1. Introduction
Internal waves are the main cause of turbulence in the thermocline and the deep ocean (Munk 1981; Gregg 1987; Garrett and St. Laurent 2002). Wave–wave interaction theory and microstructure measurements have shown that the turbulent kinetic energy dissipation rate can be estimated from the 10-m vertical shear (e.g., Gregg 1989; Polzin et al. 1995; Sun and Kunze 1999). However, the mixing rate depends further on the mixing efficiency Γ or the buoyancy flux as a fraction of the energy dissipation rate (e.g., Osborn 1980). Osborn (1980) determined Γ to be 0.2 assuming that the steady-state balance is achieved in the ocean mixing caused by shear instability and that the flux Richardson number (Rif) is restricted to a critical value to maintain steady-state turbulence. Here, Rif is defined as the ratio of the potential energy gain to the kinetic energy loss and is related to Γ by Γ = Rif/(1 − Rif) in steady-state turbulence. The value 0.2 is often used in the ocean modeling as well as in interpretations of microstructure measurements (Munk and Wunsch 1998; Wunsch and Ferrari 2004; Simmons et al. 2004; Polzin et al. 1997; Arneborg 2002), but it is likely to depend on various factors, such as the duration and intensity of high-shear events, which lead to instability and turbulence (Garrett 2001).
Because of its importance, mixing efficiency for shear-driven turbulence has been studied through in situ ocean observations, laboratory experiments, theories, and numerical simulations. In the ocean observations, Oakey (1982, 1985) defined Γ in stratified steady turbulence that can be estimated from microscale temperature gradient and velocity shear measurements. Ivey and Imberger (1991) related Γ to two dimensionless parameters (overturn Froude number, the ratio of the largest eddy length without buoyancy effect to the most energetic overturn scale, and overturn Reynolds number, the ratio of the most energetic overturn scale to the smallest length scale in turbulence) from laboratory experiments. Imberger and Ivey (1991) classified observed turbulent events in this parameter space and inferred Γ by different generation processes of turbulence. Moum (1996), Ruddick et al. (1997), and St. Laurent and Schmitt (1999) also followed the definition by Oakey (1982, 1985) and obtained similar values of Γ ≈ 0.2 ± 0.1 in turbulent patches. Seim and Gregg (1994) observed the evolution and decay of Kelvin–Helmholtz (KH) instability in a tidal channel and found highly variable Γ (0.17–1.3) in each stage of instability and a higher average value of Γ = 0.58. Gargett and Moum (1995) and Seim and Gregg (1995) pointed out the difference in the definition of Γ in Seim and Gregg (1994), and Γ should be halved for comparison with past studies. Gargett and Moum (1995) also reported Γ is around 0.2 in a tidal front.
Laboratory experiments mainly investigate the ratio of the potential energy gain to the kinetic energy loss Rif in turbulent patches. Thorpe (1968) reproduced KH instability in tilted tube experiments, and Thorpe (1973) estimated Rif over one tilting experiment and obtained 0.21 < Rif < 0.27 (0.27 < Γ < 0.36). By using the results from Thorpe (1973) and Koop (1976), Linden (1979) showed that Rif over one tilting experiment is a decreasing function of the Richardson number. In theoretical studies, Thompson (1980), inspired by the results of Thorpe (1973), showed that the maximum value of Rif in turbulence is equivalent to the critical Richardson number [Ric = 0.25; Miles (1961); Howard (1961)] from energy budgets. However, he also assumed that vertical eddy diffusivity is the same for momentum and density, that turbulence occurs in a limited region, and that turbulence always exists and has the same intensity when Ri < Ric. Winters et al. (1995) and Winters and D’Asaro (1996) further explored the energy balance in stratified turbulent events and obtained energy budgets under the Boussinesq approximation. Because this framework is easy to reproduce in a direct numerical simulation (DNS), it becomes the basis for studies using the computational approach.
Due to recent advances in computer resources, DNS has been used to study stratified turbulence. Smyth (1999), Smyth and Moum (2000), Caulfield and Peltier (2000), Staquet (2000), Smyth et al. (2001), and Peltier and Caulfield (2003) reproduced KH instability in unforced shear flow and reported the time-dependent behavior of efficiencies (their definitions of efficiencies are reviewed in Smyth et al. 2007). Caulfield and Peltier (2000) and Peltier and Caulfield (2003) introduced cumulative mixing efficiency and parameterized cumulative values during the turbulent stage as a function of Richardson number. Staquet (2000) and Smyth et al. (2001) also investigated the complex behavior of instantaneous ratios. Smyth et al. (2001) defined mixing efficiency averaged over one mixing event including a preturbulent stage, and examined dependences on the Richardson and Reynolds numbers. Smyth et al. (2007) and Carpenter et al. (2007) studied mixing efficiency in Holmboe waves. Smyth et al. (2007) suggested that net mixing efficiency is governed by the relative duration and intensity of the preturbulent and turbulent phases.
