## 1. Introduction

The work of Eady (1949) and Charney (1947) has convincingly shown that it is possible to get realistic growth rates and vertical disturbance structures by retaining only the most essential atmospheric ingredients. While Eady retained the tropopause but neglected the meridional dependence of the Coriolis parameter (the *β* effect), Charney retained the *β* effect but ignored the tropopause and the effect of the stratosphere. Green (1960) extended the Eady model by studying the effect of nonzero *β*. Afterward, many authors replaced the rigid lid by a stratosphere, and some retained the effects of compressibility and *β* (Müller 1991; Rivest et al. 1992; Juckes 1994; Harnik and Lindzen 1998).

The transient growth properties of baroclinic shear flows have been investigated extensively after Farrell (1982) showed that initial-value experiments often temporarily exhibit growth rates much larger than the fastest growing normal mode. This observation led to the theory of optimal perturbations, which are atmospheric initial disturbances that amplify maximally for finite time and according to a certain norm (Farrell 1984; Farrell and Ioannou 1996). The normal modes, on the other hand, amplify exponentially in all norms. Previous studies carried out on the *f* plane have emphasized that a (linear) resonance between interior potential vorticity (PV) perturbations and the surface edge wave plays a key role in the surface dynamics (e.g. De Vries and Opsteegh (2005, hereafter DO5) even if exponentially growing normal modes are present (De Vries and Opsteegh (2007, hereafter DO7). The inclusion of *β* in Eady-like models is known to change the stability properties (Green 1960). Not much is known of the effect of *β* to the optimal perturbation evolution, which may be related to the fact that even the discrete normal modes have a very complex structure in terms of boundary potential temperature (PT) and interior PV (Robinson 1989; Heifetz et al. 2004b).

The present study investigates to what extent the results, obtained previously in DO5 and DO7, are modified when *β* is included and, accordingly, the mean PV gradient becomes nonzero throughout the domain. We closely follow the setup of DO7 and many of the techniques presented in that paper will be used. With the possible application of explosive surface cyclogenesis, we will focus in detail on the surface dynamics but also study the role of the different growth mechanisms for the amplification in the interior.

The order of the paper is as follows. Section 2 contains a qualitative discussion of the model and the methods used. The role of the growth mechanisms in the growing normal mode is addressed in section 3. Sections 4 and 5 contain the core information regarding the optimal perturbation evolution at the surface and in the interior respectively. Concluding remarks are presented in section 6.

## 2. Theory

### a. Basic state and equations

The two models (M1 and M2) under consideration are the same as in DO7 but instead of the *f*-plane approximation, the meridional dependence of the Coriolis parameter is included by means of a *β*-plane approximation. On the *β* plane, M1 is the incompressible version of the Charney (1947) model. If *β* is set to zero, M1 becomes the semi-infinite extension of the Eady model (e.g., Thorncroft and Hoskins 1990; Chang 1992; DO5). Its basic state has a uniform shear Λ of the zonal wind * u*, and a constant buoyancy frequency

*N*

^{2}. Exponentially growing normal modes (GNM) exist only on the

*β*plane, and their properties are discussed extensively in most textbooks (Gill 1982; Pedlosky 1987). The second model (M2) is the two-layer Eady model on the

*β*plane. The basic state of M2 is formed by a two-layer troposphere–stratosphere system, in which the buoyancy frequency

*N*

^{2}(

*z*) and the shear Λ of the zonal wind

*(*u

*z*) are different in both layers. A matching condition is applied at the interface (tropopause). We have used the nondimensional values

*N*

^{2}

*= 1 (in dimensional quantities 10*

_{T}^{−2}s

^{−2}),

*N*

^{2}

_{S}= 4, Λ

*= 1 (3 m s*

_{T}^{−1}km

^{−1}), Λ

*= −1, and a tropopause at*

_{S}*z*=

*d*= 1 (10 km). Normal-mode properties of M2 with

*β*= 0 have been discussed in Müller (1991).

*β*plane

*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 as in DO7. The Bretherton (1966) formalism is used to interpret the surface boundary condition (rigid lid) and the interface condition (matching vertical velocities) in terms of PV. In this perspective

*q*and ∂

*/∂*q

*y*contain the singular contributions from the interfaces at

*z*= 0 (surface) and

*z*=

*d*(a possible tropopause):

*θ*= ∂

*ψ*/∂

*z*defines the potential temperature (PT) 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

### b. PV building blocks and propagator dynamics

*ψ*= ∫

*G*(

*z*,

*z*′)

*q*(

*z*′)d

*z*′ or via its vertically discretized analog

*β*is nonzero, as is the case in the present paper, all PVBs are active. We will further make the distinction between the PVBs residing at the singular levels (surface and tropopause, if present) and all other PVBs that are part of the continuous PV distribution. The last mentioned PVBs will also be called the “interior” PVBs, which makes a detailed comparison with the

*β*= 0 case possible.

