Internal Tide Nonstationarity and Wave–Mesoscale Interactions in the Tasman Sea

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ANNA C. SAVAGE AND AMY F. WATERHOUSE Marine Physical Laboratory, Scripps Institution of Oceanography, La Jolla, California

SAMUEL M. KELLY Large Lakes Observatory and Physics and Astronomy Department, University of Minnesota–Duluth, Duluth, Minnesota

(Manuscript received 19 November 2019, in ﬁnal form 13 July 2020)

ABSTRACT

Internal tides, generated by barotropic tides flowing over rough topography, are a primary source of energy into the internal wave field. As internal tides propagate away from generation sites, they can dephase from the equilibrium tide, becoming nonstationary. Here, we examine how low-frequency quasigeostrophic background flows scatter and dephase internal tides in the Tasman Sea. We demonstrate that a semi-idealized internal tide model [the Coupled-Mode Shallow Water model (CSW)] must include two background flow effects to replicate the in situ internal tide energy fluxes observed during the Tasmanian Internal Tide Beam Experiment (TBeam). The first effect is internal tide advection by the background flow, which strongly depends on the spatial scale of the background flow and is largest at the smaller scales resolved in the background flow model (i.e., 50–400 km). Internal tide advection is also shown to scatter internal tides from 2 vertical mode-1 to mode-2 at a rate of about 1 mW m 2. The second effect is internal tide refraction due to background flow perturbations to the mode-1 eigenspeed. This effect primarily dephases the internal tide, 2 attenuating stationary energy at a rate of up to 5 mW m 2. Detailed analysis of the stationary internal tide momentum and energy balances indicate that background flow effects on the stationary internal tide can be accurately parameterized using an eddy diffusivity derived from a 1D random walk model. In summary, the results identify an efficient way to model the stationary internal tide and quantify its loss of stationarity. KEYWORDS: Ocean; Australia; Baroclinic flows; Eddies; Internal waves; Shallow-water equations

1. Introduction 1997; Zhao et al. 2016; Zaron 2015). While these techniques accurately estimate stationary tides (i.e., the Internal tides propagate at the interfaces of density signal component phase-locked with the equilibrium layers and are the primary energy source of the global tide), they are largely unable to estimate nonstationary internal wave field (Munk and Cartwright 1966; Egbert tides, or tides whose phases wander with respect to the and Ray 2000; Wunsch and Ferrari 2004; Waterhouse equilibrium tide (Ray and Zaron 2011; Shriver et al. et al. 2014). While internal tides are generated over 2014; Zaron 2017; Nelson et al. 2019). Interactions be- rough topography, they rarely dissipate locally. Instead tween internal tides and slowly evolving (i.e., quasi- they often propagate for thousands of kilometers, dis- geostrophic) background flows can dephase internal tributing global internal wave energy (Shriver et al. tides from the equilibrium tide, and the estimated global 2012; Zhao et al. 2016; Buijsman et al. 2016). Techniques distribution of nonstationary tides suggests this process such as harmonic analysis and plane-wave fitting have is important. Mesoscale eddies can alter the propagation produced global maps of internal tide sea surface dis- of low-mode internal waves (Rainville and Pinkel 2006; placement from satellite altimetry (Ray and Mitchum Park and Watts 2006; Dunphy and Lamb 2014; Ponte et al. 2015; Dunphy et al. 2017; Duda et al. 2018) and Denotes content that is immediately available upon publica- semidiurnal tides are largely nonstationary along the tion as open access. equator and in western boundary currents (Buijsman et al. 2017; Zaron 2017; Savage et al. 2017a). While Corresponding author: Anna C. Savage, [email protected] several studies have examined the temporal variability

DOI: 10.1175/JPO-D-19-0283.1 Ó 2020 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses). Unauthenticated | Downloaded 10/02/21 10:10 AM UTC 2932 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 50 of tidal nonstationarity (Zaron and Egbert 2014; Buijsman The Coupled-Mode Shallow Water model (CSW) et al. 2017) and the local correlation between tidal non- (Kelly et al. 2016) is an internal tide process model that stationarity and the strength of the mesoscale field has been used to understand nonstationary tides. In the (Pickering et al. 2015), little is known about the domi- Gulf Stream, Kelly et al. (2016) used CSW to show nant horizontal scales of the background flows in these that a strong western boundary current such as the Gulf interactions, or whether these interactions merely Stream can refract a propagating internal tide wave. redirect/refract the internal tides, or actively dissipate Here, we use CSW to study the effect of mesoscale and/or scatter them to higher modes. Quantifying such eddies on the semidiurnal internal tide in the Tasman interactions is difficult because wave–background flow Sea, particularly in the region of the TBeam experiment. interactions are hard to observe and computationally An important feature of CSW is that it can be run with expensive to model over large spatial and temporal and without a background flow; this allows us to quantify scales. Here, we find that tidal nonstationarity is also the effect of this background flow on the propagation of dependent on the horizontal scale of the mesoscale field, the internal tide. and not only on its strength. The remainder of the paper includes a short review of In the Tasman Sea, Macquarie Ridge (the underwater tidal analysis methods (section 2), descriptions of the extension of the South Island, New Zealand) generates TBeam mooring data (Waterhouse et al. 2018) and an internal tidal beam that propagates northwest to modeling methods (section 3). Section 4 provides a the continental slope of Tasmania. This beam has a comparison of CSW with in situ observations and al- large stationary and weak nonstationary component timetry. Section 5 analyzes which background flow (Waterhouse et al. 2018). It is approximately unidirec- effects dominate internal tide nonstationarity in the tional because Macquarie Ridge is the only major source Tasman Sea by quantifying the separate effects of wave of internal tides in the Tasman sea, and the basin is advection, scattering, and refraction by the background relatively shielded from remotely generated internal flow, and contextualizes this in terms of previous studies, tides (Zhao et al. 2018). From January to March of 2015, (e.g., Zaron and Egbert 2014; Buijsman et al. 2017). This the Tasman Tidal Dissipation Experiment (TTIDE) section also aims to determine how internal tide prop- observed the open-ocean propagation of the internal agation depends on the horizontal scale of the back- tide beam and its dissipation and reflection on the ground flow. Section 6 focuses on stationary internal tide Tasman slope (Pinkel et al. 2015). Earlier in the exper- dynamics and develops parameterizations for back- iment, Johnston et al. (2015) observed a standing in- ground flow effects. Section 7 summarizes the results. ternal tide off the Tasmanian coast using gliders. The standing wave was caused by internal tide reflection at the continental slope, which was simulated in detail by 2. Tidal analyses Klymak et al. (2016). Before analyzing observations and numerical simu- The Tasmanian Internal Tide Beam Experiment lations, it is useful to review the concepts of orthogonal (TBeam) was conducted in tangent to TTIDE, and modes and tidal stationarity. aimed to study propagation of the M2 energy fluxes between the generation site and the Tasman Shelf a. Vertical modes (Waterhouse et al. 2018). Figure 8 of Waterhouse et al. Horizontal velocity, u(x, z, t) 5 [u(x, z, t), y(x, z, t)], (2018) shows the energy fluxes computed from moored and pressure p(x, z, t), can be written as a sum of or- observations from the TBeam experiment (recreated thogonal vertical modes here in Fig. 5). Though nonstationary semidiurnal energy flux at the TBeam mooring was approximately 25% ‘ 5 f of total semidiurnal internal tidal energy flux, the non- u(x, z, t) un(x, t) n(z), (1a) n50 stationary internal tide caused large short-term variability in both the magnitude and heading of the tidal ‘ p(x, z, t) 5 p (x, t)f (z), (1b) energy fluxes. While the heading of the stationary tidal n n n50 energy fluxes is relatively steady, directed from the generation site toward the Tasman Shelf (ranging be- where x 5 [x, y], z, and t are horizontal, vertical, and tween ;1408 and 1608 due to the spring–neap cycle), the time coordinates, un(x, t) and pn(x, t) are the velocity and heading of the total tidal energy fluxes varies dramatically pressure modal amplitudes, n is the vertical mode throughout the experiment. This suggests that even in a number, and fn(z) is the vertical structure function (i.e., region with a weak nonstationary tide, the nonstationary the mode) computed by solving the Sturm–Liouville tide can still significantly alter the tidal energy flux. problem

