Pith. sign in

REVIEW 4 major objections 5 minor 31 references

Simulations of the churning mode: toroidally symmetric plasma convection and turbulence around the X-points in a snowflake divertor

T0 review · 4 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read This paper argues that the churning mode, a toroidally symmetric vortex around the X-points, activates at $\beta_{pm}\simeq 0.08$ and can both add transport across the nulls and distort or flip the magnetic flux surfaces in a snowflake…

desk verdict The transport scans are a real step forward, but the claimed topology flip in Fig. 12 contradicts the model's own frozen-flux law and is likely a numerical or time-averaging artifact. read the letter →

arxiv 2505.21223 v1 pith:RDUH74ZT submitted 2025-05-27 physics.plasm-ph

D Power , M V Umansky , V A Soukhanovskii This is my paper
classification physics.plasm-ph
keywords snowflakedivertorchurningmodeX-pointtransportreducedmagnetohydrodynamicslegpowersharingmagnetictopologyMAST-Upoloidalbeta
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

In a snowflake divertor, two magnetic nulls sit close together, creating a region of very weak poloidal field. This paper argues that in that region a toroidally symmetric plasma vortex, the churning mode, switches on once the ratio of plasma pressure to poloidal magnetic pressure, $\beta_{pm}$, exceeds roughly 8%, and that this vortex then acts as an extra transport channel carrying heat across the X-points. The simulations cover realistic inter-null separations $d_{xx}$ and orientations $\theta$ for a spherical tokamak, and show the extra transport grows about linearly as $d_{xx}$ shrinks, saturating near $d_{xx} \simeq 2.5$ cm. The more consequential claim is that the vortex does not merely add diffusion: it distorts the flux surfaces around the nulls, making the share of exhaust power delivered to each divertor leg strongly sensitive to $\beta_{pm}$, $d_{xx}$ and $\theta$, and for small $\theta$ it can even flip the local magnetic topology, moving the secondary X-point across the separatrix and redirecting power from one secondary leg to another. That matters because such geometry changes would not be captured by diffusive edge models or Grad-Shafranov equilibrium reconstruction.

What carries the argument

The load-bearing object is the churning-mode model: a reduced-MHD system (vorticity eq. 3, frozen flux eq. 4, pressure advection with field-aligned conduction eq. 5) in which the curvature drive $-2/R_0\,\partial p/\partial y$ competes with the magnetic restoring force. The poloidal flux $\psi$ is passively advected, so the vortex reorganises the magnetic geometry; the size of the convective zone is set by $r_{cz} \simeq 0.81 a(\beta_{pm}\varepsilon)^{1/3}$, and the local null-region geometry is parametrised by a flux function (eq. 10) written in terms of $d_{xx}$ and $\theta$. Field-aligned thermal conduction keeps the vertical pressure gradient alive, letting the mode reach a quasi-steady turbulent state.

What would settle it

Measure the secondary-leg power ratio in a snowflake discharge with small null orientation ($\theta \sim 15^\circ$) and inter-null separation larger than the scrape-off-layer width, while ramping density so $\beta_{pm}$ crosses roughly 8–10%: the model predicts $f_2/f_3$ jumps from near zero to about 30 and the secondary X-point moves from the outer to the inner scrape-off layer, so a discharge that shows no topology change or leg-power swing would refute the claim. A second check is to compare magnetic reconstructions against strike-point motion on the tens-of-microseconds timescale, where the CM predicts primary-separatrix oscillations of 1–2% of the minor radius.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is that the churning mode, previously analysed only in the exact snowflake limit, is active and dynamically important in realistic inexact snowflakes. In near-exact configurations ($d_{xx}$ small) the mode produces additional cross-null transport for $\beta_{pm} \gtrsim 0.08$, with the transported power increasing roughly linearly as $d_{xx}$ is reduced and saturating around $d_{xx}\simeq 2.5$ cm. The simulations reproduce the total null-region transport with a local diffusive 'umbrella' model using peak $\chi_x$ up to about $10^2$ m$^2$s$^{-1}$, but they show that this diffusive model misses the main effect: the churning flow advects the poloidal flux and permanently reconfigures the flux surfaces, so the fractional power into each leg becomes a sensitive function of $\beta_{pm}$, $d_{xx}$ and $\theta$. At $\theta \sim 15^\circ$ with $d_{xx}$ larger than the SOL width, the CM can change the magnetic topology, moving the secondary X-point from the outer to the inner SOL and shifting the ratio of secondary-leg powers $f_2/f_3$ from near zero to roughly 30. The paper also derives an empirical scaling $\chi_x \simeq 1.90\,\varepsilon^{-3.10}\beta_{pm}^{2.90}\lambda_{mp}^{-1.74}$ for the peak diffusive coefficient.

