## 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 CO_{2} 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 CO_{2} (from which the doubled CO_{2} 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 CO_{2} 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

*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

*t*

_{0}, when

*T*= 0. (In practice, anomalies may be relative to a nonperturbed climate, possibly slowly varying.) The equation for the anomaly is thenwhere

*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).

*t*

_{0}to an imposed radiative forcing perturbation

*Q*(

*c*) (where

*c*is the CO

_{2}concentration normalized by the unperturbed value) is proportional to

*T,*so that

*F*

*Q*

*c*

*λT,*

*λ*is a response or inverse feedback parameter (e.g., Dutton 1995; Gregory and Mitchell 1997). It is convenient [as noted by Murphy (1995)] to represent this in terms of an effective equilibration temperature for

*c*= 2, or sensitivity,

*T*

_{e}. Since the time-mean TOA flux is balanced at equilibrium, setting

*F*= 0 and

*Q*=

*Q*(

*c*= 2) ≡

*Q*

_{2}in Eq. (2) provides the relationship between

*λ*and

*T*

_{e}:

*λ*=

*Q*

_{2}/

*T*

_{e}.

The calibration of the simple model involves specifying *C* and *T*_{e} as functions of time. Given a CO_{2} 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 CO_{2} 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 CO_{2} 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

_{2}concentration in the range considered here, the instantaneous radiative perturbation due to a change in CO

_{2}is approximately logarithmic in the concentration (Wigley 1987). The forcing function for the model can then be written as

*Q*

*c*

*Q*

_{2}

*c*

*Q*

_{2}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

*Q*

_{2}= 3.5 W m

^{−2}. However, if one uses the second year means as well, to allow an evaluation of

*Q*

_{2}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

*Q*

_{2}of ±0.2 W m

^{−2}will be assumed.

#### 2) Sensitivity

*T*

_{e}, as a function of time, given known functions

*F, T,*and

*Q,*is

*T*

_{e}

*Q*

_{2}

*T*

*Q*

*F*

*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

*C*from a known response using Eq. (1), the integral from time

*t*

_{0}of both sides can be taken, the right side being then the net energy into the system. Then

*C*is given simply byThe capacity from this integral method for the MLO run is virtually constant at the value

*C*

_{M}= 1.8 × 10

^{8}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.

*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 CO_{2} 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 CO_{2}, *F* ≈ *κT,* with *κ* a constant and the warming increasing almost linearly in time. In this case, by substituting a *T* proportional to *t* − *t*_{0} 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 CO_{2} 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

*y*=

*CT,*then on combining Eqs. (1) and (2) we get the first-order linear equationwhere

*τ*=

*CT*

_{e}/

*Q*

_{2}is a timescale. The general solution for

*T*(and allowing a nonzero initial value) can then be expressed in the formwhere

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

*G*

*t*

*u*

*C*

*t*

*μ*

*u*

*μ*

*t*

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

*τ,*then with

*Q*constant after an instantaneous doubling of CO

_{2}at

*t*

_{0}= 0, the analytic solution is

*T*

*T*

_{e}

*e*

^{−t/τ}

*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}.

_{2}concentration increasing as

*c*= exp[

*γ*(

*t*−

*t*

_{0})] after time

*t*

_{0}, 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

*Q*

*Q*

_{2}

*γ*

*t*

*t*

_{0}

*C*and

*T*

_{e}constant, there is a well-known analytic solution, namely,

*T*

*T*

_{e}

*γ*

*t*

*t*

_{0}

*τ*

*e*

^{−(t−t0)/τ}

*C.*The solution asymptotes to a line that passes through the “offset” temperature

*T*

_{O}at

*t*

_{0}, with

*T*

_{O}= −(

*T*

_{e})

^{2}

*Cγ*/[ln(2)

*Q*

_{2}]. 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

*t*

_{0}. Thus

*T*

_{O}is the cold start temperature error for the scenario starting at

*t*

_{0}(asymptotic in both “warm-up time” and in time into the future) as defined by Hasselmann et al. (1993).

## 3. Coupled model simulations

### a. Models and scenarios

The transient CO_{2} simulations that are analyzed here are from coupled GCMs that include two different atmospheric models, both run with nine vertical levels and using a spectral representation at horizontal resolution R21. Each GCM uses one of several versions of the Geophysical Fluid Dynamics Laboratory ocean model (Bryan 1969), each using the 56 lat × 64 long R21 transform grid. The simulations are summarized in Table 1, which includes further references.

