<figure><img src="image_19.png" /><figcaption>**Fig. 17** Magnetic reconnection modeled using three different codes (HVM, HVLF and PIC). Left: normalized reconnection rate $R/B_0v_A$. The inset shows the time interval corresponding to the ticker curves in the main plot. To ease the comparison, the curves in the inset are shifted in time. Right: (first row) out-of-reconnection-plane magnetic field $\Delta B_z/B_0 = (B_z - B_z(t=0))/B_0$ showing the expected Hall quadrupolar pattern; (second row) electric field parallel to the ambient magnetic field $E_\|/\tilde{E}_\|$, where $\tilde{E}_\|$ is the root mean square of $E_\|$ in the shown region; (third row) current density in the out-of-plane direction $J_z$; (fourth row) electron current density in the plane $J_e^{(\text{in-plane})}$. The superposed black or white curves are the magnetic field lines. The three columns show results from the three different models. The left column show results from HVM at the simulation time $t = 237.5 \, \Omega_{\text{cp}}^{-1}$; the center column show results from the HVLF code at the simulation time $t = 232.5 \, \Omega_{\text{cp}}^{-1}$; the right column show results from the PIC code at the simulation time $t = 235.0 \, \Omega_{\text{cp}}^{-1}$. $\Omega_{\text{cp}}$ is the proton cyclotron frequency. Adapted from Finelli et al. (2021)</figcaption></figure>

current sheet (Harris 1962b) perturbed by long wavelength magnetic field fluctuations with random phase (with $1 < |\mathbf{k}|d_p < 9$, where $\mathbf{k}$ is the wave vector of the fluctuations and $d_p$ is the proton inertial length). The size of the simulations domains is $L_x \times L_y = 24\pi d_p \times 12\pi d_p$ discretized with $N_x \times N_y = 1024 \times 512$ grid points. In the HVM and HVLF simulations, the velocity space domain in each direction $(x, y$ and $z$) is $[-6.4, +6.4] \, v_{\text{th},p}$, where $v_{\text{th},p}$ is the proton thermal speed, and it is discretized by $51^3$ grid points. Figure 17 shows results comparing the three models.

While the reconnection linear phase evolution, as well as the overall reconnection signatures and patterns, are quite similar for all three models, Finelli et al. (2021) report that the region of intense current at the centre of the current sheet is more elongated in the case of the HVLF and PIC simulations than in the HVM run. Also, the normalized reconnection rate $R/B_0v_A$ computed in the quasi-steady state is higher for the HVM ($R/B_0v_A \sim 0.06$, where $B_0$ is the upstream magnetic field and $v_A$ is the Alfvén speed) than for the HVLF and PIC simulations ($R/B_0v_A \sim 0.04$). Despite these differences, the results of all three simulations agree qualitatively. In terms of electron dynamics, which is not captured by the HVM code, the HVLF model reproduces the main features obtained with the fully kinetic treatment of the PIC code.

Magnetic reconnection and turbulence are intricately coupled in plasmas (Stawarz et al. 2024, this collection), where coherent structures such as magnetic holes, magnetic islands, and current sheets naturally develop (see, e.g., Matthaeus et al. 2015). Then, current sheet widths tend to approach the kinetic scale, leading to magnetic reconnection. Hybrid-Vlasov simulations of turbulent plasmas have been successful in modelling both turbulence-induced “standard” reconnection with ion-coupling and electron-only magnetic reconnection (Cali-fano et al. 2020; Arròs et al. 2020). A key result in the context of the interplay between re-connection and turbulence is the fact that turbulence is mediated by magnetic reconnection. More specifically, reconnection plays a key role in driving the onset of sub-ion turbulent
cascade (Franci et al. 2017; Cerri et al. 2017; Manzini et al. 2023; Adhikari et al. 2024). Hybrid-Vlasov simulations with the HVM code have played a crucial role in providing evidence for the role played by reconnection in this context.

### 8.3.2 Global Simulations

The main goal of Vlasiator is to model the solar wind–magnetosphere interaction with a hybrid-Vlasov approach. For this reason, the computational efforts have been mainly directed toward performing global simulations of the entire magnetosphere. As a consequence, a broad variety of magnetospheric plasma phenomena have been investigated using Vlasiator (notably collisionless shock (e.g., Johlander et al. 2022) and foreshock physics (e.g. Turc et al. 2023), magnetosheath jets (e.g. Suni et al. 2021), auroral proton precipitation (e.g. Grandin et al. 2020, 2023) to mention a few). As magnetic reconnection plays a key role in magnetosphere dynamics, it has been investigated in several Vlasiator studies, both at the magnetopause (Pfau-Kempf et al. 2016; Hoilijoki et al. 2017, 2019; Akhavan-Tafti et al. 2020; Pfau-Kempf et al. 2020) and in the magnetotail (Palmroth et al. 2017; Juusola et al. 2018; Runov et al. 2021; Palmroth et al. 2023). Since Vlasiator does not include an explicit resistive term, magnetic reconnection is enabled by the numerical diffusivity or resistivity.

In this section, we present key results from selected Vlasiator studies in two subsections, one devoted to magnetopause reconnection and the other focusing on magnetotail reconnection. We start with discussing 2D-3V simulations, and then present 3D-3V simulations since recent algorithmic improvements allowed running global, three-dimensional (3D-3V) hybrid-Vlasov simulations of Earth’s magnetosphere (Ganse et al. 2023).

**Magnetopause Reconnection:** Global simulations allow us to study the interaction between the solar wind and the magnetosphere, including how and to which extent dayside magnetic reconnection is affected by the solar wind and magnetosheath dynamics. Hoilijoki et al. (2017) investigate this topic by using a 2D-3V Vlasiator global simulation in the GSE polar xz plane, focusing in particular on the laminar or bursty nature of magnetic reconnection during steady solar wind conditions (Fig. 18).