These numerical studies used unforced shear as a background flow; however, turbulence induced by internal waves may be sensitive to the time evolution of the forcing fields. KH instability associated with internal wave breaking was observed directly by Woods (1968). Internal wave breaking has been studied numerically (Lelong and Dunkerton 1998a,b; Bouruet-Aubertot and Thorpe 1999; Bouruet-Aubertot et al. 2001; Fringer and Street 2003), but there are still limitations of computer resources for reproducing and quantifying turbulence due to wave breaking in large parameter ranges.
The objective of this study is to increase our understanding of mixing efficiency in the ocean by using DNS of KH billows in a time-dependent mean flow with a view toward gaining insight into internal wave breaking. We focus on the relation between mixing efficiency and time-dependent forcing fields. We reproduce time-dependent forcing by solving the equations of motion within a tilted coordinate framework and allowing the tilt angle to vary sinusoidally in time. Spatial and temporal scales are based on the interfacial wave observations of Woods (1968), who shows that interfacial gravity waves break via shear instability in observations of the Mediterranean thermocline.
Dimensional parameters that govern the flow include the maximum buoyancy frequency N, forcing frequency ω, maximum shear ∂U/∂z, molecular viscosity ν, molecular diffusivity κ, and layer half-thickness h. According to Buckingham’s Π theorem (Kundu and Cohen 2008), we can make four independent dimensionless parameters from these six dimensional numbers: the Prandtl number Pr = ν/κ, normalized forcing frequency ω/N, Richardson number Ri = N 2/(∂U/∂z)2, and Reynolds number Re = (h2/ν)(∂U/∂z). Since we explore turbulent mixing generated by KH instability over one wavelike forcing cycle, first we evaluate the mixing efficiency as a function of two dimensionless parameters that represent forcing wave properties: the minimum Richardson number achieved in one forcing cycle, Rimin, and the frequency of tilting compared with the buoyancy frequency, ω/N. These choices allow us to investigate the behavior of mixing efficiency in energetic and persistent mixing events inferred from the ocean internal wave spectrum (Garrett 2001).
In section 2, we explain our mathematical model and numerical solution methods. Our wavelike forcing field is also explained in detail. Section 3 reviews the basic concepts for quantifying a mixing event. Section 4 describes the main results. We begin by examining the main effect of the wavelike forcing, namely, the deceleration of the mean shear after instability growth has begun. We then look at mixing statistics in various parameter ranges, including high-frequency forcing following those observed by Woods (1968) and lower-frequency forcing that we expect to occur more commonly in the ocean. Dependences on the Reynolds number and on the amplitude of the initial perturbation are also explored. In section 5, we compare DNS results with microstructure measurements via the turbulent kinetic energy equation.
2. Computation method
a. Model equations in the tilted frame
b. Numerical methods
Spatial discretization is performed via the Fourier pseudospectral method. Time stepping is accomplished using a third-order Adams–Bashforth method, except for the viscosity and diffusion terms, which are evolved analytically. Grid spacing is isotropic and designed to resolve turbulence due to the breakdown of KH billows (Table 1). Further details of the DNS model are found in Smyth et al. (2005).
c. Forcing and tilting angle
d. Initial conditions and termination criteria
Nondimensional domain lengths for the streamwise, spanwise, and vertical directions are Lx/(2h0) = 13.96, L/(2h0) = 3.49, and Lz/(2h0) = 6.98, respectively. We anticipate that acceleration phases subsequent to ωt = 2π may drive additional mixing; however, by ωt = 2π, the transition layer has invariably thickened so that billows emerging in later acceleration phases will have significantly increased wavelength. As a result, the domain dimension chosen to accommodate the initial instability will no longer be appropriate. For this reason, we terminate all simulations no later than ωt = 2π. In some cases where the forcing time scale is very long, mixing is complete before ωt reaches the final value 2π. For economy, these runs are terminated when the intensity of the scalar mixing becomes the molecular level.
