ORIGIN OF TIDAL DISSIPATION IN JUPITER. I. PROPERTIES OF INERTIAL MODES

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1 The Astrophysical Journal, 635: , 005 December 10 # 005. The American Astronomical Society. All rights reserved. Printed in U.S.A. ORIGIN OF TIDAL DISSIPATION IN JUPITER. I. PROPERTIES OF INERTIAL MODES Yanqin Wu Department of Astronomy and Astrophysics, University of Toronto, 60 St. George Street, Toronto, ON M5S 3H8, Canada Received 004 July 9; accepted 005 August ABSTRACT We study global inertial modes with the purpose of unraveling the role they play in the tidal dissipation process of Jupiter. For spheres of uniformly rotating, neutrally buoyant fluid, we show that the partial differential equation governing inertial modes can be separated into two ordinary differential equations when the density is constant or when the density has a power-law dependence on radius. For more general density dependencies, we show that one can obtain an approximate solution to the inertial modes that is accurate to the second order in wavevector. Frequencies of inertial modes are limited to!<(is the rotation rate), with modes propagating closer to the rotation axis having higher frequencies. An inertial mode propagates throughout p ffiffiffi much of the sphere with a relatively constant wavelength and a wave amplitude that scales with density as 1/. It is reflected near the surface at a depth that depends on latitude, with the depth being much shallower near the special latitudes ¼ cos 1!/. Around this region, this mode has the highest wave amplitude as well as the sharpest spatial gradient (the singularity belt ), thereby incurring the strongest turbulent dissipation. Inertial modes naturally cause small Eulerian density perturbations, so they are only weakly coupled to the tidal potential. In a companion paper, we attempt to apply these results to the problem of tidal dissipation in Jupiter. Subject headings: convection hydrodynamics planets and satellites: individual (Jupiter) stars: oscillations stars: rotation waves 1. INTRODUCTION 1.1. Physical Motivation We study properties of global inertial modes in neutrally buoyant, rotating spheres. We briefly introduce our motivation for doing so, but refer readers to Ogilvie & Lin (004), as well as Wu (005, hereafter Paper II), for a more thorough introduction. Jupiter s tidal dissipation factor (dimensionless Q-value) has been estimated to be 10 5 Q ; 10 6 (Goldreich & Soter 1966; Peale & Greenberg 1980), based on the current resonant configuration of the Galilean satellites. The Q-value is inversely proportional to the rate of tidal dissipation. So far, the physical explanation for this Q-value has remained elusive, with the most reliable calculations yielding a tidal Q orders of magnitude greater than the above inferred value. Importantly, known extrasolar planets also exhibits Q-values of the same order as Jupiter (Wu 003). This is inferred from the value of the semimajor axis (0.07 AU) within which extrasolar planets are observed to possess largely circularized orbits. We call attention to two common characteristics shared by Jupiter and the close-in exoplanets: except for a thin radiative atmosphere, the interiors of these bodies are convective and they spin fast; typical spin periods are shorter or comparable to twice the tidal forcing period. This motivates us to search for an answer to the tidal Q problem that relates to rotation. In a sphere of neutrally buoyant fluid, rotation gives rise to a branch of eigenmodes: inertial modes. These modes are restored by the Coriolis force and have many interesting properties that make them good candidates for explaining the tidal Q-value. 1.. Previous Works Waves restored by the Coriolis force have been studied extensively in the oceanographic and atmospheric sciences. Various names are associated with them, e.g., Rossby waves, planetary waves, inertial waves, and r-modes (Greenspan 1968). In these 674 contexts, the waves are typically assumed to propagate inside a thin spherical shell. There have been a few studies of inertial modes 1 in the astrophysical context. Schenk et al. (00) give a fairly complete survey of the literature related to inertial modes. We refer readers to that paper for a better understanding of the nomenclature and past efforts. Here we only mention a few related early works that have particular impact on the current one. Bryan (1889) studied tidal forcing of oscillations in a rotating, uniform-density spheroid (or ellipsoid). He performed a coordinate transformation (pioneered by Poincaré) under which the oscillation equation becomes separable. This immensely facilitates our and others study of inertial modes. Lindblom & Ipser (1999) followed essentially the same approach. Papaloizou & Pringle (1981) studied inertial modes in a fully convective star with an adiabatic index ¼ 5/3 (equivalent to a n ¼ 1:5 polytrope). They realized that one could obtain the eigenfrequency spectrum fairly accurately without detailed knowledge of the eigenfunction. This is achieved using the variational principle and by expanding the eigenfunction in a well-chosen basis. They also pointed out that the fully convective case is special in that gravity modes vanish so inertial modes form a single sequence in frequency that depends only on rotational frequency. Lockitch & Friedman (1999, hereafter LF99) studied inertial modes in spheres with an arbitrary polytropic density profile. They expressed the spatial structure of each inertial mode as a sum of spherical harmonic functions and curls of the spherical harmonics. They presented some eigenfrequencies for, e.g., an n ¼ 1 polytrope. We will compare our results against theirs. Inertial modes have also been attacked numerically, via integration of the characteristics ( Dintrans & Rieutord 000; Ogilvie 1 Inertial modes (also called generalized r-modes, hybrid modes) refer to rotationally restored modes in a zero-buoyancy environment.

