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Uniform Asymptotics for the Linear Kelvin Wave in Spherical Geometry JOHN P. BOYD
AND
CHENG ZHOU
University of Michigan, Ann Arbor, Michigan (Manuscript received 22 November 2006, in final form 6 March 2007) 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 ⑀. An asymptotic approximation valid in the limit 公s2 ⫹ ⑀ k 1 is derived that generalizes the usual “equatorial wave” limit that ⑀ → ⬁ for fixed s. Just as shown for Rossby waves two decades ago, the width of the Kelvin wave is (⑀ ⫹ s2)⫺1/4 rather than ⑀⫺1/4 as in the classical equatorial beta-plane approximation.
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
1 “Hough” is pronounced “Huf” in honor of Sydney Samuel Hough, F. R. S. (1870–1922), for Hough (1897, 1898).
Corresponding author address: John Boyd, University of Michigan, 2455 Hayward Ave., Ann Arbor, MI 48109-2143. E-mail:
[email protected] DOI: 10.1175/2007JAS2356.1 © 2008 American Meteorological Society
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integer. (The case of s ⫽ 0 is a nonpropagating mode that is not relevant to our analysis.) Lamb’s parameter ⑀ is a nondimensional mean depth that is explicitly
⑀⫽
4⍀2a2 , gH
共1兲
where ⍀ 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 ⑀ 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 ⑀ → ⬁. 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 ⑀ 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 ⑀ are small.
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where ⫽ cos, is colatitude, is longitude, is the height, and is the nondimensional frequency. All variables have been nondimensionalized using a as the length scale, 2⍀ as the time scale, and H as the depth ˜ are the nondimensional “Marscale. Here U and V gules–Robert” velocities: U⫽
1 d 公1 ⫺ 2 ; 2⍀ dt
˜ ⫽ ⫺i V
1 d 公1 ⫺ 2. 2⍀ dt 共6兲
FIG. 1. The Kelvin wave lives in a two-dimensional parameter space where the horizontal axis is the square root of Lamb’s parameter ⑀ and the vertical axis is the zonal wavenumber s. When s and ⑀ are both small, the Kelvin wave fills the entire globe from pole to pole. When r ⫽ 公s2 ⫹ ⑀ is large compared to 1, the Kelvin wave is equatorially trapped, proportional to exp[⫺(1⁄2)r2] where is the sine of latitude. The horizontal axis is 公⑀ rather than ⑀ itself so that r is just the distance from the origin in this map of the parameter space. When ⑀ is large and much greater than s2, the Kelvin wave is well approximated by the equatorial beta plane. When s k 公⑀ (and not necessarily large), the velocity potential ⬇ exp(is) P ss(), where P ss is the usual associated Legendre function and the frequency ⬇ 公s(s ⫹ 1)/公⑀. 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.
Twenty years ago, Boyd (1985) showed that for Rossby waves, the parameter that controls the width of an equatorially trapped wave is not ⑀, but rather
⑀eff ⬅ ⑀ ⫹ s2.
共2兲
Furthermore, even in the barotropic limit (⑀ ⫽ 0), a Rossby wave of large zonal wavenumber s and lowlatitudinal 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), ˜ ⫺ s ⫽ 0, U ⫺ V
共3兲
˜ ⫹ 共1 ⫺ 2兲 ⫽ 0, U ⫺ V
共4兲
˜ ⫺ ⑀共1 ⫺ 2兲 ⫽ 0, sU ⫺ 共1 ⫺ 2兲V
共5兲
The factor of i ⫽ 公⫺1 accounts for a quarter-of-awavelength 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 increases from zero at the North Pole to 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 h ⫽ 共sⲐ兲 ↔ ⫽ 共Ⲑs兲h.
共7兲
Solving the longitudinal momentum equation for U in terms of the other variables reduces the set to two equations in two unknowns:
冉
s
2
冊
˜ ⫹ 2 ⫺ s V
冋
s h ⫹ 共1 ⫺ 2兲h ⫽ 0
˜ Ⲑ ⫺ 共1 ⫺ 2兲V ˜ Ⲑs ⫹ 1 ⫺ ⑀ V
2 s2
册
共1 ⫺ 2兲 h ⫽ 0.
