1. Introduction

The computational power now available to climate researchers is adequate to allow predictions of greenhouse gas–induced climate change over the coming centuries by “coupled” general circulation models (GCMs) of the atmosphere and ocean. Nevertheless, as the global mean surface temperature is a prime measure of climate change, simple energy balance models (EBMs) for this quantity continue to be used for both interpretive and predictive purposes (e.g., Schlesinger 1989; Dutton 1995; Murphy 1995; Kattenburg et al. 1996).

In most EBMs the atmospheric forcing by perturbed CO₂ concentrations, and the radiative response to the warming are assumed to be simple functions of concentration and temperature, respectively, the latter dependent on the “sensitivity” of the climate system. Given that the climate system is not equilibrated in the realistic case of “transient” change, the parameterization of the rate of heat uptake by the ocean in simple models is a major difficulty. For restricted periods this uptake is sometimes modeled using a constant-depth homogeneous ocean (a “0D model;” Dickinson and Schaudt 1998), while long-term predictions have often been made with 1D models that solve the ocean warming as a function of depth (e.g., Hoffert et al. 1980; Wigley and Raper 1992). As will be shown here, for interpretive purposes, at least, it may be adequate to simply allow the effective oceanic heat capacity in a 0D EBM to be time dependent. A simple model of this form, and its calibration, is presented in the following section, using as an example a simulation by a slab-ocean GCM with instantaneously doubled CO₂ (from which the doubled CO₂ forcing can be determined).

Uncertainties in the climate sensitivity, the parameterization of oceanic mixing by small-scale eddies, and the necessity for surface flux adjustments in most coupled models remain outstanding problems (Gates et al. 1996; Gregory and Mitchell 1997), however. In addition, it is likely that the present oceanic state, which (of necessity) provides the initial condition for most GCM simulations, is not in equilibrium with either the present or preindustrial climate forcing. The related “cold start error,” an underprediction of the temperature rise after the present in models that do not properly take into account past forcing (Hasselmann et al. 1993; Schneider 1996; Kattenburg et al. 1996), is still an issue.

The present study considers these problems by analyzing six GCM simulations with various forcing scenarios, including cases with stabilized CO₂ over several centuries, as described in section 3. A calibration of the simple model appropriate to each case is performed. The diagnosed effective heat capacity of the ocean increases almost linearly with time in most of the runs. Analytic solutions for this case are presented in section 4. The oceanic temperatures in the simulations are examined in more detail in section 5, in order to provide an understanding of the linear behavior of the GCM ocean. An analysis of the cold start error based on the simple model is given in section 6. A summary and the conclusions follow.

2. A simple energy balance model

The simple model is based on an equation for the global mean surface temperature anomaly, T (for convenience the customary Δ will be omitted), which is taken as characterizing the perturbation in heat content of the whole system, about an equilibrium assumed to exist prior to time t₀, when T = 0. (In practice, anomalies may be relative to a nonperturbed climate, possibly slowly varying.) The equation for the anomaly is then

View Expanded

where C is the effective heat capacity of the system (per unit area), and F is the globally averaged net radiative flux into the system at the top of the atmosphere (TOA).

The calibration of the simple model involves specifying C and T_e as functions of time. Given a CO₂ scenario for which these functions apply, and the forcing function Q(c), the two model equations can then be readily integrated for T and F. As our main use of the model is as a tool for interpretating GCM simulations, the calibration will be performed by using the equations in a diagnostic fashion, given a GCM response F(t) and T(t) to a certain c(t). The resulting functions will be approximated, and the consistency of the solution of this simple model with the GCM response will be analyzed.

