1. Introduction
In agreement with basic linear large-scale, low-frequency dynamics, Clarke and Dottori (2008, hereafter CD08; see also Dottori and Clarke 2009), showed that observed coastal sea level and dynamic height and satellite sea surface height (SSH) propagate westward from the California coast like nondispersive Rossby waves (RWs). This interannual and decadal variability in sea level and dynamic height is mostly of equatorial origin via the ocean coastal waveguide (Enfield and Allen 1980; Chelton and Davis 1982; Clarke and Lebedev 1999; Kessler 1990; Frischknecht et al. 2015), and huge interannual and decadal fluctuations in the California Current ecosystem are intimately connected with the equatorial signal (Chelton et al. 1982; Clarke and Lebedev 1999; CD08; Dottori and Clarke 2009; Frischknecht et al. 2015). Based on the RW westward propagation physics, CD08 used the lead time of the monthly coastal sea level to show that monthly San Diego negative sea level anomalies could predict the log of the California zooplankton 2–4 months in advance with statistically significant correlations greater than 0.6. Remote equatorial forcing and RW dynamics is thus an essential mechanism in the prediction of the large-scale California Current physics and ecosystem.
Although the independent SSH and California Cooperative Oceanic Fisheries Investigation (CalCOFI) dynamic height and coastal sea level observations seemed to agree well with the theory in that westward long RW propagation was observed, at 4.2 cm s−1 the standard first vertical mode RW speed is about double that expected. Furthermore, CD08 found that even when the standard linear theories were modified to include zonal mean flow (Killworth et al. 1997) or coupled with topography (Killworth and Blundell 1999, 2003), the resultant wave speed estimates were still far too low (1.6–2.3 cm s−1). Tailleux and McWilliams (2001) suggested that if somehow the deep ocean could be “decoupled” from the upper ocean, then the RW speed could be adjusted to agree with observations, but it was not clear to CD08 how this “bottom pressure decoupling,” or p′ = 0 bottom boundary condition could be justified. Tailleux and McWilliams (2001) proposed that standard constant-depth flat-bottom vertical modes under the influence of bottom topography or a mean flow may decorrelate and give rise to zero bottom pressure. However, LaCasce and Groeskamp (2020) pointed out that bottom pressure decoupling seems conceptually problematic, since it depends on the fast barotropic mode interacting with a much slower first baroclinic flat bottom mode. In addition, although a barotropic mode and a first baroclinic mode can be added together so that their bottom pressures add to zero, the resulting structure does not agree with observations as well as the vertical structure of a first baroclinic mode with p′ = 0 replacing vanishing vertical velocity on the ocean floor (see, e.g., Figs. 2 and 3 in de La Lama et al. 2016).
Negligible horizontal flow and p′ = 0 at the ocean floor rather than vanishing vertical velocity are consistent with observations in other far more general contexts than California (Wunsch 1997; Wortham and Wunsch 2014; de La Lama et al. 2016). Various explanations for negligible flow and bottom pressure have been offered, including distortion of the pressure and horizontal flow by the mean flow (Killworth et al. 1997; Brink and Pedlosky 2020), dissipation of the large-scale flow by bottom friction (Brink and Pedlosky 2018), modification of the bottom boundary condition by large-scale bottom slope (Killworth and Blundell 1999; LaCasce 2017), or rough bottom topography (Samelson 1992; Bobrovich and Reznik 1999; LaCasce and Groeskamp 2020). In theory rough bottom topography with bottom slope greater than about 10−3 causes the large scale flow to vanish and p′ = 0 at the bottom, and global estimates of bottom slope for topography at 1-min horizontal resolution (see Fig. 12 of de La Lama et al. 2016) show that indeed small scale bottom slope does exceed 10−3 for almost all of the ocean floor, including the region of interest off California. Furthermore, both Tailleux and McWilliams (2001) and LaCasce and Groeskamp (2020) show that the p′ = 0 bottom boundary condition does lead to Rossby wave phase speed increases comparable to the observed increases.