The simulation domain covers $x = [-94, +48] R_E$ and $z = [-56, +56] R_E$ ($R_E = 6371$ km is the Earth’s radius) and it features a 2D line dipole centered at the origin modelling the Earth’s magnetosphere dipole and scaled to match the geomagnetic dipole strength. The steady solar wind has a density $n = 1 \text{ cm}^{-3}$, a constant velocity $v_{\text{SW}} = -750 \text{ km/s}$ and a proton temperature of 0.5 MK. The interplanetary magnetic field (IMF) is directed purely southward and it has a magnitude of 5 nT. The resolution is uniform in the simulation domain; the spatial resolution is 300 km ($\sim 0.047 R_E \sim 1.3 d_{p,SW}$, where $d_{p,SW}$ is the proton inertial length in the solar wind) and the velocity space resolution is 30 km/s ($\sim 0.33 v_{\text{th,p,SW}}$, where $v_{\text{th,p,SW}}$ is the solar wind proton thermal speed). The solar wind flows into the simulation domain from the boundary at $x = +48 R_E$ with constant parameters. The boundaries of the simulation box are periodic in the out-of-plane y direction while the $-x$ and $\pm z$ boundaries apply copy boundary conditions. The inner boundary of the magnetosphere is a circle of radius $4.7 R_E$ centered at the origin and it enforces a static Maxwellian proton velocity distribution and perfect conductor field boundary conditions.

Hoilijoki et al. (2017) reported that, despite the steady solar wind conditions, magnetic reconnection at the subsolar magnetopause does not reach a steady state and it is very dynamic. Indeed, magnetic islands are constantly produced and the presence of multiple X-points is observed as is evident in Fig. 18b. The motion of the X-points appears to be mostly dictated by the outflow produced by the neighboring X-points. Hoilijoki et al. (2017) suggest that including the ion kinetic physics in the model promotes the development of a dynamic and bursty reconnection process at the dayside.
<figure><img src="image_20.png" /><figcaption>**Fig. 18** (a) Plasma β; (b) proton $V_2$. The black lines show the magnetic field lines. (c) Magnetic field strength (black solid) and plasma density (red dashed) fluctuations from the virtual spacecraft location indicated with the black dot in panel (a). The anticorrelation between magnetic field and density fluctuations is compatible with mirror mode waves. Adapted from Hoilijoki et al. (2017)</figcaption></figure>

This study investigates also the reconnection rate at the multiple simultaneous X-points and how the rate is affected by the local plasma conditions near the X-point. In particular, the presence of mirror modes (Fig. 18c) in the magnetosheath appears to affect the reconnection rate, in agreement with spacecraft observations (Laitinen et al. 2010). The dependence of the magnetopause reconnection rate upon the IMF direction is further investigated in global Vlasiator simulation by Hoilijoki et al. (2019). In particular, the run presented in Hoilijoki et al. (2017) is compared to a run with similar parameters but with a positive component of the IMF ($B_{IMF} = [3.54, 0, -3.54] nT$). The Sun-ward tilt of the IMF results in a smaller tangential field at the magnetopause, leading to a reduction of the reconnection rate with respect to the purely southward-directed IMF case. The presence of a non-zero $B_{x,IMF}$ introduces an asymmetry that impacts the reconnection process in terms of flux transfer events (FTE) size, speed and occurrence rate. In particular, FTEs are observed more frequently in the Northern Hemisphere and they are smaller in size with respect to the Southern Hemisphere.

The findings of Hoilijoki et al. (2017) have been confirmed by Pfau-Kempf et al. (2020), who analyze the reconnection process in a three-dimensional setup reproducing the magnetopause surface. In particular, Pfau-Kempf et al. (2020) report a Vlasiator simulation of the noon–midnight meridional plane which is extended to cover 7 $R_E$ in the dawn–dusk direction. The study by Pfau-Kempf et al. (2020) is the first example of a 3D-3V Vlasiator simulation of a cylindrical geometry mimicking the subsolar dayside magnetosphere. While the dimensionality is increased, the cylindrical geometry and the limited extent in the
$y$ direction allow keeping the computational cost affordable and much lower than a global 3D-3V simulation modeling the entire magnetosphere.

The simulation domain covers $x = [-16, +31] R_E$, $y = [-3.5, +3.5] R_E$ and $z = [-35, +35] R_E$ and the solar wind and IMF parameters are the same used for the 2D-3V run reported in Hoilijoki et al. (2017) and discussed above. The spatial resolution is ~0.24 $R_E$, which is larger than the resolution of 2D-3V Vlasiator simulations because of the increased computational cost of 3D-3V runs.

Identifying the magnetic reconnection site in 3D settings is not as straightforward as in 2D-3V simulations, where the local behaviour of the flux function allows to identify saddle points corresponding to X-points that are associated with reconnection sites. Hence, in this study the X-line location is estimated by combining the four-field junction method (Laitinen et al. 2006) with the identification of the locations exhibiting a flow reversal in the $z$ direction. However, the four-field junction method is insufficient for identifying multiple reconnection X-lines. Pfau-Kempf et al. (2020) find that, despite the uniform initial condition and the cylindrical symmetry in $y$, the X-line is not a straight line and it exhibits variations along the $y$ direction. It is suggested that structures in the magnetosheath break the translation symmetry along $y$. Analogously to Hoilijoki et al. (2017), Pfau-Kempf et al. (2020) point out that reconnection is bursty and patchy, with multiple reconnection sites being present at the various $z$ and $y$ locations across the magnetopause, despite the homogeneous and steady-state solar wind conditions.

**Magnetotail Reconnection:** Recently, magnetotail reconnection has been investigated in the context of a 3D-3V Vlasiator simulation investigating the dynamics of plasma eruptions (Palmroth et al. 2023). The three-dimensional simulation in both ordinary and velocity space is made possible by technological advances, notably by enabling static adaptive mesh refinement (AMR) for ordinary space (Ganse et al. 2023). With AMR, regions of high scientific interest such as the magnetotail plasma sheet are sampled with higher resolution ($0.16 R_E$) with respect to other regions in the simulations, the coarser resolution is $1.26 R_E$.

