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# Global Linear Stability of the Two-Dimensional Shallow-Water Equations: An Application of the Distributive Theorem of Roots for Polynomials on the Unit Circle

Jia Wang

## Abstract

This paper deals with the numerical stability of the linearized shallow-water dynamic and thermodynamic system using centered spatial differencing and leapfrog time differencing. The nonlinear version of the equations is commonly used in both 2D and 3D (split technique) numerical models. To establish the criteria, we employ the theorem of the root distributive theory of a polynomial proposed by Cheng (1966). The Fourier analysis or von Neumann method is applied to the linearized system to obtain a characteristic equation that is a sixth-order polynomial with complex coefficients. Thus, a series of necessary and sufficient criteria (but only necessary conditions for the corresponding nonlinear equations) are obtained by applying Cheng's theorem within the unit circle. It is suggested that the global stability should be determined by this set of criteria rather than the Courant–Friedrichs–Lewy (CFL) criterion alone. Each of the conditions has physical meaning: for instance, h + ζ > 0, |f| Δt < 1, and 0 < Δtβ^′ < 1, etc., must be satisfied as well, which helps define the model domain and the relation between damping coefficients and integration time step, where h is the undisturbed water depth, ζ the free surface elevation, f the Coriolis parameter, β^′ the sum of bottom friction coefficient and horizontal viscosity, and Δt the integrating time step. The full solution and the physical implications are given in the paper. Since Cheng's theorem was published in Chinese only and is of considerably theoretical and practical value in numerical stability analysis, the translation of the theorem is in appendix A.

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# A Nowcast/Forecast System for Coastal Ocean Circulation Using Simple Nudging Data Assimilation

Jia Wang

## Abstract

This study describes the establishment of a Nowcast/Forecast System for Coastal Ocean Circulation (NFS-COC), which was run operationally on a daily basis to provide users ocean surface currents and sea levels that vary with synoptic winds, and seasonal and mesoscale variability intrinsic to the Florida Current. Based on the requirements of users, information about possible oil spills, trajectories, etc., is also provided by NFS-COC.

NFS-COC consists of two parts: a 3D ocean nowcast/forecast circulation model, Princeton Ocean Model (POM), and a 2D trajectory model. POM is automatically run to forecast ocean variables for up to 2 days under forcing of the Florida Current inflow/outflow and the predicted surface winds, which are automatically transferred (by ftp) from a file server at the National Meteorological Center (now known as the National Centers for Environmental Prediction). The winds from the mesoscale Eta Model are called Eta winds. Then the trajectory model is run to predict the path due to 1) the POM-predicted ocean surface currents, 2) wind drift due to the predicted Eta winds, and 3) turbulent dispersion based on a random flight (Markov process) model. The predicted surface trajectories can be used to estimate the physical transport of oil spills (and other drifting or floating objects) in the Straits of Florida and many other coastal seas. A simple data assimilation scheme (nudging to the volume transport) is designed into the NFS-COC, although some powerful data assimilation methods exist for assimilating other physical variables.

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# Inertial Stability and Phase Error of Time Integration Schemes in Ocean General Circulation Models

Jia Wang and Moto Ikeda

## Abstract

Numerical finite-difference schemes of time integration in widely used ocean general circulation models are systematically examined to ensure the correct and accurate discretization of the Coriolis terms. Two groups of numerical schemes are categorized. One group is suitable for simulating an inertial wave system and geostrophic adjustment processes in the ocean with the necessary condition for stability being |F| = |f| Δt < 1 (where f is the Coriolis parameter and Δt is the integration time step in the model), such as the predictor–corrector scheme (as shown in this study), the most commonly used leapfrog scheme (as used in MICOM, POM, SPEM, and many others), Euler-centered scheme (as used in SOMS), and leapfrog scheme plus Euler-centered Coriolis terms [as used in the Geophysical Fluid Dynamics Laboratory (GFDL) model]. The other group is able to serve as a long-term climate study using a large integration time step that may violate |F| = |f| Δt < 1 by damping out inertial waves, such as the GFDL scheme plus Euler-backward Coriolis terms and the Euler predictor–corrector scheme plus an implicit treatment of the Coriolis terms used in OPYC model. Caution is made regarding the use of the Euler-forward and other schemes that produce unstable inertial waves; this problem could be serious for a calculation longer than one week. The predictor–corrector scheme is recommended as a replacement for the simple Euler-forward scheme. The explicit leapfrog and predictor–corrector schemes tend to overestimate the phase frequency, whereas the Euler schemes and implicit schemes underestimate it. To better simulate the correct phase frequency, F < 0.1 is recommended. Furthermore, an alternate use of an explicit scheme (e.g., leapfrog) and an implicit scheme (e.g., Euler backward or Masuno scheme, etc.) is strongly recommended to preserve the correct phase frequency.

