An exponent dimensions cannot give

Two of this collection’s essays are about self-similar solutions and they have very different epistemic status, which is not visible from either of them alone.

A blast wave has a radius going as $R \propto (Et^2/\rho)^{1/5}$, and the exponent is arithmetic: energy, density and time are the only dimensional quantities in the problem, there is exactly one way to combine them into a length, and the two-fifths follows. The constant depends on γ; the exponent does not, and cannot, because γ is a pure number and no exponent obtained by counting dimensions can contain one.

A shock converging on a point has a radius going as $R \propto (-t)^{0.717}$. That number is not arithmetic. It depends on γ. And an exponent that depends on a dimensionless parameter is not a dimensional result at all.

Barenblatt called these the first and second kinds, and the distinction is one of the few in this subject that changes what can be known rather than how accurately.

Why the counting fails #

The counting fails for a reason that is easy to state and easy to miss.

Sedov’s exponent comes from a conserved quantity. The energy released by the explosion is fixed, the blast wave carries all of it, and E is available as a dimensional input. That is what makes the dimensional argument work: there are exactly three inputs, three dimensions, no dimensionless group, and therefore one answer.

A shock imploding on an axis has no such quantity. Energy is being drawn in from behind as the shock converges — the flow behind it does work on it, the shock strengthens without limit, and the energy inside the converging front is not conserved and is not even bounded. The quantity that would have fixed the exponent does not exist, so dimensional analysis returns a similarity form with an undetermined exponent in it:

$$R = A(-t)^\alpha,$$

with both $A$ and $\alpha$ unknown. Guderley’s 1942 achievement was to determine α — by requiring the similarity solution to pass smoothly through a sonic point, which is a solvability condition on a nonlinear ordinary differential equation and has nothing to do with dimensions.

A shorter route to the same number #

Guderley’s eigenvalue problem is a file of its own. What is computed here instead is Whitham’s geometrical shock dynamics, which reaches the same exponent in three lines of algebra and a quadrature, and agrees with it to four figures.

The Chester–Chisnell–Whitham relation connects a shock’s Mach number to the area of the ray tube it is travelling down:

$$\frac{\mathrm dA}{A} = -\frac{\lambda(M)\,M\,\mathrm dM}{M^2 - 1},\qquad \lambda(M) = \left(1 + \frac{2(1-\mu^2)}{(\gamma+1)\mu}\right)\left(1 + 2\mu + \frac1{M^2}\right),$$

with $\mu^2 = \frac{(\gamma-1)M^2+2}{2\gamma M^2 - (\gamma-1)}$. Its content is that a shock entering a converging channel strengthens by an amount that depends only on the area ratio, which is an approximation — it neglects the disturbances that overtake the shock from behind — and a startlingly good one.

For a strong shock λ tends to a constant. With $\mu_\infty = \sqrt{(\gamma-1)/2\gamma}$ the limit is closed-form, and at γ = 1.4 it is $\lambda_\infty = 5.0743$ — checked here against λ evaluated at Mach 10⁷ to a part in 10⁸.

The exponent, in one line #

A shock converging on an axis travels down a ray tube whose area goes as $r$; on a point, as $r^2$. So $M \propto r^{-j/\lambda_\infty}$ with $j = 1$ or 2, and since $\mathrm dR/\mathrm dt \propto M$,

$$R^{1 + j/\lambda_\infty} \propto t \qquad\Longrightarrow\qquad \alpha = \frac{\lambda_\infty}{\lambda_\infty + j}.$$

At γ = 1.4 that gives 0.83537 cylindrical and 0.71729 spherical, against Guderley’s 0.834 and 0.717174. Four figures, from an argument with no eigenvalue anywhere in it.

That agreement is the well-known thing about geometrical shock dynamics and it is not the point of the figure. The point is the slope. The exponent runs from 0.790 at γ = 1.1 to 0.667 at γ = 2, a span of 0.123, and Sedov’s is a horizontal line at 0.4.

An exponent that moves with the gas cannot have come from counting dimensions, because the gas enters only through a dimensionless number. That is the cleanest available test of which kind a similarity solution is, and it needs no knowledge of the solution at all.

And γ = 2 is exact #

One value is worth pausing on. At γ = 2 the algebra gives $\lambda_\infty = 4$ exactly, so $\alpha = 4/6 = 2/3$ for the spherical case and $4/5$ for the cylindrical — round numbers, from a formula that produces nothing round anywhere else.

γ = 2 is not a real gas. It is the shallow-water analogy’s effective ratio, and shallow-water hydraulic jumps behave like γ = 2 gas shocks, which is why a converging bore in a circular tank implodes with a two-thirds power law. It is also the one case where a reader can check the arithmetic by hand.

