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Trapped diurnal internal tides, propagating semidiurnal internal tides, and mixing estimates in the California Current System from sustained glider observations, 2006–2012
Deep-Sea Research II ∎ (∎∎∎∎) ∎∎∎–∎∎∎
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Trapped diurnal internal tides, propagating semidiurnal internal tides, and mixing estimates in the California Current System from sustained glider observations, 2006–2012 T.M. Shaun Johnston n, Daniel L. Rudnick Scripps Institution of Oceanography, University of California, San Diego, 9500 Gilman Dr # 0213, La Jolla, CA 92093-0213, USA
art ic l e i nf o
Keywords: Mixing parameterizations Internal tides Seasonal cycle Cross-shore decay
a b s t r a c t From 2006–2012, along 3 repeated cross-shore transects (California Cooperative Oceanic Fisheries Investigations lines 66.7, 80, and 90) in the California Current System, 33 609 shear and 39 737 strain profiles from 66 glider missions are used to estimate mixing via finescale parameterizations from a dataset containing over 52 000 profiles. Elevated diffusivity estimates and energetic diurnal (D1) and semidiurnal (D2) internal tides are found: (a) within 100 km of the coast on lines 66.7 and 80 and (b) over the Santa Rosa-Cortes Ridge (SRCR) in the Southern California Bight (SCB) on line 90. While finding elevated mixing near topography and associated with internal tides is not novel, the combination of resolution and extent in this ongoing data collection is unmatched in the coastal ocean to our knowledge. Both D1 and D2 internal tides are energy sources for mixing. At these latitudes, the D1 internal tide is subinertial. On line 90, D1 and D2 tides are equally energetic over the SRCR, the main site of elevated mixing within the SCB. Numerous sources of internal tides at the rough topography in the SCB produce standing and/or partially-standing waves. On lines 66.7 and 80, the dominant energy source below about 100 m for mixing is the D1 internal tide, which has an energy density of the D2 internal tide. On line 80, estimated diffusivity, estimated dissipation, and D1 energy density peak in summer. The D1 energy density shows an increasing trend from 2006 to 2012. Its amplitude and phase are mostly consistent with topographically-trapped D1 internal tides traveling with the topography on their right. The observed offshore decay of the diffusivity estimates is consistent with the exponential decay of a trapped wave with a mode-1 Rossby radius of 20–30 km. Despite the variable mesoscale, it is remarkable that coherent internal tidal phase is found. & 2014 Elsevier Ltd. All rights reserved.
1. Introduction Mixing in the thermocline is produced mainly by breaking internal waves (Gregg, 1989; MacKinnon et al., 2013), which are forced predominantly by tides and winds (Alford, 2003; Ferrari and Wunsch, 2009; Garrett and Kunze, 2007). Much of the internal wave energy is in low vertical modes (i.e., in waves with horizontal wavelengths of 100 km or more) and propagates away from generation sites (Alford, 2003; Zhao and Alford, 2009). Smallerscale internal waves may break near their generation site and contribute to mixing (Gregg et al., 1986; Nakamura et al., 2010; Alford and Gregg, 2001; Klymak et al., 2006). Here, we consider the contribution of not only the freely-propagating semidiurnal (D2) internal tide (Garrett and Kunze, 2007), but also the subinertial
n
Corresponding author. Tel.: þ 1 858 534 9747. E-mail address: [email protected] (T.M.S. Johnston).
diurnal (D1) internal tide, which is topographically-trapped near its generation site. The latter topic has received much less attention in the literature. In our study area in the coastal ocean off of California, D1 internal tides are often more energetic than D2 in our observations and in previous work around the Southern California Bight (SCB) (Beckenbach and Terrill, 2008; Kim et al., 2011; Nam and Send, 2011). Poleward of 301, D1 internal waves are subinertial and evanes1 cent, since the Coriolis frequency (f) is 4 1 cycle day . Considerable mixing may arise due to topographically-trapped internal tides because they likely dissipate in the same area where they are forced (Padman and Dillon, 1991; Nakamura et al., 2010), although alongshore propagation is possible with dissipation elsewhere at the topography. Such mixing may contribute to water mass formation (Tanaka et al., 2010) and higher primary productivity (Tanaka et al., 2013). Energy loss from the D1 barotropic tide is a maximum around the Pacific rim (Egbert and Ray, 2003). A local maximum in this dissipation is found along the west coast of
http://dx.doi.org/10.1016/j.dsr2.2014.03.009 0967-0645/& 2014 Elsevier Ltd. All rights reserved.
Please cite this article as: Johnston, T.M.S., Rudnick, D.L., Trapped diurnal internal tides, propagating semidiurnal internal tides, and mixing estimates in the California Current System from.... Deep-Sea Res. II (2014), http://dx.doi.org/10.1016/j.dsr2.2014.03.009i
T.M.S. Johnston, D.L. Rudnick / Deep-Sea Research II ∎ (∎∎∎∎) ∎∎∎–∎∎∎
2
North America beyond 301N. The sink for this energy is likely the topographically-trapped D1 internal tide, which in turn dissipates turbulently. To assess the contributions of topographically-trapped D1 and freely-propagating D2 internal tides to mixing in the coastal ocean, we use an extensive dataset to estimate diffusivity using parameterizations based on observed current shear and/or isopycnal strain (Gregg, 1989; Polzin et al., 1995; Gregg et al., 2003). Six years of sustained glider observations in the California Current System (CCS) along three repeated California Cooperative Oceanic Fisheries Investigations (CalCOFI) cross-shore transects (lines 66.7, 80, and 90 in Fig. 1) provide over 52 000 profiles, at Oð10 m) vertical and 3-km horizontal resolution from the continental slope to 300–500 km offshore (Davis et al., 2008; Todd et al., 2011a, 2012). With this unique combination of spatial and temporal coverage, we can examine the seasonal and cross-shore structure of mixing estimates and internal tides. For example, D1 internal tides may control the cross-shore decay scale at the continental slope and the seasonal cycle of mixing at the northern end of the SCB. At the Santa Rosa-Cortes Ridge (SRCR), a prominent feature of the SCB, D1 and D2 internal tides have similar energies and may contribute equally to mixing. Next, we provide some background on mixing parameterizations (Section 2.1), previous observations of mixing in the SCB and other coastal locations (Section 2.2), and possible energy sources for mixing (internal waves and frontogenesis in Sections 2.3 and 2.4). In Section 3, our methods are described including a basic outline of the mixing parameterizations with further details in Appendix A. Section 4 shows an example transect from line 90 and time-mean transects (i.e., binned in depth and cross-shore distance) of the data and harmonic fits from lines 80 and 90 (line 66.7 is similar to line 80). In Section 5, depth-mean diffusivity estimates
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2. Background 2.1. Finescale mixing parameterizations Direct measurements of either mixing via tracer release or turbulent fluctuations via specialized microstructure instruments occur of necessity during infrequent research cruises of usually several weeks duration. Only recently moored microstructure measurements have become possible (Moum et al., 2013). However, finescale or Oð10m) scale profiles of shear and strain are readily available from many platforms including the gliders described in this paper. Turbulent dissipation (ϵ) and diapycnal diffusivity (K ρ ) can be estimated from these finescale measurements using parameterizations based on the idea of a downscale energy transfer via weakly interacting internal waves (Gregg, 1989; Polzin et al., 1995; Gregg et al., 2003; Kunze et al., 2006; MacKinnon et al., 2013). Assuming there is a steady state transfer, by measuring the finescale variance in shear and strain, we estimate the downscale energy transfer or dissipation at a scale much larger than the actual turbulence. The parameterizations relate ϵ to the ratio of the observed finescale variance to the variance in the empirical Garrett–Munk (GM) spectra of the internal wave field. The GM spectrum is used to compare internal wave energy levels in locations with different stratification and latitude. For example, over the SRCR, a site of internal tidal generation, some example shear and strain spectra are within a factor of three of GM (Appendix A). For further comparison, the mission-mean spectrum from glider measurements east of Luzon Strait, which averages over regions of both weak and strong internal wave activity, is consistent with a level of 4 times GM (Rudnick et al., 2013). These values appear reasonable and reinforce the well-known utility of the GM spectrum in a wide variety of conditions. Finescale parameterizations have been applied to diverse data sets over large areas with appreciable tidal and inertial signals (Kunze et al., 2006; MacKinnon et al., 2013; Whalen et al., 2012; Waterhouse et al., 2014). These methods under a variety of conditions are considered accurate to a factor of 2–4 at best (MacKinnon et al., 2013, and references therein) compared to specialized microstructure instruments, which measure the actual turbulent fluctuations (Klymak and Nash, 2009; Thorpe, 2005). In this paper, we estimate mixing in the coastal ocean using such parameterizations (Section 3.3 and Appendix A). 2.2. Previous observations of mixing in coastal waters
32o
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are binned in time or cross-shore distance and are then related to similarly averaged internal tides and mesoscale flows. A seasonal cycle is composited for line 80 (Section 5.4). A discussion and summary of our findings follow in Sections 6 and 7.
Depth [km]
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Fig. 1. Example tracks (orange) from glider missions 12801401, 12801101, and 12803001 along (a) line 66.7 and (b) lines 80 and 90 are shown over the bathymetry (blue colors) off the California coast. The origins of the cross-shore coordinate (x) are denoted by the circles near Monterey, Point Conception, and Dana Point. The Santa Rosa-Cortes Ridge (SRCR) extends roughly south-southeast from Point Conception and is a key topographic feature on line 90. Lines 66.7 and 80 have relatively narrow shelves, while line 90 extends over the rough bathymetry of the Southern California Bight. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)
With rough topography, varying stratification, and numerous local and remote sources of internal waves in our study region, considerable spatial and temporal variability of mixing is expected. Using our methods, we can examine variability in the cross-shore direction along with tidal and seasonal variability, which complements previous moored or ship-based process studies, which have greater temporal resolution, but are limited in temporal or spatial extent. Such variability in mixing over a semidiurnal tidal period, over a spring-neap cycle, and over variable topography is found in a bay situated between two distinct sources of internal tides near Oahu, Hawaii (Alford et al., 2006; Martini et al., 2007). Also the two sources of internal tides are of unequal strength and produce partially-standing internal waves. This interference pattern varies
Please cite this article as: Johnston, T.M.S., Rudnick, D.L., Trapped diurnal internal tides, propagating semidiurnal internal tides, and mixing estimates in the California Current System from.... Deep-Sea Res. II (2014), http://dx.doi.org/10.1016/j.dsr2.2014.03.009i
T.M.S. Johnston, D.L. Rudnick / Deep-Sea Research II ∎ (∎∎∎∎) ∎∎∎–∎∎∎
2.3. Internal waves Shear and strain from internal waves produce instabilities and mixing (Alford and Pinkel, 2000). Below we briefly cover some features of near-inertial, D1, and D2 internal waves relevant to the SCB and mixing. In general, shear in the internal wave band is dominated by NIW (Alford and Gregg, 2001; Pinkel, 1985). In the SCB, wind stress variance is greatest from March–June [Table 1 in Hickey, 1979]. The region near Point Conception at the northern edge of the SCB displays a local maximum in a model of wind work to inertial motions in the mixed layer, i.e. a precursor to NIW generation in the thermocline (Figure 2 in Simmons and Alford, 2012). The magnitude of this input is similar to stormier climates near 451N on the west coast of North America. Trapped D1 internal tides are often more energetic than D2 internal tides in the SCB. These waves may have considerable impact on mixing as noted earlier (Section 1) because there are likely numerous sites of generation and dissipation in close proximity at rough topography (Tanaka et al., 2013). Elsewhere, at a seamount in the North Pacific near 321N, dissipation is strong enough to dissipate the trapped D1 motions within 3 days, which makes for a strongly forced and damped system (Kunze and Toole, 1997). In the SCB, surface drifters in this area show diurnal/inertial oscillations with decay scales of 10 days (Poulain, 1990). There are numerous sources of D2 internal tides along the continental slope and over the rough topography of the SCB (Beckenbach and Terrill, 2008; Buijsman et al., 2012). Internal tidal generation may produce local mixing (Johnston et al., 2011a; Klymak et al., 2006). Propagating internal tides may encounter topography, scatter to smaller scales, and break (Johnston and Merrifield, 2003; Johnston et al., 2003; Kelly et al., 2012; Martini et al., 2011; Nash et al., 2004). 2.4. Fronts Other relevant sites of elevated mixing includes fronts (Hoskins and Bretherton, 1972; Johnston et al., 2011b), which are ubiquitous in the SCB (e.g., Davis et al., 2008; Powell, 2014). Frontogenesis can lead to mixing at depth on the dense, cyclonic side of fronts where
the gradients (i.e., shear and strain) are largest (Hoskins and Bretherton, 1972). It is unclear what are the relative contributions to mixing from frontogenesis or internal waves which are trapped, reflected, or refracted at a front (Johnston et al., 2011b). Elevated mixing at fronts can lead to enhanced primary productivity (Hales et al., 2009; Li et al., 2012). Three possible issues are noted with respect to estimating mixing at fronts. First, if mixing at fronts is not due to internal waves, then the mixing parameterizations used here do not apply. Second, there is some question of how much the internal wave field is modified in the vicinity of background shear and whether the empirical GM spectrum (Section 3.3) is applicable (Munk, 1981). Third, we note persistent fronts at locations where internal tides are strong on both lines 80 and 90, which unfortunately hampers discrimination of their effects. For the first two issues, we note our spectra resemble GM (Appendix A), which provides reasonable confidence in our approach. These parameterizations essentially use a Richardson number criterion as an indicator of mixing and have proven useful across a wide range of conditions (Section 2.1).
3. Methods 3.1. Data Since 2006/2007, three cross-shore transects have been almost continuously covered by underwater gliders to 500-m depth (Davis et al., 2008; Todd et al., 2011b). These transects follow CalCOFI lines 66.7, 80, and 90 which begin near Monterey Bay, Point Conception, and Dana Point and extend 300–550 km offshore (Fig. 1). One glider mission lasts about 100 days, with each cross-shore transect taking about 3 weeks (Fig. 2). In this analysis, over 19 000 density and velocity profiles from October 2006 to
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over the spring-neap cycle and with varying stratification at or between the sources. Models generally have greater turbulence at boundaries, but do not reproduce observed midwater column mixing with a variety of turbulence parameterizations (Wijesekera et al., 2003). In the coastal waters off Oregon, the superposition of shear from inertial and semidiurnal internal waves with shear from a coastal current in thermal wind balance leads to midwater column mixing, when none of the individual components would trigger a shear instability (Avicola et al., 2007). It is now more common for regional models to include both mesoscale and tidal phenomena, but assessment of their mixing parameterizations is lacking perhaps due to a lack of sufficient temporal and spatial extent of observations for comparison. The depth-mean mixing estimates (excluding the mixed layer but including the thermocline to a depth of 362 m) in this paper may prove useful in this respect. Previous tracer release experiments and microstructure measurements in the SCB show elevated K ρ at the steep topography, which can be as high as Oð10 4 m2 s 1 Þ (Gregg and Kunze, 1991; Ledwell and Watson, 1991; Ledwell and Hickey, 1995; Ledwell and Bratkovich, 1995). This value is much larger than away from the topography, where the mean thermocline values is K ρ 10 5 m2 s 1 (Gregg, 1987). Limited velocity measurements do not identify the processes leading to this mixing near topography, but either trapped D1 or freely-propagating D2 internal tides are likely candidates.
3
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[m] 20 0 20
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z
400 400
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x [km] Fig. 2. An example cross-shore section on line 90 from mission 12803001 for a glider heading toward shore (x ¼0) from 3–27 September 2012 shows (A) potential temperature, isopycnals (black lines, sθ ¼ 24–26:5 kg m 3 in 0.5 kg m 3 intervals), and the mixed layer depth (gray line); (B) salinity; (C) vertical displacements; and (D) strain. Gray shading denotes regions with no data, which are larger for displacement data due to the lowpass averaging in their calculation. Longitude along the line is on the upper axis of each subplot. Additional smaller tick marks on the upper axis of Fig. 2B mark positions of glider profiles.
Please cite this article as: Johnston, T.M.S., Rudnick, D.L., Trapped diurnal internal tides, propagating semidiurnal internal tides, and mixing estimates in the California Current System from.... Deep-Sea Res. II (2014), http://dx.doi.org/10.1016/j.dsr2.2014.03.009i
T.M.S. Johnston, D.L. Rudnick / Deep-Sea Research II ∎ (∎∎∎∎) ∎∎∎–∎∎∎
December 2012 are used on each of lines 80 and 90. This dataset comprises 27 glider missions on line 90 and 22 on line 80. From April 2007 to December 2012, over 14 000 profiles from 17 glider missions on line 66.7 are used. Data plots are available for these missions at http://spray.ucsd.edu. In Fig. 2, the example transect is from mission 12803001, where the identifier comprises a twodigit year (12 for 2012), a one-digit hexadecimal month (8 for August), a three-digit glider serial number (030 for Spray 30), and a two-digit mission number (01 in this case). A Spray glider moves vertically through the ocean by inflating/ deflating an external bladder, while wings provide lift to move the glider forward resulting in a sawtooth pattern in depth and time (or horizontal distance) (Sherman et al., 2001). The payload comprises (a) a pumped Sea-Bird Electronics (SBE) 41CP conductivity-temperature-depth (CTD) instrument from which potential temperature (θ), salinity (S), in situ density (ρ), and potential density (sθ ) are obtained (Fig. 2A, B); (b) a Seapoint chlorophyll a (chl a) fluorometer; and (c) a Sontek 750 kHz acoustic Doppler profiler (ADP) aligned to measure horizontal velocities in five 4-m vertical range bins (Davis, 2010; Todd et al., 2011b). Data are obtained on ascents, when the CTD experiences clean flow. For the purposes of this paper, data and times are averaged in 31 vertical bins, which are 16-m tall, non-overlapping, and centered from 10–490 m. A dive cycle to 500 m and back to the surface is completed every 3 h, which gives a Nyquist frequency of 1/6 cycle h 1 and is sufficient to sample D2 signals (Section 3.2). During a dive cycle, a glider moves about 2.7 km through the water. Vertical displacements are obtained from density differences from a lowpassed mean (Fig. 2C):
ηðz; tÞ ¼
g Δsθ
sθ 〈N 2 〉1:5d
ð1Þ
where z is the vertical coordinate (up is positive), t is time, g is gravitational acceleration, 〈 〉 denotes a running mean over the subscripted intervals i.e., 1.5 days, Δsθ ¼ sθ 〈sθ 〉1:5d is the density deviation from the 1.5-day lowpassed mean, and N2 is the buoyancy frequency. The 1.5-day mean is roughly equivalent to a 30-km mean. Due to this averaging window, η and other variables calculated below with similar averaging will vary slowly in time over periods of O (1 day or 20 km) as in Fig. 2C. Some further details of sampling with a slowly-moving glider are covered in Section 3.2. Strain, ∂z η, is calculated from a first difference (Fig. 2D). Velocity profiles are made similar to lowered acoustic Doppler current profiles from ships (Visbeck, 2002). Depth-mean currents are obtained from a combination of Global Positioning System fixes and a glider's measured attitude (Todd et al., 2009). The depth-mean current and ADP-measured, glider-relative velocities are combined in a linear system of equations that is solved by a least squares method for water velocities (Todd et al., 2011b). Referencing objectively-mapped and vertically-integrated ADPmeasured shear to the depth-mean current produces similar results (Davis, 2010). Hereafter, horizontal currents (u and v) are given in the crossshore and alongshore directions (Fig. 3A, B), which are positive onshore (x) and to the approximately north–northwest of the transects (y). The coordinates are defined separately for each of lines 66.7, 80, and 90. Glider tracks differ from these lines: (a) when depth-mean currents are strong and contrary to the desired glider course and (b) during launch and recovery of gliders near the coast. We calculate shear components, i.e. ∂zu and ∂zv, from first differences (Fig. 3C, D). We note elevated shear variance below 346–378 m due to a decrease in acoustic scatterers. Quantities related to shear variance below 362 m are excluded from further consideration, most notably (a) the D1 and D2 velocity bandpasses (Section 3.2) and (b) the diffusivity estimates, which
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x [km] Fig. 3. The example cross-shore section on line 90 (Fig. 2) shows (A) cross-shore velocity (positive onshore); (B) alongshore velocity (positive roughly north-northwestward); (C) and (D) shear components; and (E) Rossby number. Gray shading denotes regions with either excluded or no data, which is larger for Ro due to the averaging needed to calculate it. Longitude along the line is on the upper axis of each subplot.
depend on shear variance squared (Section 3.3 and Appendix A). However, we still show mean cross-shore sections of u, v, and Rossby number (Ro) with data below 362 m, since the noise is at high vertical wavenumber and thus affects shear more strongly than velocity. Using full-depth data produces minor quantitative differences in diffusivity estimates by including shear measurements with lower signal-to-noise ratio, but none of the qualitative conclusions changes. Using this cutoff produces better agreement between shear- and strain-based mixing parameterizations (method described in Section 3.3). Horizontal gradients of density and currents are calculated after smoothing the density and currents over 30 km (e.g., Fig. 3E). This procedure is needed because spectra show noise (i.e., internal waves) in density gradients and geostrophic currents at wavelengths shorter than about 30 km when calculated along depth surfaces (Rudnick and Cole, 2011). Since we only measure the ∂xv component of relative vorticity (ζ), we estimate Ro ¼ ζ =f ∂x v=f (Fig. 3E) and so the actual Ro could be twice as large if there is solid body rotation, where ∂x v ¼ ∂y u. The fluorometer is uncalibrated and suffers from offsets from mission to mission, but for any given mission the spatial pattern is qualitatively accurate even though the magnitude is not. Fluorometer voltage is multiplied by 3 to obtain an uncalibrated number similar to chl a concentrations in mg m 3 to illustrate relative changes. Mixed layer depth is calculated with Δsθ ¼ 0:1 kg m 3 from the near-surface value (Johnston and Rudnick, 2009). 3.2. Energy density and fluxes With relatively slow-moving gliders ( 20 km/day), we estimate tidal and near-inertial signals similar to Johnston et al.
