A review of plasma acceleration and detachment mechanisms in propulsive magnetic nozzles Open Access

In contemporary times, space technology is seamlessly integrating into our daily lives.¹ Global satellite navigation, weather prediction, and satellite communications have become essential elements of human existence. Moreover, the public is increasingly intrigued by ambitious large-scale space exploration endeavors, such as lunar missions, Mars colonization, and deep space exploration. These technological advances and breakthroughs are underpinned by progress in multiple domains, including propulsion systems, space weather research, and orbital mechanics. Propulsion technology stands as a crucial pillar in this overall scheme. Carrier rockets are responsible for transporting spacecraft to their intended orbits, relying on powerful chemical rocket engines capable of generating thrust in the hundreds of tons to counteract Earth's gravitational pull. In contrast to ground launches, spacecraft operate in microgravity environments, where smaller, low-power propulsion systems can effectively manage their movement. To sustain the complex operations required for long-term orbital missions, including attitude control, orbital maneuvers, and deep space exploration, there is a pressing need for innovative, highly efficient, and durable propulsion technologies for spacecraft. Electric propulsion systems, emerging as a mature and reliable option, are gaining widespread adoption in the space sector due to their distinctive advantages over traditional propulsion systems. These advantages include higher specific impulse, greater efficiency, longer operational lifespans, and lighter system masses.² 

The history of modern rockets, launch vehicles, and spacecraft can be traced back to the theoretical work of Konstantin Tsiolkovsky.² Tsiolkovsky's equations, which describe the fundamental principles of rocket flight in space, are among the most important formulas in the field of astronautics and laid the theoretical foundation for rocket engines. In 1911, he first proposed a concept that can be considered the precursor to electric propulsion, suggesting that it might be possible to use electricity to impart high velocities to the particles expelled by a rocket.² The basic concept of electric propulsion involves using charged particles as propellants, which are accelerated using electrical energy to achieve significantly higher exhaust velocities compared to chemical propulsion.³ This dramatically improves the efficiency of propellant mass utilization, typically measured by specific impulse (Isp), which is the ratio of thrust to propellant mass flow rate. In space missions, where mass is constrained by the payload capacity of launch vehicles, the high specific impulse of electric propulsion offers a revolutionary advantage over chemical propulsion, making it highly potential for spacecraft orbit control.³ However, the application of electric propulsion has been limited by the availability of electrical power available in space, and it only began to emerge in the 1960s. After half a century of development, electric propulsion has become one of the most important space propulsion technologies, widely used in large-scale space missions such as the low-orbit mega constellation Starlink.^4,5 In 2012, NASA identified electric propulsion as a priority technology for future development.⁶ 

Magnetic nozzle guide plasma flow through converging-diverging or diverging magnetic fields, converting internal energy and non-axial kinetic energy into axial kinetic energy. This is an important plasma acceleration scheme. Compared to electrostatic acceleration, magnetic nozzle can effectively confine high-temperature and high-density plasmas by regulating magnetic fields, making them one of the preferred acceleration methods for high-power electric propulsion in the future.^7,8 In 1969, magnetic nozzle were first discovered by Anderson⁹ to be capable of generating plasma jets with Mach number $∼3$⁠. They are named so because of their functional and configurational similarities to the de Laval nozzle in chemical engines. As shown in Fig. 1, a magnetic nozzle is created by converging-diverging or diverging magnetic fields generated by electrified coils or permanent magnets. In electric propulsion research, the current focus is primarily on the expansion acceleration section of the magnetic nozzle, specifically the supersonic region. Expansion magnetic nozzle can be considered as inverse magnetic mirror configurations. Plasma plumes expand along the magnetic field axis and are coaxial (⁠ $u∥B$⁠). The plasma accelerates and gradually escapes the magnetic field, becoming supersonic fluid. Compared to the de Laval nozzle, in the magnetic nozzle, the radial confinement of the plasma is not provided by solid metal walls but by a magnetic field structure, thereby avoiding issues such as wall ablation and wall flow losses that occur during prolonged operation. However, the interaction between the magnetic field and the plasma under real conditions is significantly more complex than the interaction between solid walls and high-temperature combustion gases. The plasma contains multiple charged components, including heavy ions and electrons, which exhibit significant electrostatic and viscous interactions. These interactions between particles, as well as the electromagnetic interactions between particles and the magnetic field, together constitute the flow expansion process of the plasma within the magnetic nozzle. In this context, the interaction between the propellant and the nozzle no longer involves only thermal pressure but also includes electromagnetic forces between the magnetic field and the charged particles. Importantly, due to the closed nature of the magnetic field lines, the magnetic field lines at the edges of the magnetic nozzle curve upstream after expansion and return to the spacecraft itself. As a result, the electromagnetic interaction of the plasma at the downstream region becomes a negative effect that impedes the detachment of the plasma from the magnetic field. From a research perspective, this operational model presents many challenges, making it difficult to directly analyze the effective surface and flow loss factors of the magnetic nozzle.

Over the past five decades, thanks to the continuous innovation and breakthroughs in computational efficiency and experimental techniques, our understanding of the internal physical mechanisms within magnetic nozzles has made significant progress. We gradually unraveled the mechanisms behind particle acceleration and magnetic detachment within magnetic nozzles. Additionally, magnetic nozzles have been widely applied in various electric propulsion devices, such as applied-field magnetoplasmadynamic thruster (AF-MPDT), radio frequency sources, and vacuum arc thrusters, leading to numerous experimental and diagnostic studies. However, in practical research, we find that the conclusions regarding the acceleration and magnetic detachment mechanisms in magnetic nozzles are not entirely consistent, with some research results even presenting contradictory findings. Factors such as the plasma source, magnetic field strength, and magnetic field structure can all significantly impact the acceleration and detachment processes within the magnetic nozzle. Therefore, conducting a systematic review of the acceleration and detachment mechanisms of magnetic nozzles under various propulsion scenarios is crucial for the understanding and optimization of the magnetic nozzle.

This review will outline the latest advancements in the research on acceleration and magnetic detachment within magnetic nozzles, as well as the challenges and issues that magnetic nozzle research still faces. The purpose of this study is to contribute to the literature on magnetic nozzles by not only describing the fundamental principles and well-established physics of PMNs but also by introducing new insights, methodologies, and concepts brought about by modern research approaches. Additionally, it will offer some new perspectives on the future development of magnetic nozzle research. The structure of this article is as follows. In Sec. II, we first introduce the definition of the PMN and its applications and development trends in different plasma sources. Then, in Sec. III, we summarize the acceleration mechanisms in PMNs, categorizing them into ambipolar electric field acceleration, generalized Hall acceleration, and inverse magnetic mirror acceleration. In Sec. IV, we review the development process of magnetic detachment mechanisms in PMNs, focusing on the disagreements among different researchers in collisionless detachment studies. Finally, in Sec. V, we conclude this review and connect the research with broader applications in PMNs, providing suggestions for future research directions.

