On the Observability of Bottom Topography from Measurements of Tidal Sea Surface Height

of whether features of the ocean bottom topography can be identified from measurements of water level is investigated using a simplified one-dimensional barotropic model. Because of the nonlinear dependence of the sea surface height on the water depth, a linearized analysis is performed concerning the identification of a Gaussian bump within two specific depth profiles, (1) a constant depth domain, and, (2) a constant depth domain adjoining a near-resonant continental shelf. Observability is quantified by examining the estimation error in a series of identical-twin experiments varying data density, tide wavelength, assumed (versus actual) topographic correlation scale, and friction. For measurements of sea surface height that resolve the scale of the topographic perturbation, the fractional error in the bottom topography is approximately a factor of 10 larger than the fractional error of the sea surface height. Domain-scale and shelf-scale resonances may lead to inaccurate topography estimates due to a reduction in the effective number of degrees of freedom in the dynamics, and the amplification of nonlinearity. A realizability condition for the variance of the topography error in the limit of zero bottom depth is proposed which is interpreted as a bound on the fractional error of the topography. Appropriately designed spatial covariance models partly ameliorate the negative impact of shelf-scale near-resonance, and highlight the importance of spatial covariance modeling for bottom topography estimation

Topographic and Frictional Controls on Tides in the Sea of Okhotsk

The sensitivity of barotropic tides to bottom topography and frictional parameters has been studied in a model for the Sea of Okhotsk. This region was chosen because of the paucity of bathymetry data and the possibility of using satellite altimeter data to better identify the bottom topography using variational inverse methods. The sensitivity was studied using both the direct and adjoint sensitivity. In the former approach, perturbations to the nominal model were applied to examine their impact; in the latter approach, the sensitivities were computed using the adjoint of the tangent linearization of the dynamical model. It is found that small-scale coastal near-resonant amplification controls the tidal dynamics, and the sensitivity of the solutions is dominated by topography in these regions, far exceeding the influence of other factors. Consequently, the tidal dynamics and resonant amplification creates a non-local relationship between water level and bottom topography and leads to a linear dependence of measurements upon a very few degrees of freedom. The results indicate severe limitations on inverse approaches for identification of topography, and add to the rationale for the collection and sharing of high quality bathymetry data to enable improved ocean modeling

Aliased Tidal Variability in Mesoscale Sea Level Anomaly Maps

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

Ocean and Ice Shelf Tides from CryoSat-2 Altimetry

A new empirical model of ocean tides has been developed for the Weddell Sea, south of 668S, between 908W and 08, using six years of radar altimeter data from the CryoSat-2 satellite mission. Because of its long groundtrack 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 CryoSat2 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

On the Observability of Bottom Topography from Measurements of Tidal Sea Surface Height

of whether features of the ocean bottom topography can be identified from measurements of water level is investigated using a simplified one-dimensional barotropic model. Because of the nonlinear dependence of the sea surface height on the water depth, a linearized analysis is performed concerning the identification of a Gaussian bump within two specific depth profiles, (1) a constant depth domain, and, (2) a constant depth domain adjoining a near-resonant continental shelf. Observability is quantified by examining the estimation error in a series of identical-twin experiments varying data density, tide wavelength, assumed (versus actual) topographic correlation scale, and friction. For measurements of sea surface height that resolve the scale of the topographic perturbation, the fractional error in the bottom topography is approximately a factor of 10 larger than the fractional error of the sea surface height. Domain-scale and shelf-scale resonances may lead to inaccurate topography estimates due to a reduction in the effective number of degrees of freedom in the dynamics, and the amplification of nonlinearity. A realizability condition for the variance of the topography error in the limit of zero bottom depth is proposed which is interpreted as a bound on the fractional error of the topography. Appropriately designed spatial covariance models partly ameliorate the negative impact of shelf-scale near-resonance, and highlight the importance of spatial covariance modeling for bottom topography estimation

