A model of rotating convection in stellar and planetary interiors: II -- gravito-inertial wave generation
DDraft version September 23, 2020
Typeset using L A TEX twocolumn style in AASTeX62
A model of rotating convection in stellar and planetary interiors: II - gravito-inertial wave generation
K. C. Augustson, S. Mathis, and A. Astoul AIM, CEA, CNRS, Universit´e Paris-Saclay, Universit´e Paris Diderot, Sarbonne Paris Cit´e, F-91191 Gif-sur-Yvette Cedex, France
ABSTRACTGravito-inertial waves are excited at the interface of convective and radiative regions and by theReynolds stresses in the bulk of the convection zones of rotating stars and planets. Such waves havenotable asteroseismic signatures in the frequency spectra of rotating stars, particularly among rapidlyrotating early-type stars, which provides a means of probing their internal structure and dynamics.They can also transport angular momentum, chemical species, and energy from the excitation regionto where they dissipate in radiative regions. To estimate the excitation and convective parameterdependence of the amplitude of those waves, a monomodal model for stellar and planetary convectionas described in Paper I is employed, which provides the magnitude of the rms convective velocity as afunction of rotation rate. With this convection model, two channels for wave driving are considered:excitation at a boundary between convectively stable and unstable regions and excitation due toReynolds-stresses. Parameter regimes are found where the sub-inertial waves may carry a significantenergy flux, depending upon the convective Rossby number, the interface stiffness, and the wavefrequency. The super-inertial waves can also be enhanced, but only for convective Rossby numbersnear unity. Interfacially excited waves have a peak energy flux near the lower cutoff frequency whenthe convective Rossby number of the flows that excite them are below a critical Rossby number thatdepends upon the stiffness of the interface, whereas that flux decreases when the convective Rossbynumber is larger than this critical Rossby number.
Keywords:
Convection, Instabilities, Turbulence, Waves – Stars: Evolution, Rotation INTRODUCTIONGravito-inertial waves (hereafter GIWs) are low-frequency internal gravity waves (hereafter IGWs) thatpropagate in the stably stratified regions of rotatingstars and planets (Dintrans & Rieutord 2000). Theypropagate under the simultaneous restoring action of thebuoyancy and Coriolis forces. Such waves are currentlydetected at the surface of rapidly rotating intermediate-mass and massive stars thanks to high-precision aster-oseismology (e.g., Neiner et al. 2012; Moravveji et al.2016; Van Reeth et al. 2018; Christophe et al. 2018;Aerts et al. 2018, 2019, and references therein). More-over, GIWs and IGWs have been detected throughmultiple observational techniques in the atmosphere,interior, and oceans of Earth (e.g., Melchior & Ducarme1986; Gerkema et al. 2008; Gubenko & Kirillovich 2018;Maksimova 2018), and the atmospheres of Mars (e.g.,Gubenko et al. 2015), Jupiter (e.g., Young et al. 1997;Fletcher et al. 2018), Titan (e.g., Hinson & Tyler 1983),and Venus (e.g., Tellmann et al. 2012; Ando et al.2018). In intermediate-mass and massive stars, GIWs
Corresponding author: K. C. [email protected] and IGWs constitute a powerful probe of the chemicalstratification and the radial differential rotation at theboundary between the convective core and the radia-tive envelope (e.g., Van Reeth et al. 2016; Ouazzaniet al. 2017; Van Reeth et al. 2018; Christophe et al.2018; Li et al. 2019). While propagating in the convec-tively stable zones of stars and planets, they are able totransport angular momentum, energy, and chemicals tothe regions where they dissipate through thermal dif-fusion (e.g., Schatzman 1993; Zahn et al. 1997; Mathiset al. 2008; Mathis 2009), co-rotation resonances (e.g.,Goldreich & Kumar 1990; Alvan et al. 2013), and non-linear wave breaking (e.g., Rogers et al. 2013; Rogers& McElwaine 2017). Thus, GIWs, alongside magneticfields, provide a possible explanation for the weak ra-dial differential rotation revealed by space-based helio-seismology and asteroseismology observations of stellarradiative zones across the Hertzsprung-Russel diagram(e.g., Garc´ıa et al. 2007; Beck et al. 2012; Deheuvelset al. 2012; Mosser et al. 2012; Deheuvels et al. 2014;Kurtz et al. 2014; Benomar et al. 2015; Saio et al. 2015;Murphy et al. 2016; Spada et al. 2016; Van Reeth et al.2016; Aerts et al. 2017; Fossat et al. 2017; Gehan et al.2018). Indeed, IGWs have been shown to be poten-tially efficient at angular momentum redistribution inthe radiative core of the Sun (e.g., Talon & Charbonnel a r X i v : . [ a s t r o - ph . S R ] S e p Augustson et al.
Outline
The model of convection derived in Paper I is em-ployed to estimate the GIW energy flux into the stableregion adjacent to convective zones. The general frame-work of the convection model is briefly summarized in otating Convection: Penetration & Gravito-inertial Waves §
2. GIWs and their excitation mechanisms are brieflyreviewed in §
3. Following the arguments of Press (1981)and Andr´e et al. (2017), the interfacial generation ofGIWs and their associated energy flux is assessed in §
4. Subsequently, in §
5, an estimate is given for theenergy flux of GIWs excited by Reynolds stresses usingthe convection model. A summary of the results andperspectives are presented in § HEAT-FLUX MAXIMIZED CONVECTIONMODEL2.1.
Hypotheses and Localization
A self-consistent and yet computationally tractabletreatment of stellar and planetary convection has beena long sought goal, with many such models having beenemployed in evolution models. One such model basedupon a variational principle for the maximization of theheat flux (Howard 1963) and a turbulent closure assump-tion for the velocity amplitude (Stevenson 1979) hasbeen expanded upon in Paper I (Augustson & Mathis2019). In the context of GIW excitation, one needs toascertain the amplitude of the velocity field that ex-cites the waves both through Reynolds stresses actingthroughout the bulk of the convection zone on both theevanescent super-inertial waves and propagating inertialwaves and also exciting them directly through thermalbuoyancy in the region of convective penetration.To that end, a local region is considered as in PaperI, where a small 3D section of the spherical geometryis the focus of the analysis. This region covers a por-tion of both the convectively stable and unstable zonesas shown in Figure 1, where the set up is configuredfor a low mass star with an external convective enve-lope. One may exchange these regions when consideringa more massive star with a convective core. In this lo-cal frame, there is an angle between the effective gravity g eff and the local rotation vector that is equivalent tothe colatitude θ . The Cartesian coordinates are definedsuch that the vertical direction z is anti-aligned withthe gravity vector, the horizontal direction y lies in themeridional plane and points toward the north pole de-fined by the rotation vector, the horizontal direction x is equivalent to the azimuthal direction. The angle ψ inthe horizontal plane defines the direction of horizontalwave propagation χ .While the details of the derivation of the heat-fluxmaximized rotating convection model may be found inPaper I, it is necessary to recall a few of the relevantresults as they are applied in subsequent sections. Theheuristic model is local such that the length scales of theflow are much smaller than either the density or pres-sure scale heights, thus ignoring the global-scale flows,which will be the focus of a forthcoming paper. Thedynamics are further considered to be in the Boussi-nesq limit. This localization of the convection there-fore consists of an infinite layer of a nearly incompress-ible fluid with a small thermal expansion coefficient Figure 1.
