Length Scales of Turbulence
in Stably Stratified Mixing Layers*

W.D. Smyth and J.N. Moum

* summary of a paper of the same title published in Physics of Fluids
(vol. 12, p. 1327-1342).

 

INTRODUCTION

The proper interpretation of ocean microstructure measurements requires that we be able to estimate the age of an observed mixing event. Does a particular observation represent active mixing, or the decayed remnant of a more powerful event? Because of the limitations of present observational technology, it is not possible to make more than a single measurement of any event; we must rely instead on indirect inference.

Here, we describe a sequence of direct numerical simulations of mixing events typically observed in the ocean thermocline. The competing influences of shear and stratification on the turbulence evolution are examined in detail. Finally, we demonstrate the utility of two measurable diagnostics for estimation of the age of a mixing event.

SIMPLE MODELS OF MIXING LAYER GROWTH

Figure 1: The layer thickness, h, increases in time via turbulent entrainment. The rate of increase depends on the relative strengths of background shear and stratification, as quantified by the bulk Richardson number, Ri.

Consider the evolution of a horizontally homogeneous turbulent layer such as that sketched in figure 1. The layer thickness h increases in time due to turbulent entrainment. Effects of background shear and stratification may be understood by considering two limiting cases, shear only and stratification only (figure 2). In each case, we assume that the background shear and stratification are uniform.

Figure 2: Idealized scenarios for mixing layer evolution. (a) Sheared environment with no stratification. (b) Stratified environment with no shear. Arrows indicate flow transitions: 1 = Largest eddies deformed by buoyancy; 2 = Dissipation-range eddies deformed by buoyancy.

h = layer depth as in figure 1
= Corrsin scale, above which eddies are deformed by shear
= Ozmidov scale, above which eddies are deformed by stratification
= Kolmogorov scale; = approximate upper limit of dissipation range

In the sheared case, eddies larger that the Corrsin scale,, are deformed by the background shear so as to set up Reynolds stresses that feed energy into the turbulent fluctuations. As turbulence becomes more intense, grows while the Kolmogorov scale (, the size of the smallest eddies) decreases. As a result, the energy spectrum exhibits a monotonically spreading band of eddy sizes within which eddies are too small to be deformed by the background shear and too large to be damped by viscosity, i.e. an inertial subrange.

In the case of pure stratification, layer growth is eventually arrested due to buoyancy effects. Because there is no energy source to balance losses to buoyancy and viscosity, turbulence decays in time. The Ozmidov scale (, above which eddies are deformed by buoyancy) may be very large at t=0, but it decreases in time. Meanwhile, the Kolmogorov scale increases. In contrast to the sheared case, the inertial subrange (bounded by and ) shrinks in time and ultimately vanishes. The point at which buoyancy effects first impact the largest eddies is labelled "1" in figure 2; the point at which buoyancy effects are first felt in the dissipation subrange is labelled "2".

When both shear and stratification are present, turbulence evolution depends on the value of the bulk Richardson number (Ri, the squared ratio of stratification to shear). If Ri is less than some critical value, the flow evolution is qualitatively similar to the case of pure shear. If Ri exceeds the critical value, stratification dominates and the flow evolution resembles the case of pure stratification. The critical value of Ri is near 1/4, the critical value for linear instability (Rohr et al. 1988, Holt et al. 1992).

 

TURBULENCE IN A STRATIFIED SHEAR LAYER

In the ocean, shear and stratification are usually not uniform, but rather tend to be localized in stratified shear layers. Turbulence in this class of flows arises as the result of the growth and breaking of Kelvin-Helmholtz billows (figure 3). The turbulence life cycle exhibits similarities with both of the simple cases discussed above. Initially, turbulence grows as in the sheared case. Later, however, layer growth is arrested and turbulence decays as in the stratified case. The fundamental difference between this scenario and the uniform gradient cases described above is that the bulk Richardson is no longer constant in time, but instead increases. If the initial value of Ri is less than 1/4, turbulence grows. Eventually, though, Ri must grow to exceed 1/4. Beyond that point, stratification dominates and turbulence decays.


Figure 3: Isotherms in cross-sections of breaking Kelvin-Helmholtz billows, shown at three
successive stages of flow evolution. (a) Laminar rollup and pairing of 2D Kelvin-Helmholtz vortices.
(b) Fully developed turbulence. (c) Decayed turbulence, remnant gravity waves.

The turbulence life cycle has been examined in a sequence of eight simulations, in which the initial value of Ri was varied between 0.08 and 0.16, and the Prandtl number was varied between 1 and 7. The evolution of the layer depth is shown in figure 4a. In each run, an initial period of rapid growth of h was followed by a brief decrease as the billows collapsed. This point coincided with the transition to turbulence. Subsequent growth of the shear layer was much slower than the initial growth, and was eventually arrested by buoyancy effects.




Figure 4: Shear layer thickening as a function of time (scaled by initial shear). (a) the layer depth, h.
(b) the bulk Richardson number. The horizontal line indicates the critical value, 1/4.

