Local Predictability in a Simple Model of Atmospheric Balance

Local predictability in a simple model of atmospheric balance

G. Gyarmati1, I. Szunyogh2, and D.J. Patil2 1Department of Meteorology, Eotv¨ os¨ Lorand´ University, Budapest 2Institute for Physical Science and Technology and Department of Meteorology, University of Maryland, College Park, Maryland

Camera-ready Copy for Nonlinear Processes in Geophysics Manuscript-No. 1044

Offset requests to: I. Szunyogh Institute for Physical Science and Technology University of Maryland, CCS Building College Park, MD, 20742-2431 Nonlinear Processes in Geophysics (2002) ???:1–19 Nonlinear Processes in Geophysics c European Geophysical Society 2002

Local predictability in a simple model of atmospheric balance

Abstract. The 2 degree-of-freedom elastic pendulum equa- eriksen, 2000; Trevisan et al., 2001; Ziehmann et al., 2001) to tions can be considered as the lowest order analogue of in- more realistic atmospheric models with higher teracting low-frequency (slow) Rossby-Haurwitz and high- complexity (e.g., Lacarra and Talagrand, 1988; Houtekamer, frequency (fast) gravity waves in the atmosphere. The 1991; Molteni and Palmer, 1993; Vukicevic, 1993; Buizza strength of the coupling between the low and the high fre- and Palmer, 1995; Ehrendorfer and Errico, 1995; Vannitsem quency waves is controlled by a single coupling parameter, and Nicolis, 1997; Szunyogh et al., 1997; Patil et al., 2001). ¡ , defined by the ratio of the fast and slow characteristic time The aim of the present paper is to further investigate this scales. subject for a low-order, non-dissipative analogue to the at- In this paper, efficient, high accuracy, and symplectic struc- mospheric governing equations that can maintain motions of ture preserving numerical solutions are designed for the elas- distinctly different time scales and can accommodate an ana- tic pendulum equation in order to study the role balanced dy- logue to balanced atmospheric motions. namics play in local predictability. To quantify changes in The first low-order model to study interactions between the local predictability, two measures are considered: the lo- motions of different time scales in the atmosphere was de- cal Lyapunov number and the leading singular value of the rived by finding the lowest order truncation of the shallow tangent linear map. water equations (Lorenz, 1986). This model, today known as It is shown, both based on theoretical considerations and the five-variable Lorenz (L5) model, is a nonlinearly coupled numerical experiments, that there exist regions of the phase system of two simple integrable mechanical subsystems: a space where the local Lyapunov number indicates exception- nonlinear pendulum and a harmonic oscillator ally high predictability, while the dominant singular value (Camassa, 1995; Bokhove and Shepherd, 1996). For appro- indicates exceptionally low predictability. It is also demon- priate choices of the control parameters the frequency of the strated that the local Lyapunov number has a tendency to rotation of the pendulum is much slower than the frequency choose instabilities associated with balanced motions, while of the oscillator. In this case, the ”slow” swinging of the pen- the dominant singular value favors instabilities related to dulum and the ”fast” resonance of the oscillator can be con- highly unbalanced motions. The implications of these find- sidered as analogues to the low frequency Rossby-Haurwitz ings for atmospheric dynamics are also discussed. waves and the high frequency gravity waves in the atmosphere. In a perfectly balanced state, the coordinates (variables) associated with ”fast” motions are slaved to the coordinates related to ”slow” motions. That is, the ”fast” compo- 1 Introduction nents can be determined from the ”slow” components at any given time instance by solving an algebraic balance equation. Although it is widely accepted that predictability varies with Lynch (1996, 2002) pointed out that a slight modification the atmospheric flow, the issue of finding a single best mea- of the coupling term in the L5 model leads to the equation of sure to quantify this variation has not yet been settled. Mea- motion for the elastic pendulum, where a mass is suspended sures of local predictability in phase space have been studied from an elastic rod which can rotate and stretch, but can not and compared for a wide variety of models with atmospheric bend. The role balanced dynamics plays in predictability can, relevance ranging from systems with few degrees of free- therefore, be addressed by studying either the L5 model or dom (e.g., Farrell, 1988, 1990; Trevisan and Legnani, 1995; the structurally equivalent elastic pendulum. In this paper, Lorenz, 1996; Legras and Vautard, 1996; Smith, 1997; Fred- for the sake of physical transparency, we study the elastic Correspondence to: I. Szunyogh pendulum equations.

1 2 Gyarmati et al.: Local predictability

To quantify changes in the local predictability of the elas- a system of ordinary differential equations,

tic pendulum we consider two different measures: the lead- ¢¤£

£ ¥ £ £

ing singular value of the tangent linear map and the local ¢¤¥§¦©¨ (1)

£ ¥ ¥

Lyapunov number. These two measures (and/or some of their £

derivatives) have already been compared in a series of papers where the state vector, , and the vector ﬁeld, ¨§ ,

have components.

(e.g., Trevisan and Legnani, 1995; Szunyogh et al., 1997; £

The time evolution of an inﬁnitesimal perturbation, , ¥ Reynolds and Errico, 1999; Samelson, 2001; Ziehmann et al., £ 2001). The main motivations for revisiting the issue for the to the nonlinear trajectory at is determined by the

elastic pendulum are twofold: (1) Since the elastic pendu- associated tangent-linear equation,

¢ £

£ ¥ £ £ ¥!" £

lum has Hamiltonian structure, its singular value spectrum

¦© ¨§ ¦

¢¤¥

(2) is symmetric. This symmetry can be exploited to study the relationship between singular values and the local Lyapunov

where #¨ is the Jacobian matrix for the vector-valued func- ¥

number in a more transparent manner than in a dissipative £

tion ¨ (e.g. Ott, 1993). The solution of the tangent- ¥

system. (2) Studies with complex models based on the atmo- £

linear equation deﬁnes the tangent-linear map $ , for spheric primitive equations (Szunyogh et al., 1997; Thorpe, which

1996; Errico, 2000; Montani and Thorpe, 2002) found that

£ ¥ £ ¥ % £

for a common choice of the coordinates, unbalanced motions (3) ¥ played an important role in local predictability when mea- £

