Baroclinic stability for a family of two-level, semi-implicit numerical methods for the 3D shallow water equations

• Published in: International journal for numerical methods in fluids Vol. 54; no. 3; pp. 237 - 268
• Main Authors: Rueda, Francisco J., Sanmiguel-Rojas, Enrique, Hodges, Ben R.
• Format: Journal Article
• Language: English
• Published: Chichester, UK John Wiley & Sons, Ltd 30.05.2007; Wiley
• Subjects: 3D shallow water; baroclinic terms; Earth sciences; Earth, ocean, space; Exact sciences and technology; Hydrology; Hydrology. Hydrogeology; stability analysis; two-level semi-implicit methods; two-level semi-implicit methods; stability analysis; digital simulation; 3D shallow water; Modeling; Euler scheme; Predictor corrector method; Baroclinic instability; Boussinesq equation; Navier Stokes equation; Continuity equation; Free surface flow; estuaries; lakes; Shallow-water equations; baroclinic terms
• ISSN: 0271-2091; 1097-0363
• DOI: 10.1002/fld.1391

The baroclinic stability of a family of two time‐level, semi‐implicit schemes for the 3D hydrostatic, Boussinesq Navier–Stokes equations (i.e. the shallow water equations), which originate from the TRIM model of Casulli and Cheng (Int. J. Numer. Methods Fluids 1992; 15:629–648), is examined in a simple 2D horizontal–vertical domain. It is demonstrated that existing mass‐conservative low‐dissipation semi‐implicit methods, which are unconditionally stable in the inviscid limit for barotropic flows, are unstable in the same limit for baroclinic flows. Such methods can be made baroclinically stable when the integrated continuity equation is discretized with a barotropically dissipative backwards Euler scheme. A general family of two‐step predictor‐corrector schemes is proposed that have better theoretical characteristics than existing single‐step schemes. Copyright © 2006 John Wiley & Sons, Ltd.