The 3D-3V simulation domain covers $x = [-111, +50] R_E$ and $y, z = [-58, +58] R_E$. The simulation parameters and initial conditions (IMF, solar wind density and speed) are the same adopted in (Hoilijoki et al. 2017). However, since this is a 3D-3V run, the Earth's dipole is 3D and the inner boundary is a sphere of radius $4.7 R_E$, while the $\pm y$ boundaries apply copy boundary conditions, as the $\pm z$ boundaries. Differently from (Hoilijoki et al. 2017; Palmroth et al. 2018a; Juusola et al. 2018), where Ohm's law included only the Hall term, this run includes the electron pressure gradient term as well. A polytropic closure is adopted for electrons, $\mathbf{P}_e = p_e \mathbf{I}$ and $p_e = n^\gamma T_e$, where $\mathbf{P}_e$ is the electron pressure tensor, $p_e$ is the scalar pressure, $T_e$ is the electron temperature and $n$ is the density. The polytropic index $\gamma$ is set to $5/3$ (adiabatic).

Palmroth et al. (2023) focus on the investigation of magnetotail, revealing complex dynamics where magnetic reconnection and kinking instability co-exist in the magnetotail current sheet. In particular, in Fig. 19 it is shown that both processes are required to induce a global topological reconfiguration of the magnetotail, with the formation of a tail-wide plasmoid which is released and rapidly moves tailward. As mentioned above, the identification of the reconnection sites is challenging in 3D systems since we cannot rely on the identification based on the flux function. The reconnection site (X-line) in the magnetotail is identified by a combination of magnetic field and velocity proxies. X-lines and O-lines are identified as the locations where both $B_r = B_z = 0$, where $B_r$ is the radial magnetic field component. The quantity $\partial B_z / \partial r$ allows us to distinguish between X-lines ($\partial B_z / \partial r > 0$ in the magnetotail) and O-lines ($\partial B_z / \partial r < 0$ in the magnetotail). The locations where an X-line is co-located with a $v_x$ reversal (diverging plasma flow) are identified as reconnection sites.
<figure><img src="image_21.png" /><figcaption>Fig. 19 Evolution of the magnetotail current sheet in a 3D-3V Vlasiator simulation. The panels show the current sheet surface (defined as $B_r = 0$) at different times, $t = 1300$ s (a), $t = 1400$ s (c), $t = 1470$ s (g). The color of the surface corresponds to the current density $J$. The yellow line indicates the flow reversal between the Earthward and tailward reconnection outflow. The magenta and green lines are locations where $B_r = 0$ and $B_z = 0$ and correspond to X-lines and O-lines (differentiated using the sign of $\partial B_z / \partial r$, which is positive at the X-lines and negative at the O-lines). The primary reconnection line is where the X-line (magenta) and flow reversal (yellow) contours are approximately co-located. The background grid shows the coordinates but also the magnetic-field topology: the black grid shows areas where the magnetic field is directed northward, and the white grid shows the areas where it is southward-directed. Adapted from Palmroth et al. (2023)</figcaption></figure>

Palmroth et al. (2023) further confirm that reconnection is ongoing at those locations by showing reconnection signatures such as the Hall electric field and ion demagnetization. In the 3D-3V run, a dominant tail-wide reconnection X-line is found at $X \sim -15 R_E$ throughout the simulation. The dominant X-line is very dynamic and new X-lines and O-lines with limited extent in the Y-direction are constantly formed.

# 9 The Rice Convection Model

## 9.1 Introduction:

As a reminder, the equations in this section are written in SI units. The Rice Convection Model (RCM), by definition, never includes the physics of reconnection. However, reconnection is a microscale/mesoscale process. Only a small fraction of the magnetic flux in the magnetosphere is included in reconnection at any given time. Including the RCM allows us to discuss how the reconnection process impacts the rest of the magnetosphere, specifically the inner magnetosphere, where the magnetic field lines are closed. Since the RCM's outer boundary condition comes from the plasmasheet, any process in the tail, such as reconnection, can impact the inner magnetosphere. Specifically, reconnection can generate low entropy bubbles that move toward the Earth at high speeds and can be a significant source of transport of plasma and magnetic field from the tail to the inner magnetosphere. In addition, the RCM helps quantitatively predict the plasmas from the inner magnetosphere which can reconnect on the dayside.

## 9.2 Assumptions and Equations

The physics behind the RCM can be found in detail in Toffoletto et al. (2003) and Wolf (1983), and a detailed discussion of the use of the RCM can be found in Wolf et al. (2016) and Toffoletto et al. (2003), and ring current models are described by Toffoletto (2020). In the RCM, the distribution of magnetospheric particles is assumed to be isotropic, that is divided up into multiple energy channels. For each channel, a key variable is the isotropic energy invariant

$$|\lambda_k| = W_k(\mathbf{x}, t) V^{2/3} \qquad (77)$$
where $W_k$ is the particle kinetic energy, including bounce and gyro motion, $k$ is the energy channel label, and the sign of $\lambda_k$ is positive for positive ions and negative for electrons. The flux tube volume is

$$V = \int_{sh}^{nh} \frac{ds}{B(\mathbf{x}, t)} \qquad (78)$$

where the integral extends along the field line from the southern to the northern ionosphere.

The motions of magnetospheric particles in the inner magnetosphere are assumed to be governed by

Drift velocity $\ll$ bounce motion $\ll$ gyro motion

The RCM calculates the bounce-averaged drift velocity, including gradient, curvature, and $E \times B$ drifts, i.e.,

$$\mathbf{v}_k = \frac{\left(\mathbf{E} - \frac{1}{q_k} \nabla W_k(\mathbf{x}, t)\right) \times \mathbf{B}(\mathbf{x}, t)}{B(\mathbf{x}, t)^2} \qquad (79)$$

where $q_k$ is the charge of a particle of species k. Inertial drift is assumed to be negligible. The quantity $\eta_k(\mathbf{x}, t)$ is defined as the number of particles per unit magnetic flux for particles of a specific chemical species and a specific value of energy invariant. It follows a conservation law (Wolf 1983)

$$\left[ \frac{\partial}{\partial t} + \mathbf{v}_k(\lambda_k, \mathbf{x}, t) \cdot \nabla \right] \eta_k = -L(\eta_k) \qquad (80)$$

where $L$ is the loss rate of particles due to precipitation and charge exchange. The classic RCM neglects particles flowing up from the ionosphere to the magnetosphere. In the early 2000's Stan Sazykin and Darren DeZeeuw implemented a grid-based scheme using the CLAWPAK package (De Zeeuw et al. 2004; Mandli et al. 2016) that was more robust but a bit more diffusive than the earlier Lagrangian scheme.

