## 1. Introduction

Zonal flows with a constant vertical shear of velocity and stably stratified density along the vertical and meridional directions (Eady flow) are the subject of plenty of geophysical and meteorological research because they are ubiquitously found in the atmosphere and ocean. Perturbations to this type of flows can be divided into two main classes: symmetric (independent of the streamwise coordinate) and nonsymmetric (with streamwise dependence). The theory of symmetric perturbations is quite well developed by now. Necessary and sufficient conditions for symmetric stability are formulated in fundamental papers such as Fjørtoft (1950), Ooyama (1966), and Xu (1986a). The linear theory of symmetric instability is presented in Stone (1966), Bennetts and Hoskins (1979), Emanuel (1979), and Weber (1980), among others. In particular, in these papers it is shown that flows with a constant vertical shear are symmetrically unstable if Richardson number Ri < 1. Subsequent evolution of symmetric instability leads to the emergence of billows in the plane perpendicular to the flow. Intensive development of symmetric instability theory is connected with the problems of generation of Hadley cells in the atmosphere, frontal cloud bands, band structure of the Jupiter’s atmosphere, etc. (Stone et al. 1969; Miller 1984; Miller and Antar 1986; Xu 1986b; Gu et al. 1998). There are also a number of important results obtained in the theory of nonlinear symmetric instability (cf. Cho et al. 1993; Mu et al. 1996; Kalashnik and Svirkunov 1996, 1998; Mu et al. 1999; Lu and Shao 2003).

Both meteorological and hydrodynamical research during the last two decades has revealed novel and important aspects regarding the dynamics of nonsymmetric perturbations in shear flows. Below we first outline the major breakthrough in this direction. It has long been established that nonsymmetric perturbations in shear flows (e.g., coherent structures) gain mean flow energy due to linear mechanisms. In other words, extraction of energy by perturbations can be analyzed using linearized governing dynamical equations (cf. Butler and Farrell 1992, 1993; Farrell and Ioannou 1993a,b,c; Foster 1997; Farrell and Ioannou 1998; Bakas et al. 2001). However, the operators existing in these linearized equations, when treated by means of a modal approach (i.e., spectral expansion of perturbations in time), are nonnormal because of the shear of mean velocity field. As a result, corresponding eigenfunctions are nonorthogonal and strongly interfere (Reddy et al. 1993), thereby very much complicating the understanding of perturbation dynamics. For this reason, the classical modal (or spectral) method of hydrodynamics is usually unable to predict and characterize finite-time–transient phenomena (existing even in the simplest shear flows) such as the emergence of coherent structures; they are simply overlooked in the spectral analysis. As an example of the importance of finite-time phenomena particularly in a planetary boundary layer, we would like to mention the work by Foster (1997), in which the perturbation amplification was studied in a neutrally stratified laminar Ekman boundary layer for Reynolds numbers typical of planetary boundary layers. In this work, it was shown that the optimal perturbations can grow because of nonnormality at least an order of magnitude larger than the growth of the most unstable normal mode.

The nonnormality of the operators results in two novel channels of energy exchange in the linear theory of smooth (without inflection point) shear flows. Through the first channel, perturbations transiently exchange energy with a mean flow. This channel is well perceived both by the hydrodynamical (cf. Gustavsson 1991; Reddy and Henningson 1993; Schmid and Henningson 2001) and meteorological (cf. Butler and Farrell 1992; Farrell and Ioannou 1993a,b,c, 2000; Bakas et al. 2001) communities. It forms the basis for the linear theory of the emergence of coherent structures (optimal perturbation theory; Farrell 1988; Farrell and Ioannou 1996) because these structures are generally determined by the perturbation that transiently grows the most, or by optimal perturbation (Butler and Farrell 1993; Farrell and Ioannou 1993a,c, 1998; Bakas et al. 2001). The second channel—linear mode coupling—ensures energy exchange among different modes/types of perturbations—for example, between vortices (in meteorology known as slowly varying, quasigeostrophic, balanced motions) and waves in hydrodynamical flows (nonresonant phenomenon of emergence of waves from vortices in smooth shear flows; Chagelishvili et al. 1997; Farrell and Ioannou 2000; Chagelishvili 2002) and in zonal meteorological shear flows (cf. Vanneste and Yavneh 2004; Kalashnik et al. 2006; Heifetz and Farrell 2007; Olafsdottir et al. 2008; Vanneste 2008; Bakas and Farrell 2009b). This second channel of transient energy exchange phenomena, as important as the first one, is not yet fully appreciated. Traditionally, coupling among different perturbation modes is associated with nonlinear effects [see the review by Riley and Lelong (2000) and references therein] whereas, as shown here and in the above papers, it persists even in the linear theory provided there is an appreciable shear of mean velocity field. These two basic linear phenomena—transient growth of perturbations and coupling among different modes—originating from the nonnormality must be taken into consideration to fully understand perturbation dynamics.

However, symmetric perturbations do not display the abovementioned phenomena induced by the nonnormality. All these arise only in the case of nonsymmetric perturbations and can be grasped by discarding the spectral–modal analysis and adopting so-called *nonmodal* approach. The first formulation of this approach consists in the transformation from the laboratory frame of reference to the frame comoving with a flow and then studying the evolution of *spatial Fourier harmonics*, or Kelvin modes of perturbations without any spectral expansion in time. This method, originally devised by Lord Kelvin (1887), has had a wide distribution since 1990 (cf. Craik and Criminale 1986; Farrell 1987; Criminale and Drazin 1990; Butler and Farrell 1992; Chagelishvili et al. 1997; Bakas et al. 2001; Vanneste and Yavneh 2004; Kalashnik et al. 2006; Bakas and Ioannou 2007; Bakas and Farrell 2009a,b). Recently, it has been mathematically proven by Yoshida (2005) that the Kelvin mode approach represents a canonical/optimal technique for stability studies of flows with constant velocity shear.

