The original Budyko–Sellers type of 1D energy balance models (EBMs) consider the Earth system averaged over long times and apply the continuum mechanics heat equation. When these and the more phenomenological box models are extended to include time-varying anomalies, they have a key weakness: neither model explicitly nor realistically treats the conductive–radiative surface boundary condition that is necessary for a correct treatment of energy storage.

In this first of a two-part series, I apply standard Laplace and Fourier techniques to the continuum mechanics heat equation, solving it with the correct radiative–conductive boundary conditions and obtaining an equation directly for the surface temperature anomalies in terms of the anomalous forcing. Although classical, this equation is half-ordered and not integer-ordered: the half-order energy balance equation (HEBE). A quite general consequence is that although Newton's law of cooling holds, the heat flux across surfaces is proportional to a half-ordered (not first-ordered) time derivative of the surface temperature. This implies that the surface heat flux has a long memory, that it depends on the entire previous history of the forcing, and that the temperature–heat flux relationship is no longer instantaneous.

I then consider the case in which the Earth is periodically forced. The classical case is diurnal heat forcing; I extend this to annual conductive–radiative forcing and show that the surface thermal impedance is a complex valued quantity equal to the (complex) climate sensitivity. Using a simple semi-empirical model of the forcing, I show how the HEBE can account for the phase lag between the summer maximum forcing and maximum surface temperature Earth response.

In Part 2, I extend all these results to spatially inhomogeneous forcing and to the full horizontally inhomogeneous problem with spatially varying specific heats, diffusivities, advection velocities, and climate sensitivities. I consider the consequences for macroweather (monthly, seasonal, interannual) forecasting and climate projections.

Ever since Budyko (1969) and Sellers (1969) proposed a simple model describing the exchange of energy between the Earth and outer space, energy balance models (EBMs) have provided a straightforward way of understanding past, present, and possible future climates. The models usually have either zero or one spatial dimension, respectively representing the globally or latitudinally averaged meridional temperature distribution (for a review, see McGuffie and Henderson-Sellers, 2005, and North and Kim, 2017).

The fundamental EBM challenge is to model the way that imbalances in incoming shortwave and outgoing longwave radiation are transformed into changes in surface temperatures. In an energy-balanced climate state, the vertical flux imbalances are transported horizontally. Here I am primarily interested in the anomalies with respect to this state. When an external flux (forcing) is added, some of this anomalous imbalance is radiated to outer space, while some is converted into sensible heat and conducted into (or out of) the subsurface. This latter flux accounts for both energy storage and surface temperature changes as well as attendant changes in longwave emissions. EBMs avoid explicit treatment of this critical surface boundary condition, treating it phenomenologically in ways that are flawed; in this two-part paper, I show how they can easily be improved with significant benefits: first, the (idealized) homogeneous case (Part 1) and then the general horizontally inhomogeneous (2D) case (Part 2; Lovejoy, 2021).

First, consider box EBMs with zero horizontal dimensions as a model of the mean
Earth temperature. These are based on two distinct assumptions, namely (a) that the
rate that heat (

The basic physical problem is that anomalous radiative flux imbalances partly lead to heat conduction fluxes into the subsurface and partly to changes in longwave radiative fluxes. The part conducted into the subsurface is stored and may re-emerge, possibly much later. Starting with the heat equation, realistic and mathematically correct treatments involve the introduction of a vertical coordinate and the use of conductive–radiative surface boundary conditions (BCs). If one considers the horizontally homogeneous 3D problem in a semi-infinite medium with these mixed BCs and linearized longwave emissions, the problem is classical and can be straightforwardly solved using Laplace and Fourier techniques. Mathematically it turns out that the key is the surface layer that defines the surface vertical temperature gradient. The influence of the subsurface is only over a thin layer of the order of a few diffusion depths where most of the energy storage occurs. This depth depends on the specific heat per volume and the diffusivity, and it is estimated to be typically of the order of 100 m for the ocean (depending on its turbulent diffusivity) and less over land (see Appendix A, Part 2).

