A Buoyancy–Vorticity Wave Interaction Approach to Stratified Shear Flow

AUGUST 2008 HARNIK ET AL. 2615

N. HARNIK AND E. HEIFETZ Department of Geophysics and Planetary Sciences, Tel Aviv University, Tel Aviv, Israel

O. M. UMURHAN Department of Geophysics and Planetary Sciences, Tel Aviv University, Tel Aviv, and Department of Physics, The Technion, Haifa, Israel, and City College of San Francisco, San Francisco, California

F. LOTT Laboratoire de Meteorologie Dynamique, Ecole Normale Superieure, Paris, France

(Manuscript received 4 September 2007, in final form 19 November 2007)

ABSTRACT

Motivated by the success of potential vorticity (PV) thinking for Rossby waves and related shear flow phenomena, this work develops a buoyancy–vorticity formulation of gravity waves in stratified shear flow, for which the nonlocality enters in the same way as it does for barotropic/baroclinic shear flows. This formulation provides a time integration scheme that is analogous to the time integration of the quasigeostrophic equations with two, rather than one, prognostic equations, and a diagnostic equation for streamfunction through a vorticity inversion. The invertibility of vorticity allows the development of a gravity wave kernel view, which provides a mechanistic rationalization of many aspects of the linear dynamics of stratified shear flow. The resulting kernel formulation is similar to the Rossby-based one obtained for barotropic and baroclinic instability; however, since there are two independent variables—vorticity and buoyancy—there are also two independent kernels at each level. Though having two kernels complicates the picture, the kernels are constructed so that they do not interact with each other at a given level.

1. Introduction ter (Drazin and Reid 1981). These conditions are quite easily obtained from the equations governing each type Stably stratified shear flows support two types of of instability, but their physical basis is much less clear. waves and associated instabilities—Rossby waves that We do not have an intuitive understanding as we do, for are related to horizontal potential vorticity (PV) gradi- example, for convective instability, which arises when ents, and gravity waves that are related to vertical den- the stratification itself is unstable (e.g., Rayleigh– sity gradients. Each of these wave types is associated Bernard and Rayleigh Taylor instabilities). with its own form of shear instabilities. Rossby wave There are two main attempts to physically under- instabilities (e.g., baroclinic instability) arise when the stand shear instabilities, which have gone a long way PV gradients change sign (Charney and Stern 1962), toward building a mechanistic picture. Overreflection while gravity wave–related instabilities, of the type de- theory (e.g., Lindzen 1988) explains perturbation scribed for example by the Taylor–Goldstein equation, growth in terms of an overreflection of waves in the arise in the presence of vertical shear, when the Rich- cross-shear direction, off of a critical level region. Un- ardson number at some place becomes less than a quar- der the right flow geometry, overreflected waves can be reflected back constructively to yield normal-mode growth, somewhat akin to a laser growth mechanism. Corresponding author address: Nili Harnik, Dept. of Geophys- ics and Planetary Sciences, Tel Aviv University, P.O. Box 39040, Since this theory is based on quite general wave prop- Tel Aviv 69978, Israel. erties, it deals both with gravity wave and vorticity wave E-mail: [email protected] instabilities. A very different approach, which has been

DOI: 10.1175/2007JAS2610.1

JAS2610 2616 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 65 developed for Rossby waves, is based on the notion of associated with the “divergent” part of the flow, are counterpropagating Rossby waves1 (CRWs; Bretherton filtered out. From the gravity wave perspective, how- 1966; Hoskins et al. 1985). Viewed this way, instability ever, the dynamics can be nondivergent, when viewed arises from a mutual reinforcing and phase locking of in three dimensions. Moreover, as we show later on, such waves. This explicit formulation applies only to gravity waves involve vorticity dynamics, and this part Rossby waves. of the dynamics has the same action-at-a-distance fea- Recently, the seemingly different overreflection and tures as quasigeostrophic dynamics. CRW approaches to shear instability have been united This paper presents a vorticity–buoyancy view of for the case of Rossby waves. A generalized form of gravity waves, examines how this interplay between CRW theory describes the perturbation evolution in vorticity and buoyancy affects the evolution of strati- terms of kernel–wave interactions. Defining a local vor- fied shear flow anomalies, and explores its use as a basis ticity anomaly, along with its induced meridional veloc- for a kernel view. Our general motivation in developing ity, as a kernel Rossby wave (KRW), Heifetz and Meth- a vorticity-based kernel view of the dynamics is quite ven (2005, hereafter HM) wrote the PV evolution equa- basic: in a similar way in which the KRW perspective tion in terms of mutual interactions between the has yielded mechanistic understanding for barotropic KRWs, via a meridional advection of background PV, and baroclinic shear flows we expect that finding the in a way that was mathematically similar to the classical corresponding gravity wave building blocks will provide CRW formulation. The KRWs are kernels to the dy- a new fundamental insight into a variety of stratified namics in a way analogous to a Green function kernel. shear flow phenomena, such as a basic mechanistic Harnik and Heifetz (2007, hereafter HH07) used this (rather than mathematical) understanding of the cross- kernel formulation to show that KRW interactions are shear propagation of gravity wave signals, the necessary at the heart of cross-shear Rossby wave propagation conditions for instability, the overreflection mecha- and other basic components of overreflection theory, nism, and the nonmodal growth processes in energy and in particular, they showed that overreflection can and enstrophy norms. be explained as a mutual amplification of KRWs. The paper is structured as follows. After formulating Given that CRW theory can explain the basic com- the simplified stratified shear flow equations in terms of ponents of Rossby wave overreflection theory (e.g., vorticity–buoyancy dynamics (section 2), we examine wave propagation, evanescence, full, partial, and over- how the interaction between these two fields is manifest reflection) using its own building blocks (KRWs), and in normal modes in general (section 3a), in pure plane overreflection theory, in turn, can rationalize gravity waves (section 3b) and in a single interface between con- wave instabilities, it is natural to ask whether a wave– stant buoyancy and vorticity regions (section 3c). We then kernel interaction approach exists for gravity waves as go on to develop a kernel framework in section 4, for a well. Indeed, it has been shown that a mutual amplifi- single interface (section 4a), two and multiple interfaces cation of counter-propagating waves applies also to the (sections 4b and 4c), and the continuous limit (section interaction of Rossby and gravity waves, or to mixed 4d). We discuss the results and summarize in section 5. vorticity–gravity waves (Baines and Mitsudera 1994; Sakai 1989). These studies did not explicitly discuss the 2. General formulation case of pure gravity waves, in the absence of back- We consider an inviscid, incompressible, Boussinesq, ground vorticity gradients. In this paper we present a 2D flow in the zonal–vertical (x–z) plane, with a zonally general formulation of the dynamics of linear stratified uniform basic state that varies with height and is in shear flow anomalies in terms of a mutual interaction of hydrostatic balance. analogous kernel gravity waves (KGWs). We show how We start with the momentum and continuity equa- this formulation holds even when vorticity gradients, tions, linearized around this basic state: and hence “Rossby-type” dynamics, are absent. Du 1 Ѩp At first glance, Rossby waves and gravity waves seem ϭϪ Ϫ ͑ ͒ wUz ␳ Ѩ , 1a to involve entirely different dynamics. The common Dt 0 x framework used to describe Rossby waves is the quasi- Dw 1 Ѩp ϭ Ϫ ͑ ͒ b ␳ Ѩ , 1b geostrophic (QG) one, in which motions are quasi- Dt 0 z horizontal. In this framework, gravity waves, which are Db ϭϪwN 2, ͑1c͒ Dt 1 Since Rossby waves propagate in a specific direction, set by Ѩu Ѩw the sign of the mean PV gradient, their propagation is either with ϩ ϭ 0, ͑1d͒ or counter the zonal mean flow. Ѩx Ѩz AUGUST 2008 HARNIK ET AL. 2617 x), u ϭ (u, w)isthe ity and displacement perturbations, we can determineץ/ץ)t) ϩ Uץ/ץ) where (D/Dt) ϵ perturbation velocity vector and its components in the the vertical velocity associated with the vorticity per- zonal and vertical directions, respectively; U and Uz are turbation. We note that the vertical velocity is nonlocal, the zonal mean flow and its vertical shear, respectively; in the sense that it depends on the entire vorticity p is the perturbation pressure; ␳␱ is a constant reference anomaly field. Given the vertical velocity field, it will ϵϪ␳ ␳ density; b ( / 0)g is the perturbation buoyancy; locally determine how the displacement field evolves, ץ ␳ץ ϵϪ ␳ 2 N (g/ 0)( / z) is the mean flow Brunt–Väisälä and along with the displacement field, will also deter- frequency, with ␳ and ␳ as the perturbation and mean mine how the vorticity field evolves. This scheme of flow density, respectively; and g is gravity. We note that inverting the vorticity field to get a vertical velocity (via 2 ϭ N bz, and use this in further notation. a streamfunction), then using this velocity to time inte- We now take the curl of Eqs. (1a) and (1b), assume grate the vorticity and displacement fields, inverting the that buoyancy anomalies arise purely from an advec- new vorticity to get the new velocity, and so on, is simi- tion of the basic state, and express the buoyancy per- lar in principle to the forward integration of the QG turbation in terms of the vertical displacement ␨ (not to equations, only we now have two, instead of one dy- be confused with the vertical component of vorticity). namical variable. This yields an alternative set of equations for the vor- Phrased this way, nonlocality enters the dynamics in ticity and displacement tendencies: the same was as it does in QG, and in the CRW formulation. We will use this to develop analogous KGWs, Dq Ѩ␨ ϭϪ Ϫ ͑ ͒ and a corresponding view of gravity wave dynamics. wqz bz Ѩ , 2a Dt x More explicitly, introducing a streamfunction ␺,so Ϫ ץ ץ ϭ ץ ␺ץ ϭ ץ ␺ץϭϪ D␨ that u ( / z), w ( / x) and q ( w/ x) -z) ϭٌ2␺, and for a single zonal Fourier compoץ/uץ) ϭ w, ͑2b͒ Dt nent with wavenumber k of the form e ikx, w ϭ ik␺, and 2 ץ 2␺ץ ϭϪ ␨ ͑ ͒ ϭϪ 2␺ ϩ b bz , 2c q k ( / z ). We can then express the inversion of q into w via a Green function formulation: ϭϪ 2 ץ ץ Ϫ ץ ץ ϭ where q ( w/ x) ( u/ z) and qz Uzz. Ex- pressed in this way, the dynamic evolution of anomalies is reduced to a vorticity equation and a trivial kinematic w͑z͒ ϭ ͵ q͑zЈ͒G͑zЈ, z, k͒ dzЈ, ͑3͒ relation for particle displacements, with vertical veloc- zЈ -z2) ϭ ik␦(z Ϫ zЈ)G(zЈ, z, k) deץ/2Gץ) ity mediating between the vorticity and displacement where Ϫk 2G ϩ fields. We see that vorticity evolves either by a vertical pends on the boundary conditions and on the zonal advection of background vorticity gradients, or via a wavenumber. For example, for open flow, which we displacement (buoyancy) term. The latter expresses the assume here for simplicity, the Green function is tendency of material surfaces (which are surfaces of constant density) to flatten out, creating vorticity in the i G͑zЈ, z͒ ϭϪ eϪk| zϪzЈ|. ͑4͒ process. This is shown schematically for a sinusoidal 2 perturbation of an interface between two layers of constant density (in Fig. 2). Concentrating on the zero Other boundary conditions will change G(zЈ, z, k) (see ␨ Ͼ point at which the surface tilts upward to the east ( x HM for some explicit examples), but will not affect the 0), the tendency of gravity to flatten the interface will main points of this paper. Substituting (3) into Eqs. (2a) raise the surface just west of this point and sink it on its and (2b), using (4), we get the following set of equa- east, resulting in clockwise motion of the surface at the tions which describe the evolution of perturbations of zero point, and a corresponding production of negative zonal wavenumber k: vorticity. We further note that for nondivergent flow, a knowl- Dq i ϭ ͵ ͑ Ј͒ Ϫk| zϪzЈ| Ј Ϫ ␨ ͑ ͒ edge of w is sufficient to determine the entire flow field, qz q z e dz ikbz , 5a Dt 2 zЈ which is directly related to q via a standard vorticity inversion. The following picture emerges: given vortic- D␨ i ϭϪ ͵ q͑zЈ͒eϪk| zϪzЈ| dzЈ. ͑5b͒ Dt 2 zЈ 2 In a 3D system, the component of vorticity perpendicular to the x–z plane would be defined as minus our definition. We chose For practical purposes Eqs. (5a) and (5b) can be dis- this convention to highlight the analogy to the case of horizontal cretized into N layers to obtain a numerical scheme (see barotropic flow. appendix A). 2618 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 65

