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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 Courantâ€“Friedrichsâ€“Lewy (CFL) criterion alone. Each of the conditions has physical meaning: for instance, h + Î¶ > 0, |f| Î”t < 1, and 0 < Î”tÎ²^â€² < 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, Î¶ the free surface elevation, f the Coriolis parameter, Î²^â€² the sum of bottom friction coefficient and horizontal viscosity, and Î”t 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| Î”t < 1 (where f is the Coriolis parameter and Î”t is the integration time step in the model), such as the predictorâ€“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| Î”t < 1 by damping out inertial waves, such as the GFDL scheme plus Euler-backward Coriolis terms and the Euler predictorâ€“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 predictorâ€“corrector scheme is recommended as a replacement for the simple Euler-forward scheme. The explicit leapfrog and predictorâ€“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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# The Impact of Constrained Data Assimilation on the Forecasts of Three Convection Systems during the ARM MC3E Field Campaign

Jia Wang
and
Minghua Zhang

## Abstract

A constrained data assimilation (CDA) system based on the ensemble variational (EnVar) method and physical constraints of mass and water conservations is evaluated through three convective cases during the Midlatitude Continental Convective Clouds Experiment (MC3E) of the Atmospheric Radiation Measurement (ARM) program. Compared to the original data assimilation (ODA), the CDA is shown to perform better in the forecasted state variables and simulated precipitation. The CDA is also shown to greatly mitigate the loss of forecast skills in observation denial experiments when radar radial winds are withheld in the assimilation. Modifications to the algorithm and sensitivities of the CDA to the calculation of the time tendencies in the constraints are described.

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# A Constrained Data Assimilation Algorithm Based on GSI Hybrid 3D-EnVar and Its Application

Jia Wang
and
Minghua Zhang

## Abstract

Data assimilation (DA) at mesoscales is important for severe weather forecasts, yet the techniques of data assimilation at this scale remain a challenge. This study introduces dynamical constraints in the Gridpoint Statistical Interpolation (GSI) three-dimensional ensemble variational (3D-EnVar) data assimilation algorithm to enable the use of high-resolution surface observations of precipitation to improve atmospheric analysis at mesoscales. The constraints use the conservations of mass and moisture. Mass constraint suppresses the unphysical high-frequency oscillation, while moisture conservation constrains the atmospheric states to conform with the observed high-resolution precipitation. We show that the constrained data assimilation (CDA) algorithm significantly reduced the spurious residuals of the mass and moisture budgets compared to the original data assimilation (ODA). A case study is presented for a squall line over the Southern Great Plains on 20 May 2011 during Midlatitude Continental Convective Clouds Experiment (MC3E) of the Atmospheric Radiation Measurement (ARM) program by using ODA or CDA analysis as initial condition of forecasts. The state variables, and the location and intensity of the squall line are better simulated in the CDA experiment. Results show how surface observation of precipitation can be used to improve atmospheric analysis through data assimilation by using the dynamical constraints of mass and moisture conservations.

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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 (Ï‰ 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 Ï‰ 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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# Joint Boost to Super El NiÃ±o from the Indian and Atlantic Oceans

Jia-Zhen Wang
and
Chunzai Wang

## Abstract

Super El NiÃ±o has been a research focus since the first event occurred. On the basis of observations and models, we propose that a super El NiÃ±o emerges if El NiÃ±o is an early-onset type coincident with the distribution of an Atlantic NiÃ±a (AN) in summer and a positive Indian Ocean dipole (IOD) in autumn, conditions referred to as the Indo-Atlantic Booster (IAB). The underlying physical mechanisms refer to three-ocean interactions with seasonality. Early onset endows super El NiÃ±o with adequate strength in summer to excite wind-driven responses over the Indian and Atlantic Oceans, which further facilitate IAB formation by coupling with the seasonal cycle. In return, IAB alternately produces additional zonal winds U over the Pacific Ocean, augmenting super El NiÃ±o via the Bjerknes feedback. Adding AN and IOD indices into the regression model of U leads to a better performance than the single NiÃ±o-3.4 model, with a rise in the total explained variances by 10%â€“20% and a reduction in the misestimations of super El NiÃ±os by 50%. Extended analyses using Coupled Model Intercomparison Project models further confirm the sufficiency and necessity of early onset and IAB on super El NiÃ±o formation. Approximately 70% of super El NiÃ±os are early-onset types accompanied by IAB and 60% of early-onset El NiÃ±os with IAB finally grow into extreme events. These results highlight the super El NiÃ±o as an outcome of pantropical interactions, so including both the Indian and Atlantic Oceans and their teleconnections with the Pacific Ocean will greatly improve super El NiÃ±o prediction.

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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 (1953â€“95) and the National Centers for Environmental Predictionâ€“National Center for Atmospheric Research (NCEPâ€“NCAR) reanalysis dataset (1958â€“99), possible feedbacks of sea ice variations in the Greenland and Barents Seas to the atmosphere are investigated. Winter (Februaryâ€“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 (Î¸ 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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# 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 30Â°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 25Â°â€“28Â°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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# Influence of a River Plume on Coastal Upwelling Dynamics: Importance of Stratification

Zhaoyun Chen
,
Yuwu Jiang
,
Jia Wang
, and
Wenping Gong

## Abstract

Satellite images show that the Pearl River plume is entrained into the upwelling front in the northeastern South China Sea. To understand the processes and extend to other coastal zones, an idealized numerical model is used to investigate the upwelling dynamics in response to the arrival of the river plume. Upon forcing by an upwelling-favorable wind, the model reproduces the upwelling frontal jet with a stratified water column, which takes the river plume far away from the mouth of the estuary. The river plume introduces additional upwelling and downwelling at its inshore and offshore sides (defined as plume-related secondary upwelling circulation), respectively. For the initially unstratified water column, the plume-related secondary upwelling circulation is stronger and extends to deeper water than for the stratified condition. The surface boundary layer thins and the offshore current intensifies in the river plume. The variations in wind-driven current over the deep-water shelf in different stratified conditions are modulated by the vertical profiles of the eddy viscosity, which are shown by a one-dimensional numerical model. Offshore transport is reinforced when the head of the river plume arrives. Thereafter, it is changed by the cross-shore baroclinic geostrophic component of velocity, due to alongshore density variation by the river plume. The horizontal gradient of stress on the two sides of the river plume is responsible for the plume-related secondary upwelling circulation owing to different stress decay scales inside and outside the river plume.

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