## 1. Introduction

In a previous study, Dong and Colucci (2005, hereafter referred to as DC2005) identified two mechanisms that each acted alone, but rarely in concert, to force the weakening of midtropospheric westerlies associated with the analyzed Southern Hemisphere (SH) blocking cases. These mechanisms are the advection of a meridional gradient in potential vorticity (PV), referred to hereafter as the advection forcing, and the interaction between deformation and the PV gradients, referred to as the interaction forcing. It was found that the advection and interaction forcings generally opposed each other such that the geostrophic westerlies weaken during block onset when one forcing (usually the advection forcing) contributes more negatively to the geostrophic zonal wind tendency than the opposing positive contribution from the other forcing. Furthermore, this opposition between the advection and interaction forcings was also found in the nonblocking cases studied for comparison. In these cases, one forcing or the other usually contributed more positively to the geostrophic zonal wind tendency than the other opposing negative forcing.

The purpose of the present contribution is to offer a theoretical explanation for this observational finding. The present study is organized as follows. In section 2, the advection and interaction forcings from the diagnostic model of DC2005 are comparatively evaluated with an idealized analytical model of the geopotential height field, and the opposition between the two forcings is analytically derived. In section 3, four regimes of zonal and meridional wavenumbers are identified and special focus is placed upon two of these regimes. A synoptic interpretation of the two opposing block-onset forcing mechanisms is provided in section 4. Section 5 specifically addresses how this opposition works in the nonblocking events. The opposing effect between two block-onset mechanisms is further discussed in the context of the zonal propagation of barotropic Rossby wave in section 6. Finally, a summary is given in section 7.

## 2. Analytical model

*F*

_{adv}, while the bracketed second and third terms denote the interaction forcing

*F*

_{inter}. The advection forcing, which does not directly include deformation, indicates that the advection of equatorward increasing cyclonic PV (or equatorward decreasing anticyclonic PV) could force a local weakening of the geostrophic westerlies or increasing geostrophic easterlies associated with block onsets. The interaction forcing, which is the net effect of interaction between deformation and PV, indicates that eastward increasing PV embedded in a cyclonically sheared flow or equatorward increasing PV coincident with a stretching (diffluent) flow could each force a weakening in the geostrophic westerlies. The variables in Eq. (1) have the same meanings as in DC2005. It is found from DC2005 that, in the diagnosed blocking and nonblocking cases,

*F*

_{adv}and

*F*

_{inter}usually have opposite signs and thus contribute oppositely to the geostrophic zonal wind tendency ∂

*u*/∂

_{g}*t*.

*F*

_{adv}and

*F*

_{inter}, we model the geopotential height field

*z*, following Holton (2004), as

For midtropospheric systems over midlatitudes of the SH, the above parameters have the following characteristic scales:

*z*_{0}∼ 5300 m is the typical 500-hPa geopotential height at 50°S. This latitude is taken as the zero coordinate (*y*= 0) in the present study.*b*=*f*_{0}*U*_{0}/*g*∼ −10^{−4}is the geopotential height gradient with latitude, where the Coriolis parameter*f*_{0}∼ −10^{−4}s^{−1}is evaluated over midlatitudes of the SH, the zonal mean wind*U*_{0}∼ 10 m s^{−1}, and gravity*g*∼ 10 m s^{−2}.*B*∼ 20 m is the amplitude of the geopotential height perturbation referenced at the zero coordinate (50°S).*k*= 2*πn*/*L*, where_{x}*n*is the zonal wavenumber and*L*= 2_{x}*πR*cos(*θ*). Here*θ*= 50°S is the reference latitude and*R*≃ 6.37 × 10^{6}m is the earth’s radius.*l*= 2*πm*/*L*, where_{y}*m*is the meridional wavenumber and*Ly*=*πR*.

*q*= (

*g*/

*f*

_{0})∇

^{2}

*+*

_{p}z*f*+

*f*

_{0}

*g*∂[(1/

*σ*)(∂

*z*/∂

*p*)]/∂

*p*,

**V**

*= (*

_{g}*g*/

*f*

_{0})

**k**×

**∇**

*, and beta-plane approximation ∂*

_{p}z*f*/∂

*y*=

*β*= 10

^{−11}m

^{−1}s

^{−1}, the two forcings shown in Eq. (1) can be written as

By assuming that geopotential height is a two-dimensional field described by Eq. (2), its dependence on pressure (*p*) is not considered. In other words, the present analytical study is carried out in the barotropic framework such that the vertical differential term (stratification term) of quasigeostrophic PV (QGPV) does not contribute to Eqs. (3) and (4).

