A critical humidity threshold for monsoon transitions

Introduction Conclusions References


Introduction
Monsoon rainfall is the major prerequisite of agricultural productivity in many tropical and subtropical regions of the world, and its variability has been affecting the livelihoods of a large share of the world's population from ancient civilizations until today (e.g.Parthasarathy et al., 1988;Kumar et al., 2004;Auffhammer et al., 2006;Zhang et al., 2008;Rashid et al., 2011).Proxy records show evidence of abrupt and strong monsoon shifts during the last two glacial cycles (Burns et al., 2003;Wang et al., 2005aWang et al., , 2008) ) and the Holocene (Gupta et al., 2003;Hong et al., 2003;Wang et al., 2005b;Rashid et al., 2011) in India, the Bay of Bengal, and East Asia.In many instances in the past, periods of strong monsoon rainfall thus appear to have alternated with periods of prolonged drought, with comparatively rapid transitions between the two.
Both spatial patterns and temporal evolution of continental monsoon rainfall are influenced by a number of physical processes (Hahn and Shukla, 1976;Webster et al., 1998;Krishnamurthy and Goswami, 2000;Clark et al., 2000;Kucharski et al., 2006;Goswami et al., 2006;Goswami and Xavier, 2005;Dash et al., 2005;Ramanathan et al., 2005;Wang, 2005;Yang et al., 2007) as well as characteristics of vegetation (Meehl, 1994;Claussen, 1997;Robock et al., 2003) and topography (Liu and Yin, 2002).Though these details are crucial for the specific behavior of different monsoon systems, and their significance will vary from region to region, there exist defining processes fundamental to any monsoon dynamics (e.g.Webster, 1987a,b).These are the advection of heat and moisture during the monsoon season and the associated rainfall and release of latent heat.While differential heating of land and ocean in spring is important for the initiation of the monsoon season, land surface temperatures drop substantially after the onset of heavy precipitation, diminishing the surface temperature gradient.The monsoon circulation over the continent is thereafter predominantly sustained by the release of latent heat and subsequent warming of the atmospheric column over land (Webster et al., 1998).Using a complex conceptual model, Zickfeld et al. (2005) found that a monsoon circulation that is sustained by the moisture-advection feedback can undergo abrupt changes in response to changes in the land surface albedo.Knopf et al. (2006) however showed that the threshold albedo in this model is very far away from modern conditions and is highly uncertain due to the dependence on various model parameters.
Here we apply a minimal conceptual model that comprises the heat and moisture budgets of an idealized monsoon circulation.It reflects the dominant role of the selfamplifying moisture-advection feedback during the monsoon season, which is supported by observations Levermann et al. (2009).We find that the model yields a threshold behaviour with respect to the atmospheric humidity over the ocean adjacent to the monsoon region.Below the threshold, the advection and release of latent heat is not sufficient to sustain monsoon rainfall over land.It is important to note that globally, rainfall associated with the intertropical convergence zone will naturally be sustained even without continental monsoon rainfall.Furthermore the seasonal reversal of cross-equatorial winds, driven by the seasonal change in hemispheric insolation, is not affected by our analysis.The question addressed here is which conditions are necessary in order to sustain a rainy season over land which goes beyond the zonal-mean Figures

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Full dynamics of the intertropical convergence zone.The basic dynamics captured in our model thus form a necessary condition for continental monsoon rainfall to exist.
We describe the conceptual model in Sect. 2 and analyze its implications in Sect.3. In Sect.4, the critical threshold is estimated for four major monsoon regions.In Sect.5, we apply the model to abrupt monsoon changes on orbital timescales.Section 6 concludes.

