Geometric approximations

Analytic tokamak theory seldom works in the real torus. It works in a slab, a cylinder or an expanded torus, and it carries a few coefficients over from the real equilibrium. This page sets out those reductions using the literature’s names. The formulas are in vaft.formula.geometry.

The toroidal parent geometry itself (flux surfaces, the Shafranov shift, shaping, $q$ as a winding number, and $\theta$ versus $\theta^*$) is drawn in the tokamak-geometry section of Scientific diagrams. How VAFT keeps coordinates, projections, representations, derived quantities and solver conventions of one real equilibrium apart is on Equilibrium representations.

Two axes, not one hierarchy

Geometry says what space the model lives in: slab, cylindrical or toroidal. Ordering says which scales are assumed small: local ($|x| \ll r$), large aspect ratio ($\epsilon = a/R_0 \ll 1$) or neither (global). The two are independent. A slab may carry $q$, $\hat s$, $R_0$ or a $1/R_0$ curvature inherited from a torus and still be a slab. Large aspect ratio is an ordering, not a geometry: some large-aspect-ratio theories stay toroidal.

Conventions used throughout

• Perturbations are $e^{i(m\theta - n\phi)}$, as in vaft.formula.helical_phase. Both mode numbers are positive and the helicity is in the minus sign.
• $q$ is a magnitude, whatever sign a COCOS gives it. $\hat s$ keeps the sign of $d|q|/dr$: reversed shear ($\hat s < 0$) flips the sign of $L_s$.
• A torus is straightened at $R_0$ with $z = R_0\phi$ and $B_z \leftrightarrow B_\phi$. Along that $z$, the harmonic has wavenumber $-n/R_0$.
• A sheared slab is $\mathbf B = B_0(\hat{\mathbf z} + (x/L_s)\hat{\mathbf y})$, with $z$ along the field at $x = 0$ and $y$ the binormal in the surface. This field-aligned $z$ is not the straightened-torus $z = R_0\phi$.
• The local slab of a surface $r_0$ is that frame, with $x = r - r_0$. There the harmonic has:
  • $k_y = m/r_0$;
  • $k_z = (m - nq_0)/(q_0R_0)$, which is zero on a rational surface;
  • $L_s$, which is signed: positive shear gives $L_s = -q_0R_0/\hat s < 0$.

  local_slab_from_cylinder returns $(k_y, L_s, k_z)$ in the order sheared_slab_parallel_wavenumber takes them. shear_length_from_q_R0_s returns the magnitude $qR_0/|\hat s|$.

Straight slab

• Coordinates: $(x, y, z)$: radial, binormal, along $\mathbf B$.
• Field: $\mathbf B = B_0\hat{\mathbf z}$, uniform.
• Ordering: local. Gradients are taken over a region small against every equilibrium scale.
• Formula: $\tilde f \propto e^{i(k_xx + k_yy + k_zz)}$, and $k_\parallel = \mathbf k\cdot\mathbf B/B = k_z$ (slab_parallel_wavenumber).
• Keeps: wave propagation and local density or temperature gradients.
• Drops: magnetic curvature and magnetic shear, unless they are added separately.
• Used for: drift waves, and local dispersion relations before shear or curvature enter.

Sheared slab

• Field: $\mathbf B = B_0(\hat{\mathbf z} + (x/L_s)\hat{\mathbf y})$ (sheared_slab_field), or equivalently $B_y(x) \simeq B_y’x$ near a resonant surface.
• Shear: $\hat s = (r/q)\,dq/dr$ (vaft.formula.shear_from_r_q). Under the local, large-aspect-ratio ordering, $|L_s| = qR_0/|\hat s|$ (shear_length_from_q_R0_s).
• Formula: $k_\parallel(x) \simeq k_z + k_y x/L_s$ (sheared_slab_parallel_wavenumber). The perturbation is field-aligned on one surface and $|k_\parallel|$ grows linearly away from it.
• Keeps: the resonance and its localisation.
• Drops: global geometry, and curvature unless it is added.
• Used for: the inner layer of tearing modes (FKR, Rutherford), drift waves with shear, and the local limits of gyrokinetics.
• Related configurations in the same frame: the reversing field of a Harris current sheet, $B_y = B_0\tanh(x/a)$ (harris_sheet_field, harris_sheet_current_density), whose centre is a sheared slab with $B_y’ = B_0/a$ and no guide field, and the current-free X-point $\psi = B’(x^2 - y^2)/2$ (x_point_flux). These configurations, not geometries, are drawn in the field-configuration section of the diagrams guide.

