The dynamics of mid-ocean ridge hydrothermal systems: Splitting plumes and fluctuating vent temperatures
ABSTRACT We present new, accurate numerical simulations of 2D models resembling hydrothermal systems active in the high-permeability axial plane of mid-ocean ridges and show that fluid flow patterns are much more irregular and convection much more unstable than reported in previous simulation studies. First, we observe the splitting of hot, rising plumes. This phenomenon is caused by the viscous instability at the interface between hot, low-viscosity fluid and cold, high-viscosity fluid. This process, known as Taylor–Saffman fingering could potentially explain the sudden extinguishing of black smokers. Second, our simulations show that for relatively moderate permeabilities, convection is unsteady resulting in transiently varying vent temperatures. The amplitude of these fluctuations typically is 40 °C with a period of decades or less, depending on the permeability. Although externally imposed events such as dike injections are possible mechanisms, they are not required to explain temperature variations observed in natural systems. Our results also offer a simple explanation of how seismic events cause fluctuating temperatures: Earthquake-induced permeability-increase shifts the hydrothermal system to the unsteady regime with accompanying fluctuating vent temperatures. We demonstrate that realistic modelling of these high-Rayleigh number convection systems does not only require the use of real fluid properties, but also the use of higher order numerical methods capable of handling high-resolution meshes. Less accurate numerical solutions smear out sharp advection fronts and thereby artificially stabilize the system.
- [Show abstract] [Hide abstract]
ABSTRACT: Simulation of geothermal systems is challenging due to coupled physical processes in highly heterogeneous media. Combining the exponential Rosenbrock-Euler method and Rosenbrock-type methods with control-volume (two-point flux approximation) space discretizations leads to efficient numerical techniques for simulating geothermal systems. In terms of efficiency and accuracy, the exponential Rosenbrock-Euler time integrator has advantages over standard time-discretization schemes, which suffer from time-step restrictions or excessive numerical diffusion when advection processes are dominating. Based on linearization of the equation at each time step, we make use of matrix exponentials of the Jacobian from the spatial discretization, which provide the exact solution in time for the linearized equations. This is at the expense of computing the matrix exponentials of the stiff Jacobian matrix, together with propagating a linearized system. However, using a Krylov subspace or Léja points techniques make these computations efficient.The Rosenbrock-type methods use the appropriate rational functions of the Jacobian of the ODEs resulting from the spatial discretization. The parameters in these schemes are found in consistency with the required order of convergence in time. As a result, these schemes are A-stable and only a few linear systems are solved at each time step. The efficiency of the methods compared to standard time-discretization techniques are demonstrated in numerical examples.Advances in Water Resources 03/2013; · 2.78 Impact Factor - SourceAvailable from: Shamik SarkarInternational Journal of Multiphase Flow 07/2014; · 1.94 Impact Factor
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ABSTRACT: A new numerical model that integrates hydrodynamics, thermodynamics, and the chemistry of mineral formation in deepwater hydrothermal vents is developed. Transport and spread of plume fluid and the minerals formed are simulated in three stages: plume dynamics stage that is momentum and buoyancy driven; transition to far-field conditions as a gravity current; and far-field conditions where the mineral particles move according to advection–diffusion governed by ambient currents and settling velocities that eventually lead to bed deposition. Thermodynamics include the change in plume temperature and its related properties. Chemical reactions due to hot vent fluid mixing with cold entrained ambient water into the plume change its properties and the behaviour. The model considers the formation of several types of minerals. Model simulations and field measurements compare reasonably well for plumes in Atlantic and Pacific Oceans. Results for a simulation in East Pacific Rise 21°N are presented.Journal of Hydraulic Research 03/2014; 52(1). · 1.35 Impact Factor
Page 1
The dynamics of mid-ocean ridge hydrothermal systems:
Splitting plumes and fluctuating vent temperatures
Dim Coumoua,⁎, Thomas Driesnera, Sebastian Geigera,1, Christoph A. Heinricha,
Stephan Matthäib
aInstitute of Isotope Geochemistry and Mineral Resources, ETH Zurich, 8092 Zurich, Switzerland
bDepartment of Earth Science & Engineering, Imperial College London, RSM Building, Exhibition Road, London, SW7 2AZ
Received 18 November 2005; received in revised form 17 February 2006; accepted 23 February 2006
Available online 17 April 2006
Editor: H. Elderfield
Abstract
We present new, accurate numerical simulations of 2D models resembling hydrothermal systems active in the high-permeability
axial plane of mid-ocean ridges and show that fluid flow patterns are much more irregular and convection much more unstable than
reported in previous simulation studies. First, we observe the splitting of hot, rising plumes. This phenomenon is caused by the
viscous instability at the interface between hot, low-viscosity fluid and cold, high-viscosity fluid. This process, known as Taylor–
Saffman fingering could potentially explain the sudden extinguishing of black smokers. Second, our simulations show that for
relatively moderate permeabilities, convection is unsteady resulting in transiently varying vent temperatures. The amplitude of
these fluctuations typically is 40 °C with a period of decades or less, depending on the permeability. Although externally imposed
events such as dike injections are possible mechanisms, they are not required to explain temperature variations observed in natural
systems. Our results also offer a simple explanation of how seismic events cause fluctuating temperatures: Earthquake-induced
permeability-increase shifts the hydrothermal system to the unsteady regime with accompanying fluctuating vent temperatures. We
demonstrate that realistic modelling of these high-Rayleigh number convection systems does not only require the use of real fluid
properties, but also the use of higher order numerical methods capable of handling high-resolution meshes. Less accurate numerical
solutions smear out sharp advection fronts and thereby artificially stabilize the system.
© 2006 Elsevier B.V. All rights reserved.
Keywords: convection; numerical simulation; Taylor–Saffman instability; viscous fingering; black smoker
1. Introduction
Since the discovery of sub-seafloor hydrothermal
systems in the 1970s, numerous manned and unmanned
dives [1–5] have given insight into the underlying
physical and chemical processes. Continued observa-
tion over the last decades revealed highly transient and
complex flow systems. The sub-seafloor physics
include the vigorous convection of chemically complex
Earth and Planetary Science Letters 245 (2006) 218–231
www.elsevier.com/locate/epsl
⁎Corresponding author.
