The altimetry products used for a comparison with theory were produced by Ssalto/Duacs and distributed by Aviso, with support from CNES. The data we used are the multi-mission (i.e., up to four satellites at a given time, e.g., TOPEX/Poseidon, Jason 1, Jason 2, Envisat) gridded sea surface heights, sampled on a 1/4∘ × 1/4∘ Cartesian grid once a week from 1 January 1993 to 31 December 2012. These data are improved compared to those used in previous studies since the combination of data from several, present-day satellites enables high-precision altimetry in both time and space at finer resolutions. More details on the way the SSHA data are produced by Aviso can be found at http://www.aviso.altimetry.fr/duacs/.

The SSHA time series of each grid point in this region were low-pass filtered in the present study by performing a 5-week-running average to eliminate short-term variability such as storms, tides (including the fortnightly component) and other variations of periods less than 1 month. Though this filtering leaves parts of the high-frequency signals in the averaged signal, these parts are minute since the window contains many cycles of the high-amplitude signals such as the M2 tides. Calculations with wider windows of 27 and 53 weeks (done to examine the possible contribution of longer-term variability such as seasonal winds) yielded qualitatively identical results (see details in Sect. 3 below).

The basis for estimating the speed of westward propagation of SSHAs is time–longitude (Hovmöller) diagrams of the SSHA field at fixed latitude. In this diagram the westward propagation is evident from the left-upward tilt of constant SSHA values; i.e., same color contours, and the angle between this tilt and the ordinate is directly proportional to the speed of westward propagation. The diagram provides a time series of the SSHA changes at fixed longitude and a longitude variation series at any particular time so fast Fourier transforms can be easily calculated in time and longitude to yield the frequency and zonal wavenumber spectra of observed SSHAs.

Three objective methods are employed in the literature for calculating the phase speed of waves from time–longitude diagrams. The first method is the frequently used (e.g., Chelton and Schlax, 1996; Chelton et al., 2003; Tulloch et al., 2009) Radon transform that is used in image processing for detecting structures on any digital image (see details in, e.g., Jain, 1989). The Radon transform of a two-dimensional function f(x, y) that describes the intensity of an image at (x, y), such as SSHA values in a given (longitude, time) domain, is the integral of f(x, y) along a line L inclined at an angle θ relative to the ordinate (i.e., θ±90∘ relative to the abscissa) and displaced a distance s from the origin. For each angle θ we sum the squares of the values of the integrals along all lines having the same θ (i.e., having different distance s). The angle at which this sum of squares attains its maximum is the most accurate estimate for the orientation of structures with the same SSHA value on the time–longitude diagram. The tangent of this preferred θ is proportional to the sought westward propagation speed. Note that in order to minimize the effect of few very high entries on the sum of squares, we apply the Radon transform to a modified time–longitude diagram where the signal is scaled on the [0,1] interval and the mean of the scaled signal is subtracted.

The second method is a relatively new algorithm (Polito and Liu, 2003; Barron et al., 2009) that constitutes an adaptation of Radon transform to a propagating wave. In this method the variance of amplitude values is calculated along the same lines. For each angle θ we average the variances along all lines at different distances s and the westward propagation speed is then determined by the tangent of the angle θ at which the mean of variances is minimal. The third method commonly used (e.g., Zang and Wunsch, 1999; Osychny and Cornillon, 2004) to obtain the observed phase speed is the application of the 2D FFT to the time–longitude diagram to get a frequency–wavenumber (i.e., ω, k) diagram of the signal's amplitude. The phase speed is obtained by locating the values of ω and k where the amplitude is maximal (i.e., maximum spectral coefficient) and calculating C=ω/k at this point of maximum spectral coefficient. Alternatively, the directionality of the spectral coefficients in the (ω, k) diagram can be found by sweeping over all lines that pass through the origin and inclined at angles ranging from 0∘ to 180∘ relative to the abscissa (i.e., ω/k lines). The value of C is then determined as the slope of the line of maximal sum of squares of all spectral coefficients (“total energy”).

A comparison between the three methods was made using synthetic signals (De-Leon and Paldor, 2017). Based on the insight gained from the study of synthetic signals, an estimation of the observed phase speed is accepted here only when an isolated peak (characterized by the point at which the derivative changes sign at a clearly defined sharp peak and maintains the same sign in bands that are at least 3∘ wide on either side of the peak) is evident in at least two of the three methods and the phase speeds that correspond to these peaks agree by better than 10 %.

Note that the observed phase speed is obtained from the Hovmöller diagrams in units of 1/4∘ longitude per week, which is converted to units of 1 cm per second by multiplying the observed phase speed by 4.6cos⁡ϕm (where ϕm is the latitude of observation).