Since 1969, when Anderson first demonstrated experimentally that a convergent-divergent coaxial magnetic field could produce supersonic plasma flows, researchers around the world have combined various plasma sources or electric thrusters with magnetic nozzles over the past half-century. Given that the magnetic field itself cannot do work, the premise for magnetic nozzles to have propulsive significance is that the plasma at the inlet cannot be strictly cold plasma. Only when the plasma has a certain amount of internal energy can the magnetic nozzle function effectively. The concept of the propulsive magnetic nozzle (PMN) was proposed by the teams of Merino and Ahedo.¹⁰ They argued that the plasma at the inlet of the magnetic nozzle should at least possess a finite electron temperature, i.e., $Te$ > 0. For different plasma sources, the composition, density, temperature, flow velocity, magnetic induction, and other parameters of the plasma in the magnetic nozzle can vary, and these differences may affect the physical mechanisms of plasma acceleration and detachment from the magnetic field within the nozzle.

To better evaluate and analyze the physical mechanisms within the magnetic nozzle, we have compiled the historical development of plasma sources applied in PMNs. Given the multitude of thrusters that can leverage magnetic nozzles for acceleration, as well as the countless experiments involved, this paper focuses on the inventory of representative PMNs applications that are closely related to magnetic nozzle mechanisms. Table I lists representative PMN studies in chronological order.

In the period from the 20th to the early 21st century, research on magnetic nozzles primarily focused on electrode discharge-type propulsion devices. These devices are characterized by high propellant flow rates and the capability to operate in high-power pulsed or steady-state discharge modes. The main representatives of this type of electric propulsion include arc heater and AF-MPDT. Between the two, the AF-MPDT, as a high-power electric propulsion system, continues to be a subject of extensive research due to its potential for enhanced performance.

In 1970, Kuriki et al.¹¹ studied the flow characteristics of plasma generated by arc heaters in a converging-diverging magnetic nozzle. They found that the acceleration of ions downstream of the magnetic nozzle occurs partly through electrostatic acceleration and partly through aerodynamic acceleration. They also proposed that the energy equation for ions and electrons is interconnected through the electric potential. These findings provided early insights into the role of magnetic nozzles in plasma dynamics. After the 1960s and 1970s, the development of AF-MPDT began. AF-MPDT are equipped with magnetic nozzles, and the additional magnetic field not only significantly influences the discharge process within the discharge chamber but also enhances the propulsion performance through the acceleration effect of the magnetic nozzle. In 1991, Myers et al.¹² conducted diagnostic studies on the magnetic nozzle plumes of several AF-MPDTs. They used measurement methods such as optical emission spectroscopy, quantitative imaging, and electrostatic probes. The results showed that the electron density and temperature in the plume, as well as the electric field structure, are highly sensitive to the presence of the magnetic nozzle. The addition of the magnetic nozzle strongly constrains the plume, and calculations of the Hall parameter revealed a strong coupling between the plasma flow and the magnetic field lines. However, in the downstream region of the magnetic nozzle, the density distribution of the plume indicated that the flux surfaces of the plasma do not fully align with the magnetic flux tubes, suggesting that the plume can effectively detach from the magnetic nozzle, though the specific reasons for this detachment were not discussed.

The performance and mechanisms of AF-MPDT were further analyzed in subsequent studies. In 1992, Sasoh et al.¹³ analyzed the thrust generation mechanism of AF-MPDT based on performance tests, and they found that the high performance of AF-MPDT is closely related to the significant electromagnetic effects produced by the additional magnetic field. In 2002, Inutake et al.¹⁴ studied the influence of the magnetic nozzle on the MPD plume using pulse-type magnetic plasma arc (MPD Arcjet) discharge. They measured the electron temperature and density using Langmuir triple probes and the ion Mach number using a Mach probe. The experimental results showed that, compared to the operation of the MPD Arcjet alone, the addition of the magnetic nozzle increased the ion Mach number from 1 to nearly 3. The distribution of the Mach number revealed that the plasma expansion and energy conversion processes in the MPD magnetic nozzle are more consistent with isentropic processes rather than magnetic moment conservation.

Early explorations of magnetic nozzles in electrode discharge propulsion applications have validated the potential of magnetic nozzle acceleration. These studies collectively confirm the critical role of magnetic nozzles in high-power electric propulsion systems, particularly in enhancing plasma acceleration and plume confinement. Given that electrode discharge thrusters at the time commonly employed high-power discharge modes, their ion temperatures may have been significant compared to electron temperatures, which likely contributed to the early observations by Kuriki et al. of ion aerodynamic acceleration within magnetic nozzles.

Entering the 21st century, the plasma sources used in magnetic nozzle experiments have gradually transitioned from large electrode discharge devices to electrodeless discharge devices represented by radio frequency (RF) devices. There is a significant variation in the operating conditions of electrodeless discharge devices. For example, RF and electron cyclotron resonance (ECR) plasma sources typically have power levels below 1 kW, while the helicon plasma thruster (HPT) operates at power levels ranging from kW to tens of kW. There are even expectations for power levels in the hundreds of kW to MW in the Variable Specific Impulse Magnetoplasma Rocket (VASIMR). In this new era, the application and research of magnetic nozzles exhibit diverse trends, and a rich variety of mechanisms underlying magnetic nozzle operations are being gradually unveiled.

The HPT or helicon double layer thruster (HDLT) uses helicon wave discharge to produce high-density, highly ionized plasma.^49,93,94 HDLT is a special type of HPT that can form a current-free double layer (CFDL) structure at the end of the discharge chamber,⁹⁵ which is beneficial for plasma acceleration. Takahashi and his team have been dedicated to experimental research on magnetic nozzles with RF sources, starting from 2007. They have conducted extensive studies on magnetic nozzles, focusing on electron thermodynamics and transport processes within the nozzle,^{27,34,39,41,42} as well as azimuthal currents and induced magnetic fields.^32,35,96 In collaboration with Imai et al., they have explored the mechanism of thrust vector deflection using asymmetric magnetic nozzles.^43,97,98 Additionally, Takahashi et al. have worked with Lafleur and Boswell to conduct magnetic nozzle experiments using permanent magnet HDLT, measuring and studying the mechanism of electromagnetic thrust generation.^26,30,32 Experimental findings indicate that the thrust of HDLT consists of two components: electron pressure in the helicon source and Lorentz force generated in the magnetic nozzle by electron diamagnetic drift current and magnetic field.²⁶ 

Between 2013 and 2016, Little et al. at Princeton University^54,56,58 also conducted experimental studies on magnetic nozzle plumes using a helicon source. They discovered a potential well that constrained the ions. When this potential well disappeared downstream, the magnetic nozzle lost its constraining effect on the plasma. Over the past two decades, numerous researchers have uncovered many new mechanisms of charged particle transport within magnetic nozzles in the context of HPT applications. These findings are crucial for understanding the acceleration and detachment processes in magnetic nozzles.

