Editorial Type:
Article
Article Type:
Research Article

Uniform Asymptotics for the Linear Kelvin Wave in Spherical Geometry

John P. Boyd University of Michigan, Ann Arbor, Michigan

Search for other papers by John P. Boyd in
Current site
Google Scholar
PubMed Close
and
Cheng Zhou University of Michigan, Ann Arbor, Michigan

Search for other papers by Cheng Zhou in
Current site
Google Scholar
PubMed Close
Print Publication:
01 Feb 2008
DOI:
https://doi.org/10.1175/2007JAS2356.1
Page(s):
655–660
Full access

Abstract

The Kelvin wave is the gravest eigenmode of Laplace’s tidal equation. It is widely observed in both the ocean and the atmosphere. In the absence of mean currents, the Kelvin wave depends on two parameters: the zonal wavenumber s (always an integer) and Lamb’s parameter ϵ. An asymptotic approximation valid in the limit s2 + ϵ ≫ 1 is derived that generalizes the usual “equatorial wave” limit that ϵ → ∞ for fixed s. Just as shown for Rossby waves two decades ago, the width of the Kelvin wave is (ϵ + s2)−1/4 rather than ϵ−1/4 as in the classical equatorial beta-plane approximation.

Corresponding author address: John Boyd, University of Michigan, 2455 Hayward Ave., Ann Arbor, MI 48109-2143. Email: jboyd@umich.edu

Abstract

The Kelvin wave is the gravest eigenmode of Laplace’s tidal equation. It is widely observed in both the ocean and the atmosphere. In the absence of mean currents, the Kelvin wave depends on two parameters: the zonal wavenumber s (always an integer) and Lamb’s parameter ϵ. An asymptotic approximation valid in the limit s2 + ϵ ≫ 1 is derived that generalizes the usual “equatorial wave” limit that ϵ → ∞ for fixed s. Just as shown for Rossby waves two decades ago, the width of the Kelvin wave is (ϵ + s2)−1/4 rather than ϵ−1/4 as in the classical equatorial beta-plane approximation.

Corresponding author address: John Boyd, University of Michigan, 2455 Hayward Ave., Ann Arbor, MI 48109-2143. Email: jboyd@umich.edu

Keywords: Kelvin waves; Waves, atmospheric; Dynamics

1. Introduction

Laplace showed more than two centuries ago that the free oscillations of a layer of homogeneous fluid and uniform depth on a rotating, spherical earth are governed by a trio of nonlinear partial differential equations that are usually called the “Laplace tidal equations” or the “nonlinear shallow water wave equations.” When linearized about a state of rest, these equations have eigenmodes that are commonly called “Hough” functions.1 The slowest eastward-traveling wave has been given the special name of the “Kelvin wave” because of many striking similarities to the coastally trapped waves analyzed by Lord Kelvin in the nineteenth century (Thomson 1880). The Kelvin wave has enormous practical importance as reviewed in sources as diverse as Chapman and Lindzen (1970), Majda (2003), and Andrews et al. (1987).

Longuet-Higgins (1968) carried out a magisterial study of the Hough functions in general and the Kelvin wave in particular nearly 40 years ago. Even so, there are some lacunae in the theory, which we fill below.

The Kelvin wave depends on two parameters; see Fig. 1. The zonal wavenumber s is always a positive integer. (The case of s = 0 is a nonpropagating mode that is not relevant to our analysis.) Lamb’s parameter ϵ is a nondimensional mean depth that is explicitly
i1520-0469-65-2-655-e1
where Ω is the angular frequency of the earth’s rotation in radians per second, a is the radius of the planet, and g is the gravitational constant, which is 9.8 m s−2 for the earth, and H is the mean depth of the fluid. As explained in Chapman and Lindzen (1970), Majda (2003), and Marshall and Boyd (1987), Laplace’s equations can be profitably employed for continuously stratified (rather than homogeneous) fluids if the depth H is interpreted as the “equivalent depth” of a given baroclinic mode. Thus, to describe all possible varieties of Kelvin waves in a three-dimensional stratified ocean or atmosphere, one needs to solve Laplace’s tidal equations for a very wide range of ϵ ranging from very small (for the “barotropic” or nearly-barotropic waves) to very large (for high-order baroclinic modes).

