1. Introduction
Two very different classes of models have been used to understand turbulence and mixing in stratified geophysical flows. One approach attempts to extrapolate the physics of unstratified turbulence into the stratified regime. Such “stratified turbulence” models (Mellor and Yamada 1982; Luyten et al. 1996) work well if the stratification is not too strong and the flow remains highly turbulent, but fail if the stratification is too strong (Simpson et al. 1996). A second approach attempts to extrapolate the physics of internal waves into a partially turbulent regime. Such “wave–wave interaction” models (Müller et al. 1986) work well in the weakly turbulent ocean thermocline (Gregg 1989; Polzin et al. 1995; Winkel 1998) but have not been well tested in other regimes. The two classes of models are fundamentally different in their physical assumptions and mathematical forms and will yield very different results if applied to the same flow.
The two different classes of models are designed to operate in different large-scale Richardson number regimes. Wave–wave interaction models assume that the flow can be expressed as the sum of interacting internal waves, each of which is a solution to the inviscid linear internal wave equations. Energy transfer and fluxes occur through the interactions of these waves. This approach is appropriate at high large-scale Richardson numbers, where the waves are stable and interact only weakly. Stratified-turbulence models generally parameterize fluxes by assuming a local turbulent eddy viscosity, or similar closure, whose strength depends on the local time and length scales of the flow. This approach is appropriate at low Richardson number where the flow is mostly turbulent and waves play only a minor role (Henyey 1989). A transition from the high Richardson number wave physics to the low Richardson number turbulent physics should occur at some intermediate Richardson number. We will call this the wave to stratified turbulence transition, or W–T transition. It is the focus of this paper.
We will study the W–T transition from an oceanographic viewpoint, considering how the physics of a broadband, high Richardson number flow, dominated by internal waves, such as that in the open ocean thermocline, changes as its energy increases. The analysis is based on the observed shapes of oceanic and atmospheric spectra and existing parameterizations of the dissipation rate. We first (section 2) consider the Lagrangian frequency spectrum of velocity, a recently measured quantity. We then (section 3) consider the Eulerian wavenumber spectrum of shear, a quantity that is well measured and modeled in the oceanic thermocline. Neither spectrum can maintain its low energy form at high energy, implying that a transition must occur. We then quantify the transition (section 4) and find that both spectra lose their low energy form at the energy level at which the rms velocity reaches approximately the lowest mode phase speed. We compare these predictions with oceanic observations and traditional turbulence closure models (section 5) and finally (section 6) summarize and discuss the limitations of the analysis.
2. Lagrangian frequency spectrum
Lien et al. (1998) and D’Asaro and Lien (2000) describe frequency spectra (Φu(ω), Φυ(ω), Φw(ω)) [m2 s−1] of velocity (u, υ, w) [m s−1] as a function of Lagrangian frequency ω [s−1], as observed by neutrally buoyant floats in regions of both strong and weak turbulence. On the basis of their observations, we propose the spectral form shown in Fig. 1. The model spectrum is the sum of internal wave and turbulent components.
Figure 2 shows the evolution of the model Φw(ω) as a function of internal wave energy. For typical oceanic, that is, Garrett–Munk (GM) energy levels (spectrum A), the internal wave spectrum has far more energy than the turbulence spectrum. This results in a sharp drop in the spectrum near N. This drop is commonly observed, but is particularly sharp when observed from neutrally buoyant floats (Cairns 1975; D’Asaro and Lien 2000; Kunze et al. 1990), that is, for Lagrangian spectra. The ratio ΔWT =
3. Eulerian wavenumber spectrum
a. Spectral shape
The spectrum, ΦSS(m) [m s−2] of horizontal shear S [s−1] as a function of vertical wavenumber m [m−1] in the ocean thermocline has been measured by many investigators using vertical profilers (Gregg 1991) that resolve the shear down to dissipation scales. Measurements have been mostly at near-GM energy levels. The basic form, first proposed by Gargett et al. (1981), is shown in Fig. 3. There are three wavenumber bands: a low wavenumber “internal wave” band, a high wavenumber “turbulence” band, and an intermediate “−1” band named for its spectral slope. We will extrapolate this form to energy levels much higher than the GM level using wave–wave interaction theory, and show how it eventually fails.