The VASIMR is another well-known application of the PMN. VASIMR consists of three stages: a helicon source, an ion cyclotron resonance heating (ICRH), and a magnetic nozzle.⁹⁹ The ICRH stage is capable of heating ions to temperatures far exceeding the electron temperature. For a fully operational three-stage VASIMR, Glover et al.¹⁸ demonstrated that heating ions using ICRH significantly increases the ion velocity within the magnetic nozzle. When using low atomic mass propellants such as deuterium, the specific impulse of VASIMR increases almost linearly with the power of the ICRH.¹⁰⁰ This indicates that as ICRH power increases, the proportion of ion thermal energy contribution in the acceleration process within the magnetic nozzle also rises. This finding highlights the importance of ion heating in enhancing the performance of PMNs, particularly in achieving higher exhaust velocities and specific impulses.

Since 2019, the application of PMNs has exhibited a trend toward miniaturization, with plasma sources adopting lower-power technologies such as electron cyclotron resonance (ECR), inductively coupled plasma (ICP), and micro cathode arc thrusters (⁠ $μ$-CAT). In 2019, Correyero et al., in collaboration with the French Aerospace Laboratory,⁷² investigated the relationship between voltage drop and flow rate in ECR magnetic nozzles while also measuring electron cooling rates. Collard, Wachs, and Sheehan et al. have been studying the magnetic nozzles of radio frequency (RF) thrusters since 2014, exploring mechanisms such as ambipolar acceleration,¹⁰¹ particle detachment,⁶⁹ nozzle efficiency,⁷⁰ and anomalous electron transport.¹⁰² Since 2020, an increasing number of researchers have joined the study of magnetic nozzles. Sekine et al. began investigating magnetic nozzles in inductively coupled RF discharge mode thrusters,^79,80 while Boni et al.^88,89 focused on the electron thermodynamics in ECR magnetic nozzles. In 2024, Qi et al.⁹² applied permanent magnet magnetic nozzles to $μ$-CAT, utilizing triple probes to measure the thruster plume and successfully demonstrating the effectiveness of magnetic nozzles in electron cooling and thrust enhancement.

From the review of these experimental studies, certain patterns and trends can be summarized. Early experimental research on magnetic nozzles often did not employ plasma sources commonly used in electric propulsion, nor were they intended for propulsion purpose. From the 1990s to the early 2000s, experimental research on magnetic nozzles began to focus on plasma sources or thrusters utilizing arc discharges. These plasma sources featured high discharge power and strong ionization capabilities, thus typically using lighter propellants such as $H2$⁠, He, and Ar. In the 21st century, with the development of electrode-less plasma sources, the application of magnetic nozzles has diversified. High-power thrusters such as HPT and VASIMR continue to use traditional inert gases such as Ar as propellants, while miniaturized RF and ECR thrusters have shifted toward more efficient but costlier propellants such as Xe. Up until now, magnetic nozzles have found extensive application in the domain of electric propulsion, driven by advancements in experimental methodologies and diagnostic technologies. Nonetheless, the widespread adoption of magnetic nozzles is also contingent upon the rapid progress in theoretical investigations elucidating the complex physical processes involved, including plasma acceleration and detachment.

The acceleration of plasma in magnetic nozzles depends on the magnetic confinement of the plasma. Generally, in PMNs, electrons should at least remain magnetized, while ions are typically in an unmagnetized or finitely magnetized state, depending on the magnetic confinement. Research on the acceleration mechanisms in magnetic nozzles began in the 1960s when Kosmahl¹⁰³ applied Gerwin's model and found that if the initial plasma beam diameter did not exceed one-fifth of the electromagnetic solenoid diameter, the applied magnetic field could potentially confine the plasma.^104,105 Due to the limitations of experimental techniques at the time, Bowditch probe diagnostics could not accurately measure the plasma characteristics within magnetic nozzles, making it impossible to compare theoretical simulations with experimental results.¹⁰⁶ 

In the 1990s, with the rapid development and application of various electromagnetic plasma acceleration devices, research on magnetic nozzle acceleration mechanisms attracted increasing attention from researchers. From our review of magnetic nozzle applications, we observe that electric propulsion devices employing magnetic nozzles span six orders of magnitude in power, from watts to megawatts. As power increases, the plasma energy at the magnetic nozzle inlet grows, usually transitioning the magnetic nozzle from being electron-driven to ion-driven, accompanied by changes in the internal acceleration mechanisms. In this chapter, we will review the history of research on plasma acceleration mechanisms within magnetic nozzles, analyze the primary acceleration mechanisms, and discuss their applicable ranges.

Ambipolar electric fields are electric potential structures generated due to the separation of ions and electrons in plasma, resulting in non-uniform plasma distribution. They reflect the interactions between ions and electrons. Therefore, the essence of ambipolar electric field acceleration is not to increase the overall kinetic energy of the plasma but an ion acceleration mechanism. Since ions and electrons in the plasma are always constrained by the electric field and move together, the plasma can diffuse downstream of the magnetic nozzle only after the ions are accelerated. Therefore, we believe that understanding the ion acceleration mechanism is also an important part of the plasma acceleration process. From 2004 to 2008, Arefiev and Breizman et al.^107,108 established a two-dimensional electromagnetic field model based on the VASIMR magnetic nozzle configuration to study the motion patterns and energy conversion of charged particle flows. Through theoretical analysis, they proposed that the thermal energy of electrons can be converted into the kinetic energy of ions through ambipolar electric fields, which is referred to as ambipolar electric field acceleration. In 2011, Longmier et al.¹⁰⁹ set up the VX-200i experimental system, as shown in Fig. 2, using Ar (argon) with a mass flow rate of 25 mg/s and an electron temperature of 20 eV as the propellant for thrust generation. They measured the potential drop structures within the magnetic nozzle with scales ranging from $104$ to $105$ Debye lengths (⁠ $λD$⁠). The maximum plasma velocity was approximately 4.1 times the speed of sound, verifying the conclusion that ion velocity indeed increases continuously due to the influence of ambipolar electric fields.^51,109 Additionally, the decrease trends of the electron temperature and the plasma potential were basically consistent.

