Long-term measurements of volcanic gas emissions conducted during the last decade suggest that under certain conditions the magnitude or chemical
composition of volcanic emissions exhibits periodic variations with a period
of about 2 weeks. A possible cause of such a periodicity can be attributed
to the Earth tidal potential. The phenomenology of such a link has been
debated for long, but no quantitative model has yet been proposed. The aim of
this paper is to elucidate whether a causal link between tidal forcing and variations in volcanic degassing can be traced analytically. We model the
response of a simplified magmatic system to the local tidal gravity
variations and derive a periodical vertical magma displacement in the conduit
with an amplitude of 0.1–1 m, depending on the geometry and physical state of
the magmatic system. We find that while the tide-induced vertical magma
displacement presumably has no significant direct effect on the volatile
solubility, the differential magma flow across the radial conduit profile may
result in a significant increase in the bubble coalescence rate at a depth of
several kilometres by up to several multiples of 10 %. Because bubble coalescence
facilitates separation of gas from magma and thus enhances volatile
degassing, we argue that the derived tidal variation may propagate to a
manifestation of varying volcanic degassing behaviour. The presented model
provides a first basic framework which establishes an analytical
understanding of the link between the Earth tides and volcanic degassing.
Introduction
Residual gravitational forces of the Moon and the Sun deform the Earth's
surface and interior periodically and thus lead to the so-called Earth tides.
The tidal potential can be modelled as the result of the interference of an
infinite number of sinusoidal tidal harmonics with precisely known
frequencies and amplitudes . At the Equator,
the tidal potential varies predominantly with a semi-diurnal periodicity. The
amplitude of the semi-diurnal cycle is modulated within the so-called
spring–neap tide cycle with a periodicity of 14.8 d caused by the
interference of the lunar semi-diurnal tide and the solar semi-diurnal tide.
The peak-to-peak amplitude of the associated semi-diurnal gravity variations
is aastrost=2.4µms-2 during spring tide and aastront=0.9µms-2 during neap tide and is at
an intermediate level at other times of the cycle. At midlatitudes, the tidal
potential varies predominantly with diurnal periodicity, and at other
latitudes both periodicities mix. The spring–neap tide cycle is, however, manifested everywhere and has maximum variability at the Equator
. The tidal potential firstly gives rise to a periodical
elevation of the Earth's crust with a semi-diurnal peak-to-peak variation of
up to about 50cm (maximum at the Equator), and secondly all
crustal compartments exhibit an additional semi-diurnal gravity variation by
up to 1.16⋅aastrost. This gravity
variation typically has no effect on the rigid solid crust but can cause
fluid movement, e.g. prominently manifested in the form of ocean tides
.
Evidence for tidal impacts on volcanism has been gathered by numerous
empirical studies, which detected a temporal proximity between tidal extrema
and volcanic eruptions or
seismic events or
found a correlation between the spring–neap tide cycle and variations in
volcanic deformation or variations in
the volcanic gas emissions.
The tide-induced stress variations (∼ 0.1–10 kPa) appear to be
negligibly small in comparison to tectonic stresses (∼ 1–100 MPa) or stresses caused by pressure and temperature gradients
within a shallow magmatic system (∼1MPa). The rate of tidal
stress change can, however, be around 1 kPa h-1 and thus potentially
exceeds stress rates of the other processes by 1 to 2 orders of magnitude
. Furthermore, these subtle stress
variations may cause an amplified volcanic reaction, when, for example, the tidal
variations cause a widening of tectonic structures ,
a periodic decompression of the host rock , a
variation in the host rock permeability , self-sealing of hydrothermal fractures , or a
mechanical excitation of the uppermost magmatic gas phase .
First studies on the covariations in tidal patterns and volcanic gas emissions hypothesised a possible tidal
impact on the observed sulfur dioxide (SO2) emission fluxes at
Masaya and Kilauea . Since the 2000s,
automatic scanning networks based on UV spectrometers e.g.
have provided multi-year time series of volcanic gas emissions of
SO2 and bromine monoxide (BrO). The availability of such data sets
enabled extensive investigation of long-term degassing variations.
Correlation with the long-term tidal patterns has been reported for the
SO2 emission fluxes of Villarrica and Llaima
and the BrO/SO2 molar ratios in the gas plume of Cotopaxi
. Another possible but less significant correlation has
been reported for the SO2 emission fluxes of Turrialba with
a periodicity somewhere between 9.1 and 16.7 d;. Furthermore,
reported a periodicity of roughly 16 d in the SO2 emission fluxes
of Redoubt retrieved from the satellite-based Ozone Monitoring Instrument (OMI) – the authors proposed that this periodicity was, however,
an artefact of the satellite orbit rather than a tidal signal. In addition,
correlation with the long-term tidal patterns have been reported for the
diffuse radon degassing of Terceira and Stromboli
.
Cycles in volcanic degassing patterns are not unique to periodicities which
match the tidal potential. Many studies reported periodic volcanic degassing
patterns with periods of minutes e.g.. In contrast, observations of long-term
periodicities are rare. Besides the above-mentioned, roughly biweekly
periodicities, periodic long-term pattern with periodicities of 50 and
55 d have been observed in the SO2 emission flux of Soufrière Hills
and Plosky Tolbachik , respectively.
In view of the growing number of studies revealing similar biweekly
patterns in volcanic activity, this paper investigates whether causality between the tidal potential and variations in the volcanic degassing is
analytically traceable in a comprehensible way. High-temperature gas
emissions of persistently strong, passively degassing volcanic systems are
commonly thought to be fed by sustained magma convection reaching the
uppermost portions of the volcanic conduit, where volatile-rich low-viscosity
magma ascends through essentially degassed magma of higher viscosity, which
in turn descends at the outer annulus of the conduit . Magma ascent rates associated with such convective
flow typically vary roughly between 1 and 100 mh-1 and thus are orders of magnitudes larger
than what we can derive for potentially tide-induced vertical magma
displacement rates of at most 0.6m within 6 h (if not further
amplified). A comprehensive model of the tidal impact on the magma motion
thus requires a coupling of the convective and the tide-induced transport
mechanisms.