E-mail address: coumou@erdw.ethz.ch (D. Coumou).
1Current address: Department of Earth Science & Engineering,
Imperial College London, RSM Building, Exhibition Road, London,
SW7 2AZ.
0012-821X/$ - see front matter © 2006 Elsevier B.V. All rights reserved.
doi:10.1016/j.epsl.2006.02.044
Page 2
fluids, super-critical phase separation of saline fluids
and high-temperature fluid–rock interaction including
dissolution and precipitation of minerals [6,7]. The
observational data posed a number of key scientific
questions, one of them concerns the temperature of
black smoker effluents [8]: Even though magma-
chambers have temperatures of ∼1200 °C, maximum
temperatures of vent fluids are never greater than
∼400 °C. In addition, transient and spatial variations in
black smoker temperatures remain a key question of
active current debate [9]. Some black smokers have
effluent temperatures, which remain constant over
decades [10,11], while others show strong variation
or extinguish without obvious reason [12,2,13,14].
Spatially, effluent temperatures from black smokers can
vary on the scale of tens of meters.
Numerical modelling can help to elucidate the sub-
seafloor physics and explain observational data. For two
reasons, the general approach has been to model
convection in a two-dimensional open-top porous
medium. First, at many locations deep circulation is
believed to be primarily two-dimensional, in the plane
of the spreading axis where fracture permeability is
greatest [15,7]. Second, the absence of a sediment layer
above the axis allows for free flow of fluids in and out of
the oceanic crust. Due to the extreme physical and
chemical complexity, however, all numerical studies
have to make simplifications, even in the two-
dimensional case.
Most studies took a Boussinesq approximation in
which fluid density variations are considered only in the
buoyancy term [16–19]. In addition to the Boussinesq
assumption, Rosenberg et al. [16] linearized density
variations as a function of temperature and assumed
viscosity to be constant. Despite these simplifications,
they correctly showed that discharge areas are much
smaller than recharge areas and suggested that hydro-
thermal convection at the ridge axis might never reach a
steady state. Wilcock [17] took a steady-state approach
adding non-linear terms for density, viscosity, and heat
capacity of the fluid as well as including non-
Boussinesq terms. He showed that by doing so,
maximum venting temperatures are up to 50% higher
than those obtained with uniform fluid properties and
the flow pattern includes much more recirculation.
Further, Wilcock [17] argued that the exit temperature in
open-top hydrothermal systems reaches a limit of 0.65
times the bottom temperature. Rabinowicz et al. [20],
taking a steady-state approach as well, showed that this
factor is independent of the Rayleigh number. By taking
an inclined base of the porous medium, Rabinowicz et
al. [20] were able to increase exit temperatures to a
maximum of 0.85 times the bottom temperature. The
steady state approach of Rabinowicz et al. [20] and
Wilcock [17] is limited, however, to low Rayleigh
number calculations.
Employing a second order accurate finite volume
scheme, a Boussinesq approximation and a constant
viscosity, Cherkaoui et al. [18] modelled time-depen-
dent, high-Rayleigh number, open-top convection
systems. They described the character of convection
for Rayleigh numbers up to 1100. They showed that at
such high Rayleigh numbers unsteady flow and heat
transport display a sequence of bifurcations from
periodic to quasi-periodic to chaotic patterns. Straus
et al. [21] showed that, compared with a Boussinesq
approach with linearized properties, including the non-
Boussinesq terms and using non-linear water properties
greatly enhances the instability of convection. Recent-
ly, Jupp and Schultz [8] included the non-Boussinesq
terms as well as realistic fluid properties in open-top
convective system. Based on simulations using the
HYDROTHERM pure H2O software package [22],
they argued that maximum black smoker effluent
temperatures of ∼400 °C can be explained solely by
the non-linear, temperature and pressure dependent
properties of H2O.
All these numerical studies have greatly contributed
to our understanding of fluid flow patterns in open-top
thermal convection systems. They were, however, not
able to explain the observed spatial and transient
variations in black smoker temperatures. A steady-
state approach [17,20] is limited to low-Rayleigh
number systems. Frequently, the thermal expansivity,
compressibility, viscosity and heat capacity were
assumed to have constant values [18,19,23,24], al-
though for the temperature and pressure range encoun-
tered in mid-ocean ridge hydrothermal systems they
vary highly non-linearly over several orders of magni-
tude and peak near the critical temperature. This greatly
enhances the instability of convection in a water-
saturated porous medium [25].
Jupp and Schulz [8] did include all non-linearities
but nevertheless found steady-state temperature solu-
tions. A first order accurate Finite Difference method
[22] and relatively coarse meshes were used in their
study. This numerical approach smears out sharp
advection fronts, which numerically stabilizes the
convective system and might result in artificial
steady-state solutions.
In this study we account for the full non-linearity of
the properties of H2O, including the effects of thermal
expansion and compression on flow transients. In
addition, we perform the calculations on a high-
219D. Coumou et al. / Earth and Planetary Science Letters 245 (2006) 218–231
Page 3
resolution mesh employing a numerical method that is
second order accurate in space [26–28]. This allows us
to detect new dynamic features that potentially explain a
range of salient observations on MOR hydrothermal
systems.
We will show that for reasonable permeability
values, convection is much more vigorous than
previously reported. The unsteady behavior of sub-
seafloor convection can by itself explain temperature
variations measured on the seafloor. The permeability
structure does not necessitate any geometric complex-
ity, nor transient events such as cracking or magma
addition are required to produce temperature varia-
tions in the range of the observations. We will argue
that the non-linearity of fluid properties not only
explains the temperature of vent fluids [8], but also its
transient variability. However, events such as earth-
quakes may have a profound influence on the
permeability structure [29] and by increasing perme-
ability shift the systems from steady to unsteady
convection.
Second, we will demonstrate that due to the large
viscosity variations, the front of uprising thermal
plumes may become unstable and can split into two
plumes. This phenomena, known as Taylor–Saffman
fingering, has to our knowledge never been reported
within the context of thermal convection in hydrother-
mal systems. We show under which conditions this
phenomenon occurs and discuss the numerical
requirements.
