Nonlinear Rossby waves off California



[1] A new interpretation of SSH anomalies propagating in the California Current System as weakly nonlinear Rossby waves (RWs) is suggested. Satellite altimetry and float data were used to extract annual and semi-annual components of RWs from a multi-scale altimetry signal and estimate their kinematic characteristics. Different propagation regimes for the waves were identified by propagation speed, wave steepness and length of spatial phase coherence (SPC). A transition from a SSH field dominated by waves to a turbulent-like field was detected in the saturation regime. The recurrence period for wave behavior was estimated as about 105–120 (195–210) days for the semiannual (annual) component. The propagation speed and length of SPC decreased when wave steepness increased, and westward propagation halted during the saturation regime.

1. Introduction

[2] Rossby waves (RWs) play an important role in forming the general circulation of the World Ocean [Pedlosky, 1987]. These waves have a clear signature in satellite altimetry signals which show westward propagating SSH anomalies in all ocean regions except the Antarctic Circumpolar Current, Kuroshio and Gulf Stream where the anomalies can propagate eastward [e.g., Quartly et al., 2003]. There have been several attempts to interpret propagating SSH anomalies in terms of linear baroclinic RWs [e.g., Tokmakian and Challenor, 1993; Chelton and Schlax, 1996], nonlinear eddies [Chelton et al., 2007], and RWs and geostrophic turbulence theory [Tulloch et al., 2009]. However, the physical interpretation of propagating SSH signals in areas where they can be represented as a superposition of processes sharing similar spatio-temporal scales is difficult and should be combined with a multi-scale analysis of the signals.

[3] SSH, RAFOS and Argo float data have been used in this study to interpret propagating SSH anomalies off California in terms of nonlinear RWs. It is shown that these waves propagate in different regimes and with different speeds. Although the waves effectively transported tracers, their transport features depended strongly upon the propagation regime. Transitions from a SSH field dominated by the waves to a turbulent-like SSH field was detected within one of propagation regimes. Wave- and eddy-dominated fields were distinguished from one another using a new criterion: length of spatial phase coherence (SPC) of the SSH signal. This new interpretation of SSH anomalies as nonlinear waves is needed to understand and explain westward transport off California and possibly other subtropical Eastern Boundary Current regions.

2. Data and Analysis Methods

[4] The SSH anomaly field was produced by the AVISO Project (Archiving Validation and Interpretation on Satellite Data in Oceanography) for the period from Oct. 10, 1992 to May 23, 2007 [Collecte Localisation Satellites, 2006]. When possible, satellite altimetry data were compared to RAFOS and Argo float data collected within the same time period [Margolina et al., 2006] ( RAFOS floats were deployed between 350 and 500 m. The Argo floats were at a depth of about 1000 m when they were not profiling and transmitting data.

[5] SSH anomaly observations were used to understand spatio-temporal structure of RWs and specify their kinematic characteristics at the ocean surface. Float data at two additional levels, 350 m and 1000 m, were collected well below the strongest velocities for the California Current (California Undercurrent) which occur near surface (at about 100 m). Therefore, the float trajectories contained smaller contributions from upper ocean processes and were used for alternative estimates of the tracer transport in addition to estimates from SSH anomalies.

[6] A visual analysis of SSH anomaly snapshots and RAFOS float trajectories clearly demonstrated that (a) the SSH anomaly field was multi-scale (as an example, Figure 1a shows at least two different spatial scales), (b) there was a recurrence in behavior of the anomalies (anomaly amplitudes were clearly quasi-periodic but their phases need not return to their original values at the end of each cycle), (c) SSH anomaly patterns can be thought of as representing interference of different scale processes (when these processes were in phase (out of phase), anomaly amplitudes were strongly amplified (reduced or nearly zero)), and (d) temporal intermittency was observed in the behavior of floats (at times, the floats stopped drifting westward).

Figure 1.

Features propagating in SSH anomaly field. (a) The field on Jun. 15, 2005; extracted features corresponded to (b) annual and (c) semi-annual RWs. The annual (semi-annual) anomalies contain approximately 8% (32%) of the SSH signal. The contour interval is 1 cm.

