Three-dimensional (3D) magnetic nulls are abundant in the solar atmosphere, as has been firmly established through contemporary observations. They are established to be important magnetic structures in, for example, jets and circular ribbon flares. Although simulations and extrapolations support this, the mechanisms behind 3D null generation remain an open question. Recent magnetohydrodynamic simulations demonstrated magnetic reconnections to be responsible for both generating and annihilating 3D nulls. However, these simulations began with initial magnetic fields already supporting preexisting nulls, raising the question of whether magnetic reconnection can create nulls in fields initially devoid of them. Previously, this question was briefly explored in a simulation with an initial chaotic magnetic field. However, the study failed to precisely identify locations, topological degrees, and natures (spiral or radial) of nulls, and it approximated magnetic reconnection without tracking the magnetic field lines in time. In this paper, these findings are revisited in light of recent advancements and tools used to locate and trace nulls, along with the tracing of field lines, through which the concept of generation/annihilation of 3D nulls from chaotic fields is established in a precise manner.

Contemporary observations,1–4 particularly in the context of circular ribbon flares, unequivocally indicate the existence of nulls in the solar atmosphere.5–7 Numerous extrapolations and simulations further back this up.8–25 Nevertheless, the generation of three-dimensional (3D) nulls is still an unresolved problem and merits further attention. Toward this goal, recent magnetohydrodynamic (MHD) simulations by Maurya et al.,26,27 (hereafter called YRD1 and YRD2) demonstrated magnetic reconnection to be responsible for the 3D null generation and their eventual annihilation. This process of null pair generation is distinctive in comparison to null bifurcations,28,29 for details refer to YRD1. YRD1 initiates the simulation with a preexisting potential null, while YRD2 advances the proposal by executing a data-based simulation where the initial magnetic field is extrapolated using vector magnetogram data of a solar active region (AR11977). Although in both the studies null generation was ubiquitous, the initial magnetic field supported preexisting nulls. A natural question is then whether magnetic reconnection can generate 3D nulls from a magnetic field having no such nulls initially. The plausibility of such a scenario has been briefly explored in the simulation used in the study by Nayak et al.,30 where the initial magnetic field was chaotic and devoid of any 3D null. Although that study demonstrated the generation of magnetic nulls, it failed to precisely identify their location, topological degree, and nature (spiral or radial as described in the study by Parnell et al.31) using presently available standard tools like the upgraded null detection technique described in YRD2. Moreover, a claim of magnetic reconnection demonstrated by change in field line connectivity requires strict maintenance of the involved magnetic field lines, which was approximated in the study by Nayak et al.30 by keeping the initial point of field line integration constant at every instant, whereas a more precise requirement is to follow the reconnecting field lines as they advected with plasma flow (in the ideal MHD region). For completeness, it is then indispensable to revisit the findings of Nayak et al.30 in light of recent understanding and tools developed in YRD1 and YRD2 and put the idea of the generation of 3D nulls from the chaotic field on a firmer footing.

