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Solving the sea level equation, part I, theory
No full textPreface
This booklet is devoted to the study of some theoretical and practical aspects
of the so–called ”Sea Level Equation” (SLE), an integral equation that predicts the time–dependent shape of the equipotential surface of a deformable
body subject to surface forces. In the field of global geodynamics, the SLE
serves as a tool for computing the postglacial sealevel variations and other
observable quantities, taking as an input the shapes and chronology of the
Pleistocene ice–sheets.
Our first purpose was simply to collect various sparse notes and to translate in simple words the theory of the SLE for the PhD students attending
my lessons of ”Global Geodynamics” at the University of Bologna. However,
in Part II we also provide details on the numerical discretization of the SLE
and a freely available Fortran 90 code (SELEN) that anyone can use to solve
the SLE on his own computer. We hope that the material presented will
facilitate the work of colleagues at their first approach to Glacial Isostatic
Adjustment (GIA), and perhaps also more experienced geophysicists willing
to benchmark their own codes. As far as we know, this is the first time that
a sealevel equation solver is made freely publically available.
The development of the theory of the SLE is based on a number of approximations. First, the Earth is assumed to be radially stratified and incompressible, and the various layers are characterized by a linear viscoelastic
rheology. This is a widely diffused approximation, but recent work has been
done to include non–Newtonian rheologies and lateral viscosity variations in
spherical Earth models (see e. g. [4, 26]). Second, it is assumed that the
ocean function is constant, that implies fixed shorelines. Third, we totally
neglect the effects of rotation on the GIA–induced sealevel variations. The
reader is referred to [7] for the theoretical details concerning the rotational
feedback and for the numerical evaluation of its consequences. In view of
the approximations listed above, this booklet provides a zeroth–order model
for the postglacial sealevel changes, that can be considerably refined in the
future, hopefully with the aid and the contribution of other investigators.
The future releases of this document will benefit from the feedback of the
readers of this first edition. Please feel free to write to [email protected]
for questions, comments, and suggestions.
GS, February 8, 2005
Perturbations in the Earth's rotation induced by internal density anomalies: implications for sea-level fluctuations
No full textThe effects of internal mass anomalies on the Earth's rotation are analyzed within the framework of linearized Liouville equations and Maxwell rheology for the mantle. Our approach is appropriate for a simplified modeling of subduction. Sea-level fluctuations induced by long-term rotational instabilities are also considered. -from Author
TABOO - User guide
Full text linkIn this manual we describe the general purpose of TABOO, its structure, and
its applications. A very basic knowledge of the Fortran 90 language and
of Unix is required. The mathematical theory behind TABOO is given in a
separate theory document (hereafter referred as to TD) which is released
with these instructions
Generalized Maxwell Love numbers
Full text linkBy elementary methods, I study the Love numbers of a homogeneous, incompressible,
self–gravitating sphere characterized by a generalized Maxwell rheology, whose
mechanical analogue is represented by a finite or infinite system of classical Maxwell
elements disposed in parallel. Analytical, previously unknown forms of the complex
shear modulus for the generalized Maxwell body are found by algebraic manipulation,
and studied in the particular case of systems of springs and dashpots whose
strength follows a power–law distribution. We show that the sphere is asymptotically
stable for any choice of the mechanical parameters that define the generalized
Maxwell body and analytical forms of the Love numbers are always available for
generalized bodies composed by less than five classical Maxwell bodies. For the
homogeneous sphere, “real” Laplace inversion methods based on the Post–Widder
formula can be applied without performing a numerical discretization of the n–th
derivative, which can be computed in a “closed–form” with the aid of the Fa`a di
Bruno formula
Using the Post-Widder formula to compute the Earth's viscoelastic Love numbers
No full textThe post-glacial or post-seismic relaxation of a Maxwell viscoelastic earth, 1-D or slightly laterally heterogeneous, can be calculated in a normal-mode approach, based on an application of the propagator technique. This semi-analytical approach, widely documented in the literature, allows to compute the response of an earth model whose rheological parameters vary quite strongly with depth, at least as accurately and efficiently as by 1-D numerical integration (Runge-Kutta). Its main drawback resides in the need to identify the roots of a secular polynomial, introduced after reformulating the problem in the Laplace domain, and required to transform the solution back to the time domain. Root finding becomes increasingly difficult, and ultimately unaffordable, as the complexity of rheological profiles grows: the secular polynomial gradually gets more ill behaved, and a larger number of more and more closely spaced roots is to be found. Here, we apply the propagator method to solve the Earth's viscoelastic momentum equation, like in the above-mentioned normal-mode framework, but bypass root finding, using, instead, the Post-Widder formula to transform the solution, found again in the Laplace domain, back to the time domain. We test our method against earlier normal-mode results, and prove its effectiveness in modelling the relaxation of earth models with extremely complex rheological profiles
Post–glacial sea-level in the Mediterranean Sea: Clark's zones and role of remote ice sheets
No full textThe theory behind TABOO - a posT glAcial reBOund calculatOr
No full textThis is an attempt to collect in a single document the basic traits of the the-
ory describing the deformations of the Earth under peculiar surface loads:
the ice sheets. The book has a mainly pedagogical purpose. It is written in a
simple way, and an e®ort is made to avoid the sentence it can be shown that.