This article is structured as follows. In Sections 2–4
the governing equations, numerical method and model
setup are described. Section 5 summarizes the
numerical results. A detailed discussion on the
observed plume splitting phenomena is given in
Section 6. It also discusses possible implications for
natural MOR hydrothermal systems and Section 7
presents the final conclusions.
2. Governing equations
Volumetric flow through porous media is usually
described by Darcy's law [30]:
v ¼ −k
lf
ðjp−qfgÞð1Þ
Where v is the Darcy velocity, k the permeability, μf
the fluid's dynamic viscosity, p the pressure, ρf the
fluid's density and g the gravitational acceleration
vector. A mass balance for a single-phase fluid in a
porous medium can be expressed in terms of the
continuity equation:
/Aqf
At
¼ −jdðvqfÞð2Þ
Here we assume that the compressibility of the rock
is orders of magnitude lower than that of the fluid. ρfis
a function of temperature (T) and pressure (p). Writing
the partial derivative of Eq. (2) in terms of p and T, and
inserting Eq. (1) into Eq. (2) gives the following
expression for the pressure field:
/qf bf
Ap
At−afAT
At
??
¼ jdqf
k
lf
ðjp−qfgÞ
??
ð3Þ
where αfis the fluid's thermal expansivity and βfits
compressibility. Assuming local thermal equilibrium
between rock and fluid, the advection–diffusion
Table 1
List of symbols used
Physical parameterValueUnit
αf
Fluid expansivity
−1
qf
Aqf
Ap
3.0
1×10−6
Eq. (7)
Eq. (7)
Eq. (8)
EOS [32](*)
EOS [32](*)
2700
0.1
D/(2H)
EOS [32]
880
–
Aqf
AT
°C−1
βf
Fluid compressibility
1
qf
(*)
Pa−1
ϵ
κ
λ
λc
λm
μf
ρf
ρr
ϕ
A
cpf
cpr
D
Exponential term
Thermal diffusivity
Width of instabilities
Cut-off λ
λ of maximum growth
Fluid viscosity
Fluid density
Rock density
Porosity
Aspect ratio of convection cell
Fluid isobaric heat capacity
Rock isobaric heat capacity
Distance between adjacent
upflow zones
Gravitational acceleration
Vertical extent model
Permeability
Thermal conductivity
Pressure
Temperature
Darcy velocity
Dimensionless time
1-D vertical velocity
Critical velocity
EOS = equation of state.(*)Unless stated differently.
–
m2s−1
m
m
m
Pa s
kg m−3
kg m−3
–
–
J (kg °C)−1
J (kg °C)−1
m
g
H
k
K
p
T
v
t*
U
vc
|g|=9.8
1000(*)
10−14(*)
2.0
Eq. (3)
Eq. (4)
Eq. (1)
–
–
Eq. (5)
m s−2
m
m2
W (m °C)−1
Pa
°C
m s−1
–
m s−1
m s−1
220D. Coumou et al. / Earth and Planetary Science Letters 245 (2006) 218–231
Page 4
equation for the temperature field at single phase
condition can be written as:
ð/qfcpfþ ð1−/ÞqrcprÞAT
¼ jKjT−jd ðqfcpfvTÞ
With cpfbeing the specific heat and K the thermal
conductivity. This formulation neglects the adiabatic
term and this scheme thus assumes that cpfis constant
over a single timestep. For single phase fluids, however,
this error is small [31]. Table 1 summarizes all symbols
used in this paper.
At
ð4Þ
3. Numerical method
We employ the IAPS-84 equation of state of pure
water [32] to determine μf, αf, βf, cpf, ρfand enthalpy.
Eqs. (3) and (4) are strongly non-linear, as all fluid
properties are non-linear functions of p and T. They have
mixed hyperbolic (advective) and parabolic (diffusive)
character. We solve the system of equations within the
framework of the object-oriented C++ code Complex
System Platform, CSP [33], using a mixed finite
element–finite volume (FE–FV) approach [27,28]. FV
methods are robust in solving advection-type problems,
whereas FE methods perform better on diffusion-type
problems. In the FE–FV method, the pressure Eq. (3) is
solved decoupled from the energy conservation Eq. (4).
An explicit FV method is used to solve the advective
part of Eq. (4), while an implicit FE method, employing
the SAMG multigrid solver [34], is used to solve the
parabolic equations of heat and pressure-diffusion.
Hence, the coupled Eqs. (3) and (4) are solved
sequentially but the best suited numerical method is
applied on the respective sub-equations. Temperature
diffusion is calculated first. Thereafter, advection of heat
is solved using the velocities from the previous timestep.
Now the fluid properties are updated and the pressure
field is calculated. From the pressure field new Darcy
velocities (Eq. (1)) are determined. Geiger [27,28]
subjected this FE–FV-scheme within CSP to numerous
analytical and numerical benchmark tests to verify its
accuracy and computational efficiency.
4. Model setup
Unless stated otherwise, we integrate Eqs. (3) and (4)
on a 3600 m by 1000 m rectangular geometry with a
mesh of 50,000 uniform, triangular elements. Perme-
ability k as well as porosity ϕ are kept constant during
the simulations. The top boundary represents the
seafloor at roughly 2.5 km water depth resulting in a
constant pressure p=25 MPa. To allow hot fluids to vent
freely through the top boundary, we use a mixed thermal
boundary condition [17,8,19,35]. In finite elements
along the top boundary experiencing upflow, the vertical
temperature gradient is set to zero, allowing uninhibited
outflow of hydrothermal fluid of any temperature.
Boundary elements experiencing downflow take in
water of a fixed temperature of 10 °C. Fluids can leave
or enter the model only through its top boundary. All
other boundaries are no-flow boundaries. The bottom
boundary is kept at a fixed temperature of 1000 °C.
Initially, the porous medium is saturated with 10 °C
water at hydrostatic pressures.
Though highly simplified, this geometry approx-
imates a vertical plane of high permeability along the
ridge axis with a continuous magma-chamber under-
neath. 2-D models have been used as a matter of
numerical convenience. However, as permeability is
expected to be higher along the axis compared to off-
axis [36], convection is likely primarily two-dimension-
al [37,15]. Seismic studies have identified magma lenses
stretching continuously for several kilometers at fast
spreading ridges [38]. On the other hand, precipitation
of anhydrite in narrow, along axis, recharge zones might
limit along-axis convection [39].