Load-bearing premise

The central assumption is that the poloidal magnetic field is passively swept along by the plasma with no magnetic feedback, so if the field resists the vortex's distortion, the predicted flux-surface rearrangement, power redirection and topology flip would not occur.

Editorial extensions

If this is right

  • The churning mode is a concrete, testable mechanism for the enhanced transport across the X-points inferred in snowflake experiments.
  • A simple local diffusive model can predict total transport across the null region (with $\chi_x$ up to $\sim 10^2$ m$^2$s$^{-1}$), but it cannot predict leg-by-leg power sharing because it misses the CM-driven flux-surface distortion.
  • Leg power fractions are strong functions of $\beta_{pm}$, $d_{xx}$ and $\theta$, so the divertor's heat load distribution can change dramatically between inter-ELM and ELM conditions.
  • For small $\theta$ and large $d_{xx}$, the CM can flip the magnetic topology and redirect exhaust power from a high-field-side secondary leg to a low-field-side one, a behaviour a fixed-equilibrium reconstruction would miss.
  • The near-exact snowflake is unlikely to give even four-leg mixing; across the scanned conditions the lower-left leg consistently receives the smallest fraction, matching TCV and DIII-D observations.

Reading between the lines

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

  • If the topology flip occurs in experiments, magnetic reconstructions that assume flux surfaces are frozen would systematically misattribute the power asymmetry to other causes; tracking strike-point positions on the $\sim 20$ $\mu$s fluctuation timescale could reveal CM activity.
  • The strong $\beta_{pm}^{2.9}$ scaling implies the CM's transport and geometry effects are negligible at low beta but dominant at ELM-like pressures, so time-averaged and peak heat loads across legs may respond very differently to ELMs.
  • A toroidal-field-direction reversal experiment would separate the CM (insensitive to $B_t$ direction) from strong $\mathbf{E}\times\mathbf{B}$ drift effects, which depend on both magnitude and sign; the paper suggests this test but leaves it for future work.
  • The model omits atomic physics and radiation, so the quantitative leg fractions and topology threshold are not directly comparable to experiment; adding radiation could shift the $\beta_{pm}$ threshold and the $f_2/f_3$ ratio.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 5 minor

Summary. The paper presents two-dimensional (toroidally symmetric) reduced MHD simulations of the churning mode (CM) in snowflake divertors, using the model of Umansky and Ryutov extended with field-aligned thermal conduction and implemented in BOUT++. Scans over inter-null separation d_xx, orientation angle theta, and poloidal beta beta_pm show that the CM drives additional transport across the X-point region for beta_pm >~ 0.08, with transport increasing as d_xx decreases. The authors propose a diffusive 'umbrella' model and an empirical scaling for the added diffusion coefficient, and they report that this model reproduces the total transport but not the distribution of exhaust power among divertor legs. A further reported result is that the CM distorts flux surfaces and, for small theta and large d_xx, can induce a change in magnetic topology that redirects exhaust power from one secondary leg to another. The paper also discusses implications for magnetic reconstruction and experimental signatures of the CM.

