A Unified Phenomenology of Ion Heating in Low-$\beta$ Plasmas: Test-Particle Simulations

Ions within the turbulent low-$\beta$ solar corona and wind are observed to be continously heated, with more energization perpendicular to the local magnetic field than parallel [1, 2, 3]. Additionally, heavy ions are heated more than protons [4, 5, 6] and protons more than electrons [7], despite the low-frequency gyrokinetic model predicting that electron heating dominates at low $\beta$ [8, 9]. Many mechanisms have been proposed to explain these observations, including stochastic heating by uncorrelated turbulent fluctuations [10, 11], wave-particle interactions leading to quasi-linear heating [12, 13, 14], and others [15, 16, 17]. Despite this previous work, aspects of the underlying processes remain unclear, representing an important deficiency for building understanding of coronal heating and solar-wind acceleration.

Extending previous studies [18, 19, 20, 21, 22, 23], in this Letter we present a unified description of turbulent heating of ions in both balanced turbulence (with equal amplitudes of inwards and outwards Elsässer fluctuations $\bm{z}^{\pm}$) and imbalanced turbulence (where one dominates: $|\bm{z}^{+}|\gg|\bm{z}^{-}|$). Noting that stochastic heating (SH) and cyclotron-resonant quasi-linear heating (QLH) can be considered as two limits of similar physical processes and motivated by the principle of “critical balance”, we propose that the ion-heating rate is controlled by the power in scales at which turbulent fluctuations reach their maximum frequency, which may not be at ion-gyroradius scales in the presence of a steep ion-kinetic “transition-range” drop in the spectrum. We test this proposal with high-resolution simulations of test particles interacting with low-$\beta$ magnetized turbulence. Our proposed scaling accurately captures measured proton and minor-ion heating rates in turbulence across a wide range of conditions (balanced, imbalanced, with and without a “helicity barrier”). These results clarify the physics of collisionless turbulent heating with clear application to both in-situ observational analyses and theoretical models of astrophysical plasmas [24].

Before continuing we define important symbols, with species index $s$ either ions or protons ($s=\text{i}$ or p): $\bm{E}$ and $\bm{B}$ are the electric and magnetic fields; $k_{\|}$ and $k_{\perp}$ are the components of the wavevector $\bm{k}$ parallel and perpendicular to $\bm{B}$; $\rho_{s}=v_{\text{th},s}/\Omega_{s}$ is the thermal gyroradius with $v_{\text{th},s}=\sqrt{2T_{s}/m_{s}}$ and $\Omega_{s}=|q_{s}|B/m_{s}c$ the ion thermal speed and gyrofrequency; $q_{s}=Ze$ and $m_{s}=Am_{\text{p}}$ are the ion charge and mass in units of proton quantities; $\beta_{s}=v^{2}_{\text{th},s}/v_{\text{A}}^{2}$ is the ion beta; $c$ is the speed of light; $\delta u_{\lambda}$ is the rms $\bm{E}\times\bm{B}$ velocity at scales $k_{\perp}\lambda\sim 1$; and $v_{\text{A}}$ is the Alfvén speed.

Turbulence Theory.—Turbulence near the Sun is imbalanced, with $|\bm{z}^{+}|\gg|\bm{z}^{-}|$. Recent work [25, 26, 27] argues that such turbulence forms a “helicity barrier”, which, due to its effect on turbulent fluctuations near $\rho_{\text{p}}$-scales, is a key ingredient in understanding ion heating. The effect occurs because low-$\beta$ gyrokinetics conserves energy and a “generalized helicity” [28, 25], which must be simultaneously cascaded with constant flux to be in a statistical steady-state. At scales larger than $\rho_{\text{p}}$, helicity is the cross-helicity which cascades to small perpendicular scales. At scales smaller than $\rho_{\text{p}}$, helicity becomes magnetic helicity, which would inverse cascade in imbalanced turbulence. For balanced turbulence, with zero helicity, energy is able to cascade to small perpendicular scales. However, in imbalanced turbulence only the balanced (zero-helicity) portion of the energy cascade is let through to small perpendicular scales; the remainder is blocked from cascading and trapped at scales above $\rho_{\text{p}}$ by the helicity barrier. This causes the amplitude of the turbulence to grow in time and, through critical balance [29], reach and dissipate at small parallel scales. Hybrid-kinetic imbalanced turbulence simulations [26] show that fluctuations at these scales become oblique ion-cyclotron waves, heating ions through quasi-linear interactions and allowing the turbulence to saturate. The helicity barrier is well supported observationally, explaining the steep “transition-range” drop in fluctuation spectra [30, 31] and other features [32, 33].

