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Ren-Chieh Lien
and
Peter Müller

Abstract

Small-scale oceanic motions consist of vortical motion and internal waves. In a linear or weakly nonlinear system these two types of motions can be unambiguously separated using normal-mode decomposition in which the vortical mode carries the linear perturbation potential vorticity, whereas the gravity mode does not. Normal-mode decomposition can be easily achieved using the fields of horizontal divergence, relative vorticity, and vortex stretching. An attempt to estimate these three fields is made using the Internal Wave Experiment (IWEX) measurements. Estimates of horizontal divergence and relative vorticity using the three-point array are attenuated at horizontal scales smaller than the size of the array and mutually contaminated at the horizontal separation scale of the sensors. Estimates of vortex stretching using vertically separated vertical displacement measurements are also attenuated at small vertical scales.

The observed frequency spectra represent oceanic wavenumber frequency spectra subjected to array response functions as spectral windows. In principle, wavenumber frequency spectra can be obtained by applying inverse transformations provided that frequency spectra at different array sizes and vertical separations are measured. The IWEX array does not have a sufficient spatial resolution to reliably perform all of the necessary inverse transformations. Spectral estimates are compared with the GM-76 internal wave spectrum model. Observed frequency spectral estimates of horizontal divergence agree well with the GM model at small Rossby numbers in the entire internal wave frequency band and at moderate Rossby number O(1) in the low-frequency regime (ω<1 cph). In contrast, frequency spectra of estimated relative vorticity agree with the GM model only at small Rossby numbers in the low-frequency regime (ω≤0.1 cph). Since the calculation of horizontal divergence and relative vorticity spectra for the GM-76 model employs the dispersion relation of linear internal waves, the observed discrepancy could be due to either the failure of the linearity assumption or the existence of small-scale vortical motion. Spectral estimates of vortex stretching are well explained by the GM model, suggesting that fluctuations of vortex stretching are dominated by the gravity mode at vertical scales greater than O(34 m), the smallest resolvable vertical scale in this analysis.

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Ren-Chieh Lien
and
Thomas B. Sanford

Abstract

Twenty Electromagnetic Autonomous Profiling Explorer (EM-APEX) floats in the upper-ocean thermocline of the summer Sargasso Sea observed the temporal and vertical variations of Ertel potential vorticity (PV) at 7–70-m vertical scale, averaged over O(4–8)-km horizontal scale. PV is dominated by its linear components—vertical vorticity and vortex stretching, each with an rms value of ~0.15f. In the internal wave frequency band, they are coherent and in phase, as expected for linear internal waves. Packets of strong, >0.2f, vertical vorticity and vortex stretching balance closely with a small net rms PV. The PV spectrum peaks at the highest resolvable vertical wavenumber, ~0.1 cpm. The PV frequency spectrum has a red spectral shape, a −1 spectral slope in the internal wave frequency band, and a small peak at the inertial frequency. PV measured at near-inertial frequencies is partially attributed to the non-Lagrangian nature of float measurements. Measurement errors and the vortical mode also contribute to PV in the internal wave frequency band. The vortical mode Burger number, computed using time rates of change of vertical vorticity and vortex stretching, is 0.2–0.4, implying a horizontal kinetic energy to available potential energy ratio of ~0.1. The vortical mode energy frequency spectrum is 1–2 decades less than the observed energy spectrum. Vortical mode energy is likely underestimated because its energy at vertical scales > 70 m was not measured. The vortical mode to total energy ratio increases with vertical wavenumber, implying its importance at small vertical scales.

