by Makarieva A.M., Gorshkov V.G., Nefiodov A.V., Chikunov A.V., Sheil D., Nobre A.D., Li B.-L.
New questions and ideas about hurricanes and their power.
Predicting the intensity of tropical cyclones, hurricanes and typhoons is a recognized challenge. We are grateful to Judith for this further opportunity to discuss our work with the Climate Etc. readers. (Some of you may remember previous posts about our work here, here, here and here). In this post we discuss how tropical cyclones fuel themselves by their motion, how this determines their intensity and we note some of the more general implications for atmospheric and land-cover sciences.
UPDATE: link to full paper
Hurricanes and their rainfall
Intense tropical cyclones — hurricanes and typhoons — involve violent winds, torrential rain and low atmospheric pressures. (Please note that hurricanes and typhoons reflect the same atmospheric phenomena — the label used depends only on where they occur.)
It is known that local evaporation accounts for less than a quarter of ongoing rainfall within the hurricane’s rainfall area (about 600-800 km in diameter and comprising the distinct vortex of clouds visible from space). Remarkably, despite the danger associated with hurricane rainfall, where the other three-quarters of the rain come from remains uncertain.
Fig. 1. An approximate water budget for a hurricane (square): mean evaporation 0.5 mm/h, mean rainfall 2 mm/h and imported moisture 1.5 mm/h within 400 km from the center. Mean tropical evaporation in a hurricane-free environment is about 0.2 mm/h. Also shown are relative humidity H and surface pressure ps at different radial distances r from the hurricane center.
In our recent study (in press in Atmospheric Research, accepted manuscript available from here, discussed in Physics World here) we argue, based on an analysis of available data, that hurricanes extract pre-existing moisture from the atmosphere as they move through it. Thus, the storm’s so-called propagation velocity — its velocity relative to the rest of the atmosphere — is key.
Our result has implications for understanding hurricane intensity too. It is widely believed that a hurricane derives its energy extracting heat from the ocean. Theory tells us that the power of such a steady-state process is constrained by Carnot efficiency (see https://en.wikipedia.org/wiki/Carnot_cycle) multiplied by the oceanic heat flux. In contrast we argue that a hurricane derives its energy from the water vapor previously accumulated in the atmosphere (this vapor represents a major store of potential energy). The power of such a system is not constrained by Carnot efficiency: while the potential energy is limited to what is stored in the atmosphere, there is no thermodynamic limit on the rate that this can be released (as in an avalanche).
Before discussing the storm intensity in more detail, it is relevant to examine why water budgets have received such limited attention.
Why rainwater origins were not studied – our hypothesis
In 1960, in an influential paper, Malkus and Riehl (1960) [hereafter MR] proposed that rainfall does not matter. They wrote (our emphasis):
“… variations in the rate of import, condensation and export of normal tropical air will not lead to variations in surface pressure because the ascent path, and therewith the density of the vertical column, is entirely determined by the θE of the rising air. A storm will not deepen if simply more water is condensed at θE=350oA in the core; it can do so only if there is an additional heat source so that condensation will occur at θE greater than 350oA.”
For those unfamiliar with the meterological terminology, equivalent potential temperature θE is a measure reflecting, besides local temperature and pressure, the amount of water vapor in the air. Essentially, MR say that a surface pressure difference driving the hurricane-force winds can form only if the air rising in the hurricane core carries more water vapor than the ambient air. This excess water vapor — provided by oceanic evaporation as the air moves towards the hurricane core — represents the (main part of the) “additional heat source”.
In other words, rainfall does not matter, but evaporation does. This statement is somewhat puzzling. Indeed, for the surface pressure to fall, something must be removed from a hydrostatic atmospheric column, not added — meanwhile evaporation adds water vapor to the atmosphere.
We use our new approach to explain MR’s ideas as follows. Consider the mean tropical troposphere which we will here approximate as having a mean lapse rate of 6 K/km, surface temperature 26 oC and surface pressure 1015 hPa. We thus know the dependencies of pressure p and temperature T on altitude z. Relative humidity at the surface is about 80% (see, e.g., Table 5 of Jordan 1958 — the data used by MR).
Let’s compare this environmental pressure profile pe(z) with the pressure profile ph(z) the hurricane air would have if it were rising adiabatically from the surface with the same pressure and temperature, but with a higher relative humidity — 100%.
