Estimates of quantities in a Hall effect geodynamo theory

Estimates of quantities in a Hall effect geodynamo theory 261 ESTIMATES OF QUANTITIES IN A HALL EFFECT GEODYNAMO THEORY A. de Paor School of Electrical, Electronic and Mechanical Engineering, National University of Ireland, Dublin (UCD), Belfield, Dublin 4, Ireland e-mail:annrao[email protected]e Summary Currents, resistances, dynamo constant, Hall voltage coefficient and inductances are estimated for the author’s geodynamo theory incorporating the Hall Effect. It is concluded that the Hall Coefficient in the bulk liquid core of the Earth is approximately 1.512x10-1, orders of magnitude greater than in normal liquid metals. The ordering effect of enormous pressure is a possible cause. 1. INTRODUCTION de Paor [1] published a radical theory of the Earth’s magnetic field and Sunspots, in which the idea of the self-excited dynamo [2] is integrated with the Hall Effect [3]. Conventional models of the geodynamo are very complicated and are simulated on supercomputers [4]. The author’s treatment may be understood from two first order nonlinear ordinary differential equations, describing currents in RL circuits, resulting from mass circulation of the liquid outer core, with coupling provided by Hall voltage generators. Other theories are silent on the most prominent secular variation of the Earth’s magnetic field, a cycle of declination taking about 624 years, and an associated cycle of dip. These follow directly from the author’s theory [1], although the calculations need to be corrected, because of the accidental use of dip poles rather than geomagnetic poles. Relationships between these poles were presented in [5]. The author’s theory was criticised [6] on two grounds. The first was a misconception of the Hall Effect as a perfect orthogonal axis power transfer mechanism and clarification was given [7], leading to correction of the classical expression for the twospecies Hall Coefficient [8]. The second criticism was that the Hall Coefficient is orders of magnitude too small to support the field. This was based on invocation of the resistivity of liquid steel at 1550 0 C and atmospheric pressure, and on the electron density in iron at room temperature and pressure. The pressure in the Earth’s liquid core however has a mean value of 232.3x10 9 Pa, and the mean temperature is 4650K. The pressure “ordering energy” per atom is about 17eV, compared to the thermal “disordering energy” of 0.4eV per atom. So dominant is pressure that the core solidifies at the centre, and so a dramatic effect on electromagnetic properties cannot be ruled out. Nobody has yet measured the Hall Coefficient under the pressures and temperatures involved, and the matter must remain open until that can be done. Allen [6] suggested a value of -4.7x10 -7 , whereas the author’s current estimate is +1.512x10 -1 . The author’s theory is filled out here by estimation of values for the currents involved, resistances, dynamo and Hall voltage constants, inductances, power relationships, and the Hall Coefficient. All units used here are S.I. 2. ESTIMATION OF QUANTITIES The basic equations of the theory [1] are 2 ahaddd dd dahaa aa ikikiR dt diL iikiR dt diL −+−= +−= ω (1) The first current, i a , sustained by the power k h i a 2 i d injected from the i d circuit, encircles the solid inner core, flowing around an annular path of depth 2r 1 , whose inner radius is that of the solid inner core, r 1 = 1.217x10 6 m [9]. The outer radius of the annulus is not quite that of the liquid core, r 2 = 3.485x10 6 m, but of a postulated transition zone at the core-mantle boundary, assigned the value 6 1213 10258.3][9.0 ×=−+= rrrr (2) Variation of current density with radius in the annulus is [1]       − =)(2 1 )( 13 3 2 rr ir r rj a (3) The contribution of j(r) to the Earth’s magnetic dipole moment (area by current) is drrrjrdm 1 2 2)( π = (4) Integrating this from r 1 to r 3 gives the total dipole moment as 262 Advances in Electrical and Electronic Engineering a irrm 31 π = (5) Currently [11] 22 10835.7 ×=m (6) and so 9 1029.6 ×= a i (7) The increment of power dissipated between r and dr is drr rdrrrj dP 1 2 1 2 2]2)([ πρ = (8) Integrating this from r 1 to r 3 and equating to P = R a i a 2 gives )](2[ )]([ 131 13 rrr rr R a − + = ρπ (9) The resistivity estimate [9],  = [1/6]x10 -5 , gives 12 10717.4 − ×= a R (10) The resistance of a frustum with base area A b , apex area A a and height h is ][ )]/ln(.