Abstract

In this study, singular vectors (SVs) are calculated for tropical cyclone (TC)–like vortices on an f plane and β plane using a barotropic model, and the structure and time evolution of the SVs are investigated. In the f-plane study, SVs are calculated for TC-like vortices that do and do not satisfy a necessary condition of barotropic instability of normal modes, in which the vorticity gradient changes sign. It is found that, in the case where the initial vortices do not meet the condition, 1) the SVs are tilted against the shear of the background angular velocity as found earlier by Nolan and Farrell, indicating the growth of SVs through the Orr mechanism; 2) the leading singular value increases with the maximum tangential wind speed Vmax and decreases with the radius of the maximum wind (RMW); and 3) the locations of SVs move outward with increasing RMW, Vmax, and the optimization time. In the case where the initial vortex allows for barotropic instability, the SV is initially tilted against the background shear and exhibits transient growth for a limited period. At a certain time during the initial growth, the SV “locks in” to a normal mode structure and remains in that structure so that it may grow exponentially with time.

In contrast to the SVs on an f plane, the azimuthal distribution of the SVs on a β plane becomes more asymmetric, and the extent of the asymmetry increases as the strength of the beta gyres increases. On the β plane, all first and second SVs calculated in this study have an azimuthal wavenumber-1 structure at the optimization time, regardless of whether the vorticity gradient of initial TC-like vortices changes sign and the TC-like vortices include the beta gyres at initial time. It is found that when the first and second SVs are used as ensemble initial perturbations, the linear combination of the initial first and second SVs can shift the vortex toward any direction at the optimization time. This is true even when SVs with a low horizontal resolution are used as initial perturbations, as in the European Centre for Medium-Range Weather Forecasts (ECMWF) and Japan Meteorological Agency (JMA) ensemble prediction system. Such wavenumber-1 perturbations could be useful for generating sufficient spread among the tropical cyclone tracks in ensemble forecasts.

1. Introduction

Singular vectors (SVs)—that is, fast-growing perturbations over a finite time interval (optimization time)—are widely used in the tropical cyclone (TC) forecasting and research communities. The most well-known use of SVs is as a generator of initial perturbations in an ensemble prediction system (EPS). For example, the European Centre for Medium-Range Weather Forecasts (ECMWF) adopts TC-targeted diabatic (moist) SVs to create ensemble initial perturbations around TCs (Barkmeijer et al. 2001; Puri et al. 2001; Leutbecher and Palmer 2008). The typhoon EPS at the Japan Meteorological Agency (JMA) also computes moist SVs, which are targeted for TCs in the western North Pacific basin only (Yamaguchi et al. 2009b). Another well-known use of SVs is sensitivity analysis guidance for adaptive observations (Palmer et al. 1998; Buizza and Montani 1999). Since analysis errors in initial conditions for numerical weather prediction (NWP) may project onto growing perturbations such as SVs, analysis errors that evolve into large forecast errors are expected to decrease by observing the areas of the growing perturbations and assimilating the observations. As a result, better forecast performance can be expected compared with forecasts that do not include the additional targeted observations. Yamaguchi et al. (2009a) conducted an observing system experiment (OSE) to verify the usefulness of JMA SVs for targeting observations of a TC. The OSE was performed for Typhoon Conson (2004) using dropwindsonde observations from the Dropwindsonde Observation for Typhoon Surveillance near the Taiwan Region (DOTSTAR; Wu et al. 2005). DOTSTAR observations were found to have a positive impact on the track forecast of Conson, and the observations within the sensitive region as identified by the SVs were sufficient to predict the northeastward movement of Conson. Aberson et al. (2011) conducted OSEs for TC track forecasts using three types of sensitivity analysis guidance: ensemble variance (Aberson 2003), ensemble transform Kalman filter (e.g. Bishop 2001), and total energy SVs. They found the superiority of the SVs though the sample size was small. SVs were also used for TC targeting missions during The Observing System Research and Predictability Experiment (THORPEX) Pacific Asian Regional Campaign (T-PARC) and Tropical Cyclone Structure 2008 (TCS-08) field campaigns. Dropsondes were deployed where the vertically accumulated total energy of SVs was relatively large. Using the observational data obtained during T-PARC and TCS-08, Harnisch and Weissmann (2010) showed that observations in the vicinity of the TC were more useful on average than those in remote areas or in the storm itself.

In a previous study on SVs in the vicinity of TC-like vortices, Nolan and Farrell (1999) illustrated the nonmodal growth of SVs through the Orr mechanism (Orr 1907) using a barotropic model. The SVs are initially tilted against the shear of the background angular velocity and obtain eddy kinetic energy through wave–mean flow interaction. The growth continues until the upshear property vanishes. Peng and Reynolds (2006) showed that initial SVs of the U.S. Navy Operational Global Atmospheric Prediction System (NOGAPS) appear about 500 km from the centers of TCs, where the radial potential vorticity gradient changes sign (a necessary condition of barotropic instability of normal modes). Such a condition was found to be created by surrounding synoptic features. Peng et al. (2009) identified exponentially growing perturbations (not SVs) in the outer region of a TC-like vortex that satisfies the necessary condition of barotropic instability of normal modes. The upshear tilt of the asymmetric vorticity causes energy transfer continuously from the symmetric mean flow to the asymmetric disturbances. In addition, Yamaguchi and Majumdar (2010) illustrated that baroclinic energy conversion in a vortex can be a mechanism for SVs to grow in the vicinity of TCs. Streamfunction and temperature perturbations in a cylindrical coordinate system centered on the TC collocate, with the temperature perturbation 90° ahead of the streamfunction perturbation. Consequently, the perturbation kinetic energy can grow with time through the conversion of azimuthal mean available potential energy into eddy available potential energy and then the conversion of eddy available potential energy into eddy kinetic energy.

