PSFC/JA-16-30 Turbulent Fluctuations During Pellet Injection into a Dipole Confined Plasma Torus

We report measurements of the turbulent evolution of the plasma density profile following the fast injection of lithium pellets into the Levitated Dipole Experiment (LDX) [Boxer et al., Nat. Phys. 6, 207 (2010)]. As the pellet passes through the plasma, it provides a significant internal particle source and allows investigation of density profile evolution, turbulent relaxation, and turbulent fluctuations. The total electron number within the dipole plasma torus increases by more than a factor of three, and the central density increases by more than a factor of five. During these large changes in density, the shape of the density profile is nearly “stationary” such that the gradient of the particle number within tubes of equal magnetic flux vanishes. In comparison to the usual case, when the particle source is neutral gas at the plasma edge, the internal source from the pellet causes the toroidal phase velocity of the fluctuations to reverse and changes the average particle flux at the plasma edge. An edg...


I. INTRODUCTION
Plasmas confined by dipole magnetic fields have unique stability and transport properties that govern plasma dynamics both in planetary magnetospheres and in laboratory plasmas confined by strong dipole magnets.In magnetospheres, low-frequency fluctuations driven by the solar wind or excited by planetary rotation drive plasma radial transport.This transport can be either radially inward or outward depending upon the source of particles.
When the particle source is external, transport is inward as occurs, for example, when solar-wind disturbances increase the number of trapped ring current protons 1,2 .When the particle source is internal, the direction of particle transport reverses, becoming outward.For example, Jupiter's planetary wind is fueled by Io's volcanic plumes 3,4 .Centrifugally-driven interchange modes are believed to cause the outward transport of relatively cool and dense plasma-filled flux-tubes and the inward motion of flux-tubes containing relatively less dense but higher-temperature plasma 5 .Remarkably, whether or not the plasma source is internal or external, radial transport of magnetospheric plasma results in centrally peaked pressure and density profiles with steep gradients near the limits for pressure-driven and rotationallydriven interchange instability [6][7][8] .When the frequencies of turbulent fluctuations are well below the particle cyclotron and bounce frequencies, inward (or outward) radial transport also energizes (or de-energizes) plasma through betatron heating and Fermi acceleration that is associated with the conservation of cyclotron and bounce adiabatic invariants [9][10][11][12] .This heating is incorporated into space weather models 13 and occurs even when turbulence creates chaotic radial particle motion 14 .
In the laboratory, the inward turbulent particle pinch also occurs when toroidally-confined magnetized plasma is produced and sustained by external heating and particle sources.However, the magnetic field lines in a laboratory levitated dipole plasma torus are not open to the poles as in planetary magnetospheres.In the laboratory, turbulent radial transport always dominates confinement; whereas, in planetary magnetospheres, radial transport dominates only during those active periods when the rates of radial transport exceed collisional and wave-driven losses to the planetary poles.
The magnitude of the turbulent pinch and the degree of central density peaking depends on the radial variation of differential magnetic flux-tube volume, δV ≡ dl/B, defined by an integration along the magnetic field and the magnetic field strength, B. Low-frequency turbulent transport in a magnetized plasma torus tends to equalize the particle number contained within tubes of equal magnetic flux.The average density profile scales roughly as n ∝ 1/δV , and the outward particle flux is proportional to the gradient ∂(nδV )/∂ψ.Here, ψ the poloidal magnetic flux, |∇ψ| ∼ B p R, for a tokamak-like torus, or the total magnetic flux for an axisymmetric dipole, B = |∇ψ| |∇ϕ|.For example, in tokamaks, the flux-tube volume is proportional to the safety factor, δV ∝ q, and density peaking scales as 1/q [15][16][17] .
Similar density peaking has been observed in stellarators 18 .By comparison, a significantly larger inward pinch occurs when the plasma torus is confined by a strong levitated current ring 19,20 .In this case, the magnetic field resembles the inner dipolar regions of planetary magnetospheres, and the flux-tube volume increases rapidly with radius, δV ∼ L 4 , with L the equatorial radius.In a dipole magnetic field, the condition n ∝ 1/δV ∼ 1/L 4 is equivalent to the Melrose condition for marginally stability to centrifugally-interchange motion 7 and also equivalent to the condition for a "stationary" density profile.For a "stationary" density profile, the turbulent radial particle flux and the particle number gradient, ∂(nδV )/∂ψ, vanish 21 .
Previous observations of the inward particle pinch in a plasma torus confined by a levitated magnetic dipole used neutral gas at the outer edge as the ionization source.Lowfrequency fluctuations caused radial diffusion from the outer particle source and created a strong inward turbulent pinch 19,20 .Unlike the Earth's magnetosphere, where low-frequency fluctuations are externally-driven by the solar wind, in the laboratory, low-frequency fluctuations are internally-driven, and turbulence is excited and sustained by continuous plasma heating and neutral gas ionization.
Like the density profiles of trapped magnetospheric particles, which are ultimately lost to the Earth's surface, the steady-state density profiles in the laboratory never become truly "stationary".A "stationary" density profile would have n ∝ 1/δV ∼ 1/L 4 .Interferometer measurements in the Levitated Dipole Experiment (LDX) 22 , show the density profile scales as n ∝ 1/L α , with 2.5 < α < 3.7 depending upon the radial profile of plasma heating.
By switching the plasma heating power completely on and off, the rate of the inward pinch was directly measured 19 and found to be equal to rate predicted by Birmingham 12 using the measured turbulent electric fields at the plasma edge.The outer extent of the ionization source was inferred by measuring the profile of visible light emission 23 .When the central plasma density increased, either by increased heating power or by neutral gas puffing, the particle source moved further outward, even though the overall density profile shape, ∂ ln(nδV )/∂ψ, was unchanged.
An interesting question is whether or not the turbulent particle flux reverses when a sufficiently large particle source is internal to the plasma, as appears in Jupiter's magnetosphere due to it's volcanic moon Io.Another interesting question is whether or not the structural and spectral characteristics of the low-frequency turbulent fluctuations change in response to changes to the particle fueling location and to the radial profiles of plasma density and temperature.
Here, we report the first answers to these questions by measuring the evolution of the plasma density and the electrostatic potential fluctuations during fast injection of solid lithium pellets into an otherwise steady-state high-temperature plasma torus confined within the LDX device.
As the pellet passes through the plasma, it provides a significant internal particle source that lasts for about 15 ms.The total electron number within the dipole plasma torus increases by more than a factor of three, and the central density increases by more than a factor of five.The shape of the density profile becomes more centrally peaked, and the gradients of the particle number nearly vanish, ∂(nδV )/∂ψ ≈ 0, or slightly reverse.
Pellet injection also significantly changes the turbulent fluctuations as measured by an array of Langmuir probes located near the plasma's outer boundary 24 .The toroidal phase velocity of the turbulence reverses directions during pellet injection: fluctuations propagate toroidally in the ion drift direction during pellet injection; whereas, before and after pellet injection, when edge particle fueling dominates, the fluctuations propagate in the electron drift direction.Although measurements from the probe array are not easy to interpret because electron temperature fluctuations cannot be independently measured 25,26 , the estimated turbulent particle flux, both before and after the pellet, is inward and consistent with previous measurements made with the interferometer array 19 .During pellet injection and using the same method to interpret the probe array, the direction of the particle flux reverses.
These observations are consistent with theoretical and computational expectations of tur-bulent transport due to drift-interchange and entropy modes in a dipole-confined plasma with relatively cool ions, T e T i , as is believed to be the case in the LDX device.With a central heating source, the central plasma pressure increases and becomes unstable when the pressure profile gradients, ∂p/∂ψ, exceed the usual MHD condition 6 , − d ln p/d ln δV ∼ γ = 5/3.
Broadband electrostatic interchange turbulence develops consisting of many global modes with chaotic amplitudes and phases 27 that cause intermittent transport 28 .Although the energy flux is always outward, theory and simulation show the radial particle and temperature flux can be either inward or outward depending upon the ratio of temperature to density gradient scale lengths, η ≡ d ln T e /d ln n.When η = (γ − 1) = 2/3, the MHD condition of marginal stability to pressure-driven interchange motion coincides with the "stationary" profile, where both ∂(nδV )/∂ψ ≈ ∂(pδV γ )/∂ψ ≈ 0. For a "stationary" profile at marginal stability, interchange motion leaves the profile unchanged.This condition corresponds to a state of minimum entropy production 29 .Any density and temperature perturbations that occur in "stationary" profiles represent entropy waves 30 that propagate in the direction of the curvature heat flux.
When η > 2/3, the inner, high pressure region is "less dense" (nδV is smaller) and "hotter " (T e δV γ−1 is higher) than in the outer plasma.This is the usual case with external fueling and central heating.Entropy mode turbulence results, and particle transport is inward while the temperature flux is outward.When η < 2/3, the inner, high pressure region is "denser " and "cooler " than the outer plasmas.This is the case with pellet injection.
Pellet injection creates an internal particle source and cools the central electrons.The directions of the turbulent particle and temperature flux reverse.Non-linear simulations that utilized the GS2 31 code showed this transport dependency on η is robust 32,33 .Also, the relationship between η and the direction of particle and thermal transport is consistent with basic principles of turbulent equipartition 21 provided the turbulent electric field has a sufficiently broad spectrum and sufficiently short correlation time.
The following six sections present the results and interpretations from these experiments.
First, the LDX experiment, diagnostics, and lithium pellet injector (LPI) are described, emphasizing magnetic geometry, plasma heating, and fluctuation and density measurements.
Although temperature diagnostics are not available at LDX, the discharge described in this report is similar to those described by Davis and co-authors in Ref. 34  During pellet injection, the probability of outward moving dense flux-tubes increase, which create time periods when the average particle flux nearly vanishes or becomes directed outward.In Sec.VI, the frequency and wavenumber dispersion of the measured turbulence is compared with a linear gyrofluid theory 36,37 .A relatively simple dispersion relation results for the special case of interchange and entropy modes driven by warm electrons and cold ions.The electron's collisionless curvature heat flux provides the underlying physical explanation for the reversal of the toroidal phase velocity of the fluctuations during lithium pellet injection.Finally, we summarize these results, point-out opportunities for further study, and discuss the importance of these measurements to the understanding of magnetically-confined high-temperature plasmas.

