The only sub-aerial manifestation of the Mid-Atlantic Ocean Ridge (MAR) is located on Iceland and is defined by epicentres of Earthquake (Einarsson, 2008) [1] and the zone of active volcanism (Figure 1). The plate

boundary across Iceland is slow spreading rate (~2 cm∙yr⁻¹). The ridge segments across Iceland are characterised by central volcanos with associated fissure swarms. The volcanic systems have been grouped into the Western (WVZ), Eastern (EVZ) and Northern Volcanic Zones (NVZ) (Figure 1).

The plate spreading style in Iceland has been fitted using an elastic dislocation model by several authors (e.g., LaFemina et al., 2005; Árnadóttir et al., 2006, 2009; Scheiber-Enslin et al., 2011; Geirsson et al., 2012), [2]-[6] a stretching model (e.g., Pedersen et al., 2009) [7] and a temperature-dependent, non-linear rheology model (Islam et al., 2013) [8]. The applied rheology is elastic in the elastic half-space in a dislocation model and a uniform elastic-viscoelastic half-space in stretching model where strain rate is linear to stress. All models are constrained to fit the observed surface deformation. However, the pattern of subsurface flow and the rheology beneath Iceland can be predicted by advection-convection modeling. It may provide new insights into a nature of plate tectonic motion in MAR system. Here we investigate advection-convection geodynamical models using temperature-dependent Newtonian rheology for plate spreading as measured by Global Positioning System (GPS) in EVZ, Iceland.

Our geodynamic modeling is based on a cooling the oceanic lithosphere. To explore subsurface temperature distribution and mass-flow pattern, COMSOL Multiphysics 4.3b finite-element software package is used. The simulation for subsurface mass-flow is conducted by coupling of a thermal structure and the flow of rheology. All implemented models are two dimensional (2D). The domain of these models is 100 km in depth and 200 km length crossing the rift axis in perpendicular direction of both sides (Figure 3). The plate spreading is applied as advection. All models are executed using temperature dependent Newtonian rheology which is further compared with a uniform viscosity of 1 × 10¹⁹ Pa∙s.

The viscosity of Newtonian rheology is dependent on temperature. Temperature dependent viscosity is calculated by

where η₀ is the reference viscosity of 1 × 10¹⁸ Pa∙s, Q is the activation energy of 250 kJ∙mole⁻¹, R is the gas constant, T_a is the temperature of ridge axis in asthenosphere and T is the subsurface temperature, both in K (Behn et al., 2007) [29]. This gives a linear approximation of non-linear rheology (Christensen, 1983) [30]. The reference viscosity is obtained from Pagli et al. (2007) [31] and, Jones & Maclennan (2005) [32], who suggest the viscosity of rheology beneath Iceland is order of 10¹⁸ Pa s. The Equation (2) gives viscosity higher than 10²² Pa s in temperature less than 700˚C. Any viscosity higher than 10²² Pa∙s is fixed to 10²² Pa∙s.

First, we calculated a steady-state temperature distribution in the solid subsurface using the thermal structure suggested by Durbyshire et al. (2000) [18], Kaban et al. (2002) [13] and Björnsson (2008) [17] who from seismogenic studies in Iceland. The temperature is used as a boundary condition for the thermal model which is applied at the surface and the ridge axis (Figure 3(A)). A constant temperature 1300˚C is set in the interval 30 - 100 km depth, is used for the asthenosphere (Turcotte & Schubert, 2002) [33]. The temperature distribution in solid materials is controlled by the basic energy-balance equation as (COMSOL Multiphysics, 2013) [34],

where ρ is the material density, C is the heat capacity at constant pressure of 1000 J∙kg⁻¹∙K⁻¹, u is the advection velocity, T is the temperature and k is the thermal conductivity = 1.9 W∙m⁻¹∙K⁻¹. Average density of lithosphere- asthenosphere is set to 3300 kg∙m⁻³ (e.g., Behn et al., 2007; Roland et al., 2010) [29] [35]. The temperature distribution in the model is presented in Figure 4(A).

We use the 700˚C (973 K) isotherm from Turcotte & Schubert (2002) [33] to demarcate the boundary between the lithosphere and asthenosphere, for hot and dry rheology. Since the rheology of newly formed upwelling Earth crust beneath Iceland is very hot and dry (Barnhoorn et al., 2011) [36]. Therefore, above the 700˚C isotherm, the left and right vertical boundaries of the models are allowed for advection at a rate of 7 mm∙yr⁻¹, which corresponds to the total spreading rate of 14 mm∙yr⁻¹ (Figure 3(B)). Temperature 700˚C isotherm at 12 km depth is used for boundary condition since as the 12 km elastic depth is found by elastic dislocation modes (e.g., LaFemina et al., 2005; Geirsson et al., 2012) [2] [6], and a similar depth is obtained by the 700˚C isotherm applying a temperature-dependent, non-linear rheological model along the same profile (Islam et al., 2013) [8].

