## 1. Introduction

Stability theories have traditionally been applied to midlatitudinal atmospheric shear flows to understand extratropical cyclogenesis and forecast errors. In both cases, the interest lies mainly in the growth rate of the disturbances rather than in the propagation rate. Nevertheless, investigation of the structure of analysis and forecast errors in the Pacific jet shows that, on average, the speed of the peak of the error packet is close to the speed of the mean jet-level zonal wind, suggesting that the leading edge spreads downstream faster than the zonal wind (Hakim 2005). To explore this possibility, and to develop a better fundamental understanding of the spreading rate of localized disturbances in complicated, time-dependent flows, we consider the problem in an idealized framework, where the disturbance structure may be controlled.

The general problem of disturbance growth in space and time is reviewed by Huerre and Monkewitz (1990), who describe growth and displacement with respect to fixed and moving observers. Flows for which a fixed observer sees exponential growth for all time are referred to as “absolutely unstable” because the disturbance eventually affects the entire domain. Flows for which a fixed observer sees only transient growth, followed by decay, are referred to as “convectively unstable” because the disturbance affects only a finite region before traveling away.

The development of localized disturbances on a baroclinic jet was considered by Simmons and Hoskins (1979), who studied the response through an asymptotic analysis. They found that for the Eady model, downstream disturbances travel along the tropopause with a limiting speed close to the zonal flow of the basic-state jet at that level. Farrell (1983) confirmed that the spreading rate of the disturbance approaches the zonal-mean flow toward the leading edge, and proved that the peak of the wave packet moves at the group speed of the unstable waves; therefore, the wave packet grows spatially in time. This analysis places a finite limit on how fast a disturbance can spread downstream, which is the zonal wind at the tropopause.

Moving closer to realistic jets, Snyder and Hamill (2003) analyzed the development of disturbances on a time-dependent turbulent baroclinic jet in a three-dimensional quasigeostrophic (QG) model. They computed the Lyapunov vectors and exponents, which are analogous to the eigenvectors and eigenvalues for the stability of stationary states. The key property of this analysis is that any perturbation initialized far in the past will converge asymptotically in time to the leading Lyapunov vector with a growth rate given by the corresponding leading Lyapunov exponent. Snyder and Hamill (2003) found that disturbances maximize at the ground and tropopause with secondary contributions from the interior of the troposphere. The instantaneous structure of the Lyapunov vectors reveals the concentration of the perturbations at locations where the ambient tropopause potential temperature gradients are large. Moreover, the scales of these perturbations are comparable to the features of the ambient unperturbed flow. They hypothesized that, regardless of the form of the initial perturbation, the perturbed solution eventually differs from an unperturbed solution only in locations where gradients in the basic-state flow are large.

Stevens and Hakim (2005) analyzed a problem intermediate to linear time-independent flows and nonlinear time-dependent flows by performing Floquet analysis on a time-periodic, nonparallel shear flow consisting of a baroclinic jet with a superimposed neutral wave. The Floquet vectors are equivalent to the Lyapunov vectors for the time-periodic flow, describing the asymptotic behavior of the system. They found that adding a neutral wave of sufficient amplitude to a time-independent basic state promotes localized disturbance spreading both upstream and downstream at a rate faster than the zonal wind limits given in Farrell (1983) and Simmons and Hoskins (1979). This behavior can be understood by considering the limiting case of a basic state with meridionally independent jets, such as a sine wave in the zonal direction. In this case, a localized disturbance may spread in the zonal direction if it projects onto the unstable Floquet mode, even though there is no zonal flow in the basic state.

Our goal in this paper is to address the basic question of the spreading of spatially localized disturbances in time-dependent baroclinic jets. The remainder of the paper is outlined as follows. Section 2 describes the method, including the quasigeostrophic model and initial-condition specification. Results are described in section 3, and section 4 provides a discussion and concluding summary.

## 2. Method

A nonlinear quasigeostrophic model is used in this study to simulate the spatial growth of disturbances in a turbulent baroclinic jet stream. The objective here is not to reproduce the full atmospheric general circulation, but to focus on the dry dynamics of the midlatitudinal troposphere. The two-surface quasigeostrophic model described in the section below is intermediate in complexity and computational demand between numerical weather prediction models and simple two-layer models. The results of Snyder and Hamill (2003) indicate that, for our purposes, the two-surface geometry is a reasonable approximation. The model is discussed subsequently, followed by a description of the initial-value problem and a discussion of the localized disturbance.