Ion Heating.—Two widely studied ion-heating mechanisms are stochastic heating and cyclotron-resonant quasi-linear diffusion. Stochastic heating involves ions being heated by uncorrelated kicks from fluctuations at scales $k_{\perp}\rho_{\text{i}}\sim 1$. This leads to a diffusion perpendicular to $\bm{B}$ in velocity space, generally heating the plasma [34, 35, 10]. The theory explains ion heating well in low-$\beta$ balanced turbulence simulations [19, 36, 21, 22], and has also been applied to ion heating within the imbalanced near-Sun environment [37, 38, 39, 40, 41]. Quasi-linear theory predicts that ions and waves interact strongly if the waves’ Doppler-shifted frequencies are resonant with the ion gyrofrequency [42, 43, 44]. This causes ions to diffuse ions in velocity space along contours of constant energy in the frame moving at the wave’s phase velocity, heating the plasma (however, note that ions may also be heated by non-resonant interactions with waves [45, 35]).

While traditionally considered as separate mechanisms, we argue that SH and QLH should instead be considered as two limits of a continuum controlled by the nonlinear broadening of the fluctuations’ frequency spectrum. Unbroadened fluctuations—waves with a single frequency $\omega$ at a particular $\bm{k}$—cause QLH-like behavior; fluctuations that are broadened to a level comparable to $\omega$ itself, which thus have significant power in low-frequency fluctuations [46], cause SH-like behavior. This reveals a deep connection to the turbulence imbalance: in imbalanced turbulence $\bm{z}^{+}$-fluctuations have linear frequencies that exceed their nonlinear rate (the QLH-limit); in balanced turbulence the two are comparable [29] (the SH-limit). A smooth transition between these limits occurs as the imbalance is adjusted.

To further illustrate the connection, we argue that SH and QLH rates scale similarly with turbulent amplitude, even though SH is traditionally discussed in the context of low-frequency turbulence, while QLH is generally considered to involve high-frequency fluctuations ($\omega\sim\Omega_{\text{i}}$). SH requires turbulent amplitudes $\delta u_{\rho_{\text{i}}}$ comparable to $v_{\text{th,i}}$ (generally $\delta u_{\rho_{\text{i}}}/v_{\text{th,i}}\gtrsim 0.1$ [10, 19]). To understand the amplitude at which QLH will be activated, we use the principle of “propagation critical balance” (in which parallel scales are determined by the wandering of magnetic field lines by perpendicular perturbations [47, 48, 46]), giving $\omega\sim k_{\|}v_{\text{A}}\sim k_{\perp}\delta u_{\lambda}$ throughout the inertial range. Thus, in order to reach $\omega\sim\Omega_{i}$ and activate QLH requires $\Omega_{i}\sim\delta u_{\rho_{\text{i}}}/\rho_{i}\implies\delta u_{\rho_{\text{i}}}\sim v_{\text{th,i}}$, where we used the fact that the ion-eddy interaction is strongest at $\rho_{\text{i}}$ scales because $\omega$ increases with $k_{\perp}$ but interactions with $k_{\perp}\rho_{\text{i}}>1$ fluctuations are suppressed [49] (there may still be contributions from such eddies, however [21, 14]). This means QLH requires similar turbulent amplitudes as SH to occur; likewise, for critically balanced turbulence, the amplitudes needed to cause SH imply turbulent frequencies that approach $\Omega_{\text{i}}$, although due to eddies decorrelating, there remains significant power in $\omega<\Omega_{i}$ fluctuations in balanced turbulence [46]. The argument also implies that heating is strongest at the smallest parallel scale $l_{\|}\sim 1/k_{\|}\sim v_{\text{A}}/\Omega_{\text{i}}$ with $\lambda\lesssim\rho_{\text{i}}$.

