The effects of a time-varying fluid velocity have been included in the estimation of frozen-flux core surface flow. This has been done to improve the quality, performance, and accuracy of kinematic analyses of the geomagnetic secular variation. The derived flows consist of a reference velocity plus a steady acceleration and thus evolve linearly with time. The method features simultaneous solution for the initial radial geomagnetic field component at the core-mantle boundary and the subadjacent, time-varying fluid motion. The trade-off between time-averaged spatial complexity of the derived flows and misfit to the weighted Definitive Geomagnetic Reference Field models for 1945--1980 is explored. At fixed flow complexity, simultaneous solution for both initial geomagnetic field and steadily accelerating flow gives substantial reductions in the square-weighted residuals per degree of freedom when compared with previous solutions for either initial field and steady flow or steady flow alone. The apparent timescale for flow change varies nonmonotonically with flow complexity between about 23 and 41 years; the single maximum value is associated with unit negative slope of the log-log trade-off curve and is used to help identify a preferred solution. This solution has an rms speed of but 7.540 km yr-1 and an rms acceleration of 0.183 km yr-2, or 1.84¿10-13 m s-2. The latter indicates extremely weak net forces upon the liquid outer core. ¿ American Geophysical Union 1995