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\fBVIII. THERMAL EVOLUTION OF THE PLANETS\fR
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8.1 HEAT TRANSFER
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Given that the planet formed, the issue at hand is to understand how it evolved to its present state and how it may still evolve in the future. The primary effect is the evolution of the earth's temperature distribution: its \fIthermal history\fR. \fBHeat is the geological lifeblood of planets\fR.
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To study this, we need to get some basic concepts:
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\fIHEAT\fR is a measure of the internal \fIenergy\fR of a body
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\fITEMPERATURE\fR is a measure of the heat contained and the ability to transfer heat
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For example, if we change the temperature of a body by $DELTA T$, the heat change per unit volume is
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DELTA U ~=~rho ^ C sub {p} ^ DELTA T
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where $C sub {p}$ is the specific heat per unit mass, this tells us how much the object will heat up for a given temperature change, and $rho$ is the density (mass per unit volume).
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Heat has the units of energy (joules, ergs, or calories), and $C sub {p}$ has units of energy/degree-mass (joules/\(deC kg).
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Some unit relationships:erg = (dyne)(cm)
= $10 sup {-7}$ joules
calorie = 4.2 joules
joule = N m
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A useful physical analogy is Feynman's - heat is like the amount of water in a towel, while temperature is the "wetness." Given two towels of the same "wetness" the bigger one contains more water. A small towel can be much "wetter" but contain less water than a big towel.
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Heat can flow from one place to another by several methods:
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\fIConduction\fR - heat is transferred by molecular collisions through a solid object. If you touch something hot, the heat is transferred into your fingers by conduction.
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\fIConvection\fR - heat is transferred by motion of the material - for example, by a hot fluid. As the bottom of the pot heats up the water starts to convect, as hot water rises, transporting heat upward. Solid rocks convect at temperatures found in the mantle.
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\fIRadiation\fR - heat is transferred by electromagnetic waves, the way the sun heats the earth. This effect isn't that important in solid rock, but is crucial at the planet's surface.
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\fB8.2 HEAT CONDUCTION\fR
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Heat flows from a hot object (one at higher temperature) to a colder one (lower temperature) according to \fIFourier's Law of Conduction\fR
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q~=~-k {dT} over {dz}
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where q is heat flux, flow of heat \fIper unit time\fR and \fIunit area\fR,
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and k is thermal conductivity. Materials with k large conduct well - low k materials insulate.
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Example:
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move left 0.25 "T$+ DELTA T$" below
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"$dt over dz ~ =$ temperature gradient" at 2,0
"=${T-(T + DELTA T)} over {L}$ = $- { DELTA T} over L$" at 2,-0.4
"$q= k { DELTA T} over L$" at 2,-0.8
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Notice that the minus sign is set up to make heat go the right way - from hot to cold. (Imagine the consequences if the minus sign weren't there - warm objects would spontaneously heat up!)
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In the 1800s it was recognized that temperature in mines and caves increased with depth:
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typically ~~ {dT} over {dz}~=~ \(ti 25 ~ \(deC/km
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conductivities were measured to show k \(ti2 W/(m \(deC) $->$ (Watt=Joule/s) so, q = \(ti50 mW/$m sup 2$ or (in other words) \(ti1 x $10 sup {-6} cal over {cm sup 2~s}$
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you may see this written in "heat-flow units" (HFU)
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1 HFU= $10 sup {-6}$ cal/($cm sup 2$ s)
= 42 ${mW} over m sup 2$
(Note that these are per unit time)
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On continents, heat flow is typically \(ti60 mW/$m sup 2$ and rock thermal conductivities are \(ti3 W $m sup {-1} K sup {-1}$ so,
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{dT} over {dz}~=~q over k~=~{6~x~10 sup {-2}} over 3~=~ 2~x~10 sup {-2} {K} over m
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=~20~ {K}/km ~ is ~ the ~ temperature ~ gradient.
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\fB8.3 THE HEAT EQUATION\fR
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How does the heat, and hence temperature, inside an object change with time? Assume only two methods of change: conduction and production
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\fIHEAT INSIDE = HEAT CONDUCTED IN OR OUT + HEAT PRODUCED\fR
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rho ^ C sub {p} ^ dz { partial T} over { partial t} (z,^t)~=~ q(z)~-~q(z~+~dz)~+~ rho ^ H ^ dz
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difference in heat heat production (phase
going in and out changes, radioactive)
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\fIThe Conduction Term\fR
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q(z) ~-~ q(z~+~dz)~=~q(z)~-~ left ( q(z)~+~{ dq} over { dz} dz right )~=~-{ dq} over { dz} dz
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so - we have temperature as a function of position (z) and time (t) in T(z,t)
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Using Fourier's law of heat conduction
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q~=~-k {{ partial T} over { partial z}}
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{{ dq} over { dz}} dz ~=~ left ( ^ - k {partial sup {2} T} over {partial z sup 2} right ) ^ dz ~~ (if ~ k ~ is ~ constant)
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~=~ -k ^ dz {{ partial sup {2} T} over {partial z sup 2}}
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ASIDE $->$ Heat produced inside, assume H = heat production rate per unit mass, then
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rho H dz
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(we'll discuss later how heat is produced in the Earth, but here's an answer: radioactive)
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Equating our equations (dividing by dz) with our new conduction term gives
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rho C sub {p} {partial T (z,t)} over {partial t}~=~k {partial sup 2 T (z, t)} over {partial z sup 2} ~+~rho H
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This animal is called the 1-dimensional (only z coordinate) heat equation with conduction (we didn't worry about convection) IT IS VERY IMPORTANT! Since it tells how the temperature changes with time and position.
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\fB8.4 GEOTHERMS\fR
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Let's use the heat equation to find temperature in the earth-assuming that the temperature isn't changing with time. (this is called a \fIsteady state\fR $({partial T} over {partial t}$ = 0) solution).
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We know the heat flow at the earth's surface $- ^ q sub s$, and the temperature at the surface $T sub s$.
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"$q sub s$ upward" at 0.5,-0.6
"or $- ^ q sub s$ down" at 0.5,-0.75
"z" at -0.1,-0.5
"$q sub s$" at 0,0.3
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The heat equation is then
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0~=~k {partial sup 2 T} over {partial z sup 2}~+~rho H
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we then integrate with respect to z
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-k {partial T} over {partial z} ~+~ C sub 1~=~ rho H z
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$C sub 1$ is a constant of integration - to find it consider z = 0, and remember that the heat flow at the surface is
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- ^ q sub s~=~-k {partial T} over {partial z}~~at~~z~=~0
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so, $C sub 1~=~q sub s$
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Now we have
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q sub ...