Analytical theory is used to examine the linear response of a meridionally unbounded stratified ocean to large-scale, low-frequency wind forcing. The following results, applied mainly to the equatorial Pacific, were obtained.
(i) Provided that the wind stress curl vanishes at large distance from the equator, a general Sverdrup solution is valid in the quasi-steady (frequency ω → 0) limit. The meridionally averaged zonal flow toward the western boundary layer is zero so that there is no net mass flow into the boundary layer and the large-scale boundary condition is therefore satisfied. This solution predicts a zero pycnocline response in the eastern equatorial Pacific. It therefore predicts that, for the eastern equatorial Pacific, a slow weakening of the equatorial trade winds will not lead to long-term El Niño conditions there.
(ii) Consistent with observations and other previous work, for finite but small frequencies there are two modes of equatorial motion. One is a “tilt” mode in which the equatorial sea level and thermocline are tilted by the in-phase zonal wind stress and the other is an equatorial warm water volume (WWV) mode in which the discharge of equatorial warm water (negative WWV anomaly) lags the wind stress forcing by a quarter of a period.
(iii) The amplitude of the WWV mode approaches zero like ω1/2. Therefore, as ω → 0, the equatorial solution reduces to the tilt mode.
(iv) The WWV mode is not due to a dominant meridional divergence driven by the wind, as suggested by some previous work. Meridional and zonal divergence approximately cancel. Reflection of energy at both ocean boundaries together with the strong dependence of long Rossby wave speed on latitude is crucial to the existence of the disequilibrium WWV mode. Because higher-latitude Rossby waves travel so much more slowly, the Rossby waves reflecting from the western ocean boundary are not in phase. This gives rise to a reflected equatorial Kelvin wave and a WWV that is not in phase with the wind stress forcing.
(v) Observations from past work have shown that much low-frequency wave energy, particularly westward propagating Rossby wave energy poleward of about 5°N and 5°S, is damped out before it reaches the western ocean boundary. In this way dissipation likely has a strong influence on the equatorial Kelvin wave reflection and hence the disequilibrium WWV.
Equatorial warm water volume (WWV), originally discussed theoretically by Cane and Zebiak (1985; see also Zebiak and Cane 1987; Jin 1997), was found observationally by Meinen and McPhaden (2000) to be a key ingredient in El Niño–Southern Oscillation (ENSO) dynamics. This theoretical and observational work suggests that ENSO involves a recharge and discharge of WWV along the equator and that the cyclic nature of ENSO results from a disequilibrium between zonal winds and zonal thermocline depth. The “disequilibrium” is associated with anomalous WWV both leading and lagging standard ENSO indices by about a quarter of the period of the oscillation. This amounts to a lag of several months on ENSO time scales and, for decadal periodicity, about 2–3 yr (Hasegawa and Hanawa 2003a).
One might expect rapid ocean adjustment to wind forcing near the equator where equatorial Kelvin and Rossby waves propagate quickly. If so, then why should there still be such large lags, corresponding to about one-quarter of the period of the oscillation? Jin (1997) obtained such a large lag in his conceptual ENSO model, but his results depended on a parameterized western boundary condition. Li (1997) also explained the large lag using a parameterized conceptual model and, in his case, assumed that no mass flux from the western boundary entered the equatorial WWV region. Clarke et al. (2007) emphasized the importance of wind stress curl to WWV dynamics and suggested how the curl might generate an equatorial ocean response with one-quarter period lag.
