Interactions between vortex tubes and magnetic-flux rings at high kinetic and magnetic Reynolds numbers

The interactions between vortex tubes and magnetic-flux rings in incompressible magnetohydrodynamics are investigated at high kinetic and magnetic Reynolds numbers, and over a wide range of the interaction parameter. The latter is a measure of the turnover time of the large-scale fluid motions in units of the magnetic damping time, or of the strength of the Lorentz force in units of the inertial force. The small interaction parameter results, which are related to kinematic turbulent dynamo studies, indicate the evolution of magnetic rings into flattened spirals wrapped around the vortex tubes. This process is also observed at intermediate interaction parameter values, only now the Lorentz force creates new vortical structures at the magnetic spiral edges, which have a striking solenoid vortex-line structure, and endow the flattened magnetic-spiral surfaces with a curvature. At high interaction parameter values, the decisive physical factor is Lorentz force effects. The latter create two (adjacent to the magnetic ring) vortex rings that reconnect with the vortex tube by forming an intriguing, serpentinelike, vortex-line structure, and generate, in turn, two new magnetic rings, adjacent to the initial one. In this regime, the morphologies of the vorticity and magnetic field structures are similar. The effects of these structures on kinetic and magnetic energy spectra, as well as on the direction of energy transfer between flow and magnetic fields, are also indicated.

Figure 1

Left: Vorticity magnitude isosurfaces at level equal to 0.38 times its maximum value, for a homogeneous, isotropic, incompressible Navier-Stokes turbulence at Taylor Reynolds number ${\text{Re}}_{\lambda }\approx 100$. Results taken from an accompanying incompressible (non-MHD) turbulence calculation. Although sheetlike structures are present, we predominantly see linear vortices. Right: The initial configuration consists of a straight vortex tube, and a magnetic-flux ring, in a periodic box. Both isosurfaces correspond to levels equal to half of the corresponding maximum field-magnitude values.

Figure 2

$N=2.5\times {10}^{-3}$. Left: Magnetic-flux ring configurations for three different times: $t=0$ (undisturbed ring), $t=31.45{\tau }_{\mathrm{d}}$ (intermediate ring, just started forming a spiral), and $t=106.25{\tau }_{\mathrm{d}}$ (ring warped into a spiral structure). Here, $t=0$ graph simultaneously shows isosurfaces for five values spanning the range $0.5–1$ of maximum magnetic field magnitude, $t=31.45{\tau }_{\mathrm{d}}$ graph simultaneously shows isosurfaces for five values spanning the range $0.55–1$ of maximum magnetic field magnitude, and $t=106.25{\tau }_{\mathrm{d}}$ graph simultaneously shows isosurfaces for eight values spanning the range $0.8–1$ of maximum magnetic field magnitude. Right: Tube-ring configuration at $t=106.25{\tau }_{\mathrm{d}}$: the ring is wrapped around a virtually undisturbed vortex tube, with straight vortex lines. For the magnetic field, isosurfaces for ten values spanning the whole range between minimum and maximum magnetic-field magnitudes are shown simultaneously, while for the vorticity field, a single isosurface corresponding to half of the maximum vorticity magnitude is shown. Moreover, vortex and magnetic field lines (that are computed via the Runge-Kutta-Fehlberg numerical integration method) are also depicted. The field-lines color code is the same for both vorticity and magnetic-field lines. Smaller magnitudes correspond to blue color and larger magnitudes to red color.

Figure 3

$N=2.5\times {10}^{-3},t=106.25{\tau }_{\mathrm{d}}$. Left: Magnetic field spiral, and superposed field lines. Highest field curvature is associated with the, initially, closest and furthest from the vortex tube parts of the ring, and highest field-magnitude values are located within the flattened arms of the spiral. Right: Magnetic-field isosurfaces (low opacity), and current lines: relatively weak current engulfs the magnetic field structure, and the highest current values are associated with current loops formed within the structure. In both figures, isosurfaces for ten values spanning the whole range between minimum and maximum magnetic field magnitudes are shown simultaneously. The field lines are computed via the Runge-Kutta-Fehlberg numerical integration method.

Figure 4

$N=2.5\times {10}^{-3}$. Left: Scaled $\tilde{B}$ spectrum at $t=106.25{\tau }_{\mathrm{d}}$. Right: Evolution of (scaled) $\tilde{B}$ energy with time. From lower to upper curves, we move from low to high interaction parameter $N$ values.

