Propagation and Breakdown of Internal Inertio-Gravity Waves Near Critical Levels in the Middle Atmosphere

Transcription

1 February 1984 M. D. Yamanaka and H. Tanaka 1 Propagation and Breakdown of Internal Inertio-Gravity Waves Near Critical Levels in the Middle Atmosphere By Manabu D. Yamanaka* Institute of Space and Astronautical Science, Meguro-ku, Tokyo 153, Japan and Hiroshi Tanaka Water Research Institute, Nagoya University, Chikusa-ku, Nagoya 464, Japan (Manuscript received 28 December 1982, in revised form 6 December 1983) Abstract Behaviors of internal inertio-gravity waves (IIGW) near Jones' critical levels are studied theoretically in view of a possible origin of turbulence layers in the middle atmosphere. The inertial effect associated with the earth's rotation cannot be neglected when time constant of the wave is large. Assuming that the vertical shear and Coriolis factor are constant, exact solutions of IIGW are obtained from inviscid and linear equations. The asymptotic expressions are derived by means of the Liouville-Green method developed by Olver (1974) which leads to an exact dispersion relation near the critical levels. Two important features about critical level problem of IIGW are found from the dispersion relation: valve effect across the Jones' critical levels in somewhat different sense from Grimshaw (1975, 1980), and presence of a pair of turning levels between both Jones' critical levels. Coupling these features, we predict that IIGW is absorbed or reflected by the Jones' critical levels depending on the direction of wave-front. The absorption rate and the thickness of turbulence layer produced by critical level breakdown increase as the wave-fronts tend to direct to the zonal direction, on the other hand IIGW is substantially reflected when they direct to the meridional direction. With increase of basic Richardson number the turning levels approach asymptotically the critical levels, so that turbulence layers inside the critical levels become thinner than those outside them. These features vanish in the case of non-inertial gravity waves. The relation between IIGW and turbulence layers is calculated to compare with the turbulence layers observed in the stratosphere and to have information on IIGW's propagating upwards to the mesosphere and the thermosphere. In general, thickness of the turbulence layers associated with IIGW's is thinner than that associated with non-inertial gravity waves for common mesoscale wavelength domain. 1. Introduction Recent observational studies of the middle and lower stratosphere have found frequent presence of turbulence layers of m in thickness (Barat, 1975a, 1982; Cadet, 1977; Woodman et al., 1981). The most possible origin of the turbulence layers seems to be related to critical level breakdown of propagating internal gravity waves generated in the troposphere below. Theoretical * Graduate student of Nagoya University. investigations on this mechanism have so far omitted the inertial effect by the earth's rotation to simplify the mathematical treatment (Geller et al., 1975; Tanaka, 1975, 1982, 1983b; Klostermeyer, 1980). On the other hand some observations have revealed that the typical horizontal wavelength of stratospheric gravity waves ranges from 100km to 1000km (Sawyer, 1961; Hearth et al., 1974; Cadet and Teitelbaum, 1979). A several number of investigators have treated the propagation of IIGW in shear flows (Jones, 1967; Lindzen, 1970; Grimshaw, 1975, 1980;

2 2 Journal of the Meteorological Society of Japan Vol. 62, No. 1 Miyahara, 1976, 1981; Kitchen and McIntyre, 1980; Olbers, 1981). However, these works are not always sufficient in treatment of the behaviors of IIGW in the vicinity of the critical levels. The pioneering work of Jones (1967) includes exact Frobenius solutions for a homogeneous rotation but never shows the local structures of IIGW explicitly. To complement the deficiency, Grimshaw (1975) and Olbers (1981) used the WKB method, but it fails close to the critical levels. Lindzen (1970), Miyahara (1976, 1981) and Kitchen-McIntyre (1980) treated the waves in the equatorial region. They were obliged to omit some terms in the equations to simplify the theoretical treatment, otherwise to solve the equations numerically. Oversimplification of the governing equations happens to miss important properties of the solutions in the vicinity of singularities like critical levels. Any finite difference method would break near the critical levels due to rapid changes of velocities and other quantities. Therefore, more exact and sophisticated mathematical manipulations are necessary to clarify the mechanism of propagation and breakdown of IIGW near the Jones' critical levels. Turbulence layers produced by gravity wave breakdown in the middle atmosphere are important from two points of view; diffusion of momentum, heat and minor constituents (Barat, 1975b, 1982; Cadet, 1977), and friction on the mean flow. Recently the latter is considered inevitable to induce the mesopause reversal of zonal wind and the inverse temperature gradient near the mesopause region (Lindzen, 1981; Matsuno, 1982; Holton, 1982). As one of the present authors (Tanaka, 1983a) pointed out, the wave momentum flux divergences of long mesoscale IIGW's are too small to contribute to decelerate the mesopause wind. This fact emphasizes the need to study how the IIGW's damp so effectively during propagation in the middle atmosphere. Fortunately we found that the Liouville-Green method developed by Olver (1974), which can be called "extended Liouville-Green method" or "extended WKB method", is applicable to the critical level problem of IIGW. The conventional WKB method breaks for above problem. The most striking feature of IIGW propagation across the Jones' critical levels is the valve effect, of which possibility was found by Grimshaw (1975, 1980) in a slightly different situation. We advanced the concept of valve effect with sophisticated mathematical treatments in this paper. Connection between solutions near the Jones' critical levels of IIGW and those in far regions, i.e., solutions of non-inertial gravity wave, reveals how Coriolis effect works to absorption mechanism. 2: Equations and solutions 2.1 Governing equations* The basic field is in geostrophic, hydrostatic and hence thermal-wind equilibria: where we assume Provided that the atmosphere is inviscid, adiabatic and Boussinesq, the momentum, thermodynamic and continuity equations for perturbations are given by where D/Dt*t+u*/*x. Note that the second term of the left-hand side of Eq. (3d) comes from the basic thermal-wind equilibrium (1c). Eqs. (3a)-(3e) complete a closed system for five dependent variables u, *, w, *, * and can be reduced to an equation without omission: Notations of symbols are summarized in Appendix A.

