A simple conceptual model of surface specific humidity change over land is described, based on the effect of increased moisture advection from the oceans in response to sea surface temperature (SST) warming. In this model, future q over land is determined by scaling the present-day pattern of land q by the fractional increase in the oceanic moisture source. Simple model estimates agree well with climate model projections of future (mean spatial correlation coefficient 0.87), so over both land and ocean can be viewed primarily as a thermodynamic process controlled by SST warming. Precipitation change is also affected by , and the new simple model can be included in a decomposition of tropical precipitation change, where it provides increased physical understanding of the processes that drive over land. Confidence in the thermodynamic part of extreme precipitation change over land is increased by this improved understanding, and this should scale approximately with Clausius–Clapeyron oceanic q increases under SST warming. Residuals of actual climate model from simple model estimates are often associated with regions of large circulation change, and can be thought of as the “dynamical” part of specific humidity change. The simple model is used to explore intermodel uncertainty in , and there are substantial contributions to uncertainty from both the thermodynamic (simple model) and dynamical (residual) terms. The largest cause of intermodel uncertainty within the thermodynamic term is uncertainty in the magnitude of global mean SST warming.
Future changes in surface humidity under global warming could affect human comfort (Willett and Sherwood 2012), labor capacity (Dunne et al. 2013), and even, for extreme warming scenarios, the habitability of some land regions (Sherwood and Huber 2010; Pal and Eltahir 2016). Humidity changes are also closely linked to changes in surface evaporation, transpiration, soil moisture, mean and extreme precipitation, and clouds (e.g., Emori and Brown 2005; Fasullo 2012; Sherwood and Fu 2014; Byrne and O’Gorman 2015; Chadwick 2016; Kamae et al. 2016).
Over the oceans, energy balance considerations (described in Held and Soden 2000; Schneider et al. 2010) suggest that surface specific humidity (q) should increase approximately in line with surface air temperature under fixed relative humidity (RH), which implies increases of around 7% degree−1 of local warming under the Clausius–Clapeyron equation (Held and Soden 2006). Climate model projections agree well with this prediction (Held and Soden 2006; Sherwood et al. 2010; O’Gorman and Muller 2010), with only small RH increases projected over the oceans. Observed and modeled trends of historical surface q change over the oceans are also consistent with this Clausius–Clapeyron rate of increase (Dai 2006).
Over land it is far less obvious how surface humidity will respond to warming, as the moisture supply from the land surface is often limited, and the energy balance arguments that apply to oceanic are much less relevant. Net moisture transport from ocean to land increases under warming in future model projections (Zahn and Allan 2013), but the exact implications of this for humidity change are unclear. Observed trends from the period of 1973–99 are consistent with a local Clausius–Clapeyron response under fixed RH in many land regions, but the relationship between local temperature and q change breaks down in drier regions (Dai 2006; Willett et al. 2010). Over the more recent period of 1999–2015 there is evidence of significant decreases in RH over land (Simmons et al. 2010; Willett et al. 2014, 2015), causing this local Clausius–Clapeyron model to be brought into question. Agreement between the in situ record and reanalyses is good (Willett et al. 2015), but natural variability over the observed period and the sparseness of the observational network over many land regions means that care must be taken when interpreting these trends. Climate models do not reproduce the observed recent decreases in RH when run in coupled atmosphere–ocean configuration over the same period, but do slightly better when forced with observed SSTs (R. J. H. Dunn et al. 2016, unpublished manuscript).
In end-of-twenty-first-century climate model projections there is a clear reduction of surface RH over many land regions, corresponding to an overall sublocal–Clausius–Clapeyron increase in q over land (O’Gorman and Muller 2010). It has been suggested this is due to the increased land–sea temperature contrast (e.g., Joshi et al. 2008), which outstrips the ability of moisture supply—originating from the less warm oceans—to maintain constant RH over the warmer land surface (Rowell and Jones 2006; O’Gorman and Muller 2010; Simmons et al. 2010). O’Gorman and Muller (2010) made a rough estimate of how land RH would decrease in response to this enhanced land warming if specific humidity change were identical between land and ocean, and this estimate is consistent with GCM projections of the change in global land RH. However, they noted that this homogenization of between land and ocean does not actually occur, and so more work is needed to understand humidity change over land, particularly on regional scales. Berg et al. (2016) demonstrated that land–atmosphere feedbacks can substantially alter the change in land RH under warming, and so the advection of moisture over land is not the only important process in determining changes in relative humidity.
In the present study we extend these ideas of how surface q over land might respond to increased q over the oceans. We start from the following principles: 1) on climate time scales, the vast majority of atmospheric moisture over land originates from the oceans (Simmons et al. 2010; Gimeno et al. 2010, 2012; Van der Ent and Savenije 2013) and 2) the adjustment time scale of moisture over land in response to transport from the oceans is much faster than the warming time scale of the surface oceans under climate change. These underlying principles are combined with a number of simplifying assumptions to build a simple conceptual model of how surface specific humidity over land might respond under climate change to increased moisture advection from the oceans, driven by warmer SSTs. The simple model is then compared with climate model projections.
Section 2 describes the climate model data used in this study. Section 3 presents the basic concept of the simple model and applies it to climate model projections, followed by a more detailed derivation. The model assumptions are considered in detail in section 4. Section 5 analyses intermodel uncertainty in GCM projections of future , and relates it to the simple model and its residual. Section 6 describes how the new simple model can be integrated into a decomposition of tropical precipitation change. Finally, a summary and some conclusions are presented in section 7.
