Sonin Ch. 14 — Agent-Focused Summary for the E_\text{rep}(h) Derivation

What this document is

A working extraction from Sonin (2016), Chapter 14 (“Elements of a theory of quantum turbulence”), selected and annotated for the specific problem of determining the functional form of the repulsive energy E_\text{rep}(h) in the counter-rotating shear layer between co-rotating chirality sheets. This is the final piece of the five-chapter toolkit (Ch. 3, 6, 8, 9, 14) needed to complete the three-term energy balance for the inter-sheet spacing h.

1. Classical Turbulence Scaling: The Template

1.1 The Kolmogorov cascade

Energy is pumped at the global scale L, cascades through the inertial range without dissipation, and is absorbed by viscosity at the dissipation scale l_d. The key assumption is locality: energy transfer at scale l depends only on parameters at that scale.

The energy density in wave-number space (the Kolmogorov–Obukhov -5/3 law, Eq. 14.2):

e(K) \sim \left(\rho g_\varepsilon^2\right)^{1/3} K^{-5/3}

where g_\varepsilon = -dE/dt is the energy flux (dissipation rate). The dissipation scale (Eq. 14.3):

l_d \sim \left(\frac{\rho\nu^3}{g_\varepsilon}\right)^{1/4}

The total energy is concentrated at large scales (Eq. 14.5):

E \sim \rho^{1/3}(g_\varepsilon L)^{2/3}

1.2 Turbulence decay

When energy input stops, the energy decays as (Eq. 14.8):

E \sim \frac{\rho L^2}{t^2}

Relevance: This 1/t^2 decay law will appear again in the quantum turbulence context. The decay rate sets the timescale over which the shear layer can maintain its turbulent energy content.

2. Vinen’s Theory of the Quantum Vortex Tangle

2.1 The single parameter: vortex line density \mathcal{L}

The turbulent tangle is characterized by one parameter: the vortex line length per unit volume \mathcal{L} [dimensions: \text{m}^{-2}]. The self-similar tangle hypothesis says \mathcal{L} determines everything — the inter-line spacing l_0, the curvature radius, and the velocity fluctuations:

l_0 \sim \mathcal{L}^{-1/2}

2.2 Vinen’s equation (Eq. 14.12)

\boxed{\frac{d\mathcal{L}}{dt} = \alpha_V \mathcal{L}^{3/2}|\boldsymbol{v}_n - \boldsymbol{v}_s| - \chi\kappa\mathcal{L}^2}

Two competing processes:

Process Rate Physical origin
Growth \alpha_V \mathcal{L}^{3/2}|\boldsymbol{v}_n - \boldsymbol{v}_s| Mutual friction stretches vortex rings; ring growth rate \sim \alpha|v_\text{rel}| (Eq. 14.9). Parameter \alpha_V \sim \alpha (mutual friction) up to geometric O(1) factor.
Decay -\chi\kappa\mathcal{L}^2 Mutual annihilation of vortex lines; dimensional argument from \kappa and l_0. Decay rate \sim \alpha\kappa/R for shrinking ring. Parameter \chi also depends on \alpha.

2.3 Steady-state line density (Eq. 14.13)

Setting d\mathcal{L}/dt = 0:

\boxed{\mathcal{L}_\text{ss} = \frac{\alpha_V^2}{\chi^2\kappa^2}|\boldsymbol{v}_n - \boldsymbol{v}_s|^2}

The line density grows as the square of the counterflow velocity. This is the key result for E_\text{rep}(h): the turbulent tangle density in the shear layer is determined by v_\text{rel}^2.

Physical consequence: The mutual friction force per unit volume is proportional to \mathcal{L} \cdot |v_\text{rel}|, so the steady-state force goes as |v_\text{rel}|^3 — the Gorter–Mellink cubic law. This is the dissipative force that the shear layer exerts against compression.

2.4 Decay without driving (Eq. 14.15)

When the counterflow is stopped:

\mathcal{L}(t) = \frac{1}{\chi\kappa(t - t_0)}

This 1/t decay is for the mutual-friction-dominated regime (counterflow turbulence). At late times, the decay switches to a t^{-3/2} law (Eq. 14.21), characteristic of the Kolmogorov cascade regime.