3. Mixing diagnostics
4. Results
Here, we explain how Γc depends on forcing fields. First, we connect our work with previous studies by showing how mixing is suppressed in the deceleration phase. Second, we investigate the dependences of Γc on Rimin and ω/N with fixed Re0 and b. Third, we explain further the dependences of Γc on Re0 and b. Last, we focus on low-frequency forcing fields with higher Re0.
a. Effects of mean flow deceleration
The present experiments differ from previous simulations of the unforced case (e.g., Smyth et al. 2001) mainly because the mean shear that drives instability exists only for a limited time before it is decelerated by the external forcing. The mean flow is accelerated from the beginning of the run until ωt = π, and then decelerated from ωt = π to ωt = 2π. Note that the mean flow does not change sign. In the acceleration phase, KH billows grow much as they do in the unforced case. The subsequent deceleration, however, can have dramatic effects on the disturbance evolution as it reduces the mean flow kinetic energy available to drive the disturbance. We examine the effects of the deceleration phase by comparing runs with and without deceleration. To omit deceleration, we simply leave the tilt angle (8) as zero for ωt > π.
Figure 1 shows the results for the cases with Rimin = 0.08 and ω/N = 0.05 (AD7 and A1 in Table 1). After Ri drops below 1/4, a train of KH billows emerges. The computational domain accommodates two such billows as shown in Fig. 1a. The billows pair (Fig. 1b), then break and become turbulent (Figs. 1c and 1d). The effect of deceleration is clear in Fig. 1c. With no deceleration phase (A1, bottom frames in Fig. 1), the paired KH billow is sheared into two parts by the strong mean flow. With deceleration (AD7), the billow relaxes to a more circular shape. As the mean flow decelerates toward zero, the billow collapses into a field of weakly turbulent gravity waves (Fig. 1d, top), whereas without deceleration (A1), the mean shear generates sustained turbulence (Fig. 1d, bottom).
Energy budgets can help to explain these mixing events. KH instability occurs after the acceleration (Fig. 2a); during this weakly nonlinear regime, the mean flow is diminished due to deceleration and ceases to support the KH billows. Thus, the maximum potential energy is diminished between tnd = 12 and 13 (Fig. 2b). The subsequent turbulence Rei after tnd = 13 is also diminished as is the diapycnal flux
If Rimin is larger, KH billows will grow more slowly (e.g., Hazel 1972) and be suppressed by mean flow deceleration. If ω/N is larger, the faster arrival of the deceleration will suppress the growth of the KH billows. Therefore, the relative timing of the billow growth and mean flow deceleration will be important for a mixing event.
b. High-frequency forcing
In the classic observations of Woods (1968), KH billows were generated by gravity waves propagating on thin layers of strong stratification and were related to existing theories of shear instability (Miles and Howard 1964). The observed gravity waves had wavelength λ ≈ 10 m and the thickness of the interface was 2h ≈ 0.1 m. Using (5), the normalized frequency of the observed waves can be estimated as ω/N ∼ 0.1. Although our wavelike forcing does not strictly correspond to interfacial gravity waves, we choose ω/N = 0.1 as a reference with Rimin = 0.07 (AD6) and 0.08 (AD13) corresponding to the lower limit of Richardson number in Woods (1968) (Fig. 3 and Table 1). When Rimin is larger than this, there is no full development of KH billows due to the slower growth of the billows. Then,
In both cases, the deceleration phase of the high-frequency forcing strongly affects the initial growth of the KH billows (Fig. 4) and there is no pairing. Mixing is not completed within one forcing cycle. Both Rec and
c. Effects of decreased forcing frequency
In our parameter space (Fig. 3), smaller Rimin or smaller ω/N tends to make larger Rec and
Over the range of parameter values shown in Fig. 3, Γc remains within about 20% of 0.3. When preturbulent mixing is important and turbulence is suppressed by the deceleration phase, as mentioned in section 4b, Γc is higher. When Rimin is smaller, this higher Γc appears at larger ω/N because Γc is controlled by the relative timing of two events: the growth of KH billows and the arrival of the deceleration phase. The boundary between the pairing and nonpairing cases seems to be defined by this relative timing (Fig. 3). This is analogous to the boundary between the mixing and nonmixing cases, because the pairing instability is the subharmonic mode of the primary KH instability (e.g., Klaassen and Peltier 1989). For Rimin = 0.07 and ω/N = 0.07 (AD3 in Table 1), KH billows can grow faster and pair, then deceleration suppresses turbulence and makes Γc higher.
