Editorial Type:
Article
Article Type:
Research Article

The Relationship between the Speed and the Latitude of an Eddy-Driven Jet in a Stirred Barotropic Model

Joseph Kidston Climate Change Research Centre, The University of New South Wales, Sydney, New South Wales, Australia, and Princeton University, Princeton, New Jersey

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G. K. Vallis Princeton University, Princeton, New Jersey

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Print Publication:
01 Nov 2012
DOI:
https://doi.org/10.1175/JAS-D-11-0300.1
Page(s):
3251–3263
Full access

Abstract

A stirred barotropic model on a sphere is used to investigate the relationship between the speed and the latitude of an eddy-driven jet. When the wind speed is increased in the model, the jet shifts poleward, despite the fact that the stirring of vorticity remains statistically constant. The cause is found to be increasing meridional shear that results from increasing the wind speed in a meridionally confined region and reduces the absolute vorticity gradient on the flanks of the jet. This has two related consequences. The first is that wave propagation is discouraged, as a turning latitude is created where the absolute vorticity gradient tends to zero. On the sphere, this occurs first at high latitudes, thereby shifting wave dissipation toward the equator. The reduced high-latitude dissipation causes a poleward shift of the jet. The second consequence occurs when the vorticity gradient actually becomes negative, in which case the waves may overreflect where an instability is present, providing a high-latitude source of pseudomomentum. This may further encourage the jet to shift poleward.

The relevance of the barotropic dynamics to more realistic atmospheres is unclear, but the intermodel variability of the poleward shift of the jet in response to increasing CO2 across a suite of state-of-the-art GCMs is consistent with the barotropic dynamics, suggesting that further investigation is warranted.

Corresponding author address: J. Kidston, Climate Change Research Centre, Level 4 Matthew’s Building, University of New South Wales, Sydney NSW 2052, Australia. E-mail: j.kidston@unsw.edu.au

Abstract

A stirred barotropic model on a sphere is used to investigate the relationship between the speed and the latitude of an eddy-driven jet. When the wind speed is increased in the model, the jet shifts poleward, despite the fact that the stirring of vorticity remains statistically constant. The cause is found to be increasing meridional shear that results from increasing the wind speed in a meridionally confined region and reduces the absolute vorticity gradient on the flanks of the jet. This has two related consequences. The first is that wave propagation is discouraged, as a turning latitude is created where the absolute vorticity gradient tends to zero. On the sphere, this occurs first at high latitudes, thereby shifting wave dissipation toward the equator. The reduced high-latitude dissipation causes a poleward shift of the jet. The second consequence occurs when the vorticity gradient actually becomes negative, in which case the waves may overreflect where an instability is present, providing a high-latitude source of pseudomomentum. This may further encourage the jet to shift poleward.

The relevance of the barotropic dynamics to more realistic atmospheres is unclear, but the intermodel variability of the poleward shift of the jet in response to increasing CO2 across a suite of state-of-the-art GCMs is consistent with the barotropic dynamics, suggesting that further investigation is warranted.

Corresponding author address: J. Kidston, Climate Change Research Centre, Level 4 Matthew’s Building, University of New South Wales, Sydney NSW 2052, Australia. E-mail: j.kidston@unsw.edu.au
Keywords: Barotropic flows; Dynamics; Nonlinear dynamics; Rossby waves; Waves; atmospheric

1. Introduction

The eddy-driven surface westerlies and associated storm tracks are a prominent feature of the general circulation of Earth’s atmosphere, and they are an important influence on ocean circulation and the global climate. One of the robust circulation changes in simulations of climate with increasing greenhouse gases (GHGs) is a poleward shift of the storm tracks and surface westerlies (e.g., Fyfe et al. 1999; Meehl et al. 2007a). Idealized general circulation models (GCMs) shift their westerlies poleward in response to the large-scale thermodynamic changes that occur under increasing GHGs, and this encourages the view that the shift is a robust response and that idealized models contain the same underlying dynamics. In a dry GCM these large-scale changes include raising the tropopause height and/or heating the tropical upper troposphere (Williams 2006; Lorenz and DeWeaver 2007; Haigh et al. 2005; Butler et al. 2010). In a moist GCM, increasing the surface temperature also acts to heat the tropical upper troposphere via latent heat release, and shifts the jets poleward (Frierson et al. 2006; Lu et al. 2010). Dry (and moist) GCMs also shift their surface westerlies poleward in response to cooling the polar stratosphere, which is analogous to a realistic atmosphere’s response to depleted polar stratospheric ozone (Haigh et al. 2005; Polvani and Kushner 2002). As such, idealized GCMs are seen as providing a reductionist tool for understanding the dynamics that alter the latitude of the surface westerlies in response to large-scale changes in the mean state.

All of the mean-state changes mentioned above increase the speed of the upper-level jet that is associated with the surface westerlies. Heating of the tropical troposphere increases the meridional temperature gradient in the midlatitudes, and the thermal wind increases the speed of the jet above. A reduction in polar stratospheric ozone has the same affect by cooling high latitudes. An increase of the tropopause height increases the zonal wind speed at the height of the tropopause if the vertical shear and surface winds are approximately unchanged. This has led some investigators to ask whether faster jets are inherently located closer to the pole, and to investigate the underlying dynamics by making changes to the speed of the jet in idealized models (Chen et al. 2007; Chen and Zurita-Gotor 2008). In this study we use a similar approach with a barotropic model. This model has often been used as the base of a model hierarchy of the midlatitude circulation, as it displays realistic nonlinear wave–mean flow interaction (Vallis et al. 2004; Kidston and Vallis 2010; Barnes et al. 2010) but is relatively simple to understand.

The dynamics of the eddy-driven jet in this model are simplified by the fact that the flow is a single layer of incompressible nondivergent fluid. The zonally averaged zonal momentum equation for this flow is
e1
where the overbars represent the zonal averages and primes departures therefrom. The first term on the rhs is the meridional flux of vorticity and in this framework is the same as the convergence of the meridional flux of zonal momentum. The second term on the rhs is the momentum damping, and when is given by a linear damping on the zonal-mean momentum, in a statistically steady time average (shown by angle brackets), we have
e2
where τm is the time scale for the linear drag on the zonal-mean momentum. In the absence of forcing, the absolute vorticity of fluid parcels is conserved, allowing us to write
e3
where the lhs is the material derivative of absolute vorticity and the rhs represents any sources S and sinks D. Multiplying Eq. (3) by the eddy vorticity and using the fact that the flow is nondivergent gives an equation for the enstrophy:
e4
where is the sum of the planetary and relative meridional vorticity gradients. The full nonlinear equation governing the time-mean, zonal-mean zonal wind is then
e5

The first two terms on the rhs of Eq. (5) are the sources and sinks of pseudomomentum or wave activity that result from adding and removing vorticity in a forced-dissipative framework; these terms will be called and , respectively. The third term on the rhs is a triple eddy product term, and in the quasilinear framework this is ignored. Thus, in the quasilinear framework, where there is a net source of wave activity (), is positive and an eddy-driven jet is generated. The triple product term on the rhs of Eq. (5) is related to the divergence of the flux of enstrophy and therefore to sources and sinks of enstrophy, although the physical meaning is not entirely clear. As discussed in more detail below, only wave–wave interactions give rise to nonzero fluxes of enstrophy.

