## 1. Introduction

It is well known that the stability properties of the inviscid Eady (1949) model depend to a large extent on the formulation of the boundary conditions. In the original setup proposed by Eady, rigid lids are prescribed at the earth surface and at the level of the tropopause. Potential temperature (PT) anomalies propagate along such rigid lids and are therefore called edge waves (Davies and Bishop 1994). If the conditions are favorable, baroclinic instability sets in as a sustained interaction between the two edge waves (e.g., Gill 1982; Pedlosky 1987).

The qualitative agreement of the growing Eady wave with observed structures of growing extratropical cyclones has triggered numerous modifications of the original model. The Eady model can be extended for instance by removing the upper rigid lid (Thorncroft and Hoskins 1990; Chang 1992; Bishop and Heifetz 2000). In this case the upper-level edge wave is absent and there is only one growing normal mode (exhibiting linear growth), formed by the resonance between the surface edge wave and an interior potential vorticity (PV) anomaly residing at the steering level of the surface edge wave (Thorncroft and Hoskins 1990; Chang 1992). Another approach is to replace the rigid lid by a more or less realistic stratosphere (Weng and Barcilon 1987; Müller 1991; Rivest et al. 1992; Juckes 1994; Ripa 2001). In this approach, the troposphere is specified by a constant buoyancy frequency *N* ^{2} and a constant vertical shear Λ of the zonal wind * u*. The shear and the buoyancy frequency have different values in the stratosphere and a matching condition is invoked to satisfy continuity requirements. The introduction of a second layer with different shear and buoyancy frequency, replacing the rigid lid, modifies the amplitude and possibly also the sign of the mean PV gradient at the level of the tropopause. The Charney and Stern (1962) condition requires a sign reversal in the mean PV gradient for baroclinic instability to occur. It is therefore expected that the stability properties of the extended two-layer Eady models will (slightly) differ from the conventional Eady model in which the rigid lid is retained (Müller 1991).

The authors mentioned above have investigated the normal-mode stability properties of various extensions of the Eady model. What is presently lacking for the two-layer Eady model is a detailed investigation of the nonmodal growth properties and the optimal perturbation evolution. Nonmodal growth is defined as temporal or sustained growth resulting from the superposition of more than one normal mode. Optimal perturbations are defined as disturbances that amplify optimally for finite time according to a chosen norm. The idea of nonmodal growth is old and goes back at least to the work of Orr (1907). Mainly since the work of Farrell (1982), it has been realized that transient nonmodal growth can play an important role in the initial development of perturbations. This has resulted in a substantial literature on the subject in which authors have investigated nonmodal, and finite-time optimal growth using both numerical and analytical methods with or without taking into account the continuous spectrum^{1} explicitly (Farrell 1982, 1984, 1989; Rotunno and Fantini 1989; Warrenfeltz and Elsberry 1989; Joly 1995; Barcilon and Bishop 1998; Snyder and Joly 1998; Fischer 1998; Hoskins et al. 2000; Bishop and Heifetz 2000; Badger and Hoskins 2001; Heifetz and Methven 2005). Studies specifically devoted to the Eady model have affirmed and emphasized the importance of growth mechanisms other than traditional normal-mode baroclinic instability (Mukougawa and Ikeda 1994; Morgan 2001; Morgan and Chen 2002; Kim and Morgan 2002). A detailed investigation of the different growth mechanisms for the semi-infinite Eady model has been made by De Vries and Opsteegh (2005, hereafter DO5). In DO5 it is found that the finite-time optimal growth at the surface could be explained to a large extent by the occurrence of the simple resonance between interior PV and the boundary edge wave at the surface. The extension to include nonzero *β* while keeping the mean PV gradient zero in the interior, as in Lindzen (1994), has been treated in De Vries and Opsteegh (2006), confirming the importance of resonance. The growth due to resonance is linear in time. Therefore the resonance effect may be more rapid initially than exponential growth from standard baroclinic instability. Nevertheless the resonance effect has not received much attention in the literature (Chang 1992; Bishop and Heifetz 2000; Jenkner and Ehrendorfer 2006).

The aim of the present study is therefore to investigate whether the resonance mechanism as found in DO5 is also important for the surface development in the presence of exponentially growing normal modes. We will investigate in which way the optimal perturbation results, obtained for the Eady model with or without upper rigid lid, are modified when replacing the rigid lid by an unbounded stratospheric layer with reversed shear and higher buoyancy frequency. We use a Green’s function approach that is entirely based on the well-known PV perspective (Bretherton 1966; Hoskins et al. 1985). This Green’s function approach can be generalized easily to include the *β* effect or even highly realistic vertical profiles with smoothly varying buoyancy frequency and zonal wind.

Motivated by the aim mentioned above, we pay attention to the following questions: What are the differences in terms of kinetic energy growth between optimal perturbations of the one-layer semi-infinite version of the Eady model and the present more realistic two-layer model? Can we quantify the importance of normal-mode growth and compare it to the other growth mechanisms, such as growth due to resonance between interior PV and boundary edge waves, and growth due to PV unshielding? Is the tropopause PV wave amplified mostly by the tropospheric PV, by the surface PT (exponential instability) or by the stratospheric PV? All that is known presently for the two-layer Eady model is that the asymptotic longtime evolution will be characterized by the phase-locked interaction of the surface PT and tropopause PV wave forming a pair of counterpropagating Rossby waves (Hoskins et al. 1985; Heifetz et al. 2004).

The paper is organized as follows. A general overview of the model and its normal modes is presented in section 2. The PV perspective and the Green’s function and propagator formalism are introduced in sections 3 and 4. The way to construct and analyze finite-time optimal perturbations is discussed in sections 5 and 6. Sections 7–10 discuss the results, followed by some concluding remarks in section 11.

## 2. Mean flow and perturbation dynamics

The basic state is formed by a two-layer troposphere–stratosphere system as in Müller (1991). The tropopause, defined as the material interface between the troposphere and the stratosphere, is represented by an infinitely thin region, but the extension to a tropopause of finite width is straightforward. The zonal wind * u*(

*z*) is in thermal wind balance with the meridional temperature gradient. The shear Λ of

*and the buoyancy frequency*u

*N*

^{2}are specified separately (but constant) in both layers. A matching condition for the streamfunction and the vertical velocity is applied at the interface. Continuity of the basic-state zonal wind and potential temperature requires that the tropopause has a meridional tilt. This tilt, however, does not play a role in the stability analysis at leading order (Rivest et al. 1992; Juckes 1994). Finally, the Coriolis parameter has been approximated by a constant.

*β*included in the derivations below. The linearized, two-dimensional perturbation evolution is determined by quasigeostrophic potential vorticity (PV) dynamics:

*q*(

*) is the perturbation (basic state) PV and*q

*υ*= ∂

*ψ*/∂

*x*the meridional velocity, where

*ψ*is the perturbation streamfunction. Variables have been nondimensionalized using conventional scalings (Table 1). The boundary and interface conditions are included in Eq. (1) adopting the Bretherton (1966) formalism. In Bretherton’s view,

*q*and ∂

*/∂*q

*y*contain the singular contributions from the surface at

*z*= 0 and the tropopause at

*z*=

*d*

*θ*= ∂

*ψ*/∂

*z*defines the potential temperature and the subscripts (

*S*,

*T*) indicate stratosphere and troposphere respectively. The continuous parts of the PV are related to the streamfunction and to the basic state by

*(*u

*z*) and

*N*

^{2}(

*z*), the surface and tropopause regions attain a finite width, specified by relatively large values of mean PV gradient compared to the ambient mean PV gradients of the interior troposphere. Since the model is unbounded in the vertical the amplitudes of the perturbations are required to vanish as

*z*→ ∞. Clearly, the

*f*-plane Eady model with the upper rigid lid is a special case (i.e., the limit where

*N*

^{2}

_{S}→ ∞) of the general two-layer Eady model described above. In this particular limit, the surface and tropopause mean PV gradients have the same amplitude but opposite sign. In the more general situation the mean PV gradient at the tropopause may attain any amplitude [Eq. (3)]. It can then be expected that the normal-mode stability properties (as well as the vertical structure of the normal modes) are modified (Müller 1991; Rivest et al. 1992; Ripa 2001).

### a. Modal solutions: Discrete and continuous spectrum

Figure 1 shows the real and imaginary part of the phase speed *c* = *c _{r}* +

*ic*of the discrete normal modes (NM) of the two-layer Eady model for two different basic states. Figure 1a shows results for the two-layer analog of the one-layer semi-infinite Eady model, obtained by choosing the shear and the buoyancy frequency ratios such that

_{i}*(*q

_{y}*d*) = 0 [Eq. (3)]. Therefore the discrete NMs are neutral. One of them is a pure surface edge wave with zero PV at the tropopause. The second NM has both nonvanishing PV at the tropopause and nonzero PT at the surface. Because

*(*q

_{y}*d*) = 0, its propagation speed is given by

*(*u

*z*=

*d*) = 1, independent of

*K*.

