a. Observations

Schott et al. (1994) and Reppin et al. (1999) reported biweekly current oscillations in the meridional velocity (υ) field from a moored ADCP array located on the equator at 80.5°E from December 1990 through March 1992. An equatorial wave fit to the observations showed that much of the variance was explained by antisymmetric Yanai waves. (The equatorial u field was weak but not zero, indicating that some Kelvin waves were also present.)

Murty et al. (2002) and Sengupta et al. (2004) reported biweekly oscillations in υ on the equator at 83° and 93°E from current meters deployed by the National Institute of Oceanography (NIO) beginning in December 2000. From a comparison of records at the two locations, Sengupta et al. (2004) noted the existence of wave groups with westward phase propagation, estimating their horizontal wavelength to be 2000–6000 km. Spectra in the biweekly band decreased markedly with depth (by a factor of 3 between the current meters at depths of 106 and 26 m at 93°E; see their Fig. 2), with power in the 20–30-day and 30–60-day bands becoming ever more apparent. This surface trapping suggests that the biweekly variability is surface driven. Consistent with this idea, upward phase (downward energy) propagation between υ records at different depths sometimes occurred.

Most recently, Masumoto et al. (2005) reported a prominent biweekly spectral peak in equatorial υ from a moored ADCP at 90°E. The ADCP measurements provide a detailed picture of the vertical structure of the variability in the upper 400 m. Consistent with the NIO moorings, they show that the biweekly signal is surface trapped within the upper 100 m, with lower-frequency (20–30 day) variability dominating at depths greater than 125 m.

It is noteworthy that a similar biweekly signal is present in the western Pacific Ocean. Based on Tropical Ocean and Global Atmosphere Coupled Ocean–Atmosphere Response Experiment (TOGA COARE) data, Zhu et al. (1998) concluded that it was a first-baroclinic-mode Yanai wave with westward phase speed and a 4000-km wavelength.

b. Dynamics

There is considerable variability in the wind field at periods near 2 weeks in the western Pacific and eastern Indian Ocean (e.g., Chatterjee and Goswami 2004), suggesting that the biweekly ocean variability could be wind forced. The wind variability is attributable to a distinctive oscillation known as the 10–20-day mode or quasi-biweekly mode (QBM). Chen and Chen (1993) discussed its structure and propagation during the summer monsoon of 1979. It had a double-cell structure, with both cells having either low or high pressure, one cell centered from 15° to 20°N and the other centered on the equator. After their formation, both cells moved coherently westward with a zonal wavelength of about 6000 km and a westward propagation speed of 4–5 m s⁻¹. Based on National Centers for Environmental Prediction–National Center for Atmospheric Research (NCEP–NCAR) reanalysis data from 1992 to 2001, Chatterjee and Goswami (2004) showed that the QBM exists throughout the year, and argued that it is an l = 1 Rossby wave shifted north and south in the summer and winter, respectively, in response to the vorticity distribution of the background circulation. Using an intermediate (4½ layer) atmospheric model, they concluded that the QBM is an instability caused by moisture convergence within the planetary boundary layer, with baroclinic instability and evaporation–wind feedback providing secondary contributions. Sengupta et al. (2004) report a similar QBM in Quick Scatterometer (QuikSCAT) data from 1999 to 2002 (their Fig. 12), except that its wavelength was only 3000–4500 km.

Figure 1 is a plot of the spectra for τ^y and (ω/β)τ^x_y at three locations on the equator. [Spectra of (ω/β)τ^x_y are plotted rather than τ^x itself since that field directly forces Yanai waves; see the discussion of (8) below.] The spectra show significant power in the biweekly (10–18 day) band, but there is also considerable power throughout the intraseasonal (10–50 day) band, a consequence of the presence of other types of intraseasonal oscillations.

Sengupta et al. (2001) used an ocean general circulation model (GCM: see section 2) to investigate the origin of oceanic intraseasonal variability. They obtained three solutions, one forced by daily NCEP wind stresses (control run) and the other two by seasonal (periods of 90 days and longer) and intraseasonal (periods shorter than 90 days) NCEP winds. A time series of zonal transport from 3.5° to 5.6°N from their control run simulated the observed biweekly variability present in the Schott et al. (1994) data very well, suggesting that it was generated deterministically by the wind. Indeed, Yanai waves with periods of 10–18 days were prominent in both the control and intraseasonal runs but absent in the seasonal run, confirming that these signals were wind forced in the model.

