Here is a number that should bother you. In the solar wind near the Earth’s orbit the proton density is about $5$ per cubic centimetre and the temperature is around $10^{5}$ K. Work out how far a proton travels before Coulomb collisions turn it appreciably off course and you get something like $10^{13}$ cm, most of the way to an astronomical unit. Yet stream-interaction regions, shocks and flux ropes in that wind are routinely modelled with fluid equations, often on scales comparable to this or smaller. The hot gas that fills a galaxy cluster is worse. There the mean free path is some twenty kiloparsecs, comparable to the turbulent eddies and cold fronts that people model with magnetohydrodynamics every day. By the standard we apply to air or water, these plasmas have no business behaving like fluids.1The numbers in this article are representative, order-of-magnitude values. A Coulomb mean free path depends on the species, the temperature and the relaxation process used to define it. I use the thermal estimate common in the cluster literature, $\lambda_{\rm mfp}\approx23\,{\rm kpc}\,(T/10^{8}\,{\rm K})^{2}(n_e/10^{-3}\,{\rm cm^{-3}})^{-1}$ (Sarazin 1986), with the Coulomb logarithm adjusted for each environment. That gives about $10^{13}$ cm for the solar wind (with $L=1$ AU), $2\times10^{12}$ cm for the warm ionised medium ($L=100$ pc) and $20$ kpc for the cluster ($L=200$ kpc). Other conventions give values a few times smaller. In a real application $L$ should be the relevant gradient scale, not the size of the whole system. The NRL Plasma Formulary (2023) lists species-dependent collision rates and plasma scales.
They behave like fluids anyway. Magnetohydrodynamics predicts the structure of the solar wind, the propagation of Alfvén waves through it, the shapes of cold fronts in clusters, the collimation of astrophysical jets and the stability of accretion discs, and it does so well enough that its failures are informative rather than catastrophic. The electromagnetic field obviously has something to do with this. The question is what, exactly, the field supplies, and what else is needed before a closed set of fluid equations can be justified.
Start with why a neutral gas needs collisions at all, since the requirement is easy to take for granted. A fluid description assigns a density, a velocity and a pressure to each point and claims that their evolution closes on itself. That is a strong claim. The underlying object is a distribution function $f(\mathbf{x},\mathbf{v},t)$ in six-dimensional phase space, and to shrink it down to a handful of local fields you need some way of writing the higher velocity moments in terms of the ones you keep. Collisions provide it. They push the local distribution toward a Maxwellian on a timescale $\nu^{-1}$. If that is short compared with the time over which the flow changes, and the mean free path $\lambda$ is short compared with the scale $L$ over which gradients vary, the distribution stays close to Maxwellian everywhere and the fluid equations close. The controlling parameter is the Knudsen number $$\mathrm{Kn}=\frac{\lambda}{L},$$ and the local collisional fluid limit needs $\mathrm{Kn}\ll1$, along with collisions that are fast compared with the dynamics. For air at room temperature and atmospheric pressure $\lambda$ is about $70$ nanometres, so a flow that varies over $10$ cm has $\mathrm{Kn}\sim7\times10^{-7}$. Nobody worries about this for a river. People do worry about it in microfluidics and in the thin upper atmosphere.2The transport coefficients of a neutral gas come out of the same expansion: viscosity and thermal conduction are first-order corrections to the Maxwellian. Kinematic viscosity and thermal diffusivity both scale as the mean free path times the thermal speed; dynamic viscosity and conductivity carry extra factors of density and heat capacity. Once $\mathrm{Kn}$ is no longer small the Chapman–Enskog expansion stops being controlled and you need kinetic theory.
In the solar wind and the cluster, $\mathrm{Kn}$ is $0.1$ or larger. What rescues the situation is that a plasma can organise its motion collectively, through the electromagnetic field, instead of through binary encounters. To see how, it helps to build the relevant scales from scratch, because the relations between them do most of the explaining.
