Water Cycling Between Ocean and Mantle: Super-Earths Need Not be Waterworlds

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by  Nicolas B. Cowan

Draftversion January6,2014
PreprinttypesetusingLATEXstyleemulateapjv.8/13/10
WATER CYCLING BETWEEN OCEAN AND MANTLE: SUPER-EARTHS NEED NOT BE WATERWORLDS
Nicolas B. Cowan1 & Dorian S. Abbot2
Draft version January 6, 2014
ABSTRACT
Large terrestrial planets are expected to have muted topography and deep oceans, implying that
most super-Earths should be entirely covered in water, so-called waterworlds. This is important
because waterworlds lack a silicate weathering thermostat so their climate is predicted to be less
4
stable than that of planets with exposed continents. In other words, the continuously habitable zone
1
forwaterworldsismuchnarrowerthanforEarth-likeplanets. Aplanet’swaterispartitioned,however,
0
betweenasurfacereservoir,theocean,andaninteriorreservoir,themantle. Platetectonicstransports
2
water between these reservoirs on geological timescales. Degassing of melt at mid-ocean ridges and

n serpentinization of oceanic crust depend negatively and positively on seafloor pressure, respectively,
a providing a stabilizing feedback on long-term ocean volume. Motivated by Earth’s approximately
J
steady-statedeepwatercycle,wedevelopatwo-boxmodelofthehydrosphereandderivesteady-state

3 solutions to the water partitioning on terrestrial planets. Critically, hydrostatic seafloor pressure is
proportionalto surface gravity,so super-Earthswith a deepwatercycle will tendto store morewater
] in the mantle. We conclude that a tectonically active terrestrial planet of any mass can maintain
P
exposed continents if its water mass fraction is less than ∼ 0.2%, dramatically increasing the odds
E
that super-Earths are habitable. The greatest source of uncertainty in our study is Earth’s current
.
h mantle water inventory: the greater its value, the more robust planets are to inundation. Lastly, we
p discuss howfuture missions cantestour hypothesis by mapping the oceans andcontinents ofmassive
- terrestrial planets.
o
r Subject headings: planets and satellites: composition — planets and satellites: interiors — planets
t and satellites: oceans — planets and satellites: physical evolution — planets and
s
a satellites: surfaces — planets and satellites: tectonics
[

1 1. FEEDBACKORLUCK? mantle (McGovern & Schubert 1989).
v
There are two classes of explanations for Earth’s
The stochastic delivery of water combined with its
0
geologically stable surface character (Kasting & Holm
2 low density relative to rock leads to the generic expec-
1992): blind luck in water delivery (Raymond et al.
7 tation that many terrestrial planets should be entirely
2004) and ocean–mantle fluxes (McGovern & Schubert
0 coveredin water, so-called waterworlds (Raymond et al.
1989), or the existence of a stabilizing feedback
. 2004; Morbidelli et al. 2012).3 Massive terrestrial plan-
1 (Kasting & Holm 1992; Holm 1996; Abbot et al. 2012).
ets, “super-Earths,” are more likely to be waterworlds:
0 The partitioning of water is a key factor in regulat-
a planet’s mass scales faster than its surface area, so
4 ing planetary climate: surface water is the source of
bigger planets ought to have deeper oceans, while their
1 atmospheric water vapor, provides thermal inertia, and
v: i(nKcirteeaesetdals.u2r0fa0c9e; Agrbabvoitty&prSowduitczeesrs2h0a1l1lo)w. ocean basins helps transport heat. Moreover, clement surface con-
Xi Detrital zircons indicate that Earth has had both dtaitinioendsoovnerEgaeortlohgaicraelthiympeotbhyesaizseildicatotehwaevaethbeereinngmfeaeidn--
oceans and exposed continents for roughly 4.5 Gyrs
r back (Walker et al. 1981). The silicate weathering ther-
a (Wilde et al. 2001), while stratigraphy of sedimentary mostat requiresexposed continents because their chemi-
deposits suggests that the average height of continen-
cal weathering is strongly temperature-dependent. Crit-
tal surfaces above sea level —freeboard— has remained
ically, the traditional conception of the habitable zone
approximatelyconstantsinceatleast2.5Ga(Wise1974;
assumes a silicate weathering thermostat (Kasting et al.
