2. Experimental and simulation methodologies

The Big Red Ball (BRB) at the Wisconsin Plasma Physics Laboratory is a versatile plasma confinement device well-suited to the study of high-$\beta$ and flow-dominated systems. The capabilities of the BRB are discussed in more detail in other works (Forest et al. Reference Forest2015; Olson et al. Reference Olson, Egedal, Greess, Myers, Clark, Endrizzi, Flanagan, Milhone, Peterson, Wallace, Weisberg and Forest2016; Flanagan et al. Reference Flanagan, Milhone, Egedal, Endrizzi, Olson, Peterson, Sassella and Forest2020; Endrizzi et al. Reference Endrizzi, Egedal, Clark, Flanagan, Greess, Milhone, Millet-Ayala, Olson, Peterson, Wallace and Forest2021) and the experimental set-up of the Parker Spiral solar wind experiments and initial findings are detailed by Peterson et al. (Reference Peterson, Endrizzi, Beidler, Bunkers, Clark, Egedal, Flanagan, McCollam, Milhone, Olson, Sovinec, Waleffe, Wallace and Forest2019). A depiction of the experimental configuration of the BRB for these experiments as well as the computational model of the experiment are shown in figure 1. The summary of the experimental methodology for generating the Parker Spiral in the BRB is as follows: Thermally emissive lanthanum hexaboride cathodes are used to generate a warm, dense, unmagnetized plasma atmosphere ($T_e \sim 7$ eV, $n_e \sim 4\times 10^{17}$ m$^{-3}$). A permanent dipole magnet is placed inside this background plasma with two ring electrodes located near its north and south poles. Current is driven from molybdenum anodes in the plasma atmosphere into the dipole magnetosphere and collected on the polar electrodes. This current path is visualized in figure 1(b) in the context of the NIMROD simulation domain. These cross-field currents generate a torque on the plasma which causes the magnetosphere to rotate, thereby twisting the magnetic field into a Parker Spiral.

Figure 1. Experimental (a) and computational (b,c,d) configurations. The Big Red Ball is shown in (a) along with the local cylindrical coordinate system, dipole magnet and diagnostics used for mapping the magnetosphere. The two-dimensional (2-D) finite element mesh with current injection boundary conditions and initial vacuum field configuration are shown in (b). Panels (c) and (d) show the location of probes within the NIMROD simulation for outputting high-time-resolution field measurements. Both the initial magnetic field configuration (c) and time-averaged magnetic field configuration after the current injection has reached steady state (d) are shown to demonstrate the probe positions relative to where the plasmoid formation process occurs.

At the interface between the closed field lines of the magnetosphere and the open field lines of the Parker Spiral, periodic reconnection occurs, which ejects axisymmetric plasmoids, much like the density structures observed in the heliospheric current sheet which emanate from streamer top reconnection. A number of diagnostics were employed to map the 2-D time dynamics of the Parker Spiral including two arrays of three-axis Hall sensors, and an array of triple probes and 2-D Mach probes for measuring density, temperature and flows in the plasma. One of the Hall sensor arrays was kept stationary and displaced azimuthally from the 2-D scanning plane to provide phase reference measurements critical for the reconstruction of the plasmoid dynamics.

The computational domain and vacuum magnetic field for the accompanying NIMROD simulations are shown in figure 1. Figure 1(b) represents the hemispherical mesh in cylindrical coordinates, which extends down to $R=5$ cm rather than to the experiment's support rod at $R=2$ cm. This is to allow for a small dipole magnet to be placed outside the computational domain and to facilitate the boundary condition manipulation to model current injection/extraction in a manner representative of the experiment. By prescribing $B_\varphi$ along the boundary as a function of time, we can set the normal component of $J$, or the current into and out of the vessel. In all the presented simulation work that follows, the current injection linearly ramps from zero up to a prescribed steady-state value. The ramp duration in some of the initial simulations was 1 ms, but was shortened to 100 $\mathrm {\mu }$s to reduce the required simulation time for some of the higher current injection cases that have smaller time steps. For all simulations discussed in this work, there are zero heat flux and zero particle flux conditions applied to the boundary as well as a no-slip boundary condition on the velocity. All simulations were performed with experimental parameters of $n_{e} = 4\times 10^{17}$ m$^{-3}$, $T_{e} = 7$ eV, $T_{i} = 0.5$ eV, which give viscous and resistive diffusivities of $\nu = 50$ m$^{2}$ s$^{-1}$ and $\eta = 35$ m$^{2}$ s$^{-1}$, respectively. We also note that in the experiments and simulations the ion skin depth is set to $d_i = 70$ cm. In terms of resolution, the simulations were performed with 2400 bicubic poloidal elements which provides centimetre-scale resolution in the current sheet.

