A Buoyancy–Vorticity Wave Interaction Approach to ...harnik/publications/Harnik-etal-08-JAS.pdf · Stably stratified shear flows support two types of waves and associated instabilities—Rossby

A Buoyancy–Vorticity Wave Interaction Approach to Stratified Shear Flow

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 flowphenomena, 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. Thisformulation provides a time integration scheme that is analogous to the time integration of the quasigeo-strophic equations with two, rather than one, prognostic equations, and a diagnostic equation for stream-function through a vorticity inversion.

The invertibility of vorticity allows the development of a gravity wave kernel view, which provides amechanistic rationalization of many aspects of the linear dynamics of stratified shear flow. The resultingkernel 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 indepen-dent kernels at each level. Though having two kernels complicates the picture, the kernels are constructedso that they do not interact with each other at a given level.

1. Introduction

Stably stratified shear flows support two types ofwaves and associated instabilities—Rossby waves thatare related to horizontal potential vorticity (PV) gradi-ents, and gravity waves that are related to vertical den-sity gradients. Each of these wave types is associatedwith its own form of shear instabilities. Rossby waveinstabilities (e.g., baroclinic instability) arise when thePV gradients change sign (Charney and Stern 1962),while gravity wave–related instabilities, of the type de-scribed for example by the Taylor–Goldstein equation,arise in the presence of vertical shear, when the Rich-ardson number at some place becomes less than a quar-

ter (Drazin and Reid 1981). These conditions are quiteeasily obtained from the equations governing each typeof instability, but their physical basis is much less clear.We do not have an intuitive understanding as we do, forexample, for convective instability, which arises whenthe stratification itself is unstable (e.g., Rayleigh–Bernard and Rayleigh Taylor instabilities).

There are two main attempts to physically under-stand shear instabilities, which have gone a long waytoward building a mechanistic picture. Overreflectiontheory (e.g., Lindzen 1988) explains perturbationgrowth in terms of an overreflection of waves in thecross-shear direction, off of a critical level region. Un-der the right flow geometry, overreflected waves can bereflected back constructively to yield normal-modegrowth, somewhat akin to a laser growth mechanism.Since this theory is based on quite general wave prop-erties, it deals both with gravity wave and vorticity waveinstabilities. A very different approach, which has been

Corresponding author address: Nili Harnik, Dept. of Geophys-ics and Planetary Sciences, Tel Aviv University, P.O. Box 39040,Tel Aviv 69978, Israel.E-mail: [email protected]

AUGUST 2008 H A R N I K E T A L . 2615

DOI: 10.1175/2007JAS2610.1

© 2008 American Meteorological Society

JAS2610

developed for Rossby waves, is based on the notion ofcounterpropagating Rossby waves1 (CRWs; Bretherton1966; Hoskins et al. 1985). Viewed this way, instabilityarises from a mutual reinforcing and phase locking ofsuch waves. This explicit formulation applies only toRossby waves.

Recently, the seemingly different overreflection andCRW approaches to shear instability have been unitedfor the case of Rossby waves. A generalized form ofCRW theory describes the perturbation evolution interms of kernel–wave interactions. Defining a local vor-ticity anomaly, along with its induced meridional veloc-ity, as a kernel Rossby wave (KRW), Heifetz and Meth-ven (2005, hereafter HM) wrote the PV evolution equa-tion in terms of mutual interactions between theKRWs, via a meridional advection of background PV,in a way that was mathematically similar to the classicalCRW formulation. The KRWs are kernels to the dy-namics in a way analogous to a Green function kernel.Harnik and Heifetz (2007, hereafter HH07) used thiskernel formulation to show that KRW interactions areat the heart of cross-shear Rossby wave propagationand other basic components of overreflection theory,and in particular, they showed that overreflection canbe explained as a mutual amplification of KRWs.

Given that CRW theory can explain the basic com-ponents of Rossby wave overreflection theory (e.g.,wave propagation, evanescence, full, partial, and over-reflection) using its own building blocks (KRWs), andoverreflection theory, in turn, can rationalize gravitywave instabilities, it is natural to ask whether a wave–kernel interaction approach exists for gravity waves aswell. Indeed, it has been shown that a mutual amplifi-cation of counter-propagating waves applies also to theinteraction of Rossby and gravity waves, or to mixedvorticity–gravity waves (Baines and Mitsudera 1994;Sakai 1989). These studies did not explicitly discuss thecase of pure gravity waves, in the absence of back-ground vorticity gradients. In this paper we present ageneral formulation of the dynamics of linear stratifiedshear flow anomalies in terms of a mutual interaction ofanalogous kernel gravity waves (KGWs). We show howthis formulation holds even when vorticity gradients,and hence “Rossby-type” dynamics, are absent.

At first glance, Rossby waves and gravity waves seemto involve entirely different dynamics. The commonframework used to describe Rossby waves is the quasi-geostrophic (QG) one, in which motions are quasi-horizontal. In this framework, gravity waves, which are

associated with the “divergent” part of the flow, arefiltered out. From the gravity wave perspective, how-ever, the dynamics can be nondivergent, when viewedin three dimensions. Moreover, as we show later on,gravity waves involve vorticity dynamics, and this partof the dynamics has the same action-at-a-distance fea-tures as quasigeostrophic dynamics.

This paper presents a vorticity–buoyancy view ofgravity waves, examines how this interplay betweenvorticity and buoyancy affects the evolution of strati-fied shear flow anomalies, and explores its use as a basisfor a kernel view. Our general motivation in developinga vorticity-based kernel view of the dynamics is quitebasic: in a similar way in which the KRW perspectivehas yielded mechanistic understanding for barotropicand baroclinic shear flows we expect that finding thecorresponding gravity wave building blocks will providea new fundamental insight into a variety of stratifiedshear flow phenomena, such as a basic mechanistic(rather than mathematical) understanding of the cross-shear propagation of gravity wave signals, the necessaryconditions for instability, the overreflection mecha-nism, and the nonmodal growth processes in energyand enstrophy norms.

The paper is structured as follows. After formulatingthe simplified stratified shear flow equations in terms ofvorticity–buoyancy dynamics (section 2), we examinehow the interaction between these two fields is manifestin normal modes in general (section 3a), in pure planewaves (section 3b) and in a single interface between con-stant buoyancy and vorticity regions (section 3c). We thengo on to develop a kernel framework in section 4, for asingle interface (section 4a), two and multiple interfaces(sections 4b and 4c), and the continuous limit (section4d). We discuss the results and summarize in section 5.

