Glacial isostatic adjustment constrains dehydration stiffening beneath Iceland

Earth and Planetary Science Letters 359-360 (2012) 152–161

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Glacial isostatic adjustment constrains dehydration stiffening beneath Iceland ¨ Peter Schmidt a,n, Bjorn Lund a, Tho´ra A´rnado´ttir b, Harro Schmeling c a

Department of Earth Sciences, Uppsala University, Villav¨ agen 16, 752 36 Uppsala, Sweden Nordic Volcanological Center, Institute of Earth Sciences, University of Iceland, Sturlugata 7, IS-101 Reykjavı´k, Iceland c Institute of Earth Sciences, Section Geophysics, J.W. Goethe-University, Frankfurt am Main, Germany b

a r t i c l e i n f o

a b s t r a c t

Article history: Received 6 March 2012 Received in revised form 26 September 2012 Accepted 15 October 2012 Editor: L. Stixrude Available online 10 November 2012

During melting in the upper mantle the preferred partitioning of water into the melt will effectively dehydrate the solid residue. Linear extrapolation of laboratory experiments suggests that dehydration can produce a sharp viscosity contrast (increase) of a factor 500 across the dry solidus. In this study we show that the suggested magnitude of dehydration stiffening in a plume–ridge setting is incompatible with the present glacial isostatic adjustment (GIA) in Iceland. Using GPS observations of current GIA in Iceland, we find that the data are best fit by a viscosity contrast over the dry solidus in the range 0.5–3. A viscosity contrast higher than 10 requires a mantle viscosity below the dry solidus lower than 4-8  1018 Pa s, depending on the thickness of the dehydrated layer. A viscosity contrast of 100 or more demands a mantle viscosity of 1018 Pa s or less. However, we show here that a non-linear extrapolation of the laboratory data predicts a viscosity contrast as low as a factor 3–29, assuming conditions of constant strain rate to constant viscous dissipation rate. This is compatible with our GIA results and suggests that the plume–ridge interaction beneath Iceland is governed by a non-linear rheology and controlled by a combination of kinematic and dynamic boundary conditions rather than buoyant forces alone. & 2012 Elsevier B.V. All rights reserved.

Keywords: glacial isostatic adjustment Iceland dehydration stiffening rheology viscosity

1. Introduction It has been known since the early 1970s that water will significantly reduce the creep strength of olivine (Blacic, 1972), which is the major mineral constituent of the Earth’s mantle and therefore controls mantle viscosity. Water does, however, behave as an incompatible element in olivine and dissolves 100–1000 times easier in a basaltic melt (e.g. Tenner et al., 2009). Upon melting of the mantle, the solid residue will therefore be effectively dehydrated (Hirth and Kohlstedt, 1996). Laboratory experiments show that the increase in mantle viscosity in response to a reduction in water content (e.g. Mei and Kohlstedt, 2000a, 2000b; Karato and Jung, 2003) is significantly greater than the viscosity decrease due to small amounts of melt (Mei et al., 2002). As a result, the net effect of melting in an initially wet mantle will be an increase in viscosity due to dehydration stiffening (Karato, 1986), unless a sufficient source of water is available to keep the solid residue wet, such as in the mantle wedge above a subducting oceanic plate (Mei and Kohlstedt, 2000b). Partial melting beneath a mid-ocean ridge or in a mantle plume is, as a

n

Corresponding author. E-mail address: [email protected] (P. Schmidt).

0012-821X/$ - see front matter & 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.epsl.2012.10.015

consequence, predicted to generate a sharp viscosity contrast of a factor 5007 300 over the dry solidus (Hirth and Kohlstedt, 1996; Ito et al., 1999), for typical mid ocean ridge basalt (MORB) source water contents (Hirth and Kohlstedt, 1996), 125775 ppm. In order to study the effects of dehydration stiffening in the field we require both a geographic location where mantle melting is occurring and some means of accessing viscosity information at mantle depths. Both these requirements are met in Iceland, as the island is situated on top of a mantle plume, cross-cut by the midAtlantic ridge and covered to about 11% by melting glaciers. The ice caps in Iceland have been retreating since the end of the 19th ¨ century (Bjornsson and Pa´lsson, 2008), producing ongoing glacial isostatic adjustment (GIA) with maximum current uplift velocities in excess of 20 mm/yr (A´rnado´ttir et al., 2009). Since the GIA process to first order depends on the viscosity structure of the mantle, we here have a unique opportunity to investigate the presence and magnitude of dehydration stiffening in a plume– ridge setting using field observations. In this study we build on previous GIA investigations in Iceland (A´rnado´ttir et al., 2009; Schmidt et al., submitted for publication) and extend these using three-dimensional (3D) models of the viscosity structure of the uppermost Icelandic mantle. We investigate to what extent our model is sensitive to 3D structures in general and a viscosity contrast due to dehydration stiffening in

P. Schmidt et al. / Earth and Planetary Science Letters 359-360 (2012) 152–161

particular. We then study if the suggested magnitude of dehydration stiffening beneath Iceland is compatible with geodetic observations. In Section 2 we briefly discuss the rheology of the upper mantle, followed in Section 3 by a short review of available constraints on the Icelandic upper mantle relevant to this study. Our GIA modeling approach and the GPS data are presented in Section 4. In Section 5 we model the effects of dehydration stiffening, using both a conceptual model of the 3D viscosity structure and models derived from self-consistent geodynamic models of plume–ridge interaction. We finish the paper with discussions and conclusions in Sections 6 and 7.

