1. Introduction
The question of “spontaneous” emission of inertia–gravity waves (IGWs), which is being actively discussed in the literature, starting from the pioneering paper (O’Sullivan and Dunkerton 1995), in the context of realistic (e.g., Plougonven and Snyder 2007) or idealized (e.g., Dritschel and Viúdez 2007) numerical simulations, is a question of the limits of decoupling of fast (waves) and slow (vortices) motions in geophysical fluid dynamics and, therefore, a question of the limits of balanced models based on such decoupling. In what follows we will review some recent results on fast–slow decoupling and the absence of spontaneous IGW emission at small Rossby numbers Ro. We will, however, point out some phenomena that can be confused with spontaneous emission at small Ro. We then display dynamical mechanisms leading to violation of fast–slow decoupling and emission of IGWs at increasing Rossby numbers and attempt to classify the emission mechanisms on the basis of the fundamental parameters: Rossby and Burger numbers.
The natural context for studying fast–slow decoupling is geostrophic adjustment of localized disturbances. This fundamental process, ubiquitous in the atmosphere and oceans, consists of the relaxation of any perturbation toward the state of geostrophic equilibrium (balance). Such process is an obvious source of IGW because any initially unbalanced disturbance is getting rid of its unbalanced part by emitting waves. This IGW emission is primary. As will be shown below, a much weaker secondary emission also exists and may be quantified. At least at small Rossby numbers, the balanced (vortex) and unbalanced (IGW) parts of the flow are well identified and are well separated dynamically. However, the presence of the secondary IGW emission means that this separation is not complete. At large Rossby numbers, the dynamical separation of balanced and unbalanced motions may be strongly violated because of qualitatively new phenomena, such as ageostrophic instabilities of balanced flows or Lighthill radiation, both leading to IGW emission.
The paper is organized as follows: We first review the fast–slow decoupling in the framework of the geostrophic adjustment at small Rossby numbers in a hierarchy of GFD models of increasing complexity (rotating shallow water, multilayer rotating shallow water, and continuously stratified primitive equations), showing the limits of fast–slow separation in each regime, with special emphasis on the unbalanced part which, under some circumstances, may be misinterpreted as “spontaneously emitted” IGWs. We then give an overview of new dynamical mechanisms arising at large Rossby numbers and their role in fast–slow coupling and IGW emission.
The presentation is mostly based on the works of the author with collaborators and is not an exhaustive review. The relevant references may be found in the papers cited below.
2. Nonlinear geostrophic adjustment and fast–slow splitting at small Ro
In this section we give a brief review of the analytic results on nonlinear geostrophic adjustment of localized initial disturbances at small Rossby numbers.
a. Analytic approach: The method
The method (Reznik et al. 2001; Zeitlin et al. 2003; see Reznik and Zeitlin 2007 for a review) consists in considering an initial-value problem for an arbitrary localized initial disturbance treated by perturbation theory in Rossby number Ro. The technique uses asymptotic expansions in Ro and time averaging. Smallness of the Rossby number is always assumed, which does not necessarily mean that nonlinearity is small. In fact, the nonlinearity may be arbitrary, provided that the geostrophic balance is verified at the lowest order. The typical spatial scales are either of the order of or much greater than the Rossby deformation radius Rd. Multiple time scales are used: t0 ∼ (1/f ), t1 ∼ (1/Rof ), . . . , where f is the mean value of the Coriolis parameter [the midlatitude tangent plane approximation is being used; for adjustment in the equatorial tangent plane, see Le Sommer et al. (2004)]. In this approach, the slow dynamics appears from the removal of resonances in the fast one by fast-time averaging.
rotating shallow water (RSW),
2-layer RSW (2RSW) with a flat bottom and rigid lid boundary conditions, and
hydrostatic primitive equations (HSPEs) with a flat bottom and rigid lid boundary conditions.
