Abstract

A series of experiments has been performed using an idealized model of the global atmosphere to study the role eddies play in communicating changes in the zonal mean state between the Tropics and extratropics. When an oscillatory heating perturbation centered about the equator is imposed, the author found a poleward-propagating zonal wind anomaly emanating from the Tropics into the midlatitudes when the heat source oscillates with a period of around 25–100 days. At higher frequency, most of the zonal wind perturbation is confined within the Tropics, while at lower frequency, the main signal occurs in the midlatitudes.

The angular momentum budget and Eliassen–Palm cross sections have been examined. The results suggest that eddies act to communicate changes in the Tropics into the midlatitudes in at least two ways. First, changes in zonal mean zonal wind in the Tropics lead to a shift in the eddy angular momentum divergence pattern. Second, heating in the Tropics changes the temperature gradients between the Tropics and midlatitudes, giving rise to changes in the amplitude of eddy fluxes and hence eddy momentum divergence. Both effects act to damp the perturbation in the Tropics, as well as to transmit the tropical perturbation poleward into the midlatitudes. A simple three-component analytical model has been developed based on these ideas, and the model reproduces the main features observed from the numerical model experiments.

Low-frequency (period 200 days and longer) variability excited by tropical heating has been examined further. When the perturbation is a single heat source centered on the equator, the author found that the main response appears to be a standing oscillation in the midlatitudes, with very weak poleward-propagating signal. However, when the author added a heating source at 15° latitude with the opposite phase, an apparently significant poleward-propagating signal from the Tropics into the extratropics was obtained. Analyses suggest that this poleward-propagating signal may just be an illusory superposition of two largely standing oscillations located side by side, each with relatively weak poleward propagating tendency of its own.

1. Introduction

Chang (1995, hereafter C95) studied the influence of Hadley circulation intensity changes on extratropical climate using an idealized model. The results suggested that increasing the intensity of the Hadley circulation by latitudinal concentration of heating within the Tropics may lead to temperature increase in the winter high latitudes, together with an equatorward shift of the midlatitude westerly jet. Chang (1995) also suggested that the effects produced by the increase in the Hadley circulation due to changes in the heating distribution are equivalent to those caused by a westerly acceleration anomaly in the Tropics, and this, together with the fact that the temperature changes were observed to be due to changes in the transport by the mean meridional circulation (MMC) in opposition to changes in the eddy heat fluxes, suggested that the momentum budget instead of the heat budget might be the controlling factor.

Chang (1995) further suggested that since in the control experiment the eddy momentum fluxes act to displace the subtropical jet poleward from its position implied by the eddy-free Held and Hou (1980) solutions, stronger westerly acceleration in the Tropics means that the eddy momentum fluxes will not be as effective in displacing the jet poleward; hence, the jet position in the perturbation experiments is more equatorward when compared to that in the control experiment. Here, we want to understand dynamically how this equatorward shift of the midlatitude westerly jet can be attained as a response to forcings in the Tropics. Chang (1995) suggested that the observed shift in the midlatitude jet in the model simulations could be associated with anomalous westerly acceleration in the Tropics due to the increase in Hadley intensity. However, an equatorward shift of the jet involves not only an acceleration of the jet toward the equatorward flank of the jet, but also deceleration toward the poleward flank of the jet. It is not clear what aspects of the eddy–mean flow interactions lead to the deceleration.

In C95, we examined the steady-state response to time-independent tropical forcing. To examine how the midlatitude response is set up, in this paper we will conduct experiments to study the response of the midlatitude model atmosphere to time-varying tropical forcings to examine how the tropical forcing is transmitted into the midlatitudes. Investigating the time-varying problem is in fact interesting in its own regard. Studies of the axial angular momentum balance of the earth’s atmosphere (e.g., Rosen 1993; Dickey et al. 1992) suggested that there are interannual variations in extratropical relative angular momentum values on ENSO timescales that seem coherently linked to antecedent tropical anomalies, suggestive of poleward propagation of zonal mean feature from the Tropics to well into the midlatitudes possibly excited by tropical heating anomalies. Recent analyses of the variability in the Hadley circulations by Oort and Yienger (1996) suggested the Hadley circulation is stronger during El Niño years as opposed to La Niña years. However, it is not clear why the atmosphere can exhibit such slow timescale propagating signals. Hence it would be interesting to investigate whether tropical forcings in our model can excite zonal mean perturbations that propagate poleward into the midlatitudes and to study the dynamics of such anomalies.

In the following sections, we will use the idealized model to investigate the issues discussed above. In section 2, we will present some results on the model response to a sudden onset of tropical heating. The angular momentum budget will be diagnosed in detail. Then in section 3 the response to oscillatory forcing will be discussed. The response to low-frequency forcing will be discussed in more detail in section 4. Finally, in section 5, we will construct a simple analytical model based on insights obtained from the previous sections to see whether we can represent the main features observed in the numerical experiments using ideas from eddy–mean flow interactions. We will then conclude with some discussion in section 6.

2. Transient response to a sudden onset of tropical heating perturbations

In C95, we investigated the steady-state response to time-independent tropical heating anomalies by comparing the statistically steady climate from the perturbation run with the climate from the control run. Here we will investigate the transient response with two types of time dependence, first a sudden onset of perturbation heating and second, in section 3, we will study the response to oscillatory heating perturbations.

The numerical model used here is the same as that used in C95, except that here we have used only five levels in the vertical instead of 10. We have found that reducing the number of levels does not change the results qualitatively. However, the sensitivity of the response to the forcing is changed somewhat. We have decided to keep fewer levels because we want to conduct series of transient experiments to understand the dynamics of eddy–mean flow interactions better, and keeping 10 levels would strain our computational resources. Five levels appear to be the minimum needed to be retained because experiments with different numbers of levels suggest that a further reduction in vertical resolution will result in the model climate transiting easily into an equatorial superrotating state (e.g., Saravanan 1993) when tropical heating (or acceleration) anomalies are applied as forcing.1

The model is a 5-level, primitive equation spectral model on a sphere, with a spectral cutoff of R21. Some experiments have been repeated using 10-level R21 and T30 models, and the results are very similar. The only“physics” used in the model are radiative damping toward an effective radiative–convective equilibrium temperature structure with a radiative timescale of 20 days and surface friction at the lowest level using simple Rayleigh friction with a damping timescale of 2 days. We also include biharmonic diffusion to control small-scale noise, with a damping timescale of 1.2 days for the highest wavenumber retained (total wavenumber 42). For more details of the model, please refer to C95 and references therein.

The same equilibrium temperature profile used in C95—basically a −ΔH(yy0)2 profile, where y is sine of the latitude, y0 is the latitude of maximum temperature, and ΔH the equator-to-pole temperature difference [see Eq. (5) of C95]—is used here for the control experiment. The values of y0 used corresponds to a latitude of 10°N and ΔH equals 50 K. Hence the imposed pole-to-equator radiative equilibrium temperature difference is 69 K in the Southern Hemisphere and 34 K in the Northern Hemisphere, roughly corresponding to Southern Hemisphere winter conditions. While the model is fully three-dimensional, all imposed forcings and boundary conditions are zonally symmetric. As a reference to discussions in the following sections, the time-averaged zonal mean zonal wind from the control climate is shown in Fig. 1a, while the equilibrium response to a concentration in tropical heating [following Hou and Lindzen (1992), the perturbation basically consists of perturbation heating centered at the thermal equator, flanked on the two sides by cooling such that the total perturbation heating integrated over the entire domain is zero; see C95 for further detail], with peak heating amplitude of 1° day−1, is shown in Fig. 1c. The heating perturbation is shown in Fig. 1b. The response shown in Fig. 1c corresponds to an equatorward shift of the jet in the Southern Hemisphere. The results are basically the same as those discussed in C95 (a 10-level model was used in that study). Below, we will try to dissect how this response is dynamically attained by studying transient instead of equilibrium responses.

Fig. 1.

(a) Climate of the control experiment as represented by zonal mean zonal wind. (b) Heating perturbation applied in the perturbation experiment. (c) Zonal mean zonal wind anomaly when the heating perturbation is applied. Contour intervals are 5 m s−1 in (a), 0.1° day−1 in (b), and 4 m s−1 in (c).

