Assessing the Equatorial Long-Wave Approximation: Asymptotics and Observational Data Analysis

H. Reed Ogrosky Department of Mathematics, University of Wisconsin–Madison, Madison, Wisconsin

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Samuel N. Stechmann Department of Mathematics, and Department of Atmospheric and Oceanic Sciences, University of Wisconsin–Madison, Madison, Wisconsin

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Abstract

Equatorial long-wave theory applies where a small horizontal aspect ratio between meridional and zonal length scales is assumed. In an idealized setting, the theory suggests that (i) meridional wind is small, (ii) geostrophic balance holds in the meridional direction, and (iii) inertio-gravity waves are small in amplitude or “filtered out.” In this paper a spectral data analysis method is used to quantitatively assess the spatial and temporal scales on which each of these aspects of long-wave dynamics is observed in reanalysis data. Three different perspectives are used in this assessment: primitive variables, characteristic variables, and wave variables. To define each wave variable, the eigenvectors and theoretical wave structures of the equatorial shallow-water equations are used. Evidence is presented that the range of spatial and temporal scales on which long-wave dynamics holds depends on which aspect of the dynamics is considered. For example, while meridional winds are an order of magnitude smaller than zonal winds over only a very narrow range of spatial scales (planetary wavenumber ), an examination of meridional geostrophic balance and inertio-gravity waves indicates long-wave dynamics for a broader range of scales (). A simple prediction is also presented for this range of scales based on physical and mathematical reasoning.

Supplemental information related to this paper is available at the Journals Online website: http://dx.doi.org/10.1175/JAS-D-15-0065.s1.

Corresponding author address: H. Reed Ogrosky, Department of Mathematics, University of Wisconsin–Madison, 480 Lincoln Dr., Madison, WI 53706-1325. E-mail: ogrosky@math.wisc.edu

Abstract

Equatorial long-wave theory applies where a small horizontal aspect ratio between meridional and zonal length scales is assumed. In an idealized setting, the theory suggests that (i) meridional wind is small, (ii) geostrophic balance holds in the meridional direction, and (iii) inertio-gravity waves are small in amplitude or “filtered out.” In this paper a spectral data analysis method is used to quantitatively assess the spatial and temporal scales on which each of these aspects of long-wave dynamics is observed in reanalysis data. Three different perspectives are used in this assessment: primitive variables, characteristic variables, and wave variables. To define each wave variable, the eigenvectors and theoretical wave structures of the equatorial shallow-water equations are used. Evidence is presented that the range of spatial and temporal scales on which long-wave dynamics holds depends on which aspect of the dynamics is considered. For example, while meridional winds are an order of magnitude smaller than zonal winds over only a very narrow range of spatial scales (planetary wavenumber ), an examination of meridional geostrophic balance and inertio-gravity waves indicates long-wave dynamics for a broader range of scales (). A simple prediction is also presented for this range of scales based on physical and mathematical reasoning.

Supplemental information related to this paper is available at the Journals Online website: http://dx.doi.org/10.1175/JAS-D-15-0065.s1.

Corresponding author address: H. Reed Ogrosky, Department of Mathematics, University of Wisconsin–Madison, 480 Lincoln Dr., Madison, WI 53706-1325. E-mail: ogrosky@math.wisc.edu

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