Cooling flow regime of a plasma thermal quench

A large class of Laboratory, Space, and Astrophysical plasmas is nearly collisionless. When a localized energy or particle sink, for example, in the form of a radiative cooling spot or a black hole, is introduced into such a plasma, it can trigger a plasma thermal collapse, also known as a thermal quench in tokamak fusion. Here we show that the electron thermal conduction in such a nearly collisionless plasma follows the convective energy transport scaling in itself or in its spatial gradient, due to the constraint of ambipolar transport. As the result, a robust cooling flow aggregates mass toward the cooling spot and the thermal collapse of the surrounding plasma takes the form of four propagating fronts that originate from the radiative cooling spot, along the magnetic field line in a magnetized plasma. The slowest one, which is responsible for deep cooling, is a shock front.

A signature property of a large class of magnetized and unmagnetized plasmas in the Laboratory, Space, and Astrophysical systems is the extremely low collisionality that can be due to high plasma temperature T e or low plasma density n e , or a combination of the two [1]. For example, a fusion-grade plasma in a tokamak reactor has T e ∼ 10 − 20 kilo-electronvolts (KeV) and n e ∼ 10 19−20 per cubic meter, which result in a mean-free-path λ m f p ∼ 10 4 meters (m), while the toroidal length of the confinement chamber is merely 20-30 m [2][3][4]. In the earth's radiation belt, the electron λ m f p can be as long as 10 11 m with electron energy from tens of KeV to MeV and an n e ∼ 10 4 m −3 [5][6][7][8]. At the even grander scale of clusters of galaxies, the intracluster hot gas has n e ∼ 10 2 − 10 4 m −3 and T e ∼ 2 × 10 7 − 10 8 K [9][10][11][12][13], so λ m f p is in the order of tens of kilo-parsec to mega-parsec.
A whole class of problems arises if a localized cooling spot is introduced into such a nearly collisionless plasma. This could be structure formation in a galaxy cluster where a radiative cooling spot is driven by increased particle density [9] or an event horizon of a black hole that provides an absorbing boundary for plasmas [14]. A satellite traversing the earth's radiation belt can be a sink for plasma energy and particles [15,16]. In a tokamak reactor, solid pellets that are injected into the fusion plasma for fueling and disruption mitigation [17][18][19], provide localized cooling due to a combination of energy spent on phase transition and ionization of the solid materials, and the radiative cooling that is especially strong when high-Z impurities are embedded in the frozen pellet. Even in the absence of pellet injection, large-scale magnetohydrodynamic instabilities can turn nested flux surfaces into globally stochastic field lines that connect fusion core plasma directly onto the divertor/first wall [20][21][22][23], causing a thermal collapse via fast parallel transport along the field lines within a short period of time that can range from micro-seconds to milliseconds [24]. An outstanding physics question is how a thermal collapse of the surrounding plasma, commonly known as a thermal quench in tokamak fusion, would come about in such a diverse range of applications.
The most obvious route for the thermal collapse is via elec-tron thermal conduction along the magnetic field line that intercepts the cooling spot, for which the Braginskii formula [25] would produce an enormous heat flux [9,11] if there is a sizeable temperature difference ∆T = T 0 − T w ∼ T 0 between the cooling spot (T w ) and the surrounding plasma (T 0 ), Here v th,e = T 0 /m e is the electron thermal speed and L T the distance or field line length over which the temperature drop ∆T is established. For a nearly collisionless plasma, the temperature collapse necessarily starts with Knudsen number K n ≡ λ m f p /L T 1, a regime in which the free-streaming limit [26] of is supposed to apply in lieu of Braginskii, with α ≈ 0.1 [27]. The pressure-gradient-driven plasma flow V i along the magnetic field line is limited by the ion sound speed c s , so the convective electron energy flux is bounded by n e c s T 0 . The scaling of q e with v th,e in Eq. (2) suggests that the electron energy flux would be dominated by conduction as normally v th,e c s in a plasma of comparable electron and ion temperatures. In such a conduction-dominated situation, the much colder but denser cooling spot would be rapidly heated up by the electron thermal conduction from the surrounding hot plasma, and as the result, it can become over-pressured and the original cooling spot, say an ablated pellet in a tokamak, tends to expand into the surrounding plasma, yielding an outflow.
