Radial drift of plasma blobs in a toroidal magnetic field with fully kinetic and reduced fluid models

In curved magnetic geometries, field-aligned regions of enhanced plasma pressure and density, termed ‘blobs,’ move as coherent filaments across the magnetic field lines. Coherent blobs account for a significant fraction of transport at the edges of magnetic fusion experiments and arise in naturally-occurring space plasmas. This work examines the dynamics of blobs with a fully kinetic electromagnetic particle-in-cell code and with a drift-reduced fluid code. In low-beta regimes with moderate blob speeds, good agreement is found in the maximum blob velocity between the two simulation schemes and simple analytical estimates. The fully kinetic code demonstrates that blob speeds saturate near the initial sound speed, which is a regime outside the validity of the reduced fluid model.


Introduction
Plasma blobs are magnetic field-aligned filaments characterized by increased density and pressure relative to the ambient plasma.In tokamaks, blobs may be formed by the velocity shearing of edge instability streamers [1,2], and blobs are observed to move as coherent structures radially outward from the plasma edge towards the outer wall [3][4][5][6][7].This outward transport of coherent blobs results in a bursty [8] convective (rather than diffusive) type of transport, and it may account for a substantial fraction of edge transport in magnetic confinement devices [9][10][11].Given their importance in edge transport, blobs have also been studied in focused laboratory experiments [12,13].Similar blob dynamics occurs in Original Content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence.Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.natural plasma environments, including blobs associated with solar flares [14] and ionospheric disturbances [15][16][17].
The dynamics of plasma blobs have been studied with fluid simulations and, to a limited extent, in particle codes [18][19][20].Particle-in-cell (PIC) codes offer a high-fidelity kinetic description of blob phenomena, but this high fidelity comes at a high computational cost.As a result, various approximations have been used in previous PIC models of plasma blob dynamics that reduce the computing requirements.For example, PIC blob calculations have been carried out in an electrostatic limit appropriate for low-β regimes in either multiple spatial dimensions [20] or in one spatial dimension while including collisional effects [18].Previous fully electromagnetic PIC calculations [19] considered blob dynamics in the two-dimensional (2D) drift plane perpendicular to the magnetic field.Those previous PIC calculations, however, had a limited scale separation between the electrons and ions (the Debye length was artificially set equal to the ion sound Larmor radius, and the total blob size was ∼5 Debye lengths).
Here, we reconsider 2D blob dynamics in the plane perpendicular to the magnetic field with the electromagnetic PIC code vector particle-in-cell (VPIC) [21].Note that the 2D model neglects parallel dynamics related to sheath currents at the ends of a magnetic flux tube that terminate on a wall [11,[22][23][24][25] as well as drift waves that have finite wave-numbers along the magnetic field [26].Nevertheless, we present the first relatively large-scale electromagnetic PIC simulations with plasma blobs comparable to or larger than characteristic kinetic length scales (such as the ion inertial length) and separated from the electron kinetic scales (including the Debye length, electron skin depth, and electron Larmor radius).Because our kinetic simulations approach the larger fluid scales, it is reasonable to compare our results to a fluid model.Here, we compare the maximum blob radial drift velocities found in our PIC simulations to reduced magnetohydrodynamic (MHD) fluid simulations performed with the BOUT++ code [27] as well as to simple analytical scalings based on a reduced fluid theory.Although computationally challenging, the parameters between the PIC code and reduced MHD code are matched exactly for many runs.An ensemble of PIC simulations is performed over a range of parameters ranging from a limit where the fluid assumptions are well-satisfied to examples where kinetic effects modify the blob dynamics.Our kinetic simulations are the first to demonstrate an asymptotic regime where the blob speed approaches the sound speed, a limit where the simple assumptions of reduced fluid models break down.
This paper is organized as follows: section 2 offers a brief review of blob dynamics from a simplified fluid description.Section 3 describes the simulation codes and model set-ups used in this study.The results of the numerical simulations are described in section 4, where several scalings for blob motion are tested by varying the blob parameters in the simulations.We include an investigation of how the blob radial drift scales with a bulk fluid velocity of the blob parallel to the magnetic, an effect not considered in previous blob models.A brief discussion of the results and prospects for future studies in section 5 concludes the paper.