In order to better compare results from simulation with the experimental measurements, the NIMROD code was modified in order to take a list of $R$, $Z$ coordinates for placing history nodes, or probes. For the simulations in this study, four probes were placed in the current sheet at $R=25, 40, 55, 70$ cm and $Z=0$ cm, as shown in figure 1. The probe locations relative to the vacuum magnetic flux configuration and the flux distribution near the end of a simulation can be seen in figures 1(c) and 1(d), respectively. These probes provide outputs for the solution fields after every time step to allow for higher frequency phenomena to still be captured at a few select locations in the simulation. All NIMROD simulations presented in this work are axisymmetric.

3. Observation of streamer top reconnection and plasmoid formation in experiment and two-fluid simulations

While the basic operating parameters, mean magnetic field and plasma flows that develop in both the experiment and simulations are presented by Peterson et al. (Reference Peterson, Endrizzi, Beidler, Bunkers, Clark, Egedal, Flanagan, McCollam, Milhone, Olson, Sovinec, Waleffe, Wallace and Forest2019), this article sheds more light on the details of the fluctuation measurements as well as their scaling properties. Previous work demonstrated that, during the generation of the Parker Spiral, two fluid effects are critical as the system size is small compared with the ion skin depth, the electrons are the only magnetized species and $T_e \gg T_i$. The consequence of this is a radially outward flow of electrons that advects the dipolar magnetic field into a Parker Spiral and simultaneously generates an inward electric field via the Hall effect. This Hall electric field drives accretion of ions into the magnetosphere where a density and pressure gradient begin to build until a loss of equilibrium occurs. In both experiment and simulation, this loss of equilibrium occurs in the current sheet associated with the Parker Spiral where $\beta > 10$ (Peterson et al. Reference Peterson, Endrizzi, Beidler, Bunkers, Clark, Egedal, Flanagan, McCollam, Milhone, Olson, Sovinec, Waleffe, Wallace and Forest2019) and increases up to $\beta \sim 50$ further downstream.

Measurements of the magnetic field as well as the plasma density show interesting fluctuations whose frequency depends on the amount of current injected. As shown in figure 2(a), the vertical component of the magnetic field, $B_Z$ (the poloidal component perpendicular to the current sheet), is non-axisymmetric and highly uncorrelated in the initial, high-current phase of the experiment (Phase I). However, as shown in figure 2(a.ii), these magnetic fluctuations become coherent as the current injection falls with the characteristic time scale set by the RC circuit of the current injection system ($\sim 100$ ms). In addition, the $B_Z$ fluctuations are bi-polar and thus suggestive of closed loops of magnetic flux disconnected from the inner magnetosphere of the rotating plasma. A spectrogram typical of these magnetic fluctuations in the current sheet can be seen in figure 2(b), which shows the turbulent nature of the high-current Phase I and coherent mode Phase II with a high degree of correlation between the plasmoid frequency and current injection of the system. This scaling relationship will be discussed in subsequent sections. Performing phase correlation measurements between the stationary Hall array and the scanning array over many discharges allows us to reconstruct the plasmoid dynamics – both magnetic field and density – through conditional averaging. This reconstruction method first identifies 500 $\mathrm {\mu }$s time windows for every shot that exhibits the highest degree of shot-to-shot similarity as measured by a single stationary Hall probe. A centre frequency of 20 kHz was used for this process as it showed the best correlation shot-to-shot between 1.1 and 1.15 s, as shown in figure 2(b). Once the best time window from each shot is selected, a phase shift is calculated for each shot using the stationary reference probe which allows us to align in time the many different shots at different locations and build up a poloidal map of the 2-D plasma dynamics. This process reveals periodic reconnection occurring near the top of the streamer structures resulting in plasmoids being ejected into the Parker Spiral current sheet as shown in figure 3. Figure 3(a) shows the $B_Z$ measurement as well as ion density fluctuations $\tilde {n}_i$ as measured by the probe located at ($R=42$ cm, $Z=-8$ cm) and shown as a cyan dot in figure 3(b–d). We believe the up-down asymmetry of the current sheet, or ‘droop’, is likely due to preferential current draw to an anode in the southern hemisphere that was larger than the others, which is not captured in simulations since the current injection is uniform over the range of $\pm 15^{\circ }$ latitude. Figure 3(b–d) show subsequent time steps of 10 $\mathrm {\mu }$s through one half-period of the reconnection process, which exhibits a periodic build up of plasma density (and pressure) inside the streamer leading to field line stretching, reconnection and plasmoid ejection.

