Mixed Rossby–Gravity Wave–Wave Interactions

JANUARY 2019 E D E N E T A L . 291

CARSTEN EDEN AND MANITA CHOUKSEY Institut fur€ Meereskunde, Universitat€ Hamburg, Hamburg, Germany

DIRK OLBERS Alfred Wegener Institute for Polar and Marine Research, Bremerhaven, and Center for Marine Environmental Sciences, University of Bremen, Bremen, Germany

(Manuscript received 15 April 2018, in ﬁnal form 18 October 2018)

ABSTRACT

Mixed triad wave–wave interactions between Rossby and gravity waves are analytically derived using the kinetic equation for models of different complexity. Two examples are considered: initially vanishing linear gravity wave energy in the presence of a fully developed Rossby wave field and the reversed case of initially vanishing linear Rossby wave energy in the presence of a realistic gravity wave field. The kinetic equation in both cases is numerically evaluated, for which energy is conserved within numerical precision. The results are validated by a corresponding ensemble of numerical model simulations supporting the validity of the weak-interaction assumption necessary to derive the kinetic equation. Since they are generated by nonresonant interactions only, the energy transfers toward the respective linear wave mode with vanishing energy are small in both cases. The total generation of energy of the linear gravity wave mode in the first case scales to leading order as the square of the Rossby number in agreement with independent estimates from laboratory experiments, although a part of the linear gravity wave mode is slaved to the Rossby wave mode without wavelike temporal behavior.

1. Introduction the interactions are known to transport energy pre- dominantly to larger scales (e.g., Rhines 1975; Salmon The dynamics of rotating stratified flows are often 1998). Internal gravity waves, on the other hand, evolve at described using linear Rossby and inertia–gravity waves much shorter temporal scales and also often have small (hereinafter called gravity waves), which have well- spatial scales down to a few meters, which renders them known properties. The interactions between these lin- hard to observe as well as to resolve in models. Olbers ear waves generated by the nonlinear terms in the (1976), Pomphrey et al. (1980), Müller et al. (1986),and equations of motions, however, are less well known but Henyey et al. (1986), for example, have discussed how can profoundly affect the evolution of the flow field. The wave–wave interactions of gravity waves in the ocean nonlinear wave–wave interactions within the gravity can lead to an energy transfer within the gravity wave and Rossby wave components along with the mixed in- spectrum to smaller scales through processes such as teractions between the two wave modes are the subject induced diffusion (e.g., Müller et al. 1986) and parametric of this paper. We discuss the energy transfers by weak subharmonic instability (e.g., Sutherland 2010). gravity, Rossby, and mixed wave–wave interactions in Rossby wave–wave interaction is often discussed and models of different complexity. understood as an energy flux from one wavenumber Rossby waves evolve at temporal scales much longer to its neighboring wavenumber, that is, local in wave- than the inertial period. Wave–wave interactions of number space, generated by the nonlinearities in the Rossby waves are typically characterized by motions model under investigation. Similar to the equivalent with spatial scales on the order of the deformation radius energy flux in the theory of isotropic turbulence by and larger, in both the atmosphere and the ocean, and Kolmogorov (1941), scaling arguments can then lead to an inference of the magnitude and wavenumber de- Corresponding author: Carsten Eden, carsten.eden@uni- pendency of the flux and the energy spectrum of Rossby hamburg.de waves (Salmon 1998). Such an energy flux can also be

DOI: 10.1175/JPO-D-18-0074.1 Ó 2019 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses). Unauthenticated | Downloaded 09/26/21 12:46 PM UTC 292 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 49 diagnosed from observations or models (Frisch 1995) instability (Plougonven and Snyder 2007). Spontaneous and shows indeed a direction toward large scales (Scott generation of internal gravity waves from the balanced and Wang 2005). On the other hand, gravity wave–wave flow has been studied in laboratory (e.g., Williams et al. interactions are usually discussed in the framework 2005, 2008) and numerical studies (e.g., Molemaker of resonant triad interactions, where three waves of et al. 2005; Sugimoto and Plougonven 2016; Chouksey different wavenumbers and frequencies—satisfying a et al. 2018), where it was shown that this process can certain relation given below—exchange energy. Such transfer energy from the balanced flow to gravity waves. triad interactions can generate therefore an energy It was speculated that this process may represent a sig- transfer between rather different wavenumbers, that is, nificant energy sink for the balanced flow in the ocean nonlocal in wavenumber space. Although the nonlocal (e.g., Williams et al. 2008; Brüggemann and Eden 2015), triad interaction and the local energy flux framework which would modify our view of the energy cycle in the are in principle equivalent, the former presents more ocean and which would need consideration in parame- information since all waves can participate in the triad terizations for ocean models. Here we also discuss the interactions forming a complicated network. The net opposite scenario, the generation of Rossby waves or effect of these complex triad interactions can result in an balanced flow by gravity waves, which might have im- energy transfer in either upscale or downscale direction, portance for the energetics of the wave field. which is then diagnosed as the local energy flux in This study is structured into four sections. Section 2 wavenumber space. presents a theoretical layout of an evolution equation Understanding the triad of interactions between grav- for the wave energies in spectral space—the so-called ity waves, Rossby waves, and their mixed interactions kinetic equation or scattering integral—that is derived therefore can contribute to better understanding of and discussed in Fourier space for a quasigeostrophic the dynamics of atmosphere and ocean. Wave–wave in- layer model, for a single-layer reduced-gravity model, teractions have been discussed for the case of Rossby and for vertically resolved models using the weak- waves (e.g., Kenyon 1964; Longuet-Higgins et al. 1967; interaction assumption. The derivation is detailed Connaughton et al. 2015) and gravity waves (e.g., Olbers mainly for the reduced-gravity model but also holds for 1976; Nazarenko 2011) based on the weak-interaction the other models. From the general kinetic equation, assumption and the kinetic equation formulated for Rossby wave, gravity wave, and mixed Rossby–gravity geophysical applications first by Hasselmann (1962).The wave interactions are derived. Section 3 demonstrates wave–wave interaction between the two principal wave numerically the generation of linear gravity waves from modes in this framework, however, has attracted less in- balanced flow and vice versa and validates the results terest so far, an exception being Lelong and Riley (1991). with a numerical model. Section 4 presents a summary In a general derivation for models of different complex- and discussion of the results. ity, we here also allow for the resonant and nonresonant interactions of all wave modes present, that is, mixed triad 2. The kinetic equation or scattering integral interactions between (internal) gravity waves and Rossby waves, referred to as Rossby–gravity mixed interactions. We use as an algebraically simple model a reduced- Although usually separated by vastly different time gravity model for a single layer. All variables and also the scales, it is known that both wave modes can interact differential operators are scaled with a velocity scale U, with each other (e.g., Müller 1977; Lighthill 1978; Ford length scale L, and time scale 1/V,whereV denotes the et al. 2000). A general framework for the interaction magnitude of the constant Coriolis parameter. This yields between long and short waves has been set by Benney › 1 1 = 52 = (1977), investigating nonlinear differential equations tu f Hu h Rou u and that allow, in the linearized case, for long and short wave › h 1 c2= u 52Ro= hu, (1) solutions. The theory can be applied to the resonant t interaction of two short internal gravity waves with a where Ro 5 U/(VL) is the Rossby number, u denotes1 long Rossby wave. The resonance condition of such a the scaled layer velocity, and h is the layer thickness triad of waves is an approximate equality of the long perturbation scaled by g0/(VLU), with g0 being reduced wave phase velocity and the short wave group velocity. gravity. Furthermore, More recently, spontaneous emission or loss of balance (e.g., Vanneste 2013) has attracted considerable atten- tion and refers to the process of gravity wave generation 1 by a balanced flow spontaneously in the absence of any The vector Hu denotes anticlockwise rotation of the vector u by p/2; that is, u 5 (2y, u) for u 5 (u, y). external forcing, which often happens during baroclinic H

Unauthenticated | Downloaded 09/26/21 12:46 PM UTC JANUARY 2019 E D E N E T A L . 293 qffiffiffiffiffiffiffi also yields Eq. (2) but with the three-dimensional defini- c 5 g0h/(VL) 5 Ro/Fr tion of potential vorticity given by is the scaled phase velocity of gravity waves without q 5 =2c 1 N2/f 2› c. rotation, h is the mean layer thickness, and zz qffiffiffiffiffiffiffi Fr 5 U/ g0h a. The quasigeostrophic layer model in Fourier space Before turning to the more complex models, the denotes the Froude number. The scaled Coriolis pa- quasigeostrophic layer model is discussed first because rameter f 5 1 was kept for reference. The model is of its simpler algebraic form. The approach taken here, considered on a double-periodic domain. however, is analogous to the other models. A Fourier For Ro ; Fr 1 the quasigeostrophic approximation ansatz given by is valid and yields in this scaling ð‘ ^ ikx › 1 b› c 52 =c = c(x, t) 5 dkc(k, t)e (4) tq x RoH q (2) 2‘

