Sonin Ch. 3 — Agent-Focused Summary for the Inter-Layer Spacing (h) Derivation

What this document is

A working extraction of equations and physical reasoning from Sonin (2016), Chapter 3 (“Vortex array in a rotating superfluid: elasticity and macroscopic hydrodynamics”), selected and annotated for the specific problem of deriving the repulsive energy functional E_\text{rep}(h) between co-rotating vortex sheets separated by a counter-rotating shear layer, and for validating the identification of \alpha_{mf} as the inter-layer coupling coefficient.

1. The HVBK Framework: What We Can Use Directly

1.1 The macroscopic Euler equation for a vortex array

The coarse-grained Euler equation for a superfluid containing a dense vortex array is (Eq. 3.9):

\frac{\partial \boldsymbol{v}}{\partial t} + \tilde{\boldsymbol{\omega}} \times \boldsymbol{v}_L = -\nabla \mu

where \tilde{\boldsymbol{\omega}} = \nabla \times \boldsymbol{v} is the averaged vorticity (equal to the vortex line density times \kappa), \boldsymbol{v}_L is the vortex velocity, and \mu is the chemical potential. The vorticity continuity equation follows by taking the curl (Eq. 3.10):

\frac{\partial \tilde{\boldsymbol{\omega}}}{\partial t} + \nabla \times [\tilde{\boldsymbol{\omega}} \times \boldsymbol{v}_L] = 0

Relevance to our problem: This is the starting point for the dynamics on each side of the shear layer. Within a given co-rotating sheet, the vortices form an array described by these equations. The inter-layer region is where \tilde{\boldsymbol{\omega}} reverses sign (counter-rotating layer), and the HVBK framework breaks down in its standard form there — that’s what makes the E_\text{rep} calculation non-trivial.

1.2 The Magnus force as the inter-layer coupling mechanism

Sonin defines (Eq. 3.22) the force per unit volume on the vortex array when vortices move relative to the fluid:

\boldsymbol{f} = -\rho \tilde{\boldsymbol{\omega}} \times (\boldsymbol{v}_L - \boldsymbol{v})

This is the Magnus force density. It’s the product of the circulation quantum \kappa times the 2D vortex density, times the relative velocity. This is the same structure as the polar-jet coupling in our model — the energy transfer between layers is mediated by this force acting across the boundary where \tilde{\boldsymbol{\omega}} changes sign.

1.3 The elastic force on vortices (Eq. 3.30)

In the absence of mutual friction, the general equation of vortex motion is:

-\rho \tilde{\boldsymbol{\omega}} \times (\boldsymbol{v}_L - \boldsymbol{v}) = \nabla_j \frac{\partial E_v}{\partial \nabla_j \boldsymbol{u}} - \nabla_j \left(\frac{\partial E_v}{\partial \nabla_j u_i}\right) \nabla u_i

The left-hand side is the Magnus force; the right-hand side is the elastic restoring force from deformations of the vortex array. Key point: The elastic force is derived from a Lagrange variation (not Euler), because the force acts on a specific vortex, not at a fixed point in space. Sonin carefully distinguishes these (Eqs. 3.24–3.29). This distinction will matter when we compute forces on vortices at the boundary of the shear layer.

2. The Three Elastic Moduli and Their Physical Meaning

For the triangular vortex lattice, the elastic energy density is (Eq. 3.40):

E_v = \frac{C_{44}}{2}\left(\frac{\partial \boldsymbol{u}}{\partial z}\right)^2 + \frac{C_{11}}{2}(\nabla \cdot \boldsymbol{u})^2 + \frac{C_{66}}{2}\left[\left(\frac{\partial u_x}{\partial y} + \frac{\partial u_y}{\partial x}\right)^2 - 4\frac{\partial u_x}{\partial x}\frac{\partial u_y}{\partial y}\right]

The three moduli in the HVBK theory are (Eq. 3.72):

Modulus Value Physical meaning
C_{44} (tilt) 2\rho \nu_s \Omega Line tension force; resists bending of vortex lines out of the z-direction
C_{11} (compression) -\rho\kappa\Omega / 8\pi Resists compression of the vortex lattice in-plane; negative but does not cause instability because longitudinal displacements are coupled to fluid velocity
C_{66} (shear) \rho\kappa\Omega / 8\pi Tkachenko shear rigidity; resists shear deformation of the lattice

The line tension parameter is \nu_s = (\kappa/4\pi)\ln(r_v/r_c), where r_v is the intervortex distance and r_c is the core radius.