Coordinate system adopted for the models ofrotating convection and gravity-wave excitation, showing (a)the global geometry and f-plane localization, (b) the f-planegeometry, and (c) the direction χ in the horizontal plane ofthe f-plane. The orange tones denote a convective region andthe yellow tones denote a stable region for late-type stars,and vice versa in early-type stars. α T = − ∂ ln ρ/∂T | P that is confined between two in-finite impenetrable boundaries differing in temperatureby ∆ T = T ( z ) − T ( z c ), with the lower boundary locatedat z and the upper boundary at z . In this model, it isassumed for this model that T ( z ) < T ( z c ) and that theboundaries are separated by a distance (cid:96) = z − z c , asin Figure 1, where z c is the point of transition betweenthe convectively stable and unstable regions.The recent motivation behind the development of theconvection model arose from the numerical work ofK¨apyl¨a et al. (2005) and Barker et al. (2014), where itwas found that the rotational scaling of the amplitudeof the temperature, its gradient, and the velocity fieldcompare well with those derived in Stevenson (1979).Moreover, the experimental work of Townsend (1962)and the analysis of Howard (1963) have shown that aheat-flux maximization principle provides a sound ba-sis for the description of Rayleigh-B´enard convection,leading to its use here. Thus, two hypotheses underliethe convection model: the Malkus conjecture that theconvection arranges itself to maximize the heat flux andthat the nonlinear velocity field can be characterizedby the dispersion relationship of the linearized dynam-ics. Constructing the model of rotating convection thenconsists of three steps: deriving a dispersion relation-ship that links the normalized growth rate ˆ s = s/N ∗ to q = N ∗ , /N ∗ , which is the ratio of superadiabatic-ity of the nonrotating case to that of the rotating case(where N ∗ = | gα T β | is the absolute value of the squareof the Brunt-V¨ais¨al¨a frequency), and to the normalizedwavevector ξ = k /k z , maximizing the heat flux withrespect to ξ , and assuming an invariant maximum heatflux that then closes this three variable system.2.2. Dispersion Relationship and Flux Maximization
For rotating convection, one may show that for impen-etrable and stress-free boundary conditions the solutions
Augustson et al. of the equations of motion are periodic in the horizon-tal, sinusoidal in the vertical, and exponential in time,e.g. v z = v sin [ k z ( z − z c )] exp ( i k ⊥ · r + st ), where k ⊥ is the horizontal wavevector, s is the growth rate, r isthe local coordinate vector, and v is a constant velocityamplitude. To satisfy the impenetrable, stress-free, andfixed temperature boundary conditions, it is requiredthat the vertical wavenumber be k z = nπ/(cid:96) . The in-troduction of this solution into the reduced linearizedequation of motion yields the following dispersion rela-tionship that relates s to the wavevector k as (cid:0) s + κk (cid:1) (cid:0) s + νk (cid:1) k + gα T βk ⊥ (cid:0) s + νk (cid:1) + 4 ( Ω · k ) (cid:0) s + κk (cid:1) = 0 . (1)This equation may be nondimensionalized by dividingthrough by the appropriate powers of N ∗ and k z , leadingto the definition of additional quantitiesˆ s = sN ∗ , ξ = 1 + a = k k z , a = k x k z + k y k z = a x + a y , (2) K = κk z N ∗ , V = νk z N ∗ . Introducing these into the dispersion relationship yields (cid:0) ˆ s + Kz (cid:1)(cid:16) ξ (cid:0) ˆ s + V ξ (cid:1) + O (cos θ + a y sin θ ) (cid:17) − (cid:0) ξ − (cid:1)(cid:0) ˆ s + V ξ (cid:1) = 0 , (3)with 4( Ω · k ) /N ∗ = k z O (cos θ + a y sin θ ) where O = 4Ω N ∗ , (4)where Ω is the bulk rotation rate of the system.The characteristic velocity v of the nonrotatingand nondiffusive case is derived from the growthrate and maximizing wavevector in that case, with s = 3 / | g α T β | = (3 / N ∗ , , β being the ther-mal gradient, g being the effective gravity, and where k = (5 / k z with k z = π/(cid:96) . This leads to v = s k = √ N ∗ , k z = √ π N ∗ , (cid:96) . (5)Thus, the definition of the convective Rossby numberRo c is Ro c = v (cid:96) = √ N ∗ , π Ω , (6)which implies that O = 2Ω N ∗ = v N ∗ Ro c (cid:96) = √ N ∗ , πN ∗ Ro c . (7) The superadiabaticity for this system is (cid:15) = H P β/T ,meaning that N ∗ = | gα T T (cid:15)/H P | , where H P is thepressure scale height. The potential temperature gra-dient in the nonrotating and nondiffusive case is as-certained from the Malkus-Howard turbulence model(Malkus 1954; Howard 1963), which yields a value of N ∗ , . It is also useful to compare the timescales relativeto N ∗ , . Letting the ratio of superadiabaticities be q = N ∗ , /N ∗ , (8)all parametric quantities have the following equivalen-cies O = q √ π Ro c = qO ,K = q κk z N ∗ , = qK , (9) V = q νk z N ∗ , = qV . So, the dispersion relationship (Equation 3) and theheat flux may be written as (cid:0) ˆ s + K qξ (cid:1)(cid:16) ξ (cid:0) ˆ s + V qξ (cid:1) + O q cos θ (cid:17) − (cid:0) ξ − (cid:1)(cid:0) ˆ s + V qξ (cid:1) = 0 , (10) F = F q (cid:20) ˆ s ξ + V q ˆ s (cid:21) , (11)where F = (cid:104) ρ (cid:105) c P N ∗ , / (cid:0) gα T k z (cid:1) .To ascertain the scaling of the superadiabaticity, thevelocity, and the horizontal wavevector with rotationand diffusion, an additional assumption is made to closethe system. This assumption is that the maximum heatflux is invariant to any parameters: max [ F ] = max [ F ] so the heat flux is equal to the maximum value max [ F ] obtained for the nonrotating case, which fits with the as-sumption that the energy generation of the star is notstrongly effected by rotation.In the case of planetary and stellar interiors, the vis-cous damping timescale is generally longer than the con-vective overturning timescale (e.g., V (cid:28) N ∗ , ). Thus,the maximized heat flux invariance is much simpler totreat. In particular, the heat flux invariance conditionunder this assumption is thenmax [ F ]max [ F ] = 256 (cid:114) (cid:20) ˆ s q ξ + V ˆ s q (cid:21) max ≈ (cid:114)
53 ˆ s q ξ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) max = 1 , (12)implying that ˆ s = ˜ sqξ + O ( V ) , (13)where ˜ s = 2 / / − / and max [ F ] = 6 / (cid:112) / F follows from the definition of the flux and the maximiz-ing wavevector used to define v above in Equation 5. otating Convection: Penetration & Gravito-inertial Waves a = k x /k z Maximizing horizontal wavevector k z = π/(cid:96) Maximizing vertical wavevector K = κk z /N ∗ , Normalized thermal diffusivity O = √ / (5 π Ro c ) Normalized Coriolis coefficientRo c = √ N ∗ , / (10 π Ω ) Convective Rossby numberˆ s = s/N ∗ Normalized growth rate v = √ N ∗ , (cid:96) / (5 π ) Velocity of the nonrotating case V = νk z /N ∗ , Normalized viscosity q = N ∗ , /N ∗ Ratio of buoyancy timescales ξ = k /k z Normalized wavevector
Table 1.
Frequently used symbols in the convection model.
Rotational Scaling of Superadiabaticity, Velocity,and Wavevector
The assumption of this convection model is that themagnitude of the velocity is defined as the ratio ofthe maximizing growth rate and wavevector. With theabove approximation, the velocity amplitude can be de-fined relative to the nondiffusive and nonrotating casescales without a loss of generality as vv = k s sk = 5 √ N ∗ N ∗ , ˆ sξ / = 5 √ sqξ / = (cid:18) (cid:19) ξ − . (14)So only the maximizing wavevector needs to be found inorder to ascertain the relative velocity amplitude. Forreference, the symbols that will be frequently used fromthis section are listed in Table 1.With all the equations in hand, the horizontalwavevector may be seen to be the roots of thefourteenth-order polynomial, ξ (cid:0) V ξ +˜ s (cid:1) (cid:2) V K ξ (cid:0) ξ − (cid:1) +˜ sξ ( V + K ) (cid:0) ξ − (cid:1) +˜ s (cid:0) ξ − (cid:1)(cid:3) − θ π Ro (cid:2) s ( K − V ) + 3˜ s ξ +˜ s ( K + 5 V ) ξ + 3 K V ξ (cid:3) = 0 , (15)whereas the superadiabaticity is defined as (cid:15)(cid:15) = (cid:0) ˜ s + K ξ (cid:1)(cid:16) π Ro ˜ s ξ (cid:0) ˜ s + V ξ (cid:1) +6 cos θ (cid:17) π Ro ˜ s ( ξ −
1) (˜ s + V ξ ) . (16)For the study of adiabatic GIWs, the nondiffusivemodel is employed where V → K →
0, lead-ing to 2 ξ − ξ −
18 cos θ π Ro ˜ s = 0 , (17) and (cid:15)(cid:15) = 25 π Ro ˜ s ξ + 6 cos θ π Ro ˜ s ( ξ − . (18)So, to ascertain the maximizing wavenumber, and thusthe velocity and superadiabaticity, of the motions thatmaximize the heat flux one supplies the colatitude θ andthe convective Rossby number of the flow Ro c . Nowthat the quantities related to the convection model havebeen defined, the impact of rotation on the convectiveexcitation of gravito-inertial waves can be characterized. GRAVITO-INERTIAL WAVESWhen examining the excitation of GIWs, the regionof interest is near the radiative-convective interface. Asa first step toward a coherent global treatment of GIWexcitation, the forthcoming analysis will share the sameCartesian geometry as the convection model, which isdepicted in Figure 1 where the stable region is now alsoconsidered. For compactness, one may introduce thetwo components of the rotation vector along the verticaldirection z and the latitudinal direction y as f = 2Ω cos θ, and f s = 2Ω sin θ. (19)As depicted in Figure 1, the waves to be considered prop-agate along a direction with an angle ψ in the horizontal x − y plane, the latitudinal component of the rotationvector has two images in this plane with f sc = 2Ω sin θ cos ψ, and f ss = 2Ω sin θ sin ψ. (20)In this analysis, both components of the rotation vec-tor are kept in the equations of motion, as opposed tothe so-called traditional approximation that considersonly its vertical component in the Coriolis accelerationin order to yield a separable dynamical system. How-ever, in the near-inertial frequency range, nontraditionaleffects act as a singular perturbation. Specifically, thephase of the wave has a vertical dependence that isabsent under the traditional approximation. Also, asshown in Gerkema & Shrira (2005), when consideringa non-constant stratification, sub-inertial GIWs can betrapped in regions of weak stratification. This behaviordoes not arise in the traditional approximation.The near-inertial wave dynamics are quite sensitive tovariations in the effective Coriolis parameters f and f s ,which could arise from a locally strong vortex. For in-stance, the low Rossby number, quasi-geostrophic flowsthat likely exist deep in stellar interiors and that im-pinge upon stable regions could transform a near-inertialwave from the super-inertial regime into the sub-inertialregime. The wave would suddenly find itself trapped in awaveguide, leading to a strong interaction between the Augustson et al. near-inertial waves and large-scale motions. Such no-tions will be considered in a forthcoming investigationof global-scale dynamics.Following Gerkema & Shrira (2005) and Mathis et al.