Similarity Structure of Unsteady Compressible Two-Fluid Plasma Boundary Layers

1. Introduction

1.1. Background

The similarity solutions of the laminar boundary layer—Blasius, Falkner–Skan, Hiemenz, Rayleigh–Stokes—remain the fixed points around which the subject is organised [1,2,3,4,5]. Compressibility was absorbed into that framework by Howarth [6] and Dorodnitsyn [7], whose density-weighted normal coordinate converts the compressible momentum equation into something very close to its constant-density counterpart, and by Illingworth [8] and Stewartson [9], who extended the construction to flows with a pressure gradient. With the Chapman–Rubesin viscosity law [10] and the Crocco energy integral [11,12] this machinery yields the compressible Falkner–Skan–Dorodnitsyn system and, at a stagnation point with strong wall heat transfer, the variable-property solutions of Cohen and Reshotko [13]. The unsteady problem is older and harder: Stewartson’s analysis of the impulsively started plate [14,15] supplies a single variable $\tau \propto t/x$ interpolating between a diffusive Rayleigh layer and a convective Blasius layer, and reductions of this two-parameter type continue to be developed [16,17,18], with the Williams–Rhyne chart [19] providing the device that carries a solution through the region a forward march cannot enter.

For electrically conducting fluids the corresponding programme was carried out for single-fluid magnetohydrodynamics. Greenspan and Carrier [20], Sears [21], Sears and Resler [22] and Gribben [23] established the structure of aligned-field MHD boundary layers, in which an ambient field parallel to the free stream is swept into the layer and the character of the flow changes qualitatively as the Alfvén number passes through unity. The compressible version was treated by Rossow [24], Bleviss [25] and Bush [26], and the magneto-aerodynamic framework was set out by Resler and Sears [27]; general accounts of the single-fluid theory are given by Shercliff [46] and Davidson [45].

Plasmas, however, are not single fluids. Once the layer thickness approaches the ion inertial length $d_i=c/{\omega }_{pi}$, ions and electrons decouple: the field remains frozen to the electron fluid while the ions slip across it. The two-dimensional signature of this Hall regime is an out-of-plane field of quadrupolar symmetry, generated by the in-plane electron current and mediated by dispersive whistler dynamics. It was predicted by Sonnerup [28] and Terasawa [29], identified as the controlling ingredient of fast reconnection by Mandt, Denton and Drake [30], confirmed across the model hierarchy in the GEM challenge [31], analysed asymptotically by Uzdensky and Kulsrud [32], and measured in the laboratory and in space [33,34,35]. Any two-fluid boundary-layer theory must produce it. The four-field system that carries it is standard [36,37]; what is not standard is what happens to it when the density is allowed to vary.

1.2. What Is Missing, and What This Paper Supplies

What the literature does not contain is the two-fluid boundary layer at variable density. This is not a cosmetic extension, for three reasons that the body of the paper makes precise.

The first is that compressibility acts on the Hall coefficient itself. The quantity $\lambda =1/{\mu }_{0}ne$ that multiplies every Hall term is inversely proportional to the number density, so in a layer whose wall is four times hotter than the free stream the Hall coefficient varies by the same factor across the layer, and the local ion inertial length $d_i\propto n^{-1/2}$ by its square root. Neither variation can be scaled away. Whether the Hall physics is thereby amplified or suppressed is a question the constant-density theory cannot even pose.

The second is that variable density resurrects a term that constant-density theory annihilates. At uniform n the electron-pressure contribution to the induction equation is the curl of a gradient and vanishes identically. At variable n it is the Biermann battery [39,40], the canonical mechanism for generating magnetic field from nothing in astrophysical [41,42] and laser-produced plasmas. Since the Biermann source in two dimensions is directed out of the plane, it competes directly with the Hall stretching term for control of the quadrupole—the very structure that the two-fluid theory exists to predict.

The third is that compressibility and Hall physics impose competing similarity requirements. Constancy of the Hall group demands a layer of fixed thickness, because $d_i$ is an absolute length; constancy of the compressibility parameter demands a uniform external stream. In the single-fluid compressible theory only the second constraint is felt; in the incompressible two-fluid theory only the first. Together they are incompatible, and the resulting dichotomy dictates the entire architecture of the paper.

Contributions

1.

A closed unsteady master system for the compressible two-fluid layer with two temperatures (Section 5, Section 6), containing the compressible single-fluid and the constant-density two-fluid problems as exact limits.

2.

A weight-structure theorem (Section 3): under the Dorodnitsyn map the Hall operator, the magnetic force per unit mass and the Lorentz driving of the out-of-plane velocity are invariant, whereas the pressure gradient and the resistive terms are not.

3.

A complete account of the Biermann term (Section 5.3): excluded exactly from the in-plane flux equation, excluded again by similarity at uniform pressure, and surviving only as a streamwise-pressure-gradient source ${\theta }_{e,\eta }/2M^2$ that dominates the Hall quadrupole at low Alfvén number.

4.

An exactness dichotomy (Theorem 2) and its two resolutions.

5.

A kinematic change-of-type theorem for the unsteady system (Theorem 3) and a proof that the Alfvénic degeneracy is not displaced by compressibility (Theorem 4), both confirmed numerically.

We do not attempt a stability analysis. The base states derived here are the prerequisite for one, and the competition between the compressible Mack modes of high-speed layers [47,48] and the whistler branch identified below seems to us to warrant a separate treatment rather than an appendix.

2. Compressible Two-Fluid Model and Exact Planar Reductions

2.1. Species Equations and Closure

We take a quasineutral plasma of singly charged ions of mass $m_i$ and electrons of mass $m_e$, with common number density n and mass density $\rho =m_{i}n$. The species momentum equations are

$$\begin{array}{cc}\hfill m_{i}n\left({\partial }_t{\mathbf{u}}_i+{\mathbf{u}}_i·\nabla {\mathbf{u}}_i\right)& =-\nabla p_i+en(\mathbf{E}+{\mathbf{u}}_i\times \mathbf{B})-\nabla ·{\mathbf{\pi }}_i+{\mathbf{R}}_{ie},\hfill \end{array}$$

(1)

$$\begin{array}{cc}\hfill m_{e}n\left({\partial }_t{\mathbf{u}}_e+{\mathbf{u}}_e·\nabla {\mathbf{u}}_e\right)& =-\nabla p_e-en(\mathbf{E}+{\mathbf{u}}_e\times \mathbf{B})-\nabla ·{\mathbf{\pi }}_e-{\mathbf{R}}_{ie},\hfill \end{array}$$

(2)

with ${\mathbf{R}}_{ie}$ the friction force and ${\mathbf{\pi }}_s$ the viscous stresses. We adopt the standard boundary-layer closure: electron inertia is neglected, electron viscosity is neglected relative to ion viscosity, the ion stress is isotropic with dynamic viscosity $\mu$, and the friction is resistive, ${\mathbf{R}}_{ie}=-en{\eta }_r\mathbf{J}$ [43,44]. Dropping $m_e$ in (2) and using ${\mathbf{u}}_e=\mathbf{u}-\mathbf{J}/ne$ with $\mathbf{u}\equiv {\mathbf{u}}_i$ gives the generalised Ohm’s law [38]

$$\mathbf{E}+\mathbf{u}\times \mathbf{B}={\eta }_r\mathbf{J}+{\displaystyle \frac{1}{ne}}\left(\mathbf{J}\times \mathbf{B}-\nabla p_e\right),$$

(3)

in which the second and third terms on the right are the Hall and electron-pressure contributions. Throughout we write

$$\lambda \equiv {\displaystyle \frac{1}{{\mu }_{0}ne}},{\eta }_m\equiv {\displaystyle \frac{{\eta }_r}{{\mu }_0}},d_i={\displaystyle \frac{c}{{\omega }_{pi}}}={\displaystyle \frac{1}{ne}}\sqrt{{\displaystyle \frac{\rho }{{\mu }_0}}},$$

(4)

so that $\lambda \sqrt{{\mu }_0\rho }=d_i$ and ${\lambda }^2{\mu }_0\rho =d_i^2$. In contrast to the constant-density theory, $\lambda$ and $d_i$ are now fields: at constant pressure across the layer,

$${\displaystyle \frac{\lambda }{{\lambda }_r}}={\displaystyle \frac{{\rho }_r}{\rho }}=\omega ,{\displaystyle \frac{d_i}{d_{ir}}}={\omega }^{1/2},\omega \equiv {\displaystyle \frac{T}{T_r}},$$

(5)

with T the mixture temperature defined below and the subscript r denoting a constant reference state. Equation (5) is the seed of most of what follows.

Because the two species carry separate temperatures, the equation of state is $p=n{k}_B(T_i+T_e)=\rho RT$ with

$$T\equiv \frac{1}{2}(T_i+T_e),R={\displaystyle \frac{2k_B}{m_i}},c_p=\frac{5}{2}R,\gamma =\frac{5}{3}.$$

(6)

It is worth pausing on (6): the density weight that controls the Dorodnitsyn map is set by the mean of the two temperatures, so the geometric reduction of the compressible problem is itself a dual-temperature statement. Nothing analogous occurs in the single-fluid theory, and nothing at all occurs in the constant-density theory.

