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  • View in gallery

    (a) Geographic locations of the mooring array with contours denoting the topography (m). (b) The velocity measurements on M1, M2, and M3. The solid lines represent the sampling interval of ADCP, while the circles denote the current meters. The dashed lines bound the depth range analyzed in this study.

  • View in gallery

    (a) Lag correlation of normal strain Sn = UxVy, shear strain Ss = Uy + Vx, and relative vorticity ς = VxUy at 245 m to those at various depths. (b) The time lag (h) associated with the highest correlation. (c) The time-mean , , and . The values shown are the average over M1, M2, and M3.

  • View in gallery

    (a) Time-mean energy exchange rate P derived from Eq. (1), (b) the time series of P averaged within 245–450 m derived from Eq. (1), and (c) the time series of OW parameter averaged within 245–450 m. The red and blue lines in Fig. 3b correspond to the energy exchange during the periods with positive and negative OW parameter, respectively.

  • View in gallery

    PDFs of P during (a) positive and (b) negative OW parameter conditions. The numbers in brackets are the 90%-confidence intervals computed from the bootstrap method.

  • View in gallery

    (a) Partition of P in the frequency domain derived from Eq. (2) and (b) the frequency spectrum of horizontal velocity (thick gray line) and that multiplied by the factor (ω2f2)/(ω2 + f2) (thin black line).

  • View in gallery

    (a) Scatterplot of vertical mean −(〈uu〉 − 〈υυ〉) vs vertical mean Sn in the period with negative OW parameter. The solid line is the linear regression with its 90%-confidence interval (gray dashed). (b) As in (a), but for −2〈〉 vs Ss.

  • View in gallery

    Mean θw and its 90%-confidence interval computed from the bootstrap method for different wave frequencies.

  • View in gallery

    Time series of vertical mean Re(λ) (solid red line) and Re(λ′) (dashed blue line) within 245–450 m. The shaded regions denote the wave-capture-favoring period.

  • View in gallery

    PDF of P during the (a) wave-capture-favoring periods and (b) remaining positive-OW periods.

  • View in gallery

    PDF of θw (a) during the wave-capture-favoring periods, and simulated by the ray-tracing experiment with (b) realistic and (c) idealized settings.

  • View in gallery

    Snapshot (22 Sep 2012) of the OW parameter derived from the sea level anomaly using the geostrophic relation. The sea level anomaly is obtained from the SSALTO/DUACS multimission altimeter products of AVISO.

  • View in gallery

    The vertical wavenumber spectrum of range-averaged ADCP records within 245–450 m at M1, M2, and M3. The grey dashed line denotes the estimate for φn(m).

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Observed Energy Exchange between Low-Frequency Flows and Internal Waves in the Gulf of Mexico

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  • 1 Key Laboratory of Physical Oceanography, Ocean University of China, Ministry of Education, and Qingdao National Laboratory for Marine Science and Technology, Qingdao, China
  • | 2 Key Laboratory of Physical Oceanography, Ocean University of China, Ministry of Education, and Qingdao National Laboratory for Marine Science and Technology, Qingdao, China, and Department of Oceanography, and Department of Atmospheric Sciences, Texas A&M University, College Station, Texas
  • | 3 Department of Oceanography, and Geochemical and Environmental Research Group, Texas A&M University, College Station, Texas
  • | 4 Key Laboratory of Physical Oceanography, Ocean University of China, Ministry of Education, and Qingdao National Laboratory for Marine Science and Technology, Qingdao, China
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Abstract

A long-term mooring array deployed in the northern Gulf of Mexico is used to analyze energy exchange between internal waves and low-frequency flows. In the subthermocline (245–450 m), there is a noticeable net energy transfer from low-frequency flows, defined as having a period longer than six inertial periods, to internal waves. The magnitude of energy transfer rate depends on the Okubo–Weiss parameter of low-frequency flows. A permanent energy exchange occurs only when the Okubo–Weiss parameter is positive. The near-inertial internal waves (NIWs) make major contribution to the energy exchange owing to their energetic wave stress and relatively stronger interaction with low-frequency flows compared to the high-frequency internal waves. There is some evidence that the permanent energy exchange between low-frequency flows and NIWs is attributed to the partial realization of the wave capture mechanism. In the periods favoring the occurrence of the wave capture mechanism, the horizontal propagation direction of NIWs becomes anisotropic and exhibits evident tendency toward that predicted from the wave capture mechanism, leading to pronounced energy transfer from low-frequency flows to NIWs.

Denotes content that is immediately available upon publication as open access.

© 2018 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Zhao Jing, jingzhao198763@sina.com

Abstract

A long-term mooring array deployed in the northern Gulf of Mexico is used to analyze energy exchange between internal waves and low-frequency flows. In the subthermocline (245–450 m), there is a noticeable net energy transfer from low-frequency flows, defined as having a period longer than six inertial periods, to internal waves. The magnitude of energy transfer rate depends on the Okubo–Weiss parameter of low-frequency flows. A permanent energy exchange occurs only when the Okubo–Weiss parameter is positive. The near-inertial internal waves (NIWs) make major contribution to the energy exchange owing to their energetic wave stress and relatively stronger interaction with low-frequency flows compared to the high-frequency internal waves. There is some evidence that the permanent energy exchange between low-frequency flows and NIWs is attributed to the partial realization of the wave capture mechanism. In the periods favoring the occurrence of the wave capture mechanism, the horizontal propagation direction of NIWs becomes anisotropic and exhibits evident tendency toward that predicted from the wave capture mechanism, leading to pronounced energy transfer from low-frequency flows to NIWs.