However, most of the low-frequency variability in the ocean is forced by the wind at the sea surface, and the variability seen at the bottom does not reach the bottom instantly or locally. In that case the bottom pressure perturbation and associated flow generated by the wind would be zero at the bottom and p′ = 0 would be an appropriate bottom boundary condition. We suggest that this is the case for the interannual Rossby wave signal off California. Specifically, we present theory showing that the wind-driven interannual equatorial El Niño ocean signal does not reach the ocean floor by the time it reaches the eastern boundary, and that therefore there is no remote wind-driven energy at the deep ocean floor. Consequently, p′ = 0 at the eastern boundary ocean floor at the equator. At these low frequencies the flow is quasigeostrophic, and, since there is no flow into the eastern boundary, p′ is constant spatially along the boundary and so vanishes at the bottom at California.
Tailleux and McWilliams (2001) and LaCasce and Groeskamp (2020) showed that globally the p′ = 0 bottom boundary condition results in increased long RW speeds, and in the deep sea off California the speeds are approximately in agreement with the observed CD08 speeds. For the case of constant depth water, long Rossby waves propagate westward at a speed proportional to (f)−2, which implies that over the California Current region stretching from about 27° to 39°N the long RW speed increases southward by a factor of 1.9. In addition, in the southern part of the California Current System (CCS) the bottom topography is shallower near the coast than in the northern part, and this may also affect the RW speed measured from the coast asymmetrically. To examine in more detail how the RW speed varies over the CCS region, we decided (section 2) to update the CD08 analysis with higher-resolution calculations of the locally constant depth RW speeds in both the w′ = 0 and p′ = 0 cases.
As has already been discussed and shown theoretically and observationally in the literature, the large-scale low-frequency signal south of about San Francisco is of equatorial origin (Enfield and Allen 1980; Chelton and Davis 1982; Clarke 1983). We briefly review this result with up-to-date satellite SSH data in section 3 and confirm that, within 100 km of the coast from 35°S to 35°N, the deep-sea interannual equatorial signal dominates the SSH and therefore the near-surface pressure signal. The equatorial theory and physical discussion in section 4 suggest that this low-frequency surface signal should propagate downward at such a shallow angle that there is no baroclinic bottom pressure at the eastern ocean boundary, and that therefore that p′ = 0 rather than w′ = 0 is a more efficient bottom boundary condition to use for the vertical modes in this case. In section 5, we discuss why the first baroclinic vertical mode should dominate the wind-driven ocean response. Section 6 briefly examines the applicability of our results to regions other than California, and concluding section 7 summarizes the main results.
2. Long Rossby wave speed estimates off California
To examine the influence of p′ = 0 on the bottom with higher resolution than previous analyses, including the effect of shallower water on the bottom boundary condition, we calculate theoretical long RW speeds in the same way as CD08 with updated data. Climatological values of
Locally computed nondispersive long RW speeds for the first baroclinic mode are displayed in Fig. 1 within the CCS (27°–37°N, 115°–125°W) for depths greater than 3 km under the conditions (i) w′ = 0 at z = −H, (ii) w′ = 0 at z = −H with
Contour plot of the local first baroclinic mode Rossby wave speed in deep waters (>3 km) under the conditions (a) w′ = 0 at z = −H, (b) w′ = 0 at z = −H with
Citation: Journal of Physical Oceanography 54, 3; 10.1175/JPO-D-23-0024.1
Average theoretical wave speeds for the California Current region were estimated in a similar way to CD08 by grouping and averaging theoretical lags in 130-km zonal distances found from local first vertical mode wave speeds, and then estimating the average speed via linear regression (see Fig. 2). Linear regressions of these lags and distances for either the w′ = 0 and p′ = 0 bottom boundary conditions give the estimated zonal theoretical speeds summarized in Table 1 and Fig. 2. Figure 2a displays the regression estimate of the longwave RW speed when lags for the first 260 km are excluded. When w′ = 0 is the bottom boundary condition, the estimated RW speed is 2.41 cm s−1 (green line) for
Theoretical Rossby wave speeds for the first baroclinic mode for (a) the exclusion (“Standard”) and (b) inclusion (“Shallower Lags Included”) of grouped lags for the first 260 km from the coast. In each panel, zonal distance is regressed on theoretical lags grouped by 130-km zonal distance increments away from the California coast based on either bottom boundary condition w′ = 0 and p′ = 0 with and without baroclinic zonal mean flow inclusion (Killworth et al. 1997). Vertical error bars encompass the 130-km zonal distance per increment, while horizontal error bars are a standard deviation of the lags used to average within each increment. Note: the 95% confidence intervals found in Table 1 capture the variability depicted by the error bars on the wave speed estimates.
