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    Depth–time maps of the zonal component of shear obtained during the HOME Farfield Experiment. An eight-beam Doppler sonar suspended at a depth of 375 m below the Research Platform FLIP produced these data. (a) The data presented in an Eulerian frame. Slowly varying bands of low-frequency shear are seen. They are advected vertically by the semidiurnal tide. Classical “WKB stretching” will increase the magnitude of the deep shear relative to the shallow shear and alter apparent vertical wavelengths. It does not address the issue of vertical advection. (b) The same data are presented as a function of time and isopycal density; that is, shear fluctuations are displayed on a set of isopycnals whose mean depths are separated by fixed Δz (1 m here). The ordinate gives mean isopycnal depth. In this semi-Lagrangian frame, the effects of advective distortion are greatly reduced. The vertically sloping crests indicate diapycnal phase propagation. The crests progress with remarkable steadiness, suggesting a multiday passage time for many of the wave groups. Despite the baroclinic tidal activity at the site, the shear is predominately near inertial. The noisy portion of the shallow Eulerian record results from sonar side-lobe hits on FLIP’s hull and the sea surface. The near-surface SL record cannot be defined in the mixed layer. The entire Farfield record extends for 27 days.

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    (a) A wavenumber–frequency spectrum of Arctic Ocean shear estimated from the SHEBA Doppler sonar. The white rectangle indicates the smoothing used to achieve 150 degrees of freedom. The left quadrants of the spectrum correspond to anticyclonic rotation in time. The lower-left and upper-right quadrants represent upward phase propagation, which for the internal-wave component of the shear implies downward energy propagation. (b) The associated normalized spectrum N(κz, σ) = E(κz, σ)/∫E(κz, σ) . First (white) and second (black) spectral moments are indicated. Color contours mimic an hourglass pattern. (c) The Garrett and Munk (1975) spectrum of (internal wave) shear fails to capture the essential structure of the observations. In an N frame (vs the present Eulerian frame) the disagreement would be much less striking. (d) Modeled log10 shear spectral density for August record, contoured vs linear wavenumber and observed frequency. The contour interval is 5 dB. The model gives the shear variance distribution that results when line spectra at intrinsic inertial and vortical (zero) frequency are broadened by Doppler shifting. The modeled spectrum is smoothed by convolution with the rectangular block shown in white in the lower left of the figure. Variance associated with waves propagating through a steady shear will both migrate in wavenumber and move diagonally across the spectral quadrants (black arrow). Time changes in sensor or depth-mean velocity lead to variance migration at constant κz (white arrow). Propagation through a depth-varying buoyancy profile should cause no spectral alteration, because the data have been WKB stretched prior to spectral analysis. Traditional estimates of the ratio of clockwise shear to counterclockwise shear variance are formed from spectral levels at like |σ| and |κz| in diagonally opposite quadrants.

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    Model sensitivity studies. Drawings show hypothetical “observed” Eulerian frequency spectra corresponding to an initial narrowband (blue line) spectrum that is subject to (a) horizontal (green), (c) random vertical (red–violet), and (e) deterministic (black) vertical advection. The logarithm of shear variance density is plotted vs linear frequency. Note that the shape of the cusp in (c) reflects the shape of the associated correlation function of vertical advection, as specified in Eqs. (17) and (18). (b), (d), (f) The observed Arctic shear frequency spectrum for August–September 1998 is given by the blue line. This is modeled as a sum of contributions from intrinsically vortical (ω = 0) and inertial (ω = −f ) spectral lines (Δω = 0). The vortical contribution to the modeled frequency spectrum is given in green, the near-inertial contribution is in red, and the sum spectrum is in black. In (b), the effect of lateral Doppler shifting alone is plotted as a solid line at the best-fit aspect ratios B1 and B2 and at aspect ratios that are smaller by a factor of 2 (dashed). In (d) and (f), the lateral Doppler shift values are held at their best-fit levels and vertical Doppler effects are added. In (d), τη is set at 2 h and ση is fixed at 1.25 m (solid line) and 0.625 m (dashed). In (f), ση is set at 1.25 m, and τη is given for 2 h (solid) and 1 h (dashed). There is no need to account for deterministic tidal advection [as in (e)] in the western Arctic. The frequency spectra in (b), (d), and (f) are computed from integrals over wavenumber of the observed (Fig. 2a) and modeled (Fig. 2d) wavenumber–frequency spectrum, for σ < 0.

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    Observed frequency spectra of (a) shear (cph) (blue line) and (b) velocity and shear (blue lines) formed from integrations of E(κz, σ) over wavenumber. Only the anticyclonic portion of each spectrum is shown in the logarithmic plots in (b). The black line gives the corresponding model fits, including both wave (red) and vortical (green) contributions. (c) The associated vertical wavenumber spectrum of observed shear (blue lines) showing the modeled (black), inertial (red), and vortical (green) constituents, integrated over both positive and negative frequencies. The vortical spectrum approximates a κ0z form. It is isotropic with respect to the sign of the wavenumber, as expected for a nonpropagating phenomenon. At wavenumbers greater than about 0.05 cpm, Doppler-shifted inertial waves dominate the σ = 0 band, rendering the vortical–inertial separation imprecise. The wave spectrum is band limited in wavenumber, with an excess of variance associated with upward phase propagation (κz < 0). At the smallest wavenumbers |κz|, the model accounts for only 50% of the observed variance. The difference results from an inability of the model to advect large-scale motions to high encounter frequencies. The inclusion of a second inertial field, with larger aspect ratio and larger scale, would improve the fit considerably.

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    (a), (b) Cross sections of the observed wavenumber–frequency spectrum of shear for the August record (blue) and model predictions (black) at frequencies −f, −2f, . . . , −5f for anticyclonic motions. The modeled inertial spectral level is set equal to the observed level at σ = −f. The model accurately replicates the frequency bandwidth of the spectrum at vertical scales <25 m. It fails dramatically at scales >50 m. The energy observed at low wavenumber and high frequency significantly exceeds that which results from Doppler smearing of the inertial and vortical peaks. The mismatch would be reduced if the assumed inertial peak was assigned an intrinsic frequency bandwidth Δω. (c), (d) Cross sections of the modeled vortical (green) and inertial (red) components of the shear field at observed frequencies 0, −f, . . . , −5f for the August record.

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    Contour plot of the logarithm of the normalized wavenumber–frequency spectrum N(κz, σ) for August–September 1998, plotted vs linear frequency and wavenumber. Both (top) the observations and (bottom) the model depict the increasing frequency bandwidth of the spectrum with increasing wavenumber. The observations have variance at low wavenumber and intermediate frequency, outside the hourglass, that is not reproduced in the model.

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    Wavenumber–frequency spectra of shear obtained in (top) Eulerian and (bottom) semi-Lagrangian frames, from the HOME Farfield data of Fig. 1. Spectra are presented at 30 degrees of freedom. White reference lines are plotted at −f ± nM2, for small integer |n|. The spectral variance associated with the harmonic lines is distributed through a broad range of intermediate wavenumbers in the Eulerian frame.

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    (a) Frequency spectra of anticyclonic shear and velocity formed by integration of the 2D spectrum EM(κz, σ) of Fig. 7 over wavenumbers −0.1 < κz < 0.1 cpm. Eulerian and SL estimates exhibit greater differences in shear than in velocity, which emphasizes the largest vertical scales. The Eulerian spectra are reduced at low- and high-frequency extremes relative to the SL. Deterministic tidal advection causes both the appearance of harmonics and the elevation of the midfrequency region of the Eulerian spectrum. Reference lines indicate the presence of advective harmonics associated with f ± nM2. Note that these are distinct from the intrinsic tidal harmonics at frequencies ±nM2 that are barely visible in the Eulerian and SL velocity spectra but are totally absent in the shear. Other harmonics associated with diurnal motion are also apparent. Both SL spectra display a precutoff rise and a high-frequency cutoff at the buoyancy frequency. (b) Vertical wavenumber spectra based on the integration of the 2D spectrum E(κz, σ) over frequencies −7.5 < σ < 7.5 cph. The spectra are band limited, rising as κ1/2z to κ1z for κz < 0.025 cpm and falling as κ−1z for κz > 0.05 cpm. The advection model operates on a wavenumber-by-wavenumber basis and does not transport variance across wavenumbers. The observations, under WKB stretching, reflect this property. The similarity in wavenumber spectra seen in (b) is not obvious in Figs. 1a,b.

