Characteristics of anomalous deterministic transport in steady plane viscous flows

TL;DR

This work analyzes deterministic transport of passive tracers in steady, two-dimensional, spatially periodic incompressible flows and shows that stagnation-point singularities govern unbounded variance growth even in the absence of diffusion. By mapping the Lagrangian dynamics to special-flow constructions over circle maps and validating with numerical simulations, it characterizes how different singularities of the passage time T(u) produce distinct transport regimes, including logarithmic (for generic saddles) and power-law (for degeneracies) growth, as well as subdiffusive and superdiffusive trends in more extreme cases. The authors derive explicit exponents and asymptotic forms for the variance and higher moments, employing simplified models such as equidistant-point and CTRW frameworks to capture the observed trends and decorations tied to the rotation number’s continued fraction. These results provide quantitative predictions for deterministic transport in laminar flows and have implications for mixing, dispersion, and the design or analysis of flow systems with controlled stagnation structures. Overall, the paper links local stagnation-point geometry to global transport exponents through rigorous modeling and numerical evidence, advancing understanding of anomalous transport in steady advection.

Abstract

We consider transport of passive particles in steady laminar plane flows of incompressible viscous fluids. While drifting along the streamlines, the particles experience alternating accelerations and slowdowns. For an ensemble of particles, recurring slow passages across the vicinities of stagnation points affect the transport and result in the unbounded growth of the ensemble variance. This growth is logarithmic in case of generic stagnation points and has a power-law character in the presence of degeneracies. We interrelate quantitative characteristics of the variance growth with the singularities of the passage time and derive explicit estimates for the transport exponents.

Figures (10)

• Figure 1: Plane steady velocity field with periodic boundary conditions, stagnation points, confined eddies and unbounded jets. The field has been generated by the time-independent spatially periodic force with Fourier harmonics up to the 4th order. The Figure is taken from Kogler
• Figure 2: Flow patterns, defined by Eq.(\ref{['stat']}) at $\alpha=(\sqrt{5}-1)/2,\,\beta=1$ ("golden mean" rotation number) and different combinations of $f$ and $\nu$. (a) $\nu=1$, $f=0.9<f_{\rm cr}$; (b) $\nu=1$, $f=f_{\rm cr}=1.17557$; (c) $\nu=1$, $f=1.5>f_{\rm cr}$; (d) $\nu=0$, $f=f_{\rm cr}=0.61803$. Crosses: saddle stagnation points, filled circles: non-hyperbolic stagnation points; dashed red lines in (c): closed streamlines. Patterns (a) and (c) are robust; patterns (b) and (d) contain degenerate stagnation points.
• Figure 3: Temporal growth of variance for different flow patterns. (a) pattern and parameter values from Fig. \ref{['fig:4patterns']}a; (b) pattern and parameter values from Fig. \ref{['fig:4patterns']}c; (c) pattern and parameter values from Fig. \ref{['fig:4patterns']}b; (d) pattern and parameter values from Fig. \ref{['fig:4patterns']}d. Notation $\{n_0,n_1,n_2,n_3\ldots\}$ indicates first entries in the continued fraction expansion $\rho=n_0+1/(n_1+1/(n_2+1/(n_3+...)))$. Except for $\rho=1/e$, all shown continued fractions are periodic.
• Figure 4: "Decoration" around the power laws $t^{\alpha}$ with $\alpha=0.67\pm 0.01$ for the curves from Fig. \ref{['fig:4laws']}d. Horizontal axis: time $t$ in units of the average time $\langle T\rangle$ for one turn around the torus along the $x$-direction. Vertical dotted lines: numerators of the best rational approximations (see Appendix \ref{['app:approx']}). (a): the golden mean rotation number $\rho=\left(\sqrt{5}-1\right)/2=\{0,1,1,1,\ldots\}$. (b): rotation number $\rho=\left(\sqrt{3}-1\right)/2=\{0,2,1,2,1,2,1\ldots\}$. (c): rotation number $\rho=1/\hbox{e}=\{0, 2, 1, 2, 1, 1, 4, 1, 1, 6, 1,\ldots\}$.
• Figure 5: Self-similarity of decoration for the curve from Fig. \ref{['fig:decorations']}a: growth of variance in the ensemble of particles carried by the flow (\ref{['velocity_field']}) with the golden mean rotation number $\sigma$. Parameter values: $\alpha=(\sqrt{5}-1)/2,\,\beta=1,\,\nu=0,\,f=f_{\rm cr}=0.61803$. Horizontal axis: time $t$ in units of the average time $\langle T\rangle$ for one turn around the torus along the $x$-direction. Vertical dotted lines: Fibonacci numbers $F_n$. (a) the overall diagram for $t/\langle T\rangle \leq F_{24}$=46368. (b) magnified rectangular area from (a): interval between $F_{22}$ and $F_{23}$. (c) magnified rectangular area from (b); (d) magnified rectangular area from (c).