We present quantitative modeling results for the effects of surface relief on hydrothermal convection at ocean-spreading centers investigating how vent site locations and subsurface flow patterns are affected by bathymetry induced sub-seafloor pressure variations. The model is based on a 2-D FEM solver for fluid flow in porous media and is used to simulate hydrothermal convection systematically in 375 synthetic studies. The results of these studies show that bathymetric relief has a profound effect on hydrothermal flow: bathymetric highs induce subsurface pressure variations that can deviate upwelling zones and favor venting at structural highs. The deviation angle from vertical upwelling can be expressed by a single linear dependence relating deviation angle to bathymetric slope and depth of the heat source. These findings are confirmed in two case studies for the East Pacific Rise at 9°30′N and Lucky Strike hydrothermal fields. In both cases, it is possible to predict the observed vent field locations only if bathymetry is taken into account. Our results thereby show that bathymetric relief should be considered in simulations of submarine hydrothermal systems and plays a key role especially in focusing venting of across axis hydrothermal flow onto the ridge axis of fast spreading ridges.
 Mid-ocean ridges with their intense magmatic and hydrothermal activities are good examples where we can study the effects of bathymetry on submarine hydrothermal flow. At both, fast and slow spreading ridges, formation of new oceanic crust is controlled by a combination of magmatic, tectonic and hydrothermal processes. Fast spreading ridges, like the East Pacific Rise, are likely to be in steady state and have a stable melt lens beneath the ridge axis [Baran et al., 2005; Morgan and Chen, 1993]. Their topography is characterized by an axial high on the order of hundred meters in height whereas the lateral extension may reach several kilometers [Carbotte and Macdonald, 1994] and hydrothermal venting appears to be exclusively located directly on-axis [Baker and German, 2004]. Slow-spreading ridges are more complex and undergo shifting periods of magmatic and tectonic phases [Escartín et al., 2008]. They are most likely not in steady state and a stable melt lens has so far only been imaged beneath the magmatically active Lucky Strike segment on the Mid-Atlantic Ridge [Singh et al., 2006]. The bathymetry of slow-spreading ridges show mostly wide, deep median valleys with highly variable relief in excess of 1000 m that can be up to 10 km in width [e.g.,Coakley and Cochran, 1998; Dick et al., 2003; Kleinrock et al., 1997; Searle et al., 1998; Smith and Cann, 1992; Tolstoy et al., 2001]. Vent fields have been reported both on-axis (e.g., 5°S [Koschinsky et al., 2008], Lucky Strike [Dusunur et al., 2009]) as well as off-axis (e.g., Logatchev [Schmidt et al., 2007], TAG [deMartin et al., 2007]). Direct observations indicate that magmatic hydrothermal systems at mid-ocean ridges often vent at bathymetric highs. For example, the Lucky Strike vent-field is located on the ridge axis above the Axial Magma Chamber (AMC) on a central volcanic cone. At fast spreading ridges, the situation is somewhat simpler as most hydrothermal venting occurs on-axis on top of the axial high [Baker and German, 2004]. At the East Pacific Rise (9°40′N, 104°17′W), for example, a band of seismicity above the magma chamber illustrates that hydrothermal activity is aligned along the ridge axis [Tolstoy et al., 2008].
 At both fast- and slow-spreading ridges, the locations of hydrothermal venting are likely to be controlled by a multitude of parameters including permeability structure, heat source location and crustal accretion mode. Previous studies have addressed submarine hydrothermal flow using numerical models of different complexities. Single phase studies were carried out by numerous researchers [Coumou et al., 2008, 2009b; Fontaine and Wilcock, 2007; Theissen-Krah et al., 2011; Wilcock, 1998]. More sophisticated multiphase studies on convection patterns were recently performed as well [Coumou et al., 2009a]. However, the effect of bathymetric relief was largely neglected in these studies. The impact of topography on subaerial systems is well known [Ingebritsen et al., 2006]: groundwater flow is directed from relative highs in the water table to relative lows so that venting is likely to occur in structural lows and recharge occurs in structural highs. The opposite occurs in submarine hydrothermal systems where venting occurs in bathymetric highs. The physics of the focusing effect of bathymetric relief on hydrothermal venting into the highs can be explained by closer inspection of the temperature and pressure fields of submarine hydrothermal systems and has been already described by previous work [Ingebritsen et al., 2006]. Pressure at the seafloor is controlled by water depth, while the pressure inside the crust is additionally influenced by the density of the pore fluids. Steady state conductive geotherms mimic the bathymetric relief and are consequently deflected upwards below bathymetric highs. As fluid densities are strongly temperature dependent, this thermal structure controls the pressure field and results in relative pressure lows below structural highs. As a consequence, a lateral force will be exerted driving convection from the high toward the low pressure area (found beneath the bathymetric high) and any thermal plume will be forced to ascend toward and vent at bathymetric highs.