II. THE LEVITATED DIPOLE EXPERIMENT (LDX)
The design and operation of the Levitated Dipole Experiment (LDX) have been described previously 19,23,34,38 .A high-temperature plasma is confined by a strong superconducting coil, which is magnetically levitated inside a 5 m diameter vacuum vessel.The levitated superconducting coil is energized to a current of 1.2 MA creating dipole-like magnetic field with a relatively large dipole moment of 0.4 MA•m 2 .A circular levitation coil is located above the superconducting dipole, applying an upward attractive force on the dipole coil for magnetic levitation.The current within the levitation coil is actively controlled, which maintains the position of the coil constant to within a few millimeters at the midplane of Fig. 1 shows a schematic of LDX along with the geometries of the Li pellet injection, the four-channel microwave interferometer, and an edge probe array.The pellet injector 39 was originally used to inject deeply penetrating lithium pellets into Alcator C-Mod discharges allowing access improved confinement regimes 40 and internal measurement of the local magnetic field 41 .In LDX, the injector was mounted nearly horizontally and aimed so that Li pellets would reach an inner most radius within 0.75 m < L < 0.85 m and pass through the plasma with nearly a straight line trajectory.
The evolution of the density profile is measured with a 60 GHz, four-channel interferometer 22 .The four channels cross the plasma within the horizontal plane of the dipole (i.e.z = 0) and reach innermost tangency radii of L = 0.77, 0.86, 0.96, and 1.25 m.Using assumptions of axisymmetry and of uniform density along a magnetic field-line, the fourchannel interferometer is used to compute the plasma density profile and total electron number within the plasma torus 19 .
An array of 24 Langmuir probes, equally spaced along a 90 deg toroidal angle, is used to measure various properties of edge turbulence 24 .The probe array is located below the plane of the dipole, (R, z) = (1.0 m, −0.55 m), and the probe tips are inserted inside the plasma edge by ≈ 0.02 m.As originally configured, all 24 probes measured the floating potential, V f .Estimates of the radial particle diffusion coefficient were made by computing the azimuthal component of the electric field.This required the simplifying assumption that fluctuations of the floating potential, Ṽf , are proxy for fluctuations of the plasma potential, Φ.The difference between two adjacent floating potential probes is then an estimate for the azimuthal electric field, Ẽϕ ≈ −R −1 ∆ Φ/∆ϕ.The radial particle diffusion coefficient (valid for both the magnetosphere 11 and the laboratory 19 ) is D = R 2 Ẽ2 ϕ τ cor , where τ cor is the autocorrelation time of the electric field fluctuations.Each probe is recorded at 125 ksps, allowing accurate calculation of the statistical properties of the turbulence including τ cor and the average RMS value of the turbulent toroidal electric field.With random electric field fluctuations, the average radial particle flux depends upon the radial gradient of the particle number, Γ = −D ∂(nδV )/∂ψ.For these Li pellet experiments, the probe array was reconfigured to make an estimate for the local instantaneous particle flux.As shown in Fig. 1, the probe array was positioned on the side of the torus opposite from the trajectory of the Li pellet.Alternating probes in the array were replaced by ion saturation probes, and a subset of these were used to observe changes in the edge fluctuations before, during, and after pellet injection.Estimates of the local, instantaneous radial particle flux were made using the method described by Carreras and co-authors 35 .The voltage difference between two floating potential probes, separated toroidally by ∆ϕ = 8 deg, is used to estimate the azimuthal electric field, R Ẽϕ ≈ ∆Φ/∆ϕ, and the ion saturation current measured from a third probe in between the two floating potential probes is used as a proxy for the local plasma density.By ignoring the influence of electron temperature fluctuations, a crude estimate for the local particle flux is the product Although probe measurements of ion saturation current and floating potential are relatively simple edge diagnostics, estimates of particle flux using the probe array are not definitive because we are unable to make simultaneous measurements of electron temperature fluctuations.As discussed by Gennrich and Kendl 25 and Kočan and co-authors 26 , the assumption of the equivalency of the floating and plasma potential fluctuations tends to over estimate particle flux.On the other hand, because the phase shift between the density and temperature fluctuations can be small, the direction of the radial particle flux may still be correctly estimated 42 .While recognizing uncertainties in the use of the edge probe array to estimate radial particle flux, these edge measurements are significant for several reasons.They show changes in the toroidal dispersion of the edge fluctuations, indicate relative changes to the turbulent particle flux, Γ , averaged over time and toroidal angle, and show a similarity between the PDFs measured in dipole-confined plasmas and plasmas within tokamaks and stellarators 35 .
In addition to those diagnostics shown in Fig. 1a, two fast video cameras (Vision Research, Phantom v7.1) and two photodiode arrays were used to measure the visible light from the Li pellet as it passes through the plasma 43 .The sightlines for these imaging diagnostics are indicated by shaded regions in Fig. 1b.As described by Davis 34  Quasi-stationary deuterium plasmas, lasting several seconds, are produced with multifrequency electron cyclotron resonance heating (ECRH) and with the same discharge programming previously described 34 .For the experiments reported here, approximately 25 kW of power was injected into the vacuum vessel at four frequencies: 2.45 GHz (4 kW), 6.4 GHz (3 kW), 10.5 GHz (10 kW) and 28 GHz (8 kW).
Neutral deuterium gas was injected throughout the discharge, which provided an ionization source near the plasma's outer edge.The pressure within the vacuum vessel was maintained nearly constant, 0.13 mPa, with deuterium gas puffing, but rises slightly by 10% during a brief 25 ms period near pellet injection.Because the plasma acts like a particle "pump," continuous particle fueling is required to maintain a constant plasma density and to prevent the excitation of fast electron interchange instabilities 45 .An accounting for particles in LDX has not been completed, but, based upon the relatively long period of outgassing observed after each discharge, we hypothesize that the cooled surface of the levitated cryostat acts as a temporary "getter pump" for inward moving plasma.
Using magnetic equilibrium reconstruction and multiple x-ray measurements, Davis 34 computed the plasma pressure for these conditions.High temperature confined plasma consists of a small population of very energetic electrons (50-100 keV) within a much larger population of warm thermal electrons.Without pellet injection, peak density reaches 0.5 × 10 18 m − 3. The plasma stored energy is 350 J.The population of energetic electrons, with average energy near 60 keV, makes up at least half of the stored energy but less than 5% of Turbulence During Pellet Injection the plasma density.The temperature profile of the warm, thermal electrons is estimated from the measured pressure and density profiles.The central electron temperature reaches about 0.5 keV.At the plasma pressure peak (L ≈ 0.8 m), the normalized ion sound gyroradius is ρ * s ≡ ρ s /L ≈ 0.04, and the normalized ion inertial length is λ * i ≡ c/ω pi L ≈ 0.4.The relatively low plasma density, high magnetic field strength, and long ion inertial length means the low-frequency fluctuations in LDX are electrostatic.Including the pressure from the energetic electrons, the magnetic reconstructions show the peak plasma beta reaches 10% at L > 0.9 m.
The presence of a small population of energetic electrons is important during pellet injection.For LDX, the average magnetic drift frequency of the 60 keV energetic electrons within the central high-pressure core is 450 kHz (while being only 3.8 kHz for the thermal electrons).The average range of energetic electrons within solid lithium is about 0.2 mm.
Because the energetic electrons drift quickly around the torus, about half of the energetic electrons within the dipole plasma torus will come into contact with pellet as it passes through the plasma.

