J.J. Stoker - Water waves. The mathematical theory with applications (796980), страница 69
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Thus we begin with the classical hydrodynamical equations. The equations of motion in the Eulerian form arc taken:Sinceit isdesired that this section should be asself contained,dudvdv(10.11.1)dw}"77==at~dpa~OZ+ 6 FM ~OS+with d/dt (the particle derivative) defineel by the operator d/dtu dfdxv d/dyw d/dz.
In these equations u v re are the velocityrelativeto our rotating coordinate system, p is the prescomponentsthesure, Qdensity, >/<%) etc. the components of the Coriolis forcedue to the rotation of the coordinate system, anei pg is the force ofgravity (assumed to be constant). These equations hold in both the1+warm+9and the cold9but it is preferable to distinguish the eiethetwo different layers; this is done herependentthroughout by writing u' v', w' for the velocity components in thewarm air and similarly for the other elependent quantities.airair,quantities in9We n6wintroduce an assumption whichdynamic meteorologyphere,i.e.that the airiscommonly madeinin discussing large-scale motions of the atmosis incompressible.
In spite of the fact that suchWATEE WAVES382an assumption rules out thermodynamic processes, it does seem ratherreasonable since the pressure gradients which operate to create theflows of interest to us are quite small and, what is perhaps the decisivepoint, the propagation speed of the disturbances to be studied is verysmall compared with the speed of sound in air (i.e. with disturbancesgoverned by compressibility effects). It would be possible to considerthe atmosphere, though incompressible, to be of variable density.for the purpose of obtaining as simple a dynamical model asitseems reasonable to begin with an atmosphere having apossibleHowever,constant density in each of the two layers.
As a consequence of theseassumptions we have the following equation of continuity:u*+v y +w, =(10.11.2)0.The equations (10.11.1) and (10.11.2) together with the conditionsof continuity of the pressure and of the normal velocity componentson the discontinuity surface, the condition wat the ground,appropriateproblemdoubtlessly yield a mathematicalthe solution of which would furnish ainitial conditions, etc.call itProblemIreasonably good approximation to the observed phenomena. Unfortunately, such a problem is still so difficult as to be far beyond thescope ofknown methodscomputation. Thusstillof analysis including analysis by numericalfurther simplifications arc in order.Oneof the best-founded empirical laws in dynamic meteorology isthe hydrostatic pressure law, which states that the pressure at anypoint in the atmosphere is very closely equal to the static weight ofthe column of air vertically above it. This is equivalent to saying thatthe vertical acceleration terms and the Coriolis force in the thirdequation of (10.11.1) can be ignored with the result(10.11.3)This=-&.also the basis of the long-wave or shallow water theory ofsurface gravity waves, as was already mentioned above.
Since theiscomponent of the acceleration of the particles is thus ignored,follows on purely kinematical grounds that the horizontal components of the velocity will remain independent of the vertical coordinateverticalitwas the case at the initial instant t = 0. Since weassume an initial motion with such a property, it followsthat we havez for all time if thatdoin factLONG WAVES IN SHALLOW WATERu(10.11.4)Thefirst=u(x, y,vt),=v(x, y,wt),two of the equations of motion(10.11.1)continuity (10.11.2) therefore reduce to=px+F(x)vv y=py+F(y)+v y =ux=0.*and the equation ofvu y(10.11.5)8830,where we use subscripts to denote partial derivatives and subscriptsenclosed in parentheses to indicate components of a vector.
TheCoriolis acceleration terms are now given byf(10.11.6)[FF(x)'*'(y}==2co sin2wvq>\sin(p= Avu =Awwhen use is again made of the fact that w = 0. (The latitude angle 9?,was indicated earlier, is assumed to be constant.) We observe oncemore that all of these relations hold in both the warm and cold layers,and we distinguish between the two when necessary by a prime on thesymbols for quantities in the warm air. It is perhaps also worth mentioning that the equations (10.11.5) with F (x} and F (y) defined byas(10.11.6) arc valid for all orientations of the x, j/-axes; thusit isnotnecessary to assume (as we did earlier, for example) that the originalstationary front runs in the east-west direction.Wehave not so(10.11.3).Tofarendthismadeitisuse of the hydrostatic pressure lawuseful to introduce the vertical heightfull/i(o% y, t) of the discontinuity surface between the warm and coldlayers and the height h'h'(x, y,t) of the warm layer itself (seeFigure 10.11.3).
Assuming that the pressure p' is zero at the top ofh=thewarmlayerwe(10.11.7)findby integratingp'Or, y,z, t)=for the pressure at any point in thein similar fashion:(10.11.8)* Itp(xty, z, t)==q'g(h'(10.11.3):e 'g(h'warm-h)-air.+z)In the cold airQg(h-we have,z)infer that we assume the vertical displacementsa peculiarity of the shallow water theory in general whichresults, when a formal perturbation series is used, because of the manner inwhich the independent variables are made to depend on the depth (cf.