Fig. 1.

(a) Climate of the control experiment as represented by zonal mean zonal wind. (b) Heating perturbation applied in the perturbation experiment. (c) Zonal mean zonal wind anomaly when the heating perturbation is applied. Contour intervals are 5 m s−1 in (a), 0.1° day−1 in (b), and 4 m s−1 in (c).

a. Response to a sudden onset of perturbation forcing

An ensemble of 40 experiments have been run using the time series from the control experiment as initial conditions. At day 0, the heating perturbation shown in Fig. 1b, which is equivalent to a concentration of heating near the thermal equator that gives rise to increased intensity in the Hadley circulation, is abruptly applied and maintained henceforth, and the transient development of the flow is shown in Fig. 2a. It can be seen that right from the start, apart from the acceleration of the zonal wind around 15°S due to the increased Hadley intensity, there is deceleration seen emanating from around 35°S. As time progresses, both positive and negative anomalies become stronger and displace poleward, with the positive anomaly finally settling down around 40°S, and the negative perturbation around 55°S, and a quasi-steady state is reached after 100 days or so (only the first 50 days are shown in Fig. 2).

Fig. 2.

(a) Temporal evolution of zonal wind anomaly (zonal and vertical averaged) when the heating perturbation shown in Fig. 1b is applied at time 0. (b) Same as in (a) except that the heating perturbation is replaced by a monopole perturbation of heating alone centered at the equator. (c) Same as in (b) except that the applied perturbation is a cooling perturbation centered at 15°S. Contour intervals 0.5 m s−1 in (a) and 1 m s−1 in (b) and (c). All negative perturbations are shaded.

Fig. 2.

(a) Temporal evolution of zonal wind anomaly (zonal and vertical averaged) when the heating perturbation shown in Fig. 1b is applied at time 0. (b) Same as in (a) except that the heating perturbation is replaced by a monopole perturbation of heating alone centered at the equator. (c) Same as in (b) except that the applied perturbation is a cooling perturbation centered at 15°S. Contour intervals 0.5 m s−1 in (a) and 1 m s−1 in (b) and (c). All negative perturbations are shaded.

With the form of the forcing (shown in Fig. 1b), it is easy to understand how the heating will increase the intensity of the Hadley circulation (as discussed in Hou and Lindzen 1992) and give rise to an acceleration around 20°S. However, it is not immediately clear why we see deceleration around 35°S nearly right from the start. To investigate this further, we integrated a zonal mean (2D) version of the model, using time mean eddy fluxes obtained from the control experiment as additional forcing on the zonal mean momentum and thermodynamics equations. As expected, integration of the 2D model using the prescribed radiative forcing for the control run gives rise to a climate exactly equal to that of the 3D control run shown in Fig. 1a. However, when we apply the perturbation heating shown in Fig. 1b as additional forcing, while keeping the eddy fluxes unchanged, the steady-state response of the zonal mean model is shown in Fig. 3a. Apart from an area of acceleration around 20°S, we see that the perturbation forcing also induces deceleration around 35°S, even without changes in the eddy fluxes. Inspecting the mean meridional circulation (MMC, not shown here), induced by the perturbation forcing, we see that apart from a direct cell induced by the heating–cooling doublet (enhancement of the Hadley circulation), an indirect cell is also induced near the poleward flank of the perturbation due to the cooling anomaly. Hence it appears that the response due to the zonal mean circulation directly induces an acceleration–deceleration doublet in the Tropics and subtropics, and that the response of the eddies is to displace this doublet poleward into the subtropics and midlatitudes, respectively, resulting in an equatorward shift of the midlatitude jet in the 3D numerical experiments.

Fig. 3.

Results from integration of a 2D model of the zonal mean flow, with eddy forcings taken from the 3D control experiment as additional forcing. All panels show zonal mean zonal wind anomalies after a steady state has been achieved. The perturbation forcings are as follows. (a) Heating perturbation shown in Fig. 1b. (b) Heating only centered at the equator. (c) Cooling only centered at 15°S. Contour interval 4 m s−1.

Fig. 3.

Results from integration of a 2D model of the zonal mean flow, with eddy forcings taken from the 3D control experiment as additional forcing. All panels show zonal mean zonal wind anomalies after a steady state has been achieved. The perturbation forcings are as follows. (a) Heating perturbation shown in Fig. 1b. (b) Heating only centered at the equator. (c) Cooling only centered at 15°S. Contour interval 4 m s−1.

How do the eddies act to shift the zonal wind anomalies poleward? To examine that, we would like to analyze a simpler situation in which we only have either an acceleration or deceleration “monopole” instead of a dipole. It turns out that a positive heat source centered at the equator not flanked by cooling (i.e., a heating monopole) will give rise to an anomalous direct MMC and zonal wind acceleration around 25°S, with no deceleration to its poleward flank, as shown in Fig. 3b, which shows the results from the zonal mean (2D) model with the applied perturbation (perturbation heating centered at the equator) while keeping the eddy forcings the same as in the control experiment. Conversely, a monopole negative heat source (i.e., cooling) centered around 15°S gives rise to deceleration around 30°S, with no significant acceleration at its poleward flank (Fig. 3c). Note that apart from Fig. 3, which shows results using a 2D model, all other figures show results from 3D model experiments.

The corresponding transient response of the 3D model for ensemble experiments in which the monopole sources are suddenly turned on at time zero are shown in Figs. 2b and 2c. In Fig. 2b, we see that immediately after the onset of the forcing (heating centered at the equator), acceleration can be seen peaked around 25°S. As time goes on, the peak shifts poleward to nearly 40°S, the amplitude increases gradually, and the response approaches that of the steady state shortly after day 50. For the response to cooling centered around 15°S (Fig. 2c), the opposite happens, except that the signal is shifted slightly poleward due to the more poleward position of the imposed forcing.

The steady-state response of zonal mean flow in the full 3D model (allowing the eddies to fully adjust to changes in forcing) to these monopole heating perturbations are shown in Figs. 4a,b. The results suggest that if we just impose an acceleration monopole in the Tropics, there will just be an acceleration of the midlatitude jet. Similarly, with a monopole deceleration forcing in the Tropics, the midlatitude jet will be decelerated. The eddies mainly act to move the region of acceleration/deceleration poleward.

Fig. 4.

(a) Zonal mean zonal wind anomaly when the heating perturbation with heating only centered at the equator is applied. (b) Same as in (a) except that the applied perturbation is a cooling perturbation centered at 15°S. (c) Sum of (a) and (b). Contour intervals 2 m s−1 in (a) and (b) and 1 m s−1 in (c).

Fig. 4.

(a) Zonal mean zonal wind anomaly when the heating perturbation with heating only centered at the equator is applied. (b) Same as in (a) except that the applied perturbation is a cooling perturbation centered at 15°S. (c) Sum of (a) and (b). Contour intervals 2 m s−1 in (a) and (b) and 1 m s−1 in (c).

When the two are applied together, they form an acceleration–deceleration doublet in the Southern Hemisphere subtropics similar to that due to the concentration of heating discussed above (see the Southern Hemisphere portion of Fig. 1b). The sum of the results from the two panels is shown in Fig. 4c. It appears that the sum of the two experiments is equivalent to an equatorward shift of the midlatitude jet, similar to that seen in C95 and shown in Fig. 1c. However, the amplitude of the combined response shown in Fig. 4c is weaker than that produced by the simultaneous action of two heating and cooling perturbations (see Fig. 1c), which suggests that the response may not be entirely linear.

b. Examination of angular momentum budget

The equation governing the time tendency of the vertical and zonal mean zonal wind can be written as follows:2

 
formula

Here, [X] denotes zonal mean, starred quantities denote deviations from the zonal mean (eddy component), and an overbar denotes vertical (mass weighted) mean. Also, us is the wind at the lowest model level and τF is the surface friction timescale used in the model. The term on the left is the time tendency. The rhs terms are the contributions by the eddy momentum flux convergence, the transport of momentum by the MMC, and surface friction. Here, the tendency due to Coriolis acceleration is exactly zero when the vertical mean is taken, since the net poleward mass flux is restricted to be zero. We have also performed some experiments using another model3 that allows for nonzero meridional mass flux, but the Coriolis term was never found to be important in all cases that we have examined.