If both Rimin and ω/N are small, we could expect that a mixing event may be completed within the acceleration phase (Fig. 4). We further explore this situation with different initial conditions and will show that Γc ≈ 0.2. In preparation for this, we first examine the effects of different projected Reynolds numbers Re0 and the amplitude of the initial perturbation b showing that higher Re0 and b intensify the development of KH billows and allow us to avoid the interface thickening in the low-frequency forcing.
d. Sensitivity to Re0 and b
In addition to Rimin and ω/N, Γc depends on the projected Reynolds number Re0 and the amplitude of the initial perturbation b. Furthermore, the fact that the value for Re0 used in our DNS is smaller than typical oceanic values gives some doubt as to the generality of our results for the ocean. To examine the sensitivity to these additional parameters, we add DNS runs with different values of the projected Re0 = 500 (ADR1) and Re0 = 800 (ADR2) and different b (AD− with b = 0.05 and AD+ with b = 0.2) in Table 1. We use Rimin = 0.08 and ω/N = 0.07, where the deceleration has a significant effect on the mixing at Re0 = 300 (Fig. 3 and AD10 in Table 1).
As Re0 becomes higher, KH billows grow faster and pairing can occur. Higher Re0 can accelerate secondary instability (Klaassen and Peltier 1985); therefore, efficient mixing during pairing and breakdown is enhanced and is stronger than in Re0 = 300 runs (Table 1). ADR2 has the larger Γc (Table 1) because the deceleration suppresses mixing just after breakdown. Although mixing continues after one forcing cycle in higher Re0 runs, tilting is stopped after one forcing cycle.
For AD−, the smaller initial disturbance amplitude b makes the onset slower; thus, mixing is suppressed by deceleration at an earlier phase in billow evolution. For AD+, the larger b makes the onset faster than AD10 and promotes pairing and subsequent intense, and less efficient, turbulence. Thus, the effects of preturbulent mixing become less. This reduced importance of preturbulent mixing is also found in Holmboe wave simulations (Smyth et al. 2007). The result is that Γc becomes smaller and the intensity of the mixing is similar to that in AD7.
Next, we conduct the higher Re0 = 500, larger b = 0.2, and longer duration of forcing (ω/N = 0.05) runs (with deceleration, ADR+, and without deceleration, AR+, in Table 1) to see whether the effects of deceleration are reduced. KH billows start to develop at tnd = 9.5. There is a little difference in the maximum potential energy in the developing stage because billows can grow before the deceleration affects the mixing (Fig. 5b). During and after breakdown, Rei is high because of the faster growth of the secondary instability (12 < tnd < 13 in Fig. 5c);
e. Highly unstable, low-frequency forcing
When both Rimin and ω/N are small, mixing by fully developed turbulence is not suppressed by the deceleration, and we therefore expect to recover the canonical turbulent mixing efficiency Γc = 0.2. To test this speculation, we examine a case with low Rimin = 0.03 and low ω/N = 0.025. We also use increased values of Re0 and b to promote the rapid onset of turbulence (ADRL+ in Table 1).
KH billows start to develop and break before Rimin is achieved (tnd < 15 in Figs. 6a and 6b). Intense mixing is almost completed before deceleration begins at tnd = 20 (Figs. 6c and 6d). Because onset occurs during acceleration, the maximum potential energy is larger than before (tnd = 14.2 in Fig. 6b). Turbulence becomes intense after breakdown and the maximum value is achieved after breakdown (tnd > 16 in Fig. 6c). However, the larger
From the above results, we may infer ocean mixing that is due to a low-frequency wave with high Reynolds number as follows. The high Reynolds number can accelerate the onset of KH billows and the subsequent transition to turbulence. This process leads to efficient and intense mixing around breakdown, but this faster transition also makes the duration of turbulence longer, which tends to cancel the efficient mixing. Finally, mixing efficiency becomes ≈0.2. The lack of this long duration of turbulence may explain why both laboratory experiments (Thorpe 1973) and DNS (e.g., Smyth et al. 2001) had higher mixing efficiency (Γc ≥ 0.3). We may need to be careful not to assume 0.2 for the ocean mixing forced by high-frequency internal waves, because in that case a large fraction of the mixing can be accomplished in the highly efficient preturbulent phase.
