Geodynamo Between Reversals
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In
physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
, the dynamo theory proposes a mechanism by which a celestial body such as
Earth Earth is the third planet from the Sun and the only astronomical object known to harbor life. While large volumes of water can be found throughout the Solar System, only Earth sustains liquid surface water. About 71% of Earth's surfa ...
or a
star A star is an astronomical object comprising a luminous spheroid of plasma (physics), plasma held together by its gravity. The List of nearest stars and brown dwarfs, nearest star to Earth is the Sun. Many other stars are visible to the naked ...
generates a
magnetic field A magnetic field is a vector field that describes the magnetic influence on moving electric charges, electric currents, and magnetic materials. A moving charge in a magnetic field experiences a force perpendicular to its own velocity and to ...
. The dynamo theory describes the process through which a rotating, convecting, and
electric Electricity is the set of physical phenomena associated with the presence and motion of matter that has a property of electric charge. Electricity is related to magnetism, both being part of the phenomenon of electromagnetism, as described by ...
ally conducting fluid can maintain a magnetic field over
astronomical Astronomy () is a natural science that studies celestial objects and phenomena. It uses mathematics, physics, and chemistry in order to explain their origin and evolution. Objects of interest include planets, moons, stars, nebulae, galaxi ...
time scales. A dynamo is thought to be the source of the
Earth's magnetic field Earth's magnetic field, also known as the geomagnetic field, is the magnetic field that extends from Earth's interior out into space, where it interacts with the solar wind, a stream of charged particles emanating from the Sun. The magnetic f ...
and the magnetic fields of Mercury and the
Jovian planets The giant planets constitute a diverse type of planet much larger than Earth. They are usually primarily composed of low-boiling-point materials (volatiles), rather than rock or other solid matter, but massive solid planets can also exist. Ther ...
.