### a. Numerical model

*q*(QGPV hereafter) equation satisfiesVariables are nondimensionalized as defined in Table 1. The geopotential

*ϕ*provides a geostrophic streamfunction for the horizontal velocity components

*u*= −

*ϕ*and

_{y}*υ*=

*ϕ*, and through hydrostatic balance relates to the potential temperature by

_{x}*θ*=

*ϕ*, where subscripts denote partial differentiation. In the constant-PV configuration, the dynamics are driven solely by the horizontal advection of potential temperature on the rigid horizontal boundaries at

_{z}*z*= 0 and

*z*=

*H*. The disturbance thermodynamic equation applied at the rigid lids (

*w*= 0) at the ground and tropopause iswhere the boldface variables represent basic-state quantities and

*J*(

*a*,

*b*) =

*a*−

_{x}b_{y}*a*is the Jacobian. We shall refer to the first term on the right-hand side as basic-state advection, the second term as disturbance propagation, and the third term as nonlinear advection. To control the buildup of grid-scale noise, a

_{y}b_{x}*θ*field with the diffusion coefficient

*ν*chosen such that the smallest scale resolved by the model is damped with an

*e*-folding time of 20 model time steps. The radiative restoration of midlatitude baroclinicity is modeled as a relaxation process whereby disturbances are damped with an

*e*-folding time scale

*τ*of 15 days relative to a basic state containing a baroclinic jet zone given by (Hoskins and West 1979)Here,

*l*is a constant (1.1343) that defines the jet zone and

*μ*controls the jet concentration; we set

*μ*= 1 (note that the Eady jet is recovered for

*μ*= 0). The model domain is chosen to be 20

*L*(20 000 km) long in

*x*, 11.08

*L*(11 080 km) wide in

*y*, and 1

*H*(10 km) deep in

*z*, with a dealiased spectral resolution of (128, 64) (

*x*,

*y*) waves. The basic state is constructed by localizing the Hoskins–West jet in the middle of the domain and extending the limiting values of the jet zone to the edges of the domain.

Scaling parameters and dimensional values.

### b. Initial-value problem

The initial state is obtained by integrating the model for approximately *t* = 2450 time units (475 days) starting from a zonal jet specified by (3) plus a small-scale Gaussian disturbance. Given this initial state, solutions for interval *T* = 21 time units (92.6 h) are obtained with and without the addition of a localized disturbance; we refer to these as the perturbed and control solutions, respectively. The localized disturbance is introduced at the center of the domain on the tropopause and its structure is described in the following section. The difference between the perturbed and the control solutions gives the spatiotemporal evolution of the initially localized disturbance. An ensemble of 5000 experiments is generated by repeating this procedure every *T* = 21 time units (92.6 h).

### c. Localized disturbance

The structure of the localized initial disturbance has a dramatic influence on the spreading characteristics of the disturbance. Since the model is closed on the rigid horizontal boundaries by the thermodynamic equation [(2)], an obvious choice for the disturbance field is *θ*, which results from specifying the Neumann boundary condition *θ* = *ϕ _{z}* =

*f*(

*x*,

*y*) on the Laplacian given by (1). Another way of constructing the disturbance is by specifying the streamfunction, that is, the Dirichlet boundary condition given by

*ϕ*=

*f*(

*x*,

*y*).

*a*, amplitude

*a*, and a finite distance at which the function goes to zero

_{o}*r*= 1. It is important that the function go to zero at a finite distance, which is not possible for commonly used functions of exponentials, such as the Gaussian, which has nonzero tails that extend over the domain; that is, the disturbance cannot be completely localized.

*S*and

*T*represent the surface and tropopause, respectively.

The localized potential temperature distribution produces nonlocalized geopotential and meridional velocity fields, whereas the localized geopotential distribution produces a localized potential temperature and meridional velocity distribution (Fig. 1). Note that for localized geopotential, the potential temperature field is localized but not single sign as in the localized potential temperature field. In terms of the dynamics [(2)], we expect that for the localized potential temperature disturbance, the nonlocal velocity field will project onto **Θ** gradients, resulting in nonzero disturbance propagation. On the other hand, when the disturbance is absolutely localized through geopotential localization, the disturbance propagation term is initially numerically zero far from the disturbance and therefore only the basic-state advection term in (2) is responsible for spreading this initially localized disturbance.

## 3. Results

### a. Single experiment

We randomly draw an ensemble member as specified in the previous section to illustrate disturbance spreading characteristics. The basic state for the experiment is characterized by a north–south potential temperature gradient that is associated with a meandering westerly jet near the center of the domain. The disturbance localized in potential temperature spreads rapidly in this state, both downstream and upstream, mainly to locations where the magnitude in the gradient of the basic flow is large (Fig. 2). Examination of Figs. 2b–d reveals negative (positive) perturbations downstream (upstream) of the initial disturbance location, consistent with the clockwise induced flow around a “warm” potential temperature anomaly advecting the gradient in the basic flow.

