This work explores the inverse problem of what are the precursors to a given blocking anomaly in climatological flow over the Atlantic and Pacific Oceans, respectively. Blocking anomaly in geopotential height field is specified as a dipole structure which is dominated by a strong positive anomaly centred at about 60°N and a weak negative anomaly south of it. The method of conditional nonlinear optimal perturbation (CNOP) is applied to investigate the above problem by using a T21L3 quasi-geostrophic model and its tangent linear and adjoint versions.
As early as Berggren et al(1949), a notion that baroclinic synoptic-scale disturbance might act to maintain a planetary-scale blocking circulation had drawn strong attention. Subsequent to this pioneering work, a lot of case-studies discussed the role of an individual intense cyclone during the onset of blocking (e.g. Colucci, 1985, 1987; Li et al, 2001). Nakamura and Wallace (1993) documented the more robust features of such baroclinic eddies in composites based upon a large number of blocking events. The results also show that only one or two pairs of migratory cyclone and anticyclone appear to be associated with the onset of the blocking patterns. Liu et al(1996) studied the maintenance of the blockings over the North Pacific, the North Atlantic Ocean and Alaska individually, and found that the relative importance of transient eddies is different. Nakamura et al(1997) pointed out that the quasi-stationary wave train plays a more important role than transient eddies in the blocking over Europe, based upon a 30-year record of 500 hPa height and sea-level pressure, while it is contrary over the Pacific. Following that, Frederiksen (1998) explored the precursors to the Pacific–North American blockings and the North Atlantic ones with explicit matrix inversion using a two-level tangent linear model with time-dependent basic states taken from observations. The results are that the precursors are small-scale wave-train disturbances upstream of the blockings. Li et al(1999) described the dynamics of adjoint sensitivity perturbations that excite block onsets over the Pacific and Atlantic Oceans, which are carried out in a hemispherical primitive-equation, θ-coordinate, two-layer model.
Other possible preconditions for block onset have also been explored. Colucci (2001) investigated the local preconditioning of the mid-tropospheric planetary-scale flow prior to the onset of a blocking episode during January 1985, and pointed out that the planetary-scale preconditioning is anomalously diffluent, or characterized by anomalously negative planetary-scale, geostrophic stretching deformation. Dong and Colucci (2005) examined the relative importance of interactions between deformation and potential vorticity (PV) as a block-onset mechanism in 30 cases of atmospheric blocking over the Southern Hemisphere. Great progress has been made on the preconditions of blocking onset. However, weather forecasts during periods when the atmospheric flow changes from strong zonal flow to blocking or vice versa still frequently suffer from a rapid loss of predictability (Frederiksen et al, 2004). Therefore, how to better capture the onset of blocking is a principle task for weather forecasting. To that end, Mu and Jiang (2008) presented the approach of conditional nonlinear optimal perturbation to seek the perturbations triggering blocking onset, pointing out that the nonlinear optimization approach is superior in finding the initial optimal perturbations triggering blocking onset in a given ensemble. However, trying out this idea in the generation of initial perturbations for ensemble prediction, particularly for the weather events of transitions, is a challenge.
In the present study we will explore the precursors to blocking anomalies in climatological flows by the theory of conditional nonlinear optimal perturbations (Mu et al, 2003; Duan et al, 2004; Mu and Jiang, 2008), which is rooted in the belief that the blocking onset is dominated by complicated nonlinear systems (Shutts, 1983; Luo et al, 2001). The corresponding nonlinear optimization problem is constructed according to the blocking index proposed by Liu (1994). The climatological state is adopted as the background flow in this work, which can be referenced to many other studies (e.g. Molteni and Palmer, 1993; Li and Ji, 1997; Li et al, 1999).
The outline of this paper is as follows. In section 2, we summarize the model and present the corresponding nonlinear optimization problem. Section 3 presents the numerical results. The precursors to blocking anomalies over the Atlantic and Pacific Oceans are compared. The energy source for precursor development and a possible mechanism for block onset are considered in section 4. Data analysis is verified in section 5. The conclusions of this study are summarized in section 6.
2. Model description and nonlinear optimization problem
We summarize below the properties of the numerical model and outline the nonlinear optimization problem associated with the block onset.
