Data assimilation errors due to incomplete accounting of balanced versus unbalanced moisture

Abstract The notion of balance has played an important role in data assimilation. However, traditional ideas of balance have been based on dry dynamics, without moisture. Recently, the notion of balanced moisture has been formulated based on moist dynamics. Here, the goal is to investigate data assimilation strategies with moisture, and to assess the impact of accounting for moisture's balanced and unbalanced components. To illustrate the main ideas, examples are presented using a simple model consisting of linear, moist, shallow‐water equations with stochastic forcing, and a Kalman filter in its stationary state for a data assimilation method akin to 3D‐Var. If a traditional treatment of moisture is used, which does not properly decompose moisture into its balanced and unbalanced components, then the error in the recovered moisture is increased by up to 15% or more. These errors are most significant on smaller spatial scales, where inertia–gravity wave variance is larger, and in the presence of larger vertical gradients of the background moisture state, and in cases with larger observational noise. Underlying these results are some subtle theoretical aspects of the moist dynamical core and its dynamical operator, which is not symmetric; instead, it is self‐adjoint with respect to a non‐diagonal inner product, which is associated with conservation of moist energy. From the results here, and given the longstanding difficulties with data assimilation and forecasting of precipitation, it is possible that balanced versus unbalanced moisture may play a role in these difficulties, among many challenging aspects of water vapor, clouds, and precipitation.