**A study of dissipative, shoaling high-frequency internal waves in
shallow water**

*James Lerczak, Clinton Winant and Myrl Hendershott*

Scripps Institution of Oceanography, CCS 0209

University of California at San Diego

La Jolla, California 92093-0209

http://www-ccs.ucsd.edu/iwaves/

The evolution of shoreward propagating, high-frequency, mode one internal waves was studied in a nearshore region where water depths ranged from 30 to 15 m. The data for this study were obtained from upward-looking acoustic doppler current profilers (ADCPs) deployed off of Mission Beach, California in 1996 and 1997 (Table 1). The ADCPs were rigidly mounted on the bottom, allowing vertical velocities to be unambiguously distinguished from the more energetic horizontal velocities. The bathymetry was approximately planar, with an onshore bottom slope, , of approximately 0.01. Data were analyzed over the period 26 September to 23 October (26.2 days) in 1996 and 4 September to 18 October (44.0 days) in 1997.

Ten frequency bands, having equal widths on a logarithmic scale, between
0.0167 and 0.3 cycles per minute (T = 60 to 3.33 min) were studied. Within
this range, coherences between vertical velocity, *w*, and horizontal
velocities, ,
were high. This range contained approximately
of the mid-depth *w* signal variance. The ADCP signal to noise variance
ratio was approximately 20, where the noise level was estimated to be the
high-frequency noise floor of *w*. Vertical velocities were studied
because, when measured from a stable platform, they are a very sensitive
index of high-frequency internal wave variability (Marsden *et al.*,
1994), and are less likely to be contaminated by other dynamics, such as
barotropic eddies, than horizontal velocities. While *w* of long-wavelength
waves is much weaker than ,
the maxima of
of mode one waves occur near the surface and near the bottom. These regions
of the water column were not well-covered by the ADCPs, which sampled the
center 67 to 75% of the water column. In contrast, the single maximum of
*w*
of mode one waves occurs near the thermocline. This maximum was always
detected by the ADCPs.

To isolate the mode one internal wave variability, the vertical velocities
at each mooring were decomposed into complex empirical orthogonal functions
(cEOFs) (Wallace and Dickinson, 1972) within each of the ten frequency
bands for consecutive blocks of data 2^{12} minutes (2.84 days)
in duration. Some of the results of the cEOF analysis are briefly summarized
here. From 70 to 90% of the variance in each band was explained by the
first cEOF mode, ,
indicating the high frequency wavefield was dominated by mode one waves.
The vertical structure of
was consistent with mode one waves, having a single, mid-column maximum.
The average depth of this maximum varied approximately linearly from 7
to 12.5 m as water depth changed from 15 to 30 m. The relative phases between
and horizontal velocities in the upper and lower water column were consistent
with onshore propagation (
leading *w* by 90* ^{o}* near the surface, and lagging

The variance of the amplitude of as a function of frequency and water depth was calculated. For linear waves, this variance is an estimate of , where is the vertical velocity amplitude of the mode one internal wave at the depth of maximum variance within the frequency band . Since cEOFs were estimated for consecutive ensembles of length 2.84 days, a smoothed time series of the internal wave variance, summed over the frequency range, was obtained (Fig. 1). The time series were quite coherent from mooring to mooring, and the variance clearly decreased with decreasing water depth. The frequency spectra of cEOF amplitude at each mooring had a peak at approximately 0.11 cpm (T = 9 min) (Fig. 2). The amplitude of this peak decreased with decreasing water depth, however, there was no obvious frequency shift in the peak or change in the spectral shape. The variance, summed over the frequency range, and averaged over ensembles, decreased nearly monotonically with decreasing water depth (Fig. 3).

This decrease is well described by a simple shoaling model with linear
dissipation in which, for time independent propagation of waves up a beach
with constant slope, ,
the divergence of the energy flux is balanced by dissipation. Thus

(1) |

in which *E* is the depth-integrated, time-averaged energy density,
and the substantial derivative is taken in a frame moving at the waves
group velocity, ,
so that

(2) |

For a fluid with constant stratification, the amplitude, ,
of the wave consequently varies with water depth according to

(3) |

where *N* is the buoyancy frequency of the fluid. For inviscid
progagation (*r* = 0), the amplitude is inversely proportional to
water depth. The variance would then increase by a factor of 4, as a wave
propagates from a water depth of 30 m to 15 m. This was clearly not observed
in our data. When linear dissipation is included in the model, the amplitude
may increase or decrease as the wave shoals, depending on the size of *r*.
We estimate *r* by minimizing the mean square error between the observed
variance at each depth, and the estimated variance using the above shoaling
model, the spectrum at 30 m (Fig. 2), and an assumed buoyancy frequency
of 0.263 cpm obtained from the estimated long-wave phase speed at different
water depths ( ).
The resulting estimates for *r* were 2.0
x 10^{-4} and 1.6 x
10^{-4}s^{-1}for 1996 and
1997, respectively. These values correspond to decay time scales of 1.4
and 1.8 hours, respectively, which are comparable to the typical time (
hours) it takes a wave packet to pass from the 30 m to the 15 m moorings
(cross-shore distance = 1.5 km). Such time scales are consonant with those
use by Pringle and Brink, in prep., in somewhat deeper water depths.

*1998-09-08*