Magnetic and velocity fluctuations from nonlinearly coupled tearing modes in the reversed field pinch with and without the reversal surface

In the Reversed Field Pinch (RFP), velocity and magnetic field fluctuations combine to produce mean (toroidally and poloidally symmetric) emf's $⟨ṽ×B̃⟩$ that have an important effect on the evolution of the current profile.^1–4 These emf's are sometimes referred to as a Magnetohydrodynamic (MHD) dynamo effect because of the analogies to the dynamo problem in astrophysics where velocity and magnetic field fluctuations couple to induce the growth and sustainment of large scale magnetic fields.^5–7 Recent work has shown that two fluid effects can also be important in the RFP leading to an emf from $⟨j̃×B̃⟩$ as well.^8–10 

We report here on experimental and computational studies of the velocity fluctuations associated with tearing modes in the RFP. These modes are always in the nonlinear regime where interactions of the mode with the equilibrium fields and with other modes are important. In this regime, the velocity fluctuations observed at a given wavenumber are not simply proportional to the magnetic fluctuations with the same wavenumber as would be the case in the linear regime. Instead, the velocity fluctuations at a given wavenumber and the emf's produced by them are strongly influenced by the activity of other wavenumbers to which the mode is nonlinearly coupled. In this sense, just as nonlinear interactions between electrostatic waves are important for modifying turbulent transport in tokamaks and other devices,¹¹ so too nonlinear interactions between tearing modes play a central role in determining the strength of the fluctuation-induced emf which modifies the current density profile in the RFP. In particular, we show that the emf produced throughout most of the plasma volume by core-resonant tearing modes with the poloidal mode number m = 1 is strongly impacted by the presence of m = 0 modes, resonant in the edge at the reversal surface where the toroidal magnetic field changes sign. These m = 0 modes are a key intermediary in the nonlinear coupling between the core-resonant m = 1 modes^12,13 and, as we show here, the ability of the m = 1 modes to generate a fluctuation-induced emf is dependent not only on their own magnetic fluctuation amplitudes, but also on the amplitude of the m = 0 modes.

To examine the importance of m = 0 modes on the m = 1 mode behavior, we performed dedicated experiments in the Madison Symmetric Torus (MST),¹⁴ some with the m = 0 resonant surface in the plasma and others where it was excluded. Tearing mode velocity fluctuations are measured directly by fast Doppler spectroscopy in both plasma types and are correlated with magnetic field fluctuations to determine whether a significant emf is produced. We also present results from visco-resistive MHD simulations using the DEBS code.¹⁵ The code was adapted to include new boundary conditions and an artificial line-of-sight velocity diagnostic permitting side by side comparisons of simulated and experimental results. Both simulation and experiment confirm that m = 1 velocity fluctuations and the associated emf depend strongly on the amplitude of the m = 0 modes. When the reversal surface is removed from the plasma, m = 0 modes are suppressed but m = 1 magnetic fluctuations are not. Although m = 1 magnetic fluctuations are not reduced, the m = 1 velocity fluctuations and the associated emf are greatly reduced when m = 0 modes are suppressed.

We describe the results in two main sections. In Sec. II, the experimental methods and results are described. Section III contains the simulation methods and results. Finally, in Sec. IV we offer a summary and concluding discussion.

MST generates and confines a large toroidal plasma with a major radius R = 1.5 m and a minor radius a = 0.52 m. The magnetic configuration is normally that of a reversed field pinch but this can be modified by suitable control of the poloidal and toroidal magnetic field programming. Standard RFP operation allows the toroidal magnetic field at the wall to reverse direction relative to the average toroidal field, yielding a safety factor profile $q(r)=rBϕ/RBθ$ as shown by the solid line in Figure 1. Modes with the poloidal mode number m = 1 and the toroidal mode number n = 6, 7, 8, … are resonant in the core of the discharge where $q=m/n$⁠. The m = 0 modes with n = 1, 2, 3, … are resonant at the reversal surface where the toroidal magnetic field changes sign and q = 0.

