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\part{The oceans}
\chapter{The ocean lithosphere}
The age of the oceanic crust, as reflected in the magnetic striping of the ocean
floor, increases with distance away from the mid-ocean ridges, indicating that the
ridges are the site of generation of new oceanic crust. The volcanic rocks extruded at
the surface of the ridges are exclusively basalt (mid-ocean ridge basalt or MORB)
which, together with there sub-volcanic intrusive equivalents - gabbros and sheeted
dykes, comprise the entire oceanic crust. The total thickness of the oceanic crust
generated by mafic igneous activity at the ridges is typically about
5-7 km. The structure of oceanic crust and parts of the subcrustal lithosphere can be directly
observed in some ancient orogenic belts where fragments of the oceanic lithosphere have been
obducted to form {\em ophiolites} during collision processes, for example the Semail
ophiolite in Oman.
\begin{figure}[h,t,b]
\label{oceanlith}
\vspace{1.5in}
\hspace{0.3in}
\special{epsf=:gdfigs:chap7:Fig7_1.eps scale=0.77}
\caption{\protect\scriptsize{Schematic structure of the ocean lithosphere. The ocean lithosphere
consists of about 6-7 km thick crust (heavy stiple), and the mantle lithosphere
(intermediate stiple) which thickens with age (and hence distance away from the ridge as
indicated by arrows). Light stiple show the region of decompression melting beneath the
ridge. Stream lines in the asthenosphere may be largely decoupled from the motion of the
overlying lithosphere (see Section 5.5), although asthenosphere must undergo
decompression immediately beneath the ridge.}} \end{figure}
\section{Age, bathymetry and heatflow}
In ocean lithosphere younger than about 80 Ma there is a remarkable correspondence
between age of the ocean crust, the depth to the sea floor (bathymetry) and the heat flow
through the lithosphere (Fig. \ref{oceanlith}); with bathymetric depth increasing, and
surface heatflow decreasing, with the $\sqrt{Êage}$. This correspondence between age,
bathymetry and heatflow is due to time dependent changes in the thickness of the
lithosphere. Two models for the ocean lithosphere have been proposed in order to account
for this relation: the {\it half-space model} and the {\it thermal plate model}.
\subsubsection{The half-space model}
The cooling of ocean lithosphere after formation at a ridge can be treated as a thermal
conduction problem (see Chapter 3) in which a non-steady state condition (the situation
at the ridge) gradually decays towards a thermally equilibrated state (as the ocean
lithOSphere slides away from the ridge). The analogy (Fig \ref{half}a) can be made
with the cooling of a semi-infinite half space, which is given by:
\begin{equation}
\label{eq:halfs}
\frac{T_{z}Ê-ÊT_{m}}{ T_{s}Ê-ÊT_{m}} =
erfc\left(\frac{z}{2\,\sqrt{\kappa\, t}}\right)
\end{equation}
where $T_{z}$ is the temperature at depth $z$, $T_{s}$ is the
temperature at the surface interface of the semi-infinite half space (which in this case
is the temperature of ocean water and is taken to be $0^{o}$C), $T_{m}$ is the
temperature of the half space in the initial condition and which is
maintained at infinite distance for all time (in our case the temperature
of the deep mantle, {$1280^{o}$C}), {$\kappa$} is the thermal diffusivity
and {$t$} is time (the error function, {$erf$}, and its compliment, {$erfc$},
arise commonly in analytical solutions to the heat equation and related differential equations which
employ similarity variables).
The behaviour of the error function, and hence Eqn \ref{eq:halfs}, is illustrated in
Figure \ref{half}b. As {$z$ tends to $\infty$ or {$t$} to 0 then:
\begin{displaymath}
erfc \left(\frac{z}{2\,\sqrt{\kappa\, t}} \right) \rightarrow 0
\end{displaymath}
and {$T_{z}$} approaches {$T_{m}$}. As {$z$} tends to 0 or {$t$} to $\infty$
then
\begin{displaymath}
erfc\left(\frac{z}{2\,\sqrt{\kappa \, t}} \right) \rightarrow 1
\end{displaymath}
and {$T_{z}$} approaches {$T_{s}$}, providing
\begin{displaymath}
\left(\frac{z}{2\,\sqrt{\kappa \,t}} \right) < 2
\end{displaymath}
\begin{figure}[h,t,b]
\label{half}
\vspace{2.00in}
\special{epsf=:gdfigs:chap7:Fig7_2.eps scale=0.65}
\caption{\protect\scriptsize{Schematic thermal structure of the ocean lithosphere treated as a problem of the cooling
of a semi-infinite half space. (a) shows the thermal structure at the
ridge ($t_{0}$) where asthenosphere at temperature $T_{m}$ is juxtaposed with ocean
waters at temperature $T_{s}$. The thermal structure at successive distances away
from the ridge where cooling of the initial temperature discontinuity in the
semi infinite half space has lead to thickening of the ocean lithosphere is
shown by the curves $t_{1}$ and $t_{2}$. (b) shows the error function
($erf$) and complimentary error function $(erfc = 1 - erf)$. }}
\end {figure}
From the half space model, theoretical predictions about the temporal evolution of
lithospheric thickness, heat flow and bathymetry can be derived from the basic equations
governing the thermal evolution of the lithosphere. Following Turcotte and Schubert (1982, p.
