On the propagation and dissipation of gravity-wave ... - CiteSeerX

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On the propagation and dissipation of gravity-wave spectra through a realistic middle atmosphere by C.D. WARNER and M.E. MCINTYRE Centre for Atmospheric Science, Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge, England, CB3 9EW

16 February 1996

(To appear in the Journal of the Atmospheric Sciences)

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ABSTRACT We investigate the one-dimensional propagation of a spectrum of gravity waves through a realistic middle atmosphere, separating as far as possible the propagation-invariant aspects from the more empirical wave-breaking and other nonlinear aspects. The latter are parameterized by a simple broadband spectral saturation criterion, but the conceptual framework allows for other wave-breaking parameterizations. An upward-propagating initial or \launch" spectrum is prescribed in the lower stratosphere. The propagation aspects are handled with careful attention to the mappings and their Jacobians between spectral spaces. Results for several test cases produce realistic behaviour, including cases where some of the waves are back-re ected, as in the summer stratosphere, with much of the spectrum propagating conservatively through substantial altitude ranges. Any launch spectrum can be used in the computational scheme; for de niteness we concentrate on the model spectrum of Fritts and VanZandt, but also carry out sensitivity tests in which the shape and total energy are varied. Other sensitivity tests include varying the steepness of the saturation criterion. The shapes and magnitudes of the computed pro les of wave-induced force, as a function of altitude, are sensitive to some of these changes, especially to the asymptotic shape of the launch spectrum at the smallest values of vertical wavenumber m, about which there is little direct observational evidence. However, the maxima and minima of the pro les are located at similar altitudes in each case. Besides pointing to ways of improving gravity-wave parameterization schemes for general circulation models, the results may help to tighten observational constraints on spectra for small m.

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1. Introduction The transport of momentum and angular momentum by gravity waves is important for the global circulation of the middle atmosphere, particularly the mesospheric branch, as is well known. At altitudes where gravity waves are breaking or otherwise dissipating, they deposit momentum ux, causing an altitude-dependent wave-induced force exerting a negative or retrograde torque in the winter mesosphere and a positive or prograde torque in the summer mesosphere. The response is a reduced zonal wind in the mesosphere and a wave-driven \gyroscopic pumping" (e.g., Holton et al. 1995) that feeds air from the summer mesosphere and tropical upper stratosphere into the winter mesosphere, in solstice conditions, and hence downward into the winter stratosphere. General circulation models (GCMs) must try to represent such eects correctly, especially because eects of the wavedriven pumping can be signi cant at much lower altitudes, both on tropical upwelling and on polar downwelling and hence, for instance, on the photochemical history of air from the troposphere and on temperatures and chemistry in the Antarctic polar vortex (e.g., Haynes et al. 1991, 1996; Garcia and Boville 1994; Holton et al., op. cit.; McIntyre 1995). Gravity waves cannot generally be resolved in GCMs, and so the eects of the waves must be parameterized. How to do this is a major problem because real gravity-wave elds are far more complicated than can be modeled in detail, not least the non-orographic gravity waves now known to be important. But the non-orographic, or presumably non-orogarphic, gravity-wave energy spectra that are derived from observations of temperature and horizontal velocity have a slope at large vertical wavenumber m that often, though not always, tends to be roughly independent of time, place and altitude (e.g., among many publications, Allen and Vincent 1995; Bacmeister et al. 1996; Dewan 1994; Dewan and Good 1986; Fritts and Lu 1993; Fritts and VanZandt 1993; Smith et al. 1987; VanZandt 1982). It seems plausible that this observed spectral slope can be partly explained in terms of saturation by wave breaking (Dewan and Good, op. cit.; Smith et al., op. cit.), often thought of as resulting from local convective or Kelvin{Helmholtz instabilities (e.g., Fritts and Rastogi 1985) and probably involving, in reality, not only such instabilities but also a whole range of nonlinear eects including triad interactions and various Doppler-spreading

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eects (e.g., Bretherton 1969; McEwan 1971; Broutman and Young 1986; Hines 1991a, 1991b, 1991c, 1993, submitted 1996). The wave breaking, regardless of detail, tends to limit the growth of the large-m part of the energy spectrum to a ceiling roughly / m?3. The limitation seems robust in the sense that a shallower slope, say m?2 or less, over a substantial range of wavenumbers, would imply large-m wave components whose amplitudes exceed thresholds to known wave-breaking mechanisms by enormous factors, well outside the bounds of physical plausibility. This argument assumes that the large-m components are more like propagating waves than fully developed three-dimensional turbulence. Such turbulence would show m?5=3 energy spectra and would correspond to a far stronger stirring of the atmosphere, implying unrealistically small coarse-grain Richardson numbers, Ri; m?5=3 is not observed except at vertical scales small enough to permit locally small Ri. Practical GCM spectral gravity-wave parameterization schemes such as those suggested by Fritts and Lu (1993), and by Hines (submitted 1996), have so far tended to describe gravity-wave spectral evolution in terms of the one-dimensional propagation and saturation of a spectrum through sequential altitude increments, i.e., in terms of upward propagation only, neglecting, for instance, nonhydrostatic eects and downward back-re ection. Coriolis eects are also neglected, partly or wholly. The upward evolution of the spectrum, and the extent to which gravity-wave breaking occurs, are modeled using various further simplifying assumptions, incorporating the empirical aspects in various ways. In the work of Fritts and Lu, an invariant spectral shape is imposed even under conservative Doppler-shifting. In this paper we look closely at what is involved in describing the spectral evolution in such one-dimensional schemes, using fewer approximations and assumptions. In particular we try to separate clearly, as far as possible, the theoretically justi able from the empirical aspects of our knowledge of gravity wave propagation and dissipation, under conditions typical of the middle atmosphere. The resulting conceptual framework can accommodate a hierarchy of wave-breaking models, and should be useful in future studies. By separation we mean conceptual separation | keeping clear the distinction between well understood

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aspects such as linear, conservative propagation, and ill-understood aspects such as the multifarious nonlinearities involved in wave breaking. In the rst place, we use more fully what is known theoretically about linear, conservative propagation, avoiding the neglect of Coriolis and nonhydrostatic eects. Conservative propagation can reasonably be assumed to dominate the evolution of waves that are propagating fast and not breaking, as may happen under certain conditions such as Doppler-shifting to large intrinsic wave frequencies !^ by the background wind, signi cant for instance in the extratropical summer stratosphere. Conservative propagation can lead to large changes in spectral shape and, when !^ increases as far as the buoyancy frequency N (Brunt{Vaisala frequency), also to back-re ection of some spectral components that might otherwise contribute to the wave-induced force. In the second place, when the waves are breaking, especially when breaking strongly, we want to bring in what is known empirically. In this paper, the empirical model of wave breaking is simply to impose as an upper bound the saturated portion of the spectral shape, usually but not always taken to incorporate the m?3 slope already mentioned. Other, more complicated wave-breaking models, such as that proposed by Hines (1991a, 1991b, 1991c, 1993, submitted 1996) and its variant elaborated by Medvedev and Klaassen (1995), could be used, but the choice of nonlinear eects to be included or excluded is a profoundly dicult question to which no de nitive answer is presently available | as well illustrated, for instance, by the work of Broutman and Young (1986) on unsteady Dopplerspreading dynamics. In this paper we stay with the simple saturation model but test the sensitivity of our results to changes in the spectral slopes of the saturation criterion. Not surprisingly, the shapes and magnitudes of the computed pro les of wave-induced force, as a function of altitude, are aected to varying degrees by such changes. But the altitudes of maxima and minima of the pro les are relatively insensitive to changing the steepness of the saturation criterion, including an m?1 \vertical cuto" saturation spectral shape and replacement of the usual !^ ?5=3 dependence with !^ ?1, the latter possibly a more consistent model for a saturated energy spectrum, where !^ is again the intrinsic wave frequency. The use of m?1 partially mimics Hines' suggestion that wave breaking can be parameterized as annihilating all components with vertical wavenumbers m exceeding some value.

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The framework developed here is exible enough, then, to accomodate future developments in both theory and observation. The spectrum is initiated from a \launch altitude" in the lower stratosphere. For this purpose we use the empirical model spectrum of Fritts and VanZandt (1993), though our sensitivity tests suggest that alternative shapes of launch spectrum (e.g., Sidi 1993) could have been used to obtain wave-induced force pro les with maxima and minima at similar altitudes, albeit with dierent detailed magnitudes and shapes. The greatest launch-shape sensitivity is to the asymptotic shape for small m, a point that could be exploited to put observational constraints on the small-m behavior. The gravity-wave motions are described in terms of the energy content of each propagationinvariant spectral element and the associated vertical ux of pseudomomentum. Each such spectral element, which we can think of as a notional \generalized wave packet" or set of wave packets, has its intrinsic frequency !^ Doppler-shifted by the variation in background wind with altitude z. Each spectral element's vertical wavenumber m varies both with !^ and with the buoyancy frequency N (Brunt{Vaisala frequency) of the background state. The dispersion relation and pseudomomentum ux are used in their full forms incorporating Coriolis and non-hydrostatic eects; the latter are signi cant for the ltering of the wave eld by back-re ection in the extratropical summer stratosphere. The paper is laid out as follows. Section 2.1 describes our choice of launch spectrum. Section 2.2 establishes notation and recalls the relevant wave-propagation theory. Section 3 establishes the computational scheme for one-dimensional spectral evolution. Section 4 describes the results for a selection of examples including the summer and winter CIRA 1986 model atmospheres, and an example of real winds from temperatures centred on 28 November, day 331, of 1991 at 37.5N latitude and 0.0 longitude). Section 5 gives brief details of some sensitivity tests, including the eect of azimuthal directional resolution. Section 6 presents concluding remarks.