The location of the reversal surface can be controlled in the experiment by adjusting the value of $Bϕ(a)$⁠. We use a programmable power supply to feed back on the current in the toroidal field coils in order to maintain a target value of edge toroidal magnetic field. If the target value is set to be positive, then the reversal surface is excluded from the plasma and the q profile takes the form shown by the dashed line in Fig. 1. The entire profile is shifted upward and there is no longer any point where q = 0 and hence no resonant surface for the m = 0 modes. One will observe that the locations of the core-resonant m = 1 modes shift radially outward and the m = 1, n = 5 mode becomes resonant in the nonreversed plasmas.

The magnetic mode amplitudes and phases are measured with a toroidal array of coils. An array of 32 coils measures the poloidal magnetic field and an array of 64 coils measures the toroidal magnetic field in these experiments. Fourier decomposition of these data provide the amplitude and phase for each n number as a function of time. Because the arrays are at only one poloidal location, the measurement of each n includes contributions from all m numbers. However, the m = 0 and m = 1 modes are known to be the largest fluctuations in the spectrum and these dominate the measurements. In addition, the polarization of these modes is such that the m = 0 contribution to the poloidal magnetic field is very small. Hence, in most cases the n = 1–4 toroidal magnetic field fluctuations are dominantly m = 0 modes and the n = 5,6,7, … poloidal magnetic field fluctuations are dominantly m = 1 modes. This is a well-tested assumption which we employ throughout our analysis of the experimental data.

Impurity ion velocity measurements are routinely made in MST using fast Doppler spectroscopy.^3,4,17,18 This is done with both passive observation of impurity emission and active stimulation of emission via Charge Exchange Recombination Spectroscopy. For the data shown here, only passive line-integrated velocity measurements will be used. All measurements discussed here result from emission at 343.4 nm. Depending on plasma conditions, this emission can come from a combination of $O+5$ ions and $C+5$ ions.¹⁹ These both produce a set of closely spaced lines near 343.4 nm that overlap into one composite structure in MST. For the plasmas we are considering the oxygen emission dominates the passive measurements.

By measuring the Doppler shift of the emission line with good time resolution (10 μs cadence), we obtain time-resolved measurements of the line-integrated velocity component along the line of sight. Figure 2 shows the viewing geometry for our experiment. The line of sight lies in a poloidal plane and crosses the plasma on the outboard side. The tangency radius of the chord is r/a = 0.37 meaning that the magnetic flux surface tangent to the chord midpoint (illustrated by the circle in the figure) has a radius of r/a = 0.37. The emission profile is known to be very broad in radius with emission extending from the center of the plasma out to about r/a = 0.7 and then falling gradually to zero near the wall. This emission profile, coupled with the projection of a plasma velocity vector onto the line of sight, means that the measurement is most sensitive to poloidal velocities (as opposed to radial) near the middle of the chord where emission is strong. This particular chord was chosen because its sensitivity is large in the vicinity of the rational surfaces for the largest core-resonant modes.

To separate true velocity fluctuations from noise and to give information on the mode number content, the velocity measurements are correlated in time with magnetic fluctuation data from the toroidal array of coils. Assuming that the fluctuating current goes to zero near the wall, then $∇×B̃=0$ implies that $B̃θ$ and $B̃ϕ$ will be perfectly in or out of phase at the wall. Requiring $∇·B̃=0$ then implies a $±π/2$ phase shift between $B̃θ$ and $B̃r$⁠. These phase relationships have been verified experimentally. From MHD simulations and experimental measurements, it has also been observed that the phase of the radial magnetic field fluctuation for the tearing modes is virtually constant as a function of minor radius. We use all of this information to shift the phase of the poloidal magnetic fluctuation measured by the coil array so that we obtain a signal which is proportional to the radial magnetic field fluctuation at the midpoint of the velocity measurement chord. The radial component of the magnetic field is desirable because it is the component that combines with the measured poloidal velocity fluctuation to make the $⟨ṽ×B̃⟩$ emf that drives current along the mean magnetic field.