Based on the argument above, we propose to extend the usual SH formula for the ion heating rate per unit mass [10],

to also describe QLH in imbalanced turbulence. Here, $\hat{c}_{1}$ and $\hat{c}_{2}$ are empirical constants and the exponential suppression factor accounts phenomenologically for the conservation of magnetic moment at small amplitudes and/or the exponentially small number of particles resonant with lower-frequency fluctuations (in reality, this suppression factor presumably depends on imbalance and other parameters). A key change, however, is needed to correctly describe the helicity barrier, with the definition of $\xi_{\text{i}}$ extended from its usual one ($\xi_{\text{i}}=\delta u_{\rho_{\text{i}}}/v_{\text{th,i}}$) to instead capture the scale at which the fluctuations reach their highest frequency. Thus $\hat{\xi_{\text{i}}}\equiv\delta u_{\lambda}/v_{\text{th,i}}$, where $\delta u_{\lambda}$ is the turbulent amplitude at perpendicular scales $\lambda\gtrsim\rho_{\text{i}}$ corresponding to the smallest parallel scale $l_{\|}$, estimated from the maximum of $\delta u_{\lambda}/\lambda$ via propagation critical balance: $v_{\text{A}}/l_{\|}\sim\delta u_{\lambda}/\lambda\sim k_{\perp}\tilde{z}^{+}_{k_{\perp}}$. Here, $\tilde{z}^{+}_{k_{\perp}}$ is the rms value at scales $k_{\perp}\lambda\sim 1$ of $\tilde{\bm{z}}^{+}$, the generalized Elsässer eigenmode of the FLR-MHD model (introduced below), which reduces to its MHD counterpart $\bm{z}^{+}$ at $k_{\perp}\rho_{\text{i}}\ll 1$.

For turbulence without a helicity barrier (such as balanced turbulence or imbalanced RMHD), the energy cascade is able to dissipate at perpendicular scales smaller than $\rho_{\text{p}}$, forming a smooth spectrum with continuously increasing $\delta u_{\lambda}/\lambda$. Based on the argument above, we expect ions to be heated by fluctuations at scales $\lambda\sim\rho_{\text{p}}$, and Eq. (1) reduces to the normal SH formula with $\hat{\xi}_{\text{i}}=\delta u_{\rho_{\text{i}}}/v_{\text{th,i}}$. However, with a helicity barrier, the resulting steep spectral break causes $k_{\perp}\tilde{z}^{+}_{k_{\perp}}$ to peak at an intermediate scale $\lambda>\rho_{\text{p}}$ before dropping off steeply at smaller scales. We thus use $\lambda\sim\text{max}(\rho_{\text{i}},1/k^{\text{nl}}_{\perp})$ where

the average value of $k_{\perp}$ weighted by $k_{\perp}\tilde{z}^{+}_{k_{\perp}}$ over scales $k_{\perp}\rho_{\text{i}}\leq 1$, is used as a proxy for the scale at which the heating is most efficient. This also takes into account that minor ions interact with fluctuations at scales larger than the spectral break, due to their larger gyroradii.

To verify this proposal, we now present numerical simulations of test protons and minor ions interacting with both balanced and imbalanced turbulence with or without a helicity barrier.

