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Magnetic reynolds number and model-zgeodynamoA. P. Anufriev a & I. Cupal ba Geophysical Institute , Bulgarian Academy of Sciences , Sofia,Bulgaria , 1113b Geophysical Institute , Czechoslovak Academy of Sciences ,Prague, Czechoslovakia , 141 31Published online: 19 Aug 2006.

To cite this article: A. P. Anufriev & I. Cupal (1991) Magnetic reynolds number andmodel-z geodynamo, Geophysical & Astrophysical Fluid Dynamics, 60:1-4, 261-268, DOI:10.1080/03091929108220006

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Geophys. Astrophys. Fluid Dynamics, Vol. 60, pp. 261-268 Reprints available directly lrom the publisher Photocopying permilled by license only

c 1991 Gordon and Breach Science Publishers S A Printed in the United Kingdom

MAGNETIC REYNOLDS NUMBER AND MODEL-Z GEODYNAMO

A. P. ANUFRIEV

Geophysical Institute, Bulgarian Academy of Sciences, I I13 Sojia, Bulgaria

and

I. CUPAL

Geophysical Institute, Czechoslovak Academy of Sciences, 141 31 Prague, Czechoslovakia

(Received 12 October 1990; in jnal form 24 April 24 1991)

The magnetic Reynolds number (MRN) is used in estimating the quantities appearing in the current theory of the nearly symmetric dynamo. In kinematic theory, all expansions can be developed in terms of the MRN, but this seems to be less advantageous in the case of hydromagnetic theory and in particular the model-Z. An attempt has been made to replace the role of the MRN in the nearly symmetric expansion by the amplitudes of the individual quantities, where the amplitudes of the non-axisymmetric quantities are considered to be small compared with those of the azimuthal quantities. The amplitude of the axially asymmetric velocities and the amplitude of Archimedean buoyancey play essential roles here.

The expansions of the theory are valid only when certain assumptions are made about the field amplitude, and the velocity and buoyancy forces, even though the amplitude of the resulting a-effect is unaffected. The a-effect can be altered only by rapid changes in the non-symmetric velocities in the generation region.

KEYWORDS: Magnetic Reynolds number, model-Z.

1. INTRODUCTION

Braginsky (1 965) introduced nearly symmetric expansions where all quantities can be expressed in terms of W-+, where W = L V / v is the magnetic Reynolds number (MRN) and L, V and y~ denote a characteristic length, a characteristic azimuthal velocity and a magnetic diffusivity respectively. There can be no essential objection to L being the radius of the Earths core, and q = l / p , where p and G are the permeability and the conductivity of the core respectively. The question is, however what is V? In linear kinematic theory, V can be selected in arbitrary way. It can, for example, be assumed to be of the same order as the westward drift of geomagnetic field; this is not the case in hydromagnetic theory.

Braginsky (1978) and Braginsky and Roberts (1987) solved a particular model-Z characterized by a large azimuthal thermal wind, and this led to large values of the MRN. Braginsky (1978) also concluded that the model azimuthal velocity is about 4 to 8 degree/year, which is more than one order of magnitude higher than the velocity of the westward drift. Braginsky (1978) therefore called W an effective MRN and also argued that the observed magnetic field drift does not directly indicate the fluid velocity, but reflects the velocities of MAC waves: 9 is assumed to be about

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262 A. P. ANUFRIEV A N D I . CUPAL

100 in model-Z, and thus V is of the order of the speed of westward drift and is not a characteristic azimuthal velocity in the model-Z. In addition to this problem, the solution of model-Z also led to a relation between poloidal and azimuthal quantities that is different from that assumed in the nearly symmetric expansions of Braginsky.

Probably, the introduction of 92 into the nearly symmetric expansions is not suitable in case of hydromagnetic theory because it depends on a characteristic velocity which cannot be arbitrary selected as in the case of the linear kinematic theory.