Conceptual model
We use the minimal conceptual model presented by Levermann et al. (2009) (see illustration in Fig. 1).It is based on the observation that, during the rainy season, the regional-scale moist static energy balance of the atmospheric column is dominated by latent heating due to precipitation, which is balanced by advective as well as radiative cooling.According to NCEP/NCAR reanalysis data (1948-2007;Kistler et al., 2001), this holds for all major monsoon regions (Levermann et al., 2009).The moist static energy balance can thus be approximated by where ∆T is the atmospheric temperature difference between land and ocean, and P is the mean precipitation over land (in kg m −2 s −1 ).Latent heat of condensation is L = 2.6 × 10 6 J kg −1 and volumetric heat capacity of air C p = 1.295J m −3 K −1 .The ratio = H/L between vertical extent H of the lower troposphere and the horizontal scale L of the region of precipitation enters due to the balance of the horizontal advective heat transport and the vertical fluxes of net radiative forcing R and precipitation P .Note that this is a model of the monsoon season only, and includes no annual cycle.The above balance therefore neglects the contribution from sensible heating, which is important at the onset stage but relatively weak once heavy rainfall has started to cool the land surface (Fig. 2).Consequently, this model does not capture any interseasonal means that situations in which no solution of the model with positive W can be found will be considered as parameter and forcing combinations for which no monsoon rainfall can be sustained.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 (Petoukhov, 1982;Webster, 1987a;Brovkin et al., 1998): (2) Reanalysis data confirm that this is a valid first-order approximation (Fig. 3).Neglecting the effect of evaporation over land (which will be discussed below) and associated soil moisture processes, precipitation has to be balanced by the net landward flow of moisture where q o and q L are specific humidity over ocean and over land, respectively.Consistent with reanalysis data (Fig. 4) and theoretical considerations (Petoukhov, 1982;Petoukhov et al., 2000), continental rainfall is assumed to be proportional to the mean specific humidity within the atmospheric column: Note that the proportionality constants α and β have explicit physical interpretations.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 (Petoukhov et al., 2000).While Eqs. (1) to (4) are the basic relations necessary to capture the moisture-advection feedback, Eq. ( 4) can be made more realistic by considering an offsets in q L , which will be discussed further below.3) and (4) it follows that This equation represents the heat budget of the conceptual monsoon circulation in terms of latent heat.The two terms on the r.h.s. represent the loss of heat from the land region by advection of warm air and by radiation, respectively (note that R < 0).Their sum must be balanced by latent heat as provided by the inflow of humid air from the ocean, namely Lβq o .The term 1 + β/ ρW incorporates the fact that the latent heat has to be transported from ocean to land by means of advection; the lower the advective velocity W , the higher the specific humidity q o that is necessary to provide the required amount of latent heat to the atmospheric column over land.Using Eq. ( 2) and the relations w ≡ W ρ/β; r ≡ R • αρ/(C p β 2 ), and l ≡ αρ 2 Lq o / C p β , the non-dimensional form of Eq. ( 5) is obtained: where l is proportional to q o .This is the governing equation of the conceptual model.
Its solution is determined entirely by the only free parameter r.The physical part (l ≥ 0, w ≥ 0) of the solution of Eq. ( 6) is shown in Fig. 5 for the case r = −0.05,where the thick red line denotes a stable solution and the thin red line an unstable one.The advective (dashed line) and radiative (dotted line) terms are also plotted separately to show how the solution is obtained as the sum of these two contributions (to illustrate this, the figure is organized with the control parameter l on the y-axis).In the case r ≡ 0, only the advective part of the solution remains (i.e. the red line would collapse onto the dashed line).The y-axis in Fig. 5 can be interpreted as the demand in latent heating that results from the loss of heat from the land region due to radiation and advection.Figures

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Full It turns out that no physical solution exists below a critical threshold l c (horizontal line in Fig. 5), which corresponds to a critical value of specific humidity over the ocean, q c o .When q o falls short of this value, the supply of moisture is not sufficient to maintain the monsoon circulation driven by the moisture-advection feedback.No conventional monsoon circulation can thus develop in a climate where q o < q c o .
Equation ( 6) can also be expressed in terms of non-dimensional precipitation p ≡ P/(q o β), which is directly related to the wind through p = w/(1 + w). (7) The solution in terms of p has a similar shape as in terms of w (Fig. 6), while dimensional precipitation P scales approximately linearly with l as long as l is sufficiently above the threshold l c (Fig. 7).While we do not expect to find this quasi-linear relation perfectly reflected in observations, NCEP/NCAR reanalysis data show that seasonal mean precipitation and specific humidity over ocean are correlated to some extent (Fig. 8).