5. Results
5.1. Convection patterns at early times: splitting
plumes
Splittingofhot,risingplumeshasbeenobservedinall
simulationswithsufficientlylargepermeability.Herewe
discuss simulation results for k=10−14m2(Figs. 1 and
3). Plumes start to form after a relatively short period
(∼40 years) of vertical thermal diffusion. Initially these
plumes have similar widths but due to merging, a
variation in plume widths arises. Since the upward
Fig. 1. Temperature field of a segment of simulation 2 (Table 2) after
(a) t=125 years and (b) t=130 years. The straight vertical line in (a)
represents the position of the thermal profile plotted in Fig. 7.
221D. Coumou et al. / Earth and Planetary Science Letters 245 (2006) 218–231
Page 5
velocity of a plume is a function of its width [40], some
plumes grow faster than others. Plumes travelling faster
than their direct neighbors typically spread in the
transverse direction, thereby shielding their neighbors.
Shielded small plumes eventually merge with larger
ones. The cross-flow accompanying the spreading of the
largerplumes, causes the thermal gradientat their tops to
become steeper [41]. A steeper gradient is hydrodynam-
ically less stable since viscosity and density change over
a shorter distance. In short, simultaneously the plume
itself becomes broader and its front less stable. This can,
when the front is steep enough and the width of the
plume isbroadenough,result intheplumetosplit. Fig.1
shows a typical case of plume splitting.
We have performed a series of simulations to
investigate the plume splitting phenomenon. Some
selected simulations covering a range of physical
parameters are summarized in Table 2. They indicate
that plume splitting occurs when two conditions are
satisfied. First, convection needs to be vigorous, i.e.
k=10−14m2. Second, the large viscosity contrast
between cold and hot fluids is essential. The initial
condition of a completely cold layer underlain by a
1000 °C hot layer gives a very large initial viscosity
contrast. To determine the influence of these initial
conditions, a simulation was run which started with a
linear geotherm (simulation 4, Table 2). This gave
similar results in terms of convective behavior including
plume splitting. Table 3 summarizes the most important
results for different numerical schemes. Based on these
results, we will show in section 6 that the observed
phenomena are caused by Taylor–Saffman instabilities
[42] rather than a numerical artifact.
5.2. Convection patterns at late time: temperature
variations
In this section we discuss the characteristics of
convection for the standard geometry (3600×1000 m)
Table 2
Simulations (1)
#Dimension
x×z
(km)
k
(m2)
Fluid properties
EOS, simplification
ConvectionVenting Plume
splitting
Mode Aspect ratioT
(°C)
Area
(m)
1
2
3
4
5
6
7
8
9
10
11
12
3.6×1.0
3.6×1.0
3.6×1.0
3.6×1.0
3.6×1.5
7.2×2.0
3.6×1.0
3.6×1.0
3.6×1.0
3.6×1.0
3.6×1.0
7.2×2.0
10−13
10−14
10−15
10−14
10−14
10−14
10−14
10−14
10−14
10−14
10−14
10−14
[32]
[32]
[32]
[32]
[32]
[32]
[32], βf=0
[32], αf=0 (Eq. (3))
[32], μf=3×10−4
[32], μf=3×10−5
Boussinesq(*)
Boussinesq(*)
Unsteady
Unsteady
Steady
Unsteady
Unsteady
Unsteady
Unsteady
Unsteady
Unsteady
Unsteady
Unsteady
Unsteady
–
0.22
0.26
0.22
0.20
0.18
(?)
(?)
0.2/0.3
0.14
0.3
0.23
374±20
377±20
387
377±20
375±20
375±20
(?)
(?)
50–400
50–400
50–500
50–450
∼60
∼60
∼150
∼60
∼60
∼100
(?)
(?)
∼200
∼100
∼250
∼400
Yes
Yes
Yes
Yes
Yes
Yes
Yes
CSP simulation results for different model geometries, permeability and used fluid properties. All results were obtained employing the same
numerical method: 70 m2elements with linear interpolation functions, second order FV transport scheme and first order time-stepping. For
parameters not listed here, values have been used as given in Table 1. Except for simulation 4, which has an initial linear geotherm, all simulations
started with cold water everywhere within the 2D-domain.(*)The Boussinesq approximation is used as well as constant μfand αf. (?) Simulations not
run long enough for accurate determination.
Table 3
Simulations (2)
# FV
scheme
1st/2nd
order
FE spatial
discretization
Temporal
discretization
Plume
splitting
Size
(m2)
Interpolation Time
stepping
1st/2nd
order
13
14
15
16
17
18
19
20
21
22
23
1
1
2
2
2
2
2
2
2
1
1
12
70
70
70
70
70
70
70
70
600
600
Linear
Linear
Linear
Quadratic
Linear
Linear
Quadratic
Linear
Quadratic
Linear
Quadratic
CFL
CFL
CFL
CFL
0.1×CFL
Iterative
Iterative
CFL
CFL
CFL
CFL
1
1
1
1
1
–
–
2⁎
2⁎
1
1
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
CSPsimulationresultsfor different numericalmethods.Weuse several
temporal discretization schemes as well as different FE interpolation
functions to test for which numerical scheme plume splitting is
observed.All simulations useda 3600×1000 m geometry, k=10−14m2
and included all non-linearities in H2O [32]. Properties as given in
Table 1 have been used.
⁎Predictor-corrector scheme.
222D. Coumou et al. / Earth and Planetary Science Letters 245 (2006) 218–231
Page 6
and k=10−14m2(simulation 2 in Table 2). Plume
splitting only occurs when hot fluid displaces cold
fluid. Up to times of ∼2000 years, convection is highly
unstable with plumes collapsing and new plumes
forming continuously. Within this time frame, plume
splitting has a profound effect on the overall convec-
tion pattern, making it much more irregular. At later
times (t=2000 years), the system becomes more stable
with regularly spaced plumes. Their positions remain
relatively constant over time. Still, parts of the system
episodically become unstable, leading to the occasional
collapse of plumes and rise of new ones. Fig. 2 shows
such a situation, where part of the system has become
unstable. Clearly, plume splitting can be observed
again. This “semi-steady” convection pattern is typi-
cally sustained for thousands of years. We define the
aspect ratio of convection cells as A=D/(2H), where D
is the average horizontal distance between neighboring
plumes and H the vertical extent of the model. Typical
values for A are ∼0.22. Close to the top, upward
travelling, pipe-like plumes show necking. Though
plumes have average widths of ∼200 m, at the top their
width typically is only ∼60 m.
