Spectral analysis using orthonormal functions with a case study on the sea surface topography

A primitive equation ocean model has been applied to the study of mesoscale ocean dynamics and prediction using sea surface height information derived from a satellite‐borne altimeter. Results from a model of the Gulf of Mexico were directly compared with the altimeter data from GEOS 3 and Seasat reported by Marsh et al. (1984) and in situ hydrographic data analyzed by Maul and Herman (1985). In the eastern Gulf the amplitude and position of relative maxima of sea surface height variability were found to be similar for model, altimeter, and in situ data. In the central Gulf the model and the altimeter‐derived sea surface height variability maps were similar but differed considerably from the in situ results. The mean sea surface in the model and from the in situ data also agreed well in the eastern Gulf, but there was little agreement in the central Gulf. While there is no geoid adequate for determining an independent altimetric estimate of the mean dynamic height of the sea surface in the Gulf by differencing the geoid with the altimeter‐derived mean sea surface, the model dynamic mean sea surface can be subtracted from the mean surface of Marsh et al. (1984) to obtain a new estimate of the Gulf geoid, which may be the best geoid available in that region for several years. A benchmark experiment (“truth”) was integrated to statistical equilibrium and compared with results from four experiments in which the model was initialized with fields modified from archived benchmark data. The experiments differed only in the initialization fields. Each experiment was integrated for 100 days with inflow transport remaining constant throughout the integration. The idealized experiments were initialized geostrophically (1) with the exact sea surface and pycnocline height fields, (2) with only the exact sea surface heights and a pycnocline assumed to compensate it such that the deep pressure perturbations vanished, (3) just as in case 2 but with a geoid error component added on small spatial scales, and (4) just as in case 3 except the geoid error model included an additional contribution in strong geoid gradient regions. The sequence of four numerical experiments showed that (1) geostrophic initialization with exact sea surface and pycnocline height information provided accurate forecasts to 100 days, (2) even when only the sea surface height information was provided to the numerical model for geostrophic initialization and deep pressure perturbations assumed to vanish at the initial time, the forecasts were superior to persistence or climatology over the 100‐day forecast period, and (3) errors in the geoid on spatial scales comparable to the grid resolution of the model did not seriously degrade the forecast, even in dynamically active regions with large gradients in the geoid height where instability processes were observed to occur.

Abstract. We use output from a freely running NEMO model simulation for the equatorial Pacific to investigate the utility of linearly removing the local influence of vertical displacements of the thermocline from variations in sea surface height. We show that the resulting time series of residual sea surface height, denoted ηnlti, measures variations in near-surface heat content that are independent of the local vertical displacement of the thermocline and can arise from horizontal advection, surface heat flux, and diapycnal mixing processes. We find that the variance of ηnlti and its correlation with sea surface temperature are focused on the Niño4 region. Furthermore, ηnlti averaged over the Niño4 region is highly correlated with indices of central Pacific El Niño–Southern Oscillation (CP ENSO), and its variance in 21-year running windows shows a strong upward trend over the past 50 years, corresponding to the emergence of CP ENSO following the 1976/77 climate shift. We show that ηnlti can be estimated from observations, using satellite altimeter data and a linear multi-mode model. The time series of ηnlti, especially when estimated using the linear model, show pronounced westward propagation in the western equatorial Pacific, arguing for an important role for zonal advective feedback in the dynamics of CP ENSO, in particular for cold events. We also present evidence that the role of the thermocline displacement in influencing sea surface height increased strongly after 2000 in the eastern part of the Niño4 region, at a time when CP ENSO was particularly active. Finally, the diagnostic is easy to compute and can be easily applied to mooring data or coupled climate models.

