A Volumetric Study of Flux Transfer Events at the Dayside Magnetopause

The numerical approach involves solving a set of resistive MHD equations in three dimensions using the MHD module of the PLUTO code (Mignone et al. 2012). The module solves a set of conservation laws in the form of single-fluid MHD equations given as

$$\begin{eqnarray}\begin{array}{rcl}\displaystyle \frac{\partial \rho }{\partial t}+{\rm{\nabla }}.(\rho {\boldsymbol{v}}) & = & 0\\ \displaystyle \frac{\partial (\rho {\boldsymbol{v}})}{\partial t}+{\rm{\nabla }}.\left[\rho {\boldsymbol{v}}{\boldsymbol{v}}-{\boldsymbol{B}}{\boldsymbol{B}}\right]+{\rm{\nabla }}\left(p+\displaystyle \frac{{{\boldsymbol{B}}}^{2}}{2}\right) & = & 0\\ \displaystyle \frac{\partial {\boldsymbol{B}}}{\partial t}+{\rm{\nabla }}\times (c{\boldsymbol{E}}) & = & 0\\ \displaystyle \frac{\partial {E}_{t}}{\partial t}+{\rm{\nabla }}.\left[\left(\displaystyle \frac{\rho {{\boldsymbol{v}}}^{2}}{2}+\rho e+p\right){\boldsymbol{v}}+c{\boldsymbol{E}}\times {\boldsymbol{B}}\right] & = & 0,\end{array}\end{eqnarray} \tag{ 1 }$$

where ρ is the mass density, v is the gas velocity, p is the thermal pressure and B is the magnetic field. A factor of 1/$\sqrt{4\pi }$ has been absorbed in the definition of B . E_t is the total energy density, which can be described as

$$\begin{eqnarray}&&{E}_{t}=\rho e+\displaystyle \frac{\rho {{\boldsymbol{v}}}^{2}}{2}+\displaystyle \frac{{{\boldsymbol{B}}}^{2}}{2}.\end{eqnarray} \tag{ 2 }$$

An ideal equation of state provides the closure as ρ e =p/(γ − 1), wherein γ is the ratio of specific heats having a value of 5/3. The electric field, E, is composed of a convective and a resistive component, and is defined as

$$\begin{eqnarray}&&c{\boldsymbol{E}}=-{\boldsymbol{v}}\times {\boldsymbol{B}}+\displaystyle \frac{\eta }{c}{\boldsymbol{J}},\end{eqnarray} \tag{ 3 }$$

where η is the resistivity and J = c∇ × B is the current density. To detach the dependence of the reconnection process on the numerical resistivity of the system, we incorporate an explicit resistivity in the domain having a value of η/μ₀ = 1.27 × 10⁹(m² s⁻¹). It is expected that the inclusion of an explicit resistivity will broaden the current layer at the magnetopause (Komar 2015). The chosen explicit resistivity was motivated by its outcome that the full width at half maxima (FWHM) of the current layer be resolved by no less than six to seven grid cells. This, in turn, also ensures that the entire span of the magnetopause current sheet is always resolved by at least 12 to 14 grid cells. Due to the high computational costs, we have refrained from performing parameter dependence studies on the value of this explicit resistivity. The flux computations have been performed with the second-order accurate Harten–Lax–vanLeer (HLL) solver, and the solenoidal constraint (∇ · B = 0) is enforced by coupling the induction equation to a Generalized Lagrange Multiplier (GLM) and solving a modified set of conservation laws in a cell-centered approach (Dedner et al. 2002; Mignone et al. 2012).

Additionally, we have adopted a formalism where the total magnetic field B inside the domain is treated in a split configuration given as

$$\begin{eqnarray}&&{\boldsymbol{B}}(x,y,z,t)={{\boldsymbol{B}}}_{0}(x,y,z)+{{\boldsymbol{B}}}_{1}(x,y,z,t),\end{eqnarray} \tag{ 4 }$$

where B₀ is a curl-free, time-invariant, background magnetic field, and B ₁ behaves as a deviation. For such a configuration, the energy depends only on the deviation B ₁, which turns out to be computationally convenient when dealing with magnetically dominated systems.

The above set of resistive-MHD equations are solved in a Cartesian box enclosed by − 30 R_E ≤ X ≤ 130 R_E , − 100 R_E ≤Y ≤ 100 R_E , and − 100 R_E ≤ Z ≤ 100 R_E , where R_E denotes the radius of Earth. The domain has a base resolution of 64 ×80 × 80 grid cells, which is refined using a block-structured AMR strategy. The AMR strategy only refines the regions where a higher resolution is required (e.g., the dayside magnetopause), keeping the remaining computational grid coarse. This strategy therefore proves to be extremely beneficial in cases with a high disparity of scale sizes, and it saves a significant amount of computational expense. The details of the AMR implementation in PLUTO can be found in Mignone et al. (2012). The mesh is adaptively refined up to four refinement levels in a hierarchical manner where the first two and the last two AMR levels have respective refinement ratios of 2× and 4×. This leads to an effective resolution of 4096 × 5120 × 5120, which sets the smallest grid size at Δx = Δy = Δz = 0.039R_E . The refinement criterion is set to be the current density J , which dictates that regions with sharp gradients in J will be recursively refined to finer resolutions.

An internal boundary (IB) is set at radii r ≤ 4 R_E , inside which the values of the MHD variables are kept fixed. A thermal pressure of 6.69 × 10⁻³ nPa and a density proportional to the local magnetic field strength are prescribed inside the IB. The IB acts as a sink, and any MHD variable that flows in to the IB is overwritten by these fixed values. The magnetic field is considered to be purely dipole-like inside the IB, and the aforementioned density profile prevents the Alfvén speed from attaining large values at the IB, which assists in limiting the simulation time step. Plasma-β is defined as the ratio of the gas pressure to the magnetic pressure. In low-β plasmas (generally found in regions of high magnetic field strength in global MHD simulations), negative pressure values may arise due to truncation errors while calculating the thermal pressure, generally expressed as a difference of two very large numbers (following Equation (2)). To ensure that such unphysical values of pressure do not arise in the domain, the equation of entropy is also added to the set of conservation laws given by Equation (1). Even though the energy equation is used everywhere in the domain, the pressure is computed from the entropy after AMR operations such as coarse-to-fine prolongations and restrictions (Mignone et al. 2012). This procedure is one of the many techniques that can be used to ensure pressure positivity in simulations where the magnetic energy is the dominant contribution to the total energy density (Lyon et al. 2004; Ridley et al. 2010; Li et al. 2021).