Examination of the budget for all cases suggested that the MMC term is in general smaller than the other terms. The MMC term is not entirely negligible, however, and its main action in the Tropics is to shift the tendency due to surface friction slightly poleward, while in the midlatitudes it is basically a reaction to the eddy-forcing term and acts to diminish its effects slightly. Hence the main balance is between the eddy momentum flux transport, surface friction, and time tendency. While the heating perturbation directly drives an anomalous MMC in the Tropics, the MMC by itself does not give rise to any change in the barotropic component of the zonal wind. However, the MMC drives a baroclinic zonal wind component due to the Coriolis term, and surface friction acting on this baroclinic component will give rise to changes in the barotropic component (see Chang 1996). The question then becomes, how do changes in [u] in the Tropics affect the eddy momentum fluxes, which then act to transmit the changes in the Tropics into the midlatitudes?

1) Monopole forcing

To see how the eddies respond to changes in the zonal mean circulation in the Tropics, we examined the momentum budget for the sudden onset of monopole heating perturbation case (the case shown in Fig. 2b and discussed above). However, it turns out that the zonal mean state response is very gradual, and because the eddy statistics are very noisy, it is difficult to arrive at any significant conclusion by examining this case.

Because the eddies do not feel the heating perturbation directly (the heating perturbation is zonally symmetric and hence only directly forces the zonally symmetric part of the circulation), they only feel the changes in the zonal mean state (mainly zonal wind and temperature distributions) induced by the forcing. To magnify the change in tropical zonal mean state in order to amplify the eddy response, we performed the following set of experiments. Another ensemble of transient experiments is performed, again using conditions from the control experiment at different times as initial conditions. Instead of just turning on the forcing at time zero (as in the case shown in Fig. 2b), we add changes in the zonal mean state that are consistent with the perturbation forcing, assuming that the eddy component remains unchanged. Changes in the zonal mean state can be obtained by integration of the zonal mean model and is just equal to the response shown in Fig. 3b. Hence, effectively, what we did was equivalent to turning on the perturbation forcing, first allowing the zonal mean circulation to adjust directly to the perturbation forcing while keeping the eddies fixed and then, after that adjustment, allowing the eddies to adjust to the new zonal mean state and forcing, in turn modifying the zonal mean state to reach a final equilibrium. The evolution of the vertical and zonal mean zonal wind anomaly for this case is shown in Fig. 5a. Comparing to Fig. 2b, we see that now at time zero the zonal wind anomaly (due to the “direct” adjustment of the zonal mean circulation to changes in the forcing) is already very significant and is centered around 25°S and that the adjustment of the eddies acts to shift this zonal wind anomaly poleward during the first 20 days or so while also strengthening it slightly, eventually approaching the final equilibrium state after about 50 days.

Fig. 5.

(a) Same as Fig. 2b except that at time 0, apart from the application of the heating perturbation, the zonal wind perturbation (as well as temperature perturbation) shown in Fig. 3b is added on as well. (b) Same as in (a), except that a dipole heat source is applied. Contour interval 2 m s−1. All negative perturbations are shaded.

Fig. 5.

(a) Same as Fig. 2b except that at time 0, apart from the application of the heating perturbation, the zonal wind perturbation (as well as temperature perturbation) shown in Fig. 3b is added on as well. (b) Same as in (a), except that a dipole heat source is applied. Contour interval 2 m s−1. All negative perturbations are shaded.

For this case, because the initial changes in zonal mean state in the Tropics are much larger compared to the case shown in Fig. 2b, changes in eddy forcing are also much less noisy and easier to interpret. In Fig. 7, we show the anomalous Eliassen–Palm (EP) cross section, averaged over days 8–12 (the plotting convention for the vectors follows that of Edmon et al. 1980).4 The corresponding plots for the control experiment are shown in Fig. 6 for comparison. Note that in Fig. 7, all fields shown are anomalies (except for the critical latitude, which is computed using the total wind), that is, differences from those of the control experiments shown in Fig. 6.

Fig. 7.

Same as in Fig. 6 except for anomalies for the sudden onset of heating experiment with a monopole heat source at the equator. The contour intervals are 2 m s−1 in (a), 1 K 1000 km−1 in (b), and 0.5 m s−1 day−1 in (c).

Fig. 7.

Same as in Fig. 6 except for anomalies for the sudden onset of heating experiment with a monopole heat source at the equator. The contour intervals are 2 m s−1 in (a), 1 K 1000 km−1 in (b), and 0.5 m s−1 day−1 in (c).

Fig. 6.

All panels shown are for the control experiment. Vectors in (a)–(c) show Eliassen–Palm flux vectors. Contours represent (a) zonal mean zonal wind (contour interval 5 m s−1), (b) meridional temperature gradient (contour interval 2 K 1000 km−1), and (c) EP flux divergence (contour interval 1 m s−1 day−1). Thick lines in (c) mark the critical latitude of the dominant eddies. (d) Vertical and zonal mean zonal wind acceleration anomaly (m s−1 day−1) due to divergence of EP flux.

Fig. 6.

All panels shown are for the control experiment. Vectors in (a)–(c) show Eliassen–Palm flux vectors. Contours represent (a) zonal mean zonal wind (contour interval 5 m s−1), (b) meridional temperature gradient (contour interval 2 K 1000 km−1), and (c) EP flux divergence (contour interval 1 m s−1 day−1). Thick lines in (c) mark the critical latitude of the dominant eddies. (d) Vertical and zonal mean zonal wind acceleration anomaly (m s−1 day−1) due to divergence of EP flux.

Let us first examine the EP cross sections from the control experiment (Fig. 6). The top two panels show the EP flux vectors, together with the time-averaged zonal mean zonal wind and meridional temperature gradient. We can see the characteristic upward flux in the midlatitude lower troposphere, close to where the meridional temperature gradient is maximum. These fluxes turn equatorward in the upper-troposphere subtropics. The divergence of EP flux (expressed in terms of zonal wind acceleration) is shown in Fig. 6c. We can see strong divergence near the lower boundary near 40°S and convergence of EP fluxes in much of the mid- and upper troposphere, with maxima near 55° and 35°S. Both the orientation of the flux vectors and the divergence pattern resemble observed patterns shown in Edmon et al. (1980). For later reference, in Fig. 6c the critical latitude for the most dominant wave (zonal wavenumber 6 with a phase speed of 11 m s−1 at 45°S) is also shown by thick lines, and in Fig. 6d the vertically averaged acceleration due to the EP fluxes (i.e., eddy momentum flux convergence, since divergence of EP flux due to eddy heat fluxes equals zero when boundary contributions are included in the vertical average) is plotted. Figure 6d clearly shows the poleward transport of momentum by the eddies.

The anomalies in EP fluxes, zonal wind, meridional temperature gradients, EP divergence, and vertical mean acceleration for the perturbation forcing experiment are shown in Fig. 7. Note the differences in the scale of the vectors and contour intervals between Figs. 6 and 7. At first glance, the anomalies in EP vectors appear to resemble the vectors seen in the control experiment (Fig. 6). However, several differences emerge when one examines the anomalous vectors and the divergence pattern closely. First, the orientation of the anomalous vectors are tilted more toward the horizontal direction when compared to the vectors in the control case. Figure 7c also shows that the anomalous divergence pattern, instead of being negative all through the upper troposphere, is positive near 40°S and negative south of 35°S. These two observations together suggest more equatorward propagation of waves and a shifting of the upper-tropospheric EP convergence pattern equatorward. Comparison of the critical latitude of the dominant waves in the perturbation experiment (thick line in Fig. 7c) to that in the control experiment (Fig. 6c) shows that due to the increase in zonal wind within the Tropics, the critical latitude in the upper troposphere has shifted equatorward by about 10° latitude in the upper troposphere. Hence the largely horizontal EP flux anomaly in the upper troposphere, and its associated dipole in the EP divergence anomaly, can be interpreted as the result of eddies that originally break and dissipate near 40°S now being able to penetrate farther equatorward because of the equatorward displacement of the critical latitude.

