Emergence and interacting hierarchies in shock physics

Abstract

It is argued that explanations of shock waves display explanatory emergence in two different ways. Firstly, the use of discontinuities to model jumps in flow variables is an example of “physics avoidance”. This is where microphysical details can be ignored in an abstract model thus allowing us access to modal information which cannot be attained in principle in any other way. Secondly, Whitham’s interleaving criterion for continuous shock structure is an example of the way different characteristic scales interact in shock dynamics. To fully explain the shock structure one must take account of these different scales, and by doing so explanations of shock structure have irreducible aspects. Lastly, the implications of this explanatory irreducibility are examined in the context of explanatory indispensability arguments used in realism debates elsewhere in the philosophy of science. It is concluded that explanatory emergence on its own only supports an epistemic form of emergence. Yet this epistemic emergence is fully objective.

Appendices

Appendix 1: Example MHD shock

As a concrete example consider an ideal magnetohydrodynamic (MHD) gas with a magnetic field B. We can ignore displacement currents and assume perfect conductivity, hence (in natural units):

$$ \mathbf{E}=-\mathbf{v}\times \mathbf{B},\mathbf{j}=\nabla \times \mathbf{B} $$

The energy density is given by:

$$ e=\frac{P}{\gamma -1}+\frac{1}{2}\rho {v}_x^2+\frac{{\mathbf{B}}^2}{2} $$

The gas pressure is $P=\frac{a^2\rho }{\gamma }$ (a is the sound speed and γ is the polytropic index). This gives a set of conserved variables and fluxes:

$$ \mathbf{u}=\left(\begin{array}{c}\hfill \rho \hfill \\ {}\hfill \rho {v}_x\hfill \\ {}\hfill \rho {v}_y\hfill \\ {}\hfill \rho {v}_z\hfill \\ {}\hfill e\hfill \\ {}\hfill {B}_y\hfill \\ {}\hfill {B}_z\hfill \end{array}\right)\kern1em \mathbf{f}=\left(\begin{array}{c}\hfill \rho {v}_x\hfill \\ {}\hfill \rho {v}_x^2+P+\frac{{\mathrm{B}}^2}{2}-{B}_x^2\hfill \\ {}\hfill \rho {v}_x{v}_y-{B}_x{B}_y\hfill \\ {}\hfill \rho {v}_x{v}_z-{B}_x{B}_z\hfill \\ {}\hfill \left(e+P+\frac{{\mathbf{B}}^2}{2}\right){v}_x-{B}_x\mathbf{v}\cdot \mathbf{B}\hfill \\ {}\hfill {v}_x{B}_y-{v}_y{B}_x\hfill \\ {}\hfill {v}_x{B}_z-{v}_z{B}_x\hfill \end{array}\right) $$

An MHD gas can support three distinct waves (the eigenvalues of the Jacobian): slow (c_s), fast (c_f) and intermediate (c_i).

$$ \begin{array}{l}{c}_{f,s}=\frac{1}{\sqrt{2}}{\left[{v}_A^2+{a}^2\pm \sqrt{v_A^4+{a}^4-2{v}_A^2{a}^2 \cos \left(2\theta \right)}\right]}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\hfill \\ {}\kern5em {c}_i={v}_A \cos \left(\theta \right)\hfill \end{array} $$

Where θ is the angle between the direction of propagation and the upstream magnetic field, and ${v}_A=\frac{\left|\mathbf{B}\right|}{\sqrt{\rho }}$ is the Alfven speed. From the jump conditions it is possible to show that:

$$ \left[\left(\frac{v_x^2}{c_i^2}-1\right){B}_z\right]=0 $$

Where B _z is the transverse component of the magnetic field and v _x is the flow velocity in the shock frame parallel to the direction of propagation.

Defining the flow velocity as (1): c _f  < v _x , (2): c _i  < v _x  < c _f , (3): c _S  < v _x  < c _i , (4): v _x  < c _s then six types of MHD shock are possible: fast (1 → 2), slow (3 → 4), and intermediate (1 → 3, 2 → 3, 1 → 4, 2 → 4).