To emulate the dipolar magnetic field of an Earth-like planet, the configuration for B ₀ in Equation (4) is prescribed as a magnetic dipole, placed at the origin of this box, having an equatorial field strength of 3 × 10⁻⁵ T at 1R_E . The axis of the dipole is tilted by 5° with respect to the Z-axis, with its south pole leaning toward the Sun. This small tilt helps provide a preferred direction for the propagation of transient reconnection events, such as FTEs on the magnetopause surface. The entire domain is initially filled with a very low density of 0.5 amu cm⁻³. Thereafter, a solar wind inflow is prescribed at the left X boundary, with a modest plasma inflow speed of v_sw = 400 kms⁻¹, a density of ρ_sw = 5 amu cm⁻³, and a pressure of P_sw = 3 × 10⁻² nPa. An interplanetary magnetic field with components B_y(sw) = 5 nT and B_z(sw) = −5 nT is also prescribed at the inflow boundary; it then propagates into the domain with the solar wind. The input conditions correspond to typical solar wind parameters at 1 au (Gosling 2014; Klein & Vech 2019), and similar inflow conditions have previously been used in the literature for numerical studies of FTEs (Sun et al. 2019). Once the inflow is established, the sonic (M_S ) and the Anfvénic Mach numbers (M_A ) are calculated to be M_S ∼ 4.9 and M_A ∼ 5.9, respectively. The remaining five boundaries are set to have Neumann boundary conditions, which dictate that all MHD variables have zero gradient across these boundaries.

As the simulation starts, the solar wind plasma flows in from the left X-boundary and interacts with the magnetic field of the dipole, compressing it on the dayside and stretching it out on the nightside, forming a paradigmatic magnetosphere as seen in panel (a) of Figure 1. The panel shows a volumetric rendering of the simulated magnetosphere

Zoom In Zoom Out Reset image size

During the early phase of the simulation, the evolving mesh refines the regions of strong gradients in current density as directed by the AMR refinement criteria. The dayside magnetopause region is expected to attain a local maxima of the current density, due to the southward IMF of the incoming flow. We find that this region is indeed adaptively refined to the finest level, and in due course, the polar cusps are refined as well. The magnetopause eventually attains a nearly steady state after t ∼ 960 s (in units of physical time) as shown in panel (b) of Figure 1. In Figure 1, the resolution of the base grid (Δx = Δy = Δz = 2.5R_E ) in each of the three dimensions is refined to 1.25 R_E , 0.625 R_E , 0.156 R_E , and 0.039 R_E by the AMR levels 1, 2, 3, and 4, as denoted by the legend. The white circle centered at the origin represents the internal boundary.

First, we identify the dayside magnetopause in our simulated magnetosphere as an isosurface of the maximum value of the current density J within −8 R_E ≤ Y ≤ 8 R_E and −5.5 R_E ≤ Z ≤ 5.5 R_E . This exercise has been done during a quiet time when the surface is seemingly stationary and devoid of any transient phenomena. As a validation step, we then compared this detected surface to the analytical model of Cooling et al. (2001), which considers the magnetopause to be a paraboloid of revolution. We confirm that the two are well in agreement. In particular, we compared the X coordinate of a sample of more than 10⁵ points from the simulated magnetopause surface to their corresponding analytical positions and found an excellent agreement with a correlation coefficient of 0.99 and a root mean square (rms) deviation of ∼0.1R_E across the entire surface. The detected magnetopause, in addition to serving as a validation step, also has multiple additional uses, as will be evident in the following sections.

Due to the presence of a significant magnetic shear, as a consequence of the southward IMF, the dayside magnetopause exhibits a local maximum of the current density (J) and thereby turns out to be a prime location for magnetic reconnection. The following paragraphs describe the process of magnetic reconnection from a magnetohydrodynamic perspective, in order to elucidate how the process occurs within a resistive MHD scenario. Magnetic reconnection occurs in resistive-MHD simulations, due to the presence of an explicit resistivity in the induction equation (Equation (3)), which facilitates dissipation. Within the context of MHD, regions of high current density locally enhance the contribution of the second term in Equation (3), leading to the formation of diffusion regions at ion scales, where the frozen-in constraint of the magnetic field lines breaks down.

Panel (a) of Figure 2 shows a cross-sectional slice (2D representation) across the axis of a typical FTE formed due to magnetic reconnection at the magnetopause. The magnetospheric dipolar field lines are marked in solid red color, whereas the solar wind magnetic field lines on the magnetosheath side are marked in solid blue color. Two ion diffusion regions marked as "IDR" are shown in the schematic. These IDR are the sites where the magnetic field lines of the magnetopause and the magnetosheath reconnect (dotted lines of the corresponding colors). As represented in a 2D projection, the change in topology of the magnetic field lines within the IDR causes a loop-like magnetic field configuration in the region between the two IDR marked in solid black field lines. The green arrows pointing into the IDR and the orange arrows pointing away from the IDR represent the reconnection inflows and exhausts, respectively. A portion of the hot plasma exhausts from the IDR creates a plasma bulge at the boundary layer. Such a plasma bulge in addition to the characteristic loop-like magnetic field topology constitutes an FTE. In 3D, however, the magnetic field topology is generally more complex. The IDR can extend in the third dimension (into and out of the plane in panel (a) of Figure 2) up to an arbitrary length, and the loop-like magnetic field lines then exhibit the configuration of a helical rope called a flux rope. Such sporadic, patchy reconnection at the magnetopause boundary layer gives rise to multiple FTEs, which materialize in the form of transient flux ropes.

Zoom In Zoom Out Reset image size

Panel (b) shows a slice of a typical flux rope along its axis on the boundary layer (blue curved surface), where the black dotted line represents the flux-rope axis and the solid black arrows represent the field line configuration of the portion of the flux rope above the boundary layer. For such a slice, any constituent helical magnetic field line of the flux rope intersects the boundary layer twice. The magnetic field lines in the upper half of the flux-rope axis are seen to be moving out of the boundary layer, whereas the field lines on the lower half are seen to be moving into the boundary layer. In situ probes generally measure the B_N component at the boundary layer, which is the component of the total magnetic field perpendicular to the boundary layer. Typical in situ FTE detections therefore constitute a bipolar signature in the B_N component as the FTE passes through the probe (negative due to the lower portion of the FTE and positive due to the upper portion, or vice versa). Such a bipolar signature can also be correspondingly identified in the B_X component of the magnetic field in the Geocentric Solar Magnetospheric (GSM) coordinate system, though one must note that modeling smaller-scale reconnection signatures such as the quadrupolar Hall fields or lower hybrid drift waves in addition to particle distribution functions in reconnection regions requires more sophisticated and computationally intensive numerical frameworks such as Hall-MHD, hybrid-Vlasov, or particle-in-cell modeling approaches (Chen et al. 2017; Hoilijoki et al. 2019; Akhavan-Tafti et al. 2020b).

