Variability in the width of the tropics and the annular modes

Authors


Corresponding author: J. Kidston, CCRC, University of New South Wales, Sydney, Australia. (j.kidston@unsw.edu.au)

Abstract

[1] The correlation between unforced variability in the latitude of the edge of the Hadley cell (ΦHC) and latitude of the surface westerlies (ΦEDJ) is examined using a simplified moist general circulation model (GCM) and a suite of state of the art GCMs. The correlation can be determined by the time-mean separation of the two features. When the separation is small, there is a positive correlation, and as the separation between them increases, the correlation reduces. In the simplified model, a weak negative correlation emerges at large separations.

[2] The location of the anomalous meridional mass flux associated with variations in the latitude of ΦEDJ, relative to the climatological Hadley cell position, determines the extent to which ΦHC is influenced by changes in ΦEDJ. Changes in the latitude of ΦEDJ are driven by anomalous eddy momentum flux convergence, and these are approximately balanced by the Coriolis torque on the meridional flow, as expected under quasi-geostrophic scaling. Under changing time-mean climates, the anomalous flow associated with ΦEDJ variability translates location so that it is approximately fixed relative to the time-mean ΦEDJ. This means that the influence of ΦEDJ variability on ΦHC varies as a function of the time-mean separation of the features.

[3] Initial indications are that the same causal relationship holds in a suite of state of the art GCMs and that this explains the seasonal variation in the correlation between ΦHC and ΦEDJ.

1 Introduction

[4] The Hadley cell and the mid-latitude storm track are characteristic features of Earth's atmosphere. The thermally direct meridional overturning of the Hadley cell gives rise to a tropical climate zone with wet deep tropics and dry subtropics. In the mid-latitudes, the baroclinic eddies that give rise to the eddy-driven jet, Ferrel cell, and surface westerlies also play a dominant role in determining local climate characteristics.

[5] It has recently emerged that variability in the latitude of the edge of the Hadley cell (ΦHC) and the latitude of the storm tracks and surface westerlies (ΦEDJ) is correlated [Kang and Polvani, 2011]. This is particularly true in the Southern Hemisphere (SH) during summer. The finding holds in both the reanalysis and a suite of state of the art GCMs. During periods when the two features are correlated, a ratio was found in the variability of ΦEDJ to ΦHC of 2:1. This same ratio was found in the forced response to increasing greenhouse gasses (GHGs), again in the SH during summer [Lu et al., 2008].

[6] Further work on this phenomenon has often been addressed through the viewpoint of the El Niño southern oscillation (ENSO) teleconnections. ENSO is also known to affect both ΦHC and ΦEDJ in the SH during austral summer [L'Heureux and Thompson, 2006; Seager et al., 2005; Fogt and Bromwich, 2006; Lu et al., 2008].

[7] These relationships have led to research on the dynamical links between ΦHC and ΦEDJ, with several researchers positing that the speed of the thermally driven subtropical jet, located at the edge of the Hadley cell, affects ΦEDJ through the influence of the background wind speed on the dissipation of equatorward-propagating eddies [Robinson, 2002; L'Heureux and Thompson, 2006; Seager et al., 2005; Lu et al., 2008].

[8] It is known that excursions of ΦEDJ are sustained by anomalous momentum flux convergence (math formula) [e.g., Limpasuvan and Hartmann, 2000; Lorenz and Hartmann, 2001; Kidston et al., 2010]. To first order, the anomalous math formula forms a dipolar structure, with a node at approximately the latitude of the time-mean eddy-driven jet (math formula). Here we find that the anomalous math formula is approximately balanced by Coriolis torque on the meridional flow. This is expected under quasi-geostrophic scaling, and this may cause ΦHC to change.

