Near-inertial wave critical layers over sloping bathymetry
GGenerated using the official AMS L A TEX template—two-column layout. FOR AUTHOR USE ONLY, NOT FOR SUBMISSION! J O U R N A L O F P H Y S I C A L O C E A N O G R A P H Y
Near-inertial wave critical layers over sloping bathymetry L IXIN Q U ∗ AND L EIF
N. T
HOMAS
Department of Earth System Science, Stanford University R OBERT
D. H
ETLAND
Department of Oceanography, Texas A&M University
ABSTRACTThis study describes a specific type of critical layer for near-inertial waves (NIWs) that forms when isopy-cnals run parallel to sloping bathymetry. Upon entering this slantwise critical layer, the group velocity of thewaves decreases to zero and the NIWs become trapped and amplified, which can enhance mixing. A realisticsimulation of anticyclonic eddies on the Texas-Louisiana shelf reveals that such critical layers can form wherethe eddies impinge onto the sloping bottom. Velocity shear bands in the simulation indicate that wind-forcedNIWs are radiated downward from the surface in the eddies, bend upward near the bottom, and enter criticallayers over the continental shelf, resulting in inertially-modulated enhanced mixing. Idealized simulationsdesigned to capture this flow reproduce the wave propagation and enhanced mixing. The link between theenhanced mixing and wave trapping in the slantwise critical layer is made using the ray-tracing and an anal-ysis of the waves’ energetics in the idealized simulations. An ensemble of simulations is performed spanningthe relevant parameter space that demonstrates that the strength of the mixing is correlated with the degree towhich NIWs are trapped in the critical layers. While the application here is for a shallow coastal setting, themechanisms could be active in the open ocean as well where isopycnals align with bathymetry.
1. Introduction
Processes that drive enhanced mixing near the slopingseafloor have received increased attention in recent yearsdue to their potential role in shaping water mass trans-formation and diapycnal upwelling (Ferrari et al. 2016;McDougall and Ferrari 2017; Callies and Ferrari 2018).One such process is critical reflection of inertia-gravitywaves (IGWs) which occurs when wave rays align withbathymetry such that upon reflection, wave energy is fo-cused near the bottom, leading to bores, boluses, vortices,turbulence, and mixing (Cacchione and Wunsch 1974;Kunze and Llewellyn Smith 2004; Chalamalla et al. 2013).The phenomenon has almost exclusively been studied withthe internal tides in mind since they carry a significantfraction of the energy in the oceanic internal wave fieldand because many continental slopes are near-critical fortidal frequencies (Cacchione et al. 2002). Near-inertialwaves (NIWs) carry a comparable amount of energy andhave a power input into them that is similar to the internaltides (Alford 2003; Ferrari and Wunsch 2009), but theyhave not been considered as key players in driving mixingvia critical reflection on sloping topography. It seems rea- ∗ Corresponding author address:
Lixin Qu, Department of EarthSystem Science, Stanford University, 473 Via Ortega, Stanford, CA.USAE-mail: [email protected] sonable to neglect NIWs in this regard, because accordingto classical internal wave theory, NIWs propagate at veryshallow angles and therefore would only experience criti-cal reflection off nearly-flat bathymetry, which would notresult in much wave amplification. However, classical in-ternal wave theory does not account for the modificationof wave propagation by background flows.In particular, baroclinic, geostrophically-balanced flowscan greatly alter the propagation pathways of NIWs, re-sulting in rays with slopes s ray = s ρ ± (cid:115) ω − ω min N (1)(where ω is the frequency of the wave, ω min is the mini-mum frequency allowable for IGWs, and N is the squareof the buoyancy frequency) that are symmetric aboutisopycnals of slope s ρ , which are tilted in baroclinic flows(Mooers 1975; Whitt and Thomas 2013). The minimumfrequency of IGWs tends to be close to the inertial fre-quency f , thus NIWs with ω ≈ f propagate along rays thatrun nearly parallel to isopycnals, i.e. s ray ≈ s ρ . This opensthe possibility that NIWs can experience critical reflectionoff sloping bathymetry when isopycnals are aligned withthe bottom slope, α . However, when NIWs approach aregion where their frequency is equal to ω min , their groupvelocity goes to zero, and rather than reflecting, the waves Generated using v4.3.2 of the AMS L A TEX template a r X i v : . [ phy s i c s . a o - ph ] S e p J O U R N A L O F P H Y S I C A L O C E A N O G R A P H Y F IG . 1. Section of potential density (contoured every 0.5 kg m − )across the Texas-Louisiana shelf (the location of which is indicated bythe black line in the upper panel) from observations made on June 14,2010 as part of the Mechanisms Controlling Hypoxia study. can be trapped and amplified in critical layers (Kunze1985) which are slantwise if s ρ (cid:54) = α = s ρ , can give rise to enhanced near-bottom mix-ing associated with NIWs entering such slantwise criticallayers.It is not unusual to find flows in the ocean with isopy-cnals that follow bathymetry. Dense overflows, suchas those found on the western Weddell Sea margin, inthe Denmark Strait, or over the Iceland-Faroe Ridge,for example, naturally generate bottom-intensified along-isobath currents where the isopycnals that encapsulatetheir dense waters blanket topographic features (Muenchand Gordon 1995; Girton et al. 2001; Beaird et al. 2013).Isopycnals can also be aligned with bathymetry by upslopeEkman flows associated with the Ekman arrest of currentsflowing opposite to the direction of Kelvin wave propaga-tion (Garrett et al. 1993). The Florida Current is an exam-ple of such a flow and indeed has isopycnals that tend toparallel the continental slope off of Florida (Winkel et al.2002). Wind-forced coastal upwelling can also result inisopycnals paralleling the bottom, and there is evidencethat NIWs are amplified in critical layers during periodsof upwelling but not during downwelling (Federiuk andAllen 1996).Another example of flow that can meet the s ρ = α cri-terion are the currents on the inshore side of the anti-cyclonic eddies that form in the Mississippi/Atchafalaya river plume on the Texas-Louisiana shelf. High-resolutionhydrographic sections made on this shelf as part of theMechanisms Controlling Hypoxia study (e.g. Zhang et al.2015) illustrate the structure of the density field associatedwith these eddies (Fig. 1). Density surfaces form a bowl-like structure within the anticyclones while near the bot-tom isopycnals create a stratified layer that shoals towardsthe shore with s ρ ≈ α .During the summer time the anticyclones on the Texas-Louisiana shelf coincide with strong near-inertial cur-rents driven by the diurnal land-sea breeze which is near-resonant since the diurnal frequency is close to f (Zhanget al. 2009). Therefore if these near-inertial currents cre-ate downward propagating waves, then the anticycloneswould provide the ideal conditions for critical reflectionof NIWs over sloping topography. Realistic simulationsof the circulation and wave field on the Texas-Louisianashelf suggest that these conditions are indeed met. Wewill describe these simulation in section 2 and use themto motivate theoretical analyses (section 3) and idealizedsimulations (section 4) aimed at understanding the under-lying physics behind the phenomenon. With this coastalscenario as an example, the ultimate goal of this study isto build the link between wave trapping within a slantwisecritical layer and the enhanced bottom mixing that it caninduce. We will end the article with discussions of theparameter dependence of NIW trapping and the enhanceddiapycnal transport in bottom critical layers (section 5),which will be followed by a summary of our conclusions(section 6).