The flux tube content $\eta_k$ is related to the thermodynamic pressure $P$

$$PV^{5/3} = \frac{2}{3} \sum_k \eta_k |\lambda_k| \qquad (81)$$

while the flux tube content is related to the plasma distribution function $f_k(\lambda)$ as

$$\eta_k = \frac{4\pi 2^{1/2}}{m_k^{3/2}} \int_{\lambda_{min}}^{\lambda_{max}} |\lambda|^{1/2} f_k(\lambda) d\lambda \qquad (82)$$

where $(\lambda_{max} - \lambda_{min})$ is the width of the invariant energy channel associated with species k. The species k is defined for a given chemical species (usually $e^-$, $H^+$, $O^+$), and the specific value of the energy invariant.

The electric field can be expressed as the sum of a potential component and an inductive component

$$\mathbf{E} = -\nabla\Phi - \mathbf{v}_{induction} \times \mathbf{B} \qquad (83)$$
In the RCM, the inductive electric field is included implicitly through time-dependent magnetic field mappings. The inductive magnetic field in the ionosphere is assumed to be zero there; however, it is not zero in the magnetosphere.

There are two more complications in the electric field:

1. In the classic RCM, we assume the electric field is perpendicular to the magnetic field.

2. There are two coordinate systems used in the classic RCM. One moves with the Earth as it rotates, and the potential in that system is labeled $\Phi_i$. The other does not rotate with the Earth, is approximately an inertial system, and is labeled $\Phi$.

We can translate from one system to the other in the ionosphere using the formula

$$ \Phi = \Phi_i - \frac{\omega_E B_0 R_E^3 \sin^2(\theta_i)}{R_i} \quad (84) $$

where $\omega_E$ is the angular rotation rate of the Earth, $B_0$ is the magnetic field at the Earth's equator, $R_E$ is the radius of the Earth, $\theta_i$ is the colatitude, and $R_i$ is the radius of the ionosphere. Equation (84) applies to the equatorial plane, but it applies only to a dipole magnetic field. To compute $\Phi$ in the equatorial plane of the magnetosphere, the RCM calculates $\Phi$ by mapping between the ionosphere to the equatorial plane, assuming $\Phi$ is constant along each field line.

In the thin-shell approximation, the equation for the conservation of current is $(\nabla \cdot \mathbf{J} = 0)$ can be written

$$ \nabla_i \cdot \left[ \overleftrightarrow{\Sigma} \cdot \left( \nabla_i \Phi_i \right) \right] = \left( J_{\parallel nh} - J_{\parallel sh} \right) \sin(I) \quad (85) $$

where $\overleftrightarrow{\Sigma}$ is the field-line integrated conductivity tensor due to both hemispheres, $I$ is the dip angle of the magnetic field in the ionosphere, and $J_{\parallel nh} - J_{\parallel sh}$ is the ionospheric field-aligned current density.

The Vasyliunas (1970) equation, which is based on force balance

$$ \mathbf{J} \times \mathbf{B} - \nabla P = 0 \quad (86) $$

is given by

$$ \frac{J_{\parallel nh} - J_{\parallel sh}}{B_i} = \frac{\hat{b}}{B} \cdot \nabla V \times \nabla P \quad (87) $$

which relates field-aligned currents in the ionosphere to pressure gradients in the magnetosphere, and $B_i$ is the magnetic field at the southern- and northern-ionospheric footprints of the field line (assumed the same). The derivation makes use of the fact the right-hand side of Eq. (87) can be evaluated anywhere along the field line. The RCM equations are solved on a fixed ionospheric grid that has variable grid spacing in latitude to better resolve the auroral zone. The RCM grid is time-dependent in the equatorial plane, ranging from just inside the magnetopause on the dayside to $10 - 20R_E$ on the night side.

## 9.3 Inputs, Boundary, and Initial Conditions and Outputs

*The magnetic field model:* For many years, the RCM assumed a constant magnetic field, but, beginning about 2000 the RCM used a time-dependent semi-empirical model such as the Tsgyanenko models (1989, 1995, 2003). The RCM can also use the Hilmer and Voigt
(1995) magnetic field model. In classic RCM runs, the magnetic field is not designed to be consistent with Ampere's law and equation (86). Ways of including Eq. (86) consistently in the RCM are described in Sect. 9.4. In the ionosphere, where the magnetic field is assumed to be dipolar, the magnetic field is

$$ \mathbf{B}_i = -\frac{\mu_0 \mathbf{M}_E - 3\hat{r}(\mathbf{M}_E \cdot \hat{r})}{4\pi R_i^3} \quad (88) $$

where $\hat{r}$ is the radial unit vector in the ionosphere, and $\mathbf{M}_E$ is the magnetic moment, which is in the southern direction, and

$$ B_i = \hat{r} \cdot \mathbf{B}_i \quad (89) $$

**Ionospheric conductance:** Ionospheric conductance has two major drivers: Solar heating (e.g., the Sheffield University Ionosphere Plasmasphere (SUPIM) model, Bailey et al. 1997). The second is Auroral heating. The standard treatment uses the electron precipitating energy flux and average energy (Robinson et al. 1987).

**Loss models:** Separate models are needed for electrons and ions. The simplest electron loss model assumes a fixed fraction of strong pitch-angle scattering, often between 33% and 67% (Schumaker et al. 1989). That procedure is reasonable for the plasma sheet but over-estimates the electron loss rate in the inner magnetosphere. A slightly more sophisticated model (Chen and Schulz 2001) is somewhat more realistic. For the Ion loss model, there are many theoretical models of ion charge exchange. The overall ion loss rate for an energy and L value is calculated using an algorithm developed by James Bishop (Freeman et al. 1993; Bishop 1996).