The line-averaged velocity measurement is correlated with this radial magnetic field fluctuation. The velocity signal is also correlated with a second reconstructed magnetic field signal whose phase has been shifted in time by $π/2$⁠. This allows us to extract the correlated velocity fluctuation amplitude (independent of the relative phase between $vθ$ and B_r) and the relative phase for each n number. Such a technique yields the “pseudospectrum” of the velocity fluctuations and has been employed previously to make measurements of the tearing mode velocity fluctuations.^3,4

The pseudospectrum method as implemented here is well suited to extracting the part of the velocity fluctuation that is coherent with the edge magnetic fluctuation. This coherent part is the part that contributes to the $⟨ṽ×B̃⟩$ emf. For these global tearing type fluctuations, MHD simulations suggest that most of the velocity fluctuations in the plasma should be coherent with the magnetic fluctuations. In fact, the observed coherence between v and B in the region we report on here is observed to be fairly high (above 0.5) and the incoherent part of the velocity fluctuation is consistent with expectations for random noise from photon counting statistics. In MHD simulations discussed below, where the random noise source is absent, the coherence is even higher.

Figure 3 shows four representative measurements from a typical reversed discharge (left column) in MST and from a typical nonreversed discharge (right column). The plasma current for these data was about 400 kA. The edge safety factor q is negative in the reversed discharge and slightly positive in the nonreversed discharge.

Let us first focus on the characteristics of the reversed discharges in the left column of Fig. 3. The m = 0 magnetic fluctuations in reversed discharges appear as sharp bursts coincident with sawtooth crashes. Core-resonant m = 1, n = 6 magnetic field fluctuations also exhibit a sawtooth cycle in these discharges with maximum amplitude occurring at or slightly before the sawtooth crash. At each sawtooth crash, the toroidal magnetic flux increases as a result of the redistribution (flattening) of the global current density profile.

As the figure shows, nonreversed discharges (right column) have different magnetic fluctuation activities. There is still a sawtooth cycle apparent in the m = 1, n = 6 mode amplitude and the average magnitude of this mode is about the same as for reversed discharges. However, the m = 0 fluctuations are strongly suppressed when the reversal surface is removed. Interestingly, the increase in toroidal magnetic flux at sawtooth crashes is also strongly suppressed. Indeed it is generally true in MST operation that the amplitude of the m = 0 modes is strongly correlated to the amount of toroidal flux generation at the sawtooth crash.

To obtain quantitative measurements of the behavior of any physical observable throughout a typical sawtooth crash, we assume each sawtooth is an independent and identically distributed instance of a statistical ensemble. We then compute the ensemble average of signals over many similar sawtooth events. The average behavior of the m = 0, n = 1 mode and m = 1, n = 6 mode in MST is illustrated in the left two panels of Fig. 4. The time axis has been adjusted so that t = 0 corresponds to the time of maximum toroidal flux increase and the time range for these plots is for a small window of ±1 ms around the sawtooth crash. As noted earlier, both modes peak near the sawtooth crash in reversed discharges. In nonreversed discharges, the m = 0 mode is suppressed but the m = 1 mode continues to undergo a sawtooth cycle with comparable average amplitude. Although the peak and average m = 1 amplitudes are similar in the two cases, the temporal behavior is slightly different. In the nonreversed case, the m = 1 modes show a smoother time evolution compared to the reversed case where there is a distinctive spike near t = 0.