the Rossby number Ro = (U/fL), where U and L are typical velocity and horizontal scales, respectively;
the nonlinearity λ [i.e., the normalized typical deviation of the isopycnal (ocean) or isentropic (atmosphere) surfaces from the equilibrium positions]; and
the Burger number Bu = (R2d/L2), where Rd is the deformation radius.
the quasigeostrophic (QG) regime, in which Ro = ϵ(small), Bu = O(1) with small deviations of the isopycnal (isentropic) surfaces, and
the frontal geostrophic (FG) regime, in which Ro = ϵ(small), Bu = O(Ro) with large deviations of the isopycnal (isentropic) surfaces.
b. Fast–slow splitting in RSW
In this subsection we present the results (Reznik et al. 2001) of the above-described approach applied to the RSW model. They corroborate the results of earlier studies on fast–slow splitting (Warn et al. 1995; Embid and Majda 1996; Babin et al. 1997a), with notable differences of (i) considering localized initial conditions, (ii) going farther in asymptotic expansion in Ro, and (iii) considering the FG regime as well as the standard QG. The advantage of using the open boundary conditions and working with localized initial disturbances in all models below is that, in addition to allowing us to assess the radiation efficiency of the adjustment process, we get rid of most of the wave–wave resonances [studied in detail in Embid and Majda (1996) and Babin et al. (1997a)]. The only resonances which do give contributions below are those corresponding to so-called “catalytic” interactions (Lelong and Riley 1991).
1) The QG regime
Thus, in the leading order in Ro the standard QG dynamics for the balanced part of the initial perturbation is recovered, and IGWs run out of the location of the perturbation. Balanced and unbalanced motions are noninteracting. Once the first-order correction is taken into account, which is necessary for applying slow dynamics at times longer than 1/Rof, the standard QG for balanced motion is “improved.” This may be interpreted as “waves” giving corrections to the slow manifold, although there is no fast-motion drag in the slow-motion dynamics. However, the first-order correction [the first term on the rhs of (5)] to the fast motion gives a secondary IGW emission due to fast–slow interaction.
A preliminary conclusion from this analysis is that although fast–slow (balanced–unbalanced) splitting is not complete, the effect of secondary emission is small at small Ro. To have such secondary emission, the fast component should be present in initial conditions. Its appearance, thus, is not spontaneous, although it may be confused with such if the initial conditions are not properly posed. It is to be noted that as in all other models/regimes considered below, the consistency of the approach relies on nongeneration of scales essentially smaller than L (leading to the violation of the small Rossby number hypothesis) in times less than 1/Ro2f because of the intrinsic dynamics of the improved QG equation.
2) The FG regime
c. Fast–slow splitting in 2RSW
In this section we present the results (Zeitlin et al. 2003) of the application of the general method in the case of two-layer RSW model, which is the simplest one taking into account the baroclinic effects.
1) The QG regime
2) The FG regime
Fast motion consists of inertial oscillations with ω ≃ f. They split out from the slow motion and exert no drag upon this latter. The slow motion concerns both vortex and wave components and consists of the evolution of the barotropic (P) and baroclinic (η) pressure fields and the slow modulation of the envelope of the inertial oscillations
The modulation of the wave field is thus guided by the pressure field in both subregimes, whereas the pressure evolves without feeling any influence of the waves. The same conclusion as in the FG regime in RSW is valid for the second subregime (layers of essentially different depth), namely, that guiding of the envelope of the fast motion by the slow one may give amplification of the former in specific locations. The dynamical separation of slow and fast components is, thus, incomplete, in spite of the absence of any fast-motion drag on the slow one. The manifestations of the fast motion are, nevertheless, not spontaneous in the sense that the presence of the fast component in the initial conditions is required. An even more drastic “non-decoupling” of slow and fast components arises in the first subregime (layers of comparable depth). Although there is still no drag of the fast motion on the slow one, the envelope of the inertial oscillations evolves at the same slow time scale as the underlying (guiding) slow field. Again, the manifestations of the fast field on the background of the slow one are not spontaneous, although the focusing of the inertial oscillations field due to guiding may lead to confusion with such, as in the RSW case.
d. Fast–slow splitting in HSPE
In this section the results (Zeitlin et al. 2003) of the application of the general method to hydrostatic primitive equations are presented. Again, they corroborate the earlier results (Embid and Majda 1998; Babin et al. 1997b), with the distinction of treating localized initial conditions, and thus wave emission, including higher-order corrections in Ro, and treating the FG regime.
