J.J. Stoker - Water waves. The mathematical theory with applications (796980), страница 68
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The three differentialequations of Problem III are probably capable of yielding reasonablyaccurate approximations to the frontal motions under consideration,but they are still rather difficult to deal with, even numerically,principally because they involve three independent variables*: suchequations are well known to be beyond the scope of even the mostmodern digital computing machines as a rule.
Consequently, stillfurther simplifying assumptions are made in order to obtain a theoryin fourcapable of yielding some concrete results through calculation.At this point, two different approaches to the problem are proposed.One of them, by Whitham [W.12], deals rather directly with thethree differential equations of Problem III.Twoof these equationsare essentially the same as those of the nonlinear shallow water theorytreated in the preceding sections of this chapter. These two equationswhich refer to motions in vertical planes -can therefore be integrated.
Afterwards the transverse component of the velocity is foundby integrating alinear first order partial differential equation. In thisway aquite reasonable qualitative description of the dynamics offrontal motions can be achieved, at least in special cases, which isgood agreement with many of the observed phenomena. However,this theory has a disadvantage in that it docs not permit a completeinnumerieal integration because of a peculiar difficulty at cold fronts.(The difficulty stems from the fact that a cold front corresponds inthis theory to what amounts to the propagation of a bore down thedry bed of a stream a mathematical impossibility.
If one had ameans of taking care of turbulence and friction at the ground,itwouldperhaps be possible to overcome this difficulty.) Nevertheless, thequalitative agreement with the observed phenomena is an indicationthat the three differential equations furnishing the basic approximatetheory from which we start i.e. those of our Problem III have inthem the possibility of furnishing reasonable results.The author's method (ef. [S.24]) of treating the three basic differential equations is quite different from that of Whitham, but it unfortunately involves a further assumption which has the effect oflimiting the applicability of the theory.
The guiding principle was that* The work of Freeman[F.9, 10] is based on a theory which could be considered as a special case of Problem III in which the Coriolis terms due to therotation of the earth are neglected and the motion is assumed at the outset todepend on only one space variable and the time. The idea of deriving the theoryresulting in Problem III occurred to the author while reading Freeman's paperand, indeed, Freeman indicates the desirability of generalizing his theory.WATER WAVES378two independent variables should benumberofbutthatthefound,dependent variables need not be sodifferential equations in onlyruthlessly limited. Finally, it is highly desirable to obtain differentialequations of hyperbolic type in order that the theory embodied in thecharacteristics becomes available in formulating and solvconcreteingproblems.
These objectives can be attained by makingquite a few further simplifying assumptions with respect to the mechanics of the situation. The result is what might be called Problemmethod ofIV. The theory formulated in ProblemIVisembodiedina system offour nonlinear first order partial differential equations of hyperbolicnumericaltype in four dependent and two independent variables.Aintegration of these equations is possible, but the labor of integratingthe equations is so great that only meagre results are so far available.Once Whitham's theory and Problem IV have been formulated,is led once more to consider dealing with Problem III numericallyonein spite of the fact that there are three independent variables in thiscase; in Problem IV, and also in the theory by Whitham, for thatmatter, the basic idea is that variations in the ^/-direction are lessrapid than those in the ^-direction, and thus a finite difference schemetwo space variables and the time might be possible under suchinspecial circumstances.WormColdTT?7GroundWormFig.
10.11.1.Astationary frontWeproceed to the derivation of the basic approximate theory. Tobegin with, a certain steady motion (called a stationary front) is takenas an initial state, and this consists of a uniform flow of two super-imposed layers of cold and warm air, as indicated in Figure 10.11.1.s-axis is taken positive upward* and the x, t/-plane is a tangentThe*Here we deviate from our standard practice of taking the t/-axis as theconform to the usual practice in dynamic meteorology.This should cause no confusion, since this section can be read to a large extentvertical axis, in order toindependently of the rest of the book.LONG WAVES IN SHALLOW WATER379plane to the earth. The rotation of the earth is to be taken intoaccount but, for the sake of simplicity, not its sphericity a commonpractice in dynamic meteorology.