III. GENERAL OBSERVATIONS OF PELLET INJECTION
Fig. 2 shows measurements of the dipole plasma torus with a Li pellet injected near t ≈ 6 s.The pellet passed through the plasma in approximately 15 ms while traveling at approximately 175 m•s −1 .The pellet injection is accompanied by a drop ∼ 62% in plasma diamagnetism (as measured by the outer flux loop) and more than a five-fold increase in the line-density measured with the innermost interferometer channel (shown with the black color in the top frames of Fig. 2.) When the plasma density is taken to be axisymmetric and uniform along each magnetic field line, the interferometer measurement implies a three-fold increase of electron number, rising from 2.5 × 10 18 to more than 7.5 × 10 18 electrons.The increase in electron number is estimated to be ≈ 10% of the number of atoms within the pellet.
About 25 ms after the pellet has passed through the plasma, the plasma density has nearly returned to the profile before pellet injection.The plasma energy, as indicated by the outer diamagnetic flux loop, returns to the pre-pellet level with two time scales: an initial time when the diamagnetism rises with a time near, but slightly longer than, the global energy confinement time, 14 ms, and then a second longer time, representative of the time required to re-create the small population of 60 keV, energetic electrons.
In addition to the increase in electron number and the decrease in plasma diamagnetism (proportional to the total plasma energy), the pellet's passage through the plasma causes bright visible light emission seen by the photodiode arrays and high-speed cameras and bright x-ray emission seen with a NaI detector.Many channels of the photodiode array saturate, as does the x-ray intensity detector.The plasma light emission returns to the pre-pellet level shortly after the passage of the Li pellet.In contrast, the x-ray intensity and 137 GHz radiometer signal do not return to the pre-pellet level for a much longer time, about 0.9 s, which is another measure of time required to reheat the small population of energetic electrons.
Two high-speed video cameras (512 × 256 8-bit images captured at 10 kfps) record the pellet injection as it passes through the plasma.These recordings are described and available as Supplementary Material.
The four-channel interferometer shows the evolution of the plasma density profile.A close-up of the line-density and photodiode measurements is shown in Fig. 3(a).Fig. 3(b) shows the plasma density profile reconstructed at four times during pellet injection.The rapid increase of line density, at t = 6.024 sec in Fig. 3(a), indicates a significant internal particle source.At the same time, oscillations of the inner line density channels suggest that the density is not axisymmetric during the initial rapid density increase.Measurements do not exist to determine the toroidal variations of the plasma density following pellet injection, but the probe array (discussed in Sec.IV) shows a relatively prompt current change at t = 6.025 sec, a rapid increase in ion saturation current beginning at t = 6.031 sec, and finally a gradual relaxation at the same rate as seen by the interferometer.These measurements suggest that pellet injection causes a very large disturbance to the plasma for the short time when the pellet first enters through the plasma (6.023 < t < 6.031) followed by a period lasting about 20 msec when the plasma torus is relatively axisymmetric and the plasma density gradually relaxes.
Although detailed measurements of plasma temperature and dynamics during Li pellet injection are not available, a general picture is summarized below.
The lithium pellet passes through the plasma in 15 msec causing the loss of about onehalf of the energetic electron energy and cooling the thermal electrons.The 200 J loss of equilibrate, although the degree of thermal equilibration is not known.With cooler electrons and higher density, the normalized ion sound gyroradius is ρ * s ∼ 0.02, and the normalized ion inertial length remains unchanged, λ * i ∼ 0.4, due to the higher mass of lithium ions.
Fig. 3c shows the radial profile of the electron number within flux-tubes, nδV , computed from the four-channel interferometer with the assumption of uniform density along field lines.As the plasma density increases, the radial gradient of electron number decreases (and may reverse).Except for a region near the inner most interferometer channel, the plasma density is "stationary" to interchange mixing, corresponding to the particle number per flux-tube being nearly constant, ∂(nδV )/∂ψ ≈ 0.
Finally, it is noteworthy, and perhaps even remarkable, that the plasma torus undergoes a significant and seemingly violent perturbation due to the pellet yet recovers quickly and relaxes back to the same quasi-steady conditions that existed prior to pellet injection in a short time, about 25 ms.

