Equatorial Oscillations in a Middle Atmosphere GCM

Charles McLandress

University of Toronto, Toronto, Canada

FIGURES

Introduction

Until recently middle atmosphere general circulation models (GCMs) have been unable to simulate long-period zonal mean wind oscillations in the tropics like the quasi-biennial oscillation (QBO). The inability of first principles models to generate the QBO can be attributed in part to Brewer-Dobson upwelling in the equatorial lower stratosphere which acts to suppress the oscillation. This was demonstrated by Dunkerton [1997] who showed that in the presence of mean upwelling the total wave momentum flux which is needed to maintain the QBO is several times larger than that estimated from observed planetary-scale waves and that the additional flux most likely comes from small-scale gravity waves which are not resolved by most GCMs.

In this presentation the Doppler spread gravity wave parameterization (DSP) of Hines [1997] is used to generate equatorial oscillations. The discussion consists of two parts. In Part 1 a mechanistic model is used to demonstrate the ideas of Dunkerton [1997]. Here the DSP is used in the context of a one-dimensional model like that used by Holton and Lindzen [1972]. In Part 2 results from the Canadian Middle Atmosphere Model (CMAM) are presented. Here the relative roles of parameterized and resolved wave drag, the tendency for the period of the oscillation to lock itself to the seasonal cycle, as well as the impact of the finite-difference formulation of the parameterization scheme, are discussed.

 

Mechanistic Model

Description

The prognostic equation for the zonal mean wind ($\bar{u}$) in the one-dimensional mechanistic model is given by

\begin{displaymath} \bar{u}_t = -\, \rho_o^{-1} (\rho_o\overline{u'w'})_z -\,\bar{w}\bar{u}_z +\, \rho_o^{-1} (\rho_o\mu\bar{u}_z)_z \end{displaymath} (1)

where $\rho_o=\rho_s$ e$^{-z/H}$ and $H=7$ km. The three terms on the right hand side are the momentum flux divergence arising from vertically propagating waves, vertical advection, and diffusion. Boundary conditions of zero mean wind and zero vertical gradient of the mean wind are applied at the model bottom (17 km) and top (60 km), respectively. The diffusion coefficient is specified as

\begin{displaymath}\mu=0.3+50[1+\tanh\left({{z-60}\over{7.5}}\right)] \hspace{0.4cm}\mbox{m$^2$s$^{-1}$} \; . \end{displaymath}


The vertical wind is estimated from the CMAM (Figure 1) and is prescribed as

\begin{displaymath} \bar{w}= w_s+(1\!-\!w_s) \mbox{e}^{-(z-15)}+ (1.5\!-\!w_s) \mbox{e}^{-a(z-51)^2} \end{displaymath} (2)

where $a=(81$ km)$^{-2}$ and $w_s$ is the upwelling in the lower stratosphere which attains a maximum value of 0.2 mm/s. The second and third terms denote the mean vertical motion due to the upper portion of the tropospheric Hadley circulation and the stratospheric nonlinear Hadley cell, respectively.

The momentum flux ( $\rho_o \overline{u'w'}$) is separated into planetary wave and small-scale gravity wave components. The planetary wave fluxes are represented as in Holton and Lindzen [1972] using an eastward propagating Kelvin wave (zonal wavenumber $k=1$) and a westward propagating mixed Rossby-gravity wave ($k=4$). Both waves have phase speeds of 25 m/s and are thermally damped using the Newtonian cooling approximation. The gravity wave fluxes are computed using the DSP. The parameter settings are the same as in the CMAM (see Section 3.1), with the exception of the rms winds which are set to 3 m$^2$/s$^2$ at 17 km.

 

\includegraphics[width=0.4\textwidth]{wres_mean.epsi}

Figure 1: Annual mean residual vertical velocity $\bar{w}^*$ at the equator from CMAM computed from the six-year simulation using the uniform gravity wave source (solid line). See Section 3.2 for a description of this experiment. The dashed line is the analytical form used in the mechanistic model.