A simulation, denoted here as B1, from the Australian Bureau of Meteorology Research Centre (BMRC) uses the Kuo convection model of Colman and McAvaney (1995), for which the MLO equilibrium sensitivity is 2.1°C. A standard 12-level ocean model is attached, without any surface flux adjustment. The 1% compounding CO_{2} scenario is used in the simulation, for which the assumed forcing is shown in Fig. 2.

The other simulations are by the CSIRO Mark-2 atmospheric GCM (as in the MLO case above); and freshwater, heat, and momentum surface flux adjustments are used in coupling the ocean (Gordon and O’Farrell 1997). Both a standard 12-level ocean (denoted by “S” in Table 1) and a 21-level version with the turbulent mixing scheme of Gent and McWilliams (1990, hereafter GM) are used. Results are available for both 1% compounding (run C1) and IS92a (post 1975, denoted IS92a-T) CO_{2} scenarios (Leggett et al. 1992), runs CS, and CG. Since the first 60 yr or so of the latter scenario can be approximated as a 0.5% compounding, CS and CG provide low-growth cases to contrast with the high-growth cases, B1 and C1 (see Fig. 2). Tripled CO_{2} is reached after 182 yr in runs CS and CG. The CS run was also continued with constant *c* beyond year 128, when CO_{2} doubling is reached, with the run denoted C2 (of which years 1–128 duplicate CS).

The final simulation, C3, also uses the GM ocean. The forcing represents the IS92a radiative forcing scenario, including trace gases and notionally starting from calendar year 1881 (although in applying the GCM to this scenario, the effective CO_{2} concentration was scaled relative to the standard initial value of 330 ppmv, and an initial state based on the present climate was used, as in the previous runs). The forcing is shown in Fig. 2 using two time offsets. In the upper panel, the C3 time offset is 20 years, so that the labeled time 0 is 1900, when the effective CO_{2} exponential growth rate parameter (*γ*) changes from approximately 0.0007 to 0.0018 yr^{−1}. After 1960, which is year 0 in the lower panel, the growth rate again increases sharply to around 0.0080 yr^{−1}. At year 202 (notionally 2082 and labeled 122 in the figure), the CO_{2} concentration reaches 3 × CO_{2} and is then held constant. Simulation C3 can thus provide an additional 60-yr low CO_{2} growth rate case from (virtually) a cold start, as well as a second extended run with large growth and ending with stabilized CO_{2} forcing. Only anomalies between the runs and corresponding control runs are considered here. The control runs have a fairly steady climate (particularly those with the GM ocean), except for the non-flux-adjusted B1 case.

The global and annual mean temperature anomalies in the simulations are shown as (smoothed) functions of time in Fig. 3. While some decadal-scale variability is evident, the curves show a clear contrast in warming rates that is rather consistent with that of the forcings. The contrast between B1 and C1 is potentially consistent with the reduced equilibrium sensitivity of the BMRC MLO GCM. A delay in attaining the nearly steady warming rate achieved between years 20 and 60, or so, is apparent in all runs, as will be quantified shortly. A sharply reduced warming rate after CO_{2} stops rising is also evident in C2 and C3.

The energy input anomaly to the system for each run is shown in Fig. 4. As would be anticipated, the comparison between the time series in the upper panel is similar to that for the temperature anomalies. The C1 and B1 curves tend to have a slightly decreasing rate, however. The decreasing flux after CO_{2} stabilization in C2 and C3 provides the major contrast to the form of the temperatures, in the lower panel.

### b. Sensitivity

The effective sensitivity during the coupled model simulations, evaluated from Eq. (4) using smoothed data, is shown in Fig. 5. To omit spurious values, particularly in the early years, values are not shown if a representative uncertainty range is greater than 2°C. (The range is calculated by considering individually the uncertainty in *Q*_{2}, an error range of ±0.004°C in *T,* and a range of ±0.025 W m^{−2} in *F.*) There is an apparent initial suppression of sensitivity only in C1 and CS, when *F* is small, however. Meaningful values are not obtained in the first 25 yr of C3.

