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 108-8477, 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.
We used the United States Navy research submarine
Dolphin to study microstructure in the seasonal
thermocline off San Diego, California in 1988. 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).
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.
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.
* Corresponding author. E-mail: [email protected]
Copyright © The Oceanographic Society of Japan.
709
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. 1.
2.2 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 2 = ( dU / dz )2 + ( dV / dz )2 .
(1)
The term in angle brackets 〈 〉 represents an average over
10 seconds. The spatial derivatives of U and V were ob710
H. Li and H. Yamazaki
tained from a cubic spline after smoothing the profiles of
U and V with a five point triangular filter. A threshold
value of S2 = 2 × 10–4 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 fc = 1/2.3t int. We
set tint, the integration time, to 4 seconds. This corresponds
to a maximum length scale of 4 × 1.2 = 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 fc = 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 fc = 1/2.3t int, with tint = 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),
2
  ∂v  2
 ∂w  
ε = v 5.5  + 2   ,
  ∂x 
 ∂x  

(2 )
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 Reb
(Reb = 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
 ∂T ′ 
χ = 6 D
 ,
 ∂x 
(3)
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.
Fig. 2. Schematic diagram of the instantaneous heat flux signals corresponding to different development stages of a KH
billow.
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
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
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 24960 m horizontal distance, to select dynamically unstable portions where Ri < 1/4. One segment,
at elapsed time 1441–1444 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.
712
H. Li and H. Yamazaki
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 1441–1444 in Figs. 4i and 4j), with values
of O(10 –9) W kg –1 and O(10 –9)°C2s–1, respectively. These
values are close to the noise level of the probes
(ε noise ~ 3 × 10 –10 W kg–1, χnoise ~ 3 × 10–10°C2s–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
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
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:
Fig. 6. Temperature gradient power spectra for the KH billow in Fig. 4. Bold solid curve - Batchelor spectrum corresponding to ε = 2.2840 × 10 –9 W kg–1 and χ = 3.8548 ×
10 –9°C2s–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 = (1/2 π )( ε /N 3 ) –1/2 cpm),
Kolmogorov wavenumber (middle, k k = (1/2 π )(ε /v 3 )1/4
cpm) and the Batchelor wavenumber (right, k B =
(1/2 π)(ε/vD2) 1/4 cpm). The short vertical dashed line marks
the lower end of the sub-inertial range.
Buoyancy length scale (Lb):
( )
Lb = w 2
1/ 2
(5)
/ N.
L b gives the vertical displacement traveled by fluid particles in converting all its vertical fluctuating kinetic energy into potential energy.
Ellison scale (LE) (Ellison, 1957):
Ozmidov scale (Lo ):
(
Lo = ε / N
)
3 1/ 2
,
( 4)
where N is the buoyancy frequency. Lo 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).
714
H. Li and H. Yamazaki
( )
LE = T ′ 2
1/ 2
(
)
/ dT / dz .
(6 )
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.
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/Lo, in terms of Buoyancy Reynolds
number Reb = ε/vN2. 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 in-
tegration 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
Lb, Lo 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, Reb , to investigate whether this holds for the data from the billow
section. The ratios of Lb/Lo (Fig. 7) and LE/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,
Lb 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 Lb comes from
the two-dimensional eddy motions in low wavenumber,
Observations of a Kelvin-Helmholtz Billow in the Ocean
715
not from turbulence. This causes L b to exceed the
Ozmidov scale in the billow section data.
The Ellison scale LE, 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/Lo to be greater than unity. On the other
hand, the values of LE/Lo 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 = Jb + ε ,
(7)
where P = – uw (∂U/∂z), the mechanical production of
turbulent kinetic energy by the mean shear background,
and Jb = –(g/ ρ0) wρ ′ , the production (Jb < 0) or destruction (Jb > 0) of turbulent kinetic energy by buoyancy. This
equation can also be expressed:
1 + ρ vw S ∗ +
Jb
= 0.
ε
(8)
Here, ρvw is correlation coefficients between w and v. S*,
the shear number, is defined as:
S∗ =
Sq 2
P  q2 
= −  ,
ε
ε  uw 
( 9)
and S is the background mean shear equivalent to ∂U/∂z,
and q2 is twice the turbulence kinetic energy (Shih et al.,
2000). When the buoyancy term is ignored, the second
term must be unity, thus
−1
S ∗ ≅ − ρ vw
.
(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
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H. Li and H. Yamazaki
Fig. 9. (a) Shear number, S* = q 2S/ ε, and (b) stratification
number, S t, in terms of the Buoyancy Reynolds number
Re b = ε/vN2. 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, St, to parameterize buoyancy-generating turbulence:
( )  (dT / dz),