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    The behavior of the spectrum across the cyclonic–anticyclonic boundary at σ = 0 constrains the aspect ratio of the Hawaii broadband inertial motions [(a) Eulerian; (b) semi-Lagrangian]. With these quantities fixed, the random vertical advection amplitude and correlation time are adjusted to match the high-frequency form of the Eulerian observations [in (a) here and Figs. 10a,c)]. The SL frame is presumably advected with these vertical motions. The various harmonic lines are absent in both the SL data and the model. Shear and velocity spectra are presented in logarithmic format for (c) Eulerian and (d) SL data. The model replicates the shear spectra in both frames, except at high frequency, where intrinsically high-frequency motions [seen in (d) as the precutoff spectral shoulder] are shifted to both lower and higher frequencies [(c)]. The arrows indicate frequencies M2 and 2M2, which are visible in the velocity spectra but not in the shear. Identical background parameters are used in (a)–(d) with a few exceptions. Tidal amplitudes are set to zero in the SL frame. Modest random vertical displacement 〈η21/2 = 0.75 m and τη = 0.07 h is included in the SL model to mimic the effect of error in determining true isopycnal depth.

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    Model fits to the observed (left) Eulerian and (right) SL frequency spectrum of shear. Narrowband inertial motions (red) and broadband near-inertial motions (green) combine to form the model result (black). The widths of the inertial peak and the harmonics (Eulerian) strongly constrain the aspect ratio of the narrowband motions and the assumed magnitudes of the M2 and D1 tidal constituents. Reference lines indicate the position of the harmonics associated with M2. The unmarked peaks represent the D1 contribution. The model has no means of transferring near-inertial variance to high encounter frequency in an SL frame. The implication is that the observed high-frequency variance is in fact real and is not an artifact of the advection of near-inertial motions.

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    Normalized wavenumber–frequency spectra N(κz, σ) for (a), (b) Eulerian and (c), (d) SL analyses of HOME data. The model spectra (b), (d) replicate the hourglass nature of the observations. The Eulerian model produces an idealized approximation to the spectral harmonics, mimicking the appearance of sharp harmonics in frequency at intermediate vertical wavenumbers. Reference lines indicate the position of the M2 harmonics. The SL model fails to replicate the high-frequency variance of the observations, suggesting an intrinsic reality to these motions.

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Advection, Phase Distortion, and the Frequency Spectrum of Finescale Fields in the Sea

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  • 1 Marine Physical Laboratory, Scripps Institution of Oceanography, La Jolla, California
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Abstract

Continuous depth–time measurements of upper-ocean velocity are used to estimate the wavenumber–frequency spectrum of shear. A fundamental characteristic of these spectra is that the frequency bandwidth increases linearly with increasing wavenumber magnitude. This can be interpreted as the signature of Doppler shifting of the observations by time-changing “background” currents as well as by instrument motion. Here, the hypothesis is posed that the apparently continuous wavenumber–frequency spectrum of oceanic shear results from the advective “smearing” of discrete spectral lines. In the Arctic Ocean, lines at the inertial (ω = −f ) and vortical (ω = 0) frequencies (where f is the Coriolis frequency) account for most of the variance in the shear spectrum. In the tropical ocean, two classes of inertial waves are considered, accounting for 70% of the observed shear variance. A simple model is introduced to quantify the effects of lateral advection, random vertical advection (“fine-structure contamination”), and deterministic (tidal) vertical advection on these “otherwise monochromatic” records. Model frequency spectra are developed in terms of the probability density and/or spectrum of the advecting fields for general but idealized situations. The model successfully mimics the increasing frequency bandwidth of the shear spectrum with increasing vertical wavenumber. Excellent fits to the observed frequency spectrum of shear are obtained for the Arctic (weak advection and short-spatial-scale inertial waves) and low-latitude (strong advection and long and short inertial waves) observations. While successfully replicating the wavenumber–frequency spectrum of shear, the model does not even consider motion at scales greater than ∼250 m, the “energy containing” scales of the internal wave field. To a first approximation, the waves with the majority of the kinetic and potential energy constitute a population apart from those with the momentum, shear, and strain.

Corresponding author address: Robert Pinkel, Marine Physical Laboratory, Scripps Institution of Oceanography, La Jolla, CA 92093-0213. Email: rpinkel@ucsd.edu

Abstract

Continuous depth–time measurements of upper-ocean velocity are used to estimate the wavenumber–frequency spectrum of shear. A fundamental characteristic of these spectra is that the frequency bandwidth increases linearly with increasing wavenumber magnitude. This can be interpreted as the signature of Doppler shifting of the observations by time-changing “background” currents as well as by instrument motion. Here, the hypothesis is posed that the apparently continuous wavenumber–frequency spectrum of oceanic shear results from the advective “smearing” of discrete spectral lines. In the Arctic Ocean, lines at the inertial (ω = −f ) and vortical (ω = 0) frequencies (where f is the Coriolis frequency) account for most of the variance in the shear spectrum. In the tropical ocean, two classes of inertial waves are considered, accounting for 70% of the observed shear variance. A simple model is introduced to quantify the effects of lateral advection, random vertical advection (“fine-structure contamination”), and deterministic (tidal) vertical advection on these “otherwise monochromatic” records. Model frequency spectra are developed in terms of the probability density and/or spectrum of the advecting fields for general but idealized situations. The model successfully mimics the increasing frequency bandwidth of the shear spectrum with increasing vertical wavenumber. Excellent fits to the observed frequency spectrum of shear are obtained for the Arctic (weak advection and short-spatial-scale inertial waves) and low-latitude (strong advection and long and short inertial waves) observations. While successfully replicating the wavenumber–frequency spectrum of shear, the model does not even consider motion at scales greater than ∼250 m, the “energy containing” scales of the internal wave field. To a first approximation, the waves with the majority of the kinetic and potential energy constitute a population apart from those with the momentum, shear, and strain.

Corresponding author address: Robert Pinkel, Marine Physical Laboratory, Scripps Institution of Oceanography, La Jolla, CA 92093-0213. Email: rpinkel@ucsd.edu

1. Introduction

In this work, the effect of advection on the frequency spectrum of observations is explored through a series of simple models. The goal is to modernize and unify existing recipes for “fine-structure contamination” (Phillips 1971; Garrett and Munk 1971; McKean 1974) and then to reexamine the apparently continuous frequency spectrum of internal wave and subinertial finescale fields. The effort proceeds in the spirit of Holloway (1981), who emphasized that small-scale internal waves and quasigeostrophic structures share the same wavenumber–frequency domain, given the expected magnitude of “Doppler smearing” in the sea. Such smearing cannot, in general, be unscrambled, but the task is much easier if the spectrum consists of a few discrete lines. To what extent does a line or narrowband process yield a continuous frequency spectrum when viewed in an Eulerian frame?

This effort complements a recent thrust in atmospheric research, in which both dynamic (Hines 1991; Chunchuzov 1996, 2002) and kinematic (Eckermann 1999; Sica and Russell 1999; Klaassen and Sonmor 2006) efforts to model the vertical wavenumber spectrum of horizontal velocity have been developed. Here, the frequency spectrum associated with any given wavenumber spectrum is modeled using a purely statistical approach.

To quantify the effects of advection, it is attractive to begin in a frame in which linear motions produce sinusoidal signals in motion or tracer fields. For a continuous wavenumber spectrum of energetic internal waves, no single “ideal” observing frame exists, although some are clearly better than others (e.g., Andrews and McIntyre 1978). Here, this sensitive issue is sidestepped. The model posits the existence of a “nonexistent frame” N in which measured velocities and tracer trajectories evolve sinusoidally in space–time. A sequence of discrete transformations maps monochromatic variance in the N frame to its spectrally continuous Eulerian frame representation.

To motivate the discussion, consider depth–time maps of oceanic shear obtained during the Hawaii Ocean Mixing Experiment (HOME) Farfield Experiment south of Hawaii (Fig. 1). The upper panel gives the shear in an Eulerian frame. The shear is predominately near inertial. However, pronounced vertical displacement of shear layers is seen, associated with the semidiurnal tide. The distortion will result in tidal and harmonic peaks in the Eulerian frequency spectrum of shear. In the lower panel, the same shear field is presented as a function of time and density; that is, shear fluctuations are displayed on a set of isopycnals whose mean depths (separated by 1 m) are given by the ordinate in the figure. The shear signal is found to be predominately inertial. Continuous patterns of upward- and downward-propagating phase are now seen. However, still included in this “semi Lagrangian” (SL) record are the effects of lateral (isopycnal) advection. For wave-induced currents, lateral advective effects are comparable in magnitude to those resulting from vertical advection.

We are challenged to obtain data that are “uncontaminated” by advective effects1 and/or to quantify the effects found in our real-world observations. In an effort toward the latter goal, it is attractive to imagine an N frame populated only by near-inertial and tidal baroclinic waves and by small-scale quasigeostrophic (vortical) motions. The near-inertial and vortical motions provide all of the shear variance but have little vertical displacement. The tidal motions have the vertical displacement but supply negligible shear. Starting with this simple motion field, the goal is to predict its spectral signature in SL and Eulerian frames.

