## 1. Introduction

Singular vectors (SVs) are those structures with a norm that amplifies most rapidly over a specified time period (the optimization interval) for a given basic state.^{1} Because of the rapidly growing property of SVs, they have been proposed as an alternative means to describe the rapid growth of disturbances in the atmosphere–ocean system (e.g., Farrell 1989), and they have been used to construct the initial perturbations for ensemble prediction (e.g., Molteni et al. 1996) as well as detect regions of large sensitivity to small perturbations for the purposes of making adaptive or targeted observations (e.g., Palmer et al. 1998). In addition to these uses, SVs can be used to construct the eigenvectors of the forecast error covariance matrix for the end of the optimization interval (e.g., Ehrendorfer and Tribbia 1997).

Amplification of SVs is measured with respect to a predefined norm: the most commonly chosen norms have been streamfunction variance, potential enstrophy, and total energy. For the streamfunction variance and energy norms, SVs resemble synoptic-scale wave packets and are initially upshear tilted structures localized in the lower troposphere (e.g., Mukougawa and Ikeda 1994; Buizza and Palmer 1995; Hartmann et al. 1995; Hoskins et al. 2000). In contrast, the initial structure of SVs in the potential enstrophy norm is of large scale (e.g., Palmer et al. 1998). Joly (1995), in an examination of the stability of frontal cyclones, explored the initial structure and amplification rate for SVs in a uniform potential vorticity (PV) semigeostrophic model for a variety of norms including the enstrophy, total energy, and geopotential variance norms. Joly (1995) observed that very different SV structures could be identified depending on which norm was chosen. While it is evident from these examples that SV initial structure is dependent on the choice of the defining norm, it is not clear how the mechanisms governing SV development are dependent on norm. The question of what the fundamental development mechanisms for SVs are is not an academic one as the resolution of this question may provide insight into such practical questions as where to target observing systems to improve specific forecasts or how better data assimilation schemes may be devised.

To better understand the fundamental mechanisms for SV development, the amplification of initially upshear tilted (and untilted) structures in general (e.g., Badger and Hoskins 2001, hereafter BH01) and SVs in particular (e.g., Morgan 2001; Morgan and Chen 2002) within simple Eady-type basic states have been studied. BH01 demonstrate that initially confined perturbations, characterized by a “rich” vertical structure, may experience rapid perturbation kinetic energy growth in baroclinic shear flows. The initial growth occurs by a process BH01 refer to as “PV unshielding,” wherein the kinetic energy of an initial perturbation, distinguished by vertically layered positive and negative PV anomalies, amplifies as the baroclinic shear unshields the PV due to differential advection. BH01 observed that the longer-term sustained growth of the perturbations is attributed to a coupling of thermal waves along the upper and lower boundaries of the Eady-type domain.

Amplification of the SV perturbation streamfunction (and kinetic energy for a single wavenumber perturbation) in the adiabatic, inviscid Eady model is accomplished by two mechanisms: baroclinic superposition of PV (“the Orr mechanism”) and amplification of the upper and lower boundary temperature anomalies (BTAs) by horizontal advection attributed to either the opposing BTAs or interior PV anomalies. Using a combination of PV inversion and Eliassen–Palm (E–P) flux diagnostics, Morgan (2001) described a three-stage sequence for SV development in the Eady model for the streamfunction variance norm. The growth of the initially upshear tilted SV begins as the baroclinic shear tilts the PV upright (leading to the PV being maximally superposed). During the next stage of development, the winds attributed to the PV increase the amplitude of the initially small BTAs. Finally, as the PV is tilted downshear, the interaction of the amplified BTAs describes the subsequent development. As noted above, the initial structures of the SVs vary with the choice of norm; as a consequence, it may be anticipated that the mechanisms for amplification vary with norm as well.

In this paper, using PV and E–P flux diagnostics, the initial SV structures and their subsequent developments are diagnosed for the potential enstrophy and energy norms and the results are compared with those for the streamfunction variance norm. While for the Eady model the contribution to the potential enstrophy norm amplification [defined in (2.9)] from growth of the interior PV is zero, the PV may still play a role in the SV development by amplifying the BTAs. If the PV does not play a role, it is possible the solution of the optimization problem would “select” an initial SV structure that possesses no PV, as it would be suboptimal for an SV structure to contain a component that does not amplify or contribute to amplification. In this study, we wish to examine which of the two scenarios outlined above is valid for the potential enstrophy norm. For the energy norm [defined in (2.13)], we seek that initial perturbation that maximizes both perturbation kinetic energy and perturbation available potential energy at the optimization time. We may anticipate that because the streamfunction variance norm is equivalent to the kinetic energy norm for a single wavenumber disturbance, there may be some modifications to the structure of the initial total quasigeostrophic (QG) perturbation energy norm SV to account for the maximization of the potential energy at the optimization time.

Section 2 contains a formulation of the SV problem for various norms and a brief review of the PV inversion and E–P flux diagnostics used in describing SV evolution. Singular vector evolutions for the potential enstrophy and the energy norms are presented in sections 3 and 4. The relative importance of the initial BTAs and the interior PV during SV evolution is described for the various norms and optimization times in section 5. Section 6 contains discussion and conclusions.

## 2. Formulation and diagnostics of SVs

*S̃*are nondimensional measures of shear and stratification, respectively, and

*ψ*′ is the perturbation streamfunction. The nondimensional thermodynamic equation is applied at the lower and upper boundaries (

*z*= 0 and

*z*= 1, respectively):

*ψ*′(

*x,*

*y,*

*z,*

*t*) =

*ψ̂*

*z*)

*e*

^{σt}

*e*

^{i(kx+ly)}and vertically discretizing (2.1) and (2.2) into

*M*levels, the model equations may be written as an eigenvalue problem:

*ψ̂*

*σψ̂*

*M*×

*M*linearized dynamical operator. The

*M*eigenvectors satisfying (2.3)

*ψ̂*

^{′}

_{j}

*σ*

_{j}are the growth rates of those disturbances.