Numerical Setup.—We use the “finite Larmor radius MHD” (FLR-MHD) model [50, 28, 25], which can be formally derived from gyrokinetics in the limit $\beta\ll 1$ and $k_{\perp}d_{\text{e}}\gg 1$ (where $d_{\text{e}}$ is the electron inertial length). At $k_{\perp}\rho_{\text{i}}\ll 1$, FLR-MHD reduces to the well-known RMHD model [51]; for $k_{\perp}\rho_{\text{i}}\gg 1$ it reduces to “electron RMHD” [16, 52, 22]. The FLR-MHD equations are advanced in time using a pseudospectral method [53, 25] (see Suppl. Mat.). The simulation domain is a three-dimensional periodic grid of size $L_{\perp}=L_{z}=2\pi$ and a resolution of $N_{\perp}^{2}\times N_{z}$, where the background magnetic field is along the $z$-direction and $\perp$ represents the $x$- and $y$-directions. To test our proposed heating model across a wide range of regimes, we simulate three cases: balanced and imbalanced FLR-MHD with $\rho_{\text{p,gk}}/L_{\perp}=0.02$, and imbalanced RMHD (we assume a majority hydrogen plasma, with $\rho_{\text{p,gk}}$ the $\rho_{\text{i}}$ in the gyrokinetic model). These simulations have a resolution of $N_{\perp}^{2}\times N_{z}=1024^{2}\times 1024$ for the balanced FLR-MHD case and $1024^{2}\times 512$ for the other two cases, refined from an initial resolution of $256^{2}\times 512$ to obtain spectra that extend to scales smaller than $\rho_{\text{p,gk}}$ (see Suppl. Mat.). In the imbalanced FLR-MHD simulations we run ions at three different stages of the helicity barrier’s evolution as it grows, at times $tv_{\text{A}}/L_{z}\approx 3,6,$ and 10.

Once the refined turbulence simulations have reached a quasi-steady-state, we introduce test ions (for details on implementation, see Suppl. Mat.). For a given collection of ions, we can choose the initial $\rho_{\text{i}}/L_{\perp}$, $\beta_{\text{i}}$, and $\delta u_{\rho_{\text{i}}}/v_{\text{th,i}}$ [19]. For FLR-MHD, the choice of $\rho_{\text{i}}/L_{\perp}$ is fixed to match $\rho_{\text{p,gk}}/L_{\perp}=0.02$; we also choose this value for ions in RMHD simulations for comparison. This scale determines $\delta u_{\rho_{\text{i}}}$ via $\delta u_{\rho_{\text{i}}}^{2}=2\int_{k^{-}}^{k^{+}}\text{d}k_{\perp}\ \mathcal{E}_{\bm{E}}(k_{\perp})$, where $\mathcal{E}_{\bm{E}}(k_{\perp})$ is the perpendicular electric-field spectrum (equivalent to the $\bm{E}\times\bm{B}$ velocity spectrum in FLR-MHD) and using $k^{\pm}\equiv e^{\pm 1/2}/\rho_{\text{i}}$ in line with previous work [10, 19]. This value of $\delta u_{\rho_{\text{i}}}$ with the choice of $\delta u_{\rho_{\text{i}}}/v_{\text{th,i}}$ determines $v_{\text{th,i}}$. Due to the invariance of the quantity $v_{\text{A}}\nabla_{\|}$ in both RMHD and FLR-MHD, we are free to choose $v_{\text{A}}$ based on the values of $v_{\text{th,i}}$ and $\beta_{\text{i}}$ so long as length-scales along the background field are scaled by the same factor [46]; this rescaling is done within the ion integrator. Once these quantities are determined, the ions are uniformly distributed throughout the simulation domain and given an isotropic Maxwellian velocity distribution. For a given choice of $\rho_{\text{i}}/L_{\perp}$ and $\beta_{\text{i}}$, we initialize separate cohorts of $N=10^{6}$ particles with $0.01\lesssim\delta u_{\rho_{\text{i}}}/v_{\text{th,i}}\lesssim 0.3$.

