1. Introduction
Internal waves provide an important link between large- and mesoscale motions and turbulence-scale mixing. In the upper equatorial Pacific Ocean, the vertical flux of momentum and energy is critical to the maintenance of the equatorial undercurrent (EUC). Internal waves are known to play a significant role in that redistribution of energy and momentum and are found to be present at times of increased turbulent mixing in the thermocline of the upper eastern equatorial Pacific (Wijesekera and Dillon 1991; Hebert et al. 1992; McPhaden and Peters 1992; Moum et al. 1992; Lien et al. 1996; Mack and Hebert 1997).
Internal waves are generated by some mechanism, propagate through the fluid carrying momentum and energy, and deposit that momentum and energy either in one localized region or over some broader region, depending upon how the waves decay. The generation mechanism of the high-frequency equatorial internal waves is thought to be a type of shear instability near the base of the mixed layer (Wijesekera and Dillon 1991;Moum et al. 1992; McPhaden and Peters 1992; Skyllingstad and Denbo 1994; Lien et al. 1996; Sutherland 1996; Mack and Hebert 1997), though the details of the exact generation mechanism are still in question. In the equatorial Pacific the mean flows are primarily zonal. Shear-generated internal waves vertically transport the zonal momentum of the westward flowing south equatorial surface current, which is directly forced by the trade winds. Some studies have suggested that these waves transport significant momentum below the EUC core (e.g., Wijesekera and Dillon 1991; Sutherland 1996), while other studies have found mixed results or no deep momentum penetration (Skyllingstad and Denbo 1994; Lien et al. 1996; Mack and Hebert 1997).
The process of internal wave decay in this region is poorly understood. Internal waves have been observed to be actively overturning (e.g., Hebert et al. 1992) and have been correlated with enhanced turbulence kinetic energy (TKE) dissipation rates, ε, below the mixed layer (Moum et al. 1992; Hebert et al. 1992; Lien et al. 1996). Overturning waves, which are statically unstable, are presumably due to some type of wave destabilization and lead to a local increase in turbulent mixing. A few studies have tied internal waves directly to turbulent mixing. Moum et al. (1992) found hourly averages of ε to be significantly correlated with several internal wave properties (e.g., rms vertical displacement, wave slope, and wave energy). They argued that the internal waves extracted energy from the mean shear and, during decay, deposited it into local turbulent mixing (i.e., ε). Lien et al. (1996) found ε to be significantly correlated with shear production of TKE. They concluded that 50% of the shear production went to ε, 2%–12% contributed to the buoyancy flux, and the remaining shear production presumably was balanced by transport divergence. However, because viscous effects are negligible on internal wave scales, shear production goes into the growth of internal waves. How is the energy transferred from waves to turbulence scales? Two main mechanisms for internal wave destabilization are known: advective instability and wave-induced shear instability (Munk 1981). Moum et al. (1992) calculated the criteria for each of these instabilities for a linear wave field with observed vertical structure in a background stratified flow. Using a variety of possible phase speeds, they concluded that observed internal waves of the eastern equatorial Pacific were susceptible to both types of instabilities.
In the present study, we use high-resolution observations from a vertical array of towed thermistors to further investigate the dynamical link between internal waves and turbulence. For example, during one large internal wave event depicted in Fig. 1, an activity index AT, a quantity related to the TKE dissipation rate as described in section 2, shows a definite horizontal and vertical structure of mixing linked to internal wave displacements. Herein, we investigate the type of internal wave destabilization that leads to turbulent mixing and interpret some observations of the structure of turbulent mixing within an internal wave cycle.
In section 2 the calculation of the activity index AT from the observed horizontal temperature measurements will be described and then compared to the accurately measured ε from a vertical profiler. We will show that AT is a good indicator of the presence of turbulent mixing. Advective instability and wave-induced shear instability theories are reviewed and observations of AT within an internal wave cycle and in comparison to a combined advective–shear instability theory developed by Thorpe (1978a b) are presented and discussed in section 3. A summary and conclusions are presented in section 4.
