf{Q}_i^{(n+1)} = \\mathbf{Q}_i^{(n)} - \\Delta t \\sum_j A_{ij}\\hat{\\mathbf{F}}_{ij}^{(n+1/2)}. $$ The motion $\\vec{w}$ of this interface is uniquely defined by the velocities $\\vec{w}_i$ and $\\vec{w}_j$, and the fluxes have to be calculated with the Riemann solver in the rotated frame ($x',y'$). A detailed description of this technique along with its performance in test problems can be found in [102] .\n","tags":["翻译","astronomy:cosmology","astronomy:cosmology:simulation","numerical:hydrodynamics"],"title":"§3.7 移动网格方法"},{"content":"Meshless methods 最近，一类新的拉格朗日方法——即所谓的无网格公式——在天体物理领域得到了发展。更多细节见[102] 、[104] 、[105] ，这些工作继承了[106] 、[107] 、[108] 更早的开创性研究。简而言之，推导从积分形式 $$ \\int[u(\\vec{x},t)\\dot\\phi(\\vec{x},t) + \\vec{F}(u,\\vec{x},t)\\cdot\\nabla\\phi(\\vec{x},t) + S(\\vec{x},t)\\phi(\\vec{x},t)],d\\vec{x},dt = 0 $$ 出发，它是标量守恒律 $$ \\frac{\\partial u}{\\partial t} + \\nabla\\cdot(\\vec{F} + \\vec{a}u) = S $$ 的积分形式。其中，$u(\\vec{x},t)$是一个标量场，$S(\\vec{x}, t)$是其源项，$\\vec{F}(u,\\vec{x},t)$是在以速度$\\vec{a}(\\vec{x}, t)$运动的参考系中的通量，而$\\phi(\\vec{x},t)$是空间和时间上的任意可微函数，由此定义了随体导数$\\dot\\phi(\\vec{x}, t) = \\partial\\phi(\\vec{x}, t)/\\partial t + \\vec{a}(x,t)\\cdot\\nabla\\phi(\\vec{x}, t)$。A new class of Lagrangian methods, the so-called meshless formulations, have been recently developed for astrophysical problems. More details can be found in [102] ,[104] ,[105] , which follow earlier, pioneering work by [106] ,[107] ,[108] . In short, the derivation starts from the integral form $$ \\int[u(\\vec{x},t)\\dot\\phi(\\vec{x},t) + \\vec{F}(u,\\vec{x},t)\\cdot\\nabla\\phi(\\vec{x},t) + S(\\vec{x},t)\\phi(\\vec{x},t)]\\,d\\vec{x}\\,dt = 0 $$ of a scalar conservation law $$ \\frac{\\partial u}{\\partial t} + \\nabla\\cdot(\\vec{F} + \\vec{a}u) = S \\,. $$ Here, $u(\\vec{x},t)$ is a scalar field, $S(\\vec{x}, t)$ is its source, $\\vec{F}(u,\\vec{x},t)$ is its flux in a frame moving with velocity $\\vec{a}(\\vec{x}, t)$, and $\\phi(\\vec{x},t)$ is an arbitrary differentiable function in space and time leading to the advective derivative $\\dot\\phi(\\vec{x}, t) = \\partial\\phi(\\vec{x}, t)/\\partial t + \\vec{a}(x,t)\\cdot\\nabla\\phi(\\vec{x}, t)$.\n采用与SPH类似的做法，以一组粒子$i$进行离散（光滑长度为$h(\\vec{x})$，核函数为$W(\\vec{x},h)$），则粒子划分可写为 $$ \\psi_i(\\vec{x}) = w(\\vec{x})W(\\vec{x}-\\vec{x}_i,h(\\vec{x})), $$ 其中粒子的数密度为$w(\\vec{x})^{-1} = \\sum_j W(\\vec{x}-\\vec{x}_j,h(\\vec{x}))$。任意函数$f(\\vec{x})$的离散化可写为 $$ \\int f(\\vec{x}),d\\vec{x} \\approx \\sum_i f_i \\int \\psi_i(\\vec{x}),d\\vec{x} \\equiv \\sum_i f_i V_i, $$ 其中$V_i = \\int \\psi_i(\\vec{x}),d\\vec{x}$是粒子$i$的有效体积。因此，积分方程(3.28)的离散形式可写为 $$ \\sum_i \\int [ V_i u_i \\dot\\phi_i + V_i F^\\alpha_i (D^\\alpha\\varphi)_i + V_i S_i \\phi_i ] = 0 , . $$ 虽然原则上可采用$(D^\\alpha\\phi)_i$的SPH估计，但更方便的做法是采用[107] 建议的更精确的无网格梯度估计。结合第一项的分部积分，由此（如[102] 所示）可将$\\phi(\\vec{x}, t)$分离出来并得到： $$ \\frac{d}{dt}(V_i u_i) + \\sum_j\\left[V_i F_i^\\alpha\\psi_j^\\alpha(\\vec{x}_i) - V_j F_j^\\alpha\\psi_i^\\alpha(\\vec{x}_j)\\right] = V_i S_i , . $$ 该方程（及其对一般矢量场$\\vec{u}$的推广）与移动网格方法的有限体积方程非常相似[Fn: 但注意，方程[eq:flux]是积分形式。]，但在某种程度上也与SPH方程类似。其区别在于，不同粒子之间的相互作用由源项和通量项描述，而它们可通过求解粒子$i$和$j$之间的近似Riemann问题得到。进一步观察还可发现，实际只需粒子间连线方向上的投影，例如在中点处求解Riemann问题\\footnote{更精确的做法是在核长度$h_i$和$h_j$的等分点处求积，参见[104] 中的讨论及其中的参考文献。}。这也意味着需要将原始变量外推到中点（例如使用线性外推），且通常还需施加通量限制器（参见[102] 、[104] 、[105] 中的讨论）。Fig. 7展示了无网格方法、非结构（移动）网格和经典核加权形式（SPH）在体积划分上的差异。Using a discretization through a set of particles $i$ with a smoothing length $h(\\vec{x})$ and a kernel function $W(\\vec{x},h)$, akin to what is done in SPH, the partitioning of the particles can be written as $$ \\psi_i(\\vec{x}) = w(\\vec{x})W(\\vec{x}-\\vec{x}_i,h(\\vec{x})), $$ where the number density of particles is $w(\\vec{x})^{-1} = \\sum_j W(\\vec{x}-\\vec{x}_j,h(\\vec{x}))$. The discretization of an arbitrary function $f(\\vec{x})$ can be written as $$ \\int f(\\vec{x})\\,d\\vec{x} \\approx \\sum_i f_i \\int \\psi_i(\\vec{x})\\,d\\vec{x} \\equiv \\sum_i f_i V_i, $$ where $V_i = \\int \\psi_i(\\vec{x})\\,d\\vec{x}$ is the effective volume of a particle $i$. Therefore, the discrete form of the integral equation (3.28) can be written as $$ \\sum_i \\int [ V_i u_i \\dot\\phi_i + V_i F^\\alpha_i (D^\\alpha\\varphi)_i + V_i S_i \\phi_i ] = 0 \\, . $$ Although in principle an SPH estimate for $(D^\\alpha\\phi)_i$ could be used, it is convenient to use a more accurate meshless gradient estimate, as suggested by [107] . This, along with the integration of the first term by parts, makes it possible (as shown in [102] ) to split $\\phi(\\vec{x}, t)$ and to obtain: $$ \\frac{d}{dt}(V_i u_i) + \\sum_j\\left[V_i F_i^\\alpha\\psi_j^\\alpha(\\vec{x}_i) - V_j F_j^\\alpha\\psi_i^\\alpha(\\vec{x}_j)\\right] = V_i S_i \\, . $$ This equation (and its extension to a general vector field $\\vec{u}$) is very similar to the finite volume equation for the moving mesh method [Fn: Note that, however, Eq. [eq:flux] is in the integral form.] but also somewhat similar to the SPH equations, except that the interactions between different particles is described in the source and flux terms, which can be obtained as the solution of an approximate Riemann problem between particles $i$ and $j$. A closer inspection also reveals that only the projection on the direction between the particles is needed, e.g., the solution of the Riemann problem at the midpoint\\footnote{More accurate would be the quadrature point at an equal fraction of the kernel length $h_i$ and $h_j$, see discussion in [104] and references therein.}. This also means that the primitive variables have to be extrapolated to the midpoint, e.g., using linear extrapolation, and generally a flux limiter has to be applied (see discussion in [102] ,[104] ,[105] ). Fig.\u0026nbsp;7 illustrates the differences in partitioning the volumes between mesh-less methods, unstructured (moving) grid and classical kernel weighted formalism (SPH).\nFig.\u0026nbsp;7. 系列导航 ← 上一篇：§3.7 → 下一篇：§3.9\n","date":"2026-06-25","permalink":"/posts/2026-06-25-hydrodynamic-09/","series":["宇宙学模拟中的流体动力学方法与次网格模型"],"summary":"Meshless methods 最近，一类新的拉格朗日方法——即所谓的无网格公式——在天体物理领域得到了发展。更多细节见[102] 、[104] 、[105] ，这些工作继承了[106] 、[107] 、[108] 更早的开创性研究。简而言之，推导从积分形式 $$ \\int[u(\\vec{x},t)\\dot\\phi(\\vec{x},t) + \\vec{F}(u,\\vec{x},t)\\cdot\\nabla\\phi(\\vec{x},t) + S(\\vec{x},t)\\phi(\\vec{x},t)],d\\vec{x},dt = 0 $$ 出发，它是标量守恒律 $$ \\frac{\\partial u}{\\partial t} + \\nabla\\cdot(\\vec{F} + \\vec{a}u) = S $$ 的积分形式。其中，$u(\\vec{x},t)$是一个标量场，$S(\\vec{x}, t)$是其源项，$\\vec{F}(u,\\vec{x},t)$是在以速度$\\vec{a}(\\vec{x}, t)$运动的参考系中的通量，而$\\phi(\\vec{x},t)$是空间和时间上的任意可微函数，由此定义了随体导数$\\dot\\phi(\\vec{x}, t) = \\partial\\phi(\\vec{x}, t)/\\partial t + \\vec{a}(x,t)\\cdot\\nabla\\phi(\\vec{x}, t)$。A new class of Lagrangian methods, the so-called meshless formulations, have been recently developed for astrophysical problems. More details can be found in [102] ,[104] ,[105] , which follow earlier, pioneering work by [106] ,[107] ,[108] . In short, the derivation starts from the integral form $$ \\int[u(\\vec{x},t)\\dot\\phi(\\vec{x},t) + \\vec{F}(u,\\vec{x},t)\\cdot\\nabla\\phi(\\vec{x},t) + S(\\vec{x},t)\\phi(\\vec{x},t)]\\,d\\vec{x}\\,dt = 0 $$ of a scalar conservation law $$ \\frac{\\partial u}{\\partial t} + \\nabla\\cdot(\\vec{F} + \\vec{a}u) = S \\,. $$ Here, $u(\\vec{x},t)$ is a scalar field, $S(\\vec{x}, t)$ is its source, $\\vec{F}(u,\\vec{x},t)$ is its flux in a frame moving with velocity $\\vec{a}(\\vec{x}, t)$, and $\\phi(\\vec{x},t)$ is an arbitrary differentiable function in space and time leading to the advective derivative $\\dot\\phi(\\vec{x}, t) = \\partial\\phi(\\vec{x}, t)/\\partial t + \\vec{a}(x,t)\\cdot\\nabla\\phi(\\vec{x}, t)$.\n","tags":["翻译","astronomy:cosmology","astronomy:cosmology:simulation","numerical:hydrodynamics"],"title":"§3.8 无网格方法"},{"content":"Code comparison in galaxy and cluster simulations 前文所述的欧拉和拉格朗日方法，在理论上应用于同一问题时应给出相同的结果。为验证代码能否正确积分流体力学方程组，通常会用已知解析解的问题来测试。常见的测试问题包括激波管或球对称坍缩问题。The Eulerian and Lagrangian approaches described in the previous sections are theoretically supposed to provide the same results when applied to the same problem. To verify that codes succeed at correctly integrating the set of hydrodynamical equations, they are usually tested against problems whose solution is known analytically. In practice, these test problems are shock tubes or spherical collapse problems.\n不过，理想化的流体动力学测试——例如多相流体的相互作用 [109] ——通常会揭示不同方法所得结果之间的根本差异。造成这些差异的原因可能是基本方程的表述形式（例如经典SPH中不存在混合）、离散化方式（例如SPH表述中的体积偏差），或是数值误差的影响（例如网格代码中因重构误差而导致的平移不变性偏离）。However, idealized hydrodynamical tests like the interaction of multi-phase fluids [109] often reveal fundamental differences among results obtained with different methods. Such differences can be driven by the formulation of the underlying fundamental equations (like no mixing in classical SPH), by the discretization (like the volume bias in SPH formulations) or they can be due to the influence of numerical errors (like the departure of translation invariance in grid codes due to errors in the reconstruction).\n在宇宙学中，具有已知解析解的问题并不现实；因此，一个有意义的测试是比较不同代码在标准设置下模拟宇宙结构形成时给出的结果。例如，[110] 比较了GADGET（基于SPH）和ENZO（基于网格）代码所预测的星际介质热力学性质。另一例网格代码与SPH代码之间的对比见 [111] 。In cosmology, problems with known analytical solutions are impractical: as a consequence, a meaningful test is represented by comparing the results provided by different codes when they simulate the formation of cosmic structures in a standard set-up. As an example, [110] compares the thermodynamical properties of the IGM predicted by the GADGET (SPH-based) and ENZO (grid-based) codes. Another example of a comparison between grid-based and SPH-based codes can be found in [111] .\n因此，对模拟星系或星系团形成与演化的流体动力学代码进行详细比较，是一项极其重要的测试。一项开创性的对比工作来自所谓的Santa Barbara星系团对比项目[112] 。在该项目中，12个不同的小组各自使用基于SPH技术（七个小组）或网格技术（五个小组）的代码，从相同的初始条件出发，对星系团进行了非辐射模拟。A detailed comparison of hydrodynamical codes which simulate the formation and evolution of a galaxy or of a galaxy cluster is therefore an extremely important test. A pioneering comparison was performed within the so-called Santa Barbara Cluster Comparison Project [112] . Here, 12 different groups, each using a code either based on the SPH technique (seven groups) or on the grid technique (five groups), performed a non-radiative simulation of a galaxy cluster from the same initial conditions.\n一个更新的对比项目——所谓的nIFTy星系团模拟 [113] ——还纳入了现代SPH实现以及移动网格代码AREPO。两项研究在气体性质的很大范围内都取得了一致的结果，例如密度、温度和熵轮廓，如Fig. 8所示。两项研究均发现，在轮廓的内区，网格代码与经典粒子代码之间差异显著更大。然而，正如[113] 所示，包含显式混合处理的粒子代码产生的结果与网格代码的结果非常相似。A more recent comparison project, the so-called nIFTy galaxy cluster simulations [113] , also involved modern SPH implementations as well as the moving mesh code AREPO. A similar agreement over large ranges was obtained for many of the gas properties in both studies, like the density, temperature and entropy profiles, as shown in Fig.\u0026nbsp;8. Both studies found significantly larger differences between mesh-based codes and classical particle-based codes to be present for the inner part of the profiles. However, as shown in [113] , particle-based codes which include an explicit treatment of mixing produce results very similar to grid-based codes.\n一些代码对比活动也针对星系进行（见[114] 、[115] 、Fig. 8和§3.14）。A few code comparison campaigns have targeted galaxies as well (see e.g., [114] ,[115] , Fig.\u0026nbsp;8, and §3.14).\n值得指出，一旦纳入了额外的物理过程——如恒星形成和AGN反馈，各种数值方法所得结果的差异便不再源于流体动力学求解器本身的区别；它们更多取决于模拟这些附加过程时所采用的数值方案的具体细节[116] 。It is worth mentioning that as soon as additional physics like star formation and AGN feedback is included, the discrepancies in the results obtained by the various numerical methods are no longer driven by differences which can be ascribed to the hydrodynamical solvers: they are rather due to the details of the numerical prescriptions adopted to model these additional processes [116] .\nFig.\u0026nbsp;8. 系列导航 ← 上一篇：§3.8 → 下一篇：§3.10\n","date":"2026-06-25","permalink":"/posts/2026-06-25-hydrodynamic-10/","series":["宇宙学模拟中的流体动力学方法与次网格模型"],"summary":"Code comparison in galaxy and cluster simulations 前文所述的欧拉和拉格朗日方法，在理论上应用于同一问题时应给出相同的结果。为验证代码能否正确积分流体力学方程组，通常会用已知解析解的问题来测试。常见的测试问题包括激波管或球对称坍缩问题。The Eulerian and Lagrangian approaches described in the previous sections are theoretically supposed to provide the same results when applied to the same problem. To verify that codes succeed at correctly integrating the set of hydrodynamical equations, they are usually tested against problems whose solution is known analytically. In practice, these test problems are shock tubes or spherical collapse problems.\n不过，理想化的流体动力学测试——例如多相流体的相互作用 [109] ——通常会揭示不同方法所得结果之间的根本差异。造成这些差异的原因可能是基本方程的表述形式（例如经典SPH中不存在混合）、离散化方式（例如SPH表述中的体积偏差），或是数值误差的影响（例如网格代码中因重构误差而导致的平移不变性偏离）。However, idealized hydrodynamical tests like the interaction of multi-phase fluids [109] often reveal fundamental differences among results obtained with different methods. Such differences can be driven by the formulation of the underlying fundamental equations (like no mixing in classical SPH), by the discretization (like the volume bias in SPH formulations) or they can be due to the influence of numerical errors (like the departure of translation invariance in grid codes due to errors in the reconstruction).