## 1. Introduction

Vertically propagating gravity waves are now generally believed to provide the strong mechanical forcing that keeps the extratropical mesosphere far from radiative equilibrium. Gravity waves are also thought to play an important role in the stratosphere in helping to drive both the quasi-biennial oscillation in the Tropics (Dunkerton 1997) and the equator-to-pole residual circulation in the summer hemisphere (Alexander and Rosenlof 1996). Since the horizontal scales of these waves are, for the most part, too small to be resolved by current general circulation models (GCMs), their effects on the large-scale circulation must be parameterized if realistic simulations of the middle atmosphere are to be obtained.

The gravity wave drag (GWD) parameterization problem has received considerable attention since the pioneering work of Lindzen (1981) and the first implementation of his GWD scheme in a middle atmosphere model by Holton (1983). Over the past two decades, numerous observational studies have revealed the existence of a “saturated” vertical wavenumber *m* spectrum of horizontal wind variance that is characterized by a *m*^{−3} tail at large *m* (e.g., VanZandt 1982; Dewan et al. 1984; Tsuda et al. 1989). Its apparent universal shape has led to plenty of theoretical speculation as to the underlying physical mechanism that is responsible for this behavior (e.g., Dewan and Good 1986; Smith et al. 1987; Hines 1991a, b).

All of this activity has led to the development of a number of spectral GWD parameterizations (e.g., Medvedev and Klaassen 1995; Hines 1997a, b; Alexander and Dunkerton 1999; Warner and McIntyre 2001). The use of these parameterizations in middle atmosphere GCMs has met with a considerable degree of success. Not only do they bring about the zonal-mean zonal wind reversals in the mesosphere, they also alleviate the winter stratosphere cold bias in the Southern Hemisphere, produce an earlier breakdown of the Southern Hemisphere winter vortex, and help drive realistic stratospheric quasi-biennial and mesospheric semiannual oscillations (e.g., Manzini and McFarlane 1998; Medvedev et al. 1998; Scaife et al. 2002; Fomichev et al. 2002; Giorgetta et al. 2002; Scinocca 2002, 2003).

The key feature that differentiates these parameterizations is the mechanism by which the gravity waves are dissipated. Some of the mechanisms are built on theoretical ideas about the nonlinear processes that are believed to generate the observed saturated spectrum (e.g., Weinstock 1982; Hines 1991a, b,c, 1993), while others are formulated using a more pragmatic approach (e.g., Warner and McIntyre 1996; Alexander and Dunkerton 1999).

Given the variety of dissipation mechanisms, an obvious question to ask is which one is most correct. This is a contentious issue and an exceedingly difficult question to answer. An equally important issue involves understanding the sensitivity of the GCM response to the choice of dissipation mechanism. Here we shall focus on this more practical issue by carefully comparing the GCM response to several commonly used dissipation mechanisms. The hope is that such a comparison will provide guidance and perspective on the importance of the differences between the parameterizations. GCM simulations are required if one wants to understand the impact of the different GWD parameterizations on the climate since feedback processes between the GWD and the resolved wind and temperature fields determine the response.

The manner in which this comparison is done is critical if we are to meaningfully understand differences in the response. For instance, the most straightforward approach would be to simply use the operational versions of the parameterizations in one GCM and compare climatologies. However, this approach fails because, in addition to the impact of the different dissipation mechanisms, some unknown portion of the response will be due to differences in the properties of the parameterizations that can, in principle, be made identical (e.g., characteristics of the launch spectra).

To circumvent this problem, we take another approach in which the individual dissipation mechanisms are incorporated into a single new parameterization. In this way, the properties of the different parameterizations that *can* be made identical *are* made identical. Consequently, we are able to identify and document differences in the climate response that are due solely to the fundamental component that differentiates current operational GWD parameterizations, namely the dissipation. For our study we have chosen to examine the dissipation mechanisms employed in three nonorographic GWD parameterizations that are currently used in middle atmosphere GCMs: Hines (1997a, b), Warner and McIntyre (2001), and Alexander and Dunkerton (1999).

While the primary goal of our study is to compare and document the GCM response that is due to differences in the dissipation mechanisms themselves, another equally important goal is to understand the source of the differences in the GCM response. A central result of this study is that these response differences are simply due to systematic differences in the height at which momentum is deposited. Sensitivity experiments reveal that it is possible to eliminate these systematic differences and obtain nearly identical GCM responses from each dissipation mechanism. Consequently, it is argued that the response is largely insensitive to the precise details of the dissipation mechanisms used in current parameterization schemes.

The paper is organized as follows. In section 2 we describe the new parameterization and the manner in which the dissipation mechanisms are incorporated. In section 3 we describe the results of the offline calculations and GCM simulations using the three dissipation mechanisms in order to quantify the effects on the large-scale circulation. In section 4 we investigate the source of the differences of the GCM response by discussing a set of sensitivity experiments in which we attempt to reduce systematic differences in the height at which momentum is deposited. In section 5 we discuss the implication of our findings.

## 2. Generalized spectral parameterization

The effectiveness of our study hinges on our ability to identify differences in GWD that arise solely from differences in the dissipation mechanisms used in the parameterization schemes. This is achieved by coding these mechanisms into the same scheme, which we shall refer to as the generalized spectral parameterization (GSP). In this way we eliminate from the comparison potential differences related to such aspects of the parameterizations as the launch spectra and numerics. To accomplish this we use a modified version of the scheme developed by Scinocca (2002, hereafter S02), which is an efficient implementation of the Warner and McIntyre (1996) parameterization for nonorographic GWD. Detailed information regarding its formulation and application may be found in S02. Here, we discuss only the aspects that pertain directly to our study.

