Sonin Ch. 6 — Agent-Focused Summary for the Two-Fluid HVBK Framework

What this document is

A working extraction from Sonin (2016), Chapter 6 (“Vortex dynamics in two-fluid hydrodynamics”), focused on how the mutual friction coefficients \alpha and \alpha' derived microscopically in Chapters 8–9 enter the macroscopic dynamical equations that govern the inter-layer coupling in the vortex sheet model. This chapter is the dynamical framework that connects the microscopic \alpha_{mf} to the macroscopic E_\text{rep}(h).

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1. The Complete Two-Fluid HVBK System

1.1 The equation of vortex motion with mutual friction

This is the central equation for the entire problem. The vortex velocity \boldsymbol{v}_L is determined by (Eq. 6.33):

\boxed{\boldsymbol{v}_L = \boldsymbol{v}_{sl} + \alpha'(\boldsymbol{v}_n - \boldsymbol{v}_{sl}) + \alpha[\hat{s} \times (\boldsymbol{v}_n - \boldsymbol{v}_{sl})]}

or equivalently, projecting out the component along the vortex line (Eq. 6.34):

\boldsymbol{v}_L = \boldsymbol{v}_{sl} - \alpha'[\hat{s} \times [\hat{s} \times (\boldsymbol{v}_n - \boldsymbol{v}_{sl})]] + \alpha[\hat{s} \times (\boldsymbol{v}_n - \boldsymbol{v}_{sl})]

Here \boldsymbol{v}_{sl} is the local superfluid velocity at the vortex (distinct from the coarse-grained average \boldsymbol{v}_s), \boldsymbol{v}_n is the normal velocity, and \hat{s} is the unit vector along the vortex line.

The two mutual friction parameters: - \alpha (dissipative): Drives the vortex velocity perpendicular to the counterflow \boldsymbol{v}_n - \boldsymbol{v}_{sl}, within the plane normal to \hat{s}. This is the component that dissipates energy. - \alpha' (reactive): Drives the vortex velocity along the counterflow direction, projected into the plane normal to \hat{s}. This is the non-dissipative Hall-like deflection.

1.2 Connection to the Hall–Vinen parameters B, B'

The widely-used B, B' parameters are related to \alpha, \alpha' by (Eq. 6.35):

\alpha = \frac{\rho_n}{2\rho}B, \qquad \alpha' = \frac{\rho_n}{2\rho}B'

1.3 Connection to the Kopnin parameters d, d'

The force-based parametrization (Eq. 6.36–6.37):

\alpha = \frac{d}{d^2 + (1-d')^2}, \qquad 1 - \alpha' = \frac{1-d'}{d^2 + (1-d')^2}

At weak mutual friction: \alpha \approx d and \alpha' \approx -d'.

From Ch. 9: The Kopnin–Kravtsov force gives d \propto \omega_0\tau/(1+\omega_0^2\tau^2) and 1-d' \propto 1/(1+\omega_0^2\tau^2), which maps to \alpha_{mf} = \frac{1}{2}\sin 2\delta_0 via \delta_0 = \arctan(\omega_0\tau).

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2. The Force Balance and Dissipation

2.1 The mutual friction force per unit volume

The mutual friction force (Eq. 6.32):

\boldsymbol{f}_{fr} = -\rho_s\alpha[\hat{s} \times [\tilde{\boldsymbol{\omega}} \times (\boldsymbol{v}_n - \boldsymbol{v}_{sl})]] - \rho_s\alpha'[\tilde{\boldsymbol{\omega}} \times (\boldsymbol{v}_n - \boldsymbol{v}_{sl})]

This is the force on the vortices from the normal component. By Newton’s third law, an equal and opposite force acts on the normal component.

2.2 The total force on vortices

The net force on vortices combines the elastic force (from deformations) and the mutual friction force (Eq. 6.29):

\boldsymbol{f}_s = \boldsymbol{f}_{el} + \boldsymbol{f}_{fr} = -\rho_s\tilde{\boldsymbol{\omega}} \times (\boldsymbol{v}_L - \boldsymbol{v}_s)

The left side is the Magnus force. This is the master equation: it says that any deviation of the vortex velocity from the average superfluid velocity produces a Magnus force, which must be balanced by elastic + friction forces.

2.3 The dissipation function

Energy dissipation occurs only through mutual friction (Eq. 6.31):

R = -\frac{1}{2}(\boldsymbol{v}_{sl} - \boldsymbol{v}_n) \cdot \boldsymbol{f}_{fr}

This is the energy transfer rate per unit volume between the superfluid and normal components. For the inter-layer problem, this is the rate at which energy is pumped into the shear layer by the coupling between co-rotating sheets.