\n","tags":["翻译","astronomy:cosmology","astronomy:cosmology:simulation","numerical:hydrodynamics"],"title":"§3.9 星系与星系团模拟中的代码对比"},{"content":"Sub-resolution modelling of astrophysical processes 在宇宙学流体动力学模拟中，涉及星系形成和演化的天体物理过程跨越了巨大的动力学范围。它们从约$\\sim$Mpc尺度——引力不稳定性驱动暗物质组分演化与暗晕等级式并合之处——一直延伸到约$\\sim$pc和$\\sim$亚pc尺度——黑洞吸积和恒星形成等过程发生之处，途经约$\\sim$kpc尺度——例如星系风将恒星反馈能量分配到周围介质之处。 因此，对次网格物理的需求至关重要：次分辨率模型描述的是发生在宇宙学流体动力学模拟分辨率极限以下的过程，但这些过程会影响在显式解析尺度上演化的模拟结构。次分辨率方案通常借助相当简单的解析或理论模型，辅以恰当选择并经校准以重现观测的参数，对相当复杂的过程进行唯象描述（另见§3.16小节）。The astrophysical processes involved in the formation and evolution of galaxies in cosmological hydrodynamical simulations span a huge dynamical range of scales. They indeed vary from the $\\sim$Mpc scales where gravitational instabilities drive the evolution of the DM component and the hierarchical assembly of haloes, down to the $\\sim$parsec and $\\sim$sub-parsec scales, where processes like BH accretion and star formation take place, going through the $\\sim$kpc scales, where e.g. galactic winds distribute the stellar feedback energy to the ambient medium. The call for sub-grid physics is thus essential: sub-resolution models account for processes that occur below the resolution limit of cosmological hydrodynamical simulations, but that affect the evolution of the simulated structure on scales that are explicitly resolved. Sub-resolution prescriptions usually resort to rather simple analytical or theoretical models, and/or to the phenomenological description of rather complex processes, through a suitable choice of parameters that are calibrated to reproduce observations (see also sub-section §3.16).\n数值天体物理学常受到的一种批评，便是建模物理过程时所使用的参数数量过多，特别是次分辨率模型中的参数。然而，仅靠微调参数并无法重现观测或预期结果。结果更主要地取决于过程的恰当参数化方式。例如：在建模恒星形成时（见§3.12小节），结果主要由恒星形成率（SFR）与实际燃料（如可用的分子气体或冷气体质量）之间的关系所决定，而非上述两个量之间比例常数的精确取值。 因此，参数空间探索的目的并非为模拟所要比较的观测提供最佳拟合。它更旨在深化对模型依赖于不同未知量的理解，并对约束较弱的物理量做出预测。A common criticism that has been often raised to numerical astrophysics deals with the number of parameters that are employed when modelling physical processes, in particular when referring to the parameters that enter sub-resolution models. However, observed or expected results cannot be reproduced by just fine-tuning parameters. Results are rather determined by proper parametrizations of processes. For instance: when modelling star formation (see sub-section §3.12), results are mainly driven by the way in which the star formation rate (SFR) relates to the actual fuel (e.g. the mass of molecular or cold gas available), rather than to the exact value of the proportionality constant between the two aforementioned quantities. Parameter space exploration is thus not meant to provide the best fit to observations that simulations want to compare with. It rather aims at enabling a better understanding of the dependence of a model on different unknowns and at making predictions for loosely constrained physical quantities.\n在接下来的小节中，我们将回顾对星系形成和演化影响最为显著的天体物理过程。它们均自洽地包含在最先进的宇宙学流体动力学模拟中。In the next sub-sections, we will review the most important astrophysical processes that significantly affect galaxy formation and evolution. They are all self-consistently included in state-of-the-art cosmological hydrodynamical simulations.\n系列导航 ← 上一篇：§3.9 → 下一篇：§3.11\n","date":"2026-06-25","permalink":"/posts/2026-06-25-hydrodynamic-11/","series":["宇宙学模拟中的流体动力学方法与次网格模型"],"summary":"Sub-resolution modelling of astrophysical processes 在宇宙学流体动力学模拟中，涉及星系形成和演化的天体物理过程跨越了巨大的动力学范围。它们从约$\\sim$Mpc尺度——引力不稳定性驱动暗物质组分演化与暗晕等级式并合之处——一直延伸到约$\\sim$pc和$\\sim$亚pc尺度——黑洞吸积和恒星形成等过程发生之处，途经约$\\sim$kpc尺度——例如星系风将恒星反馈能量分配到周围介质之处。 因此，对次网格物理的需求至关重要：次分辨率模型描述的是发生在宇宙学流体动力学模拟分辨率极限以下的过程，但这些过程会影响在显式解析尺度上演化的模拟结构。次分辨率方案通常借助相当简单的解析或理论模型，辅以恰当选择并经校准以重现观测的参数，对相当复杂的过程进行唯象描述（另见§3.16小节）。The astrophysical processes involved in the formation and evolution of galaxies in cosmological hydrodynamical simulations span a huge dynamical range of scales. They indeed vary from the $\\sim$Mpc scales where gravitational instabilities drive the evolution of the DM component and the hierarchical assembly of haloes, down to the $\\sim$parsec and $\\sim$sub-parsec scales, where processes like BH accretion and star formation take place, going through the $\\sim$kpc scales, where e.g. galactic winds distribute the stellar feedback energy to the ambient medium. The call for sub-grid physics is thus essential: sub-resolution models account for processes that occur below the resolution limit of cosmological hydrodynamical simulations, but that affect the evolution of the simulated structure on scales that are explicitly resolved. Sub-resolution prescriptions usually resort to rather simple analytical or theoretical models, and/or to the phenomenological description of rather complex processes, through a suitable choice of parameters that are calibrated to reproduce observations (see also sub-section §3.16).\n","tags":["翻译","astronomy:cosmology","astronomy:cosmology:simulation","numerical:hydrodynamics"],"title":"§3.10 天体物理过程的次分辨率建模"},{"content":"Gas cooling 热气体冷却出现在方程equation:firstlaw的右端，同时也出现在描述亚网格模型中可能存在的不同气体相之间质量与能量流动的方程中。Hot gas cooling enters the right-hand side of equation equation:firstlaw, as well as the equations describing mass and energy flows among different gas phases which may be present in sub-resolution models.\n在宇宙学模拟中，关注的重点通常是维里温度超过$\\sim 10^4$ K的结构。冷却函数$\\Lambda(u,\\rho)$标准实现中的常见假设是气体为光学薄且处于电离平衡。 通常还假设三体冷却过程不重要，从而将处理限定在二体过程。对于具有H和He原初组分的等离子体，这些过程包括：H$^0$和He$^+$的碰撞激发，H$^0$、He$^0$和He$^{+}$的碰撞电离，H$^+$、He$^+$和He$^{++}$的标准复合，He$^+$的双电子复合，以及自由-自由发射（轫致辐射）。 碰撞电离和复合速率仅依赖于温度。因此，若不存在电离背景辐射，所得速率方程可解析求解，由此得到的冷却函数$\\Lambda(u)/\\rho^2$如Fig. 9左图所示。 反之，若存在电离背景辐射，速率方程则可通过迭代求解。注意，对于典型的宇宙学辐射背景（例如，来自恒星形成星系和类星体的紫外背景，见[118] 、[119] 、[120] 、[121] ），冷却函数的形状可能发生显著改变，特别是在低密度下。更详细的讨论见星系形成章节和[17] 。In cosmological simulations, the focus is usually on structures whose virial temperature exceeds $\\sim 10^4$ K. Common assumptions in standard implementations of the cooling function $\\Lambda(u,\\rho)$ is that the gas is optically thin and in ionization equilibrium. It is also usually assumed that three-body cooling processes are unimportant so as to restrict the treatment to two-body processes. For a plasma with primordial composition of H and He, these processes are: collisional excitation of H$^0$ and He$^+$, collisional ionization of H$^0$, He$^0$, and He$^{+}$, standard recombination of H$^+$, He$^+$, and He$^{++}$, dielectric recombination of He$^+$, and free--free emission (bremsstrahlung). The collisional ionization and recombination rates depend only on temperature. Therefore, should an ionizing background radiation be absent, the resulting rate equations can be solved analytically. This leads to a cooling function $\\Lambda(u)/\\rho^2$ as illustrated in the left panel of Fig.\u0026nbsp;9. On the other hand, in the presence of ionizing background radiation, the rate equations can be solved iteratively. Note that for a typical cosmological radiation background (e.g., UV background from star-forming galaxies and quasars, see [118] ,[119] ,[120] ,[121] ), the shape of the cooling function can be significantly altered, especially at low densities. For a more detailed discussion, see, for example, 星系形成章节 and [17] .\nFig.\u0026nbsp;9. 此外，金属的存在将使气体冷却的途径大大增多。由于此时计算冷却函数的计算量极大，宇宙学模拟通常依赖预计算的表格化冷却函数。例如，Fig. 9右图显示了来自[123] 的表格化冷却函数，温度在$10^5$ K以上，对应不同的气体金属丰度，同时保持不同金属元素的比例固定为太阳值。 如今模拟中常采用的一种改进方法是使用Cloudy程序（[124] ），它允许用户针对不同的紫外强度网格以及各种不同的化学元素，分别表格化冷却和加热率。通过这种方式，冷却率可针对任意化学 % 组分自洽地计算。 注意，几乎所有的实现都将上述速率方程（乃至气体的冷却）作为与流体动力学处理解耦的\u0026quot;子时间步\u0026quot;问题来求解，这相当于假设密度在整个时间步内固定不变。 此外，出于实际原因，底层流体动力学模拟的时间步通常既不受冷却时标的控制，也与冷却时标无关。这些近似所引入的不确定性尚未得到深入探索，显然为未来的研究留下了空间。Additionally, the presence of metals will drastically increase the number of possible processes by which gas can cool. As it becomes computationally very demanding to calculate the cooling function in this case, cosmological simulations usually rely on a pre-computed, tabulated cooling function. As an example, the right panel of Fig.\u0026nbsp;9 shows the tabulated cooling function from [123] , for temperatures above $10^5$ K, for different metallicities of the gas, keeping the ratios of different metal species fixed to solar values. A refinement that is nowadays often exploited in simulations consists in using the `Cloudy` code ([124] ), which allows users to tabulate the cooling and heating rates, individually for a grid of UV intensities and for various different chemical elements. In this way, cooling rates are self-consistently calculated for arbitrary chemical % compositions. Note that almost all the implementations solve the above rate equations (and therefore the cooling of the gas) as a ``subtime step'' problem, decoupled from the hydrodynamical treatment. This translates in assuming that the density is fixed across the time step. Furthermore, the time step of the underlying hydrodynamical simulation is in general, for practical reasons, not controlled by nor related to the cooling time scale. The resulting uncertainties introduced by these approximations have not yet been deeply explored and clearly leave room for future investigations.\n对于维里温度低于$10^4$ K的暗晕中第一批天体的形成，电离平衡假设不再成立。在这种情况下，必须考虑非平衡反应，即在宇宙学演化过程中求解每种物质各能级的平衡方程。 在没有金属的情况下，主要的冷却剂是H$_2$和H$_2^+$分子（见[125] ）。HD分子也可发挥重要作用。当存在金属时，可用的反应更多，其中一些可对$10^4$ K以下的冷却函数产生显著贡献。这一效应在Fig. 9右图中$T\u0026lt;10^4$ K的部分清晰可见。更多细节见星系形成章节、参考文献[126] 、[122] 及其中的参考文献。For the formation of the first objects in halos with virial temperatures below $10^4$ K, the assumption of ionization equilibrium no longer holds. In this case, the non-equilibrium reactions have to be considered, by solving the balance equations for the individual levels of each species during the cosmological evolution. In the absence of metals, the main coolants are H$_2$ and H$_2^+$ molecules (see [125] ). HD molecules can also play a significant role. When metals are present, many more reactions are available and some of these can contribute significantly to the cooling function below $10^4$ K. This effect is clearly visible in the right panel of Fig.\u0026nbsp;9 for $T\u003c10^4$ K. For more details, see 星系形成章节, Refs. [126] ,[122] , and references therein.\n系列导航 ← 上一篇：§3.10 → 下一篇：§3.12\n","date":"2026-06-25","permalink":"/posts/2026-06-25-hydrodynamic-12/","series":["宇宙学模拟中的流体动力学方法与次网格模型"],"summary":"Gas cooling 热气体冷却出现在方程equation:firstlaw的右端，同时也出现在描述亚网格模型中可能存在的不同气体相之间质量与能量流动的方程中。Hot gas cooling enters the right-hand side of equation equation:firstlaw, as well as the equations describing mass and energy flows among different gas phases which may be present in sub-resolution models.\n在宇宙学模拟中，关注的重点通常是维里温度超过$\\sim 10^4$ K的结构。冷却函数$\\Lambda(u,\\rho)$标准实现中的常见假设是气体为光学薄且处于电离平衡。 通常还假设三体冷却过程不重要，从而将处理限定在二体过程。对于具有H和He原初组分的等离子体，这些过程包括：H$^0$和He$^+$的碰撞激发，H$^0$、He$^0$和He$^{+}$的碰撞电离，H$^+$、He$^+$和He$^{++}$的标准复合，He$^+$的双电子复合，以及自由-自由发射（轫致辐射）。 碰撞电离和复合速率仅依赖于温度。因此，若不存在电离背景辐射，所得速率方程可解析求解，由此得到的冷却函数$\\Lambda(u)/\\rho^2$如Fig. 9左图所示。 反之，若存在电离背景辐射，速率方程则可通过迭代求解。注意，对于典型的宇宙学辐射背景（例如，来自恒星形成星系和类星体的紫外背景，见[118] 、[119] 、[120] 、[121] ），冷却函数的形状可能发生显著改变，特别是在低密度下。更详细的讨论见星系形成章节和[17] 。In cosmological simulations, the focus is usually on structures whose virial temperature exceeds $\\sim 10^4$ K. Common assumptions in standard implementations of the cooling function $\\Lambda(u,\\rho)$ is that the gas is optically thin and in ionization equilibrium. It is also usually assumed that three-body cooling processes are unimportant so as to restrict the treatment to two-body processes. For a plasma with primordial composition of H and He, these processes are: collisional excitation of H$^0$ and He$^+$, collisional ionization of H$^0$, He$^0$, and He$^{+}$, standard recombination of H$^+$, He$^+$, and He$^{++}$, dielectric recombination of He$^+$, and free--free emission (bremsstrahlung). The collisional ionization and recombination rates depend only on temperature. Therefore, should an ionizing background radiation be absent, the resulting rate equations can be solved analytically. This leads to a cooling function $\\Lambda(u)/\\rho^2$ as illustrated in the left panel of Fig. 9. On the other hand, in the presence of ionizing background radiation, the rate equations can be solved iteratively. Note that for a typical cosmological radiation background (e.g., UV background from star-forming galaxies and quasars, see [118] ,[119] ,[120] ,[121] ), the shape of the cooling function can be significantly altered, especially at low densities. For a more detailed discussion, see, for example, 星系形成章节 and [17] .\n","tags":["翻译","astronomy:cosmology","astronomy:cosmology:simulation","numerical:hydrodynamics"],"title":"§3.11 气体冷却"},{"content":"Star formation 恒星形成是一个关键的天体物理过程，涉及一系列层次递进的物理过程：来自大尺度（$\\sim$Mpc）的气体吸积、气体冷却产生中性氢（HI）和分子氢（H${2}$）（$\\sim$kpc尺度）、巨分子云（GMCs；$\\sim$10$\\div$100 pc）的形成、H${2}$ 碎裂并吸积为团块（$\\sim$1 pc）和核心（$\\sim$0.1 pc），以及随后核心收缩形成恒星（$\\sim$10$^{-8}$ pc）。大量综述对这些复杂过程以及多相恒星形成 ISM 的性质和不同恒星形成模式的描述进行了详尽的讨论，例如[127] 、[128] 、[129] 、[130] 。 恒星形成是致密冷气体云引力坍缩的结果。由于恒星形成实际发生的亚pc尺度无法为最先进的宇宙学流体动力学模拟所分辨，因此这是一个亚分辨率过程。从宇宙学模拟所涉及的质量和尺度的数量级来估计，气体元素的质量通常在 $10^4 \\div 10^7$ M$_{\\odot}$ 范围内（它们无法分辨GMCs），其相关的软化长度也与数百pc相当或更大（取决于分辨率）。恒星粒子在质量和力分辨率上与气体元素相当，并被处理为SSPs。总而言之，宇宙学模拟中的恒星形成旨在有效捕捉经历了辐射冷却的冷气体向无碰撞恒星粒子的转化。 