content_image/419166.jpg content_image/419180.jpg content_image/419162.jpg content_image/419181.jpg content_image/419178.jpg content_image/419161.jpg content_image/419163.jpg content_image/419164.jpg content_image/419177.jpg content_image/419179.jpg content_image/419165.jpg
In the PROMISE experiment the azimuthal field is B$_{φ}$ ∝ 1 /R . In this paper we will also consider the generalization to B$_{φ}$ = a$_{B}$R + b$_{B}$/R , where the extra term a$_{B}$R corresponds to an axial electric current running through the fluid as well , and hence opens the possibility of current-induced (Tayler) in- stabilities. The resulting (nonaxisymmetric) instabilities may also be modified by adding either a differential rotation or a n axial magnetic field. 

One might suppose that adding an axial field would be im- portant only if its amplitude is of the same order as that of the azimuthal field. Chandrasekhar [10] showed that for Ω = 0 , a sufficiently strong axial field will always suppress any ax- isymmetric instabilities of an azimuthal field, by derivingthe stability condition 

where I > 0 and ξ$_{R}$ is the (purely real) radial eigenfunc- tions. (Note how (6) reduces to (4) for B$_{z}$ = 0 .) However, we will show that the influence of B$_{z}$ cannot be ignored even for rather small values. 

We will find that, depending on the magnitudes of the im- posed differential rotation and magnetic fields, the field ma y either stabilize or destabilize the differential rotation, and the most unstable mode may be either the axisymmetric Taylor vortex flow (the SMRI or HMRI), or the nonaxisymmetric AMRI, or the nonaxisymmetric Tayler instability. In com- bined axial and azimuthal fields, we will also show that the nonaxisymmetric modes differ between m and − m , corre- sponding to left and right spirals. As first pointed out by [11 ], if the imposed field has both axial and azimuthal components, the system no longer exhibits ± z symmetry. For axisymmet- ric modes, the consequence of this is that what were previ- ously stationary modes (SMRI) become oscillatory, traveling wave modes (HMRI). For nonaxisymmetric modes, breaking the ± z symmetry of the basic state breaks the ± m symmetry of the instabilities. Physically this corresponds to the fact that modes spiraling either in the same or the opposite sense to the spiral structure of the basic state are indeed different. This ± m symmetry breaking is also a convenient distinguishing feature between the AMRI and the Tayler instabilities; for the AMRI the most unstable mode spirals in the opposite sense to the imposed field, for the Tayler instabilities in the same sense. 

Finally, in order to produce benchmarks for the application of incompressible 3D MHD codes, in this work we will focus primarily on magnetic Prandtl number Pm = 1 . 

We are interested in the linear stability of the background field B = (0 , B$_{φ}$ ( R ) , B 0 ) , with B 0 = const, and the flow U = (0 , R Ω( R ) , 0) . The perturbed state of the system is described by 

<img src='content_image/419166.jpg'>

## II. THE EQUATIONS 

<img src='content_image/419180.jpg'>

Developing the disturbances into normal modes, the solutions of the linearized MHD equations are considered in the form 

where F is any of the velocity, pressure, or magnetic field disturbances. 

The governing equations are 

<img src='content_image/419162.jpg'>

and 

where $^{u}$is the perturbed velocity, $^{b}$the magnetic field, p the pressure perturbation. ν is the kinematic viscosity and η the magnetic diffusivity. 

<img src='content_image/419181.jpg'>

The stationary background solution is 

where a Ω , b Ω , a$_{B}$ and b$_{B}$ are constants defined by 

<img src='content_image/419178.jpg'>

with 

<img src='content_image/419161.jpg'>

<img src='content_image/419163.jpg'>

<img src='content_image/419164.jpg'>

<img src='content_image/419177.jpg'>

<img src='content_image/419179.jpg'>

<img src='content_image/419165.jpg'>

