Inertial-range dynamics and scaling laws of two-dimensional magnetohydrodynamic turbulence in the weak-field regime

We study inertial-range dynamics and scaling laws in unforced two-dimensional magnetohydrodynamic turbulence in the regime of moderately small and small initial magnetic-to-kinetic-energy ratio $r_0$, with an emphasis on the latter. The regime of small $r_0$ corresponds to a relatively weak field and strong magnetic stretching, whereby the turbulence is characterized by an intense conversion of kinetic into magnetic energy (dynamo action in the three-dimensional context). This conversion is an inertial-range phenomenon and, upon becoming quasisaturated, deposits the converted energy within the inertial range rather than transferring it to the small scales. As a result, the magnetic-energy spectrum $E_{\mathit{b}}\left(k\right)$ in the inertial range can become quite shallow and may not be adequately explained or understood in terms of conventional cascade theories. It is demonstrated by numerical simulations at high Reynolds numbers (and unity magnetic Prandtl number) that the energetics and inertial-range scaling depend strongly on $r_0$. In particular, for fully developed turbulence with $r_0$ in the range $[1/4,1/4096]$, $E_{\mathit{b}}\left(k\right)$ is found to scale as $k^{\alpha }$, where $\alpha \gtrsim -1$, including $\alpha >0$. The extent of such a shallow spectrum is limited, becoming broader as $r_0$ is decreased. The slope $\alpha$ increases as $r_0$ is decreased, appearing to tend to $+1$ in the limit of small $r_0$. This implies equipartition of magnetic energy among the Fourier modes of the inertial range and the scaling $k^{-1}$ of the magnetic potential variance, whose flux is direct rather than inverse. This behavior of the potential resembles that of a passive scalar. However, unlike a passive scalar whose variance dissipation rate slowly vanishes in the diffusionless limit, the dissipation rate of the magnetic potential variance scales linearly with the diffusivity in that limit. Meanwhile, the kinetic-energy spectrum is relatively steep, followed by a much shallower tail due to strong antidynamo excitation. This gives rise to a total-energy spectrum poorly obeying a power-law scaling.

Figure 1

Schematic description of dynamo and antidynamo wave triads $\mathit{k}=\ell +\mathit{m}$ ($\ell =\left|\ell \right|<m=\left|\mathit{m}\right|$) responsible for energy conversion and transfer. The arrows indicate the direction of energy transfer.

Figure 2

Representative initial magnetic potential field (left) and spectrally localized (center) and fully developed (right) initial vorticity fields used in the simulations, presented here for $r_0=1/256$.

Figure 3

Kinetic energy $E_{\mathit{u}}\left(t\right)$, magnetic energy $E_{\mathit{b}}\left(t\right)$, and their ratio $r\left(t\right)=E_{\mathit{b}}/E_{\mathit{u}}$ versus $t$ for numerical simulations with resolutions $1024\times 1024$ (dotted), $2048\times 2048$ (dashed-dotted), $4096\times 4096$ (dashed), and $8192\times 8192$ (solid) and $r_0=1/4,1/16,1/64,1/256,1/1024$. The rows are in decreasing order of $r_0$.

Figure 4

Kinetic-, magnetic-, and total-energy dissipation rates ${\epsilon }_{\mathit{u}}\left(t\right)$, ${\epsilon }_{\mathit{b}}\left(t\right)$, and $\epsilon \left(t\right)$, respectively, versus $t$ for numerical simulations with resolutions $1024\times 1024$ (dotted), $2048\times 2048$ (dashed-dotted), $4096\times 4096$ (dashed), and $8192\times 8192$ (solid) and $r_0=1/4,1/16,1/64,1/256,1/1024$. The rows are in decreasing order of $r_0$.

Figure 5

Evolution of the exponential energy dissipation rates ${\epsilon }_{\mathit{u}}/E_{\mathit{u}}$, ${\epsilon }_{\mathit{b}}/E_{\mathit{b}}$, and $\epsilon /E$ for the highest-resolution runs, with $r_0=1/4$, $1/16$, $1/64$, $1/256$, and $1/1024$ given by the dashed-double-dotted, dotted, dashed-dotted, dashed, and solid lines, respectively.