The low wavenumber band is dominated by internal waves (Müller et al. 1978). We assume a white normalized shear spectrum, ΦSS(m)/N2 with a level ϕiw [m−1], consistent with numerous observations (Gregg et al. 1993; Polzin et al. 1995). Typically, the vertical wavenumber m varies with depth owing to the change of N with depth. This effect can be mostly removed by using a “WKB-stretched” wavenumber (Leaman and Sanford 1975), which we use in this analysis. The lower end of the internal wave band is set by the WKB-stretched wavelength of the gravest internal wave mode m1. For the open ocean, this is set by the thermocline depth; we assume the GM76 value m1 = 2π/b [m−1] with b = 1300 m. For the continental shelf, this is set by the water depth; we will use b = 100 m.
In the Garrett–Munk spectrum the shear spectral slope changes from white to +2 below a vertical mode number j*. GM76 uses j* = 3. Levine et al. (1997) show evidence that j* is large for low-energy internal wave fields. Measurements under a storm (D’Asaro 1985) and on the continental shelf (M. Levine 1999, personal communication) suggest that j* is small for high energy internal wave fields. The data therefore suggest that at energy levels substantially above the GM level, j* probably falls below 1 and the spectrum is nearly white down to the lowest mode. We therefore ignore the j* factor in GM76 and retain a white shear spectrum all the way to the lowest mode.
b. Internal wave bandwidth
Figure 4 shows the evolution of the vertical wavenumber spectrum with increasing energy assuming that (14) is true. We will measure the internal wave bandwidth by Δmiw = m1/mc. At the GM level (spectrum
The model wavenumber spectrum becomes singular as mc approaches m1; that is, as Δmiw approaches one. This will mark the W–T transition. In this regime, however, (13) and (14) are no longer equivalent, and the relationship between ϕiw and mc becomes uncertain. The obvious assumption is to replace (14) by ϕiwmc(1 − Δmiw) = Frc with the same Frc. However, as Δmiw approaches one, this assumption causes the “−1” spectral level to rise far above the observed “Saturation” level and causes mb to greatly exceed mO, as shown by spectrum
There are several possible ways to justify this choice and still maintain the physical simplicity of a critical value of the Froude function (13). One might suppose that Frc decreased as Δmiw approached one, that is, that a narrowband internal wave field might break with less shear than a wide-band one. It seems unlikely that Frc could change enough to make this effect important. Alternatively, the internal wave spectral slope might become steeper at high energy, as suggested by the dotted “?” line in Fig. 4. This would produce a peak in the spectrum near mc and thus decrease the importance of m1 in the Froude function (13). Both Fritts and VanZandt (1993) and Duda and Cox (1989) find spectra with peaks near mc, supporting this idea. This would not eliminate the problem, however, just delay it to a larger value of Δmiw. The experimental evidence and theoretical support for (13) is not very strong, and probably less than that in favor of a universal “−1” spectral level, so there is probably no reason to be alarmed if (13) is violated. For the purposes of this paper, it is probably best to view the use of (14) near the W–T transition as an extrapolation of the relations appropriate for the ocean thermocline into a higher energy regime and expect that many of the predicted features will be only qualitatively accurate.
c. Variances
We will ignore the braced terms in (16) and (17) since their inclusion leads to much greater algebraic complexity with little gain of insight. Formally, the braced terms can be neglected only well below the W–T transition where Δmiw is small. However, many of the assumptions of our analysis, specifically the assumed Eulerian wavenumber spectral shape, the estimation of mc, and the parameterization discussed in section 4a, become uncertain near the W–T transition. Different assumptions can change the functional forms of (16) and (17), and subsequent results, in different ways. As in section 3b, the most consistent approach is to view our results as an extrapolation of relations appropriate well below the W–T transition, that is, those with the braced termed ignored, into a higher energy regime.
4. Quantification
a. Dissipation closure
b. The W–T transition
The three estimates yield results identical within the errors in the various coefficients and the substantial theoretical uncertainties described in section 3. Thus at the W–T transition ΔWT = 1, that is, the turbulent and internal wave spectra merge in the Lagrangian frequency spectrum; Δmiw ≈ 1, that is, the internal wave bandwidth in the Eulerian wavenumber spectrum becomes small; and σU ≈ c1, that is, the rms velocity equals the phase velocity of the lowest mode; and the linear and quadratic parameterizations of ε converge. This is the main result of the paper.