3. The buoyancy–vorticity interaction in normal In this subsection we examine the vorticity–buoyancy modes interplay for such waves. Plugging the plane wave vorticity anomalies field [Eq. (8)] into the integrals on the The interaction between the vorticity and buoyancy rhs of Eqs. (5a) and (5b), assuming a constant infinite fields has to have specific characteristics for normal basic state, yields the following equations for q and ␨ modes, which by definition, have a spatial structure [where we have used relation (A5)]: which is fixed in time. Understanding how the two fields arrange themselves so that they evolve in concert qz ͑U Ϫ c͒q ϭ q Ϫ b ␨, ͑9a͒ yields some understanding of the mechanistics of grav- K 2 z ity waves and their propagation, and of related instabilities. Moreover, the kernel view, which we present in q ͑ Ϫ ͒␨ ϭϪ ͑ ͒ section 4, is based on the normal-mode solutions of a U c 2 , 9b K single interface. where K 2 ϭ k 2 ϩ m2, is the total square wavenumber. a. General background flow This indeed yields neutral q and ␨ structures, which We first note a few general properties of the normal- propagate with the following (flow relative) phase mode solutions to Eqs. (2a) and (2b) [or (5a) and (5b)]. speed:

Multiplying Eq. (2b) by qz, and adding to Eq. (2a), to 2 w q Ϯ qz qz bz get rid of , we get an equation that involves only c Ϫ U ϭϪ Ϯ ͱͩ ͪ ϩ . ͑10͒ and ␨: 2K 2 2K 2 K 2

D Ѩ␨ We also obtain the following ␨–q relation: ͑q ϩ q ␨͒ ϭϪb . ͑6͒ Dt z z Ѩx qϮ ϭ ͓K 2͑cϮ Ϫ U͔͒␨Ϯ. ͑11͒ We assume a normal-mode solution of the form Ϫ e i(kx ct), where the zonal phase speed is allowed to be Note that (cϮ Ϫ U) is either positive or negative, and ␨ ϭ ϩ complex, c cr ici, and we write the vorticity and and q are either in phase or antiphase. The Ϯ super- displacement in terms of an amplitude and phase, as script thus denotes the sign of cϮ Ϫ U, and the sign of ␣ ␪ follows: q ϭ Qe i , ␨ ϭ Ze i . Equating the real and the correlation between q and ␨. The results are also ϭ imaginary parts of (6) gives consistent with Eqs. (7a) and (7b) (with ci 0). We note that for pure plane wave solutions, we need bzci tan͑␪ Ϫ ␣͒ ϭ , ͑7a͒ to assume qz and U are both constant. To allow an ͑ Ϫ ͒ Ϫ ͓͑ Ϫ ͒2 ϩ 2͔ bz cr U qz cr U ci analogy with single interface solutions of the next section, we assume both U and q are nonzero and con- 2 ͓͑ Ϫ ͒ Ϫ ͔2 ϩ 2 2 2 z Q ͕ cr U qz bz ci qz ͖ ͩ ͪ ϭ . stant, though strictly speaking this can only be valid if Z ͑ Ϫ ͒ Ϫ ͓͑ Ϫ ͒2 ϩ 2͔ 2 ϩ 2 2 ͕bz cr U qz cr U ci ͖ bzci there is an external vorticity gradient analogous to the QG ␤ (i.e., q ϭ U ϩ⌫). Since no such ⌫ exists for the ͑7b͒ z zz vertical direction (though this might be relevant for ϭ plasma flow with a background magnetic field vertical We see that for neutral normal modes (ci 0), q and ␨ are either in phase or antiphase, while for growing/ gradient), this derivation should only be treated as a ␨ thought experiment, done to get at the essential mecha- decaying normal modes (ci 0) q and have a different phase relationship, which, for a given complex zonal nistics of vorticity–buoyancy interplay in the presence of buoyancy and vorticity gradients. phase speed, varies spatially with U, bz, and qz. More- ϭ Under the right conditions, Eq. (10) reduces to the over, if there are no vorticity gradients (qz 0), q and ␨ have to be in quadrature at the critical surface (where well-known Rossby–gravity plane wave dispersion re- ϭ lations. When b ϭ 0 we get a pure vertical Rossby cr U). z plane wave:3 b. Plane waves The most common textbook example of gravity 3 We are using the term Rossby waves in its most general sense waves is that of neutral infinite plane waves of the fol- of vorticity waves on a vorticity gradient, even though Rossby lowing form: waves generally refer to PV perturbations on a meridional PV gradient, while here we have vorticity anomalies on a vertical ϭ i͑kxϩmzϪ␻t͒ ␨ ϭ ␨ i͑kxϩmzϪ␻t͒ ͑ ͒ q q0e ; 0e . 8 vorticity gradient. AUGUST 2008 HARNIK ET AL. 2619

