## Linear stability analysis for the differentially heated rotating annulus.(English)Zbl 1206.76021

Summary: We use linear stability analysis to approximate the axisymmetric to nonaxisymmetric transition in the differentially heated rotating annulus. We study an accurate mathematical model that uses the Navier-Stokes equations in the Boussinesq approximation. The steady axisymmetric solution satisfies a two-dimensional partial differential boundary value problem. It is not possible to compute the solution analytically, and thus, numerical methods are used. The eigenvalues are also given by a two-dimensional partial differential problem, and are approximated using the matrix eigenvalue problem that results from discretizing the linear part of the appropriate equations. A comparison is made with experimental results. It is shown that the predictions using linear stability analysis accurately reproduce many of the experimental observations. Of particular interest is that the analysis predicts cusping of the axisymmetric to nonaxisymmetric transition curve at wave number transitions, and the wave number maximum along the lower part of the axisymmetric to nonaxisymmetric transition curve is accurately determined. The correspondence between theoretical and experimental results validates the numerical approximations as well as the application of linear stability analysis. A linear stability analysis is also performed with the effects of centrifugal buoyancy neglected. Along the lower part of the transition curve, the results are significantly qualitatively and quantitatively different than when the centrifugal effects are considered. In particular, the results indicate that the centrifugal buoyancy is the cause of the observation of a wave number maximum along the transition curve, and is the cause of a change in concavity of the transition curve.

### MSC:

 76E07 Rotation in hydrodynamic stability 76D05 Navier-Stokes equations for incompressible viscous fluids 76E20 Stability and instability of geophysical and astrophysical flows 76U05 General theory of rotating fluids

ARPACK
Full Text:

### References:

 [1] Barcilon V, J. Atmos. Sci 21 pp 291– (1964) [2] Chandrasekhar S, Oxford University Press (1961) [3] Chossat P, Springer-Verlag (1994) [4] Christodoulou K, J. Sci. Comp. 3 pp 355– (1988) · Zbl 0677.65032 [5] Dijkstra H, Computers and Fluids 24 pp 415– (1995) · Zbl 0848.76056 [6] Farrell B, J. Atmos. Sci. 53 pp 2025– (1996) [7] Fein J, Geophys. Fluid Dynam. 5 pp 213– (1973) [8] Fultz D, Adv. Geophys. 7 pp 1– (1961) [9] Ghil M, Springer-Verlag (1987) [10] Govaerts W, SIAM (2000) [11] Hide R, Adv. Geophys. 24 pp 47– (1975) [12] Hide R, Geophys. Astrophys. Fluid Dynam. 10 pp 121– (1978) [13] Hignett P, Quart. J. Roy. Met. Soc. 111 pp 131– (1985) [14] James I, Quart. J. Roy. Met. Soc. 107 pp 51– (1981) [15] Koschmieder E, Geophys. Astrophys. Fluid Dynam. 10 pp 157– (1978) [16] Lehoucq R, SIAM (1998) [17] Lewis G, University of British Columbia (2000) [18] Lewis G, SIAM J. Appl. Math. 63 pp 1029– (2003) · Zbl 1027.37051 [19] Lu H, Geophys. Astrophys. Fluid Dynam. 75 pp 1– (1994) [20] Maslowe S, Springer-Verlag pp pp. 181–228– (1985) [21] Miller T, J. Atmos. Sci. 48 pp 811– (1991) [22] Pedlosky J, Springer-Verlag (1987) [23] Trefethen L, SIAM Rev. 39 pp 383– (1997) · Zbl 0896.15006 [24] Trefethen L, Science 261 pp 578– (1993) · Zbl 1226.76013 [25] Williams G, J. Fluid Mech. 49 pp 417– (1971) · Zbl 0239.76109
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.