History of theory

When William Gilbert published ''
de Magnete ''De Magnete, Magneticisque Corporibus, et de Magno Magnete Tellure'' (''On the Magnet and Magnetic Bodies, and on That Great Magnet the Earth'') is a scientific work published in 1600 by the English physician and scientist William Gilbert. A h ...
'' in 1600, he concluded that the Earth is magnetic and proposed the first hypothesis for the origin of this magnetism: permanent magnetism such as that found in
lodestone Lodestones are naturally magnetized pieces of the mineral magnetite. They are naturally occurring magnets, which can attract iron. The property of magnetism was first discovered in antiquity through lodestones. Pieces of lodestone, suspen ...
. In 1919,
Joseph Larmor Sir Joseph Larmor (11 July 1857 – 19 May 1942) was an Irish and British physicist and mathematician who made breakthroughs in the understanding of electricity, dynamics, thermodynamics, and the electron theory of matter. His most influent ...
proposed that a
dynamo file:DynamoElectricMachinesEndViewPartlySection USP284110.png, "Dynamo Electric Machine" (end view, partly section, ) A dynamo is an electrical generator that creates direct current using a commutator (electric), commutator. Dynamos were the f ...
might be generating the field. However, even after he advanced his hypothesis, some prominent scientists advanced alternative explanations. Einstein believed that there might be an asymmetry between the charges of the
electron The electron ( or ) is a subatomic particle with a negative one elementary electric charge. Electrons belong to the first generation of the lepton particle family, and are generally thought to be elementary particles because they have no kn ...
and
proton A proton is a stable subatomic particle, symbol , H+, or 1H+ with a positive electric charge of +1 ''e'' elementary charge. Its mass is slightly less than that of a neutron and 1,836 times the mass of an electron (the proton–electron mass ...
so that the
Earth's magnetic field Earth's magnetic field, also known as the geomagnetic field, is the magnetic field that extends from Earth's interior out into space, where it interacts with the solar wind, a stream of charged particles emanating from the Sun. The magnetic f ...
would be produced by the entire Earth. The
Nobel Prize The Nobel Prizes ( ; sv, Nobelpriset ; no, Nobelprisen ) are five separate prizes that, according to Alfred Nobel's will of 1895, are awarded to "those who, during the preceding year, have conferred the greatest benefit to humankind." Alfr ...
winner
Patrick Blackett Patrick Maynard Stuart Blackett, Baron Blackett (18 November 1897 – 13 July 1974) was a British experimental physicist known for his work on cloud chambers, cosmic rays, and paleomagnetism, winning the Nobel Prize for Physics in 1948. ...
did a series of experiments looking for a fundamental relation between
angular momentum In physics, angular momentum (rarely, moment of momentum or rotational momentum) is the rotational analog of linear momentum. It is an important physical quantity because it is a conserved quantity—the total angular momentum of a closed syst ...
and
magnetic moment In electromagnetism, the magnetic moment is the magnetic strength and orientation of a magnet or other object that produces a magnetic field. Examples of objects that have magnetic moments include loops of electric current (such as electromagne ...
, but found none. Walter M. Elsasser, considered a "father" of the presently accepted dynamo theory as an explanation of the Earth's magnetism, proposed that this magnetic field resulted from electric currents induced in the fluid outer core of the Earth. He revealed the history of the Earth's magnetic field through pioneering the study of the magnetic orientation of minerals in rocks. In order to maintain the magnetic field against
ohm Ohm (symbol Ω) is a unit of electrical resistance named after Georg Ohm. Ohm or OHM may also refer to: People * Georg Ohm (1789–1854), German physicist and namesake of the term ''ohm'' * Germán Ohm (born 1936), Mexican boxer * Jörg Ohm (b ...
ic decay (which would occur for the dipole field in 20,000 years), the outer core must be convecting. The
convection Convection is single or multiphase fluid flow that occurs spontaneously due to the combined effects of material property heterogeneity and body forces on a fluid, most commonly density and gravity (see buoyancy). When the cause of the convec ...
is likely some combination of thermal and compositional convection. The mantle controls the rate at which heat is extracted from the core. Heat sources include gravitational energy released by the compression of the core, gravitational energy released by the rejection of light elements (probably
sulfur Sulfur (or sulphur in British English) is a chemical element with the symbol S and atomic number 16. It is abundant, multivalent and nonmetallic. Under normal conditions, sulfur atoms form cyclic octatomic molecules with a chemical formula ...
,
oxygen Oxygen is the chemical element with the symbol O and atomic number 8. It is a member of the chalcogen group in the periodic table, a highly reactive nonmetal, and an oxidizing agent that readily forms oxides with most elements as wel ...
, or
silicon Silicon is a chemical element with the symbol Si and atomic number 14. It is a hard, brittle crystalline solid with a blue-grey metallic luster, and is a tetravalent metalloid and semiconductor. It is a member of group 14 in the periodic tab ...
) at the inner core boundary as it grows, latent heat of crystallization at the inner core boundary, and radioactivity of
potassium Potassium is the chemical element with the symbol K (from Neo-Latin ''kalium'') and atomic number19. Potassium is a silvery-white metal that is soft enough to be cut with a knife with little force. Potassium metal reacts rapidly with atmosphe ...
,
uranium Uranium is a chemical element with the symbol U and atomic number 92. It is a silvery-grey metal in the actinide series of the periodic table. A uranium atom has 92 protons and 92 electrons, of which 6 are valence electrons. Uranium is weak ...
and
thorium Thorium is a weakly radioactive metallic chemical element with the symbol Th and atomic number 90. Thorium is silvery and tarnishes black when it is exposed to air, forming thorium dioxide; it is moderately soft and malleable and has a high ...
. At the dawn of the 21st century, numerical modeling of the Earth's magnetic field has not been successfully demonstrated. Initial models are focused on field generation by convection in the planet's fluid outer core. It was possible to show the generation of a strong, Earth-like field when the model assumed a uniform core-surface temperature and exceptionally high viscosities for the core fluid. Computations which incorporated more realistic parameter values yielded magnetic fields that were less Earth-like, but indicated that model refinements may ultimately lead to an accurate analytic model. Slight variations in the core-surface temperature, in the range of a few millikelvins, result in significant increases in convective flow and produce more realistic magnetic fields.