The two forcings in Eqs. (3) and (4) can be simply denoted as *F*_{adv} = *F*_{1a} + *F _{b}*, with

*F*

_{1a}representing the first term of

*F*

_{adv}and +

*F*the second term, and

_{b}*F*

_{inter}=

*F*

_{2a}−

*F*, with

_{b}*F*

_{2a}as the first term of

*F*

_{inter}and −

*F*the second term again. Note that a common term

_{b}*F*appears in both

_{b}*F*

_{adv}and

*F*

_{inter}but with an opposite sign. Therefore,

*F*

_{adv}+

*F*

_{inter}=

*F*

_{1a}+

*F*

_{2a}.

*F*is evaluated as follows. First, consider the ratio of

_{b}*F*

_{1a}to

*F*in Eq. (3):Henceassuming that sin(

_{b}*ly*) and sin(

*kx*) are of the same order. Applying the aforementioned parameters yieldsFinally it givesbased on the earth radius

*R*≃ 6.37 × 10

^{6}m. Hence it means that, within the specified choices of zonal wind speed and wave amplitude, the advection forcing

*F*

_{adv}is dominated by

*F*

_{1a}, rather than

*F*, as long as the meridional wavenumber is small enough.

_{b}*F*

_{2a}to

*F*in Eq. (4) can be assessed as follows:HenceFinally it givesIn other words, if [(1.56

_{b}*n*)

^{2}+ (2

*m*)

^{2}]

*m*≤ 656, within the specified choices of zonal wind speed and wave amplitude, the interaction term

*F*

_{inter}tends to be dominated by

*F*

_{2a}rather than

*F*. This criterion, along with the former criterion that the meridional wavenumber

_{b}*m*≤ 16 as shown in Eq. (5), ensures that

*F*

_{1a}and

*F*

_{2a}are leading terms in

*F*

_{adv}and

*F*

_{inter}, respectively. Thus,

*F*

_{1a}should be able to qualitatively represent

*F*

_{adv}, and

*F*

_{2a}qualitatively represents

*F*

_{inter}, provided that these criteria are met.

*m*≤ 16 and [(1.56

*n*)

^{2}+ (2

*m*)

^{2}]

*m*≤ 656, the ratio

*F*

_{inter}/

*F*

_{adv}can be approximated byIt is clear that

*F*

_{2a}and

*F*

_{1a}have opposite signs to each other, given the zonal mean flow

*U*

_{0}> 0. Further applying characteristic scales of the parameters yieldsorEquation (7) explicitly states that

*F*

_{1a}and

*F*

_{2a}are negatively proportional to each other. This further implies that within the two constraints,

*m*≤ 16 and [(1.56

*n*)

^{2}+ (2

*m*)

^{2}]

*m*≤ 656,

*F*

_{adv}and

*F*

_{inter}therefore have opposite signs. Additionally, Eq. (7) also indicates that for small zonal wavenumber

*n*and meridional wavenumber

*m*, satisfying (1.56

*n*)

^{2}+ (2

*m*)

^{2}≤ 41,

*F*

_{2a}is greater than

*F*

_{1a}in magnitude and thus

*F*

_{inter}prevails over

*F*

_{adv}. For larger wavenumbers,

*F*

_{adv}dominates over

*F*

_{inter}instead. Table 1 is a summary of combinations of wavenumbers

*n*and

*m*that cause

*F*

_{inter}to dominate over

*F*

_{adv}. This indicates that for ultralong waves with

*n*≤ 4 and

*m*≤ 3,

*F*

_{inter}appears to be dominant in magnitude.