 

Results

Results from a single multiyear integration of the mechanistic model are shown in Figure 2. Throughout the integration the planetary wave fluxes are held constant. The vertical wind in the lower stratosphere ($w_s$ in Equation 2) is slowly increased from an initial value of zero to a maximum value of 0.2 mm/s by about day 50, keeping the vertical motion at the tropopause and stratopause fixed. In the presence of the increasing upwelling in the lower stratosphere the period of the oscillation lengthens. By day 35 the oscillation ceases and a steady-state is attained. At day 50 the source level rms winds in the DSP are slowly turned on, keeping the lower stratospheric upwelling fixed. The presence of the additional momentum flux provided by the gravity waves results in the regeneration of the oscillation by day 60 and the subsequent shortening of its period as the gravity wave flux is increased.

 

\includegraphics[width=0.7\textwidth]{toy_model_u30km.epsi}

Figure 2: Zonal mean wind at 30 km computed using the mechanistic model. Planetary wave momentum flux is constant. Upwelling in the lower stratosphere is initially zero and is increased to a maximum value of 0.2 mm/s by day 50 at which point the source level rms winds of the Doppler spread parameterization are turned on.

Canadian Middle Atmosphere Model

Description

Only a brief description of the relevant features of the CMAM are outlined here. For a general description refer to Beagley et al [1997]. The version of the model discussed herein has a T32 spectral resolution and 50 levels in the vertical. This corresponds to a horizontal resolution of approximately 5$^{\circ}$ and a vertical resolution which varies from less than one kilometer in the troposphere to about 3-4 km in the stratosphere and mesosphere. The upper boundary of the model is located at about 97 km. One physical parameterization in the troposphere which plays a crucial role in determining the ease with which a middle atmosphere GCM generates long period mean wind oscillations in the tropical stratosphere is the convection scheme. In the CMAM the deep convective parameterization of Zhang and McFarlane [1995] is used.

The effects of unresolved gravity waves are parameterized using the Doppler spread parameterization. The DSP requires the specification of the vertical wavenumber ($m$) spectrum of horizontal winds at the source level which is taken to be the ground. For simplicity and convenience a wavenumber spectrum that is proportional to $m$, with a minimum allowable wavenumber of 1/(3km), is employed. The momentum flux source is assumed to be horizontally isotropic and independent of time and longitude. More details of the scheme and its impact on the CMAM can be found in McLandress [1998].

Results

The remainder of this presentation focusses mainly on results from two 6-year simulations of the CMAM. The first is one in which the source level rms horizontal winds ($\sigma^2$) of the DSP are set to the uniform value of 1.5 m$^2$/s$^2$. The second simulation is one in which $\sigma^2$ is enhanced at low latitudes, i.e., $\sigma^2(\theta)=1.5+\mbox{e}^{-(\theta/15^{\circ})^2}$. Timeseries of the zonal mean winds at 10 mb (as a function of latitude) and at 3$^{\circ}$N (as a function of height) are shown in Figures 3 and 4, respectively, for the two experiments. The simulation using the uniform rms winds exhibits a strong semi-annual variation in the upper stratosphere and mesosphere but no obvious signs of a longer period oscillation in the lower stratosphere. However, in the presence of the enhanced gravity wave momentum flux in the tropics, an oscillation with a period of approximately 18 months appears. Unlike the observed QBO, this oscillation appears to be phase locked to the annual cycle as seen in the right panel of Figure 3. Synchronization with the seasonal cycle has been observed in other numerical simulations of the QBO using large gravity wave fluxes [Dunkerton, 1997].

\includegraphics[width=0.48\textwidth]{ubar_10mb_ym5fc.epsi}

\includegraphics[width=0.48\textwidth]{ubar_10mb_ym5ff.epsi}

Figure 3: Timeseries of the monthly average zonal mean zonal wind at 10 mb (32 km) from CMAM for the simulation using the uniform gravity wave source (left) and equatorially enhanced gravity wave source (right). Contour interval is 10 m/s.

\includegraphics[width=0.48\textwidth]{ubar_3n_ym5fc.epsi}

\includegraphics[width=0.48\textwidth]{ubar_3n_ym5ff.epsi}

Figure 4: Timeseries of the monthly average zonal mean zonal wind at 3$\,^{\circ }$N from CMAM for the simulation using the uniform gravity wave source (left) and the equatorially enhanced gravity wave source (right). Contour interval is 20 m/s.