Unauthenticated | Downloaded 10/02/21 10:10 AM UTC OCTOBER 2020 S A V A G E E T A L . 2933 ! 1 1 Because variance in the nonstationary tide increases › f 1 f 5 0, with f (0) 5 f (2H) 5 0, z N2 nz c2 n nz nz with record length (Nash et al. 2012; Ansong et al. 2015), 0 n nonstationary tides calculated over only a few months (2a) are an underestimate of the multidecadal nonstationary tide. Because stationary tides are determined by least and the orthogonality condition squares regression, they satisfy the ‘‘normal equation’’ ð 0 (see Wunsch 2006) and the stationary and nonstationary 1 f f 5 d m n dz mn , (3) signals are orthogonal with respect to time averag- H 2 H ing, e.g., where z subscripts indicate partial derivatives, N0(x, z)is s r 5 the buoyancy frequency, cn(x) are the eigenspeeds, H(x) un pn 0, (7) is depth, and dmn is the Kronecker delta (e.g., Kundu 2004). 2 where the overbar indicates a time average. In practice, N0 is computed from an observed density profile and the system is solved using a finite-difference or a spectral method (Kelly 2016; Early et al. 2020). The 3. Observations and model vertical velocity modes, Fn, are defined in appendix and F 5 f related to the horizontal velocity modes via nz n. a. TBeam Exploiting the orthogonality of the modes, the vertical- mode amplitudes are A full water-column mooring was deployed from ð the Research Vessel (R/V) Revelle at 44.58S, 1538E 0 5 1 f on 10 January and recovered on 28 February 2015. un(x, t) n(z)u(x, z, t) dz, (4a) H 2H While this study only takes advantage of the mooring ð 0 data, full details of the observations collected during 5 1 f the TBeam campaign (including shipboard measure- pn(x, t) n(z)p(x, z, t) dz. (4b) H 2H ments from the R/V Falkor from 16 January through 13 February 2015) can be found in Waterhouse et al. b. Stationary and nonstationary tides (2018). On the mooring, pressure measurements were collected every minute using six Sea-Bird SBE37s and To separate tidal and nontidal variability, the an RBR-Concerto at 37, 285, 493, 975, 1680, 2707, and time series of the modal amplitudes, u (x, t)and n 4726 m, with temperature observations made at 31 p (x, t), are bandpassed around the M tidal fre- n 2 other depths along the mooring. The velocity obser- quency v 6 0:4cpd, retaining both the stationary M2 vations were collected using upward-facing 300- and semidiurnal constituents and the phase-shifted in- 75-kHz ADCPs at 38- and 748-m depths, sampling ternal tides. After bandpassing, least squares fits to every 10 min. The direct correlation between temper- tidal harmonics determine the M and S constitu- 2 2 ature and density observed from the CTD measure- ents. The sum of the M and S time series is the 2 2 ments made at eight stations discussed in Waterhouse stationary tide1 et al. (2018) made for a straightforward conversion M S s 5 2 1 2 from the high vertical resolution temperature obser- un un un , (5a) vations to density. Mooring knock-down events were M S s 5 2 1 2 pn pn pn . (5b) accounted for by interpolating all velocity and density observations to a vertical coordinate system. The first

The nonstationary tide is the difference between the two pressure modal amplitudes (p1 and p2)andveloc- bandpassed and stationary tidal time series ity amplitudes (u1 and u2)werecomputedbydirect projection of the density proﬁles, respectively (see r 5 2 s un un un , (6a) Waterhouse et al. 2018). Due to the vertical resolution of the observations, only the lowest two modes could be r 5 2 S pn pn pn . (6b) reliably computed, and are thus the only modes analyzed here and in Waterhouse et al. (2018). b. Coupled-Mode Shallow Water model 1 The stationary tide can alternatively be deﬁned from ensemble averages of tidal constituents over many short time windows or CSW solves the evolution equations for the vertical- 5 using the response method, but the results are nearly identical mode amplitudes of tidal transport Un(x, t) Hun and (Munk and Cartwright 1966; Kelly et al. 2015). pressure pn(x, t) using

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‘ staggered second-order Adams–Bashforth time-stepping U 1 (u =)U 1 f k^ 3 U 52H=p nt mn m n n algorithm. The vertical structure functions (f and F )and m50 n n ‘ eigenspeeds (cn) are computed once at each location 2 using the spectral method of Kelly (2016) with the HTmn pm , m50 21-yr mean stratiﬁcation (1994–2015) from the Hybrid (8a) Coordinate Ocean Model (HYCOM; Chassignet et al.