2. Methods
a. The data
The data used in the present study were collected in April 1987 while traveling eastward along the Pacific equator from 140° to 110°W. These data consist of measurements from a towed thermistor chain, a rapid sampling vertical profiler (RSVP), and an acoustic Doppler current profiler (ADCP), as well as accompanying surface meteorological measurements [see Hebert et al. (1991) for more details on these instruments]. From the ADCP and RSVP instruments profiles of zonal and meridional velocities and of density, ρ, and ε were obtained. The ρ and ε measurements extended from near the surface to about 140-m depth; velocity measurements ranged from approximately 19-m to 140-m depth. All of these data were temporally averaged over one hour (approximately 9 km zonally with a 2.5 m s−1 average ship speed) and vertically averaged over 4 m, as outlined by Mack and Hebert (1997). The detailed (two-dimensional) structure of the internal gravity waves and turbulent mixing was measured using the towed thermistor chain. Thermistor spacing was nominally 4 m in the vertical over the upper 125 m. The zonal resolution was approximately 12.5 cm, determined from a 20-Hz sampling rate and average ship speed. The temperature resolution of individual thermistors was 0.001°C and interthermistor calibration was accurate to 0.01°C [see Moum et al. (1992) for further thermistor chain details].
While the 10-s-averaged thermistor data have been extensively studied (Moum et al. 1992; Hebert et al. 1992; Mack and Hebert 1997), the high-frequency thermistor data have previously been viewed only briefly by Hebert et al. (1992) in investigating the details of a single wave event (shown here in Fig. 1). Hebert et al. (1992) looked at temperature fluctuations scaled by the mean vertical temperature gradient for one energetic wave packet and found increased scaled temperature fluctuations associated with the overturning region of the wave. These scaled temperature fluctuations varied in intensity both vertically and within a wave cycle, although the details of exactly how the temperature fluctuations were related to the wave cycle were not examined. Their study did indicate that the turbulent mixing apparently is not uniform within an internal wave cycle. In the present study we look at turbulent mixing as it is associated with internal wave packets in greater detail and for a broader range of cases. In order to do this, we will show first that the temperature fluctuations measured from the thermistor chain are related to turbulence quantities.
b. Activity number, AT
Profiles of ε were determined from RSVP velocity shear sensor data. However, since the profiles were only once every 12 minutes, or approximately 1.8 km, they did not resolve fluctuations on the horizontal scale of internal waves, which had wavelengths of 150–250 m (Moum et al. 1992). Based upon the findings of Hebert et al. (1992) we expect that the turbulent mixing within the wave cycle can be determined using the high-frequency thermistor data. In order to do so, we need to estimate turbulence-scale quantities such as the temperature variance dissipation rate (χ, defined below) and ε from the temperature fluctuations. If the peak of the Batchelor spectrum, the theoretical spectrum for temperature gradient fluctuations, were resolved, the spectrum could be integrated to find χ directly.
First, in order to estimate the temperature gradient variance, a Hamming window and a 120th-order finite impulse response digital filter were applied to thermistor data. The filter passed 50% of the signal at 0.54 s−1 (approximately 4.6-m length scale); 5% and 95% of the signal were passed at 0.36 s−1 and 0.73 s−1, respectively. Discrete zonal differentiation was carried out at the sampling scale of 12.5 cm after converting the temperature time series to a spatial series using the constant ship speed of 2.5 m s−1. We averaged 100 data points (12.5 m) for the horizontal temperature gradient average in (5). This averaging interval was chosen large enough to give a representative mean yet small enough to provide several AT values within an internal wave cycle. To determine the mean vertical gradients, a 1-h (9-km horizontal) averaging timescale was chosen which was sufficiently large to average over wave cycles and wave events, the latter of which had timescales of about 30 min (Mack and Hebert 1997). The vertical gradient of
To show that the activity index AT is related to the turbulent kinetic energy dissipation rate, we compared overlapping occurrences of AT from the thermistor chain and ε from the profiler data (Fig. 2). Thermistor measurements were at 22 discrete depths and vertical profiles were at 677 discrete times during four days of internal wave activity. The exact times for which AT was calculated for comparison with ε were 5-s intervals centered on the profiler drop times plus a time lag equal to the time that it took the profiler falling speed of 1 m s−1 to reach the depth of each thermistor.