Using these ensemble-averaged waveforms, it is straightforward to determine the peak mode amplitude for a typical sawtooth event. In the right two panels of Fig. 4, we show this peak mode amplitude for both m = 0 and m = 1 as a function of the edge safety factor. (The plasma current was about 250 kA for the data shown in the right panels so the absolute mode amplitudes do not precisely match the peak values in the left panels where the plasma current was 400 kA.) The peak m = 1 mode amplitude is relatively insensitive to the degree of reversal, decreasing only slightly when the reversal surface is removed. The peak m = 0 mode amplitude, however, depends strongly on q(a). In fact, the peak m = 0 amplitude nearly matches the absolute value of the mean toroidal magnetic field at the wall prior to the sawtooth event. This implies that in reversed MST discharges, the m = 0, n = 1 mode grows until the magnetic island associated with the mode touches the conducting wall. When this occurs, there is one location toroidally where the toroidal magnetic field is zero because the m = 0, n = 1 mode produces a positive toroidal magnetic field at that location that just matches the mean (n = 0) negative toroidal magnetic field. This explains why the peak m = 0 mode amplitude depends linearly on the q(a) since q(a) is proportional to the mean toroidal magnetic field at the wall. The more negative q(a) is, the larger the m = 0 island can grow before it touches the wall.

Finally, in Fig. 5 we show the measured m = 1 velocity fluctuations in MST for both reversed and nonreversed discharges (with plasma current of 400 kA). The left panels show the result for reversed discharges and the right panels show the results for non-reversed discharges. Data for three different core-resonant m = 1 modes are shown. (Velocity fluctuations from m = 0 modes are not reliably measured by this technique and will not be discussed here.) The data shown are averaged over many similar sawtooth events with t = 0 representing the time of maximum toroidal flux change as before. The thick lines show the measured pseudospectral velocity fluctuation amplitude (independent of phase). The velocity has been normalized to the Alfven speed computed using the central line-averaged density and the poloidal magnetic field at the wall. The thin lines in Fig. 5 show the noise floor for the measurement, computed by correlating the line-averaged velocity signal with a magnetic fluctuation whose phase has been chosen at random for each time point.

In reversed discharges, the m = 1 velocity fluctuations peak near the time of the sawtooth crash where the m = 1 magnetic fluctuations are also peaking. The detailed temporal behavior of the velocity fluctuations does not match that of the magnetic fluctuations but both do peak at the sawtooth event. The m = 1, n = 6 magnetic fluctuations are high even before the sawtooth crash arrives and are lower afterward (see Fig. 4) but the velocity fluctuations are low on both sides of the event. The m = 1 velocities before and after the sawtooth event are at the noise level but the levels during the sawtooth crash are well-resolved.

What is most interesting is that the m = 1 velocity fluctuations are strongly suppressed in nonreversed discharges (right panels in Fig. 5). On this particular line of sight, the suppression is greatest for the n = 7 and 8 modes, which are only barely resolved above the noise floor in the non-reversed cases. However, the correlated amplitude (shown by the thick line) is reduced for all modes. (No statistically significant velocity fluctuations correlated with the n = 5 mode are observed on this viewing chord in either type of plasma.) The reduction in velocity fluctuations is especially striking because the magnetic fluctuations for these modes are not suppressed. In fact, the m = 1 magnetic fluctuations have very similar amplitude in both reversed and nonreversed discharges. Evidentally the removal of the reversal surface and the suppression of the m = 0 fluctuations is correlated with a suppression of the m = 1 velocity fluctuations, even though the m = 1 magnetic fluctuations are not suppressed.

The suppression of the m = 1 velocity fluctuations in nonreversed discharges is particularly significant because it is known from past work^3,4 that the m = 1 magnetic and velocity fluctuations in the RFP couple to produce a mean emf that acts to flatten the field-aligned current density profile. The results presented here imply that the fluctuation-produced mean emf is suppressed when the reversal surface is removed and m = 0 modes are suppressed. Figure 5 also shows the phase between the measured velocity fluctuations and the radial magnetic field fluctuation at the mid-point of the measurement chord. If the phase is near 0 or π, then the contribution of these fluctuations to the mean emf $⟨ṽ×B̃⟩$ will be the strongest. If it is near $±π/2$⁠, then the velocity and magnetic field fluctuations are out of phase and will not produce a mean emf. The measurements show that during sawtooth events in reversed plasmas, the phase is nearly $±π$ resulting in a strong emf. In nonreversed plasmas, not only are the velocity fluctuations reduced in amplitude, but the phase difference shifts to $−π/2$ implying a very small emf. Hence, the portion of the velocity fluctuations that produces an emf is even more strongly suppressed than indicated by the change in the full correlated velocity amplitudes. What is interesting is that this occurs without a reduction in the m = 1 magnetic fluctuations.