2. General formulation

We consider an inviscid, incompressible, Boussinesq,2D flow in the zonal–vertical (x–z) plane, with a zonallyuniform basic state that varies with height and is inhydrostatic balance.

We start with the momentum and continuity equa-tions, linearized around this basic state:

Du

Dt� �wUz �

1�0

�p

�x, �1a�

Dw

Dt� b �

1�0

�p

�z, �1b�

Db

Dt� �wN2, �1c�

�u

�x�

�w

�z� 0, �1d�

1 Since Rossby waves propagate in a specific direction, set bythe sign of the mean PV gradient, their propagation is either withor counter the zonal mean flow.

2616 J O U R N A L O F T H E A T M O S P H E R I C S C I E N C E S VOLUME 65

where (D/Dt) � (�/�t) � U(�/�x), u � (u, w) is theperturbation velocity vector and its components in thezonal and vertical directions, respectively; U and Uz arethe zonal mean flow and its vertical shear, respectively;p is the perturbation pressure; � is a constant referencedensity; b � �(�/�0)g is the perturbation buoyancy;N2 � �(g/�0)(��/�z) is the mean flow Brunt–Väisäläfrequency, with � and � as the perturbation and meanflow density, respectively; and g is gravity. We note thatN2 � bz, and use this in further notation.

We now take the curl of Eqs. (1a) and (1b), assumethat buoyancy anomalies arise purely from an advec-tion of the basic state, and express the buoyancy per-turbation in terms of the vertical displacement (not tobe confused with the vertical component of vorticity).This yields an alternative set of equations for the vor-ticity and displacement tendencies:

Dq

Dt� �wqz � bz

��

�x, �2a�

D�

Dt� w, �2b�

b � �bz�, �2c�

where q � (�w/�x) � (�u /�z) and qz � �Uzz.2 Ex-pressed in this way, the dynamic evolution of anomaliesis reduced to a vorticity equation and a trivial kinematicrelation for particle displacements, with vertical veloc-ity mediating between the vorticity and displacementfields. We see that vorticity evolves either by a verticaladvection of background vorticity gradients, or via adisplacement (buoyancy) term. The latter expresses thetendency of material surfaces (which are surfaces ofconstant density) to flatten out, creating vorticity in theprocess. This is shown schematically for a sinusoidalperturbation of an interface between two layers of con-stant density (in Fig. 2). Concentrating on the zeropoint at which the surface tilts upward to the east (x �0), the tendency of gravity to flatten the interface willraise the surface just west of this point and sink it on itseast, resulting in clockwise motion of the surface at thezero point, and a corresponding production of negativevorticity.

We further note that for nondivergent flow, a knowl-edge of w is sufficient to determine the entire flow field,which is directly related to q via a standard vorticityinversion. The following picture emerges: given vortic-

ity and displacement perturbations, we can determinethe vertical velocity associated with the vorticity per-turbation. We note that the vertical velocity is nonlocal,in the sense that it depends on the entire vorticityanomaly field. Given the vertical velocity field, it willlocally determine how the displacement field evolves,and along with the displacement field, will also deter-mine how the vorticity field evolves. This scheme ofinverting the vorticity field to get a vertical velocity (viaa streamfunction), then using this velocity to time inte-grate the vorticity and displacement fields, inverting thenew vorticity to get the new velocity, and so on, is simi-lar in principle to the forward integration of the QGequations, only we now have two, instead of one dy-namical variable.

Phrased this way, nonlocality enters the dynamics inthe same was as it does in QG, and in the CRW for-mulation. We will use this to develop analogous KGWs,and a corresponding view of gravity wave dynamics.

More explicitly, introducing a streamfunction �, sothat u � �(��/�z), w � (��/�x) and q � (�w/�x) �(�u /�z) � 2�, and for a single zonal Fourier compo-nent with wavenumber k of the form e ikx, w � ik�, andq � �k2� � (�2�/�z2). We can then express the inver-sion of q into w via a Green function formulation:

w�z� � �z�

q�z��G�z�, z, k� dz�, �3�

where �k2G � (�2G/�z2) � ik�(z � z�)G(z�, z, k) de-pends on the boundary conditions and on the zonalwavenumber. For example, for open flow, which weassume here for simplicity, the Green function is

G�z�, z� � �i

2e�k | z�z� |. �4�

Other boundary conditions will change G(z�, z, k) (seeHM for some explicit examples), but will not affect themain points of this paper. Substituting (3) into Eqs. (2a)and (2b), using (4), we get the following set of equa-tions which describe the evolution of perturbations ofzonal wavenumber k:

Dq

Dt�

i

2qz�

z�

q�z��e�k | z�z� | dz� � ikbz�, �5a�

D�

Dt� �

i

2 �z�

q�z��e�k | z�z� | dz�. �5b�

For practical purposes Eqs. (5a) and (5b) can be dis-cretized into N layers to obtain a numerical scheme (seeappendix A).

2 In a 3D system, the component of vorticity perpendicular tothe x–z plane would be defined as minus our definition. We chosethis convention to highlight the analogy to the case of horizontalbarotropic flow.


3. The buoyancy–vorticity interaction in normalmodes

The interaction between the vorticity and buoyancyfields has to have specific characteristics for normalmodes, which by definition, have a spatial structurewhich is fixed in time. Understanding how the twofields arrange themselves so that they evolve in concertyields some understanding of the mechanistics of grav-ity waves and their propagation, and of related insta-bilities. Moreover, the kernel view, which we present insection 4, is based on the normal-mode solutions of asingle interface.

a. General background flow

We first note a few general properties of the normal-mode solutions to Eqs. (2a) and (2b) [or (5a) and (5b)].Multiplying Eq. (2b) by qz, and adding to Eq. (2a), toget rid of w, we get an equation that involves only qand :

D

Dt�q � qz�� bz

��

�x. �6�

We assume a normal-mode solution of the forme i(kx�ct), where the zonal phase speed is allowed to becomplex, c � cr � ici, and we write the vorticity anddisplacement in terms of an amplitude and phase, asfollows: q � Qei�, � Zei�. Equating the real andimaginary parts of (6) gives

tan�� bzci

bz�cr �U� � qz��cr �U�2 � ci2�

, �7a�

�Q

Z�2

��cr �U�qz � bz�

2 � ci2qz

2�2

�bz�cr �U� � qz��cr �U�2 � ci2��2 � bz

2ci

2.