The second formulation of the nonmodal approach is a generalized stability theory (GST; Farrell and Ioannou 1996) that extends modal stability theory to comprehensively account for all transient processes in shear flows including the interaction between modes and the mean flow regardless of whether the modes are spectrally stable or not. The GST introduces a finite-time horizon over which an instability is observed. Finite-time stability analysis markedly deviates from the traditional Lyapunov stability concept (Khalil 2002) and it is not surprising that the perturbation that grows the most over a short time scale differs significantly from the spectrally least stable mode. Given a finite-time horizon, the GST is widely employed to quantitatively describe the perturbation dynamics (cf. Schmid and Henningson 2001; Schmid 2007). It reveals a remarkably rich picture of linear perturbation behavior that greatly differs from solutions of the modal approach. The GST is easily extendable to incorporate time-dependent flows, spatially varying configurations, stochastic influences, nonlinear effects, and flows with complex geometries. Heifetz and Farrell (2003, 2007) and Bakas and Farrell (2009b) used the GST in their analysis of mode coupling.

In this paper, we adopt the Kelvin mode or nonmodal approach and investigate the linear dynamics of perturbations in unbounded zonal inviscid flows in geostrophic balance having a constant vertical shear of mean velocity profile. A fluid is incompressible and density is stably stratified along the vertical and meridional (or spanwise) directions. We focus on the range Ri ≲ 1 for which the condition for symmetric instability is met (Stone 1966; Hoskins 1974). Such values of Ri can occur, for example, in fronts and jet streaks (cf. Koch and Dorian 1988; Bosart et al. 1998; Wakimoto and Bosart 2000, 2001) and in the lower stratosphere (see also Nakamura 1988; Lott et al. 2010). In fact, we investigate the transient dynamics of nonsymmetric perturbations versus symmetric instability in these flows. Our study shows that the nonnormality-induced transient dynamics (growth) of nonsymmetric perturbations can prevail over the symmetric instability for the range of parameters Ri ≲ 1 and Ro ≳ 1 studied herein. Thus, in general for such cases, one should consider both symmetric and nonsymmetric perturbations concurrently without invoking spectral analysis, as opposed to earlier papers on this subject employing the latter analysis (cf. Antar and Fowlis 1982; Miller 1985; Miller and Antar 1986; Gu et al. 1998; in the last paper the stability problems of symmetric perturbations are examined for the range 0.3 ≤ Ri ≤ 1). From the above, it follows that in studying the dynamical activity of this type of zonal shear flows, emphasis should be shifted to the nonnormality, or shear-induced, transient dynamics of nonsymmetric perturbations. The same view follows from Bakas et al. (2001), who showed that in a stratified and nonrotating zonal flow with a vertical shear, optimal perturbations undergoing largest transient amplification determine the form of observed coherent structures (hairpin vortices); they also adopted small Richardson numbers, Ri ≲ 1, in their analysis.

We show that nonsymmetric perturbations can be classified into two fundamental types/modes—inertia–gravity wave (IGW) and vortex modes—according to the value of Ertel potential vorticity (PV). IGW mode perturbations have zero PV, whereas vortex mode perturbations have nonzero PV and are nonoscillatory, corresponding to balanced motions/modes in quasigeostrophic models. Such a classification is also analogous to that accepted in the adjustment theory (Obukhov 1949; Blumen 1972; Gill 1982; Bartello 1995) and in the theory of stratified turbulence in the atmosphere and ocean [see the review by Riley and Lelong (2000), and references therein]. In the nongeostrophic case considered here, these two modes appear to be linearly coupled because of the nonnormality. Specifically, we demonstrate that vortex mode perturbations generate IGW mode perturbations, thus providing a channel of energy transfer from balanced motions to unbalanced nongeostrophic IGWs. In other words, balanced motions appear as a source of IGWs, which play an important role in transporting shear flow energy and momentum and consequently can influence the general circulation and energy balance of the atmosphere. We should mention that the generation of IGWs from balanced motions studied herein was also found in observational studies (Uccellini and Koch 1987; Guest et al. 2000; Pavelin et al. 2001; Plougonven et al. 2003).

Research of nongeostrophic instabilities in meteorological flows with vertical/horizontal shear has quite a long history, beginning with Stone (1966). In particular, Nakamura (1988), Molemaker et al. (2005), and Plougonven et al. (2005) analyzed nongeostrophic instability in Eady flows with constant vertical shear and stratification using the modal approach, thereby focusing on the spatial aspect of the dynamics and not on the transient–temporal dynamics. Specifically, they showed that the spatial coupling of unbalanced IGWs above the inertia critical levels with nearly balanced motions below these levels provides a principal mechanism for the growth of nongeostrophic instability.

To more comprehensively analyze IGWs’ energetics and dynamics, allowing for temporal evolution, the nonmodal approach has been employed in recent studies. By means of this approach, perturbation dynamics has been analyzed for different configurations: zonal flows having a vertical or horizontal shear with or without rotation. Specifically, with the GST, the emission of IGWs from a stably stratified layer with a vertical shear of zonal velocity, neglecting the Coriolis force, was studied by Lott (1997) and Bakas and Ioannou (2007) and from a linear two-dimensional Gaussian jet by Bakas and Farrell (2008). They found that because of the nonnormality of the underlying dynamics, small perturbations initially undergo transient growth tapping energy from the mean shear. In this process, they generate IGWs with horizontal wavelengths larger than the depth of the shear layer and propagating away from the layer. Bakas and Ioannou (2007) used the Kelvin mode approach to follow the propagation of IGWs. Although in all these three papers the generation and propagation of IGWs is considered, they do not focus on the interaction between IGWs and balanced PV perturbations.