2. THE NEARLY SYMMETRIC EXPANSION

Let R be the Earths angular velocity and p the average density of the core. Adopting L2/q as the unit of time, q /L as the unit of velocity (small), (2!2qpp)+ as the unit of magnetic field, 2!2qp/L as the unit of buoyancy forces (small) the following dimensionless equations can be derived in the case of the Earths core

dA aB -+VO-(vxB)=AA, -- V x (V x B) = AB, at at

1, x v = V p + (V x B) x B + F

It should be noted that similar dimensionless equations can be derived by adopting 9 q / L as the unit of velocity, (2BRqpp) as the unit of magnetic field and 2.%Rqp/L as the unit of buoyancy forces (units used by Braginsky 1965, 1978 and by Braginsky and Roberts 1987), however 92~ x B then appears in equations instead ofv x B.

The nearly symmetric expansions introduced by Braginsky (1965) are

F = 8- FI, + F, + 9- $F, where only dashed quantities depend on 4 and the effective MRN 9 plays the role of a parameter that scale the relative magnitude of the individual terms in the expansions, and is assumed to be of order 10. We hope to replace Braginskys expansions by

F = 9 + F l + + 9 F , + 5P . All quantities B, B,, B, 6, V p , V, F, P,, F are assumed to be of order 1. The amplitudes B, a,, 97, Y-, V,, V, 9, 9+, gr are, however, of different orders, and they scale the individual terms in expansions (1) differently. We shall assume B,/B

MODEL-2 GEODYNAMO 263

accordance with the nearly symmetric theory) it is also assumed that

The smaller the ratios #/@, V/V, F/.F, the better the nearly symmetric approximation. Braginsky (1965) characterized the ratios by an effective MRN and he introduced the scaling a,/@ = Vp/V = 6%- and g/g = V/V= B-*. Tough and Roberts (1968) and Soward (1971) later adopted also 9,/9=93?- and 9/9=9?-%. This scaling (very useful in kinematic theory) seems to be too rigid for hydromagnetic theory, though the nearly symmetric theory can be successfully developed under weaker assumptions ( 2 ) . Perhaps, the first relation in ( 2 ) may also be omitted. Nevertheless, we take it into account in the our considerations. The folloking notation will be also used

B = BB, B, = BpBp, u = VY-U, V, = VpVp, F, = FF,,

for the sake of simplicity.

motion implies Using ( l ) , and taking mean values over 4 denoted here by ( ), the equation of

- 9 ( V x Fp)@ - (9?)2(V x [(V x B) x B])4 - g;{V x [(V x B,) x BPI)@.

The last two terms may be omitted and, therefore,

(3)

holds true. The question is, Under what conditions can this truncation be made? The truncation is sensible when, for example,

V - 9 - B 2 (4)

Of course, the truncation may also be incorrect if (for example) B change abruptly anywhere. Then VB is large and the term V x V x B may not be of order l !

Denoting u/s by 5 and restricting ourselves to the case of a non conducting mantle [B(s,zi)=O], we find (3) implies

[ = B 2 / s 2 + ~ + f, (5)

Z I

where w=u(s,zI)/s, z,=(l-s2)*, f = s - l [ (V x FP)+ dz and, of course, (=Vc, o = V G , and f = 97. J:

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264 A. P . ANUFRIEV A N D I . CUPAL

It is possible to show that, under conditions (4), truncation in the remaining equations of the nearly symmetric hydromagnetic dynamo theory is also acceptable. The equation of motion still implies

U, = s - B, - V(SB) + 9-(&?/&?)2Fz, (6) where Fz represents the Lorentz forces of the non-axisymmetric magnetic field and under certain conditions need not be small. Soward (1971) showed that Fz can be expressed in terms of the non-axisymmetric velocities and the axisymmetric Archimedean buoyancy. It should be noted that F = O in the case of the Archimedean buoyancy and, therefore, F = FF, + 5 F is actually used in expansions (1). Introducing $ = sA the equation governing the meridional field takes the form