Curved slab

Curved slab is a family of local models, not one model. VAFT fixes no closure here: the curvature coefficient is whatever a particular model inherits from its parent torus.

• Coordinates and field: those of the straight or sheared slab.
• Curvature: toroidal curvature is added as a parameter, either an effective curvature $\boldsymbol\kappa = \mathbf b\cdot\nabla\mathbf b$ (typically $|\kappa| \simeq 1/R_0$ pointing to the major axis), or an effective gravity $g \sim v_{th}^2/R_0$.
• Ordering: local. The curvature is a constant coefficient over the region.
• Keeps: the interchange drive (the pressure gradient against the curvature) and the $\nabla B$ and curvature drifts.
• Drops: the poloidal variation of curvature (good and bad curvature sides), the geometry of the real flux surfaces, and toroidal mode coupling.
• Next more realistic model: a local toroidal (flux-tube) model, where $\kappa$ varies along the field line.
• Used for: interchange and Rayleigh–Taylor analogues, resistive-g modes, and reduced edge and SOL turbulence.

Cylinder, screw pinch, cylindrical tokamak

• Coordinates: $(r, \theta, z)$.
• Field: $\mathbf B = B_\theta(r)\hat{\boldsymbol\theta} + B_z(r)\hat{\mathbf z}$.
• Safety factor: $q(r) = rB_z/(R_0B_\theta)$ (cylindrical_safety_factor_from_r_B).
• Resonance: $k_\parallel = (m - nq)/(qR_0)$ (cylindrical_parallel_wavenumber), so $k_\parallel = 0 \Leftrightarrow q(r_s) = m/n$.
• The three are related but not synonyms:
  • A straight cylinder is only the coordinate system.
  • A screw pinch is a cylindrical equilibrium with both $B_\theta(r)$ and $B_z(r)$, so its field lines are helices. Pure $\theta$- and $z$-pinches are its limits.
  • A cylindrical (straight) tokamak is a screw pinch that is periodic in $z$ with length $2\pi R_0$ and has $B_z \gg B_\theta$, both taken from a torus.
• Keeps: radial structure, the current profile, $q(r)$ and rational surfaces.
• Drops: $1/R$ variation of $B_\phi$, toroidal curvature, inboard/outboard asymmetry, the Shafranov shift, toroidal coupling of harmonics, and shaping. The helical field lines are still curved, which is what drives Suydam interchange.
• Used for: Newcomb’s equation, the internal kink, the outer tearing problem and $\Delta’$.

Large-aspect-ratio ordering

For a circular torus $R = R_0 + r\cos\theta$, so

The second expression is vaft.formula.vacuum_toroidal_field.

Worked $O(\epsilon)$ example: toroidal coupling. Any factor of $1/R$ multiplying a harmonic $e^{im\theta}$ creates sidebands:

So in a torus an $m/n$ perturbation drags along $(m \pm 1)/n$ components of relative size $\epsilon/2$. In a cylinder it cannot.

The expansion is used in two different ways:

1. Large-aspect-ratio toroidal model. It keeps selected $O(\epsilon)$ terms:
   • curvature;
   • inboard/outboard asymmetry;
   • coupling of $m$ to $m \pm 1$.

   It is still toroidal. It keeps the $O(\epsilon)$ curvature and $1/R$ terms and drops $O(\epsilon^2)$. The Shafranov shift enters at $O(\epsilon(\beta_p + l_i/2))$ relative to $a$. Shaping is outside the circular expansion used here, although shaped large-aspect-ratio models exist (for example Miller local equilibria). Typical uses are the $s$–$\alpha$ ballooning model, toroidal Alfvén gaps and neoclassical trapping.

2. Cylindrical tokamak limit. It keeps only $O(1)$: $R \simeq R_0$, and the toroidal direction is straightened as $z = R_0\phi$. The cylindrical tokamak is the leading-order limit of the large-aspect-ratio expansion. It is not the same thing as large aspect ratio.

