Earth's inner magnetosphere is a zoo of plasma waves where electromagnetic and electrostatic emissions with distinct frequencies coexist and interact. Spacecraft observations have shown that ...whistler‐mode chorus waves, one of the key components in the magnetospheric dynamics, are often modulated by ultralow frequency (ULF) waves. Here, we investigate the effects of two typical ULF wave modes (i.e., field line resonance and mirror mode) on the nonlinear generation process of chorus waves. We report for the first time periodic excitations of lower‐ and upper‐band chorus waves near ULF wave crests and troughs, respectively. Their anticorrelated occurrence is explained by the nonlinear theory, in which the threshold amplitude of nonlinear wave growth is modified by the ULF wave field configuration and the modulated electron distributions. In this framework, the newly observed feature of chorus wave occurrence near the ULF wave crests is attributed to the antisymmetric field profiles of mirror‐mode ULF waves, which periodically modulate the threshold amplitude by modifying the second‐order derivative of the background dipole field. In addition to the second‐order derivative, the first‐order derivative of the background magnetic field is also modulated by the ULF waves to regulate the size of the chorus wave source region.
Plain Language Summary
Previous observations of wave modulation showed that chorus waves are often excited preferentially near the magnetic field troughs of ULF waves. In this study, we report for the first time the periodic occurrence of lower‐band chorus waves near the crests of ULF wave field, although upper‐band chorus waves are still excited near ULF wave troughs. This feature differs dramatically from previous observations, which is attributed to different modes of the observed ULF waves. In this event, ULF waves are identified to be drift‐mirror mode rather than field‐line resonances, which are characterized by the antisymmetric profile of wave magnetic field with respect to the equator. It is the antisymmetric profile that periodically adjusts the second‐order derivative of the background magnetic field, which in turn modifies the threshold amplitude for nonlinear growth of chorus waves and consequently fosters chorus wave excitation near ULF wave crests. We demonstrate that the electron distributions modulated by ULF waves also play a role in modifying the threshold amplitude, which contributes to the excitation of upper‐band chorus waves near ULF wave troughs. The good agreement between the theory and the observations highlights the combined effects of ULF wave field configuration and electron distributions in modulating chorus waves.
Key Points
Chorus wave generation is affected differently by two typical ultralow frequency (ULF) waves, field line resonance (FLR) mode and mirror mode
FLR mode results in lower band chorus in troughs, while the mirror mode causes lower‐ and upper‐band near the wave crest and troughs
The modulation of ULF waves on chorus is due to variation of magnetic field configuration and electron distributions
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BFBNIB, FZAB, GIS, IJS, KILJ, NLZOH, NUK, OILJ, SAZU, SBCE, SBMB, UL, UM, UPUK
Interplanetary (IP) shocks can trigger substorms when they interact the magnetosphere. In study of Hajra and Tsurutani (2018, https://doi.org/10.1016/0032-0633(77)90001-0), they reported a ...shock‐induced substorm event where the energy dissipation exceeded the energy input (ε $\varepsilon $ parameter). In this study, we examined 198 IP shock‐induced substorms from 1995 to 2021 and found 32 underpowered events where the energy dissipation exceeded the energy input calculated from IP shock sheath parameters. We also found underpowered events using other newly developed energy functions. To resolve this dilemma, we introduce the concept of dual compressions of the interplanetary magnetic field by both the IP shock and the bow shock, respectively. Based on in situ observations in the magnetosheath, we obtained a set of new parameters from the dual compressions. The stronger magnetic field resulting from the double compressions reconnects the geomagnetic field, leading to an increased input of energy into the magnetosphere. This new energy input is generally sufficient to balance the energy dissipation.
Plain Language Summary
The magnetosphere is the region controlled by the geomagnetic field. The ε $\varepsilon $ parameter estimates the magnetosphere energy input by considering the Poynting flux in the solar wind. In some cases, however, the energy dissipation exceeds the energy input calculated from solar wind parameters. Considering that the solar wind slows down to form a bow shock when it interacts the Earth's magnetosphere with the solar wind kinetic energy converted into magnetic energy, it is more appropriate to consider the Poynting flux in the magnetosheath. Therefore, we suggest that the ε $\varepsilon $ parameter should be calculated in the magnetosheath. By taking the values of the magnetic field and velocity in the magnetosheath after dual compressions by IP shock and Bow shock for IP shock‐induced substorm events, we obtain larger energy input compared with the ε $\varepsilon $ parameter. The new energy input is generally sufficient to balance the energy dissipation.