1) The QG regime
The same conclusions as in the RSW and 2RSW cases concerning fast–slow splitting and IGW emission hold in this case.
2) The FG regime
e. Discussion
We have seen in this section that although traditional balanced QG and FG models are proved to be self-consistent because of the absence of fast-motion drag on the slow motion (however, corrections leading to the improved versions are necessary to safely apply the models at longer time scales), the fast motion itself does feel the slow component. Both secondary emissions and guiding phenomena can lead to manifestations of wave activity in some locations at some times. To distinguish them from truly spontaneous emissions (see below), an accurate analysis of initial conditions and thorough identification of dynamical regimes are needed. The persistence of the quasi-inertial oscillations highlighted in the FG regimes in all models has a simple physical explanation: the group velocity of quasi-inertial waves is close to zero, and they are difficult to evacuate. This was repeatedly observed in both direct numerical simulations (e.g., Bouchut et al. 2004) and in experimental studies of the geostrophic adjustment (Stegner 2007).
3. Obstacles for the fast–slow splitting at large Ro and IGW emission
a. Lighthill radiation
The following observation is almost trivial: any motion that stirs the fluid at frequencies larger than the IGW threshold frequency f cannot avoid generating waves. This is also true for vortex motions, but the Rossby number should necessarily exceed unity in this case. Wave emission by vortices in the acoustic context was considered in the pioneering paper by Lighthill (1952). We remind below of the Lighthill radiation mechanism in nonrotating fluids in the technically simplest case of point vortices (following Gryanik 1983) and then make extension to the rotating case (following Zeitlin 2007). A more elaborated study with similar results was done in Ford et al (2000). The singular character of point vortices is not important in this context; basically the same results can be obtained for distributed vorticity configurations, like the elliptic Kirchhoff vortex (Zeitlin 1991). The extension from shallow-water to full primitive equations is also rather straightforward (Plougonven and Zeitlin 2002).
1) The nonrotating case
2) The effects of rotation
We therefore see that nonstationary vortex motions do spontaneously emit waves via the Lighthill radiation mechanism. This happens, however, in the region of parameters Ro > 1, Bu ≫ 1, far from that considered in the previous section. In addition, in this parameter regime it is the cyclogeostrophic (gradient wind) balance, rather than geostrophic one, that is relevant for localized vortex systems. Note also that the intensity of the radiation is weak, ∼M4. An important consequence of Lighthill radiation is its backreaction (Gryanik 1983; Zeitlin 1991; Plougonven and Zeitlin 2002), leading to a slow change of vortex parameters due to the radiative energy loss.
b. High Ro and baroclinicity: Symmetric/inertial instabilities of geostrophic motions
We show in this and the following subsections that in the presence of baroclinic effects, even in their simplest form, the balanced motions at high Ro may be subject to instabilities with respect to unbalanced perturbations. The subsequent development of such instabilities gives rise to secondary IGW emission. The first example is symmetric/inertial instability. We consider it in the “dry” case. Although it is known that dry symmetric instability is hard to achieve in midlatitudes (Bennets and Hoskins 1979), it gives an interesting example of ageostrophic instability of the balanced motion arising at large Ro. A peculiar feature of the inertial instability is its relation with the fast modes trapped by the slow one, which are destabilized beyond the threshold of the instability.