One important class of Kelvin waves is the equatorially trapped waves, which occur at the limit ϵ → ∞. The usual derivation assumes that s is fixed as this limit is taken. Here, we show that the structure and speed of the Kelvin wave are significantly modified when s is large. Our approximation is uniformly valid for large s and/or ϵ regardless of the size of the smaller parameter. The new approximation turns to be surprisingly accurate outside its formal range of validity in the region where both s and ϵ are small.

Twenty years ago, Boyd (1985) showed that for Rossby waves, the parameter that controls the width of an equatorially trapped wave is not ϵ, but rather
i1520-0469-65-2-655-e2
Furthermore, even in the barotropic limit (ϵ = 0), a Rossby wave of large zonal wavenumber s and low-latitudinal mode number is confined to low latitudes. The new asymptotic approximation shows that the same is true for the Kelvin wave.

2. A new asymptotic approximation

a. Derivation

The linear tidal equations can be written, as in Longuet-Higgins (1968),
i1520-0469-65-2-655-e3
i1520-0469-65-2-655-e4
i1520-0469-65-2-655-e5
where μ = cosθ, θ is colatitude, λ is longitude, ϕ is the height, and σ is the nondimensional frequency. All variables have been nondimensionalized using a as the length scale, 2Ω as the time scale, and H as the depth scale. Here U and Ṽ are the nondimensional “Margules–Robert” velocities:
i1520-0469-65-2-655-e6
The factor of i = −1 accounts for a quarter-of-a-wavelength phase differences between the north–south velocity and the other fields; the minus sign ensures that V is positive for a northward flow, as is the usual meteorological convention (but opposite to the velocity as defined using a physicist’s longitude–colatitude spherical coordinates since θ increases from zero at the North Pole to π at the South Pole).
It is convenient to define a new pressure/height unknown so that for the Kelvin wave, U will approximately equal the new unknown h where
i1520-0469-65-2-655-e7
Solving the longitudinal momentum equation for U in terms of the other variables reduces the set to two equations in two unknowns:
i1520-0469-65-2-655-e8
Figure 2 shows how this pair of equations is simplified to yet another set in (14), which is solved exactly below.

The first approximating principle is that when either s or ϵ is large, the Kelvin mode will be confined very close to the equator. Equatorial confinement was demonstrated for large ϵ by Longuet-Higgins; Fig. 3 shows that even for ϵ = 0, the Kelvin wave becomes more and confined to low latitude as the zonal wavenumber s increases. It follows that in the region where the wave has significant amplitude, the neighborhood of μ = 0, it will be a good approximation to replace (1 − μ2) by 1.