### a. Launch spectrum and conservative propagation

*ϕ*is therefore governed by the dispersion relation (e.g., Gill 1982):where

*N*is the buoyancy frequency,

*m*is the vertical wavenumber,

*ω̂*is the intrinsic frequency (

*ω̂*=

*ω*−

*kU*),

*ω*is the (ground based) frequency, and

*k*and

*U*are the magnitudes of the horizontal wavenumber and wind vectors, respectively, projected onto this azimuth.

*o*denotes the launch level,

*m*∗ is a characteristic vertical wavenumber, and

*p*and

*B*are constants. We use

*m*∗ = 2

*π*/(2 km) and

*p*= 3/2 in all calculations. The value of

*B*is determined by normalizing the launch spectrum to a specified amount of upward momentum flux in each azimuth, which is a fundamental free parameter in the problem. Note that the launch spectrum (2) is azimuthally isotropic in intrinsic frequency, or equivalently intrinsic phase speed.

While the energy density *Ê*_{o}(*m*, *ω̂*, *ϕ*) is used to specify the properties of the launch spectrum, it proves advantageous to employ the momentum flux density *ρF*(*k, ω*, *ϕ*) to describe the evolution of the wave field with height. This follows from the fact that *ρF*(*k, ω*, *ϕ*) is conserved for steady conservative propagation, and therefore changes only as a consequence of dissipation. Following S02 the momentum flux density is related to the energy density through wave-action relations and independent wave variables (*k*, *ω*) are used in place of (*m*, *ω̂*) via a Jacobian transformation.

The momentum flux density *ρF*(*k*, *ω*, *ϕ*) is discretized using four horizontal azimuths (the cardinal directions), *n _{k}* horizontal wavenumbers, and

*n*frequencies. To reduce the values of

_{ω}*n*and

_{k}*n*, a coordinate stretch is used to increase resolution at small

_{ω}*k*and small

*ω*. The parameter settings for this stretch are given in S02.

A schematic of the momentum flux density at the launch level is illustrated in Fig. 1a for a single azimuth. It is bounded by the frequencies *ω* = *f* + *kU _{o}* (i.e.,

*ω̂*=

*f*) and

*ω*=

*N*+

_{o}*kU*

_{0}(i.e.,

*ω̂*=

*N*) and the horizontal wavenumbers

_{o}*k*= 0 and

*k*=

*k*

_{max}= 1 × 10

^{−2}m

^{−1}. Note that, while (1) implies wavelike disturbances (i.e., real

*m*) for all

*ω̂*≠ 0, it is meant to be a midfrequency approximation (i.e.,

*f*≪

*ω̂*≪

*N*). We have therefore limited its value at launch to

_{o}*f*≤

*ω̂*≤

*N*to conform to the more physical bounds of nonhydrostatic rotational wave dynamics. The dashed line,

_{o}*ω*=

^{C}_{o}*kU*, in Fig. 1a corresponds to the wavenumbers and frequencies of critical levels at the launch level (for H-R wave dynamics); its significance will be explained in the following section.

_{o}### b. Critical-level and nonlinear dissipation

In all operational GWD parameterizations, wave dissipation can be separated into two components. The first, which is common to all schemes, is critical-level dissipation, which we will also refer to as critical-level filtering. Its application is illustrated schematically in Fig. 1b, where we show *ρF*(*k*, *ω*, *ϕ*) on the first vertical level (denoted by the subscript 1) above the launch level. As depicted in this azimuth, the winds have changed from *U _{o}* to

*U*

_{1}, where

*U*

_{1}>

*U*. The wavenumbers and frequencies in the region labeled A have undergone critical-level filtering between these two levels. This may be interpreted visually as the

_{o}*ω*line pivoting from

^{C}*ω*to

^{C}_{o}*ω*

^{C}_{1}and removing the momentum flux from all wavenumbers and frequencies it encounters. Integrating

*ρF*over region A yields the amount of momentum deposited by the wave field to the flow as a consequence of critical-level dissipation in this azimuth.

Wavenumbers and frequencies in the shaded region of Fig. 1b survive critical-level dissipation and are free to conservatively propagate upward to the next vertical level. However, because of the decrease in ambient density, these waves will eventually attain sufficiently large amplitudes that nonlinear effects become important. These waves are then potentially subject to turbulent dissipation through the onset of instability. This is the second type of dissipation, which we will refer to simply as nonlinear dissipation.

The Hines, Warner and McIntyre, and Alexander and Dunkerton approaches all differ in their choice of nonlinear dissipation. The manner in which these three dissipation mechanisms are included in the GSP will now be discussed.

#### 1) Hines dissipation

In terms of the formulation of the GSP, the Hines dissipation mechanism assumes that nonlinearity within the wave field may be modeled by simply enhancing the horizontal wind *U* in the dispersion relation (1) with an rms measure of the wave-induced horizontal wind perturbations *U*_{rms}. Dissipation ensues because this enhancement causes critical-level filtering to extend to additional wavenumbers and frequencies.

*U*

_{tot}is written aswhere

*σ*is the rms wind of waves directed into the current azimuth,

*σ*is the rms wind of all waves, and Φ

_{T}_{1}and Φ

_{2}are empirical constants. Here we use Φ

_{1}= 1.5 and Φ

_{2}= 0.3, which are the midpoint values suggested by H97. We shall refer to this as Hines dissipation.

