Tridimensional to bidimensional transition in magnetohydrodynamic turbulence with a guide field and kinetic helicity injection

We study the transition in dimensionality of a three-dimensional magnetohydrodynamic flow forced only mechanically when the strength of a magnetic guide field is gradually increased. We use numerical simulations to consider cases in which the mechanical forcing injects (or not) helicity in the flow. As the guide field is increased, the strength of the magnetic field fluctuations decreases as a power law of the guide field intensity. We show that for strong enough guide fields the helical magnetohydrodynamic flow can become almost two-dimensional. In this case, the mechanical energy can undergo a process compatible with an inverse cascade, being transferred preferentially towards scales larger than the forcing scale. The presence of helicity changes the spectral scaling of the small magnetic field fluctuations, and affects the statistics of the velocity field and of the velocity gradients. Moreover, at small scales the dynamics of the flow becomes dominated by a direct cascade of helicity, which can be used to derive scaling laws for the velocity field.

Figure 1

Top: Kinetic energy as a function of time for simulations in set A, without helicity. The peak of kinetic energy increases with the strength of the guide field, with simulation A0 (dotted) having $B_0=0$ and simulation A8 (solid) having $B_0=8$. Bottom: Kinetic energy as a function of time for runs A2 (without helicity and with $B_0=2)$ and B2 (with helicity, same $B_0)$.

Figure 2

Dimensionless ratio of r.m.s. magnetic field fluctuations to the amplitude of the guide field, ${\left(2E_b\right)}^{1/2}/B_0$, as a function of $B_0$, for the simulations in Table 1 with ${512}^3$ grid points, and for several simulations with the same configuration but with spatial resolution of ${128}^3$ grid points. A power law resulting from a best fit to the data is shown as a reference.

Figure 3

Top: Kinetic energy spectrum for simulations in set A (without helicity). Bottom: Kinetic energy spectrum (solid) and helicity spectrum normalized by $k_f$ (dotted) for simulations in set B (with helicity). The spectra have been shifted vertically for better visualization, and the slopes indicate several power laws as references (see text for details). The isotropic spectrum $E_v\left(k\right)$ is shown for runs A0, A2, and B2; in all other cases we show $E\left(k_{\perp }\right)$.

Figure 4

Diagram showing a typical flux (either of kinetic energy, magnetic energy, or kinetic helicity) as a function of $k$ for simulations with ${\mathbf{B}}_0\ne 0$. In the diagram we show several characteristic values used for the analysis: the scale injection $k_f$ in which the flux changes sign, the maximum value of positive flux ${\mathrm{\Pi }}^{+}$ (i.e., of flux towards small scales), the minimum value of negative flux $-{\mathrm{\Pi }}^{-}$ (i.e., of inverse flux), and the value of the flux when $k\to k_{\max },\mathrm{\Delta }\mathrm{\Pi }$.

Figure 5

Top: Flux of kinetic energy for runs in set A (without helicity), for different values of $B_0$, and time-averaged for long times on the inverse cascade scales. Bottom: Same for runs in set B (with helicity). The flux is shown as a function of $k$ for run A0 and as a function of $k_{\perp }$ for all other runs. In both cases, ${\mathrm{\Pi }}^{+}$ decreases as $B_0$ is increased, negative values of the flux are observed for $k<k_f$ for large values of $B_0$, and $\mathrm{\Delta }\mathrm{\Pi }$ decreases towards zero.

Figure 6

Ratio of maximum direct helicity flux ${\mathrm{\Pi }}_h^{+}$ to the maximum of direct kinetic energy flux ${\mathrm{\Pi }}_v^{+}$ normalized by $k_f$, as a function of $B_0$.

Figure 7

Top: Compensated spectrum $H\left(k_{\perp }\right)E\left(k_{\perp }\right)/k_{\perp }^{-4}$ for run B8 (with mechanical helical forcing, and $B_0=8)$. Bottom: Spectrum of helicity $H\left(k_{\perp }\right)$ in the same run, compensated by $k_{\perp }^{-3/2}$.

Figure 8

Relative helicity spectrum $h\left(k_{\perp }\right)$ for runs B4 and B8 (both with kinetic helicity, and respectively with $B_0=4$ and 8). A slope $k_{\perp }^{-1}$ is shown as a reference. Note the relative helicity spectrum is shallower than $k_{\perp }^{-1}$ everywhere except in the dissipative range.

Figure 9

Kinetic energy dissipation rate $\left(2\nu \mathrm{\Omega }\right)$ as a function of $B_0$, for simulations with and without helical forcing. The inset shows the kinetic energy dissipation rate normalized by the energy injection rate $\left(\epsilon \right)$ as a function of $B_0$. As expected for the case with negligible magnetic fluctuations, for large $B_0$ this ratio approaches unity, as energy can only be dissipated by velocity fluctuations.

Figure 10

Top: Power spectrum of magnetic field fluctuations for simulations with zero helicity (runs in set A). Bottom: Same for the two simulations with strongest guide field, runs $\text{A}8\left(B_0=8,{\alpha }_h=0\right)$ and $\text{B}8\left(B_0=8,{\alpha }_h=\pi /4\right)$. A slope $k_{\perp }^{-1}$ is shown as a reference.

Figure 11

Top: Probability density functions (PDFs) of the parallel spatial derivative (i.e., the spatial derivative in the direction of the guide field) of a component of the velocity perpendicular to the guide field $\left({\partial }_{\parallel }{v}_{\perp }\right)$, for all runs. Bottom: PDFs of the component of the velocity parallel to the guide field $\left(v_{\parallel }\right)$ for all runs.