Simple blob dynamics
The basic mechanism that drives the motion of a plasma blob is a polarization caused by opposite electron and ion curvature and grad-B drifts in an inhomogeneous magnetic field.This polarization sets up an electric field, and the blob subsequently E × B drifts across field lines.Plasma filaments in a tokamak edge are typically well-described by a reduced low-beta MHD model [9,11] that assumes the plasma is well-magnetized, with gradients lengths scales larger than the gyro-radius and time variations slow compared to the gyro-frequencies.In addition, the bulk flows are usually sub-sonic.Under these conditions, the quasi-neutrality condition ∇ • J = 0 may be expressed as where J dia = (B × ∇P)/B 2 is the diamagnetic current (which can be related to the guiding center drifts [28,29]), J pol = (m i n/B 2 ) Ė⊥ is the polarization current, and J || is the parallel current along the magnetic field.For simplicity, we consider plasma blobs in a toroidal vacuum field, B = (B 0 R 0 /R)ê θ , and as mentioned above, we neglect the effects of parallel currents flowing into a sheath.The balance of the diamagnetic and polarization currents then reduces to: where P = n(T e + T i ) is the total plasma pressure, the bulk motion of the blob is given by u E = E ⊥ × B/B 2 , and the total convective derivative is D/Dt ∼ ∂/∂t + u E • ∇.Assuming the blob has a characteristic scale size δ ⊥ , the blob density is n b , the background plasma density is n 0 , and the total time deriv- yields an estimated blob radial drift speed of: where c s = (T i + T e )/m i is the ion sound speed and R c is blob radial location (and local radius of curvature of the magnetic field).Equation (3) gives the blob speed in the so-called inertial limit [30,31] neglecting parallel currents [26], and there are experimental observations in this regime consistent with this scaling [32].The main goal of this paper is to compare 2D kinetic and fluid blob simulations to the simple velocity scaling of equation ( 3) with dependencies on the blob size, electron beta, ion and electron temperature, and relative background plasma densities.

Methods: plasma blob simulations
Simulations of blob motion in the 2D drift plane perpendicular to the magnetic field were carried out using two different plasma modeling codes.Reduced MHD simulations were performed with the code BOUT++ [33], and fully kinetic electromagnetic PIC calculations were performed with the VPIC code [21].The two codes were compared against each other and to the predictions of the MHD scaling of equation (3).Below, we briefly describe the simulation schemes and set-ups for simulating blob motion.
In each simulation, the initial density profile is specified by a uniform background density n 0 added to a Gaussian profile for the blob with amplitude n b and width δ ⊥ : For blobs in a purely toroidal magnetic field, the radial location of the blob center R c is also the local radius of the curvature of the magnetic field.As described below, this local radius of curvature enters as a free parameter in the reduced MHD equations we solve in BOUT++.The electron and ion temperatures are assumed uniform (with the same value in the background and blob regions), with cold ions assumed in the reduced MHD model and separate electron temperature T e and ion temperature T i in the PIC simulations.In the following, we normalize parameters based on characteristics at the blob center.The reference magnetic field B 0 is evaluated at R = R c , the blob electron β e is defined as β e = 2µ 0 n b T e /B 2 0 , the ion cyclotron frequency is ω ci = eB 0 /m i , and the ion sound Larmor radius is ρ s = c s /ω ci = m i (T e + T i )/eB 0 .
In both simulations schemes, the blob accelerates radially outward driven by pressure and magnetic field gradients.To track the blob position and radial velocity, we compute a radial position of the blob center of mass, R COM .This is defined as: where R I is the radial coordinate of cell with index I, n I is the cell's total plasma density, the sums are taken over 2D cell indices I where n I ⩾ n thr , and n thr is a threshold density defining the blob.The threshold density was typically set as n thr = 1.5 × n 0 , with the exception of low-amplitude blobs (n b < n 0 ) that used a lower threshold cutoff (see the appendix for a table A1 of simulation parameters).The threshold was set to eliminate the influence of the background density which could skew the center of mass position.The radial velocity is then calculated over a sample time ∆t as u R = ∆R COM /∆t.The results were rather insensitive to the choice of threshold so long as it was sufficiently high cut off most of the fluctuations in the background density, particularly in the PIC simulations that are susceptible to particle noise.