Fig. 2. Pressure Δp(z) = ph(z)-pe(z) and temperature ΔT(z) = Th(z)-Te(z) differences between hurricane air rising adiabatically and the mean tropical environment. The red square marks the altitude where air densities are equal, the blue circle marks the altitude where pressures are equal. (Cf. Fig. 1c in Makarieva et al. 2017b and Fig. 2 in Makarieva et al. 2015a).
We can see that the hurricane has a greater pressure in the troposphere. The reason is that as the air rises and cools, the water vapor condenses. The more water vapor condenses, the more latent heat is released. As there is more water vapor in the hurricane’s air, its temperature drops with altitude more slowly than in the surrounding air: or, to put it simply, the hurricane is warmer. Thus, since the scale height of the exponential pressure distribution is proportional to temperature, the air pressure drops with height more slowly in the hurricane than it does in the environment outside the storm.
So far we have not created (i.e., explained) any pressure drop at the surface. Now the key point: Let us imagine that this mid-altitude pressure surplus drives air away from the hurricane core. Since the surface pressure reflects the weight of air in the column, as soon as some air is removed, the surface pressure drops. But the pressure surplus diminishes as well, since the hurricane pressure at any altitude is proportional to surface pressure. Thus, gradually decreasing surface pressure in the hurricane we observe that at a certain moment the pressure surplus disappears altogether:
Fig. 3. Pressure difference between hurricane and environment versus altitude as dependent on the surface pressure drop.
Once this happens, we obtain an “unperturbed top” for our hurricane — an altitude where pressure, density and temperature in the hurricane and environment coincide.
This is the gist of the conventional hurricane intensity theory: pull the bottom of the pressure difference curve in Fig. 3 to the left until the pressure surplus disappears, and you get the maximum hurricane pressure drop at the surface. In this context “maximum” presumably refers to two things: (1) the fact that the rising air has 100% relative humidity at the surface and thus contains the maximum possible amount of water vapor and (2) the fact that once the unperturbed top is obtained, we, for some reason, cannot pull the curve any further to the left.
This approach ignores rainfall and other fluxes: it only requires a difference in the water vapor content between hurricane air and the wider environment. Since this difference is provided by evaporation, other factors such as rainfall are, this approach suggests, irrelevant.
Thus obtained surface pressure drop appears to be highly sensitive to minimal variations of atmospheric parameters in the vicinity of the unperturbed top. This sensitivity of the surface pressure estimates to parameters set at the top of the troposphere was discussed by Holland (1997) and, from a different perspective, by Makarieva et al. (2015a). However, the main problem with this approach is that it is not specified why an unperturbed point should exist at all in the presence of hurricanes. Why cannot we pull the pressure difference distribution even further to the left reaching greater surface pressure drops?
Maximum potential intensity and the Carnot cycle
In MR’s approach maximum potential intensity (MPI) is a unique function of the relative humidity H in the hurricane core (provided the environmental pressure profile and surface temperature Ts are known). The temperature and pressure of the unperturbed top is also set by H. Emanuel (1986) modified MR’s approach by adding one more free parameter to the MPI calculation. In his approach the environment is represented not by a fixed pressure profile, but by an isotherm and an adiabat (see curves DE and EF, respectively, in Fig. 1). Varying the altitude where the isotherm and the adiabat meet (i.e. point E), for a given H in the hurricane core it is possible to arrange an unperturbed top at any desirable altitude, pressure level and temperature. The temperature of the unperturbed top has become a free parameter called the outflow temperature To.
With two isotherms (one at the surface and another in the upper atmosphere, curves FB and DE in Fig. 1) and two adiabats (BD and EF) it became possible to interpret the hurricane as a Carnot cycle working on an isothermal oceanic surface and receiving heat and moisture from the ocean.
But it is a peculiar kind of Carnot cycle. Emanuel’s approach retained an essential property of MR’s approach: no mechanical work is performed in the upper atmosphere. Indeed, when an air parcel of a given mass rises from the surface in the hurricane core to reach the unperturbed top, then flows along the unperturbed top to the outer environment where it descends back to the surface, such an air parcel does not generate any mechanical work along its path (see Appendix A in Makarieva et al. 2017b for details). This is because the change of its potential energy along this path is zero, and no kinetic energy is generated either because at the unperturbed top there is no horizontal pressure gradient (by definition). Likewise, Emanuel’s hurricane is assumed to produce zero work in the upper atmosphere. Thus the total work produced by this Carnot cycle is equal to the work at the lower isotherm (surface); this mechanical work takes the form of the kinetic energy of the hurricane.
