temperature have occurred over the past 10,000 years (i.e., the period over which we have the best data and during which the planet's albedo-related boundary conditions have probably varied little) (Röthlisberger, 1986; Grove, 1988). Since this evidence is indirect and based on limited geographical coverage, we can only offer speculative estimates of the magnitude of these annual global-mean temperature changes. Most probably, the fluctuations have been within a band 0.5°C wide (Wigley and Kelly, 1990). Dating and data uncertainties do not allow us to compare the recent warming with similar warmings in the past in any useful quantitative way, so we cannot say whether the twentieth-century warming is occurring at an unprecedented rate.
An alternative approach is to try to estimate 50-year time-scale variability using either a statistical or a deterministic model. In both cases, information from the observed, instrumental record of global-mean temperature changes over the past 130 years is used to estimate the spectrum of natural variability over the full range of frequencies.
In the statistical approach, the data are fitted to a standard-time series model of some sort (such as an nth-order autoregressive process or some more complex model), and the fitted model is used to extend the spectrum to lower frequencies (e.g., Bloomfield and Nychka, 1992; Kheshgi and White, 1993).
A deterministic approach has been used by Wigley and Raper (1990, 1991). Here, a simple climate model is employed to extend the spectrum of observed variability: The method used is to force the model stochastically with white-noise forcing, the magnitude of which is chosen to match the observed high-frequency variability (1- to 10-year time-scale). It should be noted that this approach yields information only on passive internal variability, i.e., variability that arises in the absence of ocean circulation changes. Low-frequency variability may also occur, for example, through changes in the ocean's thermohaline circulation, as evidenced by recent coupled ocean/atmosphere general-circulation model (GCM) results (e.g., Washington and Meehl, 1989; Manabe et al., 1991; Cubasch et al., 1992) and by much longer simulations with an ocean GCM (Mikolajewicz and Maier-Reimer, 1990) and with a two-dimensional ocean model (Mysak et al., 1993). It has been speculated that such circulation changes may be coupled to surface temperature changes, leading to feedbacks that would amplify low-frequency climate variability (e.g., Gaffin et al., 1986; Piehler and Bach, 1992; Harvey, 1992). Model simulations of passive internal variability, therefore, may only represent a lower bound to overall internally generated variability.
An example of the output from a simulation of passive internal variability is given in Figure 1. For a simple linear first-order model (e.g., a one-box ocean model) white-noise forcing as input is transformed to red-noise temperature output, with an enhancement of variability at low frequencies. For more complex models (such as that used here), the output spectrum is still basically reddish in character, in that there is a marked enhancement of the power at low frequencies.
The passive internal variability modeling approach gives results very similar to those of the statistical model of a fractionally integrated white-noise process that is preferred by Bloomfield and Nychka (1992) (see Figure 2). The deterministic modeling approach yields additional physical insights. It shows that high-frequency variability (≤ 10 years) is virtually independent of the climate sensitivity (ΔT2×), and mainly reflects the magnitude of the forcing and the heat capacity of the upper mixed layer of the ocean (Wigley and Raper, 1991). For similar reasons, the response to short-lived forcing events, such as those of volcanic eruptions, is also independent of ΔT2×. This is why interannual variability can tell us nothing directly about ΔT2×. At low frequencies (≥ 30 years), however, the temperature-response spectrum of passive internal variability depends critically on ΔT2×.
If we had evidence of large natural climate excursions in the past on the 50-year time scale, there would be two