Our conceptual model aims to provide the first step by investigating the
purely tide-induced transport mechanism acting on the low-viscosity inner magma
column, neglecting any interference between the magma ascent and the tidal
mechanism, i.e. the model ignores the magma convection in the column. We
model the response of such a quasi-static magmatic system (volcanic conduit
connected to a laterally more extended deeper magma reservoir) to
tide-induced gravity variations analogously to the response of a classical
mercury thermometer to temperature variations: the tide drives a periodical
expansion of the magma in the reservoir, which leads to a periodical vertical
displacement of the low-viscosity magma column in the conduit.
We derive the temporal evolution and amplitude of the vertical magma
displacement across the radial conduit profile and examine its impact on the
bubble coalescence rate. In order to introduce our novel approach
comprehensibly, the modelled processes and conditions are as simplified as
suitable; the major simplifications are listed in Appendix A. All findings in
this paper are derived analytically. The quantitative model estimates are
presented for two exemplary magmatic systems. These examples are intended to
match simplified versions of the Villarrica (39.5∘ S) and Cotopaxi
(0.7∘ S) volcanoes, where covariation between outgassing activity and
Earth tidal movements has been observed previously . The associated model parameter sets are listed in
Table . Further, all quantitative estimates are
presented for the spring tide, and the consequences of the contrast between
spring tide and neap tide are discussed in the last part of this paper.
Tide-induced magma displacement in the conduitModel set-up
Panels (a) and (b): sketch of the model set-up. The model compartments
are indicated by white boxes but not depicted to scale. (b) The
semi-diurnal tide causes a radial magma displacement profile in the conduit
with different amplitudes during spring tide and neap tide.
(c) Concept of the tide-enhanced bubble coalescence: two bubbles
which are initially close to each other (see “without tide”) exhibit
differential vertical tide-induced displacements, which enhances the chance for
bubble coalescence (here “at low tide”).
We model the magmatic system analogously to established convection models
, with the exception that the
descending high-viscosity magma annulus is assumed to be not affected by the
tide-induced dynamics and therefore is considered as an effective part of the
host rock, while “conduit” refers in our model exclusively to the ascending
low-viscosity magma column. We assume the conduit to be a vertically oriented
cylinder with length Lc, radius Rc, and cross-sectional area Ac=π⋅Rc2, which is confined by the penetrated host rock (and high-viscosity
magma annulus), connected to a deeper, laterally more extended magma
reservoir with volume Vr and centre of mass at a depth Dr, and either
exhibiting an open vent or capped by a gas-permeable solid plug
(Fig. ). The magmatic melt in the conduit is modelled
as a mixture of a liquid phase and a gas phase having a mean density
ρmelt, which varies with pressure and thus depth, a constant
kinematic bulk viscosity ν, and homogeneous local flow properties. The
magma compressibility β(ϕ) strongly depends on the gas volume
fraction ϕ and lies between the compressibility β(0)=2⋅10-10Pa-1 of volatile-rich rock and the compressibility
β0(1)≈p-1 of an ideal gas see, e.g.,.
The magmatic melt in the reservoir is modelled to be volatile-rich but
hosting no gas phase of significant volume and thus having a constant
compressibility βr≈β(0). Further, the quasi-static
condition implies a steady-state density stratification within the magma and
also with respect to the host rock no neutral
buoyancy;. In this equilibrium, we assume a constant
hydrostatic pressure gradient (∇p)vert.
Response of the host rock on tidal stresses
Magma pathways are often located at intersection points of large-scale fault
systems or in fault transfer
zones e.g., where the surrounding host rock geometry is
relatively sensitive to directional changes in pressure. The vertical and
horizontal components of the tidal force exert additive shear tension on the
host rock, potentially causing a compression of the host rock
or a differential slip between both sides of the fault
system . Both mechanisms can cause an increase in the areal
conduit cross section. Connected to the magma reservoir, such an increasing
conduit volume is accompanied by decompression and thus causes magma to
flow from the reservoir to the conduit, which pushes the initial magma column
in the conduit upwards until the initial hydrostatic pressure gradient is
re-established. Vice versa a relative decrease in the areal conduit cross
section leads to an effective descent of the initial magma column in the
conduit. For a given periodic area increase ΔAc, the amplitude
Δzhr of this additive elevation-descent cycle of the centre of
mass of the initial magma column is given by
Δzhr=Lc2⋅ΔAcAc+ΔAc≈Lc2⋅ΔAcAc.
The quantitative scale of tide-induced conduit cross section variations is
presumably hardly accessible. The theoretical horizontal components of the
tide-induced ground surface displacement are up to about ±7cm. Slip-induced dilation of faults with widths in the
sub-centimetre range thus appear to be plausible. For illustration, a conduit
radius increase by ΔRc=1mm would result in an additive
vertical centre of mass displacement by Δzhr=0.33m for
Villarrica and Δzhr=0.13m for Cotopaxi. As a remark,
these mechanisms do not require a cylindrical conduit and fault–slip
mechanisms would lead to an unidirectional area increase rather than
a homogeneous radial increase. Furthermore, the tide could also cause a
variation in the host rock permeability . This mechanism and its possible interference with the concept presented here
are ignored in our model.
Tide-induced magma expansion in the reservoir
The semi-diurnal tide causes a sinusoidal variation in the gravitational
acceleration with angular frequency ωsd=1.5⋅10-4rads-1 and amplitude (equals the half
peak-to-peak amplitude) a0st= 1.4 µms-2 during
spring tide and a0nt=0.5µms-2 during neap
tide. Besides those host rock mechanisms triggered by the tidal stresses,
these tide-induced gravity variations may also cause a periodical elevation
of the magma in the inner conduit.