The growth of SVs has been studied in many contexts. However, in comparison to the weight of the prior work on SVs for midlatitude dynamics (e.g., Mureau et al. 1993; Buizza 1994; Buizza and Palmer 1995; Hoskins et al. 2000), the basic properties of SVs targeted for TCs are still not fully understood. In addition, it is hypothesized that the structures and locations of SVs would be highly dependent on initial vortex and temperature profiles and configurations of SV calculations such as the optimization time (Buizza 1994), the norm by which the growth of SVs is optimized (Kim and Morgan 2002; Kuang 2004; Kim and Jung 2009), the resolution of the numerical model (Buizza et al. 1997), and the design of a tangent-linear model and its adjoint such as the representation of moist processes (Ehrendorfer et al. 1999; Hoskins and Coutinho 2005; Kim and Jung 2009). The choice of those configurations may be critical in determining with which physical processes the computed SVs are associated (e.g., TC motion, intensity, genesis, etc.). To provide a fundamental understanding of the dynamical mechanisms of SV growth in TCs, it is worthwhile to compute SVs under idealized conditions using simplified barotropic and baroclinic models. As a first step in a series of such studies, SVs are computed in a nondivergent barotropic framework on f and β planes. A nondivergent barotropic model is a good starting point for understanding the basic properties of SVs in the vicinity of TCs.

In this study, SVs are calculated for TC-like axisymmetric vortices without azimuthal dependence (see section 3a for more details) using a numerical model of two-dimensional fluid flow: a barotropic version of the Spectral Element Ocean Model (SEOM; Iskandarani 2008; Iskandarani et al. 1995). The following issues are first investigated on an f plane: 1) sensitivity of SVs to maximum wind speed Vmax and radius of maximum wind (RMW) of initial TC-like vortices and 2) sensitivity of SVs to the optimization time. In addition, SVs are calculated for initial vortices that satisfy the necessary condition of barotropic instability. Following this, the SV computations are extended to a β plane to understand how the SVs affect the modification of the track of initial TC-like vortices. In a β-plane study, the impact of SV resolution on the track modification is also investigated because, in the operational SV-based EPS in ECMWF and JMA, the resolution of SVs is much lower than that of the forecast model.1

This paper is organized as follows. Section 2 describes the methodology of SV computations. Sections 3 and 4 describe SVs on an f plane and β plane, respectively. Section 5 presents a summary.

2. Methodology

a. General overview of SVs

The evolution of a state vector u of a general n-dimensional dynamical system can be written as

 
formula

where t is time and f(u) is a function to describe the time evolution of u. The evolution of an infinitesimal perturbation to this system can be written as a linear dynamical system

 
formula

where y is a state vector of the perturbation and is the tangent-linear operator. The linearized dynamical matrix is the Jacobian matrix, written as

 
formula

By introducing a positive-definite Hermitian operator , or a norm operator, the energy of the system can be written as

 
formula

where T denotes a matrix transpose. Then the following useful change into generalized velocity coordinates is performed:

 
formula
 
formula

so that

 
formula
 
formula

where 〈,〉 represents an inner product.

One can obtain fast-growing perturbations over the optimization time t in the tangent linear system in Eq. (7) by singular vector decomposition (SVD). Consider the normalized growth of a perturbation x:

 
formula

By introducing the propagator

 
formula

the growth can be written as

 
formula
 
formula
 
formula

where

 
formula

Thus the SVs of , or the eigenvectors of , give a set of perturbations with amplification factors (singular values) of σ in the system, and the SV with the largest value of σ corresponds to the fastest-growing perturbation (leading SV).

When are not a function of time ( are autonomous operators), the propagator can be obtained by explicitly solving Eq. (7) (Farrell and Ioannou 1996a):

 
formula

When are a function of time ( are nonautonomous operators), the propagator can be written as

 
formula

where t0 = 0 and t = kδt (Coddington and Levinson 1955; Farrell and Ioannou 1996b).

b. SV computations in a nondivergent barotropic framework

The governing equation in this study is a nondivergent barotropic vorticity equation with a viscosity term:

 
formula

where ζ is the relative vorticity, v is the horizontal velocity vector, υ is the meridional velocity, β is the latitudinal variation of the Coriolis parameter, and ν is the kinematic viscosity. We set β to a value of 2.2103 × 10−11 m−1 s−1, corresponding to 15°N.

In this study, we use a fully nonlinear model of two-dimensional, incompressible flow. The model is SEOM, a spectral element model presented by Iskandarani (2008) and Iskandarani et al. (1995). The domain is set to a 2000-km square box that is discretized into 20 × 20 square spectral elements of 100-km size. A fifth-degree polynomial is used to approximate the solution in each element, yielding an average effective grid spacing of 20 km. The boundary condition is periodic in the zonal direction and ψ = 0 (impenetrable) at the northern and southern boundaries, where ψ denotes streamfunction. The kinematic viscosity is set to 75 000 m2 s−1 to avoid numerical noise in the solutions.

Kinetic energy norms are used to evaluate the growth of the SVs. Using streamfunction and vorticity, the kinetic energy over the model domain is written as

 
formula

where represents integration over the model domain. To obtain the kinetic energy, a Green’s function operator is constructed:

 
formula

where ψ and ζ represent state vectors of streamfunction and vorticity, respectively. In this case, . Thus, is equivalent to as defined in Eq. (4).

To construct the propagator for SV computations, Eq. (17) is solved for an initial TC-like vortex by adding a point vortex perturbation at each discretized grid point of the numerical model. On an f plane, an initial TC-like vortex does not evolve with time. Thus are considered autonomous operators, and the procedure to construct the propagator is as follows:

  1. Run a numerical model from a certain initial condition for only one time step.

  2. Calculate the tendency, ∂u/∂t, at each grid point of the model.

  3. Run the model again for only one time step by adding a perturbation at a certain grid point.

  4. Calculate the tendency, ∂u/∂t, at each grid point of the model.

  5. Take the difference between the results of 2 and 4. The difference approximates one column of the Jacobian matrix .

  6. Repeat steps 3–5 for all grid points to complete the Jacobian matrix.

Once the Jacobian matrix is created, the propagator can be obtained through Eqs. (6) and (15), and then the SVD can be performed.