Significance. If correct, the results would identify the churning mode as a significant extra transport mechanism in snowflake divertors, with practical consequences for divertor design, power-load prediction, and magnetic equilibrium reconstruction. The study is the first to simulate the CM in non-exact snowflake geometries with finite d_xx and theta and to include parallel thermal conduction, and it is candid about model limitations (e.g., no atomic physics, no radiation, no self-consistent Spitzer-Härm conduction). The transport trends across the beta_pm and d_xx scans are internally consistent, and the paper offers a concrete, falsifiable prediction (leg-power redirection for small theta) that could be checked experimentally. However, the headline topology-change claim conflicts with the model's own advection law, and the diffusive-model validation is in-sample, so the overall significance is contingent on resolving those issues.

major comments (4)
  1. [Sec. IV.H and Fig. 12] Under Eq. (4), dψ/dt = 0, ψ is a Lagrangian invariant in an incompressible flow, and the advection map is a diffeomorphism. Therefore the critical points of ψ (the two nulls) are preserved, and the secondary null cannot move from the outer SOL to the inner SOL without crossing the primary separatrix, which would require a topological reconnection that the ideal model has no terms to allow. The paper does not distinguish between the instantaneous field, whose topology is rigorously invariant, and the time-averaged field, whose critical points can differ from those of the instantaneous field; it also reports no grid-convergence study that would rule out numerical diffusion of ψ in the finite-difference advection. Because the abstract's headline result ('change in topology, redirecting exhaust power') rests on this, the paper must either identify a physical mechanism for the change (e.g., numerical reconnection) or reframe the claim as a change in time-averaged flux-surface geometry with the instantaneous topology preserved.
  2. [Sec. V, Eq. (23) and Fig. 14] The validation of the diffusive 'umbrella' model is in-sample. The inferred χ_x is obtained from P_out via Eq. (22), Eq. (23) is fit to those same inferred values from the simulations, and Fig. 14 then compares the umbrella model using Eq. (23) against the same simulations. This is a curve fit, not an independent predictive test, so the abstract's statement that a diffusive model 'is shown to predict the total transport across the null points' overstates what is demonstrated. An out-of-sample test, such as a leave-one-device-out cross-validation or a comparison on a d_xx/θ case not used in the fit, is needed to support the predictive claim.
  3. [Sec. III and Sec. IV (all quantitative results)] No grid-convergence or domain-size studies are reported, and time-averaged quantities such as P_out, f_i, and χ_x are presented without error bars or fluctuation amplitudes (the only exception is the KDE in Fig. 10). Given that the parallel-conduction operator uses a simplified field-line-map method (Appendix A assumes b is constant within the bounding box) and that the model contains strong anisotropy, the quantitative claims—the threshold β_pm ≈ 0.08, the approximately linear scaling in d_xx, and the leg-power fractions in Table I—are not yet supported. The authors should provide a resolution scan and a domain-size scan to establish numerical robustness.
  4. [Sec. III, Eq. (10)] The local ψ form in Eq. (10) is a Taylor expansion around the nulls and is used here for d_xx up to 20 cm, far beyond its expected range of validity. The normalization A = A0 (d_xx,0 / d_xx)^3 is an ad hoc construction, not derived from a global equilibrium. Since the d_xx = 20 cm cases are the ones used for the topology-change result in Fig. 12, the paper should justify the use of Eq. (10) at large d_xx or replace it with a global equilibrium that is valid over the full simulation domain.
minor comments (5)
  1. [Throughout] There are several typographical errors: 'bteween' in Sec. I, 'sensistive' in Sec. VI, 'incercepts' in Appendix A, 'equlibrium' in Sec. III, and a duplicated 'in' in Sec. I ('There is a discussion of the results in in sec. VII').
  2. [Fig. 10 caption] The caption says 'Pi where l = 1,2,3,4', but the summation index should be i, not l.
  3. [Sec. III, footnote 1] The footnote stating that self-consistent Spitzer-Härm conduction is numerically unstable in the hot core is an important limitation and should be mentioned in the main text, not only in a footnote.
  4. [Sec. V, Eq. (23)] The units of the coefficient 1.90 in Eq. (23) are not specified; the authors should state the units or give the scaling in dimensionless form.
  5. [Table II] The table lists NSTX and NSTX-U parameters but the text says these devices were not simulated; either remove those rows or clarify their purpose.

