Inertial modes in slowly rotating stars: An evolutionary description

PHYSICAL REVIEW D 66, 123001 ~2002!

Inertial modes in slowly rotating stars: An evolutionary description

Loıc Villain* and Silvano Bonazzola†

Laboratoire Univers et The´ories (F.R.E. 2462 du CNRS), Observatoire de Paris, F-92195 Meudon Cedex, France~Received 28 March 2002; published 3 December 2002!

We present a new hydrocode based on spectral methods using spherical coordinates. The first version of thiscode aims at studying time evolution of inertial modes in slowly rotating neutron stars. In this paper, weintroduce the anelastic approximation, developed in atmospheric physics, using the mass conservation equationto discard acoustic waves. We describe our algorithms and some tests of the linear version of the code, and alsosome preliminary linear results. We show, in the Newtonian framework with a differentially rotating back-ground, as in the relativistic case with the strong Cowling approximation, that the main part of the velocityquickly concentrates near the equator of the star. Thus, our time evolution approach gives results analogous tothose obtained by Karino, Yoshida, and Eriguchi@Phys. Rev. D64, 024003~2001!# within a calculation ofeigenvectors. Furthermore, in agreement with the work of Lockitch, Andersson, and Friedman@Phys. Rev. D63, 024019~2001!#, we found that the velocity seems to always get a nonvanishing polar part.

DOI: 10.1103/PhysRevD.66.123001 PACS number~s!: 97.60.Jd, 02.70.Hm, 04.40.Dg

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I. INTRODUCTION

In 1998, Andersson@1# discovered that in any relativistirotating perfect fluid,r-modes1 are unstable via the couplinwith gravitational radiation. This is just a particular casethe Chandrasekhar-Friedman-Schutz~CFS! mechanism dis-covered in 1970~see@4# and @5#!. However, a characteristiof r-modes is the fact that this instability is expected whever the speed of rotation of the star~see also Friedmanet al.@6#!. This discovery triggered important activity in the netron stars~NS! community since such inertial instabilitiemight contribute to the explanation of the observed relatslow rotation rates of young and recycled NS~see Bildsten@7#!. Moreover, such an instability could in principle makeNS efficient sources of gravitational waves~GW! for obser-vation by the ground interferometric detectors under cstruction or their direct descendants. Nevertheless, as vanot always well-known physical processes take place whmight inhibit the growth ofr-modes in real NS, their relevance is still questionable at this time. More details canfound in the reviews by Andersson and Kokkotas@8# and byFriedman and Lockitch@9#.

The present study was originally motivated by the resuannounced by Lindblomet al. in 2001 @10#. These authorscomputed, in the highly nonlinear regime, the evolutionatrack of the CFS instability ofr-modes in a Newtonian NSwith a magnified approximate post-Newtonian radiationaction ~RR! force. In their calculation, they found a rapgrowth of the mode, until strong shocks appeared aquickly damped it. A recent article by Arraset al. @11# asserts

*Email address: [email protected]†Email address: [email protected] this paper we shall use the terminology of the hydrocomm

nity: inertial modes are the oscillatory modes of a fluid whosestoring force is the Coriolis force.R-modes @or purely toroidal~axial! modes# then form a subclass of inertial modes. The lawere discovered by Thomson in 1880@2#. The reader can find ashort point of history in Rieutord@3#.

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that this phenomenon was just an artifact linked with thuge RR force. Yet, we found those results so interestingso surprising that we decided to try to reproduce and beunderstand them with a different approach. The codeLindblom et al. @10# was written in cylindrical coordinateswith a 3D finite difference scheme and they studied fasttating stars. We chose to create a spectral code in sphecoordinates and to begin with a slowly rotating NS.

The use of spectral methods was motivated by thethat they can provide a deep insight in describing the turlence generated by quadratic terms. This is analogouwhat is done with Fourier analysis in the study of homogneous turbulence~e.g., Lesieur@12#!. Furthermore, we chosea slowly rotating NS for several reasons that will be eplained in the following.

Since NS are known for their quite rapid rotation asince inertial modes have for restoring force the Corioforce, ana priori reasoning would probably lead to the coclusion that the only—or most—interesting case of inertmodes to study is the case of modes in a fast rotatingHowever, the final answer is not so easy to decide, as amobservational data and among models of cooling of pulsmany things in fact support slowly rotating newborn singNS. For instance, the estimations of the rotation periodsbirth of the two historical x-rays and radio pulsars whoages are known exactly~the 820 year old 65.8 ms periopulsar in SNR 3C58@13,14# and the 948 year old 33 mperiod Crab pulsar! are, respectively, 60 ms@13# and 14 ms@15#. Moreover, compared with the assessment of the angmomentum of isolated supergiant stars~whose core collapsewill give type II supernovæ and then potentially NS!, thesenumbers show that important losses of angular momenare expected to happen during the stellar evolution. Furtmore, this is the same concerning the giant phase ofmassive stars that give white dwarfs. Indeed, the obserrotation periods of 19 white dwarfs range between 12.1 mand 12 days with a median value of about 1 h@16# but, if theSun shrank without losing any angular momentum intowhite dwarf of the same mass with a radius of 5000 km,rotation period would be of a few minutes only. In the stel

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LOIC VILLAIN AND SILVANO BONAZZOLA PHYSICAL REVIEW D 66, 123001 ~2002!

evolution community, several ways to explain these weangular momenta can be found and the final answer isnot clear. But a quite accepted idea is that magnetic fieprobably play the key role via magnetic braking type mecnisms ~Schatzman@17#!. And whatever the actual mechanism, the current conclusion in this community is that a huamount of angular momentum is supposed to be lost duthe stellar evolution for main sequence stars of any mThen, the most difficult thing to explain in stellar evolutiophysics seems to be the reason why these rotation periobirth ~of the orders of 1 h for white dwarfs and of 1 ms foNS! are sosmall ~e.g., Spruit@18# and Spruit and Phinney@19#!: stellar evolution scenarios do not expect baby NS tofast rotating.

However, it can be mentioned that some isolated fasttating NS are found. But, for the most part, the current idethat these stars have been recycled in binary systemsthese systems, where fast rotating NS exist, the NSthought to have been spun up by the accreted matter.recent discoveries of accreting millisecond puls~XTEJ0929-314, SAXJ1808.4-3685, XTEJ1751-305, and1636-53! whose rotation frequencies are, respectively, 1401, 435, and 581 Hz@20# ~corresponding to respective periods of 5.4, 2.5, 2.3, and 1.7 ms! support the above scenarioYet, there is also an example of a single millisecond pulassociated with a supernova remnant: PSR J0537-6whose period is 16 ms, whose age is about 5000 yr,which is supposed to have had an initial period of a few~see, for instance,@21#!. Hence, in spite of all the abovarguments, the existence of rapidly rotating baby NS shonot be completely excluded. Moreover, a fast rotating baNS with very weak magnetic field and consequently veweak losses of angular momentum due to magnetic braklike mechanisms during the supergiant phase would escto the observations as pulsars but would be very interesas GW sources.

Nevertheless, such a discussion does not make cleaimportant point. Indeed, whatever the value of the frequenit does not directly tell if a pulsar is fast rotating or if it inot. What determines in a particular work, if a NS canregarded as~quite! slowly rotating is the relative importancof the deformation for this study. In a more general framwork, this question is settled by the ratio between the angvelocity of the pulsar and its Keplerian angular velociEven for the fastest rotating known pulsars~that are in binarysystems! this ratio is less than a third, since the Kepler frquency is around 1 ms~see@22#!. Furthermore, what is im-plied in the hydrodynamics of a NS is not this ratio, butsquare. Hence, in appropriate units, this factor is less t10% of theCoriolis forcefor most of the NS. Anyway, thereis a last crucial issue. In a Newtonian star, even if this facis a few percent correction in the equation of motion, itfundamental since it creates a coupling between the pand the axial parts of the velocity. Yet, in a relativistic stthe situation is completely different as noticed by Kojim@23#. Indeed, in a relativistic rotating star, the frame draggterm has the same qualitative result: it makes a coupbetween the polar and the axial parts of the velocity. Butappropriate units, this coupling is scaled by the ratio betw

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the angular velocity and the Keplerian angular velocity anot by the square of this ratio. Hence, even for the fasrotating known NS, the deformation introduces a kindsecond order correction that can be neglected in aapproach.2

Thus, our choice of using the slow rotation limit as a firstep was motivated by all the astrophysical reasons mtioned above: even for the millisecond pulsar, the slow rotion approximation is still quite good. But, secondly, wwanted to look for a possible saturation due to nonlincoupling that may occur before a highly nonlinear regimereached. To better understand such a phenomenon,thought it was probably wiser to begin with an easier sitution in which there are not several effects with consequenof the same orders of magnitude. Thus, we began to buinonlinear hydrodynamics code using the Newtonian theof gravity and the slow rotation approximation. But, oncelinear Newtonian code had been written~the first step to anonlinear version!, upgrading it to a general relativity~GR!linear code withstrong Cowling approximation3 was quiteobvious and we chose it for a second approach to ineinstabilities. Indeed, as was already mentioned aboKojima first noticed in 1998@23# that the frame draggingphenomenon makes the relativisticr-modes quite differentfrom the Newtonianr-modes. But concerning this linear relativistic study, we also decided to begin with the slow rotatilimit to try to get a better understanding of the sphericrelativistic case before putting what can be seen as a seorder correction.

This article is organized as follows. In Sec. II we startdescribing the physical conditions we chose in the Newtian case. Then we write the Navier-Stokes equations~NSE!in dimensionless form, and the RR force we introducedthe linear study. This enables us to calculate the associcharacteristic numbers and to observe that the corresponnumerical problem is a stiff one with typical time scalbeing orders of magnitude different. We discuss someproximations that can be made in Sec. III. The most imptant of them is the anelastic approximation. Section IV eplains the basic tests of the code with, for instance, the linr-modes of typel 5m in a rigidly rotating spherical fluid.This admits an easy and analytical solution of the Euequation ~EE! in the divergence-free approximation. Wdemonstrate that, thanks to spectral methods, the velo

2As an example, we verified that, for a rotation frequency of 3Hz in a star of 1.7M ( with a fully relativistic code for stationaryconfiguration of rotating stars, the coupling between the spherharmonic terms due to the drag effect is one order of magnitlarger than the coupling due to the deformation of the star.

3In December 2001, during a workshop onr-modes which tookplace at the Meudon site of the Paris Observatory, Carter suggeto call the strong Cowling approximation the approximation iwhich all the coefficients of the metric are frozen. This name wchosen to contrast with what should be called theweakCowlingapproximations where some perturbations of the metric are allowSee, for instance, the work of Ruoffet al. @24,25# and Lockitchet al. @26# using the Kojima equations@27#.

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INERTIAL MODES IN SLOWLY ROTATING STARS: . . . PHYSICAL REVIEW D 66, 123001 ~2002!

profile is exactly determined and preserved within the rouoff errors. We also show that the error in the conservationthe energy is only due to the time discretization and thavanishes as the cube of the time stepDt. Finally, this sectionends with tests of the anelastic approximation and of theforce we adopted. Section V deals with results obtaineddifferentially rotating NS. As expected, purer-modes are re-placed with inertial modes which are still CFS unstable, evif they are no longer purely axial. Moreover, we show thathe amplitude of the difference of angular velocity betwethe interior and the surface of the star is sufficiently larthese modes quite rapidly develop a kind of ‘‘singularity’’the first radial derivative of the velocity. There is then a kiof concentration of the motionnear the surface and mainlnear the equatorial plane. Our time evolution approach tprovides results of the same kind of those obtainedKarino et al. in 2001 @28# with a modes calculation. A viscous term has to be added in order to regularize the solubut it does not change the qualitative result. We shall discthis result in the conclusion. In Sec. VI we give some firesults for the GR case in a slowly rotating NS with tstrongCowling approximation. In this framework the abovconclusions still hold. More results in GR will be given infuture article that is in preparation. Section VII contains tconclusion and discussion. A detailed description of themerical algorithm is given in the Appendixes.