Measures that quantify the local predictability at are

¥ £

sured by the dominant singular value. In addition, Szunyogh £

(+*,() ( traditionally deﬁned by the expansion rate () of

et al. (1997) demonstrated that the local Lyapunov number £ (-.( a properly selected initial perturbation vector , where measured instabilities associated with nearly balanced dy- denotes the Euclidean norm. The difference between the lo- namics. That paper also conjectured that the difference be- cal predictability measures (in this paper the local Lyapunov

tween the two measures of local predictability and their re- number and the dominant singular value) is in the selection £ lationship to the balanced dynamics may be due to the fact of the initial perturbation vector . that the local Lyapunov number measures predictability for trajectories on the attractor of the system, while the dominant 2.2 Dominant singular value singular value can potentially be associated with trajectories

which are initially off the attractor (e.g., Anderson, 1995; Singular Value Decomposition (SVD) (e.g., p. 70-72 Golub

£ ¥ /

Legras and Vautard, 1996; Trevisan and Pancotti, 1998). In and Van Loan, 1983) guarantees that for $ there

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132 1687

this paper, we will demonstrate that the relationship between exist two sets of orthonormal basis vectors, 0 and

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0 9 9 7 : :

6 2 6

the predictability measures and the role of balanced dynam- 2 , and a set of real, non-negative scalars,

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: :'= :

; ; 6

ics is no different for the elastic pendulum than for dissipative ( 2<; ), such that

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atmospheric models. This suggests, that in terms of balanced >

:? 93? 1? (4)

dynamics, it is not the existence of an attractor, but the pres-

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ence of multiple time scales that plays the key role in the The scalars :@2 are the singular values, while

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132 1687 0 9 2 936A7 differences between the predictability measures. (The elastic 0 and are, respectively, the right (ini-

pendulum has multiple time scales but no attractor since it is tial) and the left (evolved) singular vectors of the matrix that

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a non-dissipative system). represents the tangent linear map . The outline of the paper is as follows. Section 2 is a brief It is important to emphasize that the SVD of a linear op-

overview of the measures of local predictability considered erator is not independent of the choice of the local coordi-

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B BC= B 6 here. Section 3 presents the basic dynamical equations and nates (components of ), 2 ; (or of the choice of gives a short description of the geometry of the phase space. the inner product if the coordinates are ﬁxed and the inner A new family of symplectic structure preserving numerical product is varied). In the atmospheric sciences, the general solutions for the elastic pendulum is presented in section 4 practice is to choose the coordinates so that the square of the (technical details are provided in the Appendix), while the norm generated by the Euclidean inner product becomes a main results of the paper on local predictability are presented quantity with an energy dimension (Talagrand, 1981; Buizza

in section 5. Section 6 offers some conclusions. et al., 1993).

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132 1687

Since 0 provide an orthonormal basis, any initial £

perturbation, , can be written as 6

2 Measures of local predictability D

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1? 1?

(5) ?4E

2.1 The tangent-linear map 2GF

I - where - denotes the Euclidean inner product, which

Equations that typically arise in atmospheric dynamics (and combinedF with (3) and (4) gives

in numerical weather prediction), after the partial differential

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equations that govern the atmospheric motions have been dis- &

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: 1 9 (6)

cretized in space (e.g., Kad´ ar´ et al., 1998), can be written as

?4E

2 F Gyarmati et al.: Local predictability 3

This means that when the uncertainty in the prediction is measured by the Euclidean norm the largest possible am-

plification of the uncertainty is defined by the first singu-

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lar value :C2 . The dominant (ﬁrst) singular value,

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:C2 , can, therefore, be considered as a measure of

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local predictability at . (A closely related predictability

measure is the ﬁnite time Lyapunov exponent,

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2@L4M

: 2 , deﬁned by Ziehmann et al. (2001) in a slightly different form). The dominant singular value,

of course, is dependent not only on the phase space location

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, but also on the choice of the coordinates, and the time r θ interval, . l

Finally, it should be noted that in atmospheric sciences the 0

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predictability at is typically measured by the dominant

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singular value associated with the time interval 0 rather

¥P§ ¥ 7 than 0 . In this paper, we deﬁne the dominant singular value based on the past evolution of the trajectory so the results can be directly compared to the local Lyapunov number. m None-the-less, our main conclusions on the role of multiple time scales and balanced dynamics are equally valid for both

deﬁnitions. Fig. 1. Schematic picture of the elastic pendulum. The length of the un- [

stretched rod is XZY , the distance between the origin and the point mass, , ] is \ , and the angle of the pendulum is . The dashed line shows the circle,

2.3 Local Lyapunov number along which the pendulum would move if the rod was non-elastic.

¥ £ ¥

As goes to inﬁnity, all typical perturbations, in equa-

£R

tion (3), turn into the direction, Q , parallel to the ﬁrst left to be recomputed. This shows, on the one hand, that while

singular vector deﬁned for an inﬁnitely long time interval, £R

£ S the local Lyapunov number measures expansion (or contrac-

9 Q 2 . The vector is usually referred to as Lyapunov tion) associated with the same physical process regardless vector. It is important to emphasize that, although the choice of the coordinates, the dominant singular value equation (4) involves the SVD (which is dependent on the co- can emphasize different physical processes depending on the ordinates), the Lyapunov vector is independent of the choice choice of the coordinates. On the other hand, the dominant of the coordinates (Legras and Vautard, 1996; Trevisan and

singular value is a truly local quantity, depending only on

¥^_ £ ¥

Pancotti, 1998). For a short interval of time, , the speed of £

the trajectory connecting and , while the local

the separation between trajectories can be characterized by Lyapunov number is also dependent on the entire trajectory ¥!" the local Lyapunov number £

leading to .

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T £

( Q (

¦ ¥!"

£ (7)

( (/Q 3 The elastic pendulum

(A closely related predictability measure is the ﬁnite sam-

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2/V 3.1 Basic equations

¦ ple Lyapunov exponent, U , deﬁned by Ziehmann et al. (2001) in a slightly different form.) The local

The elastic pendulum is a point mass, ` , suspended by a

Lyapunov number is always positive with values of greater V than one indicating diverging trajectories, and values of weightless elastic rod, with an unstretched length of and smaller than one indicating trajectories that are converging spring constant a , which may stretch but not bend (see Fig- ure 1). The angular oscillation frequency of the pendulum

in the phase space. For any given choice of the coordinates,

f ¦ed * the upper bound for the local Lyapunov number (deﬁned by in the limit of small displacement is bPc , where

the related Euclidean norm in (7)) is equal to the dominant f is the gravitational acceleration. When the displacement is ¥ /

£ not small the frequency of the rotational motion can be sig-

singular value, 2 , while the lower bound is equal to

£ ¥ / b

niﬁcantly different from c (e.g., Tabor, 1989). The elastic

the smallest singular value, :6 . The upper bound is

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d a*h`

frequency is b!g , and the coupling parameter is

reached when the ﬁrst right singular vector, 132 , and

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b *+bPg

deﬁned by the ratio c . In this paper, only values

Q k

the Lyapunov vector, are parallel, since for the & ¡ji given choice of coordinates the ﬁrst right singular vector de- are considered. ﬁnes the direction of the fastest possible growth with respect For this simple physical system, the nondimensionalized to the Euclidean norm. equations of motions can be written as a system of nonlinear It is important to notice, that when the coordinates are ordinary differential equations (Lynch, 1996, 2002)

changed the new Lyapunov vector can be easily computed by ¢l

=/r =

2 q

¦ m'n@ po a coordinate transformation, but the right singular vector has ¢¥ (8)

4 Gyarmati et al.: Local predictability

m N As%t l!