Figure 3 is a depiction of the temporal evolution of FTEs that form in our simulation. The subplots show a zoomed-in view to distinctly feature the regions of interest. The columns represent different quantities, whereas the rows represent different time snapshots. The corresponding physical times are encased in boxes to the left of each row in Figure 3. Panels (a) and (b) show the X–Z slices at y = 4.0. Panel (a) shows the magnitude of current density ∣J∣, highlighting the bow shock and the magnetopause surface. One can also see two distinct FTEs on the magnetopause surface in panel (a), denoted by the arrows. The FTEs can also be similarly seen in panel (b), which represents the pressure on the same slice as panel (a). Panel (c) shows a projection of the B_x component on the magnetopause surface. The B_x component shows bipolar features due to the specific magnetic field configuration within the FTEs. One can see these signatures corresponding to the two FTEs of panels (a) and (b) at the location marked by the arrows in panel (c). The dotted vertical line in panel (c) marks the Y coordinate corresponding to the slices shown in panels (a) and (b). The next row (panels (d), (e), and (f)) are plotted after a time gap of ∼64 s. The FTE that was located at the top in panels (a), (b), and (c) has now moved out of the plot, and the other FTE has visibly grown in size. It can also be seen that the bipolar B_x signature of the other FTE has strengthened as well. The bottom row is plotted after another 64 s. It shows that this FTE has now significantly grown in size, and one can also see the initial stages of a new FTE forming at z ∼ − 2.5. Such transient features are persistent throughout our simulation and remain in the region resolved by the finest grid size for a significant amount of time, allowing for a detailed study of their evolution. We note here for clarity that the FTEs shown in Figure 3 are representative of the multiple transient features produced in our simulations and are separate from the ones that have been extensively analyzed later in this paper. We also highlight the fact that the prescription of a southward IMF at the inflow boundary and the use of an explicit resistivity to imitate the dissipation of magnetic energy in regions of high magnetic shear are sufficient conditions to initiate magnetic reconnection, leading to the production of FTEs within our simulation. Figure 4 is a representation of the magnetic field line connectivities near the magnetopause surface at t ∼ 3364 s. To avoid overcrowding the plot, we have only plotted a small fraction of the traced magnetic field lines. The field line connectivities show clear signatures of magnetic reconnection wherein the incoming solar wind magnetic field lines and the Earth's magnetospheric field lines reconnect to produce magnetic fields that are connected to the solar wind at one end and the ionosphere/ northern or southern hemisphere at the other end. These lines are labeled as "SW—NH" and "SH—SW," denoting "Solar Wind to Northern Hemisphere" and "Southern Hemisphere to Solar Wind" connectivities, respectively. One can also see a glimpse of a helical magnetic flux rope (dark blue field lines) toward the right side of the picture, as marked by the arrow. This flux rope is, in fact, a depiction of the early stages of one of the FTEs that will be analyzed later in this paper (namely FTE-3; see the following section for nomenclature).

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Due to the lack of control over the exact location on the magnetopause where the reconnection occurs in the simulation, the FTEs generally tend to have an arbitrary extent along each coordinate, which makes it challenging to determine their complete three-dimensional structure. In this study, we have used an agglomerative hierarchical structure finding algorithm to perform volumetric detection of the FTEs within our simulation. A detailed methodology of the volumetric detection has been described in the Appendix for completeness. Using this process as described in the Appendix, we have detected and tracked several FTEs moving along the magnetopause surface with time. From the tens of detected FTEs, we select three FTEs, named FTE-1, FTE-2, and FTE-3, for a further detailed analysis. The FTEs were chosen so that they originate and travel along different directions on the magnetopause surface. This is to ensure a broader coverage of the magnetopause and to ascertain that any bias on the FTE properties arising due to their location could be eliminated. One of the principal magnetic field signatures for FTE identification used in in situ observations is the bipolar B_N component of the magnetic field (Russell & Elphic 1978; Jasinski et al. 2016). The B_N component, being a part of the boundary normal L–M – N coordinate system, represents the projection of the total magnetic field along a direction that is normal to the magnetopause surface. All the FTEs formed in our simulations universally show such distinct bipolar B_N signatures. Panel (a) of Figure 5 shows, in hatched line shading, a projection of the azimuthal extent of the detected volume of FTE-2 obtained from the structure detection algorithm, superimposed on the B_N component at the magnetopause surface. The detected structure is seen to span the entire azimuthal extent (the extent on the Y – Z plane) of the bipolar structure (red–blue junction) correctly. Two slices across this FTE along the red and blue dashed lines in panel (a) are shown in panels (b) and (c), where the shaded regions correspond to the detected FTE structure. Such a validation exercise has been carried out for each of the three FTEs for each time step by visually comparing the detected structure to the B_N component signature in the X – Z plane, as well as along multiple X – Z slices, to ascertain that the entirety of the detected structures indeed lie within the FTE thermal pressure bulge. For all the FTEs detected, our volumetric identification is found to provide a fairly accurate detection within a margin of at most two grid cells inside the FTE structure. We note here that, upon a closer look at panel (a) of Figure 5, one might posit that there exist positive and negative B_N regions that lie outside the detected FTE volume. However, a detailed inspection of panels (b) and (c) reveals that such regions at the top and bottom of the FTE slice are in fact the transitions between the ellipsoidal FTE bulge and the magnetopause layer, and therefore excluding these segments in reality only isolates the cardinal regions to be the dominant portions of the detected FTE structures. We also note that, toward the later stages of FTE evolution, the thermal pressure within the FTE drops as one moves toward the flux-rope core. Such a pressure profile has also been seen in observations (Akhavan-Tafti et al. 2019a). This variation is, however, less prominent during the early stages of FTE evolution.

Zoom In Zoom Out Reset image size

Figure 6 shows the motions of the centers of mass of the selected FTEs with time, as seen when looking from the Sun toward the magnetopause. The FTEs travel along the magnetopause surface, which dictates that their motion along the X direction is governed by the curvature of the magnetopause surface, which is insignificant within the bounds of the plot. The dominant motion of the FTEs is therefore in the Y – Z plane. The center-of-mass motions of the three FTEs can be considered representative of the general motions of the FTE volumes along the magnetopause. The multiple scatter points for each FTE correspond to the different time snaps over which the FTEs were detected and tracked.

Zoom In Zoom Out Reset image size

Further details of the selected FTEs have been summarized in Table 1, where t_start is the global time (in seconds) when the FTEs were first detected by the algorithm. The column titled Δt represents the time duration over which each FTE was tracked. The average speeds of the FTEs (〈v〉) in km s⁻¹ are shown in the fourth column, and the fifth column gives the general locations of the selected FTEs. The average speeds have been estimated from the total distances traveled by the FTEs in the time interval Δt.

As stated before, due to constraints associated with spacecraft trajectories, it is generally difficult to extract precise information about the azimuthal extent of the FTEs, which makes the inference of the total FTE volume from observational data extremely challenging. From a numerical perspective, however, the volumetric detection paves the way for such a study. Figure 7 shows the time evolution of the volume of the selected FTEs in our simulation. The volumes are given in units of ${R}_{E}^{3}$. For comparative purposes, the abscissa denotes time in ${t}_{\det }$, which stands for the "time of first detection." This corresponds to the time elapsed after the FTE was first volumetrically detected in the simulation. For each FTE, ${t}_{\det }=0$ thus corresponds to the global time given in the second column of Table 1.