Citation: Journal of Physical Oceanography 54, 3; 10.1175/JPO-D-23-0024.1
Summary of estimated westward-propagating nondispersive Rossby wave speeds (cm s−1) in the CCS (27°–37°N, 115°–125°W) for the first baroclinic mode for both the standard vanishing vertical velocity (w′ = 0) and vanishing baroclinic pressure (p′ = 0) bottom boundary conditions for both zero and nonzero mean zonal flow. Theoretical estimates were computed using the results shown in Fig. 1 and a linear regression analysis of grouped monthly lags in 130-km zonal distance increments from the coastline in the same manner as CD08 (Fig. 2). Observed wave speeds were obtained from hydrographic-derived dynamic height and satellite altimetry by CD08. The inclusion of a truncated
The choice to exclude theoretical lags within 260 km from the coast, as did CD08, is due to the prominence of shallow waters (<3 km) within 30°–35°N (Fig. 1). This region was included in CD08’s RW speeds estimated from the satellite altimetry and hydrographically derived dynamic height observations, but theoretical calculations with that region included only slightly changed the results. Specifically, under the p′ = 0 bottom boundary condition the estimated RW speed is 4.08 and 4.33 cm s−1 with and without
Averaged zonal geostrophic velocity in the CCS (27°–37°N, 115°–125°W) based on WOA13-V2 climatological data for the years 1955–2012 for depths greater than or equal to 3 km.
Citation: Journal of Physical Oceanography 54, 3; 10.1175/JPO-D-23-0024.1
3. The eastern Pacific Ocean boundary signal at interannual and lower frequencies
a. California Rossby waves are predominately of equatorial origin
Figure 4 shows that interannual anomalies in coastal sea level at San Diego, California (32.7°N, 117.2°W), to be strongly correlated with the sea level at Santa Cruz Island, Ecuador (0.6°S, 90.3°W), and the Niño-3.4 SST index, respectively. Past work has also documented that interannual and lower-frequency coastal sea level signals from the equator are prominent along the eastern Pacific boundary in both Southern and Northern Hemispheres (Bretschneider and McLain 1976; Chelton 1980; Enfield and Allen 1980; Chelton and Davis 1982; Strub and James 2002).
Monthly time series for interannual anomalies for coastal sea level measurements at San Diego, California (blue line; 32.7°N, 117.2°W), Santa Cruz, Ecuador (green line; 0.6°S, 90.3°W), and interannual anomalies of the Niño-3.4 SST index (averaged over the box 5°S–5°N, 170°–120°W). Each time series is created by removing the seasonal cycle and filtered using an 11-point symmetric interannual low-pass filter (Trenberth 1984). Each time series is continuous from January 1970 to December 2016 except for the Santa Cruz station where infrequent sampling becomes commonplace after May 2003. The maximum correlation between every time series occurs at lag-zero and is r = 0.72 [rcrit(99%) = 0.48] between San Diego vs Santa Cruz sea levels, r = 0.75 [rcrit(99%) = 0.42] for San Diego sea level vs Niño-3.4 SSTs, and r = 0.74 [rcrit(99%) = 0.49] for Santa Cruz sea level vs Niño-3.4 SSTs. All critical correlation coefficients, here and elsewhere, are computed using Ebisuzaki’s (1997) nonparametric statistical significance test.