IV. LOW-FREQUENCY TURBULENT FLUCTUATIONS
Low-frequency turbulent fluctuations are measured with the four-channel microwave interferometer and the edge probe array.For times before (t < 6 s) and after (t > 6.1 s) Li pellet injection, these fluctuations are similar to previously reported measurements 19 and studies 43 .The interferometer channels show the low-frequency fluctuations extend throughout the plasma, and the largest amplitude modes have long radial and azimuthal wavelengths.The edge probe measurements show the fluctuations rotate toroidally with short azimuthal correlation lengths, ∼ 16 deg, and the intensity of the electric field fluctuations are similar to those previously reported 19 .azimuthal structures of the turbulent modes and the ensemble power spectra as a function of frequency 27,43 .Before and after pellet injection, when long time records of turbulence are measured, the ensemble power spectrum, ensemble cross coherence, κ 2 , and ensemble cross phase, α can be accurately calculated using standard methods (e.g.Ref. 27).

V. OBSERVATION OF TURBULENT TRANSPORT
Direct measurements of turbulent particle flux in a dipole-confined plasma were made previously in two ways: (i ) by observing the time evolution of the line density profile that resulted from the turbulent pinch fueled at the edge with neutral gas 19 or (ii ) by global measurement of plasma convection using a polar detector array for whole-plasma imaging 28 .
The density profile evolution determined the inward particle flux to equal the product of a diffusion coefficient, D = R 2 Ẽ2 ϕ τ cor , with the gradient of the particle flux-tube number, ∂(nδV )/∂ψ.Whole plasma imaging of interchange convection was seen to be "bursty", consisting of outward moving and field-aligned dense "filaments" and inward moving lessdense "filaments".
For the experiments reported here, these previously-used methods to determine the particle flux could not be used.The particle source due to the fast-moving pellet was not well-known, and whole-plasma imaging was not available.Instead, we report evidence of turbulent transport measured using probes from the edge probe array consisting as alter-pairs of nearby floating potential and ion saturation probes.The ensemble correlations were computed using overlapping 1.6 ms windows during a 30 ms time interval from 6.02 ≤ t ≤ 6.05.Although both the floating potential and ion saturation current fluctuations are less coherent, the cross-phases show that the toroidal directions of the fluctuations reverse.

V. OBSERVATION OF TURBULENT TRANSPORT
Direct measurements of turbulent particle flux in a dipole-confined plasma were made previously in two ways: (i ) by observing the time evolution of the line density profile that resulted from the turbulent pinch fueled at the edge with neutral gas 19 or (ii ) by global measurement of plasma convection using a polar detector array for whole-plasma imaging 28 .
The density profile evolution determined the inward particle flux to equal the product of ϕ τ cor , with the gradient of the particle flux-tube number, ∂(nδV )/∂ψ.Whole plasma imaging of interchange convection was seen to be "bursty", consisting of outward moving and field-aligned dense "filaments" and inward moving lessdense "filaments".
For the experiments reported here, these previously-used methods to determine the particle flux could not be used.The particle source due to the fast-moving pellet was not well-known, and whole-plasma imaging was not available.Instead, we report evidence of turbulent transport measured using probes from the edge probe array consisting as alter- nating floating potential and ion saturation current probes that span a toroidal angle of ∆ϕ = 24 deg.Each probe "triplet" of two floating probes on either side of the ion saturation probe were used in the same way as described by Carreras and co-authors 35 .The voltage difference between two floating potential probes is a measure of the azimuthal electric field, E ϕ , and the ion saturation probe is used as an estimate of the local plasma density.The ratio of the distribution before pellet to that during injection is shown in Fig. 9(d).
Pellet fueling is associated with a three-fold increase in the high-speed outward flows while leaving the probably of inward transport largely unchanged.
These measurements are similar to previous estimates of the radial particle flux at the edge of tokamak and stellarator discharges using same three probe technique 35 .As seen previously, the distribution of the radial particle flux is non-Gaussian, intermittent, and "bursty."The PDFs of the particle flux from these earlier tokamak and stellarator measurements can also represent the PDF's from the dipole plasma torus.The so-called "two-field" PDF for the fluctuation-induced turbulent flux (Eq.7 of Ref.