Formal definition

Dynamo theory describes the process through which a rotating, convecting, and electrically conducting fluid acts to maintain a magnetic field. This theory is used to explain the presence of anomalously long-lived magnetic fields in astrophysical bodies. The conductive fluid in the geodynamo is liquid iron in the outer core, and in the
solar dynamo The solar dynamo is a physical process that generates the Sun's magnetic field. It is explained with a variant of the dynamo theory. A naturally occurring electric generator in the Sun's interior produces electric currents and a magnetic field ...
is ionized gas at the
tachocline The tachocline is the transition region of stars of more than 0.3 solar masses, between the radiative interior and the differentially rotating outer convective zone. This causes the region to have a very large shear as the rotation rate change ...
. Dynamo theory of astrophysical bodies uses
magnetohydrodynamic Magnetohydrodynamics (MHD; also called magneto-fluid dynamics or hydro­magnetics) is the study of the magnetic properties and behaviour of electrically conducting fluids. Examples of such magneto­fluids include plasmas, liquid metals, ...
equations to investigate how the fluid can continuously regenerate the magnetic field. It was once believed that the
dipole In physics, a dipole () is an electromagnetic phenomenon which occurs in two ways: *An electric dipole deals with the separation of the positive and negative electric charges found in any electromagnetic system. A simple example of this system ...
, which comprises much of the
Earth's magnetic field Earth's magnetic field, also known as the geomagnetic field, is the magnetic field that extends from Earth's interior out into space, where it interacts with the solar wind, a stream of charged particles emanating from the Sun. The magnetic f ...
and is misaligned along the rotation axis by 11.3 degrees, was caused by permanent magnetization of the materials in the earth. This means that dynamo theory was originally used to explain the Sun's magnetic field in its relationship with that of the Earth. However, this hypothesis, which was initially proposed by
Joseph Larmor Sir Joseph Larmor (11 July 1857 – 19 May 1942) was an Irish and British physicist and mathematician who made breakthroughs in the understanding of electricity, dynamics, thermodynamics, and the electron theory of matter. His most influent ...
in 1919, has been modified due to extensive studies of magnetic
secular variation The secular variation of a time series is its long-term, non-periodic variation (see decomposition of time series). Whether a variation is perceived as secular or not depends on the available timescale: a variation that is secular over a timescale ...
,
paleomagnetism Paleomagnetism (or palaeomagnetismsee ), is the study of magnetic fields recorded in rocks, sediment, or archeological materials. Geophysicists who specialize in paleomagnetism are called ''paleomagnetists.'' Certain magnetic minerals in rock ...
(including polarity reversals), seismology, and the solar system's abundance of elements. Also, the application of the theories of
Carl Friedrich Gauss Johann Carl Friedrich Gauss (; german: Gauß ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician and physicist who made significant contributions to many fields in mathematics and science. Sometimes refer ...
to magnetic observations showed that Earth's magnetic field had an internal, rather than external, origin. There are three requisites for a dynamo to operate: *An electrically conductive fluid medium *Kinetic energy provided by planetary rotation *An internal energy source to drive convective motions within the fluid. In the case of the Earth, the magnetic field is induced and constantly maintained by the convection of liquid iron in the outer core. A requirement for the induction of field is a rotating fluid. Rotation in the outer core is supplied by the
Coriolis effect In physics, the Coriolis force is an inertial or fictitious force that acts on objects in motion within a frame of reference that rotates with respect to an inertial frame. In a reference frame with clockwise rotation, the force acts to the ...
caused by the rotation of the Earth. The Coriolis force tends to organize fluid motions and electric currents into columns (also see Taylor columns) aligned with the rotation axis. Induction or creation of magnetic field is described by the
induction equation The induction equation, one of the magnetohydrodynamic equations, is a partial differential equation that relates the magnetic field and velocity of an electrically conductive fluid such as a plasma. It can be derived from Maxwell's equations an ...
: :\frac = \eta \nabla^2 \mathbf + \nabla \times (\mathbf \times \mathbf) where u is velocity, B is magnetic field, ''t'' is time, and \eta=1/(\sigma\mu) is the
magnetic diffusivity The magnetic diffusivity is a parameter in plasma physics which appears in the magnetic Reynolds number. It has SI units of m²/s and is defined as:W. Baumjohann and R. A. Treumann, ''Basic Space Plasma Physics'', Imperial College Press, 1997. :\e ...
with \sigma electrical conductivity and \mu permeability. The ratio of the second term on the right hand side to the first term gives the
magnetic Reynolds number In magnetohydrodynamics, the magnetic Reynolds number (Rm) is a dimensionless quantity that estimates the relative effects of advection or induction of a magnetic field by the motion of a conducting medium to the magnetic diffusion. It is the ...
, a dimensionless ratio of advection of magnetic field to diffusion.


