## 1. Introduction

In Alford and Zhao (2007, henceforth Part I), time series of depth-integrated energy *E* and flux **F** were computed at 80 historical moorings for the first two baroclinic modes. They describe a globally propagating field of near-inertial and semidiurnal internal waves. In this paper we seek more insight into these wave fields by examining the quantity **ĉ**_{g} ≡ **F***E*^{−1}, which for linear waves is equal to the group velocity **c*** _{g}*. Knowledge of the group velocity proves very powerful, because it yields internal-wave intrinsic frequency via the dispersion relation—enabling, for example, estimation of horizontal wavenumber from single-point measurements. Where the “true” intrinsic frequency is known (such as for the semidiurnal internal tides), comparison of

**ĉ**

_{g}with the theoretical value of

**c**

*provides a sensitive test of the linearity of the wave field.*

_{g}In addition, uni- versus multidirectional wave fields can be distinguished from single-point measurements, since multidirectional fluxes also cause **ĉ**_{g} to differ from **c*** _{g}*. Without rotation,

**ĉ**

_{g}= 0 for the extreme case of a perfect standing wave (two waves of equal amplitude propagating in opposite directions). However, away from the equator transverse velocities give rise to alternating bands of transverse flux (and hence nonzero group speed; Nash et al. 2004), complicating single-point measurements. Still, we show that the latitudinal dependence of the free and standing-wave curves is sufficiently different to distinguish between the two cases for latitudes equatorward of about 35°.

We explore these concepts for internal tides and near-inertial waves here. First the theory is presented; then specific examples are provided to demonstrate the technique. The latitudinal dependence of group speed is compared with that expected from theory. The excellent agreement in the case of the semidiurnal tides proves the validity of the moored flux and energy estimates, as well as indicating that 1) low-mode internal tides propagate according to linear dynamics and that 2) deep-ocean standing waves are relatively rare (at least for |latitude| < 35°). Both of these are vital aspects of the global internal-tide field and have implications for its dissipation. For near-inertial waves, various group velocities are typically seen (consistent with the presence of many groups, each having traveled from a different latitude and thus displaying a different frequency “blueshift”). A discussion follows.

## 2. Linear theory

*ω*and wavenumber

**k**propagates at the group velocity

**x**= 0, energy at a particular wavenumber is found at a distance

**c**

*rather than*

_{g}t**c**

*, where*

_{p}t**c**

*is the phase velocity.*

_{p}*ω*≪

*N*propagating through a Boussinesq fluid, the governing equations can be expressed as

*f*and

*N*(

*z*) are the inertial and buoyancy frequencies and

*ω*is the vertical velocity. Linear internal waves

*w*(

*x*,

*y*,

*z*,

*t*) =

*ŵ*(

*z*)exp[−

*i*(

**K**·

**x**−

*ω*

**t**)] propagating horizontally through a uniform medium exist only if the dispersion relation is satisfied,

**K**= (

*k*,

_{x}*k*) is the horizontal wavenumber and

_{y}*c*is the eigenspeed of the

_{n}*n*th mode.

**K̂**is the unit vector in the direction of wave propagation), so that

**c**

_{g}·

**c**

_{p}=

**c**

^{2}

_{n}(Alford et al. 2006, henceforth A06).

In the presence of advective nonlinearity, (2) is still valid when modified by the advective velocity. However, strong nonlinearity (as for internal solitary waves) is sufficient to invalidate (2), as well as the dispersion relation (4). Some subtleties in computing and interpreting **c*** _{g}* for a nonlinear wave field are discussed in Chang and Orlanski (1994).

Stratification and currents affect linear wave propagation, but do not invalidate (2), provided their scale is larger than that of the waves so the Wentzel–Kramers–Brillouin (WKB) approximation can be made (Vanneste and Shepherd 1998). Shear levels typical of the open ocean affect the modal shapes *ŵ*(*z*) only slightly (Pinkel 2000) and therefore do not significantly affect propagation of the low modes (Peters 1983). Stratification changes alter *c _{n}* (and thus

**c**

*and*

_{p}**c**

*; A06; Part I), and depth-independent currents modify the dispersion relation and therefore the expression for*

_{g}**c**

*(Rainville and Pinkel 2006, henceforth RP06). These corrections to*

_{g}**c**

*(≈5%–10%) are sufficient to cause substantial wave refraction (A06; RP06; Part I), but are small relative to the latitudinal dependence. Thus, comparison of observed latitudinal dependence with that expected from (5), as done here, constitutes a test of linear internal-wave theory.*

