M. RESHETNYAK, P. HEJDA

58

poles. Later we will discuss some other scenarios which

yield similar results.

For negative C, when the geostrophy breaks due to

the relative intensification of convection in the equatorial

plane, we get the opposite result, see Figure 1(c): the

regime of the frequent reversals observed in Figure 1(c)

in [7]. In this case force F is directed from the poles and

the equilibrium point at the poles becomes unstable. The

new minimum of the potential energy at the equator leads

to the appearance of a new attractor, so that for C = -5

the axial dipole fluctuates with zero mean value and

maximal amplitude M~0.4. This state corresponds to the

equatorial dipole.

1

1

()sin( ()),

N

e

i

i

t

N

t (5)

so that |Me|~1 and does not undergo reversal. Similar

behavior of the magnetic dipole is observed on Neptune

and Uranus; for more details see, e.g. [21].

Now we consider ψ = -cos22

1

θ for which the corre-

sponding force F = -2sin4θ changes sign in each hemi-

sphere, see the example with the second-order zonal

spherical harmonic -, where is associated Leg-

endre polynomials, in Figure 1(e) in [7].

0

4

Pm

l

P

For C > 0, the potential barrier is in the middle lati-

tudes, which prohibits the reversals as observed in Fig-

ure 1(e) in [7]. In one of our runs only one reversal was

observed for C = 2. The other important point is the ex-

istence of the stable point at the equator, where M = 0.

We could then assume a regime with two attractors: near

the poles and at the equator. This regime is really ob-

served in Figure 1(d). The inverse transition was not

observed. Inspection of the state with small M at the be-

ginning of the run leads to a quite large estimate of the

equatorial dipole amplitude Me, see Figure 1(d). Thus, in

principle, this regime can also be related to the giant

planets’ dynamo as well.

The last example corresponds to the case C < 0, when

in addition to the attractors at the poles (related to rota-

tion ), two new attractors appear at the middle latitudes.

The variation of C leads to regimes C = -1 with fre-

quent reversals, observed in Figure 1(f) in [7]. Moreover,

we can get regimes, see Figrue 1(e), when the magnetic

pole stays at the high latitudes, however, ||M

(the

partial synchronization of the spins). There are some

jumps to the unstable state M = 0 with the dipole at the

equator. Note that the decrease of the spatial scale of

leads to the increase of its amplitude C.

4. Conclusions

To conclude, let us stress that the toy models do not pre-

tend to compete with the known 3D dynamo models.

However, toy models are being developed for better un-

derstanding of physics of the processes. The main point

is that model of the spins does have a background based

on our present knowledge of the flow structure in the

core. The system of the cyclones in the core, which act

like the individual magnets, produces the net magnetic

flux observed outside the volume of generation. These

individual magnets can interact with each other, feel the

direction of angular rotation and are disturbed by the

thermal-compositional convection forces. The domino

model is based on the function decomposition related to

the geostrophic state. It is the geostrophy, which is re-

sponsible for the cyclone formation along the axis of

rotation.

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