In Part 1, I considered the zero-dimensional heat
equation, showing quite generally that conductive–radiative surface
boundary conditions lead to half-ordered derivative relationships between
surface heat fluxes and temperatures: the half-ordered energy balance
equation (HEBE). The real Earth, even when averaged in time over the
weather scales (up to

I then consider the full inhomogeneous problem with horizontally varying diffusivities, thermal capacities, climate sensitivities, and forcings. For this I use Babenko's operator method, which generalizes Laplace and Fourier methods. By expanding the inhomogeneous space–time operator at both high and low frequencies, I derive 2D energy balance equations that can be used for macroweather forecasting, climate projections, and studying the approach to new (equilibrium) climate states when the forcings are all increased and held constant.

In Part 1, I showed that when the surface of a body exchanges heat both conductively and radiatively, its flux depends on the half-order derivative of the surface temperature (Lovejoy, 2021). This implies that energy stored in the subsurface effectively has a huge power-law memory. This contrasts with the usual phenomenological assumption notably used in box models (including zero-dimensional global energy balance models) that the order of the derivative is an integer (one) and that, on the contrary, the memory is only exponential (short). The result directly followed by assuming that the continuum mechanics heat equation was obeyed and the depth of the media was of the order of a few diffusion depths; for the Earth, this is perhaps several hundred meters. The basic result was a classical application of the heat equation barely going beyond the results that Brunt (1932) already found “in any textbook”.

A consequence was that although Newton's law of cooling is obeyed, the
temperature obeyed the half-order energy balance equation (HEBE) rather than
the phenomenological first-order energy balance equation (EBE). When applied
to the Earth, the HEBE and its implied long memory explain the success of
both climate projections through 2100 (Hebert, 2017; Lovejoy et al., 2017; Hébert et al., 2020) and macroweather (monthly, seasonal) temperature forecasts (Lovejoy et al., 2015; Del Rio Amador and Lovejoy, 2019, 2021a, b). I also considered the responses to periodic forcings, showing that surface heat fluxes and temperatures are related by a complex thermal impedance (

Although in Part 1 I discussed the classical 1D application of the heat equation to the Earth's latitudinal energy balance (Budyko–Sellers models, especially their ad hoc treatment of the surface boundary condition), the discussion was restricted to zero horizontal dimensions. In this Part 2, I first (Sect. 2) extend the Part 1 treatment to systems with homogeneous properties but with inhomogeneous forcings, first in the horizontal plane (Sect. 2.1, 2.2), then – following Budyko–Sellers – latitudinally varying on the sphere (Sect. 2.3). The homogeneous case is quite classical and can be treated with standard Laplace and Fourier techniques; it leads to the (horizontally) generalized HEBE: the GHEBE. Although the GHEBE has a more complex (space–time) fractional derivative operator that is unlike anything I know of in the literature, like the HEBE, it can nevertheless be given precise meaning via its Green's function.

In Sect. 3, I derive the inhomogeneous GHEBE and HEBE needed for applications. This is done by using Babenko's method (Babenko, 1986), which is essentially a generalization of Laplace and Fourier transform techniques. The challenge with Babenko's method is to interpret the inhomogeneous space–time fractional operators. Following Babenko, this is done using both high- and low-frequency expansions respectively corresponding to processes dominated by storage and horizontal heat transport. The long-time limit describes the new energy balance climate state that results when the forcing is increased everywhere and held fixed: for the model this corresponds to equilibrium. I also include several appendices focused on empirical parameter estimates (Appendix A), the implications for two-point and space–time temperature statistics (when the system is stochastically forced with internal variability; Appendix B), and finally (Appendix C) the changes needed to account for the Earth's spherical geometry, including the definition of fractional operators on the sphere.

In Part 1 I recalled the heat equation for the time-varying temperature
anomalies (

In Part 1, I nondimensionalized the zero-dimensional homogeneous operators by nondimensionalizing time by the relaxation time

When

We can now use the basic Laplace shift property,

For the surface we obtain simpler expressions for the diffusive impulse
and step responses (see Eq. 35, Part 1).

Since the advection term has this simple consequence, below, take

To better understand the impulse response, Fig. 1 shows this surface

The surface impulse response function (

The surface step response (time) and Dirac (space) function (

A comparison of the spatial impulse response Green's functions for
equilibrium with surface forcing via conduction only; i.e.,

Just as the zero-dimensional HEBE was derived by showing that it had the same
Green's function as the

The above shows that even with the purely classical integer-ordered
Budyko–Sellers type of heat equation, surface temperatures already obey
long-memory, half-order equations. However, it is not certain that the
classical heat equation is in fact the most appropriate model.
Straightforward generalizations to fractional heat equations, where

If

Let us now investigate the equilibrium state. Since d

The

To study the convergence to equilibrium, consider a simple model of a
surface hot spot where the forcing is confined to a unit circle,
turned on at

This is the step response in time and (circular) step in space for
conductive–radiative forcing. Lines for

The response to a unit intensity forcing in the unit circle. The
temperature as a function of nondimensional time is given for different
distances from the center: top (

Space–time contours for unit circle forcing as a function of nondimensional time (left to right) and nondimensional horizontal distance (vertical axis).

