## Abstract

Variability of the wind field over the equatorial Indian Ocean is spread throughout the intraseasonal (10–60 day) band. In contrast, variability of the near-surface *υ* field in the eastern, equatorial ocean is concentrated at biweekly frequencies and is largely composed of Yanai waves. The excitation of this biweekly variability is investigated using an oceanic GCM and both analytic and numerical versions of a linear, continuously stratified (LCS) model in which solutions are represented as expansions in baroclinic modes. Solutions are forced by Quick Scatterometer (QuikSCAT) winds (the model control runs) and by idealized winds having the form of a propagating wave with frequency *σ* and wavenumber *k _{w}*. The GCM and LCS control runs are remarkably similar in the biweekly band, indicating that the dynamics of biweekly variability are fundamentally linear and wind driven. The biweekly response is composed of local (nonradiating) and remote (Yanai wave) parts, with the former spread roughly uniformly along the equator and the latter strengthening to the east. Test runs to the numerical models separately forced by the

*τ*and

^{x}*τ*components of the QuikSCAT winds demonstrate that both forcings contribute to the biweekly signal, the response forced by

^{y}*τ*being somewhat stronger. Without mixing, the analytic spectrum for Yanai waves forced by idealized winds has a narrowband (resonant) response for each baroclinic mode: Spectral peaks occur whenever the wavenumber of the Yanai wave for mode

^{y}*n*is sufficiently close to

*k*and they shift from biweekly to lower frequencies with increasing modenumber

_{w}*n*. With mixing, the higher-order modes are damped so that the largest ocean response is restricted to Yanai waves in the biweekly band. Thus, in the LCS model, resonance and mixing act together to account for the ocean's favoring the biweekly band. Because of the GCM's complexity, it cannot be confirmed that vertical mixing also damps its higher-order modes; other possible processes are nonlinear interactions with near-surface currents, and the model's low vertical resolution below the thermocline. Test runs to the LCS model show that Yanai waves from several modes superpose to form a beam (wave packet) that carries energy downward as well as eastward. Reflections of such beams from the near-surface pycnocline and bottom act to maintain near-surface energy levels, accounting for the eastward intensification of the near-surface, equatorial

*υ*field in the control runs.

## 1. Introduction

Oceanic intraseasonal oscillations have been reported at various frequencies along the equator in the Indian Ocean [see Schott and McCreary (2001) for a recent review]. Much of the discussion of this variability has focused on signals at periods of 20–60 days, which appear to be generated either by oceanic instabilities (20–30 days: Kindle and Thompson 1989; Woodberry et al. 1989; J. Vialard 2004, personal communication) or are wind-forced by Madden–Julian oscillations (30–60 days). More recently, prominent variability in the biweekly (10–18 day) band has also been noted.

### 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*at three locations on the equator. [Spectra of (

^{x}_{y}*ω*/

*β*)

*τ*are plotted rather than

^{x}_{y}*τ*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.

^{x}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*. On the other hand, spectra of equatorial

^{y}*υ*in their solutions intensified and sharpened considerably to the east, whereas those for

*τ*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

^{y}*υ*from west to east, whereas that for

*τ*decreases. Thus, the relationship between

^{y}*υ*and

*τ*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.

^{y}### 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*? What are the influences of local and remote forcing? What processes account for the sharpness and eastward intensification of the biweekly spectral peak in

^{y}*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*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.

^{y}## 2. The ocean models

The GCM is the Modular Ocean Model, version 2.2 (Pacanowski 1996). The model domain is the Indian Ocean basin north of 30°S and from 30° to 110°E. Its horizontal grid is variable with a resolution of about 0.33° north of 5°S. There are 19 vertical levels, 6 in the upper 100 m and 4 below 1000 m. The coefficients of horizontal eddy diffusivity and viscosity are 2000 m^{2} s^{−1}, and vertical mixing is based on the scheme of Pacanowski and Philander (1981). Surface temperature and salinity are relaxed to the observed seasonal cycle. There is a sponge layer at 30°S where temperature and salinity are relaxed to their Levitus (1982) climatologies, and the eastern boundary is closed everywhere so that there is no Indonesian Throughflow. Bottom topography is a somewhat smoothed version of the 5′ Gridded Earth Topography (ETOPO5) data (National Oceanographic and Atmospheric Administration 1988).