The second case is a realistic setup (to be used in the remaining part of the paper) in which the zonal wind attains a maximum at the tropopause. This maximum is accompanied by a jump in the buoyancy frequency. The sign change of the shear and the jump of the buoyancy frequency across the tropopause create a mean PV gradient at the tropopause whose absolute value is larger than the mean PV gradient at the surface. For a range of wavenumbers the flow is unstable and a growing normal mode (GNM) and a decaying normal mode (DNM) exist. A short-wave cutoff appears naturally from a consideration of the Rossby height. Absent in the conventional Eady model with the upper rigid lid, also a long-wave cutoff is found in Fig. 1b (Müller 1991). The two-layer Eady model with equal but opposite mean PV gradient [obtained by setting Λ* _{S}* = 0 in the stratosphere, cf. Eq. (3)] is the analog of the rigid-lid Eady model. The long-wave cutoff does not occur in this limit (Müller 1991; Rivest et al. 1992).

Apart from the discrete NMs, there exists (for inviscid flows) an infinite number of so-called continuum modes (CM), which are modal solutions specified by a delta-function distribution of PV at one interior level and a specific combination of boundary PT at the surface and PV at the tropopause [see, e.g., the recent work of Jenkner and Ehrendorfer (2006) for an application to the conventional Eady model with upper rigid lid]. It is only by including the continuous spectrum that the evolution of arbitrary initial perturbations is correctly described. By using PV as the fundamental quantity to formulate initial-value experiments, the CMs are incorporated in a natural way.

## 3. Using PV in initial-value experiments

### a. PV building blocks

^{2}(PVBs), defined as zonally wavelike PV anomalies that reside at a stipulated level

*h*≥ 0:

*N*is the total number of PVBs,

*h*is the vertical position of the PV,

_{j}*η*(

_{j}*t*) is the (real) phase of the PVB at level

*j*, and

*Q*(

_{j}*t*) its amplitude. In the limit where

*N*→ ∞, a continuous PV distribution is obtained. In principle all PVBs in Eq. (5) have time-dependent amplitudes

*Q*(

_{j}*t*). In practice however,

*Q*(

_{j}*t*) will be time-dependent only if the mean PV gradient is nonzero at level

*j*. This motivates us to discriminate between active and passive PVBs, depending on whether or not they may amplify in time. In the present

*f*-plane study, the active PVBs reside at the surface and the tropopause [see Eq. (3)] and we label them with subscripts

*B*and

*T*, respectively. A generalization in which

*β*is included renders all PVBs to be active because the mean PV gradient is nonzero everywhere in that case. Note, however, that a regularization is required in that case [see Eqs. (7)–(9) below].

### b. Green’s functions

*k*as

**q**

*(*

_{j}*t*) =

*q*(

*z*,

_{j}*t*) is the PV at level

*j*, [𝗨]

_{ij}=

*(*u

*z*)

_{i}*δ*represents the mean flow,

_{ij}*δ*is the Kronecker symbol, and [𝗤

_{ij}**y**]

_{ij}=

*(*q

_{y}*z*)

_{i}*δ*the mean PV gradient. The matrix

_{ij}**Δ**is necessary to properly weight the discrete versus the continuous contributions to the mean PV gradient for a given discretization.

^{3}It is defined as a diagonal matrix with entries

*i*,

_{B}*i*are the levels of the surface and tropopause, respectively, and

_{T}*δz*is the distance between the levels used to approximate the interior. Finally [𝗚]

*=*

_{ij}*G*(

*z*,

_{i}*z*) in Eq. (6) is the Green’s function defined through

_{j}*(*

_{z}G*z*,

*h*)|

_{z}_{=0}= 0 at the surface, ∂

*(*

_{z}G*z*,

*h*)|

_{z}_{→∞}= 0 as

*z*→ ∞ and ∂

*(*

_{z}G*z*,

*h*)|

_{z=d+}

*N*

^{−2}

_{s}= ∂

*(*

_{z}G*z*,

*h*)|

_{z=d−}

*N*

^{−2}

_{t}if a tropopause is modeled [i.e., zero PV at the tropopause in the sense of Eq. (2)]. Physically,

*G*(

*z*,

*h*) represents the streamfunction

*ψ*(

*z*) attributable to a PVB with unit PV amplitude at height

*h*. In general, the inversion, Eq. (4), depends nontrivially on

*N*

^{2}(

*z*), but in the present paper the computation of

*G*(

*z*,

*h*) is straightforward (appendix B). Figure 2 displays

*G*(

*z*,

*h*) as a function of the height

*h*of the PV (horizontal axis) and

*z*(vertical axis) for the semi-infinite one-layer Eady model (hereafter M1) with

*N*

^{2}

_{t}= 1, and for the two-layer Eady model (hereafter M2) with a tropopause at

*d*= 1 (10 km) and

*N*

^{2}

_{S}= 4

*N*

^{2}

*. We have chosen a zonal wavenumber*

_{T}*k*= 1.55, which corresponds to the most unstable wave in M2 with Λ

_{S}= −1 (Λ

*= 1,*

_{T}*N*

^{2}

_{t}= 1 and

*N*

^{2}

_{S}= 4). The Green’s function is symmetric in its arguments,

*G*(

*z*,

*h*) =

*G*(

*h*,

*z*), which results from making the Boussinesq approximation

^{4}(Robinson 1989). For a given

*h*,

*G*(

*z*,

*h*) attains maximum amplitude at

*z*=

*h*in M1, decaying exponentially away from the PV. Furthermore, in M1 a low-level PVB is associated with a stronger maximum wind speed than a PVB of the same amplitude at higher altitudes. In M2 the exponential decay is much faster in the stratosphere than in the troposphere because

*N*

^{2}

_{s}= 4

*N*

^{2}

_{t}.

## 4. Propagator dynamics

*t*) = exp(𝗔

*t*), which we will call the P-propagator (note that the matrix exponential is defined via its Taylor expansion). What is the information contained in a component [𝗣(

*t*)]

*? It is the contribution from the PVB at level*

_{ij}*j*at

*t*= 0 [i.e., an initial condition

**q**(0) with all components zero except for component

*j*, which is unity] to the PVB at level

*i*at time

*t*. Clearly, [𝗣 (

*t*)]

*contains information about the excitation of all the other PVBs (initially by the PVB at position*

_{ij}*j*and later by all other active PVBs that are excited). A more interesting type of propagator (from the point of view of cyclogenesis) can be constructed by making use of the identity

**= 𝗚 ·**

*ψ***q**in Eq. (10) above. One then obtains

*t*) = 𝗚 · 𝗣(

*t*) = 𝗚 exp(𝗔

*t*). A component [𝗪(

*t*)]

*(which we will call the W-propagator)*

_{ij}^{5}gives the contribution of the PVB at time

*t*= 0 at level

*j*to the streamfunction at level

*i*at time

*t*. In other words, the components [𝗪(

*t*)]

*are associated with the time evolution initiated by individual PVBs. This does involve the subsequent excitation of the active PVBs after*

_{ij}*t*= 0.

### a. Propagator maps and the structure of PVB interactions

Figures 3 and 4 display contour maps of |𝗪(*t*)| (constructed using *k* = 1.55). To interpret these figures, remember that a vertical cross section at *h* = *i* (on the horizontal axis) produces the absolute value of the streamfunction (in contours) as a function of height *z* (vertical axis). At initial time |𝗪(*t*)| is equivalent to the (symmetric) Green’s function because no advection of the mean PV gradients has occurred yet. Gradually symmetry is lost as time progresses. In M1 (Fig. 3), the maximum propagates along the bottom, from the left corner (surface) to the right (higher levels), and asymptotes to the steering level of the surface edge wave (*z _{s}* = Λ

*/*

_{T}*k*= 0.65). Physically, this propagation indicates that the strongest wind maxima can be found initially by putting a PVB (of unit PV) near the surface. If we wait longer, it is more favorable (in the sense of obtaining a stronger wind maximum) to locate the PVB near the steering level of the surface PVB. For the PVB positioned near the steering level, the wind maximum propagates rapidly from its initial position at the level of the PVB down toward the surface at time

*t*= 2.5 (Figs. 2a and 3a). The nonzero surface mean PV gradient allows the interior PVB to excite the active surface PVB, which is a manifestation of the resonance effect (DO5). Upper-level PVBs hardly excite any surface PVB. This can be concluded from the fact that at

*t*= 10 the bottom-right part in Fig. 3d is still almost identical to the same part in Fig. 3a.

A similar investigation can be performed for M2 (Fig. 4). Since the mean PV gradient is now nonzero, not only at the surface but also at the tropopause, M1 and M2 start to diverge after some time because of the inevitable excitation of the active tropopause PVB. For short times (up to *t* = 2.5; i.e., one day in dimensional units) and with the purpose of getting the strongest winds at the surface, the PVB is best positioned just above the surface (similar to M1). For longer times the best position of the initial PVB is again the midtroposphere (in fact, the steering level of the GNM is approached from below). In a similar way, the strongest winds at the tropopause level can be obtained by letting the PVB approach the steering level from above. The wind maxima obtained in M2 after two and four days are significantly stronger than in M1.

