Semiclassical Structure of the Advection--Diffusion Spectrum in Mixed Phase Spaces

TL;DR

This work analyzes the spectrum of the two dimensional advection–diffusion operator in flows with mixed phase space at very large $\,\mathrm{Pe}$. By computing leading eigenpairs of the one period transfer operator for the Chirikov standard map and interpreting them via a semiclassical lens with $\hbar_{\mathrm{eff}}=\sqrt{D}$, the authors uncover a robust organization into island-localized diffusive modes, core-localized advective modes, and inter-island tunneling hybrids. They show that diffusive modes obey $\gamma^{(p)}_k \sim D\alpha_p^{-2} k^2$ while advective modes follow $\gamma^{(\phi)}_k \sim \sqrt{D}\beta_p^{1/2} k$, with the onset of advective modes and inter-island hybridization governed by geometric length scales and area ratios; as $\mathrm{Pe}$ grows, advective contributions migrate to higher indices, yet multiple slow channels persist, precluding a single mode from dominating finite-time dynamics. The results provide a geometry driven framework for predicting spectral ordering, mode multiplicities, and the presence of avoided crossings, offering a bridge between Hamiltonian phase-space structure and transport in non-turbulent flows with mixed dynamics.

Abstract

We examine the spectral structure of the two-dimensional advection-diffusion operator in flows with mixed phase space at very large Peclet number. Using Fourier discretization combined with symmetry reduction and Krylov-Arnoldi methods, we compute on the order of one hundred leading eigenpairs reliably in the asymptotic, weak-diffusion regime. While the principal eigenvalue is asymptotically diffusive and localized on the largest regular region, the broader spectrum exhibits a rich organization controlled by local Lagrangian phase-space geometry. In particular, exponential mixing in chaotic regions rapidly suppresses correlations, whereas algebraic mixing in integrable regions generates long-lived coherent structures that dominate the slow and intermediate parts of the spectrum. We identify three distinct classes of eigenmodes: advective modes associated with transport on invariant tori, diffusive modes and, within the duffusive branch, tunneling modes arising from weak coupling between dynamically separated regular regions. Drawing on a semiclassical analogy, we assign quantum-number-like labels to these families and predict the appearance, scaling, and ordering of their sub-spectra directly from the Hamiltonian phase-space structure. The coexistence of these families implies that no uniform control of the spectral gap exists across the full spectrum: although the slowest mode is diffusive, arbitrarily small gaps arise between competing families at higher mode numbers. As a result, finite-time advection-diffusion dynamics is generically governed by persistent modal competition rather than single-mode dominance, even at asymptotically large Peclet number.

Figures (13)

• Figure 1: Local integrable structure of the standard map at $K=1.7$. (a) Effective Hamiltonian $H_{\mathrm{eff}}(x,y)$ reconstructed on the period--1 island, shown as filled contours. (b) Poincaré section of the map, with chaotic trajectories in gray and selected invariant curves outlining the period--1 (red), period--2 (blue) and period--6 (black) island chains. (c) Effective Hamiltonian $H_{\mathrm{eff}}(x,y)$ reconstructed on a single lobe of the period--6 chain shown zoomed on the black box in panel (b). In panels (a) and (c), $H_{\mathrm{eff}}$ is normalized to $[0,1]$ and displayed only on the corresponding regular region.
• Figure 2: Evolution of two localized initial conditions under the advection-diffusion operator for $K = 1.7$, $D=10^{-6}$. The two rows correspond to $\rho_1=\partial_x G$ (top) and $\rho_2=\partial_y G$ (bottom), where $G$ is the same Gaussian centered at $(x_0,y_0)$. Columns show the initial condition and the fields after $n=50$, $250$, and $1200$ periods.
• Figure 3: Time evolution of the $L^2$ norm of the scalar fields shown in Fig. \ref{['Fig3']}. Solid curves correspond to $\mathrm{Pe}=10^6$, dashed curves to $\mathrm{Pe}=10^5$. Blue curves show $\rho_1=\partial_x G$; red curves show $\rho_2=\partial_y G$.
• Figure 4: Leading eigenvalues of the one-period advection-diffusion operator for $\mathrm{Pe}=10^5,\,10^6,$ and $10^7$, plotted in polar form using $\theta=\arg(\lambda)\in[0,\pi]$ (folded by complex-conjugate symmetry) and $\gamma=-\log|\lambda|$, normalized in each panel by the leading decay rate $\gamma_1$. Solid vertical lines mark the symmetry directions $\theta = m\pi/3$ ($m=0,1,2,3$), corresponding to the roots of unity associated with the dominant period-1, 2, and 6 island chains. Colored circles denote diffusive eigenmodes $\psi^{p}_{m,k}$, grouped by island family $p$ ($p=1,2,6$) and clustered along these symmetry directions, forming discrete diffusive ladders. Red squares identify advective modes $\phi$, which lie away from the symmetry axes and populate the spectral gaps between diffusive branches. Dashed vertical lines indicate the independently computed advective phases obtained from the mean rotation number $\rho(I)$ of invariant tori within the period-1 island.
• Figure 5: Three families of diffusive modes for $K = 1.7$, $D = 10^{-6}$. Top row: Period-1 modes $\psi^1_{0,k}$ at $k = 2, 3, 4, 6$. Bottom two rows: Period-2 modes $\psi^2_{m,k}$ at the same $k$-values in both symmetry classes ($m = 0, 1$) (For clarity, Period-2 modes are unwrapped on the torus and plotted in ($\tilde{x} = x +\pi/2, \tilde{y} = y -\pi)$).