Pointwise vanishing velocity helicity of a flow does not preclude magnetic field generation

Pointwise zero velocity helicity density is shown not to prevent steady flows from acting as kinematic dynamos. We present numerical evidences that such flows can generate both small-scale magnetic fields as well as, by the magnetic $\alpha$-effect or negative eddy diffusivity mechanisms, large-scale ones. The flows are constructed as curls of analytically defined space-periodic steady solenoidal flows, whose vorticity helicity (i.e., kinetic helicity) density is everywhere zero.

Figure 1

Helicity spectrum seminorms ${\sum }_{M-1<\left|\mathbf{m}\right|\le M}\left|H_{\mathbf{m}}\left(\mathbf{v}\right)\right|$ (vertical axis) vs. $M$ (horizontal axis) for a sample flow Eq. (3) constructed for a potential $\mathbf{w}$ from family L [9] (see the definition in Sec. 3.

Figure 2

Graphs of $U_1$ (solid lines), $U_2$ (dashed lines) and $U_3$ (dotted lines, vertical axes) as functions of the respective Cartesian coordinate variable $x_i$ (horizontal axis) used to construct the zero velocity helicity sample flows (13.1) and (13.2) for ${\mathrm{V}}_1$ (a) and ${\mathrm{V}}_2$ (b) family potentials $\mathbf{w}$ (see (3)), (14.1) and (14.2), respectively.

Figure 3

Maximum slow-time growth rates Eq. (7) of large-scale modes generated by the $\alpha$-effect in sample flows Eq. (3) of zero velocity helicity for $\mathbf{w}$ from families ${\mathrm{V}}_1$ (a), ${\mathrm{V}}_2$ (b), and L (c) as functions of the molecular diffusivity $\eta$. Insets in panels (b) and (c) show zooms of the graphs near the critical diffusivity (indicated by thin vertical lines) for the onset of the small-scale dynamo action. The resolution of computations is ${128}^3$ (solid dots) or ${256}^3$ (hollow circles) Fourier harmonics. Large solid dots (red in the electronic version of the paper) in (a) show computations in which the ${128}^3$ harmonics resolution is insufficient.

Figure 4

Graphs of $10{max}_{\left|\mathbf{q}\right|=1}\mathrm{Re}{\lambda }_1\left(\mathbf{q}\right)$ (dashed line), ${max}_{\left|\mathbf{q}\right|=1}\mathrm{Im}{\lambda }_1\left(\mathbf{q}\right)$ (solid line) and their ratio (dotted line) as functions of the molecular eddy diffusivity $\eta$ (horizontal axis). Small circles show the computed values. Thin vertical line is the asymptote for the maximum growth rate ${max}_{\left|\mathbf{q}\right|=1}\mathrm{Re}{\lambda }_1\left(\mathbf{q}\right)$ located at the critical molecular diffusivity $\eta ={\eta }^{\mathrm{cr}}$, for the onset of the small-scale dynamo.

Figure 5

Dominant growth rates ${\gamma }_{\mathrm{sm}}$ of small-scale modes versus minimum eddy diffusivity Eq. (26) in the primary cosine flows Eqs. (24) (filled circles) and (25) (hollow circles, red in the electronic version of the paper) for $\eta =0.01$ (a) and 0.02 (b).