The most basic fact about a plasma is that it hides its own charges. Drop a test charge $q$ into a neutral plasma of density $n$ and electron temperature $T$. The electrons are light and rearrange quickly, so let them reach thermal equilibrium in the resulting potential while the much heavier ions stay put as a uniform neutralising background. Away from the test charge, Poisson’s equation reads $$\nabla^{2}\phi=-4\pi\rho_q=-4\pi e\left(n_i-n_e\right),$$ with $n_i=n_0$ fixed and the electrons Boltzmann-distributed, $n_e=n_0\exp(e\phi/k_BT)$. For potentials small compared with the thermal energy, $|e\phi|\ll k_BT$, expand the exponential to first order, so that $n_i-n_e\simeq-n_0e\phi/k_BT$ and $$\nabla^{2}\phi=\frac{4\pi n_0e^{2}}{k_BT}\,\phi\equiv\frac{\phi}{\lambda_D^{2}}.$$ This is a screening equation, not Laplace’s equation, and its spherically symmetric solution around a point charge is $$\phi(r)=\frac{q}{r}\,e^{-r/\lambda_D},\qquad \lambda_D=\sqrt{\frac{k_BT}{4\pi n_0e^{2}}}.$$ The Coulomb potential dies off exponentially beyond $\lambda_D$, the electron Debye length.3I use Gaussian cgs units throughout. The symbol $n$ is the electron number density, equal to the proton density in a quasineutral hydrogen plasma; $\rho_q$ is charge density, while $\rho$ in the fluid equations below is mass density. At the origin the screening equation has the point source $-4\pi q\delta^{(3)}(\mathbf r)$, which fixes the prefactor of the Yukawa potential. If the ions are allowed to respond too, they add their own screening term, and in general $\lambda_D^{-2}=\sum_s 4\pi n_sq_s^{2}/k_BT_s$. For singly charged ions at the electron temperature this gives $\lambda_D=\sqrt{k_BT/8\pi ne^{2}}$, smaller by $\sqrt2$. Which one applies depends on the timescale: faster than the ion plasma period, the ions haven’t had time to move. I stick with the electron Debye length because it is the one tied to the plasma frequency and to the classical Coulomb cutoff below. The literature uses both, so check which one a quoted number assumes.
In practical units, $\lambda_D\simeq743\,\sqrt{T[\mathrm{eV}]/n[\mathrm{cm^{-3}}]}$ cm. The warm ionised component of the interstellar medium, with $n\simeq0.3$ and $T\simeq8000$ K, has a Debye length of about $11$ metres. The solar wind at $1$ AU comes out at about $10$ metres, and a coronal plasma with $n\simeq5\times10^8\,\mathrm{cm^{-3}}$ and $T\simeq10^6$ K at about $3$ millimetres. Set against the systems these plasmas fill, those numbers are absurdly small. On scales much larger than $\lambda_D$, and for dynamics slow compared with the plasma’s response, quasineutrality holds to excellent accuracy. That is the first thing a plasma does that a neutral gas cannot: it polices its own charge separation.
How many particles take part in that policing? Count the number inside a Debye sphere, $$N_D=n\cdot\frac{4\pi}{3}\lambda_D^{3}.$$ For the warm interstellar medium this is about $2\times10^{9}$, for the solar wind at $1$ AU about $2\times10^{10}$, and for an intracluster plasma with $n\simeq10^{-3}\,\mathrm{cm^{-3}}$ and $T\simeq10^8$ K about $4\times10^{16}$. These are enormous numbers, and they are the defining feature of the plasmas in question. Each particle interacts, weakly, with billions of others at once through their overlapping Coulomb fields, and no single neighbour matters much. A plasma with $N_D\gg1$ is called weakly coupled, and the name is precise: each pairwise interaction is weak, while the collective ones are not.
The second thing a plasma does is oscillate. Take a slab of plasma and shift all the electrons by a small distance $x$ relative to the ions, which stay put because they are nearly two thousand times heavier. The shift leaves a surface charge density $\sigma=nex$ on one face and the opposite on the other, and the field between them, $E=4\pi nex$, pulls the electrons back. Newton’s law for an electron reads $$m_e\ddot{x}=-eE=-4\pi ne^{2}x,$$ a harmonic oscillator with $$\omega_p=\sqrt{\frac{4\pi ne^{2}}{m_e}}.$$ In practical units $\omega_p\simeq5.6\times10^{4}\sqrt{n[\mathrm{cm^{-3}}]}$ radians per second, so in the warm interstellar medium the plasma frequency is about $5$ kHz and the period about $0.2$ ms. Any charge separation is undone on a timescale of order $\omega_p^{-1}$. In this idealised model the electrons overshoot and keep oscillating; it takes damping, which we have left out, to make them settle.
These two quantities are not independent, and the connection between them is the first real insight. The electron thermal speed is $v_{\rm th,e}=\sqrt{k_BT/m_e}$, and dividing it by the plasma frequency gives $$\frac{v_{\rm th,e}}{\omega_p}=\sqrt{\frac{k_BT}{m_e}}\cdot\sqrt{\frac{m_e}{4\pi ne^{2}}}=\sqrt{\frac{k_BT}{4\pi ne^{2}}}=\lambda_D.$$ The Debye length is the distance a thermal electron covers in one response time $\omega_p^{-1}$ (not in a full period, $2\pi/\omega_p$). Screening and oscillation are the same balance between thermal motion and electrostatic restoring force, one seen in space and the other in time.