Key Points
The energy dissipation during some IP shock‐induced substorms exceeded the energy input by using various energy coupling functions
The dilemma can be solved by the dual compressions of interplanetary magnetic field by IP shock and bow shock, which result in more energy input
A set of new parameters based on the dual compressions are obtained, the new energy input is large enough to provide sufficient energy input
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BFBNIB, FZAB, GIS, IJS, KILJ, NLZOH, NUK, OILJ, SAZU, SBCE, SBMB, UL, UM, UPUK
We examine the drift‐resonant particle dynamics for toroidal ultralow frequency (ULF) waves in a pure dipole background geomagnetic field. We confirm that the resonant condition originally believed ...to apply only for poloidal ULF waves, mωd=ω, also applies for compressional toroidal waves. The predicted phase relationships have been confirmed from Van Allen Probes observations. Their good agreement provides the first observational evidence for the drift resonance condition controlled by the compressional toroidal ULF wave. Moreover, we extend the drift resonance theory into a nonlinear regime. The resulting particle motion can be described by a modified pendulum equation with solutions depending on the wave number m. For high‐m toroidal waves, the resonant islands become asymmetric to perturb the particle trajectories within each potential well and consequently increase the trapping widths in both energy and L‐shell. We further carry out test‐particle simulations to show the evolution of electron distribution functions when they interact with either toroidal or poloidal waves. These findings demonstrate that toroidal ULF waves, like their poloidal counterparts, play an important role in magnetospheric particle dynamics.
Plain Language Summary
In the magnetosphere, ultralow frequency (ULF) electromagnetic waves in the mHz range are usually categorized into poloidal and toroidal modes. It has been widely accepted that poloidal ULF waves, characterized by electric field oscillations in the azimuthal direction, play a key role in accelerating and transporting charged particles in the inner magnetosphere. On the other hand, toroidal waves have long been considered incapable of accelerating particles since their radial electric field is perpendicular to the azimuthal drift velocity of inner magnetospheric particles unless their drift paths are significantly distorted by dayside compression of the magnetosphere. Here we show that even in a pure dipole field, the toroidal waves can still resonate with energetic particles because of the nonzero curl of the wave electric field in association with the compressional magnetic field oscillations, and the resonant condition remains the same as its poloidal counterpart. The predictions of compressional toroidal wave‐particle drift resonance show good agreement with spacecraft observations. Further analysis shows that the particle nonlinear behavior in the toroidal wave field can be described by a modified pendulum equation. Therefore, it is important to consider not only the poloidal but also the toroidal modes in the study of ULF wave‐particle interactions.
Key Points
Compressional toroidal ultra low frequency (ULF) waves have the same drift resonance condition as poloidal waves in a pure dipole model
Drift resonance between electrons and compressional‐toroidal ULF waves is identified from Van Allen Probes observations
A modified pendulum equation is derived to describe the nonlinear drift resonance between toroidal waves and charged particles
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The excitation of electrostatic and/or electromagnetic waves in the plasma universe is often associated with anisotropic velocity distributions of charged particles. In Earth's inner magnetosphere, ...this anisotropy can gradually develop as particles injected from the magnetotail drift around the Earth at different speeds depending on their energy and pitch angle. Here, we show that the perpendicular‐moving and bouncing ions can be separated more abruptly near the injection front. These pitch‐angle filters are localized magnetic dip structures formed by the diamagnetic behavior of the injected particles, which can trap perpendicular‐moving ions and allow bouncing ions to overtake. The resulting ion anisotropy facilitates the rapid generation of electromagnetic ion cyclotron (EMIC) waves, which in turn can largely reshape the Van Allen radiation belts. This scenario is examined by case and statistical observations, together with numerical simulations that reproduce most of the observational signatures, to support the causal relationship between magnetic dips, anisotropic ion distributions, and localized excitation of EMIC waves. Our study highlights the important roles of magnetic dips in the inner magnetospheric dynamics, as pitch‐angle filters of the injected ions and traveling hotspots of EMIC wave activities.
