fluid dynamics - Non-linear waves and shock formation

In the cases of non-linear acoustics, why is shock formation unlikely when the dispersion is strong when compared to the non-linearity of the wave?

Answer

The nonlinear term or steepening term, $\left( \mathbf{V} \cdot \nabla \right) \mathbf{V}$, determines the rate of steepening of a wave. This can be balanced/offset by loss terms like dispersion (e.g., $\propto \ \beta \ \partial_{x}^{3} v$), diffusion, viscosity (e.g., $\propto \ \nu \ \partial_{x}^{2} v$), resistivity, friction (e.g., $\propto \ \zeta \ v$), etc. If the loss term dominates over the nonlinear term, then the wave cannot steepen as there is too much damping. If the loss term balances the nonlinear term and is irreversible, a shock wave can be initiated as the steepening wave approaches a gradient catastrophe (i.e., point where wave breaking occurs).

You can think of dispersive effects as acting to "spread out" the spatial scales in the wave while the nonlinear term acts to "focus" the spatial scales. In fully nonlinear systems, the loss terms are often dependent on the gradient in the spatial scale of the wave front. This means that the terms are often small/negligible until the spatial scales of the wave front become small, thus the gradients large.

Excerpt
- from: Sagdeev, [1966] in Rev. Plasma Phys. Vol. 4, pg. 21 (of 69):

Dissipative effects (viscosity, thermal conductivity, etc.) introduce irreversibility and increase the order of the derivatives by an odd number (for example, viscosity implies the addition of a term containing a second derivative in the Euler equation in gas dynamics). Dispersion effects, on the other hand, do not affect reversibility and increase the order of the derivatives in the equations by an even number.

Interestingly, dispersion is not an irreversible term because it increases the order of the derivatives in the equations by an even number. The other loss terms I mentioned increase the order of the derivatives in the equations by an odd number, thus introducing irreversibility. Therefore, dispersion can regulate a shock once formed but it cannot initiate a shock.

A link to Sagdeev's article can be found here. Though the link does not contain the PDF file, unfortunately.

An example is a dispersive sound wave that has a dispersion relation approximated by: $$ \omega\left( k \right) \simeq k \ C_{s} - \beta \ k^{3} $$ where $C_{s}$ is the phase speed of the mode (here the sound speed), $\beta$ is the dispersive parameter (often assumed to be small), $k$ is the wavenumber, and $\omega$ is the wave angular frequency. In general, a sound wave would steepen in the absence of losses because $C_{s}$ is proportional to the wave amplitude.

For instance, in the animation below one can see that the initially symmetric soliton pulse begins to steepen (i.e., the gradient scale length decreases asymmetrically) as it propagates. This is because the peak of the pulse propagates faster than the lower amplitude parts of the pulse. Thus, the higher amplitude parts of the wave can out run the lower amplitude parts, which results in a sort of pile-up in the direction of propagation. If no losses were present, the higher amplitude(faster) parts of the pulse could overrun the lower amplitude(slower) parts, resulting in a multi-valued solution. This is the gradient catastrophe or wave breaking, to which I eluded above.

In a collisionless plasma, the dispersive term can actually regulate the steepening of a shock front. This occurs because the shock wave is a nonlinearly steepened dispersive fast/magnetosonic wave (which happens to be a compressive version of a whistler mode wave under the right conditions). The dispersive nature of the magnetosonic-whistler wave is such that smaller(larger) wavelengths(frequencies) propagate faster than the converse. So as the mode steepens, the smaller gradient scale length of the steepened edge causes the edge to couple to shorter and shorter wavelengths. The end result can be a wave train of magnetosonic-whistler waves upstream of the shock ramp. If those waves carry energy/momentum away from the shock front, then they will act as a loss mechanism that can limit the steepening of the shock ramp.

In the case of one of the forms of the Korteweg-de Vries equation, we can write: $$ \partial_{t} v + \left( C_{s} + v \right) \partial_{x} v + \beta \ \partial_{x}^{3} v = 0 $$ where $\partial_{j}$ = $\partial/\partial \ j$ (i.e., I am not using covariant vs. contravariant forms here, just being lazy), $\partial_{x}^{3}$ = $\partial_{xxx}$ = $\partial^{3}/\partial \ x^{3}$ or the third partial derivative with respect to $x$, $C_{s}$ is the phase speed of the mode, and $\beta$ is a constant affecting the rate of the dispersive loss. If $\lvert \beta \ \partial_{x}^{3} v \rvert > \lvert \left( C_{s} + v \right) \partial_{x} v \rvert$ here, then the dispersive losses can limit (or even prevent) the steepening of the wave.

Note that the Korteweg-de Vries-Burgers equation includes an extra loss term $\propto -\nu \ \partial_{x}^{2} v$, where $\nu$ is a kinematic viscosity.

Here is an example of nonlinear wave steepening that I created using Mathematica. This was in response to a question I posted on the Mathematica Stack Exchange site here. The pulse is meant to represent a nonlinear magnetosonic wave using the method described by Shukla et al., [2004].

You can see that the initial soliton-like solution starts to distort and the gradient scale length of the right-hand side decreases. Since Mathematica is too "smart" for its own good sometimes (at least for those who are not Mathematica whisperers), the code reaches a "stiff solution" before the wave breaks or shocks, depending on the how one treats the loss terms.

• Coroniti, F.V. "Dissipation discontinuities in hydromagnetic shock waves," J. Plasma Phys. 4, 265, doi:10.1017/S0022377800004992, (1970).


• Krasnoselskikh, V.V., B. Lembège, P. Savoini, and V.V. Lobzin "Nonstationarity of strong collisionless quasiperpendicular shocks: Theory and full particle numerical simulations," Phys. Plasmas 9, 1192-1209, doi:10.1063/1.1457465, (2002).


• Shukla, P.K., B. Eliasson, M. Marklund, and R. Bingham "Nonlinear model for magnetosonic shocklets in plasmas," Phys. Plasmas 11(5), pp. 2311-2313, doi:10.1063/1.1690297, (2004).


• Tidman, D.A., and T.G. Northrop "Emission of plasma waves by the Earth’s bow shock," J. Geophys. Res. 73, 1543–1553, doi:10.1029/JA073i005p01543, (1968).