# Land Fraction Diversity on Earth-like Planets and Implications for their
Habitability

###### Abstract

A balanced ratio of ocean to land is believed to be essential for an Earth-
like biosphere and one may conjecture that plate-tectonics planets should be
similar in geological properties. After all, the volume of continental crust
evolves towards an equilibrium between production and erosion. If the interior
thermal states of Earth-sized exoplanets are similar to the Earth’s – a
straightforward assumption due to the temperature-dependence of mantle
viscosity – one might expect a similar equilibrium between continental
production and erosion to establish and, hence, a similar land fraction. We
will show that this conjecture is not likely to be true. Positive feedback
associated with the coupled mantle water – continental crust cycle may rather
lead to a manifold of three possible planets, depending on their early
history: a land planet, an ocean planet and a balanced Earth-like planet. In
addition, thermal blanketing of the interior by the continents enhances the
sensitivity of continental growth to its history and, eventually, to initial
conditions. Much of the blanketing effect is however compensated by mantle
depletion in radioactive elements. A model of the long-term carbonate-silicate
cycle shows the land and the ocean planet to differ by about 5 K in average
surface temperature. A larger continental surface fraction results both in
higher weathering rates and enhanced outgassing, partly compensating each
other. Still, the land planet is expected to have a substantially dryer,
colder and harsher climate possibly with extended cold deserts in comparison
with the ocean planet and with the present-day Earth. Using a model of
balancing water availability and nutrients from continental crust weathering,
we find the bioproductivity and the biomass of both the land and ocean planet
to be reduced by a third to half of Earth’s. The biosphere on these planets
might not be substantial enough to produce a supply of free oxygen.

Astrobiology

Potsdam-Institute for Climate Impact Research, Potsdam, Germany International
Space Science Institute, Bern, Switzerland Corresponding Author
<EMAIL_ADDRESS>
##  1 Introduction

In the absence of observable unique biomarkers, the search for life beyond the
solar system focuses on the search for habitable planets (e.g., Segura
Kaltenegger, 2010). The concept of the circumstellar habitable zone (CHZ)
around a host star has long been established (Hart, 1979; Huggett, 1995) and
developed further (see e.g., Cockell . 2016 and Ramirez 2018 for reviews). The
CHZ is defined as the zone of orbits in which a planet would have surface
temperatures that allow the presence of liquid water, thus recognizing the
importance of water for life as we know it (e.g., Brack ., 2010). The
boundaries of the habitable zone depend on the luminosity of the star but also
on the atmosphere pressure at the surface of the planet and on atmospheric
chemistry, in particular, on the concentration of the atmosphere greenhouse
gases carbon-dioxide and methane (e.g., Lammer ., 2009). More recent
discussions of planetary habitability go beyond stellar luminosity and
atmosphere chemistry to include other planetary properties such as the ratio
between land and water covered surface areas, diversity of geologic provinces,
magnetic fields, and processes such as plate tectonics that drive the
planetary engine and its evolution (e.g., Heller Armstrong, 2014; Schulze-
Makuch ., 2020; Glaser ., 2020; Lingam Loeb, 2018, 2019, 2021).

Having both emerged continents and oceans on Earth is an outcome of plate
tectonics. About 40% of its surface is covered with continental crust of which
87% is emerged; 13% of the area is continental shelf area and covered by
water. Emerged land provides easy access to solar energy and weathering of
continental crust rock provides essential nutrients for the biosphere. As a
result, most of the biomass is found on continents and their shelves while the
deep ocean biomass is much lower (e.g., Behrenfeld Falkowski, 1997; Gray,
1997; Kallmeyer ., 2012), even though the net primary production NPP from
oceans is similar to that from land (Field ., 1998).

The importance of a suitable balance between land and ocean covered surface
areas for life as-we-know-it has recently been emphasized by Lingam Loeb
(2019, 2021) and Glaser . (2020). The required balance is a consequence of the
need for water on the one hand and nutrients such as phosphorous and nitrogen
on the other. On a land dominated planet with minimal water, the biological
productivity should be low due to water limitations. Ocean-dominated planets
could be stymied by the lack of nutrients. If phosphorous is the productivity
limiting nutrient (e.g., Kamerlin ., 2013) then the increased weatherability
of subaerial continental rock by slightly acidic rain-water gives land
surfaces an edge over the ocean floors. The build up of free oxygen in the
atmosphere for more advanced life is likewise favoured by continental crust
weathering that produces the sediments in which organic carbon is buried and
removed from oxidization. Lingam Loeb (2019, 2021) discuss a simple
parameterization of the NPP and the biomass as functions of the land-ocean
surface ratio. They find both functions to be maximized at the present-day
Earth ratio and to decrease by an order of magnitude as one goes to land
dominated (1-2$\sigma$) or ocean dominated worlds. We note that the Earth-like
land-ocean ratio would tend to maximise the extent of continental shelf areas
where life will profit from solar insolation and the riverine entry of
dissolved nutrients. Lingam Loeb (2019, 2021) also find that oxygen can build
up in worlds that have a water coverage between roughly 25 and 85%. Thus,
planets with a balance of landmasses and oceans might be potentially conducive
to hosting rich biospheres. In addition to the land-ocean distribution, the
surface temperature will matter for the bioproductivity as has been emphasised
recently by Heller Armstrong (2014), Lingam Loeb (2018) and Schulze-Makuch .
(2020).

It is well established that the concentration of carbon-dioxide in the Earth’s
atmosphere is buffered by the long-term carbonate-silicate cycle (e.g.,
Kasting ., 1993), thereby stabilising the atmosphere temperature and clement
conditions. Negative feedback in the cycle causes the atmospheric CO2 to
attain an equilibrium between carbon outgassing and recycling (e.g., Lehmer .,
2020; Isson ., 2020). Plate tectonics is a central element of the modern
cycle, exchanging CO2 between the atmosphere and the mantle reservoirs through
volcanism and subduction and the rate of outgassing is a function of the
interior properties of the planet. The surface temperature is buffered as the
rate of removal of CO2 from the atmosphere through rain and weathering of
continental crust – the latter a product of plate tectonics – increases with
increasing temperature (e.g., Walker ., 1981; Kasting ., 1993; Foley, 2015).
Weathering of continental crust results in carbon fixation and its rate
depends on the continental surface area and the topography. Water is the main
agent for transporting CO2 from the atmosphere to the surface and for
transporting weathering products including nutrients from the continents to
the oceans and eventually to the subduction zones, where CO2 is then returned
to the mantle. Like CO2, water is cycled by plate tectonics through the mantle
and mantle outgassing may have been a major source of the present ocean mass
(e.g., Elkins-Tanton, 2011). The oceans provide the single most important
thermal reservoir for buffering the surface temperature (e.g., Bindoff .,
2007; Cheng ., 2019; von Schuckmann ., 2020) as well as they provide a sink
for atmospheric CO2 on short timescales (e.g., Pierrehumbert, 2010).

The operation of plate tectonics, like on present-day Earth, is vital to the
long-term carbon cycle outlined above. Venus may have had surface recycling
(e.g., Crameri, 2017) but its hypsometric curve is unimodal (e.g., Rosenblatt
., 1994). While there are units resembling terrestrial rifts and ridges such
as Beta Regio, there is no clear indication of subduction zones and
continental crust, although Ishtar Terra as a high elevation physiographic
province resembles a terrestrial continent (Ivanov Head, 2011). Rather, it
seems more likely that Venus has a continuous lithosphere. While Venus may be
debated (Davaille ., 2017), the smaller planets Mars and Mercury are widely
agreed to have continuous outer shells termed stagnant-lid lithospheres.
Planetary mantle convection modelling finds stagnant lids to form naturally as
a consequence of a temperature dependent rheology of mantle rock (e.g., Stein
Hansen, 2008) while subduction and plate-like behaviour require additional
strain related mechanisms for which water may be important (e.g., Bercovici,
2003).

Uncertainties in estimates of the interior properties of exoplanets have
significant effects on assessments of their habitability (e.g., Seales
Lenardic, 2020). For example, uncertainties in the concentrations of heat
producing elements will naturally affect estimates of the outgassing rates and
the evolution of the surface temperature (e.g., O’Neill ., 2020; Oosterloo .,
2021; Unterborn ., 2022). It has even been suggested that there may be
feedback between the surface temperature and the tectonic regime (plate-
tectonics versus stagnant lid) that could result in irreversible bifurcation
between the two modes (Weller ., 2015; Weller Lenardic, 2018). Another key
uncertainty in exoplanet evolution is the initial mantle temperature.
Theoretical considerations and simple thermal history models suggest that
differences in the initial mantle temperature should become insignificant
after $\sim$1 Gyr of evolution due to the temperature-dependence of mantle
viscosity (e.g., Tozer, 1967), at least for planets less massive or as massive
as Earth (Kruijver ., 2021). Therefore, one might conclude that the climate at
4.5 Gyr should be largely independent of the initial mantle temperature. This
conjecture, however, neglects feedback associated with the growth of
continental crust, which we will explore in this paper.

Even for Earth-sized planets with plate tectonics, it is not a given that the
surface would have a balanced ocean to land coverage similar to Earth. Höning
Spohn (2016) and Höning . (2019a) have argued that – depending on initial
conditions – the evolution of a plate tectonics Earth-like planet would rather
result in a planet mostly covered by continents or mostly covered by oceans
(Fig. 1). In their analysis, the continental crust and the water cycles are
feedback cycles with either one or three fixed points depending on whether
negative feedback or positive feedback dominates, respectively. Of the three
fixed points, two would be stable (pertaining to the continent and the ocean
planet) and one conditionally stable (the Earth). These differences in the
outcome of planetary evolution are likely to be mirrored in the climate and
possibly the biomass. The present climate of such a planet would retain a
memory of the planet’s very early evolution, since the equilibrium between CO2
outgassing and silicate weathering would depend on the fraction of emerged
continents and on the thermal state of the planet’s interior. Their rates of
change in turn would depend on the properties of the coupled continent-water
system.

Figure 1: Graphical illustration of the Land Planet, Earth-like land/continent
mix, and Ocean Planet. Credit: Europlanet 2024 RI/T. Roger

The dependence of the strengths of positive and negative feedback on details
of continental growth mechanisms complicates an extensive analysis of
evolution pathways of (exo)planets (Höning Spohn, 2016). Therefore, in the
present paper we will parameterise feedback using four dimensionless
quantities. We will show evolution pathways that depend on the assumed values
of these quantities and then comprehensively discuss reasonable parameter
choices and evolution scenarios. We extend our earlier models by analysing the
effects of insulating continents delaying mantle cooling and the opposing
effect of a depletion of mantle radioactive isotopes with continental growth.
Furthermore, we calculate average surface temperatures for the model planets
using a model of the carbonate-silicate cycle and estimate the bioproductivity
and -mass following Lingam Loeb (2021). We will present our models in a 3D
phase space spanned by continental coverage, mantle water concentration and
mantle temperature as variables.

In the following section, we begin by discussing plate tectonics from the
point of view of complex non-linear system theory. We will then discuss fixed
points of the system and their stability depending on dimensionless parameters
that describe the strengths of positive and negative feedback. Coupling the
model to a thermal evolution model of the mantle, we will show how fixed
points evolve with time and steer the evolution of the planet and its
habitability (Section 3). This section is followed by a discussion and
conclusions.

##  2 Mantle thermo-chemical evolution and continental growth as a nonlinear
system

Plate tectonics and its role in geology has been described in a large number
of publications (e.g., Sleep, 2015). In a nutshell, the Earth’s surface is
covered by seven major plates that move horizontally across the surface driven
by solid-state convection in the mantle. The plates consist of oceanic crust
and lithosphere and continental crust and lithosphere. Oceanic crust is
produced by pressure release partial melting of mantle rock underneath mid-
oceanic ridges and is returned to the mantle in subduction zones. The
subduction zones are the places where modern plate tectonics produces
continental crust by partially melting oceanic crust, subducted sediments, and
mantle rock in the presence of subducted water reducing their solidus
temperature (e.g., Campbell Taylor, 1983).

