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  • Zhang, Z., 1988: The linear study of zonally asymmetric barotropic flows. Ph.D. thesis, University of Reading, 177 pp. [Available from University of Reading, Dept. of Meteorology, P.O. Box 243, Earley Gate, Reading RG6 6BB, United Kingdom.].

  • Zou, X., A. Barcilon, I. M. Navon, J. Whitaker, and D. G. Cacuci, 1993: An adjoint sensitivity study of blocking in a two-layer isentropic model. Mon. Wea. Rev.,121, 2833–2857.

  • View in gallery

    Upper-layer streamfunction of the mean blocking pattern. Contour interval 1.6 × 106 m2 s−1. Dashed contours for negative values. The boundary circle is the equator and interior circles are 30° and 60°N.

  • View in gallery

    Time series of the blocking index (light line) and maximal sensitivity index (dark line) between days 1400 and 2000. The geographical region considered is the semihemisphere 0°–90°W–180°. Values of the maximal sensitivity are in units of 100 × m−2 s.

  • View in gallery

    Time evolution of the streamfunction of the control run between days 1896 and 1901. For each day shown, upper and lower map display the upper- and lower-layer streamfunction, respectively. Contour interval 5 × 106 m2 s−1 for upper layer and 3 × 106 m2 s−1 for lower layer. The boundary circle is 20°N and interior circles are 30° and 60°N. Fields have been rotated 90° to the west with respect to Fig. 1.

  • View in gallery

    Streamfunction of day 1900 (left panels) and day 1903 (right panels) when the initial condition (flow of day 1896) is perturbed with the maximal perturbation, initially normalized to have a maximum of 10 m s−1 in the wind field. Top panels display the upper layer and lower panels the lower layer. Contour interval 5 × 10 m2 s−1 for the upper layer and 3 × 10 m2 s−1 for the lower layer. The boundary circle is 20°N.

  • View in gallery

    Streamfunction of the real phase (left panels) and imaginary phase (right panels) of the FGNM of day 1896. Top panels display the upper layer and lower panels the lower layer. Contour interval used for upper layer is two times the (arbitrary) contour interval for lower layer. Zero contour omitted, and dashed contours for negative values. The boundary circle is the equator and interior circles are 30° and 60°N.

  • View in gallery

    (Continued) Streamfunction of the FGNM of day 1896 for (σit) = 34.4°.

  • View in gallery

    As in Fig. 5 except for the adjoint of the FGNM computed in the energy norm. Contour interval is arbitrary.

  • View in gallery

    As in Fig. 4, except for the flow of day 1900 when the initial perturbation is the real phase of the adjoint of the FGNM computed in the energy norm.

  • View in gallery

    Time evolution of the total energy in the semihemisphere 0°–90°W–180° for the optimal phase of the FGNM (solid), the optimal phase of the adjoint of the FGNM computed in the energy norm (long dashed), and for the first regional SV of the energy norm (short dashed). Light lines are for integrations on the T20 tangent linear model with resting basic state and dark lines for nonlinear integrations on the T31 model. For each case, the plot represents the loge of the ratio of the regional energy at time t to that at initial time. In the nonlinear integrations the initial perturbations were normalized to have an initial regional energy of 1 m4 s−4 K−1, which corresponded to maximum perturbation winds in the range of 8–10 m s−1. To understand the energy units, we note that as shown in de Pondeca (1996), the Exner-layer thickness plays the role of a density in the expression for the total perturbation energy.

  • View in gallery

    As in Fig. 5, except for the first regional SV (left panel) and fifth regional SV (right panel) of day 1896 computed in the energy norm. Contour interval is arbitrary.

  • View in gallery

    Percentage of the total perturbation energy of the maximal perturbation on day 1896 explained by each of the first 40 regional SVs of day 1896 optimized over three days and computed in the energy norm. The regional SVs were taken at initial time.

  • View in gallery

    As in Fig. 10, but for calculations at the final time (day 1899). The regional SVs at the optimization time were obtained by applying the time-independent T20 propagator to the regional SVs at initial time.

  • View in gallery

    As in Fig. 10, but for the case in which the initial condition is the model flow of day 1584.

  • View in gallery

    As in Fig. 10, but for the case in which the the initial condition is the model flow of day 2059.

  • View in gallery

    Streamfunction of the maximal perturbation at initial time, that is, day 1896 (left panels) and final time, that is, day 1899 (right panels). The upper-layer contour interval is 1.5 × 104 m2 s−1 for the initial time and 1.2 × 105 m2 s−1 for the final time. For both days, the lower-layer contour interval is half that of the corresponding upper layer. Zero line omitted and dashed contours for negative values. The boundary circle is the equator and interior circles are 30° and 60°N.

  • View in gallery

    As in Fig. 5, except for the first regional SV of the energy norm at the optimization time. The time-independent T20 propagator was applied to the regional SV at initial time. Contour interval is arbitrary.

  • View in gallery

    Energy (per unit mass) spectrum of the first regional SV of the energy norm at initial time (solid line) and final time (dashed line). Units are m2 s−1. At both times the perturbation was normalized to a unit energy norm. A Hanning filter was used twice on the data to smooth the spectra.

  • View in gallery

    As in Fig. 5, except for the first (left panel) and second (right panel) regional SV of the L2 norm. Contour intervals used for the upper layers are 1.5 times the arbitrary contour intervals for the lower layers.

  • View in gallery

    As in Fig. 10, except for the regional SVs of the L2 norm.

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An Adjoint Sensitivity Study of the Efficacy of Modal and Nonmodal Perturbations in Causing Model Block Onset

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  • 1 Department of Meteorology and Geophysical Fluid Dynamics Institute, The Florida State University, Tallahassee, Florida
  • | 2 Mesoscale and Microscale Meteorology Division, National Center for Atmospheric Research, Boulder, Colorado
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Abstract

With a blocking index as the response function, the adjoint sensitivity formalism is used to assess the impact of normal modes, adjoint modes, and regional singular vectors on prediction of block onset in a two-layer model. The authors focus on three blocks excited by perturbing the model’s state vector at times preselected using the maximal perturbation that defines the direction in phase space associated with the largest possible change in the response function. The sets of normal modes, adjoint modes, and regional singular vectors (using the total energy or the L 2 norm) are computed on instantaneous basic-state flows for the preselected times and sensitivity results are presented for a time window of 3 days.

When ordered by decreasing values of the growth rates of the normal modes, the authors find that some distant normal modes and adjoint modes can produce larger changes in the response function than some of their leading counterparts. In contrast, the sets of regional singular vectors contain easily identifiable subsets of structures associated with relatively large changes in the response function. The largest changes are produced by less than the first 20 regional singular vectors. Some of these individual regional singular vectors capture the onset of the block when used as perturbations to the initial condition in a nonlinear model integration, a result of the importance for ensemble forecasting. It is found that the first five most explosive regional singular vectors of the energy (L 2) norm explain over 20% (60%) of the norm contained in the maximal perturbation at initial time.

Despite the failure of all individual normal modes to excite the block, as opposed to adjoint modes and regional singular vectors, the authors argue that, paradoxically, the normal mode concept remains a viable tool to explain the dynamics of block onset.

Corresponding author address: Dr. Albert I. Barcilon, Geophysical Fluid Dynamics Institute, The Florida State University, 18 Keen Bldg., Tallahassee, FL 32306-3017.

Email: barcilon@gfdi.fsu.edu

Abstract

With a blocking index as the response function, the adjoint sensitivity formalism is used to assess the impact of normal modes, adjoint modes, and regional singular vectors on prediction of block onset in a two-layer model. The authors focus on three blocks excited by perturbing the model’s state vector at times preselected using the maximal perturbation that defines the direction in phase space associated with the largest possible change in the response function. The sets of normal modes, adjoint modes, and regional singular vectors (using the total energy or the L 2 norm) are computed on instantaneous basic-state flows for the preselected times and sensitivity results are presented for a time window of 3 days.