(2014), the linearized equations of motion used to con-struct the convection model above are extended into theradiative region to study the coupling of the convec-tion with both the gravity and inertial waves present inboth regions. Specifically, these equations are Boussi-nesq and in the Emden-Cowling approximation (Emden1907; Cowling 1941), where the gravitational potentialperturbations are ignored, with ∂ t v x − f v y + f s v z = − ∂ x p, (21) ∂ t v y + f v x = − ∂ y p, (22) ∂ t v z + f v x = − ∂ z p + b, (23) ∂ x v x + ∂ y v y + ∂ z v z = 0 , (24) ∂ t b + N R ( z ) v z = 0 , (25)where the buoyancy is b = − g eff ρ (cid:48) ( r , t ) /ρ . One mayeliminate the pressure, buoyancy and the horizontal ve-locities to yield an equation of motion for the verticalcomponent of the velocity as (cid:104) ∂ t ∇ + 4 ( Ω ·∇ ) + N R ∇ ⊥ (cid:105) v z = 0 , (26)where ∇ ⊥ is the horizontal Laplacian and N R ( z ) is theBrunt-V¨ais¨al¨a frequency in the radiative zone. If onethen further considers monochromatic GIWs with a fre-quency ω that propagates along the direction character-ized by the angle ψ in the horizontal plane and a coor-dinate χ = x cos ψ + y sin ψ along that direction as inFigure 1 with a solution of the form v z ( r , t ) = w ( r ) e iωt ,one obtains the Poincar´e equation for the GIWs (cid:0) N R − ω + f ss (cid:1) ∂ χ w + 2 f f ss ∂ χz w + (cid:0) f − ω (cid:1) ∂ z w = 0 . (27)Nominally, this is a nonseparable equation. However, itmay be transformed when assuming the following spatialform of the solution w = (cid:98) w ( z ) e ik ⊥ [ χ + δ (Ro w ) z ] , (28)as in Gerkema & Shrira (2005), where k ⊥ is the wavevec-tor along χ , Ro w = ω/ is the wave Rossby number,and δ (Ro w ) = sin θ cos θ sin ψ Ro − cos θ (29)is the phase shift linking the horizontal and vertical di-rections. Yet the above form of the solution leads to ahomogeneous Schr¨odinger-like equation in the verticalcoordinate as ∂ z (cid:98) w + k V ( z ) (cid:98) w = 0 , (30) where k V ( z ) = k ⊥ (cid:34) N R − ω ω − f + (cid:18) ωf ss ω − f (cid:19) (cid:35) . (31)Similar to the nonrotating case, this permits the use ofthe method of vertical modes to find the modal func-tions (cid:98) w j that satisfy the appropriate boundary condi-tions. Indeed, it can be shown that solutions of the formof Equation 28 constitute an orthogonal and completebasis (Gerkema & Shrira 2005).In convectively stable regions where rotation is impor-tant, GIWs may propagate if their frequency falls withinthe range between ω − and ω + , ω ± = 1 √ (cid:114) N R + f + f ss ± (cid:113) ( N R + f + f ss ) − (2 N R f ) , (32)whereas in convection zones one has that k = k ⊥ (cid:103) Ro w − − − Ro w cos θ ) , (33)where the local wave Rossby number is (cid:102) Ro w = Ro w (cid:112) cos θ +sin θ sin ψ . (34)At the pole in a convectively stable region, this im-plies that the frequency must be between 2Ω and N R for the wave to propagate, where the Brunt-V¨ais¨al¨a fre-quency is typically much larger than the rotational fre-quency in the radiative core of late-type stars and theradiative envelope of early-type stars (e.g., Aerts et al.2010). More generally, at other latitudes, the hierar-chy of extremal propagative wave frequencies satisfy theinequality ω − < < N R < ω + . As these waves prop-agate, the Brunt-V¨ais¨al¨a frequency varies, for instanceit becomes effectively zero in the convection zone. Thisimplies that waves in the frequency range ω < areclassified as sub-inertial GIWs in stable regions, becom-ing pure inertial waves in convective regions. Waves inthe frequency range ω ≥ are classified as super-inertial GIWs in the stable region, which in contrast tosub-inertial GIWs become evanescent in the convectiveregion. Figure 2 in Mathis et al. (2014) provides a con-cise visual reference of the hierarchy of frequencies, towhich the reader is referred. INTERFACIAL GRAVITO-INERTIAL WAVEENERGY FLUX ESTIMATESThere are many models for estimating the magnitudeof the gravity wave energy flux arising from the wavesexcited by convective flows. One of the first and moststraightforward of such estimates is described in Press otating Convection: Penetration & Gravito-inertial Waves π/k c and a single time scale for the convection 2 π/ω c that also selects the depth of the transitional interfacewhere N R ( r ) = ω c for gravity waves, where ω c = ω / √ ξ with ω = 2 πv /(cid:96) , which lends itself well to the aboveconvection model. This approach yields a wave energyflux proportional to the product of the convective ki-netic energy flux and the ratio of the wave frequency tothe Brunt-V¨ais¨al¨a frequency in the nonrotating case forgravity waves.The convective model established above captures someaspects of the influence of rotation on the convectiveflows. Therefore, the impact of the Coriolis force on thestochastic excitation of GIWs can be evaluated. In thiscontext, recent work has established an estimate of theGIW energy flux (Andr´e et al. 2017). It can be usedto estimate the rotational scaling of the amplitude ofthe wave energy flux arising from the modified prop-erties of the convective driving. From Equation 61 ofAndr´e et al. (2017), the vertical GIW energy flux canbe computed from the horizontal average of the productof the vertical velocity and pressure perturbation that,given the linearization of the Boussinesq equations formonochromatic waves propagating in a selected horizon-tal direction, can be evaluated to be F z = 12 ρ ω − f ωk ⊥ k z v w , (35)where v w is the magnitude of the vertical velocity of thewave. Moreover, the solution for the vertical velocityimplies that the dispersion relationship is k z k ⊥ = (cid:34) N R − ω ω − f + (cid:18) ωf ss ω − f (cid:19) (cid:35) , (36)where f and f ss are defined above in Equation 20. Notethat a reference table is given to help identify the manyparameters in this section (Table 2).Following P81, further assumptions are necessary tocomplete the estimate of the wave energy flux. The con-vection is turbulent. So the fluctuating part of the ve-locity field is of the same order of magnitude as theconvective eddy turnover velocity v ≈ ω c /k c , which im-plies that convective pressure perturbations are approxi-mately P c = ρ v . Assuming that the pressure is contin-uous across the interface between the convectively stableand unstable regions, the horizontally-averaged pressureperturbations of the propagating waves excited at the χ = cos ψx + sin ψy Horizontal position δ (Ro w ) = sin θ cos θ sin ψ Ro − cos θ Horizontal phase shift∆ j = (cid:82) z c z j dk ⊥ | k V | Vertical phase shift k ⊥ = cos ψk x + sin ψk y Horizontal wavevector k V Vertical wavevector f = 2Ω cos θ V. Coriolis frequency f s = 2Ω sin θ H. Coriolis frequency f ss = 2Ω sin θ sin ψ P. H. Coriolis frequency f sc = 2Ω sin θ cos ψ P. H. Coriolis frequency N R Brunt-V¨ais¨al¨a frequency ω c = √ vN ∗ , / (5 πv ) Convective frequency ω e Eddy timescale ψ Angle in horizontal planeRo w = ω/ = 5 πσ Ro c S/ √ (cid:102) Ro w = Ro w (cid:112) cos θ +sin θ sin ψ Local Rossby number S = N R /N Interface stiffness S Convective source σ = ω/N R Normalized frequency
Table 2.
Frequently used symbols in the models ofinterfacially-excited and Reynolds-stress-induced wave exci-tation. The abbreviations used in the table are V. for Verti-cal, H. for Horizontal, and P. H. for Projected Horizontal. interface must then be equal to the turbulent pressureon the convective side of the interface. Those pressureperturbations follow from the solution for the vertical ve-locity and the nondiffusive Boussinesq equations (Andr´eet al. 2017). For plane wave solutions, the magnitude ofthose perturbations can then be written as P = ρ v w k ⊥ ω (cid:20) ω f sc + (cid:0) ω − f (cid:1) k z k ⊥ (cid:21) . (37)Note however, the pressure matching condition fails forthese modes at the pole for sub-inertial waves ( ω → f ).The reason is that the propagation domain of sub-inertial GIWs excludes the pole and it becomes increas-ingly concentrated toward the equator for faster rotationrates (e.g., Dintrans & Rieutord 2000; Prat et al. 2016).Using the dispersion relationship, and equating the twopressures, yields the following equation for the verticalwave velocity v w = ωk ⊥ v (cid:2) ω f s + (cid:0) N R − ω (cid:1) (cid:0) ω − f (cid:1)(cid:3) − , (38)Therefore, the wave energy flux density becomes F z = 12 ρ v k ⊥ ω (cid:2) ω f ss + (cid:0) N R − ω (cid:1) (cid:0) ω − f (cid:1)(cid:3) ω f s + ( N R − ω ) ( ω − f ) . (39) Augustson et al.
