1. Introduction

Observations of downward-propagating anomalies in the annular mode index (Baldwin and Dunkerton 1999) have spurred much recent research into the dynamical link between the stratosphere and the troposphere. Recent studies (Plumb and Semeniuk 2003; Christiansen 2003; S. Hardiman 2005, personal communication) have investigated the propagation of the anomalies from high in the stratosphere down to the tropopause, but have not accounted for the communication of annular mode index anomalies from the tropopause to the ground, which seems to occur by a fundamentally different process. Referring to the height–time picture of the downward-propagating NAM index (Baldwin and Dunkerton 2001, their Fig. 1), we note that whereas the descent of anomalies to the tropopause follow a broad slanted path, indicating a slow descent of long-lived anomalies, the communication of anomalies from the tropopause to the underlying troposphere appears to occur on a much faster time scale, with sporadic, short-lived anomalies appearing throughout the troposphere while the tropopause anomaly persists. That the response of the troposphere is mediated by baroclinic eddies is suggested by several facts: the time scale of a few days over which tropospheric anomalies occur following the descent of a stratospheric anomaly to the tropopause is consistent with the time scale of baroclinic instability and the tropospheric response is concentrated near the ground and the tropopause, consistent with the structure of Eady-like baroclinic modes. Recent work strengthens the suggestion that baroclinic instability is sensitive to the state of the stratosphere: Polvani and Kushner (2002) found a response in the tropospheric annular mode to different stratospheric polar vortex strengths in long integrations with a dynamical core, and that the response was mediated by synoptic eddies (Kushner and Polvani 2004); Charlton et al. (2004) documents synoptic-scale tropospheric responses to different stratospheric conditions in models, and Limpasuvan et al. (2004) documents synoptic-scale responses to stratospheric sudden warmings in observations; Limpasuvan and Hartmann (2000) found that synoptic eddies were an important driver of the tropospheric annular mode index; and Wittman et al. (2004, hereafter W04) shows surface annular mode responses to changing polar vortices in life cycle experiments. This paper focuses specifically on the role of vertical shear in the lower stratosphere, since, as we will show, changes to the vertical shear at the top boundary have a simple effect on the linear growth of baroclinic eddies, which may determine the nonlinear development of baroclinic eddies and thus the tropospheric mean flow.

To investigate the role of baroclinic instability in communicating stratospheric anomalies to the surface, we return to the simplest model of baroclinic instability, the Eady problem. Despite its simplicity, the Eady problem constitutes a remarkably powerful description of baroclinic instability, and is well suited to investigating the effect of tropopause flow anomalies because the instability relies on the resonance between counterpropagating Rossby waves at the ground and the tropopause (Heifetz et al. 2004), changes to conditions at the tropopause will result in changes to the properties of the upper wave, and thus whole growing mode. The Eady problem has also been shown to apply in cases where growth is nonmodal (Rivest and Davis 1992).

We start with a modified Eady problem, in which we find the effect of increasing stratospheric shear is to increase the phase speed of the growing baroclinic modes, and to increase the growth rate of the lower synoptic wavenumbers. Because the Eady problem supplies no information about the meridional effect of the instability on the mean flow, which we need to determine if we wish to understand the origin of the annular mode signal, we turn next to the linear problem in primitive equations on the sphere. Here we find that the effect of stratospheric shear on the growth rates and phase speeds is qualitatively the same as in the one-dimensional model, but that the structure of the normal modes, and the acceleration of the mean flow, do not change in a way that is consistent with the observed annular mode signal. Thus, in the next section, we consider the finite amplitude effect of baroclinic instability in life cycle experiments. Here we find that increasing stratospheric shear leads to enhanced acceleration of the jet on the poleward side, consistent with an increased annular mode response. We also find that with an initial perturbation of intermediate wavenumber, it is possible to change the life cycle from an LC1 to an LC2 by increasing the stratospheric shear. The transition from LC1- to LC2-type behavior raises new issues in understanding the dynamics underlying the transition, since we are able to induce the transition without changing the structure of the baroclinic jet in the troposphere. Finally, we examine reanalysis data and find evidence that the relationship between stratospheric shear and the annular mode response suggested by the life cycle experiments obtains.