Circularity Check

1 steps flagged · score 4.0 of 10

Core CM simulations are self-contained; only the umbrella-diffusion 'prediction' is an in-sample fit (Eq. 23 fitted to chi_x inferred from P_out via Eq. 22), so circularity is partial and confined to Sec. V.

  1. fitted input called prediction [Sec. V (Transport scaling and assessment of reduced models), Eqs. (22)-(23) and Fig. 14]
    "By varying the null pressure and χ⊥ we can simulate a range of βpm and λmp. The result is that we find χx ≃ 1.90ε −3.10β 2.90 pm λ −1.74 mp (23) fits the inferred values ofχx well with a root mean square error of 8%. We may use this fit to estimate χx in the umbrella diffusive (UD) model and compare predictions of Pout to the turbulent calculation: this is shown in fig. 14a."

    Equation (22) is the inversion χx ≃ [Poutλx/(p0Ld) − χ⊥] 2A/π(r∗cz)^2, so every inferred χx is a monotone function of the simulated Pout. Equation (23) is then fit to those inferred χx values, and Fig. 14a uses Eq. (23) inside the UD model to compute Pout and compare it with the same turbulent Pout that produced the fit. The reported 'prediction' of total transport is therefore an in-sample consistency check, not an independent validation of the umbrella model. This does not affect the main CM results (threshold, dxx scaling, leg-power splitting), which are obtained by solving Eqs. (3)-(5) with fixed inputs; it is a partial circularity confined to the reduced-model validation.

full rationale

The central simulations are not circular: Eqs. (3)-(5) are evolved in BOUT++ with fixed inputs (dxx, θ, βpm, λx, ψ0), and the CM onset, transport scaling, leg-power sharing, flux-surface distortion, and topology-change claims are emergent outputs of that dynamical system. References [15] and [16] supply the model and the theoretical convective-zone radius rcz, but the paper tests rather than assumes their quantitative predictions (e.g., Fig. 8 shows inferred χx falling below the Eq. (18) prediction). The only genuinely circular step is the diffusion-model validation in Sec. V: χx is inferred from Pout via Eq. (22), a regression for χx is fitted to those inferred values (Eq. (23)), and Fig. 14a then presents the resulting UD-model Pout as a 'prediction' against the same Pout that generated the fit. That is an in-sample agreement, not an independent test; the paper itself acknowledges the UD model fails on the leg-resolved quantity finner. Since this circularity does not carry the paper's main physics claims, the score is moderate rather than high.

Assumptions & free parameters 4 free parameters · 8 assumptions · 0 invented entities

The paper introduces no new physical entities. Its load-bearing content rests on a reduced MHD model inherited from the authors' earlier work, a local model of the SF magnetic field, and an empirical diffusion fit. The main fitted quantities are A0 and the coefficients of the chi_x scaling law.

