a. Full AGCM and thermal forcing

The main modeling tool of this study is the Geophysical Fluid Dynamics Laboratory (GFDL) dry atmospheric dynamical core forced by a relaxation toward a given radiative equilibrium temperature profile and damped by Rayleigh damping in the planetary boundary layer. The detail of the model configuration was described in Held and Suarez (1994), and the radiative equilibrium temperature is specified as

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where and . All the profiles including the thermal perturbations used in this study are hemispherically symmetric. All integrations are performed at the horizontal resolution of R30 (rhomboidal 30 spherical harmonic truncations) with 20 evenly spaced sigma levels in the vertical.

In the perturbation experiments, diabatic perturbation is introduced by modifying the equilibrium temperature in the tropical troposphere. The detailed description for the thermal perturbations is provided in SCL. Here we only briefly describe the heating profiles corresponding to tropical upper-tropospheric warming set (TUW) data, which are designed to mimic the tropical heating profiles used in Butler et al. (2010). A perturbation of the form is added to the box term in (4). As the result, the total equilibrium temperature becomes

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In (5), defines the structure of tropical heating centered at the upper troposphere with A set to be −0.1438; is a weighting function to specify the meridional extent of thermal perturbations, with setting the boundary of the thermal perturbation and controlling the sharpness of the boundary. To test the sensitivity of the response to the widths of the tropical heating, we vary from 10° to 30° with an increment of 5°. See Figs. 1a–f for the structures of with and 30°, respectively. Note that the net heating to the atmosphere increases with increasing width. Sensitivities to the model horizontal and vertical resolutions and the type of numerical diffusions were already discussed in SCL.

The thermal forcing and response of the zonal mean temperature and zonal mean zonal wind in the full AGCM and the zonally symmetric model in (left) TUW10 and (right) TUW30 experiments. (a),(f) Equilibrium temperature; (b),(g) zonal mean temperature in full AGCM (K); (c),(h) zonal mean zonal wind in full AGCM (m s⁻¹); (d),(i) zonal mean temperature in the zonally symmetric model (K); and (e),(j): zonal mean zonal wind in the zonally symmetric model (m s⁻¹). In each panel, the climatological field is overlaid as contours.

Citation: Journal of Climate 27, 6; 10.1175/JCLI-D-13-00372.1

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The thermal forcing and response of the zonal mean temperature and zonal mean zonal wind in the full AGCM and the zonally symmetric model in (left) TUW10 and (right) TUW30 experiments. (a),(f) Equilibrium temperature; (b),(g) zonal mean temperature in full AGCM (K); (c),(h) zonal mean zonal wind in full AGCM (m s⁻¹); (d),(i) zonal mean temperature in the zonally symmetric model (K); and (e),(j): zonal mean zonal wind in the zonally symmetric model (m s⁻¹). In each panel, the climatological field is overlaid as contours.

Citation: Journal of Climate 27, 6; 10.1175/JCLI-D-13-00372.1

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The thermal forcing and response of the zonal mean temperature and zonal mean zonal wind in the full AGCM and the zonally symmetric model in (left) TUW10 and (right) TUW30 experiments. (a),(f) Equilibrium temperature; (b),(g) zonal mean temperature in full AGCM (K); (c),(h) zonal mean zonal wind in full AGCM (m s⁻¹); (d),(i) zonal mean temperature in the zonally symmetric model (K); and (e),(j): zonal mean zonal wind in the zonally symmetric model (m s⁻¹). In each panel, the climatological field is overlaid as contours.

Citation: Journal of Climate 27, 6; 10.1175/JCLI-D-13-00372.1

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b. Zonally symmetric model

The zonally symmetric model of the GFDL dynamical core is constructed for two purposes. First, by comparing the full response from the zonally symmetric response to a same thermal forcing, one can cleanly separate the effect of the eddy feedbacks from the direct thermally forced response as in Kim and Lee (2001). Second, to examine the change of the propagation characteristics of the linear waves under WKB assumptions, one should not impose the wind change simulated by the full model because it already includes the effect of eddies. A better practice would be to isolate the thermally direct component of the response in the absence of the eddy feedbacks and use it to perturb the mean state. As the spatial structure of the direct thermally forced response in the zonally symmetric model remains qualitatively similar in time, we will use the equilibrated change for the diagnostic of the refractive index.

In the zonal symmetric configuration, only the zonal mean part (spectral zonal wavenumber 0) is integrated forward in time and the effect of eddies is added as a constant external forcing to the primitive equations. The eddy forcing is diagnosed as the negative tendency of applying the zonal mean fields to the governing equations and integrating forward by one time step. See Kushner and Polvani (2004) and SCL for more details on the construction of the zonally symmetric model and the computation of eddy forcing. The eddy forcing helps to reproduce the same mean state as the full model. This is important for the nonlinear axisymmetric Hadley cell response to thermal forcing, which can depend on the mean state of the atmosphere (e.g., Held and Hou 1980).