The rest is simple: since work W on the warmer isotherm is a function of the surface pressure difference Δps, while heat input Q is a function of Δps and ΔH, relating W = kQ via Carnot efficiency k=(Ts–To)/Tsprovides an equation on Δps as a function of k and ΔH. The intensity of such a cycle can be considered a maximum in that sense that all kinetic energy is produced at the warmer isotherm – i.e. at the surface where the hurricane develops. Nothing is left for the upper atmosphere.
The sensitivity problem persists and the question why work in the upper atmosphere should be zero, lying at the heart of this approach, remains unanswered.
Before Holland (1997) exposed the high sensitivity of MPI to the outflow temperature, DeMaria and Kaplan (1994) had compiled intensity estimates of North Atlantic hurricanes to find that maximum intensity for each surface temperature agrees favorably with Emanuel’s MPI — provided some dependence is postulated between the surface and outflow temperatures. Since the outflow temperatures are defined and measured with far less certainty than surface temperatures, it was difficult to test this dependence empirically. Since then, MPI is considered as a plausible upper limit to hurricane intensity.
Reports of intensity beyond the MPI are rare but they do occur (see Montgomery et al. (2006)discussing Hurricane Isabel in 2003); to our knowledge, sporadic mentions of these intensity “over-achievers” are not systematized.
More recent developments: The gravitational power of precipitation
When in 2000 Pauluis, Balaji and Held proposed that the gravitational power of precipitation makes a significant contribution to the atmospheric power budget, nobody in the hurricane world seemed to notice. (The fact that it was the 21st century before the atmospheric sciences noticed rain as something beyond latent heat, is interesting in itself. Even now the global gravitational power of precipitation has only been estimated by only one group, ours — and we have recently submitted this estimate to publication.)
Another advance occurred in 2015 when a team of Japanese physicists estimated that in hurricanes, too, a significant part of the circulation power goes to raise the rainwater. They noted that hurricane intensity (i.e. their kinetic energy) must be significantly influenced by this drain on their energy substantially reducing estimates. Thus Sabuwala et al. (2015) published corrected intensity estimates for the 1999-2010 hurricane seasons — all by 10-30% lower than the conventional MPI. Since most hurricanes never reach their MPI, accounting for the gravitational power of precipitation has driven the corrected intensity estimates closer to observations. The authors interpreted this as a positive thing.
However, this achievement has raised a problem: what about those most intense hurricanes that have been previously shown, by DeMaria and Kaplan (1994) and others, to match the uncorrected (i.e. overestimated) MPI? Since those hurricanes do raise rainwater too, they are apparently over-achievers – i.e. total mechanical work performed by them goes beyond the Carnot cycle output.
Fig. 4. This is modified Fig. 1 of DeMaria and Kaplan (1994) (small black dots and thin curves) where we conservatively (and very approximately) applied the intensity corrections from Fig. 4a of Sabuwala et al. (2015) (large red dots and thick curve). Hurricanes above the red curve are potential “over-achievers” (1-5% of all observations).
Thus, now we do not just have hurricane Isabel possessing an apparently higher intensity than MPI (see Montgomery et al. 2006). We have a whole population of them.
The findings of Sabuwala et al. indicate conceptual problems with the MPI approach, which could be expected to have stirred a community of theorists. Yet, two years after publication in a high profile journal, this illuminating study has only one citation as reported by CrossRef — that was by our team.
To summarize, the problems with the current MPI approach are
- High sensitivity to key parameters (outflow temperature), which makes possible easy tuning and hinders verification by observations
- No theoretical rationale for the key dynamic proposition: the existence of unperturbed top in the approach of MR and zero mechanical work for z > 0 in the approach of Emanuel
- Most intense hurricanes go beyond MPI (an explanation for this pattern is provided in Section 6 of our Atmos Res paper)
The use of numerical models to understand hurricanes deserves a comment. To generate a hurricane, we must remove air to reduce surface pressure and generate the center of the storm. This requires air flowing out from the hurricane center. In modern hurricane models this radial motion is governed by parameters of turbulent friction — these parameters are fitted empirically to provide a realistic pressure profile (e.g., Bryan and Rotunno 2009). Such models — by their construction — are unable to test whether (and to what degree) the pressure surplus associated with a higher water vapor content in the hurricane core can produce the pressure gradients required to drive an intense hurricane. Likewise, switching rainfall “on” and “off” in such fitted-parameter models sheds little light on how rainfall influences hurricane dynamics.