*ψ*

^{′}

_{arb}

**Λ**

_{t}is a diagonal matrix with

*e*

^{σjt}

**a**is the vector of projection coefficients

*a*

_{j}, and

*P*

^{t}

_{t=0}

**Λ**

_{t}𝗫

^{−1}=

*e*

^{A}

^{t}, is the forward tangent propagator from time 0 to

*t.*

For the calculation to follow, the model was discretized into *M* = 51 levels, with a basic state characterized by vertical shear of 3 m s^{−1} km^{−1} in a troposphere of 10-km depth, a Brunt–Väisälä frequency of 10^{−2} s^{−1}, a Coriolis parameter of 10^{−4} s^{−1}, and zero interior meridional PV gradients. The nondimensional time *t* = 1 corresponds to 9.3 h. The nondimensional wavenumber *k* = 1 corresponds to a wavelength of approximately 3142 km. The meridional wavenumber *l* is taken to be zero. The Eady model shortwave cutoff *k*_{c} = 2.4. Henceforth, we define longwave (shortwave) perturbations as those perturbations with zonal wavenumber *k* < *k*_{c} (*k* > *k*_{c}).

### a. Optimization problem

*t*=

*τ*

_{opt}. In this study, we choose the initial and final norms to be the same. We define the amplitude of the streamfunction in some norm

*C*:

*C*norm, and the superscript ‘H’ indicates the conjugate transpose. The norm without any specification will be taken to be the

*L*

_{2}norm. The constrained optimization problem seeks to maximize the Rayleigh quotient

*λ*

^{2}(the amplification factor),

*t*=

*τ*

_{opt}. It may be shown that the maximum of this ratio is realized when

*ψ*′(0) is the leading SV of the matrix operator

*P*

^{t}

_{0}

*C*norm, that is,

*ψ*′(0) satisfies

#### 1) Streamfunction variance norm

*ψ*

*t*

^{2}

*ψ*

*t*

*ψ*

*t*

Thus for the streamfunction variance norm, 𝗖 is the identity matrix. The streamfunction variance norm is proportional to the kinetic energy norm for a single wavenumber streamfunction perturbation. The streamfunction variance norm in this model is called the *L*_{2} norm from now on because the statevector of the model is streamfunction.

#### 2) Potential enstrophy norm

*q*′ is the interior perturbation PV and

*L*

_{x}is the wavelength in the zonal direction.

*ψ*′, the potential enstrophy is

*Q*represents the potential enstrophy norm. The

*q*

^{′}

_{tot}

*q*′,

*θ*′} is the statevector of perturbation PV, which includes both the interior PV (

*q*′) as well as BTAs (

*θ*′), where { } denotes the vector. The interior PV, defined for 0 <

*z*< 1, is given by

Thus for the potential enstrophy norm, 𝗖 = 𝗟^{H}𝗟, where the elliptic operator 𝗟 is defined by 𝗟*ψ*′ = {*q*′, *θ*′}.

#### 3) Total QG disturbance energy norm

*ψ*

*t*

^{2}

_{E}

*ψ*

*t*

*ψ*

*t*

### b. PV inversion diagnosis

A diagnosis of SV development in the Eady model requires identifying the interactions between interior PV anomalies at a given height in the Eady domain with anomalies at other heights and with the BTAs. For this purpose, piecewise PV inversion is used and the PV is partitioned into three parts: lower BTA, interior PV, and upper BTA. Based on this partition, we may attribute the SV streamfunction into those parts attributed to the BTAs, *ψ*^{′}_{θ}*ψ*^{′}_{T}*ψ*^{′}_{B}*ψ*^{′}_{q}*ψ*^{′}_{T}*ψ*^{′}_{B}*ψ*^{′}_{q}*ψ*^{′}_{q}*q*^{′}_{sv}*ψ*^{′}_{θ}*ψ*^{′}_{θ}*θ*^{′}_{sv}*q*^{′}_{sv}*θ*^{′}_{sv}*ψ*^{′}_{sv}

Singular vector growth due to the advective amplification of the BTAs on a given boundary is diagnosed whenever the phase difference between the meridional winds attributed to either the interior PV or the BTAs along the opposing boundary is within 90°. The meridional velocities *υ*^{′}_{q}*υ*^{′}_{θ}*υ*^{′}_{q}*ψ*^{′}_{q}*x* and *υ*^{′}_{θ}*ψ*^{′}_{θ}*x.* The *υ*^{′}_{θ}*υ*^{′}_{T}*ψ*^{′}_{T}*x*)] and that attributed to the lower BTA [*υ*^{′}_{B}*ψ*^{′}_{B}*x*)]. The both BTAs may be calculated as *θ*^{′}_{B}*ψ*^{′}_{θ}*z*|_{z=0} and *θ*^{′}_{T}*ψ*^{′}_{θ}*z*|_{z=1}. The growth of an SV due to baroclinic PV superposition may be diagnosed whenever the magnitude of either the SV streamfunction or meridional velocity attributed to the interior PV increases (i.e., whenever ‖*ψ*^{′}_{q}*υ*^{′}_{q}

### c. E–P flux diagnosis

An alternate diagnosis of the development of the Eady SVs may be afforded by an E–P flux perspective (e.g., Edmon et al. 1980). For application to the Eady model with *l* = 0, the E–P flux vector has only a vertical component. Growing disturbances in the Eady model (i.e., those disturbances with increasing perturbation energy) are characterized by an upward-directed E–P flux vector (i.e., a positive zonally averaged meridional heat flux).