For a linear rheology the viscosity is defined as

s Z¼ _ 2e

ZðsÞ ¼

2. Upper mantle rheology

n

e_ ¼ As

C kOH

  H exp  RT

ð2Þ

whereas in the case of a non-linear rheology an effective viscosity can be defined using Eq. (2), noting that the effective viscosity will vary with the stress–strain state of the system. Combining Eqs. (1) and (2) we can derive analytical expressions for the effective viscosity as a function of stress, strain rate or viscous dissipation rate, D ¼ e_ s

Zðe_ Þ ¼

As olivine is the major mineral constituent of the upper mantle, the rheology of olivine is expected to control upper mantle rheology. Although the predominant secondary upper mantle mineral, pyroxene, may have slightly higher effective viscosities than olivine (Hitchings et al., 1989), numerical modeling (Tullis et al., 1991) as well as experiments (Kohlstedt and Zimmerman, 1996; Zimmerman and Kohlstedt, 2004) indicate that the viscosity of the upper mantle is indeed dictated by the viscosity of olivine, in both the diffusion and dislocation creep regimes. Using laboratory experiments, significant efforts have therefore been made to understand the creep behavior of olivine (e.g. Chopra and Paterson, 1981; Mackwell et al., 1985; Karato, 1986; Mei and Kohlstedt, 2000a, 2000b; Mei et al., 2002; Karato and Jung, 2003). For our purpose the olivine creep law can be expressed as

153

ZðDÞ ¼

sðn1Þ

ð3Þ

B

e_ ð11=nÞ

ð4Þ

B1=n Dðn1Þ=ðn þ 1Þ

ð5Þ

B2=ðn þ 1Þ

where for simplicity we have defined B ¼ 2AC OH expðH=RTÞ. Eqs. (3)–(5) are equivalent for a given stress–strain state. As we will study changes in the water content, we define a ratio of water contents before and after dehydration, ROH ROH ¼

C OH,before Bbefore ¼ C OH,after Bafter

ð6Þ

Using Eqs. (3)–(5) we can derive the following relationship between the ratio of water contents and the ratio of viscosities before and after dehydration, RZ

Zbefore Bafter ¼ ¼ R1 OH Zafter Bbefore

At constant s :

RZ ¼

At constant e_ :

RZ ¼ ROH

At constant D :

RZ ¼ ROH

1=n

ð7Þ

ð8Þ

ð1Þ

where e_ is the strain rate, A is the pre-exponential factor, s is the deviatoric stress, COH is the hydroxyl content in olivine, n and k are the stress and hydroxyl exponents, respectively, H is the activation enthalpy, R is the universal gas constant, and T is the temperature. Typically the hydroxyl exponent is found to be about 1.2 (e.g. Hirth and Kohlstedt, 2003; Karato and Jung, 2003), but for simplicity we shall make the assumption k¼1 and note that COH translates directly to the water content in olivine. We will further neglect differences in the activation enthalpy due to the presence of water, as this will not have a major impact on our study. For diffusion creep the stress exponent is found to be 1 and we refer to diffusion creep as a linear rheology. In the dislocation creep regime the stress exponent is in the range 3–4, resulting in a non-linear rheology. In large parts of the upper mantle, the pressure, temperature and deviatoric stress state is close to the transition between diffusion and dislocation creep and the governing creep process is still debated (Ranalli, 1998). We note that the deviatoric stresses and strain rates used in the laboratory to measure the constants and exponents of the creep law are many orders of magnitude larger than those expected in the mantle. Hence, conclusions drawn for mantle rheology involve significant extrapolation (some 10 orders of magnitude). Further, the method used by early investigators to measure the hydroxyl content in their experiments was recalibrated by Bell et al. (2003) and Koga et al. (2003), who found that water concentrations had been underestimated by a factor of 2–4. Compensating the estimate by Hirth and Kohlstedt (1996) for this reduces the expected viscosity contrast due to dehydration stiffening to about 1707100 for MORB source water contents.

2=ðn þ 1Þ

ð9Þ

Eqs. (7)–(9) indicate that if dehydration occurs under different mechanical conditions, the effective viscosity contrast will be different. To predict the viscosity contrast due to dehydration therefore requires explicit knowledge of the mechanical state.

3. Observational constraints on the upper mantle beneath Iceland The viscosity structure of the upper mantle beneath Iceland is dominated by the presence of a mantle plume, inferred to be ¨ located beneath the NW part of Vatnajokull (e.g. Fedorova et al., 2005, see also Fig. 1). Tomographic models and geochemical studies of Iceland indicate an almost vertical, close to cylindrical plume conduit of radius 100–120 km, with an excess temperature of 150–200 K (see review in Ruedas et al., 2007). The excess temperature is expected to result in a viscosity contrast between the plume and the background mantle of a factor of 10–50, at constant stress conditions and assuming an activation enthalpy in the range 300–500 kJ mol  1 (Korenaga and Karato, 2008), as well as an increased depth to the dry solidus by 45–60 km, placing it at about 100–120 km depth (Hirth and Kohlstedt, 1996; Ito et al., 1999). Less is known about the structure of the plume head, although surface wave tomography indicates a plume head extending some 1700 km along and approximately 600 km perpendicular to the ridge, with a possible base at 150–200 km depth (Allen et al., 2002; Pilidou et al., 2005; Delorey et al., 2007). This places the dry solidus and viscosity contrast due to dehydration stiffening in the plume head. Given that the indicated dimensions of the plume head is significantly larger than Iceland itself, the viscosity

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change due to dehydration is expected to manifest itself as a high viscous layer (HVL) underlying the lithosphere beneath Iceland (e.g. Ito et al., 1999). The water content of the plume source is found to be very high compared to typical MORB source water contents. Studies of basalt from the north Atlantic volcanic province (Jamtveit et al., 2001), as well as studies of basaltic glasses from submarine/ subglacial eruptions, sampled along the Mid-Atlantic ridge from the Reykjanes ridge SW of Iceland all across the island to the Theistareykirarea in the North (Nichols et al., 2002), indicate water contents as high as 920 ppm in the Icelandic mantle source.

