On the Theory of Bulk Viscosity of Cold Plasmas and Thermodynamics of Alkali-Noble Gas Cocktails

This paper derives an explicit expression for the bulk viscosity of cold plasmas, demonstrating that it can vastly exceed shear viscosity and that the Mandelstam-Leontovich approximation is exact under these conditions, while also providing thermodynamic formulas for alkali-noble gas mixtures and discussing their implications for solar atmospheric heating and laboratory verification.

The Gist

The Big Picture: Why Does the Sun Glow?

Imagine the Sun's atmosphere (the chromosphere) as a giant, invisible ocean of gas. Scientists have long wondered: How does this gas get so hot? The surface is relatively cool, but the layers above it are scorching. One theory suggests that sound waves traveling through this gas act like a heater, dumping their energy into the gas and warming it up.

But for sound waves to heat the gas, they need to get "stuck" or absorbed. Usually, sound passes right through a gas without losing much energy. This paper argues that in the Sun's atmosphere, there is a hidden "friction" that stops the sound waves and turns them into heat. This friction is called Bulk Viscosity.

The Main Characters: Shear vs. Bulk Viscosity

To understand this, let's use two analogies:

1. Shear Viscosity (The Honey): Imagine stirring a pot of honey. It resists your spoon sliding sideways. This is shear viscosity. It's the "stickiness" we usually think of when we talk about fluids. In the Sun's gas, this is like the gas molecules bumping into each other as they slide past one another.
2. Bulk Viscosity (The Squeezed Sponge): Now, imagine a sponge. If you squeeze it, it resists getting smaller. If you let it go, it bounces back. But if you squeeze it really fast, the sponge doesn't have time to let the air out, and it feels harder. If you squeeze it slowly, the air escapes easily.
   • Bulk viscosity is the resistance a gas feels when it is being compressed and expanded (like a sound wave squeezing it).
   • In most gases, this resistance is tiny. But in the Sun's "cold" plasma (a gas where atoms are just starting to break apart into electrons and ions), this resistance is massive.

The Paper's Discovery: The authors found that in the Sun's atmosphere, this "squeezing friction" (Bulk Viscosity) is millions of times stronger than the "sliding friction" (Shear Viscosity). It's the dominant force that stops sound waves.

The Secret Mechanism: The Ionization Switch

Why is the squeezing friction so strong in the Sun? It's because of a chemical reaction happening inside the gas: Ionization.

• The Analogy: Imagine a crowd of people (atoms) holding hands. When you squeeze the crowd (compress the gas with a sound wave), the pressure and heat rise. Suddenly, some people let go of each other and start running around wildly (the atoms break apart into electrons and ions). This takes a lot of energy.
• The Lag: When the wave expands and the crowd relaxes, the people try to grab hands again (recombine). But they can't do it instantly. There is a time delay.
• The Result: Because the crowd is busy letting go and grabbing hands, the gas absorbs the energy of the squeeze. The sound wave loses its energy to this chemical "dance," turning into heat.

The authors calculated exactly how long this "dance" takes (the relaxation time) and found it matches a classic physics formula perfectly.

The "Cold" Plasma Surprise

The paper focuses on "Cold" plasmas. In physics, "cold" doesn't mean freezing; it just means the temperature is low enough that the atoms aren't fully broken apart yet. They are in a delicate state of being half-atom, half-electron.

The authors used a specific recipe for the Sun's gas (mostly Hydrogen and Helium, with a pinch of Carbon, Magnesium, etc.) and solved the math to show:

1. The Sound Waves Die: The sound waves get absorbed very quickly because of this ionization "dance."
2. The Heating Works: This absorption is strong enough to explain how the Sun's lower atmosphere gets heated up to the transition region.
3. The Formula is Simple: Even though the math is complex, the result follows a simple, predictable pattern (called the Mandelstam-Leontovich approximation). It's like a perfect clockwork mechanism.

Why Should You Care?

1. Solving a Solar Mystery: This helps explain why the Sun's atmosphere is hotter than its surface. It suggests that sound waves are the "combustion chamber" that heats the gas before magnetic waves take over higher up.
2. Lab Experiments: The authors suggest we can test this on Earth. If we mix alkali metals (like Sodium) with noble gases (like Neon) in a lab, we can create a "solar cocktail" to measure this bulk viscosity and confirm the theory.
3. A New Tool: They provided a clear, usable formula for scientists to calculate how much heat sound waves generate in any similar gas.

Summary in One Sentence

This paper proves that in the Sun's atmosphere, sound waves get "stuck" not because the gas is sticky, but because the gas is busy breaking apart and reassembling itself, creating a massive amount of friction that turns sound into the heat that warms our star.

Technical Summary

1. Problem Statement

The paper addresses a significant gap in plasma physics: the calculation of bulk viscosity ($\zeta$) in cold, partially ionized plasmas from first principles.

• Context: In many physical systems (e.g., solar chromosphere, ocean water, dissociated gases), bulk viscosity dominates over shear viscosity ($\eta$). However, calculating $\zeta$ is notoriously difficult due to the complexity of internal degrees of freedom and chemical reactions (ionization-recombination).
• Specific Challenge: While shear viscosity is well-understood, bulk viscosity is often neglected or treated phenomenologically. The authors aim to derive an explicit, analytical expression for bulk viscosity in cold plasmas where the temperature $T$ is much lower than the first ionization potential ($I_a$) of the constituent atoms ($T \ll I_a$).
• Motivation: The primary application is understanding acoustic heating in the solar chromosphere. The authors investigate whether sound waves, damped by bulk viscosity, can provide sufficient energy to heat the solar atmosphere up to the transition region.

2. Methodology

The authors employ a rigorous theoretical framework combining statistical mechanics, kinetic theory, and hydrodynamics.