While the position of plumes is relatively stationary,
the temperature of fluids leaving the top boundary
varies with time. Their mean temperature is 377 °C but
oscillations with an amplitude of ∼40 °C occur (Fig.
3). The period of oscillation is on the order of decades.
Temperatures reach maxima of ∼395 °C and, although
typically minimum fluid temperatures are ∼365 °C,
they can drop as low as ∼250 °C. This is a direct
consequence of the instable convection patterns. Each
of the high-temperature upflow zones behaves in a
pulsating manner. In regular intervals, relatively hot
volumes of water are mobilized at the bottom boundary
and travel upward quickly.
Apart from this relatively regular oscillation, two
sharp drops in temperature, for periods of ∼10 years,
can be observed in Fig. 3. They are caused by a locally
increased inflow of cold seawater pushing down the
thermal plume. The plume retains its horizontal
position but does not reach the top boundary anymore
for a relatively short time. The process is caused by the
counter balance of recharge and discharge. The
discharge region temporarily becomes an inflow region
while neighboring plumes vent at an increased rate.
5.3. Influence of permeability
For k=10−13m2(simulation 1, Table 2), flow is
extremely irregular and plume splitting occurs contin-
uously. After 2000 years of simulation, no “semisteady
state” has evolved. Only for relatively short periods
(∼decades) fluids vent at the same location on the
seafloor. Afterwards the plume becomes unstable and
vanishes in the convection system. Within such a
period the exit temperature of the plume still fluctuates
between ∼365 and ∼395 °C, very similar to the
k=10−14m2-case. The period of oscillation is much
shorter, however. For k=10−13m2it is of the order of
Fig. 2. Temperature T of simulation 2 (Table 2) after t=3000 years. Part of the system has become unstable and plume splitting can be observed.
Fig. 3. Temperature T (simulation 2, Table 2) at a location where hot
fluidsexit thetop boundarymeasuredoveratime periodof 1000years,
calculated by taking the mean temperature of a number of FE nodes in
a 50 m length at the top boundary.
223D. Coumou et al. / Earth and Planetary Science Letters 245 (2006) 218–231
Page 7
years, compared to decades when k=10−14m2. The
upflow zones are narrow and generally of the order of
∼100 m wide. This is considerably smaller than the
200 m wide plumes at k=10−14m2. Flow focussing
near the top boundary, however, is less pronounced,
such that the outflow areas are of a similar size
(∼60 m).
At k=10−15m2(simulation 3, Table 2), plume
splitting does not occur at all. The evolution of the
convection pattern is completely different. After 1500
years of vertical thermal diffusion, convection starts
and a steady-state convection system is established at
t=8600 years. The convection cells have an aspect
ratio of A=0.26. The temperature of fluid exiting the
top boundary is 387 °C and remains constant through
time. The convection systems become the hottest at
k=10−15m2, which is in good agreement with
simulation results by Hayba and Ingebritsen [43] for
continental magmatic-hydrothermal systems.
5.4. Influence of magma-chamber depth
We have also conducted a number of simulations to
test the effect of larger horizontal and vertical model
dimensions (simulations 5–6, Table 2). Increasing the
vertical dimensions of the model decreases the aspect
ratio A of convection cells. In other words, plumes
become more elongated. The temperature of fluids
exiting the top boundary for models with larger vertical
dimensions (simulations 5–6, Table 2) is very similar to
those with z=1000 m. One might expect that due to
increased loss of heat by diffusion and expansion as the
fluids have to travel over a larger distance, temperatures
would be lower. However, the larger pressure at the
bottom causes the upwelling temperatures to be higher.
As argued by Jupp and Schultz [44], upwelling
temperatures will be such that the total power output
of the convection cell is maximized. They showed that
the upwelling temperature at which power output is
maximized, increases with increasing pressure. Hence,
the higher bottom pressures in simulations 5–6 (Table 2)
will cause upwelling temperatures to be hotter, which
appears to compensate the heat loss due to diffusion and
expansion along the extended upwelling path.
5.5. Influence of viscosity
A number of simulations have been run to determine
theinfluenceofsimplifiedassumptionsforμf(simulations
9–12, Table 2). Making μfconstant suppresses plume
splitting.Thereisstillasensitivitytoit:Whenμf=3×10−5
Pa s (simulations 10, Table 2) relatively narrow plumes
form.Forlarger μfthe widthofplumesbecomes larger.In
all cases (simulations 9–12, Table 2), no necking of
plumes near the top boundary is observed. This results in
larger areas where fluids exit the system, with tempera-
tures ranging spatially from 50 to 500 °C. Although
convection is unstable, the pulsating behavior in the
upflow zones, as observed in simulations using all non-
linear fluid properties, is far less pronounced. Fig. 4
compares exit temperatures of simulation 2 with those of
simulation 11, which uses a Boussinesq approximation, a
linear dependency of the density on temperature and a
constant viscosity. In simulation 2 all fluids exit the top
boundary at temperatures between 350 and 400 °C,
reflecting the transient variability. For simulation 11,
fluids exit the top boundary at temperatures ranging from
50 to 500 °C, reflecting spatial rather than transient
variability.
6. Discussion
6.1. Theory of splitting plumes
Plume splitting, or viscous fingering, can occur
when a less viscous fluid displaces a more viscous one.