Citation: Journal of Physical Oceanography 54, 3; 10.1175/JPO-D-23-0024.1
At interannual frequencies the no normal flow boundary condition implies that the quasigeostrophic deep-sea pressure signal should be spatially constant and in phase along the boundary, so that an empirical orthogonal function (EOF) analysis of the SSH in the deep sea near the coast should result in complete dominance by the leading EOF whose spatial structure function would be constant along the deep-sea eastern boundary (Clarke and Shi 1991). We tested this using zonally averaged (1° longitude) JPL Making Earth Science Data Records for Use in Research Environments (MEaSUREs) SSH monthly anomalies off the eastern Pacific boundary from 35°S to 35°N (Fig. 5, top panel). Figure 5 shows that the first EOF does indeed dominate with 86% of the explained variance (Fig. 5, middle), but the structure function shows that the equatorial signal has an overall tendency to decrease poleward. This is due to several factors, including the increasing influence of other secondary signals like alongshore wind forcing (Enfield and Allen 1980), variations in coastline direction (Clarke and Shi 1991; Li and Clarke 2004), and dissipation of the large-scale signal due to eddies (Giunipero and Clarke 2013). Even though the EOF structure function is not spatially constant but decays poleward, it is clear from the leading EOF that the deep-sea signal near the surface and coast is dominated by the equatorial signal. Consistent with previous work, Fig. 5 (lower panel) shows that the principal component time series for the eastern boundary sea level signal is strongly correlated with the El Niño SST index Niño-3.4 {r = 0.81 [rcrit(99%) = 0.52]}. In other words, in agreement with theory and past analyses there is a strong, baroclinic interannual El Niño pressure signal at the surface that reaches California.
(top) Hovmöller time series of filtered (Trenberth 1984) interannual JPL MEaSUREs (Zlotnicki et al. 2019) SSH anomalies (cm) zonally averaged 1° off the eastern Pacific boundary for the latitudinal range 35°S–35°N. Our zonal average was taken to be 1° off the coast to capture the boundary signal. (middle) The primary EOF structural function of the zonally averaged interannual JPL SSH with percentage of explained variance (PEV = 86%). (bottom) Time series of the first principal component (blue line) and Niño-3.4 (orange line) with a maximum autocorrelation at lag 0 at r = 0.81 [rcrit(99%) = 0.52]. An interannual 11-point symmetric low-pass filter (Trenberth 1984) was utilized to capture the dominant interannual signal. This EOF analysis is an extension of Clarke (1992) who used 4 coastal stations from 15°S to 10°N to argue coastal pressure is spatially constant at low frequencies.
Citation: Journal of Physical Oceanography 54, 3; 10.1175/JPO-D-23-0024.1
b. Westward propagation of the boundary signal
Coastal interannual and lower-frequency sea level signals of equatorial origin have been observed to propagate offshore like long RWs as far south as Chile (Vega et al. 2003; Challenor et al. 2004; Ramos et al. 2008; Vergara et al. 2017) and as far north as California (Herrera Cervantes and Parés-Sierra 1994; Ramp et al. 1997; Fu and Qiu 2002; CD08; Frischknecht et al. 2015). Such a large-scale nondispersive westward-propagating signal from an eastern boundary in the form of RWs is in accordance with low-frequency theory (Cane and Sarachik 1977; McCreary 1977; Schopf et al. 1981; Cane and Moore 1981; Clarke 1983, 1992; Grimshaw and Allen 1988; Clarke and Shi 1991), which states that RWs are reflected wave solutions of incident longwave equatorial Kelvin waves (EKWs) at interannual frequencies. Additionally, it has been argued that local alongshore wind stress, especially in California (Clarke and Lebedev 1999; Clarke 1992; Dottori and Clarke 2009; Frischknecht et al. 2015) is a secondary contributor to the interannual coastal sea level signal compared to the boundary signal generated remotely at the equator.
Previously it has been assumed that this interannual pressure signal has the vertical structure of a standard first baroclinic mode with w′ = 0 as a deep-sea bottom boundary condition. But in section 2 we confirmed that the California RW propagation from the coast is far better described by a first vertical mode with p′ = 0 as the bottom boundary condition. To understand the vertical structure of this signal, we next consider its origin.
c. Physics of the interannual equatorial eastern boundary deep-sea pressure signal
The only way low-frequency, large-scale energy can reach the eastern equatorial Pacific boundary is via EKWs, since these are the only large-scale waves with eastward group velocity. Such waves are generated by low-frequency zonal equatorial wind stress surface forcing west of about 150°W (see Fig. 6), and so east of 150°W the eastward group velocity signal can be approximated as freely propagating EKWs. In fact, analysis shows (see, e.g., Clarke 1992) that the eastern boundary sea level is
(top) Regression of anomalous July–November surface wind (arrows) and outgoing longwave radiation (OLR) onto an equatorial SST anomaly index (anomalous July–November SST averaged over 6°S–6°N, 180°–80°W). Wind vectors are only shown for those grid points whose u or υ correlations with the SST index exceed 0.4 in absolute value. The OLR contour interval is 10 W m−2 °C−1 of the SST index. The zero contour is darkened, and the positive contour is dashed. Values less than −20 W m−2 °C−1 are shaded. The wind regressions are based on the period 1946–85 and the OLR regression on 1974–89 (1978 missing). (bottom) As in the top panel, but for December–February [redrawn from Deser and Wallace (1990)].