Two transport closures are needed, both direct analogues of Chapman–Rubesin:

$$\rho \mu =C{\rho }_r{\mu }_r,\rho {\eta }_m=C_m{\rho }_r{\eta }_m^{\left(r\right)},\rho {\kappa }_s=C_s{\rho }_r{\kappa }_s^{\left(r\right)}.$$

(7)

We write $\widehat{\nu }=C{\nu }_r$, $\widehat{\eta }=C_m{\eta }_m^{\left(r\right)}$, ${\widehat{\chi }}_s=C_s{\chi }_s^{\left(r\right)}$ with ${\chi }_s={\kappa }_s/\left(\frac{3}{2}n{k}_B\right)$, and define

$$P_m={\displaystyle \frac{\widehat{\nu }}{\widehat{\eta }}},{\mathrm{Pr}}_s={\displaystyle \frac{\widehat{\nu }}{{\widehat{\chi }}_s}}.$$

(8)

The viscous closure is the classical one and needs no further comment. The magnetic closure ${\eta }_m\propto T$ is a genuine modelling assumption and is the weakest link in the chain: Spitzer resistivity varies as $T^{-3/2}$, so (7)₂ is not a fit to a real plasma but a device that makes the resistive operator conservative in the transformed plane. A one-parameter generalisation costs nothing—if ${\eta }_m/\widehat{\eta }={\omega }^{1+s_m}$ then every resistive term below acquires the weight ${\omega }^{s_m}$ and no structural statement changes— and we return to the point in Section 10. All computations use $s_m=0$.

2.2. Four Exact Statements About the Planar Problem

The following results delimit which terms can appear at all. Three of them are inherited unchanged from the constant-density theory; the second is reversed by compressibility, and that reversal is the physical heart of this paper.

(i) The Hall force does not act on the mixture.

Adding (1) and (2), the friction forces cancel by Newton’s third law and the electromagnetic forces sum to $en({\mathbf{u}}_i-{\mathbf{u}}_e)\times \mathbf{B}=\mathbf{J}\times \mathbf{B}$, so that with $m_e\to 0$ and $p=p_i+p_e$,

$$\rho \left({\partial }_t\mathbf{u}+\mathbf{u}·\nabla \mathbf{u}\right)=-\nabla p+\mathbf{J}\times \mathbf{B}+\nabla ·{\mathbf{\pi }}_i.$$

(9)

No term proportional to $\lambda$ survives. The argument nowhere uses $\nabla ·\mathbf{u}=0$, so it carries over verbatim to compressible flow: the Hall force is internal to the electron–ion system and transfers momentum between species, not to the mixture. Adding a separate Hall force to (9) double counts $\mathbf{J}\times \mathbf{B}$. The Hall effect enters the dynamics only through (3).

(ii) The Biermann term survives, and only out of plane.

At constant n the electron-pressure term in (3) is a pure gradient, $\nabla (p_e/ne)$, and its curl vanishes identically. At variable n it does not:

$$\nabla \times \left({\displaystyle \frac{\nabla p_e}{ne}}\right)=\nabla \left({\displaystyle \frac{1}{ne}}\right)\times \nabla p_e=-{\displaystyle \frac{k_B}{en}}\nabla n\times \nabla T_e,$$

(10)

where we used $p_e=n{k}_B{T}_e$, so that $\nabla n\times \nabla p_e=n{k}_B\nabla n\times \nabla T_e$. This is the Biermann battery. Two restrictions apply immediately. In a z-independent geometry ${(\nabla p_e)}_z\equiv 0$, so the electron-pressure term makes no contribution whatever to the z-component of Ohm’s law and hence none to the in-plane flux equation: like the Hall term, and for the same topological reason, the Biermann term is an out-of-plane effect. And (10) requires misaligned density and electron-temperature gradients. We show in Section 5.3 that similarity itself imposes a further restriction, leaving the streamwise pressure gradient as the only surviving driver.

(iii) A strictly planar field contains no Hall physics.

Let $\mathbf{B}=\nabla \times \left(\psi \widehat{\mathbf{z}}\right)$ be strictly planar, $B_z\equiv 0$. Then ${\mu }_0\mathbf{J}=(0,0,-{\nabla }^2\psi )$ is purely out of plane, so ${(\mathbf{J}\times \mathbf{B})}_z=J_x{B}_y-J_y{B}_x=0$ and the Hall term drops out of the flux equation identically; equivalently ${\mathbf{u}}_e·\nabla \psi =\mathbf{u}·\nabla \psi$ because ${\mathbf{J}}_{\perp }=0$. Solenoidality is all that is used, so again the statement is density-independent. Restoring $B_z$ gives ${\mu }_0{\mathbf{J}}_{\perp }=({\partial }_y{B}_z,-{\partial }_x{B}_z,0)$ and

$${\mathbf{u}}_e·\nabla \psi =\mathbf{u}·\nabla \psi +\lambda {\mathbf{B}}_{\perp }·\nabla B_z.$$

(11)

A model that discards $B_z$ has discarded the Hall effect, however many factors of $\lambda$ its equations appear to contain.

(iv) $B_z$ magnetic pressure cancels from streamwise momentum.

With ${\mu }_0{J}_z\simeq -{\partial }_y{B}_x$, ${\mu }_0{J}_x={\partial }_y{B}_z$, ${\mu }_0{J}_y=-{\partial }_x{B}_z$, the transverse Lorentz component is ${(\mathbf{J}\times \mathbf{B})}_y=-{\partial }_y[(B_x^2+B_z^2)/2{\mu }_0]$, so the boundary-layer y-momentum balance gives

$$\Pi \equiv p+{\displaystyle \frac{B_x^2+B_z^2}{2{\mu }_0}}=\Pi (x,t).$$

(12)

Eliminating p from the streamwise component, the terms $\pm B_z{\partial }_x{B}_z/{\mu }_0$ cancel identically and

$$-{\partial }_{x}p+{(\mathbf{J}\times \mathbf{B})}_x=-{\partial }_x\Pi +{\displaystyle \frac{B_x{\partial }_x{B}_x+B_y{\partial }_y{B}_x}{{\mu }_0}}.$$

(13)

The same cancellation occurs in the free stream, so the elimination of $\Pi$ is unaffected by $B_z$. As in the constant-density case, this is the mechanical counterpart of (i): the Hall effect reaches streamwise momentum only indirectly, by modifying $B_x$ and $B_y$.

2.3. Energy Pathways

From (3),

$$\mathbf{J}·\mathbf{E}=-\mathbf{J}·(\mathbf{u}\times \mathbf{B})+{\eta }_r{\left|\mathbf{J}\right|}^2+{\displaystyle \frac{\mathbf{J}·(\mathbf{J}\times \mathbf{B})}{ne}}-{\displaystyle \frac{\mathbf{J}·\nabla p_e}{ne}}.$$

(14)

The Hall contribution vanishes identically because $\mathbf{J}·(\mathbf{J}\times \mathbf{B})=0$: the Hall electric field is perpendicular to the current and does no work, at variable density exactly as at constant density. Any formulation exhibiting an explicit “Hall heating” term has mis-assigned the electron work. The electron-pressure term, by contrast, is no longer a pure surface contribution: since $\nabla ·\mathbf{J}=0$,

$$\nabla ·{\mathbf{u}}_e=\nabla ·\mathbf{u}+{\displaystyle \frac{\mathbf{J}·\nabla n}{n^{2}e}},$$

(15)

so the electron fluid compresses at a different rate from the ion fluid and $p_e$ does compressional work against that difference. Scaling shows the extra rate to be $O\left({\epsilon }_H^2\right)$ relative to $\nabla ·\mathbf{u}$, with ${\epsilon }_H$ defined in (35), so it enters at the same order as the Hall back-reaction on the in-plane field and is retained only there.

The species internal-energy equations are then

$$\begin{array}{cc}\hfill \frac{3}{2}n{k}_B{\displaystyle \frac{D{T}_i}{Dt}}& =-p_i\nabla ·\mathbf{u}+{\partial }_y\left({\kappa }_i{\partial }_y{T}_i\right)+\mu \left[{\left({\partial }_{y}u\right)}^2+{\left({\partial }_y{u}_z\right)}^2\right]+Q_{\Delta },\hfill \end{array}$$

(16)

$$\begin{array}{cc}\hfill \frac{3}{2}n{k}_B{\displaystyle \frac{D_e{T}_e}{Dt}}& =-p_e\nabla ·{\mathbf{u}}_e+{\partial }_y\left({\kappa }_e{\partial }_y{T}_e\right)+{\eta }_r{\left|\mathbf{J}\right|}^2-Q_{\Delta },\hfill \end{array}$$

(17)

with $D_e/Dt={\partial }_t+{\mathbf{u}}_e·\nabla$ and the equipartition exchange $Q_{\Delta }=\frac{3}{2}n{k}_B(T_e-T_i)/{\tau }_{\Delta }$. Viscous heating is assigned to the ions and Ohmic heating to the electrons, as the closure requires; the compressional terms are new relative to the constant-density theory and, as Section 9 shows, are not small.

2.4. Boundary-Layer form and Free-Stream Compatibility

With x streamwise and y wall-normal, and with $\Pi$ eliminated through (12), the system to be reduced is

$$\begin{array}{cc}\hfill & {\partial }_t\rho +{\partial }_x\left(\rho u\right)+{\partial }_y\left(\rho v\right)=0,\hfill \end{array}$$

(18)

$$\begin{array}{cc}\hfill & \rho {\displaystyle \frac{Du}{Dt}}={\rho }_e{\displaystyle \frac{D{U}_e}{Dt}}+{\displaystyle \frac{B_x{\partial }_x{B}_x+B_y{\partial }_y{B}_x-B_{xe}{\partial }_x{B}_{xe}}{{\mu }_0}}+{\partial }_y\left(\mu {\partial }_{y}u\right),\hfill \end{array}$$

(19)

$$\begin{array}{cc}\hfill & {\partial }_t\psi +\mathbf{u}·\nabla \psi +\lambda {\mathbf{B}}_{\perp }·\nabla B_z={\eta }_m{\partial }_y^2\psi +E_0\left(t\right),\hfill \end{array}$$

(20)

$$\begin{array}{cc}\hfill & {\displaystyle \frac{D{B}_z}{Dt}}+B_z\nabla ·\mathbf{u}={\mathbf{B}}_{\perp }·\nabla \left(u_z+\lambda {\partial }_y{B}_x\right)+{\partial }_y\left({\eta }_m{\partial }_y{B}_z\right)+S_B,\hfill \end{array}$$

(21)

$$\begin{array}{cc}\hfill & \rho {\displaystyle \frac{D{u}_z}{Dt}}={\displaystyle \frac{{\mathbf{B}}_{\perp }·\nabla B_z}{{\mu }_0}}+{\partial }_y\left(\mu {\partial }_y{u}_z\right),\hfill \end{array}$$

(22)

together with (16)–(17). Here $E_0\left(t\right)$ is a spatially uniform applied electric field arising from the residual gauge freedom $\psi \to \psi +c\left(t\right)$, and

$$S_B=-{\displaystyle \frac{k_B}{en}}\left({\partial }_{x}n{\partial }_y{T}_e-{\partial }_{y}n{\partial }_x{T}_e\right)$$

(23)

is the Biermann source from (10). The term $B_z\nabla ·\mathbf{u}$ in (21) is compressible in origin; combining it with the material derivative,

$${\displaystyle \frac{D{B}_z}{Dt}}+B_z\nabla ·\mathbf{u}=\rho {\displaystyle \frac{D}{Dt}}\left({\displaystyle \frac{B_z}{\rho }}\right),$$

(24)

which identifies $B_z/\rho$, not $B_z$, as the quantity transported by the flow—a reformulation we shall find is exactly what the reduced equations want.