The trapped wave theory in mid-latitudes applies without any modification to domains of large meridional extent (so the β-plane approximation applies) that are bounded on their equatorward side by a wide zonal boundary. As shown in Fig. 1 such a nearly zonal boundary exists in the Indian Ocean south of Australia. The domain of study extends from the south coast of Australia at about 31.5∘ S to only about 45∘ S since south of this latitude, the SSHA field is strongly affected by the nearly 2000 km wide, fast and strongly meandering Antarctic Circumpolar Current (ACC).

The Coriolis frequency on the β-plane, expanded linearly about some latitude, ϕ0, is given by f(y)=f0+βy=2Ωsin⁡ϕ0+2Ωacos⁡ϕ0⋅y, where Ω is the frequency of Earth's rotation about its polar axis, a is Earth's radius and y=(ϕ-ϕ0)×a (where ϕ is the latitude) is the north coordinate.

In a channel on the mid-latitudes β-plane where the Coriolis frequency is expanded near ϕ0=ϕm, the latitude of observation, the fastest baroclinic phase speed (in units of meters per second) of harmonic Rossby waves is (see Cushman-Roisin, 1994; Vallis, 2006): Charm=-βk2+l2+f02g′H′=-2Ωacos⁡ϕmk2+l2+2Ω2sin⁡2ϕmg′H′, where k and l (the latter is denoted in other studies by n) are the zonal and meridional wavenumbers of the Cartesian coordinates, respectively, g′ is the reduced gravity and H′ is the weighted depth of the two (or more) layers that make up the baroclinic ocean; therefore, (g′H′)12 is the speed of gravity waves. For sufficiently long waves when both k and l can be neglected this phase speed reduces to Charm=-βf02g′H′=-g′H′cos⁡ϕm2Ωasin⁡2ϕm. In contrast to the harmonic wave theory, which is fully described in many textbooks, the application of the trapped wave theory requires some more detailed explanation. In this theory, the waves are trapped next to a single wall that marks the equatorward boundary of the domain and the meridional variation of the wave's amplitude is given by the regular (at infinity) Airy function, Ai, which oscillates (but is not periodic in contrast to harmonic/sinusoidal oscillations) in the (-∞, 0) interval and decays to zero faster than exponential in the (0, +∞) interval (see e.g., Abramowitz and Stegun, 1972). The phase speed of trapped waves in a mid-latitude channel is (see Eq. 6 of Gildor et al., 2016) Ctrap=-βk2+f02g′H′+ζn×2f0βg′H′23-2f0βg′H′×yw, where ζn is the absolute value of the nth zero of Ai and yw is the location of the equatorward wall. Following the studies of Paldor and Sigalov (2008) and De-Leon and Paldor (2009), we expand here the Coriolis frequency near ϕ0=ϕw, where ϕw is the latitude of the equatorward boundary of the domain so yw=0 there, in which case the last term in the denominator of Eq. (3) vanishes (in contrast to Gildor et al., 2016, where the wall was placed at yw=-L/2, where L is the channel width). In addition, the boundary condition at ϕw in Gildor et al. (2016) is the vanishing of the meridional velocity, whereas in the present application Fig. 2b implies that the meridional derivative of the height field vanishes at ϕw. Thus, ζn in Eq. (3) should be replaced in the present application by ξn – the absolute value of the nth zero of the derivative of Ai (see the discussion following Eq. 5 below). The resulting expression for the phase speed of the first baroclinic mode of sufficiently long trapped waves (i.e., for zonal wavenumber k=0 and meridional mode number n=0 for which ξ0=1.0188, see p. 478 of Abramowitz and Stegun, 1972) is Ctrap=-βf02g′H′+1.0188×2f0βg′H′23=-2Ωacos⁡ϕw2Ω2sin⁡2ϕwg′H′+1.0188×2Ω2ag′H′sin⁡2ϕw23. Note that in contrast to the planar harmonic theory where l is a meridional wavenumber (measured in units of m-1), which cannot be determined when no channel exists (and the same is true for the zonal wavenumber, k), in the trapped wave theory n is a non-dimensional mode number that counts the number of zeros of the eigenfunction inside the meridional domain.

The trapped wave theory is valid when the meridional range is larger than 2+ζnag′H′4Ω2sin⁡2ϕw13 (see Eq. 7 of Gildor et al., 2016). For n=0, for typical values of (g′H′)12 of 2 to 3 m s-1 and ϕw=30∘ this condition is satisfied when the domain is wider than about 500 km. Accordingly, the harmonic theory applies only in unrealistically narrow channels that are only a few hundred kilometers wide (see also Fig. 3 in Paldor and Sigalov, 2008).

