Spectral characteristics of velocity and vorticity fluxes in an ...

Spectral characteristics of velocity and vorticity fluxes
in an unstratified turbulent boundary layer
R.-C. Lien and T. B. Sanford
Applied Physics Laboratory and School of Oceanography, College of Ocean and Fishery Sciences,
University of Washington, Seattle
Abstract. Wavenumber spectral characteristics of the velocity and vorticity
fluxes in an unstratified turbulent boundary layer are presented. The
observed vertical and streamwise velocity spectra agree with empirical
forms found in the atmospheric boundary layer. Spectral ratios of 4/3
between the vertical and streamwise velocity spectra and the agreement
between the observed vorticity flux quad spectrum and that of isotropic
turbulence suggest local isotropy at scales smaller than Z. The normalized
cospectrum of the momentum flux agrees remarkably well with the empirical
form found in the atmospheric boundary layer. In the inertial subrange the
momentum flux cospectrum shows a clear spectral slope of –7/3. The observed
composite vorticity flux cospectrum has most of its variance at the streamwise
wavenumber k x =(1–10) Z −1 and has a spectral slope of –7/3 in the inertial
subrange. The –7/3 spectral slope is consistent with a dimensional argument,
assuming that the vorticity flux cospectrum is proportional to the gradient of
the mean vorticity, and depends on the turbulence kinetic energy dissipation
rate ε and the wavenumber. A model turbulent vorticity flux cospectrum is
constructed based on the shape of observed spectra, a –7/3 spectral slope in
the inertial subrange, and the similarity scaling of the vorticity flux in an
unstratified turbulent boundary layer. The turbulence vorticity flux is directly
related to the divergence of turbulence momentum flux, the force exerted by
turbulence on the mean flow. Therefore our proposed empirical cospectral
form of the vorticity fluxes might be useful for turbulence parameterization
in numerical models.
1. Introduction
Extensive studies of turbulence spectral properties
have been conducted in the atmospheric boundary layer,
e.g., the 1968 Kansas experiment and the 1973 Minnesota
experiment [Kaimal and Wyngaard, 1990]. Understanding
of the spectral and cospectral properties of
turbulence is needed to interpret and correct measurements
of turbulent energy and fluxes. These properties
are also useful for parameterizations of turbulent fluxes
in numerical models.
Universal spectral forms of turbulence velocity, momentum
flux, and heat flux have emerged from comprehensive
measurements and theoretical studies of the atmospheric
boundary layer. On the basis of dimensional
arguments, Wyngaard and Co´te [1972] suggested a –7/3
power law for the turbulent momentum flux 〈u ′ w ′ 〉 and
heat flux 〈w ′ θ ′ 〉 cospectrum in the inertial subrange.
Evidence of the –7/3 slope has been found in the atmospheric
surface layer [Kaimal et al., 1972; Kader and
Yaglom, 1991]. Antonia and Zhu [1994] found a large
wavenumber range with a –5/3 slope for the heat flux
cospectrum before the inertial subrange slope of –7/3.
They suggested that the observed slope of –5/3 might
be due to a strong anisotropy of their measured turbulence
field. Horst [1997] presented a simple correction
formula for turbulence fluxes estimated from a fastresponse
anemometer. The formula depends on both
the sensor response time and the true heat flux cospectrum,
particularly the wavenumber of the cospectral
1

LIEN AND SANFORD: SPECTRA OF VELOCITY AND VORTICITY FLUXES 2
peak. These formulas are needed to correct the observed
flux cospectra so as to obtain correct estimates
of fluxes.
There have been relatively few studies of turbulence
spectral properties in the oceanic boundary layer compared
to studies in the atmospheric boundary layer.
Gross and Nowell [1985] found a slope of –5/3 of velocity
spectra in the inertial subrange in an unstratified
oceanic tidal boundary layer. Their observed momentum
flux cospectra have a significant fraction at wavelengths
greater than the distance from the boundary,
suggesting that significant portion of turbulence stress
might be due to flattened-out eddies. Bowden and Ferguson
[1980] studied turbulence structure in the bottom
boundary layer in the eastern Irish Sea. They also observed
the -5/3 inertial-subrange spectral slope at high
wavenumbers. The ratio of streamwise velocity spectra
to vertical velocity spectra approaches 3/4 at k x ≥ 2πZ,
where k x is the streamwise wavenumber and Z is the
height above the bottom, suggesting isotropic turbulence.
McPhee [1994] proposed that the turbulence mixing
length scale can be estimated as the inverse of the
wavenumber of the peak of vertical velocity spectrum.
Their velocity spectra observed beneath multiyear pack
ice in the western Weddell Sea agree with the empirical
spectrum in the atmospheric boundary layer formulated
by Busch and Panofsky [1968].
In the turbulent boundary layer the turbulent advective
flux of vorticity plays an important dynamical role.
Laboratory experiments and numerical simulations of
boundary layer turbulence have shown coherent vortical
motions in turbulent boundary layers [Robinson, 1991].
These vortical motions are important for momentum
transfer between inner and outer boundary layers [Kim
et al., 1971; Willmarth and Lu, 1972; Blackwelder and
Eckelmann, 1979; Ranasoma and Sleath, 1992]. The
sweeping process of the momentum flux is related to
the creation of new vortices at the boundary [Bernard
et al., 1993].
Indeed, there exists a kinematic relation between the
divergence of the momentum flux and the advective
fluxes of vorticity. The divergence of the turbulence momentum
flux represents the force that turbulence exerts
on the mean flow. Therefore the turbulence vorticity
flux is directly related to the turbulence force on the
mean flow. Klewicki et al. [1994] have conducted extensive
studies of advective vorticity fluxes. They found
that the divergence of the momentum flux is balanced
by two components of the advective fluxes of vorticity,
one associated with the gradient of vorticity and the
other associated with the change of eddy scales from
boundaries. This balance is consistent with Tennekes
and Lumley’s [1972] dimensional argument. The advective
flux of vorticity is also important in the surface
boundary layer, where the turbulence vorticity advected
by Stokes drift is the main driving mechanism of Langmuir
circulation [Li and Garrett, 1995].