*θ*

^{′}

_{q}

*ψ*

^{′}

_{q}

*z.*

The four interaction terms on the right-hand side of (2.15) are denoted as *q*′ − *q*′, *q*′ − *θ*′, *θ*′ − *q*′, and *θ*′ − *θ*′, respectively. The *q*′ − *q*′ term may be used to diagnose the interior PV superposition mechanism. The *q*′ − *θ*′ term is associated with horizontal advection of BTAs by winds attributed to interior PV anomalies. The *θ*′ − *q*′ term is associated with the interaction between interior PV anomalies with the winds attributed to BTAs. The *θ*′ − *θ*′ term can be used to diagnose the mutual interactions between the BTAs on opposing boundaries.

## 3. Evolution of the potential enstrophy SVs

Since the SV evolution is different for shortwave (*k* < *k*_{c}) and longwave (*k* > *k*_{c}) SVs (e.g., Mukougawa and Ikeda 1994; Morgan 2001), the SV evolution is described for different wavelengths for each norm. The wavenumbers *k* = 1 and *k* = 5 are chosen for the purpose. The time necessary for the shear flow to render the initially upshear tilted PV vertical is defined as *τ*_{Orr}. The intermediate case of SV development for *τ*_{opt} around *τ*_{Orr} for the *L*_{2} norm [as described in Morgan (2001)] is discussed to clarify how *τ*_{Orr} changes depending on the choice of norm.

### a. SV development for k = 1 < k_{c} and τ_{opt} = 4.2

The sequence of Figs. 1a, 1c, 1e, and 1g shows the streamfunction^{2} at the times *t* = 0, 0.5, 1.8, and 4.2. The streamfunction is initially characterized by an absence of vertical tilt with maximum amplitude located at the boundaries (Fig. 1a). Shortly after the initial time, the streamfunction acquires and maintains a westward tilt with height. In the potential enstrophy norm, the streamfunction amplifies by a factor of 80.07 by *τ*_{opt} and resembles the unstable mode of the Eady model for this wavenumber. Figures 1b, 1d, 1f, and 1h show the evolution of the potential temperature (PT). The only dynamically important feature of the PT distribution is its distribution along the upper and lower boundaries. Initially the magnitude of the PT is largest along the boundaries and the PT is downshear tilted (Fig. 1b). The phase difference between the upper and lower BTAs is approximately 163°. As the SV evolves, this phase difference decreases to 20° at *τ*_{opt} and remains constant beyond this time (Figs. 1d, 1f, and 1h)—an additional indication that a growing normal mode structure dominates the disturbance structure shortly after the initial time. The decrease in the phase difference between the upper and lower BTAs may be readily interpreted from a PV perspective: that the phase difference between the upper and lower BTAs is nearly 180° at the initial time implies that the circulations attributed to BTAs are of the same sense. As a consequence of this configuration, the BTAs on the upper (lower) boundaries aid in the eastward (westward) propagation of the BTAs on the lower (upper) boundaries. This propagation continues until the BTAs become phase-locked (e.g., Hoskins et al. 1985).

Figure 2 shows the time evolution of the total and partitioned components of the vertical component of the E–P flux. The *q*′ − *q*′ interaction term is approximately six to seven orders of magnitude smaller than the total term (shown in Fig. 2e). Additional experiments reveal that the amplitude of the interior PV becomes smaller as the vertical resolution of the discretization increases. This result suggests that the interior PV may be numerical noise. The *θ*′ − *θ*′ term is constant with height in the interior and shows upward transport of wave activity for whole period (Fig. 2d). The meridional heat flux *υ*′*θ*′*θ*′ − *θ*′ term (Fig. 2e).

In summary, the longwave SV evolution in the enstrophy norm is characterized by an amplification of the BTA due to the initially transient and then sustained mutual interaction of the BTAs. There is no interior PV in this SV to make a contribution to the SV amplification.

### b. SV development for k = 5 > k_{c} and τ_{opt} = 1.8

Figures 3a, 3d, 3g, and 3j show the streamfunction at the selected times *t* = 0, 0.76, 1.8, and 3. In contrast to the *L*_{2} norm, the initial streamfunction structure for the same wavenumber and optimization time resembles that of the streamfunction associated with a combination of upper and lower boundary edge waves, rather than the initially highly upstream-tilted structure seen for the *k* = 5 *L*_{2} norm SV. In terms of potential enstrophy, the amplitude of the streamfunction is 1.08 at *τ*_{opt}. At *t* = 0, the PV is tilted upshear without significant structure in the interior (Fig. 3b). The initial PV structure for this norm is maximized close to the steering level for the edge waves while for the *L*_{2} norm the maximum amplitude is in midtroposphere. By *τ*_{opt}, the PV is already tilted downshear indicating that, as with the longwave SV for the potential enstrophy norm, *τ*_{Orr} is much shorter than the *τ*_{Orr} for the *L*_{2} norm [cf. our Fig. 3h with Fig. 3h in Morgan (2001)]. The PT initially possesses no vertical tilt and has large BTAs (Fig. 3c). For all times, the PT resembles a superposition of the upper and lower edge waves (Figs. 3f, 3i, and 3l).

