Evidence of equatorial Rossby wave propagation obtained by deep mooring observations in the western Pacific Ocean

Abstract

We investigated long-term zonal flow (U) oscillations in the deep layer in the western Pacific Ocean by comparing measurements obtained from deep mooring observations near the equator with a high-resolution model. For the comparison, we used mooring data at 2,000 m and below north and south of the equator obtained from preceding studies. In our model results, vertically propagating equatorial annual Rossby waves were detected in the velocity and pressure anomaly (P _a) fields. P _a had a multi-cell meridional structure symmetric about the equator. The corresponding meridional distribution of U, averaged along the wave propagation, was the same as that previously obtained from Argo float data, suggesting the existence of the multi-cell structure in P _a in the real ocean. A harmonic analysis of the simulated U indicated that the equatorial annual Rossby waves excited over the central and eastern equatorial Pacific propagated down to below 2,000 m in the western equatorial Pacific. Decomposition of the annual U harmonics indicated that the multi-cell structure in pressure was not a manifestation of a single meridional mode but a composite of several lowest meridional modes, suggesting that the cell widths may change according to relative mode intensities. The mooring data north and south of the equator was consistent with the simulated multi-cell structure in P _a of the equatorial Rossby waves, although some discrepancies remained. We consider that the observed variation in U is a manifestation of the Rossby wave propagation in the real ocean. The present study is the first to recognize the existence of equatorial annual Rossby waves below 2,000 m in the western Pacific Ocean from direct current observations.

Appendices

Appendix 1: Vertically propagating equatorial Rossby waves

In the appendices, the variables U, V, P are used as the meridional structure functions of the zonal velocity u, meridional velocity v, and pressure p, respectively.

Let us assume that wave solutions of a quantity q (any of the variables u, v, p, w, or ρ) of the linearized equations on the equatorial β-plane have the form

$$q(x,y,z,t) = Q(y)Z(z){\text{e}}^{i(kx - \sigma t)} ,$$

(1)

where Q and Z are the meridional and vertical structure functions of q, k the zonal wavenumber, and σ the frequency. For forced waves with σ and k given, the vertical structure of the vertical velocity w is expressed as

$$\frac{{{\text{d}}^{2} W}}{{{\text{d}}z^{2} }} + \frac{{N_{b}^{2} }}{gh}W = 0,$$

(2)

where W(z) is the vertical structure function of w; N _b the Brunt–Väisälä frequency; g the gravity acceleration; and h the equivalent depth, which is a function of σ and k (Philander 1978). For vertically propagating waves, N ² _b /gh must be positive (h is positive). Taking N ² _b /gh = m ², m may be the local vertical wavenumber under the WKB approximation in the vertical direction to address the nonhomogeneous Brunt–Väisälä frequency N _b (z).

For linearized equations of motion with the continuity equation, an equation of V is obtained under the equatorial β-plane approximation as

$$\frac{{{\text{d}}^{2} V}}{{{\text{d}}y^{2} }} + \left( {\frac{{\sigma^{2} }}{gh} - k^{2} - \frac{\beta k}{\sigma } - \frac{{\beta^{2} y^{2} }}{gh}} \right)V = 0,$$

(3)

where β is the meridionally changing rate of the Coriolis parameter at the equator. This equation has solutions that are bounded at large values of |y| provided that

$$\frac{{\sigma^{2} }}{gh} - k^{2} - \frac{\beta k}{\sigma } = \frac{\beta }{{\sqrt {gh} }}\left( {2l + \, 1} \right)\quad l = \, 0, \, 1, \, 2, \ldots$$

(4)

so that the eigenvalue (gh)^1/2 for each l value is

$$\sqrt {gh_{l} } = \frac{{ - \beta \left( {2l + 1} \right) \pm \sqrt { \beta^{2} (2l + 1)^{2} + 4 \left( {\frac{\beta k}{\sigma } + k^{2} } \right)\sigma^{2} } }}{{2(k^{2} + \frac{\beta k}{\sigma })}} .$$

(5)

Here, l = 0 with a positive sign corresponds to the Rossby-gravity modes (l = 0 with a negative sign leads to a physically unrealistic solution). A positive sign for l > 0 corresponds to the inertia-gravity modes, and the negative sign for l > 0 to the Rossby waves. Another solution with V ≡ 0 is obtained, which corresponds to the Kelvin wave. Here, we are concerned with the Rossby modes. The eigenfunctions for V are

$$V_{l} \left( y \right) \, = \phi_{l} \left( {\xi_{l} } \right),$$

(6)

where $\phi_{l} \left( {\xi_{l} } \right)$ is the lth order Hermite function

$$\phi_{l} \left( {\xi_{l} } \right) \, = \pi^{ - 1/4} \left( {2^{l} l!} \right)^{ - 1/2} H_{l} \left( {\xi_{l} } \right) \, \exp \left( {{-}\xi_{l}^{2} /2} \right)$$