qz (5b) yield the following dispersion relation (cf. Baines ͑cϪ Ϫ U͒ ϭϪ , ͑12a͒ Rossby K 2 and Mitsudera 1994): where the dispersion relation is equivalent to the hori- ⌬q ⌬q 2 ⌬b cϮ Ϫ U ϭϪ Ϯ ͱͩ ͪ ϩ . ͑14͒ zontal Rossby (1939) ␤-plane wave (we ignored the re- 4k 4k 2k ϭ dundant meaningless null solution). When qz 0 the two pure gravity plane waves are obtained: We see that there are two normal-mode solutions, one eastward and one westward propagating, relative to the ͌ Ϯ bz mean flow, denoted by the superscript, respectively. ͑cϮ Ϫ U͒ ϭϮ . ͑12b͒ Gravity K The dispersion relation reduces to the single back- ϭ ground gradient solutions as expected: plugging qz 0 c. Single interface solutions yields the well-known gravity wave dispersion relation for the pair of gravity wave normal modes (to be found To understand how the buoyancy and vorticity fields in standard textbooks, e.g., Batchelor 1980): interact and evolve in normal modes, and how their dynamics affects their phase speed, it is illuminating to ⌬ Ϯ b examine the simple case of an interface between two c Ϫ U ϭϮͱ , ͑15͒ 2k constant vorticity and buoyancy regions. The solution ϭ to this problem was already obtained by Baines and whereas plugging bz 0 we get the Rossby wave dis- Mitsudera (1994). Nonetheless, we present it here to persion relation at a single interface (e.g., Swanson et emphasize the vorticity–buoyancy interaction mecha- al. 1997): nistics. Moreover, the normal-mode solutions to the ⌬q single interface will serve as the basis for a kernel view c Ϫ U ϭϪ , ͑16͒ of stratified shear flow anomalies (section 4), as was 2k done for Rossby waves. and another null solution c ϭ U in which all fields are ϭ We examine the evolution of waves on a fluid with zero. Note that when bz 0, the buoyancy perturbation ϭϪ ␨ two regions of constant vorticity–buoyancy separated at is zero, and Eq. (2a) yields the relation q qz , which ϭ an interface at z z0. The vorticity–buoyancy gradients is similar to the buoyancy–displacement relation in the are then presence of a density gradient.5 The fact that q and ␨ ϭ ⌬ ␦͑ Ϫ ͒ ͑ ͒ are directly tied this way means there is only one inde- qz q z z0 , 13a pendent dynamical equation and one normal-mode so- ϭ ⌬ ␦͑ Ϫ ͒ ͑ ͒ bz b z z0 , 13b lution. ⌬ ϭ Ͼ Ϫ Ͻ ⌬ ϭ Coming back to the general case of nonzero bz and where q q(z z0) q(z z0), and b Ͼ Ϫ Ͻ ϭ qz, it is easy to verify from the dispersion relation in b(z z0) b(z z0). Equation (2c) implies that b ˜␦ Ϫ ˜ ϭϪ⌬ ␨ (14) that when viewed from the frame of reference b (z z0) for this basic state, so that b b . Note that b is proportional to minus the density, so that a moving with the mean flow, the westward mode propagates faster than the pure gravity wave while the east- stable stratification implies positive bz, and an upward interface displacement is associated with a positive den- ward mode propagates slower than the pure gravity sity anomaly, or a negative buoyancy anomaly. Simi- wave. This makes sense, a priori, since the additional ϭ ␦ Ϫ Rossby-type dynamics tends to shift the pattern west- larly, from Eq. (2a), we see that q q˜ (z z0). The ␦-function form of q and b makes sense since a vertical ward. displacement will induce anomalies only at the inter- We can obtain a more mechanistic understanding by face.4 Note, however, that even in the absence of vor- examining the normal-mode structure. Since the two ϭ variables evolve differently—q evolves as a result of ticity gradients (qz 0), a delta function in vorticity will be produced, due to the jump in buoyancy. In any case, horizontal interface slopes and mean vorticity advec- ␨ the vertical velocity for this anomaly is simply w ϭ tion by the vertical velocity anomaly, while evolves Ϫ(i/2)q˜ [cf. Eqs. (3) and (4)]. directly from the vertical velocity, which is tied by defi- Using the above definitions, and assuming normal- nition to the vorticity—the normal modes have to as- mode wave solutions of the form e ik(xϪct), Eqs. (5a) and sume a specific structure, which allows them to propagate in concert. This structure can be obtained from Eq. (5b) and the definition of w: 4 Note that we are ignoring the possibility of neutral normal modes associated with the continuous spectrum, for which which ϭ ϭ 5 ϭ q 0 and b 0intheqz 0, bz 0 regions. Equations (7a) and (7b) also yield this relation for bz 0. 2620 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 65

FIG. 1. Schematic illustration of the normal-mode solution to a pure vorticity jump, on the zonal– vertical plane. The wavy line denotes the interface, the arrows denote the vertical velocity, and the sign ␨ ϭ of q and at the crests is marked. The shaded regions on the top plot (at time t t0) denote regions of positive vorticity generation and negative displacement generation. This vorticity and displacement generation pattern shifts the wave westward relative to the mean flow (denoted by the thick horizontal ϭ ϩ⌬ arrow). The wave position at a quarter period later is shown below (marked t t0 t). For this case there is only the negatively correlated ␨ Ϫ mode (see text for details), which is a westward-propagating Ͼ Rossby wave (for qz 0).

q˜ Ϯ ϭ 2k͑cϮ Ϫ U͒␨Ϯ. ͑17͒ shifted a quarter wavelength to the west of the vorticity ϭ pattern, which, according to Eq. (2a) (with qz 0), will Note that q and ␨ are either in phase or in antiphase [as increase q to the east of the positive q center, and de- expected from Eq. (7a)], depending on the phase crease it to the west, resulting as well in an eastward propagation direction. Note also that Eq. (17) is con- shifting of the pattern. For the negatively correlated ϭ Ϫ sistent with Eqs. (7a) and (7b), with ci 0. mode (␨ ), the displacement remains the same but the It is easiest to understand mechanistically how these vorticity, and associated vertical velocity fields flip modes propagate by first considering the pure vorticity signs. As a result, the wave pattern will be coherently and pure density jumps, and then their combination. shifted westward rather than eastward (see Fig. 2, bot- The wave on a pure vorticity jump (illustrated in Fig. 1) tom). propagates via the well-known Rossby (1939) mecha- For the combined vorticity–buoyancy jump (Fig. 3), nism: the vertical velocity is always shifted a quarter the phase relations between q, ␨ Ϯ, and w are the same wavelength to the east of the vorticity field (Fig. 1). For as in the pure buoyancy jump case, but now, the vertical a positive qz, this tends to shift the pattern westward, as velocity also produces vorticity via advection of qz. The a result of the vertical advection of the background direct effect of this is only on the vorticity evolution. vorticity. Note that since q and w are in quadrature, the Looking for example at ␨ ϩ (Fig. 3, top), the w–␨ phase wave cannot change its own amplitude. relation will shift ␨ ϩ to the east. At the same time, the Looking at the ϩ wave on a pure density jump (il- vorticity field evolves via two processes, according to lustrated in Fig. 2, top), the vertical velocity is shifted a Eq. (2a). The advection of background vorticity gradi- quarter wavelength to the east of ␨ ϩ (since q and ␨ are ent by the vertical velocity tends to decrease q˜ to the in phase) so that w is upward to the east of the interface east of its positive peak (the interface crest), but the ridges. This induces a positive displacement anomaly to interface slope tends to increase the vorticity there. the east, resulting in an eastward shifting of the dis- Thus, the vertical velocity tends to shift the pattern placement pattern relative to the mean flow. At the westward (the Rossby wave mechanism) while the ␨ x)is gradient tends to shift it eastward (the gravity waveץ/␨ץ) same time, the west–east interface slope AUGUST 2008 HARNIK ET AL. 2621