_{g}As described in Part I, mode-1 and -2 eigenspeeds **c*** _{n}* are calculated for each mooring location from climatological ocean stratification profiles from the

*World Ocean Atlas 1994*(Levitus and Boyer 1994). Group and phase speed are then computed from (5) and (6) for the tides assuming

*ω*=

*M*

_{2}. As shown in RP06’s Fig. 6, spatial dependence of group speed is mostly due to latitude, but also affected by stratification. Group speed is maximum at the equator, decreasing to zero at the turning latitude of 75°. Phase speed equals group speed at the equator (where waves are nondispersive), but increases to infinity at the turning latitude. These theoretical estimates are compared with our observational estimates of |

**ĉ**

_{g}| in the next section.

Wind-generated near-inertial internal waves, by contrast, have a frequency close to the inertial frequency at which they were generated. They then propagate toward the equator (Alford 2003; Part I), so that their observed frequency is greater than the local inertial frequency owing to the *β* effect. Their group velocity (which depends on the difference between *ω* and local *f* ) is therefore unknown, since their generation latitude is not generally known. Furthermore (as will be shown), it is expected to vary strongly over a yearlong record, since individual near-inertial events passing a particular mooring may originate from different latitudes. In the near-inertial case, we will compare our results with the group velocity expected for *ω*/*f* = 1.05, 1.15, 1.25, spanning frequencies typically observed in current-meter records (Fu 1981).

## 3. Observational estimates

### a. Semidiurnal band

The technique, first used in Alford et al. (2006), exploits 1) the strong connection between energy and energy flux and 2) variability in each quantity (Part I). It is demonstrated here for the semidiurnal case. All wave-period-averaged values of flux magnitude *F _{i}* are scatterplotted versus energy

*E*for each mode for a typical record in the northeast Pacific Ocean (Fig. 1, left panels). The probability density function (PDF) of the ensemble of individual |

_{i}**ĉ**

_{g}| estimates,

*F*/

_{i}*E*, is plotted at right.

_{i}Observed *E* and *F* values (gray dots) display considerable variability (owing to their spring–neap cycle as well as lower-frequency modulation), but are strongly correlated (as shown in Part I) for both modes 1 (upper) and 2 (lower). Their slope is close to the theoretical value from (5), indicated with the dashed line. This motivates calculation of |**ĉ**_{g}| as the slope of the bin-averaged quantity *dF*/*d E*. First, the mean energy (dots) and the standard deviation (horizontal bars) are estimated in each flux bin. Then, the slope (solid line) and its 95% confidence limits (right panels, upper right) are estimated by linear regression.

Noise-free energy and flux values for free waves of varying strengths would all fall on a line with the theoretical slope |**c*** _{g}*| (dashed line). The PDF of group velocity would be a

*δ*-function at the theoretical value (right panel, dashed line). By contrast, |

**ĉ**

_{g}| for perfect standing waves (appendix) varies spatially between 0 and |

**c**

*|*

_{g}^{standing}four times per wavelength, where |

**c**

*|*

_{g}^{standing}varies from zero at the equator to >

**ĉ**

_{g}at high latitude. The phase variations observed in Part I should shift the standing-wave pattern back and forth past the mooring, leading to a broader distribution of |

**ĉ**

_{g}| values, with a mean indicated by the thin line. The tight correlation between

*E*and

*F*and narrow peak of the PDF of |

**ĉ**

_{g}| are consistent with free waves, but at this latitude the expected mean value of |

**c**

*|*

_{g}^{standing}is too close to |

**c**

*| to distinguish between the two.*

_{g}At other locations, multidirectional fluxes appear to cause more contamination of the estimated group velocity. An extreme example from the North Atlantic Ocean is shown in Fig. 2. Here, the spread (gray dots, black error bars) and mean value of energy (black dots and line) are greater for each given flux than in the previous example, both as expected for partly standing waves. In addition, the binned quantity *E**F*) displays an *x* intercept, *E**F* = 0), nearly 2 times as large as in the previous example. Correspondingly, the observational estimate |**ĉ**_{g}| is only about one-half of that of the theoretical value; it is in fact much closer to the expected standing-wave value for this latitude (thin black). However, |**c*** _{g}*| still appears to limit the range of observed values (dashed line); few values are observed greater than the theoretical value, as seen in the PDF at right.

