Origin of Sub-Kilometer Waves in Micro-WRF Simulations

When the community Weather Research and Forecasting (WRF) model was used with highly nested horizontal resolution to simulate the urban boundary layer in NYC during a summer 2010 heat wave, it generated horizontal roll vortices of questionable origin (Gutiérrez et al., 2012; Bornstein et al., 2012). This work characterized the existence of these structures and the numerical configuration of WRF that produced them, but did not provide an explanation for their occurrence. That grid schemes used for atmospheric or oceanic simulations can produce spurious wave behavior has been known since Charney and Phillips (1953). Typically, models have used nested rectilinear grids in simulation systems of the atmosphere to increasing resolution locally. This creates internal lateral boundaries between regions of different grid resolution, with a necessary result that small scale flows resolvable in the high resolution grid region may not be resolvable in the low resolution region. Thus, small-scale fluctuations in the high resolution region may go unstable as the acoustic energy propagates back and forth in the virtual cavity created by the nest boundaries. While these waves may saturate rather than be purely unstable, they nevertheless are strictly a manifestation of the numerical approach used for the problem. The source of the fluctuations is the fact that at a grid boundary it is impossible to match both the value and gradient exactly. Dealing with the transmission / absorption / reflection of atmospheric flow at the internal boundary is a major numerical problem to be overcome when creating an atmospheric or oceanic simulation system. The prior simulations reported by Gutiérrez and Bornstein et al., and other similar simulations used WRF with horizontal grid resolution nesting down to as little as 111 m. These simulations all exhibit high amplitude, but stable wavelike structures during unstable atmospheric conditions that some have characterized as roll clouds. While roll clouds do exist, the magnitude of the vorticity of some of these structures is such that they do not appear to be physically realistic, and thus some have used the term Smagorinsky waves to describe them, under a supposition that they are related to the Smagorinsky horizontal turbulent closure scheme in WRF, however similar behavior has been noted in simulations using large eddy simulation (LES) or k- closure schemes. The question, therefore, is whether these structures are real features, hitherto unobserved due to their small scale, or whether they are fictitious features that are a manifestation of the numerical grid and/or methods used for the simulations. If the latter, the question then arises is what is the origin of the numerical waves. While historic studies of gridding have looked at the stability or instability caused by grid geometry (such a study demonstrated the need for staggered grids to couple the mass / pressure and momentum equations), the potential for instability also exists due to differences in physical properties (such as the index of refraction change that occurs at the internal boundary caused by grid changes). A third possibility is that this is a parametric instability due to the time-varying changes that a wave sees as it moves from low to high to low resolution. While less commonly considered, parametric instabilities may be of interest because, in addition to changing the index of refraction for acoustic waves, the change in grid resolution changes the effective turbulent diffusivity due to the Smagorinsky closure scheme. Thus, the term, Smagorinsky wave may be apt, though indirectly. Determining whether these waves are real or an artifact of the numerical method is important for boundary layer simulation of, for example, air quality; hence this paper will discuss the overall problem and present several options for the creation of these waves, which, hopefully, will lead to a solution to the above questions.