‘ 2009), and global bathymetry (Smith and Sandwell Hp H dc2 nt 1 = 1 n = 52= 1997). The (static) topographic coupling coefficients 2 2 umn pm 2 Un Un cn cn m50 cn are computed using centered differences and numer- ‘ ical integration. Prescribed TPXO surface tide trans- 1 T U , nm m ports (U ) force the model through the topographic m50 0 coupling terms, allowing us to utilize both highly ac- (8b) curate surface tides and long time steps (as permitted where t subscripts indicate time derivatives and f is the by the Courant–Friedrichs–Lewy condition; Kundu inertial frequency (see appendix). Equations (8a) and 2004). To eliminate some complicated standing waves, (8b) are analogous to the shallow-water equations de- we limit surface-tide forcing to the Macquarie Ridge and scribing the evolution of horizontal momentum and neglect the (weak) generation sites in the northern part of surface displacement, respectively. The equations for the domain, allowing us to focus on the fate of the internal n . 0 are forced by prescribing surface-tide velocities U0 tide that radiates from Macquarie Ridge. computed a priori (forcing terms involving p0 are neg- CSW requires a dissipation scheme for numerical ligible; see Kelly 2016). The terms with summations stability. The exact form and implementation of the couple modes and produce scattering. The Tmn terms scheme is somewhat arbitrary and solely designed to are the topographic coupling coefficients, and umn are have minimal impact on the dynamics of interest. To this the background flow coupling coefficients [i.e., pro- end, we stabilize the simulations with a weak horizontal n 5 2 21 jections of the depth-dependent quasigeostrophic cur- numerical viscosity, N 27.5 m s (Bryan et al. 1975), rent u(x, z, t)] and a flow-relaxation region (sponge) along the domain boundary (Lavelle and Thacker 2008) and in shallow ð 0 locations with poor ‘‘wave resolution’’ (see Adcroft et al. 5 1 f =f 21 Tmn(x) n m dz (m ), (9a) 1999). We do not analyze the dynamics of our numerical H 2 H viscosity, but we report viscous dissipation simply to show ð 0 that it is an order of magnitude weaker than the advective 5 1 f f 21 umn(x, t) u(x, z, t) m n dz (m s ), (9b) terms in most of the model domain (section 5a). H 2H The time-dependent background flow terms, u and ð mn 0 dc2, are updated once per day by interpolating snapshots d 2 5 1 d 2 F F 2 22 n cn(x, t) N (x, z, t) n n dz (m s ). (9c) from a global 1/128 nontidal HYCOM simulation (ex- H 2H periment 91.1; HYCOM.org; Chassignet et al. 2009)to

The term Tmn is zero where the bottom is flat and the 1/208 CSW grid. The power spectral density of sur- stratification is horizontally uniform (i.e., where =f 5 face relative vorticity in the horizontal plane (z/f , where 0); umn is diagonal where quasigeostrophic currents are z 5 dy/dx 2 du/dy) shown in Fig. 1 shows a peak near barotropic, and dense where geostrophic currents are l 5 200 km. Temporally averaged surface vorticity and d 2 baroclinic. The term cn (derived in appendix) accounts eddy kinetic energy (EKE) highlight a region in the for changes in propagation (i.e., refraction) by evolving southern part of the study region, associated with the stratification, and does not couple modes. Antarctic Circumpolar Current (ACC). There is also a re- CSW is numerically implemented on a spherical 1/208 gion of strong relative vorticity and EKE along the eastern C-grid using a finite-volume formulation and eight ver- coast of southern Australia, associated with the East tical modes. Test simulations (not shown) established Australian Current (EAC). However, in the interior of the that the open-ocean mode-1 and mode-2 solutions, basin, where the internal tidal beam has been observed, which we focus on here, are insensitive to increased vorticity and EKE are relatively weak and small scale. resolution. Higher resolution improves the accuracy of The model equations (8) in CSW are slightly different coastal tides and topographic scattering, but those aspects than those derived by Kelly et al. (2016). The new equa- have been examined using other models (D. Brahznikov tions (i) invoke the horizontal geometric approximation and H. Simmons 2017, personal communication; Klymak and (ii) rely on a time-scale separation rather than an et al. 2016). Transport and pressure evolve using a amplitude separation (see appendix for the new derivation

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FIG. 1. (a) Background flow surface kinetic energy computed from HYCOM and averaged from 1 Jan to 1 Mar 2015. (b) Background flow surface relative vorticity (z/f) computed from HYCOM and averaged over the same time period. (c) Wavenumber spectral density (computed as in Savage et al. 2017b) of surface kinetic energy and relative vorticity computed from daily snapshots of HYCOM. Colored rectangles show the various background flow scales examined in section 5b; red shows scales of 600–800 km, purple shows scales of 400–600 km, blue shows scales of 200–400 km, and yellow shows scales of 50–200 km. and a complete list of simplifying assumptions). The term background flow variability are critical for our central re- ‘‘geometric approximation’’ is borrowed from optics and sults. The geometric approximation is formally invalid means that the propagation medium changes slowly over where quasigeostrophic eddies and internal tides have many wavelengths.2 Here, the horizontal geometric ap- comparable horizontal length scales, but we employ it ev- proximation omits terms that include explicit horizontal erywhere because it stabilizes the (linearized) dynamical derivatives of the background flow. Because the back- system. Including the explicit terms in the momentum ground flow is in geostrophic balance to first order (see equations that involve background flow strain would permit appendix), this approximation also neglects terms that exponentially growing waves that quickly destabilize the explicitly include vertical shear, which would need to be numerical model (e.g., see the dispersion relations derived balanced by (neglected) horizontal buoyancy gradients. by Kunze 1985; Zaron and Egbert 2014). Nonlinear terms This approximation does not require the background could be included to cap this exponential growth, but these flow be spatially uniform. In fact, vertical and horizontal terms would greatly complicate the model, negating CSW’s greatest strengths: simplicity and numerical efficiency. Scaling the mode-1 internal tide dispersion relation sup- 2 This is a similar constraint to the WKB approximation, which ports our use of the horizontal geometric approximation. assumes the amplitude and wavenumber of the solution changes Zaron and Egbert (2014) derived a formula for the relative slowly over many wavelengths (Bühler 2009). changes in mode-1 phase speed:

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FIG. 2. Mean amplitudes during 2015 of nondimensional changes in mode-1 phase speed dcp/cp due to 1/128 HYCOM background flow (a) stratification, (b) advection, and (c) horizontal shear, as defined by (10), which was derived by Zaron and Egbert (2014). The color bar has a log10 scale.

dc 1 jdc2j pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ju j 1 f 2 z stationary component of surface displacement agrees with p ’ 1 1 1 2 f 2/v2 11 1 11 , (10) 2 v2 satellite observations (Fig. 4). Specifically, the model and cp 2 c1 c1 2 f data have statistically equivalent amplitude, phase, direc- where the terms on the right-hand side represent the tion, and wavelength, when both datasets are fit to a plane contributions of background flow stratification, advec- wave using data subsampled at the eight stations occupied tion, and vertical vorticity, respectively. While deriving during TBeam (Waterhouse et al. 2018). The model and (10), Zaron and Egbert (2014) neglected vertical vari- satellite data also agree well with in situ observations (cf. ability in the background flow to avoid modal coupling, Fig. 4 here with Fig. 6 in Waterhouse et al. 2018). thus the only nonzero term in the background flow b. Mode-1 and mode-2 tidal energy fluxes summation is in u11 [see (8a)]. The root-mean-square (rms) magnitude of each term in (10) was computed A comparison of the mode-1 tidal energy fluxes com- using (9) with daily snapshots of 1/128 HYCOM output puted from both the observations and model demon- during all of 2015. The annual mean terms were an order strates the internal tide variability due to a background of magnitude smaller than the RMS values because the field (Fig. 5). The observational fluxes (black lines) are background flow is primarily associated with unsteady recalculated as in Fig. 8 of Waterhouse et al. (2018).The eddies. The HYCOM analysis indicates that phase speed dashed black line represents the stationary tidal energy changes in the Tasman Sea, like the North Pacific (Zaron fluxes calculated from observations, and the solid black and Egbert 2014), are dominated by stratification and line represents the total tidal energy fluxes. The beating advection, while shear effects are an order of magnitude between the M2 and S2 tidal constituents creates the smaller (Fig. 2). These results indicate that the horizontal fortnightly oscillation (i.e., the spring–neap cycle) evident geometric approximation (i.e., neglecting shear) does not in both the stationary and nonstationary energy fluxes. significantly degrade predictions of internal tide propa- Though the magnitude of the nonstationary tidal flux is gation through the HYCOM mesoscale in this region. small compared to the stationary flux, most of the variability in both the magnitude and direction of the total tidal energy flux is due to the nonstationary tide (Waterhouse 4. Comparison of CSW to altimetry and moored et al. 2018). For comparison with the observations, tidal observations energy fluxes computed from CSW simulations run with (yellow lines) and without (purple lines) background flow a. M amplitude and phase from CSW 2 are also shown in Fig. 5. For the model run without back-

