Co-Rotating Helical Basis

• Co-Rotating Helical Basis is an orthonormal framework attached to a helix that renders helical patterns stationary and simplifies phase-dependent formulations.
• It enables dimensional reduction by converting complex 3D problems into 2D or 1D formulations, significantly reducing computational costs in fluid dynamics and vortex analyses.
• The method leverages coordinate maps, Serret-Frenet frames, and rotational transformations to enforce helical symmetry in boundary elements and stability analyses.

A co-rotating helical basis is an orthonormal frame or coordinate representation attached to a helix and transported so that the helix, or a helical pattern imposed on the flow, is stationary in the chosen frame. In the cited formulations, co-rotation is implemented either by a helical symmetry operator $H(\varphi)$, by a rotation of the $\{\hat e_\theta,\hat e_z\}$-plane through a fixed helix-angle, or by a simultaneous rotation about the $z$-axis and translation along $z$. In each case the construction is used to remove explicit helical phase dependence, enforce helical periodicity, or make a time-dependent helical potential time-independent, thereby enabling dimensional reduction or a simpler spectral formulation (Liu et al., 2013, Castillo-Castellanos et al., 2021, Cao et al., 8 Nov 2025, Okulov, 2017).

1. Helical symmetry and the co-rotating frame

For a perfect infinite helix, the fundamental symmetry is invariance under a combined rotation about the $x_3$-axis and translation along $x_3$. In the boundary-element formulation of helical Stokes flow, this symmetry is encoded by

$$H(\varphi)\colon \mathbf{x}\mapsto R^3(\varphi)\,\mathbf{x}+ \begin{pmatrix} 0\ 0\ \frac{\Gamma}{2\pi}\varphi \end{pmatrix}, \qquad R^3(\varphi)= \begin{pmatrix} \cos\varphi & -\sin\varphi & 0\ \sin\varphi & \cos\varphi & 0\ 0 & 0 & 1 \end{pmatrix},$$

where $\Gamma=2\pi R\cot\theta$. Under $H(\varphi)$, any point on the helix is carried into an “identical” point one turn higher (Liu et al., 2013).

A related construction appears in rotating helical media. In the helical-container formulation, one introduces

$$r'=r,\qquad \phi=\theta+\Omega t,\qquad z'=z+v_z t,\qquad v_z=\frac{\delta\omega}{2k}=\frac{\ell\Omega}{k},$$

so that the laboratory-frame helical potential $\{\hat e_\theta,\hat e_z\}$0 becomes

$\{\hat e_\theta,\hat e_z\}$1

and is therefore time-independent in the rotating-translating frame (Okulov, 2017).

In the Euler setting of co-rotating helical vortices, the same idea is imposed by the change of variables

$\{\hat e_\theta,\hat e_z\}$2

With the helical-orthogonality condition $\{\hat e_\theta,\hat e_z\}$3, the 3D vorticity-stream-function system reduces to a 2D transport system on $\{\hat e_\theta,\hat e_z\}$4, and in rotating-frame form becomes stationary (Cao et al., 8 Nov 2025).

2. Orthonormal triads adapted to a helix

One canonical realization of the co-rotating helical basis is the Serret-Frenet frame of the helical centerline. For a helix of radius $\{\hat e_\theta,\hat e_z\}$5 and pitch $\{\hat e_\theta,\hat e_z\}$6,

$\{\hat e_\theta,\hat e_z\}$7

and

$\{\hat e_\theta,\hat e_z\}$8

The corresponding unit tangent, normal, and binormal are

$\{\hat e_\theta,\hat e_z\}$9

$z$0

These three vectors form an orthonormal “body-fixed” frame which rotates about $z$1 at the same rate as the helix (Liu et al., 2013).

A second realization is obtained by rotating the local cylindrical basis through a fixed helix-angle. For a right-handed uniform helix

$z$2

with

$z$3

the helical basis is

$z$4

At each $z$5, $z$6 is an orthonormal, right-handed triad. The geometric torsion is

$z$7

In the same formulation, a co-rotating frame spinning about $z$8 at

$z$9

(minus for a right-handed helix) renders the helix stationary (Castillo-Castellanos et al., 2021).

