This article was downloaded by: [University of Saskatchewan Library] On: 20 September 2012, At: 12:01 Publisher: Taylor & Francis Informa Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House, 37-41 Mortimer Street, London W1T 3JH, UK Geophysical & Astrophysical Fluid Dynamics Publication details, including instructions for authors and subscription information: http://www.tandfonline.com/loi/ggaf20 Nonlinearities and Linearities in Internal Gravity Waves of the Atmosphere and Oceans Colin Hines a a 15 Henry Street, Toronto, M5T 1W9, Canada Version of record first published: 24 Sep 2010. To cite this article: Colin Hines (2002): Nonlinearities and Linearities in Internal Gravity Waves of the Atmosphere and Oceans, Geophysical & Astrophysical Fluid Dynamics, 96:1, 1-30 To link to this article: http://dx.doi.org/10.1080/03091920290018826 PLEASE SCROLL DOWN FOR ARTICLE Full terms and conditions of use: http://www.tandfonline.com/page/terms-and-conditions This article may be used for research, teaching, and private study purposes. Any substantial or systematic reproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in any form to anyone is expressly forbidden. The publisher does not give any warranty express or implied or make any representation that the contents will be complete or accurate or up to date. The accuracy of any instructions, formulae, and drug doses should be independently verified with primary sources. The publisher shall not be liable for any loss, actions, claims, proceedings, demand, or costs or damages whatsoever or howsoever caused arising directly or indirectly in connection with or arising out of the use of this material.
Transcript
This article was downloaded by: [University of Saskatchewan Library]On: 20 September 2012, At: 12:01Publisher: Taylor & FrancisInforma Ltd Registered in England and Wales Registered Number: 1072954 Registeredoffice: Mortimer House, 37-41 Mortimer Street, London W1T 3JH, UK
Geophysical & Astrophysical FluidDynamicsPublication details, including instructions for authors andsubscription information:http://www.tandfonline.com/loi/ggaf20
Nonlinearities and Linearities in InternalGravity Waves of the Atmosphere andOceansColin Hines aa 15 Henry Street, Toronto, M5T 1W9, Canada
Version of record first published: 24 Sep 2010.
To cite this article: Colin Hines (2002): Nonlinearities and Linearities in Internal Gravity Waves ofthe Atmosphere and Oceans, Geophysical & Astrophysical Fluid Dynamics, 96:1, 1-30
To link to this article: http://dx.doi.org/10.1080/03091920290018826
PLEASE SCROLL DOWN FOR ARTICLE
Full terms and conditions of use: http://www.tandfonline.com/page/terms-and-conditions
This article may be used for research, teaching, and private study purposes. Anysubstantial or systematic reproduction, redistribution, reselling, loan, sub-licensing,systematic supply, or distribution in any form to anyone is expressly forbidden.
The publisher does not give any warranty express or implied or make any representationthat the contents will be complete or accurate or up to date. The accuracy of anyinstructions, formulae, and drug doses should be independently verified with primarysources. The publisher shall not be liable for any loss, actions, claims, proceedings,demand, or costs or damages whatsoever or howsoever caused arising directly orindirectly in connection with or arising out of the use of this material.
(Received 11 May 2001; In final form 1 August 2001)
The spectrum of internal gravity waves in the atmosphere and oceans is sufficiently intense that nonlinearinteractions must occur, if these waves are analyzed in Eulerian coordinates as is usually done. As it happens,however, if these waves are analyzed in Lagrangian coordinates the most important nonlinearity can beentirely avoided: it is an Eulerian mathematical construct only, not a physical process. The mathematicalbasis for this assertion is developed here, and some of its consequences are discussed. Among the latter isa questioning of the validity of standard Eulerian eikonal methods of calculating ray paths and related func-tions in a multiwave environment, discussed in an appendix.
Internal gravity waves occur throughout Earth’s atmosphere and oceans. Initial recog-nition of their pervasive presence in the middle and upper atmosphere was accompaniedby recognition of the role that nonlinearities must play, particularly when, aswas often the case, many waves were present in superposition (Hines, 1960, SectionIV.5). Individual waves were less prone to the problems of nonlinearity because oftheir being almost transverse oscillations. For many years such waves provided materialfor study within a linear framework sufficient to occupy researchers almost fully. Witha few isolated exceptions, nonlinear effects in atmospheric waves were pretty wellignored until emphasized by Weinstock (1976) in response to an increasing supply ofobservations that described the breadth and intensity of the middle-atmosphere spec-trum. At much the same time, the importance of nonlinearities in oceanic internalwaves was made clear (Holloway, 1980, 1981; Munk, 1981). Both regimes havespawned a substantial body of work investigating the consequences of nonlinearities,virtually all of it developed with the use of Eulerian coordinates.
ISSN 0309-1929 print: ISSN 1029-0419 online/01/010000-00 � 2002 Taylor & Francis Ltd
DOI: 10.1080/03091920290018826
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What has not been generally appreciated is that, to some extent, the role playedby nonlinearities is dependent on the nature of the coordinate system chosen forthe description of the waves. Specifically, the adoption of Eulerian coordinates hasrendered the description of certain waves strongly nonlinear, while those same waves –or, rather, their counterparts (see below) – remain virtually linear in a correspondingLagrangian description, a fact first emphasized in this context by Allen and Joseph(1989). The converse can also occur: certain waves that permit a linear Eulerian des-cription are strongly nonlinear in a Lagrangian description. Clearly, there are someadvantages to be gained by the use of linear theory when it is available.
The present paper is designed to establish the conditions in which unambiguous non-linearity occurs in both descriptions, and those in which the dichotomy arises and opensthe way to an advantageous choice of coordinate system.
Experience with earlier reviewers of this theme, both formal and informal, indicatesthat a disclaimer is necessary at this point. The equations of fluid dynamics areinherently nonlinear, and no linearizing approximations can alter that fact. Suchapproximations have proven fruitful in the past, however, producing the vast majorityof the literature on waves in fluids. When I write here of waves in one coordinatesystem or the other being linear, and when I establish criteria for the validity of linear-ization, I am by no means denying that some residual effects of nonlinearity mayremain for subsequent treatment. Instability and resonant wave–wave interactionsmay occur, for example, and may well be relevant to some problem at hand. I am,instead, directing attention to the fact that a linear approach may be expected toprovide a legitimate starting point and valid consequences, however much these may(or may not) want refinement once initial developments have been completed andguidance obtained. The issue is one that is passed over almost without questionwhen linearized Eulerian equations are adopted as a starting point, but it seems torequire explicit recognition when linearized Lagrangian equations come under examin-ation. I will be asserting in due course that certain conditions permit linearizationin Eulerian and/or Lagrangian coordinates, always with the provisos of this paragraphimplied. Skeptics may prefer to read such assertions in a reverse fashion: if any of theconditions for linearization is not met for a chosen coordinate system, then significantnonlinear effects are virtually guaranteed to be operative in that system.
My assertion of the dichotomy of nonlinearities according to coordinate systemhas occasionally elicited a response such as, ‘‘Nonsense! A wave is either linear orit is not: its nature cannot depend on your choice of coordinate system.’’ The onlyvalid reply to this reaction is, ‘‘Not at all: it depends on what is meant by ‘a wave’and by ‘linear’.’’
One commonly deals with ‘‘a plane wave’’, even when it is known that the wave inpractice may be but a local portion of a spherical wave, for example, and one drawsconclusions that are locally useful. The wave in question has no knowledge of whetherwe are going to analyze it as a plane wave or as a spherical (or other) wave: it simplygoes about its business and we try to represent its behavior as best we can. Similarly, areal wave disturbance has no knowledge of whether we will use Eulerian or Lagrangiancoordinates to describe it: it will behave as it is obliged to behave by physical laws, andwe are left to analyze its behavior in whatever coordinates we may prefer. If ithappens that one coordinate system requires the use of nonlinear mathematicsto describe its own idealization of the real wave, whereas the other does not, that isof no interest to the real wave but only to us.
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A distinction must be drawn in the meaning of ‘‘a plane wave’’ as it applies in thetwo coordinate systems. A plane wave in Eulerian coordinates is one whose surfacesof constant phase are planar in Eulerian coordinates, whereas a plane wave inLagrangian coordinates is one whose phase surfaces are planar in Lagrangian coordi-nates. Since Lagrangian coordinates are somewhat distorted as viewed from Euleriancoordinates, just as Eulerian coordinates are somewhat distorted as viewed fromLagrangian coordinates, a single ‘‘Eulerian plane wave’’ is somewhat different from asingle ‘‘Lagrangian plane wave’’ having the same wavenumbers and frequency: neithermaps exactly onto the other. But, when the dynamic system as a whole produces onlyminor deformations of either coordinate system as seen by the other, the mapping tendsto be remarkably good (just as the mapping of a spherical wave into a plane waveis remarkably good at sufficient distance from the center of the spherical wave, overa region whose dimensions are small in comparison with that distance). An observerof the atmosphere or oceanic interior would be hard pressed to say, on any given occa-sion, whether an observed apparently plane wave would be better represented by (andso analyzed by) an Eulerian or a Lagrangian plane wave, much less make thedistinction when dealing with the more usually observed or inferred multiwave systems.
Further, and more to the point here, the question at issue will not be the linearity ofa particular plane wave on its own, but whether or not a particular plane test waveoscillates and propagates in a fashion that is (virtually) independent of the presenceor absence of a broad spectrum of other waves: whether or not the usual linear polar-ization relations and dispersion relation can be used to characterize it in the coordinatesystem of choice, despite the presence of those other waves. It is in this specific sensethat the term ‘‘linear’’ and its relatives will be employed here. Some readers mayhave to adapt to this meaning from some other meaning that is more commonlyused in their own specializations.
In practice, one tends to use the coordinate system that is thought to be the moreconvenient. And, in practice, this almost invariably turns out to be the Euleriansystem if only because most of our observing systems provide data as functions ofEulerian coordinates. But the essential point of this paper is that there are other circum-stances in which Lagrangian coordinates prove to be the more convenient. These occur,in particular, when the Lagrangian waves remain linear (in the sense described) whiletheir Eulerian counterparts become nonlinear. Whether or not linear Lagrangian resultscan be mapped to produce corresponding Eulerian results is a question whose answerdepends on the wanted information and on the circumstances. It is a question that forsome purposes is irrelevant – e.g., when the observing system is borne by the fluid –and, when not irrelevant, can be postponed for treatment in its appropriate context.
