## 1. Introduction

There has been a recent revival of interest in formulating a theory for internal gravity waves in a Lagrangian frame of reference, rather than the usual Eulerian frame of reference. Following the work of Allen and Joseph (1989), Hines (2001), and Chunchuzov (2002) have argued that the nonlinear terms in the Lagrangian form of the equations are, in some cases, of less importance than the corresponding nonlinear terms in the Eulerian form of the equations. In effect, the implication is that simply using a coordinate transformation from the Eulerian frame to the Lagrangian frame can remove some of the effects of nonlinearity. This is an interesting, albeit provocative, suggestion. It is of course true that, in some special cases, a system of partial differential equations can be totally or partially explicitly linearized by a coordinate transformation, but this is not so in general for the transformation from Eulerian to Lagrangian coordinates.

The above studies are concerned with an internal wave field in a background composed of other internal waves. In each of these studies, the Lagrangian internal wave dispersion relation is taken to have the same functional form as the well-known Eulerian internal wave dispersion relation, but without the Doppler shifting term [see, for instance, (2.3) of Allen and Joseph (1989) or (59) below]. Hence, such a Lagrangian dispersion relation loses a crucial dependence on the background flow and, in particular, cannot account for refraction by the background flow. Thus, in particular, Hines (2002) has suggested that under appropriate conditions (see section 5), there are consequent simplifications for the ray tracing of internal waves if the rays are expressed in Lagrangian coordinates.

These issues have motivated us to reexamine the structure of the internal wave dispersion relation in a Lagrangian reference frame vis-à-vis the corresponding internal wave dispersion relation in an Eulerian reference frame. Hence, in this paper, we explicitly and systematically derive the Lagrangian dispersion relation for small-amplitude internal waves in the presence of a general background flow of finite amplitude. For simplicity, we consider two-dimensional flow and ignore the effects of the earth's rotation. Two derivations are given, which lead to the same result. The first involves a transformation of the dispersion relation from Eulerian to Lagrangian coordinates (section 3a), and the second involves a wave-packet asymptotic analysis of the Lagrangian equations of motion (section 3b). A general, slowly varying, space- and time-dependent background is treated first. Then, for illustration, we specialize to a horizontal background flow that varies only in the vertical direction (section 4). The main conclusion from our analysis is that there is a complete equivalence between the Eulerian and Lagrangian wave dispersion relations and that internal wave rays that refract in an Eulerian frame will refract in a Lagrangian frame, and vice versa.

## 2. The Eulerian dispersion relation

The Eulerian coordinates are **x** = (*x,* *z*) with *z* vertical. We use an overbar to denote quantities associated with the background. For example, the background velocity **u****x**, *t*) = (*u**w**ρ***x**, *t*). The background flow is incompressible and satisfies the equations for conservation of density, but we allow for the possible presence of arbitrary body forces in the momentum equations to maintain the background flow.

*u,*

*w,*

*ρ,*and pressure

*p*are

*g*is the acceleration due to gravity. In the momentum equations {⋯} denote terms that can be neglected at leading order since they involve derivatives of

*u*

*w*

*X*=

*ϵx,*

*Z*=

*ϵz,*

*T*=

*ϵt,*where

*ϵ*is a small parameter characterizing the slow variation of the background with respect to the wave field, so that

*u*

*u*

*X,*

*Z,*

*T*), etc. The derivatives are then

*O*(

*ϵ*) with respect to the wave frequency. But it is important here to note that

*g*

*ρ*

_{x}/

*ρ*

*g*

*ρ*

_{z}/

*ρ*

*O*(1) quantities with respect to the square of the wave frequency. The reason for this is that for these internal waves the wave frequency scales with (

*g*/

*H*)

^{1/2}, where

*H*is a scale height for the density stratification. To avoid excessive notation, we will not formally introduce these slow variables into the derivation, although their presence is always understood.