These conclusions result from the extrapolation of results at small
c. Physics
The W–T transition, as computed here, marks the energy level where the basic assumptions of GM-based wave–wave interaction theories fail. At open ocean energy levels and water depths almost all of the energy is in the internal wave wavenumber band and can be reasonably represented by the sum of internal wave modes. The transfer of energy from these scales to the smaller-scale turbulence is due to wave–wave interactions. Henyey et al.’s (1986) model of this (19) assumes the interaction of individual “test” waves with the “internal wave” band of wavenumbers in Fig. 3. At the W–T transition, mc and m1 become nearly equal, this band becomes small, and the wave–wave interaction theory becomes ill posed. In addition, the Richardson number of the lowest wavenumbers, that is, m1, becomes order 1 on average, and their velocity approaches the fastest wave phase speed. Statistical fluctuations will reduce the Richardson number to below critical (¼) in places, resulting in localized shear instability of the large-scale motions. The large-scale motions thus become locally unstable to shear instability and can directly transfer energy to turbulence. Alternatively, the large-scale waves will start to break since their velocity is comparable to their phase speed. This is the physics of stratified turbulence. Thus, the W–T transition marks a change from energy transfer controlled by wave–wave interaction to that controlled by instability and turbulence.
d. The transition energy level
For deep ocean depths (m1 = 2π/1300 m, mcGM = 2π/10 m) (26) predicts the W–T transition at 130 times the GM spectral level, (27) predicts an internal wave horizontal kinetic energy at the transition of (0.7 m s−1)2, and (28) predicts ε = 3 × 10−5 W kg−1 at the transition. Values of ε this large are rarely, if ever, observed in the open ocean thermocline (Gregg 1998). It appears to remain almost always below the W–T transition.
In shallow water, the low modes are excluded, and there is much more shear per unit kinetic energy. Consequently, the W–T transition occurs at lower energy. Equivalently, the lowest mode phase speed c1 is lower for the same (assumed constant) stratification, so lower water velocities are needed match c1. In 100 m of water, (26) predicts the W–T transition at 10 times the GM spectral level, (27) predicts an internal wave horizontal kinetic energy at the transition of (0.06 m s−1)2, and (28) predicts ε = 2 × 10−7 W kg−1 at the transition. Moum and Nash (1999, personal communication) report ε values much larger than this over rough topography on the Oregon shelf. Gregg et al. (1999) report substantially larger values on the shelf near Monterey Bay. The flows in these locations are energetic enough to be clearly above the W–T transition. Flows at other locations on the shelf are clearly less energetic and lie below the transition.
5. Parameterizations
a. Quadratic
There is ample evidence for quadratic scaling of ε with energy in the ocean thermocline (Gregg 1989; Polzin et al. 1995; Winkel 1998) and considerable theoretical justification (Winters and D’Asaro 1997; Müller et al. 1986). In this regime the Richardson number is above critical and the flow is mostly laminar. In places, the Richardson number falls below critical, creating localized turbulence and mixing. The frequency and intensity of these mixing events are controlled by the supply of energy from the wave–wave interactions at larger scale (Polzin 1996; Pinkel and Anderson 1997), which therefore control the overall rate of mixing.
b. Linear
c. Stratified turbulence models
Equation (31) uses k, the total kinetic energy, rather than its vertical component
Thus the high stratification limit of the BPR model reproduces many of the basic equations of oceanic mixing. Despite this, there are fundamental conceptual differences between these stratified turbulence models and the model presented here. The stratified turbulence models claim to represent only the “turbulent” velocity fluctuations. Thus k in (31) is the turbulent kinetic energy. Here, we propose that much of the energy and all of the anisotropy in stratified turbulence is due to its internal wave component, as is discussed in more detail by D’Asaro and Lien (2000) and Hanazaki and Hunt (1996). Internal waves, in this view, are an intrinsic part of stratified turbulence. Thus
BPR find that their model greatly underpredicts ε in the stratified thermocline. Their solution is to continue to use (31), but to set an arbitrary lower limit k >
d. N scaling of diffusivity
We have emphasized how the dynamics of a stratified flow vary with energy. Alternatively, one can view the decay rate of kinetic energy as a function of stratification at fixed energy, as shown in Fig. 5b. Below the W–T transition a “+2” slope is found (22); above the transition the decay rate is constant (11).