 This effect has, to some extent, been addressed by earlier studies [e.g., Fisher et al., 1990; Hartline and Lister, 1981; Lowell, 1980]. Fisher et al.  studied the influence of topography on the southern flank of the Costa Rican Rift and found that subcritical convection in the study area results in wide zones of fluid recharge occurring in the troughs accompanied by narrow discharge zones at the ridges. A similar result was also obtained by Fisher et al. who found that basement relief and differential sediment thickness play a greater role than bathymetry in enhancing passive, off-axis convection. More recently,Kawada et al. studied the influence of seamount aspect ratio and heat flow on fluid flow patterns within and around the seamount. They established that seamounts with high aspect ratio contain multiple convection cells irrespective of the crustal heat flow. On the other hand, small seamounts can act as a discharge zone when heat flow is low and act as a recharge zone if heat flow is high. However, a systematic study on bathymetry-driven fluid flow is still missing. Here we present a 2-D single phase model that is especially tailored to address this topic. In addition to a number of synthetic studies, we use observations from sites across the East Pacific Rise (9°30′N) and the Lucky Strike vent-fields as real-world counterparts in order to understand how bathymetry affects hydrothermal convection patterns.
2. Hydrothermal Convection Model
2.1. Governing Equations
 We have developed a hydrothermal flow model that is based on Darcy flow and resolves for pure water convection in porous media. The model formulation assumes that only one fluid phase is present at any spatial location and is, therefore, restricted to single phase flow of pure water during which phase separation does not occur. This simplification is justified as most mid-ocean spreading centers are located at water depths in excess of or at least close to the critical end-point of pure water (∼220 bar).
Equation (1) describes mass conservation in porous media, considering the fluid is compressible and porosity is constant
where is the Darcy fluid velocity, ρf is fluid density, t is time and ϕ corresponds to matrix porosity (see Table 1 for complete list of variables and values). The velocity is defined by Darcy's Law
Here k is the rock permeability, μf is the dynamic viscosity of the fluid, P is the fluid pressure and is the gravitational acceleration. Substituting equation (2) into equation (1) and by splitting density changes into their pressure and temperature parts, we obtain
where αf is fluid thermal expansion coefficient and βf is fluid isothermal compressibility.
Table 1. List of Model Parameters
Value (Case Studies)
Density of fluid
Based on PROST
Based on PROST
Dynamic viscosity of fluid
Based on PROST
Based on PROST
Specific heat capacity of fluid
Based on PROST
Based on PROST
J kg−1 K−1
Density of matrix
Specific heat capacity of matrix
J kg−1 K−1
Porosity of matrix (synthetics)
Permeability of matrix
5 × 10−15
W m−1 K−1
Thermal expansion coefficient
 The energy equation is described by
where λ corresponds to effective thermal conductivity, T to temperature and cp to specific heat capacity, which accounts for advective and conductive heat transport in addition to heat storage both in the fluid and the rock matrix. Subscript m and fdenote matrix and fluid properties, respectively. Pressure-volume work and viscous dissipation are neglected in the energy equation [Garg and Pritchett, 1977]. Fluid properties of density, viscosity and specific heat capacity are treated as functions of pressure and temperature and are computed at every time step using the PROST library [Bauer, 1998] for the IAPWS-84 equation of state.
2.2. Numerical Implementation
 The governing equations are solved using a 2-D finite element approach in combination with an advection scheme. The solution is computed for every time step in sequential order: (1) thermal diffusion is resolved, (2) temperature is advected using the Darcy velocities, (3) fluid properties are evaluated, (4) new fluid pressure and porous flow velocities are then computed.