Monthly data from the first ensemble members of 28 phase 5 of the Coupled Model Intercomparison Project (CMIP5; Taylor et al. 2012) models were used, listed in Table 1. The present-day period was taken as a climatological mean over the years 1971–2000 of each model’s historical run, and the future period was a 2071–2100 mean from the high-emissions representative concentration pathway 8.5 (RCP8.5) scenario. All model data were regridded to a common resolution of 2.5° for comparison with one another.
3. A simple model of oceanic moisture advection over land
a. Simple model
Our simple conceptual model represents the change in moisture over land that occurs in response to advection of increased q from the oceans, under SST warming. It assumes the circulation remains unchanged, and so can be seen as the thermodynamic component of q change over land. It takes the form of a scaling of the present-day pattern of land q by the fractional increase in the oceanic moisture source:
where is the time-mean surface (2 m) specific humidity at a given point over land, is the time-mean surface specific humidity at the oceanic moisture source for that land point, and subscripts p and f denote present-day and future climates, respectively.
From this, we can predict the spatial distribution of from the fractional change in over the ocean, combined with the present-day pattern of . We need to specify which region of ocean we are using to represent (the oceanic moisture source) for each land point, and we choose to use the time-mean zonal mean of ocean q at the latitude of each land point. From Eq. (1), it can be seen that the sensitivity of estimates of to the choice of source region depends on the spatial uniformity of . This turns out to be fairly zonally uniform over the oceans at most latitudes (Figs. 1b, 2b), so the choice of zonal mean values appears to be reasonable.
Simple model estimates (hereafter ) are shown in Figs. 1 and 2, and they perform well in capturing the GCM-projected patterns of over land, with a mean spatial correlation coefficient (between simple model estimates and GCM projections) of 0.87 across CMIP5 models. This is particularly true for the multimodel mean (Fig. 1), but they also perform well for individual models (HadGEM2-ES is shown in Fig. 2), with a few regional exceptions that vary by model and that we will examine later in more detail. The pattern of is determined mainly by the pattern of present-day in each model (spatial correlations between and actual GCM are similar over land to those between and ), while the magnitude is determined by the size of SST-warming-driven q increases over the oceans. This is because the circulation is assumed to be unchanged, so the simple model simulates the effect of future oceanic moisture increases being transported over land by the present-day circulation. This assumption of an unchanged circulation will be examined in detail, and relaxed, in section 3d. Variations in zonal mean (Figs. 1b, 2b) make a much smaller contribution to the pattern of , except at very high northern latitudes where there is a strong amplification of surface warming and fractional moistening over the oceans.
The simple model provides a better estimate of GCM q change than an alternative simple model of assuming that local RH remains constant over land, and so q increases with the Clausius–Clapeyron (CC) response to local warming (Figs. 1e,f, 2e,f). This alternative model provides an accurate description of q change over the ocean and can be considered a type of null hypothesis of how q might change over land. It is calculated from the local 2-m temperature increase using the August–Roche–Magnus formula (Lawrence 2005), assuming the percentage change in q is equal to the percentage change in saturation vapor pressure under fixed relative humidity. Skill scores comparing the two methods for each model are shown in Table 1, with the new simple model of moisture advection performing better in almost all cases. On average the RMSE is reduced by a third, and the magnitude of the bias is halved compared to the local CC estimate. The sign of the bias also changes from positive to negative.
This simple conceptual model is derived by considering moisture advection from the oceans in a Lagrangian airmass framework. We now demonstrate this derivation in first a basic then a more detailed manner.
b. Basic derivation: Sum over air masses
At any given point in time, surface q at a grid point is broadly equivalent to the moisture contained in the low-level air mass that lies over the point. We generalize this to climate time scales, so that time-mean q at some grid point over land is determined by a weighted sum over a number of distinct air masses that bring moisture to that point. Weights correspond to the proportion of time that the circulation brings each air mass to the point. The moisture carried to a point by each air mass depends on the moisture at the oceanic source (where the air mass crosses the coastline) combined with the net moisture change of the air parcel between the coast and the land point. The magnitude of this moisture change depends on the amount of evaporation into the air mass and vertical mixing of moisture out of the air mass’s boundary layer on the way. We treat these moisture changes collectively as properties of the path of the air mass.
This can be formulated as
where is the time-mean 2-m specific humidity at a given point over land, i is used to sum over air masses, is the specific humidity of air mass i at the land point in question, and is the specific humidity at the oceanic source of each air mass. The term is the weight attached to each air mass, corresponding to the proportion of time that air at the land point comes from that air mass; is a function of fractional moisture change along the path of the air mass, in general varying between 1 and 0 (though values above 1 are possible); and is the distance traveled by each air mass along its path between the ocean source (the coast) and the land point. The function F is often a monotonic decreasing function between the coast and the land point, though there are exceptions to this, particularly in tropical forest regions (see section 4b). Savenije (1995) used a similar simple model to examine moisture recycling over West Africa, with F in that case being an exponentially decreasing function of distance from the ocean. For the purpose of the current derivation, the exact form of this function turns out not to be important.
Under climate change, sea surface temperatures (SSTs) increase and, as a result, specific humidity in the low-level air above the oceans also increases, approximately in line with the Clausius–Clapeyron equation (see Figs. 1e,f). From Eq. (2), q at a land point in the present-day and future climates can be written as
where p denotes present-day climate, and f denotes future climate. Changes in caused by increasing can be thought of as thermodynamic changes, and those associated with changes in w, F, or d can be loosely thought of as “dynamical” changes. As well as potential changes in circulation and vertical mixing along each path, these dynamical changes will include changes in evaporation and transpiration, some of which may be unrelated to circulation change. It should also be noted that it is not possible for d to change independently of F, whereas F could change independently of d.
