Optimal velocity fields for instantaneous magnetic field growth

Abstract

We consider a variant of the kinematic dynamo problem. Rather than prescribing a velocity field and searching for high-growth magnetic fields via an eigenvalue problem, we treat the seed magnetic-field structure as given and ask which velocity field maximally enhances its instantaneous growth. We show this second problem has an elegant formulation in terms of variational calculus. Upon simultaneously constraining the velocity's kinetic energy and enstrophy, the Euler-Lagrange equation leads to a forced Helmholtz partial differential equation (PDE) for the optimal velocity field. For the special case of fixed kinetic energy and unconstrained enstrophy, the optimal velocity field everywhere opposes the divergence-free projection of the Lorentz force. In the more general setting, the optimal velocity field can be found through numerical solution of the forced Helmholtz PDE. We construct 2.5-dimensional numerical examples to support the theoretical findings.

Figures (5)

• Figure 1: Numerical test with $k^2_{\text{max}}=1$ and a randomly selected seed magnetic field. (a) Visualization of the solenoidal magnetic field ${\mathbf{B}}$ selected through random sampling of Fourier coefficients; color shows the out-of-plane component of the vector field. (b) The corresponding optimal velocity field ${\mathbf{u}}_{\text{opt}}$ determined through the numerical solution of \ref{['PDE2']}. This combination of ${\mathbf{B}}$ and ${\mathbf{u}}_{\text{opt}}$ yields a magnetic-energy growth rate of $\dot{M}_{\text{opt}}=1.05$. (c) Histogram of $\dot{M}$ values produced by the same ${\mathbf{B}}$ and an ensemble random fields ${\mathbf{u}}_{\text{rand}}$ selected from the same space as ${\mathbf{u}}_{\text{opt}}$. The vertical dashed lines show $\pm \dot{M}_{\text{opt}}$ and confirm that ${\mathbf{u}}_{\text{opt}}$ yields the optimal magnetic-energy growth rate for the specified ${\mathbf{B}}$.
• Figure 2: Numerical test with $k^2_{\text{max}}=2$ and a randomly selected seed magnetic field. (a) Visualization of the randomly selected magnetic field ${\mathbf{B}}$, and (b) the corresponding optimal velocity field ${\mathbf{u}}_{\text{opt}}$ with spectral truncation applied. This combination of ${\mathbf{B}}$ and ${\mathbf{u}}_{\text{opt}}$ yields $\dot{M}_{\text{opt}}=0.61$. (c) Histogram of $\dot{M}$ values produced by the same ${\mathbf{B}}$ and an ensemble random fields ${\mathbf{u}}_{\text{rand}}$ selected from the same space as the truncated ${\mathbf{u}}_{\text{opt}}$. The vertical dashed lines show $\pm \dot{M}_{\text{opt}}$ and confirm that ${\mathbf{u}}_{\text{opt}}$ yields the optimal magnetic-energy growth rate for the specified ${\mathbf{B}}$.
• Figure 3: Numerical test with $k^2_{\text{max}}=1$ and with ${\mathbf{B}} = {\mathbf{B}}_{\text{opt}}$ numerically optimized for maximal $\dot{M}$. (a) Visualization of the numerically optimized ${\mathbf{B}}_{\text{opt}}$, and (b) the companion optimal velocity field ${\mathbf{u}}_{\text{opt}}$. This combination of ${\mathbf{B}}_{\text{opt}}$ and ${\mathbf{u}}_{\text{opt}}$ produces $\dot{M}_{\text{opt}}=1.41$. (c) Histogram of $\dot{M}$ values produced by the same ${\mathbf{B}}_{\text{opt}}$ and an ensemble random fields ${\mathbf{u}}_{\text{rand}}$. The vertical dashed lines show $\pm \dot{M}_{\text{opt}}$ and confirm that ${\mathbf{u}}_{\text{opt}}$ yields the optimal magnetic-energy growth rate.
• Figure 4: Numerical test with $k^2_{\text{max}}=2$ and with ${\mathbf{B}} = {\mathbf{B}}_{\text{opt}}$ numerically optimized for maximal $\dot{M}$. (a) Visualization of the numerically optimized ${\mathbf{B}}_{\text{opt}}$, and (b) the companion optimal velocity field ${\mathbf{u}}_{\text{opt}}$. This combination of ${\mathbf{B}}_{\text{opt}}$ and ${\mathbf{u}}_{\text{opt}}$ produces $\dot{M}_{\text{opt}}=1.83$. (c) Histogram of $\dot{M}$ values produced by the same ${\mathbf{B}}_{\text{opt}}$ and an ensemble random fields ${\mathbf{u}}_{\text{rand}}$. The vertical dashed lines show $\pm \dot{M}_{\text{opt}}$ and confirm that ${\mathbf{u}}_{\text{opt}}$ yields the optimal magnetic-energy growth rate.
• Figure 5: Numerical test with $k^2_{\text{max}}=4$ and with ${\mathbf{B}} = {\mathbf{B}}_{\text{opt}}$ numerically optimized for maximal $\dot{M}$. (a) Visualization of the numerically optimized ${\mathbf{B}}_{\text{opt}}$, and (b) the companion optimal velocity field ${\mathbf{u}}_{\text{opt}}$. This combination of ${\mathbf{B}}_{\text{opt}}$ and ${\mathbf{u}}_{\text{opt}}$ produces $\dot{M}_{\text{opt}}=2.07$. (c) Histogram of $\dot{M}$ values produced by the same ${\mathbf{B}}_{\text{opt}}$ and an ensemble random fields ${\mathbf{u}}_{\text{rand}}$. The vertical dashed lines show $\pm \dot{M}_{\text{opt}}$ and confirm that ${\mathbf{u}}_{\text{opt}}$ yields the optimal magnetic-energy growth rate.