Ion-Acoustic Surface Wave

• Ion-acoustic surface waves are plasma modes that occur at the interface between a dense, quasi-neutral region and a rarefied, strongly magnetized plasma, exhibiting distinct ion-sound and fast magnetosonic characteristics.
• The wave follows a closed-form dispersion relation, with frequencies bounded by the ion gyrofrequencies and emerging when dense plasma dynamic pressure balances external magnetic pressure.
• Energy analyses indicate that kinetic energy dominates within the dense plasma while enhanced external electric fields amplify the wave signature, proving critical for experimental validations.

An ion-acoustic surface wave is a collective mode that propagates along the interface between a dense, quasi-neutral plasma region and a rarefied, strongly magnetized plasma. This mode emerges specifically when the dynamic pressure of dense, supersonic plasma flows becomes comparable to the undisturbed magnetic pressure, forcing the magnetic field outward from the plasma bulk. The ion-acoustic surface wave is characterized by a frequency band bounded by the ion gyrofrequencies inside ($\Omega_{i\rm in}$) and outside ($\Omega_{i\rm ext}$) the dense plasma and exhibits unique energy and field localization properties. Its underlying physics links isotropic ion sound in the dense phase to fast magnetosonic perturbations in the external, magnetically dominated regime (Koryagin et al., 10 Dec 2025).

1. Physical Model and Existence Conditions

The canonical system comprises two semi-infinite, isothermal plasma half-spaces meeting at a flat boundary $x=0$. The "inside" ($x<0$) is the dense plasma with ion density $n_{\rm in}$ and isothermal electron temperature $T_e$, hosting a weak residual field $B_{\rm in}$, such that electron pressure ($p_{e\,\rm in}=Z n_{\rm in} T_e$) exceeds the local magnetic pressure ($p_{B\,\rm in}=B_{\rm in}^2/(8\pi)$). The "outside" ($x>0$) features a rarefied plasma ($n_{\rm ext} \ll n_{\rm in}$), permeated by a strong field $B_{\rm ext} \gg B_{\rm in}$, so that magnetic pressure ($p_{B\,\rm ext}=B_{\rm ext}^2/(8\pi)$) is of the same order as the internal electron pressure.

Ions are cold ($T_i \ll T_e$), and the ion-sound speed is $c_s = \sqrt{Z T_e/m_i}$. The regime of interest imposes the hierarchy:

$$c_{A\,\rm in} = \frac{B_{\rm in}}{\sqrt{4\pi m_i n_{\rm in}}} \ll c_s \ll c_{A\,\rm ext} = \frac{B_{\rm ext}}{\sqrt{4\pi m_i n_{\rm ext}}}$$

with ion gyrofrequencies $\Omega_{i\,\rm in} = e B_{\rm in}/(m_i c)$ and $\Omega_{i\,\rm ext} = e B_{\rm ext}/(m_i c)$. The relevant surface wave satisfies $$\Omega_{i\,\rm in} \ll \omega \ll \Omega_{i\,\rm ext}$$.

For the surface mode to exist, the dynamic pressure of dense plasma must approximately balance the external magnetic pressure:

$$n_{\rm in} m_i v_i^2 \approx \frac{B_{\rm ext}^2}{8\pi}$$

This ensures the magnetic field is expelled from the dense bundle, confining ion sound waves inside and fast magnetic sound externally (Koryagin et al., 10 Dec 2025).

2. Dispersion Relation and Frequency Spectrum

The surface ion-acoustic wave is constructed from bulk solutions on either side of the interface:

• Inside (Ion-Sound Domain):

The wave potential $$\Phi_{\rm in}(x) \sim e^{i k_\parallel z + \kappa_{\rm in} x - i \omega t}$$ satisfies

$$\omega^2 = c_s^2(k_\parallel^2 + \kappa_{\rm in}^2)$$

with decaying amplitude ($\text{Re}\,\kappa_{\rm in} > 0$).

• Outside (Fast Magnetosonic Domain):

The analogous wave satisfies

$$\omega^2 = c_{A\,\rm ext}^2(k_\parallel^2 + \kappa_{\rm ext}^2)$$

Boundary matching at $x=0$ enforces four conditions:

1. Total pressure balance,
2. Frozen-in normal electron displacement continuity,
3. Continuity of normal electric induction $D_n$,
4. Continuity of tangential electric field $E_\parallel$.

