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Physics comparison: gravity and force and acceleration, isostasy and pressure and buoyancy, heat and temperature. Gravity was application of a fundamental force law (of the 4 fundamental forces).

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Physics comparison: gravity and force and acceleration, isostasy and pressure and buoyancy, heat and temperature
  • Gravity was application of a fundamental force law (of the 4 fundamental forces).
  • The gravitational attractive force between masses causes motion that is determined by Newton’s three laws of motion (Newtonian Mechanics).
  • Isostasy uses the concept of a ideal-fluid in a gravitational field that produce the force called ‘weight’. This leads to the concept of stress or more simply pressure. This lead to the concept of a buoyancy force.
  • Heat and temperature is a very different physics that applies Newtonian mechanics to systems with 1024 molecules (atoms). A statistical approach is required as no one can solve for the motion of 1024 molecules. Happily, we have discovered that in thermal/chemical equilibrium thermodynamic averages of many-particle system is well-defined so that macroscopic averages like temperature, heat content are defined by simple equations.
Red=hot; blue=cold.
  • Is this in thermal equilibrium ?
  • How is equilibrum attained ?
first law of thermodynamics
First Law of Thermodynamics

The first law of thermodynamics is the application of the conservation of energy principle to heat and thermodynamic processes:

microscopic internal energy
Microscopic (Internal) Energy

Internal energy involves energy on the microscopic scale. For an ideal monoatomic gas, this is just the translational kinetic energy of the linear motion of the "hard sphere" type atoms , and the behavior of the system is well described by kinetic theory. However, for polyatomic gases there is rotational and vibrational kinetic energy as well. Then in liquids and solids there is potential energy associated with the intermolecular attractive forces. A simplified visualization of the contributions to internal energy can be helpful in understanding phase transitions and other phenomena which involve internal energy.

work energy in transition
Work (energy in transition)
  • The amount of mechanical work done can be determined by an equation derived from Newtonian mechanics: Work = Force x Distance moved in force direction
  • It can also be described as the product of the applied pressure and the displaced volume: Work = Applied pressure x Displaced volume
  • The unit of work is joule, J, which is defined as the amount of work done when a force of 1 Newton acts for a distance of 1 m in the direction of the force.
  • 1 J = 1 Nm
  • Solid earth, liquid oceans, and gaseous atmosphere are all heat engine that do work on the planet (create kinetic energy) due to disequilibrium associated with lateral variation in temperature.
  • Solid earth: hot bottom, cold top: convection moves heat to cool solid planet.
  • Liquid ocean: hot at equator (why?), cold at poles, pole ward flow moves heat from tropics to polar regions.
  • Gaseous atmosphere: hot at equator , cold at poles, pole ward flow moves heat.
mechanical equivalent of heat
Mechanical Equivalent of Heat

Heat flow and work are both ways of transferring energy. As illustrated in the heat and work example, the temperature of a gas can be raised either by heating it, by doing work on it, or a combination of the two.

In a classic experiment in 1843, James Joule showed the energy equivalence of heating and doing work by using the change in potential energy of falling masses to stir an insulated container of water with paddles. Careful measurements showed the increase in the temperature of the water to be proportional to the mechanical energy used to stir the water.

heat and work example
Heat and Work Example

This example of the interchangeability of heat and work as agents for adding energy to a system can help to dispel some misconceptions about heat. I found the idea in a little article by Mark Zemansky entitled "The Use and Misuse of the Word 'Heat' in Physics Teaching". One key idea from this example is that if you are presented with a high temperature gas, you cannot tell whether it reached that high temperature by being heated, or by having work done on it, or a combination of the two.

heat e g steam engines and society
Heat (e.g., steam) engines and Society
  • Cellular metabolism: heat released in cells
  • Electricity : heat engine, e.g., coal fired plant making steam to spin turbine
  • Cars: heat production via compustion of air/gas mixture.
  • House heating: burning natural gas or electricity
  • Solid/liquid/gaseous earth envelopes are all heat engines.
  • Energy as burning stuff to make heat to drive heat engines to extract motive force
Steam turbine energy (chemical/heat/mechanical/electrical): burn coal, make heat to make steam that spins a turbine to spin an electrical generator
newton s cooling law
Newton’s cooling law

Q = Thermal energy (Joules) h = heat transfer coeficient

A = area of the heat being transferred T = Temperature of the object

Tenv = Temperature of the environment

ΔT(t) = T(t) − Tenv : time-dependent thermal gradient between environment and object

