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Trapped planetary (Rossby) waves observed in the Indian Ocean by satellite borne altimeters

Using 20 years of accurately calibrated, highresolution observations of sea surface height anomalies (SSHAs) by satellite borne altimeters, we show that in the Indian Ocean south of the Australian coast the low-frequency variations of SSHAs are dominated by westward propagating, trapped, i.e., non-harmonic, Rossby (Planetary) waves. Our results demonstrate that the meridional-dependent amplitudes of the SSHAs are large only within a few degrees of latitude next to the southern Australian coast while farther in the ocean they are uniformly small. This meridional variation of the SSHA signal is typical of the amplitude structure in the trapped wave theory. The westward propagation speed of the SSHA signal is analyzed by employing three different methods of estimation. Each one of these methods yields speed estimates that can vary widely between adjacent latitudes but the combination of at least two of the three methods yields much smoother variation. The estimates obtained in this manner show that the observed phase speeds at different latitudes exceed the phase speeds of harmonic Rossby (planetary) waves by 140 to 200 % (which was also reported in previous studies). In contrast, the theory of trapped Rossby (planetary) waves in a domain bounded by a wall on its equatorward side yields phase speeds that approximate more closely the observed phase speeds in the study area.

observations as a surface manifestation of the first baroclinic mode of planetary (also known as Rossby) waves that propagate westward (i.e. their phase speed is negative) in the ocean thermocline. Recent studies (e.g., Chelton et al., 2007;Chelton et al., 2011), however, argue that the SSHA features belong to mesoscale eddies and not surface manifestations of planetary waves in the thermocline but this change of view has no effect on the estimate of the westward propagation speed since these eddies propagate westward at the phase speed of long Rossby waves (Chelton et al., 2011; see also Nof, 1981 for 5 theoretical estimate of eddy migration rate on the -plane).
The quantification of the rate of westward propagation of the observed SSHA features is based on the construction of time-longitude (also known as Hovmöller) diagrams at a given latitude. The slopes of contours on these diagrams are proportional to the propagation speed of the SSHA features. These slopes can be calculated using methods that are commonly employed in image processing such as the Radon transform (or its more recent alternativethe variance method) 10 and the Two Dimensional Fast Fourier Transform (2D FFT) which are described in details below.
Previous studies of the westward propagation of SSHA in mid-latitudes have all yielded rates of westward propagation that are faster than the phase speeds predicted by the harmonic planetary wave theory (see below for details). Explanations for these underestimates by the harmonic theory were proposed which are based on considerations that involve either the addition of mean zonal flows in the equations (Killworth el al., 1997 and see also Colin de Verdière and Tailleux, 2005, who 15 emphasized the curvature effect of the mean flow rather the mean flow itself) or the influence of the bottom topography (Tailleux and McWilliams, 2001) while Killworth and Blundell (2005) applied a combination of these two effects. LaCasce and Pedlosky (2004) argued that due to baroclinic instability the wave structure is changed and becomes more barotropic so it propagates faster and no mean flow is required. Along similar lines, Hochet et al. (2015) suggested that the assumption that observations are of the first baroclinic mode cannot be made a-priori, but the vertical structure is predicted from the 20 altimetry data. Thus, they found that in some regions the vertical structure is more barotropic than baroclinic so the theoretical phase speed is larger and no discrepancy exists between theory and observations. By incorporating physical elements that are not included in the simple linear wave theory of the Shallow Water Equations (e.g. velocity shear, nonlinear terms, topography, mean flows and juxtaposing barotropic and baroclinic modes) these (and other) past studies were successful in bridging some of the discrepancies found between the observed SSHA propagation speeds and the phase 25 speeds of harmonic wave theory.
In contrast to the phase speed, other wave characteristics such as the meridional variations of the SSHA amplitudes (which are predicted by the harmonic theory to be sinusoidal) have never been verified in these past studies. The reason is that in the framework of the harmonic theory (see more details below) the central latitude,  0 , which determines the origin of the y-(meridional) coordinate, is determined by the latitude of observation. Thus, observations of SSHA at adjacent latitudes 30 Ocean Sci. Discuss., doi:10.5194/os-2017Discuss., doi:10.5194/os- -3, 2017 Manuscript under review for journal Ocean Sci. Discussion started: 15 February 2017 c Author(s) 2017. CC-BY 3.0 License.
cannot be compared to one another since their y-dependencies are determined by the same equations but with different origins so the same y-coordinate denotes different points in the two sets of equations.
The interpretation of these SSHA observations has employed the harmonic theory of westward propagating, low frequency, waves that assumes the existence of a zonal channel that bounds the north-south extent on the -plane. Under these assumptions zonally propagating wave solutions of the Shallow Water Equations can be constructed and explicit 5 expressions can be derived for both the zonal phase speed of the waves and the spatial structure of their amplitudes. The emerging spatial structure of the waves is oscillatory (harmonic) in both the zonal and meridional coordinates i.e., the waves simply oscillate with wavenumber k in the zonal direction and wavenumber l in the meridional direction (Pedlosky, 1982;Cushman-Roisin, 1994;Vallis, 2006).
An alternative to the traditional harmonic theory is the trapped wave theory which was developed on the mid-latitude -10 plane by Paldor et al. (2007) and Paldor and Sigalov (2008). In this theory the meridional variation of the wave's amplitude is not harmonic and is given instead by the regular (at infinity) Airy function, Ai, that oscillates in the (−∞, 0) interval (but the amplitudes and periods are not uniform as they are in sinusoidal oscillations) and decays to zero faster than exponential in the (0, +∞) interval (see e.g., Abramowitz and Stegun, 1972). Therefore, the requirement of two channel walls of the harmonic theory is replaced in the trapped wave theory by a single wall that marks the equatorward boundary of the domain. 15 In sufficiently wide meridional ranges the phase speed of the trapped waves is higher than that of the corresponding harmonic waves by a factor of 2 to 4.
The current study employs the available series of SSHA observations sampled on a 1/4 spatial grid which are compared to the theoretical phase speeds and meridional structures of the height field using the trapped, and harmonic, wave theories.
The comparisons provide a measure of the relevance of the trapped and harmonic wave theories to the observed SSHA fields 20 in the Indian Ocean. This paper is organized as follows: Section 2 provides details of the observations and methods used for estimating the observed phase speed and in Sect. 3 we compare the theoretical and observational meridional variation of the height field in the Indian Ocean south of the Australian coast. Section 4 describes theoretical expressions for the phase speed and meridional structure of the height field of the harmonic and trapped wave theories that are compared with SSHA 25 observations in the region of interest in Sect .5. The paper ends in Sect. 6 with summary and discussion of the findings.