Tidal heating supporting a dynamo

Tidal forces between celestial orbiting bodies cause friction that heats up their interiors. This is known as tidal heating, and it helps keep the interior in a liquid state. A liquid interior that can conduct electricity is required to produce a dynamo. Saturn's Enceladus and Jupiter's Io have enough tidal heating to liquify their inner cores, but they may not create a dynamo because they cannot conduct electricity. Mercury, despite its small size, has a magnetic field, because it has a conductive liquid core created by its iron composition and friction resulting from its highly elliptical orbit. It is theorized that the Moon once had a magnetic field, based on evidence from magnetized lunar rocks, due to its short-lived closer distance to Earth creating tidal heating. An orbit and rotation of a planet helps provide a liquid core, and supplements kinetic energy that supports a dynamo action.


Kinematic dynamo theory

In kinematic dynamo theory the velocity field is ''prescribed'', instead of being a dynamic variable: The model makes no provision for the flow distorting in response to the magnetic field. This method cannot provide the time variable behaviour of a fully nonlinear chaotic dynamo, but can be used to study how magnetic field strength varies with the flow structure and speed. Using
Maxwell's equations Maxwell's equations, or Maxwell–Heaviside equations, are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, and electric circuits. ...
simultaneously with the curl of
Ohm's law Ohm's law states that the current through a conductor between two points is directly proportional to the voltage across the two points. Introducing the constant of proportionality, the resistance, one arrives at the usual mathematical equat ...
, one can derive what is basically a linear eigenvalue equation for magnetic fields (), which can be done when assuming that the magnetic field is independent from the velocity field. One arrives at a critical ''
magnetic Reynolds number In magnetohydrodynamics, the magnetic Reynolds number (Rm) is a dimensionless quantity that estimates the relative effects of advection or induction of a magnetic field by the motion of a conducting medium to the magnetic diffusion. It is the ...
'', above which the flow strength is sufficient to amplify the imposed magnetic field, and below which the magnetic field dissipates.


Practical measure of possible dynamos

The most functional feature of kinematic dynamo theory is that it can be used to test whether a velocity field is or is not capable of dynamo action. By experimentally applying a certain velocity field to a small magnetic field, one can observe whether the magnetic field tends to grow (or not) in response to the applied flow. If the magnetic field does grow, then the system is either capable of dynamo action or is a dynamo, but if the magnetic field does not grow, then it is simply referred to as “not a dynamo”. An analogous method called the ''
membrane paradigm In black hole theory, the black hole membrane paradigm is a simplified model, useful for visualising and calculating the effects predicted by quantum mechanics for the exterior physics of black holes, without using quantum-mechanical principles or ...
'' is a way of looking at
black hole A black hole is a region of spacetime where gravitation, gravity is so strong that nothing, including light or other Electromagnetic radiation, electromagnetic waves, has enough energy to escape it. The theory of general relativity predicts t ...
s that allows for the material near their surfaces to be expressed in the language of dynamo theory.


