Nonlinear decay of rmodes in rapidly rotating neutron stars

Wolfgang Kastaun

International School for Advanced Studies (SISSA), via Bonomea 265, Trieste 34136, Italyand Istituto Nazionale di Fisica Nucleare (INFN), Via Enrico Fermi 40, 00044 Frascati (Rome), Italy

(Received 26 September 2011; published 20 December 2011)

We investigate the dynamics of rmodes at amplitudes in the nonlinear regime for rapidly but uniformly

rotating neutron stars with a polytropic equation of state. For this, we perform three-dimensional

relativistic hydrodynamical simulations, making the simplifying assumption of a fixed spacetime. To

excite r modes, we linearly scale exact eigenfunctions to large amplitudes. We find that for initial

dimensionless amplitudes around three, r modes decay on time scales around ten oscillation periods,

while at amplitudes of order unity, they are stable over the evolution time scale. Together with the decay, a

strong differential rotation develops, conserving the total angular momentum, with angular velocities in

the range 0:5 . . . 1:2 of the initial one. We evolved two models with different rotation rates and found

slower decay for the more rapidly rotating one. We present r-mode eigenfunctions and frequencies, and

compare them to known analytic results for slowly rotating Newtonian stars. As a diagnostic tool, we

discuss conserved energy and angular momentum for the case of a fixed axisymmetric background metric

and introduce a measure for the energy of nonaxisymmetric fluid oscillation modes.

DOI: 10.1103/PhysRevD.84.124036 PACS numbers: 04.30.Db, 04.40.Dg, 95.30.Sf, 97.10.Sj

I. INTRODUCTION

The r modes of neutron stars are a subclass of inertialmodes, i.e. modes where the Coriolis force is the mainrestoring force, which have purely axial parity in the slowrotation limit. In the limit of slowly and rigidly rotatingNewtonian stars, the r-mode eigenfunction and frequencyare analytically known. Its frequency, as for all inertialmodes, is proportional to the angular velocity� of the star.The rmodes are current dominated, that is the amplitude ofthe density perturbations is smaller than the velocity per-turbations by a factor�. Because of this and also due to thelower frequencies, they are weak emitters of gravitationalwaves compared to modes of same energy for whichpressure is the dominant restoring force.

The extension of the r-mode solution to the case of rapidrotation and/or general relativity (GR) is an ongoing field ofresearch. In the slow rotation approximation, GR equationshave first been derived by [1,2]. In [3], it was shown thatthey are valid only for nonbarotropic equations of state(EOS), and that the equations for barotropic stars are quali-tatively different. Further corrections to the equations werepointed out by [4]. The equations in [2,4] for nonbarotropicstars contain singular points, whose interpretation is stillunder debate [3,5–12]. In particular, [5] provides argumentsthat the slow rotation approximation breaks down near thesingular points, and including higher-order terms wouldlead to valid physical solutions. However, there is still nomathematical proof that regular r modes of nonbarotropicstars in GR exist under all circumstances. For the barotropiccase, i.e. an EOS where the pressure is a function of densityalone, GR counterparts of the Newtonian rmode have beenfound by [6]. Those solutions are hybrid modes containingboth polar and axial parity perturbations.

The rapidly rigidly rotating case has been investigated inGR only with the simplifying assumption of a fixed space-time (relativistic Cowling approximation). In [13], thetwo-dimensional partial differential equations governingthe eigenfunctions for the case of a barotropic EOS havebeen solved numerically by using finite differences. It wasfound that the ratio of oscillation and rotation frequenciesis decreasing with increasing ratio of rotational to bindingenergy. In [14], the r modes and pressure modes ofrapidly but rigidly rotating polytropic stars are investi-gated using time evolution of the linearized equations fora prescribed � dependency. The method is extended toinclude the nonbarotropic case in [15], and differentialrotation in [16].The r mode is of astrophysical interest since in GR it is

generally subject to unstable growth, emitting gravitationalradiation, as shown by [17,18]. The reason is that ther-mode wave pattern is generally counterrotating in thecorotating frame, but corotating in the inertial frame,which is the condition for the CFS mechanism(Chandrasekhar, Friedman, Schutz, see [19,20]) to be op-erational. Because of the CFS instability, r modes are apossible source for gravitational wave astronomy, butmight also explain the limitation of observed neutron starrotation rates. It is thus important to know at which ampli-tudes the r-mode instability saturates. There are severaleffects which might suppress the instability or limit theamplitude to small values, including nonlinear couplings[21–23], winding up of magnetic field lines by differentialrotation associated with the r mode [24–26], and bulkviscosity [27] enhanced by the presence of hyperons. Thestudies mentioned so far apply to newborn neutron stars,which are still hot enough to prevent superfluidity, whichfurther complicates the picture; see [28–30].

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Assuming the r mode has an instability window, i.e. arange of temperature and rotation rate where the CFSinstability is not suppressed, it would be important toknow the behavior at very high amplitudes, typically ex-pressed as a dimensionless quantity proportional to theratio of velocity perturbation to rotational velocity at theequator. Using numerical simulations of uniformly rotatingstars in the Newtonian framework, it was found by [31] thatformation of shocks and breaking of surface waves set in atdimensionless amplitudes around three. These simulationsuse a gravitational backreaction force more than 4000times stronger than in GR in order to excite rmodes withintime scales short enough for numerical evolution. Thislarge driving force was still present during the reportedonset of wave breaking. In a similar simulation [23,32], butswitching off the artificial driving force after amplitudes inthe range 1:6 . . . 2:2 had been reached, a catastrophic decayof the r modes was found. This decay was attributed tomode coupling effects, due to the observed growth ofsecondary modes. In full GR, a numerical simulation ofan r mode with unit amplitude was performed by [33], butno decay apart from numerical damping was found on theevolution time scale of 26 oscillation periods.

The r modes are also linked to differential rotation inmany ways. For the case of rapidly and differentiallyrotating barotropic Newtonian stars, [34] found that ther-mode eigenvalue problem for a polytropic Newtonianstar becomes difficult to solve numerically for degrees ofdifferential rotation large enough to cause corotation pointsof the wave pattern. On the other hand, [35] found regularsolutions for a Newtonian, incompressible thin shell modelwith a similar amount of differential rotation. The exis-tence of regular r-mode solutions for differentially rotatingstars is thus likely, but remains unproven. For uniformlyrotating stars, the presence of r modes implies a certainamount of differential rotation, which is of second order inthe mode amplitude. In [24,25], this effect was estimatedusing only the first-order eigenfunctions. The importancefor the evolution of a magnetic field via winding up of fieldlines was demonstrated in [24,26]. Complementary to this,it was shown by [36] that the second-order corrections tothe eigenfunction of the velocity field necessarily contain adifferentially rotating part, at least for slowly and rigidlyrotating Newtonian stars with barotropic EOS. As arguedin [37,38], the resulting second-order contributions to theangular momentum have to be considered for the timeevolution of the r-mode instability. A similar, but distincteffect concerns the interaction with gravitational radiation.For a toy model consisting of a spherical shell, it wasshown by [39] that the gravitational backreaction directlyinduces differential rotation instead of causing only auniform spin-down. Further, the aforementioned studies[31,32] of nonlinear r-mode decay also observed the for-mation of significant differential rotation during the decay.These studies support the view that the CFS instability of

the r mode will cause a certain amount of differentialrotation, even if the r-mode amplitude saturates at smallvalues.In the present work, we extend the previous studies in

mainly two directions. First, we extract r-mode eigenfunc-tions and frequencies for rapidly (but rigidly) rotatingrelativistic stars under the simplifying Cowling approxi-mation. We are using a fundamentally different approachthan [13,14], and also discuss the properties of the eigen-functions in detail, in particular, their energy and estimatedgravitational radiation. Second, we investigate the decayof high-amplitude r modes and the formation of differen-tial rotation already found in [31,32]. We are howeverusing the relativistic Cowling approximation instead ofNewtonian gravity for the evolution and scaled exactr-mode eigenfunctions as initial perturbation, instead ofexciting r modes by using artificially large backreactionforces. Linear eigenfunctions scaled to large amplitude arenot necessarily more realistic, but the comparison willreveal whether those effects depend on details of the initialdata.