Eriksson 1999). This is remarkable given the growth of
1993). Barring the existence of a seafloor weathering
continentalcrustovertime(Harrison2009),andthecon-
feedback,waterworldswillnothaveasilicateweathering
tinuous two-way exchange of water between ocean and
thermostat and should have a much narrower habitable
zone(Abbot et al.2012). Dryplanetscouldhaveawider
1CenterforInterdisciplinaryExplorationandResearchinAs- habitable zone in principle (Abe et al. 2011), but may
trophysics(CIERA),DepartmentofEarth&PlanetarySciences,
lack the plate tectonics (Mian & Tozer 1990) and phys-
Department of Physics & Astronomy, Northwestern University,
2145SheridanRoad,Evanston, IL,60208, USA ical erosion (West et al. 2005) necessary to maintain a
[email protected] silicate weathering thermostat.
2Department of the Geophysical Sciences, University of
Chicago, 5734SouthEllisAvenue, Chicago,IL,60637, USA
3 Matsui&Abe (1986) proposed that the mass of Earth’s hy- 1.1. Water Capacity of Earth’s Mantle
drosphere was set early-on by magma ocean buffering of a steam
The exact water inventory of Earth’s interior is cur-
atmosphere, independent of the water content of Earth’s building
blocks. Butthishypothesisdoesnotexplainthelong-termstability rently unknown, but is thought to be comparable to the
ofEarth’ssurfacewaterreservoir. surface reservoirs. The mantle contains water both in
2 Cowan & Abbot
hydrousandnominallyanhydrousminerals(Hirschmann
!"#$%&
2006; Hirschmann & Kohlstedt 2012). The Fe core may !$# 23-0$.!%&
also contain primordial water (Abe et al. 2000), which
,$*$)(&
we ignore in the present analysis. !#
%
Inoue et al. (2010) determined maximum water mass *+,-+".!%& !#
"
fractions of 0.7% (olivine, shallower than 410 km),
-#/$**1%/&
3.3%(wadsleyite,410–520km), 1.7%(ringwoodite,520– ! #
660 km), and 0.1% (perovskite, below 660 km). Com- & ’$%()#& /0$%1(#&
bining 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 cur- Fig.1.— A schematic of our hydrosphere model. Degassing of
rent surface reservoir. Measurements of electrical con- melt decreases with seafloor pressure, while hydration of ocean
crustincreases withseafloor pressure,providingapotential stabi-
ductivity(Dai & Karato2009)suggestthatEarth’sman-
lizingfeedback.
tlecontains1–2oceansworthofwateratpresent-day,but
otherestimateshavedifferedbyafactorofafewineither critical,whichoccursatnearlythepressuresandtemper-
direction (e.g., Huang et al. 2005; Smyth & Jacobsen atures in hydrothermal systems today (Kasting & Holm
2006; Khan & Shankland 2012). 1992). If Earth’s water started in the mantle and
gradually degassed to the surface, then this pressure-
1.2. The Deep Water Cycle dependence could explain why the oceans stabilized at
their current depths. A planet starting with very deep
There is a two-way flux of water between ocean and
oceans (Korenaga 2008), however, would not be able
mantleonEarth. Oceancrustformsatmid-oceanridges
to effectively subduct water and would remain a water-
bydepressurizationmeltingofmantle,releasingvolatiles
world.
into the ocean. Plate tectonics then drives the ocean
In order to solve the problem of regassing water
crusttowardsubductionzones. Hydrothermalalteration,
into the mantle, Holm (1996) suggested that the re-
principallyserpentinization,producesoceancrustthatis
duced efficiency of hydrothermal heat exchange beneath
5–10% water by mass. During subduction, much of the
deepoceanswouldresultinhighermantletemperatures,
wateris bakedoutofthe subducting slab,producing ex-
greaterplatevelocitiesandthereforeefficientsubduction
plosive volcanism like Mount St. Helens. Some of the
of hydrated crust. This mechanism may not work quan-
water, however,is subducted deep into the mantle, clos-
titatively, however, because hydrothermal heat transfer
ing the loop (for a review of the relevant geochemistry,
only accounts for 30–50% of heat flux through oceanic
see Arndt 2013).
crust(Stein et al.1995). Therearealsoqualitativeprob-
The current H O degassing at mid-ocean ridges is ap-
2 lems with this regassing argument: first of all, the same
proximately 2 × 1011 kg/yr (Hirschmann & Kohlstedt
logic should apply to planets with very shallow oceans,
2012), while the regassing flux at subduction zones
potentiallynegatingthesupercriticalcirculationhypoth-
is estimated to be 0.7–2.9×1012 kg/yr (Jarrard 2003;
esis. Moreover,hydratingthemantlelowersitsviscosity,
Schmidt & Poli 1998). Given the order-of-magnitude
furtherincreasingplatevelocitiesandmakingthismech-
agreement of these figures, and the large associated un-
anism a destabilizing feedback. Lastly, although rapid
certainties,Earth’s currentocean volume is typically as-
subduction may aid regassing, a hotter mantle almost
sumed to be in a steady state (McGovern & Schubert
certainlyhindersit(Bounama et al.2001). Inshort,itis
1989). Moreover, realistic parameterizations of mantle
not clearthat reducedheat transportthroughthe ocean
convection predict water fluxes of 1011–1013 kg/yr in crust leads to net regassing of water.