We can then explore what would happen to in the presence of thermodynamic increases, if the dynamical components were to remain fixed. In this case w, F, and d are assumed to be constant between present and future climates, and so from Eqs. (3) and (4),
We now assume that, within each present and future climate period, the spatial and temporal variation in q over the ocean sources that supply moisture to any given land point is relatively small, so that for both present and future climates. Equation (5) then simplifies to
which is a very simple model of moisture change over land in response to increased advection from the oceans, and identical to Eq. (1).
c. A more detailed derivation: Lagrangian path-integral approach
This simple model appears to work well in most regions, despite the fact that there actually are significant changes in circulation, vertical mixing, and evaporation over many land regions under warming (e.g., Chou et al. 2009; Seager et al. 2010; Chadwick et al. 2013). Thus, specific humidity change over land is mainly determined by the magnitude of moisture increases over the oceans, combined with the pattern of present-day ocean-to-land moisture transport. This is consistent with the results of Zahn and Allan (2013), who examined future changes in moisture transport across continental coastlines and found that these are primarily an enhancement of the pattern of present-day ocean–land moisture transport.
This dominance of thermodynamic processes is in complete contrast to the response of land precipitation to warming, where the pattern of change is dominated by dynamical circulation change and/or local evaporation change (Chadwick et al. 2013; Roderick et al. 2014; Greve et al. 2014; Oueslati et al. 2016). As precipitation over land would be on average around 40%–60% lower without any moisture recycling (Van der Ent et al. 2010; Rodell et al. 2015), it is also not obvious why any changes to this recycling would be less influential on surface q change than changes in ocean moisture sources.
To understand why our simple model of q change is successful, we now use a Lagrangian path-integral approach to examine in more detail how moisture loss occurs along the path of each air mass and how this might change under warming. This approach is in some ways similar to that of Savenije (1995), but with differences in its purpose and assumptions, including the use of an atmospheric boundary layer moisture budget instead of a budget for total atmospheric precipitable water.
For a given air mass, we can write its total boundary layer moisture content , at some point along its path over land, as a path integral of moisture loss between the source of the air mass and the land point:
where is the total boundary layer moisture content at the oceanic source of the air mass. We assume a well-mixed boundary layer of fixed column mass, such that the 2-m can be recovered from simply by dividing by the total (fixed) mass of the boundary layer air column, and similarly for from . The term is the boundary layer specific humidity; is the total mass of air mixed vertically between the boundary layer and free troposphere; is a measure of free-tropospheric specific humidity, all at point x along the path; and is the total moisture flux (evaporation and transpiration) from the surface to the boundary layer at point x on the path. We neglect the smaller contributions from reevaporation of falling hydrometeors and precipitation originating within the boundary layer.
Now, assuming that boundary layer q is on average much greater than that in the free troposphere, then , and
Over land, the Mezentsev–Choudhury–Yang equation (Mezentsev 1955; Choudhury 1999; Yang et al. 2008), derived from the Budyko framework (described in, e.g., Roderick et al. 2014), gives a curvilinear relationship between E and P:
where is the upper energy limit on evaporation, defined as the liquid water equivalent of the net downward radiative flux at the surface, and n is a catchment properties parameter that modifies the partitioning of P between E and runoff. Figure 3 (for HadGEM2-ES), and Fig. 2 of Roderick et al. (2014) (for the CMIP5 ensemble mean), show the mean shape of the relationship between E and P in GCMs, and this does not noticeably change between present-day and future climates. Note that these figures combine land points over a range of different values of and n, and so are illustrative of the Budyko curve in general, rather than of any particular combination of parameter values at a given grid point. GCMs do not directly use the Budyko framework [Eq. (9)] to calculate E; it is instead an emergent property of the simulated relationship between E and P (Roderick et al. 2014).
So, for given fixed values of and n at some grid point, E can be written as a function G of P:
It has been suggested the pattern of evaporation change over land in response to warming is in general a response to changes in moisture availability, governed by changes in precipitation, rather than either changes in energy availability at the surface or changes to the catchment properties parameter (Roderick et al. 2014). Making this assumption, the response of E to warming can be modeled using a single curvilinear function at each grid point. Although Eq. (10) is derived in an Eulerian frame of reference, a very similar relationship between E and P should also hold at each point x along the climate-mean path of an air mass. So, we can write
Now, in the present climate, we can define at each point x as some value between 0 and 1, such that . We then make the approximation that in the future climate, , where is unchanged from present-day values at each point. In other words, we use to approximate a section of a curvilinear function , at each point x, as a linear function. This is a reasonable assumption in moisture-limited regions, as values of E and P lie along a section of G that is quasi-linear (Fig. 3), but may be less so in more moisture-abundant regions, where it is likely to overestimate the change in E from any given change in P. This linear approximation can be used to predict local changes in E from changes in P (Fig. 4 shows this for HadGEM2-ES), and to first order gives an accurate estimate of . This assumption is examined in more detail in section 4e. Combining this approximation with Eq. (8) for present-day and future climates gives
We now make the assumption that, for climate means, the moisture source of precipitation at each grid point is dominated by local vertical advection of moisture from the boundary layer, rather than convergence of moisture within the free troposphere, so (Held and Soden 2006; Chadwick et al. 2013). This has been shown to be a good approximation in GCMs within tropical regions (Chadwick et al. 2013; Kent et al. 2015), where convective rainfall dominates, but is likely to be less accurate outside of the tropics. Substituting this into Eq. (13) gives
where m is the mass of a boundary layer air column in each section of the path (assumed constant). So, to recover Eq. (1)—after also averaging over different air masses—we need to show that is constant between present-day and future climates. From Eq. (17) this is equivalent to showing
To show this, we now approximate the integral in Eq. (19) as a sum over very small distances in the path of the air mass, so we want to demonstrate
where the subscript j denotes values at the jth step along the path. We do this by using a proof by induction. We first demonstrate that Eq. (20) is constant for the first step of the path from the oceanic source, and then show that if it is constant for the nth step of the path, it must also be constant for the (n + 1)th step. If this is true, then it must be constant for all steps along the path, including the final one at our land point l, and so Eq. (20) is satisfied. We will also show which conditions need to be met for this proof to be valid.
