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extremely simplified, cartoon-like equation, sum-
marizing many partial reactions that are variously
important at different pH, can be written as
boundary layer. Since the thickness of the boundary
layer decreases with flow rate, this provides a mech-
anism for a causal link between flow rate and pre-
cipitation rate (Buhmann & Dreybrodt 1985). This
idea has been confirmed experimentally (Liu &
Dreybrodt 1997; Dreybrodt et al. 1997) and by com-
parison with field measurements (Dreybrodt et al.
1992; Liu et al. 1995). Wooding (1991) presents a
particularly interesting analysis of the growth of
individual travertine and ice terraces, using a
similar conceptual model. However, this model pri-
marily applies to turbulent flow, and can not explain
pattern
Ca
2þ
þ 2HCO
3
OCO
2
þ CaCO
3
þ H
2
O
In the classical model for calcite precipitation
and dissolution in karst settings by Buhmann &
Dreybrodt (1985), it is therefore assumed that
precipitation/dissolution rate is stoichiometrically
equal to outgassing/ingassing. However, this holds
true only under equilibrium conditions and at large
scales. In the presence of non-homogeneous advec-
tion and diffusion, local, small-scale variations in
outgassing at the water-atmosphere interface are
not echoed by corresponding local variations in pre-
cipitation rate at the bottom, at least in deep water
relative to flow rate (Hammer et al. 2008).
One promising application of the outgassing
hypothesis is the case of thin, laminar film flow,
forming microterracettes on steep surfaces and
ridges on stalactites. Ogawa & Furukawa (2002)
studied the problem of ridge formation on icicles,
which are analogous to stalactites but with precipi-
tation being dependent on heat loss rather than
CO
2
loss. They explained the characteristic ridge
wavelength as a result of two competing processes.
Thermal diffusion in air involves steeper tempera-
ture gradients and therefore faster heat transport
around protrusions, encouraging the formation of
structure at small wavelengths (so-called Laplace
instability). However, thermal transport in the
flowing water film makes the temperature distri-
bution more uniform, suppressing small wave-
lengths. Ueno (2003) carried out a similar
analysis, but emphasized the role of gravitational
and surface tension forces on the water film to
explain the suppression of short wavelengths.
Several authors have invoked the Bernoulli
effect as a mechanism for increased outgassing
under rapid flow, leading to faster precipitation
(Chen et al. 2004; Veysey & Goldenfeld 2008).
This effect refers to the lower fluid pressure associ-
ated with higher flow velocities. Hammer et al.
(2008) showed that even at a very high flow rate
of 1 m/s, this effect would lead to only about 0.5%
pressure drop at the water-air interface. The corre-
sponding 0.5% decrease in dissolved gas concen-
tration under equilibrium conditions, according to
Henry's law, is unlikely to have a major effect on
precipitation rates.
Mixing in the water column will strongly
increase precipitation rates by efficiently bringing
solutes to and from the calcite surface. In the case
of turbulent flow, we can assume almost complete
mixing in the turbulent core away from the traver-
tine surface. Precipitation rates will then be
limited by diffusion of solutes through the laminar
formation
at
the
smallest
scales
under
shallow, slow laminar flow.
Hammer et al. (2008) developed a detailed,
mechanistic model of carbonate precipitation on
an imposed obstruction in shallow 2D laminar
flow, with the aim of understanding the localization
of precipitation on terrace rims (Fig. 8). This model
included hydrodynamics, diffusion of solutes, solute
carbonate chemistry, precipitation kinetics and out-
gassing, and compared well with laboratory exper-
iments. A Laplace instability (Ogawa & Furukawa
2002) caused enhanced precipitation in regions of
high convex curvature. In addition, precipitation
was high in the shallow, high-velocity region on
top of the obstruction, but also in a position down-
stream from the obstruction. Dramatic experimental
increase of outgassing in local positions in the
model had no effect on precipitation patterns.
Hammer et al. (2008) concluded that advection is
of central importance, by bringing ions to and
away from the calcite surface. This was also
implied in the model of Goldenfeld et al. (2006),
which includes a term in the precipitation model
for flow rate normal to the surface. In addition,
solute concentration gradients set up in a pool are
geometrically compressed when advected through
the shallow region over the rim. This results in
steeper gradients and therefore faster vertical diffu-
sion, leading to enhanced precipitation. Additional
possible mechanisms include ballistic deposition
of carbonate particles onto the rim from suspension
(cf. Eddy et al. 1999) and biological effects (Chafetz
& Folk 1984).
Statistical properties
The quantitative morphology (morphometrics)
and size distribution of travertine terraces as a func-
tion of parametres such as slope, flux and water
chemistry have not yet been studied in detail. It
is possible that shape or size descriptors could be
useful proxies for the reconstruction of, for
example, paleoflux. However, statistical properties
of
terraces
in
individual
settings
have
recently
been subject to investigation.
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