Estimation of the critical threshold q c o
The critical point [l c , w c ] (or [q c o , w c ]) will vary for different monsoon systems.It is determined by the non-dimensional radiation r via The critical l can then be computed from and the critical humidity threshold q c o via the definition of l .Within the limitations of this minimal conceptual model, we estimate q c o for four different monsoon regions.We use seasonal mean precipitation P , radiation R, land-ocean temperature difference ∆T , and specific humidity over ocean q o from the NCEP/NCAR reanalysis to compute time Introduction

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Full 2 , assuming applicability of the model and stationary statistics within the observational period .Because the observational period is subject to significant anthropogenic global warming, we remove a linear trend from all reanalysis data to get a closer approximation of a stationary climate.From α(t), β(t) and r(t), q c o can then be obtained for each year via Eqs.( 8), ( 9) and the definition of l .Note that q c o is independent of , the only quantity that is not constrained by data.The resulting q c o distribution (Fig. 9, blue) is much lower than the observed distribution of q o (black) in the Bay of Bengal, West Africa and China, while in India the distributions are closer.The blue pin marks the q c o estimate that is obtained from the time-mean values α(t), β(t) and R(t).
The spread in the distribution of q c o is due to substantial variability in α(t), β(t) and r(t) throughout the reanalysis period.The interannual variability in the dimensional net radiation R is about 15-20 % during the reanalysis period, depending on the region.On longer timescales, R can be expected to be rather stable because of the negative longwave radiation feedback according to the Stefan-Boltzmann law.However, the factors α and β may also vary over time.Moreover, in reality the basic relations of Eqs. ( 2) and (4) are blurred by higher-order physical processes that are not represented in our idealized model, limiting our ability to determine α and β (cf.Figs. 3 and 4).Depending on the relative importance of actual variability and observational uncertainty, the distribution of q c o can be interpreted either (i) as a noisy estimate of a stationary critical threshold, or (ii) as a probability distribution of an interannually varying threshold.
In order to obtain more realistic estimates of q c o , we extend Eq. ( 4) by an offset q o L that terrestrial humidity q L needs to exceed before precipitation is initiated (as suggested by the correlation in Fig. 4): After replacing q o by q o − q o L in the definitions of l and p, the non-dimensional Eqs. ( 6)-( 9) remain unchanged.
As above, we estimate q c o for this refined version of the model, obtaining the parameter q o L from linear regression of the corresponding reanalysis data (Fig. 4).Note that, due to Eq. ( 10), q c o now also depends on .Since we have no direct observation of , we choose such that α(t) matches the α that we observe as the slope of the linear regression between W and ∆T (Fig. 3).The results for q c o are shown in Fig. 9 (red), where again the pin marks the estimate from mean quantities.The consideration of the offset q o L generally yields a distribution of q c o which is narrower and closer to, while still clearly seperate from, the present-day range of q o .Only for the Indian region, the distribution of q c o overlaps with that of the observed q o ; however, when considered pointwise, q c o (t) is still lower than q o (t) for all years.Our conceptual monsoon model offers a simplified but robust mechanism to explain such sort of behaviour.It shows that the moisture-advection feedback implies a threshold q c o that seperates two regimes: one where a conventional monsoon circulation can exist, and one where it cannot.We therefore speculate that orbital-timescale variations in NHSI and the associated surface temperature changes might have affected evaporation at the ocean surface such that average humidity over the ocean persistently crossed the threshold, thus critically altering the moisture supply for the adjacent monsoon region and triggering a transition between the two regimes.In the following we apply our model to demonstrate how such variations in q o could have led to monsoon variations consistent with those observed in the proxy record.In doing so, we assume that the values of α, β, , and q o L estimated for modern climate from reanalysis data also hold for the penultimate glacial period; and that R was also comparable during that period to its modern value.In reality, R might have also varied in phase with NHSI, however this variation would have been damped by the stabilizing long-wave radiation feedback.Moreover, a variation of R along with NHSI would act to exacerbate the threshold effect, moving the threshold towards higher values when insolation, and thus the inferred q o , is low (cf.Fig. 11).Therefore, neglecting variations in R yields a conservative result with respect to the volatility of the system.