Figure 2. Time histories and frequency content of magnetic signals measured in the Parker Spiral current sheet by three-axis Hall sensors in the experiment. The difference between the turbulent, non-axisymmetric Phase I and axisymmetric, single mode Phase II can be seen in (a) where two probes at the same radius, but different azimuthal angles, have very different mean field values as well as high-frequency characteristics in Phase I (a.i.), but these are nearly identical in Phase II (a.ii). A spectrogram of one of these time histories (b) shows broadband fluctuations in Phase I, followed by a coherent downward chirping spectrum of laminar plasmoid ejection.

Figure 3. Plasmoids are ejected from the helmet streamer cusp at a frequency of 20 kHz when the current injection is 150 A in the experiment. (a) The magnetic field ($B_Z$) and density fluctuation ($\tilde {n}_i$) signals as measured by the probe in the current sheet at $R=42$ cm and denoted by the teal dot in panels (b), (c) and (d). (b), (c) and (d) The flux map and density perturbation map at three successive 10 $\mathrm {\mu }$s time steps corresponding to the magenta, cyan and yellow vertical lines in panel (a), respectively. These panels show how the magnetic field at the streamer cusp expands outwards with higher density plasma until reconnection occurs, which releases a plasmoid into the current sheet.

In addition to plasmoids observed experimentally, they also manifest in extended-MHD simulations when the Hall term and electron pressure gradient term are used in Ohm's law. Two-dimensional cross-sections of the magnetic flux and density fluctuations from the NIMROD simulations as well as the time resolved signals in the current sheet, as would be measured by probes in the experiment or satellites in space, are shown in figures 4 and 5. The time histories of both experimental and laminar plasmoids can be seen in movies published in previous work (Peterson et al. Reference Peterson, Endrizzi, Beidler, Bunkers, Clark, Egedal, Flanagan, McCollam, Milhone, Olson, Sovinec, Waleffe, Wallace and Forest2019), whereas the evolution of the turbulent plasmoids in figure 5 can be viewed in the supplementary movie available at https://doi.org/10.1017/S0022377821000775. Figure 4 represents a moderate current injection level of 400 A in the simulation which is slightly above the threshold necessary to observe plasmoids. As a result, we observe relatively large single plasmoids emitted with a very regular frequency. We refer to these plasmoids as laminar since the magnetic flux evolves very smoothly with regular ejection of similarly sized plasmoids. On the other hand, figure 5 represents a high-current drive case of 1000 A where we see plasmoids of many sizes interacting in a thinner current sheet resulting in a more turbulent medium downstream.

Figure 4. Laminar plasmoid formation in an axisymmetric NIMROD simulation with 400 A of injected current. (a) The time history of $B_Z$ and ion density fluctuations, as measured by a probe at $R=70$ cm denoted by the teal dot in (b), shows the periodic ejection of high-density plasmoids that are roughly similar size over time.

