# The origin of spurious high frequency waves in numerical weather prediction

I am currently reading this paper on digital filters and their relevance to meteorological models Digital Filter Initialization . My question relates to this following text - "The requirement to modify meteorological analysis to avoid spurious high frequency oscillations has been known from the beginning of numerical weather prediction". Is it possible to get a mathematical illustration of how this problem originally arises(i.e. spurious high frequency oscillations) in numerical weather prediction ? I am not sure whether I can edit this question but I would like to restrict the domain of the question to high frequency oscillations arising in high resolution simulations and one that involves a LES. Numerically I have four nested grids - my country, a state within that and a district within that state and then a small valley inside that district. My question relates to the high frequency oscillations inside the innermost grid. Let us assume model independence for this question.

I don't have a mathematical illustration at the moment (though I might edit this answer if I come across something in my notes). But the basic problem is sound waves. If you make a numerical model of the full Navier-Stokes equations, that model has to allow sound waves to propagate. That implies that the time step used has to be incredibly short -- under 5 minutes for sure for a grid size of 100 km, and preferably much shorter than that. So sound waves are fast-moving, high frequency, and don't really contribute to weather. That's why weather prediction models tend to use the primitive equations: these equations are very similar to the Navier-Stokes, but sound waves aren't solutions to these equations.

Hopefully that's a start. Feel free to ask for a clarification on how this filtering is done in practice. Like I said, I'll look through my notes for something, I remember doing a problem set involving this.

Edit: Here are some notes from David Randall on the anelastic and Boussinesq approximations. These notes are in more of a classroom format, but at the end they derive the link between the continuity equation and sound waves.

Briefly, sound waves arise from variations in density that are independent of variations in enthalpy/temperature. The continuity equation in general is $\frac{\partial \rho}{\partial t} + \nabla\cdot (\rho \vec{u}) = 0$ in which density variations can be accompanied by contraction and expansion of the fluid. The anelastic equations filter out sound waves by replacing this dynamic equation with a diagnostic one, removing the time derivative: $\nabla\cdot(\rho_0 \vec{u}) = 0$ where $\rho_0$ is a background density profile. No more time derivative for density means no more expansion and contraction except under changes in enthalpy.

This is how sound waves are filtered out. I realize that I didn't read the question closely enough, though. If you're working with an LES, at that fine of a spatial resolution, you're probably 1) already using the anelastic equations, and 2) properly resolving a lot of the high-frequency oscillations anyway. As @milancurcic mentioned, gravity waves become an issue as the next-highest frequency waves. These notes go through the problem of gravity waves in a shallow-water model, and this article (behind a paywall) shows how a good choice of initialization can help eliminate those waves.

• Also the article by Peter Lynch seems to hint that removal of gravity waves cannot be justified always and that high frequency waves can contribute to mesoscale development of systems. Any thoughts on that ? – gansub Jul 14 '15 at 2:45
• a 5 minute timestep seems incredibly long to me but I am used to working with much finer grids. For a 250 m grid resolution I use a 1.5 s large time step and 7 acoustic time steps per large time step (roughly 0.21 s) in a fully compressible LES. For a reference on filtering acoustic waves from N-S, look at the derivation of the anelastic approximation. – casey Jul 14 '15 at 15:13
• The spurious waves that the OP (and the linked document in the question) refers to are $2\Delta x$ gravity waves, rather than sound waves. They can occur either due to single-cell forcing in the mass field, or as a spurious numerical mode in the differencing schemes. I hope your answer can address this. See the seminal report by Mesinger and Arakawa, twister.caps.ou.edu/CFD2013/Mesinger_ArakawaGARP.pdf, and/or Dale Durran's numerical methods book. – milancurcic Jul 14 '15 at 15:36
• gansub: As far as I know, gravity waves can be important in the breakup of frontal systems. This is included in some LES as an energy backscatter from the small scales. casey: 5 minutes at 100 km is ~.75 s at 250 m. milancurcic: You're absolutely right, I didn't read the question closely enough. If you write up another answer on gravity waves in particular, that should be the accepted answer. – Jareth Holt Jul 15 '15 at 12:41

The answer before me is correct, but that is not the only reason why digital filters are used. After the data assimilation procedure occurs, the resulting model state may not be in equilibrium. For example, if geopotential height (or surface pressure) is updated in the assimilation procedure, but wind is not (or vice versa), then the assimilated fields are out of quasi-geostrophic balance. This can also be applied to the hydrostatic approximation. Being that the resultant fields are out of balance, gravity waves may occur in the model, but may not actually occur physically. The digital filter reduces most of these gravity waves.

If you have a Met Ed account, this will help explain http://www.meted.ucar.edu/nwp/pcu1/d_adjust/6_0.htm.