where

B, = V x (s-$l+), vp = V x ( s - ~ l+ ) , up = /U,

AS($/s) = V2($/s) - $/s3 = s-A-$ = s-V*(V$ - s-$lS), YU, = ZU, + SU,. Here ct is expressed in terms of the non-axisymmetric velocity through up. If Vu,- 1 then ct - 1. In our notation a= (nY/%)ct and the amplitude of the a-effect is small. On solving for example model-Z, however, an a-effect of large amplitude in the layer is required before the generation of B, from B is sufficient. There is then a good opportunity of introducing a layer of thickness 6 much smaller than 1 in which some quantities change sharply (Braginsky, 1975). Generally, the estimation of cl in the layer is complicated. The &averaged terms in the expression for E include the product of two mutually perpendicular velocity components whose gradient perpendicular to the 8-layer cannot be too large because V-i=O. The estimate Vup-6-+, and hence c(.-8-, therefore seems to be correct. However, the a-layer in model-Z has some special features, e.g. the concentration of a-effect in the equatorial area, and the estimates are probably rather too low. We adopt

Similar considerations lead also imply that

in (6). Introducing 6, (6) and (7) in the layer therefore take the form

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MODEL-Z GEODYNAMO 265

where now also E and Fr are of order 1 in the layer. In the layer, condition (4), which enables small terms in our equations to be omitted, takes the form

The question is, Under what conditions can the term Fr be omitted from (6) or (6a). This will be discussed in another paper.

Using the nearly symmetric expansions (l), the equation governing the azimuthal magnetic field reads

dB - + SV;V(S-B) = AsB + (V[ x V$)$. at

3. VISCOUS COUPLING

In analyzing core-mantle coupling, a condition for the geostrophic part, o, of the azimuthal velocity can be found. Generally, both viscous and electromagnetic coupling should be taken into account, but, we shall assume that the viscous forces dominate; electromagnetic coupling will therefore be neglected. This means that only the Ekman layer plays a role, and we obtain

Ib; us dz = E sw/(4z )*, where kinematic viscosity.

=d,/L; here 6,= (v/Q)* is the thickness of the Ekman layer and v is the

Assuming that F , can be omitted, (6) and (9) imply that

where T = j; BB dz is the Maxwells tension. Braginsky and Roberts (1987) also used (9) with, however, ~ = 9 d , / L replacing on the right-hand side of (9) and (10) in their scaling.

4. COMPARISONS AND ESTIMATIONS

Comparing (3)-( 10) with the corresponding equations of Braginsky (1 978) or Braginsky and Roberts (1986), we see that only the constant coefficients differ from one set of equations to the other. This makes it possible to use their solution in our

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266 A. P. ANUFRIEV AND I . CUPAL

nearly symmetric expansions (1) which do not contain the effective MRN, 92, as the scaling factor. Braginsky and Roberts (1989) used a for the amplitude of Archimedean buoyancy and y for the amplitude of a-effect. We shall usef, and a, to denote their amplitudes of Archimedean buoyancy and a-effect respectively, to avoid possible confusion. When we compare the two sets of equations, we must also allow for the fact that their quantities measure amplitudes and that therefore their Cl denotes our Vco, their B, denotes our BB, and their B, denotes our @,BP. The set of equations (3)-(10) is formally the same as these of Braginsky and Roberts (1989), but their $ and vp are scaled to be of the same order as B and u respectively. Thus, their solution cannot be directly used. Braginsky (1975) noted that (2)-( 10) allow the following formal transformation of the quantities for an arbitrary constant q

Of course, the transformation must respect nearly symmetric expansions (1 ), especially conditions (2). If the solution of (3) (10) does not satisfy conditions (2) it is not made correct by applying the above transformation. The value of q can be obtained comparing (9) with that of Braginsky and Roberts (1987) where their E = %d,/L replaces our E' = 6,/L and thus q = 9-* = 10. Denoting their amplitudes by subscript 0 we easily arrive at

In fact (1 1) also imply

where the quantities on the left-hand side denote the solution of Braginsky and Roberts (1987) and, therefore, conditions (2) are satisfied because Bp?-B0 in their solution. It should be noted that the solution of model-Z leads to different values Fpo/Bo and V",,/Y, depending on f,, a, and E . We can estimate the solution of Braginsky and Roberts for their &=0.01, d=O.l,fchanging from -270 to 0 and a changing from -200 to 350 in the layer. It allows us to assume thatf,,a, - 100. The solution of model-Z implies that (Y'/Y)'-6-0.1 in the layer and thus non-axisymmetric velocity components are relatively large at the core-mantle boundary. Of course, V'/V is much smaller in the volume of the core.