VPIC
The kinetic simulations in this paper used a version of the VPIC code [21] in 2D R − Z cylindrical geometry [34].In particular, the 2D R − Z grid used for the simulation domain is a poloidal cut of a toroidal system, and it is assumed that ∂/∂θ = 0 (where θ is the angular variable in cylindrical coordinates).VPIC employs a standard PIC algorithm with explicit timestepping, including a finite-difference time-domain electromagnetic field advance on a staggered Yee-Mesh [35] and a particle pusher based on a cylindrical coordinate version of the Boris-Leapfrog method [36].The charge and current densities are accumulated on a logical space mesh using a modified version of the charge-conserving Villasenor-Buneman method [37], which preserves Gauss' law and the solenoidal condition on the magnetic field.With this scheme, energy is conserved for timesteps ∆t → 0, and total energy was conserved in our simulations to within a fraction of a percent.
As is commonly done in PIC simulations, a reduced ionto-electron mass ratio of m i /m e = 100 is used to reduce the computational cost [19].The PIC simulations employ a ratio of the electron plasma frequency to gyrofrequency of ω pe /ω ce = 1.The effects of collisions are neglected and left for future study.The initial magnetic field is a purely toroidal vacuum field B = (B 0 R c /R)ê θ , where B 0 is the magnetic field strength at the initial blob position R = R c .Because we compare the PIC simulations to a fluid model taken in the cold ion limit, we set T i /T e = 0.1 in the VPIC simulations.Finite ion gyroradius effects became discernible, however, especially in a scaling study in which both T e and T i were increased until the total plasma β was no longer low.

BOUT++
BOUT++ is a fluid modeling framework designed to simulate plasmas in the tokamak edge region using a variety of drift-reduced fluid models [27].We use a two-field model for the blob density and vorticity to compare with the kinetic PIC simulations.The equations are given by: where ω is the vorticity, ϕ the electric potential, n the density, b the magnetic field unit vector, ρ s is the sound radius.L || is the parallel connection length, and in the work here parallel transport effects are neglected and this term is set to zero.The toroidal geometry is not modeled directly, but rather the local curvature of the magnetic field is treated by setting R c as a parameter in the equations ( 6) and ( 7).This approximation is appropriate for the blobs we model with a large aspect ratio R c /δ ⊥ .The above reduced model we solve in BOUT++ assumes cold ions, and therefore ion finite Larmor radius gyro effects are not treated.In order to compare to the collisionless PIC simulations, the BOUT++ runs used small numerical viscosity and diffusion coefficients of 1 × 10 −6 to approach the inviscid limit.These small coefficients primarily served to allow the simulations to run with a reasonable time step at late stages in the non-linear evolution of the blobs, and tests with larger dissipation coefficients up to 1 × 10 −3 made practically no difference in the maximum blob speeds observed.

Qualitative comparison of numerical results
We first show typical example cases of 2D blob simulations in each of the modeling schemes.Figure 1 shows the density profile at four different times of (left) a VPIC kinetic PIC simulation and (right) a comparable reduced MHD BOUT++ simulation.Both schemes show very similar blob motion in the initial phases of evolution, and the maximum radial drift speed is nearly identical in the two models.Both schemes produce a 'mushrooming' effect observed in previous modeling and experiments [12,20,22,38].While the general early-time motion is similar between the two model types, there are differences in the nonlinear evolution of the blobs between the two models.Both cases show vortical density structures typical of secondary Kelvin-Helmholtz fluid instabilities, which eventually appear as trailing arms behind the blobs.The filaments of the density thin down to kinetic scales in the PIC model, and these density perturbations dissipate into the background plasma.In the reduced MHD model, on the other hand, the vortical density trails remain coherent even as they turn over multiple times, and the spiraling eddies continue to travel with the blob.These slowermoving eddies of density contribute to an apparent slowing of the blobs in the reduced MHD model compared to the kinetic PIC model at later times.The blobs in the kinetic model are also susceptible to additional instabilities absent in the reduced fluid treatment.Based on the dominant wavelengths and location of flute-like modes on the edge of blobs in the PIC model, we see evidence of the lower hybrid drift instability [39,40].These are pressure-gradient driven, mainly electrostatic modes that contribute to dissipating the blob.In addition, the kinetic blob model shows considerably more asymmetry, likely owing to finite ion gyroradius effects [41].Additional evidence of gyromotion contributing to the asymmetry is that the PIC blob center of mass drifts towards the left in this simulation, while the blob drifts in the opposite direction (to the right) when the magnetic field direction was reversed in test runs.