164-165, p. 181-182) the thickness of the lithosphere, $z_{l}$, as a
function of age, $t$, is given by:
\begin{equation}
\label{eq:lith}
z_{l} = 2.32\, \sqrt{Ê\kappaÊ\,t}Ê
\end{equation}
The depth of the ocean floor beneath the ridge crest, $w$, at time, $t$, after formation is given by:
\begin{equation}
\label{eq:water}
w = \frac{2Ê\,\rho_{m}\,Ê\alpha \,ÊT_{m}}{\rho_{m}Ê-Ê\rho_{w}}
\sqrt{Ê\frac{\kappaÊ\,t}{Ê\pi}}
\end{equation}
where $\alpha$ is the thermal coefficient of expansion and $\rho_{m}$ and $Ê\rho_{w}$
are the density of mantle and water, respectively. The surface heat flow, $q_{s}$, at
time, $t$, after formation is given by:
\begin{equation}
\label{eq:heatf}
q_{s} = k\,T_{m}\, \frac{ 1}{\sqrt{\pi \,\kappaÊ\,t}}
\end{equation}
where $k$ in the thermal conductivity.
Some interesting consequences arise from the behaviour described by these equations.
For example, Equation \ref{eq:water} shows that the average depth of the ocean
is proportional to its crustal age. Given a constant volume of sea water, a change in the average age and hence depth of the
oceans must result in a change in sea level, which is reflected in the geological record by the
extent of ocean onlap on the continents. The average age of the oceans is inversely proportional
to the rate of sea floor spreading, and directly proportional to the square root of the mean age of
subduction. The mean depth of the oceans, $\omega$, as a function of the average
age of subduction, $t$, is given by
\begin{equation}
\label{eq:depth}
\omega = \frac{1}{\tau}\,\int_{0}^{\tau}\, wÊ \,dt
\end{equation}
substituting for $w$ in Eqn 2.2 gives:
\begin{equation}
\label{eq:depth2}
\omega = \frac{4\,
\rho_{m} \, \alpha_{\upsilon}\,ÊT_{m}}{3Ê\,\left(\rho_{m}-\rho_{w}\right)}
\sqrt{\frac{\kappa\, \tau}{\pi}}
\end{equation}
Secular variations in
the rate of sea floor spreading, reflected in the mean age of subduction may
therefore have important implications to the average height of the oceans.
Indeed, this explanation has been used to account for high ocean stands
during the Cretaceous (when sea level may have been up to 300 m higher than
today) which correspond with periods of fast ocean floor spreading (as
indicated by analysis of ocean floor magnetic anomalies).
\subsubsection{The thermal plate model}
The semi-inifinite half space model predicts continuous cooling (albeit at a rate that
gradually decays with time) and therefore thickening of the lithosphere through time
(Figure \ref{model}). While the predictions are in remarkable agreement with the observations on
bathymetry and heat flow in young ocean lithosphere these relationships appear to break down in
ocean lithosphere older than about 80 Ma, when the thermal structure of the oceanic lithosphere
appears to be stabilised.
\begin{figure}[h,t,b]
\label{model}
\vspace{1.9in}
\special{epsf=:gdfigs:chap7:Fig7_3.eps scale=0.68}
\caption{\protect\scriptsize{Thermal structure of the ocean lithosphere predicted by the
semi-infinite half space model (a) and by the thermal plate model (b).}}
\end{figure}
The semi-infinite half space model assumes that the half space is not convecting. In the
earth the deep mantle is convecting, with the consequence that a convective heat flux is
provided at the top of the convecting layer (see Chapter 7). The thermal plate model accounts
for this apparent time independent behaviour of old oceanic lithosphere by assuming that the
convection in the subjacent mantle provides sufficient heat to the base of the cooling lithosphere
to stabilise the cooling once a critical thickness is reached, the observations suggest this critical
thickness is about 125 km corresponding to the thickness of ~80 Ma old lithosphere (Figure
\ref{model}b). Simply stated, the oceanic plate structure is thermally stabilised when the convective
heat supply to the base of the lithosphere balances heat lost through the lithosphere.
