Superfluid Rivers in Spinning-down Neutron Stars

We study the motion of neutron superfluid vortices in a spinning-down neutron star, assuming axisymmetry of the flow and ignoring the motion of vortices about the rotation axis. We find that the vortex array, if initially rectilinear, is soon substantially deformed as the star spins down; vortices are swept outward by the Magnus force, accumulating in regions of the inner crust where they pin, accompanied by significant bending of the vortex array. As the star spins down to below a spin rate of ∼20 Hz (twice the spin rate of the Vela pulsar), the Magnus and pinning forces gradually compress the vortex array into dense sheets that follow spherical shells. In some cases, the vortex array bends on itself and reconnects, forming one or more tori of vortex rings that contain superfluid “rivers” with significant angular momentum. Vortex sheets are likely to form near the base of the inner crust in the regime of nuclear pasta.


INTRODUCTION
The manner in which the neutron superfluid (nSF) spins down in a neutron star (NS) is likely related to various observed spin irregularities. Spin glitches in NSs are generally attributed to sudden transitions, driven by stellar spindown, between different metastable states of pinned vorticity in the inner crust (Anderson & Itoh 1975) or the outer core (Ruderman et al. 1998; see Haskell & Melatos 2015 for a review). Pulsars also exhibit long-timescale stochastic spin irregularities, known as timing noise (see Hobbs et al. 2010 and references therein). Timing noise could arise from a number of superfluid-related processes such as turbulence (Greenstein 1970;Melatos & Link 2014;Lasky et al. 2015), stochastic transport of vorticity (Alpar et al. 1985;Jones 1990a), vortex avalanches (Cheng et al. 1988;Warszawski & Melatos 2008), and magnetospheric processes unrelated to superfluid dynamics (Cheng 1987a,b;Urama et al. 2006;Lyne et al. 2010).
Knowledge of the structure of the superfluid flow -in particular, columnar and laminar versus turbulent -is essential for the development of realistic models of NS spin-down and timing irregularities. Ruderman & Sutherland (1974) argued long ago that the array of superfluid vortices in a spinning-down NS is effectively very rigid and so remains rectilinear despite the presence of forces on vortices that vary with position in the star; they concluded that the vortex array will evolve without significant bending or tangling -a superfluid version of the Taylor-Proudman theorem of classical hydrodynamics -precluding the development of superfluid turbulence. As a result of the work of Ruderman & Sutherland (1974), and also perhaps as an understandable simplification of a complex problem, much work on the hydrodynamics of the NS interior has assumed that the vortex array remains rectilinear as the NS evolves (e.g., Alpar et al. 1984aAlpar et al. , 1985Link & Epstein 1996;Larson & Link 2002;Antonelli & Pizzochero 2017). Other work has assumed that the nSF finds itself in a turbulent state, and has explored the consequences of this assumption (e.g., Greenstein 1970;Andersson et al. 2007;Lasky et al. 2013;Melatos & Link 2014;Haskell et al. 2020). A number of interesting simulations of superfluid turbulence in NSs introduce laminar-turbulent transitions by hand (e.g., Peralta et al. 2006).
In this paper we address the evolution of the superfluid vortex array in axisymmetry, accounting for vortex pinning in the inner crust. We find that for realistic vortex pinning, the Taylor-Proudman theorem is strongly violated. Instead, in regions where the pinning strength has strong gradients, vortices accumulate into dense spherical sheets. For some distributions of the pinning force, the spherical sheets bend and reconnect to make tori with with superfluid rivers flowing inside them. Generally, we suspect that the vortex array evolves into complex configurations that could become tangled, with an accompanying superfluid flow that could be turbulent.
The plan of this paper is as follows. In §2 we derive the equations of motion for superfluid vorticity. In §3, we derive a superfluid version of Taylor-Proudman Theorem, and show that the theorem is typically violated when pinning forces are present. In §4 we describe the motion of a single vortex. In §5 we describe the global dynamics in axisymmetry. In §6 we present a simple global model for vortex mobility that is easily solved. In §7 we present our simulations. In §8 we discuss our results.

