## 1. Introduction

The idea that there is an inherent finite range of predictability for certain fluid flows originated with Lorenz (1969, hereafter L69). This idea is based on a conception of how a small initial difference (or “error”) between two solutions to the fluid-motion equations may grow with time. According to this conception, flows with many scales of motion in which smaller-scale error spreads to larger scales and in which the error-doubling time decreases with decreasing scale have a finite range of predictability: Attempts to extend the time for which the error is less than a given tolerance by resolving finer scales (and thereby reducing the initial error) will be thwarted as the finer-scale error will grow faster, spread upscale, and thus nearly offset the expected gain in the range of predictability from the initial error reduction. The L69 theory of limited predictability is now a well understood and accepted part of the canon of dynamical meteorology. Less well understood and accepted is the model upon which L69 based its conclusions. The present paper revisits the L69 model, acknowledges known deficiencies, and then remedies them with a new application of the L69 model showing that that model can be a useful analytical tool for understanding atmospheric predictability.

The L69 model for the time evolution of the error energy spectrum is based on the two-dimensional vorticity equation (2DV). Required as input for the L69 model is the energy spectrum of the basic flow within which the errors grow. L69 considers energy spectra with power-law behavior (*K*^{−}* ^{p}*, where

*K*is wavenumber magnitude) for high wavenumbers and discusses results for

*p*= 5/3, 7/3, and 3. L69 found that for

*p*< 3, the error-doubling time decreases with scale and that the upscale spreading of initially small-scale error provides an effective limit to the predictability of such flows; for flows with

*p*≥ 3, L69 concludes that there is no such limit.

At the time of the L69 writing, it was just becoming apparent that the observed energy spectrum of the large-scale (presumably two dimensional) flow has a power-law behavior most consistent with *p* = 3 (Wiin-Nielsen 1967). Moreover it had only recently been discovered that the energy spectra downscale of a large-scale forcing is, according to the 2DV, also one with *p* = 3 (Kraichnan 1967). Hence, in retrospect, it is clear that *p* = 3 is the only consistent choice as an input spectrum for the 2DV-based L69 model, so, for the type of flow considered in L69, the L69 model can only provide an example of a turbulent flow with unlimited predictability. Subsequent work by Leith and Kraichnan (1972) using a different spectral closure model for both the three- and two-dimensional Navier–Stokes equations showed limited predictability for energy-cascading three-dimensional turbulence (*p* = 5/3) and unlimited predictability for enstrophy-cascading two-dimensional turbulence (*p* = 3), as foreshadowed by L69.

Since three-dimensional isotropic homogeneous turbulence is not a good model for atmospheric motions with scales much larger than a few kilometers and since the two-dimensional vorticity equation neglects important baroclinic effects, the authors were led to consider the predictability of solutions to the surface quasigeostrophic equations (SQGs), which include baroclinic effects but to a great extent share the mathematical simplicity of the 2DV (Held et al. 1995). An interesting feature of the SQG is that the kinetic energy spectrum downscale of a large-scale forcing is a power law with *p* = 5/3 (Blumen 1978), so, based on the physical reasoning of L69, one might expect limited predictability for such a flow. We show here that the L69 stochastic model for the error energy spectrum can, with minor modification, be applied to SQG turbulence (e.g., Pierrehumbert et al. 1994) and that the modified L69 model indicates limited predictability for that flow.

The L69 stochastic model, being relatively simple, then allows one to follow the spectral error-energy dynamics in SQG that produces limited predictability and compare those dynamics to the spectral error-energy dynamics in 2DV that lead to unlimited predictability. The application of the L69 error-growth model in each case is now consistent with the assumed basic-flow kinetic energy spectrum. The application of the L69 model to SQG also shows that the distinction between flows with limited and unlimited predictability is determined by the kinetic energy spectrum of the basic flow (i.e., *p* = 3 versus *p* = 5/3) much more than the dynamics (SQG or 2DV) that govern the error growth, consistent with the scaling arguments of Lilly (1972), which consider only the kinetic energy spectrum. Although the typical concern in predictability studies is the upscale spreading of error, the present analysis also identifies the critical role that downscale error spreading plays in distinguishing flows with limited predictability.

Analysis of atmospheric kinetic energy spectra indicates a “−3” spectrum at synoptic scales with a transition to a “−5/3” spectrum within the mesoscale (Nastrom and Gage 1985). The latter finding continues to motivate theoretical explanations (see Skamarock 2004 for a review) and speculations on the implications of this spectrum for atmospheric predictability (Straus and Shukla 2005). Although the SQG system exhibits a −5/3 spectrum, it is at present unknown whether this spectrum has any relation to that observed in the atmospheric mesoscale (but see Tulloch and Smith 2006). From the point of view of the present work, the predictability properties of SQG (with a −5/3 spectrum downscale from a large-scale forcing) represent a physically defensible counterpoint to the predictability properties of 2DV (with a −3 spectrum downscale from a large-scale forcing), both of which may now be analyzed and compared using the L69 model.

Given the trivial nature of the generalization of the L69 model to SQG, the following section 2 will both review and extend the L69 stochastic model. Section 3 first shows numerical results reproducing Lorenz’s original calculations, followed by new results using the extended L69 model for SQG and 2DV. Section 4 provides an analysis of the spectral error-energy dynamics in each case. Section 5 contains a discussion and summary of the results.