5 =2c 2 c 2 is applied to Eq. (2), with wavenumber vector 5 (k , k ), where potential vorticity q /LR and the geo- k x y strophic streamfunction is deﬁned by =c 5 u, and where position vector x 5 (x, y), and complex wave amplitudes H ^ ^ the Rossby radius L 5 c/f 5 1 is kept for reference. c(k, t). Since the streamfunction c is real, c(k) 5 R ^ Here, a small variation of the Coriolis parameter V on the c*(2k) is required. The ansatz yields in Eq. (2) same order of magnitude as Ro and Fr was introduced, ð represented by the reference parameter b ; Ro, but V dkeikx(› c^ 1 ivRc^) 2 t is kept constant in the reduced-gravity model. ð ð 2 1 22 As a more complex and realistic model we also con- ^ ^ Hk 1 k2(k2 LR ) i(k 1k )x 5 Ro dk dk c c e 1 2 , (5) 1 2 1 2 2 1 22 sider the primitive equations, for which Boussinesq k LR and hydrostatic approximation are applied to the full c^ 5 c^ system. Using as time scale, horizontal and vertical with the notation n (kn) and the Rossby wave vR 52b 2 1 22 length scale 1/V, L, and H, respectively, and scaling frequency kx/(k LR ). Multiplying with 2 horizontal velocity u as U, vertical velocity w from exp( ik0 x) and integrating over x yields the continuity equation as UH/L, pressure p from geo- ð ð 2 1 22 V Hk k (k L ) strophic balance as UL, and buoyancy b from hydro- › c^ 1 ivRc^ 5 Ro dk dk c^ c^ 1 2 2 R t 0 0 0 1 2 1 2 2 1 22 static balance as VUL/H yields k0 LR 3 d(k 1 k 2 k ), (6) › 1 1 = 52 = 1 › 1 2 0 tu f Hu p Ro(u u w zu), and with the delta function › b 1 N2w 52Ro(u =b 1 w› b) (3) t z ð d 5 p 2 2ikx with the diagnostic relations ›zp 5 b and = u 1 ›zw 5 0; (k) 1/(2 ) dxe . the (scaled) stability frequency N 5 Ro/Fr 5 LR/L,which is kept constant in the vertical; the Froude number The right-hand side of Eq. (6) contains the effect of 5 ~ 5 ~ V Fr U/c; and the Rossby radius LR c/ . The unscaled the nonlinear terms on the right-hand side of Eq. (2). gravity wave speed without rotation becomes in this The wave amplitude c^ still contains the regular wave ~ ~ 0 model c~5 NH,whereN denotes the unscaled stability oscillation in time, which can be removed by setting frequency. Here, f 5 1 was again kept for reference. For ^ 2ivRt the limit of Ro 1 (but ﬁxed relation Ro ; Fr, L ; L) c 5 0 R (k0, t) [a0(t)/f ]e and introducing again a small variation in the Coriolis parameter, the quasigeostrophic approximation of Eq. (3) (frequently called ‘‘interaction representation’’; see, vR 5 vR e.g., Nazarenko 2011). Here, 0 (k0) represents the frequency of the Rossby wave for the wavenumber

vector k0 and a0(t) 5 a(k0, t) is the amplitude of the wave 2 The reason is that the double-periodic boundary conditions for at k0. The wave amplitude is allowed to change in time, which b 6¼ 0 are possible for the quasigeostrophic model but not for the reduced-gravity model. Closed meridional boundaries could be and the scaling of the amplitude a0 by f is introduced to used instead, which would put an additional constraint on the wave match a0 with total wave energy in the reduced-gravity amplitudes in the next section. and primitive equation models below.

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Substituting the interaction representation in Eq. (6) now a vector equation. Standard linear algebra methods, yields described below, are used to cast it into a scalar equation ð ð similar to Eq. (6). 2 1 22 (0) Hk k (k2 LR ) The matrix A has three eigenvalues v 5 0 and › a 5 Ro dk dk a a 1 2 t 0 1 2 1 2 2 1 22 v(6) 56 2 1 2 2 1/2 v(6) f (k0 LR ) (f c k ) . Two of them, , correspond (0) 2i(vR1vR2vR)t to gravity waves and the other one, v , corresponds 3 d(k 1 k 2 k )e 1 2 0 , (7) 1 2 0 to Rossby waves, but here with vanishing frequency R 5 vR 5 vR instead of v since we have assumed a constant Coriolis with an a(kn)and n (kn). Equation (7) is a parameter f. It is shown in appendix B that by tendency equation for the amplitude a0 of the Rossby wave at wavenumber k . Only if the right-hand side including a small variation in f the vanishing eigenvalue 0 v(0) 5 vR of Eq. (7) is nonzero, that is, if there are wave–wave of the system gets ﬁnite and . But the magni- v(0) v(6) (triad) interactions by the nonlinear terms in Eq. (2), tude of remains still much smaller than j j for A will the amplitude a of the Rossby wave change in time. Ro 1. Related to the eigenvalues, the matrix has 0 s s By renaming the integration variables, Eq. (7) can be left and right eigenvectors P and Q [discussed also by, rewritten as e.g., Hasselmann (1970), Leith (1980), and Olbers et al. ð ð (2012)] given by 2i(vR1vR2vR)t › a 5 dk dk a a D d(k 1 k 2 k )e 1 2 0 , s A 5 vs s A s 5 vs s 5 6 t 0 1 2 1 2 k0,k1,k2 1 2 0 P P and Q Q , s 0, . (12) (8) Any state vector ^z can be expressed by the eigenvec- with the Rossby wave interaction coefﬁcients tors as

s s s s 2 2 2 ^z(k, t) 5 å g (t, k)Q (k), g (t, k) 5 P (k) ^z, Hk k (k k ) 5 5 1 2 2 1 s50,6 Dk ,k ,k Dk ,k ,k Ro 2 22 , (9) 0 1 2 0 2 1 2f (k 1 L ) 0 R s 5 0, 6. (13) which are made symmetric with respect to the last two arguments. The symmetric version, Eq. (8), turns out Using the decomposition equation [Eq. (13)] in Eq. (11) s below to be more convenient to use than Eq. (7). and scalar multiplication with P (k) yields a scalar evolution equation for gs(k) similar to Eq. (6) of the Rossby b. The reduced-gravity model in Fourier space wave case. But before doing so, the wave oscillation in An equation analogous to Eq. (8) can be derived time is removed by setting for the reduced-gravity model, but for both Rossby s 5 s ivs(k)t and gravity wave amplitudes, instead of Rossby wave g a (t, k)e , amplitudes only as in Eq. (8). We apply a Fourier to obtain an evolution equation for the wave amplitude ansatz to the reduced-gravity wave model in Eq. (1) as as before for the Rossby wave case. It is given by given by ð ð ð‘ s s s s ,s ,s › 0 52 å å 1 2 0 0 1 2 ikx ta0 i dk1 dk2 a1 a2 Ck ,k ,k 5 ^ 5 6 5 6 0 1 2 u(x, t) dku(k, t)e (10) s1 0, s2 0, 2‘ s s s i(v 1 1v 2 2v 0 )t 3 d(k 1 k 2 k )e 1 2 0 (14) with complex wave amplitudes u^(k, t)withu^(k) 5 1 2 0 ^ 2 u*( k), and similar for h. This yields in Eq. (1) after with s 5 0, 6, the notation as 5 as(k ), vs 5 vs(k ), 2 0 n n n n multiplication with exp( ik0 x) and integration and the exchange coefficients C0, which follow using over x the decomposition [Eq. (13)] in the vector function ð ð s ‘ ‘ N and scalar multiplication with P (k0). The interac- › ^ 5 A ^ 2 d 1 2 N tion coefficients can be made symmetric in the last two tz0 i z0 i dk1 dk2 (k1 k2 k0) , (11) 2‘ 2‘ indices by renaming the integration and summation ^ variables as before going from Eq. (7) to Eq. (8) in with the system matrix A(k0), the state vector z(k0, t) 5 ^ the Rossby wave case. The symmetric interaction co- (u^0, ^y0, h0), and the vector function N, which are given s0,s1,s2 ^ efficients Ck ,k ,k are given in appendix A,replacing in appendix A.ThetermiA z0 contains all linear terms s ,s ,s 0 1 2 C0 0 1 2 .Since in Eq. (1) while the integral involving the vector k0,k1,k2 N function contains all nonlinear terms. Equation (11) 0,0,0 5 Ck ,k ,k iDk ,k ,k , (15) is analogous to Eq. (6) of the Rossby wave case but is 0 1 2 0 1 2