The second approximating principle is that for all ϵ, the frequency σ is close to its beta-plane value s/ϵ with a relative error that falls rapidly with increasing s or ϵ. To be precise, Longuet-Higgins has demonstrated numerically that σ/(s/ϵ) varies monotonically between its ϵ = 0 limit (1 + 1/s) and ϵ = ∞ limits (of 1), implying that
i1520-0469-65-2-655-e9
For fixed s and large ϵ, Longuet-Higgins showed that ζ ∼ 1/(4ϵ). It follows that for either large s or ϵ, it is a good approximation to replace σ by s/ϵ.
There are subtleties in the two terms that require both approximations simultaneously (marked by hollow arrows). In the upper left of the diagram, the frequency approximation suggests
i1520-0469-65-2-655-e10
When ϵ ≤ s2, the μ2 term can be neglected compared to the one in the parentheses. The subtlety is that when ϵ → ∞ for fixed s, the usual rules of the equatorial beta-plane apply, and in this limit, the scale of μ is O(ϵ−1/4) so that the term in μ2 is emphatically not small compared to 1. However, in this same limit, Ṽ tends to zero as O(ϵ−3/4) relative to h as shown by Longuet-Higgins (1968). Therefore, the replacement (s/σ2)μ2 − Ṽ ≈ −sṼ yields small errors relative to the other terms in the equation for s or ϵ or both large.
In the last term in the second equation (bottom right of the figure), the factor
i1520-0469-65-2-655-e11
consists of a (μ independent) constant plus a term quadratic in μ2, which we might naively think could be neglected. However, when σ is equal to its limiting value as ϵ → ∞, the constant in the braces is zero. It follows that, no matter how large ϵ is, it is never a good approximation to neglect the μ2 part of this factor. However, in the limit ϵ → ∞, Ṽ diminishes rapidly compared to h, and μ2 will be small in the equatorial region where the wave has most of its amplitude. It follows that although 1 − ϵσ2/s2 is small compared to one, this term is not negligible. Therefore, writing
i1520-0469-65-2-655-e12
we shall treat this factor with near cancellations as
i1520-0469-65-2-655-e13
Everywhere else in (8), 1 − μ2 → 1 and σ → s/ϵ, yielding the approximate equations:
i1520-0469-65-2-655-e14
The exact solution of this simplified pair of equations is
i1520-0469-65-2-655-e15
i1520-0469-65-2-655-e16
i1520-0469-65-2-655-e17
There is one further refinement that is helpful for a small zonal wavenumber s. All of the unknowns U, Ṽ, and ϕ have expansions in spherical harmonics, and all spherical harmonics of a given wavenumber have a common factor of (1 − μ2)s/2 that forces all these fields to have a root of order s/2 at each pole. [This property is true of all scalars in spherical geometry when expanded in a longitudinal Fourier series as discussed in Boyd (2001).] For small s and ϵ, the Gaussian factors of μ in (15) and (16) do not enforce these zeros. It is therefore desirable to make the following replacement:
i1520-0469-65-2-655-e18
Because (1 − μ2)s/2 = exp[(s/2) log(1 − μ2)] ≈ exp[−(s/2)μ2] for small μ, these two expressions in (18) are indistinguishable when either s or ϵ or both are large, but separation of the s/2 order zeros at the poles yields a better approximation when s and ϵ are small.

b. Results

Converting back to the original variables gives the final asymptotic approximation, uniformly accurate when either s or ϵ or both are large:
i1520-0469-65-2-655-e19
i1520-0469-65-2-655-e20
i1520-0469-65-2-655-e21
i1520-0469-65-2-655-e22
For fixed s and large ϵ, this frequency approximation simplifies to the formula derived nearly 40 years ago by Longuet-Higgins (1968), σ = s/ϵ{1 + (¼ϵ)}.

The price for the simplicity of these approximations is that their derivation is not based on systematic power series expansions, but rather more on the sort of mathematical banging-on-pipes that engineers do without apology (“if the dam holds, hurrah!”) and that physicists dignify with the fine-sounding German word “ansatz.” In this instance, no apologies are necessary because the full parameter space is only two dimensional (Fig. 1) and we have a highly accurate numerical method to compute “exact” solutions to compare with the approximations throughout the whole of parameter space, and thus validate the approximations with a thoroughness that even a mathematician can accept.

3. Results and numerical plots of errors

The maximum relative errors in the new approximation for the frequency σ for all ϵ ∈ [0, ∞] are 6.1% [s = 1], 2.5% [s = 2], 1.2% [s = 3], and 0.70% [s = 4], and in general O[1/(8s2)]. It is remarkable that an approximation derived for large s and/or ϵ is in fact rather accurate even for small s and ϵ.