We divide the domain of M2 in to four regions called surface, troposphere, tropopause, and stratosphere [indicated by subscripts (*B*, trop, *T*, stra)]. The PVBs are also labeled into these four groups. In M1 we distinguish only between the surface PVB (denoted by *B*) and all remaining PVBs (denoted by PV). Figure 1 shows schematically in which way a positive PVB (denoted by the encircled plus symbol) gives rise to PV tendencies at all other levels on the *β* plane. Although everyone may be familiar with this figure, at least since Hoskins et al. (1985), we have included it mainly to stress that an existing PVB amplifies the surface PVB lying at most *π* eastward, but destroys the existing interior PVBs lying eastward.

*t*), called the P-propagator:

*t*) = exp(𝗔

*t*) with 𝗔 = −

*ik*(𝗨 +

**Δ**· 𝗤

**y**· 𝗚) the matrix appearing in the vertically discretized analog of Eq. (1), which reads

**q̇**= 𝗔

**q**. The matrices 𝗨,

**Δ**, and 𝗤

**y**are further defined in DO7. The propagator 𝗣(

*t*) forwards in time arbitrary initial conditions [expressed in terms of the complex-valued initial PVB amplitudes

**q**(0)]. By multiplying Eq. (6) from the left with the Green’s function 𝗚 of Eq. (5), we arrive at

*t*) = 𝗚 · 𝗣 (

*t*) is called the W-propagator. The W-propagator translates an initial condition to streamfunction values at finite time. By plotting the absolute value of the W-propagator at various instants in time, we get insight in the dynamics of initial-value experiments in which one single PVB of unit amplitude is used (DO7). This has been done in Fig. 2 for M1 and M2 at four instants in time (

*t*= 5 corresponds to roughly two-dimensional days). The figures are interpreted as follows. For a given time

*t*and a particular choice of the height of the initial PVB (i.e., a value

*h*on the horizontal axis), the contours above

*h*give the streamfunction amplitude at time

*t*. At initial time (not shown), the results are symmetric around the line

*z*=

*h*and identical to the absolute value of the Green’s function. Due to the advection processes taking place after

*t*= 0, the figures start to develop asymmetrically. Already at

*t*= 2.5 (one-dimensional day) the maximum streamfunction amplitudes are found near the surface, even if the initial PVB is positioned in the middle or upper troposphere. The results are similar to the

*β*= 0 results reported in DO7 although slightly more growth occurs with

*β*(both in M1 and M2) and maximal amplitudes are obtained for PVBs at lower altitudes. This relates directly to the lower position of the critical level of the GNM in the presence of

*β*. More discussion can be found in DO7.

### c. Isolating growth mechanisms

*i*,

*c*(

_{i}*t*), is composed of similar terms, but contains an additional term stemming from the basic-state zonal wind at level

*i*. This term leads to passive advection of the PVB at level

*i*if the mean PV gradient is zero at level

*i*. In short, the evolution of a PVB at level

*i*is described by

^{1}

*α*= (

*B*, trop,

*T*, stra) and

*U*=

_{i}*(*u

*z*=

*i*) the mean zonal wind at level

*i*. Note that all coefficients

*A*

_{i,α}and

*C*

_{i,α}defined above are zero if the mean PV gradient is zero at level

*i*. The first subscript indicates the level, the second its origin (e.g.,

*A*

_{i,B}is the contribution to the growth rate of the PVB at level

*i*due to the surface PVB).

*i*, we write

*ψ*(

_{i}*t*) = Ψ

*(*

_{i}*t*) exp[

*iϵ*(

_{i}*t*)]. The growth rate of the streamfunction should be partitioned in a different way than the growth rate of the PVBs (DO7). The main reason is that the Orr effect is an additional growth mechanism, and relates to the instantaneous difference in phase speed between all the PVBs. Trying to subdivide the Orr term in multiple terms breaks the Galilean invariance of the results. By separating off the Orr part explicitly, we are able to relate the remaining terms to the different growth mechanisms. We write (using vector notation)

*α*,

*α*′ = (

*B*, trop,

*T*, stra). Each term [

**f**

_{α,α′}] represents only 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). Certain combinations of

**f**

_{α,α′}can be interpreted easily in terms of the following growth mechanisms:

Generalized Orr: This is the term

*γ*^{Orr}_{i}(*t*) related to the growth due to the instantaneous phase-speed difference between the individual PVBs.*B*–PV interaction: This is the interaction between the surface PVB and the PVBs belonging to the continuous PV distribution. It is computed as**f**_{trop,}+_{B}**f**_{B}_{,trop}+**f**_{stra,}+_{B}**f**_{B}_{,stra}. A further distinction can be made by subdividing the interior PVBs into tropospheric and stratospheric PVBs. If*β*= 0, this process is identical to*B*resonance discussed in DO7.*T*–PV interaction: Similar as*B*–PV interaction but for the tropopause PVB, computed as**f**_{trop,T}+**f**_{T}_{,trop}+**f**_{stra,T}+**f**_{T}_{,stra}. If*β*= 0, this process is identical to*T*resonance discussed in DO7.*B*–*T*interaction: This term involves the interaction between the PVBs at the surface and the tropopause and is computed as**f**_{B,T}+**f**_{T,B}. If*β*= 0, this interaction represents the complete normal-mode growth.PV interaction: Absent if the mean PV gradient is zero in the interior of the troposphere and stratosphere. It is computed here as

**f**_{trop,trop}+**f**_{stra,stra}+**f**_{trop,stra}+**f**_{stra,trop}and therefore includes two self-interaction terms.