To illustrate the model and the diagnostic approach, we examine in this section the idealized doubled CO₂ experiment of Watterson et al. (1997), which uses the Commonwealth Scientific and Industrial Research Organisation (CSIRO) GCM coupled to a simple mixed layer ocean—the “MLO” case. The simulated monthly and global mean temperature anomalies (relative to a control run, and smoothed by a 37-month running mean) as the model equilibrates to an instantaneous CO₂ doubling are shown in Fig. 1. The GCM equilibration sensitivity, based on the mean temperature of years 36–65 is 4.34°C.

a. Calibration

1) Forcing

For variations of CO₂ concentration in the range considered here, the instantaneous radiative perturbation due to a change in CO₂ is approximately logarithmic in the concentration (Wigley 1987). The forcing function for the model can then be written as

QcQ₂c(3)

although the Q₂ value must allow for stratospheric adjustment (Mitchell and Johns 1997) and should be averaged over an annual cycle. Using the first year means of F and T in the MLO run in Eq. (2), and assuming λ takes its final equilibrium value, gives Q₂ = 3.5 W m⁻². However, if one uses the second year means as well, to allow an evaluation of Q₂ on the assumption that λ is unchanged in these years, then the value is 3.8 W m⁻². The smaller value agrees with that estimated for the earlier Mark 1 version of the model by Watterson et al. (1999), from a diagnostic evaluation that included an equilibration of the upper-stratospheric temperature (σ = 0.017). But it is possible that this underestimates the full stratospheric adjustment. As a compromise, we will use the value 3.7 W m⁻² here, for both the MLO case and the coupled GCM runs. An uncertainty for Q₂ of ±0.2 W m⁻² will be assumed.

2) Sensitivity

From Eq. (2), the diagnostic equation for T_e, as a function of time, given known functions F, T, and Q, is

T_eQ₂TQF(4)

Given the variability of F, it is necessary to use smoothed time series of F and T to obtain useful values of T_e. The result for the MLO case is shown in Fig. 1. Since F approaches zero, T_e tends to the equilibrium warming. However, over the first few years it is evidently somewhat lower than this. Murphy (1995) found a similar effect in a coupled model run and attributed it to variation in the shortwave components of the cloud and albedo feedbacks. The causes of such variations, which are mainly in the initial years in the runs analyzed in this paper, need not be considered in detail here.

3) Heat capacity

In order to diagnose C from a known response using Eq. (1), the integral from time t₀ of both sides can be taken, the right side being then the net energy into the system. Then C is given simply by

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The capacity from this integral method for the MLO run is virtually constant at the value C_M = 1.8 × 10⁸ J m⁻² s⁻¹ K⁻¹. Most of this capacity is due to the mixed layer ocean, which averages 70 m deep (it is 50 m, except in and near regions with ice, where it is 150 m). As we will see, in the coupled model runs C varies markedly.

It is useful to note that in most, if not all, previous studies using a 0D model the heat capacity of the system is assumed to be constant, and the heat equation is written as

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In the diagnostic approach, the “alternative” capacity C_A can be evaluated as a function of time from this equation, although it is poorly defined as equilibration is approached. Naturally, the result is similar to C in the MLO case, but it is quite different if C varies. While Eq. (6) can be used with some success in interpreting GCMs (e.g., Watterson 1997), physically, if the effective capacity varies, then heat input is required, even if the temperature is constant. As this contradicts Eq. (6), Eq. (1) appears more appropriate here.

Some authors have evaluated the heat capacity in other, although related, ways. Murphy (1995) used a ratio of mean energy flux and temperature change, which produces C for a period starting when CO₂ first changes. However, if applied to later and sufficiently short periods (as in Keen and Murphy 1997), the result will be close to C_A. Gregory and Mitchell (1997) noted that in their coupled model runs with 1% compounding CO₂, F ≈ κT, with κ a constant and the warming increasing almost linearly in time. In this case, by substituting a T proportional to t − t₀ into (4) and (5) one obtains heat capacities that increase, with C_A ≈ κt and C half this function; we return to this case later. Hasselmann et al. (1993) approximated a doubled CO₂ warming result for a coupled model as the sum of two exponential functions, in effect constructing an EBM with two simultaneous timescales or constant capacity parameters.

b. Solutions

If we let y = CT, then on combining Eqs. (1) and (2) we get the first-order linear equation

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where τ = CT_e/Q₂ is a timescale. The general solution for T (and allowing a nonzero initial value) can then be expressed in the form

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where G is the Green’s function [used as in Hasselmann et al. (1993)] given by

GtuCtμuμt(9)

and μ is the integrating factor for Eq. (7),

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Note the appearance of C in these expressions, which results from the conversion of the solution for y to one for T. Given a solution for T, the result for F follows from simple substitution in Eq. (2).