Now for collisions, which in a plasma are stranger than they look. Two charges scatter through the Coulomb interaction, and the impact parameter that gives a large deflection is set by equating the potential energy at closest approach to the thermal kinetic energy, $$\frac{e^{2}}{b_{90}}\sim k_BT\qquad\Longrightarrow\qquad b_{90}\sim\frac{e^{2}}{k_BT}.$$ In the warm interstellar medium that is about $2\times10^{-7}$ cm, a tiny target. If large-angle encounters were the whole story, the cross-section would be $\pi b_{90}^{2}$ and the mean free path would be gigantic. But the Coulomb force is long-ranged, and a particle passing at $b\gg b_{90}$ still picks up a small deflection, of order $b_{90}/b$. These small kicks are far more numerous, since the number of encounters with impact parameter near $b$ grows as $b\,db$, and they add up as a random walk in angle. Summing the mean-square deflection rate over impact parameters gives, up to numerical factors, $$\frac{d\langle\Delta\theta^{2}\rangle}{dt}\sim nv\int\left(\frac{b_{90}}{b}\right)^{2}b\,db=nvb_{90}^{2}\ln\left(\frac{b_{\max}}{b_{\min}}\right).$$ The integral diverges logarithmically at both ends. It is cut off below at $b_{90}$, where the small-angle approximation fails, and above at $\lambda_D$, where screening switches the interaction off. The effective cross-section is therefore $\sigma\sim b_{90}^{2}\ln\Lambda$ with $$\Lambda=\frac{\lambda_D}{b_{90}},$$ and the mean free path is $$\lambda_{\rm mfp}=\frac{1}{n\sigma}\sim\frac{(k_BT)^{2}}{ne^{4}\ln\Lambda}\propto\frac{T^{2}}{n\ln\Lambda}.$$ The logarithm changes slowly, so the scaling is usually quoted as $T^2/n$. Numerical prefactors differ between treatments by up to an order of magnitude depending on which species and which relaxation process you follow, so trust the scaling more than the coefficient.4The standard results are the Spitzer collision times, which distinguish electron–electron, electron–ion and ion–ion relaxation and give different coefficients for momentum transfer, energy exchange and temperature equilibration. The classical estimate $b_{\min}\sim b_{90}$ also has to be replaced by the de Broglie wavelength when that is larger, as it is for electrons in the hottest cluster gas. Notice, too, that the large-$b$ divergence is cut off by screening, so the Debye length enters the collision rate itself. Even in principle, collisions in a plasma are not independent of its collective behaviour.
The $T^{2}/n$ scaling settles half of the paradox we started with. Hot, thin plasmas have long mean free paths because both factors push the same way. Whether a given plasma counts as collisionless still depends on the length and time scales in the problem, and a great many astrophysical problems land on the collisionless side. What the scaling doesn’t explain is why those plasmas behave like fluids.
The explanation starts from a relation that is easy to miss. Put the electron Debye length into the classical estimate for $\Lambda$ and use $k_BT=4\pi ne^{2}\lambda_D^{2}$: $$\Lambda=\frac{\lambda_D}{b_{90}}\sim\frac{\lambda_Dk_BT}{e^{2}}=\frac{\lambda_D\cdot4\pi ne^{2}\lambda_D^{2}}{e^{2}}=4\pi n\lambda_D^{3}=3N_D.$$ So the Coulomb logarithm is, give or take the choice of cutoffs, the logarithm of the number of particles in a Debye sphere. That ties together two facts we had been treating separately. Large $N_D$ means that individual Coulomb interactions are weak and that the collective field is smooth and well defined. It does not stop collisions mattering over long enough times. Whether a particular problem is collisional is decided by $\nu/\omega$ and $\lambda_{\rm mfp}/L$, not by $N_D$. The astrophysical plasmas here are deep in the weakly coupled regime, with $\ln\Lambda$ between about $20$ and $40$.
So Coulomb relaxation is slow, yet the plasma still reacts to charge separation within $\omega_p^{-1}$ and over $\lambda_D$. That is a kind of local coupling, but it only constrains the charge density. It says nothing about how momentum gets around. For that we need the magnetic field.