2 TIDES IN JUPITER. I. 675 & Lin 004), the finite-difference method (Savonije & Papaloizou 1997), and the spectral method Ogilvie & Lin (004) This Work There are two purposes to this paper. First, it lays down the foundation for Paper II, where we discuss our attempt to solve the tidal Q problem. Second, it presents a new series of exact solutions to inertial modes in spheres with a power-law density profile, as well as approximate solutions for spheres with an arbitrary ( but smoothly varying) density profile. Our approach centers on the ability to reduce the partial differential equation governing fluid motion in a rotating sphere to ordinary differential equations. This semianalytical approach produces results that are both easily reproducible and have clear physical interpretations. The mathematics are concentrated in x. We then expose properties of inertial modes that are relevant for its interaction with the tidal perturbations (x 3). A discussion section (x 4) follows in which we consider the validity of our assumptions. Readers interested in the tidal-dissipation problem alone are referred to x 5 for a brief summary of the features of inertial modes upon which we build our theory of tidal dissipation (Paper II). Readers interested in getting only a favor of what inertial modes look like are referred to Figures 5 and 6. The accompanying description (x 3.4), as well as a comparison between inertial modes and the more commonly known gravity and pressure modes (x 3.5), may prove useful as well.. INERTIAL-MODE EIGENFUNCTIONS This section documents our effort in obtaining semianalytical eigenfunctions for inertial modes. The relevant equation of motion is first introduced in x.1. We then deal with increasingly more complicated and more realistic cases. This proceeds from simple uniform-density spheres (x.), to power-law density spheres (x.3), and lastly to spheres with realistic planetary density profiles (x.4)..1. Equations of Motion Consider a planet spinning with a uniform angular velocity of 6 pointing in the z-direction. In the rotating frame, the equations for momentum and mass conservation read ẍ þ 6 < ẋ ¼:p0 þ :p 0 : 0 ; 0 þ :=(x) ¼ 0; where x is the displacement vector while p 0, 0, and 0 are the Eulerian perturbations to the pressure, density, and gravitational potential, respectively. We ignore rotational deformation to the hydrostatic structure, as well as the centrifugal force associated with the perturbation. Both these terms are smaller by a factor (/ max ) than the terms we keep, with max being the breakup spin rate of the planet. This factor is 10% for Jupiter and much smaller for close-in extrasolar planets, which likely have reached spin synchronization with the orbit. We restrict ourselves to adiabatic perturbations, and hence the Lagrangian pressure and density perturbations are related to each other by p p ¼ 1 ; ð1þ ðþ ð3þ where the adiabatic index 1 ¼ ln p/ ln j s and is related to the speed of sound by 1 ¼ c s /p. The interiors of giant planets are convectively unstable, with a low degree of superadiabaticity, as is guaranteed by the fact that the convective velocity is fairly subsonic. This allows us to treat the fluid as neutrally buoyant. Setting the Brunt-Väisälä frequency to zero, we obtain d dr ¼ dp 1 p dr : Given the following expression that relates the Lagrangian and the Eulerian perturbations in a quantity X, X ¼ X 0 þ x =:X ; equations (3) and (4) combine to yield p 0 p ¼ 0 1 ; and the right-hand side of equation (1) can be simplified to :p0 þ :p 0 :0 ¼: p0 : 0 ¼:! ; ð7þ where we have introduced a new scalar ¼ 1 p 0! þ 0 ¼ 1! cs 0 þ 0 ð4þ ð5þ ð6þ ; ð8þ with! being the mode frequency in the rotating frame. We adopt for all variables the following dependence on time (t)and on the azimuthal angle (): X / exp ½i(m!t)Š. Modes with denotation (m,!)and(m,!) are physically the same mode, so we restrict ourselves to! 0, with m > 0 representing a prograde mode and m < 0 representing a retrograde one. In solving for the inertial-mode eigenfunction, we adopt the Cowling approximation (negligible potential perturbations associated with the fluid movement) and ignore any external potential forcing (e.g., the tidal potential), so 0 ¼ 0. The former is justified in x 4.1, and the latter is relevant as we are interested in free oscillations. Equation (8) yields 0 ¼! c s : ð9þ Following convention, we define the following two dimensionless numbers:! ; q ¼ 1 : ð10þ Inertial modes satisfy 0 < 1. Equations (1) and () can now be recast as :=x þ! c s x þ iq(e z < x) ¼ : ; ¼ e r = x H ð11þ ¼ g cs (e r = x); ð1þ where H dr/d ln is the density scale height and H ¼ c s /g (eq. [4]), with g being the local gravitational acceleration.

3 676 WU Vol. 635 Fig. 1. Ellipsoidal coordinates for ¼ 0:75. Left: Equidistant curves of constant x 1 (solid curves) and constant x (dashed curves) in a meridional plane. They become dotted curves outside the surface of the planet, which corresponds to either x 1 ¼ or jx j¼, or both (at the special latitude jcos j ¼). The equidistant curves are most closely packed near the surface at this special latitude. Over the rest of the sphere, the x 1 axis runs similarly as the cylindrical radius, while the x axis runs largely parallel to the rotation axis. Right: Equidistant curves of constant spherical radius in the x 1 -x plane. Operating on equation (11) with e z = and e z <, respectively, and replacing e z = x and e z < x using the resultant equations, we obtain the following relationship between x and : x ¼ 1 1 q (1 iqe z < q e z e z = ): : ð13þ We substitute this equation into equation (1) to eliminate x and to acquire the following key equation for : 9 q z ¼ 1 H r q cos mq (1 q )! z r cs : ð14þ Here is the zenith angle and cos ¼ z/r, with z being the height along the rotational axis. Let! be the cylindrical radius. The partial derivatives here are to be understood as /r ¼ j/rj, / ¼ j/j r, /z ¼ j/zj!, and /! ¼ j/! j z. In general, the above partial differential equation is not separable in any coordinates and only fully numerical solutions can be sought. The displacement vector in spherical coordinates is related to as (eq. [13]) r ¼ 1 1 q r ¼ q r ¼ 1 1 q mq r mqz r! im! q cos z þ q sin