SUPERFLUID DYNAMICS
A superfluid rotates by establishing an array of quantized vortices of microscopic cross section. The circulation about a vortex is fixed according to where v s (r, t) is the superfluid velocity at location r at time t; κ = 2πℏ/m is the circulation where m is the mass of the fundamental boson in the system, twice the neutron mass in the case of a neutron superfluid. An important distinction between a superfluid and a classical fluid is that vortices in a superfluid are physical entities that can move at their own velocity distinct from that of the superfluid, interact with their environment, and pin or experience physical drag from other parts of the system. Regardless of how superfluid vortices move, each vortex always carries a quantum of circulation according to eq. 1.
Consider an incompressible superfluid. The momentum equation in the inertial frame is where µ is the chemical potential, ϕ is the gravitational potential, f is the force per unit volume acting on the superfluid from vortex pinning, mutual friction, and vortex tension, and ρ s is the mass density of the superfluid. It is convenient to work in the frame corotating with the crust at angular velocity Ω(t) and we denote quantities in this frame with the superscript *. The velocity is In the rotating frame, the momentum equation is Taking the curl, and using a vector identity 1 , gives Here ω is the total vorticity measured in the inertial frame and ω * ≡ ∇ × v * s is the vorticity in the rotating frame. In a quantum fluid, vortices are persistent structures to which we can assign a local velocity v * v in the rotating frame. Vortex motion obeys the conservation law, The force f contains contributions from mutual friction and bending forces of the vortex array. Comparing eqs. 5 and 7 gives the vortex equation of motion up to the gradient of an arbitrary scalar potential, which we absorb into the definition of µ. The right-hand side is the Magnus force (per unit mass). For f = 0, the vortex lines move with the superfluid, while for f ̸ = 0, vortices are forced to move locally at a different velocity v We can write eq. 8 as an equation of motion for a single vortex by defining the areal density of vortices n v related to the vorticity through where κ has magnitude κ in the direction of the local vorticity ω. Noting that the force per unit length on a single vortex is 3. SUPERFLUID TAYLOR-PROUDMAN THEOREM For steady rotation of a classical fluid in the limit that the Coriolis force is much larger than the inertial force, the fluid flow does not depend on position along the axis of rotation. If a solid body is slowly pulled through the fluid, the fluid must flow around the body, and a vertical Taylor column is established along the axis of rotation. In a rotating superfluid the analogous effect is seen experimentally as vortices tend to become straight and align with the rotation axis in a rapidly rotating vessel. We review the derivation of the Taylor-Proudman for a rotating fluid. We then show that the theorem is violated in the presence of pinning forces in a quantum fluid.
In a NS, the spin-down torque acts on the crust and is communicated to the nSF over a long time scale t sd , so that Ωt sd ≪ 1. We therefore consider very slow variations in ω when a spatially-dependent force f is present. Eq. 7 gives The Taylor-Proudman theorem applies when the Coriolis force dominates both the inertial force in eq. 4 and ρ −1 s f . In terms of vorticity, the Coriolis force dominates the inertial force when 2Ω ≫ |ω * |. In this limit, with Ω =ẑΩ whereẑ is a unit vector along the rotation axis, eq. 11 becomes Typically in a classical fluid ρ −1 s f is determined by the properties of, for example, an obstacle affecting the flow. In the limit of fast rotation the right-hand side goes to zero and we arrive at the Taylor-Proudman theorem; all