We consider the statistical properties of solutions of the stochastic
fractional relaxation equation and its fractionally integrated extensions
that are models for the Earth's energy balance. In these equations, the
highest-order derivative term is fractional, and it models the energy storage processes that are scaling over a wide range. When driven stochastically, the system is a fractional Langevin equation (FLE) that has been considered
in the context of random walks where it yields highly nonstationary
behaviour. An important difference with the usual applications is that we instead consider the stationary solutions of the Weyl fractional relaxation
equations whose domain is

An additional key difference is that, unlike the (usual) FLEs – where the highest-order term is of integer order and the fractional term represents a scaling damping – in the fractional relaxation equation, the fractional term
is of the highest order. When its order is less than

Since these processes are Gaussian, their properties are determined by their second-order statistics; using Fourier and Laplace techniques, we analytically develop corresponding power series expansions for fRn and fRm and their fractionally integrated extensions needed to model energy storage processes. We show extensive analytic and numerical results on the autocorrelation functions, Haar fluctuations and spectra. We display sample realizations.

Finally, we discuss the predictability of these processes which – due to
long memories – is a

Over the last decades, stochastic approaches have rapidly developed and have spread throughout the geosciences. From early beginnings in hydrology and turbulence, stochasticity has made inroads in many traditionally deterministic areas. This is notably illustrated by stochastic parameterizations of numerical weather prediction models, e.g. Buizza et al. (1999), and the “random” extensions of dynamical systems theory, e.g. Chekroun et al. (2010).

In parallel, pure stochastic approaches have developed primarily along two distinct lines. One is the classical (integer-ordered) stochastic differential equation approach based on the Itô or Stratonovich calculus that goes back to the 1950s (see the useful review by Dijkstra, 2013). The other is the scaling strand that encompasses both linear (monofractal, Mandelbrot, 1982) and nonlinear (multifractal) models (see the review by Lovejoy and Schertzer, 2013) that are based on phenomenological scaling models, notably cascade processes. These and other stochastic approaches have played important roles in nonlinear geoscience.

Up until now, the scaling and differential equation strands of stochasticity have had surprisingly little overlap. This is at least partly for technical reasons: integer-ordered stochastic differential equations have exponential Green functions that are incompatible with wide-range scaling. However, this shortcoming can – at least in principle – be easily overcome by introducing at least some derivatives of fractional order. Once the (typically) ad hoc restriction on integer orders is dropped, the Green functions are based on “generalized exponentials” that in turn are based on fractional powers (see the review by Podlubny, 1999). The integer-ordered stochastic equations that have received the most attention are thus the exceptional, non-scaling special cases. In physics they correspond to classical Langevin equations; in geophysics and climate modelling, they correspond to the linear inverse modelling (LIM) approach that goes back to Hasselmann (1976) and later elaborated notably by Penland and Magorian (1993), Penland (1996), Sardeshmukh et al. (2000), Sardeshmukh and Sura (2009) and Newman (2013). Although LIM is not the only stochastic approach to climate, in two recent representative multi-author collections (Palmer and Williams, 2010; Franzke and O'Kane, 2017), all 32 papers shared the integer-ordered assumption (a single exception being Watkins, 2017; see also Watkins et al., 2020).

Under the title “Fractal operators”, West et al. (2003) review and emphasize that, in order to yield scaling behaviours, it suffices that stochastic
differential equations contain fractional derivatives. However, when it is
the time derivatives of stochastic variables that are fractional –
fractional Langevin equations (FLEs) – then the relevant processes are generally non-Markovian (Jumarie, 1993), so that there is no
Fokker–Planck (FP) equation describing the corresponding probabilities. Even in the relatively few cases where the FLE has been studied, the fractional
terms are generally models of viscous damping, so that the highest-order terms are still integer-ordered (an exception is Watkins et al., 2020, who mention “fractionally integrated FLE” of the type studied here but without investigating its properties). Integer-ordered terms have the
convenient consequence of regularizing the solutions, so that they are at least root mean square continuous; in this paper the highest-order derivatives are fractional, so that when the highest-order terms are

An additional obstacle is that – as with the simplest scaling stochastic model, fractional Brownian motion (fBm, Mandelbrot and Van Ness, 1968) – we expect that the solutions will not be semi-martingales and hence that the Itô calculus used for integer-ordered equations will not be applicable (see Biagini et al., 2008). This may explain the relative paucity of mathematical literature on stochastic fractional equations (see however Karczewska and Lizama, 2009). In statistical physics, starting with Mainardi and Pironi (1996), Metzler and Klafter (2000) and Lutz (2001) helped with numerics; the FLE (and a more general “Generalized Langevin Equation”, Kou and Sunney Xie, 2004; Watkins et al., 2019) has received a little more attention as a model for (nonstationary) particle diffusion (see West et al., 2003, for an introduction, or Vojta et al., 2019, for a more recent example). These technical aspects may explain why the statistics of the resulting processes are not available in the literature.