For the first step,
which can be rearranged to give
The value of has already been assumed to be constant between present and future climates, as has m, and the distance is taken to be unvarying between present and future climates. The only further assumption we need to make for this to be constant is that the amount of vertical mixing M at each step in the path remains constant between present-day and future climates.
For the second part of our proof, we first assume is constant between present-day and future climates. We then seek to show that if this is the case, is also constant. So,
which is indeed constant under the same assumption about M as applied to the first step, with the additional assumption that the route and length of the path traveled by the air mass remain unchanged. So, by proving by induction that Eq. (19) is true (under certain assumptions), we have recovered our original simple model of Eq. (1). Consequently, we now have a much better physical understanding of the assumptions that are needed for to be constant and so for our simple model to be accurate.
d. Why the simple model works
Some interesting points come out of the more detailed formulation in Eq. (17) that serve to limit the impact on of any changes in M in a future climate. First, the effect of any future changes in M on q at a given grid point are damped by multiplication by , with α quite close to 1 in moisture-limited regions, and generally above 0.5 in most regions (see Fig. 3). Physically, this is because local evaporation, precipitation and vertical mixing are linked, so the effect of any changes in M and P on q are opposed by changes in E, and vice versa (see Fig. 4).
Second, as is a path integral, the effect of any small-scale future changes in M on will generally be limited, as these will only affect a small proportion of the path of an air mass. Only consistent changes in the amount of vertical mixing along a large proportion of the path will make a substantial difference to the path integral, and hence to . These two damping effects on the influence of changes in vertical mixing (i.e., circulation) are likely to be the main reasons why thermodynamic processes dominate the pattern of change. In contrast, neither of these damping effects will apply to precipitation change, which is therefore much more affected by any changes in local circulation.
We can also consider the potential effect on of the slowdown of the tropical circulation predicted by GCMs (Held and Soden 2006; Vecchi and Soden 2007). Tropical circulation change under warming can be considered the sum of a large-scale slowdown of the circulation and spatial shifts in the regions of convection (Chadwick et al. 2013). We now explore the consequences of relaxing our assumptions so that M is allowed to respond in future to the circulation slowdown, but still not to convective shifts. We first make the assumption that the amount of time-dependent vertical mixing at each grid point is proportional to the local convergence of horizontal winds within the boundary layer . This assumption is appropriate for vertical mixing associated with convection, but may be less so for shallower mixing between the boundary layer and free troposphere. So now, the change in due to the weakening circulation scales with , which itself scales with the change in (as we are still assuming no shifts in the pattern of convection and convergence).
Returning to the Lagrangian formulation of Eq. (17), if the change in time-dependent over each section x of the path does scale with the decrease in the speed of the air mass over that section , the total amount of M during section x will remain the same. In that case, the weakening of the tropical circulation would have no effect on the total vertical mixing of the air mass during its path, and therefore no impact on . This can be understood if it is considered that in a Lagrangian framework, the moisture change within an air mass is not dependent on its speed of travel, only on the total mixing and evaporation during its path. This further explains the differing responses of q and P to SST warming, as the weakening circulation opposes the thermodynamic increase of (Seager et al. 2010; Chadwick et al. 2013) but not of .
By considering the sum over air masses in Eq. (2), we can try to predict the circumstances under which changes in circulation might have more or less of an effect on the time-mean . If air masses have relatively similar sources, and moisture loss over their paths, then any future change in the weights w (i.e., the amount of time that each air mass is present over a certain grid point) will not have a large impact on . In fact, the presence of multiple air masses that contribute to the time-mean will serve to buffer from the effect of changing circulation (M or path) in any particular air mass.
However, if a grid point is influenced by air masses with very different properties, even small changes in weights or circulation paths could have a strong influence on . For example, in monsoon regions, any seasonality changes that extend or contract the length of the dry season, or a northward or southward shift in the maximum extent of the monsoon flow, could have large consequences for , particularly at grid points on the margins of the monsoon.
4. Model assumptions
This simple moisture advection model contains quite a few assumptions, which we examine here. We also consider the conditions under which the simple model is likely to be less accurate.
a. Constant circulation under warming
The largest assumption of the simple model is that the circulation is unchanged in a future climate, such that the path of each air mass and the vertical mixing along it (F) remains unchanged and the relative contribution from each air mass to the mean moisture at a given land point (w) remains the same. In practice, GCM projections do not conform to this assumption, with substantial circulation changes over many land regions in response to warming. These circulation changes dominate the pattern of precipitation change over land, so the question here is why the same is not true of specific humidity change, and why our simple model of thermodynamic moisture increase from oceanic sources performs so well.
As described in section 3d, there are two main reasons why this is the case, both of which are apparent from Eq. (17). The effect of changes in M on are mitigated by two factors: 1) The correlation between changes in M and changes in E, reflected in the factor . This substantially reduces the impact of any local changes in M on local q. 2) The fact that depends on a path integral of vertical mixing, so any localized change in M is unlikely to have a large effect on the path integral as a whole. Neither of these mitigating factors is true for precipitation change, which is therefore much more sensitive to local circulation change.
The assumption of unchanged circulation can be relaxed to allow the tropical circulation to weaken, without changing the pattern of convergence and convection. Because this does not affect the total mixing over a path, only the speed at which the mixing takes place, it also has no effect on in our simple model. Again, this is not the case for precipitation change.