To help fix ideas, it will be well to think of two ‘‘test’’ waves low in a simply stratifiedatmosphere. Here and throughout, the atmosphere is nonrotating, nondissipative, andstationary and isothermal except for the effects of some background wave system thatis present, be it linear or nonlinear. The Lagrangian coordinates of a fluid parcel willbe defined as the Eulerian coordinates that the parcel would have had in the absenceof all waves. (An azimuthally isotropic spectrum may be assumed, if required bypurists, to remove any Stokes-drift effect.) One of the test waves is an Eulerian planewave, the other a Lagrangian plane wave, and both are propagating upward. The dyna-mical system induced by the background waves is sufficiently weak low in the atmos-phere that the two coordinate systems, and hence the two waves, are virtuallyindistinguishable there. But, because of the diminution of gas density at greater heights,
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the relative deformations of the two coordinate systems become significant there.The crucial question to be answered is, ‘‘Do both test waves come to be altered bythe other waves (i.e., become nonlinear in the sense defined here) at much the sameelevation, or do they not?’’ In an oceanic context, the corresponding question wouldarise as the spectrum of background waves became progressively more intense, in athought experiment if not in reality.
The two test waves may be identified, while both are still linear, by common valuesof wavevector k and frequency !. They differ then only because the phase variation ofthe Eulerian wave is given by k E x�!t, where x is a three-dimensional Eulerian coor-dinate and t is time, whereas that of the Lagrangian wave is given by k E r�!t, where r
is a three-dimensional Lagrangian coordinate. The two methods of analysis seek toobtain suitably chosen (but different) wave variables as functions of (x, t) and (r, t)respectively. Each does so for whatever background system of waves may be present,whether linear or nonlinear (though for some purposes we shall assume quasi linearity),plus a test wave unique to itself but defined by common values of k and !.
The analyses proceed in a straightforward fashion. They will be conducted as for theatmosphere, though a corresponding development can be made readily for the ocean.In Section 2 we enquire into the conditions that the background wave system andour Eulerian test wave must satisfy if the test wave is to propagate free from significantnonlinear effects as determined in an Eulerian coordinate system. In Section 3 we dothe same for the Lagrangian test wave in a Lagrangian coordinate system. In Section4 we compare the two sets of conditions and establish their overlaps and their contrasts.In Section 5 we draw conclusions relevant to practical application in the middle atmos-phere and the ocean. The most important of these is that the predominant nonlinearityof internal gravity waves in these media is an Eulerian nonlinearity exclusively: nocorresponding nonlinearity occurs in a Lagrangian analysis.
Appendix A identifies relations and approximations employed in the main text forlinearized Eulerian waves. Appendix B does the same for linearized Lagrangian waves.One application of the present results, of significance in its own right, is presented inAppendix C: the propagation of Lagrangian wave packets, their departure from theray paths described by Eulerian eikonal analysis, and the doubts that must thereforebe attached to the latter.
2. EULERIAN ANALYSIS
We shall employ as our basic Eulerian variables the density
�� � �g expð�x3=H þ Rþ �Þ, ð2:1Þ
pressure
pp � pg expð�x3=H þ Pþ pÞ, ð2:2Þ
and velocity
vv � V þ v, ð2:3Þ
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in which �g is unperturbed ground-level density, pg is unperturbed ground-levelpressure, x3 is height above ground and H is atmospheric scale height. R is the contri-bution from the background wave system to the logarithm of the density, and � is thefurther contribution that arises upon introduction of the test wave, both as measured at(x, t). Similarly with P and p, while V and v are the corresponding contributions to thevelocity. (It will later be found that rms R � 1 and rms P � 1 are conditions that mustbe met for linearization of the test wave to be valid, whereupon all of R, �, P and pbecome equivalently fractional fluctuations from the unperturbed density and press-ure.) We shall employ the ratio of specific heats � (¼ 1.4 for air), the gravitational accel-eration g, the speed of sound C whose square is given by C2 � �gH ¼ �pg=�g, and theunperturbed buoyancy frequency N whose square is given by N2 ¼ ð� � 1Þg2=C2.
Here, j (¼ 1, 2, 3) is a coordinate index (with 3 upward), a subscribed comma denotesdifferentiation with respect to the coordinate whose index follows it, and summationover a repeated index within a term is implied.
The background wave system must satisfy this same equation with the test-wavecontributions set equal to zero. We subtract the resultant relation from (2.5), ignoreproducts of test-wave variables, and so obtain
@�=@tþ Vj�, j þ vjR, j � v3=H þ vj, j ¼ 0: ð2:6Þ
We shall assume that the test wave behaves as a linear wave, free from significantmodification by the background wave system, and subsequently determine the con-ditions that are required to validate this assumption, in a standard, a posteriori fashion.These will be conditions imposing upper bounds on the intensity of the backgroundsystem, which system is represented in (2.6) by Vj and R, j. Though implicitly the con-ditions should apply over all space-time, they will be taken to be represented adequatelyby corresponding conditions on various rms magnitudes, as these are evaluated locallyat a given height of interest by horizontal and/or temporal averaging (taken to be equiva-lent). We must acknowledge in advance, then, that even if our deduced conditions onrms values are met, there may be isolated regions of space-time where the assumedlinearity breaks down: regions of ‘‘rogue wave’’ behavior. The Eulerian and Lagran-gian systems are subject to such breakdown equally, though not necessarily with thesame consequences. The rms magnitudes will be denoted by angle brackets.
This now has the form of a homogeneous linear equation in test-wave variables, albeitwith some terms arising from nonlinearity. Additional equations of this form
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will follow from other basic equations, and the entire set could be solved in a standardfashion to produce polarization relations and a dispersion relation. These would all beof the usual linearized form, but with additional terms. Our objective is to determine theconditions under which all such additional terms will be small, in the usual sense ofsatisfying some gross inequality, to the point of being negligible for operationalpurposes.
It is instructive to pursue this end objective, one governing equation at a time,by asking that the additional terms of each equation should be small relative to thedominant terms. This procedure is not entirely safe, in that cancellations can removesome initially dominant terms from the polarization relations that are deduced fromcombinations of the governing equations, but provision has been made for thisdanger (off line) as will be indicated where relevant. We shall assume approximate hor-izontal isotropy of the background system so that, for example, hR, 1i and hR, 2i aretaken to be of comparable magnitude; a condition imposed on one is tacitly taken tobe imposed on the other, and it is unnecessary to write both explicitly.
With linearity postulated for the test wave, subject to the a posteriori check that ourconditions for validity will imply, we can adopt a complex representation and anticipatea waveform exp½x3=2Hþ iðkEx � !tÞ� for its field variables. (Note that � and p, asdefined above, approximate to the fractional density and pressure variations, and sohave the same linearized waveform as v). We shall deal only with real k. For conveni-ence we write k � ixx3=2H ¼ K (where xx3 is a unit upward vector), so the waveform is
and we take ! to be positive. Without loss of generality, since our azimuths are unspe-cified, we take k2 ¼ 0. Though v2 then vanishes in the absence of nonlinear interaction,we must anticipate a non-zero v2 in the presence of such interaction. Our relatedcriterion for the insignificance of nonlinearity will be that jv2j � jv1j.
Equation (2.7) already yields a polarization relationship – that between � and v –without any need for union with other governing relations. It is seen to be virtuallyidentical to the fully linearized relationship provided only that
hVjKji � hVEKi � !, ð2:9Þ
obtained from the terms in �, and that
hR, jvji � hJREvi � jK�Evj, ð2:10Þ
obtained from those in v, where K� � k þ ixx3=2H is the complex conjugate of K. Theseconditions become necessary conditions for our test wave to be treated legitimatelyas being uninfluenced by the other waves – conditions for linearity, as defined here,in the treatment of the test wave.
The first of these, (2.9), is simply a general form of the usual statement that Dopplershifting (or spreading) imposed on the test wave by the background wave system shallbe small in comparison with the frequency of the test wave. This statement, expressedin this or an equivalent way, is a standard statement as a condition necessary for thelegitimate neglect of interaction of the test wave with the other waves as described
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in the chosen Eulerian coordinate system. It is a clearly understood statement andmay be viewed as prototypical: similar statements that are about to be made providefurther necessary conditions under the same interpretation, with the same measureof legitimacy. All are adopted regularly, if only implicitly, without apparent qualm.Corresponding to them are other statements, developed in Section 3, intended forapplication in a Lagrangian coordinate system. Absent any relevant argument to thecontrary, it would appear that those statements should be accorded the same measureof acceptance as these are, for the corresponding purposes.
It will be seen that (2.9) is as much a condition on the test wave as it is on thebackground: it is wave-specific. It is found to be violated in an important part of thespectrum of waves in the middle atmosphere and in the oceans. Much more will besaid about it in Section 5.
With respect to (2.10): in most circumstances of interest, there will be no correlationbetween the test wave and the background waves, such as could produce cancellationbetween the horizontal and vertical contributions to the term on the left hand side.Accordingly, each of those contributions must itself satisfy the gross inequality: in prac-tice, both hR, 1ijv1j and hR, 3ijv3j must be � jK�Evj. (It is necessary that hR, 2ijv2j alsosatisfy this gross inequality, of course. But with jv2j � jv1j, it would do so automaticallyonce the condition on hR, 1i was met, in view of our assumption of rough horizontalisotropy of the background system.) Allowance must be made for cancellationswithin jK�Evj, however. These cancellations can be deduced from the linear theorythat is applicable to the test wave, whose relevant relations are contained inAppendix A.
After reduction, in most circumstances of interest – including a restriction such that!2 does not approach too near to N2 – the first of these two conditions demands that
hR, 1i � jk1=K3Hj � !=C, ð2:11Þ
while the second demands that
hR, 3i � 1=H: ð2:12Þ
(Numerical factors of order 1 have been and will be ignored. See Appendix A for theirnature.)