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Eric Kunze
and
Ren-Chieh Lien

Abstract

Microstructure measurements in Drake Passage and on the flanks of Kerguelen Plateau find turbulent dissipation rates ε on average factors of 2–3 smaller than linear lee-wave generation predictions, as well as a factor of 3 smaller than the predictions of a well-established parameterization based on finescale shear and strain. Here, the possibility that these discrepancies are a result of conservation of wave action E/ω L = E/|kU| is explored. Conservation of wave action will transfer a fraction of the lee-wave radiation back to the mean flow if the waves encounter weakening currents U, where the intrinsic or Lagrangian frequency ω L = |kU| ↓ |f| and k the along-stream horizontal wavenumber, where kUkV. The dissipative fraction of power that is lost to turbulence depends on the Doppler shift of the intrinsic frequency between generation and breaking, hence on the topographic height spectrum and bandwidth N/f. The partition between dissipation and loss to the mean flow is quantified for typical topographic height spectral shapes and N/f ratios found in the abyssal ocean under the assumption that blocking is local in wavenumber. Although some fraction of lee-wave generation is always dissipated in a rotating fluid, lee waves are not as large a sink for balanced energy or as large a source for turbulence as previously suggested. The dissipative fraction is 0.44–0.56 for topographic spectral slopes and buoyancy frequencies typical of the deep Southern Ocean, insensitive to flow speed U and topographic splitting. Lee waves are also an important mechanism for redistributing balanced energy within their generating bottom current.

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Eric A. D’Asaro
and
Ren-Chieh Lien

Abstract

Simultaneous measurements of temperature, salinity, their vertical gradients, and the vertical gradient of velocity across a 1.4-m-long Lagrangian float were used to investigate the accuracy with which the dissipation of scalar variance χ can be computed using inertial subrange methods from such a neutrally buoyant float. The float was deployed in a variety of environments in Puget Sound; χ varied by about 3.5 orders of magnitude. A previous study used an inertial subrange method to yield accurate measurements of ɛ, the rate of dissipation of kinetic energy, from this data. Kolmogorov scaling predicts a Lagrangian frequency spectrum for the rate of change of a scalar as Φ D σ/Dt (ω) = β s χ, where βs is a universal Kolmogorov constant. Measured spectra of the rate of change of potential density σ were nearly white at frequencies above N, the buoyancy frequency. Deviations at higher frequency could be modeled quantitatively using the measured deviations of the float from perfect Lagrangian behavior, yielding an empirical nondimensional form Φ D σ/Dt = β s χ H(ω/ω L ) for the measured spectra, where L is half the float length, ω 3 L = ɛ/L 2, and H is a function describing the deviations of the spectrum from Kolmogorov scaling. Using this empirical form, estimates of χ were computed and compared with estimates derived from ɛ. The required mixing efficiency was computed from the turbulent Froude number ω 0/N, where ω 0 is the large-eddy frequency. The results are consistent over a range of ɛ from 10−8 to 3 × 10−5 W kg−1 implying that χ can be estimated from float data to an accuracy of least a factor of 2. These methods for estimating ɛ, χ, and the Froude number from Lagrangian floats appear to be unbiased and self-consistent for ɛ > 10−8 W kg−1. They are expected to fail in less energetic turbulence both for instrumental reasons and because the Reynolds number typically becomes too small to support an inertial subrange. The value of β s is estimated at 0.6 to within an uncertainty of less than a factor of 2.

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Ren-Chieh Lien
and
Eric A. D'Asaro

Abstract

This study tests the ability of a neutrally buoyant float to estimate the dissipation rate of turbulent kinetic energy ɛ from its vertical acceleration spectrum using an inertial subrange method. A Lagrangian float was equipped with a SonTek acoustic Doppler velocimeter (ADV), which measured the vector velocity 1 m below the float's center, and a pressure sensor, which measured the float's depth. Measurements were taken in flows where estimates of ɛ varied from 10−8 to 10−3 W kg−1. Previous observational and theoretical studies conclude that the Lagrangian acceleration spectrum is white within the inertial subrange with a level proportional to ɛ. The size of the Lagrangian float introduces a highly reproducible spectral attenuation at high frequencies. Estimates of the dissipation rate of turbulent kinetic energy using float measurements ɛ float were obtained by fitting the observed spectra to a model spectrum that included the attenuation effect. The ADV velocity measurements were converted to a wavenumber spectrum using a variant of Taylor's hypothesis. The spectrum exhibited the expected −5/3 slope within an inertial subrange. The turbulent kinetic energy dissipation rate ɛ ADV was computed from the level of this spectrum. These two independent estimates, ɛ ADV and ɛ float, were highly correlated. The ratio ɛ float/ɛ ADV deviated from one by less than a factor of 2 over the five decades of ɛ measured. This analysis confirms that ɛ can be estimated reliably from Lagrangian float acceleration spectra in turbulent flows. For the meter-sized floats used here, the size of the float and the noise level of the pressure measurements sets a lower limit of ɛ float > 10−8 W kg−1.

