Hybrid approach
Hybrid-kinetic approaches are widely used to model ion-scale phenomena in space plasmas. Hybrid codes differ from fully kinetic particle-in-cell (PIC) codes in that the electrons are modeled as a fluid that can be considered even massless, while the electric field is not advanced in time, but instead calculated at the new time level from the advanced ion quantities and the magnetic field. (Electric field is quasistatic, \(\omega_p\ll\omega_c\)). In this article, we concentrate on such hybrid models with massless electrons, beginning with a discussion of the basics of a simple hybrid code algorithm. We then show examples of recent use of hybrid codes for large-scale space plasma simulations of structures formed at planetary bow shock—foreshock systems, magnetic reconnection at the magnetopause, and complex phenomena in the magnetosheath due to the interaction of kinetic processes associated with the bow shock, magnetic reconnection, and turbulence.
A hybrid code in the context of plasma physics generally refers to a computational model in which some of the plasma species are treated kinetically and others as fluids, while the electric and magnetic fields can be considered as either electrostatic or electromagnetic. Most often in hybrid codes employed in space plasmas all the different ion species are treated kinetically using particle-in-cell methods, the electrons are modeled as a fluid with massless electrons, and Maxwell’s equations are solved in the radiation-free (i.e., no light waves) limit. Even in this limited space, there is still freedom in how to treat the fast phenomena associated with electron dynamics.
In space plasmas in particular, hybrid codes occupy a unique niche, intermediate between fluid codes that are most useful for modeling large portions of the solar wind or the interaction of the solar wind with the Earth’s magnetosphere and fully kinetic treatment of all the plasma components, i.e., all ion species and electrons, that are resolved down to the smallest electron scales. Instead, hybrid codes model a portion of the solar wind-magnetosphere boundary or a small region in the solar wind or the magnetosphere on spatial and temporal scales on which ion dynamics dominates. The effects of the larger fluid scales can be modeled through boundary conditions, while the small-scale effects of electrons are added in through models for the electron pressure tensor, and high-frequency (e.g., electronion lower-hybrid or higher electron-electron) wave effects are included through a resistivity or other transport coefficients. But as computers have become larger and faster, fully kinetic codes can now model system sizes and time scales that previously were only accessible to hybrid codes, and hybrid codes in turn can now model systems large enough that could once only be studied by fluid models.
With larger computational domains, longer times, and more realistic initial/boundary conditions, the emphasis is now on investigating the interactions between various physical processes, rather than focusing on a single physics issue.(这就是multiscal的相比于hybrid code不一样的地方，multiscale可以比较方便的描述不同scale的physical process的交互，而hybrid code只能够描述单尺度的问题。) This has been motivated in part by the existence of better spacecraft data, involving very high-resolution simultaneous data from multiple satellites. One example is the MMS (Magnetosphere Multi-Scale) mission [2] whose four closely-spaced satellites have fully resolved ion and have begun to explore electron scales related to magnetic reconnection layers. In addition, extended hybrid models that include electron inertia effects, new quasi-neutral models, and better Vlasov methods have become more prevalent, which allow investigating short-scale electron processes that are now becoming accessible in space and laboratory observations of magnetic reconnection.
This article on hybrid codes has several purposes. The first purpose is to serve as a tutorial for newcomers to the field of computational space plasma physics by providing a brief introduction to hybrid code methodology. It includes a review of the basics of the hybrid physics model, the underlying fundamental equations that are used, and a simple example of the physics implementation into a working algorithm.
Multiscale approach
Plasmas are by their intrinsic nature multiscale and multi-physics. Electrons and ions respond on very different scales by virtue of their different masses, and even electron-positron plasmas might have multiple masses (in the laboratory frame) when relativistic velocities are present and the range of Lorentz factors is wide. Plasmas have many waves and eigenmodes, ranging many orders of magnitude in space and in time.For hydrogen plasmas, the mass ratio is 1836, and its square root determines the range of scales between election and ion inertial lengths and plasma frequencies.(质量比的开方是电子和离子的惯性尺度比以及静电振荡比)The cyclotron frequencies differ by the mass ratio(质量比是电子和离子的回旋频率比).The species beta \(\beta_s\), the ratio of the species’ pressure with the magnetic energy, can also vary over orders of magnitude in different space and astrophysical plasmas, determining the ratio between a species inertial length and its gyro-radius or its cyclotron and plasma frequency.(从这里可以看出，等离子体的无量纲量很多，所以会出现很多不同的regime). In the case of the electrons, the Debye length is usually much smaller than the electron inertial length because of the low thermal speed compared with the speed of light: \(\lambda_{De}/d_e = v_{the}/c\).(\(\lambda_{De}= v{the}/\omega_{pe}, \delta_{e} = c/\omega_{pe}\), 所谓的惯性尺度就是趋肤深度，即电磁波能够扩散的尺度，如果这个尺度趋于0，那么不存在磁扩散，磁场frozen在等离子体中。退一步讲，如果电子的skin length趋于0，而离子不为0，则磁场frozen在电子上，而不是在离子上，离子与电子解耦。)
Implicit methods prevent unresolved scales from producing numerical instabilities that become a source of unphysical energy. Provided stability is ensured, using an implicit method is like using a reduced model that eliminates a physical process but with one critical difference: if one desires to reintroduce the process, one just need to increase the resolution.This approach then allows the user to conduct convergence studies varying the domain size and the time interval. It is an instrument of great power that needs to be used with experience, properly selecting the scales resolved.(没看懂在讲什么)
Temporal Adaptation
Time is discretized in steps. We consider here some general properties applicable to a vast class of methods if not all.Time discretization can be either explicit,when one time step is directly computed from the previous time step without requiring iterations or solution of linear or nonlinear coupled sets of equations, or implicit, when the time step to be calculated appears in the calculation and an iteration (either linear or nonlinear) is needed. We review below the two methods and consider their advantages.
Model adaptation
The most critical aspect of coupling fluid and kinetic models is the strategy for handling the interface conditions. Fluid models are structurally incapable of receiving the information from the kinetic models: for example, they can’t handle waves that are not present in the fluid models or can’t handle high-energy particle beams. We then use two types of boundary conditions, semi-open or linked with fluid. The semi-open boundary conditions are based on the perfectly matched layer [26, 83] approach where a damping term is added to the Maxwell’s equations to absorb outgoing waves.(这个做法挺有意思的，因为如果所有的域都需要解Maxwell equation的话，那么永远都会有光速的限制，怎么样拿掉光速的限制，或者怎么样处理EM波到MHD波的变化，是很重要的。在Two-fluid方法里面Hakim他们也求解了Maxwell equation，但是如何过渡到MHD regime，不需要去求解Maxwell equation呢？Lepenta的方法似乎是做domain interface，然后在不同的domain求解不同的方程，然后尝试求解好界面上的过度。)