Flows in a gravitationally stratified convective mediumtend to have an extent in the direction of gravity thatis much larger than their extent in the transverse direc-tions. Therefore, the horizontal wavenumber of the con-vective flows is much greater than the vertical wavenum-ber. This implies that k ⊥ ,c ≈ ω c /v . For efficient waveexcitation, the frequency of the wave needs to be close tothe source frequency (Press 1981; Lecoanet & Quataert2013), which means that the horizontal scale of thewaves will be similar to that of the convection. Moregenerally, there will be a distribution of excitation effi-ciency as a function of the wave frequency ω , which maybe peaked near the convective overturning frequency ω c . However, since this distribution is unknown, thefull frequency dependence is retained. This assumptionsimplifies the wave energy flux to F z ≈ ρ v ω (cid:2) ω f ss + (cid:0) N R − ω (cid:1) (cid:0) ω − f (cid:1)(cid:3) ω f s + ( N R − ω ) ( ω − f ) . (40)In this case, the nonrotating wave energy flux estimatefound in P81 can be recovered when letting Ω → F ≈ ρ v ω N R − ω ) . (41)Finally, taking the ratio of the two energy fluxes tobetter isolate the changes induced by rotation, assum-ing that the Brunt-V¨ais¨al¨a frequency is not directly im-pacted by rotation, and at a fixed wave frequency ω , onehas that F z F ≈ (cid:18) vv (cid:19) ω (cid:8)(cid:0) N R − ω (cid:1)(cid:2) ω f ss + (cid:0) N R − ω (cid:1) (cid:0) ω − f (cid:1)(cid:3)(cid:9) ω f s +( N R − ω ) ( ω − f ) . (42)To make this a bit more parametrically tractable, onecan normalize the wave frequency as σ = ω/N R , andcast the rotational terms into a product of the stiffness ofthe transition S = N R /N , with the convective Rossbynumber of the convection zone as defined above in § F z F ≈ (cid:18) vv (cid:19) (cid:2) Ro − sin θ + (cid:0) σ − − (cid:1)(cid:0) − Ro − cos θ (cid:1)(cid:3) − (cid:104)(cid:0) σ − − (cid:1) Ro − sin θ sin ψ + (cid:0) σ − − (cid:1) (cid:0) − Ro − cos θ (cid:1)(cid:105) , (43)where Ro w = ω/ = 5 πσ Ro c S/ √ (cid:102) Ro w = Ro w / (cid:112) sin θ sin ψ + cos θ .This is depicted in Figures 2 and 3, where the coloredregion exhibits the magnitude of the logarithm of the en-ergy flux ratio and the energy flux itself. An interfacialstiffness of S = 10 is chosen as it is a rough estimateof the potential stiffness in most stars, being the ratio - - - - - - - - - Ro c σ Log F z / F - - - - - - - Figure 2.
Convective Rossby number dependence of the ra-tio of the interfacial gravito-inertial wave energy flux excitedby rotating convection relative to the nonrotating case F z /F for the nondiffusive convection model near the equator, withan interface stiffness of S = 10 and a horizontal direction of ψ = π/
2. The red dashed line indicates the lower frequencycutoff σ − , the green dashed line indicates the upper cutofffrequency of σ = 1 since the wave energy flux is being com-pared to a non-rotating case, whereas the blue dashed lineindicates a wave Rossby number of (cid:102) Ro w = 1. The verticaldashed orange line indicates the critical convective Rossbynumber. of the buoyancy time-scale in the stable region to theconvective overturning time. The choice of latitude de-termines the width of the frequency band of sub-inertialwaves, where it is a minimum at the pole and maxi-mum near the equator. This is due to the presence ofa critical latitude of the gravito-inertial waves, wheresub-inertial waves become evanescent (cos θ c = Ro w2 ).The direction of ψ = ± π/ ω ± , there are no sub-inertial waves in theprograde or retrograde propagation case ( ψ = { , π } ,respectively), whereas the super-inertial waves may stillpropagate with roughly the same frequency range. Thewhite region corresponds to the domain of evanescentwaves for a given convective Rossby number with fre-quencies below the lower cut-off frequency ( σ − , dashedred line) for propagating GIWs. At frequencies above otating Convection: Penetration & Gravito-inertial Waves - - - - - - - - - Ro c σ Log F z - - - - - - - Figure 3.
Convective Rossby number dependence of the in-terfacial gravito-inertial wave energy flux normalized by thenon-rotating convective flux excited by rotating convection F z for the nondiffusive convection model, with parametersas in Figure 2, showing the scaling of the flux for σ > this threshold there is a frequency dependence of theenergy flux ratio until reaching the upper cut-off where σ = ω/N R = 1, which arises due to the domain ofvalidity when comparing GIW to gravity wave energyfluxes. Indeed, gravity waves may propagate if ω < N R ,whereas super-inertial GIWs may propagate even when N R < ω < ω + . The transition between super-inertialand sub-inertial waves is demarked with the dashed blueline, with super-inertial waves for (cid:102) Ro w > (cid:102) Ro w <
1. Here, interfacially-excitedsuper-inertial waves exhibit both a frequency and con-vective Rossby number dependence. Specifically, thewave energy flux decreases algebraically with frequencyat a fixed convective Rossby number and have a re-duced energy flux for convective Rossby numbers be-low unity. The interfacially-excited sub-inertial wavespossess a small frequency domain at a fixed convec-tive Rossby number over which they are propagative.The sub-inertial wave energy flux increases with decreas-ing convective Rossby number until a critical convec-tive Rossby number Ro c, crit = √ / (5 πS ) as depictedby the vertical dashed orange line in Figure 2. Belowthis critical convective Rossby number, the sub-inertialwave energy flux decreases and their frequency domainis further restricted until it vanishes entirely and thereare no propagative super-inertial waves . The effect ofthe stiffness is to lower (raise) the value of the criticalconvective Rossby number for larger (smaller) values of S , which corresponds to the ratio of the buoyancy time-scale in the radiative zone to the convective overturn- age [yr] − − − − − − R o c M ? [ M (cid:12) ] =0.70.91.01.1 1.21.31.41.5 Figure 4.
Variation of the convective Rossby number atthe base of the convective envelope of low-mass stars (from0.7 to 1.5 M (cid:12) ) along their evolution. The critical convectiveRossby numbers are shown for which a potential increase ofthe excitation rate of GIWs (when compared to the one ofpure IGWs) is expected. The purple dashed-dot line cor-responds to the case of their interfacial excitation and thedashed gray line to the case of the excitation triggered byReynolds stresses. ing time. This may have important consequences forthe wave-induced transport of angular momentum dur-ing the evolution of rotating stars. In particular, theconvective Rossby number can vary by several orders ofmagnitude over a star’s evolution from the PMS to itsultimate demise (e.g. Landin et al. 2010; Mathis et al.2016; Charbonnel et al. 2017). Moreover, it can varyinternally as a function of radius due to the local am-plitude of the convective velocity and due to transportprocesses, angular momentum loss through winds, andstructural changes that modify the local rotation rate(Mathis et al. 2016). Figure 4 presents the variation ofthe convective Rossby number at the base of the con-vective envelope of low-mass stars (from 0.7 to 1.5 M (cid:12) )throughout their evolution. These convective Rossbynumbers have been calculated using grids of stellar mod-els that take into account rotation computed with theSTAREVOL code (Siess et al. 2000; Palacios et al. 2003;Decressin et al. 2009; Amard et al. 2016). The detailsof the micro- and macro-physics used for these gridsare described in Amard et al. (2019). The dot-dashedpurple line provides the value of the critical convectiveRossby number (Ro c, crit = √ / (5 πS ), shown here for S = 10 , for which an increase of the interfacial excita-tion of GIWs can be expected. For all the stars consid-ered here, which have a median initial rotation (i.e. 4.5days), this should happen during their PMS.0 Augustson et al.
Figure 5.
