Güneş çevriminin dinamik modellenmesi
Başlık çevirisi mevcut değil.
- Tez No: 39138
- Danışmanlar: PROF.DR. ERDOĞAN ŞUHUBİ
- Tez Türü: Doktora
- Konular: Mühendislik Bilimleri, Engineering Sciences
- Anahtar Kelimeler: Dinamik modelleme, Dinamo teorisi, Güneş enerjisi, Dynamic modelling, Dynamo theory, Solar energy
- Yıl: 1993
- Dil: Türkçe
- Üniversite: İstanbul Teknik Üniversitesi
- Enstitü: Fen Bilimleri Enstitüsü
- Ana Bilim Dalı: Belirtilmemiş.
- Bilim Dalı: Belirtilmemiş.
- Sayfa Sayısı: Belirtilmemiş.
Özet
ÖZET Güneşin gözlemlenen 22 -yıllık yaklaşık - periyodik manyetik aktivitesi, ortalama elan elektrodinamiği çer çevesinde ve diferansiyel rotasyon ile ©(-etkinliğinden kaynaklanan dinamik etkinmeler gözönüne alınarak ince lendi. iCurulan modeller, kısmi türevli -diferansiyel denklemler olan alan denklemlerinden tek bir mod üzerinde yapılan kesintiyle elde edilen.kuvadratik nonlisecrite sahibi adi diferansiyel denklem sistemleri ni içermektedir. îki ayrı modelde elde edilon bu denklem "takımlarının uygun parametre aralıkları içinde limit- çevrim çö'zümleri bulunduğu gerek nümerik analizle, gerek normal form analiziyle gösterilmiştir. Yukarıda adı geçen her iki dinamik etkinliği kapsayan ikinci model, limit- çevrim ve torlar gibi stabil çözümlerin yanısıra belli parametre aralıklarında güneşin düşük aktivite 4tvirle» rine (grand -minima) karşılık gelecek şekilde kuvazi- periyodik ya da kaotik çözümler de içerir. Gözlemlerin en belirgin özelliklerini bu şekilde kapsayan bu tarz basitleştirilmiş modeller, bize nonlineer denklem sistem lerinin davranış zenginliğini kanıtlar ve bugün için analizlerine yetecek matematik bilgisine sahip olmadığı mız ana denklemlerin daha geniş kapsamlı çözüm zengin-' ligine işaret ederler. fv.)
Özet (Çeviri)
SUMMARY The solar magnetic activity, in addition to the 22-yeax cycle, shows certain non linear characteristics as evidenced by the unpredictability of sunspot numbers from one period to the next and the famous“grand minima”, times when the magnetic activity is very low for several periods of the cycle. The kinematic the ory of the solar dynamo is insufficient for explaining these higher-order features of the magnetic activity. In this work, we have tried to obtain these nonlinear features by building sim ple dynamical models. These models incorporate feedback from the generating mechanisms. The kinematical theory of magnetic field generation in electri cally conducting fluids is based on obtaining growing solutions of the induction equation: ^ = VA(uAB) + r/V2B (1) where u,is the velocity field of the fluid, B. is the magnetic field and r\ is the mag netic diffusivity, assumed constant. Roughly we can say that the first term on the right, the convective term, is responsible for giving generation i. e. growth of the magnetic field for appropriate assumed velocity fields while the second term gives diffusion. When the u-field is assumed to be given independently of the B-field, this equation is a linear evolution equation for the B-field. In this case, we say we have dynamo action if we obtain exponentially-growing solutions for B. Obtaining stationary solutions ( steady or oscillatory ) is only possible for very special u-fields, but then these are incapable of amplifying seed-fields. This is the impasse of the linear theory. Of course, in general a growing mag netic field will begin to act on the fluid motions after a while and modify them through the Lorentz force, so the u-field can no longer be assumed as given, but has to be calculated by solving the Navier-Stokes equation containing the Lorentz force term. This makes the problem nonlinear. Considering that the Navier-Stokes eqations are in general unsolvable at the present state of the art even in absence of the Lorentz force, this is a formidable problem. It is only recently being tackled and then generally by rather oblique approaches» Ours is going to be one such approach of truncating the partial differential equations. We shall work in the framework of Mean-Field Electrodynamics. This is an (vi)approach to the induction equation that has been developed to deal with tur bulent velocity fields, which is generally the case in astrophysical contexts. The basis of this theory is the separation of the fields into their mean and fluctuating parts. Since the effects we are concerned with have to do with the large-scale field of the sun, we shall use the averaged form of the induction equation: BB ^- = VA(u0ABQ) + VA(aB0) + r]eV2B0 (2) where uq and B0 are the averaged velocity and magnetic fields, the term with the a-effect is derived from the < u A6.