Motivated by the experimental results, we undertook numerical studies with the visco-resistive MHD code DEBS. Our primary goal was to see if the features observed in experiment with removal of the reversal surface are also captured by the code. In fact, many of these features are reproduced in the numerical simulations giving us confidence that the dominant physics is well-described by visco-resistive MHD and generic to plasmas in the reversed field pinch magnetic configuration. The computation also allows us to examine the velocity fluctuations more completely than can be done within the limitations of the experimental diagnostics.

DEBS solves the MHD equations in 3D using a semi-implicit algorithm for time advance.¹⁵ The geometry is that of a periodic cylinder with a conducting wall boundary and we have matched the MST aspect ratio (R/a = 1.5/0.52 = 2.88) in all of the runs shown here. We have run the code with the pressure gradient set to zero and a flat density profile that is held constant in time for simplicity. DEBS operates in physical coordinates for the radial direction and uses Fourier components to describe poloidal and toroidal variations. The runs reported here were performed with 100 radial points, 11 poloidal modes, and 171 toroidal modes. We have verified that increasing the resolution in any of these dimensions does not significantly affect the results reported here.

The resistivity profile we use is fixed in time and has been chosen to match the experimentally measured resistivity profile in MST.²⁰ The viscosity profile is flat and constant in time. The Lundquist number $S=μ0vAa/η$⁠, where v_A is the Alfven speed and η is the resistivity, quantifies the ratio of the resistive diffusion time to the Alfven transit time. It was 10⁵ for all of the runs shown here, about a factor of 10 lower than the experiments reported above. The magnetic Prandtl number $P=ν/η$ is the ratio of the resistive diffusion time to the viscous diffusion time. It was 40 for these runs which is somewhat higher than the number inferred from the classical collisional viscosity in MST but lower than P calculated using the viscosity inferred from transient momentum transport experiments in MST.²¹ We have done numerical simulations at other S and P values and although mode amplitudes and detailed mode behavior can differ, the main results we are reporting here are not affected. Our choice of these parameters yields data with recognizable sawtooth oscillations and runs that execute in tolerable amounts of CPU time.

Most previous studies using DEBS for RFP modeling have used a perfectly conducting boundary and held the toroidal magnetic flux constant. To facilitate comparisons with experiment, we implement a toroidal field circuit model for the boundary condition on $Eθ(a)$ that incorporates feedback in order to achieve a target $Bϕ(a)$⁠. The circuit parameters are chosen to match those of MST. The target value of $Bϕ(a)$ is specified as a function of time, allowing us to control the edge boundary condition in a way similar to the MST experiments. For normal reversed plasmas, the target $Bϕ(a)$ is negative. For nonreversed runs, the plasma is first allowed to reverse and then the edge $Bϕ$ is brought positive just as is done in the experiment.

Figure 6 shows sample outputs from typical reversed (q = −0.035) and nonreversed (q = +0.005) simulations. Although the sawtooth oscillations are not as clean in the simulations as in the experiment (compare to Fig. 3) there are still cyclical oscillations present in the m = 0 and m = 1 mode amplitudes. With the boundary condition we have chosen, the sawtooth crash events in the code are also accompanied by an increase in the toroidal magnetic flux. One will note that just as in experiment, the m = 0 mode amplitude is suppressed in the simulation when the reversal surface is removed from the plasma but the m = 1 mode amplitude stays about the same on average. Also similar to experiment is the suppression of the strong toroidal flux jump at the sawtooth crashes in the nonreversed case.

