Scaling laws for dynamical plasma phenomena

Similarity arguments and dimensional analyses in the studies of gas and fluid flows have been used for about 150 years, starting with the works by Stokes, Reynolds, Sommerfeld, and Richardson, and discussed in classical monographs by Bridgman¹ and Sedov.² These monographs provide an excellent introduction to dimensional analysis and scaling theory and contain a number of specific examples.

The use of these techniques in plasma physics is relatively new. Among the first studies in this area are papers by Lacina,³ Kadomtsev,⁴ and Connor and Taylor,⁵ where dimensional analysis was used to find confinement scalings in fusion devices. In particular, Connor and Taylor have found dependences of the confinement time on the plasma parameters for a variety of plasma physics models. In more recent history, a renewed interest in scaling issues has emerged from the needs of high-energy-density laboratory astrophysics.⁶ In this arena, one needed to find ways of replicating phenomena occurring on the scale of light-years (as, e.g., in the problem of supernova (SN) ejecta plowing into an ambient medium 1000 years after SN explosion) to the scales of a millimeter (for an imitation experiment with intense lasers). Initial focus was on the magneto-hydrodynamic (MHD) similarities,^7,8 since MHD was (and is) thought to be adequate for describing large-scale astrophysical flows. Later, similarities for collisionless processes, in particular, those related to the formation and evolution of collisionless shocks, were addressed in Refs. 9 and 10.

Dynamical similarities can also be used in the studies of fusion plasmas created in pulsed plasma systems. As an example one can mention Ref. 11, addressing the scalability of liner-imploded plasmas.¹² The motivation of scaling analyses here is in that the corresponding experiments on large-scale facilities are expensive and difficult to diagnose. So, a possibility of a scaled-down, less expensive, and, possibly, easier to diagnose experiment is attractive.

Before going any further, we illustrate some specifics of “dynamical plasma systems” using as an example the study of perturbation growth at an accelerating/decelerating interface of two fluids subject to the Rayleigh-Taylor instability. One starts with a low-amplitude waviness of a sharp interface and ends up with a gradually thickening turbulent layer made of a finely grained mix of materials that were initially situated at the opposite sides of the interface, with a thickness of the mixed layer greatly exceeding the initial wavelength of the perturbations. This process is common in a number of astrophysical and laboratory systems, and is a subject of continuing theoretical and experimental studies, see review.¹³ 

The question that one can ask is can one extrapolate from studying the temporal evolution of one such system between dramatically different initial (sharp boundary) and final (fine mix) states to another system of this kind. This is, apparently, an initial-value problem and one must, therefore, consider conditions under which these two systems evolve along the same path (up to the scaling transformation). This is what makes scaling for the dynamical systems more challenging than scaling between two stationary systems, like a classic example of the Reynolds scaling between a flow past two bodies of different size, or scaling for a confinement time in a steady-state fusion device with respect to its size, magnetic field, and heating power. We will return to some commonalities between the dynamic and static scalings later in this paper.

A scaling analysis is usually started from identifying dimensionless parameters in equations and initial conditions. This is usually the simplest and most straightforward part of the problem. Sometimes it is considered as the only step needed to establish the scalability. In reality, this is just a first step. The further steps include:

1. Assessment of the temporal range during which the system obeys this similarity. The most complex issue here is related to the evaluation of time for possible development of small spatial scales, which can be much smaller than the initial scale. These small scales may introduce new effects, which were absent in the initial setting. In particular, viscous and resistive effects in MHD may come into play, while they were unimportant initially (Sec. II C). Or, in the initially collisionless turbulence, inverse cascading may lead, after some time, to the formation of large scales, with the size exceeding the collisional mean-free-path (Sec. V C). Other time constraints may appear due to the inevitable presence of the boundaries in the laboratory experiment, whereas the astrophysical system would not have such boundaries.

2. Identification of the initial states for which the number of dimensionless parameters characterizing the system can be reduced or even become zero. In the latter (and highly desirable) case, there are no constraints on the initial input parameters of the scaled system except geometrical similarity. This happens, e.g., in the scaling of a strong shock propagating through a non-uniform medium (Sec. II D).

3. In the systems whose macroscopic evolution is significantly affected by microturbulence which cannot be directly controlled (e.g., arising from thermal fluctuations), one has to evaluate characteristic times for its development leading to the onset of the macroscopic effect in question (e.g., formation of a collisionless shock). In this case, even if the microturbulence itself falls under particular scalability transformation, the onset of its macroscopic effects may take a longer time than the experiment can provide.

4. The scaling transformation allows usually for a very broad range of parameters of the scaled system vs. the initial one (e.g., parameters of the scaled laboratory experiment vs. those of some astrophysical phenomenon). The parameters may be different by orders of magnitude. Then, it may happen that some new effects (unaccounted for in the proposed scaling) show up in the simulation experiment, in particular, the equation of state (EOS) may change significantly compared to the assumed one. To avoid that problem, the materials in the experiment should be chosen carefully. If, for example, there is a reason to believe that an astrophysical matter can be characterized by the power-law EOS, the materials with approximately the same EOS power-law exponents should be used in the laboratory model. Choosing materials with appropriate equations of state has been an important step in designing successful laboratory experiments (e.g., those summarized in a review paper⁸).

We present more substantive discussion of these issues in Secs. II and III, using as an example an ideal-MHD scaling (Sec. II).

Mentioned above are general issues that have to be addressed for any scaled experiment. There are also specific issues arising in the course of designing a particular experiment. Those include selection of materials, spatial and temporal scales, type of the driver, suitability of the experimental configuration for the use of desired diagnostics, and others. Discussing this very important part of developing a good scaled experiment goes beyond the scope of this paper. We will, however, present later a few examples of successful laboratory experiments.

Now, we come back to dimensional analysis, needed for the identification of mutually independent dimensionless scaling parameters. It can be performed in different ways. The most common one (e.g., Ref. 2) is based on performing a simple transformation of variables and unknown functions, x→ax, t→bt, $ρ$→c$ρ$⁠, etc. One then substitutes these new variables into the original equations and identifies conditions that leave the transformed equations identical to the original ones. This approach has been used, in particular, in the problem of confinement time scaling in fusion devices in Ref. 5 and in the studies of astrophysics-relevant similarities in Ref. 14.

In this paper, we follow a slightly different approach, where the scaling is established by a direct inspection of equations and initial conditions, as was done, in particular, in Refs. 10 and 15. We find this approach more transparent for the studies of dynamical similarities although it does not differ conceptually from the aforementioned one.

It is worth noting that the aforementioned similarities are quite general in that they cover the entirety of motions allowed by the dynamical equations. In particular, if self-similar solutions are allowed in a complete set of equations, these solutions will remain in all scaled systems. Usually, some additional symmetry is required for self-similarity to exist (e.g., Ref. 16).

This paper is largely based on the studies which included the author's participation although there exist many other studies in this broad area of science; a number of them are referenced throughout this paper. We also refer the reader to several reviews, Refs. 13 and 17–19, where further references can be found.

By the nature of a paper based on the author's presentation at the APS DPP Meeting, the content closely follows the material presented at the meeting. Section II contains a derivation of scaled equations for ideal MHD. The Euler similarity is discussed, alongside with reduced, parameter-less similarities (e.g., “a strong drive similarity”^7,15). Section III contains discussion of extensions needed for inclusion of dissipative processes, radiation, and non-polytropic equations of state. Section IV is concerned with the similarities for plasma confinement in imploding liners. In Sec. V, we touch upon collisionless similarities and their applications to collisionless shocks and particle acceleration; self-similar solutions in the problem of a cloud of suprathermal electrons expanding in a dense background plasma are presented. In all cases, we address not only the trivial problem of reducing equations to a non-dimensional form but also some (or all) of the issues 1–4 mentioned above. Section VI contains a brief discussion of the relations between experiments, similarity analyses, and numerical simulations. Finally, Sec. VII summarizes the results presented in this review.

One of the most widely used dynamical similarities is a similarity for the phenomena describable by ideal magnetohydrodynamics (MHD). It is usually called the “Euler similarity.”^7,8 We characterize here this similarity in a way that: (1) highlights its main features, (2) allows one to identify various important limiting cases, and (3) highlights the points that are sometimes overlooked.

The set of ideal MHD equations for a polytropic gas is^20–22 

Here, $ρ$ is the density, p is the pressure, v is the fluid velocity, and B is the magnetic field. The energy equation is based on the assumption that the matter is polytropic, i.e., the energy density $ε$ is related to the pressure as

where C is a constant (related to the adiabatic index by $γ$ = 1 + 1/C).

Initial state is characterized by spatial distributions of density, pressure, velocity, and magnetic field: $ρ|t=0,p|t=0,v|t=0,B|t=0.$ We also introduce a characteristic spatial scale L* for this initial state. As we shall see shortly, the presence of the dynamical similarity requires that the initial states for the systems that we are comparing are geometrically similar. Therefore, one can introduce L* as a distance between two arbitrary chosen points in this initial configuration. It is of course advisable to choose this scale judiciously, so that it would reflect the prevalent scale of the anticipated motion.

The next step is to provide the dimensionless description of the initial state. We measure the distance in the units of L* and present the initial state in terms of the dimensionless coordinate $r′=r/L*$ in the following form:

Here, the multipliers marked by an asterisk represent the value of the initial density, pressure, velocity, and the magnetic field at some points of the initial configuration. Specific choice of these points is formally arbitrary, but one can try to choose them so that L*, ρ*, p*, v*, and B* would be representative for the problem under consideration. For example, in the problem of the interaction of shocks and clumps (see Sec. V D in Ref. 17), the L* and $ρ*$ could be the initial radius of the clump and its density, and v* could be the shock velocity. More examples will be presented below.

