Kink instability: evolution and energy dissipation in Relativistic Force-Free Non-Rotating Jets

Alexander A. Philippov; Chandra B. Singh; Jordy Davelaar; Omer Bromberg

REVIEW 4 major objections 5 minor 1 cited by

Kink instability: evolution and energy dissipation in Relativistic Force-Free Non-Rotating Jets

T0 review · 4 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Kink instability in relativistic jets dissipates magnetic energy and drives the plasma toward a force-free Taylor state.

desk verdict A genuinely useful simulation study of kink-driven dissipation, but the Taylor-state minimal-energy claim is not as clean as the abstract suggests. read the letter →

arxiv 1908.08620 v2 pith:SNQEH5NZ submitted 2019-08-22 astro-ph.HE

Omer Bromberg , Chandra B. Singh , Jordy Davelaar , Alexander A. Philippov This is my paper · ORCID

classification astro-ph.HE

keywords kinkinstabilityrelativisticjetsmagneticreconnectionforce-freeplasmaTaylorrelaxationenergydissipationMHDcurrentsheets

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper uses three-dimensional relativistic MHD simulations to establish that kink instability in a magnetized, force-free, non-rotating plasma column is an efficient way to dissipate magnetic energy. The instability grows as a helical $m=-1$ kink mode, compresses current sheets between its lobes, and drives magnetic reconnection that converts field energy into heat at a rate $dU_{B_\varphi}/dt \approx -0.1\,U_{B_\varphi}/\tau$, where $\tau \approx 20\pi P_0/v_A$ is the linear growth time. The simulations show the column then relaxes to a state close to a force-free Taylor state, with the pitch parameter at the marginal-stability value $\alpha R_j = 3.176$. In the increasing-pitch case about 50% of the initial magnetic energy is available for dissipation and about 40% is dissipated by the end of the run, implying a near-minimal-energy relaxed state. These results matter because they give quantitative expectations for how and how fast relativistic jets, twisted coronal loops, and magnetar magnetospheres can lose magnetic energy.

What carries the argument

The argument runs on the pitch profile $P(r)=rB_z/B_\varphi$, which decides whether a resonant surface exists for the fastest-growing $m=-1$ kink mode. With increasing pitch the resonant surface creates an internal kink whose lobes inverse-cascade through mergers, pumping energy into turbulence; with decreasing pitch there is no resonant surface and the field breaks apart more violently. The relaxed state is modeled as a Taylor state, a force-free configuration with $\mathbf{j}=\alpha\mathbf{B}$ for constant $\alpha$, whose cylindrical form is $B_z=B_0J_0(\alpha r)$ and $B_\varphi=B_0J_1(\alpha r)$. The final energy is obtained from conservation of the zero-gauge helicity $K$ and axial flux $\Psi$, closed by the linear stability bound $\alpha R_j=3.176$. This machinery converts the turbulent, three-dimensional relaxation problem into a small set of algebraic equations for the final field strength, twist, and radius.

What would settle it

Compute the helicity $K$ and axial flux $\Psi$ inside the dissipation radius $R_j$ as functions of time; if $K$ drops by more than about 10 percent in a way that does not vanish when the box is enlarged or resolution increased, the Taylor-state energy closure fails. Similarly, if a longer, higher-resolution run of the increasing-pitch case does not asymptote to $\alpha R_j \approx 3.176$ with roughly half the initial energy dissipated, the marginal-stability energy budget is wrong.

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Extended reading notes

Core claim

The central discovery is that the kink instability in a relativistic force-free column dissipates magnetic energy through reconnection that is driven by compression of the growing kink lobes, and that the system relaxes to a force-free Taylor state. The measured toroidal-field dissipation rate is $dU_{B_\varphi}/dt \approx -0.1\,U_{B_\varphi}/\tau$, with growth time $\tau \approx 20\pi P_0/v_A$, consistent with the sideways expansion velocity of the kink mode pushing field lines together. In the increasing-pitch configuration, the final state has $\alpha R_j$ close to 3.176, the marginal-stability value from linear theory, and the energy budget indicates that roughly half the initial energy is available for dissipation with about 40 percent already dissipated. In the decreasing-pitch configuration the dissipation is faster and larger (60 percent dissipated, about 75 percent available), so the system is still relaxing. The paper argues that helicity and axial flux are conserved to about 10 percent and that these conserved quantities, together with the stability criterion, close the equations that predict the final energy of the relaxed state.