Unauthenticated | Downloaded 09/26/21 12:46 PM UTC JANUARY 2019 E D E N E T A L . 295 the Rossby wave model in Eq. (8) is recovered by re- For the primitive equation model, a wave ansatz stricting the double sum in Eq. (14) to s0 5 s1 5 s2 5 0. analogous to Eq. (16) is applied to u, p,andb.This Equation (14) is therefore analogous to Eq. (8) but leads after the same manipulations as explained above contains now a double sum over all three wave modes, to an amplitude equation analogous to Eq. (14),with including the triad interaction between them. the only difference being that the integrals are taken over K instead of k. The analogous deﬁnitions of A, N, c. The vertically resolved models in Fourier space and ^z, together with eigenvalues of A and the corre- The derivation of a corresponding amplitude equation sponding interaction coefﬁcients Cs0,s1,s2 , are given in K0,K1,K2 for the primitive equation model Eq. (3) and its quasi- appendix C. geostrophic version is very similar to the single-layer models. For the vertically resolved quasigeostrophic d. Wave energy and weak-interaction assumption model the wave ansatz The sum of kinetic energy and potential energy of the ð ‘ waves (averaged over one wave period) in the reduced- ^ i(kx1mz) c(x, z, t) 5 dKce (16) gravity model is related to 2‘

^ 2 2 is applied, where K 5 (k, m) denotes the three- 1 1 jhj 1 jasj ( u^ 2 1 ^y 2) 1 5 å , (19) dimensional wavenumber vector and m is the vertical j j j j 2 s 2 2 c 2 s n wavenumber. This yields an amplitude equation for the vertically resolved quasigeostrophic model where ns results from the normalization of the eigen- ð ð vectors and is defined in appendix A. The square of › 5 the wave amplitudes as is thus related to wave en- ta0 dK1 dK2a1a2DK ,K ,K 0 1 2 ergy. This definition of energy is the same in the 2i(vR1vR2vR)t 3 d(K 1 K 2 K )e 1 2 0 , (17) primitive equation model, but with potential energy 1 2 0 ^ 2 given by jbj /(2N2) in this case. The derivation of the R 2 2 2 2 with the Rossby wave frequency v 52bkx/(k 1 f m /N ) energy equation below is completely analogous for and with the Rossby wave interaction coefficients the reduced-gravity model and the primitive equation model, as well as for the quasigeostrophic counterparts. 2 1 2 2 2 2 2 2 2 2 2 Hk k (k f m /N k f m /N ) We will sketch the derivation here for the reduced- D 5 Ro 1 2 2 2 1 1 . K0,K1,K2 2 1 2 2 2 2f (k0 f m0/N ) gravity model only. (18) To obtain an evolution equation for the square of the s0 amplitudes we multiply Eq. (14) by (an )*, we multiply The algebraic structure of Eq. (8) and Eq. (17) is thus the the complex conjugate of Eq. (14) for k0 / kn (but s0 same, except for the integrals taken over k instead of K. keeping s0)bya0 , and we add both, which yields

ðð " s s s s s s s s s ,s ,s i(v 1 1v 2 2v 0 )t › 0 0 52 å 1 2 0 0 1 2 d 1 2 1 2 0 t[a0 (an )*] i dk1dk2 a1 a2 (an )*Ck ,k ,k (k1 k2 k0)e s ,s 0 1 2 1 2 # s s s s s s s ,s ,s * 2i(v 1 1v 2 2v 0 )t 2 (a 1 a 2 )*a 0 C 0 1 2 d(k 1 k 2 k )e 1 2 n . (20) 1 2 0 kn,k1,k2 1 2 n

For kn 5 k0,Eq.(20) describes the evolution of the total turbulence. By invoking the weak-interaction assumption energy of the wave at k0—besides a scaling factor as described by, for example, Hasselmann (1962, 1966) s nn/2—and for kn 6¼ k0, the equation describes the evolu- and Nazarenko (2011), however, the triple covariances on tion of covariances between the waves of type s0 at the right-hand side of Eq. (20) can be cast into a form different wavenumbers. However, to evaluate this evo- involving the energies of the other waves of different lution, triple covariances of the amplitudes need to be wavenumber and/or mode. The result is a closed system known, which enter on the right-hand side of Eq. (20). of equations involving the energies of all modes that is Formulating equations for such triple covariances would called the kinetic equation or scattering integral. involve quadruple covariances and so forth, resulting in The lengthy derivation starting from Eq. (20) is only s the typical closure problem of nonlinear systems or sketched here: the amplitudes an are decomposed into a

Unauthenticated | Downloaded 09/26/21 12:46 PM UTC 296 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 49 power series in the small parameter Ro. The zero-order Eq. (14), the first-order amplitudes can be expressed variables in this power series are assumed to be time to first order by integrals of products of two zero- independent; thus it is assumed that the amplitudes are order amplitudes. Using this, the triple covariances in only weakly changing. Furthermore, it is assumed that Eq. (20)—which vanish for the zero-order amplitudes— the zero-order variables are statistically independent can then be cast to first order into quadruple co- and can be described by Gaussian statistics. Note that variances of the zero-order variables. Because of this second assumption is used by Hasselmann (1966) the assumption of Gaussian statistics, the quadruple co- but can also be replaced by a weaker assumption on variances can in turn be written as products of two the statistics of the amplitudes (Nazarenko 2011). The second-order covariances, which are related to the total s first assumption on the weakly changing amplitudes wave energy En. is, however, essential to the procedure. Using then The resulting kinetic equation is given as

ð hi 2 2 s0 4 s0,s1,s2 s1 s2 s1 s2 s0,s2,s1 * s0 s1 s0 s1 s2, s1,s0 s0 s2 s0 s2 s1, s2,s0 › E 5 dk å C n n E E C 2 n n E E C 2 2 n n E E C 2 t 0 s0 1 k ,k ,k 1 2 1 2 k ,k ,k 0 1 0 1 k , k ,k 0 2 0 2 k , k ,k n s ,s 0 1 2 0 2 1 2 1 0 1 2 0 0 1 2 ð s s s s s s s ,s ,s s ,s ,2s 3D v 1 1 v 2 2 v 0 2 0 å 1 1 å 0 0 2 2 1 1 D 1 ( 1 2 0 ) 4E0 dk n1 E1 Ck ,k ,0C0,k ,2k (s f ) c.c. , (21) 1 56 56 0 0 1 1 2 s1 s2

v 5 vs1 1 vs2 2 vs0 5 Where If the condition 1 2 0 0 is met, there is a ð resonant triad interaction. Such resonant interactions s are most important in the long run, but nonresonant s 5 s 5 å s m s E (kn) En (1/2) s dkmhan(am )*i/nn m interactions can also become important for short time intervals. The last term in Eq. (21) is never resonant and is the total energy of the wave of mode s at wave- is related to inertial oscillations. For k 5 0, a further 5 2 0 number kn,withk2 k0 k1,andwherec.c.denotes nonresonant contribution related to inertial oscillations the complex conjugate of the proceeding terms. For the has to be added, which is not given here. vertically resolved model, replace kn with Kn.The time-dependent function D(v, t)isgivenby e. Energy conservation

Integrating the kinetic equation Eq. (21) over k0 and ð summing over s yields total energy conservation. Re- t ivt 0 0 1 2 e D(v, t) 5 eivt dt0e2ivt 5 i , limD(v, t) 5 pd(v). naming one of the integration variables and changing v /‘ 2 2 0 t v s 2 52vs s 5 s the sign of another, using ( k) (k), n2n nn, (22) 2s 5 s and E2n En, yields

2 3 ð ð ð ð s ,s ,s s ,2s ,s s ,s ,2s C 0 1 2 * 2 1 0 1 0 2 s s s s ,s ,s s s 4 k ,k ,k Ck ,2k ,k Ck ,k ,2k å› 0 5 å 2 1 0 1 2 d 2 2 2 1 0 1 2 2 2 1 0 2 1 0 2 5 dk0 tE0 4 dk0 dk2 dk1n2 n1 Ck ,k ,k (k0 k1 k2)E2 E1 s s s s s ,s ,s 0 1 2 0 2 1 0 1 2 0 n0 n2 n1

s s s 3D v 0 2 v 1 2 v 2 1 5 ( 0 1 2 ) c.c. 0. (23)

The term in brackets in the second line vanishes for this model is not a second-order quantity as for the the coefficients of the primitive equation model and primitive equation model, but involves cubic terms. Only the layered and vertically resolved quasigeostrophic for a linearized version of the reduced-gravity model models, such that total energy of the models is indeed the energy gets a second-order quantity. Thus, wave conserved. Here only the first two lines of Eq. (21) energy and full energy differ in the reduced-gravity have been used, the last term related to inertial os- model, and Eq. (23) does not describe the total energy cillations balances with the term at k0 5 0. of the full reduced-gravity model as is the case for However, the coefficients of the reduced-gravity the primitive equation and the quasigeostrophic models. model do not vanish in the term in brackets in the sec- This nonconservative effect is, however, only a few per- ond line of Eq. (23). The reason is that the total energy of cent at maximum in the numerical evaluations below.