Spontaneous breakdown of a topological supersymmetry

Kinematic dynamo can be also viewed as the phenomenon of the spontaneous breakdown of the topological supersymmetry of the associated stochastic differential equation related to the flow of the background matter. Within stochastic supersymmetric theory, this supersymmetry is an intrinsic property of ''all''
stochastic differential equation A stochastic differential equation (SDE) is a differential equation in which one or more of the terms is a stochastic process, resulting in a solution which is also a stochastic process. SDEs are used to model various phenomena such as stock p ...
s, its interpretation is that the model’s phase space preserves continuity via continuous time flows. When the continuity of that flow spontaneously breaks down, the system is in the stochastic state of ''deterministic chaos''. In other words, kinematic dynamo arises because of chaotic flow in the underlying background matter.


Nonlinear dynamo theory

The kinematic approximation becomes invalid when the magnetic field becomes strong enough to affect the fluid motions. In that case the velocity field becomes affected by the Lorentz force, and so the induction equation is no longer linear in the magnetic field. In most cases this leads to a quenching of the amplitude of the dynamo. Such dynamos are sometimes also referred to as ''hydromagnetic dynamos''. Virtually all dynamos in astrophysics and geophysics are hydromagnetic dynamos. The main idea of the theory is that any small magnetic field existing in the outer core creates currents in the moving fluid there due to Lorentz force. These currents create further magnetic field due to Ampere's law. With the fluid motion, the currents are carried in a way that the magnetic field gets stronger (as long as \; \mathbf \cdot ( \mathbf \times \mathbf ) \; is negative). Thus a "seed" magnetic field can get stronger and stronger until it reaches some value that is related to existing non-magnetic forces. Numerical models are used to simulate fully nonlinear dynamos. The following equations are used: *The induction equation, presented above. *Maxwell's equations for negligible electric field: :: \nabla \cdot \mathbf = 0 :: \nabla \times \mathbf = \mu_0 \mathbf *The continuity equation for
conservation of mass In physics and chemistry, the law of conservation of mass or principle of mass conservation states that for any system closed to all transfers of matter and energy, the mass of the system must remain constant over time, as the system's mass can ...
, for which the Boussinesq approximation is often used: :: \nabla \cdot \mathbf = 0, *The Navier-Stokes equation for conservation of
momentum In Newtonian mechanics, momentum (more specifically linear momentum or translational momentum) is the product of the mass and velocity of an object. It is a vector quantity, possessing a magnitude and a direction. If is an object's mass an ...
, again in the same approximation, with the magnetic force and gravitation force as the external forces: :: \frac = -\frac\nabla p + \nu \nabla^2 \mathbf + \rho' \mathbf + 2 \mathbf \times \mathbf + \mathbf \times \mathbf \times \mathbf + \frac\mathbf \times \mathbf ~, :where \,\nu\, is the kinematic
viscosity The viscosity of a fluid is a measure of its resistance to deformation at a given rate. For liquids, it corresponds to the informal concept of "thickness": for example, syrup has a higher viscosity than water. Viscosity quantifies the inte ...
, \,\rho_0\,is the mean density and \rho' is the relative density perturbation that provides buoyancy (for thermal convection \;\rho' = \alpha \Delta T\; where \,\alpha\, is
coefficient of thermal expansion Thermal expansion is the tendency of matter to change its shape, area, volume, and density in response to a change in temperature, usually not including phase transitions. Temperature is a monotonic function of the average molecular kineti ...
), \,\Omega\, is the rotation rate of the Earth, and \,\mathbf\, is the electric current density. *A transport equation, usually of heat (sometimes of light element concentration): :: \frac = \kappa \nabla^2 T + \epsilon :where is temperature, \;\kappa = k / \rho c_p \; is the thermal diffusivity with thermal conductivity, \,c_p\, heat capacity, and \rho density, and \,\epsilon\, is an optional heat source. Often the pressure is the dynamic pressure, with the hydrostatic pressure and centripetal potential removed. These equations are then non-dimensionalized, introducing the non-dimensional parameters, : R_\mathsf = \frac \;,\quad E = \frac \;,\quad P_\mathsf = \frac \;,\quad P_\mathsf = \frac where is the
Rayleigh number In fluid mechanics, the Rayleigh number (, after Lord Rayleigh) for a fluid is a dimensionless number associated with buoyancy-driven flow, also known as free (or natural) convection. It characterises the fluid's flow regime: a value in a certai ...
, the Ekman number, and the
Prandtl Ludwig Prandtl (4 February 1875 – 15 August 1953) was a German fluid dynamicist, physicist and aerospace scientist. He was a pioneer in the development of rigorous systematic mathematical analyses which he used for underlying the science of ...
and magnetic Prandtl number. Magnetic field scaling is often in
Elsasser number The Elsasser number, Λ, is a dimensionless number in magnetohydrodynamics that represents the ratio of magnetic forces to the Coriolis force. \Lambda = \frac where σ is the conductivity of the fluid, ''B'' is the magnetic field A magnetic ...
units B = (\rho \Omega/\sigma)^\frac\;.