Local sheared-slab reduction

Near a surface $r_0$, with $x = r - r_0$ and $|x| \ll r_0$, expand $q \simeq q_0 + q_0’x$ and take the field-aligned frame: $z$ along $\mathbf B(r_0)$, $y$ the binormal. The harmonic $e^{i(m\theta - n\phi)}$ has there

(local_slab_from_cylinder). Along the straightened torus the same harmonic has $-n/R_0$; along the field it has $k_z$. On a rational surface $q_0 = m/n$, so the resonance $q(r_s) = m/n$ becomes $k_\parallel(0) = k_z = 0$ and the shear becomes $k_\parallel’ = k_y/L_s = -k_y\hat s/(q_0R_0)$. The test suite checks that this slab reproduces the cylinder’s $k_\parallel$ to first order in $x$.

A local slab does not need a cylinder as an intermediate step. It can be expanded from a toroidal equilibrium directly. What defines it is locality relative to the equilibrium scale, not large aspect ratio. Turbulence and interchange models often keep toroidal coefficients in a slab, such as $|L_s| \simeq qR_0/|\hat s|$ and, in curved slabs, $\kappa \simeq 1/R_0$.

The same field line in each geometry

Toroidal	Cylindrical	Sheared slab

MHD mode representations across geometries

The same instability is described differently in each geometry, and MHD vocabulary mixes several classification systems in one name. “$m = 1$”, “kink”, “internal”, “ideal”, “tearing parity”, “low-$n$” and “resonant” are answers to different questions. Schematically,

and two modes with the same name can differ in any of these. This section maps mode families across the slab, cylindrical and toroidal reductions above. It does not claim that every mode has a unique descendant in the next geometry.

mhd_mode_geometry_map keeps four kinds of relation apart, and draws each with its own arrow:

Fourier representation and resonance

With the conventions above, a perturbation is

Straightening the torus at $R_0$ gives $z = R_0\phi$ and $k_z = -n/R_0$. The field-aligned local slab of a surface (local_slab_from_cylinder) has $k_y = m/r_0$, and along the field the harmonic has $k_\parallel = (m - nq)/(qR_0)$ (cylindrical_parallel_wavenumber). Hence

which is mode_number_mapping. These relations exist once, in vaft.formula.geometry and vaft.formula.stability.helical_phase; the map quotes them and implements nothing new.

Mode numbers are morphology, not instability

In a cylinder, $m = 0$ is a sausage, $m = 1$ a kink (a rigid helical shift) and $m \ge 2$ higher helical distortions (cylindrical_mode_morphology). These are Fourier morphologies. They are not a taxonomy of instabilities. “$m = 1$ means kink instability” says too much. It is more accurate to say that $m = 1$ has kink-like cylindrical morphology, and that whether an unstable kink branch exists depends on the equilibrium and on the eigenproblem posed. The $m = 0$ sausage instability of a Z-pinch is driven by the pressure gradient against the curvature of $B_\theta$, so the map files it under the pressure-driven family. A strong $B_z$ stabilises it, and it has no tokamak descendant on the map.

Pressure- and curvature-driven family

• Curved slab. A pressure gradient against a prescribed field-line curvature acts like a heavy fluid above a light one, with the curvature playing gravity: a magnetic Rayleigh–Taylor analogue.
• Cylinder. Flute or interchange perturbations have $k_\parallel \simeq 0$ and so bend field lines as little as possible. Suydam’s criterion is their local stability limit. Interchange names a mechanism, not a mode number.
• Torus. Three branches are kept apart:
  • interchange / Mercier: the localised pressure–curvature limit; Mercier’s criterion is the toroidal continuation of Suydam’s;
  • ballooning: high $n$, modified by field-line bending and magnetic shear and localised on the bad-curvature side (ballooning_curvature_drive, ballooning_harmonic_envelope, ballooning_eigenfunction, s_alpha_ballooning);
  • infernal: low $n$, pressure-driven, where magnetic shear is weak.

Current-driven family

The internal/external distinction already exists in the cylinder (internal_external_kink; Kruskal–Shafranov is a cylindrical external-kink criterion). The torus continues both:

• Internal kink: a global ideal mode confined inside an internal rational surface, the $m = 1$ mode inside $q = 1$. In a torus its leading cylindrical $\delta W$ vanishes and the toroidal (Bussac) terms decide stability. Finite resistivity at $q = 1$ gives the resistive kink.
• External kink: a global ideal mode that displaces the plasma boundary (internal_external_kink, plasma_vacuum_wall, kink_mode).
• Peeling: an edge-current-driven, edge-localised, external-kink-like branch. It is not simply the same mode as the external kink.
• Resistive-wall mode (RWM): the external-kink branch modified by a finite-conductivity wall. It grows on the wall time instead of the Alfvén time.
• Peeling–ballooning: the coupled edge branch of peeling and ballooning, the picture behind the peeling_ballooning operational diagram.

