## Abstract

The problem of fluid motion in the form of inertial waves in an incompressible inviscid fluid contained in a rotating sphere is governed by the Poincaré equation, a second-order hyperbolic partial differential equation. Its explicit general analytical solution in terms of a double Poincaré polynomial was found by Zhang *et al*. (Zhang *et al*. 2001 *J. Fluid Mech*. **437**, 103–119), describing the pressure *p*_{mnK} and the velocity *u*_{mnK} of spherical inertial waves, where the triple indices (*mnK*) are indicative of the azimuthal, vertical and radial structures, respectively. On the basis of the general explicit solution, we reveal a new intriguing integral property of the spherical inertial wavesfor all possible values of *m*, *l* and *n*, where *M* and *K* are related to the degree of the double Poincaré polynomial and denotes the complex conjugate of *u*_{mlM}. A mathematical proof of the vanishing of the integral involving the construction of two auxiliary recurrence relations is presented. Furthermore, a comparison with the corresponding integral for rotating cylinders is made, showing a fundamental difference between the two systems.

## 1. Introduction

Many geophysical and astrophysical problems are concerned with the classical fluid dynamical problem of wave motion in a nearly inviscid incompressible fluid contained in a sphere rotating with constant angular velocity *Ω* (e.g. Poincaré 1885; Chandrasekhar 1961; Greenspan 1968; Gubbins & Roberts 1987). As a consequence of rotation, oscillatory fluid motion can be maintained in a homogeneous fluid that rotates at constant angular velocity. This type of oscillatory motion, usually referred to as inertial waves, is ubiquitous in rotating fluid systems. It can be excited and maintained, for example by thermal instabilities (e.g. Zhang 1995), by precession (e.g. Gans 1970), by tidal effects (e.g. Kerswell 1994), by planetary liberation (e.g. Noir *et al*. in press) and by differential rotation (e.g. Kelley *et al*. 2007). A strong zonal flow can also be generated by the oscillatory motion through the nonlinear interaction of inertial waves (e.g. Zhang & Liao 2004; Tilgner 2007). In rotating giant planets for which the tidal forcing frequencies are typically comparable to the spin frequency of the planets, it has been argued that inertial waves provide an effective mechanism for tidal dissipation (e.g. Ogilvie & Lin 2004). There also exist geomagnetic variations of non-axisymmetric magnetic flux, dominated by a single wavenumber at the core surface in the equatorial region, that are likely to be indicative of a magnetically modified equatorially trapped inertial wave (Zhang 1993; Finlay & Jackson 2003).

In the inviscid limit, the oscillatory fluid motion, described by the velocity ** u**=

**(**

*u***)exp(i2**

*x**σt*) and the pressure

*p*=

*p*(

**)exp(i2**

*x**σt*) with

*σ*being the half frequency scaled by

*Ω*, is usually in the form of azimuthally travelling waves. The pressure

*p*(

**) of the inertial waves is governed by the well-known dimensionless Poincaré equation, a hyperbolic partial differential equation(1.1)which was derived more than a century ago by Poincaré, where is the unit vector parallel to the axis of rotation and the half frequency**

*x**σ*is bounded 0<|

*σ*|<1. The Poincaré equation (1.1) is solved subject to the inviscid boundary condition(1.2)on the container's bounding surface characterized by the normal vector . After obtaining the pressure

*p*from the Poincaré equation (1.1), the velocity

**for inertial waves can be expressed as(1.3)The Poincaré equation (1.1) together with the boundary condition (1.2) and the relationship (1.3) between**

*u**p*and

**constitute the boundary value problem for inertial waves in rotating systems.**

*u*To describe solutions of the Poincaré equation (1.1), it is convenient to introduce a triple-index notation by writing *p*→*p*_{mnk}, ** u**→

*u*_{mnk}and

*σ*→

*σ*

_{mnk}. In this subscript notation, the spatial structure of an inertial wave is signified by a triple set of numbers: the first index,

*m*, is indicative of the azimuthal structure; the second subscript,

*n*, represents the vertical structure; and the third index,

*k*, reflects the radial structure of the wave.