(7)

and ξ _l  = y/λ _l , λ _l  = (c _l /β)^1/2, c _l  = (gh _l )^1/2, and H _l (ξ _l ) is the lth order Hermite polynomial. Corresponding eigenfunctions for U(y) and P(y) are as follows:

$$U_{l} \left( y \right) \, = iU_{l}^{\prime } \left( y \right) \, = i\frac{{\sqrt {\beta c_{l} } }}{\sigma }\left[ {\frac{{\sqrt {(l + 1)/2} }}{{1 - kc_{l} /\sigma }}\phi_{l + 1} (\xi_{l} ) + \frac{{\sqrt {l/2} }}{{1 + kc_{l} /\sigma }}\phi_{l - 1} (\xi_{l} )} \right]$$

(8)

$$P_{l} \left( y \right) \, = iP_{l}^{\prime } \left( y \right) \, = i\frac{{c_{l} }}{g}\frac{{\sqrt {\beta c_{l} } }}{\sigma }\left[ {\frac{{\sqrt {(l + 1)/2} }}{{1 - kc_{l} /\sigma }}\phi_{l + 1} \left( {\xi_{l} } \right) - \frac{{\sqrt {l/2} }}{{1 + kc_{l} /\sigma }}\phi_{l - 1} \left( {\xi_{l} } \right)} \right],$$

(9)

where “i” is the imaginary unit. The meridional structure functions for several lowest modes of U _l ′ and P _l ′ are presented in Fig. 16. U _l ′ is maximum at the equator in each symmetric mode and P _l ′ is maximum in its own outermost zones. The set of eigenfunctions U _l ′ are not orthogonal, but that of eigenfunctions V _l are orthogonal with a weighting factor σ ²–β ² y ²:

$$\int_{ - \infty }^{\infty } {\left( {\sigma^{2} {-}\beta^{2} y^{2} } \right)V_{m} \left( {\xi_{m} } \right)V_{n} \left( {\xi_{n} } \right)dy = \lambda_{n} \left[ {\sigma^{2} {-} \, \left( {n + 1/2} \right)c_{n} \beta } \right]\delta_{mn} ,}$$

(10)

where δ _mn is the Kronecker δ (i.e., δ _mn  = 1, if m = n and δ _mn  = 0 if m ≠ n). The set of eigenfunctions P _l ′ are orthogonal (Wunsch 1977):

$$\int_{ - \infty }^{\infty } {P_{m}^{\prime } \, \left( {\xi_{m} } \right)P_{m}^{\prime } \left( {\xi_{n} } \right)dy = \frac{{\beta c_{n}^{3} }}{{2g^{2} \sigma^{2} }}\lambda_{n} \left[ {\frac{n + 1}{{(1 - kc_{n} /\sigma )^{2} }} + \frac{n}{{(1 + kc_{n} /\sigma )^{2} }}} \right]\delta_{mn} .}$$

(11)

Fig. 16

Meridional structure functions of a symmetric modes of zonal velocity, b antisymmetric modes of zonal velocity, c symmetric modes of pressure, and d antisymmetric modes of pressure. For symmetric modes a, c the l = 1 mode is shown in black, 3 in red, 5 in green, and 7 in blue. For antisymmetric modes b, d the l = 2 is in black, 4 in red, and 6 in green. The vertical scales are arbitrary

Appendix 2: Decomposition of annual harmonics of zonal flow into meridional modes of the equatorial Rossby wave

The 5-year-averaged monthly mean data of zonal flow u, meridional flow v, and pressure p, represented by q = q (x, y, z, t), are expanded by the temporal Fourier series as follows:

$$q\left( {x,y,z,t} \right) = A_{q0} \left( {x,y,z} \right)/2 + \sum\nolimits_{n = 1}^{6} {\left[ {A_{qn} \left( {x,y,z} \right) \, \sin \left(\frac{2\pi n}{T}t\right) \, + B_{qn} \left( {x,y,z} \right) \, \cos \left(\frac{2\pi n}{T}t\right)} \right],}$$