II. EQUATIONS AND NUMBERS

A. The barotropic case in Newtonian gravity

Cooling calculations~e.g., Nomoto and Tsuruta@29#, Ya-kovlev et al. @30#! showed that several minutes are enoughenable the NS matter to fall far below its Fermi temperatuwhich is roughly 1010 K. Furthermore, in the Newtoniannonlinear hydrodynamics of a not too young slowly rotatiNS, it is not worth trying to use a very sophisticated equatof state~EOS!. We shall therefore adopt a barotropic EOWith those assumptions, we have

P5P@n#, ~1!

where P is the pressure andn the mass density, and thPoisson equation for the gravitational potentialU is

DU54pGn. ~2!

The Newtonian Navier-Stokes equations~written in the iner-tial frame for a rigidly rotating NS! are

~] t1V]w!WW 1~WW •¹W !WW 12VW `WW 1¹W H

51

n F¹W ~¹W h•WW !1¹W `~WW `¹W h!1hDWW 2WW Dh

1¹W F S z1h

3 D¹W •WW G G1AW . ~3!

Here we take (r ,q,w) for a spherical system of coordnates in the inertial frame. In this system,q50 coincideswith the direction of the rotation axis of the NS, which

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parallel to the angular velocity:VW 5VeW z . WW is the velocityin the corotating frame, i.e. the part that is added to thvelocity of the rigidly rotating background when amodeispresent. Note that both velocities can be of the same ordethe nonlinear case. This is the reason why we shall notthe term Eulerian perturbation for WW . Otherwise, H5*(1/n)(dP/dn)dn is the enthalpy,h and z are, respec-tively, the dynamical shear and bulk viscosity coefficienand finallyAW contains any externalacceleration~or forceperunit of mass!. The modifications that are needed in the caof the linear study with differential rotation or in the relativistic linear case are, respectively, discussed in Secs. VVI.

In what follows, AW is mainly the effective gravitationaacceleration, i.e. the gradient of the difference betweencentrifugal potential12 V2r2 and the gravitational potentiaU, where r is the distance from the rotation axis. In thlinear regime, we will sometimes introduce an RR acceletion ~cf. Blanchet@31#! of the form

ARR j54

c2v i] jY i ,

where

Y i54

45

G

c5e i jkxjxmSkm(5)@ t# ~4!

and

Skm@ t#5epq(kE d3xnxm)xpvq

with (km)5 12 (km1mk), v i being the full velocity and the

superscript (5) the fifth time derivative, which cannoteasily calculated in a numerical work~see Rezzollaet al.@32#!. Note that this formula is valid only if written withCartesian components of the tensors and this is the rewhy we did not really make a distinction between contra acovariant components. Finally, we insist on the fact that san acceleration does not include any mass multipole but othe current quadrupole that is the most important coefficifor the emission of GW by axial modes in a slowly rotatinNS.

To complete the system of equations, we need to addmass continuity equation:

~] t1V]w!n1WW •¹W n1n¹W •WW 50. ~5!

Taking into account that the EOS is a barotropic one,preceding equation can be written in a slightly different wabut we will discuss it in Sec. III.

As far as inertial modes are concerned, the typical tiscale and length scale are 2pV21 and R, the characteristiclength of the background star. The velocity associated wthose values,RV, can be very far from the characteristvelocity of the mode. Another velocity, scaling that of thmode, must be introduced. But instead, we introducea, a

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pure number that is defined as the ratio of these two velties. Otherwise, the typical mass is obviously the star’s,M !

;1.4M ( . Bearing all this in mind, we define the followindimensionless variables:

j51

Rr , t5tV, VW 5

1

aVRWW ,

d@jW #51

n0n@rW#, s@jW #5

1

nn0VR2 h@rW#, ~6!

b@jW #51

ln0VR2z@rW#, eW z5

1

VVW ,

H@jW #51

aV2R2H@rW# and SW @jW #5

1

aV2RAW @rW#.

Thus the motion equations are written@with same conven-tions than in Eq.~3!# as

~]t1]w!VW 1a~VW •¹W !VW 12eWz`VW 1¹W H

5n

d H ¹W ~¹W s•VW !1¹W `~VW `¹W s!1sDVW 2VW Ds

1¹W F S l

nb1

s

3D¹W •VW G J 1SW ,

where the¹W andD operators are now performed with dimesionless variables (j,q,w) instead of (r ,q,w).

Here, we have introduced the following notation.a a pure number, characteristic of the initial amplitude

the mode, but also of the ratio of the nonlinear term andCoriolis force. With our conventions,a is twice the usual

Rossby numberand VW is a vector whose norm att50 isequal to 1 in a given point on the equator.

n05M ! /( 43 pR3). For the full slow rotation limit ap-

proximation~only terms linear inV and spherical shape!, Rwould be the radius of the star and thenn0 the mean density

s ~hear! andb ~ulk! dimensionless functions, andn andlpure numbers, all chosen in such a way that if there isviscosity,s andb are equal to 1 in a specific position. Typcally, for a single fluid model, this would be the center of tstar. Nevertheless, it is worth pointing out that to buildmore realistic model of NS, several different layers couldmade to coexist. In this case, there would be differentn andl, hugely depending on the shell~see, for instance, Haenset al. @33#!. With our definitions,those numbersn andl aretwice the usual Ekman numbers, and they then quantify theratios between the viscosities and the Coriolis force.

H andSW , respectively, the dimensionless enthalpy anddimensionless external accelerations, both scaled by theverse ofa.

Concerning the latter quantities, we cut them in two paand use for variables the difference between their prevalues and their values in the steady state. For the bground parts, we then have

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SW 052¹W U1¹W~jsinq!2

2awith ¹W H05SW 0 . ~7!

This equality enables the background to be a solutionthe NSE and can be solved separately. From now untilend, we consider that this condition is realized, and we foraboutSW 0 and H0 ~except in the mass conservation equatiwhere the latter still appears as anexternalparameter!.

Finally, some words about the Newtonian gravitationfield. Inertial modes are current oscillations associated wsmall density variations. In this context, it is natural to uthe Cowling approximation~Cowling @34#! which consists offorgetting fluctuations of the gravitational field~see the rel-evance of this approximation for Newtonianr-modes in Saio@35#!. We will apply it for the nonlinear study. Neverthelesfor the Newtonian linear study, we shall see later thatpending on the way mass conservation is treated, it maynot necessary~see Sec. III for more details!. In this case, wehave

SW 5SW 01sW with sW 52¹W dU ~8!

wheredU is the Eulerian perturbation of the dimensionlegravitational potential.

B. Dimensionless equations of motion for the sphericalcomponents of the velocity

The equations of motion for the spherical componentsthe velocity in the orthonormal basis associated w(j,q,w) are given by

]tVr1]wVr1~VW •¹W !Vr21

j~Vq

2 1Vw2 !22 sinqVw1]jh

5n

d H ~]js!~2]jVr !1sFDVr22

j2 S 1

sinq]q~sinq Vq!

11

sinq]wVw1Vr D G2]js~¹W •VW !

1]jF S l

nb1

s

3D¹W •VW G J 1s r ,

]tVq1]wVq1~VW •¹W !Vq11

j~VqVr2Vw

2 cotq!

22 cosqVw11

j]qh

5n

d H ~]js!S ]jVq11

j~]qVr2Vq! D

1sFDVq11

j2 S 2]qVr2Vq

sin2@q#2

2 cosq

sin2@q#]wVwD G

11

j]qF S lb

n1

s

3D¹W •VW G J 1sq , ~9!

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]tVw1]wVw1~VW •¹W !Vw11

j~VwVr1VwVq cotu!

12~sinqVr1cosqVq!11

j sinq]wh

5n

d H ~]js!S ]jVw11

j sinq~]wVr2sinqVw! D

1sFDVw11

j2 S 2

sinq]wVr1

2 cosq

sin2@q#]wVq

2Vw

sin2@q#D G1

1

j sinq]wF S lb

n1

s

3D¹W •VW G J 1sw ,

where

D f 51

j]j

2~j f !11

j2 sinq]q~sinq]q f !1

1

j2 sin2q]w

2 f

is the scalar Laplacian, while

~VW •¹W ! f 5Vr]j f 1Vq

1

j]q f 1

1

jsinqVw]w f

and

¹W •VW 51

j2]j~j2 Vr !1

1

j sinq]q~sinqVq!1

1

j sinq]wVw .

In the above equations, we assumes to only depend onjin the (j,q,w) system of coordinates. This assumptionlinked with the slow rotation limit we should apply fromnow. This approximation supposes that the star is slorotating compared with its Kepler frequency. As explainedthe Introduction, observational data make this assumpcredible, while known pulsars seem to rotate with a velocsmaller than a third of their Kepler velocity. In the loweframework, only terms linear inV are kept, and the background star is supposed to have preserved its spherical shYet, to improve the slow rotation approximation, one cana step further and take into account the deformation ofstar, assuming it is a small order effect~or small quantumnumberl effect!. To make this improvement, it is sufficiento introduce a new variablex@j,q# that is equal to the unityat the surface, that coincides with isosurfaces of pressenthalpy, and density, and that decomposes onm50 Leg-endre functions.4 With our algorithm, the easiest way to proceed would be to keep the spherical basis for the vectors,to express all spatial operators depending on (j,q,w) as thesum of the new operators depending on (x,q,w) and ofterms to interpret asapplied accelerations. More details canbe found in Bonazzolaet al. @36–38#. This will not be donein this paper, for reasons that were also explained inIntroduction. Finally, note that we do not really solve the

4Restricting to the casem50 is sufficient for an isolated NS, buin a binary system,m52 should also be included.

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equations. Indeed, we use the Helmholtz theorem~see, forinstance, Morse and Feshbach@39#! that says that any vectoof R3 can be in a unique way written as the sum ofdivergence-free vector and of the gradient of a potentgiven its normal component over the boundary:

VW 5¹W c1¹W `VW . ~10!

Instead of working with the above equations@Eq. ~9!#, weuse the equation on the scalar potentialc and the equationson the q and w components of the divergence-free vect¹W `VW . Most of the time, these equations cannot be reacanalytically. The way to proceed is then to separate thetential and the divergence-free parts of the initial equatioby numerically solving Poisson-like equations obtainedtaking their divergence. The reader can find in the Appdixes more details about these algorithms.

C. Characteristic numbers

To better understand what are the dominant processethe dynamics of NS, it is worth estimating characteristic timscales and associated numbers. The acoustic time, i.eduration of the acoustic waves travel across the NS, is bythe shortest. For a typical NS, it is about 231024 s.5 As weare dealing with slowly rotating NS, the period of rotatioshould be more than;231023 s ([500 Hz). It followsthat this time is at least one order of magnitude greater tthe acoustic time. It is then the same for the typical periodinertial modes, their frequency being~at linear order! propor-tional to V.

Another time scale is the viscous damping time associawith viscosity:

Tv;n0R2

max@h,z#

or

Tv;1

2 max@Es ,Eb#V,

where Es and Eb are, respectively, the Ekman number fshear and bulk viscosities. In other words, the Ekman nuber must be interpreted as characteristic of the ratio betwthe period and the viscous time. This number is typically lethan 1027 and the viscous time is more than 7 ordersmagnitude greater than the period~see Cutler and Lindblom@40# for more precise values!. A flow will be said ‘‘rotationdominated’’ if the above Ekman and Rossby numberssmall compared to the unity. Note that another usual hyddynamical number appears with them: the Reynolds numprodrome of turbulence. In a rotating fluid, it is definedthe ratio between Rossby and Ekman numbers, or influid by the ratio between the nonlinear and the viscoterms.

5This time is of the same order of magnitude as the inverse offrequency of the fundamental pressure mode, the so-calledp-mode.

1-5

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TABLE I. Characteristic numbers implied in the dynamics of inertial modes of rotating NS. Note thadefinition of the Chandrasekhar number is adapted to be of the order of 1 for linearl 5m52 r -modes.

