## 1. Introduction

*t*

*D*

*Dt*

*w*

*z.*

*w*∂/∂

*z*term, are present as well as those associated with the intrinsic

*D*/

*Dt.*This Doppler shifting can be reduced by examining quantities in a semi-Lagrangian or isopycnal-following frame (Pinkel and Anderson 1997). The resultant spectra are much more indicative of intrinsic processes (Anderson 1993). For example, Kunze et al. (1990) found that observed shear frequency spectra could be almost entirely explained by vertically advected near-inertial motions.

Typically, the intrinsic and advective motions have broad and overlapping frequency ranges. A common “narrowband” situation occurs when near-inertial shear layers are advected by tidal displacements. But, in midlatitudes the nearness of the inertial and tidal frequencies makes even this situation difficult to interpret.

This paper seeks to provide insight into fine-structure contamination by demonstrating the mechanism in a narrowband (two wave) situation. Measurements of velocity, shear, and isopycnal displacement at 6.5°S are presented, where the Coriolis frequency (*f* = 1/4.4 cpd) is well below the dominant (*K*_{1}, diurnal) tidal frequency. This spectral separation allows the tidal heaving of the near-inertial shear layers to be seen clearly. The mechanism is evident in both depth–time maps and Eulerian frequency spectra, which show sharp peaks at *f* ± *K*_{1}. The time series and spectra are well modeled by a simple kinematic two-wave model. As expected, the shifted peaks are absent when the shear is examined in the isopycnal-following frame.

The theoretical framework and the two-wave model are described in the next section, followed by a brief description of the data collection and processing in section 3. Depth–time maps and frequency spectra of shear are presented and compared to model predictions in section 4. Conclusions follow.

## 2. Theory

### a. Semi-Lagrangian frame

*ρ*(

*z*)

*ρ.*The coordinate

*ζ*defines the semi-Lagrangian (s-L) or isopycnal-following frame, and, in the at-rest ocean, is identical to the Eulerian depth

*z.*Both coordinates are taken positive downward.

*z*

*ρ,*

*t*

*ζ*

*ρ*

*η*

*ρ,*

*t*

*η*(

*ρ,*

*t*) is the instantaneous upward displacement of the isopycnal with density

*ρ.*Now, an Eulerian measurement of shear (say) at a particular depth records the shear on whatever isopycnal is presently at that depth. This is the essence of fine-structure contamination. Equation (3) is the mathematical equivalent of this statement, and relates the s-L and Eulerian frames. Transformation to the s-L frame removes the variability associated with vertical structure that is swept past fixed-depth sensors by isopycnal displacements.

### b. Two-wave tidal heaving model

^{1}The rotary velocity field,

*w*

_{I}≡

*u*+

*iυ,*of a near-inertial wave (frequency

*ω*

_{o}≈

*f*) is given in the isopycnal-following frame by

*w*

_{I}

*ζ,*

*t*

*w*

_{o}

*e*

^{i(ωot+moζ)}

*m*

_{o}and

*ω*

_{o}positive,

^{2}the real and imaginary parts of

*w*

_{I}yield the zonal and meridional velocity components of a counterclockwise-polarized (Southern Hemisphere) inertial wave with upward phase propagation (Fig. 1, gray lines).

^{3}

*η*

_{o}displaces the isopycnals at the diurnal frequency,

*K*

_{1}:

*η*/∂

*z*= 0), and therefore

*η*(

*t,*

*z*) =

*η*(

*t,*

*ζ*).

*w*

_{I}

*z,*

*t*

*w*

_{o}

*e*

^{i(ωot+moz)}

*e*

^{imoη}

*m*

_{o}

*η*

_{o}/2 ≪ 1), then

*w*

_{I}

*z,*

*t*

*w*

_{o}

*e*

^{i(ωot+moz)}

*im*

_{o}

*η*

*ω*

_{o}±

*K*

_{1}. The ratio of the spectral amplitude, Φ, of the Doppler-shifted peaks (last two terms) to that of the original (first term) is then

*ζ*/∂

*z*= 1 + ∂

*η*/∂

*z.*For small strain, ∂

*η*/∂

*z*≪ 1, then ∂

*w*

_{I}/∂

*z*≈ ∂

*w*

_{I}/∂

*ζ,*and Eulerian and semi-Lagrangian shear differ only via advection and not by straining. The analysis for shear is thus identical to that for velocity.

## 3. Data

Data for this study was collected during October/November 1998 aboard the Indonesian Vessel *Baruna Jaya IV,* as part of the ARLINDO Microstructure experiment. The ship repeated 36-km legs centered at 6.5°S, 128°E, in the central Banda Sea. The goal of the study was to quantify the mixing occurring in the Indonesian Throughflow. These results are described in Alford et al. (1999).

The shear field was dominated by a downward-propagating near-inertial wave of amplitude *S*_{max} ≈ 0.02 s^{−1}, which was responsible for three-fourths of the observed mixing (Alford and Gregg 2001). Specifically, dissipation rate and diapycnal diffusivity were both coherent at the 95% confidence level with inertial-band shear and Froude number.