2. Realistic simulations of NIW-eddy interactions onthe Texas-Louisiana shelf
Here, we present results from the TXLA model, a real-istic simulation on the Texas-Louisiana shelf in the north-ern Gulf of Mexico, that highlights the interaction ofNIWs with anticyclones in a coastal region with slopingbathymetry (Zhang et al. 2012). In the northern Gulf, theMississippi and Atchafalaya rivers create a large region ofbuoyant, relatively fresh water over the Texas-Louisianashelf. The river plume front is unstable to baroclinic in-abilities during summertime, due to a pooling of fresh wa-ter over the Louisiana shelf by weak upwelling winds anda lack of storm fronts, which generates a rich field of ed-dies (Hetland 2017; Qu and Hetland 2020). As illustratedin the TXLA model output, the eddies are characteristi-cally fresh (buoyant) anti-cyclones, surrounded by strongcyclonic filaments at their edges (Fig. 2a and 2b). In ad-dition, since storms are infrequent in the summertime andwinds are generally mild, the diurnal land-sea breeze be-comes an important forcing mechanism (Fig. 2c). Not-ing that this region is near the critical latitude, 29 ◦ N,the diurnal land-sea breeze is nearly resonant with the lo-cal inertial frequency, such that the land-sea breeze drives
O U R N A L O F P H Y S I C A L O C E A N O G R A P H Y F IG . 2. (a and b) Snapshots of surface salinity and normalized relative vorticity ζ f from the TXLA simulation. (c) Time series of the zonal (red)and meridional (blue) component of the wind stress at the red dot marked in (a) and (b). (d) Surface velocity hodograph at the red dot from Jun 12to 14. (e) Time series of volume-averaged (in the green dashed box in panel f) TKE dissipation rate ε . (f, g, h, and i) Sections of ε , ζ f , dudz , and dvdz along the black line marked in (a) and (b). The time of the snapshots is 7:00, Jun 13, 2010, indicated by the dashed grey line (c) and (e). significant near-inertial oscillations, with peak clockwiserotating velocities of around 0.5 m s − (Fig. 2d). Thereare indications that these oscillations at the surface be-come downward propagating NIWs that radiate away fromthe offshore edge of anticyclones towards the shoalingbathymetry. Namely, bands of vertical shear in the zonaland meridional velocities descend from the offshore edgeof the eddy and bend upwards with isopycnals near thebottom on the inshore side of the eddy (Fig. 2h and 2i).The shear bands propagate upward (not shown) indicatingupward phase propagation and hence suggesting down-ward energy propagation.Interestingly, dissipation is enhanced near the bottomwhere the waves are approaching (marked by the greenbox in Fig. 2f) with values that are comparable to the dis-sipation near the surface. The turbulent kinetic energy(TKE) dissipation rate ε is diagnosed via the k − ε tur-bulence closure scheme; higher ε indicates strong mix- ing. The bottom dissipation pulses over an inertial pe-riod (Fig. 2f) indicating a relationship between the en-hanced dissipation and the resonantly forced near-inertialmotions. We explore the underlying physics behind thisrelationship using theory and idealized simulations in thenext two sections.
3. Theory
In this section, we develop a simple theoretical modelto interpret and link the three key features revealed bythe TXLA simulation: 1) downward propagation of near-inertial energy from the surface; 2) upward bending ofshear bands near the bottom; 3) enhanced dissipation inthe stratified layer over the bottom. The theoretical modelthat integrates these key elements is schematized in Fig. 3and is elaborated on below. J O U R N A L O F P H Y S I C A L O C E A N O G R A P H Y F IG . 3. Schematic of the theoretical model. Trapping and amplifi-cation of inertial waves within the slantwise critical layer formed whenisopycnals run parallel to the bottom slope results in enhanced mixing. a. Downward propagation of near-inertial wave from thesurface In our theoretical model, the wind oscillates at the lo-cal inertial frequency (such as the diurnal land-sea breezeat the latitude of 29 ◦ ), so inertial waves with ω = f areresonantly forced. In the absence of a background flow,the minimum frequency of IGWs is f , therefore the slopeof rays, (1), for these inertial waves is zero (since isopyc-nals are flat when there are no currents), and wave energycannot propagate vertically. In the presence of a back-ground flow, u , that follows the thermal wind balance: M = f ∂ u / ∂ z = − ∂ b / ∂ y , and that has a vertical vortic-ity ζ = − ∂ u / ∂ y , the minimum frequency of IGWs is ω min = (cid:113) f e f f − M / N . (2)where f e f f = (cid:113) f ( f + ζ ) is the effective inertial frequency(Mooers 1975; Whitt and Thomas 2013). Consequently,in regions of anticyclonic vorticity ω min < f and thereforeinertial waves have rays with non-zero slopes, allowingfor vertical wave propagation. This results in enhanceddownward energy propagation of NIWs in anticyclones, aphenomenon that is known as the ”inertial chimney” effect(Lee and Niiler 1998).However, to vertically propagate, NIWs need to acquirea finite horizontal wavelength and a non-zero horizontalwavenumber. The horizontal wavelength of NIWs can bereduced due to the presence of vorticity gradients, via theprocess of refraction (Young and Jelloul 1997; Asselin andYoung 2020). Gradients in ζ set up lateral differencesin wave phase since near-inertial oscillations separated ashort distance from one another oscillate at slightly dif-ferent frequencies. As a result, the near-inertial motionsdevelop a horizontal wavenumber whose magnitude in-creases linearly with time at a rate that is proportional tothe gradient in f e f f van Meurs (1998). As illustrated in F IG . 4. Initial conditions for the density (upper panel) and the along-shore velocity (middle panel) in the base run. The parameters of thebase run are listed in Tab. 1. Initial distribution of f ef f (lower panel).The boundaries of the anomalously low-frequency regime and the crit-ical layer are marked by the green dashed lines and the orange lines,respectively. Fig. 3, we envision that such refraction is active at the off-shore edge of the anticyclone near the maximum in veloc-ity where the vorticity gradient is maximum and f e f f = f .Therefore it is here where resonantly-forced inertial waveswill develop a horizontal wavenumber and radiate downinto the anticyclone. b. Reversal of vertical energy propagation in the anoma-lously low-frequency regime The upward bending of the shear bands at depth on theinshore side of the anticyclone seen in the TXLA sim-ulation (e.g. Fig. 2h and 2i) suggests that the verticalpropagation of the surface-generated NIWs changes signat depths well above the bottom. Such a reversal of ver-tical energy propagation not due to bottom reflections ispossible in background flows with baroclinicity. This fol-lows from the expression for the slope of wave rays (1)