FIG. 2. Schematic illustration of the normal-mode solutions to a pure buoyancy jump. Shown are (top) the negatively correlated, westward-propagating ␨ Ϫ and (bottom) the positively correlated, eastward- propagating ␨ ϩ. The schematics are similar to Fig. 1, with the shaded regions denoting regions of negative displacement generation and positive/negative vorticity generation in the top/bottom. mechanism). Since the normal-mode solution requires Equation (18) for the phase speed expresses a kine- the vorticity and displacement fields to evolve coher- matic relation. Equation (17) further incorporates the ently, the buoyancy gradient effect must win. For a relation between w and q˜ assuming 2D nondivergent given background flow, and a given wavelike interface flow and open-boundary conditions [w ϭϪ(i/2)q˜]. displacement, this will only happen if the vertical ve- Both equations hold for a general interface. The basic- locity (and hence the vorticity anomaly) will be small state structure, characterized by qz and bz, affects the enough. This constraint on the size of the w anomaly, normal-mode structure and dispersion relation via the per given unit ␨ anomaly, further implies that the phase vorticity equation, which determines the vorticity-per- speed is small, since from Eq. (2b) we have displacement “production ratio.” For the pure buoyancy jump case, this production ratio is equal for the i w c Ϫ U ϭ , ͑18͒ two modes, resulting in a symmetric dispersion relation. k ␨ For the pure vorticity jump case, we can only have a where we should note that in our case iw/␨ is a real negatively signed production ratio for a positive vortic- number, since w and ␨ are Ϯ␲/2 out of phase (in fact, ity gradient and hence only westward-propagating growth requires w and ␨ to not be in quadrature). waves.6 Similar arguments hold for the negatively correlated kernel (␨ Ϫ), but now the vertical velocity is shifted ␲/2 4. A kernel view to the west of the crests, so that the crests are shifted The CRW view of shear instability, as a mutual am- westward (Fig. 3b). At the same time the vertical ad- plification and phase locking of vorticity waves, was vection of vorticity now works with the buoyancy gradient to create a negative q˜ anomaly to the west of the 6 As opposed to b , which must be positive in stably stratified crest, resulting as well in a westward shift of the vortic- z flow, q can be either positive or negative. In general, the Rossby ity pattern. Correspondingly, the resulting vorticity z ϩ wave propagation is to the left of the mean vorticity gradient, anomaly is larger than in the ␨ case, hence w/␨ and the ϩ Ϫ hence for negative qz we obtain a positive c U and a positive phase speed are larger (in absolute value). ␨–q correlation. 2622 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 65

FIG. 3. As in Fig. 2, but for the combined buoyancy–vorticity jump. Note that the phase speed, as well as vertical velocity arrows, denote both the direction and relative magnitude, so that ␨ Ϫ propagates westward faster, and has a larger vertical velocity, than the pure buoyancy mode of Fig. 2, while ␨ ϩ propagates eastward slower, with a weaker vertical velocity (per unit displacement). first formalized for the idealized case of two interfaces lies elsewhere. If, on the other hand, the basic state has with oppositely signed vorticity jumps Bretherton additional vorticity jumps, new vorticity anomalies will (1966), and the instability was rationalized in terms of be created at these jumps by the far-field vertical ve- the interaction between anomalies on the two inter- locity of the q(z0) vorticity anomaly. These new anoma- faces. For this case, each interface on its own supports lies, will in turn induce a far field w, which is not nec- a neutral normal mode, but when both interfaces exist, essarily in quadrature with q(z0), allowing a change in interaction between the waves may yield instability. amplitude, as well as phase. The full flow evolution is Davies and Bishop (1994) formalized this for the Eady thus viewed as an interaction of multiple single- model, by mathematically expressing how the cross- interface kernels, where by kernel we mean the delta shear velocity induced by a wave at one interface af- function of vorticity at a given interface along with the fects the phase and amplitude of the wave at the other vertical velocity that it induces. In other words, the ker- interface, via its advection of the basic-state vorticity. nel at z0 is the normal-mode solution of a mean flow Unstable normal-mode solutions then emerge when the that has a single vorticity jump at z0. two waves manage to resist the shear and phase lock in This view is useful because it can be generalized to a configuration that also results in a mutual amplifica- continuous basic states by taking the limit of an infinite tion. number of jumps spaced at infinitesimal intervals, as This approach also holds for more than two jumps, in formulated in HM. So far, besides rationalizing shear which case the flow field evolution is expressed as a instability, a kernel-based view of Rossby waves has multiple wave interaction, as follows. A pure vorticity yielded physical insight into basic phenomena like ϭ jump (let’s say at z z0) supports a Rossby wave for cross-shear wave propagation, wave evanescence and which q and w are in quadrature, and since vorticity in reflection (HH07), and various aspects related to non- this case is only affected by vorticity advection, this normal growth (HM; de Vries and Opsteegh 2006, wave only propagates zonally (its phase changes) but it 2007a,b; Morgan 2001; Morgan and Chen 2002). does not amplify. Since there are no vorticity gradients With the goal of developing similar insight into grav- except at the jump, it does not create vorticity anoma- ity wave phenomena, we wish to develop a KGW for- AUGUST 2008 HARNIK ET AL. 2623 mulation of stratified shear flow anomalies. Following In the simplified case of a pure buoyancy jump, we note the Rossby wave example, we start with examining the that (cϩ Ϫ U) ϭϪ(cϪ Ϫ U), and these equations reduce evolution of anomalies on multiple basic-state jumps to (both vorticity and buoyancy). We define our kernels to be the normal-mode solutions to an isolated buoyancy– 1 q˜ ␨ϩ ϭ ͫ␨ ϩ ͬ, ͑21a͒ vorticity jump, then examine the evolution and mutual 2 2k͑cϩ Ϫ U͒ interaction of these kernels on two and multiple jumps, and finally we take the continuous limit. 1 q˜ ␨Ϫ ϭ ͫ␨ ϩ ͬ. ͑21b͒ Ϫ Ϫ a. The general time evolution on a single jump 2 2k͑c U͒

