## 1. Introduction

In his often-cited paper on the effects of geostrophic flow on near-inertial internal wave propagation, Kunze (1985, hereafter K85) showed that the geostrophic vorticity *ζ* shifts the lower bound of the internal wave band from the inertial frequency *f* to an effective inertial frequency *f*_{eff} = *f* + *ζ*/2, a result established earlier, albeit with more restrictive assumptions (Fomin 1973; Mooers 1975; Weller 1982). The intuitive explanation for this result is that “a wave experiences the fluid’s as well as Earth’s rotation” (K85). However, because Earth’s rotation rate Ω gives rise to inertial oscillations with frequency *f* = 2Ω sin*θ* (where *θ* is latitude), which is twice the rotation rate around the local vertical direction, one could naively expect geostrophic vorticity to shift the frequency by *ζ*, which is twice the rotation rate of fluid parcels relative to Earth. K85 argued that the *ζ*/2 shift results from the use of rectilinear instead of polar or spherical coordinates in his treatment of the problem. Kunze et al. (1995, hereafter K95) and Kunze and Boss (1998, hereafter KB98) used polar coordinates to treat the somewhat different problem of near-inertial waves trapped within the core of an axisymmetric vortex and found that the lower bound of the trapped internal wave band was shifted to *f* + *ζ*, seemingly confirming the explanation given by K85.

We find this explanation unsatisfactory, because the frequency shift of near-inertial waves in geostrophic flow should not depend on the type of coordinates used to solve the problem. The purpose of this note is to demonstrate that the fundamental difference between the results obtained by K85, K95, and KB98 arises in fact from the different rotation rates of the frames of reference chosen to describe the motions. Durran (1993) showed that, although inertial oscillations on the North Pole *f* plane have a frequency *f* when described in a reference frame rotating with Earth, they have a frequency *f*/2 when described in an inertial (nonrotating) reference frame. We extend in section 2 the approach of Durran (1993) to include a solid-body rotating vortex centered on the North Pole. This provides the simplest conceptual framework to understand the fundamental difference between the results obtained by K85, K95, and KB98, as discussed in section 3. Although it is more difficult to visualize inertial oscillations in alternative reference frames for an *f* plane at an arbitrary latitude than for the polar plane, there is no difference in the physics or the mathematics. An extension to arbitrary background flows is given in the appendix.

## 2. Inertial oscillations in solid-body rotating flow on the polar *f* plane

Let us investigate inertial oscillations on the North Pole *f* plane, in the presence of a mean solid-body rotating vortex centered on the North Pole, with constant relative vertical vorticity *ζ*. We use three different frames of reference to describe the motions: (i) an inertial (i.e., nonrotating) frame, (ii) a frame rotating with the vortex flow, and (iii) a frame rotating with Earth.

### a. Inertial reference frame

*d*/

*dt*)

_{I}is the material derivative in I (following the fluid parcel’s motion relative to I),

**x**is the position of the fluid parcel relative to the North Pole,

**F**

_{g}is the gravitational force per unit mass,

*ρ*

_{0}is density (assumed constant here), and

*p*is pressure (e.g., Gill 1982, p. 72).

**Ω**

_{a}= [(

*f*+

*ζ*)/2]

**k**, where

*f*= 2Ω is the Coriolis parameter at 90°N (assumed greater than |

*ζ*|) and

**k**is the vertical unit vector. The gravitational and pressure gradient forces therefore combine to yield the required centripetal acceleration,The free surface of the fluid is warped relative to the geoid, causing the familiar horizontal (i.e., along the geoid) pressure gradients responsible for geostrophic flows and allowing fluid parcels to rotate at a rate different from Earth’s (Fig. 1).

Suppose the circular motion of a fluid parcel around the North Pole described above is momentarily disrupted (e.g., by a sporadic wind event for a surface water parcel) but the force having caused the disruption subsequently vanishes. Assume the impulsive forcing is of large horizontal extent, so that subsequent horizontal pressure and velocity gradients are negligible. After the disruption, the fluid parcel’s inertial acceleration is again given by (2), and the new motion of the fluid parcel is a counterclockwise ellipse with frequency (*f* + *ζ*)/2. This is the motion of inertial oscillations in solid-body rotating flow, as seen by an observer fixed in I.