O U R N A L O F P H Y S I C A L O C E A N O G R A P H Y s ray =
0, which happens whenthe wave’s frequency equals the local effective inertialfrequency ω = f e f f . In a flow that is baroclinic, since ω min < f e f f waves can propagate past the point where ω = f e f f and when they do so, s ray and the vertical com-ponent of their group velocity changes sign. In this region,the wave’s frequency is less than f e f f but greater than ω min , i.e. ω min < ω < f e f f . This is the so-called anoma-lously low-frequency regime defined by Mooers (1975),where NIWs are characterized by unusual behavior. Inparticular, the vertical components of the group and phasevelocities can be in the same direction in the anomalouslylow-frequency regime (Whitt and Thomas 2013). This isobserved in the TXLA simulation, since the shear bandsthat bend upwards near the bottom on the inshore side ofthe eddy (thus fluxing energy to the shallows) also propa-gate upwards in time, indicating a positive phase velocity.This process is schematized in Fig. 3 for inertial waves.The location where the wave rays start to bend is where s ray = f e f f = f . After passing the bending location, s ray increases from zero so that the wave rays bend up-wards. At the same time, the waves enter the anomalouslylow-frequency regime (where f e f f > f ), and, by the the-ory, the phase velocity should have the same sign as thegroup velocity so that the phase also propagate upwards. c. Trapping in a slantwise critical layer As NIWs enter the anomalously-low frequency regime,their frequency approaches ω min . At the location or loca-tions where ω = ω min , the magnitude of the group velocity | c g | = N ω | m | (cid:115) ( ω − ω min )( + s ray ) N (3)( m is the wave’s vertical wavenumber) goes to zero (Whittand Thomas 2013). These locations can be either turningpoints or critical layers depending on the geometry of thecontour where ω min = ω . This contour is known as theseparatrix and if it is aligned with wave rays, waves can-not radiate away from this boundary and are trapped, thusforming a critical layer. From (1), wave rays run parallelto isopycnals at the separatix since ω = ω min there, hencean alignment of the separatrix with isopycnals marks thelocations of critical layers. In weakly baroclinic anticy-clones for example, critical layers are nearly flat and format the base of the vortices where the vorticity increaseswith depth (Kunze 1985). In strongly baroclinic currentsin contrast, NIW critical layers tilt with isopycnals andtend to be found in regions of cyclonic vorticity (Whitt andThomas 2013). Such slantwise critical layers can form instratified layers over sloping bathymetry, as we demon-strate below. F IG . 5. (a and b) Across-slope sections of dudz and dvdz from the baserun. (c) Ray-tracing solution based on the initial conditions of the baserun; the rays are colored by the group velocity (normalized by its max-imum value on the ray) and the initial locations of the rays are denotedby the black stars. (d) Across-slope section of TKE dissipation rate ε and the control volume used in the energy budget (green dashed box).Time series of mean dissipation rate (e) and turbulent buoyancy flux κ N (f) in the control volume. Three inertial periods are shown in (e)and (f), and the vertical sections are made at t=105 Hr. Here, we introduce a specific type of slantwise crit-ical layer for inertial waves with ω = f over slopingbathymetry. This critical layer forms in a stratified layerwith isopycnals that run parallel to bathymetry, as illus-trated in Fig. 3, mimicking the layers that have been seenin the observations and simulations of the flows on theTexas-Louisiana shelf (Fig. 1 and 2). The tilted isopyc-nals induce a horizontal buoyancy gradient and, assumingthat the flow is in geostrophic balance, create a thermalwind shear ∂ u ∂ z = − f ∂ b ∂ y = − ˜ N f sin θ , (4)where ˜ N is the ”stratification” in the rotated coordinatesand θ is the bottom slope angle which is assumed to bepositive (see Appendix A for the derivation). Correspond-ingly, this thermal wind flow induces a finite Richardsonnumber, Ri g = ∂ b ∂ z (cid:18) ∂ u ∂ z (cid:19) − = f ˜ N sin θ tan θ , (5)where the vertical buoyancy gradient ∂ b ∂ z is related to ˜ N by ∂ b ∂ z = ˜ N cos θ (see Appendix A). Furthermore, the J O U R N A L O F P H Y S I C A L O C E A N O G R A P H Y F IG . 6. Time series of the control-volume integrated terms from theKE equation (11). The dominant terms are shown in the upper paneland less significant terms are in the lower panel. geostrophic flow is also horizontally sheared because ofthe sloping bathymetry. The horizontal shear in u can bewritten as ∂ u ∂ y = − ˜ N f sin θ tan θ . (6)(see Appendix A), yielding a cyclonic vorticity and hencea positive Rossby number Ro g = ζ / f = − ∂ u ∂ y / f = ˜ N f sin θ tan θ . (7)In this stratified layer the minimum frequency of IGWs isexactly inertial: ω min = (cid:113) f e f f − M / N = f (cid:113) + Ro g − Ri − g = f , (8)since the contributions from vorticity (7) and baroclin-icity (5) cancel, however the effective inertial frequency f e f f = f (cid:112) + Ro g is superinertial. Hence inertial wavesin this layer enter the anomalously low-frequency regimeas ω min = f < f e f f . Moreover, since ω min is uniform andequal to f in this layer, the separatrix runs parallel toisopycnals and the criterion for a slantwise critical layeris met. Finally, from (3) it is clear that the group veloc-ity is equal to zero in the layer and should cause inertialwaves to be trapped and amplified there, which could driveenhanced mixing. The enhanced bottom dissipation in theslantwise stratified layer exhibited in the TXLA simulation(Fig. 2f) suggests that this mechanism is active there. Wetest these theoretical ideas in a more controlled environ-ment than the TXLA model using idealized simulations,as described in the next section.