Figure 5

Left: Scaled spectra of kinetic energy dissipation rate $\epsilon [N=2.5\times {10}^{-3}$ (lower curve), $N=2.5\times 10$ (middle curve), $N=2.5\times {10}^3$ (upper curve)]. Middle: Scaled spectra of ${\tilde{\mathbf{J}}}^2$ (proportional to magnetic-energy dissipation) $[N=2.5\times {10}^{-3}$ (lower curve), $N=2.5\times 10$ (middle curve), $N=2.5\times {10}^3$ (upper curve)]. Right: Scaled cross helicity ${\mathrm{H}}^{\mathrm{C}}=\mathbf{u}·\mathbf{B}$ versus scaled time $[N=2.5\times {10}^{-3}$ (horizontal curve), $N=2.5\times 10$ (curve close to the horizontal), $N=2.5\times {10}^3$ (strongly varying curve)]. From small to large interaction parameter values, the corresponding times are $t=106.25{\tau }_d,t=102.38{\tau }_d,t=58.29{\tau }_d$.

Figure 6

Probability density functions of ${\lambda }^{*}$ parameter for $N=2.5\times {10}^{-3}$ (left), $N=2.5\times 10$ (middle), and $N=2.5\times {10}^3$ (right). From small to large interaction parameter values, the corresponding times are $t=106.25{\tau }_d,t=102.38{\tau }_d,t=58.29{\tau }_d$.

Figure 7

Probability density functions of vorticity/strain-rate-eigenvector cosines for $N=2.5\times {10}^{-3}$ at time $t=106.25{\tau }_{\mathrm{d}}$. From left to right: $\cos (\mathbf{\omega },{\mathbf{\lambda }}_1),\cos (\mathbf{\omega },{\mathbf{\lambda }}_2)$, and $\cos (\mathbf{\omega },{\mathbf{\lambda }}_3)$.

Figure 8

Left: Probability density function of scaled enstrophy source $\frac{\langle {\omega }_i{\omega }_j{S}_{ij}\rangle }{\langle {\omega }_i{\omega }_i\rangle {\langle S_{ij}{S}_{ij}\rangle }^{1/2}}$. Right: Probability density function of vorticity and vortex-stretching vector cosine $\cos (\mathbf{\omega },\mathbf{W})$. Both results are for $N=2.5\times {10}^{-3}$ at time $t=106.25{\tau }_{\mathrm{d}}$.

Figure 9

Probability density functions of magnetic-field and strain-rate-eigenvector cosines for $N=2.5\times {10}^{-3}$ at time $t=106.25{\tau }_{\mathrm{d}}$. From left to right: $\cos (\mathbf{B},{\mathbf{\lambda }}_1),\cos (\mathbf{B},{\mathbf{\lambda }}_2)$, and $\cos (\mathbf{B},{\mathbf{\lambda }}_3)$.

Figure 10

Left: Probability density function of scaled magnetic-energy source $\frac{\langle B_i{B}_j{S}_{ij}\rangle }{\langle B_i{B}_i\rangle {\langle S_{ij}{S}_{ij}\rangle }^{1/2}}$. Right: Probability density function of magnetic-field and magnetic-field stretching vector cosine $\cos (\mathbf{B},\mathbf{M})$. Both results are for $N=2.5\times {10}^{-3}$ at time $t=106.25{\tau }_{\mathrm{d}}$.

Figure 11

$N=2.5\times 10,t=102.38{\tau }_{\mathrm{d}}$. Left: Under the influence of the vortex-tube, the magnetic-flux ring forms a spiral. In contrast to the small interaction number case, the spiral arms are not flat but present well-formed bulges at their boundaries. Right: The magnetic field lines are smooth (without any reversals) within the spiral arms. The highest field values are attained within the bulges. In both figures, isosurfaces for ten values spanning the whole range between minimum and maximum magnetic field magnitudes are shown simultaneously. The field lines are computed via the Runge-Kutta-Fehlberg numerical integration method.

