## 1. Introduction

Gravity waves are a ubiquitous feature of the ocean and are generated by a variety of processes. One of them, the generation by spontaneous wave emission (e.g., Vanneste 2013), has recently attracted considerable attention. The process refers to the generation of gravity waves by balanced flow spontaneously in the absence of any external forcing, which appears to happen during baroclinic or barotropic instability (e.g., Plougonven and Snyder 2007; Hien et al. 2018; Chouksey et al. 2018), and is thought to be similar to sound generation in aerodynamics (Lighthill 1978; Ford et al. 2000), a process referred to as Lighthill radiation. Although it is intuitively clear to a physical oceanographer what balanced flow and gravity waves are, the exact definition of slow balanced flow and fast gravity waves is surprisingly difficult, since it is complicated by the presence of the nonlinear terms coupling both motions with each other. It is known that such nonlinearities can generate wavelike structures, which are nevertheless part of the balanced flow, the so-called ageostrophic balanced modes or slaved modes (e.g., Warn et al. 1995; McIntyre and Norton 2000; Kafiabad and Bartello 2018). The wavelike structures often seen in numerical simulations of lateral and vertical shear instabilities of the balanced flow could therefore be misinterpreted as spontaneous wave emission by the balanced flow, where in fact they are part of the balanced flow without the occurrence of spontaneous emission.

The most familiar model for the balanced mode is the result of the quasigeostrophic approximation, a first-order asymptotic expansion in the limit of small Rossby (Ro) and Froude number. Since the zero-order geostrophic velocity is free of horizontal divergence in that expansion, any significant horizontal divergence or vertical velocity seen in model simulations or observations is often interpreted as a gravity wave signal. It is, however, clear that the first-order ageostrophic vertical velocity—which can be calculated, for example, from the so-called omega equation (Hoskins et al. 1978)—although of first order in Ro and thus usually small, could also contribute to the observed vertical velocity. The ageostrophic vertical and horizontal first-order velocities in the quasigeostrophic approximation correspond to the ageostrophic balanced mode or slaved wave mode (to first order in Ro), and it is shown in Warn et al. (1995) how this concept of balanced models and wave modes can be extended to higher orders in Ro. This concept allows in principle to uniquely define the slow balanced flow and to differentiate the “true” fast gravity wave signal from that to arbitrary precision in terms of orders of Ro. The limit for infinitive order is called the slow manifold (and the difference to the full state the fast manifold), but it remains unclear if this limit exists (Lorenz 1980; Leith 1980; Lorenz 1986; Ford et al. 2000). In fact, for simple analogs to the equation of motions of the ocean and atmosphere it can be demonstrated that such an asymptotic limit does not exist (Vanneste 2013). In that case, the expansion in Ro is expected to diverge at large orders of Ro, although the lower orders of the expansion would still be useful for practical analysis.

In practice, however, the differentiation of waves and balanced flow in numerical models turns out to be rather difficult even at low order, such that the issue of the actual existence of the slow manifold appears to be of less concern. McIntyre and Norton (2000) applied a method to construct balanced models valid to higher order in Ro to a numerical model of the shallow water equations on a (half) sphere. They find that the balanced model simulation agrees surprisingly well with a model simulation using the full equations including the waves, such that they conclude that spontaneous wave emission remains unimportant in that simulation. To our knowledge, similar higher-order balance models have not yet been applied to numerical simulations of vertically resolved models. Therefore, we perform here a similar analysis as in McIntyre and Norton (2000) but for simulations of lateral shear instability in an idealized configuration applicable to the ocean for a range of values of Ro using a novel decomposition method at the discrete level of the model valid up to fourth order in Ro, and also extend this to the more realistic case of a vertically resolved model having also vertical shear instability.

In the next section, the theoretical background of the concept to differentiate waves from the balanced flow is outlined, while in section 3 the method is applied to numerical models of lateral and vertical shear instability. The last section provides a summary and discussion of the results.

