Greenhouse gases make Earth's surface hotter than
it would be if the planet were simply a blackbody radiator. That additional warming is an important
driver of hurricanes.
The tropics have
generally the most benign climates found on Earth, with gentle breezes and small daily and seasonal
temperature variations. Why, then, do tropical climates breed the most destructive wind storms
known? This brief tutorial explains the paradox and presents an overview of hurricane physics.
The greenhouse effect
Of the solar energy that streams to Earth,
about 30% is reflected by clouds or the surface, and an additional small percentage is directly
absorbed by atmospheric water—either gaseous or condensed in clouds. The radiation that
escapes reflection or absorption in the atmosphere is absorbed by the surface, which transmits
energy upward both by radiation and in vast convective currents whose visible manifestations
are the beautiful cumulus and cumulonimbus clouds that ply the tropical skyscape. The outgoing
photons have much longer wavelengths than the incoming photons, since Earth's surface temperature
is far lower than the Sun's. The outgoing IR radiation is strongly absorbed by clouds and by trace
amounts of certain gaseous components of the atmosphere, notably water vapor, carbon dioxide,
and methane. Those constituents reradiate both upward and downward. Remarkably, the surface
receives on average more radiation from the atmosphere and clouds than direct radiation from the
Sun. The warming of the surface by back radiation from the atmosphere is the greenhouse effect.
Because of it, Earth's surface temperature is some 35 K higher than its effective blackbody temperature.
That difference makes hurricanes possible.
The relatively high surface
temperature also means that atmospheric radiation exports entropy to space. The reason is that
the atmosphere is heated at approximately the surface temperature, but it cools at the much lower
effective emission temperature of Earth. In equilibrium, the planet must generate entropy, and
the vast majority of that entropy is produced in the atmosphere, mainly through the mixing of the
moist air inside clouds with the dry air outside them and through frictional dissipation by falling
raindrops and snowflakes. Were it not for moisture in the atmosphere, the entropy would have to
be produced by frictional dissipation of the kinetic energy of wind. The resulting air motion would
be too violent to permit air travel.
Water in the atmosphere
thus has a paradoxical effect on climate. It is far and away the most important greenhouse molecule
in the atmosphere and is responsible for a surface temperature increase that requires the production
of entropy. On the other hand, mixing and irreversible processes associated with precipitation
absorb most of the entropy production and spare people from violent winds. But not always.
A Carnot engine
In the part of the tropics where the sea
surface is warm enough and the projection of Earth's angular velocity vector onto the local vertical
axis is large enough, random small-scale convective currents sometimes organize into rotating
vortices known as tropical cyclones. In computer models of the tropical atmosphere, such organization
can happen spontaneously, but usually only if a combination of ocean temperature and rotation
is somewhat higher than those observed in nature. In subcritical conditions, some trigger is necessary
to initiate the vortices, and in the terrestrial atmosphere tropical cyclones only develop from
preexisting disturbances of independent origin. In mathematical parlance, tropical cyclones
may be said to result from a subcritical bifurcation of the radiative–convective equilibrium
state. About 10% of them develop in the Atlantic Ocean, where the disturbance is often a 100-km-scale
"easterly wave" that forms over sub-Saharan Africa and then moves westward out over the Atlantic.
When its maximum wind speed exceeds 32 m/s, it, by definition, becomes a hurricane.
The convective core of
a tropical cyclone may be many tens to hundreds of kilometers across, orders of magnitude greater
than the few hundred meters' width of an ordinary cumulus cloud. The core's small surface-to-volume
ratio, together with the strong stability to horizontal displacement afforded by the inertial
stability of its rotation, greatly reduces mixing between cloudy moist air and clear dry air. In
a strong tropical cyclone, entropy production by the mixing of dry and moist air is virtually shut
down, and dissipation of the wind's kinetic energy takes over as the primary mechanism for producing
entropy. Most of the dissipation occurs in a turbulent atmospheric boundary layer within a few
hundred meters of the ocean surface.
The mature hurricane is
an almost perfect example of a Carnot heat engine whose working fluid may be taken as a mixture of
dry air, water vapor, and suspended condensed water, all in thermodynamic equilibrium. The engine
is powered by the heat flow that is possible because the tropical ocean and atmosphere are not in
thermal equilibrium. This disequilibrium arises because, thanks to the greenhouse effect, the
ocean must lose heat by direct, nonradiative transfer to the atmosphere to balance the absorption
of solar radiation and back radiation from the atmosphere and clouds. The heat transfer is accomplished
mostly by evaporation of water, which has a large heat of vaporization. To maintain substantial
evaporation rates, the air a short distance above the sea surface must be much drier than would be
the case were it in equilibrium with the sea.
The Figure illustrates
the four legs of a hurricane Carnot cycle. From A to B, air undergoes nearly isothermal expansion
as it flows toward the lower pressure of the storm center while in contact with the surface of the
ocean, a giant heat reservoir. As air spirals in near the surface, conservation of angular momentum
causes the air to rotate faster about the storm's axis. Evaporation of seawater transfers energy
from the sea to the air and increases the air's entropy.
Once the air reaches the
point where the surface wind is strongest—typically 5–100 km from the center of the
hurricane—it turns abruptly (point B in the Figure) and flows upward within the sloping ring
of cumulonimbus cloud known as the eyewall. The ascent is nearly adiabatic. In real storms the air
flows out at the top of its trajectory (point C in the Figure) and is incorporated into other weather
systems; in idealized models one can close the cycle by allowing the heat acquired from the sea surface
to be isothermally radiated to space as IR radiation from the storm outflow. Finally, the cycle
is completed as air undergoes adiabatic compression from D to A.
The rate of heat transfer
from the ocean to the atmosphere varies as vE, where v is the surface wind speed and
E quantifies the thermodynamic disequilibrium between the ocean and atmosphere. But
there is another source of heat; the dissipation of the kinetic energy of the wind by surface friction.
That can be shown to vary as v3. According to Carnot, the power generation by
the hurricane heat engine is given by the rate of heat input multiplied by the thermodynamic efficiency.
If the storm is in a steady
condition, then the power generation must equal the dissipation, which is proportional to v3.
Equating dissipation and generation yields an expression for the wind speed:
Here Ts
is the ocean temperature and To is the temperature of the outflow. Those temperatures
and E may be easily estimated from observations of the tropics, and v as given by
the equation is found to provide a good quantitative upper bound on hurricane wind speeds. Several
factors, however, prevent most storms from achieving their maximum sustainable wind speed, or
"potential intensity." Those include cooling of the sea surface by turbulent mixing that brings
cold ocean water up to the surface and entropy consumption by dry air finding its way into the hurricane's
core.
The thermodynamic cycle
of a hurricane represents only a glimpse of the fascinating physics of hurricanes; more complete
expositions are available in the resources given below. The transition of the tropical atmosphere
from one with ordinary convective clouds and mixing-dominated entropy production to a system
with powerful vortices and dissipation-driven entropy production remains a mysterious and inadequately
studied phenomenon. This may be of more than academic interest, as increasing concentrations
of greenhouse gases increase the thermodynamic disequilibrium of the tropical ocean–atmosphere
system and thereby increase the intensity of hurricanes.
Kerry
Emanuel is a professor of atmospheric sciences at the Massachusetts Institute of Technology
in Cambridge.
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