Here we find that as the jet is increased in speed, the latitude of the jet shifts poleward, even with the stirring of vorticity statistically unchanged. The cause appears to be the decrease in β* on the flanks of the jet that is associated with increasing the speed of the jet in a meridionally confined region. This has two related repercussions that encourage a poleward shift of the jet. The first is that poleward-propagating waves turn at high latitude and subsequently dissipate on the equatorward flank of the jet. The second is that waves may actually overreflect, providing a high-latitude source of pseudomomentum.

2. Methods

The barotropic model is publicly available through the National Oceanographic and Atmospheric Administration’s Geophysical Fluid Dynamics Laboratory’s Flexible Modeling System and is the same as is used in Vallis et al. (2004), Barnes et al. (2010), and Kidston and Vallis (2010). Damping is provided by a linear damping (on the momentum) and hyperviscoscity. Thus the equation that is integrated is Eq. (3), with
e6
where U is the vector velocity (U = ui + υj). There is also a Roberts damping term, although this is small. Large-scale dissipation is provided through the first term on the rhs, which represents a linear damping of momentum, and requires the curl to be taken to provide the damping in vorticity space. The momentum damping time scale for the eddies τe and the zonal-mean τm can be treated separately, so that
e7

The source term is a stochastic vorticity stirring created in spectral space and is the same as used in Vallis et al. (2004) and Kidston and Vallis (2010). Only total wavenumbers 4–12 are excited, and waves with zonal wavenumber k less than 4 are masked. The stirring is meridionally localized by applying a Gaussian mask that is centered at 40° and has a half-width of 10°.

Results are presented for the experiments summarized in Table 1. There are three sets of nonlinear experiments: a fully nonlinear experiment and two quasilinear experiments. In the fully nonlinear experiments the amplitude of the stirring is set to 7 × 10−11 s−2, which with the damping values used typically gives RMS eddy velocities on the order of 10–50 m s−1. The value of τe is set to 6 days, and τm is set to either 6 or 24 days. These experiments are named NonLin_Strong and NonLin_Weak, respectively.

Table 1.

Summary of experiments. The name is that used in the text. Parameters shown are S, the magnitude of the vorticity stirring used in integrating Eq. (3); τm and τe, the damping time scale used in Eq. (7) for the zonal mean and the eddies, respectively; M, the number by which the eddy vorticity advection was multiplied to produce the quasilinear runs (see text); and the linearity, which states whether Eq. (3) was linearized before integrating. Variables shown are the percentage of days that contain a β* reversal within the domain.

Table 1.

In the quasilinear experiments, the amplitude of S is decreased by four orders of magnitude, but the effect of the eddies on the mean flow is increased so that the small-amplitude eddies generate a realistic amplitude jet, as in Barnes and Hartmann (2011). The zonal-mean momentum tendency due to eddy vorticity advection is increased by calculating the value at each time step and then multiplied by a large number M. In the first set of quasilinear experiments M is fixed at 107, and τm is varied from 0.5 to 100 days. The eddies are relatively weakly damped with τe set to 6 days. These experiments are named QLin_τm. In the second set of quasilinear experiments the damping time scales are fixed and the value of M is varied between 107 and 109. Here the eddies are relatively strongly damped, with τm and τe both set to 2 days. These experiments are named QLin_M.

In addition to these three sets of nonlinear experiments, a fully linear experiment is carried out in which the equations of motion are linearized about a zonal-mean flow, and small-amplitude waves are allowed to propagate and dissipate in a statistically steady state. To keep the background flow steady the damping on the zonal-mean flow is removed, so that the dissipation is set to
e8

3. Wave propagation

a. The relationship between the speed and latitude of the eddy-driven jet

As seen in Fig. 1a, the magnitude of increases from NonLin_Strong to NonLin_Weak, and the maximum value is higher in NonLin_Weak. The latitude of the jet Φjet is taken as the latitude of the maximum and is found by evaluating a quadratic fit to across the two latitudes either side of the maximum. This is marked by the vertical line, and it is shifted slightly poleward in NonLin_Weak.

Fig. 1.
Fig. 1.

(a) The time-mean, zonal-mean zonal wind for the two fully nonlinear experiments, NonLin_Strong (dashed) and NonLin_Weak (solid). The vertical lines mark the latitude of the maximum , which is taken as the latitude of the jet Φjet. (b) The variation of Φjet with the maximum value of for the suite of quasilinear experiments named QLin_M (see text). (c) As in (b), but for the suite of quasilinear experiments named QLin_τm (see text). (d) The time-mean, zonal-mean eddy vorticity flux (thick dashed for NonLin_Strong and thick solid for NonLin_Weak) and the value of normalized by the value of τm (same lines but thin, which are indistinguishable from the thick lines).

Citation: Journal of the Atmospheric Sciences 69, 11; 10.1175/JAS-D-11-0300.1

Chen et al. (2007) also found a poleward shift of the surface westerlies when the damping on the zonal-mean momentum was reduced in a multilayered primitive equation model. The explanation offered in Chen et al. (2007) was that the increase in at the latitude of the jet causes the critical latitude of propagating waves to be located closer to the jet core. This causes the dissipation of equatorward-propagating eddies to shift poleward [the analogy with Eq. (5) is that shifts poleward]. The argument that the westerlies shift poleward relies on the idea that the easterly torque exerted in the upper troposphere by wave breaking/dissipation is associated with a thermally direct meridional overturning (by the theory of downward control); the downwelling branch of this meridional overturning gives adiabatic compression and heating at low levels, and there is a high meridional temperature gradient on its poleward flank; by shifting this low-level heating poleward, the latitude of baroclinic instability may shift poleward [analogously shifting in Eq. (5)], causing a wholesale poleward shift of the westerlies (G. Chen 2009, personal communication). The model used in this study is barotropic and the stirring is constant, and so even if increasing the speed of the jet does cause to shift poleward, there is no avenue for the mechanism implicit in Chen et al. (2007) to cause the poleward shift of .