Denotes content that is immediately available upon publication as open access.

© 2018 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Zhao Jing, jingzhao198763@sina.com

1. Introduction

The energy exchange between low-frequency flows and internal waves has been proposed to play an important role in the ocean kinetic energy budget (Ferrari and Wunsch 2009). On one hand, the energy transfer from low-frequency flows to internal waves can be a significant energy sink of ocean general circulation. On the other hand, it may also provide a substantial portion of energy for the internal wave field in the ocean interior. To maintain the abyssal stratification, 2 TW of energy is required to furnish the diapycnal mixing (Munk and Wunsch 1998). Much of the oceanic mixing away from boundaries occurs through internal wave breaking. Energy is input into the internal wave field primarily by the tides and the winds (Wunsch and Ferrari 2004). Tidal dissipation in the open ocean is estimated to be about 1 TW (Jayne and St. Laurent 2001; Egbert and Ray 2001), while computed wind work on near-inertial internal waves (NIWs) ranges from 0.3 to 1.4 TW (Alford 2003; Watanabe and Hibiya 2002; Jiang et al. 2005; Rimac et al. 2013; Liu et al. 2017). However, given that only a small portion (~30%) of near-inertial energy flux can radiate into the ocean interior (Furuichi et al. 2008; Zhai et al. 2009; Alford et al. 2012), the near-inertial wind work and tidal dissipation alone are not sufficient to supply the energy required to furnish the diapycnal mixing. It may be possible that the energy transfer from low-frequency flows to internal waves in the subthermocline and deep ocean also plays an important role.

Indeed, a few observational studies in the Gulf Stream reveal a permanent energy transfer from low-frequency flows to internal waves in the subthermocline and deep ocean (Frankignoul 1976; Ruddick and Joyce 1979; Brown and Owens 1981; Polzin 2010). Many mechanisms have been proposed to account for this permanent energy exchange (e.g., Bretherton 1966; Müller 1976; Bühler and McIntyre 2005; Danioux et al. 2012; Thomas 2012; Vanneste 2013; Xie and Vanneste 2015; Grisouard and Thomas 2016; Wagner and Young 2016). In some mechanisms (e.g., Bühler and McIntyre 2005; Thomas 2012; Grisouard and Thomas 2016), the energy transfer from low-frequency flows to internal waves is only a transient process followed or accompanied by rapid internal wave dissipation. In other mechanisms (e.g., Müller 1976; Xie and Vanneste 2015; Wagner and Young 2016), internal wave dissipation is not directly triggered. Nevertheless, both cases provide an energy pathway to furnishing diapycnal mixing.

Observed evidence for energy exchange between low-frequency flows and internal waves is quite limited and mainly confined to the Gulf Stream region. It remains unclear whether similar results also hold in other parts of the global ocean. Furthermore, although many energy exchange mechanisms have been proposed, their validity and efficiency are still poorly assessed in reality. The long-term mooring array deployed in the northern Gulf of Mexico provides a unique opportunity to estimate the energy exchange between low-frequency flows and internal waves (Fig. 1) and to explore the underlying mechanism responsible for this energy exchange. The study is organized as follows. Data and methodology are given in section 2. Energy exchange between low-frequency flows and internal waves is estimated in section 3 based on mooring data. In section 4, we explore the underlying mechanism to the extent the available data permit. Finally, conclusions are provided in section 5.

Fig. 1.
Fig. 1.

(a) Geographic locations of the mooring array with contours denoting the topography (m). (b) The velocity measurements on M1, M2, and M3. The solid lines represent the sampling interval of ADCP, while the circles denote the current meters. The dashed lines bound the depth range analyzed in this study.

Citation: Journal of Physical Oceanography 48, 4; 10.1175/JPO-D-17-0263.1

2. Data and methodology

a. Mooring data

In July of 2012, six moorings, henceforth referred to as M1 through M6, were deployed in the northern Gulf of Mexico (Fig. 1a; Spencer et al. 2016). Each mooring was equipped with a Teledyne Rowe-Deines Instruments (RDI) Broadband 75-kHz Long-Ranger ADCP to measure horizontal current velocity profiles at 30-min intervals. Each ADCP was fitted inside a 43'' diameter syntactic foam flotation mount. The ADCPs were set to record 37 vertical bins 16 m long, yielding a maximum vertical range of 592 m. The instruments were set to record 12 acoustic pings per ensemble. Data were then tested against community standard practices for quality-controlled data, and quality assurance protocols were applied. These protocols include range checking, statistical outliers, stuck values, linear trends or offsets, first-order difference, and data acceleration. Additionally, internal ADCP quality control metrics such as error velocity, vertical velocity, beam coherence, percent good data, and echo/backscatter intensity were used to provide quality assurance of the data used. Horizontal velocity estimate data that do not pass quality protocols are flagged as suspect and not used in the subsequent analyses. The noise level of the current record is estimated to be around 0.01 m s−1 (see appendix A for the estimation details).

Only M1, M2, and M3 are used in this study as they are close to each other and sample the water column between 245 and 450 m (Fig. 1b), making it possible to compute the horizontal velocity gradient of low-frequency flows. There are also several single-point current meters below the ADCPs to measure the velocity in the deep ocean (Fig. 1b). The profiles of velocity are interpolated onto 5-m regular grids for analysis. There were 26 CTD profiles collected during the deployment and recovery of moorings. They are used to compute the stratification.