We note that, in the longer forcing run, there could be an additional pairing if our boundary conditions allowed it, and that could alter the value of Γc. To simulate this would require more memory than is available. Instead, we address the issue indirectly by examining a case in which the first pairing is suppressed. An auxiliary simulation was conducted with Rimin = 0.08, ω/N = 0.05, and Re0 = 300, similar to AD7, but with Lx halved so as to suppress the paring (AD7x in Table 1). The result was that Γc was increased from 0.26 (AD7) to 0.32 (AD7x) because of the weaker turbulence after breakdown. This indicates that suppression of pairing results in larger Γc. Therefore, if pairing was allowed in ADRL+, we expect that Γc would be smaller.
We add two runs to see the effects of forcing parameters on the above results. One run has the same parameters except smaller Re0 (= 325) (ADL+ in Table 1). The other run has an intermediate Rimin (= 0.05) and ω/N (= 0.035) with the same Re0 (= 500) (ADRI+ in Table 1) to see effects of Rimin and ω/N. In ADL+, the pairing instability is not reproduced and one KH billow is absorbed in the interface [cf. “draining instability” in Klaassen and Peltier (1989)], but forcing accelerates the mean shear during the turbulent phase. We note that
5. Comparison with observations
The [LO]/h0 versus [LT]/h0 diagram in Fig. 7a describes how the mixing is modified by the forcing. When the rollup of KH billows is suppressed, both [LT]/h0 and [LO]/h0 are small. When the pairing instability occurs, both [LT]/h0 and [LO]/h0 are larger than in the no-pairing cases. However, for some pairing cases, mixing is not completed within one forcing cycle; thus, [LT]/h0 is large, but [LO]/h0 is small. Here [ROT] becomes smaller than for the no-pairing cases. For the long forcing runs, [LO]/h0 can have larger values within one forcing cycle, then [ROT] is comparable to in situ observations (Dillon 1982; Crawford 1986). In Fig. 7b the Γc versus [ROT] diagram also shows the dependences on the forcing fields in terms of the energy budget, as in (37). When the deceleration phase suppresses a mixing event, [ROT] is small and Γc is high. The value of [RBT]2 is around 0.07. If deceleration does not suppress turbulence until late in the event, [ROT] is higher, Γc is smaller, and [RBT]2 is around 0.15.
These results suggest the importance of the acceleration in the turbulence stage to reproducing the observed values in DNS. Since this scaling shows clear dependences on the forcing fields, application of this scaling to in situ microstructure measurements will be useful for furthering our understanding of the mixing efficiency. The in situ observations could give us some idea of the high Reynolds number turbulence, which we cannot explore by using present-day computers. We also need to compare the statistics of turbulent events and the effects of different sampling methods in the ocean and DNS (e.g., Wijesekera and Dillon 1997; Smyth et al. 2001).
6. Summary
We have investigated the dependence of mixing efficiency on forcing fields via direct numerical simulation (DNS) of Kelvin–Helmholtz (KH) instability. We used a tilted coordinate system to reproduce time-varying forcing fields with a wavelike oscillation, so that the mean shear that drives instability exists only for a limited time before it is decelerated. The cumulative mixing efficiency Γc is defined as buoyancy flux as a fraction of the energy dissipation rate, both averaged over one forcing cycle. We described Γc as a function of the projected minimum Richardson number Rimin, the normalized forcing frequency ω/N, the projected Reynolds number Re0, and the amplitude of the initial perturbation b. We showed sensitivities of Γc on these parameters as well as the details of one mixing event. We also found a limiting case in which the canonical value of the mixing efficiency 0.2 is recovered. The time-averaged Thorpe scale and the Ozmidov scale are derived from the turbulent kinetic energy equation following Dillon (1982) and Garrett (2001), and are compared with past microstructure measurements.
A mixing event that results from shear instability mixes via two distinct mechanisms that occur in sequence (e.g., Smyth et al. 2001, 2007). The first is preturbulent mixing, in which a growing, wavelike disturbance exerts a persistent, compressive strain on the scalar gradients, sharpening the gradients and thereby accelerating the diapycnal scalar flux. This mixing occurs with relatively little dissipation of the kinetic energy and is in this sense highly efficient. The second mechanism is turbulence, which develops as the instability grows beyond some amplitude threshold. In contrast to preturbulent mixing, turbulence exerts a strain field that varies rapidly in time. As a result, scalar gradients do not have time to adjust to the optimal orientation for compression (Smyth 1999). This rapidly fluctuating turbulent strain is also effective at dissipating kinetic energy, with the net result that instantaneous mixing efficiency is reduced to somewhere near the canonical value 0.2. Our central conclusion here is that, in a forced mixing event, forcing governs mixing efficiency by altering the relative duration and intensity of the preturbulent and turbulent phases.