Specifically, for ease of illustration of how the wind stress curl might generate WWV, consider the simple case of a two-layer ocean model of upper-layer depth h and upper-layer velocity (u, υ), where u is the eastward velocity and υ the northward velocity. Suppose we integrate the upper-layer continuity equation
over an equatorial strip bounded in the north by y = Δy, in the south by y = −Δy, in the east by the eastern ocean boundary x = L, and in the west by x = 0, a longitude just east of the western ocean boundary layer. Using the boundary condition of no normal flow at the eastern ocean boundary we obtain, from the divergence theorem,
In (1.2) υN is the northward velocity at y = Δy, υS is the northward velocity at y = −Δy, and
In other words, (∂/∂t)(WWV) is equal to the northward transport into the box along its southern edge plus the southward transport along its northern edge. If Sverdrup balance were to hold at y = Δy and y = −Δy, then (∂/∂t)(WWV) would be in phase with the wind stress curl forcing; that is, WWV would lag the wind stress curl forcing by a quarter of a period. Actually, Sverdrup balance does not hold at y = ±Δy because the balance there is really that for long Rossby waves forced by the wind stress curl. However, Clarke et al. (2007) show that using the long Rossby wave balance instead of Sverdrup balance does not alter the result that (∂/∂t)(WWV) is proportional to the low-frequency wind stress curl forcing; that is, it takes a quarter of a period for the WWV to respond to low-frequency wind forcing. The detailed physics of this result is discussed at the end of section 3 of Clarke et al. (2007; see also section 5.6 of Clarke 2008).
then we would expect that, as the frequency ω → 0, the WWV would become very large like curl forcing/ω. This is questionable; it seems more likely that, as frequency ω → 0, (1.1) would reduce to
that is, the meridional divergence is balanced by the zonal convergence. For our box model this would mean that the meridional transport, say, out of the box would be balanced by the zonal transport into it and the WWV would not change; no WWV disequilibrium with associated large lags would exist.
Analysis of observations by Meinen and McPhaden (2001) support the notion that zonal convergence cannot be neglected in the WWV balance. Clarke et al. (2007) also found from their box model that the zonal transport at the western edge of the box was not negligible. On the other hand, Bosc and Delcroix (2008) concluded that during 1992–2006, a large part of the WWV changes resulted from net convergence and divergence of meridional transports. From another point of view, Jin (2001) suggested that WWV fluctuations are closely linked to low-frequency damped ocean basin modes.
In view of the above uncertainty, it would be useful to know what analytical linear theory says about the equatorial ocean response to low-frequency wind forcing with nonzero wind stress curl. Does it suggest that zonal convergence can be neglected at low frequencies? Does it even predict the basic elements of the low-frequency response observed? Specifically, does it predict the two equatorial modes explaining most of the low-frequency thermocline and sea level—namely, a “tilt” mode (Fig. 1) in phase with the zonal wind stress tilting it and a WWV mode (Fig. 2) in disequilibrium with the wind stress in the same way as in the observations? If there is a disequilibrium WWV mode, does it grow as ω → 0 or is there a quasi-steady (in phase) response to the wind at low enough frequency?
Previous analytical theory has either parameterized the linear ocean model (e.g., Li 1997; Jin 1997) or has used wind forcing that either does not vary longitudinally (Cane and Sarachik 1981; Emile-Geay and Cane 2009) or meridionally (Neelin and Jin 1993), or has concentrated on coupled dynamics (Cane et al. 1990). The approach here will try to address the questions of the previous paragraph using an ocean model forced by a wind stress that varies both zonally and meridionally.
The ocean model is assumed to be unbounded meridionally. In reality the North Pacific is partially closed by the Aleutian Islands at its northern end. However, low-frequency equatorial ocean signals appear to be dissipated before they reach the end of both the northern and southern eastern ocean boundaries (Enfield and Allen 1980; Pizarro et al. 2001; Li and Clarke 2007), so a meridionally unbounded ocean seems reasonable.
In section 2 the reader is reminded of the linear large zonal scale, low-frequency governing equations, and their boundary conditions. Section 3 establishes that a quasi-steady linear ocean response does, indeed, exist. This suggests that at low enough frequency there is no disequilibrium WWV mode. The low-frequency analytical solution derived in section 4 is used in section 5 to show that the linear wind-forced ocean response does consist of a tilt and disequilibrium WWV mode, the latter, in agreement with section 3, approaching zero as the frequency approaches zero. Section 5 also analyses the physics of the disequilibrium WWV mode. A discussion of the low-frequency theory and observations in section 6 suggests that dissipation plays a major role in determining the disequilibrium WWV amplitude. Section 7 contains some concluding remarks.