The instruments were deployed from July 2012 through June 2014 with a short servicing in July 2013. Only the data collected before the servicing time are used. During this period, mooring layover is minimal with a root-mean-square (RMS) vertical excursion of 0.36 m for M1, 0.30 m for M2, and 0.30 m for M3. In particular, the ADCPs always stayed within 1 m of their target depth except that M1 underwent a rise of 3 m in June 2013.

b. Isolating the internal waves and low-frequency flows

The horizontal velocity of internal waves (u, υ) is obtained by high-pass filtering the velocity records to remove frequencies below the local inertial frequency f. The half-power point is chosen as 0.8f following Brown and Owens (1981). The horizontal velocity of low-frequency flows (U, V) is attained by low-pass filtering with a half-power point of 6 inertial periods (IPs). We note that changing the half-power point from 2 to 10 IPs makes no appreciable impact.

The horizontal velocity gradient of low-frequency flows is computed by taking the difference of the low-pass-filtered velocities among M1, M2, and M3. Because of the distance (~40 km) between the moorings, the velocity gradient of submesoscale low-frequency flows cannot be resolved. Therefore, our analysis applies to the energy exchange between internal waves and low-frequency flows at larger than mesoscale.

The observed low-frequency flows are vertically coherent in the upper 200–800 m (Figs. 2a,b). But their strength decreases roughly linearly as depth increases (Fig. 2c). In the upper 245–450 m, the low-frequency flows are characterized by a Rossby number of 0.05, estimated using the observed RMS velocity. The variance of convergence/divergence in the low-frequency flows is considerably smaller than that of strain or vorticity (Fig. 2c), suggesting that the low-frequency flows are in a quasigeostrophic regime (Pedlosky 1987). In the deeper region, the ratio of convergence/divergence variance to that of strain or vorticity increases. This may be partly due to the errors resulting from vertical interpolation (Fig. 1a) or topographically induced ageostrophic flows at greater depth. For this reason, we confined our analysis to the depth range between 245 and 450 m that is fully covered by the ADCP measurements.

Fig. 2.
Fig. 2.

(a) Lag correlation of normal strain Sn = UxVy, shear strain Ss = Uy + Vx, and relative vorticity ς = VxUy at 245 m to those at various depths. (b) The time lag (h) associated with the highest correlation. (c) The time-mean , , and . The values shown are the average over M1, M2, and M3.

Citation: Journal of Physical Oceanography 48, 4; 10.1175/JPO-D-17-0263.1

c. Energy exchange between low-frequency flows and internal waves

In this study, we address the energy transfer rate from low-frequency flows to internal waves through the horizontal wave stress due to the limited data. Provided that low-frequency flows satisfy geostrophic approximation, the corresponding energy transfer rate is given by
e1
(Polzin 2010), where Sn = UxVy and Ss = Uy + Vx are the normal and shear components of strain of low-frequency flows (the subscripts denote the spatial derivatives), respectively. The represents the running mean over a 3-IP interval. The 3-IP interval is used because it is sufficiently longer than the time scale of internal waves but still short enough to allow low-frequency flows to remain quasi invariant. With the aid of the Parseval theorem, Eq. (1) can be reexpressed in the frequency domain:
e2
where Φuu and Φυυ are the frequency spectra of u and υ, Φ is the cospectrum between u and υ, and Re is the real operator. The spectra and cospectra of internal-wave horizontal velocity are computed for every 3-IP transform interval with half overlapping. A Hanning window is applied before Fourier transform to smooth the spectral estimates.

d. Azimuth of horizontal wave vector of internal waves

A horizontally isotropic internal wave field does not exchange energy with the low-frequency flows through the horizontal wave stress as both 〈uu〉 − 〈υυ〉 and 〈〉 vanish in this case. A nonzero value for P can only occur when the low-frequency flows impose a permanent influence on the azimuth θ of horizontal wave vector of internal waves. According to the polarization relation of internal waves, θ can be estimated from the following relation:
e3

Measurement errors in the ADCP records may contaminate the azimuth estimates. Using Monte Carlo simulations in which noise is assumed to be normally distributed with a standard deviation of 0.01 m s−1, we estimate that the RMS error in θ ranges from 12° to 20° and becomes larger with increasing wave frequency. The larger RMS error for high-frequency internal waves is mainly due to their smaller current amplitude, causing low signal-to-noise ratios.

3. Results

Figure 3a displays the vertical distribution of time mean P computed from Eq. (1) using the mooring data. The value of P is almost systematically positive throughout 245–450 m (Fig. 3a). The vertical mean P is (4.1 ± 0.6) × 10−11 m2 s−3 (the standard error denotes the 90%-confidence interval computed from a bootstrap method),1 indicating a permanent energy transfer from low-frequency flows to internal waves. The time series of P reveals that the instantaneous energy exchange rate can be both positive and negative (Fig. 3b). There appears to be an association between the value of P and the sign of Okubo–Weiss parameter (Provenzale 1999) defined as OW = 4(VxUyUxVy) (Fig. 3c). A permanent energy transfer from low-frequency flows to internal waves appears only to occur in the positive OW parameter condition. This is verified by the difference of the probability density functions (PDFs) of P between positive and negative OW parameter conditions (Fig. 4). When the OW parameter is positive, the PDF is skewed rightward, corresponding to a positive energy transfer rate from low-frequency flows to internal waves. The mean P in this case is 7.2 × 10−11 m2 s−3; its 90%-confidence interval is (6.5–8.0) × 10−11 m2 s−3. In contrast, when the OW parameter is negative, the PDF is close to symmetric and the mean P value, estimated at −0.2 × 10−11 m2 s−3, is not significantly different from zero. Therefore, a permanent energy transfer only occurs in periods when the strain dominates relative vorticity in the low-frequency flows.