The RCM needs boundary conditions both at its outer (large L) boundary and at its lowest (low L) boundary. The large-L boundary depends on MLT as well as UT. Note that the large-L boundary can't be aligned with the grid, except in a few cases. At this boundary, the number $\eta_s$ which is the number of particles for a given type s per weber of magnetic flux, is needed as well as the potential distribution, which can be a simple function of solar wind conditions or an empirical model such as the Weimer model (1985).

The low-L boundary (low-latitude) is set at least a few degrees latitude from the equator. Given the aligned-dipole assumption, the latitudinal current density should, in principle, be zero at the equator. The RCM and many other models use a thin-wire approximation to represent the region near the equator, to provide a boundary condition

$$ \frac{\partial J_{\theta}}{\partial \theta} + S \frac{\partial J_{\phi}}{\partial \phi} = 0 \quad (90) $$

which was derived by Blanc and Richmond (1980). Here J is the current density, $\theta$ is the colatitude coordinate, and $\phi$ is the longitudinal coordinate. S is a function of $\phi$ that was defined by Blanc and Richmond (1980).

**Initial conditions:** These are needed to provide the value of the initial value of $\eta_k$, which is a function of grid location and time. The RCM can be initialized with an empty value and run for a period of time to fill in the inner magnetosphere. Alternatively, the RCM uses the Spence et al. (1989) model for the initial pressure distribution. Earlier versions of the RCM assumed a Maxwellian distribution, but more recent versions have the option to assume a kappa distribution (e.g., Yang et al. 2015).

**RCM outputs:** The main RCM outputs are the electric potential ($\Phi$), field-aligned currents ($J_{\parallel nh} - J_{\parallel sh}$), the distribution function ($\eta_s$) and moments (pressure and density) within the RCM modeling region in the ionosphere and the magnetospheric equatorial plane.
## 9.4 Generalizations of the RCM

There have been many modifications to the RCM, particularly since 2000.

**Other planets:** Tom Hill and several of his students modified the RCM to be appropriate for Jupiter and Saturn. Jupiter's moon Io has volcanoes that loft neutrals and positive ions into the inner magnetosphere, and there is also a similar effect at Saturn's moon Enceladus. Centrifugal force is stronger than gravity near Io, and the region beyond Io is consequently interchange unstable. In the simulations, plasma develops finger-like structures, moving outward because of centrifugal force and azimuthally because of Coriolis force. The clearest magnetospheric signature of interchange transport occurs in the inner magnetosphere of Saturn, where the hot plasma injection-dispersion structures are evident (Hill et al. 2005 and references therein).

**CRCM (Comprehensive Ring Current Model):** This model (Fok et al. 2001) was similar to the classic RCM, except that it used a much more complete equation for the distribution function. Whereas the classic RCM assumed an isotropic pitch-angle distribution, CRCM assumes conservation of the first and second invariant. Additional terms account for precipitation and charge-exchange losses and pitch-angle scattering. CIMI (Comprehensive Inner Magnetosphere Model) includes radiation belt electrons and the plasmasphere (Fok et al. 2014).

**RCM-E:** In the classic RCM, the magnetic field was not required to satisfy the force balance equation $J \times B = \nabla P$ but $P = (2/3)V^{-\gamma} \sum_k |\lambda_k| \eta_k$ and $\eta_k$ was based on the theoretical equation (80), with $\gamma = 5/3$. The RCM-E (equilibrium) is run for a small time step (typically 1–5 minutes), and a modified MHD code, called the “friction code”, recalculates the magnetic field in order to make it approximately consistent with $J \times B = \nabla P$ (Lemon et al. 2003). For conditions of strong convection, the time development of the RCM-E would cause the magnetic field to be tail-like and more like a substorm growth phase. If $PV^\gamma$ was constant on the nightside large L-boundary, it became difficult to form a realistic strong ring current (Lemon et al. 2004). The RCM-E usually exhibited the pressure balance inconsistency (Erickson and Wolf 1980). In other words, the more theoretically consistent model became a less realistic representation of observations. Other modelers have developed models that are variants of the RCM-E — e.g., Chen et al. (2012) or the RAM-SCB model (Zaharia et al. 2006) — that use an alternative ring current and force balance model. The solution to the pressure balance inconsistency turned out to be bursty bulk flows (BBFs), which are localized regions of the inner and middle plasma sheet (Angelopoulos et al. 1992) that flow rapidly earthward. The flow bursts, which are also often called “bubbles”, often move very fast (typically 400 km/s). These flows correspond to regions of low $PV^\gamma$ (Pontius and Wolf 1990). Angelopoulos et al. (1992) found that BBFs account for a large fraction of the total earthward flow in the plasma sheet. BBFs usually terminate about the inner edge of the plasma sheet, although some of the fast flows penetrate the ring current (Gkioulidou et al. 2014; Yang et al. 2016). Lemon et al. (2004) produced a substantial ring current injection by reducing the $PV^\gamma$ at the RCM’s outer boundary over a limited region of local time, simulating a ring current injection during a storm. Yang et al. (2014a) showed a possible relation to the streamers observed in the polar cap and bubbles in the plasma sheet, and Yang et al. (2014b) argued that this effect could account for pressure balance inconsistency and that during storms low entropy flux tubes could account for up to 60% of the ring current (Yang et al. 2015). See also the Sect. 9.5 below.

**RCM-I (RCM-Inertial):** The main problem with using RCM-E to represent BBFs is they move so fast that the assumption of force balance is not valid. Yang et al. (2019) developed a more complex version of the RCM that includes inertial effects in a very approximate way. Equation (85) is replaced by a much more complex expression that involves a $\partial\Phi/\partial t$ term.
## 9.5 Large, Coupled Models That Include RCM

Single fluid, global MHD models have become powerful tools in recent years (e.g., Lyon et al. 2004; Raeder et al. 2001, Zhang et al. 2019). However, these models do not capture all the important physics in the inner magnetosphere where gradient and curvature drifts become important but are neglected in MHD. Coupling these models is a daunting task as the modeling regions overlap in space and information is fed back and forth between them. There are different physics assumptions associated with each model: the RCM model assumes slow flow and force balance and neglects waves, while MHD does not. However, MHD does not include energy-dependent drifts that are important in the inner magnetosphere. There have been several successful efforts to couple the RCM with global MHD, which provides many of the inputs used by the RCM (boundary and initial conditions) such as the magnetic field, plasma density, and pressure as well as the ionospheric potential. In return, the RCM provides the density and pressure that are derived from computing the moments from the RCM distribution function that have been subject to energy-dependent drifts.