The relationship between and Φjet is summarized in Fig. 1b for the suite of experiments named QLin_M. As increases, Φjet shifts systematically poleward. The same behavior is also seen in Fig. 1c for the suite of experiments named QLin_τm, although the changes in Φjet are larger in QLin_τm.

It appears that the variation of Φjet with (the slope of the line) is less for very weak and very strong jets. The sensitivity of Φjet to an increase in is highest for realistic values of , in the 4–15 m s−1 range, and it seems to saturate at higher and lower speeds. Nonetheless, at values of comparable to Earth’s midlatitude westerlies, as the jet speeds up it also shifts poleward, even though the stirring of the flow is unchanged.

As seen in Eq. (2), the vorticity fluxes in this model completely determine the value of . This is confirmed in Fig. 1d, where the value of is indistinguishable from the value of . The same relationship is seen in the quasilinear experiments (not shown), and this confirms the applicability of Eq. (2) to these integrations. Therefore, understanding the poleward shift of Φjet when the speed of the jet increases is synonymous with understanding the changes in in this model.

b. Vorticity flux balance

Using Eq. (5), it should be possible to determine which terms in the equation are important for the change in and thereby to gain some insight into the dynamics. The barotropic model is particularly suited to this because both S and D are known. However, in reality β* is often small and crosses zero, as seen in Table 1. The smallness of β* on the poleward flank of the jet is a central tenet of this study and will be discussed in more detail below. Where β* is close to zero, Eq. (5) is poorly behaved. The corresponding equation with β* in the numerator is
e9
and this can be examined to determine the relative importance of the triple product term. The terms in Eq. (9) are calculated offline, from daily averages of wind speeds, vorticity, and the vorticity tendency due to stirring/dissipation. Figure 2a shows the lhs of Eq. (9) for the run NonLin_Strong (solid line). The dash-dotted line is the sum of the first two terms on the rhs of Eq. (9) and represents the value that would be assumed to balance the lhs under quasilinear approximations. These terms clearly do not balance the lhs, implying that the third term on the rhs of Eq. (9) is nonnegligible (i.e., wave–wave interactions that give rise to enstrophy fluxes are important). The dashed line in Fig. 2 shows the entire rhs of Eq. (9) and comes much closer to balancing the lhs. Possible causes of the small differences between the dashed and solid lines include the loss of subdaily covariance, the finite differencing approximation used when calculating the derivatives, the Roberts contribution to D (which has been ignored), and the rhomboidal truncation in the model.
Fig. 2.
Fig. 2.

(a) The balance of terms in Eq. (9): the lhs (solid), the first two terms on the rhs (dash-dotted), and the full rhs (dashed) for NonLin_Strong. (b) As in (a), but for the QLin_τm run with τm = 6 days (see text).

Citation: Journal of the Atmospheric Sciences 69, 11; 10.1175/JAS-D-11-0300.1

The fact that the triple product term on the rhs of Eq. (9) is so large makes the prospect of understanding the differences in between NonLin_Strong and NonLin_Weak rather daunting, and this motivates the quasilinear integrations. A similar analysis of Eq. (9) for a quasilinear integrations is shown in Fig. 2b (it is the run from QLin_τm with τm set to 6 days). The dash-dotted line, representing the first two terms on the rhs, balances the lhs very well, confirming that the flow is quasilinear. The other quasilinear runs also show a similar balance (not shown). In the quasilinear runs the eddy kinetic energy is small and maximizes at around 10−5 m2 s−2 near the latitude of the jet (not shown). This implies that the RMS eddy velocities are on the order of 3 × 10−3 m s−1 and are small compared with the mean flow (Fig. 1), and so it is not surprising that the triple product term on the rhs of Eq. (9) is negligible.

c. Dynamics

The two integrations that provide the most parsimonious situation for understanding the underlying dynamics are the QLin_M experiments with M = 1 and M = 2. This is because (i) they are quasilinear, (ii) they are strongly damped, and (iii) the value of β* is always positive (Table 1), and so dynamic instabilities are unlikely to be important. As such Eq. (5) is well behaved and can be used to evaluate the vorticity flux changes. The balance of terms in Eq. (5) for QLin_M (M = 1) is shown as the thick lines in Fig. 3a, and there is good agreement between the lhs and the first two terms on the rhs (solid and dashed lines, respectively).

Fig. 3.
Fig. 3.

(a) Thick lines indicate the balance of terms in Eq. (5) for the quasilinear experiment QLin_M (M = 1) (black solid line is the lhs and black dashed line is the first two terms on the rhs). Gray lines indicate the same terms, but for the difference between the experiments QLin_M with M = 2 and M = 1. (b) The balance of terms in Eq. (10), for the differences between the same experiments as shown in (a): the solid line is the lhs and is the same as the gray dashed line in (a), and the dashed line is the rhs. (c) The two terms on the rhs of Eq. (10): the solid line is the first term, and the dashed line is the second term. (d) The numerator of the first term on the rhs of Eq. (10): the solid line is the total term, the dashed line is the change in , and the dash-dotted line is the change in .

Citation: Journal of the Atmospheric Sciences 69, 11; 10.1175/JAS-D-11-0300.1

If were to remain unchanged when M is doubled, the only change to would be an increase by a factor of 2 everywhere. The difference in between the QLin_M runs with M = 2 and M = 1 is shown as the thin solid line in Fig. 3a and reveals that increases at high latitude, as expected when the jet has shifted slightly poleward. There is a decrease in at the latitude of the jet. The thin dashed line shows that the changes in the first two terms on the rhs of Eq. (5) capture the changes in fairly well. Thus, Fig. 3a shows that there are qualitative changes in in response to a doubling of M that cause the jet to shift poleward, and that these can be understood by the behavior of the first two terms on the rhs of Eq. (5).

For small changes, the change in net source of wave activity can be approximated as
e10
where the first term on the rhs is due to the changes in the source and sink regions, and the second is due to the change in the vorticity gradient. Both sides of Eq. (10) are shown in Fig. 3b, and the agreement shows that the assumptions underlying the equation are relatively good, and so we can further decompose the changes in between the two runs.

The two terms that contribute to the rhs of Eq. (10) are shown separately in Fig. 3c. The second term on the rhs changes as a direct result of changing M, due to the changes in that change the relative vorticity component of β*. At the core of the jet the increase in and increase in β* gives a negative contribution to . As is seen in the dashed line in Fig. 3c, it is this term that indeed gives the reduction in at the core of the jet.

The positive changes in on the flanks of the jet seen in Fig. 3b are not directly caused by the changes in β* (cf. dashed lines in Fig. 3b and 3c). Instead, the increases in are caused by the changes in and represent a shift to more of a net source region on the flanks of the jet. This increase is much larger on the poleward flank than the equatorward flank (Fig. 3c, solid line), which is the reason the jet shifts poleward.

