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- Author or Editor: Edward D. Zaron x
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Abstract
Assimilating hydrographic observations into a planetary geostrophic model is posed as a problem in control theory. The cost functional is the sum of weighted model and data residuals. Model errors are assumed to be spatially correlated, and hydrographic station data are assimilated directly. Searches in state space and data space, for minimizing the cost functional, are compared to a direct matrix inversion algorithm in the data space.
State-space methods seek the minimizer of the cost functional by performing a preconditioned search in an N-dimensional space of state or control variables, where N is approximately 650 000 in the present calculations. Data-space methods solve the Euler–Lagrange equations for the extremum of the cost functional by working in an M-dimensional dual space, where M is the number of measurements. The following four solvers are compared: (i) an iterative state-space solver, with a naive diagonal matrix preconditioner; (ii) an iterative state-space solver, with a sophisticated preconditioner based on the inverse of the model’s dynamical operators; (iii) an iterative data-space solver, with no preconditioning; and (iv) a direct, M × M matrix inversion, data-space solver. The best solver is the iterative data-space solver, (iii), which is approximately 10 times faster than the sophisticated preconditioned state-space solver, (ii), and 100 times faster than the direct data-space solver, (iv).
Abstract
Assimilating hydrographic observations into a planetary geostrophic model is posed as a problem in control theory. The cost functional is the sum of weighted model and data residuals. Model errors are assumed to be spatially correlated, and hydrographic station data are assimilated directly. Searches in state space and data space, for minimizing the cost functional, are compared to a direct matrix inversion algorithm in the data space.
State-space methods seek the minimizer of the cost functional by performing a preconditioned search in an N-dimensional space of state or control variables, where N is approximately 650 000 in the present calculations. Data-space methods solve the Euler–Lagrange equations for the extremum of the cost functional by working in an M-dimensional dual space, where M is the number of measurements. The following four solvers are compared: (i) an iterative state-space solver, with a naive diagonal matrix preconditioner; (ii) an iterative state-space solver, with a sophisticated preconditioner based on the inverse of the model’s dynamical operators; (iii) an iterative data-space solver, with no preconditioning; and (iv) a direct, M × M matrix inversion, data-space solver. The best solver is the iterative data-space solver, (iii), which is approximately 10 times faster than the sophisticated preconditioned state-space solver, (ii), and 100 times faster than the direct data-space solver, (iv).
Abstract
A new empirical model of ocean tides has been developed for the Weddell Sea, south of 66°S, between 90°W and 0°, using six years of radar altimeter data from the CryoSat-2 satellite mission. Because of its long ground-track repeat period (368 days) and its diverse measurement modes, low-rate mode (LRM) over the ocean and synthetic aperture radar interferometric mode (SARin) over ice surfaces and parts of the ocean, the CryoSat-2 data pose a number of challenges for tidal analysis. The space and time sampling properties of the exact repeat, near-repeat, and crossover ground tracks have been analyzed to discover which tides may be estimated using a combination of conventional harmonic analysis and local spatial regression. Using this information, the M2, S2, K2, N2, K1, O1, P1, and Q1 tides have been mapped for both the ocean and floating ice shelves in this domain. Validation against independent in situ data, along with comparison with existing tide models, finds that the CryoSat-2-derived tides are consistent with previous estimates and that they are more accurate than other models at the M2 and S2 frequencies. The high inclination of the CryoSat-2 orbit causes the orbit plane to precess relatively slowly, which leads to significantly less accurate estimates of the K2 tide. This purely empirical model ought to provide improved tidal corrections for studies of low-frequency variability and secular trends in ice shelf thickness, and it suggests that further increases in quantitative accuracy could be achieved by assimilation of CryoSat-2 data into dynamical tide models.