Papers by Heifetz and Farrell (2003), Vanneste and Yavneh (2004), Heifetz and Farrell (2007), Olafsdottir et al. (2008), and Bakas and Farrell (2009b) specifically address the coupling between balanced motions and unbalanced nongeostrophic IGWs in zonal shear flows. Heifetz and Farrell (2003, 2007) considered the linear dynamics of perturbations in zonal flows with a constant vertical shear and stratification using the GST. They found that for large Richardson and small Rossby numbers there is a very weak coupling between balanced perturbations and nongeostrophic IGWs; they basically evolve independently. By contrast, for intermediate Richardson and Rossby numbers on the order of unity, the coupling between these two modes is quite appreciable because of the increased nonnormality of the flow. Vanneste and Yavneh (2004), Kalashnik et al. (2006), Olafsdottir et al. (2008), and Bakas and Farrell (2009b) investigated the coupling between balanced PV (i.e., vortex mode) perturbations and IGWs in a zonal flow with a horizontal shear using the Kelvin mode approach. Vanneste and Yavneh (2004), Olafsdottir et al. (2008), and Bakas and Farrell (2009b) considered the regime of small Rossby and Froude numbers (or, equivalently, strong stratification) and, using an exponential–asymptotic technique, found that the amplitude of IGWs generated by the vortex mode is exponentially small. Bakas and Farrell (2009b) also considered the regime of Froude numbers on the order of unity (i.e., weak stratification) without rotation. Because of the increased degree of nonnormality in this regime, the coupling between the vortex and IGW modes is quite strong and hence a robust IGW generation takes place after large transient amplification of the vortex mode, as also shown in Kalashnik et al. (2006). In the present paper, performing a parallel analysis of the dynamics of symmetric and nonsymmetric perturbations in the case of vertical shear, we study the linear mode coupling as well, as in the above papers. Thus, our work is related to Nakamura (1988), Plougonven et al. (2005), Molemaker et al. (2005), and Heifetz and Farrell (2003, 2007), with the exception that we adopt the Kelvin mode approach and consider an unbounded flow. We show that for small Richardson and large Rossby numbers, the generation of IGWs by vortical perturbations can be a powerful process and the basic mechanism for this is essentially similar to that described in Bakas and Farrell (2009b) for the regime of order-of-unity Froude numbers, but for zonal flows with a horizontal shear. In the opposite case (i.e., for large Richardson and small Rossby numbers), the same type of very weak (exponentially small) IGW generation occurs as described by Vanneste and Yavneh (2004) in horizontal shear.

A somewhat analogous study of IGW generation from PV anomalies in an unbounded, stratified zonal flow with vertical shear has been carried out very recently by Lott et al. (2010), who link the modal and nonmodal studies mentioned above. Based on the properties of singular PV normal modes, IGW generation was studied for different shapes of PV anomalies. The general picture is that close to the PV anomalies perturbations are in nearly geostrophic balance, whereas beyond the inertia critical levels they have the form of vertically propagating IGWs with amplitudes rapidly decreasing as Richardson number increases. However, our analysis can be applied to the generation of IGW packets by localized PV perturbations, which can also be represented as a certain sum of Kelvin modes. A similar procedure of summation over Kelvin modes contained in localized PV perturbations was considered by Olafsdottir et al. (2008) in order to analyze the generation of localized IGW packets by the latter.

The paper is organized as follows. Mathematical formalism and mode classification are presented in section 2. The outline of symmetric instability is presented in section 3. Dynamics of nonsymmetric perturbations—the linear coupling of vortex and IGW modes and a parallel analysis of the transient growth of nonsymmetric perturbations versus symmetric instability—are presented in section 4. Conclusions are given in section 5.

## 2. Mathematical formalism

*f*plane in the Boussinesq approximation is governed by the following set of equations:where

**v**is the velocity with components

*u*,

*υ*, and

*w*along the zonal (

*x*), meridional (

*y*), and vertical (

*z*) axes, respectively;

*σ*≡ −

*gρ*′/

*ρ*

_{*}is the buoyancy, where

*ρ*′ is the deviation of density from the background value,

*ρ*

_{*}= const, and

*g*is the gravitational acceleration;

*P*≡

*p*′/

*ρ*

_{*}, where

*p*′ is the deviation of pressure from the hydrostatic value;

*f*is the Coriolis parameter;

**n**is the unit vector along the vertical

*z*axis; and

*d*/

*dt*= ∂/∂

*t*+

**v**·

**∇**is the total derivative operator. From Eqs. (1) follows the conservation of PV (Gill 1982; Pedlosky 1987):

*f*∂

*u*

*z*= −∂

*σ*

*y*is obtained. The vertical and meridional gradients of

*σ*

*N*

^{2}= ∂

*σ*

*z*(square of Brunt–Väisälä frequency) and

*S*

^{2}= −∂

*σ*

*y*, are assumed to be spatially constant.

*A*≡

*S*

^{2}/

*f*(Eady 1949):where

*q*

**v**=

**v**

_{0}+

**v**′,

*σ*=

*σ*

*σ*′,

*P*=

*P*

*P*′ in Eqs. (1), in the linear approximation, for small perturbations

**v**′,

*σ*′, and

*P*′ about the equilibrium state (4), we obtain the system (henceforth primes will be omitted):whereFrom Eqs. (5), one can easily derive the conservation law for the linearized PV:which represents the linearized form of Eq. (2) and plays an important role in the subsequent analysis. This set of equations is the same as that used by Heifetz and Farrell (2003, 2007) except that the quasihydrostatic balance approximation is relaxed here, since we consider an unbounded problem. As a result, the linearized PV is strictly conserved and has a different form than that used by these authors.

*m*(

*t*), or the Kelvin modes:where

*k*,

*l*, and

*m*(

*t*) ≡

*m*(0) −

*Akt*are the zonal, meridional, and vertical wavenumbers, respectively. Also,

*ũ*(

*t*),

*υ̃*(

*t*),

*w̃*(

*t*),

*σ̃*(

*t*), and

*P̃*(

*t*) are the amplitudes depending only on time. Note that the amplitude

*q̃*does not depend on time because of the conservation of PV. In the physical variables (

*x*,

*y*,

*z*), the representation (7) describes a harmonic plane wave with a time-dependent amplitude and phase

*θ*=

*kx*+

*ly*+

*m*(

*t*)

*z*. Because the vertical wavenumber

*m*(

*t*) is time dependent (one can say that harmonic plane waves drift in the wavenumber space), the plane of constant phase rotates and becomes nearly parallel to the horizontal (

*x*,

*y*) plane in the limit

*t*→ ∞. Substitute Eqs. (7) into Eqs. (5)–(6) and switch to nondimensional variables being used throughout the paper;

*ft*→

*t*, (

*ũ*/

*u*

_{0},

*υ̃*/

*u*

_{0},

*w̃*/

*u*

_{0}) → (

*ũ*,

*υ̃*,

*w̃*),

*σ̃*/

*fu*

_{0}→

*σ̃*,

*lP̃*/

*fu*

_{0}→

*P̃*,

*q̃*/

*u*

_{0}

*lf*

^{2}→

*q̃*, where

*u*

_{0}is a typical mean flow velocity. By introducing the notations Rossby number Ro ≡

*A*/

*f*, Richardson number Ri ≡

*N*

^{2}/

*A*

^{2}, and the normalized wavenumbers

*a*≡

*k*/

*l*,

*b*(

*t*) ≡

*m*(

*t*)/

*l*=

*b*(0) −

*a*Ro

*t*(here

*t*is the nondimensional time) for the time-dependent amplitudes, we finally obtainand the conservation of PV

