ON MASS-TRANSPORT IN NONVISCOUS, NON-SELF-GRAVITATING FLUID DISKS

We analyze the mass flow in a barotropic, nonviscous, non-self-gravitating two-dimensional fluid disk due to a mild nonaxisymmetric perturbing potential. The perturbing potential is assumed to be of the form Re {φ1(r) exp [im(θ - Ωpt) + γt]} with complex function φ1(r), pattern speed Ωp, and growth rate γ arbitrarily specified. We take advantage of the existence of two small parameters: the relative disk thickness ∈ and relative potential perturbation strength λ. We consider the flow equations in second order in λ and derive a differential equation for the mass flow rate. We demonstrate the following: 1. For γ = 0, mass flow rate Ṁ vanishes everywhere, except in the vicinity of the corotation resonance and where Oort constant B vanishes, as a consequence of Kelvin's circulation theorem. 2. The mass flow contains both a resonant and nonresonant contribution. Within a small neighborhood of corotation, of radial extent on the order of the disk thickness (∈rc, with corotation radius rc), a resonant mass flow is generated. Elsewhere and away from B = 0, a nonresonant mass flow occurs. 3. For γ < Ωp, the nonresonant mass flow rate is roughly |Ṁ| ∼ λ2 tan-2 (i)|γ\σ0rc2, for potential pitch angle i. The local flow direction, inward or outward, depends upon several competing fluid stress terms. 4. Near corotation and for ∈ ≤ tan i, the resonant mass flow rate is nonzero even for periodic potentials (γ→0+), and \Ṁ\ ∼ ∈-1λ2σ0Ω prc2. For γ(dB/σ0)/dr positive (negative), mass flows outward (inward) across corotation and shifts with a characteristic time scale ∼(∈/λ)2Ωp-1. A higher order analysis suggests that the mass flow is strongly affected by a backreaction on the vorticity gradient before shifting much of the mass in the corotation region. 5. Landau damping at corotation creates a torque in the fluid in a neighborhood of radial extent of order |γ\/(m|dΩ/dr\). In response, pressure forces cause angular momentum to be advected over a radial extent of ∼∈rc. 6. For a time-periodic potential, a mass flow occurs wherever B attains a local minimum value of zero. The mass flow rate \Ṁ\ ∼ ∈ -1λ2 tan-2 (i)σ0rc2, for driving pitch angle i. The local flow direction again depends on several competing stress terms. The spatial form of mass flow rate is similar to that at corotation. 7. For typical parameters appropriate for tightly wrapped spiral galaxies, the resonant mass flux is roughly comparable to the mass flux due to viscous stresses and shocks. The nonresonant mass flux is comparable for growth rate γ ∼ ∈Ωp. The mass flux at B = 0 is larger than the resonant mass flux by roughly tan-2 i and is the most powerful.