On a β plane, an initial TC-like vortex evolves with time because of the beta gyres; thus, are considered nonautonomous operators. The procedure to construct the propagator is the same as in autonomous operators except that the Jacobian matrix should be calculated about the evolving (time-dependent) background states. As Eq. (16) indicates, in nonautonomous operators, the Jacobian matrix needs to be created ideally at every time step or every several time steps during numerical integrations from initial time to the optimization time. In the SV calculations on a β plane in this study, the optimization times of 24 and 48 h are tested, where the Jacobian matrix is updated every 6 h for the optimization time of 24 h and every 12 h for the optimization time of 48 h. To measure the accuracy of the SVs computed when using a small number of evaluations of the time evolution operator to compute the propagator [Eq. (16)], we calculate the projections of the leading SVs onto leading SVs computed using hourly updates of the time operator. Projection values of 1 indicate the SVs are identical. As Table 1 shows, these projections reach a value of 0.99 for 6-hourly updates for the 24-h leading SV and 0.95 for 12-h updates for the 48-h leading SV. This shows that the SVs, as the time interval of keeping the Jacobian constant is shortened, actually converge and that the time intervals of 6 and 12 h are short enough.

Table 1.

The projection coefficients of SVs with the time interval of keeping the Jacobian constant for 3, 6, 12, 24, and 48 h onto SVs with the time interval of keeping the Jacobian constant for 1 h. The SVs are calculated on a β plane with an optimization time interval (OTI) of 24 and 48 h for an initial TC-like vortex of Vmax = 35 m s−1, RMW = 100 km, and b = 1.0. The number in parentheses is the number of times the Jacobian matrix is updated.

The projection coefficients of SVs with the time interval of keeping the Jacobian constant for 3, 6, 12, 24, and 48 h onto SVs with the time interval of keeping the Jacobian constant for 1 h. The SVs are calculated on a β plane with an optimization time interval (OTI) of 24 and 48 h for an initial TC-like vortex of Vmax = 35 m s−1, RMW = 100 km, and b = 1.0. The number in parentheses is the number of times the Jacobian matrix is updated.
The projection coefficients of SVs with the time interval of keeping the Jacobian constant for 3, 6, 12, 24, and 48 h onto SVs with the time interval of keeping the Jacobian constant for 1 h. The SVs are calculated on a β plane with an optimization time interval (OTI) of 24 and 48 h for an initial TC-like vortex of Vmax = 35 m s−1, RMW = 100 km, and b = 1.0. The number in parentheses is the number of times the Jacobian matrix is updated.

It should be noted that, in this study, a tangent-linear model and its adjoint are not used in the SV calculations. The Jacobian matrix is computed as a finite difference computation on an f plane while the Jacobian is assumed to be stepwise constant on a β plane. This approach is useful when a numerical model is simple, and the tangent-linear model and its adjoint are not available. However, it is almost impossible, in terms of the computational cost, to construct the Jacobian matrix and solve the eigenvalue problem of Eq. (14) explicitly when the matrix size is large, say 106 or larger. The alternative approach is to use techniques to approximately solve an eigenvalue problem of a large matrix. The Lanczos method is one of the techniques (e.g., Davidson 1975; Golub and van Loan 1996). Different from the explicit approach, each element of the propagator does not have to be specified in the Lanczos method. Instead, the eigenproblem is solved by successively applying the propagator and its adjoint (tangent-linear model and its adjoint) to perturbations. The good thing about the Lanczos method is that it gives several largest or smallest eigenvalues and eigenvectors selectively. This is particularly useful in EPS where fast-growing perturbations are of interest.

3. Singular vectors on an f plane

a. Sensitivity of SVs to Vmax and RMW

Nine numerical experiments are conducted to investigate the sensitivity of SVs to maximum tangential wind speed and the radius of maximum wind of initial TC-like vortices. The initial TC-like vortices used in this study are the same as those used in Chan and Williams (1987) and Fiorino and Elsberry (1989). The vortices are axisymmetric without azimuthal dependence and have no mean radial flow. The tangential wind profile of the vortices is written as

 
formula

where V(r) is the tangential wind, r is the radial position, Vmax is maximum tangential wind speed, rmax is the RMW, and b is a parameter that determines the shape of the profile. As b becomes larger, the tangential wind tends to zero outside the RMW more quickly.

The values of Vmax and RMW tested in the nine experiments are 20, 35, and 50 m s−1 and 100, 200, and 300 km, respectively. The value of b is fixed to 1.0, and the optimization time is also fixed to 6 h for all nine experiments. For creating the Jacobian matrix, the amplitude of each point vortex perturbation is set to 10−4 s−1. Table 2 shows the singular value of the leading SV for each experiment, and Fig. 1 shows the vorticity field of the leading SV. The following four features can be seen in Fig. 1 and Table 2:

  1. SVs are tilted against the shear of the background angular velocity as seen in Fig. 10 of Nolan and Farrell (1999), indicating the growth of SVs through the Orr mechanism.

  2. The leading singular value increases with Vmax and decreases with RMW.

  3. The locations of SVs move outward with increasing RMW.

  4. The locations of SVs move outward with increasing Vmax.

Table 2.

Singular values of the leading SVs for initial TC-like vortices with different Vmax and RMW. The value of b and optimization time is fixed to 1.0 and 6 h, respectively.

Singular values of the leading SVs for initial TC-like vortices with different Vmax and RMW. The value of b and optimization time is fixed to 1.0 and 6 h, respectively.
Singular values of the leading SVs for initial TC-like vortices with different Vmax and RMW. The value of b and optimization time is fixed to 1.0 and 6 h, respectively.
Fig. 1.

Vorticity fields of the leading initial SVs for initial TC-like vortices with different Vmax and RMW. The values of Vmax and RMW are (left) 20, (middle) 35, and (right) 50 m s−1, and (top) 100, (middle) 200, and (bottom) 300 km, respectively. The value of b and the optimization time are fixed to 1.0 and 6 h, respectively. The thick (thin) lines are for positive (negative) values, and the amplitude of vorticity vector of the SVs is set to 1 (same for the following figures).

Fig. 1.

Vorticity fields of the leading initial SVs for initial TC-like vortices with different Vmax and RMW. The values of Vmax and RMW are (left) 20, (middle) 35, and (right) 50 m s−1, and (top) 100, (middle) 200, and (bottom) 300 km, respectively. The value of b and the optimization time are fixed to 1.0 and 6 h, respectively. The thick (thin) lines are for positive (negative) values, and the amplitude of vorticity vector of the SVs is set to 1 (same for the following figures).

