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  • View in gallery

    The climatology (contours) and annular mode (i.e., EOF1; shading) of the CTL: (a) zonal wind (m s−1) and (b) temperature (K). The contour interval is 6 m s−1 for zonal wind and 15 K for temperature. Dashed lines indicate negative values.

  • View in gallery

    The forced simulations. (top) full (barotropic + baroclinic) annular mode anomaly (EXP1): (a) (m s−1) and (b) (K). (bottom) Only the barotropic component of the annular mode anomaly (EXP2): (c) (m s−1) and (d) (K).

  • View in gallery

    Eddy momentum and heat fluxes: climatology of CTL (contours) and response to the annular mode pattern (shading). (top) Response to the full (barotropic + baroclinic) annular mode anomaly (EXP1): (a) (m2 s−2) and (b) (m s−1 K). (bottom) Response to only the barotropic component of the annular mode anomaly (EXP2): (c) (m2 s−1) and (d) (m s−1 K). The contour interval is 15 m2 s−2 for eddy momentum flux and 5 m s−1 K for eddy heat flux. Dashed lines indicate negative values.

  • View in gallery

    Effect of surface friction. (a) Shading shows [see Eq. (4) for the definition of normalization {}u]. Contour lines show the streamlines of the induced secondary circulation, with solid (dashed) lines indicating clockwise (counterclockwise) circulation. (b) Shading shows the total tendency of resulting from surface friction normalized according to Eq. (4). Contour lines show , with solid (dashed) lines indicating westerly (easterly) wind. (c) Shading shows the total tendency of resulting from surface friction normalized according to Eq. (5). Contour lines show , with solid (dashed) lines indicating positive (negative) temperature. Shading in all panels has units of day−1.

  • View in gallery

    As in Fig. 4, but for the effect of surface friction on the barotropic component of the annular mode , where .

  • View in gallery

    Effect of thermal radiation. (a) Shading shows [see Eq. (5) for the definition of normalization {}T]. The induced secondary circulation is weak and because the streamlines have the same intervals as in other figures, they do not appear here. (b) Shading shows the total tendency of resulting from thermal radiation normalized according to Eq. (4). (c) Shading shows the total tendency of resulting from thermal radiation normalized according to Eq. (5). Shading in all panels has units of day−1. See the caption of Fig. 4 for more details.

  • View in gallery

    As in Fig. 6, but for the effect of thermal radiation on the barotropic component of the annular mode , where .

  • View in gallery

    Effect of meridional eddy momentum flux divergence. (a) Shading shows (calculations are conducted in spherical coordinates and is used for convenience). (b) Shading shows the total tendency of normalized according to Eq. (4). (c) Shading shows the total tendency of normalized according to Eq. (5). Shading in all panels has units of day−1. See the caption of Fig. 4 for more details.

  • View in gallery

    As in Fig. 8, but for the effect of meridional eddy momentum flux divergence in response to the barotropic component of the annular mode , where .

  • View in gallery

    Effect of meridional eddy heat flux divergence. (a) Shading shows (calculations are conducted in spherical coordinates and is used for convenience). (b) Shading shows the total tendency of normalized according to Eq. (4). (c) Shading shows the total tendency of normalized according to Eq. (5). Shading in all panels has units of day−1. See the caption of Fig. 4 for more details.

  • View in gallery

    As in Fig. 10, but for the effect of meridional eddy heat flux divergence in response to the barotropic component of the annular mode , where .

  • View in gallery

    As in Fig. 10, but for vertical eddy heat flux divergence. The induced secondary circulation is weak and because the streamlines have the same intervals as in other figures, they do not appear here.

  • View in gallery

    The meridional wind of the induced secondary circulation (m s−1): (a) and (b) sum of all induced meridional velocities calculated individually from solving the Eliassen problem.

  • View in gallery

    Contribution of different processes to the tendency budget of (top) , (middle) , and (bottom) quasigeostrophic potential vorticity QGPV. The vertical axes have units of day−1 and show (top) , (middle) , and (bottom) for tendency A. Blue (red) bars indicate the contribution to the full (barotropic-only component of the) annular mode anomaly.

  • View in gallery

    Schematic of the annular mode dynamics in the idealized atmosphere, focused on the poleward half of the annular mode anomaly (40°–60°) for convenience. Red (blue) arrows indicate positive (negative) influence.

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Quantifying the Annular Mode Dynamics in an Idealized Atmosphere

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  • 1 Department of Mechanical Engineering, and Department of Earth, Environmental and Planetary Sciences, Rice University, Houston, Texas
  • | 2 Department of Earth and Planetary Sciences, and John A. Paulson School of Engineering and Applied Sciences, Harvard University, Cambridge, Massachusetts
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Abstract

The linear response function (LRF) of an idealized GCM, the dry dynamical core with Held–Suarez physics, is used to accurately compute how eddy momentum and heat fluxes change in response to the zonal wind and temperature anomalies of the annular mode at the quasi-steady limit. Using these results and knowing the parameterizations of surface friction and thermal radiation in Held–Suarez physics, the contribution of each physical process (meridional and vertical eddy fluxes, surface friction, thermal radiation, and meridional advection) to the annular mode dynamics is quantified. Examining the quasigeostrophic potential vorticity balance, it is shown that the eddy feedback is positive and increases the persistence of the annular mode by a factor of more than 2. Furthermore, how eddy fluxes change in response to only the barotropic component of the annular mode, that is, vertically averaged zonal wind (and no temperature) anomaly, is also calculated similarly. The response of eddy fluxes to the barotropic-only component of the annular mode is found to be drastically different from the response to the full (i.e., barotropic + baroclinic) annular mode anomaly. In the former, the eddy generation is significantly suppressed, leading to a negative eddy feedback that decreases the persistence of the annular mode by nearly a factor of 3. These results suggest that the baroclinic component of the annular mode anomaly, that is, the increased low-level baroclinicity, is essential for the persistence of the annular mode, consistent with the baroclinic mechanism but not the barotropic mechanism proposed in the previous studies.