The GCM control run is forced by satellite-derived surface winds from the QuikSCAT (Liu 2002) for the period from July 1999 to December 2003, with wind stress determined from the winds using a constant drag coefficient of 0.0013. Initial conditions are from a 15-yr run of the model forced by NCEP–NCAR reanalysis daily surface winds.

Equations of motion for the LCS model are

(e.g., McCreary 1984; McCreary et al. 1996). They are a linearization of the primitive equations about a background state of rest with Brunt–Väisälä frequency *N _{b}*(

*z*), which, unless specified otherwise, is taken to be the annual-mean density field calculated from the World Ocean Database 2001 (Conkright et al. 2002) averaged from 1°S to 1°N and from 40° to 100°E. Vertical mixing is included with a coefficient of the form

*ν*=

*A*/

*N*

^{2}

_{b}, and unless specified otherwise,

*A*= 8.84 × 10

^{−4}cm

^{2}s

^{−3}. With this choice,

*ν*has realistic values in the upper ocean, varying from a maximum of 99.5 cm

^{2}s

^{−1}in the upper 100 m to a minimum of 2.34 cm

^{2}s

^{−1}in the thermocline (where

*N*

^{2}

_{b}is a maximum); however, it increases at greater depths to unrealistically large values, for example, having a value of 363.7 cm

^{2}s

^{−1}at 2000 m. The horizontal mixing coefficient

*ν*is 6 × 10

_{h}^{6}cm

^{2}s

^{−1}for the numerical version of the model and is zero for the analytic version.

Wind is introduced into the system as a body force with the vertical structure

where *z*_{1} = −10 m and *z*_{2} = *z*_{1} − 10 m. With these choices, *Z*(*z*) simulates a constant-thickness, mixed layer in which turbulence is spread uniformly throughout the upper 10 m and weakens linearly from 10 to 20 m. The body force is also consistent with the way that wind stress is introduced into the GCM, namely, as a surface stress condition on a vertical grid with a resolution of 10 m. Note that ∫^{0}_{−D }*Z*(*z*) *dz* = 1, the only constraint on the structure of *Z*(*z*).

The ocean bottom is assumed flat at a depth of *D* = 4500 m. With this restriction and the particular form of *ν* noted above, solutions can be represented as expansions in the vertical normal modes of the system, *ψ _{n}*(

*z*). The modes are the eigenfunctions of the equation

subject to the boundary conditions *ψ _{nz}*(0) =

*ψ*(−

_{nz}*D*) = 0 and normalized so that

*ψ*(0) = 1. Let

_{n}*q*be

*u*,

*υ*, or

*p*. Then,

*q*can be written as

where the expansion coefficients satisfy

*c _{n}* is the speed of an equatorial Kelvin wave, H

_{n}= ∫

^{0}

_{−D }

*ψ*

^{2}

_{n}(

*z*)

*dz*is the coupling coefficient of mode

*n*to the wind, and

*Z*= ∫

_{n}^{0}

_{−D }

*Z*(

*z*)

*ψ*(

_{n}*z*)

*dz*. The

*w*and

*ρ*fields are then known in terms of

*p*. The summation in (4) should extend to ∞, but in practice it is truncated at a finite value

_{n}*N*. In addition, the summation begins at

*n*= 1, thereby neglecting the barotropic (

*n*= 0) response; that response is weak in comparison to the baroclinic flows of interest here. Table 1 lists values of

*c*, ℋ

_{n}_{n},

*Z*, and other useful quantities for the first 10 baroclinic modes. Note that the wavelength of the

_{n}*n*= 2 Yanai wave is −4305 km, close to the wavelength of the variability in the GCM control run and to observational estimates. As we shall see (section 4), this wave is preferentially excited by the ocean.

Solutions to (5) are obtained both numerically and analytically, the analytic solutions with *ν _{h}* = 0. For the numerical solutions, the response of each mode is obtained on a staggered grid with a horizontal resolution of 0.25°. The number of modes used is

*N*= 30, and with this choice solutions are well converged. The analytic model is discussed in detail in McCreary (1984), and an expression for its Yanai wave response is derived in the appendix.

