The effective diffusion coefficient D* of a passive component, such as test particles, dye, temperature, magnetic flux, etc., is derived for motion in periodic two-dimensional incompressible convective flow with characteristic velocity v and size d in the presence of an intrinsic local diffusivity D. Asymptotic solutions for effective diffusivity D*(P) in the large P limit, with P∼ vd/D, is shown to be of the form D*=cDP1/2 with c being a coefficient that is determined analytically. The constant c depends on the geometry of the convective cell and on an average of the flow speed along the separatrix. The asymptotic method of evaluation applies to both free boundary and rough boundary flow patterns and it is shown that the method can be extended to more complicated patterns such as the flows generated by rotating cylinders, as in the problem considered by Nadim, Cox, and Brenner [J. Fluid Mech. 164, 185 (1986)]. The diffusivity D* is readily calculated for small P, but the evaluation for arbitrary P requires numerical methods. Monte Carlo particle simulation codes are used to evaluate D* at arbitrary P, and thereby describe the transition for D* between the large and small P limits.

A low‐ (four‐) dimensional state space model for the basic energy components of the night‐side magnetosphere is developed based on truncated descriptions of the collisionless microscopic (Hamiltonian) energy transfer processes occurring in the quasi‐neutral layer. The substorm trigger, due to bifurcation of the system either from magnetic reconnection or from ballooning‐mirror modes, is modeled by a fast unloading above a critical current. For constant southward interplanetary magnetic field solar wind input, the system performs oscillations with recurrence times in the range of 60–70 min. For northward IMF, the system decays to a static MHD equilibrium. For the intermediate case of a weak solar wind input, there occur uncorrelated, isolated substorms. The theoretical basis for a low‐dimensional model may be supported within the framework of self‐organized criticality since the geotail is a highly stressed global system during the growth phase of substorms. The new energy‐conserving state space model eliminates the “dripping faucet” unloading role of the Klimas et al. [1992] model and includes the parallel streaming kinetic energy in the substorm dynamics. The new model shows a wide range of dynamical behavior according to the values of the system parameters and the form of the solar wind input signal.

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A correlation dimension analysis of people’s visual experiential streams captured by a smartphone shows that visual experience is two-scaled with a smaller dimension at shorter length scales than at longer length scales. The bend between the two scales is a phase transition point where the lower scale primarily captures relationships within the same context and the higher dimensional scale captures relationships between different contexts. The dimensionality estimates are confirmed using Takens’ delay embedding procedure on the image stream, while the randomly permuted stream is shown to be space-filling thereby establishing that the two-scaled structure is a consequence of the dynamics. We note that the structure of visual experience closely resembles the structure of another domain of experience: natural language discourse. The emergence of an identical structure across different domains of human experience suggests that the two-scaled geometry reflects a general organizational principle.