4. The GIA model A glacial isostatic adjustment model consists of two parts, an earth model and a model of the ice history. By comparing the predictions of the GIA model to observations pertaining to the GIA process we can make inferences on the material properties of the Earth. Our GIA model is an extension of the model used by A´rnado´ttir et al. (2009), introducing a 3D earth model and a more detailed ice model (Schmidt et al., submitted for publication). The GIA model is constrained by the vertical component of the surface velocity of Iceland, Fig. 1a, estimated from two nation-wide GPS campaigns in 1993–2004 and available continuous GPS stations (A´rnado´ttir et al., 2009), in total 125 stations. Previous studies (e.g. Latychev et al., 2005; Whitehouse et al., 2006) have shown that the horizontal component of the surface deformation is generally more sensitive to 3D Earth structure than the vertical component. Unfortunately, the horizontal component in Iceland is significantly affected by other processes (e.g. tectonic, magmatic and thermal, see A´rnado´ttir et al., 2009) and we cannot use it here. The Earth model is implemented with the finite element method, in the commercial software Abaqus (Abaqus, 2007). We use a flat-earth approximation, neglecting self-gravity and internal buoyancy (Wu, 2004) and accounting for pre-stress advection using springs (Schmidt et al., 2012). The lithosphere is modeled as a purely elastic layer, and as our GPS data are insensitive to variations in lithospheric thickness (Schmidt et al., submitted for publication), we use an uniform thickness in the models below.

-25° 67°

-23°

-21°

-19°

-15°

-17°

We model the mantle as a Maxwell solid with densities and elastic parameters derived as volume averages from the preliminary reference Earth model (Dziewonski and Anderson, 1981). The ice model includes the five largest and eight smaller Icelandic glaciers, at a spatial resolution of 2  2 km. Previous studies (Fleming et al., 2007; A´rnado´ttir et al., 2009) have demonstrated that the ice history prior to 1890 has little impact on present day vertical displacement rates. The ice history therefore starts from steady-state conditions in 1890 (Tho´rarinsson, ¨ 1943; Bjornsson, 1979; A´rnado´ttir et al., 2009), assuming constant deglaciation rates and areal extents until present. The ¨ largest ice cap, Vatnajokull, is estimated to have lost 435 km3 of ice between 1890 and 2004 (Pagli et al., 2007). We divide ¨ Vatnajokull into 10 melting regions based on the mean annual mass balance between glacier years 1991/1992–2005/2006 ¨ (Bjornsson and Pa´lsson, 2008). Adopting a minimum melting rate of 25 cm/yr and a constant difference in melt rate between adjacent regions, the maximum modeled melt rate at the edge ¨ of Vatnajokull becomes 82 cm/yr. For all other glaciers we use a constant melt rate of 65 cm/yr. The total volume loss in the ice model over the period 1890–2004 is 658 km3. For a detailed description of the ice model, see Schmidt et al. (submitted for publication).

4.1. Best fitting 1D earth model We quantify the fit of the GIA model to the GPS observations using the normalized w2v measure, calculated as

w2v ¼

gps 1 X vi vmod i Nm i sgps i

!2 ð10Þ

where N is the number of data points, m is the number of free parameters in the GIA model, vgps and vmod are the vertical i i velocities estimated from the GPS data and the model, respectively and sgps is the corresponding uncertainties of the vertical i velocities. In addition, we use a variance reduction criterion, VR, (A´rnado´ttir et al., 2009) of 90% or more as a measure of an

-25°

-13°

-23°

-21°

-19°

-15°

-17°

-13°

66°

67°

66° 5

3

65°

65° 1 1

1

64°

64°

km 0

63°

100

63°

Vertical velocity component [mm/yr]

-9

-7

-5

-3

-1

1

3

5

7

9

11 13 15 17 19 21

Residual vertical velocity [mm/yr]

-11

-9

-7

-5

-3

-1

1

3

5

7

9

11

Fig. 1. (a) Current uplift rates in Iceland estimated from GPS campaigns in 1993 and 2004 and from continuous GPS stations, in the ITRF2005 reference frame. (b) Residual uplift rates after subtraction of the best fit 1D earth model predictions. Triangles mark the locations of campaign GPS stations, squares mark continuous GPS stations. Station symbols filled with yellow color have been excluded when computing the model fit, but are included in the residual uplift rate map. Outlines of the Icelandic glaciers are marked by black polygons. Black circle displays the extent of a 100 km radius plume conduit.

P. Schmidt et al. / Earth and Planetary Science Letters 359-360 (2012) 152–161

et al., 2005), post-rifting deformation (Hofton and Foulger, 1996) and periodic rifting (LaFemina et al., 2005).

