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Magneto-Vortex Dynamo Model in Solar convection zone.

Sergey V. Ershkov
Institute for Time Nature Explorations, M.V. Lomonosov's Moscow State University, Leninskie gory, 1-12, Moscow 119991, Russia.

Here is presented a new magneto-vortex dynamo model for modeling & predicting of a processes in Solar plasma convection zone [1-3]. Solar convection zone is located above the level r > 0,6В0,7 R, where R is a Solar radius (see Pic.1 below). A key feature of such a model is that equation of Solar plasma motion as well as equation of magnetic fields evolution - are reduced to Helmholtz's vortex equation [4], which is up-graded in according with -effect [1-3].

Pic.1. Distribution of Solar plasma angular velocities,
depending on radius of Solar zone [1-3].
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Modeling of processes in Solar convection zone:

Below are represented a key steps of modern science scheme for evolution of Solar magnetic fields [1-3]:

1.

It is the fact that a Solar inner poloidal magnetic field (see Pic.2) has to be located in Sun's core [3], below the Solar convection zone r < 0,6В0,7 R (Pic.1):

Pic.2. Solar poloidal magnetic field (magnetic dipole).

2.

Besides, a proper scheme of 'frozen-in magnetic field' plasma motion or opposite scheme of 'frozen-in plasma motion' magnetic field are assumed to be actual in Solar convection zone [5]: - It takes place when plasma crosses magnetic field lines in its motion inside the Solar convection zone [1-3]; the main reason is the great conductivity of Solar plasma [2]: once intersecting a proper magnetic field line, particles of plasma will immediately be 'catching' by it, then particles are assumed to be moving only along a proper magnetic field line.

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3.

The next important step - is a 'differential rotation' [1-3]: - Shearing of poloidal magnetic field by the Sun's differential rotation in convection zone. The Sun rotates faster at the equator than the pole (20-30%) [3]. Toroidal field is produced due to this shearing by differential rotation [6] as well as due to effect of 'frozen-in magnetic field' plasma motion (so-called effect, Pic.3):

Pic.3. Shearing of poloidal field by the Sun's differential rotation (-effect).

4.

A key point is a transforming of toroidal magnetic field ­ back into additional poloidal magnetic field in Solar convection zone: - An additional field of Coriolis force initiates so-called '-effect' [1-3] (Pic.3 b): vortex in plasma is moving poleward under the influence of Coriolis force near equator. Then collecting near the pole, such a vorticity forms an additional total projection on poloidal magnetic field due to scheme of 'frozen-in plasma motion' magnetic field.

5.

Limitation of '-effect': - We should note that there is no problem with limitation of '-effect' in above scheme. An actual mechanism of `Solar refining from extra-magnetic field' regenerates a new magnetic field due to the process of magnetic field diffusion
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in plasma convection zone as well as due to magnetic field 'frozen-in Solar wind plasma' which is leaving the Sun (a very regular process). Besides, as a result of above scheme (steps 1-4) we should obtain that a maximum of Solar activity ­ strong Solar wind, extra-flares, Sunspots with strong magnetic field - are assumed to be concentrated at the proper belt in Solar convection zone, i.e. in belt from middle latitudes of the Sun (~30 latitude in
respect to equator)

up to equator itself: namely such a Solar activity will be located

where vorticity had begun massively to collect from equator under the influence of Coriolis force. For example, one of such axisymmetric belt phenomena is well-known "Maunder's butterfly" diagram [2].

Mathematical Model of Magneto-Vortex Dynamo in Solar convection zone:

In accordance with [5], we note that equation of motion for each components of plasma should be represented in form of generalised vorticity evolution:

rot P , P m v e A / c

B



e rot A / c , rot v w



- here m ­ is the mass of particle, v ­velocity of particle, e ­ it's charge, c ­ speed of light in vacuum, B ­ vector of local magnetic field.

When the electro-magnetic part is dominating in above expression for generalised plasma vorticity , we obtain a super-position of magnetic field B ('frozen-in plasma motion v') & vortex field w = rot v, which is described by Helmholtz's vortex equation [4], to be generalised & up-graded with Coriolis-effect [3]:

w t w rot v (B w) rot w ,





B ~ v w t w rot v w rot w
- here ­ is a kinematic viscosity, ­ some numeric coefficient.

