10 0 0.5 1 1.5 2 2.5 3 −20 −15 −10 −5 log$_{10}$ t / M log 10 | 1 5 | 3.06 3.08 3.1 3.12 3.14 3.16 3.18 −21 −20.8 −20.6 −20.4 −20.2 −20 −19.8 t −9  100Kx100  80Kx80  64Kx64 0 0.5 1 1.5 2 x 10 −3 8 8.2 8.4 8.6 8.8 9 9.2  M / t 1 n 5  100Kx100  80Kx80  64Kx64 FIG. 10: The ℓ = 5 multipole projections, for initial data of a pure monopole field as functions of time. Shown are the fields with three different grid densities, 64 K × 64 (dash–dotted curve), 80 K × 80 (dashed curve), and 100 K × 100 (solid curve). The inserts show the same for segments of the full data, in addition to a reference curve ∼ t − $^{9}$. FIG. 11: The local power index 1 n 5 as a function of the time t for three grid resolutions, 64 K × 64 (dash–dotted curve), 80 K × 80 (dashed curve), and 100 K × 100 (solid curve). In Fig. 12 we show the fields for m = 1 and m = 2 modes that are created by an initial octupole and excites all odd ℓ modes. Specifically, we show the ℓ = 3 , 5 projections for both m values. We also show the full fields for both m values. Notably, for the case m = 1 the dipole mode is also excited, and eventually dominates the full field at late times. In the m = 2 case the smallest ℓ -mode that is excited is the octupole, such that the full field in the m = 2 case decays more rapidly than in the m = 1 case. We also show the local power indices in Fig. 13. The late time values of the local power indices for each projected ℓ mode approach the same values independently of the value of m , in accordance with Eq. (3). In Table IV we re-organize the data appearing in Tables I,II, and III, so that the m -independence of the power-law indices is brought out in a sharper way. Specifically, the late–time decay rates for any allowed excited ℓ ′ mode from an initial ℓ mode is independent of the value of m in all the cases we have examined.