iq! ; ð15þ ; ð16þ z : ð17þ In the following discussions (except where noted), we scale all lengths by the radius of the planet (R). Equation (14) describes the pulsation for both inertial modes and pressure modes in a uniformly rotating, neutrally buoyant fluid; gravity modes have zero frequencies in such a medium. While the compressional term [(1 q )! /cs ] represents the main restoring force for the pressure modes, it is negligible (in comparison to the Coriolis terms) for inertial modes, as the latter are much lower in frequency. We adhere to this simplification throughout our analysis and further justify it in x 4.1. We now introduce an important coordinate transform that facilitates much of our analysis. We follow Bryan (1889) in adopting a set of ellipsoidal coordinates (x 1, x ) that depend on the value of. More details on these coordinates are presented in Appendix A. Figure 1 depicts the topographic curves of (x 1, x ) in a meridional plane, as well as the curves of constant r in the x 1 -x plane, when ¼ 0:75. Generally, the x 1 -coordinate is parallel to the!-axis, while the x -coordinate is parallel to the z-axis over much of the sphere. On the spherical surface, either x 1 or x (or both at the special latitude ¼ cos 1 ) equals ; x 1 ¼ 1 at the rotation axis and x ¼ 0 at the equator. The ellipsoidal coordinates are a natural choice for studying inertial modes in a rotating sphere. Motion restored by the Coriolis force (inertial waves) would have liked to follow the cylindrical coordinates, were it not for the spherical topology of the object within which they dwell. The ellipsoidal coordinates, being a hybrid between the cylindrical and the spherical coordinates, are therefore particularly suitable for this purpose. With this set of coordinates, the left-hand side of equation (14) is transformed into 9 q z ¼ 1!!! þ 1!! þ 1 q z ( ¼ 1 x1 1 x x 1 x1 x 1 m x 1 1 x1 1 x ) x x m x 1 x : ð18þ

4 No. 1, 005 TIDES IN JUPITER. I. 677 This, as we shall see shortly, allows the separation of variables under certain circumstances... Uniform-Density Sphere..1. Formal Solution In a uniform-density sphere, the scale height H ¼1, and the right-hand side of equation (14) vanishes. Its left-hand side (eq. [18]) can benefit from the following decomposition, with i satisfying ¼ 1 (x 1 ) (x ); ð19þ D i i þ K i ¼ 0: ð0þ Here the differential operator D i is D i ¼ 1 xi m x i x i 1 xi ; ð1þ and K is a constant introduced when we separate variables. This result was first obtained by Bryan (1889), and its solutions are called Bryan s modes. In fact, the solutions to 1 and are the associated Legendre polynomials. Requiring to be finite at the rotation axis (x 1 ¼ 1), we find 1 and to be the same spherical harmonic of the first kind (Abramowitz & Stegun 197): 1 ¼ ¼ Pl m (x), with l being an integer, K ¼ l(l þ 1), and the variable x taken over the ranges x 1 ½; 1Š and x ½; Š. We explicitly require that 1 (x 1 ¼ ) ¼ (x ¼ ), so the eigenfunction needs only one normalization constant. The following boundary conditions apply. First, at the equator (x ¼ 0), even-parity modes satisfy d dx ¼ 0 ðþ x ¼0 while odd-parity modes satisfy j x ¼0 ¼ 0: ð3þ Properties of the Legendre polynomials (Abramowitz & Stegun 197) require that l þ m be an even integer in the former case and odd in the latter. Second, 1 is finite at the polar axis (x 1 ¼ 1). The numerical equivalent to this statement is best realized by introducing a variable g i that is related to i as i(x i ) ¼ 1 xi jmj=gi (x i ): ð4þ This variable satisfies (eq. [0]) 1 xi d g i dx i x i (jmjþ1) dg i dx i þ k g i ¼ 0; ð5þ where k ¼ K jmj(jmjþ1) ¼ l(l þ 1) jmj(jmjþ1). Regularity of the eigenfunction at x 1 ¼ 1 then translates into a boundary condition dg 1 dx 1 ¼ x1 ¼1 k (jmjþ1) g 1: ð6þ Our solutions show that near the rotation axis, g 1 approaches a constant while 1 approaches zero (if jmj > 0). It is straightforward to show that the displacement vector x has the same equatorial symmetry as. In this problem, the eigenfunction can be solved independently of the eigenvalue. So to determine, we need one more boundary condition. We enforce the physical condition that there be vacuum outside the planetary surface (r ¼ 1), and therefore pressure perturbation at the surface has to be zero. Written in a convenient form of p/ ¼ 0 (Unno et al. 1989), this corresponds to (using eq. [1] and ignoring the compressional term) p ¼ p 1 ¼ 1(:=x) p ¼g r ¼ 0: ð7þ We relate r to using equation (15), as well as relations presented in Appendix A: 1 r ¼ (1 )(x1 x )(1 x 1 )(1 x ) x x1 x1 ; x 1 x 1 1 x x x 1 x1 þm x (1 x1 )(1 x ) : ð8þ So requiring r ¼ 0 at the surface (x 1 ¼ or jx j¼) is equivalent to requiring that d 1 dx ¼ m 1 x1 ¼ 1 1j x1 ¼ ; d m dx ¼sgnðx Þ jx j¼ 1 j jx j¼ : ð9þ Since 1 and have the same functional form [Pl m (x)], these two equations are in fact the same thing. In actual numerical procedure, we solve for g i as opposed to i.equation(9)ismodified for g i as dg 1 dx 1 x1 ¼ ¼ (m jmj) 1 g 1 j x1 ¼ : ð30þ Using equation (4), we find that g 1 is a polynomial of order l jmj. 3 For each (l, m) pair, there can therefore be l jmjroots satisfying this boundary condition, with half of them having > 0. Another way of phrasing this is that for each eigenfunction [ (x 1 ; x ) ¼ P jmj l (x 1 )P jmj l (x )], there are l jmj eigenfrequencies. Roughly half of these are prograde modes (m > 0) and the other half retrograde modes (m < 0). Table 1 presents some example eigenfrequencies for inertial modes in a uniform-density sphere. Our results agree with those presented in Tables 3 and 4 of LF99, as well as Table 1 of Lindblom & Ipser (1999).... The Dispersion Relation In the previous section, we have introduced various eigenvalues (K, k, l, and m), as well as the eigenfrequency.whatis the geometrical meaning of these eigenvalues and what is the dispersion relation for inertial modes? Here we present a derivation that answers these questions. We first convert the independent variable from x i to i ¼ cos 1 x i (to be differentiated with the spherical angle ) so that equation (5) becomes d g i d i þ (jmjþ1) cos i sin i dg i d i þ k g i ¼ 0: ð31þ 3 One can express g i as d jmj Pl 0(x)=dxjmj,andPl 0 (x) is a polynomial of order l.