[9] The correlation between ΦHC and ΦEDJ depends on the time-mean separation of the two features. Under changing time-mean climates, the anomalous flow associated with ΦEDJ variability translates location so that it is approximately fixed relative to the time-mean ΦEDJ. When the features are close in the time mean, the reduction in math formula associated with a poleward shift in ΦEDJ occurs in the subtropics. This is balanced by poleward meridional flow, which causes ΦHC to extend poleward. When they are well separated in the time mean, the induced anomalous meridional flow lies far poleward of the climatological ΦHC, and ΦHC is unchanged.

2 Methods

[10] We use the simplified gray-radiation GCM described in Frierson et al. [2006]. The resolution is 2.8 × 2.8, with 25 vertical levels, and the incoming solar flux is set by

display math(1)

where ϕ is latitude, and the flux is only absorbed at the surface. The parameter Δ sets the latitudinal variation of insolation, with stronger meridional temperature gradients resulting from higher Δ. The model is run for Δ equal to 0.5, 0.8, 1.0, 1.2, 1.4, 1.7, and 2.0. The simulations are run for 4500 days after discarding the first 360 days spin-up.

[11] In addition to varying Δ, runs were also performed where the polar stratosphere was cooled. This is well known to shift ΦEDJ poleward [e.g. Polvani and Kushner, 2002; Kushner and Polvani, 2004; Haigh et al., 2005; Lorenz and DeWeaver, 2007]. A diabatic heating term was added to the temperature tendency of the form

display math(2)

where p is pressure, the 0 subscripts indicate reference pressure/latitude, and the h subscripts indicate the pressure/latitude scale over which the heating anomaly decays. The value of φ0 was set to ± 90°, and the value of p0 was set to 0 hPa, with φh = 20 and ph = 50 hPa. The value of C0 was varied in order to give a reasonable shift of ΦEDJ (see Table 1).

Table 1. A summary of the experiments with the simplified GCM. The first column gives the value of Δ, used in equation ((1)). The second column gives the value of C0, used in equation ((2)) for the stratospheric cooling experiments, and this was set to zero for the control experiments, whose values are bolded. The variables shown for each experiment are the global-mean surface temperature (TG), the difference in surface temperature between the equator and the pole (dyT), and the time-averaged values of the latitude of the edge of the Hadley cell (math formula) and the latitude of the surface westerlies (math formula)
ExperimentTG (K)dyT (K)math formulamath formula
Δ = 0.5C0 = 0285.324.233.422.6
C0 = 0.2324.035.823.7
Δ = 0.8C0 = 0284.731.538.024.9
Δ = 1.0C0 = 0284.137.140.326.0
 C0 = 0.1536.742.426.4
C0 = 0.1836.544.126.6
C0 = 0.18136.450.926.2
C0 = 0.18536.554.726.2
Δ = 1.2C0 = 0283.543.242.326.9
Δ = 1.4C0 = 0282.749.344.727.6
 C0 = 0.0548.945.927.7
C0 = 0.0749.046.427.7
C0 = 0.07349.248.727.6
C0 = 0.07549.852.027.5
C0 = 0.08049.650.527.4
C0 = 0.1049.654.027.4
C0 = 0.12549.957.027.3
Δ = 1.7C0 = 0281.661.150.728.5
Δ = 2.0C0 = 0280.276.057.029.2

[12] The streamfunction is computed from the zonally averaged meridional wind data. It is given by

display math

and is integrated down from the top to pressure level p′. The value of ΦHC is then indexed as the first zero crossing of the streamfunction going poleward from the equator at a pressure level roughly 500 hPa. Effectively, it is the latitude where the Eulerian meridional mass transport goes to zero. The data points around the zero crossing are fitted with a second-order polynomial, the root of which gives ΦHC. The value of ΦEDJ is taken as the latitude of the maximum surface zonal-mean zonal wind speed (ū), and again, this was found by evaluating a quadratic fitted around the maxima.