The exact treatment of this homogeneous problem confirms that Newton's law
of cooling holds but shows that the classical box model relation between
heat flux and the surface temperature is wrong: symbolically the correct
relation is

Half-order derivatives have appeared in heat and diffusion problems since at
least the 1960s (Meyer, 1960; Oldham and Spanier, 1972; Oldham, 1973; Oldham and Spanier, 1974). An equation
mathematically identical to the homogeneous

More generally, fractional derivatives and their equations (Podlubny, 1999) have a history going back to Leibniz in the 17th century, and their development has exploded in the last decades (for books on the subject, see, e.g., Miller and Ross, 1993; Podlubny, 1999; Hilfer, 2000, West et al., 2003; Tarasov, 2010; Klafter et al., 2012; Klafter et al., 2012, Baleanu et al., 2012; Atanackovic et al., 2014).

Interestingly, the explicit or implicit application of fractional derivatives to model the Earth's temperature – and more recently the energy budget – has several antecedents arising from the wide range of spatial scaling symmetries of atmospheric fields respected by the fluid equations, models, and (empirically) the atmospheric fields themselves (see the reviews in Lovejoy and Schertzer, 2013; Lovejoy, 2019a). Since this includes the velocity field – whose spatial scaling implies scaling in time – it implies that power laws should be more realistic than exponentials. At first, this led to power-law climate response functions (CRFs) (Rypdal, 2012; van Hateren, 2013; Rypdal and Rypdal, 2014; Rypdal et al., 2015; Hebert, 2017; Hébert et al., 2021). However, without truncations, pure power-law CRFs lead to divergences: the “runaway Green's function effect” (Hébert and Lovejoy, 2015), whereby a model is unstable to infinitesimal step function increases in forcing and the equilibrium climate sensitivity is infinite. These can be tamed by a high-frequency truncation (Hebert, 2017; Hébert et al., 2021) or avoided by constraining forcings to return to zero (Rypdal, 2016; Myrvoll-Nilsen et al., 2020).

However, Lovejoy (2019a) and Lovejoy et al. (2021) argued that it is not the
CRF itself but rather the Earth's heat storage mechanisms that respect the
scaling symmetry. This hypothesis implies that the corresponding storage
(the derivative term) in the energy balance equation (EBE) is of fractional
rather than integer order: the fractional energy balance equation (FEBE).
Denoting the order of the derivative term in the equation by

In spite of empirical and theoretical support, the FEBE is essentially a
phenomenological global model; in this paper I show how – at least for the

This work is divided into two parts. The first part is classical; it focuses on the homogeneous heat equation, pointing out the consequence that with semi-infinite geometry (depth) and with (realistic) conductive–radiative boundary conditions, the surface temperature satisfies the homogeneous HEBE. I relate this to the usual box models, Budyko–Sellers models, and classical diurnal heating models including the notions of thermal admittance and impedance as well as complex climate sensitivities that are useful in understanding the annual cycle. I underscore the generality of the basic (long-memory) storage mechanism. The second part extends this work to the horizontal, first to the homogeneous case (but with inhomogeneous forcing, including a direct comparison with the classical latitudinally varying 1D Budyko–Sellers model on the sphere) and then – using Babenko's method – to the general inhomogeneous case. Part 2 also contains several appendices that discuss empirical parameter estimates, spatial statistics useful for empirical orthogonal functions, and understanding the horizontal scaling properties as well as the changes needed to account for spherical geometry.

In most of what follows, the Earth's spherical geometry plays no role, and I
use Cartesian coordinates with the

Let us start with energy transport by diffusion: Fick's law

At the surface, there is an incoming energy flux

As usual,

A schematic diagram showing the correct 3D energy balance equations
with conductive–radiative surface boundary conditions.

Now decompose the heat flux and temperature into time-independent
(climatological) and time-varying (anomaly) components:

The incoming radiation at the location

Now, in the temperature equation (Eq. 3), replace

The separation into one equation for the time-invariant climate state and another for the time-varying anomalies is done for convenience. As long as the outgoing longwave radiation is approximately linear over the whole range of temperatures (as is commonly assumed in EBMs), this division involves no anomaly smallness assumptions or assumptions concerning their time averages; the choice of the reference climate depends on the application. Below, I choose anomalies defined in the standard way (although not necessarily with the annual cycle removed; Sect. 3.3); this is adequate for monthly and seasonal forecasts as well as 21st-century climate projections. However, a different choice might be more appropriate for modeling transitions between different climates including possible chaotic behaviors.