It is a well studied process in flow through porous
media especially for the case of two immiscible fluids
[45,46]. In general, when one fluid displaces another, a
combination of density and viscosity ratios as well as
the direction and magnitude of flow can conspire to
trigger fingering [47]. In the one-dimensional case of
two immiscible fluids, a critical velocity vc can be
found above which a moving plume head becomes
Fig. 4. The temperature distribution of fluid exiting through the top
boundary. The vertical axis denotes the percentage of the fluid that
traveled upward through the top boundary with a temperature given by
the horizontal axis, measured over a period of 3000 years. Black bars
show results from simulation 2 (Table 2) which takes account of all
non-linearities in fluid properties, grey bars show results from a
simulation 11 (Table 2) which employs a Boussinesq fluid.
224D. Coumou et al. / Earth and Planetary Science Letters 245 (2006) 218–231
Page 8
unstable [47]. For the special case of buoyancy driven
flow caused by temperature differences, we can find vc
when we assume that thermal diffusion is negligible.
The critical velocity vcfor an upward migrating hot
plume, displacing cold water (inset of Fig. 5) is then
given by
vc¼kgðqc−qhÞ
lc−lh
ð5Þ
In this context, vccan be regarded as the critical
interface velocity above which instabilities will get
amplified and plumes might split. The subscripts h and
c respectively refer to hot fluids, with low viscosity and
density, and cold fluids with high viscosity and density.
If the difference in density between the fluids increases,
the critical velocity increases. This, however, is of
minor importance in the case of buoyancy driven
convection systems as velocities themselves are
proportional to density differences. Viscosity variations
have a much more profound effect. Eq. (5) shows that
when viscosity differences go to zero the critical
velocity goes to infinity. Simulations employing the
non-linear fluid properties of H2O but taking μ
constant, showed, in agreement with Eq. (5), that
plume splitting does not occur (Table 2).
In Fig. 5, vcis plotted against relevant temperatures
at pressures of 25, 30 and 35 MPa. Also plotted is the
approximate Darcy velocity at the interface between
hot and cold fluid for the 1D case (inset of Fig. 5). The
vertical Darcy velocity v (Eq. (1)) of the hot fluid at
the interface can be approximated by v=kg(ρc−ρh)/
μh. Comparing this formulation with Eq. (5) shows that
vNvcif μcN2μh. This condition is already fulfilled for
an upward flowing fluid of 40 °C when the cold fluid
is 10 °C. Temperature differences are much larger in
the systems discussed here. Therefore, v is typically an
order of magnitude larger than vc.
As both v and vcscale with k, one could conclude
that viscous fingering should occur at all permeabil-
ities, in disagreement with the numerical results.
However, this is incorrect as the relative influence of
diffusion compared to advection plays a crucial role. If
diffusion is included, Eq. (5) only holds for t=0. For
later times diffusion smears out the thermal front. The
steepness of the front determines at which minimum
width instabilities can form. The steeper the front, the
smaller the minimum width of an instability will be.
Tan et al. [40] considered the one-dimensional
horizontal flow of fluid with a solute-dependent
viscosity. They find analytical solutions for t=0 and
numerical solutions for tN0. In an analogous way, we
can find first order approximations for 1D vertical flow
of hot water displacing cold water (inset of Fig. 5). We
assume that the decay of μfwith increasing T can be
written as an exponential function:
lfce−ϵT
ð6Þ
For ϵ=3 a reasonable fit of actual H2O viscosities is
achieved. Next, we take the so-called quasi-steady-state
approximation. This assumes that the growth rate of
disturbances is much faster than the rate of change in the
base state. (For a detailed discussion and complete
derivation of the formulae, see Tan et al. [40].) We can
now derive the following equations:
k0
c¼8pj
ϵU
ð7Þ
k0
m¼
8pj
ffiffiffi
ð25
p
−4ÞϵU
ð8Þ
Where λ is the width of an instability in meters. The
subscripts c and m refer to the cut-off, or minimum
width and the width of maximum growth of an
instability, respectively. In other words, λc is the
smallest plume that can form and λmis the width of a
plume that is most likely to form. The superscript 0
indicates a non-dimensional time t*=0, with
t ¼ t*j
U2
ð9Þ
In Eqs. (7)–(9), U isthe upward velocity of the plume
and κ is the thermal diffusivity.
Eqs. (7) and (8) show that, if ϵ=3, only for small κ/
U ratios (κ / Ub10 m), λcand λmhave values smaller
than the model scale. The Peclet number Pe, defined as
Fig. 5. The critical velocity above which an advection front becomes
unstable plotted against T for P=25 MPa (solid), P=30 MPa (dotted)
and P=35 MPa (dashed). The dashed-dotted line represents the Darcy
velocity for the 1D geometry as depicted in the inset.
225D. Coumou et al. / Earth and Planetary Science Letters 245 (2006) 218–231
Page 9
the ratio between advective transport and diffusive
transport, consequently has to be large (Pe≥103). For a
realistic thermal diffusivity of 10−6m2, this requires
relatively large permeabilities (k=10−14). For times
larger than t*=0, analytical solutions cannot be found.
Numerical solutions were obtained by Tan et al. [40] and
are shown in Fig. 6 for ϵ=3, U=1.0×10−7m/s and
κ=1×10−6m2/s.Fig.6 shows the relative growth rate of
instabilities for several values of t*. The maximum of
each curve corresponds to λm. The point where the
curves cross with the x-axis, i.e. instabilities have a zero
growth rate, corresponds to λc. Fig. 6 shows that for
larger times the width of instabilities will increase due to
the diffusion of the initially steep thermal front.
6.2. Quantification of splitting plumes
Fig. 6 shows that when t increases, λc and λm
increase due to diffusion of the thermal front. In
advection dominated fluid flow systems, the steepness
of thermal fronts is not only determined by the time it
has diffused, but also by the total fluid flow pattern. As
described in the previous section, plumes traveling
faster than their neighboring plumes, spread in the
lateral direction. The cross flow accompanied with this
spreading process steepens the top of a thermal front
[41], making it less stable. In other words, simulta-
neously the plume itself becomes wider and the
minimum width of instabilities becomes smaller. At
the moment the width of the plume allows more than
two instabilities to grow, the tip of the plume becomes
unstable and splits. The width of a rising plume should
therefore at least be 2λc, before it can split. This process
occurs when convection is fully evolved, hence no
direct semi-analytical comparison is possible.
In a first approximation, however, we can compare
the temperature profile perpendicular to a front, just
before plume splitting occurs, to an analytical pure-
diffusion solution:
?