The compressible magma in the reservoir is pressurised by the hydrostatic
load whose weight is proportional to the local gravitational acceleration
g. A reduction in the local gravitational acceleration by a0 leads to a
decompression and thus expansion of the magma in the reservoir by ΔVr=a0g⋅(∇p)vert⋅Dr⋅βr⋅Vr. The tidal force can accordingly lead to a periodical magma
expansion–shrinkage cycle in the reservoir with a semi-diurnal periodicity
and an amplitude modulation within the spring–neap tidal cycle of up to
ΔVr∼O(100–1000 m3).
The realisation of this additional magma volume implies a displacement and
thus compression of the host rock at the location of maximum host rock
compressibility. This is typically true for the conduit. Assuming that the
magma expansion in the reservoir ultimately and exclusively causes an
increase in the conduit volume, the volume increase causes an elevation of
the centre of mass of the initial magma column in the conduit by
Δzdec=ΔVrAc=a0g⋅(∇p)vert⋅Dr⋅βr⋅Vrπ⋅Rc2.
In the general case, the additional volume could be realised by a slight
increase in the conduit radius by ΔRdec≈Rc2⋅ΔzdecLc∼O(1mm) caused,
for example, by the tidal stresses. If the magmatic system has an open vent, the
additional volume can alternatively be realised by an elevation of the lava
lake level and thus without a host rock compression.
Analogously, the tide-induced gravity variations result in an expansion of the
initial magma column in the conduit. This effect is, however, typically
negligible compared to the reservoir effect for sufficiently large reservoirs
(volume contrast between reservoir and conduit of more than 1000; see
Table ); thus, for simplicity we neglect the effect of
the expansion of the initial magma column in the conduit.
The responses of the overall magmatic system on the tidal stresses and
tide-induced gravity variations act simultaneously and in phase with the
tidal force. The overall vertical tide-induced magma displacement in the
conduit Δzmax can thus be larger then the individual mechanisms; i.e. {Δzhr,Δzdec}≤Δzmax<Δzhr+Δzdec. In the following we focus on the reservoir expansion
mechanism only in order to keep the derivation of the model parameters
strictly analytical. The host rock mechanism is therefore reduced to
establishing the required areal conduit cross section increase of ΔRdec.
Choice of model parameters, motivated by conditions at (1) Villarrica volcano located at 39.5∘ S hosting a persistent
lava lake of basaltic composition and (2) Cotopaxi volcano located at
0.7∘ S, which preferentially erupts andesitic magma and
intermittently is capped by a solid plug. If not stated otherwise, all
numerical values in this paper are calculated with these parameters.
Model parameter Location-independent constants/assumptions Physical parameterNotationUnitValueLiterature Pure spring tide amplitudea0stms-21.4×10-6, at the Equator Semi-diurnal periodicityωsdrad s-11.5×10-4Hydrostatic pressure gradient(∇p)vertPam-12.7×104for andesitic host rock Solubility coefficient of waterKH2OPa-11×10-11Magma compressibilityβrPa-12×10-10for the magma in the deep reservoir, see Appendix B (Local) gas volume fractionϕ<ϕpercϕperc=0.3-0.7, Villarrica Cotopaxi Conduit lengthLckm2see Appendix B4see Appendix BConduit radiusRcm6see Appendix B40see Appendix BReservoir volumeVrkm335see Appendix B35see Appendix BDepth of reservoir (c.o.m.)Drkm3see Appendix B8see Appendix BKinematic viscosityνm2 s-10.14(andesitic melt)Melt densityρmeltkgm-326002500(andesitic melt)Melt weight fraction of waterCH2O0%25Max vertical tidal accelerationa0m s-20.61×a0st, at 39.5∘ Sa0st, at 0.7∘ SGravitational accelerationgms-29.81at 39.5∘ S9.78at 0.7∘ SMagma temperatureT∘C12001000Radial flow profile in the conduit
The tide-induced vertical magma displacement in the conduit is delayed and
extenuated by a viscosity-induced drag force. We access the temporal
evolution and amplitude of the tide-induced displacement via the force (per
unit mass) balance acting on the centre of mass of the magma column in the
conduit:
γ⋅z˙(t)︸drag force︷inner force=a0⋅sin(ωsd⋅t)︸tidal force-ω02⋅z(t)︸restoring force-z¨(t)︸inertial force︷external force,
where the two model parameters are the bulk damping rate γ and the
eigenfrequency ω0 of the magma column. The restoring force ensures
that the centre of mass displacement tends towards the current “equilibrium”
displacement associated with the current strength of the tidal force, i.e. a0=ω02⋅Δzmax. We further assume a Newtonian bulk drag
force proportional to the flow velocity.
The continuity condition implies that the magma flows faster in the conduit
centre than close to the boundary between the low-viscosity and high-viscosity
magma or host rock. Accordingly, we assume a no-slip condition at the conduit
boundary r=Rc and derive the analytical solution of the tide-induced
parabolic vertical displacement profile z(r,t) in the conduit:
z(r,t)=Ψ⋅1-rRc2⋅sin(ωsd⋅t-φ0)Ψ=2⋅a0(ω02-ωsd2)2+(γ⋅ωsd)2φ0=arctanγ⋅ωsdω02-ωsd2γ=8⋅νRc2ω02=a0Δzdec=g⋅π⋅Rc2βr⋅Vr⋅Dr⋅(∇p)vert,
with the radial coordinate 0≤r≤Rc, the maximum vertical magma
displacement amplitude Ψ (which equals twice the centre of mass
displacement) and the phase shift φ0 between tidal force and magma
displacement in the conduit (see Appendix C).
For Villarrica, the model implies a tidal displacement amplitude of
Ψvillst=0.45m, which lags behind the tide by
φ0,vill⋅ωsd-1=2.0h, where the
displacement is predominantly limited by drag force. For Cotopaxi, the tidal
displacement amplitude is Ψcotost=0.09m and lags by
φ0,coto⋅ωsd-1=0.2h, where the
displacement is predominantly limited by the restoring force. In comparison,
the direct tide-induced gravity variations leads to a variation in the
hydrostatic pressure of 10–100 Pa. In the context of the hydrostatic
pressure gradient, this pressure variation has a similar effect as a vertical
magma displacement by about 1 mm, thus rendering the direct tidal impact
negligible compared to the indirect mechanism derived here.