## 3. Four regimes

*n*≤ 4.1 and

*m*≤ 3.2, as shown in Fig. 1,

*F*

_{adv}can always be represented by

*F*

_{1a}and

*F*

_{inter}by

*F*

_{2a}. The negative proportionality between

*F*

_{1a}and

*F*

_{2a}therefore ensures that

*F*

_{adv}and

*F*

_{inter}have opposite signs. In addition, as long as Eq. (10) holds,

*F*

_{inter}always prevails over

*F*

_{adv}such that the sign of the total forcing,

*F*

_{tot}=

*F*

_{adv}+

*F*

_{inter}, will be determined by

*F*

_{inter}.

In the present work, we define the regime confined by Eq. (10) as Regime I_{long}, which is characterized by long waves. When wavenumbers *n* and *m* increase until violating Eq. (10) while still satisfying Eqs. (8) and (9), *F*_{1a} and *F*_{2a} are still able to effectively represent *F*_{adv} and *F*_{inter}, respectively, such that *F*_{adv} remains negatively proportional to *F*_{inter}. At this stage, however, *F*_{adv} starts dominating over *F*_{inter} in magnitude. We define this regime as Regime I_{short}, which features relatively short waves, as shown in Fig. 1. It is clear that within Regimes I_{long} and I_{short}, there is always an opposition between *F*_{adv} and *F*_{inter}. If wavenumbers *n* and *m* continue to grow such that Eq. (9) is violated, but Eq. (8) still holds, then *F _{b}* dominates over

*F*

_{2a}, but

*F*

_{1a}still prevails over

*F*. It indicates that

_{b}*F*

_{2a}is no longer able to represent

*F*

_{inter}and thus the negative proportionality between

*F*

_{1a}and

*F*

_{2a}is not necessarily associated with the opposition between F

_{ad}

*and*

_{v}*F*

_{inter}. This regime is defined as Regime II. When wavenumbers

*n*and

*m*keep growing beyond Regime II, terms +

*F*and −

_{b}*F*will be instead the dominant terms in

_{b}*F*

_{adv}and

*F*

_{inter}, respectively. In this case,

*F*

_{adv}and

*F*

_{inter}tend to cancel with each other exactly, due to exact cancellation between +

*F*and −

_{b}*F*. Therefore, the opposition between

_{b}*F*

_{adv}and

*F*

_{inter}still exists, whereas neither

*F*

_{1a}resembles

*F*

_{adv}nor

*F*

_{2a}resembles

*F*

_{inter}. This regime is defined as Regime III. The four primary regimes depicted on the wavenumbers

*n*versus

*m*plane are shown in Fig. 1.

Alternatively, four regimes are summarized as follows:

- Regime I
_{long}: |*F*_{1a}| ≥ |*F*| and |_{b}*F*_{2a}| ≥ |*F*|, with_{b}*F*_{1a}∝ −*F*_{2a}and |*F*_{2a}| ≥ |*F*_{1a}|, thus*F*_{adv}∝ −*F*_{inter}, with*F*dominating in the total forcing._{inter} - Regime I
_{short}: |*F*_{1a}| ≥ |*F*| and |_{b}*F*_{2a}| ≥ |*F*|, with_{b}*F*_{1a}∝ −*F*_{2a}and |*F*_{1a}| ≥ |*F*_{2a}|, thus*F*_{adv}∝ −*F*_{inter}, with*F*_{adv}dominating in the total forcing. - Regime II: |
*F*_{1a}| ≥ |*F*| and |_{b}*F*_{2a}| ≤ |*F*|, thus no evidence of the opposition between_{b}*F*_{adv}and*F*_{inter}. - Regime III: |
*F*_{1a}| ≤ |*F*| and |_{b}*F*_{2a}| ≤ |*F*|, thus_{b}*F*_{adv}∝ −*F*_{inter}, due to exact cancellation between +*F*and −_{b}*F*._{b}

To vividly illustrate the opposition between *F*_{adv} and *F*_{inter}, two examples drawn from Regimes I_{long} and I_{short} are shown in Figs. 2 and 3, respectively. The horizontal distributions of the idealized 500-hPa geopotential height *z*, geostrophic relative vorticity *ζ _{g}*, components and sum of

*F*

_{adv}and

*F*

_{inter}, for

*n*= 2 and

*m*= 2 are displayed in Fig. 2. At the reference latitude 50°S, the 500-hPa geopotential heights are the zonally averaged heights, namely,