 The fluid pressure equation (equation (3)) is solved using finite elements. The weak form of this equation, derived by the Galerkin method, is
where Nis linear shape function for three-node triangle elements. Bars and primes denote averaged properties on an element and transposes, respectively. Superscriptsn and o denote new and old time steps, respectively. Symbols Ω and Γ are computational domain and its boundaries, respectively. All fluid and rock properties are averaged over an element and all mass matrices are lumped. The pressure solution is obtained by implementing equation (5)into a modified version of the Matlab-based FEM solver MILAMIN byDabrowski et al. . Taking derivatives of the pressure solution provides the updated Darcy velocities at element centers. These velocities are subsequently averaged onto nodes in order to use them with a node-based semi-Lagrangian advection scheme. Note that the fluid and matrix properties are calculated on the nodes and Darcy velocities are mapped on nodes by taking equal contributions of neighboring elements. This mapping could be avoided by the use of node-centered finite-volume techniques that use velocities at element centers [Geiger et al., 2004]. We have experimented with both advection schemes and find that for single-phase calculations, the semi-Lagrangian approach does not have problems with mass conservation and allows for larger computational time steps. We have, therefore, used a semi-Lagrangian solver for the advective part of the energyequation (4) that uses the element shape functions for temperature interpolation. The diffusive part of equation (4) is again solved with the FEM scheme. This model was validated against the results of Coumou et al.  and a previous version of the model was also benchmarked against Cherkaoui and Wilcock  [Iyer et al., 2010].
3. Synthetic Studies
 This section aims at constraining the effects of bathymetric relief on hydrothermal convection. Two sets of simulations are carried out. The first set explores the effect of a single bathymetric high on a single plume and the second set of simulations takes into account the effect of multiple bathymetric highs on multiple plume formation. Simulations with a flat seafloor are used as reference cases for both sets which are then used to discuss the effects of bathymetry. The synthetic studies with bathymetric relief are referred to as non-references cases. All simulations are run until a quasi-steady state is approached; i.e., the single high and multiple high simulations are run for 10,000 and 3000 years, respectively. The fluid and matrix parameters used in the simulations with bathymetric relief are the same as those used in the corresponding reference case simulation. The matrix is homogeneous and its properties are kept constant throughout the run for all synthetic simulations. The values for matrix properties are listed inTable 1. The number of elements used in the simulations is approximately 8,300 elements per square kilometer.
3.1. Single Bathymetric High
3.1.1. Model Geometry and Setup
 The domain in the single bathymetric high simulations is 4600 m wide while the depth is varied between 1000 and 2000 m. The bathymetric high is triangular in shape and rises above the flat surface of the domain. The starting point of the bathymetric high is fixed at distance of 1450 m. The flat surface of the 2-D domain is kept at a constant pressure of 30 MPa simulating a seafloor depth of ∼3 km to ensure that phase separation does not occur in experiments with the highest elevation of the peak. The surface rising above the flat part is kept at a pressure corresponding to its depth. Temperature in the model is initialized to 10°C and the model is heated at the bottom boundary by a Gaussian heat source located at distance of 1500 m. The maximum temperature of the heat source is 1000°C and the width of Gaussian function is 1/12 of the box size in the x-direction. The defined width of the Gaussian heat source ensures that only a single plume is formed in the simulations. The side-boundaries of the box are insulating and the sides and bottom boundaries of the box are impermeable. An upper boundary condition is applied to allow the fluid to vent freely where the plume reaches the surface and allow water at 10°C to enter the domain in recharge zones. This condition is applied by calculating the direction of fluid vectors at the upper boundary nodes.
3.1.2. Reference Simulation
 A single hydrothermal plume forms at the bottom heat source and ascends vertically to reach the surface right above the center of the heat source. Plume-splitting is observed initially at the bottom of model, but the single plume is rapidly established. The plume vents at the surface after 750 years of convection when the model depth is 1500 m and reaches steady state.