## 2. Generalization of the L69 model to SQG^{1}

### a. Derivation

*ψ*is the streamfunction,

*q*the vorticity, and

_{ψ}*J*(

*A*,

*B*) ≡ ∂

*∂*

_{x}A*− ∂*

_{y}B*∂*

_{x}B*. The error ε is defined as the difference between two solutions*

_{y}A*ψ*of (2.1) corresponding to two different initial conditions. For ε small compared with

*ψ*, (2.1a) simplifies to the linear form

_{0}of streamfunction fields

*ψ*(

*x*,

*y*,

*t*) and, for each

*ψ*, a subensemble

*of error fields ε(*

_{ψ}*x*,

*y*,

*t*). L69 observes that (2.2) equally well describes the evolution of ε′ (the departure of ε from its average over

*) and, since ε′ is generally smaller than ε, (2.2) will be a better approximation for the evolution of ε′. Hence the starting point for L69’s stochastic model is (2.2) with ε′ replacing ε.*M

_{ψ}_{0}and

*form the grand ensemble*

_{ψ}*ψ*, ε′) and, for the reasons described in L69 (p. 293), if

*πD*, the dependent variables of (2.2) may be expressed as

**r**= (

*x*,

*y*) and

**K**= (

*K*,

_{x}*K*). Substituting (2.3) and (2.4) into (2.2) then gives

_{y}^{2}

*θ*while (2.1b) is replaced by

*z*is the vertical coordinate. The analogous expression in spectral form is

*Q*

**is the expansion coefficient representing**

_{K}*θ*and

*β*= 1 (n.b.

**K**=

^{1}*K*). In the 2DV case the spectral form of (2.1b) is also (2.6a) but with

*β*= 2 and the expansion coefficient representing the vorticity. Hence, it is easy to see from (2.5) and (2.6a) that the generalization of L69’s (12)–(13) to either 2DV or SQG is

*e*

_{K}*e*

_{−}

**, namely**

_{K}*ψ*and ε′ are statistically independent (except initially), for otherwise

*ψ*) and the quantity

*e*

_{K}*e*

_{−}

**would not change in time. Instead, L69 considers the equation resulting from the time derivative of (2.9),**

_{K}*ψ*and quadratic functions of ε′ remain statistically independent

^{3}so that

_{t}S

_{K}_{−}

**is quadratic by (2.1) and that ∂**

_{L}_{t}

*S*

_{K−L}= 0 by spatial homogeneity. Under the foregoing assumptions, (2.10) reduces to

*D*→ ∞), the wavenumber

**K**varies continuously; assuming that

**K**, (2.13) may be written in continuous form in terms of the energy densities

*X*′(

**K**) and

*Z*′(

**K**) (respectively, ½

*D*

^{2}

**K**

^{2}

*D*

^{2}

**K**

^{2}

*D*→ ∞), namely,

_{0}and

*that are initially isotropic; thus,*

_{ψ}*X*′(

**K**) and

*Z*′(

**K**) depend only on wavenumber magnitude

*K*(L69, p. 296).

At this point all of the important assumptions and restrictions have been made and, thus, (2.14) with the extended definition of the *A*** _{KL}** (2.8) represents the generalization of L69’s stochastic model; with

*β*= 2, (2.14) is L69’s model for 2DV; with

*β*= 1 it applies to SQG.

### b. Transformation and simplification of (2.14)

**M**=

**K**−

**L**and the coordinate transformation (

*L*,

_{x}*L*) → (

_{y}*L*,

*M*), the differentials in (2.14) transform as

*K*,

*L*,

*M*). Using the definition (2.8) of

*A*

**, (2.15), and the substitutions**

_{KL}^{4}[

*X*(

*K*),

*Z*(

*K*)] = 2

*πK*

^{2}[

*X*′(

*K*),

*Z*′(

*K*)], (2.14) becomes

*B*(

_{j}*K*,

*L*,

*M*) = 0 for

*M*lying outside the interval (|

*K*−

*L*|, |

*K*+

*L*|).

*Z*(

*K*) over the

*n*discrete resolution intervals, defined by

*a*= ln

_{k}*N*and

_{k}*N*=

_{k}*ρ*

^{k}N_{0}; the resolution factor

*ρ*determines the size of the intervals starting from the minimum wavenumber

*N*

_{0}and ending at the maximum wavenumber

*N*. Applying (2.20) to (2.16) and assuming on the rhs that

_{n}*Z*(

*K*) =

*σ*

^{−1}

*Z*(where

_{k}*σ*= ln

*ρ*) gives

*X*(

*K*) =

*σ*

^{−1}

*X*, (2.22) can be expressed as

_{k}*B*given in (2.18)–(2.19), it may still be shown that

_{j}*K*′,

*L*′) = (

*K*/

*M*,

*L*/

*M*). Finally with

To summarize, (2.30) describes the evolution of discrete portions of the error-energy spectrum; with *ρ* = 2 in (2.20), each *Z _{k}* represents an octave of that spectrum. The dynamics of the model are encapsulated in the

*C*, which as one can see from (2.29) depends on the input spectrum

_{kl}*X*, and the coefficients

_{k}*B*

_{(j)kl}, which represent the dynamics (2DV or SQG) of the originating Eqs. (2.7)–(2.8).

## 3. Numerical results

### a. Computation of C_{kl}

*C*(2.29) begins with the computation of the integrals in (2.27) defining the coefficients

_{kl}*B*

_{(j)kl}, which will be discussed in the appendix; (2.29) also requires the specification of the basic-state energy spectrum

*X*. L69 specifies

_{k}*ρ*= 2), which has the properties of vanishing at

*k*= 0 and represents a −5/3 spectrum for

*k*large.