Figure 4 compares the exact Kelvin wave on the sphere, as computed numerically, with three different asymptotic approximations for a typical pair of parameter values (ϵ = 5, s = 5). The equatorial beta-plane approximation (dots) is terrible. The asymptotic approximation that is unconstrained to vanish at the poles is much better, but not too good at high latitudes. The asymptotic approximation that is proportional to (1 − μ2)s/2, in contrast, is visually indistinguishable from the exact height field.

Figure 5 shows the error in frequency σ (upper left) and of (u, υ, ϕ) (in the L∞ norm, that is, the maximum error for any latitude) for the new asymptotic approximation in the ϵ − s plane. The frequency error is not one signed, but rather has an accidental zero along a ray s ≈ 3ϵ. This is not important, but the fact that the error is small everywhere is gratifying. The errors in the velocities and height are very small for small ϵ because of the built-in factor of (1 − μ2)s/2: u and ϕ are approximately equal to this factor for small ϵ. However, again the error is uniformly small everywhere in the two-dimensional parameter space.

4. Summary

We have derived a new asymptotic approximation for the Kelvin wave that fills the gap between the equatorial beta plane (fixed zonal wavenumber s, Lamb’s parameter ϵ → ∞) and the small ϵ, “velocity potential is Pnn(μ)” exp(isλ) approximation. The new approximation was derived under the assumption that at least one of (s, ϵ) is large, but numerically, is moderately good even when both parameters are small.

The approximation shows that the degree of equatorial confinement is not controlled by ϵ alone, but rather by the parameter
i1520-0469-65-2-655-e23
Boyd (1985) showed that the same is true for Rossby waves. A Kelvin wave of moderate zonal wavenumber s will be confined to the tropics even for ϵ = 0, a barotropic wave, as illustrated in Fig. 3.

Acknowledgments

This work was supported by the National Science Foundation through Grant OCE OCE 0451951. We thank Liang Tao for a first draft of Fig. 1. We also thank the two reviewers and the editor, Chris Snyder, for their assistance.

REFERENCES

  • Andrews, D. G., J. R. Holton, and C. B. Leovy, 1987: Middle Atmosphere Dynamics. International Geophysical Series, Vol. 40, Academic Press, 489 pp.

    • Search Google Scholar
    • Export Citation

  • Boyd, J. P., 1985: Barotropic equatorial waves: The non-uniformity of the equatorial beta-plane. J. Atmos. Sci., 42 , 1965–1967.

  • Boyd, J. P., 2001: Chebyshev and Fourier Spectral Methods. Dover, 630 pp.

  • Chapman, S., and R. S. Lindzen, 1970: Atmospheric Tides. D. Reidel, 200 pp.

  • Hough, S. S., 1897: On the application of harmonic analysis to the dynamic theory of the tides, I. On Laplace’s “oscillations of the first species” and on the dynamics of ocean currents. Philos. Trans. Roy. Soc., 189A , 201–257.

    • Search Google Scholar
    • Export Citation

  • Hough, S. S., 1898: On the application of harmonic analysis to the dynamic theory of the tides, II. On the general integration of Laplace’s tidal equations. Philos. Trans. Roy. Soc., 191A , 139–185.

    • Search Google Scholar
    • Export Citation

  • Longuet-Higgins, M. S., 1968: The eigenfunctions of Laplace’s tidal equation over a sphere. Philos. Trans. Roy. Soc. London, 262A , 511–607.

    • Search Google Scholar
    • Export Citation

  • Majda, A., 2003: Introduction to PDEs and Waves for the Atmosphere and Ocean. Vol. 9, Courant Lecture Notes, American Mathematical Society, 234 pp.

    • Search Google Scholar
    • Export Citation

  • Marshall, H. G., and J. P. Boyd, 1987: Solitons in a continuously stratified equatorial ocean. J. Phys. Oceanogr., 17 , 1016–1031.