3. Results
In this section we present a detailed view of how internal waves are linked to turbulent mixing. Two criteria for internal wave destabilization are reviewed. Next these criteria are compared to AT both vertically and within the wave cycle. Finally, AT is viewed in wave slope/shear space.
a. Internal wave destabilization criteria
The destabilization of internal waves generally leads to a cascade of wave energy to smaller scales where viscous forces dissipate turbulence energy into heat. It is thought that one or both of two instability mechanisms are responsible for internal wave destabilization and subsequent turbulent mixing: wave-induced shear instability and advective instability (e.g., Munk 1981).
The shear and advective instability criteria were previously investigated by Moum et al. (1992) using 10-s-averaged thermistor data, the same thermistor data used in the present study. They assumed a linear plane wave superimposed on a mean stratified flow. A mean vertical profile of the observed wave vertical displacement was used for the vertical structure of the plane wave. They plotted the two instability criteria for a range of possible phase speeds at three different depths and found both instability mechanisms likely to be active. However, their analysis for advective instability was flawed in that they compared only the wave-induced flow velocity, not the total flow velocity, to the phase velocity [i.e., (8) with U = 0]. As a result, they found advective instability likely at all depths for a phase speed near zero. In reality, this type of instability is expected only near a critical depth if c is within the range of U, as is expected in the upper eastern equatorial Pacific.
The general structures of Frw and OT for the fastest growing solution (Fig. 3) are consistent with the structures of 15 other wave events studied by Mack and Hebert (1997) for the same general location and time period as the present study. The eigenfunction solution, W, was scaled to be consistent with a maximum vertical displacement amplitude of 4 m, which was similar to observed amplitudes. The basic state profiles are shown in Fig. 3a and Fr is shown in Fig. 3b. To bring out best the effect of wave-induced shear Fr′ = Frw − Fr, the“perturbation” Froude number, was contoured in Fig. 3c relative to the phase profile of wave displacement crests. We should emphasize that this is only a single example of a particular basic state and a particular scaled eigenfunction. As such, it can give one a sense of where such a wave field has greater or lesser stability for each type of instability mechanism but should not be used as a definitive indicator for whether or not each instability is important. Also, it is the total Frw that is important, not just Fr′, for instability.
It can be seen for both instability mechanisms that the greatest instability with respect to the phase of the wave is associated with the crest above the critical depth and with the wave trough below the critical depth (Fig. 3). For shear instability, Frw > 2, the vertical region over which instability may occur is very broad spanning about 40 m. Likewise, the phase-shift region over which the maximum of Fr′ changes from crest to trough spans a vertical region of about 20 m. We might expect to see a similar structure of mixing with respect to the phase of the waves. Conversely, instability due to the advective mechanism is narrowly confined to a region about 5 m above and below the critical depth and there is no phase-shift region; rather, the maximum of OT simply switches from the crest to the trough across the critical depth.
b. Mixing within an internal wave cycle
Next, we consider the observed structure of turbulent mixing within a wave cycle. For example, in Fig. 1 the variability of AT appears coupled to the wave displacement phase, though the exact relation between the two is uncertain. In order to study this relationship between internal waves and turbulent mixing, we bin-averaged ÂT ≡ AT/
The method for obtaining δh and k for each internal wave packet is as follows. The observed vertical displacement, h, at each thermistor depth was determined from 10-s-averaged temperature contoured data of the thermistor chain with the assumption that isothermal surfaces represent isopycnal surfaces (Mack and Hebert 1997). The complex wave displacement, h, was constructed using h for the real part and the Hilbert transform of h for the imaginary part. The Hilbert transform is related to the original data by a 90° phase shift; that is, sines becomes cosines and vice versa (see Krauss et al. 1994 for further description of the Hilbert transform). From h the wave amplitude, |h|, phase, δh ≡ arg(h), and zonal wavenumber, k ≡ |hx/h|, were determined where hx represents a spatially discrete differentiation on a 25-m horizontal scale (10 s times 2.5 m s−1 ship speed). The Hilbert transform was applied to 128-point segments (about 21 min or 3.2 km) of h at each thermistor depth. The phase, δh, is relative to a wave crest and increases from west to east.