acceptable fit to the data ! VR ¼ 100 

1

155

2 v,mod 2 v,0

w w

ð11Þ

Here w2v,0 refers to the fit of a null-velocity model, where the modeled vertical velocities at all the stations are zero and w2v,mod is the fit of the particular GIA model. For our data set w2v,0 is 34.2, hence a model with a variance reduction better than 90% should have w2v r 3.42. Following A´rnado´ttir et al. (2009) we exclude some of the GPS stations, marked in yellow in Fig. 1, from the computation of the model fit as these are clearly affected by nonGIA processes. We do, however, include them in maps of the residual velocities. Instead of the variance reduction criterion, we could have computed confidence limits on the model parameters using regular w2 statistics. However, the confidence limits derived this way are very narrow ( 715% in viscosity estimates at 99%, 2:6s, confidence level) and in view of the fit to the data uncertainties and the spatial variation in misfit we find them unreasonably small. The small estimates may arise from an overestimation of the degrees of freedom, v (here estimated simply as Nm), or that the observations, especially at large distances from the center of uplift, are less independent than required by the statistics. The variance reduction criterion then offers a much more robust measure of the acceptable fit. The GPS data set can be surprisingly well fit by a simple onedimensional (1D) earth model with a 35 km thick elastic lithosphere on top of a viscoelastic mantle with viscosity 1019 Pa s, Fig. 1b. This model has w2v ¼ 1:74 and VR ¼94.9%. The large residuals, evident in Fig. 1b in southwestern and northern Iceland, are due to processes other than GIA (mainly rifting, A´rnado´ttir et al., 2009). Our best fit model agrees well with previous studies of the early Holocene GIA processes in Iceland (e.g. Sigmundsson, 1991), as well as studies of post-seismic relaxation (A´rnado´ttir

-11

-9

-7

5. Modeling the viscosity structure due to dehydration stiffening We model the effects of dehydration stiffening beneath Iceland in a two pronged approach; first a conceptual model of dehydration in a plume setting, and then models based on self-consistent viscosity fields from geodynamic models of plume–ridge interaction. 5.1. Conceptual plume model The conceptual model of the Icelandic mantle plume is based on the studies summarized in Section 3 and the 1D GIA model in Section 4.1. The model consist of an elastic lithosphere on top of a high viscous layer (HVL), representing the dehydrated region above the dry solidus, and a cylindrical plume conduit embedded in a uniform viscosity background mantle, Fig. 2. The model initially has six free parameters: the thickness of the elastic lithosphere, the depth to the base of the HVL, the radius of the plume conduit, and the viscosities of the HVL, plume conduit and background mantle. Tests show almost no change in the fit to the GPS data between a model with a HVL extending horizontally to infinity and a model with a HVL radius of 600 km, equal to the inferred width of the plume head perpendicular to the ridge (approximately 10 times the radius of the load). Additional tests show that a plume conduit extending deeper than 410 km has marginal effects on the model fit. Therefore, in our conceptual models the HVL is modeled as a uniform thickness layer beneath the entire elastic lithosphere and the plume conduit is terminated at the base of the upper mantle (670 km).

Residual GPS velocities [mm/yr] -3 -1 1 3

-5

5

7

9

11

25

0.10

-1.0

0.16

-0.8

0.25

-0.6

0.40

0.63

-0.4

-0.2

Viscosity ratio [η/1019] 1.00 1.58 2.51 3.98

0.0 0.2 0.4 0.6 Viscosity [log10(η/1019)]

6.31

0.8

10.00 15.80 25.11

1.0

1.2

1.4

39.81

1.6

Fig. 2. Viscosity structure and maps of residual vertical velocities for the conceptual model of dehydration stiffening in the Iceland mantle plume. (a) Best fitting model, with viscosity contrast 2, w2v ¼ 1.78 and variance reduction 94.8%. (b) Model with viscosity contrast 100, w2v ¼3.38 and variance reduction 90.1%.

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P. Schmidt et al. / Earth and Planetary Science Letters 359-360 (2012) 152–161

As the modeling requires significant computing resources we cannot perform a full grid search for optimum model parameters. Instead, we vary a single model parameter at a time. Once an optimum value for that parameter is found in terms of model fit to the GPS data, we fix that parameter and vary the next model parameter. The iterative grid search is continued until we find a set of model parameters that provide an optimized fit in all six parameters simultaneously. The resulting best fit model parameters are 2  1019 Pa s, 1  1019 Pa s and 3  1018 Pa s for the HVL, background mantle and plume conduit viscosities respectively. The elastic thickness is 30 km, the depth to the base of the HVL 100 km (HVL thickness 70 km) and the radius of the plume conduit is 100 km, Fig. 3. Hence the viscosity contrast between the HVL and the background mantle is 2 for this model. We note that the conduit radius is compatible with observational constraints and that the depth to the base of the HVL is in reasonable agreement with the estimated depth to the viscosity contrast beneath Iceland, Section 3. The elastic thickness and the viscosity of the background mantle are similar to the estimates from our uniform mantle GIA model, Section 4.1. In order to investigate the sensitivity of the optimum conceptual model to the model parameters we perturb a single parameter at the time, Fig. 3. Perturbations in the plume conduit radius or plume conduit viscosity, Fig. 3c and f, only result in small changes in the model fit. This is most likely due to the stabilizing effect of the surrounding higher viscosity mantle and the overlaying HVL, as well as the relatively small areal extent of the surface loads. Given the observational constraints on the plume conduit excess temperature (150–200 K), the viscosity contrast between the conduit and background mantle is expected to be of the order of 10–50, i.e. significantly larger than in our optimal model. However, models with plume conduit viscosity as low as 1017 Pa s only display minor changes in the model fit (Dw2v o0:1), even if the plume radius is increased to 150 km. The trends observed in Fig. 3b–f are found to be only weakly

