In this study we quantify the relationship between the aerosol optical depth increase from a volcanic eruption and the severity of the subsequent surface temperature decrease. This investigation is made by simulating 10 different sizes of eruption in a global circulation model (GCM) by changing stratospheric sulfate aerosol optical depth at each time step. The sizes of the simulated eruptions range from Pinatubo-sized up to the magnitude of supervolcanic eruptions around 100 times the size of Pinatubo. From these simulations we find that there is a smooth monotonic relationship between the global mean maximum aerosol optical depth anomaly and the global mean temperature anomaly and we derive a simple mathematical expression which fits this relationship well. We also construct similar relationships between global mean aerosol optical depth and the temperature anomaly at every individual model grid box to produce global maps of best-fit coefficients and fit residuals. These maps are used with caution to find the eruption size at which a local temperature anomaly is clearly distinct from the local natural variability and to approximate the temperature anomalies which the model may simulate following a Tambora-sized eruption. To our knowledge, this is the first study which quantifies the relationship between aerosol optical depth and resulting temperature anomalies in a simple way, using the wealth of data that is available from GCM simulations.
 Large volcanic eruptions can eject sulfur dioxide into the stratosphere where it reacts with water vapor to produce highly scattering sulfate aerosol droplets, raising planetary albedo and leading to a radiative cooling of the Earth's surface [Robock, 2000]. The local and regional cooling following large volcanic eruptions has historically been documented by interested observers in various places across the globe [Oppenheimer, 2003; Thordarson and Self, 2003] and, in recent years, the global pattern of cooling has been recorded by satellite instruments [Bluth et al., 1992; Minnis et al., 1993]. These historic and satellite-based observations have been combined by a few authors [Pollack et al., 1976; Sigurdsson, 1990; Rampino and Self, 1993] to quantify the relationship between volcanic emissions and atmospheric cooling. It is these relationships, based on extrapolating the trend from small emission masses which were, until recently, used to estimate the cooling which may occur following the largest “supervolcanic” eruptions [see, e.g., Zielinski et al., 1996; Ambrose, 1998; Oppenheimer, 2002].
 GCM simulations are an excellent tool for creating such a relationship. First, it is possible to simulate very high emissions in GCMs thereby removing the need to extrapolate the relationship, which otherwise intrinsically assumes that the response follows the same trend for smaller eruptions as it does for the largest. Second, is it possible to extract not only the global mean postvolcanic temperature anomalies from climatic noise with a good degree of reliability, but it is also possible to extract regional temperature anomalies at the grid box scale, information which is not easy to extract from observations of historic eruptions. This second point is important because observations of historic eruptions and paleoclimatic reconstructions from proxy data are primarily drawn from single point locations. The ability to analyze the temperature changes which are consistent across regions of a few model grid boxes holds great potential for facilitating a user-friendly comparison between modeling and observation data.
 This paper details the method and uses of producing user-friendly GCM-based relationships between surface temperature anomalies and volcanic emissions on both global mean and grid box scales. Section 2 outlines our method of simulating volcanic eruptions in HadCM3, a fully coupled atmosphere-ocean GCM, by changing the sulfate aerosol optical depth of the model. Section 3.1 describes the derivation of the global-mean relationship using a range of different aerosol optical depth anomalies. Section 3.2 describes the formation of regional relationships between temperature anomaly and eruption size and discusses their potential applications. Section 4 includes a discussion of the limitations of the data presented here.