3.1.3. Non-reference Simulations
 The non-reference simulations are used to study the maximum deviation of the plume from its original vent location as observed in the reference simulation in the presence of a single bathymetric high. The dimensions of the high are defined by its wavelength (half-width of the high) as well as its amplitude (elevation of the summit with respect to the flat seafloor) which in turn controls how much the plume could deviate from its original vent location. The wavelength and amplitude of the high is systematically varied from 500 to 1500 m in 250 m increments and 100 to 500 m in 50 m increments, respectively. A single plume is initially formed and vents at the seafloor, right above the center of heat source as observed in the reference simulation. The plume migrates toward the summit of the bathymetric high with time and finally reaches a steady state. The time after which plume migration begins and the time required for migration to complete before a steady state is reached depends on the wavelength and amplitude of the prescribed bathymetric high. The results show that plume deviation and migration time of the plume is inversely dependent on the wavelength and directly dependent on the amplitude (Figure 1). An increase in wavelength results in an increase in the distance of the summit from the original plume location, thereby decreasing the control of bathymetry on convection whereas an increase in amplitude directly amplifies this effect. The plume can deviate toward the summit completely for relatively low wavelengths and high amplitudes. Conversely, if the wavelength and amplitude are relatively high and low, respectively, migration of the plume stops at some distance away from the summit. We find that the lateral deviation of the plume divided by its vertical extent, a value easily converted to the plume deviation angle relative to the vertical (θ), can be expressed as a linear function of bathymetric slope (amplitude/wavelength) (Figure 2a). In the following we will refer to this value as relative plume deviation (D = lateral plume deviation/vertical plume extent). The results show a further dependence on the depth of the heat source (Figure 2a), which is systematically varied between 1000 and 2000 m in 250 m increments. The deeper the heat source, the less pronounced is the impact of bathymetry on plume deviation. This can be explained by the diminishing relative pressure difference caused by the bathymetric high as depth is increased. We find that the heat source depth has a linear influence on the coefficients of the dependence between relative plume deviation and bathymetric slope so that a single general relationship between relative plume deviation, bathymetric slope and depth of heat source can be formulated
where A is the amplitude (m), W is the wavelength (m), and z is depth of the heat source (m). Figure 2b shows good agreement between the simulation results and the functional dependence described in equation (6). A linear least squares fit yields a slope of 0.986 and a regression coefficient of 0.983 showing that equation (6) can be used to approximate the effect of a single bathymetric feature on subsurface fluid flow if the location of the heat source is known.
3.2. Multiple Bathymetric Highs
3.2.1. Model Geometry and Setup
 The domain in the 2-D synthetic simulations with multiple bathymetric highs is 3.6 km wide and 1 km deep. The bathymetric relief, in the non-reference cases, is represented by a sinusoidal wave with varying amplitudes and wavelengths.
 The turning point of the sinusoidal surface in the simulated domain is overlain by 2.5 km of seawater and the temperature in the domain is initialized to 10°C. The domain is continuously heated at the bottom at a temperature of 1000°C mimicking the effect of a stable magmatic heat source. Similar to the single bathymetric high simulations, the side-boundaries of the box are insulating and the sides and bottom boundaries of the box are impermeable. The temperature upper boundary condition is also the same as that applied to the single bathymetric high simulations. The non-reference simulations are carried out for surface wavelengths of 3600, 1800, 900, 720, 600, 514, 450, 400, 360 and 300 m corresponding to between 2 and 11 bathymetric highs, respectively, while the amplitude varies between 20 m and 300 m, by 20 m increments for all the wavelengths.
3.2.2. Reference Simulation
 In the reference simulation, several plumes are initially formed at the base of the box where the temperature gradient and subsequently, the density and viscosity gradients in the fluid are highest. These plumes coalesce during their ascent to the surface to form fewer stable upwelling zones (Figure 3) and sometimes exhibit plume-splitting phenomena similar to that seen byCoumou et al. . For the given model setup, the simulation evolves toward a quasi steady state after ∼1250 years of convection where eight plumes reach the surface at a fixed distance from each other. A number of smaller plumes are formed at the base of the domain but they are quickly assimilated in the larger, established plumes (Figure 3). The average distance between the plumes is ∼514 m. The aspect ratio of a convection cell is defined as A = D/2H, where D is the distance between two adjacent plumes and H is the depth of the domain. The aspect ratio for the convection cells in the reference simulation is therefore 0.25, which is consistent with the results of simulations performed by Coumou et al. .
3.2.3. Non-reference Simulations
 In these numerical experiments, the bathymetry is represented by a sinusoidal wave with varying amplitude and wavelength. A surface with periodic bathymetry and a constant temperature bottom boundary is selected in order to study the effect of multiple highs on multiple plumes while accounting for the interaction between different plumes.