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# Diagnosing Ocean Unstable Baroclinic Waves and Meanders Using the Quasigeostrophic Equations and Q-Vector Method

Jia Wang and Moto Ikeda

## Abstract

A three-dimensional, primitive equation model is applied to the ocean mesoscale eddies and unstable baroclinic waves across a density front in a channel under a very low viscosity environment. Current meanders are well produced. The unstable baroclinic waves are examined for flat, positive (same sense as isopycnal tilt) and negative sloping bottoms. The growth rates with flat, gentle, medium, and steep slopes and with different wavelengths (wavenumbers) are discussed. A positive slope clearly suppresses the meandering wave growth rate whose maximum slightly shifts to a lower wavenumber compared to the flat bottom. A gentle negative slope, however, favors the wave growth with the maximum shifting toward higher wavenumber. When the negative slope becomes steeper, the growth rate significantly decreases correspondingly.

Furthermore, a diagnostic analysis package for the pressure tendency and vertical velocity equations, analogous to the approaches in meteorology (ω equation and Q-vector method), is developed for the first time to reveal the physical processes and mechanisms of the unstable wave propagation in the midlatitude ocean. The weaknesses and strengths of these two diagnostic approaches are evaluated and compared to the model results. The Q-vector method is superior to the quasigeostrophic ω equation for diagnosing the vertical motion associated with the mesoscale dynamics from a hydrographic CTD array because the former has no phase error.

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# Seasonal Exchanges of the Kuroshio and Shelf Waters and Their Impacts on the Shelf Currents of the East China Sea

Jia Wang and Lie Yauw Oey

## Abstract

Previous in situ observations and modeling studies have indicated that, through mass and momentum exchanges across the shelf edge, the Kuroshio can significantly influence the shelf currents of the East China Sea (ECS). Here, instead of localized observations, this study uses 25 yr of drifter data, supported by satellite and other data to identify seasonal cross-shelf exchanges along the entire shelf edge. The authors show that Kuroshio meanders onshore from fall to winter and offshore from spring to summer, with the largest amplitude northeast of Taiwan. The influence is limited to the shelf edge when the Kuroshio meanders offshore in spring and summer. By contrast, strong onshelf intrusions and cross-shelf exchanges occur when the Kuroshio meanders onshore in fall and winter. Drifters intrude onshelf northeast of Taiwan and spread as far north as 30°N against the strong northeasterly wind. The forcing on the shelf is identified as a northward downsloping of the sea level that is steepest north of Taiwan at 25°–28°N, but which is 3 times weaker farther north. The vorticity budget computed from a numerical model indicates that intrusion during fall and winter is primarily a result of balance between onshelf advection of ambient potential vorticity and vorticity production by the along-isobath pressure gradient acting on the changing mass of water column across the continental slope.

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# Possible Feedback of Winter Sea Ice in the Greenland and Barents Seas on the Local Atmosphere

Bingyi Wu, Jia Wang, and John Walsh

## Abstract

Using monthly Arctic sea ice concentration data (1953–95) and the National Centers for Environmental Prediction–National Center for Atmospheric Research (NCEP–NCAR) reanalysis dataset (1958–99), possible feedbacks of sea ice variations in the Greenland and Barents Seas to the atmosphere are investigated. Winter (February–April) sea ice anomalies in the Greenland and Barents Seas display important feedback influences on the atmospheric boundary layer in terms of both thermodynamic and dynamic processes. The vertical gradients of potential pseudo-equivalent temperature (θ se) between 850 and 700 hPa are greater over sea ice than over open water, implying that a more stable boundary layer forms below 700 hPa over sea ice. The effects of temperature advection are shown to account for a relatively small percentage of the temperature variance in area of sea ice feedbacks. Horizontal and vertical variations of the effects of sea ice on temperature in the Nordic and Barents Seas create the potential for dynamical influences on the atmospheric boundary layer through vertical motion induced by the pressure anomalies in the boundary layer.