A nonrelativistic charged particle in a uniform magnetic field $\mathbf{B}$, with nothing else acting on it, moves freely along the field and gyrates around it with angular frequency $$\Omega_c=\frac{|q|B}{mc}$$ and radius $r_g=v_\perp/\Omega_c$. The sign of $q$ sets the sense of rotation. In practical units the proton gyrofrequency is $\Omega_{cp}\simeq9.6\times10^{3}\,B[\mathrm{G}]$ radians per second, and the electron’s is larger by the mass ratio. In the warm interstellar medium, with $B\simeq5\,\mu$G, a thermal proton moving at $v_{\rm th,p}=\sqrt{k_BT/m_p}$ gyrates on a radius of about $170$ km.
Compare that with the collisional mean free path in the same gas, about $2\times10^{7}$ km. The gyroradius is smaller by a factor of $10^{5}$. In the solar wind, with a field of about $5$ nT, the ratio is around a million. In a cluster with a microgauss field, a thermal proton gyrates on a radius of about $10^{5}$ km while travelling some $20$ kpc between collisions, a ratio of $10^{12}$ to $10^{13}$. Across the field, its orbit is the size of a large planet. Along the field, it can cross a good fraction of the cluster core before anything deflects it. The Lorentz force ties each particle to its field line far more tightly than collisions ever could, but it does nothing to stop the particle streaming along that line or its guiding centre drifting.
It is tempting to declare victory here: across the field, gyration does the job that collisions do in a neutral gas, so the fluid equations follow. That is too quick, and it pays to separate what magnetisation gives you from what it doesn’t.
What it gives you is scale separation. When the fields change slowly compared with the gyrofrequency and over distances large compared with $r_g$, the fast gyration can be averaged away, leaving guiding centres that drift across the field and stream along it. A collision then moves a particle across the field by about $r_g$ rather than $\lambda_{\rm mfp}$, although the resulting transport still depends on the scattering rate and the field geometry. That is a big simplification, and it underpins a family of reduced kinetic theories that work even without collisions.
What it doesn’t give you is closure. A fluid theory needs the higher moments written in terms of the ones it keeps, and gyration supplies no such rule. Nothing confines particles along the field, so the parallel distribution can be far from Maxwellian, carry a heat flux that no local formula captures, and trade energy with waves through Landau damping. The pressure can become anisotropic, because $p_\perp$ and $p_\parallel$ respond differently to compression. At finite $k_\perp r_g$ there are finite-Larmor-radius corrections that ordinary MHD leaves out. Scale separation gets you a gyro-averaged kinetic theory. Getting from there to single-fluid MHD takes further, separately justified approximations for the pressure, the heat flux and the moments above them. The last part of this article is about how a plasma can arrange some of those conditions for itself.5For a systematic treatment see Schekochihin et al. (2009), Astrophysical gyrokinetics. In their low-frequency, anisotropic ordering, the Alfvénic part of the turbulent cascade obeys reduced-MHD equations even at collisionless scales, while the compressive fluctuations need a kinetic description. Whether a fluid reduction is valid can depend on which fluctuations you are following.
Put the scales together and you get a hierarchy, which is the real subject of this article. For the warm interstellar medium, $$\lambda_D\sim11\ \mathrm{m}\ \ll\ r_g\sim170\ \mathrm{km}\ \ll\ \lambda_{\rm mfp}\sim2\times10^{7}\ \mathrm{km}\ \ll\ L\sim100\ \mathrm{pc},$$ spanning some seventeen orders of magnitude from the Debye length to the system size. The ordering matters more than the individual numbers. Below $\lambda_D$, charge separation has to be kept. Between $\lambda_D$ and $r_g$, low-frequency fluctuations are quasineutral but still resolve the gyration. On perpendicular scales well above $r_g$, and at frequencies well below $\Omega_c$, gyration averages out, though guiding centres still stream along the field. Only when the parallel scales are also much longer than $\lambda_{\rm mfp}$, and the evolution is slow compared with collisions, does a local collisional closure become available. So the hierarchy shows that a gyro-averaged description becomes useful far below the collisional mean free path, provided the timescales cooperate too. The step to a single fluid needs an argument about the distribution function, and the ordering of scales doesn’t provide one.