In contrast, the disturbance localized in geopotential (absolutely localized) spreads strictly downstream, and more slowly compared to the potential-temperature-localized disturbance (Fig. 3). Propagation is also constrained to locations having larger gradients in the background potential temperature field, indicating the relative importance of the disturbance propagation effect. The difference from the previous case is due mainly to the fact that the potential-temperature-localized disturbance has nonzero wind in the far field, which activates the disturbance propagation term in these locations. These qualitative impressions of individual experiments do not quantify the spreading rate of the disturbance relative to the basic flow, which motivates the introduction of a passive tracer.

### b. Tracer dynamics

*ψ*is governed byBecause of the absence of a term similar to disturbance propagation in (2),

*J*(

*ϕ*,

**Θ**), the tracer is simply advected by the mean and perturbation velocity field. As a result, tracer spread provides a reference bound to determine if the disturbance spreads faster than the ambient flow.

### c. Ensemble experiment

*T*= 21 time units (92.6 h). Simple and useful measures of the ensemble include the mean and spread. The ensemble mean is defined for the disturbance and tracer fields byrespectively, where 〈

*θ*〉 and 〈

*ψ*〉 are the mean values,

*N*gives the ensemble size, and subscript

*i*identifies individual experiments. The ensemble spread

*σ*is defined bywhich provide a measure of the variability in the ensemble solution.

Figures 4 and 5 show the time evolution of the ensemble mean and spread for potential temperature and geopotential localization, respectively, and Fig. 6 shows the same results for the tracer field. Results for potential temperature localization reveal a dipole pattern that expands to fill the domain as it moves downstream (Fig. 4). As in the individual example discussed previously, this dipole behavior may be understood from the induced flow due to the positive potential temperature anomaly on the tropopause, which is associated with clockwise circulation. Since the basic flow has lower values of potential temperature to the “north” on average, the disturbance winds contribute to negative (positive) advection downstream (upstream) of the disturbance through the disturbance propagation term. Moreover, the disturbance projects on all wavenumbers, with short waves traveling downstream faster than the long waves, which explains the asymmetry in the dipole pattern of the disturbance field. In contrast, results for the disturbance localized in geopotential show a localized wave packet that develops and spreads downstream (Fig. 5), with a dominant wavelength of about 4000–5000 km. This wavelength roughly approximates the most unstable mode for jet profile [(3)]. This response appears similar to a downstream development pattern found by Szunyogh et al. (2002) from the impact of dropsonde observations targeted to improve downstream weather forecasts.

Results for the tracer experiments show an evolution that is distinctly different from the perturbation experiments (Fig. 6). Since the evolution of the tracer is similar for both initial conditions, we show the tracer evolution only for the disturbance localized in potential temperature (Fig. 6). The tracer spreads strictly downstream, and meridionally toward the edges of the domain. This “diffusion” of the tracer field at longer times can be attributed to the fact that the baroclinic eddies and the waves on the jet transport the tracer in random directions and thus, on average, the tracer field appears diffused. The tracer spread closely follows the distribution of mean tracer, whereas for the perturbation experiments the spread extends beyond the mean field and, especially for the disturbance localized in potential temperature, grows in time as the perturbed forecasts diverge from the control. The asymmetry in the tracer field at short times can be attributed to the fact that the perturbation warm anomaly introduced in the tropopause produces anticyclonic circulation, which advects the tracer southward.

The leading edge of the tracer reflects the time-evolving maximum velocity of the background flow, located near *y* = 0 on average. Comparing Figs. 4 and 6, it appears that at least through *t* = 14 time units, the disturbance field moves downstream faster than the tracer. In contrast, a comparison of Figs. 4 and 5 shows that the leading edge of perturbation spread matches the leading edge of the tracer very closely, so that the group velocity of the leading edge of the packet is close to the ambient flow. We conclude that, for these experiments, disturbances do not travel faster than the flow and only appear to do so when a small far-field disturbance grows in time. This suggests that the linear stability properties of zonal and time independent flows may be useful in explaining the spreading behavior of disturbances in a turbulent jet with active mixing. We explore this connection in the next section.

### d. Time-independent flows

In the previous section, we found that the spreading characteristics of an absolutely localized disturbance in a time-dependent flow appeared to conform to expectations based on linear theory for time-independent flows. Here we pursue this connection by performing similar experiments for the linear-shear Eady jet and the localized jet defined by (3). A disturbance of the same structure and amplitude is added to these jets at the same location as in previous experiments.

Results for these steady jets are qualitatively similar to those for the ensemble average of experiments on the time-dependent turbulent jet. Specifically, for the potential-temperature localized disturbance, a dipole response dominates, with the disturbance apparently moving downstream faster than the tracer field (Fig. 7). For the geopotential-localized disturbance, the tracer marks the leading edge of a spreading wave packet (Fig. 8), which reflects the stability properties of the jet (Farrell 1983). This behavior is clearest for the Eady jet, which shows the development of a wave packet with the most unstable wavelength near the packet peak, and progressively shorter, faster-moving, waves as one moves toward the leading edge of the packet.