**SMWF:** The earliest successful coupling effort was De Zeeuw et al. (2004) which merged the BATS-R-US Global MHD (Tóth et al. 2005) code with the RCM. The RCM was embedded in the Global MHD code as a subroutine that later became part of the Space Weather Modeling Framework (SWMF) (Tóth et al. 2005). Each coupling exchange requires computing many field-line integrals to obtain the flux tube volume (Eq. (78)), which is used by the RCM, and the mapping of the 2D RCM quantities into the 3D domain of the MHD code, which is then used to update the MHD. This field line tracing requires using a parallelized and efficient field line tracer that exploits the nested adaptive grid used in the MHD code (e.g., De Zeeuw et al. 2004). This version of the model demonstrated that including the RCM increased the pressure in the inner magnetosphere ring current region as compared to standalone MHD. The RCM also was able to model the Region-2 currents in the ionosphere. Later versions of the SWMF also include other inner magnetosphere models such as the Comprehensive Ring current model (CRCM) (Glocer et al. 2013) and other models (Tóth et al. 2012), becoming the first coupled magnetosphere model to be used in the NASA's Community Coordinated Modeling Center (CCMC).

**LFM-RCM:** Pembroke et al. (2012) coupled the RCM to the Lyon Fedder Mobary (LFM) global MHD code (Lyon et al. 2004) that included the MIX ionosphere model (Merkin and Lyon 2010). This approach used a loose coupling scheme where the models (LFM, MIX, and RCM) ran independently as separate processes and used the InterComm software package to exchange information at pre-set intervals (Lee and Sussman 2004). In this model, RCM returned both pressure and density to the MHD code and included a simple static plasmasphere based on the Gallagher et al. (2000) empirical model. Since the RCM was only tracking the distribution function and not computing the potential as in the standalone RCM, the assumption of zero dipole tilt could be relaxed in the coupled model, allowing for more realistic simulations. The resulting coupled model was very dynamic, especially during geomagnetic storm simulations, and significantly impacted the ring current region (e.g., Wiltberger et al. 2017). To keep the code stable, the RCM boundary was restricted to regions where the field line average plasma beta was less than 1. With moderately strong solar wind driving, the coupled model produced a strong ring current and Region-2 currents.

**OpenGGCM-RCM:** Hu et al. (2010) and Cramer et al. (2017) coupled the OpenGGCM global MHD code to RCM (Raeder et al. 2001) that also includes the Coupled Thermosphere-Ionosphere Model (CTIM) (Fuller-Rowell et al. 1996). The RCM is embedded within the MHD code, where the feedback to the MHD code used a configurable
ramp-up region based on the strength of the magnetic field for numerical stability. Cramer et al. (2017) found that most of the transport of plasma to the inner magnetosphere is via low entropy bubbles, consistent with Yang et al. (2015). Raeder et al. (2016) also used the coupled model to simulate a geomagnetic storm and showed that it developed subauroral polarization streams (SAPS) from electron precipitation computed from the MHD code. Hu et al. (2011) examined the entropy profile in an idealized Open GGCM simulation and found that violations of the frozen-in-flux in MHD could lead to an entropy profile that produced a low entropy bubble that was earthward of an entropy enhancement. Such a configuration causes the bubble/blob pair to move earthward/tailward, which thins the current sheet in the region between them and can ultimately result in tearing or other configuration changes.

**MAGE:** The newest version of a global magnetosphere model is the Multiscale Atmosphere Geospace Environment Model (MAGE) that couples the RCM to the Grid Agnostic MHD for Extended Research Application (GAMERA) global MHD code (Zhang et al. 2019; Sorathia et al. 2021), the ReMIX ionosphere model (Merkin and Lyon 2010) which is a revised version of the MIX solver, and the NCAR Thermosphere-Ionosphere-Electrodynamics General Circulation Model (TIE-GCM) (Roble et al. 1988). The GAMERA MHD model is derived from the LFM model but with improved numerical algorithms and updated software designed for efficient use on modern supercomputers. The coupling to the RCM has also been significantly improved and modernized, for example, it uses a highly configurable and customizable parallel field line tracer. Other improvements include moving the RCM boundary in MAGE further from the Earth compared to the coupled LFM-RCM code, allowing more plasma from the plasmasheet to move into the RCM modeling region, and the option of a Maxwellian distribution to compute RCM distribution functions replaced with a Kappa distribution (Sciola et al. 2023). The new model also includes improved loss rate mechanisms for electrons (Bao et al. 2023) where the electron precipitation model is based on RCM-computed electron energy fluxes that are used to modify the ionospheric conductances (Lin et al. 2022a). The use of this conductance model was found to influence the formation of the SAPS channel (Lin et al. 2021). The model also includes a dynamic plasmasphere density that is tracked using a zero-energy channel in the RCM that is fed back to the MHD model (Bao et al. 2023). Pham et al. (2022) used the coupled MAGE model to investigate the impact on thermospheric density perturbations produced by traveling ionospheric disturbances. Sciola et al. (2023) found that in the MAGE model, fast magnetospheric flows associated with low entropy channels can contribute over 50% of the ring current population, consistent with Yang et al. (2015) and Cramer et al. (2017).