Continents, despite their longevity in comparison with oceanic crust, are
continuously weathered and eroded and the resulting sediments are transported
to subduction zones by surface water flow. In addition, continental crust is
eroded through friction between the continental keel and the subducting slab.
The rates of subduction and crust production are closely linked to the rate of
mantle flow, which is governed by the rheology of mantle rock. On the scale of
mantle flow, the rheology can be parameterised by an effective viscosity. The
latter depends non-linearly on mantle temperature and on the concentration of
water and – to a lesser extent – of carbon-dioxide in mantle rock. In
addition, the mantle viscosity is a function of pressure, which should be
considered for planets more massive than 2-3 Earth masses (e.g., Kruijver .,
2021). Since we restrict our analysis to Earth-mass planets, we neglect this
dependence here.

Sediments play important roles in subduction zone processes and in this paper,
they play a decisive role: The erosion rate of the continents is proportional
to their area and thus provides negative feedback limiting their growth. As
carriers of water, however, sediments contribute to water transport (Hacker
2008; Deschamps et al. 2013) in subduction zones and provide positive feedback
increasing the continental growth rate. Moreover, Sobolev Brown (2019) argue
that sediments are essential for keeping plate tectonics running by
lubricating the subducting plate.

While mid-oceanic ridges are the most prominent source regions of mantle water
entering the oceans and the atmosphere, subduction zones are the major sink
regions for surface water. Today, about $1.8\cdot 10^{12}$ kg of water enter
subduction zones each year mainly in hydrous minerals in sediments and
hydrated basaltic crust (Jarrard, 2003). Significant, but difficult to
constrain (e.g., Höning ., 2014) quantities of the subducted water are
expelled at shallow depth, the remaining water – partly stored in
serpentinites and other hydrous minerals – reaches the source region of
continental crust rock where hydrous minerals break down and release water
that is consumed upon melting. The remainder of the water is subducted deeper
into the mantle where it is eventually incorporated into nominally anhydrous
minerals (Bell Rossman, 1992). As a consequence, mantle viscosity is reduced
(Hirth Kohlstedt, 2003) and the rate of mantle convection flow increased. The
water is transported in the convection flow towards mid-ocean ridges where
part of it is outgassed into the atmosphere and oceans.

Feedback associated with the growth of continental crust can also originate
from its thermal properties. The continental heat flow per unit area of about
70 mW/m2 is known to be significantly smaller than the oceanic heat flow of
about 105 mW/m2 (e.g., Jaupart ., 2015). Moreover, the basal heat flow per
unit area from the mantle into stable continental crust is only about 15
mW/m2. These observations suggest that the continents act as thermal blankets
of the mantle reducing the heat flow and the mantle cooling rate and diverting
heat flow from sub-continental to sub-oceanic areas. The growth of continents
diverting heat flow to oceanic areas can cause immediate positive feedback
(Höning ., 2019a). But even a reduction of the overall cooling rate introduces
a sensitivity of the continental growth rate to its past history.

An opposing effect to thermal blanketing of continental crust is the
redistribution of heat sources during crust growth, which may provide negative
feedback. Radiogenic elements, because of their large ionic radii and their
valence states, are enriched in melts producing continental crust. The
concentration of heat sources in present-day continental crust is debated,
however. Estimates are based on measured heat flow values and concentrations
in mantle and crustal rock (e.g., Jaupart ., 2016). Uncertainties are due to
uncertainties in the ratio between surface heat flow and mantle radiogenic
heat production (Urey ratio) and the density of heat sources in the lower
mantle. But it is estimated that the modern continental crust contains between
25 and 40% of the total planetary inventory of heat sources (Dye, 2012;
Jaupart ., 2015, 2016). The transfer of heat sources from the mantle to the
growing continental crust effectively cools the mantle (e.g., Spohn, 1991) and
provides negative feedback by slowing down mantle convection and thus the
production rate of new continental crust.

The above description of plate tectonics suggests that the continental crust
volume, the mantle water concentration and mantle temperature are linked
through non-linear feedback cycles as illustrated in Figure 2. An increase in
mantle water concentration (blue) leads to a reduction of the mantle viscosity
and thereby to an increase in the convection speed. The increased subduction
rate will increase the mantle water concentration establishing a positive
feedback. The latter may be balanced by negative feedback associated with
mantle water outgassing. The water cycle is linked to the continental crust
cycle (green) through the water subduction rate. An increase in continental
coverage increases the weathering and erosion rates and – assuming that the
increased sedimentary volume will be subducted and contribute to water
subduction – to increased rates of mantle water regassing and continental
crust production. This feedback may be balanced by continental loss through
erosion. The thermal evolution of the mantle (red) is linked to the mantle
water cycle via the mantle viscosity and to the continental cycle via thermal
blanketing and mantle depletion of radiogenic elements. The role of sediments
in water subduction and the insulating effect of continents determine the
strength of positive feedback in the coupled system.

Figure 2: Graphical illustration of the feedback cycles linking continental
coverage (green, compare Eq. 1), mantle water concentration (blue, compare Eq.
2), and mantle temperature (red, compare Eq. 3). The evolution of the former
two is coupled through the subduction of water. Positive feedback – indicated
by ’+’ signs – results from the subduction of water reducing the mantle
viscosity and increasing the plate speed and the crust production rate.
Increasing the continental coverage results in an increased production rate of
sediments feeding additional water and rock into the melt zone from which
continental crust originates. Water outgassing, surface erosion and cooling
dampen the positive feedback and are indicated by ’-’ signs. Continental crust
growth feeds back further on the mantle temperature through thermal blanketing
and transfer of radiogenic elements from the mantle to the crust. For further
explanation see text.

The mantle temperature is subject to the negative feedback provided by the
temperature dependence of the mantle viscosity and known as the Tozer
principle (Tozer, 1967). Accordingly, an increase in mantle temperature will
decrease the viscosity which will result in an increased convection and mantle
cooling rate – all else being equal. There is no positive feedback mechanism
that could lead to runaway heating as long as the mantle heat source density
decreases with time but continental crust growth and the mantle water budget
can feedback onto the evolution of the mantle temperature. An increased
degassing of the mantle will increase the mantle viscosity and result in
reduced mantle cooling – with effects on crust growth and the water budget.
Likewise, an increase in continental surface area can reduce the surface heat
flow and the cooling rate – with possible readjustments of the convection
system that may lead to adjustments of the plate speed. But only in an extreme
case of thermal blanketing can continental growth result in temporary mantle
heating, as we will show further below.

Mathematically, the non-linear feedback system can be described by the
following set of differential equations. In writing Eqs. (1) and (2) we follow
Höning Spohn (2016) but we simplify their equations while keeping the most
important feedback terms. We complement these equations by an energy balance
equation that includes the effects of continents (Eq. 3).

$\frac{\dot{A}}{\dot{A}_{l,E}}=\underbrace{\left[\xi_{1}A^{*}+(1-\xi_{1})\,v_{p}^{*}\right]\,L_{oo,oc}^{*}f_{r}}_{\text{continental
production}}-\underbrace{\left[\xi_{2}A^{*}+(1-\xi_{2})\,L_{oc}^{*}\right]\,}_{\text{erosion}},$
(1)
$\frac{\dot{w}}{\dot{w}_{l,E}}=\underbrace{\left[\xi_{1}A^{*}+(1-\xi_{1})v_{p}^{*}\right]L_{oo,oc}^{*}f_{r}}_{\text{regassing}}-\underbrace{w^{*}v_{p}^{*}}_{\text{outgassing}},$
(2) $\begin{split}\dot{T}\,\rho_{m}c_{p}V_{m}&=-q_{m,E}\underbrace{\Delta
T_{u}^{*(1+\beta)}\sqrt{v_{p}^{*}}\left[(1-A)^{*}\xi_{3}+A^{*}(1-\xi_{3})\right]}_{\text{rel.
surface heat flow}}\\\
&+Q_{A_{E}}\underbrace{[1+\xi_{4}(1-A^{*})]}_{\text{rel. mantle heat
prod.}}+q_{core}.\end{split}$ (3)

$A$, $w$ and $T$ are the state variables of the system. $A$ is the continental
surface fraction (combined surface area of the continents divided by the
surface area of the planet), $w$ is the mantle water concentration, and $T$
the mantle temperature; $\dot{A}$, $\dot{w}$, and $\dot{T}$ are their rates of
change with time, respectively. $\dot{A}_{l,E}$ and $\dot{w}_{l,E}$ are the
present-day rates of continental loss by erosion and mantle water loss by
outgassing (in mass per time), respectively, and $f_{r}$ is the present-day
ratio between loss and gain rates ($f_{r}=1$, for equilibrium between
continental erosion and production and between mantle water gain and loss). We
generally use the asterisk to denote variables scaled to present-day Earth
values and the index $E$ to denote the latter. $v_{p}(T,w)$ is the plate speed
and a function of $T$ and $w$. It is taken to be proportional to the mantle
convection flow speed. $L_{oo,oc}(A)$ is the combined length of ocean-ocean
subduction zones $L_{oo}$ (such as the Marianas) and ocean-continent
subduction zones $L_{oc}$ (such as the subduction zone at the Peru-Chile
trench) and is a function of $A$. The length of subduction zones is calculated
from the the total length of continental margins as a function of continental
coverage (see B for details). $\Delta T_{u}$ is the temperature difference
across the upper thermal boundary layer of mantle convection, $q_{m,E}$ is the
present-day heat flow from the mantle, $Q_{A_{E}}$ is the radiogenic heat
production rate as a function of time of the present-day depleted mantle
(compare Eq. 21 in Section C). The depletion of mantle heat sources due to
continental crust growth is modelled by the term $1+\xi_{4}(1-A^{*})$, where
$\xi_{4}$ is the present-day share of the Earth’s inventory of radiogenic
elements that is in the continental crust. $q_{core}$ is the heat flow from
the core. $\rho_{m}$, $c_{p}$, and $V_{m}$ are the mantle density, specific
heat capacity, and volume, respectively. $\xi_{1}$ to $\xi_{4}$ are parameters
describing the relative weight of terms in the balance equations (compare Tab.
1). As we will discuss in more detail below, $\xi_{1}$ and $\xi_{2}$ weigh the
strengths of positive and negative feedback in continental growth, and $f_{r}$
shifts the fixed points in the phase plane.

Eq. 1 calculates the growth rate of continental crust area from a balance of
continental crust production and loss through erosion. One part of the growth
is directly proportional to the continental surface area. This part is a
consequence of the effect of sediments on continental crust production and
provides positive feedback as illustrated in Figure 2. Its relative weight is
given by the dimensionless factor $\xi_{1}$, which may take values between 0
and 1. A second contribution to the growth rate is proportional to the plate
speed, which depends on the water concentration in the mantle and the mantle
temperature, thereby coupling Eq. 1 to Eqs. 2 and 3. Its relative contribution
is given by $(1-\xi_{1})$. Loss by erosion is composed of a surface erosion
part depending on the continental surface area and provides negative feedback
to the system. The remaining part of continental erosion is related to the
friction in subduction zones (subduction erosion) and depends on the total
length of ocean-continent subduction zones. It should be noted that the latter
part is a function of the continental surface area.

The term describing mantle water regassing (first term on the RHS of Eq. 2)
mirrors the production term in Eq. 1, as both are related to the subduction of
water. The regassing term altogether provides non-linear positive feedback to
the rate of change of mantle water concentration through the dependence of the
plate speed on the latter (compare Fig. 2). Mantle water outgassing described
by the second term on the RHS of Eq. 2 is directly proportional to $w^{*}$,
which contributes negative feedback to the change of the mantle water
concentration. A step-by-step derivation of Eqs. 1 and 2 is given in A.

Eqs. 1 and 2 are coupled to the energy equation (Eq. 3) via the plate speed.
The latter equation balances the surface heat flow with the heat production
rate in the mantle by radiogenic decay and the heat flow from the core to
calculate the rate of change of mantle internal energy. The surface heat flow
is coupled to the water balance through the non-linear dependence of the
mantle convection speed on the water concentration (C) and is calculated in
terms of the present-day mantle heat flow $-q_{m,E}$. Furthermore, it is
coupled to the surface area of insulating continental crust through the term
$(1-A)^{*}\xi_{3}+A^{*}(1-\xi_{3})$. The term is written such that it will
equal unity for $A$ equal 0.4, the value for the present-day Earth,
independent of the choice of $\xi_{3}$.