Technical difficulties may also explain the apparent paradox of continuous-time random walks (CTRWs) and other approaches to anomalous diffusion that involve fractional equations. While CTRW probabilities are governed by the deterministic fractional-ordered generalized fractional diffusion equation (e.g. Hilfer, 2000; Coffey et al., 2012), the walks themselves are based on specific particle jump models rather than (stochastic) Langevin equations. Alternatively, a (spatially) fractional-ordered Fokker–Planck equation may be derived from an integer-ordered but nonlinear Langevin equation for a diffusing particle driven by an (infinite-variance) Levy motion (Schertzer et al., 2001).

In nonlinear geoscience, it is all too common for mathematical models and
techniques developed primarily for mathematical reasons to be subsequently applied to the real world. This approach – effectively starting with a
solution and then looking for a problem – occasionally succeeds, yet
historically the converse has generally proved more fruitful. The proposal
that an understanding of the Earth's energy balance requires the fractional energy balance equation (FEBE, Lovejoy et al., 2021, announced in
Lovejoy, 2019a) is an example of the latter. First, the scaling exponent of
macroweather (monthly, seasonal, interannual) temperature stochastic
variability was determined (

In parallel, the multidecadal deterministic response to external
(anthropogenic, deterministic) forcing was shown to also obey a scaling law
but with a different exponent (Hébert, 2017; Lovejoy et al., 2017;
Procyk et al., 2020, 2022; Procyk, 2021),

In the EBE, energy storage is modelled by a uniform slab of material, implying that, when perturbed, the temperature exponentially relaxes to a new thermodynamic equilibrium. However, as reviewed in Lovejoy and Schertzer (2013), both conventional global circulation models and observations show that atmospheric, oceanic and surface (e.g. topographic) structures are spatially scaling. A consequence is that the temperature relaxes to equilibrium in a power law manner. This motivated earlier approaches (van Hateren, 2013; Rypdal, 2012; Hébert, 2017; Lovejoy et al., 2017) to postulate that the climate response function (CRF) itself is scaling. However, these models require either ad hoc truncations or imply infinite sensitivity to small perturbations (Rypdal, 2015; Hébert and Lovejoy, 2015).

The FEBE instead situates the scaling in the energy storage processes; this is the physical basis for the phenomenological derivation of the FEBE proposed in Lovejoy et al. (2021), and the zeroth-order term guarantees that equilibrium is reached after long enough times. The scaling of the basic physical quantities in both time and space motivates the study of the FEBE and its fractionally integrated extensions discussed below with temperature treated as a stochastic variable. The FEBE determines the Earth's global temperature when the energy storage processes are scaling and modelled by a fractional time-derivative term. Recently, analysis of the atmospheric radiation budget has shown that, at least over some regions, the internal component of the radiative forcing may itself be scaling: this justifies the consideration of the extensions to fGn forcing.

The FEBE differs from the classical energy balance equation (EBE) in several ways. Whereas the EBE is integer-ordered and describes the deterministic, exponential relaxation of the Earth's temperature to equilibrium, the FEBE is of fractional order, and because it is both deterministic and stochastic, it unites all the forcings and responses into a single model. Whereas the stochastic part represents the forcing and response to the unresolved degrees of freedom – the “internal variability” – and is treated as a zero mean Gaussian noise, the deterministic part represents the external (e.g. anthropogenic) forcing and the forced response modelled by the total external forcing. Complementary work (Procyk et al., 2020, 2022; Procyk, 2021) uses the deterministic FEBE as the basic model for the response to external forcing, but its Bayesian parameter estimation uses the stochastic FEBE to characterize the likelihood function of the residuals assumed to be the responses to stochastic internal forcing and governed by the same equation. It thus avoids the ad hoc error models involved in conventional Bayesian parameter estimation. The result is a parsimonious, FEBE projection of the Earth's temperature to 2100 that has much lower uncertainty than the classical global circulation model alternative. This is the first time that classical general circulation model climate projections have been confirmed by an independent, qualitatively different, approach.