## 9.6 Summary

While the RCM does not model reconnection, it is impacted by it. Reconnection in the tail produces low entropy flux tubes that rapidly interchange their way toward the Earth (e.g., Wiltberger et al. 2015; Sorathia et al. 2021). Some of these flux tubes make it into the inner magnetosphere and play an important role in the formation and structure of the ring current region. It can also have ionospheric effects such as the formation of streamers. Over the years, the RCM has helped illuminate the impact of processes in the tail on the inner magnetosphere. This is especially true using the new generation coupled models that have been developed in recent years. However, there are several limitations in the models that present challenges. One is the modeling of the region in the tail where the magnetic field is transitioning from a stretched tail-like configuration to a dipole. When fast flows appear in this region, as they often do, neither MHD, which neglects gradient/curvature drifts, nor the RCM, which assumes slow flow, are applicable.
Furthermore, including the RCM in MHD models can also affect the location and effec-
tiveness of dayside reconnection as well. Since the Region-2 FACs modeled by the RCM
shields the inner magnetosphere from convection, it also strongly affects the distribution
of return flow back to the dayside boundary. The RCM has been successful in modeling
the plasmaspheric plumes that can bring dense plasmas to the dayside reconnection re-
gions (Goldstein et al. 2002; Huba and Sazykin 2014; Bao et al. 2023). Dayside SWMF
runs that include RCM all successfully place MMS within 1 $R_E$ of at an X-line (or separa-
trix) whenever clear EDRs are observed (e.g. Reiff et al. 2017).

The next generation coupled models will need to add the effect of ionospheric plasma
sources (e.g., Glocer et al. 2009a; Varney et al. 2016), which will ultimately require a multi-
fluid MHD model coupled to a ring current model and includes a model of ionospheric
outflow to track all the species in the magnetosphere.

10 Conclusions

In this paper, we have presented a “brief” overview of the large collection of computational
methods that are used to study magnetic reconnection. It should be clear to the reader of this
text, that simulating magnetic reconnection is nearly a separate field of study in and of itself.
It should be and has been (Büchner et al. 2003) the topic of entire books. The single element
to take away from this paper is that simulating a multifaceted and multiscale problem like
magnetic reconnection is not a simple endeavor. Both plasma models and simulation initial
conditions must be tuned carefully to match the goals of the study.

We have reviewed simulation methods for magnetic reconnection in space plasmas, from macroscopic MHD scales to microscopic kinetic scales. Basically, macroscopic plasma behaviors can be simulated based on fluid modes, and as we resolve smaller scale physics, kinetic models need to be implemented in simulations. In addition, we have reviewed novel approaches to incorporate multi-scale physics.

MHD simulations are useful to study large scale physics, including performing global
simulations for planetary magnetospheres. We have discussed the basic algorithm for MHD
simulations, and also how to implement test particles that follow MHD fields. Hall MHD
simulations contain Hall physics, which allows kinetic scale waves to propagate, mediating
fast reconnection. We have reviewed recent progresses of Hall MHD studies, and also the
effect of the electron inertia term and EMHD.

Hybrid PIC simulations and full PIC simulations include particle kinetic physics, where particle motions are directly solved by equations of motion of particles. For hybrid simulations, we have reviewed techniques to overcome the limitation of spatiotemporal resolution in global models, and also the implementation of electron kinetic physics into hybrid simulations. For full PIC simulations, we have reviewed simulation studies of magnetic reconnection in the magnetotail and magnetopause, particle acceleration, and shock driven reconnection.

Next, we have reviewed two novel approaches to address multi-scale physics in magnetic
reconnection: embedded PIC, and kglobal. In the embedded PIC approach, the macro-scale
region is solved using MHD equations, and local kinetic domains are embedded in the MHD
domain, where full PIC techniques are employed. In kglobal, on the other hand, equations
for the ion and electron fluids are combined with the particle equations for electrons, and
the macro-scale evolution is modified by the kinetic physics.

Finally, we have reviewed two types of other simulation techniques: Vlasov simulations
and the Rice convection model. In Vlasov simulations, kinetic effects are implemented in
simulations by solving 2D-3V or 3D-3V Vlasov equations. We have discussed recent pro-
gresses of studies by Vlasov models from local reconnection simulations to global simu-
lations. The Rice convection model is a kinetic approach to simulate physics of the inner
magnetosphere, where bounce averaged particle drift motion is taken into account.

**10.1 Outlook**

We first wish to stress that even relatively simple numerical models continue to shed light on the essential physics of magnetic reconnection, even though some of these models have been around for decades. A good example of this is the explanation of fast magnetic reconnection in collisionless plasmas—the longstanding Reconnection Rate Problem. Only in the last few years, built upon the endeavors of previous theoretical efforts, has a convincing theory that passed the cross-examination of all these numerical models (i.e., PIC, hybrid, Hall-MHD, EMHD, resistive-MHD, pair plasmas) been developed to offer the rate prediction (Liu et al. 2022). Comparing and contrasting “numerical experiments” of reconnection in these models provides invaluable, rigorous constraints to a theory. Such constraints from different models will continue to play a pivotal role in the theory development of any nonlinear phenomenon in plasmas, especially when one attempts to discern the “cause” from the “consequences”.

A major driver on the progress of simulations of reconnection has been the uncanny steady exponential increase in computing power known as Moore's law (Schaller 1997). If technology manages to continue Moore's law, current simulations will undoubtedly continue to yield major insights. For example, fully kinetic particle in cell simulations are only just now reaching around 100 ion inertial scale sizes in three dimensions. However, such massive simulations require extremely large data storage, with significant environmental implications.

Beyond increasing computational power, more efficient algorithms continue to be a
source of study. For example, GPUs have allowed a significant increase in computational
speed for modest sized systems (Bard and Dorelli 2014). For global hybrid models, novel
algorithms that drive particles based on “events” rather than time steps are another new
direction. The HYPERS global hybrid simulations (Omelchenko and Karimabadi 2012) de-
scribed in Sect. 4 is one such example.

A major thrust in the coming decade is expected to be the feedback between the multiple disparate scales associated with reconnection. One such multi-scale example is the connection between microscales and mesoscales during reconnection at the dayside of Earth's magnetosphere, where the formation of x-lines depends strongly on magnetic geometry. Coupling different models, as described in Sect. 6, is a promising way forward to address these questions, but there are significant challenges to overcome, especially in regards to boundary conditions and disparate length and time scales.