2. The linear equatorial ocean model
a. Governing equations
In standard fashion (e.g., Gill and Clarke 1974), the governing linear perturbation equations for wind-forced flow in a continuously stratified ocean of constant depth can be written as a sum of vertical modes. Using the standard notation that β is the northward gradient of the Coriolis parameter and c is the Kelvin wave phase speed for a given vertical mode, we nondimensionalize the horizontal equations for each vertical mode on a β plane by (c/β)1/2 for horizontal length, ( βc)−1/2 for time, c for horizontal velocity, and c2 for the perturbation pressure p divided by the mean water density. Then the nondimensional equations for each vertical mode take the form
In these equations and in what follows, x, y, t, u, υ, and p refer to nondimensional versions of distance eastward, distance north of the equator, time, eastward and northward velocity components, and perturbation pressure divided by the mean water density. For the nth vertical mode with perturbation pressure eigenfunction Fn(z), the forcing terms X and Y are defined by
where ρ0 refers to the mean water density, τ x and τ y to the x and y components of the wind stress, and
In (2.6) Hmix is the smallest depth below which the wind-driven turbulent stress vanishes. The coefficients bn in (2.6) were derived for a model of turbulent ocean stress in which the stress falls linearly from a value equal to the wind stress at the surface to zero at the mixed layer depth equal to Hmix; that is, the stress divergence is constant over the mixed layer and equal to τ x/Hmix and zero beneath it [see, e.g., the discussion in sections 3.2 and 3.3 of Clarke (2008)]. For realistic Hmix and Fn = 1 at the surface, reasonable values of and in the equatorial Pacific are and .
In the Pacific Ocean low-frequency wind forcing has a large east–west scale and much smaller north–south scale (see, e.g., Fig. 3). In addition, the Ekman transport is proportional to the reciprocal of the Coriolis parameter f and so varies rapidly near the equator. Consequently, the wind-forced low-frequency interior ocean response can be expected to have a much larger east–west than north–south scale. Under the low frequency (∂/∂t ≪ y), the large east–west scale (∂/∂x ≪ ∂/∂y) approximation (2.2) can be simplified (Gill and Clarke 1974) to
b. Boundary conditions
Although the western equatorial Pacific has a gappy irregular boundary, on interannual and longer time scales, Clarke (1991) and Du Penhoat and Cane (1991) showed that dynamically this boundary behaves approximately like an infinite meridional wall. For such a boundary, the boundary condition for the large-scale interior flow (Cane and Sarachik 1977) is
Based on the same analysis as Clarke and Liu (1993) but now applied to the eastern equatorial Pacific, on interannual and lower frequencies the eastern boundary behaves like a meridional wall with boundary condition
3. The quasi-steady ocean response
Substituting this result into (3.3) gives
and hence, using the boundary condition (2.9) at x = L,
An interior ocean Sverdrup solution of the form (3.4) and (3.5) has been discussed previously (see, e.g., Kessler et al. 2003). However, what does not seem to have been appreciated is that, provided that the curl of the low-frequency wind forcing vanishes at long distances from the equator, this interior solution is the complete linear large-scale solution. This is because, when the wind stress has this property, the western boundary condition (2.8) is satisfied since
Notice that, if we integrate p(x, y) in (3.8) over the model domain, then
or, after an integration by parts,
This implies that, if X is of one sign, then the integral of p over the model domain is nonzero. But for, say, the first vertical mode p is proportional to sea level and thermocline displacement, so the lhs of (3.10) being nonzero would seem to imply that mass is not conserved! However, as our model ocean is infinite, our solution remains valid.
To understand this result in more detail, first recognize that the forced solution (3.4), (3.6), and (3.8) is also valid in a bounded basin provided that the wind forcing is negligible at the northern and southern boundaries. The problem with this solution is that, because of (3.10), mass is not necessarily conserved. However, we can fix this by adding the homogenous solution of (3.1)–(3.3), namely,
to the forced solution so that the lhs of (3.10) vanishes and mass is thus conserved. If the forcing region is small compared to the size of the ocean basin, then the constant pressure solution will be small compared to the forced solution. In the infinite basin limit the constant in (3.11) is zero and the forced solution (3.4), (3.6), and (3.8) is valid by itself. Further discussion of the bounded ocean response to wind forcing when the frequency is small but nonzero and the wind forcing is independent of longitude has been given by Emile-Geay and Cane (2009).