II. ANALYTIC TOOLS

In this section, we review the analytic tools used forthis work. Throughout the article, we use the followingnotation:

� � Rest frame rest mass density

P � Pressure

� � Specific internal energy

h � 1þ �þ P

�

cs � Soundspeed

u� � Fluid 4-velocity

vi � Fluid 3-velocity

W � Fluid Lorentz factor

g�� � 4-metric of signatureð�;þ;þ;þÞ

gij � 3-metric

ffiffiffig

p �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi� detðg��Þ

qffiffiffiffi�

p �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffidetðgijÞ

q

n� � Unit normal to t ¼ const hypersurface

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� � Lapse function

�i � Shift vector

� � Angular velocity of the star

All equations assume geometric units G ¼ c ¼ 1. Greekindices run from 0 . . . 3, indices i, j, k, l from 1 . . . 3.When working in cylindrical coordinates, generally de-noted by ð#; z;�Þ, indices a, b, c run from 1 . . . 2,excluding �.

A. General properties of eigenfunctions

Although there is no analytic solution for the oscillationeigenfunctions of rapidly rotating stars, one can derivesome general properties. The following is valid for a fixedspacetime, rigid rotation, and a barotropic EOS.

For any global oscillation mode, the perturbation of afluid quantity X can be written as

Xð ~x; tÞ ¼ A<ðXð#; zÞeið!tþm�þXÞÞ; (1)

where A is a dimensionless amplitude, ! the real-valued

oscillation frequency, and X is a constant phase shift. X isthe (suitably normalized) real-valued eigenfunction.

The set of variables we use to completely specify theoscillation is f�; vig. The relative phase shifts are given by� ¼ v� ¼ 0, v# ¼ vz ¼ �

2 . For a derivation, see [40].

Because of the equatorial symmetry of the unperturbedmodels, the eigenfunctions also have a well-defined zparity, with the relations

Pz½�� ¼ Pz½v#� ¼ �Pz½vz� ¼ Pz½v��: (2)

B. Newtonian r mode

For slowly rotating stars in Newtonian theory, there existanalytic solutions for the r modes. The velocity eigenfunc-tions are given by

~v ¼ A�R

�r

R

�l<ð ~YB

llÞ; (3)

where

~Y Blm ¼ ðlðlþ 1ÞÞ�ð1=2Þ ~r� ~rYlm (4)

are the pure-spin vector harmonics of magnetic type (seee.g. [41]). The density perturbation is proportional to�2Ylþ1;l (see [42]) and the frequencies !i in the inertial

and !c in the corotating frame are

!i ¼ �ðl� 1Þðlþ 2Þlþ 1

�; !c ¼ 2�

lþ 1(5)

(see [43]). The negative sign of !i means prograde motionof the wave patterns.

To measure the r-mode amplitude during a simulation, itis common to use a scalar product with the magnetic vector

harmonic times some radial weighting function. Thismakes sense because the scalar product with other oscil-lation modes vanishes in the slow rotation limit. A conve-nient choice to measure the r-mode amplitude is given bythe magnetic current multipole moment Jll, which is alsoused to estimate the gravitational wave strain. For theNewtonian r mode, we find

Jll ¼Z

�rl ~v � ~YB?ll d3x; (6)

¼ 1

2A�R1�lei!t

Z�r2lj ~YB

llj2d3x: (7)

Hence,

A ¼ jJllj12�R1�l

R�r2lj ~YB

llj2d3x: (8)

In this work, we use the above formula to define thedimensionless r-mode amplitude also for the rotating rela-tivistic case, setting R to the circumferential equatorialradius, and evaluating the denominator for the backgroundmodel. Note this is not a covariant measure; for exactcomparison to other works, one should use invariants liketotal energy or maximum velocity. With increasing rotationrate, one can expect that the presence of other modes (withthe same m) starts contributing to the above measure aswell. This should not be a problem as long as the rmode isthe dominant one. Differential rotation and other axisym-metric perturbations do not contribute to Jll.

C. Evolution equations

We evolve the general relativistic hydrodynamic equa-tions for an ideal fluid, which in covariant form read

r�T�� ¼ 0; (9)

r�ð�u�Þ ¼ 0: (10)

The stress energy tensor T�� of an ideal fluid is given by

T�� ¼ �hu�u� þ Pg��: (11)

For numerical evolution, a 3þ 1 split is applied to obtain afirst-order system of hyperbolic evolution equations inconservation form with source terms

@0q ¼ �@ifiðq; xiÞ þ sðq; xiÞ; (12)

q � ðD; �; SjÞ; (13)

with the evolved hydrodynamic variables given by

D � ffiffiffiffi�

pW�; (14)

� � ffiffiffiffi�

p ðW2�h� P�W�Þ; (15)

Si � ffiffiffiffi�

pW2�hvi: (16)

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In flat Minkowski space (and using standard coordinates),D, �, and Si reduce to mass density, energy density notincluding rest mass, and linear momentum density. Theflux terms fi ¼ ðfiD; fi�; fiSjÞ are given by

fiD ¼ wiD; (17)

fi� ¼ wi�þ �ffiffiffiffi�

pviP; (18)

fiSj ¼ wiSj þ �ffiffiffiffi�

pPi

j; (19)

where wi ¼ �vi � �i is the advection speed relative to thecoordinate system. The source terms can be written inmany ways; the formulation we are using is discussed in[44]. Finally, the evolution equations need to be completedby an EOS of the form P ¼ Pð�; �Þ to compute thepressure.

D. Conserved quantities

Making the assumption of a fixed axisymmetric space-time not only simplifies the numerical evolution, it alsoimplies the existence of conserved fluxes. The stationarityof the metric leads to a conserved energy density, while theaxisymmetry of the metric leads to a conserved angularmomentum. Mathematically, a symmetry means the exis-tence of Killing vector. If k� denotes a Killing vector of thefixed background metric, then for any solution of evolutionEq. (9)

r�k�T�� ¼ T��r�k� ¼ 1

2ðT�� þ T��Þr�k� (20)

¼ 12T

��ðr�k� þr�k�Þ ¼ 0; (21)

where we have first used T�� ¼ T�� and then the Killingequation. Thus, any Killing vector yields a conserved fluxk�T

��. It is not required that the fluid has the samesymmetries.