^{5}Following L69, the fundamental units of length and time are chosen such that the total nondimensional energy,

^{6}

*E*= Σ

^{n}

_{k=1}

*X*

_{k}= 1, and the minimum nondimensional wavenumber

*N*

_{0}= 1.

To check the present calculation against that reported in L69, the *B*_{(j)kl} are calculated with *β* = 2 (2DV) and (3.1) is used in the calculation (2.29) of the *C _{kl}*. The result is shown in Table 1 for the first nine rows and columns; Table 2 contains the reported values shown in L69’s Table 2. For most of the elements of

*C*, the comparison is very good and inspires confidence that the present calculation indeed follows that of L69. The major discrepancy occurs on the diagonal two lines below the main diagonal where the present calculation produces values systematically lower than those reported in L69. We have made detailed checks of convergence in the numerical evaluation of the integral in (2.29) and are therefore confident in our results; however, since we do not have access to the original code used in L69, the source of the discrepancy must remain unknown.

_{kl}^{7}In any event, as will be evident from the discussion to follow, the noted differences do not produce any important sensitivity in the solution of (2.30) for

*Z*.

_{k}As discussed in the introduction, a −5/3 basic-state energy spectrum downscale from a large-scale forcing is not consistent with 2DV. To obtain a more consistent −3 spectrum for 2DV, following L69, the values obtained from (3.1) are retained for *k* < *k*_{0}, whereas for *k* > *k*_{0}, the original *X _{k}* are multiplied by

*ρ*

^{−4(k−k0)/3}; in the present case

*k*

_{0}= 3. Table 3 displays the coefficients

*C*calculated with the

_{kl}*X*modified to represent a −3 spectrum and with 2DV dynamics (

_{k}*β*= 2). The striking difference between Table 1 (−5/3) and Table 3 (−3) is the much-reduced value of the entries in the lower left-hand corner.

Finally Table 4 displays *C _{kl}* computed using the original −5/3 spectrum (3.1) but now with SQG dynamics (

*β*= 1). Comparing Table 1 (

*β*= 2) with Table 4 shows that the values are quite similar, indicating that using the more consistent SQG dynamical model makes little difference to the interaction coefficients. One further calculation (not shown) of

*C*for SQG with a (physically inconsistent) −3 spectrum produces values similar to those shown in Table 3. Thus the large difference in character between Table 3 (2DV) and Table 4 (SQG) is due to the basic-flow spectra used, rather than the dynamics (SQG or 2DV) of the error-growth model. Hence, in the following it will be understood that the basic-state spectrum is the determining factor in the error-energy evolution and that the dynamical model (SQG or 2DV) simply provides a physical rationale for choosing between a −5/3 and a −3 basic-state spectrum.

_{kl}### b. Computation of the error energy Z_{k}(t) for −5/3 (SQG) and −3 (2DV) spectra

With the *C _{kl}* computed, it is a simple matter to integrate (2.30) forward in time using the simple leapfrog technique with a time step = 0.001. As the Eq. (2.30) governing error growth is linear, some device must be employed to represent the neglected nonlinear terms that would prevent the error energy

*Z*(

_{k}*t*) from exceeding the basic-state energy

*X*. Here a similar approach to that of L69 is followed:

_{k}*Z*(

_{k}*t*) is fixed at

*X*if

_{k}*Z*(

_{k}*t*) ≥

*X*in the course of the integration. We report here on predictability experiments with

_{k}*Z*(0) =

_{k}*X*for

_{k}*k*=

*n*with

*Z*(0) = 0 otherwise, and ∂

_{k}*(0) = 0 for all*

_{t}Z_{k}*k*.

Figure 1 shows the results of the calculations with −5/3 (SQG) and −3 (2DV) spectra in terms of the energies per unit wavenumber, *K*^{−1}*Z*(*K*, *t*), *K*^{−1}*X*(*K*) for nondimensional time *t* = 0.0 − 2.0. It is clear that the −5/3 (SQG) case (Fig. 1a) displays the characteristics of a flow with limited predictability in that the initial small-scale error grows rapidly, saturates, and then grows more slowly as it reaches larger scales. This latter behavior is qualitatively similar to that reported by Métais and Lesieur (1986, their Fig. 2) using a spectral-closure model for error-energy growth in 3D isotropic homogeneous turbulence. In the −3 (2DV) case, the initial error, concentrated at the smallest scale, undergoes an adjustment, spreading to large scales but maintaining small amplitude for all scales *k* < *n*; this adjusted initial error spectrum has a peak at a large scale and subsequently grows at a uniform rate irrespective of error-energy saturation. This latter behavior is again similar to that shown in Métais and Lesieur (1986, their Fig. 9) using a spectral-closure model for error-energy growth in 2D isotropic homogeneous turbulence.

Another view of the error growth is provided in Fig. 2, where the total error energy Σ^{n}_{k=1} *Z*_{k} is plotted. In the −5/3 (SQG) case (Fig. 2a) there is a repeated sequence of growth to saturation with a diminished growth rate after each saturation event.^{8} In the −3 (2DV) case (Fig. 2b), apart from the initial adjustment period, the growth is characterized by a single growth rate.