An important but subtle EBE–FEBE difference is that, whereas the former is an

We have mentioned that the FEBE can be derived phenomenologically where the
fractional derivative of order

To obtain the HEBE, it is sufficient to follow the Budyko–Sellers approach but to avoid one of their key approximations. The Earth's atmosphere and ocean are driven by local imbalances in radiative fluxes. While
Budyko–Sellers models simply redirect this flux away from the Equator, the HEBE improvement (Lovejoy, 2021a, b) is to instead use the
mathematically correct radiative–conductive surface boundary conditions. When this is done in the classical energy transport equation, one obtains an
important

The choice of a Gaussian white noise forcing was made not so much for its theoretical simplicity as for its physical realism. Using scaling to divide atmospheric dynamics into dynamical ranges (Lovejoy, 2013, 2015a, 2019b), the main ones are weather, macroweather and climate. While the temperature variability in both space and time is generally highly intermittent (multifractal), there is one exception: the temporal macroweather regime (starting at the lifetime of planetary structures – roughly 10 d – up until the climate regime at much longer scales). Macroweather is the regime over which the FEBE applies, and it has exceptionally low intermittency: temporal (but not spatial) temperature anomalies are not far from Gaussian (Lovejoy, 2018). Responses to multifractal or Levy process FEBE forcings may however be of interest elsewhere.

This paper is structured as follows. In Sect. 2 we present the fractional relaxation equation, forced by a Gaussian white noise as a natural generalization of classical fractional Brownian motion, fractional Gaussian noise and Ornstein–Uhlenbeck processes (Sect. 2.1 and 2.2). When forced by Gaussian white noises, the solutions define the corresponding fractional relaxation motions (fRm) and fractional relaxation noises (fRn). We consider further extensions to the case where the equation is forced by a scaling noise fGn (Sect. 2.3, Eqs. 21 and 22). This is equivalent to considering the fractionally integrated fractional relaxation equation with white noise forcing. In Sect. 2, we first solve the equations in terms of Green's functions and then introduce powerful Fourier techniques that yield integral representations of the second-order statistics, including autocorrelations, structure functions (Eqs. 33 and 35), Haar fluctuations and spectra (with many details in Appendix A and in Appendix B, we derive the properties of the HEBE special case). In Sect. 3, we develop both short- and long-time (asymptotic) series expansions for the statistics (Eqs. 49 and 51), and we display and discuss sample fRn and fRm processes. In Sect. 4 we discuss the problem of prediction – important for macroweather forecasting – and derive expressions for the optimum predictor (Eq. 63) and its theoretical prediction skill as a function of forecast lead time (Eq. 68). In Sect. 5 we conclude.

We could note that the paper is somewhat complex due to the necessity of developing several approaches: Fourier for the main integral representations (Sect. 2), Laplace for the asymptotic expansions (Sect. 3), and real space for the predictability results (Sect. 4).

In the introduction, we outlined physical arguments that the Earth's global
energy balance could be well modelled by the fractional energy balance
equation. Taking

Since Eq. (1) is linear, by taking ensemble averages, it can be
decomposed into deterministic and random components with the former driven
by the mean forcing external to system

When

To simplify the development, we use the relaxation time

In dealing with fRn and fRm, we must be careful of various small and large

In the high-frequency limit, the derivative dominates, and we obtain the simpler fractional Langevin equation

Although it will turn out that Fourier techniques are very convenient for
calculating the statistics, there are also advantages to classical (real-space) approaches, and in any case they are needed for studying the
predictability properties (Sect. 4). We therefore start with a discussion
of Green's functions that are the classical tools for solving inhomogeneous
linear differential equations:

Integrating this equation, we find an equation for their integrals

For fGn, Green's functions are simply the kernels of the fractional integrals

For fRn, we now recall some classical results useful in geophysical
applications. First, these Green functions are often equivalently written in terms of Mittag–Leffler functions (“generalized exponentials”),

The impulse

In order to understand the relaxation process – i.e. the approach to the
asymptotic value 1 in Fig. 1 for the step response

Before proceeding to discuss the statistics of fRn and fRm processes, it is useful to make a generalization to the fractionally integrated processes:

In the Earth's radiative balance, such fractionally integrated fRn processes
arise in two physically interesting situations. The first is where the
forcing itself has a long memory – e.g. it is an fGn process. Whereas the
memory in a pure fRn process is purely from the high-frequency storage term, in this case, the forcing (the overall radiative imbalance) also contributes
to the memory, and this has important consequences for the predictability (Sect. 4). Although the solutions

The second situation where fractionally integrated fRn processes arise is
for the energy storage (even in the purely white noise forcing case). The
storage process is the difference between the forcing and the response:

The storage Green functions for the fractional relaxation equation (

In the above, we discussed fGn, fRn and their order 1 integrals fBm, fRm as well as fractional generalizations, presenting a classical (real-space)
approach stressing the links with fGn and fBm. We now turn to their statistics.