�7b�

We see that for neutral normal modes (ci � 0), q and are either in phase or antiphase, while for growing/decaying normal modes (ci � 0) q and have a differentphase relationship, which, for a given complex zonalphase speed, varies spatially with U, bz, and qz. More-over, if there are no vorticity gradients (qz � 0), q and have to be in quadrature at the critical surface (wherecr � U).

b. Plane waves

The most common textbook example of gravitywaves is that of neutral infinite plane waves of the fol-lowing form:

q � q0e i�kx�mz��t�; � � �0e i�kx�mz��t�. �8�

In this subsection we examine the vorticity–buoyancyinterplay for such waves. Plugging the plane wave vor-ticity anomalies field [Eq. (8)] into the integrals on therhs of Eqs. (5a) and (5b), assuming a constant infinitebasic state, yields the following equations for q and [where we have used relation (A5)]:

�U � c�q �qz

K2 q � bz�, �9a�

�U � c�� q

K2 , �9b�

where K2 � k2 � m2, is the total square wavenumber.This indeed yields neutral q and structures, whichpropagate with the following (flow relative) phasespeed:

c� � U � �qz

2K2 �� qz

2K2�2

�bz

K2. �10�

We also obtain the following –q relation:

q� � �K2�c� � U��. �11�

Note that (c� � U) is either positive or negative, and and q are either in phase or antiphase. The � super-script thus denotes the sign of c� � U, and the sign ofthe correlation between q and . The results are alsoconsistent with Eqs. (7a) and (7b) (with ci � 0).

We note that for pure plane wave solutions, we needto assume qz and U are both constant. To allow ananalogy with single interface solutions of the next sec-tion, we assume both U and qz are nonzero and con-stant, though strictly speaking this can only be valid ifthere is an external vorticity gradient analogous to theQG � (i.e., qz � Uzz � �). Since no such � exists for thevertical direction (though this might be relevant forplasma flow with a background magnetic field verticalgradient), this derivation should only be treated as athought experiment, done to get at the essential mecha-nistics of vorticity–buoyancy interplay in the presenceof buoyancy and vorticity gradients.

Under the right conditions, Eq. (10) reduces to thewell-known Rossby–gravity plane wave dispersion re-lations. When bz � 0 we get a pure vertical Rossbyplane wave:3

3 We are using the term Rossby waves in its most general senseof vorticity waves on a vorticity gradient, even though Rossbywaves generally refer to PV perturbations on a meridional PVgradient, while here we have vorticity anomalies on a verticalvorticity gradient.


�c� � U�Rossby � �qz

K2 , �12a�

where the dispersion relation is equivalent to the hori-zontal Rossby (1939) �-plane wave (we ignored the re-dundant meaningless null solution). When qz � 0 thetwo pure gravity plane waves are obtained:

�c� � U�Gravity � ��bz

K. �12b�

c. Single interface solutions

To understand how the buoyancy and vorticity fieldsinteract and evolve in normal modes, and how theirdynamics affects their phase speed, it is illuminating toexamine the simple case of an interface between twoconstant vorticity and buoyancy regions. The solutionto this problem was already obtained by Baines andMitsudera (1994). Nonetheless, we present it here toemphasize the vorticity–buoyancy interaction mecha-nistics. Moreover, the normal-mode solutions to thesingle interface will serve as the basis for a kernel viewof stratified shear flow anomalies (section 4), as wasdone for Rossby waves.

We examine the evolution of waves on a fluid withtwo regions of constant vorticity–buoyancy separated atan interface at z� z0. The vorticity–buoyancy gradientsare then

qz � �q�z � z0�, �13a�

bz � �b�z � z0�, �13b�

where �q � q(z � z0) � q(z � z0), and �b �b(z � z0) � b(z � z0). Equation (2c) implies that b �b�(z � z0) for this basic state, so that b � ��b. Notethat b is proportional to minus the density, so that astable stratification implies positive bz, and an upwardinterface displacement is associated with a positive den-sity anomaly, or a negative buoyancy anomaly. Simi-larly, from Eq. (2a), we see that q � q�(z � z0). The�-function form of q and b makes sense since a verticaldisplacement will induce anomalies only at the inter-face.4 Note, however, that even in the absence of vor-ticity gradients (qz � 0), a delta function in vorticity willbe produced, due to the jump in buoyancy. In any case,the vertical velocity for this anomaly is simply w ��(i/2)q [cf. Eqs. (3) and (4)].

Using the above definitions, and assuming normal-mode wave solutions of the form e ik(x�ct), Eqs. (5a) and

(5b) yield the following dispersion relation (cf. Bainesand Mitsudera 1994):

c� � U � ��q

4k��q

4k�2

��b

2k. �14�

We see that there are two normal-mode solutions, oneeastward and one westward propagating, relative to themean flow, denoted by the � superscript, respectively.The dispersion relation reduces to the single back-ground gradient solutions as expected: plugging qz � 0yields the well-known gravity wave dispersion relationfor the pair of gravity wave normal modes (to be foundin standard textbooks, e.g., Batchelor 1980):

c� � U � ��b

2k, �15�

whereas plugging bz � 0 we get the Rossby wave dis-persion relation at a single interface (e.g., Swanson etal. 1997):

c � U � ��q

2k, �16�

and another null solution c � U in which all fields arezero. Note that when bz � 0, the buoyancy perturbationis zero, and Eq. (2a) yields the relation q ��qz, whichis similar to the buoyancy–displacement relation in thepresence of a density gradient.5 The fact that q and are directly tied this way means there is only one inde-pendent dynamical equation and one normal-mode so-lution.

Coming back to the general case of nonzero bz andqz, it is easy to verify from the dispersion relation in(14) that when viewed from the frame of referencemoving with the mean flow, the westward mode propa-gates faster than the pure gravity wave while the east-ward mode propagates slower than the pure gravitywave. This makes sense, a priori, since the additionalRossby-type dynamics tends to shift the pattern west-ward.

We can obtain a more mechanistic understanding byexamining the normal-mode structure. Since the twovariables evolve differently—q evolves as a result ofhorizontal interface slopes and mean vorticity advec-tion by the vertical velocity anomaly, while evolvesdirectly from the vertical velocity, which is tied by defi-nition to the vorticity—the normal modes have to as-sume a specific structure, which allows them to propa-gate 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 normalmodes associated with the continuous spectrum, for which whichq � 0 and b � 0 in the qz � 0, bz � 0 regions. 5 Equations (7a) and (7b) also yield this relation for bz � 0.


q� � 2k�c� � U��. �17�

Note that q and are either in phase or in antiphase [asexpected from Eq. (7a)], depending on the phasepropagation direction. Note also that Eq. (17) is con-sistent with Eqs. (7a) and (7b), with ci � 0.