共8兲 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 ⑀ is large, the Kelvin mode will be confined very close to the equator. Equatorial confinement was demonstrated for large ⑀ by Longuet-Higgins; Fig. 3 shows that even for ⑀ ⫽ 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 ⫽ 0, it will be a good approximation to replace (1 ⫺ 2) by 1. The second approximating principle is that for all ⑀, the frequency is close to its beta-plane value s/公⑀ with a relative error that falls rapidly with increasing s or ⑀. To be precise, Longuet-Higgins has demonstrated numerically that /(s/公⑀) varies monotonically between its ⑀ ⫽ 0 limit (公1 ⫹ 1/s) and ⑀ ⫽ ⬁ limits (of 1), implying that
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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.
⫽
s
公⑀
共1 ⫹ 兲,
ⱕ 公1 ⫹ 1Ⲑs ⫺ 1 ⱕ 1Ⲑ共2s兲.
共9兲
For fixed s and large ⑀, Longuet-Higgins showed that ⬃ 1/(4⑀). It follows that for either large s or ⑀, it is a good approximation to replace by s/公⑀. 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
冉
共sⲐ 2兲2 ⫺ s ⬇ ⫺s 1 ⫺
⑀ s
冊
2 . 2
共10兲
When ⑀ ⱕ s2, the 2 term can be neglected compared to the one in the parentheses. The subtlety is that when ⑀ → ⬁ for fixed s, the usual rules of the equatorial beta-plane apply, and in this limit, the scale of is O(⑀⫺1/4) so that the term in 2 is emphatically not small ˜ tends to compared to 1. However, in this same limit, V ⫺3/4 zero as O(⑀ ) relative to h as shown by LonguetHiggins (1968). Therefore, the replacement (s/ 2)2 ⫺ ˜ ⬇ ⫺sV ˜ yields small errors relative to the other terms V in the equation for s or ⑀ or both large. In the last term in the second equation (bottom right of the figure), the factor 1⫺⑀
2 s2
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共1 ⫺ 兲 ⫽ 2
冉
1⫺⑀
2 s2
冊
⫹⑀
2 s2
2
共11兲
consists of a ( independent) constant plus a term quadratic in 2, which we might naively think could be neglected. However, when is equal to its limiting value as ⑀ → ⬁, the constant in the braces is zero. It follows that, no matter how large ⑀ is, it is never a good
FIG. 3. The latitudinal structure of u or (which are identical) for the lowest 10 zonal wavenumbers s for ⑀ ⫽ 0 (barotropic waves) and u ⫽ (1 ⫺ 2)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 | ⱕ 30°) for s ⱖ 5.
approximation to neglect the 2 part of this factor. ˜ diminishes rapidly comHowever, in the limit ⑀ → ⬁, V 2 pared to h, and will be small in the equatorial region where the wave has most of its amplitude. It follows that although 1 ⫺ ⑀ 2/s2 is small compared to one, this term is not negligible. Therefore, writing
⫽ sⲐ公⑀ ⫹ 1共⑀, s兲
共12兲
we shall treat this factor with near cancellations as 1⫺⑀
2 2
s
共1 ⫺ 2兲 ⬇ ⫺2
公⑀ s
1 ⫹ 2.
共13兲
Everywhere else in (8), 1 ⫺ 2 → 1 and → s/公⑀, yielding the approximate equations: ˜ ⫹ 公⑀h ⫹ h ⫽ 0 ⫺sV
公⑀ s
˜ ⫺V ˜ Ⲑs ⫹ V
冉
⫺2
公⑀ s
冊
1 ⫹ 2 h ⫽ 0.
共14兲
The exact solution of this simplified pair of equations is h ⫽ exp关⫺共1Ⲑ2兲公⑀ ⫹ s22兴, ˜ ⫽ V
冉公 冑 冊
1 ⫽ ⫺
⑀
s
⫺
1 1 ⫹ 2s 2公⑀
1⫹
⑀
s2
冑
1⫹
共15兲
exp关⫺共1Ⲑ2兲公⑀ ⫹ s22兴, 共16兲
⑀ s2
.