© 2019 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Pedram Hassanzadeh, pedram@rice.edu

Abstract

The linear response function (LRF) of an idealized GCM, the dry dynamical core with Held–Suarez physics, is used to accurately compute how eddy momentum and heat fluxes change in response to the zonal wind and temperature anomalies of the annular mode at the quasi-steady limit. Using these results and knowing the parameterizations of surface friction and thermal radiation in Held–Suarez physics, the contribution of each physical process (meridional and vertical eddy fluxes, surface friction, thermal radiation, and meridional advection) to the annular mode dynamics is quantified. Examining the quasigeostrophic potential vorticity balance, it is shown that the eddy feedback is positive and increases the persistence of the annular mode by a factor of more than 2. Furthermore, how eddy fluxes change in response to only the barotropic component of the annular mode, that is, vertically averaged zonal wind (and no temperature) anomaly, is also calculated similarly. The response of eddy fluxes to the barotropic-only component of the annular mode is found to be drastically different from the response to the full (i.e., barotropic + baroclinic) annular mode anomaly. In the former, the eddy generation is significantly suppressed, leading to a negative eddy feedback that decreases the persistence of the annular mode by nearly a factor of 3. These results suggest that the baroclinic component of the annular mode anomaly, that is, the increased low-level baroclinicity, is essential for the persistence of the annular mode, consistent with the baroclinic mechanism but not the barotropic mechanism proposed in the previous studies.

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Corresponding author: Pedram Hassanzadeh, pedram@rice.edu

1. Introduction

In the extratropical circulation of both hemispheres, a long-recognized dominant pattern of variability at the intraseasonal to interannual time scales is the “annular mode,” which is often derived from empirical orthogonal function (EOF) analysis of meteorological fields such as zonal-mean zonal wind or geopotential heights and has been known for decades (Kidson 1988; Thompson and Wallace 1998; Feldstein 2000). The leading EOF (EOF1) of features an equivalent-barotropic dipolar structure centered around the time-mean jet and describes north–south temporal fluctuations of the midlatitude westerlies (Nigam 1990; Hartmann and Lo 1998; Lorenz and Hartmann 2001, 2003). The zonal index, the time series [principal component (PC)] associated with the annular mode, is characterized by temporal persistence, that is, characteristic time scale longer than the synoptic time scale: the year-round decorrelation (e-folding) time scale of the zonal index is ~10 days in both hemispheres (Thompson and Woodworth 2014; Thompson and Li 2015) although there is a strong seasonal dependence with time scales of ~15 days in the Northern Hemisphere (NH) in December–January and ~20 days in the Southern Hemisphere (SH) in November–December (see Fig. 2 of Gerber et al. 2008a).

It should be noted that recently, Thompson and Woodworth (2014), Thompson and Barnes (2014), and Thompson and Li (2015) have found another dominant pattern of variability, called the “baroclinic annular mode” (BAM), in the extratropical circulation of both hemispheres. BAM emerges as the leading EOF of eddy kinetic energy (EKE) and its PC exhibits quasi periodicity at the time scales of ~20–25 (~25–30) days in the NH (SH). Boljka et al. (2018) have offered some evidence for the coupling of the barotropic annular mode (EOF1 of ) and the baroclinic variability (EOFs of EKE). In this study, we focus on the former and simply refer to it as annular mode.

Annular modes also robustly emerge as the leading pattern of variability in a broad hierarchy of models, from stochastically forced barotropic models to comprehensive GCMs (e.g., Robinson 1991; Feldstein and Lee 1998; Limpasuvan and Hartmann 1999; Vallis et al. 2004; Gerber et al. 2008b,a; Barnes et al. 2010; Zurita-Gotor et al. 2014; Sheshadri and Plumb 2017). However, the annular modes simulated in GCMs are too persistent and the decorrelation time scales can be several times larger than the observed time scales (Gerber et al. 2008a,b). The problem of too-persistent annular modes, which exists across the hierarchy of idealized to comprehensive GCMs, is a matter of concern not only because a key aspect of a leading pattern of climate variability is not correctly simulated in GCMs, but also because the fluctuation–dissipation theorem (FDT) suggests that if the internal time scale of a system is overestimated, then the system’s response to external forcings (e.g., changes in radiative forcing) is overestimated by the same factor as well (Gerber et al. 2008b; Ring and Plumb 2008; Shepherd 2014); see Simpson and Polvani (2016) and Sheshadri and Plumb (2017) for further discussions and recent updates on this subject.

To understand this component of climate variability and to potentially improve the GCMs, the persistence of the annular mode and its source have been extensively studied and debated in the past few decades. The persistence is often (but not always; see below) attributed to a positive eddy–jet feedback internal to the troposphere: eddies are altered by the zonal-mean anomaly (of the annular mode) such that the anomalous eddy fluxes reinforce the zonal-mean anomaly, thus increasing the persistence of the annular mode (e.g., Robinson 1991; Yu and Hartmann 1993; Branstator 1995; Limpasuvan and Hartmann 2000; Robinson 2000; Lorenz and Hartmann 2001, 2003). Lorenz and Hartmann (2001) formulated a simple linear feedback model for the zonal index and developed a statistical method to quantify the feedback. Using the reanalysis data, they found evidence for the existence of a positive eddy–jet feedback and quantified its magnitude in the SH and NH extratropical circulations (Lorenz and Hartmann 2001, 2003). Simpson et al. (2013) have recently proposed another statistical framework to quantify the feedback in reanalysis data.

The mechanism of this internal (to troposphere) eddy–jet feedback, however, has remained unclear. Some studies have argued for a barotropic mechanism for the feedback: the annular mode’s anomalous changes the upper-tropospheric meridional shear, thus altering the upper-level wave propagation and leading to a positive eddy–jet feedback (e.g., Gerber and Vallis 2007; Chen and Zurita-Gotor 2008; Nie et al. 2014; Lorenz 2014). Some other studies have supported a baroclinic mechanism for the feedback: the annular mode’s zonal-mean anomaly changes the lower-tropospheric baroclinicity, thus altering the low-level eddy generation and leading to a positive eddy–jet feedback (e.g., Robinson 1996, 2000; Lorenz and Hartmann 2001; Robert et al. 2017; Boljka et al. 2018). We emphasize that the two mechanisms are not mutually exclusive. We also emphasize again that the discussion here is concerned with the barotropic annular mode and the terms barotropic and baroclinic mechanisms should not be confused with barotropic and baroclinic annular modes.

Even the source of the annular mode persistence is not a settled issue and a few studies have questioned the importance of the internal eddy–jet feedback (Feldstein and Lee 1998; Byrne et al. 2016, 2017; Byrne and Shepherd 2018). In fact, Byrne et al. (2016) have shown that the statistical methods, such as the ones developed in Lorenz and Hartmann (2001) and Simpson et al. (2013), cannot distinguish between an internal eddy–jet feedback and a low-frequency external forcing, casting doubt on the evidence for the internal feedback in the SH reanalysis data. They have further argued that this external forcing is a result of the stratospheric variability (Byrne et al. 2017; Byrne and Shepherd 2018).