*υ̃*and

*P̃*from this set and reduce it to a new set of equations for three basic variables

*ũ*,

*w̃*,

*σ̃*:and the PV

Equations (10) and (11) are the basic equations of the subsequent analysis (because from now on we use only these amplitudes everywhere, tildes over them will be omitted). Note that the perturbation dynamics depends only on the normalized wavenumbers *a* and *b*(*t*), since there is no characteristic length scale in the unbounded problem considered here. As mentioned in the introduction, in this paper we focus on Richardson numbers in the range Ri ≲ 1, which may occur in fronts and jet streaks as well as in the lower stratosphere. If we take *H* = 10 km for the height of the earth’s atmosphere and the characteristic mesoscale flow velocity *u*_{0} = 10 m s^{−1}, for the shear parameter we find *A* = *u*_{0}/*H* = 10^{−3} s^{−1} and consequently for Ro = 10 ( *f* = 10^{−4} s^{−1}) (see also Gu et al. 1998). Smaller values of Rossby number, Ro ∼ 1, are also considered to see how they change dynamics. We also define the nondimensional dynamical, or shear, time as *t*_{sh} = *f*/*A* = 1/Ro.

*σ*:where the coefficients

*c*

_{1},

*c*

_{0}, and

*c*are lengthy expressions and are given in the appendix. This equation as well as the original set of Eqs. (10) are general and also describe in the limit

*a*= 0, the dynamics of symmetric perturbations. After some algebra, one can rewrite Eq. (12) in the following form:where

*h*=

*σ*/

*s*andEquation (13) is of oscillatory type with a source term on the right-hand side originating from the PV,

*q*, of perturbations. Oscillations relate to IGWs and the source term—to nonoscillatory, slowly varying, quasigeostrophic balanced motions or, in our terminology, to vortex mode perturbations (see below). Thus, PV is tied only to the vortex mode, or balanced perturbations, since IGWs always carry zero PV. Often, the essence of “object,” which is a main carrier of PV, is not analyzed in meteorological studies; this object is labeled as balanced motions. The linear mode coupling studied in this paper is a general phenomenon, inevitably taking place in large shear flows, and the generation of IGWs by vortex mode perturbations is just its particular manifestation occurring in the type of a meteorological shear flow considered here. The interest of astrophysical, hydrodynamical, and plasma communities in similar phenomena has also increased over the last several years. To reveal general features of the mode coupling phenomenon, carriers of PV (i.e., vortex mode perturbations) will be described in more detail below.

### a. Mode classification

It is possible to classify modes involved in Eq. (13) [and, therefore, in basic Eqs. (10)] from the mathematical and physical standpoints separately.

Mathematically, the general solution of Eq. (13) can be written as a sum of two parts: a general solution of the corresponding homogeneous equation (nongeostrophic oscillatory IGW mode) and a particular solution of this inhomogeneous equation. It should be emphasized that the particular solution of the inhomogeneous equation is not uniquely determined: the sum of a particular solution of the inhomogeneous equation and any particular solution of the corresponding homogeneous equation (i.e., wave mode solution) is a particular solution of the inhomogeneous equation as well; that is, the particular solution may comprise any amount of the wave mode.

Physically, Eq. (13) describes two different modes (in other words, types or branches) of perturbations:

- Nongeostrophic IGW mode
*h*^{(w)}, which is determined by a general solution of the corresponding homogeneous equation and carries zero PV. It oscillates with the instantaneous time-dependent (due to shear) frequency*ω*[*a*,*b*(*t*)]. Naturally, in the shearless limit (*A*= 0), this frequency is constant with time and reduces to the well-known expression for the IGW frequency normalized by*f*:where*b*and*m*are time-independent in this limit. Thus,*ω*[*a*,*b*(*t*)] is a more general form for the IGW frequency in the presence of shear. - Vortex mode
*h*^{(v)}, which is aperiodic, originating from the equation inhomogeneity,*qc*/*s*, and is associated with a nonoscillatory part of a particular solution of the inhomogeneous equation. In the absence of shear, this mode is independent of time and represents balanced vortical perturbations (slow manifold) satisfying thermal wind relation and having zero vertical velocity [i.e., the motion for this mode is confined in the horizontal (*x*,*y*) plane]. However, because of shear, geostrophic balance ceases to hold for vortical perturbations exactly, which results in them becoming slowly varying with time or quasigeostrophic, and consequently, as we will show below, being able to couple with IGW mode perturbations and generate them. Therefore, the form of the vortex–aperiodic mode in the presence of mean velocity shear is uniquely determined (see also section 4a); actually it is a generalization of its form in the absence of shear to the case with nonzero shear. From the above argument, it follows that the correspondence between the aperiodic vortex mode and the particular solution of the inhomogeneous equation is quite unambiguous: the vortex mode is associated only with just that part of the particular solution that does not contain any oscillations. The amplitude of the vortex mode is proportional to*q*and goes to zero when*q*= 0. As noted in the introduction, the present classification of perturbation modes into two types—IGWs and vortices—depending on the value of PV is also accepted in the classical theory of nonlinear geostrophic adjustment and in the theory of stratified turbulence in the atmosphere and ocean.

*ϵ*bywhich measures the effect of shear on the dynamics of IGWs. If

*ω*[

*a*,

*b*(

*t*)] varies only a little during the period of IGW oscillations—that is, if the time scale of

*ω*[

*a*,

*b*(

*t*)]’s variation,

*ω*[

*a*,

*b*(

*t*)]{

*dω*[

*a*,

*b*(

*t*)]/

*dt*}

^{−1}, greatly exceeds

*ω*

^{−1}[

*a*,

*b*(

*t*)]—thenmeaning that the Wentzel–Kramers–Brillouin (WKB), or adiabatic, approximation (with respect to time) is satisfied and shear plays only a minor role in the IGW dynamics. The value of

*ϵ*depends on Ri, Ro,

*a*, and

*b*(

*t*). At Ri ≫ 1 and Ro ≪ 1, the WKB condition (14) is met for all values of

*a*and

*b*(

*t*) (Fig. 1). As a result, for these values of Ri and Ro, the dynamics of IGWs and vortex mode are basically separated/decoupled; there is only an exponentially small coupling between these two modes. By contrast, as one can see in the same figure, at Ri ≲ 1 and Ro ≳ 1,

