Theory of the Spatio-Temporal Dynamics of Transport Bifurcations

The development and time evolution of a transport barrier in a magnetically confined plasma with non-monotonic, nonlinear dependence of the anomalous flux on mean gradients is analyzed. Upon consideration of both the spatial inhomogeneity and the gradient nonlinearity of the transport coefficient, we find that the transition develops as a bifurcation front with radially propagating discontinuity in local gradient. The spatial location of the transport barrier as a function of input flux is calculated. The analysis indicates that for powers slightly above threshold, the barrier location $x_b(t) \sim ( D_n t (P-P_c)/P_c)^{1/2},$ where $P_c$ is the local transition power threshold and $D_n$ is the neoclassical diffusivity . This result suggests a simple explanation of the high disruptivity observed in reversed shear plasmas. The basic conclusions of this theory are insensitive to the details of the local transport model.Comment: 21 page Tex file, 10 postscript file

Crossed product of a C*-algebra by a semigroup of endomorphisms generated by partial isometries

The paper presents a construction of the crossed product of a C*-algebra by a semigroup of endomorphisms generated by partial isometries.Comment: 22 page

Shape transformations in rotating ferrofluid drops

Floating drops of magnetic fluid can be brought into rotation by applying a rotating magnetic field. We report theoretical and experimental results on the transition from a spheroid equilibrium shape to non-axissymmetrical three-axes ellipsoids at certain values of the external field strength. The transitions are continuous for small values of the magnetic susceptibility and show hysteresis for larger ones. In the non-axissymmetric shape the rotational motion of the drop consists of a vortical flow inside the drop combined with a slow rotation of the shape. Nonlinear magnetization laws are crucial to obtain quantitative agreement between theory and experiment.Comment: 4 pages, 3 figure

Dynamics of nearly spherical vesicles in an external flow

We analytically derive an equation describing vesicle evolution in a fluid where some stationary flow is excited regarding that the vesicle shape is close to a sphere. A character of the evolution is governed by two dimensionless parameters, $S$ and $\Lambda$, depending on the vesicle excess area, viscosity contrast, membrane viscosity, strength of the flow, bending module, and ratio of the elongation and rotation components of the flow. We establish the ``phase diagram'' of the system on the $S-\Lambda$ plane: we find curves corresponding to the tank-treading to tumbling transition (described by the saddle-node bifurcation) and to the tank-treading to trembling transition (described by the Hopf bifurcation).Comment: 4 pages, 1 figur