Recently, Sanford and Lien [1999] reported velocity
and vorticity measurements in an energetic tidal channel
and estimated the turbulent enstrophy, momentum
flux, and advective vorticity flux. They proposed a new
method to estimate the friction velocity based on the
turbulence vorticity flux. Their sensor scale is 0.09 m.
Wallace and Foss [1995] reviewed attenuation problems
in vorticity measurements by numerous sensors and concluded
that in order to have accurate measurements of
enstrophy, sensor scales must be of the order of the
Kolmogorov scale. In oceanic environments and tidal
channels this requires sensors on millimeter to centimeter
scales.
Sanford and Lien [1999] defined an eddy diffusivity of
vorticity A ζ based on the ratio of the advective vorticity
flux and the gradient of the mean vorticity in analogy
to the definition of eddy viscosity A ν . Because the estimate
of A ζ is equal to A ν in the boundary layer, Sanford
and Lien [1999] argued that both the momentum and
vorticity fluxes are not significantly attenuated by the
finite sensor scale. To confirm this assertion, one needs
to know the spectral forms of the momentum flux and
the vorticity flux.
Here we present spectral properties of turbulence velocity
and vorticity fluxes observed in an oceanic tidal
boundary layer. We begin with a brief review of classical
theory of the turbulent velocity spectra and flux
cospectra for isotropic turbulence and empirical velocity
spectra and flux cospectra observed in atmospheric
boundary layers (section 2). In section 3, spectra of
the velocity and the momentum flux calculated from
measurements in an unstratified tidal channel are presented
and compared with theoretical spectral forms
and empirical forms found in the atmospheric boundary
layer. Our observed vorticity flux spectrum is discussed
in section 4. The spectral slope of the vorticity
flux cospectrum in the inertial subrange is consistent
with the prediction from the dimensional argument. An
empirical model for the vorticity flux spectrum is proposed.
In section 5 we discuss the quad spectrum of the
vorticity fluxes, scaling of the vorticity flux, and effect
of viscous dissipation and summarize our results. Sensor
response functions for the momentum and vorticity
fluxes are described in Appendix A. Fractions of resolved
momentum fluxes and vorticity fluxes as a func-

LIEN AND SANFORD: SPECTRA OF VELOCITY AND VORTICITY FLUXES 3
tion of wavenumber and distance from the boundary are
discussed in Appendix B.
2. Review of Turbulence Velocity and
Flux Spectra
2.1. Isotropic Turbulence
For isotropic turbulence the velocity wavenumber
spectra and the momentum flux cross spectra in the inertial
subrange are described by Batchelor [1959]. In the
inertial subrange the turbulence energy flux is constant
and spectral properties are independent of the external
forcing at large scales and viscous dissipation at Kolmogorov
microscales. In this wavenumber range, turbulence
energy spectrum is well described by the Kolmogorov
spectrum with a -5/3 spectral slope.
The vorticity flux cross spectra in the inertial subrange
of isotropic turbulence can be obtained based
on Batchelor’s formula. In this analysis we are interested
mostly in the streamwise wavenumber k x spectra.
After Batchelor’s three-dimensional spectra are integrated
over the vertical and spanwise wavenumbers,
the streamwise wavenumber spectra of the streamwise
velocity Φ uu (k x ), the vertical velocity Φ ww (k x ), the momentum
flux Φ uw (k x ), and the vorticity flux Φ wζy (k x )
have the following forms:
Φ uu (k x ) = 18
55 αε2/3 k −5/3
x , (1)
Φ ww (k x ) = 4 18
3 55 αε2/3 kx −5/3 , (2)
Φ uw (k x ) = 0, (3)
Φ wζy (k x ) = −iα 3
10 ε2/3 k −2/3
x . (4)
The Kolmogorov constant α is ∼1.5 [Sreenivasan, 1995];
ε is the turbulence kinetic energy dissipation rate; u and
w are the streamwise velocity and vertical velocity components,
respectively; and ζ y is the spanwise vorticity.
For isotropic turbulence the streamwise wavenumber
spectrum of vertical velocity is 4/3 of the streamwise
velocity spectrum in the inertial subrange. This has
been used to test for local isotropy [McPhee and Smith,
1976]. Another salient feature is the vanishing turbulence
fluxes, because the isotropic turbulence has no
preferred flux direction. Note that the nonzero quad
spectrum of the vorticity flux (4) for isotropic turbulence
does not involve dynamic reasons but, rather, is
a result of pure kinematics. The reality condition of
the vorticity flux implies that Φ wζy (k x ) = Φ ∗ wζ y
(−k x ),
where the asterisk indicates the complex conjugate.
Therefore the quad spectrum in the positive streamwise
wavenumber domain equals the minus of the quad
spectrum in the negative streamwise wavenumber domain,
and the integration of quad spectrum over the
whole streamwise wavenumber domain vanishes.
2.2. Atmospheric Boundary Layer Observations
The expressions for the spectra discussed in the previous
section exclude effects of the boundary and stratification,
which introduce anisotropy into the turbulence
field. In the presence of a boundary and stratification,
these spectral forms are modified. On the basis of data
taken in the 1968 Kansas turbulent boundary layer experiment,
Kaimal et al. [1972] obtained empirical spectral
forms for turbulent velocity and momentum flux
in different stability conditions. Under the neutral stability
condition the streamwise velocity spectrum Φ uu ,
the vertical velocity spectrum Φ ww , and the momentum
flux cospectrum P uw are expressed as
Φ uu (k x ) = 102 (
2π u2 ∗Z 1 + 33 k ) −5/3
xZ
, (5)
2π
[
Φ ww (k x ) = 2.1
( ) ] 5/3 −1
kx Z
2π u2 ∗Z 1 + 5.3
, (6)
2π
P uw (k x ) = − 12 (
2π u2 ∗Z 1 + 9.6 ) −7/3
2π k xZ , (7)
where the factor of 2π is used to convert their expressions
from cyclic wavenumber to radian wavenumber
and u ∗ is the friction velocity, u ∗ = (τ/ρ) 1/2 , where τ
is the bottom stress and ρ is the density.