## 2. Theoretical background

The theoretical background for the method to diagnose the model simulations following Warn et al. (1995) and Kafiabad and Bartello (2018) is outlined in this section. The focus is here on the analytical formulation, but since it is found that the numerical evaluation of the balanced modes needs exact consistency also on the discrete level of the numerical models, all relations on the discrete level which are actually used for the diagnosis are given in the appendix.

### a. The model

*f*and constant background stability frequency

*N*. The resulting primitive equations are scaled using

*L*,

*H*, and

*U*as time scale, horizontal and vertical length scale, and horizontal velocity scale, respectively. The vertical velocity scale follows from the continuity equation, the pressure scale from geostrophic balance, and the buoyancy perturbation scale from hydrostatic balance. This yields

^{1}

*h*, the parameter

### b. The vertical modes

*M*yields the vertical or baroclinic modes by its scaled eigenvalues

*M*allow to diagonalize the linear part of the system. The variables

*p*and

**u**are therefore expressed in terms of the eigenfunctions or vertical modes as

*n*are only coupled by the nonlinear terms

### c. The spectral representation

**x**,

*n*denoting the baroclinic mode number is omitted in Eq. (9) for simplicity. The matrix

*s*denotes here either the slow linear geostrophic mode (for

### d. The interaction integral

*C*result from the projection of

*s*, which is true for all kinds of models with quadratic nonlinearities.

### e. The balanced models

The spectral representation of the primitive equations given by Eq. (10) describes the evolution of the gravity wave amplitudes

*T*only since

### f. The ageostrophic balanced modes

*T*as in Kafiabad and Bartello (2018), which yields

### g. The effect of damping

In the numerical model it is necessary to include damping to suppress the gridscale noise generated by artificial numerical wave dispersion and the finite representation of the interaction integral *A* is used for momentum and buoyancy since then the only effect of the numerical damping is that the matrix

**k**. In that case, Eq. (22) and Eq. (24) are used replacing Eq. (27) to determine

**k**, but using the complex

## 3. Numerical experiments

Numerical simulation of shear instabilities of balanced flow are diagnosed in terms of ageostrophic balanced modes given by Eq. (27). A single-layer model and a primitive equation model are used. Although less realistic than the primitive equation model, we use the single-layer model here because of its reduced complexity for which the method can be easier applied and tested. Further, it is possible to generate larger lateral shear in the initial conditions in the single-layer model, since in the primitive equation model too strong shear can generate convective or symmetric instabilities during the subsequent integration, even when the initial conditions are convectively stable.

The spatial discretization of the primitive equation model is outlined in the appendix, and it follows the example of standard C-grid ocean models such as the MITgcm (Marshall et al. 1997). The discretization of the single-layer model is similar and detailed in Eden et al. (2019). The time-stepping scheme in both models is Euler forward for the first time step and afterward an Adam–Bashforth two-time level interpolation with adjusted weights to allow for a stable integration. In the primitive equation model, all vertical modes including the barotropic mode are integrated explicitly with sufficiently small time step, since we find that using an implicit method to solve for the barotropic mode to allow for a larger time step, the diagnostic of the higher-order ageostrophic balanced modes becomes inaccurate. The time step in the model is chosen according to the criteria given in the appendix.

### a. The wave diagnostic

The following method to differentiate gravity waves from balanced flow in the model solution is applied: any given model state *K* of the model state. The difference to the actual model state is interpreted as the residual gravity wave signal.

### b. The single-layer model

The single-layer model on a double periodic domain size of 10 × 5 is initialized with an unstable zonal flow with an initial profile of

We present simulations with the single layer using different parameters, from

Spatial mean

Figure 1 shows a time series of the ageostrophic balanced zonal velocity and the residual wave signal up to fourth order and the full thickness *p* at *u* denotes the actual zonal velocity in the model. Note that