## 3. Discrete normal modes in M2

If the *f*-plane approximation is made in M2, the mean PV gradient is zero everywhere except at the surface and the tropopause. An analytical normal-mode (NM) stability analysis is easily performed. Müller (1991) showed that growing normal modes (GNM) exist for a range of wavenumbers bounded by a long-wave and a short-wave cutoff wavenumber, as long as the surface and the tropopause mean PV gradients have opposite sign. If, on the other hand, the *β*-plane approximation is made, the Charney and Stern (1962) condition for instability is satisfied for all basic states with positive tropospheric shear. We have computed the NM growth rates for variable *k* and *β* for the basic state with *N* ^{2}* _{T}* = 1,

*N*

^{2}

_{S}= 4, Λ

*= 1, Λ*

_{T}*= −1 (which produces a mean PV gradient of 1.25 at the tropopause and a value of −1 at the surface) and a tropopause at*

_{S}*z*=

*d*= 1 (10 km). The results are shown in Fig. 3. Similar to the

*β*-plane extension of the Eady model with upper rigid lid by Green (1960),

*β*destabilizes the basic state and the shortwave cutoff is absent. The longwave cutoff also disappears, similar to calculations performed by Harnik and Lindzen (1998) for more realistic vertical profiles of zonal wind and buoyancy frequency.

### a. Structure of the GNM

The fact that both the wavenumber and the growth rate of the fastest GNM are almost independent of *β* (Fig. 3), at least as long as *β* is not large (*β* = 0.5 is a typical nondimensional value for the midlatitudes), suggests that the dominant mechanism producing growth of the GNM may still be qualitatively similar to the Eady edge-wave interaction. To elaborate on this idea, Figs. 4a–b shows the vertical structure of the streamfunction and PV of the fastest growing NM for three different values of *β*, all scaled to have |*ψ*(*z* = 0)| = 0.5. The phase of the surface PVB is fixed at −*π*.

#### 1) Streamfunction

The streamfunction structures in Fig. 4a are qualitatively similar to the familiar Eady wave, with streamfunction maxima at the surface and the tropopause and a minimum near the critical level. The phase of the streamfunction increases with height in all cases (and becoming constant at high altitudes), producing the characteristic westward (upshear) tilt of amplifying disturbances. Note that because of the reversed shear, the streamfunction tilts downshear in the stratosphere. For *β* = 0, the phase remains constant above the tropopause since all stratospheric PVBs of the GNM have zero amplitude.

Unlike the classic Eady model with upper rigid lid, however, the *β* = 0 streamfunction amplitude is larger at the surface than at the tropopause [because the mean PV gradient is larger at the tropopause than at the surface, see Eq. (3)] as in Müller (1991). If *β* is increased, the position of the tropospheric streamfunction minimum lowers. This minimum is found to be just above the critical level (where * u* =

*c*) of the GNM, which itself lowers for increasing

_{r}*β*.

#### 2) Potential vorticity

Figure 4b displays the amplitude and the phase of the PV at each level, which is up to a constant rescaling factor^{2} (DO7) equal to the amplitude and phase of the PVB at that level. If *β* = 0 all amplitudes of the PVBs not residing at the surface and tropopause are zero. Therefore their phases are meaningless and are indicated in Fig. 4b only to mark the phase of the tropopause PVB. For nonzero *β* the amplitudes of all PVBs are nonzero and the phases are nontrivial. The PV amplitude distribution for nonzero *β* attains a local maximum near the tropospheric critical level and decays upward.

Because the mean PV gradient at the surface is of opposite sign as the mean PV gradient in the interior, the surface PVB is *π* out of phase with the first interior PVB above the surface.^{3} The phase rapidly decreases across the critical level (causing and eastward tilt of the PV with height), then becomes roughly constant in the upper parts of the troposphere. Remarkably, the phase difference between the surface and tropopause PVB is almost equal for the three cases. In the stratosphere, the phase rapidly increases across the stratospheric critical level and approaches a constant value at high altitudes.

Figure 4c shows the phase difference between the streamfunction and the PV at all levels for the cases where *β ≠* 0. There are two regions where positive PV is associated with cyclonic motion (the surface and the region between the tropospheric and stratospheric critical level). In the remaining parts, however, positive PV is associated with anticyclonic motion. This indicates that the negative surface PV anomaly induces a circulation in the interior that is strong enough to overwhelm the local interior cyclonic circulation of the interior PV below the critical level (see DO5 for a similar situation occurring in the semi-infinite Eady model). At the position of the critical levels, the streamfunction and the PV are *π*/2 out of phase exactly (the PVBs at those levels are optimally growing).

