Billow evolution from a point source:

a mechanism for banded clouds and turbulent patches

 

Summary of:

"Kelvin-Helmholtz billow evolution from a localized source'',

W.D. Smyth,

submitted December 14th, 2003, to Q. J. R. Met. Soc.

 

 

 

Figure 1: Banded clouds over Corvallis, OR.

 

 

Introduction

 

A glance into the sky during early morning or late afternoon often reveals clouds arranged in parallel bands (figure 1). The observer lucky enough to see such clouds from the side may find that they have a vortical shape strongly reminiscent of breaking waves on a water surface (figure 2). Such clouds represent a visible example of Kelvin-Helmholtz (KH) billows. They form when two layers of air, separated by a horizontal interface, are in motion relative to one another. The vorticity between the layers rolls up to create billows aligned perpendicular to the ambient flow. The motion may be made visible to the naked eye either pre-existing clouds advected by the billowing motion, or by clouds created directly in the rising parts of the billow. KH billows are common not only in atmosphere but also in the oceans, where they are most readily observed in echosounder images.

 

 

 

 

Figure 2: Side view of Kelvin-Helmholtz billows made visible by a cloud layer over Denver, CO (Colson 1954).

 

 

 

Billow patches

 

The simplest theory for KH billows assumes that billows have infinite extent in the lateral dimensions. This idealization is not obeyed in nature. In the atmosphere, banded clouds often appear in quasi-elliptical patches (figure 3). In the oceans, KH billows are often associated with volumes of turbulent fluid whose lateral extent is finite but greatly exceeds their thickness (Smyth et al. 2001).

 

 

 

 

 

Figure 3: A patch of banded clouds over Dunstable, UK (Ludlam 1967).

 

The finite extent of a billow patch is an expression of lateral inhomogeneity, either in the background conditions that drive the instability, or in the quasi-random disturbances from which billows grew. If the background conditions are laterally homogeneous, then a localized disturbance will seed an elliptical billow patch that expands in time. Simultaneously, the billows within the patch grow, overturn, break and become turbulent. My purpose here is to give a theoretical description of the growth of a billow patch under these conditions.

 

 

Theoretical results

 

The linear theory of Kelvin-Helmholtz instability predicts that billows will take the form of plane waves and will grow exponentially at a rate (e.g. Hazel 1972). The wave vector is oriented in the horizontal, and the growth rate is a function of the wave vector components k and l (k being the component parallel to the background flow). The growth rate also depends on the background conditions, primarily through the bulk Richardson number, J, that describes the ratio of stratification to shear. Billows grow only when .

 

Based on this linear theory, one may construct an approximate solution of the initial value problem that describes the response to a point disturbance. The initial wavenumber spectrum is white, i.e. all values of k and l are present with equal amplitudes. As the billows grow, the wave vector for which the growth rate is maximized will come to dominate the pattern. That plane wave will be modulated by an elliptical envelope whose characteristics depend on the way in which decreases away from its maximum, which is in turn a function of J. The result of this calculation allows prediction of both the shape and the growth rate of the billow patch.

 

The aspect ratio of the elliptical envelope remains constant as the patch expands. That aspect ratio (defined as the ratio of the cross-stream (y) to the streamwise (x) extent of the envelope) is given by . Thus, in the limit of weak stratification ( ), the billow patch is elongated by about 40% in the direction of the background flow. At the opposite extreme (the value beyond which billows do not grow), the envelope is circular.

 

After an initial adjustment period, the axes of the ellipse grow linearly in time. The semi-major axis is closely approximated by , where is the velocity change across the transition layer. Patches grow fastest in strongly unstable (low J) conditions. However, the ratio of the expansion rate to the growth rate of the billows is greatest when J approaches 1/4. Therefore, at a given stage of billow growth, a strongly stratified patch will contain more billows than a weakly stratified one.

 

 

Validation

 

The theoretical results outlined above depend critically upon the linear perturbation theory that describes small fluctuations in flow speed and density; however, the billows ultimately grow beyond the domain in which linear theory is valid. Nevertheless, the results may be expected to be valid because envelope growth is controlled mainly by the region near the outer edge of the envelope, where billow amplitude remains small.

 

This expectation is verified in figures 4 and 5, which show results from a fully nonlinear simulation of a billow patch growing from a point disturbance. Both the growth rate and the aspect ratio of the predicted envelope compare well with the nonlinear results.

 

 

Figure 4: Sequential cutaway views from a large-eddy simulation of a patch of KH billows that evolved from a point disturbance.

 

 

Figure 5: Comparison of computed envelope growth with theoretical prediction. (a) Colors indicate density at y=z=0 for the run shown in figure 4; curves show the semimajor axis of the theoretical envelope. (b) Colors show density at z=0 for the second frame shown in figure 4; curve shows the theoretical envelope.

 

 

 

Discussion

 

Patches arising from a point source, as described above, have much in common with those observed in nature (compare figures 4 and 3). Any billow patch exhibiting an elliptical envelope may have grown from a localized disturbance. Of course, every billow patch is eventually constrained by the lateral extent of the unstable region, which is in turn determined by whatever large-scale process drives the shear. For example, Figure 6 shows a train of billows having aspect ratio far in excess of the maximum predicted here. The present results therefore indicate that that patch was constrained by inhomogeneity in the ambient flow, at least at the time the photograph was taken.

 

 

 

 

Figure 6: Composite photo of an elongated billow patch over Lake Garda, Italy (Ludlam 1967).

 

 

In summary, instability growth from a point disturbance is an interesting and possibly realistic model for billow patches in the atmosphere and oceans. Issues meriting further investigation include the potential for gravity wave radiation as the patch grows, the effects of finite patch extent on the processes leading to turbulence, and the application of the theory to other background flow profiles such as stratified jets.

 

 

References

 

Colson, D., 1954, "Wave cloud formation at Denver ", Weatherwise 7, 34-35.

 

Hazel, P., 1972, "Numerical studies of the stability of inviscid parallel shear flows " J. Fluid Mech., 51, 39-62.

 

Ludlam, F.H., 1967, "Characteristics of billow clouds and their relation to clear air turbulence", Q. J. R. Met. Soc. 93, 419-435.

 

Smyth, W.D., J.N. Moum and D.R. Caldwell, 2001: "The efficiency of mixing in turbulent patches: inferences from direct simulations and microstructure observations.", J. Phys. Oceanogr. 31, 1969-1992.