The correlation between the amplitude of the MHD turbulence in the upstream solar wind and the amplitude of the Earth's geomagnetic activity indices AE, AU, AL, Kp, ap, Dst, and PCI is explored. The amplitude of the MHD turbulence is determined by the fluctuation amplitude of the solar wind magnetic field. This turbulence effect in solar wind/magnetosphere coupling is more easily discerned when the interplanetary magnetic field (IMF) is northward, but the effect is also present when the IMF is southward. The magnitude of the effect is the same for northward and southward IMF, accounting for about 150 nT of the variability of the AE index. Tests are performed that conclude (1) that the turbulence effect is not caused by the turbulence amplitude acting as a proxy for ∣B∣ in the solar wind and (2) that reversals of the IMF from northward to southward in the turbulent fluctuations is not the cause of the correlations. An expression is derived for the total viscous-shear force on the surface of the magnetosphere; improved solar wind/magnetosphere correlations result when this expression is used. With insight from fluid-flow experiments, the turbulence effect is interpreted as an enhanced viscous coupling of the solar wind flow to the Earth's magnetosphere caused by an eddy viscosity that is controlled by the amplitude of MHD turbulence in the upstream solar wind: more upstream turbulence means more momentum transfer from the magnetosheath into the magnetosphere, resulting in more stirring of the magnetosphere, which produces enhanced geomagnetic activity indices.