SSHA Data
The altimetry products used for a comparison with theory were produced by Ssalto/Duacs and distributed by Aviso, (http://www.aviso.altimetry.fr/duacs/) 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 5 1/4°1/4° Cartesian grid once a week from 1/1/1993 to 31/12/2012. The SSHA data distributed by Aviso are based on a twenty-year reference period, 1993-2012, which integrates the evolution of the sea level in terms of trends, annual and interannual signals at small and large scales (e.g., El Niño, La Niña) in that period (and the possible bias is removed by subtracting the mean SSHA of 1993 from all data). 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 10 finer resolutions. The SSHA time-series of each grid-point in this region were low-pass filtered 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 one 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 15 such as seasonal winds) yielded qualitatively identical results (see details in section 3 below).

Methods of estimating observed phase speed of SSHA
The basis for estimating the speed of westward propagation of SSHA 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 20 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 SSHA.
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) 25 Radon transform 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  relative to the ordinate (i.e., relative to the abscissa) and displaced a distance s from the origin. For each angle  we sum the squares of the Ocean Sci. Discuss., doi: 10.5194/os-2017-3, 2017 Manuscript under review for journal Ocean Sci. values of the integrals along all lines having the same  (i.e., having different distance s). The angle at which this sum-ofsquares 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  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 5 second method is a relatively new algorithm (suggested by 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  we average the variances along all lines at allowed distances s and the westward propagation speed is then determined by the tangent of the angle  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 10 to the time-longitude diagram to get a frequency-wavenumber (i.e., , k) diagram of the signal's amplitude. The phase speed is obtained by locating the values of  and k where the amplitude is maximal (i.e., maximum spectral coefficient) and calculating C=/k at this point of maximum spectral coefficient. Alternatively, the directionality of the spectral coefficients in the (, 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., /k lines). The value of C is then determined as the slope of the line of maximal sum-15 of-squares of spectral coefficients ("total energy").
A comparison between the three methods was made using synthetic signals (De-Leon and Paldor, 2016). 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 20 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 one-quarter degree longitude per week which is converted to units of one centimetre per second by multiplying the observed phase speed by .

