# Constraint on precipitation response to climate change by combination of atmospheric energy and water budgets

Sep 3, 2020

### Energy and water budgets control on precipitation

Figure 1 presents the multi-model mean divergence terms of both the water and the energy budgets, averaged over different spatial scales. In the tropics the two divergence terms are strongly anti-correlated and show the same spatial structure (please note that div(qv) is presented with a minus sign to be consistent with Eqs. (1) and (2). The multi-model mean spatial correlation between −div(qv) and div(s) in the tropics (−30° to 30°) is 0.94 (see Supplementary Fig. S1, for the zonal mean behaviour of all models). The high correlation in the tropics demonstrates that any convergence of water vapour that generate precipitation will be accompanied by production and divergence of dry static energy. The opposite is true in the sub-tropics where there is a net divergence of water vapour and a net convergence of dry static energy. Moving poleward, div(s) becomes negative and large (in absolute magnitude), while −div(qv) becomes positive and small. At high latitudes, there is a net convergence of both water vapour and dry static energy as both are being advected from lower latitudes by eddies. However, as the amount of water vapour decreases with the decrease in temperatures towards the poles, the convergence of water vapour decreases from the mid-latitudes storm tracks to the poles. In contrast, the dry static energy convergence increases pole-wards. The zonal mean divergence terms of water vapour and dry static energy reflect the meridional advection of moist static energy15. We note that at the native resolution (upper row) the divergent terms appear small at many locations especially over land; however, they are not negligible compared to the local precipitation (see Supplementary Fig. S2 presenting the normalised divergent terms).

Averaging the water vapour and dry static energy divergence terms over increasingly larger scales (Fig. 1), we note that the spatial pattern becomes weaker and almost completely vanish at 3000 km. This weakening of the spatial pattern occurs on smaller scales for the water budget (c.f. Dagan et al.8) than for the energy budget.

Following Dagan et al.8 in Fig. 2a, b we present the length scales L for which the water budget and the energy budget are locally closed to within 10%, respectively (LWB(10%), LEB(10%)), i.e.:

$$|left( {P – E} right)|/P, < ,0.1,$$

(6)

for the water budget and

$$|(P + Q)|/P, < ,0.1,$$

(7)

for the energy budget.

LWB and LEB exhibit similar spatial features such as evident at the eastern parts of the subtropical oceans and a sharp transition around ±40°8. These patterns emerge due to the fact that at a centre of a region of negative/positive −div(qv) or div(s) (such as the eastern parts of the subtropical oceans) the required averaging scale for closure of the relevant budget is larger. In addition, at high latitudes beyond 40°, the averaging scale required to close both the water budget and the energy budgets is large due to the large role of advection of water and energy from lower latitudes.

On average LEB > LWB due to the larger variance in E compared to Q, which more effectively counteract the large variance in P (see Supplementary Fig. S3). However, at different regions the ratio between LEB and LWB changes (Fig. 2c). For example, at high latitudes LEB > LWB everywhere, while at mid-latitudes (around ±40°) LEB is generally smaller than LWB over the oceans but less so over land (for which almost at all latitudes LEB > LWB). Over the tropical oceans we note a difference between the Atlantic and the eastern part of the Pacific, for which LEB < LWB, and the west Pacific and the Indian ocean for which LEB > LWB. For the latter, the existence of the warm pool and associated cloud cover releases a significant amount of latent heat by precipitation, which cannot be compensated locally by the relatively small radiative term.

Although both budgets are contributing to constrain precipitation, we expect that the smaller scale between LEB and LWB (locally) would be the limiting factor and will determine the spatial scales of changes in precipitation in future climate. Hence, we combine LEB and LWB to a single scale (LEB+WB), which is: LEB+WB = min(LEB, LWB) (Fig. 2d).

As was shown in Dagan et al.8, the average scale required for closure of the water budget depends on the definition of closure, or to within what percentage P is close to E. The same is true for the energy budget (Fig. 3). A stricter closure requires larger scale of averaging for both the water and the energy budgets. We also note that for both budgets the scale for closure in the tropics is smaller than the global mean scale. This is again due to the large contribution of advection of water vapour and dry static energy from low to high latitudes. In addition, on the global mean LEB > LWB for all the levels of closure. As expected, the combined scale, LEB+WB, has a smaller mean for all the levels of closure (as it is defined as the minimum of LEB and LWB for each location, Fig. 3). We note that the mean values of LEB and LWB presented here based on climate models are consistent with previous estimates based on observations8,14.

The average scale of precipitation changes under climate change (LδP) is also presented in Fig. 3. The scale of precipitation changes is calculated as the scale for which the relative precipitation changes (as absolute value) is smaller than a given value, R:

$$|delta P/P|, < ,R,$$

(8)

for R in the range of 7.5–15%. We note that the level of imbalance in the different budgets for a given closure threshold is similar (in terms of water amount) to the magnitude of the relative precipitation change R, as all are normalised by the (same) local historical precipitation (Eqs. 68).

Figure 3 demonstrates that the combined scale LWB+EB is more similar to (but slightly larger than) LδP than either LEB or LWB separately. This is also demonstrated in Fig. 4 which presents the multi-model mean LδP vs. LEB, LWB and LWB+EB for different levels of closure. Figure 4 demonstrates that LEB is much larger than LδP for all levels of closure. The same is true for LWB but to a lesser extent, and the combined scale (LWB+EB) is the closest to LδP. These results demonstrate that combination of water and energy budgets provides a better constraint on the scale of precipitation changes under climate change (2500–3000 km for a 10% combined budget imbalance or relative precipitation change). Above this scale, precipitation changes approach the global mean change (usually on the order of a few percent), while below it they could be substantially larger.