## 1. Introduction

Forecasting the motion and intensity evolution of tropical cyclones (TCs) accurately is an important but challenging task. While progress has been made in the last decades, especially in predicting TC tracks, it is still often very difficult to forecast TCs reliably. The complex interaction of TCs with the midlatitude flow, which causes the recurvature of the storms and can lead to an extratropical transition (ET), is particularly problematic in this respect. The interaction processes are very sensitive to the phasing of the storm relative to the midlatitude flow (Ritchie and Elsberry 2007; Riemer et al. 2008; Riemer and Jones 2010; Scheck et al. 2011b) and can result in a reduced forecast skill (Jones et al. 2003). Moreover, these processes are associated with large forecast uncertainties downstream of the cyclone (Anwender et al. 2008; Harr et al. 2008).

Probabilistic forecasts using an ensemble prediction system (e.g., Leutbecher and Palmer 2008; Barkmeijer et al. 2001; Puri et al. 2001; Yamaguchi et al. 2009) can provide valuable additional information in such low-predictability situations. Moreover, additional observations in sensitive regions are a promising method to improve TC forecasts (Palmer et al. 1998; Buizza and Montani 1999; Harnisch and Weissmann 2010). Both these adaptive observations and the ensemble forecasts make use of sensitivity information to determine sensitive regions and to generate initial ensemble perturbations, respectively. Often singular vectors (SVs), which are defined as the fastest-growing perturbations in a given time interval and for a given perturbation norm (e.g., Diaconescu and Laprise 2012), provide this information.

The structure of SVs can be quite complex and depends on the perturbation norm, the optimization time interval (OTI), the grid resolution, and whether moist processes are included (Palmer et al. 1998; Barkmeijer et al. 2001; Puri et al. 2001; Kim and Jung 2009; Komori and Kadowaki 2010; Lang et al. 2012). Recently, the SV structure of TCs and changes in this structure arising from the interaction of a TC with the midlatitude flow was addressed in several studies. Peng and Reynolds (2006) investigated SVs for groups of straight-moving and nonstraight-moving (including recurving) TCs. For both groups, they found high sensitivity in an annulus with a radius of about 500 km from the storm center, in a region where the radial gradient of potential vorticity changes its sign. The sensitivity maximum tended to occur in inward-flow regions with respect to the storm. For the nonstraight-moving storms the sensitivity extended to larger radii and the singular values and the track errors were larger. Reynolds et al. (2009) compared SVs for recurving and nonrecurving TCs in 2006. They confirmed the existence of the high-sensitivity annulus around the storm center. During recurvature, sensitive regions developed several thousand kilometers to the northwest of the storm and the singular values increased significantly. Chen et al. (2009) found that maximum sensitivity was reached either in regions with maximum inward flow or in regions with low radial wind speed with respect to the storm center. Kim and Jung (2009) showed for Typhoon Usagi (2007) that before recurvature sensitivity close to the center of the storm dominates, whereas afterward the sensitivity at an upstream midlatitude trough becomes more important. Lang et al. (2012) investigated the influence of varying resolution and the inclusion of moist processes on the SV structure for Hurricane Helene (2006) using the European Centre for Medium-Range Weather Forecasts (ECMWF) ensemble prediction system (EPS). They found that before recurvature the initial SV structure was dominated by a spiral tilted against the shear in the vertical and the horizontal directions—a configuration that allows perturbation to gain energy by barotropic and baroclinic energy conversion. The evolved leading SV was characterized by a deep dipole structure, indicating that the fastest-growing perturbation caused a displacement of the TC. Also, for other EPSs, the leading SVs targeted on TCs often show a dipole structure (Kim and Jung 2009; Reynolds et al. 2009).

The complex structure of SVs computed for full-physics models makes it hard to identify the most important growth processes and understand their properties. For this reason, SVs and perturbation growth mechanisms for TC-like vortices have been investigated for idealized models. Nolan and Farrell (1999) demonstrated for a barotropic model that barotropic instability is not required for a significant growth of perturbations over finite times. They identified the Orr effect (Orr 1907), which allows perturbations tilted against the shear to grow, as the primary growth mechanism in this case, and showed that also vortices with barotropic unstable normal modes can benefit from this effect. Peng et al. (2009) investigated the growth of perturbations in unstable vortices and found that exponential growth results from the mutual amplification of asymmetries outside and inside the vortex core. Yamaguchi et al. (2011) performed SV calculations for TC-like vortices in a nondivergent barotropic model and investigated the influence of the vortex structure, the length of the OTI, and beta effects on the SV structure and singular values. They showed that on the *β* plane the first two leading SVs are always dominated by dipole structures, indicating a displacement of the vortex.

The aim of the present study is to investigate in a strongly idealized framework how perturbation growth is modified when TCs start to interact with the midlatitude flow. We use a nondivergent barotropic framework, in which the most important processes controlling the motion of TCs are present. From a barotropic perspective, the main influence of the midlatitudes is that they provide a background flow with horizontal shear, which causes the recurvature of TCs. To investigate perturbation growth in this situation, we compute energy norm SVs for TC-like vortices in zonal background shear flows.

In section 2, we describe the numerical methods used to compute SVs; in section 3, we introduce the vortices used in this study and discuss their SV structure (without background shear flow). The SV structure and growth mechanisms for stable vortices in horizontal background shear on an *f* plane are investigated in section 4 for different vortex profiles and shear strengths. In section 5, we consider unstable vortices, in section 6 we consider beta effects, and section 7 contains our summary and conclusions.

## 2. Numerical methods

For this study we use a nondivergent barotropic model. We employ a modified version of the explicit finite-difference code used in Scheck et al. (2011a,b) to solve these equations numerically on the *f* plane or the *β* plane. Instead of the third-order upwind scheme used in the original code, for which no adjoint exists, fourth-order centered differences are used in the current version to compute the advection terms. We employ second-order diffusion, unless noted otherwise with a constant viscosity parameter *ν* = 1000 m^{2} s^{−1}. In addition, a sixth-order diffusion term is used to suppress small-scale noise while influencing the larger-scale flow as little as possible.