Earth’s geological past, implying ocean-cycling times of Whatever geophysical processes govern Earth’s deep
108 yrs (McGovern & Schubert 1989). A significant flux watercyclepresumablyoperateonotherterrestrialplan-
imbalance would have long ago desiccated or submerged ets with plate tectonics. The partitioning of water on
the planetary surface. Earth is not only an outstanding problem in geophysics
(Fyfe 1994; Langmuir & Broecker 2012), but one with
1.3. Previous Work importantimplicationsforthesurfaceconditionsandcli-
Aplanetentirelycoveredinwatermaydevelopexposed mate of terrestrial exoplanets.
continents by losing water to space, hydrating the crust
and mantle, or reshaping continents and ocean basins. 2. HYDROSPHEREMODEL
For example, an ocean-coveredplanet with a hot strato- As shown in Figure 1, we develop a two-box model
sphere could lose water to space until continents are ex- of the deep water cycle for a terrestrial planet with
posed. The resulting vigorous silicate weathering might plate tectonics: the two water reservoirs are the ocean
draw down sufficient CO to cool the planet out of the and mantle, while the basalt and granite are important
2
moistgreenhousestate,leavingapartiallywater-covered for the transport of water and for setting the depth
planet (Abbot et al. 2012). Detailed atmospheric simu- of ocean basins. A two-box model is appropriate for
lations of such hot planets, however, indicate that the grossestimatesofwaterpartitioningoverbillionsofyears
tendency of carbon dioxide to cool the stratosphere im- (McGovern & Schubert 1989; Ueta & Sasaki 2013), but
pedes the loss of water (Wordsworth & Pierrehumbert may be too simple to simulate small changes in ocean
2013). volume on shorter timescales (Parai & Mukhopadhyay
In an alternate hypothesis, the Rayleigh number of 2012).
convecting water in hydrothermal systems at mid-ocean Weassumethatthetotalhydrospheremass,W,iscon-
ridges exhibits a sharppeak when water becomes super- stant, i.e., we neglect water loss from the top of the at-
Super-Earths Need Not be Waterworlds 3
mosphere. Isotopic evidence indicates that Earth’s hy-
drospherewasatmost26%moremassiveintheArchean TABLE 1
Model Variables&Parameters
(Pope et al. 2012).
Name Symbol Value
2.1. Water Fluxes
The principle dependent variable in our model is the pnloarnmeatalirzyedwgartaevrimtyaassfractiona ωg˜ ωg˜⊕⊕==16.2×10−4
bulk water mass fraction of the mantle, x. We assume mantlewatermassfractiona x x⊕=5.8×10−4
plate tectonics but our results should be largely insensi- densityofwater ρw 1.0×103 kg/m3
tive to plate velocity: degassing and regassing are pro- densityofgranite ρg 2.9×103 kg/m3
portionalto the spreading and subduction rates, respec- densityofbasalt ρb 3.0×103 kg/m3
tively,whichareequalinsteady-state. Wethereforecon- densityofmantle ρm 3.3×103 kg/m3
sinidge,rxt′h≡e cdhxa/ndgτe,inwhmeraenttlheewnaotenr-dpiemresnesai-oflnoaolrtiomveer,tτu,rnis- tdhhyeipdctrkhanteoesdfsmcorfeulotsctewanaitcercrfurascttion ddxbmhelt 6600.×0×5101303mm
relatedtotheareaofoceaniccrust,A ,spreadingrate,S, subductionefficiency χ 0.23
b
anTdhteheneltencghtahngoefomfimd-aonctelaenwraidtegresp,eLr,sevai-aflτoo=rotvLeSrt/uArbn.- EEaarrtthhdhyegdarsastiinognedffiepctiehncy fddh⊕⊕ 03.×9 103 m
ing is: Emaarxt.htsheiackflnoeosrsporfescsounrte.crust Pdm⊕ax 740××101703Pg˜a−1 m
A g
x′ = b (w −w ), (1) oceanbasincoveringfraction fb 0.9
Mm ↓ ↑ omcaenatnlemmasasssfrfarcatcitoinonofEarth ωfmo 02..638×10−4
whereMm isthemantlemass,w↓ isthewatercontentof normalizedoceanbasinarea f˜b 1.3
subducted crust and sediments, and w↑ is the water de- seafloor pressuredependenceb φ 2
gassed by the formation of ocean crust at MORs (both Earthmantlewatercontentb x⊕ 5.8×10−4
have units of kg/m2). This equation is similar to that maxmantlewaterfractionb xmax 7×10−3
solved by previous researchers (McGovern & Schubert Note. —aThesearethevariablesinourmodel;welistheretheir
1989; Ueta & Sasaki 2013), except that we include the nominal values for Earth. bThese are critical parameters whose
sensitivitywetestin§4.1;welisttheirnominalvalueshere.
first-order effects of seafloor pressure, as described be-
low.