Please cite this article as: Johnston, T.M.S., Rudnick, D.L., Trapped diurnal internal tides, propagating semidiurnal internal tides, and mixing estimates in the California Current System from.... Deep-Sea Res. II (2014), http://dx.doi.org/10.1016/j.dsr2.2014.03.009i
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mode-1 fits of the harmonics are used where depths exceed 362 m: η~ 0 , u~ 0 , and v~ 0 , where the tilde denotes the modal fit (Johnston et al., 2013). Data are extended using the World Ocean Atlas 2009 analysis for depths greater than 500 m (Antonov et al., 2010; Locarnini et al., 2010) and bathymetry from Smith and Sandwell (1997). The mode-1 pressure perturbation, p~ 0 , is calculated from η~ 0 (Johnston et al., 2013). Lastly, mode-1 D1 and D2 energy fluxes are obtained based on both density and velocity measurements:
(2013). Space–time confusion arises since relatively high frequency signals (i.e., internal waves with frequencies from f to N) are projected into spatial variations (Rudnick and Cole, 2011). The projection is caused by two effects (smearing because the glider moves slowly and aliasing because the glider profiles at a frequency in the internal wave band) and cannot be undone. The smearing of D1 and D2 signals with wavelengths of order 100 km is limited to observed wavenumbers of about 1/20-1/10 cycles km 1 (equation 5 in Rudnick and Cole, 2011), which is within our averaging window (following paragraph). Therefore, the slow motion of the glider is advantageous in estimating these harmonic signals. A moving 1.5-day ( 30-km) window is used, which discriminates between D1 and D2, but cannot distinguish near-inertial motions from D1 since the frequency resolution is 2/3 cycle day 1. In essence, this method is a bandpass filter. Harmonic analysis of η, u, and v using M2 and K1 frequencies is applied over the moving 1.5-day window producing the harmonics for displacement (η0 ), for example, as follows (Fig. 4):
η0 ðz; tÞ ¼ Aη cos ½ωðt t o Þ þ ϕη
½F x ðz; tÞ; F y ðz; tÞ ¼ 〈p~ 0 u~ 0 ; p~ 0 v~ 0 〉1:5d
3.3. Finescale mixing parameterizations As noted in Section 2.1, the advantage of these parameterizations is that turbulence estimates can be made from standard Doppler current and CTD profiles. The physical basis of these finescale mixing parameterizations relies on a steady downscale energy transfer from large-scale weakly interacting waves to small-scale breaking internal waves to turbulence. For a GM internal wave field, this transition occurs at roughly 10 m, but this cutoff is adjusted based on Richardson number. The white wavenumber spectrum of either shear or strain drops off at wavenumbers above the cutoff. In our case, the cutoff is placed at 1/40 cycle m 1, where noise in the ADP-measured shear becomes apparent. Details of the calculations are in Appendix A. We produce two estimates of K ρ : one from observed shear and strain (Ksh) and the other from strain alone (Kst) with an assumed shear-to-strain ratio. The observed shear-to-strain ratio is larger than the GM value and so Kst will be an underestimate by a factor of up to 2 (Appendix A). We only show ϵ calculated from both shear and strain.
The amplitude is Aη ðz; tÞ ¼ jηr þ iηi j, where ηr ¼ Aη cos ϕη and ηi ¼ Aη sin ϕη are the real and imaginary components of the fit to cos ½ωðt t o Þ and sin ½ωðt t o Þ. The phase is ϕη ðz; tÞ ¼ tan 1 ðηi =ηr Þ. The M2 and K1 periods are T ¼12.42 and 23.93 h. The radial frequency is ω ¼ 2π =T. The reference time, to, for phase calculations is 0000 Coordinated Universal Time (UTC) on 1 January 2010, roughly the midpoint of the data. Velocity fits are made in complex form, u þ iv. Inertial motions have T ¼23.2–20.9 h from 311 to 351N. Later when mean phases are shown, they are calculated from the means of the real and imaginary components. Energy density and fluxes in D1 and D2 frequency bands are calculated from amplitudes as ð3Þ
where time averages, 〈 〉t , are mission means (Lee et al., 2006). Kinetic energy (Ek) comprises the first two terms, while potential energy (Ep) is the last term. No attempt has been made to remove barotropic components, but the results of the harmonic analysis appear dominated by baroclinic structure i.e., internal waves (Fig. 4). Energy fluxes require full-depth measurements and so
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3.4. Bin means In much of the following analysis of temporal and spatial variability (Sections 4 and 5), data are binned in cross-shore
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ð4Þ
with averages over 1.5 days or 30 km. The mode fit is performed from 20–362 m in depth, even though modes are orthogonal only over the full water depth and can lead to flux errors of about 40%. Limits of the modal decomposition are explained in more detail in Johnston et al. (2013).
ð2Þ
Eðz; tÞ ¼ 〈ρ〉t ðA2u þ A2v þ 〈N 2 〉t A2η Þ=4
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[m] 15
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Fig. 4. D1 (left) and D2 (right) harmonics for (A, B) cross-shore velocity, (C, D) alongshore velocity, and (E, F) vertical displacement are sometimes larger near topography on the example cross-shore section on line 90 (Fig. 2). Gray shading denotes regions with either excluded or no data. Longitude along the line is on the upper axis of each subplot.
Please cite this article as: Johnston, T.M.S., Rudnick, D.L., Trapped diurnal internal tides, propagating semidiurnal internal tides, and mixing estimates in the California Current System from.... Deep-Sea Res. II (2014), http://dx.doi.org/10.1016/j.dsr2.2014.03.009i
T.M.S. Johnston, D.L. Rudnick / Deep-Sea Research II ∎ (∎∎∎∎) ∎∎∎–∎∎∎
6
SRCR on line 90. The mesoscale current and density structures are also described. First, we examine a single example transect on line 90 to illustrate the type of data obtained from a glider (Section 4.2). Second, we apply harmonic analysis to u, v, and η. Third, timemean transects are then examined to better identify mesoscale and internal tidal flows for lines 80 and 90 (Sections 4.3 and 4.4). Several features are noted on the example transect on line 90 including (Figs. 2–4): (a) larger displacements above the SRCR; (b) greater strain within the SCB; (c) considerable mesoscale variability in Ro; (d) a front over the SRCR; and (e) stronger D1 and D2 motions in the SCB but especially at the SRCR. The timemean transects describe the expected cross-shore structure of the CCS, confirm the features described earlier in this paragraph, and further display consistent tidal phase near these internal tidal generation sites (Figs. 5–8). These characteristics are used to distinguish NIW from D1 internal tides, identify D1 trapping, and illustrate D2 propagation.
distance and time, x and t. These results are used to describe timemean, cross-shore sections along with the seasonal cycle of the currents, density structure, internal tides, and mixing. To form time-mean cross-shore sections, bin sizes are Δx ¼ 10 km. Data located 415 km away from the nominal transect are not considered, which mainly occurs on launch and recovery, but also when the depth-mean cross-transect (i.e., alongshore) currents exceed 0.25 m s 1. Derived quantities (e.g., ∂x s or N2) are calculated first and then binned. Bin means in depth and x are shown for some data and amplitudes and phases from the harmonic fits. Bin-mean phases are calculated from the bin means of real and imaginary components. Bin means of depth-integrated fluxes and depth-mean energies are shown. The former emphasizes mode-1 content, while the latter better represents higher modes (i.e., smaller-scale vertical structure). Temporal bins are Δt ¼ 60 days, which is enough time to cover slightly more than two cross-shore transects. When binning is both in t and x, Δx is increased to 20 km. These values for x and t bins are similar to decorrelation scales used in objective maps of the same data (Todd et al., 2011b). A composite seasonal cycle for line 80 is constructed by binning data by month. A time series for line 80 is made by binning data from x¼ 100 to 0 km in 60 day increments. Lines 66.7 and 90 show little indication of seasonal cycle in K ρ and so we do not show similar calculations there.
4.2. An example from line 90 An example of a shoreward transect on line 90 during mission 12803001 from 3–27 September 2012 illustrates typical data (Figs. 2 and 3). Tick marks along the upper axis of Fig. 2B show glider dives roughly every 3 km. While the transect is not synoptic, mesoscale structure and internal waves are apparent. Increased internal wave activity over the SRCR (topography near x ¼ 190 km) produces vertical displacements seen in T, S, sθ , and η (Fig. 2A–C). Larger strain is found near the topography and within the SCB (x 4 200 km, Fig. 2D). A front is found near the SRCR, as evidenced by increased cross-shore gradients of θ, S, and sθ there (Fig. 2A, B). Both internal wave oscillations and mesoscale
4. Cross-shore and depth structure of the mesoscale and internal tides: an example section and time means 4.1. Overview In this section, we are primarily concerned with examining D1 and D2 variability near the continental slope on line 80 and the
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Please cite this article as: Johnston, T.M.S., Rudnick, D.L., Trapped diurnal internal tides, propagating semidiurnal internal tides, and mixing estimates in the California Current System from.... Deep-Sea Res. II (2014), http://dx.doi.org/10.1016/j.dsr2.2014.03.009i
T.M.S. Johnston, D.L. Rudnick / Deep-Sea Research II ∎ (∎∎∎∎) ∎∎∎–∎∎∎
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Fig. 6. On line 80, D1 (left) and D2 (right) amplitudes and phases are binned in cross-shore distance and depth to produce a time mean. Phases are obtained from binned complex amplitudes. (A, B) The complex velocity amplitudes (Section 4.3) and (C, D) the displacement amplitudes are largest near topography. Phases of (E) D1 cross-shore velocity, (F) D2 cross-shore velocity, and (G, H) displacement are shown. (I, J) Depth-mean phase differences are used to determine propagation characteristics. Gray shading indicates lack of data and the black lines show the bathymetry (Fig. 5J), but sometimes data are obtained off the mean glider track in deeper water. Longitude along the line is on the upper axis of each subplot.
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Fig. 7. Time-mean on line 90 as in Fig. 5. The location of the SRCR is indicated near x ¼ 190 km.
Please cite this article as: Johnston, T.M.S., Rudnick, D.L., Trapped diurnal internal tides, propagating semidiurnal internal tides, and mixing estimates in the California Current System from.... Deep-Sea Res. II (2014), http://dx.doi.org/10.1016/j.dsr2.2014.03.009i
T.M.S. Johnston, D.L. Rudnick / Deep-Sea Research II ∎ (∎∎∎∎) ∎∎∎–∎∎∎
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structure are seen in the currents (Fig. 3A, B). With a glider moving at 20 km/day, 5 and 10 oscillations in currents over 100 km indicate D1 and D2 motions. Both up- and downward phase propagation is present in the shear, which indicate a combination of down- and upward propagating internal wave energy (Fig. 3C, D), but shear does not appear particularly elevated near the topography in this transect. The magnitude of Ro exceeds 0.4 near the coast (Fig. 3E), but may be up to twice as large (Section 3.1). Internal tides appear larger near the SRCR and within the SCB. In particular, D1 and D2 η are larger (Fig. 4E, F). Velocities can be equally strong further offshore, especially for D1, which could be due to NIW (Fig. 4A–D). NIW currents impinging on topography would produce larger displacements near the topography as would trapped D1 internal tides (Fig. 4E), while freelypropagating D2 internal tides should show larger η radiating away from the topography higher and lower in the water column, as possibly seen in Fig. 4F. Larger D1 v and η preferentially on one side of the SRCR suggest topographic trapping, a process which is examined in more detail below. 4.3. Time-mean, cross-shore transects: lines 66.7 and 80 The time-mean, cross-shore structure displays the canonical, mid-latitude eastern boundary current system with a coastal upwelling front, a poleward undercurrent, and equatorward flow further offshore (Fig. 5) (Hill et al., 1998). Data from all missions are averaged in time to produce a time-mean, x–z section for all three lines. Lines 66.7 and 80 are qualitatively similar with coastal upwelling (shoaling isopycnals and stratification, a density front, and elevated chl a) and a northward undercurrent with anticyclonic ζ (Ro o 0) inshore of the current's core (Fig. 5A–F, H; only line 80). To avoid repetition, figures for line 66.7 are not shown. Typically 4 200 data points are in each bin, but more profiles per unit cross-shore distance are collected near shore where (a) dives are sometimes shallower than 500 m and (b) currents are stronger
and gliders attempt to stay on track by crossing the current at an angle (Fig. 5G, I). The continental slope is steep: at x ¼ 30 km the water depth is over 500 m, but by x ¼ 100 km the water depth reaches 3 km (Fig. 5J). Harmonic analyses of η, u, and v show considerable D1 and D2 motions (Fig. 6, left and right sides). We will show (a) the D1 waves near topography are consistent with trapped internal tides, which decay offshore and propagate with the topography on the right; and (b) the D2 waves propagate offshore from numerous sources, which interfere and produce partially-standing waves. The time-mean D1 amplitudes are largest (a) within 30–40 km of the coast due to trapped D1 internal tides and (b) in the upper 100 m presumably due to NIW (Fig. 6A, C). Aη exceeds 12 m in places, which is not much larger than an open ocean GM rootmean-squared η of 7 m. jAu j is the magnitude of the tidal velocity qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi amplitude, A2u þ A2v . D1 motions will be more visible as currents, since internal waves become more horizontal as ω approaches f. The spatial extent of surface-intensified jAu j suggests NIW, which are ubiquitous. Over the slope and below 50 m, Aη decays to background values over about 70 km in the cross-shore direction. This decay is consistent with the baroclinic Rossby radius of 20 km (a decay to 5% of background levels occurs over 60 km) and trapped D1 internal tides. The decay of Au is not as apparent as with Aη . D1 phases are useful for distinguishing NIW from internal tides. ϕη is relatively constant near the continental shelf and slope where amplitudes are large (i.e., in a triangular region in the lower right of Fig. 6G, where x 4 100 km). In the same small region, jAu j is also large and ϕv is also relatively constant (Fig. 6E). Over most of the water column away from the surface and topography, ϕv shows relatively little change compared with ϕη . The D1 phase differences (ϕη ϕu and ϕη ϕv ) are useful indicators of the direction of wave propagation. For a two-layer
Please cite this article as: Johnston, T.M.S., Rudnick, D.L., Trapped diurnal internal tides, propagating semidiurnal internal tides, and mixing estimates in the California Current System from.... Deep-Sea Res. II (2014), http://dx.doi.org/10.1016/j.dsr2.2014.03.009i
T.M.S. Johnston, D.L. Rudnick / Deep-Sea Research II ∎ (∎∎∎∎) ∎∎∎–∎∎∎
flow, when the lower layer's η and velocity have phase differences of 01/1801 (i.e., in phase/out of phase), the wave propagates in the positive/negative direction (see Figure 6.3 in Gill, 1982). Phase differences of 7901 imply standing waves. Depth-mean phase differences are obtained from 154–362 m to avoid NIW influence near the surface and emphasize deeper D1 internal tidal motions. The depth mean of complex amplitude for each quantity (η, u, and v) is calculated at each cross-shore distance, the angle is computed, and then we difference two such angles (Fig. 6I). With this method, regions with higher amplitudes are emphasized. For x 4 100 km, where D1 amplitudes are largest (not including the surface), ϕη ϕu is near 7 901, which yields no cross-shore energy propagation and is consistent with trapped waves (Fig. 6I). On the other hand, ϕη ϕv is near 0 (x ¼ 120 to 30 km, Fig. 6I), which implies northward phase propagation for a mode-1 wave. The combination of these phase results with elevated amplitudes near the slope indicates a topographically-trapped baroclinic wave traveling with the topography on its right. Next, D2 harmonic fits are used to illustrate cross-shore propagation in contrast to D1 above. One internal tidal generation site appears to be at the shelf edge and another, deeper one over the slope is near x¼ 80 km with waves propagating both onshore and offshore from there. Canyons, submarine ridges, and a seamount are found near 341N, 1211E (or x¼ 80 km) and are somewhat visible in Fig. 1B. D2 jAu j is large near the topography, in the upper 100 m, and also about 150 km away from the slope (Fig. 6B). This distance corresponds to a D2 wavelength and suggests a surface reflection of a tidal beam (Cole et al., 2009; Johnston et al., 2011a; Martin et al., 2006). Aη also suggests a beam-like structure emanating upward from topography near x¼ 80 km (Fig. 6D). Several regions of on- and offshore phase increase are found in ϕu and ϕη (Fig. 6F, H), which produce roughly 75 km-long sections of on- and offshore propagation, with ϕη ϕu ¼ 01 and 1801 (Fig. 6J). Onshore propagation (ϕη ϕu ¼ 0) is found for x 4 70 km consistent with generation at the slope. A standing wave (ϕη ϕu ¼ 90) is found further offshore ð 140 km o x o 70 kmÞ, which suggests at least two sources of propagating waves. 4.4. Time-mean, cross-shore transect: line 90 The time-mean, cross-shore structure along line 90 is consistent with previous observations in the SCB [e.g., Davis et al., 2008; Hickey, 1979; Todd et al., 2011b]. Inshore of the continental slope near x ¼ 260 km, the topography comprises several ridges and basins in the SCB unlike lines 66.7 and 80 which have narrow shelves (Figs. 1, 5J, and 7J). Like line 80, there is a nearshore shoaling of isopycnals and stratification at a coastal front near x ¼ 20 km, where chl a is also larger (Fig. 7A–D). Flows in most of the SCB, from the coast to x 150 km, are generally northward, while southward flows are found in the California Current offshore of the slope in the upper 100 m with the northward California Undercurrent below 100 m (Fig. 7F). Near the SRCR, a narrow region of southward flow is found along with anticyclonic Ro (Fig. 7F, H). Cyclonic Ro bands are found on either side, which may assist with D1 trapping by making the effective vorticity larger (Fig. 7H) (Kunze and Toole, 1997). Ro is small in this time mean, but values on a single section can be large (Fig. 3E). A front associated with the inshore boundary of the California Current is found above SRCR with a broader lateral extent of chl a compared to the coastal front (Fig. 7A, B, D). More profiles are obtained near the two ridges where dives are often shallower, but typically 4150 data points are in each bin (Fig. 7G, I). Similar to line 80, higher D1 and D2 amplitudes are found near the topography on line 90, but, unlike line 80, the higher amplitudes are at topography within the SCB, mainly at the SRCR and not at the coast (Fig. 8A–D). The complex topography provides
9
numerous generation sites for internal tides. Increasing phase indicates a few areas of propagating waves, but large regions of slowly-varying phase show that standing or partially-standing waves are dominant features (Fig. 8E–J). D1 amplitudes are larger on the east side of the SRCR (in the San Nicolas Basin, near x ¼ 180 km) consistent with an interpretation of trapped D1 internal tides propagating southward with the ridge on their right (Fig. 8A, C). Within 50 km of the east side of the ridge, ϕη and ϕv appear distinct from the surrounding phases (Fig. 8E, G). A small region within 10–20 km of the SRCR has ϕη ϕv ¼ 1801 (01) indicating southward (northward) phase propagation on the east (west) side of the ridge (Fig. 8I). We discount NIW currents impinging on topography as the dominant signal below the upper 50 m because they would produce large mean amplitudes on both sides of SRCR and the ridge near x¼ 50 km. D2 has larger amplitudes on both sides of the SRCR especially for displacements (Fig. 8B, D). D2 Aη is also larger near x ¼ 350 and 500 km, about one and two D2 wavelengths from the SRCR (Fig. 8D). Smaller displacements are seen at two other ridges near x¼ 50 and 100 km. As with the D1 band, D2 phases vary slowly, consistent with standing or partially-standing waves (Fig. 8F, H). Phase differences show small regions of propagating, standing, or partially-standing waves (Fig. 8J). 4.5. Summary On line 80, trapped D1 internal tides are shown with: (a) largest amplitudes near topography, (b) slowly-varying phase in the same locations, and (c) phase differences which are consistent with northward propagation, and (d) the coast is on the waves' right consistent with a trapped wave. D2 internal tides show propagation away from two sources, the shelf edge and rough topography on the slope. The phase differences show a combination of propagating and standing waves. On line 90, the amplitudes and the phases at the SRCR provide evidence for trapped D1 internal tides rather than NIW: (a) larger amplitudes on the east side of the SRCR, (b) distinct phases there, and (c) some suggestions of propagation with the ridge on the right. Numerous D2 wave sources over the complex topography of the SCB yield patterns dominated by partially-standing waves. Line 80's time-mean, cross-shore structure shows a coastal upwelling front, a poleward undercurrent, and an equatorward flow further offshore. Line 90 displays a similar time-mean, cross-shore structure offshore of the slope/SRCR. Most of the flow in the SCB is poleward with a weaker upwelling front at the coast.