2.1. Model description
For this study we use the T21, quasi-geostrophic (QG) global spectral model of Marshall and Molteni (1993). The model integrates prognostic equations for potential vorticity (PV) in three layers, which are as follows.
where J is the Jacobian of a two-dimensional field. PV is defined as
where f = 2 Ωsinϕ, R1 ( = 700 km) and R2 ( = 450 km) are Rossby radii of deformation appropriate to the 200–500 hPa layer and the 500–800 hPa layer, respectively. h is the real orographic height, and H0 is a scale height (9 km).
D1, D2 andD3 are linear operators representing the effects of Newtonian relaxation of temperature, linear drag on the 800 hPa wind, and horizontal diffusion of vorticity and temperature. S1, S2 and S3 are time-independent but spatially varying sources of PV. The index i = 1, 2, 3 refers to 200 hPa, 500 hPa and 800 hPa respectively. The exact form adopted for these operators can be referenced in the paper by Marshall and Molteni (1993).
Though the T21, QG model is a very simple dynamical one, the results of Marshall and Molteni (1993) showed that if an appropriate forcing function is employed, it is able to generate a very realistic climatology in a long nonlinear integration and, furthermore, two regimes (blocking and strong zonal flow) similar to the observed ones. Modelled and observed regimes have not only similar spatial patterns but also an almost identical distribution of the residence time.
If X denotes the column vector of PV spectral coefficients, we can formally write the spectral equations in the form
where N denotes a nonlinear matrix operator.
Assume that for fixed time T > 0 and the initial potential vorticity Q|t=0 = Q0, the propagator M is well–defined; Q(T) = MT(Q0) is the solution of equation (3) at time T.
2.2. The definition of blocking
To clarify a flow, the blocking index B induced by Liu (1994) is used to measure the resemblance of a particular circulation pattern with the blocking regime.
where φb is the blocking anomaly field represented by the stream function over the climatological mean field ψc. φd = ψ− ψc, the daily stream function anomaly field. The angle brackets denote the Euclidean inner product on a sphere, integrated over height:
where V represents integration over the whole atmosphere. A circulation pattern with B ≥ 0.5 is defined as a blocking flow following Liu (1994). It is easy to see that the larger positive B is, the more pronounced is the blocking flow.
2.3. Nonlinear optimization problem
In the following, we assume negligible model errors. The nonlinear optimization problem is constructed according to the above blocking index. Our purpose is to seek the initial perturbation, , which satisfies the initial constraint condition and makes the blocking index B acquire the maximum:
where F is an operator which transforms the potential vorticity of basic state to stream function field. Qc is the climatological potential vorticity field. q0 is a random given initial potential vorticity perturbation, which satisfies ||q0||≤ σ. σ is a presumed positive constant representing an upper-bound of the magnitude of the initial perturbation. q0 = Eφ0, where E is a linear operator, which transforms stream function perturbation into potential vorticity perturbation. E−1 is the inverse operator of linear operator E. Note that if the given initial constraint condition or optimization time interval is small, the flow triggered by at the optimization time may be only strong meridional flow. However, such initial perturbation is still defined as the precursor to blocking anomaly φb.
Given two three-dimensional potential vorticity perturbation fields, q1 and q2, the stream function squared norm is defined as follows.
Here, the initial constraint condition is ||q0||2 = [q0, q0].
In the numerical model used for this study, the adjoint M* of the tangent version of the model equations has been defined with respect to the Euclidean inner product 〈•, •〉. According to Buizza et al(1993), the adjoint operator M*S with respect to the stream function squared norm can be deduced from M*. Here, M*S = EEM*E−1E−1.
To capture the maximum of B(q0) with the initial constraint condition ||q0||≤ σ, we may calculate the minimum of
with the same constraint ||q0||≤ σ. The process to acquire the gradient of an objective function JN to initial perturbations can be referenced in Jiang et al(2008). Here, the gradient of JN with respect to initial potential vorticity perturbation in the stream function squared norm can be deduced as follows:
To obtain the nonlinear optimal perturbation numerically, the optimization algorithm of the spectral projected gradient 2 (SPG2) is employed, which calculates the least value (a local or global minimum) of a function of several variables subject to box or ball constraints. Detailed descriptions of SPG2 are given by Birgin et al(2000). The above presents the objective function and its gradient with respect to the initial perturbation q0, which are needed for the algorithm of SPG2 in the calculation of . Then, the precursors in stream function field can be acquired directly: .