*ϵ*≳ 1 for a certain range of wavenumbers that form so-called nonadiabatic regions. For instance, at Ri = 0.3 and Ro = 10, the nonadiabatic region includes wavenumbers |

*b*(

*t*)/

*a*| ≲ 1 and at Ri = 0.3 and Ro = 1, it includes wavenumbers |

*b*(

*t*)/

*a*| ≪ 1 (Fig. 1). In this nonadiabatic region, the time scales of the vortex and IGW modes become comparable, which results in a strong mode coupling and generation of IGWs from an initially balanced vortex mode. The transient dynamics of nonsymmetric perturbations is studied in detail in section 4; here we only note that the strong coupling regime described in section 4 is similar in nature to that occurring in a horizontal shear flow in another regime of order-of-unity Froude numbers also considered by Bakas and Farrell (2009b).

*u*

^{(w)},

*w*

^{(w)}and

*u*

^{(v)},

*w*

^{(v)}are found, respectively, by means of

*σ*

^{(w)}=

*s*(

*t*)

*h*

^{(w)}and

*σ*

^{(v)}=

*s*(

*t*)

*h*

^{(v)}and their corresponding time derivatives using the formulas given in the appendix [for

*u*

^{(w)},

*w*

^{(w)}, and

*σ*

^{(w)}one should simply set

*q*= 0 in these formulas]. An exact form for

*σ*

^{(v)}is given by Eq. (17) below in section 4.

In fact, the (modified) initial value problem is solved by Eq. (13) [or, equivalently, by Eqs. (10) and (11)]. The character of subsequent dynamical evolution depends on which mode of perturbation is inserted initially in Eqs. (10): a pure IGW mode without admixture of vortex mode perturbations or a pure vortex mode without admixture of IGWs. *One of the goals of the present paper is to investigate the dynamical evolution when pure vortex mode perturbations are inserted initially in Eqs. (10) and, in particular, the generation of the IGW mode by the vortex mode for small Richardson numbers,* Ri ≲ 1.

## 3. Dynamics of symmetric perturbations

*a*= 0 and time-independent

*b*) perturbations, since later we will compare the transient growth of nonsymmetric perturbations with symmetric exponential instability. In the case of symmetric perturbations, Eq. (12) is greatly simplified:The density perturbation pertaining to the vortex mode is independent of time and is given bybecause this expression is the only nonoscillatory solution of the inhomogeneous Eq. (15). Again, other corresponding variables

*u*

^{(v)},

*υ*

^{(v)},

*w*

^{(v)}, and

*P*

^{(v)}are found using formulas from the appendix. Note that vertical velocity

*w*

^{(v)}= 0, as it should be for balanced motions.

*σ*in Eq. (15) becomes negative. This happens for the range

*b*

_{1}<

*b*<

*b*

_{2}< 0, where

*b*

_{1,2}= −Ro ∓ Ro

*b*< 0, implying that emerging structures are inclined toward the vertical

*z*axis. The nonlinear development of these structures leads to inclined convective eddies in the plane perpendicular to the flow (Bennetts and Hoskins 1979). If we assume time dependence of symmetric (wave) perturbations of the form ∝exp(

*γt*), we can find the largest growth ratewhich occurs atIf we take Ri = 0.3, Ro = 10 that may occur for fronts and jet streaks, we find

*γ*

_{max}= 1.45 and the corresponding characteristic growth time is

*t*= 1/

_{c}*γ*

_{max}= 0.69, which is larger than the shear time

*t*

_{sh}= 1/Ro = 0.1, meaning that the effects of shear–nonnormality are important. We can go further and estimate the minimum possible characteristic time for symmetric instability, or equivalently the maximum possible growth rate, for various Ri and Ro. Indeed, it is easy to show that for a given Ro,

*γ*

_{max}can achieve is

*t*

_{c,min}= 1/

*t*

_{sh}/

*t*

_{c,min}= 1/

*t*

_{sh}/

*t*

_{c,min}≲ 1; that is, the shear time is smaller than the characteristic growth time of symmetric instability,

*implying that for*Ro ≳ 1 (

*and of course at the same time for*Ri ≲ 1)

*shear (nonnormality)-induced dynamics of nonsymmetric perturbations can be more important than symmetric instability.*In the next section, we will demonstrate that the transient amplification of nonsymmetric perturbations during the initial stage of evolution is indeed on the order of and even larger than the exponential growth of symmetric perturbations for such values of Ri and Ro. As we have estimated, for fronts and jet streaks typically Ro ≃ 10. More detailed observations (cf. Uccellini et al. 1984; Zack and Kaplan 1987; Koch and Dorian 1988; Kaplan et al. 1997) indicate that Rossby numbers on the order of or larger than unity can indeed occur in such flows.

## 4. Dynamics of nonsymmetric perturbations

Here we concentrate on the dynamics of nonsymmetric perturbations (with *a* ≠ 0). Below in section 4a we first describe in detail the abovementioned phenomenon of generation of IGWs from quasigeostrophic vortical perturbations and in section 4b we compare the optimal (transient) growth of nonsymmetric perturbations with the growth of symmetric instability. Both these phenomena arise from the nonnormality–shear of background zonal flow.

### a. Coupling of IGW and vortex modes

*t*= 0) separate out pure vortex mode–aperiodic perturbations classified in section 2. As noted above, such a separation is possible in the WKB regime [i.e., in the adiabatic regions at |

*b*(

*t*)/

*a*| ≫ 1 where

*ϵ*≪ 1; Fig. 1]. Using a technique similar to that of Vanneste and Yavneh (2004) for separating out balanced motions (in our case with respect to small

*ϵ*), we can find an asymptotic nonoscillatory, slowly varying, quasigeostrophic solution of Eq. (13):In fact, expression (17) is the first term of a regular perturbation expansion of slowly varying (due to shear) balanced solution in powers of small

*ϵ*, with higher-order correction terms to be found by the method described in Vanneste and Yavneh (2004). For our subsequent numerical analysis, only the first term of such an expansion [i.e., expression (17)] will be sufficient. We should note that such an asymptotic expansion technique can only yield a nonoscillatory solution—the vortex mode—and is not able to capture IGW generation (see also Vanneste and Yavneh 2004; Bakas and Farrell 2009b), so numerical integration is necessary in this case. The