**x**(

*t*). For a sufficiently small initial perturbation

*δ*

**x**

_{initial}and a sufficiently short OTI [

*t*

_{initial},

*t*

_{final}], the perturbation at the final time can be approximated by

*δ*

**x**

_{final}=

*δ*

**x**

_{initial}, where

*δ*

**x**that maximize

**s**

_{i}is the

*i*th initial singular vector and

*σ*

_{i}the corresponding singular value. During the OTI, the initial SV

**s**

_{i}evolves to

**s**

_{i}, the evolved SV. We sort the SVs by their singular values, so that

**s**

_{1}denotes the SV with the largest singular value.

The Tangent Linear and Applied Model Compiler (TAMC) package (Giering and Kaminski 1998) is used to generate the code representing ^{T} automatically from the Fortran-90 source code of the nondivergent barotropic model. The eigenvector problem Eq. (2) for the real, symmetric matrix ^{T}

## 3. Vortices on the *f* plane without shear

### a. Vortex structure

In this study we consider four vortices (see Table 1 and Fig. 1)—the GAUSS and SUD vortices used in Scheck et al. (2011a); a weaker, unstable version of the SUD vortex denoted as SUW; and vortex C from Yamaguchi et al. (2011)—in the following called YAMAC. The structure of the GAUSS and SUD vortices is very similar within a radius *r* of about 150 km. In both cases, the wind attains a maximum speed of 40 m s^{−1} at a radius of 100 km. In the SUD vortex the vorticity is negative for *r* > 180 km, which is associated with a stronger decay of the wind speed with radius, compared to the GAUSS vortex. Although the SUD vortex fulfills the necessary condition for barotropic instability, ∂*ζ*/∂*r* = 0, no unstable normal modes have been found for this vortex (Weber and Smith 1993). The vorticity distribution of the YAMAC vortex is much broader, the maximum wind speed of 22.5 m s^{−1} is reached at a radius of 210 km, and negative values of the vorticity are found for *r* > 280 km. However, in the YAMAC vortex the ring of negative vorticity around the positive core is much stronger and causes the circulation to become anticyclonic for *r* > 450 km (Fig. 1). The YAMAC vortex fulfills the instability condition ∂*ζ*/∂*r* = 0 at radius *R*_{ζ} = 400 km and is unstable. The fastest-growing normal mode has an azimuthal wavenumber-2 structure (Yamaguchi et al. 2011). The SUW vortex is described by the same tangential wind function as the SUD vortex [Eq. (11) in Scheck et al. (2011a)] but with parameters *a* = 0.5218, *b* = 0.028 86, *υ*_{0} = 41.24 m s^{−1}, and *r*_{0} = 100 km. It is also wavenumber-2 unstable and its maximum wind is similar to the one of the YAMAC vortex. However, the SUW vortex is more compact, with a radius of maximum winds of 100 km and *R*_{ζ} = 210 km, and the circulation is cyclonic at all radii.

Properties of the vortices used in this study. (left)–(right) Maximum velocity, radius of maximum winds, and radius at which the radial vorticity gradient changes sign.

The vortices considered in this study are larger than most observed TCs—an approach followed also by other idealized TC studies (e.g., Yamaguchi et al. 2011). Using relatively large vortices seems justified here, as one motivation for this work is to contribute to a better understanding of the SVs computed in operational NWP centers. The horizontal resolution of these SVs is often considerably lower than in operational forecasts (e.g., Leutbecher and Palmer 2008). Consequently, TCs are represented as larger and weaker vortices (Lang et al. 2012).

### b. SV structure and singular values

We performed SV calculations for the four vortices using different OTIs. The vortices are placed on a quadratic domain (typically 6000 km × 6000 km in size) with doubly periodic boundary conditions.^{1} The properties of the first three singular vectors are listed in Table 2. Here, *l*_{i} is the wavenumber dominating the *i*th evolved singular vector. To determine this quantity, we perform an azimuthal Fourier transform of the vorticity at the end of the OTI for a radius of 100 km; *l*_{i} is the index of the Fourier component with the largest amplitude.

Singular values *σ* and mode indices *l* for the three fastest-growing singular vectors for single-vortex *f*-plane model runs using optimization time intervals of different lengths *t*_{opt} and different vortex profiles.

Examples of the vorticity distribution of the initial and evolved leading SVs are displayed in Fig. 2. In all cases, the initial SV vorticity structure (Figs. 2a–c) is dominated by thin spiral bands of alternating sign tilted against the horizontal shear of the vortex circulation. These structures, which are also present in the calculations of Yamaguchi et al. (2011) and Nolan and Farrell (1999), allow for a transient gain in energy when they are “untilted” by the shear (Orr 1907).

In the inviscid case, the highest growth rates due to the Orr effect are to be expected for perturbations close to the radius of maximum winds (Yamaguchi et al. 2011). To avoid the untilted state being reached too early and the energy starting to decrease, very tightly wound, thin spiral bands would be required near the radius of maximum winds (RMW) for longer optimization time intervals. However, thinner structures are damped more effectively by viscosity. At larger radii, the shear is lower and thus broader structures can benefit from Orr growth and are less strongly affected by viscous damping. The optimal initial perturbations are therefore located outside the RMW and move farther out when the OTI is increased (Yamaguchi et al. 2011). This behavior is visible in the initial SVs for the GAUSS vortex for OTIs of 6 and 48 h (cf. Figs. 2a,b). Thinner structures and thus stronger damping result also from higher wavenumbers. For this reason, only wavenumbers 1 and 2 are found among the three fastest-growing SVs for the stable SUD and GAUSS vortices (see Table 2). The wavenumber-1 SVs benefit from longer OTIs, which allow for longer growth by the Orr mechanism and other processes (see section 4b). The wavenumber-2 SVs are affected more strongly by viscous damping and barely increase or even decrease for longer OTIs for the stable vortices. For all optimization times, the singular values of the SUD and the GAUSS vortices are quite similar, indicating that the differences in the vortex structures at larger radii (Fig. 1) are not decisive for the perturbation growth.