2.5 Alternative growth rate (Eq. 14.14)

An alternative form replaces the \mathcal{L}^{3/2}|v_\text{rel}| growth term with:

\left.\frac{d\mathcal{L}}{dt}\right|_{gr} \sim \alpha'\mathcal{L}|v_\text{rel}|^2

This gives the same Gorter–Mellink cubic law and has the advantage of analytic dependence on \mathcal{L}. Both forms are used in the literature; the choice affects O(1) coefficients but not the scaling with v_\text{rel}.

3. Energy Content of the Turbulent Tangle

3.1 The energy per unit volume

Each vortex line carries energy per unit length \varepsilon_0 = (\rho\kappa^2/4\pi)\ln(l_0/r_c). The energy per unit volume of the tangle is:

E_\text{vol} = \varepsilon_0 \cdot \mathcal{L} = \frac{\rho\kappa^2}{4\pi}\mathcal{L}\ln\frac{l_0}{r_c} = \frac{\rho\kappa^2}{4\pi}\mathcal{L}\ln\frac{1}{\sqrt{\mathcal{L}}\,r_c}

where the logarithmic factor \Lambda = \ln(l_0/r_c) is typically \sim 10–20 in helium and could be much larger in the substrate.

3.2 The Kelvin wave contribution (Eq. 14.22)

Including distortions of the vortex lines by Kelvin modes:

E = \frac{\Lambda\rho\kappa^2\mathcal{L}}{8\pi}\sum_p p^2|\boldsymbol{u}(p)|^2

This is the energy stored in Kelvin wave oscillations on the tangle. It adds to the baseline vortex line energy.

3.3 Total turbulent energy in the classical range (Eq. 14.20)

Connecting to the Kolmogorov picture at scales > l_0:

E \sim \rho\kappa^2 L^{2/3}\mathcal{L}^{4/3}

where L is the outer scale of turbulence. This includes the contribution from velocity fluctuations at all scales in the classical inertial range.

3.4 The energy flux (Eq. 14.18)

The energy dissipation rate (= energy flux through the cascade):

\boxed{g_\varepsilon \sim \rho\kappa^3\mathcal{L}^2}

This is the only expression of correct dimensionality constructible from \kappa and l_0 at T = 0 (no viscosity, no mutual friction). Equivalently, g_\varepsilon/\rho \sim \nu_\text{eff}\kappa^2\mathcal{L}^2 with the effective kinematic viscosity \nu_\text{eff} \sim \kappa (Vinen 2010).

Relevance: This energy flux is what feeds into the inter-layer dissipation. For the shear layer, g_\varepsilon represents the rate at which the turbulence machinery processes the kinetic energy of the counterflow.

4. Two Inertial Ranges: Classical and Quantum

4.1 The superfluid Reynolds number (Eq. 14.16)

\text{Re}_s = \frac{Lv(L)}{\kappa}

where \kappa replaces kinematic viscosity \nu. A long Kolmogorov inertial range requires \text{Re}_s \gg 1.

4.2 The classical inertial range (l > l_0)

At scales larger than the inter-vortex spacing, quantization of circulation is unimportant and the standard Kolmogorov K^{-5/3} spectrum holds. The vortex tangle has partial polarization at these scales — bundles of vortex lines form effective large-scale vorticity.

Mean-square vorticity at scale l (Eq. 14.19):

\langle\tilde{\omega}(l)^2\rangle \sim \kappa^2\mathcal{L}^2\left(\frac{l_0}{l}\right)^{4/3}

Key: Polarization is weak at large scales (\sim (l_0/l)^{2/3} \ll 1) and approaches full polarization only at l \sim l_0.

4.3 The quantum inertial range (l < l_0): Kelvin wave cascade

Below l_0, the cascade continues as Kelvin waves propagating along individual vortex lines. Two competing predictions:

Theory Spectrum Mechanism
Kozik–Svistunov (2004) e(p) \sim p^{-7/5} 6-wave interaction (weak turbulence, tilt-symmetric)
L’vov–Nazarenko (2010) e(p) \sim p^{-5/3} 4-wave interaction (static deformation breaks symmetry)

In the 3D velocity space, both give (Eq. 14.30):

e(K) \sim \frac{\rho\kappa^2}{l_0^2}K^{-1}

The K^{-1} spectrum follows from the Fourier transform of the 1/r velocity field around an isolated vortex — independent of the cascade details.

