Magnetic helicity in a periodic domain

Sometimes, ideally conserved quantities in turbulent flows can cascade to larger length scales as well as smaller length scales. Any quantity which approaches the largest length scale available will be strongly affected by the boundary conditions. Often turbulence is modelled in a domain with periodic boundary conditions. In three dimensions, such a domain is a compact manifold without boundary called a 3-torus. A 3-torus is multiply connected, unlike many other domains of interest (e.g., the interior of a sphere). This difference in topology affects the structure of the fields contained inside. For example, there may be streamlines or magnetic field lines which do not close upon themselves but which stretch across the entire domain. Such periodic lines do not exist in nonperiodic geometries. This paper asks whether the presence of periodic lines can change the dynamics of the fluid. Recently, Stribling et al. [1994] examined MHD turbulence with a mean magnetic field and periodic boundary conditions. They noted that the magnetic helicity of the fluctuating field decreases in time. In a closed domain such behavior is readily explained. In particular, the helicity of the fluctuating field only measures the self-linking of the fluctuating field. The conserved total helicity, however, also measures mutual linking between the fluctuating field and the mean field. The decrease of the fluctuating helicity indicates a transference of helicity to this linking. Unfortunately in a periodic domain this explanation does not work. We show that. in general, the total helicity is not even definable, much less conserved. The helicity exists only if all toroidal field lines close upon themselves without stretching across the domain. Even in this case helicity has unusual properties. For example, a simple sequence of reconnections can convert a flux tube with right handed twist into a tube with left handed twist. This flips the sign of the helicity. Taylor relaxation is not valid when such processes occur. We present a new helicity-like quantity H which can be defined even when the usual helicity does not exist. This quantity measures how much the toroidal field links both interior and exterior mean flux. It is an ideal invariant when the fluid velocity has no z component.