Scaling of the relative flux integrated over θ , ψ ,and frequency with respect to the ratio of the rotation rate Ωto the break-up rotation rate Ω b , with the stiffness S = 10 and a minimum Rossby number Ro b = 10 − at Ω b , showingthe peak at Ω / Ω b ≈ . The flux ratio F z /F integrated over latitude θ , prop-agation direction ψ , and frequency is shown in Figure5. This illustrates the general rotational trend of theinterfacial flux, namely that it decreases with increas-ing rotation rate. However, there is a peak at a rota-tional frequency that depends upon the choice of stiff-ness S and the Rossby number of the convection at thebreakup velocity Ro b . Thus, for stars with a modestrotation rate below approximately 0 . b , the interfa-cial or pressure-driven GIW wave flux could play a rolein transport processes that is at least as important asthe transport by IGWs. Yet, for more rapidly rotatingstars, this flux becomes fairly negligible due primarilyto the reduction in the convective velocity amplitudes.Nevertheless, given the complex and nonanalytic formof the full integral, the exploration of the parameter de-pendence of this peak will be left for future work. Asa means of comparison, consider the spherical Couetteflow laboratory experiments of Hoff et al. (2016), whereit is found that the kinetic energy of the dominant iner-tial mode increases with decreasing wave Rossby num-ber. Below a critical wave Rossby number this leads to awave breaking and an increase of small-scale structuresat a critical Rossby number, which may be similar to thelarge increase of wave energy flux for sub-inertial wavesbelow the critical convective Rossby number describedabove. REYNOLDS STRESS CONTRIBUTIONS TOGIW AMPLITUDESAs a means of comparison, the amplitude and the waveenergy flux of the GIWs may be computed exactly whenusing the convection model presented earlier, where theimpact of rotation on the waves is treated coherently. In a means similar to Goldreich & Kumar (1990) andLecoanet & Quataert (2013), although with a greaterdegree of computational complexity, one may derive thewave amplitudes for GIWs in a f-plane. As seen inMathis et al. (2014), one must first find solutions tothe homogeneous Poincar´e equation for the GIWs andthen use linear combinations of those solutions to con-struct solutions to the forced equation in the convectionzone. These equations result from writing the linearizedequations of motion in a f-plane as a single equation forthe vertical velocity W as (cid:104) ∂ tt ∇ + 4 ( Ω ·∇ ) + N ∇ ⊥ (cid:105) W = ∂ t S , (44)where S is the convective source term described in de-tail below. Note that the thermal sources derived inSamadi & Goupil (2001) have been neglected here asin Mathis et al. (2014), for they have been found to becomparatively small for gravity waves when compared toReynolds stresses (Belkacem et al. 2009b). In addition,the damping mechanisms (i.e. the radiative dampingand the damping due to convection-wave interactions)are neglected here. As pointed out in Samadi, R. et al.(2015), the value of the amplitude of stochastically ex-cited waves is proportional to the ratio of the energyinjection rate that measures the efficiency of the cou-plings of turbulent motions with waves and of the damp-ing. The focus of this work is on the energy injectionrate while getting a coherent treatment of the turbulentdamping of waves in rotating stars will be considered inforthcoming work.In the stable region, where S is assumed to vanish, itcan be shown that if one follows the methodology of con-structing normal modes as in Gerkema & Shrira (2005)then the solutions of the homogeneous Poincar´e equa-tion for GIWs may be expanded as w = w( χ, z ) e iωt . Thehorizontal coordinate χ = cos ψx + sin ψy correspondsto the distance along the direction of the wave propaga-tion with an angle ψ in the horizontal plane as seen inGerkema & Shrira (2005) and Figure 1 of Mathis et al.(2014). Therefore, as before, the solution of the forcedPoincar´e equation for GIWs in the convection zone maybe expanded as W ( χ, z, t ) = (cid:88) n A n ( t ) w n ( χ, z ) e iωt (45)= (cid:88) n A n ( t ) ψ n ( z ) e ik n ( χ + δz )+ iωt , (46)where ω is the chosen frequency, with k n being the se-quence of eigenvalues associated with it, the ψ n are theeigenmodes of the reduced Poincar´e equation (see Equa-tion 30), and A n is its amplitude. Technically, the fullvelocity field would be an integral over all frequenciesand the sum over modes associated with each frequency.However, for simplicity, this discussion will at first fo-cus on a single frequency taken to be within the band of otating Convection: Penetration & Gravito-inertial Waves (cid:88) n (cid:8) [ ∂ tt A n + 2 iω∂ t A n ] (cid:2) ∂ zz − k n (cid:3) + A n (cid:2)(cid:0) f − ω (cid:1) ∂ zz +2 if f ss k n ∂ z + k n (cid:0) ω − N − f ss (cid:1)(cid:3)(cid:9) w n e iωt = ∂ t S . (47)Noting that the second term is simply the homogeneousequation, it vanishes, leaving (cid:88) n [ ∂ tt A n + 2 iω∂ t A n ] × (cid:2) ∂ zz ψ n + 2 ik n δ∂ z ψ n − k n (cid:0) δ + 1 (cid:1) ψ n (cid:3) × e ik n ( χ + δz )+ iωt = ∂ t S . (48)Utilizing Equation 30, this becomes (cid:88) n [ ∂ tt A n + 2 iω∂ t A n ] × (cid:34) ik n δ∂ z ψ n − k n (cid:32) f − N f − ω + (cid:0) ω + f (cid:1) f ss ( f − ω ) (cid:33) ψ n (cid:35) × e ik n ( χ + δz )+ iωt = ∂ t S . (49)Assuming homogeneous Dirichlet boundary conditionson ψ n and that the change in the amplitudes at infinityare zero, with an initial condition of being zero, this canbe integrated against a single conjugate mode of index m to see that the constant amplitude is (cid:104) A n (cid:105) = i (cid:82) L dz (cid:82) ∞−∞ dχ (cid:82) ∞−∞ dt∂ t S ψ ∗ n e − ik n ( χ + δz ) − iωt ωk n (cid:82) L dz (cid:16) f − N f − ω + ( ω + f ) f ss ( f − ω ) (cid:17) | ψ n | , (50)where the normalization c n follows from the orthogonal-ity condition on the ψ n , c n δ nm = 1 L (cid:90) L dz (cid:32) ω − N f − ω + ω f ss ( f − ω ) (cid:33) ψ ∗ m ψ n . (51)where L = z − z is the depth of the domain. Theconvective source term is S = ∂ z ∇ · F − ∇ F z = ∂ z ∇ ⊥ · F − ∇ ⊥ F z , (52)where F = ∇ · ( v ⊗ v ) are the Reynolds stresses due tothe convective velocities v . This can be further simpli-fied noting the definition of the perpendicular direction,yielding S = ∂ χz F χ − ∂ χ F z , (53)= ∂ χχz (cid:0) v χ − v z (cid:1) + ( ∂ χzz − ∂ χχχ ) v χ v z . (54) The integral in the numerator of Equation 50 can beidentified as a Fourier transform of the source in timeand space. Treating it as such, it becomes (cid:90) L dz (cid:90) ∞−∞ dχ (cid:90) ∞−∞ dt∂ t S ψ ∗ n e − ik n ( χ + δz ) − iωt = ω (cid:90) L dz (cid:104) ik n ∂ z (cid:16)(cid:102) v z − (cid:102) v χ (cid:17) − k n (cid:0) ∂ zz + k n (cid:1) (cid:103) v χ v z (cid:105) ψ ∗ n e − ik n δz . (55)In turn this is a Fourier transform of a product, or a con-volution in spectral space of the Reynolds stress with aHeaviside function H that confines the convection to athe convective region and the reduced eigenmodes. Un-der this approach, the previous equation yields ω (cid:90) ∞−∞ dk (cid:48) (cid:104) k (cid:48) k n (cid:16)(cid:99) v z ( k (cid:48) ) − (cid:99) v χ ( k (cid:48) ) (cid:17) + k n (cid:0) k (cid:48) − k n (cid:1) (cid:100) v χ v z ( k (cid:48) ) (cid:3) (cid:91) Hψ ∗ n ( k n δ − k (cid:48) ) . (56)Assuming henceforth that the Brunt-V¨ais¨al¨a fre-quency is a discontinuous jump of an amplitude N = Sω c , there is an exact solution for all three waveclasses, sub-inertial, inertial, and super-inertial. Thisassumption provides an approximation of the stratifi-cation in a star, but captures its order of magnitudeeffects. This means that all integrals except the oneof the Reynolds stresses can be evaluated. The latterdepends upon the turbulence model that is chosen. Theone introduced at the beginning of this paper will beexamined here. Specifically, with this choice of N , thereduced Poincar´e equation becomes (cid:40) ∂ zz ψ n + k n α ψ n = 0 0 ≤ z < (cid:96) s ∂ zz ψ n + k n β ψ n = 0 (cid:96) s ≤ z ≤ L , (57)where α = N − ω ω − f + ω f ss ( ω − f ) , (58) β = ω f ss ( ω − f ) − ω c + ω ω − f , (59)and where ω c is the convective overturning time and (cid:96) s = z c − z is the depth of the radiative-convectiveinterface. The boundary conditions are that ψ n (0) = ψ n ( L ) = 0, and with matching conditions and momen-tum continuity at the interface leading to the dispersionrelationship. With these choices, above equations admitthe following solutions for the sub-inertial waves ψ n = − sin ( k n β ( L − (cid:96) s ))cos ( k n βL ) sin ( k n α(cid:96) s ) sin ( k n αz ) 0 ≤ z < (cid:96) s sin ( k n βz ) − tan ( k n βL ) cos ( k n βz ) (cid:96) s ≤ z ≤ L , (60)with a dispersion relationship2