In regions where circulation change is very large or the balance between air masses with different characteristics changes substantially, the actual is likely to deviate from the simple model prediction . We now examine several regions where the residual from the simple model is largest in HadGEM2-ES to see if these changes are linked to circulation change.
Figure 5 shows present-day q, , circulation and precipitation change over northern Africa and much of Europe during JJA. The moistening over the central Sahel region appears to be driven by a northward extension of the low-level southwesterly monsoon winds, which bring moist air from the Gulf of Guinea instead of the dry Saharan air transported by the present-day northeasterly winds into this region. There is a precipitation increase associated with this circulation change, but it does not extend as far northward as the signal of , probably because of the separation between the position of the low-level convergence zone and the rainband during the West African monsoon (e.g., Nicholson 2013). A reduction in large-scale descent over the Sahel and Sahara can also be seen (Fig. 5c), though it is not trivial to relate this directly to the and anomalies as it also includes a signal from the weakening tropical circulation.
A negative signal is seen over most of Europe during JJA (Fig. 5b), associated with a change in mean low-level circulation toward more easterly winds, bringing dryer air from continental Russia and Asia to the region. A similar drying is seen in precipitation change (Fig. 5b), though it is unclear if this is partly driven by the signal or is a cause of it. Changes in mean large-scale descent in this region (Fig. 5d) are less easy to relate to the mean water cycle changes, as would be expected outside of the tropics.
Figure 6 shows the strong drying signal over the eastern Amazon region during SON. This appears to be linked to a strong reduction in ascent and precipitation over this region (Fig. 6c,d), though it is not possible to establish the direction of causality. The negative signal could be driven by reduced convergence of moist air masses, reduced plant transpiration in response to increased CO2 concentrations, or a combination of both. Low-level northeasterly winds strengthen in the vicinity of the drying but not very much at the coastline itself, and so they appear to be linked to changes in the vertical circulation over land rather than to increased flow from the ocean.
As circulation change appears to account for many of the largest signals in (at least in HadGEM2-ES), this provides justification for describing the simple model residual as a dynamic component of q change, as opposed to the thermodynamic contribution of the simple model.
b. Air masses
It is assumed that the circulation bringing low-level moisture to each land grid point can be modeled as a number of distinct air masses, neglecting mixing between these air masses. An air mass was defined as having a fixed oceanic source and a fixed path with corresponding properties that determine the moisture loss along the path. In practice, each of these idealized air masses represents a group of similar air masses, with similar but not identical paths and sources, and can be thought of as being equivalent to a particular empirical orthogonal function (EOF) of the circulation over a given region. It is possible the model will work less well in regions dominated by descending air masses, as it is constructed to take account of air masses traveling at low levels and mixing with free-tropospheric air, rather than for surface air masses in which a large proportion of the air has descended from above. However, the general principle that such descending air masses would retain slightly more moisture from their oceanic source in a warmer climate might still be valid.
The neglect of horizontal mixing between air masses could potentially be important in some regions, particularly in tropical convergence zones and the extratropical storm tracks. In some cases the model formulation may be able to be adapted to take account of this. In fact, as long as the proportion of each air mass in the final mixed air mass could be estimated, the various air masses could continue to be treated separately in Eq. (2), but with reduced weights associated with each one.
In some cases the mean field actually increases from the coast into the continental interior, as can be seen over the Amazon in Fig. 6a. This can be explained within the Lagrangian airmass framework by the convergence of several moist air masses within a large convergence zone. As the air masses meet, some of the air ascends and the moisture contained within it is (on average) mostly precipitated locally, where much of that moisture reevaporates. This evaporation goes into the resulting combined air mass, composed of that part of the initial air masses that did not ascend. In this way, E can be larger than in Eq. (8) for the combined residual air mass (as M here would only contain vertical mixing from the residual air mass, not the large-scale convergence of the multiple air masses), and q can increase along its path as long as it continues to have a moisture source from converging moist air masses.
c. Oceanic moisture sources
In the derivation of Eq. (1), the spatial and temporal variation in across the oceanic sources bringing moisture to a given point over land is assumed to be small. This will often be true, as specific humidity varies relatively smoothly over the oceans, and the moisture sources for any given land point will tend to lie within the same general region of the globe. In fact, many continental and subcontinental regions receive moisture from a relatively small region of the world’s oceans (Van der Ent and Savenije 2013). So this assumption appears to be reasonable but may break down in some cases.
Given this assumption, it is necessary in Eq. (1) to specify a region of ocean as the mean moisture source for each land point. We choose to use the zonal mean at each point, and the sensitivity of to this choice depends on the spatial uniformity of . In fact, except in the high Arctic this value is fairly uniform (Fig. 1b, 2b), and so this choice of source region is unlikely to introduce major errors into the estimates of , though it could be important in some cases.
d. Boundary layer moisture budget
In our boundary layer moisture budget of Eq. (7) we neglect reevaporation from falling hydrometeors, and moisture loss from precipitation originating within the boundary layer, which for time means is reasonable compared to the other larger terms. We also assume during vertical mixing, free-tropospheric q is negligible compared to boundary layer q. For convection this is very reasonable, but may be less so for shallower mixing associated with low clouds. Although shallow mixing makes a significant contribution to drying the top of the boundary layer in some regions, it may not be as important as convection for drying the boundary layer as a whole. Therefore, it might be possible to redefine M in Eq. (7) so that such shallow mixing at the top of the boundary layer is omitted, in which case the reasons for neglecting free-tropospheric q would be clearer. Alternatively, we could consider only the moisture budget of the well-mixed part of the boundary layer. We also assume a fixed-column-mass boundary layer for our moisture budget to easily recover surface q from total boundary layer Q. It is unclear how much error this will introduce into the model, but it is unlikely to be a major factor compared to other, larger assumptions.