Energy conversion between magnetic and kinematic energy

The scalar product of the above form of Navier-Stokes equation with \;\rho_0 \mathbf\; gives the rate of increase of kinetic energy density, \; \tfrac \rho_0 u^2 c \;, on the left-hand side. The last term on the right-hand side is then \; \mathbf \cdot ( \mathbf \times \mathbf ) \;, the local contribution to the kinetic energy due to Lorentz force. The scalar product of the induction equation with (1/\mu_0)\mathbf gives the rate of increase of the magnetic energy density, \;\tfrac \mu_0 B^2\;, on the left-hand side. The last term on the right-hand side is then (1/\mu_0)\mathbf \cdot \left( \nabla \times \left( \mathbf \times \mathbf \right) \right) \;. Since the equation is volume-integrated, this term is equivalent up to a boundary term (and with the double use of the
scalar triple product In geometry and algebra, the triple product is a product of three 3-dimensional vectors, usually Euclidean vectors. The name "triple product" is used for two different products, the scalar-valued scalar triple product and, less often, the vector- ...
identity) to \; -\mathbf \cdot \left( \frac \left( \nabla \times \mathbf \right) \times \mathbf \right) = -\mathbf \cdot \left( \mathbf \times \mathbf \right) ~ (where one of Maxwell's equations was used). This is the local contribution to the magnetic energy due to fluid motion. Thus the term \;-\mathbf \cdot ( \mathbf \times \mathbf ) \; is the rate of transformation of kinetic energy to magnetic energy. This has to be non-negative at least in part of the volume, for the dynamo to produce magnetic field. From the diagram above, it is not clear why this term should be positive. A simple argument can be based on consideration of net effects. To create the magnetic field, the net electric current must wrap around the axis of rotation of the planet. In that case, for the term to be positive, the net flow of conducting matter must be towards the axis of rotation. The diagram only shows a net flow from the poles to the equator. However mass conservation requires an additional flow from the equator toward the poles. If that flow was along the axis of rotation, that implies the circulation would be completed by a flow from the ones shown towards the axis of rotation, producing the desired effect.