Abstract. We investigate the spatial and temporal variability of sea surface height (SSH) in the North Atlantic basin using satellite altimeter data from October 1992–January 2004. Our primary aim is to provide a detailed description of such variability, including that associated with propagating signals. We also investigate possible correlations between SSH variability and atmospheric pressure changes as represented by climate indices. We first investigate interannual SSH variations by deriving the complex empirical orthogonal functions (CEOFs) of altimeter data lowpass-filtered at 18 months. We determine the spatial structure of the leading four modes (both in amplitude and phase) and also the associated principal component (PC) time series. Using wavelet analysis we derive the time-varying spectral density of the PCs, revealing when particular modes were strongest between 1992–2004. The spatial pattern of the leading CEOF, comprising 30% of the total variability, displays a 5-year periodicity in phase; signal propagation is particularly marked in the Labrador Sea. The second mode, with a dominant 3-year signal, has strong variability in the eastern basin. Secondly, we focus on the Azores subtropical frontal zone. The leading mode (35%) is strong in the south and east of this region with strong variations at 3- and 5-year periods. The second mode (21%) has a near-zonal band of low variance between 22°–27° N, sandwiched between two regions of high variance. Thirdly, we lowpass filter the altimeter data at a cutoff of 30 days, instead of 18 months, in order to retain signals associated with propagating baroclinic Rossby waves and/or eddies. The leading mode is the annual steric signal, around 46% of the SSH variability. The third and fourth CEOFs, 11% of the remaining variability, are associated with westward propagation which is particularly dominant in a "waveband" between 32°–36° N. For all three cases considered above, no significant cross-correlation is found between the North Atlantic Oscillation index and the amplitude of the leading four PCs of interannual SSH variability. The only exception is an anti-correlation found over the North Atlantic basin between the NAO and the 4th PC. In the subtropical front, the East Atlantic Pattern index is anti-correlated with the leading PC for SSH variations lowpass filtered at 30 days. Further investigation of forcing mechanisms is suggested using hindcasts from ocean general circulation models.

A merged TOPEX/POSEIDON and ERS‐1/2 altimeter data set is used to estimate nonseasonal variations of sea surface height (SSH) in the Japan/East Sea (JES) from October 1992 through February 2002. The altimeter data have a good correlation (0.95) with tide gauge data in the western JES. The nonseasonal variability of the SSH is higher in the southern warm region than in the northern cold region. The root‐mean square value of the nonseasonal SSH is 5–10 cm in the southern area, whereas it is about 4 cm in the northern area. Empirical orthogonal function analysis of the nonseasonal SSH shows the first mode to be related to intraseasonal oscillations over the entire JES and the second mode to interannual path variations of the Tsushima Warm Current (TWC). The path change of the TWC is responsible for the high variability of SSH in the southern region.

A balanced Kalman filter ocean data assimilation system with application to the South Australian Sea

Publication in the conference proceedings of EUSIPCO, Trieste, Italy, 1996

Analyzing the observed currents at Xisha (110.3899°E, 17.1038°N) during May 2009 to May 2010, it is found that the kinetic energy has significant mesoscale variability, and each peak responds to large positive/negative ocean surface current curl caused by mesoscale eddies. Compared the kinetic energy with the wind stress work and the pressure work, it is also found that the barotropic pressure work which is mainly contributed by the sea surface height (SSH) corresponding to the mesoscale eddies behaves like the kinetic energy. The contribution of the mesoscale eddies to the kinetic energy can be up to 90% sometimes and reach deep level every time. Using the satellite altimeter data, the paths of mesoscale eddies contributing to the kinetic energy variability are traced back. In the winter half of the year, the mesoscale eddies propagating along the northern South China Sea shelf or across the basin from the west of the Philippines toward Xisha arrive at Xisha, influencing the kinetic energy. In the summer half of the year, the mesoscale eddies are mainly from the south, which were shed from the Vietnam coast current. And the cause for eddy shedding may be related to the relaxation of the Ekman transport anomalies.

