1. Introduction

Now an accepted fact in dynamical meteorology, the existence of an inherent finite-time barrier in predicting atmospheric flows was first conceptually shown by Lorenz (1969). Using a simple model, he estimated the predictability limit to be slightly over two weeks—a result echoed by recent studies with real-world operational models at major numerical weather prediction centers (Buizza and Leutbecher 2015; Judt 2018; Selz 2019; Zhang et al. 2019). Although advances in probabilistic prediction make it possible to extract predictable signals beyond this limit, the extended predictability mainly results from temporal averaging of the predicted fields, together with the slowly varying components of the climate system (Buizza and Leutbecher 2015). Moreover, the loss of information in probabilistic prediction is reflected in the growth of deterministic error, and under statistically stationary conditions, saturation of the error spectrum corresponds to the predicted probability distribution matching that of the climatology, so that the loss of deterministic and probabilistic predictability are matched. Indeed, empirical evidence suggests that the decay of forecast skill behaves broadly similarly across deterministic and probabilistic predictions (Buizza and Leutbecher 2015). Thus, the study of deterministic error growth can be used to understand the mechanisms limiting the range of predictability to a finite horizon and the role of multiscale interactions in error growth (Rotunno and Snyder 2008; Durran and Gingrich 2014; Sun and Zhang 2016). Such an analysis requires averaging over multiple cases to ensure robust results, which is a somewhat different notion of predictability compared to the fully probabilistic notion used today. However, since the earlier works adopted the deterministic approach with the averaging, we shall take the same approach as we revisit their works.

In his original work, Lorenz (1969) classified fluid systems into two categories:

• those whose error at any future time can be made arbitrarily small by suitably reducing the initial error and
• those whose error at any future time cannot be reduced below a certain limit unless the initial error is zero.

Although the barotropic vorticity equation with large-scale forcing produces a steeper spectral slope of −3, and unbalanced dynamics are required to produce a spectral slope of −5/3 in more realistic models (Sun and Zhang 2016), it has been shown that predictability is determined much more by the spectral slope than by the nature of the dynamics (Rotunno and Snyder 2008). Thus, it is appropriate to use the barotropic vorticity equation to study predictability with a range of spectral slopes, recognizing that this addresses only one aspect of what limits atmospheric predictability in practice.

Closely related to this system are the incompressible 2D Navier–Stokes (2D N-S) equations, whose well posedness (existence of a unique solution to the initial-value problem that depends continuously on the initial conditions) was first rigorously shown by Ladyzhenskaya also in the second half of the twentieth century (Robinson 2001). As we will see in section 4, it is not difficult to show that well posedness implies an indefinite range of predictability in the sense of Lorenz.

The present paper aims to reconcile the difference between the inherently finite predictability result of Lorenz and the indefinite predictability corollary of Ladyzhenskaya’s proof, in the context of incompressible 2D flows. Section 2 reviews Lorenz’s argument of its inherent finite-time behavior. In section 3 we reproduce Lorenz’s numerical results and discuss the predictability in the directly simulated 2D barotropic vorticity model. An account of the well posedness and indefinite predictability of the incompressible 2D N-S equations is presented in section 4, with which we reconcile Lorenz’s result of inherently finite predictability in section 5. The major findings are summarized in section 6.

2. Lorenz’s argument of inherently finite predictability

The derivation is arguably more straightforward if the domain is the whole $\mathbb{R}$² space and the variables are Fourier transformed rather than expanded in Fourier series. We have checked that this method returns the same equation as the limiting equation of Lorenz, up to a constant multiplicative factor.

As Rotunno and Snyder (2008) mentioned, the computation of $\mathsf{C}$ involves computing integrals of nearly singular functions. We have been cautious about these integrations and have made sure that our integrations for $\mathsf{C}$ are accurate, some details of which are provided in appendix B.

To time integrate Eq. (2), it is necessary that the initial conditions for Z and W are specified. Lorenz did not explicitly give an initial condition for W, but as Rotunno and Snyder (2008) assumed W(t = 0) ≡ 0 in their predictability experiments, we shall prescribe the same for our numerical simulations (section 3a). The nonlinear effects were accounted for by removing the corresponding components of Z, W, and $\mathsf{C}$ when the error KE saturated at a particular scale, whereupon an inhomogeneous forcing term was added to the right-hand side of Eq. (2) to account for the saturated scale’s contribution to the error growth at the unsaturated scales (details available in appendix C). Time integration with the resulting lower-dimensional system was carried on, until all scales became saturated. The evolution of the error KE spectrum in time is depicted in Fig. 1 (for illustrative purposes only; this is for a spectral slope of −3 whose simulation Lorenz excluded in his original work).