Citation: Journal of Physical Oceanography 54, 3; 10.1175/JPO-D-23-0024.1
d. Checking p′ = 0 at the deep-sea eastern equatorial ocean bottom boundary
We computed p′ by integrating the hydrostatic balance
To check the validity of the equatorial p′ = 0 bottom boundary condition near the deep-sea eastern equatorial boundary, we estimated equatorial p′(z, t) near the eastern equatorial Pacific boundary at two deep-sea equatorial longitudes 96° and 84°W using the above dataset. We chose longitude 96°W based on its deep-sea location and proximity to the TAO/TRITON equatorial buoy at 95°W, and longitude 84°W because it is closer to the eastern boundary and its depth is similar. An EOF analysis of the p′(z, t) data at each location showed that in both locations more than 93% of the variance was described by the first EOF mode, that is, p′(z, t) ≈ EOF1(z)φ1(t) where φ1(t) is the first principal component. Since the first EOF is such an excellent approximation to p′(z, t) at both locations, we can check the validity of the theoretical bottom boundary conditions p′ = 0 and w′ = 0 using the EOF1(z)φ1(t) approximation for p′(z, t). This implies that we can use the vertical structure function EOF1(z) to assess the vertical structure of p′(z, t). Figure 7 shows (see the blue curve in each panel) that, in both locations, EOF1(z) has a very large amplitude near the surface that decreases rapidly with depth over the top few hundred meters and is negligible at the ocean floor. This suggests that the p′ = 0 bottom boundary condition is an excellent approximation.
First vertical EOF structure functions for the interannual pressure anomalies p′(z, t) in the equatorial Pacific at 96° and 84°W from January 1996 to July 2012. Also shown are the theoretical first vertical mode structure functions for the p′ = 0 (red) and the w′ = 0 (green) bottom boundary conditions. These theoretical modes were computed using WOA13-V2 climatological data. All plots are nondimensional and the theoretical modes are a best fit to the EOF structure (see text).
Citation: Journal of Physical Oceanography 54, 3; 10.1175/JPO-D-23-0024.1
Calculations of the first vertical mode theoretical structures for the interannual pressure are shown in the right-hand side panels of Fig. 7 for the w′ = 0 bottom boundary condition (green curve) and p′ = 0 bottom boundary condition (red curve). Neither theoretical mode structure compares well with EOF1(z) in the upper part of the water column, and this is to be expected since the interannual variability there is associated with a shallow equatorial Kelvin wave beam of energy described by many vertical modes (see section 4). However, at the bottom the w′ = 0 first mode structure (green curves) is a poor match with EOF1(z) compared to the p′ = 0 first mode structure (red curves).
The above analysis has shown that to a first approximation the interannual and lower-frequency equatorial deep-sea coastal boundary pressure signal penetrates north and south along the boundary, and in the Northern Hemisphere reaches the California Current region. At the equator the assimilated data suggests that the signal has p′ = 0 at the deep-sea floor at the eastern ocean boundary, and furthermore that the interannual pressure field at the boundary is better approximated by the first vertical mode with p′ = 0 rather than w′ = 0 as the bottom boundary condition. The dominance of the p′ = 0 first vertical mode is consistent with the dominance of this mode at the deep-sea boundary and the interannual and lower-frequency RW speeds observed off California. To complete the proposed mechanism, it remains to confirm the physics of section 3c using some equatorial wave calculations, which will show that wind-driven energy at ENSO frequencies at the equator does not reach the ocean floor before it reaches the eastern ocean boundary. We will discuss this next in section 4.
