Response to step forcing for two-box model
The recent work by shows that a two-exponential response
can be fitted very well to a number of 150-year AOGCM runs with step-function
forcing. This raises the question of whether the power-law LRM response
representation is really only an inaccurate expression of a response with two
exponential timescales or vice versa. There is also an issue of whether the
AOGCMs really capture the true scaling properties of the observed response.
The two-box model couples the mixed layer to the deep ocean temperature T2
through a simple heat conduction term
C1dT1dt=-1SeqT1-κ(T1-T2)+FC2dT2dt=κ(T1-T2),
where C2 is the heat capacity of the deep ocean and κ is heat
conductivity. In the limit C2≫C1, Green's function for T1(t)
correct to lowest order in the small parameter C1/C2 is very simple and
transparent:
G(t)=Str0τtre-t/τtr+Seq-Strτeqe-t/τeqH(t).
The response to a step-function forcing F=H(t) then becomes
T1(t)=Str(1-e-t/τtr)+(Seq-Str)(1-e-t/τeq),
where we have introduced some new parameters,
Str=Seq1+κSeq,τtr=C1Str,τeq=C2Seq1-Str/Seq.
These parameters replace the heat capacities C1,2 and the heat coupling
constant κ, whose physical meaning is easy to grasp but hard to
measure directly. The meaning of the new parameters is apparent if we
consider the response to a step-function forcing. Since C1/C2≪1, we
have τtr≪τeq, and for t≪τeq
the response is completely dominated by the first term in
Eq. () and hence relaxes exponentially with the
transient time constant τtr to the new quasi-equilibrium
Str, which is referred to as the transient climate sensitivity.
However, when t approaches τeq, the second term comes into
play, and there is a new delayed response with time constant
τeq giving relaxation to the full radiative equilibrium
Seq.
From comparing the terms -T1/Seq and -κ(T1-T2) in
Eq. (), we observe that κSeq measures the
ratio between the heat flux into the deep ocean and the OLR at the early
stage of the response, i.e. when T2 is still close to 0. From
Eq. () it follows that the part of the sensitivity caused by the
slow response from interaction with the deep ocean is
Seq-Str=(κSeq)Str.
Hence, it appears that κSeq is an important parameter. If
κSeq≪1, the inclusion of the deep ocean has little
effect on the relaxation to equilibrium. If κSeq≃1 or
larger, the slow response leads to a significant rise in the temperature
after the transient equilibrium has been attained. The fast and the slow time
constants are always well separated if C1≪C2 since
τtrτeq=C1C2κSeq(1+kSeq)2≤C14C2.
Two-box vs. LRM fitting to GCM results
have considered 16 runs of different CMIP5
models with step-function forcing, and fitted the response
in the two-box model to the CMIP5-model responses. There are four fitting
parameters, and the fits are generally good. There is, however, a wide
scatter in the fitting parameters between the different models, which may be
an indication of overfitting. In Fig. the surface
temperature solution to the two-box model,
T1(t)=[Str(1-exp(-t/τtr))+(Seq-Str)(1-exp(-t/τeq))]F4×CO2,
and to the LRM model,
T1(t)=ctβT/2F4×CO2,
have been fitted to simulation results for the GMST of climate models with
step-forcing, F(t)=F4×CO2H(t). Here F4×CO2≈ 8.61 Wm-2 is the forcing associated with a
quadrupling of the atmospheric CO2 concentration. The fitting parameters
obtained are given in Table 1.
The LRM model in general gives a poorer fit on the short timescales. This is
not surprising, since the LRM response ctβT/2 has an infinite
derivative at t=0. However, a much better approximation is obtained if we
fit the LRM model only in the interval (0,100) months, but then βT
is raised to approximately 0.75. If we implement a four-parameter model with
one power law (βT≈ 0.75) up to 100 months and another
(βT≈ 0.35) for t>100 months, we obtain fits comparable to
the two-exponential model. There is a wide scatter in the model parameters
for the two-box model. Note particularly the huge values for
τeq and Seq for the CCSM4 model. The long
timescale tail is not captured by a reasonable exponential but is well
approximated by a reasonable power law. On the other hand, the scatter in the
LRM-model parameters is small. All this indicates that the two-box model may
suffer from overfitting in some cases.