  • Thomson, S., 1880: On gravitational oscillations of rotating water. Philos. Mag., 10 , 109–116. [Reprinted in 1911 in Mathematical and Physical Papers by Sir William Thompson, Baron Kelvin, J. Larmor and J. P. Joule, Eds., Cambridge University Press, 6 vols.].

    • Search Google Scholar
    • Export Citation

Fig. 1.
Fig. 1.

The Kelvin wave lives in a two-dimensional parameter space where the horizontal axis is the square root of Lamb’s parameter ϵ and the vertical axis is the zonal wavenumber s. When s and ϵ are both small, the Kelvin wave fills the entire globe from pole to pole. When r = s2 + ϵ is large compared to 1, the Kelvin wave is equatorially trapped, proportional to exp[−(½)rμ2] where μ is the sine of latitude. The horizontal axis is ϵ rather than ϵ itself so that r is just the distance from the origin in this map of the parameter space. When ϵ is large and much greater than s2, the Kelvin wave is well approximated by the equatorial beta plane. When s ≫ ϵ (and not necessarily large), the velocity potential χ ≈ exp(isλ) Pss(μ), where Pss is the usual associated Legendre function and the frequency σ ≈ s(s + 1)/ϵ. The regions of validity of these two regimes are marked by the dashed lines. The new asymptotic approximation derived in section 2 fills the gap between these two previously known limits.

Citation: Journal of the Atmospheric Sciences 65, 2; 10.1175/2007JAS2356.1

Fig. 2.
Fig. 2.

Schematic of the approximation of the exact pair of equations (middle inside the dotted rectangle) by the terms at the top and bottom of the diagram; this simplified pair of equations is then solved exactly.

Citation: Journal of the Atmospheric Sciences 65, 2; 10.1175/2007JAS2356.1

Fig. 3.
Fig. 3.

The latitudinal structure of u or ϕ (which are identical) for the lowest 10 zonal wavenumbers s for ϵ = 0 (barotropic waves) and u = (1 − μ2)s/2. The widest curve is s = 1 and the waves become more and more narrow as s increases. The dotted curves are guidelines that show that the half-width of the wave is within the tropics (|latitude| ≤ 30°) for s ≥ 5.

Citation: Journal of the Atmospheric Sciences 65, 2; 10.1175/2007JAS2356.1

Fig. 4.
Fig. 4.

The exact Kelvin mode for s = 5 and ϵ = 5 (thin curve with × symbols). The thick curve is the improved new asymptotic approximation, ϕ ≈ (1 − μ2)s/2 exp[−(½)(ϵ + s2 − s)μ2], which is graphically indistinguishable from the Kelvin wave. The dashed curve is the new asymptotic approximation without the (1 − μ2) factor, ϕ ≈ exp[−(½)ϵ + s2μ2]. The dotted curve is the classical equatorial beta-plane approximation, ϕ ≈ exp[−(½) ϵμ2].

Citation: Journal of the Atmospheric Sciences 65, 2; 10.1175/2007JAS2356.1

Fig. 5.
Fig. 5.

Log10 (errors) in the new asymptotic approximation to the Kelvin wave for (top left) the frequency and (top right), (bottom left), (bottom right) the three unknowns. The frequency error is the error in σ/(s/ϵ), which is close to 1 for all s and ϵ. The eigenfunctions are normalized by scaling the height to have a maximum of 1, so the errors are both the absolute and relative errors in this variable. Both u and υ were scaled by dividing the absolute errors by the global maximum of each velocity, and plotting these scaled variables.

Citation: Journal of the Atmospheric Sciences 65, 2; 10.1175/2007JAS2356.1

1

“Hough” is pronounced “Huf” in honor of Sydney Samuel Hough, F. R. S. (1870–1922), for Hough (1897, 1898).

Save
Cover Journal of the Atmospheric Sciences
Journal of the Atmospheric Sciences

  • Andrews, D. G., J. R. Holton, and C. B. Leovy, 1987: Middle Atmosphere Dynamics. International Geophysical Series, Vol. 40, Academic Press, 489 pp.

