Observations of a Kelvin-Helmholtz Billow in the Ocean

Transcription

1 Journal of Oceanography, Vol. 57, pp. 709 to 721, 2001 Observations of a Kelvin-Helmholtz Billow in the Ocean HUA LI and HIDEKATSU YAMAZAKI* Department of Ocean Sciences, Tokyo University of Fisheries, Tokyo , Japan (Received 23 March 2001; in revised form 20 August 2001; accepted 21 August 2001) We identified a Kelvin-Helmholtz billow from vertical turbulence velocity and instantaneous heat flux signals obtained from airfoil shear probes and thermistors mounted on a research submarine. The vertical turbulence velocity indicates that the horizontal scale of the billow was about 3.5 m. The spectral slope of the vertical turbulence velocity component is close to 2, revealing the flow is two-dimensional. We show a remarkable agreement between the length scales of the observed billow and those computed from direct numerical simulations based on similar conditions. Keywords: Kelvin-Helmholtz billow, shear instability, length scale. 1. Introduction Much evidence from numerical simulations (Klaassen and Peltier, 1985; Smyth and Moum, 2000a, b; Staquet, 2000) and laboratory experiments (Thorpe, 1973, 1985; Lawrence et al., 1991; Strang and Fernando, 2001) has shown that Kelvin-Helmholtz (KH) instability is an important mechanism creating shear-induced turbulence. However, field observations of KH instabilities are scarce, mainly due to the difficulties involved in the development of proper systems to measure the necessary parameters as well as the transitory nature of the phenomenon (De Silva et al., 1996). The most notable field effort has been the dye visualization experiment conducted in the Mediterranean by Woods (1968). The photographs from this study showed complete roll-up billows attributed to KH instabilities in the summer thermocline. Subsequently, Thorpe and Hall (1974) and Thorpe et al. (1977) used a thermistor chain to show roll-up thermal structures similar to KH billows in Loch Ness. Itsweire and Osborn (1988), however, pointed out that these thermal features can be caused by breaking internal waves. Hebert et al. (1992) also identified a similar billow feature from a thermistor chain towed in the equatorial central Pacific. Seim and Gregg (1994), observing naturally occurring shear instabilities with acoustic backscatter, provided two-dimensional pictures of billows; a free-falling microstructure profiler and ADCP simultaneously obtained the background mean flow and dynamic descriptions of turbulence within the billows. No previous effort to observe KH billows from a horizontally moving platform has been reported. * Corresponding author. hide@tokyo-u-fish.ac.jp Copyright The Oceanographic Society of Japan. We used the United States Navy research submarine Dolphin to study microstructure in the seasonal thermocline off San Diego, California in Submarines are ideal platforms to measure shear instability in a thermocline due to their stable horizontal movement. In this paper, we identify and describe a KH billow in terms of vertical velocity, temperature and local gradient Richardson number. We then use numerical simulations to compare our field observations with the Direct Numerical Simulation (DNS) results for KH billows of Smyth and Moum (2000a). 2. Field Measurements 2.1 Instrumentation Two sensor packages were mounted on a tripod on Dolphin s hull (51 m long and 5.6 m in diameter) forward of the conning tower to measure turbulence. One package sensed turbulence directly with two airfoil shear probes, two fast-response thermistors (FPO7), a Viatran pressure transducer, and three fast-response accelerometers. The accelerometers were aligned longitudinally (x), vertically (z) and athwartships (y) to monitor instrument vibration. The parameters recorded were: velocity shear signals w/ x and v/ x (measured by the two airfoil shear probes in the two directions orthogonal to submarine movement), fast temperature gradient signal, T / x, measured by the thermistors, and the three acceleration signals (sampling rate 512 Hz). Note that the v component is athwartships, and w is the vertical component with positive values upward. No u component was measured. The second package consisted of a Seabird CTD with SBE4 conductivity and SBE3 temperature sensors sampling at 64 Hz. Six FPO7 thermistors, sampling at 256 Hz, were spread along the front leg of tripod to act as a thermistor chain to monitor fine scale thermal structure. 709