With the hierarchy in hand, it’s worth writing down exactly what magnetohydrodynamics assumes, because the electromagnetic orderings and the fluid closure are separate ingredients. The equations are the usual continuity and momentum equations with a Lorentz force, plus an induction equation for the field: $$\begin{aligned}\frac{\partial\rho}{\partial t}+\nabla!\cdot(\rho\mathbf v)&=0,\ \rho\frac{D\mathbf v}{Dt}&=-\nabla p+\frac{1}{c}\mathbf J\times\mathbf B,\ \frac{\partial\mathbf B}{\partial t}&=\nabla\times(\mathbf v\times\mathbf B)+\eta\nabla^2\mathbf B.\end{aligned}$$ Here $D/Dt=\partial_t+\mathbf v\cdot\nabla$, $\rho$ is the mass density, $p$ the total scalar thermal pressure and $\eta$ a uniform magnetic diffusivity, with $\nabla\cdot\mathbf B=0$ and $\mathbf J=(c/4\pi)\nabla\times\mathbf B$. Several orderings sit behind these equations. Scales much larger than $\lambda_D$ and frequencies much lower than $\omega_p$ give quasineutrality. Scales larger than the gyroradii and frequencies below the gyrofrequencies allow gyro-averaging. Slow, nonrelativistic dynamics let us drop the displacement current. None of this gives $\mathbf E=-\mathbf v\times\mathbf B/c$. That ideal Ohm’s law is a separate approximation, which needs the resistive, Hall, electron-pressure and electron-inertia terms of the generalised Ohm’s law to be small. Keeping only the resistive term gives $\mathbf E+\mathbf v\times\mathbf B/c=(4\pi\eta/c^2)\mathbf J$, which is the form used above. None of these orderings requires $\lambda_{\rm mfp}\ll L$, and none of them closes the equations either. Moments of the kinetic equation form an infinite hierarchy in which each moment depends on the next, so scalar-pressure MHD needs an energy equation and some justified treatment of the heat flux and viscous stress. The simplest non-dissipative choice is $D(p/\rho^\gamma)/Dt=0$, with $\gamma=5/3$ for a monatomic gas. Collisions can justify it, and so, sometimes, can effective scattering or a restricted dynamical regime. A collisionless plasma is under no obligation to obey it.6Once resistivity is kept, the energy equation must include Ohmic heating, and other transport or heating terms where they matter; the adiabatic law is the ideal limit. Dropping the Hall and electron-inertia terms brings in the ion and electron inertial lengths, so the four scales in the figure above are not a complete checklist for MHD. See Fitzpatrick’s derivation of the MHD equations, and, for the Hall extension, Acheritogaray et al. (2011).
The induction equation carries most of the physics that makes magnetised plasmas behave as they do. The ratio of the two terms on its right-hand side is the magnetic Reynolds number, $$R_m=\frac{vL}{\eta},$$ where $\eta$ is the magnetic diffusivity, proportional to the electrical resistivity. In a fully ionised plasma the Spitzer value scales as $T_e^{-3/2}$, apart from the Coulomb logarithm. For the warm ionised medium at $T\simeq8000$ K this gives $\eta\sim10^{7}\ \mathrm{cm^{2}\,s^{-1}}$, and with $v\sim10\ \mathrm{km\,s^{-1}}$ and $L\sim1$ pc, $R_m$ comes out around $10^{17}$. If the other nonideal terms are small too, flux freezing is an excellent approximation on large scales: the magnetic flux through any surface moving with the fluid is conserved, and fluid elements that start on the same field line stay on the same field line. Thin current sheets are the exception, since their local $R_m$ can be far smaller. That one property accounts for much of what magnetic fields do in astrophysics, from the winding up of galactic fields by differential rotation to the launching of jets.
It also means the field is not a fixed backdrop. Currents in the plasma reshape it, and it pushes back. Tension along the field lines resists bending, the magnetic pressure $B^{2}/8\pi$ resists compression across them, and the ratio of thermal to magnetic pressure is $$\beta=\frac{8\pi p}{B^{2}},\qquad p=p_e+p_i.$$ For a hydrogen plasma with equal electron and ion temperatures, $p\simeq2nk_BT$. The warm ionised example then has $\beta\simeq0.7$. In the solar wind $\beta$ is typically of order unity, though it varies a lot. In the cluster it is in the hundreds, so magnetic pressure hardly matters to the overall force balance. The field still matters, though. It steers heat transport, sets the thresholds for the pressure-anisotropy instabilities described below, and shapes local structures.
So far this reads as a success story, so it is worth being clear where it stops. Gyration limits motion across the field, but a uniform field does nothing to stop particles streaming along it. Many of the important departures from MHD come from that asymmetry, though not all of them. Nonuniform fields reflect particles through the mirror force, and Hall effects, electron inertia and finite-Larmor-radius corrections impose limits of their own.