We introduce minor ions with mass $m_{\text{i}}=Am_{\text{p}}$ and charge $q_{\text{i}}=Ze$ by taking $v_{\text{th,i}}$ to be the same of that of protons within the same simulation, so that they have similar initial $\delta u_{\rho_{\text{i}}}/v_{\text{th,i}}$. This choice, along with $\Omega_{\text{i}}=(Z/A)\Omega_{\text{p}}$, determines the required ion gyroradius $\rho_{\text{i}}=(A/Z)\rho_{\text{p}}$ corresponding to $\rho_{\text{p}}/L_{\perp}$. Once this is determined, the process of initializing minor ions is identical to protons. Table 1 summarizes the turbulence and particle parameters studied in this Letter.

We note that because FLR-MHD is derived from gyrokinetics it can only simulate low-frequency dynamics, and does not contain the transition to ion-cyclotron waves. Although this still allows for QLH of protons by Alfvénic fluctuations, it does not capture their dispersion or polarization changes at $k_{\|}d_{\text{i}}\sim 1$ (where $d_{\text{i}}$ is the ion inertial length; see Ref. [26] for the effects of ion-cyclotron waves). This approximation is less impactful for minor ions, because their smaller gyrofrequency means they interact with lower-frequency fluctuations.

Results.—To highlight how imbalance affects the frequency spectrum of fluctuations, which was argued above to be important for controlling the heating, the left-hand plot of Fig. 1 shows the space-time Fourier transform $\mathcal{E}^{\rm tot}(k_{z},\omega)=\mathcal{E}^{+}(k_{z},\omega)+\mathcal{E}^{-}(k_{z},\omega)$, with $\mathcal{E}^{\pm}(k_{z},\omega)=\frac{1}{2}\left\langle|\tilde{\bm{z}}^{\pm}(k_{z},k_{\perp},\omega)|^{2}\right\rangle_{\perp}$; here, $\langle\cdot\rangle_{\perp}$ denotes an average over all $k_{\perp}$. In balanced turbulence, $\mathcal{E}^{\rm tot}(k_{z},\omega)$ has significant power at small $\omega$, a signature that the nonlinear and linear times are comparable [46]. In contrast, imbalanced turbulence is dominated by fluctuations at a particular frequency, as seen by the band peaked around $\omega\approx-k_{z}v_{\text{A}}$. These show that increasing imbalance reduces the nonlinear broadening of fluctuations relative to their linear frequency, causing the ion heating to transition from a SH-like to QLH-like mechanism.

The electric-field spectra $\mathcal{E}_{\bm{E}}(k_{\perp})$ and $k_{\perp}\tilde{z}^{+}_{k_{\perp}}$ for different turbulence simulations are compared on the right of Fig. 1, where we compute $k_{\perp}\tilde{z}^{+}_{k_{\perp}}$ from its spectrum: $k_{\perp}\tilde{z}^{+}_{k_{\perp}}\approx k_{\perp}\sqrt{k_{\perp}\mathcal{E}^{+}(k_{\perp})}$ [54]. In contrast to the wide inertial range of the imbalanced RMHD simulation, FLR-MHD captures the transition from an Alfvén-wave to kinetic-Alfvén-wave cascade [16], with the electric-field specturm approaching a $-1/3$ scaling at $k_{\perp}\rho_{\text{i}}\gtrsim 1$. The presence of a helicity barrier is clearly seen in the imbalanced FLR-MHD simulations via the steep drop in spectra at $k_{\perp}\rho_{\text{p}}\lesssim 1$, as observed in the solar wind [55, 31] and previous simulations [25, 26, 27]. The steep spectral slope of the helicity barrier also causes $k_{\perp}\tilde{z}^{+}_{k_{\perp}}$ to reach a maximum at $k_{\perp}\rho_{\text{p}}\lesssim 1$ scales. This maximum, and the corresponding spectral break in $\mathcal{E}_{\bm{E}}(k_{\perp})$, shifts to larger scales as the energy grows in time.