VI. COMPARISON WITH LINEAR INTERCHANGE AND ENTROPY MODE THEORY
Although electrostatic low-frequency fluctuations in dipole-confined plasma are nonlinear and turbulent, linear theory provides a useful framework to interpret the cross-phase measurements of the toroidal phase-velocity of the turbulence and to understand the physics behind the reversal of toroidal propagation during Li-pellet injection.
This section compares measurements of the phase dispersion of turbulence, Sec.IV, to the predictions of linear interchange and entropy modes.Specifically, the entropy mode dispersion relation for profiles that are marginally stable to the usual MHD interchange mode show the same dependence on η as seen experimentally.Before pellet injection, the central electron temperature is peaked, the plasma density is less peaked, and η > 2/3.In this case, unstable entropy modes propagate in the electron drift direction with a real frequency roughly proportional to the product of the azimuthal mode number and the average magnetic drift frequency, ω/m ω de > 0. When the pellet is injected, the central plasma cools and becomes more dense.In this case, η < 2/3, and the direction of entropy mode propagation reverses, ω/m ω de < 0.
The linear dispersion properties for electrostatic interchange and entropy modes in dipoleconfined plasma were previously derived by Kesner 36 , Ricci and co-authors 37 , and others 47 .
Because entropy modes cannot be described by conventional hydrodynamics 30,48 , these previous derivations were based on moments of the linearized drift-kinetic equation or gyrofluid equations.Additionally, both the ion and electron temperature effects were included.As shown previously (e.g.see Fig. 3 of Ref. 36), when the ion and electron temperatures were nearly equal, T e ∼ T i , the real frequencies of unstable entropy modes depend on η.When η < 2/3, entropy modes do not propagate toroidally.When η > 2/3, a pair of unstable entropy modes are excited: one propagating in the electron drift direction and the other in the ion direction.
For the dipole plasma reported here, the ion temperature is small, finite Larmor radius Adopting the gyrofluid approach taken by Ricci Quasineutrality in the dipole plasma torus requires ∇• J⊥ = 0, where J⊥ is the perturbed Unlike planetary magnetospheres, where plasma pressure perturbations induce field-alignedcurrents through the ionosphere, in the laboratory dipole, the quasineutrality constraint is satisfied by balancing the perturbed diamagnetic current with the ion inertial current.
The flux-tube average of the plasma continuity equation, ∂ ñ/∂t + ∇ • nv = 0, relates density perturbations to potential and potential fluctuations.Because the mode frequency is much less than the electron bounce frequency, the electron response is massless, and it's convenient to use the electron cross-field particle flux in the particle conservation equation, Using Eq. 3, the flux-tube average of plasma continuity is equivalent to the lowest moment of the bounce-averaged electron kinetic equation 49 .
The linear dispersion relation for entropy and interchange modes results combining the flux-tube averaged equations for the perturbed pressure, Eq. 1, the divergence of the plasma current, Eq. 2, and continuity using Eq. 3.
These three equations are most easily written in dimensionless form.When writing linear fluctuations in time and azimuth as (ñ, pe , Φ) ∝ exp(−iωt+imϕ) and normalizing frequency to m ω de , pressure to nT e , and electric potential to T e /e, then the linear dispersion relation is the characteristic function of the following system of three equations: As Eqs. 4-6 show, the characteristics of interchange and entropy modes depend upon four parameters: (i) the sonic Larmor radius relative to the plasma size, ρ * s ≡ √ M i T e /eB 0 L 0 , (ii ) the spatial structure of the electrostatic potential fluctuations, m 2 ⊥ , (iii ) the radial gradient of the flux-tube particle number, h n ∝ ∂(n 0 δV )/∂ψ, and (iv ) the radial gradient of the entropy density, h p ∝ ∂(P 0 δV γ )/∂ψ.The structure of the perturbed potential can be calculated numerically, as in Ref. 51, or parameterized using a local approximation, , where ψ and ϕ are the flux-tube average of the product of the plasma dielectric, n 0 M i /B 2 , with |∇ψ| 2 and |∇ϕ| 2 , respectively.
In dimensionless form, the profile functions, h p and h n , relate the differences of the bounce-averaged diamagnetic frequencies, ω * p and ω * n , with the magnetic drift frequency, ω de 36 .With ω * p proportional to the radial gradient of the electron pressure and ω * n proportional to the radial density gradient, then occurs whenever h p > 0, or ω * p > γ ω de , independent of η.Next, consider the effects of the curvature heat flux when perturbed flux-tubes have either warmer or cooler temperatures than the surrounding plasma.These perturbations will rotate toroidally faster or slower than the unperturbed plasma, leading to entropy modes.For the case when h p and h n are both zero, then η = 2/3, and stable entropy waves exists with ω = γ ± γ (γ − 1), analogous to the entropy waves discussed by Ware 30 but specialized to plasma with T e T i .
In the limit ρ * s m ⊥ → 0 and η ∼ 2/3, Eqs.4-6 describe unstable entropy modes with complex frequency given by Turbulence During Pellet Injection Eq. 9 shows the linear growth rates of entropy modes are slower than hydrodynamic interchange modes by about 3 √ ρ * s , and the magnitude of the rate of toroidal rotation is about 0.57 times the rate of linear growth.Fig. 10(a) shows the linear dispersion characteristics for these entropy modes for a dipoleconfined plasma torus having a pressure profile marginally stable to MHD interchange, h p = 0.When η > 2/3, entropy modes rotate in the electron drift direction.When η < 2/3, the toroidal phase velocity reverses.When Eqs. 4-6 are solved for larger (h p > 0) or smaller (h p < 0) pressure gradients, the electron's curvature heat flux also imparts a toroidal phase velocity to unstable interchange modes.Fig. 10 also presents a cartoon illustrating the cause for the toroidal phase-velocity reversal in terms of curvature heat flux.When η > 2/3, shown in Fig. 10(b), the core plasma is warmer and less dense than would be the case when h p and h n are both zero.
When the perturbed electrostatic potential causes radial flux-tube mixing, the perturbed pressure is small because h p = 0.However, even in the absence of the Ẽ × B compressibility term, the electron's curvature heat flux always causes heat to flow toroidally in the electron drift direction from regions of high temperature to lower temperature.When η > 2/3, the perturbed pressure creates perturbed fields that cause an outward motion of warm core plasma.Instability thus results only when the entropy mode rotates in the electron drift direction.When η < 2/3, shown in Fig. 10(c), the core plasma is cooler and more dense, and outward flux-tube convection causes this plasma to further cool and expand.In this case, the inward moving plasma is heated and compressed, and the toroidal phase of the pressure perturbation due to the curvature heat flux lags the potential perturbation.Instability then requires that the entropy mode rotates opposite to the electron drift, which again creates destabilizing perturbed fields.
Although the linearized gyrofluid equations discussed in this section help to explain the η-dependence of toroidal phase-velocity of the measured fluctuations, important processes are not described by Eqs.4-6.These include particle drift-resonance, nonlinear coupling and cascades between turbulent scales, and turbulent transport.Bounce-averaged driftresonance couples energy between particles and waves and is essential to understanding radial transport of energetic electrons 51 .Nonlinear turbulent cascade couples turbulent energy from smaller to larger scales 27 , and the linear growth rate cannot be used to determine the saturated turbulent spectrum.Although the measured broad-band drift-resonant fluc-Turbulence During Pellet Injection tuations justify a quasilinear representation of transport 19,21 , a fully self-consistent model of interchange and entropy mode driven transport requires nonlinear simulations.Computations using the GS2 31 code, corroborated the relationship between η and the direction of particle transport 32,33 .When η > 2/3, particle transport is inward.When η < 2/3, the direction of particle transport reverses.
The measured reversal of the toroidal phase-velocity of turbulence (in Figs. 5 and 6), the measured changes in average radial particle flux (in Fig. 8), and the expectations from the theory of entropy modes as described in this section, in Refs.36 and 37, and in transport models described by Refs.21, 32, and 33 lead us to conclude that the turbulent fluctuations in LDX are related to the presence of entropy modes.