## 4. Analysis

### a. The most rapidly growing error-energy spectrum

The linear equation (2.30) is for the evolution of the error-energy spectrum *Z _{k}*(

*t*). Looking for solutions of the form

*Z*(

_{k}*t*) ∼ exp(

*λt*) gives an eigenvalue problem in which the eigenvalues of the matrix

*C*,

_{kl}*λ*

^{2}

_{1}, . . . ,

*λ*

^{2}

_{n}, if positive, give the growth rates for the associated eigenvectors

**Z̃**

_{k}. The present calculation of the eigenvalues of the

*C*

_{kl}for −5/3 (SQG) and −3 (2DV) basic-state spectra finds that all of the eigenvalues are real and that there is a positive maximum; the eigenvector

**Z̃**

_{k}associated with this positive maximum thus represents the most rapidly growing error-energy spectrum.

Figures 3a,b show the error energy per unit wavenumber *K*^{−1}**Z̃**(*K*) (the curves labeled *n* = 12) associated with the eigenvector **Z̃**_{k} for the −5/3 (SQG) and −3 (2DV) cases, respectively. In the −5/3 (SQG) case, Fig. 3a indicates that the curve is steeply sloped toward higher wavenumbers (small scales), whereas in the −3 (2DV) case, Fig. 3b indicates a low-wavenumber maximum. The associated eigenvalues give a growth rate of *λ* ≈ 90 for SQG and *λ* ≈ 5.7 for 2DV.

The rapid growth of the high-wavenumber-peaked −5/3 (SQG) eigenvector implies that small-amplitude initial error would rapidly saturate at the high wavenumbers and, hence, the linear analysis would no longer pertain to the numerical solutions. However, after saturation, the highest wavenumbers are essentially eliminated from the ensuing dynamical evolution owing to the small values of *C _{kl}* for

*l*>

*k*+ 1 [hence contributions to the summation in (2.30) are negligible]. In an attempt to use the linear analysis to interpret the numerical results beyond error-energy saturation, an eigenanalysis of a truncated version of

*C*is carried out in which its last row and column are eliminated. The result of the calculation is represented by the curve labeled

_{kl}*n*= 11 in Fig. 3a, showing that the most rapidly growing spectrum is displaced toward lower wavenumbers, but maintains nearly the same form as that from the full matrix (

*n*= 12); the associated growth rate is the smaller value

*λ*≈ 56. Progressively eliminating the last rows and columns of

*C*produces analogous results (Fig. 3a; the curves labeled

_{kl}*n*= 10, 9 have associated growth rates of

*λ*≈ 35 and

*λ*≈ 21, respectively).

Subjecting the −3 (2DV) case to the same analysis as just described for the −5/3 (SQG) case, Fig. 3b indicates essentially no change in the most rapidly growing spectrum as the last rows and columns of *C _{kl}* are eliminated; the associated growth rates for

*n*= 11, 10, 9 vary little (

*λ*≈ 5.6, 5.4, 5.2, respectively).

^{9}

The foregoing analysis suggests that the evolution of the error-energy spectrum for the −5/3 case (Fig. 1a, SQG) can be interpreted as a sequence of growth with subsequent saturation of the most rapidly growing spectrum associated with *C _{kl}*, with highest wavenumber rows and columns eliminated as saturation occurs. In the −3 case (Fig. 1b, 2DV) the evolution of the error-energy spectrum can be interpreted as the projection of the initial error-energy spectrum onto the most rapidly growing error-energy spectrum.

By numerical experimentation, it has been determined that the very different character of the eigensolutions for the −5/3 (SQG) and the −3 (2DV) cases is rooted in the striking difference in the *C _{kl}* noted in section 3a: In the −5/3 (SQG) case (Table 4), large values populate the lower left-hand triangle, but not in the −3 (2DV) case (Table 3).

^{10}The implications of this difference will be explored in further detail below.

### b. Physical interpretation of C_{kl}

In this subsection, a simplified, qualitative discussion relating the nature of *C _{kl}* to the originating physical model is given; the supporting technical aspects are given in the appendix.

The off-diagonal elements of *C _{kl}* represent spreading of error energy from one scale of motion to another. The elements on the upper (lower) triangle represent upscale (downscale) spreading of error energy. The diagonal elements represent changes in error energy owing to the influence of errors of the same scale.

The matrices *C _{kl}* as shown in Tables 3 and 4 are characterized by small values in the upper right-hand corner, increasingly negative values along the main diagonal with positive values comparable in magnitude on the flanking diagonals, and large values [in the −5/3 (SQG), but not in the −3 (2DV) case] in the lower left-hand corner. Each of these characteristics corresponds to identifiable physical effects in the originating model (2.2). Equation (2.2) states that the error in the vorticity (2DV)/surface potential temperature

*θ*(SQG) varies according to the advection by the error velocity of the base-state vorticity/

*θ*(first term on the rhs) or through the advection by the base-state velocity of the error vorticity/

*θ*. In wavenumber space the aforementioned tendencies correspond directly to the two terms in brackets in (2.8), which together describe the change in error ε′ at scale

*K*by interactions between the base-state

*ψ*at scale

*M*= |

*K*−

*L*| and error ε′ at scale

*L*.

Elements in the upper right-hand corner of *C _{kl}* involve interactions with

*L*≈

*M*,

*K*≪

*L*(see the fourth paragraph of the appendix), that is, with base-state and error motions of similar scale. In this case the two terms in brackets of (2.8) [the two advective tendencies of (2.2)] nearly offset one another, and thus

*C*is very small (A.8). As noted in L69, the small values in the upper right-hand corner signify very little direct effect of small-scale error (at scale

_{kl}*L*) on the largest-scale error (at scale

*K*).