To understand the changes in , the total change is shown in Fig. 3d as the solid line, and the contributions from the source and the dissipation terms as the dashed and dash-dotted lines, respectively. There is an increase in the source term throughout the core of the jet (the cause of which will be addressed below), and there is a corresponding increase in dissipation (a negative change in corresponds to an increase in dissipation). There is a decrease in dissipation on the flanks of the jet, and the decrease is strongest on the poleward flank. Furthermore, the increase in dissipation at the core of the jet is stronger on the equatorward side than the poleward side. Thus increased dissipation is skewed toward the equator. We will show that at its most basic level, the poleward shift of the faster jet in this model is caused by the relative shift in the dissipation toward the equatorward side. (There is some sign that the increase in is also skewed toward the pole, but it is shown below that this is not always the case.)

d. A linear model

To examine the effect of the speed of the jet on wave propagation and dissipation at a more basic level, fully linear experiments were carried out where the equations of motion were linearized about a zonal-mean flow, as has been described. Two linear experiments are presented where waves were stirred and dissipated on jets whose only difference was the background wind speed. In the first experiment the value of was half the value from QLin_M with M = 4; this experiment is named FLin_Slow. In the second experiment was increased by a factor of 4, and this is named FLin_Fast.

The same metrics shown for QLin_M (M = 1) and QLin_M (M = 2) in Fig. 3 are shown in Fig. 4 for the fully linear runs. Even with linearized equations of motion, when increases by the same factor everywhere, the change in is an increase on the poleward flank of the jet and a decrease at the jet core (Fig. 4a, thin solid line). This change in is well captured by the change in the first two terms on the rhs of Eq. (5) (cf. solid and dashed lines in Fig. 4a). In turn, this change is reasonably well approximated by the rhs of Eq. (10) (Fig. 4b). The decrease in at the jet core is again caused by the changes in β* (Fig. 4c, dashed line), and the increase in on the poleward flank of the jet is caused by a shift toward a net source region (Fig. 4c, solid line). This shift toward a net source region at high latitudes is clearly caused by the increased dissipation being skewed toward the equator (Fig. 4d, dash-dotted line). The increase in the source term is relatively symmetrical about the climatological maximum at the center of the stirring, which is 40° (Fig. 4d, dashed line). Thus, although matters are confused by an increase in both the source and the dissipation terms across the core of the jet, the cause of the poleward shift of in (which would cause the jet to shift poleward were the flow not linearized) is the relative increase in dissipation at low latitude, and decrease in dissipation at high latitude.

Fig. 4.
Fig. 4.

As in Fig. 3, but for the two fully linear runs (see text).

Citation: Journal of the Atmospheric Sciences 69, 11; 10.1175/JAS-D-11-0300.1

e. Refractive index

The cause of the change in dissipation can be understood through the refractive index for linear Rossby waves propagating in a meridional shear flow. Plugging a wave solution for the streamfunction into the unforced linearized version of Eq. (3) yields an expression for the refractive index:
e11
where μ is the planetary radius multiplied by the cosine of latitude, k is the zonal wavenumber, and c = ω/k is the angular phase velocity. There are oscillatory solutions where n2 is positive, and where n2 goes to zero and becomes negative, waves must evanesce. Therefore, oscillatory solutions require that n2 is greater than zero and so
e12
where c and k are constant for individual waves, and in general a population of waves will have a range of values of c at any given k. The refractive index can be thought of as a constraint on the phase speeds that are able to propagate for a given value of k, with large negative phase speeds being unable to propagate. A phase-speed decomposition of the flow reveals that the propagating waves in these experiments generally have negative phase speeds (not shown), as is required for propagation away from the jet to a region where is negative, without encountering a critical latitude (where ).

Figure 5 explores the balance of terms in Eq. (12). The dashed lines show the lhs for waves with k = 7, a typical zonal wavenumber for the flow. The solid lines show the first term on the rhs, and so the vertical distance between the two lines represents the range of negative phase speeds that can propagate. If the phase speed is more negative than this difference, Eq. (12) does not hold and n2 becomes negative, implying that the waves will evanesce and be reflected.

Fig. 5.
Fig. 5.

(a) The rhs of Eq. (12) with k = 7 for the two fully linear runs (dashed; thick is FLin_Fast, and thin is FLin_Slow), and the first term on the rhs (solid). (b) The ratio of the maximum easterly wind speed at low to high latitude for the suite of runs named QLin_M (see text).

Citation: Journal of the Atmospheric Sciences 69, 11; 10.1175/JAS-D-11-0300.1

In FLin_Slow (thin lines), there is a relatively large window for propagation on both sides of the jet. In FLin_Fast (thick lines), the window of propagating phase speeds is much smaller, particularly at high latitude, and so waves are more likely to turn at high latitude and propagate back equatorward. To some extent the wave activity is prevented from propagating both ways and so is trapped near the core of the jet. With the enstrophy trapped near the core of the jet, increases, and this is the reason that the source of wave activity increases across the core of the faster jet. Nonetheless, as the jet speeds up, dissipation shifts preferentially to the equatorward side, which causes the jet to shift poleward.

Previous work has shown that in numerical integrations the dissipation can be crucial in determining the wave–mean flow interaction (Dickinson 1970; DelSole 2001). We do not think this is a candidate explanation in explaining the differences between the linear experiments FLin_Slow and FLin_Fast, or the quasilinear runs QLin_M, because in these experiments the damping was unaltered.

The turning latitude dynamics are well known in various contexts. The fact that turning latitudes appear on the flanks of fast jets can result in a waveguide (e.g., Hoskins and Ambrizzi 1993). The fact that β* reduces toward the poles is important for discouraging poleward wave propagation (Hoskins and Karoly 1981). Here we are highlighting dynamics due to a combination of these factors. As a jet increases in speed, the turning latitude appears first at high latitudes, and this has important consequences for the wave dissipation and subsequently the jet latitude. The dynamics are also essentially the same as those discussed in Barnes et al. (2010) in response to changing the stirring latitude in this model. That work highlighted reduced poleward propagation due to the decrease in β* when stirring is moved to higher latitudes. Here the decrease in β* is caused by the change in relative vorticity, but the effect on wave propagation is the same, and this causes the jet to shift poleward.