VII. DISCUSSION AND SUMMARY
Fast injection of lithium pellets into a plasma torus confined by a strong dipole magnet allows measurement of turbulent fluctuations during large changes in plasma density and temperature profiles.As the pellet passed through the plasma, it cools central electrons and creates a significant internal particle source.The total electron number within the dipole plasma torus increases by more than a factor of three, and the central density increases by more than a factor of five.During these large changes in density, the shape of density profile is nearly "stationary" such that the gradient of the particle number within tubes of equal magnetic flux vanishes, ∂(nδV )/∂ψ ∼ 0. Remarkably, about 25 ms after the pellet passes through the plasma, the dipole-confined plasma torus relaxes back to the same quasi-steady conditions that existed prior to the pellet injection.
Turbulent fluctuations are measured using multiple microwave interferometers and multiple edge probes.These measurement show the lithium pellet causes only small changes in the amplitude and spectrum of the fluctuations.The most intense fluctuations have long wavelength, global structures, with low-order azimuthal mode numbers, m ≈ 1, 2, 3, . ... In contrast, the pellet causes more significant changes in the toroidal propagation and various statistical properties of the turbulence.Before pellet injection, the ensemble-averaged toroidal phase velocity of the fluctuations are measured to be ω/2π ≈ m × 700 Hz in the electron's magnetic drift direction.During pellet injection, the toroidal phase velocity reverses direction, and ω/2π ≈ −m × 550 Hz.Measured with an array of Langmuir probes at the outer edge of the plasma torus, the low-frequency edge fluctuations are intermittent and "bursty".The intensity of the edge fluctuations do not change significantly, but pellet injection does change both the skewness ion saturation current fluctuations and the correlation between the ion saturation current and the local azimuthal electric field (proportional to the radial Ẽ × B velocity.)Before and after pellet injection the measurements with the probe array show the particle flux to be inward, consistent with previous observations of LDX particle transport using the microwave interferometer array 19,23 .By comparison, during pellet injection, the measured rate of outward turbulent particle flux at the edge increases as should be expected when the source of particles briefly changes from ionization of neutral deuterium at the edge to ionization of lithium deep within the plasma core.
Although we are not able to measure the evolution of the plasma's electron temperature profile nor the nature of electron temperature fluctuations, plasma profile measurements of similar discharges reported elsewhere 34 and the expectations from theory and simulation lead to the following two important conclusions: 1. Turbulent fluctuations in the dipole-confined plasma torus are related to the presence of entropy modes.This conclusion is based on the reversal of the toroidal phase velocity consistent with the theoretical prediction that the phase velocity of the entropy mode reverses when the profile parameter, η, changes from the "warm core" case when η > 2/3 to the "cool core" case when η < 2/3.Because η depends upon the gradient of particle flux-tube number, h n , and the gradient of the entropy density, h p , according to η = (γ − 1 + h p − h n )/(1 + h n ), measurement of η requires measurement of both pressure and density profiles.Before pellet injection, magnetic reconstruction 34 estimates h p ∼ 0, and the interferometer array 22 gives h n ∼ −1.This implies η 1.
Although we are unable to measure the electron temperature profile during pellet injection, this conclusion is reasonable because the pellet is expected to cause the central temperature to decrease and the plasma pressure gradient to relaxed.During pellet injection, h n ∼ 0, and h p < 0 and η < 2/3 because the central electrons cool to a larger degree than the edge.
2. In the dipole plasma torus, turbulent fluctuations naturally drive profiles to become "stationary" with centrally peaked density.This conclusion is supported by the change in the statistical properties of the "bursty" edge particle transport and is consistent with theoretical 21 and computational 33 predictions and by the robust evolution of the plasma density profile during and after pellet injection.Just as observed during the active periods of planetary magnetospheres, particle transport from low-frequency turbulent fields decreases gradients in flux-tube number, (nδV ), and drives profiles to become "stationary", h n ∼ h p ∼ 0. These profiles are marginally stable and correspond to a turbulent state of minimum entropy production in magnetized plasma 29 .
These observations reported here also support the conclusions from other studies.The fluctuations have a power spectrum that is dominated at low-frequencies by global modes with long wavelengths and decay at higher frequencies with a power-law 27 .Local turbulent transport is intermittent, consisting of inward and outward moving plasma filaments that are field-aligned and can move radially at speeds reaching 10% of the sound speed 28 .The PDF of the radial particle transport is non-Gaussian with high kurtosis and skewness.Furthermore, the PDF measured in a dipole-confined plasma torus resembles the probability distribution measured in the same way at the edges of tokamaks and stellarators 35 .
Limitations in the diagnostic measurement and heating of plasma in LDX suggest opportunities for future study.Much higher plasma density with thermalized ion and electron populations, T i ≈ T e , and with reduced energetic particle fractions should be achievable through the use plasma heating methods that are not limited, like ECRH, by microwave accessibility.
The LDX facility has 1 MW of tunable radio frequency power near the deuterium cyclotron frequency that could be used to create high density, ω pe /ω ce 1, high-temperature plasma.
Higher-density thermalized dipole-confined plasmas allows further study of the properties of entropy mode dispersion and turbulence with ion thermal and finite Larmor radius effects.
More importantly, the investigation of higher density dipole-confined plasma is interesting because a levitated dipole's magnetic field lines are axisymmetric, without field-aligned currents, and provide omnigeneous particle confinement.Many well-known low-frequency instabilities found in other toroidal configurations, e.g.kink, tearing, and ballooning modes, are not found in a dipole plasma torus.Additionally, because the dipole-confined plasma torus can operate with peak plasma beta exceeding unity 52,53 , higher density plasmas would allow the study of high-temperature magnetized plasma turbulence over a wide range of dimensionless scales, spanning astrophysics and fusion science.For example, experiments with a high density, high-beta steady-state plasma torus would allow systematic laboratory Turbulence During Pellet Injection study of turbulent transport, including electromagnetic and Alfvén wave effects, when both the normalized gyroradius and the normalized ion skin depth are small and comparable, Finally, studies with higher density dipole-confined plasma should allow study of plasma profile evolution when the inner edge is fully recycling.Presently, the turbulent inward particle pinch is sustained by outer gas fueling and, most likely, an inner "getter pump" at the cooled surfaces of the dipole's cryostat.At sufficiently high density and long pulse times, the surface of the floating cryostat will saturate and no longer absorb particles.
The net particle flux at the inner edge must vanish, causing the plasma density near the superconducting dipole to increase.The turbulent particle flux must reverse, becoming outward, and the plasma profiles are expected to become naturally centrally-peaked and nearly "stationary" with η 2/3.

