Extended Resolution Analysis of the Surface Waves.

Here is a brief description of the ongoing analysis of surface waves in connection with the HOME project. The objective of this component is to estimate the part of the forcing of LC by the waves, which involves extracting estimates of the energy, momentum, and mass-flux associated with the waves- including direction as well as frequency and wavelength.

First picture: two views of waves as measured by the "Long-Range Phased Array Doppler Sonar" (LRPADS).

On the right
is the radial orbital velocity data, the more-nearly "raw" form of the data. The waves are propagating away from FLIP (upwards in these plots). False color contours indicate surface horizontal velocities, with away (orange to red) corresponding to wave crests and towards us (green to blue) corresponding to troughs.

On the left
is a grey-scale image of the radial slope, with dark corresponding to up-slopes. Since the waves are propagating away, this corresponds to the "back" slope of the waves, while the lighter shades correspond to the forward faces of the waves. The black arrow indicates wind direction (and speed, about 10 m/s). Two things are of note here: (1) in addition to the dominant waves propagating downwind, and having ~150 m wavelengths (10s waves), you can see medium length waves propagating up to the right- they show up a little better in the slope grey-scale plot, but you can see them on the right as well, particularly along the right-hand edge of each "pie." (2) If you look very VERY closely at the greyscale just to the left of the wind arrow, you can see tiny "ripples", smaller than 20m wavelengths. These are real waves, not artifacts; you can see them propagate. However, because the time-sampling is a bit slow (every 2.5 seconds), they appear to go backwards, like the spokes of wheels in movies sometimes do. So we get a pretty good range of scales resolved- roughly two orders of magnitude.

Second picture: Time-range plot of the surface horizontal velocity in the downwind direction.

Picking a beam roughly parallel to the wind (just to left of center; see previous plot), this image shows the along-beam orbital velocity component versus range (vertical) and time (horizontal axis). This view is good for examining the phase and group velocities of the waves.

For the conditions at the time of this sample, the data quality is good to about 1300 m. There are two distinct scales of waves here: (1) The longer "dominant" waves (~10.6 s period) with phase-speeds near 15 m/s and group-speeds around 7-8 m/s; and (2) shorter but very intense groups that make "slashes" at a slightly lower angle, corresponding to 5 m/s group velocity. The longer waves have somewhat indistinct group structure, with several wave periods or lengths per group. The very short "packets" of higher frequency waves (~7-8 s period) are a bit of a surprise. The phase speed of these shorter waves is hard to pick out, as there appears to be less than a full wavelength per packet! The likely phase speed (about double the group velocity) matches the wind speed (~10 m/s). These short and intense packets undoubtedly have implications for wave breaking, etc.

This plot corresponds to the view along one of 35 directions for the duration of a single file. Data were gathered continually for 20 days, broken into 8.5 minute files. I find these time-range plots very useful for visualizing wave group effects.

Third picture: The 2D frequency-wavenumber Fourier transform of the time-range data.

This shows how knowledge that most waves are going downwind can be used to "unwrap" the aliased wave information, extending the effective resolution beyond the Nyquist limit in freqeuency (horizontal axis) versus wavenumber (vertical axis). The lines shown looping around the surface wave dispersion curves are based on a spectral-scaling estimate of the balance between wave variance and the noise floor. The left panel shows the complete (aliased) k-f spectrum, while the right panel illustrates the clipped (or "de-aliased") estimate. The temporal sampling is every 2.5 s, corresponding to a Nyquist frequency of 0.2 Hz; in contrast, the downwind branch of the wave spectrum is sensibly traced to about 0.3 Hz, 1.5 times farther out in temporal resolution. The spatial resolution is about 8 m, so the Nyquist wavenumber is about 0.062 cycles/m. This "extended spectrum" can be inverse-Fourier-transformed to provide interpolated time-space data with wave information at the higher resolution (e.g., as below).

Movie: Radial Velocity and Envelope, spatial propagation

This movie shows radial orbital velocity (left) and the corresponding wave amplitude envelope (right). Note 2 things: (1) the features on the left side propagate about twice as fast as those on the right. This is in line with the phase velocities (left) being about twice the group velocities (right). (2) While most variance is propagating slightly left of "up" (i.e., downwind), some can be seen propagating more to the right (and slightly up). This second category of "stuff" corresponds to a secondary peak in the directional frequency spectrum of the waves. In fact, the full 3D spectrum reveals about 4 such distinct "modes" at this time.

The orbital velocity envelope is qualitatively not too different from the corresponding component of the Stokes' drift, differing principally by an additional factor of u/c (or wave steepness), which is a weak function of both freqeuncy and conditions, in general.

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This page last updated 7/12/2006.