Mulgor Logo Long Wave Fourier Spectra at GeoNet Sites

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Introduction

In this webpage, I present the long-wave Fourier spectrum at each of the GeoNet sites. A definition of the Fourier spectrum and a description of how it is calculated are presented here.

A Fourier spectrum shows how the long wave energy is distributed with frequency or period (which is the inverse of frequency). A great deal can be learnt about the way long waves affect a site from the spectrum.

Two important features of a long-wave spectrum are:

These are illustrated in the figure below showing the spectra from Gisborne (left) and Tauranga (right). These are daily mean spectra which have been derived by calculating the spectrum for each day for 6 months, then averaging. The averaging removes any effects of storms or calm periods, but highlights any oscillations that occur on a regular basis.

Spectral Spikes

The above plot for the Gisborne site is typical of the spectrum from a harbour or embayment. The spikes correspond to seiche periods where the ocean is oscillating at its natural frequency. Seiche is like the sloshing of a bath tub, only on a much larger scale. The Gisborne spectrum shows a large spike at 85 min period, and a series of smaller spikes at lower periods. The longer the seiche period, the larger is the body of ocean that is oscillating, so the 85 min seiche period probably corresponds to sloshing of all of Poverty Bay out to the continental shelf. On the other hand, small seiche periods such as the one at 5 min probably corresponds to sloshing across the harbour. The spike at just over 2 min period is most probably spurious, being caused by long-period swell waves that have been aliased by sampling at 1-minute intervals.

If you want to learn more about seiches, I suggest you get a hold of this excellent reference:

Rabinovich, A.B., 2009. Seiches and harbour oscillations. In: Handbook of Coastal and Ocean Engineering (ed. Y.C. Kim) (World Scientific, Singapore), 193-236.

Spectral Slope

The slope of the spectrum is a measure of the fractal properties of the long wave signal. If the spectrum is flat (zero slope), as it is in the Gisborne spectrum between 10 and 30 min (ignoring the spikes), then the long waves are completely random. This is sometimes called white noise.

But if the spectrum drops from left to right (it has negative slope) as it does for the Tauranga spectrum, then the signal has some degree of self-similarity or scale-invariance, meaning that the signal has the same statistical properties no matter what scale you look at it.

If the slope is larger than 1, the signal is said to have "persistence", which means that the exisiting behaviour can be used to forecast future behaviour. If the slope is larger than 2, the signal is highly fractal in nature. The concept of persistence was devised by a hydrologist, Harold Hurst, who worked on the Nile River in the early part of the 20th century. He was interested in forecasting the flow of the Nile from one year to the next using the levels of Lake Victoria. In the 1950's when he was in his 70s, he published several papers on the validity of his theory for a wide variety of applications. Much of his work was scoffed at by professional mathematicians, until Mandelbrot (the father of fractals) resurrected it in the 1990's and now it is used extensively for forecasting financial markets.

Topographic Admittance

A site's topographic admittance is a function showing the effect of the topography, excluding the effect of incoming waves. The topographic admittances are presented here.

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Raoul Island Sites

The spectra for these two sites are similar. In the band between 10 and 100-minutes period, the spectra are essentially flat indicating that although there is significant energy in the band, the fluctuations are completely random. In the band between 2 and 10-minutes period, there are two peaks one at about 2.5 minutes and the other at about 4 minutes. The peaks at Fishing Reserve are broader than those at Boat Cove indicating that the seiche at Fishing Reserve is likely not related to local conditions, but propagates in from elsewhere (Boat Cove, perhaps). The peak at 2.5 minutes may be spurious, caused by aliasing of swell waves.

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North Cape

The spectrum shows seiche periods at 51.2, 33.8,.22.6, and 3.9 minutes.

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Great Barrier Island

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Waitemata Harbour

There are no distinguishable seiche periods in the spectrum. The slope is 2.97, indicating a high degree of persistence.

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Tauranga

There are no distinguishable seiche periods in the spectrum. The slope is 2.61, indicating a high degree of persistence.

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East Cape

The spectrum shows seiche periods at 21.7 and 12.5 minutes

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Gisborne

The spectrum shows seiche periods at 84.2, 56.3, 50.1, and 40.5 minutes

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Napier

The spectrum shows seiche periods at 77.4 65.3, 56.3, 47.0, 41.0, 33.1, 10.8 and 7.5 minutes.

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Castle Point

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Wellington Harbour

Wellington Harbour exhibits a strong seiche period at 26 min and two or three others between 15 and 17 minutes.

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Chatham Island

The GeoNet site at Owenga on the south side of Hanson Bay exhibits many spikes, whereas the NIWA site at Kaingaroa on the northeastern tip of the island (see the Google map below) has one predominant seiche period at about 11 minutes. My hypothesis is that the crescent shape of Hanson Bay is the reason for the multiple seiche periods at Owenga. We could sort this out with a simple mathematical model.

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Kaikoura

Kaikoura has seiche periods at 17, 28, and 36 min.

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Sumner Head

Sumner Head has no seiche periods, but Lyttelton has a strong seiche at 11 minutes, corresponding to cross-harbour sloshing.

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Port Chalmers

Port Chalmers shows a seiche period at 25 min.

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Puysegur

Puysegur shows seiche periods at 65.1 and 24.6 minutes.

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Manukau Harbour

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Topographic Admittances

In the very deep ocean (> 5,000 m), the long-wave spectrum is found to be:

where f is the frequency in cycles per minute (cpm) and A0 is typically 1x10-8 m2 cpm.

Normalising the observed spectrum by S0 removes the effect of the incoming waves, giving an estimate of the effect of the topograhy. This is defined as the topographic admittance, which is given by:

The way to interpret the topographic admittance, H, is:

The plots below show the topographic admittances of the various stations.

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Enquiries: Derek Goring

Mulgor Consulting Ltd

24 Brockworth Place

Riccarton, Christchurch

New Zealand

Phone: 64 3 343 5400

Fax: 64 3 343 5403

Email: d.goring@mulgor.co.nz