2. GCM Simulations of Large Volcanic Eruptions
 We have used the UK Met Office Atmosphere-Ocean General Circulation Model, HadCM3 to simulate generic volcanic eruptions. The atmosphere component of the model has a horizontal grid resolution of 2.5° × 3.75° latitude × longitude and 19 levels in the vertical up to a model top of 10 hPa [Pope et al., 2000] The ocean component has a horizontal grid resolution of 1.25° × 1.25° latitude × longitude and 20 levels in the vertical, with the highest resolution concentrated in the upper layers [Gordon et al., 2000]. This model has a climate sensitivity of 3–3.3 K in response to a doubling of CO2 [Gregory et al., 2004; Randall et al., 2007] which straddles the mean value of 3.2 K found in the IPCC AR4 for 23 GCMs and is well within the range of observationally derived values of 2.0–4.5 K as found, again, in the IPCC AR4 [Randall et al., 2007; Hegerl et al., 2007]. HadCM3 has been used extensively in the third and fourth IPCC Assessment Reports [Houghton et al., 2001; Solomon et al., 2008] as well as numerous other studies of the current, past, and future climate of the Earth [Stott et al., 2000; Hewitt et al., 2001; Tett et al., 2002; Johns et al., 2003; Stainforth et al., 2005]. It was used by Jones et al.  to perform the first fully coupled GCM simulation of a supervolcanic eruption. In addition to global mean changes, HadCM3 has been shown to simulate regional climate variations such as sea ice distribution, ENSO and the Asian Summer Monsoon with a fair degree of accuracy [Gordon et al., 2000; Collins et al., 2001; Turner et al., 2005]. There are, however, a couple of limitations to the model which will impact the postvolcanic temperature response. First, the model top is low at 10 hPa and, second, there is no interactive evolution of atmospheric chemistry, including ozone. The potential impacts of these limitations on the values to be presented here are returned to in section 4.
 The HadCM3 model is used to conduct 15 simulations of different sized eruptions using the same technique as [Jones et al., 2005]. These simulations are driven by increasing sulfate aerosol loading in the stratosphere at each time step to represent simplistically the processes of aerosol formation and sedimentation, which are not handled automatically by the model. The sulfate aerosol used is one of five “climatological” aerosols used by HadCM3 [Cusack et al., 1998] and as such it has a fixed unimodal log normal size distribution with a geometric radius of rg = 0.07 μm and a standard deviation of σ = 1.86 as well as fixed optical properties, a single scattering albedo of ω = 0.9999 and an asymmetry factor of g = 0.7 at a wavelength of λ = 0.55 μm [World Meteorological Organization, 1983]. These sizes and optical properties of the sulfate aerosol do not change throughout the simulation, which leads to inaccuracies in the behavior of the volcanic sulfate aerosol in the model, particularly for large eruption sizes. For example, aerosol sizes were observed to increase following the Pinatubo eruption [Sato et al., 1993; Stenchikov et al., 1998; Niemeier et al., 2009] and this in turn would tend to reduce scattering in the shortwave and increase longwave absorption, resulting in lower temperature anomalies [Timmreck et al., 2009]. Furthermore, there is no representation of the chemical or microphysical interactions of the sulfate aerosol [Pinto et al., 1989; Bekki, 1995; Timmreck and Graf, 2000] and so the lifetime of the aerosol is the same in all simulations. The impacts of these important limitations are somewhat reduced here as we drive and analyze our simulations with respect to the aerosol optical depth anomaly at 0.55 μm produced by the sulfate aerosol rather than by what type and how much of the aerosol is created. In doing so, we are simulating a simple and generic “Pinatubo-like” eruption, rather than attempting to simulate any specific historic eruptions.
 The optical depth anomalies used to drive the simulations of eruptions are designed to be a similar size and distribution to the post-Pinatubo optical depth anomalies at 0.55 μm in the Sato Index [Sato et al., 1993; Robock, 2000], as shown in Figure 1. The Sato Index was also used in four broad latitudinal bands used by Jones et al. , though here we used a higher latitudinal resolution. The mass of sulfate aerosol required in each vertical column of the model to create this required optical depth is calculated using the extinction coefficient of the sulfate aerosol at 0.55 μm and a conversion from Slingo  to account for the broad spectral bands of the model's radiation code. Once calculated, this mass of aerosol is then distributed throughout the model stratosphere with a uniform mass mixing ratio.