The control run of the LCS numerical model is forced from a state of rest by the aforementioned QuikSCAT winds. Solutions to the analytic and numerical versions of the LCS model are forced by idealized winds of the form

that is, zonally propagating signals with frequency *σ* and wavenumber *k _{w}*. For convenience, we assume that

*Y*′(

*x*, 0) =

*Y*(

*x*, 0) = 1 so that

*τ*

^{x}_{0}(

*x*) and

*τ*

^{y}_{0}(

*x*) are the amplitudes of the zonal and meridional winds along the equator and

*Y*′(

*x*,

*y*) and

*Y*(

*x*,

*y*) describe their off-equatorial structures. The test runs are integrated from a state of rest, spun up for 10 years, and the solutions shown are all taken from the final year of the integration.

## 3. Local and remote responses

In this section, we discuss the local (nonradiating) and remote (radiating) parts of the response forced by *τ ^{x}* and

*τ*winds. We first describe their basic properties, and then report their relative contributions to the GCM and LCS control runs.

^{y}### a. Local response

Analytic solution (A1) describes the *υ* field of the complete response to (1) when *τ ^{x}* and

*τ*are given by (6) and without horizontal mixing (

^{y}*ν*= 0). Each term in the double series involves the integral in (A2), which after integrating by parts can be rewritten

_{h}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} cm^{2} 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*) modes.

_{o}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*wind patches oscillating at various periods

^{y}*P*with the same values of

*A*and

*N*(

_{b}*z*) used for the control run. [The patch for

*τ*is defined as in (14) and (15) below, and that for

^{y}*τ*is similar except with

^{x}*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*<

*z*

_{2}) (Miyama et al. 2003). When

*P*= 14 days and

*n*= 5, the vertical structures of the local (

_{o}*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*and

^{x}*τ*forcing, respectively.

^{y}### 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

where *ω* = *σ* + *iA*/*c*^{2}_{n} is the sum of real frequency *σ* and a damping term *iA*/*c*^{2}_{n} due to vertical mixing;

is the wavenumber of the Yanai wave for vertical mode *n* (labeled *k*^{1}_{n0} in the appendix), a complex number when there is vertical mixing; *τ ^{x}_{y}*(

*x*, 0) is the amplitude of

*τ*on the equator; and we have used the relation

^{x}_{y}*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 ^{y}*

_{0}(

*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*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 (

_{y}*ω*/

*β*)/

*L*. With

_{y}*L*= 500 km and

_{y}*ω*=

*σ*= 2

*π*(14 days)

^{−1}, (

*ω*/

*β*)/

*L*= 0.5. Figures 1 and 2 plot spectra and standard deviations for the two forcings. The plots show that forcing by (

_{y}*ω*/

*β*)

*τ*

^{x}_{y0}is weaker than

*τ*

^{y}_{0}in the western ocean (where

*τ*itself is weaker than

^{x}*τ*), but that the forcings have similar strengths in the eastern ocean (where

^{y}*τ*is much stronger than

^{x}*τ*).

^{y}### c. Contribution to the control runs

To isolate the responses to *τ ^{x}* and

*τ*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 (

^{y}*τ*only) and zonal (

^{y}*τ*only) components of the QuikSCAT winds. Figure 3 plots time series of

^{x}*υ*at a depth of 50 m on the equator at 90°E, showing curves from the control (thick solid line) and the

*τ*-only (dashed line) and

^{y}*τ*-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

^{x}*τ*response is almost always larger than that forced by

^{y}*τ*, with

^{x}*τ*providing a smaller, but significant, contribution. It is noteworthy that the response to

^{x}*τ*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.

^{y}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*-only response is everywhere larger than the

^{y}*τ*-only response. Note that the standard deviations for the

^{x}*τ*- and

^{x}*τ*-only runs do not sum to equal that of the control run, a consequence of

^{y}*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*(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.

^{y}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.

