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Register a new userThe Joint Cascade of Energy and Helicity in Three-Dimensional Turbulence
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Three-dimensional (3D) turbulence has both energy and helicity as inviscid constants of motion. In contrast to two-dimensional (2D) turbulence, where a second inviscid invariant--the enstrophy--blocks the energy cascade to small scales, in 3D there is a joint cascade of both energy and helicity simultaneously to small scales. The basic cancellation mechanism which permits a joint cascade of energy and helicity is illuminated by means of the helical decomposition of the velocity into positively and negatively polarized waves. This decomposition is employed in the present study both theoretically and also in a numerical simulation of homogeneous and isotropic 3D turbulence. It is shown that the transfer of energy to small scales produces a tremendous growth of helicity separately in the + and - helical modes at high wavenumbers, diverging in the limit of infinite Reynolds number. However, because of a tendency to restore reflection invariance at small scales, the net helicity from both modes remains finite in that limit. The net helicity flux is shown to be constant all the way up to the Kolmogorov wavenumber: there is no shorter inertial-range for helicity cascade than for energy cascade. The transfer of energy and helicity between + and - modes, which permits the joint cascade, is shown to be due to two distinct physical processes, advection and vortex stretching.
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The statistics of the energy and helicity fluxes in isotropic turbulence are studied using high resolution direct numerical simulation. The scaling exponents of the energy flux agree with those of the transverse velocity structure functions through refined similarity hypothesis, consistent with Kraichnans prediction cite{Kr74}. The helicity flux is even more intermittent than the energy flux and its scaling exponents are closer to those of the passive scalar. Using Waleffes helical decomposition, we demonstrate that the existence of positive mean helicity flux inhibits the energy transfer in the negative helical modes, a non-passive effect.
We show that oppositely directed fluxes of energy and magnetic helicity coexist in the inertial range in fully developed magnetohydrodynamic (MHD) turbulence with small-scale sources of magnetic helicity. Using a helical shell model of MHD turbulence, we study the high Reynolds number magnetohydrodynamic turbulence for helicity injection at a scale that is much smaller than the scale of energy injection. In a short range of scales larger than the forcing scale of magnetic helicity, a bottleneck-like effect appears, which results in a local reduction of the spectral slope. The slope changes in a domain with a high level of relative magnetic helicity, which determines that part of the magnetic energy related to the helical modes at a given scale. If the relative helicity approaches unity, the spectral slope tends to $-3/2$. We show that this energy pileup is caused by an inverse cascade of magnetic energy associated with the magnetic helicity. This negative energy flux is the contribution of the pure magnetic-to-magnetic energy transfer, which vanishes in the non-helical limit. In the context of astrophysical dynamos, our results indicate that a large-scale dynamo can be affected by the magnetic helicity generated at small scales. The kinetic helicity, in particular, is not involved in the process at all. An interesting finding is that an inverse cascade of magnetic energy can be provided by a small-scale source of magnetic helicity fluctuations without a mean injection of magnetic helicity.
In this work, the scaling statistics of the dissipation along Lagrangian trajectories are investigated by using fluid tracer particles obtained from a high resolution direct numerical simulation with $Re_{lambda}=400$. Both the energy dissipation rate $epsilon$ and the local time averaged $epsilon_{tau}$ agree rather well with the lognormal distribution hypothesis. Several statistics are then examined. It is found that the autocorrelation function $rho(tau)$ of $ln(epsilon(t))$ and variance $sigma^2(tau)$ of $ln(epsilon_{tau}(t))$ obey a log-law with scaling exponent $beta=beta=0.30$ compatible with the intermittency parameter $mu=0.30$. The $q$th-order moment of $epsilon_{tau}$ has a clear power-law on the inertial range $10<tau/tau_{eta}<100$. The measured scaling exponent $K_L(q)$ agrees remarkably with $q-zeta_L(2q)$ where $zeta_L(2q)$ is the scaling exponent estimated using the Hilbert methodology. All these results suggest that the dissipation along Lagrangian trajectories could be modelled by a multiplicative cascade.
Direct numerical simulations of three-dimensional (3D) homogeneous turbulence under rapid rigid rotation are conducted to examine the predictions of resonant wave theory for both small Rossby number and large Reynolds number. The simulation results reveal that there is a clear inverse energy cascade to the large scales, as predicted by 2D Navier-Stokes equations for resonant interactions of slow modes. As the rotation rate increases, the vertically-averaged horizontal velocity field from 3D Navier-Stokes converges to the velocity field from 2D Navier-Stokes, as measured by the energy in their difference field. Likewise, the vertically-averaged vertical velocity from 3D Navier-Stokes converges to a solution of the 2D passive scalar equation. The energy flux directly into small wave numbers in the $k_z=0$ plane from non-resonant interactions decreases, while fast-mode energy concentrates closer to that plane. The simulations are consistent with an increasingly dominant role of resonant triads for more rapid rotation.
We solve the Navier-Stokes equations with two simultaneous forcings. One forcing is applied at a given large-scale and it injects energy. The other forcing is applied at all scales belonging to the inertial range and it injects helicity. In this way we can vary the degree of turbulence helicity from non helical to maximally helical. We find that increasing the rate of helicity injection does not change the energy flux. On the other hand the level of total energy is strongly increased and the energy spectrum gets steeper. The energy spectrum spans from a Kolmogorov scaling law $k^{-5/3}$ for a non-helical turbulence, to a non-Kolmogorov scaling law $k^{-7/3}$ for a maximally helical turbulence. In the later case we find that the characteristic time of the turbulence is not the turnover time but a time based on the helicity injection rate. We also analyse the results in terms of helical modes decomposition. For a maximally helical turbulence one type of helical mode is found to be much more energetic than the other one, by several orders of magnitude. The energy cascade of the most energetic type of helical mode results from the sum of two fluxes. One flux is negative and can be understood in terms of a decimated model. This negative flux is however not sufficient to lead an inverse energy cascade. Indeed the other flux involving the least energetic type of helical mode is positive and the largest. The least energetic type of helical mode is then essential and cannot be neglected.
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