Abstract
A new empirical model of ocean tides has been developed for the Weddell Sea, south of 66°S, between 90°W and 0°, using six years of radar altimeter data from the CryoSat-2 satellite mission. Because of its long ground-track repeat period (368 days) and its diverse measurement modes, low-rate mode (LRM) over the ocean and synthetic aperture radar interferometric mode (SARin) over ice surfaces and parts of the ocean, the CryoSat-2 data pose a number of challenges for tidal analysis. The space and time sampling properties of the exact repeat, near-repeat, and crossover ground tracks have been analyzed to discover which tides may be estimated using a combination of conventional harmonic analysis and local spatial regression. Using this information, the M2, S2, K2, N2, K1, O1, P1, and Q1 tides have been mapped for both the ocean and floating ice shelves in this domain. Validation against independent in situ data, along with comparison with existing tide models, finds that the CryoSat-2-derived tides are consistent with previous estimates and that they are more accurate than other models at the M2 and S2 frequencies. The high inclination of the CryoSat-2 orbit causes the orbit plane to precess relatively slowly, which leads to significantly less accurate estimates of the K2 tide. This purely empirical model ought to provide improved tidal corrections for studies of low-frequency variability and secular trends in ice shelf thickness, and it suggests that further increases in quantitative accuracy could be achieved by assimilation of CryoSat-2 data into dynamical tide models.
Abstract
A near-global model for the sea surface expression of the baroclinic tide has been developed using exact-repeat mission altimetry. The methodology used differs in detail from other altimetry-based estimates of the open ocean baroclinic tide, but it leads to estimates that are broadly similar to previous results. It may be used for prediction of the baroclinic sea level anomaly at the frequencies of the main diurnal and semidiurnal tides 
Abstract
A near-global model for the sea surface expression of the baroclinic tide has been developed using exact-repeat mission altimetry. The methodology used differs in detail from other altimetry-based estimates of the open ocean baroclinic tide, but it leads to estimates that are broadly similar to previous results. It may be used for prediction of the baroclinic sea level anomaly at the frequencies of the main diurnal and semidiurnal tides
Abstract
The spatially averaged frequency spectrum of sea level has been computed at 4 cycle-per-year resolution and a Nyquist frequency of 0.5 cycles per hour using dual-satellite crossover data from the Jason and CryoSat-2 satellite altimeter missions. The novelty of the analysis is that it reveals unambiguous peaks due to high-frequency tidal signals, even after removing the predicted barotropic tide, without the usual aliasing caused by altimeter sampling. The tidal continuum, that is, a tidal cusp, is present in the spectrum in the diurnal and semidiurnal tidal bands, and a Lorentzian model spectrum has been fit within each band to identify the properties of the non-phase-locked tidal variability. An interesting feature of the semidiurnal tidal continuum is the unambiguous presence of an inner and an outer band, characterized by different Lorentzian bandwidths of roughly (180 day)−1 and (30 day)−1. Considering different latitude ranges, it is clear that the tidal continuum is most prominent in the range from −30° to 30° latitude. Within this range, it is found that 1.05-cm2 variance is associated with the semidiurnal continuum, and slightly less than half of this variance, 0.41 cm2, is associated with the slower, (180 day)−1 bandwidth, variability. The ratio of non-phase-locked to total baroclinic variability is about 62% in this latitude band, a value that is consistent with previous model-based estimates for this quantity. Quantification of the properties of the tidal continuum poleward of 30° latitude is not possible with the present data, due to the small size of the tidal signal compared to the mesoscale variability and other sources of noise.
Abstract
The spatially averaged frequency spectrum of sea level has been computed at 4 cycle-per-year resolution and a Nyquist frequency of 0.5 cycles per hour using dual-satellite crossover data from the Jason and CryoSat-2 satellite altimeter missions. The novelty of the analysis is that it reveals unambiguous peaks due to high-frequency tidal signals, even after removing the predicted barotropic tide, without the usual aliasing caused by altimeter sampling. The tidal continuum, that is, a tidal cusp, is present in the spectrum in the diurnal and semidiurnal tidal bands, and a Lorentzian model spectrum has been fit within each band to identify the properties of the non-phase-locked tidal variability. An interesting feature of the semidiurnal tidal continuum is the unambiguous presence of an inner and an outer band, characterized by different Lorentzian bandwidths of roughly (180 day)−1 and (30 day)−1. Considering different latitude ranges, it is clear that the tidal continuum is most prominent in the range from −30° to 30° latitude. Within this range, it is found that 1.05-cm2 variance is associated with the semidiurnal continuum, and slightly less than half of this variance, 0.41 cm2, is associated with the slower, (180 day)−1 bandwidth, variability. The ratio of non-phase-locked to total baroclinic variability is about 62% in this latitude band, a value that is consistent with previous model-based estimates for this quantity. Quantification of the properties of the tidal continuum poleward of 30° latitude is not possible with the present data, due to the small size of the tidal signal compared to the mesoscale variability and other sources of noise.