 

Role of Resolved Waves

The role of the resolved waves in driving the oscillation is of obvious importance. Figure 5 shows profiles of the mean winds, Eliassen-Palm flux divergence (EPFD), and gravity wave momentum flux divergence (GWD) at two different phases of the oscillation separated by approximately one-half of a period. The dotted curve, which shows the EPFD computed using only zonal wavenumbers 1-10, indicates that the bulk of the resolved wave driving (at least using this relatively coarse temporal sampling) arises from the large-scale components of the flow. GWD is seen to be the dominant term, which is not surprising since the deep convective parameterization of Zhang and McFarlane [1995] is known to produce a very red latent heating frequency spectrum, which results in relatively weak momentum fluxes in the lower stratosphere by the resolved waves.

\includegraphics[width=0.7\textwidth]{u_gwepfd_ym5gf_mono.epsi}

Figure 5: Zonal mean zonal wind (thick solid), gravity wave drag (thin solid), Eliassen-Palm flux divergence for all zonal wavenumbers $k$ (dashed) and for $k=1-10$ (dotted) at 3$\,^{\circ }$N for July year 3 (left) and April year 4 (right) from CMAM for the simulation using the equatorially enhanced gravity wave source.

Finite Difference Considerations

The final set of results which are shown in Figure 6 demonstrate the impact of the form of the finite difference equation used to evaluate the parameterized gravity wave flux divergence. Here, one-sided differences are used in contrast to the centered differences employed in all of the previous results. In the left panel the GWD at level $l$ is computed by differencing the flux at the level above ($l+1$) and the flux at level $l$, while in the right panel fluxes at levels $l$ and $l-1$ are used. In both cases the background density appearing in the denominator is evaluated at level $l$. (The latter form is believed to be currently used in several middle atmosphere GCMs which employ the DSP.) The resulting impact on the oscillation is rather large. Thus, some care must be taken in attributing the period and amplitude of the simulated oscillation to only the magnitude of the parameterized momentum flux at the source level since different finite difference schemes can produce very different results.

\includegraphics[width=0.48\textwidth]{ubar_3n_ym5gi.epsi}

\includegraphics[width=0.48\textwidth]{ubar_3n_ym5gj.epsi}

Figure 6: Timeseries of the monthly average zonal mean zonal wind at 3$\,^{\circ }$N from CMAM for the simulation using the equatorially enhanced gravity wave source and one-sided finite differences from below (left) and from above (right). Contour interval is 20 m/s.

Summary and Conclusions

Equatorial oscillations in the middle atmosphere have been studied using the Doppler spread gravity wave drag parameterization of Hines [1997]. Results from a simple mechanistic model using a prescribed mean upwelling from the Canadian Middle Atmosphere Model demonstrated Dunkerton's [1997] conjecture that an additional source of momentum flux from small-scale gravity waves is required to drive the QBO in the presence of realistic Brewer-Dobson circulation in the equatorial lower stratosphere. Results from several simulations of the CMAM were analyzed. The main conclusions from that part of the study are:

 

Acknowledgements. I would like to express my thanks to both Stephen Beagley for help with some of the technical aspects of producing the poster version of this presentation and Ted Shepherd for helpful discussions.

 

References

Beagley, S. R., J. de Grandpré, J. N. Koshyk, N. A. McFarlane, and T. G. Shepherd, The radiative-dynamical climatology of the first-generation Canadian Middle Atmosphere Model, Atmos. Ocean, 35, 293-331, 1997.


Dunkerton, T. J., The role of gravity waves in the quasi-biennial oscillation, J. Geophys. Res., 102, 26053-26076, 1997.


Hines, C. O., Doppler-spread parameterization of gravity-wave momentum deposition in the middle atmosphere. Part 1: Basic formulation, J. Atmos. Sol. Terr. Phys., 59, 371-386, 1997.


Holton, J. R., and R. S. Lindzen, An updated theory for the quasi-biennial cycle of the tropical stratosphere, . J. Atmos. Sci., 29, 1076-1080, 1972.


McLandress, C., On the importance of gravity waves in the middle atmosphere and their parameterization in general circulation models, J. Atmos. Sol. Terr. Phys., 60, 1357-1383, 1998.


Zhang, G. J., and N. A. McFarlane, Sensitivity of climate simulations to the parameterization of cumulus convection in the Canadian Climate Centre general circulation model, Atmos. Ocean, 33, 407-446, 1995.

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