Now, we use the following normalization in the set of MHD equations (1):

and normalize the time t to $L*ρ*/p*,$

By direct substitution of Eqs. (4) and (5) into dynamical equations (1), one obtains the following set of dynamical equations in the dimensionless form:

Initial conditions for this set can be found from Eqs. (3) and (4)

or, in a more compact form

Here, Eu and $β$ are dimensionless parameters characterizing the initial state

The first is called “the Euler number,”⁷ and the second is a familiar plasma beta.

The dynamical equations (6) do not contain any free parameters, and a specificity of a particular system shows up only via initial conditions (7). If parameters Eu and $β$ are the same, and the functions $f(r′), g(r′), h(r′), k(r′)$ are the same between several systems, then the evolution of all these systems is identical up to the scale transformation. The latter means that, if one knows, for example, a density evolution in the system “1,” $ρ1=ρ1(r,t),$ then in the system “2” with the same Eu and $β$ the density will evolve as

cf. Eqs. (3)–(5).

It would be appropriate now to introduce a brief definition of the Euler similarity. We define it as a similarity between two ideal magnetohydrodynamical systems evolving from the properly scaled initial states.

The sameness of the functions f, g, h, and k means that initial states are geometrically similar. This also shows that the dimensional analysis alone is not sufficient to find a similarity; one also needs to expressly use a geometrical similarity for four functions characterizing the initial state. This circumstance is very important for the dynamical similarity that purports to describe an evolution of some plasma system to the state that may be quite different from its initial state. Also important is to realize that the normalization factors introduced in Eqs. (4) and (5) are the units for measuring the corresponding quantities, and the values of the corresponding parameters measured in these units may sometimes be both significantly greater and significantly lower than one. In particular, the evolutionary time of a system where high initial velocities are present may be significantly shorter than the time $L*ρ*/p*$ used above as a normalization unit (see Sec. II D).

As we have seen, the number of input parameters characterizing initial state is five: L*, $ρ*$⁠, p*, v*, and B*, whereas the number of constraints imposed by the dimensionality arguments is only 2, Eu = const and β = const. This provides a significant flexibility in developing scaled experiments: in the five-dimensional parameter space, the constraints Eu= const and β = const specify a three-dimensional manifold within which any point is “good” in the sense that it represents a scaled system. So, accommodating constraints on, e.g., the experimentally available energy, materials, dimensions become feasible without compromising the scalability. An advantage of having a small number of constraints can also be illustrated by considering some system where a number of parameters are equal to the number of constraints: this makes any system of this latter type unique and not allowing for a meaningful scaled experiment.

Aside from that, one can make significant further enhancements to the similarity, by including additional effects (Secs. II B and II C) and identifying the situations where the unconstrained similarity appears (Sec. II D).

One can add a possibility of the composition variation, which would mean that the parameter C in the polytropic relation (2) can vary in space in the initial state. The adiabatic index would vary accordingly. If the mutual diffusion of the materials is negligible (in the spirit of the ideal MHD approach), then parameter C remains constant for every fluid element, and so does the adiabatic index. In other words, for the polytropic fluid with varying $γ$ an equation

has to be used. The similarity between two systems requires, of course, that the initial distribution of the parameter $γ$ was the same in the systems under comparison

Allowing for the composition variation does not change the number of constraints on the input parameters (Eu = const, β = const). The possibility of varying composition is important, e.g., for the studies of interfacial instabilities and mix problems.

Shock waves are a natural part of ideal hydrodynamics²³ and ideal magnetohydrodynamics.²¹ They may form as a result of an overtaking occurring from initially smooth distribution of plasma parameters. They may also be imposed by externally driven pistons of various kinds, e.g., a shock driven by the SN ejecta into the ambient medium or a shock driven by the pusher into the ICF (Inertial Confinement Fusion) target. As the Rankine-Hugoniot jump conditions are expressed in terms of the conservation of the fluxes of matter, momentum, energy and magnetic field that stem from the set (1), these jump conditions are automatically covered by the Euler similarity (see definition in Sec. II A). An explicit check of this latter observation was described in Ref. 8.

If the shock appears in one of the systems, then it appears in the scaled system as well, at the scaled instant of time and scaled position. It has the same Mach numbers in the corresponding points of the shock front. Shocks can be a part of the initial state as well. When propagating through a non-uniform medium, the shocks may create vorticity. This process (thought to be important for magnetic field generation²⁴) is also covered by the Euler similarity.

In some cases, the similarity becomes unconstrained. We explain it for the case of initially resting, un-magnetized medium (v* = 0, B* = 0). In this case, both similarity parameters, Eu and $β,$ drop out from the consideration. There are now three dimensional parameters, L*, ρ*, p*, and three dimensionless functions $f(r′), g(r′),γ(r′)$⁠, characterizing the initial state (e.g., a material that has been instantaneously isochorically heated). Provided these functions are the same for the systems under consideration, i.e., the initial states are geometrically similar, the dimensional parameters, L*, ρ*, p*, can be chosen arbitrarily. Still for any choice of these parameters, two systems which are initially geometrically similar will remain dynamically similar (i.e., will evolve in a similar fashion). This is what we mean by the unconstrained similarity. The knowledge of the evolution of one such system (e.g., in the laboratory experiment) allows one to predict the behavior of any other system with the geometrically similar initial conditions, even if the spatial and temporal scales, the pressures, and the densities are orders of magnitude different.

The absence of constraints could be predicted also from the dimensional analysis (e.g., Ref. 2): there is no dimensionless combination that could be made of L*, ρ*, and v*. But the presence of similarity does not follow from that; rather, similarity requires that the system equations, including the initial and boundary conditions, be the same.

We have assumed that the magnetic field is weak and dropped the magnetic forces from the momentum equations. Still, the frozen-in equation in (1) obeys the similarity and allows predicting an evolution of the magnetic field in all the systems possessing geometrical similarities in the initial states. This point is important for assessing the problem of an early stage of the magnetic dynamo, before the field can create a significant back reaction on the fluid motion (a “kinematic dynamo”^25,26).

Another example is a strong-drive similarity, where the system is driven by a piston with a velocity greatly exceeding the initial sound or magneto-sound velocity. In this case, one can neglect initial plasma and magnetic field pressure. Provided that the piston dynamics is similar in both systems (see more detail below), the motion driven by the piston into the initially resting non-uniform medium of varying composition would be the same in the scaled variables. The boundary conditions at the surface of the piston are that the normal components of the fluid and piston velocities are the same (an impermeable piston).

The piston dynamics can be characterized by the equation

where function F characterizes the shape of the piston at time t. The length scale L* relates here to both the piston and medium that it is plowing into. In other words, the initial state of a gas in front of the piston is characterized by the density and composition distributions scaled to the same L*, $ρ|t=0=ρ∗f(r/L*), γ|t=0=γ(r/L*),$ with the functions f and $γ$ being the same for all systems that we are comparing. The velocity v* has to be chosen as a velocity of a certain point on the piston at the same dimensionless time $t′=tv*/L*$ for all systems.

As an illustration, consider a piston in the form of a three-axial ellipsoid, i.e.,

where a, b, and c are functions of time. The ellipsoidal shape is chosen to emphasize that the piston does not have to have any particular symmetry. More complex shapes are allowed, e.g., one can consider a rippled piston. Switching to scaled coordinates, we get [cf. Eq. (11)]

where

and the same for b′ and c′ [cf. Eq. (11)]. These functions have to be kept the same for all the systems under consideration. Provided these conditions are satisfied, each of the dimensional characteristics of the system, L*, ρ*, v*, can take an arbitrary value, without any further constraints. For a linearly expanding piston, these functions would be linear functions of t; for the piston expanding with a constant acceleration, they would be a second-order polynomial in $t′$⁠, etc. The same relates to a contracting piston, with the gas situated inside the piston.

The comment on the similarity of the weak magnetic field evolution made at the end of Subsection II D 1 remains in force here as well.

In the analysis of Sec. II, we considered a special case of a polytropic gas, i.e., a gas for which $ε$ = Cp, with C = const. This is a good assumption for a highly ionized plasma at modest densities (where the electron degeneracy and deviations from ideality are unimportant). One then has C = 3/2 and γ = 5/3. This is case 1 in Table I, which presents also a few other cases where polytropic approximation works well. Case 2 in this table corresponds to an opaque gas with a dominant black-body radiation pressure; here C = 3.²⁷ In a fully ionized non-relativistic gas with strongly degenerate electrons and non-degenerate ions C is again equal to 3/2 (case 3); in this case, a contribution of the ions to the total pressure is automatically negligible since the temperature is assumed to be much smaller than the Fermi energy. Case 4 corresponds to a gas with strongly relativistic electrons and non-relativistic ions; for hydrogen, C = 9/4 and $γ=13/9$⁠. Interestingly, $γ$ in all these cases is not too different from 5/3 = 1.67: in case 2, it is 1.33, and in case 4 (hydrogen) it is 1.44. The meaning of the parameter $α$ (the last column in Table I) is discussed later.

An assumption of C=const may break down even for an ideal gas. In particular, this happens in the case of a partially ionized gas with the ionization degree depending on temperature and density, as well as for a gas with internal degrees of freedom whose excitation depends on temperature. In the laboratory experiments, one may also deal with strongly coupled systems with strong inter-particle interactions that may also lead to non-polytropic equations of state.

However, a polytropic model may work even in these cases, provided the equation of state (EOS) can be approximated by a power law in the domain of interest. This has been noted in Ref. 28, see Vol. 1, Ch. 3, Sec. VIII of that reference. They considered EOS of the form

where A, $α$⁠, and β are constants. Using thermodynamic relation between the internal energy and EOS

one can easily show that internal energy $ε$ follows the polytropic relation $ε=p/(γ−1),$ if

or

This result shows that there is a broad class of equations of state corresponding to any given γ and specified by the choice of a parameter α. Values of α corresponding to EOS presented in Table I are shown in the right column of that table. [Note that Zel'dovich and Raizer, Ref. 28, used different notations and related internal energy to one mole of gas, whereas we relate it to a unit volume. The parameters α, β, and γ are the same in both notations.]