Figures 5 and 6 show the same evolution but with temperature as a function of
time for various distances (Fig. 5) and as contours in space–time (Fig. 6).
We see that equilibrium is largely established in the first two relaxation
times (here

It is helpful to clearly understand the similarities and differences between
the HEBE and the usual 1D (latitudinal) Budyko–Sellers (B–S) approach (see
the comprehensive monograph in North and Kim, 2017; see Zhuang et al., 2017, and Ziegler and Rehfeld, 2020, for recent applications and development). Since the B–S model
is on a sphere but with only latitudinal dependence, write the horizontal
transport term

In Part 1 (Sect. 3.1.1), the horizontal transport operator was expressed in terms of the transport coefficient

For the HEBE, the short- and long-time behaviors are as follows.

In order to make a more detailed comparison between the models, we can
follow North and Kim (2017), who consider a model with constant

This information can be used to estimate

A different (possibly additional) way of reconciling the estimates is to
consider the potentially large (multifractal) intermittency of the
diffusivities that introduces a strong scale effect. For example,
Havlin and Ben-Avraham (1987), Weissman (1988), and Lovejoy et al. (1998) show that in 1D, the large-scale effective thermal resistance

The thermal resistance is proportional to the inverse thermal diffusivity;
therefore, the effective HEBE diffusive transport coefficient at scale

The homogeneous heat equation in a semi-infinite domain is a classical
problem, and conductive–radiative surface boundary conditions naturally
lead to fractional-order operators: the HEBE and GHEBE. Although we have
seen that fractional operators appear quite naturally, their advantages are
much more compelling for the more realistic inhomogeneous equations relevant
for the Earth. I therefore proceed to derive the inhomogeneous HEBE and GHEBE using Babenko's method. The more usual application is to find the surface heat flux given a solution to the conduction equation (see, for example, Magin et al., 2004; Chenkuan and Clarkson, 2018), although the following application appears to be original. In the inhomogeneous case with

The first step in Babenko's method (see, e.g., Podlubny, 1999; Magin et al., 2004)
is to factor the differential operator as follows.

The final step is to use the fact that the conductive heat flux

By comparing this derivation with that of the homogeneous GHEBE via the
classical Laplace–Fourier transform method (Sect. 2.1), it is clear that
Babenko's method is very similar but is more general. Whereas in the
homogeneous equation, the transforms reduce the derivative operations
to algebra, the difficulty with Babenko's method is to find proper
interpretations of the fractional operators. However, in the above, I assumed that

Empirical estimates of the parameters used in this paper; see Appendix A for details.

Before discussing the inhomogeneous GHEBE, consider the case in which the
horizontal term

The FEBE is a linear differential equation that can be solved using Green's
functions (Miller and Ross, 1993; Podlubny, 1999). The solution is

The corresponding step response

The FEBE and the HEBE are examples of fractional relaxation equations; these
have primarily been discussed in the context of deterministic forcings that
start at

To understand the noise-driven HEBE, it is helpful to Fourier-analyze it
using

Alternatively, in real space, if

At global scales, the high- and low-frequency HEBE behaviors are close to
observations. For example, the global value

Although the HEBE was derived for anomalies, these were not defined as small
perturbations but rather as time-varying components of the full solution of
the temperature (energy) equation with the time-independent part corresponding to the climate state. The only point at which

An important feature of fractional differential operators is that they imply long memories; this is the source of the skill in macroweather forecasts (Lovejoy et al., 2015; Del Rio Amador and Lovejoy, 2019). The fractional term with the long memory corresponds to the energy storage process. In contrast, Lionel et al. (2014) introduced a class of ad hoc energy balance models with memory (EBMM) whose (nonfractional) time derivative depends on integrals over the past state of the system.