0.9 is the nominal degree of melt degassing on Earth
The subducted water is
today. Thepiecewisedefinitionensuresthatthedegassed
w =x ρ d (P)χ, (2) water does not exceed that present in the melt.
↓ h b h
where xh =0.05 is the mass fraction of water in the hy- 2.3. Regassing
drated crust (McGovern & Schubert 1989), ρ = 3.0×
b
103 kg/m3 is the density of basalt, d is the depth of The depth of serpentinizationmay also depend on the
h
pressure at the bottom of the ocean:
hydration in the crust (which depends on the seafloor
pressure, P), and χ = 0.23 is the fraction of volatiles σ
P
subducted deep into the mantle rather than outgassed dh(P)=min dh⊕ , db , (5)
P
viaarcandback-arcvolcanism(Ru¨pke et al.2004). The (cid:20) (cid:18) ⊕(cid:19) (cid:21)
subduction efficiency, χ, is highly uncertain but we per- where σ quantifies the pressure-dependence, d = 3×
h⊕
form a sensitivity analysis below to quantify how it and 103 m is the nominal value of the hydration depth on
other critical parameters likely affect our results (§4.1). Earth(McGovern & Schubert1989),andd =6×103 m
b
The degassed water is is the thickness of basaltic ocean crust. Hydration
of the lithospheric mantle below the oceanic crust is
w =xρ d f (P), (3)
↑ m melt d
functionally identical to the hydration of oceanic crust
where ρ =3.3×103 kg/m3 is the density of the upper (Ru¨pke et al.2004),butweconservativelyadoptthecon-
m
mantle, dmelt = 6 × 104 m is the depth of the MOR straint dh ≤db.
melting regime, and f is the fraction of water in the Ithasbeenarguedthatthedepthofhydrationdepends
d
melt that degasses rather than remaining in the ocean on the Rayleigh number of water in hydrothermal sys-
crustas it solidifies (model variables andparametersare tems(Kasting & Holm1992). If,astheyassumed,water
listed in Table 1). isgraduallydegassedfromthemantle,thisamountstoa
large positive σ; if water begins at the planetary surface
2.2. Degassing such a mechanism dictates a large negative σ.
The fraction of water in the melt that is degassed de-
2.4. Hypsometry & Isostacy
pends on hydrostatic pressure at the seafloor (Papale
1997; Kite et al. 2009). We parametrize this stabilizing We treat the planetary elevation distribution (“hyp-
feedback as: sometry”) as two δ-functions: one for ocean crust and
another for continental crust. This is a good approxi-
−µ
P mationofEarth’sstronglybimodalhypsometry(Rowley
f (P)=min f , 1 , (4)
d d⊕ P 2013), which is presumably typical of any terrestrial
" (cid:18) ⊕(cid:19) #
planet with plate tectonics (Stoddard & Jurdy 2012).
whereP isthepressureatthebottomoftheocean,P = Earth’s oceanic crust exhibits a clear age-depth rela-
⊕
4×107 Pais its currentvalue onEarth,µ>0 quantifies tion: oldercrustisdenserandsinksdeeperintotheman-
the pressure-dependence of melt degassing, and f = tle (Parsons & Sclater 1977). A larger ocean basin will
d⊕
4 Cowan & Abbot
therefore have a greater mean depth. The globally aver- on the ocean basin area, A =f A, where A is the plan-
b b
aged ocean depth, however, should remain roughly con- etary area. We adopt f = 0.9. In other words, 90% of
b
stant barring secular changes in spreading rates, which the planet is covered in water and the remaining 10% is
are beyond the scope of our zeroth-order treatement. exposedcontinent. Thisissufficientdrylandtomaintain
We assume that the modal continental height is level asilicateweatheringthermostat(Abbot et al.2012)and
with the ocean surface, which is natural because of the our results would not change dramatically if we instead
competing effects of erosion and deposition (Korenaga adopted f = 0.7 (as on modern Earth) or f = 1 (the
b b
2008; Rowley 2013). There is a maximum thickness limitingcaseofasingleinfinitesimalislandextendingout
that continents can achieve, however, beyond which a of the ocean).