We now specialize to our setups, rigidly rotating sta-tionary axisymmetric neutron stars. The coordinates areto be chosen such that @t, @� are Killing vectors, and

gtr ¼ gt ¼ 0. We then obtain conserved mass, energy,and angular momentum

M¼ZDd3x; E¼

ZUd3x; J¼

ZLd3x; (22)

where D is defined by Eq. (14) and

U ¼ ffiffiffiffi�

pT0�n

� �D (23)

¼ D

�Wh�� 1� �P

W�

�� �iSi; (24)

L ¼ S� ¼ � ffiffiffiffi�

pT��n

� ¼ DWhv�: (25)

The conservation of mass follows from Eq. (10), energyand angular momentum from Eq. (21) when using k ¼ @tand k ¼ @�. For the definition of E, we first subtracted the

conserved rest mass density D. The conservation of E, Jcan also be derived directly from the evolution Eq. (12),again without using the Einstein equations.Note that E depends on the gauge quantities � and �.

Those are not completely arbitrary, since we use coordi-nates adapted to the symmetries. Still, � is only unique upto multiplication with a constant. We use a normalizationsuch that � ! 1 at infinity. This leaves us with the freedomof adding a multiple of the spacelike Killing vector @� to

the shift vector, which corresponds to coordinate frames

rotating relative to each other. For �0j ¼ �j þ��j�, it

follows from Eq. (24) that the energies are simply relatedby E0 ¼ E���L. Although we evolve in corotatingcoordinates, we compute E in the nonrotating frame, i.e.coordinates for which �i ! 0 at infinity. With thosechoices it then follows that U ! 0 for an infinitely dilutedfluid element at rest at infinity. Hence, a system becomesunbound for E> 0.The quantities E and J have no physical meaning since

they rely on the unphysical Cowling approximation. In fullGR, the field equations ensure that the spacetime has thesame symmetries as the fluid. When resorting to the use ofa fixed spacetime in numerical simulations, the resultingconserved quantities should however be monitored to en-sure numerical correctness.Note that the numerically evolved energy density � is

not the same as the conserved energy density U. In theNewtonian limit, for example, for a fluid moving in aconstant external gravitational field, � does not containpotential energy, in contrast to U. The reason for not usingU in numerical schemes is that the corresponding evolutionequation involves the time derivative of the gauge quantity� in the fully relativistic case.How do energy E and angular momentum J defined

above relate to other well-known expressions in full GR?The Arnowitt-Deser-Misner (ADM) mass is an expressionfor total energy in full GR. In contrast to E, it cannot beexpressed as a locally conserved flux. The ADM mass isconserved in full GR, but not in the Cowling approxima-tion, since its conservation relies on the Einstein equations.E on the other hand is not even conserved in Newtonianphysics when allowing the gravitational field to change.The same differences apply to J and the ADM angularmomentum. The Komar mass is neither conserved in fullGR nor with the Cowling approximation, and cannot beexpressed as a linear combination of E and M.Interestingly, we found that J equals the Komar angularmomentum, as long as there are no singularities to con-sider. This is surprising since the Komar integrals were notderived as conserved quantities in dynamical systems, butas gauge-invariant measures for stationary systems.

E. Oscillation energy

In the following, we introduce a notation for the energyof oscillation modes in the Cowling approximation, which

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is useful as a diagnostic tool in numerical studies per-formed in a fixed spacetime, but might also serve as anorder of magnitude estimate in the general case.

For this, we expand the conserved energy E defined inthe previous section around a stationary backgroundmodel. For nonaxisymmetric modes, it is easy to showthat the linear terms cancel when integrating along the �direction, so we need to expand the energy to second orderin the amplitude. Since we want an expression that can becomputed without knowledge of the second-order correc-tions to the eigenfunction, we still assume that the pertur-bation itself scales linearly. However, this leads toconceptual problems. First, there is a freedom of choiceconcerning the set of variables used to completely specifythe system. If those variables are perturbed linearly, theresulting energy perturbation at second order depends onthis choice. Second, we found that in general the conservedmass changes as well when evaluated to second order.This implies that the state around which the model oscil-lates is not exactly the same as the unperturbed state.Unfortunately, we are interested in the energy differencefrom the stationary state reached after all the oscillationenergy is dissipated, e.g. due to numerical damping, whileconserving total mass and angular momentum profile. Forone of our models, we computed mass and energy changewhen perturbing �, vi with the r-mode eigenfunction. Byassuming that the mass is created with the average bindingenergy E=M of the background model, we estimated theambiguity in defining E to be around 50%. This effectcould be tracked down to the fact that not only the kineticenergy density, but also the conserved mass density dependon the velocity via the Lorentz factor, in contrast to theNewtonian case.

To cure these problems, we define the oscillation energyas the perturbation of the conserved energy when perturb-ing the variablesD, L, va linearly. This way, total mass andangular momentum are exactly conserved for nonaxisym-metric perturbations. To our knowledge, the following hasnot been discussed elsewhere. Wewill assume a cylindricalcoordinate system adapted to the symmetries and with�a ¼ ga� ¼ 0, and only consider background models

with va ¼ 0. For a nonaxisymmetric mode described byEq. (1), we define the mode energy as

E ¼ 1

A2

Z �1

2

@2U

@D2D2 þ 1

2

@2U

@L2L2

þ @2U

@D@LDLþ 1

2

@2U

@va@vbvavb

�d3x (26)

¼ �Z �

1

2

@2U

@D2D2 þ 1

2

@2U

@L2L2 þ @2U

@D@LD L

þ 1

2

@2U

@va@vbvavb

�d#dz: (27)

The quantities in Eq. (26) are defined with respect toCartesian coordinates, while Eq. (27) is valid in cylindricalcoordinates. Terms with Dva or Lva do not contrib-ute since they have the angular dependency sinðm�þ!tÞcosðm�þ!tÞ. Also, the corresponding second derivativesare zero. This energy depends on the normalization of theeigenfunctions, which has to be specified. The explicitexpressions to compute the energy defined above are quitelengthy. They can be found in the Appendix.We stress that the energy defined by Eq. (26) is only an

estimate for the energy of a finite-amplitude oscillation. Itwas shown in [36] that for r modes of Newtonian stars,differential rotation is an unavoidable feature. Most likely,such axisymmetric terms would contribute to the first-order expansion of the energy E and hence constitute asecond-order contribution in total, like the terms consid-ered in our definition. Without knowledge of the second-order perturbation, Eq. (26) is probably the best one can do.

III. NUMERICAL METHOD

In the following, we briefly describe our numericalmethods. For readers not familiar with general relativistichydrodynamics, we recommend the review [45].