Almost all of the propositions given here are demonstrated step{by{step,
even when they may appear obvious a priori. This is mainly done to facili-
tate the beginners in the 'art' of the postglacial rebound, but I also hope that
this transparent style of writing could be useful for more experienced inves-
tigators. The book is written according to an austere minimalism: we only
give the statements which are strictly needed to understand the basic con-
cepts. For this reason, the chapter devoted to the mathematical background
is largely biased towards the main tools, such has di®erential operators and
spherical harmonics.
The theory illustrated here is implemented in the source code taboo.f90
which is freely distributed by the Samizdat Press along with this document
and the accompanying user guide. At the core of TABOO there is the assump-
tion that the Earth is spherically layered. In the common language this means
that the problems which can be solved by TABOO are 1D problems. Nowadays
several research groups have developed more advanced codes, which account
for the 2D or even for the 3D structure of the lithosphere and the mantle.
However, these codes are not publically available to date, mainly for two rea-
sons. First, their are not totally developed, and some work is still to be done.
Second, di®erently from TABOO they are often based on numerical techniques
developed with the aid of software packages that are not publically available.
In a sense, TABOO has the aim of closing the chapter of the 1D problems
giving the chance of obtaining a portable source code and a full account of
the theory behind. It is hoped that this will encourage the developers of 2D
and 3D models to to the same with their procedures in the future.
The reader should be warned that TABOO is not a sealevel equation [4]
solver! The sealevel equation will be the subject of a separate review coming
in the next months along with a freely available code (SELEN).
The theory behind TABOO has only the purpose of collecting formulas and
results in an ordered structure. By no means the results presented here are
the product of my own research work. Rather, they constitute a theoretical
framework which has been constructed by a number of Authors in the course
of the last decades. It is not possible to mention all of the contributors to
this enormous (but sparse) work, and for this reason I must apologize for the
very poor bibliography that I have written at the end of this document. The
full set of original papers where the basic ideas have been ̄rst developed canii
be reconstructed on the basis of bibliographies of the manuscripts and books
quoted here.
While I have done my best to present a complete account of the theory
behind TABOO (and consequently a complete source code), some work is still
to be done. In particular, the present version of TABOO does not explicitely
compute relevant physical quantities such as the gravity anomalies, the stress
̄eld in the lithosphere and the rotational variations of the Earth. It is my
intention to include these topics in the next versions of the code (which will
also include ̄gures).
A ̄nal note concerning notation. I do not like to write vectors by bold
face letters, so that I use arrows throughout. I have been very pedantic in the
demonstration of the various propositions given in this document, certainly
too much for an experienced reader. This is admittedly boring, but I hope it
helps the novices, who are indeed the main target of this booklet. Since the
source code TABOO is totally accessible, I have not described in detail how
and where the single propositions are numerically implemented.
I have been involved in the research on these topics for ̄fteen years, ̄rst
as a student, and later as a teacher. Both need a place where a given formula
can be easily found and demonstrated. After all, this is the main purpose of
TABOO.
The future releases of this document (if any) will bene ̄t from the feedback
of the readers of this ̄rst edition. Please feel free to write to
[email protected]
for questions, comments, and suggestions.
Urbino, October 10, 200
The Sea Level Equation, Theory and Numerical Examples
No full textThe so–called “Sea Level Equation” is a linear, integral equation that governs the sea level changes due to the melting of the Pleistocene ice sheets and allows to model a suite of associated geophysical processes, including the postglacial rebound of the solid Earth, and the shape variations of the geoid. In this book we discuss the sea level equation for a spherically symmetric, linear viscoelastic Earth, assuming fixed shorelines and no rotational feedbacks. The theoretical aspects of the problem are presented in detail and only a basic knowledge of physics is demanded, that makes the discussion particularly oriented to non specialists of the field of global geodynamics. In the final part of the book we present some applications of the sea level equation in which we face the problem of predicting the Holocene and the present–day relative sea level variations, and the ongoing deformations of the solid Earth
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