We thus argue that there are good grounds to revisit the imperatives set out half a century ago when the knowledge about the atmosphere, and the resources allocated to study it, were way poorer than they are now. The statement that the rain does not matter should be re-considered.
Having established a linear correlation between rainfall rate and hurricane intensity Sabuwala et al. 2015 were apparently puzzled as to how to best account for what people actually think about rainfall in this field. In the Introduction, with a reference to Rodgers et al. 1994, Sabuwala et al. discussed the idea that rainfall can actually increase hurricane intensity: “latent heat is released which helps propel the updraft, which in turn increases the supply of water vapor and so forth.” However, this reasoning, as we have discussed, runs counter to the conventional paradigm, which denies any importance for rainfall intensity.Emanuel (1991) put it in this way: “Attempts to regard the condensation heat source as external lead to the oft-repeated statement that hurricanes are driven by condensation of water vapor, a view rather analogous to that of an engineer who proclaims that elevators are driven upward by the downward acceleration of counterweights.”
(It is of interest that in a closely related field monsoon scientists widely use the concept of “moisture advection feedback” – whereby the circulation intensity is assumed to be proportional to the (horizontal differences in) rainfall intensity – exactly because it is believed that latent heat release is proportional to rainfall rate and is what propels updrafts etc.)
Sabuwala et al. further noted that “from the positive correlation between P [precipitation] and V [hurricane wind speed] evinced in Figure 1d, it is tempting to argue that the effect of rainpower is to increase hurricane intensity. And yet positive correlation might not signify causality nor can the effect of P on V be understood in isolation from other aspects of a hurricane’s thermodynamics.”
However, the dependence between rainfall and hurricane intensity is not just a “positive correlation”. We have formulated a theoretical approach that quantitatively explains the empirical dependence between rainfall and hurricane intensity.
Fig. 5. Observations of Sabuwala et al. 2015 explained in the framework of condensation-induced hurricanes (Makarieva et al. 2015b).
In our approach, rather than being driven by latent heat, the hurricane is driven by condensation which releases potential energy accumulated in the form of atmospheric water vapor. As the water vapor condenses in the rising air, a non-equilibrium vertical pressure gradient forms that enhances the ascending motion. Once the hurricane air reaches the height where condensation ceases, it is propelled away by the centrifugal force — the centrifugal force overcomes the horizontal pressure gradient which diminishes with height. In other words, the air outflow can efficiently occur at the expense of the centrifugal force — it does not require any pre-existing pressure surplus shown in Fig. 2. Work output in the upper atmosphere can be negative (similar to what happens in Ferrel cells, see Makarieva et al. 2017b) — it does not have to be zero as currently assumed.
Hurricane power is determined by the work per unit time of the non-equilibrium pressure gradient of water vapor that formes during condensation. As water vapor fuels the storm, it is the availability of this vapor and the rate at which it can drawn into the cyclone that limits its intensity. Such processes are not steady-state Carnot cycles receiving heat from the ocean but rather deplete the potential energy of the pre-existing water vapor and can thus outperform a Carnot cycle. Key to hurricane intensity is the availability of water vapor in the surrounding atmosphere.
Thus hurricanes are not steady-state Carnot cycles receiving heat from the ocean – they deplete potential energy of the pre-existing water vapor as they move — as an avalanche. This explains why they can outperform a Carnot cycle.
Our view is that the same energy that powers hurricanes also drives many other air circulation patterns. This includes the main atmospheric transport of water vapor from the ocean to land. On land the store of water vapor in the atmosphere is largely created and maintained by plants. Thus forest and vegetation can ensure you a persistent flow of moist air from the ocean – e.g. the Californian drought could be eased by large-scale restoration of natural forests. Likewise, securing Amazon forests is crucial for ensuring reliable rainfall through South America, including in agricultural regions (Nobre 2014). Plants matter for climate much more than many of us are used to thinking (Ellison et al. 2017). We urge greater attention to the role and dynamics of water vapor condensation in hurricanes and in atmospheric sciences more generally.
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