Heat transport is the clearest case. Start with the classical collisional estimate, which makes the geometry easy to see. An electron travels a distance of order $\lambda_{{\rm mfp},e}=v_{{\rm th},e}\tau_e$ along the field in one deflection time $\tau_e$, while each scattering shifts its guiding centre across the field by about $r_{g,e}$. In the strongly magnetised limit $\Omega_{ce}\tau_e\gg1$ the ratio of conductivities is, up to numerical factors, $$\frac{\kappa_{e\perp}}{\kappa_{e\parallel}}\sim\frac{1}{(\Omega_{ce}\tau_e)^2}=\left(\frac{r_{g,e}}{\lambda_{{\rm mfp},e}}\right)^2\ll1.$$ These are electron quantities. When the electron and ion temperatures are similar, electrons carry most of the parallel heat flux, so the proton gyroradius from earlier doesn’t belong in this estimate. In the collisional regime the parallel conductivity is the Spitzer–Braginskii value, as long as the electron mean free path is short compared with the parallel temperature-gradient scale. In a weakly collisional plasma such as the intracluster medium, the heat flux can instead become nonlocal or be capped by kinetic instabilities, and tangled fields and turbulence change how heat moves across the mean field. So the classical ratio shows how extreme the microscopic anisotropy is; it is not a suppression factor to plug into a cluster model. What survives is that the field geometry decides where heat can go. A cold-front model with isotropic conduction misses this, and the observed sharpness of cold fronts has led to models in which the field is draped along the front.7For the classical coefficients and where they apply, see Fitzpatrick, Classical Closure Scheme. For kinetic limits on the parallel electron heat flux, see Roberg-Clark et al. (2018), Wave generation and heat flux suppression in astrophysical plasma systems. For draping at cold fronts, see Lyutikov (2006), Magnetic draping of merging cores and radio bubbles in clusters of galaxies.
A subtler failure is that the pressure stops being a single number. A scalar pressure is justified when collisions keep the velocity distribution nearly isotropic. Without them there is no reason for motion along $\mathbf{B}$ and motion across it to share a temperature. The natural description has two pressures, $p_\parallel$ and $p_\perp$, and the adiabatic invariants of gyromotion tell us how each one evolves. The magnetic moment $\mu=m v_\perp^{2}/2B$ is conserved when changes are slow, so strengthening the field raises $p_\perp$, while squeezing plasma along a flux tube raises $p_\parallel$. Following both through gives the double-adiabatic, or CGL, relations, $$\frac{d}{dt}\left(\frac{p_\perp}{\rho B}\right)=0,\qquad \frac{d}{dt}\left(\frac{p_\parallel B^{2}}{\rho^{3}}\right)=0,$$ which replace the single adiabatic law of ordinary gas dynamics. They come from taking moments of the kinetic equation in the long-wavelength, low-frequency limit, ignoring collisions, and closing the system by setting the heat fluxes to zero. That last step is the weak point.8The closure is due to Chew, Goldberger and Low (1956). Its problem is that zero heat flux is not generally justified in a collisionless plasma, where particles stream along field lines and carry energy with them. There is no universal local closure for that transport. Landau-fluid closures, which mimic collisionless phase mixing and Landau damping in the higher moments, are a useful compromise in the regimes they are built for; see Hammett and Perkins (1990).
What happens next is the most interesting part of the story, and it brings back a version of the fluid picture from an unexpected direction. Pressure anisotropy cannot grow without limit, because a plasma with $p_\parallel$ well above $p_\perp$ is unstable. Particles streaming along a curved field line exert a centrifugal force on it that helps the bend grow, and once the pressure anisotropy beats the magnetic tension the field buckles. In the long-wavelength fluid description the threshold is $$p_\parallel-p_\perp>\frac{B^{2}}{4\pi}.$$ This is the firehose instability, named for the way an unsecured hose whips about. The opposite anisotropy is unstable too. When $p_\perp$ exceeds $p_\parallel$ by enough, small dips in the field strength trap particles and deepen. That is the mirror instability, with a threshold for a simple gyrotropic distribution of approximately $$\frac{p_\perp}{p_\parallel}-1>\frac{1}{\beta_\perp},$$ where $\beta_\perp=8\pi p_\perp/B^2$. Precise kinetic thresholds depend on the species and the shape of the distribution. How fast either instability grows depends on how far past threshold the plasma has been pushed, but both can respond much faster than the large-scale flow that drives them. They settle the anisotropy in different ways. Firehose fluctuations scatter particles in pitch angle. Mirror modes at first regulate the anisotropy by trapping particles as the fluctuations grow, with strong scattering appearing later at the sharp edges of the mirror structures.