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# Dipole Anomaly in the Winter Arctic Atmosphere and Its Association with Sea Ice Motion

Bingyi Wu, Jia Wang, and John E. Walsh

## Abstract

This paper identified an atmospheric circulation anomaly–dipole structure anomaly in the Arctic atmosphere and its relationship with winter sea ice motion, based on the International Arctic Buoy Program (IABP) dataset (1979–98) and datasets from the National Centers for Environmental Prediction (NCEP) and the National Center for Atmospheric Research (NCAR) for the period 1960–2002. The dipole anomaly corresponds to the second-leading mode of EOF of monthly mean sea level pressure (SLP) north of 70°N during the winter season (October–March) and accounts for 13% of the variance. One of its two anomalous centers is stably occupied between the Kara Sea and Laptev Sea; the other is situated from the Canadian Archipelago through Greenland extending southeastward to the Nordic seas. The dipole anomaly differs from one described in other papers that can be attributed to an eastward shift of the center of action of the North Atlantic Oscillation. The finding shows that the dipole anomaly also differs from the “Barents Oscillation” revealed in a study by Skeie. Since the dipole anomaly shows a strong meridionality, it becomes an important mechanism to drive both anomalous sea ice exports out of the Arctic Basin and cold air outbreaks into the Barents Sea, the Nordic seas, and northern Europe.

When the dipole anomaly remains in its positive phase, that is, negative SLP anomalies appear between the Kara Sea and the Laptev Sea with concurrent positive SLP over from the Canadian Archipelago extending southeastward to Greenland, there are large-scale changes in the intensity and character of sea ice transport in the Arctic basin. The significant changes include a weakening of the Beaufort gyre, an increase in sea ice export out of the Arctic basin through Fram Strait and the northern Barents Sea, and enhanced sea ice import from the Laptev Sea and the East Siberian Sea into the Arctic basin. Consequently, more sea ice appears in the Greenland and the Barents Seas during the positive phase of the dipole anomaly. During the negative phase of the dipole anomaly, SLP anomalies show an opposite scenario in the Arctic Ocean and its marginal seas when compared to the positive phase, with the center of negative SLP anomalies over the Nordic seas. Correspondingly, sea ice exports decrease from the Arctic basin flowing into the Nordic seas and the northern Barents Sea because of the strengthened Beaufort gyre.

The finding indicates that influences of the dipole anomaly on winter sea ice motion are greater than that of the winter AO, particularly in the central Arctic basin and northward to Fram Strait, implying that effects of the dipole anomaly on sea ice export out of the Arctic basin become robust. The dipole anomaly is closely related to atmosphere–ice–ocean interactions that influence the Barents Sea sector.

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# Severe Ice Conditions in the Bohai Sea, China, and Mild Ice Conditions in the Great Lakes during the 2009/10 Winter: Links to El Niño and a Strong Negative Arctic Oscillation

Xuezhi Bai, Jia Wang, Qinzheng Liu, Dongxiao Wang, and Yu Liu

## Abstract

This study investigates the causes of severe ice conditions over the Bohai Sea, China, and mild ice cover over the North American Great Lakes under the same hemispheric climate patterns during the 2009/10 winter with a strong negative Arctic Oscillation (AO) and an El Niño event. The main cause of severe ice cover over the Bohai Sea was the strong negative AO. Six of seven winters with severe ice were associated with a strong negative AO during the period 1954–2010. The Siberian high (SH) in the 2009/10 winter was close to normal. The influence of El Niño on the Bohai Sea was not significant. In contrast, the mild ice conditions in the Great Lakes were mainly caused by the strong El Niño event. Although the negative AO generally produces significant colder surface air temperature (SAT) and heavy ice cover over the Great Lakes, when it coincided with a strong El Niño event during the 2009/10 winter the El Niño–induced Pacific–North America (PNA)-like pattern dominated the midlatitudes and was responsible for the flattening of the ridge–trough system over North America, leading to warmer-than-normal temperatures and mild ice conditions over the Great Lakes. This comparative study revealed that interannual variability of SAT in North America, including the Great Lakes, is effectively influenced by El Niño events through a PNA or PNA-like pattern whereas the interannual variability of SAT in northeastern China, including the Bohai Sea area, was mainly controlled by AO and SH.

