• Paldor, N., S. Rubin, and A. J. Mariano, 2007: A consistent theory for linear waves of the shallow-water equations on a rotating plane in midlatitudes. J. Phys. Oceanogr., 37 , 115128.

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  • Poulin, F. J., and K. Rowe, 2008: Comment on “A consistent theory for linear waves of the shallow-water equations on a rotating plane in midlatitudes”. J. Phys. Oceanogr., 38 , 21112117.

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  • 1 Institute of Earth Sciences, The Hebrew University of Jerusalem, Jerusalem, Israel, and Division of Meteorology and Physical Oceanography, Rosensteil School of Marine and Atmospheric Sciences, University of Miami, Miami, Florida
  • | 2 Institute of Earth Sciences, The Hebrew University of Jerusalem, Jerusalem, Israel
  • | 3 Division of Meteorology and Physical Oceanography, Rosensteil School of Marine and Atmospheric Sciences, University of Miami, Miami, Florida
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* Current affiliation: Department of Environmental Sciences and Energy Research, The Weizmann Institute of Science, Rehovot, Israel.

Corresponding author address: Dr. Nathan Paldor, Institute of Earth Sciences, The Hebrew University of Jerusalem, Jerusalem 91904, Israel. Email: nathan.paldor@huji.ac.il

* Current affiliation: Department of Environmental Sciences and Energy Research, The Weizmann Institute of Science, Rehovot, Israel.

Corresponding author address: Dr. Nathan Paldor, Institute of Earth Sciences, The Hebrew University of Jerusalem, Jerusalem 91904, Israel. Email: nathan.paldor@huji.ac.il

The main findings of Paldor et al. (2007, hereafter PRM07), which were obtained by numerical integration (shooting method) of a Schrodinger-like eigenvalue equation, are confirmed by the calculations of Poulin and Rowe (2008, hereafter PR08), who employed a different numerical algorithm (the Chebyshev collocation method) for solving the same eigenvalue problem. The solutions of the eigenvalue problem yield the characteristics of waves in a channel on the β plane in a model that consistently accounts for the variation of f (the Coriolis frequency) with y. Despite the use of different scales for nondimensionalizing the variables in the two studies (e.g., the length scale in PR08 is Rd, the radius of deformation, whereas in PRM07 it is the earth’s radius), the final nondimensional eigenvalue problems [Eq. (2.9) in PRM07 and Eq. (12) in PR08] are identical, which lends credence to the fundamental nature of the Schrodinger eigenvalue formulation for wave dynamics in geophysical fluid dynamics. It also highlights the fact that the use of different scales in nondimensionalizing the variables does not alter the solution as long as these scales are clearly defined (to enable the reverse transformation to dimensional variables).

The zonally propagating wave solutions obtained from numerical solutions of the eigenvalue problems in both PRM07 and PR08 have the same nonharmonic features for a sufficiently small radius of deformation. The newly found features are relevant to the ocean (where the radius of deformation is small compared to that in the atmosphere) and include the following: (i) The y-dependent amplitude is maximal near the channel’s south wall and decays rapidly toward the north wall (i.e., the waves are trapped), and (ii) in the nondispersive longwave range of wavenumbers, the phase speed of Rossby waves is significantly larger (by a factor of 2 to 3) than that of harmonic waves.

We thank PR08 for detecting two programming errors we inadvertently had in some of the figures in which the meridional distributions of zonal velocity, free surface height, vorticity, and divergence were plotted. When these programming errors were corrected in the plotting routines, we got the exact same plots as in PR08. The quantitatively different meridional structure of vorticity and divergence does not alter the viewpoint advocated in PRM07 that attributes the near nondivergence of Rossby waves to their small phase speed [i.e., δ/ξO(C), where δ and ξ are the flow’s divergence and vorticity, respectively; see Eq. (5.2) in PRM07] instead of assuming it ab initio.

The unified derivation of Poincare and Rossby waves from the same eigenvalue problem yields an identical meridional V-field structure (given by the eigenfunctions of the eigenvalue problem) for the two waves. This derivation also attributes the different dispersion relations of the two waves to different roots of a cubic C(E) relationship between the phase speed C and the eigenvalue E. Thus, the difference between the two waves is not a result of different meridional scales of the two waves.

REFERENCES

  • Paldor, N., S. Rubin, and A. J. Mariano, 2007: A consistent theory for linear waves of the shallow-water equations on a rotating plane in midlatitudes. J. Phys. Oceanogr., 37 , 115128.

    • Search Google Scholar
    • Export Citation
  • Poulin, F. J., and K. Rowe, 2008: Comment on “A consistent theory for linear waves of the shallow-water equations on a rotating plane in midlatitudes”. J. Phys. Oceanogr., 38 , 21112117.

    • Search Google Scholar
    • Export Citation
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