Key hierarchy: C_{44} \gg C_{11} \sim C_{66} by a large logarithmic factor \ln(r_v/r_c).

Relevance: The tilt modulus C_{44} controls the axial (along-vortex) response, which is the direction of the polar jet. The shear modulus C_{66} controls the in-plane response, which is what the counter-rotating shear layer compresses against. For the E_\text{rep}(h) calculation, we need to know which modulus dominates the restoring force when two co-rotating sheets are pushed together.

3. The Vortex Equation of Motion: The Starting Point for E_\text{rep}

The linearized equation of vortex motion with all three moduli is (Eq. 3.44):

-\rho[2\boldsymbol{\Omega} \times (\boldsymbol{v}_L - \boldsymbol{v})] = C_{44}\frac{\partial^2 \boldsymbol{u}}{\partial z^2} + (C_{11} - C_{66})\nabla_\perp(\nabla \cdot \boldsymbol{u}) + C_{66}\nabla_\perp^2 \boldsymbol{u}

For the problem at hand, the simplified form dropping C_{11} (which is always subdominant) is (Eq. 3.93):

-\rho[2\boldsymbol{\Omega} \times (\boldsymbol{v}_L - \boldsymbol{v})] = C_{44}\frac{\partial^2 \boldsymbol{u}}{\partial z^2} + C_{66}[\nabla_\perp^2 \boldsymbol{u} - \nabla_\perp(\nabla_\perp \cdot \boldsymbol{u})]

How to adapt this for the shear layer: In our geometry, the “vortex array” on each side of the shear zone has a well-defined \Omega (the sheet’s angular velocity). The counter-rotating intermediate layer is where the two arrays meet with opposite circulation. The C_{44} term provides the restoring force along z (axial/polar jet direction), while the C_{66} term provides the restoring force in the xy plane (in-plane shear). The shear layer energy will be dominated by whichever of these two responses costs more energy per unit area at the given layer thickness h.

4. The Glaberson–Johnson–Ostermeier Instability: Critical Velocity for the Shear Layer

This is directly relevant to the \delta(h) question — whether the shear layer width is set by h or by r_\text{eff}.

The GJO instability (§3.10) shows that axial superflow along vortex lines becomes unstable above a critical velocity (Eq. 3.124):

v_{cr} = \sqrt{2\Omega \nu_s}

This instability starts from waves with p \ll k — i.e., waves propagating nearly perpendicular to the vortex lines, not along them. The physical picture: axial flow along the vortex cores destabilizes Kelvin-like modes on the array.

Application to the shear layer: The relative velocity between co-rotating sheets is v_\text{rel} \sim \omega_0 h. If v_\text{rel} > v_{cr}, the intermediate counter-rotating layer will be GJO-unstable, generating quantum turbulence. This sets a maximum coherent thickness for the shear layer:

h_\text{GJO} \sim \frac{v_{cr}}{\omega_0} = \frac{\sqrt{2\Omega \nu_s}}{\omega_0}

For h > h_\text{GJO}, the shear layer is turbulent and the repulsive energy will have a different functional form than for h < h_\text{GJO}. This may be the transition between your two cases: \delta \propto h (below GJO threshold, laminar shear) vs. \delta \sim r_\text{eff} (above threshold, turbulent with core-scale structure).

5. The Superfluid Ekman Depth: Natural Length Scale for the Shear Layer

The evanescent penetration depth for axial modes below the 2\Omega gap is (Eq. 3.102):

\ell_E = \sqrt{\frac{\nu_s}{2\Omega}}

This is the superfluid Ekman depth — the distance over which slow perturbations can penetrate into the rotating superfluid along the z direction.

