Frequency-domain Empirical Orthogonal Functions Research Articles

Empirical estimation approach to the study of inter-area resonance interactions in power systems

Studies to deal with the identification of inter-area resonance interactions by fault diagnosis and historical databases at interconnected power systems have led to the development of new methods to delimit and split resonant regions and their control area regulations. This paper contributes in employing the frequency-domain empirical orthogonal functions (EOFs) for the measurement-based analysis of the inter-area resonance modes and its neighborhood. This multi-resolution empirical approach, developed for multiple-input multiple-output (MIMO) systems, is considered to contribute in understanding the spatial orientation properties and the dynamical/physical behavior of inter-area resonance interactions. Also, the method contributes much in advancing our knowledge of inter-area phase-matching patterns (PMPs) modes and their phase shifts from large data sets. In order to demonstrate the usefulness of obtaining and delimiting neighboring areas with resonant interactions in interconnected power systems, the proposed method is numerically applied in detail to the IEEE 16-generator 68-bus test system.

Winter variability in the western Gulf of Maine: Part 1: Internal tides

During the winter 1997–1998, a field program was conducted in Wilkinson Basin–western Gulf of Maine–as part of a study of winter convective mixing. The field program consisted of (1) Wilkinson basin-scale hydrographic surveys, (2) a tight three-mooring array with ∼100 m separations measured temperature and conductivity at rates of 2–15 min and (3) a single pair of upward/downward-looking pair acoustic Doppler current profiling (ADCP) instruments measured currents with 8 m vertical resolution over the 270 m water column in north-central Wilkinson basin at a rate of 10 min. The moored array measurements below the mixed layer (∼100 m depth) between 11 January and 6 February 1998 were dominated by a combination of the relatively strong semidiurnal external (depth-independent or barotropic) tide; upon which were superposed a weaker phase-locked semidiurnal internal tide and a very weak water column mean currents of about 1 cm/s southward or approximately across the local isobaths. The harmonic analysis of a vertical average of the relatively uniform ADCP velocities in the well-mixed upper 123 m of the water column, defined the external tidal currents which were dominated by a nearly rectilinear, across-isobath (326°T) M 2 semidiurnal tidal current of about 15 cm/s. The depth-dependent residual current field, which was created by subtracting the external tidal current, consisted of (1) clockwise-rotating semidiurnal internal tidal currents of about 5 cm/s below the mixed layer; (2) clockwise-rotating inertial currents; and (3) a considerably less energetic subtidal current variability. The results from both frequency-domain empirical orthogonal function and tidal harmonic analyses of the of isotherm displacement series at each of the three moorings in the 100 m array mutually confirm an approximate east-northeastward phase propagation of the dominant M 2 semidiurnal internal tide across Wilkinson Basin. Further investigation supports the idea that this winter internal tide is very likely generated by the interaction of the external tidal currents and the southwestern wall of Wilkinson Basin. The definitions of the local Wilkinson Basin external tide and phase-locked internal tides will enable us to analyze a less “noisy” set of measurements for the subtle atmospherically forced convective and wind-driven motions.

Instability of the Kuroshio in Luzon Strait: Effects of ridge topography and stratification

The spectral analyses of moored current velocities in the central Luzon Strait reveal northward (i.e., downstream of the Kuroshio) propagation of a frontal wave with a five-day period, with wave amplitude increasing northward. Estimated from both curve fitting and frequency domain Empirical Orthogonal Function methods, the characteristics of five-day variations have wave speeds ranging from 32 to 40 cm s−1, wavelengths ranging from 130 to 150 km, and e-folding time scales for growth ranging from 0.8 to 3 days. An analytical two-layer model used to explore linear stability characteristics indicates that bottom topography (two meridional ridges) is important for the Kuroshio stability characteristics in the Luzon Strait. In the two-layer model with the two ridges, the flow is stabilized for the long-wave mode but destabilized for the short-wave mode (due to increasing vertical shear in the horizontal velocity). The analytical model produces wavelengths and phase speeds for the most unstable mode which is similar to the observation, but the growth rate is underestimated. However, a spectral numerical model applied with a more realistic stratification and velocity structure does obtain faster growth rates comparable to the observations. Parameter sensitivity tests were conducted using the analytical model. The characteristics of the most unstable mode are most sensitive to the surface front location relative to the bottom topography but not sensitive to varying the density difference and thickness of the upper layer.