4. Idealized simulations a. Base run
The Regional Ocean Modeling System (ROMS) is em-ployed in this study, which is a free-surface, hydrostatic,primitive equation ocean model that uses an S-coordinatein the vertical direction (Shchepetkin and McWilliams2005). ROMS is configured to conduct an idealized simu-lation. The model domain represents an idealized coastalregion over a continental shelf with a constant slope α = × − , and with the depths ranging from 5 m to 118 m.The domain has an across-shore span of 226 km and analong-shore width of 4 km. The domain is set to be ex-tremely narrow in the along-shore direction with few gridpoints so that the variation in the along-shore direction canbe assumed to be negligibly small. The horizontal resolu-tion is 220 m × m . There are 64 layers in the verticaldirection with the stretching parameters of θ S = . θ B = .
4. The along-shore boundary conditions are set tobe periodic, and the offshore open boundary has a spongelayer that damps the waves propagating towards the openboundary. The Coriolis parameter is equal to the diurnalfrequency, i.e., f = π s − ≈ . × − s − . The windforcing is set to mimic a diurnal land-sea breeze - a recti-linear oscillating wind oriented in the across-shore direc-tion with an amplitude of 4 × − Nm − . The simulationis run for 10 days.The initial conditions correspond to an anticyclonicbaroclinic flow with a slantwise critical layer onshore anda buoyant front offshore (Fig. 4), with parameters that arebased on the realistic simulation. The critical layer hasan across-shore width of L C =
50 km, and the offshorefront has a width of L =
40 km. Note that there is a tran-sition zone with a width of L T =
20 km in the middlewhere the horizontal buoyancy gradient linearly decreasesto zero with increasing across-shore distance. The flowand density fiields have no variations in the along-shoredirection. The stratification is set to N = × − s − ,a value based on the realistic simulation, and is constantacross the domain. The density structure of the criticallayer is determined by N and α , and the flow is initially ina thermal wind balance with the density field. The densitystructure of the offshore buoyant front is determined bythe velocity structure due to the constraint of the thermalwind balance. In the horizontal direction, moving in theacross-shore direction, the surface velocity at the offshorefront increases from zero with a vorticity of ζ = − . f between L C + L T ≤ y ≤ L C + L T + L and decays exponen-tially to zero offshore and outside of this region. In thevertical direction, the velocity decays linearly to zero to-wards the bottom. The density is determined from the ve-locity field and the thermal wind balance. There is no ini-tial across-shore flow in the domain. The linear equation O U R N A L O F P H Y S I C A L O C E A N O G R A P H Y F IG . 7. (a) Initial density of the the comparative run. (b, c and d)Across-slope sections of ε , dudz , and dvdz at the same time of Fig. 5. (eand f) Time series of surface and bottom velocities; these quantities areaveraged within the across-shore distance of 100 km. The maxima andminima are denoted by red circles. (g) Time series of ε , averaged in thesame control volume of the base run; the control volume is denoted inFig. 5. of state of seawater is used in the simulation: ρ = ρ [ . − α T ( Temperature − T ) + β S ( Salinity − S )] , (9)where ρ = . kg / m , α T = . × − ◦ C − , β S = . × − psu − , T = . ◦ C , and S = . psu . Theinitial temperature is uniformly set to 25 ◦ C , and the initialsalinity is calculated from the density field. The MPDATAscheme is used for the tracer advection (Smolarkiewiczand Margolin 1998). k − ε turbulence closure scheme isused to calculate the vertical mixing, and the Canuto Astability function formulation is applied (Umlauf and Bur-chard 2003; Canuto et al. 2001). The parameters usedto configure this simulation are listed in the first row ofTab. 1.Under the resonant wind forcing, NIWs start to developin the first few inertial periods and then enhanced bot-tom mixing follows. Snapshots of the vertical shear af-ter four inertial periods reveal the presence of shear bands(Fig. 5a and Fig. 5b). The orientation of the shear bandssuggests that the NIWs are generated at the offshore front,where the gradient in relative vorticity is largest. This is consistent with the theoretical finding that the horizontalwavelength of NIWs shrinks in regions with strong vor-ticity gradients so that the waves can propagate vertically(Young and Jelloul 1997; Asselin and Young 2020). Theslantwise shear bands imply that the NIWs are verticallypropagating and bending upwards when approaching thebottom. Furthermore, mixing is enhanced within the bot-tom critical layer, which corresponds to the area that thewaves approach (Fig. 5d). Generally, this idealized sim-ulation reproduces the phenomena found in the realisticsimulation (Fig. 2).To understand the pattern of wave propagation sug-gested by the shear bands, ray-tracing is conducted byapplying the Wentzel-Kramers-Brillouin (WKB) approx-imation. The WKB approximation is only valid when thebackground flow field does not significantly change overthe scales of the waves. We will accept this approxima-tion a priori and then validate it below by demonstrat-ing a consistency with an energetics analysis. The ini-tial fields of density and velocity (fig. 4) are used for thebackground flow in the ray-tracing calculation. The proce-dure of the ray-tracing is described in Appendix B. Raysare initiated at z = − m at the offshore end of the frontwith 3 km spacing. The ray paths have a similar shapeto the shear bands and indicate that wave energy is radi-ated downwards from the surface. As the waves enter theanomalously low-frequency regime (marked in the lowerpanel of fig. 4), they bend such that the slopes of waverays is near zero. When the waves approach the criticallayer, the waves slow down and eventually get trapped as | c g | → ε is diagnosed via the k − ε turbu-lence closure scheme. The magnitude of the the turbulentbuoyancy flux is parameterized as κ N , where κ is the tur-bulent diffusivity also diagnosed from the k − ε closure.The mean values of ε and κ N are calculated within thecontrol volume (marked by the green box), and the tempo-ral variations of these measures are shown in (Fig. 5e and5f). Both ε and κ N exhibit inertial pulsing, implying thatthe bottom mixing is enhanced at the inertial frequency.This reproduces the inertial pulsing of ε found in the real-istic simulation (Fig. 2e), and strengthens the link betweenthe bottom enhanced mixing and the NIWs. b. Energetics An energetics analysis is conducted to further sup-port the ray-tracing solution by formulating a kinetic en-ergy equation from the primitive equations for a two- J O U R N A L O F P H Y S I C A L O C E A N O G R A P H Y F IG . 