Diurnal tidal currents (0.11 m s^{−1}) and displacements (9 m) were larger than their semidiurnal counterparts (0.09 m s^{−1} and 4 m). Both constituents were very low wavenumber (as often observed), resulting in very weak tidal shear.

### a. Isopycnal displacement data

Isopycnal displacement data were obtained from the Modular Microstructure Profiler (MMP), a loosely tethered vehicle equipped with microstructure and Sea-Bird CTD sensors. Profiles, spaced 20 minutes apart, were made to 300-m depth in 4-hour bursts separated by 2-hour gaps. To form a continuous time series, these data were interpolated onto a 5-m, 3-h grid, with each temporal bin containing 2 hours of data and a 1-hour gap. Isopycnal displacement was computed by first selecting a set of evenly spaced isopycnal surfaces, based on the cruise-mean density profile. Then, the instantaneous depth of each was computed from each density profile by linear interpolation.

### b. Velocity and shear data

Velocity and shear records were obtained using an RDI 150-KHz broadband acoustic Doppler current profiler mounted on the ship's bottom. Low scattering strength limited the device's useful range to about 160 m (quantitative reliability begins to fail sooner, at 120 m). The vertical resolution was 4 m. Shear estimates were computed by differencing over 8 m. Finally, velocity and shear were averaged over the same 3 hours, and interpolated onto the same 5-m grid, as the isopycnal displacement data.

Semi-Lagrangian records of shear and velocity were computed from the Eulerian records by interpolating onto the instantaneous depth of each isopycnal surface obtained from MMP.

## 4. Results

### a. Depth–time maps

The basic tidal heaving mechanism is visible in the Eulerian map of zonal shear (Fig. 2a). Isopycnal depths *z*(*ρ,* *t*), with mean spacing 10 m, are plotted in black. The downgoing near-inertial wave is evident as broad upward-sloping shear bands with wavelength 2*π**m*^{−1}_{o}*π**ω*^{−1}_{o}*πf*^{−1} = 4.4 days.^{4}

The near-inertial shear layers are advected by the (primarily diurnal) isopycnal displacement field. The s-L shear field (Fig. 2b), which has most of the advection removed, does not show this distortion. Consequently, the inertial bands are much straighter when viewed in the s-L frame. Residual deviations of s-L shear from inertial phase lines are due either to errors in interpolation and isopycnal depth, or to intrinsic diapycnal motions.

Figure 2a shows fine-structure contamination. To make this clear, a single representative Eulerian time series from 93-m depth (a horizontal slice through Fig. 2a) is low-pass filtered with a cutoff 1.5 cpd (to focus on heaving by the diurnal tide alone), and plotted in Fig. 2c. The time series clearly contains frequency components other than inertial. Good agreement is seen with the two-wave model (Fig. 2c, black line) when observed wave parameters and a visual best-fit phase (2*π**m*^{−1}_{o}*η*_{o} = 9 m, *ϕ*_{o} = *π*/2) are used.^{5}

A time series of s-L shear at the same depth (that is, shear across the isopycnal whose mean depth is 93 m) is much more sinusoidal (Fig. 2d, black line), indicating that the effects of advection are much reduced. The model (black line) is, of course, perfectly sinusoidal in the s-L frame [Eq. (4)].

### b. Frequency spectra

The frequency content of the data are examined via rotary spectral analysis (Gonella 1972). Spectra of rotary velocity *u* + *iυ,* rotary shear *u*_{z} + *iυ*_{z}, and isopycnal displacement are computed by demeaning, detrending, and Hanning and Fourier transforming each time series between 50 and 120 m and over yeardays 296.5–307.4. The transformed series are then averaged together to form one spectral estimate. For shear and velocity, spectra are computed for both Eulerian and semi-Lagrangian quantities.

The resulting spectra are plotted vs frequency in Fig. 3. The rotary spectra (Figs. 3a,c) are plotted vs negative/positive frequency, representing clockwise/counterclockwise motions. (The displacement spectrum is plotted vs positive frequency, since the positive/negative portions are redundant for a real time series.) Note that the plotting scale is linear in the abscissa but logarithmic in the ordinate. Confidence limits are indicated. The spectra are plotted as stairs to indicate the limited frequency resolution, Δ*f* = 1/(10.9 days) = 0.09 cpd. Eulerian and semi-Lagrangian spectra are plotted in red and blue, respectively.

A strong inertial peak is present in the spectrum of all three quantities. Its presence at +*f* in the rotary spectra (Figs. 3a,c) indicates its counterclockwise sense of rotation, consistent with Southern Hemisphere dynamics. It especially dominates the shear records, containing 68% of the total variance.

The internal tide is evident in the velocity and displacement fields. The *K*_{1} tide is nearly completely counterclockwise polarized, while the *M*_{2} constituent^{6} is much more nearly even.

The absence of corresponding peaks at the *K*_{1} and *M*_{2} frequencies in the Eulerian shear spectrum (Fig. 3c, red line) confirms the earlier assertion that the tidal motions are low mode. Present instead are peaks at *f* ± *K*_{1}, whose magnitudes are accurately predicted by the two-wave model [Eq. (9), black circles]. A cursory inspection of the spectrum would lead to the erroneous conclusion that the *K*_{1} tide contains strong shear. The presence of the shifted peaks in the Eulerian shear spectrum, which has no discernible tidal constituents, is instead a direct consequence of diurnal tidal heaving of near-inertial shear.