The Euler-vortex formulation also uses the Frenet-Serret language $z$0, with $z$1 the tangent, $z$2 the normal, and $z$3 the binormal. For a circular cylinder of radius $z$4,

$z$5

supplemented by the usual Frenet-Serret formulae $z$6 (Cao et al., 8 Nov 2025).

3. Coordinate maps, transported fields, and basis changes

For a helical filament of circular cross-section radius $z$7, a convenient global parametrization of the surface is

$z$8

where

$z$9

The matrix $x_3$0 tilts the circular cross-section so that it is perpendicular to $x_3$1 (Liu et al., 2013).

Because every cross-section normal to $x_3$2 is identical up to a rotation about $x_3$3, any vector field on the surface obeys

$x_3$4

With $x_3$5,

$x_3$6

Thus the entire 2D distribution of $x_3$7 is determined by its values on the single “meridian” $x_3$8. In body-fixed coordinates,

$x_3$9

and in the rotating frame all fields lose their $x_3$0-dependence: $x_3$1 This is the key algebraic step behind the dimensional reduction (Liu et al., 2013).

In the vortex-filament stability formulation, the map from cylindrical perturbations $x_3$2 to helical components $x_3$3 is a linear rotation with unit Jacobian: $x_3$4 Conversely, $x_3$5 can be recovered from $x_3$6 by the inverse rotation (Castillo-Castellanos et al., 2021).

A distinct clarification arises in the rotating-helical-container formulation. The carried-over cylindrical basis $x_3$7 remains orthonormal in $x_3$8, with metric

$x_3$9

If one instead introduces a helical-tangent direction via $$H(\varphi)\colon \mathbf{x}\mapsto R^3(\varphi)\,\mathbf{x}+ \begin{pmatrix} 0\ 0\ \frac{\Gamma}{2\pi}\varphi \end{pmatrix}, \qquad R^3(\varphi)= \begin{pmatrix} \cos\varphi & -\sin\varphi & 0\ \sin\varphi & \cos\varphi & 0\ 0 & 0 & 1 \end{pmatrix},$$0, then

$$H(\varphi)\colon \mathbf{x}\mapsto R^3(\varphi)\,\mathbf{x}+ \begin{pmatrix} 0\ 0\ \frac{\Gamma}{2\pi}\varphi \end{pmatrix}, \qquad R^3(\varphi)= \begin{pmatrix} \cos\varphi & -\sin\varphi & 0\ \sin\varphi & \cos\varphi & 0\ 0 & 0 & 1 \end{pmatrix},$$1

lies along the helix. That basis is no longer aligned with coordinate lines but is still orthonormal once properly normalized (Okulov, 2017).

4. Reduction of Stokes boundary integrals

In standard single-layer boundary-element form, the velocity at $$H(\varphi)\colon \mathbf{x}\mapsto R^3(\varphi)\,\mathbf{x}+ \begin{pmatrix} 0\ 0\ \frac{\Gamma}{2\pi}\varphi \end{pmatrix}, \qquad R^3(\varphi)= \begin{pmatrix} \cos\varphi & -\sin\varphi & 0\ \sin\varphi & \cos\varphi & 0\ 0 & 0 & 1 \end{pmatrix},$$2 is

$$H(\varphi)\colon \mathbf{x}\mapsto R^3(\varphi)\,\mathbf{x}+ \begin{pmatrix} 0\ 0\ \frac{\Gamma}{2\pi}\varphi \end{pmatrix}, \qquad R^3(\varphi)= \begin{pmatrix} \cos\varphi & -\sin\varphi & 0\ \sin\varphi & \cos\varphi & 0\ 0 & 0 & 1 \end{pmatrix},$$3

with Stokeslet

$$H(\varphi)\colon \mathbf{x}\mapsto R^3(\varphi)\,\mathbf{x}+ \begin{pmatrix} 0\ 0\ \frac{\Gamma}{2\pi}\varphi \end{pmatrix}, \qquad R^3(\varphi)= \begin{pmatrix} \cos\varphi & -\sin\varphi & 0\ \sin\varphi & \cos\varphi & 0\ 0 & 0 & 1 \end{pmatrix},$$4