Plain Language Summary
In Earth's Van Allen radiation belts, energetic particles are magnetically trapped to form a hazardous environment for spacecraft and astronauts. These particles can stay trapped for years before they are scattered toward the atmosphere by resonant interactions with plasma waves. Among these waves, the electromagnetic ion cyclotron (EMIC) waves are particularly important in the removal of energetic ions and relativistic electrons. The EMIC wave excitation is often associated with strong ion anisotropy, which gradually develops as particles injected from the magnetotail drift around the Earth at different speeds depending on their energy and pitch angle. Here, we propose a new scenario of localized EMIC wave excitation, in which the injected perpendicular‐moving and bouncing ions are abruptly separated by their associated diamagnetic dips, to develop strong ion anisotropy inside the dip structure and generate EMIC waves. This scenario is validated via case and statistical observations, together with numerical simulations that reproduce most of the observational signatures, which reveals the importance of the magnetic dips as traveling hotspots of EMIC wave activities. These insights indicate the localized and dynamical removal of energetic particles and provide key revision to the prevailing radiation belt models.
Key Points
We perform case and statistical study on the association between magnetic dips, anisotropic proton distributions and EMIC waves
Magnetic dips serve as reversed pitch‐angle filters to trap the perpendicular‐moving protons and allow the bouncing protons to overtake
The pitch‐angle filter leads to a higher ion anisotropy therein, which favors the localized excitation of electromagnetic ion cyclotron wave
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BFBNIB, FZAB, GIS, IJS, KILJ, NLZOH, NUK, OILJ, SAZU, SBCE, SBMB, UL, UM, UPUK
Recent observations in the nightside inner magnetosphere have identified a series of wedge‐like spectral structures in the energy‐time spectrograms of oxygen, helium, and hydrogen ion fluxes. ...Although the shapes and distributions of these structures have been characterized by case and statistical studies, their formation mechanism remains unclear. Here we utilize a particle tracing model to reproduce the wedge‐like structures successively observed by the twin Van Allen Probes. The model suggests that these structures originate from intermittent substorm injection, and it is the accessibility region of these injected ions that determines their shapes. This mechanism is similar to the formation of another kind of structures, the inner magnetospheric nose‐like structures, except that the wedge‐like structures are separated from the tail population by the discontinuation of ion injections. This scenario is also supported by the distribution statistics of wedge‐like structures, which provides new insights into the dynamics of the magnetotail‐inner magnetosphere coupled system.
Plain Language Summary
In Earth's inner magnetosphere, multiple plasma populations interact with one another to render a highly coupled system, with its dynamic evolution also subject to magnetospheric convection and magnetotail particle injections. The interplay of these complicated processes is manifested in spacecraft observations by various spectral structures of particle fluxes, and the understanding of their generation provides unique insights into the underlying processes. Here we focus on a specific type of spectral structures in the nightside inner magnetosphere, named the wedge‐like structures. These structures are characterized by enhanced fluxes of oxygen, helium, and hydrogen ions at increasing energies (from a few eV to several keV) with decreasing equatorial distance to the Earth. Based on a particle tracing model, we successfully reproduce the structure characteristics observed by NASA's (National Aeronautics and Space Administration) twin Van Allen Probes. The model suggests that these structures originate from transient injections associated with substorm activities, and their shapes are determined by the energy‐dependent ion accessibility into the inner magnetosphere. This scenario of wedge‐like structure formation is consistent with both case and statistical results, which sheds new light on our current understanding of the inner magnetospheric dynamics and its coupling process with the magnetotail plasma sheet.