In the aforementioned problem of clusters of galaxies, one has instead observed robust cooling flows into the radiative cooling spot that aggregate mass onto the cooling spot [9], although more recent observations reveal a more modest massaccreting cooling flow that indicates the role of various additional heating mechanisms to balance the cooling [11][12][13]. This is inconsistent with the conduction-dominated scenario mentioned above [9,11,28,29]. Extensive efforts have been made to find ways to inhibit the electron thermal conduction in the nearly collisionless plasma, for example, by tangled magnetic fields [30,31] or plasma instabilities [32][33][34], in order to reach the convection-dominated scenario, which would naturally yield the cooling flow regime of a plasma thermal quench.
In this Letter, we show that in a nearly collisionless plasma, even along the magnetic field lines, like in the case of pellet injection into a tokamak, ambipolar transport will naturally constrain the electron parallel thermal conduction in such a way that the plasma thermal collapse comes with a cooling flow toward the radiative cooling spot. The necessary constraint is on the spatial gradient of electron parallel conduction flux, which can be seen from the energy equation for the electrons along the magnetic field, Here x is the distance along the magnetic field line, n e , T e ,V e are the density, parallel temperature, and parallel flow of the electrons, and q en ≡ m e v −V e 3 d 3 v is a component of the parallel heat flux. Let the cooling flow span a length L T , one can see the convective energy transport terms follow the scaling of n e T e V e /L T . Ambipolar transport constrains , which would overwhelm the convective energy transport (∝ m −1/2 i ) to force a T e collapse and remove the pressure gradient drive that sustains the cooling flow. The condition for accessing the cooling flow regime of plasma thermal quench is thus We report in this Letter that this is indeed realized by ambipolar constraint in a nearly collisionless plasma. In the case that the cooling spot is a perfect particle and energy sink (e.g., a black hole), which can be modeled by an absorbing boundary, q en itself obtains the convective energy transport scaling, q en ∼ n e V i T e . With a radiative cooling mass, which can be modeled as a thermobath, the boundary of which recycles all particles across the boundary but clamps the temperature to a low value T w T 0 , the cold electrons thus produced can restore the free-stream scaling for q en ∼ αn e v th,e T e but its spatial gradient over the cooling flow region retains the convective energy transport scaling of Eq. (4). As the result, a robust cooling flow appears to aggregate mass towards the cooling spot.
Most interestingly, in such a cooling flow regime, the plasma thermal collapse comes in the form of propagating fronts that originate from the cooling spot with characteristic speeds. There are totally four (three) propagating fronts for the thermobath (absorbing) boundary: two of them propagate at speeds that scale with v th,e , so are named electron fronts, while the other two are ion fronts that propagate at speeds that scale with the local ion sound speed c s (the last ion front disappears for the absorbing boundary). Fig. 1 illustrates the structure of the four fronts that propagate into a hot plasma for the thermal collapse with a thermobath boundary. It is important to note that cooling of a nearly collisionless plasma produces strong temperature anisotropy, so one must examine the collapse of T and T ⊥ separately. Cooling of T e in a nearly collisionless plasma is primarily through free-streaming loss of suprathermal electrons satisfying v < − 2e(∆Φ) max /m e into the radiative cooling spot. Here (∆Φ) max is the maximum reflective potential in the plasma with ∆Φ = Φ ∞ − Φ(x) and the constant Φ ∞ the far upstream plasma potential. The precooling zone bounded by the precooling front (PF) and the precooling trailing front (PTF) has T i unchanged and V i ≈ 0, but a lowered T e , which is due to the depletion of fast electrons satisfying the Heaviside step function that vanishes for v > v c , and δ (x) the Dirac delta function. The ambipolar electric field can draw some low-energy electrons to compensate for the loss of high-energy electrons and thus maintain quasi-neutrality. This in-falling cold electron population is modeled in Eq. (5) as a cold beam that due to ambipolar electric field acceleration has the speed v c . Between the PF and PTF, the electron beam can be ignored, so whereṽ ≡ v − V e . Eq. (6) predicts a detectable decrease in T e (v c ) from T 0 for v c ≈ 2.4v th,e , suggesting an electron PF propagating at This corresponds to fast electrons with v > U PF traveling from the left boundary into the plasma, leaving behind a distribution at the PF with a void in v > U PF . The PTF comes about due to the reflecting potential (∆Φ) RF = (∆Φ) max − (Φ ∞ − Φ RF ) with Φ RF the ambipolar potential at the ion recession front (RF), which sets a lower cutoff speed v c at The deeper void now gives rise to a further reduced T e , The PTF rides these electrons that are reflected by the reflecting potential, and propagates at U PT F (< U PF ). Since the ambipolar reflecting potential must satisfy e (∆Φ) RF ∼ T 0 in a nearly collisionless plasma, U PT F ∼ v th,e and T e (U PT F ) is only mildly cooler than T 0 . Furthermore, T e and Φ vary little between the RF and PTF, since the cutoff velocity remains the same at U PT F . The ion flow remains vanishingly small ahead of the RF, so the electron cooling between the RF and PF is the result of electron conduction, which for the model f e in Eq.