The tilt against the background flow is a common structure as a growth mechanism of perturbations on two-dimensional vortices. This is because the tilting property is required for the downgradient flux of momentum (e.g., Orr 1907; Staley and Gall 1979; Gall 1983; Flierl 1988; Schubert et al. 1999).

The fact that the leading singular value increases with Vmax and decreases with RMW can be explained by an equation for the rate of change of perturbation kinetic energy K′ with zero mean radial flow (e.g., Nolan and Farrell 1999):

 
formula
 
formula

where u′ and υ′ are perturbation radial and tangential velocity, is azimuthal mean angular velocity, is azimuthal mean tangential velocity, and the overbar refers to an azimuthal average. This equation represents the growth or decay of perturbation kinetic energy through momentum fluxes. Figure 2 shows the plots of as a function of radius for the initial vortices with 1) Vmax = 50 m s−1 and RMW = 100 km, 2) Vmax = 20 m s−1 and RMW = 100 km, and 3) Vmax = 20 m s−1 and RMW = 300 km. The maximum value of appears at RMW, and the value increases with Vmax and decreases with RMW. Considering that the kinetic energy norm is used for the SV calculations, and that the SVs appear near RMW with this optimization time (6 h), it can be inferred that the increased rate of change of perturbation kinetic energy by the increasing Vmax should lead to the larger amplification factor. Similarly, the decreased rate of change of perturbation kinetic energy as the RMW increases results in the smaller singular value.

Fig. 2.

Values of as a function of radius for the initial vortices with Vmax = 50 m s−1 and RMW = 100 km, Vmax = 20 m s−1 and RMW = 100 km, and Vmax = 20 m s−1 and RMW = 300 km (b = 1.0).

Fig. 2.

Values of as a function of radius for the initial vortices with Vmax = 50 m s−1 and RMW = 100 km, Vmax = 20 m s−1 and RMW = 100 km, and Vmax = 20 m s−1 and RMW = 300 km (b = 1.0).

The reason why the locations of SVs move outward with increasing RMW can be also explained by the equation for the rate of change of perturbation kinetic energy. As Fig. 2 shows, the growth of perturbation kinetic energy can become larger around the RMW. Thus, in order for the leading SV to possess the largest amplification factor, the SV seeks out a part of the flow where the rate of change of perturbation kinetic energy is relatively large, namely around the RMW. As a result, the SV appears increasingly removed from the cyclone center with increasing RMW. Figure 3 shows the distance between the locations of maximum kinetic energy of the leading SV and the cyclone center for all nine experiments. It is clear that the SV tends to occur farther away from the cyclone center as RMW increases.

Fig. 3.

Distance between the locations of maximum kinetic energy of leading initial SVs and the cyclone center as a function of Vmax and RMW.

Fig. 3.

Distance between the locations of maximum kinetic energy of leading initial SVs and the cyclone center as a function of Vmax and RMW.

In fact, Fig. 3 shows that the locations of the SVs are not exactly at RMW, but outside of it, and their distance beyond the RMW increases with Vmax. As will be explained in the next section, both of these results are related to the choice of the optimization time.

b. Sensitivity of SVs to the optimization time

Another set of numerical experiments is conducted to understand the sensitivity of SVs to the optimization time. Using an initial TC-like vortex with Vmax = 35 m s−1 and RMW = 100 km (b = 1.0), optimization times of 6, 12, 24, and 48 h are tested. Figure 4 shows the vorticity field of the leading SV for each experiment. The locations of the SVs are found to move outward with the increasing optimization time. As discussed in section 3a, the location at around RMW is a favorable location for SVs to grow. Nevertheless, the SVs appear far away from RMW when the optimization time is long. Why is this?

Fig. 4.

Vorticity fields of the leading initial SVs with different optimization time. The optimization time is (top left) 6, (top right) 12, (bottom left) 24, and (bottom right) 48 h; Vmax, RMW, and the value of b of the initial vortex are 35 m s−1, 100 km, and 1.0, respectively.

Fig. 4.

Vorticity fields of the leading initial SVs with different optimization time. The optimization time is (top left) 6, (top right) 12, (bottom left) 24, and (bottom right) 48 h; Vmax, RMW, and the value of b of the initial vortex are 35 m s−1, 100 km, and 1.0, respectively.

With zero viscosity, a perturbation whose phase lines are tilted all the way back, becoming nearly parallel with the flow, can exist just beyond the RMW (Nolan and Farrell 1999). However, viscosity does not allow this because viscosity quickly diffuses such a tight perturbation (which would have a high radial wavenumber). In other words, the extent of the upshear tilt is limited by viscosity. A perturbation near RMW relatively quickly becomes radially aligned because the tangential shear is relatively large. Thus, in the case of a longer optimization time interval, a perturbation seeks out a part of the flow that allows the perturbation to grow for a longer period, but while experiencing less diffusion.

This fact explains the fourth finding in section 3a: the locations of SVs move outward with increasing Vmax. Since the large Vmax wraps up the initially tilted SVs relatively quickly (before the optimization time), SVs appear where the tangential wind speed is relatively slow so that the SVs can keep growing up to the optimization time.

c. SV capturing barotropic instability

The initial vortices used in sections 3a and 3b do not satisfy the necessary condition of barotropic instability of normal modes, namely that the vorticity gradient change sign. Thus, the only mechanism for SVs to grow is the Orr mechanism. As studied by Peng and Reynolds (2006), the initial SVs of NOGAPS tend to occur at around 500 km from the storm center, where the radial potential vorticity gradient changes sign. They found that such a condition can be created by the influence of surrounding synoptic features. In this section, SVs are calculated for initial vortices that meet the necessary condition of barotropic instability to examine the characteristics of SVs that capture the instability.

For this purpose, three vortices are created using a cubic Hermitian polynomial:

 
formula

where S(x) = 1 − 3x2 + 2x3 is the cubic Hermitian polynomial that satisfies S(0) = 1, S(1) = 0, and S′(0) = S′(1) = 0. Values of ζ1ζ3 and d1 and d2 for the three vortex profiles are shown in Table 3. As Fig. 5 shows, one of the vortices does not satisfy the necessary condition of barotropic instability (hereafter referred to as initial vortex A) while two of them do satisfy the condition. The initial condition that weakly (strongly) meets the necessary condition is hereafter referred to as initial vortex B (C).