The vertical velocities of turbulence are strongly affected
by the boundary, so that most of their variance
is near the scale of the height above the bottom Z. The
horizontal turbulent velocities are less affected by the
presence of the boundary, and most of their variance is
at scales significantly greater than Z.
Turbulence velocity spectra observed in the atmospheric
boundary layer have been found to be in good
agreement with the spectra predicted by the empirical
formulas of Kaimal et al. [1972] [Lee, 1996; Young,
1987]. These empirical velocity spectra were also reproduced
based on a theoretical argument of the turbulence

LIEN AND SANFORD: SPECTRA OF VELOCITY AND VORTICITY FLUXES 4
energy budget which assumed that turbulence in the
Kansas experiment was isotropic [Moraes and Goedert,
1988].
Turbulent momentum fluxes are carried mostly by
large eddies. In the inertial subrange, momentum flux
cospectra P uw do not vanish as required by the local
isotropy condition (3) but, rather, decrease rapidly with
a k −7/3 dependence [Wyngaard and Co´te, 1972]. The
–7/3 spectral slope was predicted by the dimensional
argument in which the cospectrum of momentum flux
depends only on ε and k and is linearly proportional to
the mean shear ∂ z U.
Note that the integration of the momentum flux
cospectrum over the entire wavenumber domain should
yield the total momentum flux, i.e., −u 2 ∗. The integration
of the empirical momentum flux cospectrum proposed
by Kaimal et al. [1972], however, yields a slightly
underestimated momentum flux, 0.94 u 2 ∗.
A similar empirical spectral form for the turbulent
vertical velocity was proposed earlier by Busch and
Panofsky [1968]; that is,
Φ BP
ww (k x ) = 0.54u 2 ∗Z
[ ( ) ] 5/3 −1
kx Z
1 + 1.5
. (8)
2
At low wavenumbers k x Z the spectral level of Busch
and Panofsky’s [1968] model is greater than that of
Kaimal et al.’s [1972] model. McPhee [1994] found that
his vertical velocity spectra taken in the ocean under the
ice agree with Busch and Panofsky’s empirical form.
3. Observed Spectra
3.1. Measurements, Experiment, and Data
Selection
The measurements discussed in this analysis were
taken with an electromagnetic vorticity meter (EMVM)
[Sanford et al., 1999]. The sensor scale l is ∼0.09 m.
The data were taken in Pickering Passage, Washington,
in a strong ebb tide, ∼0.8 m s −1 peak surface
tidal flow, during October 23–26, 1995. During this
experiment the flow was unstratified. Details of the
experiment and measurements and a summary of the
turbulence properties are given by Sanford and Lien
[1999]. In this paper, spectra will be calculated using
data taken when the EMVM was held at fixed depths to
eliminate possible contamination of the vertical velocity
by sensor motion and yield a clean estimate of the
streamwise wavenumber spectrum when using Taylor’s
hypothesis to convert frequency to wavenumber. There
are ∼150 min of data taken at a 20-Hz sampling rate
divided into 79 time segments. Of these, 74 segments
were taken within 8 meters above the bottom (mab).
Because the EMVM has two sensors on opposite sides
of the instrument, 158 velocity spectra and flux cospectra
are computed in this analysis. The two sensors are
separated by ∼0.25 m spanwise. Therefore these two
sensors observe the same eddies if the eddy scales are
> 0.25 m.
3.2. Velocity Spectra
The streamwise and vertical velocity spectra are calculated
using a multitaper spectral analysis with two
tapers [Percival and Walden, 1993]. The observed velocity
spectra are attenuated at small scales by the sensor
response [Sanford et al., 1999]. To correct for the
sensor effect, the observed velocity spectra have been
compensated by dividing by the sensor response functions.
The observed velocity spectra are normalized by
the measured turbulence kinetic energy dissipation rate
ε and averaged in several depth bins (Figure 1). This
normalization (equations (1) and (2)) is done to collapse
the spectra in the inertial subrange for isotropic turbulence.
The normalized streamwise velocity spectra are
nearly independent of depth and show a spectral slope
of –5/3. The normalized vertical velocity spectra are
clearly dependent on Z. At scales greater than Z, i.e.,
k x Z ≪ 1, the vertical velocity spectra are nearly white.
At scales smaller than Z, i.e., k x Z ≫ 1, the vertical velocity
spectra exhibit a –5/3 inertial-subrange spectral
slope consistent with (2).
According to the empirical spectral form observed
in the atmospheric boundary layer (equations (5) and
(6)), velocity spectra normalized by Z should depend
only on the nondimensional wavenumber k x Z and the
friction velocity u ∗ . The friction velocity was calculated
as 0.024 m s −1 by Sanford and Lien [1999]. Normalized
wavenumber spectra at various depth bins indeed
collapse to a universal form (Figure 1b). The ratios between
observed streamwise and vertical velocity spectra
converge to the value of 3/4 expected for isotropic turbulence
at k x Z ≥ 5 (Figure 1c). Saddoughi and Veeravalli
[1994] tested the local isotropy in the turbulent
boundary layer at high Reynolds numbers in a wind
tunnel and found that the energy spectra reached the
local isotropy at k x ε 1/2 S −3/2 ≈ 3. If we assume that
ε = u 3 ∗κ −1 Z −1 and that the mean shear S = u ∗ κ −1 Z −1
in the inertial sublayer, the energy spectra should exhibit
a local isotropy condition at k x Z ≥ 7.5, close to
our observed range of k x Z ≥ 5.