5 678 WU Vol. 635 TABLE 1 Inertial-Mode Eigenfrequencies in a Uniform-Density Sphere l m Parity a m =1 m = m =3 m =4 1 b... Odd Even Even Odd Odd Odd Even Even Even Even Odd Odd Odd Odd Odd Notes. Here we present eigenfrequencies as m/jmj ¼sgnðmÞ!/,where! is the eigenfrequency in the rotating frame and the rotational frequency. So positive values denote prograde modes, while negative ones denote retrograde modes (opposite to that in LF99). The eigenvalues l and m appear in the eigenfunction as ¼ Pl m (x 1 )Pl m (x ). For each (l, m ) pair, there are l jmj distinct eigenfrequencies. These results agree with those obtained by LF99. a This denotes the parity with respect to the equatorial plane. b This row shows the eigenfrequencies of pure r-modes. They are oddparity, retrograde modes satisfying! ¼ /(jmjþ1) to the first order in. Adopting a WKB approach, we express g i / exp (i R k d i ) with the wavevector k ¼ k R þ ik I, where the real part k R O(k) 3 1 and the imaginary part k I O(1). Substituting this into equation (31) and equating terms of comparable magnitudes, we find k R k; k I jmjþ 1 cos i ¼ jmjþ 1 d ln sin i : ð3þ sin i d i Together with equation (4), this gives rise to the following approximate solution for i in the WKB regime, i ¼ P m l (x i ) / 1 jsin i j 1= cos (k i þ ): ð33þ So i is an oscillating function with a roughly constant envelope. Moreover, ¼k/ þ for even-parity modes (eq. []) while ¼k/ þ / for odd-parity modes (eq. [3]). Nodes of i ¼ 0 are roughly evenly spaced in ½0; /Š space, and the value of k corresponds to twice the total number of nodes (more below). To obtain the dispersion relation, we insert the above approximate solution into the boundary condition at the surface (eq. [9]) where 0 ¼ cos 1. This yields m k sin (k 0 þ ) þ pffiffiffiffiffiffiffiffiffiffiffiffiffi cos (k 0 þ ) 0: ð34þ 1 For k 3 1, this can be approximated by m sin (k 0 þ ) k 0 þ þ n pffiffiffiffiffiffiffiffiffiffiffiffiffi 1 k ; ð35þ where n is an integer. We retain only the positive sign as m can be positive or negative. For even-parity modes, the dispersion relation for the eigenfrequency runs as! ¼ cos 0 ¼ cos n k þ m pffiffiffiffiffiffiffiffiffiffiffiffiffi 1 k þ k " # (1 þ n) m ¼ sin pffiffiffiffiffiffiffiffiffiffiffiffiffi ; ð36þ k 1 while for odd-parity, it is! ¼ cos n k þ m pffiffiffiffiffiffiffiffiffiffiffiffiffi 1 k þ k " # (1= þ n) m ¼ sin pffiffiffiffiffiffiffiffiffiffiffiffiffi : ð37þ k 1 If we define the number of nodes in the 1 ½0; 0 Š range to be n 1 and the number in the ½ 0 ; /Š range to be n,they satisfy n 1 k 0, n k(/ 0 ). So k (n 1 þ n ), as we have mentioned earlier. Moreover, equation (35) yields n n 1 þ m/(1 ) 1/ k for even modes and n n 1 þ m/(1 ) 1/ k for odd modes. Substituting these estimates into equations (36) and (37), we find the following simplified dispersion relation: ¼! sin n (n 1 þ n ) cos k k n 1 (n 1 þ n ) : ð38þ This dispersion relation has also been numerically confirmed. So for a given set of (k, m), the eigenfrequency risesasthe partition of the x 1 /x space moves from ¼ 0(whenn 1 k/, n ¼ 0) to ¼ 1(whenn 1 ¼ 0, n ¼ k/), and there are k/ discreet eigenfrequencies. Recall that k ¼½l(lþ1) jmj(jmjþ 1)Š 1/ l jmj. So for each (k, m ) pair, we find (l jmj)/ prograde eigenmodes and (l jmj)/ retrograde ones, consistent with the results found in x..1. In x 3.1 we discuss further the geometrical meaning of the eigenvalues in spherical coordinates..3. Power-Law Density Sphere.3.1. The Equation Is Separable! We are inspired by the planetary density profile to investigate inertial modes in a sphere where density depends on radius as a power law. To our pleasant surprise, the equation turns out be separable in this case. This forms the basis for much of our work when dealing with real planets. We adopt the following power-law density profile (see eq. [A]): / (1 r ) ¼ 0 x1 x : ð39þ This is not a usual choice and is different from the more typically discussed polytropes. However, near the surface, our power-law density profile behaves like a normal polytrope. Let ¼ 1 r be the depth into the planet. For T1, we find sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 1 r ¼ 1 1 (x 1 )( x ) (1 ) 1 (x 1 )( x ) (1 ) : ð40þ

6 No. 1, 005 TIDES IN JUPITER. I. 679 Fig.. Left: Logarithm of the density (log ) as a function of the logarithm of the depth [log (1 r)] for a ¼ 1 power-law model (solid curve). The density scale is arbitrary. The dotted line is the density profile for an n ¼ 1 polytrope ( p / ). The two profiles behave similarly except near the center. Right: Eigenfunctions 1 (x 1 ) and (x ) as a function of x 1 or x for an even-parity, retrograde m ¼, ¼ 0:4881 mode in the ¼ 1 model (solid curve). The global eigenfunction (x 1 ; x ) ¼ 1. The dashed line demarcates the x 1 and x boundary at ¼ 0:4881. There are five and three nodes within the x 1 and x ranges, respectively, with a corresponding pffiffiffi l ¼ (n 1 þ n ) þjmj ¼18 (see Table ). As x i approaches, the solid curve exhibits a rise in amplitude that is described by a 1/ WKB envelope. For comparison, we draw (dotted curve) the eigenfunction of a counterpart mode in the ¼ 0 (uniform density) model. So near the surface, equation (39) yields /, as is the case for a polytrope model with the polytropic index n ¼. The advantage of choosing such a profile will become clear soon. We obtain the following expression for the density scale height H: H 1 d ln dr ¼ z r z þ! ln r! (1 ) r ¼ (x1 )( x ) ; ð41þ where equations (A) and (A6) are used. Together with equations (A4) (A6), this allows equation (14) to be recast into (again ignoring the compressional term) D 1 þ x 1(1 x1 ) x1 þ m x 1 x1 D þ x (1 x ) x þ m x x ¼ 0; ð4þ where the operator D i is defined in equation (1). So when density follows the power-law profile (eq. [39]), the equation for the inertial modes is again separable: (x 1 ; x ) ¼ 1 (x 1 ) (x ), with i satisfying D i þ x i(1 x i ) x i d þ m dx i xi i þ K ¼ E i i þ K i ¼ 0; ð43þ where we introduce a new operator E i. Note that this equation is exact (except for omitting the compressional term) for the power-law density sphere. i Numerically, it is more accurate to solve for g i ¼ i /(1 x i )jmj/. This function satisfies 1 xi d g i dx i x i (jmjþ1) dg i þ x i(1 xi ) dx i xi þ k jmjx i xi þ m xi g i ¼ 0; dg i dx i ð44þ where k ¼ K jmj(jmjþ1). The boundary conditions presented in x..1 apply here as well, with the exception of equation (6), which should be modified to dg 1 dx 1 x1 ¼1 ¼ k þ ½ðm jmjþ= ð1 ÞŠ g (jmjþ1) 1 : ð45þ So unlike the uniform-density case, both the inertial-mode equation and the boundary conditions now depend explicitly on the value of as well as the sign of m. This breaks the eigenfunction degeneracy existing in the uniform-density case: each eigenmode now has a unique eigenfunction. In Figure we plot the resulting eigenfunction for an m ¼ mode in a ¼ 1 model. We also contrast the density profile of our ¼ 1 model with the conventional n ¼ 1 polytrope model ( p / ). Details of how we construct the -models are presented in Appendix B. Table lists some sample eigenfrequencies for jmj ¼ modes in various -models. The number of eigenmodes remains conserved when varies, with a close one-to-one correspondence between modes in different density profiles. This makes mode typing trivial. We also compare our results against those of LF99 obtained for an n ¼ 1polytrope..3.. WKB Envelope While the eigenfunction inside a uniform-density sphere has a roughly constant envelope (eq. [33]), in the power-law

