The effect of extratropical wave forcing on tropical upwelling

R. K. Scott, P. H. Haynes

Department of Applied Mathematics and Theoretical Physics, Cambridge, UK

FIGURES

Abstract

1. Introduction

Crucial to our understanding of chemical and radiative properties of the middle atmosphere is an understanding of the mean meridional circulation and its dynamical and radiative causes. By mean meridional circulation we mean a zonally averaged, long-term, systematic motion of air parcels. This circulation plays an important role in carrying chemical species away from source regions, as well as creating significant departures from local thermodynamic equilibrium. See, eg., WMO (1985), Andrews et al. (1987) and a recent review in WMO (1999).

Because of the angular momentum structure of the atmosphere, dominated by the contribution from the Earth's axial rotation, a mean meridional velocity at a particular extratropical location cannot persist without a systematic zonal momentum force acting at the same location. This force is required for parcels of air to cross the nearly vertical isosurfaces of angular momentum. Such effects have been recognized since the works of Dickinson (1968,1969), and discussed more recently in Held & Hou (1980), and Haynes et al. (1991). The latter derived an expression for the zonally symmetric, steady state vertical velocity, which, under time-periodic conditions, can be extended to

$\begin{displaymath} \langle w \rangle = \frac{1}{\rho_0\cos\phi} \frac{\partial}... ...{\langle m_\phi\rangle} \right\}_{\phi=\phi(z')} dz' \right\}. \end{displaymath}$ (1)

(1)

Here $\langle\cdot\rangle$ denotes an annual average, the primes denote the time varying component, ${\mathcal J}=\partial(\psi',m')/\partial(\phi,z)$ is the Jacobian determinant, and the integral is along a contour of constant angular momentum. Thus, the force has a ``downward control'' on the meridional circulation.

Despite a good understanding of the mechanisms involved in driving the observed mean meridional circulation in the extratropics, we still do not have a clear idea of those involved at low latitudes, where contours of angular momentum deviate significantly from the vertical, and where the dynamical link between the zonal mean velocity and temperature fields is weaker. In particular there is still no complete explanation of the non-zero time-averaged upward motion observed in low latitudes in the lower stratosphere (hereafter ``tropical upwelling'') Some progress has been made recently by Plumb & Eluszkiewicz (1999), who concluded that the mean meridional circulation was predominantly linear even at low latitudes and that artificial viscosity was ultimately responsible for the tropical upwelling that they obtained in an idealized model. In the present work we examine the relative importance to tropical upwelling of several different effects, namely the seasonal cycle (transience), nonlinearity, and low latitude diffusive type forces (whether real or artificial). While we agree with Plumb & Eluszkiewicz (1999) that the tropical upwelling response has some linear dependence, we show that nonlinear effects nevertheless play an important role.

The outline of the work is as follows. In Section 2, we give a brief description of the model and diagnostics used later, followed by an example of a meridional circulation obtained with a model in which waves are included explicitly. We consider both seasonally varying and steady state calculations. In Section 3, we compare the results of Section 2 with the circulation obtained by forcing a zonally symmetric model with the zonal momentum force obtained from the wave-one model. In Section 4, we consider an idealization of the form of the wave forcing to investigate in more detail the importance of nonlinearity, diffusion, and transience. In section 5, we give a short summary of the main results.

2. The meridional circulation in a zonally asymmetric model

The model used in this section is the same as that described in Scott & Haynes (2000b) based on the primitive equations on the sphere. There are 31 vertical levels equally spaced in $\log p$ , where p is the pressure, and a spectral representation in the horizontal comprising 31 meridional modes, and the first (wavenumber-one) zonal mode. The model is forced radiatively by including Newtonian relaxation to a zonally symmetric radiative basic state that may be either constant in time or time-varying, the latter case providing an approximation to the annual cycle. Waves are forced in the model by applying a wave-one perturbation to the geopotential height at the lower boundary, in the nothern hemisphere only. Horizontal hyperdiffusion and vertical diffusion are included for model stability, with values of 4/day on the smallest wavenumber and 0.1m/s/s respectively.

