МагнитнотоМагнитното поле на поле на ЗемятаЗемята

Теория и компютърни Теория и компютърни симулациисимулации

Физика на Земята, атмосферата и космоса – съвременни проблеми

CompassCompass

Chinese were the first ones to use properties of magnetic iron needleChinese were the first ones to use properties of magnetic iron needle to point to point out the directions. First information on the use of magnetic compass in Europe out the directions. First information on the use of magnetic compass in Europe relates to the end of XII century. relates to the end of XII century.

In 1600, A.D. William Gilbert, Queen Elizabeth's physician, shed light on the In 1600, A.D. William Gilbert, Queen Elizabeth's physician, shed light on the mystery by showing that "mystery by showing that "the terrestrial globe itself is a great magnetthe terrestrial globe itself is a great magnet."."

Is the Earth a giant magnet?

where magnetic line is as follows

Magnetic poles of the Earth move!Magnetic poles of the Earth move!

1741 Hiorter and Anders Celsius 1741 Hiorter and Anders Celsius note that the polar aurora is accom-note that the polar aurora is accom-panied by a disturbance of the mag-panied by a disturbance of the mag-netic needle. netic needle.

1820 Hans Christian Oersted disco-1820 Hans Christian Oersted disco-vers electric currents create magnetic vers electric currents create magnetic effects. André-Marie Ampère deduces effects. André-Marie Ampère deduces that magnetism is basically the force that magnetism is basically the force between electric currents. between electric currents.

1859 Richard Carrington in England 1859 Richard Carrington in England observes a solar flare; 17 hours later a observes a solar flare; 17 hours later a large magnetic storm begins.large magnetic storm begins.

During 16-17 centuries English During 16-17 centuries English discovered that declination of the discovered that declination of the Earth’s magnetic field changed. It can Earth’s magnetic field changed. It can be treated as moving of the magnetic be treated as moving of the magnetic poles.poles.

Sun is also a magnetSun is also a magnet

The SunThe Sun

SunspotsSunspots

A A sunspotsunspot is an area on the Sun's surface (photosphere) is an area on the Sun's surface (photosphere) that is marked by intense magnetic activity, which that is marked by intense magnetic activity, which inhibits convection, forming areas of reduced surface inhibits convection, forming areas of reduced surface temperature.temperature.

Sun and Earth magnetic fields Sun and Earth magnetic fields and solar windsand solar winds

Now we know that the Sun magnetic field and that of the Earth interact via solar wind.

The Earth interiorThe Earth interior

Maxwell and Ohm’s equations

Here B and E are the intensities of the magnetic and electric fields and j is the space density of the electric current. Values are respectively the density of the electric charges, light velocity and conductivity of the liquid.

Magneto-hydrodynamic approximationThe main feature of the magneto-hydrodynamic approximation is the smallness of parameter . The electrical field in this approximation is small (E ~ (v/c) B) and so the third Maxwell equation converts into

Nevertheless, E plays an essential role in the Ohm’s equation

since both terms in its rhs are of the same order. Combining then the Maxwell and Ohm’s equations we obtain the Induction equation:

where is the magnetic diffusivity.

Divergences

Curls. (Magnetic Field)

Here we remind about sunspots

Induction equation 1

forms. Here η is the magnetic diffusivity and

in the Euler

and Lagrange

is the so called substantial time derivative i.e. derivative in respect of a moving liquid element and

Simplification: Induction equation without flow

Induction equation without flow 2

The same process can be seen from another point of view. The electrical current supporting the field disappears and so the magnetic field vanishes.

Induction equation without flow 3

What about the typical time for the Earth

How long is the typical time of attenuation in a body with typical size ?Simple estimation:

?

Direct solution shows that this time is almost 10 times overestimated:

Simplification 2: Induction equation with uniform flow

If the time intervals are small, then this equation takes the form:

This means that a solid body when moving progressively, transfers its magnetic field. Moreover, the rotation of a solid body does not change its field either. So the Earth moving in its orbit and rotating at its axis keeps its magnetic field.