Exponentially decay in time solution:  

T(t) = TA + (TH-TA) e-kt

diffusion is a molecular scale phenomena everywhere
Diffusion is a molecular scale phenomena everywhere

Solid state atomic diffusion

Thermal (heat) diffusion

1-dimensional random walk 2-D random walk

Diffusion equations

Chem concentration: Fick’s Law

Heat: Fourier’s Law

Current: Ohm’s Law

Porous flow: Darcy’s Law

Heat/concentration/charge or pressure gradients drive flow of energy down the gradient.

heat transfer requires t 0
Heat transfer requires ∆T ≠ 0

Conduction is diffusion of atomic vibrations at the atomic scale when there is a thermal gradient. Conduction can never ever be stopped except by the perfect insulator idealization.

Convection is the movement of heat by material flow. Convection happens when the thermal variations creates buoyancy forces sufficient to overcome the viscous strength of the mantle rock. The earth is a very viscous fluid that flows at rates up to 10 cm/yr as manifest by plate velocities.

Radiation is the movement of heat as infrared electro-magnetic waves and hence can propagate through free-space and transparent substances.

Confusing: alpha and beta radioactivedecay is called radiation but is not EM-waves, but nuclear fragments. The heat comes from the huge kinetic energy of the nuclear fragments being transfer into lattice vibrations.

heat and temperature
Heat and temperature

Discuss what you think the answer is to the following question: Which object contains the most heat, a boiling pot of water or a gigantic iceberg?

Temperature is not energy, but a measure of energy intensity.

Q = C * T where Q is heat and C is specific heat material property.

Temperature is defined at a point (intensive property) and energy is defined over a volume (extensive property).

Extensive: mass, volume, entropy, energy.

Intensive: temperature, chemical potential, density, pressure.

P * V = n *R *T

In the study of transport phenomena, flux is defined as the amount that flows through a unit area per unit time[1] Flux is a vector.

Momentum flux, the rate of transfer of momentum across a unit area (N·s·m−2·s−1).

Heat flux, the rate of heat flow across a unit area (J·m−2·s−1).

Chemical flux, the rate of movement of molecules across a unit area (mol·m−2·s−1). (Fick's law of diffusion)

Energy flux, the rate of transfer of energy through a unit area (J·m−2·s−1). The radiative flux and heat flux are specific cases of energy flux.

quiz answers
Quiz answers

Heat is kinetic/potential energy at atomic scale. The MKS unit for energy is Joules. A Joule has units of a force times a distance (Newtons*meters) which is definition of work. Thus, energy is the capacity to do work. Raising a 1 kg object 1 m in a 10 m/s2 gravitational acceleration field requires that 10 Joules of work by performed on the object.

Temperature is NOT energy, but a measure of energy intensity. Temperature is mean kinetic and potential energy of atoms/molecules. The MKS temperature unit is Kelvin (but Celsius and Farenheight are related). Temperature is a measure of the tendency of an object to spontaneously give up energy to its surroundings. When two objects are in contact, the one that tends to spontaneously lose internal energy is at the higher temperature. When object have the same temperature, they are in thermal equilibrium. At zero degrees Kelvin, all thermal motion ceases.

Simple relation between heat and energy (ignoring whether at constant pressure or temperature) is Q (J) = C (J/K) * T (K) where Q is heat and C is heat capacity. To raise the temperature of 1 kilogram of water 1° Kelvin requires 4200 Joules of energy. Different substances have different heat capacities proportional to their valence electron behavior and crystal lattice structure.

Which object has more heat energy: an iceberg at -2° C or a cup of boiling water at 100° C ?

While the iceberg is at lower temperature than the boiling water, the much larger volume of the iceberg with respect to the boiling cup of water, means it has more total heat energy (Joules).

Q = C * T where C is proportional to the volume of the object.

If a temperature contrast exists, heat will be transferred from the high to low temperature by conduction and radiation and maybe convection if the force/strength ratio in the mantle is sufficient to drive heat advection faster than diffusion.This inexorable process of heat transfer only ends when the temperature contrasts is no more (∆T=0° ), which defines thermal equilibrium

WHY MUST HEAT FLOW? All forms of energy (e.g., heat), are relentlessly driven to spread out (disperse) their energy into larger volumes over time. This energy spreading out drives a system toward thermal equilibrium. This truism is called the second law of thermodynamics which says entropy always increase in time.