Condition (2.11) is seen to be wave-specific, just as (2.9) was. In contrast, (2.12) is acondition that must be met by the background waves alone, regardless of theparameters of the test wave. It requires that the background system not be close tobreaking via convective instability.
Our second basic equation is that for adiabatic change of state, which may bewritten as
ð@=@tþ vv EJÞ ln½pp���� � ¼ 0: ð2:13Þ
Expansion as in (2.5), followed by addition of � times (2.5) and isolation of thetest-wave components, leads to a relation similar to (2.6) but now involving Pand p rather than R and �; the only further change is a multiplication of the vj, jterm by �. The conditions for negligible nonlinearity are then (2.9), as before, and
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conditions (2.11) and (2.12) with P replacing R and small changes to the omittednumerical factors:
hP, 1i � jk1=K3Hj � !=C, ð2:14Þ
hP, 3i � 1=H: ð2:15Þ
Since j pj < j�j for all linear waves, and j pj � j�j for most, it is to be expected inpractice that hPi < hRi and hP, ji < hR, ji even for nonlinear waves, hence that nonew conditions have been produced.
Our third basic equation is that for horizontal acceleration in the azimuth oftest-wave propagation:
ð@=@tþ vvEJÞvv1 þ ���1pp, 1 ¼ 0: ð2:16Þ
The second term contains the factor ðpg=�gÞE exp½P� R�E exp½p� ��. Under our pre-sumption of linearity, the test-wave exponential in this factor may be approximatedas ð1 þ p� �Þ. The background-only contribution may then be identified and sub-tracted from (2.16), and products of the test-wave variables may be ignored asbefore. The resultant equation for test-wave variables is quickly seen to differ fromthe linearized equation unless (2.9) is again met, and unless
hP� Ri � 1: ð2:17Þ
This condition is almost uniformly accepted as being met in practice, and would almostcertainly be met in the middle atmosphere if (2.12) is met. With it being expected thathPi < hRi as above, (2.17) implies that hPi and hRi separately must be � 1.
Upon introducing rms values, adopting the assumed waveform and then applying(2.9) and (2.17) to remove the associated nonlinearities, we find the residual test-wave equation to be
The net coefficient of v1 is effectively unchanged from its linear value if
hV1, 1i � !: ð2:19Þ
We would expect hV1, 2i to be no larger than hV1, 1i, and so the term in v2 is negligibleunder this same condition. The term in v3 may likewise be neglected if
hV1, 3i � j!v1=v3j � N: ð2:20Þ
Our assumption of rough horizontal isotropy implies that hV1, 3i roughly equals hV2, 3i.Condition (2.20) will then require that the Richardson number of the background wavesystem, as defined by
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Ri ¼ N2=½hV1, 3i2 þ hV2, 3i
2�, ð2:21Þ
should be large compared to unity if the test wave is to escape nonlinear interactions.The condition Ri � 1 will be recognized as a condition that requires the backgroundsystem not to be close to dynamic instability. (The standard condition for marginalinstability actually requires a somewhat different Ri, one that includes wave-inducedalterations of N2, but the distinction is irrelevant when the condition demanded bythe gross inequality is met.)
Finally, the hP, 1i nonlinearity in (2.18) is negligible if
hP, 1i � jk1p=ðp� �Þj � !=C, ð2:22Þ
a condition that simply repeats (2.14).Our next basic equation is that for vertical acceleration,
ð@=@tþ vv EJÞvv3 þ ���1pp, 3 þ g ¼ 0: ð2:23Þ
This may be expanded as before. Zero-order terms occur here but are immediatelyremoved via the specification H ¼ pg=g�g already anticipated. Steps analogous tothose that produced (2.18) now lead to
Application of (2.15) renders the term in hP, 3i negligible. The net coefficient of v3 iseffectively unchanged from its fully linear equivalent if hV3, 3i � !, but this is in facttoo stringent a condition to impose because the dominant terms of (2.24) are thosethat carry the pg=�g factor. If the hV3, 3i term is taken to be small in comparison witheither of these, the condition
hV3, 3i � ½pg=�g�jK3p=v3j � N2=! ð2:25Þ
is obtained. Nevertheless, since hV3, 3i would typically be of the same order as hV1, 1i in ahorizontally isotropic linear wave system and may be expected to remain so in practiceeven in a nonlinear system, and since we already require (2.19), it seems likely that thetoo-stringent condition would in fact be met whenever the test wave avoids nonlinearinteraction with the background.
The remaining nonlinear terms in (2.24) are negligible if jv2j � jv1j and
hV3, 1i � ½pg=�g�jK3p=v1j � N: ð2:26Þ
In practice, the latter condition will certainly be met if (2.20) is met.Finally, following steps analogous to those leading to (2.18) once again, the residual
equation for acceleration transverse to the azimuth of propagation is
Condition (2.19), applied with rough horizontal isotropy, effectively reduces the coeffi-cient of v2 to �i!. The remaining terms all have coefficients consisting of nonlinear con-tributions exclusively. Both v3 and ( p� �) may be expressed as multiples of v1 fromlinear theory, producing a total of three terms in v1. The requirement jv2j � jv1j thendemands that their coefficients all be � !. It will be found that, given rough horizontalisotropy, no new conditions result.
Though the foregoing steps have treated the basic equations one at a time, it may beshown readily that the conditions thus obtained are the same as those that wouldemerge from the final dispersion relation, upon demanding that the additional (‘‘non-linear’’) terms be small in comparison with the term N2k2
1 in that equation as it iswritten in Appendix A. Some of the conditions, but not all, are rendered more demand-ing in the vicinity of ! � N if the comparison is made with the full k1 term, given byðN2 � !2Þk2
1.In summary, linearity of the Eulerian test wave requires that (2.9), (2.11), (2.12),
(2.14), (2.15), (2.17), (2.19), (2.20), (2.25) and (2.26) be met. Of these, (2.12) requiresthat the background wave system not be close to convective instability and (2.20)requires that it not be close to shear instability.
The number of conditions can be reduced in practice if the background systemis taken to be quasi linear for purposes of relating its rms values to one another. Forexample, conditions on P and its gradients may be subsumed within those on R andits gradients, as already noted. Further, given rough horizontal isotropy and a typicalspectrum of middle-atmosphere gravity waves, hRi � hV1i=C, hR, 1i � hV1, 1i=C andhR, 3i � hV1, 3i=C. Under these circumstances, the conditions on hR, 1i and hR, 3i canbe subsumed within those on hV1, 1i and hV1, 3i, while (2.17) can be replaced by
hV1i � C: ð2:28Þ
This in turn can be subsumed in (2.20) if we accept that hV1, 3i must be at least of orderhV1i=H because of amplitude growth with height. We take (2.25) to be represented ade-quately by (2.19) and, in practice, (2.26) by (2.20). This limits our Eulerian conditions,then, to (2.9), (2.19) and (2.20). It can be shown that these three are also the relevantconditions for a gravitationally stratified, incompressible medium whose density scaleheight is gN�2, such as the ocean is often taken to be. They will be carried forwardinto Section 4 for comparison with the corresponding Lagrangian conditions.
3. LAGRANGIAN ANALYSIS
We adopt three-dimensional Lagrangian coordinates r for fluid parcels, given by theirthree-dimensional Eulerian coordinates x in the absence of any wave system – i.e.,given by the x of their ‘‘home’’ position, about which they oscillate. A given parcel re-tains the same r for all time. Lagrangian analysis employs, as one of its field variables(replacing vv), the displacement ss of the parcel from its home position: if the parcelnamed r is displaced by ss ¼ ssðr, tÞ at time t, then its Eulerian position at time t is
x ¼ r þ ssðr, tÞ: ð3:1Þ
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As with vv in the preceding section, we present ss as the sum of a displacement S producedby the background wave system and a displacement s produced by the test wave:
ss � S þ s: ð3:2Þ
The Cartesian coordinates of these vectors – such as ss1, ss2, ss3 – are components definedin Eulerian space, even though they are to be found as functions of Lagrangian coor-dinates and are to be differentiated with respect to those coordinates.
We express our solutions to the governing Lagrangian equations in terms of ss, orits constituent parts, and
�� � �g exp½�r3=H þ Rþ �� ð3:3Þ
and
pp � pg exp½�r3=H þ Pþ p�, ð3:4Þ
in a notation analogous to that of Section 2 but now relating to evaluation at (r, t). Weanticipate that the test-wave portion of these solutions, represented by the lower-casecharacters, will be of the form
As before, for convenience we adopt k2 ¼ 0 and ! positive.It should be noted that the R, P, � and p employed here are perturbations exp-
erienced by the moving fluid parcel, not at a fixed Eulerian coordinate as were theircounterparts in Section 2. They are inherently different from the latter, in conse-quence. However, the unperturbed ground-level values �g and pg are the same asbefore.
In contrast to ss and its constituents, the wavevectors k and K, which permit differen-tiation to be exhibited conveniently in the case of test waves, have components definedby the Lagrangian coordinates. They are not in general identical to the k and K of‘‘corresponding’’ Eulerian waves, though they play the identical role for purposes ofdifferentiation; if a system of waves is actually mapped from one coordinate systemto the other, different symbols should be employed for clarity.
The equation for adiabatic change of state of the fluid parcel, which is the simplestof the Lagrangian equations, is
ð@=@tÞ ln½ pp���� � ¼ 0, ð3:6Þ
which becomes
ð@=@tÞðPþ p� �R� ��Þ ¼ 0: ð3:7Þ
This ensures first, for the background system alone, that P ¼ �R; and then, afterremoving the background terms from (3.7), that p ¼ ��. There is no potential fornonlinear interaction in these relations, and so no conditions result from them.
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The equations of acceleration, written in their Eulerian component form, combinewith matrix elements @xi=@rj (whose determinant is the Jacobian of the transformation)into the Lagrangian form (e.g., Lamb, 1945, Art. 13)
where a subscribed comma followed by a number represents differentiation with respectto the numbered Lagrangian spatial coordinate, and each t following a subscribedcomma denotes a differentiation with respect to time.