Using the time of maximum toroidal flux increase as a marker, we average over similar sawtooth events in the simulation data just as was done for the experimental data. The resulting measurements of the average m = 0, n = 1 and m = 1, n = 6 magnetic mode evolution are shown in the right hand panels of Fig. 7 alongside the experimental data already shown before in Fig. 4. Once again we see that the m = 0 modes are suppressed in nonreversed discharges but the m = 1 modes are not. The peak amplitudes of the fluctuations are also larger in the simulation than in the experiment, with m = 1 modes about a factor of two higher in simulation. (Note the difference in scale on the m = 1, n = 6 panels.)

As was done in experiment, we collected data from a set of simulations with q(a) ranging from reversed to nonreversed. This data is shown alongside the experimental data in Fig. 8. The general trend in both experiment and simulation is similar for reversed discharges. However, the strict linear dependence of peak m = 0 amplitude on edge q is absent in the simulation. Hence, the associated physical interpretation of experimental results where the mode grows until the island touches the wall does not seem to fully apply in the code. The m = 0 modes are suppressed in the code with the reversal surface removed but the suppression is somewhat more complete in the experiment. In addition, the peak m = 1 amplitude tends to become larger in more nonreversed simulations but the modes amplitudes are constant or even slightly reduced in the experiment.

Some of the detailed differences between mode behavior in simulation and experiment are apparently due to physics beyond the MHD model. For example, the factor of two difference in mode amplitude between computation and experiment appears to be resolved in the two-fluid computation.¹⁰ Other differences between code and experiment may be due to simplifications we have made within the MHD model such as the fixed resistivity profile or zero pressure gradient assumptions. Examination of these detailed differences is not the main focus of this paper and is an ongoing area of research.

To enable us to compare velocity fluctuations in the simulations with those measured in MST, we have created an artificial diagnostic in the code. This diagnostic performs a line average of the total velocity fluctuation (all modes combined) along the same line of sight used in the experiment. We then perform a correlation between this artificial measurement and the edge value of a particular mode's magnetic fluctuation. This allows us to manufacture a pseudospectral measurement of velocity fluctuations in the code in a way that parallels the process used for analysis of MST data.

One difference in the code is that there is not much toroidal rotation of the modes because the physics that generates the rotation (presumably linked to particle and momentum transport) is not included. In the experiment, toroidal rotation is important for the pseudospectral measurement because it enables a single measurement location to effectively scan across the entire range of mode phases as the mode rotates past the measurement. To compensate for this lack of rotation in the code, we average the correlation results over the toroidal angle. This is effectively like scanning the single measurement position over all mode phases in the code by moving the measurement rather than the plasma. Doing this toroidal average also reduces the number of sawtooth events needed for good statistics relative to that required in the experiment.

Figure 9 shows the resulting m = 1 pseudospectral velocity fluctuations measured in the simulations for reversed and nonreversed discharges. These data can be directly compared with the experimental measurements shown in Fig. 5 as both velocities have been measured and normalized in the same fashion. Although the amplitude of the velocity fluctuations is clearly different, the simulation and experimental results look similar in many respects. In both cases, the m = 1 velocity fluctuations are suppressed when the reversal surface is removed and m = 0 modes are suppressed. The phase between v and B in the code is about zero for reversed discharges implying a strong emf produced by the m = 1 modes. The much reduced velocity fluctuations in nonreversed simulations imply a similarly reduced emf. Hence, just as in the experiment, the strength of the m = 1 velocity fluctuations and the emf they produce is much more strongly connected to the m = 0 fluctuation activity than it is to the m = 1 magnetic fluctuation activity.

