On the numerical integration of a randomly forced system: variation and feedback estimation

Simulated climate variables in a simple energy balance model subject to linearly increasing external forcing (due to increasing greenhouse gas emissions) and random internal forcings have been studied for more accurate climate prediction. The numerical method for such a system requires careful treatment of random forcings. Mathematical analyses show that the effect of random forcings should be diminished in the numerical integration method by the reciprocal of the root of the integration time step $ \left( {1/\sqrt {{\Delta t}} } \right) $ , which we call an attenuator. Our simulations consistently show that the attenuator desirably reduces variances of simulated climate variables and eliminates overestimation of the variances. However, the attenuator tends to bias the estimates of the climate feedback parameter obtained from a simple regression analysis of simulated variables toward unrealistically low values. This is because the reduced random forcings amplify the negative effect of a warming trend due to greenhouse emissions (when added to random forcing) on feedback estimation. Without the attenuator, the estimated feedback is much more accurate. The bias induced from the attenuator was largely resolved for the feedback estimation by the methodology of Lindzen and Choi (Asia-Pacific J Atmos Sci 47(4):377–390, 2011), which minimizes the negative effect of the warming trends by isolating short (few months) segments of increasing and decreasing temperature changes.