Zoom In Zoom Out Reset image size

All three selected FTEs initially show an increasing profile in their volume over time. The scatter points in the plot represent the exact values obtained from the analysis, which are then overplotted with cubic splines to highlight the trend. The volume grows, stabilizes, and eventually develops a tendency to decrease when the FTEs are farther away from the subsolar point. Of the three selected FTEs, FTE-3 grows to the largest size, with a final volume of 2.42 ${R}_{E}^{3}$, while the FTE-2 is the one with the smallest maximum size, namely 0.82 ${R}_{E}^{3}$. FTE-1 attains a moderate maximum volume of 1.47 ${R}_{E}^{3}$. We note here that conventional in situ satellite observations measure the flux-rope "size" as an equivalent diameter of a circular FTE cross section traversed by the probe. However, this can lead to significant variations in the modeled radius, depending on the relative trajectory of the FTE and the probe. For example, we find that a cross section of FTE-1 across the midpoint of its azimuthal extent yields a nearly elliptical slice. For a probe passing perfectly through the FTE core (impact parameter = 0), there exist two extreme possibilities. The FTE size measured by the probe along the longest path across the cross section gives a diameter of 1.2 R_E , whereas the shortest path detects a diameter of 0.58 R_E . Size inferences from in situ observations are prone to such large variations, and multipoint observations are therefore essential because they rely on more sophisticated reconstruction techniques to infer the FTE sizes, albeit at a much higher computational cost (Sonnerup et al. 2004, 2006; Trenchi et al. 2016).

A salient feature of the temporal evolution of FTE volumes is that the volumes do not increase indefinitely with time. This is relevant from the perspective of the size distribution of the FTEs observed at the Earth's magnetosphere. All observations consistently agree that the size distribution of the FTEs falls off exponentially with increasing FTE size, indicating that extremely large FTEs are less probable. In addition to the reason that the location of observation (near the subsolar point or further away) can inherently bias the measured FTE size distribution, the fact that the FTE growths are stunted beyond a certain point may be a reasonable cause for this distribution.

We now examine, in more detail, the primary reason for the growth of the FTEs. Akhavan-Tafti et al. (2018) have summarized, within a magnetospheric context, that there can be three primary reasons for FTE growth. (a) The subsolar magnetosheath has a larger thermal pressure as compared to the flanks, and an FTE can grow via adiabatic expansion when it travels away from the subsolar region. (b) The continuous reconnection process along the length of the evolving FTE can feed hot plasma into the flux rope, thereby increasing its size. (c) Analogous to the scenario of plasmoid merging in 2D, multiple smaller flux ropes may coalesce together in a self-similar fashion, giving rise to a larger flux rope. The process of coalescence has additional complexities associated with it, wherein the angles between the axial components of the two flux ropes influence the merging process. Previous observations of planetary magnetospheres have indeed confirmed that any one of the three processes may serve as the dominant cause of FTE growth (Jasinski et al. 2016; Akhavan-Tafti et al. 2019a). In order to have a robust inference of the mechanism of FTE growth, it is preferable to study the individual FTE structures as they evolve. Presently, due to constraints associated with in situ observations, it is only feasible to study the temporal evolution of the thermodynamic characteristics of an individual FTE from a numerical perspective. To probe into this, we examine the P–V (Pressure–Volume) characteristics of the three FTEs, focusing on the dependence of the volume-averaged pressure of the FTEs on their corresponding volumes. For this purpose, we select the regions of steepest descent within the P–V curves for each FTE. These sets of selected points for the corresponding FTEs are highlighted as encircled points in Figure 7.

Of the three reasons of FTE growth cited above, in the case of FTEs 1, 2, and 3, island coalescence is not the cause of the growth of FTE volume, as such a mechanism would have clear signatures of merger in the temporal profile of the B_N component of the magnetic field. As such signatures were not seen for the selected FTEs, the more probable causes therefore are either continuous reconnection, adiabatic expansion or a combination of the two. For adiabatic expansion to be the principal cause of FTE growth, one expects the pressure to fall according to the equation P ∝ V^−γ . For a monoatomic ideal gas, γ = 5/3, therefore a sharp decrease in pressure is expected as the FTEs expand.

The scatter points in Figure 8 plots, for each FTE, the set of encircled points of Figure 7 in a P–V diagram. The ordinate corresponds to the thermal pressure averaged over the FTE volume in nanopascals (nPa), and the abscissa represents the FTE volume in ${R}_{E}^{3}$. The solid lines represent the corresponding power-law fits. FTEs 1, 2, and 3 fit profiles of P ∝ V^−0.134, P ∝ V^−0.063, and P ∝ V^−0.336, respectively. For reference, we have also included the adiabatic expansion profiles (P ∝ V^−1.67) corresponding to the three FTEs passing through the first data point of each FTE in Figure 8. They are represented by the appropriately colored dashed lines. It is clear that the curves obtained from the simulation are incompatible with the expected P ∝ V^−1.67 profile by a substantial margin. Such a consistently weak decrease essentially means that any drop in the thermal pressure is being replenished rapidly as the FTE expands. This replenishment can be attributed to the hot plasma exhausts associated with the ongoing reconnection in the neighboring X-lines entering the flux rope (Drake & Swisdak 2014). We therefore assert that the process of ongoing continuous reconnection is the dominant cause of FTE growth for FTEs 1, 2, and 3.

Zoom In Zoom Out Reset image size

Converting our P–V relation to a pressure–diameter relation assuming V ∝ diameter² for a perfectly cylindrical flux rope of constant length yields a value of P ∝ (diameter)^−0.35, where the exponent is averaged over the three FTEs. The pressure–diameter relation is a dependence that can be inferred directly from observations. An ensemble study on pressure–diameter relations of a set of 55 FTEs observed at the Earth's magnetopause by Akhavan-Tafti et al. (2019a) have also led to a similar relation of P ∝ (diameter)^−0.24, based on which they have concluded that continuous reconnection can be the dominant cause of FTE size increase at the Earth's magnetosphere. A caveat of the pressure–diameter relation is the inherent assumption that the length of the flux rope remains constant throughout the evolution; however, we find from our simulation that this constraint is too strict and the lengths of the flux ropes generally increase as they evolve. Even so, our results are similar to those of Akhavan-Tafti et al. (2019a) and concur with their assertion that the dominant cause of FTE growth at the Earth's magnetopause, under generic conditions, is the process of continuous magnetic reconnection. We also note here that the process of FTE growth by coalescence is indeed seen in certain FTEs from our simulation (but not in FTEs 1, 2, and 3). However, we find that such FTEs show a rapid jump in the FTE volume over time, in contrast to the relatively smooth profile as seen in Figure 7. We also emphasize a deduction here that, for FTE-1 and FTE-3, the steepest descents of the P–V curves occur toward the later phases of FTE evolution (see Figure 7). Prior to this, the drop in pressure was slower. This transition may be considered an indication that, as the FTEs evolve, the contribution of adiabatic expansion to the volume growth of the FTEs increase. Thus, under the condition that any active reconnection processes have subsided, toward the very late stages of FTE evolution, the FTEs will eventually drift toward an expansion that is purely adiabatic.