2 Fig. 1. Schematic diagram of instrument layout on the research submarine Dolphin. A frontal view of the velocity and temperature probes is shown at the top-right (scale in meters). An RD Instruments 1.2 MHz acoustic Doppler current profiler (ADCP) with one-meter vertical resolution was mounted on the hull 4.74 m in front of the tripod center. The ADCP averaged five pings per second. The mean ADCP velocity components U, V, and W have the same coordinate directions as the turbulence velocity components u, v, and w. The speed of the submarine relative to the ambient water was estimated from the U component speed averaged over 4 bins. For the data presented here, this speed was about 1.2 m s 1. The instrument layout is shown in Fig Experiment background The experiment was conducted on March 10, 1988, off the coast of San Diego. Dolphin submerged at 0904 UTC (2047 PST) and the experiment was completed at 1133 UTC. The data came from a part of the dive (1200 seconds) when the submarine was between 30 and 40 m deep. The vertical hydrographic structure obtained from four quasi-profiles during two ascending and descending dives of the submarine clearly showed that the effects of convective cooling were limited to the upper 30 m (Yamazaki, 1996). As the water depth exceeded 800 m, the effects of topography are ignored. The detailed hydrographic background can be found in Yamazaki and Osborn (1993). The local buoyancy frequency, N, was estimated from the thermistor chain data as discussed in Yamazaki and Osborn (1993). The local mean shear S was estimated from the ADCP data as follows: S = ( du / dz) + ( dv / dz). ( 1) The term in angle brackets represents an average over 10 seconds. The spatial derivatives of U and V were obtained from a cubic spline after smoothing the profiles of U and V with a five point triangular filter. A threshold value of S 2 = s 2 was chosen to identify high shear regions. The turbulence velocities, w and v, and temperature fluctuation, T, were reconstructed from w/ x, v/ x and T + T/ x. The method to reconstruct these signals is similar to that used by Wolk and Lueck (2001), in which turbulence velocity is extracted by using an integration operator on shear signals. The integration is achieved by low pass filtering the time series with a first order Butterworth filter with cut-off frequency f c = 1/2.3t int. We set t int, the integration time, to 4 seconds. This corresponds to a maximum length scale of = 4.8 m at a cut-off frequency of 0.1 Hz, as the submarine moves at about 1.2 m/s. The FPO7 temperature signals were first calibrated against the SBE-3 temperature sensor. To improve the FPO7 s resolution, we combined the signal with its scaled derivative (Mudge and Lueck, 1994). The enhanced temperature signal is then recovered by applying a first order Butterworth filter with cut-off frequency f c = 1/2π to the combined signals. The anti-derivative of the gradient temperature (to get the temperature fluctuation T ) is then obtained by applying a high pass filter with a cut-off frequency of f c = 1/2.3t int, with t int = 4 seconds. The intensity of turbulence was investigated by computing the turbulence kinetic energy dissipation rate using the semi-isotropic formula of Yamazaki and Osborn (1990), v 2 w 2 ε = v , ( 2) x x 710 H. Li and H. Yamazaki

3 where v is the kinematic viscosity. Although the formula gives the lower bound of axisymmetric turbulence and may cause underestimation of the dissipation rate, Eq. (2) is within 10% of the isotropic estimate (Yamazaki and Osborn, 1990) for large buoyancy Reynolds number Re b (Re b = ε/vn 2 ). According to a recent theoretical investigation, this formula may underestimate the dissipation rate up to 30% (C. Rehmann, personal communication). The spatial response correction of the shear probe is used to recover the lost variance in high wavenumbers (Ninnis, 1984). We also estimated the temperature variance dissipation rate by using the following isotropic formula: 2 χ = 6D T, () 3 x where D is the molecular diffusivity of heat. Because FP07 thermistors do not resolve the full spectrum of gradient temperature at high wavenumber, the estimated χ is usually less than the true value. To correct for the lost variance, we fitted the spectrum of the temperature gradient in the inertial sub-range to the universal Batchelor spectrum using the estimated kinetic energy dissipation rate. Details of computing χ are given in Appendix. 3. Field Observations of the KH Billow 3.1 Conceptual model of a KH billow based on heat flux signals Previous investigations of KH instability attempted to identify billows by looking for their thermal structures within the background temperature field (Thorpe and Hall, 1974; Thorpe et al., 1977; Hebert et al., 1992; De Silva et al., 1996). However, the thermal structure of a KH billow is difficult to differentiate from internal wave breaking (Itsweire and Osborne, 1988). The problem can be presented as the following question: Do KH billows have unique features, which can be measured and/or computed, that distinguish them from other geophysical flows? To identify such features, we performed a thought experiment using an idealized KH instability process. Suppose a two-layer system is subjected to a strong shear. Before the onset of KH instability, vertical velocity and temperature measured at the interface show no heat flux (Fig. 2a). In the early stage, when small amplitude waves grow and become unstable, gradient heat flux is generated (Fig. 2b), and the paired gradient heat flux segments are separated by a region of zero flux. After the instability forms a rolled billow, counter-gradient heat fluxes Fig. 2. Schematic diagram of the instantaneous heat flux signals corresponding to different development stages of a KH billow. appear, but with relatively low wavenumber fluctuating structures (Fig. 2c). Thus, up to this stage, both the dissipation of kinetic energy, ε, and of temperature