"Dissipation range geometry and scalar spectra in sheared, stratified turbulence",

by W.D. Smyth,

**INTRODUCTION**

Ocean mixing processes depend critically on the physics of the dissipation subrange. In this project, direct numerical simulations of breaking Kelvin-Helmholtz billows (figure 1) provide a database for an intensive study of the alignment statistics that reveal the mechanics of turbulence and mixing.

Figure 1: *Stages
in the evolution of a dynamically unstable stratified shear layer
as revealed by isosurfaces of the enstrophy field. (a) Laminar
rollup and pairing of 2D Kelvin-Helmholtz vortices. (b) 3D secondary
instability triggers the transition to turbulence. (c) Fully developed
turbulence. (d) Turbulence has decayed, leaving multiple vortex
sheets and weak gravity waves.*

The ultimate objective is to compute Batchelor's universal constant, *q*, through which the geometry of the turbulent strain field governs the equilibrium spectrum of scalar gradient fluctuations. A practical application of the scalar gradient spectrum arises in attempts to estimate the turbulent kinetic energy dissipation rate, e , from temperature microstructure. This technique was standard until shear probes came into common use, and is finding renewed importance in measurements of turbulence via remote sensing. The success of such measurements depends on the accuracy of the value used for *q *(more so because the resulting estimate of e is proportional to ) *. *Estimates have ranged from 2 to 12, and it has been suggested that the value is not universal after all (Gargett 1985).

**EIGENVALUES
OF THE STRAIN TENSOR**

The local behavior of the strain tensor, , can be characterized by its eigenvalues (or principal strains), a *, **b** *and g , and corresponding eigenvectors. The three eigenvalues sum to zero (in incompressible flow); a is positive and represents dilation, g is negative and represents compression, and *b* can take either sign. The principal strains are related to the kinetic energy dissipation rate by , where n is the kinematic viscosity.

Figure 2: *Evolution of the volume-averaged principal
strains for a representative simulation. The eigenvalues are normalized
by the Kolmogorov strain rate, . The shaded curve
represents the buoyancy Reynolds number.*

Evolution of the principal strains, scaled
by , is shown in figure
2 for a typical simulation. Also shown
is the buoyancy Reynolds number, , in which *N
*is the buoyancy frequency. At early and late times, *b* approaches zero, while a and g approach their two-dimensional values, ± 1/2. During the intervening period of strong turbulence (shown by elevated values of ), the averaged eigenvalues depart from their two-dimensional limits, with b becoming positive.

Figure 3:

At large
, the statistics of the
principal strain (figure
3) approach a state that is familiar
from previous studies (both experimental and numerical) of homogeneous,
isotropic turbulence (e.g. Ashurst *et al.* 1987, Tsinober
*et al.* 1992, She *et al.* 1991). In particular the
modes fall very close to the standard ratio for
isotropic flow. These results confirm that the geometry of the
strain field at high Reynolds number displays a common form in
widely disparate flow types.

**ALIGNMENT STATISTICS**

Much has been learned about the dynamics of turbulence through studies of the alignment between the vorticity and the local principal axes of the strain field. The importance of vortex stretching would lead one to expect that vorticity would tend to align with the eigenvector corresponding to the extensional strain a*. *It turns out, however, that vorticity aligns more closely with the intermediate eigenvector - that corresponding to b (e.g. Ashurst *et al*. 1987).

Here, I find the same result (figure 4), even though my large-scale flow geometry is very different from the idealized model used in the previous studies.

**THE SCALAR
GRADIENT SPECTRUM**

Under the Batchelor (1959) scaling, the scalar gradient spectrum becomes universal when the wavenumber is scaled by the Batchelor scale, , where is the molecular diffusivity and is the rate at which scalar gradients are amplified by the strain field.

Batchelor made two assumptions in order to representin terms of measureable quantities.

1. First, he assumed that is proportional to . This left the problem of estimating the proportionality constant.

2. Although could not be measured, g could. To take advantage of this, Batchelor assumed that the scalar field was in equilibrium with the strain field, so that the scalar gradient would be aligned with the compressive strain, in which case, . This resulted in the following parameterization
for the effective strain rate:

Based on measurements of g, Batchelor estimated a value of 2.0 for the constant .