The idea that FTEs are flux ropes formed due to multiple X-line reconnection at the Earth's magnetopause is widely agreed upon and is supported by multiple observations and simulations (Paschmann et al. 1982; Lv et al. 2016; Chen et al. 2017; Sun et al. 2019; Mejnertsen et al. 2021). In this section, we analyze, in detail, the magnetic properties within the selected FTEs.

3.5.1. Force-free Nature of FTEs

As stated in Section 3.1, the FTEs formed in our simulation take the form of twisted helical magnetic flux ropes. Magnetospheric flux ropes are often modeled to be magnetically "force-free" structures. A completely force-free configuration mandates that all currents are field-aligned (in addition to the pressure gradient being negligible), i.e.,

$$\begin{eqnarray}&&| J\times B| =0;\quad J=\alpha (r)B,\end{eqnarray} \tag{ 5 }$$

where α is a function of position (r). The special case of α having a constant value is referred to as the linear force-free flux-rope model, which leads to the Lundquist solution to Equation (5) (Lundquist 1951; Burlaga 1988). Hereafter, the terms "linear force-free" and "constant-α force-free" have been used congruently with each other. The Lundquist solution to the above set of equations in cylindrical (r, ϕ, z) coordinates is given by

$$\begin{eqnarray}\begin{array}{rcl}{B}_{r} & = & 0\\ {B}_{\phi } & = & {B}_{0}{J}_{1}(\alpha r)\\ {B}_{z} & = & {B}_{0}{J}_{0}(\alpha r),\end{array}\end{eqnarray} \tag{ 6 }$$

where B₀ is the axial magnetic field component of the flux rope having a maxima at the flux-rope center, and J₀ and J₁ are the Bessel functions of the first kind. Such a linear force-free configuration is of particular interest as it represents a minimum energy configuration of twisted magnetic field lines, and therefore are the expected final states of flux-rope evolution (Woltjer 1958). Owing to its simplicity, the constant-α force-free flux-rope model has been widely used to deduce the physical parameters of magnetospheric flux ropes, such as FTE size, flux contents, and the impact parameter of the spacecraft with respect to the flux-rope core.

The following segments of this section have two objectives. First, we examine the temporal evolution of the perpendicular currents within the FTE volumes in order to investigate the magnetic configurations of FTEs 1, 2, and 3. This is to discern if the currents within FTEs drift toward a field-aligned configuration as has been widely regarded in observations. Thereafter, we proceed to further examine the validity of a linear force-free flux-rope model within the context of the selected FTEs.

From an observational perspective, flux ropes that are in their "old/mature" stages are generally treated as magnetically force-free. This is in view of the fact that such a condition is an evolutionary acquirement and not an inherent constraint on the formation mechanism. A mandate of the force-free assumption is that all currents must be field aligned, i.e., ∣J × B∣ ∼ 0. With regard to the first objective, to study the evolution of perpendicular currents within an FTE, we define a quantity ${ \mathcal F }$, which is mathematically given as

$$\begin{eqnarray}&&{ \mathcal F }={\left\langle \left|\displaystyle \frac{J\times B}{{B}^{2}}\right|\right\rangle }_{\mathrm{vol}},\end{eqnarray} \tag{ 7 }$$

where 〈 〉 _vol denotes volume average. The term ∣J × B∣ is directly derived as the Lorentz force, which is then normalized to the square of the field magnitude ∣B∣². An increasing field-aligned current would naturally mean ${ \mathcal F }\to 0$. We note here that this study does not pertain to the force balance within FTEs, but rather it highlights the evolution of the non-field-aligned currents within the structure.

Figure 9 plots the volume-averaged ${ \mathcal F }$ of the individual FTEs with respect to time. For comparison, the time is again represented in terms of ${t}_{\det }$ as defined in Section 3.4. The colors denote the separate FTEs as per the legend. The quantity ${ \mathcal F }$ is found to decay exponentially with time as seen by the exponential fits of the form ${ \mathcal F }\propto {e}^{\chi {t}_{\det }}$. The exponential fits are denoted by the appropriately colored solid lines in Figure 9. This exponentially decaying behavior is consistent across all three selected flux ropes. The corresponding exponential functions are given by ${ \mathcal F }\propto {{\rm{e}}}^{-0.54{t}_{\det }}$, ${ \mathcal F }\propto {{\rm{e}}}^{-0.24{t}_{\det }}$, and ${ \mathcal F }\propto {{\rm{e}}}^{-0.57{t}_{\det }}$ for FTEs 1, 2, and 3, respectively. This result is a demonstration of the widely accepted notion that old/mature magnetospheric FTEs have currents that are largely field-aligned (∣J × B∣ ∼ 0). Furthermore, looking at the trajectory of the FTEs in our simulation, one would expect that observations that are conducted farther away from the generation region of FTEs would find flux ropes that have more field-aligned currents than any observations close to the generation region.

Zoom In Zoom Out Reset image size

Observations of magnetospheric FTEs have been extensively modeled using the linear (constant-α) force-free flux-rope model (Eastwood et al. 2012; Akhavan-Tafti et al. 2018, 2019a) motivated by the analysis that (∣J × B∣ ∼ 0) within the FTE structure. Even though we find that the currents within the simulated flux ropes rapidly evolve toward a field-aligned form, which does not automatically guarantee that such a configuration will conform with a constant-α state. Here, we investigate the validity of a linear force-free model within the capacity of our simulations. Deriving from Equation (5), the force-free parameter "α" can be written as

$$\begin{eqnarray}&&\alpha =\displaystyle \frac{J\cdot B}{{B}^{2}}.\end{eqnarray} \tag{ 8 }$$

As an investigation of the validity of a constant-α approximation for a magnetospheric flux rope, we probe into the distribution of the α values calculated at all the grid cells constituting the FTE volume.

