Two Problems in Computational Wave Dynamics: Klemp-Wilhelmson Splitting at Large Scales and Wave-Wave Instabilities in Rotating Mountain Waves



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Two problems in computational wave dynamics are considered: (i) the use of Klemp-Wilhelmson time splitting at large scales and (ii) analysis of wave-wave instabilities in nonhydrostatic and rotating mountain waves. The use of Klemp-Wilhelmson (KW) time splitting for large-scale and global modeling is assessed through a series of von Neumann accuracy and stability analyses. Two variations of the KW splitting are evaluated in particular: the original acousticmode splitting of Klemp and Wilhelmson (KW78) and a modified splitting due to Skamarock and Klemp (SK92) in which the buoyancy and vertical stratification terms are treated as fast-mode terms. The large-scale cases of interest are the problem of Rossby wave propagation on a resting background state and the classic baroclinic Eady problem. The results show that the original KW78 splitting is surprisingly inaccurate when applied to large-scale wave modes. The source of this inaccuracy is traced to the splitting of the hydrostatic balance terms between the small and large time steps. The errors in the KW78 splitting are shown to be largely absent from the SK92 scheme. Resonant wave-wave instability in rotating mountain waves is examined using a linear stability analysis based on steady-state solutions for flow over an isolated ridge. The analysis is performed over a parameter space spanned by the mountain height (Nh/U) and the Rossby number (U/fL). Steady solutions are found using a newly developed solver based on a nonlinear Newton iteration. Results from the steady solver show that the critical heights for wave overturning are smallest for the hydrostatic case and generally increase in the rotating wave regime. Results of the stability analyses show that the wave-wave instability exists at mountain heights even below the critical overturning values. The most unstable cases are found in the nonrotating regime while the range of unstable mountain heights between initial onset and critical overturning is largest for intermediate Rossby number.