Figure 2 illustrates how the ion heating rate $Q_{\perp}$ is calculated from the evolution of the ensemble-averaged perpendicular energy per unit mass $\langle w^{2}_{\perp}\rangle$ for protons from $\text{F-6}\beta_{\text{p}}0.05$ with $0.02\leq\delta u_{\rho_{\text{p}}}/v_{\text{th,p}}\leq 0.2$. Here $w_{\perp}$ is the component of $\bm{w}\equiv\bm{v}-\bm{u}_{\bm{E}\times\bm{B}}$, the thermal velocity of the protons after subtracting the local $\bm{E}\times\bm{B}$ velocity, perpendicular to $\bm{B}$. Protons experience an initial heating transient lasting around one gyroperiod as they “pick up” the local $\bm{E}\times\bm{B}$ velocity (also seen in Ref. [19]). Because of this, we calculate $Q_{\perp}$ by a linear fit to $\langle w^{2}_{\perp}\rangle$ via $Q_{\perp}=0.5(\langle w^{2}_{\perp\text{,f}}\rangle-\langle w^{2}_{\perp\text{,}0}\rangle)/(t_{\text{f}}-t_{0})$. Here $t_{\text{f}}$ is either the end of the simulation or when $\langle w^{2}_{\perp\text{,f}}\rangle=1.2\langle w^{2}_{\perp\text{,}0}\rangle$; we choose this as $Q_{\perp}$ decreases as $\langle w^{2}_{\perp}\rangle$ increases [10, 19]. For $\delta u_{\rho_{\text{p}}}/v_{\text{th,p}}\lesssim 0.05$, the initial heating transient exhibits oscillations, so to remove the effect of these from the measurement, we choose the start time $t_{0}\Omega_{\rm i}/2\pi=5.25$ for $\delta u_{\rho_{\text{i}}}/v_{\text{th,i}}<0.02$, $3.25$ for $0.02\leq\delta u_{\rho_{\text{i}}}/v_{\text{th,i}}\leq 0.05$, and $0.75$ for $\delta u_{\rho_{\text{i}}}/v_{\text{th,i}}>0.05$ (see Suppl. Mat.).

Figure 3 compares the dependence of $Q_{\perp}$ on $\delta u_{\rho_{\text{i}}}/v_{\text{th,i}}$, the measure commonly used in SH theory, to our proposed definition $\hat{\xi}_{\text{i}}=\delta u_{\lambda}/v_{\text{th,i}}$. We calculate $k^{\text{nl}}_{\perp}$ in Eq. (2) using $k_{\perp}\tilde{z}^{+}_{k_{\perp}}$ from Fig. 1. It is clear that the standard normalization is inadequate for describing $Q_{\perp}$, particularly for cases with a helicity barrier. These show greater heating than expected from the power in fluctuations at these scales, indicating the true $\hat{\xi}_{\text{i}}$ is being underestimated. Minor ions show similar heating rates in both imbalanced FLR-MHD and RMHD, due to their gyroradii lying at scales larger than $\rho_{\text{p}}$ and close to or above $1/k^{\text{nl}}_{\perp}$. In contrast, when plotted against our proposed $\hat{\xi}_{\text{i}}$, the data are well described by Eq. (1) with best fit parameters $\hat{c}_{1}=1.11,\hat{c}_{2}=0.10$ (fits of individual simulations to Eq. (1) comparing $\delta u_{\rho_{\text{i}}}/v_{\text{th,i}}$ and $\hat{\xi}_{\text{i}}$ are listed in Table 1); our value of $\hat{c}_{2}$ is smaller than previous test particle simulations [10, 19] but similar to recent hybrid-kinetic simulations [22]. That Eq. (1) holds despite the difference in simulation conditions, from turbulence that is balanced, imbalanced, with or without a helicity barrier, and for both protons and minor ions, helps verify our unified model of ion heating. We note that the worst argreement is seen for imbalanced RMHD, which is also unphysical, neglecting the effect of $\rho_{\text{p}}$ on the turbulence.

