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We review recent mathematical results on the theory of ideal MHD turbulence. On the one hand, we explain a mathematical version of Taylor's conjecture on magnetic helicity conservation, both for simply and multiply connected domains. On the other hand, we describe how to prove the existence of weak solutions conserving magnetic helicity but dissipating cross helicity and energy in 3D Ideal MHD. Such solutions are bounded. In fact, we show that as soon as we are below the critical L-3 integrability for magnetic helicity conservation, there are weak solutions which do not preserve even magnetic helicity. These mathematical theorems rely on understanding the mathematical relaxation of MHD which is used as a model of the macroscopic behaviour of solutions of various nonlinear partial differential equations. Thus, on the one hand, we present results on the existence of weak solutions consistent with what is expected from experiments and numerical simulations, on the other hand, we show that below certain thresholds, there exist pathological solutions which should be excluded from physical grounds. It is still an outstanding open problem to find suitable admissibility conditions that are flexible enough to allow the existence of weak solutions but rigid enough to rule out physically unrealistic behaviour.

Ольга Владимировна Шереметьева, кандидат технических наук, научный сотрудник, лаборатория моделирования физических процессов, Институт космофизических исследований и распространения радиоволн ДВО РАН (Камчатский край, c. Паратунка, Российская Федерация), sheremeteva@ikir.ru. Анна Николаевна Годомская, педагог дополнительного образования, естественнонаучный отдел, Муниципальное бюджетное учреждение дополнительного образования ≪Центр ≪Луч≫ (г. Елизово, Российская Федерация), anna_antonenko@mail.ru. O.V. Sheremetyeva1, A.N. Godomskaya2 1Institute of Cosmophysical Research and Radio Wave Propagation FEB RAS, Paratunka, Russian Federation 2Municipal Budgetary Institution of Supplementary Education “Center “Luch”, Yelizovo, Russian Federation E-mails: sheremeteva@ikir.ru, anna_antonenko@mail.ru В рамках модели αΩ-динамо рассматривается магнитогидродинамическая система (далее МГД-система) с введенной аддитивной поправкой интенсивности α-эффекта в виде функции Z(t). Изменение интенсивности α-эффекта со временем определяется показательным ядром J(t) функции Z(t). Проведен предварительный анализ влияния изменения значений массовой плотности внешних сил и интенсивности α-эффекта на значения магнитного поля и поля скорости. Для упрощения численной модели МГД система перемасштабирована и в качестве единицы времени принято время диссипации магнитного поля, которое составило порядка 10⁴ лет. Выбор временной единицы определяется необходимостью исследования поведения магнитного поля на больших временных масштабах. С целью сокращения количества варьируемых параметров МГД-система обезразмерена таким образом, что управляющими параметрами выступают число Рейнольдса, которое несет информацию о крупномасштабном генераторе, и амплитуда α-эффекта, характеризующая турбулентный генератор. Результаты численного моделирования режимов генерации магнитного поля отражены на фазовой плоскости управляющих параметров, и исследуется вопрос о динамике изменения картины на фазовой плоскости в зависимости от времени ожидания, определяемого масштабным коэффициентом показательного ядра J(t) функции Z(t). In the framework of the αΩ-dynamo model a magnetohydrodynamic system (MHDsystem) with an included additive correction of the α-effect intensity in the form of a function Z(t) is considered. The variation of the α-effect intensity with time is determined by the exponential kernel J(t) of the function Z(t). A preliminary analysis of the changes influence in the values of the mass density of external forces and the α-effect intensity on the values of the magnetic field and the velocity field is carried out. To simplify the numerical model, the MHD-system was rescaled and the time of the magnetic field dissipation, which is of the order of 10⁴ years, was taken as a unit of time. The time unit choice is determined by the need to study the magnetic field modes on a large time scales. In order to reduce the number of variable parameters, the MHD-system was made dimensionless so that the control parameters are the Reynolds number, which carries information about the large-scale generator, and the amplitude of the α-effect, which characterizes the turbulent generator. The results of numerical simulation of the modes of magnetic field generation are displayed on the phase plane of the control parameters. The question of the dynamics of the change in the pattern on the phase plane depending on the waiting time determined by the scale factor of the exponential kernel J(t) of the function Z(t) is investigated

The low-mode model αΩ-dynamo is used in this paper to simulate the modes of magnetic field generation with insignificant changes in the velocity field of a viscous fluid. In the framework of those model the α-effect intensity is regulated by the process that is included in the magnetohydrodynamic system (MHD-system) as an additive correction as a functional Z(t) depended on the magnetic field energy. Function that determines damped oscillations with variable damping frequency and constant damping coefficient, taken equal to one, is selected as kernel J(t) of functional Z(t). The research of the behavior of the magnetic field is carried out on large time scales, therefore, a rescaled and dimensionless MHD-system with the unit of time iquel the time of the magnetic field dissipation (104 years) for numerical calculations is used. The control parameters of the system are the Reynolds number and the amplitude of the α-effect, that include information about the large-scale and turbulent generators, respectively. Numerical simulation of the magnetic field generation modes was carried out for the values of the damping coefficient b = 1 and frequency a = 0.1, 0.5, 1, 5, 10. According to the results of numerical simulation, an increase in the values of the damping frequency, when the damping coefficient is equal to one, is characterized by a decrease in the inhibitory effect of the process Z(t) on the α-effect and an increase in the region of divergence of the magnetic field on the phase plane of the control parameters. In a comparative analysis with the results of the authors’ work, where the change of the α-effect intensity was determined by the function Z(t) with an exponential kernel and the same value of the damping coefficient, the following differences were noted: an increase in oscillations in both a magnetic and a velocity fields, the appearance of a chaotic regime of magnetic field generation at the value of the damping frequency equal to one, and also insignificant narrowing of the region of α-effect suppression at values of the damping frequency increasing to one.

This work investigates the solvability, regularity and vanishing viscosity limit of the 3D viscous magnetohydrodynamic system in a class of bounded domains with a slip boundary condition.