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2. Functional form for, and conservative propagation of, a gravity-wave spectrum 2.1 Wave-energy spectrum at launch altitude In most of what follows, the wave-energy spectrum at the launch altitude in the lower stratosphere will somewhat arbitrarily be chosen for convenience, though not from necessity, to have a standard form identical to that of Fritts and VanZandt (1993), except in the tropics where a slight adaptation is made to avoid a singularity at the equator. The energy spectrum is a function of vertical wavenumber m, intrinsic frequency !^ and azimuthal direction of propagation :

E^ (m; !^ ; ) = E0 A(m) B(^!) () ;

(1)

with E0 a constant and A(m), B(^! ) and () all normalized so as to integrate to unity, implying that Adm, Bd!^ as well as d are dimensionless, with  in radians. The normalized vertical wavenumber spectrum is given by s =ms+1 m A(m) = A0 (s; t) 1 + (m=m )s+t ; (2)  where A0(s; t) = [(s + t)=] sin[(s + 1)=(s + t)] (3) and m, s and t are positive constants chosen empirically; m is the \characteristic" vertical wavenumber, which is close to, but not quite equal to, the wavenumber corresponding to maximum A(m). In this paper, as in Fritts and Lu (1993), s = 1 and t = 3 are taken as standard, making A0(s; t) = 4=. For a launch altitude in the lower stratosphere, 2=m will typically be 1 to 3 km (e.g., Fritts and VanZandt, op. cit.; Allen and Vincent 1995; Bacmeister et al. 1996). The normalized intrinsic-frequency spectrum, B(^!) is nonzero only in a range !^min < !^ < N where it is given by B(^!) = B0(p) !^ ?p ; (4) where

(

p?1 [1 ? (^!min=N )p?1 ]?1; for p 6= 1 (p ? 1)^!min : B0(p) = 1= ln(N=!^min ); for p = 1

(5)

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N is the buoyancy frequency, p is another empirical constant, and !^min = max(f; !^eq) ;

(6)

with f the Coriolis or inertial frequency and !^ eq a notional \characteristic equatorial minimum frequency", given by !^eq = eqLeq where Leq = (N= eq m)1=2 , the characteristic equatorial radius of deformation, with eq = 2:3  10?11m?1s?1, the equatorial planetary vorticity gradient. In this paper, as in Fritts and Lu (1993), the value p = 5=3 is taken as standard, though recent observational studies, e.g., Murayama (1993), suggest that a seasonally varying value of p, going to summer time values smaller than 5/3, may be more realistic. Values down to p = 1 are possible without gross violation of broadband saturation thresholds, as further 2=3 (1 ? (^ explained in section 3.2 below. When p = 5=3, B0(p) = (2=3) !^min !min =N )2=3 )?1. This paper assumes isotropy for the azimuthal-direction spectrum at launch, expressed by () = 0 = 21 : (7)

Finally, following Fritts and Lu (1993), we take for de niteness 2 (8) E0 = N m2 : as the standard value of E0, where is an empirical constant whose exact value cannot be regarded as greatly signi cant, but is usually taken, from typical observations, to be of the order of 10?1. Our use of such numbers implicitly assumes that the observed spectrum is dominated by upward-propagating waves, which is probably not accurately true; but the uncertainty is no worse than a factor of 2. This gives

N 2!^ ?p : (9) E^ (m; !^ ; ) = A0(s; t)B0 (p)0 m4 m 4 + m  Integration with respect to  trivially multiplies this expression by 2, and then with respect to m and !^ gives total wave-energy per unit mass, with dimensions of velocity squared. For the standard values s=1, t=3, p=5/3 we have, from (3), (5), and (7), 2=3 (1 ? (^ A0 (1; 3)B0(5=3)0 = 1:351  10?1!^min !min =N )2=3 )?1 :

(10)

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2.2 Conservative propagation of a gravity-wave spectrum The rst step is to look carefully at the eect on a gravity-wave spectrum of conservative propagation through realistic background-wind, density and buoyancy-frequency pro les. For this purpose, we regard the spectrum as a linear superposition of plane-wave spectral components, each of which has wavevector k = (k; l; ?m), with zonal component k, meridional component l and vertical component ?m, to make m > 0 for upward propagating waves with !^ > 0. The horizontal projection of k, or horizontal wavevector, is k0 = (k; l; 0) = k0(cos ; sin ; 0), with magnitude k0 = (k2 + l2)1=2 > 0. Each such plane wave spectral component has an absolute frequency, or frequency relative to the ground, denoted by !0. From standard wave theory (e.g., Gill 1982), the dispersion relation can be written 2 2 2 2 !^ 2 = f2 m 2 + N2 k0 2 ; (11a) k0 + m k0 + m or alternatively as m2 = N 2 ? !^ 2 ; (11b) k2 !^ 2 ? f 2 0

showing that f < !^ < N and expressing the fact that !^ is the frequency intrinsic to the local wave dynamics, i.e., the frequency Doppler-shifted to an observer moving with the local background wind U. The Doppler shift is given by

!^ = !0 ? k0
U = !0 ? k0U ;

(12)

where U is the component of U in the direction of k0. Because by de nition k0 > 0 and, for upward propagating waves, !^ > 0, a positive vertical shear @U=@z > 0 means that !^ will decrease in magnitude as altitude z increases, corresponding to an approach toward critical-level conditions. Because of Eqs. (11), a spectral component can be completely speci ed by a set of just three parameters. One such set consists of the azimuthal propagation direction , the magnitude k0 of k0, and the absolute frequency !0. For well known reasons (e.g., Lighthill 1978), these three parameters have the useful property of being invariant under conservative wave propagation in a time-independent background. We shall call the corresponding

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spectral space the (k0; !0; ) spectral space. Another such set of three parameters consists of , the vertical wavenumber m, and the intrinsic frequency !^ . The last two parameters are not propagation-invariant, but are more directly related to the intrinsic wave dynamics, and are the parameters commonly used in de ning model spectra like (9). We shall call the corresponding spectral space the (m; !^ ; ) spectral space. As in (9), we use a hat above the symbol for a spectral density to indicate that it is a function whose domain is the (m; !^ ; ) spectral space. We adopt the sign convention !^ > 0 for all waves, as well as k0 > 0, implying that m > 0 for upward-propagating waves since k = (k; l; ?m). In order to calculate the wave-induced force in a way that correctly takes account of Coriolis eects, we avoid the Reynolds-stress or u0 w0 type of approximation, and work directly in terms of wave pseudomomentum and pseudomomentum ux, the appropriate generalization of the Eliassen{Palm ux to a zonally asymmetric background. For theoretical justi cation see, e.g., McIntyre (1981, 1993 and refs.). In terms of E^ (m; !^ ; ), the spectral density for the horizontal component of the wave pseudomomentum itself is given by p^ = E^ (m; !^ ; )k0=!^ . The spectral density of the vertical ux of horizontal wave pseudomomentum, or, for brevity, \pseudomomentum ux", 0F^ p say, is then given by the group velocity rule

0F^ p = 0cgzp^ = 0cgzE^ (m; !^ ; )k0=!^ ; where the upward component cgz of the group velocity is given by ?


?

(13)


!^ 1 ? !^ 2=N 2 1 ? f 2 =!^ 2 (14) cgz = ?@ !^ =@m = m (1 ? f 2 =N 2 ) with the present sign convention. This is most easily veri ed by logarithmic dierentiation of (11b). Therefore, with the use of (11b), and suppressing explicit reference to the dependence on altitude z, we have

0F^ p(m; !^ ; ) = 0cgzE^ (m; !^ ; )k0=!^ = 0 N!^ E^ (m; !^ ; ) 2 =N 2 )1=2 (1 ? f 2 =!^ 2)3=2  k0  (1 ? ! ^  (1 ? f 2=N 2 ) k0 sgn(m) :

(15)

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In the case of the standard launch spectrum, (15) with (9) and (10) gives 2=3

0F^ p(m; !^ ; ) = 1:351  10?10 1 ? (^!!^min=N )2=3  min 2 2 1 = 2 2 ) (1 ? f =!^ 2)3=2   N (1 ? !^ =N (1 ? f 2 =N 2 ) m  m4 + m4 !^ ?2=3 (i cos  + j sin ) ; 

(16)

where the last factor represents the unit vector k0=k0 in terms of unit vectors i, j pointing toward  = 0 and =2 respectively. Figure 1 shows the (m; !^ ) dependence of 0F^ p in the standard case. In a one-dimensional model the wave-induced force per unit volume is the one-dimensional divergence of the total pseudomomentum ux

0Fptot = 0 = 0

Z 2

0 Z 2

0

d d

Z N !^ min Z N !^ min

d!^ d!^

Z

1

?1 Z

1

?1

dm F^ p(m; !^ ; ) dm jF^ p(m; !^ ; )j

 (i cos  + j sin ) sgn(m) ;

(17)

equivalent to a wave-induced force per unit mass

G(z) = ??0 1@ (0Fptot)=@z :

(18)

