• Andrew I.L. Williams
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An informal reply to Vecchi and Soden (2007)

Vecchi and Soden (2007) (hereafter VS07) is a historically important paper, for a number of reasons. To my mind, it was one of the first papers which demonstrated that the strength of the tropical overturning circulation decreases in response to climate warming, and a major component of this weakening occurs in zonally-asymmetric circulations such as the Pacific Walker Circulation. These ideas were not unprecedented at the time, and many of the initial insights into the response of convective mass transport to warming go back to Betts’ 1998 essay (Betts, 1998). The arguments of Betts (1998) were reframed and repopularized by Held and Soden (2006), and VS07’s primary contribution was that they explored these ideas in an ensemble of full-complexity climate models and showing that circulation weakening in response to warming is a robust prediction of GCMs.

VS07 not only popularized the idea of tropical circulation weakening under warming, but also proposed a mechanism for this which has remained lodged in the minds of climate scientists for decades since. To explain the weakening of tropical overturning circulations, they appeal to an argument first made by Betts (1998) and later popularized by Held and Soden (2006). In short, the argument says that because convection transports moisture from the boundary layer to the free-troposphere, where it condenses as precipitation, we can (in the global-mean) write:

$$P=M_{c} \, q_{BL}$$

Where q_BL is the boundary layer specific humidity, P is the surface precipitation rate and M_c is the convective mass flux at cloud-base (the interface between the boundary layer, where moisture is sourced, and the free-troposphere where it condenses as precipitation). Now, under global warming we find that boundary layer saturation specific humidity (q_BL) increases rapidly (~6-7 %/K) and that the precipitation rate (P) increases much more slowly (~2-3%/K). The reason for this mismatch is because global-mean precipitation is constrained by the atmospheric energy budget, whereas boundary layer specific humidity just increases following the Clausius-Clapeyron relation. Now, if Eq. (1) is to hold while P is increasing much more slowly than q_BL, it requires that the convective mass transport between the boundary layer and the free-troposphere must decrease at ~5%/K.

This weakening of cloud-base Mc is what VS07 argues to be the driver of a weakening tropical circulation under warming. As evidence for this claim, they plot changes in the inferred M_c, calculated by plugging global-mean P and q_BL into Eq. (1), against changes in the 500hPa vertical pressure velocity (\(\omega_{500}\)) integrated over ascending regions (\(\omega^{+}_{500}\)). This is shown in Fig. 4a of their paper, reproduced below.

At first glance, this seems quite promising! Indeed, VS07 themselves conclude that "...this further supports the basic premise that the slower increase of precipitation relative to water vapor should lead to a weakening of the circulation." However, one of the points of this brief note is to highlight a key flaw in this figure.

To see this, notice that the units on the x- and y-axis are simply percentages. This is because VS07 do not normalize these changes by the different amounts of temperature change across models. This approach neglects the fact that global-mean temperature changes may independently affect both \(\omega^{+}_{500}\) and M_c, and changes in these quantities need not be linked. For example, a rising tropopause under global warming is expected to cause an increase in the ‘gross moist stability' of the tropics, which (all else equal) will cause a weakening of tropical overturning circulations. Additionally, changes in precipitation and water vapor are both robustly linked to global-mean temperature.

A simple way to understand Fig. 4a of VS07 is to imagine two quantities (\( X\) and \( Y\)), both of which scale in some way with global-mean temperature change such that \(\Delta X \sim \alpha \Delta T\) and \(\Delta Y \sim \beta \Delta T\). In essence what VS07 have plotted in their Fig. 4a is \(\Delta X \sim \alpha \Delta T\) and \(\Delta Y \sim \beta \Delta T\), for various values of \(\Delta T\). In this case you are guaranteed to get a correlation, with slope \(=\beta/\alpha\). Indeed, a strong correlation would be expected between any two variables which scale with changes in global-mean temperature (e.g., tropical circulation strength strength and Arctic sea-ice extent). Clearly, this would not suggest a physical link between Arctic sea-ice extent changes and changes in the strength of the tropical overturning circulation.

In climate science, it is common knowledge (nowadays) that coupled models have a variety of different transient and equilibrium climate sensitivities and thus warm differently over a given time interval when subject to identical forcing. As a result, it has become standard practice to consider fractional changes in a quantity of interest per degree of global warming (or to consider changes at a certain global warming level). This is done precisely to avoid the error which has occurred in Fig. 4a of VS07.