It is easiest to understand mechanistically how thesemodes propagate by first considering the pure vorticityand pure density jumps, and then their combination.The wave on a pure vorticity jump (illustrated in Fig. 1)propagates via the well-known Rossby (1939) mecha-nism: the vertical velocity is always shifted a quarterwavelength to the east of the vorticity field (Fig. 1). Fora positive qz, this tends to shift the pattern westward, asa result of the vertical advection of the backgroundvorticity. Note that since q and w are in quadrature, thewave cannot change its own amplitude.

Looking at the � wave on a pure density jump (il-lustrated in Fig. 2, top), the vertical velocity is shifted aquarter wavelength to the east of � (since q and arein phase) so that w is upward to the east of the interfaceridges. This induces a positive displacement anomaly tothe east, resulting in an eastward shifting of the dis-placement pattern relative to the mean flow. At thesame time, the west–east interface slope (�/�x) is

shifted a quarter wavelength to the west of the vorticitypattern, which, according to Eq. (2a) (with qz � 0), willincrease q to the east of the positive q center, and de-crease it to the west, resulting as well in an eastwardshifting of the pattern. For the negatively correlatedmode (�), the displacement remains the same but thevorticity, and associated vertical velocity fields flipsigns. As a result, the wave pattern will be coherentlyshifted westward rather than eastward (see Fig. 2, bot-tom).

For the combined vorticity–buoyancy jump (Fig. 3),the phase relations between q, �, and w are the sameas in the pure buoyancy jump case, but now, the verticalvelocity also produces vorticity via advection of qz. Thedirect effect of this is only on the vorticity evolution.Looking for example at � (Fig. 3, top), the w– phaserelation will shift � to the east. At the same time, thevorticity field evolves via two processes, according toEq. (2a). The advection of background vorticity gradi-ent by the vertical velocity tends to decrease q to theeast of its positive peak (the interface crest), but theinterface slope tends to increase the vorticity there.Thus, the vertical velocity tends to shift the patternwestward (the Rossby wave mechanism) while the gradient tends to shift it eastward (the gravity wave

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 signof q and at the crests is marked. The shaded regions on the top plot (at time t � t0) denote regions ofpositive vorticity generation and negative displacement generation. This vorticity and displacementgeneration pattern shifts the wave westward relative to the mean flow (denoted by the thick horizontalarrow). The wave position at a quarter period later is shown below (marked t � t0 � �t). For this casethere is only the negatively correlated � mode (see text for details), which is a westward-propagatingRossby wave (for qz � 0).


mechanism). Since the normal-mode solution requiresthe vorticity and displacement fields to evolve coher-ently, the buoyancy gradient effect must win. For agiven background flow, and a given wavelike interfacedisplacement, this will only happen if the vertical ve-locity (and hence the vorticity anomaly) will be smallenough. This constraint on the size of the w anomaly,per given unit anomaly, further implies that the phasespeed is small, since from Eq. (2b) we have

c � U �i

k

w

�, �18�

where we should note that in our case iw/ is a realnumber, since w and are ��/2 out of phase (in fact,growth requires w and to not be in quadrature).

Similar arguments hold for the negatively correlatedkernel (�), but now the vertical velocity is shifted �/2to the west of the crests, so that the crests are shiftedwestward (Fig. 3b). At the same time the vertical ad-vection of vorticity now works with the buoyancy gra-dient to create a negative q anomaly to the west of thecrest, resulting as well in a westward shift of the vortic-ity pattern. Correspondingly, the resulting vorticityanomaly is larger than in the � case, hence w/ and thephase speed are larger (in absolute value).

Equation (18) for the phase speed expresses a kine-matic relation. Equation (17) further incorporates therelation between w and q assuming 2D nondivergentflow and open-boundary conditions [w � �(i/2)q].Both equations hold for a general interface. The basic-state structure, characterized by qz and bz, affects thenormal-mode structure and dispersion relation via thevorticity equation, which determines the vorticity-per-displacement “production ratio.” For the pure buoy-ancy jump case, this production ratio is equal for thetwo modes, resulting in a symmetric dispersion relation.For the pure vorticity jump case, we can only have anegatively signed production ratio for a positive vortic-ity gradient and hence only westward-propagatingwaves.6

4. A kernel view

The CRW view of shear instability, as a mutual am-plification and phase locking of vorticity waves, was

6 As opposed to bz, which must be positive in stably stratifiedflow, qz can be either positive or negative. In general, the Rossbywave propagation is to the left of the mean vorticity gradient,hence for negative qz we obtain a positive c� � U and a positive–q correlation.

FIG. 2. Schematic illustration of the normal-mode solutions to a pure buoyancy jump. Shown are (top)the negatively correlated, westward-propagating � and (bottom) the positively correlated, eastward-propagating �. The schematics are similar to Fig. 1, with the shaded regions denoting regions ofnegative displacement generation and positive/negative vorticity generation in the top/bottom.


first formalized for the idealized case of two interfaceswith oppositely signed vorticity jumps Bretherton(1966), and the instability was rationalized in terms ofthe interaction between anomalies on the two inter-faces. For this case, each interface on its own supportsa neutral normal mode, but when both interfaces exist,interaction between the waves may yield instability.Davies and Bishop (1994) formalized this for the Eadymodel, by mathematically expressing how the cross-shear velocity induced by a wave at one interface af-fects the phase and amplitude of the wave at the otherinterface, via its advection of the basic-state vorticity.Unstable normal-mode solutions then emerge when thetwo waves manage to resist the shear and phase lock ina configuration that also results in a mutual amplifica-tion.

This approach also holds for more than two jumps, inwhich case the flow field evolution is expressed as amultiple wave interaction, as follows. A pure vorticityjump (let’s say at z � z0) supports a Rossby wave forwhich q and w are in quadrature, and since vorticity inthis case is only affected by vorticity advection, thiswave only propagates zonally (its phase changes) but itdoes not amplify. Since there are no vorticity gradientsexcept at the jump, it does not create vorticity anoma-

lies elsewhere. If, on the other hand, the basic state hasadditional vorticity jumps, new vorticity anomalies willbe created at these jumps by the far-field vertical ve-locity of the q(z0) vorticity anomaly. These new anoma-lies, will in turn induce a far field w, which is not nec-essarily in quadrature with q(z0), allowing a change inamplitude, as well as phase. The full flow evolution isthus viewed as an interaction of multiple single-interface kernels, where by kernel we mean the deltafunction of vorticity at a given interface along with thevertical velocity that it induces. In other words, the ker-nel at z0 is the normal-mode solution of a mean flowthat has a single vorticity jump at z0.