若满足若干判据，气体粒子或网格便可形成恒星：流动必须局部汇聚（$\\nabla \\cdot \\mathbf{v} \u0026lt; 0$）且Jeans不稳定（即声波穿越时标超过动力学时标）；气体密度需超过临界密度阈值（以防止在极高红移的低密度气体中形成恒星）；以及氢原子的局部数密度需超过阈值数密度（见下文）。上述条件的详细信息见星系形成章节（另见[17] ）。 [131] 的奠基性工作推动了宇宙学模拟中对恒星形成 ISM 的描述和恒星形成数值建模的发展。他们引入了一个多相 ISM 模型，该模型考虑了恒星形成和恒星反馈。该模型认为，热气体和冷气体成分在致密气体粒子内以压强平衡的方式共存，即所谓的多相粒子。由于分子云无法被分辨，冷相和热相的性质被假定为代表 ISM 小体积上的平均值。多相粒子内的冷相为恒星形成提供了储库，同时通过热气体冷却得到补充。假设驱动恒星形成的冷气体中有一部分被瞬时用于加热热相并蒸发一部分冷相，以此模拟短寿命大质量恒星以超新星形式爆发的过程。通过积分描述热气体和冷气体相能量演化的方程，便可计算出恒星形成率。模型的参数经过校准以重现 Schmidt-Kennicutt 关系[132] 、[133] 。 该模型假设冷气体成分具有恒定温度（$\\sim 10^3$ K）。此外，它以宁静、自调节的恒星形成为特征，因为冷却作用对冷相的补充被超新星加热导致的蒸发所平衡。在此模型中，ISM 可用有效状态方程来描述：两相气体粒子的有效压强具有由热相和冷相性质所约束的函数形式，并且由于自调节恒星形成，预期在时间上保持恒定。 在最先进的宇宙学模拟中广泛采用的若干亚分辨率模型都受到上述模型的启发，或大致基于该模型。 该模型的一个有趣变体认为，气体粒子能够形成恒星的密度阈值依赖于金属丰度，其假设是气体金属丰度越高，暖中性气体转变为冷分子气体所需的密度便越低[134] 、[41] 。 Springel 和 Hernquist 模型的一个关键特征是采用恒星形成气体的有效状态方程。该思想源于这样一种认识：调节 ISM 的小尺度效应（例如热不稳定性、湍流、热传导）在短时标内建立了 ISM 的自调节平衡态。在类似的机制下，ISM 的平均温度或能量可近似为仅依赖于暖/冷气体密度的函数（不依赖状态方程或不依赖多相分辨率元素的模拟示例分别见[135] 、[136] 和[137] ）。 如上所述，宇宙学背景下大多数结构形成的数值模拟仍远不能分辨 GMCs 的结构，也远不能从第一性原理出发建模恒星形成。它们反而依赖于各种将 SFR 与可用恒星形成气体联系起来的亚分辨率处方。这通常体现为考虑冷而致密的气体（$T \\lesssim \\mbox{a few} \\times 10^4$ K；$n\\gtrsim 0.1$-$10$ cm$^{-3}$），并假设局部 SFR 正比于其密度除以一个时标（该时标接近冷气体的动力学时标）。上述用于选择恒星形成气体的密度和温度阈值实际上反映了 ISM 中 HI 的物理性质。而 SFR 与密度的正比关系则大致受 Schmidt-Kennicutt 定律[132] 、[133] 的启发。 采用这一方法的最先进宇宙学模拟包括 Illustris[40] 、Magneticum[138] 、NIHAO[137] 、Illustris-TNG[139] 、Massive-Black[39] 和Fable[140] 模拟、Horizon-AGN系列[141] 、[142] 、DIANOGA模拟[143] 、[144] ，以及SLOW模拟[145] 。 在通过[146] 的理论模型估算分子气体丰度、进而推算 SFR 的宇宙学模拟中，值得提及的有 MUFASA 和 SIMBA 模拟[147] 、[44] 、FIRE 系列[148] 、[149] 以及 FIREBOX[150] 。 其他对 H$_{2}$ 进行亚分辨率处理的宇宙学模拟示例包括[151] 、[122] 、[152] 、[153] 、[154] 、[155] 、[156] 、[157] 、[158] 、[159] 、[160] 。由于上述模拟针对单个暗晕或较小体积，它们能够对恒星形成分子气体实现更精细的建模。 接下来，我们简要概述最先进宇宙学模拟为选择恒星形成气体和建模恒星形成而采用的不同处方。下列条件通常作为本小节开头所述条件的补充。 Illustris、Magneticum、Illustris-TNG 和 SLOW 模拟均基于一个密度阈值（$n_{\\ast} \\simeq 0.1$ cm$^{-3}$）来选择恒星形成气体，并假设一个恒定的耗散时标将冷气体转化为恒星并估算 SFR（基于[131] ）。 NIHAO 模拟以及 ERIS 和 GIGA-ERIS 模拟[161] 、[162] 则同时采用密度和温度阈值来选择恒星形成气体。 依赖 MUPPI 亚分辨率模型（例如[163] 、[155] 、[135] 、[164] 、[165] 、[166] 、[160] ）选择多相气体的模拟也采用了密度和温度阈值：恒星形成随后从冷气体中的分子部分进行，分子比例通过[167] 或[168] 的处方估算；最后，耗散时标取为冷气体的动力学时标。MUPPI 中更先进的建模利用了对尘埃形成和演化的处理[165] 、[166] 来预测 H$_{2}$ 的演化[169] ，进而用于估算局部 SFR。 类似地，MUFASA 和 SIMBA 模拟采用基于 H$_{2}$ 的恒星形成处方，遵循[146] 公式（认为密度超过阈值的气体是恒星形成的），并假设耗散时标为局部动力学时标。FIRE 模拟中也采用了类似的假设来估算冷相中的分子气体。 EAGLE 和 FLAMINGO 模拟系列[41] 、[6] 假设了一个金属丰度依赖的恒星形成阈值[134] 以及一个温度阈值：为了将恒星形成与分子气体联系起来，它们利用 Cloudy 辐射转移代码[124] 来确定从暖原子气体相到冷中性气体相的转变。在这些模拟中，SFR 依赖于压强而非密度。 有趣的是，HORIZON 模拟系列（例如[142] ）——它从高于给定密度阈值的气体网格中选择恒星形成气体——具有局部变化的恒星形成效率，这与几乎所有上述模拟不同，后者假设冷/分子气体转化为恒星粒子的比例是恒定的。 [170] 、[171] 讨论了选择恒星形成气体的不同判据如何对模拟星系产生强烈影响。 恒星形成通常根据随机模型来实现：该过程不是对每颗恒星分别描述，而是以与实际 SFR 一致的期望值随机进行。例如，若概率 $p = m_{gas}/m_{\\star} (1- {exp}(m_{\\star}/m_{gas}))$ 超过一个随机数，则生成一个新的恒星粒子[131] 。 作为改进，可以假设存在若干恒星代[172] ，这样新生成的恒星粒子以其来源气体元素质量的一定份额作为它们的初始质量。 系列导航 ← 上一篇：§3.11 → 下一篇：§3.13\n","date":"2026-06-25","permalink":"/posts/2026-06-25-hydrodynamic-13/","series":["宇宙学模拟中的流体动力学方法与次网格模型"],"summary":"Star formation 恒星形成是一个关键的天体物理过程，涉及一系列层次递进的物理过程：来自大尺度（$\\sim$Mpc）的气体吸积、气体冷却产生中性氢（HI）和分子氢（H${2}$）（$\\sim$kpc尺度）、巨分子云（GMCs；$\\sim$10$\\div$100 pc）的形成、H${2}$ 碎裂并吸积为团块（$\\sim$1 pc）和核心（$\\sim$0.1 pc），以及随后核心收缩形成恒星（$\\sim$10$^{-8}$ pc）。大量综述对这些复杂过程以及多相恒星形成 ISM 的性质和不同恒星形成模式的描述进行了详尽的讨论，例如[127] 、[128] 、[129] 、[130] 。 恒星形成是致密冷气体云引力坍缩的结果。由于恒星形成实际发生的亚pc尺度无法为最先进的宇宙学流体动力学模拟所分辨，因此这是一个亚分辨率过程。从宇宙学模拟所涉及的质量和尺度的数量级来估计，气体元素的质量通常在 $10^4 \\div 10^7$ M$_{\\odot}$ 范围内（它们无法分辨GMCs），其相关的软化长度也与数百pc相当或更大（取决于分辨率）。恒星粒子在质量和力分辨率上与气体元素相当，并被处理为SSPs。总而言之，宇宙学模拟中的恒星形成旨在有效捕捉经历了辐射冷却的冷气体向无碰撞恒星粒子的转化。 若满足若干判据，气体粒子或网格便可形成恒星：流动必须局部汇聚（$\\nabla \\cdot \\mathbf{v} \u003c 0$）且Jeans不稳定（即声波穿越时标超过动力学时标）；气体密度需超过临界密度阈值（以防止在极高红移的低密度气体中形成恒星）；以及氢原子的局部数密度需超过阈值数密度（见下文）。上述条件的详细信息见星系形成章节（另见[17] ）。 [131] 的奠基性工作推动了宇宙学模拟中对恒星形成 ISM 的描述和恒星形成数值建模的发展。他们引入了一个多相 ISM 模型，该模型考虑了恒星形成和恒星反馈。该模型认为，热气体和冷气体成分在致密气体粒子内以压强平衡的方式共存，即所谓的多相粒子。由于分子云无法被分辨，冷相和热相的性质被假定为代表 ISM 小体积上的平均值。多相粒子内的冷相为恒星形成提供了储库，同时通过热气体冷却得到补充。假设驱动恒星形成的冷气体中有一部分被瞬时用于加热热相并蒸发一部分冷相，以此模拟短寿命大质量恒星以超新星形式爆发的过程。通过积分描述热气体和冷气体相能量演化的方程，便可计算出恒星形成率。模型的参数经过校准以重现 Schmidt-Kennicutt 关系[132] 、[133] 。 该模型假设冷气体成分具有恒定温度（$\\sim 10^3$ K）。此外，它以宁静、自调节的恒星形成为特征，因为冷却作用对冷相的补充被超新星加热导致的蒸发所平衡。在此模型中，ISM 可用有效状态方程来描述：两相气体粒子的有效压强具有由热相和冷相性质所约束的函数形式，并且由于自调节恒星形成，预期在时间上保持恒定。 在最先进的宇宙学模拟中广泛采用的若干亚分辨率模型都受到上述模型的启发，或大致基于该模型。 该模型的一个有趣变体认为，气体粒子能够形成恒星的密度阈值依赖于金属丰度，其假设是气体金属丰度越高，暖中性气体转变为冷分子气体所需的密度便越低[134] 、[41] 。 Springel 和 Hernquist 模型的一个关键特征是采用恒星形成气体的有效状态方程。该思想源于这样一种认识：调节 ISM 的小尺度效应（例如热不稳定性、湍流、热传导）在短时标内建立了 ISM 的自调节平衡态。在类似的机制下，ISM 的平均温度或能量可近似为仅依赖于暖/冷气体密度的函数（不依赖状态方程或不依赖多相分辨率元素的模拟示例分别见[135] 、[136] 和[137] ）。 如上所述，宇宙学背景下大多数结构形成的数值模拟仍远不能分辨 GMCs 的结构，也远不能从第一性原理出发建模恒星形成。它们反而依赖于各种将 SFR 与可用恒星形成气体联系起来的亚分辨率处方。这通常体现为考虑冷而致密的气体（$T \\lesssim \\mbox{a few} \\times 10^4$ K；$n\\gtrsim 0.1$-$10$ cm$^{-3}$），并假设局部 SFR 正比于其密度除以一个时标（该时标接近冷气体的动力学时标）。上述用于选择恒星形成气体的密度和温度阈值实际上反映了 ISM 中 HI 的物理性质。而 SFR 与密度的正比关系则大致受 Schmidt-Kennicutt 定律[132] 、[133] 的启发。 采用这一方法的最先进宇宙学模拟包括 Illustris[40] 、Magneticum[138] 、NIHAO[137] 、Illustris-TNG[139] 、Massive-Black[39] 和Fable[140] 模拟、Horizon-AGN系列[141] 、[142] 、DIANOGA模拟[143] 、[144] ，以及SLOW模拟[145] 。 在通过[146] 的理论模型估算分子气体丰度、进而推算 SFR 的宇宙学模拟中，值得提及的有 MUFASA 和 SIMBA 模拟[147] 、[44] 、FIRE 系列[148] 、[149] 以及 FIREBOX[150] 。 其他对 H$_{2}$ 进行亚分辨率处理的宇宙学模拟示例包括[151] 、[122] 、[152] 、[153] 、[154] 、[155] 、[156] 、[157] 、[158] 、[159] 、[160] 。由于上述模拟针对单个暗晕或较小体积，它们能够对恒星形成分子气体实现更精细的建模。 接下来，我们简要概述最先进宇宙学模拟为选择恒星形成气体和建模恒星形成而采用的不同处方。下列条件通常作为本小节开头所述条件的补充。 Illustris、Magneticum、Illustris-TNG 和 SLOW 模拟均基于一个密度阈值（$n_{\\ast} \\simeq 0.1$ cm$^{-3}$）来选择恒星形成气体，并假设一个恒定的耗散时标将冷气体转化为恒星并估算 SFR（基于[131] ）。 NIHAO 模拟以及 ERIS 和 GIGA-ERIS 模拟[161] 、[162] 则同时采用密度和温度阈值来选择恒星形成气体。 依赖 MUPPI 亚分辨率模型（例如[163] 、[155] 、[135] 、[164] 、[165] 、[166] 、[160] ）选择多相气体的模拟也采用了密度和温度阈值：恒星形成随后从冷气体中的分子部分进行，分子比例通过[167] 或[168] 的处方估算；最后，耗散时标取为冷气体的动力学时标。MUPPI 中更先进的建模利用了对尘埃形成和演化的处理[165] 、[166] 来预测 H$_{2}$ 的演化[169] ，进而用于估算局部 SFR。 类似地，MUFASA 和 SIMBA 模拟采用基于 H$_{2}$ 的恒星形成处方，遵循[146] 公式（认为密度超过阈值的气体是恒星形成的），并假设耗散时标为局部动力学时标。FIRE 模拟中也采用了类似的假设来估算冷相中的分子气体。 EAGLE 和 FLAMINGO 模拟系列[41] 、[6] 假设了一个金属丰度依赖的恒星形成阈值[134] 以及一个温度阈值：为了将恒星形成与分子气体联系起来，它们利用 Cloudy 辐射转移代码[124] 来确定从暖原子气体相到冷中性气体相的转变。在这些模拟中，SFR 依赖于压强而非密度。 有趣的是，HORIZON 模拟系列（例如[142] ）——它从高于给定密度阈值的气体网格中选择恒星形成气体——具有局部变化的恒星形成效率，这与几乎所有上述模拟不同，后者假设冷/分子气体转化为恒星粒子的比例是恒定的。 [170] 、[171] 讨论了选择恒星形成气体的不同判据如何对模拟星系产生强烈影响。 恒星形成通常根据随机模型来实现：该过程不是对每颗恒星分别描述，而是以与实际 SFR 一致的期望值随机进行。例如，若概率 $p = m_{gas}/m_{\\star} (1- {exp}(m_{\\star}/m_{gas}))$ 超过一个随机数，则生成一个新的恒星粒子[131] 。 作为改进，可以假设存在若干恒星代[172] ，这样新生成的恒星粒子以其来源气体元素质量的一定份额作为它们的初始质量。 系列导航 ← 上一篇：§3.11 → 下一篇：§3.13\n","tags":["翻译","astronomy:cosmology","astronomy:cosmology:simulation","numerical:hydrodynamics"],"title":"§3.12 恒星形成"},{"content":"Stellar evolution and chemical enrichment 化学演化是宇宙学流体动力学模拟中星系演化的自然结果。化学演化模型已纳入宇宙结构形成的宇宙学模拟中，以恰当地研究化学增丰过程：星际介质（ISM）和星系周介质（CGM）中金属的分布记录了关于过去恒星形成和反馈历史的宝贵信息，是流体动力学模拟的一项关键特征（例如 [173] ,[174] ）。与明确分辨单颗恒星的观测不同，宇宙学模拟通过星粒子对星族进行粗略采样。大多数最先进的宇宙学流体动力学模拟（涵盖大宇宙体积和单个星系）中的星粒子质量，取决于分辨率，通常介于 $10^8$ 到 $10^3$ M$_{\\odot}$ 之间（例如 [36] ,[40] ,[39] ,[175] ,[41] ,[176] ,[139] ,[6] ,[177] ,[178] ,[179] ,[180] ,[155] ,[135] ,[181] ）。在这些模拟中，星粒子是分辨率元素：每个星粒子代表一个简单星族（SSP），即一组具有相同初始金属丰度的同时代恒星的集合。每个星粒子的初始化学组分与其母气体元素相同，并由初始质量函数（IMF）表征。Chemical evolution is a natural outcome of galaxy evolution in cosmological hydrodynamical simulations. Models of chemical evolution have been included in cosmological simulations of cosmic structure formation to properly address the study of the chemical enrichment process: the distribution of metals in the ISM and in the circum-galactic medium (CGM) encodes valuable information of the past history of star formation and feedback, and is a crucial feature of hydrodynamical simulations (e.g. [173] ,[174] ). At variance with observations that explicitly resolve individual stars, cosmological simulations provide a coarse sampling of stellar populations by means of star particles. The majority of state-of-the-art cosmological hydrodynamical simulations of both large cosmological volumes and individual galaxies have star particles whose mass typically ranges between $10^8$ and $10^3$ M$_{\\odot}$ according to resolution (e.g. [36] ,[40] ,[39] ,[175] ,[41] ,[176] ,[139] ,[6] ,[177] ,[178] ,[179] ,[180] ,[155] ,[135] ,[181] ). In these simulations, star particles are resolution elements: each of them represents a simple stellar population (SSP), i.e. an ensemble of coeval stars that share the same initial metallicity. Every stellar particle initially shares the chemical composition of the gas element from which it has been originated, and is characterized by an initial mass function (IMF).\nIMF $\\phi (m)$ 决定了单位质量区间内的恒星数目：The IMF $\\phi (m)$ determines the number of stars per unit mass interval: $$ \\phi (m) = \\beta m^{- \\alpha} \\,\\,\\,,\n该式定义在给定质量范围 $[M_{inf}, M_{sup}]$ 内。系数 $\\alpha$ 设定了幂律在整个质量范围（或其内各质量区间）的斜率。对每个质量区间，归一化常数 $\\beta$ 由以下条件确定：在整个质量范围上 $\\int m , \\phi (m) , dm = 1$，并且在各质量区间的边界处保持连续。$$ within a given mass range $[M_{inf}, M_{sup}]$. The coefficient $\\alpha$ sets the slope of the power law over the mass range or in the different mass intervals within the mass range. For each mass interval, a normalization constant $\\beta$ is computed by imposing that $\\int m \\, \\phi (m) \\, dm = 1$ over the global mass range and continuity at the edges of subsequent mass intervals.\n常用的 IMF 包括单斜率（[182] ,[183] ）或多斜率（[184] ,[185] ,[186] ）幂律，但也存在更复杂的形状（例如 [187] ）。读者可参阅 [188] ,[189] ,[190] ,[191] ,[192] ,[193] ,[164] 以了解不同 IMF 的详细信息，并理解采用不同 IMF 如何影响宇宙学模拟中的金属分布。Commonly assumed IMFs consist of single- [182] ,[183] or multiple-slope [184] ,[185] ,[186] power law, but more sophisticated shapes have been proposed (e.g. [187] ). We refer the reader to [188] ,[189] ,[190] ,[191] ,[192] ,[193] ,[164] for details about different IMFs and to appreciate how adopting different IMFs can have an impact on metal distribution on cosmological simulations.\nIMF 直接决定了不同初始质量恒星之间的相对比例，进而影响不同类型恒星贡献的元素相对丰度。恒星演化理论预言，所有质量大于 $M_{up}=8$ M$_\\odot$ 的恒星将以核坍缩超新星或 II 型超新星（SNe II）的形式结束其生命（参见 [194] 及其参考文献）。在此假设下，可计算每个星粒子能向周围气体释放的总能量（通常为每次超新星 $10^{51}$ erg）。爆发为 SNII 的大质量恒星，其典型寿命一般不超过模拟的典型时间步长；在此近似下，死亡恒星在所谓的\u0026quot;瞬时循环近似\u0026quot;中注入反馈能量和金属，即在同一时间步长内完成。另一方面，Ia 型超新星（SNe Ia）普遍认为源自白矮星的热核爆炸（参见 [195] 及其参考文献）。这类爆炸相对于星粒子创建时间有显著延迟，因为双星系统中的白矮星需要从伴星吸积物质并达到触发热核燃烧的质量阈值。渐近巨星支（AGB）恒星经历显著的恒星质量损失，并对重元素的核合成做出重要贡献（参见 [196] 及其参考文献）。The IMF directly regulates the relative ratio between stars of different initial mass, thus affecting the relative abundance of elements contributed by different types of stars. Stellar evolution predicts that all stars with masses larger than $M_{up}=8$ M$_\\odot$ end their life as core collapse or type-II supernovae (SNe II) (see [194] and references therein). Under this assumption, the total amount of energy (typically $10^{51}$ erg per SN) that each star particle can release in the surrounding gas can be calculated. Within the approximation that the typical lifetime of massive stars which explode as SNII does not exceed the typical time step of the simulation, feedback energy and metals are injected by the dying star in the so-called ``instantaneous recycling approximation'', i.e. in the same time step. On the other hand, type-Ia SNe (SNe Ia) are believed to arise from the thermonuclear explosion of white dwarfs (see [195] and references therein). They lead to an explosion which is significantly delayed with respect to the time of creation of the star particle, as the white dwarf in a binary system has to accrete matter from the companion and reach the mass threshold for the onset of thermonuclear burning. Stars in the asymptotic giant branch (AGB) experience significant stellar mass loss and contribute significantly to the nucleosynthesis of heavy elements (see [196] and references therein).\n具有精确恒星演化和化学增丰模型的宇宙学流体动力学模拟，可计算逐渐老化并最终爆发为超新星的恒星数量，以及污染周围 ISM 的金属量。除设定 IMF 外，常用的模型（例如 [172] ）还需恒星寿命、延迟函数和恒星产额表，以计算主要由 SNe Ia、SNe II 和 AGB 星留下的化学-能量印记。Cosmological hydrodynamical simulations which feature an accurate model for stellar evolution and chemical enrichment can evaluate the number of stars aging and eventually exploding as SNe, as well as the amount of metals polluting the surrounding ISM. Besides assuming an IMF, commonly adopted models (e.g. [172] ) include stellar lifetimes, delay functions and table of stellar yields to calculate the chemo-energetic imprint mainly left by SNe Ia, SNe II, and AGB stars.\n化学演化模型采用依赖于质量的寿命函数（即质量为 $m$ 的恒星死亡时的年龄函数，例如 [197] ,[198] ,[199] ）来考虑不同质量恒星的演化时标。初始质量高于 $M_{up}$ 且低于 $M_{sup, SNII}=40$ $M_\\odot$ 的恒星，假定以核坍缩超新星的形式结束生命；而质量超过 $M_{sup, SNII}$ 的恒星，则假定直接坍缩为黑洞，因此不会对进一步的化学增丰或恒星反馈能量做出贡献。Chemical evolution models account for the evolutionary timescales of stars with different masses by adopting mass-dependent lifetime functions (i.e. functions that describe the age at which a star of mass $m$ dies, e.g. [197] ,[198] ,[199] ). Stars with an initial mass larger than $M_{up}$ and lower than $M_{sup, SNII}=40$ $M_\\odot$ are assumed to end their life exploding as core-collapse SNe, while stars that are more massive than $M_{sup, SNII}$ are assumed to implode in BHs directly, and thus do not contribute to further chemical enrichment or stellar feedback energy.\n假定在整个质量范围中，一定比例（通常为常数）的恒星处于双星系统内，并作为 SNe Ia 的前身。采用 SNe Ia 的恒星演化情景（例如 [200] 的模型），结合寿命函数和延迟时间分布函数，可计算 SN Ia 的爆发率。