(12)

$${\text {where}}\, A_{q0} \left( {x,y,z} \right) = \frac{2}{T}\sum\nolimits_{m = 1}^{12} {q(x,y,z, \, (m{-}\frac{1}{2})\Delta t)\Delta t,} \begin{gathered} A_{qn} \left( {x,y,z} \right) = \frac{2}{T}\sum\nolimits_{m = 1}^{12} {q\left( {x,y,z, \, \left( {m{-}\frac{1}{2}} \right)\Delta t} \right) \, \sin \left( {\frac{2\pi n}{T}\left( {m{-}\frac{1}{2}} \right)\Delta t} \right)} \Delta t,{\text{ and}} \hfill \\ B_{qn} \left( {x,y,z} \right) = \frac{2}{T}\sum\nolimits_{m = 1}^{12} {q\left( {x,y,z, \, \left( {m{-}\frac{1}{2}} \right)\Delta t} \right) \, \cos \left( {\frac{2\pi n}{T}\left( {m{-}\frac{1}{2}} \right)\Delta t} \right)} \Delta t, \hfill \\ \end{gathered}$$

and $\Delta t$ and T mean 1 month and 1 year, respectively. In the above calculations, q(x, y, z, t) is regarded as zero on land. We are concerned with the annual component, so we write A _q  = A _q1 and B _q  = B _q1. On the other hand, the wave solutions of the linearized equations in the equatorial region with low frequencies and low wavenumbers are assumed to be expressed by superposition of the equatorial Kelvin waves and a series of meridional Rossby modes (Appendix 1) (McCreary 1984):

$$q\left( {x,y,z,t} \right) = \sum\nolimits_{l = 1}^{\infty } {q_{l} \left( {x,y,z,t} \right)} = \sum\nolimits_{l = 1}^{\infty } {C_{ql} \left( {x,z,t} \right)Q_{l} \left( {\xi_{l} } \right),}$$

(13)

where C _ql (x, z, t) and Q _l (ξ _l ) are the coefficient of the lth meridional mode of q and its eigenfunction, respectively, representing U _l ′, V _l , and P _l ′ (Appendix 1). Here, the Kelvin mode is conveniently expressed by suffix “0” (i.e., V ₀ ≡ 0), but hereafter we omit this term. The Kelvin mode tends to be absorbed by the critical layer absorption (Kessler and McCreary 1993) and it might be limited to shallow layers, if at all it exists. Assuming that the wave solution q(x, y, z, t) in Eq. (13) is equivalent to that in Eq. (12), q(x, y, z, t) in Eq. (13) is substituted into the third and the fourth of Eq. (12), taking into account n = 1,

$$\begin{gathered} A_{q} \left( {x,y,z} \right) \, = \sum\nolimits_{l = 1}^{\infty } {A_{ql} \left( {x,z} \right)Q_{l} \left( {\xi_{l} } \right){\text{ and}}} \hfill \\ B_{q} \left( {x,y,z} \right) \, = \sum\nolimits_{l = 1}^{\infty } {B_{ql} \left( {x,z} \right)Q_{l} \left( {\xi_{l} } \right)} , \hfill \\ \end{gathered}$$

(14)

$$\begin{aligned}\text{where}\;A_{ql} \left( {x,z} \right) &= \frac{2}{T}{\sum\limits_{m = 1}^{12}} {q_{l}} \left(x,z, \left(m-\frac{1}{2}\right)\Delta{t}\right)\sin \left(\frac{2\pi}{T}\left(m-\frac{1}{2}\right)\Delta {t}\right)\Delta {t} \;{\text{and}}\\B_{ql} \left( {x,z} \right) &= \frac{2}{T}{\sum\limits_{m = 1}^{12}} {q_{l}} \left(x,z,\left(m-\frac{1}{2}\right)\Delta{t}\right)\cos \left(\frac{2\pi}{T} \left(m-\frac{1}{2}\right) \Delta{t}\right)\Delta{t}.\end{aligned}$$

In an idealized state, A _ul  ≡ A _pl  ≡ B _vl and B _ul  ≡ B _pl  ≡ −A _vl .

To obtain the coefficients A _ql and B _ql from Eq. (14), the orthogonality relationships in Eqs. (10) and (11) can be used for v and p, respectively. For u, however, its eigenfunctions U _l ′s are not orthogonal. However, a least squares method (LSM) described later, can be used to determine the coefficients A _ul and B _ul . (The two methods, i.e., the method using the orthogonality relationship and that using the LSM, produce the same values for v and p). We attempted the two methods and found that the coefficients A _ul and B _ul for u obtained by the LSM were the most consistent for both the recomposed u and p fields. The v field contains considerable small-scale noise, which was reflected in A _vl and B _vl . For the p field, the standing oscillation modes with the same frequency seemed to contaminate the calculated results for A _pl and B _pl for the propagating wave modes.

In this study, the LSM method assumes that the dominant zonal wavelength L is 11,500 km (103° in longitude) (see below for an explanation of how it was determined) and that the highest order of the meridional mode is 7. The coefficients A _ul and B _ul for u were obtained by the LSM. The matrix associated with the normal equation based on the eigenfunctions U _l ′s in the method was nonsingular and we obtained stable solutions. The meridional integration of eigenfunctions was made between 25°S and 25°N, and those with l ≤ 7 are nearly zero at both ends (Fig. 16). Values of h, c, and λ for each mode l with L = 11,500 km are given in Table 4. This table also shows the local vertical wavelength D = 2π/m, where m = (N ² _b /gh)^1/2 is the local vertical wavenumber (Appendix 1) at 1,000, 2,000, 3,000, and 4,000 m.