The field variables are now expanded as in (3.2)–(3.4). Upon insertion in (3.8)–(3.10),they yield as the final term ½pg=�g�½P, 1 þ p, 1� exp½P� Rþ p� ��, ½pg=�g�½P, 2 þ p, 2� �
exp½P� Rþ p� �� and ½pg=�g� ½�H�1 þ P, 3 þ p, 3� exp½P� Rþ p� �� respectively.The unperturbed relation H ¼ pg=�gg is obtained from the resultant (3.10), upon takingall fluctuations to vanish, just as in the Eulerian system. Relations appropriate to thebackground alone may then be obtained by taking the test-wave fluctuations tovanish (or, at least, to be of no significance to the background waves).
We next approximate exp½p� �� as ð1 þ p� �Þ in the final terms, in keeping with theassumption of linearity, and subtract the background-only relations from the rewritten(3.8)–(3.10). Second-order terms in test-wave variables are ignored, and the residualrelations for the test wave are rewritten with the use of rms values for the backgroundfluctuations. We then explore the test-wave relations to determine the conditions underwhich the contributions involving the background waves are negligible.
The common factor hexp½P� R�i ð¼ hexp½ð� � 1ÞR�iÞ in the final term of each resul-tant equation must be negligibly different from 1, for negligible nonlinear interaction,hence
hP� Ri � 1, hPi � 1, hRi � 1: ð3:11Þ
This condition, when met, permits the exponential to be replaced by 1 in the equationsfor the test wave. It is convenient henceforth to write P as �R, p as �P, ð� � 1Þ as ~��,P�R as ~��R, and p� � as ~���, thereby removing P and p from explicit appearance.
In these, the rms angle brackets that should enclose background parameters have beenomitted for simplicity.
Finally, conservation of mass requires J�� to remain constant (Lamb, 1945, Art. 14)at its unperturbed value �g exp½�r3=H�. Here J is the Jacobian determinant of thecoordinate transformation,
J � det½@xi=@rj�, ð3:15Þ
to be evaluated from (3.1). In the absence of all waves, of course, this J ¼ 1. In the pres-ence of background waves only, J is the three-dimensional determinant Jb whose i, jcomponent is �ij þ @Si=@rj, where �ij ¼ 1 if i ¼ j, otherwise¼ 0. Mass conservationthen requires Jb exp½R� ¼ 1.
With both background and test waves included, (3.15) must be expanded in full,although products of test-wave variables may be dropped as before. In the conservationequation itself, �� may be replaced by ½1 þ ���g exp½�r3=H þ R�, and products of � withother test-wave variables may be dropped.
Finally, the background-only relation Jb exp½R� ¼ 1 may be subtracted from theresultant equation. These steps lead to
where again angle brackets denoting rms values have been omitted for simplicity.We now consider the conditions, beyond (3.11), under which the test wave can
remain free from nonlinear interaction with the background system. These conditionsare established, as before, by requiring that terms containing background parametersbe small in comparison with the terms that do not – or, at least, in comparison withthe dominant terms that do not.
It is evident from (3.16) that linearity of the test wave will require, among otherconditions,
hS1, 1i � 1, hS3, 3i � 1, ð3:17, 18Þ
since each of these left-hand sides is found added to 1. We could add the conditionhS2, 2i � 1, but that follows from (3.17) under our assumption of rough horizontalisotropy. We further assume that, if (3.17) is met, then
hS1, 2i � 1, ð3:19Þ
is also met, as would certainly be the case if the background system were linear.Likewise, hS2, 1i � 1. Indeed, in all cases we will assume that any individual 2 subscriptcan be replaced by a 1 so that, for example, the condition
hS2, 3ihS3, 2i � 1, ð3:20Þ
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if satisfied, would also imply that hS1, 3i hS3, 2i � 1, etc. That (3.20) is indeed required isapparent from the term in k1s1 found in (3.16). (It might be argued that the less restric-tive demand hS2, 3S3, 2i � 1 is adequate; but other required conditions will soon implythat, if they are met, the more restrictive demand just made will also be met.)
From the full term in s1 in (3.16), it is also evident that we require
hS3, 1i � jk1=K3j � !=N: ð3:21Þ
Similarly, from the full term in s3 in (3.16) we require
hS1, 3i � jK3=k1j � N=!: ð3:22Þ
If both of these conditions are met, then (3.20) is also met; both (3.20) and all thatfollow from it may be eliminated from the list of required conditions.
Inspection of the remaining nonlinear terms in the coefficients of s1 and s3 in (3.16)will reveal that all have now been accommodated by application of (3.17), (3.18), (3.21)and (3.22) and their respective relatives. As with the Eulerian condition on jv2j, we shallrequire that js2j � js1j. In conjunction with the assumption of rough horizontalisotropy, the term in s2 in (3.16) then adds no further condition.
On turning to (3.12) and (3.14), we note that g and S3, tt appear only in the combina-tion ðgþ S3, ttÞ, and so linearity of the test wave requires
hS3, tti � g: ð3:23Þ
This condition corresponds to the quasistatic approximation that is almost invariablytaken to apply in middle-atmosphere wave systems.
We note that in (3.12) the dominant terms are those in k1gs3 and k1C2�. (This may
be established with the use of the Lagrangian polarization relations for linear waves,given in Appendix B.) By comparison of the term in S1, tt with either of these, we cometo require
hS1, tti � gjs3=s1j � !C: ð3:24Þ
The term in S3,1 produces a condition that is less restrictive than (3.21) and so may beignored, while that in R, 1 produces a condition that is less restrictive than one about tobe obtained, namely (3.26). The terms in s2 make no additional demands.
It is tempting to demand, from (3.14), that hR, 3i � jK3j; but that would be inade-quate to the need. This is because the K3 terms are effectively cancelled out when thedispersion relation is calculated, thereby giving greater prominence to R, 3 in thatrelation than it appears to have here. It is this same problem that led to our skippingthe R, 1 term in (3.12). The difficulty is avoided if (3.12) is multiplied by K3 andsubtracted from k1 times (3.14) to produce
in place of (3.14). (Terms associated with s2, which are of no consequence, have beencompletely ignored here to avoid clutter. Likewise, (3.17) and (3.18) have been takeninto account to remove clutter.)
and (3.21) and (3.22) once again.In summary, Lagrangian linearity requires (3.11), (3.17), (3.18), (3.21)–(3.24), (3.26)
and (3.27). We can dispose of some of these by again taking the background system tobe quasi linear for purposes of relating rms values to one another (see Appendix B).Thus, (3.11) is implicit in (3.27) if growth with height on a scale of order H is admitted.Likewise, (3.18) implies hS3i � H; and then, since hS3, tti < N2hS3i in internal gravitywaves, (3.23) is implicit in (3.18). In a linear system, j�j � js3j=H, so (3.26) may be sub-sumed within (3.21) and (3.27) within (3.18). Similarly, (3.18) may be taken to imply(3.17). Finally, (3.24) is contained within (3.21) via the following chain, in which � indi-cates an approximate equality on the basis of quasi linearity of the background and/orneglect of factors of order 1:
hS1, tti ¼ hV1, ti � NhS3, ti ¼ NhV3i < NHhV3, 3i � NHhV1, 1i
� N2HhS3, 1i � NChS3, 1i � !C: ð3:28Þ
This limits our Lagrangian conditions, then, to (3.18), (3.21), and (3.22). It can beshown that these are also the relevant conditions for a gravitationally stratified, incom-pressible medium whose density scale height is gN�2. They will be carried into Section 4for comparative purpose.
4. COMPARISON OF CONDITIONS FOR LINEARITY
We begin our comparisons with (2.20) and (3.18). The first of these requires thathV1, 3i � N, while the second requires that hS3, 3i � 1. However, in a linear wave at fre-quencies somewhat less than N, jv1j � Njs3j and so jv1, 3j � Njs3, 3j (see Appendix A). Ifwe take the background waves to be quasi linear as before, we obtain hV1, 3i � NhS3, 3i
and the two required conditions are seen to be virtually identical. Each coordinatesystem requires, in effect, that the Richardson number evaluated for itself shall be� 1 if nonlinear action upon the test wave is to be avoided. This is a condition onthe background waves alone, independent of the parameters defining the test wave.
In much the same way – i.e., assuming quasi linearity of the background and sohV1, 1i � NhS3, 1i – conditions (2.19) and (3.21) are found to be virtually equivalent.These are wave-specific conditions, requiring that the wave frequency be not toosmall. Representative values suggest that waves with periods no greater than severalhours would remain linear, so far as this condition is concerned. Waves at mid-latitudeinertial frequencies might come to be suspect under this criterion, but our analysis hasnot attempted to deal with the Coriolis effects they would entail.
There remain for discussion the Eulerian condition (2.9) and the Lagrangiancondition (3.22), which we copy for comparative purposes here:
jhVEKij � !, hS1, 3i � jK3=k1j � N=!: ð4:1, 2Þ
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Both of these are wave-specific, but there the similarly ends. From one point of view,they make quite different demands on the background if a chosen test wave is toremain free from nonlinear interaction. From another, given an actual backgroundsystem, they impose nonlinearity on two quite different portions of the spectrum ofpotential test waves.
The Eulerian condition is understood as arising from Doppler shifting or spreading,as already noted. The Lagrangian condition may be understood by noting that asteady, shearing background wind cannot be accommodated readily in theLagrangian analysis of waves, for it produces a secularly increasing deformation ofthe coordinate system. This would be represented in the Lagrangian equations by asecularly varying analogue to S1, 3, directly proportional to t, which would preventthe basic equations from having solutions with the osciallatory waveform assumed.Condition (4.2) is breached by test waves of high frequency – sufficiently high, itwould seem, that the deformation of the coordinate system by the backgroundwaves appears to them to be a secular deformation, and some response must bemade. The response takes the form of nonlinear interaction.
A comparison of the conditions for wave-specific nonlinearirty or linearity is given inschematic form in Fig. 1, which is drawn for representative values that are cited in itscaption. The horizontal and vertical axes are jk1j and jK3j respectively, each normalizedvia division by N=hV1i. (Note the difference of scales.) The internal waves that are ofinterest here are excluded from the horizontally hatched band at the bottom, whoseupper edge lies at jK3j ¼ 1=2H and represents waves that have k3 ¼ 0.
