J.J. Stoker - Water waves. The mathematical theory with applications (796980), страница 70
Текст из файла (страница 70)
The differential equations for Problem III can, finally, be ex-pressed in the form:+ t<u x + ru v + gJvt+ uv f +( 1/?,-f (U/t),wt1|(10.11.16)Problem IIIby using the formulasthe influence of thevv v++(!'/*)for h'xandwarm(>=h'vl/t,0.given in (10.11.15).We note thatair expresses itselfthrough its density g'(10.11.16) undoubtedly havevelocityof u, v, and h are givenvaluesthedeterminedsolutionsonceuniquely=at the initial instant t0, together with appropriate boundary conditions if the domain in #, y is not the whole space, and such solutionsmight reasonably be expected to furnish an approximate descrip-andu'.itstion of thetions areThe three equationsdynamics of frontal motions.* Unfortunately, these equa-stillquite complicated.They could beintegrated numerically* Theseequations are in fact quite similar to the equations for two-dimenmotion of a compressible fluid with h playing the role of thedensity of the fluid.sional unsteadyWATER WAVES388only with great difficulty even with the aid of the most modern highspeed digital computers -mostly because there are three independentvariables.Consequently, one casts about forstillother possibilities, either ofspecialization or simplification, which might yield a manageabletheory.
One possibility of specialization has already been mentioned:if one assumes no Coriolis force and also assumes that the motion isindependent of the ^-coordinate, one obtains the pair of equationsut+ uu x + gl-\h x=(Hfc).=(10.11.17)which are identical with the equations of the one-dimensional shallowwater gravity wave theory. These equations contain in them thecalledpossibility of the development of discontinuous motionsbores in sec. 10.7 and this fact lies at the basis of the discussions byFreeman [F.10] and Abdullah [A.7].
In such one-dimensional treatments,it isclear thatit isin principle not possible to deal with thebulges on fronts and their deformation in time and space, since suchproblems depend essentially on both space variables x and y. Anotherpossibility would be a linearization of the differential equations(10.11.16) based on assuming small perturbations of the frontal surface and of the velocities from the initial uniform state. This proceduremight be of some interest, since such a formulation would take care ofthe boundary condition at the ground, while the existing linear treatments of this problem do not.
However, our interest here is in a nonlinear treatmentwhich permits of large displacements of theOne such possibility, devised by Whitham [W.12],fronts.involves essentiallythe integration of the first and third equations for u and h as functionsof x and t, regarding y as a parameter, and derivatives with respect toy as negligible compared with derivatives with respect to x, and assuming initial values for v; thisisfeasibleby the method of characteristics.Afterwards, v would be determined by integrating the second equation considering u and h as known, and this can in principle be donebecause the equationditions.As statedisa linearearlier, thisfirstorder equation under these con-procedure furnishes qualitative resultswhich agree with observations.
In addition, the discussion can becarried through explicitly in certain cases, by making use of solutionsof the type called simple waves, along exactly the same lines as insec. 10.8 above.turn, therefore, to this first of two proposedWeLONG WAVES IN SHALLOW WATERapproximate treatments of ProblemIII, as389embodiedin equations(10.11.16).Thebasic fact from whichWhithamstartsisthat the slope a=hyof the discontinuity surface is small initially, as we have already seenin connection with the second equation of (10.11.15), and the fact thata fraction of the earth's angular velocity, and is expected to remainin general small throughout the motions considered. Since the Coriolisforces are of order a also (since they are proportional to A) it seemsclear that derivatives of all quantities with respect to y will be small ofa different order from those with respect to x\ it is assumed thereforethat u y9 h y and v y are all small of order a, but that ux and h x are finite.Furthermore we can expect that the main motion will continue to beAisa motion in the ^-direction, so that the i/-component v of the velocitywill be small of order a while the ^-component u remains of coursefinite.