*σ*

^{(v)}(

*t*) given by (17), together with corresponding

*u*

^{(v)}(

*t*),

*υ*

^{(v)}(

*t*),

*w*

^{(v)}(

*t*), and

*P*

^{(v)}(

*t*), constitutes the vortex mode. It easy to verify that this general asymptotic form for the vortex mode has negligible vertical velocity compared to the horizontal one [this does not in general hold in the moment of wave generation when (14) is violated; see Figs. 2 and 3]; consequently the motion for this mode occurs mostly in the horizontal (

*x*,

*y*) plane as is also typical of balanced motions. In the adiabatic regions, vortex and IGW modes evolve independently from each other, which allows us to analyze them separately. So, if initially the vortex mode with |

*b*(0)/

*a*| ≫ 1 is inserted into Eqs. (10), in the beginning it will evolve (in the first approximation) according to the asymptotic solution (17). However,

*b*(

*t*), varying with time, gradually decreases and starts to approach and cross nonadiabatic/unstable regions (see arrows in Fig. 1); where condition (14) is no longer met,

*ϵ*becomes on the order of unity or larger, implying also the breakdown of the quasigeostrophic approximation (asymptotic expansion) there. This results in a mixing/coupling between vortex and IGW modes.

To gain a better insight into what happens in such nonadiabatic regions, we numerically integrate Eqs. (10) using as an initial input, the pure vortex mode without a mixture of IGWs. In other words, as the initial conditions in Eqs. (10) at *t* = 0, we take *σ*^{(v)}(0) given by Eq. (17) in the adiabatic region at |*b*(0)/*a*| ≫ 1 and the corresponding *u*^{(v)}(0), *w*^{(v)}(0). Figures 2 and 3 show a typical dynamical picture of initially imposed vortex mode evolution at Ri = 0.3, Ro = 1, and Ro = 10, when the dynamics can be nonadiabatic during a certain time interval. At the beginning, being in the adiabatic region (at |*b*(*t*)/*a*| ≫ 1), the imposed vortex mode extracts energy from the mean shear flow because of the nonnormality and slowly grows algebraically, but it retains its aperiodic nature (arrow 1 in Fig. 1). Then follows the nonadiabatic–unstable region [at |*b*(*t*)/*a*| ≲ 1; arrows 2 and 3], where the transient amplification of the vortex mode changes from an algebraic to exponential type and gains more strength. At this stage, the linear coupling of the vortex and IGW modes comes into play; as a consequence, simultaneously oscillations begin to emerge as seen in Figs. 2 and 3; that is, we observe the emergence of IGWs. Thus, it turns out that the linear dynamics of the vortex mode perturbation is accompanied by IGW generation. In the unstable region, where the time scales of the vortex and wave modes are comparable, the modes are not quite well separable/distinguishable, we have some mix of aperiodic and oscillatory modes. At later times, when *b*(*t*) leaves the nonadiabatic–unstable regions (arrow 4) and gets further away at *b*(*t*)/*a* ≪ −1, the adiabatic approximation (14) holds again and the dynamics of the vortex and IGW modes become decoupled and end up cleanly separated. Initially, the vertical velocity of newly generated IGWs experiences large transient amplification as horizontal velocities do, but with time it becomes much smaller than the horizontal one; that is, the wave motion occurs predominantly in the horizontal plane at |*b*(*t*)| → ∞ (Figs. 2 and 3).

We can formally divide the energy evolution into two phases. The first phase relates to the transient amplification (due to both the nonnormality and the unstable regions) of the originally imposed vortex mode and simultaneous excitation (and also further amplification) of the corresponding IGW mode. The second asymptotic phase relates to the newly generated IGWs with the energy oscillating about a constant value. The contribution of the vortex mode energy to the total perturbation energy gradually falls off after transient amplification and wave generation and is zero at asymptotically large times. By contrast, the IGWs retain their energy after the transient amplification event and are not decaying at large times.

For large Richardson (Ri ≫ 1) and small Rossby (Ro ≪ 1) numbers, the adiabatic parameter *ϵ* is small all over the (*a*, *b*) plane (Fig. 1). This implies that the dynamics is adiabatic/quasigeostrophic at all times and therefore almost no (exponentially small) coupling between vortex and IGW modes is expected, in agreement with the results of Heifetz and Farrell (2003), Vanneste and Yavneh (2004), and Vanneste (2008). In this situation, the vortex mode dynamics is rather simple: it gradually dies down after having experienced transient amplification accompanied by very weak IGW generation (Fig. 4). In the context of hydrodynamical flows, the nature of the linear wave generation phenomenon (also called conversion of vortices into waves) is described in detail for the simplest compressible shear flow in Chagelishvili et al. (1997), Farrell and Ioannou (2000), and Chagelishvili (2002). In conclusion, the linear coupling among different modes due to the nonnormality of perturbation dynamics is typical of zonal shear flows. This in turn clearly demonstrates the inevitability and importance of coupling between balanced motions (vortex mode) and unbalanced IGWs (despite its exponential smallness in certain cases) and hence the nonexistence of only a purely balanced or slow manifold whenever background shear is present. This implies that the generation of IGWs by balanced motions cannot by any means be neglected in quasigeostrophic models.

Figure 5 shows the evolution of the perturbation energy for an initially imposed vortex mode at different values of Richardson number for a fixed Ro = 10. We see that for Ri = 0.5 and Ri = 1, the perturbation dynamics is nonadiabatic during a certain time interval and resembles that in Fig. 2. As a consequence, we observe IGW generation—energy does not decay; rather, it oscillates around a constant value at large times. By contrast, at larger Richardson numbers Ri = 10 and Ri = 40, as expected, the dynamics is adiabatic. Accordingly, an exponentially small IGW excitation takes place that is not sufficient to extract and retain energy. Therefore, the quasigeostrophic vortex mode, after transient amplification, decays gradually, returning all its energy to the mean flow. [Of course, even for small Richardson numbers, if initial *a*, *b*(0) is such that it lies outside a nonadiabatic region and during evolution *b*(*t*) does not cross it, we will not see any noticeable mode coupling.] Note also that with increasing Ri and decreasing Ro, the amount of transient amplification decreases. The nonadiabatic–unstable regions in Fig. 1 shrink with increasing Ri at a fixed Ro and, conversely, with decreasing Ro at a fixed Ri. As a result, in Figs. 2 and 3, the transient amplification of the vortex mode energy before wave appearance and therefore the amplitude of subsequently generated IGWs for Ro = 10 are higher than that for Ro = 1. Similarly, in Fig. 5, the transient amplification of the vortex mode energy for Ri = 0.5 is higher than that for Ri = 1 and both are smaller than that for Ri = 0.3 in Fig. 2.