Table 3.

Parameters for initial vortex profiles for initial vortices A, B, and C. Initial vortex A does not satisfy a necessary condition of barotropic instability. Initial vortex B (C) satisfies the condition weakly (strongly).

Parameters for initial vortex profiles for initial vortices A, B, and C. Initial vortex A does not satisfy a necessary condition of barotropic instability. Initial vortex B (C) satisfies the condition weakly (strongly).
Parameters for initial vortex profiles for initial vortices A, B, and C. Initial vortex A does not satisfy a necessary condition of barotropic instability. Initial vortex B (C) satisfies the condition weakly (strongly).
Fig. 5.

Profiles of vorticity as a function of radius for initial vortices A, B, and C. Initial vortex A does not satisfy a necessary condition of barotropic instability. Initial vortex B (C) satisfies the condition weakly (strongly).

Fig. 5.

Profiles of vorticity as a function of radius for initial vortices A, B, and C. Initial vortex A does not satisfy a necessary condition of barotropic instability. Initial vortex B (C) satisfies the condition weakly (strongly).

As in section 3b where the SV dependency on the optimization time is investigated, optimization times of 6, 12, 24, and 48 h are tested for each initial vortex. Table 4 shows the singular value of the leading SV for each experiment. In the case where the barotropic instability is not allowed, the location of the SV shifts outward as the optimization time increases (Fig. 6). As is already seen in Fig. 4, this is a characteristic of SVs for a vortex that does not satisfy the necessary condition of barotropic instability. Meanwhile, in the case where the necessary condition of barotropic instability is strongly met, the SVs occur at around r = 400 km, where the radial vorticity gradient changes sign for all optimization times greater than 6 h (Fig. 7). This indicates that the SVs capture barotropic instability. In fact, as Table 4 shows, the singular values for initial vortex C are much larger than those for initial vortex A, especially for the longer optimization time. It can be inferred that, for a relatively short optimization time, the transient growth by the Orr mechanism is dominant in the growth of SVs while the exponential growth by barotropic instability plays a primary role for a longer optimization time. Note that even in the case of barotropic instability, the initial SVs have an upshear property, implying the existence of the Orr mechanism during the early stage of the growth. This result is identical to that shown in Fig. 17 of Nolan and Farrell (1999), where the stability of a two-celled vortex is examined. For initial vortex B, which satisfies the necessary condition weakly, the results are similar to initial vortex A, whose initial condition does not allow for barotropic instability. This indicates that vortex profiles that satisfy the necessary condition do not always lead to the instability because the condition is necessary, but not sufficient. In fact, the structure of the evolved SVs for initial vortex B is not similar to those of initial vortex C, where two vorticity waves interact with each other across the location where the radial gradient of vorticity changes sign.

Table 4.

Singular values of leading SVs for initial vortices A, B, and C with different optimization times.

Singular values of leading SVs for initial vortices A, B, and C with different optimization times.
Singular values of leading SVs for initial vortices A, B, and C with different optimization times.
Fig. 6.

Distance between the locations of the maximum kinetic energy of leading initial SVs and the cyclone center as a function of optimization time and initial vortex profile.

Fig. 6.

Distance between the locations of the maximum kinetic energy of leading initial SVs and the cyclone center as a function of optimization time and initial vortex profile.

Fig. 7.

As in Fig. 4, but for initial vortex C.

Fig. 7.

As in Fig. 4, but for initial vortex C.

The evolution of SVs in a tangent linear model is explored, and how this evolution changes depending on initial vortex structures is investigated. Figure 8 shows the evolution of an azimuthal wavenumber-2 SV for initial vortices (left) A and (right) C at (top) 0, (middle) 24, and (bottom) 48 h, respectively. The optimization time is 24 h. The wavenumber-2 SVs for initial vortices A and C are the third and first SVs, respectively, and the reason why the wavenumber-2 structure is selected is that it is the most unstable normal mode. In the case that barotropic instability is not allowed, the upshear tilt at the initial time almost vanishes at the optimization time, and the perturbation takes on a downshear property after the optimization time (the first and second SVs of initial vortex A, which have an azimuthal wavenumber-1 structure, show the same growing and decaying mechanisms as the third SV). Meanwhile, in the case where the barotropic instability is allowed, the SV is initially tilted like initial vortex A, but the evolved SV at the optimization time has two wave perturbations across the location where the vorticity gradient changes sign, and the perturbation outside is located downstream with respect to the perturbation inside. Furthermore, this phase relationship between the inside and outside perturbations does not change even after the optimization time. This result indicates that an SV capturing the barotropic instability is initially tilted against the background shear and grows through transient growth during the early period, after which it keeps the same structure. At a certain time during the initial growth, the SV “locks in” to a normal mode structure and stays there so that it can grow exponentially with time. This result of the normal mode-like evolution is identical to that in Peng et al. (2009), which illustrated the existence of a normal mode perturbation with an azimuthal wavenumber-2 structure when barotropic instability is allowed.

Fig. 8.

Time evolution of azimuthal wavenumber-2 SVs (vorticity fields) in a tangent-linear model for initial vortices (left) A and (right) C, for (top) initial time, (middle) optimization time (24 h), and (bottom) after optimization time (48 h). The amplitude of vorticity vector of the evolved SVs is normalized by the initial amplitude (same for the following figures).

Fig. 8.

Time evolution of azimuthal wavenumber-2 SVs (vorticity fields) in a tangent-linear model for initial vortices (left) A and (right) C, for (top) initial time, (middle) optimization time (24 h), and (bottom) after optimization time (48 h). The amplitude of vorticity vector of the evolved SVs is normalized by the initial amplitude (same for the following figures).

d. Sensitivity tests

A sensitivity test is conducted to evaluate the finite difference approximation. In this sensitivity test, the amplitude of point vortex perturbations is changed from 10−4 s−1 to 10−5, 10−6, and 10−7 s−1, respectively. The additional SV computations are performed for an initial TC-like vortex with Vmax = 35 m s−1, RMW = 100 km, and b = 1.0, with an optimization time of 6 h. It is found through the sensitivity test that the structure of SVs is not sensitive to the amplitude. The projection coefficients of the calculated leading SVs onto the original SV are 1.00.