Unlike the wave-energy, the pseudomomentum ux is invariant under conservative propagation. So too is its spectral density, provided that we view it in the (k0; !0; ) spectral space, whose elements dk0 d!0 d are themselves propagation-invariant. That is, 0Fp(k0; !0; ), but not 0F^ p (m; !^ ; ), is propagation-invariant. This is the relevant generalization of the Eliassen-Palm theorem and follows from well known theoretical principles, as expressed by general theories of \wave-activity conservation" (e.g., Bretherton and Garrett 1968; McIntyre 1980 and refs.; Shepherd 1990). The relation between Fp and F^ p can be found from the fact that contibutions to Fptot must be obtainable by integrating either function. Therefore, for corresponding spectral elements dm d!^ d and dk0 d!0 d,

0F^ p (m; !^ ; )dm d!^ d = 0Fp (k0; !0; )dk0 d!0 d = 0F^ p (m; !^ ; ) J dk0 d!0 d ;

(19)

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dm d!^ d = J dk0 d!0 d ;

(20)

where the spectral Jacobian J is de ned by

J = @@((km;; !!^ ;;)) = @@((km;; !!^ )) 0 0 0 0 

@m = @k 0 Therefore





!0

@ !^ @!0





@m ? @! 0 k0



F^ p = J ?1Fp / J ?1

 k0

@ !^ @k0

 !0

:

(21) (22)

under conservative propagation. We evaluate J from the dispersion relation (11b) and the Doppler relation (12). The result can be reduced to a simple form that depends on U only implicitly through the Doppler-shifting of !^ , hence m:

J = km : 0

(23)

This simplifying feature is due to the special way U enters, via (12) only. U is a constant for the purpose of calculating J = @ (m; !^ )=@ (k0 ; !0), which by the standard properties of Jacobians is equal to @ (m; !^ )=@ (k0 ; !0 ? k0U ) = @ (m; !^ )=@ (k0 ; !^ ) = (@m=@k0)!^ = m=k0 by (11b). Figure 2b and 2c illustrate the behaviour of typical spectral elements in (k0; !0) space and their counterparts in (m; !^ ) space under Doppler-shifting. U , the component of background wind in the k0 direction, is changed from 0 ms?1 to 10 ms?1, toward critical-level conditions; as already noted, we de ne this to be positive shear. It Doppler-shifts !^ and m toward bottom right along the constant-k0 curves in (m; !^ ) space, Fig. 2a, implied by (11a) or (11b). Each rectangular element in (k0; !0) space is invariant under the change of U while the corresponding element in the (m; !^ ) space moves, changes shape and increases in area in accordance with (20), (23). This was checked independently by direct numerical estimation of the areas of the spectral elements shown in Fig. 2. Note also that the corresponding shrinkage of j0F^ pj is like m?1, not nearly enough to avoid an m?3 or steeper saturation threshold if m continues to increase. This correctly expresses the

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physical reality that the waves must break before critical-level conditions are reached, just as ocean-beach waves break before they reach dry land.

3. The Single-Column Numerical Model 3.1 Working assumptions We implicitly regard the troposphere and the neighborhood of the tropopause as a background source of a spectrum of non-orographic waves. Any launch spectrum can be used in the computational scheme; for de niteness, we use mainly the standard energy spectrum (9) with (10), launched at z=19 km, above the tropopause jets, and consider the modi cations that occur as the waves are propagated through conditions of changing background wind U(z), density 0(z), and buoyancy frequency N (z). As already indicated in (17) onwards, our numerical model is quasi-one-dimensional in the usual sense, like a GCM infrared radiation scheme. More precisely, we assume that we are parameterizing gravity-wave eects over a large enough surface area, of the order of a GCM grid size or larger, to justify ignoring horizontal propagation. To this extent, our numerical model is like the much simpler parameterizations used in GCMs. Departures from one-dimensional modeling would seem unjusti able given our present ignorance of the ne-scale geography of real wave sources. Wave propagation and breaking are treated as follows, keeping things simple as outlined in the introduction. We treat each nite spectral element in (k0 ; !0) space as an independently propagating \generalized wave packet" up to the moment of wave breaking. So we model spectral evolution as a sequential process of upward conservative propagation through each altitude increment, followed by wave breaking (if required) in that altitude increment, or back-re ection downwards if !^ is Doppler-shifted past N , or both. Thus there is a downward as well as an upward ux. The downward ux is handled by assuming that back-re ection is a conservative process, producing a standing wave in the vertical making zero net contribution to (17) and hence to the wave-induced force. In physical reality, some of that downward ux may well be re ected back up, but we count it as already included in the assumed, observationally-motivated, upward propagating launch spectrum.

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If the saturation threshold of pseudomomentum ux spectral density is exceeded, then wave breaking is assumed to be occurring and the pseudomomentum spectral density is simply chopped down to the saturation threshold, to be speci ed in section 3.2. This is done separately for each spectral element
k0
!0
. In other words, we apply a broadband saturation threshold in the simplest possible way. This is simpler, in particular, than the saturation scheme proposed by Hines (1993, submitted 1996), which makes the threshold depend on the spectral shape. We could use a saturation criterion more like that of Hines, but, for the reasons indicated in section 1, there are unresolved questions about how close to reality either scheme might be. A further simpli cation in our scheme is that we neglect the contribution of any back-re ected waves to the saturation threshold. In summary, then, our model propagates each nite spectral element conservatively, and independently of all the other elements, unless its pseudomomentum- ux spectral density exceeds the \saturation" threshold or its intrinsic frequency moves outside the permitted range (f < !^ < N ). If !^ exceeds the buoyancy frequency, N , then the spectral element is assumed to have been re ected back down and hence not to contribute to the wave-induced force above launch altitude. Back-re ection can and does result both from changes in background N and from Doppler-shifting by changes in background wind U. Conversely, if !^ decreases toward f then saturation must occur, annihilating the waves before !^ reaches f . To avoid a singularity in the tropics we would need to assume, slightly arbitrarily, that the !^min of (6) replaces f for this purpose, though this remains to be tested because tropical cases are not considered in this paper. Implicit in the numerical model is the fact that waves propagating in negative background wind shear (i.e., away from critical-level conditions) tend to be nearly conservative. We have not discussed infrared radiative damping, but such damping can be shown to be unimportant here (Haynes and Ward 1993; O. Buhler, personal communication). Conversely, waves propagating in positive background wind shear, toward critical-level conditions, tend to be saturation limited.

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3.2 Saturated spectral threshold or chopping function The saturation threshold E^  E^ S in most of the examples presented here is that given by VanZandt and Fritts (1989), which is equivalent to taking p = 5=3 in

E^ S(m; !^ ) = A0(s; t)B0 (p)0 N 2!^ ?pm?3 ;

(24)

though we shall use slightly dierent values of  10?1. The choice of p = 5=3 yields the same !^ ?5=3 dependence of the saturation energy spectrum as in the standard launch spectrum (9) with (10). We regard this choice of saturation threshold as mainly empirical, motivated by observed spectra. However, the m?3 factor has some theoretical support, as mentioned in section 1. For instance, following Dewan and Good (1986), one can plausibly make an assumption of \broadband self-similarity", saying that there is no distinguished vertical length scale m?1 throughout a large range of m and hence that, in order to go from spectral densities to typical amplitudes, one requires spectral integration not over xed intervals in m, but rather over octaves or other logarithmic increments. This would be true, for instance, if the wave eld consisted of many wave packets of greatly varying sizes, each having a xed number of wave crests. Dewan and Good (1986) and Smith et al. (1987) combine this idea with simple estimates, assuming f 2  !^  N 2, for the wave amplitude threshold associated with the possible convective instability of a monochromatic gravity wave: ju0j  N=jmj : (25) As they point out, this holds also, in order-of-magnitude terms, for low frequencies near f , even though its physical meaning is then the threshold for Kelvin{Helmholtz rather than convective instability, involving the Richardson number Ri  1=4 hence a slightly dierent numerical factor: Ri = N 2=j@u0=@zj2 = N 2 =jmu0j2  1=4. The nal step is to equate the variance corresponding to the square of (25) to the integral of power spectral density over a logarithmic increment in vertical wavenumber. This predicts E^ S / m?3. The frequency dependence is more problematical, and less clearly constrained by observations. Dewan (1994) argues for the !^ ?5=3 frequency dependence of the standard model energy spectrum by assuming a Kolmogorov cascade, in place of broadband saturation across the range of !^ . On the other hand, as with m, one can make an argument from

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broadband self-similarity in !^ , saying that there is no distinguished time scale !^ ?1. Such an argument predicts not !^ ?5=3 but !^ ?1, corresponding to the !^ 0 dependence for a single monochromatic wave at its saturation threshold, that is, to the fact that (25) depends on m alone and not on !^ . On the other hand, from what is known about the physics of wave breaking, such power laws certainly oversimplify matters, at least near the highest frequencies, !^  N . To keep the number of examples manageable, we chose somewhat arbitrarily to use !^ ?5=3 as the standard assumption but also use !^ ?1, with suitable renormalization, in one of the sensitivity tests in section 5. Furthermore, and again somewhat arbitrarily, we neglect the contributions of back-re ected waves to the saturation threshold. There is probably some compensation here because the assumed threshold may well be too low near !^ = N . The saturation threshold (24) has the form of an isotropic, i.e., -independent, ceiling on E^ (m; !^ ). In reality, this ceiling would not only be fuzzy, but it would be neither rigid nor isotropic. Rather, it would probably tend to move upwards for angles in, for instance, a single narrow sector when wave directions are con ned to that sector. Our assumption that it is rigid and isotropic is a deliberate oversimpli cation, but arguably a reasonable one in the present state of knowledge.