variance, χ, should be rather low. At an advanced stage secondary instabilities may develop, as mentioned in Thorpe (1987) (Fig. 2d). The billow can grow to some extent until the stabilizing buoyancy forces are sufficient to suppress them; the billow will subsequently break down into turbulent patches (Fig. 2e). At this stage, the instantaneous heat flux signals show high wavenumber structures and we expect both ε and χ to have high values. The turbulent patch resulting from the collapsed billow is difficult to differentiate from turbulence generated by other mechanisms. Using heat flux signals during the early stage of growing billows which showed a paired gradient heat flux separated by a zero flux region, and the combined necessary condition of a critical Richardson number, we should be able to identify the onset of KH billows. 3.2 Identifying oceanic KH billows Using this conceptual model we applied the following rules to identify an early stage billow: (1) Identify a water column where the local gradient Richardson number falls below 1/4; Observations of a Kelvin-Helmholtz Billow in the Ocean 711

4 Fig. 3. (a) Temperature isotherms; contour intervals are 0.05 C. (b) Shear contours; two levels are shown. The thick line both in (a) and (b) indicates the path of the turbulence package. (2) Identify the distinctive heat fluxes signal depicted in Fig. 2b, namely a paired gradient signal with zero flux in the middle. We investigated all the available data segments, equivalent to m horizontal distance, to select dynamically unstable portions where Ri < 1/4. One segment, at elapsed time second, showed vertical turbulence velocities with a ramp structure (Fig. 4b). This segment was in a well-defined shear layer at the depth where we obtained turbulence data (Fig. 3b). This spot is close to the crest of an internal wave (Fig. 4f), a position favorable for enhanced shear instability. Yamazaki (1996) used the same data to argue that these internal waves were generated by an intrusion originating from the gravitational collapse of mixing water adjacent to this region. The Richardson numbers around this spot were about 0.2, slightly less than the critical value, therefore shear instability is likely to occur from a small disturbance. Since the sharp increase in velocity signal at 1442 seconds is characteristic of shear-induced deformation of a fluid, the vertical velocities are not likely to be associated with internal waves, but with a feature like a KH billow. This is because waves cannot generate sharp jumps in a velocity field unless they are highly nonlinear. This implies that the identified feature is a nonlinear growing internal wave that represents the initial stage of the KH billow. The ramp structure in the velocity signal is not reflected in the temperature signal (Fig. 4g). There are, however, abrupt changes in temperature structure prior to the velocity ramp and at its trailing edge. We also compared the heat flux across the ramp structure to that expected for a KH billow in the early stages of development (Fig. 2b). As Fig. 4h shows, the instantaneous heat flux signal within this segment is similar to that of Fig. 2b. The left hand peak is clearly a single peak, but the right hand side comprises multiple peaks. Co-existing processes probably hinder the appearance of a clear peak. Thus, both the local Richardson number and the heat flux signals meet the two conditions expected for an early stage KH billow. The lack of a ramp structure in the horizontal turbulence velocities (Fig. 4d) in this region is consistent with the two-dimensional structure expected during the initial stage of a KH billow. Note that the observed v direction, namely y, is parallel to the core axis of the KH billow. We conclude that the data represent the initial stage of a KH billow. If we judge the presence of the billow only from the heat flux signals, the left hand edge starts at about 1441 s and ends at the right hand edge at about 1445 s. But at 1444 s, the temperature ramp structure abruptly changes slope and the vertical velocity signals approach zero. In addition, at 1445 s the local Richardson number increases to above the critical value. Thus we estimate that the billow portion is between elapsed time 1441 and 1444 seconds. As the conceptual model (Fig. 2) suggests, both the kinetic energy dissipation rate ε and the temperature variance dissipation rate χ are low within the billow section (elapsed time in Figs. 4i and 4j), with values of O(10 9 ) W kg 1 and O(10 9 ) C 2 s 1, respectively. These values are close to the noise level of the probes (ε noise ~ W kg 1, χ noise ~ C 2 s 1 ), thus the billow section data are consistent with our conceptual model developed in the previous section. 4. Properties of Observed KH Billow 4.1 Horizontal scale of the KH billow Given the time of 3 seconds required to pass through the ramp structure (1441 to 1444 s) and assuming that the probe measured the core of the billow, the approximate horizontal size of the billow is about 3.5 m (submarine speed was about 1.2 m s 1 ). We do not know which direction or what portion of the billow the probe passed through, so if our assumption is correct, our estimate is an upper bound. On the other hand, if we are measuring just a part of the billow, our estimate can be an underestimated value. Three observed reasons suggest our estimate is reasonable: 1) we confirmed paired heat flux signals with zero flux in the middle; 2) the observed v component within the billow is almost zero; 3) the observed section locates at a sharp density interface (Fig. 3). Also, note that our estimate is comparable to the billow size revealed from a numerical experiment using a similar 712 H. Li and H. Yamazaki