Figure 5. Turbulent plasmoid formation in an axisymmetric NIMROD simulation with 1000 A of injected current. (a) The time history of $B_Z$ and ion density fluctuations, as measured by a probe at $R=70$ cm denoted by the teal dot in (b), shows the high variability of plasmoids in both frequency and size.

The observation of plasmoids in the experiment as well as in the NIMROD simulations begs the question of whether this occurrence is coincidental or if the same physical processes are driving plasmoid formation in both cases. Importantly, theoretical and computational studies (Bhat & Loureiro Reference Bhat and Loureiro2018), as well as experiments (Hare et al. Reference Hare, Suttle, Lebedev, Loureiro, Ciardi, Burdiak, Chittenden, Clayson, Garcia, Niasse, Robinson, Smith, Stuart, Suzuki-Vidal, Swadling, Ma, Wu and Yang2017), have demonstrated that plasmoid formation can occur at values of the Lundquist number much below the usual $10^{4}$ required in resistive MHD when ion-scale kinetic effects are not negligible, as is the case here. We now turn our attention to a discussion of the properties and scaling relationships for these observed plasmoids.

5. Heuristic model of plasmoid evolution and extrapolation to solar streamers

A relatively simple heuristic model can be constructed for the plasma expansion in the high-$\beta$ transition region between the hydrostatic equilibrium of the closed flux corona and the hydrodynamic equilibrium of open field lines outside the HCS. The geometry of such a system is shown in figure 9. Writing the momentum equation in terms of the magnetic curvature vector, ${\kappa }$, we obtain the dynamic equation for the plasma expansion:

(5.1)\begin{equation} \rho\frac{\textrm{d}\boldsymbol{v}_\perp}{\textrm{d}t} ={-}\boldsymbol{\nabla}_\perp{p} + \frac{B^{2}}{\mu_0}\boldsymbol{\kappa} - \boldsymbol{\nabla}_{{\perp}}\frac{B^{2}}{2\mu_0}. \end{equation}

Figure 9. A cartoon drawing of helmet streamer topology with indications of the zones where equilibrium exists; namely, deep in the streamer where there is a hydrostatic solution and outside the current sheet where Parker's original hydrodynamic solution is valid. The interface between the two exhibits no equilibrium due to the particle and heat sources present in the corona. This loss of equilibrium is caused by the extremely high values of $\beta$ in this region and result in quasi-periodic plasma blobs which are driven by strong pressure gradients. The length scale $L_c$ shown above is the critical length scale of the current sheet that develops from field line stretching before reconnection occurs.

Using the equation of state $p = \rho c_s^{2}$, rewriting the right-hand side of (5.1) in terms of the plasma $\beta = 2\mu _0 p / B^{2}$, and dotting both sides with the curvature vector, ${\kappa }$, gives

(5.2)\begin{equation} \boldsymbol{\kappa}\boldsymbol{\cdot}\frac{\textrm{d}\boldsymbol{v}_\perp}{\textrm{d}t} ={-}\frac{c_s^{2}}{p}\boldsymbol{\kappa}\boldsymbol{\cdot}\boldsymbol{\nabla}_\perp [p(1 + 1/\beta)] + \frac{2c_s^{2}}{\beta}\kappa^{2}.\end{equation}

The left-hand side of this equation can be taken to define a characteristic time scale for the driven loss of equilibrium: ${\kappa }\boldsymbol {\cdot } {\textrm {d}\boldsymbol {v}_\perp }/{\textrm {d}t} \sim -\gamma _\textrm {dr}^{2}$. This can be understood as the time required to form a plasmoid of radius $R_c = \kappa ^{-1}$ due to the acceleration provided by the net force on the right-hand side of (5.2). As discussed before, the values of $\beta$ in the current sheet are very large for both the experiment and simulations and are in the range of $10 < \beta < 50$ throughout the region of interest, denoted as the high-$\beta$ dynamic region in figure 9. Therefore, ignoring terms of order $\beta ^{-1}$ results in

(5.3)\begin{equation} \gamma_\textrm{dr}^{2} \sim \frac{c_s^{2}}{p}\boldsymbol{\kappa}\boldsymbol{\cdot}\boldsymbol{\nabla}{p}.\end{equation}

This characteristic time scale can be understood as the time for a sound wave to traverse a distance equivalent to the geometric mean of the pressure gradient scale length and radius of curvature, or equivalently the free-fall time of a plasma parcel under the action of a pressure gradient and adverse magnetic field curvature.