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The evolution of the error KE spectrum (red; from bottom to top) in the Lorenz (1969) model, with a −3-slope background KE spectrum (appearing in this figure as a −2 slope, as the KE per unit logarithmic wavenumber is plotted instead of the KE spectral density), as in Rotunno and Snyder (2008). The initial condition for this run is Z(K = 20) = 2⁻⁴⁰, Z = 0 for all other K, and W = 0 for all K.

Citation: Journal of the Atmospheric Sciences 76, 12; 10.1175/JAS-D-19-0057.1

This result is echoed by arguments on dimensional grounds (Vallis 1985; Lilly 1990). Assuming that t_K in Eq. (3) depends only on the wavenumber k and the one-dimensional KE spectral density E(k) of the background flow, one has t_K ~ [k³E(k)]^−0.5 as this is the only way the physical units of k and E(k) can combine to give the dimension of time. With k ~ 2^K, one obtains t_K ~ 2^−βK, with β = 2/3, 1/3, and 0 for the spectral slopes of −5/3, −7/3, and −3, respectively, the same result as Lorenz’s.

3. Numerical simulations

We performed a series of numerical simulations, first on the Lorenz model (2) followed by a forced-dissipative version of the full 2D barotropic vorticity system (1), to see whether indefinite predictability is indeed achieved with a KE spectral slope of −3 as Lorenz hypothesized.

a. Lorenz’s model

Rotunno and Snyder (2008) solved for the growth of the error KE spectrum for a background spectral slope of −p where p = 3. To assess the range of predictability in Lorenz’s framework, we extended their calculations to study the relationship between K and t_K.

Having computed the matrix $\mathsf{C}$ as in Rotunno and Snyder (2008), we solved the linear matrix system (2) explicitly, that is, by writing out the general solution in terms of the eigenvalues and eigenvectors of

$\left(\begin{array}{cc}\mathbf{0}& \mathsf{I}\\ \mathsf{C}& \mathbf{0}\end{array}\right)$

and projecting the initial condition onto such an eigenspace to determine the constants of the general solution. This exact approach is a good and easy alternative to the numerical schemes used by Lorenz (1969) and Rotunno and Snyder (2008) and its extension by Durran and Gingrich (2014). Details of the solution procedure can be found in appendix C.

Figure 1 shows the evolution of the error for the −3 spectrum as in Rotunno and Snyder (2008), and Fig. 2 shows the saturation times t_K as a function of the scale K. Note that in Fig. 2 t_K is plotted instead of t_K − t_K+1 against K, but the choice makes little difference when β > 0 since if t_K − t_K+1 is proportional to 2^−βK, then so is t_K [cf. Eq. (3)]. It is clear that the saturation times t_K scale as 2^−βK with a small but positive β (0.05) along the inertial range, so that the sum in Eq. (3) is still finite for p = 3, contrary to Lorenz’s prediction. Indeed, arguing in the same way as Lorenz, our result indicates inherently finite predictability for a −3 spectrum, which is contrary to Lorenz’s hypothesis, although we acknowledge that β = 0.05 is just marginally away from the critical value of zero. We did, however, recover Lorenz’s result for the case of a −5/3 spectrum (not shown).

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Saturation times of various scales (red) for the same model run as in Fig. 1. The blue curve shows a line of fit with β = 0.05.

Citation: Journal of the Atmospheric Sciences 76, 12; 10.1175/JAS-D-19-0057.1

b. Forced-dissipative 2D barotropic vorticity equation

The difference between our value of β and Lorenz’s for the −3 spectrum deserves particular attention because it amounts to a qualitative contrast between inherently finite and indefinite ranges of predictability. To further investigate this, we performed direct numerical simulations (DNS) on this p = 3 spectrum in the form of identical-twin experiments (pairs of runs that only differ in the initial condition), and assessed the predictability following Lorenz’s methodology with necessary adaptations.

First of all, Eq. (1) had to be restricted to a doubly periodic domain and be made forced-dissipative:

${\displaystyle \frac{\partial \theta }{\partial t}}+J\left(\psi ,\theta \right)=f+d,\theta =\mathrm{\Delta }\psi .$(4)

The forcing and dissipation, however small, are necessary for generating statistically stationary KE spectra in the DNS. To generate a −3 spectral slope, following standard practice (Maltrud and Vallis 1991), forcing was applied at the large scale: f(t) was chosen to be an independent white-noise process for each 2D wavevector whose scalar wavenumber k falls in the narrow band (±10%) around k = 20. The dissipation d was a highly scale-selective hyperviscosity d ~ −Δ⁶θ.