3.3 Computational schemes Grids of spectral elements
k0
!0
 and
m
^!
 are de ned for each of n azimuthal directions j = 2j=n (j = 1; ::n), with
 = 2=n. In this paper we take n =2, 4 or 8. The model deals with contributions to (17) from individual sectors
, with () piecewise constant in sectors. At each altitude z there is a contribution 0Fj (z), say, from the j th azimuthal sector of directions j ?
=2 <  < j +
=2 centred on j and of size
, to the total pseudomomentum ux 0Fptot(z). This contribution is just the right hand R side of (17) with  replaced by j and d replaced by 2 sin(
=2). We chose, somewhat arbitrarily, to keep the product 2 sin(
=2)=
 constant for the values of
 used here. This arti ce allowed us to compare directly the spectra for diering numbers of azimuthal directions without the need for renormalisation. We note that 2 sin(
=2)=
 ! as
 ! 0 and we chose 2 sin(
=2)=
 = 1:047  10?1.

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In summary, the saturation threshold (24) requires that, for each j ,

0Fj  0F^ pS(m; !^ )

(26a)

where by de nition 2 2 1=2 2 2 3=2 0F^ pS (m; !^ ) = A0 (s; t)B0(p)0 0N (1 ? !^ =N(1 ?) f 2(1=N?2f) =!^ ) (26b)  m13 !^ ?q  2 sin(
=2) : This is the appropriate chopping function for 0Fj . In the standard case, q = p ? 1 = 2=3, because of the factor !^ in (15), and similarly q = 0 in the case where the energy spectrum / !^ ?1. Note that 0F^ pS decreases exponentially with altitude because of the factor 0 = 0(z). We can visualize the imposition of the saturation threshold as lowering a chopping-function-shaped grid over the pseudomomentum ux spectrum, slicing o any bits of pseudomomentum ux that stick through the grid thereby forcing the spectrum at suciently large m to coincide with the saturated spectral shape. Indeed, the shape of the large-m pseudomomentum ux spectrum can dier signi cantly from the saturated shape only if it has propagated through regions with sucient negative background shear to keep a signi cant part of the spectrum below breaking amplitude, despite the factor 0. Conversely, positive shear tends to provoke saturation. This is illustrated in Fig. 3, which illustrates the eect of chopping; cf. section 4.1.

Values of Fp and F^ p are initialized at some lower-stratospheric launch altitude, typically 19 km. The spectral elements and their Fp and F^ p values are propagated through a series of altitude increments through changing background wind, density and buoyancy frequency as described in sections 3.1 and 3.2. For each azimuthal sector (j ? 21
; j + 21
), values of jFpj are stored on a regular rectangular grid in (k0; !0) spectral space. However, as illustrated in Fig. 2, a rectangle in (k0; !0) space does not transform to a rectangle in (m; !^ ) space, and so we use linear interpolation together with (22) and (23) to derive jF^ pj values on a regular rectangular grid in (m; !^ ) spectral space. We call this the \k0!0 scheme". As an occasional check, we also used a second, less convenient but more accurate scheme in which jF^ pj values on a regular rectangular grid in (m; !^ ) space at a speci c nal altitude

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are obtained without interpolation. This is done by initialising on a non-rectangular, sheared (k0; !0) grid that will correspond to the rectangular grid of (m; !^ ) values at the nal altitude. This \m!^ scheme" produces results that closely resemble those from the k0!0 scheme, but resolve better the small-m, small-^! corner, and the large-m, large-^! corner, of spectral space, at the cost of having to re-run for each nal altitude. As a further occasional check, we also used a third scheme in which jFpj values are obtained on a \stretched" grid in (k0; !0) space. Using a stretched grid allowed us to concentrate resolution at small k0 values without neglecting much larger k0 values. Again, this stretched grid scheme produces results that closely resemble those from the (k0; !0) scheme, but resolve better the small-m, small-^! corner and the large-m, large-^! corner of spectral space at the cost of using a spectral element that can vary substantially in area with location in both the (m; !^ ) and the (k0 ; !0) spectral spaces. Such checks are desirable because of the severe numerical resolution diculties that arise from the Doppler-shearing of spectral elements illustrated in Fig. 2. The contribution 0Fj (z) from the j th azimuthal sector to the total pseudomomentum

ux 0Fptot(z) has upward and downward parts, say Fj (z) = F"j (z) + F#j (z). The downward part represents the back-re ected ux from all altitudes above z and, with our sign convention k = (k; l; ?m), corresponds to the negative half of the integration with respect to m in (17). It proves convenient to compute F"j (z) and F#j (z) separately for all z and subsequently to compute the wave-induced force as n X @ (0 F"j ) @ (0 F#j ) G(z) = Gj = ? 1 + @z @z 0 j =1 j =1 n X

!

;

(27)

which is an azimuthally discretized version of (18). The zonal and meridional components 0Fx and 0Fy of 0Fptot(z) and the zonal and meridional components Gx and Gy of G(z) can then be calculated. They are (0Fx; 0Fy ) = and

n X j =1

G(z) = (Gx; Gy ) =

0jFj j (cos j ; sin j )

n X j =1

jGj j(cos j ; sin j ) :

(28) (29)

?17?

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The total wave-energy is given by

E^ tot(z) =

Z 2

0

d

Z N !^ min

d!^

Z

1

?1

dm E^ (m; !^ ; ) ;

(30)

where E^ (m; !^ ; ) is obtained from F^ p(m; !^ ; ) using (15). At each altitude z there is a contribution E^ j (z), say, from the j th azimuthal sector to the total wave-energy E^ tot(z). This contribution is just the right hand side of (30) with  replaced by j and, again R somewhat arbitrarily, for computational convenience, d replaced by 2 sin(
=2) rather than
; the dierence is best thought of as another discretization error arising from nite spectral elements
k0
!0
, and insigni cant in comparison with the empirical uncertainity in . E^ j (z) has upward and downward parts, say E^ j (z) = E^ j"(z)+ E^ j# (z), the latter representing the wave-energy associated with back-re ected ux from all altitudes and again corresponding to negative m. Thus the total wave-energy is greater than that due to upward propagating waves alone, being the scalar sum of positive-de nite contributions from the upward and downward parts. Conversely, the total pseudomomentum ux is less than that due to upward propagating waves alone, being the vector sum of signed contributions from the upward and downward parts.

4. Results All the illustrations presented here were computed with the k0!0 scheme and use 2 sin(
=2)=
 = 1:047  10?1. The CIRA 1986 model atmosphere examples use !min = f = 9:3744  10?5 rad s?1, corresponding to 40 N, and the real-winds examples use !min = f = 8:8782  10?5 rad s?1, corresponding to 37.5N. The spectra are shown using shaded-surface plots in (m; !^ ) space. Some of the detailed spectral shapes are visibly aected by the numerical resolution problems already referred to, arising from the shearing of spectral elements and from the very wide dynamic range of numerical magnitudes associated with conservative propagation over a dozen or more pressure scale heights, dominated by the smallest values of m  m . Unless otherwise stated, m = 2= 2 km = 3:142  10?3 rad m?1, and plots showing the low-m range extend to m = 6  10?4 rad m?1. Unless otherwise stated, the !^ range of each shaded surface extends from 0 to N (z). Some of the plots, e.g., Figures 5a and 5b, show spurious zero values near the high-m, high-^! corner because of the nite range of k0 values that can be handled numerically.

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4.1 Eect of changing only the background wind shear on the launch spectrum The rst example illustrates the eect of background shear alone. Figure 3 shows, in (m; !^ ) spectral space, with m running from 0 to 0.03 rad m?1, (a) the launch spectrum (16), (b) the resulting spectrum after conservative propagation through an increase in zonal background wind of 2 ms?1 (positive background shear), (c) the nal spectrum after wave breaking with chopped spectral elements highlighted, and (d) the same but oriented so as to show more clearly the chopped spectral elements. The chopping has taken a slice o the tail of the spectrum, as it should.

4.2 July, 40N, CIRA 1986 model atmosphere temperature and density pro les but no background wind The next example is for an arti cial model atmosphere with the July, 40N, CIRA 1986 model atmosphere temperature and buoyancy-frequency pro les shown in Fig. 4, but with no background wind. The standard spectrum was launched from an altitude of 19 km. Figure 5 shows shaded-surface plots of the low-m range of the pseudomomentum

ux spectrum at the following altitudes: 19 km, 69 km, 86 km and 102 km. These altitudes (see arrows in Fig. 4) were chosen to facilitate comparison with the corresponding case in which the background wind is included, as on the left of Fig. 4. Figure 5 shows how the parametrized wave breaking causes a progressive chopping-down of the high-m portion of the spectra. Note that, in Fig. 5a, the spectral peak near m = m is out of sight, about ve m-axis lengths to the left. Figures 5b{5d show successively more severe chopping, bringing the spectral peak back into the range shown. Small irregularities show the limitations of numerical range and resolution. The spectra exhibit a high-^! cuto due to back-re ection from the minimum in N (z) just above 60 km (right of Fig. 4). This shows clearly in Figs. 5c and 5d. The high-m, high-^! corner of Fig. 5b shows the spurious zero values mentioned at the start of section 4.