### b. Summary and the range of applicability

The main importance of the W-propagator contour maps at different instants of time is, that these maps may answer the question of where in the atmosphere a vertically localized amount of PV (i.e., the initial PVB), will produce in time the strongest winds at a specific level (*z* in the figures). Hence, the W-propagator relates directly to the subject of optimal perturbations to be discussed in the next sections.

**q̂**

_{i}is a vector of length

*N*with all zeros except at position

*i*at which [

**q̂**

_{i}]

_{i}= [

**q**(0)]

_{i}and

*is the streamfunction associated with the time-evolved*

**ψ̂**_{i}**q̂**

_{i}In other words, knowledge of W gives information of how PV at each level evolves in the system.

We illustrate the range of applicability by another example relevant to cyclogenesis. The surface development occurring in the models initiated with a single PVB of unit amplitude can be compactly summarized by computing |**W**_{1}* _{j}*(

*t*)| as a function of time and the height

*h*of the PVB (Fig. 5). For M1 the optimal position of the PVB gradually approaches the steering-level from below. Although PVBs near the surface are optimal for short times, their phase speed differs too much from the surface edge wave and from

*t*≥ 4 destructive interference occurs. For M2 the same story holds for small times. Low-level PVBs generate the strongest surface winds. The steering level becomes more and more the favorable height for longer times. In the long-time run, the GNM dominates, and the initial position of the PVB is less important. Single PVBs in the upper stratosphere are not very efficient in generating strong surface winds in short time. It takes at least four days (

*t*= 10) for a single PVB located at

*h*= 2 to produce surface winds comparable to the winds associated with an initial PVB of the same amplitude near the surface (follow the 0.4 contour).

## 5. Finite-time optimal growth

### a. Construction of optimal perturbations

Given the PVB-decomposed set of dynamical Eqs. (6), and its solution in terms of either **q**(*t*) or ** ψ**(

*t*), it is straightforward to construct a perturbation that produces optimal growth for finite time in a certain norm. Such a norm-dependent optimal perturbation is called a singular vector (SV). Commonly investigated norms are potential enstrophy (PE), kinetic energy (KE), and total quasigeostrophic energy (TE), and it should be stressed that the norm can be different at initial and final time. For instance, SVs optimizing PE are found from the eigenanalysis of [𝗣

^{†}(

*t*) · 𝗣(

*t*)] (the † symbol indicates taking the conjugate transpose).

*j*at which

**W**

_{1}

*(*

_{j}*t*) attains its maximum at a given time. An initial PVB of unit amplitude at that particular level is in fact also the result of computing the SV constructed with one PVB in the initial PV distribution and a PE norm at initial time and a surface kinetic energy norm (SKE) at time

*t*. Keeping in mind the application to rapid surface cyclogenesis, we compute the SVs in the KE norm,

^{6}which, for fixed horizontal wavelength, is proportional to the streamfunction variance or L2 norm. These are found from the generalized eigenvalue problem

**W**

*: =*

_{τ}**W**(

*t*=

*τ*). The SV is given by the eigenvector

**q**(0) of the above equation with the largest eigenvalue

*λ*. The largest eigenvalue represents the dimensionless growth factor of the KE, Γ

_{KE}= KE(

*t*)/KE(0). For the basic state M2 we use Λ

*= 1,*

_{T}*N*

^{2}

*= 1, Λ*

_{T}*= −1, and*

_{S}*N*

^{2}

*= 4 as in section 4. In M1 we take Λ*

_{S}*= 1,*

_{T}*N*

^{2}

*= 1. PVBs up to*

_{T}*z*= 3 (30 km) are taken into account and we use 61 equally spaced levels. The optimization time

*t*= 5 (47 h) is chosen as well as

*k*= 1.55. This is the wavenumber producing the fastest GNM for basic state M2. Note that the SV constructed for this wavelength will not necessarily produce the largest transient growth because the surface and tropopause PVB approach a so-called hindering configuration as the GNM is reached (Heifetz and Methven 2005). Singular vectors computed for a smaller wavelength in which the discrete NMs are neutral may reach (temporarily) a helping configuration, which may enhance the growth. The present study therefore assesses to what extent the other growth mechanisms can compete with the most rapid modal growth from normal-mode baroclinic instability.

## 6. Diagnosing the optimal perturbation evolution

*ψ*(

*t*) into four parts. From bottom to top we have

*ψ*associated with the active surface PVB,

_{B}*ψ*

_{trop}associated with the tropospheric passive PVBs,

*ψ*associated with the active tropopause PVB, and finally

_{T}*ψ*

_{stra}representing all PVBs above the tropopause:

*ψ*

_{PV}for the streamfunction associated with all PVBs, which are part of the continuous distribution (in the present paper, the PVBs contributing to

*ψ*

_{PV}are all passive), and

*ψ*

_{NM}as a notation for the streamfunction associated with the discrete NMs. The surface dynamics is diagnosed from the projections of the different components of the streamfunction on the SKE (surface kinetic energy):

## 7. Isolating growth mechanisms

A major goal of the present study is to investigate which growth mechanisms play a key role in the SV surface dynamics. The Green’s function approach allows an unambiguous investigation of the growth mechanisms. For the semi-infinite Eady problem reported in DO5 it is found that a resonance between the surface edge wave (the active PVB at the surface) and interior, passive PVBs near the steering level of the edge wave is crucial for the surface development, even if a large number of interior PVBs is included. In the present study, exponential instability (the sustained interaction between the two active PVBs) is an additional growth mechanism. We will investigate in which way the results of DO5 are modified by the presence of the exponential growth mechanism.

### a. Growth rate and phase speed of the PVB

*γ*(

^{q}_{i}*t*) and the phase speed

*c*(

_{i}*t*) of the PVB at level

*i*:

*U*=

_{i}*(*u

*z*=

*i*), [𝗦𝗶(

*t*)]

*= sin[*

_{ij}*η*(

_{i}*t*) −

*η*(

_{j}*t*)], and [𝗖𝗼(

*t*)]

*= cos[*

_{ij}*η*(

_{i}*t*) −

*η*(

_{j}*t*)]. These equations generalize the results of appendix A. The antisymmetry of [𝗦𝗶(

*t*)]

*prohibits any PVB, whether it is passive or active, to contribute to its own growth rate. Physically this result is due to the fact that the PV and its associated wind field are*

_{ij}*π*/2 out of phase for a single PVB. The instantaneous growth rate of the surface PVB is formed by contributions from 1) the tropospheric PVBs, 2) the active tropopause PVB, and 3) the stratospheric PVBs. Using a similar notation for the tropopause PVB we get

*A*

_{i,j}with

*i*= (

*B*,

*T*) and

*j*= (

*B*, trop,

*T*, stra) contains all terms from group

*j*contributing to the growth rate of PVB

*i*[i.e., a partial sum of terms in Eq. (16) above]. We can follow each of the components in time and study which group of PVBs contributes mostly to the growth rate of the active PVBs.

*C*

_{i,j}(

*t*) is the contribution to the instantaneous phase speed of PVB

*i*due to the group of PVBs labeled

*j*[i.e., a partial sum of terms in Eq. (17) above]. In the GNM configuration both

*c*and

_{B}*c*are equal and time-independent in all norms (on the

_{T}*f*plane the contributions involving the interior of the troposphere and stratosphere are zero).

### b. Growth rate of the streamfunction

^{7}we write

*ψ*(

*z*,

*t*) = Ψ(

*z*,

*t*) exp [

*iϵ*(

*z*,

*t*)] and repeat the steps that led to Eq. (16) to obtain

*F*

^{tot}

*(*

_{ij}*t*) = −

*k*[

*t*)]

_{ij}[𝗚 · (𝗨 +

**Δ**· 𝗤

**y**· 𝗚)]

_{ij}, where [

*t*)]

_{ij}= sin[

*ϵ*(

_{i}*t*) −

*η*(

_{j}*t*)] (note the tilde) relates to the phase difference between the PVB at level

*j*and the streamfunction at level

*i*. The next logical step seems to subdivide Eq. (22) into contributions associated with the different PVBs similar to Eqs. (18)–(19). However, such a partitioning produces ambiguous results because single contributions

*F*

^{tot}

*/Ψ*

_{ij}Q_{i}_{i}depend on the (arbitrarily chosen) speed of the frame of reference. In appendix C we elaborate on this.

*F*

^{Orr}

*(*

_{ij}*t*) = −

*k*[

*t*)]

_{ij}[𝗚 · 𝗖𝗿(

*t*)]

_{ij}, with [𝗖𝗿(

*t*)]

*=*

_{ij}*δ*(

_{ij}c_{j}*t*) and

*c*(

_{j}*t*) the instantaneous phase speed of the PVB at level

*j*[Eq. (17)]. In (23)

*F*

^{Orr}

*(*

_{ij}*t*) contains the two separate contributions:

*F*

^{rem}

*(*

_{ij}*t*) contains the remaining growth processes:

*=*

_{ij}*δ*. Therefore

_{ij}*F*

^{rem}

*(*

_{ij}*t*) is independent of the speed of the (arbitrarily chosen) frame of reference. Furthermore, it can be verified that

*F*

^{rem}

*(*

_{ij}*t*) = 0 at time

*t*if all PVBs have zero phase difference. In that case, the instantaneous growth at that time occurs solely via the Orr mechanism. The equation above also shows that if

*= 0 everywhere, the only streamfunction growth that may occur is via the (kinematic) Orr mechanism [because 𝗨 = 𝗖𝗿 at all levels and*q

_{y}*F*

^{rem}

*(*

_{ij}*t*) = 0 identically].

