2 Model

To examine electromagnetic power balance in an RFP, we develop a simple Poynting's theorem model for an incompressible, resistive-MHD plasma with cylindrical flux surfaces and a resistive boundary. We write each quantity (such as pressure) as $p=p_0+p_1$, the sum of mean (flux-surface average or equilibrium) and fluctuating parts, respectively, where $\langle p \rangle = p_0$ and $\langle p_1 \rangle = 0$. We define the flux-surface average of a vector quantity (Moffatt Reference Moffatt1978; Krause & Rädler Reference Krause and Rädler1980; Tsui Reference Tsui1988; Ji et al. Reference Ji, Almagri, Prager and Sarff1994; Ji Reference Ji1999) as the vector sum of the flux-surface averages of its scalar components (Rädler Reference Rädler2007) in cylindrical coordinates. For example, the average current density $\boldsymbol {J}$ is written $\langle \boldsymbol {J}\rangle = \langle J_r\rangle \hat {\boldsymbol {r}} + \langle J_\phi \rangle \hat {\boldsymbol {\phi }} + \langle J_z\rangle \hat {\boldsymbol {z}}$. Note that the key results of this section can also be found by an alternative treatment using only flux-surface averages of scalars. The flux-surface average of a product of a single fluctuating quantity and a mean quantity is zero (e.g. $\langle \boldsymbol {v}_1\times \boldsymbol {B}_0\rangle = 0$), while the average of a product of two or more fluctuating quantities can be non-zero (e.g. $\langle \boldsymbol {v}_1\times \boldsymbol {B}_1\rangle$).

We begin by stating Poynting's theorem

(2.1)\begin{equation} \frac{1}{\mu_0}\int\left(\boldsymbol{E}\times\boldsymbol{B}\right)\boldsymbol{\cdot}\mathrm{d}\boldsymbol{A} +\frac{1}{\mu_0}\int\boldsymbol{B}\boldsymbol{\cdot}\dot{\boldsymbol{B}} \,\mathrm{d}V +\int\boldsymbol{E}\boldsymbol{\cdot} \boldsymbol{J} \,\mathrm{d}V = 0, \end{equation}

which represents the balance between the Poynting flux through a surface, the change of magnetic energy in the volume contained by the surface, and volumetric power transfer, which includes Ohmic dissipation. Assuming a fluid velocity $\boldsymbol {v}$ with no mean part and a non-fluctuating resistivity $\eta$ for simplicity, Ohm's law is given by

(2.2)\begin{equation} \boldsymbol{E}+\boldsymbol{v}_1\times\boldsymbol{B} = \eta \boldsymbol{J}, \end{equation}

where the flux-surface average of the second term is the dynamo EMF, $\langle \boldsymbol {v}_1\times \boldsymbol {B}_1 \rangle$, and the fluctuating part is $\boldsymbol {v}_1\times \boldsymbol {B}_0+\boldsymbol {v}_1\times \boldsymbol {B}_1-\langle \boldsymbol {v}_1\times \boldsymbol {B}_1 \rangle$. Since Maxwell's equations are linear, Poynting's theorem (2.1) holds separately for the mean and fluctuating parts of the electromagnetic quantities. Substitution of the electric fields from the respective Ohm's laws (2.2) into the power transfer terms provides results that are then flux-surface averaged, becoming

(2.3)\begin{align} \frac{1}{\mu_0}\int \left(\boldsymbol{E}_0\times\boldsymbol{B}_0\right)\boldsymbol{\cdot}\mathrm{d}\boldsymbol{A} +\frac{1}{\mu_0} \int\boldsymbol{B}_0\boldsymbol{\cdot}\dot{\boldsymbol{B}}_0 \,\mathrm{d}V +\int\eta J_0^2\,\mathrm{d}V -\int\langle \boldsymbol{v}_1 \times \boldsymbol{B}_1 \rangle \boldsymbol{\cdot} \boldsymbol{J}_0\,\mathrm{d}V = 0 \end{align}