Toward this objective, the following presents a brief discussion of the initial magnetic field. The field is constructed by superposing two Arnold–Beltrami–Childress (ABC) fields,32 each satisfying the linear force-free equation
× B = λ B ,
(1)
having solution
B x = A sin λ z + C cos λ y ,
(2)
B y = B sin λ x + A cos λ z ,
(3)
B z = C sin λ y + B cos λ x
(4)
and being represented as
B = B 1 + d 0 B 2 .
(5)
The constant d0 relates the amplitudes of the two superposed fields. In Cartesian coordinates, the components of B are
B x = A ( sin λ 1 z + d 0 sin λ 2 z ) + C ( cos λ 1 y + d 0 cos λ 2 y ) ,
(6)
B y = B ( sin λ 1 x + d 0 sin λ 2 x ) + A ( cos λ 1 z + d 0 cos λ 2 z ) ,
(7)
B z = C ( sin λ 1 y + d 0 sin λ 2 y ) + B ( cos λ 1 x + d 0 cos λ 2 x ) .
(8)
Equation (1) is an eigenvalue equation of the operator ( × ), eigenfunctions of which form a complete orthonormal basis when eigenvalues λ are real.33 Further simplification of Eqs. (6)–(8) can be made by selecting λ 1 = λ 2 = λ, rendering
B x = 0.5 A sin z + 1.5 C cos y ,
(9)
B y = 0.5 B sin x + 1.5 A cos z ,
(10)
B z = 0.5 C sin y + 1.5 B cos x
(11)
for the selection d 0 = 0.5 and λ = 1. The resulting Lorentz force
( J × B ) x = B 2 sin 2 x 2 A C sin y cos z ,
(12)
( J × B ) y = C 2 sin 2 y 2 A B cos x sin z ,
(13)
( J × B ) z = A 2 sin 2 z 2 B C sin x cos y
(14)
can be utilized to drive the plasma from an initial static state to develop dynamics. Importantly, B is chaotic, and a detailed discussion can be found in the studies by Kumar et al.34 and Nayak et al.30 Also important are the relative magnitudes of the constants A, B, and C. For instance, if A = B = 1, an increasing C makes the volume occupied by chaotic field larger—a conclusion derived in the study by Kumar et al.,34 which can be used as a measure of chaoticity. Later on, C is simply called the chaoticity. For the simulations executed here, notable is in the range 0 C 0.3142, for A = B = 1, for which B is entirely devoid of any magnetic nulls. Consequently, using the B as an initial condition provides the unique opportunity to explore null generation from a state having no preexisting nulls—the objective of this communication; along with understanding null dynamics in an environment of chaotic magnetic field, which is left as a future exercise.
The simulations are carried out using the magnetohydrodynamic numerical model EULAG-MHD,35 idealizing the plasma to be thermodynamically inactive and incompressible. The governing MHD equations are
ρ 0 ( v t + ( v · ) v ) = p + 1 4 π ( × B ) × B + ν 0 2 v ,
(15)
· v = 0 ,
(16)
B t = × ( v × B ) + D B ,
(17)
· B = 0
(18)
in standard notations and CGS system of units. The constants ρ0 and ν0 are uniform density and kinematic viscosity, respectively. Although not strictly applicable in the solar corona, the incompressibility is invoked in other works also.36,37 With details in Smolarkiewicz and Charbonneau35 (and references therein), salient features of the EULAG-MHD applicable for this work are summarized here. Crucial to the model is the spatiotemporally second-order accurate, non-oscillatory, forward-in-time, multidimensional, positive-definite advection transport algorithm MPDATA.38 The governing prognostic equations (15) and (17) are both solved in the Newtonian form with total derivatives of dependent variables and the associated forcings forming the left- and right-hand sides, respectively (see Sec. 2.1 in Smolarkiewicz and Charbonneau35 for a discussion). This guarantees the identity of null preservation as the associated forcing of the induction equation vanishes at the nulls to the accuracy of the field solenoidality [Eq. (18)], which is high.35 Another important aspect is the proven dissipative nature of the MPDATA.39–41 This dissipation is intermittent and adaptive to the generation of under-resolved scales in field variables for a fixed grid resolution. Using this dissipation property, the MPDATA removes under-resolved scales by producing locally effective residual dissipation of the second order in grid increments, enough to sustain the monotonic nature of the solution in advective transport. The D B in Eq. (17) represents the dissipative property of MPDATA—we do not employ any explicit diffusion term. The advantage of the MPDATA scheme resides in minimizing the computational cost, while tending to maximize its effective Reynolds number.42 The consequent magnetic reconnection is then in the spirit of ILESs that mimic the action of explicit subgrid-scale turbulence models, whenever the concerned advective field is under-resolved, as described in Margolin et al..43 Such ILESs performed with the model have successfully simulated regular solar cycles by Ghizaru et al.44 and Racine et al.,40 with the rotational torsional oscillations subsequently characterized and analyzed in the study by Beaudoin et al.45 The simulations carried out here also utilize the ILES property to initiate magnetic reconnections already shown by Kumar et al.46 