three components of v * s become independent of z. For a quantum fluid ρ −1 s f might be due to mutual friction, which scales as the vortex density and hence as 2Ω in the limit of fast rotation; then ∂v * s /∂z does not tend to zero as in a classical fluid. In cylindrical coordinates (ϱ, ϕ * , z), and assuming axisymmetry, eq. 5 becomes in the limit of fast rotation wheref ≡ ρ −1 s f and all quantities are functions of the coordinates (ϱ, z). If f ϱ = f z = 0, v * ϕ * is independent of z implying that the vortex array remains rectilinear as it moves (Ruderman & Sutherland 1974); the array does not bend. Generally, though, f ϱ ̸ = 0, and v * ϕ * becomes dependent upon z; the array bends as it moves. The final term in eq. 14 becomes non-zero as the array bends. From eq. 14, we can estimate the angle by which the vortex lattice deviates locally from straight: The deformation of the vortex lattice scales as Ω −1 for a force that is proportional to the vortex density. Conversely, lattice deformations can be large in the limit of slow rotation, as the simulations we present later in this paper show. It is instructive to consider the Taylor-Proudman limit for a quantum fluid in terms of vortex motion. The vortex motion in the steady state satisfies, from eq. 7 In the limit of fast rotation with 2Ω ≫ |ω * |, In steady state, the vortex motion becomes independent of the axial coordinate z, the lateral vortex velocity is nondivergent and independent of the axial coordinate, and the axial velocity decouples from the lateral velocity (see also Holm 2001). Hence, in this steady-state limit of fast rotation, the vortices straighten and become parallel to the rotation axis. The Taylor-Proudman theorem generally does not hold if vortices are immobilized in parts of the fluid volume (as opposed to only at boundaries) by pinning forces. In this case the pinning force follows from eq. 8 with v * v = 0: Now ∂ω/∂t = 0 because the vortices are immobilized and the vortices cannot adjust their configuration to make v * s independent of z for an arbitrary pinning force. If ρ −1 s f ϱ varies along z, for example, so will v * ϕ * , implying that the vortex array is not rectilinear. This is the situation we explore further in this paper.
Finally, we note that the force f contains a contribution from rigidity of the vortex lattice, which is typically tiny compared to pinning forces; see next section.

MOTION OF A SINGLE VORTEX
Let the vortex shape, measured with respect to the z axis, be given by the two-dimensional displacement vector u(z, t) =xu x +ŷu y , so the vortex velocity is v * v = ∂u/∂t. For small bending angles, so that |∂u/∂z| ≪ 1, eq. 10 can be written (see, also, Schwarz 1977Schwarz , 1978 The first term is the Magnus force. The second term is the force (per unit length) from bending the vortex; T v is the vortex tension. The last term F L is the force due to interaction of the vortex with the nuclear lattice. The second and third terms are F v appearing in eq. 10. The tension is typically where Λ is a factor of 3-10; T v ∼ 1 MeV fm. As explained below, vortex tension is important against pinning forces only over length scales below ∼ 10 3 fm. The force of the lattice on the vortex has two contributions: (i) a static, conservative contribution and (ii) a velocitydependent, damping force due to the excitation of lattice phonons by the moving vortex that carry energy away from the vortex (Epstein & Baym 1992;Jones 1992). To treat these two contributions, we take the simple form where v * c is the velocity of the crust. The first term gives the static force; V is the potential, and the gradient is perpendicular to the vortex. The second term is the drag force, and η is the drag coefficient. A non-dissipative, velocity-dependent contribution to f L is possible, but shall not be considered here; see Sedrakian et al. (1999).