Equation (3.8) shows that p = 0 at x = L; that is, there is no disturbance in pressure and, therefore, in sea level and thermocline displacement at the eastern ocean boundary. In other words, according to this quasi-steady theory, quasi-steady El Niño conditions should not occur in the eastern equatorial Pacific. This result is consistent with the observed very low-frequency response of the equatorial Pacific pycnocline to equatorial wind forcing. Specifically, observational evidence suggests that the (westward) equatorial trade winds have been weakening since the 1970s (Clarke and Lebedev 1996; Bunge and Clarke 2009) and that the observed pycnocline placement in the eastern equatorial Pacific is much smaller than that in the western equatorial Pacific (see Fig. 3 of McPhaden and Zhang 2002). The small eastern, compared to western, equatorial Pacific thermocline response to very low-frequency (global warming time scale) equatorial wind forcing is also found by coupled ocean–atmosphere general circulation models (see Fig. 4 of Vecchi et al. 2006; Fig. 12 of Vecchi and Soden 2007). This result is not just of theoretical interest; it implies, for example, that the predicted further weakening of the equatorial trades during the twenty-first century (Vecchi and Soden 2007) will not result in long-term El Niño–like conditions and loss of fisheries in the eastern equatorial Pacific because the thermocline will not be significantly displaced there.
Although the above evidence suggests that the ocean response to wind forcing at very low global warming frequencies is quasi steady, this does not seem to be the case at higher decadal frequencies. Specifically, since decadal sea level oscillations of equatorial origin are observed on the eastern Pacific Ocean boundary (Clarke and Lebedev 1999), p is nonzero at x = L and the quasi-steady solution (3.8) is invalid. This is consistent with the analysis of Hasegawa and Hanawa (2003a), who present evidence for an unsteady decadal WWV mode. Does linear theory predict that even decadal frequencies are not low enough for a quasi-steady ocean response? In the next three sections, we will address this issue by constructing and analyzing a time-dependent low-frequency solution.
4. The low-frequency analytical solution
We will suppose that the zonal wind stress forcing is nonzero between x = a and x = b and that it is negligible poleward of a northern and southern latitude. The wind forcing will be taken to be proportional to exp(iωt); if desired we could sum all the Fourier components to obtain a solution for general low-frequency wind forcing.
Although the formulas that we will derive only require vanishing X at large latitudes, it is useful to obtain some explicit results using an idealized wind stress. Major features of the structure and time dependence of the low-frequency wind forcing in the equatorial Pacific can be approximately represented by the idealized wind forcing (see Figs. 3 and 4)
where x = a corresponds to 150°E and x = b to 140°W. The y dependence has an e-folding decay scale of 7° of latitude away from the equator with dimensional μ−1/2 = 7/2 ≈ 4.95 degrees of latitude. Notice that the function A in the region a ≤ x ≤ b can be written in the more compact form
is an eastward coordinate with origin centered at the center x = (a + b)/2 of the forcing.
Figures 3 and 4 show that the real world forcing has two extra contributions of opposite sign not represented in (4.1) and (4.2): a very small contribution in the eastern equatorial Pacific and another contribution about half the size of (4.1) centered at about 18°S instead of being centered on the equator. As the problem is linear, these contributions could be added in, if desired, but here we will just focus on the response to (4.1) and (4.2).
a. The solution east of the forcing region (b ≤ x ≤ L)
The only wave that carries energy east of the forcing region is the equatorial Kelvin wave. At the eastern boundary x = L the p and u fields for this equatorial Kelvin wave are of the form
where ψ0( y) = π−1/4 exp(−y2/2) is the zeroth-order Hermite function and B is a constant to be determined later in section 4d. Note that even though (4.3) looks as if it refers to a single equatorial Kelvin wave and the following text treats it this way, the amplitude B in (4.3) actually results from the sum of two equatorial Kelvin waves; one is generated by the wind forcing and propagates eastward from the forcing region, while the other results from Rossby wave reflection at the western boundary. At the eastern boundary the total Kelvin wave will, in general, not be in phase with the wind forcing, so B in (4.3) will be complex.