The manner in which Hines dissipation acts in the GSP is illustrated in Fig. 1c, where the addition of *U*_{rms} causes *ω ^{C}* to pivot from

*ω*

^{C}_{1}to

*ω*

^{Hines}

_{1}. Integrating the momentum flux density over the wavenumbers and frequencies between

*ω*

^{C}_{1}and

*ω*

^{Hines}

_{1}(i.e., region B) yields the amount of momentum deposited by the wave field to the background flow in this azimuth as a consequence of Hines dissipation. The remaining wavenumbers and frequencies in the shaded region of Fig. 1c are free to conservatively propagate upward to the next vertical level.

The validation of Hines dissipation in the GSP is accomplished using offline calculations by comparing the computed wind tendencies to those computed using a version of the H97 parameterization that is employed operationally in GCM simulations (McLandress 1998).^{1} The resulting GWD profiles are found to be virtually identical, indicating that Hines dissipation has been correctly implemented in the GSP. Details of this comparison are provided in appendix A.

#### 2) WM dissipation

The Warner and McIntyre dissipation mechanism (Warner and McIntyre 1996, 2001) assumes that the nonlinear dissipation may be modeled by limiting the wave energy density at large vertical wavenumbers to the observed *m*^{−3} functional form.

*k*–

*ω*space asand constraining

*ρF*≤

*ρF*

^{sat}. This shall be referred to as WM dissipation. After the application of critical-level dissipation, this constraint is applied to the remaining spectral elements in each azimuth (i.e., the shaded region of Fig. 1b). In (4) we have introduced the nondimensional constant

*C**. Previous applications (e.g., Warner and McIntyre 1996, 2001; S02; Scinocca 2003) have assumed that the saturated spectrum (4) is equivalent to the launch spectrum at asymptotically large

*m*; this is recovered by setting

*C** = 1. The introduction of

*C** allows the normalization of the saturated spectrum to be specified independently of the launch spectrum. In appendix B we compare (4) to the amplitude of the observed

*m*

^{−3}wind-variance spectra and obtain the estimate 10 ≤

*C** ≤ 30. The utility of

*C** in the present study will be discussed shortly.

*ω̂*=

*ω*. In this case (4) may be rewritten aswhere

*D*is a proportionality factor that includes quantities such as

*ρ*and

*N*, which vary in the vertical. The launch momentum flux density becomesThe configuration at the launch level is illustrated in Fig. 2a, which shows the dependence on

*k*at fixed

*ω*of

*ρF*and

_{o}*ρF*

^{sat}

*for two different values of*

_{o}*C**.

^{2}When

*C** = 1,

*ρF*

^{sat}

*is asymptotically equivalent to*

_{o}*ρF*at large

_{o}*k*, indicating that the tail of the launch spectrum is saturated. In the case where

*C*> 1,

^{*}*ρF*

^{sat}

*is offset vertically from*

_{o}*ρF*, indicating that the launch spectrum is not saturated.

_{o}Figure 2b illustrates how the saturation bound induces dissipation in the wave field at a higher elevation (in this case the first level above launch). As the spectrum propagates up to this level, the momentum flux density *ρF*_{1} remains identical to the launch value *ρF _{o}* for all

*k*and

*ω*. The saturated spectrum, however, is not conserved and so decreases in amplitude with height as a result of the decrease of density. This manifests itself as the downward shift of the

*ρF*

^{sat}

_{1}curves in Fig. 2b, with the

*C** = 1 curve intersecting the launch spectrum at

*k*=

*k*. Application of the saturation bound (5) then means that the momentum flux density is set equal to

_{c}*ρF*

^{sat}

_{1}to the right of

*k*. Consequently, an amount of momentum flux equal to area A in Fig. 2b is deposited to the flow in this azimuth.

_{c}From Fig. 2b we see that the saturated curve *ρF*^{sat}_{1} does not intersect *ρF _{o}* when

*C** > 1. However, the two will eventually intersect at a higher elevation where the density is smaller. Increasing

*C** therefore increases the height at which momentum is deposited; this effect will be examined in section 4.

#### 3) AD dissipation

The central assumption of the Alexander and Dunkerton (1999, hereafter AD99) parameterization scheme is that nonlinear dissipation may be equally well modeled by depositing all of the launch momentum flux of a spectral element at the altitude of the initial onset of instability. This was motivated by a desire to obtain a simple analytic mapping of the momentum flux density between the launch level and all other vertical levels.

AD99 chose to use the Lindzen (1981) saturation criterion for discrete wavenumbers and frequencies to determine the onset of instability. In principle, however, any dissipation criterion may be used for this purpose. Here we have chosen to use the WM criterion. We shall refer to this implementation as AD dissipation. While this is not identical to what was used in AD99, it allows a straightforward evaluation of AD99’s central assumption as a perturbation to the WM dissipation mechanism.

The application of AD dissipation in the GSP is illustrated in Fig. 2c for the example discussed in relation to WM dissipation. In this case, all wavenumbers to the right of *k _{c}* initially satisfy the WM dissipation criterion between the launch level and level 1. Consequently, the momentum flux density of each spectral element to the right of

*k*is set equal to zero and an amount of momentum flux equal to area B is deposited to the flow in this azimuth.

_{c}## 3. Response to dissipation mechanisms

In this section we will address the first of our two goals, namely, the comparison of the GCM response to the three nonlinear dissipation mechanisms. We will begin with a discussion of the offline calculations since they are helpful in interpreting the fully interactive GCM results that follow. For conciseness we shall refer to the results using these dissipation mechanisms as simply Hines, WM, or AD.