Numerical solutions of eqs. 20 and 22 for various approximations to the nuclear lattice show that the vortex often responds with a slip-stick character to an applied force ρ s κ × v * s (Link & Levin 2022). For zero applied force, the vortex settles into a pinned configuration whose solution is given by The response of a pinned vortex to a slowly increasing applied force depends on the properties of the lattice, its orientation, whether the vortex-lattice interaction is attractive or repulsive, and the drag coefficient η. We now summarize the characteristics of vortex motion in the case of low drag, in the sense η/ρ s κ ≪ 1. If the lattice is regular and repulsive, for most lattice orientations the vortex is able to move through the propagation of kinks. As the force becomes large, the vortex enters a ballistic regime in which the conservative contribution to the lattice force and the tension contribution average nearly to zero. In this limit, the vortex motion is given by For some lattice orientations, the kinks hang up and the vortex remains pinned. As the force is increased further for this case, the vortex unpins suddenly at a critical value of the force, with Kelvin waves propagating along its length. If the force is then reduced, the vortex eventually repins quite suddenly, but typically at a value of the force that is smaller than the value of the force at which the vortex unpinned. This hysteretic attribute of vortex motion is due to the fact that the vortex is an extended object that supports internal vibrational degrees of freedom. Impurities in the lattice tend to prevent the propagation of kinks, giving a well-defined transition from pinned to ballistic at a critical value of the force. If the lattice is regular and attractive, there is a well-defined transition from pinned to ballistic for most lattice orientations. If the lattice is strongly disordered, or if the pinning sites are randomly positioned as in a glass, then a well-defined transition from pinned to ballistic occurs whether the lattice is attractive or repulsive. Typical values for the critical force are 10 16 dyn cm −1 to 10 17 dyn cm −1 , corresponding to critical superfluid velocities of v crit = |v * s | of 10 −5 c to 10 −4 c. The motion in the ballistic regime is given by solution to eq. 24: Locally, the vortices move at an angle θ with respect to the vector v * s ; θ is sometimes referred to as the "dissipation angle". In the limit of low drag, η/ρ s κ << 1 and θ ≃ η/ρ s κ, giving The vortex moves nearly with the superfluid, with slow drift along v * s ×κ. In the limit of high drag, η/ρ s κ >> 1 and The solutions described above apply for low drag. In the limit of high drag, an unpinned vortex moves nearly with the crust, with slow drift along v * s ×κ. We refer to this state of unpinned vortex motion as "vortex drift" to distinguish the motion from that of the low-drag, ballistic regime.
For vortex motion in the inner crust, the drag coefficient is typically γ ≡ η/ρ s κ ∼ 10 −3 (Epstein & Baym 1992;Jones 1992), and depends on velocity. 2 In the outer core, SFn vortices are predicted to acquire magnetization due to entrainment of protons by the neutrons that circulate around the vortices (Alpar et al. 1984b). Electrons scatter with the magnetic moments of the SFn vortices. The relaxation time for relative motion between the neutrons and electron-proton plasma is (Alpar et al. 1984b) τ v ∼ P (s) s, where P (s) is the spin period of the system in seconds. The dissipation angle is related to the relaxation time at spin rate frequency Ω through (Link 2014) According to this estimate, vortex motion in the core is in the low-drag regime. However, neutron vortices are also expected to pin to the far more numerous flux tubes of the outer core (see, e.g., Srinivasan et al. 1990;Ruderman et al. 1998;Jones 1991), restricting the mobility of the neutron vortices; in order to move, the vortices must cut through flux tubes; the dissipation associated with this process is poorly understood. The bending force per unit length -the first term in eq. 20 -is ∼ T v /R v , where R v is the radius of curvature of a bent vortex. For a pinning force of 10 17 dyn cm −1 ≃ 10 −3 MeV fm −2 , the pinning force will dominate the bending force for R v > ∼ 10 3 fm = 10 −10 cm. Vortex tension is therefore completely negligible for the problem of macroscopic fluid flow in a NS, though it plays an essential role in pinning and slip-stick dynamics, as described in more detail in Link & Levin (2022).

GLOBAL DYNAMICS
Consider a vortex that is initially pinned in a spinning-down NS with a nSF flow velocity v * s in the frame of the crust. As the stellar crust spins down, the superfluid flow speed past the pinned vortex will approach the local critical value for unpinning. At zero temperature, the vortex unpins only when the flow speed becomes critical. For low-drag vortex motion, the vortex moves nearly with v * s when the vortex is unpinned, while for high-drag vortex motion, the vortex drifts slowly transverse to the v * s (along v * s ×κ; see eq. 27). At finite temperature, thermal excitations cause a pinned vortex to slowly "creep" when the flow velocity is slightly subcritical (Alpar et al. 1984a;Link et al. 1993;Sidery & Alpar 2009;Link 2014). The dissipation angle for motion through creep can differ from the value given by tan θ = η/ρ s κ in the ballistic regime. As we show in a forthcoming publication (Link & Levin 2023), for low-drag thermal creep the vortex has comparable velocity components along and transverse to v s , while for high-drag thermal creep the vortex motion is nearly transverse to v * s , as for zero temperature.