3 February 1984 M. D. Yamanaka and H. Tanaka 3 Here we assume a sinusoid of w in both horizontal directions and a neutral mode: where k is positive, and l and * are real. Substituting (5) into Eq. (4), we have which makes Eq. (9) change as where intrinsic (Doppler-shifted) frequency * is used for independent variable instead of altitude z like This equation has three singularities at Z=0, 1 and *. Roots of indicial equations for the singularities are around Z=0, Modified Richardson number J' is defined by around Z=*, where J is the Richardson number for twodimensional flow. Eq. (6) or similar equations were also derived by Eady (1947), Jones (1967), Tokioka (1970) and Grimshaw (1975) but have never been solved perfectly. From a mathematical point of view, Eq. (6) is a Fuchsian type equation with four regular singuarities at *=-*, 0, * and *. For convenience of the subsequent analyses we reduce Eq. (6) to a standard form: around Z=1. Eq. (14) can be reduced to a Gauss' hypergeometric equation where W is related to w in Eq. (14) like and a, b and c are by using the following transformation: In Eq. (9), *2 and *2 are written as The exact solutions of Eq. (6) can be obtained with the aid of (10), (13), (15a), (15c) and (17a): Hereafter we assume that The omitted cases here, *=0 and l=0, will be discussed in Appendices B and C, respectively. The last assumption, J'*1/4, implies a stability condition of the basic flow for three-dimensional disturbances. where W is the linear combination of two members of Kummer's 24 solutions of the hypergeometric equation (16) under the assumptions (12). Two linearly independent solutions in the neighborhood of *=0, * and * are respectively given by 2.2 Exact solutions In order to obtain the exact solutions of Eq. (9), we try to reduce the singularities. We at first introduce a transformation of independent variable

4 4 Journal of the Meteorological Society of Japan Vol. 62, No. 1 with the definitions of Here note that * in (20a) and * in (20b) are different notations from (*1, *2) in (15b) and (*1, *2) in (15c). Kummer's connection formulae of hypergeometric functions lead to The solutions w1(*) and w2() have logarithmic branch points at respectively. The singularities (22) correspond to the critical levels of IIGW originally pointed out by Jones (1967). Branch connection of the singular solutions leads to The other choice, i.e., *i*-0, is omitted on the basis of a physical consideration (cf. Booker and Bretherton, 1967). 2.3 Asymptotic solutions Here we shall consider asymptotic expressions of the exact solutions (19a)-(19f), assuming the following form: Note that in above derivation 6 is replaced by +i*i by incorporating infinitely small damping, * i.e., where P and Q are both real functions of *. Such an asymptotic form (25) is convenient to relate the mathematical behavior of the solutions to the vertical structure of IIGW in the subsequent chapters. The standard form of the equation for IIGW,