5. Depth-mean mixing estimates and internal tides: spatial and temporal variability 5.1. Overview Within the context of the internal-wave, frontal/mesoscale, and mean flows (Section 4), depth-mean K ρ and ϵ estimates from all available glider profiles are presented for each of the three lines (Section 5.2). Then, bin means in time and space are used to better describe the temporal and spatial variability of these estimates. The cross-shore structure is described in the mean (Section 5.3). On line 80, patterns are found over the 4 6year record along with a seasonal cycle (Section 5.4). No seasonal cycle is found on the other lines. In particular, we illustrate these patterns of variability in the mixing estimates, internal tidal energies and fluxes, gradients and density at fronts, and chl a using depth means. This section covers three analyses: (a) the time- and depth-mean cross-shore
Please cite this article as: Johnston, T.M.S., Rudnick, D.L., Trapped diurnal internal tides, propagating semidiurnal internal tides, and mixing estimates in the California Current System from.... Deep-Sea Res. II (2014), http://dx.doi.org/10.1016/j.dsr2.2014.03.009i
10
T.M.S. Johnston, D.L. Rudnick / Deep-Sea Research II ∎ (∎∎∎∎) ∎∎∎–∎∎∎
structure for all lines, (b) the depth mean on line 80 for the region near the coast (x 4 100 km) shown as a time series, and (c) the depth mean on line 80 shown as a composite annual cycle (i.e., x versus month). These three analyses rely on depth means over somewhat different depths and so we summarize for each calculation the relevant depths and reasons before proceeding with the results. For analysis a, depth-mean K ρ and ϵ are averaged from below the mixed layer to 362 m, the depth of the deepest reliable shear measurements (Section 3.1 and Appendix A). sθ and ∂x sθ emphasize fronts by averaging from 10 to 90 m. Chl a is averaged over the same depth range for the same purpose and also because there is little signal below that depth. Depth-mean rather than depth-integrated E is calculated to emphasize high-mode internal waves. Its depth mean is from 58 m up to a maximum of 362 m to minimize NIW influence and to more closely compare with mixing estimates. Depth-integrated mode-1 fluxes are calculated from data which covers 10–362 m. For analyses b–c, averaging intervals are as noted above, but sθ and N2 depth means are from 106 to 202 m to show the annual cycle in upwelling. Further details on the binning are in Section 3.4.
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5.2. Depth-mean mixing estimates To demonstrate the resolution and extent of the data, we show depth-mean diffusivity estimates from all 33 609 profiles on all three lines where both shear and strain are available (Fig. 9). The spatial patterns on all three lines are consistent with locations of elevated D1 and/or D2 energy either near the shelf/slope or over the rough topography of the SCB (Figs. 6A–D and 8A–D). With a 1.5-day window, there is some smearing of these patterns over 30 km, a typical distance covered by a glider in this time and so this pattern of mixing near topography is likely more localized in space than we describe below. On line 90, the overall pattern of the parameterized dissipation and diffusivity shows highest values over the SRCR with ϵ 4 10 8 W kg 1 and K ρ 410 4 m2 s 1 from both shear–strain and strain-only estimates (Fig. 9C). Secondary maxima are seen at x 100 and 50 km near two other ridges (Figs. 1B, 7J, and 9C). Another, intermittent local maximum is near x 350 km at a possible D2 downgoing beam (Figs. 8D and 9C). Ksh estimates from lines 66.7 and 80 display an offshore decay to background values within 100 km of the coast (Fig. 9A, B). Diffusivity estimates appear more sporadic on line 66.7. Sampling density near the coast is sometimes lower because gliders are deployed and recovered from coastal locations off of the nominal lines, which are excluded from consideration because jyj 4 15 km (Section 3.4 and Appendix A). Thus, typically 2 of 4 transects on one mission will closely follow the line near the coast. Also estimates are not made, when a profile extends less than 96 m below the mixed layer depth, a condition which is more likely to occur in shallow coastal waters. Patterns of ϵ are almost identical to Ksh, which in turn are very similar to Kst and so we only show Ksh at this point. Section 5.3 will further demonstrate this similarity. The spatial patterns of Ksh and Kst agree within a factor of 2, which reflects a factor of 2 difference in the correction for shear-to-strain ratio from observations and GM (h in Appendix A). This result suggests that the independent measurements of shear and strain are in reasonable agreement (Section 3.3 and Appendix A). Since these K ρ estimates follow the statistics of ð∂z u2 þ ∂z v2 Þ2 (Appendix A), the probability distribution functions roughly resemble a lognormal distribution (figure not shown). The ratio log(Ksh/Kst) has a mean value of log(1.7) and a standard deviation of log(1.6) and the ratio of Ksh/Kst is o 2 with a standard deviation of o 1.
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5.3. Depth- and time means of the cross-shore transects In this section, we use the depth- and time-mean, cross-shore structure of mixing, internal tides, density, and chl a to assess the energetics and relevance of our mixing estimates. First, on lines 66.7 and 80, the offshore decay of Ksh is similar from values near the coast of 10 4 m2 s 1 to 10 5 m2 s 1 offshore (x o 100 km, Fig. 10A). Kst is within one standard deviation of Ksh, but appears to under-/overestimate off-/onshore. A similar decay is found on both lines 66.7 and 80 for ϵ (Fig. 10B). Due to the similarity with line 80 in the cross-shore mean, most of the description of Ksh below covers only line 80. As in Section 4.3, on line 80, the coastal upwelling front is near x ¼ 50 km, where maxima in chl a and its
Please cite this article as: Johnston, T.M.S., Rudnick, D.L., Trapped diurnal internal tides, propagating semidiurnal internal tides, and mixing estimates in the California Current System from.... Deep-Sea Res. II (2014), http://dx.doi.org/10.1016/j.dsr2.2014.03.009i
T.M.S. Johnston, D.L. Rudnick / Deep-Sea Research II ∎ (∎∎∎∎) ∎∎∎–∎∎∎
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Fig. 10. On line 80 data are averaged in time to show the mean cross-shore structure. (A) Depth-mean diffusivity (including Ksh from line 66.7) and (B) dissipation estimates (including ϵ from line 66.7) are largest near the coast. Gray shading denotes one standard deviation. (C) The depth-mean from 10–90 m of potential density is maximum at an upwelling front near shore, where (D) the chl a mean from 10–90 m and its standard deviation are also larger. (E) D1 and D2 depth-mean energies are largest at the coast and (F) D2 mode-1 cross-shore and alongshore energy fluxes decrease toward shore. Number of data in each bin is uniform offshore but (G) decreases for turbulence estimates near shore and (H) increases for density data near shore. Longitude along the line is on the upper axis of each subplot.
variance are seen (Fig. 10C, D). Depth-integrated D1 mode-1 energy flux is small ( o100 W m 1 ) and displays no clear pattern, which is expected for an evanescent internal wave (figure not shown). The depth-integrated D2 mode-1 energy flux converges shoreward from x ¼ 120 km (Fig. 10F), while the depth-mean energy densities increase for D1 and D2 shoreward of x¼ 50 and 100 km (Fig. 10E). D1 energy density is 2 greater than D2. Data coverage as noted earlier drops off closer to shore (Fig. 10G, H), yielding more uncertain mixing estimates there (Fig. 10A, B). In summary, turbulence estimates and D1 energy peak in the same location. This result along with that in Section 4.3 indicates trapped D1 internal tides are the dominant source of mixing on line 80. The decay away from topography of the D1 signal is consistent with a trapped wave. The mode-1 baroclinic Rossby radius (¼ c/f, where c is the mode-1 phase speed) within about 100 km of the coast varies from 12 to 28 km. An exponential decay in x reaches 5% of the value at the coast over a distance of 35–85 km. The larger distances are consistent with the decay scales of D1 E and K ρ on line 80, while the smaller distances are appropriate for the shallower waters in the SCB on line 90 as we will show later in this section. For line 80, we make rough estimates of the energy budget within 100 km of the coast. The energy lost from D1 and D2 internal tides should balance the dissipation. We neglect any contribution from NIW or fronts. We argue the most energetic D1 component is due to trapped internal tides. These topographically-trapped waves propagate northward with the coast on their right. If we assume 10% of their depth-mean energy within 100 km of the coast is locally dissipated over a tidal cycle (1 day) that yields a depth-integrated dissipation in roughly 1000m deep water as 0.1 5 J m 3 1000 m/86 400 s 6 m W m 2. The D2 mode-1 flux convergence: is roughly ΔF x =Δx ¼ 480 W m 1 =120 km ¼ 4 m W m 2 . The very crude estimate of energy lost by D1 and D2 internal tides is comparable to an order
of magnitude estimate of the depth-integrated dissipation (calculated from depth-mean ϵ in Fig. 10B): ρϵH ¼ 103 kg m 3 1 10 8 W kg 103 m ¼ 10 m W m 2 . Many of the factors included in this paragraph could be increased and decreased easily by factors of 2 or more given their variations within 100 km of the coast. Therefore, we conclude as an order of magnitude estimate, that there is 10 m W m 2 energy lost from the D1 and D2 bands to produce the estimated dissipation of 10 m W m 2. On line 90, elevated mixing estimates appear where D1 and D2 energy densities are largest over the SRCR (x ¼ 250 to 150 km, Fig. 11A, B, and E). There may be contributions from a persistent front (Fig. 11C). Ksh has a maximum of 3 10 5 m2 s 1 across the SRCR and the slope. A decay to background values of o 10 5 m2 s 1 in deep water is found (x o 300 km, Fig. 11A). The offshore decay occurs over 100 km similar to lines 66.7 and 80, but the onshore decay is more sudden (near x¼ 150 km, Fig. 11A, B, and E). This result is consistent with an offshore increase in Rossby radius and extent of D1 trapped internal tides. Again Kst is an underestimate compared to Ksh in deep water. Similar features are found in ϵ with slightly elevated values over other ridges at x¼ 100 and 50 km (Fig. 11A, B). We also note a persistent density front near x 200 km, where chl a and its variability are elevated (Fig. 11C, D). Depth-integrated D2 mode-1 baroclinic energy fluxes are small in the SCB and increase offshore of the continental slope (x o 250 km, Fig. 11F). Data coverage again drops off closer to shore, but increases over the ridges (Fig. 11G, H). To assess our chosen depth limits based on objective measures of data quality (Section 3.1 and Appendix A), we recalculate K ρ from data up to a maximum depth of 202 m instead of 362 m as is done elsewhere in this paper. Results for the cross-shore mean on line 90 remain similar: the peak values of K ρ near the SRCR are unchanged, while values further offshore are slightly lower. The overall pattern in Fig. 11A and our conclusions are unchanged.
Please cite this article as: Johnston, T.M.S., Rudnick, D.L., Trapped diurnal internal tides, propagating semidiurnal internal tides, and mixing estimates in the California Current System from.... Deep-Sea Res. II (2014), http://dx.doi.org/10.1016/j.dsr2.2014.03.009i
T.M.S. Johnston, D.L. Rudnick / Deep-Sea Research II ∎ (∎∎∎∎) ∎∎∎–∎∎∎
10
4
122o
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[W kg ]
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25.5 25
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chl a
[kg m 3]
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E [J m ]
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E [J m ]
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st
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D
1
D
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D2: EP EK
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3 nKsh n
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n
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K
3
sh
[kg m ]
K
N2 2 [ 10 5 s ]
10
ET [J m 3]
K [m2 s 1]
Fig. 11. On line 90, data are averaged in time to show the mean cross-shore structure as in Fig. 10, but elevated mixing estimates, internal tidal energy, and a front are found near the SRCR (x 190 km).
2012
100 0
2008
2010
2012
Fig. 12. On line 80, data are averaged within 100 km of the coast to produce a time series showing seasonal variability in (A) diffusivity estimates with gray shading denoting one standard deviation, (B) potential density averaged from 106 to 202 m in depth, (C) depth-mean D1 and D2 energy densities, (D) buoyancy frequency averaged from 106 to 202 m in depth, and (E) depth-mean D1 and D2 potential and kinetic energy densities. (F) The number of density (gray) and diffusivity (black) data points in each bin is shown.
5.4. Temporal variability on line 80 While temporal variations are seen over the record on lines 66.7 and 90, here we focus on a seasonal cycle in mixing, D1 energy, and density on line 80. Neither mixing nor energy density shows a seasonal cycle on lines 66.7 and 90. The seasonal cycle in mixing is found near the coast. Data from x¼ 100 to 0 km are binned in 60-day intervals (Fig. 12). Diffusivity estimates are higher in the summers of 2008, 2009, 2011, and 2012 (Fig. 12A). Kst has a small maximum in summer 2010, where there is missing Ksh data. An additional spring maximum is seen in 2012. Both Ksh and Kst display similar phasing and magnitude (within one standard deviation except in summer 2012). In 2007/2008, when Ksh
is unavailable, Kst is low. Similar patterns are seen in ϵ (figure not shown), which indicates the variability in K ρ is not due to the lower N2 in upwelling season (Fig. 12A, D). s is high during upwelling (Fig. 12B). The depth-mean D1 baroclinic energy is often largest in summers mainly due to the seasonal cycle in D1 Ep and, to a lesser extent, Ek (Fig. 12C, E). D1 Ep provides a continuous record, whereas Ek has some gaps. D2 Ek and Ep are generally weaker and a seasonal cycle is not evident (Fig. 12C, E). Usually a minimum of 50 samples is obtained for each 60-day bin (Fig. 12F). Even though each year displays differing amplitudes of K ρ and other quantities, a seasonal cycle of mixing is composited from the 4 6year record (Fig. 13). Larger depth-mean K ρ is found from May to August within 40 km of the coast (Fig. 13A, C) somewhat
Please cite this article as: Johnston, T.M.S., Rudnick, D.L., Trapped diurnal internal tides, propagating semidiurnal internal tides, and mixing estimates in the California Current System from.... Deep-Sea Res. II (2014), http://dx.doi.org/10.1016/j.dsr2.2014.03.009i
T.M.S. Johnston, D.L. Rudnick / Deep-Sea Research II ∎ (∎∎∎∎) ∎∎∎–∎∎∎
123o
121oW
123o
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121oW
Nov Ksh
Nov
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3
[m s ] 4 Aug 10 May
10
[kg m ] 26.5
6
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Feb
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N2 [ 10 8
1
[m s ] Aug 10 4 May
10
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s ] Aug May
0
Feb
Feb
Nov D1 E
D2 E 3
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[J m ] 7
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Nov 3
[J m ] 4
1
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nKsh 150
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x [km]
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x [km]
Fig. 13. On line 80, a composite seasonal cycle is constructed by binning observations by month and cross-shore distance. Hovmöller diagrams show: (A) depth-mean diffusivity estimates from shear and strain, (B) potential density averaged from 106 to 202 m in depth, (C) depth-mean diffusivity estimates from strain, (D) buoyancy frequency averaged from 106 to 202 m in depth, and (E) depth-mean D1 and (F) D2 energy densities (Note the different color scales.). The number of (G) density and (H) diffusivity data points in each bin are shown. Longitude along the line is on the upper axis of each subplot. Gray/blue shading indicates a lack of data in Fig. 13A–F/G-H. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)
coincident with higher D1 depth-mean energy from May to July (Fig. 13E). Depth-mean D2 energy peaks from April to June within 60 km of the coast, but is at least 2 weaker than D1 (Fig. 13F). Upwelling season displays higher s and lower N2 within 40–60 km of the coast from March to August (Fig. 13B, D) and begins before the diffusivities have peaked. Typically about 50 data points are in each bin, but that value increases near the coast for s and decreases for K ρ (Fig. 13G, H). In summary, peak mixing estimates on line 80 from May to August (a) coincide roughly with peak D1 energy from May to July; (b) occur after the peak D2 energy density; (c) occur 2 months after the onset of upwelling in March; and (d) occur 2 months after the onset of maximum wind variance in March at this latitude (Hickey, 1979) and presumably a maximum in NIW generation. This result provides further confirmation that trapped D1 internal tides are dominant energy source for mixing here.
6. Discussion 6.1. Finescale estimates of depth-mean diffusivity and dissipation from glider profiles Mixing in the thermocline is driven mainly the breaking of internal waves. Assuming a steady downscale transfer of energy, mixing parameterizations use observed shear and strain at Oð10 mÞ scales to estimate mixing. Shear–strain diffusivity estimates (Ksh) and strain-only estimates (Kst) using an assumed shear-to-strain ratio have been calculated from data obtained by autonomous underwater gliders along three sustained transects in the California Current System from 2006 to 2012 (Fig. 9). The timeand depth-mean diffusivity estimates are largest near topography: (a) K sh 10 4 m2 s 1 near the coast on lines 66.7 and 80 (Fig. 10A) and (B) 3 10 5 m2 s 1 over the SRCR in the SCB on line 90 (Fig. 11A).