continent flows under its own weight (Rey & Houseman Giventheseassumptions,themaximumvolumeofsur-
2006). The Himalayan plateau on Earth has a crustal facewaterthatcouldbeaccommodatedbyanEarth-size
thickness of 70 km and appears to be near this limit planet is 5.2×1018 m3, or a mass of 3.7M .
o
(England & McKenzie1982). Weconservativelyadopta
maximum continental thickness of 3. STEADY-STATESOLUTIONS
−1 Wefindsteadystatesolutionstothewaterpartitioning
g
dmax =70 km , (6) on a planet by setting the upward and downward water
g (cid:18)g⊕(cid:19) fluxes equal to each other,
where g is the surface gravity of the planet, g⊕ is that xρmdmeltfd(P)=xhρbdh(P)χ. (12)
of Earth, and we have adopted the gravity-dependence
of Kite et al. (2009). Using the mass-radius relation of The degassing efficiency, fd, and hydration depth, dh,
Valencia et al. (2007a), a gravity of 3g corresponds to dependonseafloorpressure,P,whichintimatelydepends
⊕
a 10M⊕ super-Earth. on ocean water depth, dw: P =gρwdw.
The thickness of granitic continents, d , is related to Oceandepthcanbeexpressedintermsofmantlewater
g
the thickness of the other layers by: content,
W −xM
m
d = , (13)
d =d +d +d , (7) w
g o b m A ρ
b w
where do is the depth of the ocean basin and dm is the but it is instructive to factor out the size-dependent
depth to whichthe continentrootextends into the man- terms and define intensive quantities:
tle (Figure 1). This parameterization assumes that the
tops of continents are at sea level (zero freeboard). This M ω−xfm
d = , (14)
is merely shorthandfor a freeboardthat is muchsmaller w A f ρ
(cid:18) (cid:19) b w
than the depth of ocean basins.
Isostaticbalancedictatesthatthe pressurebeloweach where ω = W/M is the planetary water mass fraction
vertical column must be equal: and fm = Mm/M = 0.68 is the planetary mantle mass
fraction. Note that the term in parentheses is propor-
ρwdo+ρbdb+ρmdm =ρgdg, (8) tional to gravity: M/A∝M/Rp2 ∝g. We may therefore
write the normalized ocean depth as
wheretheρ =1.0×103kg/m3andρ =2.9×103kg/m3
w g
are the density of water and granite, respectively. d ω−xf
w m
=g˜ , (15)
ofCisoomstbaitnicinbga(la7n)caen:d (8) yields the following expression dw⊕ ωof˜b
where d = 4 km is the average depth of Earth’s
d (ρ −ρ )+d (ρ −ρ )=d (ρ −ρ ). (9) w⊕
o m w b m b g m g oceans, g˜ = g/g is the normalized planetary gravity,
⊕
Given the maximal crustal thickness, dmgax, one can ωo = Mo/M⊕ = 2.3 × 10−4 is the fractional mass of
derive the maximum depth of water-filled ocean basins: Earth’ssurfacewater,andf˜b =fb/fb⊕ =1.3istheocean
basin covering fraction of the planet divided by that of
dmax(ρ −ρ )−d (ρ −ρ )
dmax = g m g b m b . (10) Earth.
o ρ −ρ The normalized seafloor pressure is therefore
m w
For our fiducial Earth-size parameters, dmoax = 11.4 km. P =g˜2 ω−xfm. (16)
Note that the thickness of oceanic crust is likely also P⊕ ωof˜b
inversely proportional to g (Sleep 2012). We conserva-
tively adopt d =6 km regardless of planet mass, which We substitute (16) into (12) and solve for the steady-
b
produces shallower ocean basins for massive planets. statemantlewaterfractionontheintervalx∈[0,ω/fm],
Thedensitiesofgraniteandbasaltarenearlythesame where the upper-limit ensures that the mantle does not
(ρ ≈ρ )andthemaximumcrustalthicknessfarexceeds contain more water than the planet as a whole. There
g b
the thickness of oceanic crust (d ≪ dmax), so we can is also a petrologicalupper limit to how much water the
b g
approximate (10) as mantle can sequester, however. For Earth, that limit
appears to be x = 7×10−3 (a dozen oceans, §1.1).
max
dmoax ≈11.4 km gg −1. (11) InGciavseens twhheesrteeaxd>y-xstmaatxe,mwaensteletwxa=texrmcoaxn.tent,itistriv-
(cid:18) ⊕(cid:19) ial to compute the depth of surface oceans using (15).