### b. Reference frame rotating with the vortex flow

**Ω**

_{a}relative to I. Given any two frames of reference, A and B, where the angular velocity of B relative to A is

**, the rates of change of any vector**

*ω***Q**in A and B are related byLetting A be the inertial frame I and B the rotating frame F and applying (3) first to the position vector,

**Q**=

**x**, and then to the velocity vector in I,

**Q**= (

*d*

**x**/

*dt*)

_{I}, yields the familiar expressionwhere

**V**

_{F}= (

*d*

**x**/

*dt*)

_{F}is the velocity of inertial oscillations in F. Combining (4) and (2) yieldsHere, (

*d*/

*dt*)

_{F}can be considered as the material derivative or the Eulerian derivative in F, because in the inertial limit the velocity field has no horizontal gradient.

Equation (5) is similar to that satisfied by inertial oscillations in a background flow at rest with respect to Earth, except that *f* is replaced by *f* + *ζ* here. The motions of fluid parcels are clockwise circles, with frequency *f* + *ζ* corresponding to the angular frequency of the velocity vector at a fixed point in F or following the fluid parcels. This is the motion of inertial oscillations in solid-body rotating flow, as seen by an observer fixed in F. The frequency measured in F is twice that measured in I and has the opposite sign (clockwise versus counterclockwise), because an observer fixed in F follows a circle, whereas a fluid parcel follows an ellipse, both at the same frequency relative to I. This is illustrated for *ζ* = 0 in Fig. 2, in Durran (1993, Fig. 3), and in Stommel and Moore (1989, Fig. 3.1).

### c. Reference frame rotating with Earth

**V**

_{F}, which is a function of time but not of position. Therefore, the measured inertial oscillation is still just

**V**

_{F}but viewed from the new reference frame E. As E rotates at −

*ζ*/2 relative to F, the application of (3) with

**Q**=

**V**

_{F}yieldsCombining (6) and (5) yieldsHere again, (

*d*/

*dt*)

_{E}can be considered in the inertial limit as the material derivative or the Eulerian derivative in E. The trajectories of fluid parcels are clockwise circles, with frequency

*f*+

*ζ*/2 corresponding to the angular frequency of the velocity vector at a fixed point in E or following the fluid parcels, superimposed on the mean vortex flow. This is the motion of inertial oscillations in the presence of solid-body rotating flow, as seen by an observer fixed in E. The frequency measured in E is

*ζ*/2 less than that measured in F, because E rotates at −

*ζ*/2 relative to F, and the frequencies correspond to the angular frequencies of the velocity vector relative to the rotating frames of reference.

## 3. Discussion

The main result of the previous section is that the trajectory and frequency of inertial motions depend on the rotation rate of the frame of reference chosen to describe them. It does not depend on the type of coordinates (e.g., Cartesian versus polar), as shown by our exclusive use of vectorial notation in section 2. There are three key points to understand the different frequencies measured in the three different frames of reference considered. First, inertial motions on the polar *f* plane in the presence of a solid-body rotating vortex centered on the North Pole are counterclockwise ellipses with frequency (*f* + *ζ*)/2 in an inertial reference frame, due to the warping (relative to the geoid) of the free surface associated with the vortex flow, which causes fluid parcels to rotate at a rate different from Earth’s (Fig. 1). This is the fluid’s rotation “felt” by the inertial motions. Second, inertial motions appear as clockwise circles with frequency *f* + *ζ* in a reference frame rotating with the vortex flow at angular velocity (*f* + *ζ*)/2, because a rotating observer follows a counterclockwise circle at the same frequency as the fluid parcel’s relative to I. This is analogous to the case of inertial oscillations in a background flow at rest with respect to Earth (Fig. 2). Third, noting that, in the inertial limit, the frequency in the rotating reference frame corresponds to the rotation rate of the inertial current vectors relative to that frame [Eq. (5)], it is straightforward to obtain a frequency of *f* + *ζ*/2 in a reference frame rotating with Earth at −*ζ*/2 relative to the reference frame rotating with the vortex flow.