If one approximates the MLO case using a simple model with constant sensitivity and capacity, and hence constant τ, then with Q constant after an instantaneous doubling of CO₂ at t₀ = 0, the analytic solution is

TT_ee^−t/τ(11)

This is shown in Fig. 1, where the timescale τ is 6.6 yr. The curve approximates the smoothed GCM series, with the slightly slower approach to the equilibrium value of the latter being mainly due to the initially suppressed T_e.

A more relevant case is for a CO₂ concentration increasing as c = exp[γ(t − t₀)] after time t₀, as, for example, the 1% compounded scenario featured in Kattenburg et al. (1996) (Fig. 6.4), for which γ = 0.0099 yr⁻¹ and the doubling time is 70 yr. The radiative perturbation is then

QQ₂γtt₀(12)

For the case of C and T_e constant, there is a well-known analytic solution, namely,

TT_eγtt₀τe^{−(t−t₀)/τ}(13)

Thus, after an initial delay the temperature rises almost linearly in time. Interestingly, the asymptotic warming rate is independent of C. The solution asymptotes to a line that passes through the “offset” temperature T_O at t₀, with T_O = −(T_e)²Cγ/[ln(2)Q₂]. For an idealized simulation started far in the past, and over which time the forcing grows at the same rate, the solution will have assumed this asymptotic linear form by t₀. Thus T_O is the cold start temperature error for the scenario starting at t₀ (asymptotic in both “warm-up time” and in time into the future) as defined by Hasselmann et al. (1993).

3. Coupled model simulations

4. The simple model calibrated to the coupled GCMs

Based on the GCM runs considered, one can anticipate that a simple model with a constant sensitivity, but increasing heat capacity, will be relevant. The analytic solutions presented in section 2b, for a constant C, are evidently not useful. While the simple model can readily be numerically integrated, it is instructive to consider analytically the case of a linear C, which appears to approximate the capacity in all the runs except C3.

Considering only runs B1, C1, CS, and CG for the present, the analytic solutions [from Eqs. (17), (2)] for T and F are shown in the upper panels of Figs. 7 and 8. The linear form C_U is used in all cases. The sensitivity differs between the runs, following the previous discussion; the constant values used are given in Table 2 (and with γ = 0.0054 for CS and CG). The analytic results reproduce quite well the (smoothed) GCM values (Figs. 3 and 4). The rms errors of the analytic temperatures over all years of the smoothed GCM series are rather small (Table 2), particularly for the shorter runs. Errors for the cases CS and CG (and C3, below) would be much smaller for a more select period. There is some cool bias in the later years of CS and CG, which can be attributed to the slightly excessive capacities from the linear form at that time. The GCM warming rates determined from linear regression over years 20–60 are well matched by the analytic form (Table 2). The offset temperatures determined from this linear fit (i.e., the value at year 0) also compare well. (The larger analytic asymptotic results are shown for comparison.) The skill of the simple model over this range of cases suggests that for the cold start case, the analytic solution (17) appears to provide an immediate estimate of the transient warming function from a GCM for any exponential CO₂ scenario, assuming the climate sensitivity is known.