East of x = b there is no forcing, so the solution will consist of the sum of the equatorial Kelvin wave and its Rossby wave reflections. Cane and Moore (1981) showed that in this region at low frequencies the p, u, and υ fields are of the form
Physically (4.8) describes a pressure field propagating westward at a nondimensional speed y−2 (dimensional speed βc2f −2). This westward propagating wave is a long Rossby wave outside the equatorial waveguide but inside the equatorial waveguide, where the equatorial Kelvin wave is nonnegligible, the westward propagation results from the sum of the eastward propagating equatorial Kelvin wave and the westward propagating Rossby waves. This sum describes a wave whose westward speed increases as y → 0 until, at the equator, the speed is infinite, the adjustment is instantaneous, and pE is independent of x.
Since the Rossby wave westward propagation speed varies strongly with y as it propagates westward, meridional phase gradients develop and the meridional wavenumber is no longer negligible (as it is at the coast where geostrophy and vanishing u imply py = 0). The meridional wavenumber causes the Rossby waves to bend toward the equator, and the simple westward propagation described by (4.8) is generalized to the refraction pattern (4.4) (Schopf et al. 1981; Cane and Moore 1981).
b. The solution west of x = b (0 ≤ x ≤ b)
where the subscript F denotes the directly forced part of p, u, and υ west of x = b. Since p = pE, u = uE, and υ = υE for b ≤ x ≤ L,
c. Determination of the directly forced solution
a balance consistent with (4.13) near the equator. Since (4.13) seems valid for both large and small y, following Fedorov (2009, manuscript submitted to J. Climate; see also Liu 2003; Emile-Geay and Cane 2009), I will use it as the governing equation for the directly forced low-frequency flow. In appendix A I check that the error made in using (4.13) is small. Fedorov (2009, manuscript submitted to J. Climate) has previously used (4.13) to obtain expressions for the wind-forced ocean response to low-frequency wind and has also examined the coupled ocean–atmosphere ENSO problem analytically. Here we focus on the ocean response to the forcing described by (4.1) and (4.2).
For this forcing X is proportional to exp(iωt) and, as pF vanishes at x = b, the solution of (4.13) is
Since the pF field is known, the uF field can be determined from (2.7). We have
The directly forced pressure solution, pF in (4.15), is evaluated in appendix A for the X defined in (4.1) and (4.2). In appendix A the pF in (4.15) is the lowest-order term p(0) of a perturbation expansion and as such is given by (A3), (A5), and (A7). Since pF has been determined, the velocity field υF can be found from (4.17) and the uF field either from (2.7) or by using the formulas (4.1) and (4.2) for X in (4.16).
d. Determination of the Kelvin wave amplitude B
The above analysis has determined the directly forced solution pF, uF, and υF. Our complete solution (4.9) would be known if we knew the Kelvin wave amplitude B that enters pE, uE, and υE [see (4.4)–(4.6)]. This amplitude can be determined from the one remaining boundary condition, the western large-scale boundary condition (2.8). Using (4.9), this boundary condition can be written
we have that
where uF is evaluated at x = 0.
When the first term on the rhs of (4.23) is integrated with respect to y by parts, it cancels with the second term so that
It will prove convenient to define
so that, by (4.32), pE can be written
5. Equatorial tilt and warm water volume modes
and with k and ξ defined in (4.2d,e).