Real-world observations depart from the N-frame ideal through advective effects, which appear as phase distortions of the sinusoidal signals; that is, sEul = s0 exp[iκ · (x + Vt) − ωt]. There are four cases to consider: small versus large phase distortion (∫|κ · V| dt > vs < π) and stochastic versus deterministic distortion velocity V. The models introduced here relate the probability density function and/or the spectrum of the advecting velocity field V to the spectrum of the observed quantity. The approach follows a path suggested by Papoulis (1984). The focus is on the temporal autocovariance function R(τ), which, for a line spectral process (in the N frame), takes the form RN(τ) = 〈s2〉 exp(−iωτ). When this motion field is observed in a realizable frame, advective effects modify the correlation function in a multiplicative fashion: Robs(τ) = RN(τ) × (lateral advection modification) × (vertical advection modification). The corresponding observed frequency spectrum is given by the N-frame spectral line repeatedly convolved with functions that account for these effects.

2. Background: The wavenumber–frequency spectrum of shear

The horizontal currents associated with near-inertial internal waves rotate both in time and in depth and are naturally described by the two-dimensional rotary power spectrum. To estimate the spectrum, currents are represented as complex data (ueast + north) and are Fourier transformed in both depth and time. The resulting rotary spectrum consists of four independent quadrants, specified by the signs of the associated wavenumbers and frequencies. In the convention used here, the negative frequency quadrants are associated with anticyclonic rotation in time. The shear spectral maximum occurs at the inertial frequency σ = −f, where f = 2Ω sin(ϕ). Here, Ω is the earth’s rotational frequency and ϕ is latitude.2

To illustrate the signature of advection, consider a wavenumber–frequency spectrum of shear obtained from a down-looking 161-kHz sonar deployed in the western Arctic Ocean (79°N, 157°W) as an aspect of the Surface Heat Budget of the Arctic Ocean (SHEBA) experiment. A 19.3-day shear record in August–September 1998 is “Wentzel–Kramers–Brillouin (WKB) stretched” and Fourier transformed in depth (52–273 m) and in time to produce this spectral estimate (see Pinkel 2005). When the spectrum is contoured on linear frequency and wavenumber axes (Fig. 2a) the apparent frequency bandwidth
i1520-0485-38-2-291-eq1
is seen to increase linearly with increasing wavenumber. Here E(κz, σ) is the wavenumber–frequency spectrum of shear (s−2 cpm−1 cph−1), with κz being the vertical wavenumber and σ being the observed frequency. Frequency σ (κz) is the first spectral moment—the “mean frequency.”
The bandwidth pattern is more apparent when the spectrum is normalized to have identical variance in each wavenumber band:
i1520-0485-38-2-291-eq2
(Fig. 2b). Here a characteristic “hourglass” pattern is seen in the spectral contours. The first moment of the spectrum identifies the inertial frequency as the principal peak. The second moment increases linearly with vertical wavenumber. This pattern is also found in midlatitude open-ocean spectra (Sherman and Pinkel 1991; Fig. 7, top).

Note that the normalized spectrum N(κz, σ) is common to all vertical derivatives or integrals of the shear, including, for example, the velocity field, Thus the focus of this study is on N(κz, σ) and the attempt to model it in a meaningful way.

It is the author’s conjecture that, in an N frame, the dominant shear in these observations is near-inertial and vortical (quasigeostrophic; Holloway 1981; Müller 1988). Each band is assumed to have a fixed aspect ratio = κH/κz. Thus any given advective velocity field V will Doppler “shift” the intrinsic lines through a range of frequencies that increases linearly with increasing wavenumber magnitude.

3. Background: Doppler shifting

It is of value to review several of the many possible views of Doppler shifting. In the landmark studies of wave propagation through spatially varying, steady shear fields (e.g., Booker and Bretherton 1967; Bretherton and Garrett 1968), a principal finding is that the waves are not Doppler shifted. Wave frequency σ, as determined by a stationary observer, does not change as the packet propagates vertically through a time-invariant shear. Wave energy does vary, and the concept of wave action, A = E/ω, is introduced as an analog of energy that is conserved. Here E represents the total energy associated with a wave packet and ω is the intrinsic frequency of the packet—the frequency as determined by an observer drifting with the background flow at the altitude/depth of the packet.

It is now appreciated that much of the shear in the ocean is associated with near-inertial waves, rather than mesoscale motions. The “background shear” often propagates more rapidly than the vertical group velocity of smaller internal waves (Broutman 1984). With shears that vary in both space and time, there is no reference frame in which all waves are seen to fluctuate sinusoidally. Changes in observed wave frequency σ, intrinsic wave frequency ω, and energy are typically linked in a manner that renders the prediction of spectral evolution challenging (e.g., Warner and McIntyre 1996).

In this work, with apologies to the pioneers of wave physics, I consider processes that alter the apparent frequency of observed wave groups σ and neglect all subsequent consequences of space–time variability in the environment. This “kinematic Doppler shifting” is an accurate model when the ocean background is unchanging in time but the instrument location migrates through the ocean at velocity V.

The justification for introducing such a “retro” model of Doppler shifting is that it leads to a view of the oceanic wave field that is simpler than previously thought, although the simplicity can only be observed in the physically reasonable, but unfortunately nonexistent, N frame.

Is this exercise of value? Legitimate dynamical investigations often posit the existence of an “initial” or “background” spectrum. The subsequent evolution of the spectrum (e.g., Warner and McIntyre 1996), energy flow through the spectrum (Polzin 2004), or refraction of a “test wave” (Henyey et al. 1986) is then quantified. If a reasonable background can, in fact, be represented by a few discrete spectral lines, the modeling task might be simpler. The relations developed here can be used to compare initial and final model states as they would be seen in a real-world reference frame, where neither state might appear monochromatic.

4. Background: A rotary spectral view of wave evolution

As a final preliminary, it is useful to review the signature of kinematic and dynamic Doppler shifting in a rotary spectrum. At the inertial frequency, wave particle orbits are circular, confined to the horizontal plane. The orbit rotation is in an anticyclonic sense, at frequency −f. At higher wave frequencies, horizontal particle orbits become progressively more elliptical. The variance of a wave packet becomes divided between positive and negative frequencies. For |ω| ≫ | f |the variance division approaches an even split. However, the variance is divided across diagonally opposite quadrants of the spectrum such that the sign of the vertical phase speed cz = σ/κz is consistent (Fig. 2).

As a thought experiment, consider a near-inertial wave, ω = σobs ∼ −f < 0, that propagates downward from the sea surface through a localized shear layer and into a steady, horizontally opposing current. Prior to passing through the shear layer, all wave variance is found in spectral quadrant III (σobs < 0, κz < 0). Following transmission through the layer, wave intrinsic frequency |ω| is increased, wave orbits become more ellipsoidal, and wave variance is found near both negative and positive f. The variance has made a transition diagonally across the spectrum with, ideally, no change in the observed wave frequency σobs.

As a contrasting case, consider a near-inertial packet propagating through a steady shear-free ocean. The observing platform, for example, sea ice, advects randomly relative to the ocean below. Advection velocities |V| of 0.1–0.25 m s−1 are common in the Arctic, and these speeds are comparable to the horizontal phase speeds of the waves. If the ice “overruns” a wave packet, the apparent frequency σobs can be Doppler shifted to zero and can even change sign. The apparent sense of vertical phase propagation and temporal rotation both reverse. In the extreme limit |V| ≫ |cwave|, the packet is effectively “frozen in space.” The apparent frequency, direction of temporal rotation, and direction of vertical propagation depend on the horizontal scale of the packet and its orientation relative to the direction of traverse.

This kinematic Doppler shifting spreads spectral variance between horizontally adjacent quadrants of the rotary spectrum. The same random advective path will result in a greater Doppler shift for smaller-scale waves than for larger. The hourglass pattern of Fig. 2b could be a signature of this process, provided that the aspect ratio B = kH/κz is approximately constant across all wavenumber bands—that is, the waves share a common intrinsic frequency.

5. A model of advective phase distortion

The frequency bandwidth of N(κz, σ) appears to increase linearly with increasing wavenumber magnitude. Can an extremely simple advection model capture the essential features of this spectrum?

The approach is fairly primitive. Consider a sinusoidal signal that is advected by a number of mutually independent processes {Vj}:
i1520-0485-38-2-291-e1
The temporal covariance of this takes the form
i1520-0485-38-2-291-e2
The Fourier transform of this covariance yields the modeled frequency spectrum of the signal: a delta at frequency ω that is sequentially convolved with the Fourier transforms of the advective terms in the covariance.

The rules of the model are as follows:

  1. All of the observed shear variance is concentrated in a few discrete spectral lines in the N frame. For the Arctic observations used here, lines at the intrinsic inertial (ω1 = −f ) and vortical (ω2 = 0) frequencies are hypothesized. For the low-latitude observations, two families of intrinsic inertial waves (ω1 and ω2 ∼ −f ) are hypothesized, with differing aspect ratios.

  2. The observed shears are distorted by larger-scale currents and by the motion of the observing platform. The east, north, and vertical components of the advecting current are assumed to be independent. Although some degree of self-advection (the advection of waves by waves) is tolerable, the mathematical independence of the advecting and advected motions is required.