Because of the symmetry between the upper and lower boundaries, we focus on the development of the lower BTAs. Figure 4a shows the evolution of the magnitudes of the lower boundary meridional velocities attributed to the upper BTA (*υ*^{′}_{T}*υ*^{′}_{q}*υ*^{′}_{T}*υ*^{′}_{q}*υ*^{′}_{q}*υ*^{′}_{T}*υ*^{′}_{q}*t* = 0.76, which corresponds to the approximate time at which the PV is nearly vertical. The phase difference between *υ*^{′}_{q}*θ*^{′}_{B}*τ*_{opt} (Fig. 4b). The favorable meridional advection of lower BTA by *υ*^{′}_{T}*υ*^{′}_{T}

Figure 5 displays the structure and evolution of the E–P flux. Upward-directed E–P fluxes associated with the *q*′ − *q*′ term are maximized near the upper and lower steering levels of the neutral Eady waves until *t* ∼ 0.1 (Fig. 5a), while downward-directed fluxes are maximized at time *t* = 1.5. The distribution of the E–P fluxes associated with the *q*′ − *q*′ term are consistent with the superposition of like-signed PV anomalies as the time of maximum superposition, *τ*_{Orr} = 0.76, occurs between the maximum upward and downward fluxes. It is noted that the superposition occurs well before *τ*_{opt} as seen in the longwave case. The *q*′ − *θ*′ term is comparable to or larger than the *θ*′ − *θ*′ term (Fig. 5b). The meridional heat fluxes associated with the *q*′ − *θ*′ term are maximized near time *t* = 1 and remain positive along the boundaries until *t* = 2. Due to the transient nature of the interactions between BTAs along opposing boundaries, the *θ*′ − *θ*′ term is characterized by a vertically uniform, alternating upward- and downward-directed E–P flux (positive and negative meridional heat fluxes; Fig. 5d). The total flux is also characterized by an alternating upward- and downward-directed E–P flux (Fig. 5e). Maximum values of the flux are found at the boundaries of the domain while minimum values of the flux are found in the middle of the domain. The total flux is dominated by contributions from the *q*′ − *θ*′ and *θ*′ − *θ*′ terms.

The shortwave SV development for this optimization interval is different from that for the longwave SV and follows the three-stage SV development (Morgan 2001): initially the development is characterized by an upward flux of wave activity that is maximized in the vicinity of the steering level close to the boundaries. Next, a favorable interaction between the BTA and the PV anomalies near the steering level of the edge waves characterizes the SV growth. Finally, transient growth and decay occur as the edge waves along the opposing boundaries travel past one another.

## 4. Evolution of the total QG disturbance energy SVs

### a. SV development for k = 1 < k_{c} and τ_{opt} = 4.2

Figures 6a, 6d, 6g, and 6j show the streamfunction at the selected times *t* = 0, 2.6, 4.2, and 6. The streamfunction is initially nearly vertical with maximum amplitude in the midtroposphere (Fig. 6a). We may infer from the streamfunction structure that the BTAs are relatively small and that the winds are stronger at the boundaries than those of the initial *L*_{2} norm SV (cf. with Fig. 1a in Morgan 2001). After *t* = 2.6, the disturbance resembles the unstable (growing) normal mode (Fig. 6d). In terms of energy, the streamfunction amplifies by a factor of 47.25 by *τ*_{opt}, before which time the maximum amplitude shifts to the boundaries (Fig. 6g). Unlike the streamfunction, the PV initially leans strongly upshear and, like the streamfunction, has maximum amplitude in the midtroposphere (Fig. 6b). At *t* = 2.6, the PV is nearly vertical (Fig. 6e). By *τ*_{opt}, the PV is already tilted downshear (Fig. 6h). Figures 6c, 6f, 6i, and 6l show the evolution of the PT. As inferred above, at *t* = 0, the PT has small magnitude along the boundaries. As the SV evolves, the maxima in the PT anomalies shifts to the boundaries. By *t* = 2.6, the thermal structure acquires a downshear tilt indicating that a growing normal mode structure has begun to dominate the developing disturbance (Fig. 6f).

From Fig. 7a, it is seen that after *t* = 0.4, the larger of the two magnitudes of the advecting velocities is *υ*^{′}_{T}*υ*^{′}_{q}*υ*^{′}_{T}*t* = 2 (Fig. 7b), the phase difference between *υ*^{′}_{T}*θ*^{′}_{B}*υ*^{′}_{T}*t* > 2, the phase difference between *υ*^{′}_{T}*θ*^{′}_{B}

Time–height cross sections of the E–P flux are shown in Fig. 8. The *q*′ − *q*′ term reaches a maximum at approximately *t* = 1.2, then diminishes to zero by *t* = 2.6—the time of maximum superposition (Fig. 8a). The *q*′ − *θ*′ term is characterized by an upward-directed E–P flux for the entire period and is maximized at the lower boundary at approximately *t* = 4.6 (Fig. 8b). The *θ*′ − *θ*′ term is constant with height in the interior, and upward transport of wave activity from lower to upper boundary is diagnosed for the period *t* > 2, consistent with the phase difference between *θ*^{′}_{B}*υ*^{′}_{T}*υ*′*θ*′*t* < 2), the *υ*′*θ*′*q*′ − *q*′ term and *q*′ − *θ*′ term. For 2 < *t* < 4, both the *q*′ − *θ*′ term and *θ*′ − *θ*′ term are important and after *t* = 4, the *υ*′*θ*′*θ*′ − *θ*′ term.

The longwave SV evolution in the energy norm follows the three-stage SV development described for the *L*_{2} norm; however, the baroclinic superposition of PV and amplification of BTAs occurs earlier in the evolution of the SV (compared with that of the *L*_{2} SVs) so that after a comparatively short time, the mutual interaction of the upper and lower BTAs becomes the dominant mechanism for SV growth. In fact, for the *k* = 1, *L*_{2} norm SV normal mode growth begin after the optimization time *τ*_{opt} = 4.2, for this norm the normal mode growth commences at *t* = 2.