CSW simulated M2 internal tides interacting with ground flow, the magnitude of the total tidal energy flux is HYCOM background flows during all of 2015 (and one nearly identical to the magnitude of the stationary tidal month of spinup). The simulation reveals a distinct mode-1 energy flux. For the model run with background flow, there beam, with a surface displacement of 1 cm, propagating is a considerably larger difference between the stationary from Macquarie Ridge toward Tasmania (Fig. 3). The and total tidal energy flux amplitudes. This is shown more

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FIG. 3. Internal tide beam in the Tasman Sea. Contours show the (a) depth of the Tasman Sea (km) and (b) amplitude (cm) and (c) phase

(8) of the stationary M2 mode-1 sea surface height from the CSW model. TBeam observations from January to March 2015 include eight CTD-LADCP stations (white diamonds) and one full-depth mooring (red circle). The internal tide generation site is along Macquarie Ridge, south of New Zealand (NZ) in (a). Mainland Australia (AUS) is noted in (a). Black contours indicate the 1000-m isobath. clearly by examining the ratio of total tidal energy flux Though the velocities from CSW compared well with amplitude to stationary tidal energy flux amplitude, plotted the observations (not shown), the pressure estimated in in the bottom panel of Fig. 6. In the CSW simulation run CSW is approximately half of the pressure measured from with no background flow, the ratio of total to stationary the TBeam mooring. As internal tide beams are com- tidal energy flux amplitude is approximately equal to one posed of interfering or superimposed plane waves, they for the entire time series, whereas for the TBeam mooring are sensitive to the details of phasing, direction, and gen- observations as well as the CSW simulation forced by a eration. While the CSW phasing and sea surface height background flow, the ratio of total to stationary tidal energy amplitude described here agrees well with altimetry in flux amplitude varies over time. In the TBeam mooring general (Fig. 4), small errors in generation location could observations, this ratio ranges from less than 0.5 to greater cause large differences between the model and in situ than 2, whereas this ratio tends to be closer to one for the observations in some locations. This difference in pressure CSW simulation (ranging from approximately 0.6 to 1.5). causes the model to underestimate the energy fluxes This demonstrates that not only are the TBeam energy compared to the mooring data. While the magnitude of fluxes larger than those calculated in CSW, but also that the energy fluxes calculated from both simulations is CSW is not fully capturing the fraction of total tidal energy about half of the magnitude of the fluxes calculated from that is nonstationary. the TBeam mooring, the ratio of stationary tidal energy

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FIG. 4. (left) Satellite and (right) CSW model stationary mode-1 M2 surface displacement (top) amplitude and (bottom) phase. The data are quantitatively compared by fitting a plane wave to the surface displacement (satellite or CSW) subsampled at the eight locations occupied during TBeam (circles) and estimating mean amplitude, total flux, heading, and wavelength. The TBeam mooring is labeled with an 3. See Waterhouse et al. (2018) for the fitting method and uncertainty estimates. Note that the in situ data at each station are individually fit to the first vertical mode, which has a theoretical horizontal wavelength. However, the horizontal plane wave fit to mode-1 SSH amplitude, over the entire in situ array, was performed at many horizontal wavelengths to determine the best fit (see Fig. 5 in Waterhouse et al. 2018). The satellite analyses are presented in Zhao et al. (2018).

fluxes to total tidal energy fluxes is better predicted by the stratification at the generation site due to the in- model. In fact, throughout the 60-day time series shown, cluded background flow. the CSW simulation including a background flow is al- Variability in the heading of the tidal energy fluxes, most always able to accurately predict when the magni- shown in the middle panel of Fig. 5, supports the idea that tude of the total energy flux is larger or smaller than the an evolving background flow is the primary cause of stationary energy flux, shown in the bottom panel of Fig. 5. mode-1 tidal nonstationarity. Again, in the CSW simula- This, in conjunction with the negligible difference be- tion without background flow forcing, the heading of the tween the total and stationary tidal fluxes in the CSW total tidal energy fluxes is nearly identical to the heading simulation without background flow forcing suggests that of the stationary tidal energy fluxes, while there is more in an open basin, low-frequency dynamics modulate the variability in the heading of the total tidal energy fluxes amplitude of internal tide energy fluxes. Specifically, in an computed from the CSW simulation with background open basin where internal wave interactions with ba- flow forcing. However, unlike the temporal variability of thymetry are limited, nonstationary internal tides can be energy flux magnitude, CSW is unable to accurately re- explained almost entirely by perturbations in internal produce the temporal variability of the heading of the tide propagation, rather than perturbations in inter- tidal energy fluxes. This seems to imply that while the nal tide generation due to time-varying stratification. magnitude of tidal fluxes is highly correlated with back- These differences in energy flux magnitude between ground flow interactions, the heading of the total flux is simulations could also result from differences in more sensitive to other dynamics that are unresolved in

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21 FIG. 5. (top) Amplitude (kW m ) and (middle) heading (8) of the semidiurnal mode-1 energy flux. (bottom) Ratio of total to stationary tidal energy flux amplitudes. Observations from the TBeam mooring (Waterhouse et al. 2018) include the total (solid black) and stationary (dashed black) energy flux amplitude and heading. Modeled energy flux from CSW include those with background flow (total, solid yellow; stationary, dashed) and without background flow (total, solid purple; stationary, dashed purple). The CSW results without background flow are nearly identical, rendering the purple dashed line indistinguishable from the purple solid line.