*u*

*ψ*

_{z}

*w*

*ψ*

_{x}

*ψ,*

*ρ*

*a*

**x**

*t*

*e*

^{iθ(x,t)}

*a*is the amplitude,

*θ*is the phase for a wave packet, and c.c. denotes complex conjugate. More formally,

*a*=

*a*(

*X,*

*Z,*

*T*) is slowly varying on the same scale as the background, whereas the phase

*θ*=

*θ*(

*X,*

*Z,*

*T*)/

*ϵ*is rapidly varying. The Eulerian wavenumber and frequency are then defined by

*ω̂*

^{2}

*k*

^{2}

*N*

^{2}

*kmM*

^{2}

*k*

^{2}

*m*

^{2}

*ω̂*

*ω*

*k*

*u*

*m*

*w*

*N*

^{2}= −

*g*

*ρ*

_{z}/

*ρ*

*M*

^{2}= −

*g*

*ρ*

_{x}/

*ρ*

*M*= 0.

*ρ*

*ρ*

*ρ*

_{0}(

*z*−

*ζ*

*ρ*

_{0}(

*z*) is the undisturbed density field (in the absence of any flow) and

*ζ*

*ζ*

_{t}

*u*

*ζ*

_{x}

*w*

*ζ*

_{z}

*w*

*ρ*

_{0}with respect to its argument (

*z*−

*ζ*

*ω̂*

^{2}

*N*

^{2}

*kk̂*

*k*

^{2}

*m*

^{2}

*N*=

*N*(

*z*−

*ζ*

*ω*=

*ω*(

**k**,

**x**,

*t*) in (15) and are given by

*d*/

*dt*is the rate of change following the ray at the local group velocity ∇

_{k}

*ω*= (

*ω*

_{k},

*ω*

_{m}). Along an Eulerian ray, we also have

*dω*/

*dt*=

*ω*

_{t}.

## 3. The Lagrangian dispersion relation

**x**′ = (

*x*′,

*z*′), the equations of motion are (e.g., Lamb 1932)

### a. Derivation by transformation of the Eulerian dispersion relation

This derivation is based on the mapping between Eulerian and Lagrangian coordinates, whose Jacobian is given in (21). The Lagrangian momentum equations (19)–(20) are not explicitly used in this derivation.

*ξ,*

*ζ,*

*ξ*

*ζ*

*ξ*

*ζ*

*ξ,*

*ζ*would invalidate the linearization procedure as applied in both the Eulerian and Lagrangian frames. In particular, their retention would lead to their occurrence in the Lagrangian dispersion relation, clearly inconsistent with a linear theory. For the rest of this subsection, we therefore neglect

*ξ,*

*ζ*in (22)–(23).

*ξ*

*ζ*

*t*= 0. Formally, the background displacements depend on the slow variables

*X*′ =

*ϵx*′,

*Z*′ =

*ϵz*′,

*T*=

*ϵt,*which are the Lagrangian counterparts of the Eulerian slow variables

*X*=

*ϵx,*

*Z*=

*ϵz,*

*T*=

*ϵt*defined in section 2. Note that the displacements are formally of

*O*(1/

*ϵ*), although of course their first derivatives are

*O*(1). As in the Eulerian case, we will not explicitly use these slow variables here, but their presence is always understood.

*k,*

*m,*

*ω*but also the Eulerian derivatives

*ζ*

_{x},

*ζ*

_{z}that occur in the definition of

*k̂*in (16). Starting with this latter transformation we first note that

*k̂*in (16) becomes

*k̂*

*k*

*m*

*ζ*

_{x′}

*k*

*ξ*

_{x′}

**k**,

*ω.*The Lagrangian wavenumber and frequency will be denoted by

**k**′,

*ω*′. We start with the phase

*θ*in (8), which transforms from Eulerian to Lagrangian coordinates according to

*θ*

**x**

*t*

*θ*

**x**

**x**

*t*

*t*

*θ*

**x**

*t*

_{x′}, ∂

_{z′}), we have

**k**

*θ*

*ω*

*θ*

^{′}

_{t}

*θ*

^{′}

_{x′}

*θ*

_{x}

*x*

_{x′}+

*θ*

_{z}

*z*

_{x′}and a similar expression for

*θ*

^{′}

_{z′}

**k**

^{T}

**k**

*k*′ =

*k̂*; that is,

*k̂*is in fact the Lagrangian horizontal wavenumber.

*ω̂*

^{2}

*N*

^{2}

*kk*

*k*

^{2}

*m*

^{2}

*ω̂*

*k,*

*m*from (39)–(43) gives the Lagrangian dispersion relation

*ξ*

*ζ*

*ω*′ =

*ω*′(

**k**′,

**x**′,

*t*) in (45), the Lagrangian ray equations are

*dω*′/

*dt*=

*ω*

^{′}

_{t}

*x*′,

*z*′.