### b. SV development for k = 5 > k_{c} and τ_{opt} = 1.8

Figures 9a, 9d, 9g, and 9j show the streamfunction at the selected times *t* = 0, 1.63, 1.8, and 4. The streamfunction is characterized initially by an upshear tilt with maximum amplitude in the upper and lower troposphere near the steering levels of the neutral Eady modes (Fig. 9a). In the energy norm, the streamfunction amplifies by a factor of 7.43 by *τ*_{opt}, and the maximum amplitude shifts to the boundaries (Fig. 9g). The PV is tilted upshear initially with maximum amplitude nearest the steering levels of the upper and lower neutral Eady modes. The PV is maximally superposed at *t* = 1.63 (Fig. 9e). The PT is also initially tilted upshear (Fig. 9c). Around *τ*_{opt} and time intervals beyond, the PT is maximized along the upper and lower boundaries (Figs. 9f, 9i, and 9l).

The *υ*^{′}_{q}*t* = 1.6 close to the time of maximum superposition *τ*_{Orr} (Fig. 10a). The phase difference between *υ*^{′}_{q}*θ*^{′}_{B}*t* < 2.75) (Fig. 10b). Amplification of the BTAs due to advections attributed to the interior PV is therefore maximized just prior to the time of *τ*_{Orr}.

Figure 11 displays the structure and evolution of the E–P flux. The magnitude of the *q*′ − *q*′ term is initially maximized near both the steering levels for the neutral modes, but reaches a maximum in the midtroposphere prior to *t* = 1.63 (Fig. 11a). Downward-directed E–P fluxes are maximized at time *t* = 1.9. As was seen in the longwave case, this antisymmetric distribution of the wave activity fluxes associated with the *q*′ − *q*′ term is consistent with the superposition and subsequent disjunction of like-signed PV anomalies. The *q*′ − *θ*′ term is large compared with the *q*′ − *q*′ term and explains most of the total E–P flux distribution (cf. Fig. 11b with 11e). The *q*′ − *θ*′ fluxes are maximized at *t* = 1.8 and remain positive in most of the domain until *t* = 2.8. The period of positive *q*′ − *θ*′ fluxes corresponds to that period within which the phase difference between *υ*^{′}_{q}*θ*^{′}_{B}*θ*′ − *θ*′ term is characterized by a vertically uniform, alternating upward- and downward-directed E–P flux (Fig. 11d). The total E–P flux increases over the time interval 0 < *t* < 1.8 in the upper and lower troposphere (Fig. 11e). Maximum values of the flux are found at the upper and lower boundaries at *τ*_{opt}. For *t* > 1.8, the flux is characterized by convergence (divergence) at the steering levels of the lower (upper) neutral Eady mode. The distribution of E–P flux divergence implies an increase in wave activity at both steering levels.

In summary, the PV diagnosis and the E–P flux diagnosis indicate that the shortwave SV evolution in energy norm follows the three-stage SV development very well.

## 5. Relative importance of the initial BTAs and interior PV

The relative importance of the initial BTAs and interior PV to the SVs' evolution in various norms can be demonstrated by eliminating either the BTAs or the PV in the initial SV structure, and allowing the modified structure to develop. Because the longwave potential enstrophy SV has no interior PV, it is not discussed in this section.

The potential enstrophy norms of the shortwave SV streamfunction, the perturbation with only initial PV (*ψ*^{′}_{qonly}*ψ*^{′}_{θonly}*ψ*^{′}_{θonly}*ψ*^{′}_{qonly}*τ*_{opt} < 0.5), but becomes smaller than the *ψ*^{′}_{qonly}*ψ*^{′}_{θonly}*ψ*^{′}_{qonly}*ψ*^{′}_{qonly}

The energy norms of the SV streamfunction, the perturbation with only initial PV (*ψ*^{′}_{qonly}*ψ*^{′}_{θonly}*τ*_{opt} = 0.5 and 1.8, the initial amplitudes of the *ψ*^{′}_{qonly}*ψ*^{′}_{qonly}*ψ*^{′}_{θonly}*ψ*^{′}_{qonly}*τ*_{opt} = 0.5 (Fig. 12g) and the *ψ*^{′}_{qonly}

A similar experiment for the *L*_{2} norm is shown in Morgan (2001). The results for the shortwave SVs in *L*_{2} norm are very similar to those for the energy norm. The longwave SVs in *L*_{2} norm shows *ψ*^{′}_{qonly}*ψ*^{′}_{θonly}*τ*_{opt} = 0.5, 1.8, and 4.2, even though the *ψ*^{′}_{qonly}*τ*_{Orr}.

Figure 13 shows the relationship between *τ*_{opt} and the magnitude of the initial BTAs and interior PV. For the shortwave SVs in potential enstrophy norm the magnitude of the initial BTAs is larger than that of the initial interior PV for very short *τ*_{opt}. For *τ*_{opt} > 1.5, the initial interior PV is larger than the initial BTAs and increases with respect to *τ*_{opt}. For the energy and *L*_{2} norms, the magnitudes of the initial interior PV increase as a function of *τ*_{opt} for the small and intermediate optimization times (i.e., dimensionally from several hours to a few days) and the initial interior PV is dominant for any *τ*_{opt}.

## 6. Discussion and conclusions

### a. Mechanisms for growth

In this paper, the initial structures and subsequent evolution of Eady model SVs for the potential enstrophy and energy norms have been diagnosed and the results compared with those for the *L*_{2} norm. Based on the results of piecewise PV inversions and a partitioning of the E–P fluxes associated with the SVs, it has been demonstrated that the initial structure of SVs is dependent on the choice of norm, wavenumber, and optimization interval. In addition, a relationship between the SV initial structure and the mechanisms for growth has been described: For those SVs for which the initial interior PV is important, the relevance of the three-stage SV development scenario described by Morgan (2001) is affirmed. The relative importance of the initial PV and BTAs to the SVs evolution in the potential enstrophy, energy and *L*_{2} norms are summarized in Table 1. Ordered by the norms studied (proceeding from left to right in Table 1), for those SVs that contain PV, the initial interior PV is of increasing importance for the potential enstrophy, total perturbation energy, and *L*_{2} norms, respectively. Excluding short optimization times for the potential enstrophy norm, the importance of initial interior PV is greater for short wavelength SVs compared with long wavelength SVs. For increasing optimization time (proceeding from top to bottom in Table 1), the importance of the initial interior PV increases as a function of optimization time for synoptically relevant optimization time intervals (0.5 < *τ*_{opt} < 5), for those SV structures that contain PV. Below we briefly summarize and interpret the results separately for each norm studied.