Where that scattering happens, it acts like an effective collision. It breaks conservation of the magnetic moment and reshuffles pitch angles at a rate set by the fluctuations, not by Coulomb encounters. Trapping offers a second way to cap the anisotropy without randomising every orbit straight away. The plasma is, in effect, responding to its own distribution function: driving pushes it past a stability boundary, fluctuations grow, and their feedback holds it near marginal stability. At high $\beta$ those boundaries allow only a small fractional difference between $p_\perp$ and $p_\parallel$, although even that small difference can rival the magnetic tension. It isn’t exact isotropy, and it doesn’t fix the heat flux or the higher moments. But it is one important reason a weakly collisional plasma can behave more like a fluid than Coulomb collisions alone would suggest. Where this effective scattering dominates, the corresponding mean free path can be far shorter than the Coulomb one. A full MHD closure still needs its own justification.9The difference between firehose scattering and mirror trapping is shown in Kunz, Schekochihin and Stone (2014), Firehose and Mirror Instabilities in a Collisionless Shearing Plasma. A similar self-regulating loop appears in cosmic-ray transport: cosmic rays streaming along the field excite waves that scatter them. If wave damping is weak, their bulk drift is held near the Alfvén speed; stronger damping lets them stream faster. The loops are analogous, not identical; see Thomas and Pfrommer (2019), Cosmic-ray hydrodynamics.
Two other places show different limits of the simplest fluid picture. The first is magnetic reconnection. Ideal MHD preserves magnetic connectivity, yet solar flares, magnetospheric substorms and accretion-disc coronae all involve changes in field topology. In collisionless reconnection, current sheets thin down to ion or electron kinetic scales, where the terms dropped from ideal MHD become important and the field is no longer frozen in. The large-scale flow can stay fluid-like while a small nonideal region decides how the connectivity changes. Fluid theory isn’t abandoned entirely, though: resistive MHD reconnects field lines perfectly well in a collisional plasma, without the layer reaching kinetic scales.10For the resistive-fluid picture, see Fitzpatrick’s plasma-physics notes, chapter Magnetic Reconnection.
The second is high-energy particles. The gyroradius grows with perpendicular momentum, $r_g=p_\perp c/(|q|B)$, so an ultrarelativistic proton at $10^{15}$ eV in a $1\,\mu$G field has a gyroradius of about a parsec: $$r_g\simeq1.1\,\mathrm{pc}\left(\frac{E}{1\,\mathrm{PeV}}\right)\left(\frac{B}{1\,\mu\mathrm G}\right)^{-1}\sin\alpha,$$ where $\alpha$ is the pitch angle between the momentum and the field. At $10^{20}$ eV the same formula gives around $100$ kpc, several times the diameter of the Galactic disc. Particles like these cannot be folded into the thermal fluid. Their transport has to be treated kinetically, or through a cosmic-ray fluid or diffusion model with its own scattering closure. The same plasma can therefore be a fluid for its thermal particles and a scattering medium for its suprathermal ones. There is no contradiction in that. The fluid approximation is always a statement about a particular population on a particular scale.11The coefficient follows from $pc\simeq E$ and $r_g=p_\perp c/(eB)$ in Gaussian units. Cosmic-ray fluid models with an explicit scattering closure are developed, for example, by Thomas and Pfrommer (2019).
Underneath all of this sits $N_D$, the number of particles in a Debye sphere. When it is large, pair interactions are weak and collective behaviour is well defined, and $\ln\Lambda\simeq\ln(3N_D)$ links the two without settling whether any given problem is collisional. Space is nearly empty, but that doesn’t make each particle a free agent. Dilution weakens binary collisions while leaving the collective field free to organise the plasma, as long as the plasma’s own scales stay small compared with the system. Whether that organised motion can be reduced to a fluid then comes down to scale separation, and to how the kinetic degrees of freedom that remain are kept in check.