The study domain in the Indian Ocean
The trapped wave theory in mid-latitudes applies without any modification to domains of large meridional extent (so the -

(b)
deviation is calculated, while the effect of winds of periods shorter than 5 weeks are filtered out by our low-pass filter. In order to examine the possible effect of longer term winds (seasonal to annual) the calculations were repeated with windows of 27 weeks and 53 weeks. These calculations yielded very similar results to those obtained with the 5 weeks window but with slight decrease in the amplitude of main signal near the coast and minute changes in the structure far from it. For the same reason this coastal peak cannot be associated with a mean long-coast current since such a current will not show up on a 5 map of temporal standard-deviation. Also, this variability cannot be associated with a coastal-trapped Kelvin waves since the observed phase speeds of 0.02 to 0.03 m s -1 (see Sect. 5.2 below) are two orders of magnitude smaller than the 2.5 to 2.8 m s -1 phase speed of Kelvin waves in this region.
The mean over all longitudes ( Fig. 2(

b), thick blue line) of the 41 individual latitudinal cross-sections (thin light-blue lines)
is compared with analytical expressions (described in the next section) for the meridional structure of the height field of both 10 the trapped wave theory (dashed red line) and the harmonic theory (dotted green line). The decay rates with latitude of both observed and trapped wave theoretical curves are similar in contrast to the flat curve of the harmonic theory. An unexplained minor secondary peak is found near 36 on the observed curve (also evident near 36 S, 125 E in panel (a)) and upon examining a larger SSHA map it turns out that this secondary peak is an eastward extension of the Leeuwin current that flows poleward along the west coast of Australia between March and July (Godfrey and Ridgway, 1985).

Application of wave theories to observations
The relevance of the trapped wave theory to observations can be best assessed by comparing the theoretical phase speed and 10 meridional structure of the waves with the observations such as those described above. In addition, it is also natural to compare the observations with phase speed and meridional structure of the harmonic planar theory and use the observations to assess the applicability of each of these theories.

Explicit expressions for the phase speed of the westward propagating waves
The Coriolis frequency on the -plane, expanded linearly about some latitude, , is given by 15 , where is the frequency of Earth's rotation about its polar axis, a is Earth's radius and (where is the latitude) is the north coordinate.
In a channel on the mid-latitudes -plane where the Coriolis frequency is expanded near , the latitude of observation, the fastest baroclinic phase speed (in units of metre per second) of harmonic waves is (see Cushman-Roisin, 1994;Vallis, 2006): 20 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 so (g'H') ½ is the speed of gravity waves. In 3D models where the density varies continuously with depth, (g'H') ½ is the separation constant between the horizontal and vertical equations (see Chapman and Lindzen, 1970). For sufficiently long waves when both k and l can be neglected this phase speed reduces to : 25 Ocean Sci. Discuss., doi:10.5194/os-2017-3, 2017 Manuscript under review for journal Ocean Sci. Discussion started: 15 February 2017 c Author(s) 2017. CC-BY 3.0 License.
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. Following the studies of Paldor and Sigalov (2008), De-Leon and Paldor (2009) and Gildor et al. (2016) we expand the Coriolis frequency near , where is the latitude of the equatoward boundary of the domain. The results shown in Fig. 2(b) clearly indicate that the meridional derivative of the height field vanishes at . Formally, the expression for the phase speed in the present set-up is obtained by setting L=0 in Eq. (6) of 5 Gildor et al. (2016) (the reason is that in the latter study the wall was placed at y=-L/2 while in the present study the wall is located at y=0). In addition, the boundary condition at in the present application differs from that in Gildor et al. (2016) in which the boundary condition was the natural condition of the vanishing of the meridional velocity at . Instead, as shown in Fig. 2 .0 88 .2 82 .820 (see P. 478 of Abramowitz and Stegun, 1972). The resulting expression for the phase speed of the first baroclinic mode of trapped waves is: For zonal wavenumber k=0 and meridional mode number n=0 i.e. for .0 88 this phase speed reduces to: .0 88 Note that in contrast to the planar harmonic theory where l is a meridional wavenumber (measured in units of m -1 ) which 15 cannot be determined when no channel exists (as well as the zonal wavenumber, k), in the trapped wave theory n is a nondimensional 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 (see Eq. (7) of Gildor et al., 2016) which is satisfied for n=0 and typical values of (g'H') ½ of 2 to 3 m s -1 and  w =30 when the domain is wider than about 500 kilometres. Accordingly, the harmonic theory applies only in unrealistically narrow channels that are only a few 20 hundred kilometres wide (see also Fig. 3 in Paldor and Sigalov, 2008).