When ordered by decreasing values of the growth rates of the normal modes, the authors find that some distant normal modes and adjoint modes can produce larger changes in the response function than some of their leading counterparts. In contrast, the sets of regional singular vectors contain easily identifiable subsets of structures associated with relatively large changes in the response function. The largest changes are produced by less than the first 20 regional singular vectors. Some of these individual regional singular vectors capture the onset of the block when used as perturbations to the initial condition in a nonlinear model integration, a result of the importance for ensemble forecasting. It is found that the first five most explosive regional singular vectors of the energy (L 2) norm explain over 20% (60%) of the norm contained in the maximal perturbation at initial time.

Despite the failure of all individual normal modes to excite the block, as opposed to adjoint modes and regional singular vectors, the authors argue that, paradoxically, the normal mode concept remains a viable tool to explain the dynamics of block onset.

Corresponding author address: Dr. Albert I. Barcilon, Geophysical Fluid Dynamics Institute, The Florida State University, 18 Keen Bldg., Tallahassee, FL 32306-3017.

Email: barcilon@gfdi.fsu.edu

1. Introduction

The inability of numerical models to simulate blocking events with the frequency, persistence, and amplitude of those observed in the real atmosphere results in poor predictive skills into the medium and extended range forecasting (e.g., Tibaldi and Molteni 1990). Much has been said about the dynamical nature of the blocked state. Pioneering theoretical research dates from as far back as the 1940s (e.g., Namias 1947; Berggren et al. 1949; Elliot and Smith 1949). However, a unified picture is still far from being achieved. Proposed theories encompass a wide range of dynamical frameworks, such as multiple-equilibria states (e.g., Charney and DeVore 1979; Charney and Strauss 1980; Charney et al. 1981; Mitchel and Derome 1983), solitary waves (e.g., McWilliams 1980; Baines 1983; Haines and Marshall 1987), forced Rossby waves (e.g., Tung and Lindzen 1979a, b), and barotropic and baroclinic instability (e.g., Frederiksen 1982, 1983, 1984, 1989; Frederiksen and Bell 1990).

The current research was first motivated by the conceptually simple theory of barotropic and baroclinic instability for the onset of blocking. In essence, this theory views the onset of blocking-type anomalies as the result of structural changes undergone by transient amplifying cyclone waves. While these structural changes are explained through the operation of nonlinear effects, the observed blocking patterns are accounted for by the normal modes of the instantaneous basic-state flows. Additional motivation came from recent renewed interest in the theory of optimal linear growth of nonmodal form (e.g., Farrell 1984, 1988, 1989). Such an approach is rooted in the alternative belief that some aspects of the atmospheric dynamics may be regarded as an optimal barotropic–baroclinic development. Perturbations obtained through optimization of growth in a well-defined sense include the singular vectors (SVs) (see also Borges and Hartmann 1992; Buizza et al. 1993; Buizza and Palmer 1995; Buizza and Molteni 1996) and the adjoint modes (e.g., Zhang 1988). Finally, motivation came also from the prospect of using the technique of the adjoint sensitivity analysis as a tool to assess the role of different perturbations in blocking (e.g., Hall and Cacuci 1983; Zou et al. 1993; Rabier et al. 1996). To clarify the last idea, let us consider a response function, R, defined at a selected time, t > t0, in terms of the model state vector x(t). Here t0 is the reference time. The first-order change δR in the response function, brought about by adding the perturbation δx(t0) to the model’s initial condition x0(t0), can be evaluated as the inner product:
i1520-0469-55-11-2095-e1
In (1), δx(t) is the departure from the reference trajectory x0(t) and (∂R/∂x)x0(t) is the gradient of the response function evaluated on that reference trajectory at time t. Equation (1) follows from simple calculus and vector algebra considerations. The advantage of the adjoint sensitivity analysis resides in the fact that (1) can be evaluated without explicitly solving for δx(t > t0). One can show that a single backward integration of the adjoint model, starting with the initial condition (∂R/x)x0(t), yields an adjoint vector q(t0) that is sufficient to evaluate the changes δR with respect to any given changes in x0(t0). Thus, we have a convenient method of inferring the relative importance of a very large number of changes in the model’s initial condition.

The aim of the present article is to use the technique of the adjoint sensitivity analysis to perform a comparative study of the impact of normal modes, adjoint modes, and regional SVs on the onset of blocking. Implicit is our belief that nonmodal growth may be relevant in atmospheric blocking. The identification of sets of perturbations capable of producing the transition to the blocked state is of both theoretical and practical importance. It sheds light onto the mechanisms of blocking and is helpful in the construction of ensembles of perturbations in operational ensemble forecast. Such ensembles are intended to account for the probability distribution function of the initial analysis error (e.g., Molteni et al. 1996). Out of the three case studies that we performed with a two-layer primitive equation model, we shall present detailed results for one of the case studies and discuss only briefly the other two. In our quest to seek block onset, we will use individual normal modes, adjoint modes, and regional SVs as perturbations to the model’s initial condition when the prevailing model flow is classified as zonal. We will limit the number of nonlinear integrations by selecting only those perturbations associated with significant changes in a selected response function. Such pinpointing of the most relevant perturbations will be done with the help of the adjoint sensitivity analysis. All perturbations, including the regional SVs, will be calculated on time-invariant basic-state flows 2–4 days prior to the time in which we seek to excite the block. We believe the approximation of time-invariant basic states to be a good first approximation to tackling our problem. This view is supported by the relative success of the theory of barotropic and baroclinic instability for the onset of blocking (e.g., Frederiksen 1982, 1983, 1984, 1989; Frederiksen and Bell 1990), which was originally formulated for time-invariant basic states. An extension of our results to include calculations of finite-time normal modes and adjoint modes (Frederiksen 1997) and regional SVs of time-evolving basic states is a natural follow-up to this work. As will be discussed in section 9 (Fig. 8), for our two-layer primitive equation model at truncation T31, the linear approximation is valid up to roughly 60 h. It also compares well with the results of Buizza (1995a) found with the European Centre for Medium-Range Weather Forecasts primitive equation model, when the perturbations were formed from SVs computed at T21L19 and characterized by initial amplitudes comparable with analysis error estimates. In an earlier study, Errico et al. (1993) found on the dry MM4 limited-area model the linear approach to accurately estimate the nonlinear perturbation evolution out to 72-hour forecasts. In that study, the initial amplitudes of the perturbations were also comparable with those of typical analysis errors. We empirically choose the value of 4 days to be our maximum time window in the sensitivity studies. The underlying assumption is that perturbations pinpointed by linear theory can still have a beneficial impact in the earlier stages of their nonlinear development.

We shall be mostly concerned with identifying the most adequate subsets of perturbations capable of causing the transition to the blocked state without studying the underlying dynamical mechanisms. We deem these mechanisms extremely important, and in an accompanying paper we will thoroughly discuss them in conjunction with the observed succession of synoptic events that lead to the onset of the block.

The two-layer primitive equation model is described in section 2. In section 3 we present the method of the adjoint sensitivity analysis, including the derivation of the pertaining equations. Section 4 gives a brief description of the methods of computing the normal modes, adjoint modes, and regional SVs. The specific strategy used to evaluate the sensitivities with respect to these perturbations is described in section 5. In section 6 we describe how the initial synoptic situation was chosen. Sections 7, 8, and 9 contain the results of the sensitivity studies for the normal modes, adjoint modes, and regional SVs, respectively, whereby the norm of the total perturbation energy was adopted to calculate the adjoint modes and regional SVs. Section 10 presents results for SVs calculated in the streamfunction-squared (L2) norm. The summary and conclusions are found in section 11.