*α*,

*α*′ = (

*B*, trop,

*T*, stra). Each term [

*f*

_{α,α′}] represents a portion of the streamfunction growth rate. In fact, [

*f*

_{α,α′}] is that portion of the streamfunction growth rate that stems from the amplification of PVB

*α*by PVB

*α*′ (note that

*α*and

*α*′ can also be a group of PVBs). The contributions

*f*

_{B,B}and

*f*

_{T,T}are zero because they contribute to the phase propagation only and have been included already in the Orr term. The components

*f*

_{trop,}

*and*

_{α}*f*

_{stra,}

*are zero if the mean PV gradient is zero in the interior troposphere and stratosphere. Certain combinations of*

_{α}*f*

_{α,α′}are interpreted easily in terms of the growth mechanisms.

Generalized Orr: This is the term

*γ*^{Orr}(_{ij}*t*) defined in Eq. (23), which generalizes PV unshielding of passive PVBs by incorporating also unshielding between passive and active PVBs, termed PV–PT unshielding in DO5.*B*resonance: This is the contribution from the passive PVBs to the growth of the active surface PVB;*B*resonance is computed as*f*_{B}_{,trop}+*f*_{B}_{,stra}.*T*resonance: Similar to*B*resonance but for the tropopause PVB. The*T*resonance is computed as*f*_{T}_{,trop}+*f*_{T}_{,stra}.*B*–*T*interaction: This term involves the interaction between the active PVBs at the surface and the tropopause and is computed as*f*_{B,T}+*f*_{T,B}.

Instead of identifying the mechanisms, one can also study the effect of the different (groups of) PVBs. The sum Σ_{α} f_{α,B} for instance produces the growth attributable to the surface PVB. This gives an indication of the importance of the different (groups of) PVBs for the streamfunction development. Finally, we want to emphasize that the present choice of separating the domain into four different regions, is not required. Using the above techniques one can in principle follow each individual PVB to see in which way it contributes to the growth.

## 8. Results: One single PVB

Optimal perturbations with one PVB in the initial perturbation are computed in the following way (DO5). We start with one PVB at the lowest model level, which has an amplitude such that the total initial KE is unity; Γ_{KE} at final time (*t* = 5) is subsequently computed. Then the position of the initial PVB is varied and the computation is performed again. This procedure is repeated until the highest model level is reached. More details on the computation of SVs with a limited number of PVBs can be found in appendix D. Figures 6a,b show how Γ_{KE} depends on the height of one single PVB for M1 and M2, respectively. In M1 the height at which Γ_{KE} attains its optimum (called the optimal growth level) for short optimization times is found to be above the steering level of the surface edge wave (*z _{s}* = Λ

*/*

_{T}*k*= 0.65). The optimal growth level approaches the steering level from above if the optimization time is increased. The results differ from Fig. 5 because we optimize the growth factor Γ

_{KE}rather than looking at the height at which a PVB of unit initial amplitude attains the largest surface streamfunction. For more details on M1 we refer to DO5. For M2, Fig. 6b shows that the optimal growth level is found close to the midtroposphere, almost coinciding with the steering level. For stratospheric PVBs in M2, Γ

_{KE}(

*t*) is much smaller than for tropospheric PVBs for all optimization times. From Figs. 6a,b we conclude two things. First, M2 produces larger growth factors than M1 for this value of

*k*(to remind, this is the value of

*k*producing the largest normal-mode growth rate for the given basic state), even for small optimization times. Second, for all optimization times a single PVB positioned at one of the active levels leads to lower values of Γ

_{KE}.

The time evolution of *ψ* and its various components in M2 is shown in Fig. 7. Also shown is the sum *ψ*_{NM} ≡ *ψ _{B}* +

*ψ*in the panels at the bottom. Initially the streamfunction is given entirely by

_{T}*ψ*

_{PV}of the optimally positioned PVB. Both a surface and a tropopause PVB are generated after

*t*= 0 by advection of ∂

*/∂*q

*y*. Due to the opposite sign of ∂

*/∂*q

*y*at the surface and at the tropopause, the surface and tropopause PVB, which are excited by the wind field of the interior PVB, are

*π*out of phase initially. At optimization time the surface and the tropopause PVB seem to have reached the phase-locked GNM configuration.

### a. Surface development of the streamfunction and kinetic energy

A comparison is made between the surface development occurring in M1 and M2 (Fig. 8). This is done by following in time the relative contributions to the SKE [components of Eq. (15) divided by the instantaneous SKE]. At optimization time, the optimal perturbation has produced more SKE in M2 than in M1. In both models the SKE is dominated initially by the contribution from the passive PVB. After *t* = 0, the active surface PVB is rapidly excited in both models and becomes the largest contribution to the SKE after *t* = 1. In M2 its contribution even exceeds unity as time increases beyond *t* = 3. In M2, the tropopause PVB is excited as well (dashed line) but its contribution to the instantaneous SKE remains negative throughout the evolution, confirming that the surface and tropopause PVB are in the hindering configuration (Heifetz et al. 2004). In M2, the contribution from the discrete normal-mode pair (GNM + DNM) dominates the long-time evolution.

Two basic questions rise. The first is which physical mechanism is responsible for the rapid growth of the SKE? The second question is which PVBs are important in the propagation and the growth of the active PVBs? To return to the first question (the second is addressed in the next subsection), Fig. 9 shows the contributions from the different growth mechanisms using the methods developed in section 6. For the barotropic initial condition of the one-PVB experiment, the growth rate is identically zero at *t* = 0. As time increases and the structure develops a vertical tilt, the SKE growth rate becomes nonzero. Both in M1 and M2 the SKE growth rate rapidly increases in the first hours of the development and attains a maximum around *t* ∼1.5. In M2 the growth rate then gradually decreases toward the asymptotic GNM value. In M1, *B* resonance completely determines the SKE growth rate. The Orr mechanism, in this case the unshielding between the interior PVB and the surface PVB, is rather weak. For a surprisingly long time, *B* resonance also is the most important growth mechanism in M2, in fact almost completely up to the optimization time *t* = 5. Although the contribution from the discrete normal modes in M2 could explain the largest part of the instantaneous SKE (and even the streamfunction structure) after *t* = 1.5, the growth is, in fact, not resulting from normal-mode baroclinic instability! The contribution from the traditional baroclinic *B*–*T* interaction is zero initially and increases only slowly. While *B* resonance works productively, the *T* resonance in M2 works destructively for the SKE throughout the time evolution toward *t* = 2*t*_{opt}. The Orr mechanism remains unimportant, similar to M1. The remaining issue is, to determine what causes the propagation and the rapid growth of the surface and tropopause PVB. We will focus on M2.

### b. Growth of the surface and tropopause PVB

Using Eqs. (18)–(21) we have computed the different contributions to the growth rate and the phase speed of the surface and tropopause PVB in M2 (Fig. 10). The instantaneous growth rate of the surface PVB decreases in time toward the asymptotic GNM value (Fig. 10a). The large value at initial time is caused by the small amplitude of the surface PVB at that time. The contribution from the optimally positioned interior PVB, *A _{B}*

_{,trop}(

*t*) (it is positioned in the troposphere) decreases with time. The contribution from the tropopause PVB,

*A*

_{B,T}(

*t*) determines the asymptotic growth rate. The contribution from the stratosphere is zero. It is noteworthy to mention that the contributions from the tropopause and the troposphere become equal just prior to

*t*=

*t*

_{opt}although the surface streamfunction contribution of the tropopause PVB becomes of equal amplitude as the surface contribution from the tropospheric PVB around

*t*= 2.6 (not shown). Figure 10a shows that it takes more time for the tropopause PVB to actually amplify the surface PVB. It is confirmed that the surface PVB mainly grows from

*B*resonance with the interior PVB up to

*t*= 5 (the first two days). For the tropopause PVB, a similar story holds (Fig. 10c).

