×

On the structure and growth rate of unstable modes to the Rossby-Haurwitz wave. (English) Zbl 1141.76391

Summary: The normal mode instability study of a steady Rossby-Haurwitz wave is considered both theoretically and numerically. This wave is exact solution of the nonlinear barotropic vorticity equation describing the dynamics of an ideal fluid on a rotating sphere, as well as the large-scale barotropic dynamics of the atmosphere. In this connection, the stability of the Rossby-Haurwitz wave is of considerable mathematical and meteorological interest. The structure of the spectrum of the linearized operator in case of an ideal fluid is studied. A conservation law for perturbations to the Rossby-Haurwitz wave is obtained and used to get a necessary condition for its exponential instability. The maximum growth rate of unstable modes is estimated. The orthogonality of the amplitude of a non-neutral or non-stationary mode to the Rossby-Haurwitz wave is shown in two different inner products. The analytical results obtained are used to test and discuss the accuracy of a numerical spectral method used for the normal mode stability study of arbitrary flow on a sphere. The comparison of the numerical and theoretical results shows that the numerical instability study method works well in case of such smooth solutions as the zonal flows and Rossby-Haurwitz waves.

MSC:

76E20 Stability and instability of geophysical and astrophysical flows
65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs
76B60 Atmospheric waves (MSC2010)
76B65 Rossby waves (MSC2010)
76U05 General theory of rotating fluids
86A10 Meteorology and atmospheric physics
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Pedlosky, Geophysical Fluid Dynamics (1979)
[2] Silberman, Planetary waves in the atmosphere, J Meteor 11 pp 27– (1954)
[3] Kuo, Dynamic instability of two-dimensional nondivergent flow in a barotropic atmosphere, J Meteor 6 pp 105– (1949)
[4] Lorenz, Barotropic instability of Rossby wave motion, J Atmos Sci 29 pp 258– (1972)
[5] Tung, Barotropic instability of zonal flows, J Atmos Sci 38 pp 308– (1981)
[6] Drazin, Hydrodynamic Stability pp 527– (1981)
[7] Simmons, Barotropic wave propagation and instability, and atmospheric teleconnection patterns, J Atmos Sci 40 pp 1363– (1983)
[8] Haarsma, Barotropic instability of planetary-scale flows, J Atmos Sci 45 pp 2789– (1988)
[9] Branstator, Analysis of general circulation model sea surface temperature anomaly simulations using a linear model. Part II: Eigenanalysis, J Atmos Sci 42 pp 2242– (1985)
[10] Wu, Nonlinear resonance and instability of planetary waves and low-frequency variability in the atmosphere, J Atmos Sci 50 pp 3590– (1993)
[11] Bulgakov, Loop current dynamics in laboratory experiments, Intersciences 28 pp 316– (2003)
[12] Haurwitz, The motion of atmospheric disturbances on the spherical earth, J Marine Res 3 pp 254– (1940)
[13] Wu, Non-linear structures with multivalued relationships-exact solutions of the barotropic vorticity equation on a sphere, Geophys Astrophys Fluid Dyn 69 pp 77– (1993)
[14] Tribbia, Modons in spherical geometry, Geophys Astrophys Fluid Dyn 30 pp 131– (1984)
[15] Verkley, The construction of barotropic modons on a sphere, J Atmos Sci 41 pp 2492– (1984)
[16] Verkley, Stationary barotropic modons in westerly background flows, J Atmos Sci 44 pp 2383– (1987)
[17] Verkley, Modons with uniform absolute vorticity, J Atmos Sci 47 pp 727– (1990)
[18] Neven, Quadrupole modons on a sphere, Geophys Astrophys Fluid Dyn 65 pp 105– (1992)
[19] Neven, Modons on a Sphere (1993)
[20] Neven, Linear stability of modons on a sphere, J Atmos Sci 58 pp 2280– (2001)