5 February 1984 M. D. Yamanaka and H. Tanaka 5 (a) Case for *0 The result obtained from the LGO method shows that the phase integrant P and the amplitude function Q in the vicinity of *=0 are given by Subscripts 1 and 2 correspond to the solutions (19a) and (19b), respectively. (b) Case for * In this case we have Fig. 1 Locations of turning levels for The basic Richardson number J(=N2/uz2) parameterizing *(=l/k). (a) Real turning levels (*2/*2) (b) Imaginary turning levels (*2/*2) (9), seems to be solved by the conventional WKB method. However, it does not work near the Jones' critical levels. The points * and since * is real. Fig. 1(a) shows the location of the turning points for variation of two-dimensional Richardson number of the basic fields, J, by parameterizing the ratio of both horizontal wavenumbers l/k. We find that It should be kept in mind that the expressions in (31) and (32) are valid when *0 (see Appendix C). Subscripts 1 and 2 in (31) and (32) correspond to the solutions (19c) and (19d), respectively. (a) Case for * We obtain P and Q like is always satisfied and that We also find from Fig. 1(b) that *0 for J* and * for J*. As was mentioned before application of the conventional WKB method to the present problem is mathematically invalid since it breaks near the Jones' critical levels (see Appendix D). So we use the Liouville- Green method developed by Olver (1974) (hereafter referred as LGO method), which is able to provide the asymptotic solutions near the critical levels. which correspond to the solutions (19e) and (19f), respectively, when J'*1/4. As *, (33) and (34) become the same expressions as those based on the conventional WKB method (see Appendix E). 3. Vertical propagation of IIGW in a shear flow 3.1 Local dispersion relation We have mentioned already that the conventional WKB method breaks near the critical levels of IIGW. Therefore, we must derive the

6 6 Journal of the Meteorological Society of Japan Vol. 62, No. 1 dispersion relation without using the WKB (or WKBJ) formalism. We start from the solution of Eq. (6) and reduce it to the form: Thus we can define the local vertical wavenumber as i.e., in the regions where * *, in order to relate IIGW to non-inertial gravity wave. We find from mathematical considerations shown in Appendix B that the limit * *is equivalent to the limit *0. Thus we have a local dispersion relation of IIGW far from the critical levels as the same form as (A3): where the asterisk denotes the complex conjugate. Substituting (10) into (35) with aid of (7) we have If the asymptotic form (25) is used, (35) can be written as Here any expression of (35), (35)' and (35)" is equivalent. Group velocity vector is determined from the local dispersion relation as follows: The vertical component of the group velocity is given by These results imply that IIGW can be regarded as non-inertial wave when it is propagating very far from the critical levels. It is clear that the two fundamental solutions, w1(*) and correspond to up- and downward propagating waves, respectively, infinitely far from the Jones' critical levels in the case of uz*0. When the wave is not very far from the Jones' critical levels, the local dispersion relation can be directly obtained from the asymptotic expressions (33) and (34) as follows: By using (35)" the vertical group velocity is written as Since the wave energy is accompanied by the group velocity, we hereafter call a wave with positive Wg an upward propagating wave. As will be mentioned later, we find both upward and downward propagating waves at every point in the region * *, which are included in the fundamental solutions of (31), (32) and (33), (34). In the conventional method of ray tracing, only one ray-solution is adopted on the basis of physical consideration. However, from a mathematical point of view, the special solution satisfying the boundary conditions is generally expressed by a linear combination of two fundamental solutions, i.e., upward and downward propagating waves. 3.2 Propagation of IIGW far from the critical levels Although our main objective is to look at the behavior of IIGW near the Jones' critical levels, where *, it is worthwhile to consider the propagation feature far from the critical levels, If the basic shear uz is negligible, that is, the Richardson number J is infinitely large, (39) is reduced to which is well known as the dispersion relation of IIGW in a no-shear flow. As shown in Appendix E, (39)' is also obtained by the conventional WKB method taking the limit uz *0. Thus we can safely mention that the conventional WKB method is applicable to IIGW problem as long as IIGW is propagating far from the Jones' critical levels in the middle atmosphere where J is sufficiently large (cf. Kitchen and McIntyre, 1980). 3.3 Propagation in the vicinity of Jones' critical levels The local dispersion relation of IIGW near the critical levels is given by from (31) and (32)