Fig. 3.
Fig. 3.

(a) Time-mean energy exchange rate P derived from Eq. (1), (b) the time series of P averaged within 245–450 m derived from Eq. (1), and (c) the time series of OW parameter averaged within 245–450 m. The red and blue lines in Fig. 3b correspond to the energy exchange during the periods with positive and negative OW parameter, respectively.

Citation: Journal of Physical Oceanography 48, 4; 10.1175/JPO-D-17-0263.1

Fig. 4.
Fig. 4.

PDFs of P during (a) positive and (b) negative OW parameter conditions. The numbers in brackets are the 90%-confidence intervals computed from the bootstrap method.

Citation: Journal of Physical Oceanography 48, 4; 10.1175/JPO-D-17-0263.1

The NIWs, defined as internal waves with frequencies between f and 2f, make a dominant contribution to the energy exchange rate (Fig. 5a). The important role of NIWs is partially due to their energetic horizontal wave stress as indicated by the frequency spectrum of horizontal velocity (Fig. 5b). With the aid of the polarization relations of internal waves, the decomposition of horizontal wave stress in the frequency domain is given by
e4
e5
As indicated by Eqs. (4) and (5), the horizontal stress induced by internal waves of frequency ω depends on three factors: Φuu(ω) + Φυυ(ω), the frequency spectrum of horizontal velocity of internal waves; (ω2f2)/(ω2 + f2), related to the eccentricity of the velocity hodograph of internal waves; and (k2l2)/(k2 + l2) or kl/(k2 + l2), a function of the horizontal azimuth of internal waves. The first two factors are determined by the frequency spectrum of internal waves and measure the intensity of wave stress. Because of the pronounced peak of Φuu + Φυυ around the inertial frequency, (ω2f2)/(ω2 + f2)(Φuu + Φυυ) exhibits significant enhancement in the near-inertial band (Fig. 5), contributing to the important role of NIWs in the energy exchange between low-frequency flows and internal waves.
Fig. 5.
Fig. 5.

(a) Partition of P in the frequency domain derived from Eq. (2) and (b) the frequency spectrum of horizontal velocity (thick gray line) and that multiplied by the factor (ω2f2)/(ω2 + f2) (thin black line).

Citation: Journal of Physical Oceanography 48, 4; 10.1175/JPO-D-17-0263.1

NIWs are primarily generated by the wind work on near-inertial motions in the surface mixed layer (Wunsch and Ferrari 2004). Furthermore, as the moorings are close to the critical latitude 28.9°N for the M2 tide, the subharmonic parametric instability (PSI) may also play a role in generating NIWs (DiMarco et al. 2000; Zhang et al. 2010). In both cases, the initial propagation azimuth of generated NIWs is expected to be independent from the horizontal velocity gradient of low-frequency flows. Therefore, there would be no permanent energy exchange between NIWs and low-frequency flows unless low-frequency flows had a permanent influence on the azimuth of NIWs. In the following section, we will explore the underlying mechanism responsible for this permanent influence.

4. Discussion

a. Potential mechanisms

Two mechanisms have been proposed to account for the permanent influence of low-frequency flows on the azimuth of NIWs. Müller (1976) suggested that the low-frequency flows can permanently affect the azimuth through a relaxation effect of internal waves (i.e., the restoring of disturbed internal wave field to their equilibrium state), leading to negative correlation of 〈uu〉 − 〈υυ〉 to Sn and 〈〉 to Ss. In this case, there should be a permanent energy transfer from low-frequency flows to NIWs regardless of the sign of the OW parameter (Müller 1976). This is, however, not supported by the observations (Fig. 4). Furthermore, the regression of 〈uu〉 − 〈υυ〉 versus Sn suggests that there is no evident negative correlation (Fig. 6a) when the OW parameter is negative. The estimated horizontal viscosity, given by υh = −(〈uu〉 − 〈υυ〉)/Sn, is not significantly different from zero (Fig. 6a) with a 90%-confidence interval between −7 and 7 m2 s−1. Similar is the case for the regression of −2〈〉 versus Ss (Fig. 6b). Therefore, the observations are not supportive of the mechanism proposed by Müller (1976).

Fig. 6.
Fig. 6.

(a) Scatterplot of vertical mean −(〈uu〉 − 〈υυ〉) vs vertical mean Sn in the period with negative OW parameter. The solid line is the linear regression with its 90%-confidence interval (gray dashed). (b) As in (a), but for −2〈〉 vs Ss.

Citation: Journal of Physical Oceanography 48, 4; 10.1175/JPO-D-17-0263.1

The other mechanism is the wave capture mechanism (Bühler and McIntyre 2005). To a large extent, low-frequency flows in the ocean are geostrophically balanced (Pedlosky 1987). Provided that there is a scale separation between waves and low-frequency background flows, the evolution of internal waves in geostrophic flows can be described by a set of ray-tracing equations (Lighthill 1978):
e6
e7
e8
where K = (k, l, m) is the wave vector, Ω is the dispersion relation, and d/dt is the time derivative along the ray path. It should be noted that the horizontal scale of NIWs in the subthermocline could be comparable to that of background flows. However, previous numerical simulations suggest that the ray tracing still works reasonably well even when scales of waves and background flows are comparable (e.g., Bender and Orszag 1978; Kunze 1985).