Unauthenticated | Downloaded 09/26/21 12:46 PM UTC JANUARY 2019 E D E N E T A L . 297 f. Rossby wave interaction For the vertically resolved Rossby waves, the kinetic equation is given by replacing k with K in Eq. (24) with Restricting the sums in Eq. (21) to s 5 s 5 s 5 0— n n 0 1 2 the symmetric interactions coefficients that is, to Rossby waves only—energy is already conserved and the kinetic energy equation for single-layer 0 5 2 1 2 2 2 2 2 2 2 2 2 DK ,K Hk k (k2 f m2/N k1 f m1/N ) (25) Rossby waves is recovered (Kenyon 1964; Longuet- 1 2 1 2 Higgins et al. 1967): 2 and by replacing L 2 with f 2m2/N2, f 2m2/N2, and ð 0 1 0 0 0 2 0 0 0 2 0 0 0 2 2 2 E1E2Dk ,k E0E1Dk ,k E0E2Dk ,k f m /N in the first, second, and third set of parentheses, › 0 5 0 1 2 0 1 0 2 2 E dk D 2 2 2 t 0 1 k1,k2 2 1 2 2 1 2 2 1 2 respectively, in the fraction. (L k0)(L k1)(L k2) 3 R D Dv [ ( )] (24) g. Gravity wave interaction 5 2 Dv 5 vR 1 vR 2 vR with k2 k0 k1,with 1 2 0 ,theco- Energy is also conserved restricting the sum in efficients D0 5 Rok k (k2 2 k2), and using n0 5 Eq. (21) to s 5 s 5 s 56, that is, to the gravity waves k1,k2 H1 2 2 1 n 0 1 2 2 22 1 2 R f /(L kn). The function denotes taking the real only. After several manipulations, resolving the double part of its argument. The interactions become reso- sum, and using general properties of the energy, the nant if Dv 5 0. If b 5 0, all interactions are resonant, frequency, and the coefficients, the kinetic equation but for the quasigeostrophic models, only certain in- can be rewritten in simplified form (three terms instead teractions become resonant. of eight) as

ð ð › 1 5 4 D v 1 v 2 v d 1 2 1,1,1 1,1,1 * 1 1 2 1 1 1,2,1 tE0 dk1 dk2f ( 1 2 0) (k1 k2 k0)Ck ,k ,k ½ Ck ,k ,k n1n2E1 E2 n0n1E0 E1 Ck ,2k ,k 0 1 2 0 1 2 2 1 0 n0 2 1 1 1,2,1 1D v 2 v 2 v d 2 2 1,1,2 1,1,2 * 1 1 n0n2E0 E2 Ck ,2k ,k ( 1 2 0) (k1 k2 k0)2Ck ,k ,2k ½ Ck ,k ,2k n1n2E1 E2 1 2 0 0 1 2 0 1 2

2 1 1 2,2,1 2 1 1 1,1,1 1D 2v 2 v 2 v d 1 1 n0n1E0 E1 C2k ,2k ,k n0n2E0 E2 Ck ,k ,k ( 1 2 0) (k1 k2 k0) 2 1 0 1 2 0 hi 3 1,2,2 1,2,2 * 1 1 2 1 1 2,1,1 2 1 1 2,1,1 1 Ck ,2k ,2k Ck ,2k ,2k n1n2E1 E2 n0n1E0 E1 C2k ,k ,k n0n2E0 E2 C2k ,k ,k g c.c. (26) 0 1 2 0 1 2 2 1 0 1 2 0

2 ü The corresponding expression for E0 is just the complex given in Olbers (1974, 1976) and M ller and Olbers conjugate of the terms explicitly given on the right-hand (1975) for the resonant interactions but differ in general side. For the generalization to three dimensions, just ex- for the nonresonant interactions. change k0 with K0, and correspondingly for the other h. Mixed Rossby–gravity wave interaction wavenumber vectors. The last term cannot become resonant and is usually not considered; the first term and the As a consequence of the individual energy conservation of middle term are called sum and difference interaction, Rossby and gravity wave modes, the same is true for the respectively. It can be shown that for the reduced-gravity remaining terms, that is, the mixed components of the sum model the resonance conditions for gravity waves cannot describing the interaction of Rossby and gravity waves. 6 3 5 be satisfied but that resonant sum and difference in- Considering Eq. (21) for a state with initially E jt50 0for teractions are possible for the three-dimensional (in- all k—that is, initially without gravity waves—and energy 0 56 ternal) gravity waves. Similar forms of the kinetic only in the balanced mode E yields for s0 the following: ð ð equation for nonhydrostatic internal gravity waves have 2 › 6 5 4 6,0,0 0 0 0 0 E 6 dk dk jC j n n E E been derived by, for example, Olbers (1976),andarecent t 0 1 2 k0,k1,k2 1 2 1 2 n0 overview is given in Lvov et al. (2012). The corresponding 3 R[D(v 2 vR 2 vR)]d(k 1 k 2 k ). (27) interaction coefficients C in Eq. (26) for a non- 0 1 2 1 2 0 hydrostatic model (not given) can be related to the ones This term cannot become resonant since max(jvRj) 5

bLR/2 1 and min(v0) 5 f 5 1fork0 5 0. Only if b ; 3 Similar to short surface gravity waves, it is necessary to consider Ro gets larger can the term become near reso- higher-order terms for resonant interactions, as demonstrated by nant. However, a ﬁnite amount of net gravity wave Hasselmann (1962). energy is generated during an initial adjustment, as

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0 5 demonstrated in the next section, which we interpret The reversed case is given if Et50 0 for all k in the as generation of the linear gravity wave mode by the initial state and energy only in the gravity wave mode 6 balanced ﬂow. E . Then we ﬁnd for s0 5 0 the following:

ð ð 2 2 › 0 5 4 1 1 1 1 0,1,1 D v 1 v 2 vR d 1 2 1 0,1,2 D v 2 v 2 vR 3 d tE0 0 dk1 dk2n1 n2 E1 E2 ½jCk ,k ,k j ( 1 2 0 ) (k1 k2 k0) 2jCk ,k ,2k j ( 1 2 0 ) (k1 0 1 2 0 1 2 n0 2 2 2 1 0,2,2 D 2v 2 v 2 vR d 1 1 1 k2 k0) jCk ,2k ,2k j ( 1 2 0 ) (k1 k2 k0) c.c. (28) 0 1 2