The simulations have been carried out for the aforementioned field with C { 0.15 , 0.3 } to explore null generations with an increase in chaoticity. The kinematic viscosity is set as ν = 0.010 cm2 s–1, while the spatial and temporal grid increments are Δ x = Δ y = Δ z = 0.099 73 cm along the x, y, z-axes, respectively, and Δ t = 0.016 s, in CGS units. Triply periodic boundary conditions are applied, and the grid having 64 × 64 × 64 pixels resolved on a computational grid of x, y, z  { π , π } cm in a Cartesian coordinate system, mapping a physical dimension of ( 2 π ) 3 cm3 to facilitate magnetic reconnection while optimizing the computation costs. Each simulation spans a physical time of 32 s. Figure 1 plots the number of nulls with time for different values of C, depicting an increase in the number of nulls at a given instant and its maximal value over the temporal range with an increase in chaoticity. Additionally, nulls appear earlier for larger values of C, precisely at t = ( 31 , 23 , 9 , 8) s for C = ( 0.15 , 0.2 , 0.25 , 0.3 ). Interestingly, the null generation for all C values is in bursts, most pronounced for C = 0.3, which shows three identifiable peaks at t = { 9.26 , 16.18 , 23.28 } s. A possible reason can be a sudden increase in chaoticity near the peaks, followed by its decrease. Figure 2 (Multimedia available online) verifies this ansatz by following a local flux surface traversed by a single field line for C = 0.3 in t { 16.08 , 16.29 } s, spanning the second prominent peak at t = 16.18 s. Clearly, the surface loses its coherent structure as the line becomes more volume filling and hence chaotic. At t = 16.18 s [panel (c)], which marks the second peak, the local flux surface is almost destroyed but reorganized itself at later times [panels (d)–(f)]. It has been proposed that the presence of chaotic field lines may promote the occurrence of magnetic reconnection in fields without nulls,47,48 and in this case, the increase in chaoticity is cotemporal with the generation of nulls and with reconnection (see below). The causal link remains to be fully explored in future investigations along with the plausibility of the local flux surfaces being attractors because of their repetitive destruction and reformation throughout the simulation. The subsequent retrieval of the flux surface arrests this increase in reconnection—leading to a peak in the number of nulls.

FIG. 1.

The plot shows an increase in the number of nulls at a given instant and its maximal value over the temporal range with an increase in chaoticity. The vertical axis represents the number of nulls, and the horizontal axis represents time. The plots in different colors (pink, blue, green, and red) represent the variation in number of nulls for a particular value of the chaoticity (0.15, 0.20, 0.25, and 0.30, respectively). Generation of nulls occurs earlier in time as the chaoticity C increases, i.e., t = ( 31 , 23 , 9 , 8) s for C = ( 0.15 , 0.20 , 0.25 , 0.30 ).

FIG. 1.

The plot shows an increase in the number of nulls at a given instant and its maximal value over the temporal range with an increase in chaoticity. The vertical axis represents the number of nulls, and the horizontal axis represents time. The plots in different colors (pink, blue, green, and red) represent the variation in number of nulls for a particular value of the chaoticity (0.15, 0.20, 0.25, and 0.30, respectively). Generation of nulls occurs earlier in time as the chaoticity C increases, i.e., t = ( 31 , 23 , 9 , 8) s for C = ( 0.15 , 0.20 , 0.25 , 0.30 ).

Close modal
FIG. 2.

Panels depict a sudden increase in chaoticity near the second peak in the number of nulls, followed by its decrease. A local flux surface traversed by a single field line for C = 0.3 in t { 16.08 , 16.29 } s, spanning the second prominent peak at t = 16.18. Panel (a) depicts a flux surface structure drawn at t = 16.08 s, and the traversing field line is almost perfectly tangential to the surface and becomes part of the chaotic region with the evolution [panel (b)]. Subsequently, the surface loses its coherent structure as the line becomes more volume filling and hence chaotic. At t = 16.18 s [panel (c)], which marks the second peak, the local flux surface is almost destroyed but reorganized itself at later times [panels (d)–(f)]. Multimedia available online.

FIG. 2.

Panels depict a sudden increase in chaoticity near the second peak in the number of nulls, followed by its decrease. A local flux surface traversed by a single field line for C = 0.3 in t { 16.08 , 16.29 } s, spanning the second prominent peak at t = 16.18. Panel (a) depicts a flux surface structure drawn at t = 16.08 s, and the traversing field line is almost perfectly tangential to the surface and becomes part of the chaotic region with the evolution [panel (b)]. Subsequently, the surface loses its coherent structure as the line becomes more volume filling and hence chaotic. At t = 16.18 s [panel (c)], which marks the second peak, the local flux surface is almost destroyed but reorganized itself at later times [panels (d)–(f)]. Multimedia available online.