Figure 4 shows the results for selected example runs after 3000 years model run time. All left panel plots show the temperature field and all right panel plots show corresponding histograms of fluid venting as functions of bathymetric relief. Note that the depth of the surface is normalized to the amplitude of the bathymetric relief with the turning point of the surface slope taken as the zero value. A first order observation is that venting generally tends toward bathymetric highs in the presented simulations. Both wavelength and amplitude control how effectively venting is focused into bathymetric highs. Simulations in Figures 4a–4c illustrate the effects of relief wavelength for a fixed amplitude of 40 m. The tendency of venting at bathymetric highs increases as the wavelength decreases. Figures 4c and 4i show that venting occurs exclusively in the bathymetric highs when the number of bathymetric highs is comparable to or less than the number of plumes in the reference simulation. If the wavelength is increased, venting is more widely distributed over the surface (Figures 4a and 4g).
 The amplitude of bathymetric relief also has a strong effect: as amplitude increases, venting is progressively more focused into bathymetric highs. The bottom three simulations in Figure 4 demonstrate the effect of amplitude for a fixed wavelength of 1800 m. The results show that at low amplitude venting may occur at the flanks or even bathymetric lows (Figures 4d and 4j), but the plumes increasingly migrate toward bathymetric highs when the amplitude is increased (Figures 4e, 4f, 4k and 4l). In general, stronger deviations from vertical plume ascent toward bathymetric highs are found with increasing amplitude. Plumes may vent at flanks and lows during the initial stages of the simulations but are shifted toward the highs as the plumes evolve in time.
 The preceding section showed that both amplitude and wavelength control vent field location. To further explore the focusing of venting at bathymetric highs, the time-integrated mass fraction of vent fluids exiting within the upper half of bathymetric highs is plotted as a function of wavelength and amplitude inFigure 5a. The intersections of the gridlines are the locations of data points. The mass flux is integrated over time starting after 1000 years of convection when all the plumes have reached the surface until 3000 years of convection when the simulation is in a quasi steady state. The plot summarizes the characteristics of the relationships between amplitude and wavelength discussed above. If the wavelength is slightly larger or smaller than the plume spacing in the reference case (∼514 m), a bathymetric relief of only ∼40 m amplitude is sufficient to focus venting into bathymetric highs. If the wavelength is now increased, venting also occurs at flanks or even lows. This de-focusing effect of increasing wavelength can be compensated by increasing amplitude. There is, however, a limit to this: if amplitudes become very large for small wavelengths, venting again tends to occur at flanks as well which occurs when the size of the ascending plume head is larger than the width of the bathymetric high. This behavior suggests that the focusing effects of amplitude and wavelength can be summarized by a single parameter.Figure 5b shows that all values plotted in Figure 5ado collapse onto a bell-shaped curve if plotted against slope at the turning point of the sinusoidal relief. The curve shows maxima approximately between 30° and 60° where all the surface slopes steeper or gentler will result in lower focusing of plumes at the bathymetric highs. Therefore, the curve can be split into three regions: low-angle (<30°, zone 1), medium-angle (30°–60°, zone 2) and high-angle (>60°, zone 3) surfaces. As observed, maximum venting is achieved through the bathymetric highs where the surface angle lies in zone 2. This suggests that the probability of finding vent-sites at bathymetric highs is roughly proportional to seafloor slope. The simulations were also conducted using deeper and shallower domains. The results for these simulations do not significantly differ from those presented above and the maximum volume of venting at the highs still occurs at slopes between ∼30° and ∼60° similar to the previous simulations. The results obtained fromsection 3.1 combined with equation (6) allows for a qualitative assessment of the relation between the surface slope and amount of venting shown in Figure 5. A rigorous quantitative analysis is not possible due to the influence of plume interactions and due to the difference in slope variation used in the single and sinusoidal bathymetry simulations. Nevertheless, simulations with a surface slope of less than ∼30° show less than optimal venting as the angle of deviation required for a plume farthest away from a bathymetric high to be present at that high is larger than the maximum angle of deviation calculated by equation (6) for the given parameters. Conversely, in simulations where the surface slope angle is greater than ∼30°, the angle of deviation required for any plume to reach the closest high is less than the calculated maximum angle of deviation suggesting that complete migration to a high is possible. Simulations with a surface slope greater than ∼60° show reduced venting at highs as the width of the plume head exceeds the width of bathymetric relief.