St =  LI / T ′ 2

1/ 2
(11)
where LI = Te U is the integral length scale, Te the e-folding time computed from the autocorrelation of temperature fluctuations, and U the mean speed. When St < 1,
there is sufficient available potential energy to create turbulence. In Fig. 9b, the stratification numbers St 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 1.
5.
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
r
∂u r r r r
= u × ∇ × u − ∇Π + gθkˆ + v∇ 2 u ,
∂t
(
Π=
)
(12)
p 1r r
+ u ⋅ u,
ρ0 2
(13)
where, the p is pressure, ρ0 is reference density constant.
θ is the fractional density deviation e.g. θ = –(ρ – ρ0 )/ρ0.
The pressure field Π indicates an impressibility condir
tion, ∇· u = 0. The thermodynamic value θ 0 evolves following:
∂θ r
= u ⋅ ∇θ + D∇ 2θ .
∂t
(14)
The model used periodic boundary conditions in horizontal dimensions by taking:
(
)
f ( x + Lx , y, z ) = f x, y + Ly , z = f ( x, y, z ).
(15)
Here, the Lx and Ly 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 1354. 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 = qe3 /ε . Note that qe2 is (1/3)q2 because we
followed the original length scale definition of Batchelor
(1953). The kinematic viscosity v is 1.2 × 10–6 m2s–1. From
our field data around the billow region, we obtained qe ~
0.0011 ms–1, ε ~ 1.57 × 10–9 W kg–1, hence L is about
0.90 m. Thus, the Reynolds number is about 1430.
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
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
6~22 × 10 – 1 0
≈0.0274
≈0.0125
≈0.2
≤3.5
8.6
1430
4.3~6.9 × 10 – 1 0
0.0117~0.0319
0.0051~0.094
0.09~0.19
≈2.0
7
1354
ε [W kg – 1 ]
S [s – 1 ]
N [s – 1 ]
Ri
Billow horizontal scale [m]
Pr
Re
An excellent agreement in the buoyancy length scale,
Lb , 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 LE for the billow section from both
data sets also agree (Fig. 11b). However, LE 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
718
H. Li and H. Yamazaki
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 (LE/L o) are of
the order of 10.
4) The shear number is of the order of 100.
5) The stratification number is roughly 1.
6) There is excellent agreement in KH billow characteristics between the field data and the numerical simulation results.
Schwarz, 1963; Oakey, 1982):
S( k ) = ( q / 2 )
1/ 2
χθ k B−1 D −1 f (α ),
(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:
kB =
1  ε 
2π  v 2 D 
1/ 4
( A2)
.
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:
{
f (α ) = α e −α
2
/2
∞
− α ∫ e− x
2
/2
α
}
dx ,
(A3)
in which the non-dimensional wavenumber α =
(2q)1/2 (k/kB). The temperature variance dissipation rate
for isotropic turbulence χθ satisfies
2
∞
 ∂T ′ 
χθ = 6 D∫ s(k )dk = 6 D
 .
0
 ∂x 
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 LE vs. buoyancy Reynolds number
Re b = ε v/N2. 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 10640421. 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 χθ from the Batchelor Universal
Spectrum
The Batchelor spectrum of temperature gradient fluctuation in one dimension is expressed as (Gibson and
( A 4)
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).
(
2 2
),
(A5)
τ = 0.005U −0.32 [sec].
( A 6)
HT2 = 1 / 1 + (2πτf )
where,
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
Fig. A1. Well-corrected temperature spectrum. All meaning of
lines are same as Fig. 6. Batchelor spectrum corresponds to
ε = 2.4830 × 10–8 W kg–1 and χ = 3.6079 × 10–6°C2s –1.
Fig. A2. Same as Fig. A1 but for an example of not fully corrected spectrum at high wavenumber. Batchelor spectrum
corresponds to ε = 1.1839 × 10–7 W kg–1 and χ = 7.3309 ×
10 –5°C2s –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 χθ from (Eq. (A4)).
5) Compute the universal Batchelor spectrum (Eq.
(A1)) from the estimated χθ 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 χθ 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 χθ should be
reasonable provided the Batchelor spectrum is acceptable.
We used the estimated Batchelor spectrum in order to
compute χθ.
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