The major challenge in understanding and quantifying the role of eddies in the annular mode dynamics is the lack of a complete theory that allows us to compute changes in eddy fluxes in terms of changes in the mean flow. Specifically, we need to calculate how eddy momentum and heat fluxes change in response to the anomalous of the annular mode in the low-frequency (approximately steady state) limit; see, for example, the discussion in Robinson (2000) and in particular the last sentence.1 Recently, Hassanzadeh and Kuang (2016a) have introduced a framework, based on using the linear response functions (LRFs), that can help with addressing these types of eddy–mean flow interaction problems in GCMs.

Applying this framework to an idealized GCM, the dry dynamical core with Held–Suarez physics (Held and Suarez 1994), Ma et al. (2017) showed unequivocally that there is an internal positive eddy–jet feedback in the annular mode dynamics of this idealized atmosphere, accurately quantified the feedback magnitude, and developed a new statistical method, based on low-pass filtering, to compute the feedback from data. The estimates of feedback magnitude in the GCM and SH reanalysis data were shown to be more robust with respect to the free parameters in the low-pass-filtering method of Ma et al. (2017) compared to other statistical methods. However, the problem of isolating internal feedback from external forcing in reanalysis data using statistical methods still exists.

The work of Ma et al. (2017) was limited to the zonal index (i.e., the PC1), a scalar quantity based on zonal wind. Here, we shall extend the analysis to the full latitude–pressure plane and include both zonal wind and temperature to further explore the detailed physical processes that produce the positive feedback. More specifically, in the current study we aim to use the LRF framework of Hassanzadeh and Kuang (2016a) and the same idealized setup as Ma et al. (2017) to

  1. quantify the contribution of different physical processes (eddy momentum flux, eddy heat flux, surface friction, thermal radiation, and meridional advection) to the annular mode dynamics,
  2. quantify the contribution of the eddy–jet feedback to the annular mode persistence, and
  3. examine the role of baroclinic and barotropic mechanisms in the feedback dynamics.

The remainder of the paper is structured as follows. The idealized GCM and its annular mode are discussed in section 2. The LRF framework and simulations for quantifying the eddy–jet feedback are presented in section 3. Quantifying different physical processes and the annular mode budget are discussed in sections 4 and 5, respectively. Discussion and summary are in section 6.

2. The control simulation and its annular mode

We use the Geophysical Fluid Dynamics Laboratory (GFDL) dry dynamical core, which is a pseudospectral GCM that solves the primitive equations on sigma (σ) levels. We use the Held–Suarez physics with the configuration and physical parameters identical to the ones described in detail in Held and Suarez (1994). Briefly, the model is forced by Newtonian relaxation of temperature to a prescribed equinoctial radiative-equilibrium state [Eq. (A1)] with specified relaxation rate [Eq. (A3)], where ϕ and p are latitude and pressure, respectively. Rayleigh drag with a prescribed damping rate [Eq. (A4)] is used to remove momentum from the low levels (σ < 0.7); ∇8 hyperdiffusion is used to remove enstrophy at small scales. A T63 spectral resolution with 40 equally spaced sigma levels and 15-min time steps are used to solve the equations. The above setup is referred to as control (CTL). The CTL simulation consists of 10 ensemble runs, each 45 000 days with 6-hourly outputs (to create 10 independent simulations, a different set of weak and zonally asymmetric perturbations are added to the initial condition of each simulation). The first 500 days of each run are discarded for spinup and given the hemispherical symmetry of the model, the data of the Northern and Southern Hemispheres are used together in the calculations of the time averages and EOF analysis.

The dry dynamical-core GCM has been widely used to study the extratropical circulation and its low-frequency variability (Gerber et al. 2008b; Chen and Plumb 2009; Hassanzadeh et al. 2014; Sheshadri et al. 2018; Ronalds et al. 2018) and is deemed a key member of the hierarchy of elegant climate models (Held 2005; Jeevanjee et al. 2017). The setup of the idealized GCM used here does not have a dynamically active stratosphere and provides a minimal primitive equation model to study the dynamics of the annular mode internal to the troposphere.

The model’s annular mode is obtained from the EOF analysis of daily averaged , which provides and zonal index (PC1); is obtained from regressing on the zonal index. Figure 1 shows and of this model, which closely resemble the annular mode in reanalysis data of the NH and SH; see Figs. 2 in Thompson and Li (2015) and Thompson and Woodworth (2014). Ma et al. (2017) compared the spectra, autocorrelation, and cross correlation of the anomalous zonal-mean zonal wind and eddy momentum flux projected onto EOF1 in this model and the SH reanalysis data, and found that the spatiotemporal properties of this GCM’s annular mode overall agree reasonably well with the observed ones, with the exception of a too-persistent decorrelation time scale of the zonal index (see section 6). As discussed earlier, this is a common problem among GCMs.

Fig. 1.
Fig. 1.

The climatology (contours) and annular mode (i.e., EOF1; shading) of the CTL: (a) zonal wind (m s−1) and (b) temperature (K). The contour interval is 6 m s−1 for zonal wind and 15 K for temperature. Dashed lines indicate negative values.

Citation: Journal of the Atmospheric Sciences 76, 4; 10.1175/JAS-D-18-0268.1

3. The forced simulations and eddy feedbacks

a. EXP1: Barotropic + baroclinic components of the annular mode

To accurately quantify the response of eddy momentum and heat fluxes to the annular mode’s zonal wind and temperature anomaly , we have used the framework that is introduced in Hassanzadeh and Kuang (2016a) and employed in Ma et al. (2017). In this framework, a second simulation (referred to as EXP1) is conducted in which a time-invariant, zonally symmetric forcing of and is added to the GCM such that the time-mean responses of and compared to CTL almost match and of CTL. The forcing is calculated using the GCM’s linear response function as where is a vector containing and (this vector is scaled such that the maximum of is 0.1 m s−1). We use of the exact same setup of this GCM that was previously calculated by Hassanzadeh and Kuang (2016a) using the Green’s function method. EXP1 is essentially the same as test 3 in Hassanzadeh and Kuang (2016a) and the forced experiment in Hassanzadeh and Kuang (2015).

Similar to CTL, EXP1 consists of 10 ensemble runs, each 45 000 days. With angle brackets indicating time and ensemble averaging, Figs. 2a and 2b show and , which agree well with the EOF1 of CTL (Figs. 1a,b). The long datasets of CTL and EXP1 (around 1 million days each) are needed to ensure that the weak time-mean response emerges from the noise. The distinction between the patterns in 1a and 1b and 2a and 2b should be highlighted: the former shows the pattern of variability that explains the most variance in CTL (i.e., EOF1) while the latter shows the difference between the climatologies of EXP1 and CTL (i.e., the change in long-time-averaged flow).