Explicit expressions for the meridional structure of the waves
The meridional structure of the height field of harmonic waves in mid-latitudes varies with y, the meridional coordinate, as cos(ly) (with arbitrary amplitude and possible phase) 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)  employing Eq. (4) for the phase speed.

Meridional structure of the height field 10
The meridional structure of the height field of the trapped waves curve is computed from (y) of Eq. (5) with C= C trap of Eq.
(4) with n=0=k in all of these expressions. In the calculation of C trap and (y) the value of  w was set to 31.5 S and (g'H') ½ , the speed of gravity waves, was set to 2.8 m s -1 following Fig. 2 in Chelton et al. (1998) (see also http://wwwpo.coas.oregonstate.edu/research/po/research/rossby_radius/). Note that the theoretical expression for (y) in Eq. (5) consists of two terms: Ai(y) and Ai'(y), so its derivative contains two terms as well: Ai'(y) and Ai''(y), but when we apply the boundary 15 condition, '(y( w ))=0, (i.e. maximal amplitude of (y) at the wall) we consider only the Ai'(y) term (whose zeros are well known) since the coefficient of Ai''(y) is negligible compared to that of Ai'(y) (the ratio between them is of order 10 -3 ). 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. 2

Phase speeds
An estimation of the speed of westward propagation of SSHA 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 5 Eq. (4)   The various objective methods for obtaining the phase speed from the Hovmöller diagram are now applied to the diagram in 5 Fig. 3(b). The distribution of the sum-of-squares (or standard deviation) of the Radon transform as a function of the angle  (for  values near the peak) is shown in Fig. 4(a) (solid blue curve) where the maximum is at 30 (i.e., C1.9 cm s -1 , in absolute value, hereafter). The distribution of the mean of variances as a function of the angle  is shown in Fig. 4 The frequency-wavenumber diagram obtained by applying 2D FFT to the time-longitude diagram at this latitude is shown in Fig. 4(c) 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) of the frequency-wavenumber diagram shown in Fig. 4(c) occurs at k=0.1571 which is a sufficiently small value that justifies the 15 long-wave approximation made earlier (k=0.1571 corresponds to wavelength of about 160 degrees of longitude). The frequency with maximal spectral amplitude at this wavenumber is -0.09045 so 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 4(c) 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 20 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).
The distribution of the sum-of-squares of the spectral coefficients along /k lines versus the inclination angle, arctan(C), is shown in Fig. 4(d) (blue curve) where the curve attains its maximum at arctan(C)151 i.e. 29 in terms of the Radon transform method (C1.8 cm s -1 ). 25 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, so 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 30 phase speed obtained by the variance method ( Fig. 4(b)) is much closer to that of the trapped wave phase speed than the Ocean Sci. Discuss., doi: 10.5194/os-2017-3, 2017 Manuscript under review for journal Ocean Sci. Discussion started: 15 February 2017 c Author(s) 2017. CC-BY 3.0 License. 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 match over an entire range of latitudes was also examined.  Estimates north of 35 S and between 37 S and 39 S have not satisfied the agreement criteria between methods 15 outlined in the end of Sect. 2.2. The lack of reliable phase speed estimates at these latitudes even though the amplitudes of the SSHA 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  m alone, does not provide any information on the variation of the SSHA amplitude with  m , while in the trapped wave theory 20 the variation of the amplitude with  m is determined up to an overall multiplicative constant. Regardless of whether the meridional structure of SSHA 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 three examples of Hovmöller diagrams north of 35 S in 25 which the phase speed estimates are not reliable: In panels (a) and (b) the westward propagation is clearly evident but the different methods yield conflicting estimates probably because the high sensitivity of the estimates obtained by different methods due to the nearly horizontal slope of the contours (i.e. very high phase speed) while in panel (c) the signal itself is too noisy. The implications from similar comparisons carried out every 0.5 between 33 S and 45.5 S can be summarized as follows : 25 At about 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 so 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 30 estimates that are closer to one another than 10 %, the resulting phase speed estimate can be considered reliable.  Figure 6 shows the observed and the two theoretical speeds as a function of  m between 35 S and 45.5 S. 