4. Low-frequency wind-forced equatorial dynamics and vertically propagating waves
a. Theoretical background
When the equatorial deep ocean is forced at the surface by the wind, the barotropic response is negligible because the barotropic response involves moving the total ocean water column, so much deeper than the mixed layer depth. The depth-dependent baroclinic energy, which dominates the ocean response, does not instantly reach the bottom, but propagates vertically. To model this simply, and so estimate whether this energy reaches the ocean bottom before it strikes an eastern or western ocean boundary, we use the linear, wind-forced, continuously stratified, equatorial ocean model of Gill and Clarke (1974) that is derived and summarized in Clarke and Liu (1993) and chapter 5 of Clarke (2008). The solution consists of a sum of a complete set of vertical modes Fn(z) that satisfy the linearized surface condition w′ = ηt and the bottom boundary condition w′ = 0 on the constant depth ocean floor z = −H. As mentioned above, the barotropic (n = 0) mode contributes negligibly to the wind-forced response, and here we concentrate on the baroclinic modes for which w′ = 0 (and consequently Fnz = 0) at the free surface z = 0 is an excellent approximation. Note that in this paper the subscript n on a variable refers to the nth baroclinic mode and the x, y, z, and t subscripts to differentiation by the x, y, z, and t variables, respectively. These variables have their usual coordinate meanings of distance eastward, distance north of the equator, distance upward from the ocean at rest and time, respectively.
b. Application to the equatorial Pacific
c. Summation of vertical modes to obtain EKW rays
Note from (4.24) for shallower depths and j = 0, x = S/ω − a is small enough for singularities to occur before the eastern equatorial Pacific boundary. From (4.23) this corresponds to a vertical phase propagation proportional to exp(iωt + iS/c1), that is, from the definition of S, upward phase propagation at the WKB slowly varying speed c1ω/N(z). Since the phase propagation is also c1 eastward, the phase velocity is upward and eastward with slope +ω/N(z) to the horizontal.
Figure 8 shows the ray paths of EKWs emitted at x = 0 (150°W). We assumed that at x = 0 the EKWs begin their journey at 50-m depth, the approximate climatological depth of the surface (mixed layer Ando and McPhaden 1997; de Boyer Montégut et al. 2004; Zhang and Clarke 2015). The plot for ω = 2π/2 years (solid white line) shows that even at this “high” interannual frequency the EKWs beginning at x = 0 only reach 280-m depth. For lower interannual frequencies the water depths are even shallower, far above the ocean floor. For example, the dotted white line in Fig. 8, right above the ω = 2π/2 years ray path, represents the ray path for ω = 2π/3 years, which has incidence at about 190-m depth. Therefore, it is not surprising that the p′ = 0 bottom boundary condition for the baroclinic modes is valid at interannual frequencies. This is consistent with other results that show that wind-forced vertically propagating baroclinic EKWs approximately follow the angle ω/N(z) from the surface (see, e.g., McCreary 1984; Dewitte and Reverdin 2000; Kutsuwada and McPhaden 2002; Rydbeck et al. 2019) despite experiencing variable wind forcing and complex ocean currents before reaching the eastern ocean boundary. It is also consistent with the results obtained from the assimilated data in section 3d.
Integrated EKW WKB ray paths beginning at the climatological mixed layer depth of 50 m at 150°W (de Boyer Montégut et al. 2004; Zhang and Clarke 2015). Shaded contours in the background represent the climatological buoyancy frequency as a function of depth and longitude averaged across the latitude band 2°S–2°N. The longitudinal domain extends from the eastern edge of interannual wind forcing (150°W) in the equatorial Pacific associated with El Niño events (see Fig. 6, top panel) to its eastern boundary (80°W). WKB ray paths reveal the depth of penetration for EKWs, whether forced in the interior or emitted from the western boundary, with dependence on angular frequency: ω = 2π/2 years (solid white line) and ω = 2π/3 years (dotted white line).