3. Optimal precursors to block onset
In this section, we explore the block onset by solving the above nonlinear optimization problem. The climatological flow and Atlantic blocking anomaly are specified as those in Mu and Jiang (2008). The detailed process to acquire them is as follows. Firstly, 1800 days (corresponding to 20 winters) of integration using the T21L3 QG model are performed with the initial conditions of the European Centre for Medium-Range Weather Forecasts (ECMWF) analysis of 0000 UTC 1 December 1983. The mean state of the 20 winters is defined as the climatological mean. Next, for each day, the latitudinal mean anomalies of the 500 hPa stream function at 60°N over the Atlantic and western European areas have been computed by averaging the daily stream-function anomalies at 16 grid points from about 55°W to 30°E. For this time series, the 256 days with the largest positive stream-function anomalous values have been selected, of which the mean state and the anomaly from the climatological mean are obtained. This anomaly pattern is defined as Atlantic blocking anomaly, which is dominated by a strong positive anomaly centred at about 60°N and a weak negative anomaly south of it.
In order to study Pacific blockings, we have shifted the Atlantic blocking anomaly pattern to the Pacific area over about 150°E to 236°E, which can be referenced in Oortwijn (1998) and Li et al(1999). Therefore, the results of the Atlantic and Pacific blockings can be comparatively analysed. The blocking anomalies in geopotential height field at 500 hPa over the Atlantic and Pacific Oceans are shown in Figure 1. The climatological zonal wind over the 20-year integration is given in Figure 2. In the following numerical experiments, the initial constraint condition σ = 4.0 × 105m2 s−1 is chosen, so that the amplitude of initial geopotential height perturbation at 500 hPa is within 20 geopotential metres (gpm). The results for an optimization time of 3 and 5 days are presented, respectively.
3.1. Atlantic block
Figure 2 shows the optimal precursors to Atlantic block onset for an optimization time of 3 days. One sees that the wave trains are mostly localized in the northward flanks of the Atlantic upper-level jet, which present a northeast–southwest trend. The leftover parts located in the southward flanks of the corresponding upper-level jet take on a northwest–southeast trend. The maximum amplitudes of the perturbations lie upstream of the blocking region. Additionally, the perturbations are characterized by a baroclinically amplifying pattern with a westward tilt with height. The initial optimal perturbations for an optimization time of 5 days and their nonlinear evolutions with time at 500 hPa are shown in Figure 3. We can find that the initial main wave-train structures are more upstream than those for an optimization time of 3 days. What is more, the initial optimal perturbations are less localized. From the time sequence, we can clearly see that the precursor has a smaller scale than the final anomaly. The disturbance propagates eastward across eastern North America and increases its scale. It appears that the anticyclonic eddy over southeastern North America evolves into the blocking high anomaly by day 5, and the one near Greenland incorporates and enforces the anomaly. Another anticyclonic eddy near Alaska perhaps preconditions the environment for subsequent blocking. Thus, it may be said that the structural modification of the eddies in the wave trains leads to the planetary structure that becomes associated with block onset.
3.2. Pacific block
Figure 4 shows the initial optimal perturbations to the Pacific blocking anomaly for an optimization time of 5 days and their nonlinear evolutions at 500 hPa. One sees that as well as the Atlantic block precursor perturbations, the perturbations are also mainly localized in the northward flanks of the Pacific upper-level jet, which are northeast–southwest wave trains. Their maximum amplitudes lie in the East Asia area, upstream of the corresponding blocking region, which are more upstream than those with the optimization time of 3 days (not shown here). The disturbance propagates eastward across the western Pacific Ocean and increases its scale. Similarly, we find that the anticyclonic eddy over the Bering Sea and northwest North Pacific Ocean evolves into the blocking high anomaly by day 5, and the ones near Korea and the Asian continent incorporate in turn and enforce the anomaly. However, the maximum value of the wave train for the Pacific block is larger than that for the Atlantic block. On the contrary, at the optimization time the Atlantic blocking positive anomaly is stronger than the Pacific one. Besides, the dipole structure in the south–north direction over the Pacific area is more symmetric than that over the Atlantic area. This may be to say that the precursors over the Pacific Ocean are more apt to evolve into a dipole block compared with those over the Atlantic area.
