Journal Article

Can Second‐Order Numerical Accuracy Be Achieved for Moist Atmospheric Dynamics With Non‐Smoothness at Cloud Edge?

Stechmann, Samuel N., Marsico, David H., Du, Shukai

2025

Overview

Non‐smoothness arises at cloud edge because, in moist thermodynamics, the thermodynamic properties of the atmosphere are different inside a cloud versus in clear air. In particular, inside a cloud, the vapor pressure of water is constrained by the saturation vapor pressure, which acts as a threshold. Due to this threshold, while the water vapor mixing ratio may vary continuously across cloud edge, its derivatives are not necessarily continuous at cloud edge. Similarly, non‐smoothness also arises for buoyancy and other variables. Consequently, this non‐smoothness in buoyancy and other variables can cause a degraded accuracy in computational simulations. Here we consider special treatment of numerical methods for the interface that arises from phase changes and cloud edges, in order to enhance the accuracy and potentially achieve second‐order accuracy. Numerical solutions are computed for the moist non‐precipitating Boussinesq equations as an idealized cloud‐resolving model with phase changes of water, that is, with cloud formation. Convergence tests, both spatial and temporal, are conducted to measure the numerical error as the grid spacing and time step are refined. While approximately second‐order accuracy is seen in root‐mean‐square (L2${L}^{2}$ ) error, the accuracy is degraded in the maximum (L∞${L}^{\\infty }$ ) error. Discussion is also included on theoretical issues and potential implications for numerical simulations. Plain Language Summary An important issue that arises from clouds is the difference in atmospheric properties inside the cloud versus outside the cloud. Inside a cloud, the vapor pressure of water is constrained by the saturation vapor pressure, which acts as a threshold. Due to this threshold, while water vapor, temperature, and many other common thermodynamic variables may vary continuously across cloud edge, their derivatives are not necessarily continuous. This lack of smoothness poses challenges for numerical approximations on computers. For instance, due to cloud edge, traditional numerical methods used for cloud‐resolving models may suffer a loss of accuracy. In this paper, we study the accuracy degradation that arises from the presence of clouds, and consider whether it is possible to achieve second‐order accuracy for numerical solutions in the presence of clouds and phase changes, where for instance, water vapor changes to cloud water. Key Points Numerical methods are presented for moist atmospheric dynamics and accounting for non‐smoothness at cloud edge Approximately second‐order numerical accuracy is seen in root‐mean‐square (L2${L}^{2}$ ) error, even with cloud formation Degraded accuracy is seen in maximum (L∞${L}^{\\infty }$ ) error, due to clouds

• Publisher: John Wiley & Sons, Inc,American Geophysical Union (AGU)
• Subject: Accuracy / Atmosphere / Atmospheric dynamics / Boussinesq approximation / Buoyancy / Cloud formation / cloud latent heating / Clouds / cloud‐resolving models / cloud–clear air interface / Finite volume method / interface / Interfaces / large‐eddy simulations / Mixing ratio / moist atmospheric dynamics / moist convection / Numerical analysis / Numerical simulations / Phase changes / Saturation vapor pressure / second‐order accuracy / Thermal properties / Thermodynamic properties / Thermodynamics / Vapor pressure / Vapour pressure / Water vapor / Water vapour