Terrestrial heat power: 44 TW

Flux: 0.04 W/m2

Solar mean heat power:

174 pW or 174,000 TW

Flux: 680 W/m2

Solar input 20,000 times more than 44 TW

Heat flow:

Red (high 200 mW)

Blue (low 20 mW)

temperature at 3 km depth
Temperature at 3 km depth

This map is made by solving the heat flow equation for dT(z)/dz:

q (W/m2 ) = K * dT(z)/dz

geologic phenomena and heat flow
Geologic phenomena and heat flow
  • Yellowstone hydrothermal geyser fields
  • Oil/gas maturation
  • Volcanoes
  • Earthquakes
  • Hydrothermal mineralization
  • Plate tectonics
  • Geothermal power production
  • Mid-ocean ridges
  • Subducting slabs
  • Metamorphism of rocks
chapter 17 geothermics
Chapter 17 Geothermics
  • The earth is hot at the core-mantle boundary ( about 4000° C) and cold at its surface (19 ° C) and very cold in space (-270 ° C). This large temperature contrast means the earth is in thermal dis-equilibrium and must be transferring heat into space.
  • The internal earth is cooled by convection that delivers heat to the base of the lithosphere where the heat is then conducted through the lithosphere. This heat flux warms the atmosphere and finally the atmosphere is always radiating heat into space.
  • Note: the Earth primarily cools itself by creating new lithosphere at ridges that is pulled across the ocean basin and finally subducted deep into the mantle to cool it.
  • The solid earth is a heat engine because work is done moving the plates and raising the mountains and pulling apart the ridges/rifts. The ocean-atmosphere is also a heat engine.
  • In the last 4 Ga, the earth’s cooling has been about 100 ° C per Billion years.
  • The earth is predicted to reach thermal equilibrium with space in about 10 Ga.
t hree heat transport processes
Three heat transport processes

There are three ways to transfer heat

Conduction: heat in a solid is transferred via diffusion of crystal lattice vibrations. Conduction can never be stopped if a temperature difference exists between two regions. Examples?

Convection: heat is moved (advected) by different density parcels flowing up and down in the earth’s gravity field. This only occurs if a parcels buoyancy force (Archimedes principle) is sufficiently large with respect to the strength (viscosity) of the mantle. Examples?

Radiation: heat moves as infrared electromagnetic waves at speed of light. All matter at non-zero temperatures radiates energy. Examples?

Radiative heat flux from Planet earthEnergy (N*m or J) Power=energy/time (J/s or W) Flux = Power/area (J/s-m2 or W/m2 )
Conductive heat transport in lithosphere and convective heat transport in mantle and fluid outer core

The lithosphere is strong and hence translates as a quasi-rigid block over the asthenosphere, therefore heat cannot be convected across the lithosphere and instead heat is conducted across the lithosphere.

The mantle dominantly moving heat via convection, although heat conduction is always occurring. But the 1-10 cm/yr mantle flow rates moves (advects) the heat much faster than heat conduction!

The temperature with respect to depth (geotherm) in the lithosphere is near a straight line which is called the conductive geotherm: dT(z)/dz = 30°C/km. The lithospheric geotherm will have curvature when there is significant heat production (e.g., in continental crust).

The adiabat geotherm is about dT/dz = 0.3-0.5°C/km; this temperature rise with depth is because each parcel of heat is being compressed into a smaller volume due to the pressure increase with depth.

adiabatic thermal gradients
Adiabatic thermal gradients
  • adiabatic - a process where a parcels temperature changes due to an expansion or compression, but no heat is added or taken away from the parcel.
  • Adiabatic temperature gradients do NOT drive heat flow.

Mantle Potential temperature

heat flux and amount of heat power
Heat flux and amount of heat power

Little-q Heat flux: W/m2

Power (W) per area (m2 )

Energy (J) per time (s) per area (m2)

Big-Q Heat Power: W

Energy (J) per time (s)

Geotherm with 0° C at surface (z=0 m)

Heat flux through area A



Conductive lithosphere

∆z =



measuring thermal conductivity
Measuring thermal conductivity

Because it is very hard to measure mW scale heat fluxes at earth’s surfacee, a tactic is to measure the temperature gradient in a borehole and the thermal conductivity of the rocks in the borehole.

Heat flux is calculated: q = k*dT/dz (W/m2 )

The thermal conductivity of a rock can be measured by heating one side of the metal rod and cooling the other side and measuring the thermal gradients in the metal and rock-sample regions.