The composites of normalized spectra of the observed

LIEN AND SANFORD: SPECTRA OF VELOCITY AND VORTICITY FLUXES 5
vertical and streamwise velocity are compared with empirical
spectra suggested by Kaimal et al. [1972] and
Busch and Panofsky [1968] (Figure 2). Because the momentum
flux 〈u ′ w ′ 〉 is nearly constant in the bottom 9
m [Sanford and Lien, 1999], only spectra at Z < 9 m
are used for constructing the composite spectra. The
streamwise spectrum agrees with the empirical form of
Kaimal et al. [1972] within the 95% confidence interval,
except at the highest wavenumbers, where the observed
spectrum falls below the empirical spectrum. The discrepancy
at high wavenumbers might be due to sensor
attenuation. Although the sensor effect has been corrected
by the response function, it is possible that the
correction was not adequate owing to the smearing effect
of the varying mean streamwise velocity. Our composite
vertical velocity spectrum agrees with the empirical
form suggested by Busch and Panofsky [1968] within
the confidence interval, similar to McPhee’s [1994] results.
At low wavenumbers the observed spectrum is
slightly greater than the empirical spectrum found by
Kaimal et al. [1972] in the atmospheric boundary layer.
3.3. Momentum Flux Cospectra
We compute the momentum flux cospectra from observations,
normalize them by Z, and form a composite
cospectrum as a function of the normalized wavenumber
k x Z (Figure 3). Again, only spectra calculated at Z <
9 m are used to construct the composite spectrum. The
observed composite momentum flux cospectrum agrees,
within a 95% confidence interval, with the empirical
spectrum [Kaimal et al., 1972] based on the estimated
friction velocity of 0.024 m s −1 [Sanford and Lien, 1999].
At small scales, i.e., k x Z ≥ 20, the observed cospectrum
falls below the empirical cospectrum, although
they still agree with each other within the confidence
intervals. The sensor response attenuates the observed
cospectra at small scales. The sensor response function
for the momentum flux cospectrum of the EMVM measurements
is derived in Appendix A. We correct the
observed spectra by the sensor response function and
recalculate the composite momentum flux cospectrum.
The corrected momentum flux cospectrum is slightly
greater than the uncorrected cospectra at large k x Z,
but the difference is insignificant. In other words, our
estimates of the momentum flux are not significantly affected
by the finite size of the EMVM sensor in the high-
Reynolds-number flow of an unstratified tidal channel.
Indeed, on the basis of the empirical momentum flux
cospectrum and the EMVM response function, we conclude
that the EMVM estimates capture more than 94%
of the total momentum flux at 0.5 mab (Appendix B).
Φ u Z −1 & Φ w Z −1 (m 2 s −2 )
10 −2
10 −3
10 −4
10 −5
10 −6
10 −7
Φ u (ε¯ / ε) 2/3 (m 3 s −2 )
10 −2 15.9 m
(a)
10 −3
10 −4
10 −5
k x Z
5.3 m
1.9 m
0.5 m
10 −1 10 0 10 1 10 2
10 −1 10 0 10 1
(b)
k x (m −1 )
Φ u / Φ w
10 1 (c)
10 0
10 −1
10 −1 10 0 10 1 10 2
k x Z
Figure 1. (a) Observed spectra of streamwise velocity
(solid curves) and vertical velocity (dashed curves)
averaged in various depth bins as a function of streamwise
wavenumber k x . The labels indicate the height
above the bottom Z of each depth bin. The shading
represents the 95% confidence interval. These spectra
have been normalized by the observed turbulence
kinetic energy dissipation rate ε. (b) Observed spectra
and wavenumbers normalized by Z (using equations
(5) and (6)). The horizontal and vertical velocity spectra
are collapsed to their corresponding universal forms.
The thick solid and dashed curves represent the mean
normalized streamwise and vertical velocity spectra, respectively.
(c) Ratios between observed streamwise velocity
spectra and vertical velocity spectra. Dots are
ratios of spectra at different depth bins. Crosses are
ratios of mean spectra. Vertical bars across symbols
denote the 95% confidence interval of the F test. The
horizontal line in Figure 1c indicates the 3/4 value expected
for isotropic turbulence (equations (1) and (2)).

LIEN AND SANFORD: SPECTRA OF VELOCITY AND VORTICITY FLUXES 6
Fractions of the resolved momentum flux as a function
of wavenumber are discussed in Appendix B to provide
other experimentalists with references for interpreting
their observations.
Φ u / (u *
2
Z) & Φ w / (u *
2
Z)
10 0
10 −1
10 −2
10 −3
10 −4
10 −1 10 0 10 1 10 2
10 1
Figure 2. Comparison of the observed streamwise velocity
spectrum and vertical velocity spectrum with empirical
spectra found in the atmosphere boundary layer.
The thick solid curve is the composite observed streamwise
velocity spectrum. The thin solid curve is the
empirical streamwise velocity spectrum suggested by
Kaimal et al. [1972]. The thick dashed curve is the
composite observed vertical velocity spectrum, and the
thin dashed curve is Kaimal et al.’s [1972] empirical vertical
velocity spectrum. The thin solid curve with open
circles is the empirical vertical velocity spectrum proposed
by Busch and Panofsky [1968]. Shadings denote
the 95% confidence interval ofthe observed spectra.
−P uw / (u *
2 Z)
10 0
10 −1
10 −2
10 −3
10 −4
−7/3
(a)
10 −5
10 −1 10 0 10 1 k x Z
10 2
−k x Z P uw / (u *
2 Z)
0.3
0.25
0.2
0.15
0.1
0.05
(b)
0
10 −1 10 0 10 1 k x Z
10 2
Figure 3. Comparison of the observed momentum flux
cospectrum with the empirical cospectral form found
by Kaimal et al. [1972] in the atmospheric boundary
layer. (a) Comparison of observed and model spectra.