## 4. Selection mechanism

The property that the spectrum of near-surface *υ* in the GCM solution is sharper than that of the forcing in the eastern ocean (Fig. 1) suggests that an oceanic selection mechanism favors the biweekly period. In this section, we use analytic solution (8) to investigate the dynamics of this selection and present a set of numerical solutions to the LCS model that illustrate the responses at different forcing frequencies. We discuss solutions both with and without vertical mixing, finding that biweekly variability is favored only when mixing is sufficiently strong.

### a. Analytic spectra

For convenience, we assume the forcing is by *τ ^{y}* only and that

*τ*

^{y}_{0}(

*x*) =

*τ*

_{0}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

where Δ*k = k _{w}* −

*k*, and

_{r}*k*=

_{r}*σ*(

*c*

^{−1}

_{n}−

*β*/|

*ω*|

^{2}) and

*k*= (

_{i}*A*/

*c*

^{2}

_{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*forcing is the same as (10) except that the factor in parentheses is replaced by 2(

^{x}_{y}*τ*

^{x}_{y0})

^{2}

*Z*

^{2}

_{n}/(|

*ω*|

^{2}H

_{n}).]

A related expression is the maximum value of 𝒮* _{n}* anywhere

*along*the equator:

where *x̂* is the smallest positive root of

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 (

^{^}

_{S}

_{n}/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, (

^{^}

_{S}

_{n}/H

_{n})/

*P*, where

*P*= 2

*π*/

*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

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*) and drops off when it does not: for example, when |Δ

_{w}*k*| =

*π*/

*L*and 2

*π*/

*L*,

^{^}

_{S}

_{n}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*| to

_{w}*π*/

*L*, the response being narrow (broad) when |

*k*| ≫

_{w}*π*/

*L*(|

*k*| ≲

_{w}*π*/

*L*). In our forced problem, |

*k*| ≈ 2

_{w}*π*/

*L*and the response is broad. For example, with

*L*= 2

*π*/|

*k*| = 6000 km it follows that

_{w}^{^}

_{S}

_{n}is greater than 40% of its maximum provided that −12 000 km < Λ

*< −4000 km, where Λ*

_{n}*= 2*

_{n}*π*/

*k*.

_{n}The left panel in Fig. 5 plots ^{^}_{S}_{n} from (13) for the *n* = 1–10 modes when *τ*_{0} = 0.1 dyn cm^{−2}, an amplitude roughly equal to that of observed *τ ^{y}* (Fig. 2), and

*L*= 2

*π*/

*k*= 6000 km. For each

_{w}*n*,

^{^}

_{S}

_{n}has a peak centered on the frequency where

*k*=

_{n}*k*, and the peaks shift monotonically toward lower frequencies as

_{w}*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 ℋ

_{2}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 _{n}k*, for the equatorial Kelvin wave, and they are plotted using the nondimensional variables,

*σ*′ =

*σ*/(

*βc*)

_{n}^{1/2}and

*k*′ =

*k*/

*α*

_{n}_{0}. 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}| = 2

*π*/(

*α*

_{n0}

*L*) and using the values for

*L*and

*k*stated in the previous paragraph; note that with these choices the right-hand edge of the band is located at

_{w}*k*′ = 0 for all the modes, the left-hand edge is at 2

*k*, and

_{w}*k*′ =

*k*′

_{w}midway between the two edges. According to (13), the Yanai wave response for a particular mode is significant (

^{^}

_{S}

_{n}> 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,

and a meridional structure,

where *θ* is a step function, *x _{m}* = 50°E,

*L*= 10°,

_{x}*L*= 20°, and

_{y}*τ*

_{0}= 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*is large enough for (

_{y}*α*

_{n}_{0}

*L*)

_{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}* = 2

*π*(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 ^{^}_{S}_{n}(*σ*) from (11) and (12) when there is vertical mixing (*A* = 8.84 × l0^{−4} cm^{2} 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} cm^{2} 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.