For the potential enstrophy norm, the initial SV structure and the subsequent mechanisms for amplification are wavenumber dependent. Of the two amplification scenarios suggested in the introduction, the scenario in which the solution of the optimization problem “selects” an initial structure with no PV is valid for longwave potential enstrophy norm SVs. For these longwave SVs, the initial SV structure consists of thermal perturbations on opposing boundaries that are nearly 180° out of phase. The thermal anomalies on one boundary enhance the edge wave propagation on the opposing boundary so that the edge modes may relatively quickly become phase locked. The resulting configuration then allows for normal mode growth to commence.

In contrast with the longwave SVs, for shortwave potential enstrophy norm SVs' interior PV does play a role in amplification. This PV, initially upshear tilted, is concentrated about the steering levels of the neutral Eady models. While the evolution of the interior PV and BTAs for this norm follows the three-stage evolution described for the *L*_{2} norm by Morgan (2001), amplification of the BTAs by the interior PV is the dominant mechanism for development. The necessity of interior PV for the shortwave SVs for the potential enstrophy norm is simply explained: At wavenumbers beyond the Eady model shortwave cutoff, the BTAs cannot interact to produce a sustained amplification of potential enstrophy (boundary PT variance) for the optimization times longer than the interaction time (time for which the upper wave passes the lower) *t* = 1.5; consequently, SVs at these wavenumbers must be composed of both BTAs and PV. For very short optimization times, the transient interaction between the BTAs is a more efficient mechanism for development than either the interior PV superposing or the interior PV amplifying the BTAs. As the optimization time lengthens, the PV becomes increasingly concentrated at the steering level of the neutral modes and increasingly upshear tilted. This allows for a longer near-resonant interaction (with associated near-linear amplification of the BTAs) between the BTAs and the interior PV.

For the total energy norm, the initial SV structure is dependent on wavenumber while the growth mechanisms are not. As with the *L*_{2} norm, the importance of the initial interior PV relative to the BTAs increases with increasing optimization time. The initial structure of SVs in this norm for all wavenumbers differs from that of the *L*_{2} norm in that the perturbation wind at the boundaries is not negligible. As a consequence, initial development of SVs in the total energy norm is associated with both an amplification of the BTAs and a baroclinic superposition of interior PV. Following the time of maximum superposition *τ*_{Orr} as with the *L*_{2} norm, the SV structure and evolution are largely governed by the mutual interactions between upper and lower BTAs for the longwave SVs, and the interactions between PV nearest the steering level of the neutral Eady modes and the BTAs for the the shortwave SVs.

As suggested in the introduction, because the *L*_{2} norm is equivalent to the kinetic energy norm for a single wavenumber disturbance, modifications of the *L*_{2} norm initial structure are necessary to account for the maximization of the sum of the kinetic and potential energies at the optimization time. That the PV for the total energy norm is initially less tilted than the PV of the *L*_{2} norm (and as a consequence already tilted downshear for optimization times larger than *τ*_{Orr}) is consistent with the fact that for a plane wave PV disturbance, the potential energy is maximized just prior to and just following *τ*_{Orr} (refer to BH01's Fig. 4e). This configuration allows for the simultaneous maximizing of both the perturbation kinetic energy and potential energy in the interior due to both the interior PV and the amplified BTAs.

### b. Implication of results for targeted or adaptive observation

As noted in the introduction, in the context of adaptive or targeted observing, SVs have been used to identify those regions in space for which additional observations, if properly assimilated, may improve a subsequent forecast. These regions may be considered sensitive in that changes to the initial conditions of a numerical weather prediction model (i.e., the analysis increments) that have a projection onto the initial SV structures in these regions will have a particularly large effect on a forecast as opposed to changes made elsewhere. Palmer et al. (1998) note that the sensitive regions identified subjectively using PV may disagree with those identified by objective SV techniques. In particular subjective (PV) diagnosis typically indicates that the regions of maximum sensitivity are in the vicinity of the large quasi-horizontal PT gradients along the dynamic tropopause and near the earth's surface. SV calculations, on the other hand, indicate that the lower and midtroposphere are most sensitive dynamically.

Palmer et al. (1998) argue that these differences in the identification of sensitive regions come from that fact that subjective PV diagnosis emphasizes lateral advection of PV (or advection of PT along the dynamic tropopause), while objective SV diagnosis emphasizes transport of wave activity from regions of small PV gradient (i.e., the troposphere) to regions of large PV gradient (i.e., the tropopause). Provided that wave activity is conserved, the potential enstrophy associated with an initially weak perturbation will increase if the perturbation wave activity is transfered from the lower to midtroposphere to the tropopause.

Morgan (2001) demonstrated that, for the *L*_{2} norm SVs, amplification was in fact associated with the transport of wave activity from the lower troposphere to the tropopause as initially upshear-tilted PV structures were baroclinically superposed. The current study demonstrates that SV amplification for the total perturbation energy and for the shortwave potential enstrophy norm SVs is also associated initially with the superposition of interior PV and concomitant upward transport of wave activity, though for the longwave total energy norm, the upward transport of wave activity from the lower boundary to the upper boundary, associated with mutual amplification of BTAs, also contributes to SV amplification. For the longwave potential enstrophy norm, this study suggests a means of reconciling the apparent discrepancy between subjective (PV) sensitivity diagnosis and objective (SV) sensitivity diagnosis. In the midlatitude troposphere, horizontal gradients of PV along isentropic surfaces are relatively small—much like the vanishing horizontal gradients of PV in the Eady model. As a consequence, subjective diagnoses of sensitive regions are equivalent to objective potential enstrophy SV diagnoses (for longwave SVs) as both approaches identify those regions of the flow for which horizontal advective amplification of PV waves is possible. Despite the lack of interior PV for this norm, SV amplification for this norm also indicates an upward transport of wave activity from the lower troposphere (the surface) to the upper troposphere (the tropopause).