3 Results

3.1 Simplified GCM

3.1.1 Empirical Relationships

[13] As Δ increases, the climate moves from summer-like to winter-like conditions. The area-weighted global-mean surface temperature (TG), shown in Table 1, drops by 5 K over the range of Δ considered, while the pole-equator surface temperature difference (dyT) increases by over 50  K. In qualitative agreement with Lu et al. [2008] and Walker and Schneider [2006], both math formula and math formula (where math formula indicates the time average) increase as Δ increases (Table 1).

[14] In order to investigate the unforced variability of ΦHC and ΦEDJ, their values were averaged over 30-day periods, and these values will be labeled with a monthly superscript. A scatter plot of the values of ΦHCmo against math formula shows that the correlation between the two features decreases as the conditions move from summer-like to winter-like (Figure 1a). When Δ is small, math formula and math formula are highly correlated, with R = 0.8. As Δ increases, the correlation decreases and actually becomes slightly negative when Δ is large.

Figure 1.

(a) Scatter plot of math formula and math formula for each of the runs where Δ was altered (see text). (b) The correlation coefficient for math formula and math formula plotted as a function of the separation of math formula and math formula. The data from the runs with polar stratospheric cooling are plotted in blue, and those from the uncooled runs are plotted in red.

[15] One of the changes that occurs as Δ is increased is that the separation of math formula and math formula increases, as can be inferred from Table 1 and is shown by the horizontal lines in Figure 2a. We hypothesize that the correlation between ΦHC and ΦEDJ depends on their separation. To test this, the features can be separated by a mechanism other than changing Δ. As seen in Table 1, cooling the polar stratosphere in this model shifts math formula much more than math formula and acts to separate the two features. The only run where there is a large math formula response is the run with Δ = 0.5. When math formula and math formula are relatively well separated in the control run, as when Δ = 1.0, 1.4, the Hadley cell shows minimal response to polar stratospheric cooling. The stratospheric cooling does not extend to the surface, so dyT is essentially unchanged, implying that the thermodynamic changes in the mean state induced by stratospheric cooling are different to when Δ is altered.

Figure 2.

The slope of the regression with math formula as a function of latitude for (a) the upper tropospheric vertical average (| … |, see text) of the zonal-mean zonal wind (math formula), (b) the eddy momentum flux convergence (math formula), and (c) the product of the Coriolis parameter and the zonal-mean meridional wind speed (math formula). In Figure 2a the latitude of math formula is marked with a circle and dashed line, and math formula is marked with a square and dashed-dotted line, with the horizontal lines showing the separation between the two features.

[16] The relationship between the separation of math formula and math formula and the correlation between math formula and math formula is summarized in Figure 1b. The correlation coefficient is shown for all of the runs in Table 1, and there are two points for each run (one for each hemisphere). In the experiments with polar stratospheric cooling (blue) and the runs where only Δ was altered (red), the correlation is well determined by the time-mean separation, and both sets of experiments follow a similar relationship. The correlation between the features becomes small or negative at similar separations. For the runs where only Δ was changed, this occurs at Δ = 1.7 or 2.0. The runs with Δ = 1.4 and polar stratospheric cooling populate the same space as the runs with Δ = 1.7 or 2.0 that have no stratospheric cooling.

3.1.2 Dynamics

[17] The regression of time-varying flow characteristics onto math formula gives an indication of the anomalies associated with a poleward displacement of math formula. The regression with the upper tropospheric zonal-mean zonal wind (math formula, where | … | indicates a density-weighted vertical average from 215 to 315 hPa) shows that for each value of Δ, when the surface westerlies are displaced poleward, an approximately dipolar math formula anomaly exists (Figure 2a). The node of each dipole is roughly coincident with the value of math formula for the corresponding value of Δ (marked with a vertical dotted line of the same color). This behavior is well known and reflects the fact that the annular modes are equivalent barotropic [e.g. Lorenz and Hartmann, 2001]. The corresponding value of math formula is marked with a vertical dashed line.

[18] It is well established that eddies sustain the anomalous flow associated with annular mode excursions [Limpasuvan and Hartmann, 2000; Lorenz and Hartmann, 2001]. The regression between math formula and math formula supports this (Figure 2b). The patterns closely follow those of the math formula anomalies. The eddies drive the observed math formula anomalies, with other forces eventually balancing the momentum equation.