We consider now applying the simple model to more complex scenarios. In run C2 after year 128, time t_e, when the forcing is stabilized, the diagnostic results can still be approximated by the linear C and a constant sensitivity (although the slightly higher value in Table 2 is appropriate). An analytic form for the warming for this period is available, also. With t_e in place of t₀, the general solution [Eq. (8)] simplifies to

View Expanded

where again r = 1/a, and δ = [τ(t)/τ(t_e)]^(−r−1). Joining the linear and constant forcing solutions at t_e we get the solution for the C2 case shown in the lower panels of Figs. 7 and 8. (Note the difference to the CS case due to the different assumed sensitivity.) Again the analytic temperature is fairly accurate (Table 2). The analytic F has a sharper peak at t_e, although the extra energy that this represents is small. An interesting aspect of the analytic result is that T approaches a constant, Q(t_e)/[λ(1 + a)], that is lower (for a > 0) than the equilibrium value for steady C. Here F approaches a nonzero value. This energy input is then solely feeding the steadily increasing heat capacity. The GCM warming rate does not slow as much, evidently due to a slackening of the growth rate of C (Fig. 6).

Although the more complex forcing scenario of run C3 can be approximated in a similar piecewise fashion (with three linear Q segments, then one constant segment), the solution cannot be approximated as a sequence of segments calculated using (16), due to the clearly nonlinear C. However, there is some resemblance of C itself to a sequence of linear segments offset in turn by 20 yr (starting 1900) and 80 yr (starting 1960), suggestive of components that override the previous ones. In fact, the C3 forcing is well approximated as the sum of three components of the C2 form. The first starts at year t_0,1 = 0 with t_e,1 at year 20. The second is nonzero after t_{0, 2} = 20 and constant after t_e,2 = 80, and the third is nonzero after t_0,3 = 80 and constant after t_e,3 = 202. Let us propose, then, that the GCM warming can be approximated as the sum of three solutions A_i, (i = 1, 2, 3) of the C2 form, each the result of a forcing component. For each component, it is assumed that C has the linear form C_U with the appropriate t₀. The sensitivity is assumed to be the value in Table 2 throughout, fairly consistent with Fig. 5.

The three component solutions A_i of the C3 simple model, along with the relevant C functions, are shown for the early years in Fig. 9. The net T, shown in Figs. 7 and 9, is quite similar to the GCM result (Fig. 3), as indicated in Table 2. The function F obtained by substituting the net T and the net approximate forcing into Eq. (2) is given in Fig. 8 and compares well with Fig. 4. An overestimate of the warming soon after t_{0, 3} is partly due to the excessive forcing then of the prescribed linear fit. The larger errors after year 380 in both warming and flux occur for the same reason as the later bias in C2. That is, the net or effective C, evaluated by substituting T and F into Eq. (5), exceeds the GCM result after about year 380 (not shown). Earlier, as shown in Fig. 9, around each successive t₀ the effective C makes a transition from one linear segment to another, rather like the GCM result. Thus the simple model that is proposed to be appropriate to this (initially) realistic scenario has a rather complicated effective heat capacity function. In any case, some indication of warming for even complex scenarios is apparently gained by this form of simple model. Limitations of the approach are considered in section 6.

5. Ocean heating

What causes the effective heat capacity in the coupled model simulations to increase, almost in a universal way (at least for the cases examined here with linear forcing and from a cold start), irrespective of the sensitivity and the forcing growth rate? The calculation of C involves two quantities: the TOA radiative energy input to the system, and the global mean surface temperature anomaly T. Assuming that a GCM conserves energy accurately, the energy input should be equal to the increase in the energy content of the system, which is predominantly the ocean heat anomaly. The distribution of this heat anomaly is the focus here. (Further analysis of the mechanisms of the heat transport is being undertaken by A. Hirst, personal communication 1998.)

Oceanic temperatures in the runs C1, CS, C2 and C3 have been examined in detail. Typical of previous studies (e.g., Manabe et al. 1991), the early warming is largest near the surface. Zonal means of the warming at later periods from the C2 and C3 simulations, shown in Fig. 10, indicate that the depth of significant warming increases at most latitudes. While this penetration of warming is larger at regions of downwelling and convection in higher latitudes (than for regions where diffusion provides the main downward flux of heat; Cai and Gordon 1998), given that these latitudes represent a relatively small fraction of the global ocean, the mean warming profile T_w(z), averaged over all ocean points at each depth z, is representative of much of the ocean. Examples of such profiles are given in Fig. 11. For all the profiles shown before year 200, the mean warming is greatest in the top 50 m. The top (first) level ocean temperatures T₁ from selected profiles are given in Table 3, along with T.

