# 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.

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**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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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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