### b. Derivation by an asymptotic analysis of the Lagrangian equations

*ξ,*

*ζ*

*a*

**x**

*t*

*e*

^{iθ′(x′,t)}

**k**′ = ∇′

*θ*′ and

*ω*′ = −

*θ*

^{′}

_{t}

*ξ*or

*ζ*from these equations, we obtain

*ω*

^{2}

*N*

^{2}

*kk*

*k*

^{2}

*m*

^{2}

## 4. A steady horizontal background flow

*u*

*ξ*

*u*

*z*

*t,*

*ζ*

*k*

*k*

*m*

*m*

*k*

*u*

_{z′}

*t.*

*ω*

^{2}

*k*

^{2}

*N*

^{2}

*k*

^{2}

*m*

*k*

*u*

_{z′}

*t*

^{2}

The background shear *u*_{z′} is small compared with *ω*′since, when expressed in terms of the slow variables, it is *ϵ**u*_{Z′}. This does not imply, however, that we can neglect *k*′*u*_{z′}*t* in comparison with *m*′ in (58). Indeed, in terms of the slow variables this term is *k*′*u*_{Z′}*T* and is *O*(1) with respect to the small parameter *ϵ.* Alternatively we could estimate *k*′*u*_{z′}*t* as *k*′*u**c*_{g}. Here *c*_{g} is the vertical component of group velocity (which for this problem has the same value in both the Eulerian and Lagrangian frames). We have used the estimates that along a ray path *u*_{z′}*z*′ ∼ *u**z*′/*t* ∼ *c*_{g}. Thus, *k*′*u*_{z′}*t* is an *O*(1) quantity and of the same order as *m*′.

It is clear from the study of Hartman (1975) that the term *k*′*u*_{z′}*t* in (58) can be important for the description of rays in the Lagrangian frame. He considers the particular case of constant *N* and constant background shear *u*_{z′}. Then *m*′, as well as *k*′, are constant along a Lagrangian ray, but *ω*′ is not constant. This follows from the Lagrangian dispersion relation (58), which depends explicitly on *t* but not on *z*′ when *u*_{z′} is constant. For this particular case, the Lagrangian rays are straight lines in (*x*′, *z*′), but the group velocity changes along the ray due to the change in *ω*′. The Lagrangian ray reflects from the same turning point height (where *ω*′ → *N*) and asymptotes toward the same critical layer height (where *ω*′ → 0) as for the corresponding Eulerian ray.

Note that the coordinates used by Hartman (1975) are Lagrangian in the sense that we have defined here. Hartman's coordinates are *x*′ = *x* − *u**z*)*t* and *z*′ = *z,* that is, the position that moves with the velocity of the background flow. As the notation indicates, these are the same as our Lagrangian coordinates *x*′ = *x* − *ξ**z*′ = *z* − *ζ*

## 5. Discussion

*ω*

^{2}

*N*

^{2}

*k*

^{2}

*k*

^{2}

*m*

^{2}

*ξ*

*ζ*

Allen and Joseph (1989) used Lagrangian coordinates to describe a spectrum of internal waves, with each wave component of the Lagrangian spectrum satisfying (59). When this Lagrangian spectrum is transformed to the Eulerian frame, the resulting high-wavenumber components of the Eulerian spectrum were found to be nonwavelike. That is, these components did not satisfy the Eulerian dispersion relation (15). Our work suggests a possible factor contributing to this result, in that Allen and Joseph's Lagrangian dispersion relation (59) and the Eulerian dispersion relation (15) are not equivalent to each other under the Lagrangian-to-Eulerian coordinate mapping except in the limit of no background shear. For these high wavenumbers, this limit is not appropriate because the high wavenumbers are subjected to strong background shears resulting from the longer waves. A comprehensive discussion of this issue of the high-wavenumber Eulerian spectrum can be found in the recent review by Fritts and Alexander (2003).

Hines (2002) examines another issue. He considers a “test wave” in a background of waves and asks the question: when can the interaction of the test wave with the background be ignored? Hines then finds that the criteria for neglecting the interaction with the background depends on the test-wave wavenumber in different ways according to whether one uses an Eulerian or a Lagrangian coordinate system (see Fig. 1 of Hines 2002).