So, what happens if we (re-)calculate Fig. 4a from VS07, but now we correctly normalize each model by its warming rate (to account for the confounding impacts of differing ECS and TCR on the correlation)? As it happens, this was already done for us by Su et al. (2019). The resulting figure, reproduced below, shows that while a strong correlation between these variables might exist when calculating fractional changes per decade (right panel), in a similar manner to VS07, this correlation disappears entirely when one correctly normalizes by the modeled warming rates over this period (red markers in the left panel).

Why didn’t this result cause a stir? I think it is mostly because this figure from Su et al. (2019) only appears in the Supplemental Information, and isn’t really talked about much in the paper itself. Nevertheless, I myself found this to be quite an important finding, which is why I’m writing this note!

There are a number of different issues with the VS07 argument. For example, does '(\( P = M_{c} q_{BL}\)' actually hold? Previous studies have suggested that it does not (Schneider et al., 2010; Jeevanjee, 2023) and we are currently about to submit a paper which tests this scaling in cloud-resolving models (spoiler: doesn’t work there either). It is thus not a good idea to use 'P=Mq' to reason about changes in convective mass flux with warming. However, I want to focus on something else here...

Another error in the VS07 argument, which is more insidious, is the equating of changes in a mass flux with changes in vertical velocity. Convective mass flux (M_c) is defined as:

$$M_{c}= \rho A_{c} w_{c}$$

Where \( \rho\) is the air density, \( w_{c}\) is the speed of convective updrafts and \( A_{c}\) is the fraction of the domain (e.g. the tropics) which is covered by these convective updrafts. The units of a mass flux are kilograms per square-meter per second (kg/m2/s); on the other hand vertical velocities are measured in meters per second (m/s). Thus, even if ‘P=Mq’ did hold up in models and thus gave us a robust constraint on the behaviour of convective mass fluxes with warming, this need not translate into a constraint on upwards vertical velocities! In constraining Mc we constrain the product of the area of convection (\( A_{c}\)) and the strength of convective ascent (\( w_{c}\)). Thus, there is not a one-to-one relationship between changes in \( M_{c}\) and changes in \( w_{c}\) (which we may think of as being more closely linked to the tropical overturning circulation).

It’s worth pointing out that the Held and Soden (2006) paper makes this point already, when they say that ‘...a reduction in the mass exchange in the tropics does not necessarily entail a proportional reduction in the strength of the mean tropical circulation’ (and they give examples where changes in \( M_{c}\) would not be associated with changes in the mean circulation). It appears that this nuance has been lost in the proceeding years.

A good example of the non-uniqueness of the \( M_{c}-w_{c}\) relationship comes in the form of the strengthening of the Walker Circulation over the historical period, despite a decrease in Mc. This was shown most nicely by Sandeep et al. (2014), and the key figure from their paper is reproduced below. In red/blue are the trends in convective mass flux, and in black are the trends in the Pacific Walker Circulation (PWC). AMIP-style simulations with prescribed sea-surface temperatures were conducted in two models, which correspond to the top and bottom panels.

Over the historical period, particularly the last 40 years or so, we have seen the tropical Pacific move towards a ‘La Niña-like’ state, which simply means that the East-West SST gradient in the tropical Pacific has strengthened (like it does during a La Niña event). It is well documented that this has led to a contraction of the area of the tropics which is covered by deep convection, and thus we may this of tropical circulation trends as being a competition between decreases in convective mass (which on their own would act to decrease \( w_{c}\) and a contraction of the convective area (which acts to offset the influence of mass flux decline). When framed from this perspective, the concurrent decline of convective mass flux and strengthening of the tropical circulation is not that confusing (although the reasons for the ‘La Niña-like’ SST trends are still a point of great contention).

Nevertheless, a number of papers have viewed the recent Walker Circulation trends as being in opposition to a mass flux-based argument (e.g. Heede et al. 2020; Watanabe et al. 2023), which I argue is not the case. As far as I can tell, this apparent contradiction in the literature can be linked squarely back to the VS07 paper and their choice to scatter changes in a mass flux against changes in vertical velocities in their Fig. 4a.