7 680 WU Vol. 635 density case behaves differently. Following the same operation as in x.., one finds sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k R k þ (m jmjcos i ) jcos i k; j k I jmjþ 1 cos i cos i sin i sin i jcos i j ¼ d h ln sin jmjþ1= i þ ln cos i i = ; ð46þ d i and i TABLE Eigenfrequencies of jmj ¼ Inertial Modes for Various Power-Law Density Profiles l m Parity =0.0 =0.1 =0.5 =1.0 =.0 1 jsin i j 1= jcos i j = cos (k i þ ): ð47þ So, the eigenfunction has an amplitude that rises toward the surface (when jx i j!). In fact, its envelope scales with density as 1 ¼ 1 (x 1 ) (x ) / ½(x1 )( x / p 1 ffiffiffi : )Š= p = k ( LF99) a 1 b... Odd Even Even Odd Odd Odd Even Even Even Even Odd Odd Odd Odd Odd Notes. Eigenfrequencies are again presented as m/jmj ¼sgnðmÞ!/, with positive values denoting prograde modes and negative ones denoting retrograde modes. Here we define l ¼ (n 1 þ n ) þjmj, where ¼ 0 for even parity and 1 for odd parity, and n 1 and n are the number of nodes in the x 1 and x ranges, respectively. When ¼ 0, this l matches that appearing in the Legendre polynomial (Pl m ). The number of eigenmodes is conserved when varies, with a close one-to-one correspondence between modes in different density profiles. a For comparison, we list results calculated by LF99 (their Table 5) using a series expansion method for a polytrope model p /. Not surprisingly, their eigenfrequencies fall somewhere in between our results for the ¼ 1and ¼ models. b This row shows pure r-modes. Both the frequency and the eigenfunction of these modes do not depend on the equation of state, to the lowest order in. ð48þ Figure shows one example of such a WKB envelope. We find that the dispersion relation is also modified slightly from the case of a uniform-density sphere. But qualitative features of equation (38) remain..4. Approximate Solution for Realistic Density Profiles The density profile inside a planet typically traces out two different polytropes: near the surface, the gas can be approximated as an ideal gas composed of diatomic molecules, with a mean degree of freedom of 5, so 1 ¼ ln P/ ln j s ¼ 7/5 and the correct polytrope number is n ; in the interior of the planet, Coulomb pressure and electron degeneracy modify the equation of state and raise 1 to while reducing the polytrope number to n 1. This motivates us to model Jupiter s density profile using two power laws of the form in equation (39): ¼ 1 in the interior and ¼ 1:8 near the surface, with the transition occurring at a radius r 0:98 (see Paper II). This reproduces profiles in realistic Jupiter models (Guillot et al. 004). The presence of a core or a phase transition complicates this picture. We discuss them in more detail in Paper II. In general, equation (14) is not separable under these density profiles. However, we discover an approximate solution that is accurate to the second order in wavenumber [O(1/k ) with k 3 1], for the case when the density profile is a power law near the surface and smoothly varying in the interior. This is largely inspired by the WKB result in x.3.. We first introduce a fiducial density surf that satisfies equation (39), with a power-law index taken to be that for the true density near the surface. We also define X ¼ surf / so that X 1 near the surface and deviates from unity toward the center. The inverse of the density scale height can be written as 1 ln ¼d ¼ d ln X þ (1 ) r H dr dr (x1 )( x ) : ð49þ Introducing a variable t, t ¼ x1 x ¼ 1 1 r ; ð50þ one finds d ln X dr ¼r (1 ) d ln X dt : ð51þ Now repeating the same calculations that led to equation (4), we obtain (E 1 E ) d ln X d ln t ; (1 x 1 )x 1 (x1 þ (1 x )x ) x 1 ( x ) þ m(x 1 x ) x t ¼ 0; ð5þ where the operator E i is defined in equation (43). Here, since /x i k, we cannot ignore the terms in the square brackets if we want to be accurate to O(1/k ), the final goal of our procedure. Instead, we experiment with the following decomposition for, ¼ p ffiffiffiffi X rffiffiffiffiffiffiffiffiffi surf 0(x 1 ; x ) ¼ 0(x 1 ; x ): ð53þ This is inspired by the WKB envelope presented in equation (48). Formally expressing E i ¼ a i /x i þb i /x i þ c i,where the meanings of a i, b i,andc i are clear from equation (43), we find E i ¼ ffiffiffiffi p p ffiffiffiffi pffiffiffiffi p X 0 X ffiffiffiffi X X Ei 0 þ a i þ a i x i x 1 xi þ b i 0; x i ð54þ

8 No. 1, 005 TIDES IN JUPITER. I. 681 where p ffiffiffiffi X ¼ x i x i x i pffiffiffiffi d ln X X d ln t : ð55þ In the WKB region, let : ik and /z ik z ; equation (14) yields k q k z 0; ð59þ A straightforward but lengthy derivation shows that equation (5) canberecastintoanequationfor 0, (E 1 E ) 0 þ (d 1 d ) 0 ¼ 0; ð56þ where the coefficient d i is ( d i ¼ 1 p pffiffiffiffi 1 xi ffiffiffiffi X þ d ln X X x i x i d ln t ; x i(1 xi ) pffiffiffiffi ) X xi m d ln X x i xi : ð57þ d ln t Here refers to the power-law index for surf, or that for near the surface. The benefit of the transformation introduced in equation (53) is that d i contains no derivatives on. Near the surface, X ¼ const and d i ¼ 0, as expected, and equation (56) is separable. But in deeper regions of the planet, X is a complex function of x 1 and x and equation (56) is not separable. However, if X is a smoothly varying function with a scale length being the radius of the planet, 4 one can show that in the WKB region, d i 0 0 and is 1/k smaller than the E i 0 term. So if we ignore the d i terms, we only introduce errors of order O(k )T1. It is worth pointing out that the non-wkb region occurs near the surface where our solution is exact. In conclusion, we can adopt the following approximate solution for : ¼ p ffiffiffiffi X 0 ¼ rffiffiffiffiffiffiffiffiffi surf 1(x 1 ) (x ); ð58þ with i (x i ) satisfying E i i þ K i ¼ 0. This solution is exact near the surface and is accurate to O(1/k ) in the WKB region. The latter attribute indicates that the approximate solution describes accurately both the envelope and the phase of the actual inertial mode eigenfunction. 3. PROPERTIES OF INERTIAL MODES Having discussed methods to obtain inertial-mode eigenfunctions in various density profiles, here we focus on studying general properties of inertial modes. This include the WKB properties, the normalization relationship, and the density of modes in the frequency spectrum. Moreover, we attempt to give readers a graphical impression of what inertial modes look like both inside and outside the planet. Lastly, we compare inertial modes against well-known gravity and pressure modes to gain intuition for this branch of eigenmodes WKB Properties We first derive the WKB dispersion relation for inertial modes. We then determine the confines of the WKB region and end with a general derivation for the WKB envelope of inertialmode amplitude. 4 In the case of Jupiter, X ¼ 1 near the surface and gradually rises to larger values toward the center. or k z /k 1. This is to be compared with result from a more careful derivation (x..) that shows sin (n /k). Since the x -axis is largely along the z-axis (Fig. 1), k z n /R. So we have k k/r (n 1 þ n )/R, with R being the planet radius (normalized to be 1). Such a dispersion relation implies that the mode frequency (! ¼ ) does not depend on the number of wiggles in a mode, but rather on the direction of wave propagation. Modes that propagate close to the rotation axis have higher frequencies (! ) than those that propagate close to the equator (! 