 As a cautious note it should be added that these results are somewhat dependent on initial conditions. The presented simulations all assume an initially cold domain so that the initial pressure regime is not influenced by bathymetry. Only when a plume reaches a bathymetric feature, the pressure field is affected by bathymetric relief. Conversely, if an initial geothermal gradient is assumed, the initial pressure field is already affected by bathymetric relief. However, since we have largely restricted ourselves to the analysis of steady states, which are only weakly dependent on initial conditions, our results keep their generality.
4. Case Studies
 The preceding section has shown that bathymetric relief can have strong controls on vent site location. We will now use these insights to analyze how venting at the fast spreading EPR at 9°N and the slow-spreading Lucky Strike segment are influenced by bathymetric relief.
4.1. East Pacific Rise Vent Field (9°30′N)
4.1.1. Geological Setting
 The EPR is characterized as a fast spreading ridge with an average full spreading rate of 110–150 mm/yr [Foulger, 2010]. The bathymetry at 9°N at EPR shows relatively smooth relief with an axial high corresponding to the ridge axis [e.g., Toomey et al., 1994; Tolstoy et al., 2008]. A nearly continuous shallow melt lens was imaged at a depth of ∼1.5 km below the seafloor [Detrick et al., 1987]. The melt lens is extremely narrow with a thickness of 10–50 m and width of ∼1 km [Kent et al., 1993]. Seismic tomography suggests that a “slot” of hot partially molten crust exists below the ridge axis with a width of about ∼6 km [Dunn et al., 2000]; the width of this hot zone widens in the mantle region to nearly 15 km. Fisher has argued that this special shape of hot crust can be explained by off-axis hydrothermal cooling, which is consistent with the ideas on pervasive hydrothermal circulation developed at the Oman ophiolite [Nicolas et al., 2003]. However, if this pervasive off-axis (and across axis) hydrothermal flow does exist, it apparently does not result in off-axis venting; nevertheless, further off-axis surveys are essential to confirm this. All currently known vent sites are located directly on axis [Haymon et al., 1991] including the sites at 2570 m water depth at 9°30′N [http://www.interridge.org/irvents/content/epr-9-30n]. Here we test if hydrothermal fluids rising along off-axis margins of the hot central temperature anomaly are likely to be focused onto the ridge axis by bathymetric relief.
4.1.2. Model Setup
 The East Pacific Rise (9°30′N) vent site was modeled based on a depth converted map derived from studies by Dunn et al.  and a topographic sketch by Carbotte and Macdonald . We modeled two scenarios: one with a flat top and one with the actual bathymetric relief. The temperature is set to 10°C in the matrix and the pressure is set to hydrostatic values initially. A hot slot (1000°C) is assumed to start from 1.6 km below the seafloor and extends up to depth of 8 km (bottom of model). The width of this high temperature increases rapidly to up to ∼20 km at the bottom of modeling domain. Relatively smooth bathymetric variations of up to 400 m occur within the modeling domain. The maximum elevation of the seafloor is located just above the melt lens. The sides of model are insulated and the sides as well as the bottom of the model are impermeable. The surface pressure varies with topography where the highest point of the domain lies at a depth of 2.57 km below sea level. The matrix permeability structure at the East Pacific Rise and at fast spreading ridges in general is not well known and few studies give quantitative estimates. Theissen-Krah et al. provided estimates for the permeability structure above the melt lens and suggest that this permeability must decrease off-axis - potentially as a consequence of alteration and mineral precipitation reactions. We largely follow the permeability model ofTheissen-Krah et al. and extend it by higher permeability values in the deeper near- to off-axis crust to allow for deep hydrothermal flow as discussed above. We achieve this by employing a permeability model, in which the surface permeability decreases exponentially with depth to the anomalously hot zone. For numerical convenience this is done by using a steady state diffusion temperature field and defining matrix permeability as a function of this initial temperature field. The computed permeability is kept fixed though and does not evolve by temperature variations during simulations. The surface permeability at the ridge axis is assumed to be ∼6 × 10−15 m2, which is consistent with the estimates of Theissen-Krah et al. . The resulting permeability field is scaled by a lateral exponential decrease in permeability from 2000 m away from the ridge axis. This results in the following permeability model
where x is the absolute distance from 2000 m away from the ridge axis (m), k* is the revised permeability which is used as a substitute for k in equations (2), (3) and (5), and T the initial steady state conductive temperature (°C). This permeability model is plotted in Figure 6and is consistent with the observed melt lens depth and allows for pervasive off-axis hydrothermal flow. However, our static model setup is not suited for making estimates on permeability so that it remains our best guess on the actual permeability structure at EPR. Hopefully more data in combination with dynamical models will help to further constrain the deep permeability at ridges in the future.