If the potential enstrophy norm were chosen for the purposes of targeted observations, despite the presence of tropospheric SV streamfunction perturbation (and the implied thermal and wind perturbations) structure, only the tropopause and surface PT need be sampled (to the extent that lateral gradients of PV are small). Any sampling strategy that focused on enhanced tropospheric observations or any data assimilation scheme that only created analysis increments in the troposphere based on those targeted observations, would be ineffective in reducing a subsequent forecast error in the sense of potential enstrophy norm.

## Acknowledgments

The authors wish to thank Dr. Chris Snyder of the National Center for Atmospheric Research for helpful comments regarding this work. We have benefited from valuable comments provided by three anonymous reviewers. This work was supported by the National Science Foundation Grant ATM-9810916.

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Time–height cross sections of E–P flux (a) *q*′ − *q*′, (b) *q*′ − *θ*′, (c) *θ*′ − *q*′, (d) *θ*′ − *θ*′, and (e) total for *k* = 1, *τ*_{opt} = 4.2 potential enstrophy SV

Citation: Journal of the Atmospheric Sciences 59, 21; 10.1175/1520-0469(2002)059<3099:DOSVSA>2.0.CO;2

Time–height cross sections of E–P flux (a) *q*′ − *q*′, (b) *q*′ − *θ*′, (c) *θ*′ − *q*′, (d) *θ*′ − *θ*′, and (e) total for *k* = 1, *τ*_{opt} = 4.2 potential enstrophy SV

Citation: Journal of the Atmospheric Sciences 59, 21; 10.1175/1520-0469(2002)059<3099:DOSVSA>2.0.CO;2

Time–height cross sections of E–P flux (a) *q*′ − *q*′, (b) *q*′ − *θ*′, (c) *θ*′ − *q*′, (d) *θ*′ − *θ*′, and (e) total for *k* = 1, *τ*_{opt} = 4.2 potential enstrophy SV

Citation: Journal of the Atmospheric Sciences 59, 21; 10.1175/1520-0469(2002)059<3099:DOSVSA>2.0.CO;2

Singular vector (*k* = 5, *τ*_{opt} = 1.8) streamfunction structure for selected times for the potential enstrophy norm: (a) *t* = 0, (d) *t* = 0.76, (g) *t* = 1.8, and (j) *t* = 3; (b), (e), (h), (k) PV structure at the same times, and (c), (f), (i), (l) potential temperature structure at the same times, respectively

Singular vector (*k* = 5, *τ*_{opt} = 1.8) streamfunction structure for selected times for the potential enstrophy norm: (a) *t* = 0, (d) *t* = 0.76, (g) *t* = 1.8, and (j) *t* = 3; (b), (e), (h), (k) PV structure at the same times, and (c), (f), (i), (l) potential temperature structure at the same times, respectively

Singular vector (*k* = 5, *τ*_{opt} = 1.8) streamfunction structure for selected times for the potential enstrophy norm: (a) *t* = 0, (d) *t* = 0.76, (g) *t* = 1.8, and (j) *t* = 3; (b), (e), (h), (k) PV structure at the same times, and (c), (f), (i), (l) potential temperature structure at the same times, respectively

For *k* = 5, *τ*_{opt} = 1.8 potential enstrophy SV, the time evolution of the (a) magnitudes of the meridional velocities attributed to the upper boundary potential temperature anomaly (*υ*^{′}_{T}*υ*^{′}_{q}*υ*^{′}_{T}*υ*^{′}_{q}*θ*^{′}_{B}*υ*^{′}_{T}*θ*^{′}_{B}*θ*^{′}_{B}*υ*^{′}_{q}*θ*^{′}_{B}*θ*^{′}_{B}*υ*^{′}_{T}*υ*^{′}_{q}*θ*^{′}_{B}

For *k* = 5, *τ*_{opt} = 1.8 potential enstrophy SV, the time evolution of the (a) magnitudes of the meridional velocities attributed to the upper boundary potential temperature anomaly (*υ*^{′}_{T}*υ*^{′}_{q}*υ*^{′}_{T}*υ*^{′}_{q}*θ*^{′}_{B}*υ*^{′}_{T}*θ*^{′}_{B}*θ*^{′}_{B}*υ*^{′}_{q}*θ*^{′}_{B}*θ*^{′}_{B}*υ*^{′}_{T}*υ*^{′}_{q}*θ*^{′}_{B}

For *k* = 5, *τ*_{opt} = 1.8 potential enstrophy SV, the time evolution of the (a) magnitudes of the meridional velocities attributed to the upper boundary potential temperature anomaly (*υ*^{′}_{T}*υ*^{′}_{q}*υ*^{′}_{T}*υ*^{′}_{q}*θ*^{′}_{B}*υ*^{′}_{T}*θ*^{′}_{B}*θ*^{′}_{B}*υ*^{′}_{q}*θ*^{′}_{B}*θ*^{′}_{B}*υ*^{′}_{T}*υ*^{′}_{q}*θ*^{′}_{B}

