U.S. Dept. of Commerce / NOAA / OAR / PMEL / Publications

Forcing of intraseasonal Kelvin waves in the equatorial Pacific

William S. Kessler and Michael J. McPhaden

Pacific Marine Environmental Laboratory, NOAA, Seattle, Washington

Klaus M. Weickmann

Climate Diagnostics Center, NOAA, Boulder, Colorado

J. Geophys. Res., 100(C6), 10,613-10,631 (1995)
This paper is not subject to U.S. copyright. Published in 1995 by the American Geophysical Union.

4. An Ocean-Atmosphere Feedback

We have shown that the commonly observed intraseasonal Kelvin waves in the equatorial Pacific are generated by the fluctuations of winds and tropical convection associated with the MJO. In particular, the waves exhibit low-frequency modulation due to annual and interannual variations of west Pacific convection, which itself is a signal propagating from further west. Since the MJO life cycle is sensitive to the distribution of warmest SST over both the Indian and Pacific Oceans, and to the planetary atmospheric circulation, the oceanic signal must be taken to be a manifestation of a global phenomenon, and not simply internal to the Pacific. Although it is easy to see how low-frequency variations of SST in the Pacific can affect the MJO, which has its intense convection signals over the warmest SST, we now ask whether the MJO events themselves could have a role in the interannual variations of the Pacific. Such a process would require a nonlinear coupling between the relatively high intraseasonal frequencies and a rectified low-frequency response.

One mechanism that might produce this interaction is suggested by the 500- to 1000-km intraseasonal bumps on the SST contours in Figure 5, which occur both in the eastern and western Pacific. SST variability at this timescale, in the absence of corresponding atmospheric forcing, points to zonal advection by the intraseasonal Kelvin waves as a possible explanation. In view of the fact that convection and westerly winds follow the warmest water eastward, this could provide a mechanism by which intraseasonal variability in the ocean can feed back to affect the atmosphere. Since the atmosphere can respond to SST forcing (by shifting the location of convection) much more rapidly than the ocean responds to changing winds, each eastward advection event can draw subsequent convection further east.

4.1. An Example of Intraseasonal Westerly/SST Interaction

An apparent example of this process occurred during the basin-wide warming of the El Nio of 1991-1992. Figure 12a shows a detail of the SST zonal section from Figure 5 for the period July 1991 through April 1992. Overlaid on the SST contours and shading are, first, the zero contour of the zonal winds (same data as Figure 4) showing the advance-and-retreat eastward expansion of westerlies, and second, the Kelvin wave propagation lines from Figure 3 showing the four downwelling waves (associated with eastward current anomalies) observed in 20C depth during the onset of the 1991-1992 warm event. The second and third westerly events, in November 1991 and January 1992, each extended about 2000 km further east than the previous one; in the intervening times the westerlies retreated west of the date line (Figure 12a). The warmest water (marked by the 29.5C contour) also moved east in pulses following the westerly winds, with the first 29.5C water east of the date line observed in October 1991, then a pause until December when the east edge of this warm water suddenly moved to 155W. Similar abrupt warmings also occurred in the central Pacific following the passage of the downwelling waves (e.g., the 28C and 29C isotherm in Figure 12a). It is clear from Figure 12a that not all the warming of the 1991-1992 event was produced by Kelvin advection; for example, the first warming began in September 1991, probably before such a signal could develop by Kelvin advection, and the final basin-wide warming occurred late March 1992, again with a timing that does not seem consistent with an advective mechanism. Hayes et al. [1991a] also found no evidence for a zonal advective effect on SST associated with the passage of an intraseasonal Kelvin wave at 110W in January 1987.