A fraction (usually constant) of stars relative to the whole mass range is assumed to be located in binary systems that are progenitors of SNe Ia. Assuming a stellar evolution scenario for SNe Ia (e.g. according to the model by [200] ) and adopting lifetime functions and delay time distribution functions, the rate of SN Ia explosions can be computed.\n采用恒星产额追踪演化并最终爆发的恒星所产生的不同重金属。产额表示质量为 $m$、初始金属丰度为 $Z$ 的恒星产生的不同金属元素 $i$ 的抛出质量，即 $p_{Z_{i}}(m,Z)$。通常，至少需要三个主要过程的预测：AGB 恒星的持续质量损失、SNe II 和 SNe Ia。依赖于质量和金属丰度的恒星产额表已纳入最先进的宇宙学流体动力学模拟的化学演化模型中。恒星产额预测仍存在不确定性，主要原因是恒星演化模型中星风质量损失过程尚未充分理解。常用的恒星产额集包括：SNe Ia 的 [201] ；AGB 星的依赖于质量和金属丰度的产额 [202] ,[203] ,[204] ；SNe II 的依赖于质量和金属丰度的产额 [205] ,[206] ,[207] ,[208] 。读者可参阅（例如 [209] ,[210] ）获取解析形式的全面综述。The production of different heavy metals by stars that evolve and eventually explode is followed by adopting stellar yields. They represent the ejected mass of different metal species $i$ produced by a star of mass $m$ and initial metallicity $Z$, i.e. $p_{Z_{i}}(m,Z)$. In general, predictions for at least three main processes are needed: the continuous mass loss of AGB stars, SNe II and SNe Ia. Tables of mass- and metallicity-dependent stellar yields are incorporated in the chemical evolution models of state-of-the-art cosmological hydrodynamical simulations. Predictions for stellar yield are still affected by uncertainties, mainly due to the poorly understood process of mass loss through stellar winds in stellar evolution models. Commonly adopted sets of stellar yields include: [201] for SNe Ia, mass- and metallicity-dependent yields by [202] ,[203] ,[204] for AGB stars, mass- and metallicity-dependent yields by [205] ,[206] ,[207] ,[208] for SNe II. We refer the reader to (e.g. [209] ,[210] ) for a comprehensive review of the analytic formalism.\n最后需指出，将模拟结果与单颗恒星观测进行比较时，通常基于以下假设：模拟中星粒子的金属含量在统计上再现了该星粒子所采样 SSP 的平均金属丰度。更精确的技术见 [211] ,[212] 的介绍。As a final caveat, we note that comparisons of results from simulations to observations of single stars are usually performed under the assumption that e.g., the metal content of a star particle in simulations statistically reproduces the mean metallicity of the SSP that the star particle samples. More accurate techniques are introduced in [211] ,[212] .\n系列导航 ← 上一篇：§3.12 → 下一篇：§3.14\n","date":"2026-06-25","permalink":"/posts/2026-06-25-hydrodynamic-14/","series":["宇宙学模拟中的流体动力学方法与次网格模型"],"summary":"Stellar evolution and chemical enrichment 化学演化是宇宙学流体动力学模拟中星系演化的自然结果。化学演化模型已纳入宇宙结构形成的宇宙学模拟中，以恰当地研究化学增丰过程：星际介质（ISM）和星系周介质（CGM）中金属的分布记录了关于过去恒星形成和反馈历史的宝贵信息，是流体动力学模拟的一项关键特征（例如 [173] ,[174] ）。与明确分辨单颗恒星的观测不同，宇宙学模拟通过星粒子对星族进行粗略采样。大多数最先进的宇宙学流体动力学模拟（涵盖大宇宙体积和单个星系）中的星粒子质量，取决于分辨率，通常介于 $10^8$ 到 $10^3$ M$_{\\odot}$ 之间（例如 [36] ,[40] ,[39] ,[175] ,[41] ,[176] ,[139] ,[6] ,[177] ,[178] ,[179] ,[180] ,[155] ,[135] ,[181] ）。在这些模拟中，星粒子是分辨率元素：每个星粒子代表一个简单星族（SSP），即一组具有相同初始金属丰度的同时代恒星的集合。每个星粒子的初始化学组分与其母气体元素相同，并由初始质量函数（IMF）表征。Chemical evolution is a natural outcome of galaxy evolution in cosmological hydrodynamical simulations. Models of chemical evolution have been included in cosmological simulations of cosmic structure formation to properly address the study of the chemical enrichment process: the distribution of metals in the ISM and in the circum-galactic medium (CGM) encodes valuable information of the past history of star formation and feedback, and is a crucial feature of hydrodynamical simulations (e.g. [173] ,[174] ). At variance with observations that explicitly resolve individual stars, cosmological simulations provide a coarse sampling of stellar populations by means of star particles. The majority of state-of-the-art cosmological hydrodynamical simulations of both large cosmological volumes and individual galaxies have star particles whose mass typically ranges between $10^8$ and $10^3$ M$_{\\odot}$ according to resolution (e.g. [36] ,[40] ,[39] ,[175] ,[41] ,[176] ,[139] ,[6] ,[177] ,[178] ,[179] ,[180] ,[155] ,[135] ,[181] ). In these simulations, star particles are resolution elements: each of them represents a simple stellar population (SSP), i.e. an ensemble of coeval stars that share the same initial metallicity. Every stellar particle initially shares the chemical composition of the gas element from which it has been originated, and is characterized by an initial mass function (IMF).\n","tags":["翻译","astronomy:cosmology","astronomy:cosmology:simulation","numerical:hydrodynamics"],"title":"§3.13 恒星演化与化学增丰"},{"content":"Stellar feedback 恒星反馈是结构形成宇宙学模拟的关键组成部分。在其众多关键作用中，它可防止高红移处的过度冷却，在星际介质中分配能量，将金属驱出恒星形成区，并触发星系外流，从而确保形成中的星系与其周围星系周介质之间的持续相互作用。 鉴于驱动和发射星系风的pc尺度物理过程尚不完全清楚，且远未达到能在宇宙学模拟中直接实现的程度，这些模拟不得不借助唯象预设来捕捉星系外流的效应。从驱动外流并支配其运动学的能量来源来看，有两种常见途径：星系风可以是能量驱动的，也可以是动量驱动的。关于这两种情景的细节，建议读者参考星系形成章节，此文为本小节的具体主题提供了补充信息。 本概述远非完备：更深入的讨论可见于[213] 、[155] 、[214] 、[215] 等文献。Stellar feedback is a key component of cosmological simulations of structure formation. Among its many crucial roles, it prevents overcooling at high redshift, distributes energy in the ISM, drives metals out of the star formation sites, and triggers galactic outflows, which guarantee a continuous interaction between the forming galaxy and its surrounding CGM. Since the understanding of the pc-scale physics responsible for launching and driving winds is still partial and however far from being implemented directly in cosmological simulations, these simulations have to resort to phenomenological prescriptions to capture the effects of galactic outflows. As for the sources of energy that power outflows and govern their kinematic, there are two commonly pursued approaches: winds can be either energy-driven or momentum-driven. For details about these two possible scenarios, we refer the reader to 星系形成章节, which provides complementary information on the specific topic of this sub-section. This outline is far from being complete: deeper insight can be gained e.g. from [213] ,[155] ,[214] ,[215] .\n宇宙学模拟中已提出多种恒星反馈方案：星际介质中超新星爆发的能量注入，其数值描述方式确实对最终结果有强烈影响。不同的亚分辨率模型能否有效捕捉反馈能量注入，其可靠性究竟如何，目前仍存争议。A variety of stellar feedback schemes has been proposed in cosmological simulations: the numerical description of the energy injection from SN explosions in the ISM has indeed a strong impact on final results. How reliable different sub-resolution models are in capturing an effective description of feedback energy injection is still debated.\n超新星能量可由恒星形成气体元素以热能或动能形式分配至周围环境。将反馈能量以热形式注入——如[17] 的先驱模型——其局限在于：反馈通常无法将能量有效输运至远离释放区域之处。提供能量的气体元素，其周围温度通常为$\\sim 10^5$ K，密度足够高，几乎立即辐射能量，因而冷却时标很短。因此，反馈无法有效抵消冷却，也难以防止过度的恒星形成。为克服这一弱点，[216] 提出了随机热反馈模型，该模型将接收热反馈能量的气体加热到足够高的温度，以保证反馈有效。具体来说，上述模型假设超新星爆发释放的热能通过一个选择判据注入周围介质：接受反馈的气体粒子必须加热到一个阈值温度，该阈值温度是模型的一个参数（其值通常接近$10^7$ K）。这样的温度升高确保受热粒子的冷却时标长于其声波穿越时标，从而使受热气体在辐射掉全部能量之前能够有效离开恒星形成区。值得指出，在这个模型中恒星反馈实际上是通过热通道实现的：然而，其有效结果是星系风，因为热能转化为动量，进而产生外流。例如，EAGLE和FLAMINGO模拟便采用了这种恒星反馈模型[41] 、[6] 。SN energy can be distributed by star-forming gas elements to the surroundings in the form of thermal or kinetic energy. Injecting feedback energy as thermal, as in the pioneer model of [17] , may have the limitation that the feedback usually does not result effective in driving energy far from regions where it is released. Gas elements surrounding those that provide energy have a typical temperature of $\\sim 10^5$ K and density high enough to radiate energy almost immediately, their cooling time being short. As a result, feedback is not effective in counterbalancing cooling and in preventing excessive star formation. To overcome this weakness, a stochastic thermal feedback model has been proposed [216] , where gas receiving thermal feedback energy is heated up to a temperature high enough to guarantee the effectiveness of feedback. Specifically, the aforementioned model assumes that thermal energy released by SN explosions is injected in the surrounding medium using a selection criterion: the gas particles that experience feedback have to be heated up to a threshold temperature, which is a parameter of the model (whose value is usually close to $10^7$ K). Such a temperature increase ensures that the cooling time of a heated particle is longer than its sound-crossing time, so that heated gas can effectively leave the star formation site before it radiates all the energy away. It is worth noting that stellar feedback is actually implemented as a thermal channel in this model: however, the effective outcome is a galactic wind, because thermal energy is converted into momentum and hence outflows originate. This model for stellar feedback is for instance adopted by the EAGLE and FLAMINGO simulations [41] ,[6] .\n另一种方式则是将恒星反馈能量以动能形式提供，用于加速周围气体元素（例如[131] 、[217] ），将它们从其原始位置踢出。它们最终可以热化并辐射掉能量，但按其构造，这比热情景要晚。为增强动能反馈方案的有效性，接收能量并采样星系外流的粒子（通常称为{\\it{风粒子}}）通常与流体动力学相互作用解耦：这样，它们就不会在获得能量后很快被阻止和热化。Otherwise, stellar feedback energy is provided in the form of kinetic energy and used to boost the velocity of surrounding gas elements (e.g. [131] ,[217] ), which are kicked from their original position. They can eventually thermalise and radiate energy away, but later than in the thermal scenario by construction. To enforce the effectiveness of kinetic feedback schemes, particles that receive energy and sample galactic outflows (usually referred to as {\\it{wind particels}}) are often decoupled from hydrodynamic interactions: in this way, they are prevented from being halted and from thermalising energy soon after they have been provided with.\n另一种恒星反馈模型是所谓的冲击波反馈[218] 、[219] 。在该方案中，符合条件的粒子被赋予热反馈能量，但在短时间内（通常为数十Myr）被阻止冷却。这一预设的物理动机源于星际介质中超新星遗迹的演化[220] 。当一颗爆发的超新星驱动冲击波时，它首先经历自由膨胀阶段，随后进入绝热阶段（Sedov-Taylor阶段），其中辐射损失可以忽略，然后才进入辐射阶段。暂时禁用冷却模拟了未分辨的绝热阶段，其持续时间估计约为$30$ Myr；此后，气体被允许再次冷却。另一种解释是，关闭冷却可理解为超新星爆发释放的能量在未分辨尺度上产生湍流，并在数十Myr内部分耗散，从而阻碍气体冷却[221] 、[222] 。在宇宙学模拟中实现时，这种反馈预设能够有效避免过度的恒星形成，并成功产生一个双相星际介质，其气体成分在局部不处于压力平衡。An alternative stellar feedback model is represented by the so-called blast-wave feedback [218] ,[219] . Within this scheme, eligible particles are provided with thermal feedback energy, but are then prevented from cooling for a short period of time (typically few tens of Myr). The physical motivation behind this prescription stems from the evolution of a SN remnant in the ISM [220] . As soon as an exploding SN drives a blast wave, this undergoes a first phase of free expansion, followed by an adiabatic stage (the Sedov-Taylor phase) where radiative losses are negligible, before entering the radiative phase. Temporarily disabling cooling mimics the unresolved adiabatic phase, whose duration is estimated to be of order $30$ Myr; afterwards, gas is allowed to cool again. Alternatively, the switch off of cooling can be explained by assuming that the energy released by SN explosions generates turbulence at unresolved scales and is partially dissipated over few tens of Myr, thus hindering gas cooling [221] ,[222] . When implemented in cosmological simulations, this feedback prescription results effective in avoiding excessive star formation and also succeeds at producing a two-phase ISM, whose gas components are not in pressure equilibrium locally.\n大质量恒星影响星系气体储库的反馈过程不仅源于超新星爆发后的能量沉积和动量注入，还源于大质量恒星爆发前的电离效应（通常称为早期恒星反馈；例如[179] ）以及星风（例如[223] 、[224] ）。 具体来说，[179] 提出的早期恒星反馈代表了一种UV电离源，可提高周围气体的温度，并提供加热和压力支撑。这一反馈通道对恒星形成区的星际介质进行预处理，有助于超新星有效调节恒星形成。在他们的模拟中，[179] 展示了早期恒星反馈如何帮助抑制高红移处的恒星形成（另见NIHAO模拟以了解早期恒星反馈如何运作[137] 、[225] ）。Feedback processes through which massive stars can affect the reservoir of gas of a galaxy stem not only from the energy deposition and momentum injection following SN explosions, but also from the ionizing effect that massive stars have before exploding (usually referred to as early stellar feedback; e.g. [179] ) and from stellar winds (e.g. [223] ,[224] ). Specifically, the early stellar feedback by [179] represents a UV ionization source, that increases the surrounding gas temperature and supplies heating and pressure support. This feedback channel pre-processes the star-forming ISM and facilitates the effectiveness of SNe at regulating star formation. In their simulations, [179] show how early stellar feedback helps in suppressing SF at high z (see also the NIHAO simulations to appreciate how early stellar feedback operates [137] ,[225] ).\n此外，SNe II释放的能量预期不会在宇宙时间中或在所有恒星形成区中保持不变，理论模型预测，在几乎原始或弱增丰环境中爆发的超新星会为星际介质提供更大量的恒星反馈能量。 根据恒星形成区星际介质的物理性质来区分恒星反馈结果的想法已在宇宙学模拟中得到探索，并提出了一些有效的预设。其基本思想是采用非恒定值的恒星反馈效率（即每个超新星提供的能量中，实际作为反馈能量耦合到周围气体的比例）。In addition, the energy released by SNe II is not expected to be constant across cosmic time nor in all the star-forming regions, and theoretical models predict that SNe exploding in almost pristine or weakly enriched environments provide the ISM with a larger amount of stellar feedback energy. The idea of differentiating the outcome of stellar feedback according to the physical properties of the star-forming ISM has been already pursued in cosmological simulations, and a few effective prescriptions have been proposed. The basic idea consists in adopting a non-constant value of the stellar feedback efficiency (i.e. the fraction of energy provided by each SN that is actually coupled to the surrounding gas as feedback energy).\nEAGLE模拟引入了一个依赖金属丰度和密度的恒星反馈效率[41] 、[226] 。其中，上述效率随气体金属丰度增加而降低，随气体密度增加而升高。采用这种参数化有两方面原因：$i)$ 辐射损失预期随金属丰度增加而增加；$ii)$ 高密度恒星形成区中的能量损失可能使恒星反馈效率过低，必须加以补偿。 