2 q

¦ po

¢¤¥ the largest) are also locations of the largest potential decay

£ ¥ £ ¥ /

:'6 :

¢ 2

r ( ). This also implies that there

¦ m'u ¢¤¥ are no phase points, where the largest singular value can be

¢ smaller than one . This is in contrast to the behavior observed

KCv K

u N N sAl

m &

r r

= = = =w/x

2 q 2%q 2 2%q

¡ ¡ ¡

¡ for dissipative systems (Smith et al., 1999; Ziehmann et al.,

¦ m o

¢¥ n 2001).

The units of mass, length and time are deﬁned, respectively, For the elastic pendulum there are two generalized coor-

m mCu n

by the mass of the bob, ` ; the length of the unstretched rod, dinates ( and ) and two generalized momenta ( and ),

V o ; and the inverse of the rotational frequency, *+bPc . The which means that (8) is a 2 degree-of-freedom Hamiltonian dynamical variables are deﬁned in a polar coordinate system system. Thus, the phase space is 4-dimensional (n=4); how- and rescaled so that they are all of the same order for the typ- ever, due to the conservation of energy, the trajectories are

ical values of the initial condition. Namely, l is the angle be- conﬁned to a 3-dimensional space (energy shell). Moreover,

¡ ¡i o tween the rod and the vertical ( ¦y is the angle at rest) and when is small, or more formally when , the

the angular momentum, m'n , is positive for counter clockwise Kolmogorov-Arnold-Moser (KAM) theorem ensures that re- rotation. In addition, it is assumed that the amplitude of the mains a positive measure of trajectories on 2-dimensional elastic motion is small. More precisely, for the cases consid- invariant tori embedded in the 3-dimensional energy shell

ered here the distance between the point mass and the ﬁxed (Lynch, 1996, 2002). The exceptions are chaotic trajecto-

r r

2%q

¡ ¦{o

end of the rod is z and is of the same order as ries that ﬁll the 3-dimensional space between the invariant

l ¢ ¢¥ ¢ ¢¥

2%q

¡ ¡

m m'u|¦ }¦

n *

and . The scaling z¤* ensures tori. The invariant tori deﬁne the analogue of the atmospheric ¡ that m'u is of the same order as the other variables. slow manifold. As increases an increasing number of tra-

Using the generalized coordinate and momentum repre- jectories become chaotic and the invariant tori gradually dis-

¦ ¦ ZmCn m

sentation, i.e. introducing ~ and ; and appear.

~ deﬁning the Hamiltonian by the total energy of the Lynch (1996, 2002) showed numerically that most invari- system ant tori remain practically intact for such large values of ¡

as 0.2. Images of the invariant tori in the Poincare sections,

K K

/ s,l N o N N %N o

r r r

= w/x = = = =

2%q 2

¡ ¡ ¡

m m'u

o ¦ m'n o m

(9) plotted for the ( , ) and the ( , ) planes, are continuous (typically closed) curves, while images of the chaotic orbits the equation of motion ( 8) can be expressed in symplectic ﬁll areas between the tori, but in ﬁnite time numerical com- Hamiltonian form putations appear as sets of disconnected dots (examples are

¢£ shown in Fig. 4-7).

¦ R ¢¥

(10)

Here, ~ , 4 Numerical solutions

|¦ (11) 4.1 A new family of numerical solutions

and the symbols and stand, respectively, for the zero and For most initial conditions, the elastic pendulum equation the unity matrices. cannot be solved analytically as an explicit integral, hence For the elastic pendulum, as for any Hamiltonian system, only approximate numerical solutions can be obtained. The there are two independent conservation laws errors introduced by the numerical integration scheme and

the errors due to uncertain initial conditions (chaos) are gen-

¢ ¢ b

erally considered to be two independent sources of forecast ¢¥"¦y, ¢¤¥¦y, (12) error. It should be noted though, that the non-integrable na-

ture of the equations and the chaotic behavior of the Hamil-

| ~ where b is a differential two-form and the sym- tonian system they describe are closely related (e.g., Tabor,

bol stands for the exterior-derivative. (e.g. Arnold, 1989; Weintraub, 1997). The ﬁrst conservation law is the 1989). well-known conservation of energy, while the second conser- The numerical integration schemes can introduce both lo- vation law is the conservation of symplectic structure. These cal integration errors and structural errors into the solutions. conservation laws provide a symmetric singular value spec- The local integration error, deﬁned as the quantitative accu- trum described in the following subsection. racy of the solution for a time step, is proportional to the time step and the order of the integration scheme. Decreasing 3.2 The geometry of the phase space (increasing) the time step has the same effect as increasing (decreasing) the order of the scheme. When enhanced preci-

Due to the conservation of the symplectic structure, the sin- sion is needed, usually, increasing the order of the scheme is

£ ¥ /

¦ ?

gular values are in pairs deﬁned by the relation : cheaper than decreasing the time step. The possibilities are

£ ¥ /

?4 2

6 . This means that phase space locations of not unbounded though since the time step cannot be larger