$\xi_{3}$ is defined as the present-day ratio between the oceanic heat flow
and the mantle heat flow. It is a measure of the efficiency of the thermal
blanketing of the mantle by continental crust. We discuss a case of perfectly
insulating continents, for which $\xi_{3}=1$, as well as a case for which
$\xi_{3}=0.85$ and for which the percentage of mantle heat flow passing
through continental crust is about 15% (Jaupart ., 2016). Neglecting
continental insulation altogether requires setting $\xi_{3}=0.6$, which equals
the share of the surface area of oceanic crust. A detailed derivation is given
in Section D. Other parameter values pertaining to the present-day Earth are
collected in the tables in F.

Table 1: Parameters $\xi_{i}$ that control the feedback in the coupled system. These parameters are defined for the present-day Earth and kept constant throughout the evolution. We explore wide ranges of values of $\xi_{i}$ since these carry large uncertainties. Symbol |  Definition | Explored Values
---|---|---
$\xi_{1}$ |  Present-day fraction of the continental production rate that is directly proportional to the area of the continents | 1/3, 2/3
$\xi_{2}$ |  Present-day fraction of the continental erosion rate that is directly proportional to the area of the continents | 1/3, 2/3
$\xi_{3}$ |  Present-day share of heat flow through oceanic crust | 0.6, 0.85, 1
$\xi_{4}$ |  Present-day share of heat sources in continental crust | 0, 1/4
$f_{r}$ |  Present-day ratio between the rate of continental production and that of continental erosion | 1, 0.775

##  3 Results

The state variables $A$, $w$ and $T$ span a phase space – a space in which all
possible states of the system are represented by unique combinations of the
state variables – in which solutions can be discussed. Equilibrium conditions
within the phase space are described by $\dot{A}=0$ and $\dot{w}=0$. These
conditions define two surfaces in phase space that intersect in lines towards
which the system will tend to evolve.

We begin by introducing the continent-water system in a phase plane evaluated
at a representative present-day mantle temperature (Section 3.1). Following
this, we discuss how the phase planes and the state variables evolve with
temperature in 3D phase space in Section 3.2. In addition, we show time
histories of selected quantities. Implications for the long-term carbon cycle
are presented in Section 3.3 and the effect of thermal blanketing on the
system is shown in Section 3.4.

###  3.1 Phase planes at constant present-day mantle temperature, fixed
points, and stability

Figure 3 presents phase planes spanned by $A$ and $w$ at a constant
representative present-day mantle temperature of 1750 K for various
combinations of values of $\xi_{i}$ and $f_{r}$. The exact choice of the value
of the present-day temperature is not critical and does not affect our results
in any significant way because the mantle viscosity is scaled accordingly to
reach this temperature at 4.5 Gyr. While values of $\xi_{i}$ parameterize the
strength of the feedback, $f_{r}$ describes the ratio between the present-day
Earth gain rates (continental production and mantle water regassing) and loss
rates (continental erosion and mantle water outgassing) and therefore shifts
the fixed points in the phase plane.

Figure 3: Phase planes defined by the mantle water concentration and
continental coverage at the present-day mantle temperature. Shown are local
arrows pointing into the directions $dA/dw$ of evolution of $A$ and $w$. In
addition, graphs of the integrals of $\dot{A}=0$ (red) and $\dot{w}=0$ (blue)
are shown. Parameters values are: (a) $\xi_{1}$ = 1/3, $\xi_{2}$ = 2/3,
$f_{r}$ = 1; (b) $\xi_{1}$ = 1/3, $\xi_{2}$ = 2/3, $f_{r}$ = 0.775; (c)
$\xi_{1}=\xi_{2}=0.5$, $f_{r}$ = 1; (d) $\xi_{1}$ = 2/3, $\xi_{2}$ = 1/3,
$f_{r}$ = 1. For further discussion see the text.

Fig. 3 displays local arrows pointing into the directions $dA/dw$ of evolution
of $A$ and $w$ and graphs of the integrals of $\dot{A}=0$ (red) and
$\dot{w}=0$ (blue). In panels a and b, with $\xi_{1}<\xi_{2}$ these intersect
in a single fixed point marking a single stable solution to the system of Eqs.
1–3 at constant mantle temperature $T$. The integral of $\dot{A}=0$ is a
single valued function $A(w)$ under these conditions. Increasing the water
content in the mantle will increase the growth rate through its dependence on
$v_{p}(T,w)$. The increase is balanced by the erosion rate increasing with the
continental surface area. The integral $w(A)$ of $\dot{w}=0$ is a single
valued function for all parameter combinations considered with a maximum value
around 50% continental coverage. The rate of water release first increases
with mantle water concentration and is balanced by increasing water
consumption in subduction zones. At around 50% continental surface area, the
curve bends over as continents begin to dominate on the surface. For
geometrical reasons, ocean-continent subduction zones and ocean-ocean
subduction zones will shrink in length while the length of continent-continent
collision zones without subduction will increase. This will reduce the
potential to subduct water. The mantle water consumption rate will then
decrease with increasing $A$ and will need to be balanced, for $\dot{w}=0$, by
a decreasing outgassing rate requiring decreasing values of $w$. The maximum
of $w(A)$ mainly results from the variation of the total length of ocean-
continent type subduction zones $L_{oc}$ with $A$, which has a maximum at
$A\approx$ 0.37 (see B). The maximum of $w(A)$ occurs at somewhat larger value
of $A$ because the effect of $L_{oc}$ decreasing with $A$ for $A>$ 0.37 is
partly balanced by the sediment subduction flux increasing with $A$ and the
effect of the latter on mantle water regassing.

At $\xi_{1}\geq\xi_{2}$, the integral $A(w)$ of $\dot{A}=0$ becomes a
multivalued function of the mantle water concentration (compare Fig. 3, panels
c and d). At small values of $A$, $v_{p}(T,w)$ increasing with $w$ dominates
crustal growth as before and $A(w)$ has a positive slope on this first branch
of the S-shaped curve. At a specific value of $A$, decreasing with increasing
$\xi_{1}$, the graph of $A(w)$ bends over and its slope becomes negative. This
occurs because the growth rate increasing with $\xi_{1}A$ can only be balanced
by erosion if $v_{p}(T,w)$ and hence $w$ decrease with increasing $A$. The
increase of $A(w)$ with negative slope continues until at some value of $A$,
which is $\sim 0.6$ in Fig. 3 panel d, the decreasing length of ocean-
continent subduction zones must be compensated by increasing values of $w$ to
keep $\dot{A}=0$. The critical value of $A$ at which the graph of the integral
of $\dot{A}=0$ resumes a positive slope increases with increasing values of
$\xi_{1}$.

The graphs of the integrals of $\dot{A}=0$, $A(w)$ and $\dot{w}=0$, $w(A)$,
intersect at three points (panel d) for $\xi_{1}>\xi_{2}$. Of these, two
represent stable equilibria at small and large continental coverage,
respectively. $A(w)$ has positive slopes at these points. The intersection
point between the two – at $A=0.4$ and $w=100$ppm here – is a saddle point,
stable with respect to perturbations in $w$ but unstable with respect to
perturbations in $A$. $A(w)$ has a negative slope at this point. In panel c,
we plot a transitory case for which $\xi_{1}=\xi_{2}=0.5$, $f_{r}$=1. Here,
the two curves just touch at $A=0.4$ and $w=100$ppm with $dA/dw=dw/dA$ and
there is a stable equilibrium point at $A=0.25$ and $w=80$ppm. It should be
noted that $\xi_{1}=\xi_{2}=0.5$ marks a bifurcation for the solutions of Eqs.
1 – 2 (Höning ., 2019a).

Figure 4: Evolution models without thermal blanketing and continental crust
enrichment in radiogenic elements. Shown are trajectories in the $A,w,T$ phase
space along with $A,w$ phase planes at constant mantle temperatures of 1850 K
and 1733 K. The latter temperature is the present-day mantle temperature in
all models shown here. Time in units of Gyr is marked by dots and labelled.
The model shown in the top panel has the same values of $\xi_{1}=1/3$,
$\xi_{2}=2/3$, and $f_{r}=0.775$ as the model in panel b of Fig. 3. In the
bottom panel $\xi_{1}=2/3$, $\xi_{2}=1/3$, and $f_{r}=1$ are chosen as in
panel d of Fig. 3. In both panels $\xi_{3}=0.6$, $\xi_{4}=0$ are chosen such
that the terms related to $A$ in Eq. 3 cancel. The initial mantle temperature
$T_{0}$ for the green curve has been chosen such that the a present-day
continental coverage of 0.4 is obtained; the purple curve starts at a 150 K
higher mantle temperature and the cyan curve at a 150 K lower temperature. The
onset time $t_{p}$ of continental growth is 0.5 Gyr.

###  3.2 Mantle cooling without thermal blanketing

In the following, we will present numerical solutions to Eqs. 1 – 3 in the
$A,w,T$-phase space (Fig. 4). We also show phase planes at the present day
mantle temperature and at 1850 K, respectively, with graphs of the integrals
of $\dot{A}=0$ and $\dot{w}=0$. The latter phase planes can be compared with
those in Fig. 3 (in particular panels b and d). The parameters $\xi_{3}=0.6$
and $\xi_{4}=0$ have been chosen such that the respective terms cancel in Eq.
3 as thermal blanketing and radiogenic depletion are neglected. In Fig. 5 we
show the state variables for the models as functions of time along with the
plate speed $v_{p}$.

Figure 5: Continental coverage, mantle water concentration and temperature and
plate speed over time (top to bottom) for the models of Fig. 4. Left: Negative
feedback dominating, $\xi_{1}<\xi_{2}$. Right: Positive feedback dominating,
$\xi_{1}>\xi_{2}$.

Model calculations are started at $t=0$ by integrating Eqs. 2 and 3 with a
given initial temperature $T_{0}$ and water content $w_{0}=100$ ppm. At a
later time $t=t_{p}$ according to the assumed onset of plate tectonics – or
more specifically, the onset of continental growth by subduction zone melting
– we include Eq. 1 in the integration with $A_{0}=0$ as starting value. The
onset time $t_{p}$ is 0.5 Gyr for the models in Figs. 4 and 5. Note that the
onset of continental growth by subduction zone melting needs to be started
that early as we grow the entire continental surface area by that mechanism.
The starting values could be easily adjusted if plate tectonics is taken to
start with the Proterozoic at $t_{p}=2.5$ Gyr and the Archean continents had
grown by a different mechanism to some area $A_{0}>0$ at the time.

We present three models in each panel of Fig. 4 that differ by the chosen
initial mantle temperature, colour coded cyan, green and purple. The initial
mantle temperature for the models belonging to the green curves (1945 K) in
both panels was chosen such that a continental coverage of 40% as on present-
day Earth results after 4.5 Gyr. For the models belonging to the purple
curves, a 150 K higher initial temperature was chosen; for the models
belonging to the cyan curves the initial temperature was chosen 150 K lower.
Otherwise the models have the same parameter values. We note that at given
values of $\xi_{1}$, $\xi_{2}$, and $f_{r}$, the values of $T_{0}$ and $t_{p}$
can be traded against each other to arrive at the Earth’s present-day
continental coverage. For instance, we could have used a lower initial mantle
temperature in combination with a later onset time $t_{p}$. We could also have
set the value of $f_{r}>1$, i.e. forced the present-day Earth continental
growth rate to exceed the erosion rate, which would, for example, have allowed
a smaller value of $T_{0}$.

Fig. 4 shows how the number of fixed points and their locations in the phase
planes change with temperature. The phase planes at 1850 K ($t\approx 1.5$ Ga)
in the top and bottom panels resemble each other by both showing a single
fixed point. This can be understood as resulting from the strong temperature
dependence of the viscosity and hence the flow speed $v_{p}$ which causes the
term proportional to $v_{p}$ to dominate the production rate in Eq. 1, almost
regardless of the chosen value of $\xi_{1}$. It is interesting to note that
the graph of the integral of $\dot{A}=0$ in the bottom panel already begins to
attain an S-shape which it does not in the top panel. However, the single
fixed point in the top panel with $\xi_{1}<\xi_{2}$ at 1850 K is at a notably
smaller continental coverage, thereby reflecting the stronger dependence of
the erosion rate on $A$ under these conditions. At $A<0.6$, the two integral
curves $\dot{A}=0$ (red) and $\dot{w}=0$ (blue) are almost independent of $w$
again reflecting the strong dependence of continental growth and mantle water
hydration on temperature through $v_{p}(T,w)$. The importance of the mantle
water concentration increases with decreasing temperature, in particular at
values of $A>0.6$ for which the length of ocean-continent subduction zones
substantially decreases with increasing $A$.