8. (upper) Across-slope sections of TKE dissipation rate ε from the ensemble runs with a stratification of N = × − s − , offshorefrontal width of L = km , and varying surface vorticity ( ζ = − . f , − . f , and − . f ). The sections were made at t=105 Hr (when thedissipation is maximum). (lower) Rays for each run; red denotes rays that enter the critical layer, and green, rays that reflect off the bottom andmove away from the critical layer. dimensional flow invariant in the x -direction: ∂ u ∂ t + u · ∇ u − f v = ∂∂ z ( ν ∂ u ∂ z ) , ∂ v ∂ t + u · ∇ v + f u = − ρ ∂ p ∂ y + ∂∂ z ( ν ∂ v ∂ z ) , − b = − ρ ∂ p ∂ z − ρ ∂ ¯ p ∂ z , ∂ u ∂ x + ∂ v ∂ y + ∂ w ∂ z = , (10)where u is the along-shore velocity, v is the across-slopevelocity, and w is the vertical velocity, p pressure, b is thewave (mean) buoyancy, and the Boussinesq approximationhas been assumed. A 24-hour high-pass filter is applied onthe pressure and buoyancy to isolate the wave fields p (cid:48) and b (cid:48) from the mean fields p and b . The reasoning for the filteris to facilitate the calculation of the wave energy flux con-vergence (which quantifies the degree of wave trapping),and which requires the wave pressure rather than the totalpressure. However, it is not necessary to filter the velocityfield. The 2.5D-like model configuration ensures ∂∂ x = u -related term drops out from the convergenceof the wave energy flux (see Eq. 11), and v and w haveno mean component and are thus the wave velocities bydefinition.A kinetic energy budget can be assessed by formulatingthe following kinetic energy equation. It is obtained bytaking dot product of the momentum equations of Eq. 10 with the velocity and applying the continuity equation: ∂ KE ∂ t = ADV + W EF + W BF + MEF + MBF + RKE + DKE , KE = ( u + v ) , ADV = − (cid:126) u · ∇ KE , W EF = − ∇ · ( p (cid:48) ρ (cid:126) u ) = − ∂∂ y ( p (cid:48) ρ v ) − ∂∂ z ( p (cid:48) ρ w ) , W BF = wb (cid:48) , MEF = − ∇ · ( ¯ p ρ (cid:126) u ) , MBF = w ¯ b , RKE = ∂∂ z ( ν ∂ KE ∂ z ) , DKE = − ν ( u z + v z ) . (11) ADV is the advection of kinetic energy.
W EF is the con-vergence of wave energy flux.
W BF is the wave buoyancyflux representing the energy transfer between wave kineticand potential energy.
MEF and
MBF are the mean energyflux convergence and buoyancy flux, and go to zero whenaveraged over times greater than a wave period.
RKE rep-resents the redistribution of kinetic energy by turbulence.
DKE represents the loss of mean and wave kinetic energyto turbulence.The energetics of the waves in the idealized simula-tion are analyzed, using Eq. 11, within the control volumemarked by the green box in Fig. 5d, where the mixing isenhanced. Each term in Eq. 11 is integrated over the con-trol volume to obtain time series (Fig. 6). (cid:82)
W EF dV rep-resents the wave energy flux coming into (positive) or go-ing out of (negative) the control volume. Since the turbu-lent viscosity parameterizes turbulent momentum fluxes,it follows that
DKE is related to the shear production
O U R N A L O F P H Y S I C A L O C E A N O G R A P H Y F IG . 9. (upper) Across-slope sections of TKE dissipation rate ε from the ensemble runs with a stratification of N = × − s − , offshoresurface vorticity of ζ = − . f , and varying frontal width ( L = km , 40 km , and 50 km ). The sections were made at t=105 Hr (when thedissipation is maximum). (lower) Ray-tracing solutions of the corresponding runs; red denotes rays that enter the critical layer, and green, rays thatreflect off the bottom and move away from the critical layer. terms in the TKE equation, and thus (cid:82) DKE dV repre-sents the portion of the Reynolds-averaged kinetic energytransferred to TKE, which would further go into turbulentmixing and dissipation. (cid:82)
RKE dV quantifies the energyremoval out of the control volume by the bottom stress,because it can be rewritten as (cid:82) A b u · (cid:126) τ b dS (where (cid:126) τ b isthe bottom stress and A b is the bottom boundary) if thestress at the top boundary of the control volume is neg-ligible. A clear inertial pulsing is found in the time se-ries of (cid:82) DKE dV , which follows the inertial pulsing of ε and κ N (Fig. 5e and 5f) and hence the inertially en-hanced mixing. The time series of (cid:82) W EF dV exhibitspeaks that lead (cid:82)
DKE dV by 1 hour, implying that theconvergence of wave energy causes the enhanced bottommixing. The inertial pulsing of (cid:82)
RKE dV suggests thatthe bottom stress removes energy when the wave energyflux converges and the flow enhances the turbulence. Allthe other terms in the KE budget are less significant (lowerpanel of Fig. 6). Overall, given the high correlation be-tween (cid:82)
DKE dV and (cid:82)
W EF dV , the process drivingthe enhanced mixing is wave trapping, consistent with theinference based on ray-tracing. c. Comparative run
To demonstrate the effect of coastal fronts on vertically-radiating NIWs, we present a comparative simulationwithout the eddy-like front to contrast the response ofNIWs and bottom mixing to the simulation with the eddy-like front (that is, the base run discussed above). The ini-tial density field is shown in Fig. 7. The difference with thesetup of the base run is the absence of the lateral buoyancygradients and hence a background flow, all other parame-ters are the same (Tab. 1).A ”two-layer” response is found to be dominant in thecomparative simulation; the surface and bottom velocitiesare out of phase by nearly 180 degrees (Fig. 7e and 7f). Such a ”two-layer” structure has been observed in manycoastal seas (Orli´c 1987; van Haren et al. 1999; Knightet al. 2002; Rippeth et al. 2002) and also some large lakes(Malone 1968; Smith 1972). This response is attributedto the presence of a coastal boundary (Davies and Xing2002). The presence of the coastal boundary yields a pres-sure gradient at depth, which drives the inertial current inthe lower layer and leads to a 180 degree phase shift withthe upper layer (Xing and Davies 2004). Consistent withthe observations, the comparative run shows that the firstbaroclinic mode dominates the response in a coastal sys-tem without fronts or currents.The differences between the comparative and base runsare summarized as follows. First, no clear slantwise shearbands exist in the interior (Fig. 7c and 7d), suggesting thatthere is no significant vertical radiation of NIWs from thesurface. Second, the dissipation does not exhibit clearinertial pulsing that has been observed in the base run(Fig. 7g). Lastly, the bottom boundary layer over theshelf is thin (Fig. 7b), and the response of the dissipa-tion is much weaker than that in the base run (Fig. 7e and7g). Overall, the comparative simulation indicates that theeddy-like coastal front is essential for the vertical radiationof NIWs and therefore the bottom enhanced mixing.
