Spectrum width is the distribution of velocities within a single radar pixel. One pixel on radar represents a volume. Within this volume can be literally millions of individual hydrometeors. Each individual hydrometeor will have its own speed and direction of movement. When turbulence exists in the atmosphere, individual hydrometeors within the radar pixel volume have vastly different radial velocities.
In order to produce a pixel of radar velocity, the radar averages the individual radial velocities with a volume sample to produce a single average radial velocity that is displayed for that pixel. In a situation where shear and turbulence is small within a pixel, the spectrum width will be small. In a situation where shear and radial velocity is large within a pixel, the spectrum width will be large. A technical way of defining spectrum width is that it is the standard deviation of the velocity distribution within a single pixel.
Spectrum width can be used to locate the center pixel of a TVS. Within the center pixel of a TVS will be a large velocity distribution (near zero at one location within the pixel and strong inbound and/or strong outbound velocities within that same pixel). Variations in the velocity distribution within a pixel occur depending on if the center of circulation is in the exact center of a pixel or near the edge of a pixel. The large distribution of velocities lead to a large standard deviation of velocities and thus a large spectrum width. Besides in association with a mesocyclone, spectrum width will show a turbulent pattern near any shear axis (such as the change in wind speed and direction along a squall line).