2. Review of the 1D problem

A comprehensive mathematical development of the 1D quasigeostrophic (QG) Eady problem with a stratosphere is given in Müller (1991) and Blumen (1979), although their work predated the discovery of the downward propagation of annular mode index anomalies from the stratosphere. Since the Eady problem is well known, here we give only an overview of the key results; Müller (1991) gives a fuller treatment.

We consider two atmospheric layers, in which both the static stability and wind shear are constant (Fig. 1). The troposphere is bounded below, at z = 0, and shares a free interface, at z = H, with the stratosphere above. The stratosphere is unbounded above, and the Brunt–Väisälä frequency of the stratosphere is fixed at its typical value of 4 times that of the troposphere (N ²_s = 4N ²_t). The stratospheric shear Λ_s is varied from −1 to +1 times the tropospheric shear Λ_t.

We nondimensionalize with the height of the tropopause H, the tropospheric shear Λ_t, the tropospheric static stability N_t, and the Coriolis parameter f₀. The relations between the nondimensional and dimensional variables are shown in Table 1; the variables shown with asterisks are nondimensional.

The nondimensional streamfunction Ψ* then obeys Laplacian equations

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and boundary conditions

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These conditions represent the disappearance of wind velocity and temperature perturbations at the top of the domain, continuity of pressure, and vertical wind fields at the interface (at z* = 1), and temperature conservation at the surface, respectively.

Normal mode solutions of (1) for λ* = −1, 0, 1 (i.e., stratospheric shear equal to −1, 0, and 1 times tropospheric shear) result in the phase speeds and growth rates shown in Fig. 2. Positive stratospheric shear shifts the growth rate curve upward and toward low wavenumbers, so that the peak wavenumber is reduced, and the growth rates of wavenumbers below the peak are increased. Negative stratospheric shear shifts the growth-rate curve downward and toward higher wavenumbers. The phase speed of the unstable waves increases monotonically with stratospheric shear at all unstable wavenumbers. It should be noted that extreme positive and negative stratospheric shears, beyond the range considered here, suppress growth rates at all wavenumbers (cf. Müller 1991).

The effect of increasing stratospheric shear may be understood by considering the background potential vorticity (PV) gradient at the tropopause: Bretherton (1966) gives the expression for the background QG potential vorticity gradient as

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so that q_0y is positive for the range of Λ_s and N ²_s that we consider. Recall that β is absent in the Eady problem. Since the phase speed of a Rossby wave is given by c − u = −q_0y/k, c − u becomes less negative with increasing stratospheric shear, and a free Rossby wave at the tropopause propagates more rapidly eastward with the mean flow.

Eady baroclinic instability may be considered as the interaction between two counterpropagating free Rossby waves that propagate on the potential vorticity gradients that exist only at the lower boundary and the tropopause (Davies and Bishop 1994; Heifetz et al. 2004, hereafter H04). When stratospheric shear is increased, the phase speed of the upper wave, which is the phase speed of a free Rossby wave and is directed westward, against the mean flow, decreases. Thus the phase speed of the coupled mode, which is equal to the mean of the phase speeds of the two edge waves [H04, their Eq. (53)], increases. Hence, increased stratospheric shear leads to higher phase speeds at all wavenumbers.

When the stratospheric shear is zero, the PV gradients at each edge are equal and opposite, as in the original Eady problem. At the wavenumber at which the growth rate is a maximum (the peak wavenumber), the two phase speeds of the top and bottom waves are equal and opposite, and the phase difference between them is π/2. The barotropic wind field associated with each edge wave is thus optimally aligned to amplify the meridional displacement of the other wave, giving the maximum growth rate. At wavenumbers less than the peak, the phase speed of each wave is higher, and without mutual interaction the two waves would tend to travel past each other. (Recall that for free Rossby edge waves, the phase speed is given by c = −q_0y/k; c is positive at the bottom edge, and negative at the top edge. Additionally, at the top edge the phase speed is Doppler shifted by the nonzero wind there.) To ensure the phase locking necessary for modal growth, the phase difference between the two waves increases (H04; Fig. 2). Correspondingly, at wavenumbers above the peak the phase speeds of the two edge waves are too low for coherence, and the phase difference must decrease, so that each wave helps the other to propagate. Away from the peak wavenumber, the growth rate decreases because some component of the wind induced by each wave helps or hinders the other’s propagation, and therefore is unavailable for growing the other wave.