References and Footnotes
- 1The numbers in this article are representative, order-of-magnitude values. A Coulomb mean free path depends on the species, the temperature and the relaxation process used to define it. I use the thermal estimate common in the cluster literature, $\lambda_{\rm mfp}\approx23\,{\rm kpc}\,(T/10^{8}\,{\rm K})^{2}(n_e/10^{-3}\,{\rm cm^{-3}})^{-1}$ (Sarazin 1986), with the Coulomb logarithm adjusted for each environment. That gives about $10^{13}$ cm for the solar wind (with $L=1$ AU), $2\times10^{12}$ cm for the warm ionised medium ($L=100$ pc) and $20$ kpc for the cluster ($L=200$ kpc). Other conventions give values a few times smaller. In a real application $L$ should be the relevant gradient scale, not the size of the whole system. The NRL Plasma Formulary (2023) lists species-dependent collision rates and plasma scales. ↩︎
- 2The transport coefficients of a neutral gas come out of the same expansion: viscosity and thermal conduction are first-order corrections to the Maxwellian. Kinematic viscosity and thermal diffusivity both scale as the mean free path times the thermal speed; dynamic viscosity and conductivity carry extra factors of density and heat capacity. Once $\mathrm{Kn}$ is no longer small the Chapman–Enskog expansion stops being controlled and you need kinetic theory. ↩︎
- 3I use Gaussian cgs units throughout. The symbol $n$ is the electron number density, equal to the proton density in a quasineutral hydrogen plasma; $\rho_q$ is charge density, while $\rho$ in the fluid equations below is mass density. At the origin the screening equation has the point source $-4\pi q\delta^{(3)}(\mathbf r)$, which fixes the prefactor of the Yukawa potential. If the ions are allowed to respond too, they add their own screening term, and in general $\lambda_D^{-2}=\sum_s 4\pi n_sq_s^{2}/k_BT_s$. For singly charged ions at the electron temperature this gives $\lambda_D=\sqrt{k_BT/8\pi ne^{2}}$, smaller by $\sqrt2$. Which one applies depends on the timescale: faster than the ion plasma period, the ions haven’t had time to move. I stick with the electron Debye length because it is the one tied to the plasma frequency and to the classical Coulomb cutoff below. The literature uses both, so check which one a quoted number assumes. ↩︎
- 4The standard results are the Spitzer collision times, which distinguish electron–electron, electron–ion and ion–ion relaxation and give different coefficients for momentum transfer, energy exchange and temperature equilibration. The classical estimate $b_{\min}\sim b_{90}$ also has to be replaced by the de Broglie wavelength when that is larger, as it is for electrons in the hottest cluster gas. Notice, too, that the large-$b$ divergence is cut off by screening, so the Debye length enters the collision rate itself. Even in principle, collisions in a plasma are not independent of its collective behaviour. ↩︎
- 5For a systematic treatment see Schekochihin et al. (2009), Astrophysical gyrokinetics. In their low-frequency, anisotropic ordering, the Alfvénic part of the turbulent cascade obeys reduced-MHD equations even at collisionless scales, while the compressive fluctuations need a kinetic description. Whether a fluid reduction is valid can depend on which fluctuations you are following. ↩︎
- 6Once resistivity is kept, the energy equation must include Ohmic heating, and other transport or heating terms where they matter; the adiabatic law is the ideal limit. Dropping the Hall and electron-inertia terms brings in the ion and electron inertial lengths, so the four scales in the figure above are not a complete checklist for MHD. See Fitzpatrick’s derivation of the MHD equations, and, for the Hall extension, Acheritogaray et al. (2011). ↩︎
- 7For the classical coefficients and where they apply, see Fitzpatrick, Classical Closure Scheme. For kinetic limits on the parallel electron heat flux, see Roberg-Clark et al. (2018), Wave generation and heat flux suppression in astrophysical plasma systems. For draping at cold fronts, see Lyutikov (2006), Magnetic draping of merging cores and radio bubbles in clusters of galaxies. ↩︎
- 8The closure is due to Chew, Goldberger and Low (1956). Its problem is that zero heat flux is not generally justified in a collisionless plasma, where particles stream along field lines and carry energy with them. There is no universal local closure for that transport. Landau-fluid closures, which mimic collisionless phase mixing and Landau damping in the higher moments, are a useful compromise in the regimes they are built for; see Hammett and Perkins (1990). ↩︎
- 9The difference between firehose scattering and mirror trapping is shown in Kunz, Schekochihin and Stone (2014), Firehose and Mirror Instabilities in a Collisionless Shearing Plasma. A similar self-regulating loop appears in cosmic-ray transport: cosmic rays streaming along the field excite waves that scatter them. If wave damping is weak, their bulk drift is held near the Alfvén speed; stronger damping lets them stream faster. The loops are analogous, not identical; see Thomas and Pfrommer (2019), Cosmic-ray hydrodynamics. ↩︎
- 10For the resistive-fluid picture, see Fitzpatrick’s plasma-physics notes, chapter Magnetic Reconnection. ↩︎
- 11The coefficient follows from $pc\simeq E$ and $r_g=p_\perp c/(eB)$ in Gaussian units. Cosmic-ray fluid models with an explicit scattering closure are developed, for example, by Thomas and Pfrommer (2019). ↩︎