 The 15 simulations consist of a sulfate-aerosol free control run, a five-member ensemble simulation of a Pinatubo-sized eruption, and a further nine simulations of larger eruptions. The 50-year control run is started under fully equilibrated initial conditions representative of January 2000, following which the climatic conditions of the simulations are not forced to follow any observational or historical records, but allowed to evolve freely in response to the model's own internal variability. A base Pinatubo-sized ensemble member is constructed from the control run by initiating the start of the eruption five years after the beginning of the simulation and allowing the model to respond freely to this sulfate-aerosol forcing for 30 more years. The remaining four ensemble members are created by initiating the start of the eruption in successive years. For the larger eruptions, the relative latitudinal and temporal distribution of aerosol optical depth anomalies at 0.55 μm remains identical to the base ensemble member while the total amount of the sulfate aerosol in the model is increased in each simulation by multiplying the aerosol optical depth in each grid box by a scaling factor (3, 5, 10, 15, 20, 40, 60, 80 and 100). Each simulation of these larger eruptions is a single run. It is possible, in light of the work by Niemeier et al. , that the optical depth anomalies which we are simulating in the largest eruptions are unobtainable large but it is beyond the scope of this study to ascertain an upper limit to the possible sulfate aerosol optical depth anomalies which can be produced volcanic eruptions. As the eruption sizes are changed by changing the optical depth anomalies then hereafter the term “eruption size” strictly describes the size of the optical depth perturbation.
Figure 2 shows the results from the five-member ensemble of HadCM3 simulations of the Pinatubo-sized eruptions as well as re-analysis data, observations published by McCormick et al.  and ENSO-corrected observations by Santer et al. . The time series of the multivariate ENSO index is also shown. The most marked difference between the HadCM3 simulations and the observed temperature anomalies are that the model overestimates the timescale of cooling in comparison to the reanalysis products, taking approximately 10 years to return to equilibrium temperatures instead of 2. Aside from this, there is a good agreement in the magnitude of the postvolcanic temperature anomalies in the ensemble simulations and the reanalyses/observations when the effects of ENSO are taken into account. While HadCM3 is capable of simulating ENSO events, the climatic conditions of the model are not forced to represent those of the 1991 Pinatubo eruption and so the large El Nino which occurred around the time of Pinatubo (and the resulting increase in temperatures) is not present in the simulations. Both McCormick et al.  and Santer et al.  have calculated and removed the effect of ENSO from their observations of surface temperature anomalies and there is a good and clear agreement between these adjusted values and the model simulations. These results give us confidence that our model simulations of generic Pinatubo-sized eruptions are sufficient for constructing relationships between aerosol optical depth and surface temperature anomaly.
 Observational data is not available to check the sufficiency of the largest eruption simulations and so instead we draw comparison with the supervolcanic simulations of Jones et al.  and Robock et al. . The temperature anomalies seen in the 100 × Pinatubo simulation are of a similar order of magnitude to those two studies. There is a very good agreement, to within a degree or so, between the simulations conducted here and the results of Jones et al. , which is expected given the similarity of the experimental setup. To compare with Robock et al. , we must first choose a metric by which to make that comparison, as their simulations are based on emissions of sulfur dioxide, rather than optical depth anomalies. We have chosen to compare eruptions which have similar incoming shortwave radiation anomalies (not shown for HadCM3) as this is a measure of changes in atmospheric transmission, though it is somewhat crude. Using this framework we find that the 2 Gt SO2 and 6 Gt SO2 simulations made by Robock et al.  are comparable to the 60 × Pinatubo and 100 × Pinatubo experiments made here respectively. These two comparisons show that the simulations of Robock et al.  are 3–4 K cooler than the simulations on HadCM3 and this temperature difference is around a degree larger in the comparison of the two bigger eruptions. This increase, although small, may be significant and is probably due to the extended lifetime of sulfate aerosol used by Robock et al. , which will be discussed again in Section 4. We are confident that our simulations broadly agree with the two previous representations of a generic supervolcano and so we will use these values to form a simple relationship between aerosol optical depth and eruption size. We must stress, however, that these relationships are constructed strictly within the framework of Pinatubo-like eruptions on HadCM3 and may not be applicable to other model simulations or the real world.
