JANUARY 2019 E D E N E T A L . 291 Mixed Rossby–Gravity Wave–Wave Interactions 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 Á u and that allow, in the linearized case, for long and short wave › h 1 c2= Á u 52Ro= Á 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.