Figure 7 suggests that apart from an increase in the equatorward penetration of the eddies, there is also an increased baroclinic eddy source, as indicated by the upwardly directed anomalous EP vectors and the low-level divergence centered around 35°–40°S. This source is located slightly poleward of the zonal wind and meridional temperature gradient anomalies, probably due to the nonlinear dependence between eddy amplitudes and temperature gradients (see, e.g., Stone and Yao 1990).5 A separate analysis of the eddy enstrophy and energy also confirms that the eddy amplitude gradually increases with time, supporting the interpretation of the upward EP flux anomaly as an increased eddy source. These two effects together contribute to the divergence/convergence dipole of the EP flux, decelerating the zonal wind directly over and toward the equatorward flank of the zonal wind anomaly and accelerating it toward the poleward flank of the zonal wind anomaly (see Fig. 7d), displacing the zonal wind anomaly poleward.

The above discussions suggest that the eddies communicate the changes in the Tropics to the midlatitudes in two ways. First, increased zonal mean zonal winds in the Tropics induced by the anomalous forcing allows eddies to penetrate farther into the Tropics before dissipating (see, e.g., Randel and Held 1991), shifting the deceleration due to divergence of eddy angular momentum flux equatorward, thus acting to damp the anomalous acceleration in the Tropics while leading to relative acceleration of the zonal flow in the subtropics and midlatitudes. In addition, the applied forcing also induces an increase in temperature gradients in the subtropics (in thermal wind balance with the zonal wind anomaly), which leads to an increase in the baroclinic eddy source. The resulting anomalous eddy propagates toward the Tropics and dissipates there close to its critical latitude (Edmon et al. 1980; Randel and Held 1991). This also results in increased acceleration in the midlatitudes and increased deceleration in the subtropics. Both of the above act to shift the anomalous zonal wind acceleration pattern in the Tropics poleward toward the midlatitudes. The two mechanisms discussed above are obviously not independent of each other; as large-scale motion in the atmosphere is close to thermal wind balance, an increase in barotropic momentum, which comes about mainly through an increase in upper-level zonal wind driven by an increase in Hadley intensity, necessarily requires an increase in the meridional temperature gradient. Hence we expect that these two mechanisms will generally act together.

2) Dipole forcing

We have also investigated the momentum balance and EP cross sections for the case with monopole cooling at 15°S, and the results are basically just the reverse of what is discussed above, except that all the changes are in the opposite polarity, and the responses are shifted about 7° latitude poleward due to the more poleward position of the heat source (see Figs. 3c and 4b). Let us now turn our attention to the case with a dipole heating perturbation (heating at equator, cooling at 15°S). This heating perturbation resembles the concentrated heating perturbation shown in Fig. 1b except that the cooling in the Northern Hemisphere is absent here. While the concentrated heating perturbation is perhaps more realistic (see discussions in section 6), a dipole is simpler to construct (just the sum of two heat sources) and the results slightly easier to interpret. Since we will only focus our attention on the winter hemisphere anyway, the differences between the heating concentration perturbation and the dipole are not significant.

As in the monopole case, an ensemble of transient experiments is performed, using conditions from the control experiment plus changes in the zonal mean state that are consistent with the perturbation forcing as initial conditions. However, we found that due to the partial cancellation between the heating and cooling perturbations, the response is much weaker than the response with just a heating monopole alone. Hence we have increased the amplitude of heating by a factor of 2 (i.e., 2° day−1 heating at the equator, and 2° day−1 cooling at 15°S) to obtain a stronger eddy response. The evolution of the vertical and zonal mean zonal wind anomaly for this case is shown in Fig. 5b. Again, we can see that the adjustment of the eddies acts to shift the zonal wind anomalies poleward during the first 20 days.

The EP anomalies corresponding to the dipole case, averaged between days 10 and 14, are displayed in Fig. 8. In response to the dipole in zonal wind and temperature gradient anomalies, the EP anomalies basically resemble the sum of the response of the two monopole cases. The pattern near 30°S resembles that from the monopole heating case, with increased penetration of eddies into the Tropics as suggested by the equatorward EP flux anomalies, upper-tropospheric EP flux divergence center near 40°S, convergence center near 20°S, and an increase in eddy source as indicated by the upward EP flux anomalies and divergence in the lower troposphere in response to an increase in subtropical temperature gradients. The response poleward of 40°S resembles that from monopole cooling applied at 15°S:downward and poleward EP flux anomalies arising from a reduction of eddy activity in response to a reduction in midlatitude temperature gradients. Such an eddy response leads to the “triplet” acceleration pattern shown in Fig. 8d, which acts to move the dipole anomaly in the zonal wind poleward, just as the dipole acceleration pattern in the monopole case (Fig. 7d) acts to displace the monopole zonal wind anomaly poleward.

Fig. 8.

Same as in Fig. 7 except for anomalies for the sudden onset of heating experiment with a dipole heat source.

Fig. 8.

Same as in Fig. 7 except for anomalies for the sudden onset of heating experiment with a dipole heat source.

3. Response to oscillatory forcing: Monopole heat source

To examine how the zonal mean momentum responds to oscillatory forcing in the Tropics instead of a sudden onset, we performed a series of experiments, using the heating monopole at the equator. In this series of experiments, the amplitude of the applied forcing oscillates sinusoidally in time, with periods of 400, 200, 100, 50, 25, and 12.5 days, respectively. For each experiment, 45 cycles were run, with the results averaged over the final 40 cycles used for analyses. For the cases with periods shorter than 100 days, the amplitude of the forcing is increased by factors of 2 in order that the response can be separated from climate noise. Comparing experiments using the same period of forcing but with amplitudes different by a factor of 2 suggested that for all experiments shown here, the response is linear w.r.t. the amplitude of the forcing to within experimental noise, and hence results from all experiments are normalized to an amplitude of 1 (the amplitude used for the steady-state experiments shown in the above section).

The responses to the forcing with periods of 50 and 200 days are shown in Figs. 9a and 9b, respectively. In the figure, we show a time–latitude plot of the vertically and zonally averaged zonal wind anomaly, averaged over 40 periods. The anomaly is formed by subtracting the time mean of the quantity shown. Note that day 0 corresponds to minimum forcing and, for clarity, two periods have been shown. Following the analyses done in C95, let us just concentrate on the Southern (winter) Hemisphere. In Fig. 9a, we see that with a period of 50 days, the response in the Tropics (around 20°S) lags that of the forcing by about 7 days or so, and there is an apparent propagation of the signal from the Tropics into midlatitudes, with the maximum response around 40°S lagging the maximum response in the Tropics by about 10 days (nearly a quarter-period). When the forcing has a period of 200 days (Fig. 9b), we see that the main response is around 40°S. For this case, we may argue that we still see a weak poleward propagation signal, especially for the negative phase, but the signal is weak. The midlatitude response now lags the forcing by only about one-sixth of a period (compared to about three-eighths of a period in the case of 50 days), and the maximum amplitude, ∼4.7 m s−1, is significantly stronger than that for the case of 50 days (∼1.2 m s−1).

Fig. 9.

Zonal and vertical averaged zonal wind anomaly (time mean subtracted) when an oscillatory heat source is applied. The heating perturbation is a monopole forcing centered at the equator. The period of the forcing are (a) 50 and (b) 200 days. Contour intervals 0.25 m s−1 in (a) and 1 m s−1 in (b). Time 0 corresponds to time of minimum forcing. All negative anomalies are shaded. The shading intervals are 0.25 m s−1 in (a) and 1 m s−1 in (b).

Fig. 9.

Zonal and vertical averaged zonal wind anomaly (time mean subtracted) when an oscillatory heat source is applied. The heating perturbation is a monopole forcing centered at the equator. The period of the forcing are (a) 50 and (b) 200 days. Contour intervals 0.25 m s−1 in (a) and 1 m s−1 in (b). Time 0 corresponds to time of minimum forcing. All negative anomalies are shaded. The shading intervals are 0.25 m s−1 in (a) and 1 m s−1 in (b).