CSW, such as interactions with other propagating internal 2 energy fluxes increases by ;50% with the addition of waves. The heading of the total observed tidal energy fluxes background flow, the average mode-1 energy flux mag- varies over 1608 over the observation period, which nitude decreases by 20%. This simultaneous loss of has been attributed to possible variations occurring at mode-1 energy and gain of mode-2 energy implies that a the generation site (Waterhouse et al. 2018). fraction of the additional mode-2 energy can be attrib- Amplitude of the mode-1 and mode-2 energy fluxes uted to intermodal scattering by the background flow. computed from CSW with and without background flow, As shown in (Rainville and Pinkel 2006), the mode-2 plotted in Fig. 6, shows that the mode-1 energy is being internal tide is more susceptible to interactions with scattered into higher modes by the background flow background flows, and therefore has a larger fraction (seen also in section 5a). Note that, in Fig. 6, the mode-2 of nonstationary variance. This can be seen easily in energy fluxes are multiplied by 10, to be more easily comparing mode-1 and mode-2 ratios of total to sta- compared with the mode-1 energy fluxes. When the tionary internal tide energy flux amplitude, as plotted background flow is added to the model, mode-2 internal in the bottom panel of Fig. 6. The total-to-stationary tide energy flux magnitude increases by an average of energy flux ratio is considerably larger in mode-2 ;50%, implying that the mode-2 internal tides are more than mode-1. Additionally, the mode-2 ratio is almost sensitive to background flow than mode 1, consistent exclusively greater than one, demonstrating that the with previous studies (Rainville and Pinkel 2006; Zaron background flow typically energizes mode-2, likely from and Egbert 2014). Although the magnitude of the mode- intermodal scattering as discussed in the previous

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21 FIG. 6. (top) Amplitude (kW m ) of the mode-1 and mode-2 semidiurnal energy flux from CSW. Mode-2 energy flux amplitudes are multiplied by 10 for scale. Dashed lines are stationary internal tide components, as in Fig. 5. (bottom) Ratio of total to stationary tidal energy flux amplitudes. paragraph (and in detail in section 5a). The ratios also the mode-2 tide is partially dependent on other fac- hint at differences in the causes of mode-1 and mode-2 tors, like subtle changes in stratification over sloping nonstationarity. While the mode-1 energy flux ampli- bathymetry. tudes without a background flow were almost indistinguishable from the stationary energy flux amplitudes (Figs. 5, 6), there are clear differences between the total 5. Analysis of internal tide advection by a mode-2 energy flux amplitudes without background flow low-frequency background flow and the stationary mode-2 energy flux amplitudes. a. Internal tide energy balance This difference suggests that, unlike the mode-1 tidal nonstationarity, which is primarily caused by the The steady-state mode-n energy balance is derived by additional background flow, the nonstationarity of computing (8a) Un/H 1 (8b)pn and time averaging

8 9 > > ‘ <> => U Hp dc2 = n 1 = n 1 n = (umn )Um umn pm pn Un 5 > H c2 > c2 m 0:>|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}n ;> |fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl}n

Amn Rn ‘ 52= 1 2 Unpn (Umpn Tnm Unpm Tmn) , (11) |fflffl{zfflffl} m50 |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} Fn Cmn

where Amn is the ‘‘advection’’ term and Rn is the and topographic internal tide generation C01, which can 2 ‘‘refraction’’ term due to wave–background flow interac- reach 100 mW m 2 (not shown). The advection term tions. The term Fn is the energy flux, and Cmn is topographic SAm1, intermodal topographic scattering SCm1,anddis- intermodal scattering. When the background flow is zero, sipation are also large near the Macquarie Ridge, because the left-hand side is zero and energy flux divergence equals local internal tide energy is enhanced by topographic topographic scattering. generation. In the center of the basin, internal tide energy Averaged over the full year of 2015, the first-order is smaller, and weak energy–flux convergence is balanced mode-1 energy balance is between energy flux divergence by the advection term and intermodal scattering, which

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FIG. 7. The total M2 mode-1 energy balance, averaged over 2015, is between (a) energy flux convergence (corrected for internal tide generation), (b) topographic scattering, (c) viscous dissipation, (d) background flow advection, (e) background flow refraction, and (f) a residual (this includes the energy tendency, numerical error, and dissipation by boundary sponges). Red indicates mode-1 energy loss in every panel except (b).

2 can reach 5 mW m 2 (Fig. 7). Dissipation (by the back- The advection term in Eq. (11) can be conveniently ground horizontal viscosity) and the refraction term R1 separated into symmetric and antisymmetric parts. The are negligible. Errors in the energy balance occur over symmetric part AS,mn 5 (Amn 1 Anm)/2 can be written rough topography and along the boundary sponge, where as a flux divergence (provided the background flow is radiating waves are artificially dissipated. Within the horizontally nondivergent), which has an area integral model domain, 57% of mode-1 generation is scattered to of zero, via the divergence theorem, provided umn 5 0on higher modes by rough topography, but most of this the boundaries. The antisymmetric part AA,mn 5 (Amn 2 scattering occurs in shallow water near Macquarie Ridge. Anm)/2 describes intermodal scattering by the back- In the open ocean, (defined as regions deeper than ground flow, because energy transfer from mode-1 to 3000 m), only 13% of mode-1 generation is scattered to mode-2 is equal and opposite to energy transfer from higher modes by rough topography. In addition, 8% of mode-2 to mode-1, etc. The self-interaction (diagonal) mode-1 generation is scattered to higher modes by the terms are zero. For the mode-1 M2 tide, SAS,m1 is equally background flow (9% occurring below 3000 m), 1% is positive and negative (i.e., it simply pushes energy 22 dissipated by numerical viscosity and the remaining 34% around), while SAA,m1 is persistently 1–2 mW m , is absorbed along the boundaries by the sponge. Overall, indicating a net sink of mode-1 energy (Fig. 8). While the energy balance analysis indicates that CSW back- this energy loss is small (8% of internal tide generation ground flow terms are weaker than topographic internal within the model domain), it is indistinguishable from tide generation, but are significant and quantifiable in the open-ocean turbulent dissipation in the Tasman Sea 2 center of the basin. (1–2 mW m 2; Waterhouse et al. 2018) and previous

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FIG. 8. Mode-1 energy advection by the background flow Amn can be separated into (a) a symmetric component AS,mn that laterally transports energy and (b) an asymmetric component AA,mn that scatters energy between vertical modes. Red indicates mode-1 energy loss. numerical estimates of intermodal scattering by the Gulf (Fig. 9a) was larger than that in mode-2 (Fig. 9b). This is 2 Stream (0–2 mW m 2; Kelly et al. 2016). to be expected, as there is generally much more energy in mode-1 internal tides than in mode-2 internal tides. b. Dependence of internal tide advection on However, as was discussed in terms of the energy flux in horizontal scale of the background flow the section 4b, the fraction of mode-2 energy advection The effects of the flow in modifying the distribution of is larger compared to the mode-1 energy advection than the internal tide energy are quantified by Amn and Rn in the fraction of mode-2 tidal energy to mode-1 tidal en- Eq. (11). For mode-1 and mode-2, background flow ef- ergy. This suggests once again that the background flow fects were largest near the generation site (south of New has a larger overall effect on mode-2 than on mode-1. Zealand) and in the southern part of the basin, where To further investigate the advection of the internal the ACC is energetic. In general, advection in mode-1 tide by the background flow, we analyzed the spatial

5 22 FIG. 9. Average advection, An mAmn (mW m ), in (a) mode-1 and (b) mode-2.