For nearly inviscid fluids, small-amplitude fluid motion in a rapidly rotating system with a characteristic length *d* and a small Ekman number *E*≪1, driven by a small, time-dependent external force *f*^{*} resulting from, e.g. precession or tide, is governed by the two dimensionless equations (Greenspan 1968)(1.4)(1.5)where ** f** represents the non-dimensional external force with |

**|≪1. The Ekman number**

*f**E*, defined as

*E*=

*ν*/

*Ωd*

^{2}with

*ν*being kinematic viscosity, is usually very small for geophysical and astrophysical bodies and provides the measure of relative importance between the typical viscous force and the Coriolis force. For the purpose of mathematical simplicity without loss of key physics, we assume that the velocity boundary condition is stress-free, so that the boundary-layer flux is small and negligible. By further assuming that the eigenfunctions of the Poincaré equation, the inertial wave modes, are complete, solutions to (1.4) and (1.5) can be expanded in terms of the inertial modes (Greenspan 1968)(1.6)where the time-dependent coefficients are to be determined. It follows that the problem of solving the partial differential equations (1.4) and (1.5) reduces to find solutions of the ordinary differential equations for the coefficients (1.7)where denotes the complex conjugate of

*u*_{mnk}. For given

*u*_{mnk}and

**, the two volume integrals, and , can usually be readily evaluated. In the case of a nonlinear problem, the additional term**

*f***.∇**

*u***should be included explicitly in the right-hand side of (1.7). Generally speaking, the volume integration is connected with viscous dissipation (e.g. Ogilvie & Lin 2004), and will be therefore referred to as the dissipation integral in this paper. At this point, it is worth mentioning that, if (1.4) and (1.5) are solved by a perturbation approach for which the leading-order problem is governed by the Poincaré equation, we also need the dissipation integral for the higher-order analysis (e.g. Zhang**

*u**et al*. 2007). Although we may call the dissipation integral, it should be pointed out, as we shall discuss further, that this integral is not always negative and, hence, of physical significance. This is because the inertial wave

*u*_{mnk}does not satisfy a physical boundary condition such as the non-slip condition. In other words, an additional surface integral is usually required for a viscously modified spherical inertial mode (e.g. Liao & Zhang 2008).

A major advantage of employing inertial modes *u*_{mnk} in the expansion (1.6) is that the rotational effect does not couple them. This offers an effective and efficient way to solve the fluid dynamical problem in rotating systems. When explicit analytical solutions *u*_{mnk} are available, there exist, in principle, no difficulties in carrying out the dissipation integral in (1.7). A crucial step towards solving (1.4) and (1.5) is then to evaluate and understand the dissipation integral in the right-hand side of (1.7).

In comparison with the problem of inertial waves in rotating spheres, it should be pointed out that the problem of inertial waves in rotating spherical shells is much more complicated. In general, the mathematical solution of inertial waves in spherical shells can be divided into two different categories. The first is, similar to that for the sphere, non-singular and marked by its high concentration in the equatorial boundary region that is largely unaffected by the presence of an inner core (e.g. Zhang 1993). In the second category, the inertial wave solution, as demonstrated by Rieutord *et al*. (2000, 2001, 2002), is singular and marked by its high concentration along an attractor formed by the periodic orbit of characteristics of the Poincaré equation in spherical shells.

This paper focuses on a new unusual and intriguing property of the dissipation integral discovered in rotating fluid spheres. In an effort to illustrate a fundamental difference of the integral between spherical and other geometries, we shall also briefly discuss the same problem in rotating circular cylinders. In what follows, we shall begin in §2 by presenting the property of the dissipation integral in rotating cylinders. The basic properties of inertial waves in rotating spheres, as well as some simple explicit examples, are discussed in §3 and a mathematical proof for the new property of the dissipation integral is provided in §4. The paper closes in §5 with a summary and some remarks.