One further piece of corroboratory evidence is the relative speed of the low- versus high-latitude easterlies. From Eq. (2), the local strength of the easterlies is given by the vorticity flux, and from Eq. (5) in the quasilinear limit (ignoring the third term on the rhs) this is given by the sources and sinks of wave activity. Therefore if dissipation moves preferentially toward the equator for faster jets, it might be expected that the low-latitude easterlies become stronger when compared with the high-latitude easterlies. A crude measure of this is simply the ratio of the strongest easterly wind speed on the equatorward side of the jet to that on the poleward side. A higher ratio means relatively stronger easterlies on the equatorward side. As is shown in Fig. 5b, as the jet shifts poleward in the suite of runs QLin_M, the low-latitude easterlies get systematically stronger than the high-latitude easterlies. This is consistent with the idea that as the jet speeds up, an increasing fraction of the wave activity is dissipated at low rather than high latitude. A similar relationship is seen for the experiment QLin_τm (not shown).

4. Barotropic instability

When the jet increases in speed and the meridional shear increases, β* often becomes negative, as seen in Table 1. The reduction in the planetary component of β* toward the pole means that the β* reversal is more likely to occur at high rather than low latitudes. In this case, poleward-propagating waves may not find merely a turning latitude and be reflected but may actually tunnel through to the region of β* < 0 and overreflect. This barotropic instability will provide a high-latitude source of pseudomomentum that will accelerate the poleward flank of the jet and may provide a further encouragement for it to shift poleward. This mechanism can be thought of as closely related to, and an extension of, the refractive index argument outlined in the previous section.

Diagnosing barotropic instability in a fully developed nonlinear flow is a challenge. One approach is to consider the behavior of individual zonal wavenumbers. Equation (4) can be written for individual wavenumber components (ζk, etc.) as
e13
where we have used the fact that the product of two eddy terms with different zonal wavenumbers is zero because the modes are orthogonal. This is not true for the triple product, and so the third term on the lhs must be summed over all wavenumbers. Indeed, this term is zero when i = j = k, implying that only wave–wave interactions contribute, and so this term is inherently nonlinear. In considering whether barotropic instability is occurring, one usually considers an unforced flow, which renders the rhs zero, and small wave amplitude (or a single dominant wavenumber), which renders the third term on the lhs negligible. This gives
e14
where we have dropped the summation because there is no cross-wavenumber term. This leads to the well-known result that for instability to occur, β* must reverse somewhere in the domain, and furthermore that a westerly torque () will be exerted in the region where β* < 0 (e.g., Holton 1992).

From Eq. (14), an indicator of barotropic instability is the globally and time-averaged value of being positive. This is shown in Fig. 6a for the experiments NonLin_Strong and NonLin_Weak and in both experiments, the low-k waves exhibit signs of instability. This is part of the explanation for the existence of these waves, as waves with k < 4 were not stirred and so result from either instability or an inverse cascade. Consistent with the increased likelihood of a β* reversal, NonLin_Weak shows signs of increased instability across a larger range of wavenumbers.

Fig. 6.
Fig. 6.

(a) The time average of as a function of k for the two nonlinear runs NonLin_Strong (solid) and NonLin_Weak (dashed). (b) The time average of summed over waves with k = 1–3 for NonLin_Strong (solid) and NonLin_Weak (dashed). (c),(d) The value of summed over the values of k indicated.

Citation: Journal of the Atmospheric Sciences 69, 11; 10.1175/JAS-D-11-0300.1

Further support for the presence of instability has been found in transient experiments (not presented here). In these experiments, S and D in Eq. (3) were set to zero. The flow was initialized with the daily mean values from the quasilinear experiments, but the eddy component of the initial flow was reduced by four orders of magnitude to suppress the cascade. In these experiments, the globally averaged eddy kinetic energy (EKE) frequently grew exponentially with time, indicating instability.

Further support for the notion that the flow is often barotropically unstable can be gleaned from comparing the values of with , as shown in Fig. 6b. The data are for NonLin_Weak and are summed over k = 1–3. These waves were not stirred into existence, and they indicate clear signs of instability (Fig. 6a). The value of for waves with k = 1–3 shows a double maximum in NonLin_Weak. The double maximum in the values of indicates that there is net wave dissipation at the core of the jet and generation on the flanks. The value of 〈β*〉 is positive everywhere (not shown), even though there is nearly always a β* reversal somewhere in the domain (Table 1). As shown in Kidston and Vallis (2010) (their Fig. 2a), these reversals occur primarily on the poleward flank of the jet. Comparing the values of with indicates that the β* reversals at high latitude are important for the generation of waves with k = 1–3. Within the region marked with vertical lines (latitudes 50°–70°), β* reversals are clearly important during periods when is positive because the product is less than the components and 〈β*〉, even though these terms are both positive.

Calculating the enstrophy cascade term for these waves as a residual [the third term on the lhs of Eq. (13)] indicates that there is an inverse cascade giving an enstrophy flux into these waves, and that there is more dissipation due to this term than due to the mean flow instability (not shown), but nonetheless instability does appear to be occurring.

The impact of increased instability, as indicated in Fig. 6a, on the vorticity fluxes is clear in Fig. 6c. The value of for waves with k = 1–8 shows a double maximum in NonLin_Weak. The negative at the core of the jet indicates that there is net wave dissipation here, even though this is where the stirring maximizes.

The increase in on the flanks of the jet between NonLin_Strong and NonLin_Weak (the difference between the two lines) is larger at high than low latitudes. This is as expected when instability exerts westerly torque in the regions of β* reversals, and these occur more frequently on the poleward flank of the jet (Kidston and Vallis 2010). This, and the fact that waves with k ≥ 9 show no region of negative torque at high latitude, encourages the poleward shift of the jet.

Comparison of Figs. 6c and 6d shows that waves with low k are primarily responsible for the total change in . Given that the changes in for the low-k waves appear to be dominated by increased barotropic instability, it is extremely difficult to evaluate the relative importance of the refractive index argument that was discussed in the previous section. The basic dynamics remain valid, as clearly β* has reduced on the poleward flank of the faster jet, but separating the importance of reflection versus over reflection may not be possible without further experiments.

5. Full GCMs

The importance of the barotropic dynamics to more realistic atmospheres is the subject of ongoing work. The question is best addressed through the analysis of the transient adjustment in baroclinic models to idealized forcings, but this is beyond the scope of the current study. However, one hint that the barotropic dynamics may be relevant to full GCMs is provided through the correlation between the change in the speed in the upper troposphere, and the change in the latitude of the westerlies, across a suite of GCMs. Figure 7 shows the change in the zonal wind speed at 300 hPa at the latitude of the maximum surface westerlies, plotted against the change in the latitude of the surface maximum for the models that contributed a 1% increase in CO2 per year experiment to the model archive of phase 3 of the Coupled Model Intercomparison Project (CMIP3; Meehl et al. 2007b). Each point represents a single model, and the data are for the Southern Hemisphere during austral summer. The changes were calculated by a linear fit through the 70 yr when CO2 was increasing. Models that increase the speed of the upper-level westerlies jet more also shift the westerlies farther poleward. The relationship also holds in the annual mean. The fit is even tighter if the speed is taken at the surface, which is sometimes referred to as the “barotropic component” of the flow. The relationship is not caused by the pseudoconservation of angular momentum, since when the change in angular momentum or the change in angular velocity is plotted on the abscissa, the correlation remains (not shown).