Panel (a) exhibits, by shading, the region of exclusively Eulerian nonlinearity repre-sented by (4.1). It also exhibits, as a broken horizontal line near the top, the lowerlimit that this region would have had if (4.1) had been read as hV1ijk1j � !, whichcorresponds to the Doppler-spread condition as it is frequently judged. Evidently thevertical component of V, which has been taken here to equal 1% of the horizontal com-ponent, can play a major part in producing nonlinearity.
Panel (b) similarly exhibits, at bottom right, the region of exclusively Lagrangiannonlinearity represented by (4.2). Panel (b) also exhibits, near its vertical axis, theregion of Lagrangian nonlinearity that is shared by Eulerian nonlinearity, as repre-sented by (3.21) and (2.19) respectively. The manifestation of this region in panel (a)is lost within the exclusively Eulerian nonlinearity (except in the immediate vicinityof k3 ¼ 0), but its boundary is exhibited by a sloping dotted line.
In all cases, the gross inequalities of the linearity conditions have been replaced byequalities. A gross inequality, say by a factor of 3, would be included if the boundarieswere retained intact but the cited rms values were decreased by a factor of 3.Alternatively, the rms values could remain intact but the shaded regions of nonlinearitywould then encroach further into the unshaded regions of linearity. The curved bound-ary in panel (a) would move to the right at its lower end, to jk1 j values a factor of3 greater than those illustrated, and downward at its upper end, to jK3 j values afactor of 3 less than those shown, with smooth continuity between the ends such asthat already shown. The two sloping boundaries in panel (b) would rotate about theorigin toward one another, by respectively decreasing and increasing their slopes bya factor of 3.
Encroachment in this fashion can also be thought of as applying to the original repre-sentation when height increases sufficiently that the background waves increase theirrms perturbations by a factor of 3. When height increases even further, of course,
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the regions of linearity of both systems are diminished and ultimately obliterated by theencroaching regions of nonlinearity. In company with these changes, the Richardsonnumber of both systems would be decreasing and could impose total nonlinearityfirst. The role of the Richardson number cannot be illustrated in Fig. 1, since it isnot wave-specific.
The essential feature of the figure – the heart of the present paper – is the fact that theregions of linearity differ as between Eulerian linearity and Lagrangian linearity; a test
FIGURE 1 Regions of test-wave |k1|, |K3| space in which nonlinear effects are imposed by the backgroundwave system (shaded), or in which nonlinear effects are unimportant (unshaded), as evaluated fromwave-specific criteria for an Eulerian wave (panel a) and for a Lagrangian wave (panel b). Both coordinatescales have been normalized by the characteristic wavenumber N=hV1i. The transitions from nonlinear tolinear have been drawn after imposing equality, rather than gross inequality, on the conditions specified in thetext. The curved line in panel a represents the uniquely Eulerian condition (4.1); the sloping line in panel b atbottom right represents the uniquely Lagriangian condition (4.2); the sloping line in both panels at upper leftrepresents the shared condition (2.19) and (3.21); see discussion in the text Section 4. The conditionsfor linearity include, beyond these, a requirement that a Richardson number associated with the backgroundwave system be � 1. The horizontally hatched strip at the bottom is unoccupied by internal waves; its upperboundary marks k3¼ 0. For this illustration, the values assumed are hS3,1i ¼ 10�3; hS1,3i ¼ 10; hV1i �
10m=s; hV3i � 0:1 m=s;N � 0:02 rad=s; H � 7 km. The corresponding hS1,1i and hS3,3i would be of order0.1, thereby producing a Lagrangian Ri � 1. A representative spectral peak, shown by the dot, has horizontalwavelength 700km and vertical wavelength 7 km, wave period 8 h. The arms of the cross represent variationsfrom this peak by factors of 2 and 1/2. Thus, a deformed ellipse drawn about the arms as axes would representa broader spectrum ranging in horizontal wavelength from 350 to 1400 km and in vertical wavelength from14km down to 3.5 km. Under the approximation ! ¼ NjK3=k1j, obtained from (A.6) for most internal gravitywaves, contours of constant frequency would be represented by straight lines radiating out from the origin ofcoordinates. Two such are found in panel b, corresponding to periods of about 52min (lower right) and 87 h(upper left) respectively. The latter is, of course, too large for Coriolis effects to be ignored, as here, exceptnear the equator; it simply gives some idea of the relevant range. The bottom and right tips of each crosscorrespond to periods of about 4 h, while the top and left tips of each cross correspond to periods of about16 h. All values selected here are roughly representative of accepted atmospheric values, at least sufficiently sofor illustrative purposes.
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wave may well propagate without significant interaction with other waves as judged inone coordinate system, even when its counterpart in the other system cannot propagatewithout such interaction. The situation is more or less symmetric: neither coordinatesystem is preferred in any absolute sense, but only when the discussion of certainwaves and their counterparts is at issue.
Though the regions of nonlinearity and linearity have been sketched here for rmsvalues that are thought to be representative of the middle atmosphere (at someheight), the primary purpose of the diagram is simply to emphasize the dichotomybetween Eulerian and Lagrangian behavior. Readers may redraw the diagram withparameters to suit their own estimates, not only for the middle atmosphere but equallyfor the oceans. The dichotomy will recur, with greater or less impact, except with theadoption of such extremely large rms values that the entire wave spectrum is subjectto significant nonlinear effects in both coordinate systems (and, probably, with morethan marginal instability as judged by Ri).
5. SAMPLE APPLICATIONS
For comparison with the illustrated regions of nonlinearity and linearity, a representa-tive middle-atmosphere spectrum is indicated in each panel of Fig. 1. Its spectral peak ismarked by a dot. A spectral range of important horizontal and vertical wavenumbers,extending by a factor of 2 above and below their values at the peak, is indicated bya cross. No claim is made that this spectrum accurately matches reality on any selectedoccasion – particularly with respect to |k1| variations, which may not even exhibit aspectral peak within the range of validity of the present gravity-wave approxima-tions – nor that it accurately matches in practice the rms values that were adoptedfor illustrative purposes. It does, however, accurately reveal the contrasting importance,in application to real spectra, of the various Eulerian and Lagrangian nonlinearities: theexclusively Eulerian nonlinearity is overwhelmingly the most important.
The potential importance of this Eulerian nonlinearity has never been in doubt. Whatis newly and explicitly established here is the absence of any correspondingly significantLagrangian nonlinearity, and the relative unimportance of such Lagrangian nonlinear-ities as do exist. This is the essential conclusion of the present analysis in applicationto the real world. [Allen and Joseph (1989) have already pointed out the absence ofa corresponding nonlinearity both in principle and in practice with regard to a modelspectrum of oceanic waves, while Chunchuzov (1996) built on their work in applicationto the atmosphere – see below. But neither study isolated the ranges of relevance of thevarious forms of nonlinearity in the present fashion or with the present degree of gen-erality.]
A significant portion of the wave spectrum in the middle atmosphere – indeed,most or all of that which constitutes the large-wavenumber ‘‘tail’’ of the Eulerianvertical-wavenumber spectrum – is widely believed to breach the exclusively Euleriancondition for linearity. It would be represented in panel (a) by an upward extensionof the branch of the cross that rises from the dot. The breach of this particular lin-earity condition is believed to be important by some (e.g., Weinstock, 1976; Hines,1991, 1996; Chunchuzov, 1996), but believed to be preferably ignored by others. Thecorresponding breach in oceanic studies has been fully recognized, and without con-troversy its importance has been accepted, ever since the fact of the breach was
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stressed by Holloway (1980,1981) and Munk (1981). The essential point to be madehere is that there is no corresponding breach of Lagrangian linearity. Variousconsequences follow from this fact.
First, with Lagrangian linearity intact, the propagation of waves can be examinedin straightforward fashion in Lagrangian coordinates, and to some extent the con-sequences can be mapped into Eulerian coordinates. This process providesEulerian results that are not easily obtained otherwise. Examples are given by theoceanographic analysis of Allen and Joseph (1989) and, following from it, theatmospheric analysis of Chunchuzov (1996). (The former validated the quasi linear-ity of the Lagrangian waves it employed, whereas the latter merely assumed it. Thevalidation was done for its specific adopted spectrum by examination of the non-linear energy terms that were automatically omitted upon assuming linear polariza-tion and dispersion relations, rather than as here for more general spectra.) Bothanalyses found that the exclusively Eulerian nonlinearity acts to produce a spectraltail at large vertical wavenumbers in the Eulerian description of the system, a tailthat does not exist (at significant amplitude) in the Lagrangian description. Onthis interpretation, most of the upper branch of the cross in panel (a) of Fig. 1represents a part of the tail that results from Eulerian nonlinearity, rather thanfreely propagating modes that satisfy a dispersion relation: our opening assumptionof linearity is not supported by the a posteriori test. Most of the upper branch ofthe cross in panel (b) might well be deleted, the better to represent Lagrangian rea-lity. This fact has further implications with respect to the Richardson number, aswill be discussed shortly.
A second example is given by Eckermann (1999). He in effect assumedLagrangian linearity via the assumption made for his input spectrum, which wastreated in an inherently Lagrangian fashion. He then numerically Fourier analyzedthe output Eulerian waveform, thereby revealing an Eulerian spectral tail of theform anticipated. Unfortunately, his assumed input spectrum would have beensubject to Lagrangian nonlinearity that was not taken into account (as was firstpointed out to me, in different terms, by G.P. Klassen and A.S. Medvedev, personalcommunication, 1999). The details of his output are therefore questionable. Hisvisual illustrations of the nature of the Eulerian nonlinear behavior are neverthelessperfectly valid as sketches of the waveform deformation that generates the tailportion of his Eulerian spectra, but it would be preferable to see them producedfrom a properly linear Lagrangian spectrum.
Finally, at least for the moment, it becomes evident that there are advantages to begained by conducting ray-tracing in Lagrangian, rather than Eulerian, coordinates.Indeed, in the simple circumstances contemplated here, wave packets as described inLagrangian coordinates undergo no interaction with the background wave systemand so follow straight lines. These straight lines can be mapped readily into theEulerian coordinate system, where they become sinuous because of the relative defor-mation of the two systems. This application is developed in greater detail inAppendix C.
It seems unlikely that the foregoing list has exhausted the areas to which applicationof uniquely Lagrangian linearity may contribute in the future, once its existence isrecognized and exploited by the imaginative.