Despite the complexity of a three-dimensional geostrophic flow, Jones (1969) and Bühler and McIntyre (2005) proposed a simple characterization of interactions between internal waves and geostrophic flows. In this theoretical framework, the evolution of the internal wave vector is determined by the OW parameter under the condition that the background buoyancy frequency and gradient of geostrophic flow are approximately constant along the ray (see appendix B for details). When the OW parameter is negative and low-frequency flow is dominated by vorticity, horizontal wave vectors of internal waves rotate with oscillating magnitude and the geostrophic flow has no permanent influences on internal waves. However, when the OW parameter is positive and low-frequency flow is dominated by strain, the azimuth of horizontal wave vectors asymptotically points toward a direction solely determined by the geostrophic velocity gradient. Along this direction, the magnitude of wave vector exhibits exponential growth at large time. As the group velocity of internal waves decreases with the increasing magnitude of wave vector, internal waves will be eventually captured by geostrophic flows [i.e., the wave capture (Bühler and McIntyre 2005) or shrinking catastrophe (Jones 1969)]. In this wave capture scenario, internal waves insert a permanent influence on the vortical component of geostrophic flows, whether they break or not. Furthermore, there is horizontally a negative stress–strain correlation for linear internal waves (i.e., a positive horizontal viscosity in a flux-gradient closure), which leads to a permanent energy transfer from geostrophic flows to internal wave field (see appendix B for details). This is distinctly different from the mechanisms proposed by Müller (1976), Xie and Vanneste (2015), and Wagner and Young (2016), which predict a permanent energy transfer from low-frequency flows to internal waves regardless of the sign of OW parameter.

Our observation that a permanent energy transfer occurs only when strain dominates relative vorticity in the low-frequency flow is more consistent with the prediction from the wave capture mechanism. In the following analysis, we attempt to explore the role of wave capture mechanism in the observed permanent energy exchange between low-frequency flows and internal waves to the extent that the available data permit.

b. Role of wave capture mechanism in the energy exchange

In the wave capture theory, the azimuth θ of horizontal wave vector asymptotically approaches θa = arctan[−(Ux + λ)/Vx] or θa + π where λ is half the square root of OW parameter (see appendix B for details). Along θa or θa + π, internal waves extract energy from low-frequency flows. We introduce θw, defined as the minimum between |θaθ| and |θa + πθ|, to measure the role of wave capture in energy exchange in the observed data. We expect θw to be uniformly distributed within 0°–90° when the wave capture mechanism has a negligible influence. If the wave capture mechanism plays a role, we expect the mean value be significantly smaller than 45°. Figure 7 shows the mean θw when the OW parameter is positive. While the mean θw for high-frequency internal waves is close to 45°, the NIWs are associated with a noticeable reduction of mean θw, implying that NIWs are no longer horizontally isotropic. This is consistent with the dominant contribution of NIWs to the energy exchange under the positive OW parameter condition.

Fig. 7.
Fig. 7.

Mean θw and its 90%-confidence interval computed from the bootstrap method for different wave frequencies.

Citation: Journal of Physical Oceanography 48, 4; 10.1175/JPO-D-17-0263.1

The mean θw for NIWs is only about 3° less than 45°, although the reduction is statistically significant at 90% level using a bootstrap method, implying that the wave capture mechanism rarely works sufficiently long to reach its eventual stage in the real ocean. In the wave capture theory, the velocity gradient of low-frequency flows is assumed to be constant along the ray. However, this assumption is difficult to hold in reality owing to the spatial and temporal variability of low-frequency flows, leading to variation of θa along the ray. If θa changes too rapidly, internal waves may not have sufficient time to respond to the change of θa. In this case, it is difficult for θw to reach its asymptotic value (i.e., 0°). Furthermore, the wave capture theory assumes low-frequency flows to be geostrophic. Otherwise, a positive OW parameter does not guarantee the occurrence of wave capture. The influence of ageostrophy can be measured by the difference of eigenvalues of ray-tracing equations (i.e., λ and λ′) derived from two different definitions for the OW parameter. The OW parameter can be alternatively defined as (ς = VxUy is the relative vorticity) in addition to 4(VxUyUxVy). These two definitions are identical when low-frequency flows are exactly geostrophic but differ significantly in the presence of strong ageostrophy.

Figure 8 shows a time series of the real part of eigenvalues and computed based on two different definitions of the OW parameter. In most periods, they differ by a small amount (less than 20%), suggesting that the low-frequency flows are approximately geostrophic. But pronounced difference also exists. As mentioned above, the wave capture mechanism is expected to work efficiently only when λ and λ′ vary slowly in both the space and time domain2 and are almost identical to each other. We identify three periods (referred to as wave-capture-favoring periods henceforth) that satisfy the above requirements by simple visual inspection of Fig. 8. These periods are marked by gray shades in Fig. 8. In these wave-capture-favoring periods, the energy transfer from the low-frequency flows to internal waves is significantly enhanced (Fig. 9a). The mean energy transfer rate increases to 17.5 × 10−11 m2 s−3. In contrast, the mean energy transfer rate reduces to 2.8 × 10−11 m2 s−3 in the remaining positive-OW periods (Fig. 9b). In particular, while the length of wave-capture-favoring periods is less than 18% of the entire period, it accounts for more than 75% of the total energy transfer.

Fig. 8.
Fig. 8.

Time series of vertical mean Re(λ) (solid red line) and Re(λ′) (dashed blue line) within 245–450 m. The shaded regions denote the wave-capture-favoring period.

Citation: Journal of Physical Oceanography 48, 4; 10.1175/JPO-D-17-0263.1

Fig. 9.
Fig. 9.

PDF of P during the (a) wave-capture-favoring periods and (b) remaining positive-OW periods.