All models in Figs. 4 and 5 begin with rapid degassing of water accompanied by
an adjustment of the mantle temperature depending on the chosen value of
$T_{0}$ even before continents start to grow. This adjustment to initial
conditions is well known from thermal evolution models (e.g., Breuer Moore,
2015). The mantle then cools and temperatures converge until differences
become insignificant after about 2 Gyr of evolution, although the models with
$\xi_{1}>\xi_{2}$ cool a bit slower reaching 1880 K about 200 Myr later. The
plate speed shown in the bottom panels of Fig. 5 evolves similarly; after
adjustment to initial conditions differences become small after about 2 Gyr of
evolution. Again, the models with $\xi_{1}>\xi_{2}$ evolve less rapidly with a
similar delay as for the mantle temperature.

Differences between models with dominating positive versus negative feedback,
respectively, are more significant for the mantle water concentration and,
even more so, for the surface coverage by continental crust. With negative
feedback dominating, $\xi_{1}<\xi_{2}$, the mantle water concentration adjusts
rapidly to initial conditions and increases steadily after a little more than
1 Gyr almost independent of the choice of initial mantle temperature to reach
a present day value of about 70 ppm, the equilibrium value being about 65 ppm.
For positive feedback dominating, $\xi_{1}>\xi_{2}$, the initial adjustment of
the variables to their initial conditions at times smaller than 0.5 Gyr
corresponds to the one in the model with dominating negative feedback. But
beyond 0.5 Gyr, the evolution paths diverge with greater and widening
differences in mantle water concentrations and continental coverage. This is
caused by the emergence of the three fixed points and the evolution paths
tending towards them.

Figs. 4 and 5 illustrate how positive feedback causes the present-day value of
continental coverage to depend on its early evolution. Although mantle
temperature values converge after a few billion years of evolution and
differences in initial conditions disappear as predicted by the Tozer
principle (Tozer, 1967), differences in the early growth history of
continental crust get emphasised if the feedback is positive as well as
differences in the net mantle regassing rate persist. If negative feedback
dominates, however, both continental coverage and mantle water concentration
tend to converge towards their single equilibrium fixed point values. It
should be noted that these equilibrium values evolve on their own with time or
temperature while the state variables follow suit, albeit with some delay in
time. It should further be noted that the continental surface area is
predicted to shrink if negative feedback dominates while it is predicted to
grow slightly or keep its level for the past 2 Gyr if positive feedback
dominates. Moreover, we find a large surface coverage by continents to imply a
wet mantle while a small coverage is related to a comparatively dry mantle.
This is due to the fact that both gain rates – continental production and
mantle water regassing – depend on the same physical parameters that control
the transport of water in subduction zones. Because we neglect the loss of
water due to atmospheric escape, the complement of the mantle water is to be
found at the surface as ocean water.

###  3.3 Implications for the long-term carbon cycle

Figure 6: CO2 degassing rate, CO2 partial pressure and surface temperature as
functions of time. The color code is the same as in Figures 4 and 5. Left:
Negative feedback dominating. Right: Positive feedback dominating.

The surface area of continental crust emerged above sea level has a direct
impact on the climate because the rates of erosion and silicate weathering
increase with increasing land area (e.g., Foley, 2015; Graham Pierrehumbert,
2020; Levenson, 2021). A planet with a larger area of emerged continents will
thus have a lower concentration of CO2 in its atmosphere (all other factors
being the same) and a lower surface temperature as compared with a planet with
a smaller land area. For details of the calculation of the CO2 degassing rate
and partial pressure and the surface temperature see E. In Fig. 6, we plot the
CO2 degassing rate, partial pressure, and surface temperature as functions of
time for the models with dominating negative (left panels) and positive (right
panels) feedback shown in Figs. 4 and 5. While differences in the degassing
rate decrease with time and are negligible after 4.5 Gyr, the partial pressure
of CO2 is lower for the model starting with 150 K higher initial mantle
temperature, in particular, if positive feedback dominates. And vice versa for
the model starting with the lower initial temperature. This observation is
consistent with the differences in continental surface area. All in all, with
dominating positive feedback, differences in the surface temperature by about
5 K result after 4.5 Gyr even though all other parameter values are the same.

###  3.4 Cooling with thermal blanketing and mantle depletion in radiogenic
elements

Thermal blanketing of the mantle by the continental crust and the depletion of
the mantle in radiogenic heat sources transferred to the crust can provide
additional feedback between the crust growth rate and its surface area. In
Fig. 7, we plot results for models with the same parameter values as in Fig. 4
but account for thermal blanketing by setting $\xi_{3}=0.85$ and $\xi_{4}=1/4$
in Eq. 3. The onset time $t_{p}$ of continental growth by subduction zone
melting for the green model was adjusted such that the continental surface
fraction grows to 0.4 in 4.5 Gyr. For models with $\xi_{1}=1/3$ and
$\xi_{2}=2/3$, $t_{p}$ was adjusted to 0.253 Gyr. For those with $\xi_{1}=2/3$
and $\xi_{2}=1/3$, $t_{p}$ was chosen to be 0.161 Gyr.

Figure 7: Same as Fig. 4 but with continents that are enriched in radiogenic
elements and that thermally blanket the mantle. Top: $\xi_{1}=1/3$,
$\xi_{2}=2/3$, $\xi_{3}=0.85$ and $\xi_{4}=1/4$, $f_{r}=0.775$, onset time
$t_{p}=0.253$ Gyr; bottom: $\xi_{1}=2/3$, $\xi_{2}=1/3$, $\xi_{3}=0.85$ and
$\xi_{4}=1/4$, $f_{r}=1$, onset time $t_{p}=0.161$ Gyr.

Figure 8: Continental coverage, mantle water content and temperature and plate
speed over time (top to bottom) for the models of Fig. 7. Left: Negative
feedback predominating. Right: Positive feedback predominating.

Figure 9:  Same as in Fig. 6, but for models with thermal blanketing and
mantle depletion of radiogenic elements upon continental crust growth and
$\xi_{3}=0.85$ and $\xi_{4}=1/4$. Left panels: Negative feedback
predominating. Right panels: Positive feedback predominating.

A comparison between Figs. 4 and 7 and between Figs. 5 and 8 shows the same
general behaviour with the models converging towards the single fixed point
when negative feedback dominates. For dominant positive feedback, the models
diverge and tend towards three distinct fixed points. There is a larger spread
between the continental growth curves for models which include thermal
blanketing, though, $0.4\pm 0.1$ versus $0.4\pm 0.05$ dimensionless units for
negative feedback and – somewhat more asymmetric – $0.4^{+0.3}_{-0.22}$ versus
$0.4^{+0.27}_{-0.18}$ dimensionless units for dominating positive feedback.

The evolution of the mantle water content is also found to depend on thermal
blanketing as can be particularly well seen when comparing the respective
panels in Figs. 5 and 8. For positive feedback and blanketing, the model with
the largest continental coverage (purple) ends with less mantle water than in
the case when blanketing is ignored. It even ends with less mantle water in
Fig. 8 (right column) than the Earth-like model with a present-day continental
coverage of 0.4. This can be understood if the mantle temperature is
considered, which shows the most prominent differences. Models with no thermal
blanketing and heat source differentiation have temperatures converging with
time, independent of the choice of the initial temperature. With blanketing
and mantle heat source depletion, mantle temperatures tend to converge
initially but then tend to diverge with the growth of continents. In
particular, the model with dominating positive feedback and with a present-day
continental coverage of $>0.6$ (purple) shows the cooling rate to decrease
after about 2.5 Gyr and the mantle temperature to start to level off. This
explains why the net water gain rate in that model decreases after about that
time. Still, the model has a present-day hotter and wetter mantle when
compared with the purple model for dominating negative feedback (left column).
The difference can be attributed to the larger contribution to water
subduction by sediments which explains why the difference in plate speed and
continental coverage is particularly pronounced for the purple models. The
plate speed, finally, evolves largely in parallel with the mantle temperature
as expected. The effect of continental insulation, which tends to reduce
cooling, exceeds that of the depletion of mantle radiogenic elements in this
model, which tends to enhance cooling.

Interestingly, for the case of dominating negative feedback (Fig. 8 left),
large continents (purple curve) result in comparatively hot and dry mantles,
with opposing effects on viscosity and plate speed. For dominating positive
feedback (Fig. 8 right), sediments from the comparatively large continents
contribute to water subduction such that a hot and wet mantle results, both
increasing the plate speed. This feeds back to continental growth such that
the difference in continental coverage at 4.5 Gyr becomes particularly
pronounced.

Figure 9 shows results pertaining to the long-term carbon cycle for the models
that include thermally insulating continents. As in previous figures, panels
on the left are for models with predominating negative feedback and those on
the right are for models with predominating positive feedback. A larger
surface fraction of insulating continents keeps the mantle warmer and
therefore results in somewhat larger degassing rates but also in a larger
weatherability and rate of CO2 removal. The differences in the CO2 partial
pressure and surface temperature between the models remain small and are even
a bit smaller than for models neglecting thermal blanketing altogether.

##  4 Discussion

In this paper, we modelled three main geologic processes as a nonlinear
coupled dynamical system: growth of continental crust, exchange of water
between the reservoirs on and above the surface (oceans, atmosphere) and in
the mantle, and cooling by mantle convection. These processes are linked
through mantle convection and plate tectonics with

  * •

subduction zone related melting and volcanism and continental erosion
governing the growth of the continents (Eq. 1),

  * •

mantle water degassing through volcanism and regassing through subduction
governing the water budget (Eq. 2), and

  * •

heat transfer through mantle convection governing the thermal evolution (Eq.
3)

The rates of these processes depend non-linearly on mantle temperature and
water concentration through the mantle viscosity (compare Eq. 26) and plate
speed $v_{p}(T,w)$ which couples the governing equations 1 \- 3. By including
a model of the carbonate-silicate cycle, we explored consequences for the
atmosphere’s CO2 partial pressure and average surface temperature as indicator
of the planet’s habitability. In addition to the coupling through
$v_{p}(T,w)$, there are immediate feedback mechanisms with the crust
production and erosion rates depending on the continental surface area and the
length of continental margins and with the mantle water outgassing rate
depending on the mantle water concentration. The former is implied by the
dependence of the melt production rate in the subduction zone on the water
supply through sediments (Höning Spohn, 2016) which in turn is directly
proportional to the continental surface area. The latter is implied by the
dependence of the mantle solidus temperature and the viscosity on the water
concentration. The mantle water concentration and the continental surface area
feed back on the mantle thermal energy balance; the former through the
dependence of $v_{p}(T,w)$ on the mantle water concentration, the latter
through the thermal blanketing effect of the continents.

The conceptual nature of our study motivated us to simplify our model beyond
the level of abstraction of our previous work (e.g., Höning Spohn, 2016;
Höning ., 2019a) by reducing the number of free parameters to four,
$\xi_{1},..,\xi_{4}$, with $0\leq\xi_{1},..,\xi_{4}\leq 1$. We still use the
present-day Earth as a reference and present-day observables as scales. In the
following, we will discuss our finding that the spread of continental coverage
on Earth-like planets is determined by the respective strengths of positive
and negative feedback in continental growth ($\xi_{1}$ vs. $\xi_{2}$, Section
4.1) and by the relationship between thermal blanketing and depletion of
radioactive isotopes upon growth of the continental crust ($\xi_{3}$ vs.
$\xi_{4}$, Section 4.2). Uncertainty in these parameter values represents the
main uncertainty in the model. Our discussion aims to provide a better
qualitative understanding of the feedback processes; we admit to lacking data
for a detailed understanding of quantitative differences. Finally, we discuss
the habitability of the model planets and their bioproductivity that result
from the various possible evolution pathways in Section 4.3.