To study the structure of the biweekly variability further, Sengupta et al. (2004) obtained additional GCM solutions forced by satellite (QuikSCAT) winds. Consistent with their previous results, time series of equatorial υ from their control run were remarkably similar to the Murty et al. (2002) observations in the biweekly (10–18 day) band. The modeled variability in this band consisted of Yanai waves with a zonal wavelength of 4000–5500 km and westward phase propagation. The authors noted a correlation between υ and equatorial τ^y (their Fig. 9), suggesting that υ was primarily forced by τ^y. On the other hand, spectra of equatorial υ in their solutions intensified and sharpened considerably to the east, whereas those for τ^y did not (Fig. 1). Figure 2 illustrates this point in another way, showing a plot of the standard deviation of the quantities in Fig. 1 along the equator, bandpassed to include only variability from 10 to 18 days. There is an obvious increase in the amplitude of υ from west to east, whereas that for τ^y decreases. Thus, the relationship between υ and τ^y is not a simple one reflecting local dynamics. Noting the similarity in wavelengths between the westward-propagating winds and oceanic response, the authors suggested that resonance may contribute to the strength of the biweekly signal.

c. Present research

In this paper, we investigate the dynamics of the oceanic biweekly variability in detail, seeking answers to questions like the following: Is the biweekly oscillation preferentially driven by τ^x or τ^y? What are the influences of local and remote forcing? What processes account for the sharpness and eastward intensification of the biweekly spectral peak in υ? How does vertical propagation of Yanai wave energy affect the response? We utilize two types of models to address these issues: a GCM and a linear continuously stratified (LCS) model. The GCM has the most complete physics and is able to reproduce Indian Ocean circulations most realistically, particularly the swift near-surface currents for which nonlinearities are significant. The LCS model reproduces the near-equatorial flow field of the GCM very well (see Fig. 3 below), it is computationally efficient, and solutions can be found both analytically (see the appendix) and numerically. As such, it provides an invaluable analysis tool.

We present details of the GCM and LCS model in section 2. In section 3, we consider the local (nonradiating) and remote (Yanai wave) parts of the responses to both τ^x and τ^y winds, finding that the local part provides background variability that is spread roughly uniformly along the equator whereas the Yanai wave part accounts for eastward intensification. In section 4, we show that the LCS model preferentially excites biweekly Yanai waves due to resonance and vertical mixing. In section 5 we discuss the vertical propagation of Yanai wave energy, concluding that eastward intensification is caused by reflections of Yanai waves from the sharp near-surface pycnocline and from the ocean bottom.

a. Local response

Analytic solution (A1) describes the υ field of the complete response to (1) when τ^x and τ^y are given by (6) and without horizontal mixing (ν_h = 0). Each term in the double series involves the integral in (A2), which after integrating by parts can be rewritten

View Expanded

The last term represents a contribution to the local part of the response since it is directly proportional to the zonal structure of the wind. Let L be the width scale of the wind. After another integration by parts, it is clear that the local term dominates in (7) provided that (|k^j_nl|L)⁻¹ ≪ 1. We define each term in (A1) that is dominated by the local part as being in “local balance” and the summation of all such terms as being the “local response.”

At biweekly periods and when there is vertical mixing (A = 8.84 × 10⁻⁴ cm² s⁻³), it follows from (A3) that the response of all the waves of the system except for Yanai waves (i.e., the waves for l > 0) are locally balanced for all values of n; moreover, the Yanai wave response becomes locally balanced for modenumbers n greater than about n_o = 5, for which (|k_n| L)⁻¹ = 0.084 (Table 1). (Figure 5, below, illustrates this property for Yanai waves in the opposite way, showing that they are significantly excited only for n ≤ 5.) Thus, it is possible to separate the local and remote responses by modenumber, the local (remote) response defined to be the contribution from the n > n_o (n ≤ n_o) modes.