This view is useful because it can be generalized tocontinuous basic states by taking the limit of an infinitenumber of jumps spaced at infinitesimal intervals, asformulated in HM. So far, besides rationalizing shearinstability, a kernel-based view of Rossby waves hasyielded physical insight into basic phenomena likecross-shear wave propagation, wave evanescence andreflection (HH07), and various aspects related to non-normal growth (HM; de Vries and Opsteegh 2006,2007a,b; Morgan 2001; Morgan and Chen 2002).

With the goal of developing similar insight into grav-ity wave phenomena, we wish to develop a KGW for-

FIG. 3. As in Fig. 2, but for the combined buoyancy–vorticity jump. Note that the phase speed, as wellas vertical velocity arrows, denote both the direction and relative magnitude, so that � propagateswestward faster, and has a larger vertical velocity, than the pure buoyancy mode of Fig. 2, while �

propagates eastward slower, with a weaker vertical velocity (per unit displacement).


mulation of stratified shear flow anomalies. Followingthe Rossby wave example, we start with examining theevolution of anomalies on multiple basic-state jumps(both vorticity and buoyancy). We define our kernels tobe the normal-mode solutions to an isolated buoyancy–vorticity jump, then examine the evolution and mutualinteraction of these kernels on two and multiple jumps,and finally we take the continuous limit.

a. The general time evolution on a single jump

To understand the evolution of an arbitrary pertur-bation field as a multiple-kernel interaction, we need tofirst understand the evolution on a single interface. Inthe case of a pure vorticity jump, there is no buoyancyanomaly and there is a single normal-mode solution[Eq. (16)],7 and correspondingly a single vorticity ker-nel. A density jump, on the other hand, supports twonormal modes [Eqs. (15) and (17)] at each interface.The � q structure of the normal modes is determinedby the requirement that their spatial structure notchange with time. Since vorticity and buoyancy are in-dependent variables (they influence each other’s evo-lution but any combination of the two can exist), twomodes are needed to represent the full anomaly field ata given level.

As with the vorticity case, we define the kernels to bethe normal modes of a single jump, thus at each levelthere are two kernels, and each of these kernels has aspecific � q structure. Taking this a step further, anarbitrary � q configuration can be uniquely dividedinto the two kernels, � and �, as follows (using thesame notation as for the normal modes):

q � q� � q� � 2k��c� � U�� c� � U��,

�19a�

� � �� , �19b�

where we have used the relation in (17). The two kernelcomponents are then obtained from q and (see ap-pendix B):

�� 1

2k��c� � U� � �c� � U��2k�c� � U�� q�,

�20a�

�� 1

2k��c� � U� � �c� � U��2k�c� � U�� q�.

�20b�

In the simplified case of a pure buoyancy jump, we notethat (c� �U)��(c� �U), and these equations reduceto

�� 12 ��

q

2k�c� � U��, �21a�

�� 12 ��

q

2k�c� � U��. �21b�

Once the projection onto the two kernels is found,the temporal evolution is readily obtained since eachkernel propagates at its own phase speed, which corre-sponds to the single jump normal mode, and there is nointeraction between the kernels. This is stated math-ematically in appendix B, using matrix notation, withthe two kernels being the eigenvectors of the propaga-tor matrix for the combined vorticity–displacement vec-tor, and the phase speeds are directly related to theeigenvalues.

b. Two interfaces

The simplest case which allows for kernel interac-tions is a mean flow with two interfaces. Such a setupwas examined by Baines and Mitsudera (1994). Herewe formulate the problem in terms of kernel interac-tions, for the most general case of buoyancy–vorticityjumps. We denote the interface locations by z1 and z2,their distance by z2 � z1 � �z � 0, and the vorticity–buoyancy jumps by �q1/2 and �b1/2, where the subscript1⁄2 denotes the interface. The four kernels (two for eachinterface) are thus denoted by �1/2. The dynamics isthen described in terms of the evolution of the kernels,as follows. Writing Eqs. (5a) and (5b) for the two in-terfaces we get a 4 � 4 matrix equation:

�

�t� � A�, �22a�

where

� � �q1

�1

q2

�2

� �22b�

and A is defined in appendix C. Using Eqs. (20a) and(20b), we can write the transformation as

� � T�, �23a�7 Again, we do not consider the normal modes associated with

the continuous spectrum.


where

� � ��1�

�1�

�2�

�2��, �23b�

and the transformation matrix T is defined in appendixC. Using the similarity transformation, we get the dy-namic equation for the kernel displacements, written inmatrix and compact form, respectively:

�

�t� � T�1AT�, �24a�

�12� � �ik��c��12 � e�k�z� c� � U

c� � c��

12

� ��c� � U�� c� � U��21�, �24b�

where 1⁄2 indicate the interface and � indicate the ker-nel, and we have used Eq. (14) for c�, to cancel outsome terms in the derivation. Note that the expressionin the inner squared brackets has a flipped 2/1 index,indicating the contribution of the opposite boundary.Note also that

�c� � c��12 ��q

2k�2

� 2�b

k �12. �25�

Equation (24b) shows, as expected, that in the absenceof interaction between the interfaces, (�/�t)�1/2 ��ik(c��)1/2, which is simply the two kernel propaga-tion equations. This confirms that the two kernels at agiven interface do not interact at all (i.e., a� kernel canonly interact with the � and � kernels of the otherinterface).

Written also in terms of displacement amplitudes (Z)and phases (�):

�12� � Z12

� e i�12�

�26�

and taking the real and imaginary parts of Eq. (24b)yields the amplitude and phase evolution equations:

Z12� � � ke�k�z� c� � U

c� � c��

12

� ��c� � U�Z��21 sin��21� � �12

� �

� ��c� � U�Z��21 sin��21� � �12

� ��, �27a�

�1k�12� � c12

� �e�k�z

Z12� � c� � U

c� � c��

12

� ��c� � U�Z��21 cos��21� � �12

� �

� ��c� � U�Z��21 cos��21� � �12

� ��. �27b�

As expected, Eq. (27a) indicates that each mixedbuoyancy–vorticity kernel on its own is neutral. Its am-plitude can grow only due to advection of mean buoy-ancy and vorticity gradients by the vertical velocity in-duced by the kernels of the opposite interface. Thisadvecting velocity is proportional to the inducing ker-nel displacement amplitudes and the nature of the in-version between vertical velocity and displacement.Hence, it decays with the distance between the inter-faces (indicated by the Green function). Only the com-ponents of the inducing velocity that are in phase withthe induced kernel’s amplitude yield growth (the sineson the rhs result from the fact that the displacementand vertical velocity of a kernel are �/2 out of phase).Equation (27b) indicates that the instantaneous phasespeed of each kernel, ��/k, is composed of its naturalspeed without interaction [Eq. (14)], and an effect ofthe opposing interface kernels. The latter only arisesfrom that part of the induced vertical velocity which isin phase with the kernel (and hence the cosines on therhs).

