Generation of a magnetic field by a double vortex in a rotating cylinder

In this paper we use simulations of the magnetohydrodynamic equations coupled with heat to show the generation of magnetic field by the dynamical interaction of a pair of vortices in a fluid electrically conducting within a cylindrical domain nonhomogeneously heated from below, setting in a rotation frame. For large enough rotation rates we show that the formation of a pair of vortices inside the primary whirl gives rise to a magnetic field. The magnetic field has a strong horizontal component, and the magnetic lines go from one vortex to the other.

Figure 1

Physical setup.

Figure 2

Stability diagram for ${\text{Pr}}_m=0.1$, varying the Ekman number Ek and $\text{Ra}/{\text{Ra}}_c$ where ${\text{Ra}}_c$ is the critical Rayleigh number for the onset of convection. The shaded region corresponds to the zone where the dynamo is found.

Figure 3

Spectral decomposition of the kinematic energy $E_K$ and the magnetic energy $E_M$ with the azimuthal wave number $k$ for different values of Ra and $\text{Ek}=0.012$. (a) $\text{Ra}=2000$; (b)$\text{Ra}=7700$; (c) $\text{Ra}=10000$; (d) $\text{Ra}=20000$.

Figure 4

Evolution of modal kinematic energies $E_K^0,E_K^1, E_K^2$ and modal magnetic energies $E_M^0,E_M^1$ with the Rayleigh number Ra for $\text{Ek}=0.012$. Velocity field $u_r-u_{\varphi }$ in the horizontal plane at $z=0.39$ at a particular instant of time. (a) $\text{Ra}=2000$; (b) $\text{Ra}=3000$; (c) $\text{Ra}=10000$; (d) $\text{Ra}=20000$.

Figure 5

(a) Time series for $u_{\varphi }(0.11,0.39)$ at $\text{Ra}=10000$ and $\text{Ek}=0.012$, showing a periodic behavior. (b) Time series for $B_{\varphi }(0.11,0.39)$ at $\text{Ra}=10000$ and $\text{Ek}=0.012$, showing a periodic behavior.

Figure 6

(a) Time series for $u_{\varphi }(0.11,0.39)$ at $\text{Ra}=20000$ and $\text{Ek}=0.012$, showing a periodic behavior. (b) Time series for $B_{\varphi }(0.11,0.39)$ at $\text{Ra}=20000$ and $\text{Ek}=0.012$, showing a periodic behavior.

Figure 7

Average values of volume integral of each term in Eq. (4) as a function of the Rayleigh number Ra and $E=0.012$. (a) Pressure force $\mathbf{\nabla }p$. (b) Viscous force ${\nabla }^2\mathbf{u}$. (c) Inertial force $(\mathbf{u}·\mathbf{\nabla })\mathbf{u}$. (d) Coriolis force $\mathbf{F}$. (e) Buoyancy force $\text{Ra}T{\mathbf{e}}_z$. (f) Lorentz force $\frac{1}{{\text{Pr}}_m}\left(\mathbf{B}·\mathbf{\nabla }\right)\mathbf{B}-\frac{1}{2{\text{Pr}}_m}\mathbf{\nabla }{\mathbf{B}}^2$.

Figure 8

(a) Velocity field $u_r-u_{\varphi }$. (b) Magnetic field $B_r-B_{\varphi }$ and contour of pressure $p$. Plots are done at an instant of the period, horizontal plane at $z=0.39, \text{Ra}=10000,$ and $\text{Ek}=0.012$.

Figure 9

(a) Contour of the $L_2$ norm of $\mathbf{B}$ in the horizontal plane. (b) Contour of the $L_2$ norm of $\mathbf{B}$ in the vertical plane. (c) Contour of the radial component of $\mathbf{B}, B_r$ in the horizontal plane. (d) Contour of the azimuthal component of $\mathbf{B}, B_{\varphi }$ in the horizontal plane. (e) Contour of the vertical component of $\mathbf{B}, B_z$ in the horizontal plane. (f) Streamlines of $\mathbf{B}$ in the horizontal plane. (g) Streamlines of $\mathbf{B}$ in the vertical plane. Plots are done at an instant of the period, horizontal plane at $z=0.39, r-z$ plane at the angle $\varphi$ sectioning the two vortices, $\text{Ra}=10000$ and $\text{Ek}=0.012$.

Figure 10

Three-dimensional contour representation of the axial vorticity (red contour) and the $L_2$ norm of the magnetic field $\mathbf{B}$ (yellow contour) for $\text{Ra}=10000$ and $E=0.012$ at a particular instant of the period. The red contour value is 51, and the yellow contour value is 3.5.

Figure 11

(a) Velocity field $u_r-u_{\varphi }$. (b) Magnetic field $B_r-B_{\varphi }$ and contour of pressure $p$. Plots are done at an instant of the period, horizontal plane at $z=0.39, \text{Ra}=20000$, and $\text{Ek}=0.012$.

Figure 12

(a) Contour of the $L_2$ norm of $\mathbf{B}$ in the horizontal plane. (b) Contour of the $L_2$ norm of $\mathbf{B}$ in the vertical plane. (c) Contour of the radial component of $\mathbf{B}, B_r$ in the horizontal plane. (d) Contour of the azimuthal component of $\mathbf{B}, B_{\varphi }$ in the horizontal plane. (e) Contour of the vertical component of $\mathbf{B}, B_z$ in the horizontal plane. (f) Streamlines of $\mathbf{B}$ in the horizontal plane. (g) Streamlines of $\mathbf{B}$ in the vertical plane. Plots are done at an instant of the period, horizontal plane at $z=0.39, r-z$ plane at the angle $\varphi$ sectioning the two vortices, $\text{Ra}=20000$ and $\text{Ek}=0.012$.

Figure 13

Three-dimensional contour representation of the axial vorticity (red contour) and the $L_2$ norm of the magnetic field $\mathbf{B}$ (yellow contour) for $\text{Ra}=20000$ and $\text{Ek}=0.012$ at a particular instant of the period. The red contour value is 60, and the yellow contour value is 4.