*z*

_{0}= 5300 m. In the SH, relatively low heights are associated with negative geostrophic vorticity while relatively high heights are associated with positive vorticity, as shown in Figs. 2a,b. Additionally, Figs. 2d,e,g,h clearly show that |

*F*

_{1a}| is about an order of magnitude greater than |

*F*|, and |

_{b}*F*

_{2a}| is also about an order of magnitude greater than |

*F*| such that |

_{b}*F*

_{1a}| qualitatively represents |

*F*

_{adv}| and |

*F*

_{2a}| represents |

*F*

_{inter}|. With

*n*= 2 and

*m*= 2, this case falls into Regime I

_{long}, where it is predicted that

*F*

_{adv}∝ −

*F*

_{inte}

*, with*

_{r}*F*

_{inter}dominating over

*F*

_{adv}. This can readily be seen in Figs. 2c,f,i that an opposition exists between

*F*

_{adv}and

*F*

_{inter}, with

*F*

_{inter}resembling the total forcing

*F*

_{adv}+

*F*

_{inter}.

Another example in which *n* = 5 and *m* = 2, falling into Regime I_{short}, is displayed in Fig. 3. As in the above example from Regime I_{long}, here |*F*_{1a}| is about an order of magnitude greater than |*F _{b}*|, and |

*F*

_{2a}| is about an order of magnitude greater than |

*F*| such that

_{b}*F*

_{1a}effectively resembles

*F*

_{adv}and

*F*

_{2a}resembles

*F*

_{inter}. The negative proportionality between

*F*

_{1a}and

*F*

_{2a}therefore is able to account for the opposition between

*F*

_{adv}and

*F*

_{inter}. However, attributable to |

*F*

_{1a}| ≥ |

*F*

_{2a}|,

*F*

_{adv}dominates in the total forcing

*F*

_{adv}+

*F*

_{inter}instead, as shown in Figs. 3c,f,i. This is consistent with what was previously discussed regarding Regime I

_{short}.

So far, four regimes in which the dominance and proportionality of *F*_{adv} and *F*_{inter} may vary have been identified. However, the primary focus of the present study is on the midlatitude weather systems containing planetary- and synoptic-scale waves that have zonal wavenumber *n* ≤ 20 and meridional wavenumber *m* ≤ 5, in general. The study area of interest would therefore fall into the Regimes I_{long} and I_{short}, where the opposition between *F*_{adv} and *F*_{inter} is successfully derived, based upon the idealized analytical model.

To clearly illustrate the dominant zonal and meridional wavenumbers associated with midlatitude weather systems, one blocking event and one nonblocking event are chosen here and examined by spectral analysis. The July 1999 blocking event, as diagnosed by DC2005, is once again chosen for the present study. The gridded SH 500-hPa heights are from the National Centers for Environmental Prediction–National Center for Atmospheric Research (NCEP–NCAR) daily average reanalysis (Kalnay et al. 1996). Figure 4a is the 500-hPa geopotential height field on 25 July 1999, which is the block-onset day. A high–low dipole pattern is situated over the southern Pacific Ocean, with the split-flow configuration that leads to considerable meridional structure in the 500-hPa height field. Following Colucci et al. (1981), a two-dimensional Fourier transform is applied to the 500-hPa height on 25 July 1999 and the spectrum is obtained, denoted by |*A*(*n*, *m*)|, as shown in Fig. 4b. Here the spectrum represents the amplitudes of perturbation in the 500-hPa height field. Note that |*A*(0, 0)| (the amplitude of the area-averaged height) and |*A*(0, 1)| (the amplitude of zonally averaged wave with wavenumber 1 in the meridional direction) are excluded from this figure due to the overwhelming dominance. Four peak amplitudes are identified from the plot, namely, |*A*(1, 1)|, |*A*(1, 2)|, |*A*(3, 2)|, and |*A*(7, 1)|, with the first two wave components as the leading contributions. This result is consistent with the two-dimensional Fourier analysis results obtained by Colucci et al. (1981), who found that the incipient stages of split-flow blocking episodes are associated with an increase in amplitude in two-dimensional wave components having zonal wavenumbers 1 and 2 and meridional wavenumber 1. Colucci et al. (1981) also stated that at the time of observed split-flow blocking, the wave component with zonal wavenumber 3 and meridional wavenumber 2 achieves peak amplitude. Therefore, it is obvious that wavenumbers associated with the peak amplitudes in the July 1999 blocking event fall into either Regime I_{long} or I_{short} and thus the opposition between *F*_{adv} and *F*_{inter} is effective for all these dominant wave components.

