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  • J. math. fluid mech. 11 (2009) 60901422-6928/09/010060-31c 2007 Birkhauser Verlag, BaselDOI 10.1007/s00021-007-0248-8

    Journal of Mathematical

    Fluid Mechanics

    Incompressible Viscous Fluid Flows in a Thin Spherical Shell

    Ranis N. Ibragimov and Dmitry E. Pelinovsky

    Communicated by G. Iooss

    Abstract. Linearized stability of incompressible viscous fluid flows in a thin spherical shell isstudied by using the two-dimensional NavierStokes equations on a sphere. The stationary flowon the sphere has two singularities (a sink and a source) at the North and South poles of thesphere. We prove analytically for the linearized NavierStokes equations that the stationary flowis asymptotically stable. When the spherical layer is truncated between two symmetrical rings,we study eigenvalues of the linearized equations numerically by using power series solutions andshow that the stationary flow remains asymptotically stable for all Reynolds numbers.

    Mathematics Subject Classification (2000). 76D05, 76E20, 34B24, 34L16.

    Keywords. NavierStokes equations on a sphere, associated Legendre equation, asymptoticstability of stationary flow, numerical approximation of eigenvalues.

    1. Introduction

    The NavierStokes (NS) equations for an incompressible viscous fluid are the fun-damental governing equations of fluid mechanics. In many cases, exact solutionscan be constructed to these equations [10] and spectral and nonlinear stability ofthese exact solutions can be analyzed [11]. Our work addresses stability of exactsolutions for the NS equations in spherical coordinates.

    The three-dimensional NS equations in a thin rotating spherical shell describelarge-scale atmospheric dynamics that plays an important role in the global climatecontrol and weather prediction [21, 22]. It was rigorously proved by Temam &Ziane [28] that the average of the longitudinal velocity in the radial directionconverges to the strong solution of the two-dimensional NS equation on a sphereas the thickness of the spherical shell goes to zero. The latter model has been usedin geophysical fluid dynamics since the middle of the last century [19].

    The treatment of the geometric singularity in spherical coordinates has formany years been a difficulty in the development of numerical simulations foroceanic and atmospheric flows around the Earth. Blinova [4, 5] represented solu-tions in the inviscous case by the eigenfunction expansions in spherical harmon-ics. Vorticity equations were considered by Ben-Yu with the spectral method [3].

  • Vol. 11 (2009) Incompressible Viscous Fluid Flows in a Thin Spherical Shell 61

    More recent work of Furnier et al. [13] applied the spectral-element method tothe axis-symmetric solutions (see [17, 23, 29] for other applications of the spectralmethods in spherical coordinates). Finite-element approximations of the vectorLaplaceBeltrami equation on the sphere were studied by Simonnet [27]. Finally,point vortex motion on a sphere was modeled by ordinary differential equationsfor vortex centers in Boatto & Cabral [6] and Crowdy [9].

    We address the three-dimensional NS equations for an incompressible viscousfluid,

    u

    t+ (u )u u + p = 0, x , t R+,

    u = 0, x , t R+,u|t=0 = u0, x ,

    (1.1)

    in a thin spherical shell = {x R3 : 1 < |x| < 1 + } with 0, subject to theboundary conditions

    u n = 0, ( u) n = 0, x . (1.2)Here u : R+ 7 R3 is the velocity vector, p : R+ 7 R is the ratio of thepressure to constant density, is the kinematic viscosity, n is the normal vectorto the boundary of the spherical shell and u0 : 7 R3 is a given initialcondition. Although Coriolis and gravity forces may be dynamically significantin oceanographic applications, our model is considered in a non-rotating referenceframe and without external forces.

    We employ the spherical coordinates (r, , ) with the velocity vector u =urer + ue + ue, where (er, e, e) are basic orthonormal vectors along thespherical coordinates. For completeness, we write explicitly the three-dimensionalNS equations (1.1) in spherical coordinates [1]:

    urt

    + ururr

    +ur

    ur

    +u

    r sin

    ur

    u2 + u

    2

    r

    = pr

    +

    (

    ur +2

    r

    urr

    +2urr2

    )

    ,

    ut

    + urur

    +ur

    u

    +u

    r sin

    u

    +urur

    u2 cot

    r

    = 1r

    p

    +

    (

    u +2

    r2ur

    ur2 sin2

    2 cos r2 sin2

    u

    )

    ,

    ut

    + urur

    +ur

    u

    +u

    r sin

    u

    +urur

    +uu cot

    r

    = 1r sin

    p

    +

    (

    u +2

    r2 sin

    ur

    +2 cos

    r2 sin2

    u

    ur2 sin2

    )

    ,

    1

    r2

    r

    (

    r2ur)

    +1

    r sin

    (sin u) +

    1

    r sin

    u

    = 0,

  • 62 R. N. Ibragimov and D. E. Pelinovsky JMFM

    where

    =1

    r2

    r

    (

    r2

    r

    )