The first (sum interaction) term and the last term within different grid, we were not able to conserve energy in the the square brackets can hardly become resonant, but the calculation of the kinetic equation.Ð Ð Energy conservation second (difference interaction) term can become reso- is checked by evaluating dk›tE/ dkj›tEj and is given 6 6 nant. It is possible to show that C0, , 5 0 holds for the k0,k1,k2 within the numerical precision error for all examples coefficients of the reduced-gravity model. The gravity that we checked for single-layer Rossby waves, the wave mode is then completely separated, but this is not reduced-gravity model, and the vertically resolved the case for the primitive equation model. Internal model, except for the gravity waves in the reduced– gravity waves can thus also generate balanced flow. gravity wave model because of the abovementioned The difference interaction term in Eq. (28) is analogous complication of total energy that does not match wave to the mechanism discussed by Benney (1977); here, how- energy in that model. [Using a modified form of the ever, on the level of the kinetic equation rather than of the nonlinear terms in Eq. (1), that is, governing differential equations. In Benney’s discussion, the gravity wave partners 1 and 2 of the resonantly inter- › 1 f 1 =h 52 : = 1 = acting triad are short, the generated Rossby wave 0 is a long tu Hu 0 5Ro( uu u u) and wave. A one-dimensional sketch of the resonance condition › h 1 c2= u 520:5Ro(= hu 1 u =h), (29) is shown in Fig. 1 of Williams et al. (2003). The gravity wave t energy, propagating with the appropriate group velocity, yields total energy identical to the total energy of the moves with the phase speed of the long Rossby wave and linear waves. Only for nondivergent flow with = u 5 0 energy can be transferred. Note, however, that we here do do the nonlinear terms in Eq. (29) become identical to not restrict the triad interaction to be resonant nor do we the ones in the full system of Eq. (1). The corresponding confine the discussion to scale-separated interacting triads. interaction coefficients derived from Eq. (29) (not shown) then indeed satisfy Eq. (23) up to numerical precision. These coefficients were used as a consistency 3. Numerical evaluations check for the numerical algorithm for the kinetic equa- a. Generation of linear gravity waves by tion of the reduced-gravity model.] balanced flow A power law with p 5 2 yields a so-called stationary spectrum for E0 with › E0 5 0. Energy becomes sta- To demonstrate the generation of linear gravity waves t from the balanced mode, we consider a certain given tionary since the kinetic equation [Eq. (24)] for the 5 spectral shape of the energy of the single-layer Rossby Rossby waves can be shown to vanish identically for p waves and initially no gravity waves; that is, 2(Kenyon 1964), which is also true for the numerical evaluation of Eq. (24) within numerical precision. In a 2 forced dissipative model, however, the spectrum cannot E0(k, t) 5 E [1 1 (jkjL)p] 1 R be stationary, and scaling laws suggest p 5 3 and an in- 6 and E 5 0. The constant E determines the total en- verse energy cascade (e.g., Salmon 1998). The numerical R 5 ergy. The kinetic equation is evaluated numerically on evaluation of the kinetic equation [Eq. (24)] for p 3 an equidistant grid in k (or K) space with an odd number and L 5 1ona403 40 spatial domain for different grid of grid points in all (positive and negative) directions resolutions is shown in Fig. 1. The Rossby wave fre- R including the point k 5 0. It is found that the wave- quency v is set in this case to zero, which results in a s number grid needs to be of this shape to allow for exactly symmetry of ›tE with respect to the wavenumber angle. s matching triads, for which all three wavenumbers can be Therefore, we show ›tE integrated over wavenumber exchanged. If that exchange is not possible by using a angle in Fig. 1 and the following figures in this section.

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Ð FIG. 1. (a) EnergyÐ distribution of the balanced mode E0(k, f) df and (b) the corresponding energy transfer within the 0 R balanced mode ›t E (k, f) df given by Eq. (24) for v 5 0, Ro 5 0.05, b 5 0, and p 5 3. Here, k denotes wavenumber 2 modulus and f is wavenumber angle, with k 5 k(cosf,sinf). The energy follows a power law of k 3 for large k. The solid, dashed, dotted, and dash–dotted lines in (b) correspond to calculations using 1272, 2552, 5112 and 10232 grid points, respectively.

All interactions become resonant and the kinetic where z(k) is a complex normal number with 0 mean equation [Eq. (24)] simply becomes a linear function of and a variance of 1. Different realizations of z are used for time. Figure 1 shows energy gain (loss) at small (large) k each ensemble member using a random number generator. indicative of an inverse energy cascade for all grid res- For consistency, we use eigenvectors Qs (and Ps)appro- olutions. The magnitude of the energy gain at low k priate to the discrete model equations given in appendix D depends on the grid resolution; for the grid with 1272 instead of the analytical version of the eigenvectors Qs grid points it is roughly half as large as for the grid with (and Ps) given in appendix A. The discretization of the 10232 grid points. At highest resolution, however, the nonlinear terms follows the energy-conserving scheme by magnitude of the maximal energy gain appears to con- Sadourny (1975) in the momentum equation, and we use verge. The region of energy loss extends farther toward second-order differences for the thickness advection. En- larger k for higher grid resolution, but the energy gain ergy density Es is diagnosed from the model ensemble 2 s s for k . 9 for the grid with 127 grid points (and k . 18 using the eigenvector P , and the energy transfer ›tE is for 2552 grid points) is an artifact of the numerical calculated from finite differences of Es. evaluation on a grid with finite k, since it moves to larger k Damping in the model is necessary for a stable in- using higher resolution. Note that these dependencies on tegration and given by biharmonic friction and diffusion grid resolution are similar to those of numerical models and by the time stepping scheme which is chosen as a (see below). The effect of nonvanishing vR is an asym- quasi-second-order Adam–Bashforth interpolation with 0 metry of ›tE with respect to the wavenumber angle (not adjusted weights to allow for stable simulations of shown). The region of energy gain is shifted toward smaller gravity waves of highest frequency. Both damping ef- zonal wavenumber, while its meridional wavenumber fects decrease the energy proportional to Es since both range stays similar to the region of energy gain in Fig. 1a). effects are linear. To eliminate the damping effect from This means that a zonalization of the energy by the b effect the diagnosed energy transfers in the model, we subtract 0 can be seen, as first described by Rhines (1975). from ›tE diagnosed from the model ensemble, the 0 To validate the results of the kinetic equation, we use corresponding ›tE diagnosed from an identical model an ensemble of numerical model simulations with the ensemble but without nonlinear terms, that is, setting same parameters as used for the evaluation of the ki- Ro 5 0 in Eq. (1). Since the damping effect is linear and netic equation. Each ensemble member is initialized acts in a very similar way in the ensemble of the linear with the same energy distribution and the full model, we can isolate the energy transfers due to the nonlinear terms in this way and can compare 0 5 1 p 21 E j 5 E [1 (jkjL) ] them with the predictions from the kinetic equation. t 0 R 0 Figure 2 shows that the diagnosed ›tE in the model 6 and E jt50 5 0 but with a random phase. Specifically, we ensemble for three different times has indeed the are using as initial condition for the model the Fourier expected shape of an inverse energy cascade. As for ^ 5 0 1/2 0 1/2 0z › 0 transform of the state vector z(k)jt50 (2E ) (n ) Q , the prediction by the kinetic equation, tE increases

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Ð › 0 f FIG. 2. Balanced mode energy transferÐ t E d in a model ensemble with 1000 members at three different times initialized and 0 diagnosed as described in the text, and ›t E df predicted by the kinetic equation. Parameters are chosen as in Fig. 1. The dotted, dashed, and dash–dotted lines correspond to a model with 1272, 2552, and 10232 grid points, respectively, and the solid lines to an evaluation of Eq. (24) with 1272 grid points. linearly in time, and also at a similar rate. The energy energy level in E0, it is getting stronger (weaker) for p , 3 transfer depends in the model, however, on the details of (p . 3) and is for p 5 2 approximately 2 times as large as the discretization of the nonlinear terms and in partic- in Fig. 3c. To test whether the nonconservation feature 0 ular on model resolution. At small wavenumbers, ›tE is for total energy in the reduced-gravity model modiﬁes the lower than predicted by the kinetic equation using the gravity wave generation shown here, we repeat the 0 same grid, but ›tE in the model increases by increasing evaluation in Fig. 3 with the model given by Eq. (29),in the model resolution. The region of energy loss is also which energy is conserved exactly. This does not change shifted toward smaller k relative to the prediction by the the results. Including the b effect also does not change the kinetic equation but appears to converge to the pre- results in terms of the gravity wave generation. 6 6 diction using higher resolution. Figure 4 shows ›tE and the increase of E inte- With the kinetic equation [Eq. (21)], we can also de- grated over k in the model ensemble. The time evolution 6 scribe the generation of linear gravity wave energy by of ›tE is very similar to the prediction of the kinetic nonresonant interactions due to the mixed components equation, although the amplitudes are larger such that 6 in the ﬁrst sum of Eq. (21). The energy gain of the gravity the total increase in E is also larger than predicted. 6 mode E is given for this case by Eq. (27), and similar Different from the energy transfer of the balanced 0 6 for the balanced mode E . Figures 3a and 3b show the mode, ›tE depends only weakly on grid resolution in 6 energy gain of E0 and E by the mixed components as a the model but strongly on details of the discretization function of time and wavenumber k. Both functions (not shown). It is therefore likely that the mismatch in oscillate in time with smaller frequencies (but larger the amplitude of the wave generation between model amplitude) for smaller k. There is also a net total energy and prediction is due to the model numerics. gain integrated over k and time for the gravity wave 6 b. The role of the slaved gravity wave mode model E and a corresponding loss of energy for the 6 balanced mode E0. The largest energy gain of E shows Part of the generated energy in the total linear gravity up near k 5 1, that is, close to the Rossby radius. The wave mode in the experiments discussed in the previous effect also shows up for different spectral laws and even section is related to the so-called slaved gravity wave for the stationary spectrum p 5 2. For the same total mode (e.g., Leith 1980; Vanneste 2013). To show this,

Ð › f 0 6 FIG. 3. Energy transfer t Ed (a) toward (if positive) balancedÐ modeÐ EÐ and (b) toward wave mode E by the mixed components in t 0 0,6 0 0 the ﬁrst sum of Eq. (21). (c) The integrated cumulative energy gain dt dk df›tE (t , k) for the balanced mode E (blue line) and the 6 0 wave mode E (red line). Parameter are chosen as in Fig. 1.