5 Fig. 4. Hydrographic conditions of the region within which the KH billow was observed. Vertical dash lines show the billow region. (a) Local gradient Richardson number. (b) Vertical velocity fluctuation w. (c) Gradient velocity w/ x. (d) Horizontal velocity fluctuation v. (e) Gradient velocity v/ x. (f) Temperature contours. The thick solid line denotes turbulence package depth. (g) Temperature fluctuation T. (h) Instantaneous heat flux wt. (i) Turbulence kinetic energy dissipation rate ε. (j) Temperature variance dissipation rate χ. dynamic condition. The numerical simulation is introduced in the next section. 4.2 Velocity and temperature spectra of KH billow formation Kolmogorov s universal equilibrium theory states that the energy of stationary, homogenous and isotropic turbulence forms a cascade from large scale eddies to smaller eddies. The spectrum in the inertial subrange between the largest scales, where the energy is injected, and small scales, where viscosity can no longer be ignored, follows a 5/3 power law. When the energy-containing scale of turbulence exceeds the Ozmidov scale, however, stratification suppresses flow in the vertical direction, resulting in anisotropic flow. Osborn and Lueck (1984) and Yamazaki (1990) present the effects of buoyancy forces on the shape of the power spectrum. What is the shape of the power spectrum for KH billows at the onset of generation? Figure 5 shows the velocity spectrum for the observed billow. The slope of the vertical velocity spectrum at low wavenumbers is close to 2, rather than 5/3. The slope value of 2 suggests that the onset of the KH billow is indeed indistinguishable from a nonlinear internal wave mode. The horizontal velocity (v, perpendicular to the direction of submarine motion and parallel to the axis of the billow) spectrum slope is much flatter, with a power level comparable to a non-turbulent water column in which the dissipation rate is no more than O(10 9 ) W kg 1. Hence the vertical velocity spectrum indicates that the flow state is two-dimensional turbulence. This is consistent with our conceptual model for the initial stage of the KH billow. The temperature gradient spectrum shows that the potential energy associated with the KH billow is in the low wavenumber range. Although the spectral level in mid-range wavenumbers agree with the Batchelor spectrum computed from the observed χ and ε, the high Observations of a Kelvin-Helmholtz Billow in the Ocean 713

6 Fig. 5. Turbulence velocity power spectra for the KH billow in Fig. 4. Thin solid curve - vertical velocity; dashed curve - horizontal velocity; bold thick curve - horizontal velocity power spectrum from an adjacent calm area with ε < 10 9 W kg 1. The straight solid line corresponds to a slope of 2 and straight dot line is a slope of 5/3. wavenumber range does not have the expected shape. Thus the temperature field does not follow the Batchelor spectrum. Furthermore, the low wavenumber range considerably exceeds the theoretical level (Fig. 6). We conclude that the Batchelor spectrum scaling is not applicable to the temperature field for the onset of the KH billow. 4.3 Length scales associated with the KH billow We exploited several length scales associated with both the billow and non-billow sections of our data to identify features unique to the billow in those length scales. These are the Ozmidov scale, the buoyancy length scale, and the Ellison scale, defined as follows: Ozmidov scale (L o ): / 3 Lo = ( ε / N ) 12, ( 4) where N is the buoyancy frequency. L o represents the largest possible turbulence scale allowed by buoyancy forces. Any motion larger than this scale would be restricted to oscillating or wave-like motions. This argument for this is based on potential energy (vertical displacement) and so it only restricts the vertical eddy scale. The horizontal scale is not restricted by the Ozmidov scale. Although the Ozmidov scale is often used to estimate the turbulent energy containing scale, strictly speaking, this scaling is only justified for decaying turbulence within a region of uniform stratification (Fernando and Hunt, 1996). Fig. 6. Temperature gradient power spectra for the KH billow in Fig. 4. Bold solid curve - Batchelor spectrum corresponding to ε = W kg 1 and χ = C 2 s 1 ; dashed curve - spectrum for the original temperature; thin solid curve - spectrum after correction (see methods). The short solid vertical lines denote the Ozmidov scaling (left, k o = (1/2π)(ε/N 3 ) 1/2 cpm), Kolmogorov wavenumber (middle, k k = (1/2π)(ε/v 3 ) 1/4 cpm) and the Batchelor wavenumber (right, k B = (1/2π)(ε/vD 2 ) 1/4 cpm). The short vertical dashed line marks the lower end of the sub-inertial range. Buoyancy