Invoking helical symmetry,

$$H(\varphi)\colon \mathbf{x}\mapsto R^3(\varphi)\,\mathbf{x}+ \begin{pmatrix} 0\ 0\ \frac{\Gamma}{2\pi}\varphi \end{pmatrix}, \qquad R^3(\varphi)= \begin{pmatrix} \cos\varphi & -\sin\varphi & 0\ \sin\varphi & \cos\varphi & 0\ 0 & 0 & 1 \end{pmatrix},$$5

it suffices to enforce the boundary integral only on the reference cross-section $$H(\varphi)\colon \mathbf{x}\mapsto R^3(\varphi)\,\mathbf{x}+ \begin{pmatrix} 0\ 0\ \frac{\Gamma}{2\pi}\varphi \end{pmatrix}, \qquad R^3(\varphi)= \begin{pmatrix} \cos\varphi & -\sin\varphi & 0\ \sin\varphi & \cos\varphi & 0\ 0 & 0 & 1 \end{pmatrix},$$6. The resulting one-dimensional integral equation is

$$H(\varphi)\colon \mathbf{x}\mapsto R^3(\varphi)\,\mathbf{x}+ \begin{pmatrix} 0\ 0\ \frac{\Gamma}{2\pi}\varphi \end{pmatrix}, \qquad R^3(\varphi)= \begin{pmatrix} \cos\varphi & -\sin\varphi & 0\ \sin\varphi & \cos\varphi & 0\ 0 & 0 & 1 \end{pmatrix},$$7

or, after defining the modified Stokeslet

$$H(\varphi)\colon \mathbf{x}\mapsto R^3(\varphi)\,\mathbf{x}+ \begin{pmatrix} 0\ 0\ \frac{\Gamma}{2\pi}\varphi \end{pmatrix}, \qquad R^3(\varphi)= \begin{pmatrix} \cos\varphi & -\sin\varphi & 0\ \sin\varphi & \cos\varphi & 0\ 0 & 0 & 1 \end{pmatrix},$$8

$$H(\varphi)\colon \mathbf{x}\mapsto R^3(\varphi)\,\mathbf{x}+ \begin{pmatrix} 0\ 0\ \frac{\Gamma}{2\pi}\varphi \end{pmatrix}, \qquad R^3(\varphi)= \begin{pmatrix} \cos\varphi & -\sin\varphi & 0\ \sin\varphi & \cos\varphi & 0\ 0 & 0 & 1 \end{pmatrix},$$9

In this way the original 2D surface integral collapses to a 1D line integral over $\Gamma=2\pi R\cot\theta$0. The method was assessed against experimental measurements of the motility of model helical flagella of various ratios of pitch to radius, together with predictions from resistive-force theory and slender-body theory, and it was also shown that reliable convergence requires appropriate treatment of the singularities in the kernel. The stated consequence is an enormous reduction of computational cost without loss of geometric detail (Liu et al., 2013).

5. Vortex-filament formulations and long-wave stability

In the filament model for closely spaced co-rotating helical vortices, the co-rotating helical basis is used to simplify the bookkeeping of perturbation expansions and the enforcement of helical periodicity. Once the base flow has been traced out as a helix of radius $\Gamma=2\pi R\cot\theta$1, pitch $\Gamma=2\pi R\cot\theta$2, and frame-rotation $\Gamma=2\pi R\cot\theta$3, every filament node carries its local helical triad $\Gamma=2\pi R\cot\theta$4. In these coordinates the base velocity is purely axial along the helix,

$\Gamma=2\pi R\cot\theta$5

and the linearized kinematic condition is written with components expressed in the helical basis: $\Gamma=2\pi R\cot\theta$6 The Floquet-Bloch ansatz in the arclength-like coordinate $\Gamma=2\pi R\cot\theta$7 becomes a Fourier expansion in $\Gamma=2\pi R\cot\theta$8,

$\Gamma=2\pi R\cot\theta$9

Because $H(\varphi)$0 is tangent to the base filament and $H(\varphi)$1 is binormal to the local stripe of the double helix, symmetric-and-anti-symmetric pairing modes appear naturally as the $H(\varphi)$2- and $H(\varphi)$3-components of the displacement. The corresponding stability analysis identified several kinds of instability modes, including local pairing of consecutive turns of the helical pattern, pairing of consecutive turns of vortex pair for densely braided patterns, and modes that modify the separation distance between the vortices in each pair and amplify specific linear wavelengths (Castillo-Castellanos et al., 2021).