As anticipated, the heat capacity evaluated using the ocean heat anomaly of each profile (not shown) agrees well with that determined from the TOA flux in section 3. A corresponding ocean depth is

View Expanded

(in discrete form), where z_b is the bottom depth and w(z) is the ratio of the ocean area at z to that of the globe. This is the depth of a homogeneous ocean with this heat capacity, assuming that the ocean area is that of the model ocean surface [with w(0) = 0.703], and that the water temperature anomaly is T. Then D ≈ C/[c_ow(0)], where c_o = 4.19 × 10⁶ J m⁻³ K⁻¹ is the specific heat. Like C, D increases during each run (Table 3).

An effective ocean heat capacity, for which the heat anomaly is divided by T₁, rather than T, provides an alternate effective depth that depends on the profiles alone:D_o = D/r, where r = T₁/T (Table 3). This depth is typically a little larger than D, as in Table 3. As a simple description of the increasing universal heat capacity, it is perhaps best to use D_o with a typical r value of, say, 0.75, rather than D. Then the capacity C_U is that of a homogeneous ocean of depth initially 200 m, and increasing by 4.3 m each year.

Three other depth statistics determined from the profiles are also given in Table 3: the mean depth of the heat [the “effective depth” of Manabe et al. (1991)] given by

View Expanded

and the depths within which 50% and 90% of the heat is contained, D₅₀ and D₉₀, respectively. For an idealized wedge-shaped profile, linear in z down to a depth z_h to which the heating has reached (a fair approximation to those of Fig. 11), then D_o = 0.5z_h, D_h = z_h/3, D₅₀ ≈ 0.29z_h, and D₉₀ ≈ 0.68z_h. (Note also that for linear profiles maintaining a constant slope in time, the alternative capacity C_A would be double C, consistent with previous comparisons.) For a profile uniform with depth down to z_h, D_h = D₅₀ = 0.5D_o.

In the top panel of Fig. 11 the CS profiles are scaled by 2.1, which is the approximate ratio of T relative to the C1 values over years 10–70, and also that of the CO₂ forcing. This scaling allows one to see that the warming at depth in the two profiles is rather linear in the surface flux for these times [as noted by Cai and Gordon (1998)]. Thus not only is D_o the same, the distribution with depth of the globally averaged heat anomaly in the ocean is similar. This would follow from an ocean in which the anomalous vertical heat transport is linear in the surface warming amplitude, for example, a simple diffusive ocean. The ordering by magnitude of the depth statistics in Table 3 is consistent with that for the linear profile, except that D_h is relatively large in the early periods. This is due to the high-latitude warming at large depth, which has a large effect on this statistic, and also D₉₀.

The rate of increase in D_h for each run is smaller than that of D₅₀ in later periods, consistent with the enhanced subsurface warming that is evident in Fig. 11. The peak warming for C2 at year 268 is actually at the 370-m level (the value at 215 m is almost the same). As Fig. 10 shows, a subsurface maximum occurs then at most high latitudes, indeed a cooling occurs at the surface at the most polar latitudes. Cai and Gordon (1998) attribute this to a reduction in the meridional overturning, due to a freshening of the polar upper ocean by increased freshwater input. The effect is more marked in the longer C3 run (Fig. 10), for which the subsurface maximum extends over most latitudes in the later years and reaches a typical depth of 500 m. Manabe and Stouffer (1994) demonstrated a similar effect in extended runs of their GCM. They noted that the reduction of upward advection of cold water in low latitudes produces the subsurface maximum there. The ordering of D_h and D_o reverts to the wedge result at later times. The growth of D_o is enhanced by the stabilization of CO₂, and the resulting drop in the surface warming rate, as is clear from the contrast between CS year 183 and C2 year 268, and also the later pairs from C2 and C3 in Table 3.