When projections are limited to 2200 CE, there is no practical difference
between using a power-law response kernel (the LRM model) and the
two-exponential kernel (the two-box model). This is illustrated in
Fig. , where we compute the response for the
exponential CO2 concentration model with τC=33 years and
the two-box model parameters corresponding to the GISS-E2-H model and the
CNRM_CM5 models. The parameters for the two models differ significantly, but
the projections are almost identical. Moreover, they are very similar to the
projections in Fig. a, where the temperature response is produced
by the LRM model with τC=33 years and βT=0.35. This
demonstrates that the mathematical divergence of the solution
Eq. () for a step-function forcing has little impact on the
projection up to 2200 CE for the forcing scenarios considered here. The
advantage of the power-law kernel is that it provides a more parsimonious
description (fewer fitting parameters), which provides a more precise
parameter estimation.
Parameters estimated by fitting Eqs. () and
() to the climate model responses to an abrupt quadrupling of
atmospheric CO2 shown in Fig. . The table shows the
parameters obtained by the Mathematica routine FindFit.
Model
τ1 (Months)
τ2 (Months)
Str (km2 W-1)
Seq (km2 W-1)
c
βT
GISS-E2-H
26
663
0.29
0.46
0.14
0.32
BNU-ESM
46
729
0.46
0.69
0.21
0.33
CCSM4
49
4.1×1010
0.33
3.9×106
0.10
0.40
CNRM_CM5
38
390
0.37
0.58
0.20
0.31
MPI-ESM-LR
34
1061
0.46
0.75
0.20
0.33
Blue curves: fit of the two-exponential response
to the climate model responses to an abrupt quadrupling of atmospheric CO2
concentration. Red curves: fit of the LRM-scaling response. The expressions
fitted are found in the caption of Table 1 and the coefficients estimated are
shown in this table.
The evolution of the GMST according to the two-box model for the CO2 concentration scenarios shown in
Fig. a and c. Panel (a):
τC=33 years and the two-box parameters for the GISS-E2-H given
in Table 1. Panel (b): τC=33 years and the two-box
parameters for the CNRM_CM5 model given in Table 1.
Divergences, causality and initial conditions
If G(t) is a power law, the integral over prehistory t∈(-∞,0) may
lead to paradoxes, such as divergences of the integral. The solution to the
paradox is to interpret the power law as an approximation, for instance to a
superposition of exponential response kernels. For a white-noise forcing this
corresponds to an aggregation of Ornstein–Uhlenbeck (OU) processes, which are
known to have the potential to produce a process that is a very good
approximation to a fractional Gaussian noise (fGn) up to the timescale
corresponding to the OU process with the greatest correlation time
.
The scaling properties on scales of decades and longer arise from the heat
transport within the oceans. This transport exhibits a maximum response time,
which will provide an upper (exponential) cut-off of the power-law response
function, but the characteristic time of this cut-off may be centuries or
millennia. state in their abstract: “Scaling up to
decades is demonstrated in observations and coupled atmosphere–ocean models
with complex and mixed-layer oceans. Only with the complex ocean model the
simulated power laws extend up to centuries.”
If we do not treat the power law as an approximation, we have to deal with the
divergences of the integral
ΔT(t)=∫-∞tG(t-t′)F(t′)dt′,
where G(s)=sβT/2-1. If we consider the unit step-function forcing
F(t)=H(t) and βT≠0, the integral is
ΔT(t)=limϵ→0+∫ϵt(t-t′)βT/2-1dt′=limϵ→0+∫ϵtsβT/2-1s=limϵ→0+2βTtβT/2-ϵβT/2.
Clearly ΔT(t) diverges as t→∞ if βT>0, but
it also diverges if βT<0 (as ϵ→0+). For
βT=0 there is a logarithmic divergence in both limits.
For physically meaningful results the βT>0 case requires some sort of
cut-off (e.g. an exponential tail) for sufficiently large t, and the
βT<0 case requires an elimination of the strong singularity of G(s)
at s=0. As shown in Appendix , AOGCMs in the CMIP5 ensemble
with step-function forcing indicate a power-law response for large s at
least up to 150 years (and the GISS-E2-R model up to 2000 years) with
βT≈ 0.35, so βT>0 is the case of interest for the
global temperature response. The AOGCMs are also well approximated by an
exponential response in the limit s→0 (for s up to a few
years), so an exponential truncation in this high-frequency limit is also
appropriate.