The B, of Braginsky and Roberts, changes from - 1.5 to 1.5, + from -0.8 to 0, B, from -4.8 to 2, Bs from -4.8 to 4.8, (' from -900 to 140, and x from - 3.6 to 3.6.

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MODEL-2 GEODYNAMO 261

It should also be mentioned that the higher values of x and B, are concentrated mainly at the Core-Mantle Boundary. This allows us to find the orders of magnitude of the dimensionless amplitudes of our quantities:

B ~ = I O - ~ , a -10 , 9-104, v-104, ~ ~ - 1 , E1-10-4.

Adopting a=4 x 105[fl-m-], p = 1.25 x 10-6[Hm-] and L=3.5 x 106[m], we find q=2[m2s-] and q/L=5.7 x 10-7[ms-]; therefore

The MRN, R, of the model, which is not effective MRN 9, is therefore

R = LV/q = 9915.

Some may argue that el is too small, but this is a consequence of the scale using MRN. Braginsky and Roberts defined their E = (6,/L)9 assuming that 9- 100. We have obtained [see (9)] c1 = SJL, and thus E~ - corresponding to 6, = 850[m]. Finally, we estimate dimensioned geostrophic velocity amplitude cod:

cod = Yq/L2 = 1.6 x 10-9[s- 1 = 2.8 deglyear,

which is one order greater than the observed westward drift.

5. CONCLUSIONS

To solve some concrete model-Z, with viscous core-mantle coupling, requires the following input parameters to be chosen :

f, -amplitude of Archimedean buoyancy, a, - amplitude of a-effect, E ~ viscous core-mantle coupling parameter, 6 -thickness of the current layer.

These parameters define a class of nearly symmetric solutions of model-Z for which expansions (1) are valid. More precisely, the parametersf,, a,, E , 6, selected for the model define the ratio V/V in the layer but not in the volume of the core because the volumetric cc-effect is set zero. This allows us to assume any suitable ratio for V/V (and also for B/B) in the volume satisfying conditions (2). The smaller this ratio, the better (3)-(10) approximates the solution, and so the smaller is the departure from axial symmetry. After choosing for example v/V in the volume, no degree of freedom is available, and only one solution of model-Z for the non-axisymmetric terms can be found. Although the MRN, R, of the model can also be calculated, it is not true that V/V can be replaced by R - t and VP/V by R-. Rather, the estimate can be made by using an artificial effective MRN, 9-100 though the solution

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need not be bound strictly to this. The solution must also satisfy conditon (4) or (4a), which defines the relation between the amplitudes of the buoyancy forces, the velocity, and the magnetic field. If this condition fails, then the truncation leading to (3)-(10) may be not correct. A large MRN for the model and a large geostrophic velocity remain in all cases the principal characteristics of model-Z.

References

Braginsky, S . I., Self-excitation of a magnetic field during the motion of a highly conducting fluid, .%Jtief

Braginsky, S . I., Nearly axially symmetric model of the hydromagnetic dynamo of the Earth I, Geomugn.

Braginsky, S. I., Nearly axially symmetric model of the hydromagnetic dynamo of the Earth, Geomagn.

Braginsky, S . I., and Robers, P.H. Model-Z Geodynamo. Geophys & Astrophys. FIuid Dyn. 38,327-349

Soward, A. M., Nearly symmetric kinematic and hydromagnetic dynamos, J . Math. Phys.

Tough, J . G. , Roberts, P. H., Nearly symmetric hydromagnetic dynamos, Phys. Earth and Planet. Inter.

Physics J.E.T.P. 20, 726-735 [Engl. trans.] (1965).

und Aeron. 15, 122-128 [Engl. trans.] (1975).

und Aeron. 18, 255-231 [Engl. trans.] (1978).

(1987).

12, 19W1906 (1971).

1 , 288-296 (1968).

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