To investigate the structure and amplitude of the local response, we obtained a set of test solutions to the LCS model forced by idealized τ^x and τ^y wind patches oscillating at various periods P with the same values of A and N_b(z) used for the control run. [The patch for τ^y is defined as in (14) and (15) below, and that for τ^x is similar except with Y replaced by yY.] At sufficiently low (quasi-steady) frequencies, the local responses to the two wind fields are a cross-equatorial Ekman flow and an equatorial roll, respectively, confined largely to the body-force layer (z < z₂) (Miyama et al. 2003). When P = 14 days and n_o = 5, the vertical structures of the local (n > 5) responses are more complicated, due to the excitation of near-resonant, albeit highly damped, gravity waves (see the discussion of Fig. 6). More important for our purposes, their amplitudes are weaker than those of the remote (n ≤ 5) responses, the ratios of the maximum values of |υ| at 50 m between the local and remote responses being 0.27 and 0.10 for τ^x and τ^y forcing, respectively.

b. Remote response

As shown in section 4, the remote response forced by biweekly winds consists almost entirely of Yanai waves. It follows from (A1) and (A11) that the υ field associated with Yanai waves along the equator is

View Expanded

where ω = σ + iA/c²_n is the sum of real frequency σ and a damping term iA/c²_n due to vertical mixing;

View Expanded

is the wavenumber of the Yanai wave for vertical mode n (labeled k¹_n0 in the appendix), a complex number when there is vertical mixing; τ^x_y(x, 0) is the amplitude of τ^x_y on the equator; and we have used the relation ϕ_n0(0) = π^−1/4. Note that the response is determined by a zonal integral of the forcing, an indication of the nonlocal nature of the underlying dynamics.

According to (A11), Yanai waves are forced both by τ^y₀(x) and (ω/β)τ^x_y0(x). Although the two forcing terms have different forms, they nonetheless drive biweekly variability roughly equally in the Indian Ocean. Let L_y denote the meridional scale of the winds. Then, for the same wind strength the magnitudes of the two forcing terms differ by the scale factor (ω/β)/L_y. With L_y = 500 km and ω = σ = 2π (14 days)⁻¹, (ω/β)/L_y = 0.5. Figures 1 and 2 plot spectra and standard deviations for the two forcings. The plots show that forcing by (ω/β)τ^x_y0 is weaker than τ^y₀ in the western ocean (where τ^x itself is weaker than τ^y), but that the forcings have similar strengths in the eastern ocean (where τ^x is much stronger than τ^y).

c. Contribution to the control runs

To isolate the responses to τ^x and τ^y forcing in the GCM and LCS control runs, we carried out two test experiments with each of the models, similar to the control runs but forced by the meridional (τ^y only) and zonal (τ^x only) components of the QuikSCAT winds. Figure 3 plots time series of υ at a depth of 50 m on the equator at 90°E, showing curves from the control (thick solid line) and the τ^y-only (dashed line) and τ^x-only (thin solid line) runs from the GCM (top panel) and LCS models (bottom panel). Each of the time series has been filtered to include only variability in the biweekly (10–18 day) band. The amplitude of the τ^y response is almost always larger than that forced by τ^x, with τ^x providing a smaller, but significant, contribution. It is noteworthy that the response to τ^y is larger in the eastern ocean even though the amplitudes of the two forcings are roughly equal (right panel in Fig. 1), another indication of nonlocal dynamics. The similarity of corresponding curves between the two models is also striking, demonstrating that the physics of the biweekly variability is fundamentally linear.

Figure 4 plots standard deviations of υ at 50 m along the equator for the preceding solutions, again filtered to include only biweekly variability. As for the time series, curves for the control and τ^y-only runs are similar and the τ^y-only response is everywhere larger than the τ^x-only response. Note that the standard deviations for the τ^x- and τ^y-only runs do not sum to equal that of the control run, a consequence of υ in the two test runs not being completely in phase (Fig. 3). Note also that the standard deviations for all solutions tend to increase eastward across the basin, the exception being a decrease in the central ocean in the LCS solution forced by τ^y (see section 5). The cause of this difference is not clear. A likely possibility is advection of the near-surface density field in the GCM, which in turn can alter the structure of the mixed layer and near-surface currents.