Fig. 7.
Fig. 7.

The relationship between the change in wind speed in the upper troposphere above the surface westerlies and the change in their latitude, across a range of GCMs (see text). Each data point is a single model that submitted a run to the experiment in the CMIP3 archive that has CO2 increasing at 1% yr−1.

Citation: Journal of the Atmospheric Sciences 69, 11; 10.1175/JAS-D-11-0300.1

Thus, just as with the barotropic model, the change in the speed of the jet is related to the magnitude of the poleward shift, across a suite of GCMs. This motivates continuing research into whether the barotropic dynamics discussed in the previous sections are the cause of the poleward shift of the westerlies in response to increasing GHGs. It should be noted that the change of jet latitude with jet speed is much greater in the CMIP3 than in the barotropic model, suggesting that other, potentially more important mechanisms may be at work in shifting the jet. However, it could also be the case that the barotropic dynamics are important in the full GCMs, but that there is a positive feedback on the jet movement, with the wave source region being encouraged to move poleward because of self-maintenance dynamics, as described in Robinson (2006) and Kidston et al. (2010), that may follow the shift in dissipation encouraged by the barotropic dynamics. This is the subject of ongoing work.

6. Discussion and conclusions

We have used a stirred barotropic model to show that even with statistically constant stirring, when the jet is forced to speed up, it moves poleward. Increasing the speed of the jet increases the meridional wind shear and reduces β* on the flanks of the jet, where the relative vorticity gradient is negative. This reduces the window of phase speeds that are able to propagate to the flanks of the jet and creates turning latitudes. This is a well-known phenomenon and is the reason that fast jets become waveguides (e.g., Hoskins and Ambrizzi 1993). We have shown that on the sphere, the turning latitude appears first at high latitudes, and this causes the dissipation to shift preferentially to the equatorward side. This appears to be the most parsimonious explanation of why the net source of wave activity moves poleward, and thus the jet shifts poleward, when the speed of the flow is increased in this model.

A further consequence of increasing the speed of the jet manifests when the value of β* becomes not only small, but actually negative. In this case waves may overreflect, or exhibit barotropic instability. This can provide a high-latitude source of pseudomomentum and can further encourage the net source of wave activity to shift poleward. It is possible that this explains the greater sensitivity of jet latitude to jet speed in the weakly damped integrations. The importance of this is left for further work.

The relevance of the quasilinear barotropic dynamics to realistic atmospheres is not clear. On the one hand, the barotropic model is very different from a realistic situation where wave activity is generated through baroclinic instability. On the other hand, the basic mechanism of the turning latitude appearing on the poleward flank of a faster jet seems so simple that it may be relevant to realistic flows. Across a suite of state-of-the-art GCMs forced with increasing GHGs, the magnitude of the poleward shift of the surface westerlies is well correlated with the increase in the speed of the jet. It remains to be seen whether the same dynamics relating jet speed to latitude that were diagnosed for the barotropic model are indeed relevant to the more realistic situation.

Acknowledgments

This work was partly supported by DOE Grant DE-SC0005189 and NSF Grant AGS-1144302. We acknowledge the modeling groups, the Program for Climate Model Diagnosis and Intercomparison (PCMDI) and the WCRP’s Working Group on Coupled Modelling (WGCM) for their roles in making available the WCRP CMIP3 multimodel dataset. Support of this dataset is provided by the Office of Science, U.S. Department of Energy.

REFERENCES

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    • Search Google Scholar
    • Export Citation

  • Barnes, E. A., D. L. Hartmann, D. M. W. Frierson, and J. Kidson, 2010: The effect of latitude on the persistence of eddy-driven jets. Geophys. Res. Lett., 37, L11804, doi:10.1029/2010GL043199.

    • Search Google Scholar
    • Export Citation

  • Butler, A., D. W. J. Thompson, and R. Heikes, 2010: The steady-state atmospheric circulation response to climate change–like thermal forcings in a simple general circulation model. J. Climate, 23, 3474–3496.

    • Search Google Scholar
    • Export Citation

  • Chen, G., and P. Zurita-Gotor, 2008: The tropospheric jet response to prescribed zonal forcing in an idealized atmospheric model. J. Atmos. Sci., 65, 2254–2271.

    • Search Google Scholar
    • Export Citation

  • Chen, G., I. M. Held, and W. A. Robinson, 2007: Sensitivity of the latitude of the surface westerlies to surface friction. J. Atmos. Sci., 64, 2899–2915.

    • Search Google Scholar
    • Export Citation

  • DelSole, T., 2001: A simple model for transient eddy momentum fluxes in the upper troposphere. J. Atmos. Sci., 58, 3019–3035.

  • Dickinson, R. E., 1970: Development of a Rossby wave critical level. J. Atmos. Sci., 27, 627–633.

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    • Search Google Scholar
    • Export Citation

  • Fyfe, J. C., G. J. Boer, and G. M. Flato, 1999: The Arctic and Antarctic oscillations and their projected changes under global warming. Geophys. Res. Lett., 26, 1601–1604.

    • Search Google Scholar
    • Export Citation

  • Haigh, J. D., M. Blackburn, and R. Day, 2005: The response of tropospheric circulation to perturbations in lower-stratospheric temperature. J. Climate, 18, 3672–3685.

    • Search Google Scholar
    • Export Citation

  • Holton, J., 1992: An Introduction to Dynamic Meteorology. 3rd ed. Academic Press, 511 pp.

  • Hoskins, B. J., and D. J. Karoly, 1981: The steady linear response of a spherical atmosphere to thermal and orographic forcing. J. Atmos. Sci., 38, 1179–1196.

    • Search Google Scholar
    • Export Citation

  • Hoskins, B. J., and T. Ambrizzi, 1993: Rossby wave propagation on a realistic longitudinally varying flow. J. Atmos. Sci., 50, 1661–1671.

    • Search Google Scholar
    • Export Citation

  • Kidston, J., and G. K. Vallis, 2010: Relationship between eddy-driven jet latitude and width. Geophys. Res. Lett., 37, L21809, doi:10.1029/2010GL044849.