To examine the generality of these results, estimates at all locations are plotted versus latitude (Fig. 3, dots). Theoretical (free and standing) and observed values are computed for each site and plotted for the first (top) and second modes (bottom). The 95% confidence limits are plotted with vertical gray lines. Continuous curves are constructed for each mode by computing the mean of all values in each 5° latitude bin (dashed). Theoretical values |**c*** _{g}*| are also computed at each location (not shown), and their 5° binned average is plotted (gray). To extend the theoretical curves to latitudes lacking data, the theoretical group speed is also computed from climatological values along 220°E (black). The theoretical curves computed at 220°E and from the mean of the in situ values differ only slightly, indicating that zonal stratification differences are relatively unimportant. As discussed in the introduction, a maximum is seen at the equator near 2.8 m s

^{−1}, where

**c**

*=*

_{g}**c**

*=*

_{n}**c**

*(nondispersive waves), decreasing toward the poles (reaching zero at the turning latitude of 75°).*

_{p}The |**ĉ**_{g}| estimates (symbols) and their 5° mean (dashed) exhibit excellent agreement with the theoretical free wave curve for both modes at all latitudes. The degree of agreement is also seen by scatterplotting observed versus theoretical values at each location (Fig. 4). The observed group velocity at each site closely parallels the expected theoretical value. Fewer than 19% and 27% of mode-1 and -2 values, respectively, differ from the theoretical value by more than 25%.

Figure 4 indicates that |**ĉ**_{g}| rarely exceeds |**c*** _{g}*| by more than 20%. However, it is evident that some values, particularly near 20°–30°N, are significantly below it, and more in line with the value |

**c**

_{g}|

^{standing}expected for standing waves (appendix, thin black). As noted above, the magnitude of the standing-wave and free-wave predictions are too close to discern between them for latitudes poleward of about 35°.

Many of the low values occur in the North Atlantic, of which the second example shown (Fig. 2) is an example. A possible interpretation is that more multidirectional fluxes are present in that basin, consistent with the larger number of sources (Part I) and its smaller size. Even including these, sites exhibiting unidirectional fluxes (e.g., Fig. 1), and hence good agreement between |**ĉ**_{g}| and |**c*** _{g}*|, are much more common than highly multidirectional sites (e.g., Fig. 2). We interpret the resulting close replication of the expected latitudinal dependence for |latitude| < 35° as strong evidence that internal tides propagate 1) according to linear theory and 2) in only one direction at a given location and time. Poleward of this, |

**ĉ**

_{g}| is still consistent with linear theory, but standing and free waves cannot be distinguished.

As indicated in the introduction, knowledge of group velocity affords calculation of horizontal wavelength, via (5), from these single-point measurements (Fig. 5). Bin-averaged theoretical values *λ* = |**K**|^{−1} (where *K* is in cyclic units) at the mooring sites (dashed) and at 220°E (black) are ≈ 150–200 km for mode 1, and 50–60 km for mode 2. Without the effects of stratification, wavelength from (5) would increase to infinity at the turning latitude; the reduced stratification at high latitude mitigates this effect. [However, the WKB solutions break down close to the turning latitude, invalidating (5) and necessitating Airy function solutions (Munk 1981). These have finite horizontal wavelength.] The calculated values, *λ̂* (asterisks and circles), and their binned average (gray) follow the theoretical values reasonably well. The locations where multidirectional fluxes yield underestimated group velocity result in overestimated wavelength by similar factors (up to a factor of 2).

### b. Near-inertial band

A corresponding flux/energy plot for a typical near-inertial record at 39°N is shown in Fig. 6. Greater spread is seen than in the tidal case (Fig. 1). As indicated above, this is likely due to the lack of an expected single theoretical value: individual near-inertial packets’ group velocity depends on the distance from their generation site. Loci of points with given slopes can be identified with specific time periods, consistent with this interpretation. Evidently, these represent the passage of specific near-inertial “groups,” presumably generated by storms at different latitudes. In this record, specific events demonstrating group velocities for *ω* = 1.02*f*, 1.05*f*, and 1.2*f* are seen (dashed lines), spanning typically observed values (e.g., Fu 1981; Alford and Gregg 2001). The latitudes associated with these values of *ω*/*f* are 40°, 41.3°, and 49°, respectively—all within the storm track—and corresponding to distances of ≈100, 200, and 1100 km, respectively, from the mooring site. In principle, these events could be ray-traced backward in time and space to their generation sites, and identified with specific wind events (D’Asaro 1991; Garrett 2001; Alford and Gregg 2001). However, this has not been attempted here.