In order to simplify the problem, starting with Budyko (1969) and
Sellers (1969), the usual approach to obtaining

A schematic diagram showing the Budyko–Sellers 1D energy balance equation obtained by latitudinal averaging and by redirecting the vertical imbalance away from the Equator.

To see how this works, return to Eq. (4) for the climatological component and
put

The usual Budyko–Sellers type of models then average

In the more popular Seller's version, the basic horizontal transport is due
to the eddy thermal diffusivity, the

The final step to obtaining the energy equation is to take the divergence:

No matter how the climate temperature equation is solved, the equation for
the time-dependent anomaly temperature remains as in Eq. (14). I now rewrite it in
a way that brings out the critical mathematical properties. Since

The roles of the various terms are clearer if the equation is
nondimensionalized. For this, note that if we include the boundary
conditions, the anomaly temperature is entirely determined by the
dimensional quantities

In order to understand the classical origin of fractional derivatives, it is
helpful to consider the homogeneous Sellers-type (diffusive transport) heat
equation, where

Using the dimensional parameters in Eqs. (20) and (21), we can write the equations
as

If the advection is chosen appropriately (as in Eq. 24 below), then we may write
the horizontal transport operator in the form

Before proceeding, it is useful to get a feel for typical values of the
parameters in the equations. In Sect. 2.3 and Appendix A of Part 2, I
combine these parameter estimates with analyses of monthly space–time
temperature anomalies in order to analyze which terms in the equations are
dominant at different timescales; the following are order-of-magnitude
estimates. The basic parameters are the horizontal diffusivity

Volumetric specific heat

Climate sensitivity

Relaxation time

Horizontal diffusivity

Vertical diffusivity

Diffusion depth

Diffusion length

Diffusive-based velocity parameter

Nondimensional advection velocity

With

Performing a Laplace transform (LT) of the heat equation, we obtain

Taking the inverse Laplace transform of Eq. (33) we obtain the integral
representation:

For the surface, the integral (Eq. 34) can be expressed with the help of
higher mathematical functions.

For long times after an impulse, the response

The nondimensional temperature as a function of nondimensional time
for various nondimensional depths with a step forcing:

Contours of nondimensional temperature as a function of
nondimensional time and depth after a step function forcing (

For future reference, I give the corresponding step response

If we take this as a model of the global temperature, we can use the ramp
Green's function to estimate the ratio of the equilibrium climate response
(ECS) to the transient climate response (TCR); we find TCR

It is interesting to compare this with the classical surface boundary
condition when the system is forced by the surface temperature; an
alternative – periodic surface heat forcing – is discussed in Sect. 3.3.
If the surface (

These classical Green's functions provide useful comparisons with the
conductive–radiative BCs. For example, integrating Eq. (34) with respect to
time and simplifying, we obtain the following.

The difference

Let us now introduce the

Most applications of fractional derivatives are for forcings that start at

This HEBE for the surface temperature could be regarded as a significant nonclassical example of the Mori–Zwanzig formalism (Gottwald et al., 2017; Mori, 1965; Zwanzig, 1973, 2001) and empirical model reduction formalisms (Ghil and Lucarini, 2020), whereby memory effects arise if we only look at one part of the system, ignoring the others. In the HEBE, the surface temperature is analogously expressed directly in terms of the forcing, ignoring the subsurface degrees of freedom. Although such memories are usually considered exponential and hence small, the HEBE shows that the classical continuum heat equation has, on the contrary, strong power-law memories. This points to serious limitations on conventional dynamical systems approaches to climate science that assume that the dynamical equations are integer-ordered with exponential memories. The HEBE shows that the (fundamental) radiatively exchanging components of the climate system will generally be characterized by long memories associated with fractional rather than integer-ordered derivatives. I develop this insight elsewhere.

Phenomenological models of the temperature based on the energy balance across a homogeneous surface may represent either the whole Earth or only a subregion. The former are global zero-dimensional energy balance models (sometimes called global energy balance models or GEBMs; see the review in McGuffie and Henderson-Sellers, 2005), whereas the latter may represent the balance across the surface of a homogeneous subsection: a “box”. The boxes have spatially uniform temperatures that store energy according to their heat capacity, density, and size. Often several boxes are used, mutually exchanging energy, and the basic idea can be extended to column models. Since the average Earth temperature can be modeled either as a single horizontally homogeneous box or by two or more vertically superposed boxes, in the following, “box model” refers to both global and regional models.