    • Search Google Scholar
    • Export Citation

  • Boyd, J. P., 1985: Barotropic equatorial waves: The non-uniformity of the equatorial beta-plane. J. Atmos. Sci., 42 , 1965–1967.

  • Boyd, J. P., 2001: Chebyshev and Fourier Spectral Methods. Dover, 630 pp.

  • Chapman, S., and R. S. Lindzen, 1970: Atmospheric Tides. D. Reidel, 200 pp.

  • Hough, S. S., 1897: On the application of harmonic analysis to the dynamic theory of the tides, I. On Laplace’s “oscillations of the first species” and on the dynamics of ocean currents. Philos. Trans. Roy. Soc., 189A , 201–257.

    • Search Google Scholar
    • Export Citation

  • Hough, S. S., 1898: On the application of harmonic analysis to the dynamic theory of the tides, II. On the general integration of Laplace’s tidal equations. Philos. Trans. Roy. Soc., 191A , 139–185.

    • Search Google Scholar
    • Export Citation

  • Longuet-Higgins, M. S., 1968: The eigenfunctions of Laplace’s tidal equation over a sphere. Philos. Trans. Roy. Soc. London, 262A , 511–607.

    • Search Google Scholar
    • Export Citation

  • Majda, A., 2003: Introduction to PDEs and Waves for the Atmosphere and Ocean. Vol. 9, Courant Lecture Notes, American Mathematical Society, 234 pp.

    • Search Google Scholar
    • Export Citation

  • Marshall, H. G., and J. P. Boyd, 1987: Solitons in a continuously stratified equatorial ocean. J. Phys. Oceanogr., 17 , 1016–1031.

  • Thomson, S., 1880: On gravitational oscillations of rotating water. Philos. Mag., 10 , 109–116. [Reprinted in 1911 in Mathematical and Physical Papers by Sir William Thompson, Baron Kelvin, J. Larmor and J. P. Joule, Eds., Cambridge University Press, 6 vols.].

    • Search Google Scholar
    • Export Citation
  • <sc>Fig</sc>. 1.
    View in gallery
    Fig. 1.

    The Kelvin wave lives in a two-dimensional parameter space where the horizontal axis is the square root of Lamb’s parameter ϵ and the vertical axis is the zonal wavenumber s. When s and ϵ are both small, the Kelvin wave fills the entire globe from pole to pole. When r = s2 + ϵ is large compared to 1, the Kelvin wave is equatorially trapped, proportional to exp[−(½)rμ2] where μ is the sine of latitude. The horizontal axis is ϵ rather than ϵ itself so that r is just the distance from the origin in this map of the parameter space. When ϵ is large and much greater than s2, the Kelvin wave is well approximated by the equatorial beta plane. When s ≫ ϵ (and not necessarily large), the velocity potential χ ≈ exp(isλ) Pss(μ), where Pss is the usual associated Legendre function and the frequency σ ≈ s(s + 1)/ϵ. The regions of validity of these two regimes are marked by the dashed lines. The new asymptotic approximation derived in section 2 fills the gap between these two previously known limits.

  • <sc>Fig</sc>. 2.
    View in gallery
    Fig. 2.

    Schematic of the approximation of the exact pair of equations (middle inside the dotted rectangle) by the terms at the top and bottom of the diagram; this simplified pair of equations is then solved exactly.

  • <sc>Fig</sc>. 3.
    View in gallery
    Fig. 3.

    The latitudinal structure of u or ϕ (which are identical) for the lowest 10 zonal wavenumbers s for ϵ = 0 (barotropic waves) and u = (1 − μ2)s/2. The widest curve is s = 1 and the waves become more and more narrow as s increases. The dotted curves are guidelines that show that the half-width of the wave is within the tropics (|latitude| ≤ 30°) for s ≥ 5.

  • <sc>Fig</sc>. 4.
    View in gallery
    Fig. 4.