It is worth noting that Eq. (4) would also be the vorticity form of the incompressible 2D N-S equations, that is,

${\displaystyle \frac{\partial \mathbf{u}}{\partial t}}+\left(\mathbf{u}\cdot \nabla \right)\mathbf{u}=-\nabla p+f\left(x,t\right)+\nu \mathrm{\Delta }\mathbf{u},\nabla \cdot \mathbf{u}=0,$(5)

if d were chosen to be d = νΔθ, ν > 0. We would have liked to run these DNS on the 2D N-S equations that will be discussed in section 4, but the buildup of KE at the smallest scales as a numerical artifact was so strong that we had to either increase ν—which would substantially shorten the inertial range and thus reduce the reliability of our conclusions—or choose a more scale-selective dissipation. We opted for the latter, as is standard practice in simulations of 2D turbulence (Maltrud and Vallis 1991).

We performed five pairs of identical-twin experiments on Eq. (4) by varying the random seed that generated the preperturbation (original) initial condition. Within each pair, notably, the realizations of the large-scale stochastic forcing f(t) in the control and perturbed runs were identical. The model was pseudospectral with a truncation wavenumber of k_t = 512, in which the J(ψ, θ) term was computed in the physical domain via a pair of fast Fourier transforms with the spectral dealiasing filter proposed by Hou and Li (2007). The original initial condition for each of the five cases was an already-developed homogeneous and (approximately) isotropic turbulence with a clean logarithmically corrected −3 spectrum in the inertial range (Fig. 3), which has been shown to be a more accurate description of the large-scale-forced 2D turbulent spectrum for finite inertial ranges (Bowman 1996).

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KE spectrum (averaged over the five cases) of the initial condition (red) and logarithmically corrected −3 reference spectra E(k) ~ k⁻³[log(k/k_r)]^−1/3 (k_r = 10 in black and k_r = 20 in green), where E(⋅ ⋅ ⋅) is the one-dimensional KE spectral density.

Citation: Journal of the Atmospheric Sciences 76, 12; 10.1175/JAS-D-19-0057.1

The perturbations were introduced spectrally at each of the 2D wavevectors k for a specified value of k_p = |k|. A random phase shift independently drawn from a uniform distribution was applied on a predetermined part γ ∈ [0, 1] of the spectral coefficients $\widehat{\psi }\left(\mathbf{k}\right)$ and thus $\widehat{\theta }\left(\mathbf{k}\right)$, where the hat indicates Fourier coefficients. It can be shown that γ(k_p) and $\mathbb{E}\left[E_e\left(k_p\right)\right]$, the expected value of the one-dimensional error KE spectral density at wavenumber k_p, are related by $\mathbb{E}\left[E_e\left(k_p\right)\right]=2{\gamma }^{2}E\left(k_p\right)$. By specifying the relative error $\mathbb{E}\left[E_e\left(k_p\right)\right]/E\left(k_p\right)$, we could work out γ and thus generate the perturbation fields, to which we added the original initial conditions to obtain the perturbed initial conditions.

Since our truncation wavenumber k_t = 512 corresponds to K = 9 of Lorenz’s paper, there would only be 8 values of t_K − t_K+1, among which only 4 or 5 would be in the inertial range. It would be inaccurate to determine β from such a few data points, so we have transformed Lorenz’s argument to incorporate information from all wavenumbers k, not only from the scale K as a collection of wavenumbers.

To transform the argument, recall that t_K ~ 2^−βK when β > 0, and let T(k) be the saturation time of wavenumber k. Since k ~ 2^K and both t_K and T represent saturation times, we may write T ~ k^−β and conclude that the T should vary with k as a power law if Lorenz’s results hold. The argument will break down when β becomes zero, that is, when the threshold for indefinite predictability is reached.

In this study, the perturbations were introduced at k_p = 256. The saturation threshold was chosen to be 1.315 times the KE spectral density of the control flow, or equivalently 0.6575 times the maximum permissible error energy, in accordance with Lorenz (1969). We applied sensitivity tests and found that the results are largely insensitive to the saturation threshold. Figures 4 and 5 show, respectively, the evolution of the error KE spectrum, and the saturation times T across different wavenumbers k that fit the T ~ k^−β relationship for β = 0.24, averaged over the five cases. (The five cases exhibited very similar qualitative behavior, showing that our results are robust to initial conditions, hence justifying the use of averaging to obtain smoother results.) Based on the transformed version of Lorenz’s argument, our result also suggests inherently finite predictability for a (logarithmically corrected) −3 spectrum, this time with greater confidence as β is further away from zero.

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Evolution of the error KE spectrum (magenta and blue, bottom to top) for an initial perturbation (blue dot) at k_p = 256. The magenta curves are for t = 0.3, 0.6, …, 2.7 and the blue curves are for t = 3, 6, …, 66. The background KE spectra at t = 0, 3, 6, …, 66, scaled by a factor of 2, are shown in red (top to bottom), with the reference spectra in black and green as in Fig. 3. The spectra are averaged over the five cases.