In addition, as an illustration of a nonblocking event, the 500-hPa height field on 20 August 1999 is presented in Fig. 4c. No dominant blocking high is apparent over the block-onset region of the July 1999 blocking event (Fig. 4a). The spectrum of the 500-hPa height field on 20 August 1999 is similarly achieved by applying the two-dimensional Fourier analysis and is shown in Fig. 4d. Once again, |*A*(0, 0)| and |*A*(0, 1)| are excluded due to their overwhelming dominance. Four primary modes of waves are found, |*A*(5, 1)|, |*A*(1, 2)|, |*A*(3, 1)|, and |*A*(8, 1)|, with descending intensity. This confirms that wavenumbers associated with peak amplitudes in a nonblocking configuration also fall into Regime I_{long} or I_{short} such that the opposition between *F*_{adv} and *F*_{inter} is valid for these primary wave components.

## 4. Synoptic interpretation

Comparisons of Eq. (12) to Eq. (3) and Eq. (13) to Eq. (4) immediately unveil direct dynamical meaning of the forcing terms. In Eqs. (12) and (3) *F*_{1a} represents the advection of the wave-induced variations of the meridional gradient of relative vorticity by the background westerly flow, which is a linear process, and *F _{b}* stands for the advection of the wave-induced variations of the meridional gradient of relative vorticity by the wave-induced winds, which is a nonlinear process. In Eqs. (13) and (4)

*F*

_{2a}is the enhancement (weakening) of the meridional gradient of absolute vorticity by confluence (difluence) acting on the planetary vorticity meridional gradient, which is again a linear process, and −

*F*is the change of the meridional gradient of absolute vorticity by deformation acting on the wave-induced relative vorticity gradient, which is nonlinear.

_{b}In the analysis preceding Eq. (5), *F*_{1a} dominates if the background westerly flow is strong enough or if the wave amplitude is small enough. In the analysis preceding Eq. (6), it is found that *F*_{2a} dominates if the wave amplitude is small enough. Therefore, Regimes I_{long} and I_{short} both represent linear dynamics, with the former associated with long waves and the latter associated with relatively short waves, respectively, while Regimes II and III are dominated by nonlinear processes.

Exact cancelling between +*F _{b}* and −

*F*in forcing

_{b}*F*

_{adv}and

*F*

_{inter}reveals that the nonlinear process of the advection of the wave-induced variations in the meridional gradient of relative vorticity by the wave-induced winds cancels out the nonlinear effect of confluence and diffluence on the preexisting relative vorticity dipole. Thus it is immediately clear that the opposition effect between

*F*

_{adv}and

*F*

_{inter}is indeed the opposition between

*F*

_{1a}and

*F*

_{2a}, within the barotropic dynamic framework. In other words, the advection of the wave-induced variations of the meridional relative vorticity gradient by the zonal flow is opposite in sign to the effect of confluence or diffluence acting on the planetary vorticity meridional gradient, and only the relative magnitudes determine which one dominates.

Following Holton (2004), advection of relative vorticity tends to move the vorticity pattern and hence the troughs and ridges eastward, whereas advection of planetary vorticity tends to move the troughs and ridges westward against the advecting wind field. For an idealized geopotential height field with a given disturbance as shown in Eq. (2), the amplitude of the vorticity increases as the square of the wavenumber as represented by Eq. (11). As a consequence, the advection of relative vorticity dominates over planetary vorticity advection for short waves, while for long waves the planetary vorticity advection tends to dominate.