length scale (L b ): 2 12 / 5 L w / N. b = ( ) ( ) L b gives the vertical displacement traveled by fluid particles in converting all its vertical fluctuating kinetic energy into potential energy. Ellison scale (L E ) (Ellison, 1957): L T 2 12 / / dt / dz. 6 E = ( ) ( ) ( ) L E gives the vertical distance traveled by fluid particles before they return to their equilibrium level or mixing (Stillinger et al., 1983). These length scales depend on the rms values of the power spectra for turbulence velocity and temperature fluctuations. To estimate their properties, we must eliminate (1) spectral contamination due to linear internal waves in the low wavenumbers, and (2) mechanical and electrical noise in the high wavenumbers. Furthermore, (3) since the probes do not fully resolve high wavenumber power, the lost variance must be recovered by applying an appropriate response function. 714 H. Li and H. Yamazaki

7 Fig. 7. Length scale ratio, L b /L o, in terms of Buoyancy Reynolds number Re b = ε/vn 2. Solid circles represent data from the billow region. Open circles are from non-billow portions. Fig. 8. Length scale ratio, L E /L o, in terms of Buoyancy Reynolds number Re b = ε/vn 2. Solid circles represent data from the billow region, open circles are from the non-billow portion of the data. Generally, we compute rms turbulence quantities by numerically integrating one-dimensional spectra of w and T over a finite wavenumber band. The lost variance of turbulence velocity and temperature were recovered using the work of Ninnis (1984), Gregg et al. (1978) and Gregg (1999). A high pass filter at 0.5 Hz removes lowfrequency components from the signals caused by vehicle motion and temperature sensitivity of the probe (Wolk and Lueck, 2001). The upper bound of the integration range is often determined by either the Kolmogorov or the Batchelor scale. However, the lower bound is not so obvious. Internal wave power is often included in the low wavenumber part of the estimated velocity and temperature spectra. In order to separate the linear internal wave kinetic energy from turbulence kinetic energy, the Ozmidov scale is a useful index for the integration limit (Gargett et al., 1984; Yamazaki, 1990). As seen in Fig. 6, the process associated with the billow is dominated by low wavenumber components, and the level of turbulence is rather weak. Hence the corresponding Ozmidov scale is small. This is an important consideration in shear instability studies, because low wavenumber structures make unique contributions to length scales associated with displacement of fluids. The Ozmidov scaling is not applicable to such a transitional state (Fernando and Hunt, 1996). When computing the rms velocity and temperature within a billow region, low wavenumber structures due to the billow should be included in the rms estimates. Thus, care must be taken to estimate the rms values for the KH billow section. In our computation scheme, we apply a variable integration method for developed turbulence data (e.g. the integration between half the Ozmidov wavenumber scale and half the Kolmogorov wavenumber scale/batchelor wavenumber scale of the shear/temperature data). In the billow region, we retain the contribution of low wavenumber motion, which represent the billow, applying a fixed integration method with the lower limit of integration at 0.28 cpm. This wavenumber is based on the horizontal scale of the billow (roughly 3.5 m) as estimated from the vertical turbulence velocity signals. For fully developed turbulence flow, the length scales L b, L o and L E should all have the same order of magnitude (Itsweire et al., 1986). We plotted their ratios as a function of the buoyancy Reynolds number, Re b, to investigate whether this holds for the data from the billow section. The ratios of L b /L o (Fig. 7) and L E /L o (Fig. 8) for the non-billow section data are nearly unity, while those within the billow region are much larger. In general, turbulence is inhibited by buoyancy forces when an overturning scale exceeds the Ozmidov scale. This concept, however, is not applicable to KH billows because their growth is controlled by the gradient Richardson number (Smyth and Moum, 2000a). From energy considerations, L b represents the kinetic energy of the fluid movement. At the initial stage of billow formation, the kinetic energy comes mostly from low wavenumber rolling movements, rather than high wavenumber turbulence. The twodimensional eddy motion grows by extracting energy from the background mean shear until the billow collapses into turbulence. Thus the major contribution to L b comes from the two-dimensional eddy motions in low wavenumber, Observations of a Kelvin-Helmholtz Billow in the Ocean 715