Order of magnitude of the magnetic field created by Earth's dynamo

The above formula for the rate of conversion of kinetic energy to magnetic energy, is equivalent to a rate of work done by a force of \;\mathbf \times \mathbf\; on the outer core matter, whose velocity is \mathbf. This work is the result of non-magnetic forces acting on the fluid. Of those, the gravitational force and the
centrifugal force In Newtonian mechanics, the centrifugal force is an inertial force (also called a "fictitious" or "pseudo" force) that appears to act on all objects when viewed in a rotating frame of reference. It is directed away from an axis which is paralle ...
are
conservative Conservatism is a cultural, social, and political philosophy that seeks to promote and to preserve traditional institutions, practices, and values. The central tenets of conservatism may vary in relation to the culture and civilization i ...
and therefore have no overall contribution to fluid moving in closed loops. Ekman number (defined above), which is the ratio between the two remaining forces, namely the viscosity and Coriolis force, is very low inside Earth's outer core, because its viscosity is low (1.2–1.5 ×10
pascal-second The viscosity of a fluid is a measure of its resistance to deformation at a given rate. For liquids, it corresponds to the informal concept of "thickness": for example, syrup has a higher viscosity than water. Viscosity quantifies the inter ...
) due to its liquidity. Thus the main time-averaged contribution to the work is from Coriolis force, whose size is \;-2\rho\,\mathbf \times \mathbf \;, though this quantity and \mathbf \times \mathbf are related only indirectly and are not in general equal locally (thus they affect each other but not in the same place and time). The current density is itself the result of the magnetic field according to
Ohm's law Ohm's law states that the current through a conductor between two points is directly proportional to the voltage across the two points. Introducing the constant of proportionality, the resistance, one arrives at the usual mathematical equat ...
. Again, due to matter motion and current flow, this is not necessarily the field at the same place and time. However these relations can still be used to deduce orders of magnitude of the quantities in question. In terms of order of magnitude, \; J \, B \sim \rho\, \Omega\, u \; and \; J \sim \sigma u B\;, giving \;\sigma\,u\, B^2 \sim \rho\, \Omega\,u \;, or: :B \sim \sqrt The exact ratio between both sides is the square root of
Elsasser number The Elsasser number, Λ, is a dimensionless number in magnetohydrodynamics that represents the ratio of magnetic forces to the Coriolis force. \Lambda = \frac where σ is the conductivity of the fluid, ''B'' is the magnetic field A magnetic ...
. Note that the magnetic field direction cannot be inferred from this approximation (at least not its sign) as it appears squared, and is, indeed, sometimes reversed, though in general it lies on a similar axis to that of \mathbf. For earth outer core, is approximately 104 kg/m3,de Wijs, G. A., Kresse, G., Vočadlo, L., Dobson, D., Alfe, D., Gillan, M. J., & Price, G. D. (1998)
The viscosity of liquid iron at the physical conditions of the Earth's core.
Nature, 392(6678), 805.
  = 2/day = 7.3×10−5/second   and     is approximately   107Ω−1m−1 . This gives   2.7×10−4  Tesla. The magnetic field of a
magnetic dipole In electromagnetism, a magnetic dipole is the limit of either a closed loop of electric current or a pair of poles as the size of the source is reduced to zero while keeping the magnetic moment constant. It is a magnetic analogue of the electric ...
has an inverse cubic dependence in distance, so its order of magnitude at the earth surface can be approximated by multiplying the above result with giving   2.5×10−5 Tesla, not far from to the measured value of 3×10−5 Tesla at the
equator The equator is a circle of latitude, about in circumference, that divides Earth into the Northern and Southern hemispheres. It is an imaginary line located at 0 degrees latitude, halfway between the North and South poles. The term can als ...
.