Close modal

With no nulls entering or leaving the computational domain while chaoticity being directly related to the onset of current sheets34 and consequent reconnections, the underlying mechanism for null pair generation can be attributed to these reconnections. A comprehensive study of field line dynamics is carried out in the following to explore the relation between reconnection and formation/annihilation of null pairs in detail. For this purpose, the dynamics corresponding to C = 0.3 is selected as the nulls are generated earlier in time and mostly away from the boundaries of the computational domain, leading to their better tractability over time. The focus is set on the nulls generated in a pair with coordinates ( x , y , z ) { ( 0.166 , 0.034 , 0.101 ) π , ( 0.169 , 0.034 , 0.101 ) π }, at t = 8.27 s—panel (a) of Fig. 3 (Multimedia available online), as it involves spiral–spiral pair generation and annihilation, hitherto unexplored in YRD1 and YRD2. Additionally, the pair is created almost at the beginning of the null pair generation, being third in the chronology.

FIG. 3.

The evolution of nulls with time after their generation. Nulls are traced in time, and field lines in green, pink, and red are drawn near the spiral null _ 1 (SN1), spiral null _ 2 (SN2), and spiral null _ 3 (SN3), respectively. The color bar in panels depicts the magnitude of the | J | / | B |, where J and B represent current density and magnetic field, respectively. Panel (a) depicts the field lines structure near the nulls at t = 8.27 s, when two spiral nulls ( S N 1 , S N 2) first appear. With the evolution, SN1 and SN2 move away from each other [panels (b)–(e)] and SN1 changes its nature from a spiral to a radial null [panel (e)]. Subsequently, this radial null reverts back to a spiral null [panels (f)–(h)], eventually annihilates with a different spiral null _ 3 (SN3) formed in a distinct null pair generation process [panel (j)]. Multimedia available online.

FIG. 3.

The evolution of nulls with time after their generation. Nulls are traced in time, and field lines in green, pink, and red are drawn near the spiral null _ 1 (SN1), spiral null _ 2 (SN2), and spiral null _ 3 (SN3), respectively. The color bar in panels depicts the magnitude of the | J | / | B |, where J and B represent current density and magnetic field, respectively. Panel (a) depicts the field lines structure near the nulls at t = 8.27 s, when two spiral nulls ( S N 1 , S N 2) first appear. With the evolution, SN1 and SN2 move away from each other [panels (b)–(e)] and SN1 changes its nature from a spiral to a radial null [panel (e)]. Subsequently, this radial null reverts back to a spiral null [panels (f)–(h)], eventually annihilates with a different spiral null _ 3 (SN3) formed in a distinct null pair generation process [panel (j)]. Multimedia available online.

Close modal

With the experience gained from YRD1 and YRD2, the field lines are advected with the plasma flow and traced in time to reveal the magnetic field line dynamics. For this purpose, two sets of field lines (one in green and two in pink) having initial points at coordinates x , y , z { ( 0.359 , 0.061 , 0.235 ) π , ( 0.359 , 0.061 , 0.234 ) π }, away from the reconnection region are selected. The corresponding field lines are advected with the plasma flow and are traced in time [Fig. 4 (Multimedia available online)] within a subvolume { ( 0.266 , 0.018 , 0.159 ) π ( 0.580 , 0.207 , 0.434 ) π }. This subvolume being far away from the periodic boundaries of the computational domain necessary for showing up the chaotic field, presumably the selected field lines maintain their discreteness. The figure is further overlaid with a probe depicting values of | J | / | B |, located approximately at the reconnection plane identified in Fig. 3. All the three field lines (two pink and one green) are connected from region a to region b at t = 8.21 s [panel (a)]. As depicted in panels (a)–(c), the green field line gets more prominently elbow-shaped compared to the other two along with becoming co-spatial with high value region of | J | / | B |. Subsequently, across panels (c) and (d), the green and one of the two pink field lines change their connectivity from regions a to b to regions a to d and c, respectively, at t = 8.26 s—a telltale sign of magnetic reconnection. Importantly, this reconnection precedes the null pair generation and indicates a causal connection between the two. Moreover, an auxiliary simulation having a 1283 grid resolution, mapping the same physical domain, has been carried out. The result (not shown) confirms null pair generations getting delayed in time and with the onset of magnetic reconnection being deferred because of the employed ILES spirit of the simulations, further corroborating this causal connection.

FIG. 4.