If vortex motion has a significant component along v * s , the vortex array will become substantially twisted as the NS spins down. For both illustration and simplicity, we henceforth neglect vortex motion along v * s , and consider axisymmetric superfluid flow. We assume that, in the rotating frame, the vortices are always poloidally-directed. This condition is rigorously satisfied in the limit of high drag, but is not satisfied in the limit of low drag if stellar spin down causes the vortex array to make transitions between a pinned state and a ballistic state. The limitations of our treatment are discussed §8.
We use cylindrical coordinates in the inertial and rotating frames with common origin and z axes that coincide: The rotation vector of the crust Ω is alongẑ. It is convenient to work with the total circulation, defined as The quantity Φ/κ is equal to the total number of vortices enclosed by a horizontal circle of radius ϱ at height z. We now establish a theorem for the time evolution of the circulation: so long as v * s exceeds the local threshold for unpinning, the Magnus force will drive a local reduction in the circulation. Eqs. 7 and 8 give, by Stokes's Theorem, The rate of change of the circulation is equal to the flux of the total vorticity vector ω through the horizontal circle centered on the axis of symmetry, whose points have fixed poloidal coordinates (ϱ, z). If the superfluid rotates faster than the crust (viz., v * s > 0), the azimuthal component of drag force, which opposes the azimuthal motion of vortices, is along −φ * , so f ϕ * is negative. From eq. 33, the Magnus force then drives a local reduction of the circulation; this remains valid even if the vortices are directed opposite to the direction of stellar rotation.

GLOBAL MODEL FOR VORTEX MOBILITY
Based on eq. 33, we consider a simple model for the mobility of vortices: the vortex is mobile only when the Magnus force exceeds a certain position-dependent critical value, that is, when v * s exceeds v crit (r). The critical velocity v crit is a function of the mass density, which has a spherically-symmetric distribution, so v crit is a function of the spherical radius r. When this unpinning threshold is exceeded, vortex motion reduces the circulation locally. Returning to the inertial frame, where the crust velocity is ϱΩ(t)φ and the superfluid velcocity is v s = (ϱΩ(t) + v * s )φ, we have the following prescription for how the speed v s evolves with time: • So long v s (ϱ, z) < ϱΩ(t) + v crit (r), the constant-circulation surface is pinned ; Φ(ϱ, z) and v s (ϱ, z) remain fixed.
• Once v s (ϱ, z) ≥ ϱΩ(t) + v crit (r), the constant-circulation surface (representing vortices) becomes unpinned and moves under the action of the Magnus force. This motion reduces v s (ϱ, z). We assume that the timescale for the unbalanced vortex motion is much shorter than that of the pulsar spindown; in that case v s reduces to the value very close to ϱΩ(t) + v crit (r) and the vortices/constant circulation surfaces become pinned again.
Here Ω 0 is the initial angular velocity of the star, and it is assumed that the superfluid was initially co-rotating with the star. The circulation is The circulation is straightforward to calculate. The contours of constant circulation trace out vortices. The remarkable feature of eq. 34 is that the superfluid velocity is determined locally, irrespective of the history of the pulsar spindown and of the pinning strengths at other location. This feature is a consequence of the assumed axisymmetry with negligible vortex motion around the rotation axis. The neighbouring constant circulation surfaces do not feel attraction or repulsion to each other; the motion of one does not change the superfluid velocity at the other. One immediate corollary is that the strength of pinning in the core does not affect the motion of nSF vortices in the crust.