Among the cases presented in Table I, there is a case of a gas with highly relativistic electrons. Its motions will still be covered by non-relativistic hydrodynamics of Sec. II, if the ions are non-relativistic, i.e., the gas temperature satisfies condition T ≪ m_ic²/Z. In this case, the sound speed $vS∼TZ/mi$ will be much less than c, and ensuing macroscopic flows will be non-relativistic.

There are a variety of dissipative processes that may affect the fluid dynamics and would have to be added to Eqs. (1). Those are viscous dissipation, thermal conductivity, resistive diffusion of the magnetic field, and mutual diffusion of species for a gas of a varying composition. These terms appear as diffusive terms in momentum equation (viscosity), magnetic field equation (resistive diffusion), energy equation (thermal conductivity), and in the equation for the adiabatic index (mutual diffusion). Heat release due to viscosity, magnetic energy dissipation, and mutual diffusion of the species has to be added to the energy equation [see Eq. (58.6) in Ref. 29].

Relative role of dissipative processes in dynamic equations can be conveniently characterized by the ratio of dissipation time for the corresponding process to the characteristic dynamical time, L*/v. For example, the time for the viscous dissipation is $τvisc∼L*2/ν$⁠, where ν is kinematic viscosity (which is dynamic viscosity divided by the fluid mass density, e.g., Ref. 29, Eq. (15.9), and the aforementioned ratio is $Re=L*v/ν.$ It is traditionally called “the Reynolds number.” If this number is large, the viscous dissipation can be neglected in the momentum equation. In the scaling problem, one has to use a judicious choice of the characteristic velocity v. If the motion is driven by pressure gradients in an initially resting medium, then a good choice will be $v=p*/ρ*$⁠. In the case of a strong supersonic drive (Sec. II D), a more relevant choice for v will be v = v*, where v* is a velocity of a certain point on the piston.

In a number of astrophysical problems, Re is indeed very large, due to large spatial scales. In the laboratory experiments with dense plasmas, the spatial scales are many orders of magnitude smaller, but the kinematic viscosity is also small. Indeed, by the order of magnitude, it can be evaluated as $ν∼λiivTi,$ where $λii$ is a mean-free path for the ion-ion collisions and $vTi$ is an ion thermal velocity. Evaluating it roughly as $vTi∼p*/ρ*$⁠, we see that the Reynolds number can be evaluated as

We note that in the strong-drive case the post-shock fluid velocity and the post-shock ion thermal velocity can be roughly evaluated as being of order of v*. Then the estimate for the Reynolds number in the post-shock flow still remains as in Eq. (19) with the only difference that now it has to be evaluated for the post-shock temperature.

In a similar way, one can characterize the other dissipative processes. In particular, thermal conduction is characterized by the so-called Peclet number

where $χ$ is thermal diffusivity. When this number is large, the thermal conduction can be neglected. The other two processes are characterized by the mass-Peclet number, $Pemass=L*v/D,$ where D is an inter-species diffusion coefficient, and the magnetic Reynolds number $Remagn=L*v/Dmagn,$ where D_magn is magnetic diffusivity.

In astrophysical settings, the radiation transport may be significant. In its most general, non-local form, it is not amenable for simple scaling analyses (see Ref. 30). If a radiation diffusion approximation is used, it can be combined with the particulate heat transport and characterized by the Peclet number. In the case of radiative loss from optically thin medium, the loss can be described by the cooling time; in that case, one more constraint (one more dimensionless parameter) should be added to the other constraints. This case was discussed in Ref. 15. A broader discussion of these issues can be found in Ref. 14.

The Euler similarity as described in Sec. II holds if all these dimensionless parameters are very large, allowing one to neglect dissipative processes and use a set (1) of ideal MHD equations. In a number of astrophysics problems, this is correct due to very large spatial scales. We will focus for a while on the viscous effects and will discuss the other dissipative processes further in this section.

A number of laboratory astrophysics experiments properly scaled to their astrophysical counterparts have been performed during the last two decades (see, e.g., reviews^17–19). As an example, we present Fig. 1 obtained in Ref. 31, who studied the Rayleigh-Taylor instability under conditions similar to the He-H interface in the exploding supernova. The Reynolds number in both cases was high: ∼10⁹, in supernova and ∼10⁶ in the laboratory experiment, so that the Euler scaling could be applied. The details of this remarkable experiment are presented in the original paper;³¹ additional information can be found in Ref. 32. Here we mention only that the interface was machined to create a 2D perturbation structure and then driven by a strong shock propagating to the right (from the denser to a less dense material). As the shock broke through the interface and caused pressure increase in the light material, the interface decelerated (see Sec. II of Ref. 31), creating conditions for the instability to grow [Fig. 1(a)]. Note that, despite the high Reynolds number, the turbulent shearing did not destroy the coherence of the clearly visible bubbles and spikes. Figure 1(b) is discussed in Sec. VI.

In a simulation experiment with Re ≫ 1, one can be confident that viscous effects do not play a role on the global scale ∼L*. Still, the effect of finite (albeit small) viscosity may be important, as large-scale flows with Re≫1 may cause the onset of hydrodynamic turbulence and cascading to much smaller scales, as described in the Kolmogorov-Obukhov model (see Ref. 29 for a nice description). Eventually, this cascading reaches very small “dissipative” scales, where viscosity finally becomes important. The role of these smallest scales on the global flow is still a matter of active research on its own (see, e.g., Ref. 13).

For the dynamic similarities, there are, however, situations, where viscous effects can still be ignored despite development of turbulence. The point made in Ref. 15 was that cascading to smaller scales takes time,²⁹ and the system may reach a state that was a goal of a simulation experiment (and that may be entirely different from the initial state) well before dissipative scales are reached via cascading. For example, the spikes of the Rayleigh-Taylor instability reveal significant degree of coherence and extend by large distances, as shown in Ref. 31, Fig. 1. Such spikes driven at the interfaces of the elementally different layers of the collapsing supernova could reach the surface and become detectable before the cascading problem starts playing significant role.¹⁵ 

If one deals with a fully ionized, ideal, non-degenerate plasma, one can cover the similarity even if the small scales are developed and their viscous dissipation affects the global-scale dynamics. As we show momentarily, this requires imposing one more constraint on the system parameters (in addition to the constancy of Eu and β, Sec. II).

We remind the reader that in the general situation one has to consider two types of viscosity. The shear viscosity produces damping of the vortex flow with $∇×v≠0,$ whereas the bulk viscosity causes the damping of the vortex-free compressional flows, e.g., acoustic waves. We note that in a fully ionized non-degenerate plasma both the shear viscosity and bulk viscosity are of the same order of magnitude; they are power-law functions of density and temperature.⁵² For the similarity analysis, it is sufficient, therefore, to consider the terms containing one of them, say, shear viscosity. For a fully ionized, non-degenerate plasma with the ion charge Z and atomic mass A, the viscosity scales as

where $ψ$ is a constant multiplier made of universal constants and inversely proportional to the Coulomb logarithm considered here as also a constant. Note that for the Z > 1 materials containing some amount of hydrogen (like CH) viscosity increases significantly over the material containing only Z > 1 ions (like C), see detailed analysis and further references in Ref. 33. The dependence of $ν$ on p and $ρ$ remains the same, but the coefficient now depends on the composition. For simplicity, we consider only the case of a pure material.

When one substitutes this viscosity to the momentum equation with the shear viscosity terms included, one gets

where we have introduced an analog of a canonical Reynolds number

It is expressed entirely in terms of the initial conditions. The matter can be compressible. Therefore, if, in addition to parameters Eu and $β$⁠, the Reynolds number defined by Eq. (23) is held the same between two systems, they will evolve identically, up to the scaling transformation. There will be also an additional scaled term proportional to 1/Re in the right-hand side of the energy equation of the set (1). We do not write it down as it is quite lengthy, but it does obey the same similarity properties. Note that what we call the Reynolds number is entirely defined by the initial conditions of the system. It accounts for compressibility and for possible change of material (A and Z are included).

This similarity covers the aforementioned situation with developing small-scale dissipative vortices that may have an effect on the global dynamics. As usual with the dimensional analysis, one cannot predict the outcome, but what is certain is that two systems with the same values of Re, Eu, and β will evolve identically up to the scaling transformation. If one applies this scaling to the similarity of initially resting medium with no magnetic field, there remain only three input parameters, $ρ*$⁠, p*, and L* and one constraint: the same Re for the systems under comparison. This allows for a significant freedom for selecting parameters of the simulation experiment, including possible change of the material [i.e., of parameters Z and A in Eq. (23)].

The just described similarity works fine for the fully ionized non-degenerate plasma. In more general cases, where both equation of state and viscosity contain some intrinsic scales (as functions of p and $ρ$⁠), so that viscosity cannot be approximated by a power-law dependence vs. p and $ρ$⁠), this similarity does not work. However, an issue of the underlying effect of small-scale vortices on the global motion can be still addressed by a somewhat different type of similarity.