5. Discussion a. Exploring the parameter dependence of wave trappingand mixing in the critical layer
The idealized simulations are used to explore the depen-dence of wave trapping and mixing in the critical layer inthe framework of the idealized configuration. There are 8controlling parameters and they are listed in Tab. 1. To ef-ficiently explore this parameter space, we fix the externalparameters (i.e., the background rotation f , wind forcing (cid:126) τ , and bottom slope α ) as well as the dimension of thenear-shore front (i.e., the length scales of the critical layer0 J O U R N A L O F P H Y S I C A L O C E A N O G R A P H Y F IG . 10. Maximum volume-integrated convergence of the wave energy flux WEF [ (cid:82) WEF dV ] max (left), maximum TKE dissipation rate ε max (middle), and maximum turbulent buoyancy flux [ κ N ] max (right) plotted against the trapping ratio γ . The maxima are calculated over the InertialPeriod 1. The runs with N = × − s − are marked by circles, and the runs with N = × − s − are marked by stars. The larger markersize represents larger offshore frontal widths, and darker colors represent stronger anticyclonic vorticity. Gray dashed lines indicate the linearregressions. and transition zone, L C and L T ), but vary the parametersassociated with the offshore front (i.e., the frontal width L , relative vorticity ζ , and stratification N ). In otherwords, we fix the properties of the critical layer but varythe parameters that influence the propagation of waves to-wards the critical layer. By varying these parameters wecan quantify the sensitivity of the dissipation and mixingto the degree of wave trapping. A total of 18 simulations(including the base run) were performed and are listed inthe second row of Tab. 1.In concert with the ROMS simulations, ray-tracing isconducted for each run. Rays are initiated at z = − m within the offshore front across the width L , separated bya spacing of dy = km , and traced according to the proce-dure described in Appendix B. Rays either reach the criti-cal layer or hit the bottom and reflect offshore. To quantifythe wave trapping in the critical layer from the ray-tracingsolutions, we define the trapping ratio γ , the ratio betweenthe number of the rays reaching the critical layer and thetotal number of the rays. A higher value of γ indicatesa larger portion of wave energy that reaches the trappingzone and hence represents a highly trapped scenario. Thetrapping ratio is a metric that concisely captures the pa-rameter dependence of wave trapping in the critical layer.Relative vorticity modifies the minimum frequency ω min , such that stronger anti-cyclonic vorticity allowsNIWs to propagate more vertically, e.g. (1). A sub-set of the ensemble simulations run with different val-ues of the vorticity ζ but with a fixed frontal width of L = km , and stratification of N = × − s − illus-trates this physics (Fig. 8). The stronger the anti-cyclonicvorticity, the more steep the rays are, and they miss thecritical layer. For instance, in the case with ζ = − . f ,there are fewer rays reaching the critical layer (denoted byred) and more rays reflect offshore (denoted by green) thanin the other two runs with weaker vorticity. Correspond-ingly, the case with ζ = − . f has the lowest TKE dis- sipation rate, suggesting that reduced wave trapping leadsto weaker mixing.The dimension of the offshore front also modulates thewave trapping. This is illustrated in Fig. 9 for a groupof ensemble runs with various frontal widths L but withfixed relative vorticity (i.e. ζ = − . f ) and stratification( N = × − s − ). Noting that the difference in vorticityacross the jet is the same for all three cases, the propaga-tion of NIWs only depends on the geometry of the offshorefront. The ray-tracing solutions shown in Fig. 9 demon-strate that the wider fronts have fewer rays that reach thecritical layer and get trapped. This is because wider frontsmove the rays away from the critical layer. Consquently,the run with L = km has the weakest TKE dissipationrate and mixing.Finally, we quantify the relation between wave trappingand mixing using the trapping ratio γ . First, to test theskill of the parameter γ in predicting the degree of wavetrapping, we calculate the maximum (in time, over one in-ertial period) volume-integrated WEF [ (cid:82) W EF dV ] max foreach run and compare this quantity to γ . Recall that a largevalue of [ (cid:82) W EF dV ] max corresponds to strong wave trap-ping. The trapping ratio γ and [ (cid:82) W EF dV ] max are highlycorrelated ( r = .
91 and p = . × − ; left panel ofFig. 10), suggesting that γ is a skillful predictor for wavetrapping. Next, γ is compared to the maximum (over oneinertial period and within the control volume) TKE dis-sipation rate ε max and the maximum turbulent buoyancyflux [ κ N ] max (Fig. 10). The correlations between thesequantities are also robust: r = .
85 and p = . × − for ε max and r = .
84 and p = . × − for [ κ N ] max .This indicates that strong trapping of wave rays leads tohigh turbulence dissipation and mixing. Overall, the en-semble runs further support the conclusion, established inSection 4, that the enhanced bottom mixing is caused bywave trapping in the bottom critical layer. O U R N A L O F P H Y S I C A L O C E A N O G R A P H Y F IG . 11. Across-slope sections of the diapycnal velocity w d fromthe base run at t=106 Hr (a) and t=118 Hr (b). The subpanel in (b) isthe across-slope section of dissipation ε at t=118 Hr. (c) Time series ofthe diapycnal velocity averaged over the control volume (green dashedbox). Three inertial periods are shown. b. Enhanced diapycnal transport in the critical layer The amplification of NIWs by wave trapping, which el-evates mixing, also enhances diapycnal transport. To ex-amine the link between the diapycnal transport and wavetrapping in the slantwise critical layer, the diapycnal ve-locity, w d ≡ ∂∂ z ( κ N ) N , (12)is diagnosed across the idealized simulations listed inTab. 1. By using the linear equation of state of seawater(Eq. 9), one can write the diapycnal velocity w d as w d = gN [ α T ∂∂ z ( κ T ∂ T ∂ z ) − β S ∂∂ z ( κ S ∂ T ∂ z )] , (13)where ∂∂ z ( κ T ∂ T ∂ z ) and ∂∂ z ( κ S ∂ T ∂ z ) are the vertical mixingterms in the temperature and salinity equations that can beobtained from diagnostics from the ROMS model. Fromthese diagnostic we observe that the diapycnal velocity isupward in the crtical layer and enhanced when the bottommixing is elevated (Fig. 11a). Also, the diapycnal veloc-ity decays as the mixing weakens (Fig. 11b), so that thetime series of the diapycnal velocity has a similar inertialpulsing as the bottom mixing (Fig. 11c). Note that theamplitude of the diapycnal velocity can reach O ( − ) m/s, which is strong and comparable to the entrainmentvelocity near the surface induced by wind-driven turbu-lence. Furthermore, the maximum volume-averaged di-apycnal velocity w d , max (i.e. the maximum over one in-ertial period) is robustly correlated with the trapping ratio γ , with r = .