Blob size velocity scaling
To begin testing the scaling of equation ( 3), the relationship between maximum blob velocity and blob radius δ ⊥ is studied in BOUT++ and VPIC simulations.In a series of simulations, we vary the blob size δ ⊥ while keeping the other parameters fixed.In both codes, the plasma blobs are initialized at a radius of curvature R c = 1600d e = 2700ρ s , with a relative background density n 0 = 0.10 × n b .
A main prediction of the simple scaling of equation ( 3) for plasma blobs in the inertial regime is that the blob velocity should increase with the blob size, scaling as u R ∝ √ δ ⊥ .Figure 2 shows the maximum velocity of blobs of varying initial sizes, and the reduced fluid and fully kinetic models agree well on the maximum velocity for comparable runs over blob size.A monotonic increase in blob maximum velocity was observed, and the analytical scaling u R ∼ √ δ ⊥ was supported by these results.The fully kinetic simulation with the largest blob and highest maximum speed, however, began to deviate from this trend with a smaller maximum velocity than predicted.
The instantaneous blob velocity over time in the fully kinetic simulations from figure 2 are shown in figure 2. Here, it is seen that these blobs of fixed blob density n b , background density n 0 , and radial position R c , accelerate at the same constant rate, consistent with theoretical predictions [42].The larger blobs, however, accelerate over a longer time interval and reach a higher maximum speed.In general, the blobs exhibit a single local maximum velocity before decelerating and eventually dissipating.The deviation from uniform acceleration seen in the blob velocity evolution noted previously comes as the blob begins to form trailing arms, suggesting that the uniform acceleration breaks down as the coherent structure of the blob begins to dissipate.
Figure 3 shows the density profiles of some of the fully kinetic PIC runs from figure 2 at their time of maximum velocity.The blobs of different sizes have qualitatively similar shapes at the time of maximum velocity.The maximum velocity is reached in the early formation of the 'mushroom' structure, when a directional density head has formed, and just as the  vortical structures of the wake are beginning to form.Following this time, the head broadens in the poloidal Z direction and eddies form on the trailing arms of the blob such (as in the later time slices of figure 1).The blobs also display an asymmetry in the Z dimension which has been noted in previous particle simulations [19].Test runs with the magnetic field reversed showed that this asymmetry in the density and velocity also reversed direction.As noted above, this suggests the Z-directed drift is related to finite ion gyro motion.
While the parameter sweep of figure 2 agrees well with the simple scaling u R ∝ √ δ ⊥ of equation ( 3), we expect the scaling to break down outside the regime of validity of the reduced model.For example, for blobs with a smaller aspect ratio (or larger ratio δ ⊥ /R c ), the predicted velocity of equation (3) can increase above the sound speed c s .This is beyond the subsonic regime with bulk flows u < c s implicit in the reduced fluid model of equations ( 6) and (7).More basically, the blob bulk velocities must remain below the sound speed based on simple energy conservation considerations.The thermal energy of the electrons (and also ions for cases with non-negligible ion temperature), is converted into bulk kinetic energy of the drifting blob [12], placing a limit of u < 2T e /m i ∼ c s .
To explore this regime outside the scope of the reduced fluid description where the flows approach the sound speed, a similar set of PIC simulations varying the blob size δ ⊥ was also conducted for a smaller radius of curvature R c = 250d e ∼ 420ρ s .The results of this set of simulations are shown in figure 4. Similar to the previous set-up with larger R c , the expected maximum velocity scales as u R ∼ √ δ ⊥ for blobs with small sizes δ ⊥ .As the ratio of blob size to radius of curvature becomes large δ ⊥ /R c ∼ 10 −1 , however, the scaling breaks down.The blob speed is limited to not much larger than the initial sound speed, c s = (T i + T e )/m i .In the fully selfconsistent PIC model, the blob speed saturates near the sound speed c s for sufficiently large blobs, in agreement with energy conservation considerations and outside the regime of validity of the simplified reduced fluid model.

Plasma beta velocity scaling
A next test of the scaling of equation ( 3) was performed to examine the dependence of the velocity on the plasma β = 2µ 0 n b (T e + T i )/B 2 0 .The simple scaling as well as the reduced fluid model of equations ( 6) and (7) are taken in a low-β limit in which the plasma temperature only enters through a rescaling of the velocities by the sound speed c s .In the kinetic model, on the other hand, the maximum drift speed of the blobs may be studied as a function of the plasma β.A parameter sweep over β at fixed blob size δ ⊥ and radius of curvature R C was conducted in the fully kinetic PIC model (see appendix for detailed simulation parameters).Blobs were initialized with varying values of β e = 2µ 0 n b T e /B 2 0 , and the ion-to-electron temperature ratio was held fixed at T i /T e = 0.1.The expectation for the scaling of the maximum blob velocity based on equation ( 3) is that u R ∝ c s ∝ √ β.This follows because the diamagnetic currents that drive the blob polarization and radial drift scale with β for a fixed blob size and peak density.
Figure 5 shows the maximum radial drift velocity for blobs with a range of initial values of β.As expected, the blob velocity scales roughly as u R ∝ √ β (dashed curve).Although not plotted, blobs with even higher β became unstable and quickly dissipated.This effect was most pronounced when the ion gyroradius became comparable to the size of the blob.Note that the initial conditions of equation ( 4) are not an equilibrium because the background vacuum field cannot balance the pressure gradient of the blob.While low-β blobs remain coherent long enough to undergo a bulk radial drift, high-β plasma blobs are strongly out of equilibrium and can disassemble on time scales comparable to the sound crossing time.This high-β regime cannot be modeled with electrostatic PIC codes [20], which do not include the back-reaction of the plasma pressure on the magnetic field.