Figure 12

$N=2.5\times 10,t=102.38{\tau }_d$. Left: Magnetic-field isosurfaces (low opacity) shown together with vorticity isosurfaces. The highest vorticity values are located at the magnetic-field bulges. The vortex tube presents two indentations at the areas where the magnetic spiral is closest to it. Middle: Intriguing solenoidlike structure of vorticity-field lines (colored field lines) located at the magnetic field bulges (background isosurface). It indicates strong Lorentz force effects. The color code refers to the vorticity-field magnitude, with the blue color corresponding to the smaller values and the red color to the larger values. Not all vortex lines within the solenoid have equally large vorticity magnitudes with those shown here. Right: Velocity isosurface at level equal to 0.5 of maximum value. The large (cylindrical) isosurface corresponds to the (undisturbed) potential vortex-tube-induced flow. The smaller inner cylinder is within the tube's solid-body rotation region. The superposed vortex-line solenoid helps to indicate the Lorentz-force-induced vortex-structure with increased (relatively to the undisturbed vortex flow) velocity there. The field lines are computed via the Runge-Kutta-Fehlberg numerical integration method.

Figure 13

$N=2.5\times 10,t=102.38{\tau }_d$. Left: Similarly with the small interaction number case, the current lines (colored tubes) engulf the arms of the magnetic spiral. However, at places, they acquire a solenoid like structure within the vorticity solenoid structure (white tubes). The current solenoids are associated with the highest current values. Right: Isosurface of vorticity (at a level equal to 0.83 of maximum value) located within the initial tube area. Due to Lorentz force action, the magnetic spiral acts like a vortex-tube constrictor, deforming the (initially cylindrical) isosurface. The field lines are computed via the Runge-Kutta-Fehlberg numerical integration method.

Figure 14

$N=2.5\times 10,t=102.38{\tau }_d$. Left: Kinetic energy spectra for $N=2.5\times {10}^{-3}$ (faster cutoff), and $N=2.5\times 10$. In latter case, there is evidence of a high wave number $k^{-4}$ scaling regime induced by the magnetic field's reaction to the flow, which follows a steeper $k^{-5}$ regime that appears in both spectra, and corresponds to the tube core. Right: Normalized magnetic energy spectra. The $N=2.5\times {10}^{-3}$ results (triangles) have been multiplied by a large factor to compare them with the $N=2.5\times 10$ results (circles).

Figure 15

Probability density functions of vorticity and strain-rate-eigenvector cosines for $N=2.5\times 10$ at time $t=102.38{\tau }_d$. From left to right: $\cos (\mathbf{\omega },{\mathbf{\lambda }}_1),\cos (\mathbf{\omega },{\mathbf{\lambda }}_2)$, and $\cos (\mathbf{\omega },{\mathbf{\lambda }}_3)$.

Figure 16

Left: Probability density function of scaled enstrophy source $\frac{\langle {\omega }_i{\omega }_j{S}_{ij}\rangle }{\langle {\omega }_i{\omega }_i\rangle {\langle S_{ij}{S}_{ij}\rangle }^{1/2}}$. Right: Probability density function of vorticity and vortex-stretching vector cosine $\cos (\mathbf{\omega },\mathbf{W})$. Both results are for $N=2.5\times 10$ at time $t=102.38{\tau }_d$.

Figure 17

Probability density functions of magnetic-field and strain-rate-eigenvector cosines for $N=2.5\times 10$ at time $t=102.38{\tau }_d$. From left to right: $\cos (\mathbf{B},{\mathbf{\lambda }}_1),\cos (\mathbf{B},{\mathbf{\lambda }}_2)$, and $\cos (\mathbf{B},{\mathbf{\lambda }}_3)$.

Figure 18

Left: Probability density function of scaled magnetic-energy source $\frac{\langle B_i{B}_j{S}_{ij}\rangle }{\langle B_i{B}_i\rangle {\langle S_{ij}{S}_{ij}\rangle }^{1/2}}$. Right: Probability density function of magnetic-field and magnetic-field stretching vector cosine $\cos (\mathbf{B},\mathbf{M})$. Both results are for $N=2.5\times 10$ at time $t=102.38{\tau }_d$.