The meridional structure of the height field of harmonic waves in mid-latitudes varies with y, the meridional coordinate, as Acos⁡(ly+Γ) (where A is an arbitrary amplitude and Γ is a phase angle that guarantees, together with l, that the wave satisfies the boundary conditions), which for l=0 yields height and velocity fields that do not vary with y.

The meridional structure of the height field of trapped waves is (see Eq. 5 in Gildor et al., 2016 with the modifications outlined in Sect. 4.1) η(y)=H′av0g′H′-C2C×4Ω2ag′H′sin⁡2ϕw13×Ai′4Ω2ag′H′sin⁡2ϕw13×y-ξn-f(y)Ai4Ω2ag′H′sin⁡2ϕw13×y-ξn, where v0 is an arbitrary amplitude and the phase speed, C, is given by Ctrap of Eq. (3) with the modifications outlined in Sect. 4.1. Note that this theoretical expression for η(y) consists of two terms: Ai(y) and Ai′(y); therefore, dη/dy contains terms proportional to Ai(y), Ai′(y) and Ai′′(y). The Airy differential equations relates Ai′′(y) to y×Ai(y); therefore, at y=0 the Ai′′(y) term vanishes and the coefficient of Ai(y) is negligible compared to that of Ai′(y), which clarifies why the extremum of η occurs at y≈0.

The meridional structure of the height field of the trapped waves curve in the area of study is computed from η(y) of Eq. (5) with C=Ctrap of Eq. (4). In the calculation of these expressions of Ctrap and η(y), the value of ϕw was set to 31.5∘ S, k=0=n (so ξn=ξ0=1.0188), and (g′H′)12, the speed of gravity waves, was set to 2.8 m s-1 following Fig. 2 in Chelton et al. (1998) (see also http://www-po.coas.oregonstate.edu/research/po/research/rossby_radius/). The analytical expression for the meridional structure of the height field of harmonic waves for k=0 and l=0 is constant, i.e., described by a straight line parallel to the abscissa at arbitrary value of the ordinate. As shown in Fig. 2b, the curve of the trapped wave theory (dashed red line) fits the observed one (solid blue line) much better than that of the harmonic theory (dotted green line).

An estimation of the speed of westward propagation of observed SSHAs is obtained by analyzing time–longitude (Hovmöller) diagrams of the SSHA field at fixed latitude as explained in Sect. 2.2. Figure 3 shows two examples of such diagrams calculated at 36∘ S (panel a) and at 45∘ S (panel b); both are sufficiently far from any major current or continent and sufficiently far from the equatorward boundary so that the condition for the validity of the trapped wave theory derived in the paragraph following Eq. (4) is satisfied (and sufficiently far (at least 200 km) from the ACC). Also plotted on these diagrams are the two lines corresponding to the theoretical phase speeds for k=0 and n=0 of trapped wave theory (Eq. 4, dashed) and the harmonic wave theory (Eq. 2, dotted). A casual visual inspection shows that the line of trapped wave theory fits the observed tilt of SSHA features more closely than the harmonic one (especially at 45∘ S; panel b).

The various objective methods for obtaining the phase speed from the Hovmöller diagram are now applied to the diagram in Fig. 3b. The distribution of the sum of squares (or standard deviation) of the Radon transform as a function of the angle θ (for θ values near the peak) is shown in Fig. 4a (solid blue curve) where the maximum is at θ≈30∘ (i.e., C≈1.9 cm s-1, in absolute value, hereafter C is positive although the wave propagates westward). The distribution of the mean of variances as a function of the angle θ is shown in Fig. 4b (solid blue curve) where the mean of variances is minimal at θ=33∘ (i.e., C≈2.1 cm s-1). The θ values corresponding to the phase speeds of trapped waves (obtained from Eq. 4, θ≈37∘, i.e., C≈2.4 cm s-1; solid red vertical line) and to the harmonic phase speed (obtained from Eq. 2, θ≈20∘, i.e., C≈1.2 cm s-1; dashed green vertical line) are also shown in panels (a), (b) and (d) of Fig. 4.

