New one-dimensional model equations of magnetohydrodynamic turbulence

New one-dimensional model equations of magnetohydrodynamic turbulence depending on the position x and the time t are proposed considering the x component of the velocity u(x,t) and the y and z components of the magnetic field b_y(x,t) and b_z(x,t). It is found that the model equations conserve the total energy and the quantity similar to the cross helicity, and that they have a similarity solution, u(x,t)∼x/t and b_y(x,t), b_z(x,t)∼C/t in the inviscid limit. A shock type solution is obtained where both the velocity and the magnetic fields have a jump in a narrow viscous or diffusive region. Numerical simulations are also carried out and the results show that the k^-2 energy spectrum develops both for the velocity and the magnetic fields.