WATER CYCLING BETWEEN OCEAN AND MANTLE: SUPER-EARTHS NEED NOT BE WATERWORLDS

The exact water inventory of Earth's interior is currently unknown, but is thought to be comparable to the surface reservoirs. The mantle contains water in both hydrous and nominally anhydrous minerals (Hirschmann 2006; Hirschmann & Kohlstedt 2012). The Fe core may also contain primordial water (Abe et al. 2000), which we ignore in the present analysis.

Inoue et al. (2010) determined maximum water mass fractions of 0.7% (olivine, shallower than 410 km), 3.3% (wadsleyite, 410–520 km), 1.7% (ringwoodite, 520–660 km), and 0.1% (perovskite, below 660 km). Combining these estimates with the pressure dependence of olivine water content from Hauri et al. (2006) puts Earth's mantle water capacity at a dozen times the current surface reservoir. Measurements of electrical conductivity (Dai & Karato 2009) suggest that Earth's interior contains one to two oceans worth of water at present day, but other estimates have differed by a factor of two in either direction (e.g., Huang et al. 2005; Smyth & Jacobsen 2006; Khan & Shankland 2012).

There is a two-way flux of water between ocean and mantle on Earth. Ocean crust forms at mid-ocean ridges (MORs) by depressurization melting of mantle, releasing volatiles into the ocean. Plate tectonics then drives the ocean crust toward subduction zones. Hydrothermal alteration, principally serpentinization, produces ocean crust that is 5%–10% water by mass. During subduction, much of the water is baked out of the subducting slab, producing explosive volcanism like Mount St. Helens. Some of the water, however, is subducted deep into the mantle, closing the loop (for a review of the relevant geochemistry; see Arndt 2013).

The current H₂O degassing at MORs is approximately 2 × 10¹¹ kg yr⁻¹ (Hirschmann & Kohlstedt 2012), while the regassing flux at subduction zones is estimated to be (0.7–2.9) × 10¹² kg yr⁻¹ (Jarrard 2003; Schmidt & Poli 1998). Given the order-of-magnitude agreement of these figures and the large associated uncertainties, Earth's current ocean volume is typically assumed to be in a steady state (McGovern & Schubert 1989). Moreover, realistic parameterizations of mantle convection predict water fluxes of 10¹¹–10¹³ kg yr⁻¹ in Earth's geological past, implying ocean-cycling times of 10⁸ yr (McGovern & Schubert 1989). A significant flux imbalance would have long ago desiccated or submerged the planetary surface.

A planet entirely covered in water may develop exposed continents by losing water to space, hydrating the crust and mantle, or reshaping continents and ocean basins. For example, an ocean-covered planet with a hot stratosphere could lose water to space until continents are exposed. The resulting vigorous silicate weathering might draw down sufficient CO₂ to cool the planet out of the moist greenhouse state, leaving a partially water-covered planet (Abbot et al. 2012). Detailed atmospheric simulations of such hot planets, however, indicate that the tendency of carbon dioxide to cool the stratosphere impedes the loss of water (Wordsworth & Pierrehumbert 2013).

In an alternate hypothesis, the Rayleigh number of convecting water in hydrothermal systems at MORs exhibits a sharp peak when water becomes supercritical, which occurs at nearly the pressures and temperatures in hydrothermal systems today (Kasting & Holm 1992). If Earth's water started in the mantle and gradually degassed to the surface, then this pressure dependence could explain why the oceans stabilized at their current depths. A planet starting with very deep oceans (Korenaga 2008), however, would not be able to effectively subduct water and would remain a waterworld.

In order to solve the problem of regassing water into the mantle, Holm (1996) suggested that the reduced efficiency of hydrothermal heat exchange beneath deep oceans would result in higher mantle temperatures, greater plate velocities, and therefore efficient subduction of hydrated crust. This mechanism may not work quantitatively, however, because hydrothermal heat transfer only accounts for 30%–50% of heat flux through oceanic crust (Stein et al. 1995). There are also qualitative problems with this regassing argument: first of all, the same logic should apply to planets with very shallow oceans, potentially negating the supercritical circulation hypothesis. Moreover, hydrating the mantle lowers its viscosity, further increasing plate velocities and making this mechanism a destabilizing feedback. Lastly, although rapid subduction may aid regassing, a hotter mantle almost certainly hinders it (Bounama et al. 2001). In short, it is not clear that reduced heat transport through the ocean crust leads to net regassing of water.

Whatever geophysical processes govern Earth's deep water cycle presumably operate on other terrestrial planets with plate tectonics. The partitioning of water on Earth is not only an outstanding problem in geophysics (Fyfe 1994; Langmuir & Broecker 2012), but one with important implications for the surface conditions and climate of terrestrial exoplanets.