7 February 1984 M. D. Yamanaka and H. Tanaka 7 It should be noted here that the propagation direction of IIGW is highly dependent on the sign of meridional wavenumber l. We find from (40) that the local structure of IIGW in the vicinity of the Jones' critical levels is quite different from that found far from the critical levels. In fact, if ever the basic shear becomes infinitely small, (40) never approaches (39)'. This result implies that IIGW is severely modified near the critical levels. Note that such a modification never appear in the non-inertial case (see Appendix B). The modification of dispersion relation of IIGW near the Jones' critical levels may come from a gap between the two regions. This fact is also understood from analytical continuations of the exact solutions, (21a) and (21b). When 0 and l*0, we can obtain the following uz* connection formula, using (21a) and (21b), for the upward propagating wave from infinitely far region: to non-inertial gravity wave as was investigated by Booker and Bretherton (1967). The region of * where the dispersion relation (40) can be adopted, is within a convergence limit of the exact solutions (19c) and (19d), and is limited by the turning levels shown in (26): In the middle atmosphere the thickness expressed in (43) is about 1km for the wave of 2*/k= 150km and about 10km for the wave of 2*/k 1500km assuming that *=10-4sec-1, uz=10-3 sec-1 and N=10-2sec-1, In the next chapter we shall discuss the detailed behavior of IIGW near the Jones' critical levels with the aid of the dispersion relation (40). 4. Wave behaviors near Jones' critical levels 4.1 Valve effect We find valve effect* of IIGW around the Jones' critical levels from the local dispersion relation (40) and the connection formulae (23a) and (23b). The dispersion diagram on an m-*, plane is shown in Figs. 2(a) and (b) for *0 and *0, respectively. The vertical group velocity defined by (36)' is obtained by taking tangent of each curve. Application of L'hospital theorem on the dispersion relation (40) leads to one finite root of m on the Jones' critical levels: Other roots of m derived from (40) are infinite as Here w1(*) and w2(*) correspond to the incident and the reflected waves, respectively. The net reflection rate of IIGW defined far from the lower Jones' critical level is given by We have already mentioned in Section 2.2 that is regular at *=* but singular at *=-*, w1(*) and that w2(*) is regular at *=-* but singular at *=*. From the asymptotic expressions (31) and (32) we can confirm that the finite root (44) corresponds to the regular solutions w1(*) at *=f and w2(*) at *=-*, and that the infinite roots (45) correspond to two branches of the singular solution w2(*) at *=f and also to those of w1(*) at *=(*). Introducing an infinitesimal damping shown in (24) and considering the connection formulae (23a) and (23b) result in an energy gap across the Jones' critical levels as follows: Similar results can be obtained in any other situation for uz and 1. As long as we look at IIGW at infinitely far region it seems equivalent * Grimshaw (1975) pointed out that the valve effect is also found when there exists a horizontal component of the Coriolis factor even in the case of non-hydrostatic equilibrium.

8 8 Journal of the Meteorological Society of Japan Vol. 62, No. 1 Fig. 2 Local dispersion relations (*-m diagram) for uz*0 obtained by the LGO method. Thick and thin curve denote upward and downward propagating waves, respectively. Consideration on the sign of the vertical group velocity leads to an interesting result. When uz*0 and l*0, IIGW propagating upwards across the lower critical level (*=-*) or IIGW propagating downwards across the upper critical level (*=*) is attenuated by a factor (46). Oppositely propagating IIGW can pass through the critical level without any significant attenuation. This feature of propagation is called the valve effect, which never appears near the Booker-Bretherton's critical level. Such a valve effect is found in any other situation of uz and l. The valve effect associated with the Jones' critical level was pointed out by Grimshaw (1975) in a different situation. He considered non-boussinesq and non-hydrostatic fluid and demonstrated that the valve effect is found due to the horizontal component of the Coriolis factor. Recently, Olbers (1981) discussed the propagation problem of IIGW with horizontal basic shears using WKB method and predicted a valve effect on Jones' type critical lines associated with the horizontal basic shear. In spite of Grimshaw's and Olbers' findings of the valve effect, our theory never decreases the value from the following points of view. First, the valve effect appears without introducing any horizontal component of the Coriolis factor in the case of IIGW under hydrostatic equilibrium. Second, two turning levels shown in (26) are predicted inside the Jones' critical levels. Third, IIGW's propagating far from the Jones' critical levels can be connected to those propagating near the Jones' critical levels. The first and second points come from the dispersion relation (40) and the third one comes from the analytical continuation of the exact solutions shown in (41). We shall discuss the physical meanings of them in the subsequent sections. 4.2 Turning levels We have already mentioned in Section 2.3 that a pair of turning levels exist inside the Jones' critical levels. The dispersion relation (40) derived by the LGO method also suggests the existence of the turning levels. As is shown in Appendix D, we are not afraid of the validity of LGO method between the critical and turning levels as long as the assumption (12) is enforced. When l*0, IIGW is totally reflected by the turning levels after passing through the