  3. Advective distortion is proportional to κ · Vj. Modeling this effect requires some knowledge of signal wavenumber κ. Doppler sonar provides data continuous in depth and time, leading to a description of motion in terms of vertical wavenumber κz and encounter frequency σ. To treat horizontal variability, each hypothesized contributor to the intrinsic motion field (inertial, vortical, etc.) is assigned a fixed aspect ratio Bi ≪ 1. The aspect ratio represents a tunable parameter in the model.

  4. The model assigns an Eulerian frequency dependence NMi(κz, σ) to the normalized spectrum associated with each spectral line Nint = δ(ωωi) in the N frame. In modeling horizontal advection, this exercise provides an estimate of the aspect ratio Bi of each constituent of the motion field. To produce a (nonnormalized) shear or velocity spectrum, it is necessary to specify the wavenumber spectrum, Eni(κz), associated with each constituent i such that EN(κz, ω) = ΣiEni(κz) δ(ωωi).

  5. The model is applicable for arbitrary Eni(κz) and must be normalized such that ∫EMi(κz, σ) = Eni(κz). Further, the vertical wavenumber spectrum summed over all constituents, ΣiEni(κz) = ∫EM (κz, σ) = E(κz), should equal the observed vertical wavenumber spectrum. Discrepancies indicate variance not ascribable to the hypothesized lines.

  6. The partitioning of E(κz) among a number of constituents represents a subjective aspect of this model. In the Arctic observations, the relative magnitudes of Einertial(κz) and Evortical(κz) are constrained by requiring the model to match the data at the observed frequencies σ = 0 and −f. For the low-latitude observations, the partitioning of variance between “traveling” (B1 < 1) and “local” (B2B1 < 1) inertial waves is more subjective.

  7. The notation EN(κz, ωi) = Eni(κz)δ(ωωi) is used to denote the underlying intrinsic spectrum (in the N frame) of constituent i of the shear field, E(κz, σ) represents the spectrum as observed, and EMi(κz, σ) is the modeled frequency dependence associated with the ith spectral line, Eni(κz).

Under these rules, the simple Doppler-shifting model of Papoulis (1984, example 10.4) can be applied to calculate the resulting Doppler spectrum.3 The advected signal is represented as in Eq. (1), with V treated as either a random variable or a known, deterministic quantity. The goal is to produce insightful models of 〈exp(iκ · Vjτ)〉 appropriate for different advective signals j. There are three cases of interest:

  1. For random V, if excursion distances are typically large relative to a wavelength, that is, ∫|κH · V| dtπ, the random phase introduced by the Doppler shift is presumably distributed uniformly over [−π, π]. This is a reasonable model of the lateral Doppler shifting of “otherwise monochromatic” observations.

  2. The contrasting case is one in which ∫|κ · V| dtπ and a small-angle approximation can be made. This approach is appropriate when considering the vertical advection of waves by waves: the so-called fine-structure contamination problem (e.g., Phillips 1971; Garrett and Munk 1971; McKean 1974).

  3. With V deterministic (say, the M2 tidal vertical velocity), we can model the advection of inertial shears by tidal displacements (Fig. 1a).

The three cases are treated separately in the following sections.

a. Random phase distortion, ∫|κH · V| dt ≫ π, and lateral Doppler shifting

Consider a signal represented as a linear superposition of advected sinusoids in the xt domain:
i1520-0485-38-2-291-e3
where ω represents the frequency observed in an N frame, κH is the horizontal wavenumber, and V is a random horizontal advection speed, with associated probability density PV(V). The signal autocovariance is
i1520-0485-38-2-291-e4
Here it has been assumed that the phases of the process Fourier coefficients are independent of the advecting velocities V and that 〈s(κ1, ω1)s*(κ2, ω2)〉 = E(κ1, ω1)δ(ω1ω2)δ(κ1κ2) defines the process wavenumber–frequency spectrum. Each realization of this process represents a continuous segment of time of length T > τ during which the signal s(x, t) is presented with a fresh phase and an independent, constant V. With this “segment by segment” realization scenario, the ensemble average of the advection term is simply the characteristic function of the random variable V:
i1520-0485-38-2-291-e5
The observed frequency spectrum of s is the Fourier transform of the autocovariance, from Eq. (4):
i1520-0485-38-2-291-e6
Thus, the spectrum of s in the advected frame is just the convolution of the spectrum of s in an N frame with the κ-scaled probability density function of the advecting velocities. For a single, randomly advected plane wave, E(κH, ω) = E0δ(ωω0)δ(κHκ0), the observed spectrum is just
i1520-0485-38-2-291-e7
The associated covariance is identified with the characteristic function:
i1520-0485-38-2-291-e8
For Gaussian PV,
i1520-0485-38-2-291-e9
where μtot is the standard deviation of the advecting flow V.

The spectral width of a Doppler-shifted sinusoid depends on both the width of the probability density function of the advecting velocity (the variance of V, μ2tot) and the wavenumber κ0 of the sinusoid.

For advection in two dimensions, the expression is easily extended, provided that the flow components are independent. Each is associated with an 〈exp(iκ · Vjτ)〉 term in Eq. (2). For an isotropic, Gaussian advecting field, Eq. (9) is recovered, with μ2tot as the sum of the component variances.

b. Random phase distortion, ∫|κH · V| dt ≪ π, and vertical Doppler shifting

Vertical advection by high-frequency internal waves serves to distort Eulerian observations of lower-frequency fields. The advection is bounded in that isosurfaces do not random-walk arbitrarily far from their mean depths. A vertical advection model is needed to spread inertial and vortical shear across the internal wave band, to encounter frequencies comparable to the buoyancy frequency. In the special (yet common) case in which vertical advection results in phase fluctuations that are small relative to π, the model yields relationships appropriate for fine-structure contamination. For Eq. (3), one can define a vertical displacement random variable η, with associated depth difference Δη = η(t) − η(t + τ). Expanding the characteristic function of Δη for small κzΔη yields
i1520-0485-38-2-291-e10
Here, 〈Δη〉 = 0 and ρ2Δη(τ) is the variance of the displacement difference occurring over a time lag τ. To be specific,
i1520-0485-38-2-291-e11
Defining the displacement correlation function as
i1520-0485-38-2-291-e12
we write
i1520-0485-38-2-291-e13
Thus the characteristic function appropriate for small random phase distortion is
i1520-0485-38-2-291-e14
Last, the covariance function of the advected signal is, from Eq. (4),
i1520-0485-38-2-291-e15
The observed frequency spectrum is
i1520-0485-38-2-291-e16
For a single, randomly advected plane wave, E(κz, ω) = E0δ(ωω0)δ(κzκ*), the observed spectrum is just
i1520-0485-38-2-291-e17
Here,
i1520-0485-38-2-291-e18
(m2 cph−1) is the frequency spectrum of the advecting displacement η, centered at the intrinsic frequency of the distorted wave ω0.
Equation (17) describes a delta line spectrum diminished in amplitude by the factor (1 − κ2*η2〉). The “missing variance” is redistributed in a continuous “skirt” or “cusp” that takes the form of the spectrum of the advecting displacement η centered at the intrinsic line. This result applies to a wide variety of small-phase distortion problems, including the phenomenon of tidal cusps as noted by Munk and Cartwright (1968) and Colosi and Munk (2006). The covariance of this distorted sinusoid is, from Eq. (15),
i1520-0485-38-2-291-e19

c. Deterministic phase distortion: Vertical advection by baroclinic tides

In modeling deterministic advection, one can simply evaluate s(x, t) = s0 exp[i(κ · xωtκ · Vt)] for known V to determine the corresponding frequency spectrum of s (Alford 2001). For sinusoidal advection, the analytic representation of the spectrum assumes a classical form.

Consider a monochromatic signal at frequency ω that is advected vertically such that
i1520-0485-38-2-291-e20
Let ηtide(t) = ηt sin(ωtt + ϕ), where ηt is the tidal amplitude, ωt is tidal frequency, and ϕ specifies the phase of the tide relative to the start time of each realization.
The autocovariance of s is
i1520-0485-38-2-291-e21
Here, Γ(τ) = 2κzηt sin(ωtτ/2). The apparent time dependence in Eq. (21) reflects the fact that individual realizations of the signal are dependent on the phase of the advecting tide. Randomness enters this otherwise deterministic description through s0 and ϕ. From realization to realization, ϕ can be considered as uniformly distributed, and
i1520-0485-38-2-291-e22
Thus
i1520-0485-38-2-291-e23

Here, θ = ωtt + ϕ and J0(Γ) is the zeroth order Bessel function. Thus a monochromatic process in the N frame yields a multipeaked spectrum in a sinusoidally advected frame. The central peak is centered at the intrinsic frequency ω, and the harmonic pattern is set by the Fourier transform of J0(Γ).