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Ð › f 6 2 FIG. 4. (a)Ð EnergyÐ transfer t Ed toward (if positive) wave mode E in the model ensemble with 127 grid points. (b) Integrated wave energy gain dk dfE6(t, k) in the model ensemble. Also shown as a dashed line is the integrated energy gain related to the slaved wave s s mode as defined in the text. (c) Exemplary time series of h at a single point from the inverse Fourier transform of the slaved mode å6f Q s s (solid) and the total wave mode å6g Q (dashed). Parameters are chosen as in Fig. 1. consider Eq. (14) without applying the interaction rep- should be considered to be part of the balanced mode. s 5 s ^ 0 0 1 s s resentation. The state vector projection g P z is When the model is initialized with g Q ås56f Q the then governed by fast oscillation is eliminated to first order, which makes ð ð the nonlinear balancing method useful for, for example, s s s s s s ,s ,s › 0 2 v 0 0 52 å 1 2 0 1 2 initializing numerical weather forecast models with ob- tg0 i 0 g0 i dk1 dk2 g1 g2 Ck ,k ,k 5 6 0 1 2 s1,s2 0, served states. In Kafiabad and Bartello (2016, 2018) and 3 d(k 1 k 2 k ). (30) Chouksey et al. (2018), the method is used to diagnose 1 2 0 gravity wave activity in model simulations. The gravity waves are initially vanishing and then— An exemplary time series of the inverse Fourier å s s according to Hasselmann’s assumption—weakly grow- transform of the actual wave mode s56g Q in terms of 6 ing, such that g 5 O(Ro). The time derivative in the model’s height field h in one ensemble member, Eq. (30) includes the fast oscillations with frequency vs0 together with the same quantity related to the slaved s and the slow growth of the amplitude. Kafiabad and mode f , is shown in Fig. 4c. Our method to calculate the Bartello (2018) introduce therefore a fast and slow time slaved mode follows Chouksey et al. (2018), except that 5 › 5 › 1 › s the eigenvectors that we use are the ones for the discrete scale with T Rot* and t Ro T t* acting on g . system from appendix D and that the nonlinear function This yields to first order in Ro for the gravity mode s0 56 as follows, N is evaluated by integrating the model one time step, ð ð in order to be fully consistent with the discretization s › 6 2 v6 6 52 0 0 6,0,0 d 1 2 in the model. While h related to the slaved mode f is g0 i 0 g0 i dk dk g1g2Ck ,k ,k (k k k ), t* 1 2 0 1 2 1 2 0 only slowly varying, h related to the total linear wave 6 (31) mode g shows fast oscillations on top of the slow variation, but both are on the same order of magnitude. The 6 since the interaction coefficients C are also of O(Ro). total energy related to f (Fig. 4b) is about one-half of 6 The nonzero right-hand side will generate an oscillating the total energy related to g . It is likely that the same 6 6 5 fast mode g even if g jt50 0. To suppress this oscil- holds for the prediction by the kinetic equation, but here › 6 5 lation it is necessary that t*g0 0or it appears not possible to differentiate between slaved ð ð mode and wave mode. 6 [ 6 5 v 6 21 0 0 6,0,0 d 1 2 g0 f0 ( 0 ) dk dk g1g2Ck ,k ,k (k k k ). 1 2 0 1 2 1 2 0 c. Generation of balanced flow by gravity waves (32) To demonstrate the generation of balanced flow s s by the gravity waves by resonant and nonresonant in- The part å 56f Q of the state vector ^z (or its inverse s teractions, we consider the vertically resolved model Fourier transform) is also called the ageostrophic bal- since here only resonant interactions of gravity waves anced or slaved wave mode and is identical to the cor- are possible (to first order). Such interactions are be- rection in the first iteration of the nonlinear balancing lieved to generate the so-called Garrett–Munk spectrum method by Machenhauer (1977) or Baer and Tribbia 0 given in dimensional form by (1977). Since it is given by the coefficients g , which vary on the slow time scale T only, the same holds for the 1 2 6 E 5 E A(m/m )m 1B(v) slaved mode f . It is therefore not fast but slow and gm * *

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1 0 1 FIG. 5. (a),(b) Content spectra of ›tE and (c) ›tE by resonant interactions for t / ‘ using a Garrett–Munk-like spectrum for E and initially E0 5 0 for the vertically resolved model as a function of horizontal and vertical wavenumber. Wavenumbers and energies are 2 shown in their dimensional form, corresponding to Ro 50.1 and Fr 5 2 3 10 3. The white lines denote v/f 5 2, 3, 4, 5, 10, and 20, and b~ 5 2 2 2 2 3 10 11 m 1 s 1, where b~ denotes the change of the Coriolis parameter with latitude in dimensional form.

(Cairns and Williams 1976; Munk 1981), where the for nonhydrostatic waves; differences are only seen 2 2 functions A and B yield power laws m 2 and v 2 for for v / N~. 5 ~ large arguments and where the parameter m* N/c* For the generation of balanced flow by the wave 5 21 and c* 0.5 m s correspond to a wavelength of about field given by Eq. (28), only the second term can be- 630 m for N~ 5 5 3 1023 s21 and small v. The functions come resonant, and it is shown in Fig. 5c for a finite A and B are normalized such that the total energy Rossby wave frequency. The generation of balanced 23 2 22 1 ofthewavefieldisgivenbyEgm 5 3 3 10 m s . flow is orders of magnitude smaller than ›tE .Itturns The spectrum, the energy transfers within the grav- out to be proportional to b; that is, it vanishes on an f ity wave field inferred from the kinetic equation, plane where v0 5 0. Thus, similar to the generation of and its dependency on parameters are discussed in gravity waves by the balanced flow, resonant in- detail in a companion study (Eden et al. 2019)for teractions play also no significant role for the gener- the nonhydrostatic case, whereas we concentrate ation of balanced flow by gravity waves. As before, here on the interaction of the wave field with the this changes again for the nonresonant interactions in balanced flow. Eq. (28), as demonstrated next. 0 Figures 5a and 5b show the content (or variance Figure 6a shows the nonresonant energy transfer ›tE 1 preserving) spectra of the energy transfer ›tE by the givenbyEq.(28) as a function of time t. Here, method 1 resonant interactions, excluding the nonresonant in- by Eden et al. (2019) is used; that is, all resonant and teractions as a function of vertical wavenumber m and nonresonant interactions on the three-dimensional horizontal wavenumber amplitude k. Note that since wavenumber grid are calculated. Since this method 1 E is symmetric in m and isotropic in k, the figure is numerically more expensive than for the resonant 1 3 shows ›tE only for positive m and integrated over interactions, we use a smaller grid of 201 grid points horizontal wavenumber angle. The method we use to and a smaller domain size with 100-km horizontal calculate the resonant interactions only are described extent and 10-km vertical extent. Similar to the gen- in detail in Eden et al. (2019) (their method 3); it uses eration of gravity waves by the balanced flow shown in the d function in the wavenumbers in Eq. (26) to re- Fig. 3a, there is initially a strong generation of bal- solve the integrals in (k2, m2)andthed function in the anced flow by the nonresonant, mixed terms given by frequencies to resolve the integral over m1.Onlythe Eq. (28), while after a fraction of a day the generation first (sum interaction) term and the second (difference of balanced flow ceases and oscillates around zero. interaction) term in Eq. (26) then contribute to the For t / ‘ the state of Fig. 5c will be reached. Figure 6b transfer. For the numerical calculation we use a grid of shows the cumulative time integral of the generation, 8012 3 1601 grid points in wavenumber space and a which becomes stationary after about one day. For domain size of 200-km horizontal extent and 10-km comparison, the figure also shows the energy transfers 24 21 ~ 1 vertical extent, V510 s ,andN 5 50V. Note that within the gravity wave field only in terms of j›tE j 1 wavenumbers and energies are shown in the figure in related to these interactions (the integral of ›tE their dimensional form, corresponding to Ro 5 0.1 and vanishes because of energy conservation), which do 2 Fr 5 2 3 10 3. The transfers are similar to the ones not decrease and lead to a continuous increase of the shown and discussed in detail in Eden et al. (2019) time integral.