The final thicknesses attained by the different simulated shear layers varied greatly due to differences in the initial conditions. The evolution of Ri was much more consistent, with final values all between 1/4 and 1/2, despite the wide range of initial conditions (figure 4b). These results are consistent with numerous laboratory experiments on Kelvin-Helmholtz billows (e.g. Thorpe 1973, Koop & Browand 1979), which show Ri asymptoting to a value near 0.33 regardless of the initial conditions.

LENGTH SCALE EVOLUTION

The evolution of the characteristic length scales is illustrated in figure 5, for comparison with the simpler cases sketched in figure 2. As we have seen previously, the layer depth increases to a finite limiting value, as in the case of pure stratification (figure 2b). The Corrsin and Ozmidov scales vary in close proportion with one another. Both grow rapidly at early times (as in the pure shear case, figure 2a), but decay at later times (as in the stratified case, figure 2b). The evolution of the Kolmogorov scale may be interpreted similarly: the early decrease reflects growth of turbulence as in the sheared case; later increase indicates turbulence decay under the influence of stratification. Beyond approximately , both the Corrsin and Ozmidov scales are smaller than , indicating that the effects of background shear and stratification are felt in the dissipation subrange. This time is indicated by the arrow on figure 5 (cf. arrow 2 on figure 2b).


Figure 5: Turbulence length scales versus time for a representative simulation of breaking Kelvin-Helmholtz billows. Time is scaled by the shear at t=0. The thickness of the mixing layer, h, is shown in red. Blue curves represent the Ozmidov scale and (twice) the Corrsin scale. The largest scale of the dissipation subrange is shown in green. The arrow corresponds to arrow 2 on figure 2b. The black curves represent the Thorpe scale based on vertical reordering (thick curve) and on three-dimensional reordering (thin curve).

Also shown in figure 5 are two variants of the Thorpe scale, which measures the length scale of a typical turbulent fluctuation. is computed in the usual way, by sorting vertical temperature profiles. is obtained by sorting the entire three-dimensional temperature field, as suggested by Winters & D'Asaro (1996). The two are nearly equal until the buoyant-inertial-viscous transition (arrow), after which begins to decay much more rapidly than . The difference lies in the fact that is nonzero only in the presence of overturns, whereas reflects temperature fluctuations of all kinds. At late times, temperature fluctuations consist primarily of stable gravity waves, which show up in but not in .

 

AGE DIAGNOSTICS IN MICROSTRUCTURE MEASUREMENTS

The length scale ratio

The ratio of Ozmidov to Thorpe scale,, provides a useful characterization of the three-dimensional structure of a mixing event that can be estimated from one-dimensional microstructure data. Large, weakly dissipative overturns such as those shown in figure 3a have large Thorpe scale and small Ozmidov scale, i.e. small . Figure 6 shows an observed example of a turbulent patch dominated by one large overturn. In this instance,=0.4. The opposite limit (small Thorpe scale, large Ozmidov scale) is attained when the flow is dominated by small overturns with large dissipation, e.g figure 3c.


Figure 6: Temperature profile and local Thorpe displacement records for a turbulent patch
observed using the Chameleon microstructure profiler.

The application of as an age diagnostic has been complicated by the fact that we don't really know how this ratio evolves in time. Gibson (1987, 1998) has argued that newly-formed billows are strongly turbulent and should therefore have large despite also having large Thorpe scale. The late stages of turbulence evolution should be composed of "fossilized" overturns that are no longer actively mixing and therefore have small . Wijesekera and Dillon (1997) and others have argued that the evolution is the reverse: evolves from small to large values as a mixing event ages. The present results support the latter view: the evolution is from small to large .

A feature of particular interest appeared in one of the simulations shown in figure 7 (see arrow; also see figure 5). There, a new set of billows formed spontaneously during the decay phase as the result of locally small Ri. As with the original billows, this new billow was characterized by decreased .


Figure 7:
The ratio of Ozmidov to Thorpe scale as a function of time. (Time is scaled by the initial shear.)
The four ranges of shown by horizontal lines correspond to classes 1-4 in Wijesekera & Dillon's
classification of observed turbulent patches. The arrow indicates an example of secondary billow formation.

These results are not conclusive: alternative models of ocean mixing events may yield different evolutionary patterns for. Scatter at late times in figure 7 is a low Reynolds number effect that should be removed in larger-scale simulations (now underway). Nevertheless, the present results suggest strongly that small is a reliable indicator of young turbulence.

Mixing efficiency

The early and late stages of billow evolution may also be distinguished on the basis of flux Richardson number, (figure 8). This quantity is the ratio of the rate of irreversible transfer of kinetic to potential energy due to mixing to the input of kinetic energy from the mean flow, and is therefore also identified as the mixing efficiency.


Figure 8: Flux Richardson number (mixing efficiency) versus time.

In the late stages of the present simulations,approaches the "standard" value 0.2 (smaller at high Pr). In the early stages, however, is much larger than this. Young billows feature extremely sharp scalar gradients and are therefore highly efficient at mixing. It is interesting to note that this efficient mixing is actually a consequence of the absence of turbulence: the sharp scalar gradients arise because the strain field is steady and coherent.

 

CONCLUSIONS

Acknowledgement: This research was sponsored by the National Science Foundation.

 

References