2. Summary of the two-layer model

The model is hemispherical, θ-coordinate, and spectral. In each of its two layers the dependent variables are represented as a sum of spherical harmonics with triangular truncation at T31. The system of coupled nonlinear model equations is symbolically written as
i1520-0469-55-11-2095-e2
where x = (· · · , ζnm, · · · , Dnm, · · · , Δπnm, · · ·)T is the state vector. It comprises all the nonzero spectral coefficients of the isentropic vorticity ζ, the divergence D, and the Exner-layer thickness Δπ, where πcp(p/p0)R/cp. Here, p0 = 1000 mb, and the remaining symbols have their standard meanings. The model includes a zonal wavenumber-2 orography, intended to be a very coarse representation of the Northern Hemisphere topography. A similar shape was also used by Legras and Ghil (1985) in their study of blocking with a barotropic model. The ridges are located along 135°E and 45°W (see, for instance, Fig. 1 for the convention used in labeling the longitudes). Their height is 2000 m and was chosen empirically while calibrating the model to yield realistic model synoptic patterns. This issue is further critically discussed in the conclusions. The latitudinal orographic profile exhibits a maximum at latitude ϕ = 45°N and vanishes at the equator and pole. A Rayleigh damping in the lower layer with a timescale of 5 days and a horizontal internal ∇12 hyperdiffusion are also included in the model. The hyperdiffusion coefficient was chosen to give a decay rate of 3 h−1 for the smallest retained scale. The driving is achieved through a Newtonian relaxation term defined for the interface Exner function, with a relaxation timescale of 15 days. The potential temperatures in the lower and upper layer are 280 and 320 K, respectively. The upper surface is a free surface, and the equilibrium interface has a zonal structure associated with a jet in the upper layer. Furthermore, there is no mass transfer across the lower boundary. This model yields realistic blocking patterns and has been used in previous studies of blocking by Zou et al. (1993). The reader is referred to the above paper for a detailed description of the model equations found in its appendix. The linearized version of the model and its adjoint, developed in Zou et al. (1993), were also used in this investigation.

3. Method of the adjoint sensitivity analysis

a. The blocking index

The response function R, also called blocking index, is the regional projection of the daily upper-layer streamfunction anomaly onto a statistically derived“mean blocking pattern”:
RCPxxPxbL2
In (3), xb is the vector of a fixed mean blocking pattern, x is the actual state vector, and x is the climatological state vector. The symbol 〈· · · , · · ·〉L2 denotes the inner product on the sphere in the L2 norm. Here P is an operator that performs the operation of going from spectral (ζ, D, Δπ) to grid point (ψ, D, Δπ), where ψ is the streamfunction, and then setting to zero all gridpoint values except those of the upper-layer streamfunction in a specified geographical region, Σ. In the present study, Σ corresponds to the semihemisphere 0°–90°W–180°. Here C is a normalization constant, chosen such that C = 〈ψb, ψb−1L2, where ψb is the upper-layer streamfunction associated with the vector of the mean blocking pattern. The climatological mean state x was obtained by averaging over a 2000-day run with sampling twice daily, from t = 101 days to t = 2100 days. The vector of the mean blocking pattern xb was obtained by averaging over those 1000 anomalies that possessed the largest positive anomalies in the upper-layer streamfunction at 60°N. Figure 1 shows the upper-layer streamfunction of the mean blocking pattern. It suggests that the blocking ridge in this model occurs preferentially near 90°E and near 90°W, that is, upstream of the mountain ridges. We adopted the empirical rule of considering the flow to be blocked if R ≥ 0.75 and zonal otherwise. A similar blocking index was introduced for the first time by Liu (1994) and Liu and Opsteegh (1995), and used in sensitivity studies of blocking by Oortwijn and Barkmeijer (1995).

b. Adjoint sensitivity equations

We concentrate on a time window, (t0 + τ) − t0, where t0 is the reference time. In the course of this time window a perturbation, δx(t0), added to the model’s initial condition x0(t0) evolves to δx(t0 + τ). The evaluation of the change δRt0+τ in the response function that results from perturbing the model’s initial condition constitutes the primary objective of a sensitivity study. It follows from (3) that in our case
δRt0+τCPδxt0τPxbL2
As shown next, within linear theory δRt0+τ can be evaluated without explicit knowledge of the deviationδx(t0 + τ).
For a sufficiently small perturbation, δx(t), over the time window (t0 + τ) − t0 we can write as the first-order approximation
i1520-0469-55-11-2095-e5
where
i1520-0469-55-11-2095-e6
is the linear propagator between time t0 and time t0 + τ. In Eq. (6), x0(t) is a reference trajectory around which the model equations (2) are linearized and (∂F/∂x)x0(t) is the operator (matrix) of the tangent linear model.
It follows from (4) and (5) that the change in the response function can be rewritten as
δRt0+τδxt0qt0L2
where
qt0At0τ, t0CPPxb
is the adjoint vector at time t0. Here P* is the adjoint of P and A*(t0 + τ, t0) is the adjoint of the linear propagator, both calculated in the L2 norm. In practice, q(t0) is obtained by integrating the adjoint model backward from time t0 + τ to t0, starting with the initial condition (CP*Pxb). We note that q(t0) is independent of the perturbation in the state vector. Therefore, a single backward integration of the adjoint model is all it takes for the subsequent application of (7) to any arbitrary perturbation δx(t0). One can refer to q(t0) as the gradient of the response function with respect to the initial conditions.

4. The nature of modal and nonmodal disturbances

Details of the methods of calculation of the disturbances are given in de Pondeca (1996). Real matrices of maximum dimension 1340, corresponding to state vectors of a reduced truncation of T20, were used throughout. Calculations with complex matrices were avoided by treating the real and imaginary parts of the complex spectral coefficients as two distinct elements of a real state vector. The reduced truncation T20 is comparable to the R15 truncation found in various studies of blocking by Frederiksen (1982, 1983, 1989). We address further the question of truncation in the conclusions.

The normal modes (Zn) are the real parts of the solutions to the linearized autonomous model equations of the form zneσnt; that is,
i1520-0469-55-11-2095-e9
The vectors znr and zni are the real and imaginary phases of the nth normal mode, respectively, which possesses a growth rate, σnr, and a frequency, σni. The adjoint modes (Gn) can be written in a form analogous to (9). They are solutions of the adjoint equations of the linearized dynamics.

The eigenvector zn = znr + izni subtends an angle, αn, with its adjoint eigenvector gn = gnr + igni in the linear space of the perturbations. Zhang (1988) refers to the factor γ = 1/cos(αn) as the projectability of the nth normal mode. This factor is a measure of the gain in initial projection that the nth normal mode can experience.

We removed the arbitrariness in the definition of the amplitudes and phases of the eigenvectors (zn) and adjoint eigenvectors (gn) by following the procedure described by Zhang (1988), that is, by imposing that 〈znr, znrE = 1, 〈znr, zniE = 0, 〈gnr, gnrE = 1, and 〈gnr, zniE = 0. Here, 〈· · · , · · ·〉E denotes the inner product in a chosen E norm (to be discussed shortly). A similar normalization and phase definitions were followed by Borges and Hartmann (1992).

The regional SVs (νn) are the perturbations that maximize the growth of a norm in a chosen geographical region, Σ, and over a prescribed time interval, (t0 + τ) − t0. They are the eigenvectors of the self-adjoint operator [TA(t0 + τ, t0)]*E[TA(t0 + τ, t0)], where A(t0 + τ, t0) is the linear propagator between t0 and t0 + τ, and T is a local projection operator (see Pondeca 1996). The superscript “*E” denotes the adjoint operator in the chosen E norm. The eigenvalues (λ2n) of (TA)*E(TA) give the ratio of the regionally integrated norm at the optimization time to the globally integrated norm at the initial time. The square roots (λn) are known as the singular values of (TA).

We present detailed results for adjoint modes and regional SVs computed in the norm of the total perturbation energy (energy norm for short). This choice of norm is supported by the argument of Molteni et al. (1996) that SVs computed in the energy norm are a good approximation to the eigenvectors of the forecast error covariance matrix. We remind the reader that our work is partially motivated by the need to assess the efficacy of using normal modes, adjoint modes, and regional SVs in ensemble forecasting. We note that our response function, which we believe physically very sound, is coincidently defined in terms of an L2 norm. It thus appears interesting to assess how our results change as we use the L2 norm to compute those perturbations that are norm dependent. In section 10, we discuss briefly the impact of the regional SVs of the L2 norm.