Please cite this article as: Johnston, T.M.S., Rudnick, D.L., Trapped diurnal internal tides, propagating semidiurnal internal tides, and mixing estimates in the California Current System from.... Deep-Sea Res. II (2014), http://dx.doi.org/10.1016/j.dsr2.2014.03.009i
14
T.M.S. Johnston, D.L. Rudnick / Deep-Sea Research II ∎ (∎∎∎∎) ∎∎∎–∎∎∎
Both Ksh and Kst agree within a factor of 2, but Kst is usually an underestimate compared to Ksh especially where diffusivity estimates are smaller. These estimates are in general agreement with tracer release and microstructure observations showing K ρ of Oð10 4 Þ m2 s 1 near topography in the SCB (Section 2.2). The depth-integrated dissipation estimate near Point Conception on line 80 within 100 km of the coast is roughly 10 m W m 2 and balances estimated D1 and D2 energy inputs (Section 5.3). Also a local maximum of 2 m W m 2 in wind work on near-inertial motions in the mixed layer is found near Point Conception in an eddying model with realistic but annually repeating winds (Figure 2 in Simmons and Alford, 2012). This energy input is a precursor to NIW production. Considering the entire Pacific Ocean, roughly 10% propagates away as low-mode NIW. Thus most of this energy is either lost in the mixed layer, transition layer, or deposited in higher-mode waves, which may dissipate locally. This mean annual value of 2 m W m 2 near line 80 is comparable to that in the mid-latitude storm tracks near 451N and is less than estimated D1 and D2 contributions of 5 m W m 2 each. We assumed that about 10% of the energy in the trapped D1 internal tide is dissipated locally with the rest propagating alongshore, but this value could also be much higher, it is quite probable that most of the D1 energy is dissipated near its generation site. K1 and O1 barotropic tidal energy dissipation (i.e., bottom friction, internal tides, or other coastally-trapped waves) displays a local maximum on the west coast of North America and is 5 m W m 2 summed over both constituents in the TPXO tidal model (Egbert and Ray, 2003). This agreement of energy inputs from near-inertial motions and D1 and D2 internal tides with our dissipation estimate should be regarded as an order of magnitude comparison. The mixing parameterizations will fail when the downscale energy cascade is circumvented. Two possible processes which circumvent the cascade are likely not at work here. A third process which contributes to finescale gradients is also noted, but may or may not be related to internal waves. First, on wide continental shelves with water depths of ≲100 m, the large-scale, mode-1 waves may be unstable or close to instability and thus there is no cascade to smaller scales (MacKinnon and Gregg, 2003, 2005). Second, large-amplitude waves may break directly near energetic generation sites, such as Luzon Strait (Alford et al., 2011), and produce overturns of several hundred meters in extent. Third, finescale shear and strain at fronts or other mesoscale features may lead to modestly elevated levels of mixing within 10–20 km of their flanks, but the relative contributions of either frontogenesis or trapped, reflected, or refracted internal waves are unclear (Johnston et al., 2011b). In our study, the shallowest water depths are 100–200 m over steep topography, typical water depths exceed 1 km, internal wave energy levels are modest (several times GM levels), and spectra resemble GM. Therefore, we expect the finescale methods are applicable in this region. 6.2. Mesoscale flows As gradients increase during frontogenesis, mixing is likely on the dense, cyclonic side of a front where the gradients are largest (Hoskins and Bretherton, 1972). Along line 90, a persistent front between the southward California Current and northward flow in the SCB is located over the SRCR (Figs. 7 and 11). A similar result is found for line 80 near its coastal upwelling front (Figs. 5 and 10). Two pieces of evidence suggest that fronts are not the dominant mechanism: (a) maximum Ksh is located about 20–50 km inshore from the maximum density gradient, which defines the front and (b) the seasonal cycles of upwelling and Ksh on line 80 are not aligned. Mixing may be important for primary productivity in this area, where the mean and variance of chl a are higher (Fig. 10D and
11D). However, the simplest explanation for this primary productivity on line 90 is southward advection of phytoplankton from the upwelling center at Point Conception on line 80 in the southward flow of the California Current to line 90 (Fig. 7F). 6.3. Free and trapped internal waves The most energetic internal waves are D1 and D2 internal tides and NIW. Since D1 internal tides are subinertial, they are trapped to the topography, but may propagate alongshore. With our methods, we cannot distinguish among individual D1 and D2 constituents, e.g. K1, O1, or near-inertial motions. We attempt to minimize NIW influence on energy calculations by using data below 58 m. Several lines of evidence indicate the presence of trapped D1 internal tides and their influence on mixing. On line 80, (a) D1 energy exceeds D2 energy by up to a factor of 2 (Fig. 10E), (b) the seasonal maxima of mixing and D1 depth-mean energy occurs in summers (Fig. 13A, C, E), (c) the cross-shore maxima of D1 energy and K ρ are in the same location (Fig. 10A, E), (d) the cross-shore decay scale is consistent with trapped waves (Fig. 10A), and (e) the D1 phase difference between η and v is consistent with a trapped wave propagating with the coast on its right (Fig. 6, left). Phase is a particularly useful indicator of internal tidal propagation (Ray and Mitchum, 1996; Johnston et al., 2013). Surface currents from coastal radars with records long enough to distinguish D1 constituents show K1 tidal variance exceeds that of both near-inertial oscillations and M2 internal tides near Point Conception (Kim et al., 2011). However, further analysis of our data in conjunction with other datasets (sea level records from tide gauges and altimetry to identify barotropic tidal forcing and spring-neap cycles; surface wind stress from moorings, shore stations, and satellite to drive a slab model for estimates of NIW generation; coastal radars to show the spatial pattern of currents in the D1 band; and moored data to provide better temporal resolution) is planned and needed to clarify the contributions of either trapped internal tides or NIW to the D1 signal on line 80. Trapped D1 internal tides are also found on line 90 (Fig. 8, left), but numerous internal tidal generation sites in the SCB lead to standing waves. On line 90, largest D1 amplitudes are found on one side of the SRCR consistent with a southward-propagating, topographically-trapped wave. Several characteristics distinguish trapped D1 internal tides from NIW. For NIW flows encountering submarine topography, there should not be a maximum on one side of the topography in a time average. It is unlikely most of the NIW encountering the SRCR are propagating westward given the weak winds in the SCB (Hickey, 1979). NIW currents seem stronger west of the SRCR (compare upper 100 m west and east of SRCR in Fig. 8A). NIW generation is episodic and therefore consistent phase structure at depth is also unlikely. The time- and depth-mean cross-shore profile of mixing displays maxima near the continental slope on lines 66.7 and 80 and decreases to background values in 100 km (Fig. 10A) and over similar distances offshore of the SRCR on line 90 (Fig. 11A). The mode-1 baroclinic Rossby radius is 12–28 km and so trapped waves would decay to within 5% of background values in 35– 85 km. Kelvin waves are one possible topographically trapped wave, but there are many other possibilities (Llewellyn Smith, 2004; Mysak, 1980). If indeed trapped D1 internal waves are a major contributor to near-slope mixing poleward of 301, then their cross-shore extent may decrease with decreasing Rossby radius (i.e., increasing latitude in general). On the other hand, the decay distance may reflect the decay to background levels of propagating internal waves. In such cases, measurements show (a) a decay of internal waves away from their source in Luzon Strait over 100 km
Please cite this article as: Johnston, T.M.S., Rudnick, D.L., Trapped diurnal internal tides, propagating semidiurnal internal tides, and mixing estimates in the California Current System from.... Deep-Sea Res. II (2014), http://dx.doi.org/10.1016/j.dsr2.2014.03.009i
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(Rudnick et al., 2013) and (b) diapycnal diffusivity decreasing from peak values over the Hawaiian Ridge to background levels within 60 km. Since the D2 internal tidal fluxes are 20–100 less in our study region than at the Hawaiian Ridge or Luzon Strait, it seems likely that their decay to background levels should occur closer to the coast. Thus, we believe trapped D1 internal tides are the dominant contributor to mixing here. The western coast of North America may have particularly large D1 internal tides, similar to several other locations poleward of 301 around the North Pacific and North Atlantic as well as in more isolated areas off Tasmania, New Zealand, and Patagonia. Further assessment of the influence of trapped internal tides on mixing is warranted also in the Arctic, where both D1 and D2 internal waves are subinertial and may lead to considerable mixing at the margins (Padman and Dillon, 1991; Padman et al., 1992), a process which is not often considered in models of the Arctic but may have considerable impact (Holloway and Proshutinsky, 2007). 6.4. Vertical distributions and mean values of diffusivity estimates This work also demonstrates that considerable mixing in the coastal ocean occurs in the middle of the water column away from the surface and bottom boundaries. Midwater turbulence due to a superposition of M2, inertial, and thermal wind shear has been previously observed with microstructure instruments on the Oregon shelf by Avicola et al. (2007). They also note that numerical models of the coastal ocean are often unable to produce mixing away from surface and bottom boundaries. In our case, mixing estimates are elevated in midwater by internal tides. The overall mean, regional mixing estimates on each line are consistent with mean thermocline values of K ρ 10 5 m2 s 1 (Gregg, 1987), which is a background value used by many models. By using a combination of the same θ-S data from gliders and a numerical state estimate, a mean diffusivity over much of the same region was obtained by following tracers over time in the state estimate (Todd et al., 2012). Values a few times larger than the model's background value of 1 10 5 m2 s 1 were found. Data are roughly lognormally distributed with a tail for the higher coastal values. Our mean Ksh has values of 0.6, 1, and 1 10 5 m2 s 1 on lines 66.7, 80, and 90. To better describe the distribution, means of log 10 K sh which expressed as Ksh means are 0.4, 0.7, and 0.8 10 5 m2 s 1 on lines 66.7, 80, and 90. Standard deviations of log 10 K sh are about 0.3–0.4 which is equivalent to a factor of 2. When averaged on a regional basis the model values and tracer estimate are similar to our mean estimates. 6.5. Seasonal cycle in mixing estimates and diurnal energy on line 80 A seasonal cycle in D1 energy density and mixing estimates are found within 100 km of the coast on line 80, but not lines 66.7 or 90 (Figs. 12A, 13A and 13C). Depth-mean D1 energy on line 80 is maximum in spring/summer (Figs. 12C and 13E). Since these maxima overlap with upwelling season, this phasing is doubtless of biogeochemical importance for nutrient, carbon, and other tracer fluxes between deep and shallow waters (Bylhouwer et al., 2013, and references therein). At least two possible explanations exist for the D1 maximum during springs/summers: either increased NIW production due to variable winds within 100 km of the coast or increased D1 internal tide generation possibly due to seasonal stratification changes (Fig. 13D). Stratification changes due to upwelling in Monterey Canyon change the character of internal waves from progressive to standing (Hall et al., 2014). The magnitude of internal tidal generation can also change by a factor of 2, mainly due to stratification effects on the baroclinic pressure perturbation (Zilberman et al., 2011).
15
On line 80 near the coast, there also appears to be an increasing trend from 2007 to 2012 in depth-mean D1 potential energy (Fig. 12E), which again implies either stronger internal tidal generation possibly due to stratification changes and/or increasing NIW production. There also appears to be an increasing offshore extent of dense, weakly-stratified water during upwellings (over depth range from 106 to 202 m, figure not shown), but the density and stratification near the coast do not show a trend (Fig. 12B, D). No trend is seen in the mixing estimates (Fig. 12A). This region near Point Conception displays considerable seasonal variation in upwelling with an increasing trend driven by the wind stress curl (Schwing and Mendelssohn, 1997; Seo et al., 2012). Winds in this area have peak variance from March to June (Table 1 in Hickey, 1979). In a regime of increasing winds, there may be increasing wind variance. If this variance has a diurnal component, it may lead to the increasing NIW production and D1 energy seen in our observations. These speculations on the effects of stratification and wind variability will be tested in the future with data (Section 6.3), which can discriminate K1 from the near-inertial peak.
7. Summary Along sustained, high-resolution, cross-shore transects on CalCOFI lines 66.7, 80, and 90 from 2006 to 2012, mixing is estimated via finescale parameterizations using shear and strain data from autonomous underwater gliders. Our primary technical result is the application of these parameterizations to spatially and temporally extensive glider data. Our main scientific result is the importance of D1 internal tides, which are subinertial at these latitudes and therefore trapped at topography e.g., either the slope or the SRCR in the SCB. Trapped D1 internal tides (a) are often more energetic than the D2 internal tides, (b) set the cross-shore decay scale of mixing near the slope and shelf, and (c) produce a seasonal cycle in mixing along line 80.
Acknowledgments This work was supported by NOAA through the Consortium for the Ocean's Role in Climate (award number NA10OAR4320156) and the Southern California Coastal Ocean Observing System (award number NA08OAR-4320894). Spray was developed by the Instrument Development Group (IDG) at the Scripps Institution of Oceanography with funding from the Office of Naval Research. IDG operates the gliders on these sustained lines. Two gliders were purchased by the California Current Ecosystem—Long Term Ecological Research program. Two anonymous reviewers and the editor provided helpful comments to improve this paper.
Appendix A. Details of mixing parameterizations Below we detail the production of depth-mean estimates of turbulent dissipation and diffusivity following Gregg (1989), Gregg et al. (2003), and Polzin et al. (1995). The first step is to scale shear relative to N and when squared is an inverse Richardson number: ∂z u þ i∂z v Shn ¼ qffiffiffiffiffiffiffiffiffi 〈N 2 〉1:5d
ðA:1Þ
The 1.5-day mean is roughly equivalent to a 30-km mean. Data are detrended before taking spectra. Then Shn variance is computed from observed and GM vertical wavenumber spectra as follows. One spectrum is made for each profile. The minimum wavenumber, mmin, is set by the length of the record, which extends from the mixed layer depth (MLD) to at least 96 m deeper and
Please cite this article as: Johnston, T.M.S., Rudnick, D.L., Trapped diurnal internal tides, propagating semidiurnal internal tides, and mixing estimates in the California Current System from.... Deep-Sea Res. II (2014), http://dx.doi.org/10.1016/j.dsr2.2014.03.009i
T.M.S. Johnston, D.L. Rudnick / Deep-Sea Research II ∎ (∎∎∎∎) ∎∎∎–∎∎∎
cpm ]
10
[s
PSD
(PSD) of normalized shear over the appropriate wavenumbers: 2 3 Z mc 6 ∂z u þ i∂z v 7 2 〈Shn 〉 ¼ ðA:2Þ PSD4qffiffiffiffiffiffiffiffiffi 5ðmÞ dm mmin 〈N2 〉1:5 d
4
Sh: x = 167 km Sh: mean
2
z
u+i
1
z
v
16
10
CW + CCW
8
and similarly for 〈ð∂z ηÞ2 〉. Each spectrum is corrected for attenuation at higher wavenumbers from first differencing at 16 m, but not for the ADP's pulse length and binning at 8 and 4 m. These spectra are then averaged over 1.5 days (or about 30 km) to produce a mean spectrum, which has 2 degrees of freedom or possibly 6, since 1 or 2 waves are sampled over 1 day and more than 1.5–3 waves over 1.5 days. The transect-mean spectra, which are not used in our calculations have confidence intervals associated with 50 degrees of freedom (Fig. A1A–B). The energy level in the internal wave field relative to GM is a combination of the kinetic and potential energies which are measured via shear and strain:
2
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z 1
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100 z
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: x = 167 km : mean
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Sh : x = 167 km
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n
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, Sh
z
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2 〈Shn 〉2 1 þ1=Rω E^ ¼ 2 4=3 〈Shn;GM 〉2
101
0
x = 361 km
z
where Rω ¼ 〈Shn 〉=〈ð∂z ηÞ2 〉 is the observed shear-to-strain ratio. Rω is 3 for a GM wavefield at this latitude and increases with increasing frequency and latitude. Note the spectra are averaged 2
x = 167 km GM x = 361 km
GM
10
10
:
3
10
m/2
2
ðA:3Þ
10
1
[cpm]
Fig. A1. Spectra from the example cross-shore section (Fig. 2) show rolloffs for (A) shear and (B) strain near wavenumbers of 1/40 cpm. Shear spectra are the sum of the clockwise and clockwise components. The location of the profiles is identified by x. Transect mean spectra are also shown (thick lines in Fig. A1A, B). With 2 and 50 degrees of freedom, 95% confidence intervals are shown for the individual and transect-mean spectra. (C) Normalized shear spectra (red lines) and strain spectra (blue, dashed lines) averaged over 1.5 days (Section 3.3) at two locations are compared to GM shear (solid dark gray line) and strain (dashed light gray line) spectra. One location is inshore of the SRCR (x ¼ 167 km), while the other is offshore in deep water (x¼ 361 km). With 4 degrees of freedom, 95% confidence intervals are shown for the average spectra. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)
usually further to the maximum depth (i.e., 362 m): mmin/2π ¼1/ (362 - MLD) cycles per meter (cpm). The same maximum depth is used for both shear and strain for consistency, even though reliable strain measurements extend deeper. The maximum wavenumber, mc/2π ¼1/40 cpm, is set by the rolloff of the glider's shear measurements (Fig. A1A), which arises due to the processing (Section 3.1) (Polzin et al., 2002). Both shear and strain show similar mc (Fig. A1A, B). However, a smaller mc may be used based on the idea of an inverse Richardson number i.e., mc is chosen where the cumulative Shn (∂z η) variance reaches 0.66 (0.22) which is at m/2π ¼1/10 cpm for the GM spectrum (Polzin et al., 1995). Some data are excluded from consideration. Data from missions with fewer than 300 profiles are not used. Data from more than 15 km away from the nominal transect are discarded. There are occasional instrument malfunctions. These considerations reduce available sθ and velocity data to 14 357 and 11 864 profiles on line 66.7, 15 992 and 12 779 profiles on line 80, and 15 408 and 14 816 on line 90. Lastly, profiles with fewer than 6 points or 96 m below the MLD are excluded, which permits shear–strain (strain only) mixing estimates on line 66.7 from 9189 (11 253) profiles, on line 80 from 10 527 (13 480) profiles, and on line 90 from 13 893 (15 004) profiles. Variance is obtained by integrating and summing the clockwise and counterclockwise components of the power spectral density
2 over 1.5 days to produce reliable values for E^ (Fig. A1C), a point which is examined in detail by MacKinnon et al. (2013). Shearbased mixing estimates are not calculated without accompanying strain measurements. 2 Turbulent dissipation scales with E^ , N2, Rω , and latitude (α):
ϵ ¼ ϵo
〈N 2 〉1:5d ^ 2 E hðRω ÞLðN; αÞ N 2o
ðA:4Þ
where ϵo ¼6.73 10 10 W kg 1 is the background dissipation due to a typical GM internal wave field, No ¼ 5.24 10 3 rad s 1 is the reference buoyancy frequency, h is a correction for the frequency content of the internal wave field based on Rω , and L is a correction based on f and N: sffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ðA:5Þ h¼ Rω 1 L¼
f cosh f 30 cosh
1
1
ðN=f Þ
ðN o =f 30 Þ
ðA:6Þ
For a GM wavefield, Rω ¼ 3 and h ¼1, but our observed Rω ¼ 5–6 and h 0:6, which implies more energy is in the near-inertial band (which in our case is also the D1 band) than is accounted for in GM (Kunze et al., 2006). L 1 at this study's latitude and mean stratification. L increases from 0 at the equator to 2 at the poles. Above, ϵ is estimated from the combination of Sh2 and ∂z η measurements via the observed Rω , but ϵ can also be obtained solely from ∂z η via an assumed GM value of Rω ¼ 3. This method is useful because velocity is not measured as often as density. With strain only, the energy level is 2 〈ð∂z ηÞ2 〉2 1 þ1=Rω E^ ¼ 4=3 〈ð∂z ηÞ2GM 〉2
The dependence of h on Rω is modified to: sffiffiffiffiffiffiffiffiffiffiffiffiffiffi R2 2 h¼ ω 9 Rω 1
ðA:7Þ
ðA:8Þ
but Rω ¼ 3 in our case and so h¼1.
Please cite this article as: Johnston, T.M.S., Rudnick, D.L., Trapped diurnal internal tides, propagating semidiurnal internal tides, and mixing estimates in the California Current System from.... Deep-Sea Res. II (2014), http://dx.doi.org/10.1016/j.dsr2.2014.03.009i
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Lastly, turbulent diapycnal diffusivity is estimated from the finescale parameterizations of ϵ: Kρ ¼
Γϵ 〈N 2 〉1:5d
ðA:9Þ
where Γ ¼ 0.2 is a typical value of the mixing efficiency. We produce two estimates of K ρ : one from observed shear and strain (Ksh) and the other from strain alone (Kst) with an assumed Rω . We only show ϵ estimated from both shear and strain. The observed Rω ¼ 5 6 and so using Rω ¼ 3 in the calculation of Kst may produce an underestimate of 1.5–2 with respect to Ksh, which is the more accurate estimate (Polzin et al., 1995; MacKinnon et al., 2013).