Note that no “small angle” approximation has been made. In a continuous wave field, it makes sense that the high-vertical-wavenumber constituents (κzηt > π) have a different harmonic pattern than their low-wavenumber counterparts. The fact that the Fourier transform of J0(Γ) describes this intricate process at all scales is perhaps not obvious. The Hawaiian observations, with large baroclinic tidal displacements, are nearly ideal for examining this process.

6. Fitting the observations

Two sets of observations will be used to test the model and the “monochromatic spectrum” hypothesis. Doppler sonar observations from the SHEBA ice camp (79°N, 157°W; 1997–98; Uttal et al. 2002) in the western Arctic represent a situation of extremely low advection speeds, with a large separation in frequency between inertial (ω = −f ) and vortical (ω = 0) motions. A 161-kHz sonar produced velocity profiles with depth resolution of 3.7 m. The observed E(κz, σ) and N(κz, σ) (Fig. 2) are produced from a 19.3-day record over depths of 52–273 m. Data collection and processing are discussed in Pinkel (2005).

To investigate deterministic vertical advection, shear data from the HOME Farfield Experiment are examined. These data were collected from the Floating Instrument Platform (FLIP) in 2001 while moored about 430 km south-southwest of Oahu, Hawaii (18.39°N, 160.7°W; Figs. 1a,b). The HOME data were collected by an upward-/downward-directed eight-beam Doppler sonar that was suspended 375 m below the surface. The sonar profiled from the surface to ∼800 m with 4.0-m depth resolution. Sonar measurements were augmented by a CTD profiling program. Approximately 9400 profiles were obtained between the sea surface and 780 m over the five weeks of the experiment. An SBE 911 CTD was used. Profiling at 3.7 m s−1, depth resolution is of order 2 m in density. Sonar-derived shear data are interpolated at both fixed depths and fixed densities to create the Eulerian and SL records used here (Fig. 1; also Rainville and Pinkel 2006a, b). To estimate the rotary wavenumber–frequency spectrum, complex velocity data (ueast + north) from the depths 101–340 m are WKB stretched, first differenced in depth, and Fourier transformed in depth and time. The squared Fourier coefficients are convolved with a uniform (boxcar) weight in both vertical wave-numberand frequency to produce statistically stable spectral estimates.

To minimize the number of parameters, and despite excellent evidence to the contrary (Munk and Phillips 1968; Garrett 2001), the intrinsic frequency bandwidth Δω of spectral lines in the N frame is assumed to be zero. Because the model distributes variance over frequency ranges that are proportional to wavenumber, it has no mechanism for generating a finite-frequency bandwidth at the lowest wavenumbers (vertical scales ≫ 100 m). This omission is forgivable if one is focused on the description of vertical shear (Figs. 2a,d; Fig. 7). It proves disastrous when modeling velocity and displacement fields, for which the primary variance is associated with long wavelength motions. {Here, model spectra are “frequency smoothed” with the same “boxcar” convolution window that is used to increase the statistical precision of the data-derived spectral estimates. This imparts an apparent frequency bandwidth (Δσ) to the model spectrum at κz = 0. For completeness, finite Δω is included in the model summary [Eq. (32)].}

7. Model sensitivity

There are a number of key ratios that establish the effects of advection on a spectrum of undistorted motions. If the intrinsic width Δω of the spectrum in an N frame is large (small) relative to κHμtot, the effects of lateral advection will be small (large) (Fig. 3a). If Δω is small relative to the inverse correlation time of the random vertical advection τ−1η, the initially narrow spectrum will develop a cusp at its base, in the form of the spectrum of η (Fig. 3c). Weak distortion of the baroclinic tide results in cusp formation about the principal tidal lines. In the converse situation, if Δω > τ−1η, random vertical advection will further broaden the spectrum (Fig. 3e). For ease in interpreting model performance, Δω is set to zero in model–data comparisons here (Figs. 3b,d,f).

In fitting the SHEBA Arctic observations, a hypothesized Gaussian autocorrelation for the vertical displacement field, rη(τ) = exp(−τ2/τ2η), induces spectral distortions that are generally sympathetic to the observations. Near Hawaii, an exponential autocorrelation, rη(τ) = exp(−|τ|/τη), appears to replicate the data the best.

If Δω is small relative to M2, deterministic tidal advection produces replicate harmonic peaks at ω0 ± n M2, n = 1, 2, 3, . . . (Fig. 3e). These also are present if the initial spectrum is broader, but they become more difficult to identify. The replicate peaks maintain the frequency bandwidth of the intrinsic signal. Deterministic vertical advection does not result in Doppler smearing. In fitting the Hawaiian data below, the observed frequency bandwidths of the harmonic lines strongly constrain the allowable bandwidth (aspect ratio) of the primary inertial wave. A “narrowband” (locally generated?) population of inertial waves is needed to replicate these harmonics. A second, “broadband” (poleward generated?), field is required to replicate the “spectral continuum” in frequency.

8. The SHEBA shear spectrum

To model the SHEBA observations, the combined effects of two-dimensional lateral advection and random vertical advection are considered. Because lateral advection is maintained primarily by low-frequency currents while vertical advection results from mid- to high-frequency internal waves, it is reasonable to consider these as independent processes, with combined autocovariance [from Eqs. (4), (5), and (15)]
i1520-0485-38-2-291-e24
The hypothesis is that the wavenumber–frequency spectrum of shear in the Arctic can be modeled as a sum of two line spectra:
i1520-0485-38-2-291-e25
Line 1 is centered at near-inertial (ω ∼ −f ) frequency, and line 2 is at ω = 0 in the N frame. Both the waves and the vortical motions are advected by an identical random background field. Horizontal advection is specified by a Gaussian probability density PV = (2πμ2tot)−1/2 exp[−V2/(2μ2tot)], with total horizontal velocity variance μ2tot set to the observed SHEBA values (including both internal wave and subinertial signals). Vertical advection is represented by the displacement variance 〈η2〉 and a Gaussian temporal autocorrelation rη(τ) = exp(−τ2/τ2η). The correlation time τη is adjusted to match the high-frequency shear spectral form.
To apply Eq. (24) to the SHEBA data it is necessary to replace integrals over horizontal wavenumber with equivalent expressions involving vertical wavenumber and aspect ratio Bi. In terms of vertical wavenumber, the covariance corresponding to each spectral line is given by
i1520-0485-38-2-291-e26
Fourier transforming Eq. (26) with respect to τ yields the model frequency spectrum associated with each intrinsic line:
i1520-0485-38-2-291-e27
The modeled wavenumber–frequency spectrum is identified by inspection:
i1520-0485-38-2-291-e28
where
i1520-0485-38-2-291-e29
i1520-0485-38-2-291-e30

Equations (27)(30) acknowledge the fact that there is no transfer of variance across wavenumbers in this model. The observed spectrum is the sum of a purely lateral component EiH, reduced in amplitude relative to Ei by the factor (1 − 2κ2zη2〉) due to vertical advection, and a combined term Eiz. In the limit τηγ, δτη and Eiz represents the effects of strictly vertical advection.

To model the apparent frequency dependence of a Doppler-shifted line spectrum, we set μtot equal to the observed rms horizontal velocity in the depth–time measurement domain of each record and set
i1520-0485-38-2-291-e31
where E(κz, −f ) is the observed 2D spectral estimate evaluated at the inertial frequency. The modeled spectrum is thus forced to have the observed spectral density at frequency σ1 = −f. The aspect ratio B1 is then adjusted in an attempt to describe the frequency dependence of the smeared inertial ridge at all wavenumbers. The modeled inertial spectrum is then subtracted from the observed estimate, and the process is repeated for i = 2, the vortical ridge.

In the model, the aspect ratio Bi controls the height-to-width ratio of each spectral peak (Fig. 3a), and the two fine-structure parameters 〈η2〉 and τη work together to set the high-frequency spectral level (Figs. 3b,c). The aspect ratio always appears in the product B2iμ2tot. It is maintained as an independent parameter because μ2tot is specified by the observed horizontal current variance. Aspect ratios of order 10−3–10−4 are found to be consistent with the observed Doppler spreading. D’Asaro and Morehead (1991) estimate aspect ratios for the Arctic Ocean, finding comparable values for inertial motions but larger values for the vortical contribution. The parameters required to implement the model are summarized in Table 1.

Modeled wavenumber–frequency spectra corresponding to the sum of inertial and vortical ridges (August 1998; Fig. 2d) can be compared with the observations. The characteristic hourglass shape of the spectrum is recovered, as is virtually all of the significant detail. In particular, the puzzling tendency for the wavenumber of the shear spectral peak to increase with increasing observed frequency is accurately replicated.

As a quantitative test, the modeled spectra can be integrated across wavenumber (Fig. 4) and compared with the frequency spectra of the observations. While less-than-perfect fits are achieved using one free parameter (Bi) per peak plus the two background fine-structure parameters, it is clear that the essential variability of the shear spectrum is captured. The linear-scale presentations illustrate the ability to account for the dominant shears.