Figure 12. (a) Detail of SST on the equator for July 1991 through April 1992 (during the peak of the El Nio of 1991-1992). Contours and shading show SST with a contour interval of 1C, with supplemental contour/shade at 29.5C. Light contours show warmer temperatures (opposite of Figure 5). The heavy slant lines are the same Kelvin lines shown in Figure 3. The heavy contour labeled "0" is the zero line of zonal winds, showing the steplike progression of westerlies eastward over the Pacific during the onset of the warm event. (b) Model SST/wind to match the timing of Figure 12a. Output of the simple model described in section 4. The heavy curve is the eastern edge of the 29C SST and the wind patch (the result of integrating (3); see text). The light sinusoidal curve at top is the time series of winds from (1) (up is easterly, down is westerly) (winds are zero before day 0). The shading shows the region of westerly winds. Slant lines indicate maximum positive pressure perturbation to match the observed Kelvin lines in Figure 12a; east of the forced region these are Kelvin characteristics, within the forced region they move at speed 2c (see text).

Kessler and McPhaden [1995b] studied the zonal advective effect on SST at 140W during the 1991-1993 El Nio and showed that although this forcing was not the most important term in the SST balance at annual and interannual frequencies it was dominant during the period of intense intraseasonal variability at the height of the warm event. Figure 13 compares the advective terms d(SST)/dt and ud(SST)/dx at 0, 140W (d(SST)/dx is estimated by centered difference between 155W and 125W) during the same period as Figure 12a for SST and winds. The positive (warming) humps of -ud(SST)/dx in Figure 13 show the advection due to the intraseasonal Kelvin waves at 140W in October and November-December 1991 and January 1992. Clearly, the first major warming that took place in September was not the result of Kelvin advection, but the next two events are quite consistent with that interpretation, and the two terms balance closely. The final warming in March occurred before the passage of the fourth Kelvin wave and again was apparently not due to that wave. The fourth wave produced only a very weak advective signal in Figure 13 because the zonal temperature gradient at 140W was near zero at that time (Figure 12a).

The net result of successive intraseasonal waves associated with steplike eastward movement of the warmest water and westerly winds appears as a much lower-frequency signal. In this view, Figure 12a suggests that zonal advection moved the 29.5C water 1000 km or so to the east following the westerly wind event, and the next convection phase of the MJO (westerlies) responded to the new position of the warm water before it had returned west to its previous location. This process can result in a slow (relative to intraseasonal timescales), steplike progression of the warmest water and westerly winds to the east along the equator as observed in late 1991, and which has been often noted as a feature of the onset of El Nio. Note that as the warm water/westerly wind couplet migrates east, the fetch of the westerlies increases. Since the Kelvin response is proportional to the fetch [Weisberg and Tang, 1983], the process can amplify.

Figure 13. Comparison of d(SST)/dt and ud(SST)/dx at 0, 140W. The solid line shows -ud(SST)/dx, where u is taken as the 14-m (shallowest level) zonal current measured by ADCP and d(SST)/dx is estimated by centered difference between 125W and 155W. The dashed line is d(SST)/dt at 140W. Both time series are filtered with a 17-day triangle filter. Upward on the plot indicates a warming tendency for both time series.

4.2. A Simple Model of the Interaction

A simple coupled model illustrates the dynamics involved. The model is not intended to be a realistic simulation of all or even most aspects of the onset of El Nio, but simply to show that a nonlinear interaction between the oceanic intraseasonal Kelvin waves and the Madden-Julian Oscillation is possible. The model is highly idealized to represent the single mechanism of an advective feedback between intraseasonal advection of SST and the rapid response of the atmosphere to changes of location of the warm pool. This feedback may be an element of the slow eastward advance of warm SST and atmospheric convection that has been noted to occur during the onset of warm events.

