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## Apparent nonstationarity

adam — Mon, 07/22/2013 - 19:14

Visitors to this site are encouraged, if interested, to view many examples of both parametric and non-parametric analysis of tidally modulated hydrothermal time series, as well as the theory forming the basis for the nfTides code by reading the PhD dissertation by Timothy Edmund Jupp, "*Fluid Flow Processes at Mid-Ocean Ridge Hydrothermal Systems*", Cambridge University, September, 2000. Dr. Jupp's thesis covers the analysis of many of hydrothermal time series we collected on our various research cruises, as well as those published by others. The dissertation provides a strong theoretical underpinning for the astronomical forcing functions for both ocean and earth tides, the means by which one may analyze signals for tidal content, and then the use of the methods proposed for extracting information on tidal modulation of seafloor hydrothermal systems.

We were fortunate in having access to state-of-the-art instrumentation (the "MEDUSA") series that were developed in my lab, that were capable of simultaneously capturing diffusely flowing hydrothermal effluent from the seafloor, measuring the fluid temperatures and flow rates while capturing discrete chemical samples of the fluids. Our research cruises took us to the TAG hydrothermal mount, the Menez Gwen and Lucky Strike vent fields on the mid-Atlantic Ridge, and the RM24 site on the southern East Pacific Rise among other places. This provided ample motivation for developing new methods of analyzing what were inevitably relatively short (typically less than one month in duration), quite noisy and complicated time series in which tidal modulations could often be obscured by other signals. The present project that has led to the development of this web site, and the evolution of the nfTides software, is a consequence of this earlier work.

Jupp's thesis further examines both the convective patterns and the fundamental thermodynamics behind high temperature (black smoker) vent systems, and develops a theory and application for the poroelastic loading of the seafloor by tides that provides a basis for constraining the thickness and permeability of the seafloor hydrothermal layers. I highly recommend reading it as well as a number of papers that resulted from that work:

Jupp, T.E. and A. Schultz, Physical balances in subseafloor hydrothermal convection cells, J. Geophys. Res., 109, B5, May 2004, DOI: 10.1029/2003JB002697

Jupp, T and A. Schultz, A thermodynamic explanation for black smoker temperatures, Nature, 403, 880-883, 24 February 2000, DOI: 10.1038/35002552

Jupp, T.E. and A. Schultz, A poroelastic model for the tidal modulation of seafloor hydrothermal systems, J. Geophys. Res., 109, B3, March 2004, DOI: 10.1029/2003JB002583

-Adam Schultz

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The problem of fantom peaks.

On the time scale often associated with hydrothermal time series (days to weeks), the rapid change in lunar declination relative to the changes in the relative position of the Earth and Sun, has a profound impact on tidal potentials. For instance, at some point in the monthly cycle when the lunar declination is close to zero, semi-duiurnal tidal variations (which have a cos**2(declination) term will dominate over diurnal variations (with its sin(2*declination) term). So when viewed during this part of the tidal cycle, the time series may appear to be dominated by tidal processes that are primarily diurnal. At another part of the monthly cycle, when there may be a larger lunar declination, the situation reverses and the time series may appear to be modulated by predominately diurnal processes. So the apparent frequency content of the tidal signal is modulated by variations in the lunar declination. This concept extends to longer periods, where the effects of changes in solar declinaton have a similar pattern, although for now, given the typical length of hydrothermal time series, we consider shorter periods where lunar declination changes predominate.

This phenomenon can be illustrated as follows

Figure 2.3 from Jupp (2000). Fantom appearance of frequency modulation of tidal potential related to changes in the lunar declination (d). |d| is plotted over 20 days in March 1990. The darker line is the tidal potential, W, at 45 degrees N, 0 degrees E. When lunar declination is zero, the tidal potential appears to be modulated by a semi-diurnal process, whereas diurnal processes appear to dominate when the lunar declination is larger.

Were one to deploy a hydrothermal instrument on the seafloor, on say March 7th, and recover it on March 15th, and then if one carried out a conventional Fourier-based spectral analysis, one would find high spectral power at semi-diurnal frequencies. Were one to repeat the experiment, but collect the time series at some other interval within the same time 20 day time period, one would instead conclude the process under study was dominated by diurnal processes rather than semi-diurnal. This is an artefact of the non-stationary nature of the tidal potentials vs the stationarity assumption implicit in Fourier analysis.

In addition to being modulated by the lunar declination, the relative positions of the Sun, Earth and Moon also impact tidal potentials. During a new or full moon period, lunar and solar tidal potentials are in-phase (the mass of both celestial bodies are roughly aligned, leading to greater gravitational attraction relative to each point on Earth). This leads to spring tides, which are associated with larger than normal amplitudes. Conversely, at half-moon periods, the solar and lunar tidal potentials are out-of-phase, which leads to neap tides, which have smaller amplitudes than normal. The interval between spring and neap tides is roughly 7.38 days (Doodson & Warburg, 1941).

Figure 2.5 from Jupp (2000). (left) The change in the amplitude of the tidal potential (W) at the TAG hydrothermal site (26.13 deg N, 44.82 deg W) due to changes in lunar phase from spring to neap. (right) The ocean tide at the same site, as generated by the CSR code.

The figure above illustrates the phase lag between spring tides in the oceans vs spring tidees in the tidal potential. The ocean being a viscous body contained within complex boundaries, has a finite hydrodynamic response to changes in tidal potential (i.e. graviational forcing).

In order to resolve all the perfectly harmonic tidal components in a time series, one would need to collect several realizations of 18.6 years duration, which is the fundamental period at which the Sun, Earth and Moon realign into precisely the same positions. Jupp (2000) points out that a significant number of published works on the temporal variability of hydrothermal and geyser systems incorrectly compare their observations not against the actual tidal oscillations, but the slowly changing amplitude envelope of those oscillations.

Earth and Ocean Tides

In addition to the familiar ocean tidal streams (changes in horizontal flow rates of the ocean) and tidal heights, the so-called "solid Earth" also responds to the gravitational forces of the Earth, Sun and Moon (i.e. the tidal potential functions). The Earth is a plastic/elastic/poroelastic body that deforms in response to tidal forces. Classically, elastic properties alone are considered in the theory of Earth tides. Both the vertical displacement of the Earth's surface, and the volumetric dilatation of the crust at the surface may be related directly to the tidal potential as well as to the elastic parameters of the Earth materials.

It can be shown that both the surface displacement and the dilatation of the crust are proportionate to, and in phase with the tidal potential. This contrasts with the ocean tides, which (as previously mentioned) experience more complicated hydrodynamic effects that lead to phase lags relative to the tidal potential. The area of ocean tides (which includes not only changes in seasurface elevation, but changes in the ocean flow field) is extremely complex, and best left to source materials, including Jupp's thesis, particularly in respect to how they are expressed in spectral terms as measured by specific types of sensors. Suffice it to say that there will be quite distinct amplitude and phase relationships that are characteristic of processes modulated by ocean tides vs those modulated by Earth tides.

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