The sensitivity values for B1 are fairly close to the BMRC MLO equilibrium value throughout. Values for C1 and CS tend to be slightly lower than the CSIRO value. The CG values are lower again, apparently the result of inhibition of sea-ice retreat in the GM model (Hirst et al. 1996). A similar result occurs (although the sensitivity is slightly increasing) in the long C3 run. The C2 final values are also close to the MLO equilibrium value.

There is evidently some variation of the *T*_{e} values from the MLO equilibrium results, and differences between the surface warming distributions in the coupled and MLO models (not shown) may contribute to this difference. It is worth noting that the use of flux adjustment does not seem a significant factor here. This contrasts with results of CO_{2} doubling from Gregory and Mitchell (1997), who found an increase in sensitivity of 0.7°C due to flux adjustment. If anything, comparing B1 and C1, the sensitivity is relatively smaller in the flux-adjusted case (although there are other differences between the models here). In essence though, Fig. 5 indicates that the effective sensitivity during each transient run is fairly steady and typically a little lower than the corresponding MLO equilibrium result.

### c. Heat capacity

*C*defined by Eq. (5) for the coupled model simulations are shown in Fig. 6. [Here the uncertainty range is 4

*C*

_{M}(the MLO value), based on a range in the smoothed

*T*of ±0.02°C.] After year 10, the results of all five runs in the upper panel are broadly similar, although with some different variation on a decadal scale. With the time offset used for the C3 function, which in effect ignores the weak forcing of the initial 20 yr, C3 values tend to be only a little higher than the others. All the curves show a rather steady increase in capacity, from an initial value of 2 to 4

*C*

_{M}. The values continue to increase in the lower panel, although at a lower rate toward the end of each run. Interestingly, the lower rate in CS after the doubling time leads to a clear difference relative to run C2 by year 180. In any case, a reasonable approximation to all the curves, except for C3, is the straight line shown, which is (in units 10

^{8}J m

^{−2}s

^{−1}K

^{−1}, with

*t*in years)

*C*

_{U}

*t*

*t*

_{0}

The similarity of the CS and CG time series occurs despite the initial expectation that they would differ due to the alternate ocean mixing formulations in the GCM. In fact, the oceanic circulations and patterns of parameterized convection are also modified in a way that evidently compensates for the mixing rate differences, as explained by Hirst et al. (1996). The effective capacities for B1 and C1 are similar functions of time, despite the differing warming rates and the use of flux adjustment in C1 but not B1. Again, this similarity contrasts with Gregory and Mitchell (1997), who found a 50% higher capacity parameter (*κ*) in the more slowly warming, unadjusted case.

Note that the alternative form *C*_{A} also produces an increasing function, as shown for the shorter runs by Watterson (1997). The *C*_{A} values are some 50% larger than *C* at each time. The comparison is roughly consistent with our assessment of the results of Gregory and Mitchell (1997), noted previously, although the linear approximations to *C* and *C*_{A} here are initially well above zero. After the CO_{2} stabilizes in both C2 and C3, *C*_{A} increases very rapidly, as the heat uptake is relatively large compared to the surface warming rate. The distribution of the heat uptake by the ocean will be considered in section 5.

## 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.

*τ*(

*t*) ≡

*CT*

_{e}/

*Q*

_{2}=

*a*(

*t*−

*t*

_{0}) +

*b,*where

*a*and

*b*are assumed constant, the integrating factor [Eq. (10)] has the analytic form

*μ*

*t*

*a*

*s*

*t*

_{0}

*b*

*a*

^{s=t}

_{s=t0}

*G*

*t*

*u*

*b*

^{−r}

*τ*

*t*

*τ*

*u*

^{r}

*C*

*t*

*r*= 1/

*a.*

*τ*

^{r}, leads to

*T*

*T*

_{e}

*γ*

*a*

*t*

*t*

_{0}

*b*

*a*

*τ*

^{−(1+r)}

*b*

^{2+r}

*a*

*T*and

*F*[obtained from Eq. (2)] are asymptotically linear in time. The final warming rate is smaller by the factor (1 + 2

*a*) than that for the constant

*C*case [Eq. (13)]. In effect, the warming is continually being held back by the rising

*C*(as

*a*> 0). Note that the asymptotic rate is not quite linear in

*T*

_{e}, as

*a*depends on it. The offset temperature of the asymptote can be written

*T*

_{O}

*T*

_{e}

^{2}

*C*

*t*

_{0}

*γ*

*Q*

_{2}

*a*

*a*

*C*case with the initial capacity. These solutions are of the same form, but with altered coefficients, as those for the alternative model [Eqs. (2), (6)], with a linear

*C*

_{A}(Watterson 1997). Note that a linear asymptotic result was recently deduced analytically for a 1D upwelling-diffusion ocean model by Dickinson and Schaudt (1998), and analysis of such models using the present methods may be of interest.