For an ideal, purely cylindrical, constant-α force-free flux rope, one would obtain a single value of α throughout the FTE volume, i.e., a histogram of such a distribution of α would have a single peak at a constant value. The histogram obtained from our simulations, however, exhibits a significant spread of α values for the very early phases of the temporal evolution of each flux rope. This can readily be seen from the blue curve in Figure 10, which is a histogram of the values of α within FTE-1 at ${t}_{\det }\sim 0.2\,\mathrm{minute}$. The abscissa in Figure 10 represents the α values, whereas the ordinate shows the fraction of grid cells within the FTE structure corresponding to that value (bin) of α. This is expected, as the reconnection process leading to the formation of FTEs does not have any associated constraint to assume a constant-α form in terms of the magnetic field configuration. The temporal evolution of the histogram, however, exhibits the formation of a skewed distribution with a growing peak. This is evident from the three histograms depicted in Figure 10. The blue, green, and red curves have been plotted in time snapshots corresponding to ${t}_{\det }\sim 0.2\,\mathrm{minute}$, ${t}_{\det }\sim 0.5\,\mathrm{minute}$, and ${t}_{\det }\sim 1.0\,\mathrm{minute}$ of FTE-1. A closer look reveals that the peak becomes narrower and more significant as time progresses. This essentially means that, with an increase in time, an increasing number of grid cells are converging toward a constant value of α throughout the entire volume of the FTE. This trend is seen for all three FTEs under consideration. The correspondingly colored dotted vertical lines in Figure 10 represent the median values of each of the distributions. The median of the distribution is also seen to shift toward smaller magnitudes as time progresses. This drift of the median value is anticipated because, mathematically, α behaves as an inverse length scale, and a linear force-free configuration with a lower magnitude of α would have a larger diameter. As the FTEs within our simulation grow in size, the larger cross-sectional areas dictate that force-free configurations for similar sizes will have lower magnitudes of α.

Zoom In Zoom Out Reset image size

We quantify the transformation of the spread of α throughout the FTE volume in Figure 11, which shows the time evolution of the median absolute deviation on α (MAD_α ) as obtained from the distribution. The MAD_α for all three FTEs show a general decreasing trend as the FTE evolves, which characteristically means that the volumes of the FTEs gravitate toward a constant-α configuration with time. However, toward the later stages of the evolution (after ${t}_{\det }\sim 1.0\,\mathrm{minute}$), the MAD_α for FTE-1 and FTE-3 stops decreasing and stabilizes around a relatively low value. FTE-2 is seen to have an initial increase in the MAD_α during the very early stage of evolution (${t}_{\det }\lessapprox 0.5\,\mathrm{minute}$), followed by a continuously decreasing profile thereafter. The trend, however, indicates that, given enough time, the MAD_α for FTE-2 would eventually stabilize in a manner akin to FTE-1 and FTE-3. We have verified that other statistical metrics of spread, e.g., mean absolute deviation, standard deviation, etc., show very similar behaviors for all three FTEs. Succinctly stated, all three curves unanimously indicate that the FTEs do tend to drift rapidly toward a constant-α force-free state. Nevertheless, there always exists a finite deviation from a constant alpha value, which inhibits the FTEs from attaining a purely linear force-free state.

Zoom In Zoom Out Reset image size

To discern the reason for the stagnation of MAD_α , we investigate the distribution of α within FTE-1 during a time when the MAD_α value has stabilized. This temporal point is denoted by the blue encircled point on the curve corresponding to FTE-1 in Figure 11. Panel (a) in Figure 12 shows a normalized histogram of the distribution of α within the FTE-1 volume. As mentioned earlier, the distribution profile is skewed with a significant peak close to a median value. The median value α_med was calculated to be −5.9 ${{\rm{R}}}_{E}^{-1}$, which is denoted by the dashed vertical line. The two solid lines on both sides denote the ±1 MAD_α from the median value. As can be seen, a substantial portion of the peak lies within ±1 MAD_α of the median value. Panel (b) of Figure 12 shows the distribution of the value of α within a cross section of FTE-1. The cross section is a slice at y = 4.5 R_E , which is approximately halfway along the azimuthal extent of the FTE. Each point constituting the slice corresponds to the α value of the grid cell at that location. The encircled points within the cross section (points encircled in black) highlight the set of locations having an α value that lies within ±1 MAD_α of the median α. We see that the value of α for a majority of the points constituting the flux-rope core falls within ±1 MAD_α of the median value. In particular, nearly 60% of the FTE cross section falls within the ±1 MAD_α bounds. The top and the bottom peripheral regions of the FTE cross section, however, have α values that lie outside these bounds.

Zoom In Zoom Out Reset image size

To probe further into this, we look into the distribution of the z component of velocity (v_z ) within this cross section. It can be seen that there exists a significant counterstreaming flow of v_z at precisely the same regions where the values of α in panel (b) deviate outside the ± 1 MAD_α bounds. The counterstreaming plasmas inside the FTEs are, in fact, the reconnection exhausts produced by the X-line pair outside the FTE boundary that feed plasma into the FTE volume. We therefore deduce that the exhausts produced by the active X lines on the top and bottom edges of the FTEs feed in newly reconnected fluxes that do not conform to the pre-existing peaked distribution of the α value. Therefore, as long as there is an ongoing continuous reconnection process, the FTEs can never, in principle, attain a purely constant-α state, due to the persistent flux injection. The nearly steady state in the temporal evolution of MAD_α for the case of FTE-1 and FTE-3, as seen in Figure 11, is thus a sustained conflict between the flux rope tending toward a constant-α form and the newly reconnected flux disrupting this state. The core region of the flux rope is, however, found to be resilient to such perturbations and remains in a nearly constant-α state up to a reasonable extent. The plot of v_z (panel (c) of Figure 12) also attests to our inference in Section 3.4 that the process of continuous reconnection feeds hot plasma into the FTEs during their evolution.

3.5.2. Flux Content Estimates

The total flux content of FTEs is an important metric, as it can provide an estimate of the contributions of such phenomena in the transfer of magnetic flux within the Dungey cycle (Fear et al. 2019). Within the linear force-free flux-rope model, Eastwood et al. (2012) presented an analytical estimate of the flux content of a perfectly cylindrical flux rope as

$$\begin{eqnarray}&&{{\rm{\Phi }}}_{\mathrm{ana}}=\left[\displaystyle \frac{2{B}_{0}{A}_{{fr}}{J}_{1}(2.40842)}{2.40842}\right],\end{eqnarray} \tag{ 9 }$$

where B₀ is the axial field strength at the flux-rope core, A_fr is the cross-sectional area of the flux rope, and J₁ is the Bessel function of first kind and first order. The core field strength B₀ and the area of the flux rope A_fr are determined from observations via the linear force-free model fits, which are then fed into Equation (9) to estimate the flux (Eastwood et al. 2012; Jasinski et al. 2016; Akhavan-Tafti et al. 2018, 2019a).

The following exercise is meant to compare the flux transported by our simulated flux ropes to the flux enclosed by the analytical estimates of an analogous, perfectly cylindrical, linear force-free flux rope as calculated from Equation (9). Within our simulation, we estimate the flux content of the detected FTEs using the following procedure. First, we span the entire azimuthal extent of the FTE to find the y slice that shows the strongest value of the B_y (axial) component. Thereafter, we consider a slice of the FTE at this particular value of the y coordinate and calculate the flux content (${{\rm{\Phi }}}_{\mathrm{sim}}$) using the expression of Equation (10), where the summation (Σ) is taken over all the points constituting the flux-rope cross section:

$$\begin{eqnarray}&&{{\rm{\Phi }}}_{\mathrm{sim}}={\rm{\Sigma }}(| {B}_{y}| .{dA}).\end{eqnarray} \tag{ 10 }$$

We then examine the cross section of the sliced FTEs, and locate the point where the axial component (B_y ) is maximum. A region spanned by 3 × 3 grid cells around this point is then selected, and the median B_y within that region is estimated as the core axial field strength (B₀) of the flux rope. This value, in addition to the cross-sectional area of the flux rope (calculated according to the number of grid cells in the flux-rope cross section), is then used in Equation (9) to estimate the flux content of an equivalent, purely linear, force-free flux rope. The flux value thus obtained is called Φ_ana and is representative of the flux obtained for a satellite traveling through the FTE core, assuming that the cross-sectional area is determined accurately. This process is repeated for all three FTEs at times denoted by the encircled points in Figure 11, i.e., during the significantly later stages of FTE evolution when the MAD_α has stagnated.