and

(2.4)\begin{align} & \frac{1}{\mu_0}\int \left\langle \boldsymbol{E}_1\times\boldsymbol{B}_1\right\rangle \boldsymbol{\cdot}\mathrm{d}\boldsymbol{A} +\frac{1}{\mu_0} \int \langle\boldsymbol{B}_1\boldsymbol{\cdot}\dot{\boldsymbol{B}}_1\rangle \,\mathrm{d}V +\int\eta\langle J_1^2\rangle\,\mathrm{d}V \nonumber\\ & \quad -\int\langle\boldsymbol{v}_1\times\boldsymbol{B}_0\boldsymbol{\cdot}\boldsymbol{J}_1 \rangle\,\mathrm{d}V -\int\langle\boldsymbol{v}_1\times\boldsymbol{B}_1\boldsymbol{\cdot}\boldsymbol{J}_1 \rangle\,\mathrm{d}V = 0. \end{align}

Note that the product $\langle \boldsymbol {v}_1\times \boldsymbol {B}_1\rangle \boldsymbol {\cdot }\boldsymbol {J}_1$ averages to zero and so does not appear in (2.4). These two equations can be added together to yield the flux-surface average of Poynting's theorem for the total electromagnetic quantities,

(2.5)\begin{equation} \frac{1}{\mu_0}\int\left\langle\boldsymbol{E} \times \boldsymbol{B}\right\rangle \boldsymbol{\cdot} \mathrm{d}\boldsymbol{A} + \frac{1}{\mu_0}\int\langle\boldsymbol{B} \boldsymbol{\cdot} \dot{\boldsymbol{B}}\rangle \,\mathrm{d}V +\int\eta \langle J^2 \rangle \,\mathrm{d}V - \int\langle \boldsymbol{v}_1\times\boldsymbol{B}\boldsymbol{\cdot} \boldsymbol{J}\rangle \,\mathrm{d}V =0, \end{equation}

remembering that flux-surface averages of a product of one mean and one fluctuating quantity equals zero (e.g. $\langle \boldsymbol {E}_0\times {\boldsymbol {B}}_1\rangle = 0$). The expansion

(2.6)\begin{align} -\langle \boldsymbol{v}_1\times\boldsymbol{B}\boldsymbol{\cdot} \boldsymbol{J}\rangle & ={-} \langle\boldsymbol{v}_1\times\boldsymbol{B}_1\boldsymbol{\cdot}\boldsymbol{J}_0\rangle - \langle\boldsymbol{v}_1\times\boldsymbol{B}_0\boldsymbol{\cdot}\boldsymbol{J}_1\rangle - \langle\boldsymbol{v}_1\times\boldsymbol{B}_1\boldsymbol{\cdot}\boldsymbol{J}_1\rangle - \langle\boldsymbol{v}_1\times\boldsymbol{B}_0\boldsymbol{\cdot}\boldsymbol{J}_0\rangle \nonumber\\ & ={-} \langle\boldsymbol{v}_1\times\boldsymbol{B}_1\boldsymbol{\cdot}\boldsymbol{J}_0\rangle - \langle\boldsymbol{v}_1\times\boldsymbol{B}_0\boldsymbol{\cdot}\boldsymbol{J}_1\rangle - \langle\boldsymbol{v}_1\times\boldsymbol{B}_1\boldsymbol{\cdot}\boldsymbol{J}_1\rangle \end{align}

was also used. Note that (2.5) can also be derived directly from (2.1) by substituting (2.2) into the power transfer term and then flux-surface averaging. Assuming that the fluctuation evolves slowly compared with an Alfvén time, which is true of tearing modes in typical experiments (Ortolani & Schnack Reference Ortolani and Schnack1993), the fluctuation is considered to be approximately in equilibrium. If we further assume that flows are slow compared with the sound speed, which is consistent with RFP observations (Fontana et al. Reference Fontana, den Hartog, Fiksel and Prager2000), we neglect contributions of flow and viscous dissipation and treat the equilibrium as magnetostatic. Under these assumptions,