The diagonal and near-diagonal elements of *C _{kl}* involve interactions with

*L*≫

*M*,

*K*≈

*L*(see the sixth and seventh paragraph of the appendix), that is, interactions between large-scale features of the base state and small-scale features of the error field. In this case the second term in brackets of (2.8) (advection by the base state velocity of the error vorticity/

*θ*) is the dominant contributor. The well-known Orr mechanism provides a physical model for the type of upscale and downscale error growth represented by this term (Palmer 2001, and references therein). For the reasons described in the appendix the general behavior of these elements does not vary greatly between the −5/3 (SQG) and the −3 (2DV) cases.

The elements in the lower left-hand corner of *C _{kl}* involve interactions with

*L*≪

*M*,

*K*≈

*M*(see the fifth paragraph of the appendix), that is, interactions between small-scale features of the base state and large-scale features of the error field. In this case the first term in brackets of (2.8) (advection of the base-state vortictiy/

*θ*by the error velocity) is the dominant contributor. Physically one may think of small-scale features (vortices for 2DV, surface potential temperature anomalies for SQG) in the base state being shifted by a large-scale error field (see also Kida et al. 1990; Boffetta et al. 1997; Snyder and Hamill 2003). As shown in the appendix (A.13), the magnitude of the values in the lower left-hand corner of

*C*is directly traceable to the input spectrum

_{kl}*X*: −5/3 implies a relatively large amount of small-scale energy in the base state, and so large-scale errors can rapidly produce downscale error growth through these interactions.

_{k}### c. Physical interpretation of the error energy Z_{k}(t) for −5/3 (SQG) and −3 (2DV) spectra

The preceding analysis suggests, somewhat counterintuitively, that the critical difference in the upscale error-growth process between the −5/3 (SQG) case (with limited predictability) and the −3 (2DV) case (with unlimited predictability) resides in the marked difference in downscale error spreading between the two models. The eigenanalysis suggests that any initial error distribution will tend to evolve to one with a maximum in the smaller scales for the −5/3 (SQG) case and in the larger scales for the −3 (2DV) case and that this difference is directly due to the role of downscale, error-energy spreading that strongly reinforces small-scale errors in the −5/3 (SQG) case.

## 5. Summary

The original idea of L69 that flows with many scales of motion may have limited predictability is widely understood, as is the recognition that L69’s use of a −5/3 spectrum in a model based on the two-dimensional vorticity equation (2DV) for synoptic-scale turbulence is unrealistic (Straus and Shukla 2005). In the present work, the L69 model is revisited and modified to apply to the surface quasigeostrophic equations (SQGs), which are mathematically very similar to 2DV and moreover, for turbulent motions forced at large scale, have a −5/3 kinetic energy spectrum at smaller scales.

The L69 stochastic model extended to SQG is shown to produce limited predictability in SQG (−5/3) and unlimited predictability in 2DV (−3) turbulence (Figs. 1 and 2), as found by L69 by varying the input spectrum for the 2DV-based stochastic model and anticipated by the simple scaling arguments of Lilly (1972). The present work shows that the basic-state spectrum is the determining factor in the error-energy evolution with the dynamical model (SQG or 2DV) playing a secondary role. The principal virtue of using the SQG model for a −5/3 spectrum and the 2DV model for a −3 spectrum is that the choice of spectrum is defensible on physical grounds.

It is shown in the predictability experiments for the −3 (2DV) case that the L69 stochastic model reproduces in all important aspects the error-energy spectrum growth obtained from a completely different spectral closure model, see Métais and Lesieur (1986, their Fig. 9) and Fig. 1b, suggesting that the important closure assumption (2.11) is at least no worse than those used in later, more sophisticated models.

The present eigenanalysis indicates that in the −3 (2DV) case the error-energy spectrum growth (Figs. 1b and 2b) may be interpreted as the projection of the initial error-energy spectrum onto the most rapidly growing spectrum (which has a large-scale maximum; Fig. 3b). In the −5/3 (SQG) case, the eigenanalysis can describe the error growth if applied in a sequence in which the most rapidly growing spectrum (which maximizes at small scales; Fig. 3a) is recalculated with the highest wavenumbers eliminated to mimic the sequential saturation of the smallest scales that occurs in the numerical solution (Fig. 1a).

Although predictability studies are fundamentally concerned with upscale spreading of errors, the side-by-side eigenanalysis of the −5/3 (SQG) and −3 (2DV) cases points to the critical importance of downscale spreading of error energy in distinguishing the −5/3 (SQG) case, with limited predictability, from the −3 (2DV) case, with unlimited predictability. Downscale error-energy spreading is much stronger in the case with a −5/3 spectrum than it is in the case with a −3 spectrum because there is much less base-state energy in the small scales in the latter. Consequently, large-scale error produces a strong downscale feedback with a −5/3 base-state spectrum, explaining the rapidly growing, small-scale-maximized error-energy spectrum; the absence of such a feedback with a −3 base-state spectrum explains the slowly growing, large-scale-maximized error-energy spectrum in that case. The importance for error growth of large-scale error interacting with small-scale features of the base state was originally noted by Thompson (1957, p. 281).

There is a vast literature on the subject of atmospheric predictability, attacking the problem on many fronts ranging from dynamical systems theory (e.g., Bohr et al. 1998) to operational models (e.g., Simmons and Hollingsworth 2002) to information theory (e.g., DelSole 2004). The present work was motivated in part by the desire to understand the original model, which justified the idea that spurred this vast literature. But in greater part, our work here was motivated by the need for an atmospherically relevant, dynamical model that is simple enough to reveal the attributes of a flow with limited predictability. It is the authors’ belief that the generalized Lorenz model described herein meets that need.