Detailed results may be summarized as follows:
When Re0 and b are fixed, Γc varies weakly as a function of Rimin and ω/N. In this parameter space, the relation between the growth rate of KH billows and the arrival of the deceleration phase controls mixing. When Rimin and ω/N are high, there is no mixing because the deceleration phase suppresses mixing. If Rimin or ω/N is sufficiently reduced, KH billows can grow, pair, and break even though deceleration suppresses the latter part of the mixing event. Because highly efficient preturbulent mixing (Smyth et al. 2001) dominates the time-averaged diapycnal flux and less efficient turbulence after breakdown is suppressed by the deceleration, Γc has relatively high values (around 0.3).
Higher Re0 and larger b act to accelerate the initial growth and secondary instability of the KH billows (e.g., Smyth et al. 2007). KH billows can grow before the deceleration phase arrives, thus changing the intensity of the mixing as well as Γc. When both Re0 and b are higher and ω/N is smaller, the effects of deceleration become less important. However, if the forcing does not accelerate the mean flow long enough, Γc in higher Re0 runs is larger than 0.3 because of efficient mixing late in the preturbulent phase.
When Rimin and ω/N become small, KH billows onset before Rimin is achieved. Then, the intense but less efficient turbulence acts over a long time before deceleration begins and therefore contributes more to the net mixing than the highly efficient preturbulent mixing; thus, Γc is near the canonical value of 0.2, even in the case with higher Re0.
Following the length-scale arguments of Dillon (1982) and Garrett (2001), the turbulent kinetic energy equation can be written in terms of three nondimensional numbers; Γc, ROT (the ratio of the Ozmidov scale LO to the Thorpe scale LT), and RBT (the ratio of the buoyancy scale LB to the Thorpe scale) and averaged over one event. Both LO versus LT and Γc versus ROT diagrams can capture the transition from suppression to acceleration of the turbulent phase by the forcing. Those relationships are consistent with the observations in cases where turbulence is able to develop and persist before mean flow deceleration. Further tests of this scaling with in situ microstructure observations may allow us to study the dependence of mixing efficiency on forcing in high Reynolds number turbulence, which we cannot explore using present-day computers, and to verify our central conclusion in the ocean where forcing fields are more complicated.
Our study cannot investigate all of the relevant parameter ranges because of the limitations of current computer resources. Effects of high Reynolds number and multiple pairing of billows should be examined further. Effects of subsequent forcing need to be studied in high-frequency cases. An improved model of progressive interfacial waves (e.g., Woods 1968) could be attained by altering the phase relationship between the tilt angle and the mean flow such that
Acknowledgments
This work is supported by the Office of Naval Research, the Natural Sciences and Engineering Research Council of Canada, and the National Science Foundation (Grants OCE0095640 and OCE0622922). Computation has been performed on the Western Canada Research Grid (WestGrid). We gratefully acknowledge Chris Garrett for useful discussions and Jeff Carpenter at the University of British Columbia for his help with the computations. Our manuscript was greatly improved by the comments made by two anonymous reviewers.
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APPENDIX
Errors in Energy Diagnostics
The energy diagnostics and the three-dimensional density reordering computation used in this study are only strictly correct when the tilting angle is 0. Thus, we assess the errors in the diagnostics by comparing Γc from the density reordering method with the cumulative mixing efficiency Γc from the differences in the potential energies at ωt = π and ωt = 2π (end of one forcing cycle). We use run AD7 (Fig. 2) for comparison; thus, we can also assess the effects of termination before one forcing cycle is completed. The time series of Γc at each time step [t2 = t in (28)] and Γc from the potential energy difference show that there is almost no difference at ωt = 2π (tnd = 20 in Fig. A1): Γc is 0.262 and that from the potential energy difference is 0.265 (about 1% difference). We note that there is also almost no difference between Γc at tnd = 18.05 (
Density field for KH billows forced (top) with (AD7) and (bottom) without (A1) a deceleration phase. Colored values cover the middle three-fifths of the density range; higher and lower values are transparent. The actual domain length in the computation is used for the axes: tnd = (a) 11.1, (b) 12.8, (c) 14.4, and (d) 16.3.
Citation: Journal of Physical Oceanography 39, 5; 10.1175/2008JPO3927.1
Density field for KH billows forced (top) with (AD7) and (bottom) without (A1) a deceleration phase. Colored values cover the middle three-fifths of the density range; higher and lower values are transparent. The actual domain length in the computation is used for the axes: tnd = (a) 11.1, (b) 12.8, (c) 14.4, and (d) 16.3.