Outside the layer the plasma is ideal and the flux is frozen in, so $D(\mathbf{B}/\rho )/Dt=(\mathbf{B}/\rho )·\nabla \mathbf{u}$. For a steady external state this integrates to

$$B_{xe}\propto {\rho }_e{U}_e,$$

(25)

the compressible generalisation of the constant-density relation $B_{xe}\propto U_e$. The Alfvén number then obeys $M^2\equiv B_{xe}^2/{\mu }_0{\rho }_e{U}_e^2\propto {\rho }_e$, so M is constant only when the edge density is. This is the first hint of the constraint proved in Section 4.

3. The Unsteady Dorodnitsyn Reduction and Its Weight Structure

3.1. Stewartson’s Lemma

Define the Dorodnitsyn variable and map $(x,y,t)\mapsto (\xi ,\overline{y},\varsigma )$ with $\xi =x$, $\varsigma =t$ and

$$\overline{y}(x,y,t)={\int }_0^y{\displaystyle \frac{\rho (x,y^{\prime },t)}{{\rho }_r}}d{y}^{\prime }.$$

(26)

In unsteady flow $\overline{y}$ depends on t through $\rho$ and the naive stream function fails. The remedy, due to Stewartson [14], is to absorb ${\overline{y}}_{,t}$ into the transformed normal velocity: with

$$V\equiv {\overline{y}}_{,t}{|}_{x,y}+u{\overline{y}}_{,x}{|}_y+{\displaystyle \frac{\rho }{{\rho }_r}}v,$$

(27)

the material derivative and the continuity equation take the incompressible forms

$${\displaystyle \frac{D}{Dt}}={\partial }_{\varsigma }+u{\partial }_{\xi }+V{\partial }_{\overline{y}},{\partial }_{\xi }u+{\partial }_{\overline{y}}V=0,$$

(28)

so that a transformed stream function $\Phi$ exists with $u={\Phi }_{,\overline{y}}$, $V=-{\Phi }_{,\xi }$, even though ${\partial }_t\rho \ne 0$. We take (28) as given; the proof is a two-line chain-rule calculation reproduced in many places and adds nothing here.

3.2. The Magnetic Bracket

The corresponding statement for the magnetic operator does not appear to have been recorded, and it is what makes the two-fluid reduction tractable.

Lemma 1

(magnetic bracket). For any field F, with $B_x={\partial }_y\psi$ and $B_y=-{\partial }_x{\psi |}_y$,

$${\mathbf{B}}_{\perp }·\nabla F={\displaystyle \frac{\rho }{{\rho }_r}}\left({\psi }_{,\overline{y}}{F}_{,\xi }-{\psi }_{,\xi }{F}_{,\overline{y}}\right)\equiv {\displaystyle \frac{1}{\omega }}\{\psi ,F\},$$

(29)

where $\{·,·\}$ is the Poisson bracket in the transformed plane $(\xi ,\overline{y})$.

Proof. 

Since ${\partial }_y=(\rho /{\rho }_r){\partial }_{\overline{y}}={\omega }^{-1}{\partial }_{\overline{y}}$ we have $B_x={\omega }^{-1}{\psi }_{,\overline{y}}$, and $B_y=-({\psi }_{,\xi }+{\overline{y}}_{,x}{\psi }_{,\overline{y}})$. Hence

$$\begin{array}{cc}\hfill {\mathbf{B}}_{\perp }·\nabla F& =B_x\left(F_{,\xi }+{\overline{y}}_{,x}{F}_{,\overline{y}}\right)+B_y{\omega }^{-1}{F}_{,\overline{y}}\hfill \\ \hfill & ={\omega }^{-1}{\psi }_{,\overline{y}}{F}_{,\xi }+{\omega }^{-1}{\psi }_{,\overline{y}}{\overline{y}}_{,x}{F}_{,\overline{y}}-{\omega }^{-1}\left({\psi }_{,\xi }+{\overline{y}}_{,x}{\psi }_{,\overline{y}}\right)F_{,\overline{y}},\hfill \end{array}$$

and the two terms in ${\overline{y}}_{,x}$ cancel.    □

The cancellation of ${\overline{y}}_{,x}$ is the exact magnetic analogue of the cancellation of ${\overline{y}}_{,t}$ in Stewartson’s lemma, and it means that—as with the mass flux—one never needs ${\overline{y}}_{,x}$ explicitly.

3.3. What carries the density weight and what does not

Theorem 1

(weight structure). Under the map (26) with the closures (7):

(a) 

inertia and the viscous term take their incompressible form, ${\rho }^{-1}{\partial }_y\left(\mu {\partial }_{y}u\right)=\widehat{\nu }{u}_{,\overline{y}\overline{y}}$;

(b) 

the pressure-gradient term acquires the weight $\omega =T/T_r$;

(c) 

the in-plane magnetic force per unit mass, the Hall transport term, and the Lorentz driving of $u_z$ areweight-free:

$${\displaystyle \frac{{\mathbf{B}}_{\perp }·\nabla B_x}{{\mu }_0\rho }}={\displaystyle \frac{\{\psi ,B_x\}}{{\mu }_0{\rho }_r}},\lambda {\mathbf{B}}_{\perp }·\nabla ={\lambda }_r\{\psi ,·\};$$

(30)

(d) 

the resistive terms are not: ${\eta }_m{\partial }_y^2\psi =\widehat{\eta }{\partial }_{\overline{y}}\left({\omega }^{-1}{\psi }_{,\overline{y}}\right)$ and ${\partial }_y\left({\eta }_m{\partial }_y{B}_z\right)=\widehat{\eta }{\omega }^{-1}{B}_{z,\overline{y}\overline{y}}$;

(e) 

the whole right-hand side of the $B_z$ equation carries the single weight ${\omega }^{-1}=\rho /{\rho }_r$, consistent with (24).

Proof. (a) is classical: with $\rho \mu =C{\rho }_r{\mu }_r$, ${\partial }_y\left(\mu {\partial }_{y}u\right)={\omega }^{-1}{\partial }_{\overline{y}}\left(C{\mu }_r{u}_{,\overline{y}}\right)$ and division by $\rho ={\rho }_r/\omega$ leaves $\widehat{\nu }{u}_{,\overline{y}\overline{y}}$. (b) follows because dividing the pressure gradient by $\rho$ produces ${\rho }_e/\rho =\omega$. For (c), combine Lemma 1 with ${\rho }^{-1}=\omega /{\rho }_r$: the two weights cancel. The same cancellation occurs for the Hall term because $\lambda ={\lambda }_r\omega$ by (5), so $\lambda {\mathbf{B}}_{\perp }·\nabla ={\lambda }_r\omega ·{\omega }^{-1}\{\psi ,·\}$. (d) is a direct computation using ${\eta }_m=\widehat{\eta }\omega$. (e) follows on dividing (21) by $\rho$ and using (24).    □

Part (c) deserves emphasis. That the magnetic force per unit mass should be Dorodnitsyn-invariant is intelligible: $\mathbf{B}/\rho$ behaves like a material line element, so the ratio of magnetic tension to inertia is blind to the expansion. That the Hall term should be equally invariant is less obvious, and is a coincidence of exponents: the Hall coefficient grows as $\omega$ precisely as fast as the magnetic derivative operator shrinks. The practical consequence is that every Hall term below is literally identical to its constant-density counterpart, and all compressible modification of the two-fluid coupling enters through the resistive weights, through the compressional terms, and through the Biermann source.

It is worth contrasting this with the low-magnetic-Reynolds-number problem, in which an externally applied field exerts a drag $\sigma B_0^2(u-U)/\rho$ that carries the full weight $\omega$. There the Lorentz term is a body force imposed from outside and its acceleration per unit mass is enhanced in the hot, light gas; here the field is frozen to the fluid and shares its expansion, and the enhancement disappears. The two problems behave oppositely for a reason that is entirely kinematic.

Finally, the local Hall length in transformed units follows from (5): a physical length ℓ corresponds to $\ell /\omega$ in $\overline{y}$, so

$${\ell }_H\equiv {\displaystyle \frac{d_i\left(\mathrm{local}\right)}{\delta }}{|}_{\overline{y}}={\displaystyle \frac{{\epsilon }_H}{M}}{\omega }^{-1/2},$$

(31)

which is smaller in the hot near-wall gas than the constant-density estimate.