c. Large member ensemble transient runs

In addition to the 4000-day-long equilibrated experiments, for each of the thermal forcings of interest we also run 100-member ensembles of transient experiments wherein the forcing is abruptly turned on. The purpose of these transient experiments is to dissect the processes of the evolution from the initial control state to the new forced equilibrium. To form a 100-member ensemble, we first run the control experiment for 7000 days, and then from every 50th day of the last 5000 days a realization branches out. For each of the 100 realizations, the same thermal perturbation is switched on instantaneously and the model integrates for 200 days.

d. Refractive index diagnostics

The characteristics of linear waves propagating in a medium defined by the zonal mean background state are usually represented by the index of refraction, which is widely used in the dynamics community to understand the behavior of waves and their interaction with the mean flow (e.g., Charney and Drazin 1961; Matsuno 1970; Hoskins and Karoly 1981). The refractive index for a linearized QG PV equation in spherical coordinates for a disturbance with given zonal wavenumber s and frequency σ can be expressed as

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where is the zonal mean wind, is the meridional gradient of zonal mean potential vorticity, is the real part of frequency, N is the Brunt–Väisälä frequency, and F(N²) is a function of N² (see Harnik and Lindzen 2001; Seager et al. 2003). Waves propagate in regions where , are evanescent in regions where , are reflected where , and are absorbed where . A general rule of thumb is that wave packets tend to refract toward large values of in regions of wave propagation.

While can be estimated directly from the given basic state and the properties of the wave, it includes the wave propagation characteristics in both meridional and vertical directions, yet only the meridional wavenumber is relevant for the eddy momentum flux and jet shift. Here we make use of a linear QG model to diagnose directly the wave propagation characteristics of a two-dimensional zonal mean basic state (Harnik and Lindzen 2001). The model takes as input the zonal mean zonal wind and temperature with a specified eddy angular phase speed and zonal wavenumber and solves the linearized QG PV equation with the eddy geopotential height perturbation specified at the surface. The model outputs the eddy fluxes, the index of refraction, and its separation into terms related to the squares of the vertical and meridional wavenumbers ( and , respectively):

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Studies have found that and from the linear QG solution serve better the purpose for indicating wave propagation in both the vertical and meridional directions than the direct estimation of (e.g., Harnik and Lindzen 2001; Perlwitz and Harnik 2003). Analogous to the effects of , waves propagate (are evanescent) in the meridional direction in regions where , and the surface of reflects (absorbs) the waves.

e. Finite-amplitude wave activity diagnostics

To elucidate the nonlinear aspects of the eddy–mean flow interaction in the extratropical circulation response to tropical forcings, we adopt the FAWA budget diagnostics developed by Nakamura and collaborators (Nakamura and Zhu 2010; Nakamura and Solomon 2010). While finite-amplitude pseudomomentum and pseudoenergy diagnostics have also been developed according to the generalized Lagrangian mean theory (e.g., Methven 2013), which better describes a local conservation of wave characteristics, Nakamura’s formalism can better describe the eddy–zonal flow interaction and is thus more suitable for the purpose of the present study. In particular, the FAWA defined as (8) below satisfies the exact Eliassen–Palm (e.g., Andrews et al. 1987) and nonacceleration theorems

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in the absence of friction and other nonconservative effects. Note that in (7) measures the net areal displacement of the PV contours from zonal symmetry and is defined geometrically following the material surface of the QG PV as

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The equivalent latitude for a PV value Q (upper case is used to emphasize its Lagrangian property) is determined by the requirement that the area enclosed by the PV contour toward the polar cap equals the area poleward of , that is,

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As such, a monotonic relationship between and Q may be established for each hemisphere and is positive definite in the interior of the atmosphere. Moreover, a hybrid Eulerian–Lagrangian framework can be formulated for nonconservative flow as

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with the irreversible dissipation of wave activity expressed as downgradient PV diffusion. In (10) the eddy QG PV flux is a quantity in zonal average while others are Lagrangian quantities on equivalent latitudes. This, therefore, is an Eulerian–Lagrangian hybrid diagnostic system. Also, is the effective diffusivity of the irreversible mixing of tracers across material surfaces (e.g., Nakamura 1996) and can be calculated either as a residual using (10) or using the modified formula for the eighth-order hyperdiffusion used in our spectral model:

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where m = 4, and is an average near the PV contour Q (see appendix A in SCL for details). Here denotes the diabatic source/sink of wave activity at equivalent latitude and can be safely ignored in the transition behavior of interest here, which takes place long after the switch-on of the thermal forcing. It should be noted that the effective diffusivity accounts for both stirring due to the resolved advection processes and small-scale mixing that ultimately dissipates wave activity. While appears to be proportional to the numerical diffusivity of the model, effective diffusivity is fundamentally governed by the large-scale stretching and deformation of PV contours. Therefore, is largely independent of the value of numerical diffusivity especially when the stirring scales are separable from the diffusion scales (e.g., Marshall et al. 2006).