0). Such a dispersion relation can be understood by the following physical argument. We relate x to using equation (13), x i k q e z k z 1 q : ð60þ So the fluid velocity (v ¼ x/t) is perpendicular to the phase velocity (v ph ¼!/k k), or v = v ph 0. Inertial waves are largely transverse waves. A mode with its phase velocity along the equator will show fluid motion in the z-direction. As a result, it experiences little Coriolis force and has a frequency that is close to zero. In contrast, a mode with k along the z-direction will experience the strongest restoring force and will have the highest frequency. While the phase velocity of the inertial wave v ph ¼!/k k,the group velocity of the inertial wave runs as v g ¼ : k! ¼! k z e z þ! k h e h ¼ k h k 3 e z k zk h k 3 e h ¼! k (k q e z k z ); ð61þ where k ¼ k z e z þ k h e h. It is easy to show that v ph = v g ¼ 0. Now we search for the boundary that separates the WKB cavity from the evanescent cavity for inertial modes. Consider propagation in either one of the (x 1, x ) directions. Equation (46) shows that the real part of the wavevector remains fairly constant over the whole planet and is little affected by the density profile, while the imaginary part of the wavevector rises toward the surface as /jx i j. The latter becomes comparable to the former when jx i j /k. Recallingthatjx i j¼ occurs at the surface, one sees that an upward-propagating wave is reflected near the surface at a depth /k at most latitudes, except near a special latitude jcos j when the wave penetrates much higher into the envelope, to a depth of /k. We call this special latitude where x 1 jx jthe singularity belt, and it is an important region for mode dissipation. We will return to this concept in x 3.4. Lastly, we turn to study the WKB envelope of an inertial mode. Multiplying equation (1) by ẋ and simplifying the resulting expression using equations (4) and (6), we arrive at the following equation of energy conservation: t ẋ = ẋ þ p0 c s þ := p 0 ẋ ¼ 0: ð6þ The terms in the first set of parentheses are readily identified as the energy density (including both kinetic energy density and compressional energy density), while that in the second set is the energy flux carried by the wave. Both quantities look identical to those

9 68 WU Vol. 635 for nonrotating objects. This is expected since the Coriolis force is an inertial force and does not do work or contribute to energy. The relative importance between the two energy-density terms is! jxj p 0 =c s ¼ c s jxj! c s k! R c s k GM=R GM=R 3! : ð63þ Since we are interested in k 3 1 modes, and since for planets rotating well below the breakup speed, inertial mode frequency (! ) is smaller than the fundamental frequency of the planet [(GM/R 3 ) 1/ ], the above ratio is much greater than unity over much of the planet. Exceptions occur near the surface, where 1/k (! /GM/R 3 )T1/k, well outside the WKB cavity. So the energy density of an inertial mode is dominated by its kinetic part. For a standing wave, the energy density (and in this case, the kinetic energy density) remains constant in time. So different velocity components must be out of phase with each other. For instance, near the surface, j jj j 3 j r j,and and are 90 apart in phase. Take a standing wave of the form x(r; t) ¼ x(r)e i!t þ x? (r)e i!t ; ð64þ equation (60) gives rise to the following energy density (the time-independent part): while the energy flux H ¼ ẋ = ẋ þ p0 c s! jx(r)j! k q k z þ q4 k z (1 q ) j (r)j! k q 1 j (r)j ; F ¼ p 0 ẋ ¼! Im ½p 0 (r)x? (r) Š ¼! 3 Im ð x? Þ ð65þ! 3 k q e z k z q 1 j (r)j ¼ v g H; ð66þ where v g is the local group velocity as derived in equation (61). Inside the WKB propagating cavity, the energy flux is constant (no reflection). This, coupled with the fact that k is largely constant inside the planet, yields the following WKB envelope: j (r)j/ p 1 ffiffiffi : ð67þ This agrees with results derived from more detailed considerations (eqs. [33], [48], and [58]). 3.. Mode Normalization We derive a scaling for the total energy in a mode. Applying results from x 3.1 and the Jacobian defined in equation (A3), we express total energy in a mode as Z Z E ¼ d 3 r H! d 3 r jx(r)j! Z 1 q d 3 r k j (r)j! (1 ) Z x1 x dx1 dx d k j j : ð68þ Fig. 3. Mode energy as a function of n 1 for m ¼ even-parity inertial modes in a ¼ 1model(solid curve). Here, all inertial modes are normalized with (x ¼ 0) ¼ 1 and satisfy n 1 ¼ n (so 0:70). The dotted curve is a fitting power law E / n1 :65. Most of the mode energy density lies inside the WKB region. Within this region, each nodal patch (x 1 x ) contributes a comparable amount to the total energy. Since the total number of patches is n 1 ; n, we obtain E! (1 ) n 1n k j j x 0 x 1x x 1 1! (1 ) n 1n j j x 0 x 1 1 ; ð69þ where the term is supposed to be evaluated at the last crest away from x 1 ¼ 1 and x ¼ 0 (center of the sphere). Adopting the arbitrary normalization of (x ¼ 0) ¼ 1 and assuming that 1(x 1 1) does not depend on n 1, E / n 1 n if evaluated at a fixed frequency. Our numerical results, however, show ( Fig. 3) that E / n :65 1 in a ¼ 1 model when only n 1 ¼ n modes are considered (or 0:70). The small difference results from the fact that as n 1 increases, the amplitude of the last crest [ 1 (x 1 1)] increases slightly. We adopt the numerical scaling E / n :65 1 for our later studies Density of Inertial Modes For our tidal problem, it is useful to study how dense inertial modes are within a given frequency interval. More specifically, we ask, for a given frequency 0, how close is the closest resonance (smallest j 0 j) one can find among modes satisfying wavenumber k k max? Since frequencies of inertial modes do not depend on the mode order but on the direction of mode propagation, modes of very different wavenumbers can coexist in the same frequency range. Figure 4 shows how depends on k for even-parity, jmj ¼ modes in a uniform-density sphere. The mean frequency spacing between modes of the same k (or n 1 þ n for more general models) is d/dn (1 ) 1/ /k. This value decreases as approaches 1 or as k rises. The typical distance away from a

