In the preceding comment, the results of our 2008 paper (Hasha et al. 2008, hereafter HBS) are compared to those in an earlier paper (Chen et al. 2005, hereafter CDH) that contains a relevant case study in which a mountain gravity wave is refracted by horizontally inhomogeneous winds. We did not cite CDH in HBS because, to our fault, we were unaware of it.
Both CDH and HBS investigate the vertical fluxes of zonal momentum due to mountain waves. As Durran’s comment points out (Durran 2009), there is no fundamental inconsistency between CDH and HBS, even though CDH found refraction-related momentum flux differences of about 50% between a case with horizontal inhomogeneity and a case without, whereas in HBS we only found differences of about 5%. This is because the time averaging that is implicit in our computations can easily account for the factor of 10 between the instantaneous momentum flux results in CDH and the time-integrated net momentum flux results in HBS. Moreover, our results were obtained using a low-resolution climate GCM (T42), and it is an open question as to how our results would play out in a high-resolution GCM.
To investigate the discrepancy between CDH and HBS quantitatively, one would need to compute the time average of the momentum fluxes of CDH near the top and bottom of the domain. Here the time average should be over one full cycle of the basic flow oscillation in CDH. Indeed, it appears that in CDH the instantaneous momentum flux at the mountain is in fact nearly equal to the steady-state value based on the instantaneous basic flow speed there. So the time average of this bottom flux should be nearly equal to the steady-state value based on the time-averaged bottom flow. Comparing this with the time-averaged flux near the top of the domain (but away from numerical damping layers, if there are any) should then give a simple measure of the refraction-related change of net momentum flux into the middle atmosphere, which is the quantity we targeted in HBS.
On another track, any such refraction-related change in the momentum flux must be accounted for in a concomitant change in the mean flow. In Bühler and McIntyre (2005), we showed that such nondissipative mean-flow changes should be detectable in the x component of the hydrodynamic impulse of the potential vorticity (PV) distribution. In CDH’s case the basic PV is barotropic and one should be able to detect a change in its domain-integrated impulse. This would complement the analysis of the momentum budget in the follow-up paper to CDH (i.e., Chen et al. 2007).
Corresponding author address: Oliver Bühler, Courant Institute of Mathematical Sciences, 251 Mercer Street, New York, NY 10012. Email: email@example.com