1) Trapped waves and symmetric instability in 2RSW
A new phenomenon appears at large enough Ro because of the baroclinic effects: a part of initial perturbation can be trapped inside the jet in the form of fast oscillations. The first consequence of this fact is that geostrophic adjustment becomes incomplete if the initial perturbation has a projection on such trapped modes. But even more drastic effects may arise because for anticyclonic shears with high enough Rossby numbers the eigenfrequencies of the trapped modes can become imaginary, thus leading to instability. This instability, which is called symmetric because of the absence of y dependence, is an instability of a balanced flow with respect to unbalanced perturbations and thus destroys the fast–slow decoupling.
If the potential is deep enough, that is, for strong enough anticyclonic shears (i.e., large enough Ro), nonoscillatory unstable modes with ω2 < 0, appear, producing the instability.
2) A Lagrangian approach to two-layer RSW equations
3) Nonlinear saturation of symmetric instability and secondary IGW emission
Equations (50) and (51) may be directly solved numerically. We present here simulations with a special profile υI = −2 cosh−2(2x), allowing us to get explicit eigenfunctions of the linearized problem and to compare linear and fully nonlinear simulations in Fig. 1. The nonlinear saturation of the symmetric instability with secondary emission of IGWs is visible in the right panel of Fig. 1.
Thus, at large Ro and small Bu the part of the IGW spectrum is trapped in the anticyclonic shear of the balanced flow. At large enough Ro, the gravest modes destabilize and lead to the development of symmetric–inertial instability, which in turn leads to a secondary emission of IGWs that may be considered spontaneous.
4) Trapped modes and symmetric instability in HSPE
For a jet with relative vorticity lower than −1 (−f in dimensional variables), modes with ω2 < 0 arise in the trapped mode spectrum, thus giving, as in the two-layer case, the symmetric instability.
The nonlinear and/or viscous saturation of the dry symmetric instability is not sufficiently understood in the continuously stratified case (cf., e.g., Thorpe and Rotunno 1989). It is, however, plausible that it will lead to IGW emission in the stratified background.
c. High Ro and baroclinicity: Nonsymmetric unbalanced shear instabilities of balanced flows
It is known that in the simplest 2RSW model the balanced adjusted states (35) may be unstable with respect to unbalanced perturbations, in addition to the classical baroclinic instability which is the core feature of these models (cf., e.g., Sakai 1989). Unlike the previously treated symmetric instability, these instabilities are “non-symmetric”, but arise at significant Rossby numbers as well. The example treated in the pioneering work of Sakai (1989) was a linear interface profile in the channel, giving the balanced constant-velocity flow in each layer (the Phillips model). The difference between this case and all the cases treated previously in the present paper is the presence of the lateral boundary. As is well known, the boundary changes the spectrum of linear perturbations and introduces the nondispersive boundary Kelvin waves. Their presence, although important for the overall momentum and mass balance (cf. Reznik and Zeitlin 2007), does not radically change either the adjustment scenario for small Rossby numbers or the equations for balanced motion, as was shown in Reznik and Grimshaw (2002). However, conjugated with baroclinicity, the Kelvin waves produce a new ageostrophic instability (Sakai 1989). Remarkably, the growth rates of this instability, called Rossby–Kelvin (RK) because of the nature of the waves at its origin, are higher than that of the standard baroclinic one. Below, we show the results of the analysis, similar to Sakai (1989), but in the general case of layers of nonequal depth. We show in Fig. 2 the unstable regions in the phase space of the model. In addition to the classical long-wave baroclinic instability arising at small Rossby number, three other unstable regions are distinctively visible. The RK instability arises at intermediate Ro for sufficiently long waves. The Kelvin–Helmholtz (KH) instability corresponds to the upper unstable zone of high Ro. Note that Bu may be taken to be of order 1 in the calculations. The corresponding growth rates are presented in Fig. 3.