Aliased Tidal Variability in Mesoscale Sea Level Anomaly MapsAbstract
Sea level anomaly (SLA) maps are routinely produced by objective analysis of data from the constellation of satellite altimeter missions in operation since 1992. Beginning in 2014, changes in the Data Unification and Altimeter Combination System (DUACS) used to create the SLA maps resulted in improved spatial resolution of mesoscale variability, but it also increased the levels of aliased tidal variability compared to the methodology employed prior to 2014. The present work investigates the magnitude and spatial distribution of these tidal signals, which are typically smaller than 1 cm in the open ocean but can reach tens of centimeters in the coastal ocean. In the open ocean, the signals are caused by a combination of phase-locked and phase-variable baroclinic tides. In the coastal ocean, the signals are a combination of aliased high-frequency nontidal variability and aliased variability caused by erroneous tidal corrections applied to the along-track altimetry prior to objective analysis. Several low-pass and bandpass filters are implemented to reduce the tidal signals in the mapped SLA, and independent tide gauge data are used to provide an objective assessment of the performance of the filters. The filter that attenuates both the small-scale (less than 200 km) and the high-frequency (period shorter than 108 days) components of SLA removes aliased baroclinic tidal variability and improves the accuracy of tidal analysis in the open ocean while also performing acceptably in the coastal ocean.
Abstract
Sea level anomaly (SLA) maps are routinely produced by objective analysis of data from the constellation of satellite altimeter missions in operation since 1992. Beginning in 2014, changes in the Data Unification and Altimeter Combination System (DUACS) used to create the SLA maps resulted in improved spatial resolution of mesoscale variability, but it also increased the levels of aliased tidal variability compared to the methodology employed prior to 2014. The present work investigates the magnitude and spatial distribution of these tidal signals, which are typically smaller than 1 cm in the open ocean but can reach tens of centimeters in the coastal ocean. In the open ocean, the signals are caused by a combination of phase-locked and phase-variable baroclinic tides. In the coastal ocean, the signals are a combination of aliased high-frequency nontidal variability and aliased variability caused by erroneous tidal corrections applied to the along-track altimetry prior to objective analysis. Several low-pass and bandpass filters are implemented to reduce the tidal signals in the mapped SLA, and independent tide gauge data are used to provide an objective assessment of the performance of the filters. The filter that attenuates both the small-scale (less than 200 km) and the high-frequency (period shorter than 108 days) components of SLA removes aliased baroclinic tidal variability and improves the accuracy of tidal analysis in the open ocean while also performing acceptably in the coastal ocean.
Abstract
The interaction of the dominant semidiurnal M 2 internal tide with the large-scale subtidal flow is examined in an ocean model by propagating the tide through an ensemble of background fields in a domain centered on the Hawaiian Ridge. The background fields are taken from the Simple Ocean Data Assimilation (SODA) ocean analysis, at 2-month intervals from 1992 through 2001. Tides are computed with the Primitive Equation Z-coordinate Harmonic Analysis of Tides (PEZ-HAT) model by 14-day integrations using SODA initial conditions and M 2 tidal forcing. Variability of the tide is found to occur primarily as the result of propagation through the nonstationary background fields, rather than via generation site variability. Generation of incoherent tidal variability is mapped and shown to occur mostly in association with waves generated at French Frigate Shoals scattering near the Musicians Seamounts to the north of the ridge. The phase-coherent internal tide loses energy at a domain-average rate of 2 mW m−2 by scattering into the nonstationary tide. Because of the interference of waves from multiple generation sites, variability of the internal tide is spatially inhomogeneous and values of the scattering rate 10 times larger occur in localized areas. It is estimated that 20% of the baroclinic tidal energy flux is lost by adiabatic scattering (refraction) within 250 km of the ridge, a value regarded as a lower bound because of the smoothed nature of the SODA fields used in this study.