(b) Variance-preserving form of Figure 3a. The thick
solid curves are observed spectra. The thin solid curves
are the empirical spectra suggested by Kaimal et al.
[1972]. The thick dashed curves, which are nearly overlapped
by the thick solid curves, are observed spectra
corrected for the sensor response function. In the inertial
subrange the empirical momentum flux cospectrum
has a spectral slope of –7/3.
Saddoughi and Veeravalli [1994] found that the momentum
flux cospectrum dropped to zero, an indication
of local isotropy, at k x Z ≥ 25. The falloff of our observed
spectrum from the model spectrum at k x Z ≥ 20
could also be because k x Z ≥ 20 is in the wavenumber
range where the viscous dissipation becomes important.
4. Vorticity Flux Spectrum
4.1. Observed Vorticity Flux Cospectrum
The observed vorticity flux cospectra averaged in
three depth bins are shown in Figure 4 as a variancepreserving
plot. Most of the vorticity flux is at wavenumbers
of 0.2–5 m −1 . In this wavenumber range the vorticity
flux is mostly positive. Below and beyond this
range the vorticity flux cospectra are sometimes less
than zero. Because of their large error bars, the vorticity
flux cospectra in these three depth bins are not
significantly different, except that the vorticity flux in
the depth bin 4–8 m seems to be slightly larger at k x of
0.4–0.5 m −1 . Sanford and Lien [1999] observed a simi-

LIEN AND SANFORD: SPECTRA OF VELOCITY AND VORTICITY FLUXES 7
lar feature in a vertical profile of vorticity flux that was
calculated from data taken while the EMVM was vertically
profiled. The averaged vorticity flux in the depth
range 0–2 m is slightly greater than that in the 2–4 m
depth bin but not significantly so relative to the statistical
confidence interval. The variation of the estimated
vorticity flux within each depth bin is greater than the
variation among different depth bins.
To our knowledge, no empirical spectral form has
been suggested for the vorticity flux in the atmospheric
or oceanic turbulent boundary layer. Despite the great
variability of the observed vorticity flux cospectra, they
do have a common spectral shape, i.e., large and positive
at intermediate wavenumbers and smaller and
sometimes negative at small and large wavenumbers.
This commonly observed shape suggests the existence
of a universal spectral form.
Following the dimensional analysis used by Wyngaard
and Coté [1972] for deriving the momentum flux
and heat flux cospectra in the inertial subrange, we assume
that the vorticity flux cospectrum in the inertial
subrange depends on only three parameters: ε, k, and
the vertical gradient of mean spanwise vorticity ∂ z 〈ζ y 〉.
We further assume that the cospectrum is linearly proportional
to the gradient of the mean vorticity but with
an opposite sign, consistent with the concept of a downgradient
flux. The dimensional analysis yields a form
for the vorticity flux cospectrum of
k x P wζy (m s −2 )
3.5 x 10−4
3
2.5
2
1.5
1
0.5
0
−0.5
5.3 m
2.8 m
1.1 m
10 0 10 1
k x (m −1 )
Figure 4. Observed vorticity flux cospectra averaged
in three depth bins: 0–2 m (solid curve), 2–4 m (dashed
curve), and 4–8 m (dotted curve). The shading denotes
the 95% confidence interval calculated from the
bootstrap method. Labels indicate the averaged heights
above the bottom of the three depth bins.
4.2. Model Vorticity Flux Cospectrum
P wζy (k x ) ∝ −ε 1/3 ∂ z 〈ζ y 〉k −7/3
x ∝ u 2 ∗(k x Z) −7/3 . (9)
This expression is similar to the empirical forms for the
momentum and heat flux cospectra. In the inertial sublayer,
ε = u 3 ∗κ −1 Z −1 and ∂ z 〈ζ y 〉 ≈ −u ∗ κ −1 Z −2 . These
identities have been used in (9). The vorticity flux
cospectrum does not vanish in the inertial subrange,
but it decays rapidly with a spectral slope of –7/3.
In the turbulent boundary layer, large eddies are affected
by the presence of the boundary. Following the
empirical momentum flux cospectral form [Kaimal et
al., 1972], we propose that the vorticity cospectrum has
the form
P model
wζ y
(k x ) = au 2 ∗ [1 + b(k x Z)] −r . (10)
This form is consistent with our observed vorticity spectra;
that is, most of the variance is in the intermediate
wavenumbers and decays at lower and higher wavenumbers.
The dimensional argument in (9) suggests that
the spectral slope in the inertial subrange r = 7/3. Parameters
a and b depend on the magnitude of the vorticity
flux and the wavenumber of the dominant vorticity
flux, respectively; that is,
〈w ′ ζ ′ y〉 =
∫ ∞
0
dk x Pwζ model a
y
(k x ) =
b(r − 1) u2 ∗/Z, (11)
(k x Z) peak = (r − 1) −1 b −1 , (12)
where (k x Z) peak is the wavenumber of the maximum
vorticity flux variance k x P model
wζ y
.
An averaged normalized vorticity flux cospectrum in
normalized wavenumber bins k x Z is shown in Figure 5.
This spectrum is constructed by first normalizing each
of the total 148 vorticity flux cospectra by u 2 ∗ and the
corresponding wavenumbers by Z −1 and then averaging
all the 148 normalized cospectra in the normalized

LIEN AND SANFORD: SPECTRA OF VELOCITY AND VORTICITY FLUXES 8
wavenumber bins. Note that ∼80% of spectra are observed
in the bottom 4 m. The averaged, normalized
vorticity flux cospectrum shows a wavenumber-bandlimited
spectral shape similar to that of the momentum
flux cospectrum. At small, normalized wavenumbers
the lower bounds of the vorticity flux estimates
are negative and show large error bars. The variancepreserving
plot k x ZP wζy u −2
∗ shows a peak at k x Z ≈ 2.