由于较高红移处气体金属丰度较低，而恒星形成区星际介质通常达到更高密度，这样的预设会产生一个依赖红移的恒星反馈效率。后者在渐近值之间变化，这些值是模型的参数，经调谐以再现低红移观测量，例如星系恒星质量函数。A metallicity- and density-dependent stellar feedback efficiency has been introduced in the EAGLE simulations [41] ,[226] . There, the aforementioned efficiency decreases with gas metallicity while increasing with gas density. The adoption of a similar parametrization has a twofold reason: $i)$ radiative losses are expected to increase with increasing metallicity; $ii)$ energy losses in high-density, star-forming regions can make the stellar feedback too inefficient, and have to be counterbalanced. Being the gas metallicity lower at higher redshifts, when also higher densities are usually reached in the star-forming ISM, such a prescription yields a redshift-dependent stellar feeback efficiency. The latter ranges between asymptotic values, which are parameters of the model tuned to reproduce low-redshift observables, e.g. the galaxy stellar mass function.\nFig.\u0026nbsp;10. Illustris-TNG模拟[139] 也采用了类似的参数化。 他们假设恒星形成分辨单元可用的星系风能量依赖金属丰度。SNe II释放的反馈能量范围为$(0.9 - 3.6) \\times 10^{51}$ erg，具体取决于它们预期爆发的恒星形成气体细胞的金属丰度——金属丰度越低，能量预算越大。[139] 论证了依赖金属丰度的恒星反馈能量调制对恒星-暗晕质量关系和星系恒星质量函数均有影响，最终结果强烈依赖于参数的调谐。他们表明，$z=0$处一个$\\sim 10^{12}$ M$_{ \\odot}$暗晕的恒星质量可变化多达$\\sim 2$倍。A similar parametrization is adopted in the Illustris-TNG simulation [139] , too. They assume that the wind energy available to a star-forming resolution element is metallicity-dependent. SNe II release indeed a feedback energy which spans the range $(0.9 - 3.6) \\times 10^{51}$ erg, according to the metallicity of the star-forming gas cell in which they are expected to explode -- the lower the metallicity, the larger the energy budget. [139] demonstrate how the metallicity-dependent stellar feedback energy modulation has an impact both on the stellar-to-halo mass relation and on the galaxy stellar mass function, final results strongly depending on the tuning of the parameters. They show that the stellar mass of a $\\sim 10^{12}$ M$_{ \\odot}$ halo at $z=0$ can vary by up to a factor of $\\sim 2$.\n在更高复杂层面上，尽管是针对较小体积的模拟，[227] 首次在宇宙学模拟中模拟了不同星族的同时演化。在他们的运行中，恒星反馈依赖于底层气体金属丰度和星族，注入的能量和假定的恒星产额取决于星族III与星族II的状态，即对不稳定性超新星与超新星。有趣的是，[228] 包含了自洽耦合辐射转移的效应，并得出结论：来自大质量星族III恒星的辐射反馈可以为原始环境中的强大反馈提供一个可行的解释。At a higher level of complexity, though in simulations targeting smaller volumes, [227] first modelled the simultaneous evolution of different stellar populations in cosmological simulations. Their runs feature stellar feedback dependent on the underlying gas metallicity and stellar population, where the injected energy and the assumed stellar yields depend on Population III versus Population II regimes, i.e. pair-instability SNe versus SNe. Interestingly, [228] included the effect of self-consistently coupled radiative transfer, concluding that radiative feedback from massive Population III stars could be a viable justification for powerful feedback in pristine environments.\n受上述发现和理论研究（例如[229] 、[230] 的综述）的推动——这些研究表明星族III恒星的极超新星和SNe II释放的能量可能比当今宇宙高出$\\sim10$倍以上——[160] 引入了一种有效的低金属丰度反馈。在这种依赖金属丰度的恒星反馈实现中，他们考虑了弱增丰环境中极超新星和SNe II爆发的影响，具体做法是增强在几乎原始的周围介质（即平均金属丰度低于阈值的星际介质）中爆发的超新星所释放的能量。Motivated by the aforementioned findings and by theoretical studies (e.g., [229] ,[230] for reviews) that suggest that hypernovae and SNe II of Population III stars can release energy higher than in the present-day Universe by more than a factor of $\\sim10$, [160] introduced an effective low-metallicity feedback. In this implementation of metallicity-dependent stellar feedback, they accounted for the effect of the explosion of hypernovae and SNe II in weakly-enriched environments, by boosting the energy released by SNe exploding in an almost pristine ambient medium (i.e. in an ISM with average metallicity below a threshold).\n描述同一物理过程（在此特指恒星反馈和超新星触发的星系外流）的不同亚网格预设对最终结果的影响是惊人的，尽管常被忽视。 [135] 通过对一系列银河系大小暗晕的宇宙学缩放模拟，研究了恒星反馈建模对盘状星系形成和演化的影响。他们展示了模拟星系的一般性质对恒星反馈触发外流实现方式的敏感程度，比较了采用不同最先进恒星反馈模型所得的结果（见Fig. 10）。 我们建议读者参考[114] 、[115] 以了解更广泛的比较项目，即Aquila和Agora比较项目（另见Fig. 8）。 有趣的是，[215] 也展示了在EAGLE和Illustris-TNG模拟中对恒星反馈效率做出不同假设，如何反映在预测的星系恒星质量函数上（另见[226] 、[139] ）。The impact of different sub-grid prescriptions accounting for the same physical process (stellar feedback and SN-triggered galactic outflows in this specific case) on final results is striking, though often overlooked. [135] investigate the impact of stellar feedback modelling on the formation and evolution of a disc galaxy, by performing a suite of cosmological zoom-in simulations of a Milky Way-size halo. They show how sensitive the general properties of the simulated galaxy are to the way in which stellar feedback triggered outflows are implemented, comparing results obtained by adopting different state-of-the-art stellar feedback models (see Fig.\u0026nbsp;10). We refer the reader to [114] ,[115] for more extended comparison campaigns, i.e. the Aquila and the Agora comparison projects (see also Fig.\u0026nbsp;8). Interestingly, [215] also show how making different assumptions as for the stellar feedback efficiencies in the EAGLE and Illustris-TNG simulations reflects on the predicted galaxy stellar mass functions (see also [226] ,[139] ).\n系列导航 ← 上一篇：§3.13 → 下一篇：§3.15\n","date":"2026-06-25","permalink":"/posts/2026-06-25-hydrodynamic-15/","series":["宇宙学模拟中的流体动力学方法与次网格模型"],"summary":"Stellar feedback 恒星反馈是结构形成宇宙学模拟的关键组成部分。在其众多关键作用中，它可防止高红移处的过度冷却，在星际介质中分配能量，将金属驱出恒星形成区，并触发星系外流，从而确保形成中的星系与其周围星系周介质之间的持续相互作用。 鉴于驱动和发射星系风的pc尺度物理过程尚不完全清楚，且远未达到能在宇宙学模拟中直接实现的程度，这些模拟不得不借助唯象预设来捕捉星系外流的效应。从驱动外流并支配其运动学的能量来源来看，有两种常见途径：星系风可以是能量驱动的，也可以是动量驱动的。关于这两种情景的细节，建议读者参考星系形成章节，此文为本小节的具体主题提供了补充信息。 本概述远非完备：更深入的讨论可见于[213] 、[155] 、[214] 、[215] 等文献。Stellar feedback is a key component of cosmological simulations of structure formation. Among its many crucial roles, it prevents overcooling at high redshift, distributes energy in the ISM, drives metals out of the star formation sites, and triggers galactic outflows, which guarantee a continuous interaction between the forming galaxy and its surrounding CGM. Since the understanding of the pc-scale physics responsible for launching and driving winds is still partial and however far from being implemented directly in cosmological simulations, these simulations have to resort to phenomenological prescriptions to capture the effects of galactic outflows. As for the sources of energy that power outflows and govern their kinematic, there are two commonly pursued approaches: winds can be either energy-driven or momentum-driven. For details about these two possible scenarios, we refer the reader to 星系形成章节, which provides complementary information on the specific topic of this sub-section. This outline is far from being complete: deeper insight can be gained e.g. from [213] ,[155] ,[214] ,[215] .\n","tags":["翻译","astronomy:cosmology","astronomy:cosmology:simulation","numerical:hydrodynamics"],"title":"§3.14 恒星反馈"},{"content":"Supermassive black holes in cosmological simulations 观测表明，几乎每个星系在其最内层区域都寄宿着一个超大质量（$10^8 \\div 10^{10}$ M$_{\\odot}$）黑洞（SMBH）（例如，[231] ,[232] ,[233] ,[234] ,[235] ,[236] ,[237] ,[238] ,[239] ）。其中一部分黑洞表现出持续的活动性，称为活动星系核（AGN）。过去和当前活动的证据可见于椭圆星系、星系群和星系团的X射线图像，其中AGN的印记通常表现为凹陷和涟漪。在众多例子中，一个众所周知的代表是MS 0735+7421星系团的复合（X射线、射电和光学）图像（例如[240] ）：巨大的X射线空洞充满射电辐射，并被Chandra图像中清晰可见的椭圆形茧状激波包围。Observations suggest that almost every galaxy hosts a supermassive ($10^8 \\div 10^{10}$ M$_{\\odot}$) black hole (SMBH) in its innermost regions (e.g., [231] ,[232] ,[233] ,[234] ,[235] ,[236] ,[237] ,[238] ,[239] ). A fraction of these BHs exhibits ongoing activity and is called AGN (active galactic nuclei). Evidence of past and ongoing activity is observed for instance in the X-ray images of elliptical galaxies and galaxy groups and clusters, where AGN imprints often appear as depressions and ripples. Among many, a well-known example is represented by the composite (X-ray, radio, and visual) image of the MS 0735+7421 galaxy cluster (e.g. [240] ): giant X-ray cavities are filled with radio emission, and surrounded by a cocoon shock clearly visible in the Chandra image as an elliptical edge.\n在讨论AGN反馈（见§3.16）之前，我们先回顾宇宙学模拟中黑洞的主要特征，重点关注黑洞种子植入与重定位、黑洞-黑洞并合以及AGN供料。Before discussing AGN feedback (see §3.16), we are now recalling the main features of BHs in cosmological simulations, focusing on BH seeding and repositioning, BH-BH mergers, and AGN feeding.\n大多数包含黑洞及相关AGN反馈的星系和星系团形成宇宙学模拟，都基于[241] ,[242] 的开创性黑洞模型，或秉承其思路。因此，该模型为讨论宇宙学模拟中黑洞处理的几个关键方面提供了基础，且凸显了最近的改进。The majority of cosmological simulations of galaxy and galaxy cluster formation that also include BHs and the associated AGN feedback are based on the seminal BH model by [241] ,[242] , or follow its spirit. Therefore, this model offers the base to discuss a few key aspects of the treatment of BHs in cosmological simulations, and to highlight recent improvements.\n在宇宙学模拟中，通常将黑洞描述为汇粒子：这些是无碰撞粒子，可以吸收邻近元素（最初用于去除致密区域周围粒子，例如[243] ），且具有可直接与观测量关联的基本性质，如吸积率。In cosmological simulations, BHs are usually described as sink particles: these are collisionless particles that can absorb neighbour elements (originally implemented to allow the removal of surrounding particles in dense regions, e.g. [243] ) and that have fundamental properties like the accretion rate which can be linked directly to observables.\n在宇宙学模拟中，黑洞粒子于相对高红移处被植入大质量晕中，随后它们得以增长并增加其初始或{种子}质量。前述常引用的质量$M_{\\bullet}$是黑洞的理论质量，在亚分辨率层次上建模，与之相对的是其动力学质量，即黑洞粒子的实际引力质量。由于我们对第一批超大质量黑洞的形成仍缺乏坚实理解（可能的形成途径见例如[244] ,[245] ,[246] ,[247] ,[248] ），且任何种子形成情景的物理过程都远超出当前分辨率所能企及，黑洞首先按种子植入方案插入（另见[249] ）。BH particles are introduced in massive haloes at relatively high-redshift in cosmological simulations, and they are then allowed to grow and increase their initial or {seed} mass. The aforementioned, commonly quoted mass $M_{\\bullet}$ is the theoretical mass of the BH, modelled at the sub-resolution level, as opposite to its dynamical mass, i.e. the actual gravitational mass of the BH particle. As we are still lacking a solid understanding of the formation of first SMBHs (see e.g., [244] ,[245] ,[246] ,[247] ,[248] for possible pathways) and the resolution needed to take the physics of any seed formation scenario into account, BHs are first inserted according to seeding prescriptions (see also [249] ).\n种子植入方案通常假设新的黑洞（质量约为$\\sim 10^4 \\div 10^6$ M${\\odot}$的大质量种子）被植入满足某些条件且尚未包含黑洞的晕中。新黑洞的植入条件包括：$(i)$ 晕质量——通常通过FOF（朋友之友，[250] ）算法估计——大于某个阈值（例如，[40] ,[41] ,[139] ,[6] ）；$(ii)$ 恒星质量超过给定值（例如，[44] ）；$(iii)$ 恒星质量以及气体与恒星质量之比大于给定阈值；$(iv)$ 基于气体性质，即气体密度、速度弥散和/或金属丰度（例如，[251] ,[252] ,[253] ,[142] ）。植入时的黑洞质量可以是恒定的（例如，[40] ,[41] ,[139] ,[6] ,[44] ），也可以根据例如M${BH}$/$\\sigma$或M${BH}$/M${\\ast}$标度关系进行缩放（例如，[138] ）。进一步的细节涉及FOF算法所作用的粒子类型，例如可仅作用于暗物质粒子（例如，[254] ）或仅作用于恒星粒子（例如，[138] ）。此外，黑洞可植入在晕中最致密气体粒子的位置（例如，[254] ）、具有最大束缚能的恒星粒子的位置（例如，[138] ），或最接近结构质心的恒星粒子的位置（例如，[44] ）。Seeding prescriptions usually assume that new BHs (massive seeds of $\\sim 10^4 \\div 10^6$ M$_{\\odot}$) are introduced in haloes which meet some criteria and do not have already BHs. New BHs are seeded if: $(i)$ the halo mass -- commonly estimated by means of a FOF (Friend-Of-Friend, [250] ) algorithm -- is larger than a threshold (e.g., [40] ,[41] ,[139] ,[6] ); $(ii)$ the stellar mass exceeds a given value (e.g., [44] ); $(iii)$ the stellar mass and gas to stellar mass fraction are larger than given thresholds; $(iv)$ based on gas properties, i.e. gas density, velocity dispersion and/or metallicity (e.g., [251] ,[252] ,[253] ,[142] ). BH mass at seeding can be either constant (e.g., [40] ,[41] ,[139] ,[6] ,[44] ) or scaled according to e.g., the M$_{BH}$/$\\sigma$ or M$_{BH}$/M$_{\\ast}$ scaling relations (e.g., [138] ). Additional details involve the type of particles on which the FOF algorithm is performed, whcih can be for instance DM particles only (e.g., [254] ) or stellar particles only (e.g., [138] ). Besides, BH can be seeded at the position of the densest gas particle in the halo (e.g., [254] ), of the star particle with the largest binding energy (e.g., [138] ), or of the star particle closest to the centre of mass of the structure (e.g., [44] ).\n在宇宙学模拟中，每个黑洞作为独立粒子经历特定的演化，这与N体模拟中通过粒子对无碰撞流体的粗粒化表示不同。因此，黑洞的动力学无法精确捕捉，数值伪迹会影响黑洞的运动（参见例如[255] ）。黑洞的虚假位移（通常源于高密度环境中的粒子间散射和数值加热）会带来严重后果，例如人为出现的游荡黑洞、对黑洞-黑洞并合的不正确描述、黑洞在偏离其星系宿主中心的位置产生反馈。为避免这些伪迹，黑洞动力学通常通过不同方法加以控制：通过钉扎进行重定位，如钉扎在最小势能处（例如，[40] ,[41] ,[139] ,[44] ）、增强动力学质量（例如，[144] ）、增强动力学质量与动力学摩擦（例如，[138] ,[145] ）。我们建议读者参考[254] ,[256] ,[35] ,[257] ,[143] ,[258] ,[249] ,[259] ,[260] 了解细节与近期改进。In cosmological simulations, each BH undergoes a specific evolution as an individual particle, at variance with the coarse-grained representation of collisionless fluids through particles in N-body simulations. As a result, the dynamics of BHs fails to be accurately captured and numerical artefacts can affect BH motion (see e.g., [255] ). Spurious displacements of BHs, which often occur due to scattering between particles in high-density environments and numerical heating, have dramatic consequences, such as e.g., artificial presence of wandering BHs, incorrect description of BH-BH mergers, BHs producing feedback off-centre with respect to their galaxy hosts. To avoid these artefacts, BH dynamics is commonly controlled through different approaches: re-positioning via pinning on e.g. minimum potential (e.g., [40] ,[41] ,[139] ,[44] ), boosted dynamical mass (e.g., [144] ), boosted dynamical mass and dynamical friction (e.g., [138] ,[145] ). We refer the reader to [254] ,[256] ,[35] ,[257] ,[143] ,[258] ,[249] ,[259] ,[260] for details and recent improvements.