8 not from turbulence. This causes L b to exceed the Ozmidov scale in the billow section data. The Ellison scale L E, a measure of the vertical distance from equilibrium, represents the associated potential energy of the fluid. Displacements caused by low wavenumber roll-up movement in KH billows provide considerable potential energy, thus causing the length scale ratio L E /L o to be greater than unity. On the other hand, the values of L E /L o outside the billow region are scattered, with most less than unity (Fig. 8). 4.4 Kinetic energy budget For steady and homogeneous stratified shear flow, the turbulent kinetic energy equation may be written: P = J b + ε, ( 7) where P = uw ( U/ z), the mechanical production of turbulent kinetic energy by the mean shear background, and J b = (g/ρ 0 ) wρ, the production (J b < 0) or destruction (J b > 0) of turbulent kinetic energy by buoyancy. This equation can also be expressed: Jb 1+ ρvws + = 0. () 8 ε Here, ρ vw is correlation coefficients between w and v. S*, the shear number, is defined as: S = 2 Sq P 2 q =, ε ε uw ( 9) and S is the background mean shear equivalent to U/ z, and q 2 is twice the turbulence kinetic energy (Shih et al., 2000). When the buoyancy term is ignored, the second term must be unity, thus S vw ρ 1. ( 10) For fully developed stratified shear flow, Shih et al. (2000) numerically demonstrated that shear number converges to roughly 11 regardless of the initial conditions of the simulation. Because the flow field is growing by obtaining energy from the mean shear at the onset of billow formation, the energy balance expressed in Eq. (7) cannot be applied. Furthermore, no strong turbulence is generated at the initial stage. Thus the flow field within the billow region is accumulating kinetic energy with no appreciable dissipation in high wavenumbers. Based on this argument, we expect the shear number to be much larger than the steady state value suggested by Shih et al. (2000). Our data indicate the shear number during the Fig. 9. (a) Shear number, S* = q 2 S/ε, and (b) stratification number, S t, in terms of the Buoyancy Reynolds number Re b = ε/vn 2. Solid circles represent data from the billow region. Open circles are from non-billow portions. onset of KH billow formation is of the order of 100 (Fig. 9a). This high shear number of 100 suggests that the momentum flux is only 1% of the total energy. This is a property that is consistent with an internal wave, thus we may argue that nonlinearly growing internal waves represent the initial stage of the KH billows. The shear numbers, S* ~ O(10), in non-billow sections are consistent with Shih et al. (2000). There is a slight tendency, however, for S* to decline as the buoyancy Reynolds number increases. In order to investigate this declining tendency, we examined the role of potential energy in billow formation. Gerz and Yamazaki (1993) studied the generation of turbulence due to potential energy, and proposed the stratification number, S t, to parameterize buoyancy-generating turbulence: / St LI / T 2 12 dt / dz, 11 = ( ) ( ) ( ) 716 H. Li and H. Yamazaki

9 where L I = T e U is the integral length scale, T e the e-folding time computed from the autocorrelation of temperature fluctuations, and U the mean speed. When S t < 1, there is sufficient available potential energy to create turbulence. In Fig. 9b, the stratification numbers S t tend to decrease with increasing turbulence and become less than order one for higher buoyancy Reynolds numbers. Therefore, the excess potential energy can generate new turbulence and the potential energy reservoir acts as both source and sink term for the kinetic energy reservoir. As a result, the buoyancy term in the simple steady state kinetic energy balance equation cannot be considered as a one way energy cascade. This may explain why the shear number decreases as the buoyancy Reynolds number increases, because the stratification number decreases at the same time. On the other hand, at the onset of billow formation, the excess of potential energy generated from two dimensional eddy structures makes the stratification number much less than Comparison of Field Observations to Numerical Experiments The limited oceanic observations available have shown that the spatial scales of KH billows range from less than one meter (Woods, 1968) to tens of meters (De Silva et al., 1996). The horizontal scale of the billow we observed was about 3.5 m. Since no billow time series exists comparable to ours from oceanic data, we compared the dynamical properties of the observed KH billow with the Direct Numerical Simulation (DNS) of Smyth and Moum (2000a). Although a series of papers (Smyth, 1999; Smyth and Moum, 2000a, b) describes the detail of the DNS, we briefly discuss their numerical approach. The numerical simulations employ the Boussinesq equations for velocity, density and pressure in a non-rotating physical space with Cartesian co-ordinates x, y and z. r u = r r r r r u u Π gθk ˆ 2 v u, t ( ) + + ( 12) r r, 2 p 1 Π= + u u ( 13) ρ 0 where, the p is pressure, ρ 0 is reference density constant. θ is the fractional density deviation e.g. θ = (ρ ρ 0 )/ρ 0. The pressure field Π indicates an impressibility condition, r u = 0. The thermodynamic value θ 0 evolves following: θ t r 2 = u θ + D θ. ( 14 ) The model used periodic boundary conditions in horizontal dimensions by taking: f( x+ L, y, z)= f x, y+ L, z f x, y, z. 15 x ( y )= ( ) ( ) Here, the L x and L y are periodicity interval constants. The simulation is initialized with a parallel flow in which shear and stratification are concentrated in the shear layer. The mean shear is established at t = 0, then decays due to turbulent