TY - JOUR

T1 - Linear waves in a symmetric equatorial channel

AU - Erlick, C.

AU - Paldor, Nathan

AU - Ziv, B.

N1 - Copyright:
Copyright 2008 Elsevier B.V., All rights reserved.

PY - 2007/4

Y1 - 2007/4

N2 - Using a scaling that allows us to separate the effects of the gravity wave speed from those of boundary location, we reduce the equations for linear waves in a zonal channel on the equatorial beta-plane to a single-parameter eigenvalue problem of the Schrödinger type with parabolic potential. The single parameter can be written δ = (Δφ)2/α1/2, where α = gH(2ΩR)-2, Δφ is half the channel width, g is the acceleraticn due to gravity, H is the typical height of the troposphere or ocean, Ω is the Earth's rotational frequency, and R is the Earth's radius. The Schrödinger-type equation has exact analytical solutions in the limits δ → 0 and δ → ∞, and one can use these to write an approximate expression for the solution that is accurate everywhere to within 4%. In addition to the simple expression for the eigenvalues, the concise and unified theory also yields explicit expressions for the associated eigenfunctions, which are pure sinusoidal in the δ → 0 limit and Gaussian in the δ → ∞ limit. Using the same scaling, we derive an eigenvalue formulation for linear waves in an equatorial channel on the sphere with a simple explicit formula for the dispersion relation accurate to O{(Δφ)2}. From this, we find that the phase velocity of the anti-Kelvin mode on the sphere differs by as much as 10% from -α1/2. Integrating the linearized shallow-water equations on the sphere, we find that for for larger α and Δφ, the phase speeds of all of the negative modes differ substantially from their phase speeds on the beta-plane. Furthermore, the dispersion relations of all of the waves in the equatorial channel on the sphere approach those on the unbounded sphere in a smooth asymptotic fashion, which is not true for the equatorial channel on the beta-plane.

AB - Using a scaling that allows us to separate the effects of the gravity wave speed from those of boundary location, we reduce the equations for linear waves in a zonal channel on the equatorial beta-plane to a single-parameter eigenvalue problem of the Schrödinger type with parabolic potential. The single parameter can be written δ = (Δφ)2/α1/2, where α = gH(2ΩR)-2, Δφ is half the channel width, g is the acceleraticn due to gravity, H is the typical height of the troposphere or ocean, Ω is the Earth's rotational frequency, and R is the Earth's radius. The Schrödinger-type equation has exact analytical solutions in the limits δ → 0 and δ → ∞, and one can use these to write an approximate expression for the solution that is accurate everywhere to within 4%. In addition to the simple expression for the eigenvalues, the concise and unified theory also yields explicit expressions for the associated eigenfunctions, which are pure sinusoidal in the δ → 0 limit and Gaussian in the δ → ∞ limit. Using the same scaling, we derive an eigenvalue formulation for linear waves in an equatorial channel on the sphere with a simple explicit formula for the dispersion relation accurate to O{(Δφ)2}. From this, we find that the phase velocity of the anti-Kelvin mode on the sphere differs by as much as 10% from -α1/2. Integrating the linearized shallow-water equations on the sphere, we find that for for larger α and Δφ, the phase speeds of all of the negative modes differ substantially from their phase speeds on the beta-plane. Furthermore, the dispersion relations of all of the waves in the equatorial channel on the sphere approach those on the unbounded sphere in a smooth asymptotic fashion, which is not true for the equatorial channel on the beta-plane.

KW - Anti-Kelvin mode

KW - Equatorial waves

KW - Equivalent barotropic

KW - Schrödinger equation

UR - http://www.scopus.com/inward/record.url?scp=34250005206&partnerID=8YFLogxK

U2 - 10.1002/qj.44

DO - 10.1002/qj.44

M3 - ???researchoutput.researchoutputtypes.contributiontojournal.article???

AN - SCOPUS:34250005206

SN - 0035-9009

VL - 133

SP - 571

EP - 577

JO - Quarterly Journal of the Royal Meteorological Society

JF - Quarterly Journal of the Royal Meteorological Society

IS - 624 PART A

ER -