Another sensitivity test is conducted to evaluate the size of the model domain. For this purpose, the domain size is increased from a 2000- to 3000-km square box. Then the additional SV computation is performed for an initial TC-like vortex with Vmax = 50 m s−1, RMW = 300 km, and b = 1.0, whose leading SV has some amplitude near the boundaries (see the bottom right panel of Fig. 1). The optimization time is 6 h. The tilting property of the SV near the boundaries is found to become slightly tight when the domain becomes larger. However, the projection coefficient of the calculated SV onto the original SV is high (0.96). It indicates little influence of the boundaries on the SV structure.

4. Singular vectors on a β plane

SV calculations are extended from an f plane to a β plane in this section. As studied on an f plane, the sensitivity of SVs to maximum tangential wind speed and the radius of maximum wind of initial TC-like vortices are first investigated. Following this, the sensitivity of SVs to the radial profile of tangential wind of initial TC-like vortices is studied. Since the background vortex evolves (moves) with time on a β plane because of the beta gyres, and the strength of the beta gyres is sensitive to the radial profile of tangential wind, it would be expected that SVs on a β plane are not identical to those on an f plane when the beta gyres are strong. Next, ensemble experiments are conducted by using the calculated SVs as ensemble initial perturbations in order to understand how the SVs modify the track and/or structure of initial TC-like vortices. In the ensemble experiments, the impact of SV resolution on the track modification is also investigated. Finally, an SV computation is performed for initial vortex C (see Fig. 5) to see the difference from the f-plane SV as shown in the right column of Fig. 8.

a. Sensitivity of SVs to Vmax, RMW, and b

Four SV calculations are conducted on a β plane by changing the maximum tangential wind speed and the radius of maximum wind of initial TC-like vortices. As studied in Chan and Williams (1987), the beta gyres become stronger as either Vmax or RMW increases; Vmax and RMW tested in the four experiments are 20 and 35 m s−1, and 100 and 200 km, respectively. The value of b is fixed to 1.0, and the optimization time is also fixed to 48 h for all four experiments (the Jacobian matrix is created every 12 h). Note that the initial vortices do not have any beta gyres and that the SVs are computed for a trajectory in which the beta gyres are gradually spun up. Figure 9 shows the vorticity field of the leading SV for each experiment. Similar to the results on an f plane (Fig. 1), the locations of the SVs on a β plane also move outward with the increasing Vmax and RMW. Different from the SVs on an f plane, the azimuthal distribution of the SVs becomes more asymmetric, and the extent of the asymmetry increases as the strength of the beta gyres increases.

Fig. 9.

Vorticity fields of the leading initial SVs on a β plane for initial TC-like vortices with different Vmax and RMW. The values of Vmax and RMW are (left) 20 and (right) 35 m s−1, and (top) 100 and (bottom) 200 km, respectively. The value of b and optimization time is fixed to 1.0 and 48 h, respectively.

Fig. 9.

Vorticity fields of the leading initial SVs on a β plane for initial TC-like vortices with different Vmax and RMW. The values of Vmax and RMW are (left) 20 and (right) 35 m s−1, and (top) 100 and (bottom) 200 km, respectively. The value of b and optimization time is fixed to 1.0 and 48 h, respectively.

Two additional SV calculations are conducted to investigate the sensitivity of SVs to the radial profile of tangential wind of initial TC-like vortices. The value of b is set to 0.6 and 1.4; Vmax and RMW are fixed to 35 m s−1 and 100 km, respectively; and the optimization time is fixed to 48 h. Note that the strength of the beta gyres is stronger for b = 0.6 because larger values of b correspond to the tangential wind going to zero more quickly outside the RMW. Figure 10 shows the vorticity field of the leading SV for each experiment. It is found that in the case where the beta gyres are strong (b = 0.6), the asymmetric feature of the SV is more distinctive. On the other hand, the SV with b =1.4 appears more like SVs on an f plane on which the beta gyres do not develop.

Fig. 10.

As in the top right panel of Fig. 9, but the value of b is (left) 0.6 and (right) 1.4.

Fig. 10.

As in the top right panel of Fig. 9, but the value of b is (left) 0.6 and (right) 1.4.

It should be emphasized that all the above six leading SVs as seen in Figs. 9 and 10 have an azimuthal wavenumber-1 structure at the optimization time even though the extent of the azimuthal asymmetry of SVs at the initial time is different among them. Note that an azimuthal wavenumber-1 structure leads to a displacement of the vortex at the optimization time. It should also be emphasized that the second SVs also have an azimuthal wavenumber-1 structure at the final time, but the direction of the vortex displacement is expected to be orthogonal to that of the leading SVs (an example will be shown in section 4b). These results indicate that the linear combination of the initial first and second SVs can displace the vortex in any direction at the optimization time. As will be shown in section 4b, the singular values of the first two SVs are much larger than those of the subsequent SVs that have azimuthal wavenumber-2 or -3 structures at the optimization time.