9 February 1984 M. D. Yamanaka and H. Tanaka 9 Fig. 3 Vertical profiles of Re[w] of IIGW for uz*0 calculated from the asympotic solutions based on the LGO method. Jones' critical levels. Fig. 3(a) shows an example of vertical variation of Re[w] which is asymptotically approximated in terms of the LGO solutions (31) and (32). When l*0, however, GW from outside of the Jones' critical II level cannot pass through it as is shown in Fig. 3(b). The results mentioned above are confirmed by ray tracing based on the local dispersion relation (40) and the formula of group velocities (36). Strictly speaking, this method is valid in the region shown by (43). Calculations on the group velocity vector are shown in Appendix F. Schematic illustration of ray paths relative to the basic flow is shown in Fig. 4 for positive and negative *. Combined effect of the turning and critical levels leads to convergence of the rays toward the critical levels regardless of the sign of *. 4.3 Wave absorption associated with Jones' critical levels We have already mentioned that energy gap across the Jones' critical levels can be expressed as (46). In this section we shall discuss it based on analytical continuations of the exact solutions of IIGW. When uz*0 and l*0, we have the following connection formula combining (21a) with (21b):

10 10 Journal of the Meteorological Society of Japan Vol. 62, No. 1 Fig. 4 Schematic illustrations of the rays relative to the basic flow the Jones' critical level. Abscissa is defined as x1=x1-u*t where x1 is along the direction normal to the wave-front. Case for (= l/k)*0. (left) and for *(= l/k)*0 (right). * from the connection formula (41) like where w2(*) and w1(*) correspond to the waves propagating upwards and downwards, respectively, just below the lower Jones' critical level. Eventually the rate of reflection is derived as Although the definition of r' depends upon the signs of l and uz, the final result of (48) holds in any case.* We find from (48) that r'*l for 0 and r'*0 for * *. * * The rate of reflection, however, is important only near the Jones' critical levels. Usually major part of the wave energies would be attenuated till the waves reach the Jones' critical levels. In fact, the rate of absorption defined * For example, when uz*0 and, l*0, w1(*) and w2(*) correspond to upward and downward propagating waves, respectively, so that we should define r'* term of w2(*)/term of w1(*) 2. However, even in such a case we finally obtain r' * exp(-2*), which is consistent with features just under the Jones' critical level shown in Figs. 3(a) and (b). goes to zero for * *. Thus we can mention that the wave absorption associated with the Jones' critical levels plays a minor role at least quantitatively to the total absorption of the wave energy except the case that the wave source is actually located close to the critical levels. 5. Formation of turbulence layers 5.1 Local convective instability In this chapter we discuss the formation of turbulence layers caused by the critical level breakdown of IIGW. We at first obtain a general criterion of local convective instability of IIGW under Boussinesq approximation. This idea is the same as Geller et al. (1975) and Tanaka (1975, 1982, 1983b) though these are associated with non-inertial gravity wave breakdown. An air parcel perturbed by a wave moves on an isopycnic surface. The air density is conserved with respect to the wave motion. An isopycnic surface corresponds to a material surface and the undulation is given by the vertical displacement of an air parcel, *, as follows: where * depends only on * and w is the same as that in (5). Inclination of the material surface

11 February 1984 M. D. Yamanaka and H. Tanaka 11 to x-direction is written as When the inclination becomes larger than that of the wave-front, the local convective instability occurs, that is, Substituting (53) ; and (54) into the instability criterion, m =1/ *, and using the relation =k uz ZT and *0+* =k uz h, we can *+* finally obtain the turbulence layer thickness near the Jones' critical levels, Two-dimensional non-inertial internal gravity wave (*=0 and l=0) induces a local convective instability near the Booker-Bretherton's critical level (*=0). Substituting the solutions given by (A14) and the vertical wavenumber given by (A15) into the marginal state of breaking criterion shown in (51), we have the following formula of turbulence layer thickness: using the relations and Note that the turbulence layer thickness is defined by the distance between the critical level (*=0) and the breaking level. This result is the same as that obtained first by Geller et al. (1975). When viscosity, Newtonian cooling and air density reduction are incorporated, the righthand side of (52) must be changed (see Tanaka, 1982, 1983b). 5.2 Turbulence layers generated by IIGW breakdown We have found in the previous chapter that the vertical wavenumber of IIGW with negative tend to infinity as it approaches the * Jones' critical levels. Let's consider the wave solution w2(*) which corresponds to the wave propagating upwards to the Jones' critical level when uz*0. In the middle atmosphere the Richardson number J reaches 102 so that * and * take almost * and 0, respectively, from (28). From. (31), (32) and (45), the following relations can be obtained near the Jones' critical levels: Since we assume that the wave source is sufficiently far from the critical levels, we can write as This formula is valid in the case of uz*0, too, as long as *(=l/k) is negative. The region where the LGO solutions (31) and (32) are valid is given by (43). If ZT exceeds this region, the formula (55) cannot be used. The turbulence layer thicknesses associated with non-inertial gravity waves derived from (52) are nearly one order larger than those associated with IIGW's derived from (55) for the mesoscale waves in the middle atmosphere when we use a common vertical wave amplitude. For IIGW's with positive *, the turbulence layers are restricted by the presence of the turning levels. In the middle atmosphere the distance between the Jones' critical level and the turning level is very small for the mesoscale waves (less than 10m). Eventually the turbulence may saturate within the thin region. The most striking difference between (55) for IIGW and (52) for non-inertial gravity wave is that the former is dependent on the horizontal wave structure, that is, *(=l/k), and the latter is dependent on the basic stratification, that is, N. It is interesting that the turbulence layer thickness for IIGW is independent of the atmospheric stratification. It seems difficult to detect observationally which kind of wave, IIGW or non-inertial gravity wave, may contribute to form the turbulence layer actually. Turbulent breakdown associated with the Jones' critical level seems important and meaningful only when the wave source is very close to the Jones' critical level. When the wave source is remote from the Jones' critical level, conventional mechanism based on non-inertial gravity wave is likely to work since the major part of wave would have damped out till the wave reaches the Jones' critical level. Some preliminary prospects on these problems in the actual stratosphere are presented elsewhere (Yamanaka and Tanaka, 1984). 6. Conclusions In this paper we demonstrate how IIGW's propagate and break in the vicinity of the Jones' critical levels by using exact and LGO solutions.