The results for the six experiments with different periods of forcing, as well as those for the steady-state response, are summarized in Table 1. In the table, the quantities shown are the amplitudes and phase (w.r.t. the forcing) of the vertical and zonal mean zonal wind response to unit forcing, and the responses are separated into a tropical part, which is the signal at around 20°–25°S, and a midlatitude part, which represents the response at around 40°S. As expected from a forced-dissipative system, the response generally decreases with an increase in forcing frequency. Apart from that, we see that for high-frequency forcing, the response is mainly within the Tropics, around the regions where the effective acceleration due to the MMC driven directly by the heating anomalies acts, with much weaker response in the midlatitudes. However, as the forcing frequency decreases, the midlatitude response increases much faster than the tropical response, and for periods around 50 days or longer, the midlatitude response becomes stronger than the Tropical response. Looking at the phase, we see that both tropical and midlatitude responses in general lag the forcing and the phase lag increases as the frequency increases. Moreover, the midlatitude perturbation phase also lags the tropical perturbation, suggestive of poleward propagation of the perturbation, but the phase lag is rather constant (between 80° and 100° in phase) over the frequency range from 12.5 days to 100 days. There is some indication that for a period of 200 days, the tropical response actually leads the forcing.

Table 1.

Amplitude and phase relations for tropical and midlatitude perturbations forced by oscillatory monopole heat source at the equator.

Amplitude and phase relations for tropical and midlatitude perturbations forced by oscillatory monopole heat source at the equator.
Amplitude and phase relations for tropical and midlatitude perturbations forced by oscillatory monopole heat source at the equator.

In order to understand the responses shown in Fig. 9 and summarized in Table 1, we have examined the angular momentum budget (not shown here) to see whether we can get some insight from it. As discussed above for the sudden onset of forcing case, the main balance here is again between the eddy momentum flux transport, surface friction, and time tendency. In the Tropics, surface friction leads the changes in [u] and eddy forcing lags by more than 90° of phase, while the opposite case is true in the midlatitudes. Hence it suggests that the applied forcing affects [u] in the Tropics via surface friction, with eddy forcing acting as damping, while in the midlatitudes, the change in [u] is forced by the eddy forcing and damped by surface friction. This is consistent with the following scenario. The heating perturbation drives an anomalous MMC in the Tropics. The MMC by itself does not give rise to any change in the barotropic component of the zonal wind. However, the MMC drives a baroclinic zonal wind component due to the Coriolis term, and surface friction acting on this baroclinic component will give rise to changes in the barotropic component (see Chang 1996). Hence the result that [u] in the Tropics is driven by surface friction. As discussed in section 2b above, the changes in [u] in the Tropics then lead to changes in the eddy momentum fluxes due to changes in the critical latitude and eddy sources, which in turn lead to changes in [u] in the midlatitudes.

The absence of a tropical response in our numerical experiments at low frequency can be easily understood as follows. In section 2, we suggested that any response in the Tropics will trigger changes in the eddy forcing that act to damp out such changes and convey that to the midlatitudes. Hence when the forcing frequency is low (timescale of forcing much longer than the eddy response timescale and the zonal flow dissipation timescale), we expect that the eddies and the midlatitude zonal flow will always have time to react to the forcing such that the response at any time will be near equilibrium, that is, resembling the steady-state response in which the midlatitude response is much stronger than the tropical response (Fig. 4a). However, when the forcing frequency is high, the eddies (and midlatitude zonal flow) will have less time to respond to changes in the forcing, and we expect the tropical response to be much more significant, leading to the appearance of poleward-propagating signal at higher frequency. At very high frequency (period much shorter than eddy response timescale), the eddy response will be weak and the main response will remain in the subtropics (Fig. 3b).

4. Low-frequency poleward-propagating anomalies: Response to oscillating dipole heat source

In section 1, we mentioned that one of the motivations of this work was observational analyses that showed low-frequency (ENSO timescale) poleward-propagating angular momentum anomalies (e.g., Dickey et al. 1992). From the results shown in Table 1 and Fig. 9, we see that we can clearly identify poleward-propagating angular momentum anomalies in our numerical model results for oscillatory tropical heat sources with periods of 25–100 days, but for lower-frequency forcing (period ∼200 days or more), the solution simply resembles a standing oscillation in the midlatitudes with very weak response in the Tropics. There is a slight hint of poleward propagation (e.g., for the negative anomalies shown in Fig. 9b) but definitely nothing like the anomalies shown in Dickey et al. (1992).

To further explore the problem, we performed another series of experiments similar to those discussed in section 3 above, but this time we used a dipole heat source rather than a monopole heat source. The heating perturbation consists of a heat source centered over the equator and a second source at 15°S that is 180° out of phase with the first source. This forcing is the same as that used for the experiment shown earlier in Figs. 5b and 8, except that here it oscillates in time rather than having a stepfunction like time dependence as in the sudden onset experiment.

With the oscillating dipole forcing, we found poleward-propagating response for all cases, for periods ranging from 25 to 400 days. The experiments with periods of 200 and 400 days are shown in Figs. 10a,b. Unlike the case of a monopole heat source (see Fig. 9b), with a dipole heat source we can clearly identify poleward-propagating zonal wind anomalies, all the way from about 20° to 60°S. However, the amplitude of the response is significantly weaker than for the case of a monopole heat source, suggesting that there is some cancellation between the two heat sources. In addition, as the period gets longer (Fig. 10b), it appears that the higher-latitude signal and the lower-latitude signal are close to 180° out of phase.

Fig. 10.

Same as in Fig. 9 except that a dipole heat source is applied. The period of the forcing are (a) 200 and (b) 400 days. Contour intervals 0.5 m s−1. The shading intervals are the same as in Fig. 9b.

Fig. 10.

Same as in Fig. 9 except that a dipole heat source is applied. The period of the forcing are (a) 200 and (b) 400 days. Contour intervals 0.5 m s−1. The shading intervals are the same as in Fig. 9b.

To dissect this case further, in Fig. 11a we show the results when we leave out the heat source at the equator and impose only the heat source with opposite polarity at 15°S, with an oscillatory period of 200 days. We see that the results again show very little apparent poleward propagation and resemble the results shown in Fig. 9b for the case when the heat source is at the equator, the only differences being that the anomalies have opposite signs and in the present case are shifted slightly poleward due to the more poleward position of the heat source. In Fig. 11b, we show the result when we simply add up the anomalies for the two cases shown in Figs. 9b and 11a. A poleward-propagating signal resembling that shown in Fig. 10a can be seen, even though each of the two constituent signals show relatively weak poleward-propagating signals individually.

Fig. 11.

Zonal and vertical mean zonal wind anomaly when an oscillatory heat source with a period of 200 days is applied. (a) Only a source at 15°S, which is 180° out of phase to the heat source for Fig. 9b, is used as heating perturbation. (b) Sum of Figs. 9b and 11a. Contour intervals 1 m s−1 in (a) and 0.5 m s−1 in (c). The shading intervals are the same as in Fig. 9b.

Fig. 11.

Zonal and vertical mean zonal wind anomaly when an oscillatory heat source with a period of 200 days is applied. (a) Only a source at 15°S, which is 180° out of phase to the heat source for Fig. 9b, is used as heating perturbation. (b) Sum of Figs. 9b and 11a. Contour intervals 1 m s−1 in (a) and 0.5 m s−1 in (c). The shading intervals are the same as in Fig. 9b.

Let us try to follow the negative (shaded) anomalies in Fig. 11b and see how the poleward-propagating signal had come about. Around t = 0, the negative anomaly near 25°S is mainly due to the response to the heat source at the equator (Fig. 9b). As we get to t ∼ 100, the negative anomaly in the midlatitudes (near 50°S) in Fig. 11b has come mainly from the heat source at 15°S (Fig. 11a), while the heat source at the equator is now contributing to the positive anomaly in the subtropics. Hence we see that the negative anomaly at the two different times are responses to the two separate sources and should not be regarded as part of a coherent, poleward-propagating signal, even though the composite (Fig. 11b) gives such a suggestion. The results are very similar for the case with an oscillatory period of 400 days (not shown).

It is easy to understand how oscillatory heating in the Tropics at higher frequency (periods of 25–50 days) can result in poleward propagation of perturbations due to the finite response time of the midlatitude zonal flow because of its inertia when faced with changes in the eddy forcing induced by changes in the tropical zonal flow. However, it is not so clear what dynamical mechanism can be responsible for such a slow propagation of signals at much lower frequencies. Here, we suggest one possible explanation. The results shown in Fig. 10 and 11 suggest that, at least for this model and the experiments considered here, superposition of two basically standing oscillations located side by side, each with relatively weak poleward propagation tendencies of its own, can lead to an apparent significant signal of poleward propagation all the way from deep within the subtropics into the midlatitudes and beyond. A slight variation of this superposition scenario had been suggested by Chen et al. (1996). More will be said about this in section 6.