The truncation of the power-law kernels is a physical, and not a technical
mathematical issue. It is an approximation to a hierarchy of exponential
responses. With this interpretation the divergences evaporate. Below is a
more detailed outline of this philosophy in an energy balance context. Let us
take as a starting point the simple zero-dimensional EBM before linearisation
of the Stefan–Boltzmann law:
CdTdt=-ϵσST4+I(t),
where T is surface temperature in Kelvin, C is an effective heat capacity
per area of the earth's surface, σS is the Stefan–Boltzmann constant,
ϵ is an effective emissivity of the atmosphere, and I(t) is the
incoming radiative flux density at the top of the atmosphere. Let I0=I(0)
be the initial incoming flux, F(t)=I(t)-I0 is the radiative forcing,
Teq=(I0/ϵσS)1/4 is the equilibrium temperature
at t=0, ΔT(t)=T(t)-Teq is the temperature anomaly
measured relative to the initial equilibrium temperature, and ΔT0=ΔT(0) is this anomaly at t=0. Note that F here is the
perturbation of the radiative flux with respect to the initial flux I0 and
not with respect to the flux ϵσST04 that would be in
equilibrium with the initial temperature T0. The linearised EBM for the
temperature change relative to the temperature Teq (the one-box
model) is
dΔTdt=-νΔT+F(t),ΔT(0)=ΔT0,
where ν=4ϵσSTeq3/C, F(t)=F(t)/C. By
definition F(0)=[I(0)-I0]/C=0. This is Eq. () and
Eq. () with slightly different notation. The solution to the
initial value problem (i.v.p.) Eq. (), with the initial
condition ΔT(0)=ΔT0, takes the form
ΔTi.v.p.=∫0tG(t-t′)F(t′)dt′+ΔT0e-νt,
where G(s)=exp(-νs). The generalisation to a linear, causal response
model, where G(s) is not necessarily exponential, involves extending the
integration domain in Eq. () to the interval (-∞,t):
ΔTr.m.(t)=∫-∞tG(t-t′)F(t′)dt′.
From the initial condition ΔT(0)r.m.=ΔT0
Eq. () yields
ΔT0=∫-∞0G(-t′)F(t′)dt′.
For exponential response G(s)=exp(-νs), it is easy to verify that
ΔTi.v.p.(t)=ΔTr.m.(t), and
Eq. () yields the following relation between the initial
temperature anomaly and the forcing F(t) for t∈(t,0):
ΔT0=∫-∞0eνt′F(t′)dt′.
For the exponential response there is no “divergence issue” in
Eq. (). Neither is there such an issue for the
two-exponential solution to the two-box model . An
“N-box model” exhibits a response function for the temperature in each
box which is a superposition of exponentials; G(s)=∑i=1Naiexp(-νis). For the surface (mixed layer) box the temperature anomaly
takes the form
ΔTr.m.(t)=∑i=1Naie-νit∫-∞teνit′F(t′)dt′.
On the other hand, the N-box initial value problem has a solution of the
form
ΔTi.v.p.(t)=∑i=1Naie-νit∫0teνit′F(t′)dt′+∑i=1Nbie-νit,
where the coefficient bi is linearly related to the initial temperatures
of each box: bi=∑j=1NMijT0j. The condition
T̃i.v.p.(t)=T̃r.m.(t) now yields the
relations between the initial temperatures and the prehistory of the forcing:
∑j=1NMijΔT0j=ai∫-∞0eνit′F(t′)dt′fori=1,…,N.
With a white-noise forcing F(t), Eq. () is the
Itô stochastic differential equation (in physics often called the
Langevin equation). The solution is the Ornstein–Uhlenbeck (OU) stochastic
process, which in discrete time corresponds to the first-order autoregressive
(AR(1)) process. The power spectral density of this process is essentially a
Lorentzian function, which means that the
high-frequency (f≫ν) part of the spectrum has the form ∼f-2
and the low-frequency part ∼f0. This means that if the climate
response were well described by a one-box EBM we could use a power-law
response model with βT≈ 2 on timescales much shorter than the
correlation time τc=ν-1. On these timescales the
stochastic process exhibits the characteristics of a Brownian motion (Wiener
process), which is a self-similar process with spectral index β=2. This
process is non-stationary and hence suffers from the divergences that we are
worried about. However, even though the Brownian motion diverges, the OU
process does not because of the flattening of the spectrum for f≪ν.
