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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