5. An illustrative three-component model of wave–mean flow interaction between the Tropics and midlatitudes

a. Model formulation

In section 2b, by examination of the angular momentum budget and EP cross sections, we have obtained qualitative understanding about how the forcing in the Tropics can affect midlatitude climate. We can also understand qualitatively the results presented in Table 1. Basically, the forcing acts to drive an anomalous MMC that in turn drives the barotropic component of the zonal mean zonal wind through surface friction; hence it is not surprising that the phase of the tropical response lags that of the forcing and that the phase lag increases and the amplitude of the response becomes smaller as the forcing frequency increases.6 Our analyses also suggested that the midlatitude response is forced by changes in the eddy forcing and changes in the eddies are driven by the changes in the zonal mean state in the Tropics; hence it is again not surprising that the midlatitude response lags that in the Tropics. But can we understand, at least semiquantitatively, the dependence of the amplitudes and phase lags of both the tropical and midlatitude responses with changes in the frequency of forcing shown in Table 1?

To understand further the amplitudes and phase lags shown in Table 1, we have constructed a very simple analytical model to illustrate the wave–mean flow interaction between the Tropics and midlatitudes. From the analyses and discussions in section 2b, we see that the model must have at least three independent components: UT, UM, and A, representing the zonal wind anomalies in the Tropics, midlatitudes, and perturbations in eddy amplitude, respectively. From the discussions presented there, the equations governing the time tendency of the three components may be written as follows:

 
formula

and

 
formula

In the above equations, the Tropical zonal wind anomaly is driven by a time-varying source (SU, to be specified below) and dissipated by surface friction and eddy forcing. Here, τU represents a spindown timescale due to surface friction. The third term on the rhs of (2) represents eddy feedback. Motivated by the discussions in section 2b, this term comprises two terms: the first term represents increased penetration of eddy into the Tropics in response to increase in tropical zonal wind and hence is dependent on the tropical zonal wind anomaly, and the second term represents increase in zonal wind deceleration due to increase in eddy amplitudes. Both of these terms lead to an increase in the divergence of eddy momentum fluxes over the Tropics. In general, these two terms are not necessarily linear. However, if we assume small changes around the control basic state, we can linearize these eddy forcing terms about the basic state and treat them as linear. As discussed above, comparisons between experiments with forcing amplitudes differing by factors of 2 suggested that the response (hence also the changes in eddy forcing) is linear to within experimental errors. In (3), UM is driven by the same eddy forcing that acts as damping to UT in (2), with the cosine factors allowing for the effects of sphericity of the earth. Here UM is damped by surface friction. In (4), A, representing the amplitude of the eddy activity, is driven by an eddy source (SA, to be specified below) and dissipation of A is represented by the second term on the rhs.

Next, we have to specify the form of the sources in (2) and (4). Baroclinic instability theory suggests that the eddy source SA in (4) is proportional to the temperature gradient.7 To leading order, the temperature gradient is driven by the heating perturbation and damped by thermal damping. Hence the equation for SA can be written as

 
formula

Here, H(t) is the applied heating perturbation.

As for SU in (2), because the direct forcing in the experiment is heating, the forcing can only indirectly force the barotropic component of the wind through surface friction in response to the anomalous MMC driven by the heating. Hence SU is proportional to −US, where US is the surface wind component. To leading order, the equation governing the surface wind can be written as

 
formula

where τSF is damping timescale due to surface friction. To leading order, fVS is proportional to −H(t). Forcing of the barotropic component can then be written as

 
formula

where MS/MB is the ratio between mass in the lowest level and that in the entire column (∼1/5 here).

For an oscillatory heat source with a frequency, ω, the two sources (5) and (7) can be written as

 
formula

and

 
formula

Substituting (8) and (9) into (2) and (4), we have a complete set of equations governing the three components UT, UM, and A, providing that we specify the various parameters. There are a total of seven free parameters: τSF, τrad, sU, sA, τU, τA, and β. In the following paragraphs we will attempt to make an order of magnitude estimate of these parameters from simple scale analysis. Readers only interested in model results can jump directly to section 5c.

b. Estimating model parameters

1) Simple scale analysis

First, we will estimate the friction and dissipation timescales. In the numerical model, surface friction is represented by Rayleigh friction with a timescale of 2 days in the lowest level and the radiative damping timescale is 20 days. Let us first consider τU. Since the model has five equal mass layers, the spindown time for the barotropic component will be five times the Rayleigh friction timescale, or 10 days. However, the anomalies do have some baroclinic component to them, and if the surface wind anomaly is assumed to be half that of the upper troposphere as in Fig. 1c, τU should have a value close to 15 days. For τSF and τrad, initially one might expect them to have values close to those used in the numerical model, that is, 2 and 20 days, respectively. However, while both surface friction and radiative damping acts directly to damp the surface wind and temperature anomalies, they are not the only processes in action. Friction and heating both drive an anomalous MMC, which acts to modify the response (see Chang 1996), and in this simple three-component model, the MMC is not explicitly modeled and its effects has to be included as part of the damping process. In the case of surface friction, it will act to drive an anomalous MMC that partially counteracts its action, and hence we expect the effective surface friction timescale τSF to be more than 2 days. As for τrad in Eq. (5), the MMC induced by heating will act to partially offset the heating, and hence the effective damping including the effects of the MMC should be stronger than just radiative damping alone, thus we expect τrad to be less than 20 days.

Next, we will consider the eddy dissipation timescale τA. Since we are interested in eddy fluxes that are quadratic quantities, we need to consider damping of eddy energy or enstrophy. Radiative damping will damp eddy potential energy (EPE) with a timescale of 10 days (half the radiative timescale for temperature). Eddy kinetic energy (EKE) is damped both by surface friction and barotropic conversion, but we have found that energy loss due to surface friction dominates. Eddy damping timescale depends on the vertical distribution of EKE. If we again assume that upper-tropospheric eddy wind anomalies are twice that on the surface, then the EKE damping timescale due to surface friction should be about 12 days. If the ratio between upper-level and surface wind anomalies is larger, then the EKE damping timescale will be much longer. Here τA represents a weighted average between the EKE and EPE damping timescale, and we expect it to be over 10 days.

We can estimate sU by considering the direct effects of heating. Near the equator, heating is balanced by temperature change and upward mass flux. Assuming that all the heating goes into inducing an upward mass flux, then given the form and the magnitude of the heating anomaly we can easily find the maximum upward mass flux, which must equal ρ × area × H/Γ, where H is the heating rate and Γ is the vertical potential temperature gradient. This upward mass flux must be balanced by horizontal mass flux, which is equatorward in the lowest level and poleward in the upper levels. Given the amplitude of heating at 1° day−1, and width of the heating covers about 10° latitude, the heating will induce a meridional wind VS of about 1 m s−1. However, since some of the heating goes into driving temperature changes, we expect that the meridional wind anomaly driven by heating to be less than 1 m s−1, and indeed in the model we observe VS to be about half that value. Given VS, we can find US from (6), and then substitute into (7) to find SU. Using a VS of 0.5 m s−1 associated with the heating, and f at 5° latitude, we find a value of about 1/20 m s−1 day−2 for sU.

It is not obvious how to estimate β from first principle, except that if we assume that (2) also holds for the control experiment, and that the eddy deceleration term on the rhs of (2) is equally divided between the UT term and the A term, then from the zonal wind distribution and deceleration due to the eddies shown in Figs. 6a,d, we find that around 25°S, acceleration due to eddies is approximately 1 m s−1 day−1 and vertical mean wind is approximately 10 m s−1; one estimates a value of 1/20 day−1 for β. However, the partition into the two terms makes sense when one considers anomalies, but the meaning of such a partition is not clear when one considers the control experiment itself.