A. Time evolution

We evolve the 3þ 1 split hydrodynamic evolution equa-tions in flux-conservative form (12). However, we use azero-temperature EOS of the form P ¼ Pð�Þ. As a conse-quence, the system becomes overdetermined. Therefore,we do not evolve the energy density �, which becomesredundant, but compute it from mass and momentum den-sities. For details, see [44]. We stress that this approach isonly self-consistent for adiabatic evolution, and hence oursimulations are only correct in the absence of shock for-mation. Discontinuities cannot produce shock heating; in-stead, they lead to a violation of energy conservation. Asharp decrease of E is therefore an indicator for shockformation; see [46] for examples.In the absence of shocks however, the evolution be-

comes more accurate, since there is no error in the evolu-tion of the specific entropy. One particular error that isavoided this way is the formation of large-scale, low-velocity convective movements driven by entropy gra-dients caused by numerical errors. Such vortices couldeasily be confused with genuine nonlinear effects in simu-lations of high-amplitude r-mode oscillations.To evolve the above system numerically, we use the

PIZZA code first described in [44]. It is based on a high-

resolution-shock-capturing scheme, which was optimizedfor quasistationary simulations. As shown in [44], the codeis able to evolve a stationary star with high accuracy, inparticular, the rotation profile. As for all such codes, spe-cial care has to be taken to treat the stellar surface. Insteadof using an artificial atmosphere, we apply the scheme usedin [46]. The advantages are that the mass is conserved to

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machine precision and that the amplitude of oscillationsexcited by numerical noise is negligible for the applica-tions presented here. Still, the surface treatment is the mainsource of numerical damping.

We use uniform three-dimensional Cartesian grids incorotating coordinates. The outer boundaries are placedfar enough from the surface to prevent matter from leavingthe computational domain. Thus, the total mass M isconserved to machine precision. The total angular momen-tum on the other hand is subject to discretization errors,since L is not an evolved variable in Cartesian coordinates.

B. Eigenfunction extraction

In order to extract eigenfunctions numerically, we usethe mode recycling method in the form described in [46].In short, the star is perturbed using some trial perturbation,and then evolved in time numerically. The frequencies ofthe oscillations excited by the perturbation are determinedusing Fourier analysis in time at some sample point. Byselecting one frequency !0 and computing the Fourierintegrals at this frequency for any point in the star, oneobtains a first estimate of the complex eigenfunction,which is usually still contaminated by other oscillationsdue to the finite evolution time. To obtain the two-dimensional eigenfunctions, we divide the numericalcomplex eigenfunction by eim� and average over �. Thisremoves contributions with different � dependency. Next,we remove contributions with the wrong z parity by (anti-)symmetrizing. Finally, we compute and factor out theaverage complex phase (see [46] for details), while usingthe phase variance as an error measure. The result is usedas the initial perturbation in a new simulation, and thewhole process is repeated until oscillation modes otherthan the desired one are reasonably suppressed.

Obviously, the method is only effective if the initialtrial perturbation significantly excites the desired mode.To extract the r-mode eigenfunction, we choose theNewtonian eigenfunction for a slowly rotating star, givenby Eq. (3), which is close enough to the actual eigenfunc-tions of the models.

IV. STELLAR MODELS

We investigate two different uniformly rotating stellarmodels with fixed central density �c ¼ 7:9053 �1017 kgm�3 and EOS, but different rotation rates. Theirproperties are summarized in Table I. The EOS, which isalso used during the evolution, is a polytropic EOS definedby

Pð�Þ ¼ �p

��

�p

��; (28)

with polytropic exponent � ¼ 2 and the constant densityscale �p ¼ 6:1760 � 1018 kgm�3. We note that polytropic

stars are stable against convection and do not possess g

modes, i.e. modes for which buoyancy is the restoringforce. Model MB85 was computed using the RNS codedescribed in [47], while model MB70 was computed usingthe code described in [48,49]. Both codes are accurateenough for our purposes; the only reason to use differentcodes is that the latter became our standard choice once itwas available.These models are a crude approximation to real neutron

stars. Their purpose is to get a basic qualitative under-standing of the nonlinear r mode dynamics in the mostsimple case. It is likely that the inclusion of compositiongradients or differential rotation will lead to new effects.To excite nonlinear oscillations, we always use the exact

eigenfunction obtained by mode recycling. However, incontrast to the linear regime, there is some arbitrarinessinvolved regarding how to scale the eigenfunctions to largeamplitudes. Ideally, we would like initial data resembling amode naturally grown to high amplitudes, e.g. due to theCFS mechanism. Since this is not feasible, we simply scalethe perturbation of the velocity and of the specific energyand recompute the other fluid quantities consistently afterapplying the perturbation.Note also that we use Eulerian perturbations. This has

the drawback that the star is not deformed correctly closeto the surface, in particular, there is no perturbation outsidethe surface of the unperturbed star. For high amplitudes,the initial data inevitably contains small shocks at thesurface. In practice however, even high-amplitude r modesdo not induce large deformations and we did not notice thecorresponding numerical artifacts.

V. NUMERICAL RESULTS

In the following, we present our results for the r modewith l ¼ m ¼ 2. Unless noted otherwise, our simulationsuse a uniform resolution of 50 points per equatorial stellarradius, which is a reasonable compromise between accu-racy and computational cost.

A. R-mode properties

Using the methods in Sec. III B, we extracted eigenfunc-tions and frequencies of the r mode for the models inTable I. We also computed the energy of the modes definedin Sec. II E. The results are given in Table II. For our

TABLE I. Details of the stellar models. MB is the total bar-yonic mass, FR the rotation rate as observed from infinity, Rc theequatorial circumferential radius, ar the ratio of polar to equa-torial coordinate radius, ec ¼ E=MB, �c ¼ j�J=ð2EÞj, where Eand J are energy and angular momentum defined by Eq. (22).

Name MB=M� FR=Hz Rc=km ar ec �c

MB85 1.6194 590.90 15.384 0.85 �0:2094 0.02406

MB70 1.7555 792.10 17.268 0.7 �0:2159 0.04864

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models, the frequencies agree with the ones in theNewtonian slowly rotating case better than 10%. Ther-mode frequencies in the inertial frame found by [16]agree with our results better than 0.1%. Given that weuse a completely independent code based on a differentmethod, the good agreement validates the results.

The mode energy is useful to quantify what amplitudesare large in the sense that strong deformations of the staroccur. Naively, one should think that e.g. A ¼ 3 is a hugeamplitude because the velocity perturbations become com-parable to the rotational velocity at the equator. Looking atthe energy however, we find that the mode energy at A ¼ 3for model MB85 is only a fraction 3 � 10�3 of the bindingenergy E. For model MB85, the energy of the perturbationequals the stars’ binding energy at A � 57. For the moderecycling process, we used amplitudes around A ¼ 0:3,well inside the linear regime.

The eigenfunctions are visualized in Figs. 1 and 2. Asone can see, the velocity perturbations are similar to thosein the Newtonian slowly rotating case, in particular,the radial component is small. Also, the density perturba-tion is qualitatively the same as in the Newtonian case. To

quantify the differences, we decompose the velocity per-turbation into vector spherical harmonic functions

~v ¼ X1l¼0

Xlm¼�l

ðaElmðrÞ ~YElmð;�Þ þ aBlmðrÞ ~YB

lmð;�Þ

þ aRlmðrÞYlmð;�ÞerÞ: (29)

The results are plotted in Fig. 3. The dominant contributionto the velocity is the magnetic-type vector harmonic.However, there are also significant radial and polar con-tributions. The dominant component of the specific energy(not shown in the plot) is the l ¼ 3, m ¼ 2 sphericalharmonic.Although higher-order multipole moments are quite

small in the inner regions of the star, they have significantamplitudes between the polar and equatorial radius. Thisdoes not imply that small-scale structures appear close to

TABLE II. Properties of the l ¼ m ¼ 2 r mode. fc is thefrequency with respect to the corotating frame, fi the frequencyobserved from infinity in the inertial frame. The estimatednumerical accuracy of fc is 1%, not including the unknownerror due to the Cowling approximation. For comparison, wecompute the r-mode frequency fNc for slowly rotating Newtonianstars given by Eq. (5). E is the r-mode energy, defined byEq. (26), normalized to an amplitude A ¼ 1 as defined byEq. (8). E is the models’ binding energy defined by Eq. (22).