One can attempt to use similarity-constrained experiments to identify a possible effect of small viscosity (Re≫1) on the dynamical evolution of the two systems with the Reynolds number being very large in both but different by some significant factor (say, 10). This could be an experiment based on the use of the most basic similarity, in which parameters $ρ*$⁠, p*, v*, and B* characterizing initial states of the two systems would be the same, and only L* would change. This similarity, which was called a “perfect similarity” in Ref. 34, is interesting in that the absolute values of pressure, density, velocity, and magnetic field are the same in the corresponding points of the systems at any time. Then, if the finite viscosity does not play significant role in the vortices that could develop by that time, the kinematic viscosity which depends on $ρ$ and p, $v=v(ρ,p),$ will also be the same at the same points of two systems which are different in scales by a factor of 10. Thereby, the generally unknown dependence of ν on ρ and p would not affect the conclusions regarding the evolution of the global mode in either of the two systems. Conversely, if the smaller system with smaller (say, by a factor of 10) Reynolds number (still much greater than unity) evolves differently, this indicates that cascading has reached the scale where viscous dissipation became significant and started showing up at the large-scale evolution. Experiment of this kind would empirically constrain the lower bound of a Reynolds number beyond which one could be sure that the dissipative-scale vortices do not play a role in a particular dynamical experiment.

Temporal evolution of the dynamical system is determined by the evolving velocity field, v(r, t). The processes of heat transfer and inter-species diffusion, as well as magnetic field decay or growth, are occurring in this dynamical environment. This is what makes the characterization of the velocity field so important and motivates a specific interest to effects of viscous friction discussed in Secs. III B and III C. The effect of the other dissipative processes is, therefore, reasonable to characterize in relation to the viscous dissipation and corresponding dimensionless parameter, Re. It is reached by introducing a so-called Prandtl number which, for the processes of the heat transfer, is defined as the ratio of the Peclet number (20) to the Reynolds number

If the Prandtl number is large, the heat transport can be strongly affected by the large-scale vortices: the shear flow in these vortices creates a “choppy” temperature distribution with decreasing spatial scale, followed by the temperature homogenization. Conversely for the small Prandtl number, the heat conduction is largely decoupled from the hydrodynamic flow. This is a very rough outline; more detail can be found, e.g., in Ref. 29.

The molecular mix in the system of a varying composition is characterized by the material Prandtl number

For a fully ionized plasma, the ratios $ν/χ$ and $ν/D$ do not depend on p and $ρ$⁠. Therefore, if the Reynolds number defined as in Eq. (23) is constant between the systems under comparison, then so are the Peclet and mass Peclet numbers. Therefore, the similarity, described in Sec. III C and constrained by a constancy of a “Reynolds number” (23), holds with a complete inclusion of heat transfer and interspecies diffusion! This makes it a powerful tool for the studies of the effects of small-scale vortices on the global dynamics of a fully ionized plasma.

The effect on the magnetic field diffusion is characterized by the magnetic Prandtl number

The magnetic field is a vector quantity, and interplay between the turbulent advection and magnetic diffusion is subtler than in the case of heat and mass transfer. The magnetic Prandtl number plays an important role in the problem of the magnetic dynamo.^25,26 In the astrophysical settings, it may be both greater than 1 and smaller than 1 (Table II), whereas in the laboratory experiments on the magnetic field generation in liquid metals,^35–37 it is universally much smaller than 1. There is a chance that high magnetic Prandtl number can be reached in the laboratory by the use of large-scale plasma devices.³⁸ The significance of the Prandtl number was emphasized in an elegant theory study.³⁹ 

There exists a broad class of fusion-oriented devices where a dense pre-heated and pre-magnetized plasma is driven to much higher, fusion-grade densities and temperatures by means of a quasi-adiabatic compression by a heavy shell (a “liner”). The shell itself is imploded by the magnetic pressure of an axial current flowing on the outer surface of the shell (like in a Z-pinch). The implosion velocity is much smaller than the sound speed of the plasma compressed by the shell; in this respect, an implosion is quasi-static and almost adiabatic, with the deviations from adiabaticity caused by the energy losses from the plasma to the inner surface of the liner. The skin-depth for axial drive current in the liner is much smaller than the liner thickness, so that the plasma is electromagnetically decoupled from the driving current and driving magnetic field. There are many plasma configurations suitable for this approach⁴⁰ that is commonly called Magnetized Target Fusion (MTF) or Magneto-Inertial Fusion (MIF).^41–43 

To be specific, we discuss below an approach called MagLIF=Magnetized Liner Inertial Fusion,¹² where the plasma is imploded by a cylindrical liner. This configuration has been studied in the experiments⁴⁴ on the Z facility at Sandia National Laboratories.⁴⁵ The initial radius of the plasma inside the liner is a few millimeters and is reduced by a factor of 30 in the final state. The small radius means that a sufficiently high axial magnetic field is needed inside the liner to suppress heat conduction to the walls by magnetizing the electron component. An axial electron heat conduction can be left un-magnetized due to a large length-to-radius ratio. The use of the plasma with β ≫ 1 is a distinct feature of the MTF/MagLIF systems: in the opposite case, β ≪ 1, the implosion energy would be largely converted to magnetic energy, not to the plasma energy, and thereby would make the energy efficiency unacceptably low.

As the implosion is deeply sub-sonic, the plasma inside the liner will be in a mechanical radial equilibrium, i.e., the sum of the gas-kinetic pressure and magnetic pressure will not vary over the radius.⁴⁶ On the other hand, the near-wall layers of the plasma will be cooled by the thermal conduction to the wall and squeezed against the wall, creating a strong radial density gradient. As the plasma is highly conducting, the magnetic field will be at least partly entrained by the plasma and also squeezed against the conducting liner. This creates a system with a large variation of the plasma parameters and corresponding variation of the transport mechanisms, from magnetized ions and electrons in the core to the ion demagnetization closer to the walls, and finally to electron demagnetization in the coldest zone. In addition, the Nernst effect leads to magnetic field advection and redistribution with respect to the line-tying situation. The wall confinement of a dense plasma heated by electron beam has been numerically studied in significant detail in Refs. 47 and 48, based on Braginskii equations. Analyses more directly related to MagLIF concept and accounting for the liner dynamics and axial losses are presented in Refs. 49 and 50.

Although the magnetic field does not “confine” the plasma mechanically (the magnetic pressure is low), it still plays very important role in suppressing the radial thermal conduction. One of the key issues here is the possibility of development of drift turbulence and anomalous transport. If the transport reaches the Bohm value, the efficiency of the plasma implosion can be significantly reduced, as too much energy would be lost to the inner surface of the liner. As there is no comprehensive theory of this mesoscale turbulence, it is desirable to address this issue experimentally, possibly with the scaled down devices (as the Z-facility with its 2MJ drive is expensive, and the plasma diagnostics on it is difficult). It is therefore appropriate to try to apply the similarity analysis and identify possible scaled-down experiments.

The time-dependence of the inner radius of the liner, a(t), is determined by the processes occurring in the dense liner material. These processes are not governed by the same equations as the hot highly ionized plasma inside the liner. Adding to complexity of the problem is the fact that the plasma parameters in a MagLIF experiment vary significantly during the course of the implosion. So, it is hardly possible to develop a comprehensive scaling that would cover the whole “shot,” with both plasma and liner descriptions combined together. One can then try a more modest approach, where only the plasma transport processes would be considered, under the assumption that the inner liner surface follows a known “trajectory” a(t). As we will see shortly, even this reduced model is not quite amenable for the scaling exercise. Instead, one can try to characterize the plasma behavior for several time intervals during the shot. Characteristic plasma parameters for two stages of the implosion presented in Table 3.

This “piecemeal” approach has been suggested in Ref. 11. An idea was to split the whole implosion processes into several stages and, for each stage, find the scaling that would allow one to translate it to a smaller-scale experiment, with less demanding energy input. The liner here can be resting, and the initial state can be created not by implosion, but by the laser heating. If the scaled experiment would show that the plasma confinement time defined as $τW=|W/Ẇ|$ (with W being the plasma energy) is significantly longer than the characteristic compression time $τ*≡|a/ȧ|$ for this stage of a full-scale experiment, this would mean that the corresponding stage of the implosion would be passed with no significant energy loss, and further compression would be possible. Note that both a and $a ̇$ are changing in the course of implosion; the times $τ*$ used in Table III are taken from the simulated liner trajectory.¹² 

The correctly scaled experiment would allow one to find a confinement time for the corresponding stage of a full-scale experiment. What is important here is that the scaled experiment would have all the ingredients of a full-scale one, including possible contribution of mesoscale anomalous transport, which is difficult to simulate and whose experimental evaluation in the scaled experiment with a resting liner would be very helpful.

In this approach, one would have to create initial plasma with the parameters corresponding to the stage in question and then follow its decay by the heat loss in the radial and axial directions. This plasma could be produced, in particular, with a laser heating (as in the original proposal¹²), but with the energy input much less than in a full-scale experiment. For reference purpose, we present Table III (based on Refs. 11 and 12) that represents the plasma parameters for two stages of the implosion, near the beginning and near the stagnation. The parameter $τ$ is the radial sound transit time ∼a/v_Ti. The first line in Table III corresponds to the early time, pre-heated plasma, just created inside the liner by some auxiliary power source. The second line corresponds to the point of the maximum compression of the fusion fuel by the liner. These are representative parameters and are not necessarily those optimized for the best performance; they just provide guidance for the scaling analysis. The meaning of the parameters R₁-R₄ on the right side of the table will be discussed shortly.

What makes the MTF/MIF plasma different from most of the other fusion systems in terms of the anomalous heat transport by drift-wave turbulence is that a characteristic drift frequency is much less than the Coulomb collision rate (see Ref. 51). This means that anomalous transport, if present, would be driven by collisional drift waves, which can be adequately described by the two-fluid Braginskii equations.⁵² 

In the scaled experiment, we are interested in the evolution of a confined plasma over the range of parameters of order one, when the plasma loses roughly a half of the initial energy. In other words, we are interested in the scaling of a confinement time $τW=|W/Ẇ|$ between two systems (a full-scale one and its smaller model). This makes the problem amenable to a general approach used in a remarkable paper by Connor and Taylor,⁵ where among other cases a confining scaling for a high-beta collisional plasma was considered. We later comment on some differences in the confinement scalings for tokamaks and high-density MagLIF plasma.