We turn now to one of the shared conditions for linearity: that the Richardsonnumber (as evaluated for each coordinate system in turn) should be � 1.
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It might be argued by some that this condition is not met in the middle atmosphereat all: that the wave systems there render the region always dynamically unstable, andso all waves are at all times subject to nonlinear interaction. There are two responses tobe made to this argument.
First, even if true, the argument should not inhibit an examination of circum-stances in which this nonlinearity does not arise. It is only by isolating nonlineari-ties, to the extent possible, that the role of each one individually can be examinedand, perhaps, understood. Certainly model spectra can be designed such that thisnonlinearity is not operative and yet one of the others is operative, in which casewe can study the consequences of that other nonlinearity (or of its separate suppres-sion) in isolation. The immediately preceding paragraphs provide illustrations of thepoint. Conversely, the other nonlinearities can be suppressed and this one studied inisolation. Indeed, that has effectively been done already in many studies. As a resultwe understand already that the Ri nonlinearity, when found in Eulerian coordinates,must be accompanied by dynamic instability. The consequences of the instabilitymight quite overshadow those of the nonlinearity itself, of course. Or, as may bethe case, the consequences of the instability might be dominant in one portion ofthe spectrum (e.g., the smaller-scale portion) without being of great significance inanother, where the nonlinearity would nevertheless remain in effect. In any event,both the mathematics and the physics of the situation would be open to deeperprobing.
Second, the argument may not even be fully valid. This is because, in part, there arestrong reasons for believing that the reduction of the Eulerian Richardson number tothe point of instability is not caused by a quasi linear Eulerian wave system at all,but rather by nonlinear consequences associated with the Eulerian spectral tail atlarge vertical wavenumbers (Hines, 1991, 1996). In Lagrangian coordinates, as notedabove, this tail does not occur because there is no corresponding nonlinearity to pro-duce it. Accordingly, the Lagrangian Richardson number can be much larger thanthe Eulerian: the integrity of Lagrangian linearity need not be undermined even ifthat of Eulerian linearity is. This does not remove the effects of instability, of course,but it renders them an add-on to what continues to be a perfectly linear analysis ofotherwise non-interacting Lagrangian waves.
It may be said, as a general conclusion, that the availability and potential importanceof Lagrangian linearity in the absence of Eulerian linearity has only recently becomeapparent to the community (beginning with the work of Allen and Joseph). There isno means of forecasting its potential for future development of middle-atmosphereand oceanic studies.
References
Allen, K.R. and Joseph, R.I., ‘‘A canonical statistical theory of oceanic internal waves’’, J. Fluid Mech., 204,185–228 (1989).
Broutman, D., Macaskill, C., McIntyre, M.E. and Rottman, J.W., ‘‘On Doppler-spreading models of internalwaves’’, Geophys. Res. Lett., 24, 2813–2816 (1997).
Chunchuzov, I.P., ‘‘The spectrum of high-frequency internal waves in the atmospheric waveguide’’, J. Atmos.Sci., 53, 1798–1814 (1996).
Eckermann, S.D., ‘‘Influence of wave propagation on the Doppler spreading of atmospheric gravity waves’’,J. Atmos. Sci., 54, 2544–2573 (1997).
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Eckermann, S.D., ‘‘Isentropic advection of gravity waves: quasi-universal M�3 vertical wavenumber spectranear the onset of instability’’, Geophys. Res. Lett., 26, 201–204 (1999).
Flatte, S.M., Henyey, F.S. and Wright, J.A., ‘‘Eikonal calculations of short-wavelength internal-wavespectra’’, J. Geophys. Res., 90, 7265–7272 (1985).
Henyey, F.S., Wright, J. and Flatte, S.M., ‘‘Energy and action density through the internal wave field: aneikonal approach’’, J. Geophys. Res., 91, 8487–8495 (1986).
Hines, C.O., ‘‘Internal atmospheric gravity waves at ionospheric heights’’, Can. J. Phys., 38, 1441–1481(1960).
Hines, C.O., ‘‘The saturation of gravity waves in the middle atmosphere. Part I: Critique of linear-instabilitytheory’’, J. Atmos. Sci., 48, 1348–1359 (1991).
Hines, C.O., ‘‘Nonlinearity of gravity wave saturated spectra in the middle atmosphere’’, Geophys. Res. Lett.,23, 3309–3312 (1996).
Holloway, G., ‘‘Oceanic waves are not weak waves’’, J. Phys. Oceanogr., 12, 293–296 (1980).Holloway, G., ‘‘Theoretical approaches to interactions among internal waves, turbulence and fine structure’’.
In: Nonlinear Properties of Internal Waves (Ed. B. West) Amer. Inst. Phys. (1981).Jones, W.L., ‘‘Ray tracing for internal gravity waves’’, J. Geophys. Res., 74, 2028–2033 (1969).Lamb, H., ‘‘Hyrdodynamics’’, Dover Publications, New York (1945).Landau, L.D. and Lifshitz, E.M., ‘‘Fluid Mechanics’’, Addison-Wesley (1959).Lighthill, J., ‘‘Waves in Fluids’’. Cambridge University Press (1978).Munk, W., ‘‘Internal waves and small-scale processes’’. In: Evolution of Physical Oceanography (Eds. B.A.
Warren and C. Wunsch), MIT Press (1981).Weinstock, J., ‘‘Nonlinear theory of acoustic-gravity waves: 1. Saturation and enhanced diffusion’’,
J. Geophys. Res., 81, 633–652 (1976).
APPENDIX A EULERIAN LINEAR RELATIONS
When all terms originating from interaction with the background wave system areeleminated from the residual equations governing the test wave, standard polarizationand dispersion relations may be obtained from those equations. The polarization rela-tions relate complex amplitudes of the variables to one another. They can be repre-sented, except for an irrelevant and unstated but common multiple (having dimensionLT2), by
� ¼ f�i ~��gk21g þ !2K�
3 , p ¼ f�!2ðK3 þ i=�HÞg, ðA:1, 2Þ
v1 ¼ f!k1C2ðK3 þ i=�HÞg, v2 ¼ 0, v3 ¼ f�!k2
1C2g þ !3: ðA:3, 4, 5Þ
These amplitude factors must be multiplied by the exp½iðK E x � !tÞ� factor found in(2.8) to yield the respective complex variables themselves, previously assigned thesame symbols. The dispersion relation may be written as
!2jK3j2 ¼ fN2k2
1g � !2k21 þ !4=C2: ðA:6Þ
In these relations, the terms on the right that are of greatest consequence over most ofthe internal-wave spectrum are enclosed in braces. The approximations that have beenmade in Section 2 result from ignoring the other terms on the right and ignoring factorsof order 1 (after setting � ¼ 1:4), except that some such factors remain implicit withinthe symbols H and N. For example, the magnitude of the factor ðK3 þ i=�HÞ in (A.2)and (A.3) is evaluated as jK3j for order-of-magnitude purposes. For jk3j � 1=2H, ofcourse, this could be further approximated as jk3j, but to retain generality this stephas not been employed. One can judge the importance of omitted terms via direct com-parison with retained terms, for any chosen k,!.
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It may be noted that a unique situation arises in the vicinity of k3 ¼ 0, !2 ¼ ð�=2ÞN2.There, the right-hand side of (A.1), taken in its entirety in conjunction with (A.6),vanishes and there is no Eulerian density perturbation whatever. In this vicinity, it isimpossible for the nonlinear correction to be small in comparison with the linear beha-vior, insofar as density fluctuations are concerned. This anomaly does not affect othervariables or the dispersion relation. However the other variables are affected, in thesecircumstances, through other terms that become significant as !2 approaches N2.
In Section 4, use has been made of the fact that, in linear waves, js3j ¼ jv3=!j �jv1k1=!K3j � jv1j=N. The approximations employed in deriving this sequence are drawnfrom those above.
APPENDIX B LAGRANGIAN LINEAR RELATIONS
Steps analogous to those taken in Appendix A lead to the identical dispersion relation,namely (A.6), and to the polarization relations
wherein v1 and v3 are to be taken from (A.3) and (A.5) respectively and, for convenienceof comparison, the same arbitrary but unstated multiple has been employed as inAppendix A. These amplitude factors must be multiplied by the exp½iðKEr � !tÞ�factor found in (3.5) to yield the respective complex variables themselves, previouslyassigned the same symbols. It may be confirmed that the time derivatives ofthe Lagrangian density and pressure perturbations, unnormalized, are respectivelyequal to those of the Eulerian density and pressure perturbations, unnormalized, whenthe latter are obtained in linearized fashion ‘‘following the motion’’ (via d=dt ¼@=@tþ v3@=@x3Þ.
APPENDIX C LAGRANGIAN WAVE PACKETS: IF NOT, WHY NOT?
(a) Introduction
In the atmosphere and oceans, a multitude of gravity waves propagate in superposition.Their interactions with one another are often modeled by eikonal, or wave-packet,analyses conducted in Eulerian coordinates. Here it is shown that in many commonlymodeled circumstances an analysis conducted in Lagrangian coordinates, withsubsequent transcription into Eulerian coordinates, provides advantages of simplicityand, very likely, of legitimacy. The distinction is based upon the different regimes oflinear behavior discussed in the main body of this paper.
We first outline the role of, and some shortcomings of, the Eulerian form of analysisin application to this problem. The Lagrangian form of analysis, together with its after-the-fact conversion to Eulerian coordinates, is presented next. Its advantages (anddisadvantages) are made evident. And finally, we discuss the conflicting implicationsof the two forms of analysis and argue that each should be employed in circumstances
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favoring itself. These include, for the Lagrangian form to be favored, most of thecircumstances that have been studied to date by Eulerian methods in multiwavemodels of the atmosphere and oceans. The validity of the Eulerian analysis in suchcircumstances is thereby called into question.