Ch. 2 andwould be wrong, however, toto be zero. Thisisearly parts of the present chapter).WATER WAVES384=when the=condition of continuity of pressure, p'h, isp for zused. (The formula (10.11.8) is the starting point of the paper byFreeman [F.10] which was mentioned earlier.) Insertion of (10.11.8)Wormh'(x t y,t)PsP>h(x,y,t)Cold''^XFig. 10.11.3. Vertical height of thetwolayersin (10.11.5) and of (10.11.7) in (10.11.5)' yields the following sixequations for the six quantities u v h, u v\ h':r9u9,uu4(10.11.9)(cold air)u(10.11.10)(warmair)w =vtvu'uvu y =gh'xuvvv=gh'y=0.These equations together with the kinematic conditions appropriate=- 0,at the surfaces z = h and zh', and initial conditions at twould again constitute a reasonable mathematical problem call itProblem II which could be used to study the dynamics of frontalmotions. The Problem II is much simpler than the Problem I formulated above in that the number of dependent quantities is reducedfrom eight to six and, probably still more important, the number ofindependent variables is reduced from four to three.
These simplifications, it should be noted, come about solely as a consequence of assuming the hydrostatic pressure law, and since meteorologists have muchLONG WAVES IN SHALLOW WATER385evidence supporting the validity of such an assumption, the ProblemII should then furnish a reasonable basis for discussing the problemof frontal motions. Unfortunately, Problem II is just about as inaccessible as Problem I from the point of view of mathematical andnumerical analysis. Consequently, we make still further hypothesesleading to a simpler theory.As a preliminary to the formulation of Problem III we write downthe kinematic free surface conditions at zh and z = h' (the dynamical free surface conditions, p =at zh' and ph,p' at zhave already been used.) These conditions state simply that the====h'(z, y, t)h(x, y, t) and zon the surface zor the surfaceh =We have therefore the conditionsparticle derivatives of the functions zvanish, sincezh'=anyparticleremains onit.+ vk v + h =u'h r + v'h y + h =*x + '*; + ti - o,uh x(10.11.11)tt>tw vanishes everywhere.
It is convenient tothethirdreplaceequations (the continuity equations) in the setsinview of the fact that(10.11.9)and (10.11.10) by(10.11.12)(uh) xf(10.11.13)[u'(h-+h)] x+ h = 0,+ [v'(h' - h)] +(vh) vtyand(h'-h) t=0,which are readily seen to hold because of (10.11.11). In fact, the lasttwo equations simply state the continuity conditions for a verticalcolumn of air extending (in the cold air) from the ground up to z = h,and (in the warm air) from z = h to z = h'.We now make a really trenchant assumption, i.e. that the motionwarm air layer is not affected by the motion of the cold air layer.This assumption has a rather reasonable physical basis, as might beargued in the following way: Imagine the stationary front to havedeveloped a bulge in the ^-direction, say, as in Figure 10.1 1.4a.
Thewarm air can adjust itself to the new condition simply through aslight change in its vertical component, without any need for a changeof theandthe horizontal components. This is indicated in Figure10, 11. 46, which is a vertical section of the air taken along the linein Figure 10. 11.40; in this figure the cold layer is shown with ain u'v' 9ABquite small height which is what one always assumes. Since weignore changes in the vertical velocity components in any case, it thusseems reasonable to make our assumption of unaltered flow conditionsWATER WAVES386in the warm air.
However, in the cold air one sees readilyas indicated in Figure 10.11.4c that quite large changes in the componentsu, v of the velocity in the cold air may be needed when a frontal disturbance is created. Thus we assume from now on that u', v', h' havefor all time the known values they had in the initial steady state in//// B7/m/(o)Fig. 10.11.4.Flows inwarm andcold air layerswhich v'const. The differential equations for our Problem0, u'III can now be written as follows:ut= -g+ uux(10.11.14)= (uk) x+(vh) vg= 0,fh'xP-A;+il-^-\hA ++(i-^~\AJ -LONG WAVES IN SHALLOW WATERinwhichandh'xconst.,1yknown functions given in terms ofThe initial state, in which v' = vthe initial= 0, u == const., must satisfy the equations (10.
11. 9) and (10.11.10);state in theuh' are387warmair.once to the conditionsthis leads atA*;-1g(10.11.15)^*(e'-Ig'g\0u,U\I/Thefor the slopes of the free surfaces initially.slope h y of the disnearly proportional tocontinuity surface between the two layers isthe velocity difference u'u since g'/p differs only slightly fromitisandmadesmallunder the conditions normally enunity,quitecountered because of the factorwhicha fraction of the angularrelation for the slope h y of the stationaryA,isvelocity of the earth. Thediscontinuity surface is an expression of the law of Margulcs in meteorology.
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