The unstable SUW and YAMAC vortices behave quite differently than the GAUSS and SUD vortices. The exponential normal mode growth is initially slower than growth by the Orr mechanism. Thus, for short OTIs the singular values do not benefit strongly from the barotropic instability. As the maximum velocities are lower than in the SUD and GAUSS vortices, and for the YAMAC vortex, in addition, the RMW is larger, the growth rates for optimization time (*t*_{opt}) intervals up to 12 h are therefore lower than for the stable vortices [as in Yamaguchi et al. (2011)]. For longer time intervals, the normal-mode growth becomes dominant and for *t*_{opt} = 48 h, the singular values for the leading SVs are about 3 (SUW) to 8 (YAMAC) times higher than for the stable vortices^{2} (Table 2). The influence of the instability is also visible in the radial vorticity structure of SVs for the YAMAC (Figs. 2c,f) and SUW vortices. The initial SV vorticity is located close to the radius *R*_{ζ} where ∂*ζ*/∂*r* = 0. In the evolved SV, a phase shift is visible at this radius. The phasing between the perturbations inside and outside of *R*_{ζ} is such that the corresponding circulations lead to a mutual amplification of the perturbations. The initial SVs are characterized by spiral structure, as for the stable vortices. This indicates that perturbation growth occurs initially via the Orr mechanism and is subsequently dominated by the unstable normal mode (Yamaguchi et al. 2011; Nolan and Farrell 1999).

### c. Impact of viscosity

The singular vectors discussed so far are obtained for *ν* = 1000 m^{2} s^{−1} and a grid resolution of 18 km. The viscosity term is not just required for numerical stability but can be regarded as a strongly idealized representation of the dissipative mechanisms like turbulence. For lower viscosities, the vorticity structures in the initial singular vector wind more tightly around the radius of maximum winds and are thinner (cf. Figs. 3a and 3b). For higher viscosities, the structures in the initial SV move outward to higher radii, which results (for fixed OTI) in broader vorticity bands that are less affected by viscous dampening (Yamaguchi et al. 2011). For lower viscosities, the singular values increase considerably and higher grid resolutions are required to resolve the smaller-scale structures adequately. The singular value of the leading singular vector for the SUD vortex and an OTI of 1 day increases from 9 for *ν* = 10^{4} m^{2} s^{−1} to 22 for *ν* = 10^{2} m^{2} s^{−1} (Fig. 3c). While a grid resolution of 18 km is sufficient to reproduce the singular value from high-resolution runs for *ν* = 10^{4} m^{2} s^{−1}, the values for runs with resolutions of 2.25 and 4.5 km still differ by about 15% for *ν* = 10^{2} m^{2} s^{−1}.

In the following, we use *ν* = 10^{3} m^{2} s^{−1}, which leads to an acceptable damping of the vortices, and a grid resolution of 18 km. The test cases displayed in Fig. 3 indicate that the results should be sufficiently close to the high-resolution results. Increasing the resolution to 9 km would increase the computational costs significantly without yielding qualitatively different results, as we have tested for several cases.

## 4. Stable vortices in shear on the *f* plane

*f*

### a. Influence of shear on the flow structure

A background shear flow defined by the zonal velocity *u* = *λ*_{y} with a shear strength *λ* of the order of 10^{−5} s^{−1} modifies the flow around the vortex core significantly. To investigate the impact of these changes on the SVs, we repeat several model runs of the previous section, but with a constant *λ* subtracted from the initial vorticity distribution that causes a linear shear background. We assume now an east–west-oriented channel configuration.

Close to the center of the vortex the background shear has no significant influence. However, at a radius *R*_{λ} [defined by |*λR*_{λ}| = |*υ*_{T}(*R*_{λ})|, where *υ*_{T} is the tangential velocity profile of the vortex], background shear and vortex circulation are of comparable strength. For large radii, the influence of the vortex is negligible and the velocity approaches the solution *u* = *λ*_{y}. The character of the flow at intermediate radii depends on the sign of *λ*. For *λ* > 0 there are two stagnation points at (0, ±*R*_{λ}) where the vortex circulation and the background shear flow cancel each other exactly. These stagnation points are located on the crossing points of separatrices—curves separating areas with closed streamlines and open streamlines (shown for the GAUSS vortex in Fig. 4a). When *λ* is increased, the stagnation points move closer to the vortex center and the area with closed streamlines around the vortex becomes smaller. For *λ* < 0, there are no stagnation points, as the signs of the zonal velocity of the vortex circulation and the background flow are always the same. The streamlines that are nearly circular for *r* ≪ *R*_{λ} become elongated in the zonal direction for larger radii and finally approach parallel lines for *r* ≫ *R*_{λ} (Fig. 4b).

### b. SV structure

In the following, we will discuss the SV structure and the underlying perturbation growth mechanisms for GAUSS vortices in anticyclonic and cyclonic shear with *λ* = ±2 × 10^{−5} s^{−1} and compare them with a case with *λ* = 0.

The vorticity distributions of the leading initial SVs for *λ* = ±2 × 10^{−5} s^{−1} are shown in Figs. 5a and 5b. In both cases sensitive regions extend far from the vortex and are nearly aligned with streamlines. For cyclonic shear, the highest vorticity values are found in a spiral structure at a radius of about 400 km. Two branches of the vorticity distribution extend to larger radii west and east of the vortex. In the anticyclonic shear case the vorticity distribution consists of two pairs of thin bands with positive and negative vorticity that are aligned with the streamlines leading to the northern and the southern stagnation points. The vorticity is restricted to those separatrix branches on which the flow is directed toward the vortex center. On the northeastern and southwestern branches perturbations would be advected away from the vortex, so these branches are not preferred locations for perturbations. The structure of the second initial SVs (not shown) is very similar to the first ones. Only differences in the finescale structure ensure that the first and second initial SVs are orthogonal.