Finally, we would like to estimate sA, the term related to change in eddy momentum transport due to changes in temperature gradients. The original equator to pole temperature forcing is about 70 K. One degree per day heating is equivalent to imposing an additional 20-K temperature difference (since the radiative timescale in the model is 20 days). Hence ΔTe/Te is approximately 2/7. If eddy response is linear, with the increase in temperature forcing, the eddies will need to transport 30% more heat. Eddy response to increased temperature gradient is nonlinear (see, e.g., Stone and Yao 1990), but since the heating anomaly is mainly in the Tropics, part of the heating will be smoothed out by the MMC and the eddies will not see the full 20-K increase in temperature forcing. For an order of magnitude estimate, we just assume that the eddy response will increase by 30%; hence we expect that βA to be ∼0.3 m s−1 day−1 for the steady-state response (ω = 0). For that case, (2), (4), and (9) give the relation

 
A = τAτradsA,

and one can estimate sA given estimates of the other parameters.

2) Estimation of parameters by fitting experiment results

The simple scale analysis presented above only gives us rough estimates of what the parameters should be. In the following paragraphs, we will discuss how we actually fitted the parameters.

Apart from the quantities shown in Table 1, there are other quantities that need to be considered, including the amplitude and phase dependence of the zonal wind and eddy sources SU and SA, that is, US in the Tropics and the baroclinic component of the wind perturbation, and also the observed amplitude and phase dependence of perturbation eddy amplitudes. Hence the seven parameters are chosen as follows: τSF and τrad are chosen to fit tropical US and the baroclinic wind component observed in the experiments, respectively, and τA is chosen to fit the observed amplitude and phase dependence of perturbation eddy enstrophy. Then the remaining four parameters are chosen to fit the quantities listed in Table 1: sA/sU is chosen such that UT ∼ 0 for steady forcing, β is chosen to fit the relative amplitude of UT and UM at high frequency, and sU and τU are chosen to fit the absolute amplitude and the ratio between high-frequency response to steady-state response, respectively. Apart from the above considerations, we also require the parameters to be reasonably close to the values estimated above. We could have performed a multiparameter least squares fit to determine the parameters. However, due to the simplistic nature of the model represented by (2)–(9), we have simply chosen the parameters by inspection such that the resulting behavior of the model resembles the behavior seen from Table 1. We feel that anything more sophisticated is incompatible with the simplistic nature of the model and is probably stretching the credibility of the model a bit too much. The following parameters have been used to compute Figs. 12 and 13:

 
formula

The values of these parameters are all within a factor of 2 of the estimates from the scale analysis discussed above.

Fig. 12.

Same as Fig. 9 except for solution of the simple analytical model (2)–(9).

Fig. 12.

Same as Fig. 9 except for solution of the simple analytical model (2)–(9).

Fig. 13.

Same as Fig. 11 except for solution of the simple analytical model (2)–(9).

Fig. 13.

Same as Fig. 11 except for solution of the simple analytical model (2)–(9).

c. Solutions of the simple model

1) Monopole forcing

In Fig. 12, we have plotted the zonal wind anomalies computed using (2) and (3) with the parameters listed above. To calculate the full latitudinal dependence, we have assumed that the subtropical and midlatitude wind anomalies to be both Gaussians with half-width of 12.5° latitude, centered at 25° and 38°S, respectively. In Figs. 12a,b, we show the solutions of (2) and (3) for periods of 50 and 200 days, respectively, and the figures should be compared to Fig. 9. We see that with the above assumptions and parameters, the solution to the simple wave–mean flow interaction model appears to represent the results from the full numerical experiments shown in Fig. 9 quite well. Thus we can claim that we have at least obtained a semiquantitative understanding of the results shown in Table 1, apart from the qualitative understanding discussed above.

The model (2)–(4) can also help us understand the results shown in Table 1 better. The resulting fit using the parameters shown above suggests that at high frequency, the effects of changes in the eddy amplitude A on (2) and (3) are small and the eddies mainly act by reacting to the change in critical latitude due to changes in tropical zonal wind UT. In that case, (2) and (3) may be approximated as

 
formula

and

 
formula

respectively. Hence, we see that at high frequency (relative to 1/τU), UM is predicted by (11) to lag UT by about 90° in phase, while (10) suggests that UT will lag the forcing by up to as much as 180° in phase when the frequency is high compared to 1/τSF and β. In addition, at such high frequency, the ratio UM/UT decreases as 1ω; hence the main signal remains in the subtropics. Those are the general features observed in Table 1.

For low frequency, the effect of A becomes important. However, even if we ignore its effects, (11) suggests that at low frequency, UM/UT will approach a value of βτU times the cosine factor, which comes out to be ∼2.5 for the parameters used here. In that case, the model suggests that UMUT, which is what is seen in Table 1. In addition, the effects of A in (2) act to further reduce UT, hence at low frequency, the predominant signal is seen in the midlatitudes, with only very weak changes in zonal wind seen within the Tropics. At low frequency, (3), (4), and (9) combine to give

 
formula

This suggests that the phase lag of UM w.r.t. the applied forcing is determined by the timescale

 
τ = τU + τA + τrad,
(13)

which is approximately 50 days given the parameters listed above. This explains why even at a frequency of 200 days, UM still lags the forcing by about 60° in phase [Eqs. (12) and (13) suggest a phase lag of 57°]. This timescale also represents the response timescale of the midlatitude zonal wind associated with eddy feedback and damping, explaining the transition in the behavior of the model response as the period changes from much less than this timescale to much larger than this timescale.

2) Superposition of two heat sources

The results shown in Fig. 11 for the superposition of two heat sources can also be obtained using the simple analytical model here. In Fig. 13a, we show the results when we apply the model represented by Eqs. (2)–(4) to the case when the heat source is at 15°S and with opposite polarity, using exactly the same parameters as before, except that now the “subtropical” response is taken to be centered around 35°S and the midlatitude response at 45°S (values taken from the numerical modeling results) to represent the effects due to the forcing being at 15°S instead of at the equator. Figure 13b is just the sum of Figs. 12b and 13a, which represents the linear combination of the responses from the two cases. In this case, we see that the signal is in fact just a superposition of two signals that are basically standing oscillations of opposite polarities in the midlatitudes, with the centers of the two oscillations displaced slightly from each other. Each signal individually shows only relatively weak poleward propagation, but the superposition of the two signals show apparent poleward propagation all the way from equatorward of 20°S to about 55°S. Figure 13b confirms that the superposition of two basically standing oscillations can lead to an illusion of significant poleward propagation.

Finally, we want to stress that the simple model presented in this section is not in any way a theory or a prognostic model. It is at best a diagnostic model to help us achieve a better understanding of the numerical experiment results based on the mechanisms proposed in section 2b. The success of the model in reproducing the experiment results lends support to the consistency of the mechanistic view.

6. Summary and discussions

In this paper, we examined the response of the zonal mean zonal wind to transient heat sources in the Tropics using an idealized model, in order to clarify the role eddies play in communicating changes in the zonal mean state between the Tropics and extratropics. By imposing (zonally symmetrical) heat sources in the Tropics in an idealized numerical model of the earth’s atmosphere, we found that oscillatory sources in the Tropics can give rise to poleward-propagating angular momentum anomalies. With the parameters used in the model in this study, excitation by a single heat source centered about the equator (section 3) leads to significant poleward propagation when the heat source oscillates with a period of around 25 to perhaps up to 100 days. At higher frequency, most of the zonal wind perturbation is confined within the Tropics and at lower frequency, the main signal occurs in the midlatitudes.

In section 2b, we examined the angular momentum budget and Eliassen–Palm cross sections to understand how the eddies respond to changes in the perturbation forcing and the mean flow. The results suggest that eddies act to communicate changes in the Tropics into the midlatitudes in at least two ways. First, changes in zonal mean zonal wind in the Tropics lead to a shift in the critical latitude for the eddies, resulting in a latitudinal shift of the region where eddies dissipate and hence a shift in the divergence of angular momentum pattern due to eddy momentum transport. Second, heating in the Tropics changes the temperature gradients between the Tropics and midlatitudes. This gives rise to changes in eddy source and eddy amplitudes, this time affecting the amplitude (instead of the position) of the eddy momentum divergence pattern. Both effects damp the perturbation in the Tropics and also act to transmit the tropical perturbation into the midlatitudes. We also suggested that thermal wind balance of the large-scale flow implies that these two effects should generally occur simultaneously.