Table 2 summarizes the results of the flux calculations using the aforementioned procedures. The first and second columns represent the names of the FTEs and the global times when the flux calculations were carried out, respectively. The third and fourth columns, titled "y_beg" and "y_end", represent the azimuthal extents of the corresponding FTEs in the domain. The column titled "y_slice" represents the location where the maximum value of B_y was obtained for each FTE, and therefore corresponds to the coordinate where the FTE was sliced for the flux calculations. "B_axial" quantifies this core magnetic field strength obtained for each FTE. ${{\rm{\Phi }}}_{\mathrm{sim}}$ represents the flux values obtained from the simulations using Equation (10), and Φ_ana denotes the values obtained using Equation (9) (in mega-Webers). The last column shows the differences in the two flux values in terms of the percentage error on ${{\rm{\Phi }}}_{\mathrm{sim}}$.

Table 2. A Summary of the Parameters Associated with the FTE Flux Calculations, the Flux Values and Errors

Name	t (s)	ybeg(RE )	yend(RE )	yslice(RE )	Baxial(nT)	${{\rm{\Phi }}}_{\mathrm{sim}}(\mathrm{MWb})$	Φana(MWb)	Error(%)
FTE-1	1467	2.63	6.77	4.5	49.01	0.52	0.43	17
FTE-2	2902	3.45	5.91	4.90	32.67	0.33	0.30	9
FTE-3	3428	−7.79	−3.45	−4.83	34.45	1.52	0.92	40

Download table as:  ASCIITypeset image

The value of ${{\rm{\Phi }}}_{\mathrm{sim}}$ for FTE-1 was calculated to be 0.52 MWb, whereas the corresponding Φ_ana had a value of 0.43 MWb. We note that FTE-1 has a moderate axial field strength and the total flux content was found to be nearly consistent between the simulated and analytical values. The flux content of FTE-2 was lower than FTE-1, where the corresponding ${{\rm{\Phi }}}_{\mathrm{sim}}$ and Φ_ana were obtained to be 0.33 MWb and 0.30 MWb, respectively. The errors in flux calculated by the analytical model of FTEs 1 and 2 were ∼17% and ∼9%, respectively. The flux obtained for FTE-3 was ${{\rm{\Phi }}}_{\mathrm{sim}}$ = 1.52 MWb; however, the analytical estimate was found to be Φ_ana = 0.92 MWb. The total flux contents of all the three FTEs, therefore, were underestimated under the linear force-free assumption, with FTE-3 exhibiting a significant underestimation of ∼40%. We re-emphasize here that the flux obtained from the analytical model (Φ_ana) for a typical observation is in a scenario where it is assumed that the core magnetic field strength has been estimated correctly.

We examine the distribution of B_y (axial component contributing to the flux for the simulated FTEs) over all the points spanning the FTE-3 cross section, as a means to explore the reason for such a significant underestimation. First, we analytically construct a perfectly cylindrical ideal flux rope that is in a constant-α force-free state. The magnetic field components given by Equation (6) are therefore a solution to such a flux rope. We then extract a random sample of points that uniformly span the circular cross section of the ideal flux rope and calculate the axial component (B_z in Equation (6)) at each sample location. The uniform sampling is done to imitate a Cartesian grid spanning the cross section. Even though the value of α used for this purpose is inconsequential, we have chosen a value that represents the median of the distribution of α obtained using Equation (8) throughout all grid cells spanning the FTE-3 volume. As the axial component is the main contribution to the total flux content, the distribution of the axial component of these uniformly spaced samples within the ideal flux-rope cross section is compared to the distribution of the axial component of all the points spanning the cross section of FTE-3 in our simulation. Panel (a) of Figure 13 shows the normalized histogram for both the Lundquist solution (denoted by the gray shaded histogram) marked as "Lqst50" and the samples from the cross section of FTE-3 (denoted by the blue hatched histogram). There are two regions where the two histograms significantly differ: (a) the low axial field strength region (left portion of the histogram), and (b) the intermediate axial field strength region (near ∼21 nT). The differences between the two distributions at lower axial field strengths would have a nontrivial impact on the flux measurements, but the total flux content will be less susceptible to such types of deviations, owing to the smaller field magnitude. The significant deviation denoted by the red shaded region in panel (a) of Figure 13 at intermediate axial field strengths, however, can have a crucial impact on the flux estimation, due to the relatively higher magnitude as well as a much higher number of such points. A plot of the FTE-3 cross section at y ∼ 4.8 R_E is shown in panel (b) of Figure 13 for reference. We have verified that the points that lie within the red shaded region (B_Y ∼ 21 nT) in panel (a) of Figure 13 are located at the top and the bottom regions of the cross section of FTE-3 depicted in the panel (b). This strongly indicates that this excess is a direct implication of the newly reconnected magnetic flux being injected into the FTE volume from the X lines associated with the FTE. With an intermediate field strength of ∼21 nT, regions like these can significantly enhance the flux content of an FTE. The linear force-free flux-rope model does not take such a continuous injection of flux into account, and it has a strictly decreasing axial component as one moves away from the flux-rope center. In other words, as one moves toward the edges of the flux rope, flux injected due to the ongoing reconnection process does not allow the axial component to drop as much as predicted by the linear force-free flux-rope model. Therefore, the flux estimation performed using this model might, in these cases, largely underestimate the flux content of an FTE. As a final exercise in this context, to estimate the magnitude of the difference in ${{\rm{\Phi }}}_{\mathrm{sim}}$ and Φ_ana that can be attributed solely to this intermediate flux region, we recalculate the flux content of FTE-3 after excluding the points that constitute the red shaded region in Figure 13. We find that this reduces the percentage error by almost half and brings the difference to a rather modest ∼20%, with the flux value newly estimated to be ${{\rm{\Phi }}}_{\mathrm{sim}}=1.15\,\mathrm{MWb}$. However, within the context of our simulations, we believe that a comprehensive statistical study of a significantly higher number of simulated FTEs will be required for a robust quantification of the discrepancies in flux context estimates within the FTEs.