The frequency–wavenumber diagram obtained by applying 2D FFT to the time–longitude diagram at this latitude is shown in Fig. 4c in the range of low frequency and low wavenumber (in the rest of the frequency–wavenumber plane the amplitudes vanish). The maximum amplitude (outside k=0 since only k≠0 values yield finite westward phase speeds by ω/k) of the frequency–wavenumber diagram shown in Fig. 4c occurs at k=0.1571, which is a sufficiently small value that justifies the long-wave approximation made earlier (k=0.1571 corresponds to wavelength of about 160∘ of longitude). The frequency with maximal spectral amplitude at this wavenumber is -0.09045; therefore, the resulting phase speed of maximal spectral amplitude is -0.09045/0.1571=-0.5757 (in degrees of longitude per 4 weeks, i.e., C≈1.9 cm s-1) and this phase speed equals the phase speed obtained independently by the Radon transform. Figure 4c also compares the phase speeds of the two theories with the observed speed and it demonstrates that the phase speed of trapped waves (dashed red line) is slightly (but not significantly) closer to the observed speed (defined by both the maximum amplitudes and the directionality of the band of high amplitudes in frequency–wavenumber plane) than that of the harmonic waves (dotted light-green line). Though this red line (that corresponds to trapped waves) connects the two maximal values of the 2D FFT at the smallest ±k≠0 (and passes through the origin as expected), at larger k its fit to the location of maximal amplitude is no better than that of the line corresponding to harmonic waves.

The distribution of the sum of squares of the spectral coefficients along ω/k lines versus the inclination angle, arctan(C), is shown in Fig. 4d (solid blue curve) where the curve attains its maximum at arctan(C)≈151∘, i.e., θ≈29∘ in terms of the Radon transform method (C≈1.8 cm s-1).

For this time–longitude diagram the phase speed obtained by the variance method differs by about 11 to 14 % from that obtained by the Radon and 2D FFT methods that yield nearly identical phase speeds; therefore, according to our criteria mentioned in the end of Sect. 2.2, the latter estimate for the phase speed is accepted. However, this observed phase speed does not clearly validate any of the two theoretical phase speeds since the corresponding vertical lines in panels (a) and (d) of Fig. 4 are located at nearly the same distance on both sides of the observed peak. In contrast, the estimate of the observed phase speed obtained by the variance method (Fig. 4b) is much closer to that of the trapped wave phase speed than the harmonic one. Thus, the determination of the relevant theory that yields the correct phase speed that matches the propagation rate determined from observations cannot rely solely on the match at any particular latitude and therefore the match over an entire range of latitudes was also examined.

The implications from similar comparisons carried out every 0.5∘ between 33 and 45.5∘ S can be summarized as follows: at about a third of the diagrams analyzed the signal was too blurred or the three methods yielded three different phase speed estimates. The application of a single method over the entire range of latitudes yields estimates that occasionally vary by over 50 % between adjacent 0.5∘ latitudes; therefore, the latitudinal continuity of the phase speed rules out the use of a single method. In only one or two latitudes (out of 22) all three methods have yielded the same (up to 10 %) estimate. Our conclusion from these comparisons bolsters our criteria that only when at least two of the three methods yield phase speed estimates that are closer to one another by less than 10 %, the resulting phase speed estimate can be considered reliable.

Phase speed estimates north of 35∘ S and between 37 and 39∘ S have not satisfied the agreement criteria between methods outlined in the end of Sect. 2.2. The lack of reliable phase speed estimates at these latitudes even though the amplitudes of the SSHAs there are higher than in adjacent latitudes in which the phase speed estimates were deemed reliable (and especially north of 35∘ S) requires an explanation. In linear theories amplitudes can only be determined up to a multiplicative factor while phase speeds are determined completely. Accordingly, the harmonic theory, where the solution is determined by ϕm alone, does not provide any information on the variation of the SSHA amplitude with ϕm, while in the trapped wave theory the variation of the amplitude with ϕm is determined up to an overall multiplicative constant. Regardless of whether the meridional structure of SSHAs is determined or not it should be stressed that higher/lower amplitudes do not necessary imply that the corresponding phase speed estimates are more/less reliable and it is possible for the amplitude to be high while the phase speed estimates are not reliable (using the methods and criteria we apply) or for the phase speed to be significant where the amplitudes are small (e.g., south of 40∘ S).

Figure 5 shows the observed and the two theoretical speeds as a function of ϕm between 35 and 45.5∘ S, where reliable estimates are obtained. The theoretical trapped speed is calculated using Eq. (4) and the theoretical harmonic speed is calculated using Eq. (2). It is clear that the trapped speeds (solid red line) are closer to the observed speeds (blue dots, squares and triangles) than the harmonic speed (dashed green line). A quantitative confirmation of this qualitative conclusion can be obtained by calculating the sum of squares of the distances between the observed and theoretical speeds. This calculation shows that the trapped speeds with sum of squares that equals 3.5 are much closer to the observed speeds than the harmonic speeds where the sum of squares is 15.3, i.e., more than 4 times that of trapped waves. Since the value of 10 % agreement (shown by blue circular dots) is somewhat arbitrary, we also include in Fig. 5 estimates of 11 and 12 % agreement (light-blue triangles) and estimates that agree by 25 % (light-blue squares).