## 2. The dissipation integral in rotating cylinders

To understand the fundamental dynamics of rotating fluids in rotating planetary interiors and atmospheres, fluid motion in the form of inertial waves in rotating circular cylinders has been extensively studied, both experimentally and theoretically (e.g. Manasseh 1992; Kerswell & Barenghi 1995; Zhang *et al*. 2007). The primary purpose of our brief discussion is, however, to contrast the dissipation integral in rotating cylinders with that in rotating spheres, in an attempt to provide helpful insight into the complexity and intricacy of the relevant mathematical problem.

Consider a homogeneous fluid in a cylinder of height *d* and radius *Γd* in the limit of kinematic viscosity *ν*=0. The cylinder is rotating uniformly about its axis with a constant vertical angular velocity *Ω* and cylindrical coordinates (*s*, *ϕ*, *z*) with corresponding unit vectors employed, being parallel to the axis of rotation. Taking height *d* as the characteristic length-scale and *Ω*^{−1} as the time scale, the Poincaré equation (1.1) describing inertial waves in rotating cylinders becomes(2.1)where 0<|*σ*|<1. The geometric parameter is given by the aspect ratio *Γ*=(*Γd*)/*d*. It can be shown that the solution with separable variables to the Poincaré equation (2.1) satisfying the boundary condition (1.2) is(2.2)where *J*_{m}(*x*) denotes the standard Bessel function, *m* is the azimuthal wavenumber assumed to be positive and *ξ* is a solution of the transcendental equation(2.3)which can be solved to determine the radial wavenumbers *ξ* for any given *m*. Solutions to (2.3) can be arranged, with the aid of subscript notation, according to their sizewhere *ξ*_{mnk} denotes the *k*th smallest solution of (2.3): the integer number *k* corresponds to the number of nodes in the radial direction and *n* is the vertical wavenumber of an inertial wave. The dispersion relation between *ξ*_{mnk} and *σ*_{mnk} is given by(2.4)Upon substituting (2.2) into (1.3), we can derive the velocity of inertial waves in rotating circular cylinders(2.5)(2.6)(2.7)where an arbitrary normalization is used.

With the explicit solution of inertial waves in rotating cylinders given by (2.5)–(2.7), we can establish, by direct integration, that(2.8)and(2.9)for all possible values of *m*, *n* and *k*. The dissipation integral (2.9) for rotating cylinders, similar to that for a rotating periodic box (Ogilvie & Lin 2004), is always negative and of physical significance.

## 3. The dissipation integral in rotating spheres: examples

The impetus for studying inertial waves in rotating spherical geometry arises from the desire to understand fluid dynamics in rotating geophysical and astrophysical bodies (e.g. Aldridge & Lumb 1987; Hollerbach & Kerswell 1995; Rieutord & Valdettaro 1997). A detailed account of earlier research results on the problem can be found in the monograph by Greenspan (1968).

Consider a homogeneous viscous fluid in a sphere of radius *d* rotating uniformly about its axis with a constant angular velocity *Ω*. Spherical polar coordinates (*r*, *θ*, *ϕ*) with the corresponding unit vectors are employed, with being parallel to the axis of rotation. Taking radius *d* as the characteristic length scale and *Ω*^{−1} as the time scale, the Poincaré equation (1.1) in rotating spheres is of the form(3.1)Solutions to equation (3.1) can be divided into two different classes according to their spatial symmetries with respect to the equatorial plane at *θ*=*π*/2: equatorially symmetric waves obeying(3.2)and equatorially antisymmetric waves whose parity satisfies(3.3)Since the mathematical analyses for the equatorially symmetric and antisymmetric solutions are entirely analogous, we shall focus only on the solutions with equatorial symmetry (3.2).