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4.3 July, 40N, CIRA 1986 model atmosphere The next example is for the July, 40N, CIRA 1986 model atmosphere of Fig. 4, including zonal winds and hence Doppler shifts. For each of four azimuths, northward, westward, southward and eastward (j =90, 180, 270, 360 i.e., mathematical, not compass, convention), spectra were launched from the zero of zonal background wind at an altitude of 19 km (lowest arrow in Fig. 4). For j =90 and 270 there are no Doppler shifts (k0 normal to U), and so the results are the same as in Fig. 5. Figure 6 shows shadedsurface plots of the eastward propagating pseudomomentum ux spectra, i.e., the spectra for j =360, at the following altitudes, corresponding to the arrows in Fig. 4: 19 km (zero background wind), 69 km (maximum westward background wind), 86 km (zero background wind), and 102 km (maximum eastward background wind). Figure 7 shows shaded-surface plots of the westward propagating pseudomomentum ux spectra (j =180) at the same altitudes. Consider rst the eastward propagating spectra of Fig. 6. Between altitudes of 19 km and 69 km (Figs. 6a{6b) the background wind shear is negative, in the sense de ned in section 3.1. In negative background wind shear, the spectral elements are Doppler-shifted to larger !^ and smaller m, and the change in the spectral Jacobian results in a decrease in the area of spectral space occupied by each spectral element and therefore in an increase in the spectral density. Some limited wave breaking nevertheless occurs. Note that the range of the vertical axis of Fig. 6b, the 69 km altitude shaded-surface plot, is ten times that of the other shaded-surface plots in Fig. 6 and that the increase in spectral density is therefore quite substantial. The increase in spectral density is particularly clear for those spectral elements which were Doppler-shifted to !^  N . As !^ ! N , approaching the top left corner of Fig. 2a, the spectral elements become vanishingly small, leading to the large spike in pseudomomentum ux seen in Fig. 6b. In reality, this spike is in nitely tall, but our numerical model smears it into a broader spike of nite amplitude. Those spectral elements that are Doppler-shifted past !^ = N are considered to have been back-re ected. Such Doppler-shifting of spectra to smaller m was also observed by Eckermann (1995), in radar data obtained under similar wind conditions to those discussed here. Between altitudes of 69 km and 86 km (Figs. 6b{6c), positive background wind shear results in

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strong wave breaking. As stated in section 2.2, in positive background wind shear the spectral elements are Doppler-shifted to smaller !^ and larger m, and the change in the spectral Jacobian results in an increase in the area occupied by each spectral element and therefore in a decrease in the spectral density. The position and area of each spectral element is again close to its value at launch, but values have been drastically chopped to the left of the peak. Between altitudes of 86 km and 102 km (Figs. 6c{6d), the background wind shear continues to be positive, the spectral evolution is dominated by wave breaking, and the spectral shape at large m stays close to that of the saturated spectrum (26b), though with the peak inexorably moving to the right. Figure 6d shows irregularities due to the extreme strain on numerical resolution for !^ near f . Now consider the westward propagating pseudomomentum ux spectra of Fig. 7. Between altitudes of 19 km and 69 km (Figs. 7a{7b) the background wind shear is now positive, and strong wave breaking results in the spectral shape at large m immediately tending toward that of the saturated spectrum (26b), and immediately putting the same strain on numerical resolution for !^ near f . Between altitudes of 69 km and 86 km (Figs. 7b{7c), negative background wind shear results in a redistribution of pseudomomentum

ux by Doppler-shifting toward the low-m, high-^! corner, but almost no wave breaking. The position and area of each spectral element is again close to its value at launch but values have again been drastically chopped, mainly before 69 km. Between altitudes of 86 km and 102 km (Figs. 7c{7d), the continuing negative background wind shear is insuciently strong to avoid wave breaking, and the high-m spectral shape once again tends somewhat towards that of the saturated spectrum (26b). Figure 8a shows the total westward propagating pseudomomentum ux 0jF2(z)j. (In the notation above (27), 0jFj (z)j for j = 2 or j = 2 =180.) Figure 8b shows the total eastward propagating pseudomomentum ux 0jF4(z)j as a solid curve, and its upward part 0jF"4(z)j as a dotted curve. The westward propagating waves are initially in positive background wind shear, but between altitudes of 69 km and 85 km, where the background wind shear for these waves is strongly negative, there is hardly any wave breaking and hardly any change in total pseudomomentum ux. The eastward propagating waves show hardly any change in total pseudomomentum ux over the altitude range 19 km to 30 km (the

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background wind shear for these waves being suciently negative to avoid wave breaking). Thereafter 0jF"4(z)j decreases, both because wave breaking sets in and also because signi cant back-re ection occurs. There is strong breaking and a rapid decrease in 0jF4(z)j above 69 km (corresponding to positive background wind shear). The wave-induced force

G is plotted as a function of altitude in Fig. 8c. For this example, the wave-induced force G is zonal because meridional background winds are set to zero and Gy = 0. The peak values of the wave-induced force are 90 ms?1day?1 eastward at an altitude of 80 km, and 350 ms?1day?1 westward at an altitude of 110 km. These values appear to be reasonable in order of magnitude. The total wave-energy Etot, including upward and downward propagating waves, is plotted as a function of altitude in Fig. 8d as the solid curve marked \s.c.m.", together with an empirical pro le derived from observational studies (Fritts and Lu 1993):



N (z) Eemp(z) = E0 N (0)

1=2

exp(z=HE ) ;

(31)

where HE = 2:3H with H the pressure scale height  7 km. In this case the spectral model comes fairly close to the empirical pro le, though with a slightly smaller energy scale height. Hardly any of the westward propagating pseudomomentum ux 0jF"2(z)j is back-re ected. The solid curve in Fig. 8a would almost exactly cover the dotted curve if plotted. However 38% of the launched eastward propagating pseudomomentum ux 0jF"4(z)j is backre ected (corresponding to 8% of the launched eastward propagating wave energy), because of the signi cant Doppler-shifting to large !^ values that occurs before the spectral evolution becomes dominated by wave breaking.

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4.4 January, 40N, CIRA 1986 model atmosphere The next example is for the January, 40 N, CIRA 1986 model atmosphere of Fig. 9. For each of four azimuths, northward, westward, southward and eastward (j =90, 180, 270, 360), spectra were launched from the local minimum of zonal background wind at an altitude of 24 km (arrow in Fig. 9). We saw exactly the same pattern as before, with allowance for the sign changes in U in shaded-surface plots of the pseudomomentum ux spectra (not reproduced here). Positive wind shear caused Doppler-shifting of spectral elements to larger m and smaller !^ , and strong wave breaking occured. Negative wind shear caused Doppler-shifting of spectral elements to smaller m and larger !^ , and much less wave breaking occured. As before, spectral element areas decreased in negative wind shear, resulting in greatly increased spectral density in (m; !^ ) spectral space, particularly as !^ ! N . As before, those spectral elements that were Doppler-shifted past !^ = N were considered to have been back-re ected. Figure 10a shows the total westward propagating pseudomomentum ux 0jF2(z)j, and its upward component 0jF"2(z)j. Figure 10b shows the total eastward propagating pseudomomentum ux 0jF4(z)j. As before, spectra propagating in negative shear are characterized by little or no decrease in total pseudomomentum ux, and by back-re ection, while spectra propagating in positive shear are characterized by a much greater decrease in total pseudomomentum ux, and by no back-re ection. The wave-induced force G is plotted as a function of altitude in Fig. 10c. For this example, the wave-induced force G is zonal because meridional background winds are set to zero and Gy = 0. The peak value of wave-induced force is about 60 ms?1day?1 westward at 80{90 km altitude, and has reached 160 ms?1day?1 eastward at 110 km altitude. Again, the order of magnitude is reasonable. The total wave-energy, Etot, is plotted as a function of altitude in Fig. 10d, together with the empirical pro le (31). The spectral model again comes fairly close to the empirical pro le, though again with a slightly smaller energy scale height.

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Hardly any of the eastward propagating pseudomomentum ux 0jF"4 (z)j is back-re ected. The solid curve in Fig. 10b would almost exactly cover the dotted curve if plotted. However 24% of the launched westward propagating pseudomomentum ux 0jF"2(z)j is backre ected, because of the signi cant Doppler-shifting to large !^ values that occurs before the spectral evolution becomes dominated by wave breaking.

4.5 Real-winds example The next example is for the real-winds example of Fig. 11, which consists of zonal background wind, meridional background wind, temperature, and buoyancy frequency pro les derived from ISAMS measurements (B. N. Lawrence, personal communication). The pro les are 24-day averages centered on day 311 (28 November) of 1991, at latitude 37.5N and longitude 0.0; the inertial frequency f at 37.5N is 8:8782  10?5 rad s?1. This particular location and date was chosen as an example of a strong meridional background wind component that varied signi cantly with altitude. For each of four azimuths, northward, westward, southward and eastward (j =90, 180, 270, 360), spectra were launched from 18 km altitude, indicated by the lowest arrow in Fig. 11. The spectra propagating in each of the four azimuthal directions were examined at altitudes corresponding to maxima and minima of the zonal and meridional background wind pro les. We see the same pattern of spectral behaviour as was observed for the climatological CIRA 1986 model atmosphere examples. Positive wind shear causes Doppler-shifting of spectral elements to larger m and smaller !^ , and strong wave breaking occurs. Negative wind shear causes Doppler-shifting of spectral elements to smaller m and larger !^ , and much less wave breaking occurs. As before, spectral element areas decrease in negative wind shear, resulting in greatly increased spectral density in (m; !^ ) spectral space. As before, those spectral elements that are Doppler-shifted past !^ = N are considered to have been back-re ected. For brevity's sake, we illustrate this by considering the evolution of the northward propagating spectrum only. Figure 12 shows shaded-surface plots of the northward propagating spectra at the following altitudes corresponding to the arrows in Fig. 11: 38 km, 54 km, 60 km, and 80 km.