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 CO_{2} scenario, assuming the climate sensitivity is known.

*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

*t*

_{0}, the general solution [Eq. (8)] simplifies towhere 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 *t*_{0}. 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 *t*_{0} 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 *T*_{1} from selected profiles are given in Table 3, along with *T.*

*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*

_{o}

*w*(0)], where

*c*

_{o}= 4.19 × 10

^{6}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 *T*_{1}, rather than *T,* provides an alternate effective depth that depends on the profiles alone:*D*_{o} = *D*/*r,* where *r* = *T*_{1}/*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.

*D*

_{50}and

*D*

_{90}, 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}= 0.5

*z*

_{h},

*D*

_{h}=

*z*

_{h}/3,

*D*

_{50}≈ 0.29

*z*

_{h}, and

*D*

_{90}≈ 0.68

*z*

_{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}=

*D*

_{50}= 0.5

*D*

_{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 CO_{2} 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 *D*_{90}.

The rate of increase in *D*_{h} for each run is smaller than that of *D*_{50} 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 CO_{2}, 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 CO_{2} stabilization occurs when it does. If it occurred earlier, one might expect *C* to increase more rapidly. Even for constant CO_{2}, 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 CO_{2} 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.

_{2}. Specifically, let the system be in equilibrium at

*t*

_{0}, and let the forcing scenario be partitioned into two continuous parts:

*Q*

_{a}, which is nonzero after

*t*

_{0}, and constant after the (present) time

*t*

_{e}; and

*Q*

_{b}, which is zero before

*t*

_{e}. If the warming function for the full scenario is

*T*

_{f}, and that for the cold start run with forcing

*Q*

_{b}is

*A*

_{c}, then the cold start error function is

*E*

*t*

*A*

_{c}

*t*

*T*

_{f}

*t*

*T*

_{f}

*t*

_{e}

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} − *t*_{0} 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*(*t*_{0})] 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 *t*_{0}. 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 *A*_{3} 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 (*A*_{1} + *A*_{2} 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 *A*_{2}).

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*(*t*_{0}). The degree of equilibration of the GCM at *t*_{e} appears to be at the heart of the limitation of the method. At *t*_{0}, 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 CO_{2} 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 *Q*_{2} 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 CO_{2} 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.

## Acknowledgments

My thanks go to members of the climate modeling groups of both CSIRO Atmospheric Research and BMRC who contributed to the development of the GCMs analyzed here, and to those who performed the simulations, in particular Martin Dix (the C2 run) and Tony Hirst (C3). Helpful discussions with both Martin and Tony are acknowledged. The comments of two anonymous reviewers were also very helpful. This work contributes to the CSIRO Climate Change Research Program and is in part funded through Australia’s National Greenhouse Research Program.

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Summary of GCM simulations.

Approximated sensitivities (*T _{e}*) and rms errors of analytic temperature results for all GCM runs. Warming rates

*w*and the offset temperatures

*T*of both the GCM runs and the corresponding simple model results (denoted by the subscripts

_{O}*g*and

*s,*respectively), based on years 20–60 are given for the relevant cases (with C3 offset by 20 yr). The asymptotic offsets

*T*from the simple model analytic solutions are also given.

_{O}Global mean surface temperature anomaly (*T,* in °C), mean temperature anomaly of the top ocean level (*T*_{1}, °C), their ratio (*r* = *T*_{1}/*T*), the effective ocean depths *D* and *D _{o},* the depth of the heat anomaly

*D*and the depth statistics

_{h},*D*

_{50}and

*D*

_{90}(all in the unit m), for various years of runs CS, C2, and C3. The 5-yr means centered on the year are used here except that 21 yr are used for

*T,*if available.