Augustson et al. α tan [ k n β ( L − (cid:96) s )] + β tan [ k n α(cid:96) s ] = 0 . (61)Similarly, the super-inertial waves are ψ n = − sinh ( k n β ( L − (cid:96) s ))cosh ( k n βL ) sin ( k n α(cid:96) s ) sin ( k n αz ) 0 ≤ z < (cid:96) s sinh ( k n βz ) − tanh ( k n βL ) cosh ( k n βz ) (cid:96) s ≤ z ≤ L , (62)with a dispersion relationship α tanh [ k n β ( L − (cid:96) s )] + β tan [ k n α(cid:96) s ] = 0 . (63)Note that with sin ( k n α(cid:96) s ) in the numerator, there arecertain values of k n α(cid:96) s = mπ with m some integer wherethis solution is invalid. This provides an additional se-lection criterion on the values of S that have solutions.Note that the inertial waves already are normalized with c n = 1. The integrals for the denominator in Equation50 are very similar.Finally, from Equation 62 in Andr´e et al. (2017), thevertical wave flux for a single mode is given as F z = ρ (cid:18) f − ω ωk n (cid:19) (cid:12)(cid:12) A n ψ n ∂ z ψ n (cid:12)(cid:12) . (64)Note that this definition of the flux is slightly differ-ent from the interfacial flux, which used an approxima-tion of the pressure. Moreover, that interfacial flux isa local model with driving taking place only at the in-terface, whereas the current model assesses wave driv-ing throughout the convective zone. The definition ofthe flux given in Equation 64 is consistent with previ-ous studies of gravity wave driving in the bulk of con-vective regions (e.g., Press 1981; Goldreich & Kumar1990; Lecoanet & Quataert 2013), where it is seen thatthe flux for a discontinuous Brunt-V¨ais¨al¨a frequency is F z ≈ F c S − , where F c is the convective flux. This scal-ing can be obtained using Equation 64 if one considersthe regime of low-frequency gravity waves in the non-rotating case (where f = 0), which are described withinthe JWKB approximation, assuming that their horizon-tal velocity is v h ≈ v c = ω c /k c , where v c , ω c , and k c are the convective velocity, frequency, and wavevector,respectively, while one also sets ω ≈ ω c and k ≈ k c .Note that within these assumptions the influence of thespatial behaviour of the eigenmodes is not taken intoaccount.Thus, with the definition of the amplitude, the flux is F z = ρ (cid:0) f − ω (cid:1) | ψ n ∂ z ψ n | ω k n × (cid:12)(cid:12)(cid:12)(cid:82) L dz (cid:82) ∞−∞ dχ (cid:82) ∞−∞ dt∂ t S ψ ∗ n e − ik n ( χ + δz ) − iωt (cid:12)(cid:12)(cid:12) (cid:32)(cid:82) L dz (cid:32) f − N f − ω + (cid:0) ω + f (cid:1) f ss ( f − ω ) (cid:33) | ψ n | (cid:33) . (65) Averaging over the stable region, this becomes F z = ρ (cid:0) f − ω (cid:1) ψ n ( (cid:96) s )8 ω k n (cid:96) s × (cid:12)(cid:12)(cid:12)(cid:82) L dz (cid:82) ∞−∞ dχ (cid:82) ∞−∞ dt∂ t S ψ ∗ n e − ik n ( χ + δz ) − iωt (cid:12)(cid:12)(cid:12) (cid:32)(cid:82) L dz (cid:32) f − N f − ω + (cid:0) ω + f (cid:1) f ss ( f − ω ) (cid:33) | ψ n | (cid:33) . (66)The integral in the denominator of the wave flux forthe sub-inertial and super-inertial waves are thus D n = sec ( k n Lβ ) × (cid:34) sin ( k n β(cid:96) )sin ( k n α(cid:96) s ) (cid:32) f − N f − ω + (cid:0) ω + f (cid:1) f ss ( f − ω ) (cid:33) × (cid:18) (cid:96) s − sin (2 k n α(cid:96) s )4 k n α (cid:19) + (cid:32) f + ω c f − ω + (cid:0) ω + f (cid:1) f ss ( f − ω ) (cid:33) (cid:18) (cid:96) − sin (2 k n β(cid:96) )4 k n β (cid:19)(cid:35) , (67)for the sub-inertial waves and D n = sech ( k n Lβ ) × (cid:34) sinh ( k n β(cid:96) )sin ( k n α(cid:96) s ) (cid:32) f − N f − ω + (cid:0) ω + f (cid:1) f ss ( f − ω ) (cid:33) × (cid:18) (cid:96) s − sin (2 k n α(cid:96) s )4 k n α (cid:19) + (cid:32) f + ω c f − ω + (cid:0) ω + f (cid:1) f ss ( f − ω ) (cid:33) (cid:18) (cid:96) k n β(cid:96) )4 k n β (cid:19)(cid:35) , (68)for the super-inertial waves.The integral in the numerator can be computed ex-actly for the convection model discussed in Section 2.Specifically, using the definition of the velocities there,e.g. v z = v (Ro c ) sin (cid:18) πz(cid:96) (cid:19) e ( ik ⊥ χ + iω c t ) , (69)v χ = iπ(cid:96) k ⊥ v (Ro c ) cos (cid:18) πz(cid:96) (cid:19) e ( ik ⊥ χ + iω c t ) , (70)as follows from the continuity equation for the convec-tion model. These integrals are otating Convection: Penetration & Gravito-inertial Waves - - - - Ro c F z F ( a ) Subinertial SuperinertialS = - - Ro c ( b ) Subinertial SuperinertialS = - - Ro c ( c ) Subinertial SuperinertialS =
10 10 - - - - - SF ( d ) Figure 6.
Scaling of the gravito-inertial wave flux F z , normalized by the gravity wave flux for the non-rotating case F , whenexcited by columnar convection at the equator for waves, where the stiffness is the ratio Brunt-V¨ais¨al¨a frequency to the rotationfrequency S = N R /N is taken to be (a) 10 , (b) 10 , and (c) 10 and where (cid:96) = (cid:96) s = L/
2, and ψ = π/
2. The vertical dashedline denotes the transition between sub-inertial and super-inertial waves and the horizontal line denotes unity. (d) illustratesthe scaling of the pure gravity wave flux ( F ) normalized by the total convective flux with the stiffness parameter, showing thatthe wave flux is always below the convective flux, but that the gravito-inertial wave flux can be greatly amplified in comparison.Note that such mode amplification of GIWs has also been seen in a global model (see Figure 7 Neiner et al. 2020) π v sec (2 k ⊥ Lβ ) (cid:0) k ⊥ (cid:96) − π (cid:1) (cid:96) (cid:104) π − π k ⊥ (cid:96) ( β + δ ) + k ⊥ (cid:96) ( β − δ ) (cid:105) × (cid:2) π − π k ⊥ (cid:96) (cid:0) β + δ (cid:1) + k ⊥ (cid:96) ( β + 14 β δ + δ )+ 4 βδk ⊥ (cid:96) (cid:16) k ⊥ (cid:96) ( β − δ ) − π (cid:17) × (cos (2 k ⊥ (cid:96) ( β + δ )) + cos (2 k ⊥ (cid:96) ( β − δ ))) − (cid:16) π − k ⊥ (cid:96) ( β + δ ) (cid:17)(cid:16) π − k ⊥ (cid:96) ( β − δ ) (cid:17) cos (4 k ⊥ (cid:96) β ) (cid:105) , (71)for the sub-inertial waves where the horizontal and timeintegrals impose ω = 2 ω c and k ⊥ = k n /
2. For the super-inertial waves, this is − π v sech ( k ⊥ Lβ ) (cid:0) k ⊥ (cid:96) − π (cid:1) (cid:96) (cid:104) π + 2 π k ⊥ (cid:96) ( β − δ ) + k ⊥ (cid:96) ( β + δ ) (cid:105) × (cid:2) π + 2 π k ⊥ (cid:96) (cid:0) β − δ (cid:1) + k ⊥ (cid:96) ( β − β δ + δ )+ 16 β δ k ⊥ (cid:96) cos (2 k ⊥ (cid:96) δ ) cosh (2 k ⊥ (cid:96) β ) − βδk ⊥ (cid:96) (cid:0) π + k ⊥ (cid:96) (cid:0) β − δ (cid:1)(cid:1) × sin (2 k ⊥ (cid:96) δ ) sinh (2 k ⊥ (cid:96) β ) − (cid:16) k ⊥ (cid:96) β + ( π + k ⊥ (cid:96) δ ) (cid:17) × (cid:16) k ⊥ (cid:96) β + ( π − k ⊥ (cid:96) δ ) (cid:17) cosh (4 k ⊥ (cid:96) β ) (cid:105) . (72)These waves will attain a maximum flux near theequator, especially for low convective Rossby numberwhere the waves become increasingly equatorially fo-cused. Thus, evaluating these expressions at the equa-tor, one has a wave flux analogous to the section on in-terfacial waves, but excited by the Reynolds stresses inthe bulk of the convection zone. Figure 6 illustrates thisflux for several values of the stiffness, where each valueof the convective Rossby number is computed such thatthe dispersion relationships are obeyed, leading to its discrete nature. The sub-inertial waves have an oscilla-tory character, where some waves achieve a resonanceand have a peak in flux. The peak flux arises at mod-erate convective Rossby numbers below 1 / √
5, due to α being small and transitioning from super-inertial to sub-inertial waves. The decay of the flux at lower convectiveRossby numbers results from the weakening convectivevelocities and the increasing horizontal wavenumber ofthe convection. The peak in the super-inertial wavesalso occurs near Ro c = 1 / √
