## Abstract

We consider a problem of advection and diffusion of passive scalar and vector fields in a particular family of steady fluid flows. These flows are obtained by adding a small uniform velocity to a spatially periodic array of spiral eddies. The uniform flow,$\overline{\boldsymbol{u}}_{\text{H}}$, is taken to have the discrete form $\overline{\boldsymbol{u}}_{\text{H}}$ = $\epsilon $(M, N, 0)/(M$^{2}$ + N$^{2}$)$^{\frac{1}{2}}$, $\epsilon \ll $ 1, where M, N are relatively prime integers. The spatially periodic part,$\boldsymbol{u}^{\prime}$, may be expressed in terms of a streamfunction $\psi ^{\prime}$, $\boldsymbol{u}^{\prime}$ = ($\partial \psi ^{\prime}$/$\partial $y, -$\partial \psi ^{\prime}$/$\partial $x, K$\psi ^{\prime}$), $\psi ^{\prime}$ = sin x sin y, where K is a constant. The flow we study is therefore$\boldsymbol{u}$ = $\overline{\boldsymbol{u}}_{\text{H}}+\boldsymbol{u}^{\prime}$. Our work is motivated by applications of dynamo theory and to classical diffusion of passive scalars. The above family of flows was chosen as typical of spatially periodic flow with non-zero mean velocity,$\overline{\boldsymbol{u}}_{\text{H}}$. The flows are comparatively simple because they are independent of z. Nevertheless the projection of the streamline pattern onto the plane z = 0 can be surprisingly complex, owing to the structure of u modulo the cell of periodicity of$\boldsymbol{u}^{\prime}$. This structure accounts for our special form of$\overline{\boldsymbol{u}}_{\text{H}}$ above, which makes the tangent of the angle of inclination of the uniform current a rational number. This uniform component breaks up the eddy pattern into closed eddies whose bounding streamlines begin and end at X-type stagnation points. The set of all such streamlines define the boundaries of the open channels, which fill the regions between the closed eddies and lie near the separatrices of$\boldsymbol{u}^{\prime}$. Then, for example, when L = M+ N is even the channel structure repeats under a shift (M$\pi $, N$\pi $) in the xy plane, leading to a periodicity in channel length of order L. Analogous results apply to the case L odd. This geometry raises interesting questions regarding the advection and diffusion of fields in the irrational limit, i.e. whenM, N $\rightarrow \infty $, M/N $\rightarrow $ irrational. A basic result of this paper will be formal asymptotic expressions, for average physical quantities of interest, in the irrational limit. An asymptotic theory of advection-diffusion is exploited, based upon a separation into closed eddies, channels, and separatrix boundary layers. The fundamental assumption is that the dimensionless parameter R (a magnetic Reynolds number in the dynamo problem, a Peclet number in diffusion problems) is large, meaning that transport by molecular diffusion is nominally small compared with transport by advection. For large R, the X-type stagnation points trigger boundary layers, which for given M, N extend a distance of order L before repeating the structure. This leads to channel boundary layers of width L$^{\frac{1}{2}}$R$^{\frac{1}{2}}$, compared with eddy boundary layer of widthR$^{\frac{1}{2}}$ and channel widths of order $\epsilon $L$^{-1}$, the eddies being separated by gaps of widths order $\epsilon $. In this setting the irrational limit is taken after the above asymptotic structure is isolated by the limit R $\rightarrow \infty $. Our results consist of numerical studies for $\beta $ = $\epsilon $R$^{\frac{1}{2}}$ of order unity, and analytic asymptotic expressions derived under the condition $\beta \gg $ L$^{\frac{3}{2}}$. In the former, the eddy separation width is comparable with the eddy boundary layer width, so that we study the transition from transport dominated by boundary layers to transport dominated by boundary layers to trasport dominated by channels. In the asymptotic theory for large $\beta $, the boundary-layer contributions may be neglected and the problem reduces to the analysis of channel geometry. Even here, the condition $\beta \gg $ L$^{\frac{3}{2}}$ restricts us to a countable set of mena flow orientations. The relation between solutions for these special orientations, and their immediate neighbours with irrational tangents, is discussed. Representative results for effective diffusion of a passive scalar field, and for mean induced electromotive force in an electrically conducting fluid (the $\alpha $-effect) are presented. We also discuss the present examples in relation to the more complex problem of advention-diffusion by flow with chaotic lagrangian paths.