#### 3) Explanation

One can—at least qualitatively—explain why a PV maximum occurs at or at least near the critical level. The rationale is based on the tacit assumption that the propagation speed of each (interior) PVB is almost exclusively determined by the wind field associated with all other (interior + boundary) PVBs, a situation that will be approached as the vertical distance between the PVBs goes to zero. In this limit, tropospheric PVBs below the critical level tend to propagate westward relative to *c _{r}*, whereas PVBs above the critical level tend to propagate eastward. Therefore the phase speeds of these off-critical-level PVBs have to be adjusted to maintain a constant propagation speed

*c*at all levels. In contrast, the PVB residing at the critical level itself requires no such phase-speed modification.

_{r}How is this phase-speed modification realized? Since the required modification can only be achieved by advection of the mean (positive) PV gradient *β*, it is clear that positive PV below the critical level must be associated with anticyclonic winds (and in a similar way that positive PV must be associated with cyclonic winds above the critical layer, see Fig. 4c). This explains why the phase of the PV must tilt eastward with height across the critical layer.^{4} Moreover, the more the phase speed of the PVB needs to be modified, the smaller the amplitude of that PVB must be (a wind field of a given strength will advect smaller amplitude PVBs faster than large amplitude ones). This explains the maximum in the PV distribution, and also the fact that the PV distribution decays more rapidly upward away from the critical level than downward.

### b. Uniformity of the growth rate

Before we continue with the more complex situation of disturbance structures, which are changing in time such as the optimal perturbations, it is interesting to see in which way each PVB contributes to the growth of the other PVBs and to the growth of the streamfunction for the case of the GNM (Robinson 1989).

#### 1) Zero *β*

If *β* = 0, we know exactly what is happening regarding the PVB growth rates. The surface PVB amplifies the tropopause PVB and vice versa. In terms of the previously introduced growth mechanisms this is the pure *B*–*T* interaction. All other PVBs have zero amplitude and are not involved in the GNM.

It is less trivial to decide which of the two PVBs causes the growth rate of the streamfunction at a particular level. Figure 5 shows the contribution from the PVB at the surface and the tropopause to the GNM streamfunction growth rate at each level. Each PVB contributes most positively at the level of the other PVB and slightly negative at its own level.^{5} The reason for this is that *B* and *T* are in a so-called hindering configuration (Heifetz et al. 2004a) in the sense that each PVB acts to reduce the streamfunction anomaly, induced by the other PVB, on its own edge. At the position of the critical level, the contributions are equal.

#### 2) Nonzero *β*

For nonzero *β* the situation is more complex because all PVBs interact. It is therefore even less trivial to maintain a growth rate that is uniform with height [implying that Eqs. (9) and (11) are equal constants]. By using Fig. 1 we can directly relate the growth of the individual PVBs in the GNM to the vertical distribution of the phase. It is then clear from Fig. 4b that the surface PVB (with a phase fixed at −*π*) amplifies (and is amplified by) the PVBs roughly up to *z* = 1.45 (the point where the PVB phases cross the zero phase line). The tropopause PVB is amplified by the surface PVB and by all interior PVBs lying at most *π* eastward, which include the mid- and upper-tropospheric PVBs. While these mid- and upper-tropospheric PVBs amplify the tropopause PVB, they are themselves destroyed by the tropopause PVB (see again Fig. 1). The lower-tropospheric and all-stratospheric PVBs up to *z* = 2 for *β* = 0.5 and *z* = 1.75 for *β* = 1 contribute negatively to the tropopause PVB amplification. The PVB phase distribution attains a minimum in the upper parts of the troposphere and monotonically increases above. Therefore from this point upward each PVB contributes to the growth of the PVB above and to the decay of the PVB below (see Fig. 1). Because the PV decreases monotonically with height (the tropopause PV being the exception), there can still be net growth.

The partitioning of the *β* = 0.5 PVB growth rate into its components is displayed in Fig. 6 (the stars indicate the two nonzero components if *β* = 0). Both surface and tropopause PVB have zero contribution at their own level. Figure 6 confirms that the surface PVB amplifies PVBs up to *z* = 1.45, damping all PVBs aloft. In a region below the tropopause its contribution to the growth rate is maximized and exceeds the GNM growth rate, which shows that other PVBs should contribute negatively. The contribution from the tropopause PVB to the growth rates in the troposphere is different. Upper-tropospheric PVBs are almost in phase with the tropopause PVB (see Fig. 4b) and are therefore not amplified. Near the critical level the PV distribution shows a rapid phase change. As a result, surface- and low-level PVBs are mostly amplified by the tropopause PVB. The integrated contribution from the tropospheric PVBs to the PVB growth rate is positive at the surface and negative at the tropopause. The effect of the stratospheric PVBs to the tropospheric PVB growth rates is small. However, in the upper parts of the stratosphere (above *z* = 1.5), the PVB growth rate is completely determined by the self-interaction, whereas the contributions from the surface and tropopause PVB become negative. Note that, due to the rapid upward decay of the PVB amplitude distribution, the individual contributions to the growth rate become very large at higher altitudes.