\n黑洞通过气体吸积以及与其他黑洞并合而增长。就后一种渠道而言，当黑洞的宿主星系及其晕并合形成单一结构时，黑洞预计会发生并合。宇宙学模拟中通常采用的方法[242] ,[241] 假设，如果两个黑洞粒子之间的距离接近模拟的空间分辨率（或其小的倍数），它们会快速并合。引力软化设定的力分辨率决定了能够正确追踪引力相互作用的最小尺度。还实现了涉及并合黑洞相对速度等的额外条件[254] 。两个黑洞最终并合成单个黑洞粒子，质量相加。BHs grow because of gas accretion and mergers with other BHs. As for the latter channel, BHs are expected to merge when their host galaxies and their haloes merge to form a single structure. The commonly pursued approach [242] ,[241] in cosmological simulations assumes that two BH particles merge quickly if their distance approaches the spatial resolution of the simulation (or a small multiple of it). The force resolution set by the gravitational softening determines indeed the minimum scale above which gravitational interactions can be properly followed. Additional conditions involving e.g. the merging BH relative speed have been implemented [254] . The two BHs are eventually merged into a single BH particle, with their masses combined.\n对于AGN供料，质量为$M_{\\bullet}$的黑洞的气体吸积率按Bondi公式[261] ,[262] ,[263] 计算，并乘以所谓的增强因子$\\alpha$： $$ \\dot{M}{B} = \\frac{4 \\pi , \\alpha , G^2, M{\\bullet}^2 , \\langle \\rho \\rangle}{(\\langle c_s\\rangle^2 +\\langle v\\rangle ^2)^{3/2}} ,, . $$As for AGN feeding, gas accretion onto a BH of mass $M_{\\bullet}$ is calculated according to the Bondi formula [261] ,[262] ,[263] , multiplied by a so-called boost factor $\\alpha$: $$ \\dot{M}_{B} = \\frac{4 \\pi \\, \\alpha \\, G^2\\, M_{\\bullet}^2 \\, \\langle \\rho \\rangle}{(\\langle c_s\\rangle^2 +\\langle v\\rangle ^2)^{3/2}} \\,\\, .\n这里，$G$是引力常数，$\\langle\\rho\\rangle$、$\\langle v\\rangle$和$\\langle c_s\\rangle$是流体动力学模拟可分辨尺度上的平均值：例如，在SPH情况下，它们通过核加权估计计算得到。黑洞吸积率$\\dot{M}{B}$通常受限于Eddington吸积率$\\dot{M}{Edd}$[Fn: $\\dot{M}{Edd}=(4\\pi:G;m_p; M{\\bullet})/(\\sigma_T;c;\\epsilon_r)$，其中$m_p$为质子质量，$\\sigma_T$为Thompson散射截面，$c$为光速，$\\epsilon_r$为辐射效率，通常取$\\approx0.1$。]（但超Eddington极限的$\\dot{M}_{\\bullet}$的例子见[264] ,[44] ）。$$ Here, $G$ is the gravitational constant, $\\langle\\rho\\rangle$, $\\langle v\\rangle$, and $\\langle c_s\\rangle$ are mean values at the scale resolved by the hydrodynamical simulation: for example, they are computed using kernel weighted estimates in the case of SPH. The BH accretion rate $\\dot{M}_{B}$ is commonly capped to the Eddington acccretion rate $\\dot{M}_{Edd}$ [Fn: $\\dot{M}_{Edd}=(4\\pi\\:G\\;m_p\\; M_{\\bullet})/(\\sigma_T\\;c\\;\\epsilon_r)$, with $m_p$ the proton mass, $\\sigma_T$ the Thompson cross section, $c$ the speed of light and $\\epsilon_r$ the radiative efficiency, typically assumed to be $\\approx0.1$.] (but see [264] ,[44] for examples of $\\dot{M}_{\\bullet}$ which breaches the Eddington limit).\n增强因子$\\alpha$最初引入[241] 是为了补偿模拟中有限的分辨率，这导致黑洞附近密度偏低、温度偏高（从而低估$\\dot{M}_{B}$）。典型值为$\\alpha =100$。多项研究通过采用依赖于分辨率[265] ,[266] 、密度[267] 或压力[268] 的增强因子来改进黑洞模型。其他模拟则通过考虑吸积气体的角动量来限制黑洞吸积率[269] ,[270] 。亚kpc尺度上的黑洞吸积高分辨率模拟[271] 发现，当包含冷却和湍流时，约100的增强因子是合适的，而纯绝热吸积则表明增强因子应小一个数量级。因此，更先进的模型区分热气与冷气吸积，并对两个组分使用不同的增强因子[272] ，甚至可以摆脱人为调节因子（例如，[139] ,[270] ）。The boost factor $\\alpha$ has been originally introduced [241] to account for the limited resolution in simulations, which leads to smaller densities and larger temperatures near the BH (and thus to an underestimate of $\\dot{M}_{B}$). A typical value is $\\alpha =100$. Several studies adapt the BH model by using a boost factor which depends on resolution [265] ,[266] , density [267] , or pressure [268] . Other simulations instead limit the BH accretion rate by taking into account the angular momentum of accreting gas [269] ,[270] . High-resolution simulations of BH accretion on sub-kpc scales [271] found that a boost factor of order of 100 is suitable when including cooling and turbulence, while pure adiabatic accretion suggests boost factors smaller by an order of magnitude. Hence, advanced models distinguish between hot and cold gas accretion and use different boost factors for the two components [272] , or can even get rid of fudge factors (e.g., [139] ,[270] ).\n利用黑洞吸积率$\\dot{M}{\\bullet}$，可为模拟中的每个黑洞赋以热光度$L{\\mathrm bol}$。一个常见的假设是遵循[273] ，根据Eddington比$f_{Edd} = \\dot{M}{\\bullet}/ \\dot{M}{Edd}$（即黑洞吸积率与Eddington吸积率之比）区分高吸积态和低吸积态，并计算： $$ L_{\\mathrm bol} = \\left{ \\begin{array}{ll} 10 , ( \\epsilon_r ; c)^2 , \\dot{M}{\\bullet} \u0026amp; f{\\mathrm Edd}\u0026gt;0.1 \\ (10 , \\epsilon_r; c)^2 , f_{\\mathrm Edd}; \\dot{M}{\\bullet};; \u0026amp; f{\\mathrm Edd}\\leq0.1 \\end{array} \\right. $$ （见[274] ,[275] ）。与原始模型[242] ,[241] 不同，现代实现通常将黑洞吸积率$\\dot{M}{\\bullet}$乘以因子$(1-\\epsilon_r)$予以修正。因此，$,L{\\mathrm bol} = \\epsilon_r / (1-\\epsilon_r) , \\dot{M}_{\\bullet} ,c^2 ,$，吸积过程中辐射出去的能量得以计及。不同的吸积率方案，加之所有亚网格过程联合作用引起的ISM/IGM性质变化，通常导致各最先进模拟之间AGN光度函数预测演化的显著差异。这在Fig. 11中进行了总结，我们可以看出模拟预测在宇宙时间尺度上如何显著偏离观测到的AGN光度函数（另见[275] 中的讨论）。By exploting the BH accretion rate $\\dot{M}_{\\bullet}$, the bolometric luminosity $L_{\\mathrm bol}$ can be associated to each BH in the simulation. A common assumption consists in following [273] to distinguish between high and low BH accretion state in terms of the Eddington ratio $f_{Edd} = \\dot{M}_{\\bullet}/ \\dot{M}_{Edd}$ (i.e. the ratio between the BH and the Eddington accretion rates) and compute: $$ L_{\\mathrm bol} = \\left\\{ \\begin{array}{ll} 10 \\, ( \\epsilon_r \\; c)^2 \\, \\dot{M}_{\\bullet} \u0026 f_{\\mathrm Edd}\u003e0.1 \\\\ (10 \\, \\epsilon_r\\; c)^2 \\, f_{\\mathrm Edd}\\; \\dot{M}_{\\bullet}\\;\\; \u0026 f_{\\mathrm Edd}\\leq0.1 \\end{array} \\right. $$ (see [274] ,[275] ). In contrast to the original model [242] ,[241] , modern implementations often correct the accretion rate $\\dot{M}_{\\bullet}$ of the BH by a factor $(1-\\epsilon_r)$. As a consequence, $\\,L_{\\mathrm bol} = \\epsilon_r / (1-\\epsilon_r) \\, \\dot{M}_{\\bullet} \\,c^2 \\,$, and the energy radiated away during the accretion process is taken into account. Different prescriptions for the accretion rate, in combination with the change of the ISM/IGM properties produced by the combined action of all the sub-grid processes, typically leads to significant variations in the predicted evolution of the AGN luminosity function among the various state-of-the-art simulations. This is summarized in Fig.\u0026nbsp;11, where we can appreciate how predictions from simulations can strongly deviate from the observed AGN luminosity function across cosmic time (see also discussion in [275] ).\nFig.\u0026nbsp;11. 系列导航 ← 上一篇：§3.14 → 下一篇：§3.16\n","date":"2026-06-25","permalink":"/posts/2026-06-25-hydrodynamic-16/","series":["宇宙学模拟中的流体动力学方法与次网格模型"],"summary":"Supermassive black holes in cosmological simulations 观测表明，几乎每个星系在其最内层区域都寄宿着一个超大质量（$10^8 \\div 10^{10}$ M$_{\\odot}$）黑洞（SMBH）（例如，[231] ,[232] ,[233] ,[234] ,[235] ,[236] ,[237] ,[238] ,[239] ）。其中一部分黑洞表现出持续的活动性，称为活动星系核（AGN）。过去和当前活动的证据可见于椭圆星系、星系群和星系团的X射线图像，其中AGN的印记通常表现为凹陷和涟漪。在众多例子中，一个众所周知的代表是MS 0735+7421星系团的复合（X射线、射电和光学）图像（例如[240] ）：巨大的X射线空洞充满射电辐射，并被Chandra图像中清晰可见的椭圆形茧状激波包围。Observations suggest that almost every galaxy hosts a supermassive ($10^8 \\div 10^{10}$ M$_{\\odot}$) black hole (SMBH) in its innermost regions (e.g., [231] ,[232] ,[233] ,[234] ,[235] ,[236] ,[237] ,[238] ,[239] ). A fraction of these BHs exhibits ongoing activity and is called AGN (active galactic nuclei). Evidence of past and ongoing activity is observed for instance in the X-ray images of elliptical galaxies and galaxy groups and clusters, where AGN imprints often appear as depressions and ripples. Among many, a well-known example is represented by the composite (X-ray, radio, and visual) image of the MS 0735+7421 galaxy cluster (e.g. [240] ): giant X-ray cavities are filled with radio emission, and surrounded by a cocoon shock clearly visible in the Chandra image as an elliptical edge.\n","tags":["翻译","astronomy:cosmology","astronomy:cosmology:simulation","numerical:hydrodynamics"],"title":"§3.15 宇宙学模拟中的超大质量黑洞"},{"content":"AGN feedback in state-of-the-art cosmological simulations 观测使我们得以理解AGN反馈如何以不同的机制（如膨胀气泡或发射外流）和不同的表现形式，在各种系统中发展。此外，这是一个反复发生的过程，每一次AGN爆发都在系统中留下清晰的印记。星系内部及周围多相气体的存在，进一步增加了AGN反馈理解和建模的复杂性。多波长观测揭示星系中存在跨越广泛密度、温度和电离状态范围的气体。多相气体不仅存在于富含冷气体的旋涡星系中，也存在于椭圆星系以及星系群和星系团的最内部区域——这些环境以X射线辐射的热气体为主导。Observations allow us to appreciate how AGN feedback develops with different mechanisms (for instance inflating bubbles or launching outflows), with different appearances, in a variety of systems. Besides, it is a recurrent process, with each AGN burst leaving a clear signature in the system. Additional evidence which adds complexity to the comprehension and modelling of AGN feedback is the presence of multiphase gas in and around galaxies. Multiwavelength observations reveal the presence of gas spanning a wide range of densities, temperatures, and ionisation states in galaxies. Multiphase gas is present not only in spiral galaxies, which are systems rich in cold gas, but also in ellipticals and in the innermost regions of galaxy groups and clusters, which are environments known to be dominated by X-ray emitting, hot gas.\n观测（[276] 、[277] 等文献）也为黑洞吸积和反馈的不同状态提供了证据。在简化框架下，AGN活动通过（至少）两个不同的阶段运作：辐射模式或类星体模式，以及动能模式或射电模式（综述见[278] ）。射电模式的特征是大型射电喷流产生热的X射线空洞，而在类星体模式下，辐射由吸积盘主导，反馈能量主要以辐射形式耗散。在理论上，这一区别可通过将AGN反馈描述为两个分量——辐射和力学外流——加以建模（[273] ；[279] 在其模型中考虑了黑洞吸积的三种不同状态）。在这些模型中，与每个分量相关的能量大小取决于Eddington比$f_{Edd}$（见§3.15）。上述两种简单理论模型不仅得到观测支持，还获得其他独立解析模型的佐证（[280] 及其参考文献），这些模型展示了如何将不同的黑洞吸积率，与辐射效率的变化及向不同类型吸积盘的转变联系起来。此外，观测（[281] 、[282] ）表明辐射效率不仅与黑洞吸积率相关，还与黑洞质量相关（另见[272] ）。Observations ([276] ,[277] among other works) also provide evidence for different regimes of BH accretion and feedback. In a simplified context, the AGN activity operates through (at least) two distinct phases: radiative or quasar mode, and kinetic or radio mode ([278] for a review). The radio-mode is characterized by large radio jets generating hot X-ray cavities, whereas in the quasar-mode the emission is dominated by the accretion disc, and feedback energy is mainly radiated away. This distinction has been theoretically modelled [273] by describing AGN feedback with two components: radiation and mechanical outflows ([279] considered three different regimes for BH accretion in their model). In these models, the amount of energy associated with each component depends on the Eddington ratio $f_{Edd}$ (see §3.15). Not only are simple theoretical models like the two aforementioned ones supported by observations, but they are also corroborated by other independent analytical models ([280] and references therein), which showed how it is possible to associate different BH accretion rates to changes in the radiative efficiency and to transitions to different types of accretion discs. In addition, observations [281] ,[282] suggest that the radiative efficiency does not only correlates with the BH accretion rate, but also with the BH mass (see also [272] ).\n然而，双模式AGN反馈仅是对平滑过渡的一种近似——这种过渡已为观测（如[276] 、[283] ）和理论（如[284] 、[285] 、[280] 、[286] ）所预期。实际上，现实比上述简化框架更为复杂，AGN反馈通过多种模式运作，且这些模式常常同时发生。主要的AGN反馈机制包括：预防模式（preventive/preventative mode），即反馈阻止气体吸积或有效冷却；抛射模式（ejective mode），即从恒星形成发生的最内层区域移除气体；AGN反馈可通过湍流和星际介质加热来抑制恒星形成效率；它也可通过对尘埃的辐射压来清除黑洞附近的气体；或者AGN可以以维持模式（maintenance mode）运作，使业已熄灭的系统保持宁静。除了上述所有黑洞反馈为负反馈（即总体上抑制恒星形成率）的模式外，AGN反馈也可以是正反馈，它主要通过星际介质超压（ISM overpressurization），在特定时间段内局部增强恒星形成效率。要解决整体的AGN反馈问题，必须考虑不同的状态，始终牢记AGN及其宿主星系嵌在暗物质晕中。However, a two-mode AGN feedback is only an approximation to the smooth transition which is observed (e.g., [276] ,[283] ) and also theoretically expected (e.g., [284] ,[285] ,[280] ,[286] ). Reality is in fact more complex than the aforementioned simplified frameworks, and AGN feedback operates through a variety of modes, which often occur simultaneously. Among the main AGN feedback mechanisms, there is the preventive (or preventative) mode, where feedback prevents the gas from being accreted or from effectively cooling; the ejective mode, in which gas is removed from the innermost regions of forming structures where SF occurs; AGN feedback can suppress the SF efficiency, mainly via turbulence and ISM heating; it can operate via radiation pressure on dust, which clears out gas from the BH vicinity; or AGN can act in the maintenance mode, by keeping quiescent an already quenched system. Besides all the aforementioned modes in which BH feedback is negative, i.e. it produces an overall suppression of the SFR, AGN feedback can also be positive and locally enhance the SF efficiency for a given period of time, mainly via ISM overpressurization. Different regimes have to be accounted for to address the overall AGN feedback problem, always considering that AGN and their host galaxies are embedded in a DM halo.\nAGN的另一个关键特征是其发展跨越很大的动力学尺度范围，并影响具有不同密度和温度范围的气体演化。这可以从核区尺度一直延伸到星系团尺度，经由星际介质（ISM）、环星系介质（CGM）、星系群内介质和星系团内介质（ICM）。这种空间尺度的大范围对应于同等宽广的时间尺度范围（如[287] ）。