Load-bearing premise

The energy budget assumes the final, turbulent 3D magnetic field can be averaged into a smooth cylindrical profile with a single constant twist parameter, and that the 10% of helicity lost through the outflow boundaries is only leakage, not a sign that the conserved quantities are changing.

Editorial extensions

If this is right

Kink-unstable regions of relativistic jets lose their toroidal magnetic field on a time scale of several growth times, set by $dU_{B_\varphi}/dt\approx -0.1\,U_{B_\varphi}/\tau$.
Magnetic dissipation stops when the configuration approaches the marginal Taylor state, so a stable, partially magnetized core remains rather than complete destruction of the field.
Pitch profile controls the outcome: increasing-pitch columns dissipate about 40 percent of the initial energy in the simulated boxes, while decreasing-pitch columns dissipate about 60 percent and do so faster.
In GRB jets, efficient kink-driven dissipation near the collimation nozzle could be tied to the observed duration of the prompt emission, because the cocoon pressure drop after breakout changes the collimation conditions.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the relaxed state is generically a marginal Taylor state, the final magnetic energy of a kink-unstable column could be predicted from initial helicity and flux alone, giving a subgrid prescription for large-scale jet simulations that cannot resolve current sheets.
The same conserved-helicity closure could be tested on rotating jets; the paper notes that rotation can stabilize the column, so whether the Taylor-state end point survives rotation and shear is an open question.
The measured dissipation rate places the main energy release in the merger/inverse-cascade phase rather than in the initial linear growth, which suggests that time-resolved variability observations could be compared with the duration of this phase to infer the pitch profile.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Omer, you should know two things right away. This is the first systematic quantitative measurement of nonlinear kink dissipation in the relativistic force-free regime, and the three-stage picture—linear growth, merger inverse cascade, turbulent relaxation—is convincing. But the claim that the relaxed state is close to a Taylor minimal-energy state is shakier than the abstract suggests.

The paper does a lot well. Three pitch profiles (increasing, decreasing, coronal), two box sizes each, resolution tests from 10 to 45 cells per unit a. The linear growth rates match the analytic scalings. The dissipation mechanism is pinned down: current sheets compressed by growing kink lobes, with pressure peaks tracking the current filaments. The measured toroidal-field dissipation rate, dU_Bphi/dt ≈ -0.1 U_Bphi/τ, is a useful number. Citation pattern is solid: Appl et al. (2000) and the Taylor/Kadomtsev relaxation theory are properly credited. The GRB collimation application is speculative but flagged.

Now the soft spots, in proportion. The big one is the Sec. 6.4 energy budget. The relaxed field is not axisymmetric—Fig. 10 shows the Bz reversal is localized, and azimuthal averaging washes it out—yet the estimate assumes an axisymmetric Bessel Taylor state. Helicity K is conserved to only ~10%, and the loss is attributed to boundary leakage without being demonstrated. Since the final energy scales roughly as K²/Ψ², a 10% error in K moves the predicted final energy by ~20%, larger than the 10-point gap between the 50% available and 40% dissipated in the IP case. So 'close to minimal energy' is not cleanly supported. The Bessel fits in Fig. 9 help, but they are azimuthally averaged profiles, not a test of whether the 3D field has single-alpha structure.

Second, the dissipation-rate normalization is confusing. The abstract calls τ the linear growth time, but eq. 26 gives τ ≈ 20πP0/vA, about eight times the e-folding time from eq. 3 (1/Λmax ≈ 7.5P0/vA). Define it clearly or quote the rate per unambiguous time.

Third, a real but minor typo: eq. 16 is missing the ξ factor; the correct expression is eq. 38 in Appendix A.

Who should read this? People working on magnetic dissipation in jets, GRBs, or accretion-disk coronae. The core numerical results—mechanism and dissipation rate—look robust and deserve a serious referee. The energy-budget section needs reworking: quantify how non-axisymmetry and K uncertainty propagate into the available-energy estimate, or soften the minimal-energy conclusion. Send it to review; with those fixes it becomes a solid contribution.