### a. Offline calculations

The offline calculations provide a useful means of comparing the nonlinear dissipation mechanisms on an equal footing since the background wind and temperature profiles are fixed. The mechanisms are, in fact, most cleanly compared without mean-wind shear since critical-level dissipation associated with the background wind is absent. The results of such a calculation are presented in Fig. 3a, which shows profiles of the momentum flux (i.e., the momentum flux density integrated over all horizontal wavenumbers and frequencies in a single horizontal azimuth).^{3} The WM and AD profiles exhibit a much more rapid initial decrease than that of Hines. The initial decrease is more rapid for AD because of the complete removal of momentum flux upon the initial onset of dissipation. When Hines is employed the momentum flux profile remains nearly constant for several scale heights before it starts to drop off.

The three remaining panels in Fig. 3 are for calculations using the wintertime Committee on Space Research (COSPAR) International Reference Atmosphere (CIRA) wind and buoyancy frequency profiles (Fleming et al. 1990) shown in Fig. 4. The effect of critical-level filtering by the background eastward wind is now seen by the rapid decrease with height of the eastward component of the momentum flux (Fig. 3b). (Note that the fluxes are plotted here on a linear scale.) It is only the westward traveling waves that are able to escape the effect of critical-level filtering and propagate up into the stratosphere. As in the no-winds case, the fluxes computed using WM and AD dissipation start to decrease much lower down than that using Hines. In all cases, however, the heights at which the westward component of the fluxes starts decreasing are greater than in the no-winds case (Fig. 3a). This is a consequence of the background winds, which Doppler shift the westward traveling waves to higher intrinsic frequencies and, correspondingly, to smaller *m* through the H-R dispersion relation (1). This tends to inhibit nonlinear dissipation for all the mechanisms considered. Consequently, those waves are able to propagate to higher elevations before they dissipate.

A quantity of fundamental importance in these comparisons is the vertical derivative of the net zonal momentum flux, which we will refer to simply as momentum deposition. Profiles of this quantity are displayed in Fig. 3c for the CIRA case. The momentum deposition is conserved in the sense that its vertical integral over the full depth of the atmosphere is equal to the flux at the launch level. Consequently, the area under these curves is identical. The same cannot be said, however, of acceleration, which is the quantity that is generally used in these kinds of comparisons. Furthermore, it is the momentum deposition that mediates the interaction of the parameterization with the GCM.

Comparing the curves in Fig. 3c we see substantial differences in the peak elevation and vertical structure of the momentum deposition for each of the dissipation mechanisms. Hines peaks at the highest elevation (∼75 km), AD at the lowest (∼40 km), and WM slightly above AD (∼50 km).^{4} The profile for AD is much more sharply peaked than that for WM. This is consistent with the fact that AD dissipation results in all of a wave’s momentum flux being deposited at one altitude, while WM spreads it out vertically. A similar result was found by AD99 when comparing their scheme to that of Lindzen (1981).

The more commonly used zonal acceleration, which is the momentum deposition divided by density, is shown in Fig. 3d. The acceleration displays the expected enhancement in the mesosphere for the Hines and WM dissipation mechanisms. Although the acceleration for AD is strongest near 40 km, large relative values occur in the region above. In spite of the fact that they all have the same amount of momentum at the launch level and deposit virtually all of it within the plotted domain, the acceleration associated with Hines is several orders of magnitude larger in the mesosphere than it is for WM and AD, which have been scaled to make them visible. Clearly, momentum deposition is a more useful quantity when making such comparisons.

The systematic differences in the elevation of Hines and WM momentum deposition seen in Fig. 3 appear to be a robust feature that does not seem to depend upon the shape of the launch spectrum. This was demonstrated in the GSP using a different low-*m* form of the launch spectrum (not shown). A similar conclusion was drawn by Charron et al. (2002) for versions of the parameterizations employing a constant momentum flux density at the launch level.

### b. GCM simulations

To compare the climate responses due to the three dissipation mechanisms we use the GSP in the Canadian Middle Atmosphere Model, which is a general circulation model extending from the surface to ∼100 km (Beagley et al. 1997). The version we employ has T32 horizontal resolution and 65 vertical levels. This corresponds to a grid size of ∼6° in latitude and longitude and ∼1.5 to 2.3 km in the vertical in the middle atmosphere.

Because of computational constraints a much coarser (*n _{k}* =

*n*= 9) resolution is used in the GSP for the GCM simulations than for the offline calculations (

_{ω}*n*=

_{k}*n*= 512). Sensitivity tests, similar to those discussed in S02 (not shown), demonstrate that the zonal-mean response computed for simulations using a higher spectral resolution (

_{ω}*n*=

_{k}*n*= 16) are essentially indistinguishable from those using the coarser resolution.

_{ω}In all simulations described in this paper, the orographic GWD parameterization of McFarlane (1987) is used, in addition to the nonorographic GWD computed using the GSP. Unless otherwise stated the simulations we will discuss are all 5 yr in length, and the plotted results comprise 5-yr averages. Since the effects of nonorographic GWD are most pronounced in the extratropics during the solstice seasons when the zonal winds are strongest, we will focus only on those months.