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Trapped diurnal internal tides, propagating semidiurnal internal tides, and mixing estimates in the California Current System from sustained glider observations, 2006–2012 T.M. Shaun Johnston n, Daniel L. Rudnick Scripps Institution of Oceanography, University of California, San Diego, 9500 Gilman Dr # 0213, La Jolla, CA 92093-0213, USA
art ic l e i nf o
Keywords: Mixing parameterizations Internal tides Seasonal cycle Cross-shore decay
a b s t r a c t From 2006–2012, along 3 repeated cross-shore transects (California Cooperative Oceanic Fisheries Investigations lines 66.7, 80, and 90) in the California Current System, 33 609 shear and 39 737 strain profiles from 66 glider missions are used to estimate mixing via finescale parameterizations from a dataset containing over 52 000 profiles. Elevated diffusivity estimates and energetic diurnal (D1) and semidiurnal (D2) internal tides are found: (a) within 100 km of the coast on lines 66.7 and 80 and (b) over the Santa Rosa-Cortes Ridge (SRCR) in the Southern California Bight (SCB) on line 90. While finding elevated mixing near topography and associated with internal tides is not novel, the combination of resolution and extent in this ongoing data collection is unmatched in the coastal ocean to our knowledge. Both D1 and D2 internal tides are energy sources for mixing. At these latitudes, the D1 internal tide is subinertial. On line 90, D1 and D2 tides are equally energetic over the SRCR, the main site of elevated mixing within the SCB. Numerous sources of internal tides at the rough topography in the SCB produce standing and/or partially-standing waves. On lines 66.7 and 80, the dominant energy source below about 100 m for mixing is the D1 internal tide, which has an energy density of the D2 internal tide. On line 80, estimated diffusivity, estimated dissipation, and D1 energy density peak in summer. The D1 energy density shows an increasing trend from 2006 to 2012. Its amplitude and phase are mostly consistent with topographically-trapped D1 internal tides traveling with the topography on their right. The observed offshore decay of the diffusivity estimates is consistent with the exponential decay of a trapped wave with a mode-1 Rossby radius of 20–30 km. Despite the variable mesoscale, it is remarkable that coherent internal tidal phase is found. & 2014 Elsevier Ltd. All rights reserved.
1. Introduction Mixing in the thermocline is produced mainly by breaking internal waves (Gregg, 1989; MacKinnon et al., 2013), which are forced predominantly by tides and winds (Alford, 2003; Ferrari and Wunsch, 2009; Garrett and Kunze, 2007). Much of the internal wave energy is in low vertical modes (i.e., in waves with horizontal wavelengths of 100 km or more) and propagates away from generation sites (Alford, 2003; Zhao and Alford, 2009). Smallerscale internal waves may break near their generation site and contribute to mixing (Gregg et al., 1986; Nakamura et al., 2010; Alford and Gregg, 2001; Klymak et al., 2006). Here, we consider the contribution of not only the freely-propagating semidiurnal (D2) internal tide (Garrett and Kunze, 2007), but also the subinertial
n
Corresponding author. Tel.: þ 1 858 534 9747. E-mail address: [email protected] (T.M.S. Johnston).
diurnal (D1) internal tide, which is topographically-trapped near its generation site. The latter topic has received much less attention in the literature. In our study area in the coastal ocean off of California, D1 internal tides are often more energetic than D2 in our observations and in previous work around the Southern California Bight (SCB) (Beckenbach and Terrill, 2008; Kim et al., 2011; Nam and Send, 2011). Poleward of 301, D1 internal waves are subinertial and evanes1 cent, since the Coriolis frequency (f) is 4 1 cycle day . Considerable mixing may arise due to topographically-trapped internal tides because they likely dissipate in the same area where they are forced (Padman and Dillon, 1991; Nakamura et al., 2010), although alongshore propagation is possible with dissipation elsewhere at the topography. Such mixing may contribute to water mass formation (Tanaka et al., 2010) and higher primary productivity (Tanaka et al., 2013). Energy loss from the D1 barotropic tide is a maximum around the Pacific rim (Egbert and Ray, 2003). A local maximum in this dissipation is found along the west coast of
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Please cite this article as: Johnston, T.M.S., Rudnick, D.L., Trapped diurnal internal tides, propagating semidiurnal internal tides, and mixing estimates in the California Current System from.... Deep-Sea Res. II (2014), http://dx.doi.org/10.1016/j.dsr2.2014.03.009i
T.M.S. Johnston, D.L. Rudnick / Deep-Sea Research II ∎ (∎∎∎∎) ∎∎∎–∎∎∎
2
North America beyond 301N. The sink for this energy is likely the topographically-trapped D1 internal tide, which in turn dissipates turbulently. To assess the contributions of topographically-trapped D1 and freely-propagating D2 internal tides to mixing in the coastal ocean, we use an extensive dataset to estimate diffusivity using parameterizations based on observed current shear and/or isopycnal strain (Gregg, 1989; Polzin et al., 1995; Gregg et al., 2003). Six years of sustained glider observations in the California Current System (CCS) along three repeated California Cooperative Oceanic Fisheries Investigations (CalCOFI) cross-shore transects (lines 66.7, 80, and 90 in Fig. 1) provide over 52 000 profiles, at Oð10 m) vertical and 3-km horizontal resolution from the continental slope to 300–500 km offshore (Davis et al., 2008; Todd et al., 2011a, 2012). With this unique combination of spatial and temporal coverage, we can examine the seasonal and cross-shore structure of mixing estimates and internal tides. For example, D1 internal tides may control the cross-shore decay scale at the continental slope and the seasonal cycle of mixing at the northern end of the SCB. At the Santa Rosa-Cortes Ridge (SRCR), a prominent feature of the SCB, D1 and D2 internal tides have similar energies and may contribute equally to mixing. Next, we provide some background on mixing parameterizations (Section 2.1), previous observations of mixing in the SCB and other coastal locations (Section 2.2), and possible energy sources for mixing (internal waves and frontogenesis in Sections 2.3 and 2.4). In Section 3, our methods are described including a basic outline of the mixing parameterizations with further details in Appendix A. Section 4 shows an example transect from line 90 and time-mean transects (i.e., binned in depth and cross-shore distance) of the data and harmonic fits from lines 80 and 90 (line 66.7 is similar to line 80). In Section 5, depth-mean diffusivity estimates
Monterey Bay
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2. Background 2.1. Finescale mixing parameterizations Direct measurements of either mixing via tracer release or turbulent fluctuations via specialized microstructure instruments occur of necessity during infrequent research cruises of usually several weeks duration. Only recently moored microstructure measurements have become possible (Moum et al., 2013). However, finescale or Oð10m) scale profiles of shear and strain are readily available from many platforms including the gliders described in this paper. Turbulent dissipation (ϵ) and diapycnal diffusivity (K ρ ) can be estimated from these finescale measurements using parameterizations based on the idea of a downscale energy transfer via weakly interacting internal waves (Gregg, 1989; Polzin et al., 1995; Gregg et al., 2003; Kunze et al., 2006; MacKinnon et al., 2013). Assuming there is a steady state transfer, by measuring the finescale variance in shear and strain, we estimate the downscale energy transfer or dissipation at a scale much larger than the actual turbulence. The parameterizations relate ϵ to the ratio of the observed finescale variance to the variance in the empirical Garrett–Munk (GM) spectra of the internal wave field. The GM spectrum is used to compare internal wave energy levels in locations with different stratification and latitude. For example, over the SRCR, a site of internal tidal generation, some example shear and strain spectra are within a factor of three of GM (Appendix A). For further comparison, the mission-mean spectrum from glider measurements east of Luzon Strait, which averages over regions of both weak and strong internal wave activity, is consistent with a level of 4 times GM (Rudnick et al., 2013). These values appear reasonable and reinforce the well-known utility of the GM spectrum in a wide variety of conditions. Finescale parameterizations have been applied to diverse data sets over large areas with appreciable tidal and inertial signals (Kunze et al., 2006; MacKinnon et al., 2013; Whalen et al., 2012; Waterhouse et al., 2014). These methods under a variety of conditions are considered accurate to a factor of 2–4 at best (MacKinnon et al., 2013, and references therein) compared to specialized microstructure instruments, which measure the actual turbulent fluctuations (Klymak and Nash, 2009; Thorpe, 2005). In this paper, we estimate mixing in the coastal ocean using such parameterizations (Section 3.3 and Appendix A). 2.2. Previous observations of mixing in coastal waters
32o
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are binned in time or cross-shore distance and are then related to similarly averaged internal tides and mesoscale flows. A seasonal cycle is composited for line 80 (Section 5.4). A discussion and summary of our findings follow in Sections 6 and 7.
Depth [km]
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Fig. 1. Example tracks (orange) from glider missions 12801401, 12801101, and 12803001 along (a) line 66.7 and (b) lines 80 and 90 are shown over the bathymetry (blue colors) off the California coast. The origins of the cross-shore coordinate (x) are denoted by the circles near Monterey, Point Conception, and Dana Point. The Santa Rosa-Cortes Ridge (SRCR) extends roughly south-southeast from Point Conception and is a key topographic feature on line 90. Lines 66.7 and 80 have relatively narrow shelves, while line 90 extends over the rough bathymetry of the Southern California Bight. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)
With rough topography, varying stratification, and numerous local and remote sources of internal waves in our study region, considerable spatial and temporal variability of mixing is expected. Using our methods, we can examine variability in the cross-shore direction along with tidal and seasonal variability, which complements previous moored or ship-based process studies, which have greater temporal resolution, but are limited in temporal or spatial extent. Such variability in mixing over a semidiurnal tidal period, over a spring-neap cycle, and over variable topography is found in a bay situated between two distinct sources of internal tides near Oahu, Hawaii (Alford et al., 2006; Martini et al., 2007). Also the two sources of internal tides are of unequal strength and produce partially-standing internal waves. This interference pattern varies
Please cite this article as: Johnston, T.M.S., Rudnick, D.L., Trapped diurnal internal tides, propagating semidiurnal internal tides, and mixing estimates in the California Current System from.... Deep-Sea Res. II (2014), http://dx.doi.org/10.1016/j.dsr2.2014.03.009i
T.M.S. Johnston, D.L. Rudnick / Deep-Sea Research II ∎ (∎∎∎∎) ∎∎∎–∎∎∎
2.3. Internal waves Shear and strain from internal waves produce instabilities and mixing (Alford and Pinkel, 2000). Below we briefly cover some features of near-inertial, D1, and D2 internal waves relevant to the SCB and mixing. In general, shear in the internal wave band is dominated by NIW (Alford and Gregg, 2001; Pinkel, 1985). In the SCB, wind stress variance is greatest from March–June [Table 1 in Hickey, 1979]. The region near Point Conception at the northern edge of the SCB displays a local maximum in a model of wind work to inertial motions in the mixed layer, i.e. a precursor to NIW generation in the thermocline (Figure 2 in Simmons and Alford, 2012). The magnitude of this input is similar to stormier climates near 451N on the west coast of North America. Trapped D1 internal tides are often more energetic than D2 internal tides in the SCB. These waves may have considerable impact on mixing as noted earlier (Section 1) because there are likely numerous sites of generation and dissipation in close proximity at rough topography (Tanaka et al., 2013). Elsewhere, at a seamount in the North Pacific near 321N, dissipation is strong enough to dissipate the trapped D1 motions within 3 days, which makes for a strongly forced and damped system (Kunze and Toole, 1997). In the SCB, surface drifters in this area show diurnal/inertial oscillations with decay scales of 10 days (Poulain, 1990). There are numerous sources of D2 internal tides along the continental slope and over the rough topography of the SCB (Beckenbach and Terrill, 2008; Buijsman et al., 2012). Internal tidal generation may produce local mixing (Johnston et al., 2011a; Klymak et al., 2006). Propagating internal tides may encounter topography, scatter to smaller scales, and break (Johnston and Merrifield, 2003; Johnston et al., 2003; Kelly et al., 2012; Martini et al., 2011; Nash et al., 2004). 2.4. Fronts Other relevant sites of elevated mixing includes fronts (Hoskins and Bretherton, 1972; Johnston et al., 2011b), which are ubiquitous in the SCB (e.g., Davis et al., 2008; Powell, 2014). Frontogenesis can lead to mixing at depth on the dense, cyclonic side of fronts where
the gradients (i.e., shear and strain) are largest (Hoskins and Bretherton, 1972). It is unclear what are the relative contributions to mixing from frontogenesis or internal waves which are trapped, reflected, or refracted at a front (Johnston et al., 2011b). Elevated mixing at fronts can lead to enhanced primary productivity (Hales et al., 2009; Li et al., 2012). Three possible issues are noted with respect to estimating mixing at fronts. First, if mixing at fronts is not due to internal waves, then the mixing parameterizations used here do not apply. Second, there is some question of how much the internal wave field is modified in the vicinity of background shear and whether the empirical GM spectrum (Section 3.3) is applicable (Munk, 1981). Third, we note persistent fronts at locations where internal tides are strong on both lines 80 and 90, which unfortunately hampers discrimination of their effects. For the first two issues, we note our spectra resemble GM (Appendix A), which provides reasonable confidence in our approach. These parameterizations essentially use a Richardson number criterion as an indicator of mixing and have proven useful across a wide range of conditions (Section 2.1).
3. Methods 3.1. Data Since 2006/2007, three cross-shore transects have been almost continuously covered by underwater gliders to 500-m depth (Davis et al., 2008; Todd et al., 2011b). These transects follow CalCOFI lines 66.7, 80, and 90 which begin near Monterey Bay, Point Conception, and Dana Point and extend 300–550 km offshore (Fig. 1). One glider mission lasts about 100 days, with each cross-shore transect taking about 3 weeks (Fig. 2). In this analysis, over 19 000 density and velocity profiles from October 2006 to
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over the spring-neap cycle and with varying stratification at or between the sources. Models generally have greater turbulence at boundaries, but do not reproduce observed midwater column mixing with a variety of turbulence parameterizations (Wijesekera et al., 2003). In the coastal waters off Oregon, the superposition of shear from inertial and semidiurnal internal waves with shear from a coastal current in thermal wind balance leads to midwater column mixing, when none of the individual components would trigger a shear instability (Avicola et al., 2007). It is now more common for regional models to include both mesoscale and tidal phenomena, but assessment of their mixing parameterizations is lacking perhaps due to a lack of sufficient temporal and spatial extent of observations for comparison. The depth-mean mixing estimates (excluding the mixed layer but including the thermocline to a depth of 362 m) in this paper may prove useful in this respect. Previous tracer release experiments and microstructure measurements in the SCB show elevated K ρ at the steep topography, which can be as high as Oð10 4 m2 s 1 Þ (Gregg and Kunze, 1991; Ledwell and Watson, 1991; Ledwell and Hickey, 1995; Ledwell and Bratkovich, 1995). This value is much larger than away from the topography, where the mean thermocline values is K ρ 10 5 m2 s 1 (Gregg, 1987). Limited velocity measurements do not identify the processes leading to this mixing near topography, but either trapped D1 or freely-propagating D2 internal tides are likely candidates.
3
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[m] 20 0 20
200 400 0 200
z
400 400
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x [km] Fig. 2. An example cross-shore section on line 90 from mission 12803001 for a glider heading toward shore (x ¼0) from 3–27 September 2012 shows (A) potential temperature, isopycnals (black lines, sθ ¼ 24–26:5 kg m 3 in 0.5 kg m 3 intervals), and the mixed layer depth (gray line); (B) salinity; (C) vertical displacements; and (D) strain. Gray shading denotes regions with no data, which are larger for displacement data due to the lowpass averaging in their calculation. Longitude along the line is on the upper axis of each subplot. Additional smaller tick marks on the upper axis of Fig. 2B mark positions of glider profiles.
Please cite this article as: Johnston, T.M.S., Rudnick, D.L., Trapped diurnal internal tides, propagating semidiurnal internal tides, and mixing estimates in the California Current System from.... Deep-Sea Res. II (2014), http://dx.doi.org/10.1016/j.dsr2.2014.03.009i
T.M.S. Johnston, D.L. Rudnick / Deep-Sea Research II ∎ (∎∎∎∎) ∎∎∎–∎∎∎
December 2012 are used on each of lines 80 and 90. This dataset comprises 27 glider missions on line 90 and 22 on line 80. From April 2007 to December 2012, over 14 000 profiles from 17 glider missions on line 66.7 are used. Data plots are available for these missions at http://spray.ucsd.edu. In Fig. 2, the example transect is from mission 12803001, where the identifier comprises a twodigit year (12 for 2012), a one-digit hexadecimal month (8 for August), a three-digit glider serial number (030 for Spray 30), and a two-digit mission number (01 in this case). A Spray glider moves vertically through the ocean by inflating/ deflating an external bladder, while wings provide lift to move the glider forward resulting in a sawtooth pattern in depth and time (or horizontal distance) (Sherman et al., 2001). The payload comprises (a) a pumped Sea-Bird Electronics (SBE) 41CP conductivity-temperature-depth (CTD) instrument from which potential temperature (θ), salinity (S), in situ density (ρ), and potential density (sθ ) are obtained (Fig. 2A, B); (b) a Seapoint chlorophyll a (chl a) fluorometer; and (c) a Sontek 750 kHz acoustic Doppler profiler (ADP) aligned to measure horizontal velocities in five 4-m vertical range bins (Davis, 2010; Todd et al., 2011b). Data are obtained on ascents, when the CTD experiences clean flow. For the purposes of this paper, data and times are averaged in 31 vertical bins, which are 16-m tall, non-overlapping, and centered from 10–490 m. A dive cycle to 500 m and back to the surface is completed every 3 h, which gives a Nyquist frequency of 1/6 cycle h 1 and is sufficient to sample D2 signals (Section 3.2). During a dive cycle, a glider moves about 2.7 km through the water. Vertical displacements are obtained from density differences from a lowpassed mean (Fig. 2C):
ηðz; tÞ ¼
g Δsθ
sθ 〈N 2 〉1:5d
ð1Þ
where z is the vertical coordinate (up is positive), t is time, g is gravitational acceleration, 〈 〉 denotes a running mean over the subscripted intervals i.e., 1.5 days, Δsθ ¼ sθ 〈sθ 〉1:5d is the density deviation from the 1.5-day lowpassed mean, and N2 is the buoyancy frequency. The 1.5-day mean is roughly equivalent to a 30-km mean. Due to this averaging window, η and other variables calculated below with similar averaging will vary slowly in time over periods of O (1 day or 20 km) as in Fig. 2C. Some further details of sampling with a slowly-moving glider are covered in Section 3.2. Strain, ∂z η, is calculated from a first difference (Fig. 2D). Velocity profiles are made similar to lowered acoustic Doppler current profiles from ships (Visbeck, 2002). Depth-mean currents are obtained from a combination of Global Positioning System fixes and a glider's measured attitude (Todd et al., 2009). The depth-mean current and ADP-measured, glider-relative velocities are combined in a linear system of equations that is solved by a least squares method for water velocities (Todd et al., 2011b). Referencing objectively-mapped and vertically-integrated ADPmeasured shear to the depth-mean current produces similar results (Davis, 2010). Hereafter, horizontal currents (u and v) are given in the crossshore and alongshore directions (Fig. 3A, B), which are positive onshore (x) and to the approximately north–northwest of the transects (y). The coordinates are defined separately for each of lines 66.7, 80, and 90. Glider tracks differ from these lines: (a) when depth-mean currents are strong and contrary to the desired glider course and (b) during launch and recovery of gliders near the coast. We calculate shear components, i.e. ∂zu and ∂zv, from first differences (Fig. 3C, D). We note elevated shear variance below 346–378 m due to a decrease in acoustic scatterers. Quantities related to shear variance below 362 m are excluded from further consideration, most notably (a) the D1 and D2 velocity bandpasses (Section 3.2) and (b) the diffusivity estimates, which
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x [km] Fig. 3. The example cross-shore section on line 90 (Fig. 2) shows (A) cross-shore velocity (positive onshore); (B) alongshore velocity (positive roughly north-northwestward); (C) and (D) shear components; and (E) Rossby number. Gray shading denotes regions with either excluded or no data, which is larger for Ro due to the averaging needed to calculate it. Longitude along the line is on the upper axis of each subplot.
depend on shear variance squared (Section 3.3 and Appendix A). However, we still show mean cross-shore sections of u, v, and Rossby number (Ro) with data below 362 m, since the noise is at high vertical wavenumber and thus affects shear more strongly than velocity. Using full-depth data produces minor quantitative differences in diffusivity estimates by including shear measurements with lower signal-to-noise ratio, but none of the qualitative conclusions changes. Using this cutoff produces better agreement between shear- and strain-based mixing parameterizations (method described in Section 3.3). Horizontal gradients of density and currents are calculated after smoothing the density and currents over 30 km (e.g., Fig. 3E). This procedure is needed because spectra show noise (i.e., internal waves) in density gradients and geostrophic currents at wavelengths shorter than about 30 km when calculated along depth surfaces (Rudnick and Cole, 2011). Since we only measure the ∂xv component of relative vorticity (ζ), we estimate Ro ¼ ζ =f ∂x v=f (Fig. 3E) and so the actual Ro could be twice as large if there is solid body rotation, where ∂x v ¼ ∂y u. The fluorometer is uncalibrated and suffers from offsets from mission to mission, but for any given mission the spatial pattern is qualitatively accurate even though the magnitude is not. Fluorometer voltage is multiplied by 3 to obtain an uncalibrated number similar to chl a concentrations in mg m 3 to illustrate relative changes. Mixed layer depth is calculated with Δsθ ¼ 0:1 kg m 3 from the near-surface value (Johnston and Rudnick, 2009). 3.2. Energy density and fluxes With relatively slow-moving gliders ( 20 km/day), we estimate tidal and near-inertial signals similar to Johnston et al.