Numbers Definition Analytical expression

Ekman Ratio betweenP andTv or between the viscousterm and the Coriolis force

E;n/(nVR2)

Rossby Ratio between the typical velocities of the modeand of the background fluid or between the

nonlinear term and the Coriolis force

Ro;W/(RV)

Reynolds Ratio between the Rossby and Ekman numbers orbetween the nonlinear and viscous terms

Re;nWR/n

Chandrasekhar Ratio betweenTv andTg or between the RR forceand the viscous term

Ch;n2GR9V6/(c7n)

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The last typical time to evaluate here is the instabilrising timeTg associated with the RR force. To get an ideaits value, we come back to the dimensionless RR acceltion, formulas~4! and definitions~6!. Analytical calculationwith Vi being thel 5m linear r-mode~with time-dependentamplitude!

VW 5A@t#1

mR@jmjW`¹W Ymm# ~11!

or in the spherical orthonormal basis

VW 5A@t#U0

jm(sinq)m21 sin[mw]

jm(sinq)m21 cos[q] cos[mw]

~12!

~whereR means the real part of the complex function!, givesin dimensioned variables at the lowest order for ther-modesof type m52

] tA@ t#5GR7V6n0

c7

216p

38527A@ t#5

1

TgA@ t#. ~13!

For a typical spherical NS withR510 km, M51.4M ( ,and V;(2 p)200 Hz, Tg is something like 108 periods ofthe NS. Thus, depending on the viscosity given by the EOit will or will not be larger thanTv , and then, the inertiainstability will or will not be relevant. At this step, a newtypical number seems natural to introduce. We shall propto call ‘‘Chandra number’’6 the ratio between the viscoutime Tv and the rising timeTg . In the same spirit as theRossby and Ekman numbers, it should also quantify the rbetween the viscous and RR forces:

NCh5216p

38527

Gn02R9V6

c7 max@h,z#. ~14!

6Chandrasekhar was the first who studied the gravitational ration driven instability for thel 5m52 fundamental modes of uniform density MacLaurin spheroid in 1970. See@4#.

12300

fa-

,

se

io

Note finally that with this factor ahead of the physicparameters, we ensure the bifurcation value ofNCh to be ofthe order of the unity, at least for the linearl 5m52 r -mode.Indeed, this point is easily illustrated by looking at NSE fthe linear l 5m52 r -mode with time-dependent amplitudand a shear viscosity of the forms512j2. This viscositythat vanishes at the surface implies no need to add mboundary conditions~BC! and gives the exact~at the lowestorder for the RR force! differential equation for the ampli-tude:

] tA@ t#51

TgA@ t#S 12

2

NChD

where one easily sees that with this shape, the viscosity wthe battle against RR force forCh,2. All the previous num-bers are gathered in Table I.

To end with this short discussion, we should insist onmost important conclusion that is summarized in Table II:above figures show how stiff is the numerical problemfinding a dynamical solution of the NSE in this frameworThis is a physical situation in which several very differetime scales appear. In order to get an efficient and accucode, some approximations have to be made.

III. MASS CONSERVATION, BOUNDARY CONDITIONS,AND APPROXIMATIONS

To be consistent with the variables we chose in NSE,have to write the mass conservation equation with the dimsionless enthalpy. With the decomposition explained atend of Sec. II A, we have

H@t,jW #5H0@jW #1ah@t,jW #. ~15!

The first termH0 is the background, the secondh corre-sponds tothe modeitself, anda still quantifies the nonlin-earity. This gives

~]t1]w!h1~VW •¹W H01G@n#H0¹W •VW !

1a~VW •¹W h1G@n#h¹W •VW !50 ~16!

a-

1-6


TABLE II. Typical values of the different time scales implied in the dynamics of inertial modes of rotating NS.

Time scale Definition Analytical expression Typical value~or range!

Acoustic Travel of acoustic waves across the NSTa;

R

cs

;231024 s

Inertial NS’s period~same order as the period of the linearinertial mode! P5

2p

V

.231023 s

Viscous Damping of inertial modes due to viscosity Tv;1/(2EV) .107PGravitational Growing due to RR force Tg;c7P/(GMR4V5) ;108P

ar.

ntThn

th

esa

itoleralnayth

retoa

tethth

thiaisinhaia

hevisa

arE,

.

nokialncyin

re

ri-

steantlectres-thethe

o-

62in

a-tialion

e--tic

toticb-de-

ingn

rpo-ndt

whereG@n#5d lnH/d lnn. In the polytropic case,P5k ng,G is constant and reduces toG5g21. Exception done of thecaseg51 where the enthalpy is a logarithm. In the linecase, we have to neglect the last part of this equation, i.edo a50 in Eq. ~16!. We shall now describe some differechoices that can be made to deal with this equation.reader can find in Appendix B the algorithms to implemeall the following schemes in the framework of spectral meods.

A. Solving the exact system of equation

If one tried to solve numerically the exact Navier-Stokand mass conservation equations, one would have two mpossibilities. The first would be to employ an explicscheme. But by this way, the Courant conditions for the flowing of the acoustic waves would impose a time step vsmall compared with the period of the star. This wouldmost forbid any hope to make evolutions during duratiolong from the inertial modes point of view. The second wto proceed would be to use an implicit scheme fordivergence-free part~see Appendix B!. Indeed, this wouldmake it possible to take a time step not too small compawith the period of the NS. But here the problem would beestimate the errors done. Hence, in a first study, someproximations can be introduced to solve the problem ineasier way.

B. The divergence-free approximation

To avoid the problem of solving acoustic waves for betstudying inertial modes, the easiest solution is to usedivergence-free approximation. It consists of replacingusual mass conservation equation by¹W •VW 50. By this waythe time step can be chosen larger than in the solving ofexact system and then allows the following of the inertmodes themselves. From a numerical point of view, thisfast and robust approximation quite useful in the explorphase of the numerical work. Yet, it should be noticed tthis drastic approximation may be not too bad for inertmodes in NS. Indeed, thel 5m52 r-modes which are themost interesting for GW have a divergence-free limit in tlinear order. Furthermore, the latest figures about bulkcosity ~Haenselet al. @33# that are quite huge, could meanfast damping of all modes, except of those thatdivergence-free. Finally, note that in the linear limit of Ethis approximation gives an evolution of the modeVW that isindependent of the background star if it is rigidly rotating

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to

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e

d

p-a

ree

elagtl

-

e

C. The anelastic approximation

Yet, instead of being quite ‘‘savage’’ with the equatioand imposing the divergence-free condition, one could lofor a cleverer way of doing physics and think about inertmodes. A main feature of these modes is that their frequeis of the same order as the angular velocity of the starwhich they occur. As their damping and growing times alarger than their period~cf. Sec. II C!, one would like to takeit for a characteristic time scale. From practical and numecal points of view, it means a time unit~or time step! choicenot too small compared with the period, in order not to wamemory and computational time calculating nonrelevphysics. From a physical angle, the philosophy is to negacoustic waves by assuming that time derivatives of the psure and density perturbations do not play a key role inphenomenon. It can be done in a consistent way withanelastic approximation.

The anelastic approximation was first introduced in atmspheric physics by Batchelor in 1953@41# and then derivedfrom a rigorous scale analysis by Ogura and Phillips in 19@42# who gave it its name. In astrophysics, it appeared1976 in a paper by Latouret al. @43# concerning convectionand was then widely used in that field and others~such asstellar oscillations! where one can neglect temporal varitions of the perturbation in density but not necessarily spaones. For a recent critical approach of this approximatwithin astrophysics, one can read Dintrans and Rieutord@44#and Rieutord and Dintrans@45#.

In Eq. ~16!, the anelastic approximation consists of nglecting (]t1]w)h which is the time derivative of the enthalpy in the rotating frame. By this way, we cut acouswaves and then have

VW •¹W H01G@n#H0¹W •VW 1a~VW •¹W h1G@n#h¹W •VW !50.~17!

As a conclusion on this approximation, we would likeinsist on a particularity of the linear case. With the anelasapproximation and the linearization of all equations, we otain an equation for the mass conservation that does notpend on the Eulerian perturbation of the enthalpy. But, takthe curl of the linear EE~or NSE! equation gives an equatiowith exactly the same feature~remember that the curl of agradient is zero! with an interesting additional fact: It neithedepends on the Eulerian perturbation of the gravitationaltential. Furthermore, it is easy to verify that for a backgrouenthalpy depending only onj, the boundary condition is tha

1-7

thityes. Ihe

da

di-ddub

ouusa

isfox

ret

s

3t

whcuatiailtio

inea

inth

io

e

tha

gto

anyich

is

ed

e.astasnd

thepli-on.theultsithic-dePU

heof

es.

the

00

hical


the radial velocity should vanish at the surface. Thus,situation is that finding the Eulerian perturbation of velocshould give exactly the same result, whatever the hypothon the Eulerian perturbation of the gravitational potentialmeans,we do not need the Cowling approximation with tanelastic approximation in the Newtonian linear case. TheCowling approximation would play a role only if we wanteto find what is the enthalpy and what is the gravitationpotential in the source term of the gradient part of EE.

D. Boundary conditions

The only missing information is now the boundary contions we chose. In an actual NS, the fluid core is supposebe surrounded by a more or less rigid crust, an ocean, anatmosphere. Their physics is by itself quite a complex sject ~see, for example, Haensel@46#!. But even for a simpletoy model, for instance a crust made of only one typenuclei, depending on the temperature and on the amplitof the motion of the inner fluid, the physical state of the crcan be quite complex to describe, something like an icepon the sea~see, for instance, Lindblomet al. @47#!. There isno need to explain how difficult it would be to translate thBC in a mathematical language. This is the reason why,the EE, we then chose to begin with two different and etreme BC to get an idea of the limit cases. The first is the fsurface BC, i.e. the absence of any crust. The second ispresence of a rigid crust atj51, if we take the inner radiusof the crust for the typical lengthR. In the first case, one haHuj5150 and in the secondVr uj5150. The numerical waysto take into account those BC can be found in Appendix BFor the NSE, as was already said in Sec. II C, in orderavoid the need of more BC, we chose adegenerateshearviscosity of the forms512j2.

IV. TEST AND CALIBRATION OF THE CODE

Now that we have presented our physical framework,will focus on the code. As was already mentioned in tpreceding sections, it uses spherical coordinates and spemethods to solve NSE. More precisely, it solves the eqtions coming from the decomposition of NSE into a potenand a divergence-free part. We will not give more detahere and send the reader to the Appendixes for informaregarding the algorithms and spectral methods.The rest ofthis paper, and the discussion about numerical stabilitythe Appendixes, are devoted to the linear study. Nonlinwork is still in progress and will be described later. Further-more, in this section, we only deal with rigid rotationorder to have some analytical solutions to make tests wi

A. Conservation of the energy

The first test of the code was obviously the free evolutof the linearr-mode of typel 5m in a rigidly rotating invis-cid and incompressible fluid. The vector defined in Eq.~12!is indeed an eigenvector of EE whose frequency in the intial frame iswi52(m21)(m12)/(m11). Moreover, notethat the absence of a radial component implies that bothfree surface and the spherical rigid crust BC are autom

12300

e

ist

l

toan-

fdetck

r-ehe

.o

eetral-lsn

r

.

n

r-

eti-

cally satisfied. Yet, for all the preliminary calculations usinthe divergence-free approximation, the code was builtwork with the rigid crust BC.

