Limits on magnetic field amplification from the r -mode instability

Abstract

At second order in perturbation theory, the unstable r-mode of a rotating star includes growing differential rotation whose form and growth rate are determined by gravitational-radiation reaction. With no magnetic field, the angular velocity of a fluid element grows exponentially until the mode reaches its nonlinear saturation amplitude and remains nonzero after saturation. With a background magnetic field, the differential rotation winds up and amplifies the field, and previous work where large mode amplitudes were considered [1], suggests that the amplification may damp out the instability. A background magnetic field, however, turns the saturated time-independent perturbations corresponding to adding differential rotation into perturbations whose characteristic frequencies are of order the Alfvén frequency. As found in previous studies, we argue that magnetic-field growth is sharply limited by the saturation amplitude of an unstable mode. In contrast to previous work, however, we show that if the amplitude is small, i.e., 10 −4 , then the limit on the magnetic-field growth is stringent enough to prevent the loss of energy to the magnetic field from damping or significantly altering an unstable r-mode in nascent neutron stars with normal interiors and in cold stars whose interiors are type II superconductors. We show this result first for a toy model, and we then obtain an analogous upper limit on magnetic-field growth using a more realistic model of a rotating neutron star. Our analysis depends on the assumption that there are no marginally unstable perturbations, and this may not hold when differential rotation leads to a magnetorotational instability.

How to search for gravitational waves from r -modes of known pulsars

“…We assume that any differential rotation has a negligible effect on the mode frequency. In practice this seems likely to be true, even though the r-mode tends to generate a small amount of differential rotation analogous to Stokes drift [24]. We assume that the magnetic field's direct effect (through restoring force) on the r-mode frequency is small.…”

Magnetic field amplification by the r-mode instability

“…The magnetic field evolution in the presence of an unstable r-mode is analyzed in our recent paper [33] using second order perturbation theory for an infinitely conducting rotating star including gravitational radiation reaction forces. The results are illustrated there by a simple toy model, and confirmed by a detailed analysis of the full problem using the symplectic product formalism [2,40].…”

“…GR ΩR, where Ω = 2πν is the spin frequency of the star, and (b) an effective force associated with the second order terms in the perturbation equations. These terms have a part independent of the magnetic field with magnitude of order f GR and a part from magnetic terms with magnitude (per unit mass) f m ∼ α 2 (t) ω A ΩR [33], assuming a non-superconducting stellar interior. Here, ω A = πB 2 /ρR 2 is a typical frequency of Alfvén modes, B is an initial magnetic field, ρ is a typical density, and R is the stellar radius.…”

“…Before saturation, when the mode amplitude α(t) is exponentially growing, α(t) = α(0) exp(t/τ GR ), the displacements of fluid elements in the azimuthal direction can be estimated in the same way as the displacement of an "Alfvén" harmonic oscillator with frequency ω A evolving under the action of an exponentially growing external force f GR + f m (with timescale τ GR /2). The result is [33]…”

“…and its growth rate ǫm ∼ 4α 4 (t) τ −1 GR ρ Ω 2 R 2 , again using the harmonic oscillator analogy, or a more detailed mathematical analysis of the full problem [33]. The energy density of r-modes can be estimated as ǫ r ∼ α 2 (t) ρΩ 2 R 2 (c.f.…”