We estimate L in Equation 1 for a scenario in which global mean temperatures rise 2.0 °C. We compute the multi-model mean zonally-averaged climatology for twenty general circulation models collected by the Intergovernmental Panel on Climate Change (IPCC)32. We compare a baseline climatology T(y) computed for the immediate future 2011–2030 against a more distant climatology over 2080–2099, the period on which IPCC projections are primarily focused. In the following analysis, we separately examine surface temperatures over the oceans and over the continents since we assume that populations can only inhabit one or the other of these environments. We also limit our analysis to 50°S-50°N latitude. The polar regions require special consideration because no displacement can go further poleward than the poles themselves; furthermore, the populations of human and most species become small near the poles under the present climate.

In Fig. 1, panels A and B plot the initial temperature profiles T(y) for the oceans and continents, respectively, while panels C and D plot the local derivative dT(y)/dy, which is smoothed for clarity. As we expect, dT(y)/dy is basically zero close to the equator and increases in magnitude in middle latitudes. Panels E and F plot the change in temperature ΔT(y) in the 2.0° scenario, which is relatively constant over latitude except for far southern regions that warm somewhat less.

Figure 1 Zonally-averaged surface temperature patterns averaged over 20 climate models. (A) Zonal mean surface temperature T(y) of ocean pixels, 2011–2030. (B) Same, but for continental pixels. (C) Derivative dT(y)/dy for panel A (grey, smoothed is black). (D) Same, but for panel B. (E) Zonal mean ocean surface temperature change ΔT(y) between 2011–2030 and 2080–2099. (F) Same, but for continents. Full size image

In panels A and B of Fig. 2, we compute the characteristic length scale L(y) using the meridional profiles shown in Fig. 1. Because the temperature change ΔT is roughly constant with latitude while temperature gradients approach zero near the equator, L increases to large values both in the tropical ocean and on tropical continents.

Figure 2 Theoretical migration length scales and actual temperature-preserving displacements are extreme in the tropics. (A) Length scale for temperature preserving displacement (Equation 1) in the ocean, computed with zonal mean profiles from Fig. 1. (B) Same, but for the continents. (C) Distributions of the shortest actual temperature-preserving migration for pixels in each 2° latitude bin of the oceans. Circles are medians, boxes are inter-quartile ranges, vertical lines are ranges, dots are outlying observations and the black line connects mean values. (D) Same, but for continental pixels. Full size image

The zonally symmetric model in Equation 1 implicitly constrains population movements to occur along the north-south axis and assumes that local temperature gradients do not vary over the course of a migration. We relax these two assumptions by computing the shortest actual distance that populations must move to preserve their average temperature. For each pixel i, we locate the nearest pixel j that exhibits a future mean temperature that is equal to or lower than the initial temperature at i, subject to the constraint that j is not oceanic if i is continental and vice versa. We plot the distribution of these distances for each 2° latitude band in panels C and D of Fig. 2. The displacement distances thus computed tend to be substantially larger for initial positions in the tropics than for those in the middle latitudes, consistent with our simpler length scale analysis. For several latitude bands near the equator, more than 75% of oceanic locations require that populations must migrate more than 1000 km to preserve their average surface temperatures. On the continents, more than 25% of locations in a broader latitude band near the tropics require that populations move more than 1000 km. Both in the ocean and on continents, displacements exceeding 2000 km appear in a narrow band near the equator.

While the structures shown in panels C and D of Fig. 2 match those in panels A and B quite well, there are some deviations that can be understood by examining the map of our calculated displacement distances shown in Fig. 3A. For example, the North-South asymmetry in the dispersion of oceanic displacements (Fig. 2C) is due to large movements required by populations initially in the north Indian Ocean, where the Asian continent prevents northward movements; and large distances arise in the southern continents (Fig. 2D) because the Southern Ocean prevents continuous southward movements from New Zealand and the southern tips of Africa, South America and Australia. The map also reveals the local influence of topography (on the continents) and coastal upwelling (in the oceans), both of which are important because these features perturb local temperature gradients relative to the zonal mean.

Figure 3 The length of minimum-distance temperature-preserving displacements and their impact on population density. The shortest temperature-preserving migration is computed for each pixel under 2 °C of global mean warming. Populations that are initially in the ocean (on land) are constrained to remain in the ocean (on land). Striped appearance over some regions occurs because the combined climate models vary in spatial resolution. (A) Logarithm of the minimum distance that an organism must travel to maintain the average temperature of its environment, plotted as a function of the organism’s initial location. (B) The percent change in population density that occurs if a hypothetical population were initially distributed uniformly over the globe and all members of that population undertake the minimum-distance temperature-preserving displacement in (A). Maps created by authors using Matlab. Full size image

A logical consequence of greater displacements of tropical populations than others is the extreme concentration of populations at the margins of tropical regions. To illustrate this, we simulate a population which is initially distributed uniformly around the globe but whose members follow the temperature-preserving displacements in Fig. 3A without experiencing any population growth. The resulting population density is shown in Fig. 3B. In the middle latitudes, population densities are largely unchanged because populations at each location shift poleward at roughly the same rate, analogously to many cars all moving forward together at the same speed. In contrast, the large displacements in the tropics lead to an almost complete evacuation of the equatorial band, with the displaced populations accumulating in tropical margins where the speed of migration rapidly slows. This is analogous to the traffic jam that occurs when a highway accident brings fast moving cars to an abrupt halt. The effect on population densities in tropical margins is dramatic, in both the oceans and on the continents, as population densities climb to above 400% of their initial concentrations. If populations were actually to concentrate this quickly in what are already exceptionally arid environments, we would expect there to be many adverse consequences in both natural and human systems, such as an accelerated transmission of infectious diseases or conflict over scarce resources.