Citation: Journal of Physical Oceanography 54, 3; 10.1175/JPO-D-23-0024.1
5. Finite-thickness mixed layer and the dominance of the first vertical mode
When the surface mixed layer is finite, the singular “rays” at the equator are no longer infinitesimally thick but have a vertical scale comparable to the mixed layer depth, and are more appropriately called equatorial “beams.” This has been shown numerically by McCreary (1984). Mathematically, when the mixed layer thickness is finite bn is no longer independent of n, but decreases rapidly with increasing n, so that the ocean response is dominated by the lower-order modes. For example, in the formulation by Zhang and Clarke (2017), bn decreases like n−2. This, and the stronger shears and dissipation associated with the higher vertical modes, is the likely reason for the dominance of the first vertical mode seen observationally (see Fig. 7) in the interannual pressure field.
6. Vertical wave propagation, bottom roughness, and negligible bottom pressure
As mentioned in the third paragraph of the introduction, previous analysis has suggested that the ocean floor is dynamically rough, and such bottom topography can explain why p′ = 0 on the ocean floor and the consequent increase in wave Rossby wave speed. We have suggested that this is not the physical reason for the faster Rossby wave speed off California, because the interannual wave variability has not yet reached the ocean floor there and therefore has not interacted with the rough bottom topography. What we have not done is to check how far from the coast our mechanism is valid, and to what extent the vertical propagation mechanism for bottom p′ = 0 applies elsewhere.
a. How far does the Rossby wave propagate westward and vertically before reaching the ocean floor?
b. Application to other latitudes and frequencies
Equation (6.3) shows that the extent of influence of the vertically propagating Rossby waves increases like β f−2, so although x(−H) is only about 824 km at 33°N, at 5° latitude x(−H) is wider than the Pacific Ocean basin. This is also true within the 5°S to 5°N equatorial waveguide, for there vertically propagating equatorial Kelvin waves and first meridional mode equatorial Rossby waves dominate the ocean response and cross the Pacific without striking the bottom. Evidence for this is provided by Dewitte and Reverdin (2000). Note that the Indian and Atlantic Ocean basins are much narrower than the Pacific, and so x(−H) does not have to be as large to guarantee that the interannual Rossby waves do not reach the ocean floor before being stopped by a boundary. In those basins most of the interannual Rossby wave variability will not reach the ocean floor in the latitude band from 15°S to 15°N. Note that Eq. (6.3) shows that x(−H) is proportional to ω−1, so the range of latitudes for which decadal variability does not reach the ocean floor is about 3 times that of interannual variability.
7. Summary and concluding remarks
Our goal in this paper has been to understand physically why longwave RWs propagating westward from the California coast propagate at a much faster speed than we would expect. We also wanted to understand why this propagation speed agrees with the first vertical mode RW speed with the w′ = 0 bottom boundary condition replaced by the p′ = 0 bottom boundary condition (Tailleux and McWilliams 2001).
Past work, consistent with our updated analysis in section 3, has shown that long RWs propagating westward from the California coast at interannual and lower frequencies result from incident EKWs, the only way such energy can reach the eastern ocean boundary from the interior ocean. The baroclinic EKW energy propagates at such a shallow angle ω/N(z) to the vertical that it is unlikely to reach the ocean bottom before it encounters the deep-sea eastern boundary of the equatorial Pacific. Consequently, p′ = 0 is an appropriate bottom boundary condition at the equator at the deep-sea eastern Pacific boundary, a result consistent with our analysis of the assimilation data in section 3d and the approximation of these data by the first vertical mode with the w′ = 0 bottom boundary condition replaced by p′ = 0. Theory suggests (see Clarke and Shi 1991) that the motion at these low frequencies is quasigeostrophic perpendicular to the boundary, and so argues for a p′ = 0 bottom boundary condition at the deep-sea boundary off California. Our updated detailed analysis in section 2 confirms that such a deep-sea boundary condition and associated first vertical mode gives offshore RWs that propagate westward at the observed speed.
Data availability statement.
The data we used are available from the following sources: SSH—Zlotnicki et al. (2019); hydrographic data—Locarnini et al. (2013) and Zweng et al. (2013); monthly sea level—Permanent Service for Mean Sea Level (PSMSL; http://www.psmsl.org/data/obtaining/); monthly Niño-3.4 index—https://stateoftheocean.osmc.noaa.gov/sur/pac/nino34.php; HYCOM assimilation data—https://www.hycom.org/data/glbu0pt08/expt-19pt1.
APPENDIX
WKB Expression for Wind Forcing Coefficient bn and Its Hmix → 0 Limit
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