Citation: Journal of the Atmospheric Sciences 76, 12; 10.1175/JAS-D-19-0057.1

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Saturation times T at different wavenumbers k (red) for an initial error at wavenumber k_p = 256, averaged over the five cases. The black curve shows a line of fit with β = 0.24.

Citation: Journal of the Atmospheric Sciences 76, 12; 10.1175/JAS-D-19-0057.1

4. Aspects from PDE theory: The incompressible 2D Navier–Stokes equations

A very different approach to the problem of inherently finite or indefinite predictability is via use of the analytic theory of partial differential equations (PDEs). The incompressible 2D N-S Eqs. (5), where we shall drop the word “incompressible” for the remainder of the paper, are always useful as a pedagogical first step toward understanding and modeling the motion of real fluid flows in the atmosphere. As such, the analytical properties of the 2D N-S problem have been extensively studied. Building on these analytic results, we now consider their implications for predictability.

5. Reconciling the contradiction with Lorenz

At first glance, the indefinite predictability derived in section 4 seems to contradict Lorenz’s result in section 2 for any p < 3. However, we have not discussed the role of the KE spectral slope in L(T) that, as we will see in the following, reconciles the conflict.

This concept, known as “asymptotic ill posedness,” was put forward by Palmer et al. (2014) as they argued that whether the three-dimensional Navier–Stokes system is well posed is practically irrelevant to the well-established theory of inherently finite predictability. We have now extended the discussion to the 2D N-S system and given a mathematical basis to the concept in our context.

When p > 3, the second integral in relation (9) does not appear to diverge as k₀ → k₂. This means one may indeed approximate ${\Vert \nabla u_s\Vert }^2$ by the ${\Vert \nabla u_m\Vert }^2$ in relation (9) with a sufficiently large value of k₀. So would L(T) of inequality (6) be approximated without regard to the model resolution, making it possible for $\Vert w\left(t\right)\Vert \le \epsilon$ by making $\Vert w\left(0\right)\Vert$ small enough in scale and thus achieving indefinite predictability.

For models and systems with multiple inertial ranges, only the range immediately before the viscous range pertains to our argument concerning the large-k₀ behavior. This applies to the real atmosphere where p = 5/3 (Nastrom and Gage 1985). Since k_t for atmospheric models is smaller than k₂ by “at least seven or eight orders of magnitude” (Palmer et al. 2014), the crucial assumption to our discussion (k_t ≪ k₂) is satisfied and we conclude that atmospheric predictability is indeed practically inherently finite.

6. Conclusions

Half a century on since Lorenz’s pioneering result of inherently finite atmospheric predictability, we revisited his original argument by (i) rerunning his simplified model of the 2D barotropic vorticity equation, (ii) directly simulating the full model, and (iii) comparing his conclusions with the well posedness of the 2D N-S equations as proven by Ladyzhenskaya.

Although his main result—that atmospheric predictability is inherently finite because the KE spectral slope is shallower than −3—has now become an “accepted part of the canon of dynamical meteorology” (Rotunno and Snyder 2008), the details behind the conclusion were reassessed. For the −3 spectrum, we saw a substantially different β in the DNS (section 3b) than in the simplified model (section 3a), which may be an indication that the model is inadequate in simulating error growth. In both cases, nevertheless, the hypothesis of indefinite predictability (β = 0) for p = 3 based on linear extrapolation (section 2) was refuted.

The 2D N-S equations that closely relate to the 2D barotropic vorticity equation were used to address the predictability problem from a more rigorous perspective. The forced-dissipative system was shown to be indefinitely predictable regardless of the spectral slope (section 4). However, we found that p = 3 serves as a cutoff between practically inherently finite and indefinite predictability by noting how quickly the initial error has to be brought down with increasing resolution in order to maintain the bound for the error at future times (section 5). This echoes Lorenz’s original conclusions except for the borderline case p = 3 itself, in which case our result of inherently finite predictability agrees with our own computations of Lorenz’s model and the DNS.

Until recently, KE spectra in global weather forecast models had only resolved the synoptic-scale −3 range. As model resolutions start to extend into the −5/3 range, the strong constraints on the range of predictability envisaged by Lorenz will become visible (Judt 2018). However, the limits on predictability arising from initial errors on the large scales will also limit predictability in practice (Durran and Gingrich 2014), and the interplay between the two could be an interesting area to explore.

By providing another approach to attacking the problem of predictability (via the analytic theory of the 2D N-S equations), we look forward to similar results on more atmospherically relevant PDEs such as the surface quasigeostrophic equations (Held et al. 1995), and a more active contribution from mathematicians on this topic.