The magnetic reconnections leading to the generation of two spiral nulls. Two pink and one green magnetic field lines are traced in time from ideal plasma elements in the vicinity of the footpoint marked “a.” The color bar represents the same quantity | J | / | B | as mentioned in Fig. 3. At t = 8.21 s, all field lines connect from region a to region b [panel (a)]. Subsequently [panels (a)–(c)], the green field line develops more prominent elbow shape, but connectivity remains unchanged. Across panels (c) and (d), a green and one pink field line change connectivity—telltale sign of reconnection. Subsequently, the second pink field line also changes connectivity [panels (d) and (e)] and generates two spiral nulls at t = 8.27 s [panel (e)]. The spontaneously generated two spiral nulls are elaborated in panel (f). Multimedia available online.

FIG. 4.

The magnetic reconnections leading to the generation of two spiral nulls. Two pink and one green magnetic field lines are traced in time from ideal plasma elements in the vicinity of the footpoint marked “a.” The color bar represents the same quantity | J | / | B | as mentioned in Fig. 3. At t = 8.21 s, all field lines connect from region a to region b [panel (a)]. Subsequently [panels (a)–(c)], the green field line develops more prominent elbow shape, but connectivity remains unchanged. Across panels (c) and (d), a green and one pink field line change connectivity—telltale sign of reconnection. Subsequently, the second pink field line also changes connectivity [panels (d) and (e)] and generates two spiral nulls at t = 8.27 s [panel (e)]. The spontaneously generated two spiral nulls are elaborated in panel (f). Multimedia available online.

Close modal

The null pair is generated at t = 8.27 s [panel (e)], their structure elaborated in panel (f), along with changes in field line connectivity. In detail, the second pink field line connects regions a–d instead of a–b, while the first pink field line now connects a–d [cf panels (d) and (e)]. An important further endeavor would be to identify separators and investigate their role in reconnections occurring in the post null pair generation phase, but left as a future exercise as achieving the involved numerical technicality is challenging and the exercise is not within the central scope of this paper. Nevertheless, a recent work by Parnell49 in this direction is worth mentioning here as an example for setting up the tone for future numerical simulations in this direction. The work demonstrates the importance of intra- and inter-cluster separators in the context of clusters of magnetic nulls and suggests that reconnection is not taking place in cluster—which can further be explored in the context of pair production of magnetic nulls.

The eigenvalues of the Jacobian matrix B at each null are calculated, and it is found that the imaginary part of the eigenvalues is non-zero for each of the two nulls, which are associated with the current along the spine of nulls. Consequently, the fan field lines of each null become spiral due to non-zero current along the spine; hence, both nulls are spiral nulls (hereafter referred to as SN1 and SN2, respectively). These nulls are traced in time, and field lines are drawn from the close vicinity of the nulls, as shown in Fig. 3. The nulls move away from each other after their generation [cf panels (a)–(e)]. The topological details of the nulls are illustrated in Fig. 5(a). The fan field lines (in green) of SN1 are directed away from the null, resulting in a topological degree of −1, whereas the fan field lines (in pink) of SN2 are directed toward the null, making a topological degree +1. The generation of nulls in pair satisfies the conservation of net topological degree.

FIG. 5.

The topological details of the two spontaneously generated spiral nulls at t = 8.46 s [panel (a)] and the null pair at t = 8.86 s [panel (b)]. The fan field lines (in green) of spiral null _ 1 (SN1) are directed away from the null, resulting in a topological degree of −1 [panels (a) and (b)], whereas the fan field lines (in pink) of spiral null _ 2 (SN2) are directed toward the null, making a topological degree +1. The fan plane field lines of spiral null _ 3 (SN3) (in red) are directed toward the null point, making topological degree +1 [panel (b)]. With time, these two nulls get annihilated.

FIG. 5.

The topological details of the two spontaneously generated spiral nulls at t = 8.46 s [panel (a)] and the null pair at t = 8.86 s [panel (b)]. The fan field lines (in green) of spiral null _ 1 (SN1) are directed away from the null, resulting in a topological degree of −1 [panels (a) and (b)], whereas the fan field lines (in pink) of spiral null _ 2 (SN2) are directed toward the null, making a topological degree +1. The fan plane field lines of spiral null _ 3 (SN3) (in red) are directed toward the null point, making topological degree +1 [panel (b)]. With time, these two nulls get annihilated.