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 trapped speeds with sum 5 of squares that equals 3.5 are much closer than harmonic speeds where the sum of squares is 15.3 i.e. more than four times that of trapped waves. In some latitudes no observed values are given since they have not passed the criteria outlined above for determining the observed phase speed (i.e. an agreement between two or more of the various methods). Since the value of 10 % agreement (shown by blue circular dots) is somewhat arbitrary, we also include in Fig. 6 estimates of 11 and 12 % agreement (light-blue triangles) and estimates that agree by 25 % (light-blue squares).  The phase speed of harmonic waves decreases monotonically with the latitude of observation,  m , as is evident from Eqs (1)-(2). In contrast, the phase speed of trapped waves depends on  w only (i.e. the latitude of the zonal boundary) and is independent of  m . Our analyses of the propagation speeds of SSHA signals show that the rate at which the observed speed decreases with  m (the trend of the data in Figure 6 is 0.12 cm s -1 deg -1 ) exceeds the rates of decrease of both harmonic 10 (where the trend is 0.09 cm s -1 deg -1 ) and trapped (no trend) phase speeds. In contrast, the values of the observed phase speeds are much closer to the trapped phase speeds than to the harmonic speeds.
Colin de Verdière and Tailleux (2005) argue that the addition of mean flows affects the propagation speed of Rossby waves via its curvature: increase (decrease) of the westward phase speed for eastward (westward) surface mean flow.
However, in the domain of the Indian Ocean studies here it is not clear whether or not a mean flow exists (in contrast to west 15 of Australia where a subtropical gyre has been observed, see e.g. Stramma and Lutjeharms, 1997) and what it is its direction (some of the flows vary seasonally, see Wyrtki, 1973) so even if the numerical values of parameters such as Richardson number or buoyancy could be somehow estimated it is still unclear whether the mean flow increases or decreases the phase speed.
We should note that the simple choice, here and in many other prior studies, to interpret that observed SSHA 20 propagation as that of the first baroclinic mode is not the only possible choice. Other choices of a single mode to fit the observation require detailed analyses of the hydrography while a (linear) combination of several modes (including the fast barotropic mode) with weights that are tuned so as to fit the observed speed can yield a better fit (see Hochet et al., 2015). Chelton et al. (2007) and Chelton et al. (2011) argue that most of the observed SSHA features in the global ocean are nonlinear mesoscale eddies whose propagation speed is close to the phase speed of long harmonic Rossby waves (but linear 25 eddies move much faster). The nonlinearity in those studies is determined by a combination of second order spatial derivatives of the SSHA that are used in the calculation of the Okubo-Weiss parameter. Since no derivatives can be computed by the methods in the present study it is impossible to use these methods to directly determine whether the SSHA features examined in the present study are linear or not. However, all observed propagation speeds calculated here move faster than the phase of long harmonic Rossby waves (see Fig. 6) which implies that only linear (in the sense defined in 30 Ocean Sci. Discuss., doi: 10.5194/os-2017-3, 2017 Manuscript under review for journal Ocean Sci. Discussion started: 15 February 2017 c Author(s) 2017. CC-BY 3.0 License. Chelton et al., 2007) eddies that propagate faster than the phase speed of long harmonic Rossby waves exist in the Indian Ocean south of Australia.
As was concluded in De-Leon and  an estimation of the observed phase speed using one method only, is not reliable in most of the observed signals. On the other hand, even when estimates of the observed speed of at least two methods agree with each other, a comparison of the observed speed and the theoretical speeds varies in accordance with the 5 method used for obtaining the observed speed. For example, in Fig. 4 the observed speed obtained by the variance method (panel b) is much closer to the trapped speed than to the harmonic speed while the observed speed obtained by the Radon and 2D FFT methods does not fit either the trapped or the harmonic speed preferentially. These differences between different methods point to the low accuracy/reliability of existing SSHA data.
As mentioned in the introduction, many studies compared observations of Rossby waves in the ocean with the harmonic 10 Rossby waves. However from a theoretical point of view, the harmonic theory is valid only in domains narrower than a few hundred kilometres so it is not clear why one should expect the harmonic speed to match the observed speed at unbounded domains. The case of Australia is unique since the trapped wave theory applies there while no other place exists that has a sufficiently wide, nearly straight, zonal coast line and meridional extent of the ocean that spans over 10 poleward of the equatorward boundary. In cases of narrower straight zonal coast line such as Puerto Rico the trapped wave theory is 15 inapplicable. In unbounded domains of the world ocean the trapped wave theory does not apply straightforwardly and additional theoretical considerations have to be developed.