Abstract
The interaction of the dominant semidiurnal M 2 internal tide with the large-scale subtidal flow is examined in an ocean model by propagating the tide through an ensemble of background fields in a domain centered on the Hawaiian Ridge. The background fields are taken from the Simple Ocean Data Assimilation (SODA) ocean analysis, at 2-month intervals from 1992 through 2001. Tides are computed with the Primitive Equation Z-coordinate Harmonic Analysis of Tides (PEZ-HAT) model by 14-day integrations using SODA initial conditions and M 2 tidal forcing. Variability of the tide is found to occur primarily as the result of propagation through the nonstationary background fields, rather than via generation site variability. Generation of incoherent tidal variability is mapped and shown to occur mostly in association with waves generated at French Frigate Shoals scattering near the Musicians Seamounts to the north of the ridge. The phase-coherent internal tide loses energy at a domain-average rate of 2 mW m−2 by scattering into the nonstationary tide. Because of the interference of waves from multiple generation sites, variability of the internal tide is spatially inhomogeneous and values of the scattering rate 10 times larger occur in localized areas. It is estimated that 20% of the baroclinic tidal energy flux is lost by adiabatic scattering (refraction) within 250 km of the ridge, a value regarded as a lower bound because of the smoothed nature of the SODA fields used in this study.
Abstract
The generalized inverse of a regional model is used to estimate barotropic tidal dissipation along the Hawaiian Ridge. The model, based on the linear shallow-water equations, incorporates parameterizations for the dissipation of energy via friction in the bottom boundary layer and form drag due to internal waves generated at topographic slopes. Sea surface height data from 364 orbit cycles of the Ocean Topography Experiment (TOPEX)/Poseidon satellite mission are used to perform inversions at eight diurnal and semidiurnal tidal frequencies. It is estimated that the barotropic M 2 tide loses energy at a rate of 19 GW, of which 88% is lost within 250 km of the ridge, presumably via conversion to the internal or baroclinic tide. Uncertainty in the assumed model error and wave drag in the forward model suggest that M 2 dissipation values from 18 to 25 GW are consistent with the altimetric observations. Other barotropic tidal constituents are estimated to lose a total of 5.7 GW. The spatial distribution of barotropic dissipation along the ridge is similar to that inferred from three-dimensional primitive equation models, and it is largely insensitive to details of assumed model and data errors. Dissipation at semidiurnal frequencies is most intense at the French Frigate Shoals with lesser, but significant, contributions at other sites. Diurnal tidal dissipation is concentrated to the east of the French Frigate Shoals, at the Gardner Pinnacles. Further work with three-dimensional models will be necessary to determine the fate of the energy that is removed from the barotropic tide.
Abstract
The generalized inverse of a regional model is used to estimate barotropic tidal dissipation along the Hawaiian Ridge. The model, based on the linear shallow-water equations, incorporates parameterizations for the dissipation of energy via friction in the bottom boundary layer and form drag due to internal waves generated at topographic slopes. Sea surface height data from 364 orbit cycles of the Ocean Topography Experiment (TOPEX)/Poseidon satellite mission are used to perform inversions at eight diurnal and semidiurnal tidal frequencies. It is estimated that the barotropic M 2 tide loses energy at a rate of 19 GW, of which 88% is lost within 250 km of the ridge, presumably via conversion to the internal or baroclinic tide. Uncertainty in the assumed model error and wave drag in the forward model suggest that M 2 dissipation values from 18 to 25 GW are consistent with the altimetric observations. Other barotropic tidal constituents are estimated to lose a total of 5.7 GW. The spatial distribution of barotropic dissipation along the ridge is similar to that inferred from three-dimensional primitive equation models, and it is largely insensitive to details of assumed model and data errors. Dissipation at semidiurnal frequencies is most intense at the French Frigate Shoals with lesser, but significant, contributions at other sites. Diurnal tidal dissipation is concentrated to the east of the French Frigate Shoals, at the Gardner Pinnacles. Further work with three-dimensional models will be necessary to determine the fate of the energy that is removed from the barotropic tide.