The opposition effect between *F*_{1a} (*F*_{adv}) and *F*_{2a} (*F*_{inter}) is schematically illustrated in Fig. 5 in terms of long waves and relatively short waves. Figure 5a depicts an idealized configuration of geostrophic flow that features a long wavelength at initial time *t* = *t*_{0}. Under the forcing *F*_{1a}, the geostrophic flow pattern would move eastward responding to the advection of the background flow; however, under the forcing *F*_{2a}, the geostrophic flow pattern would move westward responding to the conservation of absolute vorticity. Thus, at *t* = *t*_{0} + *δt* over the diffluent region, *F*_{2a} would favor a weakening in the westerly flow (∂*u _{g}*/∂

*t*< 0), whereas

*F*

_{1a}would favor a strengthening of the westerly flow (∂

*u*/∂

_{g}*t*> 0). Since

*F*

_{2a}tends to dominate over

*F*

_{1a}for relatively long waves, a net weakening in the geostrophic westerly flow therefore results over the diffluent region. This scenario can be characterized as the interaction case demonstrated in DC2005.

On the other hand, Fig. 5b depicts an idealized configuration of geostrophic flow that features a short wavelength at initial time *t* = *t*_{0}. With the geostrophic flow pattern moving eastward under forcing *F*_{1a} and moving westward under forcing *F*_{2a}, at *t* = *t*_{0} + *δt* over the confluent region, *F*_{1a} would favor a weakening in the westerly flow (∂*u _{g}*/∂

*t*< 0), whereas

*F*

_{2a}would favor a strengthening of the westerly flow (∂

*u*/∂

_{g}*t*> 0). Due to the dominance of

*F*

_{1a}for relatively short waves, a net weakening of geostrophic westerly flow would therefore take place over the confluent region. This scenario corresponds to the advection case investigated in DC2005.

## 5. The nonblocking events

It is also observed from the DC2005 study that there is an opposition between *F*_{adv} and *F*_{inter} in nonblocking events as well. Regardless of this opposition, the onset of blocking did not occur in the nonblocking events. The reason, as hypothesized by DC2005, is that one forcing contributes more positively to the geostrophic zonal wind tendency than the opposing negative contribution from the other forcing. To verify it, two nonblocking cases, chosen on 15 and 20 August 1999, are now examined in detail.

In the first nonblocking case, the advection forcing acts as the block-onset forcing mechanism contributing to a local weakening of the geostrophic westerlies, while the interaction forcing acts as the block-opposing mechanism with a confluent flow contributing to the increasing geostrophic westerlies. The 500-hPa height fields over the five consecutive days, starting on 11 August 1999, are shown in Figs. 6a–e. It is clear that no blocking structure can be identified on 15 August 1999 over 60°–40°S and 165°E–175°W, as shown in Fig. 6e. Therefore this day is defined as a nonblocking day. On 11 August 1999, which is 4 days prior to the nonblocking day, a strong confluent flow is observed over the region of interest as outlined in Fig. 6a. Just like in the advection case of DC2005, 5 days before the block onset shown in their Fig. 3a, here a strong confluent flow over the region of interest is associated with an equatorward increasing cyclonic potential vorticity advection (or poleward increasing anticyclonic potential vorticity advection). As seen in Table 2, the advection forcing consistently contributes to a local weakening of the geostrophic westerlies over the five consecutive days, while the interaction forcing mostly opposes the weakening of the geostrophic westerlies. Although on the first 2 days the advection forcing prevails over the interaction forcing in magnitude, the dominance of two forcings reverses during the remaining 3 days and this leads to the overall dominance of the interaction forcing, which opposes a local weakening of the geostrophic westerlies. This can be readily seen from the time-averaged total calculated and analyzed geostrophic zonal wind tendencies in Table 2. Therefore, unlike in the advection case in DC2005, no block onset is observed on 15 August 1999.

In the second nonblocking case, the interaction forcing acts as the block-onset mechanism while the advection forcing opposes it. Figures 7a–e depict the time evolution of this nonblocking event for five consecutive days, starting on 16 August 1999. Over the same region as in the first nonblocking event, no blocking high is identified on 20 August and this day is regarded as a nonblocking day. A diffluent flow is dominant on 16 August 1999, 4 days prior to the nonblocking day, resembling the interaction case in DC2005, 5 days prior to the block onset. The lower panel of Table 2 shows that the interaction forcing consistently favors the local weakening of the geostrophic westerlies while the advection forcing opposes it. The contribution from the interaction forcing overwhelms the advection forcing for the first 2 days, but the interaction forcing yields to the advection forcing for the remaining 3 days. This leads to an overall dominance of the advection forcing and thus results in a local strengthening of the geostrophic westerlies in the time-averaged total-calculated and analyzed fields. Hence, unlike in the interaction case of DC2005, no block onset takes place on 20 August 1999.