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(The launch spectrum is that used in sections 4.2{4.4, so is not plotted). Between altitudes of 16 km and 38 km (launch spectrum and Fig. 12a), positive background wind shear results in strong wave breaking. Between altitudes of 38 km and 54 km (Figs. 12a{12b), negative background wind shear results in only limited wave breaking. Between altitudes of 54 km and 60 km (Figs. 12b{12c), more positive background wind shear results in further strong wave breaking and the spectral shape at large m tends towards that of the saturated spectrum (26b). Between altitudes of 60 km and 80 km (Figs. 12c{12d), further negative background wind shear results in only limited wave breaking, however, the enhancement of spectral density at large !^ and small m that can result from Doppler-shifting in negative background wind shear, and from the corresponding reduction in spectral element areas, is particularly clear at this altitude where a distinct spike is again visible. Figures 13a, 13b, 13c and 13d show the total pseudomomentum uxes 0jF2(z)j, 0jF4(z)j, 0jF1(z)j, and 0jF3(z)j propagating in each of the four azimuthal directions j = 2; 4; 1; 3 (j =180, 360, 90, 270 ) as solid curves. Figures 13a and 13d also show the upward propagating parts 0jF"2(z)j and 0jF"3(z)j as dotted curves. Again, the same general behaviour was observed as for the climatological CIRA 1986 model atmosphere examples. We illustrate this by considering the northward propagating case (Fig. 13c) in more detail. Up to an altitude of 38 km, the northward propagating waves are in positive background wind shear, there is strong wave breaking and 0jF1(z)j decreases sharply. Between altitudes of 38 km and 54 km, the background wind shear is negative, relatively little wave breaking occurs and 0jF1(z)j decreases more slowly. Between altitudes of 54 km and 62 km, the background wind shear is positive, strong wave breaking occurs and 0jF1(z)j decreases sharply. Finally, between altitudes of 62 km and 80 km, the background wind shear is negative, little wave breaking occurs and 0jF1(z)j continues to decrease, but more slowly. Figure 13d has a complementary shape re ecting the opposite signed shear felt by southward propagating waves. Figures 14a and 14b show. respectively, the zonal component Gx and the meridional component Gy of the wave-induced force G as functions of altitude. The peak value of Gx is 38 ms?1day?1 eastward at 72 km altitude. The peak values of Gy are 6.3 ms?1day?1

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northward at 56 km altitude, and 26 ms?1day?1 southward at 74 km altitude. The magnitude of the zonal component of wave-induced force is similar to the magnitude of zonal components of wave-induced force achieved with the CIRA 1986 model atmosphere examples of sections 4.2{4.4. The magnitude of the meridional component of wave-induced force is somewhat smaller. The total wave-energy, Etot, is plotted as a function of altitude in Fig. 14c, together with the empirical formula (31). The spectral model again comes close to the empirical pro le, although again with a slightly smaller energy scale height. Indeed, it is closer to the empirical pro le in this, arguably more realistic, case. Hardly any of the northward propagating and eastward propagating pseudomomentum

ux is back-re ected. The solid curves in Figs. 13b and 13c would almost exactly cover the dotted curves if plotted. However 70% of the launched westward propagating pseudomomentum ux is back-re ected, and 20% of the launched southward propagating pseudomomentum ux is back-re ected, because of the signi cant Doppler-shifting to large !^ values that occurs before the spectral evolution becomes dominated by wave breaking.

5. Sensitivity tests 5.1 Sensitivity to choice of launch spectrum and to choice of saturation function Sensitivity to the choice of launch spectral shape, launch spectral density and steepness of the saturation function has been tested in various ways. For brevity, we restrict attention to ve cases, identi ed by the letters A, B, C, D, and E, all for the July, 40N, CIRA 1986 model atmosphere shown in Fig. 4. Case A, the control case, is the standard case with m = 2=2 km, which has been discussed in detail in section 4.3. Case B has an identical launch spectral shape but with double the magnitude, i.e., (16) is multiplied by 2. Case C has m = 2=1 km, hence, for the small values of m that matter most, about one sixteenth of the magnitude of control case A. Case D uses the standard launch spectrum but a dierent and very drastic chopping function, with an m?1 saturation shape, de ned such that the regions of the (m; !^ ) plane subject to chopping by the standard chopping function, (26b), with p = 5=3, q = 2=3, have their magnitudes reduced all the way to zero.

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This applies for instance to the black regions in Figs. 3c and 3d. Also set to zero are the magnitudes for all spectral elements with the same value of !^ and larger values of m. This use of m?1 partially mimics Hines' suggestion that breaking can be parameterized by insisting that all gravity waves with vertical wavenumbers exceeding a certain threshold can be considered to have broken. Case E uses the standard launch spectrum with yet another chopping function, namely (26b), with p = 1, q = 0, which may be closer to physical reality for the reasons noted below (24). Figures 15a and 15b show respectively the total westward propagating pseudomomentum ux 0jF2(z)j and total eastward propagating pseudomomentum ux 0jF4(z)j, as functions of altitude. Relative to the control case A, the double-magnitude case B and the !^ ?1 case E show generally larger values, and the double-m case C generally smaller. The control case A and the m?1 case D have the same launch spectra, but the much more severe wave-breaking criterion used in case D results in smaller values at all subsequent altitudes than for case A. The dierences between these four cases span more than a decimal order of magnitude. But the vertical pro les are all qualitatively similar. The wave-induced force G, shown as a function of altitude in Figs. 15c and 15d, reveals a similar picture, with wave-induced force magnitudes again varying between the test cases by over a decimal order of magnitude. The altitude ranges of maximum eastward wave-induced force (80 km) and of maximum westward wave-induced force (110 km) are almost the same in each case. This insensitivity of pro le shape to major changes in the saturation model and the launch spectrum, holds promise for improved GCM gravity-wave parameterization schemes.

5.2 Sensitivity to azimuthal directional resolution This is tested using the real-winds example of Fig. 11. We increase the number of azimuthal directions from n = 4 to n = 8. Only the four new directions j =45, 135, 225, 315 need to be computed, because the results for the other four are already known from section 4.5.

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Figures 16a and 16b show respectively the zonal component Gx and meridional component Gy of the wave-induced force G as function of altitude, and Fig. 16c shows the total wave-energy, Etot, plotted as a function of altitude for the eight-direction case. The four-direction cases from Fig. 14 are superimposed. We see that the magnitudes of Gx are generally similar. Above about 40 km altitude, the eight-direction case yields a Gx which is slightly larger in magnitude than it is for the four-direction case. Gy is smaller in magnitude for the eight-direction case than it is for the four-direction case above about 30 km altitude. The total wave-energy for the eight-direction case is seen to be smaller than for the four-direction case up to about 70 km altitude. Above 70 km altitude, the total wave-energy for the four-direction case is smaller than for the eight-direction case. In summary, in this real-winds example at least, the dierences between the eight-direction and four-direction resolutions are small in comparison with the variations seen in the other sensitivity tests.

6. Concluding remarks The single-column numerical model described and tested here is suciently exible to permit any launch spectral shape to be used, and to permit alternative saturation schemes. We have tested it on typical examples, and it produces results that look encouragingly realistic. The tests included sensitivity to the choice of launch spectrum, to the steepness of the model saturation criterion, and to azimuthal directional resolution. Although the magnitudes of the gravity-wave-induced force vary signi cantly from test case to test case, as illustrated in Fig. 15, the shapes of the vertical pro les are broadly similar. Many other sensitivity tests could have been carried out. For example, we could have tested our neglect of contribution from back-re ected, downward propagating waves to saturation, perhaps in conjunction with more tests of an !^ ?1 chopping function. Again, and more importantly, we expect signi cant changes in the wave-induced force magnitude to result from varying s in (2), or from choosing a dierent shape altogether. Recent observational studies by Allen and Vincent (1995) lend further support to our choice of standard launch spectrum and saturation function, as regards its m-dependence at least. Allen and Vincent found a typical characteristic wavelength 2=m of 2.5 km for spectra in the lower stratosphere and observed saturation pro les consistent with a slope / m?3.

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They assumed !^ ?5=3 as the best estimate from the literature. However, they were unable to comment on the value of the small-m launch spectrum slope s, as this region corresponds to the largest vertical wavelengths and is therefore very dicult to observe. It is certain that choosing larger values for s, for example s = 2, would signi cantly reduce the large drag values at the highest, thermospheric altitudes, because signi cantly less pseudomomentum ux would be carried by the spectral elements, with the smallest values of m, that reach these altitudes. It may be that this fact, together with empirical E^ pro les (31), will help answer a longstanding question: what values of s best t the real atmosphere? Future work will tackle such questions and will use the single-column numerical model as a testbench for assessing the eects of other saturation schemes and other launch spectra, and to address other outstanding questions. As just mentioned, determination of s to t empirical wave-energy pro les like that of (31) appears to provide a new and promising approach to constraining values of s. This may, in the end, require non-Boussinesq and sharply-variable background eects to be treated more accurately than at present.

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Acknowledgements We thank M Joan Alexander, Julio Bacmeister, Oliver Buhler, David Fritts, Bryan Lawrence, Norman McFarlane, Theodore Shepherd, John Thuburn and an anonymous referee for helpful comments and correspondence. This work received support from the Natural Environment Research Council through the UK Universities' Global Atmospheric Modelling Programme, and through Grant number N00014-92-J-2009 administered by the US Naval Research Laboratory.