As for the *l* = 1 cases without shear, the vorticity distribution of the first and second evolved SVs (Figs. 5c,d) shows a dipole structure with the maximum amplitude located approximately at the radius of maximum wind, where the steepest vorticity gradient of the vortex profile is found. The main effect of the leading SVs as optimal perturbations is thus to displace the vortex. In the anticyclonic case, the vortex is displaced approximately in the northeast–southwest direction (Fig. 5d) and for the cyclonic case in the northwest–southeast direction (Fig. 5c). The displacement direction of the second SV is orthogonal to that of the first SV in all cases.

Anticyclonic shear causes the singular value of the leading SV to increase considerably from 21 to 47 (Table 3). In the cyclonic shear case, the singular value is increased also, but only to a value of 36. The second SVs do not benefit from shear at all; their singular values are nearly the same as in the case without shear.^{3} These singular values imply that for an anticyclonic background shear flow much more energy is required to displace a cyclone in a northwest or southeast direction than in a northeast or southwest direction. For cyclonic shear, the opposite is true and the increase in the growth rate caused by the shear is somewhat weaker. To interpret these results, we investigate in the following the underlying perturbation growth mechanisms.

Singular values *σ*_{i} and displacement direction *α*_{i} for the two fastest-growing singular vectors for GAUSS-vortex *f*-plane model runs with different values of the background shear *λ*.

### c. Perturbation growth mechanisms

*ζ*′ is governed by the linearized vorticity equation,

The linear evolution of the leading SV for the cases with anticyclonic and cyclonic shear and the corresponding case without shear displayed in Fig. 6 shows that one can distinguish between two different and quite clearly separated vorticity components inside and outside a separation radius *R*_{s} (solid circle). At all times, a separation radius between about 200 and 300 km can be found at which the vorticity perturbation is negligible.^{4} The outer perturbation component (*r* > *R*_{s}) evolves from the initial SV vorticity distribution. As *r* > *R*_{s}, perturbation energy growth can only be achieved by the redistribution of perturbation vorticity. The inner component (*r* < *R*_{s}) is characterized by a dipole mode that is not present at the initial time but grows rapidly and starts to dominate after a few hours. As there is no advection of perturbation vorticity across *R*_{s}, the dipole mode in the vortex inner core must develop because of the last term of Eq. (3). The growing dipole mode can be interpreted as an increasing displacement of the vortex by a steering flow. The growth of the dominating dipole part of the perturbation can thus be understood by an analysis of the steering flow.

**v**

_{pert}and a term that accounts for the steering by the background shear flow

**v**

_{shear}. An approximation for the vortex displacement is thus given by

**v**

_{shear}has only a zonal component and is proportional to the meridional displacement of the vortex and the shear strength.

For both the cyclonic and the anticyclonic shear cases the contribution of **v**_{shear} to the zonal vortex displacement is larger than the contribution by the outer perturbation (cf. dashed and dotted lines in Figs. 7a and 7c). The vortex displacement by the background shear is thus an important factor that contributes to the larger singular value of the leading SV. The time evolution of the meridional displacement is very similar for the cases with *λ* = 0 and *λ* = 2 (Figs. 7d,e) and the time evolution of the zonal displacement for *λ* = 0 (Fig. 7b, solid line) is similar to the displacement by the outer perturbation for *λ* = −2 (Fig. 7a, dotted line). This indicates that for cyclonic background shear, the displacement by **v**_{shear} alone is responsible for the increased singular value.

For anticyclonic shear, a second effect—the more efficient displacement of the vortex by the outer perturbation—is responsible for a further increase in the singular value. The displacement by the outer perturbation (Figs. 7c,f) starts earlier than in the case without background shear (Figs. 7b,e) and the meridional displacement distance, *λ* = 0 reaches a maximum after *t* ≈ 40 h and decreases subsequently (Fig. 7e). These two differences are related to the changes in the flow structure due to the anticyclonic background shear that affect the growth of the outer perturbation.

As in the case without background shear, the initial vorticity distribution of the outer perturbation consists of thin vorticity bands tilted against the shear (Fig. 6c) and the perturbation grows by untilting (Orr 1907) and the separation of vorticity structures of different sign (“unshielding”). For anticyclonic shear, these processes are facilitated by the diverging streamlines in the vicinities of the two stagnation points. Here, positive and negative vorticity perturbations are separated efficiently (Fig. 6, right column). After 12 h, the vorticity of the outer perturbation has accumulated in four relative compact features near the stagnation points. The circulation associated with this distribution is already sufficiently strong to cause a discernible displacement of the vortex. In contrast, the unshielding in the other two cases can only be accomplished by a much slower process: the unwinding of the spiral structures by the shear of the vortex circulation (Fig. 6, left and middle columns). The positive and negative vorticity spirals are sufficiently unwound such that a significant vortex displacement can start only after 24 h (Fig. 7, left and middle columns).

The displacement of the vortex in anticyclonic shear is facilitated also owing to the fact that the direction of **v**_{pert} changes slower than in the other two cases. For *λ* = −2 × 10^{−5} and *λ* = 0 s^{−1}, the direction of the flow across the vortex caused by the outer perturbations changes by about 180° in the last 24 h (Fig. 6, left and middle columns). For this reason, the meridional displacement is reduced in the last 10 h (Figs. 7d,e) and the growth of the total displacement is slowed down. In the anticyclonic shear case, the advection of the outer perturbation proceeds more slowly because most of the time the bulk of the perturbation vorticity is close to one of the stagnation points, where the angular velocity is rather low. Consequently, the displacement direction changes only by about 90° in the last 24 h (Fig. 6, right column), and zonal and meridional displacement components increase continuously up to the end of the OTI (Figs. 7c,f). The flow structure with the two stagnation points thus allows not only for an earlier start, but also for a longer duration of the displacement process.