There are different kinds of linear waves in the system: Rossby waves propagating because of the equivalent beta-effect produced by the inclined interface between the layers, Kelvin waves appearing because of the presence of lateral boundaries, and IGWs. The dynamical origins of the displayed instabilities are direct resonances between these waves propagating in each layer (Sakai 1989): respectively, Rossby–Rossby for the standard baroclinic instability, Rossby–Kelvin for the RK instability, and Kelvin–Kelvin, IGW–Kelvin, or IGW–IGW for KH instability. The wave patterns corresponding to the maximum growth rates of RK instabilities in Fig. 3 (indicated by dots) are presented in Fig. 4. One clearly sees the Rossby wave and the Kelvin wave patterns, respectively.
Although “pure” RK instability is produced by the presence of boundaries, other ageostrophic instabilities persist in the absence of boundaries. Apart from the classical KH one, characterized by the highest Rossby numbers, the Rossby–gravity instability (which can be seen as descending from the KH region branch in Fig. 2 and corresponding to the resonance between the Rossby and the IG wave) is, in principle, not sensitive to the presence of lateral boundaries. (Its typical growth rates are of the same order as for RK; not shown).
The simulations were initiated with a slightly perturbed front with perturbation at various wavenumbers k and with various Ro. A breeding procedure was applied to isolate the most unstable mode, if any, for each set of parameters. As a result, the unstable modes corresponding to those previously identified in the two-layer model were found, with very close growth rates. In particular, the RK instability was clearly seen. Figure 6 shows pressure and velocity fields vertically averaged below and above the front, respectively. A typical structure of the Rossby wave (geostrophic wind along the isobars) can be clearly seen below the front, whereas a typical Kelvin wave pattern (wind parallel to the lateral boundaries with pressure extrema on the boundaries) can be seen above the front. The growth rate of the instability was estimated from the kinetic energy plots. The growth rates along the line Ro = k/5 in the parameter space, corresponding to the section in Fig. 3 for the two-layer model, is presented in Fig. 7. The two are remarkably close. The subsequent evolution of the instability leads to its saturation, accompanied by secondary IGW emission (Gula et al. 2007).
d. Discussion
Thus, at high Ro and Bu the fast–slow splitting is violated because of the Lighthill radiation, although the intensity of the emitted wave field is weak. Note that our analysis of this effect was essentially two-dimensional, and the question of three-dimensional instabilities proper to such regimes arises (cf., e.g., Deloncle et al. 2007).
At high Ro and small or moderate Bu, (i) trapped subinertial waves appear; (ii) growth of trapped modes (symmetric/inertial instability) takes place in sufficiently strong anticyclonic shears (its nonlinear stage, saturated or not, gives rise to the IGW emission); and (iii) shear RK, KH, and related instabilities arise at sufficiently strong shears, thus giving rise, at the nonlinear stage, to the IGW emission.
Symmetric and shear instabilities are fast, as compared to slow baroclinic instability; therefore they violate the fast–slow decoupling. They also produce spontaneous secondary IGW emissions. The ageostrophic shear instabilities are not specific to the layered models; they are present as well in continuously stratified realistic configurations.
4. Summary and discussion
The following tentative classification thus arises in the parameter space of the atmospheric and oceanic flows with regard to decoupling balanced (slow) and unbalanced (fast IGW) motions and spontaneous IGW emission.
At small Ro and small or O(1) Bu,
1) corrections to standard balanced models at longer times do exist and may be interpreted as wave modifications of the slow manifold, but there is no fast-motion drag on the slow motion; primary IGW emission corresponding to evacuation of unbalanced part of a given initial disturbance is not sensitive to the balanced part;
2) secondary IGW emission at Bu ∼1 exists because of fast–slow interactions, but the presence of initial fast field is necessary; fast–slow coupling leads to guiding and possible focusing of quasi-inertial waves by the balanced flow at Bu ≪ 1.