\section{Force balance on the ocean ridge}
For young ocean lithosphere the cooling of a semi infinite half space provides an
acceptable approximation and therefore Eqns \ref{eq:halfs} - \ref{eq:water} can be used as the basis
to calculate the force balance on the ocean ridge. The isostatic compensation of the oceanic
lithosphere causes the youngest ocean to form a high, albeit submerged, mountain range standing out
above the abyssal plains. Such profound topographic gradients necessarily lead to substantial
horizontal buoyancy forces (Chapter 2), termed the ridge push. In this section we provide the
methodology for calculating the magnitude of the ridge push.
\begin{figure}[h,t,b,p]
\label{ocean}
\vspace{3.7in}
\hspace{0.7in}
\special{epsf=:gdfigs:chap7:Fig7_4.eps scale=0.85}
\caption{\protect\scriptsize{Ridge push is the force resulting
from isostatically induced topographic gradients in the oceanic lithosphere.
(a) shows a diagramatic representation of the force balance between
a ridge and ocean lithosphere at age $t = t_{1}$. (b) shows the density
distribution appropriate to (a). The dashed line shows the vertical
density structure at $t = 0$. The solid line shows the vertical density
structure at $t = t_{1}$. (c) shows the graphical solution to the force balance.}
}
\end{figure}
Refering to Figure \ref{ocean}a the ridge push, $F_{R}$, operating on the oceanic
lithosphere of age, $t_{1}$, and depth below the ridge crest, $w$, is given
by:
\begin{equation}
\label{eq:ridgef}
F_{R} = F_{1} - F_{2} - F_{3}
\end{equation}
which is equivalent to solving Eqn 2.3, as shown diagramatically in Figure \ref{ocean}c.
Note that $F_1$ corresponds to the outward push of the asthenosphere beneath the
mid-ocean ridge while $F_2$ and $F_3$ correspond, respectively, to the push of the
water column and the old ocean lithosphere inward against the ridge.
\begin{figure}[h,t,b]
\label{push}
\vspace{1.9in}
\hspace{0.5in}
\special{epsf=:gdfigs:chap6:Fig6_5.eps scale=0.55}
\caption{\protect\scriptsize{
Ridge push, $F_{R}$, plotted as function of depth of ocean beneath ridge crest,
$w$.}}
\end{figure}
The quantitative evaluation of Eqn \ref{eq:ridgef} is given in the Appendix A.3. The
solution of Eqn \ref{eq:ridgef} for any depth, $w$, below the ridge crest is shown in
Figure \ref{push}, assuming the following physical properties $\alpha = 5$ x $10^{-5}$,
$\rho_{m} = 3300$ kg m$^{-3}$, $\rho_{w} = 1000$ kg m$^{-3}$, $T_{m} = 1250^{o}$C and $g
= 10$ m s$^{-2} $.
\begin{figure}[h,t,b]
\label{geotherm}
\vspace{3.0in}
\hspace{1in}
\special{epsf=:gdfigs:chap7:Fig7_6.eps scale=0.65}
\caption{\protect\scriptsize{The oceanic lithosphere is characterised by its conductive geothermal
gradient. The thermal gradient of the asthenosphere which is relatively well
mixed (probably due to convection) is largely the isentropic temperature
(adiabatic) gradient of the mantle due to volume and heat capacity ($C_{p}$)
changes with changes in pressure (depth) The temperature at which this
adiabatic temperature gradient extrapolates to the earth's surface is refered to
as the potential temperature. Note that with the lithosphere of normal
thickness (125 km), the solidus of the mantle peridotite nowhere intersects the
geotherm but that on rifting of the lithosphere, the decompressed
asthenosphere's adiabat intersects the solidus at about 40 km depth. }
} \end{figure}
\section{Formation of the oceanic crust}
The ridge push resulting from the topography of the ocean floor, and the density
structure within the oceanic lithosphere, provides (along with slab pull) one of the primary
driving forces for lithospheric motion. The ridge push effectively maintains the constant
rupturing of the oceanic lithosphere, and its separation on either side of the ridges. An
important result of this rupture of the lithosphere at the ridges relates to the decompression of
the underlying asthenosphere.
