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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.

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