5. Methodology

We start by expanding the model’s initial condition x0(t0) in terms of the normal modes, adjoint modes, or regional SVs,
x0t0Bc
where, for convenience, x0(t0) is a real vector and B = [ϕ1, ϕ2, · · · , ϕN, Δx0(t0)] is a real matrix. The column vectors ϕ1, ϕ2, · · · , ϕN in (10) are the vectors of the real and imaginary phases of the normal modes, or of the adjoint modes, or they are the N singular vectors. For normal modes and adjoint modes N is the dimension of the matrix of the tangent linear model L = (∂F/∂x)x0(t), cast in real representation. This follows from the fact that the eigenvalues of the corresponding eigenvalue problems are nondegenerate. The dimension of L is N = 1340 for the T20 truncation. For regional SVs N < 1340 due to the introduction of a projection operator. Here Δx0(t0) is a residual, which stems from the fact that the perturbations are calculated with a reduced truncation, and, in the case of the regional SVs, also from the fact that these structures do not form a complete basis set in the T20 truncation. In (10), c = (c1, c2, · · · , cN, cN + 1)T is a real (N + 1) column matrix made up of the N real expansion coefficients and the additional element cN+1 = 1. We note that nonstationary normal modes and adjoint modes are associated with two real expansion coefficients (one for each phase), while the singular vectors are associated with just one expansion coefficient, a consequence of the self-adjointness of (TA)*E(TA).
By writing δx0(t0) = c, one can rewrite the expression for the change in the response function at time t0 + τ [Eq. (7)] as
δRt0+τx0δcδcBTqL2
where the superscript T denotes the transpose. Equation (11) displays the use of the vector of expansion coefficients, c, as the control variable. Different indices of sensitivity can be conveniently defined in terms of the expansion coefficients. In this paper we chose to work with a rather simple index, the relative sensitivity to the nth expansion coefficient, which is defined as
i1520-0469-55-11-2095-e12
Computing the relative sensitivity (12) is literally equivalent to perturbing the initial state with one phase of a given normal mode or adjoint mode, or perturbing it with one regional SV, integrating the linearized model with the time-evolving basic state, and evaluating the change in the response function (at the final time) per unit norm of the initial disturbance. The inclusion of ‖ϕnE in the denominator of (12) is formally equivalent to normalizing all basis vectors to having a unit norm before performing the expansion (10). Our method of computing the sensitivities, as presented in this section, may appear to some as a dispensable mathematical exercise. This is because the expansion coefficients in our problem do not possess any particular physical meaning. The method is, however, applicable to other problems. For instance, one can think of expanding an initial analysis error in terms of the regional SVs (e.g., Buizza et al. 1997). In that case it may be useful to define an index that reflects the change in the response function per percent change of individual expansion coefficients.

6. Choice of the synoptic situation

We intend to explain the buildup of blocks as the result of the growth of perturbations on a background flow. It is therefore only natural to perform studies on a synoptic situation that is initially free of blocks. Clearly, there is a conceptual difficulty with performing this study on flows that already contain the block, that is, also contain the perturbation. Some authors have partially bypassed this caveat by first using a filter to remove those events in the flow with time- and spatial scales characteristics of blocks (e.g., Buizza and Molteni 1996). On one hand, we do not wish to have a block in the control run but, on the other hand, we wish to be assured that a block can be excited by adding an optimally designed perturbation to the flow. This, of course, does not guarantee that our normal modes, adjoint modes, and regional SVs will be able to excite the block. However, it provides a way of selecting the periods of time in the control run where we should concentrate our investigation.

We identified those periods of time when a block is likely to be excited by plotting the time series of the blocking index and of the maximal sensitivity index (Fig. 2), as originally proposed by Oortwijn and Barkmeijer (1995). The maximal sensitivity index at time t0 is defined as the L2 norm of the adjoint vector q(t0):
Smaxt0qt0L2
We note that, within the linear context, q(t0) is the perturbation that induces the largest possible change in the response function for perturbations scaled to have the same initial L2 norm. This follows from the Cauchy–Schwartz inequality applied to Eq. (7). It is now clear why we refer to Smax(t0) as the maximal sensitivity index. In this sense, we shall also refer to q(t0) as the maximal perturbation. The time series of Smax(t0), shown in Fig. 2, was obtained with the value of τ = 3 days. The operator P of Eq. (3) was chosen to define the semihemisphere 0°–90°W–180°. We see from Fig. 2 that the flow of day 1896 is a good initial condition to perturb. The prevailing flow is zonal, while the maximal sensitivity exhibits a pronounced peak on day 1896. This indicates a good likelihood of exciting a block if we perturb (appropriately) the flow of day 1896. Figure 3 shows the upper- and lower-layer streamfunction of the control run between days 1896 and 1901. The failure in entering the blocked state is translated by values of the response function that lie below 0.75 (Fig. 2).

The final step in choosing a suitable model synoptic flow consists of perturbing the flow of day 1896 with the auxiliary perturbation q(t0). Due to the linear character of the problem that leads to q(t0) only the direction of this vector is known. Its amplitude must be chosen by adopting a reasonable criterion. The maximal perturbation q(t0) was found to excite a block in the course of a 3–4-day nonlinear run, when normalized to have initial maxima in the wind field of roughly 5–10 m s−1. We took the value of 10 m s−1 as a reasonable upper limit in our experiments. It corresponds to the observational maximum zonal wind anomalies of Dole and Black (1990) prior to their composite onset. Initial maxima in the range of 5–10 m s−1 are associated in our model with a linear evolution up to at most 60 h. The structure of the maximal perturbation will be discussed in section 9. Another meaningful criterion of choosing the initial amplitude of q(t0), worth trying in future work, consists of viewing the mean blocking pattern as a final forecast error and solving the (linear) pseudo-inverse problem to obtain the initial error (Buizza et al. 1997). The amplitude of the latter could be used to scale q(t0).

Figure 4 shows the total upper- and lower-layer streamfunction of days 1900 and 1903, when the perturbation was initially normalized to have a maximum of 10 m s−1 in the wind field. The sequence of synoptic events starting on day 1896 (not shown) reveals that the flow forms a Rex-type block, which evolves in less than 48 h into a Ω-type block. The blocking event last for more than 10 days. The realism of this model block was also confirmed by a comparison with an observational block recently studied by Colucci and Bresky (1997). In an accompanying paper we discuss the mechanisms of the onset of the block, whereby we hypothesize that a Thorncroft–Hoskins–McIntyre LC1-type cycle (1993)“preconditions” the large-scale flow allowing for intense instability of normal-mode form.

With the certainty that a block could be excited by adding a perturbation to the flow of day 1896, we performed a comparative study of the perturbations that we tentatively assumed to be instrumental in the buildup of blocks. We defined the blocking index Rt0+τ successively on days 1898, 1899, and 1900 and calculated its sensitivity with respect to the normal modes, adjoint modes, and regional SVs of the flow of day 1896. The optimization time of the regional SVs was chosen to coincide with the time window of the sensitivity analysis. The growth of norm was maximized in the semihemisphere 0°–90°W–180°. The results concerning the three sets of disturbances were found to be consistent for all three time windows (2, 3, and 4 days). Only results pertaining the time window of 3 days will be presented.

7. Sensitivity to the normal modes

Table 1 shows the relative sensitivity (RS) for the real and imaginary phases of the first seven normal modes of day 1896. The projectabilities, growth rates, and periods are also shown. The energy norm was used here to normalize the perturbations [see denominator of Eq. (12)]. As we see, the value of |RS| corresponding to one of the phases of a normal mode can be larger by one order of magnitude or more than the value associated with the other phase. This is the case of the two phases of normal modes 1, 2, 3, and 5. The conclusion that follows appears very intuitive. In general, the information of the initial phase is crucial in asserting whether or not a normal mode is capable of producing appreciable changes in the direction of phase space that can lead to a block. The term “appreciable” is used in a relative sense, for the exact effects of the perturbations on the flow evolution can only be determined by an explicit integration of the model. A natural step to follow is thus to evaluate the relative sensitivities (RS)opt for the optimal phases of the normal modes. These are the phases associated with the largest possible values of RS. Such calculations are particularly meaningful in view of the fact that the results for the normal modes (and adjoint modes) are to be compared with those for the regional SVs. The latter are perturbations that do not contain any phase ambiguity. The definition of the relative sensitivity RS, as given by (12), was introduced for the real and imaginary phases of the normal modes (and adjoint modes) separately. A generalization to any arbitrary phase follows from the meaning of RS itself, as explained in section 5. The relative sensitivity for an arbitrary phase represents the change in the response function when that phase with a unit norm is added to the initial condition x0(t0). Given the values of the relative sensitivities for the real phase, (RS)r, and for the imaginary phase, (RS)i, it follows from (7), (9), and the normalization described in section 4 that for an arbitrary phase
i1520-0469-55-11-2095-e14
The extrema of (14) satisfy ∂(RS)/∂(σit) = 0, which leads to a third-order algebraic equation for tan(σit). We avoided solving a third-order equation by simply varying (σit) in fixed steps and using relation (14) to find the approximate maximum of RS. Table 1 also shows the values of the relative sensitivity, (RS)opt, corresponding to the optimal phase of each of the first seven normal modes. The maximum percent change between the value of |RS| associated with the phase (real or imaginary) of largest |RS| and the value of (RS)opt is found for the seventh normal mode. It amounts to 19%. Tables 2 and 3, to be presented shortly, show that this value can in some cases reach 30%.