10 No. 1, 005 TIDES IN JUPITER. I. 683 Fig. 4. Frequencies of even-parity, jmj ¼ inertial modes (shown here as sgn ) as a function of wavenumber k (n 1 þ n ). Modes above the dotted line are prograde modes, and retrograde if below. These frequencies are obtained using a uniform-density model, but the overall features persist for more general models. Note the asymmetry between the frequencies of prograde and retrograde modes; this follows from eqs. (36) and (37). resonance j 0 j is half this value. When we include all modes with k < k max, the closest resonance likely has j 0 j (1 ) 1/ /k max /k max What Do Inertial Modes Look Like? In this section, we provide a graphical impression of what inertial modes look like both inside and on the surface of planets. This may be helpful for readers as inertial modes are drastically different from modes in a nonrotating sphere, which can be described by a product of a radial function and a single spherical harmonic function. In the interior of a planet, an inertial mode produces alternating regions of compression and expansion, much like gravity or pressure modes do, except that for inertial modes, these regions are lined up along the (x 1, x ) coordinates. Figure 5 depicts what the Eulerian density perturbation ( 0 ) and the perturbation velocity in the rotating frame look like in a meridional plane, for an example inertial mode (n 1 ¼ 5, n ¼ 3, and m ¼). There are three noteworthy features. The first is that the largest perturbations, as well as the steepest spatial gradients in these quantities, are to be found near the surface, especially near the angle jcos j. The second feature is that velocity near the surface is purely horizontal. The radial component vanishes as is required by the boundary condition (eq. [7]). The third feature is that the velocity patterns inside the planet take the form of vortex rolls. Inertial modes produce largely incompressible and largely rotational motion (j:<xj3j:=xj). Figure 6 shows a surface view of the density perturbation for the same mode. For this retrograde mode, the pattern rotates retrogradely on the planet surface. First note the number of nodal patches on the surface. For a p- org-mode in a nonrotating star, the number of radial nodes does not show up in the surface Fig. 5. Meridional look at the inertial mode presented in Fig.. This mode is symmetric with respect to the equator and retrograde (m ¼), with n 1 ¼ 5and n ¼ 3. The planet model has a density profile ¼ 1. The left panel shows the Eulerian density perturbation ( 0 ), using both gray scale and equidistant contours. Lighter regions (and dashed contours) stand for 0 < 0, while darker regions (and solid contours) represent 0 > 0. The dotted curves indicate the (x 1, x ) coordinates. Counting the number of nodes along each coordinate, one p ffiffi recovers n 1 and n. The right panel shows the fluid velocity (v r and v components only) in the rotating frame as arrows, with the size of the arrows proportional to v.notethatvr vanishes at the surface, as is required by the boundary condition (eq. [7]). Both the mode amplitude and the wavevector remain relatively constant over much of the planet, but rise sharply toward the surface. This rise is most striking near the special angle jcos j ¼ ¼ 0:4881, marked here by straight lines.

11 684 WU Vol. 635 Fig. 7. Same as Fig. 5, except for an m ¼ r-mode. This mode has ¼ 1/(jmjþ1) ¼ 0:3333 and, in our notation, n 1 ¼ n ¼ 0andl ¼jmjþ1 ¼ 3. There is no internal nodal point, and the radial velocity is zero everywhere. This mode exhibits odd symmetry toward the equator. Fig. 6. Surface density perturbation ( 0 )forthesamemode(m ¼, n 1 ¼ 5, n ¼ 3) as in Fig. 5. The short vertical line marks the rotation axis. In both hemispheres, there is a latitudinal belt within which density perturbation attains the largest amplitude and varies with the steepest gradient. This lies around jcos j ¼, and we call it the singularity belt. The surface pattern of the mode also discloses the quantum numbers: from the pole to the equator, there are n 1 nodal patches above the singularity belt and n nodes below it. pattern. For inertial modes, however, the values of n 1, n, and m (and even ) are all clearly embedded in the surface pattern. The surface pattern tells all. Second, note the presence of a belt (in both hemispheres) near jcos j ¼ where the mode exhibits both the largest perturbation as well as the largest gradient of perturbation (see also Fig. 5). This we call the singularity belt, and it is a feature unique to inertial modes. It is located where both x 1 and jx j1/k, corresponding to a region with a depth 1/k and an angular extent 1/k.Thisregionwillturnout to be very important for tidal dissipation (Paper II). For comparison, we present a meridional look for the m ¼ r-mode in Figure 7; r-modes are a special branch of inertial modes, and we discuss them in x Comparison with Gravity and Pressure Modes Modes restored by pressure or buoyancy ( p-org-modes) are familiar to astronomers. To help understand inertial modes, we capitalize on this familiarity by discussing the differences between these modes and inertial modes. Inertial modes are more analogous to g-modes than to p-modes. Frequencies of inertial modes are higher if their direction of propagation is more parallel to the rotation axis (!/ k z /k, z being the rotation axis). Their frequencies are independent of the magnitudes of the wavevector and are constrained to!<. Similarly, higher frequency gravity modes propagate more parallel to the potential surface (!/N k h /k, h being the horizontal direction), satisfying!<n. Both inertial waves and g-waves are transverse waves, i.e., their group velocities (direction of energy propagation) are perpendicular to their phase velocities. The p-modes, in comparison, experience stronger restoring force and hence higher frequencies if their wavelengths are shorter. Phase velocities and group velocities of p-modes are identical. Due to their low-frequency nature, both g-modes and inertial modes cause little compression in their propagating cavities. As a result, the energy of g-modes is dominated by the (gravitational) potential and kinetic terms, which are alternately important for half a cycle, each contributing on average half to the total energy. Inertial modes live in neutrally stratified medium; their energy is dominated by kinetic energy alone. As a result, while g-modes (and p-modes) suffer dissipation from radiative diffusion and viscosity, inertial modes are only sensitive to viscosity. As p- or g-waves propagate toward the surface, their wavelengths typically shorten, resulting in the largest dissipation. An inertial mode, however, propagates in its WKB cavity with a roughly constant wavelength, except near the singularity belt (r R and cos 1 ), where its wavelength shrinks drastically. This is where we expect the largest dissipation to occur. Lastly, each p-org-mode in a nonrotating star has an angular dependence that is described by a single spherical harmonic function, P m l (; ). In contrast, the angular dependence of each inertial mode is composed of a series of spherical harmonic functions. This implies that while for p- and g-modes only the l ¼, jmj ¼ branch can be excited by a potential force of the form P (such as the lowest order tidal force), for inertial modes every jmj ¼ even-parity mode can be excited. In this sense, the frequency spectrum of inertial modes is dense and the probability of finding a good frequency match (forcing frequency mode frequency) is much improved over the nonrotating case (x 3.3). 4. FURTHER DISCUSSION 4.1. Justifying Assumptions In our effort to obtain a semianalytical solution for the inertial modes (x ), we have made a string of simplifying assumptions. We justify them here. In the analysis throughout this paper, we have ignored the compressional term [(1 q )! /cs in eq. (14)]. This is also called the anelastic approximation (ignoring 0 ), and is often adopted when studying subsonic flows in stratified medium. This assumption is justified by equation (63) and discussions around it, where we show that the inertia term dominates over the compressional term in most of the planet except well above the WKB propagating region. Hence the compressional term does not significantly affect either the frequency or the structure of an inertial mode.