The great advantage of spectral methods is to changelinear spatial operation in linear algebra calculations, whcan be done exactly. As the velocity of Eq.~12! has a verysmall number of coefficients in the reciprocal space, iteasy to verify~for example by directly looking at the timeevolution of those coefficients! that there are only errors duto round-off and to time discretization. In Fig. 1 is illustratethe time evolution ofVq at the equator for them52 linearr-mode. The spatial lattice was of the shape (83634) for(Nr ,Nq ,Nw)7 with 100 time steps per period of the modThe duration of this run was chosen for the evolution to lexactly 100 times the expected period. This calculation wdone on a DEC Alpha Station with a 500 MHz processor atook 217 s of CPU time. Comparison of the amplitude atfirst step and at the last step shows the growth of the amtude due to errors that come from the time discretizatiThis is easier to see in Fig. 2 where the time evolution oferror in energy appears. The different curves illustrate resobtained with the same grid and duration, but in runs wdifferent numbers of time steps per oscillation. Here are ptured results for 100, 150, and 200~see also the associatepower spectra in Fig. 3!. The last two calculations on thsame computer took, respectively, 269 and 342 s of Ctime.

In Fig. 4, we drew for several runs the relative error in tenergy per oscillation of the mode, versus the numbertime steps per oscillation. Both are in logarithmic scalThis error appears to exactly vary asDt3 and then as the

7In fact, symmetries of the spherical harmonics are used andeffective value ofl max is roughly twice the number of points inq.

FIG. 1. Time evolution of theq component of velocity on theequator for the linearr-mode of typem52 in an incompressibleand rigidly rotating fluid. The run is supposed to last exactly 1times the period of this mode which is 3/(4V). The beating phe-nomenon which can be seen is due to the resolution of the graptool. See the power spectrum in Fig. 3.

1-8

my

illeeerdar

ali-

eri-l totive

a

va-toical

late

o-

rgyed

toens toFig.

reen

rors

ocil

tlyvan-the

and


inverse of the cube of the number of steps per period~regres-sion slope of23.01). This is due to the second order schewe use to solve the NSE or EE. Indeed, when the energcalculated, the error of the orderDt2 done on the velocity ismultiplied by a source term linear inDt.

As we are now only dealing with the linear code, we wnot discuss the stability of the nonlinear code, nor give tpictures corresponding to modes with azimuthal numbdifferent from 2. Indeed, there is a coupling between diffent values ofm only in the nonlinear versions of NSE anEE. The modes that are the most interesting for GW

FIG. 2. Time evolution of the relative error in the energy befoany improvement of the conservation of energy. The differstraight lines correspond to runs with the same duration~100 peri-ods of them52 linearr-mode!, same spatial grid (83634 points!but different time steps. We see the linear variation of these erwith time. It shows that with the exception done of round-off erro~which are so small that they do not appear here!, the main error isdeterministic and then well controlled.

FIG. 3. Power spectra of the component drawn in Fig. 1 andthe same component calculated with 150 or 200 steps per ostion. See also Fig. 2.

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strs-

e

modes of typem52 and in the following, we will alwayschoose this kind of initial data. Nevertheless, the same cbration tests for the other linearr-modes of typel 5m weredone and gave the same power law relations betweendE/Eand the number of steps per oscillation. Finally, we also vfied that the error in the phase of the modes is proportionathe square of the number of steps per period. The relaerrors in the phase for them52 mode with 100, 150, and200 steps per oscillation were, respectively, 1.6431023,7.3331024, and 4.1231024. It is easy to verify that thelogarithms of these numbers are on a straight line withslope very close to 2.

We now shall see how we tried to improve the consertion of energy. For the freer-mode, the energy is expectedbe exactly conserved. Furthermore, we know our numererror in the velocity is of the order ofDt2. Remember that itcomes from the second order scheme used to calcusource terms:

Sj 11/253Sj2Sj 21

2. ~18!

The basic idea is thus to modify this quantity, in the cefficients space, with an additional term of the order ofDt2,in such a way to retrieve the conservation of the enewithout changing the error in the velocity. We then modifithe Eq.~18! and took

Sj 11/25~31«!Sj2~11«!Sj 21

2~19!

where« is of the order ofDt2. It is calculated using valuesof the energy at the last two instants and in such a way‘‘impose’’ on the energy to be conserved. Obviously, whthere are forces such as viscosity or RR, their power habe taken into account in the energy balance equation. In

t

rs

fla-

FIG. 4. In this picture clearly appears the fact the error exaccomes from the adopted numerical scheme with an error thatishes as the third power of the time step. The relative error andnumber of steps per oscillation are both in logarithmic scalesthe regression slope is then23.0160.06%.

1-9

lenr-

rtio

einar

d

aW

anrsnicoue

udthnb

yhen

nh

d,e-lWe

. 9n-

twi

ialth

im-outthe

nnRRows.


5 are the same curves as in Fig. 2 but in logarithmic scaWe added the result of this ‘‘improved conservation of eergy’’ for the run with 100 steps per oscillation. As the corection is local in time~done at almost each time step!, itleads to a time-independent error that should be able tomain the same as long as one could wish. This calculatook 320 s of CPU time.

B. Modes driven to instability in the rigidly rotating case

After trying to improve the conservation of energy for thfree case, we did tests with an RR force included. Forstance, we switched on the RR force in order to drive towinstability the linearr-mode of typem52. As has alreadybeen implied in Sec. II C, we did it by using a modifieversion of the formulas~4! ~see this section!. For these cal-culations, we compared numerical results with analytical cculations, both reached with the same approximation.took for the background star a homogeneous NS withM51.4M ( , R510 km, andV;(2p)500 Hz. In Fig. 6 ap-pear two time evolutions of the ratio between the energythe initial energy for 100 oscillations of the mode. The ficalculation was done without the enhanced conservatioenergy and the second with this improvement. The analytcalculation gives for the ratio a final value of the order1.0575 whose difference with the improved numerical val1.0582, is less than 1023.

The next step was to switch on the RR force, but withothe linearr-mode for initial data, even if we still assumethat the fluid was divergence-free. Instead, we chose forinitial velocity a random Gaussian noise with a first momeequal to 0 and a second moment equal to 1. It shouldnoticed that in this case and others where the initial energnot only distributed within quite low frequency modes, tabove method to improve the conservation of energy islonger appropriate and we shall not use it. Moreover, evethis method does not change the angular momentum w

FIG. 5. Here are pictured the same curves as in Fig. 2, bulogarithmic scales. We added the curve corresponding to a run100 steps per period of them52 linearr-mode with the correctionof the conservation of energy described toward Eq.~19!.

12300

s.-

e-n

-d

l-e

dtofalf,

t

eteis

oifen

applied to them52 part of the velocity, it can if one naivelyapplied it to them50 part. But, as we already mentionehere we only deal with the linear code and mainly with vlocities of typem52. And in this case, we exactly controthe error done, without changing the angular momentum.drew in Figs. 7 and 8 the time evolutions of ther and qcomponents of the velocity and their power spectra. In Figis illustrated the time evolution of one of the two indepe

inth

FIG. 6. Time evolution of the ratio between energy and initenergy in an inviscid and incompressible rigidly rotating fluid wian RR force corresponding to what should exist in a NS withM51.4M ( , R510 km, andV.(2p)500 Hz. The first curve corre-sponds to the basic code and the second one to the code withproved conservation of energy. The expected final value is ab1.057. The difference between this analytical calculation andcorrected numerical result is less than 1023.

FIG. 7. Time evolution of the radial component of velocity in ainviscid and incompressible rigidly rotating fluid with Gaussianoise for initial data and associated power spectrum. A hugeforce acts, but as expected, no polar counterpart of the mode gr

1-10

erthd

ex

.hitegtw

is

e

anoxo

gii-uis

n

BEq

afostWc-can

dedie

ns.tic

t is

me

w.hefinalthenc-

n

a

nd. 7.

o-u-d 8.

trum.not

the


dent components of theSi j tensor, with the associated powspectrum. These calculations were done with a lattice ofshape (123834), with 200 steps per oscillation and laste200 times the period of them52 r -mode. The included RRforce corresponds to what should exist in a NS withM51.4M ( , R510 km, and V.(2p)1050 Hz. Thus, thiscalculation is not really physical~due to the use of the slowrotation approximation! but just aims at giving a qualitativeidea of what should happen with physical values. Aspected, there is an axial mode~no radial velocity! driven toinstability with exactly the frequency (w5 4

3 V) and coeffi-cients (l 52) of the r-mode. Moreover, a single look at Fig9 shows that even when the velocity is mainly noisy, ttensor that plays the key role in the instability is qusmooth. In this figure, we also drew a zoom correspondintV,50 and the associated power spectrum. It showsfrequencies in this component of the tensor. One of themthe unstabler-mode with 3w/V54 and the other~with3w/V;2.70) disappearswith a longer run, as the scaleadapted to the growing mode. Yet, even in the spectrumthe full evolution, a trace of it can still be seen. We achievexactly the same features with othersnoisyinitial conditions,for instance aDirac kick into the NS.8 We also did the samecalculations with other spatial lattices~up to 6434834),and this result did not change at all.

C. Test of the anelastic approximation

The last calculations we did in a rigid and Newtonibackground were to test the effect of the anelastic apprmation. The idea was to compare the previous resultstained with the divergence-free approximation and a ricrust BC with results coming from the same initial condtions but with the anelastic approximation and the free sface BC. For the rest of this section, the background starpolytrope withg52.

First, we took for initial data the linearl 5m52 r -modeand let it freely evolve. Here again, the BC do not play arole. The spatial lattice was (83634) with 100 time stepsper period of the mode. The time evolution ofVq at theequator~same as in Fig. 1! did not show any difference withthe divergence-free case, even for the power spectrum.this is quite obvious to understand when looking at both~17! and the EE. Indeed, we see that the linearr-mode, whichis divergence-free and has no radial component, is stilleigenvector of this system of equations. Then taking itinitial data in the divergence-free case or in the anelaapproximation should give exactly the same evolution.verified ~see Fig. 10! that the radial component of the veloity does not grow and neither does its divergence. Thisculation took 344.6 s of CPU time without any optimizatioand with a very basic solving of the anelastic equation.

The next step was the driving toward instability of a mowith noise for initial data. We took exactly the same contions ~duration, lattice, initial data! as in the divergence-fre

8We mean an initial velocity equal to 0 anywhere except inarbitrary point.

12300

e

-

e

toois

ofd

i-b-d

r-a

y

ut.

nrice

l-

-

case, with the exception done of the boundary conditioFor these calculations and all that follows with the anelasapproximation, we will use the free surface condition thaautomatically satisfied~see the Appendixes for more details!.The Figs. 11, 12, and 13 that show, respectively, the tievolutions ofVr , of Vq , and of a component ofSi j , give akind of summary of the results:

The radial velocity still remains noisy and does not groThe q component grows in the same way as in t

divergence-free case. There is a difference between theamplitudes that comes from the fact that the coupling toRR force depends on an integral on the whole star of a fution that is proportional to the density~and then depends o

n

FIG. 8. Time evolution of theq component of velocity on theequator in an inviscid and incompressible rigidly rotating fluid aassociated power spectrum with the same initial data in FigHere, thel 5m52 linear r-mode is clearly driven to instability bythe RR force. As expected, its frequency isw5

43 V. Others peaks

are just marks of the initial noise.

FIG. 9. Time evolution of one of the two independent compnents of theSi j @ t# tensor that appears in the RR force. This calclation was done during the same run as the results in Figs. 7 anWe can also see the almost monochromatic associated specNevertheless, there is another frequency in this tensor that isdriven to instability. It can be seen more easily in the zoom oftime evolution or in the corresponding power spectrum.

1-11

n

lyth

iocne

ce

a

k-fors-

dlyhysoff ay-re

star

ity

apth

tyh-is

. 11.

ionR

o-thehere

ure.


the EOS!. Note that the power spectra have the same secoary peaks as in the divergence-free case.

The spectrum of theSi j tensor shows that there is onone unstable mode which has again the frequency oflinear r-mode.

The conclusion is then that the anelastic approximatgives results very close to those obtained in the divergenfree case. This is due to the fact that, in spite of the preseof all modes in the initial conditions, the only growing modis the r-mode that has no radial velocity and is divergenfree.