Citation: Journal of Physical Oceanography 48, 4; 10.1175/JPO-D-17-0263.1

During the wave-capture-favoring periods, the PDF of θw for NIWs decreases monotonically as θw increases (Fig. 10a), suggesting a highly anisotropic wave field. This provides further evidence that the wave capture mechanism is operating in the real ocean and responsible for the energy exchange between low-frequency flows and NIWs. However, it should be noted that even in the wave-capture-favoring periods it seems difficult for θw to reach its asymptotic value (i.e., 0°). The probability of θw < 10° is only about 15%. The low probability might be partially attributed to the measurement noise. Moreover, it implies that the wave capture mechanism could only be partially realized and have difficulty reaching its eventual stage in the real ocean. To examine whether the PDF shown in Fig. 10a is consistent with the partial realization of wave capture mechanism, we further performed ray-tracing experiments.

Fig. 10.
Fig. 10.

PDF of θw (a) during the wave-capture-favoring periods, and simulated by the ray-tracing experiment with (b) realistic and (c) idealized settings.

Citation: Journal of Physical Oceanography 48, 4; 10.1175/JPO-D-17-0263.1

c. Ray-tracing experiments

As NIWs are primarily generated by surface wind forcing (Wunsch and Ferrari 2004), the modeled NIWs are initially injected at 245 m (the shallowest depth sampled by the ADCPs) and radiate downward. The initial intrinsic wave frequency is chosen randomly between f and 2f using the observed frequency spectrum of internal wave horizontal velocity as its PDF (Fig. 5b) following Henyey et al. (1986). Similarly, the initial vertical wavenumber is generated using the observed vertical wavenumber spectrum of near-inertial current as its PDF. Then the initial horizontal wave vector magnitude kh can be uniquely determined based on the dispersion relation while the initial azimuth is assumed to satisfy the uniform distribution within 0°–360°.

The stratification is computed from the CTD profiles collected during the servicing of the moorings. However, using a constant stratification rather than the observed one does not show a significant impact on the simulated PDF of θw, implying that the variation of stratification along the ray does not significantly affect modeled wave propagation. The geostrophic flows between 245 and 450 m are modeled as follows:
e9
e10
where UT and VT represent the horizontal structure of geostrophic flow and F(z) represents the vertical structure. The F(z) is set to be a linear function decreasing from unity at 245 m to 0.7 at 450 m, which is consistent with the observations derived from the ADCPs (Fig. 2). The UT and VT are derived from the sea level anomaly using the geostrophic relation:
e11
e12
where η is the merged sea level anomaly measured by satellites, g is the gravity acceleration, and α is a scaling factor accounting for the difference of geostrophic flow intensity between the sea surface and 245 m. Here α is chosen in such a way that the variance of UT and VT is the same as the observed value at 245 m.

Figure 11 displays a snapshot of the surface OW parameter computed from the SSALTO/Data Unification and Altimeter Combination System (DUACS) multimission altimeter sea level anomaly produced and distributed by the Copernicus Marine and Environment Monitoring Service (CMEMS; http://www.marine.copernicus.eu). Given that the altimeters might not work well in the coastal region, the initial injection position for the modeled waves is chosen to mimic the situation in the wave-capture-favoring periods rather than to be the exact location of moorings. This leads to an initial injection position around 92°W, 28°N (referred to as site A henceforth). It is characterized by a λ of ~1 × 10−6 s−1 with weak spatial variation (The value of λ varies within a factor 2 in a 1° × 1° box centered at site A). As low-frequency flows propagate westward at a speed of about 0.1 m s−1 (not shown), λ at site A remains almost unchanged within O(10) days. These features are consistent with the condition during the wave-capture-favoring periods (Fig. 8). Sensitivity tests suggest that choosing other initial projection positions does not have a substantial impact on the results of the ray-tracing experiment as long as these positions mimic the situation in the wave-capture-favoring periods.

Fig. 11.
Fig. 11.

Snapshot (22 Sep 2012) of the OW parameter derived from the sea level anomaly using the geostrophic relation. The sea level anomaly is obtained from the SSALTO/DUACS multimission altimeter products of AVISO.

Citation: Journal of Physical Oceanography 48, 4; 10.1175/JPO-D-17-0263.1

In the ray-tracing experiment, 200 test waves are generated. The PDF of θw does not vary substantially when the number of waves becomes 100 or larger. Therefore, 200 waves should provide a sufficient sample size to compute the statistics. For each test wave, the ray tracing is ended when the wave 1) propagates into regions of a negative OW parameter, 2) breaks determined by the criterion |m| > π/10 rad m−1, or 3) radiates out of the lower bound (i.e., 450 m). A normally distributed random noise with a standard deviation of 15° is added to the records of azimuth of test waves to account for the measurement errors in the observations.

The simulated PDF of θw is qualitatively consistent with the observed one (Fig. 10b). All exhibit a decreasing trend, implying a tendency of θ to approach θa or θa + π. The probability of simulated θw < 10° is about 17%, similar to 15% derived from the observations. It should be noted that the relaxation effects of internal waves are filtered out in the ray-tracing equations. Furthermore, absorption of the waves at the critical layer does not occur in the simulation as the vertical shear of geostrophic flows is weak. Finally, the dispersion relation used in the ray-tracing experiments does not take into consideration the modification of the lower bound of internal wave frequency by relative vorticity of background flow (Kunze 1985). Therefore, the trapping mechanism proposed by Kunze (1985) is also excluded from the ray-tracing experiments. Because of this design, the wave capture mechanism is the only mechanism that can have a permanent influence on θw. Therefore, the similarity between the simulated and observed PDF of θw provides further evidence that the wave capture mechanism indeed plays a role during the wave-capture-favoring periods and is responsible for permanent energy exchange between the low-frequency flows and NIWs in the northern Gulf of Mexico.