Making use of (10), the budget for the QG EP flux divergence can be derived as

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This diagnostic equation relates the EP flux divergence/convergence, which serves as a source/sink for the zonal momentum in a transformed Eulerian mean (TEM) sense, to the wave activity tendency, the irreversible dissipation of wave activity, and the diabatic sink/source of wave activity. The leading-order balance in (11) will aid to reveal the key mechanisms for the eddy forcing for the shift of the eddy-driven circulation. Caution should be used in interpreting the budget within the deep tropics, where the QG approximations are no longer valid.

a. Full response

As reported in SCL, all the tropical upper tropospheric thermal forcings lead to a poleward displacement of the eddy-driven jet and a weakening and expansion of the Hadley overturning circulation. In addition, the magnitude of these features increases with the width of the tropical warming (see Fig. 3 in SCL). Figures 1b and 1g compare the full equilibrium temperature and zonal wind response between the narrowest (TUW10) and broadest (TUW30) cases of the TUW group. Despite the different widths of the heating source, the full temperature responses are similar in structure between the two cases: the warming is broadened by the dynamic adjustment in such a way that the meridional temperature gradients are enhanced near 35°–40° latitude at the level of heating; the warming bulges downward in both cases at around 20°, a latitude where the descending motion of the climatological overturning circulation is the strongest; the warming signal also extends downward to the lower troposphere near 40°–45°, largely resulting from the adiabatic warming associated with the anomalous eddy-driven downward motion there (not shown). Great similarity also exists in the zonal wind response between the TUW10 and TUW30 cases (Figs. 1c,h), both projecting strongly on the leading EOF pattern (i.e., the annular mode pattern) of the internal variability. The zonal wind response is in close thermal wind balance with the zonal mean temperature anomalies (Figs. 1c–h). The main difference between the two warming cases is more in magnitude than structure, implicative of a similar dynamical mechanism operating in both.

b. Axisymmetric response

Contrasting the axisymmetric temperature response (Figs. 1d,i) against the full response (Figs. 1b,g), one can see that some of the aforementioned features are rooted in the thermally direct response. The similar meridional spread of the warming between the narrow and broad cases reflects the key nature of the equatorial dynamics: the small Coriolis parameter there cannot maintain much temperature gradient and the Schneider–Held–Hou-type (Schneider 1977; Held and Hou 1980) Hadley cell adjustment always fixates the forced temperature gradient at the edge of the tropics (see also Satoh 1994). In the absence of eddy adjustment, the subtropical maximum warming at levels below the thermal forcing is now more conspicuous, stretching downward and poleward between 20° and 40° along the descending motion of the Lagrangian or TEM tropical overturning circulation (e.g., Tanaka et al. 2004). The zonal wind response in the axisymmetric model is in close thermal wind balance with the temperature field and is also very similar in structure between the narrow and broad warming cases, both characteristic of a tilted dipole between 15° and 40° latitudes (Figs. 1e,j). Note that in the absence of eddy feedback, an upper tropospheric thermal forcing produces little impact on the surface wind. These wind and temperature anomalies in the axisymmetric runs will next be used to perturb the background fields in the wave refractive index analysis.

a. Zonal wind and eddy momentum flux convergence

It is clear from the linear QG wave refraction analysis above that the full response of the circulation to tropical upper-tropospheric warming cannot be explained by the altered wave propagation characteristics resulting from the thermally direct part of the change of the mean wind. To elucidate the irreversible processes of the eddy adjustment that lead up to the full response, for both the TUW10 and TUW30 cases, we conduct 100-member ensemble simulations. Given the interhemispherical symmetry in a statistical sense, the Southern Hemisphere data are reflected about the equator to the Northern Hemisphere so as to double the sampling size. Therefore, all the ensemble mean results presented below are based on an ensemble size of 200.

Figures 4a–d show the evolution of the anomalous zonal wind at 875-hPa level in the transient ensemble experiments with the full model for the TUW10 and TUW30 warming cases, respectively. Owing to the vibrant internal variability of the midlatitude winds, the ensemble mean signal is still noisy. This is especially true for the case of the narrow warming wherein the forced signal is weak and subject to the disruption of internal noise. Nevertheless, the transitional behavior of the forced response from an equatorward to a poleward westerly enhancement is clearly discernable. It is also clear that this transition is directly forced by the poleward migrating eddy momentum convergence, with the eddy forcing leading the wind by approximately 5 days. The lead–lag relationship between the phases of the eddy momentum forcing and the near-surface zonal wind is clearer when both are projected onto the leading EOF pattern (i.e., the annular mode or zonal index) of the 875-hPa wind of the control run (Figs. 4c,f). Although no obvious equatorward-to-poleward transition is found in the transient response to the TUW30 forcing, the initial zonal wind response is still characteristic of a weak equatorward enhancement, consistent with the linear wave refraction analysis. By the end of the transient run, the broad warming produces about 5 times as large a shift as the narrow warming. Note that the scale of the projection onto the annular mode for the broad warming case is 5 times as large as that for the narrow warming case.

