高等大气动力学Rossbywave

1、Physics of Rossby Wave,Yihua Lin AOPP, Clarendon Laboratory University of Oxford, UK 4th, June, 2008,2,Wave Rossby wave in atmosphere 03 March 2005 12Z.,9,10,The 2 small black circles off the coast of the Americas correspond to sea level highs on the trailing edge of Kelvin waves. The leading edge o

2、f this wave group has bounced off the coastline, creating Rossby waves whose rising and falling sea levels are marked by solid and dashed lines, respectively.,In the July image, the circles are moving west with the Rossby waves. The X marks a relative sea level low caused by a Kelvin wave moving eas

3、t.,In December, the Rossby waves continue to move westward. The shapes of the solid and dashed lines indicate that the Rossby waves are moving away from the Americas faster at the equator than at higher latitudes.,White and red indicated higher than average levels; purple and magenta indicate lower

4、than average levels. These scenes show eastward-moving Kelvin waves and westward-moving Rossby waves. Superimposed black circles show how the elevated sea surface moves east in April, then west as Rossby waves during July and December.,Ocean,11,Rossby Waves The most important class of large-scale at

5、mospheric waves are called planetary waves, or Rossby waves. These waves are characterized by oscillations in the rotational part of the horizontal wind that are parallel to the horizontal gradient in the potential vorticity.,Mechanism of Rossby Wave,12,Restoring force is the north-south gradient of

6、 background potential vorticity (f/H). That gradient can be due to either the variation in f with latitude, or to a variation in H (“topographic Rossby waves” and when trapped near the coast “continental shelf waves”). Note that Rossby waves are tranverse waves, that is the particles move perpendicu

7、lar to the direction of propagation. Conserves PV = (f+)/H.,13,Sketch: northern hemisphere,14,Imagine that something pushes a water column perpendicular to the planetary vorticity. What will happen? The parcel will conserve total potential vorticity (f+/H). As it moves into a region of higher f, it

8、will tend to decrease relative vorticity a tendency to clockwise motion. Likewise if it were displaced towards lower relative vorticity, it will increase relative vorticity a tendency towards counterclockwise motion.,15,Now lets look at the Coriolis forces on the induced circulation pattern.,Size of

9、 restoring force (for a given displacement) is proportional to = df/dy. PV equation: D/Dt + v = 0,16,The same thing will hold true on a slope pushed towards shallower water will require decreased relative vorticity and vice versa so shallower water is the analog of northward over a flat bottom and t

10、here are analogous waves generated on strong topographic slopes called topographic Rossby waves. Hf.,17,In last section, we derived the properties of Rossby waves. Our key findings was that Rossby waves rely on the -effect for their existence. We also discussed the mechanism for Rossby wave propagat

11、ion through potential vorticity conservation.,Rossby wave propagation,18,Potential vorticity equation Consider small-amplitude disturbances to a uniform zonal background flow (U, 0, 0) for which the streamfunction is and the quasigeostrophic potential vorticity is The streamfunction for the total fl

12、ow (background plus small disturbance) is thus and the potential vorticity,19,Where We now substitute the above in the quasigeostrophic potential vorticity equation and neglect terms that are quadratic in the disturbance quantities: The first term represents the rate of change of the perturbed poten

13、tial vorticity following the mean flow, and the second term represents the advection of mean potential vorticity by the perturbed flow.,20,Dispersion relation We now look for plane-wave solutions, We shall also assume constant stratification, i.e., N2 = constant.This gives the dispersion relation fo

14、r quasigeostrophic Rossby waves: As in last section, the -effect is crucial for these Rossby waves. Setting = 0 gives = Uk, which implies c(x) p = c(x) g = U, i.e., the waves are merely carried along by the background flow, U.,21,For the barotropic motion, we get the simpler dispersion relation for

15、Rossby waves (on a motionless state, U = 0) This function is plotted as below (Note that l/= x/(x2 + 1), where x = k/l.),The function x/(x2 + 1).,22,1. /k = /(k2 + l2) 0 for k/l 1, and d/dk 0 for k/l 1: the group velocity of Rossby waves is eastward for zonally short waves, westward for zonally long

16、 waves. 4. The magnitude of the group velocity (judge by the slope of Fig. on last slid) is, typically, greater for the westward-propagating long waves than for the eastward-propagating short waves.,23,Illustrating the Rossby-wave mechanism, in terms of conservation of potential vorticity by moving

17、fluid blobs in the case in which the waves are independent of height.,24,In the special case where l = 0, the blobs move purely in the northsouth direction. Consider a line of blobs, labelled A, B, C, etc., initially lying along a line of latitude y = y0; see Figure on last slide. Suppose that these

18、 blobs are displaced into the sinusoidal pattern indicated by the solid wavy line: blob A moves southwards, so its value of increases, as indicated by the anticlockwise arrow in the figure. By the PV inversion process this induces an anticlockwise rotation in the local velocity field, as indicated b

19、y the circular arrow; in particular, blob B is encouraged to move further south. The value of associated with blob B itself increases, inducing anticlockwise rotation near B, which tends to move C southwards and A northwards again. Applying this kind of argument to each blob, we find that, after a s

20、hort time, the pattern of the blobs has moved westwards, to the position indicated by the dashed wavy line, even though each individual blob only oscillates northsouth. A self-sustaining, westward-moving Rossby-wave pattern emerges, as expected from the theory given above.,25,More generally, the zon

21、al phase speed of the waves is and hence the wave crests and troughs move westward relative to the background flow, although they can move eastward relative to the ground for a sufficiently strong background flow. The latter explains why mid-latitude atmospheric weather systems generally propagate e

22、astward.,26,Vertical propagation For given real values of k and l (e.g., imposed by land-sea contrasts or orography), we obtain vertical propagation when m is real and nonzero. Vertical propagation therefore corresponds to m2 0, i.e., where the critical background velocity depends on the horizontal

23、wavelengths of the wave. Thus for vertical propagation we must have,27,In particular, for stationary waves, whose crests and troughs do not move relative to the ground such that c(x)p = 0, we obtain the Charney-Drazin criterion Stationary waves propagate vertically only in eastward background flows

24、(U 0) that are not too strong (U Uc). Moreover, since Uc increases with increasing horizontal wavelength, long waves propagate vertically under a wider range of eastward flows than short waves.,28,This is consistent with observations: ( on p = 50mb, McLandress 2005),In the winter stratosphere, winds

25、 are eastward and stationary Rossby waves have large horizontal scales.,In the summer stratosphere, winds are westward and stationary Rossby waves are absent.,29,Vertically-trapped Rossby waves Returning to the dispersion relation, if k2 + l2 /(U c(x)p ) then m is imaginary. If we set m = i, where i

26、s real and positive, then the wave solutions are of the form i.e., the waves propagate horizontally but not vertically, and decay with height. Equivalently, this happens if For stationary waves (c(x)p = 0), vertical trapping occurs if the backgroundwinds are westward (U Uc).,30,Vertical modes In the

27、 ocean, Rossby waves are strongly constrained by the surface and bottom boundaries. We can seek Rossby wave solutions with modal structures in the vertical. Thus we seek solutions of the form Substituting the above into the potential vorticity equation gives This separable solution only works if both the larger scale of these will propagate rapidly westward. Before long, they will reach the western boundary of the ocean where they will be reflected. Unlike g