Moisture-Advection Feedback in Monsoon Dynamics.

The seasonal evolution of the continental heat budget for different monsoon systems (Fig. 2) shows that sensible heat flux from the land surface increases during spring and heats up the atmospheric column prior to the rainy season. The onset of heavy rainfall (red vertical lines in Fig. 2) is associated with a drop in surface temperature on land, and consequently, sensible heat flux reduces drastically. During the monsoon season, latent heat release dominates the atmospheric heat content, whereas net radiative fluxes are relatively constant throughout the year, reflecting the stabilizing long-wave radiative feedback. In response to the latent heat release, thermal energy is transported out of the region through large-scale advection and synoptic processes. The main dynamical driver of the monsoon is therefore the positive moisture-advection feedback (Fig. 1 A): The release of latent heat from precipitation over land adds to the temperature difference between land and ocean, thus driving stronger winds from ocean to land and increasing in this way landward advection of moisture, which leads to enhanced precipitation and associated release of latent heat. In the following, we seek to capture this feedback in a minimal conceptual model.

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Fig. 2.

Seasonal heat contributions to the atmospheric column over different continental monsoon regions in NCEP/NCAR reanalysis data (35). Radiative heating of the land surface in spring enhances sensible heat flux from the ground (‘Sensible’). During the rainy season, latent heat release dominates the heat budget (‘Latent’). Radiative heat flux comprises all radiative fluxes in and out of the atmospheric column (‘Radiative’). The excess heat is transported out of the continental monsoon region though large-scale advective and synoptic processes (‘Convergence’). Error bars give the standard deviation from the 60 years for which data is available (1948–2007). Regions from which values were taken are defined in the ( SI Appendix). The red and blue vertical lines emphasize the months of maximum sensible heat flux and latent heat flux, respectively.

Minimal Conceptual Model for Abrupt Monsoon Transitions.

For this purpose, consider the heat-balance equation of the monsoon season (Fig. 2, for example at blue vertical line). where latent heat release and net radiation into the atmospheric column, R, balance heat divergence, and the relatively weak contribution from sensible heat transport from the land surface to the atmospheric column has been neglected. ΔT is the atmospheric temperature difference between land and ocean. Latent heat of condensation is L = 2.6 · 10⁶ J/kg and volumetric heat capacity of air at constant pressure C _p = 1,295 J/m³/K. P is the mean precipitation over land (in kg/m²/s). The ratio ɛ = H/L between vertical extent H of the lower troposphere and the horizontal scale L of the region of precipitation (Fig. 1) enters because of the balance of the horizontal advective heat transport and the vertical fluxes of net radiative influx R and precipitation P. A length scale for the coastline drops out. Note that no annual cycle is included in the model. Only budgets for the rainy season are considered. Consequently, this model does not capture any interseasonal or any interannual dynamics. Equations are only valid for landward winds, W ≥ 0.

Assuming dominance of ageostrophic flow in low latitudes, the landward mean wind W is taken to be proportional to the temperature difference between land and ocean (33, 36, 37): This assumption of a linear relation between the two quantities is supported by National Centers for Environmental Prediction/National Center for Atmospheric Research (NCEP/NCAR) reanalysis data (Fig. 3) with correlation coefficients above 50% for all regions. There is significant scatter in some plots, reflecting the fact that other processes may be relevant for the monsoon dynamics in the corresponding regions. A possible offset, as observed in some regions, does not alter the model behavior qualitatively. This offset is discussed together with other possibly relevant processes in the SI Appendix. Here we seek to capture only processes relevant to the self-amplification feedback. Neglecting the effect of evaporation over land and associated soil-moisture processes in the continental moisture budget, precipitation has to be balanced by the net landward flow of moisture where q _O and q _L are specific humidity over ocean and land, and ρ = 1.3 kg/m³ is mean air density.

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Fig. 3.

Landward zonal wind versus temperature difference between land and ocean during monsoon season [NCEP/NCAR reanalysis data (35)]. The lines show best linear regression with correlation r.

Note that evaporation is clearly an important process for the moisture budget (e.g. (38)) and is omitted in Eq. 3 only for the sake of clarity. Including evaporation does not change the model behavior qualitatively (see SI Appendix). It does, however, shift the value of the critical threshold, as we will show in the next section when applying our model to data. In the minimalistic spirit of this section, we omit the effect of evaporation here because it is not of first order to the problem. Consistent with reanalysis data (Fig. 4) and theoretical considerations (36, 39), continental rainfall is assumed to be proportional to the mean specific humidity within the atmospheric column The effect of an offset between these quantities does not change the model behavior qualitatively (see SI Appendix). This set of assumptions (Eqs. 1 – 4) yields the dimensional governing equation of the model Note that through the linear relation of Eq. 2, this equation can equally be understood as an expression for the temperature difference beween land and ocean ΔT, which might be more useful for some applications. Introduction of nondimensional variables w ≡ Wρ/β and p = P/(q _Oβ) results in the nondimensional equation which depends on two parameters only: The dimensionless net radiative influx r ≡ R · ɛαρ²/(C _pβ²) and a measure for the relative role of latent and advective heat transport Large l corresponds to a strong influence of moisture advection (scaling as Lq _Oβp) on the continental heat budget compared with heat advection by large-scale and synoptic processes (scaling as C _pβ² w ²/(αɛρ²)). The nondimensional precipitation is directly related to the wind through p = w/(1 + w).