(2.7)\begin{equation} {-}\left\langle\left(\boldsymbol{v}_1\times\boldsymbol{B}\right)\boldsymbol{\cdot}\boldsymbol{J}\right\rangle = \left\langle \boldsymbol{v}_1\boldsymbol{\cdot}\boldsymbol{\nabla} p_1 \right\rangle. \end{equation}

The proof for (2.7) follows. Since $p_1 = p - \left \langle p\right \rangle$, we have

(2.8)\begin{equation} \boldsymbol{\nabla} p_1 = \boldsymbol{\nabla} p - \boldsymbol{\nabla}\left\langle p\right\rangle = \boldsymbol{\nabla} p - \left\langle \boldsymbol{\nabla} p\right\rangle, \end{equation}

where the second equality follows from the definition of a flux-surface average of a vector (Rädler Reference Rädler2007). With a magnetostatic equilibrium, this can be rewritten as

(2.9)\begin{equation} \boldsymbol{\nabla} p_1 = \boldsymbol{J}\times\boldsymbol{B} -\left\langle\boldsymbol{J}\times\boldsymbol{B}\right\rangle = \boldsymbol{J}_0\times\boldsymbol{B}_1 + \boldsymbol{J}_1\times\boldsymbol{B}_0 + \boldsymbol{J}_1\times\boldsymbol{B}_1 - \left\langle\boldsymbol{J}_1\times\boldsymbol{B}_1\right\rangle, \end{equation}

where each term that is the flux-surface average of a product including a single fluctuating quantity equals zero and is neglected. Dotting this with $\boldsymbol {v}_1$ and flux-surface averaging the result gives

(2.10)\begin{equation} \left\langle \boldsymbol{v}_1\boldsymbol{\cdot}\boldsymbol{\nabla} p_1 \right\rangle = \left\langle \boldsymbol{v}_1\boldsymbol{\cdot} \left(\boldsymbol{J}_0\times\boldsymbol{B}_1\right)\right\rangle + \left\langle \boldsymbol{v}_1\boldsymbol{\cdot} \left(\boldsymbol{J}_1\times\boldsymbol{B}_0\right)\right\rangle + \left\langle \boldsymbol{v}_1\boldsymbol{\cdot} \left(\boldsymbol{J}_1\times\boldsymbol{B}_1\right)\right\rangle \end{equation}

where we have recognized that $\left \langle \boldsymbol {v}_1\boldsymbol {\cdot }\left \langle \boldsymbol {J}_1\times \boldsymbol {B}_1\right \rangle \right \rangle =0$. If we expand the final integrand in (2.5) and use vector identities, we find that

(2.11)\begin{equation} {-}\left\langle \boldsymbol{v}_1\times\boldsymbol{B}\boldsymbol{\cdot} \boldsymbol{J}\right\rangle = \left\langle\boldsymbol{v}_1\boldsymbol{\cdot} \boldsymbol{J}_0 \times \boldsymbol{B}_1 \right\rangle + \left\langle\boldsymbol{v}_1\boldsymbol{\cdot} \boldsymbol{J}_1 \times \boldsymbol{B}_0 \right\rangle + \left\langle\boldsymbol{v}_1\boldsymbol{\cdot} \boldsymbol{J}_1 \times \boldsymbol{B}_1 \right\rangle = \left\langle \boldsymbol{v}_1\boldsymbol{\cdot}\boldsymbol{\nabla} p_1 \right\rangle \end{equation}

as needed. Inserting (2.7) into (2.5), the flux-surface averaged Poynting's theorem becomes

