GRAVITY WAVES IN FV3

https://docs.google.com/presentation/d/11SLZtjkZ1OIuCnYC4vnZ_uX4PntS-U6YgLSbQvkMd-U/edit?usp=sharing

Result 2: 

We were also interested in the flux of smaller scale perturbations. When the flow hits the mountain, a perturbation is generated in the full velocity vector (as well as in the temperature/pressure). This allows wavelike perturbations to propagate into the atmosphere according to its dispersive properties. A perturbation will propagate conservatively, unless the perturbation feature is diffusing or otherwise breaking. By momentum balance, this would have the effect of imparting a drag forcing on larger scale flows.  

The procedure to do this is somewhat involved and the specific technique used here is experimental. To get a sense of the general approach see [3] (in particular the Reynold's stress method). For the plot below, we focus on perturbations at the size scale of the mountain, and isolate the flow features larger than this by applying a Gaussian filtering in Fourier space with spectral cutoff of about 7 degrees. We use a formulation of the vertical velocity in units of Pa/s and the zonal velocity to measure the vertical zonal perturbation flux (see [4] for more details) . To measure the interactions of this smaller scale flow with the background flow, we take the gradient with respect to the sigma coordinate. This is probably the most untested aspect of this approach. The general philosophy is to directly probe the momentum tendency at each level, but it has not been rigorously verified if this a valid approach or if we would expect momentum flux to be conserved across sigma levels in the absence of wave breaking. If given time, it would also be better to measure the wave perturbation using global techniques, such as Helmholtz decomposition or a TEM formulation. However, the code we had on hand was made to measure wave fluxes in a WRF box domain (and a bird in the hand is better than two in the bush). 

The qualitative findings here are consistent with our understanding of wave properties and the implications of Result 1 in this experiment. The plot below gives horizontal-time mean profiles of the perturbation flux and sigma-gradient (see google slides or email  for gif of time development of these profiles). There are three main observations here. 1. The structure of the flux profile does not appear to vary much between resolution cases; 2. The sigma-gradient gets spikier (more variability) with increasing resolution and this effect is pronounced higher aloft; 3. The decay of the sigma-gradient is increased for lower resolution, and so we see more breaking higher aloft for higher resolutions.

The increased variability in the sigma-gradient may be either due to the fact that we are better capturing wave breaking or it may be an adverse noise/diffusive effect caused by increasing vertical resolution. However, since the perturbation flux profile is consistent across resolutions, this suggests that increasing the resolution does not significantly nor adversely modify the resolved flow features (also corroborated by Result 1). This suggests that the breaking might be physical, but there is not a clear explanation yet as to why the amplitude of the gradient seems to change at the same level between different resolution cases. The fact that this effect is more pronounced higher aloft support that the vertical resolution is relevant for wave breaking. There is also augmented drag aloft compared to the lowest vertical resolution, and the two higher resolutions seem to agree on the magnitude of the gradient in this region. This might be reflective of the fact that wave vertical structures are better resolved, and so can be sustained to higher altitudes before reach instability. This also suggests possible convergence on a "solution" to the wave breaking profile. 

Conclusion and next steps 

The analysis done in this case highlights the relevance of vertical resolution to the breaking of gravity waves. Differences in simulations with different vertical resolutions start at the region of wave breaking, but this will spread out and impact the full flow field over time. The gradient across sigma levels of vertical zonal perturbation propagation shows that higher resolutions might produce more pronounced breaking and breaking higher aloft. 

Some next steps/follow-up questions are: 

1. Verify these results with a more rigorous implementation of a GW momentum flux and drag calculation. 

2. Does the vertical resolution affect the momentum forcing from these mountain generated gravity waves? -> measure the contribution to the global momentum budget for different vertical resolutions

3. Would increasing resolution lead to convergence on vertical profiles or other flow properties? 

4. FV3 has a particular Lagrangian vertical coordinate. How does modulating the vertical resolution change the results for other vertical coordinate systems? 

5. For different vertical resolutions, maybe we can get an idea of the realism of the breaking by incorporating measurements of wave stability, such as the Richardson number. Is there agreement with the measured location of wave breaking and the location predicted by flow stability parameters?

6. What are the spectral properties of the waves excited by the topography? How does this change at the wave propagates and might refract/break? Is there any dependence on the vertical resolution?

Characterizing the resolved gravity wave response in a dynamical core can provide insight on the vertical resolutions necessary to produce realistic flows. It can also be used to estimate how different choices of vertical resolution would affect the variability of the climate state through cross-scale flow interactions. Additionally, this may also inform the design of sub-grid scale GW parameterizations, as we could gain a better idea of the lower limit of what waves are resolved in the model. 

References: 

[1] https://www.cora.nwra.com/~alexand/publications/FrittsAlexander03.pdf 

[2] https://agupubs.onlinelibrary.wiley.com/doi/10.1029/2024MS004473?af=R 

[3] https://agupubs.onlinelibrary.wiley.com/doi/full/10.1029/2022MS003585 

[4] https://agupubs.onlinelibrary.wiley.com/doi/full/10.1029/2024GL108541 

If you have any questions or would like access to the Reynold's stress code, reach out to Isabella Dula at dula@stanford.edu. 

EXP4

EXP5

Conclusion/next steps