Assume that the initial state of the ocean has a warm pool extending eastward from the western boundary. Let sinusoidally oscillating, zero-mean zonal winds occur only to the west of a particular value of SST, say the 29C isotherm. The frequency and phase of the surface winds is assumed to be fixed by upper atmosphere waves oscillating at a Madden-Julian timescale, but their longitudinal extent is determined by the SST. For simplicity, we assume that the winds do not vary in longitude within the wind patch (from the western boundary to the 29C SST isotherm), but are zero outside the patch. In the ocean, Kelvin waves forced by the oscillating winds advect the 29C patch edge. We assume a simple ocean dynamics such that the ocean response to winds is that the forced ocean current is directly proportional to the wind integrated over the Kelvin wave characteristic. Other than zonal advection, due to Kelvin wave passage or to wind forcing directly, there are no processes that affect SST in this model.

This model can be formulated as follows. The western boundary is at x = 0. Let x = a(t) mark the (time varying) east edge of the 29C SST/wind patch. Then the wind field is


where b is the (constant) amplitude of the wind and the frequency. Note that uatmos has zero mean. The ocean current at the patch edge is now taken to be directly proportional to the wind integrated over the patch along the Kelvin wave characteristic. The integral sums the forcing felt by a wave since it left the western boundary.


where b* is the (constant) coupling efficiency, c is the Kelvin wave speed, and ta is the arrival time of Kelvin characteristics at the patch edge. The coupling efficiency b* scales the speed of the current generated by a given wind, and is a tunable parameter in this model. For simplicity of notation, we combine bb* = B, which has units of (time)-1 and incorporates the effects of both wind strength and coupling efficiency; thus B represents the net forcing amplitude felt by the ocean. Since the wind does not vary (in x) for 0 x a(t), the integration can be performed in time alone from t = ta - a/c (the time a characteristic leaves x = 0) to t = ta, as indicated in (2). Since the patch edge position a(t) is changed only by zonal advection, we identify uocean = da/dt = rate of change of position of the patch edge. Carrying out the integral in (2),


This is a nonlinear (because a(t) appears in the argument to the cosine on the right-hand side) ordinary differential equation, which can be easily integrated numerically. We see from (3) that da/dt is zero for a = 0 (no patch) or for a = 2c/ (a patch with width the same as the wavelength of a Kelvin wave of frequency ), so these are limiting equilibrium positions where the motion stops, but the motion can be of either sign between these two locations.

Reasonable values of the model parameters can be chosen as follows. We have established that the Kelvin wave speed is c = 2.4 m s-1 and the central intraseasonal frequency is about = 2/(60 days). The wind forcing amplitude B can be estimated from the wind-forced linear zonal momentum equation


where = acDua2 (subscripts a and o here indicate atmosphere and ocean, respectively, and cD is the drag coefficient), and H is the thickness of the wind-forced layer. Taking usual values a = 1.2 kg m-3, o = 1025 kg m-3, and cD = 1.5 10-3, and estimating the amplitude of the wind events during late 1991 as up to 5 m s-1, and the wind-forced layer to be 50 m thick, (4) gives the amplitude |ut| 9 10-7 m s-2. According to (2), the modeled |ut| = cB; setting this equal to the result obtained from (4) gives B 4 10-7 s-1, which is the value used to construct the example shown in Figure 12b. This value implies a timescale B-1 = 29 days, consistent with 60-day oscillations.

Note that the estimate of the tunable parameter B from (4) is proportional to the wind speed squared and also that the choice of the wind-driven layer depth is somewhat arbitrary. In any case, the model has a simple parameter space, and the response is qualitatively the same for all values 0 < B < . For all positive values of B the only change in response is the length of time until the patch edge reaches equilibrium at a = 2c/ (and note that the equilibrium value is not a function of the wind forcing parameter B, but only of the unambiguous quantities c and ). For (unrealistically) large values of B the solution can jump to an integer multiple of 2c/ but then resumes identical behavior approaching the new equilibrium position. The model is also not sensitive to the starting phase of the wind. In the example discussed in section 4.3 and shown in Figure 12b we have started the winds at the beginning of their westerly phase at time zero (winds are zero before t = 0), but if the winds are started easterly at t = 0, the process takes several more oscillations before rapid growth occurs, but the eventual result is the same.