£ ¥ /

the largest potential growth (locations where :C2 is than what is necessary to resolve the fastest possible changes Gyarmati et al.: Local predictability 5

in the modeled system. This principle sets an upper bound on Table A1. These computations, as any others reported in this the order of the integrator that can be efficiently used to in- paper, are carried out at 8-byte arithmetic precision. tegrate the model at a given level of accuracy. For the elastic The rms relative energy error is plotted as function of work pendulum equation this bound is set by the time scale of the for each numerical solution in Figure 2, where it can be seen oscillator. Because the time unit is equal to the time scale of that different schemes can be optimal depending on the re- the slow motion, it can be expected that the largest sufficient quired accuracy of the solution. In general, the lower order time step would decrease as ¡ decreases. integration schemes are more efficient in producing lower Structural errors of the numerical solutions are distortions accuracy solutions, while the higher order methods are more in the geometric structure (qualitative behavior) of the sys- efficient when high precision is required. tem (Sanz-Serna, 1992). Structural errors can be severe even Since the main concern of this paper is the local predictabil- if the integration scheme provides high local accuracy. A ity of the modeled physical system, it is desirable to increase theorem by Ge and Marsden (1988) states, that a numeri- the local accuracy of the solutions as much as possible. Hence, cal solution with a constant time step cannot preserve both all predictability experiments are carried out by scheme 12, the symplectic structure and the energy for a non-integrable which is an eighth order method. For a 0.01 time unit step,

Hamiltonian system. That is, the numerical solutions of the which is used in all computations reported hereafter, this

K 2

elastic pendulum will be inevitably burdened by structural scheme provides solutions with an error not larger than o+ . errors. For this paper, we choose to search for solutions that preserve the symplectic structure. The motivations to do so

are that (i) structure preserving schemes conserve the sym- 5 Predictability experiments

£ ¥ % £ ¥ %

: :

?5 2 metry 6 for the singular values; and (ii) structure preserving schemes are known to be 5.1 Local predictability more efﬁcient in conserving the energy than energy preserving schemes in conserving the structure. As mentioned before, both local predictability measures in-

The elastic pendulum is a non-separable Hamiltonian sys- vestigated here (the dominant singular value and the local

~ #2 ~

tem, which means that a decomposition Lyapunov number) are dependent on the choice of the time

= does not exist. Therefore, standard structure preserv- interval . Ziehmann et al. (2001) demonstrated by examples ing Runge-Kutta methods (e.g. Sanz-Serna and Calvo, 1994) that the ﬁnite time Lyapunov exponent (a derivative of the cannot be applied. To the best of our knowledge, the only dominant singular value) and the ﬁnite sample Lyapunov ex- way to preserve the structure of a non-separable Hamiltonian ponent (a derivative of the local Lyapunov number) are both system is to construct a symmetric composition scheme. In strongly dependent on the selection of the short time interval.

the Appendix, a family of new ﬁrst-, second-, fourth-, sixth-, Therefore, in computing the predictability measure, our ﬁrst and eighth-order numerical solutions is constructed for the task is to select an appropriate . In order to make our re-

elastic pendulum based on the general results of McLachlan sults comparable to those in the Figure 1 of Ziehmann et al. (1995). (2001); the dependence on is characterized by the distri-

bution of the ﬁnite time Lyapunov exponent, J (as deﬁned

£ ¥ /

: 4.2 Selection of the optimal scheme in section 2.2) for the dominant singular value, 2 , and by the distribution of the ﬁnite sample Lyapunov expo-

The global structural accuracy of the numerical solution is nent, U (as deﬁned in section 2.3) for the local Lyapunov

T £ ¥ /

measured by the rms relative energy error, number, . In Figure 3, the distributions of and

¦ o

U are shown for (equal to the time step of the nu-

& & ® &

¦ o ¦ ¦eo !

merical integrator), , , and time

o P© ©

ª?

¤ unit intervals. For each value of the coupling parameter, the

¦ ¢£

¥ ¦ «

(13)

?5E 2¨§ distributions are shown for a single selected initial condition near the separatrix of the pendulum component of the sys- which is the relative root-mean-square distance between the tem. (Since the elastic pendulum has no attractor, the results constant energy surface of the real system (known from the are, strictly speaking, valid only for the trajectory containing initial conditions) and the numerical solution for a long tra- the selected initial condition.) As it will be shown later, the

jectory. Here, ¥ is the number of time steps taken along the neighborhood of the separatrix is an important region of local

trajectory, is the energy in the numerical solution after the low predictability. The distributions are strikingly similar for

& &

¦ . .

,o ¦¯ o th time step, and . ¦y and in all panels of Figure 3, while for the In order to compare the efficiency of the different integra- two longer time intervals there are important differences in tors, a work is defined to be the estimated number of function the distributions. Interestingly, these differences, especially evaluations needed to integrate the model for a unit time in- for the finite sample Lyapunov exponent, are much smaller terval. Then one typical initial / condition is selected for each for the two extreme (smallest and largest investigated) val- of the four different values of the coupling parameter, ¡ , con- ues of the coupling parameters. We decided to choose the

sidered later, and the numerical solutions are computed for a largest which provides distributions consistent with those

& ,o 2000 time unit interval by each integration scheme listed in for ( ¦° ), the shortest possible time scale for the dis- 6 Gyarmati et al.: Local predictability

0 0 10 10 1 1 2 2 −2 −2 10 3 10 3 4 4 5 5 −4 −4 10 6 10 6 7 7 8 8 −6 9 −6 9 10 10 10 10 11 11 −8 12 −8 12 10 10 Energy error Energy error

−10 −10 10 10

−12 −12 10 10 a) b) −14 −14 10 10 1 2 3 4 5 6 7 1 2 3 4 5 6 7 10 10 10 10 10 10 10 10 10 10 10 10 10 10 Work Work 0 0 10 10 1 1 2 2 −2 −2 10 3 10 3 4 4 5 5 −4 −4 10 6 10 6 7 7 8 8 −6 9 −6 9 10 10 10 10 11 11 −8 12 −8 12 10 10 Energy error Energy error

−10 −10 10 10

−12 −12 10 10 c) d) −14 −14 10 10 1 2 3 4 5 6 7 1 2 3 4 5 6 7 10 10 10 10 10 10 10 10 10 10 10 10 10 10 Work Work

Fig. 2. The rms energy error as function of work (see section 4.2) for the different integration schemes listed in Table A1. The values of the coupling parameter, ± , are 0.025 (panel a), 0.25 (panel b), 0.325 (panel c), and 0.4 (panel d). Gyarmati et al.: Local predictability 7

a) b)

Λ Σ

c) d)

Λ Σ

e) f)

Λ Σ

g) h)

Λ Σ ³ Fig. 3. Each left hand side panel shows the distribution of ² , while each right hand side panel shows the distribution of for a selected trajectory near the

separatrix of the pendulum. The samples were collected for trajectories integrated for 1000 time units. The coupling parameters, ± , are 0.025 (panels a and b), ·+º

0.25 (panels c and d), 0.325 (panels e and f), and 0.4 (panels g and h). The blue, green, red, and black lines show, respectively, the distributions for ´¶µ¸·+¹ ,

