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

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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