A key aspect of these models is the rate at which energy is stored and at
which it is exchanged between the boxes. Stored heat energy is transferred
across a surface, and it is generally postulated that its flux obeys Newton's
law of cooling (NLC). The NLC is usually only a phenomenological model; it
states that a body's rate of heat loss is directly proportional to the
difference between its temperature and its environment. In these
horizontally homogeneous models, it is only the heat energy per area (

A schematic showing Newton's law of cooling (NLC) that relates the
temperature difference across a surface to the heat flux crossing the
surface,

While the HEBE and box models both obey the NLC, the relationships between
the surface heat flux

The HEBE and box heat transfer models can conveniently be compared and
contrasted by placing them both in a more general common framework. Define
the

Over the interval

At a theoretical level, the advantage of the HEBE is that unlike, the box
models, it is a direct consequence of the standard (energy-conserving)
continuum heat equation combined with standard energy-conserving surface
boundary conditions. It is therefore natural to ask if the

To summarize, we are currently in the unsatisfactory position of having zero-
and one-dimensional (box and Budyko–Sellers) energy balance equations,
neither of which satisfy the correct radiative–conductive surface boundary
conditions. For the box models, the consequence is that the energy storage
processes have rapid (exponential) rather than slow (power-law) relaxation.
For the Budyko–Sellers models, the consequence is that, at best, they are 1D,
and even with this restriction, their time-dependent versions have
derivatives of the wrong order (Part 2, Sect. 2.3). In comparison, the
zero-dimensional HEBE is a consequence of correcting the Budyko–Sellers
boundary conditions. It satisfies the NLC and corrects the order

Up until now, I have discussed forcing that is turned on at

The first applications of Fourier techniques to the problem of radiative and
conductive heat transfer into the Earth was by Brunt (1932) and
Jaeger and Johnson (1953), who considered the (weather regime) diurnal cycle. I
already mentioned that Brunt (1932) also considered step function heat
forcing, which he claimed might be a plausible model of the diurnal cycle
near sunset or sunrise. However, in zero-dimensional models, the long-time
temperatures after step heat flux forcings are divergent (but not in 2D
models; see Part 2) so that later in his paper Brunt considered periodic
diurnal heat flux forcing with no net heat flux across the surface and used
Fourier methods instead. In this classical diurnally forced problem, the
periodic temperature response lags the forcing by a phase shift of

Following Brunt (1932) and Jaeger and Johnson (1953), consider the response
to a single Fourier component forcing (this is equivalent to Fourier
analysis of the equation). In this case, assuming a periodic temperature
response and substituting this into the 1D heat equation (time
and depth, i.e., the dimensional version of Eq. 22), we find that the
variation of amplitude with depth is

So far, this approach has only been applied to weather scales (the diurnal cycle). Let us now apply the same approach but with an eye to longer macroweather timescales, notably the annual cycle. The climate sensitivity is an emergent macroweather quantity determined by numerous feedbacks that over weather scales are quite nonlinear but over macroweather scales are considerably averaged (and, at least for GCMs, are already fairly linear; Hébert and Lovejoy, 2018). In any event, for the annual cycle we use radiative–conductive boundary conditions rather than the pure conductive ones used by Brunt.

Using conductive–radiative surface BCs with external forcing

Note that I could have deduced Eq. (59) directly by Fourier analysis of the
HEBE using

The temperature phase lag (in months, the negative of argument of
the complex climate sensitivity) using the complex climate sensitivity and
annual cycle forcing (i.e., with

Same as Fig. 7 except for the amplitude of the complex climate
sensitivity to annual cycle forcing (i.e., with

Figures 7 and 8 compare the phases and amplitudes of

In contrast, if the Brunt (1932) classical heat forcing result is used,
we obtain

Although a complete analysis with modern data is out of our present scope,
we can get a feel for the realism of this approach by using the zonally
averaged (North and Coakley, 1979) Sellers model discussed in the review
(North et al., 1981, updated in North et al., 1983) wherein most of the
Earth follows the EBE phase lags of