## Acknowledgments

The authors thank R. Morss and M. Waite for their comments on the first draft of this manuscript.

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,*Tellus***19****,**540–559.

## APPENDIX

### Analytical Properties of Ckl

*C*defined by the summation in (2.29) contains the factor

_{kl}*N*

^{2}

_{m}

*X*

_{m}. As remarked in footnote 5,

*X*∼ 2

_{m}^{m(1−p)}if the energy per unit wavenumber

*X*(

*K*)

*K*

^{−1}∼

*K*

^{−}

*; with the definition*

^{p}*N*= 2

_{m}*, therefore*

^{m}*m*, whereas for a −3 spectrum it is constant. Equation (A.1) reflects that

*C*fundamentally represents the advection and/or straining of errors by the basic-state flow; since both of the latter scale as a velocity divided by a length scale of the basic-state flow, it is clear these effects scale as

_{kl}*K*

*X*(

*K*)

*K*

^{3}[

*X*(

*K*)

*K*

^{−1}]

*K*

^{(3−p)/2}in the present notation. As one might suspect, (A.1) plays the determining role in producing the distinctive properties of

*C*attaching to SQG (

_{kl}*p*= 5/3) and 2DV (

*p*= 3), respectively.

The matrix *C _{kl}* also involves

*B*

_{(1)}

*and*

_{kl}*B*

_{(2)}

*, where the latter are integrals over octaves of*

_{kl}*B*(

_{j}*K*′,

*L*′, 1) defined by (2.27). Figure A1 shows

*B*(

_{j}*K*′,

*L*′, 1) for SQG [

*β*= 1 in (2.18) and (2.19)] and 2DV [

*β*= 2 in (2.18)–(2.19)]; it is clear that the

*B*(

_{j}*K*′,

*L*′, 1) do not vary much with

*β*, so the following remarks apply to either model.

*C*(

_{kl}*k*≠

*l*), (2.28)–(2.29) give

*B*

_{(1)}

_{k}_{−}

*,*

_{m}

_{l}_{−}

*(see 2.27) are defined. Thus the coefficients*

_{m}*B*

_{(1)}

_{k}_{−}

*,*

_{m}

_{l}_{−}

*may be identified with the grid boxes shown in Figs. A1a,b, and the summation (A.2) may be viewed as the sum along the diagonal string of boxes starting at*

_{m}*B*

_{(1)}

_{k}_{−1,}

_{l}_{−1}and continuing upward and leftward toward the box

*B*

_{(1)}

_{k}_{−}

*,*

_{n}

_{l}_{−}

*(an example for*

_{n}*n*= 3 is given in Fig. A1a). The region where the diagonal string of boxes crosses the region of

*B*

_{1}(

*K*′,

*L*′, 1) ≠ 0 signifies where triad interactions are possible, that is, where

*M*in the latter interval.

*C*(

_{kl}*l*≫

*k*), it is clear from Figs. A1a,b that the only nonzero coefficients in (A.2) are from the boxes represented in

*B*

_{(1)}

_{k}_{−}

_{l}_{,0}and

*B*

_{(1)}

_{k}_{−}

_{l}_{+1,1}, representing wave interactions for which

*L*′ ≈ 1,

*K*′ ≪ 1 (

*L*≈

*M*,

*K*≪

*L*) (e.g., the light gray boxes in Fig. A1a); in this limit (

*l*≫

*k*) these can be evaluated analytically. From (2.27) with the definition (2.18)

*l*≫

*k*[

*j*≪ −1 (

*J*′ ≪ 1)], the integrand is nonzero for 1 <

*L*′ < 1 +

*J*′; letting

*L*′ = 1 +

*δ*, where

*δ*≪ 1, allows (A.5) to be approximated by

*j*≪ −1

*B*

_{(1)}

_{j}_{,0}. Hence (A.7) and (A.1) show that

*l*≫

*k*, since for any combination of (

*β*,

*p*) considered herein, the quantity in parentheses is positive. For 2DV (

*β*= 2) the small values of

*B*

_{(1)}

_{j}_{,0}and

*B*

_{(1)}

_{j}_{,1}can be attributed to (A.5), which is small because the integral is over values of

*L*′ ≈ 1; the factor (1 −

*L*′

*)*

^{β}^{2}is therefore small [

*L*≈

*M*in (2.18)] as is the interval of possible triad interactions (A.3) [|

*K*−

*L*| ≈ |

*K*+

*L*| in (2.17)]. For SQG (

*β*= 1) there is an extra factor of

*J*′

^{2}in the integrand of (A.4) that makes

*B*

_{(1)}

_{j}_{,0}and

*B*

_{(1)}

_{j}_{,1}even smaller.

*C*(

_{kl}*k*≫

*l*), it is clear from Figs. A1a,b that the only nonzero coefficients in (A.2) are from the boxes represented by

*B*

_{(1)0,}

_{l}_{−}

*and*

_{k}*B*

_{(1)1,}

_{l}_{−}

_{k}_{+1}, representing wave interactions for which

*K*′ ≈ 1

*L*′ ≪ 1 (

*K*≈

*M*,

*L*≪

*M*) (e.g., the medium gray boxes in Fig. A1a); in this limit (

*k*≫

*l*), these too can be evaluated analytically. From (2.27), with the definition (2.18),