The HEBE is the GHEBE limit where horizontal transport effects due to the horizontal divergence of heat fluxes are dominated by temporal relaxation processes and are ignored. Although this spatial scale depends on the timescale, Appendix A estimates that at monthly timescales, this spatial scale is less

Considering the spatial part of the fractional operator, we see that it is
weighted by the effective transport velocity

Keeping only the spatial terms leading in the small parameter

The HEBE applies to timescales sufficiently short and to spatial scales
sufficiently large that the horizontal temperature fluxes are too slow to be
important, and they are neglected. The first-order correction (Eqs. 57, 58)
makes a small improvement by giving a more realistic treatment of the small-scale horizontal transport. However, a long time after performing a step
increase in the forcing, the time derivatives vanish and a new climate state
is reached. If the temperature followed the pure HEBE, the spatial
equilibrium temperature distribution would be determined by setting the HEBE
time derivative to zero:

To understand the long-time behavior, return to the GHEBE but perform a
(long-time) binomial expansion of the half-order operator assuming that the
transport terms dominate.

Applying Eq. (62) to the case

To obtain an approximate solution, let us now assume that

The horizontal heat flux divergence redistributes the energy fluxes locally,
but since the GHEBE is linear, it should not affect the overall (global)
energy balance. Let us check this by direct calculation of the globally
averaged temperature. Averaging Eq. (67), we obtain

Up until now, at macroweather and climate scales, the Earth's energy balance has been modeled using two classical approaches. On the one hand, Budyko–Sellers models assume the continuum mechanics heat equation, classically yielding a 1D latitudinally varying climate state. On the other hand, there are the zero-dimensional box models that combine Newton's law of cooling with the assumption of an instantaneous temperature–storage relationship. Both models avoid the critical conductive–radiative surface boundary condition. The former ignores heat storage, redirecting radiative imbalances meridionally away from the Equator; the latter postulates a surface heat flux that is not simultaneously consistent with the heat equation and energy conservation across a conducting and radiating surface (Part 1).

This two-part paper re-examined the classical heat equation with classical
semi-infinite geometry. In the horizontally homogeneous case (Part 1,
Lovejoy, 2021), the fundamental novelty is the treatment of the
conductive–radiative boundary conditions; here (Part 2), it is the use of
Babenko's method to extend this to the more realistic horizontally
inhomogeneous problem. In both cases, the semi-infinite subsurface geometry
is only important over a shallow layer of the order of the diffusion depth
where most of the storage occurs (roughly estimated as

The most robust result was obtained by using standard Laplace and Fourier techniques. It was shown quite generally that the surface temperatures and heat fluxes are related by a half-order derivative relationship. This means that if Budyko–Sellers models are right in that the continuum mechanics heat equation is a good approximation of the Earth averaged over a long enough time, a consequence is that the energy stored is given by a power-law convolution over its past history. This is a general consequence of the conductive–radiative surface boundary conditions in semi-infinite geometry and is very different from box models that assume that the relationship between the temperature and heat storage is instantaneous. Although the system itself is classical, this result may be viewed as a nonclassical example of the Mori–Zwanzig mechanism in which system parameters that are not modeled explicitly (here, the subsurface temperatures) imply long (power-law) memories for the modeled parameters (here, the surface temperatures). This is in contrast to the conventional short-memory (exponential) assumption. It implies that any part of the Earth system that exchanges energy both radiatively and conductively into a surface should be modeled with fractional rather than integer-ordered derivatives. A far-reaching consequence is that classical dynamical systems approaches based on integer-ordered differential equations are not necessarily pertinent to the climate system.

If we ignore the horizontal divergence of heat flux (Part 1), an immediate consequence of half-order storage is that the temperature obeys the half-order energy balance equation (HEBE) rather than the classical first-order EBE. The HEBE is a special case of the fractional EBE (FEBE) that can be obtained either by replacing the classical continuum mechanics heat equation by the fraction heat equation (discussed elsewhere) or derived phenomenologically by assuming scale invariance of the energy storage mechanisms (Lovejoy, 2019b; Lovejoy et al., 2021). Depending on the space–time statistics of the anomaly forcing, the HEBE justifies current-based macroweather (monthly, seasonal) temperature forecasts (Lovejoy et al., 2015; Del Rio Amador and Lovejoy, 2019, 2021a, b) that are effectively high-frequency approximations to the FEBE. Similarly, the low-frequency (asymptotic) power-law part can produce climate projections with significantly lower uncertainties than current general circulation model (GCM)-based alternatives (Hebert, 2017; Hébert et al., 2021) and work in progress directly using the HEBE (Procyk et al., 2020).

When the system is periodically forced, the response is shifted in phase, and borrowing from the engineering literature, the surface is characterized by a complex thermal impedance that we showed is equal to the (complex) climate sensitivity. In Part 1, I gave evidence that this quantitatively explains the phase lag (typically of about 25 d) between the annual solar forcing and temperature response.