Only in the fourth order (Figs. 1g,h) do small differences between

Figure 3a shows the residual wave energy

### c. The primitive equation model

The primitive equation model is configured similar to the single-layer model on a double periodic domain of size 5 × 5 × 1 resolved by 255 × 255 × 40 grid points, in the zonal, meridional, and vertical direction, respectively. The model is initialized with a zonal velocity shaped in the horizontal identical to the single-layer model, but here with the first vertical mode as vertical structure. The initial velocity, pressure, and buoyancy fields are balanced to first order in Ro; higher-order balancing does not change the results. Since we use scaled equations, the (scaled) background stratification (or Brunt–Väisälä frequency) is one and the Coriolis parameter is also one. Compared to the single-layer model, the amplitude of the initial zonal velocity *u* is reduced to 0.2 to exclude the occurrence of static instabilities with *u* and show up locally in the simulations. The scaled first baroclinic Rossby radius is

The scaled barotropic Rossby radius

As for the single-layer model, we present simulations for different Ro with a range from

As in Table 1, but at the surface of the primitive equation model. The last line shows the same quantities but for the simulation with a larger amplitude of 1 in the zonal velocity of the initial condition instead of 0.2.

Figure 4 shows the ageostrophic balanced vertical velocity up to fourth order, that is, from

The apparent wave signal at fourth order (i.e., in ^{2}, although we use the discretized equations consistent on the grid level to eliminate numerical inaccuracy (we found that small inconsistencies on the discrete level can lead to first-order inaccuracies in the flow decomposition). Similar to Fig. 3a, however, for

The situation changes when allowing for amplitudes of the zonal velocity in the initial conditions larger than 0.2. In this case, however, static instabilities with

Figure 5 shows the ageostrophic balanced vertical velocity up to third order, *w*, *w* a wave signal is present which dominates in the second and higher orders over the balanced mode which is shown in Figs. 5a, 5c, and 5e. The wave crests in Figs. 5b, 5d, and 5f are oriented roughly perpendicular with a wavelength of about 0.2–0.3 with a larger vertical mode than the initial conditions. The emergence of the wave signal shows up for all Ro we have tested (0.02–0.3) and can be related to the regions of static instabilities with (or close to)

Experiment with the primitive equation model without strong lateral but vertical shear, where predominantly baroclinic instability shows up, behaves the same as the above discussed experiments. For amplitudes in vertical shear that do not generate static instabilities, hardly any wave signal is detected, only with increased shear and occasional static instabilities a clear wave signal similar to Fig. 5 is generated.

## 4. Summary and discussion

This study is an elaborate attempt to find spontaneous gravity wave emission during lateral and vertical shear instability of balanced flow. We have designed a novel numerical tool to uniquely differentiate the slow balanced flow from the fast gravity waves up to fourth order based on an asymptotic expansion in the Rossby number Ro as suggested in, for example, Warn et al. (1995) and Kafiabad and Bartello (2018). Here, we apply the method the first time up to fourth order and in a model of growing instabilities to answer the specific question of spontaneous wave emission and its dependency on Ro, whereas the method was applied to a model of decaying isotropic turbulence by Kafiabad and Bartello (2018) only up to second order in Ro. The concept is applied here on the discrete level of the numerical models since we find that this high accuracy is necessary to allow for orders of the expansion larger than one. In principle, our tool allows for arbitrary precision, in practice this limit is hampered by numerical (lack of precision) and mathematical (lack of asymptotic limit) issues. Unlike the available approximate methods to differentiate between waves and balanced flow, such as the Lagrangian frequency method (Shakespeare and Taylor 2015) or the optimal potential vorticity balance method (Viúdez and Dritschel 2004), a Fourier transform of the model state is necessary in Warn et al.’s approach. The application of our decomposition tool is therefore more straightforward to idealized settings, whereas the approximate methods can be more readily applied to realistic scenarios but at the cost of a less accurate decomposition and perhaps a misinterpretation of the wavelike signals. We plan a comparison of the performance of the different methods in idealized models to transfer the results and to use the methods in the realistic ocean models as well.

