### Abstract

Previous research on the direct effect of atmospheric aerosols on climate has estimated the average radiative forcing per unit sulfate mass, and has used this average to calculate the magnitude and spatial distribution of sulfate forcing. In this paper, we posit that radiative forcing is often a nonlinear function of sulfate mass concentration. In contrast to measures of average forcing, we introduce the concept of 'marginal forcing', which is defined as the change in radiative forcing for an incremental change in sulfate concentration. A multi-component, size-resolved aerosol box model is used, which couples an aerosol chemical equilibrium model with a model for calculating radiative forcing based on Mie theory. The results for a typical nonurban continental aerosol show that total aerosol mass and radiative forcing are nonlinear functions of sulfate concentration. This nonlinearity is mainly due to the chemical interaction of sulfate with volatile inorganic components of the aerosol (ammonium, nitrate, and water). As a result, the marginal forcing varies significantly as a function of sulfate concentration; from - 550 to + 20 W (g SO4 2-)-1 at a relative humidity (RH) of 80%. Estimates of marginal forcing are strongly sensitive to RH. Absolute marginal forcing also decreases significantly with total nitrate concentration, increases with total ammonia concentration, and generally increases with temperature. We estimate that the bias in assuming a constant average forcing may cause overestimates in local continental aerosol radiative forcing by up to 50%, and in the marginal forcing by a factor of two or more. This bias is greatest at intermediate sulfate concentration, high RH, high total nitrate concentration, low total ammonia concentration( ≤ 2 μg m-3), and low temperature. Previous research on the direct effect of atmospheric aerosols on climate has estimated the average radiative forcing per unit sulfate mass, and has used this average to calculate the magnitude and spatial distribution of sulfate forcing. In this paper, we posit that radiative forcing is often a nonlinear function of sulfate mass concentration. In contrast to measures of average forcing, we introduce the concept of `marginal forcing', which is defined as the change in radiative forcing for an incremental change in sulfate concentration. A multi-component, size-resolved aerosol box model is used, which couples an aerosol chemical equilibrium model with a model for calculating radiative forcing based on Mie theory. The results for a typical nonurban continental aerosol show that total aerosol mass and radiative forcing are nonlinear functions of sulfate concentration. This nonlinearity is mainly due to the chemical interaction of sulfate with volatile inorganic components of the aerosol (ammonium, nitrate, and water). As a result, the marginal forcing varies significantly as a function of sulfate concentration, from -550 to +20 W(g SO4 2-)-1 at a relative humidity (RH) of 80%. Estimates of marginal forcing are strongly sensitive to RH. Absolute marginal forcing also decreases significantly with total nitrate concentration, increases with total ammonia concentration, and generally increases with temperature. We estimate that the bias in assuming a constant average forcing may cause overestimates in local continental aerosol radiative forcing by up to 50%, and in the marginal forcing by a factor of two or more. This bias is greatest at intermediate sulfate concentration, high RH, high total nitrate concentration, low total ammonia concentration (≥2 μg m-3), and low temperature.