• Thermodynamic Framework:

  • They model the plasma as a "cocktail" of alkali metals (e.g., Hydrogen) and noble gases (e.g., Helium).
  • They utilize the Saha equation to determine equilibrium ionization degrees ($\alpha$) and concentrations of neutral atoms, ions, and electrons.
  • Thermodynamic variables (enthalpy $w$, internal energy $\varepsilon$, heat capacities $C_p, C_v$) are derived explicitly as functions of temperature and density, accounting for ionization energy.

• Kinetic Approach:

  • The core of the derivation is the kinetic equation for the time evolution of ionization degrees ($n_{i,a}$) driven by ionization-recombination processes ($e + A \leftrightarrow A^+ + 2e$).
  • They utilize the Wannier near-threshold ionization cross-section ($\sigma \propto \epsilon^w$), which allows for an analytical calculation of the ionization rate $\beta(T)$ for cold plasmas.
  • The system is linearized around equilibrium to study small harmonic oscillations in density ($\rho$), temperature ($T$), and ionization degree ($\alpha$).

• Derivation of Bulk Viscosity:

  • The authors introduce a generalized complex polytropic index $\hat{\gamma}(\omega)$, which relates pressure and density oscillations in the frequency domain.
  • By solving the linearized kinetic equations, they derive the frequency dependence of $\hat{\gamma}(\omega)$.
  • The bulk viscosity $\hat{\zeta}(\omega)$ is extracted from the imaginary part of the polytropic index using the relation: $\hat{\zeta}(\omega) \propto -\frac{\gamma''(\omega)}{\omega}$.

3. Key Contributions

A. Explicit Analytical Formulas

The paper provides explicit, closed-form expressions for:

• The relaxation time ($\tau$) governing the ionization-recombination kinetics.
• The bulk viscosity $\zeta_0$ in the low-frequency limit.
• The generalized polytropic index $\hat{\gamma}(\omega)$ for binary alkali-noble gas mixtures.

B. Validation of the Mandelstam-Leontovich (ML) Approximation

A major theoretical finding is that for cold plasmas, the Mandelstam-Leontovich approximation (which assumes a single relaxation time $\tau$) is not just an approximation but an exact result.

• The frequency dependence of the bulk viscosity follows a Drude-like (or Cole-Cole) form:
  $$\hat{\zeta}(\omega) \approx \frac{\zeta_0}{1 - i\omega\tau}$$
• Numerical simulations for a solar-like plasma cocktail (H, He, C, Mg, Si, Fe) show that the exact kinetic solution perfectly overlaps with the ML single-time-constant curve.

C. Thermodynamics of Alkali-Noble Cocktails

The authors systemize the thermodynamics for binary mixtures (e.g., Hydrogen-Helium), providing analytical solutions for heat capacities and sound velocities that account for the non-adiabatic effects of ionization. They show how the ionization degree breaks standard relations for constant chemical compounds.

4. Key Results

• Magnitude of Bulk Viscosity:

  • The calculated bulk viscosity is many orders of magnitude larger than the shear viscosity.
  • The Prandtl number for bulk viscosity ($P_{\zeta/\eta} = \zeta_0 / \eta$) reaches values up to $10^{11}$ in the solar chromosphere. Neglecting bulk viscosity in this regime is described as "neglecting a giraffe on the background of a hydrogen atom."

• Frequency Dependence and Damping:

  • Low Frequencies ($\omega \tau \ll 1$): The damping rate $k'' \propto \omega^2$.
  • High Frequencies ($\omega \tau \gg 1$): The damping rate becomes frequency-independent (dispersionless), $k'' \approx \text{const}$.
  • There exists a critical frequency $f_c$ where bulk viscosity transitions from dominating to being negligible compared to shear viscosity.

• Solar Chromosphere Application:

  • Using the AL08 (Avrett & Loeser) solar model, the authors calculate that at heights $h < 2.1$ Mm, bulk viscosity is the dominant mechanism for sound wave damping.
  • The relaxation time $\tau$ is calculated to be approximately 10.5 days (corresponding to a very low frequency), implying that for typical acoustic waves in the chromosphere, the system is in the low-frequency limit where bulk viscosity is maximized.
  • The results suggest that acoustic heating via bulk viscosity is a viable and potentially dominant mechanism for heating the lower solar chromosphere, provided the acoustic wave spectrum has significant power at low frequencies.

• Laboratory Applicability:

  • The derived formulas are applicable to laboratory plasmas, specifically "alkali-noble gas cocktails" (e.g., Na-Ne), suggesting a path for experimental verification.

5. Significance

1. Theoretical Breakthrough: This work is one of the few instances where bulk viscosity is derived from first principles (ab initio) using the Wannier cross-section and kinetic theory, rather than relying on phenomenological models.
2. Solar Physics: It provides a robust quantitative argument for the inclusion of bulk viscosity in models of solar atmospheric heating. It challenges previous models that may have underestimated acoustic heating by ignoring the massive contribution of ionization-recombination dissipation.
3. Unification of Concepts: The paper successfully bridges the gap between microscopic kinetic theory (ionization rates) and macroscopic hydrodynamics (viscosity, sound damping), showing that the Mandelstam-Leontovich approximation is exact for this specific physical regime.
4. Experimental Guidance: By providing explicit formulas for alkali-noble mixtures, the authors offer a roadmap for laboratory experiments to confirm these theoretical predictions, potentially simulating solar heating mechanisms in controlled environments.

In conclusion, Varonov and Mishonov demonstrate that in cold, partially ionized plasmas, the bulk viscosity is a dominant transport coefficient driven by ionization kinetics. Their derivation confirms that the standard single-relaxation-time approximation is exact for these systems, offering a powerful tool for modeling acoustic heating in astrophysical and laboratory plasmas.