12 12 Journal of the Meteorological Society of Japan Vol. 62, No. 1 Our results are summarized as follows: (i) When the inertial effect associated with the earth rotation is incorporated, an IIGW has a pair of critical levels of Jones type (*=*) and a pair of turning levels (*=*). The latters are located inside the formers. WKB method breaks near the critical levels so that LGO method developed by Olver (1974) was applied to obtain a reasonable dispersion relation (40). (ii) When an upward propagating IIGW from below arrives at the lower critical level, a valve effect appears. If wave-front of IIGW runs from northeast to southwest (*0), the wave is absorbed partly just below the Jones' critical level and is reflected partly by the factor of exp (- 22* * ) where *=l/k. On the other hand, when the wave-front runs from northwest to southeast (*0), the wave once passes through the critical level and is reflected totally by the turning level. Then it goes down to the critical level and is partly attenuated just above the critical level and is reflected partly by the same factor as above. (iii) Connection formulae based on the exact solutions on the singularities (*=0, *, *) lead to reflection rates of two categories: one is approximately given by exp (- 2*) in the far regions from the critical levels and the other is approximately given by exp (- 2* * ) near the critical levels. The former is characterized by non-local and hence gradual attenuation or reflection but the latter is local near the Jones' critical levels. Strictly speaking, the former must include the latter as is shown in (42). However, the contribution of the local attenuation or reflection to the non-local one becomes very small when * is much larger than unity like in the middle atmosphere. (iv) Generation of the turbulence layers is caused by the breakdown of IIGW just outside the Jones' critical level in the case of *0. The turbulence layer thickness of IIGW is proportional to Acknowledgements The present investigation is a part of Master thesis of the first author (MDY) at the Nagoya University in We wish to thank Professor A. Ono for encouragements and Professor T. Matsuno for valuable discussions and suggestions throughout preparation of this paper. Thanks are extended to Drs. F. P. Bretherton, R. Grimshaw (one of the referees), K.-K. Tung, Y. Matsuda and S. Miyahara for suggestive comments and critics. We are also indebted to Yumiko Suzuki for typing the manuscript. Appendix A Notations The notations used in this paper are listed below, otherwise defined in each case: x, y, z : eastward, northward and upward coordinates t : time u : basic zonal velocity dependent on z u, *, w : eastward, northward and upward components of perturbed velocity,p : basic density and pressure * which is about one order smaller than that of non-inertial gravity wave for the mesoscale domain. Therefore, the turbulence layers caused by the Jones' critical level breakdown might be smeared by the thicker turbulence layers caused by the non-local breakdown. It would be difficult to detect such turbulence layers observationally in the middle atmosphere.