Referee Report

4 major / 5 minor

Summary. The paper presents 3D relativistic MHD simulations of the kink instability in force-free, non-rotating plasma columns, considering increasing-pitch (IP), decreasing-pitch (DP), and coronal (CO) configurations. It identifies magnetic reconnection in compressed current sheets as the principal dissipation mechanism, measures a toroidal-field dissipation rate dU_Bφ/dt ≈ −0.1 U_Bφ/τ with τ ≈ 20π P0/v_A, finds that the relaxed state is close to a force-free Taylor state with αR_j near 3.176 for the IP and DP cases, and estimates that about 50% of the initial energy is available for dissipation in the IP case, with 40% actually dissipated by the end of the simulation. The paper also derives constraints on kink-driven dissipation in relativistic jets and twisted coronal loops.

Significance. If the quantitative results hold, this paper provides the first detailed relativistic MHD characterization of the nonlinear kink instability's energy dissipation, with direct implications for magnetic energy conversion in GRB jets, AGN jets, and coronal loops. The reported linear growth rates match analytic theory, the convergence tests show that the dissipation rates and energies are numerically converged, and the independent Bessel-function fits to the final magnetic profiles provide non-circular evidence for Taylor relaxation. These elements give the paper substantive value despite the concerns discussed below.

major comments (4)

[Sec. 8 / Eq. (26), Sec. 2 Eq. (3)] Equation (26) defines τ ≈ 20π P0/v_A as 'the growth time of the linear instability,' but the linear growth rate quoted in Eq. (3) is Λ_max = 0.133 v_A/P0, giving an e-folding time of approximately 7.5 P0/v_A. The factor in Eq. (26) is about 8.4 times larger. Because the abstract and Sec. 8 use τ to normalize the dissipation rate, this inconsistency makes the headline quantitative result ambiguous. Please either correct the numerical factor, or define τ as the nonlinear-dissipation timescale and derive its value from the simulations rather than calling it the linear growth time.
[Sec. 4, Eq. (16) vs. Appendix A, Eq. (38)] The two expressions for Υ(R_j) disagree: Eq. (16) contains the integrand [J_0(ξ)^2+J_1(ξ)^2] dξ without the factor ξ, while the derivation in Appendix A (Eqs. 34 and 38) requires the factor ξ in the integrand. As printed, Eq. (16) is dimensionally inconsistent because the boundary term J_0(αR_j)J_1(αR_j)R_j has units of length if α has units of inverse length, whereas the integral is dimensionless. Since the final-energy estimates in Sec. 6.4 depend on Eqs. (14) and (15) through Υ, the correct form of Υ must be stated and used consistently throughout the manuscript.
[Sec. 6.4 and Sec. 6.3, Fig. 10] The claim that the system is close to a minimal-energy Taylor state is not cleanly supported by the current analysis. The final state is non-axisymmetric: Fig. 10 shows a localized B_z reversal that azimuthal averaging washes out. Additionally, the zero-gauge helicity K is conserved only to ~10% in all configurations, with the drop attributed to boundary leakage but not quantitatively verified. In the IP case the prediction of 50% available energy versus 40% actually dissipated is separated by only 10 percentage points, comparable to the stated uncertainty in K. A sensitivity test (e.g., recomputing the Taylor-state energy using K ± 10%) or an independent measure of how well the final field minimizes the energy would be needed to make this conclusion robust.
[Sec. 6.4 / Fig. 7] The dissipation radius R_j is used to close the system (Eqs. 14 and 15) and to compute the available energy, but the criterion by which R_j is measured from the simulations is never stated. The final results (αR_j, available/dissipated energy fractions) depend on this choice. Please define R_j operationally (e.g., the radius where the azimuthally averaged B_z changes sign, or where the current density falls below a threshold) and assess the sensitivity of the conclusions to that definition.

minor comments (5)

[Abstract] 'We constraint the energy' should read 'We constrain the energy.'
[Sec. 4] In the sentence 'A third condition can come comes from constraining the final α,' the word 'comes' is duplicated; remove the second occurrence.
[Sec. 5] Please define η_N explicitly as a random number drawn from a uniform distribution in [−1, 1] before using it in the velocity perturbation expression.
[Sec. 6.3] In the sentence 'In the case of the IP case this is partly due to the averaging,' the phrase 'In the case of the IP case' is redundant; 'In the IP case' suffices.
[Fig. 5] The inset comparing the growth rates is very small and difficult to read; consider a separate panel or a larger inset for clarity.