V. DIFFERENTIAL ROTATION

As we already mentioned it in the Introduction, Kojimfirst noticed~see Kojima@23#! that relativisticr-modes may

FIG. 10. Time evolution of the radial component of the velocin the equatorial plane inj50.5 with the linearr-mode of typem52 for initial data. The background star is a rigidly rotatingg52 polytrope and the calculation is done with the anelasticproximation and without any RR force. The run lasts 100 timesperiod of ther-mode. There should be no radial velocity~cf. text!and we see it comes only from round-off errors~double precisioncalculation with initial values of the order of the unity!.

FIG. 11. Time evolution of the radial component of velociwith Gaussian noise for initial data and its power spectrum. Tbackground is a polytrope withg52 and the anelastic approximation is used. As in Fig. 7 a huge RR force acts, but as the starrigidly rotating no polar counterpart of the mode grows.

12300

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e

ne-ce

-

be quite different from what they are in a Newtonian bacground due to frame dragging. Yet, since the main reasonthis difference is the modification of the NSE and the posible appearance of what is called acontinuousspectrum~seeRuoff et al. @24,25# and Beyeret al. @48#!, the same mayappend even in the Newtonian case if the star is not rigibut differentially rotating. And there are several reasons wa NS may not be in rigid rotation: first, the birth conditionof the NS themselves; second, the nonlinear couplingmodes; then, a possible drift induced by the existence omagnetic field. As we are here only dealing with linear hdrodynamics and tests of the code, we will not give modetails about those processes~but will send the reader to thefollowing articles for more details: Spruit@49#, Rezzollaet al. @50, 51#, and @52#, and Schenket al. @53#!. What wehave done is just to take as a given that the background

-e

e

FIG. 12. Time evolution of theq component of velocity and theassociated power spectrum for the same calculation as in FigThe only difference with the divergence-free case@cf. Fig. 8# is thefinal amplitude. Yet, this is not due to the anelastic approximatbut to the effect of the EOS on the coupling coefficient of the Rforce.

FIG. 13. Time evolution of one of the two independent compnents of theSi j @ t# tensor corresponding to the same run asresults in Figs. 11 and 12. Here can really be seen the fact that tis only one unstable mode with the frequency of the linearr-modeand that the anelastic approximation does not change this feat

1-12

oheetn

ed

ecifi

in-IIne

f

-ala

st t

.

b

omal

lema

cetio

he

s

co

o

no

ti-

si-w-

nlythewhat-

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atweeng-

inise

lawhe


is differentially rotating with quite an arbitrary law and tlook at the influence of this law on the existence of tmodes. Nevertheless, instead of asking the question ‘‘Is thany r-mode left in a differentially rotating NS?’’ that is nowell defined, we decided to try to answer to two different amore precise questions.

~i! Is there anything growing when an RR force is applion noise in a differentially rotating background?

~ii ! What does happen to the linearr modes if they arechosen to be the initial data in such kind of background?

Some light on these questions are in the following stions, but we shall begin with some words about the modcations implied on the equations by differential rotation.

A. Modifications

Assuming differential rotation of the background starvolves slight modifications of what we said in the Sec.First, terms coming from the spatial derivatives of the noconstantVW @j,q,w# must be added in the NSE. Second, wshall now specify what exactlyV means in the definition odimensionless variables@cf. Eq. ~6!#.

ConcerningV, we chose to take for a time unit the inverse of its value at the equator. This is uniquely determindue to the fact we have always assumed that the rotationis of the form V@r ,q,w#5V†r sin@q#‡ or of the formV@r ,q,w#5V@r #. The first case corresponds to what mube this law for the background to be stable with respecthe Newtonian EE, and the second case is in a wayinspiredby GR even if it is not a solution of the full Newtonian EEFor more details see Sec. VI.

In the dimensionless NSE, the modifications induceddifferential rotation are quite simple. First,]w is replacedwith V@r ,q#]w whereV@r ,q# is the dimensionless profileof rotation. Then, we have to add new terms coming freWwr sin@q#VW•¹W (V@r,q#) that are in the spherical orthonormbasis

U 0

0

Vrr sin@q#] rV1Vq sin@q#]qV.

~20!

B. Noise with huge RR force

Once again, our goal was to stay as close as possibthe basic and well understood situation to minimize the nuber of unknowns. Thus, we took noisy initial data, put it indifferentially rotating background, switched on the RR forand looked for what was to happen. In this way, the queswas not to look for the existence ofr-modes, but simply tolook for the existence of modes driven to instability by tRR force in a nonrigidly rotating background.

The only difference with the basic study done in the caof rigid rotation was that we had to choose a law forV. Thefirst choice was very simple and corresponded to a baground stable with respect to the Newtonian EE. It wasthe form

V@r ,q#5W~11bnr 2 sin@q#2! ~21!

12300

re

d

--

.-

tew

to

y

to-

n

e

k-f

wherebn was a constant depending on the run andW cal-culated to haveV@1,p/2#51. As explained above, we alstried a lawinspiredby GR:

V@r ,q#5W~11bgrr2! ~22!

or laws coming from the already quoted article by Kariet al. @28#:

V@r ,q#5WbL

2

r 2 sin@q#21bL2

~23!

or

V@r ,q#5Wbv

~r 2 sin@q#21bv2!1/2

. ~24!

Since in all these laws, none of the free parametersbcorresponds to a physical variable, we will not give quantative results. It will be done in the relativistic study~andthen in another article! where there is a parameter that phycally quantifies the way the equations are far from the Netonian EE: the compactness of the star. Here we will odiscuss in a qualitative way the results obtained in allprevious cases and give a very representative example:happens in ag52 polytrope with the anelastic approximation ~free surface! and the rotation law given by Eq.~21!with bn50.4. With the exception of these choices, the cculation was done with exactly the same conditions as inpreceding study with the noisy initial data in the previosection.

The main difference with the case of rigid rotation is ththe mode that grows is no longer purely axial. Indeed,can see on Fig. 14 that the radial velocity is now also drivto instability. Moreover, comparing power spectra in this fi

FIG. 14. Time evolution of the radial component of velocitya g52 polytrope with anelastic approximation and Gaussian nofor initial data. A huge RR force acts~something like what shouldexist in a star with angular velocity equal to 1000 Hz! and thebackground star is assumed to be differentially rotating with thecorresponding tobn50.4. We see that a polar counterpart of tmode now grows.

1-13

g

eelkninS

itceswatn

e is

ethe

ap-yalctor,he

ndtialo-rtec-larthe

aoxi-

d

theethed of

trueen

ouawo

edin

und

ialsart


ure and in Figs. 15 and 16 shows that this is really a sinmode with frequency very close of the frequency of ther-mode. Bya single mode, we mean that this is exactly thsame frequency for several physical quantities and sevpositions in the star. This is not a sufficient condition to taabout ‘‘discrete spectrum,’’ but it is a necessary conditioThis unavoidable existence of a polar part of the velocitya differentially rotating Newtonian star with barotropic EOshould be compared with the results achieved by Locket al. @26# in the relativistic framework. Indeed, in GR, thmain reason for the coupling between axial and polar partthe velocity is the frame dragging that is imitated in a Netonian framework by differential rotation. Finally, note ththe value of the frequency in the Newtonian case is depeing on the way we chose to normalizeV. Then, to summa-rize all our calculations, we can say that we observed thatthe

FIG. 15. Time evolution of theq component of velocity in thesame calculation as in Fig. 14. The associated power specshows that there is one mode driven to instability that is the samin Fig. 14 and has a frequency very close to the expected frequof the r-mode. See also Fig. 16.

FIG. 16. Time evolution of one of the two independent compnents of theSi j @ t# tensor that appears in the RR force. This calclation was done during the same run as the results in Figs. 1415. We can see the almost monochromatic associated spectrumthe same frequency as in the previous spectra of the unstable m

12300

le

ral

.

h

of-

d-

polar counter part of the mode appears as soon as therdifferential rotation, and whatever the chosen law. Yet, themore the law forV is far from the rigid case, or in a morpragmatic way the greater the free parameter is, the moreunstable mode has a polar counterpart.

C. Free evolution

The other question we asked was this: ‘‘What does hpen to linearr-modes if they are put into a differentiallrotating background?’’ The idea in taking such kind of initidata was to have something quite close to an eigenveassuming if there is one it should be quite similar to tlinear r-mode.

Once again, we did not make quantitative calculations apostponed it to the GR case. As in the case of noisy inidata driven toward instability by an RR force, the free evlution of the linearr-mode showed the growth of a polar paof the velocity. It can be seen in Figs. 17 and 18 that resptively show the time evolution of the ratio between the poenergy and the total energy and the time evolution ofradial velocity at the point of coordinates (j5 1

2 ,q5p/2,w50). This result comes from a calculation done withspatial lattice of the shape (64,48,4), the anelastic apprmation~with a free surface!, and a rotation law given by theEq. ~21! with bn50.4. For reasons that will be explainelater, we included degenerate viscosity~cf. Sec. III D! withan Ekman numberEs5531025 in order to regularize thesolution. The evolution was done to the last 30 periods oflinear r-mode with 100 steps per oscillation. In Fig. 17 wsee that, after a while, the ratio between the energy inpolar part of the mode and the total energy reaches a kin

mascy

--ndithde.

FIG. 17. Time evolution of the ratio between energy containin the polar part of the velocity and the total energy of the modethe anelastic approximation with a free surface. The backgrostar is assumed to be ag52 polytrope differentially rotating withthe law corresponding tobn50.4. A viscous term coming from thedegenerate viscosity withEs5531025 is included. At the begin-ning of the evolution, there is energy coming from the purely axinitial data~the linearr-mode! and going to the polar part. Then thienergy oscillates from the polar part of the velocity to the axial pand back with a constant mean value.

1-14

eg

n

tao

ee

h

forec-.hehe

a-n

l-festarwee

we

ti-nta

teresbeofults

le

ctsta-

tytiotit

owheit

e

o-u-and

econdfre-ures


stationary state with a coupling between different modThe existence of this ‘‘hybrid final state’’ was verified durinother runs with other physical conditions and is once agaicompare with results achieved in GR by Lockitchet al. @26#.

Concerning the existence of modes, looking simulneously at Figs. 18, 19, and 20 shows that apparentlysingle mode mainly appears in the reaction force~or in thecomponent of the corresponding tensor! even if both theaxial and polar parts of the velocity contain several modYet, the spectrum of theSi j tensor is quite noisy due to thfact that the RR force is not switched on and that the runshort. We verified this feature during other calculations. Tconclusion is that for small values of theb parameter~these

FIG. 18. Time evolution of the radial component of velociwith the anelastic approximation and a free surface. This calculawas done during the same run as in Fig. 17. The initial data arelinear r-mode that is no longer an eigenvector and radial velocquickly appears. Several different modes are present.

FIG. 19. Time evolution of theq component of velocity in thesame calculation as in Fig. 18. The curve and the associated pspectrum show that there are several modes, but only one of tappears in theSi j tensor with quite a huge amplitude. Moreover,has a frequency very close to the frequency of the linearr-mode.See also Fig. 20. Note finally that the same frequency also appin the spectrum of the radial velocity.

12300

s.

to

-ne

s.

ise

values are depending of the chosen rotation law!, the maineffect of differential rotation on a ‘‘free linearr-mode’’ is togive it a polar counterpart and to widen its spectrum. Butlarger values of the parameter, even if the velocity’s sptrum becomes very noisy, theSi j tensor is always less noisyYet, here we will not give more details about this points, tlinear Newtonian study not really being interesting from tNS point of view.

To end this section, we will mention a quite amazing feture found in all our calculations and illustrate it with aexample coming from the previous free run of a linearr-mode put in the differentially rotating background. We aways found that quite fast~in something like 15 periods othe linearr-mode! the main part of the velocity is most of thtime concentrated in a region close to the surface of theand to the equator. This is illustrated in Fig. 21 wheredrew the shape of thew component of the velocity versus thradius of the star for several values of theq angle. Note thatwe got the same results with different BC and even whenadd viscosity to regularize the solution~and this is the case inthis calculation! and to be sure this is not a numerical arfact. In Fig. 22 the shape of theq component appears for aevolution done with the same grid, viscosity, and initial dabut with bn50.8. With such a huge value of the paramethat governs the rotation law, there is a lot of different modthat appear in the velocity spectrum. But what seems tothe most important result is the very high concentrationthe velocity near the surface. This is analogous to the resachieved by Karinoet al. in 2001 @28#. In conclusion, wewill shortly discuss what is, in our point of view, a possibimportant repercussion of this result.