Before continuing, recall that the zero-dimensional theory discussed here
assumes that radiative flux imbalances are all stored; it ignores the
divergence of the horizontal heat transport, which according to
Trenberth et al. (2009) is small even though the heat fluxes may be
significant. Although, at least for temperature anomalies, I argue that this
effect is mostly important at small scales, the magnitude of horizontal heat
divergence at macroweather scales is not well known and is presumably quite
variable from place to place depending on (inhomogeneous) local transport
parameters (see Part 2). A simple way to parameterize the transport is to
maintain the assumption that the Earth has homogeneous parameters and to
assume that the transport is due to horizontally inhomogeneous forcing. In
Part 2, I show that for a horizontal wavenumber

From the data in Table 1 of North et al. (1981), we may deduce the following.

Now use these data to estimate the climate sensitivity, relaxation time

The static climate sensitivity

This first paper of two parts proposes a new 2D energy balance equation for macroweather scales: 10 d and longer. It follows the classical energy balance models pioneered by Budyko (1969) and Sellers (1969) and assumes that the dynamics can be adequately modeled by the continuum mechanics heat equation – by advection and diffusion. As reviewed in McGuffie and Henderson-Sellers (2005) and North and Kim (2017), the classical models treat the parts of the atmosphere and ocean that radiatively interact with outer space as a zero-thickness, two-dimensional surface. The complex radiative processes that occur in the vertical direction are only treated implicitly. The dimensionality is then further reduced by zonal averaging.

While this original time-independent model may be reasonable for the long-term (time-invariant) climate states, it is inadequate for treating time-varying anomalies. The key improvement in realism was by made explicitly
introducing a vertical coordinate

In this first part, I considered only homogeneous zero-dimensional models. These are completely classical, yet as far as I know, they have not been solved with conductive–radiative (linearized) boundary conditions. Using standard Laplace and Fourier techniques, I solved the full depth–time heat equation and showed that its Green's function was identical to a half-order fractional differential equation that directly gives the surface temperature. Although half-order derivatives have occasionally been used in the context of the heat equation (at least since Oldham and Spanier, 1972, 1974; Babenko, 1986), the resulting half-order energy balance equation (the HEBE) is apparently new. Mathematically, the result is a direct consequence of the heat equation, the semi-infinite medium, and conductive–radiative surface boundary conditions.

The consequences are surprisingly far-reaching. For example, the familiar
integer-ordered differential equations have exponential Green's functions and
short memories. In contrast, the more general fractional-ordered equations
such as the HEBE have Green's functions that are generalized
exponentials based on power laws and long memories. A general consequence
is that while the HEBE respects Newton's law of cooling – i.e., that heat
fluxes across a surface are proportional to temperature differences –
the relationship between this heat flux and the surface temperature is quite
different: it involves a half-order derivative rather than a first-order one.
The energy stored is no longer instantaneously determined by the surface
temperature, but rather by the entire prior forcing history. Irrespective of
the details, we thus

I also obtained general results for the Earth's response to periodic
forcings. Ever since Brunt (1932), Fourier techniques have used the heat
equation to model the Earth's temperature response when subjected to a
diurnal heat flux forcing. I extended this from the weather regime to
the macroweather regime and from diurnally periodic heat forcing to annually
periodic radiative–conductive forcing. An immediate consequence is that
the surface thermal impedance – equal to the climate sensitivity – is a
complex number whose phase determines the lag between the maximum of the
forcing (shortly following the summer solstice) and the temperature maximum.
Using a simple latitudinally averaged model with empirical parameters, I
estimated this complex climate sensitivity and showed how this could readily
account for the observed 22–25 d lag, estimating the (static) climate
sensitivity at

In Part 2, I extend these zero-dimensional results to the horizontal. I first continue to use Laplace and Fourier techniques to treat the case of homogenous Earth parameters, but with inhomogeneous forcing. Then, with the help of Babenko's method, this is extended to the full inhomogeneous problem with horizontally varying relaxation times, diffusivities, specific heats, climate sensitivities, and forcings.

The figures were produced using a standard numerical integration package.

The data in Sect. 3.3.2 were taken from the cited source.

The author declares that there is no conflict of interest.

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

This paper was edited by Anders Levermann and reviewed by Peter Ashwin and one anonymous referee.