Figure 7a shows the contributions from the different groups of PVBs to the GNM streamfunction growth rate as a function of height. The contributions from the surface and tropopause PVB resemble their *β* = 0 counterparts qualitatively in the troposphere. On the other hand, the contribution from the surface PVB becomes increasingly positive in the stratosphere, which also holds for the effect of the tropopause PVB. The summed effect of the tropospheric PVBs to the streamfunction growth rate is positive in the lower half of the troposphere, and negative everywhere aloft. The stratospheric PVBs contribute only very little to the streamfunction amplification in the troposphere, but they contribute increasingly negative in the stratosphere itself (upward propagation). Figure 7b shows the growth mechanisms involved in the GNM. We see that *B*–*T* interaction (the pure Eady interaction) is the largest positive contribution well into the stratosphere. Runner up is the contribution from the *B*–PV interaction (important in the Charney problem) that becomes the largest contribution at high altitudes. The *T*–PV interaction contributes negatively up to *z* = 2. All remaining interactions (collectively denoted by PV in the figures) contribute slightly negative in the troposphere and increasingly negative in the stratosphere.

### c. Summary

We have seen that *β* modifies the GNM streamfunction structure but leaves the growth rate of the GNM roughly unchanged. The upshear, westward tilt (and, because of the reversed shear, downshear tilt in the stratosphere) is slightly increased for nonzero *β* and the tilt is largest near the tropospheric and the stratospheric critical level. To produce the same growth rate at all levels (and in all norms), a delicate balance can be reached only by introducing a particular PV distribution with local maxima near the surface, tropopause, and the critical levels. At the surface, the largest contribution to both the streamfunction and the PVB growth rate is from the tropopause PVB. The dominating growth mechanism in the troposphere is the familiar *B*–*T* interaction of the pure Eady wave, but the coupling with the continuous PV distribution cannot be neglected especially in the stratosphere (and would be more important still if *β* is increased).

## 4. Optimal perturbation evolution

We proceed by computing the optimal perturbations for the kinetic energy norm for the two different model configurations M1 and M2. As in DO7, an optimization time *t* = 5 (two days) has been chosen. To be able to compare the forthcoming results with the *β* = 0 results in DO7, we use *k* = 1.55, which produces the fastest GNM in M2 for *β* = 0 (see Fig. 3). This choice produces a GNM with a nondimensional growth rate of 0.26 in M2. In M1, however, the GNM for this wavenumber has a growth rate of only 0.12. On the base of the GNM growth rate, we thus expect that the growth in M2 will exceed the growth in M1.

### a. One PVB

The evolutions of the one-PVB optimal perturbation streamfunction (in M2) and its different components are shown in Fig. 8. The total streamfunction gradually transits from a barotropic initial condition toward the westward-tilted structure qualitatively resembling the GNM. In many aspects the evolution is similar to the *β* = 0 case reported in DO7 (their Fig. 8). The *β* effect reduces the average zonal propagation rate of the optimal perturbation. The critical level of the GNM is found at lower altitudes if *β* is nonzero. This is reflected in the position of the optimally positioned initial PVB, which is slightly below its *β* = 0 optimal position. Absent if *β* = 0, the interior PVBs amplify in time due to the presence of a nonzero mean PV gradient. At optimization time their KE has roughly doubled. The surface and tropopause PVB amplify less than in absence of *β*.

#### 1) Growth of SKE

As noticed above the *β* effect does not have dramatic effects on the optimal perturbation evolution in the interior. Regarding the surface evolution, we zoom in on the major differences between the cases in which *β* is zero and nonzero.

In DO7, it is shown for *β* = 0 that M2 produces more SKE at optimization time than M1. Figure 9 shows that M1 and M2 for nonzero *β* have almost identical SKE at optimization time *t* = 5, even though the GNM growth rate in M2 is almost twice the GNM growth rate in M1. This already emphasizes the limited importance of “pure” normal-mode growth. However, the most significant difference with the *β* = 0 case is that the contribution from the interior PVBs to the SKE becomes negative after some time (roughly after *t* = 2.2 in M1 and after *t* = 3 in M2), whereas the contribution from the interior PVBs became zero for *β* = 0. One particular feature that was absent in the *β* = 0 case, is the possible effect of the excitation of the stratospheric PVBs in M2. It is seen, however, that they have a negligible contribution at the surface. Finally, the contribution from the tropopause PVB remains negative throughout the time evolution, similar to the *β* = 0 results.