TðzÞ ¼Tmax−Tmin
2
erfc
z
2
ffiffiffiffiffi
jt
p
?
ð10Þ
By fitting Eq. (10) to the observed thermal front, we
can determine a dimensionless diffusion time t*. Fig. 1
shows the temperature field of a typical plume just
before and after plume splitting. The solid curve in
Fig. 7 shows the thermal front just before the plume
splits. An optimal fit of this thermal front with the
analytical solution is produced for t*=2.8. From Fig. 6
we estimate λcand λmfor t*=2.8.
k2:8
cf85F10 m
ð11Þ
k2:8
mf170F10 m
ð12Þ
This is in good agreement with numerical observa-
tions. The plume, depicted in Fig. 1 splits into two
plumes having widths of ∼150 and ∼100 m. Both
plumes are larger than λc. The wider, 150 m plume
travels upward faster, as is expected because its width is
close to λm.
Fig. 6. Relative growth rate of instabilities plotted against the width of
an instability (λ) for different dimensionless times t*. The point where
the curves cross the x-axis coincides with λc. The maximum of the
curvesresembleλm.Thewidthof maximumgrowthrateincreaseswith
time due to smearing out of the front by diffusion. Note the inverted
and non-linear x-axis scale. Modified from Tan et al. [40].
Fig. 7. Temperature cross-profile of a plume at x=800 m after 125
years for different numerical schemes. The solid line shows results
from a second order accurate transport scheme using a high-resolution
(50, 000 element) mesh (simulation 15, Table 3). The dotted curve
plots results of a simulation using the same high-resolution mesh but
with a first order accurate scheme (simulation 14, Table 3). The dashed
line gives results using a first order accurate scheme in combination
with a low-resolution (6000 element) mesh (simulation 22, Table 3).
All curves have been fitted with analytical solutions, giving
dimensionless times of t*=2.8, t*=3.3 and t*=13.5 respectively.
226 D. Coumou et al. / Earth and Planetary Science Letters 245 (2006) 218–231
Page 10
Since the upward velocities typically exceed vcand
the widths of the splitting plumes are in good
agreement with our first order approximations, we
conclude that these are Taylor–Saffman fingers rather
than numerical artifacts. Their absence in numerical
models assuming constant viscosity further confirms
this conclusion.
6.3. Numerical aspects of splitting plumes
The preceding discussion raises the question why
previous numerical studies have not observed plume
splitting. With their steady-state approach, Wilcock
[17] and Travis et al. [48] could not resolve these
transient processes, which occur at high Rayleigh
numbers. Low-Rayleigh number simulations imply low
Peclet numbers, resulting in very large values for λc.
Most studies treated the fluids as incompressible and
held thermal expansion coefficient (αf) and viscosity
(μf) constant. Jupp and Schultz [8] modelled the full
non-linear properties of H2O, but still did not observe
plume splitting. As discussed in the previous section,
plume splitting occurs only if the thermal front is
sufficiently steep. Transport schemes that are only first
order accurate in space smear out thermal gradients
and hence stabilize it artificially. Therefore, only high-
er order accurate transport schemes that preserve sharp
thermal fronts can model plume splitting. Table 3
summarizes different numerical schemes that have
been tested for the standard geometry (3600×1000 m)
and k=10−14m2. Fig. 8 compares first and second
order transport schemes and two levels of mesh
refinement. It shows that by employing first instead
of second order accuracy, or by decreasing the spatial
resolution, plume fronts become more diffuse. Fig. 8a
shows the results of a second order accurate transport
scheme [26] combined with a high-resolution mesh
(∼70 m2elements with linear FE interpolation
functions, simulation 15, Table 3). The same mesh
has been used in Fig. 8b but now employing a first
order accurate transport scheme (simulation 14, Table
3). In this simulation plume splitting occurs only very
Fig. 8. Temperature field after 200 years for three different numerical schemes: (a) second order accurate transport with high-resolution mesh, (b) first
order accurate transport with high-resolution mesh and (c) first order accurate transport with low-resolution mesh.
227D. Coumou et al. / Earth and Planetary Science Letters 245 (2006) 218–231
Page 11
rarely. In Fig. 8c the spatial accuracy has further been
reduced by employing a low-resolution mesh
(∼600 m2elements with linear FE interpolation
functions, simulation 22, Table 3). This leads to even
more diffused fronts and the complete absence of
plume splitting. Fig. 7 illustrates how much less
accurate schemes smear out thermal fronts. Taking the
approach described previously, this leads to larger
dimensionless diffusion times and results in larger
values for λc and λm. Thus, a first order accurate
scheme (Fig. 8b and dotted line in Fig. 7) strongly
reduces plume splitting compared to a second order
accurate scheme. Using a low-resolution mesh in
combination with a first order accurate scheme (Fig. 8c
and dashed line in Fig. 7) eradicates the phenomenon.
We conclude that accurately modelling the full
dynamics of advection-dominated near-critical convec-
tion systems, requires (a) including the full non-
linearity of the physical properties of water and (b)
the use of high-resolution meshes, as well as (c) higher
order numerical schemes. Previous studies either used
first order accurate schemes, rather coarse spatial reso-
lutions compared to this study or constant viscosities.
As shown by Geiger et al. [27,28] the decoupled
pressure–temperature algorithm solved on a dual FE–
FV mesh, as used in this study, is not only second order
accurate in space, but, due to its high computational
efficiency, allows the use of high-resolution meshes.
6.4. Black-smoker vent temperatures
Althoughourgeneric model setup does not reflect the
complexity of real MOR hydrothermal system in any
detail, the features emerging in the simulations strongly
resemble many observations made on natural systems.
To us, this implies that the non-linear dependence of
water properties on pressure and temperature not only
determines the maximum venting temperatures [8] but
can also explain much of the variability in venting
temperature. Our simulations show that already for
relatively moderate permeability values [36] convection
is unsteady and black smokers will vent at transiently
varying temperatures. Though less pronounced, Cher-
kaoui and Wilcock [18] observed similar periodic
behavior in their high-Rayleigh number Boussinesq
simulations.