The next step in the derivation of the plasmoid frequency is to show that it is essentially this drive frequency which sets the frequency of reconnection in the current sheet. It will be shown that in this situation the reconnection rate is relatively insensitive to the details of the tearing mode growth rate. We can assume that the current sheet is lengthening at the rate given by the drive time scale $\gamma _\textrm {dr}$ because the plasma is frozen to the magnetic field. We can also assume that the lengthening of the current sheet is exponential in time, a result of a transition from quasi-equilibrium to a dynamic system. Incompressibility then requires that the forming sheet is likewise thinning exponentially, such that its thickness $a(t)$ can be described by

(5.4)\begin{equation} a(t) = a_0\,\textrm{e}^{-\gamma_\textrm{dr}t}, \end{equation}

where $a_0$ is the thickness at the beginning of the expansion.

As the aspect ratio $L/a$ of this forming current sheet increases, it becomes unstable to a progressively broader spectrum of tearing modes. As argued by Uzdensky & Loureiro (Reference Uzdensky and Loureiro2016), one of those modes – the one whose growth rate, $\gamma _\textrm {tear}(t)$, first satisfies $\gamma _\textrm {tear}(t_\textrm {crit})\sim \gamma _\textrm {dr}$ – will eventually grow to become as wide as the forming sheet, thereby disrupting it and leading to plasmoid ejection. This happens at a so-called critical time, $t_\textrm {crit}$, whereupon the sheet thickness is

(5.5)\begin{equation} a_\textrm{crit} \equiv a(t_\textrm{crit})= a_0\, \textrm{e}^{-\gamma_\textrm{dr}t_\textrm{crit}}. \end{equation}

This can be inverted to yield

(5.6)\begin{equation} t_\textrm{crit} = \gamma_\textrm{dr}^{{-}1} \ln{\left(\frac{a_0}{a_\textrm{crit}}\right)}. \end{equation}

As we can see, the critical time for reconnection to occur is essentially the same as the drive time scale, $\gamma _\textrm {dr}^{-1}$, since the ratio of the initial to the critical current sheet thickness represents only a logarithmic correction. That is, while reconnection is essential to the formation and ejection of plasmoids, the physics of the reconnection onset are such that details of the tearing instability that underlies it (such as the functional form of $\gamma _\textrm {tear}(t)$) are not essential to the prediction of the time scale associated with the plasmoid ejection.

Casting the drive time scale for $\gamma _\textrm {dr}$ into a characteristic frequency in Hz we obtain

(5.7)\begin{equation} f = \frac{1}{2{\rm \pi}}\sqrt{c_s^{2}\boldsymbol{\kappa}\boldsymbol{\cdot}\boldsymbol{\nabla}{p}/p}. \end{equation}

This time scale is associated with a high-$\beta$ pressure-curvature-driven loss of equilibrium. The characteristics of these oscillations – namely that they are electromagnetic, axisymmetric perturbations localized to the region of bad magnetic curvature with a frequency dependent on the pressure gradient – support the notion that these plasmoids are driven by a mechanism similar in nature to flute modes, but likely represent a loss of equilibrium due to particle and heat transport into the streamer rather than a linear instability. It is the frequency in (5.7) that empirically provides the unifying scaling between the experimental results and the extended-MHD simulations presented in figure 8. Computing this frequency along field lines from the density, temperature and magnetic flux measured by diagnostics in the experiment as well as the simulation shows a local maximum located at the outboard midplane where the curvature is largest. Plotting the measured plasmoid frequencies against this calculated pressure-curvature frequency gives the results in figure 8(b), which shows much better agreement between the experiment and simulations, and is consistent with the idea that these plasmoids are pressure driven.