The explicit analytical solution *p* satisfying the Poincaré equation (3.1) found by Zhang *et al*. (2001) can be conveniently written in spherical polar coordinates(3.4)where the double polynomial in the bracket is referred to as the Poincaré polynomial (Zhang *et al*. 2004), and *K* varies over all positive integers, *K*=1, 2, 3, …, and is related to the degree of the polynomial whose coefficients are given by(3.5)An arbitrary normalization is used in (3.4) for simplicity. Substitution of (3.4) into the boundary condition (1.2) yields an equation for determining the half frequencies, *σ*, of the spherical inertial waves(3.6)For any given integer *K*≥1, there exist 2*K* distinct real roots to (3.6), which can be arranged, with the aid of subscript notation, according to the size of *σ*with *σ*_{mnK} representing the *n*th smallest absolute root of (3.6). Upon substituting (3.4) into (1.3), we can derive an explicit expression for the velocity of all spherical inertial waves with equatorial symmetry (3.2) in a rotating sphere(3.7)(3.8)(3.9)for *m*=1, 2, 3, …, *K*=1, 2, 3, … and *n*=1, 2, 3, …, 2*K*. A similar solution with equatorial anti-symmetry (3.3) can be derived in the same manner.

On the basis of the general explicit solution of inertial waves given by (3.7)–(3.9), we are able to reveal a striking, unexpected and intricate property of spherical inertial waves(3.10)and(3.11)for all possible values of *m*, *l* and *n*, for which the integral (3.11) can be either negative or positive. While the negative integral (2.9) in rotating circular cylinders is of physical significance representing viscous dissipation, the physical significance of the integral (3.10) or (3.11) is unknown. Note that the special case of (3.10)(3.12)was discussed in the previous analysis (Zhang *et al*. 2001). The mathematical proof for the general case (3.10), which will be presented in §4, is complicated and lengthy. It is hence profitable to look at some simple explicit examples prior to presenting the general proof.

The two simplest equatorially symmetric solutions are obtained by letting *K*=1 and 2 in (3.4). For *K*=1, equation (3.6) gives rise to an equation for determining the half frequency(3.13)For each given wavenumber *m*, there exist two inertial-wave solutions given by(3.14)(3.15)One (*σ*_{m21}<0) propagates eastward while the other (*σ*_{m11}>0) propagates westward. Substituting (3.14) or (3.15) into expressions (3.7)–(3.9) at *K*=1, we obtain the simplest equatorially symmetric solution of spherical inertial waves in closed form. Another simple solution for equatorial symmetric waves is obtained by letting *K*=2 in (3.4). Equation (3.6) at *K*=2 gives rise to(3.16)There exist four different inertial waves for each given wavenumber *m*. A closed-form expression for *σ*_{mnk}, though lengthy, can also be obtained without difficulty. By inserting the analytical expression for *σ*_{mnk} into the velocity formulae (3.7)–(3.9) at *K*=2, we again obtain the fully explicit wave solution in closed form.

With the explicit expression of the flow velocity available, we can evaluate the dissipation integral through direct integration. To simplify the notation, we shall denote *u*_{mnK} and *σ*_{mnK} by *u*_{K} and *σ*_{K}, respectively. Of the three indices (*mnK*) for a spherical inertial wave, *K* is mathematically most significant because it is related to the degree of the double Poincaré polynomial (3.4). A straightforward but slightly cumbersome integration with *M*=2 and *K*=1 in (3.10) results in(3.17)where is a non-zero function of *σ*_{1} and *m*. By evaluating the summation in the bracket, which vanishes identically, we conclude that(3.18)for all possible wavenumbers and frequencies. Similarly, it can be shown with *M*=1 and *K*=2 in (3.11) that(3.19)which can be either negative or positive, for all possible values of the wavenumber and frequencies. For example, when *σ*_{1}=−0.1160 and *σ*_{2}=−0.5463 for *m*=2, while when *σ*_{1}=−0.1160 and *σ*_{2}=−0.8217 also for *m*=2.

For large *M* and *K*, the dissipation integral (3.10) can be expressed in terms of three extremely complicated summations with four indices that cannot be evaluated directly even with the aid of modern computers. In §4, we shall provide a general mathematical proof for the property (3.10) by adopting the strategy of constructing two auxiliary recurrence relations.