*k*≫

*l*[

*j*≪ −1 (

*J*′ ≪ 1)], the integrand is nonzero for 1 <

*K*′ < 1 +

*J*′; letting

*K*′ = 1 +

*δ*, where

*δ*≪ 1, allows (A.10) to be approximated by

*J*′

*)*

^{β}^{2}≈ 1 finally gives for

*j*≪ −1

*B*

_{(1)0,}

*. Hence (A.12) and (A.1) show that*

_{j}*k*≫

*l*. Consistent with the observation in L69 (p. 301), (A.13) shows that the coefficients in the lower left-hand corner do not depend on

*l*. Furthermore, (A.13) shows that these coefficients depend strongly on the input spectrum; for

*p*= 3 (2DV) these coefficients also become independent of

*k*, whereas for

*p*= 5/3 they increase with

*k*. In this case, although

*B*

_{1}(

*K*′,

*L*′, 1) → ∞ as

*L*′ → 0, the interval of possible triad interactions (A.3) [|

*K*−

*L*| ≈ |

*K*+

*L*| in (2.17)] goes to zero (as one can also judge from Figs. A1a,b) and consequently

*B*

_{(1)0,}

*and*

_{j}*B*

_{(1)1,}

*are finite.*

_{j}*C*,

_{s}

_{s}_{+1}, Figs. A1a,b indicate that nonzero coefficients in (A.2) will result from boxes along the diagonal up to

*m*=

*s*+ 1 (e.g., the dark gray boxes in Fig. A1a); that is,

*B*

_{(1)}

_{s}_{−}

*,*

_{m}

_{s}_{+1−}

*decrease with*

_{m}*m*.

^{A1}In view of (3.1) the factor

*N*

^{2}

_{m}

*X*

_{m}increases with

*m*[to a constant in 2DV according to (A.1)], so the maximum contribution to (A.14) is expected from the middle terms. The latter terms will generally be in the zone

*K*′ ≈

*L*′ > 1 (

*K*≈

*L*>

*M*). As the middle terms of (A.14) involve the energy-containing midsection of the base-state spectra, the near-diagonal terms are large for either SQG or 2DV (Tables 3 and 4). Similar remarks apply to

*C*

_{s}_{+1,}

*.*

_{s}*C*depend on both

_{kl}*B*

_{(1)}

*and*

_{kl}*B*

_{(2)}

*; for*

_{kl}*k*=

*l*=

*s*,

*B*

_{(2)}

*[which is of comparable magnitude to*

_{kl}*B*

_{(1)}

*] and, thus, the main diagonal is predominantly negative. That the physical origin of the diagonal terms is the same as for the near-diagonal terms can be seen by inspection of*

_{kl}*B*

_{1}(

*K*,

*L*,

*M*) (2.18) and

*B*

_{2}(

*K*,

*L*,

*M*) (2.19), which shows that they are the same for

*K*≈

*L*>

*M*.

Total error energy Σ^{n}_{k=1} *Z*_{k} for (a) SQG turbulence (−5/3 spectrum) and (b) 2DV turbulence (−3 spectrum).

Citation: Journal of the Atmospheric Sciences 65, 3; 10.1175/2007JAS2449.1

Total error energy Σ^{n}_{k=1} *Z*_{k} for (a) SQG turbulence (−5/3 spectrum) and (b) 2DV turbulence (−3 spectrum).

Citation: Journal of the Atmospheric Sciences 65, 3; 10.1175/2007JAS2449.1

Total error energy Σ^{n}_{k=1} *Z*_{k} for (a) SQG turbulence (−5/3 spectrum) and (b) 2DV turbulence (−3 spectrum).

Citation: Journal of the Atmospheric Sciences 65, 3; 10.1175/2007JAS2449.1

The most rapidly growing spectra of error energy per unit wavenumber *K*^{−1}*Z̃*(*K*) for (a) SQG (−5/3 spectrum) and (b) 2DV (−3 spectrum). All of the spectra are normalized by their respective maximums. The labels *n* = 9, 12 refer to spectra calculated with a truncation *C _{kl}* in which the wavenumbers greater than

*n*are eliminated.

Citation: Journal of the Atmospheric Sciences 65, 3; 10.1175/2007JAS2449.1

The most rapidly growing spectra of error energy per unit wavenumber *K*^{−1}*Z̃*(*K*) for (a) SQG (−5/3 spectrum) and (b) 2DV (−3 spectrum). All of the spectra are normalized by their respective maximums. The labels *n* = 9, 12 refer to spectra calculated with a truncation *C _{kl}* in which the wavenumbers greater than

*n*are eliminated.

Citation: Journal of the Atmospheric Sciences 65, 3; 10.1175/2007JAS2449.1

The most rapidly growing spectra of error energy per unit wavenumber *K*^{−1}*Z̃*(*K*) for (a) SQG (−5/3 spectrum) and (b) 2DV (−3 spectrum). All of the spectra are normalized by their respective maximums. The labels *n* = 9, 12 refer to spectra calculated with a truncation *C _{kl}* in which the wavenumbers greater than

*n*are eliminated.

Citation: Journal of the Atmospheric Sciences 65, 3; 10.1175/2007JAS2449.1

Fig. A1. The coefficients *B*_{1}(*K*′, *L*′, 1) for (a) SQG and (b) 2DV and the coefficients *B*_{2}(*K*′, *L*′, 1) for (c) SQG and (d) 2DV. Each box represents the area of the integral *B*_{(}_{j}_{)}* _{k}*,

*defined in (2.27); several of these boxes are labeled in (a) for illustration. The coordinates (*

_{l}*K*′,

*L*′) are on a logarithmic scale; the exponents in the coordinate labels (2

*, 2*

^{k}*) specify the box for the integral*

^{l}*B*

_{(}

_{i}_{)}

*,*

_{k}*. The light gray boxes indicate the coefficients involved in the calculation of*

_{l}*C*for

_{kl}*l*≫

*k*while the medium gray boxes indicate those for

*k*≫

*l*; the dark gray boxes indicate the coefficients used in the calculation of the off-diagonal element

*C*,

_{s}

_{s}_{+1}.