4. The Similarity Ansatz and an Exactness Dichotomy

4.1. The Two-Parameter Map

Following Sun [18], and now in the Dorodnitsyn plane, introduce

$$\eta ={\displaystyle \frac{\overline{y}}{\delta \left(x\right)}},\tau ={\displaystyle \frac{\widehat{\nu }t}{{\delta }^2\left(x\right)}},$$

(32)

and write the fields as

$$\begin{array}{cc}\hfill & \Phi =U_e\delta f(\eta ,\tau ),\psi =B_{xe}\delta g(\eta ,\tau ),\hfill \\ \hfill & B_z={\epsilon }_H{B}_{xe}h(\eta ,\tau ),u_z={\epsilon }_H{M}^2{U}_{e}w(\eta ,\tau ),\hfill \\ \hfill & T_s=T_r{\theta }_s(\eta ,\tau )(s=i,e),\omega =\frac{1}{2}({\theta }_i+{\theta }_e).\hfill \end{array}$$

(33)

Two derived profiles occur so often that they deserve names:

$$\mathfrak{b}\equiv {\displaystyle \frac{g_{\eta }}{\omega }}={\displaystyle \frac{B_x}{B_{xe}}},\mathfrak{G}\equiv g_{\eta }\mathfrak{b}-g{\mathfrak{b}}_{\eta }=\omega {\mathfrak{b}}^2-g{\mathfrak{b}}_{\eta }.$$

(34)

The distinction between $g_{\eta }$ and $\mathfrak{b}$ is purely compressible and is easy to lose: $g_{\eta }$ measures flux per unit mass coordinate, $\mathfrak{b}$ the physical tangential field. They coincide only where $\omega =1$.

Balancing the Hall source in (21) against advection fixes the amplitude of $B_z$ and defines the Hall similarity parameter,

$${\epsilon }_H\equiv {\displaystyle \frac{{\lambda }_r{B}_{xe}}{U_e\delta }}={\displaystyle \frac{M{d}_{ir}}{\delta }},M={\displaystyle \frac{v_{Ar}}{U_e}},$$

(35)

and balancing the Lorentz force against inertia in (22) gives $u_z\sim {\epsilon }_H{M}^2{U}_e$, as anticipated in (33). Note that the prefactors in (33) are definitions, not approximations: no expansion in ${\epsilon }_H$ has been made.

4.2. The Dichotomy

Autonomy of the reduced system requires every dimensionless group to be independent of x. The groups are the geometric pair

$$\Lambda ={\displaystyle \frac{\delta {\delta }^{\prime }{U}_e}{\widehat{\nu }}},\widehat{\beta }={\displaystyle \frac{{\delta }^2{U}_e^{\prime }}{\widehat{\nu }}},$$

(36)

together with M, $P_m$, ${\mathrm{Pr}}_s$, ${\epsilon }_H$, and the compressibility parameter $\Theta =1+\frac{1}{2}(\gamma -1)M_e^2$ that fixes the relation between static and total enthalpy at the edge.

Theorem 2

(incompatibility of Hall and compressible exactness). Within the power-law family $U_e=C_1{x}^m$ there is no member for which the Hall group ${\epsilon }_H$ and the compressibility parameter Θ are simultaneously independent of x.

Proof. 

Constancy of ${\epsilon }_H=M{d}_{ir}/\delta$ forces $\delta =\mathrm{const}$, because $d_{ir}$ is an absolute length fixed by the reference density alone and cannot be rescaled by any choice of external flow, and M is already required constant. Then $\Lambda =0$ and $\widehat{\beta }={\delta }^2{U}_e^{\prime }/\widehat{\nu }=\mathrm{const}$ requires $U_e^{\prime }=\mathrm{const}$, i.e. $m=1$ after absorbing a shift of origin (the degenerate alternative $m=0$ with $U_e^{\prime }=0$ gives the one-dimensional Rayleigh layer).

Constancy of $\Theta$ requires a constant edge Mach number. For an isentropic outer flow driven by $U_e=C_1{x}^m$ the edge temperature satisfies $T_e+U_e^2/2c_p=\mathrm{const}$, so $M_e$ is constant only if $U_e$ is, i.e. $m=0$. The same conclusion follows independently from (25): $M^2\propto {\rho }_e$ is constant only if the edge density is, which for an isentropic outer flow again requires $m=0$.

The two requirements are $m=1$ and $m=0$ and cannot both hold.    □

The theorem is sharper than its single-fluid ancestors because it involves two absolute constraints pulling in opposite directions, and it is constructive: it tells us there are exactly two things to do, and we do both.

Class B (stagnation-type, $m=1$).

Take $U_e=ax$, $B_{xe}=bx$, $\delta =\sqrt{\widehat{\nu }/a}$, so that $\tau =at$ carries no residual x dependence and ${\epsilon }_H={\lambda }_{r}b/a\delta$ is exactly constant. Exactness in $\Theta$ is recovered in the distinguished limit

$$M_e\to 0\mathrm{with}{\theta }_w=T_w/T_e=O\left(1\right)\mathrm{arbitrary},$$

(37)

in which the external flow is effectively incompressible—so ${\rho }_e$, $T_e$ and p are uniform and (25) gives $B_{xe}\propto U_e$—while the layer carries an order-unity density variation supplied entirely by wall heat transfer. This is precisely the setting of the classical variable-property stagnation-point solutions [13], and it is not a weak limit: at ${\theta }_w=4$ the near-wall density is a quarter of its free-stream value. Dissipation is $O\left(M_e^2\right)$ in this limit and is treated as a regular perturbation in Appendix C.

Class A (flat plate, $m=0$).

Take $U_e=U_{\infty }$, $B_{xe}=B_{\infty }$, $\delta =\sqrt{2\widehat{\nu }x/U_{\infty }}$, whence $\Lambda =1$, $\widehat{\beta }=0$ and $\tau =U_{\infty }t/2x$ is Stewartson’s variable. Here $\Theta$, M and $M_e$ are exactly constant and the compressible thermodynamics, including dissipation at arbitrary Mach number, is exactly similar. The Hall group is not: ${\epsilon }_H\left(x\right)=M{d}_{ir}/\delta \propto x^{-1/2}$, and similarity survives exactly through $O\left({\epsilon }_H\right)$, breaking only at $O\left({\epsilon }_H^2\right)$.

Each family is exact in the sector the other cannot reach. Together they bracket the problem.

5. Class B: The Strongly Heated Stagnation-Type Layer

5.1. Reduction

With $\delta$ constant, $\eta$ and $\tau$ are independent of x, the operator $\mathcal{D}=\eta {\partial }_{\eta }+2\tau {\partial }_{\tau }$ generated by ${\partial }_x$ never acts, and the reduction closes term by term. The kinematics are

$$\begin{array}{cccccc}\hfill u& =ax{f}_{\eta },\hfill & \hfill V& =-a\delta f,\hfill & \hfill v& =-a\delta \omega f,\hfill \\ \hfill B_x& =bx\mathfrak{b},\hfill & \hfill B_y& =-b\delta g,\hfill & \hfill B_z& ={\epsilon }_{H}bxh,\hfill \end{array}$$

(38)

and the dilatation, which will appear repeatedly, is obtained from $\rho T=\mathrm{const}$ as

$${\displaystyle \frac{\nabla ·\mathbf{u}}{a}}=\mathcal{D}iv\equiv {\displaystyle \frac{{\omega }_{\tau }-f{\omega }_{\eta }}{\omega }},$$

(39)

reducing in steady flow to $-f{(ln\omega )}_{\eta }$. The full substitutions are given in Appendix A. The result is the master system

$$\begin{array}{cc}\hfill f_{\eta \tau }& =f_{\eta \eta \eta }+f{f}_{\eta \eta }+\left(\omega -f_{\eta }^2\right)+M^2\left(\mathfrak{G}-\omega \right),\hfill \end{array}$$

(40)

$$\begin{array}{cc}\hfill g_{\tau }& ={\displaystyle \frac{1}{P_m}}{\mathfrak{b}}_{\eta }+f{g}_{\eta }-f_{\eta }g-{\epsilon }_H^2\left(g_{\eta }h-g{h}_{\eta }\right),\hfill \end{array}$$

(41)

$$\begin{array}{cc}\hfill h_{\tau }+f_{\eta }h-f{h}_{\eta }+h\mathcal{D}iv& ={\displaystyle \frac{1}{\omega }}\left[{\displaystyle \frac{1}{P_m}}{h}_{\eta \eta }+M^2\left(g_{\eta }w-g{w}_{\eta }\right)+{\mathfrak{G}}_{\eta }-{\omega }_{\eta }{\mathfrak{b}}^2+{\displaystyle \frac{{\theta }_{e,\eta }}{2M^2}}\right],\hfill \end{array}$$

(42)

$$\begin{array}{cc}\hfill w_{\tau }& =w_{\eta \eta }+f{w}_{\eta }-f_{\eta }w+\left(g_{\eta }h-g{h}_{\eta }\right),\hfill \end{array}$$

(43)

$$\begin{array}{cc}\hfill {\theta }_{i,\tau }& ={\displaystyle \frac{1}{{\mathrm{Pr}}_i}}{\theta }_{i,\eta \eta }+f{\theta }_{i,\eta }-\frac{2}{3}{\theta }_i\mathcal{D}iv+\Gamma ({\theta }_e-{\theta }_i)+O\left(E_c\right),\hfill \end{array}$$

(44)

$$\begin{array}{cc}\hfill {\theta }_{e,\tau }& ={\displaystyle \frac{1}{{\mathrm{Pr}}_e}}{\theta }_{e,\eta \eta }+f{\theta }_{e,\eta }-\frac{2}{3}{\theta }_e\mathcal{D}iv-\Gamma ({\theta }_e-{\theta }_i)+O\left(E_c\right)+O\left({\epsilon }_H^2\right),\hfill \end{array}$$

(45)

with $\Gamma =1/a{\tau }_{\Delta }$ and $E_c=m_i{U}_e^2/k_B{T}_r=2\gamma M_e^2$ the Eckert parameter of the distinguished limit (37). Setting $\omega \equiv 1$ makes $\mathfrak{b}=g_{\eta }$, $\mathcal{D}iv=0$, ${\theta }_{e,\eta }=0$, and (40)–(43) collapse to the constant-density two-fluid system; setting ${\epsilon }_H=0$ and $h=w=0$ leaves the compressible aligned-field MHD layer.