Another example of such multi-scale physics is reconnection particle energization throughout the heliosphere. A major challenge in the effort to understand the energization of electrons and ions has been the inability to explore the kinetic dynamics of particles in very large systems. Because particle-in-cell (PIC) and hybrid models have to resolve kinetic scales, which are a small fraction of typical macroscales (a factor of around 10⁻¹⁰ in the case of solar flares). The consequence is that although PIC simulations have been successful in identifying some of the dominant acceleration mechanisms of electrons and ions (Drake et al. 2006; Dahlin et al. 2014; Guo et al. 2014; Li et al. 2019a), their success in producing the extended powerlaws seen in flare and magnetotail observations has been limited. The largest PIC simulations have revealed electron powerlaws that extend only a single decade in energy (Li et al. 2019a) while observations reveal that powerlaws in flares extend across
many decades in energy (Lin et al. 2003; Vilmer 2012). The production of powerlaw distri-
butions of ions has been an even greater challenge (Zhang et al. 2021, 2024). However, PIC
modeling revealed that Fermi reflection during the growth and merger of large-scale flux
ropes dominates particle acceleration (Dahlin et al. 2016; Guo et al. 2014; Li et al. 2019a),
a result that led to a new computational model, kglobal, that combined MHD fluid and
particle descriptions while eliminating the kinetic scales that constrained the macro-scale
modeling of flares (Drake et al. 2019; Arnold et al. 2019). A major accomplishment was the
first exploration of magnetic reconnection-driven electron acceleration in macro-systems
which produced electron powerlaw distributions extending nearly three decades in energy
and revealed that the ambient guide field is the dominant control factor of the powerlaw in-
dex (Arnold et al. 2021). The success of the kglobal model motivates two major extensions:
to include particle ions so that the partitioning of energy between the two species can be ex-
plored; and to incorporate the kglobal particle algorithm into a global simulation code such
as the Adaptively Refined MHD Solver (ARMS) (DeVore 1991) flare simulation code or the
Block Adaptive Tree Solar-wind Roe Upwind Scheme (BATS-R-US) (Gombosi et al. 2002).
The resulting model would be able to simulate macroscale energy release during magnetic
reconnection with energetic electrons and produce synthetic photon spectra for comparison
with remote observations of solar flares observations.

**Funding Information** We acknowledge support from NASA grant 80NSSC20K0198 (MAS), NASA 80NSSC20K1813 (MAS), NASA 80NSSC24K0172 (MAS and PAC), NSF grant AGS-2024198 (MAS), NSF PHY-2308669 (PAC); DE-SC0020294 (PAC); NASA 80NSSC22K0323 (PAC); NASA 80NSSC23K0409 (PAC); NASA 80NSSC18K0834 (JB) and 80NSSC24K1452 (JB). We acknowledge high-performance computing support from the Derecho system (https://doi.org/10.5065/qx9a-pg09) provided by the NSF National Center for Atmospheric Research (NCAR), sponsored by the National Science Foundation. We also acknowledge the National Energy Research Scientific Computing Center (NERSC), a U.S. Department of Energy Office of Science User Facility operated under Contract No. DE-AC02-05CH11231.

Declarations

**Competing Interests** The authors have no competing interests to declare that are relevant to the content of this article.

**Open Access** This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.

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## Authors and Affiliations

Michael Shay¹ ID · Subash Adhikari²,¹ · Naoki Beesho³ · Joachim Birn⁴ · Jörg Büchner⁵ ·

Paul Cassak² · Li-Jen Chen⁶ · Yuxi Chen⁷ · Giulia Cozzani⁸,⁹ · James Drake¹⁰,¹¹ ·

Fan Guo¹² · Michael Hesse¹³ · Neeraj Jain¹⁴ · Yann Pfau-Kempf⁸,¹⁵ · Yu Lin¹⁶ ·

Yi-Hsin Liu¹⁷ · Mitsuo Oka¹⁸ · Yuri Omelchenko⁴,¹⁹ · Minna Palmroth⁸ · Oreste Pezzi²⁰ ·

Patricia H. Reiff²¹ · Marc Swisdak¹¹ · Frank Toffoletto²¹ · Gabor Toth⁷ · Richard A. Wolf²¹

<figure><img src="image_22.png" /></figure>

M. Shay
shay@udel.edu

<figure><img src="image_23.png" /></figure>

G. Cozzani
giulia.cozzani@cnrs-orleans.fr

1. Bartol Research Institute, Department of Physics and Astronomy, University of Delaware, Newark, 19716, DE, USA

2. Department of Physics and Astronomy, West Virginia University, Morgantown, 26506, WV, USA

3. Department of Astronomy, University of Maryland, College Park, 20742, MD, USA

4. Center for Space Plasma Physics, Space Science Institute, Boulder, 80301, CO, USA

5. Max Planck Institute for Solar System Research, Göttingen, 27077, Germany

6. NASA Goddard Space Flight Center, Greenbelt, 20771, MD, USA

7. University of Michigan, Ann Arbor, 48109, MI, USA

8. Department of Physics, University of Helsinki, P.O. Box 68, 00014, Uusimaa, Finland

9. LPC2E, OSUC, Univ Orleans, CNRS, CNES, Orleans, F-45071, France

10. Department of Physics, University of Maryland, College Park, 20740, MD, USA

11. Institute for Research in Electronics and Applied Physics, University of Maryland, College Park, 20740, MD, USA

12. Los Alamos National Laboratory, Los Alamos, 87545, NM, USA

13. NASA Ames Research Center, Moffett Field, 94035, CA, USA

14. Center for Astronomy and Astrophysics, Technical University Berlin, Berlin, 10623, Germany

15. Now at Advanced Computing Facility, CSC - IT Center for Science, Espoo, 02101, Uusimaa, Finland
16 Physics Department, Auburn University, Auburn, 36832, AL, USA

17 Department of Physics and Astronomy, Dartmouth College, Hanover, 03750, NH, USA

18 Space Sciences Laboratory, University of California, Berkeley, 94720, CA, USA

19 Trinum Research Inc., San Diego, 92126, CA, USA

20 Istituto per la Scienza e Tecnologia dei Plasmi (ISTP), Consiglio Nazionale delle Ricerche, Bari, I-70126, Italy

21 Department of Physics and Astronomy, Rice University, Houston, 77005, TX, USA

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