Subtropical gyre variability observed by ocean-color satellites

Analyses of sea surface height (SSH) records based on satellite altimeter data and hydrographic properties have suggested a considerable weakening of the North Atlantic subpolar gyre during the 1990s. Here we report hindcast simulations with high‐resolution ocean circulation models that demonstrate a close correspondence of the SSH changes with the volume transport of the boundary current system in the Labrador Sea. The 1990s‐decline, of about 15% of the long‐term mean, appears as part of a decadal variability of the gyre transport driven by changes in both heat flux and wind stress associated with the North Atlantic Oscillation (NAO). The changes in the subpolar gyre, as manifested in the deep western boundary current off Labrador, reverberate in the strength of the meridional overturning circulation (MOC) in the subtropical North Atlantic, suggesting the potential of a subpolar transport index as an element of a MOC monitoring system.

During the CINDY/DYNAMO field campaign, exceptionally large upper ocean responses to strong westerly wind events associated with the Madden–Julian oscillation (MJO) were observed in the central equatorial Indian Ocean. Strong eastward equatorial currents in the upper ocean lasted more than one month from late November 2011 to early January 2012. The remote ocean response to these unique MJO events are investigated using a high resolution (1/25°) global ocean general circulation model along with the satellite altimeter data. The local ocean response to the MJO events are realistically simulated by the global model based on the comparison with the data collected during the field campaign. The satellite altimeter data show that anomalous sea surface height (SSH) associated with the strong eastward jets propagated eastward as an equatorial Kelvin wave. The positive SSH anomalies then partly propagate westward as a reflected Rossby wave. The SSH anomalies associated with the reflected Rossby wave in the southern hemisphere propagate all the way to the western boundary. These remote ocean responses are well simulated by the global model. The analysis of the model simulation indicates the significant influence of reflected Rossby waves on sub-seasonal variability of Somali current system near the equator. The analysis further suggests that the reflected Rossby wave causes a substantial change in the structure of the Seychelles–Chagos thermocline ridge, which contributes to significant SST anomalies.

TOPEX/POSEDIENT-ERS satellite altimeter data along with the mean state from the Parallel Ocean Climate Model result have been used to investigate the variation of Kuroshio intrusion and eddy shedding at Luzon Strait during 1992–2001. The Kuroshio penetrates into the South China Sea and forms a bend. The Kuroshio bend varies with time, periodically shedding anticyclonic eddies. Criteria of eddy shedding are identified: 1) When the shedding event occurs, there are usually two centers of high Sea Surface Height (SSH) together with negative geostrophic vorticity in the Kuroshio Bend (KB) area. 2) Between the two centers of high SSH there usually exists positive geostrophic vorticity. These criteria have been used to determine the eddy shedding times and locations. The most frequent eddy shedding intervals are 70, 80 and 90 days. In both the winter and summer monsoon period, the most frequent locations are 119.5°E and 120°E, which means that the seasonal variation of eddy shedding location is unclear.

Over the last three decades, satellite altimetry has observed Sea Surface Height variations, providing a regular monitoring of the ocean circulation. Altimetry measurements have an intrinsic signal-to-noise ratio that strongly limits the space scales, and then the geophysical information that can be captured with this instrument. However, the recent progress made on both altimetry sensors and data processing, offers new perspectives in terms of research-oriented and operational applications. In this paper we present a methodological study that helps to better quantify the impact of this progress in terms of coastal circulation. We focus on a case study: the Northern Current, a narrow slope current (less than 60 km wide) located in the North Western Mediterranean Sea. We first use a high resolution numerical model validated with HF radars and underwater glider data to define the general characteristics of the Northern Current in terms of surface velocity and sea surface height signature. These characteristics are then compared with corresponding estimates of sea surface height velocity derived from 1-Hz altimetry data sets from three missions: Jason 2 (Ku-band LRM), SARAL (Ka-band LRM) and Sentinel-3A (SAR). The data from all missions were processed with the coastal-specific X-TRACK strategy. We show that near Toulon, the model is very close to the observations in terms of current estimates, providing a very good reference for altimetry data located in this area. The Northern Current is observed 15 km to the coast on average, with a mean core velocity of 0.44 m s−1. Its signature in sea level consists of a drop whose mean value at 6.14° E is 6.9 +/− 2.2 cm extending over 18 +/− 4 km. These variations show a clear seasonal pattern, but high frequency signals are also present most of the time. In 1-Hz altimetry data, the mean sea level drop associated with the Northern Current is overestimated by 3.6 cm for Jason 2, 0.3 cm for SARAL and 1.4 cm for Sentinel-3A. In terms of corresponding sea level variability, Jason 2 and SARAL altimetry estimates are larger than the model reference (+1.3 cm and +1 cm, respectively) whereas Sentinel-3A shows closer values (−0.4 cm). Without any sea level data filtering, the standard deviation of altimetry-derived velocity values is 3.7, 2.4 and 2.9 times too large for Jason-2, SARAL and Sentinel-3A, respectively. When filtering sea level data, the distribution of altimetry velocities tends to converge towards the model reference with a 50-km, 30-km and 40-km cutoff wavelength for Jason-2, SARAL and Sentinel-3A data, respectively.

