J.J. Stoker - Water waves. The mathematical theory with applications (796980), страница 71
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Fig. 10.11.1 ) in such a fashion that they differ fromthe constant values in the original uniform flow with a stationaryfront. According to the approximate theory based on equations (10.11.= t/ an18), this means, in particular, that in each vertical plane yinitial conditions for u(x,y09 t) and h(x,y ,t) would bex, J-planeoverthe entire #-axis, but in such a way that u and h areprescribedU Q > 0, h = A ^c =constant with values u(hence cfairly=Vkh)*everywhere except over a certain segment xl=^x^#2,asindicated in Fig. 10.11.5.
The positive characteristics C+ are drawn inthe characteristics C_ with dashed lines in this diagram,full lines,* It should, however,always be kept in mind in the discussion to followwill usually have different values in different vertical planes, particularly,const.ythat c=WATER WAVES892whichisto be interpreted as follows. Simple waves exist everywherein the x, J-plane except in the triangular region bounded by the C+and the C_ characteristic throughandcharacteristic throughBAterminating at point C; in this region the flow could be determinedAFig. 10.11.5. SimpleBwavesarisingfrominitialconditionsby the method indicated in sec.
10.2 aboveconnection with Fig. 10.2.1). The disturbance created over thesegment AB propagates both "upstream" and "downstream" after acertain time in the form of two simple waves, which cover the regionsbounded by the straight (and parallel) characteristics issuing fromA, B and C. In other words the disturbance eventually results in twonumerically, for example(in9one propagating upstream, the other downstream, and separated by a uniform flow identical with the initialu i.e. that thestate. In our diagram it is tacitly assumed that c >flow is subcritical in the terminology of water waves (subsonic ingas dynamics) otherwise no propagation upstream could occur. Wehave supposed u to be positive, i.e.
that the ^-component of the flowdistinct simple waves,\|,velocity in the cold air layer has the same direction as the velocity inthe warm air, which in general flows from the west, but it can be (andnot infrequently is) in the westward rather than the eastward direction. Since the observed fronts seem to move almost invariably to theeastward, it follows, for example, that it would be the wave movingupstream which would be important in the case of a wind to the westward in the cold layer, and a model of the type considered here inwhich the disturbance is prescribed by means of an initial conditionLONG WAVES IN SHALLOW WATER393and the flowis subcriticalimplies that the initial disturbances areaofsuchcharacterthat the downstream wave has aspecialalwaysForwindato the eastward, the reverse wouldnegligible amplitude.be the case.
All of this is, naturally, of an extremely hypotheticalcharacter, but nevertheless one sees that certain important elementspertinent to a discussion of possible motions are put in evidence.The last remarks indicate that a model based on such an initialdisturbance may not be the most appropriate in the majority of cases.In fact, such a formulation of the problem is open to an objectionwhich is probably rather serious. The objection is that such a motionhas its origin in an initial impulse, and this provides no mechanism bywhich energy could be constantly fed into the system to "drive" thewave. Of course, it would be possible to introduce external body forcesin various ways to achieve such a purpose, but it is not easy to see howto do that in a rational way from the point of view of mechanics.Another way to introduce energy into the system would be to feed itin other words formulate appropriate boundin through a boundaryary conditions as well as initial conditions.
For the case of frontsmoving eastward across the United States, a boundary conditionmight be reasonably applied at some point to the east of the highmountain system bordering the west coast of the continent, sincethese mountain ranges form a rather effective north-south barrierbetween the motions at the ground on its two sides. In fact, a coldnot infrequently seen running nearly parallel to the mounas though cold air had been deflectedtains and to the cast of themsouthward at this barrier. Hence a boundary condition applied atfrontissome point on the west seems notentirely without reason. Inanycase,weseek models from which knowledge about the dynamics of frontsmight be obtained, and a model making use of boundary conditionsshould be studied.isapplied at xWe suppose,0,therefore, that aand that theinitial=boundary conditionist0, x >condition for===cUQconst..const., cundisturbed, i.e.
uremark that we are considering the motion in a definitethat the flow(Again we=isy = y Q ) In this case we would have only a wave propagating eastward in effect, we replace the influence of the air to thewestward by an assumed boundary condition. The general situationis indicated in Fig. 10.11.6. There is again a simple wave in the regionof the #, $-plane above the straight line x(U Q + c Q )t which marksthe boundary between the undisturbed flow and the wave arisingfrom disturbances created at x = 0.
This is exactly the situation whichvertical plane.WATER WAVES394istreated at length in sec. 10.3; in particular, an explicit solution ofis easily obtained (cf. the discussion in sec. 10.4) forthe problemtwoThrough various choices of boundary conditions itarbitrarily prescribed disturbances in the values of either of theuquantitiesisorc.possible to supply energy to the system in a variety of ways.Fig.