Zoom In Zoom Out Reset image size

Finally, we present a quantification of the total closed and open dayside magnetospheric fluxes in our simulation. The closed flux has been estimated as follows. During a time when the magnetopause is devoid of any transient phenomena, we take a slice of the computational domain at z = 0 within the bounds −15 R_E ≤ y ≤ 15 R_E and − 15 R_E ≤ x ≤ 0 R_E . The bounds for x ensure that the slice only contains the dayside magnetosphere (x ≤ 0 ⇒ Dayside). The slice thus obtained is a 2D surface from which a region $\sqrt{({x}^{2}+{{\rm{y}}}^{2})}\leqslant 1\,{R}_{E}$ is excluded. This is done to ascertain that the contribution to the flux is only from the region between 1 R_E and the magnetopause boundary. As the solar wind is composed primarily of a southward IMF, all points on this surface that have B_Z > 0 (Earth's dipolar field lines) are taken to be the grid cells that belong inside the magnetopause and thus contribute to the total closed flux, while all points with B_Z < 0 (solar-wind field lines) are considered to lie outside the magnetopause and thus do not contribute to the total closed dayside flux. Adapting the approach from Akhavan-Tafti et al. (2020a), the total closed dayside magnetospheric flux content (Φ_MSP) is then estimated using the following expression:

$$\begin{eqnarray}&&{{\rm{\Phi }}}_{\mathrm{MSP}}=\sum _{\mathop{{\rm{r}}\gt 1\,{R}_{E}}\limits_{\forall \,{B}_{Z}\geqslant 0}}{B}_{Z}\cdot \mathrm{dS},\end{eqnarray} \tag{ 11 }$$

where "dS" is the area of each grid cell composing the 2D slice. The total closed dayside magnetospheric flux content thus obtained from our simulation has a value of ∼3.5 GWb. Thereafter, within a box bounded by − 15 R_E ≤ x ≤ 0 R_E , −30 R_E ≤ y ≤ 30 R_E and −15 R_E ≤ z ≤ 15 R_E , we also calculate the total dayside open flux as follows. From all points constituting the surfaces of this bounding box, we trace the magnetic field lines to determine if they reach 1 R_E . The magnetic field lines thus isolated are the open field lines that originate from the surface of the Earth and reach the domain boundaries. For all the field lines traced from the outer boundary to 1 R_E , we calculate the associated flux (Σ B _i · d S ), where "Σ" denotes summation over all such field lines, and " B _i " and "dS" are the magnetic field and the area of the grid cell at the field line origin, respectively. The total open flux of the dayside magnetosphere is thus calculated to be ∼0.35 GWb. The flux content of FTE-3 (which had the maximum flux content) turns out to be on the order of 1% of this total dayside magnetospheric open flux. As an additional note, we also find that this flux content is nearly 100 times lower than that of one of the large FTEs observed by Akhavan-Tafti et al. (2020a). However, this estimated flux content of FTE-3 is only after ∼1.6 min of FTE evolution, and as the FTEs evolve further, their flux content may tend to increase. We additionally report an estimation of the voltage across FTE-3 due to the ongoing reconnection. For this, we evaluate the initial (at ${t}_{\det }\sim 0$) flux content of FTE-3 as having a magnitude of ∼0.4 MWb. This flux difference over an evolutionary time period then quantifies a voltage (${\rm{\Delta }}{{\rm{\Phi }}}_{\mathrm{sim}}/{\rm{\Delta }}{t}_{\det }$) of ∼11.7 kV across FTE-3, due to reconnection.

First, the time evolutions of the volumes of the FTEs were investigated for the FTEs 1, 2, and 3. The trend of FTE volume growth was found to be monotonically increasing during the early phases of FTE evolution, followed by a stagnation of the growth. This stagnation can be attributed to either the relaxation of field lines within the FTEs or to a drag-induced compression as the FTEs encounter fast-moving magnetosheath flows (Eastwood et al. 2012). Thereafter, we investigate the principal cause for the growth of the FTE volumes. An inspection of the P–V curves in Figure 8 shows that the average pressure drops much more slowly in contrast to the relation P ∝ V^−1.67 expected from adiabatic expansion. Thus, we conclude that the process of continuous reconnection at the neighboring X lines injects hot plasma into the FTE volume, which does not let the pressure decrease as rapidly as is expected from an adiabatically expanding FTE. In the absence of FTE coalescence, the process of continuous reconnection is, therefore, found to be the dominant cause of FTE growth for all FTEs analyzed in this study. Altogether, we find our results to be congruent with observations by Akhavan-Tafti et al. (2019a), who arrived at a similar conclusion.

An investigation into the magnetic properties within the FTEs shows that the Lorentz force within the structures rapidly decreases. This indicates a swift tendency of any currents within the FTE volumes toward being field-aligned as the FTE evolves. As generally asserted from observations, the trend of ∣J × B∣ → 0 indicates a drift toward a magnetically force-free configuration. We therefore infer that our FTEs show a similar trend and eventually would attain this ∣J × B∣ ∼ 0, or magnetically force-free, state. We further investigate the validity of the constant-α force-free flux-rope model within the selected FTEs. We find that, during the initial stages of FTE evolution, there exists a distribution of the value of α throughout the FTE volume. It is seen that, as the FTEs evolve, this distribution exhibits a peak around a certain value, indicating the tendency of the FTEs to gravitate toward a constant-α force-free form throughout the evolution. We nevertheless observe that the process of continuous magnetic reconnection injects magnetic flux into the FTEs, which tends to divert them from a constant-α configuration.

Complemented with observations to infer the physical properties of a detected FTE, the constant-α force-free flux-rope model is often used to deduce the flux content of a detected FTE. We calculate the flux contents of FTEs 1, 2, and 3, and compare them with corresponding analytical estimates derived from the linear force-free flux-rope model from Eastwood et al. (2012). The analytical calculations are found to underestimate the flux content in all three cases, as shown in Table 2. The constant-α force-free flux-rope model does not take the aforementioned continuous flux injection into account, and it therefore underestimates the flux content by a significant margin. Our findings lean toward the argument given by Fear et al. (2017) that in situ measurements, in certain cases, do tend to underestimate the flux content of the FTEs. We note here that the study by Fear et al. (2017) incorporates additional tools such as ionospheric radar measurements to aid in flux calculations.

In essence, our study addresses the evolution of the transient FTEs at the dayside magnetopause using numerical simulations. The novelty includes the adaptation of volumetric detection techniques to isolate the irregular 3D FTE structures and study the temporal evolution of their thermodynamic and magnetic characteristics. This paper aims to bridge the interpretations obtained from in situ observations of such FTEs, generally conducted along an FTE cross section, with numerical simulations illustrating a full three-dimensional perspective. The paper highlights the detection and analysis of a subset of simulated FTEs. However, we note here that a statistical perspective with a large sample of FTEs is essential in order to have a robust assertion pertaining to such challenging issues as the volume distribution of FTEs and flux content estimates. A statistical study that addresses these issues is in progress and shall be the focus of a future article.