The principle dependent variable in our model is the bulk water mass fraction of the mantle, x. We assume plate tectonics, but our results should be largely insensitive to plate velocity: degassing and regassing are proportional to the spreading and subduction rates, respectively, which are equal in steady state. We therefore consider the change in mantle water per seafloor overturning, x' ≡ dx/dτ, where the non-dimensional time, τ, is related to the area of oceanic crust, A_b, spreading rate, S, and the length of MORs, L, via τ = tLS/A_b.

The net change of mantle water per seafloor overturning is

$$\begin{equation} x^{\prime } = \frac{A_b}{M_m}(w_\downarrow - w_\uparrow), \end{equation} \tag{ 1 }$$

where M_m is the mantle mass, w_↓ is the water content of subducted crust and sediments, and w_↑ is the water degassed by the formation of ocean crust at MORs (both have units of kg m⁻²). This equation is similar to that solved by previous researchers (McGovern & Schubert 1989; Ueta & Sasaki 2013), except that we include the first-order effects of seafloor pressure, as described below.

The subducted water is

$$\begin{equation} w_\downarrow = x_h \rho _{b} d_h(P) \chi , \end{equation} \tag{ 2 }$$

where x_h = 0.05 is the mass fraction of water in the hydrated crust (McGovern & Schubert 1989), ρ_b = 3.0 × 10³ kg m⁻³ is the density of basalt, d_h is the depth of hydration in the crust (which depends on the seafloor pressure, P), and χ = 0.23 is the fraction of volatiles subducted deep into the mantle rather than outgassed via arc and back-arc volcanism (Rüpke et al. 2004). The subduction efficiency, χ, is highly uncertain, but we perform a sensitivity analysis below to quantify how it and other critical parameters likely affect our results (Section 4.1).

The degassed water is

$$\begin{equation} w_\uparrow = x \rho _m d_{\rm melt} f_{d}(P), \end{equation} \tag{ 3 }$$

where ρ_m = 3.3 × 10³ kg m⁻³ is the density of the upper mantle, d_melt = 6 × 10⁴ m is the depth of the MOR melting regime, and f_d is the fraction of water in the melt that degasses rather than remaining in the ocean crust as it solidifies (model variables and parameters are listed in Table 1).

Table 1. Model Variables and Parameters

Name	Symbol	Value
Planetary water mass fractiona	ω	ω⊕ = 6.2 × 10−4
Normalized gravitya	$\tilde{g}$	$\tilde{g}_\oplus = 1$
Mantle water mass fractiona	x	x⊕ = 5.8 × 10−4
Density of water	ρw	1.0 × 103 kg m−3
Density of granite	ρg	2.9 × 103 kg m−3
Density of basalt	ρb	3.0 × 103 kg m−3
Density of mantle	ρm	3.3 × 103 kg m−3
Thickness of oceanic crust	db	6 × 103 m
Depth of melt	dmelt	60 × 103 m
Hydrated crust water fraction	xh	0.05
Subduction efficiency	χ	0.23
Earth degassing efficiency	fd⊕	0.9
Earth hydration depth	dh⊕	3 × 103 m
Earth seafloor pressure	P⊕	4 × 107 Pa
Max. thickness of cont. crust	$d_g^{\rm max}$	$70\times 10^3 \tilde{g}^{-1}$ m
Ocean basin covering fraction	fb	0.9
Mantle mass fraction	fm	0.68
Ocean mass fraction of Earth	ωo	2.3 × 10−4
Normalized ocean basin area	$\tilde{f}_b$	1.3
Seafloor pressure dependenceb	ϕ	2
Earth mantle water contentb	x⊕	5.8 × 10−4
Max. mantle water fractionb	xmax	7 × 10−3

Notes. ^aThese are the variables in our model; we list here their nominal values for Earth. ^bThese are critical parameters whose sensitivity we test in Section 4.1; we list their nominal values here.

Download table as:  ASCIITypeset image

The fraction of water in the melt that is degassed depends on hydrostatic pressure at the seafloor (Papale 1997; Kite et al. 2009). We parameterize this stabilizing feedback as

$$\begin{equation} f_{d}(P) = \min \left[f_{d\oplus } \left(\frac{P}{P_\oplus }\right)^{-\mu },\; 1\right], \end{equation} \tag{ 4 }$$

where P is the pressure at the bottom of the ocean, P_⊕ = 4 × 10⁷ Pa is its current value on Earth, μ > 0 quantifies the pressure dependence of melt degassing, and f_d⊕ = 0.9 is the nominal degree of melt degassing on Earth today. The piecewise definition ensures that the degassed water does not exceed that present in the melt.