Another key characteristic of AGN is that it develops across a large dynamical range of scales, and that it impacts on the evolution of gas with a range of densities and temperatures. This can occur from nuclear scales to galaxy cluster scales, moving through the ISM, the CGM, the intra-group medium and the ICM. This large range in spatial scales corresponds to an equivalently wide range in temporal scales (e.g., [287] ).\n最先进的宇宙学模拟仍远未完全捕捉到这种复杂性。从[242] 、[241] 的开创性工作开始，研究者提出了许多AGN反馈建模方案，用以在宇宙学模拟中描述超大质量黑洞（SMBH）反馈。遵循上述两篇论文，许多模拟假设，以吸积率$\\dot{M}_{\\bullet}$吸积的SMBH，其单位时间内释放的AGN反馈能量可表示为：State-of-the-art cosmological simulations are still far from fully capturing the complexity. Starting from the seminal works by [242] ,[241] , a number of prescriptions for modelling AGN feedback have been conceived to account for SMBH feedback in cosmological simulations. Following the two aforementioned papers, many simulations assume that the AGN feedback energy that a SMBH accreting at a rate $\\dot{M}_{\\bullet}$ releases per unit time can be cast as: $$ \\dot{E} = \\epsilon_f \\epsilon_r \\dot{M_\\bullet} c^2,\n其中$\\epsilon_r$和$\\epsilon_f$分别为辐射效率和反馈效率。$$ where $\\epsilon_r$ and $\\epsilon_f$ are the radiative and the feedback efficiencies, respectively.\n辐射效率$\\epsilon_r$通常取恒定值$0.1$（[241] 、[141] ），这对应于非旋转Schwarzschild黑洞上辐射有效吸积的平均值（[288] 、[289] 、[290] ）。近期研究（如[142] 、[291] ）则引入了依赖黑洞自旋的$\\epsilon_r$。A constant value for the radiative efficiency $\\epsilon_r$ is often used. It is often set to $0.1$ [241] ,[141] , which corresponds to the mean value for the radiatively efficient accretion onto a non-spinning, Schwarzschild BH [288] ,[289] ,[290] . Recent works (e.g., [142] ,[291] ) included a BH spin-dependent $\\epsilon_r$.\n反馈效率$\\epsilon_{f}$量化的是：从黑洞辐射的能量中，实际耦合给周围气体的比例。宇宙学模拟中通常采用的数值范围为$\\sim 10^{-4} - 0.2$（如[292] 、[293] 、[294] 、[35] 、[295] 、[139] 、[296] ）。该效率通常通过调谐来匹配黑洞-恒星质量关系的归一化（[231] 、[238] ）。$\\epsilon_{f}$的值可以是恒定的（如[241] 、[41] ），也可以取决于AGN模式。观测证据表明，以较低吸积率吸积的SMBH驱动更强的外流（[273] 、[279] 、[297] ），基于此，[298] 引入了类星体模式与射电模式之间反馈效率的急剧转变（另见[299] 、[300] 、[35] 、[268] 、[141] 、[44] ）。即使在同一种（射电）模式内，$\\epsilon_{f}$也可随ISM密度而变化（[301] 、[139] ）。The feedback efficiency $\\epsilon_{f}$ quantifies the fraction of energy radiated from the BH which is actually coupled to the. Values commonly assumed in cosmological simulations span the range $\\sim 10^{-4} - 0.2$ (e.g., [292] ,[293] ,[294] ,[35] ,[295] ,[139] ,[296] ). This efficiency is often tuned in order to match the normalisation of the BH to stellar mass relation [231] ,[238] . The value of $\\epsilon_{f}$ can be either constant (e.g., [241] ,[41] , or dependent on the AGN mode. Motivated by the observational evidence of more powerful outflows associated with SMBHs accreting at lower rate [273] ,[279] ,[297] , [298] introduced a steep transition of the feedback efficiency between quasar-mode and radio-mode, (see also [299] ,[300] ,[35] ,[268] ,[141] ,[44] ). Even within the same (radio) mode, $\\epsilon_{f}$ can change according to the ISM density [301] ,[139] .\nAGN反馈能量可通过不同的数值方案沉积：既可以纯粹以热能或动能形式注入，也可以用于注入气泡，或根据（例如）黑洞吸积率的不同通道来分配。在多种可能的AGN反馈模型中，值得指出的是现代宇宙学模拟所采用的若干方案。AGN feedback energy can be deposited by means of different numerical prescriptions: either purely in the form of thermal or kinetic energy, or it can be used to inject bubbles, or according to different channels depending on e.g., the BH accretion rate. Among the several possible AGN feedback models, it is worth highlighting a number of schemes that modern cosmological simulations adopt.\nFig.\u0026nbsp;12. 遵循[242] 、[241] 的原始思想，AGN反馈能量可简单地以热形式注入：通过这种方式，黑洞将AGN反馈能量（公式feedback_energy_old）分配给邻近的分辨率单元，可能以核函数加权的方式进行。接收到的热能用于增加黑洞周围粒子或网格的内能，从而使其温度升高。虽然该简单方案总体有效，但往往效率不高，且未能成功复现目前在不同红移处可获得的各种观测结果。尽管如此，该方案仍在宇宙学模拟中广泛使用，通常仅用于描述类星体模式反馈，同时辅以针对较低黑洞吸积率的补充模型。例如，[268] 、[35] 、[141] 、[139] 、[142] 、[145] 在其模拟中均依赖该模型来描述类星体模式反馈。Following the original idea of [242] ,[241] , the AGN feedback energy can be simply dumped thermally: in this way, the BH distributes the AGN feedback energy (eq. feedback_energy_old) to nearby resolution elements, possibly in a kernel-weighted fashion. The thermal energy received is used to increase the internal energy of particles or cells surrounding the BH, which increase their temperature as a consequence. While being overall effective, this simple prescription often turns out to be inefficient and does not succeed at reproducing the variety of observations currently available at different redshift. Nonetheless, this prescription is still widely used in cosmological simulations, that often adopt it to describe the quasar-mode feedback only, along with a complementary model for lower BH accretion rates. As an example, [268] ,[35] ,[141] ,[139] ,[142] ,[145] rely on this model to describe the quasar-mode feedback in their simulations.\n实际上，Magneticum和SLOW模拟套件采用双模式AGN反馈：对于高黑洞吸积率（即$f_{Edd} \u0026gt; 0.01$），SMBH经历类星体相并向周围释放热能。另一方面，它们通过热气泡的能量沉积来模拟射电模式反馈（$f_{Edd} \u0026lt; 0.01$）（遵循[302] 、[299] ）。在这些模拟中，辐射效率和反馈效率是自由参数。具体而言，射电模式反馈中的$\\epsilon_{f}$相比类星体相增大了4倍。有趣的是，在Magneticum模拟的一个子集中（[272] ），他们探索了双模式AGN反馈的不同情景，以外流取代热气泡。然而，由于分辨率有限，后一种力学通道在数值上实现为热反馈。Indeed, in the Magneticum and in the SLOW simulation suites, they assume a two-mode AGN feedback: for high BH accretion rates, i.e. $f_{Edd} \u003e 0.01$, SMBHs experience a quasar phase and release thermal energy in the surrounding. On the other hand, they model the radio-mode feedback ($f_{Edd} \u003c 0.01$) through energy deposition by hot bubbles (following [302] ,[299] ). In these simulations, radiative and feedback efficiencies are free parameters. Specifically, $\\epsilon_{f}$ is increased by a factor of 4 during the radio-mode feedback with respect to the quasar phase. Interestingly, in a subset of the Magneticum simulations [272] , they explore a different scenario for a two-mode AGN feedback, where hot bubbles are replaced by outflows. However, due to the limited resolution, the latter mechanical channel is numerically implemented as thermal feedback.\n虽然Illustris模拟（[268] 、[40] ）以类似于Magneticum的方式实现AGN反馈（即$f_{Edd} \u0026gt; 0.05$时使用热反馈，低于该值时使用类气泡反馈），但Illustris-TNG模拟大幅改进了黑洞反馈力学分量的建模（[301] 、[139] ）。实际上，除了在区分类星体和射电模式的黑洞吸积率阈值中加入黑洞质量依赖外，他们还通过向选定的气体网格以随机注入方向添加动量，显式模拟了射电模式下的动能风（详见[301] ）。有趣的是，Illustris和Illustris-TNG都通过唯象模型包含了辐射（电磁）AGN反馈。后一种反馈通道改变了晕气体的净冷却率。尽管它在任意$\\dot{M}_{\\bullet}$下均起作用，但仅在黑洞吸积率接近Eddington时才表现出显著效果。While the Illustris simulation [268] ,[40] implements AGN feedback in a similar way to Magneticum (i.e. thermal feedback for $f_{Edd} \u003e 0.05$ and bubble-like feedback below), the Illustris-TNG simulation substantially improves the modelling of the mechanical component of the BH feedback [301] ,[139] . Indeed, besides adding a BH mass dependence to the BH accretion rate threshold which distinguishes between quasar and radio mode, they explicitly model kinetic winds in the radio mode, by adding momentum with random injection directions to selected gas cells (see [301] for details). Interestingly, both Illustris and Illustris-TNG include a radiative (electro-magnetic) AGN feedback via a phenomenological model. This latter feedback channel modifies the net cooling rates of halo gas. Even though it operates whatever the $\\dot{M}_{\\bullet}$, it proved effective only for BH accretion rates close to Eddington.\nEAGLE模拟（[41] ）采用随机热反馈方案，其中累积的AGN反馈能量仅在足以将周围ISM加热到足够高的温度（$T \\sim 10^{8.5} \\div 10^9$ K）时才注入，与恒星反馈的处理类似（见第§3.14节）。这一选择也是FLAMINGO套件（[6] ）参考运行中的默认选项，尽管他们预留了采用喷流模式动能AGN反馈的可能性。在后一种情况下，若喷流开启，则根据黑洞自旋方向发射。这两个模拟套件（[41] 、[6] ）中的AGN反馈始终是单模式的。The EAGLE simulations [41] adopt a stochastic thermal feedback scheme, where accumulated AGN feedback is injected only when enough to heat the surrounding ISM up to an sufficiently-high temperature ($T \\sim 10^{8.5} \\div 10^9$ K), similarly as for the case of stellar feedback (see Secton §3.14). This choice is also the fiducial option in the reference runs of the FLAMINGO suite [6] , although they foresee the possibility to adopt instead a jet-mode, kinetic AGN feedback. In the latter case, should jets be on, they are launched according to the BH spin direction. AGN feedback in these two simulation suites [41] ,[6] is always one-mode.\nSimba模拟（[303] 、[44] 、[304] ）包含双模式AGN反馈：辐射模式和喷流模式。处于射电模式（$f_{Edd} \u0026gt; 0.02$）的黑洞驱动AGN风，其速度与黑洞质量成正比，且不改变外流气体的温度。在喷流模式期间（对于低$f_{Edd}$且$M_{\\bullet} \u0026gt; 10^{7.5}$ M$_\\odot$），外流中气体的温度则提升至晕的维里温度。两种类型的风均通过动量输入产生。当喷流模式活跃时，还存在额外的加热通道，即X射线反馈：它向黑洞周围气体释放能量，这些能量被认为源自吸积盘的X射线辐射。Simba simulation [303] ,[44] ,[304] includes a two-mode AGN feedback: a radiative and a jet mode. BHs in the radio mode ($f_{Edd} \u003e 0.02$) promote AGN winds, with velocity proportional to the BH mass, that do not alter the temperature of the outflowing gas. During the jet-mode (for low $f_{Edd}$ and if $M_{\\bullet} \u003e 10^{7.5}$ M$_\\odot$), the temperature of the gas involved in outflows is instead raised to the virial temperature of the halo. Both types of winds are produced via momentum input. When the jet-mode is active, there is an additional heating channel, namely the X-ray feedback: it releases energy into the BH surrounding gas, which is supposed to originate from X-rays of the accretion disk.\n在(New)Horizon-AGN（[300] 、[305] 、[141] 、[142] ）中，AGN反馈根据黑洞吸积率分为两种模式。在射电模式期间，黑洞驱动喷流，连续向气体释放质量、动量和总能量。而当黑洞处于类星体模式时，则仅向以黑洞为中心的球体内的气体网格提供热能。喷流模式下，黑洞喷流是双极的，以恒定速度发射，并在一个柱体内沉积质量、动量和能量。在New Horizon-AGN中，辐射效率和反馈效率均依赖于黑洞自旋（在类星体和射电模式中均是如此）。In the (New)Horizon-AGN [300] ,[305] ,[141] ,[142] AGN feedback features two modes, depending on the BH accretion rate. During the radio mode, the BH powers jets, continuously releasing mass, momentum, and total energy into the gas. On the other hand, only thermal energy in supplied to the gas cells within a sphere centred on the BH when it is in quasar mode. As for the jet mode, BH jets are bipolar, launched with a constant velocity, and they deposit mass, momentum and energy within a cylinder. In the New Horizon-AGN simulation, radiative and feedback efficiencies are BH spin-dependent (both in quasar and radio mode).\nFig. 12总结了一些现代宇宙学模拟在黑洞增长、AGN供给和反馈建模中所采用的数值方案。上述六个示意图旨在提供对各模型的简化而直观的理解和比较。Fig.\u0026nbsp;12 summarises the numerical prescriptions that some modern cosmological simulations adopt to model BH growth, AGN feeding and feedback. The six sketches in the aforementioned figure aim at providing a simplified yet immediate understanding and comparison among the various models.\n上述AGN反馈方案汇编远非完整。其他值得一提的工作还包括[306] 、[140] 、[307] 、[308] 、[309] 等。The compilation of prescriptions for AGN feedback listed above is far from being complete. Other works which is worth mentioning include e.g., [306] ,[140] ,[307] ,[308] ,[309] .\n以下就模拟中AGN反馈建模给出几点评论，涵盖：$(i)$模型参数校准；$(ii)$许多最先进模型背后的简化；以及$(iii)$不同模型之间的比较。A few remarks on AGN feedback modelling in simulations follow, about: $(i)$ model parameter calibration; $(ii)$ the simplification underlying many of the state-of-the art models; and $(iii)$ comparison among different models.\n亚网格物理，特别是模型的自由参数，通常通过调谐来重现观测量。AGN反馈效率（即$\\epsilon_{f}$和$\\epsilon_{r}$）尤其如此。大多数宇宙学模拟通过尝试重现一系列观测量来校准自由参数（可能在理论、唯象或数据所建议的范围内取值）。最著名的校准对象是黑洞-恒星质量关系（[231] 、[238] ）和星系恒星质量函数（GSMF，如[310] ），通常在红移$z=0$处。其他校准目标包括：星系质量-尺寸关系、重子或气体质量份额、金属丰度-质量关系和光度-质量关系。建议读者参阅[226] 以获取更广泛的讨论。模型校准的常见策略包括：对选定数据集进行拟合（如EAGLE）、利用与某些观测比较的趋势（如Magneticum、Illustris）、借助机器学习（如FLAMINGO），或与先前/降尺度版本的类似模拟进行比较。Subgrid physics and in particular free parameters of the models, are usually tuned to reproduce observables. This is especially true for AGN feedback efficiencies (i.e. $\\epsilon_{f}$ and $\\epsilon_{r}$). The majority of cosmological simulations calibrate free parameters (possibly within ranges suggested by theory or phenomenology or data) by attempting to reproduce a number of observables. The most popular examples are the BH to stellar mass relation [231] ,[238] and the galaxy stellar mass function (GSMF, e.g. [310] ), usually at redshift $z=0$. Other calibration targets include: the galaxy mass-size relation, baryon or gas mass fractions, metallicity-mass and luminosity-mass relations. We refer the reader to [226] for a more extended discussion. Common strategies to model calibration consist in performing a fit to selected data sets (e.g. EAGLE), to exploit trends in comparison to some observations (e.g. Magneticum, Illustris), to profit from machine learning (e.g. FLAMINGO) or from comparison to previous/scaled-down versions of similar simulations.\nFig.\u0026nbsp;13. 分辨率改变时，也存在不同的处理方法。弱收敛（与强收敛相对，见[41] ）通常是首选，对于不同的数值分辨率，常见做法是至少调整模型参数的若干数值。There are as well different approaches when resolution changes. Weak convergence (as opposite to strong convergence, see [41] ) is usually preferred, and for different numerical resolutions it is common practice to adapt at least a few values of the model parameters.\n然而，用于校准的观测数据集的选择不仅影响参数调谐。更重要的是，由此可理解不同模拟预测在各种观测面前的表现。例如，Magneticum模拟在重现星系团内介质（ICM）和星系群内介质的物理性质方面优于其他模拟（参见[311] 中的一个例子），而Illustris-TNG、Simba和EAGLE等模拟通常在预测星系群最终性质方面更胜一筹。这主要源于校准目标的不同，即Magneticum模拟针对ICM和恒星金属丰度含量加上X射线光度-质量关系，而其他模拟则针对GSMF和星系质量-尺寸关系。However, the choice of the observational data sets adopted for calibration does not impact parameter tuning only. Rather, it allows to understand the performance of different simulation predictions against various observations. For instance, the Magneticum simulation succeeds at reproducing the physical properties of the ICM and of the intra-group medium better than other simulations (see [311] for an example), while simulations like Illustris-TNG, Simba and EAGLE usually outperform the competition as for predicting final properties of the galaxy population. This mainly stems from the calibration targets, i.e. ICM and stellar metallicity content, plus X-ray luminosity-mass relation for the Magneticum simulation versus GSMF and galaxy mass-size in the others.