Despite the success of the unified model in reproducing $Q_{\perp}$, there remains important differences between balanced- and imbalanced-turbulent heating, as demonstrated in Fig. 4 which compares the proton velocity distribution functions (VDFs) $f(w_{\perp},w_{\|})$ in these two regimes. In balanced turbulence, the VDF undergoes diffusion in perpendicular velocity, as seen in the 1D VDF $f(w_{\perp})=\int\text{d}w_{\|}f(w_{\perp},w_{\|})$ which has a characteristic flattopped shape [38, 22]. In contrast, in imbalanced turbulence the VDF diffuses along constant-energy contours within the frame of the waves, a hallmark of QLH [42]. For Alfvénic fluctuations, these contours are semicircles centered on the phase speed $v_{\text{ph}}=\omega/k_{\|}=-v_{\text{A}}$. The 1D VDF exhibits a flattopped shape similar to the balanced case (dropping off at slightly earlier upon close inspection), presumably because the contours are nearly vertical at small $w_{\perp}$. This picture also allows an understanding of the $\beta_{\text{p}}$-dependent dropoff in $Q_{\perp}$ seen at large $\hat{\xi}_{\text{p}}$ in Fig. 3 (representing the largest deviation from the phenomenological model): the constant-energy contours become increasingly curved as $\beta_{\text{p}}$ increases, where $v_{\text{A}}$ approaches $v_{\text{th,p}}$, causing ions to gain parallel energy at the expense of perpendicular energy (see the Suppl. Mat.).

Conclusion.—In this Letter we provide a unified picture for describing ion heating in collisionless plasma turbulence across a wide range of regimes. The heating mechanism changes character depending on the frequency spectrum of turbulent fluctuations, transitioning from SH to QLH as the turbulence imbalance increases and reduces the relative nonlinear broadening of fluctuations. We propose a phenomenological model, controlled by the amplitude of fluctuations at the scale where $k_{\perp}\tilde{z}^{+}_{k_{\perp}}\sim v_{\text{A}}/l_{\|}$ peaks for $k_{\perp}\rho_{\text{i}}\lesssim 1$, which accurately captures the ion-heating rate across both regimes. In the absence of a transition-range break or for minor ions, this scale is usually at $k_{\perp}\rho_{\text{i}}\sim 1$ and our model reduces to the standard SH formula; however, with a transition range, caused by the helicity barrier, this scale can lie above at $k_{\perp}\rho_{\text{i}}<1$. Using high-resolution simulations of test particles interacting with turbulence, we show that our proposal works well in describing measured heating rates for both protons and minor ions in turbulence that is balanced, imbalanced, with or without a helicity barrier. The fact that this holds, despite the very different ‘arced’ VDFs that result in imbalanced turbulence, hints at the universality of the heating mechanism.

We note that the FLR-MHD model, on which our results are based, is highly simplified, missing out the transition to ion-cyclotron waves when $\omega\sim\Omega_{\text{i}}$. Additionally, we also only measure the instantaneous ion $Q_{\perp}$ from an initial Maxwellian, rather than tracking large changes of the velocity distribution function (as occurs for strong heating). Both of these issues have been addressed in a recent hybrid-kinetic study [56], which shows similar features. Nonetheless, the results of this Letter help to disentangle the signatures of ion heating and provide a base for more complex theories. The ideas presented here can also be used to understand helicity-barrier saturation on ion heating [26, 27], which is not captured by FLR-MHD and may be important for ab-initio models of coronal and solar-wind heating.

The authors would like to thank T. Adkins, B. D. G. Chandran, M. W. Kunz, N. F. Loureiro, M. Zhang, and M. Zhou for helpful discussions over the course of this work. Research is supported by the University of Otago, through a University of Otago Doctoral Scholarship (ZJ), and the Royal Society Te Apārangi, through Marsden-Fund grant MFP-UOO2221 (JS) and MFP-U0020 (RM), as well as through the Rutherford Discovery Fellowship RDF-U001004 (JS). High-performance computing resources were provided by the New Zealand eScience Infrastructure (NeSI) under project grant uoo02637.