The plasma in the initial state for some stage of the full experiment is characterized by the parameters of density n, temperature T, and magnetic field B on axis. Also given are radius a of a confining wall and the length of the plasma cylinder L, both independent of time within the piecemeal approach. Dimensional analysis shows that there are four independent dimensionless combinations that can be formed from these five dimensional parameters. We choose the following four dimensionless combinations (each of which has an obvious physical meaning):

where $ρi$ is the ion gyroradius and $λ$ is a Coulomb mean-free path. The parameters R₁ – R₄ have to be held the same between the full experiment and its smaller model. Note that the energy-exchange time between electrons and ions is much shorter than $τ*$⁠, so that the system can indeed be characterized by a single temperature T. [The energy exchange time can be evaluated using Eq. (2.17) in Ref. 52 and is ∼2 ns for the first line of Table III and ∼0.7 ns for the second line.]

Unlike toroidal confinement, the axial heat loss to the end surfaces can be an important constraining factor in our case; it enters the problem via an aspect ratio L/a. Normalizing all the plasma parameters to their values defined by the five input parameters, n, T, B, L, and a, and noting that there are five input parameters and only four constraints $R1−R4$ in Eqs. (27), one sees that there is some freedom in creating the systems that would be different in terms of the absolute values of the initial parameters, but would behave in a scalable fashion. If one chooses to characterize the system by its liner radius a, one finds that other dimensional parameters can be expressed in terms of a and four dimensionless parameters. In particular, the initial energy content (for the DT plasma) is

Therefore, by reducing the wall radius a and keeping the dimensional parameters unchanged, one can reduce the initial plasma energy and still get the confinement information scalable to an experiment with higher energy content. Unfortunately, the reduction of the required energy is quite modest due to the square root dependence vs. a. At this point, one can use physics-based arguments to relax the constraint of the constancy of the dimensionless parameters.

For given values of the four dimensionless parameters and radius a, one can get an idea of the physics governing the confinement in a particular regime (say, initial one, row 1 in Table III). It turns out that in this regime the ions are deeply un-magnetized, R₁ ≪ R₂, or $ρi≫λ.$ So, some modest reduction of the parameter R₁ would not have any noticeable effect on the plasma confinement, as the ions remain unmagnetized. As an example, consider the reduction of both a and R₁ by a factor of 2, leaving all other dimensionless parameters untouched. With that, the plasma energy decreases by a factor of 11 to ∼0.65 kJ, and the magnetic field decreases by a factor of 1.6 to 0.25 MG. This would make the simulation experiment much easier.

Consider now a possible scaling-down (to lower energies and smaller facilities) of the experiment on plasma confinement near the point of the maximum compression (line 2 in Table III). Here the ions are magnetized, R₁ ≫ R₂. On the other hand, the ratio of the drift frequency for the global mode, $ωD=(ρi/a)(vTi/a),$ to the ion collision frequency, $νi=vTi/λ$⁠, is still very small, $ωD/νi=1/R1R2≪1,$ meaning that we are in the regime of collisional drift instabilities considered in Ref. 51. This allows us to make the following steps directed mainly towards reduction of the total energy. The first step would be to reduce the parameter R₁ by a factor of 2. The plasma is still strongly magnetized, R₁≫R₂, and, at the same time, it remains in the collisional drift-turbulence regime, R₁R₂≫1. The ratio $L/λ=R4R2$ is huge, ∼400. This allows one to somewhat reduce the parameter R₄, say, by a modest factor of 2, so that the parallel physics would remain collision-dominated.

Even with reduced parameter R₄, the relative time for the axial distribution of the plasma, $Lτ*vTi$⁠, remains greater than one, meaning that the plasma outflow through the ends is small. For the radius of the simulation experiment of a=0.25 mm (vs. 0.12 mm in the full-blown experiment) which may provide better conditions for the plasma diagnostics, the required energy is 11 kJ, the magnetic field is 20 MG, the plasma density is 7.5 × 10²¹ cm⁻³, and the plasma length is 6 mm. These parameters may be attainable on the Omega laser facility⁵³ at the University of Rochester. Peak magnetic fields of more than 50 MG have already been reported by the Omega group.⁵⁴ 

In addition to providing some insights into the processes occurring in the experiment, the scaling approach delineated above can help us to experimentally address the issues of the collisional drift turbulence which may be of significance not only for the fusion-oriented experiments but also to the processes occurring in the stellar interiors.

We conclude this section with a brief comparison of the MagLIF confinement scaling vs. the tokamak confinement scaling, emphasizing issues that may go beyond the standard analysis. The first is related to the conductivity of the surrounding walls. In the case of a tokamak, the resistive wall modes contain parameters of the wall resistivity and thickness that are not covered by the plasma scalings. The wall conductivity is also important for the MagLIF plasma although for a different reason: the magnetic field is highly compressed near the wall, and the wall resistivity may cause the loss of the magnetic flux and corresponding confinement degradation. These issues have to be analyzed when determining the applicability of the scaling analysis.

Although the set of dimensionless parameters (not their values!) can be the same for both systems, the scaling for MagLIF works without the need of introducing different electron and ion temperatures: due to very high collisionality, the system can be characterized by a single energy parameter, T=T_i=T_e. In tokamaks, where the collisionality is low, the electron-ion energy exchange time is long, and it may be needed to characterize the system not with a single temperature, but with two temperatures. In the same vein, the role of high-energy particles, including those generated by the neutral beam injection, has to be assessed.

The scaling should, generally speaking, include also the symmetry considerations. For example, changing the toroidal magnetic field direction in tokamaks, with other input parameters held the same, may cause significant change in the plasma confinement, unless the device possesses an “up-down” symmetry. The corresponding symmetry constraints have been discussed in Ref. 55. If there is an axial current flowing through the MagLIF plasma, the chirality considerations may have to be addressed as well.

Dynamical collisionless scalings are of interest for a broad variety of plasma physics problems. Examples include wake-field acceleration,⁵⁶ shock acceleration,⁵⁷ high-order harmonic generation (see review⁵⁸), studies of the Coulomb explosion of molecular clusters,^59,60 physics of fast ignition,⁶¹ and other phenomena involving ultra-short-pulse lasers with a chirped-pulse amplification,^62,63 and intense particle beams.⁶⁴ Scaling papers for this research area include Refs. 9, 65, and 66.

We focus here on the problems related to astrophysics: formation of collisionless shocks via Weibel instability (Secs. V A–V C), energetic ion formation in magnetic cavities of the tower jets (Sec. V D), and self-similar expansion of hot electron clouds in stellar atmospheres (Sec. V E).

The Weibel instability first discussed in Refs. 67 and 68 is considered as a potential mechanism for collisionless shock formation in astrophysics. There exists enormous literature on theory, numerical simulations, and laboratory experiments on this subject. We mention here a few representative references on theory and simulation,^69–76 and experiments.^77–82 Further references can be found in these papers. A remarkable feature of this instability is that it can generate significant magnetic field in the situations where there was no initial magnetic field in the system, aside from the field of thermal fluctuations.

A typical setting for experiments on the Weibel instability is shown in Fig. 2. The counterpropagating streams interpenetrate each other due to a large mean-free path and generate characteristic Weibel filaments visualized by the proton radiography technique.⁸³ 

In our scaling discussion, we assume that the initial state of the system is represented by two identical, uniform, non-relativistic, counter-streaming plasma flows with sharp fronts. The electron density of each stream is n, and the velocity of each stream is u. The interpenetration begins at t = 0. A discussion of smooth fronts is presented in Ref. 10. We assume that the initial temperature of the plasma in each stream is low compared to the one that will later appear via the development of the Weibel instability, so that initial temperature does not enter the scaling analysis. At the same time, the temperature is assumed to be sufficiently high to make the Coulomb collisions between the particles within each stream (the intra-stream collisions) negligible even at the early stage of interaction. There is no regular magnetic field present in the system, while naturally present fluctuations serving as a seed for the instability are small and do not enter the scaling analysis. [Here we make an implicit assumption that fluctuations reach a non-linear stage and “forget” the initial level.]

We characterize the plasma state in the process of a two-stream interaction by electron and ion distribution functions, $fe,i(v,r,t),$ and the electric and magnetic fields E(r,t), B(r,t). In order to cover not only astrophysical but also laboratory plasmas, where one often uses plasma streams made of beryllium, carbon, and other materials, we allow the ion charge Z and atomic mass A to be different from Z=A=1. For brevity, only single ion species plasma is considered although our analysis can be extended to multi-species streams (see comments after Eq. (44) below). As the system may transition through a phase where the plasma is highly non-uniform, with large electromagnetic fluctuations, the distribution functions $fe,i(v,r,t)$ are full distribution functions, without separation of fluctuating and average parts. Early in the interpenetration, the distribution functions will be clearly double-humped, whereas at later stages the directed velocities are decreasing, and the zone of quasi-stagnation grows, with developed magnetic fluctuations and with increasing random (“thermal”) velocity of the ions. The scaling that we are going to develop will therefore be quite broad, covering all of these phenomena and allowing us to predict the behavior of the system over a long time (shorter, though, than the collision time, see below).

The characteristic spatial scale of the Weibel instability is on the order of

where $ωpi$ is an ion plasma frequency (e.g., Ref. 69). An expression for ω_pi is

where n is the electron density in each of the incoming streams (the ion density is n/Z). For non-relativistic plasmas, $l$ is much greater than the Debye radius, and therefore, the plasma fluctuations must be quasi-neutral

Here, $ρe$ and $ρi$ are the charge density of each species (electrons and ions, respectively). A more general similarity that includes charge-separation effects is discussed in Ref. 10; here we provide a more detailed analysis of the slow quasineutral mode relevant to magnetic field generation.