If we normalize both the pressure gradient scale length and the magnetic curvature by the critical current sheet length, $L_c$, we obtain two dimensionless quantities: $\ell _b^{-1} = L_c\kappa$ and $\ell _p^{-1} = L_c\boldsymbol {\nabla }{p}/p$. We can assume, in the laminar plasmoid case, that this critical length, $L_c$, is simply the plasmoid length just as reconnection occurs. Substituting these parameters into (5.7) provides us with the dimensional scaling:

(5.8)\begin{equation} f = \frac{1}{2{\rm \pi}}\frac{c_s}{L_c\sqrt{\ell_b\ell_p}}. \end{equation}

Given similar normalized length scales, $\ell _b$ and $\ell _p$, the scaling between experimental and solar wind frequencies is simply the ratio of critical length scales, or plasmoid size, and sound speeds; i.e.:

(5.9)\begin{equation} f_\textrm{sw} \sim \frac{L_{c,\exp}}{L_{c,\textrm{sw}}}\frac{c_{s,\textrm{sw}}}{c_{s,\exp}}f_{\exp}. \end{equation}

As mentioned before, we will use the plasmoid length as the proxy for $L_c$ as it is reasonable to assume for single plasmoids that the associated plasmoid is roughly the size of the current sheet just before it reconnects. For plasmoids in the simulations and experiment, we will take this length scale to be $L_{c,\exp } = 0.25$ m and in the solar wind we will take this scale to be $L_{c,\textrm {sw}} = 1 R_\odot = 7\times 10^{8}$ m, which is consistent with observations of plasmoids appearing around $2\text {--}4 R_\odot$, and having a length of $1 R_\odot$ and width of $0.1 R_\odot$ (Sheeley et al. Reference Sheeley, Lee, Casto, Wang and Rich2009). Combining these plasmoid length scales with the sound speed typical of the experiment ($c_{s,\exp } \sim 13$ km s$^{-1}$) and solar corona ($c_{s,\textrm {sw}} \sim 200$ km s$^{-1}$) gives us a simple scaling relationship between frequencies observed in the lab and in the solar wind as

(5.10)\begin{equation} f_\textrm{sw} \sim 5.5\times 10^{{-}9}f_{\exp}. \end{equation}

Therefore, the $20\text {--}40$ kHz plasmoids observed in both the experiment and simulations correspond to a plasmoid frequency in the solar wind of $110\text {--}220\ \mathrm {\mu }$Hz, or periods of $75\text {--}150$ min – in remarkable agreement with observations (Viall & Vourlidas Reference Viall and Vourlidas2015).

We note that this model also offers a natural explanation for the existence of laminar and turbulent plasmoid regimes. Increasing the drive (i.e. increasing the injected current in experiments and simulations) leads to a larger $\gamma _\textrm {dr}$. When the drive is strong enough such that $\gamma _\textrm {dr} \gtrsim v_A / L_c$, an ejected plasmoid has insufficient time to advect downstream a distance comparable to its length before the subsequent plasmoid is ejected. This leads to plasmoids of different sizes interacting downstream of the reconnection region and more stochastic behaviour. For the high-drive cases in the simulation, many of the plasmoids are small ($L_c \sim 5$ cm) and $v_A \sim 2$ km s$^{-1}$, which results in the condition $\gamma _\textrm {dr} \gtrsim 40$ kHz. This is in good agreement with the onset of the turbulent plasmoids shown in figures 5 and 7(d).

Another mechanism that may be contributing to the turbulent dynamics and to the non-axisymmetric nature of Phase I in the experiment is the transition from single plasmoid formation to multiple simultaneous plasmoid formation. To support this hypothesis, the calculation in Appendix A computes the threshold drive frequency necessary to destabilize shorter wavelength tearing modes in the current sheet. If this threshold is reached we may conclude the subsequent stochastic dynamics are those of a plasmoid chain (Uzdensky, Loureiro & Schekochihin Reference Uzdensky, Loureiro and Schekochihin2010; Loureiro et al. Reference Loureiro, Samtaney, Schekochihin and Uzdensky2012) and are responsible for the turbulence in the high-drive cases. Remarkably, as outlined in Appendix A, this threshold frequency is computed to be $f=\gamma _\textrm {dr}/(2{\rm \pi} ) \gtrsim 60$ kHz and is precisely in the range we would expect based on experimental evidence ($\sim 60\text {--}100$ kHz).