Finally, it should be noted that θw in the ray tracing simulations is not highly concentrated in the vicinity of 0° (i.e., the asymptotic value predicted from the wave capture mechanism). It might be an artifact of the random noise added to the records of azimuth of test waves. It could also imply that even in the wave-capture-favoring periods, the wave capture mechanism is difficult to be fully realized because of the slow but nonnegligible change of background flow velocity gradient along the ray path. To distinguish these two effects, we perform an idealized ray-tracing experiment in which the velocity gradient of background flow is fixed as the value at site A so that the wave capture mechanism can be fully realized. As shown in Figs. 10b,c, the PDF of θw in the idealized ray-tracing experiment is significantly more concentrated around 0° than that in the realistic ray-tracing experiment. The probability of θw < 10° is increased from 17% in the realistic ray-tracing experiment to 24% in the idealized ray-tracing experiment. The different θw features between the idealized and realistic ray-tracing experiments imply that the wave capture mechanism rarely reaches its eventual stage in the real ocean mainly owing to the change of background flow velocity gradient along the ray path.

5. Conclusions

Long-term mooring array in the Gulf of Mexico is used to examine the energy exchange between internal waves and low-frequency flows and its underlying mechanisms. Important findings of this study are summarized as follows.

  1. When the OW parameter is negative, the observed energy exchange between low-frequency flows and internal waves is almost zero. However, there is an evident energy transfer rate of O(10−10) m2 s−3 from low-frequency flows to internal waves when the OW parameter is positive. NIWs play a dominant role in the energy exchange. This is due to both their energetic wave stress and their relatively stronger interaction with low-frequency flows, compared to high-frequency internal waves.
  2. There is some evidence that the wave capture mechanism can be partially realized in the real ocean and is responsible for the observed permanent energy transfer from low-frequency flows to NIWs. In the periods favoring the occurrence of wave capture mechanism, NIWs become anisotropic and their horizontal azimuth exhibits evident tendency toward that predicted from the wave capture mechanism, leading to pronounced energy transfer from low-frequency flows to NIWs.

In the eventual stage of wave capture mechanism, the spatial scale of internal waves decreases to zero, leading to wave breaking. Accordingly, the wave capture mechanism is typically conceived as a dissipation mechanism for internal waves with the energy transfer from low-frequency flows to internal waves being a transient process. However, the observational evidence suggests that the wave capture mechanism rarely works long enough to reach its eventual stage in the real ocean probably due to the spatial and temporal variations of velocity gradient of background flows. This partial realization of the wave capture mechanism leads to a permanent energy transfer from low-frequency flows to internal waves but does not necessarily trigger wave breaking. Such a scenario is different from that in the atmosphere where the wave capture mechanism can work efficiently to reach its eventual stage [see Plougonven and Zhang (2014) and references therein]. The lower efficiency of the wave capture mechanism in the ocean than the atmosphere might be partially due to the smaller horizontal scale of background flow in the ocean that leads to more rapid change of background velocity gradient along the ray path. It might also result from the lack of clear scale separation between the background flow and NIWs in the ocean so that the ray-tracing theory is only qualitatively valid.

The role of energy transfer from low-frequency flows to internal waves in the energy budget of internal waves can be assessed based on its comparisons to the other major energy sources of internal waves. It is generally believed that winds provide a primary energy source for internal waves in the upper ocean (Wunsch and Ferrari 2004). In the northern Gulf of Mexico, the estimated wind work on internal waves is around 1 mW m−2 (Jing et al. 2015). Assuming that about 30% of the near-inertial wind work radiates downward into the region below 245 m (Alford et al. 2012; Zhai et al. 2009; Furuichi et al. 2008), the winds could provide an energy source of 0.3 mW m−2 for the internal wave field below 245 m. Meanwhile, provided that the estimated vertical mean value of P within 245–450 m is also representative in the deeper region, the vertically integrated energy transfer rate below 245 m is around 0.06 mW m−2, which is about 20% of the wind-induced energy source (~0.3 mW m−2). This ratio is probably underestimated as the moorings used in this study are not sufficiently close to resolve all the low-frequency velocity gradient. Therefore, the energy exchange between low-frequency flows and internal waves may play an important role in furnishing internal waves in the deep ocean.

Acknowledgments

Z. J. is supported by National Science Foundation of China (NSFC) Grant 41776006. P.C. acknowledges the support from the National Program on Key Basic Research Project (973 Program) Grant 2014CB745000 and the China National Global Change Major Research Project 2013CB956204. This research was made possible in part by a grant from The Gulf of Mexico Research Initiative (SA12-09/GoMRI-006) and in part by a grant from the Texas General Land Office (X0006375-PT). We thank Dr. Alexis Lugo-Fernandez and the BOEM for the loan of many of the current meters and mooring flotation used in this study. The authors thank John Walpert and the Geochemical and Environmental Research Group mooring team for performing the mooring construction, deployment, servicing, and recovery operations. The authors also thank the Captain Nicholas Allen and the crew of the R/V Pelican for their service during these operations. Data used in this study are publicly available through the Gulf of Mexico Research Initiative Information and Data Cooperative (GRIIDC) (https://data.gulfresearchinitiative.org; UDI: R1.x137.130:0006, R1.x137.130:0010, R1.x137.130:0011, R1.x137.130:0012, R1.x137.130:0011).