Background density velocity scaling
An additional set of simulations was performed to test the dependence of the maximum radial drift speed u R of blobs on the blob amplitude n b /n 0 .Following equation ( 3), the expectation is that as the blob amplitude increases, so too does the pressure gradient in the blob and the induced polarization and radial drift.Figure 6 shows the maximum velocity reached by blobs of varying amplitudes (n b /n 0 ).The expected scaling of u R ∼ n b /n 0 was supported for blob amplitudes n b /n 0 < 1 in the reduced fluid model.PIC simulations in this regime were difficult to assess because particle noise could not be reduced to acceptably low levels.In the intermediate regime, 1 ⩽ n b /n 0 ⩽ 100, the reduced fluid and fully kinetic PIC models show good agreement, although the blob velocities do not increase as quickly as the predicted n b /n 0 scaling (black dashed curve).In the fully kinetic code, it is possible to model a blob in vacuum, and the PIC simulations plateau towards a maximum drift speed in vacuum.The reduced fluid model over-predicts the maximum drift speeds in this regime for blobs in a near-vacuum background with n b ≫ n 0 .
Figure 6(b) shows the blob velocity over time from the kinetic PIC simulations runs in 6(a).The blob velocity evolution follows the same pattern as in figure 2(b) of constant acceleration before an inflection point at a maximum velocity.The time required to accelerate to a maximum velocity, however, is relatively insensitive to the blob amplitude in the PIC simulations for these blobs of fixed size, radial location, and temperature.

Parallel bulk flow velocity scaling
As a final study of blob velocity scaling, we consider the outward radial drift caused by a bulk parallel fluid velocity u || along the magnetic field.This radial drift is not included in the simple scaling of equation ( 3), which includes only the effect of the pressure gradient and not any bulk fluid velocity.Nevertheless, a bulk parallel flow of the electrons and ions will cause a charge separation electric field very similar to that induced by the diamagnetic current.The total current in a well-magnetized plasma may be understood in terms of the guiding center motions and a magnetization current [28,29].Qualitatively, we expect the bulk parallel velocity to contribute an additional current to equation (1) that is proportional to u 2  || and is related to the guiding center curvature drift.
Here, we consider both moderate bulk velocities and supersonic (u || > c s ) flow velocities, which fall outside the scope of the reduced fluid model.Locally supersonic flows in filaments are theoretically possible in edge scrape-off plasmas [43].Such extreme flows are usually considered with a 1D  model of parallel transport along the magnetic field [44].
While we neglect the important parallel transport processes that occur where such a filament reaches a divertor or other material edge, we consider how the parallel bulk flow couples to the perpendicular drift motion of the plasma blob.We ran a set of PIC simulations in which blobs with the same nominal size (δ ⊥ = 8 d e ) and amplitude (n 0 /n b = 0.1) were initialized with varying parallel bulk drift speeds in the range of u || /c s from 0 to ∼3.Based on the simple picture of the guiding center curvature drifts (∝ u 2 || ) contributing an additional current on top of the thermal diamagnetic current (J dia ∝ v 2 in equation (1), where v 2 is an average squared velocity given by the thermal spread or pressure of the particles), we find a simple functional fit for the maximum radial drift speed of the blob of the form u R ∝ 1 + u 2 || /c 2 s .This fit is plotted as dashed curve in figure 7 and it agrees well with the PIC simulation results.