For the LCS control run, it is also possible to separate the local and remote responses, defining the n > 5 (n ≤ 5) response to be the local (remote) part. The bottom panel in Fig. 4 also plots the standard deviation of equatorial υ at 50 m for the local (thin solid curve) and remote (dashed curve) responses. The local response has a value of about 3 cm s⁻¹ all across the basin, considerably weaker than the remote response. (The curves have similar structures at depths of 80 and 40 m, except that the amplitudes of the local responses are 5 and 2 cm s⁻¹, respectively.) Thus, it is the remote part of the response that increases to the east, with the local part providing a roughly uniform, and weaker, background signal. Note that the amplitudes of the remote and total variances are virtually the same, an indication of the phase difference between the local and remote contributions.

a. Analytic spectra

For convenience, we assume the forcing is by τ^y only and that τ^y₀(x) = τ₀ is constant across the width of the basin L, which allows the integral in (8) to be evaluated analytically.

Then, multiplying (8) by its complex conjugate and integrating over the water column yields

View Expanded

where Δk = k_w − k_r, and k_r = σ(c⁻¹_n − β/|ω|²) and k_i = (A/c²_n)(c⁻¹_n + β/|ω|²) are the real and imaginary parts of k, respectively. Equation (10) defines the total, depth-integrated spectrum of equatorial υ for Yanai waves, S(x, σ), as well as the contributions to S from each baroclinic mode, S_n(x, σ). [Assuming that τ^x_y0 is also constant, S for τ^x_y forcing is the same as (10) except that the factor in parentheses is replaced by 2(τ^x_y0)²Z²_n/(|ω|²H_n).]

A related expression is the maximum value of S_n anywhere along the equator:

View Expanded

where x̂ is the smallest positive root of

View Expanded

According to (10) and (11), S_n is largest at the eastern boundary, provided that x̂ > L, and otherwise achieves its maximum in the basin interior. A useful related quantity is (^ˆ_Sn/H_n)^1/2, the maximum standard deviation of the surface υ field along the equator associated with the Yanai wave for mode n [see Eq. (16) below]. The quantity, (^ˆ_Sn/H_n)/P, where P = 2π/σ is the period of the wave in days, is most closely related to the variance-preserving spectra in Fig. 1.

b. Inviscid response

Without vertical mixing, (11) simplifies to

View Expanded

Equation (13) identifies an oceanic selection mechanism in that the response is maximized when the wavenumber of a Yanai wave is equal to that of the forcing (k_n = k_w) and drops off when it does not: for example, when |Δk| = π/L and 2π/L, ^ˆ_Sn is smaller than its maximum by factors of (π/2)⁻² = 0.4 and π⁻² = 0.1, respectively. We refer to this narrowband response as a resonance, recognizing that it differs from two, more familiar, types of resonance for equatorially trapped waves, in which the wind excites waves with zero group velocity (see below) and Kelvin and Rossby waves reflect from ocean boundaries to interfere constructively in the interior ocean (Cane and Moore 1981; Jensen 1993; Han et al. 1999; Han 2005). The sharpness of the resonance depends on the relative size of |k_w| to π/L, the response being narrow (broad) when |k_w| ≫ π/L (|k_w| ≲ π/L). In our forced problem, |k_w| ≈ 2π/L and the response is broad. For example, with L = 2π/|k_w| = 6000 km it follows that ^ˆ_Sn is greater than 40% of its maximum provided that −12 000 km < Λ_n < −4000 km, where Λ_n = 2π/k_n.