The diapycnal diffusivity of mass due to internal-wave-driven turbulence is often computed from (32) using a mixing efficiency γ of about 0.2 (Ivey and Imberger 1991). For energies below the W–T transition, (19) leads to a diffusivity independent of N and proportional to the internal wave spectral level ϕiw squared. For energies above the W–T transition, (7) leads to a diffusivity proportional to
6. Summary and discussion
We hypothesize the existence of two distinct dynamical regimes for mixing in stratified flows. At low energies, flow evolution is controlled by interactions between internal wave modes. At high energies, flow evolution is controlled by instabilities of the wave modes. The “wave–turbulence” transition separating these regimes is marked by
an rms velocity approximately equal to the phase speed of the lowest internal wave mode
the merging of the internal wave and turbulence spectra near Lagrangian frequency N (Fig. 2)
the decrease of the large-scale Richardson number to near one
the reduction of the internal wave vertical wavenumber bandwidth to a small value (Fig. 4)
a change from a quadratic relationship between wave energy and dissipation rate at low energies (19) to a linear relationship at high energies (7).
The W–T transition is predicted to occur at a lower average dissipation rate and internal wave shear, but similar energy density, in shallow water than in deep water [Eqs. (26), (27), (28)]. The open ocean thermocline is likely to remain below the transition, but flow on the continental shelves may often rise above it. The upper atmosphere appears to be close to the transition, judging from the small range of wavenumbers below mc. Turbulence parameterization schemes that ignore internal waves will be accurate only for energies above the transition.
The present analysis, although aimed mostly at oceanic applications, should also be relevant to other stratified flows. The most obvious obstacle is the presence of the Coriolis frequency f in many of the expressions. This appears, first, because the internal wave energy in the ocean is dominated by motions near f. Expressions (2) and (4) describing energy spectra thus contain f. Second, the internal wave dissipation closure (18) depends most fundamentally on the horizontal wavenumber, kH, parameterized in (19) by the vertical wavenumber m times the aspect ratio of the flow, which is proportional to f/N. In neither case is f fundamental to the analysis, and a theory in which f did not appear could be formulated. Henyey (1991) gives a short description of how to proceed.
Although the analysis above has enabled us to sketch the nature of the W–T transition some features remain unsatisfactory. In particular, the results are sensitive to low-mode, low-frequency energy that dominates the internal wave energy spectrum. It seems far more likely that the high modes, that is, the shear rather than the energy, are important, but we do not have measurements of their frequency spectra. The analysis would thus be more satisfactory if it were formulated using Lagrangian and Eulerian spectra of either shear or vertical velocity. Similarly, the analysis ignores zero frequency motions, although any mean shear must be important if sufficiently large. This may explain the absence of “vortical modes” (Müller et al. 1986) in the formulation. Furthermore, peaks in energy at frequencies near N are often observed; these could significantly modify the results. The analysis depends heavily on assumptions about the “−1” wavenumber range in the shear spectrum. It is unsettling that the dynamical nature of this region remains unresolved (Holloway 1983; Hines 1991; Eckermann 1999; Gardner 1996; Pinkel and Anderson 1997). Finally, we have been unable to formulate a satisfactory theory near the W–T transition, as shown by the neglect of “braced” expressions throughout the analysis. Our results are bold extrapolations from above and below the transition. There is a clear need for measurements, particularly of vertical wavenumber spectra near and above the transition, to test the various assumptions required here.
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Model Lagrangian frequency spectra of vertical [Φw(ω)] and horizontal [ΦH(ω)] velocity as a function of Lagrangian frequency ω. The model consists of an internal wave component (f < ω < N), extending from the Coriolis frequency f to the buoyancy frequency N, and a turbulent component (N < ω < ωk), extending to the Kolmogorov frequency ωk and proportional to the turbulent dissipation rate ε.
Citation: Journal of Physical Oceanography 30, 7; 10.1175/1520-0485(2000)030<1669:TWTTFS>2.0.CO;2
Evolution of Lagrangian frequency spectrum of vertical velocity with increasing energy. Spectra
Citation: Journal of Physical Oceanography 30, 7; 10.1175/1520-0485(2000)030<1669:TWTTFS>2.0.CO;2
Model spectra of vertical wavenumber spectrum of shear.
Citation: Journal of Physical Oceanography 30, 7; 10.1175/1520-0485(2000)030<1669:TWTTFS>2.0.CO;2
Evolution of model vertical wavenumber spectra of shear with increasing energy. Spectrum
Citation: Journal of Physical Oceanography 30, 7; 10.1175/1520-0485(2000)030<1669:TWTTFS>2.0.CO;2
The W–T transition energy defined as the intersection of the wave–wave interaction and stratified turbulence model predictions of ε as a function of
Citation: Journal of Physical Oceanography 30, 7; 10.1175/1520-0485(2000)030<1669:TWTTFS>2.0.CO;2