Under these circumstances, the equations (10.11.16) can bereplaced by simpler equations through neglect of all but the lowestorder terms in a in each equation; the result is the set of equationshtvt+ uh x + hu x =+ uu x + kh x =+ uv x =kh y+AIQ'(u'\u\with the constant k defined by(10.11.19)*=A considerablesimplification has been achieved by this process, sincethe variable y enters into the first two equations of (10.11.18) only asa parameter and these two equations are identical with the equationsof the shallow water theory developed in the preceding sections of thischapter if k is identified with g and h with 77. This means that thetheory developed for these equations now becomes available to discuss our meteorological problems.
Of course, the solutions for h andu will depend on the variable y through the agency of initial andboundary conditions. Once u(xandhave been obtained,the third equation of (10.11.18), which then9y, t)h(x, y>t)they can be inserted inis a first order linear partial differential equation which, in principleat least, can be integrated to obtain v when arbitrary initial conditionsvv(x y, 0) are prescribed. The procedure contemplated can thus besummed up as follows: the motion is to be studied first in each vertical=9WATER WAVES390=constant by the same methods as in the shallow waterplane yfortwo-dimensional motions (which means gas dynamicstheorymethods for one-dimensional unsteady motions), to be followed bya determination of the "cross-component" v of the velocity throughintegration of a first order linear equation which also contains thevariable y, but only as a parameter.This is in principle a feasible program, but it presents problems toocomplicated to be solved in general without using numerical com-Onthe other handwe know fromthe earlier parts of thisfirst two equations ofsolutionsofthechapter that interesting special(10.11.18) exist in the form of what were called simple waves, andputations.these solutions lend themselves to an easy discussion of a variety ofmotions in an explicit way through the use of the characteristic formof the equations.
In order to preserve the continuity of the discussionit is necessary to repeat here some of the facts about the characteristictheory and the theory of simple waves; for details, sees. 10.2 and 10.3should be consulted.By introducing theinstead of the firstnewfunction c 2kh, replacing h,we obtaintwo equations in (10.11.18) the following equations:(2c tIu(10.11.20)Thus the quantityc=t+ 2uc x + cu x ==+ uu x + 2cc x == 0.Vkh, which has the dimensions of avelocity,the propagation speed of small disturbances, or wavelets in analogy with the facts derived in sec.
10.2. These equations can in turnisbe written in the formwhich can be interpreted to mean that the quantities u 2c are conc:stant along curves C in the #, 2-plane such that dx/dt = uu(10.11.21)-\-dx2c=const, along2c=const, along C_:C+:=u=u+c<uaxc.These relations hold in general for any solutions of (10.11.20). Under2c for example, has thespecial circumstances it may happen that usame constant value on all C_ characteristics in a certain region; in9LONG WAVES IN SHALLOW WATER391u + 2c is constant along each C + characteristic itu and c would separately by constant along each of thethat case sincefollows thatC+ characteristics, which means that these curves wouldallbe straightlines.
Such a region of the flow (the term region here being appliedwith respect to some portion of an r, -plane) is called a simple wave.It is then a very important general fact that any flow region adjacentto a region in which the flow is uniform, i.e. in which both c and u areeverywhere constant (in both space and time, that is), is a simplewave, provided that u and c are continuous in the region in question.It is reasonable to suppose that simple waves would occur in casesof interest to us in our study of the dynamics of frontal motions,simply because we do actually begin with a flow in which u and h(hence also c) are constant in space and time, and it seems reasonableto suppose that disturbances are initiated, not everywhere in the flowregion, but only in certain areas.
In other words, flows adjacent touniform flows would occur in the nature of things. Just how in detailinitial or boundary conditions, or both, should be prescribed in orderto conform with what actually occurs in nature is, as has already beenpointed out, something of a mystery; in fact one of the principalobjects of the ideas presented here could be to make a comparison ofcalculated motions under prescribed initial and boundary conditionswith observed motions in the hope of learning something by inferenceconcerning the causes for the initiation and development of frontaldisturbances as seen in nature.Oneobvious and rather reasonable assumption to begin withto havemight be that u> v, and h are prescribed at the time t =values over a certain bounded region of the upper half (y > 0) ofthe x, y-plane (cf.
![](https://s.studizba.com/z.php?f=/templates/si/i/noavatar.png&w=100&h=100&t=1"){kind=avatar}
