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Rossby wave propagation on potential vorticity fronts with finite width. (English) Zbl 1445.76092

Summary: The horizontal gradient of potential vorticity (PV) across the tropopause typically declines with lead time in global numerical weather forecasts and tends towards a steady value dependent on model resolution. This paper examines how spreading the tropopause PV contrast over a broader frontal zone affects the propagation of Rossby waves. The approach taken is to analyse Rossby waves on a PV front of finite width in a simple single-layer model. The dispersion relation for linear Rossby waves on a PV front of infinitesimal width is well known; here, an approximate correction is derived for the case of a finite-width front, valid in the limit that the front is narrow compared to the zonal wavelength. Broadening the front causes a decrease in both the jet speed and the ability of waves to propagate upstream. The contribution of these changes to Rossby wave phase speeds cancel at leading order. At second order the decrease in jet speed dominates, meaning phase speeds are slower on broader PV fronts. This asymptotic phase speed result is shown to hold for a wide class of single-layer dynamics with a varying range of PV inversion operators. The phase speed dependence on frontal width is verified by numerical simulations and also shown to be robust at finite wave amplitude, and estimates are made for the error in Rossby wave propagation speeds due to the PV gradient error present in numerical weather forecast models.

MSC:

76U65 Rossby waves
86A10 Meteorology and atmospheric physics
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[1] De Vries, H.; Methven, J.; Frame, T. H. A.; Hoskins, B. J., An interpretation of baroclinic initial value problems: Results for simple basic states with nonzero interior PV gradients, J. Atmos. Sci., 66, 864-882, (2009)
[2] Esler, J. G., Benjamin-Feir instability of Rossby waves on a jet, Q. J. R. Meteorol. Soc., 130, 1611-1630, (2004)
[3] Farrell, B. F., The initial growth of disturbances in a baroclinic flow, J. Atmos. Sci., 39, 1663-1686, (1982)
[4] Gradshteyn, I. S. & Ryzhic, I. M.2000Table of Integrals, Series and Products, 6th edn. (ed. Jeffrey, A. & Zwillinger, D.). Academic.
[5] Gray, S. L.; Dunning, C. M.; Methven, J.; Masato, G.; Chagnon, J. M., Systematic model forecast error in Rossby wave structure, Geophys. Res. Lett., 41, 2979-2987, (2014)
[6] Harvey, B. J.; Ambaum, M. H. P., Instability of surface temperature filaments in strain and shear, Q. J. R. Meteorol. Soc., 136, 1506-1513, (2010)
[7] Harvey, B. J.; Ambaum, M. H. P., Perturbed Rankine vortices in surface quasi-geostrophic dynamics, Geophys. Astrophys. Fluid Dyn., 105, 377-391, (2011) · Zbl 1521.86023
[8] Held, I. M.; Pierrehumbert, R. T.; Garner, S. T.; Swanson, K. L., Surface quasi-geostrophic dynamics, J. Fluid Mech., 282, 1-20, (1995) · Zbl 0832.76012
[9] Iwayama, T.; Watanabe, T., Green’s function for a generalized two-dimensional fluid, Phys. Rev. E, 82, (2010)
[10] Juckes, M., Instability of surface and upper-tropospheric shear lines, J. Atmos. Sci., 52, 3247-3262, (1995)
[11] Juckes, M., Baroclinic instability of semi-geostrophic fronts with uniform potential vorticity. I: an analytic solution, Q. J. R. Meteorol. Soc., 124, 2227-2257, (1998)
[12] Juckes, M., Baroclinic instability of semi-geostrophic fronts with uniform potential vorticity. II: comparison of analytic and numerical solutions, Q. J. R. Meteorol. Soc., 124, 2259-2290, (1998)
[13] Orr, W. M’F., The stability or instability of the steady motions of a perfect liquid and of a viscous liquid. Part I: A perfect liquid, Proc. R. Irish Acad. A, 27, 9-68, (1907)
[14] Pierrehumbert, R. T.; Held, I. M.; Swanson, K. L., Spectra of local and nonlocal two-dimensional turbulence, Chaos, Solitons Fractals, 4, 1111-1116, (1994) · Zbl 0823.76034
[15] Plougonven, R.; Vanneste, J., Quasigeostrophic dynamics of a finite-thickness tropopause, J. Atmos. Sci., 67, 3149-3163, (2010)
[16] Pullin, D. I., Contour dynamics methods, Annu. Rev. Fluid Mech., 24, 89-115, (1992) · Zbl 0743.76021
[17] Rivest, C.; Davis, C. A.; Farrell, B. F., Upper-tropospheric synoptic-scale waves. Part I: maintenance as Eady normal modes, J. Atmos. Sci., 49, 2108-2119, (1992)
[18] Scott, R. K.; Dritschel, D. G.; Polvani, L. M.; Waugh, D. W., Enhancement of Rossby wave breaking by steep potential vorticity gradients in the winter stratosphere, J. Atmos. Sci., 61, 904-918, (2004)
[19] Swanson, K. L.; Kushner, P. J.; Held, I. M., Dynamics of barotropic storm tracks, J. Atmos. Sci., 54, 791-810, (1997)
[20] Vallis, G. K., Atmospheric and Oceanic Fluid Dynamics, (2006), Cambridge University Press
[21] Verkley, W. T. M., Tropopause dynamics and planetary waves, J. Atmos. Sci., 51, 509-529, (1994)
[22] Zabusky, N. J.; Hughes, M. H.; Roberts, K. V., Contour dynamics of the Euler equations in two dimensions, J. Comput. Phys., 30, 96-106, (1979) · Zbl 0405.76014
[23] Zhu, D.; Nakamura, N., On the representation of Rossby waves on the \({\it\beta}\)-plane by a piecewise uniform potential vorticity distribution, J. Fluid Mech., 664, 397-406, (2010) · Zbl 1221.76053
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