5, above which it decays pri-marily due to the scaling of the denominator of the flux,which arises from the hyperbolic trigonometric functionsin the structure of the eigenmodes and it asymptotes tothe flux of pure gravity waves driven by nonrotating con-vection. When considering Figure 4 and also Figure 4in Mathis et al. (2016), where Ro c = 1 / √ c during these evolutionary phases(and in these regions).The actual value of the both the nonrotating and ro-tating fluxes are very dependent upon the value of thestiffness chosen due to the dependence on the average ofthe eigenfunction in the stable region in the numeratorand the normalization in the denominator of the flux de-rived above. The non-rotating wave flux normalized bythe total convective flux is shown in Figure 6(d). Notethat this flux F differs from that of Section 4 becausethe flux defined in Equation 64 has a complex spatialdependence. When averaged over the stable region, thisyields F ∝ Q ( S ) S − where Q ( S ) is the dependencearising from the integral of the source term and the aver-age value of | ψ n ∂ z ψ n | in the radiative region (see Equa-tion 65). However, if one makes the assumptions ex-plained in the Appendix, where the spatial dependenceof the eigenfunctions and their dispersion relationshipbecome simple and continuous (see Appendix), one re-covers F ∝ S − . Thus, if one makes similar assump-4 Augustson et al. tions for the gravito-inertial waves, then F z /F ∝ F z /F ∝ Q ( S ) S − as seenin Figure 6. Finally, both the flux of the IGWs and theGIWs are always weaker than the total convective flux. SUMMARY AND DISCUSSIONA model of rotating convection originating withStevenson (1979) has been extended to include ther-mal and viscous diffusions for any convective Rossbynumber in Augustson & Mathis (2019). The scaling ofthe velocity and superadiabaticity in terms of the co-latitude, and Rossby number are outlined in Section 2.Asymptotically at low convective Rossby number andwithout diffusion, these match the expressions given inStevenson (1979), as well as the numerical results foundin the 3D simulations of K¨apyl¨a et al. (2005) and Barkeret al. (2014).Here this rotating convection model has been em-ployed to examine the excitation of gravito-inertialwaves (GIWs) by two different channels: one by in-terfacial excitation and another by Reynolds-stressexcitation. First, the convection model is applied tothe interfacial wave excitation paradigm developed inPress (1981), where the gravity wave dynamics thereis replaced with the GIW wave dynamics computed inMathis et al. (2014) and Andr´e et al. (2017). Bothmechanisms are considered since, as seen in Lecoanetet al. (2015), both sources of wave excitation play a rolein simulations of gravity wave excitation, with the dom-inant one being due to the volume integrated Reynoldsstresses. Next, with a turbulent convective velocityspectrum in hand, more sophisticated approaches al-low for the computation of the wave energy flux inthe context of both more realistic variations in theBrunt-V¨ais¨al¨a frequency as well as in a non-interfacialparadigm that includes the Reynolds stresses through-out the convection zone. Such a step has been taken inthis paper, which builds upon the methods developed inBelkacem et al. (2009a), Lecoanet & Quataert (2013),Mathis et al. (2014), where the gravity wave and GIWexcitation amplitudes and accompanying wave energyinjection rate are computed by solving the wave equa-tion driven by a convective source term. This approachprovides a general method of computing the wave fluxthat takes into account the volumetric excitation of thewaves and that includes the region in which they are po-tentially evanescent. Specifically, to assess the influenceof the convective Reynolds stresses and of rotation onthe GIWs, a wave energy flux estimate is constructedusing an explicit computation of the amplitude for boththe super-inertial and sub-inertial waves. The con-vection model of Paper I is then invoked as means ofestimating the Reynolds stresses.In the context of the wave energy flux, distinct pa-rameter regimes have been found that depend upon the mode of excitation (either interfacial pressure pertur-bations or convective Reynolds stresses), the convectiveRossby number (or alternatively the rotation rate), andthe stiffness of the convective-radiative interface. Thevisibility of these regimes depends upon the colatitudeselected, with the distinction between them being stark-est at low latitudes near the equator and vanishing atthe poles due to impact of the Coriolis acceleration onthe frequency range over which GIWs may propagate.As depicted in Figure 2, interfacially-excited sub-inertialwaves have a peak energy flux near a critical convec-tive Rossby number, but decay below it. Interfacially-excited super-inertial waves, on the other hand, havean increasing energy flux with increasing frequency andincreasing Rossby number. As a means of comparison,the influence of convective Reynolds stresses on the waveamplitude and their energy flux has been assessed by di-rectly employing the convection model of Paper I. Thedetailed behavior of the eigenfunctions appropriate forGIWs and how they interact with the convective sourceis examined in 5. A trend similar to that of interfacialwaves is found where there is a decline in the amplitudeof the fluxes is found as the convective Rossby numberis decreased for both the sub and super-inertial waves.However, there is a large variation in the sub-inertialwave flux for a given convective Rossby number, depend-ing upon whether wave is in resonance or not, leadingto the series of peaks seen in Figure 6 where the fluxrelative to gravity waves in nonrotating convection canbe many orders of magnitude larger, but still below theconvective flux. The amplitude of the nonrotating fluxis computed using the same mathematical formalism asthe gravito-inertial waves, but utilizing the proper eigen-modes. The super-inertial waves have an increased fluxat lower Rossby numbers reaching a peak at the transi-tional Rossby number of 1 / √ § § otating Convection: Penetration & Gravito-inertial Waves A. GRAVITY WAVE FLUX IN THE NONROTATING AND LOW-FREQUENCY LIMITWith the appropriate limit and the assumptions made in Goldreich & Kumar (1990) and Lecoanet & Quataert(2013), which also are similar to those made in section 4 of this paper and in Press (1981), one can show that they areequivalent. To do this, recall that the definition of the flux given in Equation 64 is F z = ρ f − ω ωk n (cid:12)(cid:12) A n ψ n ∂ z ψ n (cid:12)(cid:12) . (A1)In the non-rotating limit this becomes F z = − ρ ω k n | A n ψ n ∂ z ψ n | . (A2)In the asymptotic limit of low-frequency gravity waves the JWKB approximation can be applied where ∂ z ψ n ≈ ik V ψ n ,so the flux becomes F z = − ρ ωk V k n | A n ψ n | . (A3)In this limit, the vertical wavenumber is approximately k V = Nω k n . (A4)So, then it can be seen that F z = − ρ N k n | A n | ψ n . (A5)Now, noting that v w = A n ψ n is the vertical velocity of the wave, one has that F z = − ρ N k n v w . (A6)Making the assumptions of Goldreich & Kumar (1990), Lecoanet & Quataert (2013), and Press (1981), one has that k n ≈ k c , ω ≈ ω c and v w ≈ ω c / ( k c N ), where the subscript c indicates the wave vector ( k c ) and overturning frequency6 Augustson et al. ( ω c ) of the convection. To obtain this expression of v w , we consider the low-frequency regime where the ratio betweenthe vertical and the horizontal components of the gravity waves’ velocity is given approximately by v w /v h ≈ ω/N . Inaddition, it is assumed, as in Equation (36) of Press (1981) and in Equation (49) of Lecoanet & Quataert (2013), thatthe horizontal wave velocity is given by v h ≈ v c = ω c /k c . Therefore, the previous expression becomes F z ≈ − ρ N k c (cid:18) ω c k c N (cid:19) = − ρ ω c k c N . (A7)Now, since u c = ω c /k c (the convective velocity), one has that F z ∝ ρ u c ω c N ∝ F c M, (A8)where M = ω c /N = S − is the Mach number or the inverse stiffness ( S ) and F c is the convective flux. Hence, underthese limits and assumptions, the flux definitions have the same scaling. Note that within these assumptions theinfluence of the spatial behaviour of the eigenmodes is not apparent because | ψ n | ≈ | e ik V z | = 1, which is not thecase for the exact solutions used in Equation 65. REFERENCES Aerts, C., Christensen-Dalsgaard, J., & Kurtz, D. W. 2010,AsteroseismologyAerts, C., Mathis, S., & Rogers, T. 2018, arXiv e-prints.https://arxiv.org/abs/1809.07779Aerts, C., Van Reeth, T., & Tkachenko, A. 2017, ApJL,847, L7, doi: 10.3847/2041-8213/aa8a62Aerts, C., Pedersen, M. G., Vermeyen, E., et al. 