Solving the resultant spectral relation leads to the closed-form surface-wave dispersion:

$$\omega^2 = 2(\sqrt{2}-1)\, c_s^2 k_\parallel^2 \approx 0.83\, c_s^2 k_\parallel^2$$

with frequency constrained to the interval $$\Omega_{i\,\rm in} \ll \omega \ll \Omega_{i\,\rm ext}$$ (Koryagin et al., 10 Dec 2025).

3. Mode Structure: Spatial and Field Characteristics

The surface mode exhibits a hybrid structure:

• Inside the bundle ($x<0$):

The wave is an "isotropic" ion sound with pressure perturbation $\delta p_{\rm in} = n_{\rm in} e \Phi_{\rm in}$. The dominant field is the longitudinal component $E_n = \kappa_{\rm in} \Phi_{\rm in}$, with a small tangential (inductive) component $$E_\parallel \approx (\Omega_{i\,\rm in} / \omega) E_n$$ that compensates for electron drift.

• Outside the bundle ($x>0$):

The mode corresponds to a transverse-polarized "magnetic sound" with magnetic-pressure perturbation $$\delta p_B = (B_{\rm ext} \delta B_\parallel) / (4\pi)$$, where $$\delta B_\parallel = -i k_\parallel \xi_{e\,n} B_{\rm ext}$$. The tangential field $E_\parallel$ far exceeds $E_n$, entering the transition layer via the impedance relation $E_\parallel = -i (\omega_{Bi}/\omega) E_n$, where $\omega_{Bi}(x)$ varies across the boundary.

A key feature is that the near-surface electric field strength is greatly enhanced outside compared to inside, driven both by the need for uniform electron drift and the near-electrostatic nature of propagating ion-sound oscillations in the external environment (Koryagin et al., 10 Dec 2025).

4. Boundary Conditions and Frequency Band

At the interface, the matching conditions uniquely determine the surface wave spectrum and spatial form. The solution exists only if both half-spaces can physically support the surface mode at a common real frequency, guaranteeing:

$$\Omega_{i\,\rm in} \ll \omega \ll \Omega_{i\,\rm ext}$$

This ensures propagating ion sound inside (unmagnetized character) and fast magnetic sound outside (strongly magnetized character), enabling the coupling through the interface.

5. Energy Partition and Field Enhancement

Analysis shows that the majority of the energy within the surface wave is contained in the ion kinetic energy in the dense region:

$$W_{\rm kin, in} \approx \tfrac{1}{2} n_{\rm in} m_i |v_i|^2$$

The corresponding electrical field energy inside,

$$W_{E,\rm in} \sim \frac{(\kappa_{\rm in} \Phi)^2}{8\pi}$$

is much less by a factor on the order of $(\omega^2/\Omega_{i\,\rm in}^2)$. Conversely, outside the plasma, the electric field is amplified by a factor $(\Omega_{i\,\rm ext}/\omega)$:

$$E_\parallel^{\rm ext} \approx (\Omega_{i\,\rm ext}/\omega) \kappa_{\rm in} \Phi$$

thus

$$W_{E,\rm ext} \sim \frac{|E_\parallel^{\rm ext}|^2}{8\pi}$$

substantially exceeds $W_{E,\rm in}$. This significant enhancement of the electric field near the surface is a robust signature observed in simulations of the mode (Koryagin et al., 10 Dec 2025).

6. Key Approximations and Physical Regime

The theory of the ion-acoustic surface wave relies on:

• Collisionless, quasi-neutral two-fluid plasma dynamics,
• Massless electron approximation (instantaneous pressure balancing and frozen-in drift),
• Neglect of displacement current within (Darwin approximation),
• Two-dimensional geometry (no $k_\tau$),
• Isothermal electrons and cold ions,
• Supersonic flow with Mach number $M=v_i/c_s>1$.

These assumptions ensure tractability and physical relevance within laboratory plasma configurations, such as those in the "Solar Wind" experimental setup, and accurately reflect the physics of surface modes in high-pressure, magnetically confined plasma systems (Koryagin et al., 10 Dec 2025).