## 5. Eastward intensification

### 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 ^{y}*

_{0}at resonance (Δ

*k*= 0) and without mixing (

*k*= 0). According to (8) and (A11), it is given by

_{i}which increases linearly across the basin. As noted above, eastward intensification is not limited to the case Δ*k* = *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*

^{y}_{0}, provided that

*τ*

^{y}_{0}(

*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

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

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*(i.e., when

_{b}*mL*≫ 1), so (18) fails whenever the pycnocline is sufficiently sharp. In that case, wave packets

_{z}*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

where *b* = 1300 m, *ζ* = 1 −|(*z*/*z _{o}*) + 1|,

*z*= 100 m, and

_{o}*θ*is a step function. The terms proportional to

*N*

_{1}and

*N*

_{2}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,

*N*

_{0}= 0.0004 s

^{−1},

*N*

_{1}= 0.018 s

^{−1}, and

*N*

_{2}= 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}* =

*N*

_{0}= 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*). 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.

_{e}The middle panel in Fig. 9 shows the response for *N _{b}*(

*z*) with

*N*

_{0}= 0.0004 s

^{−1},

*N*

_{1}= 0, and

*N*

_{2}= 0.005 s

^{−1}, a stratification with a main pycnocline. In comparison to the constant-

*N*solution in the top panel, ray paths are curved, bending more sharply with depth where

_{b}*N*(

_{b}*z*) decreases. The near-surface relative maximum centered near 80°E is stronger and thinner than its counterpart in the constant-

*N*solution, a consequence of

_{b}*N*being larger there and, hence, the vertical group velocity

_{b}*σ*being smaller; similarly, the bottom relative maximum in the central ocean is weaker and thicker because

_{m}*N*is smaller. There is a band of increased |

_{b}*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*is large (

_{b}*b*= 1300 m).

The bottom panel in Fig. 9 shows the solution for *N _{b}*(

*z*) with

*N*

_{0}= 0.0022 s

^{−1},

*N*

_{1}= 0.018 s

^{−1}, and

*N*

_{2}= 0, thereby adding a near-surface pycnocline to the constant stratification used for the constant-

*N*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-

_{b}*N*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

_{b}*for the low-order modes are much smaller than they are for the constant-*

_{n}*N*stratification).

_{b}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.

## 6. Summary

In this paper, we investigate the excitation of oceanic biweekly variability using a hierarchy of ocean models, namely, analytic and numerical versions of a linear, continuously stratified model and an oceanic GCM (section 2). Solutions forced by QuikSCAT winds (the model control runs) are remarkably similar in the biweekly band (Fig. 3), an indication of the primarily linear nature of the ocean's response. We conclude that, despite its linearity, the LCS model accurately represents most of the dynamics of the biweekly variability, and our conclusions are all founded upon solutions to this simpler system. [As reported by J. Vialard (2004, personal communication), it is only at lower frequencies that the GCM begins to develop variability from nonlinear processes.]

The biweekly response is composed of a local (nonradiating) part for high-order (*n* > 5) baroclinic modes and a remote (Yanai wave) part for the low-order (*n* ≤ 5) modes. In the LCS control run, the local response provides background variability that is spread roughly uniformly along the equator; in contrast, the Yanai wave response increases eastward across the basin and, hence, dominates the local response in the eastern ocean (Fig. 4). We infer that similar properties hold for the GCM control run.

Analytic solution (8) describes the Yanai wave response to forcing by idealized zonal and meridional wind fields of the form (6), that is, propagating waves with wavenumber *k _{w}* and frequency

*σ*modulated by spatially dependent amplitude functions. According to (8), the

*υ*field associated with mode-

*n*Yanai waves is determined by a zonal integral of

*τ*and (

^{y}*ω*/

*β*)

*τ*along the equator. For the QuikSCAT winds, both of these forcings are appreciable in the biweekly band, with (

^{x}_{y}*ω*/

*β*)

*τ*being smaller than

^{x}_{y}*τ*in the western ocean but roughly equal to it in the east (Fig. 2). Test runs to the GCM and LCS model, forced separately by the meridional and zonal components of the QuikSCAT winds, show that the biweekly response forced by

^{y}*τ*-only variability is only somewhat weaker than that forced by

^{x}*τ*-only variability (Fig. 4), supporting the analytic results.