What the exponent does not come with #

Here is the second half of the distinction, and it is the part that decides what a similarity solution is worth in practice.

A first-kind solution predicts the constant as well as the exponent, because the conserved quantity supplies it: Sedov’s $R = \xi_0(Et^2/\rho)^{1/5}$ has $\xi_0$ computable from γ alone, which is why the Trinity photographs give an energy.

A second-kind solution predicts only the exponent.

Integrating the relation from Mach 2, Mach 6 and Mach 20 gives three trajectories that spend most of their journey not on the power law at all, and then approach it.

The slopes agree to $1.0\times10^{-4}$: one exponent, from every initial condition, which is what an asymptotic result should do.

The amplitudes span 10.9. Nothing in the local problem knows which of them applies; $A$ is fixed by matching back to whatever launched the shock, which is a numerical calculation of the whole history and not a similarity argument at all.

So the practical content is this. A second-kind similarity solution tells how a quantity scales and not how large it is, and the constant has to come from somewhere else. That is a much weaker result than a first-kind solution, and it is still the only result available, because the alternative is a full numerical solution of a problem with a singularity at its centre.

Intermediate asymptotics, and where the solution applies #

There is a further limitation that both kinds share and that is easy to lose.

Neither solution holds at early times, when the initial conditions have not been forgotten, nor at late times, when something else takes over — the blast wave becomes an acoustic wave, the converging shock reflects off the origin. What each describes is an intermediate range, and the figures above make that visible: the trajectory started at Mach 2 is not on the power law until the radius has fallen by two decades.

That is why Barenblatt’s term for this whole class is intermediate asymptotics, and it is the same caution dynamic similarity needs: a scaling law is a statement about a window, and quoting it outside its window is not an approximation but a different claim.

What is happening physically at the focus #

It is worth translating the exponent into what the gas is doing, because $\alpha < 1$ is a strange statement about a shock.

$R \propto (-t)^\alpha$ with α = 0.717 means $\mathrm dR/\mathrm dt \propto (-t)^{-0.283}$: the shock accelerates without limit as it approaches the centre, and arrives at the origin with infinite speed and infinite strength in a finite time. That is the whole reason converging shocks are of interest — an implosion concentrates energy into a shrinking volume, and the concentration is a power law with no upper bound in the idealised problem.

The Mach number goes as $r^{-j/\lambda_\infty}$, which for a spherical implosion at γ = 1.4 is $r^{-0.394}$. So a shock launched at Mach 5 at the outer radius arrives at a hundredth of that radius at Mach $5 \times 100^{0.394} \approx 30$, and at a ten-thousandth at Mach 190. Those are the numbers the three trajectories in the figures are showing, and they are the reason the strong-shock limit is a good assumption over most of the journey and a poor one at the start.

The exponent’s dependence on γ has a physical reading too. A gas with a smaller γ is more compressible, the shock compresses it further, the ray tubes converge faster, and the implosion is more violent — which is the α = 0.79 at γ = 1.1. A gas with γ = 2 is stiff, the amplification is weaker, and α = 2/3.

And after the focus the shock reflects. The reflected shock runs outwards through gas that the incoming one has already processed, and its own similarity exponent is a different eigenvalue of the same problem. Nothing here computes it.

What geometrical shock dynamics leaves out #

The agreement above is remarkable and the approximation behind it is crude, which is worth being clear about.

It ignores the flow behind the shock entirely. The A–M relation is derived by applying the characteristic compatibility condition along the forward-running characteristic at the shock and assuming nothing catches up from behind. For a converging shock that is nearly true, because the flow behind is expanding away from the front, and it is why the method works so well here. For a shock decaying into a region it has already disturbed it is much worse.

It has no entropy field. A converging shock leaves behind a non-uniform entropy distribution, which is exactly what Crocco’s theorem turns into vorticity, and none of that is represented.

It is not stable. A converging cylindrical shock is unstable to azimuthal perturbations, so a real implosion does not stay circular: it develops facets and eventually a polygonal front. The exponent above is for the axisymmetric solution, and the instability is the practical reason inertial-confinement implosions are hard.

And the strong-shock limit is a limit. Everything after λ∞ assumes $M \gg 1$, which is excellent near the focus and poor at launch — which is precisely the region the three trajectories above are not on the power law in.

What stops the infinity #

The window has been bounded above — the shock is not on the power law until it has forgotten its launch — and it is bounded below as well, by three mechanisms the model contains none of.

The gas stops having a constant γ. Long before the focus, the temperature behind the shock is high enough to excite vibration, then to dissociate, then to ionise. Each of those lowers the effective ratio of specific heats, and this essay’s own curve says what that does: α rises as γ falls, so the implosion becomes more violent than the constant-γ solution predicts. The exponent is not merely inaccurate near the centre; it is drifting, which is the same difficulty a drifting exponent causes anywhere it appears.

