& Frisch (1980), by a shift of origin. It offers the advantage of the symmetric Fourier representation (1 2 ) .
Early-time behaviour of the inviscid flowThe symmetries of the TG vortex are listed in appendix A. Here we emphasize only those that help to visualize the qualitative features of the flow and those that may be important in making this flow atypical of general three-dimensional flow. First, for all times, no fluid crosses any of the boundaries x , y or z = nn, where n is an integer.Therefore the flow can be visualized as flow in the box 0 < x , y , z < n with impermeable stress-free faces. I n the following discussion, the region 0 < x, y , z < n is termed the impermeable box, as it confines the flow, while the region 0 < x , y , z < 2n is termed the periodicity box, as it reflects the periodicity of the Fourier series (1.2). Also, because of the symmetries listed in appendix A, the flow at any point in space can be inferred from its values in the fundamental box 0 < x, y, z < in.Secondly, if near each face we write the velocity field in terms of components parallel or perpendicular to the face, i.e. u = u,, + vl, then ul and au,,/an vanish on the face. This implies that the vorticity on each face is normal to that face so it may be written w = @, where fi is the unit normal. Note that g must vanish on all edges of the box where faces meet. I n contrast, a general incompressible flow will have only isolated points of vanishing vorticity. (Both velocity and vorticity also vanish for all time a t the centre x = y = z = in.)Thirdly, the vanishing of ul and au,,/an on each face also implies that the tensor V v is partly diagonal. One principal axis of the strain rate or symmetric part of this tensor is then perpendicular to the face. Furthermore, the magnitude of the strain rate along this axis determines the fractional growth rate of the normal vorticity on the face, i.e. d % = --~r * v --lnlcl. an ' Idt (2.1) 14-2