b. Ensemble experiments using SVs as initial perturbations

The left and middle columns of Fig. 11 show the first three initial and evolved SVs, respectively, for an initial vortex with Vmax = 35 m s−1, RMW = 100 km, and b = 1.0. The optimization time is 48 h. As Fig. 11 shows, the first SV has a wavenumber-1 structure at the optimization time. It implies that the initial first SV will make a displacement of the vortex at the optimization time when it is used as the initial perturbation. The second SV also has a wavenumber-1 structure at the optimization time, but the direction of the vortex displacement is orthogonal to that by the first SVs if it is used as the initial perturbation. These results indicate the feasibility of SVs as ensemble initial perturbations for TC track forecasting because an ensemble initial perturbation created by a linear combination of the initial first and second SVs can displace the vortex in any direction at the optimization time. The third SV has a wavenumber-2 structure at the optimization time, implying that the third SV will make the shape of the vortex into an ellipse with little impact on the track. The right column of Fig. 11 shows the results of three numerical experiments where each initial SV from the first to third SV is used as the initial perturbation. The kinetic energy of initial perturbation accumulated over the whole model domain is the same among the three experiments, and the maximum amplitude of the wind perturbation is less than 10% of Vmax. For example, the maximum amplitude of the wind perturbation is 2.5 m s−1 in the experiment where the initial first SV is used as the initial perturbation. As expected, the first SV shifts the vortex toward the west at the optimization time while the second SV shifts the vortex toward the north. In an experiment where the first and second SVs are added to the initial condition simultaneously, the vortex is found to be shifted toward the northwest at the optimization time (not shown). Meanwhile, the third SV makes the shape of the vortex elliptical at the optimization time with a small displacement. This indicates the importance of wavenumber-1 SVs for ensemble forecasts focusing on TC tracks. For singular values of those three SVs, the singular values of the first and second SVs are found to be much larger than that of the third SV; they are 3.77, 3.59, and 1.14, respectively. This indicates that the largest-growing ensemble perturbations are those that displace the vortex. From the above results, for ensemble forecasts for TC tracks, it would be particularly important to construct ensemble initial perturbations using SVs with large singular values that result in the displacement of the vortex.

Fig. 11.

(left) Initial SVs (vorticity fields), (middle) evolved SVs (vorticity fields), and (right) vorticity fields of 48-h numerical integrations where the initial SVs are used as the initial perturbation. Rows represent the (top) first, (middle) second, and (bottom) third SVs for an initial vortex with Vmax = 35 m s−1, RMW = 100 km, and b = 1.0. Optimization time is 48 h. In (right), the C indicates the location of the center of circulation of an unperturbed simulation, whereas the N shows the center of circulation of a simulation using an SV with opposite sign to that used to make the contour plots. Note that the axes in the right panels are different from axes in the left and center panels.

Fig. 11.

(left) Initial SVs (vorticity fields), (middle) evolved SVs (vorticity fields), and (right) vorticity fields of 48-h numerical integrations where the initial SVs are used as the initial perturbation. Rows represent the (top) first, (middle) second, and (bottom) third SVs for an initial vortex with Vmax = 35 m s−1, RMW = 100 km, and b = 1.0. Optimization time is 48 h. In (right), the C indicates the location of the center of circulation of an unperturbed simulation, whereas the N shows the center of circulation of a simulation using an SV with opposite sign to that used to make the contour plots. Note that the axes in the right panels are different from axes in the left and center panels.

As briefly mentioned in section 1, the current SV-based EPS in operational NWP centers such as ECMWF and JMA has a resolution gap between the forecast model and the SVs. For example, the horizontal resolution of moist SVs in ECMWF is T42 while that of the forecast model is T639. A question here is whether or not SVs with higher resolution could produce a larger vortex displacement compared with those with lower resolution, or the extent to which SVs with lower resolution can modify the track of the vortex when they are used as initial perturbations for a higher-resolution model.

To explore this question, two SV calculations are conducted with an optimization time of 24 h (the Jacobian matrix is created every 6 h). Initial conditions of the two experiments are created using ECMWF analyses at 500 hPa for Typhoon Sinlaku (2008) initiated at 0000 UTC 10 September 2008. The ECMWF analyses are obtained through the recently established THORPEX Interactive Grand Global Ensemble (TIGGE) database (Bougeault et al. 2010). Thus, the initial vortices include the beta gyres and the SVs are computed for a trajectory in which the beta gyres exist from the initial time. The difference between the two initial conditions is the horizontal resolution of the analyses; one is 0.5° × 0.5° and the other is 2.5° × 2.5°. Note that the resolution of SVs is the same between the two experiments. Figure 12 shows the radial profile of tangential wind of the two initial conditions. It is found that the RMW of the high-resolution analysis is about 100 km while that of the low-resolution analysis is about 300 km. The left and middle columns of Fig. 13 show the first two initial and evolved SVs for the high-resolution initial condition (hereafter referred to as H-SV), and the first two initial and evolved SVs for the low-resolution initial condition (hereafter L-SV). As with the case of previous SV calculations on a β plane, where the beta gyres are gradually spun up, the first two SVs have a wavenumber-1 structure at the optimization time with different directions. The right column of Fig. 13 shows the results of four numerical experiments where each initial SV is used as the initial perturbation for the high-resolution initial condition. Note that we are interested in the extent to which the L-SV can modify the vortex track in a high-resolution model. As before, the kinetic energy of the initial perturbation integrated over the whole model domain is the same among the four experiments. It is found that the first and second L-SVs can shift the vortex by a comparable distance to those by H-SV even though the direction of the displacement between them is not exactly the same. Given that the ensemble initial perturbations in the ECMWF and JMA EPS are created by linearly combining the computed SVs, this result supports the validity of using L-SVs in the operational EPS for TCs to some extent.

Fig. 12.

ECMWF analysis of radial profile of tangential wind for Typhoon Sinlaku at 500 hPa initiated at 0000 UTC 10 Sep 2008 for data with a horizontal resolution of 0.5° × 0.5° (solid) and 2.5 × 2.5° (dashed).

Fig. 12.

ECMWF analysis of radial profile of tangential wind for Typhoon Sinlaku at 500 hPa initiated at 0000 UTC 10 Sep 2008 for data with a horizontal resolution of 0.5° × 0.5° (solid) and 2.5 × 2.5° (dashed).

Fig. 13.

(left) Initial SVs (vorticity fields), (middle) evolved SVs (vorticity fields), and (right) vorticity fields of 24-h numerical integrations where the initial SVs are used as the initial perturbation for the high-resolution analysis of Typhoon Sinlaku. Rows represent the (row 1) first and (row 2) second SVs for the high-resolution analysis of Typhoon Sinlaku and the (row 3) first and (row 4) second SVs for a low-resolution analysis of Typhoon Sinlaku. Optimization time is 24 h. In (right), the C indicates the location of the center of circulation of an unperturbed simulation, whereas the N shows the center of circulation of a simulation using an SV with opposite sign to that used to make the contour plots. Note that the axes in the right panels are different from axes in the left and middle panels.