The gas starts radiating. A hot enough gas loses energy as light, and radiation drains exactly the energy the focusing is concentrating. That pushes the other way.

And the continuum fails. Near enough to the origin the region being compressed is comparable with a mean free path, and there is no shock to speak of — the fluid stops being one, and a jump condition derived for a continuum has nothing to jump between.

So the singularity is an artefact of a model that was never asked about the last few decades, and the useful reading of $\alpha$ is as the amplification law over the range where the gas is still a perfect gas and still a fluid.

Where this leaves the similarity method #

The distinction between the two kinds is not a technicality about exponents. It is a statement about what dimensional analysis can do.

Dimensional analysis works when the number of dimensional inputs equals the number of dimensions, and it works because a conserved quantity supplies one of the inputs. Where the conservation fails — because the region of interest exchanges the quantity with its surroundings — the input is missing, the counting is short by one, and the exponent has to be found by solving something.

The same structure occurs outside gas dynamics. The classic examples are the pressure of a fluid in a collapsing pore, the propagation of a crack, and the intermediate asymptotics of a filtration problem — in each case a quantity that would have been conserved is not, and in each case the exponent is an eigenvalue.

And the test is always the same. Vary a dimensionless parameter of the problem and see whether the exponent moves. If it does, no amount of counting will produce it.

What Guderley’s route actually does #

The short route above deserves a word about what it is short for, because the word “eigenvalue” is doing real work and the geometrical method hides it completely.

Guderley’s problem is the full similarity reduction: assume $R \propto (-t)^\alpha$, write the density, velocity and pressure as functions of $r/R(t)$ alone, and the partial differential equations become ordinary ones in that single variable. The boundary conditions are the shock jump at one end and regularity at the axis at the other, and for a general α the integral curve joining them passes through a singular point of the ordinary system — where the coefficient of the highest derivative vanishes — and becomes unbounded.

Only one value of α lets the curve through the singular point smoothly, and that value is the exponent. It is an eigenvalue in exactly the sense a buckling load is: the parameter for which a homogeneous problem has a bounded solution at all.

That is why no amount of dimensional bookkeeping produces it. The number is not a ratio of the problem’s inputs; it is a property of the solution’s analytic structure, and it depends on γ because γ is in the coefficients whose vanishing defines the singular point. Whitham’s relation reaches the same number by a different argument — a local statement at the front, integrated — and the agreement to four figures is the evidence that the crude approximation has kept whatever the eigenvalue is about.

Two more second-kind exponents in this collection #

It is worth pointing at the other places the same structure has already appeared, because none of them was labelled.

The Kolmogorov spectrum is first kind. The −5/3 comes from a dissipation rate and a wavenumber and nothing else, and it is the same for every fluid — which is why the inertial range has an exponent that can be written down before anything is solved.

The turbulent boundary layer’s log law is neither, and that is why it is hard. There is no dimensional group that produces a logarithm; what produces it is an overlap argument between two regions with different scalings, which is the matched-asymptotics machinery of the next essay rather than a similarity solution. That is the honest reason von Kármán’s constant has to be measured.

And the near-critical behaviour at a bifurcation is second kind. The amplitude of a convection roll just above threshold goes as $(\mathrm{Ra} - \mathrm{Ra}_c)^{1/2}$, and the one-half is a first-kind result from the normal form; the width of the region over which that holds is not, and depends on every dimensionless parameter of the problem.

The pattern is that a first-kind exponent survives every change of fluid and a second-kind one does not, and it is worth asking which is on offer before quoting either. The blast wave’s 2/5 will be 2/5 in argon and in xenon. The implosion’s 0.717 is a number about air.

Where the work came from #

Guderley’s paper is from 1942 and was published in German in a wartime journal; his exponent stood for decades as one of the harder numbers in gas dynamics to reproduce. Chisnell and Chester derived the area–Mach relation independently in 1955 and 1954, and Whitham gave it its general form and its name in 1958. Sedov and Taylor published the blast solution independently in 1946 and 1950, and von Neumann had it in 1941.

Barenblatt’s classification into two kinds is from the 1970s, and it is what made all of the above one subject rather than a collection of hard problems. The observation this essay turns on — that an exponent depending on γ cannot be dimensional — is his, and it is the sort of statement that seems obvious once written down and was not. The same reclassification tidied a great deal of porous-media theory, where several exponents that had been quoted as dimensional results turned out to be eigenvalues of nonlinear diffusion problems.

What this leaves #

Two kinds of similarity solution, one of which is arithmetic and one of which is an eigenvalue, with a test that separates them in one line. And a second-kind solution’s amplitude, which no similarity argument supplies.

The next essay is about joining two limits when neither holds everywhere, and about what the joining costs: one formula for both ends.