Interestingly, although the phase difference between v and B for m = 1 fluctuations shifts to $−π/2$ in nonreversed experimental cases, the phase difference remains about 0 in the code. A likely explanation for this difference is again related to the absence of mean flows in the code. If a non-uniform (in radius) mean flow profile exists and the magnetic flux surfaces are distorted by magnetic fluctuations, then the flows along the distorted flux surfaces will give rise to a toroidally and poloidally varying velocity with the poloidal and toroidal mode number matching the magnetic fluctuation. The phase between the measured $vθ$ and B_r fluctuations in this case will be $±π/2$ and such a velocity fluctuation will not produce an emf. This effect is largely absent in the code because the mean flows are small. However, in the experiment, there are known mean poloidal flows on the order of a few km/s or more that could give rise to velocity fluctuations of this sort. The emf-producing part of the velocity fluctuation apparently dominates in MST in reversed plasmas (giving a phase difference of 0 or π) but in non-reversed plasmas it becomes so small that the other non-emf-producing component of velocity fluctuation (resulting from helical distortions of the mean flow profile) determines the phase of the measurement. Hence, the emf from m = 1 fluctuations in both experiment and simulation is primarily suppressed due to a reduction in the m = 1 tearing modes' velocity fluctuation amplitudes and not due to a change in the phase between v and b.

In the experiment, the velocity fluctuation measurement is limited to line-of-sight measurements along particular chords. In the simulation, however, the velocity vector is fully known at every point. Hence, we can compose a complimentary measurement that averages the vector magnitude of the velocity fluctuation over the full plasma volume. The sawtooth average of this quantity is shown for reversed and nonreversed simulations in Fig. 10. These data clearly confirm that the suppression of the m = 1 velocity fluctuations with the reversal surface removed is not an artifact of the particular line of sight used to make the measurement. The suppression is global in nature.

The DEBS simulations show that the m = 1 velocity fluctuation, and the emf associated with it, is not simply proportional to the m = 1 magnetic fluctuation. The temporal evolution of the m = 1, n = 6 velocity fluctuation in Fig. 10 clearly does not match the evolution of the m = 1, n = 6 magnetic fluctuation in Fig. 7. The velocity fluctuation is more symmetric in the rise and fall and it also peaks slightly later in time than the magnetic fluctuation does. In fact, the velocity fluctuation appears to peak during the time when m = 1 and m = 0 magnetic fluctuations are changing most rapidly, not when either one is at a maximum. This is another piece of evidence that the emfs produced by the m = 1 modes are not simply proportional to their magnetic fluctuation amplitudes but are instead connected in an important way with the nonlinear mode coupling and energy transfer between m = 0 and m = 1 modes at the sawtooth crash.

A thorough understanding of how the nonlinear coupling and energy transfer impact the m = 1 velocity fluctuations and emf is beyond the scope of the present paper. However, we note that past studies of the RFP²² demonstrated that the nonlinear energy transfer between modes is the dominant driving/damping term for most of the modes in the system. These nonlinear driving and damping terms are influenced by resonant responses at each mode resonant surface that give rise to magnetic reconnection and magnetic island growth or decay. Hence, the part of the velocity fluctuation near the resonant surface is associated with reconnection and the velocities that carry the magnetic flux to/from the reconnection layer might be expected to be strongest not when magnetic fluctuations are the strongest but rather when the magnetic fluctuation amplitude (and hence reconnected flux) is changing most rapidly in time. This seems consistent with our observations. Isolated tearing modes are known to be driven unstable by the current density gradient and are especially sensitive to the current profile near the resonant surface. In turn they also relax the current profile near the resonant surface. However, the coupled system of modes in the RFP might operate in such a way that the full collection of modes responds in concert to the global current profile making each mode more sensitive to the full current profile than would be the case for isolated tearing modes. Removal of the m = 0 resonant surface in the edge, where nonlinearly forced m = 0 fluctuations have a resonant response, might therefore fundamentally alter the sensitivity of the core-resonant m = 1 modes to the edge current profile. This physics may be the reason why the m = 1 velocities and the associated emf's change so significantly when the m = 0 resonant surface is removed.