Fig. 13.

(left) Initial SVs (vorticity fields), (middle) evolved SVs (vorticity fields), and (right) vorticity fields of 24-h numerical integrations where the initial SVs are used as the initial perturbation for the high-resolution analysis of Typhoon Sinlaku. Rows represent the (row 1) first and (row 2) second SVs for the high-resolution analysis of Typhoon Sinlaku and the (row 3) first and (row 4) second SVs for a low-resolution analysis of Typhoon Sinlaku. Optimization time is 24 h. In (right), the C indicates the location of the center of circulation of an unperturbed simulation, whereas the N shows the center of circulation of a simulation using an SV with opposite sign to that used to make the contour plots. Note that the axes in the right panels are different from axes in the left and middle panels.

c. SV calculation for initial vortex C

As presented on the f-plane study (Figs. 7 and 8), SVs capture barotropic instability for an initial vortex that strongly satisfies the necessary condition (see initial vortex C of Fig. 5). The SV is initially tilted against the background shear and grows through the Orr mechanism during the early period, after which it locks into the modal structure to continuously grow with time. To see the difference of the SV evolution between an f plane and β plane, we conduct an SV calculation for initial vortex C on a β plane. The optimization time is set to 48 h, and the Jacobian matrix is created every 12 h. Figure 14 shows the 12-hourly time evolution of the leading SV (vorticity field) in a tangent-linear model. The initial SV is found to be quite similar to that on an f plane: an azimuthal wavenumber-2 structure with an upshear tilt at around r = 400 km where the vorticity gradient changes sign. However, the evolved SV is different from that on an f plane. The evolved SV on a β plane has the azimuthal wavenumber-1 structure while that on an f plane keeps the modal, azimuthal wavenumber-2 structure (Fig. 8). Although the evolution of the azimuthal wavenumber-2 structure can be seen in the early stage of 12 h, the azimuthal wavenumber-1 structure becomes more distinctive after 36 h. It can be inferred that the barotropic instability is swamped by the beta gyres. As time evolves, the background beta gyres move and change the background vortex so much that the wavenumber-2 mode is no longer unstable.

Fig. 14.

12-hourly time evolution of the leading SV (vorticity fields) in a tangent-linear model for initial vortex C on a β plane. Optimization time is 48 h.

Fig. 14.

12-hourly time evolution of the leading SV (vorticity fields) in a tangent-linear model for initial vortex C on a β plane. Optimization time is 48 h.

5. Summary

The basic properties of singular vectors (SVs) for tropical cyclone (TC)–like axisymmetric vortices without azimuthal dependence were investigated in a nondivergent barotropic framework on both f and β planes. In the f-plane study, it was found that 1) the leading singular value increases with maximum wind speed Vmax of initial TC-like vortices and decreases with the radius of maximum wind (RMW), and 2) the locations of SVs move outward with Vmax, RMW, and the optimization time of the SV calculations. In addition, the capture of normal mode barotropic instability by the SV was shown. In this case, the SV is initially tilted against the background shear of angular velocity and grows through the Orr mechanism for a limited period while it keeps the same structure after that time. At a certain time during the initial growth, the SV “locks in” to a normal mode structure and remains in that structure so that it may grow exponentially with time.

Like SVs on an f plane, the locations of SVs on a β plane move outward with Vmax and RMW. The difference from the f plane is that the azimuthal distribution of the SVs becomes asymmetric, and the extent of the azimuthal asymmetry increases as the strength of the beta gyres increases. This result was also confirmed by an experiment in which the radial profile of the tangential wind in initial TC-like vortices was changed. For the tangential wind that goes to zero relatively quickly outside RMW, the SVs resemble those on an f plane in which the beta gyres do not develop.

In numerical simulations where SVs were used as initial perturbations, the importance of azimuthal wavenumber-1 SVs was demonstrated for TC track forecasts. All the leading SVs calculated on a β plane in this study had an azimuthal wavenumber-1 structure at the optimization time regardless of the condition of barotropic instability and the inclusion of the beta gyres at initial time. In addition, it turned out that the second SVs also had an azimuthal wavenumber-1 structure at the final time, but the direction of the vortex displacement was orthogonal to that caused by the first SVs. These results indicate that the linear combination of the initial first and second SVs can displace a vortex toward any direction at the optimization time when they are used as ensemble initial perturbations. Following this, the impact of SV resolution on the vortex displacement was investigated. A low-resolution wavenumber-1 SV was found to cause the displacement of a vortex in a high-resolution simulation. This result indicates the potential utility of SVs for ensemble forecasts targeting TC tracks in the operational NWP centers.

In the future, we will include the background flow in the initial conditions of SV calculations to account for the steering flow of TCs. Furthermore, we will extend this study to a baroclinic model. In a barotropic model, the only way for perturbations to grow is through eddy momentum fluxes of wavelike perturbations that transport momentum across shear. In a baroclinic model, on the other hand, radial and vertical heat fluxes also become important for the growth of perturbations. As studied in Kwon and Frank (2005, 2008) and Yamaguchi and Majumdar (2010), the background temperature profile of TCs would play an important role in determining the structure of SVs. In addition, the phase relationship between wind and temperature perturbations would be of great interest because it affects eddy radial and vertical fluxes. Finally, we will investigate the growth of SVs not only from a dynamical perspective but also from a physical perspective. For this purpose, we will look into the impact of physical processes on the structure of SVs. Particular emphasis will be put on how to derive SVs that have influence on the TC tracks. Designing a proper norm might be one of the promising approaches.

Acknowledgments

The authors thank three anonymous reviewers for useful comments and suggestions. This study was conducted under the support of Office of Naval Research Grant N000140810250. The development of the nonhydrostatic version of SEOM was supported by NSF OCE-0622662. The authors thank The Observing System Research and Predictability Experiment (THORPEX) Interactive Grand Global Ensemble (TIGGE) for constructing useful and user-friendly portal sites and providing the analysis data of operational NWP systems.

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Footnotes

1

As of February 2011, the horizontal resolution of moist SVs in ECMWF is T42 while that of the forecast model is T639. In the JMA EPS, the horizontal resolution of moist SVs is T63 while that of the forecast model is T319.