VI. STRONG COWLING APPROXIMATION IN GR

NS are among the most relativistic macroscopic objemade of usual matter in the Universe. Moreover, the ins

nhey

erm

ars

FIG. 20. Time evolution of one of the two independent compnents of theSi j @ t# tensor that appears in the RR force. This calclation was done during the same run as the results in Figs. 1819. We can see there seems to be one main frequency and a sone of smaller importance. The first one has exactly the samequency as the unstable mode that appears in the previous figand calculations where the RR force was on.

1-15

toisdi

a

edbtoanofl

in

to

ap-a-l

e

uge

’’e

howeenme

nt

a-

mon-

e toec-

le-hetur-ofseic

orisa

theug

orer

thet


bility of the r-mode is due to GR. It is then quite naturaltry to studyr-modes in the GR framework. The problemthat it is quite difficult to evolve dynamically the full GR anhydrodynamics equations taking into account GW. Thisthe reason why we decided to use the strong Cowlingproximation at the beginning.

Here we will only give some preliminary results obtainin the GR case. A more complete and physical study willin a following article that is in preparation. Our goal is justshow that the above conclusions concerning the appearof a polar part of the velocity and the ‘‘concentrationmotion’’ for the Newtonian differentially rotating NS stilhold in the framework of relativistic rigidly rotating NS.

As we are studying slowly rotating NS, which rema

FIG. 21. w component of the velocity versus the radius fseveral values ofq. We can see the main part of the velocityconcentrated in a region near the surface and close to the equThis ‘‘snapshot’’ of the profile was taken at a moment whenderivative versus the radial coordinate of the velocity is quite hat the surface.

FIG. 22. q component of the velocity versus the radius fseveral values ofq. Here the concentration of the motion is highthan in the previous calculation.bn was 0.4 in Fig. 21 and is now0.8. Moreover, this calculation lasted three time longer. Neverless, note that here we did not choose to draw the profile whenderivative is huge but just drew it at the final instant.

12300

sp-

e

ce

spherical, a very convenient and good approximation isassume that the 3-space is conformally flat~see Cooket al.@54#!. Furthermore, as we are using the strong Cowlingproximation, there is no GW even without this approximtion. Then, in unitc51, the infinitesimal space-time intervacan be written

ds252~N@r #22a@r #2r 2 sin@q#2Nw@r #2!dt2

22a@r #2r 2 sin@q#2Nw@r #dtdw1a@r #2drW2

where, following the 311 formulation, N@r # is the lapsefunction andNw@r # the third contravariant component on thspherical coordinate basis of the shift vector,a@r # is theconformal factor, and finallydrW2 is the infinitesimal intervalin the 3-space. In this equation, we use the isotropic gaand the slow rotation limit to assume that all theseunknownfunctions are only depending onr ~and this is the reasonwhy, in the Newtonian study, we called ‘‘inspired by GRthe law of differential rotation in which the functions aronly depending onr ). In Hartle @55#, the system of coordi-nates is not the same as the one we use, but it is easy to sthat the conclusions are identical since the mapping betwthe two systems is quite trivial. Furthermore, from the saarticle we know thatNw@r # is of the first order inV.

To get the equations of motion, the relativistic equivaleof EE, we write the energy tensor of a perfect fluid

Tmn5~r1P!UmUn1Pgmn ~25!

wherer is the energy density,P the pressure, andUm the4-velocity of the perfect fluid. Then, we apply the conservtion of energy:

¹mTmn50. ~26!

If we define the generalized enthalpy

dH5dP

r1P, ~27!

it gives

UmUn¹nH1¹mH1Un¹nUm50. ~28!

It is well known that due to thermodynamics, the systeof equations obtained by adding the baryonic number cservation¹m(nUm) ~wheren is the baryonic density! to thosefour equations is a degenerate system. Then, we choswork with the baryonic number conservation and the projtions on the 3-space of Eq.~28!.

Following the Newtonian case, we just have to add Eurian perturbations to the rigid rotation and to linearize tequations with respect to both the amplitude of these perbation andV. For the generalized enthalpy, the definitionthe perturbation is obvious, but for the full 4-velocity, we uthe well-known results about rigid rotation of relativiststars and write it as

tor.

e

-he

1-16

onr omthE

as

rd

aif

s.i

it

-a

heas

heil-

t-ith

ep-of

km.

heed,cos-nusustndTherethe

ndua-

edde

free

2.37wastheu-

d by


Um@ t,r ,q,f#51

N@r #U11dU0@ t,r ,q,f#

S N@r #

a@r # D2

dUr@ t,r ,q,f#

S N@r #

a@r # D2 dUq@ t,r ,q,f#

r

V1S N@r #

a@r # D2 dUw@ t,r ,q,f#

r sin@q#.

~29!

Note that in this equation,dU0 is not a dynamical variablebut is determined according to the constraint thatUm is a4-velocity (UmUm521). Furthermore,dUr , dUq, anddUw are not contravariant components of a 4-vector but cvenient variables that are the components of a 3-vectothe orthonormal basis associated with the spherical systecoordinates for the flat 3-space. It enables us to writemotion equations in a way very similar to the Newtonian EIndeed, writing the 3-velocity on the orthonormal basissociated with the spherical coordinates as

VW 5UdUr

dUq

dUw

~30!

and defining on the same basis

ÃW @ t,r ,q,f#5H ~V2Nw@r # !cos@q#

2sin@q#~V2Nw@r #1A@r # !

0

~31!

with

A@r #5~V2Nw@r # !S d lnF a@r #

N@r #Gdln@r #

D 2rNw8@r #

2

where the prime is the derivative versus the radial coonate, we have

~] t1V]w!VW 12ÃW `VW 1¹W h50. ~32!

This equation is very close to the Newtonian EE, the mdifference being that the 3-vector that appears instead oVWis now depending on the coordinates~as in the case of dif-ferential rotation! but no longer parallel to the rotation axi

Concerning the baryonic number conservation, writingin a Newtonian-like way, we have in the slow rotation lim

~] t1V]w!n1S N@r #

a@r # D2

div~ nVW !50, ~33!

wheren is defined asn5nN2a, n being the baryonic number density. The natural generalization of the anelasticproximation is then

12300

-nofe.-

i-

n

t

p-

div~ nVW !50. ~34!

As we are using the strong Cowling approximation, tbackground star appears in the equations of motion only‘‘external’’ ~from the point of view of the mode! data. Forany relativistic calculation, what we do is to calculate tbackground configuration using the already existing codelustrated in Bonazzolaet al. @56# and then to use the resuling lapse, shift, conformal factor and their derivatives wrespect to the radial coordinate in our equations.

Here we will only give one single example, showing whave in this relativistic case with the strong Cowling aproximation results similar to those obtained in the casethe Newtonian differential rotation. We took ag52 poly-trope with 1.74 solar masses and a radius equal to 12.37The star was very slowly rotating~the ratio between its ki-netic energy and its mass energy was about 10216) and wetook for a time unit the inverse of the angular velocity. Tanelastic approximation with the free surface BC was usand to regularize the solution we added a degenerate visity with Es5531025 once the final equations were writtein a Newtonian way. We insist on the fact that this viscoterm does not come from relativistic calculations and jaims at regularizing the solution. In Figs. 23, 24, 25, 26, a27 exactly the same quantities as in Sec. V are plotted.conclusions for this preliminary relativistic calculation athe same as in the Newtonian case: A polar counterpart ofvelocity appears from the beginning of the evolution amost of the time, the velocity is concentrated near the eqtor and the surface.

FIG. 23. Time evolution of the ratio between energy containin the polar part of the velocity and the total energy of the mowith the strong Cowling and anelastic approximations and thesurface BC. The background star is ag52 relativistic rigidly rotat-ing polytrope with 1.74 solar masses and a radius equal to 1km. The ratio between its kinetic energy and its mass energy10216. We see that in spite of the fact that the initial data arelinear Newtonianr-mode that is purely axial, as soon as the calclation begins a polar counterpart of the mode appears as implieLockitch et al. @26#.

1-17

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lyswp.

tllehee

is

eith

lfsticaryefuldes

at

eorthaist

owpor

o-u-andwith

vious

oretionand

ual

e


VII. CONCLUSION

This article deals with a new hydrodynamical code baon spectral methods in spherical coordinates. This firstsion of the code was written to study inertial modes in slowrotating NS, but it can easily be modified to include farotation. As it was explained in the Introduction, the slorotation limit is a very good approximation for a first steThe present version of the code can be used to study bothNewtonian case and the relativistic case with the so-castrong Cowling approximation. Here, we have given tmain algorithms it uses and shown some tests of the lin

FIG. 24. Time evolution of the radial component of velocity

the point (12 ,q/2,0) for the same calculation as in Fig. 23. Thinitial data are the linearr-mode that is no longer an eigenvectand radial velocity appears very fast. The power spectrum oftime evolution shows several peaks, but one of them has the sfrequency as the modes of Figs. 25 and 26 and is the relativanalog of ther-mode. But it is no longer purely axial.

FIG. 25. Time evolution of theq component of velocity in thesame calculation as in Fig. 24. The curve and the associated pspectrum show that there is one mode that also appears in thepart with a frequency very close to the frequency of the linear-mode. See also Fig. 26.

12300

dr-

t

hed

ar

version. The way we overcame the numerical instabilityalso explained.

Furthermore, in order to work with very different timscales, we have introduced an approximation to deal wmass~or baryonic number! conservation that proved itsequite robust and useful to study inertial modes: The anelaapproximation. Even if this approximation is not a necessone and can be abandoned in further studies, it is very usto have an idea of the main properties of the inertial mo

emeic

erlar

FIG. 26. Time evolution of one of the two independent compnents of theSi j @ t# tensor that appears in the RR force. This calclation was done during the same run as the results in Figs. 2425. We can see the almost monochromatic associated spectrumexactly the same frequency as the mode that appears in the prefigures.

FIG. 27. q component of the velocity versus the radius fseveral values ofq. We can see the kind of concentration of thmotion near the surface and the equatorial plane. This calculawas done with the strong Cowling and anelastic approximationsthe free surface BC. The background star is ag52 relativisticrigidly rotating polytrope with 1.74 solar masses and a radius eqto 12.37 km. The initial data are the Newtonian linearr-mode oftype l 5m52. This figure corresponds to the velocity after a timequal to 15 oscillations of the linearr-mode of typel 5m52.

1-18

psrui-ooear-e

no-

on

at i

f aNtioinaritr

slt

-oithulit

hanoi

orue

th

in-

arin-

the

re.e

ical-ts

s.in

l

ofof

se

ries.

rchyary

iny isec-


in slowly rotating stars. The results presented in this pawere obtained in the linear case using very basic modelNS, such as the divergence-free case with a rigid c(Vr ur 5150) or ag52 polytrope with the anelastic approxmation and free surface. Deeper studies in the GR framewwith more quantitative results on the effects of EOS,stratification of the star, and of BC will follow. Despite thnaive aspect of our studies, some common features appeIndeed, whatever the approximation~divergence-free or anelastic! with different BC~rigid crust for the divergence-frecase and free surface for the anelastic approximation!, wesaw the appearance of a polar part in the mode, as soothe background is Newtonian and differentially rotating,relativistic and rigidly rotating~in the latter case, this phenomenon is due to the frame dragging!. Whether in the caseof noisy initial data with a radiation reaction force actingthem, or in the case of the free evolution of an initiallinearr-mode, from the very beginning the energy of the polar pis at least 1% of the energy of the axial part. This resulcompatible with the analytical work by Lockitchet al. @26#who proved, in the relativistic framework, the existence ononvanishing polar part of the velocity in a sequence ofwith a barotropic equation of state and decreasing rotarate. Furthermore, another interesting result that we obtais the ‘‘concentration of the motion’’ near the surface thquickly appears: after less than 15 periods of the linear-mode in the calculations done up to now. Adding viscoshas shown that this is a robust and physical feature. Moover, it seems to neither depend on the physical conditionon the BC. This phenomenon is very similar to the resufound by Karinoet al. in 2001 @28# with an eigenvectorscalculation. But, if this is really a singularity of the derivative of the velocity near the surface, as it seems to be intime evolutions, it would mean that a mode calculation wfinite difference schemes can only give quantitative rescompletely depending on the number of points. Indeed, wthis scheme, there is an intrinsic numerical viscosity tdepends on the resolution. Moreover, if this phenomenoverified in future studies in GR, the stratification of the NSthe physics of the crust could be very important to knowinertial modes are relevant for GW emission.