\begin{figure}[h,t,b]
\label{melt}
\vspace{2.6in}
\hspace{0.3in}
\special{epsf=:gdfigs:chap7:Fig7_7.eps scale=0.63}
\caption{
\protect\scriptsize{P-T diagram showing melting field of garnet peridotite and adiabatic (isentropic) decompression
paths for mantle with potential temperatures of $1280^{o}C$, $1380^{o}C$,
$1480^{o}C$ and $1580^{o}C$, respectively (after McKenzie and Bickle, 1988).}}
\end{figure}
The decompression of asthenosphere beneath spreading ridges is so rapid that there is
virtually no loss of heat per unit rock mass; the decompression is therefore isentropic. Since
small volume increases occur during isentropic decompression there is necessarily a small
decrease in the the heat content per unit volume, and hence temperature. The change in
temperature with pressure at constant entropy defines the adiabat. The entropy, $S$, volume, $V$,
pressure and temperature of a system are related by the Clausius - Clapeyron equation:
\begin{equation}
\label{eq:clas}
\frac{\Delta P}{\Delta T} = \frac{\Delta S}{\Delta V}
\end{equation}
During isentropic (adiabatic) decompression, the decrease in pressure is
accompanied by only small volume increases and thus T must decrease. This adiabatic
gradient is about $1^{o}$C/km in the solid mantle. If the system becomes partly
molten, then the change in volume with pressure is larger and T changes more quickly.
Since the temperature of the convective mantle is not constant but lies on an adiabat
we characterise it by its potential temperature ($T_{m}$), which is the projected of
the adiabat to the surface of the earth (i.e., at 1 atm).
If sufficient decompression occurs, melting of the asthenosphere will take place when the
adiabat passes through
its solidus (Figure \ref{melt} and \ref{amount}). The melt generated by this decompression has
the composition of MORB and provides the parental liquid for all igneous rocks
that make up the oceanic
crust.
The amount of melting generated due to decompression of asthenospheric mantle
beneath an active ridge segment depends entirely on the thermal structure of the asthenosphere
and the melting properties of the mantle as a function of pressure. For the present day thermal
structure ($T_{m} = 1280^{o}$C) the amount of melting during complete decompression,
amounts to a
vertical column some 7 km thick (Figure \ref{amount}). In the past, when the internal
temperature may have been considerably hotter than it is today, the column of
melt generated beneath the ridges, and hence the oceanic crust, may have been
significantly thicker than 7 km.
\begin{figure}[h,t,b]
\label{amount}
\vspace{2.6in}
\hspace{0.3in}
\special{epsf=:gdfigs:chap7:Fig7_8.eps scale=0.63}
\caption{\protect\scriptsize{
Thickness of melt (measured as a vertical column in kms) present below the given indicated depth
produced by the adiabatic decompression of garnet peridotite for different potential temperatures
(after McKenzie and Bickle, 1988). For the modern day mantle with a potential temperature of
$1280^{o}$C melting will not occur at depths les than about 45 km. Adiabatic
decompression of the modern day mantle by the complete removal of the overlying
lithospheric "lid" (for example at a spreading ridge) will result in the
generation of a 7 km pile of MORB-like melt (i.e., the oceanic crust).
}}
\end{figure}
\section{Coupling of the -spheres?}
Equation \ref{eq:water}, derived entirely from theoretical considerations, is in excellent agreement
with observed bathymetry of ocean lithosphere younger than about 80 Ma. Indeed, this remarkable
agreement between observations and the age-heatflow-bathymetry relationships predicted by
Eqns \ref{eq:halfs} - \ref{eq:water} provides one of the principal lynchpins of the plate tectonic
paradigm and one of the most persuasive lines of argument that the lithosphere is indeed thermally
stabilised. Morover, it suggests that the motion of the oceanic
lithosphere is largely decoupled from the flow in the underlying asthenosphere. There
is as yet no clear understanding of the location, or even the general planform of
mantle upwelling in the asthenosphere. Most importantly, we have shown that there is
no requirement that the ocean ridges represent the site of upwelling (Figure
\ref{oceanlith}). Wherever asthenospheric upwelling occurs it is
likely to modify the thermal structure of the overlying lithosphere, and the suggestion
is that the thermal structure of most old oceanic lithosphere has been modified to
some degree by upwelling from within the underlying mantle.