Tide-enhanced bubble coalescence
Integrated over a semi-diurnal cycle, the tides do not result in a net magma
displacement. A link from tides to degassing thus requires tide-enhanced
mechanisms which irreversibly change the state of the magmatic gas phase. Bubble
growth constitutes a predominantly exergonic and thus irreversible mechanism
because the bubble surface tension inhibits or at least damps bubble
shrinkage and dissolution . Within a tide-induced
radial displacement profile, neighbouring gas bubbles can exhibit
differential tide-induced vertical displacements potentially enhancing the
bubble coalescence rate (see Fig. c and Appendix D).
The variation in the bubble coalescence rate leads to bigger bubbles and thus
the tide can indeed modify an irreversible mechanism.
In this section, we set up a simplified formalisation of the magmatic gas
phase and the typically predominant mechanisms which govern the bubble
coalescence rate and estimate the relative tide-induced enhancement of the
bubble coalescence by a comparison with these classical mechanisms. We
consider a magma layer in the conduit at a particular depth; accordingly, the
parameters discussed in the following describe the local conditions within a
small volume of magma and should not be confused with the integrated bulk
values for the total magma column. The variation in the tide-induced
enhancement at different magma depths is discussed in the subsequent section.
Gas bubbles in magmatic melt
The dominant part of the magmatic volatile content is typically water, followed by carbon dioxide, sulfur compounds, and minor contributions from a
large number of trace gases such as halogen compounds
. For simplicity, we assume that all macroscopic
properties of the gas phase are dominated by the degassing of water, in
particular that the gas volume fraction ϕ exclusively consists of water
vapour. The volatile solubility of magmatic melts is primarily pressure
dependent, with secondary dependencies on temperature, melt composition, and
volatile speciation . The pressure dependency of the water
solubility CH2O in magmatic melt is given in a first approximation by
CH2O(p)=KH2O⋅p with the corresponding solubility
coefficients KH2Ofind an empirical formulation
in. For the local gas volume fraction ϕ(p) at a depth
associated with the pressure p, we obtain
ϕ(p)=ρmelt(p)ρgas(p)⋅CH2O0-KH2O⋅p
with the total water weight fraction CH2O0 of the magmatic melt and
the mass densities of the gas phase ρgas and of the overall melt
(liquid + gas) ρmelt.
The gas phase consists of separated bubbles as long as the gas volume
fraction is below the percolation threshold of ϕperc=0.3-0.7the variation is due to the range of different magmatic conditions;
. Bubbles typically vary in size following a power law
or a mixed power-law exponential distribution
and in shape from spherical to ellipsoidal
. While models based on polydisperse bubble size
distributions are available , a
common starting point to analyse the temporal evolution of the bubbles is
nevertheless the assumption of a monodisperse size distribution of spherical
bubbles .
We note the bubble size distribution δbsize(f∈R+)
with respect to the bubble radius (rather than the volume); i.e. the bubble
radius is given by rb=f⋅Rb with the hypothetical bubble radius
Rb(p) of a monodisperse bubble size distribution. An estimate of a
power-law bubble size distribution would require three parameters: the
exponent and the lower and upper truncation cut-off . An
estimate of a mixed power-law exponential bubble size distribution would
require at least two further parameters. The following analysis is conducted
for an arbitrary bubble size distribution; nevertheless, for a basic
quantitative estimate, we mimic a proper polydisperse bubble size
distribution by the simpler single-parametric
δ̃bsize(f;q)=1-q:f=1q:f=23,
with 0≤q<12, which represents a monodisperse distribution
except for a fraction of q bubbles which emerged from a past coalescence of
two bubbles with f=1.
Bubble motion and bubble coalescence
Diffusion-driven volatile degassing can only take place in the immediate
vicinity of a bubble and when the supersaturation pressure is larger than the
bubble surface tension . The volatile degassing rate
is thus controlled by the spatial bubble distribution as well as the bubble
size distribution . Both distributions change during bubble
rise, which is caused by a vertical ascent of the overall magma column or parcel
with velocity vmelt and a superimposed bubble buoyancy with a velocity
vbuoy which reads for a bubble with radius rb (Stoke's law):
vbuoy(rb)=2⋅g⋅rb29⋅ν⋅1-ρgasρmelt≈2⋅g⋅rb29⋅ν.
If the buoyancy velocity is negligible compared to the magma ascent, the
bubble flow is called “dispersed”; if the bubble buoyancy velocity
contributes significantly to the overall bubble ascent, the bubble flow is
called “separated” . Rising bubbles grow continuously
because of (1) decompression and (2) the increasing volatile degassing rate
due to the associated decreases in the magmatic volatile solubility and of
the bubble surface tension. Bubble coalescence accelerates the bubble growth.
Bubble coalescence requires two bubble walls to touch and ultimately to
merge. Once two bubbles are sufficiently close to each other, near-field
processes such as capillary and gravitational drainage cause a continuous
reduction in the film thickness between the bubble walls until the bubbles
merge after drainage times ranging from seconds to hours depending on the
magmatic conditions .
For small gas volume fractions, however, the initial distance between bubbles
is large compared to the bubble dimensions and the coalescence rate is
dominated by bubble transport mechanisms acting on longer length scales.
Because bubble diffusion is typically negligibly small, bubble walls can only
approach when a particular mechanism leads to differential bubble rise
velocities or by bubble growth. In magmas with a sufficiently separated
bubble flow, two neighbouring bubbles of different size can approach each
other vertically due to the differential buoyancy velocities
. In magmas with a dispersed bubble flow, in
contrast, the relative position of bubble centres remains fixed; thus, bubble
coalescence is controlled by the bubble expansion rate caused by the ascent
of the overall magma column (or affected magma parcel).