Citation: Journal of Physical Oceanography 39, 5; 10.1175/2008JPO3927.1
Density field for KH billows forced (top) with (AD7) and (bottom) without (A1) a deceleration phase. Colored values cover the middle three-fifths of the density range; higher and lower values are transparent. The actual domain length in the computation is used for the axes: tnd = (a) 11.1, (b) 12.8, (c) 14.4, and (d) 16.3.
Citation: Journal of Physical Oceanography 39, 5; 10.1175/2008JPO3927.1
Time series of a DNS run (Rimin = 0.08 and ω/N = 0.05). The solid line shows the run with a deceleration phase (AD7) and the dashed line that without deceleration (A1). The horizontal axis is dimensionless in time. Shown are the (a) tilting angle, (b) total (thin line) and background (thick line) potential energies normalized by
Citation: Journal of Physical Oceanography 39, 5; 10.1175/2008JPO3927.1
Time series of a DNS run (Rimin = 0.08 and ω/N = 0.05). The solid line shows the run with a deceleration phase (AD7) and the dashed line that without deceleration (A1). The horizontal axis is dimensionless in time. Shown are the (a) tilting angle, (b) total (thin line) and background (thick line) potential energies normalized by
Citation: Journal of Physical Oceanography 39, 5; 10.1175/2008JPO3927.1
Time series of a DNS run (Rimin = 0.08 and ω/N = 0.05). The solid line shows the run with a deceleration phase (AD7) and the dashed line that without deceleration (A1). The horizontal axis is dimensionless in time. Shown are the (a) tilting angle, (b) total (thin line) and background (thick line) potential energies normalized by
Citation: Journal of Physical Oceanography 39, 5; 10.1175/2008JPO3927.1
Results in Rimin and ω/N spaces for Re0 = 300. Numbers in columns show the cumulative buoyancy Reynolds number Rec, cumulative dimensionless diapycnal flux due to mixing
Citation: Journal of Physical Oceanography 39, 5; 10.1175/2008JPO3927.1
Results in Rimin and ω/N spaces for Re0 = 300. Numbers in columns show the cumulative buoyancy Reynolds number Rec, cumulative dimensionless diapycnal flux due to mixing
Citation: Journal of Physical Oceanography 39, 5; 10.1175/2008JPO3927.1
Results in Rimin and ω/N spaces for Re0 = 300. Numbers in columns show the cumulative buoyancy Reynolds number Rec, cumulative dimensionless diapycnal flux due to mixing
Citation: Journal of Physical Oceanography 39, 5; 10.1175/2008JPO3927.1
Schematic in Rimin and ω/N space.
Citation: Journal of Physical Oceanography 39, 5; 10.1175/2008JPO3927.1
Schematic in Rimin and ω/N space.
Citation: Journal of Physical Oceanography 39, 5; 10.1175/2008JPO3927.1
Schematic in Rimin and ω/N space.
Citation: Journal of Physical Oceanography 39, 5; 10.1175/2008JPO3927.1
Same as in Fig. 2, but for Rimin = 0.08 and ω/N = 0.05 with higher Re0 = 500 and b = 0.2. Solid line shows the run with a deceleration phase and dashed line shows that without deceleration.
Citation: Journal of Physical Oceanography 39, 5; 10.1175/2008JPO3927.1
Same as in Fig. 2, but for Rimin = 0.08 and ω/N = 0.05 with higher Re0 = 500 and b = 0.2. Solid line shows the run with a deceleration phase and dashed line shows that without deceleration.
Citation: Journal of Physical Oceanography 39, 5; 10.1175/2008JPO3927.1
Same as in Fig. 2, but for Rimin = 0.08 and ω/N = 0.05 with higher Re0 = 500 and b = 0.2. Solid line shows the run with a deceleration phase and dashed line shows that without deceleration.
Citation: Journal of Physical Oceanography 39, 5; 10.1175/2008JPO3927.1
Same as in Fig. 2, but for Rimin = 0.03 ω/N and = 0.025 with Re0 = 500 and b = 0.2. Only the case with a deceleration phase is shown. The deceleration starts at N0t/(2π) = 20.
Citation: Journal of Physical Oceanography 39, 5; 10.1175/2008JPO3927.1
Same as in Fig. 2, but for Rimin = 0.03 ω/N and = 0.025 with Re0 = 500 and b = 0.2. Only the case with a deceleration phase is shown. The deceleration starts at N0t/(2π) = 20.