The decomposition of the vortex displacement into contributions from the outer perturbation and the background shear [Eq. (4)] can be used to understand why there is a preferred direction of displacement. For the leading SV in the anticyclonic shear case discussed above, the meridional displacement is caused only by **v**_{pert} and the zonal displacement almost exclusively by **v**_{shear}. This is the combination that leads to optimal perturbation energy growth, and thus also to the largest displacement.^{5} As *λ* > 0, the vortex is displaced into the northeast or southwest quadrant. By design, the second SV must cause a displacement in the orthogonal direction. A displacement to the northwest requires that **v**_{pert} has a northward component and thus leads to an eastward directed contribution from **v**_{shear}. The latter must be compensated by a westward component of **v**_{pert} to ensure that in total a northwestward displacement results. To achieve the same displacement distance |*δx*_{c}|, the contribution from **v**_{pert} and thus also the norm of the initial perturbation must be larger for the second SV than for the first SV. For this reason, the second SV does not benefit from background shear and has a considerably lower singular values than the leading SV.

### d. Influence of vortex structure, shear strength, and optimization interval

Singular values and wavenumbers for the GAUSS and SUD vortices with different shear strengths *λ* are given in appendix B (see Tables B1 and B2 and also Fig. 8). Compared to the calculations without shear (Table 2), the dominant wavenumbers *l* are not changed, but the singular values of the leading SVs are significantly increased for sufficiently strong shear and long enough OTIs. In the most extreme case (*t*_{opt} = 2 days, *λ* = 4 × 10^{−5} s^{−1}), the singular values are increased by factors up to 7 (Fig. 8). The increase in the singular values of the leading SV is a factor of 2 higher for anticyclonic shear (*λ* > 0) than for cyclonic shear (*λ* < 0). The singular values of the second SVs, which have a wavenumber *l* = 1 structure like the leading SV, do not benefit from shear and are actually slightly smaller than the corresponding values without shear. These results are in agreement with the analysis in section 4b. The singular values of the third SVs that are dominated by wavenumber-2 structures are enhanced by up to 80% for higher shear strength with *t*_{opt}, in contrast to the behavior without shear (see Table 2).

For short OTIs, the vorticity structure of the initial SVs is very similar to the corresponding cases without shear and the singular values are only slightly modified. This is to be expected, as in these cases the vorticity is located close to the radius of maximum winds where the circulation is not strongly changed by the background shear (see Fig. 4). Without shear, the vorticity moves outward to larger radii for longer OTIs to avoid extremely thin structures and the associated strong viscous damping (see Fig. 2). With shear, moving to larger radii means that a part or most of the vorticity is located outside of the region where the streamlines are nearly circular. For longer OTIs and stronger shear, the vorticity extends therefore outward to larger radii along the ingoing separatrices. For *t*_{opt} = 2 days and *λ* = ±4 × 10^{−5} s^{−1}, significant vorticity can be found at a distance of more than 2000 km from the vortex core (Fig. 9b).

Sufficiently strong background shear flows can cause changes in the vortex structure during the OTI. These changes affect mainly the outer parts of the vorticity distribution. The GAUSS vortex considered in the previous section is a special case in this respect. For the shear values *λ* considered in this study, the area with closed, nearly circular streamlines around the center of a GAUSS vortex (see Fig. 4) extends out to radii where the vortex vorticity is negligible. Consequently, the vorticity structure remains nearly unchanged for several days. For vortices with weaker circulations (and thus reduced *R*_{λ}) or more extended vorticity profiles, the initial vorticity distribution of the vortex may extend beyond the region with closed streamlines around the vortex center. The advection of fluid along open streamlines causes a time-dependent deformation of the vortex and leads to the formation of filaments.

For the SUD vortex, only fluid with negative relative vorticity is sheared from the vortex core in this process for the shear strengths considered here. For anticyclonic shear (*λ* > 0), vorticity on streamlines outside of the region confined by the two branches of the separatrices connecting the stagnation points is removed from the vortex core. The fluid with negative relative vorticity sheared from the vortex core in this process forms filaments moving away from the vortex along the outgoing separatrix branches in the northeast and the southwest (as shown in Fig. 9a for *λ* = 4 × 10^{−5} s^{−1}). Subsequently, the fluid with negative vorticity axisymmetrizes and forms two anticyclones [see Fig. 9 in Scheck et al. (2011a) for an example]. Cyclonic shear (*λ* < 0) has a weaker influence on the vortex structure, in the sense that the area of the region with closed streamlines is larger for the same value of |*λ*|. Consequently, less fluid with negative vorticity is removed from the vortex.

The shear-induced changes in the outer part of the vorticity profile of the SUD vortex could in principle affect perturbation growth. However, a comparison of tables in appendix B (Tables B1 and B2) shows that the singular values for the GAUSS and the SUD vortices are quite similar, like in the calculations without shear (Table 2). The singular values of the first and second SV are only 10%–20% higher for the SUD vortex than for the GAUSS vortex, for all values of *λ* considered here. Thus, the structural changes caused by shear have no strong impact on the perturbation growth in the stable vortices considered here.

## 5. Unstable vortices in shear on the *f* plane

*f*

### a. Evolution of the vortex structure

For the unstable YAMAC and SUW vortices, changes in the vortex structure during the OTI are much more important than for the stable SUD vortex, as they affect the ring of negative vorticity around the positive core that is essential for barotropic instability. For sufficiently strong shear, the vorticity distribution of the vortex extends over the region of closed streamlines and thus a part of the vorticity is removed from the rest of the vortex. For the YAMAC vortex, anticyclonic (Fig. 10a) or cyclonic shear (Fig. 10b) with |*λ*| = 2 × 10^{−5} s^{−1} is sufficient to remove all of the negative vorticity from the vortex, and for *λ* > 3 × 10^{−5} s^{−1} the vortex is completely destroyed within 2 days. The more compact SUW vortex is more resistant to strong shear. Even for *λ* = ±4 × 10^{−5} s^{−1}, most of the positive relative vorticity remains inside the closed streamlines and only the negative parts are removed.