´¶µ¼·+¹ ½ ´¶µ º ´¶µ¸·+¹»º , , and . 8 Gyarmati et al.: Local predictability

crete maps generated by the numerical solutions. Thus for emphasize this region, the local Lyapunov number tends to

& o

this paper we consider ¦{ . As further results of the pa- emphasize the region (called region A hereafter) around the

& o per show, ¦ provides good structural resolution and unstable subspace of the hyperbolic ﬁxed point of the pendu- smooth changes of the local predictability measures in the lum. phase space. Another interesting feature is that the local Lyapunov num- The local predictability in the phase space is mapped by ber suggests exceptionally good predictability (decaying Lya-

Poincare´ sections (e.g. Tabor, 1989; Ott, 1993). Poincare´ sec- punov vector), even at such large values of the coupling as

& k

m n

tions are prepared for a ”slow” plane, deﬁned by: ¦y , for states when the system is moving toward the hy-

r r

mCu m'u

¦¾ , ; and for a ”fast” plane deﬁned by perbolic ﬁxed point of the pendulum. This behavior is further

¦ m , n . The ”slow” plane shows the pendulum com- investigated in the next subsection.

ponents of F the state vector at instances when the rod is nei- When the Poincare´ sections for the ”slow” plane are plot-

u u

m ther stretched nor contracted, and the point mass is moving ted using m (not shown) instead of the large

toward the origin along the radial direction. The ”fast” plane, values of the local Lyapunov number disappearF in region B. at the same time, shows the spring components of the state This indicates that the key to large Lyapunov numbers at

vector at instances when the pendulum is crossing through those locations is the negative radial velocity. This conclu- l

the ¦ line during a clockwise rotation. Each Poincare´ sion is well corroborated by the ”fast plane” Poincare´ sec-

section presented in this paper show ten trajectories evolving tions shown in the right-hand-side panels of Figures 4-7: The

& ¿

¡ ¦{o on the same energy shell deﬁned by . local Lyapunov number, especially for larger values of , in-

In Figures 4-7, Poincar/’e sections are shown for four dif- dicates exceptionally low predictability when the spring is r

ferent values of the coupling parameter and the local pre- contracted ( ) and moving toward an increasingly con-

u dictability measures are visualized by colors. Both the local tracted state (m ), and exceptionally high predictability

Lyapunov number (panels a and b) and the dominant singular when the spring isF contracted and moving toward a less con- value (panels c and d) indicate clear patterns of predictabil- tracted state. The dominant singular value, on the other hand, ity, though in certain regions of the phase space these patterns suggests that low predictability can occur for an even less can be strikingly different for the two measures. In what fol- contracted spring and regardless the direction of the radial lows, we ﬁrst focus attention on the slow plane (panels a and velocity. c). On this plane, both measures are symmetric about the origin due to the symmetry that the orientation of the angle, l , 5.2 Which measure should be trusted more?

can be arbitrarily chosen.

& ®

¡ When the coupling is weak ( ¦ ) the Poincare´ sec- The results presented so far raise two important questions: tion is reminiscent of the familiar phase portrait of a pendu- Which measure should be trusted more (i) in the regions lum. Since the system is only weakly chaotic in this case and where the local Lyapunov number indicates extremely high, the Lyapunov number is only slightly larger than one, there and the dominant singular value extremely low predictability is a balance between portions of the plane where the local or (ii) when the two measures indicate different regions as Lyapunov number is greater and smaller than one. The re- the primary area of low predictability? These questions can- gions of expansion and decay are symmetrically distributed not be answered without exploring the process that makes the on the Poincare´ section: the mirror image of a local Lyapunov Lyapunov and the ﬁrst singular vectors choose systematically

number about the m'n¸¦ axis is equal to the inverse of that different directions in certain regions of the phase space.

local Lyapunov number. The largest values of the local Lya- As it was explained in section 2.3, the Lyapunov vector,

£ ¥3§

punov number, as can be expected based on the theory, are Q , can sustain a local expansion rate similar to that

similar to the dominant singular values at the same locations, of the right singular vector, only if its projection on the ﬁrst

£ ¥ %

whereas the smallest values of the local Lyapunov number right singular vector, 1 becomes large. This can hap- are similar to the inverse of the associated dominant singular pen if and only if (i) the ﬁrst left and right singular vectors

values. These extreme values are located along (and near to) follow each other in a smooth fashion, i.e. if the angle be-

£ ¥ / £ ¥ÀN /

132 the separatrix of the pendulum, and approaching the origin tween 9 2 and remains small; and the values of the predictability measures becomes gradually (ii) the ﬁrst Lyapunov vector has some projection on the ﬁrst neutral. The dominant singular values are distributed like the right singular vector at the beginning of the period for which

local Lyapunov numbers except for that the mirror image of (i) is satisﬁed.

m ¦° an expansion about the n axis is an expansion of the A comparison of the first and the last four panels of Fig- same rate. ures 4-7 confirms that the two measures of predictability give As the coupling increases, and the system becomes in- similar results when the above two conditions are met. But creasingly chaotic, new local instabilities arise in the region the same panels also show that the first Lyapunov vector of the phase space (called region B hereafter), which is as- can become orthogonal to the first right singular vector even sociated with small values of the angle and large absolute if condition (i) is satisfied. This happens primarily at phase values of the angular momentum. Both measures agree that space locations where the first Lyapunov vector decays at a the relative importance of this region increases for increas- rate equal to the inverse of the dominant singular value, indi- ing coupling, but while the dominant singular value tends to cating that the Lyapunov vector can get ”trapped” by a series Gyarmati et al.: Local predictability 9

a) b)

c) d)

e) f)

g) h) ÂÄÃÅ¼·

Fig. 4. Left hand side panels show the Poincare´ section for the ”slow” plane ( Áµ|· , ), while right hand side panels show the Poincare´ section for the

ÂÄÇ¨Å"· µÈ·+¹ ·É½ ”fast” plane ( Æ¨µ_· , ) at a weak coupling ( ). Colors represent the local Lyapunov number (panels a and b), the dominant singular value (panels c and d), the projection of the Lyapunov vector on the first right singular vector (panels e and f), and the projection of the first left singular vector on the subsequent first right singular vector (panels g and h). 10 Gyarmati et al.: Local predictability

a) b)

c) d)

e) f)

g) h)

± É½ Fig. 5. Same as ﬁgure 4, except for µ¼·+¹ . Gyarmati et al.: Local predictability 11

a) b)

c) d)

e) f)

g) h)