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FIG. 10. As in Fig. 9, but for CSW simulations run with spatially filtered background flow fields. The top row is mode-1, while the bottom row is mode-2. The columns (from left to right) show An associated with background flow scales ranging from 50 to 200 km, from 200 to 400 km, from 400 to 600 km, and from 600 to 800 km. dependence of the wave-energy advection on the hori- the kinetic energy. Supplementing what was found in zontal scale of the background flow. To do this, we Buijsman et al. (2017), which showed that the temporal spatially bandpassed the HYCOM background flow variability of the vertically sheared flows has important velocities prior to including them in the internal tide implications for tidal nonstationarity, here we show that model. Four background flow scale ranges were chosen for spatial variability of the background flow is also im- analysis: a 50–200-km range, a 200–400-km range, a 400– portant in tidal nonstationarity. 600-km range, and a 600–800-km range; these ranges (shown as shaded regions of different colors in Fig. 1c) 6. Parameterizing the energy lost from the separated the eddy field captured by HYCOM from the stationary internal tide general, large scale background flow. The wave-energy advection calculated from CSW using the various hori- One can separate stationary and nonstationary internal zontal scalings are shown in Fig. 10. The simulations run tides by post processing CSW simulations with wave– with background flow dynamics ranging in size from 50 background flow interactions. However this method re- to 400 km contributed to the advection shown in Fig. 9, lies on an accurate time-dependent background flow, the largest contribution coming from the horizontal long model integrations, and extensive postprocessing, all scales between 200 and 400 km. Additionally, back- of which can be computationally expensive. Background ground flows of this scale have a peak in the spectral flow interactions are also difficult to observe from in situ density of surface vertical vorticity, but not in EKE measurements, because terms such as Amn and Rn re- (Fig. 1), suggesting that while background flow dynamics quire impractically high spatial and temporal resolution. with horizontal scales between 200 and 400 km are not Here, we pursue an efficient method of predicting the the most energetic in HYCOM, they have a larger am- stationary tide, and the energy lost from it, without plitude vertical vorticity (dependent on the horizontal knowledge of the instantaneous background flow (or scale of the background flow) than larger scale flows. Amn and Rn). The method isolates the evolution equa- This suggests that the wave-energy advection of the in- tions for the stationary tide and uses a closure scheme to ternal tide may be more dependent on the horizontal mimic background flow effects with an eddy viscosity scale of the background flow, and less associated with and/or diffusion. With this parameterization, the CSW

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FIG. 11. The stationary M2 mode-1 energy balance, computed from the 2015 CSW simulation, is between (a) energy flux convergence (corrected for internal tide generation), (b) background flow advection, and (c) background flow refraction. Red indicates stationary mode-1 energy loss.

s ‘ model reaches steady state after about 50 tidal cycles, Hp 1 s nt 52= s 1 s 2 d 2= 2 Un Um Tnm 2 [ cn Un] allowing one to predict the stationary tide from short cn m50 cn simulations and only a statistical description of the ‘ H background flow. 2 = s 2 [umn pm] , (12b) cn m50 Stationary tide equations where quantities in the square brackets depend on both Computing the stationary component of each term in the stationary and nonstationary flow. These terms de- Eq. (8) yields the stationary CSW equations scribe wave advection and refraction in the evolving ‘ quasigeostrophic flow. s 1 ^ 3 s 52 = s 2 s Unt f k Un H pn H pmTmn The stationary mode-n energy balance is m50 ‘ 2 = s [(umn )Um] , (12a) m50

( ) ‘ s s s U Hp p s = s n 1 = s n 1 n d 2= [(umn )Um] [umn pm] [ cn Un] 5 H c2 c2 m 0 |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}n |fflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflffl}n s s Rn Amn ‘ 52= Us ps 1 (Us ps T 2 Us ps T ) , (13) |fflffl{zfflffl}n n m n nm n m mn m50 |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} s s Fn Cmn where the left-hand side includes energy loss due to method. Also note that a nonstationary tide only de- background flow advection and refraction, respectively, velops if the background flow is time dependent. and the right-hand side includes energy flux divergence The open-ocean energy balance of the stationary and topographic effects (generation and scattering). mode-1 tide during the 2015 CSW simulation (with time- s r Note that time averaged cross terms (e.g., Unpn) are varying HYCOM background flow) is between energy zero because the stationary signals are orthogonal to the flux convergence and decay by background flow inter- s s residuals with respect to time averaging, a feature of actions (Am1 and R1, which dissipate 28% of the sta- least squares fits, ensemble averages, and the response tionary internal tide; Fig. 11). Unlike the total mode-1

Unauthenticated | Downloaded 10/02/21 10:10 AM UTC OCTOBER 2020 S A V A G E E T A L . 2945 tide, the stationary tide decays rapidly in the center of produces a displacement along the mean ray path the basin (through loss of coherence) and less than 6% of dx 56Std[dc]Dt over the time it takes the wave of the stationary internal tide power is lost at the to traverse the eddy, Dt 5 l/c1 (where l is the eddy s d 2 boundary sponges. In addition, R1 plays a major role in diameter defined above). Note that c1 is defined in attenuating (decohering) the stationary tide, while R1 Eq. (9c) so that it can be positive or negative. produced negligible changes in the energy of the total We estimated viscosity, diffusivity, and their associ- tide (cf. Figs. 7, 11). This result indicates that, over ated energy transfers K a posteriori using the stationary long distances, minor changes in mesoscale stratification n transport and pressure from the 2015 CSW simulations strongly modulate the phase (i.e., travel time) of the in- with HYCOM background flow. The parameterized ternal tide, without significantly changing its amplitude. viscosity in method 1 (Fig. 12a) is two orders of magni- Moreover, because R1 is negligible, the substantial loss of s tude larger than the numerical viscosity used to stabilize stationary energy through R is compensated by an 2 1 the model (which is n 5 27 m2 s 1, see section 3b). But, equivalent gain in nonstationary energy. N the parameterized viscosity is weaker than the diffusiv- Since the terms on the left-hand side of Eq. (13) slowly ity in method 2 (Figs. 12a,b) because the waves have remove energy from the stationary tide, we parameter- more kinetic than potential energy, a ratio that is a ize the terms in square brackets in Eq. (12) as diffusion function of latitude. Both parameterizations of mode-1 (see also Kafiabad et al. 2019). Thus, we close the energy loss (K ) are qualitatively similar to the exact equations for the stationary tide in the presence of a 1 energy loss ( As 1 Rs ), indicating that both parame- background flow with the ansatz: m1 1 terization are reasonable (cf. Figs. 11, 12). The variances ‘ 8 s 2 s explained by each parameterization (on a 1 grid, to 2 [(u =)U ] ’ n= U , (14a) 2 mn m n discount inevitable small-scale discrepancies) were r 5 m50 0.39 and r2 5 0.76 for methods 1 and 2, respectively. ‘ 1 s H H Background flow parameterization 2, based on a 2 d 2 = 2 = s ’ k =2 s 2 [ cn Un] 2 [umn pm] 2 pn , cn cn m50 cn random walk model, explained the majority of station- (14b) ary mode-1 energy loss in the 2015 CSW simulation, even without a tunable mixing efficiency. To further evaluate where n is an eddy viscosity and k is an eddy diffusivity. the skill of this parameterization, we ran a short (60 tidal Total stationary energy loss is then approximately cycles) CSW simulation using this k parameterization and no background flow. This simulation accurately re- ‘ Us Hps produced the stationary mode-1 tide from the year- 2n n =2Us 2 k n =2ps ’ As 1 Rs . (15) n 2 n mn n 2 5 8 |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}H cn m50 long simulation (r 0.82 on the native 1/20 grid), and had differences in sea surface amplitude and phase Kn that were only less than 5 mm and 6208,respectively One can propose numerous parameterizations for (Fig. 13). This agreement suggests that the k parameter- n and k with varying complexity (including formulas ization is an efficient way to incorporate background flow where viscosity is represented by a tensor), but we only interactions in global simulations of the stationary inter- examine two basic parameterizations: nal tide. pffiffiffiffi 1) n 5G El and k 5 0, where E is mean surface eddy kinetic energy (EKE) and l is the observed eddy 7. Summary diameter (Klocker and Abernathey 2014). We ap- proximate l as the mode-1 internal Rossby radius The analyses confirm the importance of wave–background