13 February 1984 M. D. Yamanaka and H. Tanaka 13 Appendix B Treatment far *=0 In the case that *=0, the governing equation (6) turns to a simple form: We have the solutions of Eq. (A1) and the resulting dispersion relation as follows: Note that (A2) are the same as a pair of the fundamental solutions around *=*, i.e., (19e) and (19f). In this case we have the formulas for analytical continuations of the solutions (A2) These lead to the well-known attenuation factor exp (- 2*) of non-inertial gravity waves across the critical level (*=0) derived originally by Booker and Bretherton (1967). From consideration in the last paragraph we find that (A4a) and (A4b) correspond to the formal connection formulae between far-away solutions for IIGW. Appendix C Treatment for l=0 The solutions and w1(*) are w2(*)not independent each other when l=0. Putting *=0 in w1(*) in (19c) and w2(*) in (19d), we have where *(*) is Psi function defined as *(*)*d in */d* (see Olver, 1974). Note that IIGW is evanescent inside both Jones' critical levels, i.e.,. Also note that the LGO method breaks * * when l=0. Appendix D Liouville-Green method developed by Olver (LGO method) The basic principle of LGO method is described below (see Olver, 1974): (i) An ordinary differential equation of second order: has the following fundamental solutions, (A8a) or (A8b), for a holomorphic region (*1, a2) by separating the coefficient of Eq. (A6), q(*): and for *(*)*0 and for *(*)*0 The other fundamental solution can be expressed by the Frobenius' method as where Separation of q into * and g is arbitrary. The approximate solutions can be obtained only when the error factor * is sufficiently small. (ii) When a boundary a2 is a regular singularity, q in Eq. (A6) and the separated functions are written as It is necessary and sufficient for convergence of

14 14 Journal of the Meteorological Society of Japan Vol. 62, No. 1 Va,*(F) that In the conventional WKB method, q cannot be separated, which is a special case of LGO method putting *=q and g=0. Theorem (ii) implies that such a conventional WKB method breaks near any singularity. Applying the LGO method shown in (A8b) we obtain the asymptotic solutions of IIGW near =-* as * Appendix E Examples of LGO method For non-inertial and two-dimensional gravity waves we have the following equation putting =0 and l=0 in Eq. (6): * Substituting above expression into (10) we have The exact solutions of Eq. (A12) are given by but the conventional WKB method provides the approximate solutions and the resulting dispersion relation The exact solutions and the WKB ones are different especially when J is close to 1/4 (cf. Bretherton, 1966; Lindzen et al., 1980). On the other hand, the LGO method provides the same solutions as exact ones: The solution (A19) is valid also for *=* as long as l is not zero. The LGO method is more effective for IIGW's than for non-inertial gravity waves. In fact, the actual values of J in the middle atmosphere is about 100, so that the error of (A15) derived by the WKB method is 0.25% or less. In the case of IIGW the local dispersion relation derived by the conventional WKB solutions for Eq. (9) is as follows: separating J/*2 in Eq. (A12) into (J-1/4)/*2 and l /(4*2). Associated dispersion relation must be written as Multiplying (*2 - *2)2 to (A20) results in The coefficient of the standard form of IIGW equation (9), can be expanded around a regular singularity *-*: The local dispersion relations based on the LGO method shown in (39) and (40) can respectively be modified as follows: + small terms. Therefore, theorem (ii) implies that, if l*0, we can separate the coefficient into two parts:

15 February 1984 M. D. Yamanaka and H. Tanaka 15 Appendix F Group velocity near Jones' critical level The local dispersion relation (40) can be modified as We can easily find from (A21a)-(A21c) that When IIGW's are propagating in the far region from the critical levels, the WKB method is of practical use (e.g., Tanaka, 1983a). Both sides of (A21a) do not coincide each other when uz*0, the left-hand side is l2*2uz2 and the righthand side is (k2+l2)f2uz2. On the other hand, both sides of (A21c) based on the LGO method are quite identical at the Jones' critical levels. The WKB method, thus, breaks in the vicinity of the Jones' critical levels. From (A21a) and (A21c), we obtain the following asymptotic expressions of the vertical wavenumber, m, around a singularity *=-*: where * is an angle between the x-coordinate and the direction normal to the wave-front and kh is defined by k=khcos*, l=khsin*. Partial differentiations of both sides of (A26) by m and kh lead to the vertical and horizontal components of group velocity relative to the basic flow, respectively, where M is given by Then the zonal and meridional components of group velocity are written, respectively, as Hence, as *-*, the LGO method gives a finite value of m. The same result is obtained around the other singularity *=*. The first term on the right-hand side of (A23a) is responsible for breakdown of the WKB method. The WKB method breaks in the region: where s is given by If we assume that J=100, we have *0.01, so that the regions defined by (A24) are very thin. However, one can never find the valve effect of GW so long as the conventional WKB method II is used. We find that the group velocity relative to the basic flow, (UgH, Wg), is perpendicular to the wavenumber vector, (kh, m) which is parallel to the wave-front. Therefore, one who moves with the basic flow observes IIGW packets moving along the wave-fronts. References Barat, J., 1975a: Etude experimentale de la structure du champ de turbulence Bans la moyenne stratosphere. Compt. Rend. Acad. Sci. Paris, B280, , 1975b: Une methode de mesure directe du taux de dissipation d'energie turbulence dans la stratosphere. Compt. Rend. Acad. Sci. Paris, B281, : -, Some characteristics of clear air turbulence in the middle stratosphere. J. Atmos. Sci., 39, Booker, J. R., and F. P. Bretherton, 1967: The critical layer for internal gravity waves in a shear flow. J. Fluid Mech., 27, Bretherton, F. P., 1966: The propagation of groups