When stratospheric shear is increased from zero, the PV gradient at the top edge is decreased, decreasing the phase speed of the top wave. To correct for the decreased phase speed, the peak wavenumber decreases, increasing the speed of both edge waves so they cohere again. The increase in peak growth rate is due to the vertical dependence of the edge waves on wavenumber—for a free Rossby edge wave the vertical structure of the streamfunction is an exponential whose thickness is inversely proportional to wavenumber [ψ(z) ∝ e^−kz], and thus at lower wavenumbers the wind induced by one wave at the other’s location increases. As pointed out in H04, the growth rate of a baroclinic normal mode depends both on the phase relationship between the two edge waves, which governs how effectively each wave is able to induce the other to grow, and the size of the PV gradient supporting each edge wave, which governs how much the edge wave grows as it is subjected to the influence of the other. For the values of shear considered here, the main effect on growth rate is due to the change in phase between the two edge waves. However, at higher shears the PV gradient at the top edge is reduced, reducing the growth rate.

The opposite occurs when stratospheric shear decreased below zero: the top edge PV gradient is increased so that the peak wavenumber increases to compensate for the increased edge wave speed. The maximum growth rate declines because the vertical extent of the edge waves decreases.

The cutoff that appears at low wavenumbers with nonzero stratospheric shears is due to the fact that mismatched PV gradients at the top and bottom result in mismatched phase speeds. The mismatch is worst at low wavenumbers (c = −q_0y/k), and the modal growth becomes impossible when the two waves can no longer compensate for the mismatch by altering their phase alignment.

The 1D QG problem provides information about the growth rate and vertical structure of the growing baroclinic wave, but yields no information about the meridional structure of the normal modes, nor the meridional transports of momentum and heat, which are crucial in determining the effect of the instability on the meridional structure of the zonal mean flow, and the effect on the annular mode index. Therefore, we turn next to a study of the analogous linear instability problem in three dimensions.

3. Toward a meridional picture: The linear problem in primitive equations

In the atmosphere, baroclinic instability arises in the meridionally confined midlatitude jet, and it is the momentum and heat fluxes into and out of the jet that change its structure and result in the annular mode signal—the hallmark of stratospheric influence—as the jet’s latitude is changed. To understand the downward propagation of the NAM signal from the stratosphere, therefore, we must investigate the meridional structure of the growing baroclinic modes and the associated accelerations of the basic state. We therefore calculate the baroclinically unstable normal modes in the three-dimensional linearized primitive equations for basic states with varying stratospheric shear.

a. Basic states

The basic-state zonal winds are specified as a function of latitude ϕ and log-pressure height, z = −H log(p/p₀), where H is a scale height of 7 km. The zonal wind is given by u(ϕ; z) = g(ϕ) f (z), where the meridional dependence

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specifies a midlatitude jet centered at 45°N, as for the tropospheric jet in W04.

The vertical dependence f (z) is specified so as to match that of the Eady problem as closely as possible (Fig. 3). Integrating the thermal wind relationship in latitude, and differentiating in height, so that

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imposes a limit on the maximum vertical curvature of the zonal wind, since ∂θ/∂z must be positive to ensure static stability. Thus, for reasonable choices of the function of integration h(z), the shear cannot change abruptly at the tropopause. To accomplish the required smooth shear profile we choose a vertical dependence of the initial zonal wind specified using the function introduced in Lindzen (1970),

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in which Λ_t,s are shears, δ_t,s are parameters controlling the curvature of the corners, and the z_t,s are heights. These parameters take the values shown in Table 2. This formulation well approximates a piecewise linear function of height with three sections, as can be seen in Fig. 3. The tropospheric shear approximates that of the standard baroclinic jet of Simmons and Hoskins (1980) and Thorncroft et al. (1993, hereafter T93). The stratospheric shear Λ_s is varied, as in the 1D problem.

The vertical profile of zonal wind has zero shear above 30 km in the stratosphere so that excessive zonal wind speeds do not necessitate small time steps for the numerical integration. Our results are robust to variations in this cut-off height, as well as the sharpness of the corners (within reasonable values).