Application to past abrupt monsoon changes
The solution of the conceptual model depends on the non-dimensional net radiation r.We choose r = r(t) − σ/2 = −106.5,where σ denotes a standard deviation (Fig. 11).This corresponds to a critical threshold q c o = 8.0 g kg −1 , which is in the upper part of the estimated q c o distribution for the China region (cf.Fig. 9).We further assume q o to vary linearly with NHSI (Fig. 12).The linear relation is chosen such that the maximum q o is close to the range of present-day observations.Finally, we assume that when the threshold q c o is crossed from below (i.e., coming from a no-monsoon regime), it takes an additional increase ∆ q to trigger the transition into the monsoon regime (Fig. 11).The EASM precipitation thus resulting from the conceptual model is shown in Fig. 10 (red line).We set P to zero during periods when the model yields no physical solution, to illustrate the idea that no conventional monsoon circulation can exist during those Figures periods, and no rainfall associated with that circulation would occur.However we would expect sources of rainfall other than the large-scale monsoon circulation to play a role, too, so that actual rainfall would not completely cease during such periods.Note that neither the hysteresis width ∆ q nor the second degree of freedom in the relation q o ∝ NHSI are constrained by data; instead they are chosen such that the result of the conceptual model matches the transition behaviour found in the proxy record, taking into account dating errors in the latter (grey bars in Fig. 10).

Discussion and conclusions
We have shown that, in a minimal conceptual model of large-scale monsoon circulation, a critical threshold q c o exists with respect to specific humidity over the ocean region upwind of the continental monsoon region.This threshold follows from the central role of the self-amplifying moisture-advection feedback, which governs the atmospheric MSE balance during the monsoon season.If q o falls short of the threshold q c o , no conventional monsoon circulation can exist over land.The model neglects any processes that are not crucial to the moisture-advection feedback, in order to isolate the consequences of this feedback.The basic dynamics captured in the model therefore form a necessary condition for the existence of continental monsoon rainfall beyond what is accounted for by the zonal-mean dynamics of the intertropical convergence zone.Our results complement those of Levermann et al. (2009), who found a threshold with respect to the net radiation R. While q o can generally be expected to be more volatile than R, the model allows for a superposition of changes in both quantities, with the one either damping or amplifying the effect of the other, depending on the direction of change.
As the model contains the physical feedback that causes the threshold behaviour, it can be used to produce meaningful first-order estimates of the threshold values.
Within the framework of the minimal model, we have estimated the critical threshold q c o for four major monsoon regions, using seasonally averaged reanalyses of regional Introduction

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Full precipitation, net radiation, specific humidity, and temperature for the past sixty years.
The resulting distribution of q c o can be interpreted either as a noisy estimate of a stationary critical threshold, or as a probability distribution of an interannually varying threshold.The degree to which either of these interpretations is valid depends chiefly on the relative importance of actual variability and observational uncertainty in the parameters α and β (see Eqs. 2 and 4), the assessment of which is beyond the scope of this study.However we have seen that the consideration of an offset in terrestrial specific humidity in Eq. ( 4) leads to a q c o distribution which is significantly narrower than with the basic version of the model, suggesting that at least some of the spread in q c o can be eliminated by making the model more realistic, and thus does not reflect actual year-to-year variability in q c o .Consequently, relevant second-order physical processes would have to be included into the model in order to obtain more robust results for q c o .Probably one of the most important missing processes is evaporation over land (e.g.Eltahir, 1998).Its effect on the heat budget would be mainly to reduce sensible heat flux to the atmosphere, which we have already neglected (Eq. 1) because it is comparatively small during the rainy season; on the other hand, its effect on the moisture budget (Eq.3) would be to stabilize the monsoon regime by recycling a part of the atmospheric humidity that is lost by precipitation.Therefore considering evaporation would tend to move the critical threshold towards lower values of q o .While the estimation of q c o could profit from a refinement of the model, the aim of this study is to demonstrate how the simple concept that the model is based on can help understanding past monsoon variations.The non-linear solution structure of the model can lead to abrupt changes in the modelled monsoon rainfall in response to smooth changes in the control parameter, q o .Changes in q o could be brought about by various factors acting on different timescales.For instance, as wind speed over the oceans increases due to global warming (Young et al., 2011), evaporation e.g. in the Arabian Sea could be affected both directly and via the amount of upwelling of cold waters at the continental margins, and thereby alter the moisture supply for the Indian Introduction