(a),(d) The transient evolutions of zonal-mean zonal wind anomalies at 875 hPa in the (left) TUW10 and (right) TUW30 experiments. Contour interval is 0.25 m s⁻¹ with the zero contours omitted. (b),(e) As in (a),(d), but for the barotropic momentum convergence anomaly. The black dashed lines in the top and middle panels denote the latitude of the climatological jet. Contour interval is 0.1 m s⁻¹ day⁻¹. (c),(f) As in (a),(d), but for the projections of the 875-hPa zonal wind and eddy momentum convergence onto the leading EOF of 875-hPa zonal wind. Note that the scales of the y axis are different by a factor of 5 between (c) and (f).

Citation: Journal of Climate 27, 6; 10.1175/JCLI-D-13-00372.1

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(a),(d) The transient evolutions of zonal-mean zonal wind anomalies at 875 hPa in the (left) TUW10 and (right) TUW30 experiments. Contour interval is 0.25 m s⁻¹ with the zero contours omitted. (b),(e) As in (a),(d), but for the barotropic momentum convergence anomaly. The black dashed lines in the top and middle panels denote the latitude of the climatological jet. Contour interval is 0.1 m s⁻¹ day⁻¹. (c),(f) As in (a),(d), but for the projections of the 875-hPa zonal wind and eddy momentum convergence onto the leading EOF of 875-hPa zonal wind. Note that the scales of the y axis are different by a factor of 5 between (c) and (f).

Citation: Journal of Climate 27, 6; 10.1175/JCLI-D-13-00372.1

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(a),(d) The transient evolutions of zonal-mean zonal wind anomalies at 875 hPa in the (left) TUW10 and (right) TUW30 experiments. Contour interval is 0.25 m s⁻¹ with the zero contours omitted. (b),(e) As in (a),(d), but for the barotropic momentum convergence anomaly. The black dashed lines in the top and middle panels denote the latitude of the climatological jet. Contour interval is 0.1 m s⁻¹ day⁻¹. (c),(f) As in (a),(d), but for the projections of the 875-hPa zonal wind and eddy momentum convergence onto the leading EOF of 875-hPa zonal wind. Note that the scales of the y axis are different by a factor of 5 between (c) and (f).

Citation: Journal of Climate 27, 6; 10.1175/JCLI-D-13-00372.1

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For the El Niño case, Harnik et al. (2010) applied the linear wave refraction analysis to all the instants of the zonal wind evolution in the ensemble response to a sudden switch-on of the El Niño forcing. They found that the poleward spread of the wind anomalies with time can be accounted for by linear wave refraction and the final equilibrium state is maintained by the wave–mean flow interaction system, of which the linear wave refraction feedback is a central part. To see if this is also the case for the TUW30 case here, we follow their approach and perform the same wave refraction analysis for every day of the 160 days of the ensemble mean wind response. The resultant eddy momentum acceleration/deceleration at upper-level troposphere between day 10 and 60 is displayed in Fig. 5b to compare with the corresponding level of full zonal wind (Fig. 5a) and full eddy momentum forcing (Fig. 5c). A very similar calculation was also performed for the transient zonal wind evolution in the National Center for Atmospheric Research (NCAR) Community Atmospheric Model, version 3 (CAM3), coupled to a slab ocean model to an instantaneous doubling of CO₂ (Wu et al. 2012, 2013). Consistent with the notion of Harnik et al. (2010), the linear wave refraction feedback acts to sustain the zonal wind anomalies that favor it. In addition, it contributes to the eddy–mean flow interaction that maintains the shifted jet at equilibrium. What is at issue is whether the linear wave refraction mechanism is responsible for instigating the eddy momentum forcing for the poleward transition of the jet. Comparing the temporal evolution of the acceleration in Figs. 5b and 5c against that of the zonal wind, we see that it is the full acceleration that leads the zonal wind, whereas the linear wave feedback is largely in phase with the zonal wind itself, with a few occasions when the former even lags the latter. In particular, during the transition time (from day 33 to 43), the burst of eddy momentum acceleration directly responsible for the poleward migration of the 252-hPa zonal wind takes place largely without the contribution from the linear refraction feedback. Based on this evidence, we can only conclude that linear wave refraction mechanism, in the absence of the change of the wave speed and/or wave length, only plays a feedback role in sustaining the wind anomalies that already exist, and falls short in serving as the initiating mechanism for the poleward transition of the jet.

The ensemble mean evolution under the TUW30 thermal forcing of (a) 252-mb zonal wind, (b) eddy momentum convergence from linear QG analysis, and (c) eddy momentum convergence produced in the full ensemble simulation. Only the period centered around the poleward transition of the eddy driven jet is shown. The black dashed line denotes the location of the climatological jet. Contour intervals are 0.1 m s⁻¹ and 0.1 m s⁻¹ day⁻¹ in (a) and (c), respectively, and arbitrary in (b).

Citation: Journal of Climate 27, 6; 10.1175/JCLI-D-13-00372.1

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The ensemble mean evolution under the TUW30 thermal forcing of (a) 252-mb zonal wind, (b) eddy momentum convergence from linear QG analysis, and (c) eddy momentum convergence produced in the full ensemble simulation. Only the period centered around the poleward transition of the eddy driven jet is shown. The black dashed line denotes the location of the climatological jet. Contour intervals are 0.1 m s⁻¹ and 0.1 m s⁻¹ day⁻¹ in (a) and (c), respectively, and arbitrary in (b).