The same zonal wind jet is specified in both hemispheres, and the temperature field is specified by thermal wind balance. The free function of integration h(z) is chosen so that the meridional mean vertical temperature profile, denoted by θ(z), corresponds to an Eady-like arrangement of two regions of constant N ², with a smooth transition at the tropopause:

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where the symbols are defined as before and parameter values are given in Table 2; g is the standard gravitational acceleration.

b. Details of the calculation

Since a linear stability analysis of the linearized primitive equations leads to a nonseparable eigenproblem, we take inspiration from Simmons and Hoskins (1976) and numerically solve an initial value problem in a modified dynamical core whose solutions converge to the required eigenmodes and associated growth rates.

We use the “Built on Beowulf” (BOB) dynamical core described in Rivier et al. (2002). We compute with a horizontal spectral resolution of T85 and 40 vertical pressure levels, arranged to give adequate coverage of the stratosphere as in Polvani and Scott (2004); the model top is located at approximately 80 km. For the linear normal mode calculations diffusion in time and space is set to zero. We specify initial conditions, as described above, to be a zonally symmetric function, and perturb the temperature field with a single Fourier mode of wavenumber k = m. The functional form of this perturbation is

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but our results are independent of the specific form of the perturbation, which governs only how fast the solution converges to the most unstable normal mode.

At every time step, each prognostic variable X is Fourier transformed in the longitudinal direction:

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where (λ, ϕ, p) are latitude, longitude, and pressure.

Linearization of the primitive equations requires two modifications. The zonal mean terms (k = 0) are held constant, and each product of two or more variables that are perturbations from the zonal mean is set to zero. We implement the first requirement by storing and resetting all of the components with k = 0, X̃₀, and the second by setting the X̃_k≠m to zero at each time step. Thus, the only Fourier coefficients that survive are those of the initial perturbation, which evolve, and those of the zonally symmetric basic state, which remain constant.

A zonally symmetric basic state typically supports many baroclinically unstable modes at each zonal wavenumber, and the initial perturbation in general projects onto all of them. However, since each mode grows exponentially with time, the solution always converges to the most unstable mode. Since the equations are linear, we rescale all of the k ≠ 0 coefficients by the same factor every time the perturbation amplitude exceeds some specified threshold, so as to keep their values suitably small. Thus, we monitor the solution, which grows exponentially with periodic rescaling, until it has converged satisfactorily to the normal mode.

We then calculate the growth rate of the eddy kinetic energy (EKE) from the perturbation quantities. The EKE is calculated at each time step, and grows as the square of the perturbation quantities. The phase speed is calculated from the difference in the phase of any of the perturbation fields at successive time steps.

4. The fully nonlinear problem

A natural extension of the previous section which allows us to investigate the nonlinear effects on the baroclinic development is to perform life cycle experiments, using the same initial conditions as before. Our approach here is similar to that of our previous study (W04), but here we vary only the stratospheric shear, and perform life cycles at zonal wavenumbers other than 6. We solve initial value problems with the fully nonlinear primitive equations, and consider the effect of the life cycles, with different stratospheric shears and different wavenumbers, on the mean flow. By considering the surface geopotential height we can relate our results to the observational work cast in the annular mode framework.

c. Life cycle results in the annular mode framework

The transition from LC1 to LC2 is the most dramatic consequence of changing stratospheric shear, but it occurs only at one wavenumber, and its actual occurrence in the atmosphere is untested. We therefore return to consideration of the tropospheric annular mode response to stratospherically modulated baroclinic life cycles.

Changes in the annular mode index for a zonally symmetric tropospheric jet correspond mostly to changes in the latitude of the jet (Wittman et al. 2005), but the annular mode is usually defined as the first EOF of geopotential height. Since our model lacks a prognostic equation for the surface value of geopotential height, following W04, we instead calculate the zonal mean surface geopotential height Z_surf from the υ-momentum Eulerian mean equation (see, e.g., Andrews et al. 1987), neglecting the vertical terms, which are small.

Since we cannot statistically derive an annular mode function from a large series of observations, as we could if we were considering reanalysis data or output from a long numerical integration, for this section our strategy will be to approximate the annular mode as a step function of latitude in surface geopotential, crossing zero at the latitude of the jet maximum:

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Thus, we can calculate the zonal mean zonal wind, calculate the zonal mean surface geopotential height Z_surf, and project onto the annular mode function and integrate with time to arrive at an annular mode index (AMI) by which to quantify the dependence of the meridional evolution of the jet during baroclinic instability:

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The position of the jet, and the annular mode index, are fixed, in the atmosphere, by a balance between forcing by baroclinic eddies and solar forcing. The solar forcing tends to establish a jet at an equilibrium position in midlatitudes, baroclinic eddies drive the jet away from this position, and solar forcing returns the jet to its original position. Our unforced life cycle experiments lack the restoring solar heating, and so we are only able to measure how far the jet is shifted from its original position by baroclinic eddies. The resulting shift, however, is suggestive of how the position of the jet responds to changing stratospheric conditions, since the time scale of the life cycle (saturation occurs after about 5 days) is much shorter than that of the restoring solar forcing [Held and Suarez (1994) uses a radiative cooling time scale of 40 days outside the Tropics]. Comparing AMI between life cycles therefore gives a measure of the effect of baroclinic eddies, modulated by different stratospheric shears, on the annular mode index. It is perhaps worth clarifying that projecting Z_surf onto the annular mode, and then comparing between life cycles, is different from our approach in W04, where we projected the difference in Z_surf between two life cycles onto the annular mode, and identified the resulting quantity as a measure of the surface response to changes in the stratosphere. Here, because Z_surf changes dramatically between LC1 and LC2 life cycles, we project the raw quantity.

Figure 10 shows the time mean of Z_surf over the 25-day duration of the life cycles. In the LC1 regime, increasing stratospheric shear leads to a deepening of the poleward branch of the surface geopotential, corresponding to a greater poleward strengthening of the zonal mean jet as a result of the instability, and projecting more strongly onto the annular mode. In the LC2 regime, changing the stratospheric shear has little effect on the surface geopotential height.

Figure 11 shows AMI, the cumulative projection of the surface geopotential height onto the annular mode. For all of the LC1 cases the signal grows rapidly as the instability approaches its peak amplitude. The ordering of the growth rates and final values of this index are the same as the growth rates and peak values of the EKE, and the final separation in the values of the projection coefficient is established at around the time of maximum EKE.

For the LC2 cases, the growth of the projection coefficient is initially comparable with that of the LC1 cases, but slows rapidly before the instability saturates, and remains at a small fraction of its value in the LC1 cases. It is also apparent that for the LC1 cases, increasing stratospheric shear increases AMI at all times, and that AMI rises most strongly just as the life cycles saturate.

To establish how this annular mode signal is produced, and what produces the differences between cases of different stratospheric shears, it is instructive to examine the total acceleration of the zonal mean zonal wind by life cycles with different stratospheric shears. The total change over the 25-day life cycle in zonal mean zonal wind for wavenumber 6 (all LC1 type) is shown in Fig. 12. As stratospheric shear is increased from −4 to +4 m s⁻¹ km⁻¹, the total acceleration of the jet at the surface on the poleward side increases from 40 to 60 m s⁻¹, and the location of the maximum acceleration remains unchanged at about 60°N. The weaker deceleration on the equatorward side of the jet is correspondingly decreased with increasing stratospheric shear from 20 to 10 m s⁻¹, also remaining in the same place. Thus increasing stratospheric shear produces a stronger annular mode signal by causing a greater acceleration of the jet, so that the poleward-displaced jet resulting from a baroclinic life cycle becomes stronger, and the increasing annular mode response to increasing stratospheric shear is a consequence of the more rapid growth and larger saturation amplitude that occur with increased stratospheric shear.

For the wavenumber-7 cases, shown in Fig. 13, which show the transition from LC1 to LC2 with increasing stratospheric shear, it is apparent that the resultant acceleration of the jet reverses sign at the transition, so that the LC2 life cycle moves the jet equatorward instead of poleward. For higher wavenumbers, all in the LC2 regime, the maximum acceleration is then only slightly sensitive to the stratospheric shear.

Several observational papers (e.g., Hartmann et al. 2000) have found a connection between high stratospheric annular mode indices and increased equatorward Eliassen–Palm (EP) flux in the region of the tropopause, corresponding to increased poleward momentum transport and poleward jet displacements. The EP fluxes in our life cycles, shown in Fig. 14, show the same dependence on stratospheric shear, with high-stratospheric shear giving rise to enhanced equatorward EP flux at the tropopause. This EP flux pattern, which is similar for both values of stratospheric shear, is that of a typical LC1 life cycle and increasing stratospheric shear increases the amplitude the instability reaches before saturating, so that the effect of increasing stratospheric shear is just to increase the amplitude of the LC1 EP flux pattern. The increased penetration of EP fluxes into the stratosphere with higher stratospheric shear is consistent with the normal mode extending farther into the stratosphere.