^{y}With the assumption that the wind amplitude is zonally uniform, the integrals in (8) can be evaluated analytically, yielding spectrum (10) and the related expression (11). Without mixing (11) simplifies to (13), which describes a narrowband (resonant) response for Yanai waves that shifts from biweekly to lower frequencies with increasing *n* (left panel in Fig. 5). The resonance happens because the zonal integrals in (8) are appreciable only when the wavenumber of the Yanai wave for mode *n*, *k _{n}*, is sufficiently close to

*k*. Plots of dispersion curves for equatorially trapped waves illustrate the Yanai wave resonance in an alternate way, and also reveal the existence of high-order baroclinic gravity wave resonances (Fig. 6). For the LCS model, the higher-order modes are damped when mixing is sufficiently strong so that the largest ocean response is restricted to Yanai waves in the biweekly band (right panel in Fig. 5). Numerical solutions to the LCS model confirm these analytic results: Without mixing, there are prominent gravity wave responses at periods of 7 and 14 days and a strong Yanai wave response at both 14 and 28 days (Fig. 7), whereas with mixing the gravity wave resonances are eliminated and the Yanai wave response is only apparent at 14 days (Fig. 8). Thus, resonance and mixing act together to account for the ocean's selecting the biweekly band. For the GCM, however, higher-order modes are likely also damped by other processes, for example, by interaction with near-surface currents or the model's low resolution below the thermocline.

_{w}As illustrated by (16), the near-resonant, inviscid, Yanai wave response associated with a particular mode *n* is eastward intensified, a property that results from the zonal integral in (8) and hence to remote forcing. When the responses of all modes are superposed, however, they form a beam (wave packet) that carries energy downward and eastward along ray paths determined from (18). Test runs to the LCS model confirm the existence of downward energy propagation and demonstrate the importance of reflections from the ocean bottom and near-surface pycnocline: When *N _{b}* is constant or has a slowly varying main pycnocline, downward propagation is so efficient that eastward intensification is eliminated (top and middle panels in Figs. 9 and 10); only when

*N*has a sharp near-surface (

_{b}*z*< −200 m) pycnocline are near-surface energy levels maintained, with eastward intensification within the region of the wind and a gradual decay east of it (middle panels in Figs. 9 and 10). These properties are also evident in the control runs of both models (Fig. 11) and they are confirmed in test runs forced by QuikSCAT winds confined to the eastern, central, and western basins (Fig. 12). We conclude that reflections from the near-surface pycnocline and ocean bottom are essential aspects of eastward intensification.

In conclusion, we have shown that the prominence of biweekly variability of the *υ* field in the near-surface, central and eastern, equatorial Indian Ocean results from the property that low-order-mode Yanai waves have large zonal wavelengths in this frequency band and so couple efficiently to the wind forcing. There are, of course, prominent intraseasonal oscillations in the Indian Ocean at lower frequencies, and similar processes to those discussed here are certainly involved in their dynamics as well. At biweekly frequencies, however, the dynamics of the ocean's response are particularly simple because only Yanai waves (and high-baroclinic-mode gravity waves) exist as free waves so that boundary reflections can be ignored; there is an apparent lack of unstable modes so that the response is highly linear. At lower frequencies, additional processes become available: for example, basinwide resonances involving wave reflections from basin boundaries can occur (Jensen 1993; Han et al. 1999; Han 2005), and oceanic instabilities can excite Yanai waves as efficiently as the wind (Kindle and Thompson 1989; Woodberry et al. 1989; J. Vialard 2004, personal communication).

## Acknowledgments

We thank Fritz Schott and Yukio Masumoto for stimulating discussions and Weiqing Han for providing us the source code of the linear, continuously stratified model. This research was supported by the Japan Agency for Marine–Earth Science and Technology (JAMSTEC) through its sponsorship of the International Pacific Research Center (IPRC). Toru Miyama was supported by funds from the MEXT RR2002 project. Debasis Sengupta acknowledges support from the Department of Ocean Development, New Delhi, India.