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Fig. 4.

Precipitation versus specific humidity over land during monsoon season [NCEP/NCAR reanalysis data (35)]. The lines show best linear regression with correlation r.

Solutions w(r) of Eq. 6 are determined entirely by a choice of the only free parameter l, which can be expressed in terms of a critical threshold of net radiative flux r _c, below which no physical solution exists (Fig. 5). The critical point (r _c,w _c) will vary for different monsoon systems. It is directly linked to the only remaining parameter l, through and therefore uniquely defines the solution w(r) of the model. The critical radiation can be computed from Thus for large l (as observed in some monsoon systems) the critical threshold is well approximated by r _c ≈ −l. Note that l is scaling like q _Oα/β where α and β have clear-cut physical meaning (39). α is essentially a function of the near-surface cross-isobar angle and thereby a function of surface roughness and static stability of the planetary boundary layer (PBL). β is governed by the characteristic turnover (recycling) time of liquid water in the atmosphere and thereby determined by static stability and vertical velocity in the PBL. Any physical solution for r > r _c is characterized by landward winds w > 0 and positive precipitation p > 0.

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Fig. 5.

Solution of the nondimensional governing Eq. 6. (Top) Nondimensional landward wind for two values of the only parameter l = 0 and l = 1. (Middle) Corresponding nondimensional precipitation. (Bottom) Nondimensional precipitation for higher values of l as observed in some monsoon systems. The functional form of the solution is not changed qualitatively. The critical threshold (w _c(l), p _c(l)) is given as the black curve in each frame.

Let us now try to understand the physical mechanism behind the threshold behavior observed in Fig. 5. In the tropics net radiative influx is negative, i.e. radiation cools the atmospheric column. During monsoon season the same is true for the advection of heat by the winds because winds blow predominantly from the colder oceanic surrounding. The release of latent heat compensates for both of these heat-loss processes. If monsoon winds get weaker, condensation and therefore latent heat release through precipitation are reduced (moisture-advection feedback, Fig. 1 A). The abruptness of the transition emerges through an additional stabilizing effect of the direct heat advection which is cooling the atmospheric column and is also reduced for reduced monsoon winds. Thus both advection-related processes, precipitative warming and thermal cooling, are simultaneously reduced and partly compensate until a threshold is reached at which condensation/precipitation cannot provide the necessary latent heat to sustain a circulation. As a consequence, land-ocean temperature difference ΔT and therewith monsoon winds break down (Fig. 1 B).

Estimate of Critical Threshold for Current Monsoon Systems.

In order to estimate the critical threshold of different monsoon systems within the limitation of this very simple model, we use time series of precipitation P, radiation R, temperature difference ΔT, and specific humidity q _O from the NCEP/NCAR reanalysis data (35) to compute time series for α(t) = (LP + R)/(ɛC _pΔT ²) and β(t) = ((LP + R) · ρP)/((LP + R)q _Oρ − C _pΔTP), assuming applicability of the model and stationary statistics within the observational period (1948-2007). Via α(t) and β(t), the parameter l(t) is known and the system is estimated for each year. As a simple test for the model, we calculate the remaining quantity that is not used for the computation of α(t) and β(t), the specific humidity over land The resulting model estimate of the specific humidity q′_L compares reasonably well (Fig. S2 in the SI Appendix) with the independently observed q _L that was used in Fig. 4 to motivate the relation between specific humidity and precipitation (Eq. 4).

Via the definition of l, we compute R _c from the time series α(t) and β(t) for each year between 1948-2007. Note that the only quantity that is not constrained by data in this computation is the parameter ɛ, which defines the ratio of vertical and horizontal scale. However, the critical threshold R _C is independent of ɛ, and thus the calculation depends only on relatively robust averaged values of precipitation, net radiation, average temperature difference between land and ocean, specific humidity over ocean, and the natural constants ρ, L, and C _p. We interpret the resulting distribution of the critical threshold R _c (Fig. 6, blue) as a noisy estimate of a stationary critical threshold.

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Fig. 6.

State of current monsoon systems with respect to the critical point computed from the conceptual model. The black distribution reflects observed fluctuations in net radiation in the different monsoon systems. The blue distribution provides the computed critical threshold from the conceptual model. The red distribution includes the effect of evaporation. Lines give a Gaussian function with the same mean and standard deviation as the corresponding discrete distribution.

Within the limitations of the model, the observed net radiation is higher than the critical threshold in the Bay of Bengal, West Africa, and China. In India, North America, and Australia, the distributions have significant overlap. Incorporating evaporation into the model shifts the distribution toward lower thresholds (Fig. 6, red), while at the same time increasing the precipitation threshold P _c. Standard bootstrapping (see SI Appendix) reveals that the estimates in Fig. 6 are already relative robust distributions, in view of the simplicity of the model approach.