Relevance: This is a natural candidate for the shear layer width \delta. If the counter-rotating intermediate layer acts as a “boundary” that the co-rotating sheets cannot penetrate, then \ell_E sets the minimum thickness of the transition region. Note that \ell_E involves the line tension parameter \nu_s, not just \kappa — this brings in the logarithmic factor and makes \ell_E larger than a naive estimate based on \kappa alone.

Numerically: \ell_E = \sqrt{\nu_s / 2\Omega} = \sqrt{(\kappa/4\pi)\ln(r_v/r_c) / 2\Omega}.

Connection to GJO: The critical velocity can be rewritten as v_{cr} = 2\Omega \ell_E, so the GJO instability threshold is reached when the relative velocity equals twice the angular velocity times the Ekman depth. This links the two length scales cleanly.

6. Energy Scales for E_\text{rep}(h)

6.1 The vortex energy density

The HVBK vortex energy density is (Eq. 3.51):

E_v = \varepsilon \frac{\tilde{\omega}}{\kappa} = \rho \nu_s \tilde{\omega} = \frac{\rho \kappa}{4\pi}\tilde{\omega}\ln\frac{r_v}{r_c}

where \varepsilon is the energy per unit length of a single vortex line multiplied by the areal vortex density. The energy per unit length is \varepsilon_0 = (\rho\kappa^2/4\pi)\ln(r_v/r_c).

6.2 The Tkachenko shear energy

For a pure shear deformation of the vortex lattice, the energy per unit area is:

E_\text{shear} = C_{66} u_{xy}^2 = \frac{\rho\kappa\Omega}{8\pi} u_{xy}^2

The Tkachenko velocity is (Eq. 3.112):

c_T = \sqrt{C_{66}/\rho} = \sqrt{\kappa\Omega/8\pi}

6.3 Scaling estimate for E_\text{rep}

The repulsive energy per unit area of the shear layer should scale as:

E_\text{rep} \sim \rho \kappa v_\text{rel} \cdot \frac{\delta}{\text{(some function of } h, r_\text{eff})} \sim \varepsilon_0 \cdot \frac{v_\text{rel}}{v_{cr}} \cdot \frac{\delta}{r_v}

From the HVBK framework, the natural energy scale is \rho\kappa\Omega \sim C_{66} \sim C_{11}, and the natural velocity scale is c_T or v_{cr}. The key question is whether \delta is:

  • Case A (\delta \propto h): The shear fills the gap. Then E_\text{rep} \propto \rho\kappa\omega_0 \xi^2, which is independent of h — a constant repulsive energy. This gives h \sim \mum (set by other terms in the energy balance).

  • Case B (\delta \sim r_\text{eff}): The vortex core controls the minimum shear scale. Then E_\text{rep} \propto \varepsilon_0 \cdot (\xi/h) \cdot (h/r_\text{eff}), which depends on h and can balance the attractive terms.

The GJO analysis suggests a transition between these two regimes at h \sim h_\text{GJO} = v_{cr}/\omega_0.

7. What Chapter 3 Does Not Provide (and Where to Look)

7.1 Mutual friction

Chapter 3 treats a perfect fluid (zero temperature, no normal component). Mutual friction — the dissipative coupling between superfluid and normal fluid mediated by vortex scattering — is not introduced until Chapters 8 and 9. For the \alpha_{mf} identification, we need:

  • Ch. 8.1: How \alpha and \alpha' enter the HVBK equations (the dissipative extension of the Magnus force equation)
  • Ch. 8.5: Partial-wave scattering analysis and the Aharonov–Bohm connection
  • Ch. 9.5–9.7: Andreev bound states, Kopnin–Kravtsov force — the microscopic origin of \alpha_{mf} = \frac{1}{2}\sin 2\delta_0

7.2 Counter-rotating vortex configurations

Chapter 3 assumes all vortices have the same circulation (\kappa > 0). The case of counter-rotating layers — where \tilde{\boldsymbol{\omega}} reverses sign — is not treated. This is the core theoretical novelty needed for the h derivation.

7.3 Turbulent vortex tangles

Chapter 3 treats ordered vortex arrays. The quantum turbulence that develops in the shear layer above the GJO threshold is treated in Chapter 14, using Vinen’s theory and Kolmogorov scaling arguments.