Topological Dissipation and the Small-Scale Fields in Turbulent Gases

It is shown that a large-scale magnetic field possesses a hydrostatic equilibrium only if the pattern of small-scale variations is uniform along the large-scale field. Thus equilibrium obtains only if the variations in the field consist of simple twisting of the lines, with the twists extending uniformly the full length of the field. Any more complicated topology, such as two or more flux tubes wrapped around each other to form a rope, or braided or knotted flux tubes, is without equilibrium, no matter what fluid pressures are applied along the individual lines of force. The result is rapid dissipation and field-line merging, which quickly reduces the topology to the simple equilibrium form. It follows from this general theorem that line merging has important consequences in turbulent fields. In spite of large magnetic Reynolds numbers of the individual eddies, the line merging reduces the small-scale fields in the turbulence below the value for equipartition with the turbulence. The effect has important astrophysical implications. It explains the absence of strong small-scale fields in the solar photosphere and in interstellar space in spite of the vigorous turbulence.