We consider the GCM response to each nonlinear dissipation mechanism for surface and tropopause launch levels, which are typically used operationally. These two cases also allow an evaluation of the impact of critical-level filtering by the tropospheric winds on the GCM response that is independent of the particular nonlinear dissipation mechanism employed. The simulations that employ a launch height near the surface use a momentum flux of 1.4 × 10^{−3} Pa in each azimuth. Those that employ a launch height near 100 hPa (∼16 km) use a momentum flux of 0.7 × 10^{−3} Pa (the same value that is used in the offline calculations). Note that the values for the momentum flux are chosen somewhat arbitrarily and are not intended to produce the most realistic response in the GCM. In all of the results shown here, the remaining momentum flux at the model top is allowed to escape. Sensitivity tests in which the momentum flux is not allowed to escape (results not shown) indicate that the extratropical response is not significantly different from the results presented here.

The zonal-mean zonal winds and momentum deposition for June, July, and August (JJA) and December, January, and February (DJF) are shown in Figs. 5 and 6 for the simulations using the surface and 100-hPa launch heights, respectively. Because of our experimental design, we know unequivocally that differences in the climatological response displayed here arise solely from differences in the nonlinear dissipation mechanisms. Ignoring for the time being the momentum deposition, inspection of these figures reveals that the wind response to Hines dissipation exhibits by far the largest changes in the mesosphere. This is most striking in the summer mesosphere, where the westerlies attain speeds of nearly 100 m s^{−1} in the case where the launch height is located near 100 hPa (Fig. 6).

The wind differences are more clearly seen in Fig. 7, which shows zonal wind profiles at 52.6°S and 52.6°N for the JJA simulations displayed in Figs. 5 and 6. For comparison purposes, the results of a simulation without nonorographic GWD (no-GWD case) are also shown. The simulation using Hines dissipation is clearly seen to depart the most from the no-GWD case, while that using AD differs the least. Figure 7 indicates that, overall, the application of nonorographic GWD produces winds at these two latitudes that are weaker than the no-GWD case.^{5}

As Manzini and McFarlane (1998) showed using an operational version of the H97 parameterization, the large wind differences that arise for these two launch heights is a consequence of critical-level filtering. When the waves are launched near the surface, a large fraction of the eastward momentum flux (which is responsible for producing the mesospheric wind reversal in the summer hemisphere) is removed by critical-level filtering in the troposphere. The stronger eastward winds in the summer mesosphere that result when the waves are launched near 100 hPa (e.g., top panels of Fig. 6) arise because there is no tropospheric filtering. This behavior occurs for all three dissipation mechanisms as can be seen from the corresponding vertical profiles of momentum deposition shown in Fig. 8. Comparing the relative magnitudes of the Hines momentum deposition in the summer and winter mesosphere, for example, we see that it is largest in winter when the waves are launched near the ground, but largest in summer when the waves are launched near the tropopause. This seasonal asymmetry in momentum deposition is a result of critical-level filtering in the troposphere.

Inspection of the momentum deposition in Figs. 5, 6 and 8 reveals systematic differences between the simulations employing Hines, WM, and AD dissipation. In accordance with the offline calculations, momentum is deposited much higher in the extratropical middle atmosphere for Hines. Since AD deposits momentum in the more massive lower atmosphere, its impact is not as strongly felt, which is consistent with the finding from Fig. 7 that the AD winds are most similar to the no-GWD case. Note that the sudden onset of momentum deposition seen in the offline results for AD (Fig. 3c) is masked here as a result of temporal and spatial averaging.

The wind fields of Figs. 5, 6 and 7 clearly show that the three dissipation mechanisms produce substantially different climatologies. This raises the interesting question whether these differences can actually be used to validate the assumptions and details of the underlying dissipation mechanisms. If this were the case then one could potentially use such GCM experiments to critically evaluate the validity of each approach. In fact, this is already assumed in a number of studies [e.g., Charron et al. (2002) conclude that the energy dissipation arising from the application of the Hines scheme is more realistic than that produced by Warner and McIntyre (2001)].

However, there is possibly a simpler explanation for the different climatologies. Rather than depending on the specific details of each dissipation mechanism, the different climatologies might simply be due to systematic differences in the altitude of momentum dissipation. That is, if parameter settings within the schemes allowed the reduction, or even the elimination, of these systematic differences then one might obtain more similar climatologies. In the next section, we explore this possibility as the explanation for the different climatologies documented here.

## 4. Sensitivity experiments

Here we discuss a set of GCM sensitivity experiments where we reduce the systematic differences in the height at which momentum is deposited. To demonstrate how this is done we begin with a discussion of offline results.

### a. Offline calculations

Figure 2 suggests that the systematic differences in the height at which momentum is deposited (Fig. 3) could be reduced by allowing *C** > 1 in the WM and AD dissipation mechanisms. Increasing *C** means that the waves must now propagate to higher altitudes before being damped (in the case of WM) or obliterated (in the case of AD). It is important to note that in changing *C** the details of the way in which the WM and AD dissipation mechanisms act are not altered.

The values of *C** > 1 used in the GCM simulations that follow are determined by roughly matching the momentum deposition profiles for WM and AD to that for Hines in the case of the CIRA winds and temperatures. This procedure results in values of *C** = 50 for WM and *C** = 200 for AD. The impact of these changes on the elevation of momentum deposition is illustrated in Fig. 9, where left and right columns correspond to winter (50°S) and summer (50°N) profiles, respectively. Results for *C** = 1 are displayed in Figs. 9a,b, while those for the modified values of *C** are shown in Figs. 9c,d. For reference, profiles for Hines have also been included in this figure. Comparison of the top and bottom rows of Fig. 9 show that the new settings for *C** reduce most of the systematic difference in the elevation of momentum deposition between the schemes.