We now turn to the different contributions to the phase speed of the surface PVB (Fig. 10b). The total phase speed remains nearly constant approaching the GNM phase speed. In absence of the interior tropospheric PVB and the tropopause PVB, the phase speed would have equaled *U*(0) + *C _{B}*

_{,}

*∼ Λ*

_{B}*/*

_{T}*k*= 0.65. The presence of the tropopause (and the excitation of the tropopause PVB) is crucial as it causes the surface PVB to propagate with a lower phase speed. A tropopause PVB in isolation would propagate at a phase speed

*U*(

*d*)

*+ C*

_{T}_{,}

*, which is much slower than the surface PVB in isolation. For the tropopause PVB the surface PVB is crucial in speeding up the tropopause PVB (thick full line in Fig. 10d). The interior, tropospheric PVB (dotted lines) does not influence the phase speeds of the surface and tropopause PVBs much, which confirms that its task is mainly to amplify the surface and tropopause PVB.*

_{T}## 9. Three PVBs and the effect of PV unshielding

The analysis of the Green’s function in Fig. 2 showed that nearby PVBs have similar vertical streamfunction structure. As a result, nearby PVBs can mask each others’ wind fields very efficiently if they are positioned out of phase. As time increases, the passive PVBs are advected by the basic state, and the existing shear unshields the wind fields associated with the individual passive PVBs. This so-called PV unshielding is known to be important in the SV evolution. By computing the SV with three PVBs in the initial structure, we are able to investigate the influence of PV unshielding (coming to expression in the Orr mechanism) for the basic state of the present paper. The position of the three initial PVBs is varied, and for each configuration the finite-time growth is computed. The configuration leading to optimal growth is subsequently analyzed (DO5). Note that the initial PVBs are allowed at all positions, including the surface and tropopause.

Figure 11 shows the time evolution of the three-PVB SV. The total streamfunction is maximized in a narrow region around the midtroposphere and is tilted against the shear. As time progresses, the total streamfunction loses some of its initial tilt and rapidly attains a large amplitude in the entire troposphere. The largest growth occurs at the surface. The three PVBs with nonzero amplitude are all positioned in the interior of the troposphere (near the steering level of the GNM to be excited) and therefore of the passive type and none of them resides at one of the active levels. Furthermore, the PVBs are positioned sufficiently out of phase and against the shear of the tropospheric basic-state wind. Potential vorticity unshielding dominates the growth in the interior initially (Morgan 2001; Morgan and Chen 2002). It serves to rapidly generate strong winds at the surface and the tropopause. These strong winds are then used primarily to excite the active PVBs at the surface and tropopause. Although the structure of the total SV streamfunction at optimization resembles the GNM configuration, it is important to emphasize that the surface and tropopause wave have not at all reached the phase-locked GNM configuration at optimization time (bottom panels). This result, which differs from the previous section, indicates that the interior PVBs play an important role in the amplification of the SV streamfunction.

### a. Surface development of the streamfunction and kinetic energy

The SKE evolution of M1 and M2 is compared in Fig. 12. In contrast to the one-PVB results, M1 exhibits (up to the optimization time) slightly more surface growth than M2. As before, the SKE can be attributed initially completely to the interior passive PVBs. The contribution from the surface PVB to the SKE becomes the largest somewhat later than in the one-PVB problem. In M2 the contribution from the tropopause PVB to the SKE again is negative throughout the evolution.

The growth mechanisms involved in the optimal evolution are shown in Fig. 13. In contrast to the one-PVB results, the Orr mechanism is the largest contribution to the SKE growth rate up to *t* = 1.5. Owing to the Orr mechanism, large streamfunction values are rapidly generated at the surface and the tropopause. These large streamfunction values will in turn lead to rapid growth due to resonance. In M1, the contribution from the Orr mechanism attains a maximum around *t* = 1. After that period, *B* resonance takes over and determines the asymptotic growth. In M2, the Orr mechanism also dominates initially and *B* resonance afterward. The contribution from *T* resonance on the surface is negative throughout the evolution (except the first few hours), which confirms the results of the previous section. The most important difference with the one-PVB evolution in M2 is that the contribution from the *B*–*T* interaction (growth of the GNM plus decay of the DNM) remains negative even long after optimization time has been reached. The fact that *T* resonance and the *B*–*T* interaction contribute negatively for such a long time explains the more vigorous surface development in M1.

### b. Growth of the surface and tropopause PVB

The instantaneous growth rates and phase speeds of the surface and the tropopause PVB confirm the observations of the previous section (Figs. 14a–d). The instantaneous growth rate of the surface PVB is reduced by the presence of the tropopause PVB during the complete time evolution toward *t* = *t*_{opt} except in the first few hours of the development. The contribution from the tropospheric passive PVBs is the dominant term in producing growth of the surface and tropopause PVB. Due to the upshear tilt of the initial PVBs, the phase speed evolution differs from the one-PVB problem. The effect of the tropospheric PVBs is to reduce the phase speed of the surface PVB and to increase the phase speed of the tropopause PVB. The contribution from the tropopause PVB to the phase speed of the surface PVB (and vice versa) is consistent with Heifetz et al. (2004). The active PVBs start to hinder phase propagation roughly after one day (*t* ≥ 2.6). All contributions from the stratospheric PVBs are still zero (none of the three PVBs reside in the stratosphere).

## 10. The continuous problem

The natural question to ask is in which way the three-PVB problem resembles the continuous problem. For the same basic state as before the SV is determined using *N* = 61 PVBs (including the two active PVBs). The evolution of the *t* = 5 SV resembles the time evolution of the three-PVB problem qualitatively well (Fig. 15). Absent in the three-PVB problem, the streamfunction in the stratosphere is tilted eastward with height initially (i.e., against the stratospheric shear). The total growth of the SV is larger than in the three-PVB problem. The surface and tropopause PVB do not reach the GNM configuration at optimization time despite the fact that the structure of the total SV streamfunction resembles the GNM qualitatively (top panels). In contrast to the one- and three-PVB cases, the active surface and tropopause PVB have nonzero amplitude at initial time. The streamfunction associated with the active PVBs (bottom panel) is shielded by the streamfunction from the interior PVBs, which is almost barotropic and *π* out of phase. As time increases, PV unshielding occurs and starts generating large streamfunction values near the surface and the tropopause. During the initial stage (*t* < 2) both the surface and the tropopause PVB retain almost identical amplitude but propagate at a different phase speed. As a result the total contribution from the active PVBs decreases (cf. bottom panels at *t* = 0 and *t* = 1.6). After *t* = 2 both the surface and tropopause PVB rapidly increase in amplitude.

### a. Surface development of the streamfunction and kinetic energy

The surface development is further analyzed in Fig. 16. Compared to the computations performed using a limited number of PVBs, the initial SKE becomes very small both in M1 and M2. This is the result of the effective canceling of individually large contributions. Similar to the three-PVB problem, the SKE at optimization time in M1 is slightly larger than in M2, a result that holds even if the optimization time is increased toward three days (not shown). As in DO5, the SKE decreases in the first hours of the development before strong surface cyclogenesis occurs (not shown). The relative contributions to the SKE reflect this initial decrease of the SKE as they strongly fluctuate, which has been the reason for omitting the initial period for *t* < 1.5 from the graphics. After *t* < 1.5 we get results similar to the three-PVB problem. Initially the largest contribution is from the passive PVBs, followed by the contribution from the surface PVB. In contrast to the one- and three-PVB problem, the tropopause PVB contributes positively for a short period, whereas the surface PVB contributes negatively up to *t* = 3. The sum of GNM and DNM (*ψ _{B}* +

*ψ*=

_{T}*ψ*

_{NM}) contributes negatively almost the complete first day of the development (up to

*t*= 2.5). In M2 we have subdivided the contribution from the passive PVBs into a part from the tropospheric PVBs and a part from the stratospheric PVBs. The contribution from the stratospheric PVBs to the SKE is completely negligible (even though most of the PVBs reside at those high levels).

To complete the analysis of the surface development Fig. 17 shows that (again we have omitted times *t* < 1.5) the Orr mechanism explains the SKE growth initially in M1 and M2. A barotropic tropospheric PV tower is formed only at optimization time (not shown). As in the previous cases, *B* resonance takes over, in M2 finally followed by the *B*–*T* interaction (after nearly two optimization times). The contribution from *T* resonance remains small.

### b. Growth of the surface and tropopause PVB

The evolution of the growth rates of the surface and tropopause PVBs shows rapid growth after initial decay (Fig. 18). After this initial period (say the first day, *t* = 2.5), the growth rates smoothly decay toward GNM values, but, in agreement with the previous sections, the asymptotic GNM values are not reached, not even after two optimization times. Generally, it can be said that both the surface and tropopause PVB amplify mostly due to resonance with the tropospheric PVBs, which have produced strong surface and tropopause winds through the Orr mechanism. The surface PVB propagates westward during the first day (roughly up to *t* = 2.8, Fig. 18b), whereas the tropopause PVB is accelerated to propagate with a speed larger than the maximum zonal wind speed of the basic state up to *t* = 3.5. The contribution from the tropopause PVB to the growth rate of the surface PVB (and vice versa) oscillates between positive and negative values, before settling in the asymptotic GNM configuration. Finally, even though there are many stratospheric PVBs their contribution to both the growth rate and phase speed of the tropopause PVB (let alone to the growth of the surface PVB) remains by far the smallest contribution.