To understand the evolution of an arbitrary pertur- Once the projection onto the two kernels is found, bation field as a multiple-kernel interaction, we need to the temporal evolution is readily obtained since each first understand the evolution on a single interface. In kernel propagates at its own phase speed, which corre- the case of a pure vorticity jump, there is no buoyancy sponds to the single jump normal mode, and there is no anomaly and there is a single normal-mode solution 7 interaction between the kernels. This is stated math- [Eq. (16)], and correspondingly a single vorticity ker- ematically in appendix B, using matrix notation, with nel. A density jump, on the other hand, supports two the two kernels being the eigenvectors of the propaga- normal modes [Eqs. (15) and (17)] at each interface. tor matrix for the combined vorticity–displacement vec- ␨ Ϫ The q˜ structure of the normal modes is determined tor, and the phase speeds are directly related to the by the requirement that their spatial structure not eigenvalues. change with time. Since vorticity and buoyancy are independent variables (they influence each other’s evolution but any combination of the two can exist), two b. Two interfaces modes are needed to represent the full anomaly field at The simplest case which allows for kernel interac- a given level. tions is a mean flow with two interfaces. Such a setup As with the vorticity case, we define the kernels to be was examined by Baines and Mitsudera (1994). Here the normal modes of a single jump, thus at each level we formulate the problem in terms of kernel interac- there are two kernels, and each of these kernels has a tions, for the most general case of buoyancy–vorticity ␨ Ϫ specific q˜ structure. Taking this a step further, an jumps. We denote the interface locations by z1 and z2, ␨ Ϫ Ϫ ϭ⌬ Ͼ arbitrary q˜ configuration can be uniquely divided their distance by z2 z1 z 0, and the vorticity– ␨ ϩ ␨ Ϫ ⌬ ⌬ into the two kernels, and , as follows (using the buoyancy jumps by q1/2 and b1/2, where the subscript same notation as for the normal modes): 1⁄2 denotes the interface. The four kernels (two for each Ϯ ϩ Ϫ ϩ ϩ Ϫ Ϫ interface) are thus denoted by ␨ . The dynamics is q˜ ϵ q˜ ϩ q˜ ϭ 2k͓͑c Ϫ U͒␨ ϩ ͑c Ϫ U͒␨ ͔, 1/2 then described in terms of the evolution of the kernels, ͑19a͒ as follows. Writing Eqs. (5a) and (5b) for the two interfaces we get a 4 ϫ 4 matrix equation: ␨ ϭ ␨ϩ ϩ ␨Ϫ, ͑19b͒ where we have used the relation in (17). The two kernel Ѩ ␩ ϭ A␩, ͑22a͒ components are then obtained from q˜ and ␨ (see ap- Ѩt pendix B): 1 where ␨ϩ ϭ ͓Ϫ2k͑cϪ Ϫ U͒␨ ϩ q˜͔, 2k͓͑cϩ Ϫ U͒ Ϫ ͑cϪ Ϫ U͔͒ q˜ 1 ͑20a͒ ␨ 1 ␩ ϭ ͑22b͒ Ϫ 1 ϩ ΂q˜ 2΃ ␨ ϭϪ ͓Ϫ2k͑c Ϫ U͒␨ ϩ q˜͔. ͓͑ ϩ Ϫ ͒ Ϫ ͑ Ϫ Ϫ ͔͒ ␨ 2k c U c U 2 ͑20b͒ and A is defined in appendix C. Using Eqs. (20a) and (20b), we can write the transformation as 7 Again, we do not consider the normal modes associated with the continuous spectrum. ␨ ϭ T␩, ͑23a͒ 2624 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 65 where As expected, Eq. (27a) indicates that each mixed ␨ϩ buoyancy–vorticity kernel on its own is neutral. Its am- 1 plitude can grow only due to advection of mean buoy- ␨Ϫ 1 ancy and vorticity gradients by the vertical velocity in- ␨ ϭ , ͑23b͒ ␨ϩ duced by the kernels of the opposite interface. This ΂ 2 ΃ advecting velocity is proportional to the inducing ker- ␨Ϫ 2 nel displacement amplitudes and the nature of the in- and the transformation matrix T is defined in appendix version between vertical velocity and displacement. C. Using the similarity transformation, we get the dy- Hence, it decays with the distance between the inter- namic equation for the kernel displacements, written in faces (indicated by the Green function). Only the com- matrix and compact form, respectively: ponents of the inducing velocity that are in phase with Ѩ the induced kernel’s amplitude yield growth (the sines ␨ ϭ TϪ1AT␨, ͑24a͒ Ѩt on the rhs result from the fact that the displacement and vertical velocity of a kernel are ␲/2 out of phase). Ϯ Ϫ Ϯ Ϯ Ϯ Ϫ ⌬ c U Equation (27b) indicates that the instantaneous phase ␨˙ ϭϪikͫ͑c ␨ ͒ Ϯ e k zͩ ͪ 1ր2 1ր2 ϩ Ϫ Ϫ Ϫ⑀ c c 1ր2 speed of each kernel, ˙/k, is composed of its natural speed without interaction [Eq. (14)], and an effect of ϫ ͓͑ ϩ Ϫ ͒␨ϩ ϩ ͑ Ϫ Ϫ ͒␨Ϫ͔ ͬ ͑ ͒ the opposing interface kernels. The latter only arises c U c U ր , 24b 2 1 from that part of the induced vertical velocity which is where 1⁄2 indicate the interface and Ϯ indicate the ker- in phase with the kernel (and hence the cosines on the rhs). nel, and we have used Eq. (14) for cϮ, to cancel out some terms in the derivation. Note that the expression A normal-mode solution of the full two-interface sys- in the inner squared brackets has a flipped 2/1 index, tem requires phase locking of the four kernels, which is indicating the contribution of the opposite boundary. obtained when the rhs of Eq. (27b) is the same and Note also that constant for the four kernels (this constant is the real normal-mode phase speed cr). Normal-mode solutions ⌬q 2 ⌬b ͑ ϩ Ϫ Ϫ͒ ϭ ͱͫͩ ͪ ϩ ͬ ͑ ͒ should also have a constant growth rate (kci) for the c c 1ր2 2 . 25 ˙ Ϯ Ϯ ϭ 2k k 1ր2 four kernels. This is obtained only if Z1/2/Z1/2 kci in Equation (24b) shows, as expected, that in the absence Eq. (27a). In a companion paper we analyze the nor- ␨Ϯ ϭ mal-mode dispersion relation and structure of pure ץ ץ of interaction between the interfaces, ( / t) 1/2 Ϫ Ϯ␨ Ϯ gravity waves in terms of this mutual phase locking ik(c )1/2, which is simply the two kernel propagation equations. This confirms that the two kernels at a interaction for the two interface problem Umurhan et given interface do not interact at all (i.e., a ϩ kernel can al. (2008, unpublished manuscript, hereafter UHH). only interact with the ϩ and Ϫ kernels of the other We note that although this interaction is more complex interface). for gravity waves than for Rossby waves because there Written also in terms of displacement amplitudes (Z) are two kernels at each interface, vertical shear will and phases (⑀): discriminate between these kernels because phase locking dominantly occurs for kernels which counter propa- Ϯ Ϯ ⑀Ϯ ␨ ϭ i 1ր2 ͑ ͒ 1ր2 Z1ր2e 26 gate with respect to the mean flow (UHH). and taking the real and imaginary parts of Eq. (24b) yields the amplitude and phase evolution equations: c. Multiple interfaces cϮ Ϫ U As a step toward a continuous formulation, we gen- ˙ Ϯ ϭϮ Ϫk⌬zͩ ͪ Z1ր2 ke ϩ Ϫ eralize the two-interface equations to an arbitrary num- c Ϫ c ր 1 2 ber of jumps: ϩ ϩ ϩ Ϯ ϫ ͕͓͑c Ϫ U͒Z ͔ ր sin͑⑀ ր Ϫ ⑀ ր ͒ 2 1 2 1 1 2 Ϯ Ϫ N ˙Ϯ Ϯ Ϯ c U Ϫ | Ϫ |⌬ Ϫ Ϫ Ϫ Ϯ ␨ˆ ϭϪ ͭ͑ ␨ˆ ͒ Ϯ ͩ ͪ k j n z ϩ ͓͑ Ϫ ͒ ͔ ͑⑀ Ϫ ⑀ ͖͒ ͑ ͒ j ik c j ϩ Ϫ ͚ e c U Z 2ր1 sin 2ր1 1ր2 , 27a Ϫ c c j nϭ1, j 1 e Ϫk⌬z cϮ Ϫ U Ϫ ⑀Ϯ ϭ Ϯ Ϯ ͩ ͪ ˙1ր2 c1ր2 Ϯ ϩ Ϫ ϩ ϩ Ϫ Ϫ k Z ր c Ϫ c ր ϫ ͓͑ Ϫ ͒␨ ϩ ͑ Ϫ ͒␨ ͔ ͮ ͑ ͒ 1 2 1 2 c U c U n . 28 ϩ ϩ ϩ Ϯ ϫ ͕͓͑c Ϫ U͒Z ͔ ր cos͑⑀ ր Ϫ ⑀ ր ͒ 2 1 2 1 1 2 Decomposing ␨ˆ into amplitude and phase, as in Eq. (26) ϩ ͓͑ Ϫ Ϫ ͒ Ϫ͔ ͑⑀Ϫ Ϫ ⑀Ϯ ͖͒ ͑ ͒ c U Z 2ր1 cos 2ր1 1ր2 . 27b yields AUGUST 2008 HARNIK ET AL. 2625