[19] Under quasi-geostrophic scaling the zonal-mean zonal momentum equation is

display math

and at monthly time scales, it is expected that RHS is roughly balanced, so that the ageostrophic meridional flow is determined by the eddy momentum flux convergence. The regression between math formula and math formula shows that for each value of Δ, the Coriolis torque on the meridional flow does indeed roughly balance the math formula anomalies (Figure 2c).

[20] The math formula anomalies seen in Figure 2c follow directly from the facts that (i) the annular modes are equivalent barotropic, so that the anomalous math formula is similar to the math formula anomalies near the surface; (ii) the annular modes are eddy driven so that the patterns of math formula anomalies closely match the math formula anomalies; and (iii) to first order quasi-geostropic scaling holds with large-scale variability, so that the math formula anomalies are balanced by math formula anomalies.

[21] The impact that these math formula anomalies have on math formula is then determined by their location relative to math formula. When the poleward math formula anomalies lie at the same latitude as math formula, they cause the zero crossing to be pushed poleward, resulting in a positive correlation between math formula and math formula. In the runs where the eddy-induced math formula anomalies near math formula are negligible, the latitude where the meridional overturning goes to zero is unaffected.

[22] It is clear by comparing the latitude of math formula (dashed lines, Figure 2c) and the latitude of the maximum math formula anomalies for each run that the potential to effect changes in math formula is progressively reduced as math formula and math formula separate in the time mean. In the runs with low Δ, the positive math formula anomalies coincide with math formula. In the runs with higher Δ, the equivalent math formula anomalies are far removed from math formula. If the location of the math formula and math formula anomalies is approximately constant relative to math formula, then it is expected that as math formula and math formula separate, the affect of the math formula anomalies on math formula reduces. This simple relationship would be expected to give rise to the behavior summarized in Figure 1b.

[23] The idea that the relative locations of math formula and the math formula anomalies (driven by math formula variability) determine that the correlation may be supported by the small negative correlation that appears at large time-mean feature separation. The math formula regression patterns actually a small positive lobe equatorward of the canonical negative values that occur on the flanks of the eddy-driven jet. To our knowledge, this feature has not been expounded in the literature, but empirically, it is present in the runs presented here. When Δ is small, this feature lies in the deep tropics. When Δ is larger, and math formula is further poleward, this feature lies in the subtropics. When this is the case, quasi-geostrophic balance may be expected to hold, and indeed, there is a corresponding small but negative value of math formula for the runs Δ = 1.7 and Δ = 2 (Figure 2c). These line up well with the value of math formula for these runs, and so would be expected to cause an equatorward displacement of math formula, resulting in a negative correlation with math formula. These negative correlations at large feature separation may provide a future test case for competing theories of the governing dynamics.

3.2 Full GCMs

[24] Kang and Polvani [2011] showed a clear seasonal variation in the correlation between ΦEDJ and ΦHC. We use this as a more realistic setting to test the idea that the time-mean separation determines the correlation between the features. We use data from the SH of the preindustrial control experiments in the CMIP3 model ensemble [Meehl et al., 2007] and the ERA interim reanalysis [Dee et al., 2011]. As in Kang and Polvani [2011], the data are interpolated onto a common grid, and the values of math formula and math formula are seasonally averaged over DJF or JJA to produce a single datum per season per 12-month period. The difference between the seasonally averaged value and math formula we call math formula, which is just the seasonal anomaly, similarly for math formula.

[25] A scatter plot showing the value of math formula and math formula is shown in Auxiliary Material Figure S1 and shows that as in Kang and Polvani [2011], there is generally a positive correlation in summer and a weaker correlation in winter.