free parameters (4)
  • A0 (poloidal flux scale) = not quoted; fitted to FIESTA equilibrium with dxx0 = 2.5 cm, Bpm = 0.3 T
    Sets the overall magnitude of psi in eq. 10 so the local null-region form matches a reference MAST-U equilibrium.
  • eq. 23 scaling coefficients = 1.90, -3.10, 2.90, -1.74
    Fit to inferred chi_x values from the CM simulations (RMSE 8%) and used in the umbrella diffusive model.
  • chi_perp (perpendicular conductivity) = 0.1 to 10 m^2/s (typical)
    Chosen by hand in each simulation to tune the SOL width lambda_x.
  • delta (boundary pressure profile amplitude) = not quoted; sets p0
    Input parameter in p = delta sqrt(1 - psi_n) on the core boundary; controls the pressure drive that sustains the vertical gradient.
assumptions (8)
  • domain assumption Reduced MHD ordering in the inverse aspect ratio epsilon
    Used to derive eqs. 3-5 from ideal MHD; section II.
  • domain assumption Incompressible, constant-density, toroidally symmetric flow
    Stated in section II; removes density evolution and parallel compression from the model.
  • domain assumption Lowest-order curvature approximation kappa ~= -x_hat/R0
    Appears in the vorticity equation (eq. 3); neglects curvature variation across the domain.
  • domain assumption Frozen-flux law dpsi/dt = 0 (ideal MHD)
    Eq. 4; the poloidal flux only advects with the flow and has no current/pressure feedback, which is what allows the flux-surface distortion in figs. 9 and 12.
  • domain assumption Local asymptotic form of psi (eq. 10) represents the SF geometry on the whole domain
    Section III; the form is a Taylor expansion around the nulls, scaled by dxx, and may be inaccurate at large dxx.
  • domain assumption Constant chi_parallel at the Spitzer-Harm value for Tsepx ~ 100 eV
    Section III and footnote 1; self-consistent Spitzer-Harm conduction was numerically unstable and not used.
  • domain assumption The umbrella diffusive model (eq. 20) is a valid reduced representation of CM transport
    Section IV.E; the model is taken from Khrabry et al. and used to estimate chi_x and compare total transport.
  • domain assumption Boundary conditions with fixed core pressure and q=0 in the SOL reproduce edge-code-like drives
    Section III; the pressure is equilibrated with conduction before enabling the CM.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Simulations of the churning mode: toroidally symmetric plasma convection and turbulence around the X-points in a snowflake divertor." pith.science (2026). https://pith.science/paper/RDUH74ZT

@misc{pith2026250521223,
  author       = {Pith},
  title        = {Pith review of: Simulations of the churning mode: toroidally symmetric plasma convection and turbulence around the X-points in a snowflake divertor},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RDUH74ZT}},
  note         = {Machine review of arXiv:2505.21223}
}
abstract

Using a reduced MHD model, extended to include field-aligned thermal conduction, we present numerical simulations of the churning mode (CM): a toroidally symmetric, non-linear plasma vortex in the vicinity of the null points in a snowflake (SF) divertor (Ryutov et al., Phys. Scr. 89 088002, 2014). Simulations are carried out across a range of inter-null separations, $d_{xx}$, and inter-null orientations, $\theta$, primarily in conditions relevant to the MAST-U tokamak. We find that, when $d_{xx}$ is small, the CM induces additional transport across the X-points when $\beta_{pm} \gtrsim 8$ %, where $\beta_{pm}$ is the ratio of the plasma pressure in the null region to poloidal magnetic pressure at the midplane. This transport also increases approximately linearly as $d_{xx}$ is reduced. A diffusive model of this transport is shown to predict the total transport across the null points, where diffusion coefficients of up to $\sim 10^2$ m$^2$s$^{-1}$ centred on a small region around the X-points are used. However, the CM also results in significant changes to the flux surfaces in the null region which is not captured by this diffusive model. The changes in magnetic geometry mean the fractional exhaust power delivered to each divertor leg is highly sensitive to $\beta_{pm}$, $d_{xx}$ and $\theta$. For small values of $\theta$, the CM can induce a change in topology, redirecting exhaust power from a secondary divertor leg on the high field side to one on the low field side. Similar behaviour is found in the fraction of exhaust power going to the inner and outer divertor. Such changes in the flux surfaces may not be captured by Grad-Shafranov solvers and so may be a source of error in the magnetic reconstruction of SF experiments. We consistently find that the fractional exhaust power going to a secondary divertor leg on the high field side is small, consistent with SF experiments.