The zonal wavelength L = 11,500 km was determined by a best fitting of composites of A _ul and B _ul for l = 1–7 with A _u and B _u in Eq. (12). Taking

$$A_{uc} \left( {x,y,z} \right) = \sum\nolimits_{l = 1}^{7} {A_{ul} \left( {x,z} \right)U_{l}^{\prime } \left( {\xi_{l} } \right)} {\text{ and }}B_{uc} \left( {x,y,z} \right) \, = \sum\nolimits_{l = 1}^{7} {B_{ul} \left( {x,z} \right)U_{l}^{\prime } \left( {\xi_{l} } \right),}$$

(15)

the minimum value of

$$S = \int {\left[ {\left( {A_{u} {-}A_{uc} } \right)^{2} + \, \left( {B_{u} {-}B_{uc} } \right)^{2} } \right]{\text{d}}V}$$

(16)

was sought with L varying, where V is the volume enclosed by 140°E–160°W longitude, 25°S–25°N latitude, and 855 m thickness (570–1,425 m in depth centered at 1,000 m). The value 11,500 km is consistent with wavelengths obtained by the horizontal phase distribution of the annual harmonics of u (A _u and B _u ) at levels 500–1,000 m (not shown here).

Getting A _ul and B _ul , amplitude and phase for each meridional mode of u shown in Fig. 9 are obtained from

$$\begin{gathered} {\text{Amp}}_{ul} \left( {x, \, z} \right) = \left( {A_{ul}^{2} + B_{ul}^{2} } \right)^{1/2} {\text{and}} \hfill \\ {\text{Phs}}_{ul} \left( {x, \, z} \right) = \tan^{ - 1} \left( {A_{ul} /B_{ul} } \right). \hfill \\ \end{gathered}$$

(17)

Then, assuming A _pl  ≡  A _ul and B _pl  ≡  B _ul , temporal variation of each pressure mode p _l (x, y, z, t) and the composite of the pressure modes p _c (x, y, z, t) are expressed as:

$$p_{l} \left( {x,y,z,t} \right) \, = \, [A_{pl} \left( {x,z} \right) \, \sin \left( {\frac{2\pi }{T}t} \right) + B_{pl} \left( {x,z} \right)\cos \left( {\frac{2\pi }{T}t} \right)]P_{l}^{\prime } \left( {\xi_{l} } \right){\text{ and}}$$

(18)

$$\begin{gathered} p_{c} \left( {x,y,z,t} \right) = \sum\nolimits_{l = 1}^{7} {p_{l} \left( {x,y,z,t} \right)} \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = A_{pc} \left( {x,y,z} \right) \, \sin \left( {\frac{2\pi }{T}t} \right) + B_{pc} \left( {x,y,z} \right)\cos \left( {\frac{2\pi }{T}t} \right), \hfill \\ \end{gathered}$$

(19)

where A _pc (x, y, z) = $\mathop \sum \limits_{l = 1}^{7}$ A _pl (x, z) P _l ′(ξ _l ) and B _pc (x, y, z) = $\mathop \sum \limits_{l = 1}^{7}$ B _pl (x, z) P _l ′(ξ _l ). They are shown in Figs. 10, 11, and 12.

Appendix 3: WKB ray path theory

For small wavenumbers and small frequencies, σ ² and k ² can be neglected in the dispersion relation Eq. (4) and it is simplified to

$$\sigma = - \frac{{k (gh_{l} )^{1/2} }}{2l + 1}.$$

Furthermore, taking N _b (z)/(gh _l )^1/2 = m _l (Appendix 1), where m _l is the local vertical wavenumber of the meridional mode l under the WKB approximation,

$$\sigma = - \frac{{k N_{b} (z)}}{{\left| {m_{l} } \right|(2l + 1)}}.$$

(20)

Following Kessler and McCreary (1993), the slope of ray paths in the (x, z) plane is defined as

$$\frac{{{\text{d}}z}}{{{\text{d}}x}} = \frac{{{\text{d}}z/{\text{d}}t}}{{{\text{d}}x/{\text{d}}t}} = \frac{{\partial \sigma /\partial m_{l} }}{\partial \sigma /\partial k} = \frac{(2l + 1)\sigma }{{N_{b} (z)}}\frac{{m_{l} }}{{|m_{l} |}}$$

(21)

Energy originating at the surface propagates downward to the west, while its phase propagates upward to the west across the band of energy. The slope is steeper for weaker stratification or higher meridional mode number l. On the other hand, the slope of lines of constant phase is −k/m _l , so that its value is the same as the right-hand side of Eq. (21). Thus, phase lines are parallel to WKB ray paths.