3. Simplified Temperature Predictions
3.1. Global Mean
Figure 3 shows the time series of global mean temperature anomalies following each of the volcanic eruptions simulated. It is clear that there is a monotonic relationship between eruption size and peak temperature anomaly, which is requisite for forming a simple relationship between volcanic aerosol optical depth and surface temperature anomaly. The maximum cooling following all eruptions greater than Pinatubo-sized are distinct from interannual variability (as represented by two standard deviations of the global mean annual surface temperature anomalies in the control run) and the maximum temperature anomaly following the largest 100 × Pinatubo eruption is −11.5 K. Though there are changes in the timing of the peak cooling between eruption sizes, the timescales of temperature decrease and recovery are broadly similar for each of the eruptions. There are two clear differences in the temperature response to the larger and smaller aerosol optical depth anomalies. The first is the marked increase in interannual variability between around 3–12 years after the eruption. This increase is due to a combination of the slow summer melting of the boreal ice cap and the enhanced growth of the austral ice cap giving enhanced cooling toward the end of the year along with a large summer warming, winter cooling response from the boreal landmasses. This difference will not have an impact on a simple relationship if that relationship is built using annual or 12-month mean values. The second difference is the slower return of global mean surface temperatures to equilibrium following larger aerosol optical depth anomalies and it occurs because of extra cooling as the exaggerated growth of the ice sheets increases planetary albedo. We take this second difference into account later when choosing an appropriate way of averaging the time series anomalies.
 As the relationship between volcanic aerosol optical depth perturbation and temperature decrease is monotonic, we can construct a simple mathematical relation between the two. This relation is found by using the following nonlinear expression which includes the exponent of the global mean maximum optical depth anomaly:
ΔT is the global mean 12-month maximum postvolcanic surface temperature anomaly for an eruption with a global mean maximum optical depth of τ at 0.55 μm. Using this expression, �� and �� are found by an iterative Matlab code which calculates a nonlinear best fit using the Levenburg-Marquardt algorithm [Seber and Wild, 2003]. With values of �� and �� found in this way, the first four coefficients and residuals of the Taylor expansion of the subsequent relationship are almost identical to the coefficients and residuals of a third order polynomial fit to the model data, indicating that the goodness of this fit is almost identical to that which would be obtained by using a third-order polynomial. This form of expression has an advantage over a simple polynomial, however, in that its components provide us with information about the evolution of temperature anomalies with respect to global mean maximum volcanic aerosol optical depth changes. �� is the “saturation temperature anomaly,” a theoretical maximum global cooling for an infinite peak magnitude of aerosol optical depth anomalies. �� defines the eruption size at which temperature anomalies approach the values of ��, as shown in Figure 4.
 To find values of �� and ��, we have averaged ΔT over the 12-month peak temperature anomalies of each simulation before using the iterative fitting code. This period is highlighted for each eruption in Figure 3. By doing this we create a simple relationship which does not encompass temperature anomalies as they return to equilibrium. Such narrow focus is desirable as we have seen that the return to equilibrium temperatures is much longer in the model than observed for Pinatubo, and it is insensitive to the lag in the time of peak cooling following the largest aerosol optical depth anomalies. Furthermore, this relation captures the largest temperature anomalies which is arguably the diagnostic of greatest broad interest.
 Naturally, the temperature anomalies do not all fit the relationship derived by equation (1) perfectly and this is quantified by the value 2σ[res], or two standard deviations of the residuals between the global mean temperature anomalies and the best-fit curve. These residuals are not normally distributed, and so 2σ[res] is not a formal measure of significance, but as the nonlinearity of equation (1) means that the formal measure of significance is complex, then we consider 2σ[res] to be a suitably simple and universal guide to the goodness of the fit.
 The best fitting curve of the simulated 12-month mean maximum global mean surface temperature anomalies with respect to global mean maximum volcanic aerosol optical depth anomaly, using the format of equation (1), is shown in Figure 5 and given by
It can be seen from the plot and equations (2) and (3) that 2σ[res] is considerably smaller than ��, which gives us confidence that this form of curve is a good fit to the global mean temperature anomalies simulated by HadCM3. As they are, however, entirely based on these HadCM3 simulations they should be applied to real-world eruptions with some caution.