with V(x) defined in (4.35), Q(x) in (5.3), and γ + iσ in (4.31). Equation (5.4) shows that the equatorial pressure field can be regarded as the sum of two “modes.” One mode, represented by T(x)eiωt, is in phase with exp(iωt) and therefore is in phase with the wind forcing. This mode can be called the “tilt” mode because the equatorial wind stress zonally tilts the equatorial pressure field of this mode in the forcing region a ≤ x ≤ b. Specifically, using the results [see(4.2c) and (5.3)] that dQ/dx ≡ A and that the slope of V(x) is gentle at ENSO and lower frequencies [see (4.35) with 2ω (x–L) small], we have
Figure 5 shows that at ENSO and decadal frequencies, the structure T(x) = Q(x) + γV(x) of the tilt mode pressure field changes sign, being positive in the eastern Pacific and negative in the western Pacific. Since Q(x) is associated with the forced solution pF, it is zero in the unforced eastern Pacific (b ≤ x ≤ L). An eastward wind stress (X > 0) tilts the sea level (and pressure) up toward the east, so, because the quasi-steady function Q(x) is zero east of x = b, it must be negative in the central and western Pacific (0 ≤ x ≤ b). An eastward wind stress generates eastward equatorial flow and, by geostrophy, an equatorial Kelvin wave with positive sea level and pressure in the eastern equatorial Pacific. Consistent with this, γV(x) > 0 [γ is positive and of order 1 at low frequencies and V(x) > 0 from (4.35)]. Thus, we see how it is that T(x) changes sign—a positive γV(x) is added to Q(x), which is negative in the western and central Pacific (0 ≤ x ≤ b) and zero in the eastern Pacific (b ≤ x ≤ L). Since V(x) decreases like ω1/2 as ω → 0, so does the positive portion of T(x) until, in the zero frequency limit, T(x) ≤ 0. We see this tendency in Fig. 5e—the decadal portion of positive T(x) is smaller than that of the interannual frequency.
The other mode in (5.7) is S(x)ieiωt. Being proportional to ieiωt, it leads the wind stress by a quarter of a period or, equivalently, −p for that mode lags the wind stress by a quarter of a period; that is, −pt for that mode is in phase with the wind stress. Because S(x) is proportional to V(x), this mode is due to pE [see (4.34)] and wave energy resulting from boundary reflection. Because pressure is proportional to the thermocline depth and V(x) is approximately constant at low frequencies (see Fig. 6), this mode contributes to the discharge and recharge of equatorial WWV.
(i) Like (5.7), the central equatorial Pacific tilt in Fig. 1 is in phase with the westerly equatorial wind anomalies as it is in phase with Niño-3.4, which is in turn in phase with these wind anomalies. The central equatorial thermocline tilt is as one might expect physically—westerly wind anomalies (positive Niño-3.4) push water eastward, resulting in an eastward increase in the thermocline depth (Fig. 1) and sea level anomaly (Fig. 7). The sharp tilts near the eastern and western boundaries seen in Fig. 1 are not consistent with (5.7) because the zonal wind anomalies there are comparatively small. As noted by Clarke et al. (2007), these severe near-boundary tilts are not seen in Fig. 7 in the observed sea level anomaly. Clarke et al. suggested that the discrepancy occurred near the eastern boundary because there the 20°C isotherm depth is not a good proxy for adiabatic variability of the thermocline depth.
(ii) The second EOF in Fig. 2 is qualitatively similar to the theoretical WWV mode in two key aspects: the mode is of one sign in longitude and −∂/∂t (second EOF) is proportional to Niño-3.4 and therefore is in phase with the wind stress. This behavior has been documented earlier on the ENSO time scale by Hasegawa and Hanawa (2003b) and approximately on the decadal time scale by Hasegawa and Hanawa (2003a).
b. Physics of the WWV mode
To understand how the disequilibrium WWV arises, first consider the quasi-steady (ω → 0) limit. Suppose a (say) positive wind stress curl exerts a torque on the ocean in the region a ≤ x ≤ b, increasing its angular momentum therein. Ocean particles increase their angular momentum by increasing their planetary vorticity by moving northward—that is, the Sverdrup balance (3.4) holds. As wind stress curl changes with latitude, the northward transport varies meridionally, so the meridional divergence υy is nonzero in the wind forcing region. Because the flow is horizontally nondivergent in the zero frequency limit [see (3.3)], ux must also be nonzero in the forcing region. However, ux is zero outside that region and, in fact, there is no zonal gradient anywhere outside the forcing region. Physically, as the forcing frequency ω → 0, all waves propagate, reach boundaries, reflect, and adjust the ocean before the forcing changes; the ocean response is quasi steady.