As in Fig. 2, except for the *k* = 5, *τ*_{opt} = 1.8 potential enstrophy SV

As in Fig. 2, except for the *k* = 5, *τ*_{opt} = 1.8 potential enstrophy SV

As in Fig. 2, except for the *k* = 5, *τ*_{opt} = 1.8 potential enstrophy SV

As in Fig. 3, except for the *k* = 1, *τ*_{opt} = 4.2 energy SV and *t* = 0, 2.6, 4.2, and 6

As in Fig. 3, except for the *k* = 1, *τ*_{opt} = 4.2 energy SV and *t* = 0, 2.6, 4.2, and 6

As in Fig. 3, except for the *k* = 1, *τ*_{opt} = 4.2 energy SV and *t* = 0, 2.6, 4.2, and 6

As in Fig. 4, except for the *k* = 1, *τ*_{opt} = 4.2 energy SV

As in Fig. 4, except for the *k* = 1, *τ*_{opt} = 4.2 energy SV

As in Fig. 4, except for the *k* = 1, *τ*_{opt} = 4.2 energy SV

As in Fig. 2, except for the *k* = 1, *τ*_{opt} = 4.2 energy SV

As in Fig. 2, except for the *k* = 1, *τ*_{opt} = 4.2 energy SV

As in Fig. 2, except for the *k* = 1, *τ*_{opt} = 4.2 energy SV

As in Fig. 3, except for the *k* = 5, *τ*_{opt} = 1.8 energy SV and *t* = 0, 1.63, 1.8, and 4

As in Fig. 3, except for the *k* = 5, *τ*_{opt} = 1.8 energy SV and *t* = 0, 1.63, 1.8, and 4

As in Fig. 3, except for the *k* = 5, *τ*_{opt} = 1.8 energy SV and *t* = 0, 1.63, 1.8, and 4

As in Fig. 4, except for the *k* = 5, *τ*_{opt} = 1.8 energy SV

As in Fig. 4, except for the *k* = 5, *τ*_{opt} = 1.8 energy SV

As in Fig. 4, except for the *k* = 5, *τ*_{opt} = 1.8 energy SV

As in Fig. 2, except for the *k* = 5, *τ*_{opt} = 1.8 energy SV

As in Fig. 2, except for the *k* = 5, *τ*_{opt} = 1.8 energy SV

As in Fig. 2, except for the *k* = 5, *τ*_{opt} = 1.8 energy SV

The time evolution of the potential enstrophy norm of singular vector streamfunction (‖*ψ*^{′}_{sv}_{Q}), *ψ*^{′}_{qonly}*ψ*^{′}_{qonly}_{Q}), and *ψ*^{′}_{θonly}*ψ*^{′}_{θonly}_{Q}) for *k* = 5: (a) *τ*_{opt} = 0.5, (b) *τ*_{opt} = 1.8, and (c) *τ*_{opt} = 4.2. The time evolution of the energy norm of singular vector streamfunction (‖*ψ*^{′}_{sv}_{E}), *ψ*^{′}_{qonly}*ψ*^{′}_{qonly}_{E}), and *ψ*^{′}_{θonly}*ψ*^{′}_{θonly}_{E}) for *k* = 1: (d) *τ*_{opt} = 0.5, (e) *τ*_{opt} = 1.8, and (f) *τ*_{opt} = 4.2. (g), (h), (i) The same as (d), (e), and (f) except for *k* = 5

The time evolution of the potential enstrophy norm of singular vector streamfunction (‖*ψ*^{′}_{sv}_{Q}), *ψ*^{′}_{qonly}*ψ*^{′}_{qonly}_{Q}), and *ψ*^{′}_{θonly}*ψ*^{′}_{θonly}_{Q}) for *k* = 5: (a) *τ*_{opt} = 0.5, (b) *τ*_{opt} = 1.8, and (c) *τ*_{opt} = 4.2. The time evolution of the energy norm of singular vector streamfunction (‖*ψ*^{′}_{sv}_{E}), *ψ*^{′}_{qonly}*ψ*^{′}_{qonly}_{E}), and *ψ*^{′}_{θonly}*ψ*^{′}_{θonly}_{E}) for *k* = 1: (d) *τ*_{opt} = 0.5, (e) *τ*_{opt} = 1.8, and (f) *τ*_{opt} = 4.2. (g), (h), (i) The same as (d), (e), and (f) except for *k* = 5

The time evolution of the potential enstrophy norm of singular vector streamfunction (‖*ψ*^{′}_{sv}_{Q}), *ψ*^{′}_{qonly}*ψ*^{′}_{qonly}_{Q}), and *ψ*^{′}_{θonly}*ψ*^{′}_{θonly}_{Q}) for *k* = 5: (a) *τ*_{opt} = 0.5, (b) *τ*_{opt} = 1.8, and (c) *τ*_{opt} = 4.2. The time evolution of the energy norm of singular vector streamfunction (‖*ψ*^{′}_{sv}_{E}), *ψ*^{′}_{qonly}*ψ*^{′}_{qonly}_{E}), and *ψ*^{′}_{θonly}*ψ*^{′}_{θonly}_{E}) for *k* = 1: (d) *τ*_{opt} = 0.5, (e) *τ*_{opt} = 1.8, and (f) *τ*_{opt} = 4.2. (g), (h), (i) The same as (d), (e), and (f) except for *k* = 5

The amplitude of the initial BTAs and interior PV with respect to the optimization times

The amplitude of the initial BTAs and interior PV with respect to the optimization times

The amplitude of the initial BTAs and interior PV with respect to the optimization times

Relative importance of the initial PV and BTAs to the SVs evolution in the potential enstrophy, energy, and *L*_{2} norms

^{1}

Strictly speaking, SVs are only the initial structures that satisfy these conditions (refer to the definition in section 2a). For brevity, however, in this paper that disturbance that arises from an initial SV perturbation will be referred to as the SV at subsequent times.

^{2}

All the variables in the figures are perturbations.