## 11. Summary and concluding remarks

The existing normal-mode studies of the two-layer Eady model have been extended by a detailed investigation of the finite-time and optimal growth properties. A number of quite generally applicable tools have been developed (in sections 4–6), which make an unambiguous investigation possible of the growth mechanisms that play a role in initial-value experiments (not necessarily optimal perturbations). The basic constituent, which makes this investigation possible, is the so-called PV building block (PVB), which takes the form of a zonally wavelike, vertically localized PV anomaly. A distinction has been made between passive and active PVBs, depending on whether the mean PV gradient at their level is zero or not, respectively. The tools have been formulated in such a way that they can be used to understand initial-value experiments for much more general basic states than the one described in the present paper (such as *β*-plane models with a more complex zonal wind profile or, after a straightforward generalization of the equations and methods, with a time-dependent basic state). The present paper can therefore also be viewed as a concrete example of the use of the tools. The companion paper (De Vries and Opsteegh 2007) discusses the incorporation of *β* in the theory. On the *β* plane all PVBs are active and interact by advecting each others’ mean PV gradient.

Regarding the optimal perturbation evolution in the two-layer Eady model, it is concluded that the optimal surface evolution differs fundamentally from the standard view on baroclinic development due to normal-mode baroclinic instability in Eady-like models (i.e., the interaction between the two edge waves at the surface and the tropopause). The main point that we want to underline is that the surface and tropopause PVBs, which are excited rapidly after the initialization, owe their existence almost exclusively to the existence of interior tropospheric PVBs, which by a linear resonance mechanism give them a large amplitude. The Orr mechanism (which relates to the instantaneous differences between the phase speeds of all PVBs) provides a very efficient mechanism to rapidly generate large streamfunction values near the surface and the tropopause, which in turn give rise to a strong resonance effect.

The effect of the stratospheric PVBs on the surface is found to be almost negligible for the surface evolution of the optimal perturbations, which suggests that one might use a one-layer version with a larger tropopause mean PV gradient instead of a (full) two-layer Eady model. This also affirms the importance of SV evolution in the classic symmetric Eady model since the basic mechanisms responsible for the growth are the same (Mukougawa and Ikeda 1994; Morgan 2001; Morgan and Chen 2002). The traditional normal-mode growth starts to exceed the growth due to resonance only after a long time (say four or five days). To illustrate this last point, Fig. 19 shows the evolution of the amplitude ratio and phase difference between the surface and tropopause PVB. This figure shows that it takes two optimization times in the three- and N-PVB optimal perturbation evolution before the surface and tropopause PVB settle into the “growing-hindering” configuration characterizing the GNM (Heifetz and Methven 2005).

As a final remark to the present study of optimal perturbation dynamics, we note that recent work of Snyder and Hakim (2005) [see also Hakim (2000)] has revealed a potential weakness of optimal perturbation, or singular vector analysis applied to cyclogenesis. Generally constructed SVs usually exhibit a significant amount of fine structure in the vertical (section 10). The question arises to what extent initial realistic precursor disturbances possess that degree of complexity. Snyder and Hakim (2005) argue for instance that a large number of SVs needs to be included to get a realistic initial structure such as a tropopause PV anomaly. This has been one of the reasons to include a discussion of the construction of SVs using a limited amount of PVBs (e.g., one or three) in the initial SV. Even for the SVs constructed in these simple cases the growth of the surface and tropopause PV anomalies is caused to a large extent by the linear resonance with the interior tropospheric PV, rather than by the classical view of normal-mode baroclinic instability.

## Acknowledgments

The present paper benefited from useful manuscript reviews by Dr. Eyal Heifetz and an anonymous reviewer. H.d.V wishes also to acknowledge Dr. John Methven, Dr. Nili Harnik, Dr. Jan Barkmeijer, Dr. Wim Verkley, and Dr. Aarnout van Delden for valuable discussions.

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## APPENDIX A

### Dynamical Equations

*B*), a tropopause Rossby wave (

*T*), and the interior PVBs. Following Davies and Bishop (1994) and Dirren and Davies (2004) the following time-evolution equations can be derived

*θ*

^{0}

_{y}= −Λ

*and*

_{T}*= Λ*q

^{d}_{y}*/*

_{T}*N*

^{2}

*− Λ*

_{T}_{S}/

*N*

^{2}

_{S}. In these equations, the superscripts indicate the height at which the Green’s function is evaluated, for example,

*ϕ*

^{0}

_{B}=

*ϕ*(

_{B}*z*= 0). The subscripts indicate the position of the PVB. The above Eqs. (A1), generalize Eqs. (6) of Dirren and Davies (2004) to include the influence of the stratosphere and an arbitrary number of interior PV anomalies [they are also the discretized analog of Eqs. (A8) and (A9) in Heifetz and Methven (2005)]. They can be used to study the linearized dynamics for arbitrary initial conditions.

## APPENDIX B

### Green’s Function

*G*(

*z*,

*h*) is different for tropospheric and stratospheric PVBs. We adopt the notation

*G*

_{trop}(

*z*,

*h*) for tropospheric PVBs and

*G*

_{stra}(

*z*,

*h*) for the stratospheric PVB. Note that

*G*

_{trop}(

*z*,

*h*) and

*G*

_{stra}(

*z*,

*h*) are defined for all

*z*. We require

*G*

_{trop}(

*z*,

*d*) =

*G*

_{Stra}(

*z*,

*d)*, where

*d*is the tropopause height. A straightforward calculation gives the Green’s function

*G*

_{trop}(

*z*,

*h*) for a tropospheric PVB

*H*(

*x*) is the Heaviside step function and

*μ*

_{t,s}=

*N*

_{t,s}

*k*

^{2}+

*l*

^{2}

*α*=

*N*/

_{T}*N*,

_{S}*γ*

_{1}= sinh (

*μ*),

_{T}d*γ*

_{2}= cosh (

*μ*),

_{T}d*γ*

_{3}sinh[

*μ*(

_{T}*d*−

*h*)], and

*γ*

_{4}= cosh [

*μ*(

_{T}*d*−

*h*)]. For a stratospheric PVB we get

*N*

^{2}(

*z*) structure, one might need to compute the Green’s function numerically. Complex profiles of the zonal wind are in general not a problem since

*does not appear in the Green’s function.*u

## APPENDIX C

### Galilean Invariant Results

In the main text of section 6b it is stated that one has to subtract the Orr mechanism when trying to attribute the streamfunction growth to particular PVBs. If not, results will be obtained that violate Galilean invariance.

*c*is the speed of the frame of reference (

*c*= 0 in the main text). Because ∂

*= 0 in the complete domain in this example, it follows that the only mechanism producing growth or decay of the streamfunction is the kinematic Orr mechanism, resulting from the difference in propagation speed of the PVBs. Ambiguities arise, however, if we try to attribute the growth to the individual PVBs because the results depend on the (subjective) choice of*q

_{y}*c*. For instance, the contribution from

*Q*

_{1}to the streamfunction growth is zero if

*c*=

*U*

_{1}, whereas the contribution from

*Q*

_{2}is zero if

*c*=

*U*

_{2}. To resolve the ambiguity, one should sum all terms contributing to the Orr mechanism together, as in Eq. (23).

## APPENDIX D

### Computing the Optimal Perturbation

**q**(0). If only a few PVBs are included (three, say) one proceeds as follows. One fixes the initial positions of the three PBs (at levels

*i*,

*j*,

*k*) and constructs the

*N*× 3 reduced matrix 𝘄(

*t*)

**W**(

*t*) is replaced by

**w**(

*t*). The next step is to vary the positions (

*i*,

*j*,

*k*) of the PVBs. The optimal perturbation is that perturbation (in terms of vertical positions of the PVBs and their amplitude and phase relation), which gives the largest overall eigenvalue.

Green’s function *G*(*z*, *h*) with *k* = 1.55 for the semi-infinite (a) one-layer Eady model and the (b) two-layer Eady model; *N* ^{2}* _{T}* = 1, and

*N*

^{2}

_{S}= 4. The horizontal axis represents

*h*, the height of the PVBs, the vertical axis represents the height

*z*. Contour interval is 0.1.

Citation: Journal of the Atmospheric Sciences 64, 3; 10.1175/JAS3867.1

Green’s function *G*(*z*, *h*) with *k* = 1.55 for the semi-infinite (a) one-layer Eady model and the (b) two-layer Eady model; *N* ^{2}* _{T}* = 1, and

*N*

^{2}

_{S}= 4. The horizontal axis represents

*h*, the height of the PVBs, the vertical axis represents the height

*z*. Contour interval is 0.1.

Citation: Journal of the Atmospheric Sciences 64, 3; 10.1175/JAS3867.1

Green’s function *G*(*z*, *h*) with *k* = 1.55 for the semi-infinite (a) one-layer Eady model and the (b) two-layer Eady model; *N* ^{2}* _{T}* = 1, and

*N*

^{2}

_{S}= 4. The horizontal axis represents

*h*, the height of the PVBs, the vertical axis represents the height

*z*. Contour interval is 0.1.

Citation: Journal of the Atmospheric Sciences 64, 3; 10.1175/JAS3867.1

Plots of |𝗪| at various instants of time (up to four days in dimensional units) in the one-layer semi-infinite Eady model. The horizontal axis represents *h*, the height of the PVBs; the vertical axis represents the height *z*. Lines of constant *h* = *h _{i}* therefore give the absolute value of the streamfunction generated at time

*t*due to an initial PVB at position

*z*=

*h*. Contour interval is 0.125.