Several features repay attention.

(i) The pressure gradient is the only weighted mechanical term. In (40) the buoyancy-like combination $\omega -f_{\eta }^2$ replaces $1-f_{\eta }^2$, exactly as in single-fluid compressible theory, while the magnetic group $M^2(\mathfrak{G}-\omega )$ contains $\omega$ only through the same channel: $\mathfrak{G}\to 1$ and $\omega \to 1$ at the edge, so the group vanishes there as it must.

(ii) The Hall terms are literally unchanged. The combination $g_{\eta }h-g{h}_{\eta }$ appearing in (41) and (43) is identical to its constant-density form—not merely analogous. This is Theorem 1(c) in action, and it is the reason the compressible problem remains tractable at arbitrary ${\epsilon }_H$.

(iii) The perfect-derivative structure of the Hall source is broken. At constant density the stretching source in (42) is ${\partial }_{\eta }(g_{\eta }^2-g{g}_{\eta \eta })$, the exact $\eta$-derivative of the magnetic group that supplies the tension in the momentum equation. Here, using $g_{\eta }=\omega \mathfrak{b}$,

$${\omega }^{-1}\left(g_{\eta }{\mathfrak{b}}_{\eta }-g{\mathfrak{b}}_{\eta \eta }\right)={\omega }^{-1}\left[{\mathfrak{G}}_{\eta }-{\omega }_{\eta }{\mathfrak{b}}^2\right],$$

(46)

so the structural link with the tension group $\mathfrak{G}$ survives but acquires a defect $-{\omega }_{\eta }{\mathfrak{b}}^2$ proportional to the density gradient. The quadrupole is therefore driven partly by the gradient of the magnetic tension, as before, and partly by the temperature gradient acting on the tangential field alone.

(iv) Compression work is an $O\left(1\right)$ term in the thermal problem. The $-\frac{2}{3}{\theta }_s\mathcal{D}iv$ terms in (44)–(45) are of the same order as conduction, not a correction to it. Summing the two equations with ${\mathrm{Pr}}_i={\mathrm{Pr}}_e$ and using (39) returns ${\omega }_{\tau }-f{\omega }_{\eta }=\frac{3}{5}{\mathrm{Pr}}_i^{-1}{\omega }_{\eta \eta }$, the standard compressible enthalpy equation, which is the check that the compressional bookkeeping is consistent.

5.2. Boundary Conditions

At an impermeable, no-slip wall, and matching to the free stream,

$$\begin{array}{cccccc}& f\left(0\right)=0,\hfill & \hfill & f_{\eta }\left(0\right)=0,\hfill & \hfill & f_{\eta }(\infty )=1,\hfill \\ & g\left(0\right)=0,\hfill & \hfill & \mathfrak{b}(\infty )=1,\hfill & & \hfill \\ & h\left(0\right)=0,\hfill & \hfill & h(\infty )=0,\hfill & & \hfill \\ & w\left(0\right)=0,\hfill & \hfill & w(\infty )=0,\hfill & & \hfill \\ & {\theta }_s\left(0\right)={\theta }_{sw},\hfill & \hfill & {\theta }_s(\infty )=1,\hfill & \hfill & s=i,e.\hfill \end{array}$$

(47)

The system order is $3+2+2+2+2+2=13$, matching the thirteen conditions. The condition $g\left(0\right)=0$ states that the wall is a magnetic flux surface, $B_y\left(0\right)=0$; the condition $\mathfrak{b}(\infty )=1$—rather than $g_{\eta }(\infty )=1$—is the one place where the distinction in (34) matters for the formulation, and using the wrong one displaces the whole magnetic solution. We adopt the insulating wall model, $J_y\left(0\right)=0$, which gives $h\left(0\right)=0$ exactly and requires no subsidiary expansion. The wall tangential field $\mathfrak{b}\left(0\right)$ and the wall current are outputs.

The role of $E_0$ is as at constant density. A uniform $E_0$ enters (41) as an inhomogeneous term $E_0/abx\delta$, which depends on x unless $E_0=0$; working with the uncurled, second-order form of the flux equation imposes this automatically.

5.3. The Biermann Source

We now evaluate (23) inside the similarity family, which is where the interesting restriction appears.

Proposition 1

(similarity annihilates the Biermann term at uniform pressure). If the pressure is uniform, so that $n=p/2k_{B}T$ with T a function of the similarity variable alone, then $\nabla n\times \nabla T_e\equiv 0$ and $S_B\equiv 0$.

Proof. 

At uniform p, n depends on position only through T, and both T and $T_e$ are functions of $\eta$ alone. Hence $\nabla n=n^{\prime }\left(\eta \right)\nabla \eta$ and $\nabla T_e=T_e^{\prime }\left(\eta \right)\nabla \eta$ are parallel and their cross product vanishes.    □

The proposition disposes of Class A immediately: a flat plate has no streamwise pressure gradient, so the Biermann battery is identically inoperative there, however strong the heating. It also shows that the effect is not simply “switched on by compressibility”; what switches it on is the streamwise pressure gradient, which tilts the isopycnals away from the isotherms.

In Class B the pressure is not quite uniform. Eliminating $\Pi$ as in (12) and evaluating at the edge, the magnetic contributions cancel between $\Pi$ and the edge momentum balance, leaving

$${\displaystyle \frac{dp}{dx}}=-{\rho }_e{U}_e{\displaystyle \frac{d{U}_e}{dx}}=-{\rho }_r{a}^{2}x,{\displaystyle \frac{p^{\prime }}{p}}=-{\displaystyle \frac{a^{2}x}{R{T}_r}}.$$

(48)

Writing $n=p\left(x\right)/2k_{B}T\left(\eta \right)$ and using ${\partial }_x{T}_e{|}_y=0$,

$$S_B=-{\displaystyle \frac{k_B}{e}}{\displaystyle \frac{p^{\prime }}{p}}{\partial }_y{T}_e={\displaystyle \frac{m_i{a}^{2}x}{2e}}{\displaystyle \frac{{\theta }_{e,\eta }}{\omega \delta }},$$

(49)

which on division by the reference scale $a{\epsilon }_{H}bx$ gives, with ${\lambda }_r^{-1}={\mu }_0{n}_{r}e$ and $M^2=b^2/{\mu }_0{\rho }_r{a}^2$,

$${\displaystyle \frac{S_B}{a{\epsilon }_{H}bx}}={\displaystyle \frac{m_i{a}^2}{2e{\lambda }_r{b}^2}}{\displaystyle \frac{{\theta }_{e,\eta }}{\omega }}={\displaystyle \frac{1}{2M^2}}{\displaystyle \frac{{\theta }_{e,\eta }}{\omega }},$$

(50)

which is the last term in (42). Three properties of this source are worth stating explicitly.

It scales correctly. $S_B\propto x$ exactly as $B_z\propto x$ does, so the Biermann battery is compatible with Class-B similarity rather than merely tolerated by it.

It does not vanish in the distinguished limit. Although (48) is itself $O\left(M_e^2\right)$, the reference scale against which it is measured is $O\left(M_e^2\right)$ in the same way, and the ratio (50) is free of $M_e$. The Biermann quadrupole is therefore a robust feature of the exactly similar Class-B layer, not a high-Mach-number curiosity.

It is independent of ${\epsilon }_H$ and diverges as $M\to 0$. Because h was normalised by ${\epsilon }_H{B}_{xe}$, a source of size $1/2M^2$ means a physical field $B_z^{\mathrm{B}}\sim B_{xe}{d}_{ir}\left|{\theta }_{e,\eta }\right|/2M\delta$. This is as it should be: the Biermann battery needs no seed field, whereas Hall stretching requires one, so at weak field the battery must win. Comparing (50) with the $O\left(1\right)$ Hall source (46) predicts a crossover at

$$M^2\simeq \frac{1}{2}\left|{\theta }_{e,\eta }\right|,$$

(51)

which Section 9 confirms at $M^2=0.53$ for ${\theta }_w=3$.

For orientation in physical terms, the ratio of the Biermann to the Hall source can be written as ${\beta }_e$, the electron plasma beta, times the relative temperature variation along a field line and a factor $O\left(M_e^2\right)$ measuring the strength of the streamwise pressure gradient. Low-$\beta$ layers make Hall quadrupoles; high-$\beta$ layers make Biermann quadrupoles.

5.4. Displacement Identity and Integral Relations

Writing $f\to \eta -\alpha$ and $g\to \eta -\beta$ as $\eta \to \infty$, the far-field limit of (41) is obtained by noting that $\omega \to 1$, $\mathfrak{b}\to 1$, ${\mathfrak{b}}_{\eta }\to 0$ there, so that

$${\beta }_{\tau }+\beta -\alpha =0,\mathrm{hence}\beta =\alpha$$

(52)

in the steady state. Compressibility does not modify this: the identity lives entirely in the outer region, where the density perturbation has died. The general solution of the far-field flux equation is $A(\eta -\alpha )$ plus a solution decaying as $exp[-P_m{(\eta -\alpha )}^2/2]$, and $\mathfrak{b}(\infty )=1$ selects $A=1$; no additive constant is admissible. Equation (52) is a sharp, falsifiable prediction of the formulation and we use it as a numerical check.

Integrating (40) once in the steady state, with the momentum and displacement thicknesses

$${\Theta }_v={\int }_0^{\infty }{f}_{\eta }(1-f_{\eta })d\eta ,\alpha ={\int }_0^{\infty }(1-f_{\eta })d\eta ,$$

(53)

and their magnetic counterparts formed with $\mathfrak{b}$, one obtains a von Kármán relation of the usual type in which the wall stress is the difference between a hydrodynamic and a magnetic thickness combination weighted by $M^2$, augmented by ${\int }_0^{\infty }(\omega -1)d\eta$—the thermal expansion integral, which is the only new term and which measures the extra momentum flux the layer must accommodate because the near-wall gas is light.