Please cite this article as: Johnston, T.M.S., Rudnick, D.L., Trapped diurnal internal tides, propagating semidiurnal internal tides, and mixing estimates in the California Current System from.... Deep-Sea Res. II (2014), http://dx.doi.org/10.1016/j.dsr2.2014.03.009i
T.M.S. Johnston, D.L. Rudnick / Deep-Sea Research II ∎ (∎∎∎∎) ∎∎∎–∎∎∎
mode-1 fits of the harmonics are used where depths exceed 362 m: η~ 0 , u~ 0 , and v~ 0 , where the tilde denotes the modal fit (Johnston et al., 2013). Data are extended using the World Ocean Atlas 2009 analysis for depths greater than 500 m (Antonov et al., 2010; Locarnini et al., 2010) and bathymetry from Smith and Sandwell (1997). The mode-1 pressure perturbation, p~ 0 , is calculated from η~ 0 (Johnston et al., 2013). Lastly, mode-1 D1 and D2 energy fluxes are obtained based on both density and velocity measurements:
(2013). Space–time confusion arises since relatively high frequency signals (i.e., internal waves with frequencies from f to N) are projected into spatial variations (Rudnick and Cole, 2011). The projection is caused by two effects (smearing because the glider moves slowly and aliasing because the glider profiles at a frequency in the internal wave band) and cannot be undone. The smearing of D1 and D2 signals with wavelengths of order 100 km is limited to observed wavenumbers of about 1/20-1/10 cycles km 1 (equation 5 in Rudnick and Cole, 2011), which is within our averaging window (following paragraph). Therefore, the slow motion of the glider is advantageous in estimating these harmonic signals. A moving 1.5-day ( 30-km) window is used, which discriminates between D1 and D2, but cannot distinguish near-inertial motions from D1 since the frequency resolution is 2/3 cycle day 1. In essence, this method is a bandpass filter. Harmonic analysis of η, u, and v using M2 and K1 frequencies is applied over the moving 1.5-day window producing the harmonics for displacement (η0 ), for example, as follows (Fig. 4):
η0 ðz; tÞ ¼ Aη cos ½ωðt t o Þ þ ϕη
½F x ðz; tÞ; F y ðz; tÞ ¼ 〈p~ 0 u~ 0 ; p~ 0 v~ 0 〉1:5d
3.3. Finescale mixing parameterizations As noted in Section 2.1, the advantage of these parameterizations is that turbulence estimates can be made from standard Doppler current and CTD profiles. The physical basis of these finescale mixing parameterizations relies on a steady downscale energy transfer from large-scale weakly interacting waves to small-scale breaking internal waves to turbulence. For a GM internal wave field, this transition occurs at roughly 10 m, but this cutoff is adjusted based on Richardson number. The white wavenumber spectrum of either shear or strain drops off at wavenumbers above the cutoff. In our case, the cutoff is placed at 1/40 cycle m 1, where noise in the ADP-measured shear becomes apparent. Details of the calculations are in Appendix A. We produce two estimates of K ρ : one from observed shear and strain (Ksh) and the other from strain alone (Kst) with an assumed shear-to-strain ratio. The observed shear-to-strain ratio is larger than the GM value and so Kst will be an underestimate by a factor of up to 2 (Appendix A). We only show ϵ calculated from both shear and strain.
The amplitude is Aη ðz; tÞ ¼ jηr þ iηi j, where ηr ¼ Aη cos ϕη and ηi ¼ Aη sin ϕη are the real and imaginary components of the fit to cos ½ωðt t o Þ and sin ½ωðt t o Þ. The phase is ϕη ðz; tÞ ¼ tan 1 ðηi =ηr Þ. The M2 and K1 periods are T ¼12.42 and 23.93 h. The radial frequency is ω ¼ 2π =T. The reference time, to, for phase calculations is 0000 Coordinated Universal Time (UTC) on 1 January 2010, roughly the midpoint of the data. Velocity fits are made in complex form, u þ iv. Inertial motions have T ¼23.2–20.9 h from 311 to 351N. Later when mean phases are shown, they are calculated from the means of the real and imaginary components. Energy density and fluxes in D1 and D2 frequency bands are calculated from amplitudes as ð3Þ
where time averages, 〈 〉t , are mission means (Lee et al., 2006). Kinetic energy (Ek) comprises the first two terms, while potential energy (Ep) is the last term. No attempt has been made to remove barotropic components, but the results of the harmonic analysis appear dominated by baroclinic structure i.e., internal waves (Fig. 4). Energy fluxes require full-depth measurements and so
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3.4. Bin means In much of the following analysis of temporal and spatial variability (Sections 4 and 5), data are binned in cross-shore
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with averages over 1.5 days or 30 km. The mode fit is performed from 20–362 m in depth, even though modes are orthogonal only over the full water depth and can lead to flux errors of about 40%. Limits of the modal decomposition are explained in more detail in Johnston et al. (2013).
ð2Þ
Eðz; tÞ ¼ 〈ρ〉t ðA2u þ A2v þ 〈N 2 〉t A2η Þ=4
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Fig. 4. D1 (left) and D2 (right) harmonics for (A, B) cross-shore velocity, (C, D) alongshore velocity, and (E, F) vertical displacement are sometimes larger near topography on the example cross-shore section on line 90 (Fig. 2). Gray shading denotes regions with either excluded or no data. Longitude along the line is on the upper axis of each subplot.
Please cite this article as: Johnston, T.M.S., Rudnick, D.L., Trapped diurnal internal tides, propagating semidiurnal internal tides, and mixing estimates in the California Current System from.... Deep-Sea Res. II (2014), http://dx.doi.org/10.1016/j.dsr2.2014.03.009i
T.M.S. Johnston, D.L. Rudnick / Deep-Sea Research II ∎ (∎∎∎∎) ∎∎∎–∎∎∎
6
SRCR on line 90. The mesoscale current and density structures are also described. First, we examine a single example transect on line 90 to illustrate the type of data obtained from a glider (Section 4.2). Second, we apply harmonic analysis to u, v, and η. Third, timemean transects are then examined to better identify mesoscale and internal tidal flows for lines 80 and 90 (Sections 4.3 and 4.4). Several features are noted on the example transect on line 90 including (Figs. 2–4): (a) larger displacements above the SRCR; (b) greater strain within the SCB; (c) considerable mesoscale variability in Ro; (d) a front over the SRCR; and (e) stronger D1 and D2 motions in the SCB but especially at the SRCR. The timemean transects describe the expected cross-shore structure of the CCS, confirm the features described earlier in this paragraph, and further display consistent tidal phase near these internal tidal generation sites (Figs. 5–8). These characteristics are used to distinguish NIW from D1 internal tides, identify D1 trapping, and illustrate D2 propagation.
distance and time, x and t. These results are used to describe timemean, cross-shore sections along with the seasonal cycle of the currents, density structure, internal tides, and mixing. To form time-mean cross-shore sections, bin sizes are Δx ¼ 10 km. Data located 415 km away from the nominal transect are not considered, which mainly occurs on launch and recovery, but also when the depth-mean cross-transect (i.e., alongshore) currents exceed 0.25 m s 1. Derived quantities (e.g., ∂x s or N2) are calculated first and then binned. Bin means in depth and x are shown for some data and amplitudes and phases from the harmonic fits. Bin-mean phases are calculated from the bin means of real and imaginary components. Bin means of depth-integrated fluxes and depth-mean energies are shown. The former emphasizes mode-1 content, while the latter better represents higher modes (i.e., smaller-scale vertical structure). Temporal bins are Δt ¼ 60 days, which is enough time to cover slightly more than two cross-shore transects. When binning is both in t and x, Δx is increased to 20 km. These values for x and t bins are similar to decorrelation scales used in objective maps of the same data (Todd et al., 2011b). A composite seasonal cycle for line 80 is constructed by binning data by month. A time series for line 80 is made by binning data from x¼ 100 to 0 km in 60 day increments. Lines 66.7 and 90 show little indication of seasonal cycle in K ρ and so we do not show similar calculations there.
4.2. An example from line 90 An example of a shoreward transect on line 90 during mission 12803001 from 3–27 September 2012 illustrates typical data (Figs. 2 and 3). Tick marks along the upper axis of Fig. 2B show glider dives roughly every 3 km. While the transect is not synoptic, mesoscale structure and internal waves are apparent. Increased internal wave activity over the SRCR (topography near x ¼ 190 km) produces vertical displacements seen in T, S, sθ , and η (Fig. 2A–C). Larger strain is found near the topography and within the SCB (x 4 200 km, Fig. 2D). A front is found near the SRCR, as evidenced by increased cross-shore gradients of θ, S, and sθ there (Fig. 2A, B). Both internal wave oscillations and mesoscale
4. Cross-shore and depth structure of the mesoscale and internal tides: an example section and time means 4.1. Overview In this section, we are primarily concerned with examining D1 and D2 variability near the continental slope on line 80 and the
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Fig. 5. On line 80, available data are binned in cross-shore distance and depth to produce the time-mean of: (A) potential density (gray lines, sθ ¼ 25 and 26 kg m 3); (B) horizontal density gradient after smoothing density over 30 km; (C) buoyancy frequency; (D) chl a; (E) cross-shore current (positive onshore); (F) alongshore current (positive roughly north–northwestward); (G) number of density data in each bin; (H) Rossby number after smoothing v over 30 km; and (I) number of velocity data in each bin. (J) The bathymetry under the mean glider track. In Fig. 5A–F and H, gray shading indicates lack of data and the black lines show the bathymetry (Fig. 5J), but sometimes data are obtained off the mean glider track in deeper water. Longitude along the line is on the upper axis of each subplot.
Please cite this article as: Johnston, T.M.S., Rudnick, D.L., Trapped diurnal internal tides, propagating semidiurnal internal tides, and mixing estimates in the California Current System from.... Deep-Sea Res. II (2014), http://dx.doi.org/10.1016/j.dsr2.2014.03.009i
T.M.S. Johnston, D.L. Rudnick / Deep-Sea Research II ∎ (∎∎∎∎) ∎∎∎–∎∎∎
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Fig. 6. On line 80, D1 (left) and D2 (right) amplitudes and phases are binned in cross-shore distance and depth to produce a time mean. Phases are obtained from binned complex amplitudes. (A, B) The complex velocity amplitudes (Section 4.3) and (C, D) the displacement amplitudes are largest near topography. Phases of (E) D1 cross-shore velocity, (F) D2 cross-shore velocity, and (G, H) displacement are shown. (I, J) Depth-mean phase differences are used to determine propagation characteristics. Gray shading indicates lack of data and the black lines show the bathymetry (Fig. 5J), but sometimes data are obtained off the mean glider track in deeper water. Longitude along the line is on the upper axis of each subplot.
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Fig. 7. Time-mean on line 90 as in Fig. 5. The location of the SRCR is indicated near x ¼ 190 km.
Please cite this article as: Johnston, T.M.S., Rudnick, D.L., Trapped diurnal internal tides, propagating semidiurnal internal tides, and mixing estimates in the California Current System from.... Deep-Sea Res. II (2014), http://dx.doi.org/10.1016/j.dsr2.2014.03.009i
T.M.S. Johnston, D.L. Rudnick / Deep-Sea Research II ∎ (∎∎∎∎) ∎∎∎–∎∎∎
200
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Fig. 8. Line 90's time-mean D1 and D2 amplitudes, phases, and phase differences as in Fig. 6.
structure are seen in the currents (Fig. 3A, B). With a glider moving at 20 km/day, 5 and 10 oscillations in currents over 100 km indicate D1 and D2 motions. Both up- and downward phase propagation is present in the shear, which indicate a combination of down- and upward propagating internal wave energy (Fig. 3C, D), but shear does not appear particularly elevated near the topography in this transect. The magnitude of Ro exceeds 0.4 near the coast (Fig. 3E), but may be up to twice as large (Section 3.1). Internal tides appear larger near the SRCR and within the SCB. In particular, D1 and D2 η are larger (Fig. 4E, F). Velocities can be equally strong further offshore, especially for D1, which could be due to NIW (Fig. 4A–D). NIW currents impinging on topography would produce larger displacements near the topography as would trapped D1 internal tides (Fig. 4E), while freelypropagating D2 internal tides should show larger η radiating away from the topography higher and lower in the water column, as possibly seen in Fig. 4F. Larger D1 v and η preferentially on one side of the SRCR suggest topographic trapping, a process which is examined in more detail below. 4.3. Time-mean, cross-shore transects: lines 66.7 and 80 The time-mean, cross-shore structure displays the canonical, mid-latitude eastern boundary current system with a coastal upwelling front, a poleward undercurrent, and equatorward flow further offshore (Fig. 5) (Hill et al., 1998). Data from all missions are averaged in time to produce a time-mean, x–z section for all three lines. Lines 66.7 and 80 are qualitatively similar with coastal upwelling (shoaling isopycnals and stratification, a density front, and elevated chl a) and a northward undercurrent with anticyclonic ζ (Ro o 0) inshore of the current's core (Fig. 5A–F, H; only line 80). To avoid repetition, figures for line 66.7 are not shown. Typically 4 200 data points are in each bin, but more profiles per unit cross-shore distance are collected near shore where (a) dives are sometimes shallower than 500 m and (b) currents are stronger
and gliders attempt to stay on track by crossing the current at an angle (Fig. 5G, I). The continental slope is steep: at x ¼ 30 km the water depth is over 500 m, but by x ¼ 100 km the water depth reaches 3 km (Fig. 5J). Harmonic analyses of η, u, and v show considerable D1 and D2 motions (Fig. 6, left and right sides). We will show (a) the D1 waves near topography are consistent with trapped internal tides, which decay offshore and propagate with the topography on the right; and (b) the D2 waves propagate offshore from numerous sources, which interfere and produce partially-standing waves. The time-mean D1 amplitudes are largest (a) within 30–40 km of the coast due to trapped D1 internal tides and (b) in the upper 100 m presumably due to NIW (Fig. 6A, C). Aη exceeds 12 m in places, which is not much larger than an open ocean GM rootmean-squared η of 7 m. jAu j is the magnitude of the tidal velocity qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi amplitude, A2u þ A2v . D1 motions will be more visible as currents, since internal waves become more horizontal as ω approaches f. The spatial extent of surface-intensified jAu j suggests NIW, which are ubiquitous. Over the slope and below 50 m, Aη decays to background values over about 70 km in the cross-shore direction. This decay is consistent with the baroclinic Rossby radius of 20 km (a decay to 5% of background levels occurs over 60 km) and trapped D1 internal tides. The decay of Au is not as apparent as with Aη . D1 phases are useful for distinguishing NIW from internal tides. ϕη is relatively constant near the continental shelf and slope where amplitudes are large (i.e., in a triangular region in the lower right of Fig. 6G, where x 4 100 km). In the same small region, jAu j is also large and ϕv is also relatively constant (Fig. 6E). Over most of the water column away from the surface and topography, ϕv shows relatively little change compared with ϕη . The D1 phase differences (ϕη ϕu and ϕη ϕv ) are useful indicators of the direction of wave propagation. For a two-layer
Please cite this article as: Johnston, T.M.S., Rudnick, D.L., Trapped diurnal internal tides, propagating semidiurnal internal tides, and mixing estimates in the California Current System from.... Deep-Sea Res. II (2014), http://dx.doi.org/10.1016/j.dsr2.2014.03.009i
T.M.S. Johnston, D.L. Rudnick / Deep-Sea Research II ∎ (∎∎∎∎) ∎∎∎–∎∎∎
flow, when the lower layer's η and velocity have phase differences of 01/1801 (i.e., in phase/out of phase), the wave propagates in the positive/negative direction (see Figure 6.3 in Gill, 1982). Phase differences of 7901 imply standing waves. Depth-mean phase differences are obtained from 154–362 m to avoid NIW influence near the surface and emphasize deeper D1 internal tidal motions. The depth mean of complex amplitude for each quantity (η, u, and v) is calculated at each cross-shore distance, the angle is computed, and then we difference two such angles (Fig. 6I). With this method, regions with higher amplitudes are emphasized. For x 4 100 km, where D1 amplitudes are largest (not including the surface), ϕη ϕu is near 7 901, which yields no cross-shore energy propagation and is consistent with trapped waves (Fig. 6I). On the other hand, ϕη ϕv is near 0 (x ¼ 120 to 30 km, Fig. 6I), which implies northward phase propagation for a mode-1 wave. The combination of these phase results with elevated amplitudes near the slope indicates a topographically-trapped baroclinic wave traveling with the topography on its right. Next, D2 harmonic fits are used to illustrate cross-shore propagation in contrast to D1 above. One internal tidal generation site appears to be at the shelf edge and another, deeper one over the slope is near x¼ 80 km with waves propagating both onshore and offshore from there. Canyons, submarine ridges, and a seamount are found near 341N, 1211E (or x¼ 80 km) and are somewhat visible in Fig. 1B. D2 jAu j is large near the topography, in the upper 100 m, and also about 150 km away from the slope (Fig. 6B). This distance corresponds to a D2 wavelength and suggests a surface reflection of a tidal beam (Cole et al., 2009; Johnston et al., 2011a; Martin et al., 2006). Aη also suggests a beam-like structure emanating upward from topography near x¼ 80 km (Fig. 6D). Several regions of on- and offshore phase increase are found in ϕu and ϕη (Fig. 6F, H), which produce roughly 75 km-long sections of on- and offshore propagation, with ϕη ϕu ¼ 01 and 1801 (Fig. 6J). Onshore propagation (ϕη ϕu ¼ 0) is found for x 4 70 km consistent with generation at the slope. A standing wave (ϕη ϕu ¼ 90) is found further offshore ð 140 km o x o 70 kmÞ, which suggests at least two sources of propagating waves. 4.4. Time-mean, cross-shore transect: line 90 The time-mean, cross-shore structure along line 90 is consistent with previous observations in the SCB [e.g., Davis et al., 2008; Hickey, 1979; Todd et al., 2011b]. Inshore of the continental slope near x ¼ 260 km, the topography comprises several ridges and basins in the SCB unlike lines 66.7 and 80 which have narrow shelves (Figs. 1, 5J, and 7J). Like line 80, there is a nearshore shoaling of isopycnals and stratification at a coastal front near x ¼ 20 km, where chl a is also larger (Fig. 7A–D). Flows in most of the SCB, from the coast to x 150 km, are generally northward, while southward flows are found in the California Current offshore of the slope in the upper 100 m with the northward California Undercurrent below 100 m (Fig. 7F). Near the SRCR, a narrow region of southward flow is found along with anticyclonic Ro (Fig. 7F, H). Cyclonic Ro bands are found on either side, which may assist with D1 trapping by making the effective vorticity larger (Fig. 7H) (Kunze and Toole, 1997). Ro is small in this time mean, but values on a single section can be large (Fig. 3E). A front associated with the inshore boundary of the California Current is found above SRCR with a broader lateral extent of chl a compared to the coastal front (Fig. 7A, B, D). More profiles are obtained near the two ridges where dives are often shallower, but typically 4150 data points are in each bin (Fig. 7G, I). Similar to line 80, higher D1 and D2 amplitudes are found near the topography on line 90, but, unlike line 80, the higher amplitudes are at topography within the SCB, mainly at the SRCR and not at the coast (Fig. 8A–D). The complex topography provides
9
numerous generation sites for internal tides. Increasing phase indicates a few areas of propagating waves, but large regions of slowly-varying phase show that standing or partially-standing waves are dominant features (Fig. 8E–J). D1 amplitudes are larger on the east side of the SRCR (in the San Nicolas Basin, near x ¼ 180 km) consistent with an interpretation of trapped D1 internal tides propagating southward with the ridge on their right (Fig. 8A, C). Within 50 km of the east side of the ridge, ϕη and ϕv appear distinct from the surrounding phases (Fig. 8E, G). A small region within 10–20 km of the SRCR has ϕη ϕv ¼ 1801 (01) indicating southward (northward) phase propagation on the east (west) side of the ridge (Fig. 8I). We discount NIW currents impinging on topography as the dominant signal below the upper 50 m because they would produce large mean amplitudes on both sides of SRCR and the ridge near x¼ 50 km. D2 has larger amplitudes on both sides of the SRCR especially for displacements (Fig. 8B, D). D2 Aη is also larger near x ¼ 350 and 500 km, about one and two D2 wavelengths from the SRCR (Fig. 8D). Smaller displacements are seen at two other ridges near x¼ 50 and 100 km. As with the D1 band, D2 phases vary slowly, consistent with standing or partially-standing waves (Fig. 8F, H). Phase differences show small regions of propagating, standing, or partially-standing waves (Fig. 8J). 4.5. Summary On line 80, trapped D1 internal tides are shown with: (a) largest amplitudes near topography, (b) slowly-varying phase in the same locations, and (c) phase differences which are consistent with northward propagation, and (d) the coast is on the waves' right consistent with a trapped wave. D2 internal tides show propagation away from two sources, the shelf edge and rough topography on the slope. The phase differences show a combination of propagating and standing waves. On line 90, the amplitudes and the phases at the SRCR provide evidence for trapped D1 internal tides rather than NIW: (a) larger amplitudes on the east side of the SRCR, (b) distinct phases there, and (c) some suggestions of propagation with the ridge on the right. Numerous D2 wave sources over the complex topography of the SCB yield patterns dominated by partially-standing waves. Line 80's time-mean, cross-shore structure shows a coastal upwelling front, a poleward undercurrent, and an equatorward flow further offshore. Line 90 displays a similar time-mean, cross-shore structure offshore of the slope/SRCR. Most of the flow in the SCB is poleward with a weaker upwelling front at the coast.