In summary, we report on both experimental and computational studies of tearing mode magnetic and velocity fluctuations in the RFP. In particular, we show that when the resonant surface for m = 0 modes is removed from the plasma, the m = 0 magnetic fluctuations are suppressed but m = 1 magnetic fluctuations remain the same. Despite this, the m = 1 velocity fluctuations are suppressed when the reversal surface is removed. These results are similar in both MST experiments and DEBS visco-resistive MHD simulations. The reduction of the m = 1 velocity fluctuation also results in a reduction in the $⟨ṽ×B̃⟩$ fluctuation-induced emf produced by these modes. Hence in this system where the tearing modes are strongly nonlinear and interact with one another, the velocity fluctuations and subsequent emfs can behave quite differently than the magnetic fluctuation amplitude alone would indicate. In fact, the rise in m = 1 velocity fluctuations at the normal RFP sawtooth crash is much greater than would be inferred by looking at the rise in the magnetic fluctuations alone.

These results contribute to our general understanding of plasma fluctuations in the nonlinear regime and have some particular implications for the RFP fusion development path. Our study suggests that the current profile relaxation provided by the tearing modes is in some sense more efficient for the case when several modes are present and strongly couple with one another. A large magnetic fluctuation amplitude does not necessarily imply a large emf because the velocity fluctuation is not simply proportional to the magnetic fluctuation. For RFP sustainment scenarios that rely on the tearing modes to relax the current profile, this is important to keep in mind. Utilizing the normal broad spectrum of tearing modes for relaxation has a negative impact on confinement but may be very efficient for current profile relaxation. Scenarios with one or a few dominant modes may be desirable for good confinement but could be less efficient for current profile relaxation if that is an important component of the sustainment method. Better understanding the mechanisms underlying our results could enable one to manipulate the mode spectrum or the relative size of v and b for given modes in ways which would be optimal for both confinement and current relaxation.

These results may also have implications for the astrophysical dynamo problem. It may be the case that not all systems with a given magnetic fluctuation spectrum will behave the same when it comes to the growth or sustainment of large scale magnetic fields. In the RFP, it appears that constraints imposed by boundary conditions affect intermediary modes like m = 0 through which significant nonlinear coupling occurs. These subtle changes in the external constraints have an impact on the emf produced by the broad spectrum of m = 1 modes which contain most of the fluctuation energy in the plasma. We have shown that this can be the case even though the m = 1 magnetic fluctuation spectrum itself is weakly modified by the external constraints. In general, the large scale constraints that organize the global behavior of a system may induce changes in the small scale fluctuations that make them more or less effective at producing the large scale emf's necessary for a dynamo.

Future work will utilize the tools developed here to better understand the differences between tearing modes observed in experiment and those observed in simulations. Such comparisons have promise both for better understanding the physics of these modes and for validation studies of visco-resistive MHD. We would also like to investigate the nature of the sawtooth cycle in nonreversed plasmas. In the standard RFP, current profile relaxation and strong mode-to-mode energy transfer facilitated by m = 0 modes are central to the understanding of why a sawtooth cycle exists. In nonreversed plasmas, it seems both of these pieces are greatly modified but a cycle persists nonetheless. It would also be interesting to explore the differences of how Ohm's law is balanced in the reversed and non-reversed cases since it appears that the emf produced from $⟨ṽ×B̃⟩$ is quite different.

This work has been supported by the U.S. Department of Energy, Office of Science, and Office of Fusion Energy Sciences under Award No. DE-FC02–05ER54814. The authors would like to thank Craig Jacobson, Karsten McCollam, John Sarff, and Carl Sovinec for useful discussions and we thank David Ennis and the rest of the MST Team for their assistance in collecting the experimental data. The authors would also like to thank Dr. Dorothy Chappell, Dean of Natural and Social Sciences at Wheaton College for supporting this project.