Now consider what happens when the frequency is small but nonzero. Because Rossby wave speed depends on latitude, when Rossby waves generated in the forcing region arrive at the western boundary and have small but finite frequency, they are not in phase and the western boundary condition (2.8) is not satisfied. This results in a reflected equatorial Kelvin wave that travels to the eastern boundary, reflects, and propagates back to the western boundary again with a strong latitudinal-dependent lag [see (4.4) and (4.7)]. When this signal reaches the western ocean boundary, its reflection further contributes to the disequilibrium of the equatorial Kelvin wave and hence the disequilibrium WWV.
We can verify the above physics mathematically by first noting [see (5.4) and (5.6)] that the disequilibrium WWV ieiωtσV(x) is proportional to the imaginary part of the factor γ + iσ in the equatorial Kelvin wave amplitude B [see (4.30)]. The amplitude B is in turn determined from the two meridional integrals in the western boundary condition (4.18). Examination of those integrals [see (4.19), (4.24), (4.1), and (4.2)] shows that they have a common factor i exp(iωt). Consequently, the imaginary part of B results from the factors exp(−iωy2ζ) and exp(−0.5iy2 tan2ωL) in the integrands of those integrals. In other words, it is the strong y dependence of westward propagation speed discussed earlier with reference to (4.4) and (4.8) that leads to the nonzero imaginary part of B and hence the disequilibrium WWV.
As pointed out in section 5a, the WWV disequilibrium mode is associated with pE and wave energy resulting from boundary reflection. This immediately implies that the meridional transport mechanism discussed in the introduction is not the major disequilibrium WWV mechanism because the meridional mechanism does not depend on ocean boundary reflection of wave energy.
The foregoing theory assumed that there was no dissipation of energy. In particular, it is assumed that the Rossby waves cross the entire Pacific without dissipation but this is not the case. It has been known for some time that strong dissipation of large-scale low-frequency energy occurs in the interior ocean and approximate dissipation rates have been estimated (e.g., Picaut et al. 1993; Qiu et al. 1997; Fu and Qiu 2002; Vega et al. 2003; Fedorov 2007). Recently Li and Clarke (2007) showed that south of about 5°S strong dissipation of the reflected low-frequency ENSO signal even occurs within about 100 km of the eastern Pacific Ocean boundary and, therefore, certainly does not reach the western ocean boundary. On the other hand, the analysis of Bosc and Delcroix (2008) suggested that the net meridional transport entering the 5°S–5°N latitude band propagates westward right across the Pacific, suggesting that dissipation is small close to the equator.
To estimate approximately how much dissipation influences the size of the equatorial WWV and the tilt mode, we consider simple linear dissipation by introducing damping terms ru and rp into the left-hand sides of the governing equations (2.1) and (2.3), respectively. Then, with forcing proportional to exp(iωt) [see (4.1)], the solution to our governing equations (2.1), (2.3), and (2.7) will be the same as before but with (iω + r) replacing iω everywhere except in the time-varying function exp(iωt). In other words, except in exp(iωt), our present solutions remain valid with i[ω − ri] replacing i[ω]—that is, ω − ri replacing ω.
Since ω is complex, the expressions for the real structure functions T(x) and S(x) in (5.5) and (5.6) change. The parameters γ and σ are still real and are still determined by (4.31) but now depend on the damping time scale as well as ω. In addition, as V(x) is complex when ω is replaced by ω − ir, we write
where VR(x) and VI(x) are the real and imaginary parts of V(x), being determined from the rhs of (4.35) with ω replaced by ω − ir. From (5.1), (5.2), and (6.1), the expressions for the real functions T(x) and S(x) in (5.4) now become
Estimates of the damping time scale r−1 vary from six months to a year (Picaut et al. 1993) to a few years (Fedorov 2007). Figures 8 and 9 show the dimensional tilt and disequilibrium WWV structure functions T(x) and S(x) at the interannual and decadal frequencies for damping time scales r−1 = 6 months, and 1 and 2 years, and the undamped case. The effect of the damping is qualitatively as one might expect. Higher damping should lead to a response more in phase with the wind stress, this tending to decrease the amplitude of the disequilibrium WWV mode (Fig. 9). The amplitude of the tilt mode is only slightly affected by damping (Fig. 8). At the decadal frequency, the tilt mode amplitude in the eastern Pacific increases with increased damping as the water is pushed eastward in phase with the wind stress.