cϮ Ϫ U N though the contribution of q(z)tow(z) is infinitesimal, ˙ˆ Ϯ ϭϮ ͩ ͪ Ϫk| jϪn|⌬z Zj k ϩ Ϫ ͚ e it is larger than the contribution of q at any other c Ϫ c ϭ j n 1, j height. Mathematically, we can get around this problem ϫ ͓͑ ϩ Ϫ ͒ ϩ͔ ͑⑀ϩ Ϫ ⑀Ϯ͒ by simply looking at the anomaly averaged over a finite ͕ c U Z n sin n j interval of width ⌬z. ϩ ͓͑ Ϫ Ϫ ͒ Ϫ͔ ͑⑀Ϫ Ϫ ⑀Ϯ͖͒ ͑ ͒ c U Z n sin n j , 29a To do this we discretize the domain as in appendix A, using Eqs. (A1) for the basic-state gradients, and define Ϯ Ϫ N Ϫk| jϪn|⌬z 1 c U e the average of any perturbation quantity f on height zj Ϫ ⑀ˆ˙Ϯ ϭ cϮ Ϯͩ ͪ ͚ k j j ϩ Ϫ Ϫ Ϯ as c c j nϭ1, j Zj ϩ⌬ ր ϫ ͕͓͑ ϩ Ϫ ͒ ϩ͔ ͑⑀ϩ Ϫ ⑀Ϯ͒ 1 zj z 2 c U Z n cos n j ˆ ϵ ͵ ͑ ͒ fj fdz. 31 ⌬z Ϫ⌬ ր ϩ ͓͑ Ϫ Ϫ ͒ Ϫ͔ ͑⑀Ϫ Ϫ ⑀Ϯ͒ ͑ ͒ zj z 2 c U Z n cos n j ͖. 29b The vertical velocity at zj induced by the vorticity Ϫ⌬ ϩ⌬ perturbation located between (zj z/2, zj z/2) and d. The continuous limit ⌬ approximated by qˆ j, for small z is Our overall goal is to develop a kernel formulation zjϩ⌬zր2 which holds for continuous basic states, as was done for i Ϫk| zϪzЈ| i wˆ ϭϪ qˆ ͵ e dzЈ ϷϪ qˆ ⌬z. j 2 j 2 j Rossby waves (HM). Apparently, this simply entails zjϪ⌬zր2 ⌬ → → ϱ taking the limit of Eq. (28), for z 0 with N . ͑ ͒ 32 However, this limit is ill defined when bz 0. To see this, we note that Eq. (28) involves writing Eqs. (5a) Substituting Eq. (32) in (2b) and seeking wavelike so- ϩ Ϫ and (5b) in terms of the kernels ␨ and ␨ . This re- lutions yields quires dividing the vorticity perturbation to the two ⌬ ϭ ͓ ͑ Ϯ Ϫ ͔͒␨ˆϮ ͑ ͒ kernels, according to Eq. (19a), using the dispersion zqˆ j 2k c Uj j , 33 relation in (14), and taking the above limit. Writing ϭ ␦ Ϫ which converges to Eq. (17) for qˆ j q˜ (z zj) and ⌬b ϭ b ⌬z, ⌬q ϭ q ⌬z, and noting that the single in- Ϯ z z ⌬z → 0. To get an expression for (c Ϫ U ), we take the terface vorticity q˜ is the amplitude of the vorticity delta j second total time derivative of Eq. (2a): function (which has units of vorticity times length) hence q˜ ϭ q⌬z, we get the following: 2 D qˆ j Dwˆ j ϭϪq Ϫ ikb wˆ . ͑34͒ Dt 2 zj Dt zj j 2 q q b ϭ ͫϪ z Ϯ ͱͩ zͪ ϩ z ͬ␨Ϯ ͑ ͒ For the chunk formulation this yields the following dis- q 2k ⌬ , 30 2 2 z persion relation: which blows up in the limit of ⌬z → 0, for b 0. When ␥ ␥ 2 ␥ z qzj qzj bzj ϭ cϮ Ϫ U ϭϪ Ϯ ͱͩ ͪ ϩ , ͑35͒ bz 0, this limit is well defined, and yields the basic j 2 2 2 ϭϪ ␨ 4k 4k 2k relation q qz . Indeed, for the Rossby wave case, the kernel formulation works for the continuous limit where the nondimensionalized number ␥ ϵ k⌬z should (HM). When bz 0, the formulation, in its current be taken small enough [for relation in (32) to hold]. It form, fails because an infinitesimal-width q, which has a is straightforward to verify that for a buoyancy– finite value (as opposed to a ␦ function q, which is vorticity staircase profile, this expression converges to infinitesimal in width and infinite in amplitude), has a (14) as ␥ → 0. Substituting Eq. (35) into (33) we obtain negligible contribution to w, and hence to the genera- the local ␨ˆ Ϫ qˆ relation: tion of ␨ [cf. Eq. (5b)]. This sets the ratio ␨ Ϯ/q to zero. ␥ 2 ␥ 1ր2 ␨ Ϯ qzj bzj Unlike the single interface case, where directly influ- b ␨ˆ ϭϮͫͩ ͪ ϩ ͬ qˆ ͑ ͒ z j j. 36 ences the time tendency of q [Eq. (5a)], and q directly 4k 2 2k 2 influences the evolution of ␨ [Eq. (5b)], here only ␨ Using this relation, and noting that produces q, so that q and ␨ cannot propagate in concert to form a normal-mode kernel structure. ␥ 2 ␥ 1ր2 qzj bzj ͑cϩ Ϫ cϪ͒ ϭ ͫͩ ͪ ϩ ͬ ͑ ͒ Because, however, the vorticity inversion is such that j 2 37 2k 2 k 2 its action at a distance decays with distance, the vertical velocity at a given location is most strongly influenced we get kernel amplitude and phase equations, which by the local vorticity anomaly. In other words, even are exactly similar to Eqs. (29a) and (29b), with ⌬b ϭ 2626 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 65