l1 5 c1/f plus 40 km, which crudely approximates flow interactions for predicting low-mode internal tides observed eddy diameters (Klocker and Abernathey in the Tasman Sea. A semi-idealized model, which in- 2014). The term G50.075 is a (tunable) spatially cludes internal tide advection and refraction by the uniform constant (sometimes called a ‘‘mixing effi- background flow, captures the character, but not details, ciency’’) that was chosen to maximize agreementÀ be-Á of the nonstationary tide observed during the TBeam s 1 s tween explicit mode-1 energy transfer Am1 R1 experiment. Energy balance analyses indicate that both and the parameterized transfer (K1) in our simulations. advection and refraction by the background flow at- 2) n 5 0 and k 5 Var[dc1]/(2l/c1), which is equivalent to tenuate the stationary tides at a combined rate of the diffusion coefficient for a one-dimensional ran- O (1 2 10) mW m22. Additional analyses indicate that dom walk, k 5 dx2/(2Dt)(Kundu 2004). As a wave (i) advection of the internal tide is heavily dependent on propagates through a stochastic eddy field, each eddy the spatial scale of the background eddy field, (ii) mode-2

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FIG. 12. (a) Eddy viscosity (see Klocker and Abernathey 2014) is slightly weaker than (b) eddy diffusion computed from a random walk model. (c),(d) Both background flow energy terms qualitatively resemble stationary energy flux 2 convergence (see Fig. 11a). The coefficient of determination r is reported for K1 and SAm,1 1 R1 on a 18 grid. tides are more sensitive to background flow dynamics First, in the open ocean, the stationary mode-1 tide than mode-1 tides (see also Rainville and Pinkel 2006), primarily attenuates by dephasing with the tidal potential, and (iii) advection by the background flow scatters not by topographic scattering, intermodal scattering by a mode-1 energy to mode-2 at a rate of O (1) mW m22 time-invariant background flow, or turbulent dissipation. (see also Dunphy and Lamb 2014; Kelly et al. 2016). Attenuation by dephasing can be accurately parameter- Nonstationary internal tides account for an ap- ized by an eddy diffusion based on local background flow preciable fraction of sea surface height variability statistics. in some regions (Savage et al. 2017a). These signals Second, internal tide advection by the background flow are likely to contaminate satellite observations of increases at horizontal scales smaller than 400 km, suggest- low-frequency submesoscale dynamics (Savage et al. ing accurate, high resolution, general circulation models are 2017b) that will be collected during the upcoming necessary for useful predictions of the nonstationary inter- Surface Water and Ocean Topography satellite al- nal tide. However, background flow strain and vorticity also timetry mission (SWOT; Fu et al. 2012). The results increase at smaller horizontal scales, so wave–background here have some implications for predicting stationary and flow dynamics that were omitted here by the geometric nonstationary internal tides: approximation may become important (e.g., Polzin 2010).

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FIG. 13. The stationary M2 mode-1 surface displacement (a) amplitude and (d) phase from the 2015 CSW simulation are well ap- proximated by a shorter simulation with (b),(e) background flow effects parameterized by k. (c),(f) The differences between the simulations are small, as quantified by the coefficient of determination, r2 5 0.82.

Last, the dominant sink of the nonstationary mode-1 Ocean Institute. We acknowledge the participating cap- tide remains largely unknown. In the Tasman Sea, we tains, technical support, and crew of the R/V Falkor and found that intermodal scattering by rough topog- the R/V Revelle, without whom data collection would not raphy (13% below 3000 m) and the background ﬂow be possible. We also thank Gregory Wagner for his (8%) were weak, and a large fraction of generated thoughtful comments and review prior to submission and mode-1 energy (34%) radiates to the domain bound- Ed Zaron for insights on parameterizing background aries. It is possible that additional dynamics primar- ﬂow effects. Finally, authors acknowledge the invaluable ily dissipate the mode-1 tide, such as wave–wave feedback from three anonymous reviewers. Mooring interactions (e.g., MacKinnon et al. 2013; Eden and and CTD data from the R/V Falkor is hosted with the Olbers 2014). BCO-DMO repository and can be found at https:// doi.org/10.26008/1912/bco-dmo.818958.1. The source code Acknowledgments. We thank Jen MacKinnon and for CSW can be downloaded from https://bitbucket.org/ Jonathan Nash for their support through TTIDE. smkelly/. Additionally, we thank Gunnar Voet and members of the SIO Multiscale Ocean Dynamics group for deploying APPENDIX the TBeam mooring. A. F. Waterhouse and A. C. Savage acknowledge funding from NSF-OCE1434722 and S. M. Derivation of the CSW Equations Kelly was supported by NSF-OCE1434352 and NASA- NNX16AH75G. We also acknowledge support for ship The Boussinesq, hydrostatic, f-plane, inviscid equa- time aboard the R/V Falkor supported by the Schmidt tions of motion in Cartesian coordinates are

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1 = 1 1 ^ 3 52= a. Leading-order background flow ut (u )u wuz f k u p, (A1a) Each field comprises a background flow and wave 0 52p 1 b, (A1b) z component, e.g., u 5 u 1 u0 respectively. The average of Eq. (A5) over fast time (denoted by an overbar) yields a b 1 u =b 1 wb 1 wN2 5 0, (A1c) t z 0 geostrophic flow = 1 5 : u wz 0 (A1d) ^ 3 52= f k u0 x~p0 , (A7a) We examine the asymptotic behavior of the system for 52 1 the small parameter (Wagner et al. 2017) 0 p0z b0 , (A7b)

w 5 0, (A7c) U j(u =)uj 0 « [ ; 1, (A2) fL jf uj = 5 x~ u0 0, (A7d) U L which is the Rossby number when and correspond i.e., a steady horizontally nondivergent flow in ther- to quasigeostrophic flow and a measure of nonline- mal wind balance. A key result is that the leading- arity when U corresponds to a wave with frequency 2 order background flow is independent of large-scale v ; f and wavenumber k ; L 1 (i.e., c 5 v/k ; fL is spatial gradients (i.e., u0 is independent of =xp ), U c ; « 0 the phase speed, so / ). Waves oscillate at time so that one can include regional stratification with- scales ~t ; f 21, so quasigeostrophic motions with time 2 2 out the corresponding geostrophic currents. Wagner t ; «f 1 5 LU 1 scales ( ) evolve slowly, and partial et al. (2017) note that the next-order background derivatives can be scaled flow evolves according to the standard quasigeostrophic › / › 1 «› equation and is unaffected by wave–background flow ~ t . (A3) t t interactions. Similarly, if waves and quasigeostrophic motions b. Wave evolution have horizontal length scales x~; L then structures 2 with large (regional) length scales x ; L« 1 have The leading-order wave equations describe linear waves. small gradients They arise by subtracting Eq. (A7) from Eq. (A5),