We present two sets of experiments, one in which the radiative basic state is constant in time, W1, corresponding to a perpetual winter (Figure 1a), the other in which the radiative basic state is time varying, W2, corresponding to an annual cycle (Figure 1b). Each set is forced with geopotential wave amplitudes h₀=210m, h₀=240m and h₀=270m. From Figure 1a we see that, althrough the increments in forcing amplitude are equal, there is a much greater increase in the tropical upwelling (and extratropical downwelling) when between h₀=240m and h₀=270m than between h₀=210m and h₀=240m. The reason is that between h₀=240m and h₀=270m the model response changes suddenly from a quiescient, southern hemisphere type evolution to a more disturbed northern hemisphere type evolution, similar to the model transitions found in Scott & Haynes (2000a). The increase in upwelling at h₀=270m is consistent with a similar increase in the vertical component of the Eliassen-Palm flux through the lower boundary (not shown).

In Figure 1b we see a similar small increase between h₀=210m and h₀=240m for the case of a time varying radiative basic state. For h₀=270m, however, a new model regime is found in which there is a biennial oscillation of the type investigated in Scott & Haynes (1998) (see also Scott & Haynes, 2000b), the two dotted lines in Figure 1b representing the meridional circulation produced by consecutive years with alternating quiescient and disturbed winters. Again, the disturbed year is characterized by larger values of the vertical component of the Eliassen-Palm flux through the lower boundary than those of the quiescient year.

In all cases note that there is significant upwelling at low latitudes, and that the magnitude is consistent with those values measured in the real atmosphere. Since the wave induced zonal momentum forcing decreases to very nearly zero south of around $10^\circ$ N, straightforward application of (1) fails to explain the upwelling response.

Figure 1: Residual vertical velocity, $\bar{w}^\star$ , at 20km altitude derived from heating rates according to $w=\bar Q/\bar\theta_{0z}$ , where Q is the diabatic heating, and $\theta_0$ is a reference potential temperature profile: (a) steady state response to wave-one forcing with amplitude h₀=210m, 240m and 270m; (b) annually averaged, seasonally varying response to wave-one forcing with amplitude h₀=210m, 240m and 270m. The dotted lines in (b) represent two consecutive annual cycles of the h₀=270m case.

3. The meridional circulation in a zonally symmetric model

We now investigate to what extent the meridional circulations produced in the wave-one model of the previous section can be reproduced in a zonally symmetric model using an applied zonal momentum force that is derived from the wave-induced momentum forcing of the wave-one model. Three sets of experiments are described. In the first, Z1, the radiative forcing is the same as that of the perpetual winter cases, W1, of Section 2 above and the zonal momentum force is the steady state wave induced zonal momentum force per unit mass produced by h₀=240m, that is

$\begin{displaymath} \mathcal{F}=\frac1{a\rho_0\cos\phi}{\bf\nabla\cdot F} \end{displaymath}$ (2)

where ${\bf\nabla\cdot F}$ is the Eliassen-Palm flux from W1 (eg. Andrews et al., 1987). In the second, Z2, the radiative forcing is the same as that of the seasonally varying cases, W1, of §2 above and the zonal momentum force is the seasonally varying wave induced zonal momentum force per unit mass produced by h₀=240m. In the third, Z3, both the radiative forcing and the zonal momentum force are constant in time and equal to the annual average of the values used in Z2.

The results are shown in Figure 2. We see that the zonal model produces similar upwelling responses in both the steady state and in the seasonally varying cases (Figure 2a,b solid and dashed lines). This is despite having smoothed the zonal force in both space and time, verifying that it is the broad scale structure that is important in determining the response. Figure 2b also shows the response to Z3 (dotted line), the steady state calculation forced with the time averaged values of the seasonally varying Z2. We see that there is a slight decrease in magnitude of the upwelling response, of about 20%, from that of Z3. Since linear theory predicts that the two responses should be the same, this suggests that there may be some nonlinear interaction taking place between the zonal momentum forcing and the seasonal cycle in radiative basic state. This will be considered more fully below.

Figure 2: As Figure 1 but for a zonally symmetric model forced with the wave induced zonal momentum force per unit mass output from W1 and W2 with h₀=240 (solid), together with the associated W1/W2 reponse (dashed): (a) steady state response; (b) annually averaged, seasonally varying response. The dotted line in (b) is Z3 (see text).