Figures

Figures reproduced from arXiv: 2505.21223 by the authors.

Figure 1
Figure 1. FIG. 1: Diagram of the magnetic topology of a SF divertor in [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2: Various time slices after the CM physics is turned on [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3: Total output power with and without the CM physics [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figures from the paper (8 more)
Figure 5
Figure 5. Figure 5: FIG. 5: Fractional output power going to each divertor leg: [PITH_FULL_IMAGE:figures/full_fig_p005_5.png]
Figure 7
Figure 7. Figure 7: FIG. 7: Increase in [PITH_FULL_IMAGE:figures/full_fig_p006_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8: Estimated values of the peak diffusive coefficient in [PITH_FULL_IMAGE:figures/full_fig_p006_8.png]
Figure 9
Figure 9. Figure 9: FIG. 9: Contours of the poloidal flux in a CM simulation [PITH_FULL_IMAGE:figures/full_fig_p007_9.png]
Figure 11
Figure 11. Figure 11: FIG. 11: Change in separatrix positions at each timestep after [PITH_FULL_IMAGE:figures/full_fig_p007_11.png]
Figure 12
Figure 12. Figure 12: FIG. 12: Change in magnetic topology due to the CM above a threshold [PITH_FULL_IMAGE:figures/full_fig_p009_12.png]
Figure 13
Figure 13. Figure 13: FIG. 13: Fraction of the total exhaust power leaving the [PITH_FULL_IMAGE:figures/full_fig_p009_13.png]
Figure 15
Figure 15. Figure 15: FIG. 15: Geometry of the line tracing process to evaluate [PITH_FULL_IMAGE:figures/full_fig_p011_15.png]

Discussion (0). Sign in to comment.

Reference graph

Works this paper leans on

31 extracted references · 31 canonical work pages

  1. [1]

    snowflake

    D. D. Ryutov. Geometrical properties of a “snowflake” divertor. Physics of Plasmas , 14(6):064502, June 2007

    work page 2007

  2. [2]

    D. D. Ryutov and V. A. Soukhanovskii. The snowflake divertor. Physics of Plasmas , 22(11):110901, November 2015

    work page 2015

  3. [3]

    D. D. Ryutov, R. H. Cohen, T. D. Rognlien, and M. V. Umansky. The magnetic field structure of a snowflake divertor. Physics of Plasmas , 15(9):092501, September 2008

    work page 2008

  4. [4]

    A snowflake divertor: a possible solution to the power exhaust problem for tokamaks

    D D Ryutov, R H Cohen, T D Rognlien, and M V Umansky. A snowflake divertor: a possible solution to the power exhaust problem for tokamaks. Plasma Physics and Controlled Fusion , 54(12):124050, December 2012

    work page 2012

  5. [5]

    Snowflake

    F. Piras, S. Coda, B. P. Duval, B. Labit, J. Marki, S. Yu. Medvedev, J.-M. Moret, A. Pitzschke, and O. Sauter. “ Snowflake ” H Mode in a Tokamak Plasma . Physical Review Letters , 105(15):155003, October 2010

    work page 2010

  6. [6]

    Power distribution in the snowflake divertor in TCV

    H Reimerdes, G P Canal, B P Duval, B Labit, T Lunt, W A J Vijvers, S Coda, G De Temmerman, T W Morgan, F Nespoli, B Tal, and the TCV Team . Power distribution in the snowflake divertor in TCV . Plasma Physics and Controlled Fusion , 55(12):124027, December 2013

    work page 2013

  7. [7]

