J.J. Stoker - Water waves. The mathematical theory with applications (796980), страница 67
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10.10.20. Breaking inducedby wind actionmaking apparatus at the right; the only difference is that a currentof air was blown from right to left in the case shown by the lowerphotograph. The breaking thus seems due entirely to wind action inLONG WAVES IN SHALLOW WATER878this ease.
Finally, Fig. 10.10.21 shows two stages in the breaking of awave in shallow water, when marked dissymmetry and the formationofwhat lookslike a jet at thesummit of the waveare seen to occur.of interest, historically and otherwise, to refer once more to thecase of symmetrical waves breaking at their crests. The wave crestsItissuch cases are quite sharp, as can be seen in the photograph shownin Fig. 10.10.18.
Stokes [S.28] long ago gave an argument, based onquite reasonable assumptions, that steady progressing waves with anangular crest of angle 120 could occur; in fact, this follows almost atonce from the Bernoulli law at the free surface when the free surface isassumed to be a stream line with an angular point. There is anotherfact pertinent to the present discussion, i.e. that the exact theory forinsteady periodic progressing waves of finite amplitude, as developedin Chapter 12.2, shows that with increasing amplitude the wavesflatten more and more in the troughs, but sharpen at the crests.waveFig. 10.10.21. Breaking of a longin shallowwaterIn fact, the terms of lowest order in the development of the free surfaceamplitude 77 as given by that theory can easily be found; the result isrf(x)for awaveof=wave lengtha cos x+a2cos 2x2n.
Fig. 10.10.22 shows the result of super-WATER WAVES8742a cos ximposing the second-order term a cos 2x on the wavewhich would be given by the linear theory; as one sees, the effect is asindicated. It would be a most interesting achievement to show rigorously that the wave form with a sharp crest of angle 120 is attainedwith increase in amplitude. An interesting approximate treatment ofthe problem has been given by Davies [D.5], However, the problemthus posed is not likely to be easy to solve; certainly the method ofLevi-Civita as developed in Ch. 12.2 does not yield such a wave formsinceit isshown there that thefree surfaceFig. 10.10.22.
Sharpening ofisanalytic. Presumably,waves at thecrestany further increase in amplitude would lead to breaking at the crestshence no solutions of the exact problem would exist for amplitudesgreater than a certain value.10.11.Gravity waves in the atmosphere. Simplified version of theproblem of the motion of cold and warm frontsIn practically all of this book we assume that the medium in whichwaves propagate is water. It is, however, a notable fact that somemotions of the atmosphere, such as tidal oscillations due to the samecause as the tides in the oceans, i.e. gravitational effects of the sunand moon, as well as certainsuch as wavelarge scale disturbances in the atmospheredisturbances in the prevailing westerlies of the middleand motions associated with disturbances on certain discontinuity surfaces called fronts, are all phenomena in which the airlatitudes,can be treated as a gravitating incompressible fluid.
In addition, oneof the best-founded laws in dynamic meteorology is the hydrostaticpressure law, which states that the pressure at any point in the at-mosphere is very accurately given by the static weight of the columnpf air above it. When we add that the types of motions enumeratedabove are all such that a typical wave length is large compared withLONG WAVES IN SHALLOW WATER375an average thickness (on the basis of an average density, that is) ofthe layer of air over the earth, it becomes clear that these problems fallinto the general class of problems treated in the present chapter. Ofcourse, this means that thermodynamic effects are ignored, andwith them the ingredients which go to make up the local weather,but it seems that these effects can be regarded with a fair approximation as small perturbations on the large scale motions in question.There are many interesting problems, including very interesting unsolved problems, in the theory of tidal oscillations in the atmosphere.These problems have been treated at length in the book by Wilkes [W.
2];weattempt to discuss them here. The problems involved inwavepropagation in the prevailing westerlies will also notstudyingbe discussed here, though this interesting theory, for which papers byCharney [C.15] and Thompson [T.10] should be consulted, is beingused as a basis for forecasting the pressure in the atmosphere by numerical means. In other words, the dynamical theory is being used forthe first time in meteorology, in conjunction with modern high speeddigital computing equipment, to predict at least one of the elementswhich enter into the making of weather forecasts.In this section we discuss only one class of meteorological problems,i.e. motions associated with frontal discontinuities, or, rather, itwould be better to say that we discuss certain problems in fluiddynamics which are in some sense at least rough approximations to theactual situations and from which one might hope to learn somethingabout the dynamics of frontal motions.
The problems to be treatedshall notunlike the problems of the type treated by Charney andThompson referred to above are such as to fit well with the precedingherematerial in this chapter; it was therefore thought worthwhile to include them in this book in spite of their somewhat speculative charac-from the point of view of meteorology. Actually, the idea of usingmethods of the kind described in this chapter for treating certain specialtertypes of motions in the atmosphere has been exploredAbdullahby a number ofFreeman[A.7],[F.10], Tepper [T.ll]).meteorologists (cf.One of the most characteristic features of the motion of the atmos-phere in middle latitudes and also one which is of basic importancein determining the weather there is the motion of wave-like disturbances which propagate on a discontinuity surface between a thinwedge-shaped layer of coldofwarmerin generalairon the ground and an overlying layerIn addition to a temperature discontinuity there is alsoa discontinuity in the tangential component of the windair.WATER WAVES376The study of such phenomena was initiatedandSolberg [B.20] and has been continuedlong ago by Bjerknessince by many others.
In considering wave motions on discontinuitysurfaces it was natural to begin by considering motions which departso little from some constant steady motion (in which the discontinuityvelocity in thetwolayers.surface remains fixed in space) that linearizations can be performed,thus bringing the problems into the realm of the classical linearmathematical physics.
Such studies have led to valuable insights,particularly with respect to the question of stability of wave motionswave length of the perturbations. (The problemsbeing linear, the motions in general can be built up as a combination,roughly speaking, of simple sine and cosine waves and it is thewave length of such components that is meant here, cf. Haurwitz[H.5, p. 234].) One conjecture is that the cyclones of the middle latitudes are initiated because of the occurrence of such unstable wavesin relation to theon a discontinuity surface.A glance at a weather map, or, still better, an examination of weather maps over a period of a few days, shows clearly that the wavemotions on the discontinuity surfaces (which manifest themselves asthe so-called fronts on the ground ) develop amplitudes so rapidly andof such a magnitude that a description of the wave motions over aperiod of, say, a day or two, by a linearization seems not feasible withany accuracy.
The object of the present discussion is to make a firststep in the direction of a nonlinear theory, based on the exact hydrodynamical equations, for the description of these motions, that can beattacked by numerical or other methods. No claim is made that theproblem is solved here in any general sense. What is done is to startwith the general hydrodynamical equations and make a series ofassumptions regarding the flow; in this way a sequence of three nonlinear problems (we call them Problems I, II, III), each one furnishinga consistent and complete mathematical problem, is formulated.One can see then the effect of each additional assumption insimplifying the mathematical problem.
The first two problems resultfrom a series of assumptions which would probably be generallyaccepted by meteorologists as reasonable, but unfortunately evenProblem II is still pretty much unmanageable from the point of viewof numerical analysis. Further, and more drastic, assumptions leadto a still simpler Problem III which is formulated in terms of threefirst order partial differential equations in three dependent and threeindependent variables (as contrasted with eight differential equationsLONG WAVES IN SHALLOW WATER377independent variables in Problem I).
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