Citation: Journal of the Atmospheric Sciences 65, 3; 10.1175/2007JAS2449.1

Fig. A1. The coefficients *B*_{1}(*K*′, *L*′, 1) for (a) SQG and (b) 2DV and the coefficients *B*_{2}(*K*′, *L*′, 1) for (c) SQG and (d) 2DV. Each box represents the area of the integral *B*_{(}_{j}_{)}* _{k}*,

*defined in (2.27); several of these boxes are labeled in (a) for illustration. The coordinates (*

_{l}*K*′,

*L*′) are on a logarithmic scale; the exponents in the coordinate labels (2

*, 2*

^{k}*) specify the box for the integral*

^{l}*B*

_{(}

_{i}_{)}

*,*

_{k}*. The light gray boxes indicate the coefficients involved in the calculation of*

_{l}*C*for

_{kl}*l*≫

*k*while the medium gray boxes indicate those for

*k*≫

*l*; the dark gray boxes indicate the coefficients used in the calculation of the off-diagonal element

*C*,

_{s}

_{s}_{+1}.

Citation: Journal of the Atmospheric Sciences 65, 3; 10.1175/2007JAS2449.1

Fig. A1. The coefficients *B*_{1}(*K*′, *L*′, 1) for (a) SQG and (b) 2DV and the coefficients *B*_{2}(*K*′, *L*′, 1) for (c) SQG and (d) 2DV. Each box represents the area of the integral *B*_{(}_{j}_{)}* _{k}*,

*defined in (2.27); several of these boxes are labeled in (a) for illustration. The coordinates (*

_{l}*K*′,

*L*′) are on a logarithmic scale; the exponents in the coordinate labels (2

*, 2*

^{k}*) specify the box for the integral*

^{l}*B*

_{(}

_{i}_{)}

*,*

_{k}*. The light gray boxes indicate the coefficients involved in the calculation of*

_{l}*C*for

_{kl}*l*≫

*k*while the medium gray boxes indicate those for

*k*≫

*l*; the dark gray boxes indicate the coefficients used in the calculation of the off-diagonal element

*C*,

_{s}

_{s}_{+1}.

Citation: Journal of the Atmospheric Sciences 65, 3; 10.1175/2007JAS2449.1

The coefficients *C _{kl}* with a −5/3 spectrum and

*β*= 2 (2DV). These coefficients represent the spreading of error energy in octave

*l*(decreasing scale from left to right) to octave

*k*(decreasing scale from top to bottom).

The coefficients *C _{kl}* with a −3 spectrum and

*β*= 2 (2DV).

The coefficients *C _{kl}* with a “−5/3” spectrum and

*β*= 1 (SQG).

^{1}

The present recapitulation of the original L69 model is meant as a guide to, but not a substitute for, the development given in L69; for a deeper discussion the reader is advised to go to the source. For ease of comparison with the original L69 model, identical notation will be used here; extension of the L69 model to SQG will require, however, a few extra symbols.

^{4}

Defined so that ∫^{∞}_{−∞} ∫^{∞}_{−∞} *X*′(*K*_{x}, *K*_{y}) *dK*_{x} *dK*_{y} = ∫^{2π}_{0} ∫^{∞}_{0} [*X*(*K*)/2*π**K*^{2}]*K dK d**θ* = ∫^{∞}_{−∞} *X*(*K*) *d* ln*K*.

^{5}

With the definitions in L69, the energy per unit wavenumber is *X*(*K*)*K*^{−1}; if the latter ∼*K*^{−}* ^{p}*, then the equation analogous to (2.20) shows that

*X*

_{k}= ∫

^{Nk}

_{Nk−1}

*X*(

*K*)

*K*

^{−1}

*dK*∼ ∫

^{ρkN0}

_{ρk−1N0}

*K*

^{−p}

*dK*∼

*ρ*

^{−k(p−1)}.

^{6}

For *n* = 21 used in L69, the normalization constant in (3.1) is *c* ≅ 0.703, but for the *n* = 12 used in this study, *c* ≅ 0.696; in order to make a direct comparison with L69, the former value was used for all the computations herein.

^{7}

It might also be that there was a transcription error in producing the published table in L69. The lower triangle of entries in Table 1 defined by 4 ≤ *k* ≤ 9, 1 ≤ *l* ≤ 6 is nearly identical to the lower triangle of entries in L69’s Table 2 defined by 4 ≤ *k* ≤ 9, 2 ≤ *l* ≤ 7.

^{8}

This behavior is a hallmark of limited predictability in systems with many time scales [cf. Fig. 1 of Aurell et al. (1996)].

^{9}

As noted, *λ* = 5.7 for *n* = 12, and so it appears that *λ* will converge to a constant as *n* increases.

^{10}

For example, replacing the lower left-hand triangle of the −5/3 case (Table 4; SQG) with the lower left-hand triangle of the −3 (Table 3; 2DV) case produces eigenvalues and eigenvectors consistent with those found for the −3 (2DV) case.

^{A1}

It can be shown that *B*_{(1)}* _{j}*,

_{j}_{+1}∼ 2

*for*

^{j}*j*≫ 1.