4.3. Solution for Realistic Parameters

The behavior of the solution is shown in Figure 12b. The patch edge moves east in pulses not dissimilar to those observed (Figure 12a) for SST and westerly winds during late 1991. Each step advances about 1000-2000 km to the east, then in intervening periods the edge moves west nearly as far back as it advanced. The SST advance lags the occurrence of westerly winds by about half a period or so, as was seen in the observations (Figure 12a). The maximum positive pressure (and zonal current) perturbation, indicated by the heavy slant lines in Figure 12b, approximately matches the analogous Kelvin wave lines determined from 20C depth (Figure 12a). (Within the forced region the combination of direct forcing and Kelvin waves results in an apparent propagation speed of 2c, as was shown by Philander and Pacanowski [1981], whose high-frequency solution u field is similar to ours but without the SST feedback). The eastward advance slows as the patch edge approaches 2c/ (in the present example 2c/ = 12,442 km), which is the point where a Kelvin characteristic traveling from the boundary to the patch edge has passed half under the westerly wind region and half under the easterly region and thus (2) integrates to zero; eventually, the oscillations of the patch edge die out at this asymptotic value. This model can only simulate the eastward growth of the coupled interaction, not the return of the warm water edge and westerly winds to the west. We emphasize that it is unlikely that the model results (such as the particular value of the equilibrium position) are relevant to the long time evolution of the ocean. Many other processes that have been ignored here will certainly modify the ocean-atmosphere interaction, particularly over longer time periods. The point of the model is to isolate a specific coupling mechanism through which it is possible to get a net low-frequency change in the ocean from zero-mean oscillating forcing. This seems to simulate reasonably well some important aspects of the changes observed near the east edge of the warm pool during 3 or 4 months of the onset stage of the El Nio of 1991-1992 (Figure 12).

The solution has a basic similarity to the results of a coupled general circulation model simulation reported by Latif et al. [1988]. They added a single 30-day westerly wind event to their model after spin up with annual cycle forcing and then let the coupled system run freely. The model responded with an initial rapid eastward shift of the SST maximum toward the central Pacific due to zonal Kelvin wave advection, then subsequently the model atmosphere developed persistent westerlies blowing toward the warmest water. This kept the central basin sea level and SST high for at least a full year after the single 30-day imposed forcing event. Although the Latif et al. [1988] model is much more complex than the present formulation, it appears that the coupled dynamics are similar in this case, with the atmosphere responding to transient eastward displacement of warm SST by developing westerlies that serve to maintain and extend the SST pattern.

4.4. Dynamics of the Interaction

The key dynamics that produce the rectified low-frequency outcome from the high-frequency forcing is that the model atmosphere responds immediately to the state of the SST, while the ocean's response to the atmosphere is lagged because it is due to an integration over forcing of finite duration. While the model is crude, this timescale difference between the two fluids is probably representative of a true distinction. The other important characteristic of the model formulation is that the strength of the model ocean response is proportional to the fetch, so that westerly winds, which advect the patch edge eastward, increase the fetch, while easterly winds reduce it. Thus during westerly periods the increasing fetch means the response increases, but during easterly periods the decreasing fetch produces a weaker signal, so each westward retreat is somewhat weaker than the eastward advances. This behavior is much like the observations in late 1991.

We noted in the introduction that the Madden-Julian Oscillation is a global phenomenon, but its surface expression is strong only over the warm-SST part of the equatorial ocean. Similarly in the model, we assume that the forces that establish the basic oscillation are entirely external to the feedback mechanism. In this representation the atmospheric dynamics of the MJO set the 60-day oscillation period, while the SST determines only the zonal length of the region in which convection and strong low-level winds develop during the phase favorable to upper-level divergence. Although the existence of the MJO probably requires a minimum size warm SST region to exist, the few-thousand-kilometer changes during a single event may be small perturbations to the global state SST felt by the atmosphere, and thus the interaction described here may not strongly affect the fundamental dynamics or frequency.