mixing, without considering the external internal wave forcing on the enhancement of local mean shear (W. D. Smyth, personal communication). The data used here are from one of six simulations with different Reynolds number and Prandtl number. The Initial Reynolds number, Re, is set to The Prandtl number, Pr, the ratio of the diffusivities of momentum and density, is set to 7. The initial stage of billow is defined as the duration from start to the time when the billow begins to collapse into turbulence. Figure 10 shows an example of the evolution of K-H billow, including billow generation, collapse and turbulent decay. The paired roll-up structure (Fig. 10a) gives the approximated horizontal size of the numerically simulated billow, roughly 2 m long. This is closed to our oceanic estimated billow (3.5 m). A billow pairing structure is a genuine feature of the weakly stratified case. When the initial minimum Richardson number lies between 0.16 and 0.25, the KH billows will appear, but without paired structure. Also, if there is significant stratification in the fluid above and below the shear layer, pairing may be inhibited (W. D. Smyth, personal communication). The Woods (1968) visualization experiment did not show such a pairing structure, possibly it is a common situation for billows in the ocean, or it may already have paired before we observed it. No paired feature has yet been observed in the ocean. For our field experiment, we estimate the Reynolds number using: Reexp = ql / v, ( 16) where L is the turbulence energy-containing eddy scale defined as L = q e 3 /ε. Note that q e 2 is (1/3)q 2 because we followed the original length scale definition of Batchelor (1953). The kinematic viscosity v is m 2 s 1. From our field data around the billow region, we obtained q e ~ ms 1, ε ~ W kg 1, hence L is about 0.90 m. Thus, the Reynolds number is about Table 1 summarizes the field and model parameters; both are similar and within suitable scaling and property values. Our comparisons therefore appear quite reasonable. Observations of a Kelvin-Helmholtz Billow in the Ocean 717

10 Fig. 10. Evolution of K-H billows from one of numerical simulations (Prandtl number, Pr = 1, Smyth et al., 2000a). Table 1. Parameter values from our field observations and for the numerical simulations of KH billows (from Smyth and Moum, 2000a). Quantity Field experiment Simulation ε [W kg 1 ] 6~ ~ S [s 1 ] ~ N [s 1 ] ~0.094 Ri ~0.19 Billow horizontal scale [m] Pr Re An excellent agreement in the buoyancy length scale, L b, is obtained for both billow and non-billow turbulence sections (Fig. 11a). This length scale for both the field and simulation non-billow turbulence data decreases as the buoyancy Reynolds number decreases. The buoyancy length scales for the billow section data are both about 0.1 m at a buoyancy Reynolds number of about 10. The Ellison scales L E for the billow section from both data sets also agree (Fig. 11b). However, L E for the simulation data from the non-billow section is much larger than the observed values. There are two reasons for this discrepancy. One is associated with the numerical data, which contains all the kinetic energy including internal waves (W. D. Smyth, personal communication). The second reason is because the field data do not resolve high wavenumber variance, so the correction that adds the unresolved variance may not be sufficient. Further investigation of why our field data appear to be lower than the numerical values will require close collaboration with the simulation efforts; this is a topic for future study. 6. Summary We found the following features for the KH billow observed during our field observations. 1) The vertical velocity component shows a 2 spectral slope. 2) The temperature gradient spectrum does not follow the Batchelor spectrum. 3) The ratios of buoyancy length to Ozmidov scale (L b /L o ) and Ellison scale to Ozmidov scale (L E /L o ) are of the order of 10. 4) The shear number is of the order of ) The stratification number is roughly 1. 6) There is excellent agreement in KH billow characteristics between the field data and the numerical simulation results. 718 H. Li and H. Yamazaki

11 Schwarz, 1963; Oakey, 1982): ( )= ( ) ( ) ( ) 12 / B 1 1 θ Sk q/ 2 χ k D fα, A1 where k is the wavenumber in cpm; q a universal constant (here set to 3.9, based on Grant et al., 1968); k B the Batchelor wavenumber in cpm defined as: k B = 1 ε 2 2π v D 14 / ( 2). A D is the molecular diffusivity, v the kinematic viscosity, and ε the kinetic energy dissipation rate. The universal non-dimensional spectral form is given by: { α } ( ) 2 2 α / 2 x / 2 f( α)= α e α e dx, A3 in which the non-dimensional wavenumber α = (2q) 1/2 (k/k B ). The temperature variance dissipation rate for isotropic turbulence χ θ satisfies 6D s k dk 6D T. A4 0 x χ θ = ( ) = 2 ( ) Fig. 11. Field data compared with the numerical simulation. (a) Length scale L b vs. buoyancy Reynolds number Re b = ε/vn 2 ; (b) Length scale L E vs. buoyancy Reynolds number Re b = εv/n 2. Triangles - simulation; Circles - field observations. Solid symbols are for the billow section. Acknowledgements The experiment was conducted by T. R. Osborn (Johns Hopkins University) with