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REFERENCES

Allen, S. J., and R. A. Vincent, 1995: Gravity wave activity in the lower atmosphere: seasonal and latitudinal variations. J. Geophys. Res., 100, 1327{1350. Bacmeister, J. T., S. D. Eckermann, P. A. Newman, L. Lait, K. R. Chan, M. Loewenstein, M. H. Prott, and B. L. Gary, 1996: Stratospheric Horizontal Wavenumber Spectra of Winds, Potential Temperature and Atmospheric Tracers Observed by HighAltitude Aircraft. J. Geophys. Res., ?, ?{?. In press. Bretherton, F. P., 1969: Waves and turbulence in stably strati ed uids. Radio Science, 4, 1279{1287. , and C. J. R. Garrett, 1968: Wavetrains in inhomogeneous moving media. Proc. Roy. Soc. London, A302, 529{554. Broutman, D., and W. R. Young, 1986: On the interaction of small-scale oceanic internal waves with near-inertial waves. J. Fluid Mech., 166, 341{358. Dewan, E. M., 1994: The saturated-cascade model for atmospheric gravity wave spectra, and the wavelength-period (W-P) relations. Geophys. Res. Lett., 21, 817{820. , and R. E. Good, 1986: Saturation and the \universal" spectrum for vertical pro les of horizontal scalar winds in the atmosphere. J. Geophys. Res., 91, 2742{2748. Eckermann, S. D., 1995: Eect of background winds on vertical wavenumber spectra of atmospheric gravity waves. J. Geophys. Res., 100, 14097{14112. Fritts, D. C., and P. K. Rastogi, 1985: Convective and dynamical instabilities due to gravity wave motions in the lower and middle atmosphere: Theory and observations. Radio Sci., 20, 1247{1277. , and W. Lu, 1993: Spectral estimates of gravity wave energy and momentum

uxes. Part II: Parameterization of wave forcing and variability. J. Atmos. Sci., 50, 3695{3713. , and T. E. VanZandt, 1993: Spectral estimates of gravity wave energy and momentum uxes. Part I: Energy dissipation, acceleration and constraints. J. Atmos. Sci., 50, 3685{3694. Garcia, R. R., and B. A. Boville, 1994: \Downward control" of the mean meridional circulation and temperature distribution of the polar winter stratosphere. J. Atmos. Sci., 51, 2238{2245.

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Gill, A.E., 1982: Atmosphere-Ocean Dynamics. International Geophysics Series, Vol 30, Academic Press, 662pp. Haynes, P. H., C. J. Marks, M. E. McIntyre, T. G. Shepherd, and K. P. Shine, 1991: On the \downward control" of extratropical diabatic circulations by eddy-induced mean forces. J. Atmos. Sci., 48, 651{678. , McIntyre, M. E., and T. G. Shepherd, 1996: Reply to Comments by J. Egger on `On the \downward control" of extratropical diabatic circulations by eddy-induced mean zonal forces'. J. Atmos. Sci., 53, ?{?. In press, January 1996. [[COPY EDITOR, WE WOULD BE GRATEFUL IF YOU WOULD PUT IN PAGE NUMBERS IF KNOWN]] , Ward, W. E., 1993: The eect of realistic radiative transfer on potential vorticity structures, including the in uence of background shear and strain. J. Atmos. Sci., 50, 3431{3453. Hines, C. O., 1991a: The saturation of gravity waves in the middle atmosphere. Part I: Critique of linear-instability theory. J. Atmos. Sci., 48, 1348{1359. , 1991b: The saturation of gravity waves in the middle atmosphere. Part II: Development of doppler spread theory. J. Atmos. Sci., 48, 1360{1379. , 1991c: The saturation of gravity waves in the middle atmosphere. Part III: Formation of the turbopause and of turbulent layers beneath it. J. Atmos. Sci., 48, 1380{1385. , 1993: The saturation of gravity waves in the middle atmosphere. Part IV: Cuto of the incident wave spectrum. J. Atmos. Sci., 50, 3045{3060. , 1996: Doppler-spread parametrization of gravity-wave momentum deposition in the middle atmosphere. Part 1: Basic formulation. J. Atmos. Terres. Phys., submitted. Holton, J. R., P. H. Haynes, M. E. McIntyre, A. R. Douglass, R. B. Rood, and L. P ster, 1995: Stratosphere{troposphere exchange. Revs. Geophys., 33(4), 403{439. Lighthill, J., 1978: Waves in uids. Cambridge University Press, 504pp. McEwan, A. D., 1971: Degeneration of resonantly-excited internal gravity waves. J. Fluid Mech., 50, 431{448.

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McIntyre, M. E., 1980: An introduction to the generalized Lagrangian-mean description of wave, mean- ow interaction. Pure Appl. Geophys., 118, 152{176. , 1981: On the \wave momentum" myth. J. Fluid Mech., 106, 331{347. , 1993: On the role of wave propagation and wave breaking in atmosphere{ocean dynamics (Sectional Lecture). Proc. XVIII Int. Congr. Theor. Appl. Mech., Haifa, Israel (Sponsored by the International Union of Theoretical and Applied Mechanics), 281{304. , 1995: The stratospheric polar vortex and sub-vortex: uid dynamics and midlatitude ozone loss. Phil. Trans. Roy. Soc. London, 352, 227{240. Medvedev, A. S., and G. P. Klaassen, 1995: Vertical evolution of gravity wave spectra and the parametrization of associated wave drag. J. Geophys. Res., 100, 25841{ 25853. Murayama, Y, 1993: Characteristics of gravity waves in the middle atmosphere revealed with the MU radar, rocketsondes and Lidar. Ph.D. thesis, Kyoto University, Japan, 142pp. Shepherd, T. G., 1990: Symmetries, conservation laws, and Hamiltonian structure in geophysical uid dynamics. Adv. Geophys., 32, 287{338. Sidi, C., 1993: Waves{turbulence coupling. Proc. of the NATO Advanced Research Workshop on Coupling Processes in the Lower and Middle Atmosphere, Loen, Norway (sponsored by NATO), 291{304. Smith, S. A., D. C. Fritts, and T. E. VanZandt, 1987: Evidence of a saturated spectrum of atmospheric gravity waves. J. Atmos Sci., 44, 1404{1410. VanZandt, T. E., 1982: A universal spectrum of buoyancy waves in the atmosphere. Geophys. Res. Lett., 9, 575{578. , and D.C. Fritts, 1989: A theory of enhanced saturation of the gravity wave spectrum due to increases in atmospheric stability. Pure Appl. Geophys., 130, 399{420.

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FIGURE CAPTIONS

FIG. 1. Shaded-surface plot of the standard pseudomomentum ux 0jF^ pj spectrum, derived from the Fritts-VanZandt energy spectrum at launch as a function of the vertical wavenumber m and the intrinsic frequency !^ . The low frequency cuto !^ = f , is close to but not equal to zero. Units are rad m?1 for m, rad s?1 for !^ and kg m?1 s?2 rad?1 for 0jF^ pj. Ranges on axes are 0 < m < 0:03 rad m?1 (cf. m = 2=2 km = 3:142  10?3 rad m?1), 0 < !^ < 0:02 rad s?1 (cf. N = 0:0229 rad s?1 and f = !^min = 9:3744  10?5 rad s?1, corresponding to the July, 40N, CIRA 1986 model atmosphere at 19 km altitude, so the frequency range plotted is slightly less than the full range), and 0 < 0jF^ pj < 400 kg m?1 s?2 rad?1. FIG. 2. (a) Curves of constant k0 in (m; !^ ) space. When N is constant, Doppler shifts move spectral elements along these curves. When N = N (z), the same picture holds if !^ is replaced by !^ =N . Units are rad m?1 for m and rad s?1 for !^ . Leftwards \over the hill" means back-re ection, implying zero contribution from that element to wave-induced force at any altitude. Rightwards to in nity would mean critical-level absorption, but the waves always break rst. The top curve has k0 = 1:0  10?1 rad m?1, the largest value; the next curve down has a value of k0 equal to half that of the top curve and so on; f = 10?4 rad s?1 and N = 10?2 rad s?1. (b), (c) Behaviour of spectral elements under a Doppler shift. Elements in (k0; !0) space are invariant; their counterparts in (m; !^ ) space change as shown when U , the component of background wind in the positive k0 direction, changes from 0 ms?1 to 10 ms?1. Areas increase like J = m=k0, the spectral Jacobian. Note the shearing of spectral elements, implying that numerical resolution is a nontrivial matter, just as in computing uid ow. FIG. 3. Eect of a small Doppler shift viewed in (m; !^ ) spectral space: (a) standard pseudomomentum ux spectrum at launch, Eq. (16). (b) pseudomomentum ux spectrum after conservative propagation through positive background wind shear (U increases by 2 ms?1, i.e., toward critical-level conditions, with m increasing and !^ diminishing). (c) spectrum after wave breaking with \chopped" pseudomomentum

ux spectral elements highlighted. (d) as (c) but oriented so as to show more clearly the highlighted spectral elements that have undergone wave breaking. The chopping annihilates all spectral elements that would otherwise reach m = 1, expressing the breaking of waves before they reach their critical levels. Ranges on axes are