85 and p = . × − (Fig. 12). This fur-ther strengthens the link between the enhancement of thediapycnal velocity and the wave trapping mechanism.Another way to quantify the diapycnal transport is totrack the diapycnal movement of a passive tracer. To thisend, a passive tracer was released in both the base runand the comparative run to contrast the bottom diapycnaltransport in simulations with and without wave trapping. The tracer is initialized in the first four sigma layers abovethe bottom with a concentration equal to one (Fig. 13aand 13b). The concentration outside of this layer is setto zero. The tracer is released at t=90 Hr and monitoredfor three inertial periods. In terms of the spatial distribu-tion, at t=106 Hr (the time of the peak w d ), the base runshows a significant reduction of the tracer at the locationwhere the diapycnal velocity is enhanced and the wavesare trapped (Fig. 13c). In terms of temporal variability,in the region with enhanced w d , the variation of the tracerconcentration suggests that the tracer is transported out ofthe bottom layer during the period (from 103 Hr to 109Hr) when the diapycnal velocity is enhanced (Fig. 13e). Incontrast, the comparative run does not show such a signif-icant reduction in the tracer concentration near the bottom(Fig. 13d,e).The tracer distribution in the density space is used asa metric for tracking the diapycnal tracer transport. Themetric is calculated as follows. Given a volume where thedensity is less than a certain density ρ , the tracer content M in this volume is equal to the volume integral of thetracer concentration C : M ( ρ , t ) = (cid:90) (cid:90) (cid:90) ρ (cid:48) < ρ C ( x , y , z , t ) dxdydz . (14)Then, the tracer distribution function is defined as ∂ M ( ρ , t ) ∂ρ such that the tracer content within a density class canbe obtained by integrating the distribution function in thedensity space as (cid:82) ρ ρ ∂ M ∂ρ d ρ . In other words, ∂ M ∂ρ indicatesthe instantaneous distribution of the tracer in the densityspace and any diapycnal tracer transport should be re-flected by the rate of change of ∂ M ∂ρ .The distribution function calculated from the base runindicates that the tracer migrates to lighter density classeswith time (see the upper panel of Fig. 14). When the di-apycnal velocity is largest (at t=106 Hr), there is a conver-gence of the tracer towards a narrow density class. This F IG . 12. The maximum volume-averaged diapycnal velocity w d , max plotted against the trapping ratio γ . The maximum is calculated overInertial Period 1. The size, shape, and shading of the markers are thesame as in Fig. 10. The gray dashed line indicates the linear regression. J O U R N A L O F P H Y S I C A L O C E A N O G R A P H Y F IG . 13. Initial tracer field in the base run (a) and the comparative run(b). Tracer field at t=106 Hr in the base run (c) and the comparative run(d). (e) Time series of the volume-averaged tracer concentration in thebase run and the comparative run. The volume used in the calculationis marked by the green dashed box. The gray dashed line in (e) denotest=106 Hr. highlights the role of the diapycnal velocity in transport-ing the tracer. Also, the convergence can be seen by con-trasting ∂ M ∂ρ at the time when w d is maximum with the oneat the initial time (see the lower panel of Fig. 14). Fur-thermore, the convergence of the tracer persists with time,confirming that the enhanced diapycnal velocity does ef-fectively transport the tracer across isopycnals.The enhancement of the diapycnal transport by wavetrapping has implications for coastal biogeochemistry andecosystems. In coastal zones, freshwater from riversstrengthens the stratification and can suppresses the ven-tilation of bottom waters. This combined with phyto-plankton blooms fueled by nutrients in the freshwater canlead to bottom hypoxia and the formation of ”dead zones”(Bianchi et al. 2010). One region where bottom hypoxiaoften occurs is the Texas-Lousiana shelf where we havedemonstrated that NIW trapping within critical layers ispotentially active. Thus mixing of the stratified bottomwaters by this process could potentially ventilate theseoxygen poor waters. In fact, intrusions of hypoxic watersemanating from slantwise stratified layers near the bot-tom have been observed on the shelf during the MCH sur-vey suggesting active mixing in these layers (Zhang et al.2015).