Citation: Journal of Climate 27, 6; 10.1175/JCLI-D-13-00372.1

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The ensemble mean evolution under the TUW30 thermal forcing of (a) 252-mb zonal wind, (b) eddy momentum convergence from linear QG analysis, and (c) eddy momentum convergence produced in the full ensemble simulation. Only the period centered around the poleward transition of the eddy driven jet is shown. The black dashed line denotes the location of the climatological jet. Contour intervals are 0.1 m s⁻¹ and 0.1 m s⁻¹ day⁻¹ in (a) and (c), respectively, and arbitrary in (b).

Citation: Journal of Climate 27, 6; 10.1175/JCLI-D-13-00372.1

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b. Irreversible eddy mixing through finite-amplitude wave activity diagnostics

Before delving into the FAWA analysis, let us first examine the temporal evolutions of EP flux in the two TUW ensemble experiments. For reference, Fig. 6a gives the climatological EP flux (vectors) and divergence (shading), together with the momentum convergence (contours). Guided by the temporal evolution of the near-surface wind and the barotropic eddy momentum forcing (Figs. 4c,f), we divide the evolution of the transient wind response into two stages—the stage prior to the shift (before day 38) and the stage of shifted jet (around day 100)—and plot the trends of the EP flux accordingly for both ensemble experiments. The most pronounced features in the EP flux divergence during the preshift stage are the reduction (red color) of the climatological convergence centered at 350 hPa and 45° latitude and the enhancement (blue color) of convergence at its equatorward and upward side (Figs. 6b,d). A similar dipole is evident in both the narrow and broad heating cases, with the latter being slightly poleward displaced. As will be explained later, the positive EP flux divergence in the upper troposphere is the result of reduced dissipation of wave activities coming from below and those that survive can reach the level of free propagation and propagate meridionally.

(a) Climatological EP flux vector, EP flux divergence (m s⁻¹ day⁻¹; shading), and eddy momentum convergence (contours) in the control experiment. The red solid contours denote the positive eddy momentum convergence and the black dashed contours the negative convergence. (b),(c) As in (a), but for the ensemble mean trend from day 1 to 38 and from day 1 to 100, respectively, in the transient experiments under TUW10 warming. (d),(e): As in (b),(c), but for TUW30 warming. The contour intervals are 1 m s⁻¹ day⁻¹ in (a), 0.2 m s⁻¹ day⁻¹ in (b) and (c), and 0.5 m s⁻¹ day⁻¹ in (d) and (e). Zero contours for eddy momentum convergence are omitted.

Citation: Journal of Climate 27, 6; 10.1175/JCLI-D-13-00372.1

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(a) Climatological EP flux vector, EP flux divergence (m s⁻¹ day⁻¹; shading), and eddy momentum convergence (contours) in the control experiment. The red solid contours denote the positive eddy momentum convergence and the black dashed contours the negative convergence. (b),(c) As in (a), but for the ensemble mean trend from day 1 to 38 and from day 1 to 100, respectively, in the transient experiments under TUW10 warming. (d),(e): As in (b),(c), but for TUW30 warming. The contour intervals are 1 m s⁻¹ day⁻¹ in (a), 0.2 m s⁻¹ day⁻¹ in (b) and (c), and 0.5 m s⁻¹ day⁻¹ in (d) and (e). Zero contours for eddy momentum convergence are omitted.

Citation: Journal of Climate 27, 6; 10.1175/JCLI-D-13-00372.1

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(a) Climatological EP flux vector, EP flux divergence (m s⁻¹ day⁻¹; shading), and eddy momentum convergence (contours) in the control experiment. The red solid contours denote the positive eddy momentum convergence and the black dashed contours the negative convergence. (b),(c) As in (a), but for the ensemble mean trend from day 1 to 38 and from day 1 to 100, respectively, in the transient experiments under TUW10 warming. (d),(e): As in (b),(c), but for TUW30 warming. The contour intervals are 1 m s⁻¹ day⁻¹ in (a), 0.2 m s⁻¹ day⁻¹ in (b) and (c), and 0.5 m s⁻¹ day⁻¹ in (d) and (e). Zero contours for eddy momentum convergence are omitted.

Citation: Journal of Climate 27, 6; 10.1175/JCLI-D-13-00372.1

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Extending the trend analysis to day 100 indicates that similar EP flux and convergence patterns progress gradually poleward with time and result in very similar patterns of EP flux and eddy momentum convergence for both heating cases (Figs. 6c,e). Now the eddy momentum convergence and divergence are located at 55° and 35° respectively with the nodal point at 43°, a perfect pattern for the poleward shift of the eddy-driven jet. Accompanying this poleward progression of the EP flux divergence dipole is the anomalous upward (downward) EP flux in the mid-to-lower troposphere at the poleward (equatorward) flank of the mean eddy-driven jet, consistent with the poleward transition of the lower-level baroclinicity represented by the maximum Eady growth rate (not shown). These low-level EP flux anomalies should be interpreted as the positive feedback (through the altered baroclinicity) to, rather than the cause for, the poleward shift of the eddy-driven jet. Similar baroclinic feedback was also noticed in the transient response to the TUW30 heating in SCL and in the aquaplanet model simulation in Chen et al. (2013). As will be demonstrated next, the evolution of the anomalous EP flux and the associated wind response can be accounted for largely by the upper-level dissipation processes.