>-term with u and b denoting the fluctuating fields under the assumption that this term is a linear functional of i?o and the turbulent velocity field lacks reflection symmetry. r)e is the effective diffusivity which adds turbulent eddy diffusivity /? to magnetic diffusivity and this eddy diffusivity term also has its origin in the < uAji >-term. In case of weak diffusion limit and homogeneous pseudo-isotropic turbulence the coefficients a and /? are found in terms of the fluctuating velocity u to be given by: T* T c* = -- < u.VAu >, /3=-<u2> (3a, b) o o with r as eddy relaxation time. The mean field of the sun is axisymmetric. When the fields are decomposed into their toroidal and poloidal components, the equations in spherical coordinates are given by: DA dt + (;fc- v) ^in6A^ = aB + w2 ~ ^b>A <*) dB B - + {rsin9up.V)( - - ) = rsin6Bp.VQ + [V A aB^ + Ve(V2 - ~T^)B (5) rsin9 where B is the toroidal magnetic field, A is the toroidal potential of the poloidal magnetic field Bp, up is the poloidal mean velocity field. The generating terms for the magnetic field are the first term on the r.h.s. of the first equation and the first and second terms on the r.h.s. of the second equation. For the sun this second term, can be neglected in comparison with the first, so we have an“au-”dynamo. The first term on the r.h.s. of the second equation gives the generation of toroidal magnetic field lines from those of the poloidal field by (vii)differential rotation.So ar-effect and differential rotation are the two generating mechanisms which in our approach to the dynamical theory will be modified as the magnetic field evolves. What is done is that simple evolution equations will be written down for the a-effect and the differential rotation. Since both these effects concern the ve locity field, consideration of the Navier-Stokes equations is in order. Looking at the Navier-Stokes equations with the Lorentz force included, we see that when the mean convective flow satisfies a balance between the viscous diffusive term and the inertia! forces like pressure, gravity, etc. The Lorentz force drives a fluctuation flow so that: Q 1 -= = [(V A5)A B].e+ + uV2u (6) at flop - ~~ where u is the azimuthal fluctuation velocity and v is the kinematic diffusiv- ity. This equation belongs to an oscillatory dynamical system governed by the Lorentz force and the viscous force of the turbulence. The torsional waves ob tained from the system will be taken for the modification of the differential rotation. As for writing an evolution equation for the a-effect, we shall have to consider the effect of the magnetic field on the turbulence. The existence of a magnetic field on large scales modifies the energy transfer of the turbulence on inertial scales by providing an important coupling between the velocity and magnetic fields through Alfven waves. At the same time, in case of weak diffusion there is a transport of magnetic energy and helicity to large scales due to the a- effect. The relaxation time of magnetic helicity at large scales is determined by the ohmic diffusion time in these scales and so cannot follow the magnetic field instantaneously. But its evolution can be given by a differential equation and this is independent of the form of turbulence spectrum when the magnetic Reynolds number is large i.e. in the weak diffusion case: - + V.(u0am) = - (Bo.V A Bo j- BQ) - - (7) 1 1 9' 2fcjfy Here uq and Bo are the mean velocity and magnetic fields, fc& is the wave-number of energy input, a* and am are the kinetic and magnetic heli cities. (viii)Equations (4), (5), (6), (7) are the basic equations for a dynamical model of the solar dynamo. They are far too complicated for a direct frontal attack. What we shall do is to employ the method of truncation frequently used in fluid dynamics in order to convert these partial differential equations to a system of ordinary differential equations. The equations are assumed to have wave-like solutions in space and they are truncated for a single spatial mode. This procedure turns them into a system of nonlinear equations in which only the derivatives with respect to time occur. We consider two models: one in which differential rotation is not taken into account, and another where both dynamical effects are accounted for. Both models contain quadratic nonlinearities. The existence of stationary