Nowadays, we have sufficient confidence
in the validity of Batchelor's theory that we reverse the logic:
we regard as known, and use measurements of the spectrum
to estimate e . However, the value of is in
fact quite uncertain (e.g. Gargett 1985). ** The hypothesis
here is that the uncertainty in the value of derives
from Batchelor's second assumption, **and the goal is therefore
to relax that assumption.

Using the numerical database, I compute directly, then assign an "effective" value of the proportionality constant, so that

. (2)

Figure 5 shows values for both and as functions of buoyancy Reynolds number. It is clear that and are very different quantities. Batchelor's constant, , is consistently close to the accepted value of 2.0. However,does not provide a good estimate of . The latter appears to asymptote to a value near 7 at large, and exceeds this value greatly at smaller .

In other words, *the rate at which scalar gradients are compressed is, on average, much slower than the ideal value *g* .* This is because the turbulent strain field
evolves too rapidly for the scalar gradient to maintain its equilibrium
orientation with the direction of maximum compression.

These data points were compiled from eight separate simulations having different initial conditions and Prandtl numbers ranging from 1 to 7.

**COMPARISONS**

Figure 6 shows comparisons between scalar gradient spectra computed from the model data and the functional form predicted by Batchelor (1959). Also shown is the functional form predicted by Kraichnan (1969) on the basis of the Lagrangian History Direct Interaction Approximation (LHDIA). In each case, a best fit value was used for the adjustable constant (denoted and , respectively).

Figure 6: *Streamwise
wavenumber spectra of the streamwise temperature gradient, plotted
using the Batchelor scaling. Colored curves show the theoretical
forms suggested by Batchelor (blue) and Kraichnan (red), with
adjustable constants as given in the legend. (a) and (c) exemplify
the good fits to the Kraichnan spectrum found when is
large.*

For cases in which , the fit to the Kraichnan spectrum is excellent (figures 6a and 6c), and matches to within statistical error. The fit to the Batchelor spectrum is not as good, and neither spectrum fits well at low (figure 6b). When , processes other than strain and diffusion influence the scalar spectrum, so that neither theoretical form is valid.

Comparison with previous observational and numerical studies reveals no inconsistency (table 1). Previous results show widely scattered values of , with no evident dependence on . (All existing observational studies correspond to , so the present results predict no trend.) Of eight previous estimates, six are smaller than our computed value and two are larger.

Source | |||

predicted value () | > 100 | 7.3 ± 0.4 | |

present spectra | 100 – 1000 | 4.9 ± 1.0 | 6.8 ± 1.4 |

Williams & Paulson (1977) | ¥ | 6 |
8^{(a)} |

Bogucki et al. (1977) | ¥ | 3.90 ± 0.25 | 5.26 ± 0.25 |

Gibson & Schwarz (1963) | ¥ | 2 | 3 |

Gargett (1985) class A | 34000-63000 | 12 |
16^{(a)} |

Oakey (1982) | 1000-2000 | 3.7 ± 1.5 |
4.9 ± 2.0^{(a)} |

Gargett (1985) class B | 50-1750 | 4 |
5^{(a)} |

Newberger & Caldwell (1981) |
NA^{(b)} |
4.95 (4.28, 6.65) | 6.5 (5.6, 8.8) |

Grant et al. (1968) |
NA^{(b)} |
3.9 ± 1.5 |
5.1 ± 2.0^{(a)} |

**Table 1: ***Comparison of numerical and observational
estimates of Batchelor's constant derived by fitting to theoretical
spectra. Superscripts: *^{(a)
}*Values of were estimated as . *^{(b)
}* could not be estimated explicitly, but
the value is much greater than .*

**CONCLUSIONS
**

- Present computer technology can accomplish
direct simulations of turbulence with
*Re*large enough to deliver results characteristic of a high-*Re*limit in stratified, sheared environments typical of the ocean thermocline.

- Turbulent Kelvin-Helmholtz billows exhibit
strain eigenvalues and alignment statistics that compare well,
at sufficiently high , with results from
laboratory experiments and simulations of very different flow
types. This suggests the existence of a universal flow geometry
in the dissipation range.

- The LHDIA form of the spectrum (Kraichnan
1968) is valid at high , provided that the
adjustable constant is replaced byto
account for the imperfect alignment between the scalar gradient
and the strain field.

- The revised value of the constant is , considerably larger than Batchelor's original estimate of 2, but in generally good agreement with observational estimates.

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

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