Although it is possible to follow Mandelbrot and Van Ness (1968) and derive many statistical
properties in real space, a Fourier approach is not only more streamlined,
but is also more powerful. The reason for the simplicity of the Fourier approach is that the Fourier transform (FT, indicated by the tilde) of the Weyl fractional derivative is symbolically

Now we can use the fact that the white noise

Application of Eq. (31) leads to

Since

Comparing Eqs. (33) and (35), we see that

The basic behaviour can be understood in the Fourier domain. First, putting

From Eq. (32), we may also easily obtain the asymptotic high- and low-frequency behaviours of the energy spectrum:

When

The above derivations were for noises and motions derived from differential
operators whose impulse and step Green functions had convergent

The fBm results are obtained by using the fGn step Green function (Eq. 13) in Eq. (35) with

We can now calculate the correlation function relevant for the fGn
statistics. With the above normalization,

Since fRm and fRn are Gaussian, their properties are determined by their second-order statistics, by

First, for the noises, we have

Integrating twice

For convenience, the leading terms of the normalized

For multidecadal global climate projections, the relaxation time has been
estimated at

The normalized correlation functions

For pure fRn processes, a useful formula is

Integrating

The normalized

Using the above results, we see that there are three limiting fRn

A useful statistical characterization of the processes is by the statistics
of their Haar fluctuations over an interval

Using Eq. (60), we can determine the behaviour of the root mean square (rms) Haar fluctuations; terms like

Using the results above for

The rms Haar fluctuation plots for the pure (

For the range of

It is instructive to view some samples of fRn and fRm processes (here we consider only

We must nevertheless be careful about the high frequencies since the impulse
response Green functions

In order to clearly display the behaviours, recall that when

fRn and fRm simulations (left and right columns respectively) for

Figure 6 shows three simulations, each of length

If we take the empirical relaxation scale for the global temperature to be

Figure 7 shows realizations constructed from the same random seed but for the
extended range

The same as Fig. 6 but for

Figure 8 shows simulations similar to Fig. 5a (fRn on the left, fRm on the
right), except that instead of making a large simulation and then degrading and zooming, all the simulations were of equal length (

This set of simulations is similar to Fig. 6 (

The same as Fig. 8 but for larger

The initial value for Weyl fractional differential equations is effectively
at

To deal with the small-scale divergences when

Now define the predictor for

To show that it is indeed the optimal predictor, consider the predictor
error

There are numerous skill indicators, but the most popular and easy-to-interpret definition of forecast skill is the minimum square skill score
or MSSS (

To survey the implications, let us start by showing the

The prediction skill (

Now consider the fRn skill: we will start by considering the pure (

Panels

To better understand the fGn limit, it is helpful to plot the ratio of the
fRn-to-fGn skill (Fig. 11, right column). We see even with quite small values

The ratio of (

Going beyond the

The 1-step

One-step pure (

Same as Fig. 14 except for

We may now consider the skill of the fGn-forced process (

Contour plots of the forecast skill, with

Ever since Budyko (1969) and Sellers (1969), the energy balance between the
Earth and outer space has been modelled by the energy balance equation (EBE) based on the continuum heat equation; see North and Kim (2017) for a recent
review and see Ziegler and Rehfeld (2020) for a recent regional application. It is
most commonly used as a model for the globally averaged temperature, where it is usually derived by applying Newton's law of cooling applied to a uniform
slab of material, a “box”. The resulting EBE is a first-order relaxation equation describing the exponential relaxation of the temperature to a new
equilibrium after it has been perturbed by an external forcing. Its first-order (

The resulting model relaxes to equilibrium much too quickly, so that to increase realism, it is usual to introduce a few interacting slabs
(representing for example the atmosphere and ocean mixed layer; the
Intergovernmental Panel on Climate Change recommends two such components;
IPCC, 2013). However, it turns out that these