Using the known conductivity of the metal, the heat flux is calculated.

Thus, the conductivity of the rock is:

K = q / (dT/dz).



thermal conductivity values
Thermal conductivity values

Good heat conductors:

Diamond, Metals

Poor heat conductors

Wood, Wool, Fiberglass

Conductivity is largely determined by ability of electrons to wandered through a substance. Good heat conductors are also good electrical conductors.

heat flow as function of oceanic plate age
Heat flow as function of oceanic plate age

The plate moves horizontally away from the mid-ocean ridge and cools as heat is conducted into the ocean bottom over 150 ma.

This cooling is manifest as the cool lithosphere getting thicker with time.

heat production heat flux increases in direction of heat flow
Heat production: heat flux increases in direction of heat flow

Note that the heat flux vectors becomes longer (bigger) towards the cooler side of the column.

This is because a constant heat production (μW/m3 ) throughout the column is assumed.

This creation of heat increase the heat flow and the thermal gradient towards the cooler end (i.e., the earth’s surface).

Thus, the distribution of heat production in the crust changes the geotherm to make it a curve and not a straight line for a steady state thermal equilibrium.

crustal heat productivity vs heat flux relationship
Crustal heat productivity vs. heat flux relationship

The continental crust is highly concentrated in radioactive elements such as Uranium. As Uranium decays, this is termed heat productivity in micro-Watts per unit volume.

By measuring the surface rock heat productivity, a linear relation between heat productivity and heat flux is often found. This means crustal heat production controls the heat flux.

The y-intercept value is the reduced heat flow (qr ) which is heat flux from mantle convection.

oceanic and continental heat flux and geotherms
Oceanic and continental heat flux and geotherms

The geotherm in the lithosphere is conductive and the geotherm in the convecting mantle is the adiabat.

Note that the conductive heat flux with respect to depth in the oceanic lithosphere is constant and hence the geotherm is a straight line.

But, the conductive heat flux in the continental lithosphere increases towards the surface and the geotherm is a curved line.

The base of the lithosphere has the same temperature at all three places of about 1300°C.

radioactive heat generation
Radioactive heat generation

Note that 40% of the Earth’s total heat flow comes from radioactive heat generation!

The other 60% of heat comes from ‘secular cooling’ of the original accretional heat when the Earth formed 4.5 Ga ago.

Also, note that 10% of the heat comes from continental crust, but only 0.15% from ocean crust.

In general, there is a decrease in heat flow with the age of continents. Why ?

46 +/- 3 TW (Terra-Watts)Terra is 1012World power consumption is 15 TW.The primary contributions to observed total surface heat are shown.Radiogenic heat productionmantle cooling heat flow from the core These three heat sources dominate the mantle energy budget, but there are substantial uncertainties in the latter two contributions.

Planetary heat budget

mantle temperature and melting
Mantle temperature and melting

By knowing the approximate temperature at which mantle rocks melt, the geotherm can be compared to the mantle solidus to see if melt should form.

temperature re equilibration
Temperature re-equilibration

So far, we have been assuming that the temperature does NOT vary in time. This is called thermal steady-state or dynamic equilibrium.

So, what if the temperature structure changes in time due to basins filling or tectonic overthrusts?

Adding sediments quickly to the top of the crust, causes a thermal disequilibrium that will be brought to thermal equilibrium over Ma time-scales.

Thrusting 20 km of the upper crust over the surrounding surface quickly, creates a disequilibrium that takes greater than 50 Ma to reach thermal equilibrium.

geothermal heat extraction and energy production
Geothermal heat extraction and energy production

Where high temperature rock is near the surface, the heat contained in the hot-rock can be mined to drive steam-turbines to make electricity.

This is done by pumping water down the injection well where it flows through the cracks in the rocks and heats up and corresponding cools down the rocks over time. The hot water is then extracted via a production well.

Note: geothermal energy is technically not a renewable resource as the water circulation extracts the heat much faster than the earth’s mW scale heat flux can replace the mined heat (energy).

atmospheric temperature changes effects on shallow geotherms
Atmospheric temperature changes effects on shallow geotherms

Periodic (daily, seasonal) changes in atmospheric temperature makes ‘temperature waves’ that propagate downward into the crust, but the attenuate exponentially with depth.

A recent 2 ° C increase in surface temperature between 1900-2000 can be measured in the borehole temperature profile and well modeled using the heat flow equations.