## 4. A mathematical proof of the vanishing integral for spheres

There are two well-known properties of the Poincaré equation in rotating spheres (Greenspan 1968): (i) the eigenvalue *σ*_{mnN} is real and satisfies(4.1)and (ii) any eigenvector–eigenvalue pair, (*σ*_{mnN},*u*_{mnN}) and , are orthogonal, i.e.(4.2)if *n*≠*n*′ or *N*≠*N*′. It is of primary importance to note that the mathematical proof for (4.1) and (4.2) is relatively simple and, more significantly, it can be provided directly from the basic governing equations *without reference to the particular geometry* of the problem or the detailed structure of inertial waves.

The integral property (3.10) is fundamentally different: it is associated with spherical geometry and, consequently, its proof requires the detailed structure of inertial waves in spherical geometry. The general mathematical proof is rather lengthy, involving three entangled summations in connection with the double Poincaré polynomial. We shall present only the proof for equatorially symmetric inertial waves since the proof for equatorially antisymmetric waves is entirely analogous.

With the explicit expression ((3.7)–(3.9)), it is not difficult to carry out the volume integration over the sphere, which yields(4.3)where _{j}, *j*=1, 2, 3 are a non-zero function of *σ*_{M} and *σ*_{N}while _{j}, *j*=1, 2, 3 in (4.3) represent the following three summations:(4.4)(4.5)(4.6)We shall prove thatfor all possible values of *M*, *N*, *m*, *σ*_{N} and *σ*_{M} with *M*≥*N*. Since the mathematical proofs for _{1}≡0 and _{3}≡0 are rather similar, it suffices to present the detailed analysis for _{2}≡0.

A key character is that the four indices (*i*, *j*, *k*, *l*) in the summations (4.4)–(4.6) are so intimately entangled that a direct summation over them even for moderate *M* and *N* is impossible. Upon recognizing the fact that the direct summation for small *M* and *N*, as shown in (3.17), is possible, an essential step towards proving that _{2}≡0 for all possible *M* and *N* with *M*≥*N*≥1 is then to establish a recurrence relationship that links the large *M*/*N* summation with the small *M*/*N* one. For this purpose, we introduce three additional indices, say *α*, *β* and *γ*, by considering a new summation involving six indices(4.7)where *L*=(*M*−*N*)≥0, 0≤*γ*≤*L* and the coefficients are non-zero and defined asThe precise values of coefficients , however, are not required in the mathematical proof. It is evident that the relationship between _{2} and is given by(4.8)At first glance, the new summation (4.7) with six indices is much more complicated than the original (4.5). It will be seen, however, that the additional indices enable us to derive two recurrence relationships, leading to the direct evaluation of the summation (4.7).

In order to decouple the entangled indices in (4.7), we first establish an important recurrence relationship(4.9)in which the left-hand side is given by(4.10)The first step in deriving the recurrence relation (4.9) is to realize that the expression (4.10) can be decomposed into the three different summations(4.11)where(4.12)(4.13)(4.14)in which the index *i* in (4.12) is shifted by 1, i.e. *i*→(*i*+1), while the index *j* in (4.14) is shifted by 1, *j*→(*j*+1). As clearly indicated by the recurrence relationship (4.9), we need to make a further rearrangement of the summation such that the index (*γ*+1) appears in the resulting summation. Evidently, the three different terms, _{j}*, j*=1, 2, 3 in (4.11) can be combined to yield a single summation(4.15)which leads to(4.16)The recurrence relation (4.9) implies that(4.17)A remarkable feature of the summation (4.10) is that, when *γ*=*L*, the indices (*i*, *j*) and (*α*, *β*) become decoupled and, more significantly, the resulting summation can be readily carried out. We obtain directly from (4.10) with *γ*=*L* that(4.18)for which the short summation in (4.18) can be easily evaluated, which gives(4.19)or . It follows then from (4.17) that(4.20)for any value of *γ* with 0≤*γ*≤*L*. It should be borne in mind, however, that the objective is to prove that , which is different from that given by (4.20). A second recurrence relationship is required to bridge the gap to achieve the final objective. Since the derivation of the second recurrence relation is largely analogous with the first and since the derivation is lengthy and cumbersome, we shall not present the relevant details.