4.4 The crossover at l_0

The Kolmogorov spectrum and the Kelvin wave spectrum match smoothly at K \sim 1/l_0 when g_\varepsilon \sim \rho\kappa^3\mathcal{L}^2 (Eq. 14.18). There is no bottleneck — the single scale l_0 governs the crossover.

Near the crossover, turbulence is always strong (not weak). The weak-turbulence Kelvin cascade spectrum is an asymptotic valid only far from l_0.

4.5 Kelvin wave cascade terminates at the core radius

The Kelvin cascade ends when sound (phonon) emission becomes efficient. Dipole radiation occurs for Kelvin wavelengths shorter than \nu_s/c_s \sim r_c (up to a logarithmic factor). The short-wavelength cutoff of the Kelvin wave cascade is the vortex core radius r_c.

The energy flux from dipole radiation (Eq. 14.37):

Q_d = \frac{1}{8}\rho\kappa^2\omega k^2 a^2

Quadrupole radiation (two kelvons → one phonon) is also possible, with Q_q \sim \rho\kappa^2\omega k^4 a^4 (Eq. 14.40), but is subdominant.

5. The GJO Instability and the Transition to Turbulence

5.1 GJO in the turbulence context (§14.8)

Sonin evaluates the GJO instability threshold for a twisted vortex bundle. The critical velocity is (adapting Eq. 3.124 to space-dependent vorticity):

v_{cr} = \sqrt{\tilde{\omega}(r)\,\nu_s}

For a twisted bundle, the stability condition becomes (Eq. 14.42):

Q^2 R^2 < \frac{8\nu_s}{\Omega R^2}

where Q is the twist parameter. Translating to mutual friction parameters (Eq. 14.43):

\frac{1 - \alpha'}{\alpha} < 3.5\sqrt{\frac{\nu_s}{\Omega R^2}}

5.2 GJO as precursor, not threshold

Important finding: Experiments show the transition to turbulence at \alpha/(1-\alpha') \sim O(1), while the GJO threshold predicts much higher values. Sonin concludes that GJO is a precursor to developed turbulence, not the transition itself. There is an intermediate regime of large vortex array fluctuations without essential reconnections.

Reconnections become essential only below \sim 0.3\,T_c (Hosio et al. 2011). The laminar regime becomes unstable at higher temperatures.

Relevance to the inter-layer problem: This modifies the picture from the Ch. 3 summary. The shear layer doesn’t switch abruptly from laminar to turbulent at h_\text{GJO}. Instead, there’s a continuous buildup of fluctuations, followed by a regime of developing turbulence, before reaching the fully developed tangle described by Vinen’s equation. The energy content in the intermediate regime may be lower than Vinen’s steady state.

6. Inhomogeneity and Diffusion

6.1 Vortex tangle diffusion

Nemirovskii and Fiszdon (1995) generalized Vinen’s equation to include spatial diffusion of the tangle:

\frac{\partial\mathcal{L}}{\partial t} = \text{(growth)} + \text{(decay)} + D_v\nabla^2\mathcal{L}

The diffusion coefficient is uncertain: D_v \approx 0.1\kappa (Tsubota et al. 2003) to D_v \approx 2.2\kappa (Nemirovskii 2010).

Relevance: In the inter-layer problem, the shear layer has finite width \delta and the tangle is inherently inhomogeneous. Diffusion sets how quickly the tangle can spread from the shear zone into the co-rotating sheets. If D_v \sim \kappa, the diffusion length over the tangle lifetime is l_\text{diff} \sim \sqrt{D_v / (\chi\kappa\mathcal{L})} \sim 1/\sqrt{\mathcal{L}} = l_0. The tangle stays confined to its own inter-line spacing — it doesn’t spread far from the shear layer.

7. Assembly: What Chapter 14 Gives for E_\text{rep}(h)

7.1 The driving velocity

In the inter-layer problem, the counterflow velocity driving the turbulence is the relative velocity between adjacent co-rotating sheets:

v_\text{rel} \sim \omega_0 h

where \omega_0 is the cell-level rotation and h is the inter-sheet spacing.

7.2 The steady-state tangle density in the shear layer

From Vinen’s equation (Eq. 14.13):

\mathcal{L}_\text{ss}(h) = \frac{\alpha_V^2}{\chi^2\kappa_q^2}\,\omega_0^2 h^2

The inter-line spacing in the tangle:

l_0(h) = \mathcal{L}_\text{ss}^{-1/2} = \frac{\chi\kappa_q}{\alpha_V\omega_0 h}

7.3 The energy per unit volume of the tangle

E_\text{vol}(h) = \frac{\rho_\text{DM}\kappa_q^2}{4\pi}\,\mathcal{L}_\text{ss}(h)\,\Lambda(h)

where \Lambda(h) = \ln(l_0(h)/r_\text{eff}) is the logarithmic factor evaluated at the h-dependent inter-line spacing.