A normal-mode solution of the full two-interface sys-tem requires phase locking of the four kernels, which isobtained when the rhs of Eq. (27b) is the same andconstant for the four kernels (this constant is the realnormal-mode phase speed cr). Normal-mode solutionsshould also have a constant growth rate (kci) for thefour kernels. This is obtained only if Z�

1/2/Z�1/2 � kci in

Eq. (27a). In a companion paper we analyze the nor-mal-mode dispersion relation and structure of puregravity waves in terms of this mutual phase lockinginteraction for the two interface problem Umurhan etal. (2008, unpublished manuscript, hereafter UHH).We note that although this interaction is more complexfor gravity waves than for Rossby waves because thereare two kernels at each interface, vertical shear willdiscriminate between these kernels because phase lock-ing dominantly occurs for kernels which counter propa-gate with respect to the mean flow (UHH).

c. Multiple interfaces

As a step toward a continuous formulation, we gen-eralize the two-interface equations to an arbitrary num-ber of jumps:

�j� � �ik�c��j � � c� � U

c� � c��

j

n�1,�j

N

e�k | j�n |�z

� ��c� � U�� c� � U��n . �28�

Decomposing into amplitude and phase, as in Eq. (26)yields


Zj� � �k� c� � U

c� � c��

j

n�1,�j

N

e�k | j�n |�z

� ��c� � U�Z��n sin��n� � �j

��

� ��c� � U�Z��n sin��n� � �j

��, �29a�

�1k

�j� � cj

� �� c� � U

c� � c��

j

n�1,�j

N e�k | j�n |�z

Zj�

� ��c� � U�Z��n cos��n� � �j

��

� ��c� � U�Z��n cos��n� � �j

��. �29b�

d. The continuous limit

Our overall goal is to develop a kernel formulationwhich holds for continuous basic states, as was done forRossby waves (HM). Apparently, this simply entailstaking the limit of Eq. (28), for �z → 0 with N → !.However, this limit is ill defined when bz � 0. To seethis, we note that Eq. (28) involves writing Eqs. (5a)and (5b) in terms of the kernels � and �. This re-quires dividing the vorticity perturbation to the twokernels, according to Eq. (19a), using the dispersionrelation in (14), and taking the above limit. Writing�b � bz�z, �q � qz�z, and noting that the single in-terface vorticity q is the amplitude of the vorticity deltafunction (which has units of vorticity times length)hence q � q�z, we get the following:

q � �� qz

2��qz

2�2

� 2kbz

�z��, �30�

which blows up in the limit of �z → 0, for bz � 0. Whenbz � 0, this limit is well defined, and yields the basicrelation q � �qz. Indeed, for the Rossby wave case,the kernel formulation works for the continuous limit(HM). When bz � 0, the formulation, in its currentform, fails because an infinitesimal-width q, which has afinite value (as opposed to a � function q, which isinfinitesimal in width and infinite in amplitude), has anegligible contribution to w, and hence to the genera-tion of [cf. Eq. (5b)]. This sets the ratio �/q to zero.Unlike the single interface case, where directly influ-ences the time tendency of q [Eq. (5a)], and q directlyinfluences the evolution of [Eq. (5b)], here only produces q, so that q and cannot propagate in concertto form a normal-mode kernel structure.

Because, however, the vorticity inversion is such thatits action at a distance decays with distance, the verticalvelocity at a given location is most strongly influencedby the local vorticity anomaly. In other words, even

though the contribution of q(z) to w(z) is infinitesimal,it is larger than the contribution of q at any otherheight. Mathematically, we can get around this problemby simply looking at the anomaly averaged over a finiteinterval of width �z.

To do this we discretize the domain as in appendix A,using Eqs. (A1) for the basic-state gradients, and definethe average of any perturbation quantity f on height zj

as

fj �1

�z �zj��z2

zj��z2

f dz. �31�

The vertical velocity at zj induced by the vorticityperturbation located between (zj � �z/2, zj � �z/2) andapproximated by qj, for small �z is

wj � �i

2qj�

zj��z2

zj��z2

e�k | z�z� | dz� " �i

2qj�z.

�32�

Substituting Eq. (32) in (2b) and seeking wavelike so-lutions yields

�zqj � �2k�c� � Uj��j�, �33�

which converges to Eq. (17) for qj � q�(z � zj) and�z → 0. To get an expression for (c�� Uj), we take thesecond total time derivative of Eq. (2a):

D2qj

Dt2 � �qzj

Dwj

Dt� ikbzjwj. �34�

For the chunk formulation this yields the following dis-persion relation:

c� � Uj � � qzj

4k2 �� qzj

4k2�2

� bzj

2k2, �35�

where the nondimensionalized number # � k�z shouldbe taken small enough [for relation in (32) to hold]. Itis straightforward to verify that for a buoyancy–vorticity staircase profile, this expression converges to(14) as # → 0. Substituting Eq. (35) into (33) we obtainthe local � q relation:

bz�j� � �� qzj

4k2�2

� bzj

2k2�12

qj. �36�

Using this relation, and noting that

�c� � c��j � �� qzj

2k2�2

� 2 bzj

k2 �12

�37�

we get kernel amplitude and phase equations, whichare exactly similar to Eqs. (29a) and (29b), with �b �


(bz/k)#, �q � (qz/k)#, and q and being the chunk-averaged fields q and .

We can now use these equations to numerically studythe evolution of buoyancy–vorticity anomalies on anarbitrary continuous profile in terms of kernel interac-tions. This formulation, like the piecewise basic-stateone, does not converge for # → 0. This lack of conver-gence stems from the fact that when the chunk sizeshrinks to zero, the vertical velocity induced by thechunk vanishes [Eq. (32)], and no displacement is in-duced. However, when we look at a finite region, themechanistic picture of buoyancy–vorticity interactions,with two chunk kernels at each discrete (finite width)level, holds. In particular, we can examine the case ofconstant bz and qz, with a pure plane wave solution ofthe form in (A3), calculate q and , and the correspond-ing � and � [cf. Eqs. (20a) and (20b)], and plug intothe chunk version of Eqs. (29a) and (29b). Equation(29a) does indeed yield zero amplitude growth, whileEq. (29b) yields the discretized version of the planewave dispersion relation in (10) using Eq. (A4) for thesum terms (not shown).