The left panel in Fig. 5 plots ^ˆ_Sn from (13) for the n = 1–10 modes when τ₀ = 0.1 dyn cm⁻², an amplitude roughly equal to that of observed τ^y (Fig. 2), and L = 2π/k_w = 6000 km. For each n, ^ˆ_Sn has a peak centered on the frequency where k_n = k_w, and the peaks shift monotonically toward lower frequencies as n increases. Note that only peaks for the n = 1–3 modes lie within the 10–18-day band (shaded region) and that at P = 14 days (thin vertical line) the response is dominated by the n = 2 mode. Almost all of the peaks increase in amplitude with n, achieving strengths considerably larger than for modes 1–3, a result of the factor of σ⁻⁴ in (13). The exception is for the n = 2 mode, which is relatively larger because H₂ is smaller (Table 1), a property traceable to the existence of a sharp near-surface pycnocline where N_b(z) is large. Without mixing, then, the ocean does not favor the biweekly band.

Figure 6 illustrates the resonance criterion graphically, plotting dispersion curves for Yanai waves (blue) and the other equatorially trapped waves (black). The curves are the roots of (A3), together with a line, σ = c_nk, for the equatorial Kelvin wave, and they are plotted using the nondimensional variables, σ′ = σ/(βc_n)^1/2 and k′ = k/α_n0. Horizontal lines (dotted red) indicate values of σ′ for the n = 1–10 vertical modes corresponding to periods P of 7 (top), 14 (middle), and 28 (bottom) days. Vertical lines (green) indicate wavenumbers k′ at the edges of the resonant band, defined by the relation |k′ − k′_w| = 2π/(α_n0L) and using the values for L and k_w stated in the previous paragraph; note that with these choices the right-hand edge of the band is located at k′ = 0 for all the modes, the left-hand edge is at 2k_w, and k′ = k′_w midway between the two edges. According to (13), the Yanai wave response for a particular mode is significant (^ˆ_Sn > 0.1) only when its red line intersects the Yanai dispersion curve between its corresponding green lines. It follows from (A2) that a similar criterion holds for other types of equatorially trapped waves.

When P = 7 days (top), the Yanai wave dispersion curve does not intersect any red line between its green lines, so no Yanai waves are strongly excited at this frequency, consistent with the absence of any appreciable response in Fig. 5. On the other hand, several gravity wave dispersion curves almost intersect red lines between the resonance boundaries; moreover, their dispersion curves are very nearly flat this close to the σ′ axis so that their zonal group velocity is nearly zero, satisfying the usual criterion for resonance. [In the analytic solution, this type of resonance occurs when k^j_nl = k^j′_nl in (A4).] Therefore, we can expect the response of these near-resonant gravity waves to be appreciable as well. In particular, note that the l = 4, n = 6 and l = 6, n = 9 gravity waves are near-resonant; their υ fields are symmetric about the equator (a property that holds for all equatorially trapped waves with even l), so they are favorably excited by τ^y winds. For P = 14 days (middle), the Yanai wave dispersion curve intersects red curves between resonance boundaries only for the n = 2 mode, indicating that this mode will dominate the response, in agreement with Fig. 5. Intersections for the n = 1 and n = 3 modes lie just outside the boundaries, however, so we can expect these waves to contribute as well (see section 5). At this period, the l = 1, n = 8 and l = 2, n = 13 gravity waves are near-resonant. For P = 28 days (bottom), Yanai waves for the n = 6–10 modes satisfy the resonance inequality, the shift toward a higher-order-mode resonant response at lower frequencies mirroring the shift of the spectral peaks in Fig. 5, and all of the gravity wave resonances occur beyond the vertical extent of the plot.

To illustrate these analytic results, we obtained a set of solutions to the LCS numerical model forced by a patch of τ^y wind (6) oscillating at various frequencies. In each case, the wind patch has the “tophat” zonal structure,

View Expanded

and a meridional structure,

View Expanded

where θ is a step function, x_m = 50°E, L_x = 10°, L_y = 20°, and τ₀ = 0.5 dyn cm⁻². With these choices, the patch is confined in the western basin from the western boundary to 60°E, and L_y is large enough for (α_n0L_y)⁻² ≪ 1 so that the approximation that leads to (A10) is valid.