### b. Nonsymmetric versus symmetric perturbations

In this section we carry out a parallel or comparative analysis of the transient dynamics of nonsymmetric perturbations and symmetric instability. Usually studies on stability problem at Ri ≲ 1 and Ro ≳ 1 consider the dynamics of symmetric and nonsymmetric perturbations using spectral methods, thereby neglecting finite-time–transient phenomena (cf. Antar and Fowlis 1982; Miller 1985; Miller and Antar 1986; Gu et al. 1998). A general conclusion from such studies is that symmetric perturbations prevail over nonsymmetric ones, which always appear to have smaller growth rates than those of the former. Consequently, symmetric perturbations are thought to be responsible for the “fate” of such flows. However, as we have estimated above, the characteristic time of transient growth of nonsymmetric perturbations or dynamical/shear time is smaller than the characteristic *e*-folding time of symmetric instability, implying the importance of transient dynamics of nonsymmetric perturbations for Ri ≲ 1 and Ro ≳ 1.

*a*,

*b*(0)] is defined as the perturbation with an unit initial energy,

*E*(0) = 1 and specially chosen initial values of

*u*(0),

*w*(0), and

*σ*(0) such that, with Eqs. (10) leads to the largest possible (or optimal) value of energy at some specified time

*t*. This energy is optimal among all possible perturbations with the same initial unit energy and fixed [

*a*,

*b*(0)]. We denote the optimal value of energy at

*t*as

*E*

_{opt}—this optimal value automatically gives the largest energy growth factor [

*E*(

*t*)/

*E*(0)] of nonsymmetric perturbations at

*t*because the initial energy is unity by our choice. The Kelvin modes of perturbations with

*a*≠ 0 drift in the (

*a*,

*b*) plane along the

*b*axis with the speed

*a*Ro, so that the optimal perturbation with the initial wavenumber

*b*(0) will appear at

*t*with wavenumber

*b*(

*t*) =

*b*(0) −

*a*Ro

*t*. As a result, the optimal energy

*E*

_{opt}at time

*t*is actually a function of

*a*and

*b*(

*t*). Since our aim to analyze the relative importance of transient amplification of nonsymmetric perturbations compared with the exponential growth of symmetric perturbations during the minimum possible characteristic time of symmetric instability

*t*= 1/

_{c}*γ*

_{max}, it is convenient to introduce the ratiowhere

*γ*

_{max}is given by Eq. (16). As noted above,

*E*

_{opt}[

*a*,

*b*(

*t*)] is now the optimal energy growth factor of nonsymmetric perturbations at

_{c}*t*=

*t*and exp(2

_{c}*γ*

_{max}

*t*) = exp(2) is the exponential growth factor of the energy of symmetric perturbations for the same time. Note that because we determine the optimal value of perturbation energy at some specified time

_{c}*t*by specially adjusting initial conditions

*u*(0),

*w*(0), and

*σ*(0) in Eqs. (10), in general we will no longer have a purely vortex mode as an initial input but rather a mixture of vortex and IGW modes, so that optimal perturbations always have some nonzero PV.

In Fig. 6, we show the isolines of *e*(*a*, *b*, *t _{c}*) in the (

*a*,

*b*) plane for Ri = 0.3 and Ro = 10. Other values of these parameters give a qualitatively similar distribution of isolines, so we only list the characteristic values in Table 1. It is seen that the areas of higher

*e*coincide with the unstable regions in Fig. 1, where

*ω*

^{2}(

*a*,

*b*) < 0. As a consequence, for large |

*a*|, |

*b*| ≫ 1, the isolines are located symmetrically on either side of the

*b*axis (the upper plot in Fig. 6). Even larger values of

*e*are achieved for smaller |

*a*|, |

*b*|, and negative

*a*, where the isolines are asymmetric with respect to the

*b*axis (the lower plot in Fig. 6). We see from Fig. 6, that for various

*a*and

*b*the values of

*e*can range from 0.5 to the maximum value

*e*

_{max}= 10.23 at

*a*= −0.68,

_{m}*b*= 0.20. In other words, the transient growth of nonsymmetric perturbations is larger than the exponential growth of symmetric perturbations over the characteristic

_{m}*e*-folding time of symmetric instability. It is also noteworthy that the values of

*a*are all negative (see Table 1), meaning that the constant phase plane of maximally amplified perturbations is inclined in the northeastern direction to the zonal one and the inclination angle is different for different Ri and Ro.

_{m}Generally, the transient growth of nonsymmetric perturbations decreases with the increase of Ri and decrease of Ro. However, the same happens with the growth of symmetric perturbations. Therefore, the maximal dominance of nonsymmetric perturbations over symmetric ones takes place at Ri = 0.9 and Ro = 10 and not at Ri = 0.3 and Ro = 10, for which the growth rate of symmetric instability is largest (see Table 1). From Table 1, we also see the dominance of transient growth of nonsymmetric perturbations even for Ri and Ro on the order of unity (see the case Ri = 0.9 and Ro = 1 for which *e*_{max} = 7.22). Thus, the parallel analysis presented above shows that for Ri ≲ 1 and Ro ≳ 1, the transient amplification of nonsymmetric perturbations prevails over symmetric instability.

The above results may be important for the further development in the nonlinear stage and ultimately for turbulence and coherent structures that may emerge in such flows. Of course, if we stay in the linear regime, symmetric instability will eventually dominate. Nonsymmetric perturbations undergo most of the growth only during a dynamical–shear time when crossing the unstable regions in the (*a*, *b*) plane, whereas symmetric perturbations do not drift and therefore always stay in the unstable area. It is expected that nonlinearity will come into play just during the dynamical time because of large transient amplification of nonsymmetric perturbations, so further exponential growth of symmetric instability may not persist. Thus, nonsymmetric perturbations will determine the onset of nonlinear regime and dominate the dynamics. For this reason, the role of the above-described coupling of IGWs and slowly varying vortex mode is also important for properly understanding the properties of turbulence and coherent structures in meteorological shear flows.