In view of that the EP flux adjustment during the period (days 1–100) of the poleward shift is characteristically the same between the TUW10 and TUW30 cases, we only show the FAWA budget for the TUW30 case in Fig. 7. The results for TUW10 only differ in magnitude and hence are not shown. The wave activity tendency (Fig. 7b) and the diabatic source (not shown) terms during days 1–100 turn out to be small and the main balance is between the PV flux trend (Fig. 7a) and the trend of the dissipation of wave activity (Fig. 7c). The dissipation term can be further decomposed into the components of PV gradient change (Fig. 7d) and effective diffusivity change (Fig. 7e), that is,

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The PV gradient and its change can be directly computed in equivalent latitude and the results are shown in Fig. 7f. In the upper troposphere, the maximum of the climatological PV gradient is located at 43° latitude, in alignment with the minimum of diffusivity and the position of the mean westerly jet. As such, the westerly jet acts as a mixing barrier in the upper troposphere in the same sense as does the core of the Antarctic Circumpolar Current in the Southern Ocean (Marshall et al. 2006; Ferrari and Nikurashin 2010). Under the TUW30 forcing, the total PV gradient is overall weakened except for a strengthening near the tropopause slightly poleward of the mean maximum. This poleward enhancement of the PV gradient (see also Fig. 8a) is consistent with what is found in the similar experiment conducted with a different AGCM by Butler et al. (2011, see their Fig. 7a), but with a key difference here that the enhancement of PV gradient contributes marginally to change of dissipation as it coincides with the climatological diffusivity minimum (cf. Fig. 8d).

The linear trend of the ensemble mean wave activity budget [see (8)] from day 1 to 100 in the TUW30 simulation. (a) Eddy PV flux, (b) negative wave activity tendency, (c) total dissipation change, (d) dissipation change because of PV gradient change, (e) dissipation change because of effective diffusivity change, (f) PV gradient change, (g) effective diffusivity change derived as residual, and (h) directly estimated effective diffusivity change. The contours in (a),(c),(d),(e) are the climatological mean PV flux. The contours in (f)–(h) are the corresponding climatological fields. The contour intervals for the climatological eddy PV flux, PV gradient, and effective diffusivity are 2 m s⁻¹ day⁻¹, 4.0 × 10⁻¹¹ m⁻¹ s⁻¹, and 4.0 × 10⁵ m² s⁻¹, respectively.

Citation: Journal of Climate 27, 6; 10.1175/JCLI-D-13-00372.1

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The linear trend of the ensemble mean wave activity budget [see (8)] from day 1 to 100 in the TUW30 simulation. (a) Eddy PV flux, (b) negative wave activity tendency, (c) total dissipation change, (d) dissipation change because of PV gradient change, (e) dissipation change because of effective diffusivity change, (f) PV gradient change, (g) effective diffusivity change derived as residual, and (h) directly estimated effective diffusivity change. The contours in (a),(c),(d),(e) are the climatological mean PV flux. The contours in (f)–(h) are the corresponding climatological fields. The contour intervals for the climatological eddy PV flux, PV gradient, and effective diffusivity are 2 m s⁻¹ day⁻¹, 4.0 × 10⁻¹¹ m⁻¹ s⁻¹, and 4.0 × 10⁵ m² s⁻¹, respectively.

Citation: Journal of Climate 27, 6; 10.1175/JCLI-D-13-00372.1

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The linear trend of the ensemble mean wave activity budget [see (8)] from day 1 to 100 in the TUW30 simulation. (a) Eddy PV flux, (b) negative wave activity tendency, (c) total dissipation change, (d) dissipation change because of PV gradient change, (e) dissipation change because of effective diffusivity change, (f) PV gradient change, (g) effective diffusivity change derived as residual, and (h) directly estimated effective diffusivity change. The contours in (a),(c),(d),(e) are the climatological mean PV flux. The contours in (f)–(h) are the corresponding climatological fields. The contour intervals for the climatological eddy PV flux, PV gradient, and effective diffusivity are 2 m s⁻¹ day⁻¹, 4.0 × 10⁻¹¹ m⁻¹ s⁻¹, and 4.0 × 10⁵ m² s⁻¹, respectively.

Citation: Journal of Climate 27, 6; 10.1175/JCLI-D-13-00372.1

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The response to TUW30 warming as a function of time and equivalent latitude. (a) Contours of the meridional gradient of the Lagrangian PV at 325 K (contour interval is 1.0 × 10⁻¹¹ m⁻¹ s⁻¹). Shadings indicate the eddy PV flux at 325 K. (b) Contours of the anomalous (contour interval is 1.0 × 10⁵ m² s⁻¹). Shadings indicate the dissipation component resulting from the change of .