SUPPLEMENTAL MATERIALS
See supplemental materials for a photograph of the Levitated Dipole Experiment (LDX) showing the Li pellet injector, a schematic of the pellet trajectory and fast camera views, and an mp4 movie showing the light emission from the pellet as it passes through the plasma.
who used magnetic equilibrium reconstruction and x-ray analysis to estimate the bulk electron temperature profile.General observations of the plasma density and total energy following lithium pellet injection are presented in the next section.The pellet passes through the plasma in about 15 ms and causes a rapid increase of electron density followed by a relaxation to the plasma profiles that existed prior to injection.Diamagnetic estimates of plasma energy content suggest that the central electron temperature decreases significantly: from approximately 500 eV before pellet injection to 50 eV immediately after pellet injection.Next, Sec.IV summarizes statistical measurements of the low-frequency fluctuations and reports the reversal of the toroidal phase velocity of the fluctuations during pellet injection.Sec.V discusses estimates of the instantaneous particle flux measured with the edge Langmuir probe array.The measured transport is intermittent with a non-Gaussian probability distribution function (PDF) not unlike previous observations from tokamak and stellarator plasmas35 .

Turbulence
During Pellet Injection the large vacuum vessel.The combination of the superconducting dipole magnet and the upper levitation coil creates a nearly axisymmetric magnetic field with a toroidal volume of closed field-lines bounded by both an outer edge and an inner edge.The outer boundary is defined by a (nominally) axisymmetric field null, where the contributions to the magnetic field from the superconducting dipole and levitation coil cancel.In cylindrical coordinates, the field null location is (R, z) = (1.1 m, 1.2 m), with z = 0 defined at the horizontal plane of the dipole magnet.The inner boundary is determined by those magnetic field lines that contact the superconducting dipole coil.The toroidal volume of closed field lines is 10 m 3 and extends outward in the horizontal plane of the dipole (z = 0) from an inner radius L = 0.68 m to an outer radius of L = 1.8 m. (We use the symbol L as both a radial distance and, also, as a field-line label at constant magnetic flux, ψ(L) ∝ 1/L.)The peak on-axis magnetic field strength is 2.2 T; the peak magnetic field strength in the plasma is 3.4 T; and the field strength at the outer boundary between closed and open field lines (i.e.L = 1.8 m) is only 7 mT.The ratio of differential flux-tube volumes near the outermost field line to the flux-tube volume near innermost field line is large, δV (1.8 m)/δV (0.68 m) ≈ 96.

FIG. 1 .
FIG. 1.A schematic drawings of the LDX experiment showing (a) a 3D view and (b) a top view of the magnetic geometry, Li-pellet trajectory, four-channel microwave interferometer, and electrostatic probe array.Additionally, the shaded regions in (b) show the sight lines for two fast video cameras and two photodiode arrays.
, various magnetic diagnostics (15 magnetic flux loops and 12 poloidal field coils), x-ray detectors, and electron cyclotron emission measurements at 137 GHz 44 are used to reconstruct the plasma pressure and estimate the fractional density and energy of energetic electrons produced by microwave Turbulence During Pellet Injection heating.The 137 GHz radiometer signal is localized near the peak of the steep hot electron profile and inside L ∼ 0.80 m.The 3rd to 12th harmonic ECE resonances at 137 GHz are in direct view of the radiometer above the dipole and via wall reflections below the dipole.The outer magnetic flux loop is useful as a measure of plasma diamagnetism that is proportional to plasma stored energy, ≈ 100 J/mV•sec.The trajectory of the lithium pellet across the plasma from the injector (LPI)39 is also shown in Fig.1.The LPI can inject approximately 5 × 10 19 lithium atoms (pellet volume ≈ 1 mm 3 ) per pellet at top speeds approaching 1 km•s −1 .For these experiments, the pellets are injected along a trajectory that enters the plasma slightly below the midplane (z ≈ −0.25 m), exit slightly above the midplane (z ≈ 0.25 m), and pass through the central region of highest plasma pressure (L ≈ 0.80 m) and between the tangency radii of the inner two interferometer channels.

8 a
Line density and ion saturation current fluctuation levels are normalized to the mean within the time-interval.The floating potential is normalized to the edge electron temperature, T e (edge) ≈ 17 eV; and the azimuthal electric field level shows the time-averaged RMS level in V/m.b For the line density, the deviation from the mean during Li pellet injection is dominated by the transient increase and relaxation from central pellet fueling.

2 FIG. 4 .
FIG. 4. Fluctuation measurements for the time period prior to pellet injection, 5.00 -6.00 sec.The top figures show the normalized fluctuation power spectrum, and the bottom figures show the ensemble cross-coherence, κ 2 .Left (a) shows the power spectra for the four line density measurements.The ensemble cross-coherence shows a relatively coherent global mode at ∼ 700 Hz representing a rotating m = 1 interchange convection cell.Middle (b) plots show the edge I sat fluctuations along with the relatively rapid decorrelation along the probe array.Right (c) plots show the edge floating potential power spectrum and angular decorrelation along the probe array.[Associateddataset available at http://dx.doi.org/10.5281/zenodo.45507]46

Fig. 4 1 , 2 ∼ 1 .FIG. 5 . 46 α 1 , 2 ≈
Fig.4illustrates these ensemble measurements made from 250 overlapping 8 msec time windows during a one-second time period before Li pellet injection.The largest lowfrequency mode is axisymmetric (m = 0) with a frequency near 100 Hz.This global mode is described in Ref.43 and appears to be a "breathing mode" related to accessibility of the microwaves, injected for heating, when the plasma density becomes large.The axisymmetric global mode does not appear to be directly related to plasma transport.The fluctuations with frequencies higher than 100 Hz are non-axisymmetric.They consist of quasi-coherent fluctuations with low azimuthal mode numbers, m = 1, 2, 3, . . .and with a power-law power spectrum at higher frequencies.At frequencies above 10 kHz, the power spectrum of the line density fluctuations decrease as |(δn/n) 2 | ∝ f −2.5 , and the power spectra of the ion saturation fluctuations and the floating potential fluctuations decrease as|(δn/n) 2 | ∼ |(δΦ/kT e ) 2 | ∝ f −3 .As reported previously19,43 , the fluctuation power spectra do not change significantly whether or not the dipole magnet is levitated.The fluctuations observed in the large LDX experiment resemble the fluctuations observed in a smaller magnetic dipole experiment where the spatial and temporal structures of the fluctuations are characterized as chaotically rotating flute-like modes27,28 .Fig. 4 also shows the ensemble cross-coherence (as defined, for example, in Eq. 3 of Ref. 27).The cross-coherence between two signals is denoted by κ 2 1,2 , where κ 2 1,2 ∈ [0, 1].When two probes are strongly correlated, κ 2 1,2 ∼ 1.The cross-coherence vanishes when two probes are uncorrelated.The cross-coherence between two probes decreases as the toroidal separation increases, and the probes are decorrelated when the toroidal separation is large, ∆φ > 24 deg.When the probes are nearby, ∆ϕ ∼ 8 deg (about 0.15 m), two nearby probes have sufficient coherence to determine the azimuthal mode numbers and azimuthal phase velocities for modes with relatively long azimuthal wavelengths.The azimuthal mode structure and phase velocity are determined by computation of the ensemble cross-phase, α 1,2 .(The cross-phase is defined, for example, in Eq. 4 of Ref. 27.)For two nearby probes, the crossphase is proportional to the product of the mode number and the probe azimuthal spacing,