Model fc=Hz fi=Hz fc=fNc E=E

MB85 361.4 820.4 0.917 3:033 � 10�4

MB70 544.8 1039.4 1.032 5:093 � 10�4

FIG. 3 (color online). Decomposition of the r-mode velocityperturbation of model MB85 into vector spherical harmonics.The vertical lines mark the polar and equatorial radius.

FIG. 1. Two-dimensional eigenfunctions �ð#; zÞ (left half) andvað#; zÞ (right half), belonging to the r mode of model MB85.

FIG. 2. Perturbations corresponding to the r mode of modelMB85, as longitude/latitude plot at fixed coordinate radiusr ¼ Re=2, with Re being the equatorial coordinate radius. Thevelocity perturbations v�, v are plotted on the left half, andthe specific energy perturbation � on the right half. Note thepatterns have a 180� periodicity in �.

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the surface. The reason is just that the star is ellipsoidal andthe spheres of constant coordinate radius start intersectingwith the stellar surface.

From the multipole decomposition of the numericallyextracted eigenfunctions, we estimated the gravitationalradiation caused by the r modes, using the multipole for-mulas for Newtonian sources from [41]. Luminosity,strain, and angular momentum loss are given in Table III.Unsurprisingly, we find that for the r modes only thel ¼ m ¼ 2 current multipole contributes significantly.The second-largest contribution comes from the l ¼ 3,m ¼ 2 mass multipole, which for model MB85 is smallerby a factor jAE2

32 =AB222 j � 0:02. The higher multipole mo-

ments are completely negligible.

B. Rotation profile

One of the most noticeable features in our simulations isthe development of strong differential rotation during theevolution of r modes with high initial amplitudes. As willbe shown in Sec. VC, this is accompanied by a rapid decayof the r mode.

To visualize the rotation profile alone without the con-tribution from the r-mode oscillation, we compute the�-averaged angular velocity. Of course, any axisymmetricoscillation would contribute to this measure as well.However, since the Fourier spectrum of the � velocity atsome sample points in the star did only show significantpeaks at the r-mode frequency and at zero frequency, wecan assume that any snapshot of the �-averaged angularvelocity is a good measure of the differential rotation.

Figure 4 shows snapshots at two different times duringand after the decay. During the decay, the angular velocityshows a two-dimensional structure. Strangely, the angularvelocity near the poles temporarily increases to more thantwice the initial value and then decreases again. At a laterstage, the angular velocity converges to a simple profiledepending roughly on the distance to the axis. In order toquantify the amount of angular momentum redistribution,we define an average change of angular momentum by

��J

J¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiRð �L� �L0Þ2d2xR�L20d

2x

s; (30)

�L ¼ 1

2�

Z 2�

0Ld�; (31)

where L0 is the value for the unperturbed model. Figure 5shows the time evolution of this measure as well as thelocal differential rotation at chosen positions.The profile shown in Fig. 4 is similar both in shape and

magnitude to the differential rotation found by [31] for aNewtonian star. A cut in the equatorial plane is shown inFig. 6. Near the axis, the rotation rate is slowed down by afactor of 2, while the equatorial rate is increased by a factor1.2. This also agrees well with the profile shown in [32] forthe Newtonian case. It differs however strongly from the

TABLE III. Gravitational radiation caused by r modes in thelinear regime. The gravitational luminosity Wgw and angular

momentum loss _Jgw are given in terms of the time scales

�J ¼ J= _Jgw and �E ¼ jEj=Wgw, where E and J are the binding

energy and angular momentum given in Table I. The values arenormalized to an amplitude A ¼ 1, with AB2

22 A, Wgw A2,_Jgw A2.

Model AB222 =ð10 MpcÞ �E=s �J=s

MB85 1:42 � 10�24 1:47 � 106 4:93 � 104MB70 3:56 � 10�24 1:64 � 105 1:04 � 104

FIG. 4 (color online). Development of differential rotation, formodelMB85 perturbedwith an rmode of initial amplitudeA ¼ 3.Shown is the profile during the r-mode decay (LEFT) and whenthe r mode is almost completely gone (RIGHT); compare alsoFig. 8. Color coded (with identical scales) is the �-averagedangular velocity in units of the angular velocity of the unperturbedmodel. The surface of the unperturbed star is marked by a thindotted line,while the surface atwhich the�-averaged density fallsbelow0.005 times the central density ismarked by a thin solid line.A small region near the axis has been excluded to avoid problemswhen transforming from the Cartesian grid.

FIG. 5 (color online). Time evolution for the same setup asFig. 4 of the �-averaged angular velocity at different samplepoints in the star. Also shown is the global average angularmomentum change defined by Eq. (30).

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profile shown in [23]. On the other hand, the latter was nota � average but a cut along the x axis. The magnitude ofdifferential rotation is large enough to cause a visibledeformation of the stellar surface, as shown in Fig. 4.

For the unperturbed models, our code is able to conservethe rotation profile with errors several orders of magnitudesmaller than the observed differential rotation. The errorsin the presence of differential rotation might be larger,because there is shear motion and because the fluid is notcorotating with the coordinates anymore, particularly nearthe stellar surface. In general, conserving the angular mo-mentum is problematic with codes based on Cartesiangrids. Therefore, we monitor the conserved angular mo-mentum J defined by Eq. (22). For model MB85 withinitial amplitude A ¼ 3, the angular momentum is decreas-ing more or less linearly, and the total loss at t ¼ 20 ms is�J=J ¼ 0:0032. At the same time, we observe an average

angular momentum change ��J=J ¼ 0:26, much more thanthe total violation of angular momentum. It is also greaterthan the total and average change of angular momentumintroduced by the initial perturbation (caused by second-order terms and surface effects), which are of the order

�J=J ¼ 0:018 and ��J=J ¼ 0:025.Although our computational resources did not permit a

full convergence test, we evolved the first 5 ms withresolutions N ¼ 37, 50, 75 points per stellar radius. Foreach, we sampled the perturbation of �-averaged angularvelocity at the end of the simulation along the equatorialplane, as shown in Fig. 7. On average, low resolution seemsto damp the differential rotation and not to cause it. Toquantify the errors, we computed the L2 norms of theresiduals and estimated the convergence order p � 1:3,which at resolution N ¼ 50 implies an average error of�� around 10% of the maximum value. We note that thenumerical evolution scheme is second-order accurate, butthe treatment of the surface is not. From Fig. 7, one can seethat the largest error of� is indeed found near the surface.

We conclude that the differential rotation is caused by aredistribution of angular momentum, and not by numericalerrors or contributions already present in the perturbedinitial data.