Figure 19

$N=2.5\times {10}^3,t=5.07{\tau }_d$. Left: The electrodynamic vorticity source creates two vortex rings that “sandwich” the initial magnetic ring (middle structure). One of the newly created vortex rings reconnects with the tube. Right: Although the vortex lines within the vortex rings are smooth and circular, the ring-tube reconnection region is characterized by highly coiled vortex lines. Such a coiled vortex line has reconnected with two vortex lines that (in the undisturbed flow field) ran parallel to the tube. The vorticity isosurface is drawn at level equal to 0.32 of the maximum value, and the magnetic-field isosurface at level equal to 0.58 of the maximum value. The field lines are computed via the Runge-Kutta-Fehlberg numerical integration method.

Figure 20

$N=2.5\times {10}^3,t=5.07{\tau }_d$. The serpentine reconnection: a vortex line belonging to a Lorentz-force induced ring in Fig. 19 approaches the tube-ring contact point in a way reminiscent of a serpentine. The vorticity magnitudes within the serpentine are orders of magnitude higher than the corresponding ones within the tube. The vorticity isosurface is drawn at a level equal to 0.32 of the maximum value. The field-lines are computed via the Runge-Kutta-Fehlberg numerical integration method.

Figure 21

$N=2.5\times {10}^3,t=13.52{\tau }_d$. Left: Magnetic field structure: the initial ring is located at the center of the structure having evolved into a sheet that is drawn toward the vortex tube. Two new magnetic rings are collocated with the two new vortex rings created by the Lorentz force. Right: Vorticity field structure: A rare example where a vortex sheet, two vortex rings, and a (deformed) vortex tube are interwoven together. The vortex sheet is collocated with the magnetic sheet in the left graphic. The two vortex rings are created by the Lorentz force and are also collocated with the two magnetic rings in the left graphic. The maximum vorticity value within the induced vortex rings is 3.35 times larger than the highest vorticity value within the initial vortex tube. In both figures, isosurfaces for ten values spanning the whole range between minimum and maximum magnetic-field (left) and vorticity-field (right) magnitudes are shown simultaneously.

Figure 22

$N=2.5\times {10}^3,t=58.29{\tau }_d$. Left: Normalized kinetic energy spectra for $N=2.5\times 10$ (circles), and $N=2.5\times {10}^3$ (triangles). In latter case, there is evidence of a high wave number $k^{-5/2}$ scaling regime. Right: Normalized magnetic energy spectra. The $N=2.5\times {10}^3$ results (upper curve, triangles) differ from the $N=2.5\times 10$ results (circles), in that the plateau region after the peak in the latter, has now evolved into a new (displaced) peak, directly followed by dissipative cutoff.

Figure 23

Probability density functions of vorticity and strain-rate-eigenvector cosines for $N=2.5\times {10}^3$ at time $t=58.29{\tau }_d$. From left to right: $\cos (\mathbf{\omega },{\mathbf{\lambda }}_1),\cos (\mathbf{\omega },{\mathbf{\lambda }}_2)$, and $\cos (\mathbf{\omega },{\mathbf{\lambda }}_3)$.

Figure 24

Left: Probability density function of scaled enstrophy source $\frac{\langle {\omega }_i{\omega }_j{S}_{ij}\rangle }{\langle {\omega }_i{\omega }_i\rangle {\langle S_{ij}{S}_{ij}\rangle }^{1/2}}$. Right: Probability density function of vorticity and vortex-stretching vector cosine $\cos (\mathbf{\omega },\mathbf{W})$. Both results are for $N=2.5\times {10}^3$ at time $t=58.29{\tau }_d$.

Figure 25

Probability density functions of magnetic-field and strain-rate-eigenvector cosines for $N=2.5\times {10}^3$ at time $t=58.29{\tau }_d$. From left to right: $\cos (\mathbf{B},{\mathbf{\lambda }}_1),\cos (\mathbf{B},{\mathbf{\lambda }}_2)$, and $\cos (\mathbf{B},{\mathbf{\lambda }}_3)$.

Figure 26

Left: Probability density function of scaled magnetic-energy source $\frac{\langle B_i{B}_j{S}_{ij}\rangle }{\langle B_i{B}_i\rangle {\langle S_{ij}{S}_{ij}\rangle }^{1/2}}$. Right: Probability density function of magnetic-field and magnetic-field stretching vector cosine $\cos (\mathbf{B},\mathbf{M})$. Both results are for $N=2.5\times {10}^3$ at time $t=58.29{\tau }_d$.