Close modal

With further evolution, the imaginary part of the eigenvalues of SN1 becomes zero, resulting in no current along spine, implying that the SN1 has lost its spirality and become a radial null and remains radial until t = 8.85 s. The average value of the current density ( | J |) in a subvolume enclosing that spiral null is calculated and found that it drops by 0.6% of its value at t = 8.4 s (the time at which null was spiral in nature). Subsequently, at t = 8.86 s, the imaginary part of the eigenvalues again becomes non-zero, implying non-zero current along the spine, causing it to revert back to a spiral null SN1 (Fig. 3). Accordingly, the average value of | J | in that subvolume enclosing the null increased by 10%. Simultaneously, SN1 approaches another spiral null [panels (f) and (g)], which is one of the spiral–spiral null pair generated earlier at t = 8.68 s and marked as spiral null _ 3 (SN3) in Fig. 3. SN1, SN2, and SN3 are traced in time, and the green, pink, and red field lines are drawn near SN1, SN2, and SN3, respectively [panels (f) and (g)]. SN1 and SN3 approach each other, and ultimately annihilate pairwise [panels (h)–(j) of Fig. 3]. Similar to Fig. 4, the annihilation coincides with a change of global field line connectivity (not shown). The spine and fan plane, along with the topological degree, are depicted in Fig. 5(b). The fan field lines (depicted in red) of SN3 are directed toward the null, making the topological degree +1, and the fan field lines (in green) of SN1 are directed away from the null, resulting in a topological degree −1. The conservation of net topological degree is self-explanatory.

This communication relates the spontaneous generation/annihilation of 3D nulls with varying levels of chaoticity in an initially chaotic magnetic field while investigating the evolution of the involved magnetic field lines. The initial magnetic fields have been derived by superposing two ABC fields, each satisfying the linear force–force condition. For the computations, C = 0.15 , 0.2 , 0.25 , and 0.3, corresponding to initial fields with increasing chaoticity. The updated trilinear 3D null detection technique (described in the study by Maurya et al.27) has been employed to locate the nulls, calculate their topological degrees, and nature (spiral or radial) based on eigenvalues. These analytically constructed initially chaotic fields do not contain any null (also checked using the null detection technique). The simulation results demonstrate a direct correlation between the chaoticity levels and the number of null generations, with higher chaoticity leading to earlier null creations and increased null count. Further to explore null generation/annihilation in more detail, the chaoticity is set at C = 0.3 as the generation of nulls started earlier in time. As an example of the null generation process, a spontaneously generated pair of spiral nulls is selected. Interestingly, one of the nulls changes its nature from spiral to radial with evolution because of the change in current along the spine of null from a non-zero value to zero, and this change is observed as the change in the imaginary part of the eigenvalues of the Jacobian matrix B calculated at null from a non-zero value to zero. Subsequently, this radial null reverts to a spiral null (as the current along the spine of null becomes non-zero) which later annihilates with a different spiral null formed in a distinct null pair generation process. It is already known that null generation and annihilation require local, non-ideal MHD effects.50 To elucidate the global impact of the creation and annihilation of nulls, the relevant magnetic field lines are traced in time and advected with the plasma flow in the ideal region. It is observed that the field lines change their connectivity from one domain to a different domain—demonstrating that the spontaneous generation (and annihilation) of 3D null point pairs leads to a change in the global field topology.

The computations were performed on the Param Vikram-1000 High Performance Computing Cluster of the Physical Research Laboratory (PRL). S.K. would like to acknowledge the support from SERB-SURE (No. SUR/2022/00569). We also wish to acknowledge the visualization software VAPOR (www.vapor.ucar.edu), for generating the relevant graphics, and Chiti et al. for developing the trilinear null detection technique used here (https://zenodo.org/record/4308622#.YByPRS2w0wc). The corresponding theory can be found in the study by Haynes and Parnell.51 

The authors have no conflicts to disclose.

Yogesh Kumar Maurya: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Supervision (supporting); Validation (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (supporting). Ramit Bhattacharyya: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Supervision (equal); Writing – original draft (equal); Writing – review & editing (equal). David I. Pontin: Conceptualization (equal); Methodology (equal); Writing – original draft (equal); Writing – review & editing (equal). Sanjay Kumar: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Funding acquisition (lead); Writing – original draft (supporting); Writing – review & editing (supporting).

The data that support the findings of this study are available from the corresponding author upon reasonable request.

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