 There are considerable differences between equation (2) and the similar relationships between volcanic emissions and temperature anomalies constructed by other authors [e.g., Pollack et al., 1976; Rampino and Self, 1992; Sigurdsson, 1990]. The relations constructed by these three authors, shown in gray on Figure 5, are formulated using observed temperature anomalies from a handful of historic eruptions. The relations between eruption size and temperature anomaly constructed by Pollack et al.  and Rampino and Self  both make the assumption that temperature anomalies scale linearly with eruption size. This linearity is clearly not the case as temperature anomalies following large eruptions are predominantly driven by radiative change and this itself is not a linear function of eruption size nor optical depth. There is some agreement between equation (2) and the simple scaling relation proposed by Sigurdsson  for small eruptions and the shape of both curves are similar, however the temperature anomalies predicted by Sigurdsson  for eruptions larger than 5 × Pinatubo are much smaller than those simulated. Whether equation (2) is a better way of predicting global mean temperature anomalies following a real-world supervolcanic eruption than the relation of Sigurdsson  is not proven, but equation (2) is the closest simple description of GCM-simulated supervolcanic temperature anomalies to date.
3.2. Regional Anomalies
 While global mean temperature changes are an important indicator of the impact of volcanic eruptions in comparison to other climate forcing mechanisms, regional changes in temperature can be used to evaluate the impact of large eruptions on local populations as well as in the proximity of paleoclimatic study sites. In order to quantify these regional changes, we repeat the fitting analysis to provide relationships between eruption size and temperature anomaly for each model grid box. In each grid box the continuous 12-month period exhibiting the greatest temperature anomalies is identified within the five years following each eruption and these anomalies are then fitted to equation (1), where τ is the global mean maximum optical depth anomaly as before, to give a value of �� and �� for that particular grid box. The resulting maps of ��, �� and the residuals from the fits show the regional-scale variability in these values and are displayed in Figure 6. Note that since the 12 month period exhibiting the greatest temperature anomalies at each point and for each eruption size may be slightly different, Figure 6 does not show the best fit coefficients at a snapshot in time following the eruption. Table 1 summarizes the best-fit values for the large-scale surface regions shown in Figure 7, along with two standard deviations of the control run annual mean temperature anomalies (2σ[ctrl]), the number of grid boxes averaged to give each value, and the percentage contribution that the values in each region make to the global mean.
Table 1. Mean Values of ��, ��, 2σ[res], and 2σ[ctrl] for the Land Surface Regions Indicated in Figure 7a
Column G shows the number of model grid boxes averaged to give these values and column %gm shows the percentage contribution of these values in each of the regions to the global mean values.
northwestern North America
northeastern North America
southwestern North America
southeastern North America
southern Greenland and Iceland
western and central Europe
northern Europe and northern Siberia
northeastern and central Asia
central and western Asia
Central America and northern South America
western South America
eastern South America
southern South America
eastern Australia and New Zealand
western and central Antarctica
 A comparison of 2σ[res] and 2σ[ctrl] in Figures 6c and 6d and in Table 1 shows that besides a few mainly tropical regions with very little interannual change, the fit to equation (1) for the majority (over 85%) of the surface has residuals with a smaller standard deviation than interannual variability. This suggests that if this relationship is used to estimate the 12-month mean maximum surface temperature anomaly for an eruption which has not been explicitly simulated in HadCM3, then the greatest source of error in that estimation, for the majority of the globe, would come from natural variability, rather than the error in the fit. This gives us confidence that these regional relations are a means to reconstruct temperature anomalies following generic Pinatubo-like eruptions simulated in HadCM3.