Based on the ideas discussed in the preceding paragraph, in section 5 we constructed a simple three-component analytical model to represent the wave–zonal-mean flow interactions between the Tropics and midlatitudes. After simple tuning of several parameters, we obtained a reasonable quantitative fit between the simple model and the results from the numerical experiments. The model suggests that at high frequency, due to the finite response time of the midlatitude zonal flow due to its inertia, the midlatitude response will lag the tropical response by about 90° in phase, which is the origin of the poleward-propagating signal. However, the simple model (as well as results of the numerical experiments) also suggests that the ratio between the midlatitude response and the tropical response varies as ω−1;hence at high frequency, only tropical perturbations are present and at low frequency the main response is in the midlatitudes. Only within an intermediate frequency range (for periods between 25 and 100 days8) is the tropical and midlatitude responses comparable in magnitude for the poleward-propagating signal to be clearly observed.

When we examined low-frequency (period 200 days or longer) variability excited by heating centered on the equator alone, we found that the main response in the zonal mean zonal wind appears to be a standing signal in the midlatitudes (Fig. 9b). This is what we should have expected in the first place, since at a frequency that is low compared to the eddy and zonal wind response timescales, we should expect the response to be in quasi equilibrium with the forcing at any moment. However, when we add a heating source at 15°S with the opposite phase, we got a poleward-propagating signal, as shown in Fig. 10a. Our analyses in section 4 suggest that, at least for this model and the experiments considered here, low-frequency angular momentum anomalies that show a significant signal of poleward propagation from the Tropics into the extratropics may just be the illusory superposition of two standing oscillations located side by side, each with a relatively weak poleward propagation tendency of its own. In such a scenario, we expect the major response in the midlatitudes to be nearly 180° out of phase with the subtropical signal, which is apparent in Fig. 10 (and Fig. 13b). It is interesting to note that Fig. 4 in Rosen (1993), as well as the corresponding figure in Dickey et al. (1992), showing the interannual anomalies in the zonal mean relative angular momentum, also show hints of such a behavior, especially in the Northern Hemisphere. Recently, Chen et al. (1996) also suggested that the apparent poleward propagation of atmospheric relative angular momentum observed in the interannual timescale results from a flip-flop alternation of the anomalous circulations associated with the cold and warm ENSO events. Their conclusion is basically similar to the one arrived at here, except that they decomposed the signal into dipoles (in space) in circulation patterns that flip-flop in time, whereas here we have chosen to decompose the signal into standing waves oscillating in time, with two waves having opposite sign occurring side by side in space. Obviously, as pointed out by Mo et al. (1997), if the subtropical and midlatitude signals are exactly 180° out of phase at all times, the superposition of the two signals will only give rise to standing oscillation, not apparent propagation (see their Plate 2f). The apparent propagation only arises here because each of the individual oscillations possesses a weak poleward-propagating tendency of its own.

In this paper, we have followed up on the hypothesis suggested by Bjerknes (1966, 1969), that in response to increased heating over the eastern equatorial Pacific during El Niño, there is a local increase in the Hadley circulation leading to the strengthening and equatorward displacement of the subtropical jet over the western Pacific. To keep the diagnostics simple, we have restricted ourselves to considering only zonally symmetrical heating perturbations. Anyway, diabatic heating anomalies derived from the NCEP–NCAR reanalysis products (Kalnay et al. 1996) do suggest that even in the zonal mean sense there is increased diabatic heating near the equator and decreased heating elsewhere over the Tropics associated with ENSO, similar to the heating concentration forcing shown in Fig. 1b, except that the cooling anomaly in the summer hemisphere is a bit stronger in the observation. Observational analyses by Oort and Yienger (1996) also suggested that the zonal mean Hadley circulation is stronger during El Niño years. With respect to the sensitivity of the response, the anomalies shown in Fig. 10 are equivalent to about twice the magnitude of the perturbations shown in Fig. 4 of Rosen (1993), while the imposed heating (maximum minus minimum) is again approximately twice the difference between the Januarys of 1983 and 1989.

Due to constraints in computational resources, we have performed most of the experiments using a model having relatively low resolution (five vertical levels, R21 in horizontal). In addition, the control climate (Fig. 1a) does not exactly resemble the winter climatology of either hemisphere (although the eddy component, as represented by the EP cross sections shown in Fig. 6, does resemble that observed). In order to assess whether the results depend on model resolution and control climate, additional experiments have been performed using a 10-level, T30 model, with control climates similar to that in Fig. 1a as well as one that more closely resembles Northern Hemisphere wintertime climatology. We found that for all cases, the steady-state response to forcing resembling a concentration of heating (Fig. 1b) is quite similar, except that if we use a more realistic basic state and slightly stronger cooling in the summer hemisphere (as observed in the difference in diabatic heating between 1983 and 1989), the jet in the summer hemisphere also exhibits an equatorward shift similar to the response of the winter jet and the model response looks more similar to observed zonal mean differences between 1983 and 1989. We have also varied the Rayleigh friction timescale at the lowest model level from 1 to 3 days and found that while the midlatitude jet position of the control climate depends on the strength of surface friction (the jet lies closer to the equator when surface friction is stronger), the equilibrium response to tropical heating perturbations are again quite insensitive to the strength of surface friction.

While the results presented in this paper do not seem to depend strongly on model resolution or the basic state, the following issues are of more concern. Recent analyses by Black et al. (1996) based on an extended time series of atmospheric angular momentum suggest that the main propagation signal in zonal mean angular momentum anomalies associated with interannual timescales is from the equator to the subtropics, while our model results only involve interactions between tropical and extratropical anomalies. The absence of a signal near the equator in our results may have been due to the absence of zonally asymmetrical forcing and boundary conditions in our experiments. In addition, here we have used a dry model, with significant static stability (about 4 K per km in potential temperature) in the Tropics. The sensitivity of the response to the forcing could change in the presence of moist effects. The next obvious step would be to extend the modeling and analyses effort presented in this paper using more realistic models of the general circulation that include at least some effects of moisture and zonal asymmetry.

Acknowledgments

The author would like to thank R. Lindzen, R. Rosen, and two anonymous reviewers for helpful comments. This work is supported by the U.S. Department of Energy’s (DOE) National Institute of Global Environmental Change (NIGEC) through the NIGEC Northeast Regional Center at Harvard University (Grant 901214-HAR, Project C), and DOE CHAMMP Grant DE-FG02-93ER61673. Financial support does not constitute an endorsement by DOE of the views expressed in this article.

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Footnotes

Corresponding author address: Dr. Edmund K. M. Chang, Center for Meteorology and Physical Oceanography, Massachusetts Institute of Technology, Room 54-1614, Cambridge, MA 02139.

1

Saravanan (1993) also suggested that it is more difficult to attain a state of equatorial superrotation in a multilevel model as opposed to a two-level model.

2

Here, we have written the equation in Cartesian form for simplicity, but in the data analyses the full spherical form has been used.

3

The dynamical core of the Geophysical Fluid Dynamics Laboratory model described in Held and Suarez (1994) has been used to conduct some comparison studies.

4

Here, although the model uses height as the vertical coordinate, instead of having the same thickness for each level a stretched height coordinate (see Ross and Orlanski 1982) is used such that there is approximately equal mass in each level; hence the coordinate system is very similar to pressure coordinate with equal pressure spacing between the levels.

5

If the dependence of eddy amplitudes on temperature gradient is quadratic or of higher power, then changes in eddy amplitudes will not only depend on changes in temperature gradients, but also on the magnitude of the original temperature gradient itself. Hence one would expect eddy source to increase most over regions where there is both a significant change in the temperature gradient as well as a large temperature gradient to start with.

6

This is basically a standard result for any forced-dissipative system.

7

In fact, baroclinic instability theory suggests that the eddy source should be proportional to at least the third power of the temperature gradient (e.g., Green 1970; Stone 1972; Held 1978; etc.) However, here we are looking at deviations from the control experiment, and as long as the perturbations are not too large, we can always linearize about the control basic state.

8

Note: The precise value of this frequency range depends on the model parameters (namely radiation and dissipation timescales) used here. However, we believe that the conclusions concerning the high- and low-frequency limits should be independent of model parameters.