4.5. Weaknesses of the Model

Several important weaknesses of the model are evident. We neglect entirely any heat exchange between the atmosphere and ocean, which is obviously crucial to the evolution of the coupled system. The present, highly-idealized formulation can only be relevant over short time scales during which the rapid advection that occurs as a result of the intraseasonal waves can be the dominant process affecting SST. Such dominance of intraseasonal Kelvin wave-mediated zonal advection on SST change can occur during warm event onset and was observed at 140W during November 1991 through February 1992 (section 4.1 and Figure 13). Second, the model implicitly has an infinite heat reservoir that allows the warm pool to expand indefinitely, determined only by dynamics, not a heat balance. However, the same result would still occur if eastward advection in the warm pool exposed cooler water to the west, if one makes the reasonable assumption that the convection and westerlies advance over the cool water to the warm pool. Similarly, SST cooling due to evaporation associated with the increase in absolute wind speeds during westerly events in the western Pacific is typically less than 1C, which is not enough to reduce the absolute temperature to below the threshold needed for deep convection. Therefore the primary feedback shown by the simple dynamics would not change if the model was made more realistic by allowing changing SST under the winds. However, such cooling may well have been the reason for the slight decrease in warm pool SST under the strong winds of January 1992 (Figure 12a). Third, in order to demonstrate without ambiguity that the slow change in the ocean can be due entirely to the coupled feedback rectification, we have postulated zero mean wind forcing. In the real event, it is observed that the onset of El Nio occurs in a regime of low-frequency westerly forcing with the higher-frequency convection events superimposed (Figure 3, top). This would tend to make the eastward advective signal stronger, but the aim here is to show that it is not necessary to have mean westerly forcing in order to get a net eastward propagation in the ocean. Fourth, to achieve maximum mathematical simplicity, we have specified that the winds do not vary in x within the wind patch (equation (1)). In fact, the wind signal propagates eastward with the convection signal at speeds of the order of 3-6 m s-1 [Rui and Wang, 1990]. Such eastward-propagating winds would result in an increased projection of the forcing onto the Kelvin mode [Tang and Weisberg, 1984; McCreary and Lukas, 1986], effectively increasing the coupling parameter b*. (In fact, when the model is run with B increased by about 20%, the match with the observations is somewhat closer).

An element of the ocean dynamics that we have ignored is the Rossby waves that would also be generated by the oscillating forcing. Rossby waves might affect the result in three ways. First, the Kelvin waves discussed here would produce Rossby waves upon reflection at the eastern boundary. However, a variety of studies [du Penhoat et al., 1992; Kessler and McCreary, 1993; Kessler and McPhaden, 1995a; Minobe and Takeuchi (Annual period equatorial waves in the Pacific Ocean, submitted to Journal of Geophysical Research, 1994)) have suggested that these waves will not survive propagation across the entire Pacific, thus we think this would not be a major element of a more complete solution. Second, the oscillating winds would generate Rossby waves directly. These waves propagate west, and so would not influence the east edge of the patch, except by producing secondary Kelvin waves upon reflection from the western boundary. The amplitude of the Rossby waves forced by the wind patch depends on the meridional shape of the wind field; the amplitude of the consequent reflected Kelvin waves depends on the mix of Rossby meridional wavenumbers and the shape of the western boundary [Clarke, 1983; McCalpin, 1987; Kessler, 1991]. The phase of the resulting Kelvin waves depends on the zonal width of the patch, and we can anticipate that as the patch length changes the Kelvin waves due to boundary reflection will exhibit varying phase compared to the directly forced waves and may be of either sign relative to the original solution. The time lag for Rossby wave propagation from the patch edge to the western boundary and then Kelvin wave propagation back to the patch edge a can be written tR = a/cR + a/cK = 4a/cK, where subscripts R and K indicate Rossby and Kelvin speeds, respectively, and we use the first meridional mode Rossby speed which is 1/3 the Kelvin speed. To get a rough idea of the effect of including the western boundary reflection in the model, the solution was recalculated with the addition of Kelvin waves due to the boundary reflection of wind-forced first-meridional-mode Rossby waves, assuming for the sake of this discussion that the Rossby u field amplitude was exactly the same as that of the directly forced Kelvin waves. This solution (not shown) is very similar to the original Kelvin wave-only solution shown in Figure 12b. The first eastward advance is slightly weaker than the original, but after about the middle of the second westerly wind period the combined-wave solution results in a somewhat faster eastward advance than the original. This can be qualitatively understood because the Rossby lag 4a/cK initially produces oppositely phased Kelvin waves that slow the advance, but the patch soon grows to a width where the lag results in boundary-reflected waves that have the same sign as the directly forced waves and thus enhance the eastward translation. This experiment suggests that the addition of Rossby waves forced by an equatorial wind patch would not alter the fundamental character of the solution to the Kelvin wave model (1)-(3).