    Vijvers, G.P

    W.A.J. Vijvers, G.P. Canal, B. Labit, H. Reimerdes, B. Tal, S. Coda, G.C. De Temmerman, B.P. Duval, T.W. Morgan, J.J. Zielinski, and the TCV Team . Power exhaust in the snowflake divertor for L - and H -mode TCV tokamak plasmas. Nuclear Fusion , 54(2):023009, February 2014

    work page 2014

  8. [8]

    Soukhanovskii, S.L

    V.A. Soukhanovskii, S.L. Allen, M.E. Fenstermacher, C.J. Lasnier, M.A. Makowski, A.G. McLean, W.H. Meyer, D.D. Ryutov, E. Kolemen, R.J. Groebner, A.W. Hyatt, A.W. Leonard, T.H. Osborne, T.W. Petrie, and J. Watkins. Developing physics basis for the snowflake divertor in the DIII - D tokamak. Nuclear Fusion , 58(3):036018, March 2018

    work page 2018

Show all 31 references
  1. [9]

    V. A. Soukhanovskii, G. Cunningham, J. R. Harrison, F. Federici, and P. Ryan. First snowflake divertor experiments in MAST - U tokamak. Nuclear Materials and Energy , 33(October), 2022

    2022

  2. [10]

    First EMC3 - Eirene simulations of the TCV snowflake divertor

    T Lunt, G P Canal, Y Feng, H Reimerdes, B P Duval, B Labit, W A J Vijvers, D Coster, K Lackner, and M Wischmeier. First EMC3 - Eirene simulations of the TCV snowflake divertor. Plasma Physics and Controlled Fusion , 56(3):035009, March 2014

    2014

  3. [11]

    Enhanced → E × → B drift effects in the TCV snowflake divertor

    G P Canal, T Lunt, H Reimerdes, B P Duval, B Labit, and W A J Vijvers. Enhanced → E × → B drift effects in the TCV snowflake divertor. Nucl. Fusion , 2015

    2015

  4. [12]

    S. S. Abdullaev, M. Jakubowski, M. Lehnen, O. Schmitz, and B. Unterberg. On description of magnetic stochasticity in poloidal divertor tokamaks. Physics of Plasmas , 15(4):042508, April 2008

    2008

  5. [13]

    W. A. Farmer and D. D. Ryutov. Axisymmetric curvature-driven instability in a model divertor geometry. Physics of Plasmas , 20(9):092117, September 2013

    2013

  6. [14]

    W. A. Farmer. Ballooning modes localized near the null point of a divertor. Physics of Plasmas , 21(4):042114, April 2014

    2014

  7. [15]

    The ‘churning mode’ of plasma convection in the tokamak divertor region

    D D Ryutov, R H Cohen, W A Farmer, T D Rognlien, and M V Umansky. The ‘churning mode’ of plasma convection in the tokamak divertor region. Physica Scripta , 89(8):088002, August 2014

    2014

  8. [16]

    M. V. Umansky and D. D. Ryutov. Toroidally symmetric plasma vortex at tokamak divertor null point. Physics of Plasmas , 23(3):030701, March 2016

    2016

  9. [17]

    Khrabry, V.A

    A.I. Khrabry, V.A. Soukhanovskii, T.D. Rognlien, M.V. Umansky, D. Moulton, and J.R. Harrison. Modeling snowflake divertors in MAST - U tokamak using UEDGE code. Nuclear Materials and Energy , 26:100896, March 2021

    2021

  10. [18]

    Shear- Alfvén dynamics of toroidally confined plasmas

    R.D Hazeltine and J.D Meiss. Shear- Alfvén dynamics of toroidally confined plasmas. Physics Reports , 121(1-2):1--164, May 1985

    1985

  11. [19]

    J. F. Drake and Thomas M. Antonsen. Nonlinear reduced fluid equations for toroidal plasmas. The Physics of Fluids , 27(4):898--908, April 1984

    1984

  12. [20]

    Non-linear dynamics and pattern formation in a vertical fluid layer heated from the side