    +1

    r2 sin

    (

    sin

    )

    +1

    r2 sin2

    2

    2

    is the Laplacian in spherical coordinates. One can check by direct differentia-tion that there exists an exact stationary solution to the three-dimensional NSequations in spherical coordinates:

    ur = 0, u =

    r sin , u = 0, p =

    2

    2r2 sin2 , (1.3)

    where (, ) are arbitrary parameters. The stationary solution (1.3) describes theflow tangential to a sphere of any given radius r. The stationary flow has two polesingularities at = 0 and = . The singularities correspond to the source andsink of the velocity vector at the North and South poles of the spherical shell :the fluid is injected at the North pole from an external source and it leaks out atthe South pole to an external sink.

    In the limit 0, the non-stationary three-dimensional fluid flow is confinedon a sphere S of unit radius parameterized by the polar (latitude) angle andazimuthal (longitude) angle ,

    S = {(, ) , 0 6 6 , 0 6 < 2} . (1.4)

    Since the velocity vector u and the pressure p in the NS equations (1.1) are cou-pled together by the incompressibility constraint u = 0, it is difficult to analyzethe full set of three-dimensional equations. A common approach to simplify theproblem is to use the artificial methods such as the pressure stabilization andprojections [26]. The error estimate of the pressure stabilization and projectionmethods is not however mathematically precise. Instead, we shall use the result ofTheorem B in [28], which states that provided the function u0(r, , ) is smoothenough, the strong global solution u(r, , , t) of the three-dimensional NS equa-tions converges as 0 to the strong unique global solution v(, , t) of thetwo-dimensional NS equations on the sphere, where

    v(, , t) = lim0

    1

    1+

    1

    ru(r, , , t)dr = (0, v, v).

    The vector v(, , t) is interpreted as the average velocity with respect to theradial coordinate r. (Other applications of the averaged method for the three-dimensional NS equations in cartesian coordinates with a thin layer and variousboundary conditions are reviewed in [16].) The averaged two-dimensional NSequations on a sphere S in spherical angles (, ) can be written in the form [28]:

    vt

    + vv

    +v

    sin

    v

    v2 cot = p

    +

    (

    Sv v

    sin2 2 cos

    sin2

    v

    )

    ,

  • Vol. 11 (2009) Incompressible Viscous Fluid Flows in a Thin Spherical Shell 63

    vt

    + vv

    +v

    sin

    v

    + vv cot

    = 1sin

    p

    +

    (

    Sv +2 cos

    sin2

    v

    vsin2

    )

    ,

    1

    sin

    (sin v) +

    1

    sin

    v

    = 0,

    where S is the LaplaceBeltrami operator in spherical angles

    S =1

    sin

    (

    sin

    )

    +1

    sin2

    2

    2.

    Note that no boundary conditions are specified for the vector v(, , t) on sphereS, while the initial condition v|t=0 = v0 on S is not written. For the purposesof our work, we rewrite the two-dimensional NS equations on the sphere S in anequivalent form:

    vt

    vsin

    +q

    =

    (

    Sv v

    sin2 2 cos

    sin2

    v

    )

    , (1.5)

    vt

    +v

    sin +

    1

    sin

    q

    =

    (

    Sv +2 cos

    sin2

    v

    vsin2

    )

    , (1.6)

    (sin v) +

    v

    = 0, (1.7)

    where q is a static (stagnation) pressure and is the vorticity:

    q = p+1

    2

    (

    v2 + v2

    )

    , =

    (sin v)

    v

    . (1.8)

    The stationary solution (1.3) corresponds to the exact stationary solution of thetwo-dimensional NS equations (1.5)(1.7) on the unit sphere S:

    v =

    sin , v = 0, q = , (1.9)

    where (, ) are arbitrary parameters. Besides analysis of the stationary flow (1.9)on the sphere S, we shall also model it on the truncated domain with no externalsource and sink singularities at = 0 and = , e.g. on the spherical layer

    S0 ={

    (, ) : 0 0, 0 2}

    , (1.10)

    where 0 < 0 0 imply spectral insta-bility of the stationary flow. If Re() < 0 for all perturbations, the stationary flow

  • Vol. 11 (2009) Incompressible Viscous Fluid Flows in a Thin Spherical Shell 65

    is asymptotically stable, while if Re() = 0 for some perturbations and Re() < 0for all other perturbations, the stationary flow is stable in the sense of Lyapunov.

    By neglecting the quadratic terms of the perturbation, we linearize the NSequations (1.5)(1.7) with the expansion (2.1) to the form:

    U +Q

    =

    (

    SUU

    sin2 2 cos

    sin2

    V

    )

    , (2.2)

    V+1

    sin2

    (

    (sin V ) U

    )

    +1