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1989 | Book

Tidal Potential Read chapter Read first chapter

Dynamics of Ocean Tides

Authors: G. I. Marchuk, B. A. Kagan

Publisher: Springer Netherlands

Book Series : Oceanographic Sciences Library

Included in: Professional Book Archive

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Table of Contents

Frontmatter
Chapter 1. Tidal Potential
Abstract
Let us consider the balance of forces acting upon a unit mass at a point A on the Earth’s surface. This mass moves in a gravitational field caused by the attracting forces of the Earth, G(A), of the Moon, the Sun and, generally speaking, of all the other perturbing bodies in the Solar System. Let us designate the attracting forces of these bodies as ∑ i ⊺ i (A); in what follows, ⊺ is a vectorial quantity. Let us introduce an inertial system of coordinates with its center at a point 0. Then Newtoon’s second law for the absolute motion of a unit mass at the point A can be written as
$$ \frac{{d_a^2}}{{d{t^2}}}{\mathbf{O}}A{\text{ = }}{\mathbf{P}}\left( A \right) + G(A){\text{ + }}\sum\limits_i {^Ti(A){\text{ + }}{\mathbf{F}}(A).} $$
(1.1.1)
where P(A) denotes pressure and F(A is the friction force; the subscript a in the operator d a 2 / dt2 indicates membership of the inertial coordinate system; time t will be defined below.
G. I. Marchuk, B. A. Kagan
Chapter 2. Methods and Results of Experimental Studies of Ocean Tides
Abstract
Direct measurements of tidal elevations are mainly performed at coastal and (comparatively few in number) island stations with the help of tide gauges or automatic level recorders. The accuracy of such measurements in approximately 1 cm. At present there exist some ten thousand stations, scores of them having accumulated continuous series of hourly elevation measurements of more than 50 years duration. Since 1964 deep-sea recording instruments with sensitive pressure transducers ensuring an accuracy of individual measurements of about 1 mm have been employed. The data deep-sea water-pressure measurements containing the values of harmonic tidal constants at 108 stations located in different parts of the World Ocean, and accumulated up to 1977, have been systematized by Cartwright et. al. [104].
G. I. Marchuk, B. A. Kagan
Chapter 3. Qualitatives Studies of the Tidal Dynamics Equations
Abstract
The spectrum of oscillations observed in the ocean covers a wide frequency range — much wider than the band (from 0.6 x 10−7 to 3.0 x 10−5 cycles/s) of forced oscillations generated by the major harmonics of the tidal potential. That is why the general hydrodynamic equations having solutions corresponding to all the components of the spectrum should be modified in much a way as not to contain any extraneous solutions and, at the same time, not to distort the solutions for the limited frequency band of interest to us. For this purpose, to exclude acoustic waves it is assumed that sea water is incompressible, and long (including tidal) waves are selected out of the family of gravitational oscillations, proceeding from the requirement of the smallness of the ratio between the vertical and the horizontal scales of motion. From the continuity equation it then follows that vertical velocities are small in comparison with the horizontal ones.
G. I. Marchuk, B. A. Kagan
Chapter 4. Free Oscillations in the World Ocean
Abstract
Let us consider free oscillations in the World Ocean, assuming the absence of friction and additional potentials of deformations. The equations of motion, describing such oscillations, can be presented as
$$ \frac{{\partial v}}{{\partial t}}{\rm{ + }}l k\, \times v{\rm{ = }} - gH \nabla \zeta ; $$
(4.1.1)
$$ \frac{{\partial \zeta }}{{\partial t}}{\rm{ + }}\nabla \cdot v{\rm{ = 0}}{\rm{.}} $$
(4.1.2)
G. I. Marchuk, B. A. Kagan
Chapter 5. Forced Tidal Oscillations in the World Ocean
Abstract
To describe forced tidal oscillations we turn again to the traditional equations of tidal dynamics, simplified as compared to (3.2.1), (3.2.2) by excluding the frictional forces, and present them as
$$ \frac{{\partial w}}{{\partial t}}{\rm{ + }}L{\rm{ }}w{\rm{ = }}L{\rm{ }}{w^ + }; $$
(5.1.1)
here w = (v.ζ); w+ = (0.ζ+);
$$ L\,{\rm{ = }}\left( {\begin{array}{*{20}{c}} {l{\rm{ }}k{\rm{ }} \times }&{gh{\rm{ }}\nabla }\\ \nabla &0 \end{array}} \right). $$
G. I. Marchuk, B. A. Kagan
Chapter 6. Tides in the Ocean-Shelf System
Abstract