Monitoring of Generation and Propagation of the Kuroshio Small Meander Using Sea Level Data along the Southern Coast of Japan

The occurrence of the small meander of the Kuroshio, generated south of Kyushu and propagating eastward, was examined using sea level data collected during 1961–1995 along the south coast of Japan. Intra-annual variation of the sea level was expanded by the frequency domain empirical orthogonal function (FDEOF) modes, and it was found that the second and third modes are useful for monitoring the generation and propagation of the small meander. The third FDEOF for periods of 10–100 days has a phase reversal between Hosojima and Tosa-shimizu with significant amplitude west of Kushimoto, and the amplitude of its time coefficient is large during the non-large-meander (NLM) period and has a significant peak when the small meander exists southeast of Kyushu. The second FDEOF for periods of 20–80 days has a phase reversal between Kushimoto and Uragami, and the amplitude of its time coefficient is large when the small meander propagates to the south of Shikoku. The third FDEOF mode allowed us to conclude that the small meander occurred 42 times from July 1961 to May 1995, most of them (38) occurring during the NLM periods. The second FDEOF mode permits the conclusion that half of the 38 small meanders reached south of Shikoku. Of these, five small meanders influenced transitions of the Kuroshio path from the nearshore NLM path; one caused the offshore NLM path and four brought about the large meander. About one-tenth of the total number of small meanders are related to the formation of the large meander.

Field observations of the wave bottom boundary layer

This paper presents a comprehensive set of velocity and suspended sediment observations in the nearshore wave bottom boundary layer, collected during the Duck94 field experiment on the Outer Banks of the North Carolina coast. Cross‐shore velocity measurements in the wave bottom boundary layer were made using five hot film anemometers, nominally spaced from 1 to 5 cm above the bed in 2 m of water depth. The time‐varying location of the seabed was estimated to roughly 1 cm with a stacked set of bed‐penetrating fiber‐optic backscatter sensors. The instrument array was intermittently located in the surf zone on the crest of a bar. The location of the bottom varied several centimeters over a 34 min data run. Even over 4 min segments of quasi‐steady statistics, occasional large waves caused short erosion and redeposition events, complicating the definition of bottom location and causing the root‐mean‐square velocity statistics to be nonzero below the mean bed location. This leads to obvious difficulties in comparisons with two, one‐dimensional time‐dependent, eddy viscosity wave bottom boundary layer models. For example, bed shears based on rms amplitude decay were lower than predicted. The observations show some evidence for a velocity overshoot region within the wave bottom boundary layer. The observations were compared with two linear eddy viscosity models. Larger estimates of a constant eddy viscosity and smaller than predicted phase leads are indicative of more rapid mixing of momentum than predicted by the models. The phase and amplitude frequency response estimated with frequency domain empirical orthogonal functions shows a nonlinear response of the wave bottom boundary layer over the incident band. These observations are among the first coherent looks at the wave bottom boundary layer under conditions of significant sediment response. They highlight the added complexity of the dynamics in natural environments.

Internal Tides near the Kermadec Ridge

Current meter data from the WOCE PCM-9 array, extending 1000 km eastward from the Kermadec Ridge northeast of New Zealand, are used to investigate the internal tide. A high degree of spatial and temporal variability in the semidiurnal tide, but not in the diurnal tide, suggests that there is a conversion from barotropic to baroclinic tidal energy at the semidiurnal periods. Comparison between observed tidal constituents and those predicted by the TPXO.3 model indicates that the model may be used as an estimate of the barotropic tide. Once the barotropic tide has been removed, the remaining semidiurnal energy propagates predominantly at the first vertical mode phase speed of 3.5 m s−1, consistent with linear theory. A frequency-domain empirical orthogonal function analysis is used to extract signals that have phase propagation consistent with a coherent internal tide. These modes represent a phase-locked semidiurnal internal tide having about 0.65 of the baroclinic energy. Tidal energies at the M2 period associated with the phase-locked tide are about 155 J m−2 over the Kermadec Ridge and 98 J m−2 over the abyssal plain east of the ridge. Total tidal power radiation along the ridge is estimated to be about 0.8 GW.

Observations on the Variability of the Gulf Stream Path between 74°W and 70°W

The Gulf Stream path varies in a wavelike fashion in the deep water downstream of Cape Hatteras, energetic within a range of wavelengths λ from 150 km to over 1000 km, and periods τ from 4 d to over 500 d. The authors determined the path from the 12°C isotherm at 400-m depth, which was obtained daily for 740 days (April 1983–May 1985) from objective analysis maps of the thermocline topography measured by inverted echo sounders (IESs). The IESs were deployed in a region 400 km long from 74° to 70°W. To find propagating and standing mode characteristics of this large array of data, frequency-domain empirical orthogonal functions and wavenumber-frequency spectra were calculated, using FFT in time combined with autoregressive (AR) modeling in space. This study reveals smooth transitions from (i) downstream propagating and growing short-period meander modes, to (ii) standing modes at intermediate period, to (iii) upstream propagating long-period modes. For periods shorter than semiannual, meanders propagate downstream with speeds monotonically increasing from 6 km d−1 for (λ, τ)∼(1000 km, 160 d), to 55 km d−1 for (λ, τ) ∼ (180 km, 3 d). Their maximum growth rates occur at (λ, τ) ∼ (260 km, 8 d), with e-folding spatial scales ∼250 km, and e-folding temporal scales ∼3 d. For periods between semiannual and annual, the entire path shifts like a standing mode with no detectable phase lag within our study region, and its envelope accounts for the local minimum ("node") that has been observed near 70°W. For periods from annual to at least 500 d, fluctuations propagate upstream at about 9 km d−1, and their amplitude decreases to the west in the direction of propagation.