APPENDIX A

Noise Level of ADCP Records and Its Impact on P

The noise level of horizontal velocity measured by ADCP can be estimated from its vertical wavenumber spectrum (Kunze et al. 2006):
ea1
where φA(m) represents the spectrum derived from the range-averaged ADCP records, φV(m) the spectrum of true horizontal velocity, φN(m) the spectrum of measurement noise, and SR(m) the spectral attenuation due to the range averaging inherent in the moored ADCP measurements, defined as
ea2
where Δzr is the bin size, Δzt the length of transmitted sound pulse, and sinc(x) = sin(πx)/(πx) (Polzin et al. 2002). For our ADCP data, Δzr = Δzt = 16 m.
In the real ocean, φV(m) is typically red whereas φN(m) is white assuming that the measurement errors are independent and identically distributed (iid) random variables. It is thus expected that φN(m) might dominate φV(m) at sufficiently large wavenumbers. Figure A1 displays derived from the range-averaged ADCP records within 245–450 m at M1, M2, and M3. Consistent with our hypothesis, is in general red at small wavenumbers and becomes almost flat for m > 0.08 rad m−1. Therefore, the value of φN(m) can be approximated as the value of within the wavenumber range where it is flat. With the estimate of φN(m) available, the noise level of ADCP records can be computed as
ea3
where σ is the standard deviation of ADCP measurement noise and mc = πz is the Nyquist wavenumber. According to the values shown in Fig. A1, σ is estimated to be about 0.01 m s−1.
Fig. A1.
Fig. A1.

The vertical wavenumber spectrum of range-averaged ADCP records within 245–450 m at M1, M2, and M3. The grey dashed line denotes the estimate for φn(m).

Citation: Journal of Physical Oceanography 48, 4; 10.1175/JPO-D-17-0263.1

The noise in the ADCP records will propagate into the estimates for the energy exchange rate P between low-frequency flows and internal waves. To evaluate this effect, we perform 200 Monte Carlo simulations. In each simulation, an iid Gaussian random noise with zero mean and σ = 0.01 m s−1 is added to the ADCP records and P is recomputed following the same procedures in section 2. For the vertical mean value within 245–450 m, it is found that the 90%-confidence interval of P derived from 200 simulations is (4.1 ± 0.2) × 10−11 m2 s−3. We remark that such an uncertainty (± 0.2 × 10−11 m2 s−3) in P due to the measurement error of ADCP records is much smaller than that (± 0.6 × 10−11 m2 s−3) resulting from the randomness of the internal waves and low-frequency flows (the latter can be estimated by applying a bootstrap method to the time series shown in Fig. 3b). Therefore, we conclude that the measurement error of ADCP records has a negligible impact on the estimates of P.

APPENDIX B

Wave Capture

Under the geostrophic approximation, Eqs. (6) and (7) can be written in the matrix form:
eb1
where Sn = UxVy is the normal component of strain, Ss = Uy + Vx the shear component, and ς = VxUy the relative vorticity. Here Sn, Ss, and ς are assumed to be constant along the ray of internal waves. As the strain matrix is symmetric, it is always possible to choose the directions of the orthogonal axes (i.e., the principal axes) of reference so that the nondiagonal elements are zero:
eb2
where variables in the principal axes are denoted by primes. Note that the total strain variance and relative vorticity are invariants under the rotational transform so that and ς′ = ς.
The horizontal wave vector can be reexpressed in polar coordinates:
eb3
where kh is the magnitude of horizontal wave vector and θ′ is the azimuth. Substituting Eq. (B3) into Eq. (B2) yields
eb4
eb5

Equation (B5) indicates that both the strain and relative vorticity contribute to the rotation of wave vector. While the vorticity-induced angular velocity is constant, the strain-induced angular velocity depends on θ′. The strain always tends to rotate the wave vector toward the direction along which the background flow converges.

When the OW parameter is negative , the strain-induced angular velocity is always smaller than the vorticity-induced angular velocity. In this case, the wave vector rotates unceasingly and its magnitude oscillates. However, strain- and vorticity-induced angular velocity can cancel each other when the OW parameter is positive . There are four fixed points for θ′ in this case, among which two are attractors ( and ) and two are repellers ( and ). It can be demonstrated that kh will increase exponentially with time at attractors but decrease exponentially at repellers. Therefore, for almost all initial conditions, θ′ asymptotically approaches either of its attractors and kh increases exponentially with time, leading to the wave capture.

In the principal axes, Eq. (2) is reduced to
eb6
Substituting Eq. (4) into Eq. (B6) yields
eb7
In the wave capture scenario, and :
eb8
It can be easily demonstrated that tan2θ′ < 1 when and tan2θ′ > 1 when . Therefore, P is always positive in the wave capture scenario, suggesting that energy of low-frequency flows is transferred to internal wave field. Finally, it should be noted that there are two attractors and for θ′ in the wave capture scenario. However these two attractors lead to the identical P as P only depends on tan2θ′. In the sense of energy transfer, we do not need to distinguish between these two attractors.

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1

This corresponds to a vertically integrated (245–450 m) energy transfer rate of 0.01 mW m−2. As a reference, the wind power on near-inertial internal waves in the same region is estimated to be around 1 mW m−2 (Jing et al. 2015).

2

The spatial variation of velocity gradient cannot be directly estimated owing to the limited data. However, a slow variation of velocity gradient in the time domain implies a slow variation in the space domain because the low-frequency flows propagate westward as a result of the β effect.

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