Figure 6

Kinetic-, magnetic-, and total-energy spectra $E_{\mathit{u}}\left(k\right)$, $E_{\mathit{b}}\left(k\right)$, and $E\left(k\right)$, respectively, for fully developed turbulence (at $t=2.5$), for numerical simulations with resolutions $1024\times 1024$ (dotted), $2048\times 2048$ (dashed-dotted), $4096\times 4096$ (dashed), and $8192\times 8192$ (solid) and $r_0=1/4,1/16,1/64,1/256,1/1024$. The rows are in decreasing order of $r_0$. The reference lines have a slope of $-1$.

Figure 7

Kinetic- (solid) and magnetic- (dashed) energy spectra for the highest resolution from Fig. 6 in decreasing order of $r_0$ from top left to bottom right.

Figure 8

Vorticity (left) and magnetic potential (right) fields at peak magnetic-energy dissipation, $t=3$, for $r_0=1/256$ for simulations evolving from a vorticity field spectrally confined within the wave-number interval $[5,7]$. The vorticity field is plotted with values between $-3{\langle {\omega }^2\rangle }^{1/2}$ and $3{\langle {\omega }^2\rangle }^{1/2}=30$, while the maximum value is $\approx 140$.

Figure 9

Vorticity (left) and magnetic potential (right) fields at peak magnetic-energy dissipation, $t=2.5$, for $r_0=1/256$ for simulations evolving from a fully developed vorticity field. The vorticity field is plotted with values between $-3{\langle {\omega }^2\rangle }^{1/2}$ and $3{\langle {\omega }^2\rangle }^{1/2}=19.5$, while the maximum value is $\approx 87$.

Figure 10

Flux of magnetic potential variance at the time of peak magnetic-energy dissipation for $r_0=1/256$ (dashed-dotted), $r_0=1/1024$ (dashed), and $r_0=1/4096$ (solid). These occur at $t=3,t=1.6$, and $t=1$, respectively, for evolution from a localized initial vorticity field (left), and $t=2.4,t=1.8$, and $t=1.1$ for evolution from a fully developed vorticity field (right).

Figure 11

Magnetic energy $E_{\mathit{b}}$, kinetic energy $E_{\mathit{u}}$, magnetic-energy dissipation rate ${\epsilon }_{\mathit{b}}$, and kinetic-energy dissipation rate ${\epsilon }_{\mathit{u}}$ for $r_0=1/256$ (dashed-dotted), $1/1024$ (dashed), and $1/4096$ (solid) for spectrally localized initial conditions.

Figure 12

Magnetic energy $E_{\mathit{b}}$, kinetic energy $E_{\mathit{u}}$, magnetic-energy dissipation rate ${\epsilon }_{\mathit{b}}$, and kinetic-energy dissipation rate ${\epsilon }_{\mathit{u}}$ for $r_0=1/4096$ for fully developed (solid) and spectrally localized (dashed) initial velocity fields.

Figure 13

Kinetic-energy spectrum $E_{\mathit{u}}\left(k\right)$ and magnetic-energy spectrum $E_{\mathit{b}}\left(k\right)$ at peak magnetic-energy dissipation for $r_0=1/256$ (dashed-dotted), $r_0=1/1024$ (dashed), and $r_0=1/4096$ (solid) for spectrally confined (left) and fully developed (right) initial velocity fields.

Figure 14

Magnetic- (left) and kinetic- (right) energy spectra for $r_0=1/4096$ at the time of maximum magnetic-energy dissipation for turbulence evolving from spectrally confined (dashed) and fully developed (solid) initial velocity fields.

Figure 15

Magnetic-energy spectra for $r_0=1/4096$ at one-third the time of maximum dissipation (dashed-dotted), half the time of maximum dissipation (dashed), and the time of maximum dissipation (solid) for turbulence evolving from spectrally confined (left) and fully developed (right) initial velocity fields.

Figure 16

Residual energy $E_{\mathit{b}}\left(k\right)-E_{\mathit{u}}\left(k\right)$ versus $k$ at peak magnetic-energy dissipation for $r_0=1/256$ (dashed-dotted), $r_0=1/1024$ (dashed), and $r_0=1/4096$ (solid), for spectrally confined (left) and fully developed (right) initial velocity fields.