Citation: Journal of Physical Oceanography 39, 5; 10.1175/2008JPO3927.1
Same as in Fig. 2, but for Rimin = 0.03 ω/N and = 0.025 with Re0 = 500 and b = 0.2. Only the case with a deceleration phase is shown. The deceleration starts at N0t/(2π) = 20.
Citation: Journal of Physical Oceanography 39, 5; 10.1175/2008JPO3927.1
(a) The [LO]/h0 vs [LT]/h0 diagram, and (b) the Γc vs [ROT] diagram. Runs with the deceleration phase are plotted. The open circle is Re0 = 300; the pairing is observed (AD1–AD4, AD7–AD9, and AD+). A cross is the same as an open circle but with no pairing (AD5–AD6, AD10–AD16, and AD−). The triangle is for Re0 = 500 or 800 (ADR1, ADR2, and ADR+). Diamonds are for the long forcing cases (ADRL+, ADL+, and ADRI+). Dotted lines are for ROT = 0.79 (Dillon 1982) and ROT = 0.66 (Crawford 1986). Solid lines in (a) are ROT = 0.5 and ROT = 1.0. Dashed line in (b) is RBT2 = 0.1, from Garrett (2001).
Citation: Journal of Physical Oceanography 39, 5; 10.1175/2008JPO3927.1
(a) The [LO]/h0 vs [LT]/h0 diagram, and (b) the Γc vs [ROT] diagram. Runs with the deceleration phase are plotted. The open circle is Re0 = 300; the pairing is observed (AD1–AD4, AD7–AD9, and AD+). A cross is the same as an open circle but with no pairing (AD5–AD6, AD10–AD16, and AD−). The triangle is for Re0 = 500 or 800 (ADR1, ADR2, and ADR+). Diamonds are for the long forcing cases (ADRL+, ADL+, and ADRI+). Dotted lines are for ROT = 0.79 (Dillon 1982) and ROT = 0.66 (Crawford 1986). Solid lines in (a) are ROT = 0.5 and ROT = 1.0. Dashed line in (b) is RBT2 = 0.1, from Garrett (2001).
Citation: Journal of Physical Oceanography 39, 5; 10.1175/2008JPO3927.1
(a) The [LO]/h0 vs [LT]/h0 diagram, and (b) the Γc vs [ROT] diagram. Runs with the deceleration phase are plotted. The open circle is Re0 = 300; the pairing is observed (AD1–AD4, AD7–AD9, and AD+). A cross is the same as an open circle but with no pairing (AD5–AD6, AD10–AD16, and AD−). The triangle is for Re0 = 500 or 800 (ADR1, ADR2, and ADR+). Diamonds are for the long forcing cases (ADRL+, ADL+, and ADRI+). Dotted lines are for ROT = 0.79 (Dillon 1982) and ROT = 0.66 (Crawford 1986). Solid lines in (a) are ROT = 0.5 and ROT = 1.0. Dashed line in (b) is RBT2 = 0.1, from Garrett (2001).
Citation: Journal of Physical Oceanography 39, 5; 10.1175/2008JPO3927.1
Fig. A1. Time series of Γc (Rimin = 0.08 and ω/N = 0.05). Horizontal axis is dimensionless in time. Thick dotted line is Γc from the potential energy difference between ωt = π and ωt = 2π. Thick dashed line is time when =
Citation: Journal of Physical Oceanography 39, 5; 10.1175/2008JPO3927.1
Fig. A1. Time series of Γc (Rimin = 0.08 and ω/N = 0.05). Horizontal axis is dimensionless in time. Thick dotted line is Γc from the potential energy difference between ωt = π and ωt = 2π. Thick dashed line is time when =
Citation: Journal of Physical Oceanography 39, 5; 10.1175/2008JPO3927.1
Fig. A1. Time series of Γc (Rimin = 0.08 and ω/N = 0.05). Horizontal axis is dimensionless in time. Thick dotted line is Γc from the potential energy difference between ωt = π and ωt = 2π. Thick dashed line is time when =
Citation: Journal of Physical Oceanography 39, 5; 10.1175/2008JPO3927.1
Parameters and results for numerical simulations. Here, Rimin is the projected minimum Richardson number; ω/N is the normalized forcing frequency; Re0 is the projected Reynolds number; a is the maximum tilt angle (°); b is the amplitude factor for the initial perturbation; Nx, Ny, and Nz are coarse-component grid numbers, where the grid numbers are doubled for a scalar variable; Rec is the cumulative buoyancy Reynolds number;