In summary, for low wavenumbers (below 6), life cycles are LC1 type, with wave breaking on the equatorward side of the jet. As stratospheric shear is increased, the saturation amplitude of the growing eddies increases, giving rise to larger total accelerations that displace the jet farther poleward, and result in larger annular mode signals. At wavenumber 7, increasing stratospheric shear changes the life cycle from an LC1 to an LC2, possibly by changing the phase speed, and thereby the location of the critical surface on the equatorward side of the jet. The LC2 cases show little sensitivity to stratospheric shear, and all life cycles at wavenumbers higher than 7 are of an LC2 type.

5. Observations

The above life cycle experiments suggest that for the lower synoptic wavenumbers, stratospheric shear is positively correlated with the annular mode index in the troposphere, and vice versa. To establish whether a similar relationship between stratospheric shear and tropospheric annular mode index can be found in the observations, we examine National Centers for Environmental Prediction–National Center for Atmospheric Research (NCEP–NCAR) data (Kalnay et al. 1996), for which the daily northern annular mode index has been calculated at each pressure level (Baldwin and Dunkerton 2001, hereafter BD01). Approximately 6000 wintertime values of the NAM index are categorized into three approximately equally sized groups by the anomaly of the wind shear at 100 hPa. Figure 15 shows the mean NAM index as a function of pressure for days with negative, neutral, and positive stratospheric shear anomalies, and the probability distributions of the 1000-hPa NAM index for the three categories of shear anomaly.

The probability distribution functions (PDFs) of 1000-hPa NAM index, p(x), are constructed using a Gaussian kernel:

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where i is over all observations, y_i are the observed NAM values, and h the width of the kernel, given by (Silverman 1986) 1.06σn^−1/5, where σ is the population standard deviation, and n is the number, of all of the NAM observations in each stratospheric shear category.

From the distributions, it is evident that the mean NAM index shifts by about 1 between the high- and low-stratospheric shear cases, comparable to the shift in NAM index between strong vortex and weak vortex events in BD01. It is also worth noting that their weak vortex events were defined as stratospheric NAM indices of less than −3 standard deviations from the mean, and the strong events had stratospheric NAM indices of above 1.5 standard deviations. Assuming a roughly normal distribution of stratospheric shears, our strong shear cases are those with shears greater than about 0.5 standard deviations, and our negative shear cases have shears less than about −0.5 standard deviations. Thus stratospheric shear appears to be a more reliable predictor of surface annular mode index than stratospheric annular mode index.

The above observational study shows that the atmosphere qualitatively behaves in a way which is at least consistent with the response to shear in life cycle experiments. A more direct and quantitative comparison between the life cycle experiments and observations may be made by comparing the change in upper-tropospheric momentum fluxes accompanying changes in shear in the life cycle experiments, and in observations.

Limpasuvan et al. (2004, their Fig. 3, left-hand column) shows the evolution of zonal mean zonal wind during a composite stratospheric sudden warming (SSW). At the peak of the SSW, the maximum lower stratospheric shear anomaly is about −1 m s⁻¹ km⁻¹. The accompanying maximum anomaly in synoptic momentum flux F_y integrated between 50° and 90°N (Limpasuvan et al. 2004, their Fig. 10e), is about 30 × 10⁶ kg s⁻². In our life cycle experiments a change of −4 m s⁻¹ km⁻¹ in lower stratospheric shear gives rise to a change of 70 × 10⁶ kg s⁻² in the 300-hPa momentum flux integrated between 50° and 90°N (Fig. 16), which, assuming a linear response, gives about a change of about 18 × 10⁶ kg s⁻² in response to a change of −1 m s⁻¹ km⁻¹ in lower stratospheric shear. Hence the response to a unit change in shear of upper-level synoptic momentum fluxes in life cycles is about two-thirds the size of the response of the observed atmosphere, which, considering the highly idealized nature of the life cycle experiments, is not a bad agreement.

We may also note that Limpasuvan gives a surface NAM response of about −0.75 to a −1 m s⁻¹ km⁻¹ change in stratospheric shear, and, while we cannot calculate a meaningful NAM index in our life cycle experiments, the change in NAM index reported by Limpasuvan in the observations is of the same sign as in our life cycles, and is significant.