4. Interpretation of the precursor development and block onset
In the above section, we obtain the precursors to block onset by solving the corresponding nonlinear optimization problem. In this section, it is necessary to explain the physical mechanism responsible for the growth of the optimal precursor disturbances and how they trigger the block onset.
4.1. Energy source for the precursors
Li et al(2001) indicated that the eddies at suitable positions can absorb energy from the basic flow efficiently, and favour triggering the internal mode, hence the maintenance of Ural blocking in the summer of 1998. Here, to explain the energy source for the optimal precursors, we first decompose the potential vorticity qi into a perturbation q′i and a basic state q̄i. Similarly, . The equation for PV perturbation is written as:
and i = 1, 2, 3. PV perturbation is as follows:
Then, multiplying (11) by − ψ′i, horizontally averaging the equation, and summing over the layers, we obtain
TE is the total (kinetic plus available potential) energy of the perturbations. It reveals that the perturbation energy can grow through barotropic (the first two terms) and baroclinic (the second two terms) extraction from the basic flow. Figure 5 shows the TE production of the optimal precursors to the Atlantic and Pacific blockings with time. It is found that for an optimization time of 3 and 5 days whether for the Atlantic or Pacific blocking, the available potential energy contributes more to the initial optimal perturbations. However, the kinetic energy dominates the structures with their evolutions. Based on equation (13), we may conclude that the energy source for the development of precursor perturbations mainly comes from the horizontal shear of the basic flow, and the baroclinic adjustment only plays a small part. In fact, from the structure of the perturbations and their positions relative to the jet, we can deduce that these initial perturbations belong to the unstable type, which make it easy to get energy from the basic state (Yang et al, 1998). Comparatively, the precursor perturbations to the Pacific blocking get more energy than those to the Atlantic blocking, in order to develop into a block, which may be due to the stronger Pacific jet.
4.2. A possible dynamical framework for block onset
To explore the effect of the precursor perturbation interactions on the large-scale block onset, we use ‘PV thinking’ to try to reveal the possible dynamical mechanism. The time-mean large-scale eddy feedback of the precursor perturbations during the optimization time interval is defined as follows (cf. Luo et al, 2001):
where J means the Jacobian operator. p represents the planetary part; in this paper, the zonal wave number 0–5 is chosen. The overbar ‘− ’ represents the time mean during the optimization time T. represents the nonlinear evolution of the optimal perturbation in stream function field at time t.
Figure 6 presents the Jp fields for Atlantic and Pacific blocks for an optimization time of 3 and 5 days respectively. It is seen that all the Jp fields have a low/high pattern upstream of the antecedent block, indicating the importance of the eddy feedback in exciting the block. The eddy-induced planetary-scale anticyclonic vorticity lies to the north of the eddy-induced planetary-scale cyclonic vorticity. That is to say, the eddy-induced planetary-scale potential vorticity seems to advect anticyclonic (cyclonic) vorticity to the high pressure (low trough) region of blocking, and thus it promotes the block onset. The conclusion obtained in this paper is consistent with that by Luo et al(2001). Furthermore, we compare the Jp fields for blocks over these two oceans, and find that the amplitude of the Jp field for the Pacific block is stronger than that for the Atlantic block for the same optimization time. Therefore, we maybe say, the Pacific block onset is easier to understand from an eddy-forcing-mechanism point of view.
5. Data analysis
The theoretical results reveal that for the Atlantic and Pacific blocking anomalies, precursors are wave-train disturbances upstream of the blocking regions, which mainly focus on the northward flanks of the corresponding upper-level jet. The low/high eddy-forcing pattern induced by the precursors may provide a possible dynamical explanation for the block onset. To further verify the above results, two blocking cases over the Atlantic and Pacific Oceans are analysed by using the National Centers for Environmental Prediction/National Center for Atmospheric Research (NCEP/NCAR) reanalysis daily and monthly data (1948–2008), respectively. The stream-function field is calculated as ψi, j = gZi, j/f0 at each grid point (i, j) from 20°N to 80°N (f0 is the Coriolis parameter at 55°N, g is the gravitational acceleration, and Zi, j is the geopotential height at each grid point). The 500 hPa field truncated by Fourier spectral analysis between zonal wave numbers 6 and 18 is crudely defined as the synoptic-scale field. Then, the Jp field can be calculated as equation (15).