Fig. 2.
Fig. 2.

The forced simulations. (top) full (barotropic + baroclinic) annular mode anomaly (EXP1): (a) (m s−1) and (b) (K). (bottom) Only the barotropic component of the annular mode anomaly (EXP2): (c) (m s−1) and (d) (K).

Citation: Journal of the Atmospheric Sciences 76, 4; 10.1175/JAS-D-18-0268.1

Figures 3a and 3b show the differences between climatology of eddy momentum and heat fluxes in EXP1 and CTL: and , where primes indicate deviation from zonal mean. Because the forcing added in EXP1 is time invariant and zonally symmetric, changes in the climatology of eddy fluxes between EXP1 and CTL are solely due to changes in the climatology of and between EXP1 and CTL; see Hassanzadeh and Kuang (2016a) and Ma et al. (2017) for further discussion. Therefore, Figs. 3a and 3b show the anomalous eddy fluxes in response to the annular mode anomaly (Figs. 2a,b) at the steady-state limit, finding which was the main difficulty in quantifying the role of eddies in annular mode dynamics as discussed in section 1. In fact, following the linear feedback framework of Lorenz and Hartmann (2001), Ma et al. (2017) calculated the eddy momentum forcing from Fig. 3a and used from Fig. 2a to compute the feedback strength:
e1
where indicates area-weighted domain (latitude–pressure) averaging. Note that all calculations in this paper are conducted in the spherical coordinates, but is used for convenience. The statistical methods of Lorenz and Hartmann (2001) and Simpson et al. (2013) yield a similar but slightly lower estimate for b (0.11–0.13 day−1 depending on the chosen free parameter) and the low-pass filtering method robustly gives the same estimate of +0.14 day−1 for low-pass time scales larger than 200 days (see Fig. 8 in Ma et al. 2017).
Fig. 3.
Fig. 3.

Eddy momentum and heat fluxes: climatology of CTL (contours) and response to the annular mode pattern (shading). (top) Response to the full (barotropic + baroclinic) annular mode anomaly (EXP1): (a) (m2 s−2) and (b) (m s−1 K). (bottom) Response to only the barotropic component of the annular mode anomaly (EXP2): (c) (m2 s−1) and (d) (m s−1 K). The contour interval is 15 m2 s−2 for eddy momentum flux and 5 m s−1 K for eddy heat flux. Dashed lines indicate negative values.

Citation: Journal of the Atmospheric Sciences 76, 4; 10.1175/JAS-D-18-0268.1

We highlight that the eddy fluxes in this paper are calculated from 6-hourly outputs to capture the medium-scale waves, which have time scales shorter than two days (Sato et al. 2000). Despite their weak climatological amplitudes, the medium-scale waves are strongly modified by the annular mode anomaly and their fluxes have a substantial contribution to the annular mode dynamics (Kuroda and Mukougawa 2011); for example, the eddy momentum forcing on the annular mode in the low-frequency limit can be underestimated by around 50% if the eddy fluxes are calculated from daily averaged wind (Ma et al. 2017, their Fig. 10). Note that daily instantaneous data yield the same results as the 6-hourly data, see the discussion in section 6 of Ma et al. (2017).

In addition to the meridional eddy fluxes, we have also calculated the responses of vertical fluxes: and , where ω is the pressure velocity. These responses will be used to quantify the contribution of vertical eddy momentum and heat fluxes to the annular mode dynamics.

In summary, the above analysis of CTL and EXP1 confirms the existence of a positive eddy–jet feedback and quantifies its magnitude in this idealized atmosphere [as originally reported in Ma et al. (2017)]. This analysis also allows us to compute the contributions of the meridional and vertical eddy momentum and heat fluxes to the annular mode dynamics in sections 5 in order to address objectives 1 and 2 that were mentioned in section 1. To better address objective 3, we conduct a second experiment EXP2 described below. We emphasize that in EXP2, the aim is to quantify the net effect of change in the meridional structure of zonal wind, which not only affects upper-level wave propagation, but also low-level wave generation; this distinction is further discussed at the end of this section.

b. EXP2: Only barotropic component of the annular mode

To further investigate the effect of the annular mode anomaly on the eddy heat and momentum fluxes, a second forced simulation (EXP2) is conducted in which the added forcing is meant to produce a time-mean response in zonal wind that matches the barotropic component of . To find this forcing, we have first calculated the mass-weighted vertical average of at each latitude, which along with zero temperature anomaly, constitutes . We then calculated . As before, EXP2 consists of 10 ensemble runs, each 45 000 days. Figures 2c and 2d show and . The barotropic component of is reproduced reasonably well (cf. Fig. 2a), although there are some small variations with pressure, due to the inaccuracies in [see Hassanzadeh and Kuang (2016a) for a discussion of the sources of these inaccuracies]. As a result, there is also a small temperature anomaly (Fig. 2d). Overall, EXP2 captures the barotropic component of EOF1 adequately for the purpose of our analysis.

Figures 3c and 3d show the differences between the climatology of eddy momentum and heat fluxes in EXP2 and CTL. As discussed earlier, these differences are the response of eddy fluxes to the (mostly) barotropic component of the annular mode shown in Figs. 2c and 2d. There is a striking difference between the response of the eddy fluxes to the (mostly) barotropic-only component of the annular mode anomaly and to the full (barotropic + baroclinic) annular mode anomaly as can be seen by comparing the two rows of Fig. 3.

As pointed out in Ma et al. (2017), the change in eddy fluxes in response to the full annular mode anomaly (EXP1) is consistent with the baroclinic mechanism discussed in Robinson (2000) and Lorenz and Hartmann (2001): increased (reduced) baroclinicity around the latitude of maximum (minimum) anomaly, 50° (30°) in Fig. 2a, which is due to increased (reduced) meridional temperature gradient around the same latitude (see Fig. 2b) leads to increased (reduced) eddy generation and at low levels with a maximum (minimum) around 50° (30°); see Fig. 3b. The upward (downward) wave propagation around 50° (30°) combined with the prevalence of equatorward wave propagation at upper levels due to spherical geometry (Balasubramanian and Garner 1997) lead to a maximum of between 40° and 50° at upper levels; see Fig. 3a. The resulting eddy forcing has its maximum (minimum) at upper levels around the latitude of maximum (minimum) anomaly 50° (30°). This pattern of eddy forcing leads to reinforcement of the annular mode anomaly with the feedback magnitude calculated in Eq. (1).