In 2010, Ahedo et al.¹⁰⁴ used two-fluid simulation model with ion unmagnetized or finitely magnetized to further validate Arefiev's theory on ambipolar electric fields. In 2016, Ahedo et al. modified the previous assumptions of unmagnetized ions limit (UMIL) or finitely magnetized ions limit conditions, pushing the study of magnetic nozzle acceleration mechanisms toward theoretical limits unreachable in conventional environments. They investigated the expansion and acceleration process of fully magnetized ions limit (FMIL) plasma in magnetic nozzles. They found that the differences in plasma characteristic parameters mainly occurred in the edge regions of the beam, and the generated thrust was similar to that under ion unmagnetized conditions. The primary factor for ion acceleration remained the influence of ambipolar electric fields.¹¹⁰ Numerous research results indicate that ambipolar electric field acceleration is widely present in PMNs. As shown in Fig. 3, it illustrates the schematic of ambipolar electric field acceleration in a magnetic nozzle. Whether ions are in UMIL or FMIL state, as long as there are rapidly moving electrons and slowly moving ions in the magnetic nozzle, ambipolar electric field acceleration will occur.

The electrostatic double layer acceleration can be viewed as a special type of ambipolar electric field acceleration. It involves the CFDL structure, which is a sudden drop in electric potential at the exit of a helicon source under certain conditions, occurring over a scale of tens of Debye lengths.¹¹² However, the appearance of the CFDL requires certain prerequisites, and such structures can not always be observed at the outlets of all helicon sources. Charles and Boswell were the first to relate this potential structure to the downstream double-peaked ion energy distribution function (IEDF).¹¹³ This type of IEDF distribution has been observed in many helicon source experiments.^95,114–116 As shown in Fig. 4(a), a typical CFDL potential structure as measured by Takahashi et al. is depicted, and the ion energy distribution function obtained by measuring IEDF using techniques such as RFEA is shown in Fig. 4(b). Downstream of the CFDL, the plasma consists of two ion components: a high-speed beam on the right and a low-energy thermal ion group on the left.

With changes in background pressure or working medium types, the shape of CFDL may change,⁴² or it may not be observable. Lieberman, Charles, and Takahashi have conducted detailed research into the mechanism of CFDL formation, suggesting that its formation is related to appropriate magnetic field strength, pressure, and thruster exit size.^45,117–119 CFDL exhibits a close relationship with HPT, and its close proximity to the plasma source makes it a distinctive acceleration mechanism in the upstream region of the HPT magnetic nozzle. The process of electrostatic double layer acceleration relies on electric fields to fundamentally convert electron thermal energy to ion kinetic energy, thereby aligning it with the broader category of ambipolar electric field acceleration in this context. However, since the physical scale of the CFDL is much smaller than that of the conventional ambipolar electric field shown in Fig. 2, the upstream and downstream plasma potentials of the CFDL can be diagnosed using an electrostatic probe, making it relatively convenient to predict the ion beam energy.

Azimuthal currents are commonly present in magnetic nozzles, and under the influence of divergent magnetic field, they generate significant axial Lorentz forces, which in turn accelerate charged particles. If all azimuthal currents are considered as generalized Hall currents,¹²⁰ this acceleration mechanism can be referred to as generalized Hall acceleration. Since the magnetic field itself cannot do work directly on the plasma, generalized Hall acceleration is fundamentally about converting non-axial kinetic energy of the plasma into an axial one.

An electron-driven magnetic nozzle refers to a PMN where the ion temperature $Ti0$ at the inlet of the magnetic nozzle is much lower than the electron temperature $Te0$ (i.e., $Ti0≪Te0$⁠). In such a scenario, the energy responsible for the acceleration of the plasma within the magnetic nozzle primarily originates from the electrons. Furthermore, the azimuthal induced current within the magnetic nozzle is predominantly carried by the electrons.^96,121 Figure 5 illustrates the principle of generalized Hall acceleration, where electrons undergoing diamagnetic drift are radially confined and axially accelerated by Lorentz forces. However, they still require the effect of ambipolar electric fields to further accelerate ions. From this perspective, ambipolar electric field acceleration includes multiple mechanisms such as axial electron thermal expansion and generalized Hall acceleration. The difference is that the former is driven by the gradient of electron pressure in the $B∥$ direction, does not require the involvement of magnetic fields, and is essentially an electrothermal acceleration. On the other hand, generalized Hall acceleration requires the participation of magnetic fields and falls under electromagnetic acceleration. Hence, it is usually considered an independent acceleration mechanism in magnetic nozzles.

In the same year, Takahashi employed a Hall probe (HP) to measure the azimuthal plasma currents in a 1 kW-class HPT magnetic nozzle and compared the results with theoretical calculations. As depicted in Fig. 6, incorporating both the diamagnetic drift current and the E × B drift current provides a better match with the experimental data. Moreover, the study's findings indicate that as the magnetic field strength increases, the E × B drift current significantly decreases. The reason for this is that under ion magnetization conditions, the E × B drift acts simultaneously on both ions and electrons, leading to a cancelation of the azimuthal current contributions from the E × B drift term.^96,114

In 2021, Chen¹²¹ and Hu,¹²³ building upon Takahashi's work, theoretically extended the azimuthal current model to include four components and conducted full-particle simulation validation. Their results indicate that the inertial-induced azimuthal current and viscous-induced azimuthal current cannot be neglected in the downstream region of the magnetic nozzle. Figure 7 shows the axial distribution of the four Hall acceleration components under 0.75 T (scaled-down transformation). The study found that for an RF thruster magnetic nozzle with $Ti0≪Te0$⁠, the diamagnetic drift effect is the dominant mechanism for Hall acceleration, contributing more than 90% to the process. In contrast, the E × B drift current, which is primarily paramagnetic, contributes negatively to Hall acceleration, consistent with Takahashi et al.'s measurements. Additionally, electron inertial effects provide a relatively minor positive contribution, while viscous effects offer a secondary negative contribution. Overall, the upstream region of the magnetic nozzle symbolizes the Hall acceleration phase, whereas the downstream region encounters a deceleration phase due to the influence of the closed magnetic field. The deceleration and detachment of downstream plasma will be discussed in detail in Chapter IV.