At large Ro and small or O(1) Bu,
1) trapping of internal waves in anticyclonic shears of the balanced flow takes place;
2) symmetric and shear instabilities (of different types) arise as a source of secondary IGW emission.
At large Ro and large Bu,
1) the IGW emission via the Lighthill radiation mechanism takes place;
2) the backreaction of this radiation slowly changes the balanced motion.
Although it is important to identify the concrete mechanisms leading to the IGW emission at high Ro, in general there is no surprise that fast–slow decoupling, well established at small Ro, ceases at large Ro because the Rossby number, by definition, is a ratio of the slow (vortex turnover) to fast (inertial) time scales. Hence, the question of spontaneous imbalance is, in fact, a question of how the balanced, small-Ro flow could organize the high-Ro regions during its evolution. The processes that are able to create small-scale and/or high-velocity structures locally in the small-Ro flow, apart from external forcing and (steep) topography, are still ill understood. One way to address this problem is to thoroughly study the improved QG–FG equations and possible (cascade) processes which would be able to produce, starting from initial configurations with a single typical scale, regions of localized high vorticity (=high Ro) within the time scale of validity of these equations. It is worth noting that the conclusion based on the improved QG models that no IGW emission is expected up to times ∼ (1/Ro2f) is consistent with the results of Plougonven and Snyder (2007), in which the typical Rossby numbers of initial configuration were rather high Ro ∼ 0.4, and the zones of high local Rossby numbers IGW emission were observed after ∼6 days of evolution of the baroclinic instability.
Acknowledgments
The presented results are based on the work done in collaboration with J. Gula, J. Le Sommer, S. B. Medvedev, R. Plougonven and G. M. Reznik, which is gratefully acknowledged. Special thanks to P. Lelong and T. Dunkerton for organizing the “Spontaneous Imbalance” Workshop with its stimulating atmosphere.
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APPENDIX
A Hierarchy of the GFD Models
The RSW model
The 2RSW model
Hydrostatic stratified primitive equations
(left) Linear and (right) nonlinear simulations of the evolution of symmetric instability in 2RSW with initial condition ∼exp(−x2) for the Lagrangian displacements ϕ(x, t). Spatiotemporal evolution of ϕ(x, t) in the domain −10 < x < 10, 0 < t < 5 with periodic boundary conditions in x is shown.
Citation: Journal of the Atmospheric Sciences 65, 11; 10.1175/2008JAS2481.1
Unstable regions in the k–Ro plane for the two-layer RSW flow in the channel with constant opposite velocities in the upper and lower layer with aspect ratio d = 0.7. Isolines of the growth rate are shown at intervals of 0.02 with the lowest value of 0.01. Calculations are made by the high-resolution collocation method. The section of the growth rates across the indicated line is shown below in Fig. 3.
Citation: Journal of the Atmospheric Sciences 65, 11; 10.1175/2008JAS2481.1
The growth rates along the section shown in Fig. 2. Note that the growth rate of the RK instability exceeds that of the baroclinic one.
Citation: Journal of the Atmospheric Sciences 65, 11; 10.1175/2008JAS2481.1
Growing wave patterns for RK instability: (left) k = 0.4, Ro = 0.4; (right) k = 2.4, Ro = 0.5.
Citation: Journal of the Atmospheric Sciences 65, 11; 10.1175/2008JAS2481.1
The distribution of the potential temperature of the front in continuously stratified fluid.
Citation: Journal of the Atmospheric Sciences 65, 11; 10.1175/2008JAS2481.1
Vertically averaged pressure and velocity distributions for the RK instability in continuously stratified fluid for Ro = 0.5 and k = 2.4 (top) above and (bottom) below the front.
Citation: Journal of the Atmospheric Sciences 65, 11; 10.1175/2008JAS2481.1
The growth rates of the (left) baroclinic and (right) RK instabilities of the front in continuously stratified fluid.
Citation: Journal of the Atmospheric Sciences 65, 11; 10.1175/2008JAS2481.1