Based on convection at the bottom boundary with no stress and, the observed vertical uplift and subsidence along the profile, eight different models are evaluated (Table 3).

^aTemperature dependent Newtonian rheology. ^bUniform rheology viscosity to 1 × 10¹⁹ Pa∙s.

Then, Laminar flow is executed in incompressible flow mode, which is governed by mass balance equation as

where ρ is the material density, η is the dynamic viscosity, u is the velocity and p is the pressure (COMSOL Multiphysics, 2013) [34]. The numerical solution follows minimization of residual of Newton method and weighted Euclidean norm for a convergence criterion. A tolerance of 1 × 10⁻⁶ is allowed to estimate relative error. A priori and posterioei error is generated by numarical solution which depends to element size. This error tends to null when element size tends to zero (Larson & Bengzon, 2009, p-27-28, 39-41) [37]. Therefore extremly fine elements (Physics-controlled mesh) in ranging 2 - 670 m (triangular), with 1.05 maximum growth rate, 0.2 resolution of curvature and 1 resolution of narrow regions are applied to minimize priori and posterioeri error.

The width of deformation zone in EVZ along the profile is estimated to be ~90 km based on where the slope’s gradient of the fitting curve (v(x)) is equal to zero or close to zero. More than 90% surface deformation occurs within this zone (Figure 2(B)).

The hypothesis of geodynamic modeling in this study is lithosphere is floating on top of asthenosphere in the MAR system (Turcotte & Schubert, 2002; Fowler, 2005) [33] [38]. The temperature structure in Figure 4(A) is the most likely subsurface thermal structure along the profile in EVZ, Iceland. The thermal distribution is similar as to the Pálmason model to describe the Icelandic crustal accretion model (Pálmason, 1986) [39].

The viscosity in the temperature dependent Newtonian rehological models was not allowd to exceed 10²² Pa∙s in the region at lower temperature (<700˚C), that is in shallow depth (Figure 4). Since empirical experiment of olivine (Turcotte & Schubert, 2002) [33] shows that maximum viscosity at low temperature is ~10²² Pa∙s. Therefore a temperature depenent rheology (Figure 4(B)) is obtained by Equation (2) that gives the best agrement with the viscosity of olivine experiment (Turcotte & Schubert, 2002) [33] and the 10²² Pa∙s iso- viscosity for ductile-brittle transition for the dry rheology (Barnhoorn et al., 2011) [36]. The viscosity in low temperature (<700˚C) behaves as in a brittle state, this is a representative of the lithosphere. The 700˚C isotherm and 10²² Pa s iso-viscosity provide a brittle region. They provide lithospheric thickness of 50 ± 3 km (depth) at a distance 100 km from active rift axis (Figure 4), this is in the same scale as the results of Bjarnason & Schmeling (2009) [15].

The sub-surface mass-flow shows distinct pattern when uniform viscosity and temperature dependent vis- cosity are applied (Figure 5). Model A, temperature dependent rheological model is stress free and convection is allowed from bottom, which results ~8.5 mm∙yr⁻¹ subsidence at the rift center and ~10.5 mm∙yr⁻¹ influx (uplift) at the 100 km bottom of ridge center (Figure 5(A)).

Model A may explore the scenario of the crustal deformation in the NVZ where a narrow graben is obsered at rift center (Pederssen et al., 2009) [7]. Model B, uniform rheological (viscosity of 1 × 10¹⁹ Pa∙s) model is also stress free and causes an ~8.5 mm∙yr⁻¹ uplifting at the rift center and ~12 mm∙yr⁻¹ magmatic influx at the ridge center of 100 km depth (Figure 5(B)). This phenomenon might be similar as so called central uplift of Iceland. High upwelling activities cause uplifting of whole Iceland (Sigmundsson, 2006; Árnadóttir et al., 2009) [4] [40].

Model C is similar as Model A but convection is not allowed, in model this results to ~9.5 mm∙yr⁻¹ subsidence at the rift center (Figure 5(C)). This model may give the scenario of Thingvellir graben, in the WVZ where magmatic plumbing is almost absent and continuous subsidence is observed from the whole Holocene (Sturkell et al., 2013) [41]. Model D is similar as model B but convection at the bottom is not allowed. It gives ~4 mm∙yr⁻¹ subsidence at the rift center (Figure 5(D)). However at ~60 km depth, mass-flow is not observed. But underplating drives away the plate from the ridge axis where the space is filled up by instrution (Björnsson, 2008) [17]. Therefore this model contradics with flow patter of Pálmason model.