Which growth mechanisms are most important for the surface development? Are they similar to the GNM results of Fig. 7b? Let us first discuss M1 and compare the results with DO7. Figure 10a shows that the *B*–PV interaction is most important throughout the time evolution. The contribution from the Orr mechanism is larger than in absence of *β* but still small. Zero in the *β* = 0 case, the contribution from PV interaction is frustrating the surface growth due to the *B* PV interaction. We now continue with M2. Also in M2, the *B*–PV interaction is the largest positive contribution to the growth rate. Similar to the *β* = 0 case, the *T*–PV interaction reduces the growth rate at the surface. The PV interaction also reduces the growth rate (cf. M1 in Fig. 10a). Finally, the *B*–*T* interaction starts to become the dominating mechanism long after optimization time has been reached, which is much later than in absence of *β*. So compared to the *β* = 0 case it takes more time for the SV to settle down into the GNM configuration, at least based on a consideration of the *B*–*T* interaction.^{6} This is reflected clearly by comparing the results above with the long-time GNM values in Fig. 7b, in which the contribution from *B*–*T* interaction is almost twice as large as the contribution from *B*–PV interaction.

#### 2) Growth of the surface and tropopause PVB in M2

Figure 11 displays the contributions to the growth rate of the surface and tropopause PVB. Qualitatively similar to the *β* = 0 situation, it is seen that the growth rates of both the surface and the tropopause PVB are dominated initially by the term from the interior PVBs. Gradually the influence of the PVB residing at the tropopause (in case of the surface PVB) or the surface (in case of the tropopause PVB) increases. Focusing on the differences with respect to the *β* = 0 case reported in DO7, we notice that it takes more time before the tropopause PVB dominates the surface PVB amplification. At the level of the tropopause, the self-amplification has rather the opposite effect. The surface PVB starts to be the largest contribution to the growth rate of the tropopause PVB after *t* = 3.5, which is earlier than in the *β* = 0 case.

The asymmetric development can be anticipated. In the presence of positive *β* there is a symmetry breaking between resonating the surface and the tropopause PVB by the interior PVB. Since the interior and surface mean PV gradient have opposite sign, the phase-lock resonance can be even more constructive (because the interior PVB may grow and amplify back the surface PVB stronger than linearly). On the other hand, since the interior and tropopause have the same mean PV gradient, the tropopause PVB is expecting to weaken the interior PVB (and to destroy the upper-resonance effect, see also Fig. 1). A further indication is Fig. 6, which shows that the interior tropospheric PVBs (in the long-time GNM configuration) contribute negatively to the growth rate of the tropopause PVB but positively to the growth rate of the PVB at the surface.

We now verify the statement, made in the previous section, that the phase speed is decreased due to the *β* effect. In the case with *β* = 0, the interior PVB did not contribute to the phase speed of the surface and the tropopause PVB. During the complete evolution the PVB was *π*/2 out of phase with the PVBs generated at the surface and the tropopause. On the *β* plane a more complex situation develops. We see in Fig. 11b that the interior PVBs lead to a reduction of the phase speed of the surface PVB. Their negative contribution is almost as strong as the contribution from the tropopause PVB. Apparently, the joint effect from the excited interior tropospheric PVBs, which are generated approximately *π* out of phase with the excited surface PVB, contributes in a similar way as the excited tropopause PVB, with which they are in phase, to the phase speed of the surface PVB. The term remains negative in the long-time SV evolution (i.e., when the GNM configuration has been reached), whereas for *β* = 0 the contribution vanished in the long-time limit (cf. Figs. 10b and 10d in DO7). In this way, the interior PVBs are responsible for the reduction of the phase speed. A similar reasoning holds for the tropopause PVB (Fig. 11d).

### b. Three and more PVBs

The optimal perturbations with three and *N* PVBs in the initial SV have been computed as well and the results are surprisingly similar to the results of DO7. Both in M1 and M2, the SKE attained at optimization time is found to be less if *β* is included (Fig. 12). The main reason for the reduction is that the PV interaction works rather destructive on the growth rate (Fig. 13). Further differences occur mainly because the structure of the asymptotically approached GNM is different for *β* zero or nonzero. As before the contribution from the stratospheric PVBs to the SKE remains completely negligible. It is interesting to note that, as in the *β* = 0 situation, M1 attains a larger SKE at optimization time than M2, although the GNM growth rate in M2 is twice that in M1. This once more is a clear illustration of the relative unimportance of the GNM in the initial nonmodal development.

## 5. Role of the growth mechanisms in the interior

Up to now, we have been investigating mainly the surface dynamics of the optimal perturbation evolution and aspects of the two-dimensional growth of the perturbation have been addressed only qualitatively. The methods developed in DO7 can be used as well to investigate the role played by the growth mechanisms in the interior. We take the one- and three-PVB optimal perturbation evolution in M2 and create contour maps of the contribution of the different growth mechanisms. The results have been displayed in Figs. 14 and 15.