The maximum ocean volume that can be accommo- The waterworldboundary is defined by equating surface
datedwhilemaintainingexposedcontinentsalsodepends waterdepth with maximaloceanbasindepth, (10). The
Super-Earths Need Not be Waterworlds 5
solidblackline inFigure2showsthe waterworldbound- In the presence of a modest stabilizing pressure-
aryforourfiducialparametersandpressuredependencies dependence (φ=1), the steady-state mantle water frac-
of µ=σ =1. tion is
It is useful to compare this waterworld boundary to ωx⊕g˜2
x= . (21)
the null hypthesis: ignoring isostatic adjustment and ω f˜ +x f g˜2
o b ⊕ m
the deep water cycle. In such a case, (15) simplifies
to d = d g˜ω/ω , which can be combined with the Inthehigh-gravitylimit,themantlecontainstheentirety
w w⊕ o
approximation (11) to obtain the waterworld limit of of the planet’s water, x=ω/f . In practice this cannot
m
ω =ω g˜−2. The null-hypothesisisindicatedbythe solid occur for water-rich planets because of the finite water
o
redline in Figure 2. By definition, Earth’ssurface reser- capacity of the mantle (x ≤ xmax). Nonetheless, the
voir(indicated by the ⊕) puts it rightatthe waterworld highergravityofsuper-Earthsbiasesthedeepwatercycle
boundary under these assumptions. in favor of mantle sequestration. Water inventory and
If one accounts for the ability of isostacy and erosion mantlecapacityarebothproportionaltoplanetarymass,
to reshape continents and keep their heads above water, producing a weak mass-dependence to the waterworld
but still neglects the deep-water cycle, one obtains the boundary.
dashed red line in Figure 2.
4. DISCUSSION
3.1. Analytic Approximation 4.1. Sensitivity Analysis
It is possible to obtain an analytic solution to (12) As notedin§1.1, the amountofwaterinEarth’sman-
if we ignore the piecewise nature of degassing, f , and tle is poorly constrained. The gray broken lines in Fig-
d
regassing, d . This analytic approximation has intuitive ure 2 show the analytic waterworld boundary if Earth’s
h
valuesowedevelopithere. Inthiscasethew↑ =w↓ can mantle water content, x⊕, is 10× greater (dashed) and
be written as: 10×smaller (dotted) than our nominal value. Note that
varyingx inthismodelismathematicallyequivalentto
⊕
−µ σ
P P varyinganyofthewaterfluxparametersin(19),manyof
xρ d f =x ρ χd , (17)
m melt d⊕ P h b h⊕ P which are uncertain (e.g., the subduction efficiency, χ).
(cid:18) ⊕(cid:19) (cid:18) ⊕(cid:19) Varying x by two orders of magnitude affects the wa-
⊕
which can be compactly expressed as terworld boundary for a 10M⊕ super-Earth by roughly
oneorderofmagnitude;thisisthedominantuncertainty
x P φ in our study.
= , (18) The maximum water capacity of the mantle, x is
x P max
⊕ (cid:18) ⊕(cid:19) not well known for high-mass terrestrial planets. The
where φ=µ+σ is the sum of pressure dependencies for waterstoragein Earth’smantle is thoughttobe concen-
mid-ocean ridge melt degassing and serpentinization of tratedinthetransitionzone(410–660kmdepth). Ifmas-
oceanic crust, and Earth’s bulk mantle water content is siveplanetscanonly sequesterwaterinathin transition
zone, then mantle water capacity scales with planetary
x = xhχρbdh⊕ . (19) arearatherthan mass,or xmax ∝g˜−1. The dotted black
⊕ ρ d f line in Figure 2 shows the minuscule effect of adopting
m melt d⊕
this scaling.
Our fiducial parameter values yield x =5.8×10−4, or On the other hand, the mantle of a super-Earth
⊕
roughly an ocean’s worth of water in Earth’s mantle. should be primarily in the form of post-perovskite
Substituting (16) into (18), we obtain (Valencia et al. 2007b), which may not have the same
water capacity as Earth’s dominant mantle rock, per-
x = g˜2 ω−xfm φ. (20) toevrswkoitrel.dWboeutnrdyasreyttiisngunxcmhaaxng=ed1iafnpdofistn-dpetrhoavtsktihtee wcaan-
x⊕ (cid:18) ωof˜b (cid:19) hold unlimited water.
