文章 on NumericAstronomy

宇宙学模拟中的流体动力学方法与次网格模型

Thu, 25 Jun 2026 00:00:00 +0000

原文信息

出处：Numerical Simulations in Cosmology: Chapter 3 — Hydrodynamic methods and sub-resolution models for cosmological simulations

作者：Milena Valentini (Universitá degli Studi di Trieste / INAF), Klaus Dolag (LMU München / MPI for Astrophysics)

arXiv：arXiv:2502.06954 页数：61 页，14 幅图

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.
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。

§3.1 宇宙学流体动力学模拟

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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).

§3.2 流体动力学与数值方法

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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.

§3.3 基本方程与技术

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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.

§3.4 欧拉（网格）方法

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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.

§3.5 自适应网格细化

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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 $m \equiv \rho , \Delta x^3 > m_{min}$，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, $m \equiv \rho \, \Delta x^3 > m_{min}$, 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.

§3.6 拉格朗日方法与光滑粒子流体动力学

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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.

§3.7 移动网格方法

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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] .

§3.8 无网格方法

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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 $\dot\phi(\vec{x}, t) = \partial\phi(\vec{x}, t)/\partial t + \vec{a}(x,t)\cdot\nabla\phi(\vec{x}, t)$.

§3.9 星系与星系团模拟中的代码对比

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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.

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).

§3.10 天体物理过程的次分辨率建模

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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).

§3.11 气体冷却

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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.

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] .

§3.12 恒星形成

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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（$\nabla \cdot \mathbf{v} < 0$）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

§3.13 恒星演化与化学增丰

Thu, 25 Jun 2026 00:00:00 +0000

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).

§3.14 恒星反馈

Thu, 25 Jun 2026 00:00:00 +0000

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] .

§3.15 宇宙学模拟中的超大质量黑洞

Thu, 25 Jun 2026 00:00:00 +0000

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.

§3.16 最先进宇宙学模拟中的AGN反馈

Thu, 25 Jun 2026 00:00:00 +0000

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.

§3.17 现状与展望

Thu, 25 Jun 2026 00:00:00 +0000

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.

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宇宙学模拟中的流体动力学方法与次网格模型

§3.1 宇宙学流体动力学模拟

§3.2 流体动力学与数值方法

§3.3 基本方程与技术

§3.4 欧拉（网格）方法

Eulerian (grid) methods

§3.5 自适应网格细化

Adaptive mesh refinement

§3.6 拉格朗日方法与光滑粒子流体动力学

Lagrangian methods and Smoothed Particle Hydrodynamics

§3.7 移动网格方法

Moving mesh methods

§3.8 无网格方法

Meshless methods

§3.9 星系与星系团模拟中的代码对比

Code comparison in galaxy and cluster simulations

§3.10 天体物理过程的次分辨率建模

Sub-resolution modelling of astrophysical processes

§3.11 气体冷却

Gas cooling

§3.12 恒星形成

Star formation

§3.13 恒星演化与化学增丰

Stellar evolution and chemical enrichment

§3.14 恒星反馈

Stellar feedback

§3.15 宇宙学模拟中的超大质量黑洞

Supermassive black holes in cosmological simulations

§3.16 最先进宇宙学模拟中的AGN反馈

AGN feedback in state-of-the-art cosmological simulations

§3.17 现状与展望

Current state and perspectives

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