Table 2 shows the values of the relative sensitivity for the real, imaginary, and optimal phases of all those normal modes associated with values of (RS)opt > 0.1. The normal modes have been ordered by decreasing growth rates. It is remarkable that normal modes distant in the spectrum can be important, in the sense defined by the relative sensitivity. A comparison of Tables 1 and 2 also shows that some leading normal modes are associated with values of (RS)opt that are much smaller than those of certain modes well distant in the spectrum. This is the case, for instance, of normal modes 3 and 7, on one side, and 104, 142, 314, 315, 351, 352, and 429, on the other side. To further argue the importance of distant normal modes in the context of blocking predictability, let us first concentrate on normal mode 142. This mode is associated with (RS)opt = 0.1384, is decaying with σr = −0.1074 day−1, has a period of 6.6 days, and has a projectability of 4.03. We compare these values with those for the fastest growing normal mode (FGNM). The latter is associated with (RS)opt = 0.1223, has a growth rate of 0.2920 day−1, a period of 47.04 days, and a projectability of 3.71. The values of (RS)opt and of the projectabilities of both modes are comparable. This indicates that there is no a priori reason to neglect the 142nd normal mode (as compared to the FGNM) in a 2–4-day initial value problem designed to capture the onset of blocking. It is interesting to note that the large period of the FGNM shows that the contribution of this mode to the blocking index comes essentially from a local growth. In contrast, the small period of normal mode 142 suggests that its contribution comes from the evolution of a portion of the normal mode initially far upstream from the region where we defined the response function. Normal modes 314, 315, 352, and 429 represent another interesting case. They are associated with values of (RS)opt that are comparable to that for the FGNM, but their projectabilities are 3–5 times as large as that of the FGNM. These modes can therefore be more important than the FGNM in the context of the aforementioned initial value problem.

The results concerning the importance of distant normal modes were found to be robust with respect to increase in the truncation to T31. This finding suggests that in theories of error growth based on normal modes one has to also carefully consider some of the distant normal modes. In particular, normal modes with a large projectability should be incorporated, for they are likely to experience a large initial gain in projection (Zhang 1988).

Figure 5 shows the upper- and lower-layer streamfunction of the real and imaginary phases of the FGNM. For later convenience, we also show this mode for (σit) = 34.4°. The optimal phase is not shown for it is almost identical to the real phase. It corresponds to (σit) = 5.3°. The FGNM has a growth rate of 0.2920 day−1, a period of 47.04 days, and a projectability of 3.71. Its real phase, which is associated with an appreciable value of the RS, shows a well-defined high–low dipole along meridian 120°W. At first, one would expect this mode to be able to excite a block. Integrations of the nonlinear model, with the model state of day 1896 perturbed with each one of the normal modes of Table 2 were performed. Both the phases listed in the third column of the table and the optimal phases were considered. Initial maximum perturbation winds ranged from 2.5 to 10 m s−1. In our experiments, no transition to a blocked state was observed within a 6-day integration. In particular, both the 13th normal mode, which is associated with the largest value of |RS|, and the FGNM failed to excite a block. The integrations were also carried out on the linearized model both with a time-evolving reference trajectory and with a resting basic state. Again, no transition to block was observed.

8. Sensitivity to the adjoint modes computed in the energy norm

Table 3 shows the values of the relative sensitivity for the real, imaginary, and optimal phases of all those adjoint modes of the energy norm for which (RS)opt > 0.2. A comparison with the values of RS for the normal modes (Table 2) shows that the blocking index is in general more sensitive to the adjoint modes than it is to the normal modes. By this we mean that the maxima of (RS)opt are larger in Table 3 than in Table 2. Also, just as for the normal modes, adjoint modes distant in the spectrum can induce changes in the response function that are larger than those due to leading adjoint modes. It is also true however that the first 15 or so adjoint modes represent a large portion of the adjoint modes that induce the largest changes in the response function. A simple explanation for this fact and for the larger maxima in the values of (RS)opt shown in Table 3 (as compared to those of Table 2) is as follows. The adjoint modes can be obtained as the result of an optimization problem, namely, the problem of finding the best initial condition of unit amplitude that most effectively excites the normal modes. Therefore, adjoint modes will be in general associated with faster growth rates than the corresponding normal modes. For adjoint modes and normal modes growing in the right geographical region of the onset of the block, we then expect the first to have a greater impact on the blocking index than the latter. A more subtle, but related, justification has to do with the connection between the adjoint modes and the SVs. This will be discussed at a later stage.

Figure 6 shows the upper- and lower-layer streamfunctions of the real and imaginary phases of the adjoint of the FGNM. The optimal phase is very close to the real phase and is therefore not shown. It corresponds to (σit)opt = 4.2°. The real phase is associated with RS = 0.3528 and the optimal phase with RS = 0.3532. This adjoint mode (as others not shown), besides having a baroclinic structure, reveals, especially in the upper layer, the “peanut–banana”-type configuration characteristic of barotropic growth. As with the normal modes, we perturbed the state vector on day 1896 with some of the structures of Table 3 and integrated the nonlinear model forward in time. It was found that some of the adjoint modes were able to excite a block within a period of 3–4 days. Figure 7 shows the upper- and lower-layer streamfunction of day 1900 when the initial condition (flow of day 1896) is perturbed with the real phase of the adjoint mode of the FGNM. The perturbation was normalized to have initial maxima of 10 m s−1 in the wind field. A well-defined block can be seen around 90°W. The synoptic evolution that leads to the onset (not shown) is very similar to that obtained when the initial disturbance is the maximal perturbation (also not shown). This shows that the onset of the block occurs in both cases through the same mechanisms. It is important to mention that the onset, as produced either by the maximal perturbation or by the first adjoint mode, occurs only when the full nonlinear model is used. Integrations of the linearized model with a time-evolving basic-state trajectory and with a resting basic state did not yield any blocks. This leads to the very interesting conclusion that although our perturbations were obtained within linear theory, a “kick” into the nonlinear regime is ultimately necessary for the onset of the block to occur.

9. Sensitivity to the regional SVs computed in the energy norm

We tested the idea of optimal linear growth by seeking perturbations that maximize the total perturbation energy in the semihemisphere 0°–90°W–180° over a time interval of 3 days. Figure 8, for which time 0 in the abscissa corresponds to day 1896, shows how that energy evolves in time for the most explosive regional SV, the optimal phase of the FGNM, and the optimal phase of the adjoint of the FGNM. Integrations were performed on the T20 tangent linear model, with resting basic state, and on the full nonlinear T31 model. In the nonlinear integrations, the initial perturbations were normalized to have an initial regional total energy of 1 m4 s−4 K−1 (see caption, Fig. 8). This value corresponded to maximum perturbation winds in the range of 8–10 m s−1. It is apparent in both the linear and nonlinear integrations that the most explosive regional SV produces the largest energy amplification over the optimization time (3 days). The FGNM is largely suboptimal, while the first adjoint mode grows almost as fast as the first regional SV. The comparable growth rates of the adjoint mode and the regional SV will be understood shortly. Figure 8 also shows that for the first 60 h or so the nonlinear integrations yield evolutions in energy that resemble fairly well those of the counterpart linear integrations. This resemblance suggests that our disturbances, which were obtained within linear theory and on a time-invariant basic state, maintain their dynamical properties in the nonlinear model for roughly 60 h. This statement is obviously connected to the initial amplitudes chosen in the nonlinear integrations.