␥ ⌬ ϭ ␥ ␨ (bz/k) , q (qz/k) , and q and being the chunk- fields. The vertical displacement equation is then the averaged fields qˆ and ␨ˆ. simple statement that vertical velocity is the time de- We can now use these equations to numerically study rivative of displacement. For normal modes, the two the evolution of buoyancy–vorticity anomalies on an fields (vorticity and displacement) assume specific arbitrary continuous profile in terms of kernel interac- phase and amplitude relations. In particular, for neutral tions. This formulation, like the piecewise basic-state modes, vorticity and displacement are either in phase one, does not converge for ␥ → 0. This lack of conver- or in antiphase with respect to each other. gence stems from the fact that when the chunk size For baroclinic/barotropic flow, PV invertibility al- shrinks to zero, the vertical velocity induced by the lows us to develop a kernel view of the dynamics, where chunk vanishes [Eq. (32)], and no displacement is in- anomalies are composed of localized kernel Rossby duced. However, when we look at a finite region, the waves, which interact with each other. This mutual in- mechanistic picture of buoyancy–vorticity interactions, teraction occurs via the nonlocal flow induced by each with two chunk kernels at each discrete (finite width) kernel’s PV anomaly. In analogy, our buoyancy– level, holds. In particular, we can examine the case of vorticity view also allows a clear separation between constant bz and qz, with a pure plane wave solution of local and nonlocal dynamics, with the nonlocal part of ␨ˆ the form in (A3), calculate qˆ and , and the correspond- the dynamics stemming from the vorticity inversion. ␨ˆ ϩ ␨ˆ Ϫ ing and [cf. Eqs. (20a) and (20b)], and plug into This makes a kernel view of the dynamics possible, and the chunk version of Eqs. (29a) and (29b). Equation a major part of this paper deals with developing it. (29a) does indeed yield zero amplitude growth, while As was done for vorticity kernels, we define our ker- Eq. (29b) yields the discretized version of the plane nels based on single interface of vorticity–buoyancy wave dispersion relation in (10) using Eq. (A4) for the normal modes. Since both vorticity and buoyancy are sum terms (not shown). involved in stratified shear flow, there are two normal modes at each level, and thus, two kernels. These are composed of an eastward-propagating wave with posi- 5. Summary and discussion tively correlated vorticity and displacement, and a westward-propagating wave with negatively correlated vor- Potential vorticity thinking, which includes the con- ticity and displacement. Though having two kernels cepts of PV inversion and action at a distance, provides complicates the picture, the kernels are constructed so a mechanistic understanding of many aspects of baro- that they do not interact with each other at a given tropic and baroclinic shear flow anomalies (Hoskins et al. 1985). In this paper we present a corresponding vor- level. ticity viewpoint, which is appropriate for gravity waves An arbitrary initial vorticity–displacement distribu- and stratified shear flow dynamics. tion can, at each level, be uniquely decomposed into the We show that vorticity dynamics is central to strati- two kernel components, each of which evolves accord- fied shear flow anomalies, which are composed of a ing to its own internal dynamics, with an influence from continuous interaction between vorticity and buoyancy kernels at other levels (via their induced vertical veloc- perturbations. Horizontal buoyancy gradients generate ities). The dynamic equations for vorticity and displace- vorticity locally, whereas vorticity induces a nonlocal ment can thus be decomposed and written in terms of vertical velocity, which, in turn, generates both fields by the evolution of the two kernels (at each level). advecting the background buoyancy–vorticity. Thus, To get a sense of multiple kernel interactions be- the interplay between vorticity and buoyancy anoma- tween many levels, we consider a flow with multiple lies fully describes the linear dynamics on stratified buoyancy–vorticity jumps, which we initially perturb at ϭ shear flows, both modal and nonmodal. Casting the dy- a single interface located at z z0. The resulting vor- namics in this way essentially provides us with a time ticity anomaly at z0 will induce a far-field vertical ve- integration scheme that is analogous to the time inte- locity, which will perturb the other jump interfaces. The gration of QG equations, with two rather than one resulting displacements of each interface are associated prognostic equations, and a diagnostic equation for with buoyancy–vorticity anomalies. Note that for the streamfunction, which is related to vorticity as in QG. case of a pure buoyancy jump, an initial interface dis- Since the buoyancy anomalies arise purely from ver- tortion can only create a buoyancy anomaly, and no tical advection of the background buoyancy field, we vorticity anomaly, because the vorticity is constant can express the buoyancy anomalies via the local ver- across the jump. The anomalies initially induced by an tical displacement, and the dynamics reduce to a mu- interface distortion evolve according to the local vor- tual interaction between vorticity and displacement ticity–displacement interaction dynamics. Though these AUGUST 2008 HARNIK ET AL. 2627

initial ␨ and q perturbations are initially in phase, their shear flow phenomena. In a companion paper, UHH, amplitude ratio is not necessarily the normal-mode one. we examine instabilities on a constant shear flow with Thus, both normal modes will be excited, and will pro- two density jumps, using the above buoyancy–vorticity ceed to propagate in opposite directions independently kernel view. Even though in this system there are no of each other. These new excited kernels will induce a vorticity gradients, and hence no vorticity waves, insta- far-field w, which will affect other interfaces. The full bility arises out of a mutual interaction and phase lock- flow evolution can be viewed as an interaction of mul- ing of counter-propagating gravity waves, in analogy to tiple pairs of single-interface kernels. the CRW view of classical models like those of Ray- We note, however, that the description of the evolu- leigh and Eady. Since much of the kernel dynamics tion in terms of a superposition of the two kernels found for Rossby waves is also present in UHH, we whose density and vorticity anomalies are either in or expect the kernel view to yield a mechanistic under- out of phase is not necessarily the most intuitive one. standing of other aspects of stratified shear flow, like For example, for a pure buoyancy interface, an initial nonmodal growth of anomalies, and basic processes displacement-only anomaly will excite the two modes like cross-shear gravity wave propagation, reflection, with equal amplitude (␨ ϩ and ␨ Ϫ are in phase but qϩ and overreflection. One of the central theorems of and qϪ are in antiphase). Such a superposition of the stratified shear flows is the Miles–Howard criterion for two kernels with equal amplitude actually gives rise to instability—that the Richardson number (Ri)be a standing wave (in the moving frame of the mean flow) smaller than 1⁄4 somewhere in the domain (Miles 1961; in which the vorticity and the density anomalies are in Howard 1961). Though it has been arrived at in various quadrature (see appendix B). We expect the kernel ways, our mechanistic (as opposed to mathematical) view to be the most intuitive in many cases (as we show understanding of it is only partial. Some mechanistic in a companion paper, UHH), while in other setups, an understanding is gained in the context of both overre- alternative standing-oscillation building block might flection and the counter-propagating wave interaction frameworks, as follows. R Ͻ 1⁄4 is also a criterion for provide a more intuitive description of the dynamics i wave evanescence in the region of a wave critical sur- (currently under investigation). face, which allows for the significant transmission of To make the kernel formulation applicable to gen- gravity waves through the critical level (Booker and eral stratified shear flow profiles, we use the multiple- Bretherton 1967; see also Van Duin and Kelder 1982, jump formulation as a basis for a continuous flow. The where this is explicitly demonstrated for the far field). continuous limit has been found to be quite subtle. Our On the one hand, the existence of an evanescent region kernel formulation is based on the assumption that is a necessary component for overreflection (e.g., clear vorticity–displacement relations exist, which are Lindzen 1988). On the other hand, as pointed out by used to define our kernels. For the pure Rossby wave Baines and Mitsudera (1994), this evanescent region case, where the piecewise setup converges nicely to the serves to divide the fluid into two wave regions, whose continuous problem, vorticity and displacement are di- waves can interact and mutually amplify. This suggests ϭϪ ␨ rectly related (q qz ). When buoyancy gradients the two approaches to gravity wave instabilities, based and corresponding buoyancy anomalies exist, the strict on overreflection and on counter-propagating wave in- ␨ relation between q and exists for jumps, but becomes teraction, may be related in a similar way to what was singular in the continuous limit. This stems from the shown for Rossby waves using the kernel approach ␨ fact that an important part of this strict –q relation is (HH07). Our gravity wave kernel formalism is one step ␨ the generation of by q via its induced vertical velocity. toward understanding this relation, and in the end we In the continuous limit, an infinitesimal localized q gen- expect the Richardson number criterion to become ␨ erates an infinitesimal w, and hence an infinitesimal . much clearer. To recover a ␨–q relation, we recover a locally induced w, by considering finite-width vorticity chunks, which Acknowledgments. Part of the work was done when generate a small, but nonzero ␨. This chunk formulation EH was a visiting professor at the Laboratoire de can be viewed as a numerical discretization approxima- Météorologie Dynamique at the Ecole Normale tion of the full ␨–q dynamic equations. Supérieure. The work was supported by the European To summarize, we present a vorticity–buoyancy Union Marie Curie International reintegration Grant framework for stratified shear flow anomalies, which MIRG-CT-2005-016835 (NH), the Israeli Science yields new insight into the mechanistics of gravity Foundation Grant 1084/06 (EH and NH), and the Bi- waves. This framework can now be applied to gain a national Science Foundation (BSF) Grant 2004087 (EH mechanistic understanding of a variety of stratified and OMU). 2628 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 65

APPENDIX A APPENDIX B

An Explicit Numerical Scheme for Time Eigenvalue Representation of the Single Interface Integration of Eq. (5) Dynamics