共17兲
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There is one further refinement that is helpful for a ˜, small zonal wavenumber s. All of the unknowns U, V and have expansions in spherical harmonics, and all spherical harmonics of a given wavenumber have a common factor of (1 ⫺ 2)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 ⑀, the Gaussian factors of in (15) and (16) do not enforce these zeros. It is therefore desirable to make the following replacement: exp关⫺共1Ⲑ2兲公⑀ ⫹ s22兴 ⬇ 共1 ⫺ 2兲sⲐ2 exp关⫺共1Ⲑ2兲共公⑀ ⫹ s2 ⫺ s 兲2兴. 共18兲 Because (1 ⫺ 2)s/2 ⫽ exp[(s/2) log(1 ⫺ 2)] ⬇ exp[⫺(s/ 2)2] for small , these two expressions in (18) are indistinguishable when either s or ⑀ or both are large, but separation of the s/2 order zeros at the poles yields a better approximation when s and ⑀ are small.
FIG. 4. The exact Kelvin mode for s ⫽ 5 and ⑀ ⫽ 5 (thin curve with ⫻ symbols). The thick curve is the improved new asymptotic approximation, ⬇ (1 ⫺ 2)s/2 exp关⫺(1⁄2)(公⑀ ⫹ s2 ⫺ s)2], which is graphically indistinguishable from the Kelvin wave. The dashed curve is the new asymptotic approximation without the (1 ⫺ 2) factor, ⬇ exp关⫺(1⁄2)公⑀ ⫹ s22]. The dotted curve is the classical equatorial beta-plane approximation, ⬇ exp关⫺(1⁄2) 公⑀2].
b. Results Converting back to the original variables gives the final asymptotic approximation, uniformly accurate when either s or ⑀ or both are large:
⫽ 共1 ⫺ 2兲sⲐ2 exp关⫺共1Ⲑ2兲共公⑀ ⫹ s2 ⫺ s 兲2兴,
˜ ⫽ V
s
冉公 冑 冊 ⑀
s
⫺
1⫹
⑀
s2
共19兲
共1 ⫺ 2兲sⲐ2
exp关⫺共1Ⲑ2兲共公⑀ ⫹ s ⫺ s 兲 兴, 2
U⫽
⫽ sⲐ公⑀ ⫺
2
s ˜, ⫹ V
1 1 ⫹ 2s 2公⑀
冑
1⫹
3. Results and numerical plots of errors 共20兲 共21兲
⑀ s2
.
cause 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.
共22兲
For fixed s and large ⑀, this frequency approximation simplifies to the formula derived nearly 40 years ago by Longuet-Higgins (1968), ⫽ s/公⑀{1 ⫹ (1⁄4公⑀)}. 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 be-
The maximum relative errors in the new approximation for the frequency for all ⑀ ∈ [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 ⑀ is in fact rather accurate even for small s and ⑀. 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 (⑀ ⫽ 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 ⫺ 2)s/2, in contrast, is visually indistinguishable from the exact height field. Figure 5 shows the error in frequency (upper left) and of (u, , ) (in the L⬁ norm, that is, the maximum error for any latitude) for the new asymptotic approxi-
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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 /(s/公⑀), which is close to 1 for all s and ⑀. 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 were scaled by dividing the absolute errors by the global maximum of each velocity, and plotting these scaled variables.
mation in the 公⑀ ⫺ s plane. The frequency error is not one signed, but rather has an accidental zero along a ray s ⬇ 3公⑀. 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 公⑀ because of the built-in factor of (1 ⫺ 2)s/2: u and are approximately equal to this factor for small ⑀. 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 ⑀ → ⬁) and the small ⑀, “velocity potential is
P nn()” exp(is) approximation. The new approximation was derived under the assumption that at least one of (s, ⑀) 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 ⑀ alone, but rather by the parameter
⑀eff ⫽ s2 ⫹ ⑀.
共23兲
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 ⑀ ⫽ 0, a barotropic wave, as illustrated in Fig. 3. Acknowledgments. This work was supported by the National Science Foundation through Grant OCE
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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. Boyd, J. P., 1985: Barotropic equatorial waves: The non-uniformity of the equatorial beta-plane. J. Atmos. Sci., 42, 1965– 1967. ——, 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
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the first species” and on the dynamics of ocean currents. Philos. Trans. Roy. Soc., 189A, 201–257. ——, 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. Longuet-Higgins, M. S., 1968: The eigenfunctions of Laplace’s tidal equation over a sphere. Philos. Trans. Roy. Soc. London, 262A, 511–607. Majda, A., 2003: Introduction to PDEs and Waves for the Atmosphere and Ocean. Vol. 9, Courant Lecture Notes, American Mathematical Society, 234 pp. 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.]