dependent on the elastic thickness of the lithosphere. In the following investigation of model fit we therefore constrain the conduit radius, the conduit viscosity and the elastic thickness to the optimum values found above. This leaves three free parameters in the model: (i) the depth to the base of the HVL; (ii) the viscosity of the HVL; (iii) the viscosity of the background mantle. Fig. 3 shows that there are significant changes in model fit associated with variations in these three parameters. We will further constrain the depth to the base of the HVL to the interval 100–150 km, in agreement with the estimated depth viscosity contrast in the plume (e.g. Ito et al., 1999). With these constraints we investigate in more detail the model fit as we vary the viscosities of the HVL and background mantle for various depths to the base of the HVL, Fig. 4. The best fit to the observed data is found for viscosity contrasts of a factor 0.5–3, Fig. 4 and Table 1. For a variance reduction better than 90%, a viscosity contrast larger than a factor 10 requires a mantle viscosity lower than 4-8  1018 Pa s depending on the thickness of the HVL, Fig. 4. Viscosity contrasts of the order 100 fit the data with a variance reduction greater than 90% only if the viscosity of the background mantle is less than 1018 Pa s. Model sensitivity tests, Appendix A, indicate that our GIA model is sensitive to a sudden increase in viscosity down to depths of 250–300 km in a 1018 Pa s mantle. The low viscosity layer associated with a viscosity contrast of 100 would therefore have to extend down past 250 km depth, which seems unlikely. 5.2. Self-consistent viscosity fields from geodynamic plume–ridge models The geodynamic plume–ridge model of Schmeling (2010) was designed to study crustal accretion and mantle melting of a ridge centered mantle plume. Here we will use it to extract selfconsistent viscosity structures. The model solves the equations of mass, momentum and energy in 2D, including melting, melt migration and extraction, using a modified version of the

5

5

4

4 VR = 90 %

3

3

2

2 1

1 10

20

30

40

50

60

70

80 0

Elastic thickness [km]

100

200

300

0

50

Depth to base of HVL [km]

100

150

200

Conduit radius [km]

5

5

4

4 VR = 90 %

χv2

χv2

VR = 90 %

VR = 90 %

VR = 90 %

3

3

2

2

χv2

χv2

VR = 90 %

1

1 2

4

6

8

10 12 14 16 18 20

Mantle viscosity [1018 Pa s]

20

40

60

80

HVL viscosity [1018 Pa s]

100

1

2

3

4

5

6

7

8

9 10

Conduit viscosity [1018 Pa s]

Fig. 3. Sensitivity of the conceptual plume model to perturbations in the six free parameters. The best fit reference model is marked by a large yellow symbol and have parameter values: elastic thickness of lithosphere, 30 km; depth to base of high viscosity layer (HVL), 100 km; plume conduit radius, 100 km; viscosity of background mantle, 1019 Pa s, viscosity of HVL, 2  1019 Pa s, viscosity of plume conduit, 3  1018 Pa s. The model fit of the reference model is 1.78. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

P. Schmidt et al. / Earth and Planetary Science Letters 359-360 (2012) 152–161

5

10

5

=9

2.5

3

2

90

10

%

20

VR

=

90

2

%

10 20 50

90

3

5

%

1

VR

=

20

5 0.

1

20

3

10

VR

10

50

5

2.5

5

4

3

20

2

5 0.

10

1

20

6

2

0%

1

= VR

100

50

50

HVL viscosity [1018Pas]

5

Mantle viscosity [1018 Pa s]

HVL viscosity [1018 Pa s]

Mantle viscosity [1018 Pa s]

157

100 [χ2]

1.0

1.5

2.0

2.5

3.0

4.0

6.0

9.0

13.0

20.0

Fig. 4. Model fit of conceptual plume models as a function of the viscosity of the background mantle and the viscosity of the high viscous layer, HVL. The depth to the base of the HVL is 100 km in (a) and 150 km in (b). All models have an elastic thickness of 30 km and a plume conduit radius of 100 km. The viscosity of the plume conduit is 3  1018 Pa s, except for models with mantle viscosity lower than this, which have identical conduit and mantle viscosities. The white contour outlines a variance reduction of 90%. Black dashed lines shows the viscosity contrast between the HVL and the background mantle, the lowermost dashed line corresponds to a viscosity contrast of 100.

Table 1 Summary of w2v model fit for selected best fit models. The last four models in the table all use viscosity fields from geodynamic models of the plume–ridge interaction. See text for detailed description of models. In figure

Model description

w2v

VR (%)

Fig. Fig. Fig. – Fig. – Fig.

Uniform mantle Optimal conceptual plume model (CPM) CPM with viscosity contrast 100 Dry mantle Dry mantle scaled by 0.1 Wet to dry mantle Wet to less wet mantle

1.74 1.78 3.38 18.18 2.12 6.28 2.22

94.9 94.8 90.1 46.8 93.8 81.6 93.5

1b 2a 2b 5a 5b

compaction Boussinesq approximation. The extracted melt is fed back into the model near the surface, simulating crustal accretion within an emplacement zone of 25 km half width and 6 km thickness. Boundary conditions include 2 cm/yr of a total spreading velocity at the sides and a localized plume like hot mantle influx at the base of the modeled region at 200 km depth. A temperature and pressure dependent non-linear mantle rheology is used, with a stress exponent of 3.5 based on laboratory measurements of single crystal olivine (Bai et al., 1991). For a detailed description of the model see Schmeling (2010). The effect of water on the rheology is modeled by changing the value of the product of the pre-exponential factor and the hydroxyl content AOH ¼ AC OH in the creep law, Eq. (1). For models incorporating the effect of dehydration, the rheology is switched to dry upon a depletion by melt of 1%, which typically occurs within 10 km of the locus of melting. The model does not take into account dehydration effects between the wet and dry solidi. Although wet melting will reduce the water content of the mantle between the wet and dry solidi, previous studies (Hirth and Kohlstedt, 1996; Ito et al., 1999) have indicated that the viscosity increase over this region is gradual and of small magnitude. The model is run until an approximative steady state has been reached. We expand the 2D effective viscosity field from the geodynamic models to 3D in the GIA model by rotation about the center of the plume, assumed to be located beneath northwestern ¨ Vatnajokull (see Fig. 1). The implemented viscosity field will therefore lack a cross-cutting spreading ridge. At depth below