Note that R depends on the local superfluid velocity \boldsymbol{v}_{sl}, not the average \boldsymbol{v}_s. The difference between local and average contains the elastic deformation information (line tension, Tkachenko shear).

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3. The Linearized Two-Fluid Equations

3.1 The complete system (Eqs. 6.40–6.45)

In the rotating frame, the full linearized system is:

Mass continuity: \frac{\partial\rho'}{\partial t} + \rho(\nabla \cdot \boldsymbol{v}) = 0

Entropy: \frac{\partial S'}{\partial t} + S(\nabla \cdot \boldsymbol{v}_n) - \frac{\chi}{T}\nabla^2 T = 0

Superfluid Euler: \frac{\partial\boldsymbol{v}_s}{\partial t} + \nabla\mu + 2\boldsymbol{\Omega} \times \boldsymbol{v}_L + \text{(viscous terms)} = 0

Momentum conservation: \frac{\partial\boldsymbol{v}}{\partial t} + \frac{\nabla P}{\rho} + 2\boldsymbol{\Omega} \times \boldsymbol{v} + \frac{\rho_s}{\rho}2\boldsymbol{\Omega} \times (\boldsymbol{v}_{sl} - \boldsymbol{v}_s) - \frac{1}{\rho}(\text{viscous terms}) = 0

Local superfluid velocity: \boldsymbol{v}_{sl} = \boldsymbol{v}_s + \nu_s\hat{z} \times \frac{\partial^2\boldsymbol{u}}{\partial z^2} + \frac{c_T^2}{2\Omega}[\hat{z} \times \nabla_\perp^2\boldsymbol{u} - 2\hat{z} \times \nabla(\nabla \cdot \boldsymbol{u})]

Vortex motion: \boldsymbol{v}_L = \boldsymbol{v}_{sl} + \alpha'(\boldsymbol{v}_n - \boldsymbol{v}_{sl}) + \alpha[\hat{s} \times (\boldsymbol{v}_n - \boldsymbol{v}_{sl})]

3.2 The centre-of-mass velocity

A crucial simplification: the centre-of-mass velocity (Eq. 6.46):

\boldsymbol{v} = \frac{\rho_s}{\rho}\boldsymbol{v}_s + \frac{\rho_n}{\rho}\boldsymbol{v}_n

and the counterflow velocity \boldsymbol{w} = \boldsymbol{v}_n - \boldsymbol{v}_s are natural variables. Much of the mode structure separates into centre-of-mass modes (both components move together) and counterflow modes (components move against each other).

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4. The Mode Structure: What Mutual Friction Does

4.1 First sound: unaffected by mutual friction

First sound involves pressure oscillations with \boldsymbol{v}_L \approx \boldsymbol{v}_s \approx \boldsymbol{v}_n (no counterflow). Mutual friction is irrelevant. Velocity c_s = \sqrt{\partial P/\partial\rho}.

4.2 Second sound: strongly affected

Second sound is a temperature/counterflow wave with velocity c_2 = \sqrt{\rho_s s^2 T / \rho_n c_V}. Mutual friction damps it because it involves counterflow \boldsymbol{w} \neq 0.

4.3 Kelvin modes at finite temperature (Eq. 6.78)

\omega = (2\Omega + \nu_s p^2)[\pm(1-\alpha') - i\alpha]

The mutual friction parameters directly enter: \alpha provides damping and \alpha' provides a frequency shift. The quality factor is (1-\alpha')/\alpha.

For the inter-layer problem: The Kelvin modes on the vortex sheets will be damped by the inter-layer mutual friction. The damping rate \alpha\omega sets a timescale for energy dissipation across the boundary.

4.4 Slow mode at finite temperature (Eq. 6.97)

\omega^2 = 4\Omega^2\frac{p^2}{K^2} + \frac{\rho_s}{\rho}c_T^2 k^2

The Tkachenko wave velocity acquires a temperature-dependent factor \sqrt{\rho_s/\rho}. This is because mutual friction locks the normal and superfluid components together at low frequencies, so the effective inertia is \rho (not \rho_s) while the restoring force (shear modulus) is still C_{66} = \rho_s c_T^2.

4.5 The crossover: centre-of-mass vs. counterflow

At low frequencies (strong mutual friction regime): modes split into centre-of-mass (both components oscillate together) and counterflow (relative oscillation, strongly damped). The Tkachenko wave becomes a centre-of-mass mode.