[ ab ab AA AAh R− = ρ (11) Eqn. (11) is applied to two components of R d , firstly the resistance R d1 presented to current flow across the annulus, perpendicular to i a . Here 13 11 13 22 22 rrh rrA rrA a b −= = = π π (12) 13 1 10073.1 − ×= d R (13) i d enters the annulus over two parallel paths, from north and south. In the transition zone hugging the core-mantle boundary, each of these is modelled by a resistive frustum with the areas      −−−= −= 2 1 2 3 2 1 2 21 2 3 2 2 2 ][ rrrrrA rrA a b π π (14) The effective height of this bent frustum is the mean radius (r 2 + r 3 )/2 subtending an angle δ:         + = + = ][ 2 arccos 2 ][ 32 1 32 rr r rr h δ δ (15) The resistivity here is taken as twice that for the bulk liquid core, giving 12 2 10174.2 − ×= d R (16) The next components are cylinders in the transition region, of area r 1 2 and physical length r 2 – r 3 . However, not all current passes through this length: we scale by 0.75. Using the higher resistivity noted above gives 14 3 10098.6 − ×= d R (17) Finally, there are two cylindrical paths coming up and down onto the solid inner core, which is assigned the same resistivity as the bulk liquid core. Each of these cylinders has height r 3 and area r 1 2 : 13 4 10835.5 − ×= d R (18) Adding these four, 12 10926.2 − ×= d R (19) The dynamo emf is e d = k d i a . If the angular velocity of fluid in the annulus at radius r is (r), and flux density B(r), the increment of dynamo emf over dr is given by the famed “Blv” law: )(.).( rrdrrBde d Ω= (20) where )](2[ )( 13 30 rrr ir rB a − = µ (21) This gives )(.]2/[ 30 rofmeanirike aadd Ω== µω (22) Taking the mean value of (r) as just less than /2 gives the estimate 1= d k (23) Estimates of quantities in a Hall effect geodynamo theory 263 Latest estimates of  [10] have a mean of 0.4 degrees per year which, in S.I. units is 10 10212.2 − ×= ω (24) In [1], there are the relations 1 2 331 −+= xxx (25) da d RR k x2 3 ω = (26) da ah RR ik x= 1 (27) The condition for self-excitation is x 3 >1: the actual value of x 3 is 29.78. The unknown here is k h , and so 20 10516.3 − ×= h k (28) From Eqn. (1), the equilibrium value of dynamo current i d is 8 10341.1 ×== h a d k R i (29) The power sustaining the field is thus 8 2 10866.1 ×== dahf iikP (30) The power injected into the system from whatever processes drive rotation of the solid inner core, k d ωi a i d , exceeds this by only 5.264x10 4 (dissipated in R d ). What these processes are the author does not know, but they are probably thermal in origin, and may come from nuclear reactions in the inner core. In [1], k h and Hall Coefficient R h are related by )( ][8 130 13 2 1 rr krrr R h h + − = µ (31) giving 1 10512.1 − ×= h R (32) This applies in the bulk liquid core, but the value in the transition region at the core-mantle boundary is orders of magnitude smaller, so that there is no significant dipolar current circulating there. Finally, we estimate the two inductances L a and L d . L a is calculated from an expression which the author originally found on Wikipedia, but he has confirmed independently:      +−= 125.02) 8 ln( 0 a r rL a µ (33) This is the inductance of a single turn of mean radius r with conductor radius a. The 0.125 indicates that the current density distribution is intermediate between uniform and concentrated in the skin. From eqn. (5) we have 6 31 10991.1 ×== rrr (34) The cross-sectional area of the annular current path is 2r 1 [r 3 – r 1 ] and if we equate this to an equivalent circular area a 2 we get [ ] 6 131 10257.1 2×= − = π rrr a (35) Applying these figures, there results 662.1= a L (36) Finally, L d is estimated by tuning it to set the period of the very lightly damped oscillation on the (i a , i d ) spiral equal to 624 years. The result is 289.0= d L (37) 3. DISCUSSION The theory presented in [1], analysed there in dimensionless units, is filled out by estimating values for quantities in S.I. units. The most startling finding is the very large, positive Hall Coefficient. The theory can neither be firmly established nor rejected until experimental measurements of this are available—and that is likely to take many years. REFERENCES [1] de Paor, A. “A theory of the Earth’s magnetic field and of Sunspots based on a self-excited dynamo incorporating the Hall Effect” Nonlinear Processes in Geophysics, 8, 265-279, 2001 [2] Larmor, J. “How could a rotating body such as the Sun become a magnet?” Reports of the British Association for the Advancement of Science, 159-160, 1919 264 Advances in Electrical and Electronic Engineering [3] Kasap, S.: Principles of electronic devices and materials. McGraw-Hill, New York, 2002 [4] Glatzmaier, G. “Geodynamo: numerical simulations” Encyclopedia of geomagnetism and paleomagnetism, edited by D. Gubbins and E. Herrero-Bervera, Springer, Berlin, 2007, 302-306 [5] de Paor, A. and Burke, E. “A new eccentric geomagnetic dipole to give the correct dip pole locations” Advances in Electrical and Electronic Engineering (Slovakia), 5, 316-318, 2006 [6] Allen, J. “On the Earth’s magnetic field and the Hall Effect” Nonlinear Processes in Geophysics, 10, 437-440, 2003 [7] de Paor, A. “Clarification of the Hall Effect as an energy transfer mechanism in a theory of the Earth’s magnetic field and Sunspots” Nonlinear Processes in Geophysics, 10, 435-436, 2003 [8] de Paor, A. “Correction to the classical twospecies Hall Coefficient using twoport network theory” International Journal of Electrical Engineering Education, 43, 346-351, 2006 [9] Melchior, P.: The physics of the Earth’s core: an introduction. Pergamon Press, Oxford, 1986 [10] Zhang, J., Song, X., Li, Y., Richards, P., Sun, X., Waldhauser, F., “Inner core differential motion confirmed by earthquake waveform doublets” Science, 309, 1357-1360, 2005 [11] Roberts, P. And Glatzmaier, G., “Geodynamo theory and simulations“ Reviews of Modern Physics, 74, 1081-1123, 2000