The pattern of mixing above the thermocline is apparent (Fig. 4). Maxima of ÂT above roughly 20-m depth, the region of the nighttime mixed layer, were associated with wave phases slightly to the east of the crests. A region where ÂT changed phased with depth was found between approximately 20 and 55 m, which is also a region of marginal mean stability based on the profile of
If one assumes that local regions of increased ÂT resulted from local wave destabilization then the structure of Fr′ and OT can be compared with the structure of ÂT. As a guide, the Fr′ and OT for the modal solution that best fit the first wave event in each 3-hour period is shown next to ÂT (Fig. 4). The patterns of Fr′ and OT are characteristic of the influence of the wave field on the mean conditions. The lack of a phase dependence of ÂT below approximately 60 m is likely due to the mean hydrographic conditions having a Froude number well below the critical value needed for instability. The enhanced wave shear cannot increase the Froude number to the critical value. Above approximately 50 m, the mean flow is marginally stable. In this depth range, the vertical and horizontal patterns of ÂT agree much better with shear instability (i.e., Fr′) than with advective instability (OT). This implies that the wave-induced shear instability mechanism is likely responsible for the observed turbulent mixing. However, the modal solutions that best fit the different wave events in a 3-h period have slightly different critical depths than shown in Fig. 4 and different amplitudes. Thus, we cannot rule out that the advective instability occurring at different depths for each wave event would not produced the observed pattern in ÂT. Analysis of ÂT for individual periods do not appear to support this idea; however, the uncertainty in ÂT does not allow us to state this conclusively.
c. Observations of wave stability
The competing effects of wave-induced shear instability and advective instability can also be gauged by considering a neutral stability curve for both of these instabilities. Thorpe (1978a,b) found a neutral stability curve for the combined effects of advective and shear instabilities using perturbation theory. He numerically solved for stable solutions with a variety of basic states consisting of a single stratification profile, N2, and a variety of parallel velocity profiles, U, each having the same vertical structure but differing magnitudes. Then, for each basic state, he determined the eigenfunction amplitude needed for (8) to be an equality, which is the marginal condition for advective instability. The neutral stability curve for a Couette profile (constant Uz and N) and a transition layer [both Uz and N2 proportional to sech2(z)] are shown in Fig. 5 for [extremum(Uz/N), k|h|] space. Thorpe termed k|h| the wave slope, where k is the horizontal wavenumber and |h| is the amplitude of wave displacement. For both curves, shear instability occurs for Fr = |Uz|/N > 2.
In order to compare Thorpe’s theory with observation, the assumption of local generation of turbulent mixing must be made. Thus, whereas Thorpe considered only the extremum value of Uz/N for a given basic state and the maximum value of k|h| to determine the stability of the given wave field, we consider the stability to be a function of both the local, averaged value
We used the activity index, AT, as an indicator of dynamic instability. Four days of continuous AT data at all depths were bin-averaged by local values of
First, AT has higher values above the stability curves in the unstable region (Fig. 6). The highest AT are found for
The second interesting feature is the distribution of the number of occurrences of certain wave slopes and Froude numbers. Remember, we only included periods when waves of wavelength 100–350 m were present (Fig. 6). The effect of waves on a locally stable region is to produce shear that oscillates the local region about
4. Summary and conclusions
High-frequency temperature data collected from a vertical array of thermistors towed from 140° to 110°W along the equator sampled the upper 125 m of the eastern Pacific Ocean at 20 Hz, or roughly 12.5 cm zonally. Since the observed temperature gradient variance did not resolve the Batchelor spectrum, we used a buoyancy-frequency-scaled pseudo Cox number, the activity index, as a measure of the turbulent mixing rate. Comparison between observed dissipation rates from vertically profiled velocity shear measurements and the activity index showed good agreement.
Two possible mechanisms for internal wave destabilization were examined: advective instability and wave-induced shear instability. The former is likely to occur only near critical depths, that is, depths where the wave phase velocity equals the mean flow velocity, and the latter is likely to occur at depths where the mean gradient internal Froude number, Fr = |Uz|/N, is near to or in excess of its critical value of 2. For waves generated in a shear flow, the critical depth and the depth of high Fr and large wave amplitude both occur near the depth of extremum mean shear. Therefore, mixing caused by each of these types of wave instabilities is expected to occur at similar depths making these two types of instabilities difficult to distinguish. Instability was associated with wave displacement crests above the critical depth and with troughs below this depth for both mechanisms based upon basic state profiles representative of conditions in the upper eastern equatorial Pacific.