± ÊÉ½ Fig. 6. Same as ﬁgure 4, except for µ¼·+¹ . 12 Gyarmati et al.: Local predictability

a) b)

c) d)

e) f)

g) h)

± Ë Fig. 7. Same as ﬁgure 3, except for µ¼·+¹ . Gyarmati et al.: Local predictability 13

of smoothly connected, rapidly decaying local directions. To very small regardless the strength of the coupling. Combin- verify this explanation, figures (not shown) similar to pan- ing these results and those obtained for the local predictabil- els (e-h) were also prepared for the fourth, instead of the ity, it can be concluded that the local Lyapunov number iden- first, singular vector. As expected, these figures confirmed tifies a region (region A) as typically least predictable, where that at locations where the Lyapunov vector was decaying, the dynamics is nearly balanced. Because in a nearly bal- the fourth left and right singular vectors followed each other anced state the ”slow” variables control the dynamics of the in a smooth fashion and the Lyapunov vector evolved along system, it is not surprising that the local Lyapunov number the direction of the fastest decaying singular vector. found a strong relationship between predictability and the It must be emphasized that the decay of the Lyapunov hyperbolic fixed point of the ”slow” pendulum component vector is not associated with isolated phase points, rather it of the system. shows sharp patterns on the Poincare´ sections. In the related To further explore the role unbalanced dynamics play in parts of the phase space, uncertainties in the solutions that are the local predictability, the unbalanced component is plotted associated with initial conditions uncertainties of the distant for the normalized Lyapunov and the first right singular vec- past, can never be growing. In other words, the Lyapunov tors (Figure 9). This figure shows that while the Lyapunov vector can sustain significant decay in certain regions of the vector (left-hand-side panels) is always nearly balanced on phase space, even though on average it is growing in a chaotic the slow plane independently of the strength of the coupling,

system. the first right singular vector is well balanced only for the m The large dominant singular values, and the large projec- very small values of the angular velocity, n . This shows, tion of the first left singular vectors on the following right on the one hand, that not only the basic state but also the singular vectors show, at the same time, that uncertainties in- perturbation (Lyapunov and right singular) vectors are well troduced more recently can grow fast in the same areas where balanced in region A, confirming that the instability in that the Lyapunov vector is decaying. These results show that region is associated with balanced dynamics. On the other there is no unique best choice for measuring the local pre- hand, the unbalanced nature of the right singular vector plays dictability in the elastic pendulum, and the proper measure obvious role in that the dominant singular value emphasizes cannot be selected without investigating the primary source region B as the location of exceptionally low predictability. of the uncertainty in the initial conditions. The results of this section show that the two measures identify different regions of the ”slow” plane as the primary 5.3 Relationship between local predictability and balanced location of low predictability, because the Lyapunov vector dynamics is always nearly balanced, whereas the right singular vector can be highly unbalanced. It can be concluded, that the rel- For the elastic pendulum, the analogue to a balanced atmo- evance of the two measures depends on whether the sources spheric state is a configuration in which the centripetal, the of uncertainties in the solutions are more related to balanced elastic, and the radial components of the gravitational forces or unbalanced dynamics. are in balance. In this case, the radial acceleration is zero,

which means that the last equation of (8) becomes a nonlin- 5.4 Discussion

r m ear diagnostic equation between , n , and . This algebraic equation is the analogue to the geostrophic balance equa- Are the results obtained with a simple Hamiltonian model tion of atmospheric dynamics. This equation allows for the have any relevance regarding atmospheric predictability? To

computation of the balanced length of the spring when the answer this question, it should be recalled ﬁrst that the adi- l

”slow” variables, and m'n , are given. When the state vector abatic form of the atmospheric governing equations has a is evolved by integrating the full system of equation (8), r natural Hamiltonian structure (e.g. Shepherd, 1990; Salmon,

is typically not equal to its hypothetical balanced value, rAÌ . 1998). Secondly, studies with weakly dissipative atmospheric rAÌ The difference between r and will be called the unbal- models found a symmetry between the dominant and the small- anced part of r and here it will be determined by computing est singular values (Reynolds and Palmer, 1998) which was

rAÌ from the slow variables ﬁrst, using the Newton-Raphson reminiscent of the spectrum of a Hamiltonian system. Though r method, and then subtracting rAÌ from . a recent study (Errico et al., 2001) found that diabatic The right-hand-side panels of Figure 8 demonstrate that mesoscale processes usually destroy this symmetry, observed r is driven by a combination of balanced motions and high synoptic scale instabilities can be reasonably well explained frequency resonances, while the left-hand-side panels of the by adiabatic models (e.g. Orlanski and Katzfey, 1991). It can same ﬁgure show the unbalanced part of r on the ”slow” be assumed, therefore, that a Hamiltonian model is a reason-

plane by colors. On these Poincare´ sections, the unbalanced able analogue to synoptic scale atmospheric motions.

component is equal to rÍÌ , since the ”slow” plane is deﬁned Our result on the potential decay of the Lyapunov vector r

by the condition ¦ . The motion is nearly balanced on shows that an error pattern generated by an earlier instabil-

& ®

¡ the slow plane when the coupling is weak ( ¦Î ), but ity has a very speciﬁc structure that may even lead to strong as the coupling increases r becomes much smaller than what decay later. Strong decay is possible only in those regions would be required for the balance, especially in region B. of the phase space where potentially large growth can oc- The unbalanced component in region A, on the other hand, is cur. This phenomenon is due to the Hamiltonian structure 14 Gyarmati et al.: Local predictability

a) b)

c) d)

e) f)

g) h)

Fig. 8. Left hand side panels show the Poincare´ section for the ”slow” plane at increasing coupling. The unbalanced component of Á is shown by colors. Right

hand side panels show the time evolution of Á (solid red line) and its balanced component (solid blue line) for a selected trajectory near the separatrix. The coupling parameters, ± , are 0.025 (panels a and b), 0.25 (panels c and d), 0.325 (panels e and f), and 0.4 (panels g and h). Gyarmati et al.: Local predictability 15

a) b)

c) d)

e) f)

g) h)