0 ^ 0 0 = / = 1 «= u ~ 1 f k 3 u 52= p , (A8a) x~ x . (A4) 0t 0 x~ 0 52 0 1 0 The leading-order equations are 0 p0z b0 , (A8b)

0 0 2 b ~ 1 w N 5 0, (A8c) 1 ^ 3 52= 0t 0 0 u0~t f k u0 x~p0 , (A5a) = u0 1 w0 5 0: (A8d) 52 1 x~ 0 0z 0 p0z b0 , (A5b) The O («) wave equations are 1 2 5 b0t~ w0N0 0, (A5c) 0 0 0 0 0 u 1 u ~ 1 (u = )u 1 (u = )u 1 w u = 1 5 0t 1t 0 x~ 0 0 x~ 0 0 0z ~ u w 0, (A5d) x 0 0z 1 0 = 0 1 0 0 1 ^ 3 0 52= 0 (u0 x~)u0 w0u0z f k u1 x~p1 , (A9a) and the O («) equations are 52 0 1 0 0 p1z b1 , (A9b) 1 1 = 1 1 ^ 3 52= u0t u1~t (u0 x~)u0 w0u0z f k u1 xp0 0 1 0 1 = 0 1 0 = 1 0 b0t b1~t u0 x~b0 u0 x~b0 w0b0z 2 = 0 0 0 0 0 x~p1 , 1 = 1 1 2 5 u0 x~b0 w0b0z w1N0 0, (A9c) (A6a) = u0 1 w0 5 0, (A9d) 52 1 x~ 1 1z 0 p1z b1 , (A6b) where spatial gradients in the waves are ignored at scales 1 1 = 1 1 2 5 0 b t b ~ u ~b w b w N 0, (A6c) = ’ 0 1t 0 x 0 0 0z 1 0 much greater than a wavelength (e.g., xp0 0). The 0 0 wave–wave advective terms [e.g., (u =x~)u ] only force = u 1 = u 1 w 5 0: (A6d) 0 0 x 0 x~ 1 1z an Eulerian mean ﬂow and second-harmonic wave ﬁeld,

Unauthenticated | Downloaded 10/02/21 10:10 AM UTC OCTOBER 2020 S A V A G E E T A L . 2949 so they can be ignored when these features are not of where the quantity in square brackets is zero because interest. dFm can be expanded in terms of Fm and is zero at the Only the leading-order background velocity fields are boundaries. Proof that the bracketed quantity is zero present in the advective terms, so waves only feel the follows from projecting Eq. (A14) onto Fn, integrating geostrophic component of the background flow. One by parts twice, substituting Eq. (A11), and using Eq. might suspect that the advective terms simplify after (A12). The projection of the nonbracketed quantities F d 2 substituting the leading-order wave balances Eq. (A8) onto n provides the formula for cn, and thermal wind relations Eq. (A7), but we have not ð 0 found this to be the case. In general, waves exchange 1 d 2 F F 1 d 2F F 5 ( cm mzz n N m n) dz 0, (A15a) energy with the geostrophic flow, even though the H 2H background flow evolves independent of waves. ð ð 0 2 0 d 2 1 N0 F F 5 1 d 2F F c. Projection of wave equations on to modes cm 2 m n dz N m n dz, (A15b) H 2H cm H 2H Recombining Eqs. (A8) and (A9) without the wave– wave terms, and making the horizontal geometric ap- where Eq. (A11) has been used. Applying the orthogo- proximation (see section 3b) yields nality condition to Eq. (A15b) yields Eq. (9c). d. Summary of approximations 0 1 = 0 1 ^ 3 0 52= 0 ut (u )u f k u p , (A10a) In addition to the starting assumptions of hydrostatic, 52 0 1 0 0 pz b , (A10b) Boussinesq, f-plane, inviscid dynamics, the derivation of the CSW equations required the following: b0 1 u =b0 1 w0(N2 1 b ) 5 0, (A10c) t 0 z 1) The Rossby number is small and waves are linear, as = 0 1 0 5 : defined by « 1. This also ensures quasigeostrophic u wz 0 (A10d) perturbations to N2 are small.A1 21 Projecting Eq. (A10) onto vertical modes yields the 2) The quasigeostrophic time scale t ; («f) is ~ 21 CSW equations, Eqs. (8) and (9), where primes have much longer than wave time scale t ; f .Thisis been dropped to lighten notation and bz has been re- the key to separating quasigeostrophic and wave named dN2(x, z, t). dynamics. x ; The vertical velocity modes Fn are necessary for de- 3) The spatial scale of the regional stratification 21 riving modal projections, and they satisfy the eigenvalue L« is much larger than the spatial scale of the problem quasigeostrophic flow x~; L. This is the key to including regionally variable stratification. N2 F 1 0 F 5 F 5F 2 5 4) The spatial scale of the quasigeostrophic flow is nzz 2 n 0 with n(0) n( H) 0, (A11) cn much larger than the wavelength of the internal tide (i.e., the horizontal geometric approximation; see and orthogonality condition section 3b). ð 5) Variable topography can be ignored in the back- 0 N2 1 0 F F 5 d ground flow terms and background flows can be 2 m n dz mn . (A12) H 2H cn ignored in the topographic scattering terms. We do not formally justify the omission of simulta- dc2 The equation for n, (9c), arises from perturbing the neous background flow and topographic effects, eigenvalue problem (A11): but topographic effects tend to dominate in local- ized regions over abrupt features, while back- N2 1 dN2 (F 1 dF ) 1 0 (F 1 dF ) 5 0, (A13) ground flow effects tend to accumulate slowly as mzz mzz c2 1 dc2 m m m m internal waves propagate thousands of kilometers where dN2 is a small observed variation in buoyancy through the open ocean. Thus, to some extent, 2 topographic and background flow effects are sep- frequency and dc and dFm are small variations that must be determined. The lowest-order balance in Eq. arable by context. (A13) is Eq. (A11). The next-order balance is d 2 d 2 2 cm N N0 A1 2 2 F 1 F 1 dF 1 dF 5 From the buoyancy equation b0t ; N w1,sob0z ; N U/( fL) 5 2 mzz 2 m mzz 2 m 0, (A14) c c c 2 m m m «N ,wherew1z ; «U/L from continuity.

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