J.J. Stoker - Water waves. The mathematical theory with applications (796980), страница 72
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If cuc xfrom the first equation of (10.11.20) that c t==0, itfollowson such a+means that c = on the particle path definedby dx/dt ~- u. At the same time the C + and C_ characteristics havethe same direction, since they are given by dx/dtuc. On theother hand, we have, again from (10.11.20), u + uu xkh x andwe see that the relation u = const, along a characteristic for whichlocus,andthis in turntcannot be satisfied unless h x = 0.
In connection with Fig.10.11.7 we have seen that the rising portion toward the east of a de-cpression in the discontinuity surface tends to flatten out, while thefalling part from the west tends to steepen and break because thehigher portions tend toThus when/?,moveand hencer,fasterand crowd the lower portions.tends to zero the tendency will be forbreaking to occur at the cold front, but not at the warm front. Theslope of the discontinuity surface at the cold front will then be infinite.However, a bore in the sense described above cannot occur since theremust always be a mass flux through a bore: the motion of the coldfront is analogous to what would happen if a dam were broken andwater rushed down the dry bed of a stream. Without considering insome special way what happens in the turbulent motion caused by suchcontinuous breaking at the ground, it is not possible to continue ourdiscussion of the motion of a cold front along the present lines, although such a problem is susceptible to an approximate treatment.Nevertheless, this discussion has led in a rational way to a qualitativeWATER WAVES398explanation for the well-known fact that a warm front does indeedprogress in a relatively smooth fashion as compared with the turbulence which is commonly observed at cold fronts.
Thus near a coldfront the height of the cold air layer may be considerably greater thanin the vicinity of thewarm front, where h ~0;consequently the speedof propagation of the cold front could be expected to be greater thanWarmColdt =Warmt=t,>ColdWarmtstFig. 10.11.8.2>t,Deformation of a moving frontwarmfront (as indicated by the dotted modifications of theshape of the cold front in Fig. 10.11.8), with the consequence that thegap between the two tends to close, and this hints at a possible ex-near theplanation for the occlusion process.^butOne mightalso look at the matterway: Supposeshownin Fig. 10.11.7.
If breaking once begins, itc0,iswaveknown thatsmall in the trough of thein thisiswellLONG WAVES IN SHALLOW WATER399the resulting bore moves with a speed that is greater than the propagation speed of wavelets in the medium in front (to the right) of it.although slower than the propagation speed in the medium behind it.Again one sees that the tendency for the wave on the cold front sideto catch up with the wave on the warm front side is to be expected onthe basis of the theory presented here.Finally we observe that the velocity of the wave near the undisturbedstationary front is UQ but well to the north it is given roughly by,ufUQ~~% vaky, whichproduce whatcenter of theTo sumisis lessthan U Q There.isthus a tendency toaround thecalled in meteorology a cyclonic rotationwavedisturbance.say that the approximate theory embodied in equations (10.11.18), even when applied to a very special typeof motions (i.e.
simple waves in each plane y =. const.), yields avariety of results which are at least qualitatively in accord withup,itseemsfair toobservations of actual fronts in the atmosphere. Among the phenomena given correctly in a qualitative way are: the change in shape ofa wave as it progresses eastward, the occurrence of a smooth wave ata warm front but a turbulent wave at a cold front, and a tendency toproduce the type of motion called a cyclone.It therefore seems reasonable to suppose that the differentialequations of our Problem III, which were the starting point of thediscussion just concluded, contain in them the possibility of dealingwith motions which have the general characteristics of frontal motionsin the atmosphere, and that numerical solutions of the equations ofProblem III might well furnish valuable insights.
This is a difficulttask, as has already been mentioned. However, an approximate theorydifferent from that of Whitham is possible, which has the advantagethat no especial difficulty arises at cold fronts, and which would permit a numerical treatment. This approximate theory might be considered as a new Problem IV.The formulation of Problem IV was motivated by the followingconsiderations. If one looks at a sequence of weather maps andthinks of the wave motion in our thin wedge of cold air, the resemblance to the motion of waves in water which deform into breakers (especially in the case of frontal disturbances which develop intooccluded fronts) is very strong.