C. R-mode decay

Simultaneously with the development of differentialrotation, high-amplitude r modes exhibit a rapid decay inamplitude, as will be shown in the following.As a measure for the decay, we use the dimensionless

amplitude A defined by Eq. (8). The results for all oursimulations are shown in Figs. 8 and 9. All simulationswere numerically stable (the shorter ones are exploratorysimulations).As one can see, the decay depends crucially on the initial

amplitude. In fact, the lines cross, which implies differentdecay rates at the same amplitude. Obviously, the decay

FIG. 7. Convergence of differential rotation. Shown is the�-averaged change of angular velocity after 5 ms for an rmode of model MB85 with initial amplitude A ¼ 3, at differentgrid resolutions.

FIG. 6. Snapshots at different times of the �-averaged angularvelocity in the equatorial plane versus coordinate radius, for thesame setup as Fig. 4. The lines end where the�-averaged densityfalls below 0.005 times the central one.

FIG. 8 (color online). Decay of the r-mode amplitude A de-fined by Eq. (8) for model MB85 perturbed with different initialamplitudes. The two curves with initial amplitudes of A ¼ 2:48and A ¼ 2:51 highlight the steep dependence of the decay timeon the initial amplitude.

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rate depends on the history of the evolution. For initialamplitudes of order unity, the decay is comparable to thenumerical decay, i.e. compatible to no physical decay atall, while for amplitudes as high as 3, the amplitudedecreases rapidly. Interestingly, for some initial amplitudesthere is a plateau phase before the catastrophic decay. Thelength of this phase increases rapidly with decreasinginitial amplitude. From our simulations, it is unclearwhether there is a critical amplitude for the onset of thedecay, or if all r modes decay eventually. The decay weobserve is qualitatively the same as the one reported in[32], where a Newtonian star is evolved, exciting an rmode by applying an artificially increased gravitationalbackreaction that is switched off at given r-modeamplitude.

The fact that the decay rate depends on the initialamplitude is a strong hint that the main cause is neitherwave breaking nor shock formation, nor any other effectwhich depends only on the current r-mode amplitude.There must be another perturbation besides the r mode toexplain the different decay rates at the same r-mode am-plitude. By comparing Figs. 5 and 8, we can see that theincrease of differential rotation is related to the r-modeamplitude. A hypothetical model which would explain theaccelerating decay and saturating differential rotation isthat differential rotation causes r-mode decay and thepresence of the r mode causes an increase of differentialrotation.

A comparison between models MB85 and MB70 seemsto indicate that faster rotation stabilizes high-amplitudemodes. For initial amplitudes of order unity, doing a com-parison is difficult since the damping time scales becomecomparable to the numerical damping time scales for bothmodels.

To answer the question whether the energy of the rmodecould be dissipated directly, e.g. via shock formation ornumerical dissipation, we now study the energy budget ofthe decay, using the tools from Secs. II D and II E.Figure 10 shows the total conserved energy as well as an

estimate for the energy in the r mode. The violation ofenergy conservation is obviously significant, but it is notsufficient to explain the amplitude decay, even if all the lostenergy is taken from the rmode. We assume that the modeenergy is at least partially transferred to the differentialrotation and deformation of the star described in Sec. VB.The only effects causing violation of energy conserva-

tion are formation of shocks in conjunction with the coldEOS, surface effects like wave breaking, and numericaldissipation. To estimate the latter, we compare simulationsof the first 5 ms at resolutions 37, 50, and 75 points perstellar radius. As can be seen in Fig. 11, the violation ofenergy conservation is obviously caused for the most partby numerical errors, although some residual physical effectcannot be ruled out. However, the comparison also showsthat the decay of the r mode is computed quite accurately.

FIG. 9 (color online). Decay of the r-mode amplitude at differ-ent initial amplitudes, for model MB70.

FIG. 10. Energy budget for a decaying rmode of model MB85with initial amplitude A ¼ 3. Shown is the loss of total con-served energy E defined by Eq. (22), as well as the loss of energyin the r mode, computed by estimating the amplitude from thecurrent multipole moment J22 and the mode energy fromEq. (26). For the points labeled ‘‘Phase ignored,’’ we took theabsolute value of the complex integrand when computing J22.

FIG. 11 (color online). Convergence of r-mode amplitude (a)and conserved energy (b), for model MB85 perturbed with an rmode of initial amplitude A ¼ 3.

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This implies that the loss of total energy, which doesdepend on resolution, is unrelated to the loss of r-modeenergy.

Still looking for the cause of the energy loss, we pro-duced a movie of the first 5 ms showing density andvelocity perturbations in the coordinate planes. We didnot notice any shock formation. The velocity field is domi-nated by the r mode. Right from the start, the density fieldis an overlay of many oscillation modes including the rmode and the quasiradial mode. The presence of othermodes is not surprising; on the contrary, it would bestrange if the simple linear scaling of the eigenfunctionused to excite high-amplitude r modes would not exciteother modes as well. Further, the density perturbation ofthe r mode itself is weaker than for pressure modes of thesame kinetic energy. We noticed an m ¼ 4 deformationwhich near the surface becomes quite nonlinear, but notenough to cause wave breaking. However, the resultingnumerical dissipation might explain part of the energy loss.In [31], wave breaking of the r mode itself was observedfor a simulation of a Newtonian star after the r-modeamplitude peaked at A ¼ 3:35. This does not contradictour results. Our maximum amplitude is A ¼ 3:3, and theamplitudes are not directly comparable due to the differentmodels and a slightly different definition of the amplitude.

In conclusion, neither numerical errors nor shock for-mation can be the reason for the decay of the r mode, andits energy has to be transferred elsewhere, in particular,into differential rotation. Wave breaking does not occurin our models, but might play a role for even largeramplitudes.

Next, we try to get a more local picture of the r-modedecay. For this, we study the integrand of the l ¼ m ¼ 2current multipole. However, we first have to eliminate theinfluence of differential rotation, because locally the cur-rent multipole integrand has a significant � componentand therefore couples strongly to differential rotation (androtation as such when the background model is not sub-tracted), although this cancels out after integration. It istherefore difficult to interpret plots based on the integranditself, as done in [32]. Instead, we integrate along the �direction to get rid of any contribution from perturbationswith angular dependency m � 2. Snapshots at differenttimes are shown in Fig. 12. While the total amplitudedecays, the pattern is deformed, but not destroyed. Thisseems to contradict [32], where a breakdown of the modepattern is reported. However, it is not obvious to whatdegree Fig. 5 in [32] is determined by the developingdifferential rotation instead of the r mode.