6. Stationary reductions in Euler flows and rotating helical media

For co-rotating nearly parallel helical vortices in the 3D incompressible Euler equations, the co-rotating frame reduces the 3D problem to a 2D semilinear elliptic problem. With the rotating-frame ansatz

$H(\varphi)$4

one is led to

$H(\varphi)$5

where

$H(\varphi)$6

The choice

$H(\varphi)$7

forces each component of vorticity to be a tube of radius $H(\varphi)$8. The leading-order bubbles lie on circles of radius $H(\varphi)$9, and clustered solutions of

$$r'=r,\qquad \phi=\theta+\Omega t,\qquad z'=z+v_z t,\qquad v_z=\frac{\delta\omega}{2k}=\frac{\ell\Omega}{k},$$0

concentrate on $$r'=r,\qquad \phi=\theta+\Omega t,\qquad z'=z+v_z t,\qquad v_z=\frac{\delta\omega}{2k}=\frac{\ell\Omega}{k},$$1 points at mutual distance $$r'=r,\qquad \phi=\theta+\Omega t,\qquad z'=z+v_z t,\qquad v_z=\frac{\delta\omega}{2k}=\frac{\ell\Omega}{k},$$2. In the original 3D variables these points generate $$r'=r,\qquad \phi=\theta+\Omega t,\qquad z'=z+v_z t,\qquad v_z=\frac{\delta\omega}{2k}=\frac{\ell\Omega}{k},$$3 helical tubes co-rotating with angular speed $$r'=r,\qquad \phi=\theta+\Omega t,\qquad z'=z+v_z t,\qquad v_z=\frac{\delta\omega}{2k}=\frac{\ell\Omega}{k},$$4. The finite-dimensional reduction is organized through a Green-function ansatz and a reduced interaction Hamiltonian $$r'=r,\qquad \phi=\theta+\Omega t,\qquad z'=z+v_z t,\qquad v_z=\frac{\delta\omega}{2k}=\frac{\ell\Omega}{k},$$5, and the construction is applied to five explicit configurations, including the $$r'=r,\qquad \phi=\theta+\Omega t,\qquad z'=z+v_z t,\qquad v_z=\frac{\delta\omega}{2k}=\frac{\ell\Omega}{k},$$6th roots of unity, an $$r'=r,\qquad \phi=\theta+\Omega t,\qquad z'=z+v_z t,\qquad v_z=\frac{\delta\omega}{2k}=\frac{\ell\Omega}{k},$$7-gon plus a center vortex, two asymmetric vortices, two concentric squares, and the “$$r'=r,\qquad \phi=\theta+\Omega t,\qquad z'=z+v_z t,\qquad v_z=\frac{\delta\omega}{2k}=\frac{\ell\Omega}{k},$$8” configuration (Cao et al., 8 Nov 2025).

In rotating helical containers, the same co-rotating idea is used to rewrite the Gross-Pitaevskii equation in time-independent form: $$r'=r,\qquad \phi=\theta+\Omega t,\qquad z'=z+v_z t,\qquad v_z=\frac{\delta\omega}{2k}=\frac{\ell\Omega}{k},$$9 with $\{\hat e_\theta,\hat e_z\}$00 and $\{\hat e_\theta,\hat e_z\}$01. In the form emphasized in most of the paper, the translation along $\{\hat e_\theta,\hat e_z\}$02 is absorbed into a redefinition of $\{\hat e_\theta,\hat e_z\}$03, yielding the usual rotating-frame term $\{\hat e_\theta,\hat e_z\}$04. The corresponding helical analog of the Hess-Fairbank effect is that the Bose condensate stays at rest in the laboratory frame, but in the co-moving helical frame appears to slide along the axis with velocity

$\{\hat e_\theta,\hat e_z\}$05

whereas classical fluids remain fully dragged by the walls. Within the present corpus, this makes clear that the co-rotating helical basis is not restricted to low-Reynolds-number hydrodynamics or vortex-filament theory; it also serves as the natural stationary frame for helical confinement and helical superflow (Okulov, 2017).