As already shown, the natural way of obtaining the necessary information on ocean tides consists in a numerical solution of the corresponding boundary value problem. Though this problem was formulated by Laplace as far back as 1775, all the attempts at solving it, particularly those where a priori information on ocean tides is not used, display such disagreement between each other and the observation data that this makes them hardly suitable for geophysical applications. The situation did not improve very much with the advent of high-capacity computers and efficient numerical methods for solving the equations of tidal dynamics. Today it remains almost the same as 18 years ago, when Munk and Zetler, in their innovative paper [204] outlining the possible pathways of studying open-sea tides wrote: “... mathematicians dealing with tides investigations are still staying ashore, making dubious suppositions on what is going on in the open sea.” One of the possible reasons for the failure of the theoretical approach to describing quantitatively the tides in the open sea is associated with disregard for shelf effects.
G. I. Marchuk, B. A. Kagan
Chapter 7. Global Interaction of Ocean and Terrestrial Tides
Abstract
Let us examine the influence exerted on ocean tides by the equilibrium effect of terrestrial tides and the loading and self-attraction effects of ocean tides, the exclusion of which means (see Section 3.2) that the Earth constitutes a perfectly rigid body, and, therefore, there is no interaction between the ocean and the terrestrial tides. It is natural to ask oneself a question: is such a supposition justified? The estimates of a self-consistent equilibrium tide in the ocean, presented in section 1.4, indicate that the opposite is true. But ocean tides are not in equilibrium, and the question, there fore, remains open. Provinding the answer to this question is the object of this chapter.
G. I. Marchuk, B. A. Kagan
Chapter 8. Energetics of Ocean Tides
Abstract
To formulate the energy equation we shall turn to the Equations (7.1.1), (7.1.2). Innerly multiplying the first one by ρ0u, the second by ρ0gζ, and adding together the resulting expressions we get
$$ \begin{array}{*{20}{l}} {\frac{\partial }{{\partial t}}\left[ {{\rho _0}\frac{{|u{|^2}H}}{2} + \rho \frac{{g{\zeta ^2}}}{2}} \right] = - {\rho _0}g\nabla \cdot uH\left( {\zeta - {\gamma _2}{g^{ - 1}}{U_2} - \hat \zeta } \right) + } \\ {{\text{ + }}{\rho _0}g\left( {{\gamma _2}{g^{ - 1}}{U_2} + \hat \zeta } \right)\frac{{\partial \zeta }}{{{\partial ^t}}} + {\rho _0}u \cdot F,} \end{array} $$
(8.1.1)
where F is the vector of friction forces normalized to the average sea-water density; and the other designation are the same as before.
G. I. Marchuk, B. A. Kagan
Chapter 9. Bottom Boundary Layer in Tidal Flow: Experimental Data
Abstract
In any viscous fluid with an arbitrarily small viscosity coefficient the current velocity in the immediate vicinity of a solid surface is known to change rapidly from zero to its value in the external flow, corresponding to the flow of an ideal fluid (fluid without viscosity). The lower part of the where noticeable changes in the velocity are taking place, caused by the influence of the ocean bottom, is known as the bottom boundary layer.
G. I. Marchuk, B. A. Kagan
Chapter 10. Bottom Boundary Layer in Tidal Flow: Theoritical Models
Abstract
We start with estimating the thickness of the bottom boundary layer in a tidal flow. For this purpose, to describe the effect of small-scale turbulence we introduce the effective coefficient of vertical turbulent viscosity v T; , constant with height, and, following Charney [72], assume that the turbulent boundary layer remains hydrodynamically stable up to the moment when its effective Reynolds number \( {R^{{\delta _T}}}{\rm{ = }}{U_\infty }{\delta _T}/{v_T}{\rm{ }}exceeds R_{cr}^{\delta l}. \) If we now define the thickness of the turbulent boundary layer as δ T = (2v T /σ)1/2 we obtain the estimate
$$ {\delta _T} \ge {\rm{ }}\frac{{2{U_\infty }}}{{\sigma R_{cr}^{{\delta _l}}}}, $$
where U ∞ is the tidal velocity amplitude outside the boundary layer; σ is the oscillation frequency.
G. I. Marchuk, B. A. Kagan
Backmatter
Metadata
Title
Dynamics of Ocean Tides
Authors
G. I. Marchuk
B. A. Kagan
Copyright Year
1989
Publisher
Springer Netherlands
Electronic ISBN
978-94-009-2571-7
Print ISBN
978-94-010-7661-6
DOI
https://doi.org/10.1007/978-94-009-2571-7