The change in eddy fluxes in response to only the barotropic component of the annular mode anomaly (EXP2) is quite different, and in particular, the response of is opposite: around the latitude of maximum (minimum) anomaly, which is the same as the corresponding latitude for the anomaly, there is reduced (increased) eddy generation and at low levels with a minimum (maximum) around 50° (30°); see Fig. 3d. This is accompanied by reduced (increased) below 400 hPa around 45° (above 500 hPa around 30°); see Fig. 3c. The resulting eddy forcing has its minimum (maximum) throughout the troposphere around the latitude of maximum (minimum) anomaly. This pattern of eddy forcing leads to weakening of the annular mode anomaly with the negative feedback magnitude calculated following Eq. (1) to be ~−0.1 day−1.

The substantial change in (Fig. 3d) is in spite of almost no change in baroclinicity (Fig. 2d); in fact, the little change in still resembles the one in Fig. 2b and is expected to lead to a response that has the same pattern, but much weaker amplitude, compared to the one shown in Fig. 3b. However, that is not the case. Suppression or intensification of eddy generation with changes in the meridional structure of zonal wind has been known to happen due to mechanisms such as the barotropic governor effect (i.e., stronger meridional shear can restrict the structure of the baroclinic waves and reduce the conversion of potential energy to eddy kinetic energy thus suppressing the eddy generation; James and Gray 1986; James 1987; Nakamura 1993) and changes in the effective beta (i.e., increase in the meridional gradient of relative vorticity increases the effective beta and suppresses eddy generation; Nakamura 1993). Understanding the mechanism(s) of the eddy flux change in Figs. 3c and 3d requires a comprehensive examination of eddy life cycle and its response to changes in the meridional shear and relative vorticity associated with the barotropic component of the annular mode and is left for future work.

As mentioned earlier, some studies have suggested that changes in the upper-level meridional shear lead to an eddy forcing that reinforces the annular mode anomaly. The analysis presented above is not refuting that mechanism, but showing that the net effect of the change in meridional shear on upper-level wave propagation and low-level wave generation is a negative feedback, likely because the suppression of eddy generation at low levels prevails over any potential increase in upper-level wave propagation. Separating these effects is beyond the scope of this paper, but it should be examined in future studies. However, to better address objective 3, we examine the budget for both full and barotropic-only annular mode anomalies in section 5.

4. Quantifying physical processes

Knowing the meridional and vertical eddy momentum and heat fluxes (section 3) and the GCM’s Rayleigh drag and the Newtonian cooling parameterizations (appendix A), the contribution of each physical process to the annular mode dynamics (i.e., the tendency balance of and ) can be quantified. However, in addition to the direct contributions, these processes can each also drive a secondary (meridional) circulation and indirectly affect the tendency balance of and . For example, the surface friction forcing affects the low-level baroclinicity through adiabatic warming resulting from the induced secondary circulation (see below). The combination of direct and indirect contributions from a forcing has been referred to as the “Eliassen response” (Ring and Plumb 2008, section 5) and can be calculated by solving the balanced “Eliassen problem” (Eliassen 1951).

To further illustrate this concept and the role of each physical process, we look at the zonally averaged primitive equations for annular mode anomalies of zonal wind and potential temperature . Defining the CTL time-mean (background) flow , radius of Earth a, and Coriolis parameter f, these equations become (Simpson et al. 2013; Hassanzadeh and Kuang 2016a)
e2
e3
On the right-hand side of Eq. (2) [Eq. (3)], the two terms on the first line represent the advection of the annular mode zonal wind (temperature) anomaly by the meridional circulation of the background flow, the terms on the fourth line represent the anomalous meridional and vertical eddy momentum (heat) flux divergence, and the term on the last line represents the Rayleigh drag on the zonal wind anomaly (Newtonian cooling of the temperature anomaly). The terms on the second and third lines of both equations represent the total effect of the induced secondary circulations, which contribute to the tendency budget by meridionally advecting the total zonal wind and potential temperature. Because we aim to quantify the contribution of each physical process to the annular mode dynamics, we need to compute the secondary circulation and the resulting total tendency in Eqs. (2) and (3) from each individual process.

This can be achieved by solving the balanced Eliassen problem: the terms on the second and third lines of Eqs. (2) and (3) are linearized around [which is justified given that and ] and the two equations are combined assuming gradient wind balance (which eliminates the left-hand sides) and setting , resulting in a linear, diagnostic equation for (see appendix B). The induced secondary circulation from each forcing in Eqs. (2) and (3), for example, , can now be computed by only including that term in the linear, diagnostic equation and solving for . Thus, we can quantify the direct and indirect (i.e., via meridional circulation) contribution of each process, including eddy momentum and heat fluxes, to the annular mode dynamics.

The formulation and development of a numerical solver for the Eliassen problem have been described in detail in appendix A of Ring and Plumb (2008). However, with little effort, the idealized GCM itself can be turned into an axisymmetric solver for this purpose, as described in appendix C. Using this solver, below we compute the induced circulation and the total contribution of each process to the tendency of the full annular mode anomaly and the barotropic-only component of the annular mode anomaly.

a. Surface friction

Figure 4 shows how surface friction affects the annular mode zonal wind and temperature anomalies. Shading in Fig. 4a depicts the normalized where is used. Hereafter, the normalization of any tendency A is conducted as
e4
e5
Given this normalization, any red (blue) shading means strengthening (weakening) of the annular mode anomaly.
Fig. 4.
Fig. 4.

Effect of surface friction. (a) Shading shows [see Eq. (4) for the definition of normalization {}u]. Contour lines show the streamlines of the induced secondary circulation, with solid (dashed) lines indicating clockwise (counterclockwise) circulation. (b) Shading shows the total tendency of resulting from surface friction normalized according to Eq. (4). Contour lines show , with solid (dashed) lines indicating westerly (easterly) wind. (c) Shading shows the total tendency of resulting from surface friction normalized according to Eq. (5). Contour lines show , with solid (dashed) lines indicating positive (negative) temperature. Shading in all panels has units of day−1.

Citation: Journal of the Atmospheric Sciences 76, 4; 10.1175/JAS-D-18-0268.1

As expected, the direct effect of surface friction is to weaken at low levels. Using the solver for Eliassen problem, the secondary circulation induced by this forcing is calculated and found to consist of a pair of counterrotating circulations with their descending branches around 40° (Fig. 4a). North of 40°, the poleward (equatorward) velocity at the lower (upper) levels leads to anomalous westerly (easterly) tendency mostly through the term in Eq. (2), which strengthens (weakens) at lower (upper) levels. The effect south of 40° can be similarly understood. Combined with the direct effect of friction, the total contribution of surface friction is weakening of the anomaly throughout the troposphere (Fig. 4b). Surface friction also affects via the secondary circulation: adiabatic warming from the descending flow around 40° strengthens and increases the low-level baroclinicity (Fig. 4c). The effect of surface friction on and is qualitatively similar (Fig. 5).