[7] To understand the spatio-temporal complexity of the altimetry signals, a double spectral approach [Ivanov and Collins, 2009] was applied to the SSH anomaly field. Following this approach, the SSH anomaly height, ζ(x,t), was decomposed into six frequency bands centered on frequencies (ωp) selected following Laskar [1993]: (1) 1–2 months. (2) 2–4 months, (3) 4–8 months, (4) 8–16 months, (5) 16–32 months and (6) longer than 32 months, i.e.,

equation image

and Daubechies wavelet transform was used for the decomposition [Ivanov et al., 2009]. Choice of the bandwidth took into account that propagating SSH features observed in the 3rd (4th) bands were not exactly the semi-annual (annual) frequency. Then, the following spatial spectral representation was used for each frequency band:

equation image

where ηp is the component of the SSH signal within the pth frequency band, x = (x, y) and x and y are zonal and meridional directions, respectively, ψm form a system of orthonormalized basis functions (M-modes) for the region of interest which generated the appropriate phase space for the analysis of altimetry signals, and apm are the spectral coefficients which minimized the least squares difference between the original SSH anomaly field and its spectral decomposition within the pth frequency band.

3. Selection of Propagating Signals

[8] Since seasonal current variability clearly contributed to SSH signals, there was a problem in selecting propagating features from total SSH variability. Since it was impossible to select the propagating features exactly, approximate methods were used (for discussion, see, e.g., Hayashi [1979]). Combinations of M-modes were identified in (2) for which the zonal component of propagation speed for SSH signals was close to zero during the 15 year period and these combinations were excluded from the analysis assuming that they were due to seasonal current variability.

[9] The zonal component (Upr) of propagation speed depended explicitly on spatial scales within the third and fourth frequency bands (i.e., dispersion played a role in the propagation process). SSH features formed as a combination of modes from 1 to 29 and for greater than 250 did not propagate westward. Therefore, it was assumed that propagating SSH anomalies dominated the spatial spectral band bounded by mlow = 30 and mupper = 250 and their contribution was neglected outside this band. Hence subsequently, if not specially indicated, summation in (2) was from 30th to 250th M-mode.

[10] Figure 1a shows an original SSH field with westward propagating features and Figure 1b (Figure 1c) shows typical spatial structure for the annual (semi-annual) component extracted from Figure 1a. The semi-annual component was dominant at this time. Crests and troughs corresponding to annual and semi-annual components were clearly separated in space even when components locally interfered. The SSH anomaly field was non-stationary and evolved so that the amplitude of the annual (semi-annual) component increased (decreased) and vice versa, i.e., robust recurrence in their behavior (with a delay equal to about 6 months) existed within the 15 year time period.

4. Criterion for Selection of Propagation Regimes

[11] Analysis of 763 two-dimensional maps characterizing spatio-temporal structure of SSH anomaly fields for these two frequency bands showed: (1) fields were dominated by waves because there was a regular recurrence in the behavior of SSH anomalies, (2) wave fronts were clearly identified, and (3) length of SPC was considerably larger than the size of the largest anomaly. For these reasons, it is suggested that these anomalies can be interpreted as a signature of RWs at the ocean surface.

[12] SSH anomalies propagated in several distinct regimes identified by (a) the zonal component of propagation speed (varied from zero to several cm/s), (b) the wave steepness, S (0<S≤2−3), and (c) the length of spatial phase coherence (Lcoh).

[13] The propagation speed was estimated from the kinematics of closed contours bounding SSH anomalies and wave fronts, i.e., curves where wave phase was constant. The closed contours were tracked using the method of Akhriev and Kim [2003] and wave fronts were identified through the wave phase calculated from the Morlet continuous wavelet transform [Kumar and Foufoula-Georgiou, 1997].

[14] Tracer transport speed (Utr) caused by SSH anomaly dynamics was estimated as

equation image

Pcc (Puc) is the probability that a particle does not leave an SSH anomaly (or was trapped by the anomaly) during time period t. Eremeev and Ivanov [1987] defined these probabilities as

equation image

where Ncc, Ncu and Nuc are the numbers of contours which, respectively, do not break, break and re-close within a propagating anomaly during time t. These contours are selected through a set of thresholds of anomaly amplitudes. Anomalies for which UtrUpr (∣UtrUpr∣ ∼ o(Upr)) weakly (effectively) transport tracers. Additionally, when RAFOS and Argo floats rotated anticyclonically with a clear SSH expression, the float trajectories were used to estimate Utr.

[15] The wave steepness was calculated as SU/βλ2, where U is root mean square velocity, λ∼ 20–25 km is the typical Rossby deformation radius for Central California waters and β is the beta parameter. Length of SPC characterized phase correlations between points transverse to the direction of wave propagation [Rabinovich and Trubetskov, 1989] and was calculated from the condition

equation image

where <…> means averaging over the region of interest; ϕ is wave phase; X, Y are longitudinal and transverse coordinates. Highest degrees of spatial phase coherency (largest coherence lengths) exist for a wave field, lowest (smallest coherence lengths) are characteristic of an eddy field. Methods for calculation of Lcoh used in this study were described in detail by Daintith [2004] and are not discussed here.