Figure 7 plots equatorial sections of standard deviations of υ when k_w = 2π (6000 km)⁻¹ and the forcing oscillates at periods of 7 (top), 14 (middle), and 28 (bottom) days. When P = 7 days, the response is dominated by the n = 6, l = 4 gravity wave resonance, as predicted in Fig. 6. Note that, since the resonance is composed of gravity waves with near-zero group velocity, the response does not extend much beyond the eastern edge of the wind (60°E). When P = 14 days, the n = 13, l = 2 gravity wave resonance is apparent in the wind region. There is also a strong low-order-mode response that extends well east of the wind, due to the excitation of Yanai waves for modes 1–3. When P = 28 days, there is a weak, very high order, gravity wave response confined to the wind region, but the overall response is dominated by contributions from higher-order (n = 6–10) near-resonant Yanai waves. The Yanai waves interfere to form a pattern with vertical, as well as eastward, energy propagation, a subject considered in section 5.

c. Viscid response

The right panel in Fig. 5 is a plot of ^ˆ_Sn(σ) from (11) and (12) when there is vertical mixing (A = 8.84 × l0⁻⁴ cm² s⁻³). As expected, curves for the higher-order modes are much weaker than their inviscid counterparts, so much so that the ocean's response now strongly favors the excitation of the n = 1–3 modes in the biweekly band. Thus, vertical mixing plays an important role in the oceanic selection mechanism, shifting the overall response toward contributions from the lower-order baroclinic modes that resonate at biweekly frequencies.

Figure 8 is a plot of equatorial sections of υ when there is vertical mixing, and the responses are all considerably weakened in comparison to their counterparts in Fig. 7. In particular, the mixing eliminates all gravity wave resonances because they are associated with highly damped, high-order baroclinic modes. (As noted in section 3a, however, these modes still contribute to the local response.) At P = 7 days, only a weak, surface-trapped, local response remains since no Yanai waves are available at this period. At P = 28 days, the Yanai wave response is weakened considerably since it is composed primarily of higher-order modes (n = 6–10), leaving a surface-trapped, local response. In contrast, at P = 14 days, the n = 1–3 Yanai wave response is hardly weakened by the mixing at all. [An interesting property of the P = 28 day response is its double-peak structure. It results from interference between Yanai waves and “evanescent” waves. The wavenumbers of evanescent waves, also given by (A3), are complex even without damping; that is, when ω = σ; they lie between the traditional gravity wave and Rossby wave bands of the dispersion curves, and are not plotted in Fig. 6. They are significantly excited only because the tophat structure of the idealized wind has sharp edges.]

To see how large A has to be to damp higher-order baroclinic modes, we carried out a suite of test runs using the LCS model with A decreased by factors of 10 and 100 and P = 7, 14, and 28 days. When A → A/10, the Yanai wave beam is well formed in the upper ocean, and gravity wave resonances begin to appear, so that it is necessary that A be of the order of our typical value (A = 8.84 × 10⁻⁴ cm² s⁻³). We have not carried out a similar suite with the GCM and so cannot confirm that vertical mixing also damps its higher-order modes. Nevertheless, given that values of ν in the GCM are considerably less in the deep ocean than they are in the LCS model (section 2), it seems likely that other processes are involved. Possibilities include interaction of waves with near-surface baroclinic currents, which can absorb or distort equatorially trapped waves (Rothstein et al. 1988; Proehl 1990), and the model's low vertical resolution below the thermocline.

a. Analytic results

Resonance would seem to provide a possible explanation for the eastward intensification of the near-surface υ fields in the GCM and LCS control runs (Fig. 4). To illustrate, consider the amplitude of equatorial υ for a Yanai wave associated with mode n when the ocean is forced by a constant τ^y₀ at resonance (Δk = 0) and without mixing (k_i = 0). According to (8) and (A11), it is given by

View Expanded

which increases linearly across the basin. As noted above, eastward intensification is not limited to the case Δk = k_i = 0: it still occurs across the entire basin provided that x̂ > L, but the increase is then no longer linear. It is also not restricted to the case of constant τ^y₀, provided that τ^y₀(x) has one sign everywhere on the equator.