The corresponding estimate of b is 3/8 based on (12).
P wζy / u *
2
(a)
10 −2
10 −3
−7/3
10 −4
10 −1 10 0 10 1
k x Z
10 2
10 −1
k x Z P wζy /u *
2
0.4
0.35
0.3
0.25
0.2
0.15
0.1
0.05
(b)
10 −1 10 0 10 1 10 2
k x Z
Figure 5. (a) Comparison of the observed vorticity
flux cospectrum with our proposed model vorticity flux
cospectrum. (b) The variance-preserving form of Figure
5a. The thick solid curve is the observed vorticity
flux cospectrum calculated using spectra obtained in
the bottom 8 m, and the thin curve is the model spectrum
(equation (13)). The thick dashed curve is the
observed vorticity flux cospectrum corrected for the sensor
response function. A spectral slope of –7/3 is shown
for comparison with the model spectrum in the inertial
subrange. Open circles represent observed vorticity flux
cospectra averaged using data in the bottom 5 m, where
the observed vorticity flux follows the similarity scaling
(Figure 7).
In high-Reynolds-number flows one might expect the
mixing coefficients of momentum and vorticity to have
a similar magnitude. Indeed, Sanford and Lien [1999]
found that the magnitudes of the estimated eddy diffusivity
of vorticity [A ζ = −〈w ′ ζ y〉(∂ ′ z 〈ζ y 〉) −1 ] and the
eddy viscosity of momentum [A ν = −〈u ′ w ′ 〉(∂ z U) −1 ]
were similar within 6 mab.
In the energetic tidal boundary layer the primary
contribution to the mean spanwise vorticity is the vertical
shear of streamwise velocity; that is, ζ y = ∂ z U
[Sanford and Lien, 1999]. In the inertial sublayer the
eddy viscosity of momentum A ν and the eddy diffusivity
of vorticity A ζ can be approximated as u ∗ κZ,
and the vertical gradient of the mean spanwise vorticity
∂ z 〈ζ y 〉 = −u ∗ κ −1 Z −2 . Therefore the turbulent vorticity
flux in the inertial sublayer can be expressed as
〈w ′ ζ y〉 ′ = −A ζ ∂ z 〈ζ y 〉 = u 2 ∗Z −1 .
Tennekes and Lumley [1972] similarly scaled the vorticity
flux 〈w ′ ζ y〉 ′ as ul∂ z 〈ζ y 〉, where u = u ∗ is the
turbulent velocity scale and l = κZ is the turbulent
length scale in the boundary layer. Therefore the scaling
of 〈w ′ ζ y〉 ′ = u 2 ∗Z −1 is also suggested by Tennekes
and Lumley [1972].
If we accept the foregoing scaling argument for the
turbulent vorticity flux, (11) can be expressed as a/[b(r−
1)] = 1. Given r = 7/3 and b = 3/8, the estimate of
a is 1/2. Therefore we have constructed the following
form for the model vorticity flux cospectrum:
P model
wζ y
(k x ) = 1/2u 2 ∗ [1 + 3/8(k x Z)] −7/3 . (13)
The model spectrum has the shape of observed spectra,
the slope of –7/3 in the inertial subrange predicted by
the dimensional analysis, and the dominant peak displayed
by the observed spectra, and the magnitude of
the vorticity flux is consistent with flows in the inertial
sublayer.
Note that when constructing the model spectrum,
we have used only one property of the observed spectrum,
the wavenumber of the maximum vorticity flux
variance. We did not force the level and shape of the
model spectrum to agree with those of the observed
spectrum. Surprisingly, the model spectrum clearly reproduces
the observed vorticity flux cospectrum (Figure
5) in the wavenumber range from the peak of the vorticity
flux variance extending more than 1 decade into
the inertial subrange where a spectral slope of –7/3 exists.
At low wavenumbers, k x Z ≤ 0.9, the observed
vorticity flux spectrum drops much more rapidly than
the proposed model spectrum.
Although the vorticity flux cospectrum and the momentum
flux cospectrum have a very similar spectral
shape, the wavenumber of the maximum flux is higher
for the vorticity flux than for the momentum flux. This
is partially because most of the turbulent momentum
is at large scales and the variance of vorticity is mostly
contributed by smaller-scale eddies.
To correct for the finite-size-sensor effect, we modify
the observed vorticity flux cospectra by their response
functions (Appendix A) and form the composite spectrum.
The corrected cospectrum is not significantly different
from the uncorrected one, except at k x Z ≥ 40,
where the corrected spectrum shows a small bump. We
suspect that this small bump is unreal and is caused
by dividing the observed spectrum by a response func-

LIEN AND SANFORD: SPECTRA OF VELOCITY AND VORTICITY FLUXES 9
tion that is small. On the basis of the EMVM response
function for the vorticity flux cospectrum (Appendix A)
and our model spectrum, the EMVM resolves > 90% of
the total vorticity flux at Z > 1 m (Appendix B).
5. Discussion and Summary
5.1. Vorticity Flux Quad Spectrum of Isotropic
Turbulence
Note that the vorticity flux quad spectrum Q wζy
of the isotropic turbulence does not vanish [Batchelor,
1959]. We calculate the composite quad spectrum of
the measured vorticity flux, correct for the sensor response
function, and compare the result with the expected
quad spectrum in the inertial subrange, as predicted
by (4) (Figure 6). The response function of the
quad spectrum is based on the theoretical quad-spectral
form of Batchelor [1959] for isotropic turbulence. The
observed vorticity flux quad spectrum shows a peak
near k x Z = 2, similar to its cospectrum, and rolls off
at a slope of –2/3, as expected from (4). At k x Z ≥ 2
the observed spectral level is one half the model level
before the correction for the sensor effect. After correction
the observed spectrum agrees much better with
the expected spectrum at k x Z > 20. This confirms the
derived response function and the local isotropy at large
k x Z.