To relate changes in E, P, and M within the boundary layer, we make the assumption that locally. In other words, for climate means, the moisture source of precipitation is assumed to be predominantly from the local boundary layer rather than from moisture convergence in the free troposphere. This has been shown to be a good approximation within the tropics in GCMs (Chadwick et al. 2013; Kent et al. 2015), where convective processes dominate. However, it is unlikely to hold as well outside of the tropics, where frontal rainfall processes involve significant moisture convergence above the boundary layer. For extratropical regions this assumption may be one of the largest causes of deviation from the simple model, as in reality E, P, and M will not be as closely locally related as they are in the model, and therefore changes in may be less damped in response to E or M changes.
e. Relationship between P and E
We use the Budyko framework [Eq. (9)] to frame E as a linear function of P that is constant in time at each grid point. This approximation is less good in moister regions, where Eq. (9) deviates more from linearity, and evaporation change is likely to be overestimated in response to precipitation change in these regions. This in turn may cause the simple model to underestimate the decrease in q when passing over moist regions. As Eq. (9) is derived in a time-mean Eulerian framework, its application to a Lagrangian airmass path framework is a further approximation. In regions where the circulation varies rapidly between air masses with different properties, evaporation and vertical mass flux may become less closely coupled within each air mass, though this should be mitigated by averaging over the various air masses to obtain total q for a given land point.
Another potential source of error caused by this assumption is that in some cases E can change independently of P under warming because of changes in energy limitation or catchment parameter n. This could potentially cause large differences between the simple model and GCM estimates in some regions. For example, in tropical forest regions the plant stomatal response to increased CO2 concentrations alters the balance between transpiration and runoff (e.g., Betts et al. 2004), which could be interpreted as a change in the value of n. The effect of this on q might be mitigated to some extent by the tendency of P and M to reduce in response to reduced evaporation, but would still be likely to weaken the coupling between E and P and therefore reduce the accuracy of the simple model. Equation (8) shows how the simple model could be sensitive to any decoupling of local changes in E and P, as the change in Q at each point along the path is a residual of the generally larger changes in E and . As with the sensitivity of the simple model to circulation change, this sensitivity is also reduced by integration of changes in E and along the whole path of the air mass.
f. Circumstances and regions where the model is likely to be less accurate
Cases where the simple model is likely to deviate from GCM projections of q change, and dynamical changes in q become important, are listed here. We also give possible examples from the HadGEM2-ES projections in Fig. 2 of regions where each case may be occurring.
Regions with large circulation change over a substantial proportion of the path(s) of their main contributory air masses, for example, northeast South America and much of Europe in HadGEM2-ES.
Regions where E can change independently from P, or the relationship between the two cannot be approximated as linear. In particular this includes tropical forest regions in GCMs that include the plant stomatal response to CO2, for example, northeast South America in HadGEM2-ES.
Extratropical regions where frontal rainfall and free-tropospheric moisture convergence are significant contributors to precipitation, for example, western Europe and northern China in HadGEM2-ES. It is unclear exactly how much this will cause the simple model to deviate from GCM projections, as some amount of local correlation between changes in M and E would still be expected.
Regions that are influenced by air masses with very different paths and eventual moisture contents, such as the margins of monsoon regions, for example, the central Sahel and India in HadGEM2-ES. Even small circulation changes in these regions could cause the balance between air masses (i.e., ) to change and potentially cause quite large changes in mean .
Regions with a number of oceanic moisture sources with very different values of , or where at the main moisture source(s) differs substantially from the zonal mean, for example, northern Europe in HadGEM2-ES. Some of these errors could potentially be reduced by choosing different, more representative oceanic source regions for each land region.
If taken at face value, the descriptions of these regions could be used to cover almost all the land surface. Clearly, the deviation of from the simple model depends on the amount by which each of these assumptions are violated in any particular region for a given GCM, and the simple model may be more sensitive to some assumptions than others. The fact that the simple model gives quite an accurate prediction of future specific humidity change over land in GCMs suggests that in general its assumptions are not unreasonable, and the main driver of over land is advection of thermodynamic moisture increases from the oceans.
5. Causes of uncertainty in future projections of q over land
Figure 7a shows the intermodel standard deviation of future land q change across the CMIP5 ensemble. We define the fractional uncertainty of some variable x as , where in this case denotes the absolute value of the CMIP5 ensemble mean change. The fractional uncertainty in (Fig. 7b) is less than 0.5 over most of the world, which is much smaller than the fractional uncertainty in other water cycle variables such as precipitation change (Kent et al. 2015). The signal to noise in projections of land is therefore relatively high. In this section we examine causes of uncertainty in and consider why projections of are better constrained than those of .
The uncertainty in associated with processes captured by the simple model (Fig. 7c), and the simple model residual (Fig. 7e), both contribute to the total uncertainty. The difference between the total variance and the sum of the variances of the simple model and the residual is quite small, meaning that covariance between these two terms can be neglected here. Simple model uncertainty is the larger term in most but not all regions and closely follows the pattern of ensemble mean present-day and simple model (Fig. 1c). The pattern of uncertainty in the residual is very different, and it is the larger term in the eastern Amazon, the Sahel, and northern Australia. The fractional uncertainty in the simple model (Fig. 7d) is quite small everywhere except Antarctica, meaning the magnitude and pattern of the thermodynamic contribution to land q change from oceanic q increases is relatively consistent across the CMIP5 models. This is particularly true within the tropics and slightly less so in the Northern Hemisphere extratropics. Fractional uncertainty in the residual (Fig. 7f) is much higher, suggesting the dynamical contribution to land q change is much less robust across models, which is consistent with the residual term from each model tending to average to near zero in the ensemble mean (Fig. 1d).