Table 4 shows the values of the relative sensitivity for all those regional SVs of day 1896 for which |RS| > 0.2. The amplification factors are also given. We see that the maxima of |RS| obtained for the regional SVs are larger than the maxima of (RS)opt for the adjoint modes (Table 3). The largest sensitivities are found for the first and fifth regional SV. Figure 9 shows the upper- and lower-layer streamfunction of both regional SVs. The structures of these regional SVs show that they grow by both barotropic and baroclinic energy conversions. The amplification factor of the first regional SV is 11.0610 and that of the fifth is 4.7263. We recall that the amplification factors squared give the ratio of the regionally integrated energy of the SV at the optimization time to the globally integrated energy at initial time. Assuming that this ratio increases exponentially, then the above values of the amplification factors correspond to growth rates of 0.8011 and 0.5177 day−1, respectively. The resemblance of the most explosive regional SV to the real phase of the adjoint of the FGNM is remarkable. Not surprisingly, this regional SV was also successful in exciting the block.

It is apparent from Table 4 that the largest changes in the response function are unequivocally produced by less than the first 20 regional SVs. To understand this result, we compute the percentage of (integrated total) energy of the maximal perturbation q(t0) at initial time t0 that is contained in each individual regional SV. We recall that the operator (TA)*E(TA) is self-adjoint. Its eigenvectors form an orthogonal basis in the sense of the hemispherically integrated energy, and therefore a Parseval-type relation (e.g., Jeffreys and Jeffreys 1956) can be written for the energy of any arbitrary perturbation projected onto the subspace spanned by these eigenvectors. In fact, such a partition of the energy among individual regional SVs can be performed also at the optimization time. In other words, these structures form an orthogonal basis at times t0 and t0 + τ (e.g., Noble and Daniel 1977). It is easy to show that if the projected perturbation and the regional SVs all have a unit norm then the square of the jth expansion coefficient represents the fraction of norm explained by the jth regional SV. Figure 10 represents a histogram of the percentage of energy of the maximal perturbation explained by each of the first 40 regional SVs at the initial time (day 1896). We see that the first few most explosive SVs dominate the expansion. The combined first 5 SVs contain 34.11% of the energy, and the combined first 40 contain 46.57%. In the remaining SVs the individual maximum was found to explain only 1.24% of the energy. The calculation was repeated at the final day of the time window, that is, on day 1899 (Fig. 11). The regional SVs at the optimization time were obtained by applying the T20 time-independent linear propagator to the regional SVs at initial time (day 1896). The optimal perturbation at the final time was obtained by integrating the T31 linearized model with time-evolving basic-state trajectory. Figure 11 shows that at the optimization time, among all the regional SVs, the first contains by far the largest percentage of energy of the maximal perturbation. This regional SV contains 18.44% of the energy and is seconded by the fifth SV, which contains 5.96%. The combined first five regional SVs contain 26.23% of the energy and the combined first 40 contain 41.63% of it. In the remaining regional SVs the individual maximum was found to explain only 1.20% of the energy. Qualitatively similar results were found for two other blocks that we studied, which were also excited by adding a perturbation to a basic-state flow initially free of blocks, as shown in Figs. 12 and 13. These represent histograms equivalent to that of Fig. 10 but for blocks excited by perturbing the flows of day 1584 and 2059, respectively, with the corresponding maximal perturbations. We note that the truncation used to calculate the regional SVs does not seem to be a matter of concern here. Indeed, although some of the features of SVs (amplification factors and distribution of norm among the wavenumbers at initial and final time) are strongly dependent on truncation (Hartmann et al. 1995; Buizza et al. 1997), so are those of the maximal perturbation. It is reasonable to assume that an increase in truncation would affect the most explosive regional SVs and the maximal perturbation in a comparable way, so that the histograms of Figs. 10 and 11 (and also of Figs. 12 and 13) would only experience little changes. Our results show that there is a close connection between the most explosive regional SVs and the maximal perturbation. This connection can be understood in light of how the optimization is achieved by these perturbations. Figure 14 shows the upper- and lower-layer streamfunction of the maximal perturbation at initial time (day 1896) and final time (day 1899) evaluated at T31 truncation. We witness an evolution from subsynoptic to synoptic scales over the time window. The growth to larger spatial scales is also characteristic of singular vectors calculated in the energy norm (as opposed to those obtained in the enstrophy norm, for instance) and reflects an upscale energy transfer, as it is known from the theory of two-dimensional turbulence. Figure 15 shows the upper- and lower-layer streamfunction of the first regional SV at optimization time. The time-independent T20 linear propagator has been applied to the regional SV at initial time. The structure at initial time is shown in Fig. 9. Since the lengthening of scale is not easily recognizable from the streamfunction patterns of this SV, we show in Fig. 16 the energy spectrum correspondent to the initial and final times. We see clearly that the perturbation loses energy at scales n ≥ 14 and gains energy on a broad range of (total) wavenumbers centered around n = 9.

The close relation between the maximal perturbation and the most explosive regional SVs also explains why the first 15 or so adjoint modes capture most of the largest changes in the response function (see section 8). This can be understood as follows. Theoretically, and as also stated by Buizza and Palmer (1995), for time-invariant basic states an optimization time of infinity will yield SVs coincident with the adjoint modes. Clearly, what is to be understood by an “infinite time interval” depends in practice on the actual degree of instability of the flow and on the perturbation itself. We found in our model with resting basic states that in some instances the FGNM could be excited in less than 3 days, if the initial condition was its adjoint. This timescale should be compared with the more than 30 days necessary to excite the FGNM if the initial perturbation is chosen through a random generator. The point is that for initial conditions that are adjoint modes, an “infinite time interval” can mean 3 days or less. In such cases we expect the spatial variance of leading adjoint modes to be fairly well captured by the most explosive regional SVs optimized over 3 days. We found in the present case study that for the first SV, the optimal phase of the first adjoint mode and the maximal perturbation an “infinite time interval” corresponded to less than 3 days. This becomes clear by noticing how the first regional SV and the maximal perturbation resemble on the third day of integration one of the phases of the FGNM. More exactly, they resemble the FGNM for (σit) = 34.4° (Figs. 5, 14, and 15). The optimal phase of the adjoint mode at the final time is not shown here, since this phase resembles the first regional SV (at all instants). We close this section by noting that the relation between the adjoint modes and the SVs is apparently what led Frederiksen and Bell (1990) to conclude that SVs were not particularly relevant in blocking as compared to the adjoint modes. Our case studies show, however, that the number of regional SVs that produce significant changes in the blocking index is restricted to less than the first 20 most explosive and that the first 5 capture most of the largest changes (Table 4). Such a clear cut, which is important in the context of an ensemble forecast, is not present in the adjoint modes. As with the normal modes, distant adjoint modes can also be important in inducing changes in the blocking index.

10. Results for the SVs computed in the L2 norm

Since we defined the response function through an L2 norm, it is interesting to assess the impact of regional SVs of this norm on block onset. The method of calculation follows that of Ehrendorfer and Errico (1995). A projection operator was introduced to reduce the state vector to the spectral coefficients of the streamfunction only. Note that for convenience the streamfunction now replaces the vorticity in the description of the state vector (see section 2). At initial time our SVs are pure streamfunction fields, as we demand that the initial divergence and layer thickness be zero. This corresponds to our definition of matrix Q of Ehrendorfer and Errico [their Eq. (2.5)]. This assumption is consistent with the fact that our mean blocking pattern is also a pure streamfunction field. We recall that the gradient of this field is the initial condition for the backward integration of the adjoint model, which yields the maximal perturbation.

For the block excited by perturbing the flow of day 1896, Table 5 shows the values of the relative sensitivity for the regional SVs of the L2 norm. The values were multiplied by 106 in order to make them comparable in magnitude to those of Table 4. Again, we see that the dominant regional SVs are associated with the largest values of |RS|. We note that a direct comparison of Table 5 with Table 4 is not possible, since the initial amplitudes of the disturbances in these tables were determined by prescribing two different norms.