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# A Numerical Simulation of Sea Ice Cover in Hudson Bay

Jia Wang, L. A. Mysak, and R. G. Ingram

## Abstract

Hibler's dynamic-thermodynamic sea ice model with viscous-plastic rheology is used to simulate the seasonal cycle of sea ice motion, thickness, compactness, and growth rate in Hudson Bay under monthly climatological atmospheric forcing and a prescribed ocean surface current field. The sea ice motion over most of the domain is driven mainly by the wind stress. Wintertime sea ice velocities are only of the order of 1–5 (× 10−4 m s−1) due to the nearly solid ice cover and the closed boundary constraint of Hudson Bay. However, the velocities rise to 0.10–0.20 m s−1 during the melting and freezing seasons when there is partial ice cover. The simulated thickness distribution in mid–April, the time of heaviest ice cover, ranges from 1.3 m in James Bay to 1.7 m in the northern part of Hudson Bay, which compares favorably with observations. The area-averaged growth rate, computed from the model is 1.5–0.5 cm day−1 from December to March, is negative in May (indicative of melting) and reaches its minimum value of −4.2 cm day−1 (maximum melting rate) in July. During autumn, the main freezing season, the growth rate ranges from 1 to 2 cm day−1. In the model, sea ice remains along the south shore of Hudson Bay in summer, as observed, even though the surface air temperatures are higher there than in central and northern Hudson Bay. A sensitivity experiment shows that this is mainly due to the pile-up of ice driven southward by the northwesterly winds. The simulated results for ice cover in other seasons also compare favorably with the observed climatology and with measurements from satellites. In particular, the model gives complete sea ice cover in winter and ice-free conditions in late summer. A series of sensitivity experiments in which the model parameters and external forcing are varied is also carried out.

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# A Three-Dimensional Numerical Simulation of Hudson Bay Summer Ocean Circulation: Topographic Gyres, Separations, and Coastal Jets

Jia Wang, Lawrence A. Mysak, and R. Grant Ingram

## Abstract

The summer ocean circulation in Hudson Bay is studied numerically using the Blumberg-Mellor model with a 27.5 km × 27.5 km horizontal grid and a realistic bottom topography. In the control run 1) monthly climatological forcing fields of wind stress, oceanic inflow/outflow, and salt and heat fluxes are used. In addition, results are presented for a number ot sensitivity experiments: 2) no topography (otherwise conditions are identical to the control run), 3) no wind forcing, 4) no oceanic inflow/outflow, 5) no heat and salt fluxes, 6) no temperature and salinity variations, and 7) without the nonlinear terms.

While the overall simulated circulation in Hudson Bay is cyclonic, the strong steering of the flow by the bathymetry is particularly noticeable. Mesoscale topographic gyres are simulated, and the separation of the coastal current due to topographic bumps occurs in several locations. The simulated circulation also has well-developed vorticity features and narrow, density-driven coastal jets along the western, southern, and eastern shores of Hudson Bay, which enhance the wind-driven alongshore current. From various sensitivity experiments, it is estimated that the total transport of 0.55 Sv (Sv ≡ 106 m3 s−1) is made up of a 0.23 Sv wind-driven transport, a 0.12 Sv density-driven transport, and a 0.2 Sv inflow/outflow induced transport. It is also found that the wind-driven circulation in Hudson Bay shows a recirculation, whereas the density-driven and inflow/outflow induced transports do not.

A one-dimensional version of the model is also used to simulate the thermohaline vertical structure over a seasonal cycle. In particular, the observed deepening of the mixed layer in fall is reasonably well reproduced by the model.

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