_{i}Citation: Journal of the Atmospheric Sciences 64, 3; 10.1175/JAS3867.1

Plots of |𝗪| at various instants of time (up to four days in dimensional units) in the one-layer semi-infinite Eady model. The horizontal axis represents *h*, the height of the PVBs; the vertical axis represents the height *z*. Lines of constant *h* = *h _{i}* therefore give the absolute value of the streamfunction generated at time

*t*due to an initial PVB at position

*z*=

*h*. Contour interval is 0.125.

_{i}Citation: Journal of the Atmospheric Sciences 64, 3; 10.1175/JAS3867.1

Plots of |𝗪| at various instants of time (up to four days in dimensional units) in the one-layer semi-infinite Eady model. The horizontal axis represents *h*, the height of the PVBs; the vertical axis represents the height *z*. Lines of constant *h* = *h _{i}* therefore give the absolute value of the streamfunction generated at time

*t*due to an initial PVB at position

*z*=

*h*. Contour interval is 0.125.

_{i}Citation: Journal of the Atmospheric Sciences 64, 3; 10.1175/JAS3867.1

As in Fig. 3 but for |𝗪| in the two-layer Eady model. Contour interval is 0.25 (note that this interval has been doubled compared to Fig. 3).

Citation: Journal of the Atmospheric Sciences 64, 3; 10.1175/JAS3867.1

As in Fig. 3 but for |𝗪| in the two-layer Eady model. Contour interval is 0.25 (note that this interval has been doubled compared to Fig. 3).

Citation: Journal of the Atmospheric Sciences 64, 3; 10.1175/JAS3867.1

As in Fig. 3 but for |𝗪| in the two-layer Eady model. Contour interval is 0.25 (note that this interval has been doubled compared to Fig. 3).

Citation: Journal of the Atmospheric Sciences 64, 3; 10.1175/JAS3867.1

Surface component |**W**_{1}* _{j}*(

*t*)| of the W-propagator, as a function of time and height of the PVB of unit amplitude, in the (a) one-layer Eady model and (b) two-layer Eady model.

Citation: Journal of the Atmospheric Sciences 64, 3; 10.1175/JAS3867.1

Surface component |**W**_{1}* _{j}*(

*t*)| of the W-propagator, as a function of time and height of the PVB of unit amplitude, in the (a) one-layer Eady model and (b) two-layer Eady model.

Citation: Journal of the Atmospheric Sciences 64, 3; 10.1175/JAS3867.1

Surface component |**W**_{1}* _{j}*(

*t*)| of the W-propagator, as a function of time and height of the PVB of unit amplitude, in the (a) one-layer Eady model and (b) two-layer Eady model.

Citation: Journal of the Atmospheric Sciences 64, 3; 10.1175/JAS3867.1

Finite-time growth rate Γ_{KE}(*t*_{opt}) at optimization time as a function of the height of the PVB (vertical axis) for different optimization times in the (a) one-layer Eady model and (b) two-layer Eady model.

Citation: Journal of the Atmospheric Sciences 64, 3; 10.1175/JAS3867.1

Finite-time growth rate Γ_{KE}(*t*_{opt}) at optimization time as a function of the height of the PVB (vertical axis) for different optimization times in the (a) one-layer Eady model and (b) two-layer Eady model.

Citation: Journal of the Atmospheric Sciences 64, 3; 10.1175/JAS3867.1

Finite-time growth rate Γ_{KE}(*t*_{opt}) at optimization time as a function of the height of the PVB (vertical axis) for different optimization times in the (a) one-layer Eady model and (b) two-layer Eady model.

Citation: Journal of the Atmospheric Sciences 64, 3; 10.1175/JAS3867.1

Vertical cross sections (zonal to height) of the evolution of the *t* = 5 optimal one-PVB problem in the KE norm. Negative values have been dashed. (top to bottom) *ψ*, *ψ*_{PV}, *ψ _{B}*, and

*ψ*and the sum

_{T}*ψ*+

_{B}*ψ*are shown. Measures of the perturbation |

_{T}*ψ*| = (∫

*ψψ**

*dz*)

^{1/2}and |

*ψ*(0)| = (∫

*δ*(

*z*)

*ψψ**

*dz*)

^{1/2}are indicated.

Citation: Journal of the Atmospheric Sciences 64, 3; 10.1175/JAS3867.1

Vertical cross sections (zonal to height) of the evolution of the *t* = 5 optimal one-PVB problem in the KE norm. Negative values have been dashed. (top to bottom) *ψ*, *ψ*_{PV}, *ψ _{B}*, and

*ψ*and the sum

_{T}*ψ*+

_{B}*ψ*are shown. Measures of the perturbation |

_{T}*ψ*| = (∫

*ψψ**

*dz*)

^{1/2}and |

*ψ*(0)| = (∫

*δ*(

*z*)

*ψψ**

*dz*)

^{1/2}are indicated.

Citation: Journal of the Atmospheric Sciences 64, 3; 10.1175/JAS3867.1

Vertical cross sections (zonal to height) of the evolution of the *t* = 5 optimal one-PVB problem in the KE norm. Negative values have been dashed. (top to bottom) *ψ*, *ψ*_{PV}, *ψ _{B}*, and

*ψ*and the sum

_{T}*ψ*+

_{B}*ψ*are shown. Measures of the perturbation |

_{T}*ψ*| = (∫

*ψψ**

*dz*)

^{1/2}and |

*ψ*(0)| = (∫

*δ*(

*z*)

*ψψ**

*dz*)

^{1/2}are indicated.

Citation: Journal of the Atmospheric Sciences 64, 3; 10.1175/JAS3867.1

Evolution of the relative contributions to the SKE for the one-PVB optimal perturbation evolution in (a) M1 and (b) M2. Also shown is log_{10}(1 + SKE) (denoted by the label SKE). Optimization time is indicated by the vertical line at *t* = 5.

Citation: Journal of the Atmospheric Sciences 64, 3; 10.1175/JAS3867.1

Evolution of the relative contributions to the SKE for the one-PVB optimal perturbation evolution in (a) M1 and (b) M2. Also shown is log_{10}(1 + SKE) (denoted by the label SKE). Optimization time is indicated by the vertical line at *t* = 5.

Citation: Journal of the Atmospheric Sciences 64, 3; 10.1175/JAS3867.1

Evolution of the relative contributions to the SKE for the one-PVB optimal perturbation evolution in (a) M1 and (b) M2. Also shown is log_{10}(1 + SKE) (denoted by the label SKE). Optimization time is indicated by the vertical line at *t* = 5.

Citation: Journal of the Atmospheric Sciences 64, 3; 10.1175/JAS3867.1

The contributions to the instantaneous SKE growth rate in terms of the different growth mechanisms that play a role are shown in panels for (a) M1 and (b) M2. These mechanisms have been defined in the main text. Optimization time is indicated by the vertical line at *t* = 5.

Citation: Journal of the Atmospheric Sciences 64, 3; 10.1175/JAS3867.1

The contributions to the instantaneous SKE growth rate in terms of the different growth mechanisms that play a role are shown in panels for (a) M1 and (b) M2. These mechanisms have been defined in the main text. Optimization time is indicated by the vertical line at *t* = 5.

Citation: Journal of the Atmospheric Sciences 64, 3; 10.1175/JAS3867.1

The contributions to the instantaneous SKE growth rate in terms of the different growth mechanisms that play a role are shown in panels for (a) M1 and (b) M2. These mechanisms have been defined in the main text. Optimization time is indicated by the vertical line at *t* = 5.

Citation: Journal of the Atmospheric Sciences 64, 3; 10.1175/JAS3867.1

Evolution in M2 of the different contributions to the instantaneous growth rate and phase speed of (a), (b) the surface PVB and (c), (d) the tropopause PVB. Optimization time is indicated by the vertical line at *t* = 5.

Citation: Journal of the Atmospheric Sciences 64, 3; 10.1175/JAS3867.1

Evolution in M2 of the different contributions to the instantaneous growth rate and phase speed of (a), (b) the surface PVB and (c), (d) the tropopause PVB. Optimization time is indicated by the vertical line at *t* = 5.

Citation: Journal of the Atmospheric Sciences 64, 3; 10.1175/JAS3867.1

Evolution in M2 of the different contributions to the instantaneous growth rate and phase speed of (a), (b) the surface PVB and (c), (d) the tropopause PVB. Optimization time is indicated by the vertical line at *t* = 5.

Citation: Journal of the Atmospheric Sciences 64, 3; 10.1175/JAS3867.1

Similar to Fig. 7 but for the evolution of the *t* = 5 optimal three-PVB problem in the KE norm. Measures of the perturbation |*ψ*| = (∫ *ψψ***dz*)^{1/2} and |*ψ*(0)| = (∫ *δ*(*z*)*ψψ***dz*)^{1/2} are indicated.