In high-power PMNs applications such as AF-MPDT and VASIMR, where ion temperature is no longer negligible, it significantly affects the azimuthal current in the magnetic nozzle. This type of magnetic nozzle, which requires consideration of ion temperatures, is referred to as an ion-driven magnetic nozzle.⁵⁹ Compared to electron-driven PMNs, ion-driven PMNs exhibit unique plasma structures such as high density conics and radial electric potential barriers.^46,124 In 2015, Merino et al. introduced a two-fluid model to explain the generation of radial electric fields in magnetic nozzles that counteract ion pressure.¹²⁵ They proposed that as the ion temperature increases, the radially expanding ions will lead to a relatively high electric potential near the MDML. As the radial electric field transitions from being positive along the radial direction to negative, which will likely cause the E × B drift current to change from paramagnetic to diamagnetic, altering its contribution to thrust from negative to positive.

In 2020, Chen et al. employed a Particle-In-Cell (PIC)model to re-analyze the radial electrostatic potential barrier and the E × B drift current in ion-driven PMNs under ion unmagnetization or ion finite magnetization state.¹²⁶ The full-particle simulation avoids the quasi-neutrality assumption inherent in fluid simulations, providing more physical data on the plasma distribution outside the MDML of the magnetic nozzle, as shown in Fig. 8(a). They proposed that the radial electric potential barrier in the magnetic nozzle arises from the radial separation of ions and electrons. The study's results indicate that as the magnetic field strength increases, the electrostatic potential barrier initially increases and then decreases [Fig. 8(b)]. During the process of magnetic field enhancement, the plasma undergoes a series of states: electron unmagnetization, electron finite magnetization, electron full magnetization, and ion finite magnetization. The radial separation of ions and electrons is initially enhanced by the magnetization of electrons, but it subsequently decreases as ions become magnetized. If the ion magnetization further increases to a fully magnetized state, both ions and electrons will be completely constrained to the magnetic field lines, causing the radial electric field separation to gradually diminish. In 2022, Andrews et al.¹²⁷ obtained computational results similar to those of Chen through PIC simulations under various boundary conditions, thereby validating the reliability of this analysis.

In summary, the azimuthal current is fundamentally a manifestation of the electromagnetic confinement of charged particles by the magnetic field. Electron generalized Hall acceleration requires the ambipolar electric field to influence ions, thereby representing the electromagnetic term in Eq. (2). Ion generalized Hall acceleration acts directly on the ions themselves, thus serving as the magnetic term in Eq. (1). Notably, in the majority of PMNs, ions are in a state of either unmagnetized or finitely magnetized conditions, with the azimuthal current in PMNs being predominantly driven by electron currents. Based on current research findings, we suggest that the diamagnetic E × B drift current could be a beneficial conversion mechanism for the radial kinetic energy of ions within the magnetic nozzle under ion finite magnetization.

Inverse magnetic mirror acceleration is another acceleration mechanism in magnetic nozzles, considered from the perspective of single particle motion. This mechanism originates from the well-known magnetic mirror physics.¹²⁸ Magnetic mirrors confine particles by converging magnetic fields, while magnetic nozzles accelerate particles through diverging magnetic fields. Under conditions of high magnetization of charged particles, movement perpendicular to the magnetic field reduces into cyclotron motion around the magnetic field lines. The magnetic moment of particles undergoing this cyclotron motion, expressed as $μm=mv⊥22B$⁠, is treated as an adiabatic invariant.¹²⁹ As illustrated in Fig. 9, within the divergent region of the magnetic nozzle, the magnetic field strength B along the axis decreases. However, since the magnetic moment $μm$ remains constant, the kinetic energy in the direction perpendicular to the magnetic field reduces, whereas the kinetic energy parallel to the magnetic field increases. This process effectively serves as an acceleration mechanism. It is worth noting that relatively strict magnetization conditions for charged particles need to be met when adopting the inverse magnetic mirror acceleration model. If only electrons are magnetized, the inverse magnetic mirror acceleration mechanism can only act on electrons. Ions need to obtain the kinetic energy of electrons through acceleration by the ambipolar electric field.

In this equation, $b̂$ represents the unit vector in the direction anti-parallel to the magnetic field. During the inverse magnetic mirror acceleration process, kinetic energy perpendicular to the magnetic field is converted into kinetic energy parallel to the magnetic field.

The detachment of plasma from the magnetic nozzle is a critical and inevitable scientific issue in PMNs. After being accelerated within the magnetic nozzle, if the plasma fails to detach from the magnetic field in a timely manner, it may return along closed magnetic field lines back to the thruster, leading to performance losses.

The fundamental nature of plasma detachment is the cross-field transport of charged particles, particularly that of the downstream plasma, which directly affects the thrust efficiency of the magnetic nozzle. Interestingly, the acceleration mechanism of the magnetic nozzle relies on its effective confinement of the plasma, while detachment inherently requires the nozzle to relax this confinement to some extent. These two aspects may appear contradictory, yet they coexist within the magnetic nozzle. For typical PMNs, the divergent magnetic field structure results in a gradual decrease in magnetic field strength along the axial direction. Therefore, from the perspective of engineering application, the magnetic nozzle should maintain strong confinement of the plasma in the upstream region and gradually reduce this confinement in the downstream region. Generally, the magnetic detachment mechanisms in PMNs can be categorized into two main types based on whether they depend on collisions: collision detachment and collisionless detachment. Research on these two mechanisms began in 1990s.

The collisional detachment mechanism in magnetic nozzles arises from mutual collisions and energy transfer between charged particles. Currently, collisional detachment in magnetic nozzles can be categorized into two types: resistive diffusion detachment and charge-related collision detachment.

In 1992, R.W. Moses theoretically demonstrated that an increase in plasma resistivity could enhance the collision frequency between charged particles, thereby facilitating detachment.¹³² This finding suggests that resistive diffusion of plasma is a potential mechanism for plasma detachment from the magnetic field. Due to the collective effects of plasma, the transverse resistive diffusion of plasma must consider both classical diffusion and Bohm diffusion. Chen¹³³ pointed out that the classical diffusion coefficient scales as $kTe−1/2/B2$⁠, while the Bohm diffusion coefficient scales as $kTe/B$⁠, making the former more sensitive to magnetic field strength. Consequently, the classical diffusion coefficient increases rapidly as the magnetic field weakens and electrons cool. However, he also noted that the Bohm diffusion coefficient is typically several orders of magnitude higher than the classical diffusion coefficient, implying that the dominant resistive diffusion mechanism in the magnetic nozzle may change as expansion proceeds.