## 6. Zonal propagation of barotropic Rossby waves

In section 2, the two forcing terms, as shown in Eqs. (3) and (4), are solved analytically based on the analytical model specified in Eq. (2). To accomplish this work, the stratification term in the quasigeostrophic potential vorticity is neglected. Therefore, the assumption being made here is that of barotropic dynamics. In a barotropic atmosphere, absolute vorticity is conserved and waves propagate eastward with phase speed *C _{x}* =

*U*

_{0}−

*β*/(

*k*

^{2}+

*l*

^{2}). A wave will therefore become stationary if

*C*= 0 or

_{x}*U*

_{0}=

*β*/(

*k*

^{2}+

*l*

^{2}). Our objective definition of blocking requires a persistence, in space and time, and therefore stationarity of geostrophic easterly flow. Our diagnostic model explains the development of easterly flow but not its stationarity.

We investigate the stationarity of the July 1999 blocking case as follows. Examination of the synoptic evolution of this blocking case during the 5-day preblocking period (shown in Fig. 3 of DC2005) reveals that a strong ridge originally located west of the block-onset region propagates eastward into the blocking development area and stalls on the block-onset day. The spectrum analysis of the 500-hPa height fields for this blocking event is presented in Table 3, which lists the first three leading wave components. It indicates that as the block-onset day approaches, the leading wave component tends to be a wave with longer wavelength.

In addition, setting the Rossby wave phase speed *C _{x}* =

*U*

_{0}−

*β*/(

*k*

^{2}+

*l*

^{2}) equal to zero, which corresponds to the stationary Rossby wave, yields the combinations of wavenumbers shown in Table 4, where the zonal flow

*U*

_{0}is estimated as the 6-day (20–25 July) zonal-averaged westerly flow along 50°S. It is found from Table 4 that A(1, 2) is the closest wave mode to a stationary wave, given

*U*

_{0}= 20 m s

^{−1}. Coincidentally, A(1, 2) is also found to be one of the leading wave modes on the block-onset day (25 July) in Table 3. Therefore, it clearly suggests that the leading nonstationary Rossby wave propagates eastward during the 5-day preblocking period and becomes a stationary wave on the block-onset day.

Moreover, Figs. 8a–f depict the evolution of zonal and meridional wavenumbers for the 5-day preblocking period and block-onset day of this blocking case. As the block-onset day approaches, zonal and meridional wavenumbers associated with the dominant wave components gradually decease. Accordingly, this once again verifies that the phase speed of the Rossby wave starts to stall on 25 July, thereby allowing its associated midlatitude geostrophic easterlies to persist and satisfy the definition of a block on 25 July. Nevertheless, we are unable to explain, with our diagnostic model, the calculated lengthening of the blocking wave. Identification of the processes responsible for the scale lengthening is left for subsequent research.

## 7. Summary

The opposition between two block-onset forcing mechanisms for the geostrophic zonal wind tendency, namely, an advection forcing *F*_{adv} and interaction forcing *F*_{inter}, which are consistently observed over midlatitudes of the SH midtroposphere for both blocking and nonblocking events in the diagnostic study of DC2005, is analytically derived with an idealized geopotential height model. Four regimes of wavenumbers, featuring varying proportionality and dominance of *F*_{adv} and *F*_{inter}, are defined. Weather systems of concern to the present study, primarily consisting of planetary- and synoptic-scale waves, would mostly fall into the first two regimes, I_{long} and I_{short}, where the opposition between *F*_{adv} and *F*_{inter} is analytically derived. A synoptic interpretation of this opposition is schematically presented within the barotropic dynamic framework. It is found that whether blocking occurs in diffluent or confluent flow depends upon the critical wavelength associated with the geostrophic flow. Blocking tends to take place in the diffluent flow of long waves in which *F*_{inter} dominates over *F*_{adv}. In addition, blocking also tends to occur in the confluent flow of relative short waves in which *F*_{adv} prevails over *F*_{inter}. Moreover, for nonblocking events, which are more frequently observed than blocking events, detailed diagnostic investigations reveal that the time-averaged block-onset forcing mechanism fails to overwhelm the opposition such that a local strengthening of the geostrophic westerlies is the result.