In order to view directly the process of turbulent mixing due to internal waves the observed turbulent mixing within a wave cycle was examined. Values of AT were bin-averaged according to the background wave displacement phase at each thermistor depth. A contour of AT normalized by the mean AT at each depth for four 3-h periods of intense wave activity revealed a pattern of mixing within the wave cycle. Maxima of normalized AT were slightly to the east of wave crests above roughly 20-m depth, the region of the nighttime mixed layer. A phase-shift region was found from about 20–55-m depth in which relatively high AT shifted eastward from wave crests in the upper part to wave troughs in the lower part. Below the phase-shift region a clear pattern of mixing in the wave cycle was not found, although there was some evidence supporting high relative mixing associated with wave troughs below 60-m depth. The general structure of AT was consistent with wave-induced shear instability assuming a marginally stable mean flow over the depth range of comparison. This structure was dissimilar to that for advective instability, which showed a narrow unstable region near the critical depth. Although we could not disprove advective instability as a mechanism for wave destabilization in the upper equatorial Pacific, we found strong evidence supporting wave-induced shear instability as the primary destabilizing mechanism of internal waves in this region.
Our results suggest that wave-induced shear instability dominates over advective instability in the upper eastern equatorial Pacific as the main mechanism for internal wave destabilization and turbulent mixing. More conclusive evidence would require observations of the wave field as it propagates and grows, destabilizes, and mixes. Because such direct observations are difficult, the next step in this work is a numerical modeling effort.
Acknowledgments
We thank Clayton Paulson and James Moum of Oregon State University for supplying the thermistor chain, RSVP, and ADCP data collected by them. This research was supported by the National Science Foundation Grant OCE-9201332, the Office of Naval Research AASERT Award N000149410697, and the State of Rhode Island.
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Logarithm of activity index AT (described in section 2) calculated from a vertical array of towed thermistors is shown in grayscale at 16.696 (GMT) days in April 1987. Low-passed temperature contours (dashed lines) are shown for the mean temperatures at every second thermistor depth.
Citation: Journal of Physical Oceanography 29, 12; 10.1175/1520-0485(1999)029<3090:MSOHFI>2.0.CO;2
Activity index, AT, from the thermistor chain vs turbulent kinetic energy dissipation rate, ε, from the vertical profiler for four days of overlapping data. The solid line is the linear regression of log(AT) = a log(ε) + b where a = 0.79 (±0.01) and b = 1.48 (±0.07). The 95% confidence limits were determined by the bootstrap method.
Citation: Journal of Physical Oceanography 29, 12; 10.1175/1520-0485(1999)029<3090:MSOHFI>2.0.CO;2
(a) Basic-state profiles U (thick line) and N2 (thin line) with a dashed line at zero for reference, (b) profile of Fr with a dashed line at the critical value Fr = 2, (c) Fr′, the wave contribution for shear instability, and (d) OT, the criterion for advective instability. For (c) and (d) the dashed lines are at the critical depth where U = real (c); an idealized wave displacement cycle is shown above each contour for reference. This is event B2 described by Mack and Hebert (1997).
Citation: Journal of Physical Oceanography 29, 12; 10.1175/1520-0485(1999)029<3090:MSOHFI>2.0.CO;2
Each row shows results for a 3-h period of wave events. The first column shows the vertical profile of mean Froude number with the dashed line at Fr = 2. The second and third columns show the phase-depth structure of Fr′ and OT for the modal solution that best fits the first wave event in the 3-h period. The dashed line shows the critical depth. [For more details of the modal solutions for these and the other wave events, see Mack and Hebert (1997).] The fourth column shows the vertical profile of
Citation: Journal of Physical Oceanography 29, 12; 10.1175/1520-0485(1999)029<3090:MSOHFI>2.0.CO;2
Neutral stability curves are for combined shear and advective instabilities for a Couette profile (constant Uz and N; thin line) and a transition layer [Uz(z) and N2(z) proportional to sech2(z);thick line]. The x axis is the extremum value of Uz/N and the y axis is the maximum wave slope defined as k|h|.
Citation: Journal of Physical Oceanography 29, 12; 10.1175/1520-0485(1999)029<3090:MSOHFI>2.0.CO;2
Index AT from all thermistors and over 4 days was bin-averaged by
Citation: Journal of Physical Oceanography 29, 12; 10.1175/1520-0485(1999)029<3090:MSOHFI>2.0.CO;2
