Methods and Data
Abstract
We quantified turbulent dissipation in the Raritan river using both
conventional methods and a novel technique, the structure function
method. The conventional methods utilized velocity data collected
from an Acoustic Doppler Velocimeter (ADV) from which dissipation
was estimated using a spectral method as well as using a boundary
layer model to estimate turbulent dissipation. The drawback to the
spectral method was that velocity was only measured at one point so
turbulence was assumed to be “frozen” through the entire water column
while the boundary layer model had to assume dynamics that may be
overly simple for a stratified estuary. The “novel” technique relied on
data from a Nortek High Resolution Doppler Profiler, which took
velocity data along a vertical beam in the water column. This data was
processed using the structure function method, which yielded a vertical
profile of dissipation rate and did not rely on the frozen turbulence
assumption. Overall, the results were very promising, however the
structure function method produced elevated results relative to the
conventional methods suggesting that additional analysis is necessary.
Aside from improving our understanding of the dynamics of the
Raritan river estuary, this project demonstrated the potential use of the
structure function method to measure turbulence from a moving
platform such as a glider, drifter, or towed vehicle.
Spring Neap
Flood
Above left: Measurements of velocity obtained from the ADV; the tides are ebb dominant. Above
right: River discharge (black), tidal range (blue), lower surface and bottom salinity. Stratification
increases with river discharge, decreases with increasing tidal range.
Spectral Method:
2/3
ø
U
P »e =
Kz
3
*
and U* = Cd u 2
where K is 0.41, z is distance from the bottom (1.5
m), and Cd is drag coefficient.
Nortek HR Doppler
Profiler
Inertial subrange
(red)
Line of best fit:
Y=-0.00032x-2.8*10-6
Introduction
Turbulence is a dispersive and dissipative phenomenon originating in
the instability of large scale motions. It is responsible for a number of
important geophysical processes such as stratification, the overturning
of the ocean circulation and even the length of the day and the distance
to the moon! According to Kolmogorov’s “cascade theory”, turbulence
is characterized by the transfer of kinetic energy from a fluid’s motion
to progressively smaller energy scales, ultimately being converted to
friction at the molecular level. The study of turbulence in the coastal
ocean is complex yet fundamental to improving our understanding of a
variety of physical estuarine processes and has many practical
applications, such as studying ocean forecasting.
Structure Function:
Red line: Best fit
Blue line: x=y
Boundary Layer Method:
According to Kolmogorov’s spectral scaling The Law of the Wall (log layer assumption)
was used. A drag coefficient (Cd) was
laws:
calculated by finding the slope of Reynolds
-5/3
S(w ) = Cw
stress vs. velocity squared. This coefficient
5
was then used to calculate turbulent shear
log(S(w )) = log(C)- log(w )
production (assumed to be equal to
3
3/2
dissipation
in
an
unstratified
boundary
æ C ö
layer).
e =ç
÷
Where S is turbulent energy per frequency (m2s-1), ω
is frequency, and C is a constant (which was
calculated from a linear fit done on the spectrum), ϵ
is turbulent dissipation (m2/s3), β is a constant (1.5),
u is velocity.
Base/Platfor
m
Red line: Best fit
Blue line: x=y
Ebb
è bu
ADV
Results
A second order structure function D(z, r)
can be defined at a location (z) using
velocity v’, so that:
D(z, r) = (v'(z) - v'(z + r))2 » s'2
D(z,r) is the mean-square of the velocity
fluctuation difference between two
points separated by a distance r.
D(z, r) = C 2ve 2/3r 2/3
Conclusions
The results show good agreement
between the spectral, boundary
layer, and structure function
methods. The best agreement was
between the spectral method and
the structure function method –
the boundary layer method was
perhaps the least accurate and
showed the most scatter. This is
probably due to the fact that the
assumptions it relied on (an
unstratified boundary layer) were
violated in this environment: a
stratified estuary.
Method
Mean ϵ Value
(m2/s3)
Spectral
1.020 x 10-7
Boundary Layer Model
3.565 x 10-7
Structure Function
6.578 x 10-7
The “novel” structure function method appears promising. It provided
highly correlated albeit somewhat elevated results (relative to traditional
methods) in as few as 64 measurements (see figure above). This
indicates the viability of mounting the Nortek HR Profiler on moving
vehicles (such as gliders) to obtain vertical turbulence measurements
using the structure function method. For example, the HR Doppler
Profiler can sample at 8Hz so a vehicle with a 15cm/s vertical velocity
accurately estimate dissipation at a vertical resolution of ~1m.