Citation: Journal of Climate 27, 6; 10.1175/JCLI-D-13-00372.1

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The response to TUW30 warming as a function of time and equivalent latitude. (a) Contours of the meridional gradient of the Lagrangian PV at 325 K (contour interval is 1.0 × 10⁻¹¹ m⁻¹ s⁻¹). Shadings indicate the eddy PV flux at 325 K. (b) Contours of the anomalous (contour interval is 1.0 × 10⁵ m² s⁻¹). Shadings indicate the dissipation component resulting from the change of .

Citation: Journal of Climate 27, 6; 10.1175/JCLI-D-13-00372.1

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The response to TUW30 warming as a function of time and equivalent latitude. (a) Contours of the meridional gradient of the Lagrangian PV at 325 K (contour interval is 1.0 × 10⁻¹¹ m⁻¹ s⁻¹). Shadings indicate the eddy PV flux at 325 K. (b) Contours of the anomalous (contour interval is 1.0 × 10⁵ m² s⁻¹). Shadings indicate the dissipation component resulting from the change of .

Citation: Journal of Climate 27, 6; 10.1175/JCLI-D-13-00372.1

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The dissipation features that matter most for the eddy momentum flux arise from the change of effective diffusivity. In particular, the enhancement (reduction) of dissipation centered at 40° and 250 hPa (50° and 400 hPa) plays the dominant role in anchoring the upward and equatorward EP flux and maintaining a poleward transport of momentum (cf. Fig. 6e). The change of effective diffusivity is also responsible for most of the subtropical PV flux anomalies, which in turn are related to the weak eddy-induced momentum acceleration near 15° and 300 hPa (Fig. 6e). This, together with the deceleration farther poleward, represents a poleward advance of the critical layer where the subtropical wave breaking usually occurs and the wave activity gets dissipated. Speculatively, critical layer feedback through the increased phase speed of waves might be involved in the subtropical adjustment (e.g., Chen et al. 2008).

The change of the inferred from (11) as a residual is shown in Fig. 7g. It is this pattern of change, acting on the background PV gradient, that accounts for most of the dissipation change in the subtropics and midlatitude (Fig. 7e). The most prominent feature in Fig. 7g is a dipole in the subtropics with a decrease centered at 20° (350 hPa) and an increase at 30° (250 hPa). Adding onto the background mean , this dipole tends to shift the center of the total effective diffusivity poleward and slightly upward. A special note of caution should be added that this residual estimate of cannot be guaranteed to be accurate given the many assumptions made in the budget. Perhaps the most critical caveat of the diagnostics above is the assumption that (11) is balanced to a good approximation, which is not guaranteed given the QG approximations. As an a posteriori cross-validation, we directly compute making use of the formula of Nakamura and Zhu (2010) for an arbitrary tracer satisfying the advection–diffusion equation (see the appendix of SCL). The result is presented in Fig. 7h. Although the pixel-to-pixel match is elusive, the qualitative agreement between the direct estimate and the residual estimate is encouraging. Most importantly, the up-and-poleward enhancement of the mixing in the subtropical upper troposphere identified above can be reasonably vindicated by the direct estimate. It follows that the insights gained from the wave activity budget above are likely to hold for the response in question, and are not susceptible to errors because of the QG approximations.

In summary, we find that the enhancement and poleward advance of the upper-level mixing is key for the equilibrating of the eddy PV flux, a somewhat different assertion from Butler et al. (2011), who argued that the thermally forced PV gradient anomalies dictate the eddy adjustment. To compare with Butler et al. (2011), Fig. 8a shows the temporal evolution of the ensemble mean QG PV flux and Lagrangian PV gradient () interpolated onto the isentropic surface near the tropopause (325 K). Qualitatively the same evolution has been derived with the direct calculation of the isentropic eddy Ertel PV flux and hence is not shown. This approach can be justified by the fact that the meridional eddy QG PV flux is a good approximation for the residual eddy Ertel PV flux in the QG limit (Plumb and Ferrari 2005). Near the tropopause level, although the anomalous PV gradient (contours in Fig. 8a) exhibits a similar dipolar structure to that of the experiment in Butler et al. (cf. their Fig. 7a), the eddy PV flux anomalies are not exactly down the anomalous PV gradient as in Butler et al. (2011) (cf. our Fig. 8a and their Fig. 7a). Instead, as shown in Fig. 8b, during the adjustment toward the equilibrium the effective diffusivity (contours) gradually strengthens and advances poleward, a process that dominates the total change of the dissipation term and hence the eddy PV flux (cf. the shading in Figs. 8a and 8b). Our result suggests that the simple depiction of the downgradient eddy PV flux put forward by Butler et al. (2011; see schematic in their Fig. 8), while valid for the particular case investigated therein, may not be adequate for portraying the full irreversible processes important for the adjustment of the midlatitude circulation to a tropical thermal forcing in general. Our result also underscores the challenge of understanding the atmospheric circulation response under global warming: for a theory to predict the response of the eddy-driven wind, it must also predict the change of the effective diffusivity.