From the jump conditions alone we can say what happens to B _z for these types of shock. For fast shocks B _z increases in magnitude but does not change sign. For slow shocks B _z decreases in magnitude but does not change sign. For intermediate shocks B _z changes sign and may increase or decrease in magnitude.

Appendix 2: Example flood waves and dissipative MHD shock

We can illustrate this we a simple example of flood waves. Imagine flood water running down a rectangular channel of constant inclination α. Call h the water depth, v the mean velocity and g gravitational acceleration. Then let g’ = g cos α and our initial values of h and v be h ₀ and v ₀ respectively.

We have conserved variables, fluxes and sources given by:

$$ \mathbf{u}=\left(\begin{array}{c}\hfill h\hfill \\ {}\hfill hv\hfill \end{array}\right),\mathbf{f}=\left(\begin{array}{c}\hfill hv\hfill \\ {}\hfill h{v}^2+\frac{1}{2}g^{\prime }{h}^2\hfill \end{array}\right),\mathbf{S}=\left(\begin{array}{c}\hfill 0\hfill \\ {}\hfill g^{\prime }h \tan \alpha -{v}^2g^{\prime}\frac{h_0}{v_0^2} \tan \alpha \hfill \end{array}\right) $$

This means we have frozen and equilibrium wavespeeds given by:

$$ \begin{array}{c}\hfill {a}_1=v+\sqrt{g^{\prime }h},{a}_2=v-\sqrt{g^{\prime }h},\hfill \\ {}\hfill b=\frac{3v}{2}\hfill \end{array} $$

For the uniform flow h = h ₀, v = v ₀ so the condition for stable uniform flow is:

$$ {v}_0-\sqrt{g^{\prime }{h}_0}<\frac{3{v}_0}{2}<{v}_0+\sqrt{g^{\prime }{h}_0} $$

From the Whitham criterion we can say that a shock in the equilibrium wavespeeds of speed Vs will not contain a sub-shock if:

$$ {v}^{(2)}-\sqrt{g^{\prime }{h}^{(2)}}<{V}_S<{v}^{(1)}+\sqrt{g^{\prime }{h}^{(1)}} $$

A simple example like flood waves only shows the interaction of two different scales. We can see how Whitham’s interleaving stability criterion can reflect a more complex interaction of multiple scales by considering the MHD system of section 3.

Let us now replace our simple expression for the electric field with a more complicated one (a realistic expression for a molecular cloud of different charged species if we neglect grain inertia):

$$ \mathbf{E}=-\mathbf{v}\times \mathbf{B}+{v}_{||}\left(\mathbf{j}\cdot \mathbf{B}\right)\widehat{\mathbf{B}}-{v}_H\left(\mathbf{j}\times \mathbf{B}\right)-{v}_P\left(\mathbf{j}\times \mathbf{B}\right)\times \widehat{\mathbf{B}} $$

Where ${v}_{\left|\right|}=\frac{1}{\sigma_{\left|\right|}},{v}_H=\frac{\sigma_H}{\sigma_H^2+{\sigma}_P^2},{v}_P=\frac{\sigma_P}{\left({\sigma}_H^2+{\sigma}_P^2\right)}$. The components of the conductivity tensor are; the conductivity parallel to the magnetic field:

$$ {\sigma}_{||}=\frac{e}{B}{\displaystyle \sum_j{n}_j{Z}_j{\beta}_j} $$

The Hall conductivity:

$$ {\sigma}_H=\frac{e}{B}{\displaystyle \sum_j\frac{n_j{Z}_j}{1+{\beta}_j}} $$

The Pederson conductivity:

$$ {\sigma}_P=\frac{e}{B}{\displaystyle \sum_j\frac{n_j{Z}_j{\beta}_j}{1+{\beta}_j}} $$

Here β _j is the Hall parameter (the product of gyrofrequency and the timescale for collisions with neutrals), Z _j is the charge of species j, n _j is the number density of species j and e and B are the electric charge and magnetic field magnitude respectively.