In a numerical simulation of lateral shear instability with a single-layer model our novel tool can be successfully applied up to fourth order, which allows us to diagnose the gravity wave signal to this order of accuracy. Only in the case of a strong lateral shear, a near inertial wave signal is detected at higher orders of Ro in the model simulations. This result of weak wave generation is in agreement to McIntyre and Norton (2000) who also reported a minor role of wave generation in a single layer appropriate to the global atmosphere. The small but finite wave generation at large Ro we found could be related to spontaneous gravity wave emission in the sense of Lighthill radiation, but since local Rossby numbers well exceed unity, symmetric instability due to the lateral shear could also be the cause of the wave signal. The amplitude of the wave signal, however, shows an exponential scaling with respect to Ro in agreement to previous suggestions of classical spontaneous wave generation (Vanneste and Yavneh 2004; Vanneste 2013).

As compared to the decomposition up to fourth order in Ro in the single-layer model, our novel tool can robustly identify wave signals at least up to third order in Ro in the vertically resolved model, where the reduction in accuracy is most likely related to the error in numerical precision. In the vertically resolved model with lateral and vertical shear small enough to avoid static instabilities, the diagnosis and the scaling behavior with respect to Ro suggest that there is a negligibly small wave signal. Only if the shear becomes strong enough to generate static instabilities the emission of waves is detected with lateral wavelengths of a fraction of the Rossby radius and a higher vertical mode than the balanced flow. This emission is localized in space and time to the occurrence of static instabilities and the waves closely resemble the wavelike signals often seen before in numerical simulations (e.g., Plougonven and Snyder 2007; Hien et al. 2018; Chouksey et al. 2018).

Relatively larger amplitudes of waves generated by convective or symmetric instabilities compared to the ones generated by spontaneous emission are also found and discussed in Chouksey (2018) in a suite of numerical experiments with different configurations and for a range of different Ro. There, however, the method to differentiate waves from balanced flow follows the one by Machenhauer (1977) or uses the omega equation, and is thus only accurate to first order in Ro, such that the amplitude of spontaneous wave emission is most likely overestimated.

We cannot rule out the possibility that in other model configurations than what we have discussed here (and in Chouksey 2018) spontaneous wave emission in the sense of Lighthill radiation plays a more important role. We have tested several different configurations including also ones with vertical shear as before but only weak lateral shear and always found the same result: hardly any wave generation in general, but enhanced wave generation when static instabilities are generated by the balanced flow. In all such cases, the wave signal is localized in space and time to the sites of static instability. We also cannot rule out the possibility that waves at much larger or smaller wavelengths than allowed by our model grid can be generated. On the other hand, the model and laboratory experiments in the literature refer to spontaneously generated waves at similar wavelength as the dynamical process under investigation, in agreement to the (true) wave signal seen in our experiments with convective and/or symmetric instabilities.

These results suggest a relatively small role of spontaneous wave emission in the classical sense of Lighthill radiation, and emphasize the role of convective or symmetric instabilities during frontogenesis for the generation of internal waves. We speculate that the internal wave signals seen and discussed in previous studies based on numerical simulations and laboratory experiments (e.g., Plougonven and Snyder 2007; Hien et al. 2018; Chouksey et al. 2018) may also be generated by convective or symmetric instabilities instead of the classical spontaneous emission mechanism.

## Acknowledgments

This paper is a contribution to the Collaborative Research Centre TRR 181 “Energy Transfer in Atmosphere and Ocean” funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation), Projektnummer 274762653.

## APPENDIX

### Discrete Relations

*i*,

*j*,

*k*denote discretization in the

*x*,

*y*, and

*z*directions, but shown only when they are relevant in the individual terms. Finite differencing operators

*x*direction, and averaging operators

*M*has eigenvalues

*n*. The eigenvectors and eigenvalues of

*b*and

*w*are

*n*, the time step

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^{1}

All vectors are horizontal. The vector

^{2}

Note that tests with balanced initializations of the primitive equation model without any subsequent instability also show the absence of a fast adjustment by gravity waves up to third order, but a fast wave signal due to an adjustment from the initial conditions in the fourth order, indicating an imperfect balanced initialization on this order. These integrations are, however, not presented here.