###  4.1 The feedback parameters $\xi_{1}$ and $\xi_{2}$ and their
implications for continental growth

The parameters $\xi_{1}$ and $\xi_{2}$ can be regarded as slide rulers in Eqs.
1 and 2 to weigh between feedback terms. We have chosen combinations of
$\xi_{1}$ = 1/3 and $\xi_{2}$ = 2/3, $\xi_{1}=\xi_{2}$ = 0.5, and $\xi_{1}$ =
2/3 and $\xi_{2}$ = 1/3 to demonstrate in Figs. 3 – 9 how the properties of
the system vary with these choices. Note that $\xi_{1}$ and $\xi_{2}$ do not
have to add up to 1, although they do in our parameter choices. The parameter
$\xi_{1}$ weighs between the importance of the continental surface area and
plate speed (that is, convection speed), respectively, for both the production
rate of continental crust and the mantle water degassing rate of the present-
day Earth.

The value of $\xi_{1}$ becomes particularly important late in the evolution,
when the mantle has cooled down sufficiently such that $v_{p}^{*}$, the plate
speed scaled to its present-day value, will be of the order of unity.
Geologically, this would be in the post-Archean. At earlier times, $v_{p}^{*}$
will have a greater effect than $A$, unless $\xi_{1}$ is chosen such that
$(1-\xi_{1})\ll 1$. As a geological interpretation, $\xi_{1}$ weighs the
importance of sediments for transporting water in the subducting slab to the
source region of continental crust for the present-day Earth. The rate of
melting and crust production in the model is directly proportional to the rate
of supply of water through oceanic crust and through the sedimentary layer on
top of the oceanic crust (see Höning . 2014 and Höning Spohn 2016 for
details), the share of the sediments in the balance increasing with $\xi_{1}$.

Höning Spohn (2016) have discussed how continental erosion, subduction of
sediments and continental rock formation are related. The ability of hydrous
sediments to contribute to water transport with the subducting plate is well
established (Hacker, 2008; Deschamps ., 2012, 2013) but difficult to quantify.
In addition, as described in Höning . (2014), low permeability sediments can
seal the hydrated crust, thereby reducing the rate of shallow dewatering upon
subduction. Stern (2011), Sobolev Brown (2019) and Chen . (2022) argue that
accumulation of sediments in subduction zones is crucial to lubricate the
subducting plate, which would effectively introduce another positive feedback
to the system that is not considered here, however. Sobolev Brown (2019) even
suggest that the persistence of plate tectonics requires a sufficient
thickness of the sedimentary layer and that an extended period of reduced
surface erosion and sediment subduction might have resulted in reduced
tectonic activity on Earth between 0.75 and 1.75 billion years before present.
Note that $L_{oo,oc}(A)$ is a constant for $A\leq 0.4$ (compare Fig. 13) and
decreases with approximately $2A$ for $A>0.55$ when continents increasingly
touch and merge (compare Eq. 20).

It may be argued that positive feedback may also originate from the variation
of $v_{p}^{*}$ with viscosity. After all, an increase in $v_{p}^{*}$ will
reduce the viscosity and increase the convection speed and, in turn, the water
regassing rate. This will be counteracted, however, by increased water
outgassing and cooling associated with the convective mantle flow. In the
present model, the effect of cooling on increasing the viscosity dominates
over the effect of an increasing water concentration. Altogether, the
contribution of convection and plate circulation is to provide negative
feedback. Other positive feedback mechanisms may exist that are independent of
erosion and sediment subduction. We will discuss the contribution of thermal
blanketing to positive feedback in the next section.

The parameter $\xi_{2}$ weighs between surface erosion and subduction erosion,
the latter depending on the length of the ocean-continent subduction zones. A
major share ($\approx 1/3$) of the global continental erosion rate has been
associated with subduction erosion (Stern, 2011). In addition, significant
shares of the loss of continental surface area have been attributed to
continental collision and lower crust delamination, leaving less than a third
for surface erosion Stern (2011). It thus seems reasonable to assume that
$\xi_{2}<0.5$. We note that the rate of subduction erosion may depend on the
rate of friction between the subducting slab and the continental keel in
addition to its dependence on the length of the subduction zone. Sediments may
reduce friction by lubricating the slab Stern (2011), thereby reducing the
rate of subduction erosion for increasing continental coverage. This would
effectively introduce another positive feedback to the system that is not
considered here, however.

The length of the ocean-continent subduction zones $L_{oc}(A)$ will naturally
increase with the size of the continents as long as the continental coverage
is smaller than $\approx 0.4$. At larger values of coverage, the subduction
zone length decreases because continents impinge upon each other and merge and
continental collision zones form. The ocean-continent subduction zones
$L_{oc}(A)$ varies non-linearly with the continental surface area in our model
(compare B). In linear approximations, it increases roughly with $3A$ for
$A<0.3$ and decreases roughly with $2A$ for $A>0.55$. Between these two values
of $A$ the dependence of $L_{oc}$ on $A$ is weak. Thus, there will be strong
negative feedback to $A$ associated with continental erosion for $A<0.3$, even
for small values of $\xi_{2}$. The feedback may turn entirely positive,
however, for large values of continental coverage, depending on the chosen
values of $\xi_{1}$ and $\xi_{2}$ if subduction erosion dominates and the
overall continental loss rate decreases with increasing $A$.

Our results show how strong negative feedback would lead to an evolution
largely independent of the starting conditions and the early history of the
planet which would imply a single stable present-day value of the continental
surface area. For strong positive feedback, however, the outcome of the
evolution may be quite different depending on starting conditions and the
early history. For exoplanets this suggests that Earth-like planets covered
largely by land (i.e. land planets) are possible as well as planets covered
mostly by oceans (i.e. ocean planets) (see Fig. 1). Höning Spohn (2016) found
that the land planet should be the most likely outcome, followed by cases that
end as ocean planets. Only a small percentage end as planets with a balanced
Earth-like share of land and ocean surfaces.

Earth’s further evolution can be discussed using the present day phase planes
shown in Figs. 4 and 7. It will depend on the location of the state variable
$A$ relative to the fixed point. For values of $A$ close to but larger than
the continental coverage at the unstable fixed point, $A$ will increase with
time at a small rate beyond its present-day value. While $A$ is growing, the
mantle water concentration will increase. With water budget conserved, this
will be accompanied by decreasing surface water reservoir and increasing area
of emerged land. This would be beneficial for the biosphere allowing for more
silicate weathering, provision of nutrients, and partly compensating for an
increasing solar luminosity. Eventually, the unstable fixed point will
overtake the trajectory and net continental growth will cease. Additional
calculations have shown that the upper stable fixed point will move towards
smaller values of $A$ together with the unstable fixed point moving towards
larger values of $A$. The two fixed points will eventually merge and then
disappear with further cooling while the continental area will slowly decrease
with time being far away from but evolving towards the remaining lower stable
fixed point.

An extreme case in which continental production and erosion would be entirely
independent of the surface area can be studied by setting $\xi_{1}=0$ and
$\xi_{2}=0$. The continent production rate will then depend only on the mantle
convection rate and the erosion only on the total length of ocean-continent
subduction zones. In addition, we neglect thermal blanketing (as in Fig. 4)
and show the results in Fig. 10. Without positive or negative feedback,
differences in early continental growth resulting from differences in initial
conditions are mostly retained throughout the evolution without being damped
or amplified with time as they would be with feedback. Continental coverage at
4.5 Gyr is found to attain similar values to those with dominating positive
feedback (compare Fig. 5, right panel), though. Mantle water concentrations
for the ocean and Earth-like planet are similar to the values found for
negative feedback but the land planet (purple curve) of the extreme model has
a comparatively dry mantle. This implies a comparatively larger surface water
mass and more flooded continental crust and can be explained as a consequence
of ignoring the contribution of sediments to water subduction. The
concentrations of atmospheric CO2 and the spread of surface temperatures are
similar to the scenario with dominating positive feedback explored above.

It is interesting to note that for planets around M-dwarfs, Tian Ida (2015)
predict a similar bimodal distribution of emerged land area, with most planets
either having their surface entirely covered with water or with significantly
less surface water than Earth. Their analysis, however, was based on a
combination of planet formation and water escape models, not explicitly
modelling continental growth.

Figure 10: Model scenario that neglects any direct dependence of continental
production and erosion on continental area ($\xi_{1}=0$, $\xi_{2}=0$). In
order for the green curve to reach continental coverage of 0.4 at 4.5 Gyr, we
set $f_{r}=0.603$. Thermal blanketing is neglected and the color code is same
as in the previous figures.

The small positive net growth rate of the continents since the Archean (e.g.,
Armstrong, 1991; Dhuime ., 2017; Rosas Korenaga, 2018; Korenaga, 2021)
provides a strong argument for positive feedback in Earth’s continental growth
history: Positive feedback can sustain a small net growth rate over an
extended period of time even in the presence of a slowly decreasing mantle
temperature as in the right columns of Figs. 5 and 8. In that case, the
unstable fixed point will be evolving slowly towards larger continental
coverage and $A$, being located slightly above this fixed point, will follow
suit. We note that equilibrium is not expected to be attained as mantle heat
production decreases with time and planets necessarily cool as has already
been discussed by Schubert . (1979). With negative feedback prevailing, the
continental production rate will decrease with decreasing mantle temperature.
As a result, the continental surface area will decrease with time as in the
left columns of Figs. 5 and 8. An overall decrease of the continental surface
over the past 2 Gyr as predicted by the negative feedback model is not
observed, however! Negative feedback evolution trajectories reaching a
continental coverage of 0.4 after 4.5 Gyr require a present-day ratio between
the rate of continental production and that of continental erosion of
$f_{r}=0.775$, implying an equilibrium continental coverage of only about 0.2
and an evolution of Earth towards what we have termed an ocean planet!

The continents may have grown in the Archean by mechanisms that differ from
those of modern plate tectonics (e.g., Condie Kröner, 2013). Our analysis
would still be applicable, because the bifurcation appears only at post-
Archean mantle temperatures and plate speeds. The onset time $t_{p}$ could be
2.5 Gyr and a value for $A$ at the time would be $>$0 along with assumed
values for $T$ and $w$. In such a scenario, the present-day continental
coverage would be sensitive to the coverage at $t_{p}$.

###  4.2 Thermal blanketing and mantle depletion through crust growth; effects
of $\xi_{3}$ and $\xi_{4}$

Thermal blanketing by continental crust can have two effects that may both act
at the same time: Blanketing can result in an overall reduction of the surface
heat flow and the cooling rate. And it can result in a redirection of mantle
heat flow away from the continents to the oceanic areas. Using stability
analysis, Höning . (2019a) have shown that thermal blanketing can lead to
runaway continental growth and even mantle heating (see also Lenardic . 2005
for related findings). The analysis, however, was based on the assumption of a
steady state mantle and a constant Urey Ratio. It also did not include the
redistribution of radiogenic elements between the mantle and the crust. In
their analysis, blanketing caused the heat flow from the mantle to be diverted
from the sub-continental to the sub-oceanic areas, resulting in an increasing
sub-oceanic mantle temperature, heat flow, plate speed and, finally,
continental growth rate. Their result can be considered as an extreme case but
suggests that positive feedback between thermal blanketing and continental
growth is, in principle, possible.

In the present model, we consider thermal blanketing in the presence of mantle
cooling and the decay of radiogenic elements. In addition, we consider the
transfer of heat producing elements from the mantle to the crust as a
consequence of melting and crust growth. The cooling and the decrease in time
of the mantle heating rate will reduce the self-reinforcement of continental
growth through thermal blanketing. We do not consider any possible redirection
of mantle flow away from sub-continental areas to sub-oceanic areas. Instead,
we use $\xi_{3}$ as a slide ruler to distribute the present-day mantle heat
flow $q_{s,E}$ between the oceanic areas and the continents. From its
definition in Eq. 33, $\xi_{3}$ is the present-day share of the oceanic heat
flow in the total heat flow from the mantle. For simplicity, this ratio is
kept constant throughout the evolution calculation. Also, we neglect any
enrichment of the oceanic crust in radiogenic elements.