For weaker shear, the closed streamlines around the vortex core extend out to radii where the vorticity in the vortex profile is very small. The removal of vorticity is thus negligible, but a different process becomes important. For cyclonic and anticyclonic shear, the closed streamlines around the vortex core are stretched and compressed in zonal direction, respectively (see Fig. 4). This quadrupole deformation leads to a wavenumber-2 vorticity perturbation, which can grow quickly owing to the unstable normal mode of the vortex. Singular values larger than 100, as found for the YAMAC vortex with *t*_{opt} = 2 days and *λ* = 0, mean that a perturbation with an initial amplitude of only 1% will become nonlinear within the OTI. Indeed, in our calculations using the YAMAC vortex, the shear-induced perturbation grows owing to a normal-mode-like instability and causes during the OTI of 2 days strong changes in the vortex structure for very small values of *λ*. In the nonlinear phase of the perturbation growth, a tripolar structure emerges, with a vorticity maximum located between two minima, as visible in Fig. 10c for a case with *λ* = 0.1. In the subsequent evolution of this case, the positive core becomes strongly deformed and elongated and finally breaks up into two parts.

For the SUW vortex, the normal modes grows slower (as indicated by the lower singular values in Table 1), but the growth is still fast enough so that also in this case the nonlinear phase of the perturbation growth is reached for very weak shear within 2 days. In contrast to the YAMAC vortex, the SUW vortex is not destroyed in the nonlinear phase but reaches a nearly stable state very similar to tripolar vortices observed in laboratory experiments (e.g., van Heijst and Kloosterziel 1989; Kloosterziel and van Heijst 1991). Similar flow structures have been observed in real TCs by Kuo et al. (1999) and Reasor et al. (2000). Between *t* = 48 (Fig. 10d) and 96 h, the structure changes barely. The ring of negative vorticity is still present in the tripolar state, but the lowest vorticity values are found only in two pronounced minima within the ring. The evolving shear-induced perturbation *δζ*(**x**, *t*) = *ζ*(**x**, *t*) − *ζ*(**x**, 0) of the vortex shows the same characteristic features as evolving SVs for unstable vortices without a sheared background flow (see, e.g., Fig. 2f). There is a wavenumber-2 perturbation with maximum amplitude just inside *R*_{ζ}, there is a phase jump at *R*_{ζ}, and there is a second amplitude maximum outside of *R*_{ζ} (Fig. 11a). In the corotating frame, the structure of the wavenumber-2 component of *δζ* is approximately constant and only the amplitude grows. These properties indicate that also in weak shear flows a normal-mode-like instability exists. The additional wavenumber-0 component inside *R*_{max} visible in Fig. 11a results from the damping by viscosity and is not related to perturbation growth. Apart from this difference, *δζ* resembles strongly the leading evolved SV for this case (Fig. 11b).

When the viscosity is increased by a factor of 10, compared to our standard value of *ν* = 1000 m^{2} s^{−1}, the growth rates for the barotropic instability are reduced so strongly that the nonlinear phase of perturbation growth is not reached within 48 h for both the SUW and the YAMAC vortex. In these cases, the vortex structure is barely affected by weak shear. In contrast, the impact of stronger shear, which leads to a removal of fluid with negative vorticity from the vortex, is not strongly affected by changes in the viscosity.

### b. Singular values and SV structure

The changes in the vortex structure discussed above have a strong impact on the growth rate of the normal mode. For short OTIs, the perturbation growth is not influenced strongly by the normal mode (Yamaguchi et al. 2011) and thus the singular values and the SV structure are very similar to the corresponding cases without shear. However, for sufficiently long OTIs, the SVs change considerably with shear. In the following, we will focus on cases with *t*_{opt} = 2 days.

Weak cyclonic or anticyclonic shear is sufficient to cause a strong decrease of the singular value *σ*_{1} for unstable vortices (Fig. 12). A shear strength *λ* = 0.1 × 10^{−5} s^{−1} causes a reduction of *σ*_{1} by about 50% for the YAMAC vortex. For the SUW vortex, which is able to retain a ring of negative vorticity for *λ* = 0.1 × 10^{−5} s^{−1} (Fig. 10d), the reduction is only 17% and the structure of the SV (Fig. 11b) for this case is still very similar to unstable cases without shear (see, e.g., Fig. 2f). However, when the OTI is chosen as 2–4 days instead of 0–2 days, the reduction of *σ*_{1} is much stronger—about 77%. This result indicates that moderate asymmetries in the vorticity distribution can have a strong impact on the growth rate of the barotropic instability. As discussed above, the nonlinear phase of the barotropic instability can be avoided by increasing the viscosity. In this case, the singular values for unstable vortices without shear background and with a weak shear background are nearly identical.

For stronger shear, the positive vorticity gradient is destroyed by shear before the initial wavenumber-2 perturbation has become nonlinear. In this case, the perturbation growth proceeds owing to the mechanisms discussed in section 4b for stable vortices. The transition from barotropic instability to shear-enhanced barotropic growth is visible in the dominant mode number *l*_{1} indicated by symbols in Fig. 12. For low values of *λ*, the *l* = 2 mode dominates, whereas for stronger shear, the optimal perturbation is of *l* = 1. This transition is also visible in the vorticity distribution of the initial SVs, which are dominated by structures around *R*_{ζ} for low shear values. These structures align with streamlines when the shear is increased.

For |*λ*| ≥ 10^{−5} s^{−1} the singular values of the leading SV for the unstable vortices increase with |*λ*|, like for the stable vortices (Fig. 12). An exception to this rule is the marked maximum of *σ*_{1} for *t*_{opt} = 2 days and *λ* = 2 × 10^{−5} s^{−1} for the YAMAC vortex. The huge perturbation growth rate is related to the fact that in this case the vortex is almost completely destroyed by shear. The very compact vortex that is left allows for extreme growth rates, as the stagnation radius is very small and perturbations are able to come close to the vortex core.