The electromagnetic fields can be expressed in terms of the scalar (⁠$φ$⁠) and vector (A) potentials

Due to a non-relativistic nature of the problem, one can use a quasi-steady-state relation between the current and the vector potential, where the terms of order of $1/c2$ are neglected. This leads to the Ampere law without displacement current

Note that the Faraday effect is included [Eq. (32)], and the presence of the vortex component of electric field (important for the Weibel instability) is thereby accounted for.

The set of equations that completely characterizes the problem for the instabilities with spatial scales much larger than the electron Debye radius consists of the electron and ion kinetic equations that allow one to determine the current and charge densities for both species, the Ampere's law (33) and the quasineutrality constraint (31).

We normalize the spatial scales to $l$ [Eq. (29)], the time to

and particle velocities to the initial velocity of the streams u

Here, the primed quantities on the left are dimensionless counterparts of r, t, and v. We normalize the electron and ion distribution function to $n/u3$ and $n/Zu3$⁠, respectively, so that, e.g., dimensional ion distribution function is related to the dimensionless one by

where the primed distribution function in the right-hand side is dimensionless. The particle distribution functions include the particles of both streams.

The current density j is then related to the dimensionless distribution functions by

Substituting this result to Eq. (33) and accounting for the definitions (29), (30) prompts the following normalization for the vector potential:

We retain here a canonical notation A for the atomic mass; it cannot be confused with the vector potential which is always a boldface A. With that, Eq. (33) becomes

It does not contain any of the four input parameters (n, u, A, Z). The “prime” sign by the differential operators here and below indicates that differentiations occur with respect to “primed” variables.

Normalizing the scalar potential $φ$ according to

one finds normalized Vlasov equations for the electrons and ions

Equations (39), (41), and (42) together with the normalized quasineutrality condition

form a closed set of dimensionless equations describing the problem under consideration, with all the information about the input encapsulated in a single dimensionless parameter

(we use the “hat” sign to distinguish this parameter from the Mach number). In other words, the dimensionless equations remain the same for the systems with any velocity u and any density n of the streams. One can also change the ion species with the only constraint that the ratio $A/Z$ should remain the same. One can, if so desired, use materials made of several elements with the same $A/Z$⁠, e.g., CD or CD₂, where the C and D both have $A/Z=2.$

The latter conclusion is important for the experimental simulation of the Weibel instability in astrophysics: presently used techniques for generating the plasma streams are based on ablating (by pulsed laser) the surfaces of two solid targets facing each other. The cryogenic targets with frozen hydrogen layers on the surface are not available yet, and one has to use targets made of C, CD_n, Be, or other light elements.⁷⁷ It goes without saying that the velocity of the stream should be sufficiently high to make the Coulomb collisions negligible, and the plasma ions in the streams would be fully stripped.

There is a corollary to this discussion: in the course of development of a truly collisionless turbulence the ions with different atomic masses (but the same A/Z) reach different “temperatures,” which scale as A (e.g., carbon will be 6 times “hotter” than deuterium). We note also that the similarity parameter (44) expressly contains the electron mass. A large mass disparity creates significant problems for numerical simulations.

Proton radiography technique⁸³ allows one to obtain time sequences of the filaments developing in the interpenetrating streams (Ref. 84, see Fig. 3). The figure contains also results of numerical simulations scaled to the experimental parameters. A good consistency between two sets of the images indicates a correct identification of the underlying physics. The bright vertical features are probably due to the Biermann battery mechanism acting near the target plates, with a subsequent advection of the thus produced azimuthal magnetic field towards the midplane.⁸⁵ 

The just described similarity, constrained by a single condition of the constancy of $M̂$⁠, is a very broad, blanket similarity. It covers the whole process of interpenetration including fine details of the fluctuation spectrum, electron heating process, ion slowing-down, and any other characteristic of any system of the interpenetrating streams with the same $M̂$⁠. Thereby, it predicts scaled relations connecting these characteristics for any two systems with the same $M̂$⁠.

Parameter $M̂$ characterizes a response of electrons vs the ions to electromagnetic field. It is very large, thereby emphasizing the importance of the electron physics in developing the instability and turbulence. The role of electrons clearly shows up in the linear analysis where they, due to their low mass (high mobility), tend to short-circuit any vortex electric field and significantly reduce the growth rate with respect to a “canonical” estimate of $ωpiu/c$⁠.^69,75 This pushes the maximum growth-rate at an early stage to shorter wavelengths lying between $c/ωpi$ and $c/ωpe$ (see, e.g., Ref. 76).

At the very beginning of the interaction, the magnetic field in the system is so small that the electron cyclotron frequency is much lower than the growth rate. During this, basically linear, phase the electrons can be considered as un-magnetized and the term $v×B$ in the electron kinetic equation can be neglected. The corresponding term in the ion equation has, however, to be retained, since this is exactly the term that drives the instability. Gradually, with the random magnetic field perturbations growing, the electrons at this early stage become quasi-isotropic and enter the next stage as a nearly isotropic gas, with the two initial electron populations merging into single isotropic electron population.

This next stage corresponds to magnetized electrons, with the electron gyro-frequency becoming significantly higher than the $1/τ$⁠. From this point on, the electrons are magnetized and their motion across the magnetic field becomes a drift motion. The motion along the field lines becomes much faster than the speeds related to the ion perturbations. Therefore, it seems reasonable to neglect the electron inertia in the electron momentum equation. To illustrate this point qualitatively, we present here the electron momentum equation for the case where the electron pressure is isotropic and can be characterized by the scalar pressure p_e

where, under the quasineutrality constraint, n=n_e=n_iZ. As mentioned, for the electron pressure exceeding m_enu², the left-hand side can be neglected yielding an electron equilibrium equation

The electron velocity can be expressed in terms of j and v_i

where v_i is an average ion velocity (i.e., for the two symmetric counter-propagating streams it is zero, but becomes finite in the course of development of the Weibel instability). The electric field has both potential and vortex component. Expressing it via Eqs. (46) and (47) and taking divergence, $∇·E=−∇2φ$⁠, one finds an equation for the electrostatic potential

The evolution of the electron pressure in the model of an ideal gas is

Or, according to Eq. (47)

Equations (48) and (50), together with the ion kinetic equation (42) and Ampere Law (33), describe the evolution of the system at the stage where electron pressure becomes higher than m_enu². One can check that, for the same way of making equations non-dimensional as before [i.e., using Eqs. (35)–(37) and (40)], one obtains a system that does not contain parameter m_e; in other words, at this stage, the dependence on the electron-to-ion mass ratio disappears from the scaled equations. Accordingly, any two systems at this stage evolve identically if only the parameter A/Z is constant, with no dependence of the electron-to-ion mass ratio. The initial electron temperature might have become a new scaling parameter, but if the final electron temperature is much higher, then this parameter drops out.

This means that if the initial electron temperature was higher than m_eu²/2, one could use a model of zero electron inertia from the outset. The electrons would evolve quasi-statically in the slowly evolving electric and magnetic fields. Equations (48) and (50) correspond the simplest model of an isotropic electron gas, but one can use also electron description with full-blown gyro-kinetic equations, where their distribution will be a function of the parallel velocity and magnetic moment μ. Then, taking moments of this distribution function, one obtains the analogs of Eqs. (46)–(48), which do not contain the electron mass. In particular, the equilibrium Eq. (46) becomes

and instead of Eq. (48), one now obtains that

This is an equation that determines a curl-free component of the electric field that enforces the quasineutrality constraint. And, again, this equation does not contain an electron mass and can be made dimensionless by the same replacement of variables as before.

The aim of this discussion is not in solving the full problem of the shock transition, but rather in identifying the situation where this transition does not depend on a small parameter of the electron mass. We see that, indeed, as soon as the electron thermal velocity becomes greater than the velocity of the streams, such description becomes possible and the further evolution occurs in the same way for any m_e. This makes, in particular, experimental results with carbon or plastic plasmas scalable to hydrogen plasmas in astrophysics! Another application is to numerical simulations of this complex phenomenon: the simulations are often made with artificial electron-to-ion mass ratio (artificially heavier electrons). Our discussion shows that they can indeed be scalable to the real mass ratio for the later part of the instability development and shock formation. The electrostatic potential (which is needed to evolve the ion distribution) is determined from Eq. (48) or (52) in this quasineutral model.

There are two caveats that have to be mentioned here. In a real system, there may be present electrostatic instabilities, concurrently with slower electromagnetic instabilities. They would not follow the same scaling and an additional dimensionless scaling parameter enters the analysis and introduces an additional similarity constraint (see Ref. 10). If the electromagnetic instability evolves on the time-scales and spatial scales typical for the Weibel instability (slow), then the electrostatic modes can hardly appear. However, if, in the course of these slow motions, small-scale current sheets or filaments with high current densities approaching electron thermal current density develop, then electrostatic modes may become a strong local player.

Depending on the parameters of the colliding streams, some types of collisions may come into play and have a non-negligible effect on the instability development and, at a later stage, on the structure of the shock waves formed in the zone of overlapping streams. As mentioned above, very early in the interaction process the electrons of the initial two streams form one population that can be characterized by its average random energy, which we call loosely an electron temperature and denote by T_e. Before the turbulent scattering of the ions on electromagnetic fluctuations leads to a significant (order one) slowing down of the initial streams, the ions in each stream can be characterized by their intra-beam “temperature,” T_i. Until a complete relaxation occurs, this temperature is smaller than the initial directed energy Am_pu²/2.

There are several types of collisions that may affect the instability: (a) inter-stream ion collisions; (b) intra-stream ion collisions; (c) electron-ion collisions; (d) electron-electron collisions. Here the term “inter-” refers to collisions between the ions of the two streams having velocities u and –u, respectively, whereas the term “intra-” refers to collisions within each stream and dependent of the ion “temperature” $Ti$ of each stream (which is initially much less than $Ampu2/2$⁠). As an electron thermal velocity becomes much higher than u early in the interaction process, the electron collision frequencies (both e-e and e-i) are determined by the electron “temperature.”