5. Depth-mean mixing estimates and internal tides: spatial and temporal variability 5.1. Overview Within the context of the internal-wave, frontal/mesoscale, and mean flows (Section 4), depth-mean K ρ and ϵ estimates from all available glider profiles are presented for each of the three lines (Section 5.2). Then, bin means in time and space are used to better describe the temporal and spatial variability of these estimates. The cross-shore structure is described in the mean (Section 5.3). On line 80, patterns are found over the 4 6year record along with a seasonal cycle (Section 5.4). No seasonal cycle is found on the other lines. In particular, we illustrate these patterns of variability in the mixing estimates, internal tidal energies and fluxes, gradients and density at fronts, and chl a using depth means. This section covers three analyses: (a) the time- and depth-mean cross-shore
Please cite this article as: Johnston, T.M.S., Rudnick, D.L., Trapped diurnal internal tides, propagating semidiurnal internal tides, and mixing estimates in the California Current System from.... Deep-Sea Res. II (2014), http://dx.doi.org/10.1016/j.dsr2.2014.03.009i
10
T.M.S. Johnston, D.L. Rudnick / Deep-Sea Research II ∎ (∎∎∎∎) ∎∎∎–∎∎∎
structure for all lines, (b) the depth mean on line 80 for the region near the coast (x 4 100 km) shown as a time series, and (c) the depth mean on line 80 shown as a composite annual cycle (i.e., x versus month). These three analyses rely on depth means over somewhat different depths and so we summarize for each calculation the relevant depths and reasons before proceeding with the results. For analysis a, depth-mean K ρ and ϵ are averaged from below the mixed layer to 362 m, the depth of the deepest reliable shear measurements (Section 3.1 and Appendix A). sθ and ∂x sθ emphasize fronts by averaging from 10 to 90 m. Chl a is averaged over the same depth range for the same purpose and also because there is little signal below that depth. Depth-mean rather than depth-integrated E is calculated to emphasize high-mode internal waves. Its depth mean is from 58 m up to a maximum of 362 m to minimize NIW influence and to more closely compare with mixing estimates. Depth-integrated mode-1 fluxes are calculated from data which covers 10–362 m. For analyses b–c, averaging intervals are as noted above, but sθ and N2 depth means are from 106 to 202 m to show the annual cycle in upwelling. Further details on the binning are in Section 3.4.
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5.2. Depth-mean mixing estimates To demonstrate the resolution and extent of the data, we show depth-mean diffusivity estimates from all 33 609 profiles on all three lines where both shear and strain are available (Fig. 9). The spatial patterns on all three lines are consistent with locations of elevated D1 and/or D2 energy either near the shelf/slope or over the rough topography of the SCB (Figs. 6A–D and 8A–D). With a 1.5-day window, there is some smearing of these patterns over 30 km, a typical distance covered by a glider in this time and so this pattern of mixing near topography is likely more localized in space than we describe below. On line 90, the overall pattern of the parameterized dissipation and diffusivity shows highest values over the SRCR with ϵ 4 10 8 W kg 1 and K ρ 410 4 m2 s 1 from both shear–strain and strain-only estimates (Fig. 9C). Secondary maxima are seen at x 100 and 50 km near two other ridges (Figs. 1B, 7J, and 9C). Another, intermittent local maximum is near x 350 km at a possible D2 downgoing beam (Figs. 8D and 9C). Ksh estimates from lines 66.7 and 80 display an offshore decay to background values within 100 km of the coast (Fig. 9A, B). Diffusivity estimates appear more sporadic on line 66.7. Sampling density near the coast is sometimes lower because gliders are deployed and recovered from coastal locations off of the nominal lines, which are excluded from consideration because jyj 4 15 km (Section 3.4 and Appendix A). Thus, typically 2 of 4 transects on one mission will closely follow the line near the coast. Also estimates are not made, when a profile extends less than 96 m below the mixed layer depth, a condition which is more likely to occur in shallow coastal waters. Patterns of ϵ are almost identical to Ksh, which in turn are very similar to Kst and so we only show Ksh at this point. Section 5.3 will further demonstrate this similarity. The spatial patterns of Ksh and Kst agree within a factor of 2, which reflects a factor of 2 difference in the correction for shear-to-strain ratio from observations and GM (h in Appendix A). This result suggests that the independent measurements of shear and strain are in reasonable agreement (Section 3.3 and Appendix A). Since these K ρ estimates follow the statistics of ð∂z u2 þ ∂z v2 Þ2 (Appendix A), the probability distribution functions roughly resemble a lognormal distribution (figure not shown). The ratio log(Ksh/Kst) has a mean value of log(1.7) and a standard deviation of log(1.6) and the ratio of Ksh/Kst is o 2 with a standard deviation of o 1.
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x [km] Fig. 9. Depth-mean diffusivity estimates (Ksh) are plotted on a Hovmöller diagram (cross-shore distance and time in years) for (A) line 66.7, (B) line 80, and (C) line 90 from 9189, 10 527, and 13 893 good profiles. For lines 66.7 and 80, estimates are largest within 100 km of the coast, while on line 90 estimates are largest over the SRCR near x ¼ 200 km and two other ridges near x ¼ 100 and 50 km. Note the differing scales in distance, time, and color. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)
5.3. Depth- and time means of the cross-shore transects In this section, we use the depth- and time-mean, cross-shore structure of mixing, internal tides, density, and chl a to assess the energetics and relevance of our mixing estimates. First, on lines 66.7 and 80, the offshore decay of Ksh is similar from values near the coast of 10 4 m2 s 1 to 10 5 m2 s 1 offshore (x o 100 km, Fig. 10A). Kst is within one standard deviation of Ksh, but appears to under-/overestimate off-/onshore. A similar decay is found on both lines 66.7 and 80 for ϵ (Fig. 10B). Due to the similarity with line 80 in the cross-shore mean, most of the description of Ksh below covers only line 80. As in Section 4.3, on line 80, the coastal upwelling front is near x ¼ 50 km, where maxima in chl a and its
Please cite this article as: Johnston, T.M.S., Rudnick, D.L., Trapped diurnal internal tides, propagating semidiurnal internal tides, and mixing estimates in the California Current System from.... Deep-Sea Res. II (2014), http://dx.doi.org/10.1016/j.dsr2.2014.03.009i
T.M.S. Johnston, D.L. Rudnick / Deep-Sea Research II ∎ (∎∎∎∎) ∎∎∎–∎∎∎
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Fig. 10. On line 80 data are averaged in time to show the mean cross-shore structure. (A) Depth-mean diffusivity (including Ksh from line 66.7) and (B) dissipation estimates (including ϵ from line 66.7) are largest near the coast. Gray shading denotes one standard deviation. (C) The depth-mean from 10–90 m of potential density is maximum at an upwelling front near shore, where (D) the chl a mean from 10–90 m and its standard deviation are also larger. (E) D1 and D2 depth-mean energies are largest at the coast and (F) D2 mode-1 cross-shore and alongshore energy fluxes decrease toward shore. Number of data in each bin is uniform offshore but (G) decreases for turbulence estimates near shore and (H) increases for density data near shore. Longitude along the line is on the upper axis of each subplot.
variance are seen (Fig. 10C, D). Depth-integrated D1 mode-1 energy flux is small ( o100 W m 1 ) and displays no clear pattern, which is expected for an evanescent internal wave (figure not shown). The depth-integrated D2 mode-1 energy flux converges shoreward from x ¼ 120 km (Fig. 10F), while the depth-mean energy densities increase for D1 and D2 shoreward of x¼ 50 and 100 km (Fig. 10E). D1 energy density is 2 greater than D2. Data coverage as noted earlier drops off closer to shore (Fig. 10G, H), yielding more uncertain mixing estimates there (Fig. 10A, B). In summary, turbulence estimates and D1 energy peak in the same location. This result along with that in Section 4.3 indicates trapped D1 internal tides are the dominant source of mixing on line 80. The decay away from topography of the D1 signal is consistent with a trapped wave. The mode-1 baroclinic Rossby radius (¼ c/f, where c is the mode-1 phase speed) within about 100 km of the coast varies from 12 to 28 km. An exponential decay in x reaches 5% of the value at the coast over a distance of 35–85 km. The larger distances are consistent with the decay scales of D1 E and K ρ on line 80, while the smaller distances are appropriate for the shallower waters in the SCB on line 90 as we will show later in this section. For line 80, we make rough estimates of the energy budget within 100 km of the coast. The energy lost from D1 and D2 internal tides should balance the dissipation. We neglect any contribution from NIW or fronts. We argue the most energetic D1 component is due to trapped internal tides. These topographically-trapped waves propagate northward with the coast on their right. If we assume 10% of their depth-mean energy within 100 km of the coast is locally dissipated over a tidal cycle (1 day) that yields a depth-integrated dissipation in roughly 1000m deep water as 0.1 5 J m 3 1000 m/86 400 s 6 m W m 2. The D2 mode-1 flux convergence: is roughly ΔF x =Δx ¼ 480 W m 1 =120 km ¼ 4 m W m 2 . The very crude estimate of energy lost by D1 and D2 internal tides is comparable to an order
of magnitude estimate of the depth-integrated dissipation (calculated from depth-mean ϵ in Fig. 10B): ρϵH ¼ 103 kg m 3 1 10 8 W kg 103 m ¼ 10 m W m 2 . Many of the factors included in this paragraph could be increased and decreased easily by factors of 2 or more given their variations within 100 km of the coast. Therefore, we conclude as an order of magnitude estimate, that there is 10 m W m 2 energy lost from the D1 and D2 bands to produce the estimated dissipation of 10 m W m 2. On line 90, elevated mixing estimates appear where D1 and D2 energy densities are largest over the SRCR (x ¼ 250 to 150 km, Fig. 11A, B, and E). There may be contributions from a persistent front (Fig. 11C). Ksh has a maximum of 3 10 5 m2 s 1 across the SRCR and the slope. A decay to background values of o 10 5 m2 s 1 in deep water is found (x o 300 km, Fig. 11A). The offshore decay occurs over 100 km similar to lines 66.7 and 80, but the onshore decay is more sudden (near x¼ 150 km, Fig. 11A, B, and E). This result is consistent with an offshore increase in Rossby radius and extent of D1 trapped internal tides. Again Kst is an underestimate compared to Ksh in deep water. Similar features are found in ϵ with slightly elevated values over other ridges at x¼ 100 and 50 km (Fig. 11A, B). We also note a persistent density front near x 200 km, where chl a and its variability are elevated (Fig. 11C, D). Depth-integrated D2 mode-1 baroclinic energy fluxes are small in the SCB and increase offshore of the continental slope (x o 250 km, Fig. 11F). Data coverage again drops off closer to shore, but increases over the ridges (Fig. 11G, H). To assess our chosen depth limits based on objective measures of data quality (Section 3.1 and Appendix A), we recalculate K ρ from data up to a maximum depth of 202 m instead of 362 m as is done elsewhere in this paper. Results for the cross-shore mean on line 90 remain similar: the peak values of K ρ near the SRCR are unchanged, while values further offshore are slightly lower. The overall pattern in Fig. 11A and our conclusions are unchanged.
Please cite this article as: Johnston, T.M.S., Rudnick, D.L., Trapped diurnal internal tides, propagating semidiurnal internal tides, and mixing estimates in the California Current System from.... Deep-Sea Res. II (2014), http://dx.doi.org/10.1016/j.dsr2.2014.03.009i
T.M.S. Johnston, D.L. Rudnick / Deep-Sea Research II ∎ (∎∎∎∎) ∎∎∎–∎∎∎
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2010
26 6
3 nKsh n
200
n
10
K
3
sh
[kg m ]
K
N2 2 [ 10 5 s ]
10
ET [J m 3]
K [m2 s 1]
Fig. 11. On line 90, data are averaged in time to show the mean cross-shore structure as in Fig. 10, but elevated mixing estimates, internal tidal energy, and a front are found near the SRCR (x 190 km).
2012
100 0
2008
2010
2012
Fig. 12. On line 80, data are averaged within 100 km of the coast to produce a time series showing seasonal variability in (A) diffusivity estimates with gray shading denoting one standard deviation, (B) potential density averaged from 106 to 202 m in depth, (C) depth-mean D1 and D2 energy densities, (D) buoyancy frequency averaged from 106 to 202 m in depth, and (E) depth-mean D1 and D2 potential and kinetic energy densities. (F) The number of density (gray) and diffusivity (black) data points in each bin is shown.
5.4. Temporal variability on line 80 While temporal variations are seen over the record on lines 66.7 and 90, here we focus on a seasonal cycle in mixing, D1 energy, and density on line 80. Neither mixing nor energy density shows a seasonal cycle on lines 66.7 and 90. The seasonal cycle in mixing is found near the coast. Data from x¼ 100 to 0 km are binned in 60-day intervals (Fig. 12). Diffusivity estimates are higher in the summers of 2008, 2009, 2011, and 2012 (Fig. 12A). Kst has a small maximum in summer 2010, where there is missing Ksh data. An additional spring maximum is seen in 2012. Both Ksh and Kst display similar phasing and magnitude (within one standard deviation except in summer 2012). In 2007/2008, when Ksh
is unavailable, Kst is low. Similar patterns are seen in ϵ (figure not shown), which indicates the variability in K ρ is not due to the lower N2 in upwelling season (Fig. 12A, D). s is high during upwelling (Fig. 12B). The depth-mean D1 baroclinic energy is often largest in summers mainly due to the seasonal cycle in D1 Ep and, to a lesser extent, Ek (Fig. 12C, E). D1 Ep provides a continuous record, whereas Ek has some gaps. D2 Ek and Ep are generally weaker and a seasonal cycle is not evident (Fig. 12C, E). Usually a minimum of 50 samples is obtained for each 60-day bin (Fig. 12F). Even though each year displays differing amplitudes of K ρ and other quantities, a seasonal cycle of mixing is composited from the 4 6year record (Fig. 13). Larger depth-mean K ρ is found from May to August within 40 km of the coast (Fig. 13A, C) somewhat
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123o
121oW
123o
13
121oW
Nov Ksh
Nov
2
1
3
[m s ] 4 Aug 10 May
10
[kg m ] 26.5
6
Aug May
26.1
Feb
Feb
Nov Kst 2
N2 [ 10 8
1
[m s ] Aug 10 4 May
10
6
5
Nov
2
s ] Aug May
0
Feb
Feb
Nov D1 E
D2 E 3
Aug
[J m ] 7
May
Nov 3
[J m ] 4
1
Aug May
1
Feb Nov Aug May
Feb
n
Nov
nKsh 150
100
0
0
Aug May
Feb
Feb 300
x [km]
0
300
0
x [km]
Fig. 13. On line 80, a composite seasonal cycle is constructed by binning observations by month and cross-shore distance. Hovmöller diagrams show: (A) depth-mean diffusivity estimates from shear and strain, (B) potential density averaged from 106 to 202 m in depth, (C) depth-mean diffusivity estimates from strain, (D) buoyancy frequency averaged from 106 to 202 m in depth, and (E) depth-mean D1 and (F) D2 energy densities (Note the different color scales.). The number of (G) density and (H) diffusivity data points in each bin are shown. Longitude along the line is on the upper axis of each subplot. Gray/blue shading indicates a lack of data in Fig. 13A–F/G-H. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)
coincident with higher D1 depth-mean energy from May to July (Fig. 13E). Depth-mean D2 energy peaks from April to June within 60 km of the coast, but is at least 2 weaker than D1 (Fig. 13F). Upwelling season displays higher s and lower N2 within 40–60 km of the coast from March to August (Fig. 13B, D) and begins before the diffusivities have peaked. Typically about 50 data points are in each bin, but that value increases near the coast for s and decreases for K ρ (Fig. 13G, H). In summary, peak mixing estimates on line 80 from May to August (a) coincide roughly with peak D1 energy from May to July; (b) occur after the peak D2 energy density; (c) occur 2 months after the onset of upwelling in March; and (d) occur 2 months after the onset of maximum wind variance in March at this latitude (Hickey, 1979) and presumably a maximum in NIW generation. This result provides further confirmation that trapped D1 internal tides are dominant energy source for mixing here.
6. Discussion 6.1. Finescale estimates of depth-mean diffusivity and dissipation from glider profiles Mixing in the thermocline is driven mainly the breaking of internal waves. Assuming a steady downscale transfer of energy, mixing parameterizations use observed shear and strain at Oð10 mÞ scales to estimate mixing. Shear–strain diffusivity estimates (Ksh) and strain-only estimates (Kst) using an assumed shear-to-strain ratio have been calculated from data obtained by autonomous underwater gliders along three sustained transects in the California Current System from 2006 to 2012 (Fig. 9). The timeand depth-mean diffusivity estimates are largest near topography: (a) K sh 10 4 m2 s 1 near the coast on lines 66.7 and 80 (Fig. 10A) and (B) 3 10 5 m2 s 1 over the SRCR in the SCB on line 90 (Fig. 11A).