200 km we assume a vertically uniform viscosity field equal to the viscosity at the base of the geodynamic model. This is a simplification as the viscosity of the mantle is expected to vary with depth due to increasing pressure and temperature. However, taking into account the counteracting effects of temperature and pressure on the mantle viscosity, the predicted (gradual) viscosity increase from 200 km to 400 km depth is a factor in the range 0.3– 7 under constant stress conditions (based on a wet mantle in the dislocation regime, with rheological parameters presented in Korenaga and Karato, 2008). If the deviatoric stress is not constant with depth the range narrows even further (see Section 2). We extract effective viscosity fields from three geodynamic models with different water contents: (i) a dry mantle; (ii) a wet mantle, switching to a dry mantle upon dehydration; (iii) a wet mantle, switching to a less wet mantle upon dehydration. In the wet mantle rheology, AOH is a factor 1000 smaller than in the dry rheology, whereas in the wet to less wet rheology AOH changes by a factor 100. The ratio in effective viscosities as a function of water then depends on the stress/strain state as discussed in Section 2. We note that all three geodynamic models predict crustal thicknesses between 19 and 27 km, in general agreement with estimates of crustal thickness in Iceland (e.g. Bjarnason and Schmeling, 2009). In contrast to our conceptual models, the geodynamic models have lateral and vertical viscosity variations in the HVL, and a plume conduit which continues through the HVL all the way up to the base of the lithosphere, Fig. 5. We extract effective viscosities from a model with non-linear rheology but use these viscosities in a linear rheology in the GIA earth model. The rational behind this is as follows. The deviatoric stresses due to the GIA process are of the order of 100 times smaller than the stresses associated with the mantle plume. Under these conditions the GIA process will effectively see a linear, but apparent anisotropic and laterally heterogeneous, rheology with viscosity governed by the deviatoric stresses of the mantle plume (Schmeling, 1987). The magnitude of anisotropy is of the order of the magnitude of the power law exponent, or less, thus we neglect this effect here. Non-linear effects on the GIA uplift might possibly be detected in the narrow region where the uplift turns to subsidence in the collapsing fore-bulge (Wu, 1995). This possibility arises as the fore-bulge in a linear rheology will migrate with time while in a non-linear rheology the fore-bulge is stationary, unless the ice

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-11

0.10

-1.0

-9

-7

Residual GPS velocities [mm/yr] -3 -1 1 3

-5

0.16

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-0.8

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-0.2

Viscosity ratio [η/1019] 1.00 1.58 2.51 3.98

0.0 0.2 0.4 0.6 Viscosity [log10(η/1019)]

5

6.31

0.8

7

9

10.00 15.80 25.11

1.0

1.2

1.4

11

39.81

1.6

Fig. 5. Viscosity structure used in GIA models and maps of residual vertical velocities for models with viscosity structure from geodynamic plume–ridge interaction models. (a) Dry model with viscosities scaled by a factor of 0.1, see text, w2v ¼2.12 and variance reduction 93.8%. (b) Wet to less wet model, see text, w2v ¼ 2.22 and variance reduction 93.5%.

margin changes (Wu, 1999). In the present day GIA uplift of Iceland, the collapse of the fore-bulge is located off-shore and hence will not affect our results. The lateral heterogeneity in the viscosity field is incorporated into the model, but anisotropy is neglected as it has been shown to have a negligible influence on the uplift (Han and Wahr, 1997). Viscosities extracted from the geodynamic model with a dry mantle are too high to fit the observed uplift velocities, Table 1. If this viscosity field is scaled by a factor of 0.1, the model fit is significantly improved, Fig. 5a and Table 1. However, such scaling destroys the self-consistency of the viscosity field. In the wet to dry model, the fit to the data is significantly improved compared to the dry model but still not acceptable, Table 1. For the weakest model, wet to less wet, we again see a significant improvement of the fit to the data and the variance reduction of the model is now within our criteria of 90%, Fig. 5b and Table 1. Noting that the model is still somewhat too stiff, we test models with slightly lower viscosities in the mantle. A model fit with w2v ¼ 1.73 (VR 94.9%) can be achieved if the mantle viscosity is decreased from 16  1018 Pa s to 5  1018 Pa s. Again, the procedure is not selfconsistent and we do not pursue it further. An alternative approach would be to reduce the pre-exponential factor of the wet rheology, which would require a simultaneous reduction of the plume excess temperature in order to keep the crustal generation rate within a reasonable limit. We note, however, that a fit to the GPS data with a variance reduction better than 90%, can only be achieved for relatively small viscosity contrasts. Of our three geodynamic models, only the wet to less wet model can fit the data with a variance reduction better than 90%. In this model the viscosity field in the HVL is rather complex. Locally, the viscosity ratio between the less wet region and the background mantle can reach values of about 40, but on the average it is less than 10, Fig. 5b, which agrees reasonably well with the results of the conceptual plume models in Fig. 4.