solutions corresponding to the solar magnetic cycle can be demonstrated by the limit- cycles in both models, found by either a method of substitution or by employing Normal Form analysis. Both of these models can give stabilization and modulation of oscillations for an extended range of the primary parameter, the dynamo number D, of the non- dimensional equations. This is in sharp contrast to purely kinematical theory. Also quasi-periodic and chaotic time behavior is possible with increasing D in the second model while the first seems too stable in this respect. However, it would also be possible to obtain chaotic behavior with this model if higher order spacial modes are taken into account in the evolution equation for the magnetic helicity. With our second model, it is possible to obtain quiet periods of magnetic activity with mean fields close to zero simulating the observed grand minima of solar activity. Numerical calculations at the supergranular scale (k ~ 10-8 m~x) put the range for the magnitude of the a-effect necessary for the stabilization of dynamo waves around the values calculated from kinematical models. While our models bring no novel restrictions on the range of parameters as calculated from the kinema^c theory, it is encouraging that such simplified dynamical approaches do not give rise to contradictions. A basic restreiction of our models is that truncation is made for one single mode and so interactions between modes is eliminated. So effects like the inverse cascade of magnetic energy, negative diffusivity, modification of field structure (i.e. spatial intermittency) are rendered outside the scope of the theory. The fact that truncation proeedurea are not unique is a shortcoming of all such (ix)low-dimensional models. On the other hand, these various dynamical systems exhibit common properties in their nonlinear behavior in different parameter ranges. Limit-cycles, tori, quasi-periodic and chaotic regimes are seen in almost all of them. If we interpret this fact as a sort of universality, it gives us hope that the original partial differential equations which cannot be directly tackled by available mathematical methods have solutions containing similar behavior. It appears from the observations on the number of sunspots that long-term predictions of their time behavior is not feasible. In this sense, a chaotic behavior exists. On the other hand, the 22-year cycle is a very regular phenomenon, even though a low-order perturbation in the period can be seen. That is, totally chaotic behavior cannot be in consideration. Now, the solutions of our systems (as well as of others) predict regular behavior for certain ranges of parameters and disordered behavior for other ranges. What one would hope for is to find solutions that would incorporate both types of behavior on different time-scales. It is still an open question whether such realistic behavior can be assured in the solutions by taking into account the effects that will arise from interactions between modes of different wave-numbers or the modification in parameters (e.g. diffusivity, dynamo number) with magnetic field strength. We must also remember that sunspots come into being when the field lines of the toroidal magnetic field, generated in the deep convection zone 105 km beneath the surface, are carried upwards to the surface by their buoyancy. This has a very complex mechanism in the frame of turbulent convection. So, at least partially, the disorder in the time dependence of sun-spot behavior may be due to complications arising from the transport of the generated field through the convection zone rather than to chaotic behavior in the field generation mechanisms. In short, we can say that such idealized models cannot be expected to give all aspects of the sun's mean magnetic field behavior. It is rather surprising that they can give the main definitive characteristics of the behavior of a medium that contains so many different kinds of instabilities. This fact attests to the richness of behavior contained in simple nonlinear dynamical systems. Considering how much richer the full partial differential equations will be, the induction equation coupled with the Navier-Stokes eqations containing a Lorentz force term, the behavior of solutions found for the truncated systems can point out the way and bring to bear constraints hi ways of dealing numerically with the full equations. (x)
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