In summary, the correspondence between tropospheric NAM index and lower stratospheric shear suggested by the life cycle experiments is corroborated by the NCEP–NCAR reanalysis data, with increased lower stratospheric shear associated with higher mean values of the annular mode index at all heights, and vice versa. The response to tropospheric momentum fluxes, by which the zonal mean jet structure is changed and the NAM signal established, is comparable in both life cycles and observations.

6. Conclusions

We have found that baroclinic development is affected by stratospheric shear, and confirmed robustly that stratospheric shear can account for the connection between the stratospheric and tropospheric NAM signals previously reported (Wittman et al. 2004).

Solving the linear instability problem in 3D primitive equations, we have found that the effect of stratospheric shear on the growth rate and phase speed is qualitatively the same as in a simple Eady-like 1D model, we have determined the meridional structure of the most unstable modes, and their dependence on stratospheric shear. Life cycle experiments have shown that changing stratospheric shear can induce a transition between LC1 and LC2 life cycles at zonal wavenumber 7. In the annular mode framework, we have found that, for LC1 life cycles, increasing stratospheric shear increases the poleward displacement of the jet by the finite-amplitude baroclinic eddies; this agrees with the observed stratosphere–troposphere NAM relationship. The change in acceleration of the zonal mean jet by baroclinic life cycles is a result of the saturation amplitude increasing with stratospheric shear—the larger the baroclinic wave grows before breaking, the greater the effect on the mean flow. Examining the connection between NAM index and 100-hPa shear in reanalysis data, we have found that higher stratospheric shears are, in fact, associated with higher tropospheric NAM indices, and vice versa.

Although we have shown that lower stratospheric shear can produce a tropospheric NAM signal by modulating baroclinic eddies, this does not rule out other mechanisms by which the stratosphere may influence the tropospheric circulation—indeed, it would be surprising if shear-modulated eddies were the only mechanism responsible for the downward propagation of NAM anomalies from the stratosphere to the troposphere. In particular, recent work (Kushner and Polvani 2004; Song and Robinson 2004; Thompson et al. 2006) suggests that downward control may also be involved in producing surface NAM responses to stratospheric disturbances. Though it may be difficult to design a clean numerical experiment that would ascribe causality uniquely to any one mechanism, or exclude any other, the weight of evidence suggests that baroclinic eddies are at least one component of the downward propagation of NAM anomalies to the surface. It should also be noted that the amplitude of the tropospheric response to changes to the stratosphere is small: as we reported in W04 tropospheric responses to changes to the tropospheric mean flow are at least 10 times larger than responses to the stratosphere, and, as such it surely remains difficult to fully disentangle what tropospheric behavior is a response to the stratosphere and what behavior constitutes internal variability. However, we have demonstrated and elucidated a plausible linkage between the stratosphere and troposphere.

All previous life cycle studies have induced transitions between LC1 and LC2 behavior by changing the structure of the jet in the troposphere, and Hartmann (2000) found explicitly that the transition could be induced by adding equatorward barotropic shear only if the shear was added near the ground. The results presented here suggest that the evolution of synoptic eddies may be affected by conditions in the lower stratosphere. By developing an objective algoriT93 to classify midlatitude wave breaking events into cyclonic and anticyclonic categories, one may be able to determine whether stratospheric shear is correlated to the prevalence of LC1 and LC2 type events in the observations, confirming the stratospheric shear-driven LC1/LC2 transition which we have found in these life cycle experiments. Such an algoriT93 would also find uses beyond the issue of stratosphere–troposphere coupling: LC1/LC2 wave breaking has been implicated in low-frequency internal tropospheric variability (e.g., Hartmann and Zuercher 1998).

In a wider context, this work adds to the growing body of evidence on the importance of the stratosphere in climate modeling and weather forecasting. A consistent poleward shift of the storm tracks has been documented in the recent Intergovernmental Panel on Climate Change (IPCC) simulations of twenty-first century climate (Yin 2005; Miller et al. 2006). the results presented here, showing that lower stratospheric shear is able to directly impact synoptic eddies (and thus the location of the midlatitude jets), suggest that a better representation of the lower stratosphere in climate models might be important. Similarly, greater attention to the lower stratosphere (e.g., in terms of higher vertical resolution about the tropopause) might help in improving weather forecasts, as suggested by Baldwin et al. (2003).