In this second part, I investigated the consequences of the divergence of horizontal heat flux, first in a homogeneous medium with inhomogeneous forcing on a plane and then on the sphere (Sect. 2), thus permitting a direct comparison with the usual Budyko–Sellers approach. In Sect. 3 I considered inhomogeneous material properties (including variable diffusion lengths, relaxation times, and climate sensitivities). While Laplace and Fourier techniques can still be used in time, they cannot be used in space. However, the extension to inhomogeneous media was nevertheless possible thanks to Babenko's powerful (but less rigorous) operator method. Whereas in Part 1, the homogeneous fractional space–time operator was given a precise meaning, here – following Babenko – the corresponding inhomogeneous operator was interpreted using binomial expansions for both the short- and long-time limits, yielding 2D energy balance models. Part 2 thus allows energy balance models to be extended to 2D, allowing the treatment of regional temporal anomalies.

The expansions depend both on the space scale and timescale as well as on a dimensional parameter: the typical horizontal transport speed (

The EBE and HEBE are the

As a final comment, I should mention that although this paper focused on the time-varying anomalies with respect to a time-independent climate state, this approach opens the door to new methods for determining full 2D climate states (generalizations of the 1D Budyko–Sellers type of climates) but also to determining past and future climates as well as the transitions between them. This is because the definition of temperature “anomalies” is very flexible. For example, we could first apply the method to determining the existing climate by fixing the forcing at current values and solving the time-independent transport equations. Then, the long-term effect of changes such as step function increases in forcing could be determined from the GHEBE anomaly equation (Sect. 3.5), which regionally corrects the local climate sensitivities for (slow) horizontal energy transport effects. Nonlinear effects that can be modeled by temperature-dependent forcings (i.e.,

In order to apply our results to the Earth, we need some idea of the
magnitudes of various terms in the equations. To start with, recall that the model is of the Earth system at macroweather and climate timescales; i.e., all relevant quantities are averaged over weather scales of

Probably the most important aspect is to estimate the relative importance of
the temporal relaxation (and storage) terms

For judging the relative importance of transport to storage, take their
ratio

Hence, the meridional contributions to the ratios

Parameter estimates from Part 1 (Sect. 3.1.2); see Sect. 2.3 for some planetary-scale estimates.

The rms fluctuations (at

Table A1 summarizes the dimensional and nondimensional parameter estimates;
the final step is to estimate values of the gradient and Laplacian terms
(Eq. A6). Since

The fluctuations are Haar fluctuations, but because

Since the north–south gradients are much stronger than the east–west ones,
it is sufficient to estimate the gradients and Laplacians by using only the

Using

Alternatively, we could estimate the timescales needed so that the critical
transport scale is 1000 km. From the same equations, I obtain estimates of
300 years (advection) and 30 000 years (diffusion). Note, however, that in the Anthropocene, for periods

In summary, from Eq. (A10), I conclude that for the larger scales

The temperature anomaly cross-correlation function (a matrix when the
temperature is discretized on a grid) is commonly used in climate science,
notably to determine empirical orthogonal functions (EOFs). These can be
determined from the HEBE (or GHEBE if needed) once a forcing model is given.
Let us first consider the climate sensitivities and relaxation times
to be deterministic characterizations of the local properties at points

The simplest model is to take spatial correlations obtained by temporally
averaging following a step function (

A convenient model of pure internal variability is to assume that the
forcing is statistically stationary in time with the following forcing
cross-correlations.

The easiest way to relate

The basic Laplace transforms in Eq. (B16) can be expressed in terms of higher
mathematical functions as follows (all for

The special case

At long enough timescales, the spatial transport of heat is important and the spherical geometry of the Earth must be taken into account. The standard way (see Sect. 2.3 and the reviews in North et al., 1981; North and Kim, 2017) is to use spherical harmonics. In Appendix 5D of Lovejoy and Schertzer (2013) these were used to define fractional integrals on the sphere, which is necessary in order to produce the corresponding multifractal cloud and topography models (see also Landais et al., 2019). Spherical harmonics are particularly convenient when the heat transport is diffusive, involving fractional Laplacians. In Sect. 3.5.2, these were defined in real space by taking the domain of integration to be a sphere. In this Appendix I discuss an alternative method of spherical fractional integration that may have theoretical and practical advantages.

The Laplacian on a sphere (

The spherical harmonics form a complete orthogonal basis so that any
function

The definition of the fractional Laplacian (Eqs. C5, C6) is adequate when
the horizontal transport coefficients are constant, but in Sect. 3.5 we
saw that, more generally, the half-order divergence operator was written as

A method of fractionally integrating the mean (

The code for the Haar fluctuation analysis (Appendix A) is available at

The data used in Appendix A are from the NOAA website:

The author declares that there is no conflict of interest.

I acknowledge discussions with Lenin Del Rio Amador, Roman Procyk, Raphael Hébert, Dave Clarke, and Cécile Penland.

This paper was edited by Anders Levermann and reviewed by two anonymous referees.