When forced by a Gaussian white noise, the FEBE is also a generalization of
fractional Gaussian noise (fGn), and its integral (fractional relaxation motion, fRm) generalizes fractional Brownian motion (fBm). More classically, it generalizes the Orenstein–Uhlenbeck process that corresponds to the

It was the success of predictions and projections with different exponents
but the same theoretically derived empirical underlying FEBE

While the deterministic fractional relaxation equation is classical, various
technical difficulties arise when it is generalized to the stochastic case:
in the physics literature, it is a fractional Langevin equation (FLE) that has almost exclusively been considered a model of diffusion of particles starting at an origin. This requires

Although EBEs were originally developed to understand the deterministic temperature response to external forcing, the temperature also responds to
stochastic “internal” forcing. While the Earth's system variability is generally highly non-Gaussian (multifractal, Lovejoy, 2018), the temporal macroweather regime modelled here is the quasi-Gaussian exception. This
paper therefore explores the statistics of the temperature response when it
is stochastically forced by Gaussian processes, both by white noise (

A key novelty is therefore to consider the fractional relaxation equation (a FLE) forced by white and scaling noises starting from

With these stationary Gaussian forcings, the solutions are a new stationary
process – fRn (

Much of the effort was in deducing the asymptotic small- and large-scale behaviours of the autocorrelation functions that determine the statistics and in verifying these with extensive numerical simulations. An interesting
exception was the

Beyond improved monthly and seasonal temperature forecasts and multidecadal projections, the stochastic FEBE opens up several paths for future research.
One of the more promising is to apply these techniques to the spatial FEBE
and generalize it in various directions. This is a follow-up on the special value

While the FEBE has already demonstrated its ability to project future
climates, these improvements will allow for the modelling of the nonlinear
albedo–temperature feedbacks needed for modelling of transitions between different past climates. Finally, FEBE-based projections have shown that, in spite of improved computer power and algorithms, conventional GCM
approaches may be suffering from diminishing returns; the GCMs in the latest IPCC assessment (AR6, 2021) are even more uncertain: a range of 2–5.5 K

In Sect. 2.4, we derived general statistical formulae for the
autocorrelation functions of motions and noises defined in terms of Green's functions of fractional operators. Since the processes are Gaussian,
autocorrelations fully determine the statistics. While the autocorrelations
of fBm and fGn are well known, those for fRm and fRn are new and are not so
easy to deal with since they involve quadratic integrals of Mittag–Leffler functions. In this Appendix, we derive the basic power law expansions as well as large

It is simplest to start with the Fourier expression for the autocorrelation
function for the unit white noise forcing (Eq. 33). First convert the
inverse Fourier transform (Eq. 66) into a Laplace transform. For this,
consider the integral over the contour

Take

“Im” indicates the imaginary part and

When

In the case

An advantage of writing

The expansion of the integrand around

Therefore, taking the term-by-term Laplace transform and using Watson's lemma,

The first terms are explicitly

For the motions (fRm), we need the expansion of

For many applications, one is interested in the behaviour of

The series for the coefficient

The second sum needed in

If

These and the following formulae are for

Each integer term of the expansion

The fGn correlation function is given by the single

When

The

Figure A1 shows some numerical results for

For

The expression for

In the special cases

This shows the logarithm of the relative error in
the

The expansion for

We can also expand the exponential integral:

The solid line is the constant term

To understand the behaviour, Fig. A2 shows the behaviour of the coefficient of the

This shows the logarithm of the relative error in the

When

It is possible to obtain exact analytic expressions for

The starting point is the Laplace expression (Eq. A2) with

Figure B1 shows plots

Figure B1 also shows the singular small

To obtain the corresponding results for

We can also work out the variance of the Haar fluctuations:

Figure B2 shows numerical results for

The logarithmic derivative of the rms Haar fluctuations of

Mathematica code for generating sample fRn and fRm processes is available on request to the author. Analysis and simulation software is available from

No data sets were used in this article.

The author is a member of the editorial board of

Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This article is part of the special issue “Centennial issue on nonlinear geophysics: accomplishments of the past, challenges of the future”. It is not associated with a conference.

I thank Lenin Del Rio Amador, Roman Procyk, Raphaël Hébert, Cécile Penland, and Nicholas Watkins for discussions. We are also grateful for an exchange with Kristoffer Rypdal. We thank anonymous referees for suggestions, including the fifth referee for encouraging comments on the Fourier approach. This work was unfunded, and there were no conflicts of interest.

This paper was edited by Daniel Schertzer and reviewed by five anonymous referees.