Substituting:

E_\text{vol}(h) = \frac{\rho_\text{DM}\kappa_q^2}{4\pi}\cdot\frac{\alpha_V^2}{\chi^2\kappa_q^2}\,\omega_0^2 h^2 \cdot \Lambda(h) = \frac{\rho_\text{DM}\alpha_V^2\omega_0^2 h^2}{4\pi\chi^2}\,\Lambda(h)

7.4 The repulsive energy per unit cell area

The shear layer has some effective width \delta(h). The repulsive energy per unit area of the inter-layer boundary is:

E_\text{rep}(h) = E_\text{vol}(h) \cdot \delta(h)

Two sub-cases for \delta(h):

Case A: \delta \propto h (shear fills the gap). If the tangle fills the entire inter-sheet gap:

E_\text{rep}(h) \propto \rho_\text{DM}\omega_0^2 h^3 \cdot \Lambda(h)

This is a strongly h-dependent repulsion that grows as h^3 (modulo the weak logarithm). It would dominate at large h and produce a stable minimum when balanced against the attractive/string terms.

Case B: \delta \sim l_0(h) \propto 1/h (tangle confined to its own inter-line spacing). From the diffusion argument (§6.1):

E_\text{rep}(h) \propto \rho_\text{DM}\omega_0^2 h^2 \cdot l_0(h) \cdot \Lambda(h) \propto \rho_\text{DM}\omega_0 h \cdot \frac{\kappa_q}{\chi}\Lambda(h)

This gives E_\text{rep} \propto h — a weaker repulsion that still grows with h.

Case C: \delta \sim r_\text{eff} (core-scale shear). If the tangle is confined to a core-scale region:

E_\text{rep}(h) \propto \rho_\text{DM}\omega_0^2 h^2 \cdot r_\text{eff} \cdot \Lambda(h)

This gives E_\text{rep} \propto h^2 — intermediate between Cases A and B.

7.5 The dissipation rate picture (alternative to energy content)

From the Ch. 6 dissipation function and Eq. 14.18:

g_\varepsilon(h) \sim \rho_\text{DM}\kappa_q^3\mathcal{L}_\text{ss}^2 \sim \rho_\text{DM}\frac{\alpha_V^4}{\chi^4}\frac{\omega_0^4 h^4}{\kappa_q}

The energy flux through the cascade scales as h^4 — very steep. This represents the power that must be continuously supplied to maintain the tangle. In steady state, this power comes from the relative rotation of the sheets.

The repulsive force per unit area (pressure) from the turbulence is:

P_\text{rep} \sim -\frac{\partial E_\text{rep}}{\partial h}

For the energy to balance the attractive terms (which scale as h^2/\xi from the Blatter framework), we need E_\text{rep} to grow faster than h^2 at large h and slower than h^2 at small h, OR we need E_\text{rep} to diverge as h \to 0.

7.7 Resolution: the repulsion comes from the compressed boundary layer, not the tangle

The Vinen tangle picture applies when the shear layer can freely develop. But at small h, the counter-rotating boundary layer is geometrically compressed: it must fit between two co-rotating sheets with separation h. The repulsion comes not from more turbulence, but from the incompressibility constraint on the counter-rotating fluid squeezed into a shrinking gap.

The relevant energy is the kinetic energy of the counter-rotating flow confined to width h:

E_\text{CR}(h) \sim \frac{1}{2}\rho_\text{DM}\,v_\text{CR}^2 \cdot h

where v_\text{CR} is the counter-rotating layer velocity. If v_\text{CR} is set by the vorticity quantum (one quantum of counter-rotation must exist in the boundary layer):

v_\text{CR} \sim \frac{\kappa_q}{h}

Then:

\boxed{E_\text{CR}(h) \sim \frac{\rho_\text{DM}\kappa_q^2}{2h} = \varepsilon_0 \cdot \frac{2\pi}{h}}

This gives E_\text{rep} \propto 1/h — diverging at small h — exactly the functional form needed to prevent collapse.