It is convenient to characterize the just introduced “temperatures” by dimensionless quantities

They can be used to characterize the frequencies of the aforementioned types of collisions in relation to the characteristic time $τ$⁠, Eq. (34). By the order of magnitude, these collision frequencies are

Here, $ϒ$ is a coefficient made of universal constants e, m_p, and m_e, and Λ (the Coulomb logarithm). The numerical factors are the same within a factor of 2; this difference is unimportant for the qualitative discussion that follows.

The first point that stands out is that the intra-stream ion collision frequency (54) for Z=6 (a typical Z for experiments with the laser-generated streams) is much higher than other frequencies, at least initially. In other words, the first factor that has to be addressed when evaluating the role of collisions in the experiment on collisionless instability is the role of the intra-stream collisions. At the crucial early stage of the interaction, until the ion temperature reaches, say, a quarter of their “shocked” temperature, this frequency is more than an order of magnitude higher than the inter-stream ion-ion collision frequency. This somewhat narrows the parameter domain for the design of the collisionless experiment. However, as shown in Ref. 76, the frequent intra-stream collisions do not qualitatively modify the nature of the ion Weibel instability; they simply make the ions in each stream highly collisional, without compromising the assumption of negligible collisions between the streams. Still, some additional effects can appear. Each of the ion streams can now be described by a set of hydrodynamic equations, with the density, average velocity, and temperature being adequate variables. It is possible at this stage that the shaking and compressing-decompressing of each of the streams by the instability may drive collisional shocks in each stream, thereby enhancing their heating.⁸⁶ Eventually, of course, as soon as the ion temperature in each stream becomes significant, the purely collisionless effects take over.

Additional subtleties are brought to the problem by the use of the flows with the CH or CD composition since the collisions between different species may have very different collision frequencies.⁸² 

If one uses the quasineutral scaling of Sec. V A, one sees that the scaling constrains only the parameter $M̂$⁠, Eq. (44), and does not constrain either n or u, which can be chosen arbitrary if the system is collisionless. On the other hand, the collisionality parameters (54)–(57) do depend on u and n in the combination n^1/2/u⁴; therefore, changing this parameter will change the role of collisionality. This is especially obvious with respect to the velocity dependence of the interaction (the fourth power of u). One can therefore experimentally try to isolate the collisional effects, which may play a prominent role in the interaction, especially at the early stages of the interpenetration. A more detailed analysis of various possibilities and the new features of the collisionless/collisional interaction will be published elsewhere.

Magnetized “tower jets” are a common occurrence in astrophysics. They are usually associated with accretion disks; the axial current goes along the jet axis “upward” from the disk surface and then returns along the surface of an ambient plasma surrounding the jet; the magnetic pressure keeps the external plasma outside the magnetic cavity surrounding the jet.^87,88 This geometry has been reproduced in the laboratory experiments with z-pinches⁸⁹ and revealed a clear morphological similarity with astrophysical jets. Figure 4 shows an evolution of the jet in one of the shots on the MAGPIE generator at Imperial College (London), Ref. 90. A characteristic “cocoon” is formed; the current path is closed via the outer shell, and the magnetic field is confined to the area inside the “cocoon.”

The protons with the energy of up to 3 MeV have been observed in this configuration.⁹¹ This energy is 30 times higher than eU, where U ∼ 100 kV is a voltage applied to drive a current. This suggests that the ions inside the cavity experience acceleration, possibly a diffusive Fermi acceleration by the rapidly varying magnetic perturbations that appear naturally in the cavity due to the kinking of the central column—the effect clearly seen in the x-ray images in Fig. 4.

Without attempting to develop a detailed acceleration model, one could evaluate the upper bound for the proton energy that can be obtained by such a mechanism. The protons are experiencing accelerations until they cross the cocoon boundary beyond which the field is absent, and the protons move freely away from the cocoon. This sets the upper bound on the maximum attainable energy. In astrophysics, this limitation is called the “Hillas criterion.”⁹² To cross the boundary whose distance from the axis we denote by a, the protons have to reach first the distance r∼ a/2 from the axis. They would leave the cavity if their gyroradius at this point exceeds ∼ a/2. The magnetic field at the radius r is $B=2I/rc$⁠, where I is the current in the central column. The proton gyroradius $ρp$ is

where $γ$ and v are the relativistic factor and velocity of the proton. This condition has to be understood as an order of magnitude estimate, as we ignore here a directional dependences and details of the magnetic field distribution. Substituting this estimate into condition $ρp>r$ for $r∼a/2,$ as explained above, we find the following scaling for the maximum attainable energy (see Ref. 19):

where I_pA is a so-called proton Alfven current

[The proton Alfven current is related to the physics of proton beams: for $I∼IpA(γ2−1)1/2$ the proton gyroradius in the magnetic field created by the beam with a relativistic factor $γ$ becomes comparable to the beam radius, see, e.g., Ref. 93.] Interestingly, the maximum achievable energy (59) depends only on the current through the central column, not on any other parameters.

When applied to the laboratory experiment of Ref. 91, the estimate (59) provides a reasonable agreement with the observed cutoff energy: for I ∼ 1MA, one has $γmax−1=2(I/IA)2≈2×10−3,$ i.e., W_max∼2 MeV. It would be interesting to find the scaling of the maximum energy vs. the axial current in the laboratory experiments. If the energy scales indeed as I² in the non-relativistic regime $(γ−1≪1),$ this would be a strong argument in favor of the model for the cutoff energy of accelerated particles. Then, if applied to the astrophysical magnetically driven jets, where the current may reach many thousands of the Alfven currents (e.g., Ref. 94), this would set a maximum energy of the protons ejected from the jet and available for the further acceleration by other mechanisms acting in the external medium.^95,96 In the astrophysical literature, jets as a source of high-energy particles have been discussed in Ref. 97.

In this section, we discuss a situation where collisionless system evolves to a state where development of microfluctuations (in this case, Langmuir fluctuations) leads to re-appearance of a hydrodynamic description, where any information of the turbulent fluctuations is folded into equations for the moments of an electron distribution function^98,99 and does not show up in the dynamical equations. In this regard, a situation here is very similar to that in the ideal hydrodynamics, where collisional processes do not show up in Eqs. (1), although it is just the frequent collisions that make these equations meaningful.

In our case, the ensuing quasi-hydrodynamical equations possess simple self-similar solutions. Solutions of this type found in Refs. 98 and 99 are used mostly in the context of fast electrons produced by energetic phenomena in the solar atmosphere and then expanding along the magnetic field and generating radio-bursts in the range of Langmuir frequencies (see, e.g., Refs. 100–102, and an overview¹⁰³). These solutions can be of a potential interest also for the propagation of the fast particles in laser-produced plasmas.

We describe here a one-dimensional problem relevant to the case where the fast electrons are propagating along a guiding magnetic field, although the similar phenomena should occur also in a 3D unmagnetized expansion. The density of the background plasma is assumed to greatly exceed the density of the hot electrons, so that the quasineutrality constraint is ensured by a minor spatial redistribution of the cold electrons. The coordinate in a propagation direction is denoted by x, and the velocity in this direction is denoted by v. We will consider the particles moving “to the right,” v>0. If the zone where the hot electrons are formed is narrow compared to the distance L over which the electrons are spread, then the initial distribution function can be presented as $Ng0vδx$⁠; here N is the total number of fast particles ejected in the positive direction of axis x per unit area in the yz plane (i.e., N has a dimension of cm⁻²), v is velocity in the x direction, function g(v) describes the shape of the distribution over v, and $δ(x)$ is Dirac delta-function. With that, the normalization condition for $g0$ is

(⁠$g0$ has a dimension of inverse velocity).

For the free propagation of the hot electrons, their distribution function at any point would be

and the density of the fast electrons, $n*(x,t)$⁠, would be

However, the distribution function (62) is that of a mono-energetic beam and is unstable with respect to excitation of Langmuir waves. The latter cause electron diffusion in the velocity space towards the origin (in v) and forming a plateau.^104,105

The continuing expansion, with the faster electrons trying to outrun the slower ones, tends to re-establish the zone of the positive derivative of the distribution function, but rapid relaxation forces it back to the plateau state. Only a small deviation from the plateau state remains, the one needed to sustain the fluctuations at the level required to keep the distribution function close to a plateau state.

The efficacy of the instability in enforcing the plateau state can be understood in the following way. The instability growth-rate $Γ$ depends on the width $Δv$ of the bump in the distribution function. For the case of interest for us, with $Δv$ comparable to the average velocity $v¯$ of fast electrons, the growth rate $Γ$ can be evaluated as

where n is the density of a background plasma (e.g., Ref. 99). We can now compare the instability growth time $τ$ to the time $L/v¯$ within which the expanding fast electrons are reaching some distance L from the source.

Before the arrival of the fast electrons, the Langmuir fluctuations are at the thermal level. Because of that, the time $τ$ contains now an additional logarithmic factor $Λ∼10$ (compared to a single e-folding time $1/Γ$⁠)

The ratio of $τ$ and expansion time is

As a representative set of parameters for the problem of the solar radio-bursts, one can take n∼10⁶ cm⁻³, $v¯∼1010cm/s$(i.e., the hot electron energy ∼30 keV), $L∼3×1010$ cm (a fraction of the height of a Solar corona), and $Λ∼20$⁠. For these set of parameters, $ε$ is smaller than unity for relative densities of the energetic particles $n*/n>1.5×10−7$⁠. On the other hand, a relative density $n*/n$ typically exceeds 10⁻⁴,¹⁰⁶ and one has $ε∼1.5×10−3≪1.$ This means that the plateau approximation provides a good description of the distribution function.