## 6. Stable vortices in shear on the *β* plane

*β*

Using an additional set of model runs and SV calculations, we investigate the influence of *β* effects on the perturbation growth. The absolute vorticity associated with the background flow in these calculations is set to *λ* gives rise to a zonal linear shear background flow as in the *f*-plane cases and *β* is set to a value corresponding to a latitude of about 30°.

The advection of absolute vorticity by the vortex circulation in combination with the meridional gradient of absolute vorticity leads to the formation of beta gyres, which cause a northwestward propagation of the vortex (e.g., Fiorino and Elsberry 1989 and references therein). The background shear flow contributes to the motion of the vortex also. For sufficiently strong anticyclonic shear, the vortex is therefore displaced in northeastward direction. Furthermore, owing to the vortex motion, fluid with negative vorticity sheared from the outer layers of the vortex is not distributed symmetrically around the vortex, as in the *f*-plane cases but forms an asymmetric distribution. The circulation associated with this distribution causes an additional displacement of the vortex. The vorticity distribution at *t* = 3 days for two cases; SUD vortices in background flows with *λ* = 0 and *λ* = 4 × 10^{−5} s^{−1} are shown in Figs. 13a and 13d.

At this point, we refrain from a detailed analysis of the vortex motion for varying shear and different vortex profiles but focus rather on the influence of the forming beta gyres on the perturbation growth. The flow structure changes rapidly during the formation of the beta gyres and the associated acceleration of the vortex but evolves only slowly after a few days. Therefore, we consider two OTIs: 0–2 and 3–5 days. In the first case, the gyres form during the OTI and in the second one, the gyres are already present and do not change much during the OTI.

The formation of the beta gyres causes some changes in the storm-relative wind field. Without shear, the main effect is that the wind field becomes asymmetric, with stronger velocity gradients to the east than to the west of the vortex center (as visible from the density of streamlines in Fig. 13a). With shear, the formation of beta gyres results in stagnation points that do not lie on the same streamline any more (see, e.g., Figs. 13d). Closed streamlines can develop around the negative vorticity anomaly forming east of the vortex. However, these differences seem not to hinder the perturbation growth, as indicated by a comparison of the singular value *σ*_{1} for different shear strengths on the *f* and the *β* planes for the SUD vortex (Fig. 14). For both OTIs, the singular values on the *β* plane are very similar to the *f*-plane results, indicating that the same growth mechanisms are at work.

In contrast to the singular values, the structure of the SVs is strongly affected by beta effects and is different for the two OTIs considered here. Without shear, the sensitive regions are not distributed symmetrically but are located at a similar radius as on the *f* plane (cf. Fig. 2b). For an OTI of 0–2 days, high sensitivity is limited to the southern half of the vortex (Fig. 13b), in agreement with the results of Yamaguchi et al. (2011) (see their Fig. 9). For an OTI of 3–5 days, the sensitivity rotates to the eastern part (Fig. 13c). A possible interpretation of this behavior is that the stronger radial gradient of the tangential velocity developing on the eastern side allows for a faster growth via unshielding and untilting in the first phase of perturbation growth. Because of the preexisting beta gyres in the 3–5-day interval, the initial perturbations on the eastern side immediately benefit from these enhanced growth conditions. For the 0–2-day interval, the optimal perturbations must initially be located farther upstream in order to experience later the stronger velocity gradient in the eastern part of the vortex that is not yet present at *t* = 0. It takes about 9 h to advect the initial perturbation from the southern to the eastern part, so that they still can benefit from enhanced growth in the first half of the OTI.

As an example for strong shear, the initial SV for *λ* = 4 × 10^{−5} s^{−1} and an OTI of 0–2 days (Fig. 13e) is very similar to the corresponding *f*-plane case (Fig. 9b). This is probably related to the fact that the beta gyres are still weak in the first half of the OTI, so that the locations of optimal perturbations are barely modified. For an OTI of 3–5 days (Fig. 13f), the flow structure and the sensitivity distribution are strongly modified. However, despite the much more complicated wind field, the location of the sensitive regions can still be described as the separatrix branches leading to the stagnation points, emphasizing the importance of these flow boundaries.

Like on the *f* plane, all evolved SVs computed on the *β* plane are dominated by dipole structures indicating a displacement of the vortex (see Table B5 in appendix B). Cyclonic shear leads to a northwest or southeast displacement and anticyclonic shear causes northeast or southwest displacement, like in the corresponding *f*-plane cases.

## 7. Summary and conclusions

In this study, we investigated the properties of singular vectors (SVs) for stable and unstable hurricane-like vortices in horizontal background shear flows on the *f* and the *β* planes using a nondivergent barotropic model. This setup can be regarded as a strongly idealized version of the interaction of a tropical cyclone with the midlatitude flow. The strongest background shear considered here corresponds to the flow in the vicinity of a jet stream.

For both *f*- and *β*-plane calculations, it was found that the singular values of the leading SV for stable vortices increase considerably with increasing shear strength and that sensitive regions extend farther out for stronger shear and longer optimization time intervals. For sufficiently strong anticyclonic shear, the initial SV vorticity structures are aligned with separatrices (i.e., streamlines crossing at stagnation points north and south of the vortex). The two leading evolved SVs show a dipole structure in the vortex core that can be interpreted as a displacement of the vortex. For the leading SV, background shear contributes to the displacement, thereby increasing the singular value. The flow structure for anticyclonic shear further facilitates the growth of the initial perturbations outside of the RMW. The shear-enhanced singular values for stable vortices can become comparable to the singular values for an unstable vortex without a sheared background. The exponential growth of normal modes in unstable vortices is strongly inhibited by weak shear. For stronger shear, SV perturbations in unstable vortices grow by the same mechanisms as in stable vortices.