6. Conclusions
A specific type of NIW critical layer over slopingbathymetry is explored in this study. When isopycnals align with sloping bathymetry in a stratified layer, a criti-cal layer for NIWs with ω = f forms. Upon entering thiscritical layer, the waves are trapped and amplified sincetheir group velocity goes to zero, and mixing is enhanced.Such slantwise critical layers form in a realistic sim-ulation of anticyclonic eddies on the Texas-Louisianashelf. The realistic simulation exhibits an inertial enhance-ment of bottom mixing where the energy from surface-generated NIWs is focused in bottom stratified layers onthe shelf. Idealized ROMS simulations reproduce thesephenomena, and ray-tracing and analyses of the waves en-ergetics support the idea that the enhanced bottom mixingis caused by the convergence of NIW energy in slantwisecritical layers. This conclusion is based on results from anensemble of simulations that cover the relevant parameterspace. The ensemble runs show that background flows thatmore effectively trap wave rays result in stronger wave en-ergy convergence in the critical layer and enhanced mix-ing. Overall, the link between enhanced mixing and wavetrapping is motivated by the realistic simulation, under-stood using the theoretical analyses, and strengthened bythe results from the idealized simulations and ray-tracingsolutions.Although the focus of this study is a particular applica-tion on the Texas-Louisiana shelf, the mechanism of NIWamplification in critical layers over sloping bathymetryshould be active in other settings. For example, anothercoastal application could be upwelling systems over con-tinental shelves, where upwelled, dense waters blanket F IG . 14. (upper) Hovmller diagram of the tracer distribution functionin the density space ∂ M ∂ρ in the base run. ∂ M ∂ρ is calculated over distancesin the cross-shore direction between 50 km to 70 km. (lower) ∂ M ∂ρ att=90 Hr (blue line) and t=106 Hr (red line). The times when ∂ M ∂ρ wasevaluated are indicated in the upper panel. O U R N A L O F P H Y S I C A L O C E A N O G R A P H Y T ABLE
1. Parameters used in the base run (first row) and ensemble runs (second row). α is bottom slope. f is Coriolis parameter. | (cid:126) τ | is theamplitude of the oscillatory, across-slope wind stress. L C , L T , and L are the length scales of the critical layer, transition zone, and offshore front,respectively. ζ is the surface relative vorticity of the offshore front. N is the stratification in the non-rotated coordinates. Only L , ζ , and N varyin the ensemble simulations, and there are a total of 18 ensemble runs. f (s − ) α | (cid:126) τ | (N m − ) L C (km) L T (km) L (km) ζ N (s − )7.27e-05 5.00e-04 4.00e-02 50.0 20.0 40.0 -0.3f 3.00e-03- - - - - (30.0, 40.0, 50.0) (-0.3f, -0.5f, -0.7f) (3.00e-03, 5.00e-03) bathymetry. Potential examples include the upwelling sys-tems over the Oregon shelf (Federiuk and Allen 1996;Avicola et al. 2007), the New Jersey inner shelf (Chant2001), the shelf off of the California coast (Nam and Send2013; Woodson et al. 2007), and the Tasmanian shelf,where recent observations suggest evidence of enhancednear-inertial energy and wave trapping in slantwise criti-cal layers (Schlosser et al. 2019). In these upwelling sys-tems, Federiuk and Allen (1996) highlight the importanceof background flows in modifying the group velocity ofNIWs and attribute the observed enhancement of near-inertial energy during periods of upwelling versus down-welling to wave trapping, similar to the mechanism that wehave described here. However Federiuk and Allen (1996)did not identified the key criterion for NIW critical layerformation–alignment of isopycnals with bathymetry–thatwe have determined from our analyses.Examples of open-ocean flows that can form such crit-ical layers include dense overflows and currents that driveupslope Ekman arrest in bottom boundary layers. One ex-ample of the latter is the Florida Current on the westernside of the Straits of Florida. On this side of the Straits,isopycnals near the bottom tend to align with the conti-nental slope suggesting the existence of a slantwise criti-cal layer, where in fact observations show that turbulencecan be enhanced in stratified layers off the bottom (Winkelet al. 2002). We plan to study the dynamics of these open-ocean NIW critical layers in future work.Diapycnal transport within these critical layers can alsobe enhanced due to turbulence driven by wave trapping.Such diapycnal transport can influence the distributionof biogeochemical tracers such as iron and oxygen andthus potentially influence coastal ecosystems. In the openocean, NIW trapping in critical layers could affect abyssaldiapycnal transport near the bottom, which could mod-ify water mass distributions and influence the meridionaloverturning circulation. Acknowledgments.
This work was funded by the SUN-RISE project, NSF grant numbers OCE-1851450 (L.Q.and L.N.T) and OCE-1851470 (R.H.D). We thank OlivierAsselin, Bertrand Delorme, Jinliang Liu, Jen MacKinnon,Jonathan Nash, Guillaume Roullet, Kipp Shearman, andJohn Taylor for very helpful suggestions when preparingthis manuscript. APPENDIX A
Rotated and non-rotated coordinates
The rotated coordinates are rotated with the angle of θ (considered as positive) to align with the sloping topog-raphy. The relation between the non-rotated and rotatedcoordinates is ˜ y = cos θ y − sin θ z , ˜ z = sin θ y + cos θ z , (A1)where the tildes denote the rotated coordinates so that ˜ y denotes the across-slope direction and ˜ z denotes the slope-normal direction. Assuming that the background buoy-ancy has a slope-normal gradient ˜ N ≡ ∂ b ∂ ˜ z and no across-slope gradient, the horizontal and vertical buoyancy gradi-ents are, then, ∂ b ∂ y = ∂ b ∂ ˜ y ∂ ˜ y ∂ y + ∂ b ∂ ˜ z ∂ ˜ z ∂ y = ˜ N sin θ , ∂ b ∂ z = ∂ b ∂ ˜ y ∂ ˜ y ∂ z + ∂ b ∂ ˜ z ∂ ˜ z ∂ z = ˜ N cos θ , (A2)and therefore M ≡ − ∂ b ∂ y = − ˜ N sin θ , N ≡ ∂ b ∂ z = ˜ N cos θ , (A3)where the sign convention for M of Whitt and Thomas(2013) is used so M is negative within the slantwise crit-ical layer schematized in Fig. 3. Similarly, the horizontaland vertical gradients of the background along-slope ve-locity u are ∂ u ∂ y = ∂ u ∂ ˜ z sin θ , ∂ u ∂ z = ∂ u ∂ ˜ z cos θ . (A4)Noting that u is in the thermal wind balance with the back-ground buoyancy, the vertical shear of u can be obtained,by using Eq.(A2), as ∂ u ∂ z = − f ∂ b ∂ y = − ˜ N f sin θ . (A5)4 J O U R N A L O F P H Y S I C A L O C E A N O G R A P H Y
Given Eq.(A4), the horizontal gradient of u can be thenwritten as ∂ u ∂ y = ∂ u ∂ z tan θ = − ˜ N f sin θ tan θ . (A6)Consequently, the vorticity Rossby number Ro g andRichardson number Ri g can be expressed as Ro g = − ∂ u ∂ y / f = ˜ N f sin θ tan θ , Ri g = ∂ b ∂ z / ( ∂ u ∂ z ) = f ˜ N sin θ tan θ . (A7)APPENDIX B Ray tracing
Rays are calculated by integrating the following equation dz r dy r = s ray = s ρ ± (cid:115) ω − ω min N (B1)where ( y r , z r ) is the position of the ray in the y − z plane.At a certain discrete location of the path ( y rn , z rn ) , it is pos-sible to calculate s ρ , ω min , and N , thus, the slope of thepath s ray can be obtained for a wave of frequency ω . Witha small change in y , δ y r = y rn + − y rn , the next vertical lo-cation of the ray z rn + can be estimated as z rn + = (cid:90) y rn + y rn s ray dy r + z rn ≈ s ray | ( y rn , z rn ) δ y r + z rn . (B2)Starting at an initial point, recursively calculating Eq. B1and B2 will trace out the path of a wave packet. References
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