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› 0 FIG. 6. (a) Nonresonant energy transfer tE given by Eq. (28) as function of time and horizontal wavenumberÐ Ð Ð k and t › 0 integrated over vertical wavenumber m. (b) Integrated cumulative total energy gain by theÐ balancedÐ Ð ﬂow 0dt dk dm tE t › 1 (blue), and the same quantity for absolute energy transfers within the gravity wave ﬁeld 0dt dk dmj tE j (red).

4. Summary and discussion energy as used for the kinetic equation shows a similar magnitude and shape of the energy transfer, and, as in Using the weak-interaction assumption by Hasselmann the kinetic equation, also a linear increase in time of the (1966)—that is, assuming that the wave amplitudes are energy transfers. This validation supports the validity of changing only slowly—kinetic equations for the non- the weak-interaction assumption necessary for the der- linear wave–wave interaction are derived that are closed ivation of the kinetic equation for the Rossby wave case. with respect to wave energy in wavenumber space. The Only nonresonant interactions can generate linear kinetic equations apply to a simple two-dimensional gravity wave energy from balanced flow, which are less layered model and the primitive equations, in both their important in the long run than the resonant interactions full versions and their quasigeostrophic versions. The between the Rossby waves but which can still generate result is identical to previous studies with respect to finite energy transfers. The evaluation of the kinetic Rossby and internal gravity wave interaction (e.g., equation of the generation of linear gravity waves by the Kenyon 1964; Longuet-Higgins et al. 1967; Olbers 1976; nonresonant interaction from a fully developed Rossby Nazarenko 2011), but here we also allow for the in- wave field shows energy transfers that are maximal near teraction of all waves present in the models, that is, (in- the Rossby radius and that are oscillating in time. In- ternal) gravity waves and Rossby waves. The latter tegrating the energy transfers in time yields a small but degenerate to the geostrophic mode with vanishing fre- positive generation of linear gravity wave energy for large quency on the f plane. t. A similar time evolution of the energy transfers related Since the interactions only within the gravity wave to the generation of linear gravity waves is seen in the field or only within the Rossby wave field conserve en- model ensemble, supporting the validity of the weak- ergy by themselves, the same is true for the mixed in- interaction assumption also for this case. Both in the teractions between linear gravity and Rossby waves in model ensemble and for the kinetic equation, the gravity the kinetic equation. We consider here the case of ini- wave generation scales as Ro2,whichcanbeseeninFig. 7. tially vanishing linear gravity wave energy in the pres- This can be easily understood by rewriting Eq. (27) as ence of fully developed Rossby wave field with a spectral ð 0 23 power law of E ; k in the layered model and the 6 6 › E 5 Ro2E2 a R[D(v )] / dkE j reversed case of initially vanishing linear Rossby wave t 0 R I 0 0 Dt ð energy in the presence of a fully developed gravity wave 1 2 cosv Dt ’ Ro2E2 dk a 0 (33) field in the primitive equation model with a spectral R 0 I v2 6 2 0 power law of E ; k 2. The kinetic equations for the two cases are numerically evaluated, for which energy is with a coupling coefficient aI(k0) derived from the di- conserved within numerical precision. mensionless integrals in Eq. (27) and assuming that E0 ’ 6 In the first case, the kinetic equation predicts for the const and E E0 for t ,Dt. The integral on the right- Rossby wave field an energy transfer indicative of an hand side of the expression for the integrated generated inverse energy cascade. A numerical model ensemble energy only depends on c (and time Dt), which is set with identical parameters and initialized with the same constant in all experiments.

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to the square of the Rossby number Ro, identical as we found, while other studies suggest more complicated relations (Vanneste and Yavneh 2004). Brüggemann and Eden (2015) find by a subjective fit increasing small-scale dissipation for larger Ro in a numerical model of baroclinic instability. In a similar model setup, Chouksey et al. (2018) find, by using the nonlinear gravity wave diagnostic tool described above, a dependency close to Ro2 for the fraction of generated gravity wave energy, which agrees with the dependency found by Williams et al. (2008).Sucha relationship of the wave energy emission on environmental parameters like the Rossby number can be used to physically constrain the coupling of eddy and wave energy reservoirs in the framework of consistent ocean model parameterizations, as suggested by Eden et al. (2014). Here we obtain also a dependency close to Ro2 of the generation of linear gravity wave energy by balanced flow. However, the comparison to the above cited studies is hampered by the generation of the so-called Ð Ð 6 FIG. 7. Integrated wave energy gain dk dfE (t, k) after t 5 10 slaved wave mode. This mode consists of gravity wave in the model ensemble (solid line and dots) and predicted by the modes that need to be added to the balanced modes in 2 kinetic equation (dashed line and dots) as a function of Ro . Also order to suppress the generation of fast modes by the å f s s shown is the energy gain related to the slaved mode 6 Q in the nonlinear terms to first order (in amplitude) in Ro. In model ensemble (solid line and stars), the energy gain related to the s s 2 s s the model ensemble, it can be shown that about one-half remainingÐ Ð waves å6(g Q f Q ) (dashed line and stars), and 6 dk dfE6(t, k)/Ro2, where E denotes the energy related to the of the generated linear wave energy can be attributed to s s 2 s s remaining waves å6(g Q f Q ) generated in a model ensemble the slaved modes. Figure 7 shows that both the genera- 0 0 1 å s s initialized with g Q 6f Q . tion of the slaved mode and the remaining wave energy also scale with Ro2. In agreement, Kafiabad and Bartello The reversed case is evaluated using the primitive (2018) find to first order (in amplitude) a dependency 6 2 equations and a wave spectrum following E ; k 2, that close to Ro2 for the fraction of ageostrophic (or gravity is, a Garrett–Munk spectrum of internal gravity waves. wave) energy generated by spontaneous generation in In this case resonant interactions between the gravity nonhydrostatic model simulations. waves and the balanced flow for initially vanishing To resolve the difficulty of the slaved modes, it is balanced flow are in principle possible but turn out to possible to initialize the model ensemble with the bal- 0 0 1 s s be orders of magnitude smaller than the resonant in- anced state g Q ås56f Q , instead of using only teractions within the gravity wave field in the numer- g0Q0, analogous to a nonlinear balancing method. ical evaluation. Similar to the reversed case, however, Here, g0 and f s are the amplitudes of the balanced nonresonant interactions can generate a small amount mode and the slaved mode, respectively. Doing so, of energy in the balanced flow, which seems to follow much less linear gravity wave energy is generated in s s the same scaling as for the reversed case of the gen- addition to the that related to ås56f P ,andthissmall eration of gravity waves from balanced flow. The val- energy generation scales with Ro4 as shown in Fig. 7.It idity of the weak-interaction assumption for internal is likely that this scaling behavior continues going to gravity wave interactions is discussed in Eden et al. higher orders in the balancing. Using the energy re- 0 0 1 s s (2019). It turns out that only for the high-frequency lated to g Q ås56f Q in the kinetic equation, 6 nonhydrostatic waves are the turnover time scales however, does not change the results much since jf j for those interactions getting smaller than the gravity jg0j,andEq.(27) still dominates the evaluation of the wave periods. kinetic equation. Gravity wave generation by balanced flow with wave- In conclusion, the kinetic equation is able to predict length close to the Rossby radius is also found by Williams the triad interactions leading to linear gravity wave (and et al. (2008) in laboratory experiments of baroclinic in- Rossby wave) energy generation correctly up to second stability and in many numerical model simulations (e.g., order in Ro but not beyond, but this is not surprising since Plougonven and Snyder 2007). Williams et al. (2008) it corresponds also to the error level in the derivation of estimate a relation of the gravity wave energy generation the kinetic equation. In the setup discussed here, the