Both observation data and AOGCMs indicate that the one-box EBM is inadequate,
but the considerations above are equally valid for an N-box model, for
which the white-noise forcing gives rise to an aggregation of OU processes
with different νi. Such an aggregation is known to be able to produce a
process with an approximate power-law spectrum with 0<β<2 on timescales
τ<νmin-1 .
specifically argue that volcanic forcing may have a scaling
exponent βF≈ 0.4, and hence the convergence criterion
β=βT+βf<1 then requires βT<0.6. One remark on this is
that the above discussion shows that the β<1 criterion is not necessary
on timescales shorter than τ<νmin-1. However, observation
indicates that β<1, so this does not invalidate the argument of
. More important is that in
recent papers the response to volcanic forcing has been subtracted from both
instrumental and multiproxy reconstruction data and from
millennium-long AOGCM simulations , and the residuals have
been analysed for β without finding a detectable influence of the
volcanic forcing on β. The same is seen by comparing control runs of
the AOGCMs with those driven by volcanic forcing .
The importance of including the prehistory of the energy-flux imbalance when
deriving projections for future change can be illustrated by considering a
prehistory consisting of volcanic forcing FV(t) only. The
particular feature of volcanic forcing is that it consists of a succession of
negative spikes in the radiation flux. If we assume that the time t=0 is in
a period with no volcanic forcing, we can for illustration think of the
forcing as a succession of negative forcing events of short duration,
randomly distributed in time with typically longer waiting times between
events than durations. Let us further assume that the climate response is so
slow that G(t) varies by a small amount over the mean waiting time. Hence,
there exist time intervals of duration Δt which are short enough for
G(t) to be nearly constant over the interval but long enough to have a
sufficient number of large volcanic eruptions to estimate a mean volcanic
forcing F‾V. This assumption is not very good in
practice, but let us use it for illustration. Under this assumption we can
approximate the integral
∫t1-Δt/2t1+Δt/2G(t-t′)FV(t′)dt′≈G(t-t1)∫t1-Δt/2t1+Δt/2FV(t′)dt=G(t-t1)F‾VΔt,
and hence from Eq. () the temperature anomaly due to the
volcanic forcing is
ΔTV(t)=F‾V∫-∞tG(t-t1)dt1=F‾V∫0∞G(s)ds=def-ΔTvolc.
This result is meaningful only if the integral ∫0∞G(s) ds is
finite, i.e. if power-law response kernels are properly truncated. The
obvious, but still interesting, observation is that volcanic forcing keeps
the temperature, when averaged over the timescale Δt, on a constant
level Teq-ΔTvolc, i.e. the time-averaged
temperature is ΔTvolc lower than the temperature at which
the climate system is in equilibrium during times with no volcanic forcing.
Assume some additional (e.g. anthropogenic) forcing FA(t), for
which FA=0 for t≤0. Then the total temperature anomaly for
t>0 would be
ΔT(t)=ΔTV(t)+ΔTA(t)=-ΔTvolc+∫0tG(t-t′)FA(t′)dt′,
implying that the temperature starts changing in response to this forcing
from a non-equilibrium initial state. However, the statistics of volcanic
forcing is more challenging than assumed above, and one has to consider the
possibility of long periods with zero forcing, longer than the largest
temperature relaxation time reflected in the response function G(t). If
such a quiet period starts at time tq, then the temperature for t>tq is
ΔT(t)=F‾V∫t-tq∞G(s)ds+∫0tG(t-t′)FA(t′)dt′,
and since the integral over the tail of G(s) is assumed to be finite (there
exists a maximum relaxation time constant τmax), the first
term on the right of Eq. () will vanish if
t>tq+τmax. In other words, if the time of observation has
been preceded by a very long period of weak volcanic forcing the additionally
forced temperature change may be unaffected by the non-equilibrium imposed by
volcanic forcing. If we consider, as another example, that “normal”
volcanic forcing is resumed at t=0 after a pause of the length of
|tq|>τmax, then ΔT according to Eq. ()
grows from zero towards the expression in Eq. () as t grows
beyond tmax. Hence, during the transient period t∈(0,τmax) there may be a volcanic cooling that counteracts
anthropogenic warming, provided there was a long pause in volcanic forcing
preceding the era of anthropogenic forcing.
The discussion made here serves to illustrate that the non-equilibrium of the
radiative flux balance at t=0 may influence the subsequent temperature
evolution and that volcanic forcing may be the source of such an imbalance.
Knowledge about the history of volcanic forcing in the time interval
(-τmax,t) can be helpful in assessing the influence of volcanic
forcing on the long-term temperature evolution in the Anthropocene. In the
present paper the implicit assumption has been made that Eq. ()
is valid, i.e. that there is no long pause in volcanic forcing in the period
extending from 1880-τmax to 2200 CE. Hence, this forcing only
represents a constant downshift of the temperature. This assumption may
deserve closer scrutiny.