Models E-H are produced by applying observed surface deformation as boundary condition (Figures 5(E)-(H)). Models E and G is temperature dependent, and F and H is uniform rheological model (Table 3). Model E gives ~11 mm∙yr⁻¹ magmatic influx ~10 km west of the ridge center in 100 km depth. It also results ~8 mm∙yr⁻¹ magmatic flow towards center of Earth as spriral sink (Hirsch et al., 2004, p 47) [42] ~80 km east of ridge center at 100 km depth (Figure 5(E)). This is due to convection as spiral source (Hirsch et al., 2004, p 47) [42] which further flows toward surface. However, Model F results in a ~5 mm∙yr⁻¹ magmatic influx ~55 km west of rift axis but there is no mass flow observed beneath lithosphere at the east of ridge center (Figure 5(F)). Subsurface mass flow in Model G approaching from east of ridge center towards west (Figure 5(G)). This phenomena is not valid since masses flow with high rate alonge and near at ridge axis according to what in Pálmason model. Model H is simmilar as Model D.

Model E give the best scenario of the EVZ. In the model, ridge axis (high convection rate) is shifted ~10 - 20 km towards west (Figure 5(E)) that is very similar with the vertical component of observation (Figure 2(C)). The tilt measuremnts also give the indication of shifting center.

The subsurface mass flow in a state when observed surface deformation is not applied as boundary condition, gives that more than 90% vertical subsidence are predicted from model within ~45 km from the rift axis (Figure 5(A)). Similarly the horizontal component of masses flow is fairly constant beoynd ~45 km from the rift. This may indicate a deformation zone of ~45 km from the rift axis in each direction, which is fairly similar with the deformation zone observed by the fitting curve of GPS observations (Figure 2(B)). Since convection in the model is allowed from the depth with no stress, mass flows as spiral source in asthenosphere (Figure 5(A)) which refill the empty space caused by advection (plate spreading).

Plate spreading in Iceland is symmetric and thus deformation field is expected to be more or less symmetrical. For example, it is observed in the Thingvellir graben in the WVZ both in the horizontal and vertical deformation (Sturkell et al., 2013) [41]. Deformation (spreading axis) in the EVZ shifts between the five parallel volcanic systems and the systems which currently centared host experience the maximum subsidence (LaFemina et al., 2005) [2]. Therefore, vertical deformation and center of spreading along the profile at the EVZ is “Ping Pong” between the five parallel volcanic systems which seem to be active in turns. When the measured displacement vector (vertical component) is applied as boundary conditions in modeling, this ping pong is predicted and the ridge center is shifted (Figure 5(E)).

High upwelling is associated centrally beneath Iceland, therefore the whole of Iceland is uplifting (Árnadóttir et al., 2009) [4], which may cause uplift at western part across the EVZ. On the other hand, the upper crust at the eastern part of the profile may be loosely connected due to several faults (Figure 2(A)). This may hinder the sharing of the strain of central uplift of Iceland.

Here we have introduced geodynamic models of advection-convection in a plate spreading environment. In the models,

・ Newtonian rheology is dependent on the subsurface temperature distribution,

・ the viscosity of Newtonian rheology beneath Iceland varies both vertically and horizontally,

・ the depth of lithosphere in the EVZ along the profile is identified to reach 50 ± 3 km at distance of 100 km from rift axis,

・ at ~45 km from the rift axis ~90% of the spreading is accommodated,

・ ~8.5 mm∙yr⁻¹ subsidence at the surface of rift center is predicted when magmatic plumbing is inactive and

・ predicted rift center is shifted ~10 - 20 km west to generate observed style surface deformation and in such case the highest magmatic influx is ~11 mm∙yr⁻¹.

This deformation zone in model depends to spreading velocity, isotherm and depth of isotherm. However, all these three parameters have more or less equal weight. This type of model is comparatively easy to construct based on subsurface temperature distribution which introduce variation of viscosity.

Geodetic GPS data for this study was taken from Geirsson et al. (2012). Geirsson, H. and his all co-authors are gratefully acknowledged. The authors thank to Halldór Ólafsson and all others for levelling and geodetic GPS field measurement. Mark Johnson is acknowledged for his valuable comments. Financial support is obtained from Herbert och Karin Jacobssons Stiftelse 11/v15 for publication. We thank NORDVULK for cooperation. The anonymous reviewers are also gratefully acknowledged.

Md. Tariqul Islam,Erik Sturkell, (2015) Temperature-Dependent Newtonian Rheology in Advection-Convection Geodynamical Model for Plate Spreading in Eastern Volcanic Zone, Iceland. Journal of Geoscience and Environment Protection,03,14-26. doi: 10.4236/gep.2015.35003