The basic properties of the barotropic SVs are in agreement with full-physics 3D SVs. The singular values increase with shear and the sensitive regions extend farther away from the vortex center. High sensitivity is found in regions with flow toward the vortex (the ingoing separatrices) or in regions with low radial velocity (the separatrix sections between the stagnation points). These agreements suggest that similar processes as in the barotropic case are important in full-physics 3D SVs. This would imply that a significant part of the error growth is based on balanced dry dynamics and not on stochastic, less predictable moist convection.

The important role that the separatrices play in the interpretation of our barotropic calculations suggests that investigating flow boundaries could also be helpful for interpreting the perturbation growth mechanisms and SV structure in full-physics 3D models. Methods to detect flow boundaries in three-dimensional time-dependent flows have recently been applied to TCs (Sapsis and Haller 2009; Rutherford et al. 2012). If only an estimate for the location of sensitive regions is required, it may be sufficient to detect these flow boundaries, which should be computationally much cheaper than computing SVs and does not require an adjoint model.

Motivated by the finding that the sensitivity is high in a region where the vorticity gradient changes sign (Peng and Reynolds 2006; Reynolds et al. 2009), it has been proposed that barotropic instability may play an important role (Peng et al. 2009). However, our results indicate that growth involving barotropic instability may be strongly inhibited by background shear. For TCs interacting with the midlatitude flow, the shear-enhanced barotropic growth mechanism discussed in this study seems to be a promising alternative explanation. The annulus of high sensitivity with a radius of about 500 km found in several studies could also result from the shear-enhanced barotropic growth, as typical stagnation point radii are in the right range. Moreover, the evolved SVs are of wavenumber 1, as often observed for leading full-physics SVs. The identification of the most important error growth process should give important hints on how to improve forecasts for TCs in shear flows. If the barotropic instability dominates error growth in this phase, it should be helpful to increase the grid resolution in the near-core region. If the shear-enhanced barotropic growth dominates, it may be more important to resolve remote sensitive regions better—for example, by increasing the size of the high-resolution grid nest around the TC.

## Acknowledgments

This study was carried out as part of the DFG Priority Program SPP 1276 MetStröm: Multiple Scales in Fluid Mechanics and Meteorology. We acknowledge the support of Chandramowli Subramanian of the Institute of Applied and Numerical Mathematics, Karlsruhe Institute of Technology, for assistance in implementing the Davidson method. We thank Simon Lang, Carolyn Reynolds, James Doyle, and Munehiko Yamaguchi for helpful discussions. The contributions of three anonymous reviewers are gratefully acknowledged.

## APPENDIX A

### Steering Flow Decomposition

To investigate the evolution of the inner (*r* < *R*_{s}) dipole perturbation component that corresponds to a displacement of the vortex, we calculate the perturbation steering flow advecting the vortex. For this purpose, we consider steering flows for a solution starting from the unperturbed initial conditions and a solution starting from the unperturbed initial conditions plus a perturbation (in the following, indicated with indices *u* and *p*, respectively). In the unperturbed case, the absolute vorticity *ζ*_{c}. We denote the vortex location, defined as the position of the maximum of the absolute vorticity, as *η*^{p}(**x**, *t*) = *η*^{u}(**x**, *t*) + *δζ*(**x**, *t*), where the perturbation *δζ*(**x**, *t*) leads in general to a different vortex location

*α*∈ {u, p}, the translation velocity of the vortex

*α*∈

*u*,

*p*. The velocity of the environmental flow is

*R*

_{s}from the vortex center yields the approximate vortex translation velocity

*X*defined as

**v**

_{pert}that depends directly on the circulation associated with the outer perturbation and a residual term

**v**

_{shear},

*R*

_{s}, and that

*ζ*

_{c}(

**x**) ≈ 0 for

**v**

_{shear}depends on changes of the environmental flow around the vortex center caused by the displacement of the vortex but does not depend directly on the flow caused by the outer perturbation. For a zonal flow with linear shear in meridional direction,

**v**

_{shear}(

*t*) =

*λ*(

*δ*

**x**

_{c}·

**j**)

**i**, where

**i**and

**j**are the unit vectors in zonal and meridional direction, respectively. Thus, the indirect displacement speed is proportional to the shear strength

*λ*and the meridional displacement caused by the perturbation. The total displacement during the OTI due to the indirect term is therefore

In Fig. 7 the time evolution of the zonal and meridional vortex displacement is compared to the time-integrated sum **v**_{pert} + **v**_{shear} [where the approximation in Eq. (A7) is used for **v**_{shear}] for the three cases of Fig. 6. The good agreement of the actual displacement (thick solid lines) with the estimated displacement (thin solid lines) indicates that the approximations made above are justified.

## APPENDIX B

### Singular Values and Dominant Modes for Vortices in Shear

Tables B1–B5 summarize the singular values and dominant modes for the various vortices in the *f* plane or *β* plane model runs.

Singular values *σ* and mode indices *l* for the three fastest-growing singular vectors for single GAUSS-vortex *f*-plane model runs with different values of the background shear *λ* and different OTIs *t*_{opt}.

Singular values and dominating wavenumber for the SUD vortex on the *β* plane for different shear strengths *λ* and different OTIs.

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^{1}

These conditions require that the integrated vorticity is zero. This is the case for SUD, SUW, and YAMAC vortices and can also be ensured for the GAUSS vortex by adding a small negative constant to the initial vorticity distribution.

^{2}

The singular values for the YAMAC vortex differ from the ones in Yamaguchi et al. (2011) because their viscosity is higher, *ν* = 7.5 × 10^{4} m^{2} s^{−1}, and they assume that the vortex is not damped during the OTI.

^{3}

The singular values *σ*_{1} and *σ*_{2} for *λ* = 0 differ slightly because of the different zonal and meridional boundary conditions. For doubly periodic boundary conditions, they are identical.

^{4}

The term *R*_{s} could be defined as the smallest positive radius where the function

^{5}

For perturbations resulting from a vortex displacement, the perturbation energy scales with *δυ*^{2} ≈ (*δx***∇***υ*)^{2} and increases by a factor *σ*^{2} during the OTI. Therefore, *σ* ∝ |*δx*|.