Conclusions
A series of simulation parameter sweeps in a reduced fluid code and in a fully kinetic electromagnetic PIC code showed good agreement between the maximum radial drift speed u R of a plasma blob and a simple analytical approximation.It was found that blobs generally experience a nearly uniform acceleration for a time until close to the maximum velocity.Qualitatively, this maximum velocity is reached as secondary instabilities set in and form mushroom-like trailing arms.Blobs of different sizes accelerate at the same rate, but larger blobs accelerate for a longer period of time leading to a higher maximum velocity.In contrast, blobs of higher amplitude (higher density of the peak compared to the background density) tend to accelerate at a faster rate.While the acceleration rate of the blob increases with blob amplitude, the time during which the blob undergoes acceleration does not increase significantly.The scaling of velocity with blob amplitude was as expected for blobs of amplitude n b /n 0 < 1, but plateaued at higher amplitudes.Generally, the reduced fluid and kinetic models agree well in the regime of validity of the fluid approximations.The first-principles PIC model showed that the blob speed is limited, however, to approximately the initial sound speed due to energy conservation.This effect is ordered out of the simplified drift-reduced fluid model.The above properties of blob motion in the low-beta, well-magnetized regime could be incorporated into statistical models of blob transport.In the inertial regime, a maximum speed of a blob of size δ ⊥ in a toroidal field at radius R c , with amplitude n b in a background of density n 0 , with sound speed c s , and with a parallel fluid flow u || would be The maximum blob speed u R would also be limited by the sound speed c s and the asymptotic speed in vacuum u vac , u R ∼ min {u Rmax , c s , u vac }.The motion of the blob could be additionally characterized by an acceleration rate a that is roughly independent of δ ⊥ /R c and an acceleration time τ that is roughly independent of n b /n 0 .Eventually, such a model could also incorporate effects related to parallel transport and sheath effects on blobs limited by material boundaries, effects that were not considered here.
The study here may serve as a basis for future kinetic modeling of plasma blobs, especially with extensions to 3D simulations.Important additional effects include finite blob extent parallel to the magnetic field, which may place blobs in a regime where the maximum velocity is determined by sheath effects at the boundaries.This may be modeled with the cylindrical VPIC code in 3D by including a full toroidal geometry or a wedge with a finite extent in the angular θ direction [45].The effects of collisions may also be included self-consistently in PIC codes [46], and collisional effects on blog propagation in multiple spatial dimensions may be studied in the future with VPIC or another PIC code [20].

Figure 1 .
Figure 1.Simulations of initially similar blobs in VPIC (left) and BOUT++ (right) showing the evolution of the density profile through time.The vertical axis shows the radial position and the horizontal gives Z.For this case, the maximum blob speed is nearly identical in the two models.While the initial phases of the blob motion are very similar, the models show differences in the late nonlinear phase of evolution.

Figure 2 .
Figure 2. (a) Maximum instantaneous velocity achieved plotted against the blob radius δ ⊥ for blobs initialized with varying radii in a reduced fluid model (red triangles) and in a fully kinetic PIC model (blue dots).The maximum velocity scales consistently with the expected √ δ ⊥ scaling of equation (3) (dashed curve).(b) Velocity over time of different sized blobs simulated in VPIC.Blobs of different sizes accelerate at the same constant rate, with larger blobs accelerating for longer and ultimately reaching a higher maximum velocity.

Figure 3 .
Figure 3. Blob density profile at time of maximum velocity for δ ⊥ = 4de (left), 8de (middle) and 16de (right).The density profile of blobs of different sizes at the time of maximum amplitude are qualitatively similar.In each case, trailing arms, or mushroom effects, are beginning to form.

Figure 4 .
Figure 4. Maximum instantaneous velocity depending on blob radius for Rc = 250de ∼ 420ρs, resulting in larger values of δ ⊥ /Rc than in figure 2. The maximum velocity initially adheres to the expected scaling, but the maximum velocity of large blobs saturates near the initial sound speed.

Figure 5 .
Figure 5. Blob maximum center of mass velocity in terms of an arbitrary standard velocity against initial plasma β in VPIC simulations.The expected scaling, u R ∼ √ β, is shows as a dashed curve.

Figure 6 .
Figure 6.(a) Maximum center of mass velocities in reduced fluid (red) and fully kinetic particle (blue) simulations runs at varying blob amplitude.(b) Center of mass velocity through time of blobs of varying amplitude in the set of PIC simulations.Blobs of differing amplitudes accelerate at different rates but reach their maximum velocity in around the same amount of time.

Figure 7 .
Figure 7. Blob maximum center of mass velocity as a function of the initial parallel drift speed u || of the plasma along the magnetic field.The dashed curve shows the approximate scaling u R ∝ √ 1 + u 2 || /c 2 s .