We also found that the complex phase, which is spatiallyconstant in the linear regime, develops a significant vari-ance during the decay. Figure 12 shows the variance at alate stage. This seems to imply that parts of the staroscillate out of phase. Note however that the local ampli-tude goes to zero at the axis roughly quadratically, and the

phase varies most strongly near the axis where it is verysensitive to contributions from secondary m ¼ 2 modes.We cannot rule out the possibility that the variance is anartifact caused by other perturbations. It is worth mention-ing that a large phase variance is reported in [31] as well,although this was under the influence of an artificially largegravitational backreaction force driving the r mode.In order to determine the importance of the phase vari-

ance for our estimate of the r-mode energy from the multi-pole moment, we recomputed the total multipole moment,but used the absolute value of the complex integrand. Therationale is that the energy, in contrast to the multipolemoment, is not sensitive to an axisymmetric phase shift. Asshown in Fig. 10, the differences are negligible.We note that the phase variance cannot be attributed to

high amplitudes, since it is still present when the amplitudehas already decayed to values in the linear regime. If weassume for a moment that the phase variance of the currentmultipole really means that the oscillation of the velocityfield is spatially out of phase, it follows that some otherperturbation must be present, somehow influencing the rmode. It is worth noting that the differential rotation profilepresent after the decay phase has the same structure as thephase variance.

D. Search for mode coupling

We now discuss possible nonlinear interactions of the rmode with secondary modes. For this, we looked at theFourier spectra of the time evolution at various samplepoints. To study the time evolution of the spectra, wecomputed separate spectra for the first and second half of

FIG. 12 (color online). Time development of the �-integratedl ¼ m ¼ 2 current multipole integrand for a nonlinearly decay-ing r mode of model MB85, with initial amplitude A ¼ 3. (a)–(c) Absolute value at different times. (d) Complex phase differ-ence to the equator, near the end of the decay. The phase wascomputed such that the usual phase jump from �� to � isavoided, keeping the phase continuous.

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the evolution. Figure 13 shows the spectra for the velocitycomponents vr, v, for an initial amplitude A ¼ 3. As onecan see, the rmode is the dominant contribution. However,there are significant peaks corresponding to oscillationmodes in the inertial mode spectrum. We cannot identifythose modes since the frequency resolution of our spectrais insufficient to distinguish modes in the dense inertialmode spectrum. Interestingly, there are also peaks in thefrequency range where the 2nd-order scalar partial differ-ential equation describing mode oscillations is of mixedelliptic-hyperbolic type, as discussed in [40]. It is unclearwhat the structure of solutions in this range would be, andif such solutions exist at all. It is however possible that thedeveloping differential rotation shifts the location of thisband. In any case, the only peaks we identified beside the rmode are the quasiradial F mode and the axisymmetric fmode, both quite insignificant for the velocity field. In thedensity, they are more visible, since the density perturba-tion of r modes in relation to the velocity perturbation issmaller than for pressure modes.

None of the secondary modes with significant amplitudeseem to grow. Only the increasing differential rotation isclearly visible in the spectra of v� (not plotted). FromFig. 13, we cannot confirm any mode-coupling effect, atleast not of the magnitude found in [23,32]. Note howeverthat the spectra in [23,32] are for an initial amplitudeA ¼ 1:6, where the decay is slower. Looking at spectrafrom our simulations at lower amplitudes, we sometimessee growing peaks, but their amplitudes are tiny comparedto the r mode. There are several different explanations forthe discrepancies. First, the mode coupling for our modelsmight saturate already during the time span covered by the

early-stage Fourier spectra, thus being indistinguishablefrom secondary oscillations excited by the initial perturba-tion. Second, the mode coupling reported in [23,32] mightnot be the only cause of r-mode decay, and just happens tobe less prominent for our models. Also, we might havechosen the wrong sample points, where an important sec-ondary mode happens to be small (although we alsostudied some global quantities like multipole moments).Our results thus cannot completely rule out mode couplingas the cause of the r-mode decay.

VI. SUMMARYAND DISCUSSION

This work provides new evidence that rmodes (with l ¼m ¼ 2) of uniformly rotating neutron stars with a baro-tropic EOS decay rapidly if their dimensionless amplitudeexceeds a model-dependent value of order unity. The speedof decay depends only on the initial amplitude, and thedecay does not stop even when the amplitude becomessmall compared to unity. The r mode decays more slowlyfor higher rotation rates (at a fixed central density).Together with the decay of the r mode, strong differentialrotation develops. The final rotation profile dependsroughly on the distance to the axis. Close to the axis, therotation is slowed down by a factor 0.5, while near theequator we observed speedups around 1.2.Our results are the first ones obtained using the relativ-

istic Cowling approximation, and are in good agreementwith previous studies [31,32] of uniformly rotating starswhich have been treated in the Newtonian framework, butwithout artificially fixing the gravitational field. Thus, thecause is unrelated both to relativistic effects and to thechanges of the gravitational field induced by the fluid.Those studies also used a different way to excite the rmode. While they used an artificially increased gravita-tional radiation reaction force to slowly drive the rmode tolarge amplitudes, we perturbed the initial data with linearlyscaled exact eigenfunctions. This is noteworthy because itimplies that the decay and the differential rotation are notsensitive to the amount and composition of other modes inthe initial data.However, we also found some differences. The afore-

mentioned Newtonian studies reported the occurrence ofeither wave breaking or strong mode coupling togetherwith the decay. We found no wave breaking, although theoscillations near the surface are definitely in the nonlinearregime. This is not a contradiction. Given the differentmodels, our maximum amplitude, although comparableto the reported wave-breaking case, was probably just notlarge enough. Nevertheless, wave breaking is not necessaryfor the r-mode decay. We also cannot confirm the presenceof significant mode coupling. However, since we onlyanalyzed the time evolution of selected sample pointsand a few multipole moments, we cannot rule it outcompletely.

FIG. 13 (color online). Fourier spectra of the time evolution ofv, vr at a sample point x ¼ y ¼ z ¼ 0:24R of model MB85,during the r-mode decay with initial amplitude A ¼ 3. Shownare two spectra corresponding to the first and second half of theevolution. For comparison, vertical lines mark the known fre-quencies of the r mode, the quasiradial F mode, and theaxisymmetric f mode. The shaded area marks the transitionalband between inertial and pressure modes (see main text).

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We also studied the energy budget of the process. Forthis, we derived a measure for the energy of fluid modes,which estimates the energy difference between a stateperturbed with the linear eigenfunction and a ground statewith the same angular momentum profile and total mass.For this, we took advantage of the fact that using anartificially fixed axisymmetric spacetime implies the exis-tence of a conserved energy and angular momentum be-sides the conserved mass. During the nonlinear decay, weobserved a significant loss of conserved energy causedmostly, if not completely, by numerical errors, as we foundfrom convergence tests. The energy loss of the r modehowever was greater, and not sensitive to the numericalresolution. We can thus conclude that the energy of the rmode is not dissipated directly, e.g. due to wave breaking,shock formation, or numerical errors. Instead, it is con-verted mostly into differential rotation, which also causes adeformation of the star, increasing the equatorial radius bya few percent.

Although the damping mechanism is unknown, it seemsplausible that the two main features, decay and differentialrotation, are linked by some nonlinear coupling. A modelthat would explain qualitatively the time evolution ofdecay and differential rotation is that high-amplitude rmodes induce differential rotation, and differential rotationdamps the rmode. The next obvious step would thus be thestudy of a rapidly and differentially rotating initial model.In the light of our results, one should establish whetherstable r modes exist in such a model at all. On the otherhand, the differential rotation we found is artificial in thesense that the gravitational field remained fixed althoughthe star was deformed. It might well be that differentialrotation with a consistent field has different effects on the rmode. This could be checked by also setting up initial datawith the spacetime of a rigidly rotating model, but a differ-entially rotating fluid profile in equilibrium in thatspacetime.