Is this strange? - Not so much: The Earth carries away the electrical cur-rents in its core. So the geomagnetic poles rotate over the geographic ones.

In the presence of uniform flow velocity, the first term in lhs of the Induction equation vanishes:

Frozen magnetic field

By analogy with Reynolds number MHD enters the so called magnetic

Reynolds number

liquid contour

magnetic line

~

. During the process of the flow distance

between two liquid points changes. Inductionequation shows in the case when magnetic field changes being proportional

mR :

Lorentz force in the case of large magnetic Reynolds number plays the of elastic force and respectively energy of the flow converts into magnetic energy bystretching of line of force of the field. In theof small magnetic Reynolds number this force converts into friction one: . When a star converts into neutrons one its density enhances with many orders of magnitude. What about its magnetic field?

Frozen magnetic field: Solar wind and magnetosphere

Momentum equation

Thus magnetic field can be created only by the flow of the conducting fluid. How can this flow be determined? From the second Newton’s law

one can obtain the momentum equation of Navie-Stocks for the flow:

It shows that a liquid parcel moves being driven by the pressure and Coriolis forces, by the Archimedean and Lorenz forces

and by the viscid force.

Here are, respectively, the flow velocity, intensity of magneticfield, density of electric current, temperature deviations, pressure, angular

velocity of the Earth and is the coefficient of thermal expansion. Acceleration, a, of the liquid parcel has the form:

,, , , ,,, , , , and

Heat transport equation in Boussinesq approximation

The Archimedean force, which is proportional to the density deviations is the driving force of convection. These deviations depend on the temperature and pressure and as the Bous-sinesq approximation is applied for liquids, their dependence on pressure is neglected. Correspondently the Archimedean force takes the form: , where are temperature deviations, is the density, g is the gravitational acceleration.

is the coefficient of thermal expansion under constant pressure.

The substantial derivative here means the times derivative in respect of a moving small element of liquid. This element loses heat due to diffusion, but it is heated by the internal source Q. So it is not the cooling teapot.

Whole set of the geodynamo equations

In most computer simulations geodynamo is supported by heat flux from the inner core boundary. The problem is to define 11 values: three vector fields and two scalar fields, .

Heat sources: soli-dification on ICBradioactive heating

Adiabatic reference stateThe Boussinesq approximation neglects the density changes of a liquid element due to its compressibility. Is this assumption adequate to the Earth’s core where density changes are of order of ~ 20% between bottom and top boundaries? Let us consider the conditions for existing of convection. The whole set of the reference state equations is given by PREM.

Liquid core (Fe)

Heat sources: solidi-fication on ICB andradioactive heating

Solid core (Fe)

and

under constant pressure. The differentials here are:

If the liquid element rises without heating, then its entropy does not change, dS = 0. So we obtain the equation for the temperature of the Adiabatic refe-rence state:

is the specific heat

Adiabatic and Archimedean cooling

What is the physical meaning of the new terms obtained above? The gradient of adiabatic temperature creates adiabatic heat flux with surface density :

The whole adiabatic heat flux increases with r:

This is possible only in the presence of the heat support .

Due to the small compressibility (~ 20%) of the liquid in the core this source happens to be approximately uniform:

Thus, the term in the equation for the super-adiabatic temperature plays the role of cooling. We call it the adiabatic cooling. Another new term, taking into account the equ-ation for the adiabatic temperature can be written in a form where is the rate of the Archimedean work. Thus this term describes cooling due to the work of theArchimedean force.

Heat fluxes in the Earth’s coreConvection exists only if the reference state is adiabatic one. The adiabatic temperature profilecreates the adiabatic heat flux (AHF) which increases with r. In absence of radioactive heating the only source for it is the super-adiabatic heat flux (SHF). Thus AHF plays the role of coling SHF. Energy of SHF can converts into other types of energy e.g. in the magnetic one. It is alive flux. Energy of AHF is dead. It is the energy which irreversible converts into heat. In the Earth’s core main part of SHF converts SHF and so value of SHF decreases ~ 20 times.