Finally,theblue linesinFigure2showthe waterworld
Thesteady-statemantlewatercontentallowsusto es-
boundary for pressure-dependences of φ = 3 (dashed)
timatetheoceandepth,via(15),whichwethencompare
and φ = 1 (dotted). (In both cases we use µ = σ =
to the maximum ocean basin depth. The solid gray line
φ/2 in the numerical model.) The precise strength of
inFigure2showsthewaterworldboundaryforournom-
the seafloorpressure dependence ofthe deep watercycle
inal parameter values and φ = 2. The small difference
affects the waterworld boundary by less than a factor of
between the numerical and analytic waterworld bound-
two.
aries (solid black and gray lines, respectively) indicates
that the piecewise definitions off (P) andd (P) do not
d h 4.2. Plate Tectonics
critically affect our results.
If φ < 0, the pressure feedback is destabilizing and Our model of the deep water cycle assumes plate
there are two physical steady-state solutions: shallow tectonics, but it is currently unknown whether super-
and deep oceans, respectively. The current state of a Earthsaretectonicallyactive. Theincreasedheatfluxof
planetwilldependoninitialconditions. Forφ≥0,there super-Earthsshouldproducevigorousmantleconvection
is a single physical root to (20) and therefore a single (Valencia et al. 2007a; van Heck & Tackley 2011), but
steady-state solution. If there is no net pressure depen- the increased strength and buoyancy of crust on super-
dence to the deep water cycle (φ = 0), then all planets Earthsmay prohibit plate tectonics (O’Neill & Lenardic
have the same mantle water content as Earth, x≡x . 2007; Kite et al. 2009). Mantle convection may even ex-
⊕
6 Cowan & Abbot
Fig. 2.— Waterworld boundary as a function of water mass fraction, ω, and normalized surface gravity, g˜ = g/g⊕: planets in the
upper-rightcorner are waterworlds, while those inthe lower-leftmaintain exposed continents. The solidblack lineshows the waterworld
boundaryforournominalparameters,includingthenegativefeedbacks associatedwithseafloorpressure. Thedottedblacklineshowsthe
waterworldboundaryifoneassumesthatwater canonlybestoredinamantle’stransitionzone. Thesolidredlineshowsthewaterworld
boundaryifoneappliestheg−1dependencetoEarth’scurrenthypsometryandpresumesthatallofaplanet’swaterresidesatthesurface.
The dashed red line accounts for the effects of erosion and isostatic adjustment (§2.4) but not the deep-water cycle. The black symbol,
⊕, denotes Earth if one only considers its surface water reservoir; the arrow indicates Earth’s probable location if one also accounts for
water present in the mantle. The gray lines indicate the analytic waterworld boundary for nominal parameters (solid), as well as Earth
mantle water content values of x⊕ = 5.8×10−3(dashed) and x⊕ = 5.8×10−5 (dotted). The blue lines show the waterworld boundary
forseafloor pressure-dependencies ofφ=3(dashed) and φ=1(dotted). A terrestrialplanet withsurfacegravityg˜=3corresponds toa
10M⊕ super-Earth(Valenciaetal.2007b).
hibit hysteresis, such that planets with identical bound- ers (differences in crustal thickness scale as g−1 for the
ary conditions may or may not have plate tectonics, de- same reason as ocean basin depth).
pendingoninitialconditions(Lenardic & Crowley2012).
Alternatively,ithasbeensuggestedthatsurfacewater 4.3. Homogeneity of the Mantle
ismoreimportantthanplanetarymassforplatetectonics
The homogeneity of volatiles in Earth’s lower man-
(Mian & Tozer 1990; Korenaga 2010). In a classic case
tle is questionable. The high 3He abundance of ocean
ofchicken-and-egg,van der Lee et al.(2008)arguethata
islands has been attributed to a poorly-mixed lower
deepwatercycleisanecessary,ifnotsufficient,condition
mantle (Kurz et al. 1982). By extension, this hypoth-
for long-term plate tectonics.
esis implies that Earth’s lower mantle may hold much
Continental crust formation is thought to be an in-
more water than what is inferred for the upper mantle.
evitable by-product of plate tectonics in the presence of
Gonnermann & Mukhopadhyay (2009) argue, however,
water (Rudnick 1995; Arndt 2013), so super-Earths are
that the 3He abundance of the mantle is consistent with
likelytohavelargevolumesofgraniticcrust. Infact,the
homogeneous composition.
large volume of continental crust combined with smaller
If the mantle is not well-mixed, then x represents the
maximal crustal thickness may lead to a planet entirely
waterfractionofthoseregionssampledbythemid-ocean
coveredincontinentalcrust. Thisdoesnotgreatlyaffect
ridgemeltingandaffectedbysubductionofoceaniccrust.