5. Propagation Regimes

5.1. Linear Regime

[16] For linear regime (LR) S ≪ 1 and duration was as long as one month. Here SSH anomalies had small amplitudes and propagated westward (W) or southwestward (SW) with speeds of about 3–5 cm/s for annual and 5–7 cm/s for semi-annual waves, respectively. This agreed with the classical theory at these latitudes [Pedlosky, 1987]. The propagation speed did not vary over space, i.e., propagation speed was practically the same in value and direction at any location of the study region.

[17] Spatial structures of annual and semiannual SSH anomalies in the LR are shown in Figures 2a and 2d. This regime was characterized by high spatial coherence (Lcoh was up to 1500 km). Recurrence time (time between two consecutive linear propagation regimes) for annual (semiannual) components was about 195–210 (105–120) days. The semi-annual wave field was more anisotropic than the annual field because it was stretched alongshore. However, annual waves had larger spatial scales.

Figure 2.

Evolution of (top) semi-annual and (bottom) annual SSH anomalies. The contour interval is 1 cm. (a, d) End of linear regime (Dec. 22, 2004; Mar. 17, 2004); (b, e) amplification regime (Jan. 5, 2005; Apr. 7, 2004) and (c, f) saturation regime (Feb. 15, 2005; Jun. 30, 2004). The semi-annual wave field is more anisotropic (stretched alongshore) than the annual wave field.

[18] Tracer transport was weak in the LR and similar to Stokes-like drift induced due to inhomogeneity of along-shore wave structure. Averaging over the recurrence period gives a Stokes-like drift speed equal to about O(1) cm/s. The drift was estimated using the exit time concept [Ivanov et al., 2008] applied to an ensemble of synthetic particles deployed into the surface circulation calculated geostrophically from the SSH anomalies.

5.2. Amplification and Saturation Regimes

[19] The duration of each of these regimes was considerably longer than that of the LR. During the amplification regime (AR) S ≤ 1 when small amplitude SSH anomalies intensified, they moved coherently with wave fronts and wave trains appeared with crests and troughs that were clearly identified.

[20] Particles trapped by these anomalies were effectively transported westward or southwestward. Propagation speed (and westward transport) varied with time: it was as large as 1–2 cm/s but slowed as wave amplitudes grew (by a factor of 3–5).

[21] The typical spatial structure of SSH anomalies in the AR is shown in Figures 2b and 2e. Here, in comparison with LR, the coherence of westward transport of particles was considerably lower and meridional drift more visible. However since Lcoh was approximately 800–1000 km, i.e., considerably larger than a typical SSH anomaly, the SSH anomaly field seemed to be dominated by weakly nonlinear waves.

[22] The propagation speeds in the zonal direction weakened with amplification as the saturation regime (SR) (i.e., with increasing S up to 2–3 and decreasing Lcoh to 250–300 km) was approached so that westward propagation halted (zero propagation speed) at saturation. A transition to a turbulence dominated field was observed in the SR when Rossby waves decayed and the SSH anomaly field was a combination of single intensive anomalies and wave packets represented a pair of eddies of opposite sign. Single intensive anomalies can be interpreted as mesoscale eddies because the spatial phase coherence was lowest for them and these anomalies moved independently of one another.

[23] Wave packets were distinguished from eddy dipoles by lack of rotation around a center. Wave packets moved in arbitrary directions (even to the east); interactions between eddies often dominated as compared to the β-effect. This was a turbulent-like regime in which there was no explicit advective westward transport although diffusion-type transport could have existed.

[24] A cessation of westward propagation was also confirmed from the analysis of the behavior of RAFOS floats. On occasion, floats which had left the poleward flowing undercurrent temporarily stopped drifting to the west.

[25] Theoretically, changing deformation radius can result in variations of the propagation speed for linear Rossby waves. However, the propagation speed of SSH anomalies did not vary off Central California in the linear regime as noted above.

[26] Mean currents can also influence Rossby wave propagation speed. For example, a halt of westward propagation of linear Rossby waves could be due to eastward zonal flow [Pedlosky, 1987]. However mean currents are weak in Central California waters and generally directed poleward or equatorward. Additionally, the propagation speed strongly correlated with amplification of SSH anomalies; in our opinion, this is a result of non-linear wave-wave interactions and is not due to mean currents. Therefore neither changing deformation radius nor mean currents can explain the observed variations in the propagation speed for the Rossby waves.

[27] Wave propagation regimes and their durations could be identified from the time evolution of the mean square SSH (MSH), 〈η2〉. LR (SR) accompanied minima (maxima) of MSH which were lower (higher) than a threshold h1(h2). For example, see Figure 3c for annual Rossby waves; the duration of amplification stage, τ2, was 6–7 months which was considerably longer than LR, τ1, (about 1 month) and SR, τ3 (1–2 months). The recurrence of propagation regimes was a robust feature for both annual and semiannual waves (Figures 3a and 3b). Since the waves were modulated by low-frequencies, τ1,τ2 and τ3 varied in time. However, τ2 > τ1 + τ3 during all wave periods.