Annual-mean group speed can be estimated as in the semidiurnal case (gray lines). In contrast to the semidiurnal case, where a single group-velocity value is expected, the meaning of mean near-inertial group velocity estimated from a yearlong record containing many isolated events is not clear. A proper analysis would consider each event separately; this is a daunting task and has not been attempted. For comparison with the semidiurnal case, we ignore these concerns and present record-mean group velocity versus latitude (Fig. 7), as in Fig. 3.

The latitude dependence of the theoretical group velocity expected for *ω* = 1.02*f*, 1.05*f*, and 1.2*f* is plotted in black (mode 1) and gray solid curves (mode 2). The group velocity of a near-inertial wave generated at midlatitude and propagating toward the equator would cross these curves, since *ω*/*f* would increase with distance from its generation site.

As expected from the example shown, the |**ĉ**_{g}| values exhibit considerable scatter in both modes 1 (asterisks) and 2 (black dots). As just discussed, these values average over all “events” occurring in the flux and energy time series. The derived record-mean group-velocity estimates are thus related in a complicated way to a weighted average of the superinertiality of the ensemble of events in each record. The 5° mean group velocity (gray) is ≈1–1.5 m s^{−1} for mode 1 and 0.3–0.7 m s^{−1} for mode 2. The associated “mean superinertiality” is, subject to these caveats, 〈*ω*〉 ≈ 1.05–1.25*f* at all sampled latitudes (PDF in Fig. 8), as typically observed.

## 4. Discussion

We have presented observational estimates of mode-1 and -2 group velocity, for both the semidiurnal (Figs. 1, 2) and near-inertial bands (Fig. 6). In the former case, the high degree of correspondence between observed and theoretical values (Fig. 3) lends confidence in the flux and energy estimates presented in Part I. While it is conceivable that single moorings could demonstrate *FE*^{−1} ratios of the correct order of magnitude, the replication of the expected latitudinal dependence can only mean that the energy and flux signals are due to a linear internal tide.

For the near-inertial signals, we lack a single theoretical estimate, since their generation latitude is generally unknown. However, their display of reasonable values of superinertiality for each event shown suggests that they, too, are linear, as concluded by D’Asaro et al. (1995). The majority of energy input from storms has been found by these authors to escape far from the forcing region as low-mode near-inertial waves. Low signal-to-noise ratios have, however, prevented their observation (D’Asaro 1991). The present identification of low-mode near-inertial “packets,” propagating equatorward from storms at the expected speed, represents substantial progress. However, the ultimate fate of this energy is still unknown, and more work is needed.

Other conclusions about the internal-tide wave field can be drawn. RP06 computed maps of theoretical group velocity and performed ray-tracing exercises; this study provides observational support for the ideas presented there. Low-mode internal tides do, apparently, propagate at the group velocity expected from linear theory, as computed here. This conclusion is supported by Part I’s favorable comparison of arrival times of spring–tide groups, at locations north of the Hawaiian Ridge, with expected values.

The good agreement with theoretical group velocity is consistent with the strong refraction inferred in Part I from phase (arrival time) variability. Refraction does modulate **c*** _{g}* but is not expected to bias the mean value at a particular location. That is, we expect refraction to impact the standard deviation but not the mean value of group velocity—as observed.

The latitudinal dependence of observed group speed is inconsistent with that expected for low latitudes (<35°), indicating that open-ocean standing waves are relatively rare. That is, the energy flux at most locations is apparently nearly unidirectional at a given time. This has implications for the dissipation of the internal-tide wave field; specifically, energy radiating outward from sources evidently does not survive reflections from the oceans’ boundaries. If it did, many more standing-wave patterns would result. Furthermore, the flux maps presented in Part I would not demonstrate such directionality. This latter argument can also be made of the near-inertial field. Thus, though we are a long way from mapping dissipation of the internal-wave field, these results at least provide a glimpse into aspects of its long-range propagation.

## Acknowledgments

This work was supported by the Office of Naval Research Young Investigator award (Grant N00014-02-1-0526), and National Science Foundation Grant OCE0424717.

## REFERENCES

Alford, M. H., 2003: Energy available for ocean mixing redistributed though long-range propagation of internal waves.

,*Nature***423****,**159–163.Alford, M. H., and M. Gregg, 2001: Near-inertial mixing: Modulation of shear, strain and microstructure at low latitude.

,*J. Geophys. Res.***106****,**16947–16968.Alford, M. H., and Z. Zhao, 2007: Global patterns of low-mode internal-wave propagation. Part I: Energy and energy flux.