The important point here is that, although we have some control over the height of momentum deposition for both AD and WM dissipation, we have no control over its vertical structure. That is controlled by the details of each nonlinear dissipation mechanism, as well as by the vertical structure of the background state. For interactive GCM simulations it is expected that the spatial structure of the momentum deposition will be further complicated by feedback processes between the form of the nonlinear dissipation and the response of the large-scale winds and temperature.

### b. GCM simulations

We repeat the GCM simulations discussed in section 2 but now using *C** = 50 for WM dissipation and *C** = 200 for AD dissipation. The resulting zonal-mean zonal winds for JJA for the 100-hPa launch case are shown in the first column of Fig. 10. For comparison purposes, the Hines results presented earlier are also shown. Ignoring the bottom row and two right columns for the time being, we see that the winds for WM and AD now bear a striking resemblance to those for Hines. All three exhibit the strong reversal in the summer mesosphere and the slight equatorward tilt of the winter mesosphere jet. Similar results are obtained for DJF (not shown).

A comparison of the momentum deposition shown in the second column of Fig. 10 reveals why the winds in these three simulations are so similar. The latitude–height distribution of momentum deposition for each simulation is essentially indistinguishable. Since the parameterization forces the GCM through the momentum deposition alone, the GCM response in these experiments must closely resemble each other.

The real surprise in all of this is the similarity of the momentum deposition. From the offline calculations we expected to have some influence over the height at which momentum is deposited but not over its spatial distribution. It appears that feedback processes in the GCM simulations conspire to reduce, rather than accentuate, differences in the nonlinear dissipation mechanisms.

The bottom row of Fig. 10 displays results from an additional simulation in which nonlinear dissipation is turned off and critical-level (CL) dissipation is allowed to act alone. Remarkably, the zonal winds and momentum deposition are virtually indistinguishable from the other three. Similar behavior occurs for DJF (not shown). This additional experiment offers one possible explanation for the similarity displayed in Fig. 10, namely that the momentum deposition (column 2) for each of the Hines, WM (*C** = 50), and AD (*C** = 200) simulations is dominated by CL dissipation and so each necessarily produces the same GCM response. If this were the case then the comparison undertaken in Fig. 10 would say little about the GCM response to differences in nonlinear dissipation mechanisms.

To investigate this possibility we have decomposed the momentum deposition displayed in column 2 of Fig. 10 into its two components namely nonlinear dissipation (column 3) and CL dissipation (column 4).^{6} Inspection of this decomposition for each of the Hines, WM (*C** = 50), and AD (*C** = 200) simulations reveals that nearly all of the momentum deposition above 50 km is dominated by nonlinear dissipation and not by CL dissipation. In each of these three simulations CL dissipation appears to dominate below this level. It is only when the nonlinear dissipation is turned off that CL dissipation effectively takes its place above 50 km (row 4, final column of Fig. 10).

Consequently, the results summarized in Fig. 10 comment directly on the sensitivity of the GCM response to differences between the nonlinear dissipation mechanisms. The conclusion is that significant differences between the various nonlinear dissipation mechanisms have little or no impact on the GCM response. The differences in GCM response identified in section 3 (Figs. 5 –8) are, therefore, due to systematic differences in the height at which momentum is deposited rather than to the details of each scheme.

In viewing Figs. 5, 6 and 10 one implicitly interprets the zonal-mean zonal wind as the basic state upon which the GWD parameterizations act. This generally affords considerable insight. For the simulation employing CL dissipation alone, however, the interpretation of CL dissipation acting on the zonal-mean wind appears to be insufficient to explain the GCM response. For example, the initiation of the summertime mesospheric wind reversal does not seem possible. Since the zonal-mean zonal winds in months preceding summer do not have a reversal in the mesosphere, the eastward traveling waves will not see critical levels and a reversal in the summer should not be possible. The same would also be true of the wintertime mesospheric wind reversal.

If the zonal-mean winds themselves cannot explain the initiation of a feature like the mesospheric wind reversal then another mechanism is required. A possible explanation involves both the zonal-mean flow and longitudinally dependent disturbances (e.g., resolved gravity waves and tides). If the amplitudes of these waves are sufficiently large, some local wind profiles will contain critical levels for eastward traveling waves that can initiate the required reversal. Since middle atmosphere GCMs exhibit enhanced wind variance at all zonal wavenumbers in the mesosphere (Koshyk et al. 1999), this explanation seems highly plausible.

To investigate this possibility, additional sensitivity experiments have been performed. In the top row of Fig. 11 we present two realizations of the zonal-mean zonal wind for JJA for the case of CL-only dissipation: the first (Fig. 11a) is the third year of the 5-yr ensemble shown in Fig. 10; the second (Fig. 11b) is a seasonal integration (initialized on 1 April of year 3) in which the GWD is computed in the GSP from the zonal-mean, rather than the local, wind and temperature fields. The expectation is that the zonal averaging will eliminate critical levels caused by longitudinal disturbances and prevent the onset of the summertime mesospheric wind reversal. This behavior is clearly borne out in Fig. 11b. The corresponding results for the same experiment, but performed now for year 3 of the WM simulation presented in Fig. 10, are displayed in Figs. 11c and 11d. As expected, with the inclusion of nonlinear dissipation, the GCM response is relatively insensitive to longitudinal disturbances.