Key Points
The wedge‐like ion structures in the nightside inner magnetosphere are reproduced with a simple model
The ions within the wedge originate from transient injections associated with substorm activities
The wedges are formed by the slow yet deep penetration of injected ions with specific magnetic moments
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BFBNIB, FZAB, GIS, IJS, KILJ, NLZOH, NUK, OILJ, SAZU, SBCE, SBMB, UL, UM, UPUK
Localized magnetic field depressions in the inner magnetosphere, known as magnetic dips, are produced by the diamagnetic motion of energetic ions injected via substorm activities. The magnetic dips, ...if deep enough, can produce a local minimum in the radial profile of the field strength to trap the injected protons. Therefore, the trapped protons would drift at the same speed as the dip propagation, which leads to the simultaneous enhancements of proton fluxes in multiple energy channels at the leading edge of the dip structure. On the trailing side, the reduction of proton fluxes shows dispersive features, which can be attributed to the energy‐dependent drift motion of the injected protons in the absence of the local field minimum. This scenario is examined based on comparisons between multi‐spacecraft observations and test‐particle simulations, and their good agreement validates the scenario to shed new light on the dynamics of the inner magnetosphere‐magnetotail coupled system.
Plain Language Summary
During magnetospheric substorms, energetic ions are injected from the magnetotail toward the inner magnetosphere, and the diamagnetic motion of these ions produces longitudinally localized field depressions named magnetic dips. Recent studies with multi‐spacecraft observations have revealed the westward propagation of the magnetic dips, which are believed to originate from the westward drift of the injected ions in the geomagnetic field. One may expect from the wide energy range of the injected ions and the energy‐dependence of their drift motion that these ions would be gradually dispersed in the longitude to cause the expansion and shallowing of the dip structure. These expectations, however, are not consistent with the observations of sustained magnetic dips associated with simultaneous ion flux enhancements at multiple energy channels. To reconcile this inconsistency, we propose the ion trapping process for magnetic dips with sufficient depths, which explains the simultaneous flux enhancements observed on the leading edge of the dip structure. On the trailing side, the ions cannot be trapped due to the weaker field depressions, which explains the observations of dispersive ion flux reduction given a discontinued and/or localized process of substorm injection. This scenario is supported by the good agreement between test‐particle simulations and multi‐spacecraft observations.
Key Points
We analyze the behavior of energetic protons associated with magnetic dip structures based on observations and test‐particle simulations
Magnetic dips with strong field depressions can trap energetic protons inside and have them drift at the same speed as the dip propagation
Observed dispersionless increases and dispersive reductions of proton fluxes on dip leading and trailing sides are reproduced by simulations
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FZAB, GIS, IJS, KILJ, NLZOH, NUK, OILJ, SAZU, SBCE, SBMB, UL, UM, UPUK
We present Van Allen Probes observations of periodic chorus wave emissions in the troughs of compressional ultralow frequency (ULF) waves. During this event, the spectral gap of chorus waves ...gradually widens as the spacecraft moves from the equatorial source region towards higher latitudes. Moreover, chorus wave intensity increases and frequency range widens after a substorm injection. We show that the periodic occurrence of chorus waves is attributed to the modulation of threshold amplitude for nonlinear growth of chorus waves by the second spatial derivative of ULF compressional magnetic field. The widening gap can be interpreted in terms of the nonlinear damping mechanism. A good agreement is also found between the nonlinear wave growth theory and the observations regarding the influence of substorm injection on the chorus. These findings support the applicability of the nonlinear theory in describing the chorus wave generation and damping, together with their modulations by ULF waves.
Plain Language Summary
In Earth's magnetosphere, plasma waves play a key role in the acceleration, transport, and loss of energetic particles in the Van Allen radiation belts. These waves usually coexist and interact with one another. For example, whistler waves in the kHz range are often modulated by ultralow frequency (ULF) waves in the mHz range, which leads to fascinating phenomena such as pulsating auroral patches. In this paper, we report a representative case study of long‐duration modulation of whistler‐mode chorus emissions by ULF waves, in which chorus waves appear periodically at the trough of the ULF compressional magnetic field. There are two other interesting phenomena including the gradual widening of the chorus wave‐power gap near half of the electron cyclotron frequency and the abrupt enhancement of the chorus wave intensity and frequency range after a substorm injection. We utilize the nonlinear growth theory of chorus emissions to propose, for the first time, that the latitudinal configuration of the compressional ULF waves plays an important role in the modulation process. The nonlinear theory is also used to investigate the effect of substorm injection on chorus emissions and to understand the widening of the wave‐power gap. They both show good agreement with the observations.