In the one-PVB optimal perturbation in Fig. 14, the growth rate of *ψ*, which is zero at initial time, rapidly increases at all levels. The largest growth rates are attained in the stratosphere a few hours after initialization. This is due to the initially small streamfunction amplitude at those levels. At the surface and the lower troposphere, however, maximal growth rates are found roughly after one day. Afterward, the growth rate gradually decreases toward the asymptotic growing normal mode (GNM) value. Near the position of the initial PVB (near the critical level just below the midtroposphere) the growth rates are lowest. The Orr mechanism is not important and gradually decays to zero at all levels as the GNM is formed. In line with the previous sections, the dominant positive contribution at low levels is due to *B*–PV interaction. At higher levels, however, this interaction works destructive, which clearly shows that the GNM configuration is reached only slowly (cf. Fig. 7 of the GNM). The *T*–PV interaction previously was seen to frustrate the growth at the surface. This is now seen to hold up to the midtroposphere. However, above the midtroposphere and well into the stratosphere, the *T*–PV interaction is the dominating process that amplifies the streamfunction. The *B*–*T* interaction is almost negligible at all levels initially and becomes important only after the optimization time. Finally, the PV interaction, which measures the effect of the amplification of the continuous PV integral (the self-interaction of all interior tropospheric and stratospheric PVBs), is important in the early stages at higher altitudes. A comparison between the SV results shown here and the GNM results presented in Fig. 7b shows that even after two optimization times, the SV has not settled completely into the GNM structure, although the growth rate of the total streamfunction has become roughly equal to the GNM value at all levels after two optimization times.

The three-PVB evolution in terms of the growth mechanisms (Fig. 15) differs from the one-PVB results only in one important aspect (note that the contour-interval has been doubled). Absent in the one-PVB problem the Orr mechanism has a strong positive effect on the streamfunction growth rate in the entire troposphere, but especially in the vicinity of the critical level where the PVBs reside. High up in the stratosphere, the Orr mechanism still contributes negatively as in the one-PVB results. The *B*–PV interaction is again important at low levels (it is a little delayed compared to the one-PVB results), but also strongly limits the growth higher up in the troposphere. For the *T*–PV interaction the situation is just the opposite, and strong positive interaction occurs in the upper troposphere and aloft. Finally, the *B*–*T* interaction is a small positive contribution in the troposphere and the effect of the PV interaction is similar to the one-PVB results.

The results of the optimal perturbation computed with *N* PVBs in the initial distribution agree well with the three-PVB problem in the troposphere (overwhelming positive contribution from the Orr mechanism initially). In the stratosphere, the results become messier because of the small amplitudes of all PVBs, which initially lead to very small streamfunction values, and thereby to large contributions to the growth rates.

## 6. Discussion

In DO5 and DO7 it has been shown that resonance between the surface PVB and interior tropospheric PVBs is important for the growth of optimal perturbations at the surface. If the initial optimal perturbation is composed of more than one PVB, the Orr mechanism (PV unshielding) is essential for the initial development. It serves to rapidly generate large surface winds. These surface winds advect the mean temperature gradient at the surface, and lead to the excitation of the surface PVB (the edge wave). If a tropopause is present, a tropopause PVB (interfacial wave) is also excited. If the optimal perturbation is composed of more than one PVB, the initial structure exhibits a significant upshear tilt. In that case, the tropopause PVB is seen to frustrate the development at the surface (up to optimization time). Eventually (beyond optimization time) the normal-mode growth starts to dominate.

In the present paper, we have included the *β* effect and we have studied in which way the optimal perturbation evolution is modified. With the introduction of the *β* effect, PVBs at all levels start to influence each others growth and phase propagation.

The effects of *β* are the largest, if the initial optimal perturbation is composed of one PVB. Because the mean PV gradient is nonzero at all levels if *β* is nonzero, PVBs (at the surface, tropopause, and the interior) are continuously being generated *π*/2 out of phase with the initial PVB. The contribution from the excited interior PVBs to the surface development is found to be negative. Their negative contribution is roughly equal to the negative contribution from the tropopause PVB and thereby leads to a small reduction of the total growth at the surface. The *β* effect is further seen to reduce the average propagation speed of the optimal perturbation. If the initial optimal perturbation is composed of more than one PVB, the results are strikingly similar to the *β* = 0 results in DO7. The Orr mechanism is hardly influenced by *β* (see also the appendix). The main effect of *β* is to slightly reduce the average phase speed of the optimal perturbation. Similar to the *β* = 0 results, the stratosphere remains completely unimportant for the surface development.

A subsequent investigation has been made of the role of the growth mechanisms in the interior. This has revealed that the Orr mechanism is initially important mainly in the troposphere, but only if the initial optimal perturbation is composed of more than one PVB, confirming the *f*-plane results in Morgan (2001). Linear resonance between the interior tropospheric PVBs and the surface PVB is responsible for the low-level development in the second stage of the perturbation evolution well beyond the optimization time. In a similar way, the resonance between the tropopause PVB and interior tropospheric PVBs is most important for the rapid growth at higher altitudes and in the stratosphere. The surface–tropopause interaction (the pure Eady wave) as well as the self-amplification of the interior PVBs only slightly modify this picture. This last point is supported by the integral of the continuous part of the perturbation PV for the *N*-PVB problem, which remains almost constant for a long time [at least up to optimization time (*t*_{opt} = 5)].