6. Discussion As noted earlier, the GPS data set we use in this study can be very well fit by a homogeneous mantle viscosity. In fact, the misfit of the best fit homogeneous model is as low as the best fit 3D model. This is not very surprising as the best fit 3D model only displays a very small viscosity contrast between the mantle and the HVL. As seen from Fig. 3c and f the conceptual model is not very sensitive to the presence of a low viscosity plume conduit. It is expected that the sensitivity to the conduit would increase if it penetrated through the HVL as indicated by the geodynamic models. Scaling of the viscosity fields from the geodynamical models indicate, however, that for a realistic plume radius the plume conduit will not have a significant effect on the misfits. This is due to a stabilizing support from the surrounding more viscous mantle as well as a shielding effect by the overlying HVL. Furthermore, although the geodynamic models predict both lateral and vertical variation of the viscosity in the dehydrated region, the misfits seem to be mainly sensitive to the mean viscosity of that region. We can therefore conclude that the data we use are not very sensitive to lateral viscosity variations beneath Iceland. This is partly due to the relatively sparse nation wide character of the data set, making it more sensitive to longer wavelength deformation signals, but also to the use of only the vertical deformation component. However, as we show in Fig. 3d and e, the misfit is highly sensitive to the vertical viscosity contrast between the mantle and the HVL which is the subject of interest in this study. The relative insensitivity of the data to lateral viscosity variations is therefore beneficial to this study. The small viscosity contrasts in the geodynamic models are a result of the non-linear rheology, in which the viscosity contrast depends on how stress and strain rate evolves upon dehydration. The extrapolation of laboratory results referred to

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above (Hirth and Kohlstedt, 1996) predicted a viscosity contrast of a factor 500 7300, herein reduced to 1707100 after a calibration of laboratory results (Bell et al., 2003; Koga et al., 2003). That extrapolation assumed a linear relationship between the water content and the viscosity, implying either a linear rheology or a constant stress regime. If instead the extrapolation is performed for constant strain rate to constant viscous dissipation rate conditions and a non-linear rheology (Eqs. (8) and (9)) the viscosity contrast would be a factor 3–12 for the water content of a typical MORB source and a stress exponent of 3.5. Including the possibility that the Icelandic plume source has seven times higher water content than MORB yields an estimated viscosity contrast of a factor of 6–29, based on the laboratory results. These estimates are in good agreement with the GIA results presented here. Glacial isostatic adjustment studies are a major contributor to our understanding of mantle viscosity and one of the few techniques which can obtain a direct viscosity measurement. As such, it is an ideal tool to test laboratory based inferences on mantle viscosity. GIA models frequently suffer from a trade-off between the viscosity estimates and the ice history. This arises when the ice history is constrained by observations of the GIA process, which in turn depends on the viscosity of the Earth. Much of this ambiguity can be reduced in the case of Iceland, where we have direct observations of the deglaciation history since the onset of the present deglaciation trend in 1890. The viscosity estimates presented here are therefore largely unaffected by the trade-off in viscosity and ice history. In addition, the Iceland GIA model and the GPS data we use are sensitive to a viscosity contrast in the mantle down to at least twice the depth of the onset of melting, see Appendix A. Our results therefore indicate that the existence of a large magnitude viscosity contrast beneath Iceland is highly unlikely. Instead we suggest that the absence of such a feature is indicative of a non-linear rheology in the uppermost portion of the Icelandic mantle, and close to constant viscous dissipation rate conditions during dehydration, assuming that the estimates of the water content are reasonable. This implies that the dynamics of the uppermost mantle beneath Iceland is to a large extent controlled by a combination of kinematic and dynamic boundary conditions (close to constant dissipation rate conditions) rather than buoyant forces alone (constant stress), as would be expected in a thermal plume. Note that by ‘‘constant’’ we mean ’’essentially the same value’’ in the hydrated and dehydrated regions. A condition close to constant viscous dissipation rate is also supported directly by the geodynamic models. In these the dissipation rate shows the least contrast between hydrated and dehydrated regions, while the stress field generally increases by about half an order of magnitude when entering the dehydrated region, and the strain rate decreases by the same amount across this boundary. A number of numerical studies have shown that dehydration stiffening has direct implications for the dynamics of the uppermost mantle. Buoyant upwelling beneath a spreading ridge (Phipps Morgan, 1997; Braun et al., 2000) or in a plume (Ruedas, 2006) is strongly inhibited in models using a linear rheology and the large viscosity contrast inferred from the laboratory (Hirth and Kohlstedt, 1996). Similar geodynamic plume models also produce an improved fit to observations such as gravity and seismological data (Ito et al., 1999; Ito, 2001). On the larger scale, dehydration stiffening also affects models of plate tectonics and the Earth’s thermal history (Korenaga, 2011). In light of the results presented here, we suggest that conclusions drawn in these studies be revisited using a viscosity contrast on the order of a factor of 10 or less, compatible with GIA observations and a non-linear rheology.

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7. Conclusions In this study we have used data on glacial isostatic adjustment in Iceland to assess how large a viscosity contrast can be present in the melting region beneath Iceland due to dehydration stiffening. We find that a viscosity contrast of a factor 0.5–3 best fit the data. Viscosity contrasts of a factor 10 require a mantle viscosity of 4-8  1018 Pa s or less, depending on the thickness of the dehydrated region, in order to fit the data to a variance reduction better than 90%. Viscosity contrasts of 100 require a mantle viscosity of 1018 Pa s or lower, down to depths of at least 250 km. Our constraints on the magnitude of dehydration stiffening are significantly lower than previous estimates based on linear extrapolation of laboratory experiments. If the laboratory data are extrapolated using a non-linear rheology under constant strain rate to constant dissipation rate conditions, the viscosity contrast would be a factor 3–29, depending on the assumed water content of the source. This is in agreement with our GIA results and we therefore suggest that the uppermost mantle beneath Iceland is governed by a non-linear rheology and that the mantle is controlled by a combination of kinematic and dynamic boundary conditions rather than buoyant forces alone, as would be expected in a thermal plume.

Acknowledgments The authors thank Greg Hirth for bringing to their attention the calibration of hydroxyl content in laboratory experiments on olivine rheology, as well as discussing how this affects previous estimates of the effect of dehydration on the viscosity of the mantle. Comments from two anonymous reviewers helped us to improve and clarify this manuscript. All figures have been prepared using the freely available software suite GMT (Wessel and Smith, 1991). The authors thank the GMT development team for their quick response and updates to the software package, necessary for producing Figs. 2 and 5.