Fig. 5.
Fig. 5.

As in Fig. 4, but for the effect of surface friction on the barotropic component of the annular mode , where .

Citation: Journal of the Atmospheric Sciences 76, 4; 10.1175/JAS-D-18-0268.1

b. Thermal radiation

Figure 6 shows how thermal radiation affects the annular mode zonal wind and temperature anomalies. Shading in Fig. 6a depicts the normalized where is used. As expected, the direct effect of thermal radiation is to weaken . The secondary circulation induced by this forcing is calculated using the solver for Eliassen problem. The induced circulation is weaker compared to the one in response to surface friction, and the resulting tendency on has a baroclinic dipolar pattern: weakening of the anomaly at the upper levels and strengthening at the lower levels (Fig. 6b). The total effect of thermal radiation is weakening of the anomaly. Overall, the magnitude of the contribution of thermal radiation to the tendency of zonal wind (temperature) is a factor of 10 (2) smaller than the contribution of surface friction. The effect of thermal radiation on and has a similar pattern but is significantly weaker as expected (Fig. 7).

Fig. 6.
Fig. 6.

Effect of thermal radiation. (a) Shading shows [see Eq. (5) for the definition of normalization {}T]. The induced secondary circulation is weak and because the streamlines have the same intervals as in other figures, they do not appear here. (b) Shading shows the total tendency of resulting from thermal radiation normalized according to Eq. (4). (c) Shading shows the total tendency of resulting from thermal radiation normalized according to Eq. (5). Shading in all panels has units of day−1. See the caption of Fig. 4 for more details.

Citation: Journal of the Atmospheric Sciences 76, 4; 10.1175/JAS-D-18-0268.1

Fig. 7.
Fig. 7.

As in Fig. 6, but for the effect of thermal radiation on the barotropic component of the annular mode , where .

Citation: Journal of the Atmospheric Sciences 76, 4; 10.1175/JAS-D-18-0268.1

c. Eddy momentum flux

Figure 8 shows how meridional eddy momentum flux divergence affects the annular mode zonal wind and temperature anomalies. Shading in Fig. 8a depicts the normalized projection onto the annular mode where is used. The direct effect of the meridional eddy momentum flux is to strengthen at the upper levels. The induced secondary circulation (shown in Fig. 8a) transfers some of the anomalous westerly (easterly) momentum to the low levels north (south) of 40°, leading to strengthening of throughout the atmosphere (Fig. 8b). The induced secondary circulation also strengthens particularly around 40° above 700 hPa. The direct and total effects of vertical eddy momentum flux divergence have been also calculated but found to be very weak (not shown).

Fig. 8.
Fig. 8.

Effect of meridional eddy momentum flux divergence. (a) Shading shows (calculations are conducted in spherical coordinates and is used for convenience). (b) Shading shows the total tendency of normalized according to Eq. (4). (c) Shading shows the total tendency of normalized according to Eq. (5). Shading in all panels has units of day−1. See the caption of Fig. 4 for more details.

Citation: Journal of the Atmospheric Sciences 76, 4; 10.1175/JAS-D-18-0268.1

The effect of meridional eddy momentum flux divergence in response to only the barotropic component of the annular mode is quite different (as discussed in section 3b) and leads to weakening of both and ; see Fig. 9.

Fig. 9.
Fig. 9.

As in Fig. 8, but for the effect of meridional eddy momentum flux divergence in response to the barotropic component of the annular mode , where .

Citation: Journal of the Atmospheric Sciences 76, 4; 10.1175/JAS-D-18-0268.1

d. Eddy heat flux

Figure 10 shows how meridional eddy heat flux divergence affects the annular mode zonal wind and temperature anomalies. Shading in Fig. 10a depicts the normalized where is used. The induced secondary circulation has equatorward (poleward) velocity at upper (lower) levels between 40° and 60°, which, mainly via the term, leads to the weakening (strengthening) of at the upper (lower) levels as shown in Fig. 10b; the opposite occurs between 20° and 40°. The total effect of meridional eddy heat flux on is to weaken the anomaly (Fig. 11c).

Fig. 10.
Fig. 10.

Effect of meridional eddy heat flux divergence. (a) Shading shows (calculations are conducted in spherical coordinates and is used for convenience). (b) Shading shows the total tendency of normalized according to Eq. (4). (c) Shading shows the total tendency of normalized according to Eq. (5). Shading in all panels has units of day−1. See the caption of Fig. 4 for more details.

Citation: Journal of the Atmospheric Sciences 76, 4; 10.1175/JAS-D-18-0268.1

The effect of meridional eddy heat flux divergence in response to only the barotropic component of the annular mode is quite different and leads to strengthening (weakening) of at the upper levels (lower levels) and strengthening of ; see Fig. 11.

Fig. 11.
Fig. 11.

As in Fig. 10, but for the effect of meridional eddy heat flux divergence in response to the barotropic component of the annular mode , where .

Citation: Journal of the Atmospheric Sciences 76, 4; 10.1175/JAS-D-18-0268.1

The direct and total effects of vertical eddy heat flux divergence have been calculated similarly and shown in Fig. 12. Although the magnitude of the total tendencies are weaker than those of the meridional eddy heat flux, there is noticeable influence on the low-level baroclinicity which is further discussed in section 5.

Fig. 12.
Fig. 12.

As in Fig. 10, but for vertical eddy heat flux divergence. The induced secondary circulation is weak and because the streamlines have the same intervals as in other figures, they do not appear here.

Citation: Journal of the Atmospheric Sciences 76, 4; 10.1175/JAS-D-18-0268.1

The direct and total effects of advection of and by meridional circulation and have been also calculated similarly (not shown). As a test for the accuracy of the calculations described above, we compare in Fig. 13 the meridional wind of the annular mode’s secondary circulation anomaly with the sum of all the individually calculated induced meridional winds that are calculated from solving the Eliassen problem. The agreement in pattern and magnitude of the two meridional velocities indicates the accuracy of the numerical procedure used to solve the Eliassen problem.

Fig. 13.
Fig. 13.

The meridional wind of the induced secondary circulation (m s−1): (a) and (b) sum of all induced meridional velocities calculated individually from solving the Eliassen problem.

