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• Author or Editor: Jia Wang
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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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# 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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# 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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# Application of a Hybrid Statistical–Dynamical System to Seasonal Prediction of North American Temperature and Precipitation

Sarah Strazzo, Dan C. Collins, Andrew Schepen, Q. J. Wang, Emily Becker, and Liwei Jia

## Abstract

Recent research demonstrates that dynamical models sometimes fail to represent observed teleconnection patterns associated with predictable modes of climate variability. As a result, model forecast skill may be reduced. We address this gap in skill through the application of a Bayesian postprocessing technique—the calibration, bridging, and merging (CBaM) method—which previously has been shown to improve probabilistic seasonal forecast skill over Australia. Calibration models developed from dynamical model reforecasts and observations are employed to statistically correct dynamical model forecasts. Bridging models use dynamical model forecasts of relevant climate modes (e.g., ENSO) as predictors of remote temperature and precipitation. Bridging and calibration models are first developed separately using Bayesian joint probability modeling and then merged using Bayesian model averaging to yield an optimal forecast. We apply CBaM to seasonal forecasts of North American 2-m temperature and precipitation from the North American Multimodel Ensemble (NMME) hindcast. Bridging is done using the model-predicted Niño-3.4 index. Overall, the fully merged CBaM forecasts achieve higher Brier skill scores and better reliability compared to raw NMME forecasts. Bridging enhances forecast skill for individual NMME member model forecasts of temperature, but does not result in significant improvements in precipitation forecast skill, possibly because the models of the NMME better represent the ENSO–precipitation teleconnection pattern compared to the ENSO–temperature pattern. These results demonstrate the potential utility of the CBaM method to improve seasonal forecast skill over North America.

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