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Full summer monsoon.For the East Asian summer monsoon (EASM), we have shown that, assuming variations in q o along orbital-timescale insolation changes, the model yields a series of abrupt monsoon transitions similar to that observed in a proxy record of the penultimate glacial period.While the additional assumption of a hysteresis is not crucial for the transition behaviour, it changes the timing of the individual transitions such that they are all consistent, within dating errors, with those found in the proxy record (physically, a hysteresis might be induced by inert climate components such as e.g.large-scale oceanic circulation or Himalayan glaciation, rather than by atmospheric processes).The idea of a threshold behaviour in monsoon circulations due to the defining mechanism of the moisture-advection feedback may thus be a useful firstorder concept for understanding past large-scale monsoon changes.The conceptual model investigated here may also serve as a basic building block that can be made more realistic by the inclusion of other relevant processes and by a more detailed estimation of the model parameters.For a complete understanding of monsoon variations on multiple timescales, of course, more complex models will have to be invoked.

Methods
NCEP/NCAR reanalysis data has been obtained as monthly-mean time series (January 1948-December 2007), and regionally aggregated as indicated in Table 1.W is averaged vertically between 850 hPa and 1000 hPa; q o between 600 hPa and 1000 hPa; q L between 400 hPa and 1000 hPa; and ∆T over the entire atmospheric column, as represented in the reanalysis data.Figures

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Full   nd W versus temperature difference between land and ocean region, ∆T , from R reanalysis data, for the major monsoon regions of India, the Bay of Bengal, West China (East Asia; see Table 1).The correlation coefficient r is indicated, as well as of a linear fit through the origin (black line).1).The correlation coefficient r is indicated, as well as the slope α of a linear fit through the origin (black line).

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Full cipitation P versus specific humidity over land, q L , from NCEP/NCAR reanalysis black line shows the result of a linear regression, the correlation coefficient r is s well as the slope β (in kg/m 2 s) and the offset in terrestrial humidity, q o L (in g/kg).Correlation between precipitation and specific humidity over the ocean from R reanalysis data.Black lines show the result of a linear regression, the correcients are indicated.o and q c o + ∆ q , where we choose ∆ q = 0.5 o and q c o + ∆ q , where we choose ∆ q = 0.5 g kg −1 as the width of an assumed hysteresis that is thought to appear when the threshold is crossed from either side (illustrated by arrows).Introduction

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Full Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | or interannual dynamics.Equations are only valid for landward winds, W ≥ 0. That Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | series for α Discussion Paper | Discussion Paper | Discussion Paper | Wang et al. (2008) presented a speleothem δ 18 O record from central China that testifies to several large and persistent changes in the strength of the East Asian summer monsoon (EASM) during the penultimate glacial period.These changes are in phase with, but much more abrupt than, precession-dominated oscillations in Northern Hemisphere summer insolation (NHSI): While the latter follow a quasi-sinusoidal cycle, the form of the monsoon changes rather resembles that of a step-function, with variations around either a strong or a weak mean state, followed by a comparatively rapid transition into the other state (cf.Fig.2binWang et al., 2008).This behaviour is especially apparent before about 160 kyr BP (Fig.10, grey line) and suggests that non-linear processes inherent to the monsoon system might have amplified changes in external forcing.In particular, the abrupt transitions might have been triggered by the mean insolation crossing a certain threshold that separated two different states of the monsoon circulationDiscussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | and North Atlantic climate during the Holocene, Earth Planet.Sc.Lett., 211, 371-380, doi:10.1016/S0012-821X(03)00207-3,2003.1738 Kistler, R., Kalnay, E., Saha, S., White, G., Woollen, J., Chelliah, M., Ebisuzaki, W., Kanamitsu, M., Kousky, V., den Dool, H. V., Jenne, R., and Fiorino, M.: The NCEP/NCAR 50-year reanalysis, B. Am.Meteorol.Soc., 82, 247-267, 2001.1740Discussion Paper | Discussion Paper | Discussion Paper |