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When a vortex part w = rot v is dominating in above expression for generalised plasma vorticity ('frozen-in plasma motion v', w ~ v), we obtain a proper equation for magnetic field B which is diffusing in Solar convection zone, such an equation is up-graded with '-effect' [3]:

Bt B rot v B rot B , w ~ v , w t w rot v B rot w
- here ­ is a magnetic viscosity.









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Indeed, let's obtain it in details:

I.

Near equator, the direction of toroidal magnetic field coincide with direction of plasma motion (Pic.3). In this case, the condition of magnetic field being 'frozen-in plasma motion' means that magnetic field is not diffusing in plasma as well as it is not being under the influence of '-effect' yet [3], but means that magnetic field is moving 'frozen-in plasma motion' around the equator (see (2)):

B ~ v B t rot v B 0 .
Besides, we should note that vortex field w = rot v (in expression for ) is diffusing according to (1) as well as it starts to be under the influence of Corioliseffect (vector multiplying is not equal to zero), but it is not yet being under the '-effect' for vortex field totally [3]:





w t w rot v w





1.1



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II. When '-effect' actually begin to arise in Solar convection zone (at final we should
have a total mechanical vorticity, to be concentrated near the ~30 of latitude in respect to equator

), it means that magnetic field begin to be 'unfrozen-in plasma motion', but

mechanical vortex ­ opposite, to be 'frozen-in plasma motion' (besides, 'w-effect' is
already over, hence vector multiplying is equal to zero in

(1)):

w ~ v , rot v w 0 , w t w rot w



.

In this case, magnetic field should also be diffusing in plasma as well as it should be under the proper influence of Coriolis-effect or '-effect' for vortex/magnetic field (besides, 'w-effect' is already over, hence vector multiplying is equal to zero in (2)):

B t B rot B





2.1

Thus, in Solar convection zone (starting at equator ­ across the middle latitude ­ up to the pole) we observe a proper evolution of generalised vorticity , taking into consideration the inter-changing of vortex (mechanical part of generalised vorticity) & magnetic field:

- on equator the magnetic field is being conserved ('frozen-in plasma motion'), but mechanical vortex is to be diffusing in accordance to (1.1), - at the middle latitude vortex is being 'frozen-in plasma motion', but

magnetic field, as well as vortex, is diffusing in accordance to (2.1).

We should also note that both of equations (2.1) - for vortex w as well as for magnetic field B (in the Sun-spots, near ~30 of latitude) - are identical one to each other in mathematical sense, but only physical essence has to be changed for our interchanging it one to each other (for the case of mechanical vortex/or for the case of magnetic field).

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Above equation (2.1) for each components of magnetic field B ­ is proved to be a type of inhomogeneous heat partial differential equation [7] in Cartesian coordinate system for 3-D case, with vorticity rot (B) as a source of internal generation of magnetic field B.

References:

1. 2.
3.

Wilson P R, Burtonclay D, Li Y. Astrophys. J. (1997) 489 395. Kitchatinov, L.L. The differential rotation of stars // Physics­Uspekhi. (2005) Vol.48, 5. P.449-467. Soloviev A.A. and Kirichek E.A. Astrophys. Bull. (2008) 65, 93 // see also material (in Russian): http://solarwind.cosmos.ru/txt/2009/presentations/The%20Sun/Solov'ev.ppt Dr. E.Kamke. Hand-book for ordinary differential equations // Moscow: "Science" (1971). Yankov V.V. About 'frozen-in plasma' magnetic field in the kinetic model of collisionless plasma // JETF lett. (1993) Vol.58, 7, P.516-519, see also (in Russian): http://www.jetpletters.ac.ru/ps/295/article_4763.pdf

4. 5.

6. 7.

Dikpati M, Gilman P A. J. Astrophys. Astr. (2008) 29, 29-39. Evans, L.C. (1998), Partial Differential Equations, American Mathematical Society, ISBN 0-8218-0772-2; see also: http://en.wikipedia.org/wiki/Heat_equation

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