Supporting Information

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1 Supporting Information Sullivan et al..73/pnas Temporal Averaging and Integration Time Step Effects on Attribution Metrics An important consideration in the attribution analysis is the effect of temporal averaging and time step. Fig. S3 shows primary attribution grids from the HITEMP- simulation, during which temporal attributions are calculated using means and variances for hourly output. In this case, updraft velocity is the primary contributor for between 26.4% (925 hpa) and 6.2% (825 hpa) of the N d grid and between 23.7% (25 hpa) and 49.2% (45 hpa) of the N i grid. These values closely resemble the DEF-G attributions. DEF-G2 simulation attributions, for which the GCM integration time step is half of the DEF-G time step, yield ξ ðnx w Þ coverage and mean values within % of the DEF-G values. Robustness across temporal averaging and integration time step make the adjoint sensitivity attribution metrics particularly useful. Sullivan et al. of8

2 Fig. S. Primary attribution grids, i.e., grid cells colored according to the input variable whose temporal attribution, ξ ðnx Þ, is largest at (A) 825 hpa, (C) 875 hpa, xj and (E) 925 hpa for liquid droplets and at (B) 25 hpa, (D) 35 hpa, and (F) 45 hpa for ice crystals. Values are taken from the DEF-Gyr simulation, a yearlong GEOS-5 simulation at 2 spatial resolution, using the Phillips et al. (2) heterogeneous nucleation spectrum. One daily averaged set of inputs and sensitivities is recorded per day. Then the annual variance of these inputs and mean of these sensitivities are used in the calculation of Eq.. Grid cells and time points for which new hydrometeor formation is negligible, i.e., dn d < cm 3 and dn i < L, are filtered out; regions of negligible cloud hydrometeor formation over the month are shown in white. Sullivan et al. 2of8

3 Fig. S2. As in Fig. S but secondary attribution grids, i.e., grid cells colored according to the input variable whose temporal attribution is second largest. Values are taken from the DEF-Gyr simulation. Sullivan et al. 3of8

4 Fig. S3. As in Fig. S but primary attribution grids with values taken from the HITEMP- simulation. Sullivan et al. 4 of 8

5 Fig. S4. As in Fig. S but primary attribution grids with values taken from the HITEMP-2 simulation. Sullivan et al. 5 of 8

6 o N a 8 o N b o N o 4 o S cm o N o 4 o S o S 6 o W 2 o W 8 o W 4 o W o 4 o E 8 o E 2 o E 6 o E 2 8 o S 6 o W 2 o W 8 o W 4 o W o 4 o E 8 o E 2 o E 6 o E o N c.5 8 o N d 2 4 o N 4 o N.5 o 4 o S cm 3.5 o 4 o S.5 8 o S 8 o S 6 o W 2 o W 8 o W 4 o W o 4 o E 8 o E 2 o E 6 o E 6 o W 2 o W 8 o W 4 o W o 4 o E 8 o E 2 o E 6 o E Fig. S5. Comparison of the input dust and sulfate aerosol number concentrations, N dust and N sulf, between GEOS-5 (A and C) and CAM5. (B and D) simulations, all shown in log space: (A) N dust from the second bin of the GOCART aerosol module in the DEF-G at 25 hpa, (B) N dust from the accumulation mode of the MAM3 aerosol module in DEF-C at 232 hpa, (C) N sulf from GOCART in DEF-G at 25 hpa, and (D) N sulf from MAM3 in DEF-C at 232 hpa. For DEF-G, aerosol mass from GOCART is converted to number, assuming a volume mean radius for dust in the second bin of.4 μm and for sulfate from Jensen et al. (36) of.566 μm (r g =.2 μm, σ g = 2.3). Dust density, ρ dust, is taken to be 2.5 g m 3, and sulfate density, ρ sul, is taken to be.84 g m 3. For DEF-C, both aerosol mass and number are tracked within MAM3. Lognormal size distributions are assumed for the Aitken, accumulation, and coarse modes, with fixed σ g of 2.3,.8, and.6, respectively. Sullivan et al. 6of8

7 8 o N a o N b o N o 4 o S s cm L o N o 4 o S 2 8 o S 5 8 o S 6 o W 2 o W 8 o W 4 o W o 4 o E 8 o E 2 o E 6 o E 6 o W 2 o W 8 o W 4 o W o 4 o E 8 o E 2 o E 6 o E c 8 o N 4 o N o 4 o S s cm L d 8 o N 4 o N o 4 o S 5 8 o S 5 8 o S 6 o W 2 o W 8 o W 4 o W o 4 o E 8 o E 2 o E 6 o E 6 o W 2 o W 8 o W 4 o W o 4 o E 8 o E 2 o E 6 o E Fig. S6. Comparison of the ice crystal number sensitivity to updraft velocity, N i = w (in seconds per centimeter per liter), between GEOS-5 and CAM5. simulations: (A) N i = w from the DEF-G simulation with the Phillips et al. (2) nucleation spectrum at 25 hpa and after filtering out grid cells and time points where dn i < L ; DEF-C simulation at 232 hpa with the (B) Phillips et al. ref. 2, (C) Barahona and Nenes ref. 2 CNT, and (D) Phillips et al. ref. 22 nucleation spectra. Sullivan et al. 7of8

8 8N.5 a 8N b 4N cm s.5 4S 4N.5 4S.5 8S 8S 6W 2W 8W 4W 8N 4E 8E 2E 6E c 2 cm2 s 2 4S 6W 2W 8W 4W 8N 4E 8E 2E 6E 4E 8E 2E 6E d 4S 8S.5 e 6W 2W 8W 4W 8N f 4N 2 4S.5 L E 8E 2E 6E 4N 4E 8E 2E 6E S 5 8N.5 4N 5 6W 2W 8W 4W.5 8S 6W 2W 8W 4W 4E 8E 2E 6E 4N 4S 8S 6W 2W 8W 4W Fig. S7. Comparison of the input updraft velocity, w, its variance, σ 2w, and output newly formed ice crystal number concentration, Ni, between GEOS-5 (A, C, and E) and CAM5. (B, D, and F) simulations, all shown in log space: w from (A) DEF-G and (B) DEF-C (centimeters per second); σ 2w from (C) DEF-G and (D) DEF-C (square centimeters per seconds squared); and Ni from (E) DEF-G and (F) DEF-C (per liter). GEOS-5 maps are shown at 25 hpa, after filtering out grid cells and time points where dni < L, i.e., only when nonnegligible hydrometeor formation occurs. CAM maps are shown at 232 hpa. Additional regions of C are omitted when σ 2w < 5 cm2 s 2. Sullivan et al. 8 of 8