Kelvin waves reflected off a sloped boundary

We are studying the effect of a sloped boundary on the reflection of equatorial Kelvin waves. When a first-mode, equatorial Kelvin wave hits an eastern boundary, some energy reflects as a first mode Rossy wave, and some energy goes north and south as coastal Kevlin waves. We are looking for the partitioning of energy in the coastal waves as a function of the frequency of the incoming wave and the north/south slope of the eastern boundary. To do this, we will run a simple, linear, 1.5 layer reduced-gravity model. To excite equatorial Kelvin waves of a specific frequency, 1/T, we force the model with a wind patch. The wind patch is in the western side of the basin, and decays exponentially away from the center. The figure below shows a schematic of the model set-up:

The wind patch is centered at 25E, on the equator. The energy flux through section I, from 15S to 15N, will be compared to the energy flux through II and III, at 15N and 15S, respectively. The angle of the coast, theta, will be varied. The model uses a C-grid, with a 1 degree (longitude) by 0.5 degree (latitude) resolution. The initial upper layer thickness is 300m, corresponding to a first-mode phase speed of 2.63 m/s. The reduced gravity is 0.0229 m/s2 in these experiments. The model domain extends to 30N and 30S (click here to see the problem with the smaller domain), where solid walls are applied, and the incoming signal is damped. The western boundary is open. The model is linear, and friction is only applied within 4 degrees of the northern, southern and western boundaries.

The applied wind stress is only zonal, and varies from easterly to westerly with a specified period. The figure below shows the wind vectors in panel c. The amplitude along the equator is given in panel a, and the amplitude along 25E (center of the patch) is given in panel d. The time modulation, for the example of 30-day period forcing, is given in panel b.

The model was run for 15 years to get a period solution. The animations below show the model upper layer thickness variations (ULT-ULT_init) for a 90-day period in year 15. The angle (theta) of the coast varies from 0 (straight north/south) to 5 to 10 to 20 to 30.

Energy flux calculations Zonal energy flux was computed through section I (see figure above), and meridional energy flux was computed through sections II and III:

Here, the reduced gravity (g') is 0.0229 m/s2, the first mode phase speed (c) is 2.62 m/s, and (x1,x2) are the limits of the sections II and III, and (y1,y2) are the limits of section I. In this particular case, we integrate line I along 70E from 15S to 15N (y1=-15, y2=15), and lines II and III for 20 degrees beginning at the eastern boundary. In these cases, the forcing period is 30 and 45 days, but the integrals above were made over 6 (4) forcing periods (180 days). The results of the integrations over 180-day periods for the last five years of model integration are as follows:

30-day forcing 45-day forcing


%E-flux north (II)

%E-flux south (III)

%E-flux north(II) %E-flux south (III)




50 50







32 68




35 65




25 66

The figure below shows the percent energy flux through section II as a ratio of the zonal energy flux through section I (percent outgoing to the north/incoming from the west). The integration was done over 180 days (for 30-day forcing) and the results given for the last 5 years of model integration.

The model experiments were then repeated with a 60-day wind forcing. In the animation below, the model runs for all 15 years (5400 days of animation, so it may take a while!!)

Upper layer thickness anomalies at 70E for the 60-day wind forcing. In the animation below, the model runs for all 15 years (5400 days of animation)

In this case, for some reason the model is not exactly periodic after 15 years. The figure below shows the upper layer thickness anomalies at 80E for fifteen years (Please NOTE: The three panels below were scanned; the left panel is for the first 5 years, the middle panel is the next five, and the right panel is the last 5 years. Unfortunately the y-scale is different in each):

The panels above show that the model is not periodic after 15 years.