An important distinction between these results and the nonlinear ensemble solutions is the absence of meridional diffusion of the disturbance and the tracer field; here, the disturbance and tracer both remain in the zonal jet. This results from the absence of turbulent baroclinic eddies, which transport tracer and disturbance in the meridional direction, and from far-field gradients in potential temperature, which provide the locus for disturbances to propagate away from the centerline. These results lead us to hypothesize that the nonlinear time-dependent turbulent flow problem can be linearized into a superposition of a time-independent linear flow plus a diffusion process (e.g., Farrell and Ioannou 1996).

### e. Conditional sampling

Here we explore the sensitivity of the primary results to the method. In particular, we consider the possibility that specific flow configurations, or disturbance locations, may allow disturbances to travel faster than the ambient flow. First, the ensemble results discussed above were conditionally sampled to look for the response of the disturbance with respect to its location in different synoptic phenomena (i.e., spreading characteristics when the disturbance was located initially in a ridge or a trough or on the jet). For the case on the jet, the response was also sampled for fast- and slow-moving jets within the ensemble. These studies did not reveal responses significantly different from those discussed previously, and the tracer leading edge confines the spread of the absolutely localized disturbance in all cases considered.

The second set of sensitivity tests involved new experiments that vary the size, amplitude, and location of the initial disturbance. Variations in the size and amplitude of the initial disturbance did not yield results substantially different than those explained in the previous sections; however, changing the location of the disturbance produced significantly different results depending on the proximity of the disturbance to the mean jet. In particular, when the absolutely localized disturbance is sufficiently far removed from the jet so that its induced flow does not reach the jet, the disturbance may still affect the jet if it is intercepted by another feature on the jet, such as a trough. An example is shown in Fig. 9a for a disturbance located south of the jet axis. Note that disturbance spreading (at long times) is no longer contained by the leading edge of the tracer. As the disturbance moves downstream and northward toward the jet, its flow projects onto the larger gradients near the jet, which propagates the disturbance downstream. In contrast, the tracer is constrained by advection from the basic state and the perturbation winds and remains trapped away from the jet. We note that if the disturbance is located far from the jet, it is unreachable by features on the jet, which implies a limited meridional region where such behavior may occur.

## 4. Concluding discussion

The objective of this study was to determine whether a localized disturbance in a turbulent baroclinic jet may spread spatially faster than the zonal wind, which is the bound from linear theory. A large ensemble of initial-value problems was conducted with a quasigeostrophic model to determine the average response of a localized disturbance on an ensemble of initial states of the turbulent jet. Spreading relative to the ambient flow was measured against a passive tracer having the same spatial distribution as the disturbance field.

Results show a strong dependence on the structure of the localized initial condition. When the disturbance is localized in potential vorticity (tropopause potential temperature), the disturbance evolves into a dipole that spreads both upstream and downstream faster than the tracer field. We attribute this behavior to the small, but nonzero, far-field wind associated with the disturbance, which advects large ambient potential vorticity gradients. When this far-field wind is eliminated by localizing in geopotential (“absolutely localized”), the disturbance evolves into a spreading wave packet that moves strictly downstream at a speed bounded by the tracer field. This result apparently does not involve the mechanism for faster-than-flow spreading in wavy jets described by Stevens and Hakim (2005), although the mean wave amplitude in our study is below the critical threshold suggested by Stevens and Hakim (2005).

Packet properties for the spreading absolutely localized disturbance, such as 4000–5000-km waves near the packet and shorter waves toward the leading edge, compare qualitatively well with solutions for time-independent jets such as Eady’s. Comparing the evolution of the localized disturbance in the turbulent flow with that in time-independent flows leads to the hypothesis that the nonlinear time-dependent turbulent flow can be linearized into a superposition of time-independent parallel flow and a “diffusion” process. Formally, the nonlinear problem may be effectively linearized by parameterizing the nonlinear terms as a stochastic process (e.g., Farrell and Ioannou 1996), but we have not performed such an analysis.

Conditional sampling of the ensemble did not reveal states that allow absolutely localized disturbances to spread faster than the flow, except in the case where the disturbance is located away from the time-mean jet axis but sufficiently close to interact with fluctuations in the jet. In this case, the disturbance may be advected toward the jet and interact with gradients in these features, allowing the disturbance to propagate into the jet while the tracer is trapped off the jet. Familiar examples of this type of behavior involve extratropical transition, where tropical cyclones recurve toward the jet stream, and the analogous approach toward the jet by mesoscale vortices from the arctic (Hakim and Canavan 2005).

## Acknowledgments

This work represents a portion of the first author’s master’s thesis research conducted at the University of Washington, and was supported by grants from the National Science Foundation (AGS-0552004) and the Office of Naval Research (N00014-09-1-0436).

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