## 5. Discussion

The central finding of our study is that the GCM response is surprisingly insensitive to the details of the nonlinear dissipation mechanisms employed in current nonorographic GWD parameterizations. This conclusion is supported by the remarkable similarity of the spatial distribution of momentum deposition, and hence the GCM response, in the sensitivity experiments where *C** > 1 is employed in the WM and AD dissipation mechanisms (Fig. 10). While the physical validity of *C** = 50 or *C** = 200 might be questioned (see appendix B), this is not relevant since the point of this exercise is to increase the height at which momentum is deposited without changing the details of how it is deposited. In allowing *C** > 1, the same differences between the three nonlinear dissipation mechanisms that are operative in the *C** = 1 set of experiments are also operative in *C** > 1 set of experiments.

The GCM response arises from feedbacks between the resolved winds and temperatures and the momentum deposition from the parameterized waves. It is therefore expected that these feedback processes should be sensitive to the details of how momentum is being deposited, such as the wavenumber and frequency dependencies of the dissipation mechanisms or the criteria used to initiate dissipation. For example, the dissipation at any one spectral element for Hines depends explicitly on the properties of the waves at all other spectral elements, whereas WM and AD dissipation have no such dependence. However, the set of experiments using *C** > 1 suggest that these feedback processes and the resultant GCM response are surprisingly insensitive to such details.

The insensitivity of the GCM response to the details of the nonlinear dissipation mechanisms is further underscored by the experiment where all nonlinear dissipation is turned off and critical-level dissipation alone is allowed to act. Once again, the momentum deposition and wind response are nearly indistinguishable from that produced when the Hines and modified WM and AD dissipation mechanisms are active.

These results do not suggest that CL dissipation acting alone can mimic the action of nonlinear dissipation mechanisms in all circumstances. As we demonstrate, it requires the presence of resolved waves in the GCM to initiate the onset of wind reversals. Furthermore, there are no tunable parameters when CL dissipation acts on its own. It produces one GCM response, which happens to mimic the response to the Hines scheme. While this might at first seem surprising, it appears that the physical mechanism that is invoked in the formulation of Hines dissipation is analogous to that which operates in the GCM experiments employing only CL dissipation. This may be understood as follows.

As discussed earlier, the Hines dissipation mechanism treats nonlinearity by enhancing the basic-state wind with an rms measure of the wave-induced winds (*U*_{rms}). This provides dissipation by enhancing critical-level filtering. In this formulation *U*_{rms} represents a measure of the wind perturbations due to *unresolved* waves in the (parameterized) spectrum and it increases exponentially with height because of the exponential decrease of ambient density.

For the CL-only experiments, the impact of the dissipation on the zonal-mean winds cannot be understood simply in terms of critical-level filtering by those winds. To understand its impact one must take into account the spatial and temporal fluctuations about the time- and zonal-mean winds produced by resolved waves in the GCM. From the perspective of the zonal-mean flow, these fluctuations may also be characterized by a *U*_{rms}. In this case *U*_{rms} represents a measure of the wind perturbations due to *resolved* waves in the GCM and it also increases exponentially with height because of the decrease of density.

This interpretation of the CL-only dissipation provides some explanation as to why the experiments employing Hines dissipation and CL-only dissipation produce similar GCM responses. In this regard, the CL-only experiment may be viewed as being analogous to the Doppler spread theory underlying the Hines dissipation mechanism. A caveat is that since middle atmosphere GCMs differ in the amount of resolved wave activity in the mesosphere (Koshyk et al. 1999), it is expected that the impact of CL-only dissipation would be model dependent.

The central assumption of AD99 is that one incurs no detrimental effects by depositing all the launch momentum flux of a spectral element to the flow at the initial onset of instability. The fact that the GCM response is so similar between the WM (*C** = 50) and AD (*C** = 200) simulations, which use the same criterion to determine the onset of instability, may be interpreted as support for this assumption.

Taken together, the results of this study have several implications. One is that GCMs cannot be used in a straightforward manner to validate these dissipation mechanisms. Another is that efforts might be better invested in other aspects of the parameterization problem, such as the properties of the source spectra, which should have more impact on the GCM response than the details of the nonlinear dissipation.

Future work on this problem will be directed toward investigating the sensitivity of wave driven oscillations such as the quasi-biennial oscillation (QBO) to the use of the Hines, WM, and AD dissipation mechanisms. Preliminary investigation suggests that a similar insensitivity to the details of nonlinear dissipation is obtained in GCM experiments using nonorographic GWD to generate a QBO-like oscillation.

A portable version of the GSP that employs the three nonlinear dissipation mechanisms presented here is freely available online at http://www.cccma.bc.ec.gc.ca/~jscinocca/. This version is based on the simpler hydrostatic nonorographic GWD parameterization described in Scinocca (2003).

## Acknowledgments

The authors thank T. Shepherd, S. Kharin, O. Saenko, and two anonymous reviewers for valuable comments on the manuscript and/or for helpful discussions. C. McLandress acknowledges support from the Global Chemistry for Climate Project, which is funded by the Natural Sciences and Engineering Research Council of Canada, the Canadian Foundation for Climate and Atmospheric Sciences, and the Canadian Space Agency.

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## APPENDIX A

### Validation of Hines Dissipation in the GSP

The implementation of Hines dissipation in the GSP is identical to that derived by H97. This is validated by performing an offline comparison of the GSP and an operational version of the Hines parameterization (McLandress 1998). To do this, two modifications to the GSP must be made. First, the spectral form of the launch spectrum is changed from the generalized Desaubies spectrum (2) to that used in the operational version, which employs a vertical wavenumber (*m*) spectrum of the horizontal-wind variance that increases linearly with *m*. In the GSP this corresponds to a launch momentum flux density that is independent of *k* and *ω*.