We consider a smooth basic state with vorticity q(z) For a single interface Eqs. (5a) and (5b) can be writ- and buoyancy b(z), which we discretize into N layers ten in the matrix form: ⌬ ϭ ⌬ ϭ with a vertical width z so that zn n z and n 1, 2,...,N (in principle ⌬z can be a function of z as well, ⌬q kU Ϫ k⌬b depending on the complexity of the profiles, but for Ѩ q˜ 2 q˜ simplicity we choose it constant). The discretized mean ͩ ͪ ϭϪi ͩ ͪ, ͑B1͒ Ѩt ␨ ΂ 1 ΃ ␨ vorticity–buoyancy gradients are then kU 2 ͑ ͒ Ϫ ͑ ͒ q znϩ1ր2 q znϪ1ր2 q ϵ ; zn ⌬z whose solution is ϩ Ϫ ͒ b͑z ͒ Ϫ b͑z ͒ q˜ ϩ 2k͑c U Ϫ ϩ ϵ nϩ1ր2 nϪ1ր2 ͑ ͒ ͩ ͪ ϭ ␨ ͫ ͬe ikc t b . A1a,b 0 zn ⌬z ␨ 1

Ϫ The discrete form of Eqs. (5a) and (5b) is then 2k͑c Ϫ U͒ Ϫ ϩ ␨Ϫ ͫ ͬe Ϫikc t, ͑B2͒ 0 i N 1 ϭϪ ͑ ϩ ␨ ͒ ϩ Ϫk| jϪn|⌬z⌬ q˙ j ik qjUi bzj j qzj ͚ qne z, 2 Ϯ nϭ1 where c is defined by Eq. (14). Hence, at t ϭ 0: ͑A2a͒ ␨ϩ ͑ ϩ Ϫ ͒ ͑ Ϫ Ϫ ͒ Ϫ1 q˜ 0 2k c U 2k c U 0 N ͩ ͪ ϭ ͫ ͬ ͩ ͪ i Ϫ ␨ ␨˙ ϭϪ ␨ Ϫ Ϫk| jϪn|⌬z⌬ ͑ ͒ ␨ 110 j ikUi j ͚ qne z. A2b 0 2 nϭ1 1 The plane wave dispersion relation in Eq. (10) can be ϭ 2k͓͑cϩ Ϫ U͒ Ϫ ͑cϪ Ϫ U͔͒ recovered, for instance, when we take constant values of q ϭ q and b ϭ b , assume a discretized plane Ϫ zn z zn z 1 Ϫ2k͑c Ϫ U͒ q˜ 0 wave solution of the following form: ϫ ͫ ͬͩ ͪ, ͑B3͒ ϩ ␨ Ϫ1 2k͑c Ϫ U͒ 0 q q ͩ nͪ ϭ ͩ 0ͪ i͑kxϩmn⌬zϪ␻t͒ ͑ ͒ ␨ ␨ e , A3 in agreement with Eqs. (20a) and (20b). n 0 Note that for the case of a pure density jump, an and recall that in the limit N → ϱ and ⌬z → 0weget ϭ initial displacement-only anomaly (q˜ 0 0), which is N Nր2 what we get in response to an induced w, will excite the ͑imnϪk| jϪn|͒⌬z ͑imnϪk| jϪn|͒⌬z ␨ ϩ ϭ ␨ Ϫ ϭ ͚ e ⌬z ϭ ͚ e ⌬z two normal modes with equal amplitude: 0 0 nϭ1 n ϭϪNր2 1 ␨ ⁄2 0 [cf. (B3)]. Using (B2) and (15), this gives a standing wave (in the frame of reference of the mean flow), for 2k imz → e j. ͑A4͒ which q˜ and ␨ are actually in quadrature: K 2 This converges to the following integral relation which k⌬b q˜ ϭ ␨ ͌2k⌬b sinͩͱ tͪ sin͓k͑x Ϫ Ut͔͒, ͑B4a͒ was used to derive Eq. (9) by plugging the plane wave 0 2 vorticity anomalies field [Eq. (8)] into the integrals on the rhs of Eqs. (5a) and (5b), assuming a constant infi- k⌬b ␨ ϭ ␨ cosͩͱ tͪ cos͓k͑x Ϫ Ut͔͒. ͑B4b͒ nite basic state: 0 2 ϱ Ϫ | Ϫ Ј| ͵ q͑zЈ͒e k z z dzЈ APPENDIX C zЈϭϪϱ ϱ ϭ i͑kxϪ␻t͒ ͵ imzЈ Ϫk| zϪzЈ| Ј Matrix Formulation of the Two-Interface Problem q0e e e dz zЈϭϪϱ The matrix A is obtained from direct substitution in 2k ͐ Ј Ϫk| zϪzЈ| Јϭ ϭ i͑kxϩmzϪ␻t͒ ͑ ͒ Eqs. (5a) and (5b) [recall that zЈ q(z )e dz q0e . A5 Ϫ Ϫ 2 k| z zj| ϭ ␦ Ϫ K q˜ je for q q˜ j (z zj)]: AUGUST 2008 HARNIK ET AL. 2629

⌬q ⌬q kU Ϫ 1 k⌬b Ϫ 1 e Ϫk⌬z 0 1 2 1 2 1 1 kU e Ϫk⌬z 0 2 1 2 A ϭϪi . ͑C1͒ ⌬q ⌬q Ϫ 2 e Ϫk⌬z 0 kU Ϫ 2 k⌬b ΂ 2 2 2 2΃ 1 1 e Ϫk⌬z 0 kU 2 2 2

The q˜ Ϫ ␨ to ␨ ϩ Ϫ ␨ Ϫ transformation matrix T is obtained directly from Eqs. (20a) and (20b):

1 2k͑cϪ Ϫ U͒ ͭ ͮ ͭ Ϫ ͮ 00 ͓͑ ϩ Ϫ ͒ Ϫ ͑ Ϫ Ϫ ͔͒ ͓͑ ϩ Ϫ ͒ Ϫ ͑ Ϫ Ϫ ͔͒ 2k c U c U 1 2k c U c U 1 1 2k͑cϪ Ϫ U͒ ͭ Ϫ ͮ ͭ ͮ 00 ͓͑ ϩ Ϫ ͒ Ϫ ͑ Ϫ Ϫ ͔͒ ͓͑ ϩ Ϫ ͒ Ϫ ͑ Ϫ Ϫ ͔͒ 2k c U c U 1 2k c U c U 1 T ϭ . 1 2k͑cϪ Ϫ U͒ 00ͭ ͮ ͭϪ ͮ 2k͓͑cϩ Ϫ U͒ Ϫ ͑cϪ Ϫ U͔͒ 2k͓͑cϩ Ϫ U͒ Ϫ ͑cϪ Ϫ U͔͒ ΂ 2 2΃ 1 2k͑cϪ Ϫ U͒ 00ͭϪ ͮ ͭ ͮ ͓͑ ϩ Ϫ ͒ Ϫ ͑ Ϫ Ϫ ͔͒ ͓͑ ϩ Ϫ ͒ Ϫ ͑ Ϫ Ϫ ͔͒ 2k c U c U 2 2k c U c U 2 ͑C2͒

The similarity transformation is thus

TϪ1AT ϭϪik

cϩ Ϫ U cϩ Ϫ U cϩ 0 e Ϫk⌬zͩ ͪ ͑cϩ Ϫ U͒ e Ϫk⌬zͩ ͪ ͑cϪ Ϫ U͒ 1 ϩ Ϫ Ϫ 2 ϩ Ϫ Ϫ 2 c c 1 c c 1 cϪ Ϫ U cϪ Ϫ U 0 cϪ Ϫe Ϫk⌬zͩ ͪ ͑cϩ Ϫ U͒ Ϫe Ϫk⌬zͩ ͪ ͑cϪ Ϫ U͒ 1 ϩ Ϫ Ϫ 2 ϩ Ϫ Ϫ 2 c c 1 c c 1 ϫ . cϩ Ϫ U cϩ Ϫ U e Ϫk⌬zͩ ͪ ͑cϩ Ϫ U͒ e Ϫk⌬zͩ ͪ ͑cϪ Ϫ U͒ cϩ 0 cϩ Ϫ cϪ 1 cϩ Ϫ cϪ 1 2 ΂ 2 2 ΃ cϪ Ϫ U cϪ Ϫ U Ϫe Ϫk⌬zͩ ͪ ͑cϩ Ϫ U͒ Ϫe Ϫk⌬zͩ ͪ ͑cϪ Ϫ U͒ 0 cϪ ϩ Ϫ Ϫ 1 ϩ Ϫ Ϫ 1 2 c c 2 c c 2 ͑C3͒

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