5. Summary and discussion

Potential vorticity thinking, which includes the con-cepts of PV inversion and action at a distance, providesa mechanistic understanding of many aspects of baro-tropic and baroclinic shear flow anomalies (Hoskins etal. 1985). In this paper we present a corresponding vor-ticity viewpoint, which is appropriate for gravity wavesand stratified shear flow dynamics.

We show that vorticity dynamics is central to strati-fied shear flow anomalies, which are composed of acontinuous interaction between vorticity and buoyancyperturbations. Horizontal buoyancy gradients generatevorticity locally, whereas vorticity induces a nonlocalvertical velocity, which, in turn, generates both fields byadvecting the background buoyancy–vorticity. Thus,the interplay between vorticity and buoyancy anoma-lies fully describes the linear dynamics on stratifiedshear flows, both modal and nonmodal. Casting the dy-namics in this way essentially provides us with a timeintegration scheme that is analogous to the time inte-gration of QG equations, with two rather than oneprognostic equations, and a diagnostic equation forstreamfunction, which is related to vorticity as in QG.

Since the buoyancy anomalies arise purely from ver-tical advection of the background buoyancy field, wecan express the buoyancy anomalies via the local ver-tical displacement, and the dynamics reduce to a mu-tual interaction between vorticity and displacement

fields. The vertical displacement equation is then thesimple statement that vertical velocity is the time de-rivative of displacement. For normal modes, the twofields (vorticity and displacement) assume specificphase and amplitude relations. In particular, for neutralmodes, vorticity and displacement are either in phaseor in antiphase with respect to each other.

For baroclinic/barotropic flow, PV invertibility al-lows us to develop a kernel view of the dynamics, whereanomalies are composed of localized kernel Rossbywaves, which interact with each other. This mutual in-teraction occurs via the nonlocal flow induced by eachkernel’s PV anomaly. In analogy, our buoyancy–vorticity view also allows a clear separation betweenlocal and nonlocal dynamics, with the nonlocal part ofthe dynamics stemming from the vorticity inversion.This makes a kernel view of the dynamics possible, anda major part of this paper deals with developing it.

As was done for vorticity kernels, we define our ker-nels based on single interface of vorticity–buoyancynormal modes. Since both vorticity and buoyancy areinvolved in stratified shear flow, there are two normalmodes at each level, and thus, two kernels. These arecomposed of an eastward-propagating wave with posi-tively correlated vorticity and displacement, and a west-ward-propagating wave with negatively correlated vor-ticity and displacement. Though having two kernelscomplicates the picture, the kernels are constructed sothat they do not interact with each other at a givenlevel.

An arbitrary initial vorticity–displacement distribu-tion can, at each level, be uniquely decomposed into thetwo kernel components, each of which evolves accord-ing to its own internal dynamics, with an influence fromkernels at other levels (via their induced vertical veloc-ities). The dynamic equations for vorticity and displace-ment can thus be decomposed and written in terms ofthe evolution of the two kernels (at each level).

To get a sense of multiple kernel interactions be-tween many levels, we consider a flow with multiplebuoyancy–vorticity jumps, which we initially perturb ata single interface located at z � z0. The resulting vor-ticity anomaly at z0 will induce a far-field vertical ve-locity, which will perturb the other jump interfaces. Theresulting displacements of each interface are associatedwith buoyancy–vorticity anomalies. Note that for thecase of a pure buoyancy jump, an initial interface dis-tortion can only create a buoyancy anomaly, and novorticity anomaly, because the vorticity is constantacross the jump. The anomalies initially induced by aninterface distortion evolve according to the local vor-ticity–displacement interaction dynamics. Though these


initial and q perturbations are initially in phase, theiramplitude ratio is not necessarily the normal-mode one.Thus, both normal modes will be excited, and will pro-ceed to propagate in opposite directions independentlyof each other. These new excited kernels will induce afar-field w, which will affect other interfaces. The fullflow evolution can be viewed as an interaction of mul-tiple pairs of single-interface kernels.

We note, however, that the description of the evolu-tion in terms of a superposition of the two kernelswhose density and vorticity anomalies are either in orout of phase is not necessarily the most intuitive one.For example, for a pure buoyancy interface, an initialdisplacement-only anomaly will excite the two modeswith equal amplitude (� and � are in phase but q�

and q� are in antiphase). Such a superposition of thetwo kernels with equal amplitude actually gives rise toa standing wave (in the moving frame of the mean flow)in which the vorticity and the density anomalies are inquadrature (see appendix B). We expect the kernelview to be the most intuitive in many cases (as we showin a companion paper, UHH), while in other setups, analternative standing-oscillation building block mightprovide a more intuitive description of the dynamics(currently under investigation).

To make the kernel formulation applicable to gen-eral stratified shear flow profiles, we use the multiple-jump formulation as a basis for a continuous flow. Thecontinuous limit has been found to be quite subtle. Ourkernel formulation is based on the assumption thatclear vorticity–displacement relations exist, which areused to define our kernels. For the pure Rossby wavecase, where the piecewise setup converges nicely to thecontinuous problem, vorticity and displacement are di-rectly related (q � �qz). When buoyancy gradientsand corresponding buoyancy anomalies exist, the strictrelation between q and exists for jumps, but becomessingular in the continuous limit. This stems from thefact that an important part of this strict –q relation isthe generation of by q via its induced vertical velocity.In the continuous limit, an infinitesimal localized q gen-erates an infinitesimal w, and hence an infinitesimal .To recover a –q relation, we recover a locally inducedw, by considering finite-width vorticity chunks, whichgenerate a small, but nonzero . This chunk formulationcan be viewed as a numerical discretization approxima-tion of the full –q dynamic equations.

To summarize, we present a vorticity–buoyancyframework for stratified shear flow anomalies, whichyields new insight into the mechanistics of gravitywaves. This framework can now be applied to gain amechanistic understanding of a variety of stratified

