Phase dynamics and their role determining energy flux in hydrodynamic shell models

Abstract

The transfer of energy and other conserved quantities across scales, also known as flux or spectral flux, is a central aspect of out-of-equilibrium systems such as turbulent hydrodynamic flows. Despite its role in the few predictive theories that exist, a dynamical understanding of what determines said flux (and its direction in scale) has yet to be established. In this study, we work towards this understanding by investigating how the dynamics of complex Fourier velocity phases influence the flux of conserved quantities in hydrodynamic shell models. The phase dynamics, like energy evolution, are influenced by contributions from all neighboring triads, making the full problem intractable. Instead, we assume that the dynamics of the triad phases are determined solely by the so-called self-interaction term and treat the other neighboring triad terms as noise. This transforms the phase dynamics into that of a noisy phase oscillator, which we solve analytically to predict phase statistics. We validate this assumption with a suite of shell model simulations. Our results give us analytical predictions for the energy flux, when the energy spectrum is given. We prove that all shell models that conserve energy along with a sign-indefinite quadratic quantity (this includes three-dimensional turbulence analogues) undergo a forward energy cascade, and further show that the phase dynamics prevent the shell model analogue of two-dimensional turbulence from forming an inverse cascade of energy.

Figures (7)

• Figure 1: Statistics of $\widetilde{\xi}_n$, the 'noise' variable measured from phase-only shell model runs of a 2D-like case (top row, (a) and (b)) and a 3D-like case (bottom row, (c) and (d)). The PDFs both show an approximately Gaussian shape(dashed red line), with a maximum at zero and a fall-off towards zero. Both autocorrelation functions have time-scales of order one. That of the 2D-like case, (b), shows a decaying and oscillating autocorrelation function, hinting at colored harmonic noise. Each plot shows statistics for $n=N/4$ (blue), $n=N/3$ (orange), and $n=N/2$ (green). The overlap of these curves demonstrates the $n$-independent statistics of the re-scaled variables.
• Figure 2: A plot of the predicted PDF of $\theta$, Eq. (\ref{['eq:PDF_theta']}), based on our model for the triad phase dynamics, Eq. (\ref{['eq:adler']}), assuming white noise. Multiple values of $\mathcal{K}/D_{\mathrm{eff}}$ are shown, with red color corresponding to positive $\mathcal{K}/D_{\mathrm{eff}}$, blue to negative $\mathcal{K}/D_{\mathrm{eff}}$, and black to $\mathcal{K}/D_{\mathrm{eff}} = 0$. Notice how the position of the maximum value of the PDF, $\pm \pi/2$, changes sign depending on the sign of $\mathcal{K}/D_{\mathrm{eff}}$.
• Figure 3: Measured PDFs, (a), and $D_{\mathrm{eff}}$ fits, (b), for our suite of 2D-like runs with positive-definite $H$. Panel (a) shows a demonstrative sample of PDFs of $\theta_n$, averaged in $n$, observed in our runs, with $\mathcal{K}/D_{\mathrm{eff}}$ positive (red), negative (blue), and zero (black). PDF fits for the white noise model (dashed lines), based on Eq. (\ref{['eq:PDF_theta']}), and the colored harmonic noise model (dot-dashed lines), based on Eq. (\ref{['eq:HN_PDF']}), are also shown. Panel (b) shows the measured values of $D_{\mathrm{eff}}$ based on fits to Eq. (\ref{['eq:PDF_theta']}) for all values of spectral slope $\alpha$ and second invariant exponent $\gamma$. The masked data (in white) represents runs where the PDFs are sufficiently flat such that fitting for $D_{\mathrm{eff}}$ was not possible.
• Figure 4: Time- and $n$-averaged alignment strength $\langle \sin(\theta)\rangle$, (a), and rescaled energy flux $\langle \Pi_n \rangle/(k_n \rho^3_n)$, (b), for our suite of 2D-like phase-only model runs. Panel (a) shows the alignment, which is both positive (red) and negative (blue). Lines of zero alignment predicted by our analytical model for the triad phase dynamics are seen as white dashed lines, corresponding to solutions to $\mathcal{K}=0$. Panel (b) shows the rescaled energy flux, which can also be both positive (red) and negative (blue-white). we have re-scaled the negative flux to be able to distinguish it from zero. The colorbar scale for positive flux in (b) is $2.5 \times 10^{-3}$. The dashed white lines are the same zero-alignment lines in (a), whereas the dot-dashed line corresponds to a third zero-flux line, based on the coefficient for the flux in Eq. (\ref{['eq:flux_en_final']}).
• Figure 5: Measured PDFs, (a), and $D_{\mathrm{eff}}$ fits, (b), for our suite of 3D-like runs with sign-indefinite $H$. Panel (a) shows a demonstrative sample of PDFs of $\theta_n$, averaged in $n$, observed in our runs, with $\mathcal{K}/D_{\mathrm{eff}}$ only being positive (red) and zero (black). PDF fits for the white noise model (dashed lines), based on Eq. (\ref{['eq:PDF_theta']}), and the colored harmonic noise model (dot-dashed lines), based on Eq. (\ref{['eq:HN_PDF']}), are also shown. Panel (b) shows the measured values of $D_{\mathrm{eff}}$ based on fits to Eq. (\ref{['eq:PDF_theta']}) for all values of spectral slope $\alpha$ and second invariant exponent $\gamma$. The masked data (in white) represents excluded runs with either flat PDFs ($\alpha=0$) or with non-chaotic dynamics (found for low values of $\gamma$).