Figure 7 presents the blocking case over the Atlantic Ocean. On 12 February 1994, a weak blocking high occurs at longitude 0°, and on 14 February it developed into a dipole blocking whose positive anomaly centred at about 67°N. We find that the synoptic waves on 12 February focus on the northward flanks of the climatological Atlantic upper-level jet. The eddy forcing induced by the synoptic waves presents a low/high pattern between 50°N and 70°N, west of 20°W upstream of the initial blocking. Figure 8 presents the blocking case over the Pacific Ocean, which occurs at 140°W on 27 February 1995 and matures on 5 March, whose positive anomaly centred at about 60°N. We also find that the synoptic waves at initial time focus on the northward flanks of the climatological Pacific upper-level jet. The eddy forcing induced by the synoptic waves presents a low/high pattern between 45°N and 60°N, west of 140°W upstream of the initial blocking. Note that the low/high eddy-forcing pattern may not be as evident as that obtained by the above numerical results. However, this can be attributing to the fact that the synoptic waves obtained by space-scale filter may include other information besides the optimal initial perturbations triggering block onset. Therefore, how to better capture the information of the optimal perturbations from the analysis data is another task, which we need to overcome so as to improve our forecast skill in block onset.
6. Conclusions and discussion
In this study, we view block onset as a consequence of the evolution of localized perturbations in a background flow. We attempt to answer the following questions. What are the optimal precursors to a given prescribed blocking anomaly in climatological flows? And what are the dynamics associated with such perturbations leading to block onset? The corresponding nonlinear optimization problem is constructed, and the method of conditional nonlinear optimal perturbations (CNOPs) is applied to investigate the above problem by a T21L3 quasi-geostrophic model and its tangent linear and adjoint versions. The blocking anomalies over the Atlantic and Pacific Oceans are explored comparatively.
It is found that for both the Atlantic and Pacific blockings, when the optimization time is 3 days, the precursors are baroclinic synoptic-scale wave trains, which are located upstream of the blocking region. When the time is extended to 5 days, the perturbations are less localized in the zonal direction and the maximum values are more upstream than those for an optimization time of 3 days. Comparatively, the precursors to Pacific block are more apt to form dipole blocks, while the precursors to Atlantic block are more apt to form blocking highs.
By calculating the total energy production, we find that the available potential energy contributes more to the precursors. However, the kinetic energy dominates the structures with their evolutions. That is to say, the energy source for the precursor development mainly comes from the horizontal shear of the basic flow, and the baroclinic adjustment plays a small part. This may explain why the northeast–southwest precursors are mainly located in the northward flanks of the corresponding upper-level jet.
Additionally, we find that the eddy-induced planetary-scale potential vorticity has a low/high pattern upstream of the antecedent block anomaly for both the Atlantic and Pacific blockings, which shows the importance of the eddy feedback with such structures in exciting the block onset. Therefore the eddy-induced planetary-scale potential vorticity seems to advect anticyclonic (cyclonic) vorticity to the high pressure (low trough) region of blocking, and thus it promotes the blocking onset. Comparatively, the eddy feedback pattern for the Pacific blocking is stronger that that for the Atlantic blocking. That is to say, the Pacific blocking is easier excited by the optimal wave-train structures.
Up to now, we know that the forecasting of block onset has been difficult. Certainly, if the eddy-feedback mechanism is the important one, just as Luo et al(2001) said, the optimal precursors may provide some guideline for us in medium-range forecasting. However, how to better capture the optimal precursors is another task. Meanwhile, we also recognize that there are many other mechanisms (such as planetary Rossby wave breaking: de Pondeca et al, 1998a, 1998b) that lead to block onset. This is also the reason why we can not capture all the block onsets in the present numerical models.
This research is supported by the National Natural Science Foundation of China (Grant Nos. 40905023 and 40633016) and State Key Laboratory of Severe Weather, Chinese Academy of Meteorological Sciences Program for Basic Research of China (No. 2008LASWZI01). The authors are grateful to the two anonymous reviewers and the editor for their valuable comments and suggestions.