A schematic overview of the different stages of growth of the KE optimal perturbations (Fig. 16) summarizes the main results [see Morgan (2001) for a similar schematic]. In this figure we have ignored the effect of the stratospheric PVBs (which was small at low levels) and the effect of *β* (which was marginal if multiple PVBs were used). As a final note, we would like to emphasize that since the introduction of *β* has been shown not to alter the dynamics of the optimal perturbation dramatically, previous (relatively noncomplex) works on nonmodal growth using simpler basic states with zero mean interior PV gradient are more important than perhaps thought at first sight (Mukougawa and Ikeda 1994; Davies and Bishop 1994; Morgan and Chen 2002; Dirren and Davies 2004; De Vries and Opsteegh 2005; Heifetz and Methven 2005; De Vries and Opsteegh 2006).

## Acknowledgments

The authors wish to acknowledge Dr. Eyal Heifetz, Dr. John Methven, and an anonymous reviewer for useful comments and manuscript reviews.

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## APPENDIX

### PV Unshielding in the Presence of β

*β*plane (with constant shear and buoyancy frequency). We investigate the effect of

*β*on PV unshielding for the simplest nontrivial case, namely an infinite number of equal amplitude PVBs tilted uniformly with height. The instantaneous phase speed of a given PVB is determined as (DO7)

*G*denotes a Green’s function entry and

_{ij}*η*denotes the phase of the PVB at level

_{i}*i*. Taking the difference of two nearby PVBs (labeled by

*i*and

*j*≡

*i*+ 1) gives

*G*=

_{ii}*G*and the first term with

_{jj}*β*vanishes. Because the vertical tilt is assumed to be uniform with height, all subsequent terms are also zero as long as the amplitudes remain equal. To determine whether this is the case, we compute the growth rate of each PVB given an initial configuration of equal amplitude PVBs. Because the flow is also assumed to be unbounded in the vertical, we can, without loss of generality, consider the PVB at height

*z*= 0. Using the notation

*g̃*=

_{ij}*G*sin(

_{ij}*η*−

_{i}*η*), we arrive at (DO7)

_{j}*g̃*

_{0,−i}= −

*g̃*

_{0,i}(because

*G*=

_{ij}*G*) and all contributions are zero. The growth rate of each PVB remains identically zero. Thus, an unbounded PV structure with initially constant vertical tilt and amplitude remains of constant amplitude and untilts in a way that is completely independent of

_{ji}*β*. The presence of

*β*acts solely to add a constant term to the phase propagation of the PVB at level

*i*, namely

*βĝ*, which is the same for all PVBs in the unbounded domain.

_{ii}This above simple reasoning is no longer valid if at least one of the following three disturbing factors occurs: 1) the existence of a tilt that is nonuniform with height; 2) the existence of PVBs with different amplitudes; 3) the existence of a boundary, interface, vertical density gradient or height variations of the buoyancy frequency. In these cases, the Green’s function will no longer be symmetric and PVBs will interact differently with PVBs aloft than with PVBs below.

As an example, assume that the PVBs have initially equal amplitudes but that the tilt increases with height. Then |*g̃*_{j,j+1}| < |*g̃*_{i,i−1}| and *c _{j}* −

*c*will increase; that is, PV unshielding will occur more rapidly. What happens with the amplitudes? For configurations that tilt westward with height, each PVB will be amplified from below and damped by the PVBs above. If the tilt increases with height,

_{i}*g̃*

_{0,−1}+

*g̃*

_{0,1}will become negative and therefore leads to decay of the PVB at level 0. So PV unshielding occurs more rapidly but the amplitudes of the PVBs decrease. For a tilt decreasing with height, the opposite occurs. In the general case it is difficult to get definite answers.

^{2}

This scaling factor is proportional to the layer separation.

^{3}

Because both the interior PV equation and the thermodynamic equation are approximately satisfied at *z* = 0^{+} one may derive *q*(0^{+}) = −*β*Λ^{−1}* _{T}θ_{T}*(0). This relation shows that for

*β*= 1 the surface PT equals the PV at the first interior level (because

*N*

^{1}

_{t}= 1 and Λ

*= 1 have been chosen). A similar relation can be derived for the tropopause PV.*

_{t}^{4}

In other words, near the critical level the PV and the streamfunction form an X structure, where the former is tilted with the shear and the latter against it. This observation is also in line with the wave-evanescent region below the critical level, from the wave-overreflection perspective on baroclinic instability (E. Heifetz 2006, personal communication).

^{5}

Note that if the phase difference between the PVB and the streamfunction at a given level were *π* exactly, the contribution from the PVB to its own level would be zero.

^{6}

The true situation is more complex because the GNM is composed not only of *B* and *T* but also of interior PVBs. To really assess whether it takes more time for the SV to settle into the GNM, one should compute the projection on the PV of the GNM.