Abstract
One of the most fundamental uses of ocean models is for the prediction of sea level. Vertical integration of the hydrostatic equation leads to the partitioning of sea level in terms of atmospheric pressure, steric height, and bottom pressure. In an effort to validate the baroclinic wave dynamics of numerical ocean models, some researchers have compared the steric height from models with the sea level anomaly derived from satellite altimetry. The use of steric height in these comparisons captures the qualitative aspects of the baroclinic waves, but it neglects a non-negligible contribution from bottom pressure. A more accurate evaluation of baroclinic wave dynamics using sea level would involve projecting the pressure field onto orthogonal barotropic and baroclinic components to isolate the baroclinic sea level anomaly. This note illustrates the quantitative difference between steric height and baroclinic sea level, which amounts to approximately a 20% bias of steric height over baroclinic sea level, depending on location.
Abstract
One of the most fundamental uses of ocean models is for the prediction of sea level. Vertical integration of the hydrostatic equation leads to the partitioning of sea level in terms of atmospheric pressure, steric height, and bottom pressure. In an effort to validate the baroclinic wave dynamics of numerical ocean models, some researchers have compared the steric height from models with the sea level anomaly derived from satellite altimetry. The use of steric height in these comparisons captures the qualitative aspects of the baroclinic waves, but it neglects a non-negligible contribution from bottom pressure. A more accurate evaluation of baroclinic wave dynamics using sea level would involve projecting the pressure field onto orthogonal barotropic and baroclinic components to isolate the baroclinic sea level anomaly. This note illustrates the quantitative difference between steric height and baroclinic sea level, which amounts to approximately a 20% bias of steric height over baroclinic sea level, depending on location.
Identification and Reduction of Retracker-Related Noise in Altimeter-Derived Sea Surface Height MeasurementsAbstract
Data from the Jason-2 calibration/validation mission phase have been analyzed to identify the correlation between sea surface height (SSH) and significant wave height (SWH) errors. A cross-spectral analysis indicates that the SSH and SWH errors are nearly white and significantly correlated at scales from 12 to 100 km, consistent with the hypothesized error source, the waveform retracker. Because of the scale separation between the SWH signal and noise, it is possible to correct the SSH data by removing the SSH noise correlated with the SWH noise. Such a correction has been implemented using the empirical correlation found during the Jason-2 calibration orbit phase and applied to independent data from other phases of the Jason-1 mission. The efficacy of the correction varies geographically, but variance reductions between 1.6 and 2.2 cm2 have been obtained, corresponding to reductions of 20%–27% in the noise floor of along-track spectra. The corrections are obtained from and applied to conventional, 1 Hz, altimetry data and lead to improvements in the signal-to-noise ratio for identification of high-frequency narrowband processes—for example, internal tides—from these data.
Abstract
Data from the Jason-2 calibration/validation mission phase have been analyzed to identify the correlation between sea surface height (SSH) and significant wave height (SWH) errors. A cross-spectral analysis indicates that the SSH and SWH errors are nearly white and significantly correlated at scales from 12 to 100 km, consistent with the hypothesized error source, the waveform retracker. Because of the scale separation between the SWH signal and noise, it is possible to correct the SSH data by removing the SSH noise correlated with the SWH noise. Such a correction has been implemented using the empirical correlation found during the Jason-2 calibration orbit phase and applied to independent data from other phases of the Jason-1 mission. The efficacy of the correction varies geographically, but variance reductions between 1.6 and 2.2 cm2 have been obtained, corresponding to reductions of 20%–27% in the noise floor of along-track spectra. The corrections are obtained from and applied to conventional, 1 Hz, altimetry data and lead to improvements in the signal-to-noise ratio for identification of high-frequency narrowband processes—for example, internal tides—from these data.