International audience

The marine geoid defines the hydrostatic equilibrium shape that sea level would take in the absence of tides, currents and winds. The geoid is not directly observable, but its height above a reference ellipsoid may be calculated from a model of the Earth’s gravity field. Satellite altimeters measure the instantaneous sea surface height, which is the sum of the geoid plus the dynamic topography associated with the ocean’s responses to tidal and atmospheric forcings. In the early days of altimetric oceanography, knowledge of the geoid was poor and so altimeters are usually placed in exact repeat orbits to facilitate observation of the temporal variations in sea surface height. Recent gravity field models have geoid height errors around 5 cm and geoid slope errors around 1.5 μrad. Errors in the models are primarily at full wavelengths shorter than 20 km, so that some dynamic topography signals may now be directly observed. While satellite gravimetry from missions such as GRACE has improved the long wavelength geoid, there is considerable geoid power at wavelengths too short to be measured by gravimetry at orbital altitude. Determination of the geoid at mesoscale and shorter wavelengths has come from altimetry. Careful filtering has allowed to separate the geoidal height and ocean dynamics signals in the altimeter data, and thereby to refine the marine geoid. Verification of this comes from comparison of marine gravity field models against ship gravimetry. Further improvements in the accuracy of the marine geoid will require a satellite altimeter mission with improved signal-to-noise ratio and a spatially dense (less than 5 km) network of groud tracks. One may hope that CryoSat-2 will offer some improvement.

Measurement of the angular power spectrum of the cosmic microwave background is most often based on a spherical harmonic analysis of the observed temperature anisotropies. Even if all-sky maps are obtained, however, it is likely that the region around the Galactic plane will have to be removed as a result of its strong microwave emissions. The spherical harmonics are not orthogonal on the cut sky, but an orthonormal basis set can be constructed from a linear combination of the original functions. Previous implementations of this technique, based on Gram—Schmidt orthogonalization, were limited to maximum Legendre multipoles of lmax≲50, as they required all the modes have appreciable support on the cut-sky, whereas for large lmax the fraction of modes supported is equal to the fractional area of the region retained. This problem is solved by using a singular value decomposition to remove the poorly supported basis functions, although the treatment of the non-cosmological monopole and dipole modes necessarily becomes more complicated. A further difficulty is posed by computational limitations — orthogonalization for a general cut requires operations and storage and so is impractical for lmax≳200 at present. These problems are circumvented for the special case of constant (Galactic) latitude cuts, for which the storage requirements scale as and the operations count scales as . Less clear, however, is the stage of the data analysis at which the cut is best applied. As convolution is ill-defined on the incomplete sphere, beam-deconvolution should not be performed after the cut and, if all-sky component separation is as successful as simulations indicate, the Galactic plane should probably be removed immediately prior to power spectrum estimation.