a grant provided by the Office of Naval Research. This study was supported by a Grant-in-Aid for Scientific Research (C) under grant number We thank W. D. Smyth for generously providing the DNS data for our comparison as well as thoughtful suggestions and cooperation. We also thank the reviewers for their useful comments. We are indebted to L. Haury for his editing efforts and conscientious advice on early drafts of the manuscript. Our appreciations also extend to F. Wolk and J. Mitchell for their constructive suggestions. Appendix: Computing Temperature Variance Dissipation Rate χ θ from the Batchelor Universal Spectrum The Batchelor spectrum of temperature gradient fluctuation in one dimension is expressed as (Gibson and The overbar denotes the statistical average over a spatial domain. Temperature spectra at high wavenumbers are often incompletely resolved due to the temporal response limitations of FP07 thermistors. To recover the lost variance, we applied the following single pole filter (Gregg et al., 1978; Gregg, 1999). where, ( ) ( ) HT 2 = 1/ 1+( 2πτf) 2 2, A τ = U [ sec ]. ( A6) U is the mean speed of the submarine. The correction provided by Eq. (A5) should give a reasonable temperature gradient spectrum for low dissipation rate turbulence. To estimate the entire temperature gradient variance, we recursively fit the Bathchelor spectrum as follows: 1) Estimate the rate of kinetic energy dissipation ε from each 2-second segment of the shear data by applying the Ninns response function (Moum et al., 1995). 2) Compute the temperature gradient spectrum for each corresponding 2 seconds of data. Observations of a Kelvin-Helmholtz Billow in the Ocean 719

12 Fig. A1. Well-corrected temperature spectrum. All meaning of lines are same as Fig. 6. Batchelor spectrum corresponds to ε = W kg 1 and χ = C 2 s 1. Fig. A2. Same as Fig. A1 but for an example of not fully corrected spectrum at high wavenumber. Batchelor spectrum corresponds to ε = W kg 1 and χ = C 2 s 1. 3) Use the response function (Eq. (A5)) to recover the unresolved variance in high wavenumbers. 4) Since the universal Batchelor spectrum contains 90% of the total variance below half the Batchelor wavenumber, we integrate the observed temperature gradient spectrum up to half the Batchelor wavenumber to obtain the first estimate of the temperature gradient variance. Then estimate χ θ from (Eq. (A4)). 5) Compute the universal Batchelor spectrum (Eq. (A1)) from the estimated χ θ and known ε. Compare the computed spectrum shape with the observed temperature gradient spectrum in the inertial sub-range; if both estimates agree within 5%, the estimate is accepted. If not, the estimated χ θ is modified by a factor of 5% from the initial guess. Repeat (4) and (5) until agreement is reached. The response function provides a reasonable correction up to almost half the Batchelor wavenumber (Fig. A1). On the other hand, the correction is insufficient beyond 30 cpm (Fig. A2), but the estimated χ θ should be reasonable provided the Batchelor spectrum is acceptable. We used the estimated Batchelor spectrum in order to compute χ θ. References Batchelor, G. K. (1953): The Theory of Homogeneous Turbulence. Cambridge University Press, 197 pp. De Silva, I. P. D., H. J. S. Fernando, F. Eaton and D. Herbert (1996): Evolution of Kelvin-Helmholtz billows in nature and laboratory. Earth and Planetary Letters, 143, Ellison, T. H. (1957): Turbulent transport of heat and momentum from an infinite rough plane. J. Fluid Mech., 2, Fernando, H. J. S. and J. C. R. Hunt (1996): Some aspects of turbulence and mixing in stably stratified layers. Dyn. Atmos. Oceans, 23, Gargett, A. E., T. R. Osborn and P. W. Naysmyth (1984): Local isotropy and the decay of turbulence in a stratified fluid. J. Fluid Mech., 144, Gerz, T. and H. Yamazaki (1993): Direct numerical simulation of buoyancy-driven turbulence in a stably stratified fluid. J. Fluid Mech., 249, Gibson, C. H. and W. H. Schwarz (1963): The universal equilibrium spectra of turbulent velocity and scalar fields. J. Fluid Mech., 16, Grant, H. L., B. A. Hughes, W. M. Vogel and A. Moilliet (1968): The spectrum of temperature fluctuations in turbulent flow. J. Fluid Mech., 34, Gregg, M. C. (1999): Uncertainties and limitations in measuring ε and χ. J. Atmos. Oceanic Technol., 16, Gregg, M. C., T. Meagher, A. Pederson and E. Aagaard (1978): Low noise temperature microstructure measurements with thermistors. Deep-Sea Res., 25, Hebert, D., J. N. Moum, C. A. Paulson and D. R. Caldwell (1992): Turbulence and internal waves at the equator. Part II: Details of a single event. J. Phys. Oceanogr., 22, Itsweire, E. C. and T. R. Osborn (1988): Microstructure and vertical velocity shear distribution in Monterey bay. p In Small Scale Turbulence and Mixing in the Ocean, ed. by J. C. J. Nihoul and B. J. Jamart, Elsevier. Itsweire, E. C., K. N. Helland and C. W. Van Atta (1986): The evolution of grid-generated turbulence in a stably stratified fluid. J. Fluid Mech., 162, Klaassen, G. P. and W. R. Peltier (1985): The onset of turbulence in finite-amplitude Kelvin-Helmholtz billows. J. Fluid Mech., 155, H. Li and H. Yamazaki

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