In Search of Batchelor's "q" *

W.D. Smyth


*Summary of
"Dissipation range geometry and scalar spectra in sheared, stratified turbulence",
by W.D. Smyth, Journal of Fluid Mechanics 401, p. 209 (1999)



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).



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: Statistics of the normalized principal strains, taken from a typical simulation at high .

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.



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.


Figure 4: Statistics of the orientation between the vorticity and the principal strains. If the orientations were random, all three cosines would have root-mean-square average .




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

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.


Figure 5: The proportionality constants appearing in (1) and (2), as functions of the buoyancy Reynolds number.

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



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.


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 .




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



Ashurst, W., A. Kerstein, R. Kerr and C. Gibson, 1987: "Alignment of vorticity and scalar gradient with strain rate in simulated Navier-Stokes turbulence", Phys. Fluids 30, 2343-2353.

Batchelor, G.K. 1959: "Small scale variation of convected quantities like temperature in turbulent fluid", J. Fluid Mech 5, 113-133.

Bogucki, D., J. Domaradzki and P. Yeung 1997: "Direct numerical simulations of passive scalars with Pr>1 advected by turbulent flow", J. Fluid Mech. 343, 111-130.

Gargett, A. 1985: "Evolution of scalar spectra with the decay of turbulence in a stratified fluid", J. Fluid Mech. 159, 379-407.

Gibson, C. and W. Schwarz 1963: "The universal equilibrium spectra of turbulent velocity and scalar fields", J. Fluid Mech. 16, 365-384.

Grant, H.L., B. Hughes, W. Vogel and A. Moillet 1968: "The spectrum of temperature fluctuations in turbulent flow", J. Fluid Mech. 34, 423-442.

Kerr, R. 1987: "Histograms of helicity and strain in numerical turbulence", Phys. Rev. Lett. 59, 783-786.

Kraichnan, R. 1968: "Small scale structure of a scalar field convected by turbulence", Phys. Fluids 11, 945.

Newberger, P.A. and D. Caldwell 1981: "An inertial subrange in microstructure spectra", J. Geophys. Res. 86, 4264-4268.

Oakey, N. 1982: Determination of the rate of dissipation of turbulent energy from simultaneous temperature and velocity shear measurements", J. Phys. Oceanogr. 12, 256-271.

She, Z.-S., E. Jackson and S. Orszag 1991: "Structure and dynamics of homogeneous turbulence: models and simulations", Proc. Roy. Soc. London 434, 101-124.

Tsinober, A., E. Kit and T. Dracos 1992: "Experimental investigation of the field of velocity gradients in turbulent flows", J. Fluid Mech. 242, 169-192.

Williams, R. and C. Paulson 1977: "Microscale temperature and velocity spectra in the atmospheric boundary layer", J. Fluid Mech. 83, 547-567.