When viewed on logarithmic scales (Fig. 4b), the fit to the shear spectrum is also seen to be good. The modeled spreading of vortical shear into the internal wave frequency band is very similar to the Polzin et al. (2003) estimate (their Fig. 6). Yet with the same parameters, our model underestimates the velocity spectrum at noninertial frequencies by a factor of 3–10. Under the rules of the model, there is no mechanism for shifting long wavelength motions to observed frequencies that are far from intrinsic. These long-wavelength motions have a large velocity-to-shear ratio and constitute the intrinsic part of the wave field (Munk 1981). Intrinsic waves propagate at horizontal phase speeds that are large relative to the rms currents and interact adiabatically with their surroundings. It is striking that one can account for “all” of the observed shear variance in the Arctic without significantly constraining the spectrum of velocity, kinetic energy, potential energy, and so on. In turn, an additional field of intrinsic waves could be specified to match the observed velocity spectrum of Fig. 4b. This addition would make an imperceptible contribution to the shear spectrum.

As a product of this fitting exercise, independent estimates of the vortical and inertial wavenumber spectra Ei(κz) are produced (Figs. 5c,d) for both upward- and downward-propagating motions. At all frequencies, there is near-vertical symmetry in the vortical spectrum. The vertical asymmetry in the wave spectrum stems from the predominantly downward propagation of near-inertial energy in the SHEBA deep-water observations (Fig. 6).

9. The HOME Farfield shear spectrum

To investigate deterministic vertical advection, shear data from the HOME Farfield Experiment are examined (Figs. 1a,b). Both semidiurnal and diurnal tidal energy pass through the site (18°20.1′N, 160°42.8′W) after propagating ∼430 km from the generation region west of Oahu.

Shear observations from 101 to 340 m are WKB stretched and Fourier transformed in depth and time to form a wavenumber–frequency spectral estimate (Fig. 7, top panel). Corresponding time series obtained along a set of reference isopycnals (Fig. 1b) are also processed, to produce a semi-Lagrangian spectral estimate (Fig. 7, bottom panel). Both spectral estimates display a pronounced inertial peak that spreads continuously to higher anticyclonic frequencies and across zero to the cyclonic side of the spectrum. The Eulerian estimate exhibits a much broader frequency spread than its SL counterpart.

When integrated over frequency, the resulting vertical wavenumber spectra of shear are essentially identical in the two frames (Fig. 8b). When integrated over wavenumber, the resulting frequency spectra (Fig. 8a) are very different. In an Eulerian frame, the frequency spectrum displays a pronounced inertial peak and a slightly elevated midfrequency range with harmonic peaks at frequencies f ± nM2 and f ± mD1 for small integer n and m. At high frequency, the Eulerian spectrum decreased uniformly, with no sign of a cutoff at the buoyancy frequency. The corresponding SL spectrum has a greater concentration of variance at the low- and high-frequency extremes, with a midrange level that is clearly lower than its Eulerian counterpart. There are no harmonic peaks. The buoyancy cutoff is pronounced. There is no sign of a semidiurnal tidal peak in either shear spectrum. The corresponding velocity spectra share the behavior of the shear, with a semidiurnal tidal peak added.

Can the SL and Eulerian spectra be modeled by a few spectral lines in the N frame using a common set of environmental parameters? I repeat the modeling exercise of the previous section, attempting to replicate the frequency dependence at each vertical wavenumber using as few intrinsic constituents as possible.

Given the low inertial frequency at the site, it is difficult to separate inertial and subinertial constituents. From examination of the rotary spectrum (Figs. 9a,b) as it makes a transition from cyclonic to anticyclonic frequencies, I infer that the principal shear contributors are inertial. However, two distinct inertial populations are required to model the observations. A local, narrowband population is needed to contribute the very narrow harmonic peaks seen in the frequency spectra of Figs. 8a, 9a and 10a,c. A second, larger aspect ratio, traveling or broadband population is required to fill in the intermediate frequencies of the spectrum. Somewhat arbitrarily, one-half of the observed inertial variance is assigned to each of these fields. More elaborate partitioning schemes produced a moderately improved fit. The high-frequency variance of the Eulerian spectrum is supplied by the random vertical advection of these two near-inertial fields (Figs. 10a,c). An exponential correlation function for the random vertical displacement, rη(t) = exp(−τ/τη), proves most appropriate for these low-latitude observations, as opposed to the Gaussian used to model the Arctic spectrum.

To model the harmonics seen in the Eulerian spectrum, the autocovariance is multiplied by
i1520-0485-38-2-291-eq3
[Eq. (23)] before Fourier transformation. Here D1 refers to the diurnal frequency. For each constituent, the single parameter ηt establishes the relative height of all harmonic peaks. Distinct peaks are seen only if the frequency separation between harmonics is greater than the width of the fundamental inertial peak, as distorted by random lateral and vertical advection. Thus, the existence of harmonics is an indication of narrowband waves (ω ∼ −f ) that have not been broadened by random advection.

The principal peaks are fit by the M2 displacement model, but a surprising number of smaller peaks are replicated by the addition of diurnal advection (Figs. 8a, 9a and 10a,c). The model has no ability to adjust the relative height of each harmonic peak. Thus, individual mismatch is an indication of model skill. The families of harmonics are surprisingly well fit by the Fourier transform of a Bessel function. The addition of deterministic vertical advection also results in the convex form of the Eulerian spectrum at intermediate frequencies seen in Figs. 10a,c, relative to the more concave SL spectrum of Figs. 10b,d.

In the SL frame (Figs. 10b,d), the effects of vertical advection are presumably absent. If the model is accurate, one should be able to fit the SL frequency spectrum with the identical inertial constituents used in the Eulerian fit. This effort proves successful at low frequency and unsuccessful at high frequency. Even when model parameters are specifically tuned, there is no way to maintain the narrow width of the observed inertial peak and simultaneously advect sufficient variance to high encounter frequencies. The implication is that the observed SL high-frequency motions have an intrinsic reality and are not manifestations of advectively smeared inertial currents. In the Eulerian frame, these high-frequency waves are presumably advected through a broad range of encounter frequencies.

The Hawaii observations can also be characterized in terms of the normalized wavenumber–frequency spectrum N(κz, σ). In Figs. 11a,c (observed) and Figs. 11b,d (modeled), normalized spectra are plotted for the Eulerian and SL frames. The hourglass nature of the Eulerian spectrum is captured by the model, using the two classes of advective inertial waves.

An idealized version of the harmonic structure is also produced by the model. The model emphasizes a key feature of the data—that the motions responsible for the harmonics have short rather than long vertical scales. Usually, narrow spectral lines are associated with rapidly propagating motions that are not Doppler smeared by the propagation environment. Here, the sharpness of the baroclinic tidal line induces harmonic peaks in small (vertical) scale inertial waves, whose vertical phase speeds are comparable to or slower than tidal vertical particle velocities.

In the SL frame, agreement between observed and modeled N(κz, σ) is again good. This is due, in part, to the truncation of the plots at ±1 cph, avoiding the failure of the model at high frequency seen in Figs. 10b,d. The hourglass pattern is much narrower than in the Eulerian frame, and the inertial peak is more prominent.

10. Summary and speculation

The Eulerian shear spectrum E(κz, σ) has a characteristic form that is similar in the low-energy, low-advection western Arctic and the tropical, highly advective HOME Farfield. A useful metric of this form is N(κz, σ), the normalized wavenumber–frequency spectrum. The normalized spectrum is a common metric of shear, velocity, lateral displacement, and so on and is a sensitive indicator of the frequency bandwidth of the spectrum as a function of vertical wavenumber.4 Eulerian spectra demonstrate a frequency bandwidth that increases linearly with vertical wavenumber, corresponding to an hourglass form for N(κz, σ).5

The conjecture is that the hourglass is a signature of advective processes acting on otherwise monochromatic signals. It motivates the development of a family of models for the autocovariance of shear as a function of vertical wavenumber and temporal lag:
i1520-0485-38-2-291-e32
Here,
i1520-0485-38-2-291-eq4

The advective terms modify the basic covariance (an intrinsic frequency bandwith Δω is added here to increase generality). At τ = 0, all terms assume unity value, such that the variance in each wavenumber band is unaltered. The vertical wavenumber spectrum of shear is the same in advected and unadvected frames (Fig. 8b), despite very different space–time appearances (Fig. 1).

The assumption of small distortion, κzηπ, is invoked only in the case of random vertical advection, which leads to fine-structure-contamination/tidal-cusps. This assumption is not necessary, but it leads to the surprising relationship between the form of the cusp and the power spectrum of the deformation displacement (Fig. 3b). The apparently continuous rotary power spectrum of shear in the thermocline is modeled well as a sum of a few cusps, centered at ω1 = 0 and ω2 = −f (Arctic) or ω1 = ω2 = −f (low latitude).