16 16 Journal of the Meteorological Society of Japan Vol. 62, No. 1 of internal gravity waves in a shear flow. Quart. J. Roy. Meteor. Soc., 92, Cadet, D., 1977: Energy dissipation within intermittent clear air turbulence patches. J. Atmos. Sc., 34, , and H. Teitelbaum, 1979: Observational evidence of internal inertio-gravity waves in the tropical stratosphere. J. Atmos. Sci., 36, Eady, E. T., 1949: Long waves and cyclone waves. Tellus, 1, Geller, M. A., H. Tanaka, and D. C. Fritts, 1975: Production of turbulence in the vicinity of critical levels for internal gravity waves. J. Atmos. Sci., 32, Grimshaw, R., 1975: Internal gravity waves: critical layer absorption in a rotating fluid. J. Fluid Mech., 70, , 1980: A general theory of critical level absorption and valve effects for linear wave propagation. Geophys. Astrophys. Fluid Dynamics, 14, Hearth, D. F., E. Hilsenrath, A. J. Krueger, W. Nordberg, C. Prabhakara, and J. S. Theon, 1974: The global structure of the stratosphere and mesosphere with sounding rockets and with remote sensing techniques from satellites. Structure and Dynamics of the Upper Atmosphere, P. Verniani (ed.), Elsevier, Holton, J. R., 1982: The role of gravity wave induced drag and diffusion in the momentum budget of the mesosphere. J. Atmos. Sci., 39, Jones, W. L., 1967: Propagation of internal gravity waves in fluids with shear and rotation. J. Fluid Mech., 30, Kitchen, E. H., and M. E. McIntyre, 1980: On whether inertio-gravity waves are absorbed or reflected when their intrinsic frequency is Doppler-shifted towards *. J. Meteor. Soc. Japan, 58, Klostermeyer, J., 1980: Computation of acousticgravity waves, Kelvin-Helmholtz instabilities, and wave-induced eddy transport in realistic atmospheric models. J. Geophys. Res., 85, Lindzen, R. S., 1970: Internal equatorial planctaryscale waves in shear flow. J. Atmos. Sci., 27, , 1981: Turbulence and stress due to gravity wave and tidal breakdown. J. Geophys. Res., 86, B. Farrel, -, and K. K. Tung, 1980: The concept of overreflection and its application to baroclinic instability. J. Atmos. Sci., 37, Matsuno, T., 1982: A quasi-one-dimensional model of the middle atmosphere circulation interacting with internal gravity waves. J. Meteor. Soc. Japan, 60, Miyahara, S., 1976: Wave absorptions at critical levels in laterally bounded rotating fluids. J. Meteor. Soc. Japan, 54, : A note on the -, behavior of waves around the inertio-frequency. J. Meteor. Soc. Japan, 59, Olbers, D. J., 1981: The propagation of internal waves in a geostrophic current. J. Phys. Oceanogr., 11, Olver, F. W. J., 1974: Asymptotics and Special Functions. Academic Press, 572pp. Sawyer, J. 5., 1961: Quasi-periodic wind variations with height in the lower stratosphere. Quart. J. Roy. Meteor. Soc., 87, Tanaka, H., 1975: Turbulent layers associated with a critical level in the planetary boundary layer. J. Meteor. Soc. Japan, 53, : Application of WKB -, theory to turbulence layers in the vicinity of critical levels. J. Meteor. Soc. Japan, 60, a: Momentum flux -, divergences associated with inertio-gravity and internal gravity waves in the middle atmosphere with viscosity. J. Meteor. Soc. Japan, 61, , 1983b: Turbulence layer thickness in the stratosphere under the presence of viscosity and Newtonian cooling. J. Meteor. Soc. Japan, 61, Tokioka, T., 1970: Non-geostrophic and non-hydrostatic stability of a baroclinic fluid. J. Meteor. Soc. Japan, 48, Woodman, R. F., P. K. Rastogi, and T. Sato, 1981: Evaluation of effective eddy diffusive coefficients using radar observations of turbulence in the stratosphere. Handbook for MAP, 2, Yamanaka, M. D., and H. Tanaka, 1984: Multiple "gust layers" observed in the middl e stratosphere. Dynamics of the Middle Atmosphere, J. R. Holton and T. Matsuno (ed.) TERRAPUB, Tokyo,

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