Causes of uncertainty in the simple model term can be examined using Eq. (1), which can be rewritten with as a function of , , and :
Here the overbar denotes averaging over air masses, not over models. We now examine the uncertainty that occurs in when each of the 3 variables on the rhs of Eq. (26) is allowed to vary in turn across the CMIP5 ensemble, keeping the other two constant at the value of the CMIP5 ensemble mean. So for example, the uncertainty in associated with varying is assessed by calculating the standard deviation across models of Eq. (26) at each grid point, with allowed to vary for each model, but with the same fixed ensemble mean values of and used for every model. In this way, the contribution of intermodel uncertainty in the present-day fields of q over land and ocean, and the change in q over oceans, to overall uncertainty in the simple model can be assessed.
Figure 8 shows the standard deviation of each component. There is a negligible contribution from uncertainty in (Fig. 8a), though it is possible that this would increase if the simple model were altered to use more localized ocean q sources rather than zonal means. There is a larger contribution from (Fig. 8b), which is the biggest term in the Sahara and a few other regions, presumably because present-day biases in q are most varied across models in these regions. The dominant contribution to intermodel spread in the simple model is from uncertainty in (Fig. 8c), the pattern of which is very similar to the overall uncertainty in the simple model (Fig. 7c). The covariance between these terms is assessed by computing the difference between the total simple model variance and the sum of the variances of these three components, and is relatively small. This is shown in standard deviation units (Fig. 8d) by taking the square root of the absolute value of this difference between variances.
We can further break down the part of the uncertainty due to intermodel variations in . In this case, we repeat the calculation that produced Fig. 8c, but first normalize each model’s by multiplication by a factor of (ensemble mean change in global mean SST)/(model’s own change in global mean SST). Assuming quasi linearity of with global mean SST change, this normalization removes the part of the intermodel spread that is associated with uncertainty in global mean SST change. The resulting reduced uncertainty is shown in Fig. 8f. This represents uncertainty in the simple model due to model differences in the zonal mean distribution of SST change and variations in the small amount of RH change over the oceans. The remainder, taken as the square root of the difference of the two variances associated with Figs. 8c and 8f, is shown in Fig. 8e and represents the uncertainty in the simple model due to uncertainty in global mean SST change.
So, in summary, intermodel uncertainty in the simple model and its residual both contribute to overall uncertainty in over land. The contribution from the simple model is larger than that of the residual over the majority of global land, but the regions with the largest overall uncertainty correspond to those where the residual term is very uncertain. The single largest cause of uncertainty in the simple model term is uncertainty in global mean SST change, which controls the magnitude of the increase in moisture that is advected from ocean to land. In most regions this is more important for future uncertainty than differences in model biases in the present-day distribution of q, though this balance could be different for future scenarios with less SST warming. Fractional uncertainty in may be relatively low, compared to that in , because both the ensemble mean change and intermodel uncertainty are strongly tied to changes in global mean SST. Therefore, q change over land can be viewed primarily as a global-scale thermodynamic change, which is relatively well constrained compared to the regional dynamical changes that dominate precipitation change (Kent et al. 2015).
6. Relationship between q change and tropical precipitation change
Precipitation change under warming is influenced by the increase in boundary layer moisture, so our simple model of q change also has implications for understanding precipitation change over land. In the tropics, where rainfall is dominated by convection, precipitation at each grid point can be represented as , where M is mass flux from the boundary layer to the free troposphere (Held and Soden 2006; Chadwick et al. 2013). This leads to a decomposition of precipitation change into thermodynamic and dynamic components (Chadwick et al. 2013):
where is the thermodynamic component of precipitation change associated with moisture change at constant M, is the dynamic component of precipitation change associated with circulation change at constant q, and is the nonlinear cross term. In fact, a proxy for convective mass flux is used in this calculation, but this has been shown to be closely related to actual model convective mass flux (Chadwick et al. 2013; Kent et al. 2015).
The variable can be further decomposed into a term associated with the weakened tropical circulation and a term associated with spatial shifts in convection . Here , and , where is defined to be constant throughout the tropics. As previously found, the pattern of (Fig. 9a) is largely determined by the pattern of spatial shifts in convection (Fig. 9b), with the magnitude of changes modulated by the weaker residual between the other components (Chadwick et al. 2013).
In the standard form of this decomposition, is also separated into two further terms, and . The term is the increase in q that would be expected under a Clausius–Clapeyron increase in moisture in response to local warming, with fixed RH. Its residual is associated with local RH change. A substantial term is present over many land regions (Figs. 1e,f and 2e,f).
The new simple model of q change suggests that a more physical separation of can be obtained by instead separating into components corresponding to the simple model and its residual. We define these as and , where is the change in q over land due to advection of q from the ocean in the simple model, and . The full precipitation decomposition can then be written in two alternate ways:
All components from the two alternative decompositions are shown in Fig. 9. As expected, is much smaller than because of the improved representation of moisture change provided by compared to . The majority of the thermodynamic change in precipitation is captured by the new simple model term ; is only influential in a few regions (which vary by model), where actual deviates substantially from the simple model estimate.
Kent et al. (2015) found that intermodel uncertainty over both land and ocean is dominated by because of strong anticorrelations across models between the patterns and magnitudes of and , and also between and over land. In light of the new simple model of q change, this negative intermodel covariance between and over land can be understood as an artifact of decomposing using the unrealistic assumption of a Clausius–Clapeyron response to local warming. Models that warm more over a given land region will have a larger but will also have a larger decrease in because of the inability of the actual moisture increase (advected from the relatively cooler oceans) to keep up with the increase in local atmospheric moisture-holding capacity.