Relative contribution of the tidal mechanism (magnitude given by
Htide) on the bubble coalescence rate for a purely separated
bubble flow (magnitude given by Hbuoy) depending on the
reference bubble radius Rb and the degree of polydispersity
q. The reference bubble radius is reciprocally linked to the depth of the
particular magma layer.
Comparison of bubble coalescence mechanism
The proposed tide-induced bubble transport mechanism is compared in the
following with the classically predominant bubble transport or approaching
mechanisms in order to estimate the relative contribution of the tidal
mechanism on the overall coalescence rate. We access the (absolute) strength
of a particular transport mechanism by its “collision volume” Hi (see
Appendix D). The tidal mechanism is noted by Htide. For
comprehensibility, we focus on a comparison of the tidal mechanisms with the
two “end-member” scenarios of a purely separated (Hbuoy) and a purely
dispersed (Hdisp) bubble flow, respectively. A more comprehensive
formulation of the classically predominant bubble transport or approaching
mechanisms has been proposed, e.g. by .
For a separated bubble flow, the relative tidal contribution on the bubble
coalescence rate depends reciprocally on the reference bubble radius Rb
and on the degree of polydispersity q (Fig. ). For q=0.1-0.4, the tidal mechanism contributes at least 10% to the overall
bubble coalescence rate for a range of reference bubble radii of Rb= 32–65 µm for Villarrica and Rb= 37–78 µm
for Cotopaxi. For comparison, obtained from basalt
decompression experiments mean bubble radii of (at most, depending on the
volatile content) 23 µm for a pressure of 100 MPa (∼
depth of 3.7 km) and of 80 µm for a pressure of 50 MPa
(∼ depth of 1.9 km) and concluded an extensive bubble coalescence rate
at depth associated with 50–100 MPa. Similarly,
obtained from rhyolite decompression experiments mean bubble radii of
15 µm for a pressure of 100 MPa (∼ depth of 3.7 km) and
of 30 µm for a pressure of 40 MPa (∼ depth of 1.5 km).
For andesitic magma, the dependency of the bubble size on the pressure is
presumably between the values for the basaltic and the rhyolitic magma. We
conclude that the tidal mechanism can significantly contribute to the bubble
coalescence rate in magma layers at a depth greater than 1 km,
associated with bubble radii of 30–80 µm. In contrast, the
tidal contribution becomes negligible at shallow levels once the bubble radii
are in the millimetre-range which corresponds to the bubble size range at
which bubbles efficiently start to segregate from the surrounding melt.
For a dispersed bubble flow, the relative tidal contribution on the bubble
coalescence rate depends reciprocally on the magma ascent rate, hardly on the
gas volume fraction ϕ, but it depends approximately linearly on the volatile
content CH2O0 of the magma (Fig. ). The tidal
contribution causes an enhancement of the bubble coalescence rate equivalent
to the enhancement caused by an increase in the magma ascent velocity by
about 0.5mh-1 for Cotopaxi and 2.5mh-1 for
Villarrica for the CH2O0 listed in Table . For
comparison, the magma ascent velocities in passively degassing volcanic
systems vary roughly between 1 and 100 mh-1. The tidal mechanism can accordingly contribute by at least
several percent but potentially up to several multiples of 10% to the overall
bubble coalescence rate. For gas volume fractions exceeding the minimum
percolation threshold of ϕperc≈0.3, the model assumption of
independent spherical bubbles increasingly loses its validity.
Discussion and conclusions
Our model implies a tide-induced periodical vertical magma displacement in
the conduit within every semi-diurnal cycle in the order of 0.1–1 m due to
magma expansion in the reservoir. At Villarrica, the modelled vertical magma
displacement of 0.45 m implies a periodic variation in the lava lake level
whose areal cross section is about 10 times larger than for the
conduit; of about 5 cm. At Cotopaxi, the modelled vertical
magma displacement of 0.09 m may apply additive stress on the solid plug.
Relative contribution of the tidal mechanism (magnitude given by
Htide) on the bubble coalescence rate for a purely dispersed
bubble flow (magnitude given by Hdisp) depending on the gas
volume fraction and the initial water weight fraction of the magmatic melt.
The corresponding values for ϕ are calculated with Eq. ()
assuming an ideal gas and magma temperatures of 1200 ∘C for
Villarrica and 1000 ∘C for Cotopaxi. The relative tidal contribution
is displayed as the equivalent to an enhancement of the magma ascent rate, which would have the same effect on the bubble coalescence rate. The model
increasingly loses validity above the percolation threshold of
ϕperc≈0.3.
We linked this magma displacement to bubble coalescence and compared the
relative strength of the tide-induced bubble transport mechanism with respect
to the classically predominant bubble transport mechanisms in magmas hosting
a purely separated or a purely dispersed bubble flow. For both scenarios, we
found that the tidal contributions to the overall bubble coalescence rate can
be in the order of at least several percent up to several multiples of 10 % at a
depth of several kilometres. At shallower depth, the direct tide-induced
contribution to the overall bubble coalescence rate is rather negligible
because the classical transport mechanisms become more efficient.
The tide-enhanced bubble coalescence rate at greater depth can nevertheless
affect the gas phase in the overlying shallower layer because the
additionally coalesced bubbles have a larger buoyancy velocity as well as a
reduced surface tension and can thus stimulate on the one hand enhanced
volatile degassing from the melt phase to the gas phase and on the other hand
enhanced bubble coalescence rates in overlying layers
. These enhancements can ultimately cause the
percolation of the gas phase at a somewhat greater depth compared to the
tide-free scenario. In consequence, the magma becomes gas-permeable at this
greater depth potentially causing enhanced volcanic gas emissions
. The additional contributions from this
greater depth to the volcanic gas emissions may also slightly shift the
chemical composition of the overall gas emissions towards the chemical
composition of the gas phase at this greater depth when compared to the
tide-free scenario .