In this state, the system will be evolving in a self-similar mode: any intermediate state could be considered as an initial state for the next step of the cloud expansion. In particular, the density should have the form $n*=(1t)h(x/t).$ The factor of 1/t accounts for the conservation of the total number of fast particles.

General solutions of this type for an arbitrary initial distribution function have been found in Refs. 98 and 99. Here we describe a solution for the case where the initial distribution function has a step-wise shape

with v₀ being the maximum velocity of the energetic particles. The density distribution corresponding to the free expansion, Eq. (63), has in this case a step-wise shape

(see a blue curve in Fig. 5).

According to discussion above, the relaxed distribution function will remain a plateau, but the height of the plateau will change vs. time and distance. We denote the height by the yet unknown function h(x,t), so that the distribution function is

For this distribution function, the density $n*$ and average velocity u of the fast electrons are

The continuity equation then yields

The solution has to have a form of $h=(1/t)h(ξ),ξ=x/(tv0).$ The function $hξ$ satisfies equation

so that the density distribution in the expanding cloud is

There is a divergence at the upper bound of the density distribution. It is caused by the singular (step-wise) nature of an initial velocity distribution (66). For the smooth distribution function, where one has to perform a more complicated analysis,⁹⁸ this singularity is smoothed in a natural way. In this paper, a spectrum of Langmuir oscillations was also derived. Here we simply truncate the function (74) at $x=(v0−δv)t/2,$ where $δv$ represents the width of the transition zone where the distribution function becomes zero. As the total number of particles per unit area is equal to N, integrating the density over x from zero to $x>v0t/2,$ we find the constant multiplier in Eq. (74):$const=N/ln(v0/δv).$ The density distribution is shown in Fig. 5 by a red line. To better fit the figure, the function (74) is multiplied by a factor of 1/2. The area under the blue and red curves is the same (accounting for the factor of 1/2 for the red curve) and equal to the initial number of particles moving to the right just after the burst that created the cloud of hot electrons.

We see that the cloud expansion is slowed down approximately by a factor of 2 compared to a free expansion. This result looks paradoxical, as the distribution function at any x<v₀t/2 still extends to v=v₀, and it may seem that the faster electrons with v>v₀/2 should outrun the expansion front and show up at larger distances than x=v₀t/2. The solution to this paradox is in the presence of the fast relaxation process: as soon as the electrons overtake the front, a strong instability develops, and they rejoin the marginally stable distribution.

This self-similarity solution can be constructed for the arbitrary initial distribution function monotonically decreasing from its value at v=0. Interestingly, two such solutions for different initial distribution functions are, generally speaking, not similar to each other. The similarity would additionally require that the initial distribution functions are similar as a function of v. Conversely, if two dynamical systems are similar, and one of them has a self-similar solution, the other will have such solution, scaled to the first one.

The scaling considerations may be helpful for developing reliable computational tools for simulating processes that are sometimes impossible to observe directly. We illustrate this by an example of investigating an instability leading to interpenetration of materials at the Helium-Hydrogen interface inside an exploding supernova.^6,17 The expected plasma parameters indicate that the hydrodynamic description is valid, and the Reynolds number is large (cf. Sec. III B). Even in the ensuing model of ideal hydrodynamics, the simulation is not simple because the flows driven at the initially rippled interface are quite complex, and verification of the underlying codes is needed (see a general overview of validation and verification in fusion reseach¹⁰⁷).

One can attempt to experimentally reproduce a scaled model of the phenomenon in question (in this case—a formation of bubbles and spikes and their deep interpenetration) and then compare the experimentally measured temporal history with the simulation based on the corresponding set of equations. This approach has been efficiently implemented, for example, in Ref. 31, where the Omega laser facility⁵³ was used to drive a shock in a setting scaled to the supernova. The ensuing complex structures of bubbles, spikes and shock front are shown in Fig. 1(a) in Sec. III B.

This experiment was simulated with the FLASH code,^108,109 and the results are shown in the grey-scale image in Fig. 1(b). One sees that the simulation reproduces the finest details of the experimental images, deep into the nonlinear stage of the instability, where the length of the bubbles and spikes exceeds the wavelength of the initial ripples by roughly a factor of 5, and their initial amplitude by a factor of 50–100.

More recently a scaled experiment designed to imitate instabilities driven by an expanding supernova ejecta around a young supernova remnant was performed.¹¹⁰ Here the CRASH code was used,¹¹¹ and it successfully reproduced experimental results. The success of numerical simulation of the scaled experiment indicates the validity of the hydrodynamical model and its numerical implementation for this class of problems.

Another important implication of these results is that a validated code can be safely applied to predicting the dynamics for a range of variations: different spectral composition of the initial ripples, their amplitudes, different elemental combinations, shock strengths, and other parameters, thereby producing predictive results for a broader set of mix models based on ideal hydrodynamics and thereby producing a broader characterization of the phenomenon. This capability is important since the scaled experiments are costly, and one cannot test all the variations that affect the outcome. Of course, one needs to be careful with these variations; for significant departures from initial conditions new effects may enter the problem (e.g., radiative transport may become significant, or equations of state may strongly deviate from the base model). But a code carefully validated against a scaled experiment by replicating a “typical” setting can be quite helpful.

To summarize the steps in this approach: first, the experiment has to be designed to correctly (with scaling parameters kept the same and initial conditions being geometrically similar) scale the initial system (e.g., a rippled interface between two layers in the exploding supernova, with a strong shock breaking through the interface); then, comparing the outcome of the scaled laboratory experiment with the results of numerical simulations, one can gain confidence in the code performance; then, one would be able to use this code to simulate a large variety of the similar problems that do not require inclusion of a new physics and do not qualitatively deviate from the geometry of the experimental system. This would allow one to circumvent the need of experimental study of all these variations and rely on the use of the codes. Of course, one has to be cautious that the variations do not extend to new regimes with physics not covered by the underlying equations and dimensionless parameters.

We have considered the use of scaling and similarity techniques for a variety of dynamical systems describing transitions of plasma objects between the initial and final states, which can be very different from one another. The phenomena considered correspond to a broad range of plasma models, from ideal magnetohydrodynamics to collisionless systems and collisionless microturbulence. The main conclusion that can be drawn from our analyses is that the formal use of similarity techniques has to be accompanied by a thoughtful analysis of the applicability of basic assumptions. This comment is true for any similarity exercise, but becomes especially important for the dynamical similarities, where the initial object evolves to an entirely different one in terms of dimensions, shapes, and plasma parameters. New spatial and temporal scales may appear in the system, whose interplay may involve new effects and change the relative role of various processes. So, the analysis of the dynamical similarities goes well beyond simply casting a set of equations in a dimensionless form.

As an example, one can refer to the problem of interaction between interpenetrating plasma streams, where the role of electrostatic, non-quasineutral fluctuations, and quasineutral electromagnetic modes may change significantly over time. The same happens with the contribution of collisions to an apparently collisionless flow. Evaluating the time for the onset of these processes (e.g., the appearance of the dissipative-scale vortices in the Rayleigh-Taylor mix problem, see, e.g., Refs. 7, 8, 13, and 32) is an important part of delineating the range of validity of a certain similarity.

The scope of a similarity analysis can vary enormously. On one end is a complete mapping of one dynamical system to another one, which has more modest (sometimes by many orders of magnitude) input parameters, but evolves identically, up to the scaling transformations, to the first one. This can be achievable for the systems describable by the ideal MHD equations, allowing for shock waves (“Euler similarity”). This similarity can sometimes be extended to include dissipative processes, like viscosity and magnetic diffusivity.

In other cases, straightforward similarity scaling is impossible because of the presence of external factors (like a liner radius vs time dependence, a(t), in a MagLIF approach) that are not covered by the same processes as the imploded plasma. Here one can attempt to find a way of describing the plasma energy losses for several consecutive stages of the implosion and thereby identify the acceptable implosion path (Sec. IV B).

Attempting to address a physics problem by the use of a similarity analysis leads to a better understanding of the processes involved and to identifying the most important ones for a particular system. Even if details of these processes are not fully understood, one can sometimes identify important and measurable parameters that can be scaled to laboratory experiments. An example is scaling of the maximum proton energy produced in a cocoon of the tower jet vs. the jet current,¹⁹ see Sec. V D. Another example is the scaling of the filament sizes for the Weibel instability in the experiments with a wide variation of the input parameters.^81,82,84

Similarity analyses are helpful in the development of reliable codes for a quantitative description of the processes hidden from direct observations, like the processes occurring inside the exploding supernova. Here, if one believes that equations governing a particular process are available, one can benchmark the code against the properly scaled laboratory experiments (see Sec. III C, Ref. 31) and use the code with confidence for modeling the same process in a natural environment, varying in a reasonable range the initial conditions (e.g., the wavelength and the amplitude of initial ripples, and the shock strength) and thereby providing a broad description of the model.

In conclusion, the similarity technique is not a magic wand, but it is certainly one of the useful instruments in the toolbox of physics research.

The author acknowledges many years of collaboration in the area of laboratory astrophysics with his colleagues from many institutions, especially H.-S. Park, B. A. Remington, J. S. Ross (LLNL), and R. P. Drake (University of Michigan, Ann Arbor).

The author thanks the members of the LLNL Fusion Theory Group, especially B. A. Cohen, R. H. Cohen, A. Friedman, L. LoDestro, and T. D. Rognlien for general support, illuminating discussions, and friendly criticisms.

The author is grateful to the anonymous referee for a number of valuable recommendations.

This work was performed under the Auspices of the U.S. Department of Energy by Lawrence Livermore National Security, LLC, Lawrence Livermore National Laboratory, under Contract No. DE-AC52-07NA27344.

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