2019,A&A, 624, A75, doi: 10.1051/0004-6361/201834762Alexakis, A., & Biferale, L. 2018, PhR, 767, 1,doi: 10.1016/j.physrep.2018.08.001Alvan, L., Brun, A. S., & Mathis, S. 2014, A&A, 565, A42,doi: 10.1051/0004-6361/201323253Alvan, L., Mathis, S., & Decressin, T. 2013, A&A, 553,A86, doi: 10.1051/0004-6361/201321210Alvan, L., Strugarek, A., Brun, A. S., Mathis, S., & Garcia,R. A. 2015, A&A, 581, A112,doi: 10.1051/0004-6361/201526250Amard, L., Palacios, A., Charbonnel, C., Gallet, F., &Bouvier, J. 2016, A&A, 587, A105,doi: 10.1051/0004-6361/201527349Amard, L., Palacios, A., Charbonnel, C., et al. 2019, A&A,631, A77, doi: 10.1051/0004-6361/201935160Ando, H., Takagi, M., Fukuhara, T., et al. 2018, Journal ofGeophysical Research (Planets), 123, 2270,doi: 10.1029/2018JE005640Andr´e, Q., Barker, A. J., & Mathis, S. 2017, A&A, 605,A117, doi: 10.1051/0004-6361/201730765Augustson, K. C., Brown, B. P., Brun, A. S., Miesch, M. S.,& Toomre, J. 2012, ApJ, 756, 169,doi: 10.1088/0004-637X/756/2/169Augustson, K. C., Brun, A. S., & Toomre, J. 2016, ApJ,829, 92, doi: 10.3847/0004-637X/829/2/92 Augustson, K. C., & Mathis, S. 2019, ApJ, 874, 83,doi: 10.3847/1538-4357/ab0b3dBarker, A. J., Dempsey, A. M., & Lithwick, Y. 2014, ApJ,791, 13, doi: 10.1088/0004-637X/791/1/13Beck, P. G., Montalban, J., Kallinger, T., et al. 2012,Nature, 481, 55, doi: 10.1038/nature10612Belkacem, K., Mathis, S., Goupil, M. J., & Samadi, R.2009a, A&A, 508, 345, doi: 10.1051/0004-6361/200912284Belkacem, K., Samadi, R., Goupil, M. J., et al. 2009b,A&A, 494, 191, doi: 10.1051/0004-6361:200810827Benomar, O., Takata, M., Shibahashi, H., Ceillier, T., &Garc´ıa, R. A. 2015, MNRAS, 452, 2654,doi: 10.1093/mnras/stv1493Browning, M. K., Brun, A. S., & Toomre, J. 2004, ApJ,601, 512, doi: 10.1086/380198Brun, A. S., Miesch, M. S., & Toomre, J. 2011, ApJ, 742,79, doi: 10.1088/0004-637X/742/2/79Brun, A. S., Strugarek, A., Varela, J., et al. 2017, ApJ, 836,192, doi: 10.3847/1538-4357/aa5c40Busse, F. H. 2002, Ph. Fl., 14, 1301, doi: 10.1063/1.1455626Charbonnel, C., Decressin, T., Amard, L., Palacios, A., &Talon, S. 2013, A&A, 554, A40,doi: 10.1051/0004-6361/201321277Charbonnel, C., Decressin, T., Lagarde, N., et al. 2017,A&A, 605, A102, doi: 10.1051/0004-6361/201526724Christophe, S., Ballot, J., Ouazzani, R. M., Antoci, V., &Salmon, S. J. A. J. 2018, A&A, 618, A47,doi: 10.1051/0004-6361/201832782Clark di Leoni, P., Cobelli, P. J., Mininni, P. D., Dmitruk,P., & Matthaeus, W. H. 2014, Physics of Fluids, 26,035106, doi: 10.1063/1.4868280Couston, L.-A., Lecoanet, D., Favier, B., & Le Bars, M.2018, Journal of Fluid Mechanics, 854, R3,doi: 10.1017/jfm.2018.669 otating Convection: Penetration & Gravito-inertial Waves Cowling, T. G. 1941, MNRAS, 101, 367,doi: 10.1093/mnras/101.8.367Davidson, P. 2013, Turbulence in Rotating, Stratified andElectrically Conducting Fluids (Cambridge UniversityPress).https://books.google.fr/books?id=-QpCAQAAQBAJDecressin, T., Mathis, S., Palacios, A., et al. 2009, A&A,495, 271, doi: 10.1051/0004-6361:200810665Deheuvels, S., Garc´ıa, R. A., Chaplin, W. J., et al. 2012,ApJ, 756, 19, doi: 10.1088/0004-637X/756/1/19Deheuvels, S., Do˘gan, G., Goupil, M. J., et al. 2014, A&A,564, A27, doi: 10.1051/0004-6361/201322779Dintrans, B., Brandenburg, A., Nordlund, ˚A., & Stein,R. F. 2005, A&A, 438, 365,doi: 10.1051/0004-6361:20052831Dintrans, B., & Rieutord, M. 2000, A&A, 354, 86Edelmann, P. V. F., Ratnasingam, R. P., Pedersen, M. G.,et al. 2019, ApJ, 876, 4, doi: 10.3847/1538-4357/ab12dfEmden, R. 1907, Gaskugeln: Anwendungen derMechanischen Wrmetheorie.Emeriau-Viard, C., & Brun, A. S. 2017, ApJ, 846, 8,doi: 10.3847/1538-4357/aa7b33Fletcher, L. N., Melin, H., Adriani, A., et al. 2018, AJ, 156,67, doi: 10.3847/1538-3881/aace02Fossat, E., Boumier, P., Corbard, T., et al. 2017, A&A,604, A40, doi: 10.1051/0004-6361/201730460Fuller, J. 2017, MNRAS, 470, 1642,doi: 10.1093/mnras/stx1314Fuller, J., & Ro, S. 2018, MNRAS, 476, 1853,doi: 10.1093/mnras/sty369Gallet, F., & Bouvier, J. 2015, A&A, 577, A98,doi: 10.1051/0004-6361/201525660Garc´ıa, R. A., Turck-Chi`eze, S., Jim´enez-Reyes, S. J., et al.2007, Science, 316, 1591, doi: 10.1126/science.1140598Gehan, C., Mosser, B., Michel, E., Samadi, R., & Kallinger,T. 2018, ArXiv e-prints.https://arxiv.org/abs/1802.04558Gerkema, T., & Shrira, V. I. 2005, Journal of FluidMechanics, 529, 195, doi: 10.1017/S0022112005003411Gerkema, T., Zimmerman, J. T. F., Maas, L. R. M., & vanHaren, H. 2008, Reviews of Geophysics, 46, RG2004,doi: 10.1029/2006RG000220Goldreich, P., & Kumar, P. 1990, ApJ, 363, 694,doi: 10.1086/169376Grooms, I., Julien, K., Weiss, J. B., & Knobloch, E. 2010,Physical Review Letters, 104, 224501,doi: 10.1103/PhysRevLett.104.224501Gubenko, V., & Kirillovich, I. 2018, Solar-TerrestrialPhysics, 4, 41, doi: 10.12737/stp-42201807 Gubenko, V. N., Kirillovich, I. A., & Pavelyev, A. G. 2015,Cosmic Research, 53, 133,doi: 10.1134/S0010952515020021Hinson, D. P., & Tyler, G. L. 1983, Icarus, 54, 337,doi: 10.1016/0019-1035(83)90202-6Hoff, M., Harlander, U., & Triana, S. A. 2016, PhysicalReview Fluids, 1, 043701,doi: 10.1103/PhysRevFluids.1.043701Howard, L. N. 1963, Journal of Fluid Mechanics, 17, 405,doi: 10.1017/S0022112063001427Hurlburt, N. E., Toomre, J., & Massaguer, J. M. 1986,ApJ, 311, 563, doi: 10.1086/164796Julien, K., Knobloch, E., Milliff, R., & Werne, J. 2006,Journal of Fluid Mechanics, 555, 233,doi: 10.1017/S0022112006008949Julien, K., Rubio, A. M., Grooms, I., & Knobloch, E. 2012,Geophysical and Astrophysical Fluid Dynamics, 106, 392,doi: 10.1080/03091929.2012.696109K¨apyl¨a, P. J., Korpi, M. J., Stix, M., & Tuominen, I. 2005,A&A, 438, 403, doi: 10.1051/0004-6361:20042244Kiraga, M., Stepien, K., & Jahn, K. 2005, AcA, 55, 205Kurtz, D. W., Saio, H., Takata, M., et al. 2014, MNRAS,444, 102, doi: 10.1093/mnras/stu1329Landin, N. R., Mendes, L. T. S., & Vaz, L. P. R. 2010,A&A, 510, A46, doi: 10.1051/0004-6361/200913015Le Bars, M., Lecoanet, D., Perrard, S., et al. 2015, FluidDynamics Research, 47, 045502,doi: 10.1088/0169-5983/47/4/045502Lecoanet, D., Le Bars, M., Burns, K. J., et al. 2015,PhRvE, 91, 063016, doi: 10.1103/PhysRevE.91.063016Lecoanet, D., & Quataert, E. 2013, MNRAS, 430, 2363,doi: 10.1093/mnras/stt055Lee, U., Neiner, C., & Mathis, S. 2014, MNRAS, 443, 1515,doi: 10.1093/mnras/stu1256Lee, U., & Saio, H. 1993, MNRAS, 261, 415,doi: 10.1093/mnras/261.2.415Li, G., Van Reeth, T., Bedding, T. R., et al. 2019, MNRAS,2517, doi: 10.1093/mnras/stz2906Maeder, A., & Meynet, G. 2000, ARA&A, 38, 143,doi: 10.1146/annurev.astro.38.1.143Maksimova, E. V. 2018, Scientific reports, 8, 15952Malkus, W. V. R. 1954, Proceedings of the Royal Society ofLondon Series A, 225, 196, doi: 10.1098/rspa.1954.0197Mathis, S. 2009, A&A, 506, 811,doi: 10.1051/0004-6361/200810544Mathis, S. 2013, in Lecture Notes in Physics, BerlinSpringer Verlag, Vol. 865, Lecture Notes in Physics,Berlin Springer Verlag, ed. M. Goupil, K. Belkacem,C. Neiner, F. Ligni`eres, & J. J. Green, 23 Augustson et al.
Mathis, S., Auclair-Desrotour, P., Guenel, M., Gallet, F., &Le Poncin-Lafitte, C. 2016, A&A, 592, A33,doi: 10.1051/0004-6361/201527545Mathis, S., Decressin, T., Eggenberger, P., & Charbonnel,C. 2013, A&A, 558, A11,doi: 10.1051/0004-6361/201321934Mathis, S., Neiner, C., & Tran Minh, N. 2014, A&A, 565,A47, doi: 10.1051/0004-6361/201321830Mathis, S., Talon, S., Pantillon, F.-P., & Zahn, J.-P. 2008,SoPh, 251, 101, doi: 10.1007/s11207-008-9157-0Melchior, P., & Ducarme, B. 1986, Physics of the Earth andPlanetary Interiors, 42, 129,doi: 10.1016/0031-9201(86)90085-3Moravveji, E., Townsend, R. H. D., Aerts, C., & Mathis, S.2016, ApJ, 823, 130, doi: 10.3847/0004-637X/823/2/130Mosser, B., Goupil, M. J., Belkacem, K., et al. 2012, A&A,540, A143, doi: 10.1051/0004-6361/201118519Murphy, S. J., Fossati, L., Bedding, T. R., et al. 2016,MNRAS, 459, 1201, doi: 10.1093/mnras/stw705Neiner, C., Lee, U., Mathis, S., et al. 2020, A&A, 627, A47Neiner, C., Floquet, M., Samadi, R., et al. 2012, A&A, 546,A47, doi: 10.1051/0004-6361/201219820Ouazzani, R.-M., Salmon, S. J. A. J., Antoci, V., et al.2017, MNRAS, 465, 2294, doi: 10.1093/mnras/stw2717Palacios, A., Talon, S., Charbonnel, C., & Forestini, M.2003, A&A, 399, 603, doi: 10.1051/0004-6361:20021759Pedley, T. J. 1968, Journal of Atmospheric Sciences, 25,789,doi: 10.1175/1520-0469(1968)025 (cid:104) (cid:105)(cid:105)