10.11.6.Wavefrom conditions applied at a boundaryarisingWe proceed next to discuss qualitatively a few consequences whichis assumed that frontal disturbances can be described inconst, at leastterms of simple waves in all vertical planes y = yQover some ranges in the values of the ^-coordinate. (We shall see laterthat simple waves are not possible for all values of y. ) In this discussionwe do not specify how the simple wave was originated we simplyassume it to exist. Since we consider only waves moving eastward(i.e.
in the positive ^-direction) it follows that the straight character2c is constant (inistics are C + characteristics, and hence that uresult if it=each plane y=const.) throughout the wave;u(10.11.22)-2c== UQ -A(y)thereforeA(y),with A(y) fixed by the values UQ and(10.11.22^we havethe undisturbed flow:c (y) in2c (y).++In addition, as explained before, we know that u2c is a function ofaloneoneachcharacteristicuc; hence u and cypositivedx/dtare individually functions of y on each of these characteristics. There-=fore,the characteristic equation(10.11.23)where |xthe value of x at===+may(ube integrated to yield+ c)t,=should be thoughtof as corresponding to an arbitrary instant at which simple wavesexist in certain planes yconst.
) Now, the values of u and c on theis=t0.(The timetLONG WAVES IN SHALLOW WATER395by (10.11.23) are exactly the same as the valuessamevalueof y) at the point tthe0, xf ; therefore, if we(forforisthatcafunction0, theC(#, y) at texample,suppose,givenvalue of c in (10.11.23) is C(f, y) and the value of u is, from (10.11.22),characteristic given===+2C(f, t/). Thus the simplethe equationsA(y)wavesolution can be described= C(, y),u = A(y) +2C(f,y),= + {^(y) + 8C(fbyc((10.11.24),a?,y)}t.(Although the arbitrary function occurring in a simple wave could bespecified in other ways, it is convenient for our purposes to give thevalue of h, and hence c, at t = 0.)Wecould writedownthe solution for the "cross component", ornorth-south component, v of the velocity in this case; by standardmethods (cf.
the report of Whitham [W.12]) it can be obtained byintegrating the linear first order partial differential equation whichoccurs third in the basic equations (10.11.18). To specify the solutionof this equation uniquely an initial condition is needed; this might0.v(x, y) at the time treasonably be furnished by the values v==result is a rather complicated expression from which not muchcan be said in a general way.
One of the weaknesses of the presentattack on our problem through the use of simple waves now becomesapparent: it is necessary to know values of v some time subsequent tothe initiation of a disturbance in order to predict them for the future.It is possible, however, to draw some interesting conclusions fromthe simple wave motions without considering the north-south component of the velocity. For example, suppose we consider a motionafter a bulge to the northward in an initially stationary front haddeveloped as indicated schematically in Fig. 10.11.2.
In a plane yconst, somewhat to the north of the bulge we could expect the top ofThe=the cold air layer (the discontinuity surface, that is) as given by h(x, t)to appear, for tsay, as indicated in Fig. 10.11.7. The main fea-=tures of the graph arc that there is a depression in the discontinuityso that this surface does not touch the ground.surface, but that h>(The latter possibility will be discussed later.) Assuming that themotion is described as a simple wave, we see from (10.11.24) that the= Vkh at the point x = x is equal to the value of c whichxwas at the point x = ^ at = 0, where f(A(y) + 3C(fThatthe value c = C(f y) has been displaced to the right by anvalue ofcltis,1lfl1 ,t/)}/ 1.WATER WAVES896+amount (A(y)BC(^V y)}tr Since this quantity is greater for greatervalues of C, the graph of h becomes distorted in the manner shown in0) steepens whilst theFig.
10.11.7: the "negative region" (where h x<"positive region" (where h xcontinues to smooth out, but,Fig. 10.11.7.>if0) flattens out. The positive regionthe steepening of the negative regionDeformation of the discontinuity surfacewere to continue indefinitely, there would ultimately be more thanone value of C at the same point, and the wave, as in our discussionof water waves (cf.
sec. 10.6 and 10.7), starts to break. Clearly thelatter event occurs when the tangent at a point of the curve inFig. 10.11.7 first becomes vertical. At this time, the continuous solution breaks down (since c and u would cease to be single-valuedfunctions) and a discontinuous jump in height and velocity must bepermitted. In terms of the description of the wave by means of thecharacteristics, what happens is that the straight line characteristicsconverge and eventually form a region with a fold. Such a discontinuous "bore" propagates faster than the wavelets ahead of it (thepaths of the wavelets in the x, /-plane are the characteristics) in amanner analogous to the propagation of shock waves in gas dynamicsand bores in water.In the above paragraph we supposed that h and c were differentfrom zero, and hence the discussion does not apply to the fronts, whichLONG WAVES IN SHALLOW WATERareby397definition the intersection of the discontinuity surface withWhen cthere are difficulties, especially at cold=the ground.fronts, but nevertheless a few pertinent observations can be made,2cassuming the motion to be a simple compression wave with uconstant.WhenVkh Q andhQ=cat/=follows that0, itwith a theu=UQ2c,and sincec=of the top of the coldinitial inclination=UQu2\/<x.ky in this case.
But u thenmeasures the speed of the front itself in the ^-direction, since a particleonce on the front stays there; consequently for the speed uf of thefront we haveair layer, it follows that=ufUQ2\/o%.Thus the speed of the front decreases with y, and on this basis itfollows that a northward bulge would become distorted in the fashionindicated by Fig. 10.11.8, and this coincides qualitatively with obser(10.11.25)vations of actual fronts.Actually, things are not quite as simple as this.
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