A third way in which our neglect of Rossby waves in the model is unrealistic is that there can also be easterly wind anomalies to the east of the convection on MJO timescales, and these would generate Rossby waves carrying equatorial currents westward toward the patch edge that would oppose the Kelvin signals modeled here. It is not straightforward to model these Rossby waves in the context of a model as simple as the present one, since the zonal width of the easterly forcing is much harder to define than the width of the convective region. Also note that the much slower Rossby propagation speed (1/3 of the Kelvin speed for first-meridional-mode waves) implies that a reasonably sized easterly patch region of 4000 km would be close to its equilibrium (stationary) length 2c/, and the integral over the characteristic would thus be small (see discussion of equation (6) and Figure 14 in section 4.6). In addition, we have noted that the central Pacific intraseasonal zonal wind variability is very much weaker than that over the warm pool (see Figure 4). In sum, we conclude that our neglect of the Rossby wave forcing, while unrealistic, does not distort the fundamental feedback properties of the model.

Figure 14. Complex (frequency domain) EOF 1 of intraseasonal (30- to 80-day period) OLR along the equator in the Pacific during 1979-1993. (Top) Amplitude (W m-2) of the complex eigenvector. The zonal structure of this EOF is used to estimate the zonal length of the OLR patch (see text). (Middle) Phase relative to 125E. The slope in the western Pacific indicates eastward propagation at a speed of about 4.5 m s-1. (Bottom) Percent variance represented.

Recognizing that all these dynamic and thermodynamic weaknesses and crude approximations to the observations make the model unsuitable for realistic simulation of the coupled system in general, the extremely simple form used here was chosen for the purpose of isolating a particular process that may be relevant to the real system during a limited (but perhaps important) period.

4.6. An Application to the Spectral Offset Between Intraseasonal Winds and Ocean Response

We have shown (in agreement with previous studies) that although intraseasonal variability in the atmosphere was centered at 35- to 60-day periods (Figure 6), the ocean response was shifted toward the lower-frequency end of the band. Thermocline depth and undercurrent speed variability were very weak at periods less than 50 days and had significant amplitude at 75-day periods (Figure 6). In a review of an earlier version of this manuscript, N. Graham (personal communication, 1994) pointed out to us that this discrepancy can be explained in the context of the Kelvin wave model of section 4.2. The fundamental idea is that the Kelvin signal integrates to zero if the forced region length equals the distance a wave ray travels in one period of oscillation (2c/). For the 60-day waves discussed above, this distance is 12,442 km, but shorter-period forcing (like the 30- to 50-day spectral peaks of OLR and west Pacific zonal wind in Figure 6) may be close to the zero-integral equilibrium for typical wind patch lengths, and thus have only a weak effect on the ocean.