The quantitative results have been derived for the tidal forcing during
spring tide. In contrast, the amplitude of the tide-induced mechanism is
smaller by a factor of 3 during neap tide. Accordingly, the amplitude of the
additional tide-induced contributions to the coalescence rate varies within a
spring–neap tide cycle entailing a periodical signal with a period of about
14.8 d superimposed on the (nevertheless potentially much stronger)
tide-independent coalescence rate. For a dispersed bubble flow scenario with
rather fast magma ascent, a propagation of this superimposed signal from the
enhanced coalescence rate via a variation in the percolation depth to the
volcanic gas emissions is comprehensible. For a separated bubble flow
scenario, however, the gas bubbles may need much more time than one
spring–neap tide cycle to rise from a depth of several kilometres to the
percolation depth. Magmatic systems can, however, become permeable already at a depth of 1–3 km , i.e. where the derived
tidal effects are the strongest. In such a scenario, the tide-enhanced bubble
coalescence rate could accordingly cause enhanced degassing without a
significant delay.
In a scenario with a shallower percolation depth, the periodic pattern
could nevertheless propagate to the degassing signal because several crucial
parameters such as the mean bubble radius Rb and the gas volume fraction
ϕ typically vary rather monotonously with pressure and thus depth
, implying a depth dependency of the relative tidal
contributions to the bubble coalescence rate. Convolved along the vertical
conduit axis, the tide-enhanced coalescence rate may accordingly preserve an
overall periodicity driven by the dominant contributions from those magma
layers which are particularly sensitive to the tidal mechanism. Moreover,
this pressure dependency implies that gas contributions originating from the
particularly tide-sensitive depths are more pronounced in the subsequent
volcanic gas emissions during spring tide. Therefore, tide-induced variations
in the chemical composition within the volcanic gas plumes may be particularly manifested in the relative molar degassing ratios
e.g. associated
with these depths.
In conclusion, we traced a possible tidal impact from the tidal potential to
a magma expansion in the reservoir, to a vertical magma displacement profile
in the conduit, and to an enhanced bubble collision rate (and thus an enhanced coalescence rate), and
this ultimately motivated a link between the tide-enhanced bubble coalescence rate
and the periodical signal in the observed volcanic gas emissions.
Furthermore, illustrative quantitative calculations indicated that the proposed
tide-induced mechanism could lead to an enhancement of the bubble coalescence
rate by up to several multiples of 10%. If propagated from enhanced bubble
coalescence to a variation in the magnitude or chemical composition of the
volcanic gas emissions, a periodical spring tide signal would be large enough
to explain the observed about 2-weekly variations in volcanic gas emissions.
Nevertheless, our conceptual model only aimed at a proof of concept. Future
studies may increase the complexity of the model by, e.g., (1) lifting several
of our numerous simplifications (Appendix A), (2) incorporating macroscopic
tidal mechanisms affecting the host rock explicitly, (3) adding several
further microscopic mechanisms such as a tide-induced loosening of bubbles
attached to the conduit walls or the tidal impact on crystal orientation, and
(4) investigating possible non-linear interferences between the tide-induced
dynamics and the tide-independent magma convection flow.
Data availability
No unpublished data are presented or used.
List of applied mayor simplifications
In our model we applied several simplifications regarding the shape and
physical properties of the magma plumbing system. This we did for the sake of
clarity and, even more importantly, in order to isolate the tide-induced
effect on magma flow and degassing. To achieve this, we (1) modelled the
tide-induced magma flow in the conduit neglecting any tide-independent magma
dynamics such as magma convection, which implies an initial mechanical and
thermodynamic equilibrium between magma and adjacent host rock. The only
exception is the discussion of the impact of a constant magma ascent on the
bubble coalescence rate. (2) The expansion of the initial conduit magma is
neglected. We assume (3) a gas-tight host rock, (4) a cylindrical
volcanic conduit, (5) a no-slip condition between conduit wall and magma, and (6) homogeneous magma flow properties. (7) The viscosity of the magma in the conduit is assessed by the effective
bulk viscosity. (8) The radial tide-induced magma
displacement is neglected. Moreover, (9) bubble coalescence is modelled by
bubble collision, neglecting near-field drainage processes, bubble
deformation processes, and post-collision coalescence processes. (10) Simple
bubble size distributions are chosen, and (11) it is assumed that the
volcanic gas phase exclusively consists of water vapour.
Quantitative estimates for the geometrical model parameters
The conduit radius is a crucial model parameter. The uppermost 200 m of
Villarrica's conduit have frequently been exposed during the decades prior to
the 2015 eruption due to pronounced oscillations of the lava lake level
. The cross-sectional area of the conduit
has a radius of about 30 m , which at greater depths,
however, narrows down to a mean radius of the order of Rc=6 m as is
implied by studies based on gas emission magnitudes and
seismoacoustic properties . The active vent of
Cotopaxi was capped by an area of hot material with a diameter of 116–120 m
during the eruption in 2015 . Although missing an
empirical evidence, it is plausible that the mean conduit radius is somewhat
narrower, and therefore we assume a (rather conservative) value of Rc=40 m.
Depth and volume of the magma reservoir constitute further crucial model
parameters whose empirical estimates come with an even larger uncertainty.
Seismic observations conducted at Villarrica imply the existence of a shallow
magma reservoir with a lateral diameter of at least 5 km and a vertical
extent of about 2.5 km whose centre of mass is located at a depth of around
Dr=3 km below the summit , implying a conduit
length of about Lc=2 km. Assuming an ellipsoidal magma reservoir, this
implies a magma reservoir volume of Vr=35km3 at Villarrica.