4. The effects of nonlinearity, of model diffusion, and of the seasonal cycle

To isolate the different mechanisms potentially responsible for the low-latitude upwelling response, we consider a simple idealization of the form of the zonal body force. From here on this is given a compact form with a maximum of ${\bar F}$ at a variable latitude, ${\phi_0}$ , and decaying smoothly to zero 15 $^\circ$ on either side:

$\begin{displaymath} \mathcal{F}= \left\{\begin{array}{lcr} {\bar F} \cos^2\left(... ...rt\le20\mbox{km} \\ 0 & & \mbox{otherwise} \end{array}\right. \end{displaymath}$ (3)

Note also the z-dependence, which is zero below 20km. We choose an amplitude of ${\bar F}=-2$ m/s/day, as being physically reasonable in terms of the response produced. This is reduced by up to a factor of 100 in selected experiments below to isolate linear properties of the model response. The radiative relaxation rate is increased to 1/(10 days) and the ``annual cycle'' is reduced to 200 days to reduce the model run-time. Finally, note that the form of the radiative basic state is adjusted slightly to be antisymmetric between hemispheres; this is simply so we can use a basic state that corresponds to a resting atmosphere as the annual average of the seasonally varying basic state. Figure 3 illustrates different upwelling responses for forcing centred on ${\phi_0}=40^\circ$ , $30^\circ$ , $25^\circ$ and $20^\circ$ , for the case of steady forcing and steady radiative basic state (no seasonal cycle) corresponding to a resting atmosphere. Note the significant low-latitude upwelling obtained for ${\phi_0}\le25^\circ$ .

Figure 3: Residual vertical velocity of the zonally symmetric model forced with the compact zonal momentum force with (moving equatorward) ${\phi_0}=40^\circ$ , ${\phi_0}=30^\circ$ , ${\phi_0}=25^\circ$ , ${\phi_0}=20^\circ$ .

a. Model hyperdiffusion and vertical viscosity

In considering the latitudinal scale of an induced upwelling response to given forcing it is useful to consider simple scaling arguments. Using similar analysis to that used in Holton et al. (1995) and Garcia (1987) the linearized transformed Eulerian mean equations on the sphere, with a frictional force, $-\kappa u$ in the zonal momentum equation, leads to a steady state relation of the form

$\begin{displaymath} \frac{\partial}{\partial z}\left(\frac{1}{\rho_0}\frac{\part... ...{\sin\phi} \frac{\partial\hat{\mathcal{F}}}{\partial z}\right) \end{displaymath}$ (4)

where $\kappa$ is the friction time-scale, $\alpha$ is the radiative time-scale, and other variable are as in Holton et al. (1995). If the frictional term is replaced by a hyperdiffusion of the form $-\nabla^8u$ (as in our model), then $\kappa$ in (4) can be replaced by ${\gamma_{\max}}\delta^8$ , where ${\gamma_{\max}}$ is the rate at which the smallest wave-numbers are damped, and $\delta$ is their lengthscale (ie. the grid resolution). Balancing terms of (4) outside the forcing region leads to horizontal length scales

$\begin{displaymath} L\sim \left(\frac{\gamma_{\max}}\alpha\right)^{\frac1{10}} \... ...}}\beta\right)^{\frac15} \left(\frac2{a^2}\right)^{\frac1{10}} \end{displaymath}$ (5)

in mid latitudes and

$\begin{displaymath} L\sim \left(\frac{\gamma_{\max}}\alpha\right)^{\frac1{12}} \... ...\frac23} \left(\frac{N D_{\mbox\tiny -}}\beta\right)^{\frac16} \end{displaymath}$ (6)

in low latitudes, where $a\sin\phi$ has been approximated by a/2 and L respecitively, and where $D_{\_}=\sqrt{\min(HD,D^2)}$ , with D a vertical lengthscale. In both cases, L is the scale on which an upwelling response decays away from the forcing region. Note that the dependence on ${\gamma_{\max}}$ is very weak, so that moderate changes in the hyperdiffusion will have little effect. On the other hand a bigger effect can be expected from moderate changes in the resolution $\delta$ . Figure 4a shows the effect of changing the resolution from 31 (solid line) to 64 (dash-dotted line) meridional modes for the case of ${\phi_0}=25^\circ$ . The narrowing of the spreading of the response is in good agreement with (5) in midlatitudes but not at low latitudes, where there is little change between the two resolutions.