Key Points
The observed modulation of chorus emissions by ultralow frequency waves is explained in nonlinear theory by the periodic distortion of field configuration
The chorus waves also show a widening gap as the spacecraft moves away from the equator, which agrees with the nonlinear damping scenario
The intensity and frequency range of the chorus waves increase abruptly due to the substorm injection of energetic electrons
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FZAB, GIS, IJS, KILJ, NLZOH, NUK, OILJ, SAZU, SBCE, SBMB, UL, UM, UPUK
Ultra‐low frequency (ULF) waves contribute significantly to the dynamic evolution of Earth's magnetosphere by accelerating and transporting charged particles within a wide energy range. A substantial ...excitation mechanism of these waves is their drift‐bounce resonant interactions with magnetospheric particles. Here, we extend the conventional drift‐bounce resonance theory to formulate the nonlinear particle trapping in the ULF wave‐carried potential well, which can be approximately described by a pendulum equation. We also predict the observable signatures of the nonlinear drift‐bounce resonance, and compare them with spacecraft observations. We further discuss potential drivers of the pendulum including the convection electric field and the magnetospheric dayside compression, which lead to additional particle acceleration or deceleration depending on magnetic longitude. These drivers indicate preferred regions for nonlinear ULF wave growth, which are consistent with previous statistical studies.
Plain Language Summary
In Earth's magnetosphere, electromagnetic oscillations in the mHz frequency range referred to as ultra‐low frequency (ULF) waves have been frequently reported in space and ground‐based observations. These waves have been known to interact resonantly with magnetospheric particles during their drift and bounce motions, which plays a key role in the acceleration and transport of charged particles in the Van Allen radiation belts. Drift‐bounce resonance occurs when a resonant particle experiences repetitive patterns of the wave field over consecutive bounce cycles along its drift orbit, thus enabling a cumulative energy exchange between waves and particles. In previous studies of drift‐bounce resonance, a linearization approach is applied for simplicity. Here, we extend the conventional theory into the nonlinear regime, to formulate the nonlinear trapping of particles based on a pendulum equation and to predict the characteristic signatures of drift‐bounce resonance. Moreover, we discuss the potential drivers of the pendulum, including the magnetospheric convection and the dayside compression, which break the symmetry in the phase portrait of the particle trajectories and therefore, lead to continuous particle acceleration or deceleration depending on magnetic longitude. Our study on nonlinear drift‐bounce resonance is helpful to the understanding of the ULF wave‐particle interactions and the wave excitation processes.
Key Points
A nonlinear framework for drift‐bounce resonance is developed to describe the particle behavior in Ultra‐low frequency (ULF) waves
Nonlinear drift‐bounce resonance is characterized observationally by rolled‐up structures in particle pitch angle spectrum
Our theory predicts that dayside and dusk sectors are preferred regions for nonlinear ULF wave growth
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BFBNIB, FZAB, GIS, IJS, KILJ, NLZOH, NUK, OILJ, SAZU, SBCE, SBMB, UL, UM, UPUK
In the Earth's inner magnetosphere, charged particles can be accelerated and transported by ultralow frequency (ULF) waves via drift resonance. We investigate the effects of magnetospheric convection ...on the nonlinear drift resonance process, which provides an inhomogeneity factor S to externally drive the pendulum equation that describes the particle motion in the ULF wave field. The S factor, defined as the ratio of the driving amplitude to the square of the pendulum trapping frequency, is found to vary with magnetic local time and, as a consequence, oscillates quasi‐periodically at the particle drift frequency ωd. To better understand the particle behavior governed by the driven pendulum equation, we carry out simulations to obtain the evolution of electron distribution functions in energy and L‐shell phase space. We find that resonant electrons can remain phase trapped by the low‐m ULF waves under strong convection electric field, whereas for high‐m ULF waves, the electrons trajectories can be significantly modified. More interestingly, the electron drift frequency ωd is close to the nonlinear trapping frequency ωtr for intermediate‐m ULF waves, which corresponds to chaotic motion of resonant electrons. These findings shed new light on the nature of coherent and diffusive particle transport in the inner magnetosphere.