Citation: Journal of the Atmospheric Sciences 76, 4; 10.1175/JAS-D-18-0268.1

5. Annular mode dynamics

The blue bars in Fig. 14 show the contribution of area-weighted domain-averaged normalized total tendency of each process, calculated in the previous section, to the zonal momentum balance, potential temperature balance, and quasigeostrophic potential vorticity (QGPV) balance of the full annular mode anomaly. For the zonal momentum balance, as expected, surface friction causes a negative tendency, while meridional eddy momentum flux is the largest (and only nonnegligible) source of positive tendency. The only other nonnegligible contribution is a small negative tendency from the meridional heat flux, mainly from weakening of the upper-level wind anomaly (Fig. 10b). The contribution from other terms is negligible, not only because of cancellations between positive and negative tendencies in the domain, but also (and in fact, mainly) because the magnitudes are small.

Fig. 14.
Fig. 14.

Contribution of different processes to the tendency budget of (top) , (middle) , and (bottom) quasigeostrophic potential vorticity QGPV. The vertical axes have units of day−1 and show (top) , (middle) , and (bottom) for tendency A. Blue (red) bars indicate the contribution to the full (barotropic-only component of the) annular mode anomaly.

Citation: Journal of the Atmospheric Sciences 76, 4; 10.1175/JAS-D-18-0268.1

In the potential temperature balance, as expected, thermal radiation and meridional eddy heat flux cause negative tendencies while meridional eddy momentum flux, surface friction, and vertical eddy heat flux cause positive tendencies. Although the positive tendency from surface friction is small, its importance should be further emphasized. The large positive tendency from meridional eddy momentum flux is mostly dominated by the positive tendency in the stratosphere and to some extent, between 300 and 800 hPa around 40° (Fig. 8c). The positive tendency from surface friction, on the other hand, is concentrated below 600 hPa around 40° (Fig. 4c) and reinforces the anomalous low-level baroclinicity, which increases eddy generation and leads to the positive eddy forcing from meridional eddy momentum flux at the upper levels (Robinson 2000; Lorenz and Hartmann 2001). The tendency from the vertical eddy heat flux is also mostly concentrated below 700 hPa between 20° and 60° although there is a complex pattern of positive and negative tendencies (Fig. 12c).

The QGPV tendencies2 describe the balance of the annular mode anomaly as a system, showing that overall, the anomaly persists because the meridional eddy momentum flux balances the negative tendencies from surface friction and meridional eddy heat flux. The sum of all tendencies yields a decay rate of −0.016 day−1, or a ~62-day time scale. Note that because we are looking at the steady-state balance of EXP1, the forcing , which is calculated to have a total tendency of +0.018 day−1, is indeed balancing the net tendency from physical processes. The residual of these two tendencies, ~0.002 day−1, is a measure of uncertainty in the calculated balance.

Without the contributions from the eddy fluxes, the sum of all other tendencies yields a decay rate of −0.035 day−1, or a ~28-day time scale, indicating that the positive feedbacks from eddy fluxes that are in response to the full annular mode anomaly are increasing the annular mode persistence by a factor of ~2.3.

The red bars in Fig. 14 show the contribution of area-weighted domain-averaged normalized total tendency of each process for the barotropic-only component of the annular mode. In the zonal momentum and potential temperature balances, the effect of all processes except for meridional eddy fluxes are qualitatively the same as for the full annular mode anomaly. The signs of the tendencies from meridional eddy fluxes are opposite, consistent with the discussions and results in section 3. In the QGPV balance, the tendencies from surface friction and meridional eddy momentum flux are negative, consistent with the results in the first two panels. The contribution from the meridional eddy heat flux is also negative in spite of the positive tendencies in the first two panels. Further examination shows that this tendency in QGPV is dominated by the contribution to the zonal momentum balance, which once converted into QGPV and projected onto the annular mode anomaly, leads to an overall negative tendency.

The sum of all tendencies for the barotropic-only component yields a decay rate of −0.098 day−1, or a ~10-day time scale (taking the tendency of into account, the residual tendency is −0.010 day−1). Comparing this time scale with that of the balance without eddy fluxes, where the time scale is ~28 days, these results suggest that there is a negative eddy–jet feedback between the barotropic component of the annular mode anomalies and the eddies, which decreases the persistence of the annular mode by a factor of ~2.8.

6. Discussion and summary

Using the LRF framework of Hassanzadeh and Kuang (2016a) and the same idealized setup as Ma et al. (2017), in this paper, we make the following points:

  1. The contribution of different physical processes (eddy momentum flux, eddy heat flux, surface friction, thermal radiation, and meridional advection) to the annular mode dynamics is quantified (sections 4 and 5). This was made possible by first quantifying how eddy fluxes change in response to the annular mode anomaly (section 3).
  2. It is shown in section 5 that the eddies modified by the full (barotropic + baroclinic) annular mode anomaly exert a positive feedback and increase the annular mode persistence by a factor of ~2.3. It is further shown that the eddies modified by only the barotropic component of the annular mode anomaly exert a negative feedback and decrease the annular mode persistence by a factor of ~2.8.
  3. It is shown in section 3 that how eddy fluxes change in response to the full annular mode anomaly is drastically different from their response to the barotropic-only component of the anomaly and that a positive eddy–jet feedback exists only for the former. Results suggest that the baroclinic component of the annular mode anomaly (i.e., increased low-level baroclinicity) is essential for the persistence of the annular mode, consistent with the work of Robinson (2000) and Lorenz and Hartmann (2001).

The dynamics of the annular mode at the low-frequency (approximately steady state) limit that emerges from quantifying the physical processes is shown schematically in Fig. 15. Focusing on the poleward half of the annular mode anomaly (40°–60°) for convenience, an anomalous increase in the eddies leads to a stronger barotropic jet, which in turn suppresses the generation of the eddies. The stronger barotropic jet increases the low-level baroclinicity, which in turn, intensifies eddy generation. However, the increase in eddies reduces the baroclinicity through enhanced poleward heat flux.

Fig. 15.
Fig. 15.

Schematic of the annular mode dynamics in the idealized atmosphere, focused on the poleward half of the annular mode anomaly (40°–60°) for convenience. Red (blue) arrows indicate positive (negative) influence.