The magmatic system of Cotopaxi in contrast seems to be more complex and
hosts a rather small magma pocket (2 km3) beneath the SW flank at a
depth of about 4 km below the summit . Furthermore, seismic
observations revealed fluid movements (magma and/or hydrothermal
fluids) within a centrally located 85 km3 column spanning 2 to 14 km
depth below the summit . This fluid column is assumed to
connect the laterally offset shallow pocket with two much larger deeper magma
reservoirs, which are situated between 7 and 11 km and somewhere at a depth
greater than 16 km below the summit . For heating 85 km3 of rock, these deep-seated
magma reservoirs may be rather large. Missing any accurate volume estimate,
we estimate that the upper of the two deep-seated reservoirs hosts a magma
volume of Vr=35 km3 with a centre of mass depth of Dr=8 km. The
choice of equal reservoir volumes for both Villarrica and Cotopaxi allows
for a better comparison of the impact of varying the other volcanic
parameters. Further, we assume the small magma pocket as the lower end of the
conduit, i.e. with a conduit length of Lc=4 km.
Calculation of tide-induced conduit flow
Oscillating centre of mass displacement. After a negligible settling
time, the driven oscillator described by Eq. () oscillates with
semi-diurnal periodicity, and we obtain the general long-term solution
z(t)=z0⋅sin(ωsd⋅t-φ0)z0=a0(ω02-ωsd2)2+(γ⋅ωsd)2φ0=arctanγ⋅ωsdω02-ωsd2.
Navier–Stokes equation for periodical pipe flow. When exposed to a
constant force (per unit mass) fext0, a viscous fluid in a cylindrical
pipe with radius Rc flows with a parabolic velocity profile v0(r), 0≤r≤Rc:
v0(r)=Rc2⋅fext04⋅ν1-rRc2.
When exposed to a periodically varying and thus time-dependent external force
fext(t)=fext0⋅eiωt, the analytical solution of the
flow profile is more complicated :
v(r,t)=v0(r)‾⋅ℜ-i⋅8N2⋅eiωt⋅1-J0(-iNrRc)J0(-iN),
with the centre of mass velocity v0(r)‾ of a constant forcing
(see Eq. ), the real part ℜ[..], the imaginary unit i, the
Bessel function J0(..), and the dimensionless parameter N=ων⋅Rc. In the limit N→0, the
velocity profile asymptotically adopts the time dependency as well as the
magnitude of the external force. For N=1 the exact magnitude is already
0.98⋅fext0, and the radial profile shows hardly any deviation from
a parabolic profile. For the chosen model parameters
(Table ) and ω=ωsd, we obtain N≈0.2 and thus Eq. () reduces in very good approximation
to the familiar
v(r,t)≈Rc2⋅fext(t)4⋅ν1-rRc2.
Derivation of the equation of motion (Eq. ). The
vertical velocity of the centre of mass can be obtained as z˙(t)=z0⋅ωsd⋅cos(ωsd⋅t-φ0) from
Eq. () and as v(t)=(π⋅Rc2)-1⋅∫0Rcv(r,t)⋅2πrdr=Rc28⋅ν⋅fext(t) from Eq. (). Further, we know fext(t)=fint(t)=γ⋅z˙(t) from Eq. (). Applying
fext(t) to Eq. () reveals γ=8⋅νRc2 and ultimately the fully parameterised equation of motion in
Eq. ().
Calculation of the collision volumes
As is common for most coalescence models (including those cited above), we
consider spherical bubbles only. Two spherical bubbles with radii f1⋅Rb and f2⋅Rb (f1 and f2 drawn from δbsize(f))
collide as soon as the distance between their bubble centres is rcoal=(f1+f2)⋅Rb. We introduce the collision volume H(f1,f2;Δt) associated with a bubble with radius f1⋅Rb as the volume
enclosing all possible initial locations of the bubble centre of another
bubble with radius f2⋅Rb such that both bubbles collide (and thus coalesce) at
the latest after a time interval Δt. All bubble collision mechanisms
are derived as enhancements of the initial static collision volume
H0(f1,f2)=4π3⋅Rb3⋅(f1+f2)3,
and we consider only those bubble pairs which have not collided already in
the initial state. The absolute enhancement of the collision volume due to a
particular bubble collision mechanism divided by Δt thus gives the
enhancement of the bubble collision rate contributed by the particular
mechanism. Because the tide-induced mechanisms are derived for a semi-diurnal
cycle, the relative strengths of all coalescence mechanisms are compared with
respect to this time interval Δtsd.
The collision volumes of the different collision mechanisms are all derived
with the same approach: we fix the position of a bubble with arbitrary radius
f1⋅Rb and derive H(f1,f2;Δt) with respect to the relative
motion of another bubble with arbitrary radius f2⋅Rb. In each case
the initial collision volume H0(f1,f2) is subtracted either already
tacitly in the motivation or explicitly mathematically. Higher-order details
such as the influence of a third bubble on the numeric results are ignored.
Tide-enhanced bubble collision volume. We fix the horizontal
coordinates (r,φ)bubble1=(r0,0), 0≤r0≤Rc, of the
first bubble, where the cylindrical symmetry of the conduit allows us to pick
the azimuth angle without loss of generality and vary the horizontal
coordinates (r,φ)bubble2=(r,φ) of a second bubbles. The
horizontal distance h between the two bubbles is thus given by r2=r02-2⋅r0⋅h⋅cos(φ)+h2. Within a semi-diurnal cycle, the
peak-to-peak differential tide-induced vertical displacement of two bubbles
at the radial coordinates r and r0 is given by Δztide(r,r0)=2⋅|z0(r)-z0(r0)| (see Eq. ). The tide-induced
collision volume is then the integral of Δztide(r,r0) integrated
over a circle with radius rcoal:
D2Htide(r0)=∫0rcoaldhh∫02πdφΔztide(r,r0)D3=4⋅Ψ⋅r0Rc2∫0rcoaldhh2∫02πdφ|cos(φ)-h2r0|.