Between PTF and PF, n b ≈ 0 and U PT F ≤ v c ≤ U PF so q en does scale as the free-streaming limit of Eq. (2), but with α modulating in space as a function of v c . In fact, for v c > √ 2v th,e , one finds so the solution of the energy equation, ∂ T e /∂t = −v c ∂ T e /∂ x, reveals that v c is the recession speed of T e , reaffirming the particle picture noted earlier that the momentum space void in f e propagates upstream with a speed of v c . This large q en drives fast propagating electron fronts (PF and PTF) but produces modest amount of T e cooling for the large cutoff speed v c = U PT F . Much more aggressive cooling would need to occur as the plasma approaches the radiative cooling spot that is clamped at T w T 0 . These are facilitated by the ion fronts that provide the reflecting potential (∆Φ) RF . The RF is where n i ≈ n e starts to drop, and behind which plasma pressure gradient drives a cooling flow toward the radiative cooling spot. The main reflection potential, which is tied to the electron pressure gradient, is also behind the ion RF. An ion recession layer bounded by the RF and the cooling front (CF) is similar to the rarefaction wave formed in the cold plasma interaction with a solid surface [36,37], where the plasma parameters recede steadily with the local sound speed. What is different for the thermal quench of a nearly collisionless plasma is the large plasma temperature and pressure gradient and the nature of the heat flux. The electron flow associated with f e of Eq. (5) within the recession layer is with n e (v c ) = 1 + Erf v c / √ 2v th,e n m /2 + n b . For an absorbing boundary (n b = 0), a cutoff speed around v th,e , v c ≈ v th,e 2 ln(v th,e /V i ), is sufficient to produce a V e that matches onto the increasing ion flow, V e ≈ V i , for ambipolar transport through the recession layer. The in-falling cold electron beam reduces v c and hence produces a lower reflecting potential across the recession layer as elucidated in Eq. (12).
The physics of q en in the recession layer can be elucidated by rewriting Eq. (10) as th,e + 2 n e V e T 0 − 2n b v c T 0 − 3n e T e V e + 2n e m e V 3 e .
For an absorbing wall, n b = 0, and one finds q en itself has a convective energy transport scaling: q en ∼ n e V e T 0 . The condition for the cooling flow regime, Eq. (4), is obviously satisfied. In the case of a radiative cooling spot that produces copious amount of cold electrons, the leading order of q en ∼ −2n b v c T 0 ∝ n b v th,e T 0 follows the free-streaming limit of Eq. (2). Remarkably the plasma thermal quench still produces a cooling flow, in which case Eq. (4) is satisfied due to the collisionless cold beam in the ambipolar electric field follows flux conservation n b v c = constant, so ∂ (−2n b v c T 0 )/∂ x = 0 and the remaining terms in q en have convective energy transport scaling. The VPIC [35] kinetic simulations shown in Fig. 2 confirms that convective scaling of Eq. (4) holds in the recession layer. In other words, electron cooling in a nearly collisionless plasma is modified by ambipolarity in such a way that large T e gradient can be supported in the recession layer to drive a cooling flow. The propagation speed of the RF can be understood by ex- For the absorbing boundary, q en itself behind the recession front (RF) follows the convective scaling, which can be seen by comparing q Ab en 1600 and q Ab en 100 /4 (notice that their small difference, as seen from Eq. 13 for n b = 0, comes from the dependence of v c ≈ v th,e 2 ln(v th,e /V i ) on m i ). For the thermobath boundary, q en recovers the free-streaming formula, but its spatial gradient within the cooling flow region, which is between the cooling front (CF) and the RF, follows the convective scaling, which is illustrated by the same slope of the curve as that for the absorbing boundary when m i /m e is fixed.