Citation: Journal of the Atmospheric Sciences 76, 4; 10.1175/JAS-D-18-0268.1

Further analyses in future studies are needed to understand the implications and relevance of these results beyond the idealized atmosphere studied here. In this study, we aimed to examine the annular mode dynamics internal to the troposphere in a minimal primitive equation model of the annular mode. Thus we focused on an idealized GCM setup that is at equinox and does not have a dynamically active stratosphere or topography. The extratropical circulation in this setup is dominated by EOF1, which explains 51% of the variance compared to the 17% variance explained by EOF2. Recently, Sheshadri and Plumb (2017) have shown that if the setup is changed to mimic winter and summer seasons, then in the winter hemisphere, EOF1 and EOF2 have comparable explained variances (40% and 29%), resulting in a circulation that is dominated by a propagating annular mode in which EOF1 and EOF2 are correlated at low frequency and feedback into each other. Repeating the analysis presented here for such setup can provide insight into the dynamics of propagating annular modes.

Furthermore, including topography and other zonal asymmetries to examine the feedback from planetary waves (Rivière et al. 2016) and adding an active stratosphere to investigate the effect of stratospheric variability on the annular mode persistence (Byrne et al. 2017; Byrne and Shepherd 2018) can help with developing a deeper understanding of the annular mode dynamics in the NH and SH.

Finally, the barotropic (baroclinic) annular mode is sometimes considered as variability of eddy momentum (heat) flux without much connection with the eddy heat (momentum) flux, suggesting that the two annular modes are decoupled (Thompson and Woodworth 2014). The results presented here show the essential role of eddy heat flux and baroclinicity in the dynamics of the barotropic annular mode in the low-frequency limit, suggesting that the two annular modes are potentially coupled at such time scales, consistent with the recent findings of Boljka et al. (2018). A deeper understanding of this coupling is needed for a complete model of the extratropical low-frequency variability.

The main complication in repeating the analysis conducted here for different GCM setups is that the LRF has to be recomputed for each setup using the Green’s function method. This method requires many forced simulations of the GCM, which can be computationally cumbersome, but otherwise the Green’s function method has been successfully applied to compute the LRFs for a variety of models (Kuang 2010, 2012; Hassanzadeh and Kuang 2016a; Liu et al. 2018; Khodkar et al. 2019). The FDT, which can provide the LRF from the data of one long, unforced simulation, is an attractive alternative to the Green’s function method; however, the application of FDT to GCMs has produced mixed results (Ring and Plumb 2008; Fuchs et al. 2015; Lutsko et al. 2015; Hassanzadeh and Kuang 2016b). Khodkar and Hassanzadeh (2018) have recently introduced a new FDT framework that has shown a promising performance for a model of turbulent convection; if this framework works comparably well for GCMs, then repeating the analysis of this paper for other model setups will be significantly facilitated.

Acknowledgments

We thank David Lorenz, Ding Ma, and Aditi Sheshadri for fruitful discussions and three anonymous reviewers whose input substantially improved the manuscript. This work was supported by NSF Grant AGS-1552385 and NASA Grant 80NSSC17K0266. The simulations were run on the Odyssey cluster supported by the FAS Division of Science, Research Computing Group at Harvard University. The Rice University Center for Research Computing and XSEDE Stampede2 (via allocation ATM170020) provided additional computational resources.

APPENDIX A

Parameterizations in the Idealized GCM

The parameterizations of thermal radiation as Newtonian relaxation and planetary boundary layer as Rayleigh drag are identical to those described in Held and Suarez (1994): The equilibrium temperature profile is
ea1
where
ea2
and , , and . The Newtonian relaxation rate is
ea3
where , , and . The Rayleigh drag damping rate is
ea4
where .

APPENDIX B

Diagnostic Equation for Meridional Circulation

Here we provide further details on finding a linear diagnostic equation for , which follows the detailed derivations presented in appendix A of Ring and Plumb (2008) and appendix A of Hassanzadeh and Kuang (2016a). We start from the linearized gradient wind balance equation (Andrews et al. 1987, p. 126):
eb1
where and R is the gas constant. After linearizing Eqs. (2) and (3) around , setting , defining
eb2
eb3
and subtracting from , the left-hand sides of the two equations cancel because of Eq. (B1), and one arrives at a single equation for the streamfunction of the secondary circulation χ [the equation is the same as Eq. (A6) of Ring and Plumb (2008) once their V and Ω are set to 0]. For a given , and a given eddy momentum flux, or eddy heat flux, or surface friction, or thermal radiation term, this linear equation can be solved to find χ, from which can be computed. See appendix A of Ring and Plumb (2008) for a detailed description of such numerical solver. However, rather than computing the equations and using a numerical linear solver, we turned the GCM itself into an Eliassen solver to find χ and the meridional circulation, as discussed below in appendix C.

APPENDIX C

GCM Solver for Eliassen Problem

Several changes have to be made to the GCM to turn it into an axisymmetric solver for the Eliassen problem described in section 4 and in Ring and Plumb (2008, their appendix A):

  1. The model is initialized with a completely zonally symmetric profile .
  2. A small zonally asymmetric perturbation that is added in the GCM to hasten the baroclinic instability is removed.
  3. The Rayleigh drag and Newtonian cooling of the Held–Suarez physics are disabled.
  4. The optional corrections to conserve mass and energy are disabled.
  5. The coefficient of the Robert–Asselin time filter is set to 0.
  6. For time integration, the fully backward scheme is used (by choosing ) to help with damping/slowing down the gravity waves.
For items 4–6, see the description of the model and algorithms of the GFDL spectral dynamical core.B1

With these changes, for and that are in exact gradient wind balance, the model can be integrated for thousands of days without any change in zonal wind and temperature. However, the and obtained from the climatology of the CTL are not in exact gradient wind balance which leads to generation of waves and secondary circulations. To resolve this issue, we calculated the tendency of the meridional wind after one time step, and then from the meridional momentum equation, calculated the change in the zonal wind needed to balance this tendency:
ec1
where is computed from solving a quadratic equation. Now initializing the model with leads to a balanced system that can be integrated for thousands of days without developing any secondary circulation. Note that the issue described above is not a pitfall of our approach to solving the balanced Eliassen problem. An exact gradient wind balance is the key assumption of the Eliassen problem, and in other approaches, for example the one described in appendix A of Ring and Plumb (2008), exact gradient wind balance is assumed and linearized gradient wind balance is enforced during the formulation.

To compute the induced secondary circulation and total tendency from a forcing, for example, , this forcing is added to both hemispheres of this model and the mean , , and are computed between days 50 and 100 (the results are not sensitive to the exact time period used). These are, respectively, the induced meridional velocity of the secondary circulation and the total tendency of zonal wind and temperature resulting from this forcing.

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1

“In the absence of a complete theory that allows us to calculate eddy quantities in terms of the mean flow, the necessary next step is a careful analysis of the observed relationships between eddy fluxes and zonally averaged fields.”

2

For each process, the QGPV tendency is computed as where and are the tendencies in zonal wind and potential temperature resulting from that process.

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