This integral has to be split into two integrals at the angles where the sign
of the absolute function changes, which is the case at ±φ′=±arccos(h2r0)≈±π2:
D4Htide(r0)=16⋅Ψ⋅r0Rc2∫0rcoaldhh2sin(φ′)-cos(φ′)⋅φ′︸≈1forh≪r0D5≈16⋅Ψ⋅r0Rc2⋅rcoal33D6=4⋅Ψ⋅r0π⋅Rc2⋅H0(f1,f2).
We integrate Htide(r0) over the local spatial bubble
distribution in the conduit in order to obtain the average effect. We
parameterise the (isotropic) spatial bubble distribution by the
depth-independent δbspatial(r0)=(1+α)⋅1R⋅(r0R)α, which is a homogeneous
distribution for α=1 but with all bubbles at the conduit wall if
α→∞, respectively. For the averaged
tide-induced collision volume, we obtain
D7Htide=∫0Rσtide(r0)⋅δbspatial(r0)⋅dr0D8=1+α2+α︸distribution⋅4⋅Ψπ⋅Rc︸tidal⋅H0(f1,f2)︸scale.
The “distribution term” is 23 for an isotropic bubble
distribution and approaches unity if all bubbles are close to the host rock.
Arguably, the conditions for crystal nucleation and thus bubble nucleation
are better close to the host rock where the magma is cooler and more crystals
and thus nucleation possibilities are available. Following this reasoning but
also because we want to examine the maximum possible tidal impact, we set the
distribution term to unity. The “tidal term” contains the information on
the scale of the effective tide-induced impact. The “scale term” contains
the information on the actual bubble size distribution, highlighting that the
relative tidal enhancement is identical for any bubble size distribution, at
least in our simple model.
Buoyancy-induced bubble collision volume. Two bubbles with radii
f1⋅Rb≠f2⋅Rb have a differential rise velocity Δvbuoy=|f22-f12|⋅vbuoy(Rb), and thus their relative
distance changes during the rise. The two bubbles will collide if the larger
bubble is below the smaller and if the horizontal distance between their
bubble centres is at most rcoal. Accordingly, the buoyancy-induced
collision volume Hbuoy is a cylindrical volume with base area π⋅rcoal2 and cylinder length Δvbuoy⋅Δtsd:
D9Hbuoy(f1,f2)=π⋅rcoal2⋅|f22-f12|⋅vbuoy(Rb)⋅ΔtsdD10=3⋅|f2-f1|4⋅Rb⋅vbuoy(Rb)⋅Δtsd⋅H0(f1,f2).
For a given pair of bubbles with radii f1⋅Rb≠f2⋅Rb,
f1 and f2 drawn from δbsize(f), the ratio of the
contribution from the tide-induced and the buoyancy-induced collision
mechanisms is
HtideHbuoy=24⋅Ψ⋅νπ⋅Rc⋅|f1-f2|⋅g⋅Rb⋅Δtsd.
The bulk ratio (with respect to the local magma layer) can be obtained by a
previous and separate integration of Htide and Hbuoy over f1 and
f2 with respect to the actual bubble size distribution
δbsize(f) (rather than integrating Eq. ). For
the explicit bubble size distribution δ̃bsize from
Eq. (), we obtain the bulk collision volumes
H̃tide and H̃buoy,
D12H̃tide(q)H0(1,1)=(1+0.89⋅q+0.11⋅q2)⋅4⋅Ψπ⋅Rc,D13H̃buoy(q)H0(1,1)=(q-q2)⋅916⋅Rb⋅vbuoy(Rb)⋅Δtsd,
and thus the bulk ratio (used for the calculation of Fig. ):
H̃tideH̃buoy=60⋅(0.9+1+q2q-q2)⋅ν[m2s-1]⋅Ψ[m]Rc[m]⋅Rb[µm].
Growth-induced bubble collision volume. In magma with a dispersed
bubble flow (vbuoy≪vmelt), a rising bubble exhibits a pressure
decrease rate of
ΔpΔt=vmelt⋅(∇p)vert.
Ignoring accompanying changes in secondary parameters such as melt
temperature and magma composition and assuming for simplicity a monodisperse
bubble size distribution (thus Rb3∝ϕ), for the
enhancement of the collision volume due to a rise-driven pressure decrease by
Δp≪p0 (apply Eq. on Eq. ), we obtain
Hdisp(Δp;p0)=H0(Rb(p0-Δp))-H0(Rb(p0))=H0(1,1)⋅CH2O0-12KH2O⋅p0CH2O0-KH2O⋅p0⋅Δpp0D16+O(Δpp02],
where we assume that ρmelt is constant and ρgas follows the
ideal gas law. Inserting Eq. () in Eq. (), we
obtain
Hdisp(p0)H0(1,1)=CH2O0-12KH2O⋅p0CH2O0-KH2O⋅p0D17⋅vmelt⋅Δtsd⋅(∇p)vertp0.
The ratio of the contribution from the tide-induced and the growth-induced
collision mechanism (used for the calculation of Fig. ) is
HtideHdisp=CH2O0-KH2O⋅p0CH2O0-12KH2O⋅p0︸≈0.25-0.5⋅4⋅Ψ[m]⋅p0[MPa]Rc[m]⋅vmelt[mh-1].
Author contributions
All authors have contributed at all stages to the
development of the presented model by their critical assessment of the
model set-up and the model implications. The particular foci of the individual
authors were as follows: FD and UP conceptualised the model set-up; FD and SB
contributed expertise on the Earth tides; FD, SB, SA, and NB contributed
expertise on volcanic degassing and bubble coalescence; FD, UP, and TW
assessed the physical consistency of the model. FD wrote the paper.
Competing interests
The authors declare that they have no conflict of
interest.
Acknowledgements
The authors thank Antonella Longo, Nolwenn Le Gall and another anonymous
reviewer for their comments on the paper. We thank the Deutsche
Forschungsgemeinschaft for supporting this work within the project DFG
PL193/14-1.
Financial support
The article processing charges for this open-access publication were covered by the Max Planck Society.
Review statement
This paper was edited by Antonella Longo and reviewed by
Nolwenn Le Gall and one anonymous referee.
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