5.2. Scaling of Vorticity Flux
On the basis of the turbulent properties in the inertial
sublayer of a turbulent boundary layer, the turbulent
vertical transport of spanwise vorticity 〈w ′ ζ y〉
′
should scale as u 2 ∗Z −1 . The variation of our observed
vorticity flux cospectra at individual depths is greater
than the variation among cospectra at different depths.
Longer stationary time series of vorticity flux in the
inertial sublayer are needed to confirm this scaling. Using
data taken during the period when the EMVM was
profiled through the water column in the same experiment,
Sanford and Lien [1999] found that the vertical
flux of the spanwise vorticity scaled with Z −1 in the
bottom 3 m. To reveal the details of the vorticity flux
in the bottom few meters, we calculated the average
vorticity flux in depth bins proportional to Z −1 using
the data used by Sanford and Lien [1999]. The vorticity
flux in the bottom 3 m indeed shows a u 2 ∗Z −1
scaling, with a fitted friction velocity u ∗ of 0.026 m s −1
and a 95% confidence interval of 0.002 m s −1 (Figure
7). The estimated u ∗ agrees with the value of 0.024 m
s −1 calculated by the profile method, the eddy correlation
method, and the dissipation method [Sanford and
Lien, 1999]. This result supports the scaling of vorticity
flux we used for constructing the model vorticity
flux cospectrum. At Z > 5 m the vorticity flux does
not scale as u 2 ∗Z −1 . Indeed, the scaling should work
only in the inertial sublayer. Sanford and Lien [1999]
found two log layers in their observed streamwise velocity
profile. The lower log layer exists in the bottom 3
m with an estimated friction velocity of 0.024 m s −1 ,
a transition layer between 3 and 5 mab, and an upper
log layer between 5 and 12 m with a friction velocity of
0.043 m s −1 . The averaged, normalized vorticity flux
cospectrum calculated using spectra in the bottom 5 m
is not significantly different from that calculated using
spectra in the bottom 8 m (Figure 5) because more than
80% of observed spectra are in the bottom 5 m.
− Q wζy (m 2 s −2 )
−2/3
10 −4
10 −5
10 0 10 2
k x Z
Figure 6. Comparison of the observed composite quad
spectrum of the vorticity flux and the theoretical quad
spectrum of isotropic turbulence [Batchelor, 1959]. The
observed quad spectrum is denoted by the thick solid
curve. The thin solid curve is the theoretical quad spectrum
for isotropic turbulence in the inertial subrange.
The thick dashed curve is the observed quad spectrum
corrected for the sensor response function. The shading
denotes the 95% confidence interval.
5.3. Viscous Effects
The empirical turbulence spectra and cospectra discussed
in this analysis were aimed at explaining the
observed spectra in the inertial subrange. Beyond the
inertial subrange, molecular viscosity becomes impor-

LIEN AND SANFORD: SPECTRA OF VELOCITY AND VORTICITY FLUXES 10
tant, and the inertial subrange scaling should not work.
The inertial subrange lies in the wavenumber range of
k(ν 3 /ε) 1/4 ≪ 1, where ν is the molecular viscosity. In
Pickering Passage, ε is ∼10 −6 m 2 s −3 in the bottom
boundary layer and ν is ∼1.6 × 10 −6 m 2 s −1 . Accordingly,
the inertial subrange lies at k x ≪ 4 × 10 2 m −1 .
Corrsin [1964] and Pao [1965] showed that the peak of
the shear spectrum and the roll-off of the velocity spectrum
occur at a wavenumber of 0.2(ν 3 /ε) −1/4 . Therefore
the inertial subrange of our observed spectra in
Pickering Passage probably lies below 10 2 m −1 .
Z −1 (m −1 )
1.8
1.6
1.4
1.2
1
0.8
2 u * = 0.026 (± 0.002) m s −1
0.5
0.6
2
0.4
3
0.2
5
10
0
25
0 0.5 1 1.5 2
(m s −2 )
x 10 −3
Figure 7. Vertical profile of 〈w ′ ζ y〉 ′ averaged in depth
bins as a function of Z −1 . For reference, the equivalent
depth scale is shown on the right margin. The thin
solid curve is the observed mean 〈w ′ ζ y〉, ′ and the shading
denotes its 95% confidence interval. The thick line is
the fitted vorticity flux as a function of Z −1 ; that is,
〈w ̂ ′ ζ y〉 ′ = u 2 ∗Z −1 . The estimated u ∗ is 0.026 m s −1 , with
the 95% confidence interval of 0.002 m s −1 denoted by
the two thick dashed curves. This estimate of u ∗ agrees
with the value of 0.024 m s −1 calculated by conventional
methods [Sanford and Lien, 1999].
1
Z (m)
6. Summary
The present analysis shows that the spectral and
cospectral properties of the velocity and the momentum
flux observed in an unstratified oceanic tidal boundary
layer are in good agreement with the universal forms
found in the atmospheric boundary layer. The momentum
flux is carried by large-scale eddies, which are
fully resolved by the EMVM sensor. The vorticity flux
cospectrum shows a wavenumber-band-limited spectral
shape, similar to that of the momentum flux cospectrum,
but centered at smaller scales. A model spectrum
of the turbulent vorticity flux is proposed. The
spectral slope in the inertial subrange is determined by
a dimensional analysis following that of Wyngaard and
Co´te [1972]. The wavenumber of the maximum variance
of the turbulent vorticity flux is determined by the observed
spectrum. The spectral level is determined by
classical theories of turbulent boundary flows. The observed
vorticity flux is dominantly at k x from 1 to 10
Z −1 . The model spectrum agrees with the observed
spectrum at most wavenumbers, except at the lowest
wavenumbers where the observed spectrum falls below
the model.