Near and below 2
kHz, the lowest three azimuthal modes (m = 1, 2, 3) rotate in the ion drift direction near m × 550 Hz during pellet injection.The power spectrum of both the floating potential and ion saturation current resemble the spectrum before pellet injection (i.e.Fig. 4(b) and (c)): the largest intensity corresponds to m = 1 and m = 2 and the intensity decreases inversely proportional to a power of the frequency.

Fig. 6
Fig. 6 also illustrates the reversal of toroidal rotation during pellet injection.Fig. 6 displays the floating potential fluctuations for 10 ms periods before, during, and after pellet injection.The floating potential measured with seven probes, spanning approximately 1 m in the toroidal direction, fluctuate within the range ±20 V and propagate toroidally.Before and after pellet injection, these fluctuations propagate in the electron drift direction.By comparison, during pellet injection Fig. 6(b), the direction of toroidal rotation reverses consistent with the ensemble cross-phase shown in Fig. 5(b).

FIG. 6 .
FIG. 6. Multipoint measurement of the floating potential fluctuations along the probe array for three 10 msec intervals: (a) before, (b) during, and (c) after Li pellet injection.Each plot is shown on the same scale.The toroidal variation of the potential is in the electron drift direction before and after pellet injection, but the perturbed potential reverses direction during pellet injection.

Fig. 7 FIG. 7 .
Fig. 7 illustrates the edge probe array measurements during two 4 ms time intervals: one before pellet injection and the other during pellet injection.From top to bottom, the three frames show ion saturation current, floating potential, and azimuthal electric field

Fig. 8 FIG. 8 .
Fig. 8 shows the time and toroidal average of the edge probe measurements during a 0.1 s time interval about the pellet injection.The fluctuations are shown after a moving 1 ms time average and after toroidal averaging over three probes spanning ∆ϕ = 24 deg.Pellet injection causes the edge ion saturation current to increase by a factor of three.The root mean square magnitude of the electric field fluctuations, Ẽϕ RM S , decrease slightly from ±66 V/m to ±57 V/m.The averaged fluctuations of the particle flux, proportional to I sat Ẽϕ , are large.The plots of average particle flux shown in Fig. 8(c) are time-averaged

FIG. 9 .
FIG. 9.PDF of radial particle flux as estimated by the probe array before pellet injection (5.0 < t < 6.0) and during pellet injection (6.02 < t < 6.05).The four frames show: (a) the joint probably histogram of the ion saturation current and radial velocity, proportional to Ẽϕ , before pellet injection; (b) the same joint probably histogram computed during pellet injection; (c) the probability distribution function of the radial flux measured from a single edge probe triplet; and (d) the ratio of the PDF during pellet fueling to the PDF before pellet shows the increase in outward-directed flux at the tail of the distribution.In (c), the PDF is compared to "two-field" PDF for turbulent radial flux discussed in Ref. 35.
effects are not important, and electrostatic interchange and entropy modes are easier to describe.With T e T i , only three equations are needed: one each to describe the dynamics of fluctuations in the electric potential, Ẽ = −∇ Φ, the plasma density, ñ, and the electron pressure, pe .Fluctuations are uniform along magnetic flux-tubes (e.g.B • ∇ Φ = 0) and vary only in time and in the radial and azimuthal coordinates, (ψ, ϕ).
37 and using the notation of interchange modes introduced by Rosenbluth49 , the simplest equation that correctly models the real frequency of interchange and entropy modes in an axisymmetric dipole-confined plasma requires including the collisionless curvature heat flux of Braginskii (see, for example, Eq.21 of Ref. 50, where the curvature heat flux changes the electron pressure at a rate (2γ/eB) b × κ • ∇(p e T e ), proportional to the magnetic curvature, κ).The linearized equation for electron Turbulence During Pellet Injection pressure perturbations in axisymmetric dipole is p = n 0 T e is the equilibrium electron pressure and ∂ Φ/∂ϕ is proportional to the radial component of the fluctuating Ẽ × B velocity.Eq. 1 is derived by taking the flux-tube averaged of the gyrofluid equations in (A4) and (A5) of Ref. 37, or Eq.21 of Ref. 50, for dipole magnetic geometry and when the electrons within any flux-tube can be assumed to be Maxwellian and isotropic.Eq. 1 shows that pressure perturbations for flute-type modes are driven by two terms: the compressibility term from ideal MHD and the perturbed collisionless curvature heat flux due to the magnetic drift of electrons.Both of these terms depend on magnetic flux-tube geometry.The flux-tube average used in Eq. 1 is defined as A ≡ δV −1 dl, A/B, where δV = dl/B.Because the axisymmetric magnetic geometry is without shear, magnetic flux coordinates are orthonormal, (ϕ, ψ, χ), with B = ∇ϕ × ∇ψ = ∇χ, which simplifies fluxtube averaging.The equilibrium profiles depend only upon the radial flux coordinate, p(ψ), T e (ψ), and n(ψ) = p/T e .The total particle number on a flux-tube is nδV , and the bounce-averaged magnetic drift of electrons with a temperature, T e , is ω de = 2 κ ψ T e /e, proportional to the average magnetic curvature, 2 κ ψ = −∂ ln δV /∂ψ ∼ 4/ψ.

s m 2 ⊥
FIG. 10.The physics of entropy mode dispersion for a cold ion plasma having a pressure profile marginally stable to interchange modes.(a) Depending upon the ratio of the normalized electron temperature and density gradients, η, the real frequency of the instability reverses.(b) When η > 2/3, the entropy mode rotates with the electron magnetic drift.(c) When η < 2/3, the high pressure, but cool, central plasma further cools as it moves outward.As indicated by the dotted lines in bottom figures of (b) and (c), the unstable entropy mode rotates in the ion drift direction so as to further transport warm plasma outward.

TABLE I .
Summary of statistical moments of the fluctuations before (5.00 s < t < 6.00 s, using more than 125k samples) and during (6.02 s < t < 6.05 s, using 3750 samples) Li pellet injection.Show are the average fluctuation level and the coefficients of skewness (S) and excess kurtosis (F ).