Nonlocal contributions to the turbulent cascade in magnetohydrodynamic plasmas

We present evidence for nonlocal contributions to the turbulent energy cascade in magnetohydrodynamic (MHD) plasmas. Therefore, we revisit a well-known result derived directly from the MHD equations, i.e., the Politano and Pouquet law for the transfer of kinetic and magnetic energy in scale. We propose adding a term that accounts for nonlocal transfer and represents the influence of fluctuations from large scales due to the Alfvén effect. Supported by direct numerical simulations of homogeneous and isotropic MHD turbulence, we verify that in some plasma configurations, neglecting the additional nonlocal term might consistently overestimate energy dissipation rates and, thus, the contributions of turbulent energy dissipation potentially affecting solar wind heating, i.e., a central puzzle in space plasma physics that motivates the present work.

Figure 1

Top: Surface plot of a cut through the total local energy dissipation rate, i.e., the sum of Eqs. (1) and (2), obtained from a direct numerical simulation of homogeneous and isotropic MHD turbulence with ${1024}^3$ spatial points (here, only an area of $256\times 256$ points is shown) with ${\text{Re}}_{\lambda ,\mathrm{kin}}$ and ${\text{Re}}_{\lambda ,\mathrm{mag}}$ (see main text and Table 1 for additional information). The organization of the dissipation field into sheetlike structures is visible and is dominated by a singular sheet peaking at roughly 40 times the mean value. Bottom: Same plot as on top, but obtained from a snapshot of a direct numerical simulation of homogeneous and isotropic hydrodynamic turbulence provided by the Johns Hopkins turbulence database (JHTDB) [62], with the same spatial resolution ${1024}^3$ and ${\text{Re}}_{\lambda ,\mathrm{kin}}=418$ (again, only an area of $256\times 256$ points is shown). The dissipation field is organized more randomly than its MHD counterpart.

Figure 2

Temporal evolution of the kinetic, magnetic, and total energy dissipation rates throughout the simulation. After an initial oscillation, the dissipation rates remain fairly constant due to the applied forcing mechanism, which indicates a close-to-stationary MHD flow. The assessed statistical quantities in Table 1 were averaged for $t/T_L>5$.

Figure 3

(a) Assessment of the four-fifths law in MHD turbulence from DNS (${1024}^3$) of forced MHD turbulence simulation. The dashed line represents the right-hand side of Eq. (18), which is determined from the total energy dissipation rate ${\varepsilon }_{\mathrm{tot}}$ of the snapshot. The top curve corresponds the original PP law $S_{rrr}^{\mathbf{u}\mathbf{u}\mathbf{u}}\left(r\right)-6C_{rrr}^{\mathbf{h}\mathbf{h}\mathbf{u}}\left(r\right)$ derived in [23, 24], whereas the additional nonlocal term in Eq. (18) is closer to the dashed line. The mixed tensors verify the dissipation range behavior (22). The lowest curve represents the third-order longitudinal velocity structure function $S_{rrr}^{\mathbf{u}\mathbf{u}\mathbf{u}}\left(r\right)$. (b) Same as in (a), but compensated by the right-hand side in Eq. (18). The original PP law (top curve) overestimates the total energy dissipation rate ${\varepsilon }_{\mathrm{tot}}$ by a factor of 1.71 and peaks in front of the inertial range, whereas the inclusion of the nonlocal term in Eq. (18) leads to the correct prediction (dashed line).