As a detachment mechanism, resistive diffusion relies on particle collisions and thus is often effective in the upstream region of the magnetic nozzle, where the collision frequency is relatively high. This, however, conflicts with the strong confinement required by the acceleration mechanism of the magnetic nozzle. Therefore, resistive diffusion is widely regarded as an inefficient means of achieving plasma detachment and may adversely affect thrust performance.

Another collisional detachment mechanism that may need to be considered is charge-related collision, which includes mechanisms such as recombination collisions and charge exchange collisions. Recombination collision involves accelerated ions undergoing three-body recombination collisions to form high-speed neutral particles, allowing them to escape the influence of the magnetic field. For recombination collision to be considered an effective detachment method, it requires a sufficiently high electron-ion collision frequency. However, the collision frequency on which this mechanism relies is influenced by the electron temperature in the magnetic nozzle.⁵⁸ Experimental and numerical simulation studies have demonstrated^{11,134 135} that under typical operating conditions in PMNs, the distance over which electron cooling occurs is insufficient to generate a significant number of three-body recombination collisions. In 2009, Deline et al.^136,137 used a 200 kW plasma source to simulate the VASIMR discharge process and conducted diagnostic measurements using the electrostatic probes. The nearly constant ion current measured in the plume indicates that the recombination collision mechanism for detachment does not hold. In addition to the recombination collision mechanism, Cohen and Paluszek¹³⁸ proposed in 1998 that accelerated ions could be converted into atoms through three-body recombination collisions to detach from the magnetic field. Charge exchange collisions transfer the kinetic energy of fast-moving ions to neutral particles, thus enabling detachment. However, this mechanism fundamentally fails to reduce the total number of ions in the plasma, thereby much the remaining low-speed charged particles will be trapped by the magnetic field. For plasmas with a high degree of ionization, this collision mechanism cannot be considered an effective means of detachment.

In summary, research on collisional detachment in magnetic nozzles is currently limited, and there is relatively low enthusiasm in the magnetic nozzle research community for studying plasma collisional detachment mechanisms. Given the characteristics of propulsive magnetic nozzles, such as decreasing plasma density along the axial direction, electron cooling along magnetic field lines, and ion acceleration along the axial direction, regions with a high collision frequency are primarily concentrated in the upstream area of the magnetic nozzle. Considering the inherent operational characteristics of magnetic nozzles, we do not anticipate detachment to occur in the upstream region but are more concerned with the detachment state in the downstream section. Athough the collisional detachment theory is not applicable to the downstream detachment phase of magnetic nozzles; however, it may serve as one of the factors undermining the magnetic confinement in the upstream region and the acceleration efficiency of the magnetic nozzle. Therefore, study on this mechanism remains significant.

Collisionless detachment mechanisms in magnetic nozzles arise from the interaction between the magnetic field and the plasma. Between 1990 and 1993, Sercel and colleagues conducted research on low-power ECR thrusters and discovered that detachment could occur within the magnetic nozzle even in the absence of significant collisions. Over the past three decades, collisionless detachment research has gradually become the main focus in the study of detachment mechanisms in PMN. Collisionless detachment disregards collisional effects and instead emphasizes the interactions between charged particles and the magnetic field. From the perspectives of both the change of the original magnetic field configuration and the magnetized state of the charged particles, collisionless detachment can be categorized into two types: induced magnetic field detachment and demagnetization detachment.

Induced magnetic field detachment is a mechanism by which plasma currents either enhance or cancel local magnetic fields, thereby influencing the motion of charged particles. In 2002, Ilin and colleagues investigated the detachment mechanism in the VASIMIR magnetic nozzle. Their results showed that the thermal plasma dynamic pressure in the magnetic nozzle continuously increases until the ratio of dynamic pressure to magnetic pressure, $βk$⁠, exceeds 1, at which point the detachment phenomenon becomes significant.¹³⁹ Between 2006 and 2009, C.A. Deline and others conducted magnetic nozzle detachment demonstration experiments (DDEX) using a plasma gun as the plasma source.^{136,137 140} They employed various intrusive diagnostic methods, including Langmuir triple probes, Faraday probes, and B-dot probes to measure the macroscopic properties of the plasma. The experiments observed the occur of detachment and suggested that the magnetic frozen effect, induced by super-Alfvénic flow (⁠ $βk$ > 1), is the primary mechanism for the detachment of charged particles from the magnetic nozzle. However, the experiments did not measure the magnetic field line extension predicted by the magnetic freezing theory. Winglee and colleagues used MHD simulations to study the acceleration mechanism of the high power helicon.²¹ They found that when the velocity of the high-speed magnetized plasma flow exceeds the local Alfvenic speed, the magnetic frozen effect occurs. As shown in Fig. 10, the magnetic field lines extend downstream under the influence of the paramagnetic azimuthal current.

From 2005 to 2008, Arefiev and Breizman et al.,^141,142 along with their collaborators, developed a steady-state flow model of cold plasma to study the detachment mechanisms in the magnetic nozzle. Their theoretical research results indicated that the critical condition for plasma detachment in the magnetic nozzle is $βk$ > 1. This finding supports the experimental findings of Deline¹³⁷ and Winglee.²¹ When the plasma dynamic pressure exceeds the magnetic pressure, the plasma induces a paramagnetic current within the magnetic nozzle, thereby extending the magnetic field lines downstream and facilitating the escape of the charged particles.

From 2010 to 2014, Merino and Ahedo et al.^{134,135,10,105,106} conducted research on the detachment mechanism in magnetic nozzles using numerical simulation methods. They introduced the concept of a propulsive magnetic nozzle, which accelerates plasma approaching sonic speed to enhance the thrust of the thruster, converting internal energy into directed kinetic energy. It is clear that the cold plasma magnetic nozzle studied by researchers such as Hooper, Arefiev, and Breizman cannot be classified as a PMN, as it lacks a propulsive significance. Based on the two-fluid model, Merino and colleagues found that the PMN generates thrust by inducing a diamagnetic azimuthal current, which is completely opposite to the direction of the azimuthal current obtained by Arefiev et al. under the assumption of cold plasma. Different from the $βk$ focused on by Ilin et al., Merino et al. used the ratio $β0$ of the thermal pressure to the magnetic pressure as a magnetic nozzle inlet parameter. Their research results show that when there is a certain $β0$ at the magnetic nozzle inlet, an induced magnetic field antiparallel to the original magnetic field will be generated in the magnetic nozzle.