One possible reason for the above disagreement is the deviation from the vertical of contours of constant angular momentum in low latitudes, induced by the nonlinear zonal momentum forcing. To investigate this, a series of (nearly) linear experiments are presented in Figure 4b with forcing 1/100th that used in Figure 4b, and horizontal resolution of 31 modes (solid) or 64 modes (dash-dotted line) as above. Now, the high resolution case shows a similar narrowing of the response at both high and low latitudes, in contrast to the nonlinear case. We conclude, therefore, that while the hyperdiffusion must certainly play an important role in upwelling response, the mechanism is fundumentally nonlinear. Note also that although the linear forcing is 1/100th that of the nonlinear, the response is reduced only by a factor of 1/50. Finally is interesting to note the response when the hyperdiffusion is zero, but including a vertical viscosity of 1m/s/s. Again this gives a non-zero tropical upwelling response, though with a different form than that obtained with hyperdiffusion.

Figure 4: As Figure 3 with ${\phi_0}=25^\circ$ : (a) nonlinear forcing ( ${\bar F}=-2\times10^{-5}$ m/s/s) with 31 meridional modes (solid) and 64 meridional modes (dot-dashed); (b) linear forcing ( ${\bar F}=-2\times10^{-7}$ m/s/s) and frictionless (dotted), with hyperdiffusion only (solid), and with vertical viscosity only (dashed).

b. The seasonal cycle

When the zonal velocity basic state is not that of a resting atmosphere, angular momentum contours are again distorted from the vertical, this distortion being more pronounced at low latitudes. The distortion can have a significant impact on the upwelling response, as can be seen in Figure 5a, for a linear case with steady forcing and radiative basic state corresponding to a resting atmosphere (dotted) and to a perpetual winter (solid). In midlatitudes the change in angular momentum distribution has the effect of shifting the upwelling/downwelling response equatorwards, resulting in a response similar to that which would be obtained by retaining the resting atmosphere basic state and introducing an equatorward displacement of the forcing region. The effect at low latitudes can be then understood in terms of this apparent equatorward displacement, which results in an increased tropical upwelling (cf Figure 3). We may therefore expect that when a realistic seasonal cycle is included in the radiative basic state the upwelling response be modified. (Recall that such a modification was already apparent in Figure 2b.) Further, from (1) we see that nonlinear interactions between the time varying components of meridional velocity, v', and the latitudinal gradient of the momentum, $m'_\phi$ , in ${\mathcal J}$ can act to produce a non-zero time-averaged response away from the forcing region.

Figure 5b and Figure 5c show the change in response between the annually averaged seasonally varying response (solid) and the steady state response to forcing values equal to the annaully averaged thermal and momentum forcing (dotted), for linear and nonlinear momentum forcing respectively. Equation (1) can help explain the equatorward shift in the midlatitude seasonally varying response, however it may not be used at the equator, where angular momentum contours fail to span the vertical. Here, as in the recent study by Plumb & Eluzsliewicz (1999), it seems that the very small zonal forces generated by the model diffusion are sufficient to create an upwelling response that penetrates well into the opposite hemisphere. Figure 5bc does suggest however that the nonlinear interaction implied by (1) is sufficient to shift the apparent forcing latitude equatorwards, with the result that the small diffusive forces and hence the upwelling response are increased at low latitudes.

Figure 5: As Figure 3 with ${\phi_0}=25^\circ$ : (a) steady state response to linear forcing with a radiative basic state corresponding to a resting atmosphere (dotted) and a perpetual winter (solid); (b) annually averaged response seasonally varying forcing and heating (solid) and the corresponding steady state response to steady forcing and heating (dotted), with nonlinear forcing; (c) as (b) but with linear forcing.

5. Summary

We have used a simple mechanistic model of the stratosphere to investigate the mean meridional circulation response of an atmosphere subjected to zonal momentum and radiative forcing, with particular emphasis on the low-latitude ``tropical upwelling''. Both a zonal wavenumber-one model with self-consistent waves and a zonally symmetric model in which the momentum forcing was derived from the wave-one model, produce upwelling responses that are in good agreement with each other, as well as with vertical velocities observed in the real stratosphere.

The zonally symmetric model was subsequently used with a compact momentum forcing to investigate more cleanly the effects of model diffusion, the seasonal cycle, and nonlinearity. It was found that, while diffusion was necessary to produce an upwelling response that penetrated deep into the tropics, nonlinearity and transience each played an important role in modifying the response. In particular, the dependence of the tropical upwelling on hyperdiffusion was markedly different between the linear and nonlinear cases. Further, when a seasonal cycle in both the momentum and the radiative forcing was included, the time average response was significantly greater than the steady state response obtained with the time averaged forcings.

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