    • Search Google Scholar
    • Export Citation

  • Kidston, J., D. M. W. Frierson, J. A. Renwick, and G. K. Vallis, 2010: Observations, simulations, and dynamics of jet stream variability and annular modes. J. Climate, 23, 6186–6199.

    • Search Google Scholar
    • Export Citation

  • Lorenz, D. J., and E. T. DeWeaver, 2007: Tropopause height and zonal wind response to global warming in the IPCC scenario integrations. J. Geophys. Res., 112, D10119, doi:10.1029/2006JD008087.

    • Search Google Scholar
    • Export Citation

  • Lu, J., G. Chen, and D. M. W. Frierson, 2010: The position of the midlatitude storm track and eddy-driven westerlies in aquaplanet AGCMs. J. Atmos. Sci., 67, 3984–4000.

    • Search Google Scholar
    • Export Citation

  • Meehl, G. A., and Coauthors, 2007a: Global climate projections. Climate Change 2007: The Physical Science Basis. S. Solomon et al., Eds., Cambridge University Press, 747–845.

  • Meehl, G. A., C. Covey, T. Delworth, M. Latif, B. McAvaney, J. F. B. Mitchell, R. J. Stouffer, and K. E. Taylor, 2007b: The WCRP CMIP3 multi-model dataset: A new era in climate change research. Bull. Amer. Meteor. Soc., 88, 1383–1394.

    • Search Google Scholar
    • Export Citation

  • Polvani, L. M., and P. J. Kushner, 2002: Tropospheric response to stratospheric perturbations in a relatively simple general circulation model. Geophys. Res. Lett., 29, 1114, doi:10.1029/2001GL014284.

    • Search Google Scholar
    • Export Citation

  • Robinson, W. A., 2006: On the self-maintenance of midlatitude jets. J. Atmos. Sci., 63, 2109–2122.

  • Vallis, G. K., E. P. Gerber, P. J. Kushner, and B. A. Cash, 2004: A mechanism and simple dynamical model of the North Atlantic oscillation and annular modes. J. Atmos. Sci., 61, 264–280.

    • Search Google Scholar
    • Export Citation

  • Williams, G. P., 2006: Circulation sensitivity to tropopause height. J. Atmos. Sci., 63, 1954–1961.

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Cover Journal of the Atmospheric Sciences
Journal of the Atmospheric Sciences

  • Barnes, E. A., and D. L. Hartmann, 2011: Rossby wave scales, propagation, and the variability of eddy-driven jets. J. Atmos. Sci., 68, 2893–2908.

    • Search Google Scholar
    • Export Citation

  • Barnes, E. A., D. L. Hartmann, D. M. W. Frierson, and J. Kidson, 2010: The effect of latitude on the persistence of eddy-driven jets. Geophys. Res. Lett., 37, L11804, doi:10.1029/2010GL043199.

    • Search Google Scholar
    • Export Citation

  • Butler, A., D. W. J. Thompson, and R. Heikes, 2010: The steady-state atmospheric circulation response to climate change–like thermal forcings in a simple general circulation model. J. Climate, 23, 3474–3496.

    • Search Google Scholar
    • Export Citation

  • Chen, G., and P. Zurita-Gotor, 2008: The tropospheric jet response to prescribed zonal forcing in an idealized atmospheric model. J. Atmos. Sci., 65, 2254–2271.

    • Search Google Scholar
    • Export Citation

  • Chen, G., I. M. Held, and W. A. Robinson, 2007: Sensitivity of the latitude of the surface westerlies to surface friction. J. Atmos. Sci., 64, 2899–2915.

    • Search Google Scholar
    • Export Citation

  • DelSole, T., 2001: A simple model for transient eddy momentum fluxes in the upper troposphere. J. Atmos. Sci., 58, 3019–3035.

  • Dickinson, R. E., 1970: Development of a Rossby wave critical level. J. Atmos. Sci., 27, 627–633.

  • Frierson, D. M. W., I. M. Held, and P. Zurita-Gotor, 2006: A gray-radiation aquaplanet moist GCM. Part I: Static stability and eddy scale. J. Atmos. Sci., 63, 2548–2566.

    • Search Google Scholar
    • Export Citation

  • Fyfe, J. C., G. J. Boer, and G. M. Flato, 1999: The Arctic and Antarctic oscillations and their projected changes under global warming. Geophys. Res. Lett., 26, 1601–1604.

    • Search Google Scholar
    • Export Citation

  • Haigh, J. D., M. Blackburn, and R. Day, 2005: The response of tropospheric circulation to perturbations in lower-stratospheric temperature. J. Climate, 18, 3672–3685.

    • Search Google Scholar
    • Export Citation

  • Holton, J., 1992: An Introduction to Dynamic Meteorology. 3rd ed. Academic Press, 511 pp.

  • Hoskins, B. J., and D. J. Karoly, 1981: The steady linear response of a spherical atmosphere to thermal and orographic forcing. J. Atmos. Sci., 38, 1179–1196.

    • Search Google Scholar
    • Export Citation

  • Hoskins, B. J., and T. Ambrizzi, 1993: Rossby wave propagation on a realistic longitudinally varying flow. J. Atmos. Sci., 50, 1661–1671.

    • Search Google Scholar
    • Export Citation

  • Kidston, J., and G. K. Vallis, 2010: Relationship between eddy-driven jet latitude and width. Geophys. Res. Lett., 37, L21809, doi:10.1029/2010GL044849.

    • Search Google Scholar
    • Export Citation

  • Kidston, J., D. M. W. Frierson, J. A. Renwick, and G. K. Vallis, 2010: Observations, simulations, and dynamics of jet stream variability and annular modes. J. Climate, 23, 6186–6199.

    • Search Google Scholar
    • Export Citation

  • Lorenz, D. J., and E. T. DeWeaver, 2007: Tropopause height and zonal wind response to global warming in the IPCC scenario integrations. J. Geophys. Res., 112, D10119, doi:10.1029/2006JD008087.

    • Search Google Scholar
    • Export Citation

  • Lu, J., G. Chen, and D. M. W. Frierson, 2010: The position of the midlatitude storm track and eddy-driven westerlies in aquaplanet AGCMs. J. Atmos. Sci., 67, 3984–4000.

    • Search Google Scholar
    • Export Citation

  • Meehl, G. A., and Coauthors, 2007a: Global climate projections. Climate Change 2007: The Physical Science Basis. S. Solomon et al., Eds., Cambridge University Press, 747–845.

  • Meehl, G. A., C. Covey, T. Delworth, M. Latif, B. McAvaney, J. F. B. Mitchell, R. J. Stouffer, and K. E. Taylor, 2007b: The WCRP CMIP3 multi-model dataset: A new era in climate change research. Bull. Amer. Meteor. Soc., 88, 1383–1394.