The depth of serpentinization may also depend on the pressure at the bottom of the ocean:

$$\begin{equation} d_h(P) = \min \left[d_{h\oplus } \left(\frac{P}{P_\oplus }\right)^\sigma ,\; d_b\right], \end{equation} \tag{ 5 }$$

where σ quantifies the pressure dependence, d_h⊕ = 3 × 10³ m is the nominal value of the hydration depth on Earth (McGovern & Schubert 1989), and d_b = 6 × 10³ m is the thickness of basaltic ocean crust. Hydration of the lithospheric mantle below the oceanic crust is functionally identical to the hydration of oceanic crust (Rüpke et al. 2004), but we conservatively adopt the constraint d_h ⩽ d_b.

It has been argued that the depth of hydration depends on the Rayleigh number of water in hydrothermal systems (Kasting & Holm 1992). If, as they assumed, water is gradually degassed from the mantle, this amounts to a large positive σ; if water begins at the planetary surface, such a mechanism dictates a large negative σ.

We treat the planetary elevation distribution ("hypsometry") as two δ-functions: one for ocean crust and another for continental crust. This is a good approximation of Earth's strongly bimodal hypsometry (Rowley 2013), which is presumably typical of any terrestrial planet with plate tectonics (Stoddard & Jurdy 2012). Earth's oceanic crust exhibits a clear age–depth relation: older crust is denser and sinks deeper into the mantle (Parsons & Sclater 1977). A larger ocean basin will therefore have a greater mean depth. The globally averaged ocean depth, however, should remain roughly constant barring secular changes in spreading rates, which are beyond the scope of our zeroth-order treatment.

We assume that the modal continental height is level with the ocean surface, which is natural because of the competing effects of erosion and deposition (Korenaga 2008; Rowley 2013). There is a maximum thickness that continents can achieve, however, beyond which a continent flows under its own weight (Rey & Houseman 2006). The Himalayan plateau on Earth has a crustal thickness of 70 km and appears to be near this limit (England & McKenzie 1982). We conservatively adopt a maximum continental thickness of

$$\begin{equation} d_g^{\rm max} = 70\, \hbox{km} \left(\frac{g}{g_\oplus }\right)^{-1}, \end{equation} \tag{ 6 }$$

where g is the surface gravity of the planet, g_⊕ is that of Earth, and we have adopted the gravity dependence of Kite et al. (2009). Using the mass–radius relation of Valencia et al. (2007a), a gravity of 3 g_⊕ corresponds to a 10 M_⊕ super-Earth.

The thickness of granitic continents, d_g, is related to the thickness of the other layers by

$$\begin{equation} d_g = d_o + d_b + d_m, \end{equation} \tag{ 7 }$$

where d_o is the depth of the ocean basin and d_m is the depth to which the continent root extends into the mantle (Figure 1). This parameterization assumes that the tops of continents are at sea level (zero freeboard). This is merely shorthand for a freeboard that is much smaller than the depth of ocean basins.

Isostatic balance dictates that the pressure below each vertical column must be equal:

$$\begin{equation} \rho _w d_o + \rho _b d_b + \rho _m d_m = \rho _g d_g, \end{equation} \tag{ 8 }$$

where ρ_w = 1.0 × 10³ kg m⁻³ and ρ_g = 2.9 × 10³ kg m⁻³ are the density of water and granite, respectively.

Combining (7) and (8) yields the following expression of isostatic balance:

$$\begin{equation} d_o (\rho _m-\rho _w)+d_b (\rho _m-\rho _b) = d_g (\rho _m-\rho _g). \end{equation} \tag{ 9 }$$

Given the maximal crustal thickness, $d_g^{\rm max}$, one can derive the maximum depth of water-filled ocean basins:

$$\begin{equation} d_{o}^{\rm max} = \frac{d_g^{\rm max} (\rho _m-\rho _g) - d_b (\rho _m-\rho _b)}{\rho _m-\rho _w}. \end{equation} \tag{ 10 }$$

For our fiducial Earth-size parameters, $d_{o}^{\rm max} = 11.4$ km. Note that the thickness of oceanic crust is likely also inversely proportional to g (Sleep 2012). We conservatively adopt d_b = 6 km regardless of planet mass, which produces shallower ocean basins for massive planets.