ACKNOWLEDGMENTS

We would like to thank E. Gourgoulhon and B. Carter fcarefully reading this paper. L. Blanchet and J. Ruoff gavethe opportunity to have friendly and fruitful discussions. Wwould also like to thank the computer department ofobservatory for the technical assistance.

APPENDIX A: SPIRIT OF THE SPECTRAL METHODS

The numerical algorithm adopted to solve the NSE~or theEE! is based on the spectral methods~SM! widely used inhydro or MHD problems. Before explaining this algorithmmore details, in Appendix A we will begin with a short summary of the SM in order to make more evident the peculiity of the solving of vectorial equations like NSE. Then,Appendix B we deal explicitly with EE and aim at explain

12300

erofst

rkf

ed.

asr

rts

Sn

edt

ye-ors

ur

tshtisrf

s

e

-

ing the way we implemented the approximation done onmass conservation equation.

Our group developed algorithms and routines library~seeBonazzola and Mark@57# and Bonazzolaet al. @38#!9 allow-ing us to solve partial differential equations~PDE! in differ-ent geometries, mainly in domains diffeomorphic to a spheFirst, with an example of scalar PDE, we will look at thsingularities contained in operators expressed in spherlike coordinates@see, for instance, the different componenof the NSE: Eq.~9!# and at our choice of spectral basiThen, we shall discuss the further difficulties that arisevectorial PDE and explain the way we overcome them.

1. The scalar heat equation

Consider the heat equation:

]F

]t5aDF1S ~A1!

where

DF5]2F

]r 21

2

r

]F

]r1

1

r 2 S ]2F

]q21

cosq

sinq

]F

]q1

1

sinq2

]2F

]w2Dis the Laplacian in spherical coordinates,a the heat conduc-tivity supposed to be constant, andSa source term~that mayinclude nonlinear terms if they exist!. The idea of spectramethods is to look for the solution of the Eq.~A1! on theform

F@r ,q,w,t#5 (n,l ,m50

`

Fnlm@ t#Hnl @r #Kl

m@q#Lm@w#

~A2!

with (0<r<1;0<q<p;0<w<2p) and whereHn

l @r #,Klm@q#,Lm@w# are a well chosen complete set

functions. The problem is then to find the time evolutionthe coefficientsFnlm@ t#.

In a spherical geometry, it is quite natural to chooLm@w#5expimw and Kl

m@q#5Plm@q#, wherePl

m@q# are the

9In 1980, one of us~S.B.! started to build a library of routinesbased on spectral methods to solve PDE in different geometToday, this library contains more than 700 routines written inFOR-

TRAN 70 and 90 languages. These routines have a strict, hieraand allow us to assemble codes in modular way. We call this libr‘‘Spectra.’’ A part of this library~the highest in the hierarchy! waswritten in C11 language by J.A. Marck and E. Gourgoulhonorder to allow the use of an object oriented language. This librarcalled ‘‘Lorene.’’ The code described in this section uses the ‘‘Sptra’’ library and is written inFORTRAN 90 language.

1-19

he

ey

is

be

oe

r-

t op

in

y

n

one,hat

vity

nay

a-ge-

tral.andinof

rm-mi-

a-

con-he

C


Legendre functions. With these choices, Eq.~A1! can bewritten

]Flm@r ,t#

]t2aS ]2Flm@r ,t#

]r 21

2

r

]Flm@r ,t#

]r

2l ~ l 11!

r 2 Flm@r ,t# D 5Slm@r ,t# ~A3!

whereSlm@r ,t# are the Fourier-Legendre coefficients of tfunction S at the radiusr and the instantt.

In order to handle the singularity atr 50, we shall con-sider separately the casesl 50, l 51, andl .1.

For l 50, we use the fact that aC1 function symmetricwith respect to the inversionr→2r has its first derivativethat vanishes at least asr at r 50. Therefore, for such afunction, the term (]2F/]r 2)1(2/r )(]F/]r ) is regular. EvenChebyshev polynomialsT2n@r # have this property, and thchoice Hn

0@r #5T2n@r #(n>0) then satisfies the regularitconditions.

For l 51, it is almost the same, but the final choiceHn

1@r #5T2n11@r #.The casel .1 is more delicate to handle. Indeed, it can

shown~see Bonazzolaet al. @57#! that for F@r ,q,w,t# to bea C` function, the coefficientsFlm must vanish asr l at r50. It means that theFlm@r ,t# are symmetric with respect tthe inversionr→2r if l is even, and antisymmetric in thopposite case. We shall then distinguish the two casesl evenand l odd.

Case lÌ1 even

The functionsHnl @r #5T2n@r #2T2n12@r # are even and

vanish asr 2 at the origin. Therefore the quantity$(d2/dr 2)1(2/r )(d/dr )2@ l ( l 11)/r 2#%Hn

l @r # is regular at the originand is retained.

Case lÌ1 odd

The functions Hnl @r #5(2n11)T2n11@r #1(2n

21)T2n13@r # are anti-symmetric with respect to the invesion r→2r and vanish asr 3 at r 50. Therefore the quantity$(d2/dr 2) 1 (2/r ) (d/dr ) 2 ( l ( l 11)/r 2)%Hn

l @r # is alsoregular and completes our basis.

Note that by this way, we expand the solution with a sefunctions that satisfy minimal conditions of regularity, excefor l 50 andl 51. It means that they vanish asr 2 or r 3, inorder to make regular any term in the equation, insteadvanishing asr l as they should to form aC` function. Untilnow, in all the problems that our group treated, these mmal regularity conditions were sufficient~see Bonazzolaet al. @38# for more details!. But we shall see later that themake the Euler equations unstable.

To end with the scalar case, we shall show how Eq.~A3!can be written when the time discretization is performed a

12300

ft

of

i-

d

a second order implicit scheme used10:

Fnlmj 112a

Dt

2 (p50

`

Apnl Fplm

j 11

5Fnlmj 1DtS a

1

2 (p50

`

Apnl Fplm

j 1Snlmj 11/2D .

~A4!

Here Ajnl 5^H j

l OlHnl & is the matrix of the operatorOl

5(d2/dr 2)1(2/r )(d/dr )2@ l ( l 11)/r 2#. The index j 11/2means a quantityS computed at the timet j1Dt/2, obtainedby extrapolating this quantity using its~known! values attime t j 21 and t j : Sj 11/25(3Sj2Sj 21)/2. The left-hand side~LHS! operator can be easily reduced to a pentadiagonaland then inverted with a number of arithmetic operations tis proportional to the maximum valueNmax of N.

In the case of a space-dependent heat conductia@r ,q,w#, we can use a semi-implicit method~see Gottlieband Orszag@58#!. It consists in solving the equation

Fnlmj 112~a1br2!

Dt

2 (p50

`

Apnl Fplm

j 11

5Fnlmj 1

Dt

2~a1br2! (

p50

`

Apnl Fplm

j

1DtFSnlmj 11/21~a2a2br2!S (

p50

`

Apnl FplmD j 11/2G

~A5!

wherea andb are two coefficients that satisfy the conditioa1br2>a everywhere and that we choose in such a wthat a1br2 takes the same values asa at the boundaries.

The new matrix on the LHS is again a pentadiagonal mtrix. Boundary conditions are imposed by adding a homoneous solution from Eq.~A4! or from Eq.~A5!.11 The readercan find in the quoted literature more details on the specmethods, and on how to implement boundary conditions

To conclude, note that in the present example, we expthe solution in spherical harmonics. But it turns out thatgeneral it is more convenient to use linear combinationsChebyshev polynomials to treat the expansion inq. Thephilosophy of the spectral methods then consists of perfoing simple operations such as computing derivatives, pritives, integrals, multiplications, or divisions byr, r 2, sinq,or cosq in the coefficients space, and by making multiplic

10Because of the use of Chebyshev polynomials, the Courantditions for the explicit problem formulation are quite severe. In tpresent caseDtmax is proportional to 1/Nmax

4 ~see Gottlieb andOrszag@58#!, therefore implicit or semi-implicit formulation of theproblem is required.

11The casea@1,q,w#50 is called degenerate. In this case no Bare allowed, anda1b must be50.

1-20

thth

asegun

erwqurere

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c

e-ap

arrm

ion

lito

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en

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o

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-oodlity, weiesetet inrm

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idal


tions of functions in the configuration space. If necessary,passage to a Legendre representation is performed wimatrix multiplication.

2. Analysis in vectorial case

a. Stability of numerical vectorial PDE

We have seen how to handle coordinates singularitiespearing in scalar PDE when spherical coordinates are uBut when treating vectorial PDE, as EE or NSE, the sinlarities are more malicious: a single look at the componeof NSE @Eq. ~9!# should be sufficient to convince the readThe singular terms cannot be handled only by laying doanalytical properties, as it was done in the above scalar etion. Indeed, the dependence existing among the diffespherical components must be taken into account in ordemake singular terms compensate each others. A simpleample will illustrate this crucial point.

Consider the following constant divergence free vectoVWof Cartesian components:Vx50;Vy50;Vz51. Its sphericalcomponents areVr52cosq;Vq5sinq;Vw50. Then, wehave

divVW 5]Vr

]r1

2

rVr1

1

r S ]Vq

]q1

cosq

sinqVqD50.

A small error ~for example a round-off error! in thecomputation of components will obviously forbid an exacompensation between the two singular terms 2Vr /r and(1/r )@]qVq1(cosq/sinq)Vq#. The consequence is the cration of high order coefficients and possible instabilitiespearing in the iterative process.

Indeed, we found that the hydrocode for solving lineized EE was unstable. As expected, high frequency tewere exploding after few hundred time-steps~few periods ofthe r-mode! either in the case of the anelastic approximator in the case of the incompressible approximation (divWW50). As explained above, the main reason for this instabiwas the nonexact compensation of the singular terms ctained in the different source terms in the Poisson equat~B7! and ~B9! ~see below!.

To overcome this problem, we define two angular pottials Th andPo in a such a way that

Wq5]qPo21

sinq]wTh ~A6!

Ww51

sinq]wPo1]qTh .

If rPo and Th are any arbitrary set of regular functionwe are now sure that the corresponding componentsWq andWw will make the divergent terms appearing in the vectorPDE compensate each other. Moreover, the system of etions ~A6! can be easily inverted. First, taking the angudivergence of Wq and Ww: divqwWW 5(]q1cotq)Wq

1(1/sinq)]wWw gives DqwPo5divqwWW where Dqw5]q2

12300

ea

p-d.-ts.na-nttox-

t

-

-s

yn-ns

-

la-

r

1cotq]q1(1/sin2q)]w2 . If divqwWW is then expanded in

spherical harmonics, we immediately have

Po,lm521

l ~ l 11!~divqwWW ! lm . ~A7!