[26] To test whether this seasonal dependence reflects the same relationship as found in the simple GCM, the correlation is plotted as a function of the difference between math formula and math formula, for each model, in Figure 3. There are two points for each model: one for DJF output and one for JJA output. The points are joined by a solid line, colored green when the seasons are statistically separate, and black otherwise. Statistical independence is tested separately for the values of math formula and R. The Student's t-test is applied to the seasonal values of math formula. Confidence intervals for each of the R values were calculated in the standard way using Fisher transformations. The data are considered distinct when the 95% confidence intervals do not overlap.

Figure 3.

The same as Figure 1b but for output from the preindustrial control of the CMIP3 models, as well as the ERA interim reanalysis data set (de-trended) over the period 1979–2011. There are two points for each model (red for austral summer and blue for austral winter). The correlations were calculated using seasonal averages, giving one point for each calendar year. The individual models are listed in the legend, which also gives the number of years used for each model in brackets. Connectors are colored green where the data for the two seasons are statistically distinct (see text), and black otherwise

[27] In the intra-model seasonal variability, the slope of each line is negative. When the separation of math formula and math formula increases, the correlation between math formula and math formula is reduced. The same relationship holds in the reanalysis, although it does not pass the statistical significance test, due to the relatively short length of the time series. Another interesting but statistically insignificant data pair is the ipsl_cm4 model, which is the only model that has a higher R value during winter than summer. The seasonal change in math formula is also reversed, so that the slope of the line joining the points remains negative.

[28] There is some suggestion that the same relationship holds in the inter-model variability. Both the JJA and DJF points appear to be negatively correlated.

[29] There is approximate quantitative agreement with the idealized model: for a given feature separation, the correlation between math formula and math formula is roughly the same in the idealized runs (Figure 1b) as it is in the full GCMs. Unfortunately the full GCMs do not enter the parameter space that gives rise to negative correlations in the idealized GCM, so that aspect of the relationship remains untested. Nevertheless, the intra-model seasonal variability and the inter-model variability show a similar relationship as was found in the output of the idealized model.

4 Discussion

[30] The correlation between the latitude of the edge of the Hadley cell and latitude of the eddy-driven jet has been examined. It appears that the correlation can be determined by the time-mean separation of the two features. When they are close together, there is a positive correlation, and as the separation between them increases, the correlation decreases, with a weak negative correlation emerging at large separation in a simplified GCM.

[31] The annular modes, or variability in ΦEDJ, are eddy driven. As expected from quasi-geostrophic balance, to first order the changes in eddy momentum flux convergence are balanced by the Coriolis torque on the meridional flow. As such, a poleward displacement of ΦEDJ is associated with anomalous poleward flow in the upper troposphere on the equatorward flank of the climatological ΦEDJ. The edge of the tropics is defined as the latitude where the overturning stream function goes to zero and so the location of the poleward flow anomaly relative to the climatological ΦHC determines whether the tropics extend anomalously poleward.

[32] In some ways, this relationship is very simple and unsurprising. It follows from the notion that the pattern of eddy momentum flux anomalies associated with the annular modes is approximately fixed relative to the time-mean storm track and surface westerlies. This seems to be the case empirically, and intuitively, anomalies associated with variability about the time-mean state might be expected to translate with the time-mean feature as it changes location. It remains to be seen, however, whether simple quasi-linear inviscid Rossby wave propagation theory explains the behavior.

[33] Analysis of a suite of comprehensive GCMs and a reanalysis product showed at least qualitative agreement with the idealized GCM. The correlation between ΦEDJ and ΦHC decreases with increasing time-mean separation in both the intra-model and inter-model variability. This relationship is able to determine the seasonal dependence of the correlation between the two features.

[34] Further work aims to determine whether these dynamics are manifest in (i) the ENSO-annular mode interactions and (ii) the simulated response to increasing greenhouse gasses.

Acknowledgments

[35] This work was partly supported by the Australian Research Council Discovery Early Career Research Award ARC DE120102645 and the Australian Research Council Centre of Excellence in Climate System Science.

Ancillary