Fluctuations of Gulf Stream frontal position between Cape Hatteras and the Straits of Florida

Low‐frequency variations in the onshore/offshore position of the Gulf Stream thermal front have been examined at 10 cross‐isobath transects equally spaced along the South Atlantic Bight between the Straits of Florida and Cape Hatteras. The data were obtained from the U.S. Navy's frontal analysis charts for the 12‐year period, 1976–1988. A time domain empirical orthogonal function (EOF) analysis revealed two modes of fluctuation about the mean Gulf Stream position, which together explain the well‐known “weakly and strongly deflected” frontal position states which occur on timescales of a few months in the vicinity of the Charleston Bump. Analyses of fluctuations at the annual period revealed a 3‐month phase lag between frontal position, as shown by the most significant time domain EOF mode, and volume transport through the Straits of Florida. In addition, frequency domain EOF and cross‐spectral analyses of frontal position revealed annual waves which appear to propagate to the south (north) in the region south (north) of the bump. For transect pairs having significant coherence and nonzero phase estimates, the phase speed and wavelength were about 10 cm/s southward (7 cm/s northward) and 3000 km (2000 km) in the southern (northern) region. This suggests an annual generation of a perturbation in the vicinity of the bump which radiates away as a long annual wave and which may be related to, or part of, the mechanism which produces the annual modulation of Gulf Stream transport.

Currents off south-eastern Australia: results from the Australian coastal experiment

The Australian Coastal Experiment was conducted off the east coast of New South Wales between September 1983 and March 1984. The experiment was conducted with arrays of current meters spanning the continental margin at three latitudes (37.5�, 34.5�, and 33.0�S.), additional shelf moorings at 29� and 42�S. coastal wind and sea-level measurements, monthly conductivity-temperature-depth probe/expendable bathythermograph (CTD/XBT) surveys, and two satellite-tracked buoys. Over the continental shelf and slope, the alongshore component of the current generally exceeded the onshore component, and the subtidal (<0.6 cpd, cycles per day) current variability greatly exceeded the mean flow. Part of the current variability was associated with two separate warm-core eddies that approached the coast, causing strong (>50 cm sec-1), persistent (>8 days), southward currents over the continental slope and outer shelf. Temperature and geostrophic velocity sections through the eddies, maps of ship's drift vectors and temperature contours at 250 m, and the satellite-tracked drifter trajectories showed that these eddies were similar in structure to those observed previously in the East Australian Current region. Both eddies migrated generally southward. Eddy currents over the shelf and slope were rare at Cape Howe (37.5�S.), more common near Sydney (34.5�S.), and frequent at Newcastle (33.0�S.), where strong northward currents were also observed. Near Sydney, the eddy currents over the slope turned clockwise with depth between 280 and 740 m, suggesting net downwelling there. Repeated CTD sections also indicated onshore transport and downwelling at shallower levels; presumably, upwelling occurred farther south where the eddy currents turned offshore. Periodic rotary currents over the continental slope near Sydney and Newcastle indicated the presence of small cyclonic eddies on the flank of a much larger anticyclonic eddy. Between early October and late January, no strong southward currents were observed over the continental margin near Sydney. Data from this 'eddy-free' period were analysed further to examine the structure and variability of the coastal currents. Much of this variability was correlated with fluctuations in coastal sea-level (at zero lag) and with the wind stress (at various lags). The coherence and phase relationships among current, wind-stress, and sea-level records at different latitudes (determined from spectral analysis and frequency-domain empirical orthogonal functions) were consistent with the equatorward propagation of coastal-trapped waves generated by winds in phase with those near Cape Howe. Time-domain empirical orthogonal functions show that the current fluctuations decayed with distance from shore and with depth, as expected of coastal-trapped waves.

Further Evidence for Coastal Trapped Waves along the Peru Coast

Abstract Time series of coastal sea level during 1976–77, from 2°12′S to 17°S along the west coast of South America, show that low-frequency, ω < 0.25 cycles per day (cpd), fluctuations propagate poleward with the phase speed of baroclinic Kelvin waves (2–3 m s−1). The alongshore coherence is highest in the frequency band 0.1–0.2 cpd. Computing the frequency-domain empirical orthogonal functions (EOF) for alongshore current, from an army of current meters extending from 5°S to 15°S during March-May 1977, gives 70% of the variance in the 0.1–0.2 cpd frequency band to an EOF mode with poleward phase propagation at 2.75 m s−1. The vertical structure of the alongshore current fluctuations (0.1–0.2 cpd) over the continental slope at 5°S and 15°S is consistent with a first-mode baroclinic Kelvin wave. The current and sea-level fluctuations are coherent and propagate poleward through latitudes where their frequency equals the local inertial frequency. The fluctuations are not significantly coherent with coastal ...