 The spatial distribution of �� and �� in Figures 6a and 6b shows the sensitivity of different regions of the globe to volcanically induced temperature changes. The land-sea contrast is evident in �� and this shows, as is to be expected, that even for the very largest aerosol optical depth anomalies, the land surface cools considerably more than the oceans. The highest values of �� are found over the North African and Middle Eastern desert regions, the uniform pattern of �� over these same regions indicates that these are parts of the world which will respond most dramatically to volcanically driven temperature change. This response occurs because the desert regions have a high average insolation and predominantly clear skies, making them very sensitive to decreases in incoming shortwave radiation following a volcanic eruption. The smallest values of �� are found in the oceanic upwelling regions of the Southern Ocean, where relatively warmer surface waters are drawn up from depth and also in the North Atlantic, in both these regions saturation temperature anomalies are no more than −5 K. The North Atlantic response is particularly complex and may be due to the large increases in Meridional Overturning which occur with increasing eruption size in these simulations (see also Jones et al. ). The temperature changes predicted for Europe are more moderate than those in the North America and North Asia, which may be due to the relative warmth of the North Atlantic. There is an interesting feature of high V in southwestern North America, which indicates that the region will experience similar temperature anomalies for small as for large eruptions; the reasons for this feature are not clear.
 These best fit coefficients �� and �� can be used to populate equation (1) and predict the model temperature response to any magnitude of aerosol optical depth anomaly, including those which have not been explicitly simulated, and this approximation is within a degree or two of the simulations for the 85% of model grid boxes where the residuals are smaller than natural variability. These predictions should, however, be attributed to real historic or prehistoric eruptions with extreme caution. Figure 8 shows the predicted temperature anomalies for a simulation of an eruption with a global mean maximum aerosol optical depth anomaly of τ = 1.4 at 0.55 μm, which may be the maximum optical depth anomaly following the eruption of Mt Tambora in 1815 [Stothers, 1984]. Also shown on Figure 8 in dark black/red circles are the peak annual mean temperature anomalies in four different regions between 1816 and 1820, obtained from paleoclimatic records available at the NOAA World Data Center for Paleoclimatology (see Figure 8 caption for references). We have limited our use of these records to those which are complete between 1740 and 1840, as these allow us to calculate two standard deviations of the annual mean temperature anomalies in this 100-year period (2σ[p100]) for each record. The points surrounded by red circles indicate those records that are greater than 2σ[p100]. The agreement between the paleorecords of annual mean temperature change and the model approximation is good in the studies of the Andes (paleoclimatic data points 3 and 4) where the temperature anomalies are similar to the prediction and greater than 2σ[p100]. The temperature anomalies in the European record are lower than the prediction, however this point lies in a region where natural variability is simulated to be greater than the eruption signal. There is little agreement, however, between the model approximation and the paleo data points in Columbia, US where anomalies are predicted to be the highest in the North American continent and the paleoreconstruction shows moderate cooling which is smaller than 2σ[p100]. In addition to the mean annual values, Figure 8 also shows summer-only temperature anomalies from further paleoclimatic studies in lettered circles outlined in light gray/red. While these values cannot be directly compared to the model output because they are not annual values, they do show lower temperatures than the model on the whole.
 There are numerous reasons why the agreement between the prediction and paleoclimatic data shown in Figure 8 should not be excellent. The chief cause of this is probably the incorrect assumption that the distribution and lifetime of stratospheric sulfate aerosol following Tambora is the same as the “Pinatubo-like” model used here. Other model limitations may be the initial conditions chosen or the limited representation of sulfate aerosol chemistry. There may be inaccuracies in the paleoclimatic temperature records which the inclusion of more data would help to address. The effects of the moderately strong 1817 El Nino event [Ortileb, 2000] could also be masking part of the cooling signal from Tambora in the paleoclimatic temperature record as found by D'Arrigo et al.  for tropical sites, though it is unlikely that this effect is strong in the extratropical regions examined here [Trenberth and Caron, 2000]. To investigate these differences further, an obvious second candidate for comparison would be the well-observed anomalies of Pinatubo on which these simulations are based, but as the predicted temperature anomalies for this eruption are indistinct from natural variability a direct comparison would not be robust. Despite their shortcomings, there is still value in these approximations as guides to the possible temperature anomalies across the globe following a volcanic eruption which produces a global mean aerosol optical depth anomaly falling within the range of Pinatubo −100 × Pinatubo.