    • Search Google Scholar
    • Export Citation

  • Polvani, L. M., and P. J. Kushner, 2002: Tropospheric response to stratospheric perturbations in a relatively simple general circulation model. Geophys. Res. Lett., 29, 1114, doi:10.1029/2001GL014284.

    • Search Google Scholar
    • Export Citation

  • Robinson, W. A., 2006: On the self-maintenance of midlatitude jets. J. Atmos. Sci., 63, 2109–2122.

  • Vallis, G. K., E. P. Gerber, P. J. Kushner, and B. A. Cash, 2004: A mechanism and simple dynamical model of the North Atlantic oscillation and annular modes. J. Atmos. Sci., 61, 264–280.

    • Search Google Scholar
    • Export Citation

  • Williams, G. P., 2006: Circulation sensitivity to tropopause height. J. Atmos. Sci., 63, 1954–1961.

  • <sc>Fig</sc>. 1.
    View in gallery
    Fig. 1.

    (a) The time-mean, zonal-mean zonal wind for the two fully nonlinear experiments, NonLin_Strong (dashed) and NonLin_Weak (solid). The vertical lines mark the latitude of the maximum , which is taken as the latitude of the jet Φjet. (b) The variation of Φjet with the maximum value of for the suite of quasilinear experiments named QLin_M (see text). (c) As in (b), but for the suite of quasilinear experiments named QLin_τm (see text). (d) The time-mean, zonal-mean eddy vorticity flux (thick dashed for NonLin_Strong and thick solid for NonLin_Weak) and the value of normalized by the value of τm (same lines but thin, which are indistinguishable from the thick lines).

  • <sc>Fig</sc>. 2.
    View in gallery
    Fig. 2.

    (a) The balance of terms in Eq. (9): the lhs (solid), the first two terms on the rhs (dash-dotted), and the full rhs (dashed) for NonLin_Strong. (b) As in (a), but for the QLin_τm run with τm = 6 days (see text).

  • <sc>Fig</sc>. 3.
    View in gallery
    Fig. 3.

    (a) Thick lines indicate the balance of terms in Eq. (5) for the quasilinear experiment QLin_M (M = 1) (black solid line is the lhs and black dashed line is the first two terms on the rhs). Gray lines indicate the same terms, but for the difference between the experiments QLin_M with M = 2 and M = 1. (b) The balance of terms in Eq. (10), for the differences between the same experiments as shown in (a): the solid line is the lhs and is the same as the gray dashed line in (a), and the dashed line is the rhs. (c) The two terms on the rhs of Eq. (10): the solid line is the first term, and the dashed line is the second term. (d) The numerator of the first term on the rhs of Eq. (10): the solid line is the total term, the dashed line is the change in , and the dash-dotted line is the change in .

  • <sc>Fig</sc>. 4.
    View in gallery
    Fig. 4.

    As in Fig. 3, but for the two fully linear runs (see text).

  • <sc>Fig</sc>. 5.
    View in gallery
    Fig. 5.

    (a) The rhs of Eq. (12) with k = 7 for the two fully linear runs (dashed; thick is FLin_Fast, and thin is FLin_Slow), and the first term on the rhs (solid). (b) The ratio of the maximum easterly wind speed at low to high latitude for the suite of runs named QLin_M (see text).

  • <sc>Fig</sc>. 6.
    View in gallery
    Fig. 6.

    (a) The time average of as a function of k for the two nonlinear runs NonLin_Strong (solid) and NonLin_Weak (dashed). (b) The time average of summed over waves with k = 1–3 for NonLin_Strong (solid) and NonLin_Weak (dashed). (c),(d) The value of summed over the values of k indicated.

  • <sc>Fig</sc>. 7.
    View in gallery
    Fig. 7.

    The relationship between the change in wind speed in the upper troposphere above the surface westerlies and the change in their latitude, across a range of GCMs (see text). Each data point is a single model that submitted a run to the experiment in the CMIP3 archive that has CO2 increasing at 1% yr−1.

Fig. 1.

(a) The time-mean, zonal-mean zonal wind for the two fully nonlinear experiments, NonLin_Strong (dashed) and NonLin_Weak (solid). The vertical lines mark the latitude of the maximum , which is taken as the latitude of the jet Φjet. (b) The variation of Φjet with the maximum value of for the suite of quasilinear experiments named QLin_M (see text). (c) As in (b), but for the suite of quasilinear experiments named QLin_τm (see text). (d) The time-mean, zonal-mean eddy vorticity flux (thick dashed for NonLin_Strong and thick solid for NonLin_Weak) and the value of normalized by the value of τm (same lines but thin, which are indistinguishable from the thick lines).
Fig. 2.

(a) The balance of terms in Eq. (9): the lhs (solid), the first two terms on the rhs (dash-dotted), and the full rhs (dashed) for NonLin_Strong. (b) As in (a), but for the QLin_τm run with τm = 6 days (see text).
Fig. 3.

(a) Thick lines indicate the balance of terms in Eq. (5) for the quasilinear experiment QLin_M (M = 1) (black solid line is the lhs and black dashed line is the first two terms on the rhs). Gray lines indicate the same terms, but for the difference between the experiments QLin_M with M = 2 and M = 1. (b) The balance of terms in Eq. (10), for the differences between the same experiments as shown in (a): the solid line is the lhs and is the same as the gray dashed line in (a), and the dashed line is the rhs. (c) The two terms on the rhs of Eq. (10): the solid line is the first term, and the dashed line is the second term. (d) The numerator of the first term on the rhs of Eq. (10): the solid line is the total term, the dashed line is the change in , and the dash-dotted line is the change in .
Fig. 4.

As in Fig. 3, but for the two fully linear runs (see text).
Fig. 5.

(a) The rhs of Eq. (12) with k = 7 for the two fully linear runs (dashed; thick is FLin_Fast, and thin is FLin_Slow), and the first term on the rhs (solid). (b) The ratio of the maximum easterly wind speed at low to high latitude for the suite of runs named QLin_M (see text).
Fig. 6.

(a) The time average of as a function of k for the two nonlinear runs NonLin_Strong (solid) and NonLin_Weak (dashed). (b) The time average of summed over waves with k = 1–3 for NonLin_Strong (solid) and NonLin_Weak (dashed). (c),(d) The value of summed over the values of k indicated.
Fig. 7.

The relationship between the change in wind speed in the upper troposphere above the surface westerlies and the change in their latitude, across a range of GCMs (see text). Each data point is a single model that submitted a run to the experiment in the CMIP3 archive that has CO2 increasing at 1% yr−1.
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