Toward the end of both CS and CG, the surface warming rates increase a little. In terms of the simple model, this can be attributed to a slowing of the growth of D and C. With respect to the use of the analytic solutions for linear C, it is perhaps fortuitous that in runs C2 and C3 the CO₂ stabilization occurs when it does. If it occurred earlier, one might expect C to increase more rapidly. Even for constant CO₂, the growth in the depth statistics may eventually slow due to the finite depth of the ocean. Note that for C3 year 498 there is significant warming occurring near the bottom level (Fig. 10), which may limit further increases in D_h in particular.

6. Discussion

From the diagnosis based on the simple model equations, it would seem that a key quantity for understanding the greenhouse warming of the GCMs is the effective heat capacity of the ocean. For a linear forcing scenario starting from equilibrium, the heat capacity grows steadily over a century or so, and for longer in runs in which the CO₂ forcing is then stabilized. In two such multicentury GCM runs, the warming drifts beyond the value at the time of stabilization to lie between steady temperature solutions of two versions of the simple model: a value for the increasing C version, in which there is a constant energy flux into the deep ocean; and the higher, equilibrium anomaly of the constant depth version, which depends only on the sensitivity.

Consider first the problem for a simple model that has a specific function C in time. Then T_f is simply the sum of the two temperature anomaly functions, A_a and A_b, which satisfy the two forcing components. An illustration of such solutions for an idealized linear C case is given in Fig. 12. As for the solution for C2, A_a will, in general, drift above the value at t_e, by an amount that Keen and Murphy (1997) term the “warming commitment.” If A_b were a true cold start result, then the negative of this drift, A_a(t_e) − A_a(t), is also E(t).

Schneider (1996) considers the constant C model for which the A_b is indeed the cold start result A_c. For a linear forcing scenario, the full solution [Eq. (13)] is asymptotically linear. For t_e − t₀ sufficiently large, the asymptotic cold start error would then be close to the T_O given in section 2b (as noted there). Likewise, if the linear C model is used, T_f would still be almost linear (as in Fig. 12), so the offset relevant to A_b [Eq. (18), but with C(t_e) replacing C(t₀)] would approximate the negative of the drift of A_a. (These approximations would not apply if the forcing function were nonlinear, though.) However, one would expect a true cold start result, A_c, in this case to be of the same form as T_f, as, for example, in Fig. 12. This result is different than A_b ue to it being associated with the capacity function initialized at t_e, rather than t₀. Thus −E is not the same as the drift of A_a (e.g., Fig. 12).

For a more realistic scenario, the C3 GCM result suggests that the function C for the component A_b (which we equate to A₃ in the previous C3 analysis) should be reinitialized at something like the 200-m-deep ocean value, rather than taking the larger C(t_e) of the A_a component (A₁ + A₂ of C3), as it would in the specific C model. If the sum of such components, T_f, is valid as an approximation to the GCM result, then the cold start error, which would now be correctly ascribed to A_b, is again the negative of the drift of A_a. In the C3 case, for example, the asymptotic value of this is −0.14°C (virtually all from A₂).