Citation: Journal of the Atmospheric Sciences 64, 3; 10.1175/JAS3867.1

Similar to Fig. 7 but for the evolution of the *t* = 5 optimal three-PVB problem in the KE norm. Measures of the perturbation |*ψ*| = (∫ *ψψ***dz*)^{1/2} and |*ψ*(0)| = (∫ *δ*(*z*)*ψψ***dz*)^{1/2} are indicated.

Citation: Journal of the Atmospheric Sciences 64, 3; 10.1175/JAS3867.1

Similar to Fig. 7 but for the evolution of the *t* = 5 optimal three-PVB problem in the KE norm. Measures of the perturbation |*ψ*| = (∫ *ψψ***dz*)^{1/2} and |*ψ*(0)| = (∫ *δ*(*z*)*ψψ***dz*)^{1/2} are indicated.

Citation: Journal of the Atmospheric Sciences 64, 3; 10.1175/JAS3867.1

As in Fig. 8 but for the three-PVB optimal perturbation evolution. Also shown is log_{10}(l + SKE) (denoted by the label SKE).

Citation: Journal of the Atmospheric Sciences 64, 3; 10.1175/JAS3867.1

As in Fig. 8 but for the three-PVB optimal perturbation evolution. Also shown is log_{10}(l + SKE) (denoted by the label SKE).

Citation: Journal of the Atmospheric Sciences 64, 3; 10.1175/JAS3867.1

As in Fig. 8 but for the three-PVB optimal perturbation evolution. Also shown is log_{10}(l + SKE) (denoted by the label SKE).

Citation: Journal of the Atmospheric Sciences 64, 3; 10.1175/JAS3867.1

As in Fig. 9 but for the three-PVB optimal perturbation evolution.

Citation: Journal of the Atmospheric Sciences 64, 3; 10.1175/JAS3867.1

As in Fig. 9 but for the three-PVB optimal perturbation evolution.

Citation: Journal of the Atmospheric Sciences 64, 3; 10.1175/JAS3867.1

As in Fig. 9 but for the three-PVB optimal perturbation evolution.

Citation: Journal of the Atmospheric Sciences 64, 3; 10.1175/JAS3867.1

As in Fig. 10 but now for the three-PVB optimal perturbation evolution in M2.

Citation: Journal of the Atmospheric Sciences 64, 3; 10.1175/JAS3867.1

As in Fig. 10 but now for the three-PVB optimal perturbation evolution in M2.

Citation: Journal of the Atmospheric Sciences 64, 3; 10.1175/JAS3867.1

As in Fig. 10 but now for the three-PVB optimal perturbation evolution in M2.

Citation: Journal of the Atmospheric Sciences 64, 3; 10.1175/JAS3867.1

Similar to Fig. 7 but for the evolution of the *t* = 5 optimal N-PVB problem (*N* = 61) in the KE norm. Measures of the perturbation |*ψ*| = (∫ *ψψ***dz*)^{1/2} and |*ψ*(0)| = (∫ *δ*(*z*)*ψψ***dz*)^{1/2} are indicated.

Citation: Journal of the Atmospheric Sciences 64, 3; 10.1175/JAS3867.1

Similar to Fig. 7 but for the evolution of the *t* = 5 optimal N-PVB problem (*N* = 61) in the KE norm. Measures of the perturbation |*ψ*| = (∫ *ψψ***dz*)^{1/2} and |*ψ*(0)| = (∫ *δ*(*z*)*ψψ***dz*)^{1/2} are indicated.

Citation: Journal of the Atmospheric Sciences 64, 3; 10.1175/JAS3867.1

Similar to Fig. 7 but for the evolution of the *t* = 5 optimal N-PVB problem (*N* = 61) in the KE norm. Measures of the perturbation |*ψ*| = (∫ *ψψ***dz*)^{1/2} and |*ψ*(0)| = (∫ *δ*(*z*)*ψψ***dz*)^{1/2} are indicated.

Citation: Journal of the Atmospheric Sciences 64, 3; 10.1175/JAS3867.1

As in Fig. 8 but for the N-PVB optimal perturbation evolution. Also shown is log_{10}(l + SKE) (denoted by the label SKE).

Citation: Journal of the Atmospheric Sciences 64, 3; 10.1175/JAS3867.1

As in Fig. 8 but for the N-PVB optimal perturbation evolution. Also shown is log_{10}(l + SKE) (denoted by the label SKE).

Citation: Journal of the Atmospheric Sciences 64, 3; 10.1175/JAS3867.1

As in Fig. 8 but for the N-PVB optimal perturbation evolution. Also shown is log_{10}(l + SKE) (denoted by the label SKE).

Citation: Journal of the Atmospheric Sciences 64, 3; 10.1175/JAS3867.1

As in Fig. 9 but for the N-PVB optimal perturbation evolution.

Citation: Journal of the Atmospheric Sciences 64, 3; 10.1175/JAS3867.1

As in Fig. 9 but for the N-PVB optimal perturbation evolution.

Citation: Journal of the Atmospheric Sciences 64, 3; 10.1175/JAS3867.1

As in Fig. 9 but for the N-PVB optimal perturbation evolution.

Citation: Journal of the Atmospheric Sciences 64, 3; 10.1175/JAS3867.1

As in Fig. 10 but now for the N-PVB optimal perturbation evolution in M2.

Citation: Journal of the Atmospheric Sciences 64, 3; 10.1175/JAS3867.1

As in Fig. 10 but now for the N-PVB optimal perturbation evolution in M2.

Citation: Journal of the Atmospheric Sciences 64, 3; 10.1175/JAS3867.1

As in Fig. 10 but now for the N-PVB optimal perturbation evolution in M2.

Citation: Journal of the Atmospheric Sciences 64, 3; 10.1175/JAS3867.1

Evolution of surface and tropopause PVB for the optimal perturbation. Radial plot shows *x* = |*T*|/|*B*| cos(*η _{T}* −

*η*) and

_{B}*y*= |

*T*|/|

*B*| sin(

*η*−

_{T}*η*) as a function of time. Full, dashed, and dash–dot lines correspond to the one-, three-, and N-PVB optimal evolution, respectively. Small, medium, and large disks correspond to

_{B}*t*= 0.01,

*t*= 5 (optimization time), and

*t*= 10 (double time), respectively.

Citation: Journal of the Atmospheric Sciences 64, 3; 10.1175/JAS3867.1

Evolution of surface and tropopause PVB for the optimal perturbation. Radial plot shows *x* = |*T*|/|*B*| cos(*η _{T}* −

*η*) and

_{B}*y*= |

*T*|/|

*B*| sin(

*η*−

_{T}*η*) as a function of time. Full, dashed, and dash–dot lines correspond to the one-, three-, and N-PVB optimal evolution, respectively. Small, medium, and large disks correspond to

_{B}*t*= 0.01,

*t*= 5 (optimization time), and

*t*= 10 (double time), respectively.

Citation: Journal of the Atmospheric Sciences 64, 3; 10.1175/JAS3867.1

Evolution of surface and tropopause PVB for the optimal perturbation. Radial plot shows *x* = |*T*|/|*B*| cos(*η _{T}* −

*η*) and

_{B}*y*= |

*T*|/|

*B*| sin(

*η*−

_{T}*η*) as a function of time. Full, dashed, and dash–dot lines correspond to the one-, three-, and N-PVB optimal evolution, respectively. Small, medium, and large disks correspond to

_{B}*t*= 0.01,

*t*= 5 (optimization time), and

*t*= 10 (double time), respectively.

Citation: Journal of the Atmospheric Sciences 64, 3; 10.1175/JAS3867.1

Scaling parameters and dimensional values.

^{1}

Apart from the discrete spectrum, the so-called continuous spectrum, existing for many (inviscid) flows, is required to describe the evolution from an arbitrary initial condition correctly (Pedlosky 1964).

^{2}

In DO5 the PVB was introduced as the time evolution of the perturbation streamfunction induced by a localized PV anomaly. This explicitly includes the excitation of the discrete normal modes for *t* ≠ 0. The definitions used in the present paper are equivalent to those used in DO5 only at initial time.

^{3}

Including the weighting matrix **Δ** is essential to obtain the correct dispersion relations for cases with nonzero *β*, such as the Charney (1947) or the Green (1960) problem.

^{4}

In the compressible case, *p*(*z*)*G*(*z*, *h*) = *ρ*(*h*)*G*(*h*, *z*) holds, where *ρ*(*h*) = exp(−*z*/*H*) is the density.

^{5}

Note that singular vector studies involving norms based on kinetic energy all require some form of the W-propagator, since KE optimal perturbations are essentially given by eigenvectors of 𝗪^{†}𝗪. We postpone more details to section 5.

^{6}

SKE cannot be used as an initial norm for the general optimal perturbation, since it is a seminorm (see DO5 for more details).

^{7}

The SKE growth rate relates to the streamfunction growth rate as SKE(*t*)^{−1}SK̇E(*t*) = 2Ψ(*z*_{0}, *t*)^{−1}Ψ̇(*z*_{0}, *t*).