 The matrix properties are listed in Table 1. The number of elements used in these simulations is 1000 per square kilometer. A further assumption is that porosity is constant and that both permeability and porosity do not change during the simulations.
 The modeled temporal evolution of hydrothermal convection beneath the East Pacific Rise is shown in Figure 7(the figure dimensions are restricted to 5 × 5 km to enhance the visibility of venting region). The simulations were run for 50,000 years when a quasi-steady state is attained. The left panel plots show the result with a flat seafloor, while the right panel plots the ones with the actual bathymetric relief. In the early phases of the simulations (Figures 7a and 7c), multiple plumes form above the central slot of hot crust. The plumes later merge and form a few major upwelling zones at the top of magma lens. In both settings, small convection cells develop directly on the top of hot crust, while deep large convection cells develop off-axis with upwellings along the margins of the magma chamber and recharge zones extending several kilometers off-axis (Figure 6). A striking feature of the simulation with a flat seafloor is that venting does not occur on-axis but at two, near-axis locations (Figure 6a). This is due to the fact that upwelling hydrothermal fluids along the margins of the magma chamber are not efficiently focused onto the axis and that several convection cells develop directly above the hot crust (Figure 7). This simulation is therefore inconsistent with the existing observations that all venting occurs directly on-axis.
 In contrast, the simulation with bathymetric relief predicts exclusive on-axis venting (Figures 7b and 7d). The plumes forming above the hot crust and the upwelling along the magma chamber coalesce to form fewer plumes which lean toward the axial high ultimately forming a central plume that vents directly on-axis. This clearly shows the effect of bathymetry to form and focus the central plume at the axial high, even though the bathymetric relief is quite smooth. Off-axis venting is suppressed due to the assumption that permeability decreases with increasing distance to the ridge axis, which is consistent with the current observations at EPR. However, it should be noted that detailed off-axis surveys have not been conducted at EPR and may, in future, provide new constraints on vent-locations farther away from the ridge axis. This would require a reassessment of the overall flow pattern but will not change the outcome of this study showing the influence of bathymetry on on-axis vent-locations.
4.2. Lucky Strike Vent Field
4.2.1. Geological Setting
 Lucky strike is one of the largest hydrothermal fields located on the slow spreading Mid-Atlantic Ridge (MAR) with 21 active chimney sites at approximately 1740 m depth [http://www.interridge.org/irvents/content/lucky-strike]. The full spreading rate at Lucky-Strike vent field is ∼21 mm/yr [Dziak et al., 2004]. The field is located at 37°17′29″N, 32°16′4″W, 400 km southwest of the Azores islands and is part of a group of four main hydrothermal vent fields (Menez Gwenn, Lucky Strike, Saldanha and Rainbow). The presence of a bathymetric high and a hydrothermal field surrounding a central lava lake led to the suggestion of a shallow axial magma chamber [Fouquet et al., 1995]. Furthermore, recent data from reflection seismic, which surveyed the Lucky Strike confirmed this hypothesis [Singh et al., 2006]. An AMC was discovered around 3 km beneath the Lucky Strike volcano which spreads about 7 km along the strike of the ridge with a width of 3–4 km [Singh et al., 2006].