The densities of granite and basalt are nearly the same (ρ_g ≈ ρ_b), and the maximum crustal thickness far exceeds the thickness of oceanic crust ($d_b \ll d_g^{\rm max}$), so we can approximate (10) as

$$\begin{equation} d_{o}^{\rm max} \approx 11.4 \,\hbox{km} \left(\frac{g}{g_\oplus }\right)^{-1}. \end{equation} \tag{ 11 }$$

The maximum ocean volume that can be accommodated while maintaining exposed continents also depends on the ocean basin area, A_b = f_bA, where A is the planetary area. We adopt f_b = 0.9. In other words, 90% of the planet is covered in water, and the remaining 10% is exposed continent. This is sufficient dry land to maintain a silicate weathering thermostat (Abbot et al. 2012), and our results would not change dramatically if we instead adopted f_b = 0.7 (as on modern Earth) or f_b = 1 (the limiting case of a single infinitesimal island extending out of the ocean).

Given these assumptions, the maximum volume of surface water that could be accommodated by an Earth-size planet is 5.2 × 10¹⁸ m³, or a mass of 3.7M_o.

It is possible to obtain an analytic solution to (12) if we ignore the piecewise nature of degassing, f_d, and regassing, d_h. This analytic approximation has intuitive value, so we develop it here. In this case the w_↑ = w_↓ can be written as

$$\begin{equation} x \rho _m d_{\rm melt} f_{d\oplus } \left(\frac{P}{P_\oplus }\right)^{-\mu } = x_h \rho _{b} \chi d_{h\oplus } \left(\frac{P}{P_\oplus }\right)^\sigma , \end{equation} \tag{ 17 }$$

which can be compactly expressed as

$$\begin{equation} \frac{x}{x_\oplus } = \left(\frac{P}{P_\oplus }\right)^\phi , \end{equation} \tag{ 18 }$$

where ϕ = μ + σ is the sum of pressure dependencies for MOR melt degassing and serpentinization of oceanic crust, and Earth's bulk mantle water content is

$$\begin{equation} x_{\oplus } = \frac{x_h \chi \rho _b d_{h\oplus }}{\rho _m d_{\rm melt} f_{d\oplus }}. \end{equation} \tag{ 19 }$$

Our fiducial parameter values yield x_⊕ = 5.8 × 10⁻⁴, or roughly an ocean's worth of water in Earth's mantle.

Substituting (16) into (18), we obtain

$$\begin{equation} \frac{x}{x_\oplus } = \left(\tilde{g}^2 \frac{\omega - x f_m}{\omega _o \tilde{f}_b}\right)^\phi. \end{equation} \tag{ 20 }$$

The steady-state mantle water content allows us to estimate the ocean depth, via (15), which we then compare to the maximum ocean basin depth. The solid gray line in Figure 2 shows the waterworld boundary for our nominal parameter values and ϕ = 2. The small difference between the numerical and analytic waterworld boundaries (solid black and gray lines, respectively) indicates that the piecewise definitions of f_d(P) and d_h(P) do not critically affect our results.

If ϕ < 0, the pressure feedback is destabilizing and there are two physical steady-state solutions: shallow and deep oceans, respectively. The current state of a planet will depend on initial conditions. For ϕ ⩾ 0, there is a single physical root to (20) and therefore a single steady-state solution. If there is no net pressure dependence to the deep water cycle (ϕ = 0), then all planets have the same mantle water content as Earth, x ≡ x_⊕.

In the presence of a modest stabilizing pressure dependence (ϕ = 1), the steady-state mantle water fraction is

$$\begin{equation} x = \frac{\omega x_\oplus \tilde{g}^2}{\omega _o \tilde{f}_b + x_\oplus f_m \tilde{g}^2}. \end{equation} \tag{ 21 }$$

In the high-gravity limit, the mantle contains the entirety of the planet's water, x = ω/f_m. In practice this cannot occur for water-rich planets because of the finite water capacity of the mantle (x ⩽ x_max). Nonetheless, the higher gravity of super-Earths biases the deep water cycle in favor of mantle sequestration. Water inventory and mantle capacity are both proportional to planetary mass, producing a weak mass dependence to the waterworld boundary.

As noted in Section 1.1, the amount of water in Earth's mantle is poorly constrained. The gray broken lines in Figure 2 show the analytic waterworld boundary if Earth's mantle water content, x_⊕, is 10 × greater (dashed) and 10 × smaller (dotted) than our nominal value. Note that varying x_⊕ in this model is mathematically equivalent to varying any of the water flux parameters in (19), many of which are uncertain (e.g., the subduction efficiency, χ). Varying x_⊕ by two orders of magnitude affects the waterworld boundary for a 10 M_⊕ super-Earth by roughly one order of magnitude; this is the dominant uncertainty in our study.