(2.12)\begin{equation} \frac{1}{\mu_0}\int \left\langle\boldsymbol{E}\times\boldsymbol{B}\right\rangle\boldsymbol{\cdot} \mathrm{d}\boldsymbol{A} + \frac{1}{\mu_0}\int\left\langle\boldsymbol{B} \boldsymbol{\cdot} \dot{\boldsymbol{B}}\right\rangle \mathrm{d}V +\int\eta \left\langle J^2 \right\rangle \mathrm{d}V + \int\left\langle \boldsymbol{v}_1 \boldsymbol{\cdot}\boldsymbol{\nabla} p_1 \right\rangle \mathrm{d}V=0. \end{equation}

Since the $\phi,z$ integrations in the flux-surface average are the same integration variables as are present in the volume integral, the flux-surface average can be brought outside of the last volume integral on the left. Assuming that flows are incompressible ($\boldsymbol {\nabla }\boldsymbol {\cdot } \boldsymbol {v}_1 = 0$), the last volume integral in (2.12) can be rewritten as a surface integral so that

(2.13)\begin{equation} \frac{1}{\mu_0}\int \left\langle\boldsymbol{E}\times\boldsymbol{B}\right\rangle\boldsymbol{\cdot} \mathrm{d}\boldsymbol{A} + \frac{1}{\mu_0}\int\left\langle\boldsymbol{B} \boldsymbol{\cdot} \dot{\boldsymbol{B}}\right\rangle \mathrm{d}V +\int\eta \left\langle J^2 \right\rangle \mathrm{d}V + \left\langle\int p_1\boldsymbol{v}_1 \boldsymbol{\cdot} \mathrm{d}\boldsymbol{A}\right\rangle=0, \end{equation}

where the last integral corresponds to volumetric contributions to the power balance due to flow fluctuations, including the dynamo EMF. If this expression is examined at the plasma edge where $p_1=0$ holds, the last integral vanishes. For this case, we separate quantities into mean and fluctuating parts, giving

(2.14)\begin{align} & \overbrace{\frac{1}{\mu_0}\int \left(\boldsymbol{E}_0\times\boldsymbol{B}_0\right)\boldsymbol{\cdot} \mathrm{d}\boldsymbol{A}}^{S_0} +\overbrace{\int_\text{total}\eta J_0^2 \,\mathrm{d}V}^{P_{\text{ohm}}} + \overbrace{\frac{1}{\mu_0}\int_\text{total} \boldsymbol{B}_0 \boldsymbol{\cdot} \dot{\boldsymbol{B}}_0 \,\mathrm{d}V}^{P_{\text{mag,0}}} \nonumber\\ & \quad+\underbrace{\frac{1}{\mu_0}\int_\text{total} \left\langle \boldsymbol{B}_1 \boldsymbol{\cdot} \dot{\boldsymbol{B}}_1 \right\rangle \mathrm{d}V}_{P_{\text{mag,1}}} + \underbrace{ \frac{1}{\mu_0}\int \left\langle\boldsymbol{E}_1\times\boldsymbol{B}_1\right\rangle\boldsymbol{\cdot} \mathrm{d}\boldsymbol{A} }_{S_f} =0, \end{align}

where we have also assumed that the $\eta \left \langle J_1^2\right \rangle$ integral of Ohmic losses due to fluctuations is negligible compared with any of the other integrals. Variable names have been assigned to each term for easy reference. This equation shows the expected global power balance between terms representing the inwardly directed equilibrium Poynting flux provided by the external power supplies through the plasma's surface ($S_0$), the equilibrium Ohmic loss rate in the plasma volume ($P_{\text {ohm}}$), the change of magnetic energy (both in the mean field $P_{\text {mag},0}$ and fluctuations $P_{\text {mag},1}$) and the fluctuation-induced Poynting flux $S_{f}$. Note that the assumption of a resistive plasma boundary rather than a perfectly conducting boundary permits a non-zero tangential $E_1$ and therefore a non-zero $S_{f}$.