7. Concluding remarks
Analytical theory has been used to examine the linear equatorial ocean response to low-frequency, large zonal-scale wind forcing. Qualitatively consistent with observations, the ocean response consists of a “tilt” mode in phase with the equatorial wind stress forcing and a WWV mode that lags the wind stress forcing by a quarter of a period. Ocean boundary reflection and long westward propagating Rossby waves whose speed is strongly dependent on latitude are crucial to the existence of the disequilibrium WWV mode. Because the long Rossby waves at higher latitudes travel so much more slowly, the energy reflecting from the western ocean boundary is not in phase, and this leads to a reflected equatorial Kelvin wave and WWV that is not in phase with the wind stress forcing. As the frequency ω → 0, the disequilibrium WWV mode amplitude → 0 like ω1/2, leaving a linear Sverdrup ocean solution that satisfies the large-scale linear western boundary condition of no net mass flux into the western ocean boundary layer [see (2.8)].
For the ω → 0 quasi-steady solution there is no thermocline response in the eastern equatorial Pacific. The theory therefore predicts that, for the eastern equatorial Pacific, the gradual weakening of the equatorial trade winds since the 1970s that has been observed (Clarke and Lebedev 1996; Bunge and Clarke 2009) and has been associated with global warming (Vecchi and Soden 2007; Bunge and Clarke 2009) will not lead to long-term El Niño conditions and a loss of fisheries there. This zero thermocline response prediction is supported by both observational results (see Fig. 3 of McPhaden and Zhang 2002) and the results of coupled ocean–atmosphere general circulation models (see Fig. 4 of Vecchi et al. 2006; Fig. 12 of Vecchi and Soden 2007).
More realistic ocean dissipation, mean equatorial currents, and a gappy and irregular western ocean boundary will affect the analytical results. But an understanding of the linear theory is useful, because it is at least qualitatively consistent with observations and provides a structure from which we can learn the true dynamics.
I gratefully acknowledge the support of the National Science Foundation (Grants ATM-0623402 and OCE-0850749). I also thank Stephen Van Gorder for peforming the numerical calculations and reading the manuscript.
Estimation of the Error in the Directly Forced Solution
We will estimate the error in the directly forced solution using a perturbation approach, writing
with boundary condition p(1) = 0 at x = b.
where q satisfies the ordinary differential equation
subject to vanishing q at x = b.
The solution of (A.4) is
whereas in 0 ≤ x ≤ a west of the region of forcing, where the rhs of (A.4) is zero,
or, using (A.5),
In the main text we only use the lowest-order solution, so p(0) is written as pF there. Thus, uF in (4.16) is really u(0), so we can estimate the rhs of (A.2) using (4.16) with X given in (4.1) and (4.2). The solution for p(1) satisfying (A.2), subject to p(1) = 0 at x = b, can then be obtained.
The first-order correction p(1) is largest at the western boundary x = 0. Therefore, we can examine the maximum size of the error in approximating p by p(0) by comparing the absolute values of p(0) and p(1) at x = 0. Figure A1 shows the amplitude of the dimensional sea level at the western ocean boundary corresponding to p(0) (solid line) and p(1) (dashed line) for the first and second vertical modes at interannual and decadal frequencies. The results were obtained using the idealized forcing in Fig. 4 and the vertical mode parameters of Table A1. In all cases the first-order solution corresponding to p(1) is much smaller than the zeroth-order solution corresponding to p(0).
Evaluation of at the Western Boundary
we have, from (B.1), that
Changing the integration variable to ξ [see (4.2e)] gives
so that in the integral in (B.4)
Since is typically small over the range of ξ in the integral in (B.4), we may expand binomially to obtain
Upon substituting (B.7) into the integral and using the results
Since k = π/(b − a), we may also write (B.12) as
Corresponding author address: Allan Clarke, Department of Oceanography, The Florida State University, Tallahassee, Florida 32306-4320. Email: email@example.com