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0 < m < 0:03 rad m?1 (cf. m = 2= 2 km = 3:142  10?3 rad m?1), 0 < !^ < 0:02 rad s?1 (cf. N = 0:0229 rad s?1 and f = !^min = 9:3744  10?5 rad s?1, corresponding to the July, 40 N, CIRA model atmosphere at 19 km altitude), and 0 < 0jF^ pj < 300 kg m?1 s?2 rad?1. The peak value in (a) appears lower than in Fig. 1 because, in this much coarser resolution view, none of the wire frame wires coincide with the peak value in Fig. 1. FIG. 4. July, 40N, CIRA 1986 model atmosphere: Mean zonal background wind, temperature and buoyancy frequency pro les from 0 to 120 km altitude. Arrows show altitudes for Figs. 5, 6 and 7. The minimum in buoyancy frequency N (z) near z=60 km is 0.016 rad s?1 FIG. 5. Pseudomomentum ux spectra oriented as in Fig. 1, but showing only a narrow range of m < m, for the example of an arti cial atmosphere with no background wind and with the July, 40N, CIRA 1986 model atmosphere temperature and buoyancy frequency pro les. Shaded-surface plots in (m, !^ ) spectral space at altitudes of: (a) 19 km, (b) 69 km, (c) 86 km, and (d) 102 km (arrows in Fig. 4). Note that, in Fig. 5a, the spectral peak near m = m is out of sight, about ve m-axis lengths to the left. Note also that, because the high-m, high-^! corner of the plots (corresponding to k0 ! 1) is not resolved, the area of spectral space with zero magnitude (no numerical information) at high m, high !^ in (a) is a numerical artefact that propagates through to higher altitudes. The cuto at large !^ , most cleanly represented in (c) and (d), results from the back-re ection of spectral elements when Doppler-shifted past !^ = N (cf. 0.0016 rad s?1, the minimum N near 60 km); (b) shows a combination of this eect with the lack of numerical resolution as k0 ! 1. Ranges on axes are 0 < m < 6  10?4 rad m?1 (cf. m = 2= 2 km = 3:142  10?3 rad m?1), 0 < !^ < 0:03 rad s?1 (cf. N = 0:0229 rad s?1 at the 19 km launch altitude and f = !^min = 9:3744  10?5 rad s?1), and 0 < 0jF^ pj < 150 kg m?1 s?2 rad?1. The edge of the shading shows N (z). FIG. 6. Eastward propagating pseudomomentum ux spectra oriented as in Fig. 1, but showing only a narrow range of m < m , for the example of the July, 40N, CIRA 1986 model atmosphere. Shaded-surface plots in (m, !^ ) spectral space at altitudes of: (a) 19 km, (b) 69 km, (c) 86 km, and (d) 102 km (arrows in Fig. 4). Note that the vertical axis range is ten times larger for (b) than for (a), (c), and (d). Ranges

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on axes are 0 < m < 6  10?4 rad m?1 (cf. m = 2=2 km = 3:142  10?3 rad m?1), 0 < !^ < 0:03 rad s?1 (cf. N = 0:0229 rad s?1 at the 19 km launch altitude and f = !^min = 9:3744  10?5 rad s?1), and 0 < 0jF^ pj < 100 kg m?1 s?2 rad?1 for (a), (c), and (d) and 0 < 0jF^ pj < 1000 kg m?1 s?2 rad?1 for (b). The edge of the shading shows N (z). FIG. 7. Westward propagating pseudomomentum ux spectra oriented as in Fig. 1, but showing only a narrow range of m < m , for the example of the July, 40N, CIRA 1986 model atmosphere. Shaded-surface plots in (m, !^ ) spectral space at altitudes of: (a) 19 km, (b) 69 km, (c) 86 km, and (d) 102 km (arrows in Fig. 4). Note that the irregular shape of (b) at small !^ is a numerical artefact. A prohibitively larger number of smaller spectral elements would be needed to produce a plot showing the expected smooth variation with m. Ranges on axes are 0 < m < 6  10?4 rad m?1 (cf. m = 2= 2 km = 3:142  10?3 rad m?1), 0 < !^ < 0:03 rad s?1 (cf. N = 0:0229 rad s?1 at the 19 km launch altitude and f = !^min = 9:3744  10?5 rad s?1), and 0 < 0jF^ pj < 100 kg m?1 s?2 rad?1. The edge of the shading shows N (z). FIG. 8. For July, 40N, CIRA 1986 model atmosphere: (a) Total westward propagating pseudomomentum ux 0jF2(z)j. (b) Total eastward propagating pseudomomentum

ux 0jF4(z)j (solid curve) and upward propagating part 0jF"4 (z)j (dotted curve). The back-re ected contribution to the total eastward propagating pseudomomentum

ux is 38% of the upward contribution at the launch altitude. (c) Wave-induced force G, which is zonal because meridional background winds are set to zero. (d) Total wave-energy, Etot, including back-re ected energy (solid curve marked \s.c.m." for \single column model", i.e., present calculation). The dotted curve (marked \FL") is Fritts and Lu's empirical Etot pro le de ned by (31). FIG. 9. January, 40 N, CIRA 1986 model atmosphere: Mean zonal background wind, temperature and buoyancy frequency pro les from 0 to 120 km altitude. Arrow shows launch altitude. FIG. 10. For January, 40 N, CIRA 1986 model atmosphere: (a) Total westward propagating pseudomomentum ux 0jF2(z)j (solid curve), and upward propagating part

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0jF"2(z)j (dotted curve). The back-re ected contribution to the total westward propagating pseudomomentum ux is 24% of the upward contribution at the launch altitude. (b) Total eastward propagating pseudomomentum ux 0jF4(z)j. (c) Waveinduced force G, which is zonal because meridional background winds are set to zero. (d) Total wave-energy Etot, including back-re ected energy (solid curve marked \s.c.m." for \single column model", i.e., present calculation). The dotted curve (marked \FL") is Fritts and Lu's empirical Etot pro le de ned by (31). FIG. 11. Real-winds example (ISAMS data at 37.5N, 0E, 24-day average centred on 28 November 1991): Zonal background wind, meridional background wind, temperature, and buoyancy frequency as functions of altitude. Arrows show launch altitude, and altitudes for Figs. 12a{12d. FIG. 12. Northward propagating pseudomomentum ux spectra oriented as in Fig. 1, but showing only a narrow range of m < m, for the real-winds example (ISAMS data as in Fig. 11). Shaded-surface plots in (m, !^ ) spectral space at altitudes of (a) 38 km, (b) 54 km, (c) 60 km, and (d) 80 km (arrows in Fig. 11). Note that the range of the m axis is 10 times that of the CIRA examples. Ranges on axes are 0 < m < 6  10?3 rad m?1 (cf. m = 2=2 km = 3:142  10?3 rad m?1), 0 < !^ < 0:03 rad s?1 (cf. N = 0:0194 rad s?1 at the 18 km launch altitude and f = !^min = 8:8782  10?5 rad s?1), and 0 < 0jF^ pj < 200 kg m?1 s?2 rad?1. The edge of the shading shows N (z). FIG. 13. For real-winds example (ISAMS data as in Fig. 11): (a) Total westward propagating pseudomomentum ux 0jF2(z)j (solid curve), and upward propagating part 0jF"2(z)j (dotted curve). The back-re ected contribution to the total westward propagating pseudomomentum ux is 70% of the upward contribution at the launch altitude. (b) Total eastward propagating pseudomomentum ux 0jF4(z)j. (c) Total northward propagating pseudomomentum ux 0jF1(z)j. (d) Total southward propagating pseudomomentum ux 0jF3(z)j (solid curve) and upward propagating part 0jF"3(z)j (dotted curve). The back-re ected contribution to the total southward propagating pseudomomentum ux is 20% of the upward contribution at the launch altitude. FIG. 14. For real-winds example (ISAMS data as in Fig. 11). (a) Zonal component Gx of wave-induced force G. (b) Meridional component Gy of wave-induced force G. (c) Total wave-energy Etot, including back-re ected energy (solid curve marked \s.c.m."

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for \single column model", i.e., present calculation). The dotted curve (marked \FL") is Fritts and Lu's empirical Etot pro le de ned by (31). FIG. 15. Sensitivity test cases, all for the July, 40N, CIRA model atmosphere: case A (dotted curves) control, case B (solid curves) double-magnitude, case C (dashed) double-m, case D (dash-dotted) in nitely steep (m?1) saturation m slope, and case E (dash-triple dotted) !^ ?1 saturation !^ slope. (a) Total westward propagating pseudomomentum ux 0jF2(z)j. (b) Total eastward propagating pseudomomentum ux 0jF4(z)j. (c) Wave-induced force G, which is zonal because meridional background winds are set to zero. (d) as (c) but for a limited altitude range. FIG. 16. Azimuthal resolution test for the real-winds example (ISAMS data as in Fig. 11), comparison of 8 azimuthal directions (solid) and 4 azimuthal directions (dotted). (a) Zonal component Gx of wave-induced force G. (b) Meridional component Gy of wave-induced force G. (c) Total wave-energy Etot, including back-re ected energy.

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Figure 1 is ZY Figure 2 is ZZ Figure 3 is ZX Figure 4 is ZK July CIRA u, v, T, N Figure 5 is ZU no wind, July CIRA spectra Figure 6 is ZW eastward, July CIRA spectra Figure 7 is ZV westward, July CIRA spectra Figure 8 is ZT rhoFP, Gx, E, July CIRA spectra Figure 9 is ZJ January CIRA u, v, T, N Figure 10 is ZN rhoFP, Gx, E, January CIRA spectra Figure 11 is ZI Real winds u, v, T, N Figure 12 is ZH northward, real winds spectra Figure 13 is ZG rhoFP(N,W,S,E), real winds Figure 14 is ZE Gx, Gy, E, real winds Figure 15 is ZD rhoFP(W,E), Gx, sensitivity tests Figure 16 is ZR Gx, Gy, E, azimuthal resolution tests