6. Class A: The Compressible Flat Plate

6.1. Reduction and Master System

For Class A, ${\delta }^{\prime }=\delta /2x\ne 0$ and $\mathcal{D}$ does act. Carrying out the substitutions (Appendix B), all terms proportional to $\eta$ cancel identically in every equation—the mechanism by which the Blasius-type layer remains autonomous—and one obtains

$$\begin{array}{c}\hfill f_{\eta \eta \eta }+f{f}_{\eta \eta }-M^{2}g{\mathfrak{b}}_{\eta }=f_{\eta \tau }+2\tau \left(f_{\tau }{f}_{\eta \eta }-f_{\eta }{f}_{\eta \tau }\right)+2\tau M^2\left(g_{\eta }{\mathfrak{b}}_{\tau }-g_{\tau }{\mathfrak{b}}_{\eta }\right),\end{array}$$

(54)

$$\begin{array}{c}\hfill {\displaystyle \frac{1}{P_m}}{\mathfrak{b}}_{\eta }+f{g}_{\eta }-f_{\eta }g+{\epsilon }_H^2{\partial }_{\eta }\left(gh\right)=g_{\tau }-2\tau \left(f_{\eta }{g}_{\tau }-f_{\tau }{g}_{\eta }\right),\end{array}$$

(55)

$$\begin{array}{c}\hfill {\displaystyle \frac{1}{P_m}}{h}_{\eta \eta }+{\partial }_{\eta }\left(\omega fh\right)-M^2{\partial }_{\eta }\left(gw\right)-{\partial }_{\eta }\left(g{\mathfrak{b}}_{\eta }\right)=0\left(\mathrm{steady}\right),\end{array}$$

(56)

$$\begin{array}{c}\hfill w_{\eta \eta }+{\partial }_{\eta }\left(fw\right)-{\partial }_{\eta }\left(gh\right)=0\left(\mathrm{steady}\right),\end{array}$$

(57)

$$\begin{array}{c}\hfill {\displaystyle \frac{1}{\mathrm{Pr}}}{\theta }_{\eta \eta }+f{\theta }_{\eta }+(\gamma -1)M_e^2\left\{f_{\eta \eta }^2+{\displaystyle \frac{M^2}{P_m}}{\mathfrak{b}}_{\eta }^2+{\epsilon }_H^2\left[M^4{w}_{\eta }^2+{\displaystyle \frac{M^2}{P_m}}{h}_{\eta }^2\right]\right\}=0\left(\mathrm{steady}\right),\end{array}$$

(58)

where $\omega =\theta =T/T_e$ and we have taken $T_r=T_e$. Equation (58) is the mixture equation; the species split is obtained by adding the difference equation for ${\theta }_e-{\theta }_i$, in which the ion viscous and electron Ohmic sources appear with opposite signs and equipartition supplies the damping.

Three points. First, (54) reduces at $M=0$ to $f^{\prime \prime \prime }+f{f}^{\prime \prime }=0$ irrespective of $\theta$: the classical Mach independence of the transformed flat-plate momentum problem. With $M\ne 0$ it does not, because the magnetic term $M^{2}g{\mathfrak{b}}_{\eta }=M^{2}g{(g_{\eta }/\omega )}_{\eta }$ samples the temperature profile directly. An aligned frozen-in field therefore destroys the Mach independence of the transformed skin friction. The same conclusion is reached in the low-$Rm$ applied-field problem, but by the opposite route—there through the weight on the Lorentz drag, here through the dilution of a frozen-in field by expansion—and the mechanisms should not be conflated.

Second, the out-of-plane pair (56)–(57) is in divergence form, and remains so under compressibility with the single modification $fh\mapsto \omega fh$. Integrating once from $\eta$ to ∞ and using decay,

$${\displaystyle \frac{1}{P_m}}{h}_{\eta }+\omega fh=g{\mathfrak{b}}_{\eta }+M^{2}gw,w_{\eta }+fw=gh,$$

(59)

with $h\left(0\right)=w\left(0\right)=0$: the Hall subsystem of the Blasius-type layer is a first-order initial-value problem, a considerable simplification with no single-fluid analogue and one that compressibility does not spoil. Evaluating (59) at the wall, where $f\left(0\right)=g\left(0\right)=0$, gives $h_{\eta }\left(0\right)=0$ identically for every decaying solution: the perfectly conducting wall condition is redundant, as at constant density, and the insulating condition $h\left(0\right)=0$ is the non-degenerate choice.

Third, since ${\epsilon }_H$ enters (55)–(56) only at $O\left({\epsilon }_H^2\right)$, the Hall-generated fields of the growing layer are exactly self-similar through first order,

$$B_z={\epsilon }_H\left(x\right)B_{\infty }h\left(\eta \right),u_z={\epsilon }_H\left(x\right)M^2{U}_{\infty }w\left(\eta \right),{\epsilon }_H\propto x^{-1/2},$$

(60)

with universal profiles. Hall physics in a growing layer is a leading-edge phenomenon, significant where ${\epsilon }_H\gtrsim 1$, i.e. for

$${\displaystyle \frac{x_H}{d_{ir}}}={\displaystyle \frac{M^2{\mathrm{Re}}_{d_i}}{2C}},{\mathrm{Re}}_{d_i}={\displaystyle \frac{U_{\infty }{d}_{ir}}{{\nu }_r}},$$

(61)

in which compressibility enters only through the Chapman–Rubesin constant. In physical coordinates the picture is less tidy: the layer is thicker by $\int \omega d\eta$ but the gas is lighter and $d_i$ correspondingly larger, and the two effects partially compensate, with the residual given by (31).

6.2. The change of type is purely kinematic

Collecting the $\tau$-derivatives in (54)–(55),

$$\begin{array}{cc}\hfill \left(1-2\tau f_{\eta }\right)f_{\eta \tau }+2\tau f_{\tau }{f}_{\eta \eta }+2\tau M^2\left(g_{\eta }{\mathfrak{b}}_{\tau }-g_{\tau }{\mathfrak{b}}_{\eta }\right)& =\dots ,\hfill \\ \hfill \left(1-2\tau f_{\eta }\right)g_{\tau }+2\tau f_{\tau }{g}_{\eta }& =\dots ,\hfill \end{array}$$

(62)

and differentiating the second with respect to $\eta$ shows that $g_{\eta \tau }$ carries the same factor.

Theorem 3

(kinematic threshold). The Class-A system is well posed as an initial-value problem in τ so long as $2\tau f_{\eta }<1$ throughout the layer. Since $f_{\eta }\to 1$ at the edge, the first loss of parabolicity occurs there at

$${\tau }_{\mathrm{crit}}=\frac{1}{2},$$

(63)

independently of M, $P_m$, ${\epsilon }_H$, the edge Mach number, the Prandtl numbers and the wall thermal condition.

Proof. 

The $\tau$-principal symbol of the system (62) in the unknowns $(f_{\eta },g)$ is $(1-2\tau f_{\eta })$ times the identity, plus terms of lower order in the $\tau$-derivative. Neither M nor $\omega$, $\Theta$, $P_m$, ${\epsilon }_H$ nor the boundary data appears in that coefficient, and the supremum of $f_{\eta }$ over the layer is unity.    □

This is the analytical content of Stewartson’s obstruction, now shown to be untouched by either the field or the two-fluid physics: at $\tau =1/2$, that is at $U_{\infty }t=x$, the direction in which information propagates in $\tau$ reverses, and no forward march can succeed beyond it. The remedy is the Williams–Rhyne chart [19], $s=1-e^{-\tau }$, $N=\eta /2\sqrt{s}$, $f=2\sqrt{s}F(N,s)$, in which both ends are characteristic and the whole history from impulsive start to steady state is obtained as one boundary-value problem; at $s\to 0$ one recovers $F^{\prime \prime \prime }+2N{F}^{\prime \prime }=0$, i.e. $F^{\prime }=\mathrm{erf}N$, and at $s\to 1$ the steady system (54)–(58). Since neither field nor compressibility appears at leading order as $\tau \to 0$—the diffusion terms are $O\left({\tau }^{-1}\right)$ while all others are $O\left(1\right)$—the impulsive start is universal: the quadrupole is born from a Rayleigh state that knows nothing of M, $M_e$ or ${\epsilon }_H$. We formulate the chart here but do not compute in it; the global unsteady solution is left for future work, and we are careful below to present only steady results.

7. Two Structural Consequences

7.1. The Alfvénic Degeneracy Is Not Displaced

At constant density the aligned-field layer degenerates at $M^2=1$, $P_m=1$: the equipartitioned field-aligned state $g=f$ forces $f_{\eta \eta }=0$ and no boundary layer exists [22]. One might expect variable density to move this point, since the alignment condition $u=v_A$ becomes $f_{\eta }=M\mathfrak{b}\sqrt{\omega }$ and is no longer compatible with $g=f$.

Theorem 4.

The Alfvénic critical point of the compressible two-fluid layer is $M^2=1$, independently of the wall temperature ratio.

Proof 

(Argument). Set $f=\eta -\alpha +F$, $g=\eta -\alpha +G$ with $F,G$ small in the outer region, where $\omega \to 1$ and hence $\mathfrak{b}\to g_{\eta }$. Linearising (40)–(41) there gives

$$\begin{array}{cc}\hfill F^{\prime \prime \prime }+s{F}^{\prime \prime }-2F^{\prime }+M^2\left(2G^{\prime }-s{G}^{\prime \prime }\right)& =0,\hfill \\ \hfill G^{\prime \prime }+s{G}^{\prime }-G-s{F}^{\prime }& =0,s=\eta -\alpha ,\hfill \end{array}$$

(64)

which is exactly the constant-density far-field system, whose determinant vanishes at $M^2=1$. The transition is decided in the outer region, where the density perturbation has died; the interior, however strongly heated, cannot move it.    □

We label this an argument rather than a proof because it establishes only that the far-field character changes at $M^2=1$; that this coincides with the vanishing of the wall stress is a global statement which we verify numerically in Section 9 over ${\theta }_w\in [0.5,4]$. The result is worth stating precisely because the naive expectation— that a hot, light near-wall gas should be easier for magnetic tension to arrest—is wrong.

