Three-dimensional direct numerical simulation of free-surface magnetohydrodynamic wave turbulence

We report on three-dimensional direct numerical simulation of wave turbulence on the free surface of a magnetic fluid subjected to an external horizontal magnetic field. A transition from capillary-wave turbulence to anisotropic magneto-capillary wave turbulence is observed for an increasing field. At high enough field, wave turbulence becomes highly anisotropic, cascading mainly perpendicularly to the field direction, in good agreement with the prediction of a phenomenological model, and with anisotropic Alfvén wave turbulence. Although surface waves on a magnetic fluid are different from Alfvén waves in plasma, a strong analogy is found with similar wave spectrum scalings and similar magnetic-field dependent dispersionless wave velocities.

Figure 1

Free surface gradient at a fixed time in the steady state ($t=250$) for different values of $B$, i.e., different $V_A$: $V_A^2=$ (a) 0, (b) 25, (c) 100, and (d) 300. $B$ is along the $x$-axis. Lin-scale color bar.

Figure 2

(a)–(c) Power spectra $S(k_x,\omega )$ of surface elevations in the field direction ($x$-axis) for different $B$, i.e., different $V_A$. Log-color bar. The red-dashed lines correspond to Eq. (1) (dispersion relation) and white dash-dotted lines to nondispersive wave propagation, $\omega =V_A{k}_x$. (d)–(f) Power spectra $S(k_y,\omega )$ of waves traveling along $y$-axis. The red dashed lines correspond to $\omega =k_y^{3/2}$ and $\omega =k_y^{3/2}/2^{1/2}$. (g) and (h): Cross sections $S(k_x,k_y,{\omega }^{★})$ of the power spectrum at a fixed angular frequency ${\omega }^{★}=600$ for different fields $V_A^2=0$ and $V_A^2=300$, respectively. Red-dashed lines correspond to $\left|\mathit{k}\right({\omega }^{★}\left)\right|$ using Eq. (1).

Figure 3

Power spectra $S\left(k\right)$ of surface elevations for different $V_A$. Curves have been shifted for clarity. Dashed line: capillary-wave turbulence theory of Eq. (2) [4]. Solid line: anisotropic MHD wave turbulence theory of Eq. (3). Inset: Spectra in the field direction $S\left(k_x\right)$ (green line) and normal to the field $S\left(k_y\right)$ (blue line). $V_A^2=300$.

Figure 4

Power spectra $S\left(\omega \right)$ of surface elevations for different $V_A$. Curves have been shifted for clarity. Dashed line: capillary-wave turbulence theory in ${\omega }^{-17/6}$ from Eq. (2) using $\omega \sim k^{-3/2}$ [4]. Solid line: predictions for anisotropic MHD surface wave turbulence from Eq. (3) using $\omega \sim k^{-3/2}$. Inset: spectrum coefficients $C_k$ (square), $C_{k_y}$ (diamond) and $C_{\omega }$ (circle) versus (a) $P$ for fixed $V_A^2=300$, and (b) $V_A$ for fixed $P\approx 2.7\times {10}^{-3}$. Solid lines: best fits in $P^{1/2}{V}_A^{-3/2}$.