An Analytical Framework for Midlatitude Temperature Variability with Diabatic Heating and Finite Correlation Time

<jats:p>We present an analytical framework for calculating the first four statistical moments of the atmospheric temperature distribution, explicitly including advection and diabatic effects via the stochastic advection equation under a Gaussian wind and a Newtonian relaxation approximation. While this equation has been previously solved numerically, analytical expressions for these moments have yet to be established. Here, we derive the closed-form expressions for the moments of the atmospheric temperature distribution in terms of the statistical properties of the Gaussian wind, the relaxation time, and the time-mean temperature profile. Our theory successfully recovers previous frameworks, provides an explicit explanation of diabatic effects, and demonstrates good agreement with simulations and observations. Our work offers a computationally efficient method for calculating temperature moments, while serving as a diagnostic tool to evaluate diabatic effects in models and mathematically linking diabatic processes to the frequency of temperature extremes even if Earth's state parameters change in time.</jats:p>