(b) Eulerian Wave Packets
Eulerian eikonal analysis begins (e.g., Landau and Lifshitz, 1959; Lighthill, 1978) withthe assumption that, in a given region of space-time, one has available an unambiguousdetermination of a locally dominant wavevector k and frequency ! found from thewavefield itself by spatial and temporal differentiation. These parameters define, in asense, the central wavevector and central frequency of some wave packet whoseprogress in the course of time is to be followed through regions of varying backgroundflow. What is in fact followed is a single point that will be termed the centroid of thepacket, and the path it traces out is termed the ray path. A simultaneous determinationis made of the central wavevector and central frequency, which in general are ever-changing as the packet moves and must be continually updated (in practice, bycomputer). In some studies, changing values of the density of wave action and waveenergy are also derived. The various changes are associated, though not exclusively,with changes of the Doppler-shifted central frequency. It is assumed throughout thatthe central wavevector and central frequency satisfy a linear dispersion relation thatincorporates (in a Doppler-shift fashion) the velocity of the flow field at the packetcentroid.
Such analyses have a long history of acceptance, based not only on their mathemat-ical precepts but also on confirmatory observations in laboratory systems whosebackground flow changes at most only slowly in space-time (e.g., Broutman et al.,1997). They have come to be of common currency in fluid dynamics. They are not inany fashion being brought into question here in the case of sufficiently slow changesin the background.
But the modeling that is of interest here is concerned with a different situation: withthe propagation of a test wave packet through the fluctuating velocity field V inducedby a system of many other waves, to be termed background waves, in the absence ofany shearing or time-dependent flow other than that induced by the wave systemitself (e.g., Flatte et al., 1985; Henyey et al., 1986; Eckermann, 1997). In these circum-stances, the centroid follows a sinuous course – a wiggly ray path – through the back-ground wave system. As it does so, its central wavevector and frequency varycontinuously and must be repeatedly updated according to eikonal rules. For a validanalysis, the background waves must be changing only slowly in space and time.Various criteria can be employed in judging the permissible rates of change in thesecircumstances (e.g., Jones, 1969), but there are as yet no independent tests of legitimacysuch as a laboratory study might provide, and the criteria themselves meet only limitedobjectives (see below). To this degree, application of the Eulerian eikonal approach tothe multiwave case is open to question. A basic question does arise, concerned withnonlinearities, but will be postponed for the moment.
There may also arise a problem of principle in application, in that the circumstancesof nature that are meant to be modeled preclude the occurrence of a clearly defined,locally dominant, wavenumber and frequency with which to identify the eikonal. Thisproblem might be countered in modeling by a restriction to scale-separated test waves,
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perhaps, but even that entails limiting the applicability of the results in a correspondingfashion.
Alternatively, in these circumstances, one may fall back on the somewhat less eso-teric concept of a wave packet that is specified initially quite arbitrarily rather thanby differentiation of the wave field. It comprises a superposition of (locally) planewaves with wavenumbers and frequencies initially grouped about some chosen set ofcentral values, propagating through the background waves. Its location is specifiedas before by its centroid, defined now by the principle of stationary phase: constructiveinterference will be at a maximum at a point where all waves of the test packet are inphase. That point moves with the packet velocity, which is of the same form as thevelocity that would have been inferred from eikonal analysis. The eikonal rules forupdating the central wavenumber and frequency may then be adopted by analogy,now for the arbitrarily initially specified packet, just as if there had been no difficultyassociated with the start of the eikonal analysis. Since the initial wave field is in factnever actually differentiated to determine a locally dominant wavenumber and fre-quency, there would seem to be little lost in the transition to a wave-packet startingpoint.
A difficulty does remain, however, in that the conditions for validity of an eikonalanalysis – and, therefore, of eikonal updating – may well be breached in wanted appli-cations. For example, there is a requirement for only slow space-time variations ofthe wave-induced background V field (to permit, among other things, the legitimateuse of WKB approximations). This difficulty continues to preclude valid packetcomputations when the central wavenumber and central frequency of the wavepacket lie at values comparable to or less than those of any important portion of thebackground field. The uncertainty arising from nonlinearity also remains, as doesthat arising from dependence on an analogy to determine updating procedures.
(c) Lagrangian Wave Packets
All of the foregoing has described the situation as it applies with the use of Euleriancoordinates. The situation alters dramatically, however, if we consider instead theuse of Lagrangian coordinates as in the main text here. One may construct wave pack-ets in Lagrangian coordinates as readily as in Eulerian, but their operational useappears not to have been contemplated to date. We examine their potential here, forapplication to the multiwave case in the absence of a mean background flow. The back-ground waves will be taken to be linearized, as they are in most Eulerian studies.
We will assume here that we are within the range of linear legitimacy, as depictedin panel b of Fig. 1 of the main text, unless and until we find we are not. While in it,the governing Lagrangian equations may be treated as being linear for all wave-packet purposes. For simplicity we assume uniform stratification under gravity, withconstant background stability N2 and constant density scale height H. Both Coriolisand dissipative effects are ignored.
We adopt the packet approach, whereby we conceive of a superposition of planewaves (as defined now in Lagrangian coordinates) having wavevectors k and fre-quencies ! grouped about some central values. From the principle of stationaryphase, this superposition would produce maximum constructive interference at apoint rP that would move through Lagrangian coordinates r according to the usualrule of packet propagation: drPj=dt ¼ ð@!=@kjÞP � PLj where j is a coordinate index
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(¼ 1, 2, 3, with 3 upward), a subscribed P denotes evaluation for the packet, and PL
is the Lagrangian packet velocity through Lagrangian coordinates. The Lagrangiandispersion relation, from which @!=@kj is determined, has the same form as theEulerian dispersion relation when the latter employs the intrinsic frequency (i.e.,absent any explicit Doppler-shift effect).
In the Lagrangian frame of reference, the linearized waves with which we are dealinghave no interaction with one another: the superimposed waves that make up the packetpropagate independently of one another and of the background waves, as do the back-ground waves themselves. Accordingly, the test wave packet propagates throughLagrangian coordinates as it would in the absence of the background waves, namelywith the constant packet velocity PL. Its location in Lagrangian coordinates at time tis given simply by
rP½t� ¼ r0 þ PLt, ðC:1Þ
where r0 is rP[0]. This is, in essence, the end of the problem as it arises in Lagrangiancoordinates: there is no sinuous path to be followed (and computed), merely a straightline!
In practice, of course, we often want to find the location of the packet in Euleriancoordinates in the presence of other waves. This requires that we account for the dis-placement of the Lagrangian coordinate system relative to the Eulerian, as it is inducedby those waves.
We adopt a background spectrum consisting of individual modes identified by thesubscript b. Each of these produces a displacement sb of any selected parcel from its‘‘home’’ position, with components
The total displacement S is given by summing these sb over all b. (This is the same S as inthe main text, except that the background waves are now express taken to belinearized. The amplitude components Sbj and their corresponding phase anglesbj must be consistent with the polarization relations for each b in turn. H is thescale height of the unperturbed density variation, often taken to be infinite in theocean.)
For wave amplitudes so small that intrinsic nonlinear effects may be ignored in bothcoordinate systems, the Lagrangian spectrum of background waves is virtually identicalto the Eulerian spectrum (Allen and Joseph, 1989). The Eulerian spectrum is somewhatdeformed and does generate a significant nonlinear high-wavenumber ‘‘tail’’ as ampli-tude increases, in consequence of the Eulerian V EJ nonlinearity (Allen and Joseph,1989), but these changes of spectrum are uniformly ignored in Eulerian eikonal andwave-packet analyses. The two spectral descriptions for the background waves maythen be taken to be identical: if the waves are given by (C.2) in Lagrangian coordinates,then the corresponding spectrum in Eulerian coordinates is given by (C.2) after repla-cing r by x. Equally, if the background wave spectrum is defined first in Eulerian coor-dinates, it transcribes immediately into a Lagrangian spectrum. An Eulerian descriptionnormally provides the velocity field, rather than the displacement field, but with
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the assumption of linearity for the background waves the transition of specification istrivial.
The course followed by the Lagrangian test wave packet, though a straight line inLagrangian coordinates as given by (C.1), is made sinuous in Eulerian coordinatesby virtue of the wave-induced deformation of the Lagrangian coordinate system asseen in the Eulerian system. In the Eulerian system, the packet location is given bythe sum of two terms: by (C.1), which identifies the parcel where the packet centroidis currently located and serves equally as an Eulerian ‘‘home’’ marker, plus the displa-cement of that parcel from its Eulerian home, imposed by the background waves. Thelatter is given by insertion of (C.1) in (C.2) in place of r, followed by addition over all b.Thus, the Eulerian coordinates of the packet at time t are given by
xPj ½t� ¼ r0j þ PLjtþ exp½ðr03 þ PL3tÞ=2H�X
Sbjcfbjg ðC:3Þ
where
cfbjg ¼ cosð½kbEPL � !b�tþ kbEr0 � bjÞ: ðC:4Þ
This completes the wanted analysis.The Eulerian starting point of the packet is given by
x0j ¼ r0j þ expðr03=2HÞX
Sbj cosðkbEr0 � bjÞ: ðC:5Þ
This can be inverted readily by numerical means if a controlled input of x0 values iswanted for the systematic testing of a variety of ray paths. Or, in the atmosphericcase, one can begin with Sb values so small that the distinction between x and r is neg-ligible. Interest would then lie at greater heights at later times, for an upgoing wavepacket, where the exponential factor in (C.3) will have increased substantially.
It should be readily apparent that the proposed Lagrangian method of determiningray paths in Eulerian coordinates, given by (C.3), is vastly simpler than the Eulerianmethod of determining such paths: there is nothing to compute other than a weightedsum of cosines – specifically, no change of central wavevector or frequency. Moreover,the proposed method is perfectly applicable even when the background waves containscales comparable to and even smaller than the waves of the wave packet: there issimply no interaction to be dealt with, and the question of scales does not arise as alimit to legitimacy.
(We may anticipate a point soon to be made: a wave packet extends around itscentroid, on scales at least as large as those determined by its central wavenumberand frequency. Small-scale background waves would, in the Lagrangian view, shiftthe Eulerian location of the centroid of the packet on small scales as just described,but that shift would be meaningless insofar as the location of the packet as a wholewas concerned if it did not add up, over all small-scale waves, to something comparableto the uncertainty scale of the packet.)