These two mechanisms allow us to infer a hierarchy of stochasticity associated with the drive strength and therefore the frequency of plasmoid formation. As the drive strength increases, the system experiences a loss of equilibrium and laminar ‘single’ plasmoid ejection. This phase is followed by plasmoids of different ‘generations’ catching up with each other to generate a turbulent medium downstream, but still in the ‘single’ plasmoid regime. This phase applies particularly to the NIMROD simulations which are constrained to be axisymmetric and are below the threshold for multiple plasmoids, yet still exhibit increased turbulence at higher drive. Finally, at the highest drives obtained in the experiment, the threshold to multiple simultaneous plasmoid formation is reached as discussed in Appendix A.

This analysis can be made more quantitative by formally taking into account the geometric effects of the magnetic field and pressure gradient for helmet streamer structures in the solar wind. This process likewise results in blob periodicities in the 1–2 hour range consistent with observations (Viall & Vourlidas Reference Viall and Vourlidas2015). This model is constructed by taking the streamer-like poloidal field geometry generated during NIMROD simulations and scaling the radius where reconnection occurs to coincide with $3 R_\odot$ as a characteristic location for PDS formation (Wang et al. Reference Wang, Sheeley, Walters, Brueckner, Howard, Michels, Lamy, Schwenn and Simnett1998; Viall & Vourlidas Reference Viall and Vourlidas2015; Wang & Hess Reference Wang and Hess2018). Scaling the magnetic field strength using fits to solar wind data in the ecliptic plane according to Köhnlein (Reference Köhnlein1996), produces a plausible helmet streamer geometry at the proper scale with realistic magnetic field strength. Using fits for plasma temperature and density in the ecliptic (likewise from Köhnlein Reference Köhnlein1996) and mapping them to the respective flux surfaces produces a mock helmet streamer with plausible temperature and density profiles. A diagram of this 2-D magnetic geometry for this heuristic model is shown in figure 10(a). This model enables us to calculate the same characteristic frequency used to unify the scaling between experiment and simulation ((5.7)) along the field lines in the vicinity of the reconnection site shown as the cyan dashed lines in figure 10(a). The result of calculating this frequency along flux surfaces between the cyan dashed flux surfaces in figure 10(a) is shown in figure 10(b) and plotted as a function of field line distance away from the outboard midplane. In this context, a field line distance of 0 corresponds to the streamer top or the point of highest magnetic curvature where we might expect the highest growth rate of any pressure-curvature driven loss of equilibrium. We can see from this model that the same characteristic time scale used to unify experimental and simulation plasmoid frequencies results in a blob periodicity in the solar wind of $\sim 90$ minutes if formed at $3 R_\odot$ (figure 10b). If the PDS origin radius is increased or decreased, the PDS periodicity likewise increases or decreases, respectively, in accordance with observations (Wang & Hess Reference Wang and Hess2018).

Figure 10. The work of Köhnlein (Reference Köhnlein1996) provides doubly logarithmic fits to Helios data for density, temperature and magnetic field as a function of heliocentric distance in the ecliptic plane. Mapping these quantities onto the magnetic streamer structure shown in (a) allows us to compute $f = 1/2{\rm \pi} \sqrt {c_s^{2}{\kappa }\boldsymbol {\cdot }\boldsymbol {\nabla }{p}/p}$ along field lines just inside and just outside the reconnection radius in the same fashion as shown in figure 8(b). This results in plasmoid frequencies that are peaked at the streamer cusp as expected and produce periodicities in close agreement with in situ observations for plausible solar wind parameters, as shown in panel (b).