7.2. The hierarchy in ${\epsilon }_H$

Equations (40)–(43) generate the same three-tier hierarchy as at constant density, now with compressible coefficients. At $O\left(1\right)$, $(f,g,{\theta }_s)$ obey the compressible aligned-field MHD problem with ${\epsilon }_H=0$. At $O\left({\epsilon }_H\right)$, $(h,w)$ obey the linear, ${\epsilon }_H$-independent problem (42)–(43) evaluated on that base state, forced by (46) and by the Biermann source; the quadrupole profile is universal and its amplitude linear in ${\epsilon }_H$. At $O\left({\epsilon }_H^2\right)$ the back-reaction on the in-plane fields appears. This is the precise sense in which the Hall effect is a second-order correction to the in-plane fields but a first-order effect in its own right, and Section 9 verifies it over a sixteenfold range in ${\epsilon }_H$ at ${\theta }_w=3$.

The whistler branch that mediates the coupling, ${\omega }^2=k^2{v}_A^2(1+k^2{d}_i^2)$, survives locally with the local $v_A$ and $d_i$; expressed through (31) its similarity wavenumber is ${\epsilon }_H{\omega }^{-1/2}/M$, so the dispersive scale contracts in the hot gas. The $\omega \propto k^2$ behaviour at small scales makes the semi-discrete Hall coupling a stiff fourth-order operator, and any unsteady integration of (40)–(45) must be implicit.

8. Numerical Method

The steady Class-B problem (40)–(45) is a thirteenth-order two-point boundary-value problem. We carry the state $(f,f_{\eta },f_{\eta \eta },g,\mathfrak{b},h,h_{\eta },w,w_{\eta },{\theta }_i,{\theta }_{i,\eta },{\theta }_e,{\theta }_{e,\eta })$, using $g_{\eta }=\omega \mathfrak{b}$ so that $\mathfrak{b}$ rather than $g_{\eta }$ is a primitive variable— which enforces the correct edge condition automatically and avoids differentiating $\omega$ unnecessarily. One structural difficulty must be handled: (42) requires ${\mathfrak{G}}_{\eta }$, which contains ${\mathfrak{b}}_{\eta \eta }$; differentiating (41) expresses ${\mathfrak{b}}_{\eta \eta }$ in terms of $h_{\eta \eta }$, which (42) supplies. The coupling is linear and is eliminated in closed form,

$$\left({\displaystyle \frac{1}{P_m}}+P_m{\epsilon }_H^2{g}^2\right)h_{\eta \eta }=K,$$

(65)

with K given in Appendix D; the bracket is strictly positive, so the elimination never fails. Solution is by collocation [50] with adaptive mesh refinement (scipy.integrate.solve_bvp [51]) to a tolerance of ${10}^{-9}$–${10}^{-10}$ on $\eta \in [0,{\eta }_{max}]$ with ${\eta }_{max}=20$ and 1201–2001 initial nodes, continued in ${\theta }_w$, then $M^2$, then ${\epsilon }_H$. Continuation is not optional: without it the solver readily jumps to spurious branches with $\omega <0$ at moderate $M^2$, and all results reported here were checked for pointwise positivity of $\omega$.

The Class-A problem (54)–(58) is ninth order in the steady state and is solved by the same method with ${\eta }_{max}=14$; the analogous elimination is needed for ${\mathfrak{b}}_{\eta }$ against $h_{\eta }$.

Two remarks on accuracy. The wall quantities $f_{\eta \eta }\left(0\right)$ and $\mathfrak{b}\left(0\right)$ are converged to eight significant figures and are insensitive to ${\eta }_{max}$ beyond 16. The displacement constants are not: in the outer region the advective terms are individually $O\left(\eta \right)$ and nearly cancel, so $\alpha$ and $\beta$ inherit a truncation error that falls with ${\eta }_{max}$ rather than with mesh refinement. The identity (52) is satisfied to $5.4\times {10}^{-3}$ at ${\eta }_{max}=12$, $1.5\times {10}^{-3}$ at 16 and $8.4\times {10}^{-5}$ at 20, with mesh refinement from 1201 to 3201 nodes changing nothing at any of these; we quote $\alpha$ to five figures accordingly and regard the convergence of $\beta -\alpha$ with domain size as the meaningful check.

9. Results

Throughout, $\gamma =5/3$.

9.1. Validation

Table 1 collects the checks against independently known limits. Setting ${\theta }_{iw}={\theta }_{ew}=1$ makes ${\theta }_s\equiv 1$ an exact solution of (44)–(45), hence $\omega \equiv 1$, and the Class-B system must collapse onto the constant-density two-fluid problem; at $M=0$ it must further reduce to Hiemenz flow. We obtain $f_{\eta \eta }\left(0\right)=1.232588$ and $\alpha =0.647900$, the tabulated values. The Class-A system at $M={\epsilon }_H=0$ must reduce to Blasius in the normalisation $\delta =\sqrt{2\widehat{\nu }x/U_{\infty }}$, giving $\sqrt{2}\times 0.332057=0.469600$ and a displacement constant $1.72079/\sqrt{2}=1.216782$; we obtain $0.469600$ and $1.216781$.

The thermal side is checked against the classical compressible flat plate. At $M=0$ the transformed wall stress must be independent of $M_e$, and we find $0.469600$ at $M_e^2=0,4,16$ to all digits shown. The adiabatic recovery factor at $\mathrm{Pr}=0.72$ comes out $0.84771$ against the $\sqrt{\mathrm{Pr}}$ approximation $0.848528$; the small residual is not numerical error but the known departure of the exact solution from $\sqrt{\mathrm{Pr}}$, and its sign and magnitude agree with the standard tabulations [49].

9.2. The Heated Stagnation-Type Layer

Table 2 and Fig. Figure 1 give the base state as the wall temperature ratio is varied at fixed $M^2=0.3$, $P_m=1$, ${\epsilon }_H=0.5$.

Three effects are visible. First, heating raises the wall stress substantially, by $89\%$ between ${\theta }_w=1$ and 4. Because ${\mu }_w\propto {\omega }_w$ while ${\left({\partial }_{y}u\right)}_w\propto {\omega }_w^{-1}$, the weights cancel and $f_{\eta \eta }\left(0\right)$ is the physical skin friction in units of ${\rho }_{r}C{\nu }_r{U}_e/\delta$; the increase is real and is driven by the enhanced pressure-gradient term $\omega -f_{\eta }^2$ of (40).

Second, the wall tangential field collapses, $\mathfrak{b}\left(0\right)$ falling from $0.550$ to $0.304$. This is flux expulsion by expansion: the frozen-in field shares the dilation of the gas, so a hot layer carries the same flux over a greater thickness at lower field strength. It is the magnetic counterpart of the familiar compressible thickening, and it is the reason the magnetic tension weakens even though M is held fixed.

Third, and least expected, the quadrupole amplitude is non-monotonic, falling to a minimum near ${\theta }_w=1.5$ and then rising by a factor of nearly four by ${\theta }_w=4$, while its peak jumps from $\eta \simeq 2.8$ to $\eta \simeq 0.7$. The two branches of this behaviour are the two sources: Hall stretching, which peaks in the outer part of the magnetic layer and is weakened by flux expulsion, and the Biermann battery, which is proportional to ${\theta }_{e,\eta }$ and is therefore largest at the wall. We isolate them next.

9.3. Biermann Against Hall

Figure 2 and Table 3 compare the full solution with one in which the Biermann source of (50) is deleted and everything else retained.

The predicted structure is reproduced exactly. The Hall-only amplitude varies by only $20\%$ across a sixteenfold range of $M^2$, as it must, since ${\epsilon }_H$ and not M sets it. The full amplitude tracks $M^{-2}$ at small M, crossing the Hall value at $M^2=0.53$—to be compared with the estimate (51), which gives $M^2\simeq \frac{1}{2}\left|{\theta }_{e,\eta }\right|$ and, with the computed wall gradient at ${\theta }_w=3$, $M^2\simeq 0.5$. At $M^2=0.05$ the Biermann quadrupole exceeds the Hall quadrupole by a factor of $12.7$. Its spatial signature differs too: because it is driven by the electron temperature gradient rather than by the curvature of the magnetic profile, it peaks much closer to the wall, as Fig. Figure 2(a) shows.

This is the central quantitative result of the paper. In a compressible layer with a streamwise pressure gradient and appreciable wall heat transfer, the out-of-plane field is not in general the Hall quadrupole of reconnection theory, and interpreting a measured $B_z$ as a Hall signature without first estimating the Biermann contribution will overstate $d_i/\delta$ by whatever factor Table 3 supplies.

9.4. Alfvénic Approach and the ${\epsilon }_H$ Hierarchy

Figure 3(a) shows $f_{\eta \eta }\left(0\right)$ against $M^2$ for ${\theta }_w=0.5,1,2,4$. Every curve falls to zero at $M^2=1$, and continuation fails there in every case; the wall stress at $M^2=0.95$ is $0.3320$, $0.2626$, $0.2507$ and $0.3025$ for ${\theta }_w=0.5$, 1, 2, 4 respectively, a spread of $30\%$ against a hundredfold spread in the driving. Theorem 4 is confirmed: the critical Alfvén number is not moved by heating, only the approach to it.