Last but not least, we provided eigenfunctions and fre-quencies of r modes in the relativistic Cowling approxi-mation for rapidly rotating stars. We found excellentagreement with the frequencies found in [16], thus validat-ing those results. Within numerical accuracy, the eigen-functions are smooth. Eigenfunctions and frequencies arestill very similar to the Newtonian slow-rotation case. Wealso estimated the gravitational luminosity, wave strain,and angular momentum loss caused by rmodes, and foundthat the current multipole is still strongly dominant for therapidly rotating case. For our models, an r mode with unit-dimensionless amplitude at a distance of 10 Mpc causes awave strain of the order 10�24.

Finally, we would like to speculate a bit. Although thedecay time scale diverges quickly with decreasing ampli-tude, we have no proof that the effect vanishes completelybelow some critical amplitude. Let us assume that ther-mode growth due to the CFS instability and the decay

described in this work are balanced at some amplitude toosmall to be relevant as a source for detectable gravitationalradiation. It is plausible to assume that the r-mode energyloss is still converted to differential rotation. The effectwould then be cumulative as long as the CFS instability isactive. It might well be that the CFS instability of the rmode does not induce large-amplitude oscillations, butdifferential rotation of similar energy. This effect, whichis of course purely hypothetical, might be important for thetime evolution of the rotation profile of newborn neutronstars, and also may lead to strong amplification of themagnetic field.

ACKNOWLEDGMENTS

Our numerical computations have been performed usingthe HPC-BW cluster of the University of Tubingen, the HG1

cluster at SISSA, and the Italian CINECA cluster. We wouldlike to thank Kostas Kokkotas, John Miller, and LucianoRezzolla for useful suggestions concerning the manuscript.

APPENDIX A: COMPUTING THEOSCILLATION ENERGY

In the following, we give explicit expressions to com-pute the oscillation energy defined by Eq. (26). First, theeigenfunctions of the conserved variables are given by

D ¼ ffiffiffiffi�

pWð�þ �W2v�v

�Þ; (A1)

L¼ ffiffiffiffi�

pW2h½ð1þc2sÞv��þ�ð2W2�1Þg��v

��: (A2)

To compute the second derivatives, we first define

�UðW;�; LÞ ¼ ���Lþ ffiffiffiffi�

p ðW�ðWh�� 1Þ � �PÞ(A3)

¼ UðD;L; vaÞ: (A4)

We then write

@2U

@D2¼ @2 �U

@W2

�@W

@D

�2 þ @ �U

@W

@2W

@D2þ @ �U

@�

@2�

@D2

þ @2 �U

@�2

�@�

@D

�2 þ 2

@2 �U

@W@�

@�

@D

@W

@D; (A5)

@2U

@L2¼ @2 �U

@W2

�@W

@L

�2 þ 2

@2 �U

@W@�

@�

@L

@W

@Lþ @2 �U

@�2

�@�

@L

�2

þ @ �U

@W

@2W

@L2þ @ �U

@�

@2�

@L2; (A6)

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@2U

@D@L¼ @2 �U

@W2

@W

@D

@W

@Lþ @2 �U

@�2

@�

@L

@�

@D

þ @2 �U

@W@�

�@�

@L

@W

@Dþ @W

@L

@�

@D

�

þ @ �U

@W

@2W

@D@Lþ @ �U

@�

@2�

@D@L; (A7)

@2U

@va@vb¼ @ �U

@W

@2W

@va@vbþ @ �U

@�

@2�

@va@vb; (A8)

where we included only nonzero terms. From Eq. (A3), wecompute

@ �U

@W¼ ffiffiffiffi

�p

�ð2�hW � 1Þ; (A9)

@ �U

@�¼ ffiffiffiffi

�p

Wð�hWð1þ v2c2sÞ � 1Þ; (A10)

@2 �U

@W2¼ 2

ffiffiffiffi�

p��h; (A11)

@2 �U

@�2¼ ffiffiffiffi

�p

�W2h

�c2s�ð1þ v2c2sÞ þ v2 @

@�c2s

�; (A12)

@2 �U

@W@�¼ ffiffiffiffi

�p ð2�hWð1þ c2sÞ � 1Þ: (A13)

The functions WðD;L; vaÞ, �ðD;L; vaÞ cannot be ex-pressed in closed analytic form. However, we only needthe derivatives. By computing the derivatives of the con-served variables with respect to the primitives and theninverting the resulting linear system of equations, we arriveat

@v�

@D¼ � 1þ c2sffiffiffiffi

�p

W�fv�; (A14)

@v�

@L¼ 1ffiffiffiffi

�p

W2�hf; (A15)

@�

@D¼ 2W2 � 1ffiffiffiffi

�p

Wfg��; (A16)

@�

@L¼ � v�ffiffiffiffi

�p

hf; (A17)

where

f ¼ W2v2�ð1� c2sÞ þ g��: (A18)

It follows that

@W

@D¼ � W2v2ffiffiffiffi

�p

�fð1þ c2sÞg��; (A19)

@W

@L¼ Wffiffiffiffi

�p

�hfv�: (A20)

Using Eqs. (A14)–(A20), we obtain

@f

@D¼ �Wv2g2��ffiffiffiffi

�p

f

�2W2

�ð1� c4sÞ þ ð2W2 � 1Þ @

@�c2s

�;

(A21)

@f

@L¼ v�

W2g��ffiffiffiffi�

phf

�2

�ð1� c2sÞ þ v2 @

@�c2s

�: (A22)

Finally, we find

@2W

@D2¼

�ð1þ c2sÞ

�v2

f

@f

@D� 2W2v�

@v�

@D

�

þ v2

�1þ c2s

�� @

@�c2s

�@�

@D

�W2g��ffiffiffiffi�

p�f

; (A23)

@2W

@L2¼ Wffiffiffiffi

�p

�hf

�W2g��

@v�

@L� v�

f

@f

@L

� v�

�ð1þ c2sÞ @�@L

�; (A24)

@2W

@L@D¼ Wffiffiffiffi

�p

�hf

�W2g��

@v�

@D� v�

f

@f

@D

� v�

�ð1þ c2sÞ @�@D

�; (A25)

@2W

@va@vb¼ W3

fg��gab; (A26)

@2�

@D2¼ g��ffiffiffiffi

�p

f

�Wð2W2 þ 1Þv�

@v�

@D

� ð2W2 � 1Þ 1

Wf

@f

@D

�; (A27)

@2�

@L2¼ 1ffiffiffiffi

�p

hf

�v�

f

@f

@L� g��

@v�

@Lþ v�

c2s�

@�

@L

�; (A28)

@2�

@L@D¼ 1ffiffiffiffi

�p

hf

�v�

f

@f

@D� g��

@v�

@Dþ v�

c2s�

@�

@D

�;

(A29)

@2�

@va@vb¼ � g��

f�W2gab: (A30)

First- and second-order derivatives have of course beencomputed without assuming va ¼ 0, but the results givenhere have been evaluated at va ¼ 0 for simplicity.

WOLFGANG KASTAUN PHYSICAL REVIEW D 84, 124036 (2011)

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