For the present-day Earth, the oceanic heat flow per unit area is on average
105.4 mW/m2, while the average continental heat flow is 70.9 mW/m2 and the
total average surface heat flow per unit area is 91.6 mW/m2 (Davies Davies,
2010). Jaupart . (2015) give a somewhat smaller value for the continental heat
flow of 65 mW/m2 while earlier work resulted in about 60 mW/m2 (e.g., Williams
von Herzen, 1974). The total heat flow values, accounting for uncertainties,
are 31 – 32 TW, 14 – 15 TW, and 45 – 47 TW, respectively. The continental heat
flow includes a significant contribution from radiogenic elements in the
crust. Jaupart . (2015, 2016) give values for the basal continental heat flow
– mostly in stable continental shield areas – from the mantle of 15 $\pm$ 3
mW/m2, equivalent to 2.4 - 3.7 TW. Subtracting the latter values from the
continental heat flow gives 11 – 12.3 TW. However, this value is much too
large to be explained by continental heat production for which Jaupart .
(2015, 2016) give 6 – 7 TW, the remainder of 3 – 4 TW to be attributed to
continental margins where the heat flow can be significantly higher than in
stable continental areas (e.g., Springer Förster, 1998). Altogether, this
suggests a global present-day heat flow from the mantle of $\approx$ 38 TW, so
that values for $\xi_{3}$ between 0.8 and 0.85 should be reasonably
characterising the heat flow ratios on the present-day Earth.

To calculate a consistent value of $\xi_{4}$, the share of the heat sources in
the present-day continental crust (compare Eq. 23), we use the observed
continental heat flow and subtract the above estimate of the mantle heat flow
through the continents including continental margins of 6 – 7 TW to obtain 7 –
8 TW. Unfortunately, the mantle heat production rate is much more uncertain.
Dye (2012) cites estimates between 11 and 24 TW for the bulk silicate Earth
(mantle + crust) as resulting from cosmochemical studies. Using a mantle heat
flow estimate of 36 TW and estimates from convection modelling of 30$\%$ to
50$\%$ being attributable to radiogenic heating (Dye 2012, the so called Urey
ratio), they arrive at 19 to 27 TW, again for the bulk silicate Earth. Note
that we do not adopt the case of 80$\%$ radiogenic heating as we find that the
Urey ratio predicted from constant viscosity mantle models is unrealistic.
Jaupart . (2015) have compiled mantle and crust heat production rates from the
literature and from their own interpretations of global heat flow data and
arrive at a representative value of $11\pm 6$ TW for the mantle heating rate.
This value can be considered as a lower limit as it assumes that the
(depleted) MORB source extends through the whole mantle. Thus, values for
$\xi_{4}$ between 0.25 and 0.4 should be reasonable to assume.

In Fig. 7, we used a combination of values for $\xi_{3}$ and $\xi_{4}$ within
our range of estimates that maximises the effect of thermal blanketing. For
values of $\xi_{4}>0.25$, depletion of heat sources works increasingly against
the effect of thermal blanketing such that for $\xi_{3}=0.8$ and
$\xi_{4}=0.4$, the results are close to those neglecting blanketing as in
Figs. 4 and 5. In Fig. 11, we show the difference of the continental surface
fraction at 4.5 Gyr resulting from models with a difference in the initial
mantle temperature of 300 K (i.e. between the light blue and purple curves in
Figs. 4 and 7) for dominating negative (left) and positive (right) feedback.
As expected, the spread in continental surface area at 4.5 Gyr decreases with
increasing $\xi_{4}$ but the spread can be twice as large for cases with
dominating negative feedback and $\xi_{4}=0.25$.

Figure 11: Difference in continental coverage after 4.5 Gyr for models (light
blue and purple in Figs. 4 and 7) that differ in initial mantle temperature by
300 K as a function of $\xi_{4}$ for dominating negative (left) and positive
(right) feedback, respectively. The value of $\xi_{3}$ is kept fixed at 0.85.

Long-term heating of the mantle can only be obtained if the thermal blanketing
is made extreme. In Fig. 12, we show a case with $\xi_{3}=1$ implying
perfectly insulating continents. Here, with dominating positive feedback for a
high initial mantle temperature, we obtain heating even though the depletion
of radiogenic elements in continental crust is taken into account with
$\xi_{4}$=0.25.

All in all, we conclude that the reduction of cooling by thermal blanketing
underneath growing continental crust is partly, potentially largely
compensated by the effective cooling of the mantle associated with the
transfer of radiogenic elements from the mantle to the crust.

Figure 12: An extreme case of thermal blanketing with $\xi_{3}$=1 and
$\xi_{4}$=1/4. Top panel: $\xi_{1}$=1/3, $\xi_{2}$=2/3, $f_{r}$=1, onset time
$t_{p}$ 0.748 Gyr; bottom: $\xi_{1}$=2/3, $\xi_{2}$=1/3, $f_{r}$=1, onset time
$t_{p}$ 0.0774 Gyr.

###  4.3 CO2 outgassing, weathering, surface temperature and bioproductivity

Coupling the continental erosion rate and the plate speed to a model of the
long-term carbon cycle E, we find that the large spread in continental surface
coverage between our models results in differences of surface temperatures of
not more than about 5 $K$. The reason for this rather small difference are
opposing effects of increasing continental coverage on the atmospheric CO2
concentration: On the one hand, the weatherability and thereby the CO2 sink
increases with increasing continental coverage. On the other hand, the mantle
remains hotter with larger, insulating continents, which causes an enhanced
outgassing rate of CO2. Note that our ocean planet is a shallow water world
with an ocean depth of about 4 km for which the carbonate silicate cycle is
still operative (e.g., Höning ., 2019b; Hayworth Foley, 2020), even though the
negative feedback associated with silicate weathering would be weaker. It
should not be confused with a deep ocean water world with 10 - 1000 Earth
ocean masses for which the carbonate-silicate cycle might not work at all due
to the formation of high-pressure ice layers and potentially limited magmatic
degassing (e.g., Noack ., 2016; Kite Ford, 2018; Krissansen-Totton ., 2021).

A 5 K surface temperature difference is considerably smaller than the
excursions of the Earth’s mean surface temperature over the Phanerozoic (e.g.,
Scotese, 2016) and might be assumed to be of limited significance for the
general discussion of exoplanet habitability. However, Lingam Loeb (2018) and
Schulze-Makuch . (2020) have argued that a 5 K higher surface temperature
would significantly increase the habitability of an Earth-like world. In
combination with the differences in land and water surfaces it could have an
impact on climatic zones. Earth climate modelling suggests that the
temperature difference on land surfaces, in particular near the equator, may
be twice as large (Nazarenko ., 2022). Furthermore, the polar ice sheets will
shrink with increasing temperature. In contrast, the lower surface temperature
of the land planet would allow for substantial polar ice caps. These
differences should be reflected in the sea-level, which would also differ on
account of the non-linear thermal expansion of seawater (e.g., Widlansky .,
2020). In addition, there may be dry cold deserts in the interiors of large
continents. For the present-day Earth, 40% of the surface is continental crust
but a quarter of it (10% of the Earth’s surface) is covered with water in
shelf areas, leaving 30% land surface. Our ocean planet has a continental
surface of about 20% but given the warmer climate predicted a significant part
of it may be covered by water, leaving perhaps only 10% of land surface. On
the contrary, the land planet with about 70% continental surface area could
have a land surface twice that of Earth. All in all, we expect the ocean
planet to be comparatively warm and wet and the land planet comparatively dry
and cold. Future work with comprehensive Earth System and Global Circulation
Models is needed in order to better understand the impact of land area on the
climate of Earth-like planets together with a varying CO2 concentration in the
atmosphere.

We can estimate the combined effects of the land-ocean distribution and the
surface temperature on the bioproductivity and the biomass by using the models
of Lingam Loeb (2019) for the effect of the ratio of land to ocean surfaces
and of Lingam Loeb (2018) for the surface temperature. While the former model
weighs the land and ocean areas based on their differing specific
bioproductivities, the latter model considers the temperature effect on the
metabolic rate being governed by an activation energy. We again scale to the
present Earth and find the land planet to have a net bioproductivity (NPP) of
about 43$\%$ of Earth’s, corrected down from 50$\%$ when accounting for the
about 1.7 K lower surface temperature. The model with thermal blanketing would
have a slightly larger productivity of 47$\%$ of Earth’s due to the smaller
surface temperature correction. The larger land area will lead to a predicted
biomass of 67$\%$ of Earths, corrected down from 76$\%$, obtained when the
temperature effect is neglected. The ocean planet has a net bioproductivity of
60$\%$ of Earth’s, up from 49$\%$ when accounting for the higher surface
temperature. It has a biomass of 53$\%$ of Earth’s, up from 43$\%$, again
accounting for the higher surface temperature. The indirect effects of thermal
blanketing are too small to be relevant for this model. Further using Lingam
Loeb (2019), the land and the ocean planet would be predicted to both have
only a marginal capability of building up O2 for advanced life forms in their
atmospheres.

Undoubtedly, the biosphere on worlds with different land fractions would adapt
to their environment, and a balanced ratio of ocean-to-land accompanied by
large continental shelves is certainly beneficial for an Earth-like biosphere
(e.g., Kallmeyer ., 2012; Höning Spohn, 2016; Lingam Loeb, 2019; Stevenson
Wallace, 2021). Vice versa, life on Earth co-evolved with the environment,
shaping the atmosphere and oceans (Grenfell ., 2010; Lenton ., 2014; Algeo .,
2016). The emergence of pelagic calcifiers in the deep oceans has been argued
to have played a role in the stability of Earth’s climate (Ridgwell ., 2003)
and the colonisation of land by plants enhancing continental weathering has
presumably been crucial in keeping our planet’s biosphere flourishing in the
long term (Berner, 1998; Schwartzman, 2017). In combination, life on land and
in the oceans has been shown to help stabilising Earth’s climate on timescales
$>$100 kyr with land plants dominating on timescales $>$1 Myr and marine
calcifiers being predominantly important on shorter timescales (Höning, 2020).
This particular relative importance of life on land and in the oceans is a
direct result of Earth’s balanced ratio of oceans to land, however: On planets
with a larger continental coverage, land plants would become increasingly
important for climate stability, whereas the role of marine calcifiers would
become minor (Höning, 2020).

##  5 Conclusions

In this paper, we assessed the coupled evolution of continental growth, mantle
water cycling, and thermal evolution of Earth-like planets. In particular, we
explored how differences in initial mantle temperature and the strength of
positive and negative feedback would impact their continental growth curves
and water budgets. We coupled the model to a model of the carbonate-silicate
cycle via the plate speed and the continental surface area to assess the
habitability and the bioproductivity of these planets. Our results can be
summarised as follows:

  * $\bullet$

Positive feedback in the coupled continent-water cycle may lead to the
emergence of multiple fixed points and to substantially differing land and
ocean coverage, with an Earth-like distribution of land and ocean surfaces
being the least likely. The average depth of the oceans would be similar,
around 4 km, because differences in the ocean surface will be compensated by
differences in the distribution of water between the mantle and the surface
reservoirs. The final continental coverage and the environment of the planet
will depend on the initial conditions, in particular on the initial mantle
temperature and the onset time of plate tectonics. If early mechanisms of
continental growth differ from plate tectonics then other differences in
initial conditions may still matter. An evolution largely independent of the
early conditions is possible only if negative feedback mechanisms prevail.

  * $\bullet$

Positive feedback is mostly provided by the role of sediments in continental
growth. The rate of sediment subduction will increase with the continental
surface area and the additional water transported will enhance the continental
growth rate. In addition, sediments lubricate the subducting plate and
facilitate efficient subduction. Thermal blanketing of the mantle by
continents may retard mantle cooling, keep an enhanced continental production
rate and may cause an even larger difference in final continental coverage.
But the effects of thermal blanketing is partly, potentially largely, balanced
by the removal of heat sources from the mantle going along with crust growth
and with an enrichment of radiogenic elements in the continental crust.