To assess the specific direction of the induced magnetic field in the magnetic nozzle, Corr and Boswell et al.¹⁴³ measured the induced magnetic field in the plasma flow downstream of a HPT. They found that the original magnetic field was reduced by 2%, indicating the presence of a weak diamagnetic induced current in the downstream region. As shown in Fig. 11, in 2011, Roberson and Winglee et al.¹⁴⁴ measured the induced magnetic field downstream of a high-power HPT, finding that the magnetic field throughout the region was significantly weakened, with the reduction reaching 15 G, far exceeding previous studies. These research results suggest that the diamagnetic azimuthal current may play a significant role in the phenomenon of plasma detachment in magnetic nozzles.

We have noticed that there seem to be contradictory research results regarding the detachment caused by induced magnetic fields. Although Arefiev et al. proposed magnetic frozen detachment using an ideal cold plasma model, Deline and Winglee et al. observed magnetic frozen detachment in experiments of PMNs, suggesting that this mechanism may still be present in magnetic nozzles. It is worth contemplating that Merino and Arefiev et al. employed different inlet assumptions in their simulations of the magnetic nozzle: Merino used inlet conditions dominated by thermal pressure, whereas Arefiev used inlet conditions dominated by dynamic pressure with cold plasma. These different assumptions may lead to significant changes in the induced magnetic field in the magnetic nozzle. In 2017, Takahashi et al.³⁵ conducted an experiment using a 5 kW helicon source with a magnetic nozzle configuration. They observed a transition of the induced magnetic field from canceling the original magnetic field to enhancing it. This finding suggests that two seemingly conflicting phenomena of induced current might coexist within the application of magnetic nozzles.

In 2021, Chen et al.¹²¹ used a full-particle simulation model to calculate the plasma current throughout the magnetic nozzle, adopting inlet conditions similar to those used by Merino, with plasma parameters close to those of a conventional helicon source magnetic nozzle, a number density $ne=1018m−3$⁠, and electron temperature $Te=5$ eV. As shown in Fig. 12, the calculation result indicates the presence of significant diamagnetic azimuthal currents near the MDML at the inlet of the magnetic nozzle, while the currents in the downstream region of the magnetic nozzle are paramagnetic. This phenomenon can be analyzed using Eq. (3). When the plasma temperature at the inlet of the magnetic nozzle cannot be ignored, the induced azimuthal current in the upstream part of the magnetic nozzle is dominated by the diamagnetic drift term. As the plasma pressure gradually decreases in the radial direction, the induced magnetic field is oriented in the opposite direction to the original magnetic field. Moving to the downstream region of the magnetic nozzle, the internal energy of the electrons has almost been fully converted. The conditions here are similar to the assumptions made by Arefiev et al., where the plasma has low internal energy but high kinetic energy. In this case, the induced azimuthal current in the magnetic nozzle is dominated by the E × B drift. Given that the radial electric field of the plasma is positive, the induced magnetic field has the same direction as the original magnetic field. This discovery reveals the transformation mechanism between diamagnetic and paramagnetic induced current phenomena in PMNs. Overall, in typical PMNs, the direction of the induced magnetic field is likely to change as the internal energy of the plasma is consumed. This makes it difficult to accurately predict the role of the detachment of the induced magnetic field, and thus offering limited insights for the quantitative design of PMNs.

Demagnetization detachment refers to the phenomenon where particles become detached due to the weakening of magnetic confinement. This understanding is clear, but the acceleration mechanism of the magnetic nozzle relies on particle magnetization to function, making it very difficult to balance particle acceleration and demagnetization detachment in PMNs. The Larmor radius of charged particles and the characteristic length of magnetic field changes are the most commonly used conditions to quantify the extent to which charged particles are magnetized.¹⁴⁵ However, demagnetization is significantly different for each type of particle in the plasma, with electrons being more likely to remain magnetization compared to heavy ions in the magnetic field. Nonetheless, demagnetized ions still couple with magnetized electrons through the ambipolar electric field, so it is difficult to achieve plasma detachment by relying on the demagnetization of ions alone.

In 2019, Little and Choueiri et al.⁵⁷ utilized RF compensation probes, emissive probes, and Langmuir probes to measure the region downstream of the HPT magnetic nozzle, covering a −20° to 90° sector centered around the magnetic nozzle axis. As shown in Fig. 14, they measured a radially negative constraining electric field at the edge of the magnetic nozzle for the first time. In this case, as the coil current increased, the region where electrons can be considered magnetized extended outward, reaching the turning point. At this point, the magnetic field prevented the lateral movement of electrons, while the motion of ions was essentially unaffected. Therefore, ions at the plasma edge with a transverse field velocity might exceed the MDML interface, leading to the accumulation of positive space charges and the formation of a region with $E⊥$ < 0. This experimental result validates the influence of electron demagnetization on plasma detachment in the magnetic nozzle and describes the electron demagnetization phenomenon caused by the finite electron Larmor radius (FELR) effect. Based on this finding, Little et al. further proposed a critical magnetic field strength design criterion based on the demagnetization effect, ensuring that electrons remain magnetized before the MN turning point. In 2020, Chen et al.¹²⁶ expanded and supplemented the study of this mechanism using a PIC model, their computational results supported the experimental conclusions of Little et al.

As shown in Fig. 15, their research results indicate that electrons exhibit a trend of first outward and then inward detachment within the magnetic nozzle. They pointed out that the outward detachment of electrons in the upstream region of the magnetic nozzle is primarily driven by the inertial effect of electrons. In contrast, the inward detachment of electrons downstream of the magnetic nozzle is the combined effect of viscous and inertial detachment. The study attributed the viscous detachment mechanism in the downstream region of the magnetic nozzle to the FELR (finite electron larmor radius) effect, stating that FELR electrons orbiting around adjacent magnetic field lines exchange momentum through gyro-viscosity, leading to changes in their velocity vectors. Chen et al.'s work further refined the understanding of the electron demagnetization detachment mechanism, providing a more detailed interpretation of how electron demagnetization causes inward and outward detachment of plasma in the downstream region of the magnetic nozzle. We argued that viscous detachment is fundamentally a form of inertial detachment, albeit driven by the electron's gyration inertia rather than their macroscopic motion inertia. Based on this reasoning, all detachment mechanisms that rely on ion or electron demagnetization were collectively termed demagnetization detachment.

Unlike induced magnetic field detachment, demagnetization detachment possesses significant quantifiable attributes. Despite the absence of universally recognized design criteria for PMNs based on demagnetization metrics for ion, electron, or combined particle systems, there is a burgeoning research community dedicated to this field. Consequently, it is plausible that standardized design methodologies for PMNs could be obtained in the near future.