    Dmitry A Bratsun, Alexej V Zyuzgin, and Gennady F Putin. Non-linear dynamics and pattern formation in a vertical fluid layer heated from the side. International Journal of Heat and Fluid Flow , 24(6):835--852, December 2003

    2003

  13. [21]

    Dudson, M

    B. Dudson, M. V. Umansky, X. Q. Xu, P. B. Snyder, and H. R. Wilson. BOUT ++: A framework for parallel plasma fluid simulations. Computer Physics Communications , 180(9):1467--1480, 2009. arXiv: 0810.5757 Publisher: Elsevier B.V

    2009 arXiv

  14. [22]

    Hindmarsh, Peter N

    Alan C. Hindmarsh, Peter N. Brown, Keith E. Grant, Steven L. Lee, Radu Serban, Dan E. Shumaker, and Carol S. Woodward. SUNDIALS : Suite of nonlinear and differential/algebraic equation solvers. ACM Transactions on Mathematical Software , 31(3):363 -- 396, 2005

    2005

  15. [23]

    The field line map approach for simulations of magnetically confined plasmas

    Andreas Stegmeir, David Coster, Omar Maj, Klaus Hallatschek, and Karl Lackner. The field line map approach for simulations of magnetically confined plasmas. Computer Physics Communications , 198:139--153, January 2016

    2016

  16. [24]

    M. V. Umansky, M. S. Day, and T. D. Rognlien. On Numerical Solution of Strongly Anisotropic Diffusion Equation on Misaligned Grids . Numerical Heat Transfer, Part B: Fundamentals , 47(6):533--554, June 2005

    2005

  17. [25]

    Rognlien, J.L

    T.D. Rognlien, J.L. Milovich, M.E. Rensink, and G.D. Porter. A fully implicit, time dependent 2- D fluid code for modeling tokamak edge plasmas. Journal of Nuclear Materials , 196-198:347--351, December 1992

    1992

  18. [26]

    Local properties of the magnetic field in a snowflake divertor

    D D Ryutov, M A Makowski, and M V Umansky. Local properties of the magnetic field in a snowflake divertor. Plasma Physics and Controlled Fusion , 52(10):105001, October 2010

    2010

  19. [27]

    Khrabry, V.A

    A.I. Khrabry, V.A. Soukhanovskii, T.D. Rognlien, M.V. Umansky, D. Moulton, and J.R. Harrison. Modeling snowflake divertors in MAST - U Tokamak . Nuclear Fusion , 62(1):016007, January 2022

    2022

  20. [28]

    Soukhanovskii, J.-W

    V.A. Soukhanovskii, J.-W. Ahn, R.E. Bell, D.A. Gates, S. Gerhardt, R. Kaita, E. Kolemen, B.P. LeBlanc, R. Maingi, M. Makowski, R. Maqueda, A.G. McLean, J.E. Menard, D. Mueller, S.F. Paul, R. Raman, A.L. Roquemore, D.D. Ryutov, S.A. Sabbagh, and H.A. Scott. Taming the plasma–ma...

    2011

  21. [29]

    Günter, Q

    S. Günter, Q. Yu, J. Krüger, and K. Lackner. Modelling of heat transport in magnetised plasmas using non-aligned coordinates. Journal of Computational Physics , 209(1):354--370, October 2005. Publisher: Elsevier BV

    2005

  22. [30]

    Soler, F

    J.A. Soler, F. Schwander, G. Giorgiani, J. Liandrat, P. Tamain, and E. Serre. A new conservative finite-difference scheme for anisotropic elliptic problems in bounded domain. Journal of Computational Physics , 405:109093, March 2020

    2020

  23. [31]

    Support- Operator Finite - Difference Algorithms for General Elliptic Problems

    Mikhail Shashkov and Stanly Steinberg. Support- Operator Finite - Difference Algorithms for General Elliptic Problems . Journal of Computational Physics , 118(1):131--151, April 1995

    1995

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.