12 No. 1, 005 TIDES IN JUPITER. I. 685 Inertial modes are excited by the tidal potential through its density perturbation ( 0 ; see Paper II). Could we be removing tidal forcing of inertial modes by ignoring 0? Fortunately, no. Equation (9) states that 0 ¼! /cs. Since is nonzero, tidal forcing is zero only when cs!1. Taking the anelastic approximation does not preclude tidal coupling. It does imply, however, that tidal forcing of inertial modes is expected to be weaker than tidal forcing of the fundamental mode, by the same ratio that relates the inertia term to the compressional term. We have also ignored the potential perturbation caused by inertial modes themselves ( 0 in eq. [1]). This so-called Cowling approximation is justified here. The potential perturbation 0 is related to the density perturbation by the Poisson equation, 9 0 ¼ 4G 0 : ð70þ Relative to the inertia term in equation (1), the potential term is smaller by : 0 ẍ 4G cs k 1 k ; ð71þ where we have used the results from equations (9) and (60). So the Cowling approximation is appropriate for high-order modes (k 3 1), which are indeed the modes we are concerned with (Paper II). We have assumed that turbulent viscosity does not modify mode structure significantly. This is equivalent to assuming that the turbulent dissipation rate T!. This assumption is confirmed by the numerical study to be presented in Paper II. For numerical tractability, we have ignored the rotational deformation of the planet, as well as the centrifugal force on the perturbed motion. These introduce a 10% error in the case of Jupiter, and much less for extrasolar planets. We do not expect the general structure and dispersion relation of inertial modes to be significantly modified when these effects are taken into account. Our last simplifying assumption concerns the planet s atmosphere. This is by far our most uncertain assumption. In solving for the inertial modes, we assume that the planet is fully convective (and neutrally buoyant). However, Jupiter-like planets have surface radiative zones with varying depths depending on their surface composition and external irradiation. In the Jupiter model by Guillot et al. (004), the atmosphere is neutrally stratified up to the photosphere (1 bar in pressure), above which the temperature profile is largely isothermal with a scale height (depth) 0 km 3 ; 10 4 R. Inertial waves are reflected inward at a depth 1/k at most latitudes, and at a much shallower depth 1/k near the singularity belt (x 3.1). So modes with k 1/3 ; will have most of their WKB cavity inside the convective region and are only mildly affected by the isothermal layer; while modes with k 1/(3 ; 10 4 ) 1/ 60 will have their uppermost WKB turning point below this isothermal layer and so will not be affected at all. In Paper II we show that the modes relevant for tidal dissipation have k 60. Two concerns arise following the above discussion. The referee (D. Stevenson) has called our attention to the problem of static stability in the Jovian atmosphere. This is currently uncertain due to the presence of water condensation. A wet adiabat is less steep than a dry adiabat, as water in an upward-moving parcel condenses and releases latent heat. So the actual temperature profile will be superadiabatic for some (very moist) parcels and subadiabatic for some (very dry) parcels. Such a conditional stability makes it difficult to study the propagation of inertial modes. 5 But if the stably stratified atmosphere does extend sufficiently deep and does harbor sufficiently strong buoyancy ( Brunt-Väisälä N!), inertial modes in the interior can couple to gravity waves in the atmosphere to form hybrid modes. Since most of the mode inertia is in the interior, this will not affect much the frequencies of these modes; however, it may affect their surface structure. Moreover, inertial gravity waves in the atmosphere may propagate upward freely, break, and dissipate in the low-density environment in the upper atmosphere. This brings about an extra damping mechanism for the interior inertial modes. The second issue concerns extrasolar hot Jupiters. These planets are strongly irradiated by their host stars. Their surface isothermal layers may deepen to a depth of 10 R. So even fairly low-order inertial modes may be affected. For these two reasons, it is relevant to study the behavior of inertial modes in the presence of an isothermal layer. We plan to do so in the future. 4.. Special Case: r-modes The r-modes have been considered a potential candidate for spinning down young neutron stars and for emitting detectable gravitational waves (see, e.g., Owen et al. 1998), so much attention has been paid to this special class of inertial modes (they also appear in Tables 1 and ). What is the relation between inertial modes (also called generalized r-modes by Lindblom & Ipser 1999) and r-modes? The r-modes are purely toroidal, odd-parity, retrograde inertial modes. They are, to the lowest order in, incompressible and move only along spherical shells ( Papaloizou & Pringle 1978). Setting r ¼ 0 and :=x ¼ 0, we find that the displacement vector for r-modes can be expressed using a single stream function Q ¼ Q(r; ; ), where x ¼ :<(rq), or ¼ im sin Q; ¼ Q : ð7þ Taking the curl of equation (11) and retaining only terms to the lowest order in, wefind 1 sin Q mq þ m sin sin Q ¼ 0: ð73þ This is Legendre s equation with solution Q / Y l 0 ; m(; ) andeigenvalue ¼!/ ¼m/l 0 (l 0 þ 1), where l 0 jmj (Papaloizou & Pringle 1978). So the displacement vector x ¼ :<(rq) / r <:Y l 0 ; m and is purely axial (toroidal) in nature. Obviously, both the angular dependence and the eigenfrequency of r-modes are independent of the equation of state (compare r-mode entries in Tables 1 and ). This is expected as r-modes are restricted to moving only along spherical shells. According to this analysis, there should be infinite number of r-modes for each m-value. However, in accordance with both Lindblom & Ipser (1999) and LF99, we do not uncover any r-mode with l 0 > jmj. This is explained by LF99 (see also Schenk et al. 00). They realize that under the isentropic 5 Data gathered during the descent of the Galileo probe (Seiff et al. 1998) showed that at the entry site, the temperature profile follows the dry adiabat from the photosphere (1 bar) down to 16 bars, with some evidence (temperature measurements and the presence of gravity waves) for a stable layer from 15 to 4 bars. It is now known that the probe entered a particularly dry area, so this temperature profile may not be indicative of the whole atmosphere.