The great difference is that the wavewater takes place in the vertical plane while the wave motion in our thin layer of cold air takes place essentially in the horizontal plane. When the hydrostatic pressure assumption is made in themotioninWATER WAVES400case of water waves the resulta theory in exact analogy to gasisdynamics, and thus wave motions with an appropriate "sound speed"become possible even though the fluid is incompressible the freesurface permits the introduction of the depth of the water as a dependent quantity, this quantity plays the role of the density in gasdynamics, and thus a dynamical model in the form of a compressiblefluid is obtained.
It would seem therefore reasonable to try to inventa similar theory for frontal motions in the form of a long- wave theorysuitable for waves which move essentially in the horizontal, ratherthan the vertical, plane, and in which the waves propagate essentiallyparallel to the edge of the original stationary front, i.e. the #-axis. Inthisway one might hopeto be rid of the dependenceon the variable yat right angles to the stationary front, thus reducing the independentvariables to two, x and t; and if one still could obtain a hyperbolicsystem of differential equations then numerical treatments by finitewould be feasible. This program can, in fact, be carried outin such a way as to yield a system of four first order nonlinear differential equations in two independent and four dependent variableswhich are of the hyperbolic type.differencesOnce having decidedto obtain a long-wave theory for the horizontalthetobe followed can be inferred to a large extentplane,procedurefrom what one does in developing the same type of theory for gravitywaves in water, as we have seen in Chapter 2 and at the beginning ofthe present chapter.
To begin with it seems clear that the displacer)(x,t) of the front itself in the ^-direction should be introducedmentas one of the dependent quantitiesallthe more since this quantityanyway the most obvious one on the weather maps. To have such a"shallow water" theory in the horizontal plane requires unfortuna-isa rigid "bottom" somewhere (which is, of course, vertical intelythis case), and this we simply postulate, i.e. we assume that thet/-component v of the velocity vanishes for all time on a vertical planey=d=Figureconst, parallel to the stationary front along the #-axis (see10.11.9). The velocity v(x, j/, t) is then assumed to vary=linearly* in y, and its value at the front, yr\(x, t), is called v (x, t).The intersection of the discontinuity surface zh(x 1/, t) with thed is a curve given by zplane yTi(x, t), and we assume that the==9=discontinuity surface is a ruled surface having straight line generatorsrunning from the front, yTi(x, t), andrj(x 1), to the curve z=*9The analogous statement holds also in the long-wave theory in water (tolowest order in the development parameter, that is).LONG WAVES IN SHALLOW WATERparallel to thet/,2-plane.
Finally,we assume401(as in the shallowwatertheory) that u the ^-component of the velocity, depends on x and9/a zFig. 10.11.9. Notations foronly:u=u(x,t).Theeffect of theseProblem IVassumptionsisto yield the re-lations(10.11.26)(10.11.27)y -*?(**)^h(x,dV(JT, y, t)d-y=t),YI(X, t)~v~'""d-f,(x,t)as one readily sees. In addition, we assume that a particle that is onceon the front y - r)(x, t)always remains on it, so that the relation:=(10.11.28)v(x,t)=r\ t+ ur\ xmust hold. The four quantities u(x /), rj(x, t), h(x, t), and v(x, t) areour new dependent variables.
Differential equations for them will beobtained by integrating the basic equations (10.11.16) of Problem IIIwith respect to y from y = rj to y = d which can be done since thedependence of w, u, and h on y is now explicitly given and these three9equations together with (10.11.28) will yield the four equationswant.weBefore writing these equations down it should be said that themost trenchant assumption made here is the assumption concerningthe existence of the rigid boundary yd. One might think that as=WATER WAVES402long as the velocity component v dies out with sufficient rapidity inthe ^-direction such an assumption would yield a good approximation,but the facts in the case of water waves indicate this to be not sufficient for the accuracy of the approximation: with water waves in verydeep water the vertical component of the velocity (corresponding toour v here) dies out very rapidly in the depth, but it is neverthelessessential for a good approximation to assume that the ratio of thedepth down to a rigid bottom to the wave length is small.
However,such a rigid vertical barrier to the winds does exist in some cases ofinterest to us in the form of mountain ranges, which are often muchhigher than the top of the cold surface layer (i.e. higher than thecurve z = h(x, t) in Figure 10.11.9). In any case, severe though thisrestriction is, it still seems to the author to be worth while to studythe motions which are compatible with it since something about thedynamics of frontal motions with large deformations may be learnedin the process.
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