A problem with this explanation, however, is that the contributions from different vertical modes do not interfere constructively everywhere along the equator since their eastward phase speeds differ. Thus, the total υ field need not exhibit eastward intensification, even if the contributions from each of the modes does. In fact, typically the modes sum to form a wave packet that propagates energy downward as well as horizontally, providing a mechanism for weakening the near-surface, local response and eliminating it east of the forcing region. A further complication is that downward-propagating wave packets can reflect from the ocean pycnocline and bottom to return to the surface (McCreary 1984; Rothstein et al. 1985).

Ray theory provides a useful formalism for understanding vertical propagation. Applying the Wentzel–Kramers–Brillouin (WKB) approximation, dispersion relation (9) can be rewritten

View Expanded

where c_n is replaced by N_b/|m| and m(z) is the local vertical wavenumber of a Yanai wave in the packet. It follows that energy propagates along ray paths with the slope

View Expanded

According to (18), energy propagates vertically and eastward at a slope that depends only on σ and N_b (and the sign of m), thereby forming a well-defined beam (McCreary 1984).

The WKB method that leads to (17), however, is only valid if the vertical scale of the waves is short relative to the scale L_z of variations in N_b (i.e., when mL_z ≫ 1), so (18) fails whenever the pycnocline is sufficiently sharp. In that case, wave packets reflect from, as well as propagate through, the pycnocline (Rothstein et al. 1985). The inequality is, in fact, not satisfied for the low-order baroclinic modes of the LCS control run because N_b(z) has both a main pycnocline (−1500 m < z < −200 m) and a sharp near-surface (z > −200 m) one. We can therefore expect pycnocline reflections to be a prominent feature of our solutions.

b. Solutions for idealized forcing and N_b

To isolate effects of vertical propagation and reflections, we carried out a set of test experiments to the LCS model, forced by the same idealized wind used in Fig. 8 but with various simplified structures for N_b(z) of the form

View Expanded

where b = 1300 m, ζ = 1 −|(z/z_o) + 1|, z_o = 100 m, and θ is a step function. The terms proportional to N₁ and N₂ define a sharp, triangular-shaped, near-surface pycnocline (confined above 200 m and peaking at 100 m) and an exponential “main” pycnocline, respectively. With the choices, N₀ = 0.0004 s⁻¹, N₁ = 0.018 s⁻¹, and N₂ = 0.005 s⁻¹, N_b(z) closely resembles the stratification used for the control run.

The top panel in Fig. 9 plots a solution comparable to the one in the middle panel of Fig. 8 except that N_b = N₀ = 0.0022 s⁻¹, a constant value equal to N_b(z) for the control run averaged over the water column. Vertical propagation of Yanai wave energy is apparent along the ray paths (yellow curves) plotted in each panel (determined by integrating the equation dx/dz = θ_e). The ray paths extend from the surface in the western ocean to the ocean bottom, where they reflect to return to the surface and so on; as a result, there are shadow zones where energy cannot penetrate near the surface in the central ocean and at depth in the eastern and western regions. Relative maxima in the deep central and near-surface eastern oceans are caused by the constructive interference of downward- and upward-propagating beams.

The middle panel in Fig. 9 shows the response for N_b(z) with N₀ = 0.0004 s⁻¹, N₁ = 0, and N₂ = 0.005 s⁻¹, a stratification with a main pycnocline. In comparison to the constant-N_b solution in the top panel, ray paths are curved, bending more sharply with depth where N_b(z) decreases. The near-surface relative maximum centered near 80°E is stronger and thinner than its counterpart in the constant-N_b solution, a consequence of N_b being larger there and, hence, the vertical group velocity σ_m being smaller; similarly, the bottom relative maximum in the central ocean is weaker and thicker because N_b is smaller. There is a band of increased |υ| centered around a depth of 1000 m, due to interference between the downward-propagating and bottom-reflected beams. The property that near-surface values just east of the forcing region remain greater than 2 cm s⁻¹ indicates that some energy reflects from the main pycnocline, but the effect is weak because the vertical scale of N_b is large (b = 1300 m).