What drives the change in effective diffusivity? Simple models of transport and mixing have suggested that a stronger zonal jet tends to reduce the effective diffusivity across the jet, leading to a stronger transport barrier (Haynes et al. 2007). The initial change (from day 1 to 30) in our switch-on transient simulations, as well as the zonally symmetric model, shows a strengthening of zonal wind (Figs. 1e,j and 5a) or PV gradient (Fig. 8a) on the equatorward side of the mean jet. In the full model, however, we see that the effective diffusivity enhancement drifts from 15°–30° to 20°–35° during the first 40 days (Fig. 8b). Since the effective diffusivity has increased against the suppression by a stronger thermally driven subtropical jet, eddy stirring has to increase at the latitudes of 30°–35° so as to break the mixing barrier that is reinforced initially by thermal forcing. This latitudinal shift in eddy stirring may be explained by a change in wave characteristics or an increased equatorward wave propagation resulting from either less poleward wave propagation (Kidston et al. 2011; Lorenz 2013, manuscript submitted to J. Atmos. Sci.) or less local dissipation at the latitude of baroclinic wave generation (Chen et al. 2013; SCL). The latter is supported by an increase in wave activity amplitude prior to the poleward transition in jet latitude (not shown). It would probably require a space and time decomposition of wave activity to understand why the stirring in subtropics is increased, which is beyond the scope of this study. Nevertheless, we have demonstrated that effective diffusivity can quantify the efficiency of the irreversible mixing of PV in understanding the jet shift, a capability beyond the traditional Eulerian diagnostics.

c. Wave breaking statistics

The effective diffusivity diagnosed above reflects the mixing in a coarse-grain sense due to the chaotic advection or stirring by the resolved flow fields, which produce increasingly finer scales in the PV distribution until the small-scale dissipation takes over (e.g., Haynes and Shuckburgh 2000). In the upper troposphere, strong chaotic stirring and the resultant irreversible mixing take place where synoptic wave breaking events occur, usually at the flanks of the jet near critical latitudes. The jet, on the other hand, acts as a barrier for mixing. As such, there might exist a good correspondence between the statistics of wave breakings and the distribution of effective diffusivity. Further evidence for the notion above about the importance of changing effective diffusivity may be collected from the changing statistics of the wave breaking events.

To obtain the statistics of wave breaking, we make use of the wave breaking detection algorithm of Rivière (2009) to detect the grid points where a local reversal of the QG PV takes place. The grid point occupied by a cyclonically (anticyclonically) overturning PV filament is identified as a cyclonic (anticyclonic) wave-breaking region and is denoted as CWB (AWB). Figure 9a shows the total frequency of the AWB (solid contours) and CWB (dashed contours) from the long control experiment, measured by the fraction of the meridians subject to wave breaking for a latitudinal circle in a time average sense. Comparing to the contours in Fig. 7g, it is clear that the subtropical section of the large effective diffusivity results predominantly from the AWB and the midlatitude section from the CWB, respectively. Notably, the linear trend of the wave breaking statistics in the ensemble adjustment to the TUW30 forcing (Fig. 9b) bears a qualitative resemblance to the trend of and thus these two fields are connected: the subtropical dipole in AWB frequency implicates an upward and poleward increase of the wave breaking–related effective diffusivity near the tropopause level; meanwhile, the decrease in the CWB frequency near 50° latitude accounts for the reduction of the effective diffusivity there.

(a) Climatological latitude–height distribution of the frequency of the AWB and CWB (in unit of fraction of meridians per day). The solid (dashed) contours are for the frequency of the AWB (CWB). (b) As in (a), but for the trend of the frequency of the AWB (shadings) and CWB (contours) over the period days 1–100 in the TUW30 experiment.

Citation: Journal of Climate 27, 6; 10.1175/JCLI-D-13-00372.1

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(a) Climatological latitude–height distribution of the frequency of the AWB and CWB (in unit of fraction of meridians per day). The solid (dashed) contours are for the frequency of the AWB (CWB). (b) As in (a), but for the trend of the frequency of the AWB (shadings) and CWB (contours) over the period days 1–100 in the TUW30 experiment.

Citation: Journal of Climate 27, 6; 10.1175/JCLI-D-13-00372.1

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(a) Climatological latitude–height distribution of the frequency of the AWB and CWB (in unit of fraction of meridians per day). The solid (dashed) contours are for the frequency of the AWB (CWB). (b) As in (a), but for the trend of the frequency of the AWB (shadings) and CWB (contours) over the period days 1–100 in the TUW30 experiment.

Citation: Journal of Climate 27, 6; 10.1175/JCLI-D-13-00372.1

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Since the heating profile in the TUW30 case is designed to mimic the tropical tropospheric heating associated with global warming, a parallel may be drawn between the results of Fig. 9b and the similar analysis with the simulations by fully coupled climate models. Indeed, the reduction (increase) of the CWB (AWB) frequency at the equatorward (poleward) side of the corresponding climatological distribution is exactly what is ascribed to the changes of upper-level wave breaking statistics due to the greenhouse gas forcing simulated by the French climate models [see upper panels of Fig. 2 in Rivière (2011)].