This is the same physics as the kinetic energy of a vortex confined to a channel of width h: the velocity is forced to increase as \kappa/h by the circulation quantum, and the energy density goes as \kappa^2/h^2 while the volume goes as h, giving net energy \propto 1/h.

8. Summary: The Toolkit Is Complete

8.1 What Chapter 14 provides

Result Equation Use in h derivation
Vinen’s equation d\mathcal{L}/dt = \alpha_V\mathcal{L}^{3/2}v_\text{rel} - \chi\kappa\mathcal{L}^2 Tangle dynamics in the shear layer
Steady-state \mathcal{L} \mathcal{L}_\text{ss} = (\alpha_V/\chi\kappa)^2 v_\text{rel}^2 Line density vs. h
Gorter–Mellink force f_\text{GM} \propto v_\text{rel}^3 Cubic dissipative force at the boundary
Tangle energy flux g_\varepsilon \sim \rho\kappa^3\mathcal{L}^2 Dissipation rate in the shear layer
Kolmogorov spectrum (l > l_0) e(K) \propto K^{-5/3} Classical range above inter-line spacing
Kelvin wave spectrum (l < l_0) e(K) \propto K^{-1} Quantum range below inter-line spacing
GJO is precursor only Transition at \alpha/(1-\alpha') \sim O(1) No sharp laminar/turbulent switch
Tangle diffusion confined D_v \sim 0.1–2.2\,\kappa Tangle stays near the shear zone
Kelvin cascade cutoff At r_c via phonon emission Energy dissipation at the smallest scale

8.2 The key physical insight for E_\text{rep}

The Vinen tangle does NOT provide the repulsion at small h (the tangle weakens as h \to 0 because v_\text{rel} \to 0). Instead, the repulsion comes from the quantized counter-rotating flow squeezed into a shrinking gap: one quantum of counter-circulation \kappa_q must thread the boundary layer, forcing v_\text{CR} \sim \kappa_q/h and producing E_\text{rep} \propto 1/h.

The Vinen theory contributes at large h: it sets the energy cost of the fully developed turbulent shear layer that forms when v_\text{rel} = \omega_0 h exceeds the critical velocity. This adds a broad, slowly varying energy contribution that shapes the minimum but doesn’t prevent collapse.

8.3 The three-term energy balance can now be written

Term h-dependence Physical origin Chapter
Polar-jet string +\alpha_{mf}\varepsilon_0(h^2/\xi)\ln(\xi/h) Coupling tube energy; favors small h Blatter/Ch. 8–9
Hydrodynamic attraction -\varepsilon_0 h^2/(2\xi) Pressure-mediated inter-layer binding Blatter
Counter-rotation confinement +A\varepsilon_0\xi/h Quantized vorticity in compressed boundary Ch. 14 + Ch. 3
Tangle contribution (broad) +B\varepsilon_0\omega_0^2 h^3\Lambda/\xi Vinen steady-state turbulence energy Ch. 14

The confinement term (\propto 1/h) prevents collapse to h = 0. The tangle term (\propto h^3) may contribute at large h but is subdominant near the minimum if h/\xi \ll 1.

8.4 Remaining unknowns

The dimensionless coefficients A and B contain geometric factors from the channel geometry, the ratio \alpha_V/\chi, and the effective width of the shear zone. These are O(1) to O(10) numbers that may need to be determined by matching to the known \alpha_{mf} or by self-consistency with the two-term balance upper bound (h/\xi < 0.070).

8.5 Connection to the five-chapter impedance picture

Chapter Contribution to E_\text{rep} Role
Ch. 3 Elastic moduli C_{44}, C_{66}; Ekman depth \ell_E; GJO critical velocity Sets the stiffness of each co-rotating sheet and the instability threshold
Ch. 6 Dissipation function R; mode structure; \rho_s/\rho renormalization Provides the dynamical framework for energy transfer across the boundary
Ch. 8 Three-channel impedance; D'/D decomposition; normal circulation \kappa_n Templates the coupling structure; shows reactive force needs \kappa_n
Ch. 9 \alpha_{mf} = \frac{1}{2}\sin 2\delta_0; Kopnin–Kravtsov force; resonance at \omega_0\tau = 1 Provides the coupling coefficient and its saturation at 1/2
Ch. 14 Vinen tangle energy; Gorter–Mellink force; 1/h confinement energy Provides the repulsive energy functional form