In the initial stages of NS spin down, the star is spinning down quickly and ϱΩ 0 > ϱΩ(t) + v crit (r) everywhere in star except very close to the rotation axis. Then The vorticity vector can be expressed as ω =κn v ≡ κn v , where n v is the vector areal density of the vortex array. The vortex density n v at location (r, θ) in spherical coordinates is given by (except close to the rotation axis) As described in the next section, radial gradients in the pinning force can be large in the inner crust. In such regions (except close to the rotation axis) At high spin rate, n v is uniform and points in the z direction. As Ω(t) becomes sufficiently small, the vorticity vector begins to point along ±θ in regions where the radial gradient in the pinning force is large. This transition begins when 2Ω(t) < ∼ r −1 d{rv crit (r)}/dr, corresponding to a spin period of where σ p is the characteristic dimension over which the pinning forces varies, 0.05R is a fiducial value for the thickness of the crust, and v max crit is the maximum value of the critical velocity for pinning. Hence, as the star spins down the vorticity becomes compressed into dense sheets along ±θ, even if the vortex array is uniform initially; a sheet will be along −θ in a region with d(rv crit )/dr > 0 and alongθ in a region with d(rv crit )/dr < 0.

PINNING MODELS AND SIMULATIONS
There has been substantial disagreement over the magnitude and sign of the vortex-nucleus interaction in the inner crust. Quantum calculations (using a mean-field Hartree-Fock-Bogoliubov formulation) show a repulsive interaction with an energy of up to ∼ 3 MeV (Avogadro et al. 2007(Avogadro et al. , 2008 throughout the inner crust, while semiclassical calculations (using a local density approximation) show a repulsive interaction of ∼ 1 − 2 MeV below an average baryon density of ∼ 10 −2 fm −3 (∼ 2 × 10 13 g cm −3 ) which turns attractive with a strength of ∼ 5 MeV at higher densities (Pizzochero et al. 1997;Donati & Pizzochero 2003, 2004. A step toward resolving the controversy was made by Wlaz lowski et al. (2016), using density-functional theory (in principle, an exact approach); they find that the vortex-nucleus interaction is always repulsive in the average baryon density range 0.02 fm −3 (3 × 10 13 g cm −3 ) to 0.04 fm −3 (7 × 10 13 g cm −3 ), with a force of ∼ 1 MeV fm −1 over a range of ∼ 4 fm, corresponding to an interaction energy per nucleus of ∼ 4 MeV. The strength of pinning also depends on the symmetry of the nuclear lattice (Link & Levin (2022); recall discussion of §4). Pinning is relatively strong in a regular, attractive lattice. Pinning occurs less readily in a regular, repulsive lattice, with no sharp transition to pinning for many lattice orientations. The presence of impurities in an otherwise regular lattice, such as dislocations, nuclei with different charge than their neighbors, or mono-vacancies (missing nuclei), generally enhance pinning. Pinning in nuclear glass is generally strong. Despite these uncertainties, it is clear that the basic energy scale in the pinning interaction is ∼ 1 MeV, with spatial variations over length scales much shorter than the crust thickness of ∼ 0.05 the stellar radius. Vortices are also predicted to pin to the far more numerous flux tubes of the outer core (see, e.g., Srinivasan et al. 1990;Ruderman et al. 1998;Jones 1991).
Though the strength and distribution of the pinning force is uncertain, we can proceed with two illustrative examples for the distribution of the pinning force from which we can draw some general conclusions. In both cases we make the pinning region deeper in the star and somewhat thicker than in a real NS crust for the sake of visibility. This choice has little effect on our chief results.
Example Model 1: spherically-symmetric Gaussian. v where r = ϱ 2 + z 2 is the spherical radius coordinate, r 0 is the coordinate at which v crit takes its largest value, and σ is the width of the distribution. Example Model 2: spherically-symmetric Gaussian with sinusoidal oscillations. The purpose of this model is to show the effects of large spatial gradients in pinning strength as indicated by the calculations of Pizzochero et al. (1997) (see, also, Donati & Pizzochero 2003, 2004. where k is the wave number of spatial oscillations. We fix v max crit = 10 −4 c as a typical value for the inner crust estimated by Link & Levin (2022). We fix the initial spin rate to be 1 kHz (an unimportant choice).
