The Construction of Barotropic Modons on a Sphere

Verkley, W. T. M.

doi:10.1175/1520-0469(1984)041<2492:tcobmo>2.0.co;2

Cited by 75 publications

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“…contour surgery). The only exact solutions for distributed vortex equilibria on a sphere known to the author are those of Verkley (1984Verkley ( , 1993) (on a rotating sphere) and the recently derived solutions of Crowdy & Cloke (2003) which involve hybrid vortex-patch/point-vortex combinations modelling multipolar vortices on a non-rotating sphere.…”

Section: Introductionmentioning

confidence: 99%

Stuart vortices on a sphere

Crowdy

2004

J. Fluid Mech.

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The exact steady two-dimensional solutions of the Euler equations due to Stuart (1967) are generalized to the surface of a sphere. The solutions are parametrized by three parameters N, θ 0 and Ω max ; the integer N > 1 denotes the number of smooth vorticity extrema, with extremal vorticity Ω max , that are equally spaced in longitudinal angle around a latitude circle at latitudinal spherical polar angle θ 0 . The solutions have two equal point-vortex singularities at the north and south spherical poles. Like Stuart's, the solutions are exact and explicit. The solutions are expected to be useful in geophysical and astrophysical applications where curvature effects are important.

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Section: Introductionmentioning

confidence: 99%

Stuart vortices on a sphere

Crowdy

2004

J. Fluid Mech.

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“…Verkley(1984) generalized Thomp-son's solution and demonstrated that Y ν could be a set of eigenfunctions that contains more than only spherical harmonics. Then Eq.…”

Section: Introductionmentioning

confidence: 99%

“…RH waves have proven to be very useful to describe the large-scale wave structure of atmospheric circulation in middle latitudes (Rossby, 1939;Haurwitz, 1940). The solution modon is constructed to divide the sphere S 2 into two regions (Tribbia, 1984;Verkley, 1984Verkley, , 1987Verkley, , 1990Neven, 1992): an inner region S i centered around the pole N′, and an outer region S o separated from the inner region by a boundary circle in which Ψ, q and its normal derivative Ψ′ are continuous. Modons are considered appropriate to describe some types of atmospheric blocking events (Verkley, 1990).…”

Section: Introductionmentioning

confidence: 99%

“…To obtain the relations between the unprimed and primed coordinate of any point Q on the sphere, Verkley (1984) examined the spherical triangle NQN' and the application of the cosine rules to this triangle, deriving explicit expressions for the transformation between the two coordinate systems as given by (4) and (5).…”

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confidence: 99%

“…Recent developments show that the non-zero eigenvalues also contain substantial geometric and analytic information. The solution modon constructed by Tribbia (1984), Verkley (1984Verkley ( , 1987Verkley ( , 1990 and Neven (1993) proposed the use of eigenfunctions {Y j } as basic geometric structures.…”

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Exact solutions of the vorticity equation on the sphere as a manifold

Pérez‐García

2015

Atm

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RESUMENEl propósito de este trabajo es representar las soluciones exactas de la ecuación de vorticidad barotrópica sobre la esfera unitaria S 2 en rotación soluciones generalizadas llamadas modones. Se relacionan los métodos modernos de la teoría de funciones correcta, se esclarece la noción abstracta de mapa local, cambio de mapa y atlas. Uno de los objetivos de este trabajo es entender mejor la solución de la ecuación de vorticidad barotrópica sobre la variedad S 2 y su S 2 , g). Por lo tanto, estará disponible un tipo más general de espacio que también puede contener información geométrica y analítica sustancial sobre las soluciones a la ecuación de vorticidad barotrópica. ABSTRACTThe purpose of this paper is to represent the exact solutions of the barotropic vorticity equations (BVE) on the rotating unit sphere S 2 -ferentiable manifold. When the differentiable manifold S 2 is well understood, the abstract notion of local chart, change of chart, and atlases becomes evident. One of the aims of this paper is to better understand the solution of the barotropic vorticity equation on the manifold S 2 and its usefulness to identify the properties of the solutions on the Riemannian manifold (S 2 , g). Therefore, a more general type of space will be available, which can also contain substantial geometric and analytic information about solutions for the barotropic vorticity equation.

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Linear instability of ideal flows on a sphere

Skiba

2008

Math Methods in App Sciences

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SUMMARYA unified approach to the normal mode instability study of steady solutions to the vorticity equation governing the motion of an ideal incompressible fluid on a rotating sphere is considered. The four types of well-known solutions are considered, namely, the Legendre-polynomial (LP) flows, Rossby-Haurwitz (RH) waves, Wu-Verkley (WV) waves and modons. A conservation law for disturbances to each solution is derived and used to obtain a necessary condition for its exponential instability. By these conditions, Fjörtoft's (Tellus 1953; 5:225-230) average spectral number of the amplitude of an unstable mode must be equal to a special value. In the case of LP flows or RH waves, this value is related only with the basic flow degree. For the WV waves and modons, it depends both on the basic flow degree and on the spectral distribution of the mode energy in the inner and outer regions of the flow. Peculiarities of the instability conditions for different types of modons are discussed. The new instability conditions specify the spectral structure of growing disturbances localizing them in the phase space. For the LP flows, this condition complements the well-known Rayleigh-Kuo and Fjörtoft conditions related to the zonal flow profile. Some analytical and numerical examples are considered. The maximum growth rate of unstable modes is also estimated, and the orthogonality of any unstable, decaying and non-stationary mode to the basic flow is shown in the energy inner product.The analytical instability results obtained here can also be applied for testing the accuracy of computational programs and algorithms used for the numerical stability study. It should be stressed that Fjörtoft's spectral number appearing both in the instability conditions and in the maximum growth rate estimates is the parameter of paramount importance in the linear instability problem of ideal flows on a sphere.

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The Construction of Barotropic Modons on a Sphere

Cited by 75 publications

References 0 publications

Stuart vortices on a sphere

Stuart vortices on a sphere

Exact solutions of the vorticity equation on the sphere as a manifold

Linear instability of ideal flows on a sphere

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