It is the same to computeTh . Taking the angular curl ofWW ,we obtain

DqwTh5curlqwWW ~A8!

where curlqwWW 52(1/sinq)@]wWq2]q(sinqWw)#.To complete this procedure, we have to guarantee thatWr

can also compensate the singular terms generated bycomponentsWq andWw . A satisfactory way to proceed is ttake as a new variable the quantity divWW . Indeed, once thequantityg@r ,q,w#5divWW is known, it turns out to be easy tfind Wr . Bearing in mind that divWW 5] rWr12 Wr /r1(1/r 2)divqwWW , we have, after an expansion in sphericharmonics12

Wr ,lm51

r 2E0

r

u$ug@u,q,w# lm1 l ~ l 11!Po,lm@u,q,w#%du.

~A9!

Thus, at each time step, the strategy consists of calcuing the potentialsPo and Th , and then in calculating backthe componentsWq and Ww . The componentWr is alsocomputed at each time step by using Eq.~A9!.

Finally, note that with this procedure, the potentialsPo ,Th , and the componentWr have the correct analytical behavior on the axisez , i.e., they vanish as sinmq. Thus, theinstability of the EE hydrocode was drastically reduced, bnot completely suppressed. It still kept on growing, but abten times slower than before. The reason for this residinstability was that, at each time step, the round-off errdid not have the correct analytical properties atr 50. Indeed,they did not vanish asr l . Consequently high spatial frequency terms were still generated that did not have the ganalytical properties and a runaway toward the instabiwas once again generated. To avoid this phenomenonproject, at each time step, all the scalar quantit(div WW ) lm@r #, Po , lm@r #, andTh , lm@r # on a Legendre spacPn

l @r # in such a way to satisfy the analytical conditions. Nothat the necessity of this procedure is due to the fact thathe EE, solved with spectral methods, no dissipative te~numerical or physical! is present. Consequently the numecal instability are not damped.

12It is important to take into account the following propertiesthe spherical harmonics decomposition of the functionsPo andTh :

for a given l the product byr of the componentWr ,lm of WW , aregular vectorial function, vanishes asr l 21. The corresponding po-loidal part Po,lm behaves in the same way. The associated toropart isTo,(l 21)m and is therefore a regular function.

1-21

toEr

reth

order

ina

re-

-

oreme,


b. Solving the vectorial heat equation

Now we give more details concerning the algorithmsolve vectorial PDE with spectral methods. In solving NSwe need to solve a vectorial heat equation that can beduced to an equation of the type

]BW

]t1¹`~m¹`BW !5 JW ~A10!

where JW is the divergence-free term of the source. Wemember that it is obtained by the decomposition ofsource termJW in a potential part¹W f and a divergence-free

part JW : JW5 JW1¹W f.The Eq.~A10! can be written in the following form:

]BW

]t5mnBW 1~¹`BW !`¹W m1 JW . ~A11!

From a numerical point of view, the termSW 5(¹`BW )`¹W m

12300

,e-

-e

is considered as a source and is computed by a secondscheme using its values at the timest j 21 and t j 22.

Therefore, we have to solve the equation

]BW

]t5mDBW 1 JW1SW ~A12!

with the condition¹W •BW 50.The technique generally used was already described

Bonazzola et al. @38#. Nevertheless, we adopted hereslightly different approach which gives more accuratesults.

To solve Eq.~A12! it is convenient to introduce the poloidal and toroidal potentialsPo andTh as defined in Appen-dix A 2 a. Here we will only consider the case ofm51, asthe semi-implicit scheme which must be used in the mgeneral case is straightforward. With a second-order schewe have in the harmonic representation

erms.

Po,lmj 11 2

1

2DtS d2Po,lm

j 11

dr 21

2

r

dPo,lmj 11

dr2

l ~ l 11!

r 2 Po,lmj 11 2

2

r 2Br lm

j 11D5Po,lm

j 1DtF1

2 S d2Po,lmj

dr 21

2

r

dPo,lmj

dr2

l ~ l 11!

r 2 Po,lmj 2

2

r 2Br lm

j D 1P lmj 11/2G ~A13!

whereP is the poloidal component ofJW1SW . Note the presence of a singular term (2/r 2)Brj 11 in this equation. Bearing in mind

that divBW 50 and thatPo,lmj 11 vanishes asr l 21, it is obvious that there should be a compensation between these two t

Nevertheless, we shall slightly modify Eq.~A13! in order to obtain more easily this compensation:

Po,lmj 11 2

1

2DtS d2Po,lm

j 11

dr 21

2

r

dPo,lmj 11

dr2

l ~ l 21!

r 2 Po,lmj 11 D

5Po,lmj 1DtF1

2 S d2Po,lmj

dr 21

2

r

dPo,lmj

dr2

l ~ l 21!

r 2 Po,lmj D 2

1

r 2 ~Br lm

j 1 lPo,lmj 1Br lm

j 111 lPo,lmj 11!1P lm

j 11/2G . ~A14!

OncePo,lmj 11 is known,Br ,lm

j 11 is obtained as explained above@see Eq.~A9!, with g50].

Finally, the equation for the toroidal part ofBW reduces to an ordinary scalar heat equation:

Th,(l 21)mj 11 1

1

2DtS d2Th,(l 21)m

j 11

dr 21

2

r

dTh,(l 21)mj 11

dr2

l ~ l 21!

r 2Th,(l 21)m

j 11 D5Th,(l 21)m

j 1DtF1

2 S d2Th,lmj

dr 21

2

r

dTh,lmj

dr2

l ~ l 21!

r 2Th,lm

j D 1q ( l 21)mj 11/2 G ~A15!

whereq is the toroidal component ofJW1SW .

1-22

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ind


APPENDIX B: SOLVING EULER OR NAVIER-STOKESEQUATIONS

In this section we shall show how to implement the dferent schemes proposed in the Sec. III to handle the difent time scales appearing in the EE for a slowly rotating sIn all cases, for simplicity, we will only consider the lineaized equations. First, we will begin with the case whereexact system of equations must be solved. Then we sconsider the implementation of physical approximations afinally the problem of BC.

1. Exact Euler equations

As usual, we start by doing a decomposition of the vel

ity vectorWW in a divergence-free componentWW and a poten-tial one:

WW 5WW 1¹CW , divWW 50. ~B1!

In a second order time scheme, the divergence-free comnent of the EE is then written as

WW j 115WW j1Dt@22VW `WW 1FW #DFj 11/2 ~B2!

while the potential part is

C j 115C j2Dt@wV1h1w# j 11/2. ~B3!

In Eq. ~B2!, the subscript ‘‘DF ’’ means the divergence-frecomponent of the external force. In Eq.~B3!, the subscriptVis for the potential component of the Coriolis force 2VW

`WW , w is the potential component of some other exterforce, andh is the variation of the enthalpy. Moreover, walso have the baryonic number~or mass! conservation equation:

hj 115hj2DtF S Wr1]C

]r D ]H0

]r1GH0DCG j 11/2

~B4!

whereH0 is the enthalpy of the nonperturbed configuratioIn this explicit second order scheme, the required typivalue of the time stepDt, which guarantees the numericstability, is determined by the term] rH0 that is much largerthan the Coriolis term for a slowly rotating NS.

The implicit version of the system of Eqs.~B3!, ~B4! isthe following:

C j 115C j2DtH 1

2~hj 111hj !1@wV1h1w# j 11/2J ,

hj 115hj21

2Dt@~] rC

j 111] rCj1Wr

j 111Wrj !] rH0

1GH0~DC j 111DC j !# ~B5!

whereWrj 11 is obtained from Eq.~B2!. Solving the above

system of equations reduces to invert, for each value ofl andm, an enneadiagonal (2Nr39) matrix ~9 diagonals! where

12300

r-r.

ealld

-

o-

r

.l

Nr is the number of degrees of freedom inr. The generali-zation to the NSE is quite obvious and we shall not gmore details about it.

2. Implementation of physical approximations

As was already explained in Sec. III, for solving the EENSE, we chose to use approximations in order to better ctrol the results.

a. The divergence-free approximation

The spirit of this approximation consists of replacing tmass-conservation equation~B4! by the condition

divWW 50 ~B6!

or equivalentlyC50. The problem is then to find the enthalpy h in a such a way that the condition~B6! is satisfied.This can be done easily by solving the Poisson equation

Dh2div F j 11/250 ~B7!

reached by taking the divergence of the EE. As the gradof a harmonic function is a divergence-free vector, sucfunction can then be added to a particular solution of the~B7! in order to satisfy the boundary conditions. Onceh is

obtained, the EE has to be solved withFW 5FW 2¹W h. If vis-cous terms are present, NSE case, a vectorial type equmust be solved.

b. Anelastic approximation

As was already explained, the anelastic approximatconsists of neglecting the term] th in Eq. ~B4!. Once again,the game consists of finding the enthalpyh in a such a waythat the following equation is satisfied:

GH0div WW 1Wr] rH050. ~B8!

To find h, it is sufficient to derive with respect to time thEq. ~B8!, and to replace] tdivWW by its value reached fromthe EE. It gives

GH0Dh1] rH0] rh5GH0div FW j 11/21Frj 11/2] rH0 ~B9!

where FW contains all the force terms~Coriolis force in-cluded!.

The solution of the above equation is achieved withsemi-implicit scheme very similar to the one used to soEq. ~A1!. The problem is then reduced to solving the following equation~for simplicity, we shall consider only the casof a polytropic EOS!:

~g21!~a1br2!Dh12br] rh

5H0Dh1] r H0] rh1GH0divFW j 11/21] rH0Frj 11/2

~B10!

where a and b are two constants defined as was donesolving Eq.~A1!. The LHS operator can be easily inverte

1-23

-

he

heto

an

at

se

syd

d

ionine

n

in

facen belpyer


and Eq.~B10! can be solved by iteration~see Gourgoulhonet al. @59#!. Since the iteration at each time step is quite timconsuming, a more convenient strategy~although less accurate! consists of replacingh in the right-hand side~RHS! ofEq. ~B10! by hj 11/2.

3. Boundary conditions

It was already explained how to impose BC for tdivergence-free case. For the system of equations~B5! or theEq. ~B9!, it is worth distinguishing the two cases: either tenthalpy vanishes or does not vanish at the boundary ofintegration domain. Here we will only consider the caseEE. Indeed, for NSE we chose to take a viscosity that vishes at the surface to avoid the need of further BC.

If H0ur 51.0, the solution is quite easy: Eq.~B9! or thesystem of equations~B5! admit a homogeneous solution thcan be used to satisfy one BC.

If H0(1) vanishes, the system of equations~B5! or Eq.~B9! are degenerate and therefore no BC can be impoBut we shall show that, in this case, the correct BC areau-tomaticallysatisfied.

Indeed, consider first the case of the linearized exacttem of equations~B5!. On the surface of the nonperturbestar, we have] th1WW •¹W H0 which is the correct BC. ThesurfaceH01h50 then defines the profile of the perturbe

.

tt.

A

d B

..

12300

e

hef-

d.

s-

star. To show that the solution in the anelastic approximatalso satisfies the BC, it is more convenient to first examthe nonlinear case. Here,h is not infinitesimal, and Eq.~B8!must be replaced with the equation

G~H01h!div WW 1WW •¹W ~H01h!50 ~B11!

that is satisfied ifh is a solution of nonlinearized Eq.~B9!:

G~H01h!Dh2div FW j 11/21~¹W h2FW j 11/2!•¹W ~H01h!50.~B12!

Once again, the surface of the star is defined byH01h50.But from Eq. ~B11! we have the correct BC:WW •¹W (H01h) uH01h5050. The surface of the star is then an unknow

quantity that is determined by the relationH01h50 in solv-ing Eq. ~B12! by iteration. This technique was developeda different context in the already quoted paper@59#.

In the linear case, we haveWr50 at r 51 @cf. Eq. ~B8!#,and the enthalpy does not vanish on the nonperturbed surof the star. But once again the free surface of the star caobtained by looking for the position where the total enthaH01h50 vanishes. This surface coincides within first ordquantities with the nonperturbed surfacer 51. Note that ifg.0 the pressurePur 51}hg11ur 51 vanishes within termso(h).

-.

d

ev.

el,

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Inertial modes in slowly rotating stars: An evolutionary description