On the Temporal Variability of Low-Mode Internal Tides in the Deep Ocean

In situ measurements of internal tides are typically characterized by high temporal variability, with strong dependence on stratification, mesoscale eddies, and background currents commonly observed. Thus, it is surprising to find phase-locked internal tides detectable by satellite altimetry. An important question is how much tidal variability is missed by altimetry. We address this question in several ways. We subset the altimetry by season and find only very small changes -- an important exception being internal tides in the South China Sea where we observe strong seasonal dependence. A wavenumber-domain analysis confirms that throughout most of the global ocean there is little temporal variability in altimetric internal-tide signals, at least in the first baroclinic mode, which is the mode that dominates surface elevation. The analysis shows higher order modes to be significantly more variable. The results of this study have important practical implications for the anticipated SWOT wide-swath altimeter mission, for which removal of internal tide signals is critical for observing non-tidal submesoscale phenomena

Data Assimilation in Models with Convective Adjustment

Practical hydrostatic ocean models are often restricted to statically stable configurations by the use of a convective adjustment. A common way to do this is to assign an infinite boat conductivity to the water at a given level if the water column should become statically unstable. This is implemented in the form of a switch. When a statically unstable configuration is detected, it is immediately replaced with a statically stable one in which heat is conserved. In this approach, the model is no longer governed by a smooth set of equations, and usual techniques of variational data assimilation must be modified. In this note, a simple one-dimensional diffusive model is presented. Despite its simplicity, this model captures the essential behavior of the convective adjustment scheme in a widely used ocean general circulation model. Since this simple model can be derived from the more complex general circulation model, it then follows that many of the properties of the constrained system can be observed in this very simple scalar ordinary differential equation with a constraint on the solution. Techniques from the theory of optimal control are used to find solutions of a simple formulation of the variational data assimilation problem in this simple case. The optimal solution involves the solution of a nonlinear problem, even when the unconstrained dynamics are linear. In cases with discontinuous dynamics, one cannot define the adjoint of the linearized system in a straightforward manner. The very simplest variational formulation is shown to have nonunique stationary points and undesirable physical consequences. Modifications that lead to better behaved calculations and more meaningful solutions are presented. Whereas it is likely that the underlying principles from control theory are applicable to practical ocean models, the technique used to solve the simple problem may be applicable only to steady problems. Derivation of suitable techniques for initial value problems will involve a major research effort

Can Tidal Perturbations Associated with Sea Level Variations in the Western Pacific Ocean be used to Understand Future Effects of Tidal Evolution?

This study examines connections between mean sea level (MSL) variability and diurnal and semidiurnal tidal constituent variations at 17 open-ocean and 9 continental shelf tide gauges in the western tropical Pacific Ocean, a region showing anomalous rise in MSL over the last 20 years and strong interannual variability. Detrended MSL fluctuations are correlated with detrended tidal amplitude and phase fluctuations, defined as tidal anomaly trends (TATs), to quantify the response of tidal properties to MSL variation. About 20 significant amplitude and phase TATs are found for each of the two strongest tidal constituents, K1 (diurnal) and M2 (semidiurnal). Lesser constituents (O1 and S2) show trends at nearly half of all gauges. Fluctuations in MSL shift amplitudes and phases; both positive and negative responses occur. Changing overtides suggest TATs are influenced by changing shallow water friction over the equatorial Western Pacific and the eastern coast of Australia (especially near the Great Barrier Reef). There is a strong connection between semidiurnal TATs at stations around the Solomon Islands and changes in thermocline depth, overtide generation, and the El Niño Southern Oscillation (ENSO). TATs for O1, K1 and M2 are related to each other in a manner that suggests transfer of energy from M2 to the two diurnals via resonant triad interactions; these cause major tidal variability on sub-decadal time scales, especially for M2. The response of tides to MSL variability is not only spatially complex, it is frequency dependent; therefore, short-term responses may not predict long-term behavior

A Mathematical Model for Outgassing and Contamination

A model for the mathematical description of the processes of outgassing and contamination in a vacuum system is proposed. The underlying assumptions are diffusion in the source, convection and diffusion in the cavity, mass transfer across the source-cavity interface, and a generalization of the Langmuir isotherm for the sorption kinetics on the target. Three approximations are considered where the asymptotic behavior of the model for large time is shown as well as the dependence and sensitivity of the model on some of the parameters. Some numerical examples of the full model are then presented together with a proof of the uniqueness of the solution