Numerical models

Broadly, models of the geodynamo attempt to produce magnetic fields consistent with observed data given certain conditions and equations as mentioned in the sections above. Implementing the
magnetohydrodynamic Magnetohydrodynamics (MHD; also called magneto-fluid dynamics or hydro­magnetics) is the study of the magnetic properties and behaviour of electrically conducting fluids. Examples of such magneto­fluids include plasmas, liquid metals, ...
equations successfully was of particular significance because they pushed dynamo models to self-consistency. Though geodynamo models are especially prevalent, dynamo models are not necessarily restricted to the geodynamo; solar and general dynamo models are also of interest. Studying dynamo models has utility in the field of geophysics as doing so can identify how various mechanisms form magnetic fields like those produced by astrophysical bodies like Earth and how they cause magnetic fields to exhibit certain features, such as pole reversals. The equations used in numerical models of dynamo are highly complex. For decades, theorists were confined to two dimensional ''kinematic dynamo'' models described above, in which the fluid motion is chosen in advance and the effect on the magnetic field calculated. The progression from linear to nonlinear, three dimensional models of dynamo was largely hindered by the search for solutions to magnetohydrodynamic equations, which eliminate the need for many of the assumptions made in kinematic models and allow self-consistency. The first ''self-consistent'' dynamo models, ones that determine both the fluid motions and the magnetic field, were developed by two groups in 1995, one in Japan and one in the United States. The latter was made as a model with regards to the geodynamo and received significant attention because it successfully reproduced some of the characteristics of the Earth's field. Following this breakthrough, there was a large swell in development of reasonable, three dimensional dynamo models. Though many self-consistent models now exist, there are significant differences among the models, both in the results they produce and the way they were developed. Given the complexity of developing a geodynamo model, there are many places where discrepancies can occur such as when making assumptions involving the mechanisms that provide energy for the dynamo, when choosing values for parameters used in equations, or when normalizing equations. In spite of the many differences that may occur, most models have shared features like clear axial dipoles. In many of these models, phenomena like
secular variation The secular variation of a time series is its long-term, non-periodic variation (see decomposition of time series). Whether a variation is perceived as secular or not depends on the available timescale: a variation that is secular over a timescale ...
and geomagnetic polarity reversals have also been successfully recreated.


Observations

Many observations can be made from dynamo models. Models can be used to estimate how magnetic fields vary with time and can be compared to observed
paleomagnetic Paleomagnetism (or palaeomagnetismsee ), is the study of magnetic fields recorded in rocks, sediment, or archeological materials. Geophysicists who specialize in paleomagnetism are called ''paleomagnetists.'' Certain magnetic minerals in roc ...
data to find similarities between the model and the Earth. Due to the uncertainty of paleomagnetic observations, however, comparisons may not be entirely valid or useful. Simplified geodynamo models have shown relationships between the dynamo number (determined by variance in rotational rates in the outer core and mirror-asymmetric convection (e.g. when convection favors one direction in the north and the other in the south)) and magnetic pole reversals as well as found similarities between the geodynamo and the Sun's dynamo. In many models, it appears that magnetic fields have somewhat random magnitudes that follow a normal trend that average to zero. In addition to these observations, general observations about the mechanisms powering the geodynamo can be made based on how accurately the model reflects actual data collected from Earth.


Modern modelling

The complexity of dynamo modelling is so great that models of the geodynamo are limited by the current power of
supercomputer A supercomputer is a computer with a high level of performance as compared to a general-purpose computer. The performance of a supercomputer is commonly measured in floating-point operations per second ( FLOPS) instead of million instructions ...
s, particularly because calculating the Ekman and Rayleigh number of the outer core is extremely difficult and requires a vast number of computations. Many improvements have been proposed in dynamo modelling since the self-consistent breakthrough in 1995. One suggestion in studying the complex magnetic field changes is applying
spectral method Spectral methods are a class of techniques used in applied mathematics and scientific computing to numerically solve certain differential equations. The idea is to write the solution of the differential equation as a sum of certain "basis function ...
s to simplify computations. Ultimately, until considerable improvements in computer power are made, the methods for computing realistic dynamo models will have to be made more efficient, so making improvements in methods for computing the model is of high importance for the advancement of numerical dynamo modelling.


See also

*
Antidynamo theorem In physics and in particular in the theory of magnetism, an antidynamo theorem is one of several results that restrict the type of magnetic fields that may be produced by dynamo action. One notable example is Thomas Cowling's antidynamo theo ...
*
Rotating magnetic field A rotating magnetic field is the resultant magnetic field produced by a system of coils symmetrically placed and supplied with polyphase currents. A rotating magnetic field can be produced by a poly-phase (two or more phases) current or by a singl ...
*
Secular variation The secular variation of a time series is its long-term, non-periodic variation (see decomposition of time series). Whether a variation is perceived as secular or not depends on the available timescale: a variation that is secular over a timescale ...


References

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