The bottom panel in Fig. 9 shows the solution for N_b(z) with N₀ = 0.0022 s⁻¹, N₁ = 0.018 s⁻¹, and N₂ = 0, thereby adding a near-surface pycnocline to the constant stratification used for the constant-N_b solution in the top panel. In this case, multiple reflections of Yanai wave energy from the ocean surface and the bottom of the near-surface pycnocline strengthen the near-surface response considerably, generating a surface-trapped band of energy that extends well east of the wind. The band decays eastward primarily due to energy lost by transmission through the base of the pycnocline rather than to damping by vertical mixing, and consequently the downward-propagating, transmitted beam is much broader than it is in the constant-N_b solution. As a result, interference between the transmitted and bottom-reflected beams produces a distinct, mode-3-like, standing wave pattern in the central and eastern oceans. Last, the amplitude of the remote response is obviously larger than in the other two cases, a consequence of the low-order modes coupling more efficiently to the forcing (i.e., the values of H_n for the low-order modes are much smaller than they are for the constant-N_b stratification).

To illustrate the near-surface responses of the three solutions in Fig. 9 more clearly, Fig. 10 plots their values at z = −50 m along the equator. The amplitudes of the constant-N_b (solid line) and main pycnocline (dotted line) solutions adjust to near-constant values within the forcing region and drop off precipitously east of it. Note that the main pycnocline curve does not drop off to near zero, though, due to weak pycnocline reflections. In marked contrast, in the solution with the near-surface pycnocline (dashed line), |υ| increases linearly across the forcing region, and there is a gradual amplitude decline east of it due to energy transmission into the deep ocean.

Downward propagation of Yanai wave energy is also apparent in the P = 14 and 28 day solutions in Figs. 7 and 8, which are obtained using the realistic N_b(z). A striking feature of the P = 14 day response is the band of strong |υ| in the upper (z < −2000 m) central and eastern oceans. It results from the constructive interference of transmitted and bottom-reflected beams that are broadened by multiple reflections from the near-surface pycnocline, essentially a combination of the solutions in the middle and bottom panels in Fig. 9.

c. Solutions for QuikSCAT forcing

Figure 11 shows x–z sections of standard deviations of biweekly υ along the equator from the GCM (top panel) and LCS (bottom panel) control runs. In both solutions, there is a strong surface-trapped signal everywhere along the equator due to local forcing and to Yanai wave reflections from the near-surface pycnocline. Downward and eastward energy propagation along ray paths is not visually obvious because forcing is present everywhere along the equator. Nevertheless, its existence is supported by the shadow zone in the deep western region, above which regions of intensified standard deviation extend downward roughly along ray paths. Interestingly, bottom topography near 75°E in the GCM (absent in the LCS model) blocks eastward wave propagation, generating another shadow zone to the east of it. Note that the upper-ocean (z > −1500 m) response strengthens in the eastern ocean in both solutions because of bottom-reflected beams, but the increase in the GCM solution is considerably less, apparently a result of the topographical blocking (see below).

To illustrate the relative contributions to the near-surface υ field from forcings in various regions, Fig. 12 shows standard deviations of biweekly υ at 50 m along the equator from solutions to the GCM (top panel) and LCS models (bottom panel), plotting curves from the control runs (thick solid line) and test solutions forced by QuikSCAT winds confined to the western (x ≤ 60°E: dotted line), central (60°E < x ≤ 80°E: thin solid line), and eastern (x > 80°E: dashed line) oceans. In all the tests, there is eastward intensification in the forcing region and a gradual dropoff of energy east of it as in the dashed curve in Fig. 10, confirming that resonance, reflections from the near-surface pycnocline, and radiative loss to the deep ocean are prominent in both control runs. The influence of bottom reflections is also apparent in that the western-forced contributions (dotted curves) increase in the eastern ocean as in the idealized solutions; however, the increase in the GCM is much less (2.7 versus 6.5 cm s⁻¹), supporting the idea that blocking by bottom topography accounts for the weaker GCM response in Fig. 11.

Since individual contributions to the total variance are so similar between the two models, it is noteworthy that remote forcing increases the total variance in the eastern ocean for the LCS model but not for the GCM (cf. differences between the thick solid and dashed curves in each panel in Fig. 12). The total variance is, of course, not the sum of the individual contribution since there are phase differences among them. In the GCM, the phases are such that the contributions coincidentally interfere so that the eastern-forced and total variances are the same.