## REFERENCES

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### APPENDIX

#### Analytic Solution

Here, we obtain the Yanai wave part of the complete solution to (5) forced by wind fields (6) when *ν _{h}* = 0 and in an unbounded ocean. The solution is a modification of the McCreary (1984) solution, which assumed

*ν*≠ 0,

_{h}*k*= 0, and included boundaries. The neglect of boundaries is reasonable since the only available reflected waves in the biweekly band are evanescent (boundary trapped). See Moore and McCreary (1990) for a discussion of the effects of a slanted western boundary at periods of 30 days and longer and Kessler and McCreary (1993) for a solution that allows

_{w}*k*≠ 0. Moore et al. (1998) provides a general discussion of the excitation of Yanai waves in terms of a Green's function.

_{w}The *υ* field of the complete response can be written

where *ϕ _{nl}*(

*η*) is the

*l*th Hermite function associated with vertical mode

*n*,

*η*=

*α*

_{n}_{0}

*y*,

*α*

_{n0}= (

*β*/

*c*)

_{n}^{1/2}, and

In (A2), *k*^{j}_{nl} is one of the two roots of the quadratic equation,

with *ω* = *σ* + *iA*/*c*^{2}_{n} and *α*^{2}_{nℓ} = *α*^{2}_{n0}(2*l* + 1),

where

are the Hermite expansion coefficients of *q*(*η*), and *j* ≠ *j*′ so that *k*^{j}_{nl} and *k*^{j′}_{nl} correspond to different roots of (A3). The lower limits *L _{j}* are set either to the western (

*x*= 0) or eastern (

*x*=

*L*) edges of the basin, depending on whether the group speed or decay of wave

*k*

^{j}

_{nl}is eastward or westward, respectively. Note that (A2) consists of two parts, namely, a radiating part proportional to exp(

*ik*

^{j}

_{nl}

*x*−

*iσt*) and a local part proportional to

*τ*

^{y}_{0}(

*x*) (see section 3a).

When *ν _{h}* ≠ 0, (A3) is quartic, and the summation in (A2) goes from 1 to 4. As

*ν*→ 0, the four roots separate into two large and two small wavenumbers, the large ones satisfying

_{h}*k*

^{2}=

*iω*/

*ν*and, hence, tending to ∞ and the small ones satisfying (A3). With these replacements and in the limit

_{h}*ν*→ 0, McCreary's (1984) solution simplifies to ours.

_{h}When *l* = 0, (A3) can be factored into

the roots in the first and second factors labeled *j* = 1 and 2, respectively. The first factor in (A6) is the familiar dispersion relation for Yanai waves, whereas the second is an extraneous wave introduced by the solution technique. It also follows that

a consequence of the recursion relation for Hermite functions. With the aid of (A6) and (A7), it follows that *P*^{2}_{n0} ≡ 0 (a necessary result since the *l* = 0, *j* = 2 wave is extraneous) and that

According to (A8), Yanai waves are generated by the *l* = 0 components of both *τ ^{y}* and

*τ*.

^{x}_{y}We can simplify (A8) further by assuming that *Y*(*y*) = *Y*′_{y}(*y*) = 1 over the width scale of *ϕ _{n}*

_{0}(

*η*). This assumption is valid provided that (

*α*

_{n}_{0}

*L*)

_{y}^{−2}≪ 1, where

*L*is the meridional scale of

_{y}*τ*and

^{y}*τ*. The inequality is satisfied in our solution since

^{x}_{y}*L*is 500–1000 km for biweekly forcing by the QBM and

_{y}*α*

^{−1}

_{n0}<

*α*

^{−1}

_{10}= 350 km. Setting

*q*= 1 in (A5), we have

With the aid of (A9), it follows that

where *τ ^{x}*

_{y0}is defined to be the amplitude of

*τ*on the equator.

^{x}_{y}Using (A10), *υ _{n}*

_{0}(

*x*) reduces to

## Footnotes

*Corresponding author address:* Toru Miyama, Frontier Research Center for Global Change, Japan Agency for Marine–Earth Science and Technology, 3173-25 Showamachi, Kanazawa-ku, Yokohama, Japan. Email: tmiyama@jamstec.go.jp

* School of Ocean and Earth Science and Technology Contribution Number 379 and International Pacific Research Center Contribution Number 6749.