1 . 5 2 . 0 2 . 5 3 . 0 3 . 5x 1 0 0 0 k m

0

1 0

2 0

3 0

Adiabatic heat flux

Super-adiabatic heat flux

Whole heat flux

ICBCMB

AHF: SHF:

Boussinesq heat fluxCMB

ICB

r

Heat transport equation in Incompressible approach

The pressure term and space dependence of the referent temperature are the essential differences with BA which neglects compressibility. After some thermodynamic and some algebra we obtain the heat transport equation for the Incompressible approach:

Heating of the moving liquid element is proportional to its entropy changes.

Its entropy increases with the temperature and decreases with enhancing of pressure

What is the physical meaning of these (underlined) new terms?

Incompressible approach

Energy equations

Only the heat energy conserves in the Boussinesq approximation:

Sum of these equation yields the conservation law for the whole energy:

Boussinesq onset of the convection

Very important difference between BA and IA is the onset of convection. The amplitudes of theonset flow are small and so the quadratic terms must be neglected in the equations:

Then these equations become linear:

Their solution can be searched in an exponential form:

Then we obtain uniform algebraic equations for and . Equalizing the determinant of this system to zero, we obtain the dependence of the frequency from k. Then, using the boundary conditions, we obtain the dependence of the frequency from the heat flux on the bottom boundary. Equalizing the imaginary part of the frequency to zero, we obtain the heat flux for the onset of the convection. All this discussion is based on the uniformity of the equations. However, IA shows that they are not uniform:

Amplitudes of the convectionThus Boussinesq approximation neglects the work over the flow. Therefore, the amplitudes of the flow velocity and magnetic field are infinitesimally small in this approximation. In IA the essential part ofthe heat energy converts into magnetic one and so its amplitudes are not negligible.

Under integration over the whole space, the lhs of the last equation gives the changes of the whole mag-netic energy. The first term in rhs describes the flux of the magnetic energy across CMB. Both these terms vanish under averaging this equation over long time period. Then we obtain:

The rate of work of the Archimedean force can be estimated from the equation for adiabatictemperature. It is proportional to the rate of heat production on ICB:

It is remarkable that the efficiency of the dynamo appears to be of order of the Carnot efficiency. Fromhere we obtain the estimate for the amplitude of the magnetic field, which is in a good agreement with the direct observation.

Boussinesq and Incompressible approaches

Adiabatic cooling Taken into account. Dimini-shes activity of the heat flux in the direction from ICB to CMB

Not taken into account

Law of energy

conservation

The whole energy is

conserved

Only the internal energy is

conserved

Amplitudes of the flow All values, V, B and are defined by the heat flux.

is defined by the heat flux.

V and B are undefined.

Ohmic dissipation and Archimedean cooling

Taken into account in the heat transport equation. Redistributes heat sources.

Cannot be taken into account in the heat transport equation.

Boussinesq onset of the convection

Impossible Possible

Experiments Impossible Possible

IA BA

Boussinesq and Incompressible approaches

Адиабатично охлаждане Отчита се. Намалява актив-ността на топлинен поток в напраление от ICB към CMB

Не се отчита

Запазване на енергията Пълната енергия се запазва Запазва се само топлинната енергия

Амплитуди на решението

Всички амплитуди, V, B и се определят чрез топлинен поток.

се определя чрез топлинен

поток. V and B са неопреде-лени по принцип.

Токова диссипация и Архимедово охлаждане

Отчитат се в уравнението за топлопренасяне.

Преразпределя топлинните източниците

Не се отчитат в уравнението за топлопренасяне.

Внезапно възбуждане на конвекцията

Не е възможно. Възможно е.

Експерименти Истински IA експеримент не е възможен

Всички експерименти са от Буссинесков тип.

IA BA

Computer simulation of geodynamoComputer simulation of geodynamo

БЛАГОДАРЯ!БЛАГОДАРЯ!