In dealing with deterministic tidal advection and the presence of harmonics, the model identifies J0 as an archetypical form. Even though the fundamental M2 signal is very weak in the spectrum of shear (Fig. 8a), the M2 vertical displacement signal is robust and leads to replicates of the shear inertial peak at frequencies σ = −f ± nM2. The frequency bandwidth of the individual harmonics strongly constrains the modeled width of the inertial peak, as established by B1μtot.

The vertical displacement field distorts the phase of small-scale waves more than large-scale waves. Thus it is not surprising to see strong frequency harmonics in the intermediate wavenumber bands of the shear spectrum (Fig. 11a). In this deterministic case, advection does not smear the frequency spectrum, consistent with the observations of Alford (2001) and HOME.

To convert from a prediction of N(κz, σ) to one of shear, velocity, and so on, the model must be normalized/adjusted on a wavenumber-by-wavenumber basis. The model only supplies a wavenumber–frequency spectrum if it is supplied with an initial wavenumber spectrum. For a “predictive” model, this represents a multitude of “required parameters,” one for each wavenumber band.

However, any “fine-structure model” that applies equally well to shear, velocity, lateral displacement, and so on must be specified on a wavenumber-by-wavenumber basis. It is ultimately difficult to interpret frequency spectra of motion fields without knowledge of the associated wavenumber structure. This is not surprising.

In the modeling of horizontal advection, it was assumed that advecting velocities Vi remained constant within any given realization but varied randomly from realization to realization. This scenario facilitated presentation of the model. It is not essential to the derivation. In the presentation of the fine-structure contamination and deterministic advection models the time variability of V is unrestricted.

In the Arctic, advection velocities—in particular, in the vertical direction—are very small. Also, the frequency separation of inertial and vortical peaks is at a planetary maximum. It is thus possible to produce separate wave and vortical shear spectral estimates, as in Figs. 4 –6. At low latitudes, this distinction is difficult. The narrow width of the Eulerian harmonic peaks constrains the inertial field to be modeled as a sum of narrowband and broadband constituents. (Here the “contamination” of the Eulerian signal adds physical insight.)

A few dynamical insights can be gained from this essentially kinematic exercise. A reasonable “frequency continuum” can be generated from several suitably Doppler-shifted spectral lines. Perhaps the motion field is simpler than previously thought—composed primarily of a small number of discrete entities whose signatures are altered by advective effects.

In the Hawaii observations, the broadband inertial component has an aspect ratio that is ∼18 times the narrowband. Is this peak just a surrogate for the continuous-frequency spectrum that has been measured for the past 40 years? In fact, more is implied. In addition to the broad frequency spread, the hourglass form of the wavenumber–frequency spectrum (Fig. 11) requires explanation. If the present kinematic model is rejected, a dynamical explanation is required. The fact that both Eulerian and SL spectra are fit by the same environmental and wave-field parameters and that their difference is also well modeled (Figs. 11b,d) argues for the advective model.

Alford’s (2003) “abyssal swell” populates only the lowest-vertical-wavenumber bands in the present analysis. It dominates the frequency spectrum of horizontal velocity (Fig. 4b).These low-mode waves also have significant vertical displacement and are associated with a high-frequency peak in the vertical coherence of temperature records (Pinkel 1975; Levine 1990). They are the motions that contaminate Eulerian observations of shear, extending the observed spectrum to high frequency.

By accounting for advection with a simple model, two bands of waves are identified with near-inertial intrinsic frequency. These bands might be maintained by preferential forcing (e.g., ice motion, parametric subharmonic instability) or by the propagation environment (refractive trapping). Together with a vortical constituent, they contain most of the shear in the sea.

Acknowledgments

The author thanks Eric Slater, Lloyd Green, Mike Goldin, Chris Neely, Mai Bui, Tony Aja, Tyler Hughen, and Chris Halle of the Marine Physical Laboratory of the Scripps Institution of Oceanography for assistance in the design, construction, and operation of the sonars used in this program. Roger Anderson, John Bitters, Andreas Heiberg, and Deane Stewart of the University of Washington provided critical assistance with the Arctic operations, as did Jay Ardai of Lamont-Doherty Geophysical Observatory. The captain and crew of the CCG des Groseilliers and the R/P FLIP provided able support and superlative hospitality. The National Science Foundation sponsored both HOME and SHEBA. FLIP’s participation in HOME and the development of the Deep-8 sonar was supported by the Office of Naval Research.

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Fig. 1.
Fig. 1.

Depth–time maps of the zonal component of shear obtained during the HOME Farfield Experiment. An eight-beam Doppler sonar suspended at a depth of 375 m below the Research Platform FLIP produced these data. (a) The data presented in an Eulerian frame. Slowly varying bands of low-frequency shear are seen. They are advected vertically by the semidiurnal tide. Classical “WKB stretching” will increase the magnitude of the deep shear relative to the shallow shear and alter apparent vertical wavelengths. It does not address the issue of vertical advection. (b) The same data are presented as a function of time and isopycal density; that is, shear fluctuations are displayed on a set of isopycnals whose mean depths are separated by fixed Δz (1 m here). The ordinate gives mean isopycnal depth. In this semi-Lagrangian frame, the effects of advective distortion are greatly reduced. The vertically sloping crests indicate diapycnal phase propagation. The crests progress with remarkable steadiness, suggesting a multiday passage time for many of the wave groups. Despite the baroclinic tidal activity at the site, the shear is predominately near inertial. The noisy portion of the shallow Eulerian record results from sonar side-lobe hits on FLIP’s hull and the sea surface. The near-surface SL record cannot be defined in the mixed layer. The entire Farfield record extends for 27 days.

Citation: Journal of Physical Oceanography 38, 2; 10.1175/2007JPO3559.1

Fig. 2.
Fig. 2.

(a) A wavenumber–frequency spectrum of Arctic Ocean shear estimated from the SHEBA Doppler sonar. The white rectangle indicates the smoothing used to achieve 150 degrees of freedom. The left quadrants of the spectrum correspond to anticyclonic rotation in time. The lower-left and upper-right quadrants represent upward phase propagation, which for the internal-wave component of the shear implies downward energy propagation. (b) The associated normalized spectrum N(κz, σ) = E(κz, σ)/∫E(κz, σ) . First (white) and second (black) spectral moments are indicated. Color contours mimic an hourglass pattern. (c) The Garrett and Munk (1975) spectrum of (internal wave) shear fails to capture the essential structure of the observations. In an N frame (vs the present Eulerian frame) the disagreement would be much less striking. (d) Modeled log10 shear spectral density for August record, contoured vs linear wavenumber and observed frequency. The contour interval is 5 dB. The model gives the shear variance distribution that results when line spectra at intrinsic inertial and vortical (zero) frequency are broadened by Doppler shifting. The modeled spectrum is smoothed by convolution with the rectangular block shown in white in the lower left of the figure. Variance associated with waves propagating through a steady shear will both migrate in wavenumber and move diagonally across the spectral quadrants (black arrow). Time changes in sensor or depth-mean velocity lead to variance migration at constant κz (white arrow). Propagation through a depth-varying buoyancy profile should cause no spectral alteration, because the data have been WKB stretched prior to spectral analysis. Traditional estimates of the ratio of clockwise shear to counterclockwise shear variance are formed from spectral levels at like |σ| and |κz| in diagonally opposite quadrants.

Citation: Journal of Physical Oceanography 38, 2; 10.1175/2007JPO3559.1

Fig. 3.
Fig. 3.

Model sensitivity studies. Drawings show hypothetical “observed” Eulerian frequency spectra corresponding to an initial narrowband (blue line) spectrum that is subject to (a) horizontal (green), (c) random vertical (red–violet), and (e) deterministic (black) vertical advection. The logarithm of shear variance density is plotted vs linear frequency. Note that the shape of the cusp in (c) reflects the shape of the associated correlation function of vertical advection, as specified in Eqs. (17) and (18). (b), (d), (f) The observed Arctic shear frequency spectrum for August–September 1998 is given by the blue line. This is modeled as a sum of contributions from intrinsically vortical (ω = 0) and inertial (ω = −f ) spectral lines (Δω = 0). The vortical contribution to the modeled frequency spectrum is given in green, the near-inertial contribution is in red, and the sum spectrum is in black. In (b), the effect of lateral Doppler shifting alone is plotted as a solid line at the best-fit aspect ratios B1 and B2 and at aspect ratios that are smaller by a factor of 2 (dashed). In (d) and (f), the lateral Doppler shift values are held at their best-fit levels and vertical Doppler effects are added. In (d), τη is set at 2 h and ση is fixed at 1.25 m (solid line) and 0.625 m (dashed). In (f), ση is set at 1.25 m, and τη is given for 2 h (solid) and 1 h (dashed). There is no need to account for deterministic tidal advection [as in (e)] in the western Arctic. The frequency spectra in (b), (d), and (f) are computed from integrals over wavenumber of the observed (Fig. 2a) and modeled (Fig. 2d) wavenumber–frequency spectrum, for σ < 0.