Fig. 9. Left hand side panels show the unbalanced component of Á for the Lyapunov vector, while the right hand side panels show the same for the first right singular vector. The coupling parameters, ± , are 0.025 (panels a and b), 0.25 (panels c and d), 0.325 (panels e and f), and 0.4 (panels g and h). 16 Gyarmati et al.: Local predictability of the system. Although, in contrast to the elastic pendulum, become nonlinear. When this happens, neither the dominant there are more than one expanding phase space directions singular value, nor the Lyapunov number can provide cor- for the atmosphere, there is growing evidence that synoptic rect estimates of predictability. While almost all studies on scale atmospheric dynamics can be low dimensional in the atmospheric predictability are based on the assumption that regions of a geographically localized instability (Patil et al., the evolution of uncertainties is linear for the first two-three 2001). Furthermore, for the dominant atmospheric instabili- forecast days, the rigorous tests of Gilmour et al. (2001) re- ties, typically there exists a decaying phase space direction in vealed that the linearity assumption may not even be valid association with each growing direction. For instance, while for the entire first forecast day. We are currently in the pro- westward ”leaning” perturbation structures lead to baroclinic cess of building a three-dimensional variational assimilation growth in regions of strong vertical wind shear, eastward system for the elastic pendulum. Our plan is to use this data leaning structures are decaying at a similar rate. Likewise, assimilation system for obtaining estimates of the analysis in a region of strong horizontal wind shear eddies can either and forecast uncertainties. Then, these estimates will be used grow or decay through barotropic energy conversion depend- to verify the relevance of the linearity assumption and the ca- ing on the orientation of their axis. pability of the two predictability measures to predict likely It can be expected, that in the atmosphere there are com- forecast errors in the elastic pendulum. The results of these plex interactions between the different growing and decaying experiments will be presented in a forthcoming paper. phase space directions. For instance, it is a well accepted that most synoptic scale waves are generated by baroclinic instabilities of the jet stream and wiped out by barotropic decay 6 Conclusions in the exit region of the jet (e.g. Whitaker and Dole, 1995). The two key findings of the study are: A wave generated by an earlier baroclinic instability can decay because of the orientation of its axis, while another eddy – For the simple model considered, there exist regions can start growing simply because random noise may project where the local Lyapunov number indicates exception- onto a structure in which the axis is pointing into a direction ally high predictability, while the dominant singular value needed for a barotropic growth. In this situation, the local indicates exceptionally low predictability. This can hap- Lyapunov number would presumably indicate the decay as- pen, because the Lyapunov vector can become ”trapped” sociated with the decaying wave, while the dominant singular by smoothly connected decaying singular vectors, which vector could indicate the potential for growing errors. are orthogonal to the expanding phase space directions. Our result, that the structure of the first right singular vec- This can lead to an overestimation of predictability when tor is typically highly unbalanced suggests, that in a system the important forecast errors have sources other than for which balanced and the unbalanced motions can be de- small errors introduced in the distant past. fined, unbalanced perturbations can be expected to lead to rapid instantaneous error growth regardless if an attractor ex- – The structure of the right singular vector associated with ists or not. the dominant singular value tends to be significantly On the practical side, it is essential to explore whether the more unbalanced than the structure of the Lyapunov vec- known sources of uncertainties in the initial conditions can tor. Due to this, the local Lyapunov number has a ten- lead to the unbalanced structure required for the rapid growth dency to choose instabilities associated with balanced indicated by the dominant singular value. If not, such choices motion, while the dominant singular value favors insta- of the coordinates (inner products) should be preferred so as bilities related to highly unbalanced motion. This may to efficiently filter the unbalanced features from the dominant lead to an underestimation of predictability when there singular vectors. The feasibility of finding such coordinates are no such sources of uncertainty that could create the was demonstrated by Ehrendorfer and Errico (1995). highly unbalanced structures. Since this result was ob- The predictability measures considered in this paper can tained by a conservative (Hamiltonian) model, it indi- quantify only that part of the local forecast degradation which cates that this phenomenon can occur even if an attrac- is due to sensitivity to the initial conditions. It is important to tor, as well as off-the-attractor initial states, do not exist. note, however, that in atmospheric applications, model errors also play an important role. (In our simple model, the only Appendix A Symmetric composition schemes for the elas- model related error that has some effect on the local forecast tic pendulum accuracy is the local integration error. However, due to the high-accuracy of the integration scheme, this component is A1 Symmetric composition methods negligible compared to the chaos related forecast uncertain-

ties.) The basic existence-uniqueness theorem for ordinary differ-

£ ¥

Finally, it should be noted that the magnitude of the uncer- ential equations implies that, for smooth functions ¨§ ,

tainties in the analyses (best available estimates of the true the solution to the initial value problem (1) is deﬁned in some ¥

initial state of the atmosphere) is ﬁnite and not inﬁnitesimally neighborhood of ¦ [e.g.,][](Guckenheimer and Holmes,

small as assumed by the predictability measures. The time 1983). This solution deﬁnes a map between the points £ and

£ ¥

evolution of the forecast uncertainties, therefore, can rapidly in the n-dimensional phase space. This map, denoted

Gyarmati et al.: Local predictability 17

¥ £ ¥ ¥ H£

m3 ¨ ¦ m3 ¨ ÏB

by ÏB (i.e., ), is called the ﬂow Scheme Order Type m £

generated by the vector ﬁeld ¨ (the argument is omitted 1 1 S 1 for brevity). Symmetric composition methods can be used to 2 2 S 2 build numerical integrators when it is possible to ﬁnd a de- 3 2 SS 2 4 4 SS 3 composition 5 4 SS 5 6 4 SS 5

D 7 4 S 4

¨Ð¦ Ñ ¨

? (A1) 8 4 S 5 ?5E

2 9 6 SS 7

¥ 10 6 SS 9

¨ mP ¨ m3 ¨

Ï+B ÏB =

of the vector ﬁeld such that the ﬂows 2 , ,

¥ 11 8 SS 15

m3 ¨

.., Ï+B all can be determined analytically for a time in- 12 8 SS 17

¥ ¥

m3 ¨

Ò ÏB

terval . Ñ Then different order approximations to can be constructed by the repeated use of the partial ﬂows Table A1. List of integration schemes considered in this paper. Both sym- (Yoshida, 1990). For instance, metric (Type S) and symmetric composed symmetric steps (Type SS)

schemes of different orders with varying number of evaluations for each

¥ ¥ £ ¥ ¥

&5&4&

Ó time step (m) are considered.

mP ¨ m3 ¨ ¦ m3 ¨

Ï+B = Ï+B Ï+B

2 (A2)

£ ¥ PNWÔ ¥

is a ﬁrst order integrator, while step is ` , which means that a scheme can be further opti-

mized by choosing the computationally most expensive map

¥ o ¥ ¥ o

&4&4&Õ

¨ m3 ¨ ¦ mP

2 Ï+B Ï+B

mP ¨ as Ï+B and the computationally least expensive ones as

intermediateÑ steps.