Plain Language Summary
Ultralow frequency (ULF) waves are electromagnetic oscillations in Earth's inner magnetosphere with the frequency of a few millihertz. When the azimuthal phase velocity of the ULF waves matches the drift velocity of a charged particle, the particle experiences a constant wave electric field to have a net energy gain or loss. This process, referred to as the wave‐particle drift resonance, plays important roles in the acceleration and transport of magnetospheric particles. In this study, we analyze the role of large‐scale magnetospheric convection in the drift resonance process. We find that the nonlinear particle behavior can be described by a pendulum driven by a quasiperiodic external force controlled by the convection electric field. Therefore, the resonant particle behavior is largely determined by the ratio between the external driving frequency and the pendulum eigenfrequency, which in turn depends on the azimuthal wave number m. It is shown that magnetospheric convection can hardly affect the electron trapping motion in low‐m ULF waves, whereas for high‐m waves, the electron dynamics is largely controlled by the large‐scale convection. Chaotic motion appears in intermediate‐m waves, which corresponds to an efficient radial diffusion of electrons.
Key Points
We derive nonlinear‐driven pendulum equation for drift resonant interaction of particles and ULF waves under the convection electric field
Under strong convection, resonant electrons remain phase trapped by low‐m ULF waves but can be quickly untrapped for high‐m cases
The convection electric field causes particle diffusion when its trapping frequency is close to the drift frequency for intermediate‐m case
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BFBNIB, FZAB, GIS, IJS, KILJ, NLZOH, NUK, OILJ, SAZU, SBCE, SBMB, UL, UM, UPUK
After the launch of Van Allen Probes, the three‐belt structures of ultra‐relativistic electrons are discovered. In this study, we investigate the three‐belt structures of sub‐MeV electrons, which may ...form under different mechanism compared with those of ultra‐relativistic electrons and are worth in‐depth analysis. Based on the differential flux data from Magnetic Electron Ion Spectrometer onboard Radiation Belt Storm Probes‐B satellite, we find 54 events, in which two comparable peaks of sub‐MeV electron fluxes and a slot appear where there should be the outer radiation belt. Through the statistical analysis, the three‐belt structures of sub‐MeV electrons are found to be closely related to symmetric H‐component (SYM‐H) and Auroral Electrojet (AE) indices. The 2‐day SYM‐H minimum and AE maximum before the event have a linear trend with the remnant belt and the “second slot” locations. The L‐values of the remnant belt and the “second slot” of different energy electrons decrease as energy increases in general and show interesting characteristics during their temporal evolution. Moreover, the lifetime of the remnant belt of different energy electrons increases as energy increases. We find similarities and differences between sub‐MeV and ultra‐relativistic electrons three‐belt events, which provides a new perspective in three‐belt structure study.
Plain Language Summary
An important result of the Van Allen Probes mission was that the ultra‐relativistic electron (>2MeV) radiation belts, containing an inner belt, a slot region and an outer belt, often show a three‐belt structure. That is, two belts and a slot region appear where there should be the outer radiation belt. The dominate mechanism to form the structure is still under debate. Recently, such structure for sub‐MeV electrons was also reported in a case study. Since the physical properties of these two types of electrons are different, the characteristics and mechanism to form the structure may be different. In this work, we show that common occurrence and general characteristics of sub‐MeV electron three‐belt events. The location, formation, and temporal evolution of the structure are influenced by electron energy and magnetospheric conditions reflected by geomagnetic indices. We have also determined the relatively short lifetime of the intermediate belt (remnant belt) and explained why three‐belt events for sub‐MeV electrons are less visible.
Key Points
We report for the first time the main statistical characteristics of sub‐MeV three‐belt events using Van Allen Probe data
The occurrence and the location of the remnant belt for the sub‐MeV electrons are affected by electron energy and geomagnetic activities
The lifetime of sub‐MeV electrons in the remnant belt is much smaller, explaining why these three‐belt events are less visible
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BFBNIB, FZAB, GIS, IJS, KILJ, NLZOH, NUK, OILJ, SAZU, SBCE, SBMB, UL, UM, UPUK