Citation: Journal of Physical Oceanography 38, 2; 10.1175/2007JPO3559.1

Fig. 4.
Fig. 4.

Observed frequency spectra of (a) shear (cph) (blue line) and (b) velocity and shear (blue lines) formed from integrations of E(κz, σ) over wavenumber. Only the anticyclonic portion of each spectrum is shown in the logarithmic plots in (b). The black line gives the corresponding model fits, including both wave (red) and vortical (green) contributions. (c) The associated vertical wavenumber spectrum of observed shear (blue lines) showing the modeled (black), inertial (red), and vortical (green) constituents, integrated over both positive and negative frequencies. The vortical spectrum approximates a κ0z form. It is isotropic with respect to the sign of the wavenumber, as expected for a nonpropagating phenomenon. At wavenumbers greater than about 0.05 cpm, Doppler-shifted inertial waves dominate the σ = 0 band, rendering the vortical–inertial separation imprecise. The wave spectrum is band limited in wavenumber, with an excess of variance associated with upward phase propagation (κz < 0). At the smallest wavenumbers |κz|, the model accounts for only 50% of the observed variance. The difference results from an inability of the model to advect large-scale motions to high encounter frequencies. The inclusion of a second inertial field, with larger aspect ratio and larger scale, would improve the fit considerably.

Citation: Journal of Physical Oceanography 38, 2; 10.1175/2007JPO3559.1

Fig. 5.
Fig. 5.

(a), (b) Cross sections of the observed wavenumber–frequency spectrum of shear for the August record (blue) and model predictions (black) at frequencies −f, −2f, . . . , −5f for anticyclonic motions. The modeled inertial spectral level is set equal to the observed level at σ = −f. The model accurately replicates the frequency bandwidth of the spectrum at vertical scales <25 m. It fails dramatically at scales >50 m. The energy observed at low wavenumber and high frequency significantly exceeds that which results from Doppler smearing of the inertial and vortical peaks. The mismatch would be reduced if the assumed inertial peak was assigned an intrinsic frequency bandwidth Δω. (c), (d) Cross sections of the modeled vortical (green) and inertial (red) components of the shear field at observed frequencies 0, −f, . . . , −5f for the August record.

Citation: Journal of Physical Oceanography 38, 2; 10.1175/2007JPO3559.1

Fig. 6.
Fig. 6.

Contour plot of the logarithm of the normalized wavenumber–frequency spectrum N(κz, σ) for August–September 1998, plotted vs linear frequency and wavenumber. Both (top) the observations and (bottom) the model depict the increasing frequency bandwidth of the spectrum with increasing wavenumber. The observations have variance at low wavenumber and intermediate frequency, outside the hourglass, that is not reproduced in the model.

Citation: Journal of Physical Oceanography 38, 2; 10.1175/2007JPO3559.1

Fig. 7.
Fig. 7.

Wavenumber–frequency spectra of shear obtained in (top) Eulerian and (bottom) semi-Lagrangian frames, from the HOME Farfield data of Fig. 1. Spectra are presented at 30 degrees of freedom. White reference lines are plotted at −f ± nM2, for small integer |n|. The spectral variance associated with the harmonic lines is distributed through a broad range of intermediate wavenumbers in the Eulerian frame.

Citation: Journal of Physical Oceanography 38, 2; 10.1175/2007JPO3559.1

Fig. 8.
Fig. 8.

(a) Frequency spectra of anticyclonic shear and velocity formed by integration of the 2D spectrum EM(κz, σ) of Fig. 7 over wavenumbers −0.1 < κz < 0.1 cpm. Eulerian and SL estimates exhibit greater differences in shear than in velocity, which emphasizes the largest vertical scales. The Eulerian spectra are reduced at low- and high-frequency extremes relative to the SL. Deterministic tidal advection causes both the appearance of harmonics and the elevation of the midfrequency region of the Eulerian spectrum. Reference lines indicate the presence of advective harmonics associated with f ± nM2. Note that these are distinct from the intrinsic tidal harmonics at frequencies ±nM2 that are barely visible in the Eulerian and SL velocity spectra but are totally absent in the shear. Other harmonics associated with diurnal motion are also apparent. Both SL spectra display a precutoff rise and a high-frequency cutoff at the buoyancy frequency. (b) Vertical wavenumber spectra based on the integration of the 2D spectrum E(κz, σ) over frequencies −7.5 < σ < 7.5 cph. The spectra are band limited, rising as κ1/2z to κ1z for κz < 0.025 cpm and falling as κ−1z for κz > 0.05 cpm. The advection model operates on a wavenumber-by-wavenumber basis and does not transport variance across wavenumbers. The observations, under WKB stretching, reflect this property. The similarity in wavenumber spectra seen in (b) is not obvious in Figs. 1a,b.

Citation: Journal of Physical Oceanography 38, 2; 10.1175/2007JPO3559.1

Fig. 9.
Fig. 9.

The behavior of the spectrum across the cyclonic–anticyclonic boundary at σ = 0 constrains the aspect ratio of the Hawaii broadband inertial motions [(a) Eulerian; (b) semi-Lagrangian]. With these quantities fixed, the random vertical advection amplitude and correlation time are adjusted to match the high-frequency form of the Eulerian observations [in (a) here and Figs. 10a,c)]. The SL frame is presumably advected with these vertical motions. The various harmonic lines are absent in both the SL data and the model. Shear and velocity spectra are presented in logarithmic format for (c) Eulerian and (d) SL data. The model replicates the shear spectra in both frames, except at high frequency, where intrinsically high-frequency motions [seen in (d) as the precutoff spectral shoulder] are shifted to both lower and higher frequencies [(c)]. The arrows indicate frequencies M2 and 2M2, which are visible in the velocity spectra but not in the shear. Identical background parameters are used in (a)–(d) with a few exceptions. Tidal amplitudes are set to zero in the SL frame. Modest random vertical displacement 〈η21/2 = 0.75 m and τη = 0.07 h is included in the SL model to mimic the effect of error in determining true isopycnal depth.

Citation: Journal of Physical Oceanography 38, 2; 10.1175/2007JPO3559.1

Fig. 10.
Fig. 10.

Model fits to the observed (left) Eulerian and (right) SL frequency spectrum of shear. Narrowband inertial motions (red) and broadband near-inertial motions (green) combine to form the model result (black). The widths of the inertial peak and the harmonics (Eulerian) strongly constrain the aspect ratio of the narrowband motions and the assumed magnitudes of the M2 and D1 tidal constituents. Reference lines indicate the position of the harmonics associated with M2. The unmarked peaks represent the D1 contribution. The model has no means of transferring near-inertial variance to high encounter frequency in an SL frame. The implication is that the observed high-frequency variance is in fact real and is not an artifact of the advection of near-inertial motions.

Citation: Journal of Physical Oceanography 38, 2; 10.1175/2007JPO3559.1

Fig. 11.
Fig. 11.

Normalized wavenumber–frequency spectra N(κz, σ) for (a), (b) Eulerian and (c), (d) SL analyses of HOME data. The model spectra (b), (d) replicate the hourglass nature of the observations. The Eulerian model produces an idealized approximation to the spectral harmonics, mimicking the appearance of sharp harmonics in frequency at intermediate vertical wavenumbers. Reference lines indicate the position of the M2 harmonics. The SL model fails to replicate the high-frequency variance of the observations, suggesting an intrinsic reality to these motions.

Citation: Journal of Physical Oceanography 38, 2; 10.1175/2007JPO3559.1

Table 1.

Model parameters.

Table 1.

1

Would a totally Lagrangian measurement be truly “optimal”? Consider a vertical stack of shear-measuring Lagrangian floats released at some time t0 and followed for a subsequent month. During this period, the floats would diverge laterally by many kilometers. They would not provide as cohesive a picture of the shear field as do these contaminated SL observations.

2

Wavenumbers and frequencies are represented in radial form in the equations but are given as cyclic quantities in the figures. Temporal frequencies are given in cycles per hour in the figures rather than in cycles per second.

3
The approach is only applicable for “ergodic” statistical processes. Ergodic processes are a subclass of stationary, homogeneous processes in which a space or time average taken within a single realization converges to the ensemble average, that is,
i1520-0485-38-2-291-eq5
Here Pq(q) is the probability density function of the random variable q(x, t), with F being some function of q.
4

This is in contrast to the wavenumber bandwidth j* as a function of frequency in the various GM models (Garrett and Munk 1975, 1979; Munk 1981).

5

In cases in which the shear field has a dominant constituent, N(κz, σ) does a poor job of displaying the bandwidth variation of the minor constituents. Thus there is no obvious hourglass pattern associated with the vortical peak in Fig. 2b. A more enlightened (but more subjective) normalization would yield a better view of the minor shear constituents.

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