¥ ¥ ¥ £Ö

Õ &4&5& o o

mP ¨ mP ¨ ¨ m3

Ï+B Ï+B 2 Ï+B

£ ¥ RNWÔ ¥

Ñ A2 Application to the elastic pendulum

¦ (A3)

N N

&&)&

is a second order integrator. If a partitioning of the Hamilto-

¥ ¥

m3 R m3

Ï+B #2 ÏB

It is important to notice, that and are composite nian exist such that the ﬂows Ñ , ,

mP R

functions. The computation of for one time step involves . . . , Ï+B can be determined analytically integrators

m3 ¨ the following series of function evaluations: ﬁrst Ï+B

that preserve theÑ symplectic structure and are accurate to any

£ ¥

m3 ¨

ÏB 2

is applied to the initial condition , then Ñ is given required order can be constructed by the repeated use

¥ £ ¥

mP ¨ m3 ¨

Ï+B Ï+B 2

applied to , etc. until is appliedÑ to of the partial ﬂows (Yoshida, 1990; McLachlan, 1995). In

¥ ¥ £Ö

&4&4&

mP ¨ mP ¨

Ï+B Ï+B

Ñ in the last step. Once a proper de- what follows, the above theory is applied to the elastic pen-

¥ ¥

m3 R mP R

Ï+B #2 Ï+B

composition of the vÑ ector ﬁeld has been obtained, the sub- dulum equations. The ﬂows , , ..,

&5&

¬ ¬

¦o

a m3 scripts ( ) can be assigned in any arbitrary order to ÏB can be easily determined for a decomposition

the partial vector ﬁelds. The integration is carried forward by of the vectorÑ ﬁeld that consists of systems of differential ¥

shifting the origin to Ò after each completed time step. equations with the following two properties (i) in some of The standard way to construct higher-order schemes out of the equations the right hand side is zero, i.e the variables on is to use a symmetric composed symmetric steps (or type the left hand side of the same equations are constant (ii) the

SS) technique of the general form right hand sides of the remaining equations depend only on

¥ ¥ ¥

&4&4& &4&4& &

the constant variables. Such a decomposition of the elastic

.× .×<Ø5Ù .×

2 2

2pÚq (A4) pendulum equations can be achieved by a ’three-map’ split-

RN RN

&4&

~ ~ #2 ~

ting , where

× × ×

= 2 The art is to ﬁnd a set of coefﬁcients 6 that gives

an error term of the required order. Coefﬁcients, not only sat-

= =)Ý = 2%q

isfying this requirement but also minimizing the leading or- ¡

¦ m po

~ =

2 2 (A6)

der error term, are available for a variety of different order

= 2

schemes in McLachlan (1995). This paper also derived op- ¡

¦ m

¨= =

timized coefﬁcients for a number of symmetric (or type S

¥ ¥

Ó Ó

Ý Ý Ý

= =

2 2%q

¡ ¡

in short) methods. These schemes use and v

¦ o Ü)ÞÄß

~ = =

K K

¥ Û¥

Ó Ó

¦ as building blocks to construct schemes of the general form

The three differential equations deﬁned by (A6) are easy to

K K

¥ ¢ ¥ ¥ ¢ ¥

&5&4& &

Ó Ó Ó Ó

Ù Ù

.Ü .Ü 2

2 (A5) solve and the three elementary maps are

ÏB

¥ :

m3 ¨

The number of evaluations of Ï+B per time step is al-

= `

ways equal to for both the S and SSÑ type schemes and

¥ Náà N pâ p¥

Ý Ý Ý

2%q

¦ o . m

2 2

if the solution is needed after each time step the number of 2 (A7)

N ¥

K@v

o mP ¨

Ï+B 2

evaluation per time steps is ` for the map .

¥ RN à N pâ H¥

= =Ý =

2 q 2%q

¡ ¡

m ¦ m o m

= = = For the remaining maps the number of evaluations per time 2 18 Gyarmati et al.: Local predictability

References

mP R

= Ï+B :

Anderson, J. L., Selection of initial conditions for ensemble forecasts in a

¥ N p¥

Ý Ý

¦ . m .

= = = simple perfect model framework, J. Atmos. Sci., 53, 22-36, 1995. (A8) Arnold, V. I., Mathematical methods of classical mechanics, Second Edition, 508 pp., Springer-Verlag, New York, 1989.

Bokhove, O., and T. Shepherd, On Hamiltonian balanced dynamics and the

¥ slowest invariant manifold, J. Atmos. Sci., 53, 276-297, 1996.

mP R Ï+B : Buizza, R., J. Tribbia, F. Molteni, and T. Palmer, Computation of optimal un-

stable structures for a numerical weather prediction model, Tellus, 45A,

¥ ¡à N pâ

Ý Ý

2%q

m ¦ m . o .

2 2 2 (A9) 388-407, 1993.

Buizza, R., and T. N. Palmer, The singular vector structure of the of the

à â

¥ RN s ¥

Ý = Ý

2 2 q w)x ¡

¡ atmospheric general circulation, J. Atmos. Sci., 52, 1434-1456, 1995.

m ¦ m .

= = = 2

Camassa, R., On the geometry of atmospheric slow manifold, Physica D, ¥

Ý 84, 134-139, 1995.

Steps that do not change the value of the variables (e.g. 2

Ý Ehrendorfer, M., and R. M. Errico, Mesoscale predictability and the spec- .

2 ) are omitted for brevity and the ordering of the terms trum of optimal perturbations, J. Atmos. Sci., 52, 3475-3500, 1995.

#2 , , and is not by accident. This particular ordering Errico, R., M., The dynamical balance of leading singular vectors in a prim- provides the fastest numerical integrators. itive equation model, Q. J. Roy. Meteorol. Soc., 126, 1601-1618, 2000.

For comparing the efﬁciency of the different schemes we Errico, R. M., M. Ehrendorfer, and K. Reader, The spectra of singular values

¥ in a regional model, Tellus, 53A, 317-332, 2001.

mP R deﬁne work as the number of the evaluations of Ï+B Farrell, B., Optimal excitation of neutral Rossby waves, J. Atmos. Sci., 45, needed to integrate the model for a unit time interval. This is 163-172, 1988.

equal to ` times the number of time steps for the unit time Farrell, B., Small error dynamics and the predictability of atmospheric ﬂows,

interval. This deﬁnition was chosen because the evaluation of J. Atmos. Sci., 47, 2409-2416, 1990.

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