This conclusion is supported by the semi-Lagrangian spectra (blue lines): the peaks at *f* ± *K*_{1} are greatly attenuated. [The effect is especially visible in the shear spectra (Fig. 3c), where there are no peaks at *K*_{1}, but can also be seen in those of velocity (Fig. 3a).] The loss of these peaks in the s-L spectra is due to the absence of advective frequencies in the moving reference frame.

Results for *M*_{2} are less clear. A peak at *f* + *M*_{2} is also close to the predicted magnitude for heaving by the *M*_{2} tide (black circle), but a corresponding peak is not present at *f* − *M*_{2}. Though the data are not conclusive, a peak at *f* + *M*_{2} but not at *f* − *M*_{2} is suggestive of a freely propagating wave resulting from nonlinear interaction between *f* and *M*_{2}, as argued by Mihaly et al. (1998). A proper treatment of this issue is hampered by the weaker *M*_{2} signal and the possibility of influence from the 3-h bin length, which yields only four points per *M*_{2} tidal cycle.

## 5. Conclusions

Data and a model are presented that illustrate a simple, two-wave case of the “fine-structure contamination” problem. Depth–time maps and spectra of velocity, shear and isopycnal displacement are examined from a low-latitude site, where the Coriolis frequency is much less than the diurnal tidal frequency. As a result of this spectral separation, diurnal tidal heaving of the near-inertial motions is clearly identifiable in depth–time maps and Eulerian frequency spectra, which show “contamination” peaks at *f* ± *K*_{1}. A simple two-wave model produces good agreement with observed time series and accurately predicts the magnitude of the shifted peaks. The effect is much reduced by transforming to an isopycnal-following frame, where advection is minimized.

No new physics are introduced in this paper. It is emphasized that while nonlinear interactions between the internal tides and the near-inertial wave are not ruled out, they are not necessary to explain the observations: a purely kinematic effect is responsible for the observed distortion of the near-inertial shear layers. These observations are a reminder (hardly needed to most oceanographers) that even the simplest ocean situations can contain subtle complications that require careful interpretation.

## Acknowledgments

This work was supported by M.A.'s startup funding at the Applied Physics Laboratory. The data collection and initial analysis were supported by NSF Grant OCE9729288. I am grateful to Mike Gregg for data, guidance, and support. Conversations with Chris Garrett, Eric Kunze, Steve Mihaly, and Dave Winkel were helpful.

## REFERENCES

Alford, M., and M. Gregg, 2001: Near-inertial mixing: Modulation of shear, strain and microstructure at low latitude.

*J. Geophys. Res.*, in press.Alford, M., M. Gregg, and M. Ilyas, 1999: Diapycnal mixing in the Banda Sea: Results of the first microstructure measurements in the Indonesian Throughflow.

,*Geophys. Res. Lett***26**(17) 2741–2744.Anderson, S. P., 1993: Shear, strain and thermohaline vertical fine structure in the upper ocean. Ph.D. thesis, University of California, San Diego, 143 pp.

Gonella, J., 1972: A rotary-component method for analysing meteorological and oceanographic vector time series.

,*Deep-Sea Res***19****,**833–846.Kunze, E., M. G. Briscoe, and A. J. Williams III, 1990: Observations of shear and vertical stability from a neutrally buoyant float.

,*J. Geophys. Res***95****,**(C10). 18127–18142.Mihaly, S. F., R. Thomson, and A. B. Rabinovich, 1998: Evidence for nonlinear interaction between internal waves of inertial and semidiurnal frequency.

,*Geophys. Res. Lett***25**(8) 1205–1208.Pinkel, R., and S. Anderson, 1997: Shear, strain and Richardson number variations in the thermocline. Part I: Statistical description.

,*J. Phys. Oceanogr***27****,**264–281.

^{1}

A similar derivation of the model, with some algebra errors, may be found in Anderson (1993).

^{2}

Throughout this paper, all frequencies are taken in radian units, but plotting is done in cyclic units.

^{3}

The internal wave equations are specified in terms of *z,* not *ζ.* Equation (4) is a valid solution because tidal vertical accelerations associated with the displacements are miniscule, and do not affect the dynamics.

^{4}

Alford and Gregg (2001) found 2*π**m*^{−1}_{o}*π**m*^{−1}_{o}

^{5}

Since *m*_{o}*η*_{o}/2 = 0.35 renders (7) marginally valid, a numerical evaluation of (6) is used rather than (7). The two differ somewhat in phase, but are nearly identical in magnitude.

^{6}

The spectral resolution is not sufficient to distinguish between 2 cpd and *M*_{2}. Peaks near 2 cpd are interpreted as *M*_{2} tidal, though they could be harmonics of the diurnal tide. Likewise, frequency shifting of the near-inertial wave by the mean flow and background vorticity, as examined by Alford and Gregg (2001), is too small to be resolved spectrally.