4.2.2. Model Setup
 In order to simulate hydrothermal convection at the Lucky Strike vent field, a depth-converted seismic section across the ridge axis was digitized and meshed [Dusunur et al., 2009]. The length of the seismic line is ∼16 km and the magma chamber is buried ∼3 km below the seafloor. The depth of the modeled domain is 6 km and the depth of the seafloor is 1740 m below sea level. Similar to EPR, the initial temperature and pressure are set to 10°C and hydrostatic values, respectively. The 1000°C AMC with a maximum width of 8400 m at the bottom of model is responsible for driving hydrothermal convection. The permeability structure is unknown at the Lucky Strike vent field, but most likely influenced by bounding faults and fracture network. We nevertheless use a similar permeability model as in the preceding case study for the EPR
where x is the absolute distance from 1500 m away from the ridge axis. The temperature in equation (9) is again the hypothetical steady state conductive temperature for the given boundary conditions and the magma chamber is assumed to be impermeable. The surface permeability of the Lucky Strike vent field at the spreading center is assumed to be ∼2 × 10−15m2, which is lower than in the EPR case study. This is assumed to account for the idea that the crustal permeability at slow spreading ridges is generally lower than at their faster counterparts [Fouquet, 1997]. Figure 8illustrates the permeability structure for the simulations of Lucky Strike. Strictly speaking, the Lucky Strike vent field is situated below the critical end point of water so that boiling and condensation effects could be important. Two-phase phenomena cannot be accurately resolved in the here employed single-phase model and are neglected. Two simulations, i.e., flat surface and real bathymetric relief were performed. The number of elements in these simulations is approximately 1000 per square kilometer.
 Convection initiates by the formation of several small plumes at the top of the magma chamber in the simulations with and without bathymetric relief (Figures 9a and 9c). Note that similar to EPR, the figure dimensions for presented results are limited to 5 × 5 km to demonstrate the vent-site more clearly. In the flat surface scheme, two plumes vent at the top of magma chamber after 3000 years of convection (Figure 9a). Multiple plumes are later formed which merge with previously formed plumes and cause interference resulting in the lateral movement of the vent sites at the surface. The system reaches a quasi-steady state where three plumes are present after 50,000 years of convection (Figure 9b). Note that far off-axis venting does not appear as the temperature gradient as well as the matrix permeability is not high enough to allow convection. The flow patterns show that relatively large convection cells appear above the magma chamber (Figure 8a). Similar to the EPR case study, the simulation that neglects bathymetric effects never predicts exclusive venting at the central volcano and is therefore inconsistent with observations. Only when the effects of bathymetric relief are included, the plumes coalesce to form a single upwelling zone below the vent field on top of the central volcano. Plumes initiating away from the central vent field (Figure 9d) will also merge with this central plume and system reaches a steady state. The simulations confirm that bathymetric relief can also have strong controls on vent site location at slow spreading ridges. Here we have only looked at the special case of the Lucky Strike segment, which shows an on-axis melt lens. Due the significant bathymetric relief of slow-spreading ridges, on- as well as off-axis venting at different ridge segments may well be strongly controlled by bathymetry – in addition to detachment faulting, fracturing and phase separation in multicomponent fluids.
 We have explored the effects of bathymetric relief on hydrothermal convection in a series of synthetic numerical experiments and two case studies. The presented hydrothermal convection models with unstructured meshes allow us to resolve the effects of bathymetric relief. Venting tends towards bathymetric highs as a result of bathymetry and temperature induced pressure variations in submarine hydrothermal systems. Simulations using a single plume and bathymetric high allow us to predict the maximum deviation from vertical plume ascent as a function of seafloor relief as well as the depth to the heat source. The surface slope in simulations with multiple plumes and highs controls how efficiently venting is focused on the bathymetric highs.
 At fast spreading ridges, the axial high helps to focus venting directly onto the ridge axis. Pervasive and deeply penetrating across-axis flow, which is thought to shape the hot crust at fast-spreading ridges, can be reconciled with on-axis venting if bathymetric effects are accounted for. The Lucky Strike case study shows that also on slow-spreading ridges bathymetric relief results in focusing of hydrothermal venting at bathymetric highs – at least for systems in a magmatic phase with an axial magma chamber. Our findings demonstrate the importance of including bathymetry in simulations of submarine hydrothermal flow. In addition to the here presented results on oceanic spreading centers, bathymetric effects should also be important for other submarine vent systems like caldera structures and seamounts.
 The authors would like to thank the anonymous reviewers for the constructive comments and criticisms which greatly improved the manuscript. N. Bani-Hassan, K. Iyer and L. H. Rüpke are part of the Future Ocean - Excellence Cluster at Kiel. A. Borgia acknowledges support by the American Recovery and Reinvestment Act (ARRA), through the Assistant Secretary for Energy Efficiency and Renewable Energy (EERE), Office of Technology Development, Geothermal Technologies Program, of the U.S. Department of Energy under contract DE-AC02-05CH11231.