The maximum water capacity of the mantle, x_max, is not well known for high-mass terrestrial planets. The water storage in Earth's mantle is thought to be concentrated in the transition zone (410–660 km depth). If massive planets can only sequester water in a thin transition zone, then mantle water capacity scales with planetary area rather than mass, or $x_{\rm max} \propto \tilde{g}^{-1}$. The dotted black line in Figure 2 shows the minuscule effect of adopting this scaling.

On the other hand, the mantle of a super-Earth should be primarily in the form of post-perovskite (Valencia et al. 2007b), which may not have the same water capacity as Earth's dominant mantle rock, perovskite. We try setting x_max = 1 and find that the waterworld boundary is unchanged if post-perovskite can hold unlimited water.

Finally, the blue lines in Figure 2 show the waterworld boundary for pressure dependences of ϕ = 3 (dashed) and ϕ = 1 (dotted). (In both cases, we use μ = σ = ϕ/2 in the numerical model.) The precise strength of the seafloor pressure dependence of the deep water cycle affects the waterworld boundary by less than a factor of two.

Our model of the deep water cycle assumes plate tectonics, but it is currently unknown whether super-Earths are tectonically active. The increased heat flux of super-Earths should produce vigorous mantle convection (Valencia et al. 2007a; van Heck & Tackley 2011), but the increased strength and buoyancy of crust on super-Earths may prohibit plate tectonics (O'Neill & Lenardic 2007; Kite et al. 2009). Mantle convection may even exhibit hysteresis, such that planets with identical boundary conditions may or may not have plate tectonics, depending on initial conditions (Lenardic & Crowley 2012).

Alternatively, it has been suggested that surface water is more important than planetary mass for plate tectonics (Mian & Tozer 1990; Korenaga 2010). In a classic case of chicken-and-egg, van der Lee et al. (2008) argue that a deep water cycle is a necessary, if not sufficient, condition for long-term plate tectonics.

Continental crust formation is thought to be an inevitable by-product of plate tectonics in the presence of water (Rudnick 1995; Arndt 2013), so super-Earths are likely to have large volumes of granitic crust. In fact, the large volume of continental crust combined with smaller maximal crustal thickness may lead to a planet entirely covered in continental crust. This does not greatly affect our results, provided that the planet remains tectonically active and that some regions have thicker crust than others (differences in crustal thickness scale as g⁻¹ for the same reason as ocean basin depth).

The homogeneity of volatiles in Earth's lower mantle is questionable. The high ³He abundance of ocean islands has been attributed to a poorly mixed lower mantle (Kurz et al. 1982). By extension, this hypothesis implies that Earth's lower mantle may hold much more water than what is inferred for the upper mantle. Gonnermann & Mukhopadhyay (2009) argue, however, that the ³He abundance of the mantle is consistent with homogeneous composition.

If the mantle is not well mixed, then x represents the water fraction of those regions sampled by the MOR melting and affected by subduction of oceanic crust. Indeed, the source of MOR basalts (MORB) appears to have maintained a constant water mass fraction, x, for billions of years, suggesting that subduction of hydrated oceanic crust is depositing water in the MORB source region (Hirschmann 2006).

Our model of the deep water cycle predicts that many super-Earths have exposed continents. It will eventually be possible to test this hypothesis by observationally determining the surface character of a large number of high-mass terrestrial planets (see also discussion in Abbot et al. 2012).

Disk-integrated rotational multiband photometry of Earth, essentially the changing colors of a pale blue dot, encode information about continents, oceans and clouds (Ford et al. 2001). Such "single-pixel" observations have been used to construct coarse longitudinal color maps of Earth and Mars (Cowan et al. 2009, 2011; Fujii et al. 2011; Hasinoff et al. 2011), while simulations suggest that photometry spanning an entire planetary orbit could be used to construct a rough two-dimensional color map (Kawahara & Fujii 2010, 2011; Fujii & Kawahara 2012). Finally, Cowan & Strait (2013) showed that disk-integrated multiband photometry of a variegated planet can be inverted to obtain reflectance spectra of its dominant surface types, even if the number and colors of the surfaces are not known a priori.

The bottom line is that a 5–10 m space telescope equipped with a coronagraph or starshade could produce a coarse surface map of an Earth-analog at a distance of 10 pc (Cowan et al. 2009; Fujii & Kawahara 2012). Such low-resolution maps would be sufficient to identify the continents one expects on a tectonicaly active planet. If most super-Earths exhibit the bimodal surface character of Earth, it will suggest that they experience plate tectonics and a deep water cycle. If, instead, large terrestrial planets were all determined to be waterworlds, it would indicate that our hypothesis is wrong.