The system is now of the dissipative form:

$$ \frac{\partial \mathbf{u}}{\partial t}+\frac{\partial \mathrm{f}}{\partial x}=\frac{\partial }{\partial x}\mathbf{D}\frac{\partial \mathbf{u}}{\partial x} $$

Or linearizing:

$$ \frac{\partial \mathbf{u}}{\partial t}+\mathbf{A}\frac{\partial \mathbf{u}}{\partial x}=\mathbf{D}\frac{\partial^2\mathbf{u}}{\partial^2x} $$

If again we assume solutions of the form u = u ₀ e ^{i(ωt − kx)} we get a dispersion relation:

$$ \left|\frac{\omega }{k}\mathbf{I}-\mathbf{A}-ik\mathbf{D}\right|=0 $$

Which produces a polynomial for the form:

$$ {Q}_0\left(\frac{\omega }{k}\right)-ik\left[{v}_{||}{Q}_1\left(\frac{w}{k}\right)+{v}_p{Q}_2\left(\frac{\omega }{k}\right)\right]-{k}^2\left({v}_{||}{v}_p+{v}_p^2+{v}_H^2\right){Q}_3\left(\frac{\omega }{k}\right) $$

$$ \begin{array}{l}{Q}_0\left(\frac{\omega }{k}\right)={\displaystyle {\prod}_{i=1}^7\left(\frac{\omega }{k}-{\lambda}_{0,i}\right),{Q}_{1,2}\left(\frac{\omega }{k}\right)={\displaystyle {\prod}_{i=1}^6\left(\frac{\omega }{k}-{\lambda}_{1,2,i}\right)},}\hfill \\ {}{Q}_3\left(\frac{\omega }{k}\right)={\displaystyle \prod_{i=1}^5\left(\frac{\omega }{k}-{\lambda}_{3,i}\right)}\hfill \end{array} $$

Where λ represents characteristic wavespeeds in the limit of the system to the respective coefficients, e.g., λ ₀ are the wavespeeds of the ideal non-dissipative system.

$$ \begin{array}{l}{\lambda}_{0,i}={v}_{x,i},{v}_x\pm {c}_{f,s},{v}_x\pm {c}_i\cdot {\lambda}_{1,i}={v}_x,{v}_x,{v}_x\pm {c}_{f,s}\cdot {\lambda}_{2,i}={v}_x,{v}_x,{v}_x\pm g\pm \hfill \\ {}{\lambda}_{3,i}={v}_x,{v}_x,{v}_x,{v}_x\pm a\hfill \end{array} $$

Where:

$$ {g}_{\pm}^2=\frac{1}{2}\left[{a}^2+\frac{2{\mathbf{B}}^2{B}_x^2}{\rho \left({\mathbf{B}}^2+{B}_x^2\right)}\pm {\left[{\left({a}^2+\frac{2{\mathbf{B}}^2{B}_x^2}{\rho \left({\mathbf{B}}^2+{B}_x^2\right)}\right)}^2-\frac{4{a}^2{B}_x^2}{\rho}\right]}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\right] $$

Again we can apply the Hermite-Biehler theorem and say that the uniform flow will be stable if the roots of the imaginary part interleave with the roots of the real part, i.e., Q _1,2 interleaves with Q ₀ and Q ₃ interleaves with Q _1,2. Similarly we can apply the interleaving condition to determine if we have a sub-shock. Note now that we have three scales interacting with each other to produce the final shock structure.

Fig. 4

Schematic of a dissipative MHD shock. The figure on the left shows a smooth shock structure; on the right we have a sub-shock due to the interaction of higher-order waves. (Adapted from Pexton, M. Multi-fluid shocks, unpublished manuscript)