    The exact Kelvin mode for s = 5 and ϵ = 5 (thin curve with × symbols). The thick curve is the improved new asymptotic approximation, ϕ ≈ (1 − μ2)s/2 exp[−(½)(ϵ + s2 − s)μ2], which is graphically indistinguishable from the Kelvin wave. The dashed curve is the new asymptotic approximation without the (1 − μ2) factor, ϕ ≈ exp[−(½)ϵ + s2μ2]. The dotted curve is the classical equatorial beta-plane approximation, ϕ ≈ exp[−(½) ϵμ2].

  • <sc>Fig</sc>. 5.
    View in gallery
    Fig. 5.

    Log10 (errors) in the new asymptotic approximation to the Kelvin wave for (top left) the frequency and (top right), (bottom left), (bottom right) the three unknowns. The frequency error is the error in σ/(s/ϵ), which is close to 1 for all s and ϵ. The eigenfunctions are normalized by scaling the height to have a maximum of 1, so the errors are both the absolute and relative errors in this variable. Both u and υ were scaled by dividing the absolute errors by the global maximum of each velocity, and plotting these scaled variables.

Fig. 1.

The Kelvin wave lives in a two-dimensional parameter space where the horizontal axis is the square root of Lamb’s parameter ϵ and the vertical axis is the zonal wavenumber s. When s and ϵ are both small, the Kelvin wave fills the entire globe from pole to pole. When r = s2 + ϵ is large compared to 1, the Kelvin wave is equatorially trapped, proportional to exp[−(½)rμ2] where μ is the sine of latitude. The horizontal axis is ϵ rather than ϵ itself so that r is just the distance from the origin in this map of the parameter space. When ϵ is large and much greater than s2, the Kelvin wave is well approximated by the equatorial beta plane. When s ≫ ϵ (and not necessarily large), the velocity potential χ ≈ exp(isλ) Pss(μ), where Pss is the usual associated Legendre function and the frequency σ ≈ s(s + 1)/ϵ. The regions of validity of these two regimes are marked by the dashed lines. The new asymptotic approximation derived in section 2 fills the gap between these two previously known limits.
Fig. 2.

Schematic of the approximation of the exact pair of equations (middle inside the dotted rectangle) by the terms at the top and bottom of the diagram; this simplified pair of equations is then solved exactly.
Fig. 3.

The latitudinal structure of u or ϕ (which are identical) for the lowest 10 zonal wavenumbers s for ϵ = 0 (barotropic waves) and u = (1 − μ2)s/2. The widest curve is s = 1 and the waves become more and more narrow as s increases. The dotted curves are guidelines that show that the half-width of the wave is within the tropics (|latitude| ≤ 30°) for s ≥ 5.
Fig. 4.

The exact Kelvin mode for s = 5 and ϵ = 5 (thin curve with × symbols). The thick curve is the improved new asymptotic approximation, ϕ ≈ (1 − μ2)s/2 exp[−(½)(ϵ + s2 − s)μ2], which is graphically indistinguishable from the Kelvin wave. The dashed curve is the new asymptotic approximation without the (1 − μ2) factor, ϕ ≈ exp[−(½)ϵ + s2μ2]. The dotted curve is the classical equatorial beta-plane approximation, ϕ ≈ exp[−(½) ϵμ2].
Fig. 5.

Log10 (errors) in the new asymptotic approximation to the Kelvin wave for (top left) the frequency and (top right), (bottom left), (bottom right) the three unknowns. The frequency error is the error in σ/(s/ϵ), which is close to 1 for all s and ϵ. The eigenfunctions are normalized by scaling the height to have a maximum of 1, so the errors are both the absolute and relative errors in this variable. Both u and υ were scaled by dividing the absolute errors by the global maximum of each velocity, and plotting these scaled variables.
All Time Past Year Past 30 Days
Abstract Views 0 0 0
Full Text Views 367 209 75
PDF Downloads 158 44 2