The response functions of the vorticity and momentum
flux cospectra characterizing the sensor effect are
obtained, and the observed spectra are corrected by the
response functions. The correction does not produce
significant changes in the observed momentum and vorticity
flux cospectra. This indicates that the scale of
the EMVM sensor is sufficiently small to resolve turbulent
momentum and vorticity flux cospectra, especially
at Z ≥ 2 m.
The turbulence vorticity flux plays an important dynamic
role in the turbulence boundary layer. The similarity
scaling of the turbulence vorticity flux presented
in this analysis can be used to estimate the bed stress.
The turbulence vorticity flux is related to the divergence
of turbulence momentum flux, which represents

LIEN AND SANFORD: SPECTRA OF VELOCITY AND VORTICITY FLUXES 11
the turbulence force on the mean flow. Therefore understanding
the spectral properties of the turbulence
vorticity flux will help improve turbulence parameterization
schemes in numerical models. The advective flux
of turbulence vorticity by Stokes drift is the primary
driving mechanism for the surface Langmuir circulation.
Therefore studying properties of the advective vorticity
flux in the Langmuir circulation will lead to better understanding
of its dynamics. Further investigation of
the vorticity flux cospectrum is necessary to confirm or
revise our proposed model spectrum.
Appendix A: Appendix A: Sensor
Response Functions for Momentum and
Vorticity Flux
Response functions depend not only on the sensor
characteristics but also on the true spectral forms. To
derive the response functions of the momentum flux and
vorticity flux, we need to know their spectral forms.
For the momentum flux we use the empirical cospectral
form. For the vorticity flux cospectra we derive the
response function based on our model spectrum. For
the vorticity flux quad spectrum we use the spectral
form for the isotropic turbulence [Batchelor, 1959].
The response functions for the momentum flux cospectrum
and vorticity flux cospectrum are derived numerically
(Figure A1). The response function of the momentum
flux cospectrum shows a sharp attenuation at
k x l ≥ 0.5, where l is the EMVM sensor scale. A negative
response function also exists at k x l ≥ 2.
Response Function
1
0.8
0.6
0.4
0.2
0
Q wζ
P uw
P wζ
−0.2
10 −2 10 −1 10 0 10 1 10 2
k x l
Figure A1. Response functions of the momentum
flux cospectrum (solid curve), vorticity flux cospectra
(dashed curve), and vorticity flux quad spectra (dotted
curve).
Appendix B: Appendix B: Resolving
Momentum and Vorticity Flux
Measurements of turbulent and vorticity fluxes are
taken mostly from sensors of finite size, implying a
low-pass filtering of the turbulence field. To ensure
that measurements capture most of the turbulent fluxes,

LIEN AND SANFORD: SPECTRA OF VELOCITY AND VORTICITY FLUXES 12
the sensor size must be smaller than the scales of the
fluxes. Because the scale of the dominant vorticity flux
is smaller than the scale of the dominant momentum
flux, a sensor’s scale may be small enough to measure
the momentum flux, but not sufficiently small to measure
the vorticity flux.
We define the fractions of momentum flux and vorticity
flux that are resolved as a function of wavenumbers
as
R uw (k x ) =
R wζy (k x ) =
∫ kx
0
dk x P uw
∫ ∞
0
dk x P uw
, (B1)
∫ kx
0
dk x P wζy
∫ ∞
0
dk x P wζy
. (B2)
On the basis of the empirical momentum flux spectrum
of Kaimal et al. [1972] and our proposed vorticity flux
cospectrum, R uw and R wζy are calculated as function of
Z. At a fixed wavenumber the fraction of the resolved
fluxes is smaller closer to the bottom (smaller Z) and
the fraction of the resolved vorticity flux is smaller than
that of the resolved momentum flux (Figure B1). To
measure more than 95% of the vorticity flux, the sensor
scale has to be smaller than 0.02 m when the measurement
is taken at 1 mab and smaller than 0.1 m when the
measurement is taken at more than 2 mab. To measure
more than 95% of the momentum flux, the sensor scale
has to be smaller than 0.1 m when the measurement is
taken above 1 mab.
Fraction of Resolved Flux
1
0.8
0.6
0.4
95%
R uw Z = 5 m
0.2
(a)
0
10 −4 10 −2 10 0 10 2
k x (m −1 )
R wζy Z = 5 m
R uw Z = 1 m
R wζy Z = 1 m
Z (m)
20
15
10
5
(b)
0
10 −1 10 0 10 1 10 2
k 95% (m −1 )
Figure B1. (a) Fraction of fluxes resolved (equations
(B1) and (B2)) at 1 and 5 meters above bottom (mab).
(b) Cutoff wavenumber at which more than 95% of
fluxes are resolved as a function of Z. Solid curves are
for momentum fluxes, and dashed curves are for vorticity
fluxes.
On the basis of the empirical momentum flux cospectrum,
our model vorticity flux spectrum, and the EMVM
response functions (Figure A1), we calculate the resolved
momentum and vorticity fluxes as a function of
height above the bottom (Figure B2). The EMVM captures
more than 94% of the momentum flux and 90%
of the vorticity flux above 1 mab.
Z (m)
20
15
10
5
0
0.85 0.9 0.95 1
EMVM / & EMVM /
Figure B2. Fraction of momentum flux (solid curve)
and vorticity flux (dashed curve) resolved by the
electro-magnetic vorticity meter (EMVM) sensor as a
function of Z. Fractions were computed using the empirical
momentum flux cospectrum, our model vorticity
flux cospectrum, and the EMVM sensor response functions.
Acknowledgments. The successful observations taken
in Pickering Passage were achieved with the vital help
of John Dunlap, James Carlson, Eric Boget, and Gordon
Welsh. Discussions with Eric Kunze and Eric D’Asaro have
been very helpful. This research was supported by the Office
of Naval Research.
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This preprint was prepared with AGU’s LATEX macros v4,
with the extension package ‘AGU ++ ’ by P. W. Daly, version 1.6b
from 1999/08/19.