We propose a simple physical explanation for
temperature variations which have been observed
after seismic events at black smoker fields
[12,13,49,14]: the earthquake induced permeability-
increase [29] shifts the convection system to unsteady,
pulsating venting with accompanying temperature
fluctuations. Over time, the system might reorganize
itself, with clogging of fractures decreasing the bulk
permeability, and vent temperatures becoming stable
again. This provides a very simple explanation why
some black smokers have been observed to vent with a
constant temperature on a decade timescale [11], while
others, especially after earthquakes, show variability on
timescales of weeks [50]. Observed temperature
oscillations are typically on a smaller temporal
timescale as described here [12,2,13]. Though part of
the problem might be a lack of sufficient time series,
this might reflect that permeability values could be
higher (up to 10−11m2[51]) than used in this study.
Other proposed mechanisms to explain temperature
variations, such as periodically propagating cracking
fronts [12] or magma replenishment [14], are physi-
cally much more complex. Currently, the only available
dataset of vent-temperature over a period of many years
is the one of the EPR 9°50′N [14], which has been
monitored from 1991 till 2002. Over this period,
effluent temperatures varied from 350 to 395 °C.
Clearly those vents have been undergoing phase
separation, which is outside the scope of this study,
still the range of observed temperatures matches well
with our simulations.
6.5. Spatial dynamics
The structure of the hydrothermal system, regularly
spaced, narrow, pipelike upflow zones, correlates well
with observations. One of the few datasets available
on sub-seafloor hydrothermal structures is given by
Tivey et al. [52], who inferred such narrow pipe-like
upflow zones at the Main Endeavour Field. In contrast
to the temperature of vent fluids, our models predict
that vent locations stay relatively constant in time. For
k=10−14m2, vent fields remain essentially constant.
For k=10−13m2changes in the location of vent fields
are on the order of a decade. Plume splitting can cause
a single plume to surface at two different locations for
a relatively short period, after which one of the
plumes collapses. In another mechanism, a plume can
retreat from the seafloor due to locally increased
recharge. Those two mechanisms, thus, could either
temporarily or permanently extinguish black smokers.
The first resembles the extinguishing of a single vent
within a vent field, the second resembles the
extinguishing of a whole vent area. Whether these
processes actually occur is speculative. Not many
vents have been observed to cease venting, but this
could be because of a lack of sufficient data. A
possible candidate is Tube Worm Pillar at the EPR
228 D. Coumou et al. / Earth and Planetary Science Letters 245 (2006) 218–231
Page 12
9°50′N, which stopped venting while other nearby
vents kept or even increased in temperature [53]. Von
Damm also reports two other vents at the EPR going
extinct without an apparent cause (Von Damm,
personal communication, 2005).
6.6. Model limitations
Our isotropic, homogeneous permeability distribu-
tion is a highly idealized representation of actual ridge
crest systems. Actual permeability measurements at
ridge axis are not available and estimates point to values
in the range 10−11to 10−13m2[51], which is much
higher than permeabilities used in this study. However,
these estimates are based on a simple pipe model
assuming a Boussinesq fluid. Real water tends to
convect with the critical Rayleigh number up to 31
times lower [21]. If this effect can directly be propagated
into the pipe models, the estimates would indicate
permeabilities for typical mid ocean ridge fields that
exactly match the range studied here. Further, perme-
ability in reality is layered and anisotropic, especially
within the dike segment, where highly permeable
fractures are superimposed, and appears to be temper-
ature dependent. Basalt may become impermeable at the
brittle–ductile transition at temperatures between 700
and 800 °C [7], though some authors have argued that
hydrothermal fluid flow starts at much higher tempera-
tures of 900 to 1000 °C [54]. Also, the clogging of
fractures by mineral precipitation is an important
process affecting permeability [55, 39]. The constant
temperature boundary condition at the bottom assumes
an infinite heat supply. Though an overestimation, this
best resembles fast spreading ridges which have kilo-
meters long continuous magma chambers [38]. Down-
ward propagating cracking fronts [56] could keep fluids
in contact with fresh, hot rocks. This study demon-
strates, however, that – provided that realistic fluid
properties and accurate numerical schemes are used –
geometrically simple, isotropic permeability models
naturally predict dynamic features that closely resemble
some observations at mid-ocean ridges.
Using a pure water equation of state instead of one
for seawater simplifies the calculations considerably.
Pure water above the critical point is a single
supercritical fluid. By contrast, a NaCl–H2O fluid can
boil at temperatures and pressures far above the critical
point of pure water and separate into a high-salinity
brine and low-salinity vapor. This will probably have a
significant effect on the results presented above [57].
Extending similar model simulations to saline fluid
systems will be a subject of future research.
7. Conclusion
Higher order numerical simulations of hydrothermal
circulation in the highly permeable axial plane of mid
ocean ridges, using realistic water properties, demon-
stratethatfluidflowpatternsaremuchmoreirregularand
convection less stable than previously inferred. The
splittingofrisingthermalplumescanbeexplainedbythe
viscous, Taylor–Saffman instability at the interface
between hot, low-viscosity fluids and cold, high-
viscosity fluids. Analytical estimates on the width of
splittedplumesareingoodagreementwiththenumerical
results. While plumes can vent at the same spatial
position for thousands of years, vent temperatures can
oscillate on the timescale of years and venting can stop
temporarily.Unsteady convection(at k=10−14m2) alone
can account for temperature variations of typically 40 °C
onthetimescaleofdecadesorless.Higherpermeabilities
shorten the period of oscillation.
Our sensitivity analysis shows, that in order to
realistically model high-Rayleigh number convection of
aqueous fluids above the critical temperature and
pressure,onenotonlyneedstoincludethenon-linearities
in the fluid's properties, but also has to use higher order
transport schemes and high-resolution meshes. The use
of less accurate numerical schemes smears out sharp
gradients in fluid properties by numerical diffusion. This
artificially stabilizes the system and can hide emergent
properties like Taylor–Saffman fingering.
Acknowledgements
We thank Bob Lowell for reviewing two earlier
manuscripts and giving useful comments during discus-
sions at AGU 2005 and 2006. Also we like to thank
William Wilcock for helping to improve this manu-
script. This work was supported by the Swiss National
Science Foundation.
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