Figure 3(b) tests the hierarchy. Holding ${\theta }_w=3$, $M^2=0.3$, $P_m=1$ and reducing ${\epsilon }_H$ from $0.4$ to $0.025$, the departures of the wall quantities from their ${\epsilon }_H=0$ values follow a clean power law of slope 2, with prefactors

$${\displaystyle \frac{|\Delta f_{\eta \eta }\left(0\right)|}{{\epsilon }_H^2}}\to 1.035\times {10}^{-2},{\displaystyle \frac{\left|\Delta \mathfrak{b}\right(0\left)\right|}{{\epsilon }_H^2}}\to 2.893\times {10}^{-2},$$

(66)

constant to better than $1\%$ over the whole range, while the normalised quadrupole amplitude changes by only $0.2\%$. Both halves of the hierarchy—an $O\left({\epsilon }_H^2\right)$ back-reaction and an ${\epsilon }_H$-independent quadrupole shape—therefore survive compressibility intact.

9.5. Dual-Temperature Structure and Expansion Cooling

Figure 4 and Table 4 show the thermal problem with Braginskii-like Prandtl numbers, ${\mathrm{Pr}}_i=0.225$ and ${\mathrm{Pr}}_e=5$, at ${\theta }_{iw}={\theta }_{ew}=3$.

At $\Gamma =0$ the two species are decoupled and their profiles differ by more than a factor of two at $\eta =1$, since the ion thermal layer is thick (${\mathrm{Pr}}_i\ll 1$) and the electron layer thin. As $\Gamma$ grows the profiles are driven together and the single-temperature limit is approached, the ion wall flux rising by $41\%$ and the electron flux falling by $61\%$ while the total changes by $26\%$.

The feature we did not anticipate is the electron undershoot: at small $\Gamma$ the electron temperature falls below its free-stream value in the outer part of the layer, reaching ${\theta }_e=0.944$ at $\eta =1$, despite a wall three times hotter than the free stream and no cooling anywhere in the problem. The mechanism is the compression term of (45). With a hot wall ${\omega }_{\eta }<0$ and $f>0$, so (39) gives $\nabla ·\mathbf{u}>0$: fluid carried from the cool outer region into the hot near-wall region expands and does work against its own pressure. A species whose thermal diffusivity is too small to conduct the deficit away cools accordingly. The effect is absent from any constant-density formulation, where $\nabla ·\mathbf{u}\equiv 0$, and it is selective—it acts on the poorly diffusing species only—so it is a genuine dual-temperature signature of compressibility rather than a mixture property. Note also that the quadrupole amplitude falls by $28\%$ across the $\Gamma$ range, through the Biermann source’s dependence on ${\theta }_{e,\eta }$ alone: equipartition, which is a purely thermal process, thereby controls a magnetic observable.

9.6. The Compressible Flat Plate

Table 5 and Fig. Figure 5 give the Class-A results.

The loss of Mach independence is the headline. With $M=0$ the transformed wall stress is $0.469600$ at every $M_e$; with $M^2=0.1$, $0.3$, $0.5$, $0.7$ it falls between $M_e^2=0$ and $M_e^2=16$ by $5.5\%$, $16.5\%$, $27.5\%$ and $38.4\%$ respectively. The mechanism is transparent in Fig. Figure 5(b): the wall tangential field drops from $0.416$ to $0.090$, a factor of $4.6$, as dissipative heating expands the near-wall gas and dilutes the frozen-in flux. Weaker near-wall field means weaker magnetic tension, so the field brakes the layer less effectively and—since the aligned field reduces skin friction—the stress falls further below its field-free value the hotter the layer becomes.

The quadrupole is comparatively robust, varying by only $16\%$ over the same range and peaking near $M_e^2=9$: the reduction in $\mathfrak{b}$ that weakens the Hall source is partly offset by the outward migration of the magnetic layer, and the Biermann battery is absent by Proposition 1. Finally, the wall stress is almost independent of ${\epsilon }_H$, changing by $0.5\%$ over $0\le {\epsilon }_H\le 1.2$ at $M_e^2=4$, because in Class A the Hall feedback reaches the momentum equation only through the doubly indirect chain $h\to g\to g{\mathfrak{b}}_{\eta }$; the wall field, by contrast, responds strongly, rising from $0.220$ to $0.286$.

10. Applicability, and Where the Description Fails

The results rest on assumptions that should be separated by severity.

Transport closures. The viscous law $\rho \mu =\mathrm{const}$ is the classical Chapman–Rubesin approximation and carries the error already accepted throughout compressible boundary-layer theory. The magnetic and thermal analogues in (7) are weaker. Spitzer resistivity varies as $T^{-3/2}$ and the Braginskii conductivities are strongly anisotropic in a magnetised plasma, only partially captured by using perpendicular coefficients. A power law ${\eta }_m\propto {\omega }^{1+s_m}$ is absorbed at no cost—every resistive term acquires the weight ${\omega }^{s_m}$—and since $s_m<0$ for a real plasma the hot near-wall gas would be more conducting than our baseline, sharpening the current layer and strengthening the Hall source. The quantitative values at ${\theta }_w\gtrsim 3$ should be read as indicative of trend.

The distinguished limit of Class B. Exactness there requires $M_e\to 0$, so the dissipation terms are formally $O\left(M_e^2\right)$ and enter as the regular perturbation of Appendix C. This is the classical framework for variable-property stagnation-point layers and is not restrictive for the density ratios we explore, but it does mean that Class-B results should not be quoted for a hypersonic stagnation region without carrying the $O\left(E_c\right)$ problem explicitly.

Two temperatures, one density. We have allowed $T_i\ne T_e$ but enforced quasineutrality and a single mass density, so charge separation, ambipolar fields and the thermal force are excluded, and $\gamma =5/3$ is imposed on the mixture. Electron inertia is neglected, so the electron skin depth and the associated sub-$d_i$ structure are absent; its restoration would remove the unbounded whistler frequency and, incidentally, the stiffness noted above.

Regime of validity. The tension identified in the constant-density theory persists: with ${\epsilon }_H=M{d}_{ir}/\delta$ and ${\delta }^2\sim \nu L/U$, ${\epsilon }_H^2=M^2{\mathrm{Re}}_L{(d_{ir}/L)}^2$, so collisional plasmas large compared with $d_i$ are generically of high Reynolds number and the two requirements conflict. Compressibility does not relieve this, but it does add a route not available before: because $\lambda \propto T$, a strongly heated near-wall region has a locally enhanced Hall coefficient, so the effective two-fluid parameter is larger at the wall than the edge-based estimate suggests, by ${\omega }_w$ in $\lambda$ and ${\omega }_w^{1/2}$ in $d_i$.

Laminar, and stable by assumption. At the wall temperature ratios explored here a real high-speed layer would be interacting with transition, and nothing in this paper addresses whether the states exhibited are selected. We regard a linear stability analysis of these base states—in which the Mack modes of the compressible layer and the whistler branch of the two-fluid coupling compete—as the natural next step, and deliberately have not attempted it here.

11. Conclusions

We have derived the similarity structure of the unsteady compressible two-fluid plasma boundary layer and solved the resulting steady states.

The organising structural result is that the Dorodnitsyn map treats the two-fluid physics more kindly than one has any right to expect. Because the Hall coefficient is inversely proportional to density while the in-plane magnetic derivative carries the reciprocal weight, the Hall transport operator—and with it the magnetic force per unit mass and the Lorentz driving of the out-of-plane velocity—is invariant under the transformation, and takes exactly its constant-density form. Compressibility reaches the two-fluid coupling only through the resistive weights, through compression work, and through a term that constant-density theory does not possess at all.

That term is the Biermann battery. We showed that it can never enter the in-plane flux equation, that similarity at uniform pressure annihilates it because all thermodynamic gradients become parallel, and that it therefore survives only as a streamwise-pressure-gradient source, reducing to ${\theta }_{e,\eta }/2M^2\omega$. It is absent from a flat plate and unavoidable at a stagnation point, it needs no seed field and consequently dominates at low Alfvén number, and it generates a quadrupole that peaks much nearer the wall than the Hall quadrupole does. At $M^2=0.05$ and ${\theta }_w=3$ it is the larger of the two by a factor of $12.7$; the two are equal at $M^2=0.53$, in agreement with the analytical estimate $M^2\simeq \frac{1}{2}\left|{\theta }_{e,\eta }\right|$. Any interpretation of a measured out-of-plane field as a Hall signature in a layer with heat transfer and a pressure gradient must dispose of this contribution first.

Exact similarity of the compressible two-fluid problem is impossible: the Hall group requires a layer of constant thickness and the compressibility parameter requires a uniform external stream. The two resolutions are complementary and we developed both—a stagnation-type layer, exact to all orders in ${\epsilon }_H$ and carrying order-unity density variation supplied by wall heat transfer, and a compressible flat plate, exact in its thermodynamics at arbitrary Mach number and self-similar through $O\left({\epsilon }_H\right)$, whose Hall subsystem remains a first-order pair under the single substitution $f\mapsto \omega f$.

The unsteady system changes type at $\tau =1/2$ for purely kinematic reasons, both equations carrying the factor $1-2\tau f_{\eta }$, so Stewartson’s obstruction is untouched by the field, the Mach number, the Prandtl numbers or the wall condition; and since neither field nor compressibility is felt at the impulsive start, the quadrupole is born from a universal Rayleigh state.

Numerically: the Alfvénic degeneracy remains at $M^2=1$ for wall temperature ratios from $0.5$ to 4, because the transition is decided in the outer region where the density perturbation has died; a hot wall raises the stagnation-point skin friction by $89\%$ while expelling tangential flux and reducing the wall field by $45\%$; an aligned frozen-in field destroys the Mach independence of the transformed flat-plate skin friction, by $38\%$ at $M^2=0.7$ and $M_e^2=16$, through the dilution of the near-wall field by dissipative heating; and expansion cooling drives the electron temperature below its free-stream value when equipartition is weak, an effect that no constant-density formulation can contain.

Natural extensions are finite electron inertia; an ionisation model coupling $\sigma$ and $\lambda$ self-consistently to the thermal field; the global unsteady solution in the Williams–Rhyne chart, which the change-of-type theorem shows to be necessary rather than merely convenient; and the stability of these states, where the interaction of the whistler branch with the compressible instability modes seems likely to be delicate.