Introduction

Knowledge about strong explosive volcanic eruptions is important in many scientific fields because these eruptions may be dangerous for human life and property and therefore may also be of socio-economic consequences. Apart from the direct effects in the surrounding of an erupting volcano there is also a climatic impact caused by stratospheric aerosol clouds which can be produced by strong eruptions. The strongest volcanic eruption of the past 200 years was the Tambora eruption in April 1815. The following year is known as the "year without summer" (Stommel and Stommel, 1979). However, less powerful eruptions have an impact on the climate, too. Thus, long time series of both, volcanic activity induced perturbations of the atmosphere and related climate forcing, are needed to understand volcano-climate relationships. Moreover, in order to interpret anthropogenic climate change, it is necessary to understand natural influences like volcanism which are in competition with anthropogenic forcing. Explosive volcanism is known to have a strong influence on the temperature of the atmosphere. This problem is discussed in many papers (e.g. Hansen and Lacis, 1990; Sato et al., 1993; Jones and Kelly, 1996). In spite of that, explosive volcanic forcing is usually considered in terms of index value time series as a tool for the evaluation and analysis of climate parameter time series (Jones and Kelly, 1996; Tol and de Vos, 1998). These index value time series are the dust veil index ($DVI$) proposed by Lamb (1970, 1977, 1983), the severity index by Mitchell (1970), the volcanic explosivity index ($VEI$) by Simkin et al. (1981) or Newhall and Self (1982), the smithsonian volcanic index ($SVI$) by Schönwiese (1988), or Cress and Schönwiese (1992) and the ice core volcanic index ($IVI$) by Robock and Free (1995). The latter paper also involves a comprehensive overview of the differences between these various index time series. Most of the information provided by these index time series does not address atmospheric radiation transmission processes and therefore these time series have to be seen as only a crude approximation of volcanic forcing of the climate system. Nevertheless, in very recent investigations these index values are used to offer a statistical explanation of global temperature variations caused by explosive volcanic eruptions (e.g. Tol and De Vos, 1998). This kind of approach may be appropriate for a statistical analysis and of interest in the case of unknown physical relations between volcanic forcing and related climate parameters. But if one is interested in a more quantitative analysis of past climate variations one has to know the spatio-temporal patterns of volcanic forcing in more detail. Because the volcanic aerosol changes the radiation budget of the atmosphere, it is important to know at least its aerosol optical depth ($AOD$) as a bulk property (Rind, 1996). Since 1961 measurements of atmospheric radiation extinction are available from sites in both hemispheres (Dyer and Hicks, 1968). The NIMBUS 7 satellite provides data from polar regions since 1979 (Stratospheric Aerosol Monitor, SAM II; McCormick et al., 1979, and McCormick, 1994). The Stratospheric Aerosol and Gas Experiment (SAGE II, McCormick and Wang, 1987) provides data between about $70^{\circ}N$ and $70^{\circ}S$ since 1984. If one is interested in volcanic $AOD$ before 1961 one has to reconstruct this property from other information sources. This is done for example by Sato et al. (1993) and Stothers (1996). Sato et al. (1993) provide annual mean stratospheric $AOD$ with respect to the wavelength $\lambda=.55\mu m$ for four equal-area zones from 1850 to 1990 and take into account different kinds of information. They also give an actualized version on a 24-point latitude grid with monthly resolution from 1850 to 1999 (Sato, 1995). The last five years of the record content only an exponential decay of the last observations in 1994. Nevertheless, for the period from 1850 to 1882 they only use the rescaled index from Mitchell (1970) which has no spatial resolution. Therefore Sato et al. (1993) provide global averaged values for the time before 1890. Stothers (1996) evaluates volcanic stratospheric $AOD$ from 1881 to 1933 from pyrheliometric data at several stations. Together with the more recent index values evaluated by Sato (1995), this seems to be the most reliable spatial $AOD$ time series available today. Nevertheless, both Sato (1995) and Stothers (1996) find different spatio-temporal $AOD$ patterns, as will be shown in section 4. The disadvantage of low (or no) spatial resolution of $AOD$ patterns due to volcanic eruptions before the availability of instrumental data can be avoided by using the information of the strength, date, and location of volcanic eruptions as well as by introducing a stratospheric transport parameterization. Date and location of volcanic eruptions are available with fair accuracy from the Volcanic Explosivity Index ($VEI$, Simkin et al, 1981). However, the explosivity given by $VEI$ is not a reliable measure of the stratospheric aerosol loading for two reasons. First, stratospheric mass loading is assumed to be proportional to $10^{VEI}$ (and $VEI$ is given in integers. Thus an error in $VEI$ leads to an error of at least a factor 10 in respect to the estimated stratospheric volcanic mass loading). Second, $VEI$ says nothing about the amount of precursors to build stratospheric aerosol (Robock and Free, 1995). Nevertheless, for recent eruptions $VEI$ can be corrected with the help of other information available (see section 2). For the spatial and temporal evolution of the stratospheric aerosol cloud the season and latitude of the eruption seem to be most important (Bradley, 1988). Thus, we obtain a more detailed spatial resolution of $AOD$ of historical eruptions by using a stratospheric transport parameterization (which is introduced in section 2) and information about location, date and strength of volcanic eruptions, although the latter information is not very reliable. Furthermore, the stratospheric transport parameterization can be used for further investigations of any past volcanic eruptions known. The parameterization is calibrated with respect to different information available about stratospheric transport mechanisms and the most recent eruptions of El Chichón (1982) and Mount Pinatubo (1991). In principle, the time series of spatial patterns of $AOD$ can be used to drive a general circulation model (GCM). So far, transient calculations are carried out by Hansen et al. (1996) with a pure atmospheric GCM. In addition, coupled atmospheric-oceanic GCM simulations under January and July conditions, respectively, do exist (Graf et al., 1996). But due to its large numerical effort and computation time such GCM runs can only be case studies. Therefore, no long-term GCM calculations of the volcanic impact on climate do exist. That is why simplified models have to be used. Stenchikov et al. (1997) found from GCM calculations that the aerosol radiative forcing following the Pinatubo eruption (1991) is not sensitive to the dynamical atmospheric response to this forcing. This encourages radiative forcing calculations without using GCMs. To obtain an estimate of volcanic forcing one can use the crude approximation of Lacis et al. (1992) to get the net radiative flux change at the tropopause $\Delta F_{net}$ for the case of a uniform aerosol layer:

$$\Delta F_{net}(W/m^{2})\approx 30 \cdot\tau_{.55}$$ (1)

where $\tau_{.55}$ is the $AOD$ at the wavelength $\lambda=.55\mu m$. This approximation neglects the seasonal and latitudinal dependence of the undisturbed radiation uptake. In order to obtain a time series of volcanic forcing not as crude as when using the approximation (1) but with much less numerical effort and computer time as it is needed in a GCM or a radiative-convective model, a simple solution of the radiation-transfer equation (RTE) is introduced (see section 3). Finally, using the RTE solution with respect to the $AOD$ time series, we obtain estimates of spatio-temporal patterns of volcanic forcing in $W/m^{2}$ (section 4).