  * $\bullet$

The carbonate-silicate cycle is coupled to the evolution of the interior
through the carbon outgassing rate depending on the plate speed and to
continental growth via the weathering rate increasing with the continental
surface area. We found differences in the mean global surface temperature
between the land planet and the ocean planet of up to $\sim$ 5 K. The climate
of all three planets as well as their biospheres may have evolved quite
differently, however. The warmer ocean planet’s potentially higher specific
bioactivity may be limited by the smaller availability of critical nutrients
from weathering its smaller land surface. The cooler land planet may have a
smaller specific bioactivity further reduced by the limited availability of
water but its bioactivity would be spread over a larger area. By using the
scalings of Lingam Loeb (2018, 2019) we find the bioproductivity and the
potential biomass on these planets reduced by a third to half of Earth’s. It
is questionable whether a biosphere on these planets would be substantial
enough to produce more oxygen then is consumed by geologic activity.

  * $\bullet$

Main model uncertainties are related to quantifying the role of sediments in
continental production and to estimating the depletion of radiogenic elements
by continental crust production. Future work on these issues is required in
order to improve understanding the complexities of our own planet and the
prediction of the continental fraction on Earth-like exoplanets.

## Conflict of interest

The authors declare that they have no conflict of interest.

###### Acknowledgements.

We thank two anonymous reviewers for helpful comments and suggestions. We also
thank Hendrik Hansen-Goos for valuable insights into stochastic geometry and
Thibaut Roger for designing figure 1. Part of this research was done during a
visitor scientists stay of D.H. at ISSI. Support by ISSI is gratefully
acknowledged.

##  Appendix A Coupled continental and mantle water cycles

Key variables in the model are the surface fraction of continental crust $A$
and its rate of change $\dot{A}$, which depends on the rates of continental
production $\dot{A}_{g}$ (index $g$ for gain) and continental erosion
$\dot{A}_{l}$ (index $l$ for loss)

$\dot{A}=\dot{A_{g}}-\dot{A_{l}}.$ (4)

Analogously, for the mass rate of change of the mantle water concentration, we
have

$\dot{w}=\dot{w_{g}}-\dot{w_{l}}.$ (5)

Continental production depends both on the plate speed and on the rate of
sediment subduction, which in turn depends on the continental area (see
Section 4.1). A most simple parameterization of continental production that
captures the key physics of the system is given by

$\dot{A}_{g}^{*}=\left(\xi_{1}A^{*}+(1-\xi_{1})v_{plate}^{*}\right)L_{oo,oc}^{*},$
(6)

where the asterisk denotes parameters scaled to the present-day Earth, for
example

$\dot{A}_{g}^{*}=\frac{\dot{A}_{g}}{\dot{A}_{g,E}},$ (7)

and where the index $E$ denotes present-day Earth values. In Eq. 6, a fraction
$\xi_{1}$ of the continental production rate $\dot{A}_{g}$ is taken
proportional to the continental area whereas the rest is taken proportional to
the plate speed (derived in Section C). We account for the fact that
subduction can only occur at those convergent plate boundaries that are not of
continent-continent type by including the function
$L_{oo,oc}(A)=L_{oo}(A)+L_{oc}$(A), where $L_{oo}(A)$ and $L_{oc}(A)$ are the
total lengths of ocean-ocean type subduction zones and ocean-continent type
subduction zones, respectively (see B).

An essential part of continental erosion takes place on the surface. The rate
of surface erosion increases with the surface area but depends, in addition,
on the surface relief, the rate of rainfall, and the weatherability of the
surface. Of these, we keep for simplicity the dependence on the surface area.
There are other mechanisms of continental erosion of which we include
subduction erosion for which the subducting plate scrapes off part of the
overlying continent. The rate of subduction erosion mainly depends on the
total length of ocean-continent type convergent plate boundaries. It would
also depend on a friction coefficient but since we scale to the present Earth
and assume that parameters are constant the friction coefficient will cancel.
The subduction speed may matter but for plate speeds higher than the present
one, the dependence is likely small (Stern, 2011). Thus,

$\dot{A}_{l}^{*}=\xi_{2}A^{*}+(1-\xi_{2})L_{oc}^{*},$ (8)

where $\xi_{2}$ is a constant.

We assume that the subducted water flux (regassing) to the mantle follows the
same functional dependence on plate speed, sediment production and length of
subduction zones as the continental production rate (see also Karlsen . 2019).
This is a reasonable first-order approach, since the production rate of
continental crust rock is taken directly proportional to the water subduction
rate. The regassing rate should, of course, differ from the water subduction
rate as water is consumed upon melting and lost through dehydration. However,
we assume for simplicity that the ratio between the two remains constant. The
relative mass rates – scaled to their respective present-day Earth values –
are, therefore, equal:

$\dot{w}_{g}^{*}=\dot{V}_{g}^{*}.$ (9)

Mantle degassing, the release of water to the combined atmosphere-ocean
reservoir depends, to first order, on the rate at which mantle material is
transported into the melting region beneath mid-ocean ridges and other
volcanic units and on the mantle water concentration. We neglect degassing
through volcanism in other geologic provinces.

$\dot{w}_{L}^{*}=w^{*}v_{p}^{*},$ (10)

where $v_{p}^{*}$ is the plate speed scaled by its present-day value.

Equations 1 and 2 together with Eqs. 4 – 7 and 8 need values of
$\dot{A}_{l,E}$ and $\dot{A}_{g,E}$ and $\dot{w}_{l,E}$ and $\dot{w}_{g,E}$
for rescaling. Whereas the exact present-day rates may be of secondary
importance, qualitative differences can result from either assuming an
equilibrium or disequilibrium between present-day Earth sources and sinks.
Even though a present-day equilibrium appears reasonable to assume, we cannot
exclude that the sluggish Earth system only slowly converges towards an
equilibrium state. We therefore introduce

$f_{r}=\frac{\dot{A}_{l,E}}{\dot{A}_{g,E}}=\frac{\dot{w}_{l,E}}{\dot{w}_{g,E}},$
(11)

and note that a present-day Earth equilibrium implies $f_{r}=1$.

With Equations 4 – 11, the coupled continent–water system can qualitatively be
described. A combination of these equations result in Eqs. 1 and 2.
Particularly important are the constants $\xi_{1}$ and $\xi_{2}$, which
determine the strengths of positive and negative feedback, respectively.

##  Appendix B Length of subduction zones

Figure 13: Total length of ocean-continent (yellow) and combined ocean-ocean
plus ocean-continent (blue dotted) type subduction zones scaled to the maximum
length of subduction zones.

Calculating the total length of subduction zones, we follow Höning . (2014)
and assume that the total length of ocean-continent type subduction zones is
proportional to the total length of continental margins ($L_{oc}\sim
L_{mar}$). Modeling continents as randomly distributed spherical caps that may
overlap, we find

$\frac{L_{mar}}{4\pi
R_{E}}=n(1-A)\sqrt{\left(\frac{1}{n}\text{ln}\left[\frac{1}{1-A}\right]\right)-\left(\frac{1}{n}\text{ln}\left[\frac{1}{1-A}\right]\right)^{2}},$
(12)

where $L_{mar}$ is the length of continental margins, $R_{E}$ is the radius of
Earth. For $n$ we use the number of continents on Earth of $n=7$. Höning Spohn
(2016) have shown that $L_{mar}/L_{mar,max}$ shown in Fig. 13 will not vary
significantly with $n$ for $5\leq n\leq 9$.

Eq. 12 has been derived as follows: Assume $n$ spherical caps, each with area
$A_{0}$ and circumference $L_{0}$, Poisson randomly distributed on a sphere
with surface area $A_{surf}$. For a sufficiently large number $n$, the area
covered by spherical caps $A_{c}$ is (Mecke Stoyan, 2000; Schneider Weil,
2008)

$\frac{A_{c}}{A_{surf}}=1-e^{-\nu},$ (13)

where $\nu=n\cdot\frac{A_{0}}{A_{surf}}$, and the total length of margins
$L_{mar}$ is

$\frac{L_{mar}}{n}=L_{0}e^{-\nu}.$ (14)

Using the definition $A=\frac{A_{c}}{A_{surf}}$, Eq. 13 yields

$\frac{A_{0}}{A_{surf}}=\frac{1}{n}\ln\left(\frac{1}{1-A}\right),$ (15)

and combining Eq. 13 and Eq. 14 yields

$\frac{L_{mar}}{n}=L_{0}(1-A).$ (16)

Furthermore, consider the radius of the base of the spherical cap $r_{c}$ and
its height $h_{c}$. The radius of the sphere $R_{E}$ is then

$R_{E}^{2}=r_{c}^{2}+(R_{E}-h_{c})^{2}.$ (17)

Since $L_{0}=2\pi r_{c}$ and $A_{0}=2\pi R_{E}h_{c}$, we get

$L_{0}=2\pi\sqrt{R_{E}^{2}-\left(R_{E}-\frac{A_{0}}{2\pi R_{E}}\right)^{2}}.$
(18)

Combining Eq. 16 and Eq. 18 yields

$\begin{split}\frac{L_{mar}}{n}&=2\pi
R_{E}(1-A)\sqrt{1-\left(1-\frac{A_{0}}{2\pi R_{E}^{2}}\right)^{2}}\\\ &=4\pi
R_{E}(1-A)\sqrt{\frac{A_{0}}{A_{surf}}-\frac{A_{0}^{2}}{A_{surf}^{2}}},\end{split}$
(19)

and combining Eq. 15 and Eq. 19 results in Eq. 12.

With increasing continental area, continents increasingly overlap, which in
turn reduces the total length of subduction zones. Following our assumption of
$L_{oc}\sim L_{mar}$, the total length of ocean-continent-type subduction
zones has its maximum at a continental surface fraction of $\approx 0.37$. For
a smaller (larger) fraction we assume that the remaining length of subduction
zones is of ocean-ocean (continent-continent) type (Fig. 13).

The derivative of Eq. 12 with respect to A is

$\frac{d}{dA}\frac{L_{mar}}{4\pi
R_{E}}=\dfrac{n+2\left(n+1\right)\ln\left(1-A\right)+\ln^{2}\left(1-A\right)}{\sqrt{-\ln^{2}\left(1-A\right)-n\ln\left(1-A\right)}}$
(20)

##  Appendix C Interior thermal evolution

The calculation of the thermal evolution of the mantle using Eq. 3 is based on
the conventional thermal history models of terrestrial planets (e.g., Schubert
., 2001) and includes terms describing the heat flow though the surface, heat
production within the mantle, and the heat flow from the core.

For a constant (present-day Earth) continental crust coverage $A_{E}$, the
mantle heat production rate can be calculated following Korenaga (2008)

$Q_{A_{E}}(t)=Q_{0}\sum_{i}q^{\prime}_{i}\exp\left\\{-\frac{\log(2)}{h^{\prime}_{i}}(t-4.5)\right\\},$
(21)

where $t$ is time in Gyr, $Q_{0}$ is the present-day Earth mantle heat
production rate and $q^{\prime}_{i}$ and $h^{\prime}_{i}$ are the relative
present-day heat production rates and half-lifes of the individual radiogenic
elements 238U, 235U, 232Th, and 40K. Eq. 21 accounts for radiogenic decay but
not for the transfer of radiogenic elements to the continental crust upon
growth of the latter. We account for the depletion of the mantle due to
continental crust growth by writing

$\frac{Q_{m}(A,t)}{Q_{A_{E}}}=1+\xi_{4}(1-A^{*}),$ (22)

where $Q_{m}(A,t)$ is the mantle radiogenic heat production rate as a function
of time and continental coverage and where

$\xi_{4}=\frac{Q_{C_{E}}}{Q_{A_{E}}+Q_{C_{E}}}$ (23)

with Q${}_{C_{E}}$ the present-day heat production rate in the continental
crust. In a scenario where we neglect the transfer of radiogenic elements, we
set $\xi_{4}=0$.

