Consider a simple energy balance model for the oceanic mixed layer (+ land + atmosphere) with temperature and effective heat capacity . The net downward energy flux at the top-of-atmosphere is assumed to consist of a radiative relaxation, , plus some noise, . With imposed flux from the deep ocean to the mixed layer, :
The assumption is that there is no external radiative forcing due to volcanoes or increasing CO2, etc. Can we use a simple one-box model like this to connect observations of interannual variability in and to climate sensitivity?
[This model is central to two papers by Spencer and Braswell (SB08, SB11). I provided a (signed) review for Journal of Climate of SB08, a fact that Spencer has publicized. My advice to simplify down to this one-box model, since it captures the essence of their argument, was followed by the authors. I was unclear on how best to estimate the ratio of the variances of and in this model — or, for that matter, whether this model was the appropriate setting for fitting to the data in question — so I didn’t address these issues in the review and recommended publication to let others work this out. I mention this since several people have asked me about my role in this paper.]
This is a linear equation that one can break up into the sum of two terms , where
, and .
The key assumption is that and , and, therefore, and are uncorrelated — or and for that matter. Since we are interested in estimating from observations of and — a starting point might be regressing against , as in Forster and Gregory (2006). Defining
where brackets are a time average, the difference between and is what we are interested in:
The equation for has been used in setting equal to . One could stop here, or rewrite the last expression as
(used by Murphy and Forster (2010)– MF10– in their critique of SB08), but there is actually no need to refer to — and we don’t need an evolution equation for . SB11 use this same model, looking at observed phase lags between and to argue that noise-generated temperatures must be significant in the observations.
The equation for holds if we filter all fields to emphasize certain frequency bands, so if one has some idea of the relative spectra of and , one could design the filter to reduce the effects of .
The meaning of and assumptions about its spectrum are the source of much of the confusion about this model, I think — at least it has been the source of my own confusion. In the simple model, is the variability that you have at the TOA if is fixed. You can try to estimate this fixed-T flux variance with a GCM. I looked at a few simulations with GFDL’s AM2.1 atmospheric model with fixed SSTs when I first saw the draft of SB08. They use 1.3 W/m2 for the standard deviation of monthly means of , for a domain covering tropical oceans only, while AM2.1 with fixed SSTs produces 1.0 W/m2 for the net radiation over the same domain, which did not raise any flags for me. (The model’s global mean noise amplitude in monthly means is abut 0.6 W/m2.) In the GCM, this noise is essentially uncorrelated from month to month. In preparation for this post, I tried varying the parameters in the simple model over ranges that I thought were plausible, assuming that the decorrelation time for is no longer than a month, and could not generate cases with large enough to create significant ( > 10%) differences between and and phase lags in the right ballpark — if I push the parameters to create more noise in the temperatures (by reducing the heat capacity (depth) of the mixed layer, in particular) the TOA fluxes are too noisy. Returning to SB11, I noticed something that I missed the first time through, that they pass their “noise” through a 9-month top-hat smoother. If I do this and tune the noise variance, then things look more reasonable. But can “noise” with this spectrum be independent of (ie ENSO)? I personally can’t imagine how a model with fixed SSTs can produce TOA flux variations with this long a decorrelation time.
I think a more plausible picture is something like the following. The central and eastern tropical Pacific SSTs warm due to heat redistribution from below and relatively quickly warm the entire tropical troposphere; this stabilized atmosphere reduces convective cloudiness over the tropical Indian and Atlantic Oceans, reducing the reflection of the incident shortwave in particular, producing warming of these remote oceans that takes several months to build; the global or tropical mean temperature has a component that follows this remote oceanic response. So temperatures have a component forced by TOA flux anomalies, but these anomalies would not exist without the Pacific source of variability. (This is a caricature of the tropical atmospheric bridge described more fully by Klein et al 1999 in particular). The kind of model that is used to study this sort of thing is an atmosphere/land model, with prescribed ocean temperatures in the tropical Pacific, but with the rest of the ocean consisting of a surface mixed layer, either a uniform 50m slab of water or something more elaborate with a predicted depth, that adjusts its temperature according to the simulated surface fluxes (e.g. Lau and Nath (1999, 2003),. See the figure at the top of this post. It would be interesting to compare the results from this kind of model to tropical mean or global mean TOA flux observations.
A simple change in the box model that might capture a bit of this would be to set , where is real atmospheric noise with appropriately short decorrelation time, and is a constant that relates the remote change in TOA flux to ENSO. (Ignoring the time it takes to set up this flux response may be an oversimplification.) I would also give the spectrum of typical ENSO indices. This model generates phase lags without noise (), but adding some noise might still be useful.
Changes in tropical circulation associated with ENSO warmings are quite different from the circulation responses we expect from increases in CO2, and cloud feedbacks in particular are presumably sensitive to these circulation changes. From the perspective of a climate modeler, one thing that I would look for, as discussed in posts #12 and #15, is if this co-variability of TOA fluxes and surface temperatures provides a metric that distinguishes between GCMs with different climate sensitivities. Actually, rather than equilibrium or transient climate sensitivity, I would look instead directly at the strength of the radiative restoring in transient warming runs — the canonical 1%/year CO2 growth simulations being the simplest. (The strength of radiative restoring changes in models as the system equilibrates (post #5), and equilibrium sensitivities are typically estimated by extrapolation in any case — while the transient climate response depends on ocean heat uptake processes that play little or no role in interannual variability). If GCM simulations of this co-variability are not somehow correlated across an ensemble of models with the radiative restoring that occurs when CO2 increases, one would have to understand this, since it would suggests that the connection is not very direct. I am not aware of a paper that describes this correlation either in the CMIP3 or perturbed physics model ensembles in a comprehensive way — but I’m not aware of a lot of things.
[The views expressed on this blog are in no sense official positions of the Geophysical Fluid Dynamics Laboratory, the National Oceanic and Atmospheric Administration, or the Department of Commerce.]