\nz = 2\nz = 0\nFig.\u0026nbsp;14. 在任何亚网格模型背后的众多假设中，其中一个简化是承认：关于AGN反馈能量如何与ISM不同相耦合，我们知之甚少。[270] 研究了将AGN反馈能量分配给ISM各相的重要性。在MUPPI亚分辨率模型（[155] 、[135] ）框架内，他们展示了仅向热气体或冷气体提供反馈能量、向两相均匀分配黑洞反馈能量，以及根据热气体和冷气体的物理性质（即冷云的覆盖因子）耦合AGN反馈能量所带来的影响。他们发现，AGN反馈能量耦合到ISM各相的数值方案会影响最终的黑洞和宿主星系性质（见Fig. 13）。有趣的是，他们假设AGN反馈能量要么用于提高热气体温度，要么用于蒸发分子气体，而不是例如使多相粒子偏离平衡解（[241] ）。Among the many assumptions underlying any sub-grid model, one simplification consists in accepting that little is known about how AGN feedback energy couples with different phases of the ISM. [270] investigate how relevant it is to distribute AGN feedback energy to the individual phases of the ISM. Within the MUPPI sub-resolution model [155] ,[135] , they show the impact of supplying feedback energy to hot or cold gas only, of evenly providing the two phases with BH feedback energy, and of coupling AGN feedback energy to the hot and cold gas according to their physical properties (i.e. the covering factor of cold clouds). They find that the numerical prescriptions adopted to couple AGN feedback energy to the ISM phases affect final BH and host galaxy properties (see Fig.\u0026nbsp;13). Interestingly, they assume that AGN feedback energy is used either to increase the hot gas temperature, or to evaporate molecular gas, instead of e.g. producing a deviation from the equilibrium solution of multiphase particles [241] .\n最后，评估不同AGN供给和反馈模型之间的差异并非易事。感兴趣的读者可参阅[312] ，其中有一项参考性比较研究，探讨了黑洞种子、AGN供给和反馈各种方案的影响。在CAMELS项目（[313] ）中，不同的实现方案也在同一宇宙学模拟中进行了测试。此处，Fig. 14（及其关联动画）展示了某些最先进宇宙学模拟中采用的各种亚网格模型，在能量分布和IGM/CGM演化方面所预测的差异。Finally, assessing the differences among different models for AGN feeding and feedback is not straightforward. We refer the interested reader to [312] for a reference comparative study where the impact of various prescriptions for BH seeding, AGN feeding and feedback has been addressed. Different implementations are tested with the same cosmological simulation also in the CAMELS project [313] . Here, Fig.\u0026nbsp;14 (along with the associated movie) provides evidence of how various sub-grid models adopted in some state-of-the-art cosmological simulations predict differences as for the energy distribution and the evolution of the IGM/CGM.\n系列导航 ← 上一篇：§3.15 → 下一篇：§3.17\n","date":"2026-06-25","permalink":"/posts/2026-06-25-hydrodynamic-17/","series":["宇宙学模拟中的流体动力学方法与次网格模型"],"summary":"AGN feedback in state-of-the-art cosmological simulations 观测使我们得以理解AGN反馈如何以不同的机制（如膨胀气泡或发射外流）和不同的表现形式，在各种系统中发展。此外，这是一个反复发生的过程，每一次AGN爆发都在系统中留下清晰的印记。星系内部及周围多相气体的存在，进一步增加了AGN反馈理解和建模的复杂性。多波长观测揭示星系中存在跨越广泛密度、温度和电离状态范围的气体。多相气体不仅存在于富含冷气体的旋涡星系中，也存在于椭圆星系以及星系群和星系团的最内部区域——这些环境以X射线辐射的热气体为主导。Observations allow us to appreciate how AGN feedback develops with different mechanisms (for instance inflating bubbles or launching outflows), with different appearances, in a variety of systems. Besides, it is a recurrent process, with each AGN burst leaving a clear signature in the system. Additional evidence which adds complexity to the comprehension and modelling of AGN feedback is the presence of multiphase gas in and around galaxies. Multiwavelength observations reveal the presence of gas spanning a wide range of densities, temperatures, and ionisation states in galaxies. Multiphase gas is present not only in spiral galaxies, which are systems rich in cold gas, but also in ellipticals and in the innermost regions of galaxy groups and clusters, which are environments known to be dominated by X-ray emitting, hot gas.\n","tags":["翻译","astronomy:cosmology","astronomy:cosmology:simulation","numerical:hydrodynamics"],"title":"§3.16 最先进宇宙学模拟中的AGN反馈"},{"content":"Current state and perspectives 即使在今天，用作超大型巡天（如EUCLID、DESI、LSST）理论对照的宇宙学模拟，由于需要覆盖极大的体积，仍然基于纯引力物理（例如，[10] ,[14] ）。这些模拟通常辅以星系形成的半解析模型（SAMs）（例如，[314] ）。虽然SAMs提供了星系族群性质的逼真描述，但它们充其量只能给出ISM/IGM性质的间接信息。事实上，它们并未自洽地处理气体动力学，也只能粗略捕捉重子对结构形成的影响，而这对环境效应的研究至关重要。Even nowadays, the cosmological simulations used as theoretical counterparts for very large surveys (like EUCLID, DESI, LSST), for which extremely large volumes need to be sampled, are based on pure gravitational physics (e.g., [10] ,[14] ). These simulations are usually complemented by running semianalytic models (SAMs) of galaxy formation (e.g., [314] ). While SAMs provide a realistic description of the properties of galaxy populations, they bring at best indirect information on the properties of the ISM/IGM. In fact, they do not include a self-consistent treatment of gas dynamics and loosely capture the effects of baryons on structure formation, which is highly relevant for the study of environmental effects.\n在过去的几十年里，越来越多不同的大尺度宇宙学流体动力学模拟得以执行，其分辨率和覆盖的体积各不相同（参见第§3.1节开头的Fig. 1）。最先进宇宙学模拟的一个局限在于，不同的次网格模型在可直接观测的性质上往往预测出非常相似的结果，而它们结果之间的主要差异通常隐藏于无法直接或容易获取的性质中（如星系的IGM/CGM），和/或隐藏于它们不同的演化过程中（参见Fig. 14）。这既削弱了单个模拟的预测能力，也削弱了将模拟之间的差异归因于其所采用的不同数值方案或所包含的物理过程的可能性。将模拟预测与观测结果比较有助于约束理论建模：然而，公正地评估差异却并非易事。事实上，比较过程通常需要借助复杂技术来生成有意义的模拟观测（例如，[315] ,[211] ），并且还必须同时覆盖多波段区域和不同组分（例如，恒星、气体、尘埃……）。In the last decades, a growing number of different, large-scale, cosmological, hydro-dynamical simulations have been performed with varying resolutions and volumes covered (see initial Fig.\u0026nbsp;1 in section §3.1). One limitation of state-of-the-art cosmological simulations is that different sub-grid models often predict very similar results in direct observable properties, and the main differences among their outcome are often hidden in properties which are not directly or easily accessible (like the IGM/CGM of galaxies), and/or in the different evolution of them (see Fig.\u0026nbsp;14). This weakens the predictive power of individual simulations, and undermines the possibility to ascribe discrepancies among simulations to different numerical prescriptions adopted or to physical processes included. Comparing predictions from simulations with observations can help to constrain the theoretical modelling: however, evaluating differences fairly is not trivial. In fact, the comparison process often involves sophisticated techniques to create meaningfully mock observations (e.g., [315] ,[211] ), and also has to cover multi-wavelength regimes and different components (e.g., stars, gas, dust, ...) simultaneously.\n在过去几年中，若干改进方向一直在推进。例如，它们包括对模拟结构最内部区域进行超细化[316] ,[317] ,[318] ，以提高那些对捕获额外物理过程至关重要的区域的分辨率。 不仅在最致密区域或SMBH周围需要更高分辨率，在维里半径内提高空间细化也同样至关重要，从而更准确地捕捉CGM物理和演化（例如，[319] ）。 此外，许多工作已纳入了详细的次分辨率吸积盘建模和BH自旋建模[305] ,[320] ,[321] ,[322] ,[291] ，并在将SMBH吸积过程与AGN触发的喷流发射关联方面取得了进展[323] 。 另外，对SMBH外流和喷流的建模也有所增强[301] ,[324] ，一些有趣的实验还涉及自旋驱动的（Blandford-Znajek）AGN喷流的引入[325] ,[326] 。 到目前为止，这些改进主要通过专门且某种程度上特设的设置进行了验证，或纳入较小暗晕或较小体积的宇宙学模拟中。随着上述数值改进已纳入最先进代码，我们期待这些新模块将出现在即将到来的大体积模拟中，从而在不久的将来提高数值预测的精度。Several lines of refinement have been under development during the last years. They include, for instance, hyper-refinement in the innermost regions of simulated structures [316] ,[317] ,[318] , to increase the resolution in those regions which are crucial to better resolve to capture additional physics. Not only is higher resolution needed in the densest regions or around SMBHs, but it is fundamental also increase the spatial refinement within the virial radius, to capture CGM physics and evolution more accurately (e.g., [319] ). Also, a number of works have included a detailed modelling of sub-resolution accretion discs and BH spin modelling [305] ,[320] ,[321] ,[322] ,[291] , and have progressed in linking the accretion process onto SMBHs with the launch of AGN-triggered jets [323] . Furthermore, the modelling of SMBH outflows and jets has been enhanced [301] ,[324] , and interests experiments involve the inclusion of spin-driven (Blandford-Znajek) AGN jets [325] ,[326] . So far, these improvements have been validated mainly with dedicated and somehow ad-hoc setups, or included in cosmological simulations of smaller haloes or volumes. As the numerical advancement of the aforementioned improvements has been included in state-of-the-art codes, we expect that these new modules will be featured by upcoming large-volume simulations, and improve the accuracy of the numerical prediction in the near future.\n尽管现代宇宙学模拟的预测能力毋庸置疑，但仍存在若干不可忽视的问题。 例如，大多数最先进的宇宙学模拟假设所有超过质量阈值的暗晕都寄宿有BH，并采用大质量BH种子来促进其早期增长。此外，它们通常依赖Bondi吸积——虽经修改——来促进BH的初始增长，并使它们更易达到爱丁顿极限。While the predictive power of modern cosmological simulations is beyond discussion, still there are several caveats that cannot be overlooked. The majority of state-of-the-art cosmological simulations, for instance, assume that all haloes above a mass threshold host BHs, and adopt massive BH seeds to promote their early growth. Also, they commonly rely on Bondi accretion -- though with modifications -- to facilitate the initial growth of BHs and to make them easily reach the Eddington limit.\n此外，反馈过程的实现仍远未达到足够的复杂程度，并且通常受数值效率驱动。 同时，一些可能阻碍BH增长并降低恒星形成率（SFR）的反馈过程（例如，早期反馈、恒星辐射、恒星和AGN风、尘埃上的辐射压）常被忽略。 许多相关的物理模块仍未包含在大多数大体积宇宙学模拟的参考运行中，因为它们与现有框架的集成并不简单。例如，值得指出的是：辐射输运、宇宙线、磁场、替代标准$\\Lambda$CDM的暗物质方案、化学扩散，仅仅是其中几个例子。 对更高空间分辨率的需求使得一致考虑所有相关物理过程变得困难。In addition, the implementation of feedback processes is still far from reaching an adequate degree of complexity, and it is often driven by numerical effectiveness. Besides, several feedback processes (e.g., early feedback, radiation from stars, stellar and AGN winds, radiation pressure on dust) which could hamper BH growth and reduce the SFR are often neglected. A number of relevant physics modules are still not included in the reference runs of the majority of cosmological simulations of large volumes, as their integration in the existing framework is not straightforward. As an example, it is worth mentioning: radiative transport, cosmic rays, magnetic fields, scenarios for DM alternative to the standard $\\Lambda$CDM, chemical diffusion, just to name a few. The need for higher spatial resolution makes it difficult to consistently account for all the relevant physics.\n最后，值得回顾，我们已进入高性能计算（HPC）时代，当前和未来的HPC设施提供了持续增长的计算能力。 因此，拥有在物理过程实现方面尽可能完备、同时在计算方面也非常高效的数值代码至关重要。代码应当能够在最先进的百亿亿次级基础设施上平滑扩展，并高效利用CPU（中央处理单元）和GPU（图形处理单元）。Finally, it is worth recalling that we have now entered the era of high-performance computing, with always growing computational power available from current and future HPC facilities. It is therefore of paramount importance to have numerical codes which are not only as complete as possible as for the inclusion of physical processes implemented, but also very efficient from the computational point of view. Codes are indeed supposed to be able to smoothly scale on state-of-the-art exascale infrastructures, and to efficiently exploit CPUs (Central Processing Units) and GPUs (Graphics Processing Units).\n上述努力与数值方法的改进相结合，将使我们能够以前所未有的细节研究宇宙学结构的形成过程，并更好地将星系形成的小尺度物理过程与大尺度结构和宇宙网的演化关联起来。The aforementioned efforts, together with the improvements in numerical methods, will allow us to study the formation process of cosmological structures with unprecedented detail, and to better link the small-scale physical processes of galaxy formation to the evolution of the large-scale structure and of the cosmic web.\n系列导航 ← 上一篇：§3.16\n","date":"2026-06-25","permalink":"/posts/2026-06-25-hydrodynamic-18/","series":["宇宙学模拟中的流体动力学方法与次网格模型"],"summary":"Current state and perspectives 即使在今天，用作超大型巡天（如EUCLID、DESI、LSST）理论对照的宇宙学模拟，由于需要覆盖极大的体积，仍然基于纯引力物理（例如，[10] ,[14] ）。这些模拟通常辅以星系形成的半解析模型（SAMs）（例如，[314] ）。虽然SAMs提供了星系族群性质的逼真描述，但它们充其量只能给出ISM/IGM性质的间接信息。事实上，它们并未自洽地处理气体动力学，也只能粗略捕捉重子对结构形成的影响，而这对环境效应的研究至关重要。Even nowadays, the cosmological simulations used as theoretical counterparts for very large surveys (like EUCLID, DESI, LSST), for which extremely large volumes need to be sampled, are based on pure gravitational physics (e.g., [10] ,[14] ). These simulations are usually complemented by running semianalytic models (SAMs) of galaxy formation (e.g., [314] ). While SAMs provide a realistic description of the properties of galaxy populations, they bring at best indirect information on the properties of the ISM/IGM. In fact, they do not include a self-consistent treatment of gas dynamics and loosely capture the effects of baryons on structure formation, which is highly relevant for the study of environmental effects.\n","tags":["翻译","astronomy:cosmology","astronomy:cosmology:simulation","numerical:hydrodynamics"],"title":"§3.17 现状与展望"},{"content":"欢迎 欢迎来到 NumericAstronomy。\n我们相信自然界中的一切运动和变化中都蕴含着普适而统一的物理规律，这些规律可以用简洁而优美的数学语言来描述。 然而，现实世界如此丰富多彩，\n数值天体物理学是利用计算机通过数值方法求解支配恒星、星系和宇宙本身方程的艺术。 这个网站是什么 这个网站记录了从天文小白到天体物理数值模拟专家的学习之旅。\n内容包括：文献翻译、系列教程、综述思考、个人笔记和便捷小工具。 ","date":"2026-06-25","permalink":"/posts/2026-06-25-hello-world/","series":["从零开始的数值天体物理"],"summary":"欢迎 欢迎来到 NumericAstronomy。\n我们相信自然界中的一切运动和变化中都蕴含着普适而统一的物理规律，这些规律可以用简洁而优美的数学语言来描述。 然而，现实世界如此丰富多彩，\n数值天体物理学是利用计算机通过数值方法求解支配恒星、星系和宇宙本身方程的艺术。 这个网站是什么 这个网站记录了从天文小白到天体物理数值模拟专家的学习之旅。\n内容包括：文献翻译、系列教程、综述思考、个人笔记和便捷小工具。 ","tags":["入门","introduction","astronomy","numerical","numerical:astrophysics"],"title":"你好，NumericAstronomy"}]