Adaptively Implicit Advection for Atmospheric Flows

• Published in: Journal of advances in modeling earth systems 2024-12, Vol.16 (12), p.n/a
• Main Authors: Weller, Hilary, Kühnlein, Christian, Smolarkiewicz, Piotr K.
• Format: Article
• Language: English
• Subjects: Accuracy; Acoustic waves; Acoustics; adaptive; Advection; Atmospheric flows; Deformation; Gravity waves; implicit; Linear equations; Navier-Stokes equations; numerics; Runge‐Kutta; Sound waves; Tracers; transport; Updraft
• ISSN: 1942-2466; 1942-2466
• DOI: 10.1029/2024MS004503

Implicit time‐stepping for advection is applied locally in space and time where Courant numbers are large, but standard explicit time‐stepping is used for the remaining solution which is typically the majority. This adaptively implicit advection scheme facilitates efficient and robust integrations with long time‐steps while having negligible impact on the overall accuracy, and achieving monotonicity and local conservation on general meshes. A novel and important aspect for the efficiency of the approach is that only one iteration is needed each time the linear equation solver is called for solving the advection equation. The demonstration in this paper uses the second‐order Runge‐Kutta implicit/explicit time integration in combination with a second/third‐order finite‐volume spatial discretization and is tested using deformation flow tracer advection on the sphere and a fully compressible model for atmospheric flows. Tracers are advected over the poles of highly anisotropic latitude‐longitude grids with very large Courant numbers and on quasi‐uniform hexagonal and cubed‐sphere meshes with the same algorithm. Buoyant flow simulations with strong local updrafts also benefit from adaptively implicit advection. Stably stratified compressible flow simulations require a stable combination of implicit treatment of gravity and acoustic waves as well as advection in order to achieve long time‐steps. Plain Language Summary Weather and climate prediction models take small time‐steps in order to make predictions about the future, starting from estimates of current conditions. The smaller the time‐steps are, the more of them have to be taken to make a prediction for a given time in the future. The more time‐steps that have to be taken, the more expensive the prediction is. If the time‐steps are too big, models can not only lose accuracy, they can become unstable—inaccuracies can become so large that wild oscillations are generated and the model crashes. These instabilities are often caused by the advective transport of constituents of the atmosphere by the wind. This paper describes a method—adaptive implicit advection—for calculating atmospheric transport using longer time‐steps while maintaining stability. We show that this can be achieved with minimal additional cost, and accuracy is only lost locally, where the time‐step is large relative to the flow speed and model grid size. Key Points Implicit time‐stepping for advection enables large Courant numbers Implicit tim