In contrast, the covariance between intermodel uncertainty in and is weak (see section 5), so the two components are likely to be more physically independent of each other. Therefore, and will also be more independent of one another than and . This, combined with the more physically realistic description of q change provided by the simple model compared to the assumption of Clausius–Clapeyron moisture increases over land, suggests that the modified precipitation decomposition is an improvement on the original version over land.
7. Summary and conclusions
A simple conceptual model of surface specific humidity change over land has been derived, where future q over land is determined by scaling the present-day pattern of land q by the fractional increase in the oceanic moisture source. This simple model agrees well with GCM projections (mean spatial correlation coefficient 0.87), even at regional scales, suggesting that future land q change can be primarily considered as a response to increased moisture advection from the oceans. The increased oceanic moisture source is itself mainly a consequence of the increased moisture-holding capacity of warmer air over the oceans, and hence specific humidity change over both land and ocean can be considered to be thermodynamically controlled by SST increases. This is in contrast to changes in other water cycle variables such as precipitation and evaporation, where regional changes are dominated by circulation change. One way of looking at this is that q and T are storage variables, of moisture and thermal energy, respectively, and so are intrinsically less sensitive to circulation change than flux variables such as P and E. The new moisture advection model predicts GCM land more accurately (the mean bias is reduced by half) than an alternative model where is assumed to increase with local moisture-holding capacity over land, determined by the Clausius–Clapeyron equation under fixed RH.
These results confirm the hypothesis that a general reduction in RH over land is expected to occur simply because the land warms more than the ocean (Rowell and Jones 2006; O’Gorman and Muller 2010; Simmons et al. 2010). However, land–atmosphere feedbacks (soil moisture and plant stomatal changes) play a much greater role in amplifying land RH changes than they do in determining specific humidity change over land (Berg et al. 2016).
Residuals between actual GCM and simple model estimates are present, but their position and magnitude vary between models. In HadGEM2-ES the largest residuals appear to be associated with regions of large circulation change and so could be described as dynamical changes in q. Other possible causes of residuals include plant stomatal responses to increased CO2, oceanic moisture sources that are different from the zonal mean, and convergence of moisture above the boundary layer. Future changes in q are relatively insensitive to smaller circulation changes because of cancellations between changes in local evaporation and local vertical mixing of moisture along the path of an air mass and also because q at any point is an integral of moisture change along its entire path. Therefore, local changes in convergence, vertical mixing, and evaporation have only a limited effect on , compared to their effect on other variables such as precipitation change. The weakening tropical circulation also does not appear to have an impact on , as this only affects the speed at which air masses travel over land, not the moisture content of each air mass.
This same scaling of surface specific humidity change over land has also been independently derived in an Eulerian box-model framework (Byrne and O’Gorman 2016), which provides added confidence in the result. Comparison of the two derivations should provide additional insights into the circumstances when this scaling is or is not valid.
Future projections of over land are relatively robust across models compared to projections of other water cycle variables, but there is still intermodel uncertainty in the magnitude of changes. Intermodel uncertainty in the simple model (thermodynamic) and uncertainty in its residual (dynamical) both contribute substantially to the total uncertainty. Uncertainty in processes captured by the simple model is the larger term in most regions, but the regions of highest total uncertainty are determined by peaks in the uncertainty of the residual. Within the simple model term, the largest single contributor to intermodel uncertainty is uncertainty in the magnitude of global mean SST warming.
The relative robustness across models of over land may explain the finding of Fischer and Knutti (2012) that combined projections of changes in temperature and relative humidity extremes are much better constrained than those for the two variables independently. In fact, the combination of the two variables can be interpreted as a measure of changes in . Although we only examine mean changes in here, extreme values of are also likely to be linked to moisture advection from the oceans. They would therefore be quite well constrained by the relative consistency of SST warming across models, at least compared to the larger intermodel uncertainty in the individual values of regional T and RH change over land.
Changes in surface q also affect precipitation, by increasing the available moisture supply. In the tropics, can be decomposed into dynamical and thermodynamic terms, where the thermodynamic terms represent the impact of increased low-level moisture on precipitation. The new simple model of q change can be integrated into this precipitation decomposition over land, replacing the previous assumption of local Clausius–Clapeyron moisture increases under fixed RH, and it provides improved physical understanding of the mechanisms underlying tropical rainfall change. The robustness and simplicity of mean q change over land demonstrated here also has implications for changes in land rainfall extremes (e.g., Emori and Brown 2005; O’Gorman 2015). The moisture-increase-related part of these extreme rainfall changes can be considered similarly robust, and should scale approximately with Clausius–Clapeyron increases in oceanic q under warming.
If the mechanisms that control regional temperature change over land can be better understood, this could be combined with the simple q change model to provide a prediction of regional RH change. However, improved understanding of regional temperature change is made more complex by the coupling and feedbacks between changes in temperature, RH, rainfall, evaporation, and soil moisture. Future work will also involve testing the simple model of against observed and modeled historical trends in specific humidity, with the recent period where GCMs do not reproduce observed RH trends (R. J. H. Dunn et al. 2016, unpublished manuscript) of particular interest.
We thank Gill Martin, Hugo Lambert and Alex Todd for useful discussions. The authors were supported by the Joint UK DECC/Defra Met Office Hadley Centre Climate Programme (GA01101). We acknowledge the World Climate Research Programme’s Working Group on Coupled Modelling, which is responsible for CMIP, and thank the climate modelling groups for producing and making available their model output. For CMIP the U.S. Department of Energy’s Program for Climate Model Diagnosis and Intercomparison provides coordinating support and led development of software infrastructure in partnership with the Global Organization for Earth System Science Portals. We thank Geert Jan van Oldenborgh for making the CMIP5 RCP8.5 and historical data easily available via the KNMI Climate Explorer tool. We also thank three anonymous reviewers for their comments, which helped to improve the manuscript.