An additional insight is gained from the perspective of barotropic Rossby waves. In one diagnosed case, blocking coincided, spatially and temporally, with the lengthening and subsequent stalling of a midtropospheric wave. This allowed the forcing of easterly flow associated with that wave to persist and lead to the onset of blocking. The processes responsible for the lengthening of the dominant wave in this case are not known. Possible mechanisms include upscale energy transfer through nonlinear interactions (e.g., Tanaka and Terasaki 2006) and the preferential growth of long waves through baroclinic or barotropic processes (e.g., Simmons et al. 1983). A complete description of blocking must account for the processes rendering the incipient blocking wave, and its associated easterly flow, stationary in space and time.

Furthermore, it is not known why, on rare occasions, such as the June 1955 blocking case in Table 1 of DC2005, both advection and interaction forcings contribute to or oppose block onset. It is believed to be due to the relatively large contributions from the nonquasigeostrophic processes, which are not taken into account by the diagnostic equation applied in the present study. Moreover, it could also be due to inability of a barotropic model to capture the baroclinic processes present in this case. Resolution of this matter is left for future research.

## Acknowledgments

The authors are very grateful for the insightful suggestions from Dr. John W. Nielsen-Gammon and Dr. Gordon Swaters, the two reviewers of this article. Special thanks also go to Dr. Kaijun Liu for helpful discussion during the preparation of this article. This work was supported by the National Science Foundation Grant ATM-0220009 to Cornell University. The reanalysis data for the SH were provided by the Climate Diagnostics Center of the National Oceanic and Atmospheric Administration and analyzed with the Grid Analysis and Display System (GrADS) from the Center for Ocean–Land–Atmosphere Studies.

## REFERENCES

Colucci, S. J., , A. Z. Loesch, , and L. F. Bosart, 1981: Spectral evolution of a blocking episode and comparison with wave interaction theory.

,*J. Atmos. Sci.***38****,**2092–2111.Dong, L., , and S. J. Colucci, 2005: The role of deformation and potential vorticity in Southern Hemisphere blocking onsets.

,*J. Atmos. Sci.***62****,**4043–4056.Holton, J. R., 2004:

*An Introduction to Dynamic Meteorology*. 4th ed. Academic Press, 535 pp.Kalnay, E., and Coauthors, 1996: The NCEP/NCAR 40-Year Reanalysis Project.

,*Bull. Amer. Meteor. Soc.***77****,**437–471.Simmons, A. J., , J. M. Wallace, , and G. W. Branstator, 1983: Barotropic wave propagation and instability, and atmospheric teleconnection patterns.

,*J. Atmos. Sci.***40****,**1363–1392.Tanaka, H. L., , and K. Terasaki, 2006: Blocking formation by an accumulation of barotropic energy exceeding the Rossby wave saturation level at the spherical Rhines scale.

,*J. Meteor. Soc. Japan***84****,**319–332.

Summary of combinations of the zonal wavenumber *n* and meridional wavenumber *m* that cause *F*_{inter} to dominate over *F*_{adv}.

Advection-forcing (Adv) contributed, interaction-forcing (Inter) contributed, total quasigeostrophic (TQ), and total analyzed (TA) 12-h changes in the 500-hPa geostrophic zonal wind (m s^{−1}) averaged over the region of interest (60°–40°S and 165°E–175°W) for five consecutive days of two nonblocking events in 1999, respectively. Also shown are the time averages during the entire 5-day period.

First three leading wave components of 500-hPa height field for the 5-day preblocking period and block-onset day (25 Jul) of the July 1999 blocking event.

Combinations of zonal wavenumber *n* and meridional wavenumber *m* that lead to stationary waves with the phase speed *C _{x}* =

*U*

_{0}−

*β*/(

*k*

^{2}+

*l*

^{2}) = 0, given

*U*

_{0}= 20 m s

^{−1},

*k*= (2

*πn*)/

*L*, and

_{x}*l*= (2

*πm*)/

*L*.

_{y}