The coordinate system is assumedto be rotating about the js-axis with a constant angular velocityQ=co sin (p, with co the angular velocity of the earth and <p the latitude of the origin of our coordinate system. (The motivation for this isthat the main effects one cares about are found if the Coriolis termsare included, and that neglect of the curvature of the earth has noserious qualitative effect.) As indicated in Figure 10.11.1, the cold airlies in a wedge under the warm air and the discontinuity surfacebetween the two layers is inclined at angle a to the horizontal. Theterm "front" is always applied to the intersection of the discontinuitysurface with the ground, and in the present case we have therefore asinitial state a stationary front running along the iT-axis.
The windvelocity in the two layers is parallel to the #-axis (otherwise the discontinuity surface could not be stationary), but it will in general bedifferent in magnitude and perhaps even opposite in direction in thetwo layers. The situation shown in Figure 10.11.1 is not uncommon.For instance, the j?-axis might be in the eastward direction, the t/-axisin the northward direction and the warm air would be moving in thedirection of the prevailing westerlies. The origin of the cold air at theshall see later thatground is, of course, the eold polar regions.Wesueh configurations are dynamically eorreet and that the angle aonceis uniquely determined (and quite small, of the order of)the state of the warm air and eold air is given. (The discontinuitysurface is not horizontal because of the Coriolis force arising fromthe rotation of the earth.)We proceed next to describe what is observed to happen in manyeases once sueh a stationary front starts moving.
In Figure 10.11.2 asequence of diagrammatic sketches is given which indicate in a generalway what can happen. All of the sketches show the intersection of themoving discontinuity surface (cf. Figure 10.11.1) with the ground (thex, jy-planewith thej/-axistaken northward, theo?-axis taken eastward).The shaded areaindicates the region on the ground covered by coldwhiletheunshadedair,region is covered at the ground by warm air.Of course, the cold air always lies in a thin wedge under a thick layerofwarmair.In Figure 10.11.
2a the development of a bulge in theis indicated.* Such a bulge thenstationary front toward the north* Whatagency serves to initiate and to maintain such motions appears tobe a mystery. Naturally such an important matter has been the subject of a great(footnote continued)WATER WAVES880frequently deepens and at the same time propagates eastward with avelocity of the order of 500 miles per day.
It now becomes possible todefine the terms cold front and warm front. As indicated in Figure 10.11.26, the cold front is that part of the whole front at which cold air istaking the place ofwarmair atthe ground, and theColdFrontwarmfrontistheWarm/Front(b)(o)OccludedFront(c)Fig. 10.11.2. Stages in the(d)motion of a frontal disturbanceportion of the whole front where cold air is retreating with warm airtaking its place at the ground. Since such cold and warm fronts areaccompanied by winds, and by precipitation in various forms infact, by all of the ingredients that go to make up what one callsdeal of discussionandspeculation, but there seems to be no consistent view aboutamong meteorologists. In applying the theory derived here no attempt is madeto settle this question a priori: we would simply take our dynamical model,itassume ancondition which in effect states that a bulge of the kind justand then study the subsequent motion by integrating thedifferential equations subject to appropriate initial and boundary conditions.However, if the approximate theory is really valid, such studies might perhapsbe used, or could be modified, in such a way as to throw some light on this important and vexing question.describedinitialis initiated,LONG WAVES IN SHALLOW WATERweatheritlatitudesis381follows that the weather at a given locality in the middlelargely conditioned by the passage of such frontal dis-turbances.
Cold fronts andwarmfronts behave differently in manyways. For example, the cold front in general moves faster than thewarm front and steepens relative to it, so that an originally symme-disturbance or wave gradually becomes distorted in the mannerindicated in Figure 10.11.2c.
This process sometimes though by notricalmeans alwayscontinues until the greater portion of the cold fronthas overrun the warm front; an occluded front, as indicated in Figure10.11.2d, is then said to occur. The prime object of what follows is toderive a theoryor perhaps better, to invent a simplified dynamicalmodel capable of dealing with fluid motions of this type that is noton the one hand so crude as to fail to yield at least roughly the observedmotions, and on the other hand is not impossibly difficult to usefor the purpose of mathematical discussion and numerical calculation.much as possiblewe do not lean on the basic theory developed earlierin this book.
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