In the Lagrangian frame, the Hines criteria for negligible interaction with the background involve his quantities *S*_{1,1}, *S*_{1,3}, *S*_{3,1}, *S*_{3,3}, which correspond respectively to our quantities *ξ*_{x′}, *ξ*_{z′}, *ζ*_{x′}, *ζ*_{z′}. The conditions on *S*_{1,1}, *S*_{1,3} in Hines' (3.17), (3.22), correspond here to |*ξ*_{x′}| ≪ 1 and |*ξ*_{z′}| ≪ |*m*′/*k*′|. The conditions on *S*_{3,1}, *S*_{3,3} in Hines' (3.21), (3.18), correspond here to |*ζ*_{x′}| ≪ |*k*′/*m*′| and |*ζ*_{z′}| ≪ 1. Under these conditions, the background terms in our Lagrangian dispersion relation (45) become small, so in this respect our work is consistent with that of Hines (2002).

In an Eulerian frame of reference, the corresponding conditions under which the background flow can be ignored are readily obtained from (15). Thus, we now require the conditions under which *ω* ≈ *ω̂***k** · **u***ω̂**k̂* ≈ *k.* On using the definitions (16) and (15), these can be expressed in the form |**u***N*|*k*|/|**k**|^{2}, |*ζ*_{x}| ≪ |*k*/*m*| and |*ζ*_{z}| ≪ 1. Again, these conditions are consistent with those obtained by Hines (2002) for the neglect of the effects of the background in an Eulerian reference frame. However, we point out here that the criterion |**k** · **u**| ≪ *ω̂***k** · **u** along a ray. When this is done, the criterion is replaced by |*δ***u**| ≪ *N*/|**k**|, where *δ***u** is a measure of the spatial variability in **u**. Further, we also need to ensure that *N*(*z* − *ζ**ζ**ζ*_{x}| ≪ 1 and |*ζ*_{z}| ≪ 1, or that *N* itself is approximately constant.

Hines (2002) argues that there is an advantage in using a Lagrangian frame when, as judged by the above criteria, the interaction between the Lagrangian ray and the background is weak (see appendix C of Hines 2002). But note that the above criteria are local and wavenumber dependent. They can be used to describe the conditions under which the refraction is *locally* minimal, either in an Eulerian frame on the one hand or in a Lagrangian frame on the other hand. However, the presence of variations in the background, no matter how weak, will cause some refraction and, since the criteria for minimal refraction are wavenumber dependent, it is not clear whether the refraction can necessarily remain minimal along a ray over the time scales of interest. Indeed Hines (2002, p. 26) notes that such conditions for minimal refraction may be breached locally and need to be continually confirmed. On the other hand, our present formulation based on the full Lagrangian dispersion relation (45) is not subject to any such restriction. Numerical ray tracing based on the full Lagrangian dispersion relation (45) might help to resolve this question, and to test the criteria derived by Hines (2002) and described above.

## Acknowledgments

We thank C. Hines for his comments. S.D.E. and D.B. acknowledge support for this research from the Office of Naval Research and from NASA's Office of Space Science through the Geospace Science Program Grant W19862. D.B. also received support from the National Science Foundation Grant OCE-0117869.

## REFERENCES

Allen, K. R., and R. I. Joseph, 1989: A canonical statistical theory of oceanic internal waves.

,*J. Fluid. Mech***204****,**185–228.Chunchuzov, I., 2002: On the high-wavenumber form of the Eulerian internal wave spectrum in the atmosphere.

,*J. Atmos. Sci***59****,**1753–1774.Fritts, D. C., and M. J. Alexander, 2003: Gravity wave dynamics and effects in the middle atmosphere.

*Rev. Geophys.,***41,**1003, doi: 10.1029/2001RG000106.Hartman, R. J., 1975: Wave propagation in a stratified shear flow.

,*J. Fluid Mech***71****,**89–104.Hines, C. O., 2001: Theory of the Eulerian tail in the spectra of atmospheric and oceanic internal gravity waves.

,*J. Fluid Mech***448****,**289–313.Hines, C. O., 2002: Nonlinearities and linearities in internal gravity waves of the atmosphere and oceans.

,*Geophys. Astrophys. Fluid Dyn***96****,**1–30.Lamb, H., 1932:

*Hydrodynamics*. Cambridge University Press, 738 pp.Lighthill, M. J., 1978:

*Waves in Fluids*. Cambridge University Press, 504 pp.