shear flow phenomena. In a companion paper, UHH,we examine instabilities on a constant shear flow withtwo density jumps, using the above buoyancy–vorticitykernel view. Even though in this system there are novorticity gradients, and hence no vorticity waves, insta-bility arises out of a mutual interaction and phase lock-ing of counter-propagating gravity waves, in analogy tothe CRW view of classical models like those of Ray-leigh and Eady. Since much of the kernel dynamicsfound for Rossby waves is also present in UHH, weexpect the kernel view to yield a mechanistic under-standing of other aspects of stratified shear flow, likenonmodal growth of anomalies, and basic processeslike cross-shear gravity wave propagation, reflection,and overreflection. One of the central theorems ofstratified shear flows is the Miles–Howard criterion forinstability—that the Richardson number (Ri) besmaller than 1⁄4 somewhere in the domain (Miles 1961;Howard 1961). Though it has been arrived at in variousways, our mechanistic (as opposed to mathematical)understanding of it is only partial. Some mechanisticunderstanding is gained in the context of both overre-flection and the counter-propagating wave interactionframeworks, as follows. Ri � 1⁄4 is also a criterion forwave evanescence in the region of a wave critical sur-face, which allows for the significant transmission ofgravity waves through the critical level (Booker andBretherton 1967; see also Van Duin and Kelder 1982,where this is explicitly demonstrated for the far field).On the one hand, the existence of an evanescent regionis a necessary component for overreflection (e.g.,Lindzen 1988). On the other hand, as pointed out byBaines and Mitsudera (1994), this evanescent regionserves to divide the fluid into two wave regions, whosewaves can interact and mutually amplify. This suggeststhe two approaches to gravity wave instabilities, basedon overreflection and on counter-propagating wave in-teraction, may be related in a similar way to what wasshown for Rossby waves using the kernel approach(HH07). Our gravity wave kernel formalism is one steptoward understanding this relation, and in the end weexpect the Richardson number criterion to becomemuch clearer.

Acknowledgments. Part of the work was done whenEH was a visiting professor at the Laboratoire deMétéorologie Dynamique at the Ecole NormaleSupérieure. The work was supported by the EuropeanUnion Marie Curie International reintegration GrantMIRG-CT-2005-016835 (NH), the Israeli ScienceFoundation Grant 1084/06 (EH and NH), and the Bi-national Science Foundation (BSF) Grant 2004087 (EHand OMU).


APPENDIX A

An Explicit Numerical Scheme for TimeIntegration of Eq. (5)

We consider a smooth basic state with vorticity q(z)and buoyancy b(z), which we discretize into N layerswith 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,depending on the complexity of the profiles, but forsimplicity we choose it constant). The discretized meanvorticity–buoyancy gradients are then

qzn �q�zn�12� � q�zn�12�

�z;

bzn �b�zn�12� � b�zn�12�

�z. �A1a,b�

The discrete form of Eqs. (5a) and (5b) is then

qj � �ik�qjUi � bzj�j� �i

2qzj

n�1

N

qne�k | j�n |�z�z,

�A2a�

�j � �ikUi�j �i

2 n�1

N

qne�k | j�n |�z�z. �A2b�

The plane wave dispersion relation in Eq. (10) can berecovered, for instance, when we take constant valuesof qzn � qz and bzn � bz, assume a discretized planewave solution of the following form:

�qn

�n� � �q0

�0�e i�kx�mn�z��t�, �A3�

and recall that in the limit N → ! and �z → 0 we get

n�1

N

e �imn�k | j�n | ��z�z � n � �N2

N2

e �imn�k | j�n | ��z�z

→2k

K2 e imzj. �A4�

This converges to the following integral relation whichwas used to derive Eq. (9) by plugging the plane wavevorticity anomalies field [Eq. (8)] into the integrals onthe rhs of Eqs. (5a) and (5b), assuming a constant infi-nite basic state:

�z��

�

q�z��e�k | z�z� | dz�

� q0e i�kx��t��z��

�

e imz�e�k | z�z� | dz�

�2k

K2 q0e i�kx�mz��t�. �A5�

APPENDIX B

Eigenvalue Representation of the Single InterfaceDynamics

For a single interface Eqs. (5a) and (5b) can be writ-ten in the matrix form:

�

�t �q

�� i�kU �

�q

2k�b

12

kU ��q

��, �B1�

whose solution is

� q

�� 0

��2k�c� � U�

1�e�ikc�t

� �0��2k�c� � U�

1�e�ikc�t, �B2�

where c� is defined by Eq. (14). Hence, at t � 0:

��0�

�0�� 2k�c� � U� 2k�c� � U�

1 1 ��1� q0

�0�

�1

2k��c� � U� � �c� � U��

�� 1 �2k�c� � U�

�1 2k�c� � U�� q0

�0�, �B3�

in agreement with Eqs. (20a) and (20b).Note that for the case of a pure density jump, an

initial displacement-only anomaly (q0 � 0), which iswhat we get in response to an induced w, will excite thetwo normal modes with equal amplitude: �0 � �0 �1⁄20 [cf. (B3)]. Using (B2) and (15), this gives a standingwave (in the frame of reference of the mean flow), forwhich q and are actually in quadrature:

q � �0�2k�b sin��k�b

2t� sin�k�x � Ut��, �B4a�

� � �0 cos��k�b

2t� cos�k�x � Ut��. �B4b�

APPENDIX C

Matrix Formulation of the Two-Interface Problem

The matrix A is obtained from direct substitution inEqs. (5a) and (5b) [recall that $z� q(z�)e�k | z�z� | dz� �qje

�k | z�zj | for q � qj�(z � zj)]:


A � �i�kU1 �

�q1

2k�b1 �

�q1

2e�k�z 0

12

kU1

12

e�k�z 0

��q2

2e�k�z 0 kU2 �

�q2

2k�b2

12

e�k�z 012

kU2

�. �C1�

The q � to � � � transformation matrix T is obtained directly from Eqs. (20a) and (20b):

T � � 1

2k��c� � U� � �c� � U��

1 � 2k�c� � U�

2k��c� � U� � �c� � U��

1

0 0

� 1

2k��c� � U� � �c� � U��

1 2k�c� � U�

2k��c� � U� � �c� � U��

1

0 0

0 0 1

2k��c� � U� � �c� � U��

2� 2k�c� � U�

2k��c� � U� � �c� � U��

2

0 0 � 1

2k��c� � U� � �c� � U��

2 2k�c� � U�

2k��c� � U� � �c� � U��

2

�.

�C2�

The similarity transformation is thus

T�1AT � �ik

� �c1� 0 e�k�z� c� � U

c� � c��

1

�c� � U�2 e�k�z� c� � U

c� � c��

1

�c� � U�2

0 c1� �e�k�z� c� � U

c� � c��

1

�c� � U�2 �e�k�z� c� � U

c� � c��

1

�c� � U�2

e�k�z� c� � U

c� � c��

2

�c� � U�1 e�k�z� c� � U

c� � c��

2

�c� � U�1 c2� 0

�e�k�z� c� � U

c� � c��

2

�c� � U�1 �e�k�z� c� � U

c� � c��

2

�c� � U�1 0 c2�

�.

�C3�

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A Buoyancy–Vorticity Wave Interaction Approach to ...harnik/publications/Harnik-etal-08-JAS.pdf · Stably stratified shear flows support two types of waves and associated instabilities—Rossby