Global current-driven modes have no local-slab counterpart. They need the radial current profile that a local slab discards.

Resonant and reconnecting family

global toroidal / cylindrical harmonic
  -> q = m/n                          (mode_number_mapping)
  -> cylindrical ideal outer problem  (cylindrical_tearing_outer, Delta')
  -> local sheared-slab layer         (slab_parity, rational_surface)
  -> tearing or twisting channel      (resonant_layer_matching, tearing_layer_matching)

The tearing mode is the resistive resonant instability. In a torus, toroidicity and shaping couple several $m$ at fixed $n$ (poloidal_harmonic_coupling), so the outer problem becomes a matrix over all rational surfaces of that $n$. The NTM is a nonlinear, neoclassical extension of island evolution, driven by the loss of bootstrap current inside the island (magnetic_island). It is not the next linear eigenmode in an ideal–resistive hierarchy, which is why it hangs off the map as a branch.

Parity is orthogonal to mode number

$m$ and $n$ are not parity labels. At a resonant surface the local layer response has its own classification, tearing or twisting parity (slab_parity, slab_parity_comparison). A given $m/n$ harmonic can take part in either channel, depending on the response or eigenproblem. Toroidal coupling of several $m$ at fixed $n$ does not change this. Parity stays a separate concept. With finite pressure and curvature, the Glasser–Greene–Johnson inner layer that RDCON solves classifies its solutions as tearing and interchange parity rather than twisting.

Axisymmetric free-boundary modes

The vertical instability, and the VDE it leads to, is $n = 0$, axisymmetric, and set by elongation and the free boundary (hot_vde_sequence, cold_vde_bifurcation, vde_timescales). It does not belong to the kink family. A rigid transverse shift of a cylinder is useful intuition and nothing more. There is no exact $m = 1 \leftrightarrow$ VDE equivalence.

Orthogonal mode descriptors

Where VAFT’s stability tools sit

These tools do not share a common “instability severity” scale. Interchange, ballooning, kink and peeling should not be ranked by comparing unrelated criteria with each other. Naming the dominant dynamical mode needs a common eigenvalue or growth-rate problem. How VAFT runs these codes is on MHD stability.

Gyrokinetic micro-instabilities (ITG, TEM, ETG, MTM, KBM) are outside this MHD map. They can use the same coordinate framework later.

Where these lead

cylinder / screw pinch  -> Newcomb equation, internal kink, outer tearing (Delta')
cylinder -> sheared slab -> resistive inner layer: FKR, Rutherford, modified Rutherford
straight / sheared / curved slab -> drift waves, interchange, reduced edge turbulence, local gyrokinetics

The tearing diagrams (rational_surface, delta_prime, tearing_layer_matching) start from the cylinder and sheared slab described here. The current-sheet, X-point, reconnection and MHD-wave diagrams (current_sheet, x_point, magnetic_reconnection, shear_alfven_wave, mhd_wave_family) use the slab frame of this page. See Scientific diagrams. The scales these orderings compare, and how to name them, are Characteristic scales and orderings.

References

1. J. Wesson, Tokamaks, 4th ed., Oxford University Press (2011), Ch. 3, 6 and 8.
2. J. P. Freidberg, Ideal MHD, Cambridge University Press (2014), Ch. 9 and 11.
3. H. P. Furth, J. Killeen and M. N. Rosenbluth, Phys. Fluids 6 (1963) 459.
4. W. Horton, Rev. Mod. Phys. 71 (1999) 735.
5. C. Mercier, Nucl. Fusion 1 (1960) 47; B. R. Suydam, Proc. 2nd UN Conf. Peaceful Uses of Atomic Energy 31 (1958) 157.
6. J. Manickam, N. Pomphrey and A. M. M. Todd, Nucl. Fusion 27 (1987) 1461 (infernal modes).
7. J. W. Connor, R. J. Hastie, H. R. Wilson and R. L. Miller, Phys. Plasmas 5 (1998) 2687 (peeling–ballooning).
8. R. J. La Haye, Phys. Plasmas 13 (2006) 055501 (neoclassical tearing modes).