Unauthenticated | Downloaded 09/26/21 12:46 PM UTC JANUARY 2019 E D E N E T A L . 305 slaved mode generation, which scales also with Ro2, The symmetric interaction coefficients for the reduced- complicates the predictions of gravity wave generation. gravity model in Eq. (1) are given by Other situations such as, for example, stimulated loss of s ,s ,s s ,s ,s 0s ,s ,s 0s ,s ,s balance, with an energetic wave field coexisting with C 0 1 2 5 C 0 2 1 5 0:5 C 0 1 2 1 C 0 2 1 (A6) k0,k1,k2 k0,k2,k1 k ,k ,k k ,k ,k balanced flow, will be investigated in later studies with the 0 1 2 0 2 1

s0 kinetic equation that was derived here. n s , s s s s 5 Ro 0 fq 0 * [q 2 (q 1 k ) 1 q 1 (q 2 k )] 2 0 2 1 2 1 2 1 Acknowledgments. This study is a contribution to the s s 1 (q 1 1 q 2 ) k/c2g. (A7) Collaborative Research Centre TRR 181 ‘‘Energy 1 2 Transfer in Atmosphere and Ocean’’ funded by the German Research Foundation. The authors are grateful to two anonymous reviewers for helpful comments. APPENDIX B

APPENDIX A Effects of Variations in f To obtain the first-order correction by the variation of Reduced-Gravity Model Properties the Coriolis parameter on the vanishing eigenvalue v0 in The linear system matrix A(k), the state vector ^z(k, t), the reduced-gravity model, we rewrite the linear version and the vector function N for the reduced-gravity model of Eq. (1) as in Eq. (1) are given by › d 2 f h 1 bu 1 =2h 5 0, › h 1 f d 1 by 5 0, and 0 1 0 1 t t 0 2if 2k u^ B x C › h 1 c2d 5 0 (B1) B 2 C B C t A 5 @ if 0 ky A, ^z 5 @ ^y A, and ^ with the divergence d 5 = u and the vorticity h 5 = u. 2c2k 2c2k 0 h H 0 x y 1 Here, a small variation of the Coriolis parameter was (u^ k )u^ introduced, that is, f / f 1 by, before taking vorticity B 1 2 2 C N 5 B ^ ^y C and divergence, but we need to take f constant again in Ro@ (u1 k2) 2 A. (A1) Eq. (B1), in order to apply the Fourier transform. ^ ^ 1 h2u1 (k1 k2) Equation (B1) becomes 0 1 A The eigenvectors of the system matrix are given by 0 if ik2 B C s › ^0 1 A0 1 bB ^0 5 A0 5 B 2 C q (k) tz i( ) z 0, @ if 00A Qs 5 and Ps 5 ns[ps(k), c22] (A2) 1 2ic2 00 0 1 5 6 2k /k2 k /k2 0 for s 0, and the horizontal components B x y C B C and B 5 @ 2k /k2 2k /k2 0 A (B2) if k 1 sjvjk y x H qs(k) 5 and 000 f 2 2 s2v2 if k 2 sjvjk after Fourier transform and using the state vector H 2 ps(k) 5 ns 5 nsq s(2k) 5 nsqs(k)*. (A3) ^0 5 d^ h^ ^ T vs A0 s2v2 2 f 2 z ( , , h) . The eigenvalues of are identical to the ones for A,inparticularv0 5 0, for which we want to find s s0 A0 Orthonormality holds (i.e., P Q 5 ds,s0 ) with the first-order (nonzero) correction. The eigenvectors of are related to Qs and Ps from Eq. (A2) byalineartrans- 2v2 2 2 ^ 5 ^0 s 5 s s js f j formation T with z T z and are denoted by R T Q n (k) 5 . (A4) 2 1 2 22 1 2 s 5 s 1 (1 s )(LR k ) and S P T . To find the eigenvalues and eigenvectors ofthefullmatrix(A0 1 RoB), we expand them in the small For the special case of pure inertial waves for k 5 0 and parameter Ro (or b)usingvs and Rs as zero-order and 6 v 56f, the eigenvectors are given by vs s unknown first-order 1 and R1, which yields 0 1 2si 1 (A0 1 RoB) (Rs 1 RoRs ) 1 O(Ro2) Qs(0) 5 @ s2 A and Ps(0) 5 (si, s2, 0). (A5) 1 2 5 vs 1 vs s 1 s 1 2 0 ( Ro 1)(R RoR1) O(Ro ). (B3)

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A0 s ,s ,s s s s0, s To zero order in Ro, we find the eigenvalue problem for 0 0 1 2 5 n 0 [q 1 (k 2 k m /m )(q * q 2 1 m /m )]. CK ,K ,K 0 1 2 1 2 1 0 2 2 0 B s 1 A0 s 5 vs s 1 vs s 0 1 2 again and to first order R R1 1R R1. Multiplying with Ss yields (C2) vs 5 s B s For m 5 0 and m 5 0, the corresponding terms vanish; 1 S R . (B4) 0 1 the asterisk denotes complex conjugate. The normali- bvs s 5 The first-order correction 1 to the zero-order eigen- zation n is the same as in Eq. (A4) but for LR cm/f and values vs can now be found with the eigenvectors of A0 v as defined in this section. given by 0 1 sjvj APPENDIX D B i C B c2 C B C B C v 2 s B 2 2 2 2 C s s sj j fc Discrete Model Properties R 5 B s v 2 c k C, S 5 n 2i , ,1 , B C k2 s2v2 2 f 2 B c2f C The linear discrete equations for the reduced-gravity @ A model are given by 1 j2 j1 › u 5 f y i1 2 d1h , › y 52f u i2 2 d1h , and 1 js2v2 2 f 2j t i,j i,j x i,j t i,j i,j y i,j and ns 5 . (B5) 1 1 s2 f 2 1 k2c2 › 52 2 d2 1 d2y thi,j c ( x ui,j y i,j), (D1) 5 For s 0, this yields in Eq. (B4) indeed the familiar d1 5 2 with the finite-differencing operators x hi,j (hi11,j vR 5 v0 1 bv0 2 Rossby wave dispersion relation 1 with 6 hi,j)/Dx and d hi,j 5 (hi,j 2 hi21,j)/Dx, where Dx is the grid v0 52k /(k2 1 L22). The corrections to v are not x 1 x R spacing in the x direction, and the averaging opera- relevant here, since they are small when compared with i1 i2 tors hi,j 5 (hi,j 1 hi11,j)/2 and hi,j 5 (hi,j 1 hi21,j)/2, and the zero order. The same holds for the eigenvectors. similar for the y direction. Dissipative and nonlinear terms and the time discretization are omitted in APPENDIX C Eq. (D1)Ð but are included in the numerical model. With ^ ikx T ui,j 5 dku(k)e ,where x 5 (iDx, jDy) and similar for Primitive Equation Properties the other variables, the Fourier transform of Eq. (D1) yields For the primitive equation model the linear system matrix A(k, m), the state vector ^z(k, m, t), and the vec- 1 2 ^1 ^ 1 2 ^1 ^ › u^5 1 1 f^y 2 ik h, › ^y 521 1 f u^2 ik h, and tor function N are given by t x y x t y x y › ^52 2 ^2 ^1 ^2^y 0 1 th c (ikx u iky ), (D2) 2 2 0 1 0 if kx u^ ^1 D 1 D ^2 B C B C ik k 5 eik x x 2 D k 5 eik x x 1 k 5 A 5 B if 0 2k C ^ 5 @ ^y A with x ( x) ( 1)/ x,1x ( x) ( 1)/2, x @ y A, z , and ^1 2 5 1 ^1 ^2 (k )*, 1x (1 )*, and similar for ky , ky , and so forth. 2 2 2 2 ^ ^1 1 cmk cmk 0 p Note that k / k and 1 / 1 for D / 0. The system 0 x y 1 x x x x matrix A then becomes u^ u^ (k 2 k m /m ) B 2 1 2 1 2 1 C N 5 B ^y u^ (k 2 k m /m ) C (C1) 0 1 @ 2 1 2 1 2 1 A 2 1 2 2 ^1 0 i1x 1y f kx p^ u^ m /m(k 2 k m /m ) B C 2 1 2 2 1 2 1 A 5 B 1 2 2 ^1 C @ i1y 1x f 0 ky A (D3) 2 5 2 2 2 2 ^2 2 2 ^2 with cm N /m . The second terms in the parentheses c kx c ky 0 in N result from the vertical advection and vanish for m1 5 0, and for m 5 0 the third component of with real eigenvalues N vanishes entirely. The eigenvalues of A are given 6 1/2 v(0) 5 v( ) 56 2 1 2 2 0 by 0and (f cmk ) , and the eigen- v 5 0 and vectors are the same as for the reduced-gravity model qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi v v6 56 2 ^2 ^1 1 ^2 ^1 1 1 2 2 1 2 but depend via on m, along with the Rossby radius c (ky ky kx kx ) 1x 1y 1x 1y f (D4) LR 5 cm/f. The (asymmetric) interaction coefficients are and eigenvectors given by Qs 5 [qs(k), 1]T and Ps 5 given by ns[ps(k), c22] with horizontal components

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