The structures of the first two most explosive regional SVs at initial time are shown in Fig. 17. They are associated with amplification factors of 12.4404 and 9.1820, respectively. We note the smaller scales of these regional SVs as compared to regional SVs of the energy norm (Fig. 9). We see how extraordinary well the second regional SV resembles the maximal perturbation at initial time.

Figure 18 shows the histogram of the percentage of total norm of the maximal perturbation explained by the first 40 regional SVs of the L2 norm at the initial time. It is remarkable that the first five SVs combined explain now 62.8% of the norm, that is, much more that the 34.11% found for regional SVs of the energy norm (section 9).

As with the first regional SV of the energy norm, and not surprisingly, the first regional SV of the L2 norm was also found to be capable of exciting a block. Results for the blocks excited by perturbing the flows of days 1584 and 2059 (not shown) were found to qualitatively agree with the results presented in this section. A slight difference is that the first regional SV of these flows by far dominates the other regional SVs in the histograms correspondent to Fig. 18.

11. Summary and conclusions

We have applied the technique of the adjoint sensitivity analysis to the study of blocking and evaluated the sensitivity of a blocking index with respect to the sets of normal modes, adjoint modes, and regional SVs. The experiments were performed on a two-layer primitive equation model. On an initially free-of-block synoptic situation, the performances of the above perturbations in exciting a block were compared. Detailed results were presented for one case study, but two other cases were also considered. The results were found to be consistent for time windows of 2–4 days. A number of results can be summarized from these three cases studies.

  1. The normal modes were found to be associated with absolute values of the relative sensitivities, which where significantly smaller than those for the adjoint modes and regional SVs.
  2. The information concerning the initial phase of the normal modes (and adjoint modes) was found to be important when inferring whether or not these disturbances were able to produce significant changes in the response function 2–4 days later.
  3. When ordered by decreasing growth rates of the normal modes, some of the normal modes and adjoint modes well distant in the spectrum were found to be important in a 2–4-day initial value problem.
  4. The regional SVs of the energy norm were found to have maxima in the absolute values of the relative sensitivities that were larger than those for the adjoint modes. An important result is that the largest changes were found to be associated with less than the first 20 regional SVs, a result also valid for the regional SVs of the L2 norm. Moreover, the first five regional SVs of the energy norm explain over 20% of the total energy of the maximal perturbation both at initial and final time. At initial time, the first five regional SVs of the L2 norm explain more than 60% of the norm, consistent with the fact that the response function was defined through an L2 norm.
  5. None of the normal modes was capable of exciting a block in the course of a 6-day nonlinear run, while some of the adjoint modes and regional SVs succeeded in causing block onset. It must be stressed that a sensitivity study can only pinpoint the relative importance of the structures that we are studying but that their effect on the flow evolution can only be determined by integrating the model. We restricted the initial amplitude of the disturbances in the nonlinear model integrations by requiring for the initial perturbation wind to not exceed 10 m s−1. The maximum value of 10 m s−1 was chosen to correspond to observational maximum zonal wind anomalies prior to the onset of blocking (Dole and Black 1990). It is worth mentioning that a separate set of experiments that we performed suggests that our conclusions are valid for maxima in the initial perturbation wind field somewhat larger than 15 m s−1. However, as we further increase the initial amplitude, in the first steps of integration the flow evolution becomes increasingly unrealistic until the model finally blows up. Clearly, this is associated with the fact that the amplitude of the disturbance becomes comparable to that of the background flow.

The results presented have some interesting consequences.

  1. The relative sensitivities for the normal modes suggest that error theories based on normal modes should incorporate some of the distant (growing or decaying) normal modes. In particular, the inclusion of those normal modes with large projectabilities seems to be necessary, since a large projectability indicates the possibility of a large gain in initial projection.
  2. The relative sensitivities for the regional SVs of both the energy and the L2 norm show that the use of the first few most explosive structures to construct the system of perturbations in ensemble forecasting is likely to capture important aspects of the dynamics associated with the transition to the blocked state. In particular, we saw that the first five regional SVs capture a significant part of the total norm of the maximal perturbation.

The same results also sound a note of caution. The fact that we were not able to excite a block with any of the normal modes should not be viewed as a sign of the irrelevance of these structures in accounting for the blocks reported in this study. Indeed, one can argue that the first adjoint mode reported in this study, which was able to excite the block, hides a somewhat large-amplitude FGNM. This is seen by the fact that the FGNM possesses a projectability of 3.71. This means that the integration with the adjoint mode hides an initial FGNM with 3.71 as much amplitude as that of the integration with the FGNM itself (Zhang 1988). While an integration of the nonlinear model perturbed with the FGNM with an initial amplitude 3.71 times larger than the one adopted in this work would cause the model to blow up, the nonorthogonality of the system allows us to“hide” such a large amplitude in the adjoint mode. This hiding is achieved by putting amplitude into some of the other normal modes. This result suggests that from a dynamical point of view the onset of the block can perhaps be understood through the theory of barotropic and baroclinic instability.

Although our conclusions are based on just a few case studies, we suspect that they are general for the particular model calibration used. Indeed, virtually all of the 17 blocking events observed in a 2000-day control run seem to occur as the result of very similar synoptic evolutions. The mechanisms of blocking seem to change, however, as we change the calibration of the model. This will be explained in an accompanying paper.

A natural extension of this work would be the calculation of the sets of finite-time normal modes and adjoint modes (Frederiksen 1997) and regional SVs of time-evolving basic-state trajectories, and the study of their impact on the blocking index. Future studies should also assess the sensitivity of our results to the topography used. Zou et al. (1993) found in their studies of blocking with the same model used in this paper that the height of the mountain was the model parameter to which their blocking index was most sensitive. Also, Buizza (1995b) found on a barotropic model the SVs to be sensitive to the orography. The works of Hartmann et al. (1995) and Buizza et al. (1997), among others, also indicate that the sensitivity of our results with respect to the model truncation should be studied in detail. In this paper, we only assessed the impact of increasing the truncation in the calculation of the normal modes from T20 to T31.

Acknowledgments

The first author was sponsored by the FINNIDA/WMO/SADC Meteorology Project. Part of this research was also supported by National Science Foundation Grant ATM-9415407 and Air Force Grant AFOSR F49620-96-1-0172. We acknowledge gratefully the Mesoscale and Microscale Meteorology Division of the National Center for Atmospheric Research and the Cooperative Institute for Research in the Environmental Sciences, Boulder, Colorado, where a significant part of this work was carried out. We are also grateful to Drs. Mark Borges, Roberto Buizza, Jorgen Frederiksen, Melvin Shapiro, and Jeffrey Whitaker and two anonymous reviewers for their valuable comments.

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Fig. 1.
Fig. 1.

Upper-layer streamfunction of the mean blocking pattern. Contour interval 1.6 × 106 m2 s−1. Dashed contours for negative values. The boundary circle is the equator and interior circles are 30° and 60°N.

Citation: Journal of the Atmospheric Sciences 55, 11; 10.1175/1520-0469(1998)055<2095:AASSOT>2.0.CO;2

Fig. 2.
Fig. 2.

Time series of the blocking index (light line) and maximal sensitivity index (dark line) between days 1400 and 2000. The geographical region considered is the semihemisphere 0°–90°W–180°. Values of the maximal sensitivity are in units of 100 × m−2 s.

Citation: Journal of the Atmospheric Sciences 55, 11; 10.1175/1520-0469(1998)055<2095:AASSOT>2.0.CO;2

Fig. 3.
Fig. 3.

Time evolution of the streamfunction of the control run between days 1896 and 1901. For each day shown, upper and lower map display the upper- and lower-layer streamfunction, respectively. Contour interval 5 × 106 m2 s−1 for upper layer and 3 × 106 m2 s−1 for lower layer. The boundary circle is 20°N and interior circles are 30° and 60°N. Fields have been rotated 90° to the west with respect to Fig. 1.

Citation: Journal of the Atmospheric Sciences 55, 11; 10.1175/1520-0469(1998)055<2095:AASSOT>2.0.CO;2

Fig. 4.
Fig. 4.

