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The depiction of time-dependent vector fields is a central problem in scientific visualization. This article describes a technique for generating animations of such fields where the motion of the streamlines to be visualized is given by a second "motion" vector field. Each frame of our animation is a line integral convolution of the original vector field with a time-varying input texture. The texture is evolved according to the associated motion vector field via an automatically adjusted set of random particles. We demonstrate this technique with examples from electromagnetism.