ourresults,providedthattheplanetremainstectonically
Indeed, the source of mid-ocean ridge basalts (MORB)
activeandthatsomeregionshavethickercrustthanoth-
appears to have maintained a constant water mass frac-
Super-Earths Need Not be Waterworlds 7
tion,x,forbillionsofyears,suggestingthatsubductionof Wehaveargued,ashaveothers(Kasting & Holm1992;
hydratedoceaniccrustisdepositingwaterintheMORB Holm 1996), that the approximately steady-state wa-
source region (Hirschmann 2006). ter partitioning on Earth over geological time suggests
a seafloorpressurefeedback that regulatesthe degassing
atmid-oceanridgesand/ortheserpentinizationandsub-
4.4. Observational Constraints
sequentsubductionofoceaniccrust. Althoughoceanvol-
Our model of the deep water cycle predicts that many ume may change throughout a planet’s history because
super-Earths have exposed continents. It will eventu- ofsecularcooling,wehavetackledthezeroth-orderprob-
ally be possible to test this hypothesis by observation- lem of steady-state solutions.
ally determining the surface characterof a largenumber Notably,seafloorpressureis proportionalto a planet’s
of high-mass terrestrial planets (see also discussion in surface gravity. The enhanced gravity of super-Earths
Abbot et al. 2012). produces shallower ocean basins, but also leads to shal-
Disk-integrated rotational multiband photometry of lower oceans. The solid black line in Figure 2 shows the
Earth, essentially the changing colors of a pale blue waterworld boundary if one accounts for the pressure-
dot, encode information about continents, oceans and dependence of the deep water cycle.
clouds (Ford et al. 2001). Such “single-pixel” obser- The effects of isostacy, erosion, and deposition, com-
vations have been used to construct coarse longitudi- bined with a pressure-dependentdeepwater cycle, make
nal color maps of Earth and Mars (Cowan et al. 2009, super-Earths 80× less susceptible to inundation than
2011; Fujii et al. 2011; Hasinoff et al. 2011), while simu- they otherwise would be. Our model predicts that
lationssuggestthatphotometryspanninganentireplan- tectonically active 10M planets can maintain large
⊕
etary orbit could be used to construct a rough 2D color exposed continents for water mass fractions less than
map (Kawahara & Fujii 2010, 2011; Fujii & Kawahara 2×10−3.
2012). Finally,Cowan & Strait(2013)showedthatdisk- Exoplanets with sufficiently high water content will
integrated multiband photometry of a variegated planet be water-covered regardless of the mechanism discussed
can be inverted to obtain reflectance spectra of its dom- here, but such “ocean planets” may betray themselves
inantsurface types, even if the number and colorsof the by their lower density: a planet with 10% water mass
surfaces are not known a priori. fraction will exhibit a transit depth 10% greater than
The bottom line is that a 5–10 m space telescope an equally-massive planet with Earth-like composition
equippedwithacoronagraphorstarshadecouldproduce (Sotin et al. 2007). Planets with 1% water mass frac-
a coarsesurface map of an Earth-analogat a distance of tion,however,arealmostcertainlywaterworldsbut may
10pc(Cowan et al.2009;Fujii & Kawahara2012). Such have a bulk density indistinguishable from truly Earth-
low-resolution maps would be sufficient to identify the like planets. Given that simulations of water delivery
continents one expects on a tectonicaly active planet. If to habitable zone terrestrial planets predict water mass
most super-Earths exhibit the bimodal surface charac- fractions of 10−5–10−2 (Raymond et al. 2004), we con-
ter of Earth, it will suggest that they experience plate clude that most tectonically active planets —regardless
tectonics and a deep water cycle. If, instead, large ter- ofmass—willhavebothoceansandexposedcontinents,
restrial planets were all determined to be waterworlds, enabling a silicate weathering thermostat.
it would indicate that our hypothesis is wrong.
NBC acknowledges many insightful discussions with
5. CONCLUSIONS J.P. Townsend, S.D. Jacobsen, and C.R. Bina. The
It has long been suggested that super-Earths ought authors also had useful conversations with C. Andron-
to be waterworlds (Stapledon 1937; Kite et al. 2009; icos, H. Gilbert, R. Jeanloz, M. Manga, D. McKen-
Abbot & Switzer 2011). If one accounts for the first- zie, V.S. Meadows, D.B. Rowley, J. Rudge, S. Stein,
order effects of gravity on ocean basin depth and water D.J. Stevenson, and R. Wordsworth. E.S. Kite provided
inventory, then a 10M planet is not expected to have critical feedback on an early version of the manuscript.
⊕
exposed continents unless it has a water mass fraction TheauthorsthankN.H.Sleepforsharinganunpublished
less than 3×10−5, roughly ten times drier than Earth manuscript. DSA was supported by an Alfred P. Sloan
(solid red line in Figure 2). researchfellowship.
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