Streamfunction of day 1900 (left panels) and day 1903 (right panels) when the initial condition (flow of day 1896) is perturbed with the maximal perturbation, initially normalized to have a maximum of 10 m s−1 in the wind field. Top panels display the upper layer and lower panels the lower layer. Contour interval 5 × 10 m2 s−1 for the upper layer and 3 × 10 m2 s−1 for the lower layer. The boundary circle is 20°N.

Citation: Journal of the Atmospheric Sciences 55, 11; 10.1175/1520-0469(1998)055<2095:AASSOT>2.0.CO;2

Fig. 5.
Fig. 5.

Streamfunction of the real phase (left panels) and imaginary phase (right panels) of the FGNM of day 1896. Top panels display the upper layer and lower panels the lower layer. Contour interval used for upper layer is two times the (arbitrary) contour interval for lower layer. Zero contour omitted, and dashed contours for negative values. The boundary circle is the equator and interior circles are 30° and 60°N.

Citation: Journal of the Atmospheric Sciences 55, 11; 10.1175/1520-0469(1998)055<2095:AASSOT>2.0.CO;2

Fig. 5.
Fig. 5.

(Continued) Streamfunction of the FGNM of day 1896 for (σit) = 34.4°.

Citation: Journal of the Atmospheric Sciences 55, 11; 10.1175/1520-0469(1998)055<2095:AASSOT>2.0.CO;2

Fig. 6.
Fig. 6.

As in Fig. 5 except for the adjoint of the FGNM computed in the energy norm. Contour interval is arbitrary.

Citation: Journal of the Atmospheric Sciences 55, 11; 10.1175/1520-0469(1998)055<2095:AASSOT>2.0.CO;2

Fig. 7.
Fig. 7.

As in Fig. 4, except for the flow of day 1900 when the initial perturbation is the real phase of the adjoint of the FGNM computed in the energy norm.

Citation: Journal of the Atmospheric Sciences 55, 11; 10.1175/1520-0469(1998)055<2095:AASSOT>2.0.CO;2

Fig. 8.
Fig. 8.

Time evolution of the total energy in the semihemisphere 0°–90°W–180° for the optimal phase of the FGNM (solid), the optimal phase of the adjoint of the FGNM computed in the energy norm (long dashed), and for the first regional SV of the energy norm (short dashed). Light lines are for integrations on the T20 tangent linear model with resting basic state and dark lines for nonlinear integrations on the T31 model. For each case, the plot represents the loge of the ratio of the regional energy at time t to that at initial time. In the nonlinear integrations the initial perturbations were normalized to have an initial regional energy of 1 m4 s−4 K−1, which corresponded to maximum perturbation winds in the range of 8–10 m s−1. To understand the energy units, we note that as shown in de Pondeca (1996), the Exner-layer thickness plays the role of a density in the expression for the total perturbation energy.

Citation: Journal of the Atmospheric Sciences 55, 11; 10.1175/1520-0469(1998)055<2095:AASSOT>2.0.CO;2

Fig. 9.
Fig. 9.

As in Fig. 5, except for the first regional SV (left panel) and fifth regional SV (right panel) of day 1896 computed in the energy norm. Contour interval is arbitrary.

Citation: Journal of the Atmospheric Sciences 55, 11; 10.1175/1520-0469(1998)055<2095:AASSOT>2.0.CO;2

Fig. 10.
 Fig. 10.

Percentage of the total perturbation energy of the maximal perturbation on day 1896 explained by each of the first 40 regional SVs of day 1896 optimized over three days and computed in the energy norm. The regional SVs were taken at initial time.

Citation: Journal of the Atmospheric Sciences 55, 11; 10.1175/1520-0469(1998)055<2095:AASSOT>2.0.CO;2

Fig. 11.
 Fig. 11.

As in Fig. 10, but for calculations at the final time (day 1899). The regional SVs at the optimization time were obtained by applying the time-independent T20 propagator to the regional SVs at initial time.

Citation: Journal of the Atmospheric Sciences 55, 11; 10.1175/1520-0469(1998)055<2095:AASSOT>2.0.CO;2

Fig. 12.
 Fig. 12.

As in Fig. 10, but for the case in which the initial condition is the model flow of day 1584.

Citation: Journal of the Atmospheric Sciences 55, 11; 10.1175/1520-0469(1998)055<2095:AASSOT>2.0.CO;2

Fig. 13.
Fig. 13.

As in Fig. 10, but for the case in which the the initial condition is the model flow of day 2059.

Citation: Journal of the Atmospheric Sciences 55, 11; 10.1175/1520-0469(1998)055<2095:AASSOT>2.0.CO;2

Fig. 14.
Fig. 14.

Streamfunction of the maximal perturbation at initial time, that is, day 1896 (left panels) and final time, that is, day 1899 (right panels). The upper-layer contour interval is 1.5 × 104 m2 s−1 for the initial time and 1.2 × 105 m2 s−1 for the final time. For both days, the lower-layer contour interval is half that of the corresponding upper layer. Zero line omitted and dashed contours for negative values. The boundary circle is the equator and interior circles are 30° and 60°N.

Citation: Journal of the Atmospheric Sciences 55, 11; 10.1175/1520-0469(1998)055<2095:AASSOT>2.0.CO;2

Fig. 15.
Fig. 15.

As in Fig. 5, except for the first regional SV of the energy norm at the optimization time. The time-independent T20 propagator was applied to the regional SV at initial time. Contour interval is arbitrary.

Citation: Journal of the Atmospheric Sciences 55, 11; 10.1175/1520-0469(1998)055<2095:AASSOT>2.0.CO;2

Fig. 16.
Fig. 16.

Energy (per unit mass) spectrum of the first regional SV of the energy norm at initial time (solid line) and final time (dashed line). Units are m2 s−1. At both times the perturbation was normalized to a unit energy norm. A Hanning filter was used twice on the data to smooth the spectra.

Citation: Journal of the Atmospheric Sciences 55, 11; 10.1175/1520-0469(1998)055<2095:AASSOT>2.0.CO;2

Fig. 17.
Fig. 17.

As in Fig. 5, except for the first (left panel) and second (right panel) regional SV of the L2 norm. Contour intervals used for the upper layers are 1.5 times the arbitrary contour intervals for the lower layers.

Citation: Journal of the Atmospheric Sciences 55, 11; 10.1175/1520-0469(1998)055<2095:AASSOT>2.0.CO;2

Fig. 18.
Fig. 18.

As in Fig. 10, except for the regional SVs of the L2 norm.

Citation: Journal of the Atmospheric Sciences 55, 11; 10.1175/1520-0469(1998)055<2095:AASSOT>2.0.CO;2

Table 1.

Relative sensitivity for the first seven normal modes of day 1896. The response function was defined on day 1899. In the first column, from left, are the normal mode indices, in the second column the basis vector indices, in the third column the phase of the normal mode (R for real phase, I for imaginary phase, and S for stationary), in the fourth column the values of RS, in the fifth column the values of the optimal sensitivities, in the sixth column the projectabilities, in the seventh column the growth rates (in units of day−1), and in the last column the periods (in days).

Table 1.
Table 2.

As in Table 1, but for all normal modes for which (RS)opt > 0.1. Whenever (RS)opt > 0.1 but |RS| < 0.1 for both phases, we show the phase with the largest |RS|.

Table 2.
Table 3.

As in Table 2, but for all adjoint modes of the energy norm associated with (RS)opt > 0.2. The projectabilities, growth rates, and periods of the corresponding normal modes are also shown.

Table 3.
Table 4.

As in Table 2, but for the regional SVs of the energy norm. Shown all cases in which |RS| > 0.2. Note that no information regarding the phase is necessary for regional SVs. The fourth column (from left) contains the amplification factors of the regional SVs. We normalized all regional SVs to possess a unit norm, so that the projectability (which can formally be introduced also for SVs) is 1.

Table 4.
Table 5.

As in Table 4, but for the regional SVs of the L2 norm. The values have been multiplied by 106 to make them comparable in magnitude to those of the previous tables. Shown all cases in which |RS| > 0.3.

Table 5.

* Geophysical Fluid Dynamics Institute Contribution Number 384.

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