Using terminology analogous to the model (1)-(3), but with the patch edge fixed (no feedback), let A be the (fixed) east edge of the warm SST/wind patch. The wind field is still described by (1), substituting the constant A for the previously variable a(t). Performing the same integration as in (2), again substituting A for a(t), gives an equation analogous to (3) for the zonal current at the east edge A. Since A is fixed, this expression for u(x = A) may be written as the product of a constant amplitude and a time-varying term.


where the first term on the right-hand side is the (constant) amplitude and the second is the time-varying term. The variance of the Kelvin response at A (and thus everywhere east of A) is the amplitude squared


where the function sinc(x) [x-1 sin(x)], which equals 1 at x = 0, equals 0 at x = , and thereafter represents a decaying oscillation for increasing values of x. In (6) the variance falls to zero as the period decreases toward the value A/c (the time it takes a Kelvin wave to cross the patch), because then, in summing the integral over the wind patch, the easterly and westerly contributions to the ocean forcing cancel. Therefore the amplitude of the ocean response east of equatorial wind forcing depends on the product A, and for some values of these parameters the response can vanish.

An estimate of the zonal length scale of the intraseasonal forcing can be made using OLR, which spans the entire western Pacific (the buoy observations are not suitable for this purpose since long records are available only east of 165E). The 1979-1993 history of equatorial OLR was decomposed in complex empirical orthogonal functions (CEOFs) in the frequency domain [Wallace and Dickinson, 1972], using frequencies spanning 30- to 80-day periods to define the intraseasonal band. Figure 14 shows the amplitude, phase relative to 125E and percent variance represented by the first CEOF, as a function of longitude in the Pacific. In the western Pacific, the first CEOF expresses 50% or more of the intraseasonal variance, with amplitudes up to 15 W m-2 (Figure 14). The phase indicates eastward propagation at a speed of about 4.5 m s-1 in this region, consistent with earlier analyses [e.g., Rui and Wang, 1990]. The zonal structure of the amplitude (Figure 14, top) suggests that an e-folding zonal scale of the intraseasonal variability felt by the equatorial Pacific stretches from the western boundary of the Pacific (near 135E on the equator) to 180, or about 5000 km. Not surprisingly, a similar estimate results from considering the zonal extent of the warm pool. The Reynolds and Smith [1994] satellite blended SST field (not shown), averaged over the 10 years 1984-1993, establishes that SST warmer than 29C extends between about 130E to 175E, approximately coinciding with the region of high amplitude of the first CEOF of OLR. The 29C SST isotherm divides the warm pool, which has small zonal SST change, from the region to the east where there is a strong zonal gradient. This also gives a 5000-km estimate of the size of the convection region, which we will take as an estimate of the patch length A in (6).

Figure 15 shows the theoretical variance of zonal current east of a 5000-km patch as a function of frequency, calculated according to (6), using the Kelvin wave speed c = 2.4 m s-1 and the wind-forcing parameter B = 4 10-7 s-1 estimated in section 4.2. There is a sharp falloff of energy at periods from about 50 to 30 days, because the time it takes a wave ray to cross the 5000-km patch approaches one period of the wind. The variance falloff occurs exactly in the region where the observed OLR and wind have spectral peaks but the thermocline depth and zonal current are weak (Figure 6). It helps resolve this apparent ocean-atmosphere discrepancy by shifting the part of the wind spectrum responded to by the ocean toward lower intraseasonal frequencies that have larger zonal wavelengths relative to the wind field. Shorter wind patches would allow the ocean to react to higher-frequency wind signals, but also imply smaller overall amplitude (due to the factor A2 multiplying the right side of (6)).

Figure 15. Theoretical variance of zonal current east of a 5000-km width wind patch, calculated according to (6), shown as a function of forcing frequency. The top axis numbering shows the periods in days. Note the rapid drop in variance between 100-day and 30-day periods.

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