,*J. Phys. Oceanogr.***37****,**1829–1848.Alford, M. H., M. C. Gregg, and M. A. Merrifield, 2006: Structure, propagation and mixing of energetic baroclinic tides in Mamala Bay, Oahu, Hawaii.

,*J. Phys. Oceanogr.***36****,**997–1018.Chang, E. K. M., and I. Orlanski, 1994: On energy flux and group velocity of waves in baroclinic flows.

,*J. Atmos. Sci.***51****,**3823–3828.D’Asaro, E. A., 1991: A strategy for investigating and modeling internal wave sources and sinks.

*Dynamics of Oceanic Internal Gravity Waves II: Proc. ’Aha Huliko’a Hawaiian Winter Workshop*, Honolulu, HI, University of Hawaii at Manoa, 451–466.D’Asaro, E. A., C. E. Eriksen, M. D. Levine, P. Niiler, C. A. Paulson, and P. V. Meurs, 1995: Upper-ocean inertial currents forced by a strong storm. Part I: Data and comparisons with linear theory.

,*J. Phys. Oceanogr.***25****,**2909–2936.Fu, L-L., 1981: Observations and models of inertial waves in the deep ocean.

,*Rev. Geophys. Space Phys.***19****,**141–170.Garrett, C., 2001: What is the “near-inertial” band and why is it different from the rest of the internal wave spectrum?

,*J. Phys. Oceanogr.***31****,**962–971.Levitus, S., and T. Boyer, 1994:

*Temperature*. Vol. 4,*World Ocean Atlas 1994*, NOAA Atlas NESDIS 4, 117 pp.Lighthill, J., 1978:

*Waves in Fluids*. Cambridge University Press, 504 pp.Munk, W. H., 1981: Internal waves and small-scale processes.

*Evolution of Physical Oceanography*, B. A. Warren and C. Wunsch, Eds., MIT Press, 264–291.Nash, J., E. Kunze, K. Polzin, J. Toole, and R. Schmitt, 2004: Internal tide reflection and turbulent mixing on the continental slope.

,*J. Phys. Oceanogr.***34****,**1117–1134.Peters, H., 1983: The kinematics of a stochastic field of internal waves modified by a mean shear current.

,*Deep-Sea Res.***30****,**119–148.Pinkel, R., 2000: Internal solitary waves in the warm pool of the western equatorial Pacific.

,*J. Phys. Oceanogr.***30****,**2906–2926.Rainville, L., and R. Pinkel, 2006: Propagation of low-mode internal waves through the ocean.

,*J. Phys. Oceanogr.***36****,**1220–1236.Vanneste, J., and T. G. Shepherd, 1998: On the group-velocity property for wave-activity conservation laws.

,*J. Atmos. Sci.***55****,**1063–1068.

## APPENDIX

### Perceived Group Speed of a Standing Wave

*ω*and horizontal wavenumber

*k*, propagating along the

*x*axis in opposite directions. Each has energy flux, independent of spatial location, (1/2)

*p*

_{0}

*u*

_{0}(Fig. A1, gray), where

*p*

_{0}and

*u*

_{0}are the modal amplitudes of velocity and pressure. Along the

*x*direction, the energy flux (

*f*; the

^{ s}_{x}*s*superscript indicates a standing-wave quantity) will be zero everywhere. The spatial mean of

*f*is also zero, but Nash et al. (2004) showed that it has alternating northward–southward bands (Fig. A1a),

^{ s}_{y}*and KE*

^{s}*(Fig. A1b) are also spatially dependent (Nash et al. 2004), with offset maxima. The potential energy is*

^{s}*F*, KE, and PE leads to four maxima per wave cycle in our group velocity estimates

_{y}**ĉ**

_{g}(Fig. A1c),

*R*of the maximum observational group speed to the single-wave value is

A worst-case mooring situated in a perfect standing wave at the transverse flux maximum would thus estimate nonzero flux and hence nonzero group speed. At latitudes >≈28°, this maximum value exceeds the value expected for a free wave (Fig. A2, black). In the presence of mesoscale refraction, however, the spatial pattern moves back and forth past the mooring, leading to effective sampling of spatial phase. The observed ±90° phase changes (Part I) indicate that most moorings sample at least one-quarter spatial wave cycle, leading to an expected distribution between 0 and the maximum value, with mean value shown in Figs. A1c,d and A2 (dashed lines). If a 20% margin (similar to the error in our measurements) is needed to distinguish the free and standing cases, this implies standing and free waves at latitudes >35° cannot be distinguished via this method, though the former may still be identifiable via their broadened distribution.