It can be argued that this additive method of constructing a simple model for a scenario with linear forcing segments, and thereby evaluating the cold start error of the latter component (A_b), should be limited to cases with slopes that increase significantly. For the trivial case of Q_b having the same slope as Q_a, the specific C case above (where C for A_b is not reinitialized) is clearly more relevant. As Schneider (1996) emphasizes in his analysis of the constant C case, the cold start solution A_c is actually for a model assumed to be in equilibrium at T(t_e), unlike the original model, which is in equilibrium at T(t₀). The degree of equilibration of the GCM at t_e appears to be at the heart of the limitation of the method. At t₀, equilibrium, the value of C is small. Then during a forcing period C increases steadily, and the ocean is taken farther from equilibrium. At t_e, if the forcing slope drops to zero, C rises at least as fast as before. If the second slope is the same as the first, the growth rate of C only gradually decreases. Then the specific C simple model for A_b has a C at t_e that is larger than the equilibrium value; hence, it is not a cold start result. If, however, the new slope is much larger, C actually drops as the warming from the new forcing overwhelms the earlier warming. In this case, the ocean is for a time not far from equilibrium relative to the new forcing, and after a while the net C approaches the reinitialized function, and the method is reasonable.

In any case, the method of summing solution components (as for C3) produces a simple prediction of the cold start error that is consistent with that based on the drift or warming commitment discussed by previous authors. The error so estimated may depend on a lower and more realistic prior CO₂ growth than do the analytic offset temperatures given above, which assume an extension of the exponential growth rate back in time. For example, assuming that the 60-yr Q₂ component of IS92a is the only previous forcing, the asymptotic cold start error would be −0.15°C for the C1 case and −0.06°C for B1. Both of these are smaller estimates than the offset values in Table 2.

As discussed by Schneider (1996), these considerations should serve to emphasize the uncertainty of using GCMs that are artificially constructed to be close to equilibrium at the observed, present state of the ocean. The need for flux correction to reduce residual drifts of a control simulation is related to this artificiality, in the simple model at least. Ideally, of course, a forecast of the warming of the coming century should be made using a simulation for the full forcing scenario by a GCM that is at equilibrium at the preindustrial climatic state of the ocean.

7. Summary and conclusions

A simple EBM is proposed for the interpretation of greenhouse gas–induced climate change simulated by GCMs. The global mean surface temperature anomaly is modeled using a simple heat equation, allowing a time-varying effective heat capacity. The TOA energy flux depends on the CO₂ forcing, the temperature, and an effective sensitivity of the system. Six GCM simulations, from several models and with different forcing scenarios, are analyzed, and the capacity and sensitivity functions diagnosed. It is found that while the sensitivity is approximately constant during each run, and typically close to the appropriate equilibrium result, the capacity varies. For the runs with a linear forcing scenario and starting from an assumed equilibrium (actually, differences from control runs are used), the capacity is approximately equivalent to that of a homogeneous ocean of depth initially 200 m, and rising 4.3 m each year, independent of the forcing magnitude. An analytic solution for the simple model with this universal capacity function is given. This matches the warming and TOA flux time series from each simulation quite well, although after a century or so the warming is underestimated, as the growth in capacity actually slows.

For the run C2 with the forcing held constant after around a century of linear growth, the capacity continues to increase. A combined analytic solution for such a scenario is given. The run C3, based on the IS92a scenario, has three near-linear forcing segments, with increasing growth rate. In this case, the diagnosed capacity grows during each segment, then drops back to close to the universal capacity function reinitialized for the new segment. The simple model proposed for this case has the warming given by the sum of three solutions of the C2 type. While this is adequate, such a method will likely be deficient for more general scenarios, in which the GCM is far from equilibrium at the start of each segment, so that the reinitialized capacity is inaccurate.

The cold start problem of GCM runs initialized at an artificial equilibrium at time t_e is discussed in the light of the simple model for C3. If the simulation for the full scenario is approximately the sum of the simple model solutions for the “before” and “after” t_e forcing segments, then the cold start error of the latter component is the negative of the drift of the former component past its temperature at t_e.

There are evidently limitations of the simple EBM with an idealized ocean heat capacity function in representing GCM simulations with arbitrary forcing scenarios, and analysis of simulations from other GCMs is needed in order to assess the generality of the results. Nevertheless, based on the simulations considered here and for the scenarios commonly considered, interpretation of the warming rates and cold start errors may be aided by the use of such a simple model. Furthermore, an indication of likely warmings for a range of scenarios and model sensitivities can be readily determined—even analytically.