 An additional feature of the relation shown in equation (1) is that it can be inverted to give an expression for τ if the best-fit values of �� and �� are known. By substituting a measure of the lower temperatures expected from interannual variability (i.e., −2σ[ctrl]) into this inverted relation in place of ΔT, then we get the following expression for the global mean maximum optical depth increase at which surface temperature anomalies are predicted to exceed two standard deviations of annual mean temperature anomalies.
This value has been calculated for each grid box and Figure 9 is a map of τc. This map shows that, in HadCM3, temperature anomalies following 1–3 × Pinatubo eruptions are distinct from natural variability in the tropics (excluding the equatorial Pacific) and that only eruptions with a maximum aerosol optical depth anomaly greater than 7–10 × Pinatubo result in distinct temperature anomalies across the midlatitudes. Over large regions of Antarctica only the 20 × Pinatubo eruption or greater has a clear and distinct temperature signal in the model. These results are, of course, based solely on the simulations conducted in HadCM3 and include all the accompanying caveats, but if they can be considered an approximation to real-world behavior then they suggest that paleoclimatic and remote sensing observations of temperature anomalies following large eruptions will be most effective if they are conducted in the tropics.
4. Summary and Discussion
 We have derived a set of simple global and regional relationships between Pinatubo-like aerosol optical depth anomalies and resulting surface temperature change simulated on the GCM HadCM3. We simulated a range of aerosol optical depth anomalies from the size of Pinatubo to 100 × Pinatubo, using the Sato Index as the basis of the spatial and temporal distribution of the anomalies. A simple global mean relationship was found by fitting the maximum 12-month mean global temperature anomalies from these simulations to equation (1). The best fit coefficients were �� = −11.3 and �� = 0.164 and the residuals from the fit were small. We then used this technique to formulate individual relationships for each model grid box and the resulting spatial pattern of best-fit coefficients are shown in Figure 6. We have shown that these spatial patterns can be used to approximate model temperature anomalies at eruption sizes which have not been explicitly simulated in HadCM3. We have also inverted the simple relationship to estimate the size of volcanic eruption required for a region to experience a temperature anomaly which is distinct from the model's interannual variability. The results of this inversion suggest that paleoclimatic studies in the tropics would be most beneficial when looking for the temperature signals of volcanic eruptions.
 As discussed previously, there are a number of caveats that must be considered when applying the data from these idealized simulations to past or potential future eruptions. While the relationship derived here is valid for a wider range of optical depth perturbations than those used to extrapolate from historical eruptions, it is derived using idealized aerosol distributions based on the Pinatubo eruption alone. Thus if the optical depth perturbation from a given volcano is substantially different in microphysical properties (e.g., size, etc.), spatial distribution (resulting from a different location of volcano or different atmospheric circulation patterns), or rates of aerosol formation and sedimentation, then the relationship here may not be appropriate. For example, it was shown in section 2 that the maximum temperature anomalies in the simulations of Robock et al.  are greater than the simulations made here and that this difference increases with eruption size. This increase is probably because their model explicitly replicates the conversion of sulphur dioxide to sulfate aerosol in the stratosphere. The timescales associated with this appear to enhance the duration of the aerosol forcing in comparison with the simulations made on HadCM3. Further limitations of the model itself, such as resolution and model top, may also impact these values, though this influence is likely to be small. Despite these caveats relating to the real-world applicability of these simulations, we have demonstrated the use of a widely respected GCM to produce a simple relationship between Pinatubo-like aerosol optical depth anomalies and the maximum 12 month mean surface temperature response and the further applications of such a relationship. As this simulation has only been conducted on the UKMO model HadCM3 to date, then the results presented here are necessarily dependent on that particular model, but the technique of analysis may also be applicable to similar simulations of large radiative forcing scenarios such as geoengineering simulations or global warming experiments.
 We gratefully acknowledge the funding and support of the UK Natural Environment Research Council and the National Center for Atmospheric Science. We would like to thank the British Atmospheric Data Center and the National Center for Atmospheric Research for access to reanalysis data and the National Oceanographic and Atmospheric Administration for access to paleoclimate temperature proxy data. We would also like to thank three anonymous reviewers for their help in improving this manuscript.