Appendix A. Model sensitivity to viscosity contrasts As briefly reviewed in Section 4.1 above, a number of investigations of the viscosity beneath Iceland arrive at similar conclusions: that the viscosity is approximately two orders of magnitude lower than that generally found beneath continental areas such as Fennoscandia (e.g. Milne et al., 2004) and North America (e.g Braun et al., 2008). The post-seismic relaxation (A´rnado´ttir et al., 2005) and rifting (Hofton and Foulger, 1996; LaFemina et al., 2005) studies may arguably reflect shallow processes. It has, also, been argued that also the GIA signal is most sensitive to viscosities shallower than approximately 50 km, for the prevailing Icelandic

Fig. A1. Axi-symmetric models for the study of viscosity contrast sensitivity. (a) Reference model, (b) test model. TE is the elastic thickness of the lithosphere, D is the depth to the viscosity contrast, Zref is the reference mantle viscosity and Z is the viscosity of the mantle below the contrast in the test model.

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Rη 0

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500

500 0

40

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80 120 160 200 240 r [km]

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80 120 160 200 240 r [km]

0

40

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80 120 160 200 240 280 r [km]

Fig. A2. Minimum viscosity contrast, RZ , needed to perturb the vertical surface velocity of a reference model by more than 1 mm/yr. The contrast is shown as a function of depth to the contrast and horizontal distance from the center of the ice load. (a) Viscosity contrast for a reference model with elastic thickness of the lithosphere TE ¼ 30 km and reference viscosity Zref ¼ 1019 Pa s. (b) Reference model with TE ¼30 km and Zref ¼ 1018 Pa s. (c) Reference model with TE ¼ 30 km, a high viscosity layer of 70 km thickness and viscosity 1020 Pa s and Zref ¼ 1018 Pa s. In white regions no viscosity contrast could be found that could perturb the surface velocities by more than 1 mm/yr.

surface load conditions (Ito et al., 1999). Here we show by a perturbation approach that our GIA model and the data we use are sensitive to viscosities at significantly greater depths than 50 km. We also find support for this notion in the literature. We test the sensitivity to a viscosity contrast in the mantle using a large number of axi-symmetric models, mimicking the current load conditions in Iceland. Our models have a circular, uniform ice load of 54 km radius which has the same volumetric ¨ ice loss as Vatnajokull between 1890 and 2004. We calculate surface velocities in the time interval 1993–2004, the time span of the GPS data above. Comparisons of the axi-symmetric model results to full 3D models, with the same earth structure, show that although the absolute velocities differ somewhat, the velocity differences between two earth models, computed either axisymmetrically or in 3D, are in good agreement. The Earth models are constructed as shown in Fig. A1. Here we will present the results from three reference models with the following properties: (i) elastic thickness, TE, of 30 km and a uniform mantle viscosity, Zref , of 1019 Pa s. (ii) TE ¼30 km and Zref ¼ 1018 Pa s. (iii) TE ¼30 km, Zref ¼ 1018 Pa s and a 70 km thick high viscosity layer with viscosity 1020 Pa s at the top of the mantle. We set up a suite of test models, Fig. A1b, for each reference model, with the uniform mantle perturbed by the introduction of a higher viscosity half-space below, starting at depth D. We systematically vary the depth D and the viscosity of the half-space, defining RZ as the ratio of the half-space viscosity to the mantle reference viscosity. The vertical surface velocities predicted by the reference and test models are compared, and for each depth D we search for the smallest RZ that produces a difference in the velocities, at a radius r from the center of the load, larger than a particular threshold. Here we use a threshold of 1 mm/yr, corresponding to the general uncertainty in the GPS observations. In Fig. A2 we show which RZ is needed at different depths to produce the required 1 mm/yr difference in vertical velocities, at a distance r from the load center. We see that this varies at a particular depth with distance from the center of the load. We also note the very rapid increase in contrast needed as the limit of detectability is approached. The white regions indicate that no

tested model, up to RZ ¼ 107 , can produce the required velocity difference. At larger horizontal distance the absolute velocities decay to less than 0.5 mm/yr preventing the velocity difference from becoming larger than 1 mm/yr. Fig. A2a indicates that if the uppermost mantle viscosity is 1019 Pa s, we should be able to resolve a step increase in viscosity of a factor of 10 down to 400 km depth and a factor 5 increase down to 350 km. A lower upper mantle viscosity decreases the depth of resolution, such that for a 1018 Pa s mantle we should resolve a factor 10 viscosity increase down to 220 km and a factor 5 down to 180 km, Fig. A2b. An upper layer of high viscosity (HVL), such as the dehydrated region in this study, increases the ultimate depth of resolution but decreases how well viscosity steps are resolved in the upper part of the mantle, Fig. A2c. We see that the depth to a factor 10 contrast is approximately the same as that for the model without HVL, whereas the factor 5 contrast is shallower. As is evident from Fig. A2, the resolution depths become shallower with distance from the center of the load. Our best fitting models in Section 5 above have viscosities on the order of 1019 Pa s. The sensitivity tests carried out here indicate that we would have resolving power down to approximately 400 km depth in these models. Similar to our results, Fleming et al. (2007) found that for a 5  1018 Pa s mantle they had some resolving power down to 400 km depth when modeling current GIA in Iceland. GIA studies of the early Holocene deglaciation in Iceland (Sigmundsson, 1991) concluded that a mantle viscosity of 1019 Pa s fit the relative sea-level data. This is in agreement with the viscosities found in our study, even though the late Pleistocene/early Holocene ice sheet covered all of Iceland with a thickness in excess of 1 km, probing the viscosity structure to greater depth than this study.

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