The second modification involves a rescaling of the horizontal wavenumbers used in the GSP. The representative horizontal wavenumber *k** used in the operational version is typically set to wavelengths in the range 500–1000 km. While this technically satisfies the H-R condition *ω̂* ≠ 0 from the dispersion relation (1), it falls well outside the physical range *f* ≤ *ω̂* ≤ *N _{o}*, which is imposed on the GSP launch spectra for typical values of

*m*. To circumvent this problem, we define the launch spectrum over a limited range of wavenumbers having a mean value of

*k̃** = 2.5 × 10

^{−4}m

^{−1}and simply scale the resultant momentum deposition on each level by

*k**/

*k̃**.

The momentum flux density is discretized in each of the four cardinal azimuths using *n _{k}* =

*n*= 512 horizontal wavenumbers and frequencies. Other parameter settings are Φ

_{ω}_{1}= 1.5, Φ

_{2}= 0.3, a low-

*m*cutoff of 1/(3 km),

*k** = 5 × 10

^{−6}m

^{−1}, and

*σ*= 1.5 m s

_{T}^{−1}at the launch height that is taken to be near the surface. The results of the comparison, which employ the CIRA wind and buoyancy frequency (Fig. 4), are shown in Fig. A1. The profiles of the horizontal wind acceleration computed using the GSP and the operational scheme are essentially indistinguishable, which demonstrates that the use of Hines dissipation in the GSP exactly reproduces H97.

As discussed in H97, the correct implementation of (3) requires iteration. Convergence of *U _{rms}* occurs when the remaining waves in the shaded region of Fig. 1c produce variances

*σ*and

*σ*that are equal to those used in (3). The dashed curve in Fig. A1 represents a convergent solution of the operational version of the Hines parameterization and, as can be seen, is smoother at all elevations. Comparison with the noniterated solution indicates qualitative similarity. Figure A1 also indicates that the iteration procedure does not lower the height at which the Hines scheme initially begins to deposit momentum.

_{T}## APPENDIX B

### Amplitude of the m−3 Saturated Spectrum

*m*by the observed

*m*

^{−3}form of the wind variance density:where

*D*is a nondimensional constant. Observational estimates of this constant fall in the range of 1/6 ≤

*D*

_{obs}≤ 1/2 (e.g., Smith et al. 1987). In this appendix we shall compare the value of

*D*implied by the application of saturation in the WM dissipation mechanism to this observational estimate.

The basic assumption of the WM approach is that the launch spectrum at large *m* is saturated. Therefore, the normalization of the launch spectrum [i.e., the constant *B* in (2)], determines the normalization, *D*, of the saturated spectrum. As discussed in section 2, the normalization of the launch spectrum is determined by the specification of the total momentum flux directed into each azimuth. Our goal then is to derive an expression for *D* in terms of this total momentum flux, which can be directly compared to the observed value.

*ρF̂*[Eq. (4) of S02]:where

_{o}*c*=

_{gz}*ω̂*/

*m*is the hydrostatic group velocity. In deriving (B2) we have used relations (1) and (2).

^{7}

*F*

^{tot}

*(*

_{o}*ϕ*) is given by the integral of (B2) over all

*ω̂*and

*m*. This requires evaluation of the two integralsandresulting in the expression for the constant

*B:*At launch, the saturated wave energy density is given bywhich is the limiting form of (2) for asymtotically large

*m*. Substitution of (B5) into (B6) and integrating with respect to

*ω̂*results in the following expression for the saturated wave energy density at launch:Integrating (B7) with respect to

*ϕ*and noting that the horizontal wind variance differs from

*Ê*

^{sat}by a factor of 2 yields the following value for

*D*:where

*N*is the number of azimuths. For

_{ϕ}*m*∗ = 2

*π*/2000 m

^{−1},

*p*= 1.5,

*F*

^{tot}

*(*

_{o}*ϕ*) = 0.7 × 10

^{−3}Pa,

*N*= 4, and typical midlatitude tropopause launch values of

_{ϕ}*f*= 10

^{−4}s

^{−1},

*N*= 0.02 s

_{o}^{−1}, and

*ρ*= 0.15 kg m

^{−3}, we obtain

*D*= 1.66 × 10

^{−2}. This is 10 to 30 times smaller than the observed estimate of 1/6 ≤

*D*

_{obs}≤ 1/2. This suggests that the introduction of the factor

*C** in (4), or equivalently (B1), is not necessarily unphysical. Taking

*C** =

*D*

_{obs}/

*D*indicates that a value of 10 ≤

*C** ≤ 30 would be required for the saturated spectrum in WM to better match observations in the present application.

^{1}

As described in appendix A, this comparison is performed by changing the GSP’s launch spectrum from the generalized Desaubies used in this study to that used in operational versions of the H97 parameterization.

^{2}

Figure 2a may be interpreted simply as a constant-*ω* cross section through the shaded region of Fig. 1a for the case where *U* = 0.

^{3}

In the absence of mean winds, the spectrum remains isotropic at all heights. Although momentum is still being deposited in each azimuth as a result of nonlinear dissipation, the net eastward and northward momentum fluxes are identically zero.

^{4}

The jagged nature of the Hines profile is simply a consequence of not iterating to obtain *U*_{rms} in (3) as discussed in appendix A.

^{5}

The increase of wind speed seen for WM in Fig. 7c arises as a result of a latitudinal shift in the position of the jet axis.

^{7}

In this analysis we have implicitly assumed that *p* ≠ 1 or 2. It is straightforward to extend these results to those two cases.