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Peter Müller
,
Ren-Chieh Lien
, and
Robin Williams

Abstract

The trimoored design of the current meter army of the Internal Wave Experiment (IWEX) is exploited to calculate time series of relative vorticity, horizontal divergence, vortex stretching and (linen) potential vorticity at five different levels in the vertical. Potential vorticity characterizes the vortical mode of motion which coexists with the (internal) gravity mode (which does not carry potential vorticity). The amplitudes, and the space- and time-scales of the vortical mode, or potential vorticity field, are determined by spectral analysis. The observed variance of potential vorticity (enstrophy) is 10−6 s−2, implying a Rossby number of order 10, the energy 2 × 10−4 m2 s−2 and the inverse Richardson number 0.7. The observed frequencies are interpreted as Doppler frequencies. A low frequency “steppy” potential vorticity field is advected vertically past the sensors by internal gravity waves. The advected potential vorticity field is characterized by a vertical wavenumber smaller than 0.2 m−1 and by a +2/3 (enstrophy) or −4/3 (energy) power law with respect to horizontal wavenumber. These results are summarized in a model wavenumber-frequency spectrum. The results indicate that vortical or potential vorticity carrying motion exists at scales traditionally associated with internal gravity waves and that these small-scale vortical motions contribute significantly to the observed shear.

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Anda Vladoiu
,
Ren-Chieh Lien
, and
Eric Kunze

Abstract

Horizontal and vertical wavenumbers (kx , kz ) immediately below the Ozmidov wavenumber (N 3/ε)1/2 are spectrally distinct from both isotropic turbulence (kx , kz > 1 cpm) and internal waves as described by the Garrett–Munk (GM) model spectrum (kz < 0.1 cpm). A towed CTD chain, augmented with concurrent Electromagnetic Autonomous Profiling Explorer (EM-APEX) profiling float microstructure measurements and shipboard ADCP surveys, are used to characterize 2D wavenumber (kx , kz ) spectra of isopycnal slope, vertical strain, and isopycnal salinity gradient on horizontal wavelengths from 50 m to 250 km and vertical wavelengths of 2–48 m. For kz < 0.1 cpm, 2D spectra of isopycnal slope and vertical strain resemble GM. Integrated over the other wavenumber, the isopycnal slope 1D kx spectrum exhibits a roughly +1/3 slope for kx > 3 × 10−3 cpm, and the vertical strain 1D kz spectrum a −1 slope for kz > 0.1 cpm, consistent with previous 1D measurements, numerical simulations, and anisotropic stratified turbulence theory. Isopycnal salinity gradient 1D kx spectra have a +1 slope for kx > 2 × 10−3 cpm, consistent with nonlocal stirring. Turbulent diapycnal diffusivities inferred in the (i) internal wave subrange using a vertical strain-based finescale parameterization are consistent with those inferred from finescale horizonal wavenumber spectra of (ii) isopycnal slope and (iii) isopycnal salinity gradients using Batchelor model spectra. This suggests that horizontal submesoscale and vertical finescale subranges participate in bridging the forward cascade between weakly nonlinear internal waves and isotropic turbulence, as hypothesized by anisotropic turbulence theory.