Following Durran (1993), we restricted the theoretical treatment of section 2 to the North Pole *f* plane for simplicity. The results remain unchanged for *f* planes at any other latitude but are more difficult to visualize. The reference frame with zero angular velocity component in the local vertical direction is not an inertial frame, because it has a northward (i.e., horizontal) angular velocity component of Ω cos*θ*. Insofar as all motions remain horizontal, however, this horizontal component of angular velocity has no effect. Restricting the discussion to *f* planes also neglects the important effects of the meridional variations of *f* on the propagation of near-inertial oscillations (e.g., Garrett 2001), which are beyond the scope of this short note. Indeed, we have restrained for simplicity our treatment to the inertial limit, in which inertial oscillations do not propagate.

The case of a solid-body rotating vortex provides the simplest conceptual framework to investigate the effect of background flow on inertial oscillations. This is a very particular case, however, because the strain rate of the flow is zero. Inertial oscillations in an arbitrary background flow have their frequency affected by both the background vorticity and strain rate (see appendix): the effect of the background vorticity on the inertial frequency is

K95 and KB98 described near-inertial motions trapped within the solid-body rotating core of an axisymmetric vortex and obtained an effective inertial frequency of *f* + *ζ* “in a mean Lagrangian frame following the core’s rotation” (K95)—that is, a reference frame rotating with the vortex flow—consistent with the results of section 2b. In contrast, K85 described near-inertial motions in rectilinear geostrophic flow and obtained an effective inertial frequency of *f* + *ζ*/2 in a mean Lagrangian frame translating, but not rotating, with the geostrophic flow—that is, a reference frame rotating with Earth only—consistent with the results of section 2c. This consistency is expected because the inertial limit we have considered here is a nonsingular limit of the more general internal wave propagation problem considered by K85, K95, and KB98. When observations are analyzed in a reference frame rotating with Earth, K85’s treatment is relevant. For example, Elipot et al. (2010) tracked surface drifters through geostrophic currents inferred from satellite altimetry in the global ocean and found a frequency shift of ~0.4*ζ*, closer to K85’s result than to K95’s and KB98’s.

The ideas presented here were developed and refined over the course of many years, during which C. C. was supported by NSF Grant OCE-0426112 (P.I. P. Flament), NERC Antarctic Funding Initiative Grant AFI8/17 (P.I. K. Heywood), and a Starting Fund from ISMER. We thank two anonymous reviewers for their positive comments on an earlier version of the manuscript.

# APPENDIX

## Inertial Oscillations in Arbitrary Background Flow

**u**in a mean background flow

**u**) limit. Let us solve the problem on the

*f*plane using Cartesian coordinates. Defining the vorticity

**u**=

**u**

_{0}exp(

*iωt*) yields the dispersion relationwhere

*σ*

^{2}< 4(

*f*+

*ζ*/2)

^{2}, and the frequency of inertial oscillations is affected by the background vorticity and strain rate, whereas the background divergence causes exponential growth or decay of the solution.

*ζ*,

*σ*) = (

*U*/

*L*)(

*ζ**,

*σ**), where

*U*and

*L*are typical velocity and horizontal scales characterizing the background flow. The second term on the rhs of (A4) becomeswhere Ro =

*U*/

*fL*is the Rossby number of the background flow. The strain rate only affects the frequency to

*f*

_{eff}=

*f*+

*ζ*/2. The expression for the effective inertial frequency given in (A5) differs from those typically given in the literature (e.g., Fomin 1973; Weller 1982; Kunze 1985; with the typographical correction given by Jones 2005) by emphasizing the fact that the inertial frequency is only affected by the background vorticity and strain rate. Fomin (1973) had the square of the background flow divergence explicitly appearing in his expression for the effective inertial frequency [his Eq. (15)] and concluded that “the divergence always lowers the frequency of the inertial velocity oscillations.” It turns out that the

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