In this review, we present a historical perspective on the research into plasma acceleration and detachment mechanisms within the PMN. Over the past five decades, PMNs have been widely employed in electric propulsion systems spanning a power range from 1 W to 100 kW. The PMN's capability to adapt its magnetic field to accommodate different plasma environments, coupled with its strong confinement properties that enable high thrust density, has made it particularly attractive for high-power electric propulsion systems. Additionally, the simplicity of its structure and characteristics such as quasi-neutral diffusion offer potential for miniaturization. Applications of PMNs in low-power electric propulsion systems are currently in the exploratory stage. Extensive experimental and diagnostic studies have revealed a wide range of physical phenomena in PMNs under various plasma parameter conditions, including CFDL, inductive current effect, and radial electrostatic barrier. These phenomena have significantly advanced our understanding of the operational mechanisms of magnetic nozzles and their engineering design.

Plasma acceleration and detachment are crucial for enhancing the efficiency of PMN applications. The former critically determines how plasma converts internal energy into axial kinetic energy through electromagnetic interactions, while the latter determines whether the plasma can effectively escape after acceleration. The research into plasma acceleration and detachment mechanisms in PMNs has evolved alongside advancements in experimental diagnostic techniques and numerical simulation methods over the past two decades. Through fluid and PIC simulation methods, various acceleration mechanisms within magnetic nozzles have been increasingly correlated. For ambipolar electric field acceleration, the academic community's research and understanding have converged. This mechanism essentially involves energy transfer between electrons and ions, where electrons consume thermal energy to overcome the electric field, gradually accelerating the ions. In magnetic nozzles dominated by electron energy, ambipolar electric field acceleration is the sole energy source for ion acceleration, encompassing both the axial thermal expansion of electrons and the generalized Hall acceleration effect resulting from radial thermal expansion. Therefore, the key to the development of low-power PMNs lies in enhancing the electron energy within the plasma source, with ECR serving as a notable successful example. For high-power PMNs, as magnetic field strengths and ion temperatures increase, the acceleration mechanisms become highly complex. While ambipolar electric field and electromagnetic acceleration remain important, the thermal expansion effects of ions cannot be overlooked. The particle transport processes in high-power PMNs must be analyzed in the context of specific plasma and magnetic field conditions.

Compared to acceleration mechanisms, research on detachment mechanisms in magnetic nozzles exhibits significantly more divergence. Initially, researchers believed that the detachment of plasma from the magnetic field in magnetic nozzles was primarily governed by mechanisms such as resistive diffusion and Charge-related Collisions. However, in the latter half of the 20th century, extensive experimental and simulation data revealed that collision processes are insufficiently generated in PMNs under realistic conditions. The key to understanding the induced magnetic field detachment mechanism lies in the recognition of azimuthal induced currents within the magnetic nozzle. Early studies on magnetic nozzles predominantly employed cold plasma models, where super-Alfvénic plasma jets with high $βk$ generated paramagnetic induced currents in the magnetic nozzle, leading to magnetic freezing effects and facilitating plasma expansion downstream. However, subsequent research on plasmas with initial thermal pressure demonstrated that the induced currents in magnetic nozzles exhibited significant diamagnetic characteristics, resulting in the cancelation rather than the extension of the magnetic field. With the continuous advancement of computational efficiency, PIC models based on first principles have provided more detailed insights into the transport mechanisms of plasma in magnetic nozzles. As the thermal energy of the plasma dissipates, the induced currents within the magnetic nozzle will sequentially undergo both diamagnetic and paramagnetic processes. It is possible that both mechanisms coexist within the detachment mechanisms of the magnetic nozzle. Research on the demagnetization detachment mechanism has evolved through three stages: from the demagnetization of approximate particles to the demagnetization of ions and electrons. Although some studies suggest that ion demagnetization alone may result in a partial decoupling of the plasma from the magnetic field, more research indicates that limited electron demagnetization can enhance the efficiency of magnetic nozzles. It is crucial to recognize that the demagnetization of charged particles in a magnetic nozzle is a continuous process rather than a sudden event. The demagnetization of ions and electrons occurs continuously along the magnetic field lines, though the extent varies between the two. Compared to inductive current detachment, demagnetization detachment remains a more quantifiable metric and may become an important reference for the design of magnetic nozzles in the future.

Considering the vast application potential of PMNs in the realm of electric propulsion, advancements in the mechanisms of acceleration and detachment within magnetic nozzles are crucial for achieving reliable and efficient designs. To facilitate ongoing progress in the study of charged particle transport, plasma acceleration, and detachment mechanisms within magnetic nozzles, a range of challenges and opportunities for future research are delineated herein.

• Investigating the transport mechanisms of plasma with finite magnetic confinement at the throat of magnetic nozzles may provide theoretical support for CFDL phenomena in HDLT and electron energy filtering in ECRT.

• Further investigation into the conversion mechanisms of ion non-axial energy in magnetic nozzles, particularly under conditions of finite ion magnetization, could hold significant importance. Such conditions are likely to be prevalent in the plasma plumes of AF-MPDT and HPT operating at power levels of kW and above.

• Combining studies on plasma acceleration and demagnetization detachment in PMNs, exploring the optimal position and configuration of magnetic nozzles in plasmas, avoiding the sole reliance on the turning point of single coil magnetic nozzles as the only characteristic scale. Further developing a wider range of magnetic nozzle configurations based on this approach.

• Future advancements in high-precision, high-time-resolution electrostatic probe diagnostics, and optical diagnostics will facilitate broader research approaches and perspectives in magnetic nozzle studies. Significant theoretical advances in magnetic nozzles over the past decade have been highly dependent on experimental support.

• Further development of more advanced kinetic models or hybrid fluid/particle models, expanding the dimensions of the model and the temporal/spatial scales, could reveal additional physical details within the magnetic nozzles.

This research was supported by the National Natural Science Foundation of China (Grant Nos. 12372291, U24B2008, and 52272383).

The authors have no conflicts to disclose.

Kunlong Wu: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Funding acquisition (equal); Investigation (equal); Methodology (equal); Writing – original draft (equal); Writing – review & editing (equal). Zhiyuan Chen: Data curation (equal); Investigation (equal); Writing – review & editing (equal). Jun-xue Ren: Supervision (equal). Yibai Wang: Supervision (equal). Guangchuan Zhang: Investigation (equal). Weizong Wang: Methodology (equal). Haibin Tang: Investigation (equal); Methodology (equal); Supervision (equal).

Data sharing is not applicable to this article as no new data were created or analyzed in this study.