Open access
Anda Vladoiu
,
Ren-Chieh Lien
, and
Eric Kunze

Abstract

Shipboard ADCP velocity and towed CTD chain density measurements from the eastern North Pacific pycnocline are used to segregate energy between linear internal waves (IW) and linear vortical motion (quasi-geostrophy, QG) in 2-D wavenumber space spanning submesoscale horizontal wavelengths λx ∼ 1 – 50 km and finescale vertical wavelengths λz ∼ 7 – 100 m. Helmholtz decomposition and a new Burger-number Bu decomposition yield similar results despite different methodologies. Partition between IW and QG total energies depends on 𝐵𝑢. For Bu < 0.01, available potential energy EP exceeds horizontal kinetic energy EK and is contributed mostly by QG. In contrast, energy is nearly equipartitioned between QG and IW for Bu » 1. For Bu < 2, EK is contributed mainly by IW, and EP by QG, while, for Bu > 2, contributions are reversed. Vertical shear variance is contributed primarily by near-inertial IW at small λz , implying negligible QG contribution to vertical shear instability. Conversely, both QG and IW at the smallest λx ∼ 1 km contribute large horizontal shear variance, such that both may lead to horizontal shear instability. Both QG and IW contribute to vortex-stretching at small vertical scales. For QG, the relative vorticity contribution to linear potential vorticity anomaly increases with decreasing horizontal and increasing vertical scales.

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Eric A. D’Asaro
and
Ren-Chieh Lien

Abstract

Stratified flows are often a mixture of waves and turbulence. Here, Lagrangian frequency is used to distinguish these two types of motion.

A set of 52 Lagrangian float trajectories from Knight Inlet and 10 trajectories from below the mixed layer in the wintertime northeast Pacific were analyzed using frequency spectra. A subset of 28 trajectories transit the Knight Inlet sill where energetic internal waves and strong turbulent mixing coexist.

Vertical velocity spectra show a progression from a nearly Garrett–Munk internal wave spectrum at low energies to a shape characteristic of homogeneous turbulence at high energies. All spectra show a break in slope at a frequency close to the buoyancy frequency N. Spectra from the Knight Inlet sill are analyzed in more detail. For “subbuoyant” frequencies (less than N) all 28 spectra exhibit a ratio of vertical-to-horizontal kinetic energy that varies with frequency as predicted by the linear internal wave equations. All spectra have a shape similar to that of the Garrett–Munk internal wave spectrum at subbuoyant frequencies. These motions are much more like waves than turbulence. For “superbuoyant” frequencies (greater than N) all 28 spectra are isotropic and exhibit the −2 spectral slope of inertial subrange homogeneous turbulence. These motions appear to be turbulent.

These data suggest that stratified flows may be modeled as the sum of nearly isotropic turbulence with superbuoyant Lagrangian frequencies and anisotropic internal waves with subbuoyant Lagrangian frequencies. The horizontal velocities are larger than the vertical velocities for the internal wave component but approximately equal for the turbulent component. Vertical kinetic energy is therefore a better indicator of turbulent kinetic energy than is horizontal or total kinetic energy.

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Eric A. D’Asaro
and
Ren-Chieh Lien

Abstract

Mixing in a stratified ocean is controlled by different physics, depending on the large-scale Richardson number. At high Richardson numbers, mixing is controlled by interactions between internal wave modes. At Richardson numbers of order 1, mixing is controlled by instabilities of the large-scale wave modes. A “wave–turbulence” (W–T) transition separates these two regimes. This paper investigates the W–T transition, using observed oceanic and atmospheric spectra and parameterizations. Viewed in terms of Lagrangian (intrinsic) frequency spectra, the transition occurs when the inertial subrange of turbulence, confined to frequencies greater than the buoyancy frequency N, reaches the level of the internal waves, confined to frequencies less than N. Viewed in terms of vertical wavenumber spectra, the W–T transition occurs when the bandwidth of internal waves becomes small. Both of these singularities occur when the typical internal wave velocity becomes comparable to the phase speed of the lowest internal wave mode. At energies below that of the W–T transition, the dissipation rate varies as the energy squared; above the transition the dependence is linear. The transition occurs at lower shear and dissipation rates where the phase speed of the lowest mode is smaller, that is, in shallower water for the same stratification. Traditional turbulence closure models, which ignore internal waves, can be accurate only at energies above the W–T transition.

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