\section{Oceanic basalt chemistry}
The oceanic lithosphere is generated entirely by magmatic processes, most of these
concentrated at the mid ocean ridges, but with small, but scientifically interesting additions at
intraplate hot spots forming the ocean island and sea mount chains (eg Hawaii, the Azores,
Reunion, Iceland etc). Seafloor spreading generates about 20 km$^{3}$a$^{-1}$ of
Mid Ocean Ridge Basalt (MORB) by the decompressional fusion mechanism described
above (Figures \ref{geotherm}-\ref{amount}), making these by far the most important volcanic
provinces on Earth. From the perspective of this course we are most interested to know
what oceanic magmatism tells us about the chemistry and "ages" of their mantle
source regions.
With respect to important trace element concentrations and the abundance of radiogenic
isotopes, the basaltic rocks of the ocean islands (OIB) differ quite significantly from MORB.
As oceanic magmatism must be sampling the sub-lithospheric mantle this immediately indicates
that there are chemically distinct regions of the oceanic asthenosphere and the differences in Nd,
Sr and Pb isotopic compositions (see Figure \ref{oiso}) are such that these
separate regions must have remained isolated for timescales of the order of 2
Ga or more. This in turn places limits on the type of mantle dynamics and
convection that must have occured.
As discussed in the previous chapter, MORB has the Sr, Nd (and Pb)
isotopic characteristics, as well as incompatible trace element chemistry,
consistent with derivation from a source which underwent the extraction
of significant fractions of melt at some period in the past. Broadly speaking
the extent of depletion of Rb, U and Nd in the oceanic upper asthenosphere
as reflected by present-day MORB is consistent with being the compliment of the
continental crust. Clearly, continental crust is not being produced at the
present mid oceanic ridges and for the lithophile elements to find their way
from the sub-oceanic mantle to the continents we must invoke at least a
two-stage process. This second stage process may involve subduction
and arc formation.
Alternatively, it is possible that the main continental growth occurred in
the Precambrian and was unrelated to the Earth's present geodynamic mode.
\begin{figure}[h,t,b]
\label{oiso}
\vspace{3.0in}
\hspace{0.7in}
\special{epsf=:gdfigs:chap7:Fig7_9.eps scale=0.62}
\caption{
\protect\scriptsize{The range of radiogenic isotopic compositions (Nd, Sr, Pb and He) of
oceanic basalts. The stippled area is the compositional range of Mid Ocean Ridge
Basalts (MORB). The remaining envelope is the region of ocean island basalts
(OIB). Clearly the OIB sources of the whole Earth are not all the same (from
Allegre, 1987)}} \end{figure}
\subsubsection{Rare Gas Studies}
The atmophile inert gases, particularly He, Ar and Xe have recently
provided substantial evidence for the existence, nature and life-span of
important terrestrial reservoirs. These elements are of course strongly
concentrated in the Earth's atmosphere. Perhaps surprisingly
however, recent careful
measurements of gasses emmitted during volcanic eruptions, included in fresh
volcanic rocks (glasses) and even produced from hot springs and wells in
continental and oceanic regions, reveal that detectible amounts of these and
other gasses are still being released from the earth's interior. Furthermore
these often show significant isotopic differences from the atmospheric
reservoir indicating that emissions are tapping source regions which have been
isolated for significant portions of the Earth's history.
$^{40}$Ar is the radiogenic product of the decay of $^{40}$K and $^{36}$Ar is a
stable isotope. $^{4}$He is the product of various decay series of U and Th and
$^{3}$He is a stable isotope. The atmosphere has extremely high ratios of
$^{4}$He/$^{3}$He (722,000) and $^{40}$Ar/$^{36}$Ar (295) because it is dominated
by the accumulation of these products of radioactive decay of lithophile
elements.
MORB has high $^{4}$He/$^{3}$He ratios, though values are lower than those of the
atmosphere, and extremely high $^{40}$Ar/$^{36}$Ar. This is interpreted to
indicate that the source region of MORB was effectively purged of most
of its rare gas content early in the earth's history and that
most of the He and in particular, the Ar there now is derived from
subsequent U-Th and K decay. The existence of significant $^{3}$He does
however indicate that the early degassing of the earth's interior (probably
during core formation soon after accretion) was not total and that some
primitive reservoir in the mantle has survived. This conclusion is supported
by the $^{129}$Xe results which show that oceanic sources (OIB and MORB) are
still producing this isotope. $^{129}$Xe is the product of the decay of $^{129}$I
which has a half life of only 16 x 10$^{6}$ years and this decay scheme
therefore became extinct very early in the earth's history.
By contrast with MORB, some OIB have very low $^{4}$He/$^{3}$He and
$^{40}$Ar/$^{36}$Ar ratios, reflecting that a component of their source region is
primitive and had not undergone early degassing and melt extraction.