The straight-line motion of Lagrangian packets in Lagrangian coordinates must notbe viewed too simplistically: the conditions for valid Lagrangian linearization maycome to be breached locally, and should be confirmed continuously (as should be
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Eulerian conditions in Eulerian analyses, of course). One interesting aspect of thistesting comes from consideration of the intrinsic vertical velocity of a packet.
The Eulerian vertical velocity of the packet is found by differentiating thethree-component of (C.3) with respect to t:
PE3 ¼ PL3 þ exp½ðr03 þ PL3tÞ=2H� ð�C ��SÞ, ðC:6Þ
where
�C � �ðPL3=2HÞSb3cfb3g, ðC:7Þ
�S � �ðPLEkb � !bÞSb3sfb3g, ðC:8Þ
s{b3} is the sine equivalent of c{b3}, and summations � are over all b. The local, wave-induced vertical velocity V3 of the fluid is found by differentiating the 3-component of(C.2) with respect to t at fixed r, evaluating the result at r¼ rP, and adding over all b:
V3 ¼ exp½ðr03 þ PL3tÞ=2H�X
Sb3!bsfb3g: ðC:9Þ
When this is subtracted from (C.6), it yields the intrinsic Eulerian vertical velocity of thepacket,
where �0S is given by �S with the !b terms removed:
�0S ¼ �PLEkb Sb3sfb3g: ðC:11Þ
For a packet propagating in the 1, 3 Lagrangian plane, this result may be writtenequivalently as
~PPE3 ¼ PL3ð1 þ ½S3, 3�PÞ þ PL1½S3, 1�P, ðC:12Þ
where commas indicate differentiation with respect to the numbered Lagrangian coor-dinate that follows them and a subscribed P again indicates evaluation at the currentlocation of the packet centroid. We take PL1 and PL3 > 0 for convenience.
Some interest must attend circumstances in which this intrinsic vertical velocityvanishes – circumstances that can lead to packet obliteration in simpler back-ground flows, and might in any event be thought to be required for ‘‘reflection’’ ofthe packet if the packet could be reflected. In the event that PL3hS3, 3i � PL1hS3, 1i, sothat the last term on the right may be ignored at most points of space-time, this intrinsicvertical velocity will tend to 0 whenever the packet finds itself at a location where and
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when S3,3¼� 1. Our basic analysis has had to assume that hS3, 3i � 1 for the generalapplicability of linearization, but this does not preclude the local occurrence of‘‘rogue’’ conditions such that S3,3¼� 1 here and there, now and then. The backgroundsystem of waves is itself locally unstable at such a point in space-time – its localRichardson number would be of order 1 – and one may indeed anticipate not only non-linearity of the test wave packet but also its obliteration as part of the interaction. In theopposite extreme, when the VL1 term is important, the intrinsic vertical velocity wouldtend to 0 where and when S3, 1 ¼ �PL3=PL1 � �j!P=Nj, where !P is the central fre-quency of the packet. This again requires a ‘‘rogue’’ condition, given our assumptionthat (3.21) of the main text is met, and again it implies local nonlinear interaction ofthe wave packet with the background. In this case, any corresponding assumption ofEulerian linearity would also be breached locally.
These considerations represent just one aspect of the inquiries that are opened up toexamination – and perhaps to improved understanding – by the Lagrangian form ofanalysis.
There is a price to pay for use of the Lagrangian method, in that it provides no directinformation on such things as the variations of Eulerian wavevector, intrinsic frequencyand wave-action density as the packet progresses along the ray path. It seems inevitablethat such information can be extracted indirectly, and the net process may well continueto be simpler than the standard procedure, but that possibility is not explored furtherhere. One might expect that some of the wanted quantities would depend on theJacobian of the transition between Eulerian and Lagrangian coordinates, but eventhat could be determined as a function of r, t from the basic relation x ¼ r þ ssðr, tÞand so should not cause a problem.
(d) Conflict of Eulerian and Lagrangian Predictions
It is virtually inconceivable that the ray paths determined by the two methods will agreeexactly, even when conditions for Eulerian legitimacy (e.g., adequate scale separation)obtain. This is seen, perhaps most simply, by consideration of a Lagrangian wavepacket that is launched in the 1-azimuth from the origin of coordinates at a timewhen the displacement there is zero (i.e., when the two origins of coordinates coincide).The centroid must remain in the 1,3 plane as viewed in Lagrangian coordinates.Thereafter, whenever it passes through a fluid parcel that happens to be undisplacedin the 2-azimuth as a consequence of some quite accidental superposition of wave dis-placements, it must lie at that moment in the 1,3 plane also as viewed in Eulerian coor-dinates. This is quite true irrespective of the path through Eulerian coordinates that thepacket had taken to reach that point. It seems beyond belief that an Eulerian wavepacket, however it might be constructed to match the initial flight of the Lagrangianpacket (having regard for the fluid velocity and Doppler shift at the starting point)could possibly produce this same behavior as a general characteristic. If it did, the con-clusion would constitute a most remarkable addition to the theory of integratedEulerian ray paths. But there is the further fact that Lagrangian wave packets followwell-defined paths even when scale overlap occurs, whereas Eulerian ray paths thenbecome indeterminate. And again, Lagrangian ray paths become indeterminate whenLagrangian linearity is breached, whereas Eulerian linearity may not be breached(locally) in the same circumstances. We assume, for the remainder of the discussion,that disagreement must in general occur.
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In these circumstances of conflict, one must ask which of the two methods, if either,gives the correct ray path. One might even ask whether there is such a thing as a correctray path. These are fundamental questions if the idea of eikonals, wave packets and raypaths is to be accepted and developed at all in the multiwave case.
A possible, but only partial, escape from the quandary is provided by the fact that awave packet is, by its very nature, an entity that is extended in space-time and so has aclassical uncertainty of location. It does not exist only at its centroid, and yet only thecentroid’s motion is examined. Perhaps the extended Lagrangian packet and theextended Eulerian packet always overlap to a degree that is consistent with the intrinsicuncertainty of their positions?
Perhaps. If so it would be worth knowing, if only to focus some attention on thedegree of uncertainty that must be associated with eikonal analysis in general. And,if so, then the Lagrangian method provides a much simpler means of finding the pack-et’s location to the accuracy that this location can be defined legitimately.
But then again, perhaps not. If not, then we must face the question: can eithermethod be taken to provide a valid ray path in present circumstances; and, if so,which? And how are we to judge?
In contrast to the case of wave packets propagating through a background shear-ing flow field that changes but slowly, if at all, here there are no laboratory experi-ments by which to determine the issue. Nor can a comparison of Eulerian andLagrangian computed ray paths provide illumination: the two sets of paths willdiffer, but how are we to know which is correct (if either)? A prior, necessarily emo-tional, commitment to one as opposed to the other will not suffice. We are left,then, with an examination of the underlying bases of the two, to see if either ismore suspect than the other.
The answer appears clear, at least to the present author. All who use the Euleriananalysis would agree that, if interaction between the waves of the wave packet andthe background waves could properly be ignored, then an Eulerian packet wouldfollow a straight line in Eulerian space as determined by the packet velocity. TheLagrangian analysis demands, as evidence of its validity, nothing other than thatthe same respect be granted it. And if that cannot be done, surely it is incumbent onthe skeptics to say why not.
The Eulerian analysis is not on nearly so firm a footing. Not only is interactionbetween the Eulerian waves admitted to occur, but it is the very essence of theEulerian analysis: were there no such interaction, the ray paths would never turn outto be sinuous. This interaction is inherently a nonlinear wave – wave interaction, givenbirth by the VEJ term of the Eulerian fluid-dynamic equations, and it carries with itall the complications that nonlinearity implies. These complications are avoided in theEulerian analysis largely by ignoring them – by shifting to a moving Eulerian coordinatesystem, to be sure, but thereafter ignoring the fact that, even in this new coordinatesystem, the Eulerian equations remain nonlinear.
Justification for the procedure is given, if at all, on the basis that certain criteria forsmallness of the space-time variations of the background system should be met. In prac-tice, these criteria are not always put to the test. But even if they were, the smallness ofneglected correction terms, evaluated locally, merely establishes that the standard(linear, Doppler-shifted) dispersion relation is approximately valid. This is inadequateto guarantee the smallness of errors that might nevertheless be accumulated over sub-stantial integration times. Even local evaluation is at risk. A standard requirement
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would be that the scale of variation of the background, in the direction of the k of thecentral test wave, should be large in comparison with the scale of variation of the cen-tral test wave in that same direction – the wavelength of the central test wave. This is, inessence, the WKB test of validity. But what about the scale of variation of the back-ground in directions perpendicular to that direction: how can it be large in comparisonwith the scale of variation of the central test wave in those directions, which for a planetest wave is infinite? And, if it is admitted that the test waves can no longer be plane inEulerian coordinates, then one must rethink the theory of wave packets: the motion ofthe point of constructive interference of curved constant-phase surfaces differs from themotion of the point of constructive interference of plane surfaces that happen to belocally tangential to the curved surfaces. Moreover, none of these checks addressesthe fact that the Eulerian system of equations remains nonlinear everywhere but inthe local region of choice.
In short, the Eulerian analysis is susceptible to cumulative errors and is inherentlyinternally inconsistent. No such charge can be made against the Lagrangian analysis.Readers may place their trust where they choose.
It will be recalled that there are incipient nonlinearities even in the Lagrangiansystem. When these occur accompanied by nonlinearities in the Eulerian system,wave-packet analysis in both systems must be considered suspect to the point of aban-donment. It is also possible for nonlinearities to become significant in the Lagrangiansystem alone. The relevant circumstances arise when, where and if the vertical gradientof the horizontal displacement induced by the background waves becomes as large as Ndivided by the central frequency of the wave packet – i.e., when (4.2) of the main text isbreached. In such circumstances, use of the Lagrangian wave-packet analysis wouldhave to be abandoned. These circumstances come close to matching, from the wavepacket’s point of view, the conditions provided by a steady shearing backgroundwind. The Eulerian analysis is not in question then, and the Lagrangian analysisposes it no threat.
What is at issue here, however, is the complementary situation, when the Euleriansystem is nonlinear – when the Eulerian test waves are affected by the backgroundwaves – but the Lagrangian system is not. Reciprocity would suggest that theEulerian form of analysis should then yield place to the Lagrangian. If not, why not?