Fig. 1 .
Fig. 1.Geometry of the conceptual model, illustrating wind W , precipitation P , net radiative flux R, and atmospheric specific humidity over land (q L ) and ocean (q o ).

Fig. 3 .
Fig. 3.Wind W versus temperature difference between land and ocean region, ∆T , from NCEP/NCAR reanalysis data, for the major monsoon regions of India, the Bay of Bengal, West Africa, and China (East Asia; see Table1).The correlation coefficient r is indicated, as well as the slope α of a linear fit through the origin (black line).

Fig. 4 .
Fig.4.Precipitation P versus specific humidity over land, q L , from NCEP/NCAR reanalysis data.The black line shows the result of a linear regression, the correlation coefficient r is indicated, as well as the slope β (in kg m −2 s −1 ) and the offset in terrestrial humidity, q o L (in g kg −1 ).

Fig. 5 .
Fig. 5. Solution structure of the conceptual model as a function of the non-dimensional parameter l , which is proportional to specific humidity over the ocean, q o .For illustrative purposes, l is plotted on the y-axis.The latent heat demand (red line; bold part indicates the stable branch) results from heat loss due to net radiative flux (dotted) and advection of warm air out of the land region (dashed).

Fig. 6 .
Fig. 6.As Fig. 5 (red line), but in terms of non-dimensional precipitation p, and organized with the control parameter l on the x-axis.

Fig. 8 .
Fig. 8.Correlation between precipitation and specific humidity over the ocean from NCEP/NCAR reanalysis data.Black lines show the result of a linear regression, the correlation coefficients are indicated.

Fig. 9 .
Fig. 9. Estimate of critical specific humidity value over the ocean, q c o , from NCEP/NCAR reanalysis data for the basic minimal model (blue) and including the effect of a minimum terrestrial humidity q o L required for the onset of precipitation (red).The black histogram shows the observed distribution of q o .Pins mark the estimates obtained from time-mean parameter values.

Fig. 11 .
Fig. 11.Physical solution for EASM precipitation P from the conceptual model, for three different values of the non-dimensional parameter r: r(t) (dashed), r(t) + σ/2 (dot-and-dash), and r(t)−σ/2 (solid), where r(t) is the estimate from NCEP/NCAR reanalysis data, and σ denotes a standard deviation.Bold lines indicate the upper, stable branch.The value r(t)−σ/2 = −106.5,which corresponds to a critical threshold q c o = 8.0 g kg −1 , is used for the comparison of the model with EASM proxy data.Vertical dashed lines mark q co and q c o + ∆ q , where we choose ∆ q = 0.5 g kg −1 as the width of an assumed hysteresis that is thought to appear when the threshold is crossed from either side (illustrated by arrows).
/kg) me series of q o used for the application of the conceptual model.q o is assumed to y with northern hemisphere (65 • N) summer insolation.The horizontal dashed line

Fig. 12 .
Fig. 12.Time series of q o used for the application of the conceptual model.q o is assumed to vary linearly with Northern Hemisphere (65 • N) summer insolation.The horizontal dashed line marks q c o .

Table 1 .
Regional definitions used for data analysis.Monthly-mean NCEP/NCAR reanalysis data has been averaged over the indicated regions and seasons; Land and Ocean indicate that only terrestrial or oceanic grid points have been considered, respectively; and ∆T = T L −T o .The bottom row lists the values for the dimensionless parameter that have been used in the estimation of the critical threshold (see Sect. 4).