amining the ion dynamics in the recession layer [38,39] where we invoked the electron force balance en e E ≈ −∂ p e /∂ x and quasi-neutrality n e = Zn i with Z the ion charge, and p i,e = n i,e T i,e . Introducing a parameterization of q in ≈ σ i n i V i T i , which is known from Ref. 40, and ∂ q en /∂ x ≈ σ e ∂ n e V e T e /∂ x from Eq. (4), we obtain an universal length scale for p e,i dlnp e dx ≈ µ dlnp i dx , where µ = (3+σ e )/( It is interesting to note that U > 0 and V i < 0 have opposite sign in the recession layer where a cooling flow resides. As a result, Eqs. (14)(15)(16) have self-similar solutions with similarity variable ξ = x − Ut with U being the local recession speed. We find where c s = 3(µZT e + T i )/m i is the local sound speed of a nearly collisionless plasma with anisotropic temperatures. At the ion recession front, V i ≈ 0, so the speed of the ion recession front is For Z = 1, σ i = 1, µ = 1 and T e ≈ T i = T 0 at the recession front, we have U RF ≈ 2.8v th,i with v th,i = T 0 /m i the ion thermal speed, which agrees well with the simulation result. It is worth noting that the self-similar solution of Eq. (18) also recovers a known constraint [41] on the plasma exit flow at an absorbing boundary where a non-neutral sheath would form next to it as shown in the Supplement material.
In the absence of an absorbing boundary, the mass aggregated by the cooling flow will pile up, and the resulting backpressure can now drive a second ion front (cooling front, CF). Behind the CF, T e equilibrates with T w as shown in Fig. 3. Such a deep cooling of T e is through thermal conduction as indicated in Fig. 2. When the cooling flow runs into this nearly static plasma, the ion flow energy, which is substantial in the cooling flow, is converted into ion thermal energy via a plasma shock as shown in Fig. 3. Matching the conserved quantities across the shock while ignoring the heat flux, we find that the speed of the shock, which propagates upstream into the plasma, is simply the upstream sound speed at the shock front. The CF is the shock front, so its speed is Since the plasma temperature at the CF is considerably lower than that at the RF, we have U CF < U RF . Generally, the colder T w , the smaller U CF . The presence of the CF and the cooling zone behind it, is of fundamental importance to T i⊥ and T e⊥ cooling as the cold particles provide dilutional cooling. It is also the source of cold electrons that are accelerated by the ambipolar electric field into the recession layer and beyond, cooling down T e⊥ further upstream.
In conclusion, the thermal collapse of a nearly collisionless plasma due to its interaction with a localized particle or energy sink, is associated with a cooling flow toward the cooling spot. This applies to unmagnetized plasmas, for example, in astrophysical systems, and magnetized plasmas, for example, in earth's magnetosphere or a tokamak fusion plasma. It is the fundamental constraint of ambipolar transport, along the field line in a magnetized plasma, that limits the spatial gradient of electron (parallel) heat flux to the much weaker convective (V e ) scaling as opposed to the free-streaming (v th,e ) scaling. Such weaker scaling is essential to sustain a temperature and hence pressure gradient for driving the cooling flow toward the cooling spot over the ion recession layer. The cooling flow eventually terminates against the cooling spot via a plasma shock that converts the ion flow energy into ion thermal energy. This shock or cooling front propagates away from the cooling spot at upstream ion sound speed, and it has the most profound role in the deep cooling of the surrounding hot plasmas, especially the ions. Unlike the ions, the electrons can be cooled ahead of the recession front due an electron heat flux that follows the free-streaming limit (q en ∝ n e v th,e T e ). Interestingly this large heat flux does not imply significant cooling of T e in a nearly collisionless plasma ahead of the recession front, but induces a very limited amount of T e drop over a very large volume, because the precooling and precooling trailing fronts have propagation speeds that scale with electron thermal speed.