Assuming axisymmetry of the vorticity field, and neglecting vortex motion in the azimuthal direction, the solution to the flow problem is given entirely by eq. 34. As discussed at the end of §4, the effects of vortex tension are negligible at global scales (but important at mesoscopic and microscopic scales) and we neglect them. The subsequent flow depends only on the spin rate of the crust Ω(t) for specified v crit (r). Fig. 1 shows contours of constant circulation Φ at different values of the crust spin rate for Example Model 1. These contours follow the vortex lines. The contour interval is chosen so that a uniform vortex distribution gives equally-spaced contours. All dimensions are in units of the NS radius R ≃ 10 km. Fig. 2 shows an animation of the simulation of Fig. 1. As the star spins down, the vortex array is bent into tori of vortex rings, and the vortices are compressed into dense sheets at constant spherical radius. Within the pinning shell, vortex rings shrink and disppear. At late times a dense torus of vortices remains that sustains a flow -a superfluid "river" -about the vertical axis in the equatorial plane. Fig. 3 shows Example Model 2 with k = 100R −1 , giving three pinning maxima within a spherical shell. An animation is shown in Fig. 4. The vortex array becomes significantly deformed before a relatively short spin period of 0.03 s (the spin period of the Crab pulsar) is reached. Now vortices become compressed into multiple sheets. Shrinking tori are created in multiple locations. Bent vortices at large polar radius connect to make shrinking tori while their outer parts reconnect and exit the star. At late times, the vortex distribution freezes into multiple tori that carry flow around the rotation axis. A modest value of k = 100R −1 was selected to make easily readable plots; pinning calculations indicate that configurations with stronger gradients of v crit are possible (Pizzochero et al. 1997;Donati & Pizzochero 2003, 2004, which would entail the formation of vortex sheets at high rotation rates.   As a NS spins down, the combined effects of the pinning force (which is always comparable to the Coriolis force) and the Magnus force lead to significant bending of the vortex array. Bending of the vortex array arises generally because pinning forces hold vortices largely stationary in parts of the NS, while the Magnus force drives vortex motion where the vortex array is not pinned.
Despite the uncertainties of the distribution and magnitude of vortex pinning in a NS, we can draw a number of robust conclusions. First, the vortex distribution will deviate significantly from rectilinear by the time the spin rate goes below ∼ 20 Hz (about twice the spin rate of the Vela pulsar). Later, the vortices become compressed into dense sheets that follow surfaces of constant spherical radius. The number of sheets is equal to the number of radii in the star at which |d(rv crit )/dr| is large. If d(rv crit )/dr is first large and positive at some radius, and then large and negative at larger radius, the vortex distribution evolves into a torus that spans the two regions.
A region of particular interest is the base of the crust, where the nuclear lattice makes a transition from a lattice of spherical nuclei to nuclear pasta, finally dissolving at the boundary between the crust and the outer core. Pinning forces must vary significantly in this region, so we expect vortex sheets to be present. Dense sheets of opposing vorticity attract one another, possibly leading to reconnection of vortex sheets. If sheet reconnection does occur, the superfluid flow could become turbulent at length scales comparable to the typical sheet thickness.
Our treatment depends on the assumption that the vortex array remains poloidal in the rotating frame, thus avoiding the complications of winding of the vortex array around the rotation axis of the star. As discussed in §4, calculations of vortex drag indicate sin θ ∼ 10 −3 in the inner crust and ∼ 0.1 in the outer core. If vortices are forced by stellar spin-down to move in the ballistic regime, there will be significant winding of the vortex array, coupling the poloidal and toroidal motions of the array. It is more likely that vortices are never forced into the ballistic regime, but instead move through thermally-activated vortex creep. As we show in a forthcoming publication (Link & Levin 2023), for low-drag thermal creep the vortex has comparable velocity components along and transverse to v s , while for high-drag thermal creep the vortex motion is nearly transverse to v * s as for zero temperature. We expect that the basic picture given here of the concentration of vorticity into sense sheets still holds, but with significant twisting in the azimuthal direction. We will address this problem in a future publication.