[21] Hoskins, On the simplest example of the barotropic instability of Rossby wave motion, J. Atmos. Sci. 30 pp 150– (1973)
[22] Anderson, The instability of finite amplitude Rossby waves on the infinite beta-plane, Geophys. Astrophys. Fluid Dynamics 63 pp 1– (1992)
[23] Hoskins, Stability of the Rossby-Haurwitz wave, Quart J R Met Soc 99 pp 723– (1973)
[24] Skiba, Mathematical Problems of the Dynamics of Viscous Barotropic Fluid on a Rotating Sphere pp 1– (1989)
[25] Skiba, Liapunov instability of the Rossby-Haurwitz waves and dipole modons, Soc J Numer Anal Math Modelling 6 pp 515– (1991) · Zbl 0819.76030
[26] Skiba, Rossby-Haurwitz wave stability, Izvestiya Atmos Ocean Physics 28 pp 388– (1992)
[27] Baines, The stability of planetary waves on a sphere, J Fluid Mech 73 pp 193– (1976) · Zbl 0319.76037
[28] Skiba, On the normal mode instability of harmonic waves on a sphere, Geophys Astrophys Fluid Dynamics 92 pp 115– (2000)
[29] Skiba, Instability of the Rossby-Haurwitz wave in invariant sets of perturbations, J Math Anal Appl 290 pp 686– (2004) · Zbl 1221.76090
[30] Skiba, Spectral approximation in the numerical stability study of non-divergent viscous flows on a sphere, Numer Methods Partial Differ Eq 14 pp 143– (1998) · Zbl 0903.76074
[31] Pérez García, Simulation of exact barotropic vorticity equation solutions using a spectral model, Atmósfera 12 pp 223– (1999)
[32] Stewart, Matrix perturbation theory (1990) · Zbl 0706.65013
[33] Demidovich, Lectures on Mathematical Stability Theory pp 472– (1967)
[34] Wilkinson, The Algebraic Eigenvalue Problem (1965) · Zbl 0258.65037
[35] Merilees, The equations of motion in spectral form, J Atmos Sci 28 pp 736– (1968)
[36] Machenauer, Numerical Methods Used in Atmospheric Models 2, in: GARP Publication Series pp 124– (1979)
[37] Helgason, Groups and Geometric Analysis, Integral Geometry, Invariant Differential Operators and Spherical Functions (1984) · Zbl 0543.58001
[38] R. D. Richtmyer Principles of Advanced Mathematical Physics 2 Springer-Verlag New York 1981
[39] Topuria, Fourier-Laplace Series on a Sphere (1987)
[40] Platzman, The analytical dynamics of the spectral vorticity equation, J Atmos Sci 19 pp 313– (1962) · Zbl 0106.44103
[41] Skiba, On the spectral problem in the linear stability study of flows on a sphere, J Math Anal Appl 270 pp 165– (2002) · Zbl 1063.76023
[42] Dikii, Hydrodynamic Stability and Atmosphere Dynamics (1976)
[43] Arnold, Conditions for nonlinear stability of stationary plane curvilinear flows of an ideal fluid, Sov Math Doklady 6 pp 331– (1965)
[44] Kato, Perturbation Theory for Linear Operators (1980) · Zbl 0435.47001
[45] Rayleigh, On the stability or instability of certain fluid motions, Proc London Math Soc 11 pp 57– (1880) · JFM 12.0711.02
[46] Skiba, On the lineal stability study of zonal incompressible flows on a sphere, Numerical Methods for Partial Differential Equations 14 pp 649– (1998) · Zbl 0933.76027
[47] Skiba, Dynamics of perturbations of the Rossby-Haurwitz wave and the Verkley modon, Atmósfera (México) 6 pp 87– (1993)
[48] Liapunov, Stability of Motion (1966)
[49] Fjörtoft, On the change in the spectral distribution of kinetic energy for two-dimensional nondivergent flow, Tellus 5 pp 225– (1953)
[50] Shepherd, Symmetries, conservation laws, and Hamiltonian structure in Geophysical Fluid Dynamics, Adv Geophys 32 pp 287– (1990)
[51] Riley, Mathematical Methods for Physics and Engineering (1998)
[52] Pérez García, Tests of a numerical algorithm for the linear instability study of flows on a sphere, Atmósfera 14 pp 95– (2001)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.