A time-split nonhydrostatic atmospheric model for weather research and forecasting applications
Introduction
Efforts to simulate small-scale atmospheric flows where nonhydrostatic effects are important, such as eddies in the atmospheric boundary layer (ABL) having length scales of order a few kilometers or less, and deep convective clouds that can span the depth of the troposphere, are coincident with both the development of computational fluid dynamics and the evolution of computer technology. In the early 1960s, Lilly [22] performed simulations of ABL vortices and convective clouds using a time- and space-centered (leapfrog) integration scheme for the fully compressible 2D Euler equations. From that time to the present day, atmospheric modelers simulating small-scale motions have confronted a similar set of issues that can be categorized in two general (and overlapping) areas – how to formulate efficient solvers for small-scale nonhydrostatic low-Mach-number stratified flow, and how to model energy dissipation within these solvers. The lessons learned in these 40+ years of development have increasingly important applications outside the ongoing research simulation needs because these advances are now being brought to bear in operational numerical weather prediction. Presently, numerical simulations used to produce operational weather forecasts are being run with horizontal grid-spacings of a few kilometers and they are beginning to explicitly represent small-scale (nonhydrostatic) motions such as convective clouds.
We describe the formulation of a version of the Weather Research and Forecasting (WRF) Model called the Advanced Research WRF (ARW) in this paper. The ARW model represents the latest developments following a particular modeling approach that uses time-splitting techniques to efficiently integrate the fully compressible nonhydrostatic equations of motion. While the general approach described here was developed originally for cloud models [19], it is applicable to larger scales and has been used in a number of nonhydrostatic numerical weather prediction (NWP) models (e.g. MM5 [10], LM [8], COAMPS [14], ARPS [45]). We begin in Section 2.1 by briefly outlining the most popular approaches to designing nonhydrostatic atmospheric flow solvers, followed by a description of the ARW solver’s continuous equations (Section 2.2), temporal discretization (Section 2.3) and spatial discretization (Section 2.4).
Atmospheric flow solutions do not converge in a strict sense; the model grid spacing is Δx ∼ O(km) but the Kolmogorov scale is ∼ O(cm), hence finer structures always appear with increasing resolution. Resolving these small structures is often the primary reason for increasing spatial resolution, thus an important aspect of a solver is its ability to correctly represent structures at the resolution limits (approximately 6Δx–10Δx for gridpoint models). Hence in designing atmospheric models we seek to maintain accuracy and minimize artificial dissipation at the resolved scales while removing energy at the gridscale. In Section 3.1 we present results from the ARW model using idealized flow test cases, for which converged solutions exist, and we examine solver performance as the resolution is decreased and the main structures are only marginally resolved. Following this, simulations of observed severe weather events (Section 3.2) demonstrate the ability of the ARW model to capture important NWP phenomena, in this case tornadic thunderstorms and 2005 Hurricane Katrina. In Section 3.3 we examine some statistics of high-resolution ARW NWP forecasts demonstrating the energetics of the model, the changing dynamical nature of atmospheric flow from synoptic scales to cloud scales, and model filter performance. Section 3.4 contains a further discussion of model filtering. A summary is presented in Section 4.
Section snippets
Modeling approaches
Atmospheric flow solvers produce spatial and temporal integrations of the Euler equations, and accurate solutions for time-evolving flows (as opposed to steady state solutions) are of utmost importance for NWP and most research applications. The modes of meteorological interest in the Euler equation solutions are relatively slow – they rarely exceed Mach numbers of approximately 1/3 (for example, in the jet stream). The fast modes in the solutions are the acoustic modes and they contain no
Model performance
In designing solvers we wish to maximize solver efficiency, that is, maximize the solution accuracy for a given computational expense or minimize the computational expense for a required accuracy. This requires the ability to objectively measure solution accuracy for relevant applications of the solver, and a problem we encounter when testing models is that the model solutions for our applications do not converge – spatial grid refinement leads to the appearance of smaller-scale features in the
Summary
The ARW model is the first fully compressible conservative form nonhydrostatic atmospheric model designed for both research and operational NWP applications. The integration scheme uses time-splitting to handle meteorologically insignificant acoustic modes. The time-splitting allows for efficient integrations for the low-Mach-number flows characteristic of atmospheric flows, and the efficiency of the time-splitting scheme is maintained across a wide range of scales, from synoptic scales where (Δ
Acknowledgments
Many individuals helped to construct the ARW modeling system in addition to performing many of the simulations shown in this paper. We specifically acknowledge Jimy Dudhia and John Michalakes for their major contributions in designing and constructing the ARW system, and Wei Wang and Morris Weisman for the forecast experiment results shown in this paper.
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The National Center for Atmospheric Research is sponsored by the National Science Foundation, Boulder, Colorado, USA.