Please cite this article as: Johnston, T.M.S., Rudnick, D.L., Trapped diurnal internal tides, propagating semidiurnal internal tides, and mixing estimates in the California Current System from.... Deep-Sea Res. II (2014), http://dx.doi.org/10.1016/j.dsr2.2014.03.009i
14
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Both Ksh and Kst agree within a factor of 2, but Kst is usually an underestimate compared to Ksh especially where diffusivity estimates are smaller. These estimates are in general agreement with tracer release and microstructure observations showing K ρ of Oð10 4 Þ m2 s 1 near topography in the SCB (Section 2.2). The depth-integrated dissipation estimate near Point Conception on line 80 within 100 km of the coast is roughly 10 m W m 2 and balances estimated D1 and D2 energy inputs (Section 5.3). Also a local maximum of 2 m W m 2 in wind work on near-inertial motions in the mixed layer is found near Point Conception in an eddying model with realistic but annually repeating winds (Figure 2 in Simmons and Alford, 2012). This energy input is a precursor to NIW production. Considering the entire Pacific Ocean, roughly 10% propagates away as low-mode NIW. Thus most of this energy is either lost in the mixed layer, transition layer, or deposited in higher-mode waves, which may dissipate locally. This mean annual value of 2 m W m 2 near line 80 is comparable to that in the mid-latitude storm tracks near 451N and is less than estimated D1 and D2 contributions of 5 m W m 2 each. We assumed that about 10% of the energy in the trapped D1 internal tide is dissipated locally with the rest propagating alongshore, but this value could also be much higher, it is quite probable that most of the D1 energy is dissipated near its generation site. K1 and O1 barotropic tidal energy dissipation (i.e., bottom friction, internal tides, or other coastally-trapped waves) displays a local maximum on the west coast of North America and is 5 m W m 2 summed over both constituents in the TPXO tidal model (Egbert and Ray, 2003). This agreement of energy inputs from near-inertial motions and D1 and D2 internal tides with our dissipation estimate should be regarded as an order of magnitude comparison. The mixing parameterizations will fail when the downscale energy cascade is circumvented. Two possible processes which circumvent the cascade are likely not at work here. A third process which contributes to finescale gradients is also noted, but may or may not be related to internal waves. First, on wide continental shelves with water depths of ≲100 m, the large-scale, mode-1 waves may be unstable or close to instability and thus there is no cascade to smaller scales (MacKinnon and Gregg, 2003, 2005). Second, large-amplitude waves may break directly near energetic generation sites, such as Luzon Strait (Alford et al., 2011), and produce overturns of several hundred meters in extent. Third, finescale shear and strain at fronts or other mesoscale features may lead to modestly elevated levels of mixing within 10–20 km of their flanks, but the relative contributions of either frontogenesis or trapped, reflected, or refracted internal waves are unclear (Johnston et al., 2011b). In our study, the shallowest water depths are 100–200 m over steep topography, typical water depths exceed 1 km, internal wave energy levels are modest (several times GM levels), and spectra resemble GM. Therefore, we expect the finescale methods are applicable in this region. 6.2. Mesoscale flows As gradients increase during frontogenesis, mixing is likely on the dense, cyclonic side of a front where the gradients are largest (Hoskins and Bretherton, 1972). Along line 90, a persistent front between the southward California Current and northward flow in the SCB is located over the SRCR (Figs. 7 and 11). A similar result is found for line 80 near its coastal upwelling front (Figs. 5 and 10). Two pieces of evidence suggest that fronts are not the dominant mechanism: (a) maximum Ksh is located about 20–50 km inshore from the maximum density gradient, which defines the front and (b) the seasonal cycles of upwelling and Ksh on line 80 are not aligned. Mixing may be important for primary productivity in this area, where the mean and variance of chl a are higher (Fig. 10D and
11D). However, the simplest explanation for this primary productivity on line 90 is southward advection of phytoplankton from the upwelling center at Point Conception on line 80 in the southward flow of the California Current to line 90 (Fig. 7F). 6.3. Free and trapped internal waves The most energetic internal waves are D1 and D2 internal tides and NIW. Since D1 internal tides are subinertial, they are trapped to the topography, but may propagate alongshore. With our methods, we cannot distinguish among individual D1 and D2 constituents, e.g. K1, O1, or near-inertial motions. We attempt to minimize NIW influence on energy calculations by using data below 58 m. Several lines of evidence indicate the presence of trapped D1 internal tides and their influence on mixing. On line 80, (a) D1 energy exceeds D2 energy by up to a factor of 2 (Fig. 10E), (b) the seasonal maxima of mixing and D1 depth-mean energy occurs in summers (Fig. 13A, C, E), (c) the cross-shore maxima of D1 energy and K ρ are in the same location (Fig. 10A, E), (d) the cross-shore decay scale is consistent with trapped waves (Fig. 10A), and (e) the D1 phase difference between η and v is consistent with a trapped wave propagating with the coast on its right (Fig. 6, left). Phase is a particularly useful indicator of internal tidal propagation (Ray and Mitchum, 1996; Johnston et al., 2013). Surface currents from coastal radars with records long enough to distinguish D1 constituents show K1 tidal variance exceeds that of both near-inertial oscillations and M2 internal tides near Point Conception (Kim et al., 2011). However, further analysis of our data in conjunction with other datasets (sea level records from tide gauges and altimetry to identify barotropic tidal forcing and spring-neap cycles; surface wind stress from moorings, shore stations, and satellite to drive a slab model for estimates of NIW generation; coastal radars to show the spatial pattern of currents in the D1 band; and moored data to provide better temporal resolution) is planned and needed to clarify the contributions of either trapped internal tides or NIW to the D1 signal on line 80. Trapped D1 internal tides are also found on line 90 (Fig. 8, left), but numerous internal tidal generation sites in the SCB lead to standing waves. On line 90, largest D1 amplitudes are found on one side of the SRCR consistent with a southward-propagating, topographically-trapped wave. Several characteristics distinguish trapped D1 internal tides from NIW. For NIW flows encountering submarine topography, there should not be a maximum on one side of the topography in a time average. It is unlikely most of the NIW encountering the SRCR are propagating westward given the weak winds in the SCB (Hickey, 1979). NIW currents seem stronger west of the SRCR (compare upper 100 m west and east of SRCR in Fig. 8A). NIW generation is episodic and therefore consistent phase structure at depth is also unlikely. The time- and depth-mean cross-shore profile of mixing displays maxima near the continental slope on lines 66.7 and 80 and decreases to background values in 100 km (Fig. 10A) and over similar distances offshore of the SRCR on line 90 (Fig. 11A). The mode-1 baroclinic Rossby radius is 12–28 km and so trapped waves would decay to within 5% of background values in 35– 85 km. Kelvin waves are one possible topographically trapped wave, but there are many other possibilities (Llewellyn Smith, 2004; Mysak, 1980). If indeed trapped D1 internal waves are a major contributor to near-slope mixing poleward of 301, then their cross-shore extent may decrease with decreasing Rossby radius (i.e., increasing latitude in general). On the other hand, the decay distance may reflect the decay to background levels of propagating internal waves. In such cases, measurements show (a) a decay of internal waves away from their source in Luzon Strait over 100 km
Please cite this article as: Johnston, T.M.S., Rudnick, D.L., Trapped diurnal internal tides, propagating semidiurnal internal tides, and mixing estimates in the California Current System from.... Deep-Sea Res. II (2014), http://dx.doi.org/10.1016/j.dsr2.2014.03.009i
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(Rudnick et al., 2013) and (b) diapycnal diffusivity decreasing from peak values over the Hawaiian Ridge to background levels within 60 km. Since the D2 internal tidal fluxes are 20–100 less in our study region than at the Hawaiian Ridge or Luzon Strait, it seems likely that their decay to background levels should occur closer to the coast. Thus, we believe trapped D1 internal tides are the dominant contributor to mixing here. The western coast of North America may have particularly large D1 internal tides, similar to several other locations poleward of 301 around the North Pacific and North Atlantic as well as in more isolated areas off Tasmania, New Zealand, and Patagonia. Further assessment of the influence of trapped internal tides on mixing is warranted also in the Arctic, where both D1 and D2 internal waves are subinertial and may lead to considerable mixing at the margins (Padman and Dillon, 1991; Padman et al., 1992), a process which is not often considered in models of the Arctic but may have considerable impact (Holloway and Proshutinsky, 2007). 6.4. Vertical distributions and mean values of diffusivity estimates This work also demonstrates that considerable mixing in the coastal ocean occurs in the middle of the water column away from the surface and bottom boundaries. Midwater turbulence due to a superposition of M2, inertial, and thermal wind shear has been previously observed with microstructure instruments on the Oregon shelf by Avicola et al. (2007). They also note that numerical models of the coastal ocean are often unable to produce mixing away from surface and bottom boundaries. In our case, mixing estimates are elevated in midwater by internal tides. The overall mean, regional mixing estimates on each line are consistent with mean thermocline values of K ρ 10 5 m2 s 1 (Gregg, 1987), which is a background value used by many models. By using a combination of the same θ-S data from gliders and a numerical state estimate, a mean diffusivity over much of the same region was obtained by following tracers over time in the state estimate (Todd et al., 2012). Values a few times larger than the model's background value of 1 10 5 m2 s 1 were found. Data are roughly lognormally distributed with a tail for the higher coastal values. Our mean Ksh has values of 0.6, 1, and 1 10 5 m2 s 1 on lines 66.7, 80, and 90. To better describe the distribution, means of log 10 K sh which expressed as Ksh means are 0.4, 0.7, and 0.8 10 5 m2 s 1 on lines 66.7, 80, and 90. Standard deviations of log 10 K sh are about 0.3–0.4 which is equivalent to a factor of 2. When averaged on a regional basis the model values and tracer estimate are similar to our mean estimates. 6.5. Seasonal cycle in mixing estimates and diurnal energy on line 80 A seasonal cycle in D1 energy density and mixing estimates are found within 100 km of the coast on line 80, but not lines 66.7 or 90 (Figs. 12A, 13A and 13C). Depth-mean D1 energy on line 80 is maximum in spring/summer (Figs. 12C and 13E). Since these maxima overlap with upwelling season, this phasing is doubtless of biogeochemical importance for nutrient, carbon, and other tracer fluxes between deep and shallow waters (Bylhouwer et al., 2013, and references therein). At least two possible explanations exist for the D1 maximum during springs/summers: either increased NIW production due to variable winds within 100 km of the coast or increased D1 internal tide generation possibly due to seasonal stratification changes (Fig. 13D). Stratification changes due to upwelling in Monterey Canyon change the character of internal waves from progressive to standing (Hall et al., 2014). The magnitude of internal tidal generation can also change by a factor of 2, mainly due to stratification effects on the baroclinic pressure perturbation (Zilberman et al., 2011).
15
On line 80 near the coast, there also appears to be an increasing trend from 2007 to 2012 in depth-mean D1 potential energy (Fig. 12E), which again implies either stronger internal tidal generation possibly due to stratification changes and/or increasing NIW production. There also appears to be an increasing offshore extent of dense, weakly-stratified water during upwellings (over depth range from 106 to 202 m, figure not shown), but the density and stratification near the coast do not show a trend (Fig. 12B, D). No trend is seen in the mixing estimates (Fig. 12A). This region near Point Conception displays considerable seasonal variation in upwelling with an increasing trend driven by the wind stress curl (Schwing and Mendelssohn, 1997; Seo et al., 2012). Winds in this area have peak variance from March to June (Table 1 in Hickey, 1979). In a regime of increasing winds, there may be increasing wind variance. If this variance has a diurnal component, it may lead to the increasing NIW production and D1 energy seen in our observations. These speculations on the effects of stratification and wind variability will be tested in the future with data (Section 6.3), which can discriminate K1 from the near-inertial peak.
7. Summary Along sustained, high-resolution, cross-shore transects on CalCOFI lines 66.7, 80, and 90 from 2006 to 2012, mixing is estimated via finescale parameterizations using shear and strain data from autonomous underwater gliders. Our primary technical result is the application of these parameterizations to spatially and temporally extensive glider data. Our main scientific result is the importance of D1 internal tides, which are subinertial at these latitudes and therefore trapped at topography e.g., either the slope or the SRCR in the SCB. Trapped D1 internal tides (a) are often more energetic than the D2 internal tides, (b) set the cross-shore decay scale of mixing near the slope and shelf, and (c) produce a seasonal cycle in mixing along line 80.
Acknowledgments This work was supported by NOAA through the Consortium for the Ocean's Role in Climate (award number NA10OAR4320156) and the Southern California Coastal Ocean Observing System (award number NA08OAR-4320894). Spray was developed by the Instrument Development Group (IDG) at the Scripps Institution of Oceanography with funding from the Office of Naval Research. IDG operates the gliders on these sustained lines. Two gliders were purchased by the California Current Ecosystem—Long Term Ecological Research program. Two anonymous reviewers and the editor provided helpful comments to improve this paper.
Appendix A. Details of mixing parameterizations Below we detail the production of depth-mean estimates of turbulent dissipation and diffusivity following Gregg (1989), Gregg et al. (2003), and Polzin et al. (1995). The first step is to scale shear relative to N and when squared is an inverse Richardson number: ∂z u þ i∂z v Shn ¼ qffiffiffiffiffiffiffiffiffi 〈N 2 〉1:5d
ðA:1Þ
The 1.5-day mean is roughly equivalent to a 30-km mean. Data are detrended before taking spectra. Then Shn variance is computed from observed and GM vertical wavenumber spectra as follows. One spectrum is made for each profile. The minimum wavenumber, mmin, is set by the length of the record, which extends from the mixed layer depth (MLD) to at least 96 m deeper and
Please cite this article as: Johnston, T.M.S., Rudnick, D.L., Trapped diurnal internal tides, propagating semidiurnal internal tides, and mixing estimates in the California Current System from.... Deep-Sea Res. II (2014), http://dx.doi.org/10.1016/j.dsr2.2014.03.009i
T.M.S. Johnston, D.L. Rudnick / Deep-Sea Research II ∎ (∎∎∎∎) ∎∎∎–∎∎∎
cpm ]
10
[s
PSD
(PSD) of normalized shear over the appropriate wavenumbers: 2 3 Z mc 6 ∂z u þ i∂z v 7 2 〈Shn 〉 ¼ ðA:2Þ PSD4qffiffiffiffiffiffiffiffiffi 5ðmÞ dm mmin 〈N2 〉1:5 d
4
Sh: x = 167 km Sh: mean
2
z
u+i
1
z
v
16
10
CW + CCW
8
and similarly for 〈ð∂z ηÞ2 〉. Each spectrum is corrected for attenuation at higher wavenumbers from first differencing at 16 m, but not for the ADP's pulse length and binning at 8 and 4 m. These spectra are then averaged over 1.5 days (or about 30 km) to produce a mean spectrum, which has 2 degrees of freedom or possibly 6, since 1 or 2 waves are sampled over 1 day and more than 1.5–3 waves over 1.5 days. The transect-mean spectra, which are not used in our calculations have confidence intervals associated with 50 degrees of freedom (Fig. A1A–B). The energy level in the internal wave field relative to GM is a combination of the kinetic and potential energies which are measured via shear and strain:
2
PSD
z 1
[cpm ]
10
100 z
10
z
2
: x = 167 km : mean
2
Sh : x = 167 km
n
n
1
[cpm ]
, Sh
z
PSD
2 〈Shn 〉2 1 þ1=Rω E^ ¼ 2 4=3 〈Shn;GM 〉2
101
0
x = 361 km
z
where Rω ¼ 〈Shn 〉=〈ð∂z ηÞ2 〉 is the observed shear-to-strain ratio. Rω is 3 for a GM wavefield at this latitude and increases with increasing frequency and latitude. Note the spectra are averaged 2
x = 167 km GM x = 361 km
GM
10
10
:
3
10
m/2
2
ðA:3Þ
10
1
[cpm]
Fig. A1. Spectra from the example cross-shore section (Fig. 2) show rolloffs for (A) shear and (B) strain near wavenumbers of 1/40 cpm. Shear spectra are the sum of the clockwise and clockwise components. The location of the profiles is identified by x. Transect mean spectra are also shown (thick lines in Fig. A1A, B). With 2 and 50 degrees of freedom, 95% confidence intervals are shown for the individual and transect-mean spectra. (C) Normalized shear spectra (red lines) and strain spectra (blue, dashed lines) averaged over 1.5 days (Section 3.3) at two locations are compared to GM shear (solid dark gray line) and strain (dashed light gray line) spectra. One location is inshore of the SRCR (x ¼ 167 km), while the other is offshore in deep water (x¼ 361 km). With 4 degrees of freedom, 95% confidence intervals are shown for the average spectra. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)
usually further to the maximum depth (i.e., 362 m): mmin/2π ¼1/ (362 - MLD) cycles per meter (cpm). The same maximum depth is used for both shear and strain for consistency, even though reliable strain measurements extend deeper. The maximum wavenumber, mc/2π ¼1/40 cpm, is set by the rolloff of the glider's shear measurements (Fig. A1A), which arises due to the processing (Section 3.1) (Polzin et al., 2002). Both shear and strain show similar mc (Fig. A1A, B). However, a smaller mc may be used based on the idea of an inverse Richardson number i.e., mc is chosen where the cumulative Shn (∂z η) variance reaches 0.66 (0.22) which is at m/2π ¼1/10 cpm for the GM spectrum (Polzin et al., 1995). Some data are excluded from consideration. Data from missions with fewer than 300 profiles are not used. Data from more than 15 km away from the nominal transect are discarded. There are occasional instrument malfunctions. These considerations reduce available sθ and velocity data to 14 357 and 11 864 profiles on line 66.7, 15 992 and 12 779 profiles on line 80, and 15 408 and 14 816 on line 90. Lastly, profiles with fewer than 6 points or 96 m below the MLD are excluded, which permits shear–strain (strain only) mixing estimates on line 66.7 from 9189 (11 253) profiles, on line 80 from 10 527 (13 480) profiles, and on line 90 from 13 893 (15 004) profiles. Variance is obtained by integrating and summing the clockwise and counterclockwise components of the power spectral density
2 over 1.5 days to produce reliable values for E^ (Fig. A1C), a point which is examined in detail by MacKinnon et al. (2013). Shearbased mixing estimates are not calculated without accompanying strain measurements. 2 Turbulent dissipation scales with E^ , N2, Rω , and latitude (α):
ϵ ¼ ϵo
〈N 2 〉1:5d ^ 2 E hðRω ÞLðN; αÞ N 2o
ðA:4Þ
where ϵo ¼6.73 10 10 W kg 1 is the background dissipation due to a typical GM internal wave field, No ¼ 5.24 10 3 rad s 1 is the reference buoyancy frequency, h is a correction for the frequency content of the internal wave field based on Rω , and L is a correction based on f and N: sffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ðA:5Þ h¼ Rω 1 L¼
f cosh f 30 cosh
1
1
ðN=f Þ
ðN o =f 30 Þ
ðA:6Þ
For a GM wavefield, Rω ¼ 3 and h ¼1, but our observed Rω ¼ 5–6 and h 0:6, which implies more energy is in the near-inertial band (which in our case is also the D1 band) than is accounted for in GM (Kunze et al., 2006). L 1 at this study's latitude and mean stratification. L increases from 0 at the equator to 2 at the poles. Above, ϵ is estimated from the combination of Sh2 and ∂z η measurements via the observed Rω , but ϵ can also be obtained solely from ∂z η via an assumed GM value of Rω ¼ 3. This method is useful because velocity is not measured as often as density. With strain only, the energy level is 2 〈ð∂z ηÞ2 〉2 1 þ1=Rω E^ ¼ 4=3 〈ð∂z ηÞ2GM 〉2
The dependence of h on Rω is modified to: sffiffiffiffiffiffiffiffiffiffiffiffiffiffi R2 2 h¼ ω 9 Rω 1
ðA:7Þ
ðA:8Þ
but Rω ¼ 3 in our case and so h¼1.
Please cite this article as: Johnston, T.M.S., Rudnick, D.L., Trapped diurnal internal tides, propagating semidiurnal internal tides, and mixing estimates in the California Current System from.... Deep-Sea Res. II (2014), http://dx.doi.org/10.1016/j.dsr2.2014.03.009i
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Lastly, turbulent diapycnal diffusivity is estimated from the finescale parameterizations of ϵ: Kρ ¼
Γϵ 〈N 2 〉1:5d
ðA:9Þ
where Γ ¼ 0.2 is a typical value of the mixing efficiency. We produce two estimates of K ρ : one from observed shear and strain (Ksh) and the other from strain alone (Kst) with an assumed Rω . We only show ϵ estimated from both shear and strain. The observed Rω ¼ 5 6 and so using Rω ¼ 3 in the calculation of Kst may produce an underestimate of 1.5–2 with respect to Ksh, which is the more accurate estimate (Polzin et al., 1995; MacKinnon et al., 2013).
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Please cite this article as: Johnston, T.M.S., Rudnick, D.L., Trapped diurnal internal tides, propagating semidiurnal internal tides, and mixing estimates in the California Current System from.... Deep-Sea Res. II (2014), http://dx.doi.org/10.1016/j.dsr2.2014.03.009i