Figure 3.

Wave characteristics. (a, b) Mean-squared height of annual and semi-annual SSH anomalies, respectively. (c) Relationship between duration of different propagation regimes of annual RW (τi,i = 1,2,3, where 1,2, and 3 correspond to linear, amplification and saturation regimes, respectively); (d) 〈ηm2〉/max〈η2〉 for annual SSH anomalies where summation for 〈ηm2〉 was: from 30th to 39th M-mode (10 modes, dotted) , from 30th to 79th M-mode (50 modes, dashed) and 30th to 250th M-mode (221 modes, solid); (e) amplitudes of M-modes (cm) from 30 to 39 (solid curves) for annual SSH anomalies (dashed curve shows function of 400 · 〈η2〉/max〈η2〉); (f) phases of 38th mode (annual SSH) (dashed), 50th mode (semi-annual SSH) (dotted), the phase difference (solid) and zero level (dot-dashed). Phases are given in degrees. The double arrow shows a phase locking event for the 38th and 50th modes; time evolution of mc for (g) annual and (h) semi-annual SSH anomaly spectra. Dotted lines in Figures 3g and 3h show the mean position of centroid: 77 (84) for annual (semi-annual) SSH anomalies.

[28] Although specific mechanisms of wave amplification and decay cannot be understood from SSH data alone, there was explicit evidence of wave-wave and wave-current interactions. Mode amplitudes within each spectral band were represented as

equation image

and cases of phase locking between M-modes from the same frequency band (∣ϕ3q − ϕ3r∣ ≈ const) as well as from different frequency bands (∣ϕ3q − ϕ4r∣ ≈ const) were identified. Here, q and r are a pair of M-modes. Amplitudes ∣bpm∣ and phases ϕpm were calculated using Morlet wavelet transform [Hramov and Koronovskii, 2004].

[29] Figure 3d demonstrates similarity in the behavior of 〈η2〉 calculated for 221 (from 30 to 250) and 10 (from 30 and 39) M-modes for the 3th frequency band. This similarity allowed use of only the ten m-modes in the phase analysis. Phase locking between M-modes from same frequency band and from different frequency bands is shown in Figures 3e and 3f, respectively. It is clear from Figure 3e that maxima and minima in behavior of 〈η2〉 existed when a portion of the M-modes was in phase and/or anti-phase. Phase locking was also observed between waves occurring in different frequency bands. For example, Figure 3f demonstrated a transition to a phase locking regime between 1996 and 2003 for mode m = 38 (with a periodicity of about 9.5 months, 3rd frequency band) and m = 50 (with periodicity of about 7.5 months, 4th frequency band). Here the phase locking could have been due to nonlinear mechanisms, such as resonance interactions between wave triads (wave-wave interactions) as well as to external forcing, while decoupling could be due to wave-current interactions (they could cause shifts of wave phase). These mechanisms will be discussed in a separate paper.

[30] Although single events such as mesoscale eddy merger or decay can be visually observed for the SR (especially near shore), these events were relatively rare. This means that weak redistribution of kinetic energy between high order (smaller scales) and low order (larger scales) M-modes (inverse and direct cascades) occurring within a single frequency band did not dominate the wave field. Only limited variations in the position of the spectral centroids (mc) were observed in Figures 3g and 3h. If an inverse or direct cascade occurred, there should have been a large shift in the position of the centroid.

6. Conclusions

[31] Data analysis has shown that semi-annual and annual SSH anomalies off California can be interpreted as a signature of weakly nonlinear Rossby waves at the ocean surface because there was high length of spatial phase coherence, wave fronts were clearly identified in SSH anomaly snapshots, and explicit recurrence was found in the SSH anomaly field. The RWs propagated in several regimes with speeds that (a) were slower than long-wave theory predicted for mid-latitudes and (b) varied in time.

[32] For the LR, propagation speed was greatest but westward transport was close to zero. Propagation speed reduced during the amplification regime (westward transport reached maximum) and became zero during the saturation regime when a transition from a field dominated by Rossby waves to a field dominated by geostrophic turbulence was observed. Durations of the SR and LR were found to be shorter than for the AR.


[33] The Argo data were collected and made freely available by the International Argo Project (; Support for LI (CC) was provided by NSF grant OCE-0827527 (OCE-0827160). We thank an anonymous reviewer for constructive comments that allowed us to clarify these results.