The mantle heat flow is given by

$q_{m}=A_{surf}k\frac{T_{u}-T_{s}}{\delta_{u}},$ (24)

where $T_{u}$ and $T_{s}$ are the temperatures of the upper mantle and the
surface, respectively, $k$ is the thermal conductivity, $A_{surf}$ is the
surface area of Earth, and $\delta_{u}$ is the thickness of the upper boundary
layer and given by

$\delta_{u}=\left(\frac{\kappa\eta Ra_{c}}{g\alpha_{x}\rho_{m}\Delta
T}\right)^{\beta},$ (25)

where $\kappa$, $g$, $\alpha_{x}$, $\rho$, are the thermal diffusivity,
gravitational acceleration, thermal expansivity, and density of the mantle,
$Ra_{c}$ is the critical Rayleigh Number, $\Delta T$ is the temperature
difference across the boundary layer, and $\eta$ is the mantle viscosity given
by

$\eta_{m}=f_{w}^{-1}\,\eta_{0}\,\text{exp}\left(\frac{E_{a}+p_{m}V_{a}}{R_{g}}\left(\frac{1}{T_{m}}-\frac{1}{T_{ref}}\right)\right),$
(26)

where $\eta_{0}$ is a reference viscosity, $E_{a}$ and $V_{a}$ are the
activation energy and volume, respectively, $R_{g}$ is the universal gas
constant, $T_{m}$ and $p_{m}$ the temperature and pressure half-way through
the mantle, $T_{ref}$ a reference temperature, and $f_{w}$ accounts for the
dependence of the mantle viscosity on the concentration of water $w$. $f_{w}$
is given by

$f_{w}=\text{exp}\left(c_{0}+c_{1}\text{log}(w)+c_{2}\text{log}^{2}(w)+c_{3}\text{log}^{3}(w)\right),$
(27)

where $c_{0}$ through $c_{4}$ are constants. Following boundary layer theory
(e.g., Schubert ., 2001), the plate speed is a function of the boundary layer
thickness

$v_{p}=5.38\kappa\frac{d_{m}}{\delta^{2}},$ (28)

where $d_{m}$ is the thickness of the mantle. Scaled to the present-day Earth,
the mantle heat flow can, therefore, be written as

$q_{m}=q_{m,E}\Delta T_{u}^{*(1+\beta)}\sqrt{v_{p}^{*}},$ (29)

where $\Delta T_{u}=T_{u}-T_{s}$ and

$q_{m,E}=A_{surf}\frac{k\Delta T_{u,E}}{\delta_{u,E}}.$ (30)

The parameterisation of the effects of thermal blanketing by the continental
crust will complete the derivation of Eq. 3 and is introduced in D.

##  Appendix D Continental insulation

Considering thermal blanketing of the mantle by (partly) insulating
continents, we begin by defining $\chi$ as the ratio of the present-day
specific heat flow from the mantle through continental crust to that through
oceanic crust

$\chi\equiv\frac{q_{c}/A}{q_{o}/(1-A)}=\frac{q_{c,E}/A_{E}}{q_{o,E}/(1-A_{E})}.$
(31)

where $q_{o}$ is the heat flow from the mantle through the oceanic crust and
$q_{c}$ the heat flow from the mantle through the continental crust. We assume
that $\chi$ remains constant throughout the evolution as it depends on the
heat transfer properties of the oceanic and continental crust rather than on
the thermal state of the mantle. We acknowledge, however, that $\chi$ may
depend on the areal extent of the crust and the ratio between continental
margins and old stable continental shield areas.

The heat flow from the mantle to the surface $q_{s}$ can then be written as

$q_{s}=q_{o}+q_{c}=q_{m,E}\left((1-A)+A\chi\right)\Delta
T_{u}^{*(1+\beta)}\sqrt{v_{p}^{*}},$ (32)

The mantle heat flow $q_{m,E}$ has been defined by Eq. 30.

Since by definition

$\xi_{3}\equiv\frac{q_{o,E}}{q_{o,E}+q_{c,E}},$ (33)

or $q_{o,E}=\xi_{3}q_{s,E}$ and $q_{c,E}=(1-\xi_{3})q_{s,E}$, we can write

$\chi=(1/A_{E}-1)(1/\xi_{3}-1)=1/(A_{E}\xi_{3})-1/A_{E}-1/\xi_{3}+1.$ (34)

For the surface heat flow (Eq. 32), we get

$q_{s}=q_{m,E}\left(1+A^{*}/\xi_{3}-A^{*}-A/\xi_{3}\right)\Delta
T_{u}^{*(1+\beta)}\sqrt{v_{p}^{*}}.$ (35)

The present-day Earth surface heat flow is analogously:

$q_{s,E}=q_{m,E}\left((1-A_{E})+A_{E}\chi\right)=q_{m,E}\left(\frac{1-A_{E}}{\xi_{3}}\right).$
(36)

The relative heat surface heat flow is then

$\begin{split}\frac{q_{s}}{q_{s,E}}&=\Delta
T_{u}^{*(1+\beta)}\sqrt{v_{p}^{*}}\frac{\left(1+A^{*}/\xi_{3}-A^{*}-A/\xi_{3}\right)}{\left(\frac{1-A_{E}}{\xi_{3}}\right)}\\\
&=\Delta
T_{u}^{*(1+\beta)}\sqrt{v_{p}^{*}}\frac{\left(\xi_{3}+A^{*}-A^{*}\xi_{3}-A\right)}{\left(1-A_{E}\right)}\\\
&=\Delta
T_{u}^{*(1+\beta)}\sqrt{v_{p}^{*}}\left[(1-A)^{*}\xi_{3}+A^{*}(1-\xi_{3})\right].\end{split}$
(37)

##  Appendix E Long-term carbon cycle

The plate speed determines the flux of mantle material into the melting region
beneath mid-ocean ridges and thereby the mass rate of carbon outgassing from
the mantle into the atmosphere

$F_{out}=\frac{R_{man}}{V_{m}}f_{deg}2v_{p}L_{mor}d_{melt},$ (38)

where $R_{man}$ is the carbon reservoir in the mantle, $V_{m}$ the mantle
volume, $f_{deg}$ the fraction of upwelling mantle material that degasses,
$L_{mor}$ the total length of mid-ocean ridges, and $d_{melt}$ the depth below
mid-ocean ridges where melting begins.

The weathering flux is calculated by (Foley, 2015)

$F_{w}=F_{ws}\left(1-\exp\left(-f_{land}^{*}\frac{F_{w,E}}{F_{ws}}\left(P_{CO_{2}}^{*}\right)^{b}\left(P_{sat}^{*}\right)^{a}\exp\left(\frac{E_{a,s}}{R_{g}}\left(\frac{1}{T_{s,E}}-\frac{1}{T_{s}}\right)\right)\right)\right),$
(39)

where $P_{CO_{2}}$ is the partial pressure of CO2 in the atmosphere, $P_{sat}$
is the saturation pressure, $E_{a,s}$ is the activation energy of silicate
weathering, $T_{s}$ is the surface temperature, $f_{land}$ is the land
fraction, and $F_{ws}$ is the supply limit to weathering. Since we explicitly
calculate surface erosion in this model by setting it proportional to the
surface area of continental crust, we use $f_{land}^{*}=A^{*}$. We note that
this is a significant simplification in order to obtain first-order results
without introducing additional equations. In reality, the emerged land
fraction would also depend on additional parameters such as topography and
surface water budget. The supply limit to weathering is given by

$F_{ws}=\frac{A_{surf}f_{land}E_{max}\psi\rho_{r}}{\overline{m}_{c}},$ (40)

where $E_{max}$ is the maximum physical erosion rate (in thickness per time),
$\psi$ is the fraction of reactable cations, $\overline{m}_{c}$ is its average
molar mass, and $\rho_{r}$ is the density of regolith (Foley, 2015).

Seafloor weathering is calculated as (Sleep Zahnle, 2001; Foley, 2015)

$F_{sfw}^{*}=v_{p}^{*}\left(P_{CO_{2}}^{*}\right)^{\alpha}.$ (41)

Following classic long-term carbon cycle models (Sleep Zahnle, 2001; Foley,
2015), the weathering fluxes transfer carbon from the atmosphere to the
subduction zones considering half of that carbon is released during carbonate
precipitation on the seafloor. Of the carbon that is subducted, we assume that
another half is returned to the mantle while the rest is degassed at volcanic
arcs back into the atmosphere.

The saturation pressure is calculated as

$P_{sat}=P_{sat0}\exp\left(-\frac{m_{w}L_{w}}{R_{g}}\left(\frac{1}{T_{s}}-\frac{1}{T_{s0}}\right)\right),$
(42)

where $P_{sat0}$ and $T_{s0}$ are reference values for the saturation pressure
and surface temperature and $m_{w}$ and $L_{w}$ are the molar mass and latent
heat of water, respectively. The surface temperature is calculated as

$T_{s}=T_{s,E}+2(T_{e}-T_{e,E})+4.6\left(\left(P_{CO_{2}}^{*}\right)^{0.346}-1\right),$
(43)

where $T_{e}$ is the effective temperature and given by

$T_{e}=\left(\frac{S(1-\lambda)}{4\sigma}\right)^{\frac{1}{4}},$ (44)

where $\lambda$ is the planetary albedo, $\sigma$ is the Stefan-Boltzmann
constant, and $S$ is the solar irradiation, which is assumed to increase by a
factor of 1/3 throughout 4.5 Gyr (Gough, 1981). Parameter values are given in
Tables 2, 3, and 4.

##  Appendix F Parameter Values

Table 2: Parameter values used in the model. Carbon cycle parameters from Foley (2015). Symbol | Description | Value
---|---|---
$c_{p}$ | Specific heat capacity | 1200 J kg-1 K-1
$V_{m}$ | Mantle volume |  $9.1\cdot 10^{20}$ m3
$k$ | Thermal conductivity | 4.2 W m-1 K-1
$\kappa$ | Thermal diffusivity | 10-6 m2 s-1
$Ra_{c}$ | Critical Rayleigh Number | 1100
$g$ | Gravitational acceleration | 9.81
$\alpha_{x}$ | Thermal expansivity | $2\cdot 10^{-5}$
$d_{m}$ | Mantle thickness | 2900 km
$L_{mor}$ | Total length mid-ocean ridges | 6$\cdot 10^{7}$ m
$d_{melt}$ | Maximum depth of melting beneath mid-ocean ridges | 120 km
$A_{surf}$ | Earth surface area | 5.1$\cdot 10^{14}$ m2
$E_{a}$ | Activation energy (viscosity) | 335 kJ mol-1
$V_{a}$ | Activation volume (viscosity) | 4 cm3 mol-1
$R_{g}$ | Universal gas constant | 8.314 J mol-1 K-1
$R_{g}$ | Stefan-Boltzmann constant | 5.67$\cdot 10^{-8}$ W m-2 K-4
$E_{a,s}$ | Activation energy (weathering) | 42 kJ mol-1
$E_{max}$ | Maximum physical erosion rate | 0.01 m yr-1
$\rho_{r}$ | Density regolith | 2500 kg m-3
$\psi$ | Fraction reactable cations | 0.08
$\overline{m}_{c}$ | Average molar mass reactable cations | 32 g mol-1
$m_{w}$ | Molar mass of water | 44 g mol-1
$L_{w}$ | Latent heat of water | 2469 J g-1
$f_{deg}$ | Fraction of upwelling mantle that degasses | 0.32
Table 3: Present-day Earth parameters and reference parameters used in the model. F15: Foley (2015) Symbol | Description | Value | Reference
---|---|---|---
$P_{CO_{2},E}$ | Partial pressure of CO2 | 33 Pa |
$T_{s,E}$ | Surface temperature | 285 K |
$S_{E}$ | Solar irradiation | 1360 W m-2 |
$T_{ref}$ | Reference temperature (Eq. 26) | 2050 K |
$\eta_{0}$ | Reference viscosity (Eq. 26) | 1021 Pa s |
$P_{sat0}$ | Reference saturation pressure (Eq. 42) | 610 Pa | F15
$T_{s0}$ | Reference surface temperature (Eq. 42) | 273 K | F15
$q_{m,E}$ | Mantle heat flow (Eq. 3) | 35 TW | $A_{surf}\frac{k\Delta T_{u,E}}{\delta_{u,E}}$
Table 4: Scaling constants used in the model. F15: Foley (2015), L08: Li . (2008) Symbol | Equation | Value | Reference
---|---|---|---
$\alpha$ | Eq. 41 | 0.25 | F15
$\beta$ | Eq. 25 | 1/3 |
$a$ | Eq. 39 | 0.3 | F15
$b$ | Eq. 39 | 0.55 | F15
$c_{0}$ | Eq. 27 | -7.9859 | L08
$c_{1}$ | Eq. 27 | 4.3559 | L08
$c_{2}$ | Eq. 27 | -0.5742 | L08
$c_{3}$ | Eq. 27 | 0.0227 | L08

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