The maximum magnetic flux in an active region

The Photometric-Magnetic Dynamical model handles the evolution of an individual sunspot as an autonomous nonlinear, though integrable, dynamical system. The model considers the simultaneous interplay of two different interacted factors: The photometric and magnetic factors, respectively, characterizing the evolution of the sunspot visible area A on the photosphere, and the simultaneous evolution of the sunspot magnetic field strength B. All the possible sunspots are gathered in a specific region of the phase space (A, B). The separatrix of this phase space region determines the upper limit of the values of sunspot area and magnetic strength. Consequently, an upper limit of the magnetic flux in an active region is also determined, found to be ≈7.23 × 1023 Mx. This value is phenomenologically equal to the magnetic flux concentrated in the totality of the granules of the quite Sun. Hence, the magnetic flux concentrated in an active region cannot exceed the one concentrated in the whole photosphere.

Motion of three point vortices in a periodic parallelogram

The motion of three interacting point vortices with zero net circulation in a periodic parallelogram defines an integrable dynamical system. A method for solving this system is presented. The relative motion of two of the vortices can be ‘mapped’ onto a problem of advection of a passive particle in ‘phase space’ by a certain set of stationary point vortices, which also has zero net circulation. The advection problem in phase space can be written in Hamiltonian form, and particle trajectories are given by level curves of the Hamiltonian. The motion of individual vortices in the original three-vortex problem then requires one additional quadrature. A complicated structure of the solution space emerges with a large number of qualitatively different regimes of motion. Bifurcations of the streamline pattern in phase space, which occur as the impulse of the original vortex system is changed, are traced. Representative cases are analysed in detail, and a general procedure is indicated for all cases. Although the problem is integrable, the trajectories of the vortices can be surprisingly complicated. The results are compared qualitatively to vortex paths found in large-scale numerical simulations of two-dimensional turbulence.

Fusion Rules for Quantum Transfer Matrices as a Dynamical System on Grassmann Manifolds

We show that the set of transfer matrices of an arbitrary fusion type for an integrable quantum model obeys these bilinear functional relations, which are identified with an integrable dynamical system on a Grassmann manifold (higher Hirota equation). The bilinear relations were previously known for a particular class of transfer matrices corresponding to rectangular Young diagrams. We extend this result for general Young diagrams. A general solution of the bilinear equations is presented.