3 Experiments

Experiments were conducted in the Madison Symmetric Torus (MST) (Dexter et al. Reference Dexter, Kerst, Lovell, Prager and Sprott1991), a medium-sized RFP with major radius $R=1.5$ m and minor radius $a=0.52$ m. The plasma is surrounded by a thick, close-fitting conducting shell, and a toroidal rail limiter at the midplane extending $\sim$1 cm into the plasma determines the plasma–vacuum boundary. Time-resolved global measurements include plasma current, toroidal-field shell current, external flux loops from which are derived loop voltages and magnetic fluxes, and an edge toroidal array for mode decomposition of magnetic fluctuations. Time-resolved electron temperature profiles were obtained with Thomson scattering (Parke et al. Reference Parke, Hartog, Morton, Stephens, Kasten, Reusch, Harris, Borchardt, Falkowski and Hurst2012).

To measure $S_{f}$, a probe was inserted into the edge region of low-current ($I_\text {p} \approx 205$ kA) plasmas with $20.5$ V loop voltage and input power $S_0\sim 4.2$ MW. Central chord-average electron density is $n_{e} \approx 1\times 10^{13}\ {\rm cm}^{-3}$, while the edge region of radii $r$ sampled by the probe ($0.77< r/a<1.0$, where $r/a=0.85$ is the magnetic reversal surface location) has $n_\text {e}\sim (1\text {--}5)\times 10^{12}\ {\rm cm}^{-3}$, temperatures $T_{e}\sim T_{i} \leqslant 50$ eV and volume-average mean magnetic field $| \boldsymbol {B}_0|\approx 1100$ G. The plasma current and insertion depths were chosen to ensure probe survival and negligible plasma perturbation.

The insertable probe (Thuecks et al. Reference Thuecks, Almagri, Sarff and Terry2017) was constructed to allow measurement of poloidal and toroidal electric and magnetic field fluctuations simultaneously and in close spatial proximity so that the radial $S_f$ can be determined. Eight electrodes were spaced evenly (0.59 cm spacing) on the edges of a radially facing square and were used to measure the floating potential $V_{f}$. Electric field components in the poloidal (toroidal) direction were approximated using $E \approx -(V_{{f}2}-V_{{f}1})/(x_2-x_1)$ where electrodes are separated poloidally (toroidally) by a distance $x_2-x_1$. Here, we assumed small gradients in the electron temperature fluctuations, justified by previous comparisons of independent electrostatic measurements (Stone Reference Stone2013), allowing us to approximate the fluctuating electric field using the floating potentials rather than plasma potentials. The inductive part of the fluctuating electric field is expected to be negligibly small in the present context (Ji et al. Reference Ji, Prager, Almagri, Sarff, Yagi, Hirano, Hattori and Toyama1996; Bonfiglio, Cappello & Escande Reference Bonfiglio, Cappello and Escande2005).

Magnetic fields were measured using an internal B-dot cube located along the radial axis $0.51$ cm below the probe's electrode plane. Using this cube, $\dot {\boldsymbol {B}}$ can be measured in three orthogonal directions. The $\dot {\boldsymbol {B}}$ signal from the B-dot cube was differentially amplified and was then numerically integrated during analysis to find $\boldsymbol {B}$. Analysed data was taken during the plasma current flattop when plasma conditions remained relatively stationary.

Measured quantities are examined with respect to the quasiperiodic sawtooth cycle, with datasets from (typically) hundreds of sawteeth in many shots sliced into short time series centred around the sawtooth crashes for ensemble analysis. The fluctuating part of these quantities is then found by calculating the average time-resolved signal over many sawteeth and subtracting it from the individual sawtooth datasets.

Natural plasma rotation makes a time-resolved ensemble average for a fluctuating quantity an approximate spatial average over an equilibrium magnetic flux surface. The probe is traversed by a randomly phased extent of a rotating flux surface in each sawtooth dataset. Thus, by averaging over many sawtooth datasets we approximate a time-resolved flux-surface average. The duration of the ensemble time windows matches the typical sawtooth period, so the time average of an ensemble-averaged quantity over the window reflects its sawtooth cycle average.