Circularity Check

1 steps flagged · score 2.0 of 10

No significant circularity; the Taylor-state energy check is a conditional consistency test with independent Bessel-fit support.

self definitional [The relevant passage is in Section 6.4, around eqs. 14-15 and Fig. 11.]
"Thus, eqs. 14 and 15 can be used to evaluate the final energy in the box, assuming the system has relaxed to an axially symmetric Taylor state. To close the equations we take Rj at the end of each simulation and calculate the values of α and B0 of the corresponding Taylor state. We then compare the EM energy of the Taylor state to the actual EM energy in the box and evaluate how close the system is to a minimal energy."

The final-energy estimate is constructed from the very ansatz being tested: the relaxed field is assumed to be the axisymmetric Taylor state (eq. 7), and conservation of K and Ψ then fixes B0 and α. E_Taylor is therefore the energy of the assumed minimal-energy state by construction, so comparing it with the simulated final energy is a consistency check of that ansatz rather than an independent derivation that the system is close to a minimal-energy state. The genuinely independent support comes from the separate Bessel-function fits in Fig. 9 and the flat α profile in Fig. 8; without those, the reported 50%-available versus 40%-dissipated comparison would not by itself establish closeness to the Taylor state.

full rationale

The central dissipation claim (dU_Bφ/dt ≈ −0.1 U_Bφ/τ) is a direct numerical measurement compared with an independent linear-theory growth rate and a reconnection-velocity estimate, not a fitted prediction. The linear stability thresholds come from Voslamber and Callebaut (1962) and the relaxation framework from Taylor (1974); these are external results, not self-citations. The only self-citation in the jet-implications section, Bromberg and Tchekhovskoy (2016) for θ_diss, is not load-bearing for the simulation result. The one mild self-consistency issue is the final-energy estimate in Section 6.4, where the Taylor-state energy used to define the available dissipation is computed from the assumed final ansatz plus conserved K and Ψ, making the 50%-versus-40% comparison a consistency check rather than an independent proof of proximity to the minimal-energy state. The paper's own caveats, namely that azimuthal averaging washes out the Bz reversal in the IP case (Sec. 6.3, Fig. 10) and that K is conserved only to about 10% (Sec. 6.4), weaken the energy-budget test but are not circular reductions. The apparent missing ξ factor in eq. (16) relative to Appendix A eq. (38) is an internal-consistency and correctness issue, not a circularity. Overall, the derivation is self-contained and externally benchmarked; no step reduces to its own input by construction.

Assumptions & free parameters 3 free parameters · 7 assumptions · 0 invented entities

The central results rest on the validity of Taylor relaxation theory and the related stability thresholds in the relativistic high-sigma regime, on the representativeness of the chosen equilibrium profiles, and on the assumption that numerical resistivity captures reconnection physics. These are standard domain assumptions in astrophysical MHD, but they are load-bearing for the energy budget and dissipation-rate claims.

free parameters (3)

Initial perturbation amplitude delta v = 0.1c
Chosen to seed the kink instability; control runs with delta v=0.01c showed no difference in linear growth or non-linear evolution (Sec. 5).
Best-fit Taylor-state alpha = 0.18 [1/a] (DP), 0.12 [1/a] (IP), 0.07 [1/a] (CO)
Fitted to the averaged final Bz and Bphi profiles in fig. 9; used as evidence for relaxation to a Taylor state and for the alpha Rj comparison, not in the dissipation-rate measurement.
Growth-time normalization tau = 20 pi P0/vA (eq. 26)
Defined by hand as the growth time of the linear instability; not derived from Lambda_max in eq. 3 (1/Lambda_max is about 7.5 P0/vA). The headline dissipation rate -0.1 UBphi/tau depends on this choice.