## 5. Conclusions

In this paper, we have investigated the linear dynamics of nonsymmetric perturbations in zonal baroclinic flows in geostrophic balance having a constant vertical shear by means of the nonmodal (Kelvin mode) approach and numerical analysis. Small Richardson (Ri ≲ 1) and large Rossby (Ro ≳ 1) numbers regime has been considered, which satisfy the condition for symmetric instability. It has been shown that the important feature of nonsymmetric perturbations is the conservation of potential vorticity (PV). Depending on the value of PV, perturbations were classified into two types/modes: oscillatory inertia–gravity waves (IGWs) with zero PV and vortical–aperiodic perturbations with nonzero PV, which correspond to quasigeostrophic balanced motions.

It has been demonstrated that for Ri ≲ 1 and Ro ≳ 1, the interplay of the Coriolis parameter, mean flow shear, and stable stratification produces nonadiabatic (i.e., where the quasigeostrophic approximation breaks down) regions in the wavenumber space, which also comprise unstable regions. Both nonadiabatic and unstable regions broaden with decreasing Ri and increasing Ro. Spatial Fourier harmonics, or Kelvin modes, of perturbations with time-dependent vertical wavenumbers, crossing these regions, undergo transient exponential growth due to unstable areas and the violation of the quasigeostrophic approximation leads to the linear mode-coupling phenomenon—in particular, the generation of nongeostrophic IGWs by vortical perturbations. Spatial harmonics of vortical perturbations first extract energy from the mean flow during the process of transient amplification and then generate related spatial harmonics of IGWs and eventually die down, whereas the energy of generated IGW spatial harmonics does not decay at asymptotically large times and is constant on average. It can be said that vortical perturbations act as a mediator between IGWs and the mean flow. An analogous IGW generation by vortical perturbations (balanced motions) occurs in meteorological flows with horizontal shear and generally is specific to zonal shear flows because of the nonnormality of perturbation dynamics due to shear (Vanneste and Yavneh 2004; Kalashnik et al. 2006; Bakas and Farrell 2009b). With increasing Richardson and decreasing Rossby numbers, the intensity of the nonnormality and therefore the linear mode coupling becomes weaker, in agreement with the results of previous studies (see, e.g., Heifetz and Farrell 2003).

The occurrence of broad unstable regions in the considered baroclinic zonal flow implies that the range of wavenumbers for which spatial Fourier harmonics of nonsymmetric (vortex and IGW) perturbations undergo substantial transient amplification, much larger than the growth of symmetric instability, is quite wide (see Table 1). Accordingly, our analysis suggests that the dynamical activity of fronts and jet streaks at Ri ≲ 1 and Ro ≳ 1 should be determined by nonsymmetric perturbations and not by symmetric ones, as was accepted in earlier papers. It is noteworthy that the growth of perturbations is asymmetric in wavenumber space and the largest transient amplification occurs for negative *a*(≡*k*/*l*) (see Fig. 6). Besides, the value of *a* at which the largest amplification takes place, *a _{m}*, is determined by the Richardson and Rossby numbers (see Table 1). Hence, the constant phase plane of maximally amplified perturbations is inclined in the northeastern direction to the zonal one and the inclination angle is different for different Ri and Ro.

We would like to briefly outline a possible connection of our results with some recent related nonlinear studies. Simulations of stratified turbulence with Ro ≳ 1 demonstrate that horizontal layers with strong vertical shear of zonal velocity develop. Parameters of these layers yield local Richardson numbers on the order of unity (cf. Waite and Bartello 2004, 2005, 2006; Brethouwer et al. 2007; Lindborg and Brethouwer 2007; Riley and Lindborg 2008). Then in these shear layers local patches of three-dimensional turbulence cause a further cascade of energy to smaller scales. In this turbulent state, the dynamics of vortex and IGW modes appear to be strongly coupled and together determine the energy spectrum of turbulence. This coupling is nonlinear by nature. Our case is somewhat different from these studies in that background shear is present from the beginning and the coupling between vortex and IGW modes manifests itself in the linear approximation and is appreciable for regimes Ri ≲ 1 and Ro ≳ 1. The precise investigation of the linear and nonlinear dynamics (mode coupling) in the presence of background shear is the subject of future study. In another study by Lelong and Sundermeyer (2005), the emission of IGWs by PV anomalies was also considered in the absence of any shear flow taking into account only the Coriolis term. In their case, IGW generation occurs during the adjustment of PV perturbations and is of nonlinear origin.

The transient dynamics of nonsymmetric perturbations may facilitate the onset and maintenance of turbulence and coherent structures via a nonclassical “bypass” mechanism formulated by the hydrodynamical community in the 1990s (e.g., Trefethen et al. 1993; Gebhardt and Grossmann 1994; Baggett et al. 1995; Grossmann 2000; Reshotko 2001; Chagelishvili et al. 2002; Chapman 2002), since linear transient amplification of perturbations plays a central role in this concept. In addition, the linear coupling among perturbation modes described herein can be important in determining resulting turbulent spectra and also the structure of coherent motions.

## Acknowledgments

This work is supported by the International Science and Technology Center (ISTC) Grant G-1217. G. R. M. would also like to acknowledge the financial support from the Scottish Universities Physics Alliance (SUPA). We thank both referees for valuable comments that improved the presentation of our results.

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## APPENDIX

### Derivation of Time-Dependent Coefficients

*u*,

*w*in terms of

*σ*and

*dσ*/

*dt*using Eqs. (10c) and (11):where

*D*= Ro{

*a*

^{2}(1 + Ri

^{2}Ro

^{2}) + [

*b*(

*t*) + RiRo]

^{2}}, and then substituted these velocities into Eq. (10b). This gives for the time-dependent coefficients

*c*

_{1}[

*a*,

*b*(

*t*)],

*c*

_{0}[

*a*,

*b*(

*t*)], and

*c*[

*a*,

*b*(

*t*)] in Eq. (12) the following expressions:

*a*= 0), these coefficients become time-independent and take a simpler form:

Listed at various Ri and Ro are the maximum growth rates of symmetric instability −*γ*_{max}, corresponding characteristic *e*-folding times, *t _{c}*, and maximum values of

*a*and

_{m}*b*.

_{m}