At high frequencies (weak mutual friction regime): modes split into Kelvin (superfluid) and viscous (normal). The two components oscillate independently.

The crossover frequency is \sim B\Omega, set by the mutual friction strength times the rotation rate.

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5. The Clamped Regime (§6.7)

When the normal component is clamped (\boldsymbol{v}_n = 0), the vortex equation of motion simplifies to (Eq. 6.102):

\boldsymbol{v}_L = (1-\alpha')\boldsymbol{v}_{sl} - \alpha\hat{s} \times \boldsymbol{v}_{sl}

This is the regime relevant for: - Superfluid ^3He (high viscosity) - Neutron star interiors (normal component clamped to crust by magnetic field) - The inter-layer problem at low temperatures (if the normal component in the shear layer is pinned)

In the clamped regime, the superfluid Ekman depth \ell_E = \sqrt{\nu_s/2\Omega} is unchanged from the perfect-fluid value (Ch. 3). This is because the Kelvin mode, which determines \ell_E, involves negligible normal-fluid motion at low frequencies.

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6. What This Gives Us for the h Derivation

6.1 The dissipation rate sets the coupling energy

The dissipation function R = -\frac{1}{2}(\boldsymbol{v}_{sl} - \boldsymbol{v}_n)\cdot\boldsymbol{f}_{fr} is the energy transfer rate between superfluid and normal components per unit volume. For the inter-layer problem:

E_\text{rep} \sim \int_\text{shear layer} R \cdot \tau_\text{dwell}\,dV

where \tau_\text{dwell} is the residence time of fluid elements in the shear layer. The coupling energy is the dissipation rate times the time fluid spends in the coupling region.

6.2 The \rho_s/\rho factor matters

The Tkachenko velocity at finite temperature is c_t = \sqrt{\rho_s/\rho}\,c_T. This means the shear restoring force that resists compression of the inter-layer gap is proportional to \rho_s, not \rho. At temperatures where \rho_s \ll \rho, the Tkachenko rigidity weakens and the shear layer is softer — sheets can be pushed closer together.

6.3 Two regimes of the slow mode

The slow mode dispersion (Eq. 6.97) — which governs the large-scale dynamics of the vortex sheet structure — has two contributions:

\omega^2 = \underbrace{4\Omega^2\frac{p^2}{K^2}}_\text{inertial (classical)} + \underbrace{\frac{\rho_s}{\rho}c_T^2 k^2}_\text{Tkachenko (quantum)}

The first term is the classical inertial wave (exists without quantized vorticity). The second is the quantum Tkachenko correction (exists only because vorticity is quantized). The inter-layer spacing h is determined by the balance between these two contributions and the GJO instability threshold (from Ch. 3).

6.4 The impedance formula comes full circle

Combining all four chapters:

Chapter	Contribution	Scale
Ch. 3	Elastic moduli C_{44}, C_{66}; GJO instability; Ekman depth \ell_E	Macroscopic: vortex array
Ch. 6	Two-fluid dynamics; how \alpha, \alpha' enter force balance; dissipation function R	Macroscopic: two-fluid flow
Ch. 8	External scattering: D' = -\kappa\rho_n; impedance formula Eq. 8.7; three-channel structure	Mesoscopic: quasiparticle scattering
Ch. 9	Core bound states: \alpha_{mf} = \frac{1}{2}\sin 2\delta_0 from Kopnin–Kravtsov; resonance at \omega_0\tau = 1	Microscopic: vortex core

The full inter-layer coupling is:

1. Macroscopic dynamics (Ch. 6): The slow mode dispersion gives the wave-like response of the vortex sheet structure to perturbations.
2. Elastic restoring force (Ch. 3): The tilt modulus C_{44} and shear modulus C_{66} resist deformation of the sheets; the GJO instability limits the coherent shear layer.
3. Energy transfer (Ch. 8): The three-channel impedance formula gives the total coupling between the microscopic core physics and the macroscopic normal-fluid response.
4. Core coupling (Ch. 9): The Kopnin–Kravtsov force provides the specific value \alpha_{mf} = \frac{1}{2}\sin 2\delta_0, with \delta_0 = \arctan(\omega_0\tau) encoding the precession/relaxation competition.

The equilibrium inter-layer spacing h is then determined by minimizing the total energy, which balances the attractive inter-sheet energy against E_\text{rep}(h) built from all four contributions.