assumptions (7)

domain assumption Taylor relaxation: during turbulent relaxation, total helicity is approximately conserved and the plasma relaxes to the minimum-energy force-free state j = alpha B with constant alpha.
Invoked in Sec. 3 and used in Sec. 6.4 to compute the final energy; established for non-relativistic plasmas but asserted to hold in the relativistic high-sigma regime.
domain assumption Kink stability thresholds for Bessel force-free fields: alpha Rj = 3.176 (marginal) and 3.832 (unstable for all k below 0.272 alpha).
Used in Secs. 3, 4 and Appendix A as the closing constraint for the minimal-energy state; taken from Voslamber and Callebaut (1962).
domain assumption Linear kink growth scalings: kmax approximately 0.745/P0 and Lambda_max = 0.133 vA/P0 (eqs. 2-3).
Used to compare measured growth rates in Sec. 6.1 and to set box lengths; from Appl et al. (2000), external to this paper.
domain assumption Numerical resistivity in ideal RMHD gives a converged representation of reconnection-driven dissipation.
Reconnection is not resolved physically; the paper argues convergence at 10-45 cells/a (Appendix B). This is a standard assumption in astrophysical MHD but remains an assumption.
domain assumption Initial force-free equilibria (eqs. 17-18) represent relevant astrophysical jet and coronal-loop configurations.
The IP/DP profiles are from Mizuno et al. (2009) and the CO profile from Bodo et al. (2013); the paper states these are applicable to jets and twisted loops.
domain assumption Neglect of rotation, longitudinal velocity gradients, and jet expansion: a static periodic column captures the local kink dynamics.
Stated in Sec. 5: simulations are in the jet comoving frame, neglecting rotation and gradients; rotation can stabilize the kink (Istomin and Pariev 1996).
domain assumption Conservation of azimuthal flux and zero-gauge helicity K to about 10 percent accuracy despite outflow boundaries.
Used in Sec. 6.4 to derive the final energy; the paper attributes the about 10 percent K drop to energy leaking through the boundary. If the leakage is not purely responsible, the energy estimate has a larger error.

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Pith. "Pith review of Kink instability: evolution and energy dissipation in Relativistic Force-Free Non-Rotating Jets." pith.science (2026). https://pith.science/paper/SNQEH5NZ

@misc{pith2026190808620,
  author       = {Pith},
  title        = {Pith review of: Kink instability: evolution and energy dissipation in Relativistic Force-Free Non-Rotating Jets},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SNQEH5NZ}},
  note         = {Machine review of arXiv:1908.08620}
}

abstract

We study the evolution of kink instability in a force-free, non-rotating plasma column of high magnetization. The main dissipation mechanism is identified as reconnection of magnetic field-lines with various intersection angles, driven by the compression of the growing kink lobes. We measure dissipation rates ${\rm d} U_{B\phi}/{{\rm d}t} \approx -0.1 U_{B\phi}/\tau$, where $\tau$ is the linear growth time of the kink instability. This value is consistent with the expansion velocity of the kink mode, which drives the reconnection. The relaxed state is close to a force-free Taylor state. We constraint the energy of that state using considerations from linear stability analysis. Our results are important for understanding magnetic field dissipation in various extreme astrophysical objects, most notably in relativistic jets. We outline the evolution of the kink instability in such jets and derive constrains on the conditions that allow for the kink instability to grow in these systems.

Figures

Figures reproduced from arXiv: 1908.08620 by the authors.

**Figure 1.** The initial configuration of the three profiles tested in this work Coronal (CO,blue), increasing pitch (IP, dashed orange) and decreasing pitch (DP, dash dot green). Panels show from left to right, top to bottom: Pitch (in log scale), Bz, Bϕ, σ and plasma β (in log scale). alescence events (mergers), where in each merger the longitudinal wave number, n, is reduced by unity. This phase is seen in fig. 5 as a series … view at source ↗

**Figure 2.** The evolution of the kink instability in cases IPb, DPb and COb. Shown are values of Jz on the x-z plane. Current sheets are seen as peaked color filaments [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗

**Figure 3.** Same as fig. 2 for the thermal pressure, shown in logarithmic scale. Regions of high pressure match the peak filaments in Jz, implying that most of the dissipation is occurring in current sheets. The pressure in the right most column is in the course of becoming evenly distributed across the dissipated region. netic field is compressed by the growing amplitude of the mode. Since the volume of the current sheet is sm… view at source ↗

Figures from the paper (10 more)

**Figure 4.** Figure 4: 3D color rendering of Jz at the same times and color range as in fig. 2. Magnetic field lines are shown as white tubes [PITH_FULL_IMAGE:figures/full_fig_p008_4.png]

**Figure 5.** Figure 5: Evolution of the kink mode shown as the electric energy in the three simulated profiles: IPb, DPb, and COb. Three phases are evident: i) linear growth; ii) mode inverse cascade; iii) turbulence phase. The filled circles on the three curves, mark the times at which the …

**Figure 6.** Figure 6: The EM energy dissipation in the three profiles studied. In each profile we show the dissipation in the small and big boxes (sub-indices a and b respectively). We show the total EM energy (R (E 2 + B 2 )dV ) in the box normalized by the value at t = 0. For the CO case …

**Figure 7.** Figure 7: The EM energy density, eEM = 1 8π (E 2+B 2 ) ' B2 8π and the thermal energy density, u = T00 − ρΓ 2 c 2 ' 3p averaged over z and ϕ. Shown are the distributions at times tf from simulations IPb (top), DPb (middle) and COb (bottom). The dashed vertical lines depict the…

**Figure 8.** Figure 8: shows the radial profile of α averaged over z and ϕ, for all magnetic field profiles and box sizes discussed in this work. Shown are the initial values (in dashed line) and the values at the end of the simulations. In all large box simulations the α at the core is lowe…

**Figure 9.** Figure 9: Fitting B0J0(αr) and B0J1(αr) to Bz(r) and Bϕ(r) profiles at the end of each simulation. The best fitted α values are 0.18, 0.07 and 0.12 [1/a], for the DP coronal and IP profiles respectively. The dashed black lines mark the edges of the dissipated regions, Rj . the b…

**Figure 10.** Figure 10: The value of Bz at at e end of simulations IPb, DPb, COb, shown on a cross-sectional cut in the middle of the computational box. Field reversals are evident in the IP and DP cases but not in the CO case. The dashed red line marks Rj in each case. and α obtained from c…

**Figure 12.** Figure 12: A sketch of the collimation region of a highly magnetized relativistic jet. The jet is conical up to z ' zcoll , where it’s pressure becomes equal to the pressure of the surrounding medium. Above this point the collimated flow is affected by the contracting ”hoop stre…

**Figure 11.** Figure 11: The initial (solid blue) and final (dashed orange) EM energy in the IPb, DPb and COb distributions, compared with the estimated energy of the relaxed configuration (dot-dash green). The vertical lines track the radii of the dissipated regions. In the case of CO conf…

**Figure 13.** Figure 13: Growth rates of the kink instability, represented by the value of E 2 (left) and the EM energy dissipation rates (right) in the IP configuration. Simulations were performed in a box of size 40a × 40a × 20a with a resolution of 10, 15, 30 and 45 computational cells per…

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, " * write output.state after.block = add.period write newline

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[52]

write newline

" write newline "" before.all 'output.state := FUNCTION n.dashify 't := "" t empty not t #1 #1 substring "-" = t #1 #2 substring "--" = not "--" * t #2 global.max substring 't := t #1 #1 substring "-" = "-" * t #2 global.max substring 't := while if t #1 #1 substring * t #2 gl...
[53]

, " * write output.state after.block = add.period write newline

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[54]

write newline

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

EQ@ Mw+ Ƽ 5j [ԨhboFE TĂ )Hw. Ϲ w> Ϟݙg

thebibliography [1] 20pt to REFERENCES 6pt =0pt 10pt plus 3pt =0pt =0pt =1pt plus 1pt =0pt =0pt -12pt =13pt plus 1pt =20pt =13pt plus 1pt \@M =10000 =-1.0em =0pt =0pt 0pt =0pt =1.0em @enumiv\@empty 10000 10000 `\.\@m \@noitemerr \@latex@warning Empty `thebibliography' environment \@ifnextchar \@reference \@latexerr Missing key on reference command Each re...

2017

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