In short, by decomposing (4.7) into three different summations and then rearranging and shifting the relevant indices in the resulting summations, it can be shown that obeys the following recurrence relation:(4.21)valid for 0≤*K*≤(*N*−1) and 0≤*γ*≤*L*. Taking *K*=0 and *γ*=*L* in (4.21) gives(4.22)Applying the recurrence relation (4.21) with *K*=1 and *γ*=*L* yields(4.23)implying that(4.24)It follows from (4.20) that(4.25)for any values of *K* with 0≤*K*≤(*N*−1). Similarly, using the recurrence relation (4.21) with *K*=1 and *γ*=*L*−1 yields(4.26)Here, we have used (4.24) indicating that . By repeated application of the recurrence relation (4.21), we can show that(4.27)It follows again from (4.20) that(4.28)for any values of *K* with 0≤*K*≤(*N*−1). Obviously, this process can be repeated until *γ*=0. This implies that(4.29)for any *K* and *γ* with 0≤*K*≤(*N*−1) and 0≤*γ*≤*L*, meaning thatNote that the summations _{1}≡0 and _{3}≡0 can be proved in a similar way. In other words, the dissipation integral in rotating spheres(4.30)for all possible values of *M*, *N*, *m*, *l* and *n*.

## 5. Concluding remarks

We have studied the dissipation integral in connection with the classical Poincaré problem in rotating cylinders and spheres. While the integral (2.9) in rotating cylinders is always negative and of physical significance, it is striking and intriguing that the integral (4.3) with *M*≥*N* in rotating spheres vanishes identically. Although we are able to provide a mathematical proof that the extremely complicated summations (4.4)–(4.6) vanish identically, the physical and mathematical consequences remain to be understood. The strategy of our mathematical proof for the vanishing summations (4.4)–(4.6) involves the construction of two auxiliary recurrence relations. But the question of whether there exists a simpler mathematical proof is still open.

When *M*<*N*, our extensive calculations suggest that the integral (4.3) is non-zero but can be either negative or positive: it also has no physical significance. Although a different recurrence relation for *M*<*N* can be derived, we cannot draw, as expected, any conclusion about the size or sign of the integral from the recurrence relation. In summary, the dissipation integral (4.3) in rotating spheres is identically zero when *M*≥*N*, while the integral with *M*<*N* is either positive or negative. This raises a profound question as to why the property of the dissipation integral is mathematically and physically so fundamentally different in different geometries, e.g. cylinders versus spheres.

A fundamentally important but mathematically unanswered question is completeness of the set of eigenfunctions of the Poincaré equation given by (3.7)–(3.9). The possible completeness of the set of spherical inertial waves would open an exciting new line in the analysis of many geophysical and astrophysical fluid dynamical problems in rotating spherical systems. This is because, unlike the spherical harmonics, the rotational effect does not couple the inertial modes that are directly associated with the rotational differential operators. Consequently, they offer a more effective and efficient way to solve/understand the fluid-dynamical problem in rotating systems. If we make an assumption that spherical inertial waves given by (3.7)–(3.9) are complete, a mathematical explanation on why the integral (4.3) with *M*≥*N* vanishes while it is non-zero when *M*<*N* can be offered as follows. Provided that (3.7)–(3.9) are complete, and because (3.7)–(3.9) are in the form of polynomials, we may express as(5.1)where _{mjK} are generally non-zero complex coefficients, which leads to(5.2)When *M*≥*N*, the orthogonality property means that(5.3)for all possible values of *j* and *K* in (5.2). When 0≤*M*≤(*N*−1), we then have(5.4)However, a mathematical proof for the completeness of spherical inertial modes (3.7)–(3.9), or the corresponding double Poincaré polynomials, is too complicated to be tractable and remains a great future challenge.

## Acknowledgments

X.L. is supported by NSFC/10633030, MOSTC863/2006AA01A125, STCSM 08XD14052 and CAS KJCX2-YW-T13 grants and K.Z. is supported by UK NERC and STFC grants. We wish to acknowledge helpful comments from Dr M. Rieutord on the first version of the paper.

## Footnotes

- Received October 5, 2008.
- Accepted November 18, 2008.

- © 2008 The Royal Society